ATMOSPHERIC - Earth · 2006-05-02 · 12 Mid-Latitude Atmospheric Circulation 489 12.1 Surface...

ATMOSPHERICAND OCEANIC

FLUID DYNAMICSFundamentals and Large-Scale Circulation

Geoffrey K. Vallis

Contents

Preface xi

Part I FUNDAMENTALS OF GEOPHYSICAL FLUID DYNAMICS 1

1 Equations of Motion 31.1 Time Derivatives for Fluids 31.2 The Mass Continuity Equation 71.3 The Momentum Equation 111.4 The Equation of State 141.5 The Thermodynamic Equation 161.6 Sound Waves 291.7 Compressible and Incompressible Flow 311.8 * More Thermodynamics of Liquids 331.9 The Energy Budget 391.10 An Introduction to Non-Dimensionalization and Scaling 43

2 Effects of Rotation and Stratification 512.1 Equations in a Rotating Frame 512.2 Equations of Motion in Spherical Coordinates 552.3 Cartesian Approximations: The Tangent Plane 662.4 The Boussinesq Approximation 682.5 The Anelastic Approximation 742.6 Changing Vertical Coordinate 782.7 Hydrostatic Balance 802.8 Geostrophic and Thermal Wind Balance 852.9 Static Instability and the Parcel Method 922.10 Gravity Waves 98

v

vi Contents

2.11 * Acoustic-Gravity Waves in an Ideal Gas 1002.12 The Ekman Layer 104

3 Shallow Water Systems and Isentropic Coordinates 1233.1 Dynamics of a Single, Shallow Layer 1233.2 Reduced Gravity Equations 1293.3 Multi-Layer Shallow Water Equations 1313.4 Geostrophic Balance and Thermal wind 1343.5 Form Drag 1353.6 Conservation Properties of Shallow Water Systems 1363.7 Shallow Water Waves 1403.8 Geostrophic Adjustment 1443.9 Isentropic Coordinates 1523.10 Available Potential Energy 155

4 Vorticity and Potential Vorticity 1654.1 Vorticity and Circulation 1654.2 The Vorticity Equation 1674.3 Vorticity and Circulation Theorems 1704.4 Vorticity Equation in a Rotating Frame 1774.5 Potential Vorticity Conservation 1804.6 * Potential Vorticity in the Shallow Water System 1864.7 Potential Vorticity in Approximate, Stratified Models 1884.8 The Impermeability of Isentropes to Potential Vorticity 191

5 Simplified Equations for Ocean and Atmosphere 1995.1 Geostrophic Scaling 2005.2 The Planetary Geostrophic Equations 2045.3 The Shallow Water Quasi-Geostrophic Equations 2095.4 The Continuously Stratified Quasi-Geostrophic System 2175.5 * Quasi-geostrophy and Ertel Potential Vorticity 2265.6 * Energetics of Quasi-Geostrophy 2295.7 Rossby Waves 2325.8 * Rossby Waves in Stratified Quasi-Geostrophic Flow 237

Appendix: Wave Kinematics, Group Velocity and Phase Speed 239

Part II INSTABILITIES, WAVE–MEAN FLOW INTERACTION ANDTURBULENCE 249

6 Barotropic and Baroclinic Instability 2516.1 Kelvin-Helmholtz Instability 2516.2 Instability of Parallel Shear Flow 2536.3 Necessary Conditions for Instability 2626.4 Baroclinic Instability 2646.5 The Eady Problem 2696.6 Two-Layer Baroclinic Instability 275

Contents vii

6.7 An Informal View of the Mechanism of Baroclinic Instability 2816.8 * The Energetics of Linear Baroclinic Instability 2866.9 * Beta, Shear and Stratification in a Continuous Model 288

7 Wave–Mean Flow Interaction 3017.1 Quasi-geostrophic Preliminaries 3027.2 The Eliassen-Palm Flux 3047.3 The Transformed Eulerian Mean 3107.4 The Nonacceleration Result 3207.5 Influence of Eddies on the Mean Flow in the Eady Problem 3257.6 * Necessary Conditions for Instability 3307.7 * Necessary Conditions for Instability: Use of Pseudoenergy 334

8 Turbulence, Basic Theory 3438.1 The Fundamental Problem of Turbulence 3448.2 The Kolmogorov Theory 3468.3 Two-Dimensional Turbulence 3568.4 Predictability of Turbulence 3678.5 * Spectra of Passive Tracers 372

9 Geostrophic Turbulence and Baroclinic Eddies 3839.1 Effects of Differential Rotation 3839.2 Stratified Geostrophic Turbulence 3899.3 † A Scaling Theory for Geostrophic Turbulence 3979.4 † Phenomenology of Baroclinic Eddies in the Atmosphere and Ocean 401

10 Turbulent Diffusion and Eddy Transport 41110.1 Diffusive Transport 41210.2 Turbulent Diffusion 41310.3 Two-Particle Diffusivity 41910.4 Mixing Length Theory 42310.5 Homogenization of a Scalar that is Advected and Diffused 42710.6 † Transport by Baroclinic Eddies 43010.7 † Eddy Diffusion in the Atmosphere and Ocean 43410.8 † Thickness Diffusion 44410.9 † Eddy Transport and the Transformed Eulerian Mean 447

Part III LARGE-SCALE ATMOSPHERIC CIRCULATION 453

11 The Overturning Circulation: Hadley and Ferrel Cells 45511.1 Basic Features of the Atmosphere 45611.2 A Steady Model of the Hadley Cell 46011.3 A Shallow Water Model of the Hadley Cell 47111.4 † Asymmetry Around the Equator 47211.5 Eddies, Viscosity and the Hadley Cell 47611.6 The Hadley Cell: Summary and Numerical Solutions 480

viii Contents

11.7 The Ferrel Cell 483

12 Mid-Latitude Atmospheric Circulation 48912.1 Surface Westerlies and the Maintenance of a Barotropic Jet 49012.2 Layered Models of the Mid-latitude Circulation 50112.3 † An Example of a Closed Model 51712.4 * Eddy Fluxes and Necessary conditions for Instability 51912.5 A Stratified Model and the Real Atmosphere 52112.6 The Tropopause and the Stratification of the Atmosphere 52812.7 † Baroclinic eddies and Potential Vorticity Mixing 53312.8 † Extra-tropical Convection and the Ventilated Troposphere 538

Appendix: TEM & EP Flux for Primitive Eqs. in Spherical Coordinates 540

13 Zonally Asymmetries, Planetary Waves and Stratosphere 54513.1 Forced and Stationary Rossby Waves 54613.2 * Meridional Propagation and Dispersion 55213.3 * Stratified Rossby Waves and their Vertical Propagation 55613.4 * Effects of Thermal Forcing 56213.5 Stratospheric Dynamics 568

Part IV LARGE-SCALE OCEANIC CIRCULATION 581

14 Wind-Driven Gyres 58314.1 The Depth Integrated Wind-Driven Circulation 58514.2 Using Viscosity Instead of Drag 59314.3 Zonal Boundary Layers 59714.4 The Nonlinear Problem 59914.5 * Inertial Solutions 60114.6 Topographic Effects on Western Boundary Currents 60814.7 * Vertical Structure of the Wind-driven Circulation 61314.8 * A Model with Continuous Stratification 619

15 The Buoyancy Driven Circulation 62915.1 A Brief Observational Overview 63015.2 †Sideways Convection 63115.3 The Maintenance of Sideways Convection 63615.4 Simple Box Models 64215.5 A Laboratory Model of the Abyssal Circulation 64815.6 A Model for Oceanic Abyssal Flow 65315.7 * A Shallow Water Model of the Abyssal Flow 65815.8 Scaling for the Buoyancy-driven Circulation 661

16 Wind and Buoyancy Driven Circulation 66716.1 The Main Thermocline: an Introduction 66716.2 Scaling and Simple Dynamics of the Main Thermocline 67016.3 The Internal Thermocline 674

Contents ix

16.4 The Ventilated Thermocline 68116.5 † A Model of Deep Wind-Driven Overturning 69016.6 † Flow in a Channel, and the Antarctic Circumpolar Current 699

Appendix: Miscellaneous Relationships in a Layered Model 710

References 715Index 735

An asterisk indicates more advanced, but usually uncontroversial, material thatmay be omitted on a first reading. A dagger indicates material that is still a topicof research or that is not settled. Caveat emptor.

Notation xv

NOTATION

The list below contains only the more important variables, or instances of non-obvious notation. Distinct meanings are separated with a semi-colon. Variablesare normally set in italics, constants (e.g, π) in roman (i.e., upright), differentialoperators in roman, vectors in bold, and tensors in sans serif. Thus, vector variablesare in bold italics, vector constants (e.g., unit vectors) in bold roman, and tensorvariables are in slanting sans serif. Physical units are set in roman. A subscriptdenotes a derivative only if the subscript is a coordinate, such as x, y or z; asubscript 0 generally denotes a constant reference value (e.g., ρ0). The componentsof a vector are denoted by superscripts.

Variable Description

b Buoyancy, −gδρ/ρ0 or gδθ/θ.cg Group velocity, (cxg , c

yg , czg).

cp Phase speed; heat capacity at constant pressure.cv Heat capacity constant volume.cs Sound speed.f , f0 Coriolis parameter, and its reference value.g, g Vector acceleration due to gravity, magnitude of g.h Layer thickness (in shallow water equations).i, j,k Unit vectors in (x,y, z) directions.i An integer index.i Square root of −1.k Wave vector, with components (k, l,m) or (kx , ky , kz).kd Wave number corresponding to deformation radius.Ld Deformation radius.L,H Horizontal length scale, vertical (height) scale.m Angular momentum about the earth’s axis of rotation.M Montgomery function, M = cpT + Φ.N Buoyancy, or Brunt-Väisälä, frequency.p Pressure.Pr Prandtl ratio, f0/N.q Quasi-geostrophic potential vorticity.Q Potential vorticity (in particular Ertel PV).Q Rate of heating.Ra Rayleigh number.Re Real part of expression.Re Reynolds number, UL/ν.Ro Rossby number, U/fL.S Salinity; source term on right-hand side of evolution equation.T Temperature.t Time.u Two-dimensional, horizontal velocity, (u,v).v Three-dimensional velocity, (u,v, z).x,y, z Cartesian coordinates, usually in zonal, meridional and vertical directions.Z Log-pressure, −H logp/pR. We often use H = 7.5 km and pR = 105 Pa.

xvi Notation

Variable Description

A Wave activity.α Inverse density, or specific volume; aspect ratio.β Rate of change of f with latitude, ∂f/∂y .βT , βS Coefficient of expansion with respect to temperature, salinity.ε Generic small parameter (epsilon).ε Cascade or dissipation rate of energy (varepsilon).η Specific entropy; perturbation height; enstrophy cascade or dissipation rate.F Eliassen Palm flux, (Fy ,Fz).γ Vorticity gradient, β−uyy ; the ratio cp/cv .Γ Lapse rate.κ Diffusivity; the ratio R/cp.K Kolmogorov or Kolomogorov-like constant.Λ Shear, e.g., ∂U/∂z.ν Kinematic viscosity.v Meridional component of velocity.φ Pressure divided by density, p/ρ; passive tracer.Φ Geopotential, usually gz.Π Exner function, Π = cpT/θ = cp(p/pR)R/cp .ω Vorticity.Ω,Ω Rotation rate of earth and associated vector.ψ Streamfunction.ρ Density.ρθ Potential density.σ Layer thickness, ∂z/∂θ; Prandtl number ν/κ; measure of density, ρ − 1000.τ Wind stress.τ Zonal component or magnitude of wind stress; eddy turnover time.θ Potential temperature.ϑ,λ Latitude, longitude.ζ Vertical component of vorticity.(∂a∂b

)c

Derivative of a with respect to b at constant c.

∂a∂b

∣∣∣∣a=c

Derivative of a with respect to b evaluated at a = c.

∇a Gradient operator at constant value of coordinate a, e.g., ∇z = i ∂x + j ∂y .

∇a· Divergence operator at constant value of coordinate a, e.g., ∇z· = (i ∂x + j ∂y)·.∇⊥ Perpendicular gradient, ∇⊥φ ≡ k×∇φ.

curlz Vertical component of ∇× operator, curlzA = k · ∇×A = ∂xAy − ∂yAx .DDt

Material derivative (generic).

D3

Dt,

D2

DtMaterial derivative in three dimensions and in two dimensions, for example

∂/∂t + v · ∇ and ∂/∂t + u · ∇ respectively.DgDt

Material derivative using geostrophic velocity, for example ∂/∂t + ug · ∇.

Part I

FUNDAMENTALS OF

GEOPHYSICAL FLUID DYNAMICS

Are you sitting comfortably? Then I’ll begin.Julia Lang, Listen With Mother, BBC radio program, 1950–1982.

CHAPTER

ONE

Equations of Motion

THIS CHAPTER establishes the fundamental governing equations of motion for afluid, with particular reference to the fluids of the earth’s atmosphere andocean.1 Our approach in many places is quite informal, and the interested

reader may consult the references given for more detail.

1.1 TIME DERIVATIVES FOR FLUIDS

The equations of motion of fluid mechanics differ from those of rigid-body mechan-ics because fluids form a continuum, and because fluids flow and deform. Thus,even though both classical solid and fluid media are governed by the same rela-tively simple physical laws (Newton’s laws and the laws of thermodynamics), theexpression of these laws differs between the two. To determine the equations of mo-tion for fluids we must clearly establish what the time derivative of some propertyof a fluid actually means, and that is the subject of this section.

1.1.1 Field and material viewpoints

In solid-body mechanics one is normally concerned with the position and momen-tum of identifiable objects — the angular velocity of a spinning top or the motionsof the planets around the sun are two well-worn examples. The position and ve-locity of a particular object is then computed as a function of time by formulatingequations of the form

dxidt

= F(xi, t) (1.1)

where xi is the set of positions and velocities of all the interacting objects and theoperator F on the right-hand side is formulated using Newton’s laws of motion. For

3

4 Chapter 1. Equations of Motion

example, two massive point objects interacting via their gravitational field obey

dridt

= vi,dvidt

= − Gmj

(ri − rj)2ri,j , i = 1,2; j = 3− i. (1.2)

We thereby predict the positions, ri and velocities, vi of the objects given theirmasses,mi and the gravitational constant G, and where ri,j is a unit vector directedfrom ri to rj .

In fluid dynamics such a procedure would lead to an analysis of fluid motionsin terms of the positions and momenta of particular fluid elements, each identifiedby some label, which might simply be their position at an initial time. We callthis a material point of view, because we are concerned with identifiable piecesof material; it is also sometimes called a Lagrangian view, after J.-L. Lagrange.The procedure is perfectly acceptable in principle, and if followed would providea complete description of the fluid dynamical system. However, from a practicalpoint of view it is much more than we need, and it would be extremely complicatedto implement. Instead, for most problems we would like to know what the valuesof velocity, density and so on are at fixed points in space as time passes. (A weatherforecast we might care about tells us how warm it will be where we live, and if weare given that we don’t particularly care where a fluid parcel comes from.) Sincethe fluid is a continuum, this knowledge is equivalent to knowing how the fieldsof the dynamical variables evolve in space and time, and this is often known asthe field or Eulerian viewpoint, after L. Euler.2 Thus, whereas in the material viewwe consider the time evolution of identifiable fluid elements, in the field view weconsider the time evolution of the fluid field from a particular frame of reference.That is, we seek evolution equations of the form

∂∂tϕ(x,y, z, t) = F, (1.3)

where the field ϕ(x,y, z, t) is a dynamical variable (e.g., velocity, density, tempera-ture) which gives the value at any point in space-time, and F is some operator to bedetermined from Newton’s laws of motion and appropriate thermodynamic laws.

Although the field viewpoint will turn out to be the most practically useful,the material description is invaluable both in deriving the equations and in thesubsequent insight it frequently provides. This is because the important quantitiesfrom a fundamental point of view are often those which are associated with a givenfluid element: it is these which directly enter Newton’s laws of motion and thethermodynamic equations. It is thus important to have a relationship between therate of change of quantities associated with a given fluid element and the local rateof change of a field. The material or advective derivative provides this relationship.

1.1.2 The material derivative of a fluid property

A fluid element is an infinitesimal, indivisible, piece of fluid — effectively a verysmall fluid parcel. The material derivative is the rate of change of a property (suchas temperature, or momentum) of a particular fluid element or finite mass; thatis to say, it is the total time derivative of a property of a piece of fluid. It is alsoknown as the ‘substantive derivative’ (the derivative associated with a parcel of

1.1 Time Derivatives for Fluids 5

fluid substance), the ‘advective derivative’ (because the fluid property is being ad-vected), the ‘convective derivative’ (convection is a slightly old-fashioned name foradvection, still used in some fields), or the ‘Lagrangian derivative’.

Let us suppose that a fluid is characterized by a (given) velocity field v(x, t),which determines its velocity throughout. Let us also suppose that it has anotherproperty φ, and let us seek an expression for the rate of change of φ of a fluidelement. Since φ is changing in time and in space we use the chain rule:

δφ = ∂φ∂tδt + ∂φ

∂xδx + ∂φ

∂yδy + ∂φ

∂zδz = ∂φ

∂tδt + δx · ∇φ. (1.4)

This is true in general for any δt, δx, etc. Thus the total time derivative is

dφdt

= ∂φ∂t

+ dxdt

· ∇φ. (1.5)

If this is to be a material derivative we must identify the time derivative in thesecond term on the right-hand side with the rate of change of position of a fluidelement, namely its velocity. Hence, the material derivative of the property φ is

dφdt

= ∂φ∂t

+ v · ∇φ. (1.6)

The right-hand side expresses the material derivative in terms of the local rate ofchange of φ plus a contribution arising from the spatial variation of φ, experiencedonly as the fluid parcel moves. Because the material derivative is so common, andto distinguish it from other derivatives, we denote it by the operator D/Dt. Thus,the material derivative of the field φ is

DφDt

= ∂φ∂t

+ (v · ∇)φ . (1.7)

The brackets in the last term of this equation are helpful in reminding us that (v·∇)is an operator acting on φ.

Material derivative of vector field

The material derivative may act on a vector field b, in which case

DbDt

= ∂b∂t

+ (v · ∇)b. (1.8)

In Cartesian coordinates this is

DbDt

= ∂b∂t

+u∂b∂x

+ v ∂b∂y

+w∂b∂z, (1.9)

and for a particular component of b,

DbxDt

= ∂bx∂t

+u∂bx∂x

+ v ∂bx∂y

+w∂bx∂z, (1.10)

or, in Cartesian tensor notation,

DbiDt

= ∂bi∂t

+ vj∂bi∂xj

= ∂bi∂t

+ vj∂jbi. (1.11)


where the subscripts denote the Cartesian components and repeated indices aresummed. In coordinate systems other than Cartesian the advective derivative of avector is not simply the sum of the advective derivative of its components, becausethe coordinate vectors change direction with position; this will be important whenwe deal with spherical coordinates (and see problem 2.5). Finally, we note that theadvective derivative of the position of a fluid element, r say, is its velocity, and thismay easily checked by explicitly evaluating Dr/Dt.

1.1.3 Material derivative of a volume

The volume that a given, unchanging, mass of fluid occupies is deformed and ad-vected by the fluid motion, and there is no particular reason why it should remainconstant. Indeed, the volume will change as a result of the movement of eachelement of its bounding material surface, and will in general change if there isa non-zero normal component of the velocity at the fluid surface. That is, if thevolume of some fluid is

∫dV , then

DDt

∫V

dV =∫Sv · dS, (1.12)

where the subscript V indicates that the integral is a definite integral over somefinite volume V , although the limits of the integral will be functions of time if thevolume is changing. The integral on the right-hand side is over the closed surface,S, bounding the volume. Although intuitively apparent (to some), this expressionmay be derived more formally using Leibnitz’s formula for the rate of change of anintegral whose limits are changing (problem 1.2). Using the divergence theoremon the right-hand side, (1.12) becomes

DDt

∫V

dV =∫V∇ · v dV. (1.13)

The rate of change of the volume of an infinitesimal fluid element of volume ∆V isobtained by taking the limit of this expression as the volume tends to zero, giving

lim∆V→0

1∆V

D∆VDt

= ∇ · v. (1.14)

We will often write such expressions informally as

D∆VDt

= ∆V∇ · v, (1.15)

with the limit implied.Consider now the material derivative of a fluid property, ξ, multiplied by the

volume of a fluid element, ∆V . This situation arises when ξ is the amount per unitvolume of ξ-substance — it might, for example, be mass density or the amount of adye per unit volume. Then we have

DDt(ξ∆V) = ξD∆V

Dt+∆V Dξ

Dt. (1.16)

Using (1.15) this becomes

DDt(ξ∆V) = ∆V

(ξ∇ · v + Dξ

Dt

), (1.17)

1.2 The Mass Continuity Equation 7

and the analogous result for a finite fluid volume is just

DDt

∫Vξ dV =

∫V

(ξ∇ · v + Dξ

Dt

)dV. (1.18)

This expression is to be contrasted with the Eulerian derivative for which the vol-ume, and so the limits of integration, are fixed and we have

ddt

∫Vξ dV =

∫V

∂ξ∂t

dV. (1.19)

Now consider the material derivative of a fluid property ϕ multiplied by themass of a fluid element, ρ∆V . This arises when ϕ is the amount ofϕ-substance perunit mass (note, for example, that the momentum of a fluid element is ρv∆V). Thematerial derivative of ϕρ∆V is given by

DDt(ϕρ∆V) = ρ∆V Dϕ

Dt+ϕ D

Dt(ρ∆V) (1.20)

But ρ∆V is just the mass of the fluid element, and that is constant — it is how afluid element is defined. Thus the second term on the right-hand side vanishes and

DDt(ϕρ∆V) = ρ∆V Dϕ

Dtand

DDt

∫Vϕρ dV =

∫Vρ

DϕDt

dV, (1.21a,b)

where (1.21b) applies to a finite volume. That expression may also be derivedmore formally using Leibnitz’s formula for the material derivative of an integral,and the result also holds when ϕ is a vector. The result is quite different from thecorresponding Eulerian derivative, in which the volume is kept fixed; in that casewe have:

ddt

∫Vϕρ dV =

∫V

∂∂t(ϕρ)dV. (1.22)

Various material and Eulerian derivatives are summarized in the shaded box on thefollowing page.

1.2 THE MASS CONTINUITY EQUATION

In classical mechanics mass is absolutely conserved. However, in fluid mechanicsfluid flows into and away from regions, and fluid density may change, and an equa-tion that explicitly accounts for the flow of mass is one of the ‘equations of motion’of the fluid.

1.2.1 An Eulerian derivation

We will first derive the mass conservation equation from an Eulerian point of view;that is to say, our reference frame is fixed in space and the fluid flows through it.


Material and Eulerian Derivatives

The material derivative of a scalar (φ) and a vector (b) field are given by:

DφDt

= ∂φ∂t

+ v · ∇φ, DbDt

= ∂b∂t

+ (v · ∇)b. (D.1)

Various material derivatives of integrals are:

DDt

∫VφdV =

∫V

(DφDt

+φ∇ · v)

dV =∫V

(∂φ∂t

+∇ · (φv))

dV, (D.2)

DDt

∫V

dV =∫V∇ · v dV, (D.3)

DDt

∫VρφdV =

∫Vρ

DφDt

dV. (D.4)

These formulae also hold if φ is a vector. The Eulerian derivative of an integralis:

ddt

∫VφdV =

∫V

∂φ∂t

dV, (D.5)

so that

ddt

∫V

dV = 0 andddt

∫VρφdV =

∫V

∂ρφ∂t

dV. (D.6)

Fig. 1.1 Mass conservation in an Eulerian cuboid control volume.

1.2 The Mass Continuity Equation 9

Cartesian derivation

Consider an infinitesimal rectangular parallelepiped control volume ∆V = ∆x∆y∆zthat is fixed in space (Fig. 1.1). Fluid moves into or out of the volume throughits surface, including through its faces in the y–z plane of area ∆A = ∆y∆z atcoordinates x and x + ∆x. The accumulation of fluid within the control volumedue to motion in the x-direction is evidently

∆y∆z[(ρu)(x)− (ρu)(x +∆x)] = −∂(ρu)∂x

∆x∆y∆z. (1.23)

To this must be added the effects of motion in the y- and z-directions, namely

−[∂(ρv)∂y

+ ∂(ρw)∂z

]∆x∆y∆z. (1.24)

This net accumulation of fluid must be accompanied by a corresponding increase offluid mass within the control volume. This is

∂∂t(Density× Volume

)= ∆x∆y∆z∂ρ

∂t, (1.25)

because the volume is constant. Thus, because mass is conserved, (1.23), (1.24)and (1.25) give

∆x∆y∆z[∂ρ∂t

+ ∂(ρu)∂x

+ ∂(ρv)∂y

+ ∂(ρw)∂z

]= 0. (1.26)

Because the control volume is arbitrary the quantity in square brackets must be zeroand we have the mass continuity equation:

∂ρ∂t

+∇ · (ρv) = 0. (1.27)

Vector derivation

Consider an arbitrary control volume V bounded by a surface S, fixed in space, withby convention the direction of S being toward the outside of V , as in Fig. 1.2. Therate of fluid loss due to flow through the closed surface S is then given by

Fluid loss =∫Sρv · dS =

∫V∇ · (ρv)dV (1.28)

using the divergence theorem. This must be balanced by a change in the mass Mof the fluid within the control volume, which, since its volume is fixed, implies adensity change. That is

Fluid loss = −dMdt

= − ddt

∫Vρ dV = −

∫V

∂ρ∂t

dV. (1.29)

Equating (1.28) and (1.29) yields∫V

[∂ρ∂t

+∇ · (ρv)]

dV = 0 (1.30)

Because the volume is arbitrary, the integrand must vanish and we recover (1.27).


Figure 1.2 Mass conserva-tion in an arbitrary Eu-lerian control volume Vbounded by a surfaceS. The mass gain,∫V (∂ρ/∂t)dV is equal to

the mass flowing into thevolume, −

∫S(ρv) · dS =

−∫V ∇ · (ρv)dV .

1.2.2 Mass continuity via the material derivative

We now derive the mass continuity equation (1.27) from a material perspective.This is the most fundamental approach of all since the principle of mass conserva-tion states simply that the mass of a given element of fluid is, by definition of theelement, constant. Thus, consider a small mass of fluid of density ρ and volume∆V. Then conservation of mass may be represented by

DDt(ρ∆V) = 0 (1.31)

Both the density and the volume of the parcel may change, so

∆VDρDt

+ ρD∆VDt

= ∆V(

DρDt

+ ρ∇ · v)= 0 (1.32)

where the second expression follows using (1.15). Since the volume element isarbitrary, the term in brackets must vanish and

DρDt

+ ρ∇ · v = 0. (1.33)

After expansion of the first term this becomes identical to (1.27). This result maybe derived more formally by re-writing (1.31) as the integral expression

DDt

∫Vρ dV = 0. (1.34)

Expanding the derivative using (1.18) gives

DDt

∫Vρ dV =

∫V

(DρDt

+ ρ∇ · v)

dV = 0. (1.35)

Because the volume over which the integral is taken is arbitrary the integrand itselfmust vanish and we recover (1.33). Summarizing, equivalent partial differential

1.3 The Momentum Equation 11

equation representing conservation of mass are:

DρDt

+ ρ∇ · v = 0,∂ρ∂t

+∇ · (ρv) = 0 . (1.36a,b)

1.2.3 A general continuity equation

The derivation of continuity equation for a general scalar property of a fluid issimilar to that for density, except that there may be an external source or sink, andpotentially a means of transferring the property from one location to another thanby fluid motion, for example by diffusion. If ξ is the amount of some property of thefluid per unit volume (which we will call the concentration of the property), and ifthe net effect per unit volume of all nonconservative processes is denoted by Qv[ξ],then the continuity equation for concentration may be written:

DDt(ξ∆V) = Qv[ξ]∆V (1.37)

Expanding the left hand side and using (1.15) we obtain

DξDt

+ ξ∇ · v = Qv[ξ] (1.38)

or equivalently∂ξ∂t+∇ · (ξv) = Qv[ξ]. (1.39)

If we are interested in a tracer that is normally measured per unit mass of fluid(which is typical when considering thermodynamic quantities) then the conserva-tion equation would be written

DDt(ϕρ∆V) = Qm[ϕ]ρ∆V, (1.40)

where ϕ is the tracer mixing ratio — that is, the amount of tracer per unit fluidmass — and Qm[ϕ] represents nonconservative sources per unit mass. Then, sinceρ∆V is constant we obtain

DϕDt

= Qm[ϕ]. (1.41)

The source term Qm[ϕ] is evidently equal to the rate of change of ϕ of a fluidelement. When this is so, it is common to write it simply as ϕ, so that

DϕDt

= ϕ. (1.42)

A tracer obeying (1.42) with ϕ = 0 is said to be materially conserved. If a traceris materially conserved except for the effects of nonconservative sources then it issometimes said to be ‘semi-materially conserved’.

1.3 THE MOMENTUM EQUATION

The momentum equation is a partial differential equation that describes how thevelocity or momentum of a fluid responds to internal and imposed forces. We willderive it using material methods and informally deducing the terms representingthe pressure, gravitational and viscous forces.


1.3.1 Advection

Let m(x,y, z, t) be the momentum-density field (momentum per unit volume) ofthe fluid. Thus, m = ρv and the total momentum of a volume of fluid is given bythe volume integral

∫V mdV . Now, for a fluid the rate of change of a momentum

of an identifiable fluid mass is given by the material derivative, and by Newton’ssecond law this is equal to the force acting on it. Thus,

DDt

∫Vρv dV =

∫VF dV (1.43)

Now, using (1.21b) (with χ replaced by v) to transform the left-hand side of (1.43),we obtain ∫

V

(ρ

DvDt

− F)

dV = 0. (1.44)

Because the volume is arbitrary the integrand itself must vanish and we obtain

ρDvDt

= F, or∂v∂t

+ (v · ∇)v = Fρ, (1.45a,b)

having used (1.8) to expand the material derivative.We have thus obtained an expression for how a fluid accelerates if subject to

known forces. These forces are however not all external to the fluid itself; a stressarises from the direct contact between one fluid parcel and another, giving rise topressure and viscous forces, sometimes referred to as contact forces. Because acomplete treatment of these would be very lengthy, and is available elsewhere, wetreat both of these very informally and intuitively.

1.3.2 The pressure force

Within or at the boundary of a fluid the pressure is the normal force per unit areadue to the collective action of molecular motion. Thus

dFp = −p dS. (1.46)

where p is the pressure, Fp is the pressure force, and dS an infinitesimal surfaceelement. If we grant ourselves this intuitive notion, it is a simple matter to assessthe influence of pressure on a fluid, for the pressure force on a volume of fluid isthe integral of the pressure over the its boundary and so

Fp = −∫Sp dS. (1.47)

The minus sign arises because the pressure force is directed inward, whereas Sis a vector normal to the surface and directed outward. Applying a form of thedivergence theorem to the right-hand side gives

Fp = −∫V∇p dV (1.48)

where the volume V is bounded by the surface S. The pressure force per unitvolume is therefore just −∇p, and inserting this into (1.45b) we obtain

∂v∂t

+ (v · ∇)v = −∇pρ+ F′, (1.49)

where F′ includes only viscous and body forces.

1.3 The Momentum Equation 13

µ (kg m−1 s−1) ν (m2 s−1)Air 1.8 10−5 1.5 10−5

Water 1.1 10−3 1.1 10−6

Mercury 1.6 10−3 1.2 10−7

Table 1.1 Experimental values of viscos-ity for air, water and mercury at roomtemperature and pressure.

1.3.3 Viscosity and diffusion

Viscosity, like pressure, is a force due to the internal motion of molecules. Theeffects of viscosity are apparent in many situations — the flow of treacle or volcaniclava are obvious examples. In other situations, for example large-scale flow theatmosphere, viscosity is to a first approximation negligible. However, for a constantdensity fluid viscosity is the only way that energy may be removed from the fluid,so that if energy is being added in some way viscosity must ultimately becomeimportant if the fluid is to reach an equilibrium where energy input equals energydissipation. When tea is stirred in a cup, it is viscous effects that cause the fluid toeventually stop spinning after we have removed our spoon.

A number of textbooks3 show that, for most Newtonian fluids, the viscous forceper unit volume is equal to µ∇2v, where µ is the viscosity. Although not exact, thisis an extremely good approximation for most liquids and gases. With this term, themomentum equation becomes,

∂v∂t+ (v · ∇)v = −1

ρ∇p + ν∇2v (1.50)

where ν ≡ µ/ρ is the kinematic viscosity. For gases, dimensional arguments suggestthat the magnitude of ν should be given by

ν ∼⟨mean free path×mean molecular velocity

⟩(1.51)

which for a typical molecular velocity of 300 m s−1 and a mean free path of 7 ×10−8 m gives the not unreasonable estimate of 2.1× 10−5 m2 s−1, within a factor oftwo of the experimental value (table 1.1). Interestingly, the kinematic viscosity isless for water and mercury than it is for air.

1.3.4 Hydrostatic balance

The vertical component — meaning the component parallel to the gravitationalforce — of the momentum equation is

DwDt

= −1ρ∂p∂z

− g (1.52)

where w is the vertical component of the velocity. If the fluid is static the gravita-tional term is balanced by the pressure term and we have

∂p∂z

= −ρg, (1.53)


and this relation is known as hydrostatic balance, or hydrostasy. It is clear in this casethe pressure at a point is given by the weight of the fluid above it, provided p = 0 atthe top of the fluid. It might also appear that (1.53) would be a good approximationto (1.52) provided vertical accelerations, Dw/Dt, are small compared to gravity,which is nearly always the case in the atmosphere and ocean. While this statementis true if we need only a reasonable approximate value of the pressure at a pointor in a column, the satisfaction of this condition is not sufficient to ensure that(1.53) provides an accurate enough pressure to determine the horizontal pressuregradients responsible for producing motion. We return to this point in section 2.7.

1.4 THE EQUATION OF STATE

In three dimensions the momentum and continuity equations provide four equa-tions, but contain five unknowns — three components of velocity, density and pres-sure. Obviously other equations are needed, and an equation of state relates thevarious thermodynamic variables to each other. The conventional equation of stateis an expression that relates temperature, pressure, composition (the mass fractionof the various constituents), and density, and we may write, rather generally,

p = p(ρ, T , µn), (1.54)

where µn is mass fraction of the n’th constituent. An equation of this form is not themost fundamental equation of state from a thermodynamic perspective, an issue wevisit later, but it connects readily measurable quantities.

For an ideal gas the equation of state is

p = ρRT, (1.55)

where R is the gas constant for air and T is temperature. (R is related to theuniversal gas constant Ru by R = Ru/m where m is the mean molecular weightof the constituents of the gas. Also, R = nk where k is Boltzmann’s constant andn is the number of molecules per unit mass.) For dry air, R = 287 J kg−1 K−1. Airhas virtually constant composition except for variations in water vapour content.A measure of this is the water vapour mixing ratio, w = ρw/ρd where ρw and ρdare the densities of water vapour and dry air, respectively, and in the atmosphere wvaries between 0 and 0.03. This variation makes the gas constant in the equationof state a weak function of water vapour mixing ratio; that is, p = ρReffT whereReff = Rd(1 +wRv/Rd)/(1 +w) where Rd and Rv are the gas constants of dry airand water vapour. Since w ∼ 0.01 the variation of Reff is quite small and is oftenignored, especially in theoretical studies.4

For a liquid such as seawater no expression like (1.55) is easily derivable, andsemi-empirical equations are usually resorted to. For pure water in a laboratorysetting a reasonable approximation of the equation of state is ρ = ρ0[1−βT (T−T0)],where βT is a thermal expansion coefficient and S0 and T0 are constants. However,in the ocean the density is affected by pressure and dissolved salts: seawater isa solution of many ions in water — chloride (≈ 1.9% by weight) sodium (1%),sulfate (0.26%), magnesium (0.13%) and so on, with a total average concentrationof about 35‰ (ppt, or parts per thousand). The ratio of the fractions of these saltsis more-or-less constant throughout the ocean, and their total concentration may

1.4 The Equation of State 15

be parameterized by a single measure, the salinity, S. Given this, the density ofseawater is a function of three variables — pressure, temperature, and salinity —and we may write

α = α(T , S,p) (1.56)

where α = 1/ρ is the specific volume. For small variations around a reference valuewe have

dα =(∂α∂T

)S,p

dT +(∂α∂S

)T ,p

dS +(∂α∂p

)T ,S

dp

= α(βT dT − βS dS − βp dp),(1.57)

where the second line serves to define the thermal expansion coefficient βT , thesaline contraction coefficient βS , and the compressibility coefficient βp (equal to αdivided by the bulk modulus). These are in general not constants, but for smallvariations around a reference state they may be treated as such and we have

α = α0

[1+ βT (T − T0)− βS(S − S0)− βp(p − p0)

]. (1.58)

Typical values of these parameters, with variations typically encountered throughthe ocean, are: βT ≈ 2 (±1.5) × 10−4 K−1 (values increase with both temperatureand pressure), βS ≈ 7.6 (±0.2) × 10−4 ppt−1, βp ≈ 4.1 (±0.5) × 10−10 Pa−1. Sincethe variations around the mean density are small (1.58) implies that

ρ = ρ0

[1− βT (T − T0)+ βS(S − S0)+ βp(p − p0)

]. (1.59)

A linear equation of state for seawater is emphatically not accurate enough forquantitative oceanography; the β parameters in (1.58) themselves vary with pres-sure, temperature and (more weakly) salinity so introducing nonlinearities to theequation. The most important of these are captured by an equation of state of theform

α = α0

[1+ βT (1+ γ∗p)(T − T0)+

β∗T2(T − T0)2 − βS(S − S0)− βp(p − p0)

].

(1.60)The starred constants β∗T and γ∗ capture the leading nonlinearities: γ∗ is the ther-mobaric parameter and β∗T is the second thermal expansion coefficient. Even thisexpression has quantitative deficiencies and more complicated semi-empirical for-mulae are often used if high accuracy is needed.5 More discussion is to be found insection 1.8.2.

Clearly, the equation of state introduces, in general, a sixth unknown, temper-ature, and we will have to introduce another physical principle — the first law ofthermodynamics or the principle of energy conservation — to obtain a complete setof equations. However, if the equation of state were such that it linked only densityand pressure, without introducing another variable, then the equations would becomplete; the simplest case of all is a constant density fluid for which the equationof state is just ρ = constant. A fluid for which the density is a function of pressurealone is called a barotropic fluid; otherwise, it is a baroclinic fluid. (In this context,‘barotropic’ is a shortening of ‘auto-barotropic’, which is the original phrase.) Equa-tions of state of the form p = Cργ , where γ is a constant, are sometimes called‘polytropic’.


1.5 THE THERMODYNAMIC EQUATION

1.5.1 A few fundamentals

A fundamental postulate of thermodynamics is that the internal energy of a sys-tem in equilibrium is a function of its extensive properties volume, entropy, and themass of its various constituents.6 (Extensive means that their value depends of theamount of material present, as opposed to an intensive quantity such as tempera-ture.) For our purposes it is more convenient to divide all of these by the mass offluid present, so expressing the internal energy per unit mass, I, as a function ofthe specific volume (or inverse density) α = ρ−1, the specific entropy η, and themass fractions of its various components, or its chemical composition, which weparameterize as its salinity S. Thus we have

I = I(α,η, S), (1.61a)

or an equivalent equation for entropy,

η = η(I,α, S). (1.61b)

Given the functional forms on the right-hand sides, either of these constitutes acomplete description of the macroscopic state of a system in equilibrium, and wecall them the fundamental equation of state. The first differential of (1.61a) gives,formally,

dI = ∂I∂α

dα+ ∂I∂η

dη+ ∂I∂S

dS. (1.62)

We will now ascribe physical meaning to these differentials.Conservation of energy states that the internal energy of a body may change

because of work done by or on it, or because of a heat input, or because of a changein its chemical composition. We write this as

dI = dQ− dW + dC (1.63)

where dW is the work done by the body, dQ is the heat input to the body, anddC accounts for the change in internal energy caused by a change in its chemicalcomposition (e.g., its salinity). This is the first law of thermodynamics. It is appli-cable to a definite fluid mass, so we can regard dI as the change in internal energyper unit mass, and similarly for the other quantities. Let us consider the causes ofvariations in these quantities.

Heat Input: The heat input dQ is not the differential of any quantity, and we cannotunambiguously define the heat content of a body as a function of its state.However, the second law of thermodynamics provides a relationship betweenthe heat input and the change in the entropy of a body, namely that in an(infinitesimal) quasi-static or reversible process, with constant composition,

Tdη = dQ, (1.64)

where η is the specific entropy of the body. The entropy is a function of thestate of a body and is, by definition, an adiabatic invariant. Entropy itself isan extensive quantity, meaning that if we double the amount of material thenwe double the entropy. We will be dealing with the amount of a quantity perunit mass, so that η is the specific entropy, although we will often refer to itjust as the entropy.

1.5 The Thermodynamic Equation 17

Work done: The work done by a body is equal to the pressure times the change inits volume. Thus, per unit mass, we have

dW = p dα, (1.65)

where α = 1/ρ is the specific volume of the fluid and p is the pressure.

Composition: The change in internal energy due to compositional changes is relatedto the change in salinity by

dC = µ dS, (1.66)

where µ is the chemical potential of the solution. The salinity of a parcel offluid is conserved unless there are explicit sources and sinks, such as precip-itation and evaporation at the surface and molecular diffusion. When theseeffects do occur the internal energy of a fluid parcel changes by (1.66). How-ever, these effects are usually small, and most important effect of salinity isthat it changes the density of seawater. In the atmosphere the composition ofa parcel of air primarily varies according to the amount of water vapour in it;however, the main importance of water vapour is that when condensation orevaporation occurs, heat is released (or required) which provides an entropysource in (1.64).

Collecting equations (1.63) – (1.66) together we have

dI = T dη− p dα+ µ dS . (1.67)

We refer to this (often with dS = 0) as the fundamental thermodynamic relation.The fundamental equation of state, (1.61), describes the properties of a particularfluid, and the fundamental relation, (1.67), expresses the conservation of energy.Much of classical thermodynamics follows from these two expressions.

1.5.2 * More thermodynamic relations

From (1.67) it follows that

T =(∂I∂η

)α,S, p = −

(∂I∂α

)η,S, µ =

(∂I∂S

)η,α. (1.68a,b,c)

These may be regarded as the defining relations for these variables; it is because ofthe use of (1.67), and not just the formal expression (1.62), that the pressure andtemperature defined this way are indeed related to the internal motion of motionof the molecules that constitute the fluid. Note that if we write

dη = 1T

dI + pT

dα− µT

dS, (1.69)

it is also clear that

p = T(∂η∂α

)I,S, T−1 =

(∂η∂I

)α,S, µ = −T

(∂η∂S

)I,α. (1.70a,b,c)


In the following derivations, we will unless noted suppose that the composition ofa fluid parcel is fixed, and drop the suffix S on partial derivatives unless ambiguitymight arise.

Because the right-hand side of (1.67) is equal to an exact differential, the secondderivatives are independent of the order of differentiation. That is,

∂2I∂η ∂α

= ∂2I∂α∂η

(1.71)

and therefore, using (1.68) (∂T∂α

)η= −

(∂p∂η

)α. (1.72)

This is one of the Maxwell relations, which are a collection of four similar relationswhich follow directly from the fundamental thermodynamic relation (1.67) andsimple relations between second derivatives. A couple of others will be useful.

Define the enthalpy of a fluid by

h ≡ I + pα (1.73)

then, for a parcel of constant composition, (1.67) becomes

dh = T dη+αdp. (1.74)

But h is a function only of η and p so that in general

dh =(∂h∂η

)p

dη+(∂h∂p

)η

dp. (1.75)

Comparing the last two equations we have

T =(∂h∂η

)p

and α =(∂h∂p

)η. (1.76)

Noting that∂2h∂η∂p

= ∂2h∂p ∂η

(1.77)

we evidently must have (∂T∂p

)η=(∂α∂η

)p, (1.78)

and this is our second Maxwell relation.To obtain the third, we write

dI = Tdη− p dα = d(Tη)− ηdT − d(pα)+αdp, (1.79)

ordG = −ηdT +αdp, (1.80)

where G ≡ I − Tη+ pα is called the ‘Gibbs free energy’. Now, formally, we have

dG =(∂G∂T

)p

dT +(∂G∂p

)T

dp. (1.81)


Comparing the last two equations we see that η = −(∂G/∂T )p and α = (∂G/∂p)T .Furthermore, because

∂2G∂p ∂T

= ∂2G∂T ∂p

(1.82)

we have our third Maxwell equation,(∂η∂p

)T= −

(∂α∂T

)p. (1.83)

The fourth Maxwell equation, whose derivation is left to the reader, is(∂η∂α

)T=(∂p∂T

)α, (1.84)

and all four Maxwell equations are summarized in the box on the next page. Allof them follow from the fundamental thermodynamic relation, (1.67), which is thereal silver hammer of thermodynamics.

Equation of state revisited

The fundamental equation of state (1.61) gives complete information about a fluidin thermodynamic equilibrium, and given this we can obtain expressions for thetemperature, pressure and chemical potential using (1.68). These are also equa-tions of state; however, each of them contains less information than the funda-mental equation because a derivative has been taken, although all three togetherprovide the same information. Equivalent to the fundamental equation of state are,using (1.74), an expression for the enthalpy as a function of pressure, entropy andcomposition, or, using (1.80) the Gibbs function as a function of pressure, temper-ature and composition. (Of these, the Gibbs function is often the most practicallyuseful because the pressure, temperature and composition may all be measured inthe laboratory.) The conventional equation of state, (1.54), is obtained by elimi-nating entropy between (1.68a) and (1.68b). Given the fundamental equation ofstate, the thermodynamic state of a body is fully specified by a knowledge of anytwo of p,ρ, T , η and I, plus its composition.

One simple fundamental equation of state is to take the internal energy to be afunction of density and not entropy; that is, I = I(α). Bodies with such a propertyare called homentropic. Using (1.68), temperature and chemical potential haveno role in the fluid dynamics and the density is a function of pressure alone —the defining property of a barotropic fluid. Neither water nor air are, in general,homentropic but under some circumstances the flow may be adiabatic and p = p(ρ)(e.g., problem 1.10).

In an ideal gas the molecules do not interact except by elastic collisions, andthe volume of the molecules is presumed negligible compared to the total volumethey occupy. The internal energy of the gas then depends only on temperature, andnot on the density. A simple ideal gas is an ideal gas for which the heat capacity isconstant, so that

I = cT , (1.85)

where c is a constant. Using this and the conventional ideal gas equation, p = ρRT ,where R is also constant, we can infer the fundamental equation of state; however,


Maxwell’s Relations

The four Maxwell equations are:(∂T∂α

)η= −

(∂p∂η

)α,

(∂T∂p

)η=(∂α∂η

)p,(

∂η∂p

)T= −

(∂α∂T

)p,

(∂η∂α

)T=(∂p∂T

)α.

(M.1)

These imply:∂(T , η)∂(p,α)

≡(∂T∂p

)(∂η∂α

)−(∂T∂α

)(∂η∂p

)= 0. (M.2)

we will defer that until we discuss potential temperature in section 1.5.4. A generalideal gas also obeys p = ρRT , but it has heat capacities that may be a function oftemperature (but only of temperature — see problem 1.12).

Internal energy and specific heats

We can obtain some useful relations between the internal energy and specific heatcapacities, and some useful estimates of their values, by some simple manipulationsof the fundamental thermodynamic relation. Assuming that the composition of thefluid is constant (1.67) is

T dη = dI + p dα, (1.86)

so thatT dη =

(∂I∂T

)α

dT +[(∂I∂α

)T+ p

]dα. (1.87)

From this, we see that the heat capacity at constant volume (or constant α) cv isgiven by

cv ≡ T(∂η∂T

)α=(∂I∂T

)α. (1.88)

Thus, c in (1.85) is equal to cv .Similarly, using (1.74) we have

T dη = dh−αdp =(∂h∂T

)p

dT +[(∂h∂p

)−α

]dp. (1.89)

The heat capacity at constant pressure, cp, is then given by

cp ≡ T(∂η∂T

)p=(∂h∂T

)p. (1.90)

For later use, we define the ratios γ ≡ cp/cv and κ ≡ R/cp.For an ideal gas h = I + RT = T(cv + R). But cp = (∂h/∂t)p, and hence

cp = cv +R, and (γ−1)/γ = κ. Statistical mechanics tells us that for a simple ideal


gas the internal energy is equal to kT/2 per molecule, or RT/2 per unit mass, foreach excited degree of freedom, where k is the Boltzmann constant and R the gasconstant. The diatomic molecules N2 and O2 that comprise most of our atmospherehave two rotational and three translational degrees of freedom, so that I ≈ 5RT/2,and so cv ≈ 5R/2 and cp ≈ 7R/2, both being constants. These are in fact verygood approximations to the measured values for the earth’s atmosphere, and givecp ≈ 103 J kg−1K−1. The internal energy is simply cvT and the enthalpy is cpT . Fora liquid, especially one containing dissolved salts such as seawater, no such simplerelations are possible: the heat capacities are functions of the state of the fluid, andthe internal energy is a function of pressure (or density) as well as temperature.

1.5.3 Thermodynamic equations for fluids

The thermodynamic relations — for example (1.67) — apply to identifiable bodiesor systems; thus, the heat input affects the fluid parcel to which it is applied, and wecan apply the material derivative to the above thermodynamic relations to obtainequations of motion for a moving fluid. But in doing so we make two assumptions:

(i) That locally the fluid is in thermodynamic equilibrium. This means that, al-though the thermodynamic quantities like temperature, pressure and densityvary in space and time, locally they are related by the thermodynamic relationssuch as the equation of state and Maxwell’s relations.

(ii) That macroscopic fluid motions are reversible and so not entropy producing.Thus, the diabatic term dQ represents the entropy sources — such effects asviscous dissipation of energy, radiation, and conduction — whereas the macro-scopic fluid motion itself is adiabatic.

The first point requires that the temperature variation on the macroscopic scalesmust be slow enough that there can exist a volume that is small compared to thescale of macroscopic variations, so that temperature is effectively constant within it,but that is also sufficiently large to contain enough molecules so that macroscopicvariables such as temperature have a proper meaning. Accepting these assumptions,the expression

T dη = dQ (1.91)

implies that we may write

TDηDt

= Q, (1.92)

where Q is the rate of heat input per unit mass. Eq. (1.92) is a thermodynamicequation of motion of the fluid.

For seawater a full specification of its thermodynamic state requires a knowledgeof the salinity S, and this is determined by the conservation equation

DSDt

= S, (1.93)

where S represents effects of evaporation and precipitation at the ocean surface,and molecular diffusion. Somewhat analogously, for atmosphere the thermody-namics involve water vapour whose evolution is given by the conservation of watervapour mixing ratio

DwDt

= w, (1.94)


where w represents the effects of condensation and evaporation. Salt has an im-portant effect on the density of seawater, whereas the effect of water vapour on thedensity of air is slight.

Equation (1.92) is not a useful equation unless the entropy can be related to theother fluid variables, temperature, pressure and density. This can be done usingthe equation of state and the thermodynamic relations we have derived, and is thesubject of the following sections. An ideal gas is the simplest case with which tostart.

1.5.4 Thermodynamic equation for an ideal gas

For a fluid parcel of constant composition the fundamental thermodynamic relationis

dQ = dI + p dα (1.95)

For an ideal gas the internal energy is a function of temperature only and dI = cv dT(also see problems 1.12 and 1.14), so that

dQ = cv dT + p dα or dQ = cp dT −αdp, (1.96a,b)

where the second expression is derived using α = RT/p and and cp−cv = R. Form-ing the material derivative of (1.95) gives the general thermodynamic equation

DIDt

+ pDαDt

= Q. (1.97)

Similarly, for an ideal gas (1.96a,b) respectively give

cvDTDt

+ pDαDt

= Q, or cpDTDt

− RTp

DpDt

= Q. (1.98a,b)

Although (1.98) are equations in the state variables p, T and/or α, time derivativesact on two variables and this is not convenient for many purposes. Using the masscontinuity equation, (1.98a) may be written

cvDTDt

+ pα∇ · v = Q. (1.99)

Alternatively, using the ideal gas equation we may eliminate T in favor of p and α,giving the equivalent equation

DpDt

+ γp∇ · v = QρRcv. (1.100)

Potential temperature and potential density

When a fluid parcel changes pressure adiabatically, it will expand or contract and,using (1.96b), its temperature change is determined by

cp dT = αdp. (1.101)

As this temperature change is not caused by diabatic effects (e.g., heating), it isuseful to define a temperature-like quantity that changes only if diabatic effects are


present. To this end, we define the potential temperature, θ, to be the temperaturethat a fluid would have if moved adiabatically to some reference pressure (oftentaken to be the 1000 hPa, which is close to the pressure at the earth’s surface).Thus, in adiabatic flow the potential temperature of a fluid parcel is conserved,essentially by definition, and

DθDt

= 0. (1.102)

For this equation to be useful we must be able to relate θ to the other thermody-namic variables. For an ideal gas we use (1.96b) and the equation of state to writethe thermodynamic equation as

dη = cp d lnT − R d lnp. (1.103)

The definition of potential temperature then implies that

cp d lnθ = cp d lnT − R d lnp, (1.104)

and this is satisfied by

θ = T(pRp

)κ(1.105)

where pR is a reference pressure and κ = R/cp. Note that

dη = cpdθθ

(1.106)

and, if cp is constant,η = cp lnθ. (1.107)

Equation (1.106) is in fact a general expression for potential temperature of a fluidparcel of constant composition (see section 1.8.1), but (1.107) applies only if cpis constant, as it is, to a good approximation, in the earth’s atmosphere. Using(1.104), the thermodynamic equation in the presence of heating is then

cpDθDt

= θTQ , (1.108)

with θ given by (1.105). Equations (1.99), (1.100) and (1.108) are all equivalentforms of the thermodynamic equation for an ideal gas.

The potential density, ρθ, is the density that a fluid parcel would have if movedadiabatically and at constant composition to a reference pressure, pR. If the equa-tion of state is written as ρ = f(p, T) then the potential density is just

ρθ = f(pR, θ). (1.109)

For an ideal gas we therefore have

ρθ =pRRθ

; (1.110)

that is, potential density is proportional to the inverse of potential temperature. Wemay also write (1.110) as

ρθ = ρ(pRp

)1/γ

. (1.111)


Finally, for later use we note that for small variations around a reference statemanipulation of the ideal gas equation gives

δθθ= δTT− κ δp

p= 1γδpp− δρρ. (1.112)

* Potential temperature and the fundamental equation of state

Eq. (1.107) is closely related to the fundamental equation of state: using I = cvT ,(1.105), and the equation of state p = ρRT , we can express the entropy explicitlyin terms of the density and the internal energy, to wit

η = cv ln I − R lnρ + constant . (1.113)

This is the fundamental equation of state for a simple ideal gas. If we were to beginwith this, we could straightforwardly derive all the thermodynamic quantities ofinterest for a simple ideal gas: for example, using (1.70a) we immediately recoverP = ρRT , and from (1.70b) we obtain I = cvT . Indeed, (1.113) could be used todefine a simple ideal gas, but such an a priori definition may seem a little unmoti-vated. Of course the heat capacities must still be determined by experiment or bya kinetic theory — they are not given by the thermodynamics, and (1.113) holdsonly if they are constant.

1.5.5 * Thermodynamic equation for liquids

For a liquid such as seawater no simple exact equation of state exists. Thus, al-though (1.108) holds at constant salinity for a liquid by virtue of the definition ofpotential temperature, an accurate expression then relating potential temperatureto the other thermodynamic variables is nonlinear, complicated and, to most eyes,uninformative. Yet for both theoretical and modelling work a thermodynamic equa-tion is needed to represent energy conservation, and an equation of state needed toclose the system, and one of two approaches is thus generally taken: For most theo-retical work and for idealized models a simple analytic but approximate equation ofstate is used, but in situations where more accuracy is called for, such as quantita-tive modelling or observational work, an accurate but complicated semi-empiricalequation of state is used. This section outlines how relatively simple thermody-namic equations may be derived that are adequate in many circumstances, and thatillustrate the principles used in deriving more complicated equations.

Thermodynamic equation using pressure and density

If we regard η as a function of pressure and density (and salinity if appropriate) weobtain

Tdη = T(∂η∂ρ

)p,S

dρ + T(∂η∂p

)ρ,S

dp + T(∂η∂S

)ρ,p

dS

= T(∂η∂ρ

)p,S

dρ − T(∂η∂ρ

)p,S

(∂ρ∂p

)η,S

dp + T(∂η∂S

)ρ,p

dS. (1.114)


From this, and using (1.92) and (1.93), we obtain for a moving fluid

T(∂η∂ρ

)p,S

DρDt

− T(∂η∂ρ

)p,S

(∂ρ∂p

)η,S

DpDt

= Q− T(∂η∂S

)ρ,pS. (1.115)

But (∂p/∂ρ)η,S = c2s where cs is the speed of sound (see section 1.6). This is a

measurable quantity in a fluid, and often nearly constant, and so useful to keep inan equation. Then the thermodynamic equation may be written in the form

DρDt

− 1c2s

DpDt

= Q[ρ] , (1.116)

where Q[ρ] = (∂ρ/∂η)p,SQ/T − (∂ρ/∂S)ρ,pS appropriately represents the effectsof all diabatic and salinity source terms. This form of the thermodynamic equationis valid for both liquids and gases.

Approximations using pressure and density

The speed of sound in a fluid is related to its compressibility — the less compressiblethe fluid, the greater the sound speed. In a fluid it is often sufficiently high that thesecond term in (1.116) can be neglected, and the thermodynamic equation takesthe simple form:

DρDt

= Q[ρ]. (1.117)

This equation is a very good approximation for many laboratory fluids. Note thatthis equation is a thermodynamic equation, arising from the principle of conserva-tion of energy for a liquid. It is a very different equation from the mass conservationequation, which for compressible fluids is also an evolution equation for density.

In the ocean the enormous pressures resulting from columns of seawater kilo-meters deep mean that although the the second term in (1.116) may be small, it isnot negligible, and a better approximation results if we suppose that the pressure isgiven by the weight of the fluid above it — the hydrostatic approximation. In thiscase dp = −ρg dz and (1.116) becomes

DρDt

+ ρgc2s

DzDt

= Q[ρ]. (1.118)

In the second term the height field varies much more than the density field, so agood approximation is to replace ρ by a constant, ρ0, in this term only. Taking thespeed of sound also to be constant gives

DDt

[ρ + ρ0z

Hρ

]= Q[ρ] (1.119)

whereHρ = c2

s /g (1.120)

is the density scale height of the ocean. In water, cs ≈ 1500 m s−1 so that Hρ ≈200 km. The quantity in square brackets in (1.119) is (in this approximation) thepotential density, this being the density that a parcel would have if moved adiabat-ically and with constant composition to the reference height z = 0. The density


scale height as defined here is due to the mean compressibility (i.e., the change indensity with pressure) of seawater and, because sound speed varies only slightly inthe ocean, this is nearly a constant. The adiabatic lapse rate of density is the rate atwhich the density of a parcel changes when undergoing an adiabatic displacement.From (1.119) it is approximately

−(∂ρ∂z

)η≈ ρ0gc2s≈ 5 (kg m−3)/km (1.121)

so that if a parcel is moved adiabatically from the surface to the deep ocean (5 kmdepth, say) its density its density will increase by about 25 kg m−3, a fractionaldensity increase of about 1/40 or 2.5%.

Thermodynamic equation using pressure and temperature

Taking entropy to be a function of pressure and temperature (and salinity if appro-priate) we have

Tdη = T(∂η∂T

)p,S

dT + T(∂η∂p

)T ,S

dp + T(∂η∂S

)T ,p

dS

= cpdT + T(∂η∂p

)T ,S

dp + T(∂η∂S

)T ,p

dS. (1.122)

For a moving fluid, and using (1.92) and (1.93), this implies,

DTDt

+ Tcp

(∂η∂p

)T ,S

DpDt

= Q[T]. (1.123)

where Q[T] = Q/cp − Tc−1p S(∂η/∂S) includes the effects of the entropy and saline

source terms. Now substitute the Maxwell relation (1.83) in the form(∂η∂p

)T= 1ρ2

(∂ρ∂T

)p

(1.124)

to giveDTDt

+ Tcpρ2

(∂ρ∂T

)p

DpDt

= Q[T], (1.125a)

or, equivalently,DTDt

− Tcp

(∂α∂T

)p

DpDt

= Q[T]. (1.125b)

The density and temperature are related through a measurable coefficient of ther-mal expansion βT where (

∂ρ∂T

)p= −βTρ (1.126)

Equation (1.125) then becomes

DTDt

− βTTcpρ

DpDt

= Q[T] . (1.127)

This is form of the thermodynamic equation is valid for both liquids and gases, andin an ideal gas βT = 1/T .


Approximations using pressure and temperature

Liquids are characterized by a small coefficient of thermal expansion, and it is some-times acceptable in laboratory fluids to neglect the second term on the left-hand sideof (1.127). We then obtain an equation analogous to (1.117), namely

DTDt

= Q[T]. (1.128)

This approximation relies on the smallness of the coefficient of thermal expansion.A better approximation is to again suppose that the pressure in (1.127) varies ac-cording only to the weight of the fluid above it. Then dp = −ρgdz and (1.127)becomes

1T

DTDt

+ βTgcp

DzDt

= Q[T]T

. (1.129)

For small variations of T , and if βT is nearly constant, this simplifies to

DDt

(T + T0z

HT

)= Q[T] (1.130)

whereHT = cp/(βTg) (1.131)

is the temperature scale height of the fluid. The quantity T + T0z/HT is (in thisapproximation) the potential temperature, θ, so called because it is the temperaturethat a fluid at a depth z would have if moved adiabatically to a reference depth,here taken as z = 0 — the temperature changing because of the work done by oron the fluid parcel as it expands or is compressed. That is,

θ ≈ T + βTgT0

cpz (1.132)

In seawater, however, the expansion coefficient βT and cp are functions of pressureand (1.132) is not good enough for quantitative calculations. With the approximatevalues for the ocean of βT ≈ 2 × 10−4 K−1 and cp ≈ 4 × 103 J kg−1 K−1 we obtainHT ≈ 2000 km.

The adiabatic lapse rate is rate at which the temperature of a parcel changes inthe vertical when undergoing an adiabatic displacement. From (1.129) it is

Γad = −(∂T∂z

)η= TgβT

cp. (1.133)

In general it is a function of temperature, salinity and pressure, but it is a calcu-lable quantity if βT is known. With the oceanic values above, it is approximately0.15 K km−1. Again this is not accurate enough for quantitative oceanography be-cause the expansion coefficient is a function of pressure. Nor is it a good measureof stability, because of the effects of salt.

It is interesting that the scale heights given by (1.120) and (1.131) differ somuch. The first is due to the compressibility of seawater [and so related to c2

s , orβp in (1.59)] whereas the second is due to the change of density with temperature[βT in (1.59)], and is the distance over which the the difference between temper-ature and potential temperature changes by an amount equal to the temperature


Forms of the Thermodynamic Equation

General formFor a parcel of constant composition the thermodynamic equation is

TDηDt

= Q or cpD lnθ

Dt= 1TQ (T.1)

where η is the entropy, θ is the potential temperature, cp lnθ = η and Q is theheating rate. Appplying the first law of thermodynamics Tdη = dI + p dα gives:

DIDt

+ pDαDt

= Q orDIDt

+ RT∇ · v = Q (T.2)

where I is the internal energy.

Ideal gasFor an ideal gas dI = cv dT , and the (adiabatic) thermodynamic equation may bewritten in the following equivalent, exact, forms:

cpDTDt

−αDpDt

= 0,DpDt

+ γp∇ · v = 0,

cvDTDt

+ pα∇ · v = 0,DθDt

= 0,(T.3)

where θ = T(pR/p)κ . The two expressions on the second line are usually the mostuseful in modelling and theoretical work.

LiquidsFor liquids we may usefully write the (adiabatic) thermodynamic equation as aconservation equation for potential temperature θ or potential density ρpot and rep-resent these in terms of other variables. For example:

DθDt

= 0, θ ≈

T (approximately)T + (βTgz/cp) (with some thermal expansion),

(T.4a)

Dρpot

Dt= 0, ρpot ≈

ρ (very approximately)ρ + (ρ0gz/c2

s ) (with some compression).(T.4b)

Unlike (T.3) these are not equivalent forms. More accurate semi-empirical expres-sions that may also include saline effects are often used for quantitative applica-tions.

1.6 Sound Waves 29

itself (i.e., by about 273 K). The two heights differ so much because the value ofthermal expansion coefficient is not directly tied to the compressibility — for exam-ple, fresh water at 4° C has a zero thermal expansion, and so would have an infinitetemperature scale height, but its compressibility differs little from water at 20° C.

In the atmosphere the ideal gas relationship gives βT = 1/T and so

Γad =gcp

(1.134)

which is approximately 10 K km−1. The only approximation involved in derivingthis is the use of the hydrostatic relationship.

Thermodynamic equation using density and temperature

Taking entropy to be a function of density and temperature (and salinity if appro-priate) we have

Tdη = T(∂η∂T

)α,S

dT + T(∂η∂α

)T ,S

dp + T(∂η∂S

)T ,α

dS

= cvdT + T(∂η∂α

)T ,S

dα+ T(∂η∂S

)T ,α

dS. (1.135)

For a moving fluid this implies,

DTDt

+ Tcv

(∂η∂α

)T ,S

DαDt

= Qcv. (1.136)

If density is nearly constant, as in many liquids, then the second term on the left-hand side of (1.136) is small, and also cp ≈ cv .

The thermodynamic equations for a fluid are summarized on page 28, and thecomplete equations of motion for a fluid are summarized on page 30. Also, notethat for ideal gas (1.116) and (1.127) are exactly equivalent to (1.99) or (1.100)(problem 1.11).

1.6 SOUND WAVES

Full of sound and fury, signifying nothing.

William Shakespeare, Macbeth, c. 1606.

We now consider, rather briefly, one of the most common phenomena in fluid dy-namics yet one which is relatively unimportant for geophysical fluid dynamics —sound waves. Sound itself is not a meteorologically or oceanographically importantphenomenon, except in a few special cases, for the pressure disturbance producedby sound waves is a tiny fraction of the ambient pressure and too small to be ofimportance for the circulation. For example, the ambient surface pressure in theatmosphere is about 105 Pa and variations due to large-scale weather phenomenaare about 103 Pa, often larger, whereas sound waves of 70 dB (i.e., a loud conver-sation) produce pressure variations of about 0.06 Pa. (1 dB = 20 log10(p/ps) whereps = 2× 10−5Pa.)


The Equations of Motion of a Fluid

For dry air, or for a salt-free liquid, the complete set of equations of motion may bewritten as follows:

The mass continuity equation:

∂ρ∂t

+∇ · (ρv) = 0. (EOM.1)

If density is constant this reduces to ∇ · v = 0.

The momentum equation:

DvDt

= −∇pρ+ ν∇2v + F, (EOM.2)

where F represents the effects of body forces such as gravity and ν is the kinematicviscosity. If density is constant, or pressure is given as a function of density alone(e.g., p = Cργ), then (EOM.1) and (EOM.2) form a complete system.

The thermodynamic equation:

DθDt

= 1cp

(θT

)Q. (EOM.3)

where Q represents external heating and diffusion, the latter being κ∇2θ where κis the diffusivity.

The equation of state:ρ = g(θ,p) (EOM.4)

where g is a given function. For example, for an ideal gas, ρ = pκR/(Rθpκ−1).

The equations describing fluid motion are called the Euler equations if the viscousterm is omitted, and the Navier-Stokes equations if viscosity is included.7 Some-times the Euler equations are taken to mean the momentum and mass conservationequations for an inviscid fluid of constant density.

The smallness of the disturbance produced by sound waves justifies a lineariza-tion of the equations of motion about a spatially uniform basic state that is a time-independent solution to the equations of motion. Thus, we write v = v0 + v′,ρ = ρ0 + ρ′ (where a subscript 0 denotes a basic state and a prime denotes aperturbation) and so on, substitute in the equations of motion, and neglect termsinvolving products of primed quantities. By choice of our reference frame we willsimplify matters further by setting v0 = 0. The linearized momentum and massconservation equations are then

ρ0∂v′

∂t= −∇p′, (1.137a)

1.7 Compressible and Incompressible Flow 31

∂ρ′

∂t= −ρ0∇ · v′. (1.137b)

(Note that these linear equations do not in themselves determine the magnitude ofthe disturbance.) Now, sound waves are largely adiabatic. Thus,

dpdt

=(∂p∂ρ

)η

dρdt, (1.138)

where (∂p/∂ρ)η is the derivative at constant entropy, whose particular form is givenby the equation of state for the fluid at hand. Then, from (1.137a) – (1.138) weobtain a single equation for pressure,

∂2p′

∂t2= c2

s∇2p′, (1.139)

where c2s = (∂p/∂ρ)η. Eq. (1.139) is the classical wave equation; solutions prop-

agate at a speed cs which may be identified as the speed of sound. For adiabaticflow in an ideal gas, manipulation of the equation of state leads to p = Cργ , whereγ = cp/cv , whence c2

s = γp/ρ = γRT . Values of γ typically range from 5/3 fora monatomic gas to 7/5 for a diatomic gas and so for air, which is almost entirelydiatomic, we find cs ≈ 350 m s−1 at 300 K. In seawater no such theoretical approxi-mation is easily available, but measurements show that cs ≈ 1500 m s−1.

1.7 COMPRESSIBLE AND INCOMPRESSIBLE FLOW

Although there are probably no fluids of truly constant density, in many cases thedensity of a fluid will vary so little that it is a very good approximation to considerthe density effectively constant in the mass conservation equation. The fluid is thensaid to be incompressible. For example, in the earth’s oceans the density varies byless that 5%, even though the pressure at the ocean bottom, a few kilometers belowthe surface, is several hundred times the atmospheric pressure at the surface. Let usfirst consider how the mass conservation equation simplifies when density is strictlyconstant, and then consider conditions under which treating density as constant isa good approximation.

1.7.1 Constant density fluids

If a fluid is strictly of constant density then the mass continuity equation, namely

DρDt

+ ρ∇ · v = 0, (1.140)

simplifies easily by neglecting all derivatives of density yielding

∇ · v = 0 . (1.141)

The prognostic equation (1.140) has become a diagnostic equation (1.141), or aconstraint to be satisfied by the velocity at each instant of the fluid motion. Aconsequence of this equation is that the volume of each material fluid element


remains constant. To see this recall the expression for the conservation of mass inthe form

DDt(ρ∆V) = 0. (1.142)

If ρ is constant this reduces to an expression for volume conservation, D∆V/Dt = 0,whence (1.141) is recovered because D∆V/Dt = ∆V∇ · v .

1.7.2 Incompressible flows

An incompressible fluid is one in which the density of a given fluid element does notchange.8 Thus, in the mass continuity equation, (1.140), the material derivative ofdensity is neglected and we recover (1.141). In reality no fluid is truly incompress-ible and for (1.141) to approximately hold we just require that

|Dρ/Dt| |ρ∇ · v|, (1.143)

and we delineate some conditions under which this inequality holds below. Ourworking definition of incompressibility, then, is that in an incompressible fluid densitychanges (from whatever cause) are so small that they have a negligible affect on themass balance, allowing (1.140) to be replaced by (1.141). We do not need to assumethat the densities of differing fluid elements are similar to each other, but in theocean (and in most liquids) it is in fact the case that variations in density, δρ, areeverywhere small compared to the mean density, ρ0. That is, a sufficient conditionfor incompressibility is that

δρρ0

1. (1.144)

The atmosphere is not incompressible and (1.141) does not in general hold there.Note also that the fact that ∇·v = 0 does not imply that we may independently useDρ/Dt = 0. Indeed for a liquid with equation of state ρ = ρ0(1 − βT (T − T0)) andthermodynamic equation cpDT/Dt = Q we obtain

DρDt

= −βTρ0

cpQ. (1.145)

Furthermore, incompressibility does not necessarily imply the neglect of densityvariations in the momentum equation — it is only in the mass continuity equationthat density variations are neglected.

Some conditions for incompressibility

The conditions under which incompressibility is a good approximation to the fullmass continuity equation depend not only on the physical nature of the fluid butalso on the flow itself. The condition that density is largely unaffected by pressuregives one necessary condition for the legitimate use of (1.141), as follows. Firstassume adiabatic flow, and omit the gravitational term. Then

dpdt

=(∂p∂ρ

)η

dρdt

= c2s

dρdt

(1.146)

so that the density and pressure variations of a fluid parcel are related by

δp ∼ c2s δρ. (1.147)

1.8 * More Thermodynamics of Liquids 33

From the momentum equation we estimate

U2

L∼ 1Lδpρ0, (1.148)

where U and L are typical velocities and lengths and where ρ0 is a representativevalue of the density. Using (1.147) and (1.148) gives U2 ∼ c2

s δρ/ρ0. The incom-pressibility condition (1.144) then becomes

U2

c2s 1. (1.149)

That is, for a flow to be incompressible the fluid velocities must be less than thespeed of sound; that is, the Mach number, M ≡ U/cs , must be small.

In the earth’s atmosphere it is apparent that density changes significantly withheight. Assuming hydrostatic balance and an ideal gas, then

1ρ∂p∂z

= −g, (1.150)

and if (for simplicity) we assume that atmosphere is isothermal then

∂p∂z

=(∂p∂ρ

)T

∂ρ∂z

= RT0∂ρ∂z. (1.151)

Using (1.150) and (1.151) gives

ρ = ρ0 exp(−z/Hρ), (1.152)

where Hρ = RT0/g is the (density) scale height of the atmosphere. It is easy to seethat density changes are negligible only if we concern ourselves with motion lessthan the scale height, so this is another necessary condition for incompressibility.

In the atmosphere, although the Mach number is small for most flows, verticaldisplacements often exceed the scale height and in those cases the flow cannot beconsidered incompressible. In the ocean density changes from all causes are smalland in most circumstances the ocean may be considered to contain an incompress-ible fluid. We return to this in the next chapter when we consider the Boussinesqequations.

1.8 * MORE THERMODYNAMICS OF LIQUIDS

1.8.1 Potential temperature, potential density and entropy

For an ideal gas we were able to derive a single prognostic equation for a single vari-able, potential temperature. As potential temperature is in turn simply related tothe temperature and pressure, this is a useful prognostic equation. Can we achievesomething similar with a more general equation of state, with non-constant coeffi-cients of expansion?


22

24

26

28

30

Salinity

Tem

pera

ture

32 34 36 38

0

5

10

15

20

25

30

38

40

42

44

46

48

Salinity32 34 36 38

0

5

10

15

20

25

30

Fig. 1.3 A temperature-salinity diagram for seawater, calculated using an em-pirical equation of state. Contours are (density - 1000) kg m−3, and the tem-perature is potential temperature, which in the deep ocean may be less thanin situ temperature a degree or so (see Fig. 1.4). Left panel: at sea-level(p = 105 Pa = 1000 mb). Right panel: at p = 4 × 107 Pa, a depth of about4 km. Note that in both cases the contours are slightly convex.

Potential temperature

The potential temperature is defined as the temperature that a parcel would haveif moved adiabatically to a given reference pressure pR, often taken as 105 Pa (or1000 hPa, or 1000 mb, approximately the pressure at the sea-surface). Thus it maybe calculated, at least in principle, through an integral of the form

θ(S, T ,p;pR) = T +∫ pRpΓ ′ad(S, T ,p)dp (1.153)

where Γ ′ad = (∂T/∂p)η. The potential temperature of a fluid parcel is directly relatedto its entropy, provided its composition does not change. We already demonstratedthis for an ideal gas, and to see it explicitly in the general case let us first write theequation of state in the form

η = η(S, T ,p). (1.154)

Now, by definition of potential temperature we have

η = η(S, θ;pR) and θ = θ(η, S;pR). (1.155)

For a parcel of constant salinity, changes in entropy are caused only by changes inpotential temperature so that

dη = ∂η(S, θ;pR)∂θ

dθ. (1.156)

Now, if we express entropy as a function of temperature and pressure then

Tdη = T(∂η∂T

)p

dT + T(∂η∂p

)T

dp

= cp dT − T(∂α∂T

)p

dp.(1.157)


using one of the Maxwell relations. Suppose a fluid parcel moves adiabatically, thendη = 0 and, by (1.156), dθ = 0. That is, the potential temperature at each pointalong its trajectory is constant and θ = θ(η). How do we evaluate this function?Simply note that the temperature at the reference pressure, pR, is the potentialtemperature, so that directly from (1.157)

dη = cp(pR, θ)dθθ

, (1.158)

and dη/dθ = cp(pR, θ)/θ. If cp is constant this integrates to

η = cp lnθ + constant, (1.159)

as for a simple ideal gas (1.107).Since potential temperature is conserved in adiabatic motion, the thermody-

namic equation can be written

cpDθDt

= θTQ. (1.160)

where the right-hand side represents heating. (If salinity is changing, then theright-hand side should also include any saline source terms and saline diffusion.However, such terms usually have a very small effect.) This equation is equiva-lent to (1.116) or (1.127), although it is only useful if θ can be simply relatedto the other state variables. In principle this can be done using (1.153), and inpractice empirical relationships have been derived that express potential tempera-ture in terms of the local temperature, pressure and salinity, and density in terms ofpotential temperature, salinity and pressure (see section 1.8.2 for more discussion).

Potential density

Potential density, ρθ, is defined as the density that a parcel would have if movedadiabatically and with fixed composition to a given reference pressure pR often,but not always, taken as 105 Pa, or 1 bar. If the equation of state is of the formρ = ρ(S, T ,p) then by definition we have

ρθ = ρ(S, θ;pR). (1.161)

For a parcel moving adiabatically (i.e., fixed salinity and entropy or potential tem-perature) its potential density is therefore conserved. For an ideal gas (1.161) givesρθ = pR/(Rθ) [as in (1.110)] and potential density provides no more informationthan potential temperature. However, in the oceans potential density accounts forthe effect of salinity on density and so is a much better measure of the static stabilityof a column of water than density itself.

From (1.119) an approximate expression for the potential density in the oceanis

ρθ =(ρ + ρ0gz

c2s

). (1.162)

Although this may suffice for theoretical or some modelling work, the vertical gra-dient of potential temperature in the ocean is often close to zero and a still more ac-curate, generally semi-empirical, expression is needed to determine stability prop-erties.


Because density is so nearly constant in the ocean, it is common to subtract theamount 1000 kg m−3 before quoting its value, and depending on whether this valuerefers to in situ density or the potential density the results are called σT (‘sigma-tee’)or σθ (‘sigma-theta’) respectively. Thus,

σT = ρ(p, T , S)− 1000, σθ = ρ(pR, θ, S)− 1000. (1.163a,b)

If the potential density is referenced to a particular level, this is denoted by a sub-script on the σ . Thus, σ2 is the potential density referenced to 200 bars of pressure,or about 2 kilometers depth.

1.8.2 * More About Seawater

We now consider, rather didactically, some of the properties of the equation of statefor seawater, noting in particular those nonlinearities that, although small, give itsomewhat peculiar properties. We use a prototypical equation of state, (1.60) that,although not highly accurate except for small variations around a reference state,does capture the essential nonlinearities.9 That equation of state may be written as:

α = α0

[1+ βT (1+ γ∗p)(T − T0)+

β∗T2(T − T0)2 − βS(S − S0)− βp(p − p0)

],

(1.164)where we may take p0 = 0 and βp = α0/c′2s , where c′s is a reference sound speed.The starred parameters are associated with the nonlinear terms: β∗T is the secondexpansion coefficient and γ∗ is the ‘thermobaric parameter’, which determines theextent to which the thermal expansion of water depends on pressure. An equationof this form is useful because its coefficients can, in principle, be measured in thefield or in the laboratory, and approximate values are given in table 1.2. However, itmay not be the most useful form for modelling or observational work, because T isnot materially conserved. Let us use this equation to deduce various thermodynamicquantities of interest, and also transform it to a more useful form for modelling.

Potential temperature of seawater

It would be useful to express (1.164) in terms of materially conserved variables, andso in terms of potential temperature rather than temperature. Now, by definitionthe potential temperature is obtained by integrating the adiabatic lapse rate fromthe in situ pressure to the reference pressure (zero); that is

θ − T =∫ z(p=0)

z

(∂T∂z

)η

dz =∫ 0

p

(∂T∂p

)η

dp (1.165)

Using (1.157), the adiabatic lapse rate is(∂T∂p

)η= Tcp

(∂α∂T

)p,S= Tcpα0[βT (1+ γ∗p)+ β∗T (T − T0)]. (1.166)

Now, cp satisfies cp = T(∂η/∂T )p, so that, using the Maxwell relation (1.83),(∂cp∂p

)T ,S= T

(∂∂T

(∂η∂p

)T

)p= T ∂

2α∂T 2 . (1.167)


Parameter Description Value

ρ0 Reference Density 1.027× 103 kg m−3

α0 Reference Specific Volume 9.738× 10−4 m3 kg−1

T0 Reference temperature 283KS0 Reference salinity 35 psu ≈ 35‰cs Reference sound speed 1490 m s−1

βT First thermal expansion coefficient 1.67× 10−4 K−1

β∗T Second thermal expansion coefficient 1.00× 10−5 K−2

βS Haline contraction coefficient 0.78× 10−3 psu−1

βp Inverse bulk modulus (≈ α0/c2s ) 4.39× 10−10 m s2 kg−1

γ∗ Thermobaric parameter (≈ γ′∗) 1.1× 10−8 Pa−1

cp0 Specific heat capacity at constant pressure 3986 J kg−1 K−1

Table 1.2 Various thermodynamic and equation-of-state parameters for sea-water. Specifically, these parameters may be used in the approximate equa-tions of state (1.60) and (1.173).

9.4 9.6 9.8 10

400

200

0

Potential Temperature

Pre

ssur

e (b

ars)

9.4 9.6 9.8 1032

34

36

38

Sal

inity

0 10 20 30

400

2000.2

0.6

1

Temperature

Pre

ssur

e (b

ars)

0

S

L

Fig. 1.4 Examples of variation of potential temperature of seawater with pres-sure, temperature and salinity. Left panel: the sloping lines show potentialtemperature as a function of pressure at fixed salinity (S = 35 psu) and tem-perature (13.36° C). The solid line is computed using an accurate, empiricalequation of state, the almost-coincident dashed line uses the simpler ex-pression (1.172a) and the dotted line (labelled L) uses the linear expression(1.172c). The near vertical solid line, labelled S, shows the variation of poten-tial temperature with salinity at fixed temperature and pressure. Right panel:Contours of the difference between temperature and potential temperature,(T −θ) in the pressure-temperature plane, for S = 35 psu. The solid lines usean accurate empirical formula, and the dashed lines use (1.172). The simplerequation can be improved locally, but not globally, by tuning the coefficients.(100 bars of pressure (107 Pa or 10 MPa) is approximately 1 km depth.)


Thus, for our equation of state, we have(∂cp∂p

)T ,S= −Tα0β∗T , (1.168)

and thereforecp = cp0(T , S)− pTα0β∗T . (1.169)

The first term cannot be determined solely from the conventional equation of state;in fact for seawater specific heat varies very little with temperature (of order onepart in a thousand for a 10 K temperature variation). It varies more with salinity,changing by about −5 J kg−1 K−1 per part-per-thousand change in salinity. Thus wetake

cp0(T , S) = cp1 + cp2(S − S0), (1.170)

where cp1 and cp2 are constants that may be experimentally determined.Using (1.169) and (1.166) in (1.165) gives,

θ = T exp−α0βTp

cp0

[1+ 1

2γ∗p + β

∗TβT(T − T0)

]. (1.171)

This equation is a relationship between T , θ and p analogous to (1.105) for anideal gas. The exponent itself is small, the second and third terms in square brack-ets are small compared to unity, and the deviations of both T and θ from T0 arealso presumed small. Taking advantage of all of this enables the expression to berewritten, with increasing levels of approximation, as

T ′ ≈ T0α0βTcp0

p(

1+ 12γ∗p + T0

α0β∗Tcp0

p)+ θ′

(1+ T0

α0β∗Tcp0

p), (1.172a)

≈ T0α0βTcp0

p(

1+ 12γ∗p

)+ θ′

(1+ T0

α0β∗Tcp0

p), (1.172b)

≈ T0α0βTcp0

p + θ′, (1.172c)

where T ′ = T − T0 and θ′ = θ − T0. The last of the three, (1.172c), holds for a lin-ear equation of state, and is useful for calculating approximate differences betweentemperature and potential temperature; making use of the hydrostatic approxima-tion reveals that it is essentially the same as (1.132). Note that the potential tem-perature is related to temperature via the thermal expansion coefficient and not, asone might naïvely have expected, the compressibility coefficient. Plots of the differ-ence between temperature and potential temperature, that also give both a senseof of the accuracy of these simpler formula, is given in Fig. 1.4.

Using (1.172b) in the equation of state (1.164) we find that, approximately,

α = α0

[1− α0p

c2s+ βT (1+ γ′∗p)θ′ +

12β∗Tθ

′2 − βS(S − S0)], (1.173)

where γ′∗ = γ∗ + T0β∗Tα0/cp0 ≈ γ∗ and c−2s = c′−2

0 − β2TT0/cp ≈ c′−2

0 is a referencevalue of the speed of sound (γ∗ and γ′∗ differ by a few percent, and c2

s and c′2s differby only a few parts in a thousand). Given (1.173), it is in principle straightforward,

1.9 The Energy Budget 39

−10 0 10 20 3020

25

30

35

40

45

50

55D

ensi

ty

temperature32 34 36 38

20

25

30

35

40

45

50

55

salt

Den

sity

0 1 2 3 4 5

x 107

20

25

30

35

40

45

50

pressure

Den

sity

−10 0 10 20 300

0.1

0.2

0.3

0.4

temperature

β T ×

103

S=35, p = 5 × 107

S=35, p = 105θ=5, p = 5 × 107

θ=13, p = 105

S=35, θ=13S=35, θ=30

S=35, p = 5 × 107

S=35, p = 105

(a) (b)

(c) (d)

Fig. 1.5 Examples of the variation of density of seawater (minus 1000 kg m−3)with (a) potential temperature (K); (b) salt (psu); and (c) pressure (Pa),for seawater. Panel (d), shows the thermal expansion coefficient, βT =−ρ−1

0 (∂ρ/∂T )p,S K−1, for each of the two curves in panel (a).

although in practice rather tedious, to compute various thermodynamic quantitiesof interest; a calculation of the buoyancy frequency is given in problem 2.19. Wemay approximate (1.173) further by using the hydrostatic pressure instead of theactual pressure; thus, letting p = −g(z−z0)/α0 where z0 is the nominal value of zat which p = 0, we obtain

α = α0

[1+ g(z − z0)

c2s

+ βT(

1− γ′∗g(z − z0)α0

)θ′ + β

∗T

2θ′2 − βS(S − S0)

].

(1.174)Using z instead of p in the equation of state entails a slight loss of accuracy, butit turns out to be necessary to ensure that the Boussinesq equations maintain goodconservation properties, as discussed in section 2.4. The variation of density withpotential temperature and salinity and pressure is illustrated in Fig. 1.3.

1.9 THE ENERGY BUDGET

The total energy of a fluid includes the kinetic, potential and internal energies. Bothfluid flow and pressure forces will in general move energy from place to place, but


we nevertheless expect energy to be conserved in an enclosed volume. Let us nowconsider what form energy conservation takes in a fluid.

1.9.1 Constant density fluid

For a constant density fluid the momentum equation and the mass continuity equa-tion ∇·v = 0, are sufficient to completely determine the evolution of a system. Themomentum equation is

DvDt

= −∇ (φ+ Φ)+ ν∇2v, (1.175)

where φ = p/ρ0 and Φ is the potential for any conservative force (e.g., gz for auniform gravitational field). We can rewrite the advective term on the left-handside using the identity,

(v · ∇)v = −v ×ω+∇(v2/2), (1.176)

where ω ≡ ∇ × v is the vorticity, discussed more in later chapters. Then, omittingviscosity, we have

∂v∂t

+ω× v = −∇B, (1.177)

where B = (φ+Φ+v2/2) is the Bernoulli function for constant density flow. Considerfor a moment steady flows (∂v/∂t = 0). Streamlines are, by definition, parallel tov everywhere, and the vector v×ω is everywhere orthogonal to the streamlines, sothat taking the dot product of the steady version of (1.177) with v gives v ·∇B = 0.That is, for steady flows the Bernoulli function is constant along a streamline, andDB/Dt = 0.

Reverting back to the time-varying case, take the dot product with v and includethe density to yield

12∂ρ0v2

∂t+ ρ0v · (ω× v) = −ρ0v · ∇B (1.178)

The second term on the left-hand side vanishes identically. Defining the kineticenergy density K, or energy per unit volume, by K = ρ0v2/2, (1.178) becomes anexpression for the rate of change of K,

∂K∂t

+∇ · (ρ0vB) = 0. (1.179)

Because Φ is time-independent this may be written

∂∂t

[ρ0

(12v2 + Φ

)]+∇ ·

[ρ0v

(12v2 + Φ +φ

)]= 0. (1.180)

or∂E∂t+∇ · [v(E + p)] = 0. (1.181)

where E = K+ρ0Φ is the total energy density (i.e, the total energy per unit volume).This has the form of a general conservation equation in which a local change ina quantity is balanced by the divergence of its flux. However, the energy flux,v(ρ0v2/2+ρ0Φ+ρ0φ), is not simply the velocity times the energy density ρ0(v2/2+

1.9 The Energy Budget 41

Φ); there is an additional term, vp, that represents the energy transfer occurringwhen work is done by the fluid against the pressure force.

Now consider a volume through which there is no mass flux, for example adomain bounded by rigid walls. The rate of change of energy within that volume isthen given by the integral of (1.178)

dKdt

≡ ddt

∫VK dV = −

∫V∇ · (ρ0vB)dV = −

∫Sρ0Bv · dS = 0, (1.182)

using the divergence theorem, and where K is the total kinetic energy. Thus, thetotal kinetic energy within the volume is conserved. Note that for a constant densityfluid the gravitational potential energy, P , is given by

P =∫Vρ0gz dV, (1.183)

which is a constant, not affected by a re-arrangement of the fluid. Thus, in a con-stant density fluid there is no exchange between kinetic energy and potential energyand the kinetic energy itself is conserved.

1.9.2 Variable density fluids

We start with the momentum equation

ρDvDt

= −∇p − ρ∇Φ, (1.184)

and take its dot product with v to obtain an equation for the evolution of kineticenergy,

12ρ

DvDt

2

= −v · ∇p − ρv · ∇Φ

= −∇ · (pv)+ p∇ · v − ρv · ∇Φ. (1.185)

From (1.86) the internal energy equation for adiabatic flow is

ρDIDt

= −ρpDαDt

= −p∇ · v (1.186)

where the second equality follows by use of the continuity equation. Finally, andsomewhat trivially, the potential energy density obeys

ρDΦDt

= ρv · ∇Φ. (1.187)

Adding (1.185), (1.186) and (1.187) we obtain

ρDDt

(12v2 + I + Φ

)= −∇ · (pv), (1.188)

which, on expanding the material derivative and using the mass conservation equa-tion, becomes

∂∂t

[ρ(

12v2 + I + Φ

)]+∇ ·

[ρv

(12v2 + I + Φ + p/ρ

)]= 0. (1.189)


This may be written

∂E∂t+∇ ·

[v(E + p)

]= 0 , (1.190)

where E = ρ(v2/2 + I + Φ) is the total energy per unit volume, or the total en-ergy density, of the fluid. This is the energy equation for an unforced, inviscid andadiabatic, compressible fluid. Just as for the constant density case, the energy fluxcontains the term pv that represents the work done against pressure and, again, thesecond term vanishes when integrated over a closed domain with rigid boundaries,implying that the total energy is conserved. However, now there can now be an ex-change of energy between kinetic, potential and internal components. The quantityσ = I + pα + Φ = h + Φ is sometimes referred to as the static energy, or the drystatic energy. However, it is not a component of the globally conserved total energy;the conserved energy contains only the quantity I + Φ plus the kinetic energy, andit is it is only the flux of static energy that affects the energetics. For an ideal gaswe have σ = cvT + RT + Φ = cpT + Φ, and if the potential is caused by a uniformgravitational field then σ = cpT + gz.

Bernoullis’ theorem

For steady flow ∂/∂t = 0 and ∇ · ρv = 0 so that (1.189) may be written v · ∇B = 0where B is the Bernoulli function given by

B =(

12v2 + I + Φ + p/ρ

)=(

12v2 + h+ Φ

). (1.191)

Thus, for steady flow only,DBDt

= v · ∇B = 0, (1.192)

and the Bernoulli function is a constant along streamline. For an ideal gas in aconstant gravitation field B = v2/2+ cpT + gz.

For adiabatic flow we also have Dθ/Dt = 0. Thus, steady flow is both alongsurfaces of constant θ and along surfaces of constant B, and the vector

l = ∇θ ×∇B (1.193)

is parallel to streamlines.10 A related result for unsteady flow is given in section4.8.

1.9.3 Viscous Effects

We might expect that viscosity will always act to reduce the kinetic energy of a flow,and we will demonstrate this for a constant density fluid. Retaining the viscousterm in (1.175), the energy equation becomes

dEdt

≡ ddt

∫VE dV = µ

∫Vv · ∇2v dV. (1.194)

The right hand side is negative definite. To see this we use the vector identity

∇× (∇× v) = ∇(∇ · v)−∇2v, (1.195)

1.10 An Introduction to Non-Dimensionalization and Scaling 43

and because ∇ · v = 0 we have ∇2v = −∇×ω, where ω ≡ ∇× v. Thus,

dEdt

= −µ∫Vv · (∇×ω)dV = −µ

∫Vω · (∇× v)dV = −µ

∫Vω2 dV, (1.196)

after integrating by parts, providing v×ω vanishes at the boundary. Thus, viscosityacts to extract kinetic energy from the flow. The loss of kinetic energy reappearsas an irreversible warming of the fluid (‘Joule heating’), and the total energy of thefluid is conserved, but this effect plays no role in a constant density fluid. The effectis normally locally small, at least in the earth’s ocean and atmosphere, although itis sometimes included in comprehensive General Circulation Models.

1.10 AN INTRODUCTION TO NON-DIMENSIONALIZATION AND SCALING

The units we use to measure length, velocity and so on are irrelevant to the dynam-ics, and not necessarily the most appropriate units for a given problem. Rather, itis convenient to express the equations of motion, so far as is possible, in so-called‘nondimensional’ variables, by which we mean expressing every variable (such asvelocity) as the ratio of its value to some reference value. For velocity the referencecould, for example, be the speed of of light — but this would not be very helpfulfor fluid dynamical problems in the earth’s atmosphere or ocean! Rather, we shouldchoose the reference value as a natural one for a given flow, in order that, so far aspossible, the nondimensional variables are order-unity quantities, and doing this iscalled scaling the equations. Evidently, there is no reference velocity that is univer-sally appropriate, and much of the art of fluid dynamics lies in choosing sensiblescaling factors for the problem at hand. Non-dimensionalization plays an importantrole in fluid dynamics, and we introduce it here with a simple example.

1.10.1 The Reynolds number

Consider the constant-density momentum equation in Cartesian coordinates. If atypical velocity is U , a typical length is L, a typical timescale is T , and a typicalvalue of the pressure deviation is Φ, then the approximate sizes of the various termsin the momentum equation are given by

∂v∂t

+ (v · ∇)v = −∇φ+ ν∇2v (1.197a)

UT

U2

L∼ Φ

LνUL2 . (1.197b)

The ratio of the inertial terms to the viscous terms is (U2/L)/(νU/L2) = UL/ν,

and this is the Reynolds number.11 More formally, we can nondimensionalize themomentum equation by writing

v = vU, x = x

L, t = t

T, φ = φ

Φ, (1.198)

where the terms with hats on are nondimensional values of the variables and thecapitalized quantities are known as scaling values, and these are the approximatemagnitudes of the variables. We choose the nondimensionalization so that the


nondimensional variables are of order unity. Thus, for example, we choose U sothat u = O(U) where this should be taken to mean that the magnitude of the vari-able u is approximately U , or that u ∼ U , and we say that ‘u scales like U ’. [ThisO()notation differs from the conventional mathematical meaning of ‘order’, in whicha = O(εα) represents a limit in which a/εα → constant as ε → 0.] Thus, if there arewell-defined length and velocity scales in the problem, and we choose these scalesto perform the nondimensionalization, then the nondimensional variables are oforder unity. That is, u = O(1), and similarly for the other variables.

Because there are no external forces in this problem, appropriate scaling valuesfor time and pressure are

T = LU, Φ = U2. (1.199)

Substituting (1.198) and (1.199) into the momentum equation we obtain

U2

L

[∂ v∂t

+ (v · ∇)v]= −U

2

L∇φ+ νU

L2 ∇2v, (1.200)

where we use the convention that when ∇ operates on a nondimensional variableit is a nondimensional operator. Eq. (1.200) simplifies to

∂ v∂t

+ (v · ∇)v = −∇φ+ 1Re∇2v, (1.201)

whereRe ≡ UL

ν(1.202)

is, again, the Reynolds number. If we have chosen our length and velocity scalessensibly — that is, if we have scaled them properly — each variable in (1.201) isorder unity, with the viscous term being multiplied by the parameter 1/Re. Thereare two important conclusions:

(i) The ratio of the importance of the inertial terms to the viscous terms is givenby the Reynolds number, defined by (1.202). In the absence of other forces,such as those due to gravity and rotation, the Reynolds number is the onlynon-dimensional parameter explicitly appearing in the momentum equation.Hence its value, along with the boundary conditions, controls the behaviour ofthe system.

(ii) More generally, by scaling the equations of motion appropriately the param-eters determining the behaviour of the system become explicit. Scaling theequations is intelligent nondimensionalization.

Notes

1 Parts of the first few chapters, and many of the problems, draw on notes preparedover the years for a graduate class at Princeton University taught by Steve Garner,Isaac Held, Yoshio Kurihara, Paul Kushner and myself.

2 Joseph-Louis Lagrange (1736–1813) was a Franco-Italian, born and raised in Turinwho then lived and worked mainly in Germany and France. He made notable contri-butions in analysis, number theory and mechanics and was recognized as one of the

Notes and Problems 45

greatest mathematicians of the 18th century. He laid the foundations of the calculusof variations (to wit, the ‘Lagrange multiplier’) and first formulated the principle ofleast action, and his monumental treatise Mécanique Analytique (1788) provides aunified analytic framework (it contains no diagrams, a feature virtually emulated inWhitaker’s Analytical Dynamics, 1927) for all Newtonian mechanics.

Leonard Euler (1707–1783), a Swiss mathematician who lived and worked for ex-tended periods in Berlin and St. Petersburg, made important contributions in manyareas of mathematics and mechanics, including the analytical treatment of algebra,the theory of equations, calculus, number theory and classical mechanics. He wasthe first to establish the form of the equations of motion of fluid mechanics, writ-ing down both the field description of fluids and what we now call the material oradvective derivative.

Truesdell (1954) points out that ‘Eulerian’ and ‘Lagrangian’ coordinates, especiallythe latter, are inappropriate eponyms. The so-called Eulerian description was intro-duced by d’Alembert in 1749 and generalized by Euler in 1752, and the so-calledLagrangian description was introduced by Euler in 1759. The modern confusion ev-idently stems from a monograph by Dirichlet in 1860 that credits Euler in 1757 andLagrange in 1788 for the respective methods.

3 For example Batchelor (1967).

4 Rd and Rv are related by the molecular weights of water and dry air, Mv and Md,so that α ≡ Rd/Rv = Mv/Md = 0.62. Rather than allow the gas constant to vary,meteorologists sometimes incorporate the variation of humidity into the definitionof temperature, so that instead of p = ρReffT we use p = ρRdTv , so defining the‘virtual temperature’, Tv . It is easy to show that Tv ≈ [1+w(α−1−1)]T . AtmosphericGCMs often use a virtual temperature.

5 The form of (1.60) was suggested by de Szoeke (2003). More accurately, and withmuch more complication, the international equation of state of seawater (Unesco1981) is an empirical equation that fits measurements to an accuracy of order 10−5

(see Fofonoff 1985). Generally accurate formulae are also available from Mellor(1991), Wright (1997), and (with particular attention to high accuracy) McDougallet al. (2002). These are all more easily computable than the UNESCO formula. Theformulae of Wright and McDougall et al are of the form:

ρ = p + p0

λ+α0(p + p0)

where α0, p0 and λ are expressed as polynomials in potential temperature and salin-ity, using the Gibbs function of Feistel and Hagen (1995), which is as or more accu-rate than the UNESCO formula. Wright’s formula used are used for Fig. 1.5 and Fig.1.3. Bryden (1973) provides an accurate polynomial formula for potential tempera-ture of seawater in terms of temperature, salinity and pressure, and this is used forFig. 1.4. In most numerical ocean models potential temperature and salinity are theprognostic thermodynamic variables and an empirical equation of state is used tocompute density and potential density.

6 For a development of thermodynamics from its fundamentals see, e.g., Callen (1985).

7 Claude-Louis-Marie-Henri Navier (1785–1836) was a French civil engineer, professorat the École Polytechnique and later at the École des Ponts et Chaussée. He was aexpert in road and bridge building (he developed the theory of suspension bridges)and, relatedly, made lasting theoretical contributions to the theory of elasticity, be-ing the first to publish a set of general equations for the dynamics of an elastic solid.


In fluid mechanics, he laid down the now-called Navier-Stokes equations, includingthe viscous terms, in 1822.

George Gabriel Stokes (1819–1903). Irish born (in Skreen, County Sligo), he heldthe Lucasian chair of Mathematics at Cambridge from 1849 until his retirement.As well as having a role in the development of fluid mechanics, especially throughhis considerations of viscous effects, Stokes worked on the dynamics of elasticitiy,fluorescence, the wave theory of light, and was (rather ill-advisedly in hindsight) aproponent of the idea of an ether permeating all space.

8 Some sources take incompressibility to mean that density is unaffected by pressure,but this alone is insufficient to guarantee that the mass conservation equation canbe approximated by ∇ · v = 0.

9 Following de Szoeke (2003).

10 These results, usually known as Bernoulli’s theorem, were developed mainly byDaniel Bernoulli (1700–1782). They were based on earlier work on the conserva-tion of energy that Daniel had done with his father, Johann Bernoulli (1667–1748),and so perhaps should be known as Bernoullis’ theorem.

11 Osborne Reynolds (1842-1912) was an Irish born (Belfast) physicist who was profes-sor of engineering at Manchester University from 1868–1905. His early work was inelectricity and magnetism, but he is now most famous for his work in hydrodynam-ics. The ‘Reynolds number,’ which determines the ratio of inertial to viscous forces,and the ‘Reynolds stress,’ which is the stress on the mean flow due to the fluctuatingcomponents, are both named after him. He was also one of the first scientists tothink about the concept of group velocity.

Further Reading

There are numerous books on hydrodynamics, some of them being:

Lamb, H. 1932. Hydrodynamics, 6th edn.This is a classic text in the subject, although its notation is now too dated to make itreally useful as an introduction. Another very well-known text is:

Batchelor, G. K. 1967. An Introduction to Fluid Dynamics.This mainly treats incompressible flow. It is rather heavy going for the true beginner,but nevertheless is a very useful reference for the fundamentals.

Two other useful references are:

Tritton, D. J., 1988. Physical Fluid Dynamics, 2nd edn.

Kundu, P. and Cohen, I.M. 2002. Fluid Mechanics.Both are introductions written at the advanced undergraduate/beginning graduatelevel, and are easier-going than Batchelor. Kundu and Cohen’s book has more mate-rial on geophysical fluid dynamics.

There are also numerous books on thermodynamics, two clear and useful ones being:

Reif, F., 1965. Fundamentals of Statistical and Thermal Physics,

Callen, E. B., 1985. Thermodynamics and an Introduction to Thermostatistics.Reif’s book has become something of a classic, and Callen provides a rather moreaxiomatic approach.

An introduction to thermodynamic effects in fluids, with an emphasis on fundamental prop-erties, is provided by:

Salmon, R., 1998. Lectures on Geophysical Fluid Dynamics.


Problems

It is by the solution of problems that the investigator tests the temper of hissteel; he finds new methods and new outlooks, and gains a wider and freerhorizon.David Hilbert (1862–1943).

1.1 For an infinitesimal volume, informally show that

DDt(ρϕ∆V) = ρ∆V Dϕ

Dt, (P1.1)

where ϕ is some (differentiable) property of the fluid. Hence informally deduce that

DDt

∫Vρϕ dV =

∫Vρ

DϕDt

dV. (P1.2)

1.2 Show that the derivative of an integral is given by

ddt

∫ x2(t)

x1(t)ϕ(x, t)dx =

∫ x2

x1

∂ϕ∂t

dx + dx2

dtϕ(x2, t)−

dx1

dtϕ(x1, t). (P1.3)

By generalizing to three-dimensions show that the material derivative of an integralof a fluid property is given by

DDt

∫Vϕ(x, t)dV =

∫V

∂ϕ∂t

dV +∫Sϕv · dS =

∫V

[∂ϕ∂t

+∇ · (vϕ)]

dV, (P1.4)

where the surface integral (∫S ) is over the surface bounding the volume V . Hence

deduce thatDDt

∫Vρϕ dV =

∫Vρ

DϕDt

dV. (P1.5)

1.3 Why is there no diffusion term in the mass continuity equation?

1.4 By invoking Galilean invariance we can often choose, without loss of generality, thebasic state for problems in sound waves to be such that u0 ≡ 0. The perturbationvelocity is then certainly larger than the basic state velocity. How can we then justifyignoring the nonlinear term in the perturbation equation, as the term u′∂u′/∂x iscertainly no smaller than the linear term u0∂u′/∂x?

1.5 What amplitude of sound wave is required for the nonlinear terms to become impor-tant? Is this achieved at a rock concert (120 dB), or near a jet aircraft that is takingoff (160 dB).

1.6 Using the observed value of molecular diffusion of heat in water, estimate how long itwould take for a temperature anomaly to mix from the top of the ocean to the bottom,assuming that molecular diffusion alone is responsible. Comment on whether youthink the real ocean has reached equilibrium after the last ice age (which endedabout 12,000 years ago).

1.7 Consider the following flow:

u = Γ z, v = V sin[k(x − ct)] (P1.6)

where Γ , V , k and c are positive constants. (This is similar to the flow in the mid-latitude troposphere — an eastward flow increasing with height, with a transversewave superimposed.) Suppose that Γ z > c for the region of interest. Considerparticles located along the y = 0 axis at t = 0, and compute their position at somelater time t. Compare this with the streamfunction for the flow at the same time.(Hint: Show that the meridional particle displacement is η = ψ/(u − c), where ψ isthe streamfunction and u and c are parameters.)


1.8 Consider the two-dimensional flow

u = A(y) sinωt, v = A(y) cosωt. (P1.7)

The time-mean of this at a fixed point flow is zero. If A is independent of y, thenfluid parcels move clockwise in a circle. What is its radius? If A does depend on y,find an approximate expression for the average drift of a particle,

limt→∞

x(a, t)t

where a is a particle label and A is suitably ‘small’. Be precise about what smallmeans.

1.9 (a) Suppose that a sealed, insulated container consists of two compartments, andthat one of them is filled with an ideal gas and the other is a vacuum. Thepartition separating the compartments is removed. How does the temperature ofthe gas change? (Answer: It stays the same. Explain.) Obtain an expresson forthe final potential temperature, in terms of the initial temperature of the gas andthe volumes of the two compartments.Reconcile your answers with the first law of thermodynamics for an ideal gas fora reversible quasi-static process,

dQ = T dη = cpdθθ= dI + dW = cv dT + p dα. (P1.8)

(b) A dry parcel that is ascending adiabatically through the atmosphere will generallycool as it moves to lower pressure and expands, and its potential temperaturestays the same. How can this be consistent with your answer to part (a)?

1.10 Show that adiabatic flow in an ideal gas satisfies pρ−γ = constant, where γ = cp/cv .

1.11 (a) Show that for an ideal gas (1.116) is equivalent to (1.100). You may use theMaxwell relation (∂α/∂η)p = (∂T/∂p)η.

(b) Show that for an ideal gas (1.127) is equivalent to (1.99). You may use the resultsof part (a).

1.12 Show that it follows directly from the equation of state, P = RT/α, that the internalenergy of an ideal gas is a function of temperature only.Solution: From (1.87) and p = RT/α we have

dη = 1T

(∂I∂T

)α

dT +[

1T

(∂I∂α

)T+ Rα

]dα. (P1.9)

But, mathematically,

dη =(∂η∂T

)α

dT +(∂η∂α

)T

dα. (P1.10)

Equating the coefficients of dT and dα in these two expressions gives(∂η∂T

)α= 1T

(∂I∂T

)α

and(∂η∂α

)T= 1T

(∂I∂α

)T+ Rα. (P1.11)

Noting that ∂2η/(∂α∂T) = ∂2η/(∂T∂α) we obtain

∂2I∂α∂T

= ∂2I∂T∂α

−(∂I∂α

)T. (P1.12)

Thus, (∂I/∂α)T = 0. Because, in general, the internal energy may be consideredeither a function of temperature and density or temperature and pressure, this provesthat for an ideal gas the internal energy is a function only of temperature.


1.13 Show that it follows directly from the equation of state P = RT/α, that for an idealgas the heat capacity at constant volume, cv , is, at most, a function of temperature.

1.14 Show that for an ideal gasTdη = cvdT + pdα. (P1.13)

and that its internal energy is given by I =∫cvdT .

Solution: Let us regard η as a function of T and α, where α is the specific volume1/ρ. Then

Tdη = T(∂η∂T

)α

dT + T(∂η∂α

)T

dα

= cv dT + T(∂η∂α

)T

dα (P1.14)

by definition of cv . For an ideal gas the internal energy is a function of temperaturealone (problem 1.12), so that using (1.70) the pressure of a fluid p = T (∂η/∂α)I =T (∂η/∂α)T and (P1.14) becomes

Tdη = cv dT + p dα (P1.15)

But, in general, the fundamental thermodynamic relation is

Tdη = dI + p dα. (P1.16)

The terms on the right hand side of (P1.15) are identifiable as the change in theinternal energy and the work done on a fluid, and so dI = cv dT . The heat capacityneed not necessarily be constant, although for air it very nearly is, but it must be afunction of temperature only.

1.15 (a) Beginning with the expression for potential temperature for an ideal gas, θ =T(pR/p)κ , where κ = R/cp, show that

dθ = θT(dT −αdp), (P1.17)

and therefore that the first law of thermodynamics may be written as

dQ = Tdη = cpTθ

dθ. (P1.18)

(b) Show that (P1.18) is more generally true, and not just for an ideal gas.

1.16 Obtain an expression for the Gibbs function for an ideal gas in terms of pressure andtemperature.

1.17 From (1.113) derive the conventional equation of state for an ideal gas, and obtainexpressions for the heat capacities.

1.18 Consider an ocean at rest with known vertical profiles of potential temperature andsalinity, θ(z) and S(z). Suppose we also know the equation of state in the formρ = ρ(θ, S,p). Obtain an expression for the buoyancy frequency. Check your expres-sion by substituting the equation of state for an ideal gas and recovering a knownexpression for the buoyancy frequency.

1.19 Consider a liquid, sitting in a container, with a free surface at the top (at z = H). Theliquid obeys the equation of state ρ = ρ0[1 − βT (T − T0)], and its internal energy,I, is given by I = cpT . Suppose that the fluid is heated, so that it’s temperaturerises uniformly by ∆T , and the free surface rises by a small amount ∆H. Obtain anexpression for the ratio of the change in internal energy to the change in gravitationalpotential energy (GPE) of the ocean, and show that it is related to the scale height


(1.131). If global warming increases the ocean temperature by 4 K, what is the ratioof the change of GPE to the change of I? Estimate also the average rise in sea level.Partial solution: The change in internal energy and in GPE are

∆I = Cρ1H1(T2 − T1), ∆GPE = ρ1gH1(H2 −H1)/2 = ρ1gH1βTH1(T2 − T1)/2.(P1.19)

(Derive these. Use mass conservation where necessary. The subscripts 1 and 2denote initial and final states.) Hence ∆GPE/∆I = gβTH1/2C.

1.20 Obtain an expression, in terms of temperature and pressure, for the potentialtemperature of a van der Waals gas, with equation of state (p + a/α2)(α− b) = RT ,where a and b are constants. Show that it reduces to the expression for an ideal gasin the limit a→ 0, b → 0.

If a body is moving in any direction, there is a force, arising from theearth’s rotation, which always deflects it to the right in the northernhemisphere, and to the left in the southern.William Ferrel, The influence of the Earth’s rotation upon the relative motion ofbodies near its surface, 1858.

CHAPTER

TWO

Effects of Rotation and Stratification

THE ATMOSPHERE AND OCEAN are shallow layers of fluid on a sphere in that theirthickness or depth is much less than their horizontal extent. Furthermore,their motion is strongly influenced by two effects: rotation and stratification,

the latter meaning that there is a mean vertical gradient of (potential) density thatis often large compared with the horizontal gradient. Here we consider how theequations of motion are affected by these effects. First, we consider some elemen-tary effects of rotation on a fluid and derive the Coriolis and centrifugal forces, andthen we write down the equations of motion appropriate for motion on a sphere.Then we discuss some approximations to the equations of motion that are appropri-ate for large-scale flow in the ocean and atmosphere, in particular the hydrostaticand geostrophic approximations. Following this we discuss gravity waves, a partic-ular kind of wave motion that enabled by the presence of stratification, and finallywe talk about how rotation leads to the production of certain types of boundarylayers — Ekman layers — in rotating fluids.

2.1 THE EQUATIONS OF MOTION IN A ROTATING FRAME OF REFERENCE

Newton’s second law of motion, that the acceleration on a body is proportional tothe imposed force divided by the body’s mass, applies in so-called inertial frames ofreference. The earth rotates with a period of almost 24 hours (23h 56m) relativeto the distant stars, and the surface of the earth manifestly is not, in that sense,an inertial frame. Nevertheless, because the surface of the earth is moving (infact at speeds of up to a few hundreds of meters per second) it is very convenientto describe the flow relative to the earth’s surface, rather than in some inertialframe. This necessitates recasting the equations into a form that is appropriate fora rotating frame of reference, and that is the subject of this section.

51

52 Chapter 2. Effects of Rotation and Stratification

Figure 2.1 A vector C rotating atan angular velocity Ω. It appearsto be a constant vector in the ro-tating frame, whereas in the in-ertial frame it evolves accordingto (dC/dt)I = Ω× C.

C

C

x C

Ω

Ω

θ

λ

2.1.1 Rate of change of a vector

Consider first a vector C of constant length rotating relative to an inertial frame ata constant angular velocity Ω. Then, in a frame rotating with that same angularvelocity it appears stationary and constant. If in small interval of time δt the vectorC rotates through a small angle δλ then the change in C, as perceived in the inertialframe, is given by (see Fig. 2.1)

δC = |C| cosθ δλm, (2.1)

where the vector m is the unit vector in the direction of change of C, which isperpendicular to both C and Ω. But the rate of change of the angle λ is just, bydefinition, the angular velocity so that δλ = |Ω|δt and

δC = |C||Ω| sin θmδt = Ω× C δt. (2.2)

using the definition of the vector cross product, where θ = (π/2 − θ) is the anglebetween Ω and C. Thus (

dCdt

)I= Ω× C (2.3)

where the left hand side is the rate of change of C as perceived in the inertial frame.Now consider a vector B that changes in the inertial frame. In a small time δt

the change in B as seen in the rotating frame is related to the change seen in theinertial frame by

(δB)I = (δB)R + (δB)rot (2.4)

where the terms are, respectively, the change seen in the inertial frame, the changedue to the vector itself changing as measured in the rotating frame, and the changedue to the rotation. Using (2.2) (δB)rot = Ω × Bδt, and so the rates of change ofthe vector B in the inertial and rotating frames are related by

(dBdt

)I=(

dBdt

)R+Ω× B . (2.5)

2.1 Equations in a Rotating Frame 53

This relation applies to a vector B that, as measured at any one time, is the same inboth inertial and rotating frames.

2.1.2 Velocity and acceleration in a rotating frame

The velocity of a body is not measured to be the same in the inertial and rotatingframes, so care must be taken when applying (2.5) to velocity. First apply (2.5) tor, the position of a particle to obtain(

drdt

)I=(

drdt

)R+Ω× r (2.6)

orvI = vR +Ω× r. (2.7)

We refer to vR and vI as the relative and inertial velocity, respectively, and (2.7)relates the two. Apply (2.5) again, this time to the velocity vR to give(

dvRdt

)I=(

dvRdt

)R+Ω× vR, (2.8)

or, using (2.7) (ddt(vI −Ω× r)

)I=(

dvRdt

)R+Ω× vR, (2.9)

or (dvIdt

)I=(

dvRdt

)R+Ω× vR +

dΩdt

× r +Ω×(

drdt

)I. (2.10)

Then, noting that (drdt

)I=(

drdt

)R+Ω× r = (vR +Ω× r), (2.11)

and assuming that the rate of rotation is constant, (2.10) becomes(dvRdt

)R=(

dvIdt

)I− 2Ω× vR −Ω× (Ω× r). (2.12)

This equation may be interpreted as follows. The term on the left-hand side isthe rate of change of the relative velocity as measure in the rotating frame. Thefirst term on the right-hand side is the rate of change of the inertial velocity asmeasured in the inertial frame (or, loosely, the inertial acceleration). Thus, byNewton’s second law, it is equal to force on a fluid parcel divided by its mass. Thesecond and third terms on the right-hand side (including the minus signs) are theCoriolis force and the centrifugal force per unit mass. Neither of these are true forces— they may be thought of as quasi-forces (i.e., ‘as if’ forces); that is, when a bodyis observed from a rotating frame it seems to behave as if unseen forces are presentthat affect its motion. If (2.12) is written, as is common, with the terms +2Ω× vrand +Ω × (Ω × r) on the left-hand side then these terms should be referred to asthe Coriolis and centrifugal accelerations.1


Centrifugal force

If r⊥ is the perpendicular distance from the axis of rotation (see Fig. 2.1 and substi-tute r for C), then, because Ω is perpendicular to r⊥, Ω× r = Ω× r⊥. Then, usingthe vector identity Ω× (Ω× r⊥) = (Ω · r⊥)Ω− (Ω ·Ω)r⊥ and noting that the firstterm is zero, we see that the centrifugal force per unit mass is just given by

Fce = −Ω× (Ω× r) = Ω2r⊥. (2.13)

This may usefully be written as the gradient of a scalar potential,

Fce = −∇Φce. (2.14)

where Φce = −(Ω2r 2⊥)/2 = −(Ω× r⊥)2/2.

Coriolis force

The Coriolis force per unit mass is:

FCo = −2Ω× vR. (2.15)

It plays a central role in much of geophysical fluid dynamics and will be consideredextensively later on. For now, we just note three basic properties:

(i) There is no Coriolis force on bodies that are stationary in the rotating frame.(ii) The Coriolis force acts to deflect moving bodies at right angles to their direction

of travel.(iii) The Coriolis force does no work on a body, a consequence of the fact that

vR · (Ω× vR) = 0.

2.1.3 Momentum equation in a rotating frame

Since (2.12) simply relates the accelerations of a particle in the inertial and rotatingframes, then in the rotating frame of reference the momentum equation may bewritten

DvDt

+ 2Ω× v = −1ρ∇p −∇Φ. (2.16)

We have dropped the subscript R; henceforth, unless ambiguity is present, all ve-locities without a subscript will be considered to be relative to the rotating frame.

2.1.4 Mass and tracer conservation in a rotating frame

Let φ be a scalar field that, in the inertial frame, obeys

DφDt

+φ∇ · vI = 0. (2.17)

Now, observers in both the rotating and inertial frame measure the same value ofφ. Further, Dφ/Dt is simply the rate of change of φ associated with a materialparcel, and therefore is reference frame invariant. Thus,(

DφDt

)R=(

DφDt

)I

(2.18)

2.2 Equations of Motion in Spherical Coordinates 55

where (Dφ/Dt)R = (∂φ/∂t)R+vR ·∇φ and (Dφ/Dt)I = (∂φ/∂t)I+vI ·∇φ and thelocal temporal derivatives (∂φ/∂t)R and (∂φ/∂t)I are evaluated at fixed locationsin the rotating and inertial frames, respectively.

Further, since v = vI −Ω× r, we have that

∇ · vI = ∇ · (vI −Ω× r) = ∇ · vR (2.19)

since ∇ · (Ω× r) = 0. Thus, using (2.18) and (2.19), (2.17) is equivalent to

DφDt

+φ∇ · v = 0 (2.20)

where all observables are measured in the rotating frame. Thus, the equation forthe evolution of a scalar whose measured value is the same in rotating and inertialframes is unaltered by the presence of rotation. In particular, the mass conservationequation is unaltered by the presence of rotation.

Although we have taken (2.18) as true a priori, the individual components ofthe material derivative differ in the rotating and inertial frames. In particular(

∂φ∂t

)I=(∂φ∂t

)R− (Ω× r) · ∇φ (2.21)

because Ω × r is the velocity, in the inertial frame, of a uniformly rotating body.Similarly,

vI · ∇φ = (vR +Ω× r) · ∇φ. (2.22)

Adding the last two equations reprises and confirms (2.18).

2.2 EQUATIONS OF MOTION IN SPHERICAL COORDINATES

The earth is very nearly spherical and it might appear obvious that we should castour equations in spherical coordinates. Although this does turn out to be true,the presence of a centrifugal force causes some complications which we must firstdiscuss. The reader who is willing ab initio to treat the earth as a perfect sphereand to neglect the horizontal component of the centrifugal force may skip the nextsection.

2.2.1 * The centrifugal force and spherical coordinates

The centrifugal force is a potential force, like gravity, and so we may thereforedefine an ‘effective gravity’ equal to the sum of the true, or Newtonian, gravity andthe centrifugal force. The Newtonian gravitational force is directed approximatelytoward the center of the earth, with small deviations due mainly to the earth’soblateness. The line of action of the effective gravity will in general differ slightlyfrom this, and therefore have a component in the ‘horizontal’ plane, that is theplane perpendicular to the radial direction. The magnitude of the centrifugal forceis Ω2r⊥, and so the effective gravity is given by

g ≡ geff = ggrav +Ω2r⊥ (2.23)

where ggrav is the Newtonian gravitational force due to the gravitational attractionof the earth and r⊥ is normal to the rotation vector (in the direction C in Fig.


Fig. 2.2 Left: Directions of forces and coordinates in true spherical geometry.g is the effective gravity (including the centrifugal force, C) and its horizon-tal component is evidently non-zero. Right: a modified coordinate system,in which the vertical direction is defined by the direction of g, and so thehorizontal component of g is identically zero. The dashed line schematicallyindicates a surface of constant geopotential. The differences between thedirection of g and the direction of the radial coordinate, and between thesphere and the geopotential surface, are much exaggerated and in reality aresimilar to the thickness of the lines themselves.

2.2), with r⊥ = r cosϑ. Both gravity and centrifugal force are potential forces andtherefore we may define the geopotential, Φ, such that

g = −∇Φ. (2.24)

Surfaces of constant Φ are not quite spherical because r⊥, and hence the centrifugalforce, vary with latitude (Fig. 2.2).

The components of the centrifugal force parallel and perpendicular to the radialdirection are Ω2r cos2 ϑ and Ω2r cosϑ sinϑ. Newtonian gravity is much larger thaneither of these, and at the earth’s surface the ratio of centrifugal to gravitationalterms is approximately, and no more than,

α ≈ Ω2ag

≈ (7.27× 10−5)2 × 6.4× 106

10≈ 3× 10−3. (2.25)

(Note that at the equator and pole the horizontal component of the centrifugalforce is zero and the effective gravity is aligned with Newtonian gravity.) The anglebetween g and the line to the center of the earth is given by a similar expressionand so is also small, typically around 3×10−3 radians. However, the horizontalcomponent of the centrifugal force is still large compared to the Coriolis force, theirratio in mid-latitudes being given by

Horizontal centrifugal forceCoriolis force

≈ Ω2a cosϑ sinϑ

2Ωu≈ Ωa

4|u| ≈ 10, (2.26)

using u = 10 m s−1. The centrifugal term therefore dominates over the Coriolis


term, and is largely balanced by a pressure gradient force. Thus, if we adheredto true spherical coordinates, both the horizontal and radial components of themomentum equation would be dominated by a static balance between a pressuregradient and gravity or centrifugal terms. Although in principle there is nothingwrong with writing the equations this way, it obscures the dynamical balances in-volving the Coriolis force and pressure that determine the large-scale horizontalflow.

A way around this problem is to use the direction of the geopotential force todefine the vertical direction, and then to regard the surfaces of constant Φ as be-ing true spheres.2 The horizontal component of effective gravity is then identicallyzero, and we have traded a potentially large dynamical error for a very small ge-ometric error. In fact, over time, the earth has developed an equatorial bulge tocompensate for and neutralize the centrifugal force, so that the effective gravitydoes act in a direction virtually normal to the earth’s surface; that is, the surface ofthe earth is an oblate spheroid of nearly constant geopotential. The geopotential Φis then a function of the vertical coordinate alone, and for many purposes we canjust take Φ = gz; that is, the direction normal to geopotential surfaces, the localvertical, is, in this approximation, taken to be the direction of increasing r in spher-ical coordinates. It is because the oblateness is very small (the polar diameter isabout 12,714 km, whereas the equatorial diameter is about 12,756 km) that usingspherical coordinates is a very accurate way to map the spheroid, and if the anglebetween effective gravity and a natural direction of the coordinate system were notsmall then more heroic measures would be called for.

If the solid earth did not bulge at the equator, the behaviour of the atmosphereand ocean would differ significantly from that of the present system. For example,the surface of the ocean is nearly a geopotential surface, and if the solid earthwere exactly spherical then the ocean would perforce become much deeper at lowlatitudes and the ocean basins would dry out completely at high latitudes. Wecould still choose to use the spherical coordinate system discussed above to describethe dynamics, but the shape of the surface of the solid earth would have to berepresented by a topography, with the topographic height increasing monotonicallypolewards nearly everywhere.

2.2.2 Some identities in spherical coordinates

The location of a point is given by the coordinates (λ,ϑ, r) where λ is the angulardistance eastward (i.e., longitude), ϑ is angular distance poleward (i.e., latitude)and r is the radial distance from the center of the earth — see Fig. 2.3. (In someother fields of study co-latitude is used as a spherical coordinate.) If a is the radiusof the earth, then we also define z = r − a. At a given location we may also definethe Cartesian increments (δx,δy,δz) = (r cosϑδλ, rδϑ,δr).

For a scalar quantity φ the material derivative in spherical coordinates is

DφDt

= ∂φ∂t

+ ur cosϑ

∂φ∂λ

+ vr∂φ∂ϑ

+w∂φ∂r, (2.27)

where the velocity components corresponding to the coordinates (λ,ϑ, r) are

(u,v,w) ≡(r cosϑ

DλDt, r

DϑDt,DrDt

). (2.28)


Figure 2.3 The spherical coordi-nate system. The orthogonal unitvectors i, j and k point in the di-rection of increasing longitude λ,latitude ϑ, and altitude z. Locally,one may apply a Cartesian systemwith variables x, y and z measur-ing distances along i, j and k.

That is, u is the zonal velocity, v is the meridional velocity and w the verticalvelocity. If we define (i, j,k) to be the unit vectors in the direction of increasing(λ,ϑ, r) then

v = iu+ jv + kw. (2.29)

Note also that Dr/Dt = Dz/Dt.The divergence of a vector B = iBλ + jBϑ + kBz is

∇ · B = 1cosϑ

[1r∂Bλ∂λ

+ 1r∂∂ϑ(Bϑ cosϑ)+ cosϑ

r 2∂∂r(r 2Br )

]. (2.30)

The vector gradient of a scalar is:

∇φ = i1

r cosϑ∂φ∂λ

+ j1r∂φ∂ϑ

+ k∂φ∂r. (2.31)

The Laplacian of a scalar is:

∇2φ ≡ ∇ ·∇φ = 1r 2 cosϑ

[1

cosϑ∂2φ∂λ2 +

∂∂ϑ

(cosϑ

∂φ∂ϑ

)+ cosϑ

∂∂r

(r 2 ∂φ∂r

)].

(2.32)The curl of a vector is:

curlB = ∇× B = 1r 2 cosϑ

∣∣∣∣∣∣∣i r cosϑ j r k∂/∂λ ∂/∂ϑ ∂/∂r

Bλr cosϑ Bϑr Br

∣∣∣∣∣∣∣ . (2.33)


The vector Laplacian ∇2B (used for example when calculating viscous terms in themomentum equation) may be obtained from the vector identity:

∇2B = ∇(∇ · B)−∇× (∇× B). (2.34)

Only in Cartesian coordinates does this take the simple form:

∇2B = ∂2B∂x2 +

∂2B∂y2 +

∂2B∂z2 . (2.35)

The expansion in spherical coordinates is, to most eyes, rather uninformative.

Rate of change of unit vectors

In spherical coordinates the defining unit vectors are i, the unit vector pointingeastward, parallel to a line of latitude; j is the unit vector pointing polewards, par-allel to a meridian; and k, the unit vector pointing radially outward. The directionsof these vectors change with location, and in fact this is the case in nearly all co-ordinate systems, with the notable exception of the Cartesian one, and thus theirmaterial derivative is not zero. One way to evaluate this is to consider geometricallyhow the coordinate axes change with position (problem 2.5). We will approach theproblem a little differently, by first obtaining the effective rotation rate Ωflow, rela-tive to the earth, of a unit vector as it moves with the flow, and then applying (2.3).Specifically, let the fluid velocity be v = (u,v,w). The meridional component, v,produces a displacement rδϑ = vδt, and this give rise a local effective vector rota-tion rate around the local zonal axis of −(v/r)i, the minus sign arising because adisplacement in the direction of the north pole is produced by negative rotationaldisplacement around the i axis. Similarly, the zonal component, u, produces a dis-placement δλr cosϑ = uδt and so an effective rotation rate, but now about theearth’s rotation axis, of u/(r cosϑ). Now, a rotation around the earth’s rotationaxis may be written as (see Fig. 2.4)

Ω = Ω(j cosϑ + k sinϑ). (2.36)

If the scalar rotation rate is not Ω but is u/(r cosϑ), then the vector rotation rate is

ur cosϑ

(j cosϑ + k sinϑ) = jur+ ku tanϑr

. (2.37)

Thus, the total rotation rate of a vector that moves with the flow is

Ωflow = −ivr+ jur+ ku tanϑr

. (2.38)

Applying (2.3) to (2.38), we find

DiDt

= Ωflow × i = ur cosϑ

(j sinϑ − k cosϑ), (2.39a)

DjDt

= Ωflow × j = −iur

tanϑ − kvr, (2.39b)

DkDt

= Ωflow × k = iur+ jvr. (2.39c)


Fig. 2.4 a) On the sphere the rotation vector Ω can be decomposed intotwo components, one in the local vertical and one in the local horizontal,pointing toward the pole. That is, Ω = Ωy j +Ωzk where Ωy = Ω cosϑ andΩz = Ω sinϑ. In geophysical fluid dynamics, the rotation vector in the localvertical is often the more important component in the horizontal momentumequations. On a rotating disk, (b), the rotation vectorΩ is parallel to the localvertical k.

2.2.3 Equations of motion

Mass Conservation and Thermodynamic Equation

The mass conservation equation, (1.36a), expanded in spherical co-odinates, is

∂ρ∂t

+ ur cosϑ

∂ρ∂λ

+ vr∂ρ∂ϑ

+w∂ρ∂r

+ ρr cosϑ

[∂u∂λ

+ ∂∂ϑ(v cosϑ)+ 1

r∂∂r(wr 2 cosϑ)

]= 0.

(2.40)

Equivalently, using the form (1.36b), this is

∂ρ∂t

+ 1r cosϑ

∂(uρ)∂λ

+ 1r cosϑ

∂∂ϑ(vρ cosϑ)+ 1

r 2∂∂r(r 2wρ) = 0. (2.41)

The thermodynamic equation, (1.108), is a tracer advection equation. Thus,using (2.27), its (adiabatic) spherical coordinate form is

DθDt

= ∂θ∂t

+ ur cosϑ

∂θ∂λ

+ vr∂θ∂ϑ

+w∂θ∂r

= 0, (2.42)

and similarly for tracers such as water vapour or salt.

Momentum Equation

Recall that inviscid momentum equation is:

DvDt

+ 2Ω× v = −1ρ∇p −∇Φ. (2.43)

where Φ is the geopotential. In spherical coordinates the directions of the coordi-nate axes change with position and so the component expansion of (2.43) is

DvDt

= DuDt

i+ DvDt

j+ DwDt

k+uDiDt

+ v DjDt

+wDkDt

(2.44a)


= DuDt

i+ DvDt

j+ DwDt

k+Ωflow × v, (2.44b)

using (2.39). Using either (2.44a) and the expressions for the rates of change of theunit vectors given in (2.39), or (2.44b) and the expression for Ωflow given in (2.38),(2.44) becomes

DvDt

= i(

DuDt

− uv tanϑr

+ uwr

)+ j

(DvDt

+ u2 tanϑr

+ vwr

)

+ k(

DwDt

− u2 + v2

r

).

(2.45)

Using the definition of a vector cross product the Coriolis term is:

2Ω× v =

∣∣∣∣∣∣∣i j k0 2Ω cosϑ 2Ω sinϑu v w

∣∣∣∣∣∣∣= i (2Ωw cosϑ − 2Ωv sinϑ)+ j 2Ωu sinϑ − k 2Ωu cosϑ. (2.46)

Using (2.45) and (2.46), and the gradient operator given by (2.31), the momentumequation (2.43) becomes:

DuDt

−(

2Ω + ur cosϑ

)(v sinϑ −w cosϑ) = − 1

ρr cosϑ∂p∂λ, (2.47a)

DvDt

+ wvr+(

2Ω + ur cosϑ

)u sinϑ = − 1

ρr∂p∂ϑ, (2.47b)

DwDt

− u2 + v2

r− 2Ωu cosϑ = −1

ρ∂p∂r

− g. (2.47c)

The terms involving Ω are called Coriolis terms, and the quadratic terms on theleft-hand sides involving 1/r are often called metric terms.

2.2.4 The primitive equations

The so-called primitive equations of motion are simplifications of the above equa-tions frequently used in atmospheric and oceanic modelling.3 Three related approx-imations are involved; these are:

(i) The hydrostatic approximation. In the vertical momentum equation the gravi-tational term is assumed to be balanced by the pressure gradient term, so that

∂p∂z

= −ρg. (2.48)

The advection of vertical velocity, the Coriolis terms, and the metric term (u2+v2)/r are all neglected.

(ii) The shallow-fluid approximation. We write r = a + z where the constant a isthe radius of the earth and z increases in the radial direction. The coordinater is then replaced by a except where it used as the differentiating argument.Thus, for example,

1r 2∂(r 2w)∂r

→ ∂w∂z. (2.49)


(iii) The traditional approximation. Coriolis terms in the horizontal momentumequations involving the vertical velocity, and the still smaller metric termsuw/r and vw/r , are neglected.

The second and third of these approximations should be taken, or not, together,the underlying reason being that they both relate to the presumed small aspectratio of the motion, so the approximations succeed or fail together. If we make oneapproximation but not the other then we are being asymptotically inconsistent, andangular momentum and energy conservation are not assured [see section 2.2.7].The hydrostatic approximation also depends on the small aspect ratio of the flowbut in a slightly different way. For large-scale flow in the terrestrial atmosphere andocean all three approximations are in fact all very accurate approximations. Wedefer a more complete treatment until section 2.7, in part because a treatment ofthe hydrostatic approximation is done most easily in the context of the Boussinesqequations, derived in section 2.4.

Making these approximations, the momentum equations become

DuDt

− 2Ω sinϑv − uva

tanϑ = − 1aρ cosϑ

∂p∂λ, (2.50a)

DvDt

+ 2Ω sinϑu+ u2 tanϑa

= − 1ρa∂p∂ϑ, (2.50b)

0 = −1ρ∂p∂z

− g. (2.50c)

whereDDt

=(∂∂t+ ua cosϑ

∂∂λ

+ va∂∂ϑ

+w ∂∂z

). (2.51)

We note the ubiquity of the factor 2Ω sinϑ, and take the opportunity to define theCoriolis parameter, f ≡ 2Ω sinϑ.

The corresponding mass conservation equation for a shallow fluid layer is:

∂ρ∂t

+ ua cosϑ

∂ρ∂λ

+ va∂ρ∂ϑ

+w∂ρ∂z

+ ρ[

1a cosϑ

∂u∂λ

+ 1a cosϑ

∂∂ϑ(v cosϑ)+ ∂w

∂z

]= 0,

(2.52)

or equivalently,

∂ρ∂t

+ 1a cosϑ

∂(uρ)∂λ

+ 1a cosϑ

∂∂ϑ(vρ cosϑ)+ ∂(wρ)

∂z= 0. (2.53)

2.2.5 Primitive equations in vector form

The primitive equations may be written in a compact vector form provided we makea slight reinterpretation of the material derivative of the coordinate axes. Let u =ui+vj+0 k be the horizontal velocity. The primitive equations (2.50a) and (2.50b)may be written as

DuDt

+ f × u = −1ρ∇zp, (2.54)


where f = fk = 2Ω sinϑk and ∇zp = [(a cosϑ)−1∂p/∂λ,a−1∂p/∂ϑ], the gradientoperator at constant z. In (2.54) the material derivative of the horizontal velocityis given by

DuDt

= iDuDt

+ jDvDt

+uDiDt

+ v DjDt, (2.55)

where instead of (2.39) we have

DiDt

= Ωflow × i = ju tanϑa

, (2.56a)

DjDt

= Ωflow × j = −iu tanϑa

, (2.56b)

where Ωflow = ku tanϑ/a [which is the vertical component of (2.38), with r re-placed by a.]. The advection of the horizontal wind u is still by the three-dimensionalvelocity v. The vertical momentum equation is the hydrostatic equation, (2.50c),and the mass conservation equation is

DρDt

+ ρ∇ · v = 0 or∂ρ∂t

+∇ · (ρv) = 0. (2.57)

where D/Dt on a scalar is given by (2.51), and the second expression is written outin full in (2.53).

2.2.6 The vector invariant form of the momentum equation

The ‘vector invariant’ form of the momentum equation is so-called because it ap-pears to take the same form in all coordinate systems — there is no advectivederivative of the coordinate system to worry about. With the aid of the identity(v · ∇)v = −v ×ω+∇(v2/2), where ω ≡ ∇× v is the relative vorticity, the threedimensional momentum equation may be written:

∂v∂t

+ (2Ω+ω)× v = −1ρ∇p − 1

2∇v2 + g. (2.58)

In spherical coordinates the relative vorticity is given by:

ω = ∇× v = 1r 2 cosϑ

∣∣∣∣∣∣∣i r cosϑ j r k∂/∂λ ∂/∂ϑ ∂/∂rur cosϑ rv w

∣∣∣∣∣∣∣= i

1r

(∂w∂ϑ

− ∂(rv)∂r

)− j

1r cosϑ

(∂w∂λ

− ∂∂r(ur cosϑ)

)+ k

1r cosϑ

(∂v∂λ

− ∂∂ϑ(u cosϑ)

). (2.59)

Making the traditional and shallow fluid approximations, the horizontal compo-nents of (2.58) may be written

∂u∂t

+ (f + kζ)× u+w∂u∂z

= −1ρ∇zp −

12∇u2, (2.60)


where u = (u,v,0), f = k 2Ω sinϑ, ∇z is the horizontal gradient operator (thegradient at a constant value of z), and using (2.59), ζ is given by

ζ = 1a cosϑ

∂v∂λ

− 1a cosϑ

∂∂ϑ(u cosϑ) = 1

a cosϑ∂v∂λ

− 1a∂u∂ϑ

+ ua

tanϑ. (2.61)

The separate components of the momentum equation are given by:

∂u∂t

− (f + ζ)v +w∂u∂z

= − 1aρ cosϑ

(1ρ∂p∂λ

+ 12∂u2

∂λ

), (2.62)

and∂v∂t

+ (f + ζ)u+w∂v∂z

= −1a

(1ρ∂p∂ϑ

+ 12∂u2

∂ϑ

). (2.63)

Related expressions are given problem 2.2, and we treat vorticity at greater lengthin chaper 4.

2.2.7 Angular Momentum

The zonal momentum equation can be usefully expressed as a statement about axialangular momentum; that is, angular momentum about the rotation axis. The zonalangular momentum per unit mass is the component of angular momentum in thedirection of the axis of rotation and it is given by, without making any shallowatmosphere approximation,

m = (u+Ωr cosϑ)r cosϑ. (2.64)

The evolution equation for this quantity follows from the zonal momentum equa-tion and has the simple form

DmDt

= −1ρ∂p∂λ, (2.65)

where the material derivative is

DDt

= ∂∂t+ ur cosϑ

∂∂λ

+ vr∂∂ϑ

+w ∂∂r. (2.66)

Using the mass continuity equation, this can be written as

DρmDt

+ ρm∇ · v = −∂p∂λ

(2.67)

or

∂ρm∂t

+ 1r cosϑ

∂(ρum)∂λ

+ 1r cosϑ

∂∂ϑ(ρvm cosϑ)+ ∂

∂z(ρmw) = −∂p

∂λ. (2.68)

This is an angular momentum conservation equation.If the fluid is confined to a shallow layer near the surface of a sphere, then we

may replace r , the radial coordinate, by a, the radius of the sphere, in the definitionof m, and we define m ≡ (u+Ωa cosϑ)a cosϑ. Then (2.65) is replaced by

DmDt

= −1ρ∂p∂λ, (2.69)


where nowDDt

= ∂∂t+ ua cosϑ

∂∂λ

+ va∂∂ϑ

+w ∂∂z. (2.70)

Using mass continuity this may be written as

∂ρm∂t

+ ua cosϑ

∂m∂λ

+ va∂m∂ϑ

+w∂m∂z

= −1ρ∂p∂λ, (2.71)

which is the appropriate angular momentum conservation equation for a shallowatmosphere.

* From angular momentum to the spherical component equations

An alternative way to derive the three components of the momentum equation inspherical polar coordinates is to begin with (2.65) and the principle of conservationof energy. That is, we take the equations for conservation of angular momentumand energy as true a priori and demand that the forms of the momentum equa-tion be constructed to satisfy these. Expanding the material derivative in (2.65),noting that Dr/Dt = w and Dcosϑ/Dt = −(v/r) sinϑ, immediately gives (2.47a).Multiplication by u then yields

uDuDt

− 2Ωuv sinϑ + 2Ωuw cosϑ − u2v tanϑr

+ u2wr

= − uρr cosϑ

∂p∂λ. (2.72)

Now suppose that the meridional and vertical momentum equations are of the form

DvDt

+ Coriolis and metric terms = − 1ρr∂p∂ϑ

(2.73a)

DwDt

+ Coriolis and metric terms = −1ρ∂p∂r, (2.73b)

but that we do not know what form the Coriolis and metric terms take. To determinethat form, construct the kinetic energy equation by multiplying (2.73) by v and w,respectively. Now, the metric terms must vanish when we sum the resulting equa-tions along with (2.72), so that (2.73a) must contain the Coriolis term 2Ωu sinϑ aswell as the metric term u2 tanϑ/r , and (2.73b) must contain a −2Ωu cosφ as wellas the metric term u2/r . But if (2.73b) contains the term u2/r it must also containthe term v2/r by isotropy, and therefore (2.73a) must also contain the term vw/r .In this way, (2.47) is precisely reproduced, although the skeptic might argue thatthe uniqueness of the form has not been proven.

A particular advantage of this approach arises in determining the appropriatemomentum equations that conserve angular momentum and energy in the shallow-fluid approximation. We begin with (2.69) and expand to obtain (2.50a). Multiply-ing by u gives

uDuDt

− 2Ωuv sinϑ − u2v tanϑa

= − uρa cosϑ

∂p∂λ. (2.74)

To ensure energy conservtion, the meridional momentum equation must contain theCoriolis term 2Ωu sinϑ and the metric term u2 tanϑ/a, but the vertical momentum


equation must have neither of the metric terms appearing in (2.47c). Thus wededuce the following equations:

DuDt

−(

2Ω sinϑ + u tanϑa

)v = − 1

ρa cosϑ∂p∂λ, (2.75a)

DvDt

+(

2Ω sinϑ + u tanϑa

)u = − 1

ρa∂p∂ϑ, (2.75b)

DwDt

= −1ρ∂ρ∂r

− g. (2.75c)

This equation set, when used in conjunction with the thermodynamic and masscontinuity equations, conserves appropriate forms of angular momentum and en-ergy. In the hydrostatic approximation the material derivative of w in (2.75c) isadditionally neglected. Thus, the hydrostatic approximation is mathematically andphysically consistent with the shallow-fluid approximation, but it is an additionalapproximation with slightly different requirements that one may choose, ratherthan being required, to make. From an asymptotic perspective, the difference liesin the small parameter necessary for either approximation to hold, namely

Shallow fluid and traditional approximations: γ ≡ Ha 1 (2.76a)

Small aspect ratio for hydrostatic approximation: α ≡ HL 1. (2.76b)

where L is the horizontal scale of the motion and a is the radius of the earth. Forhemispheric or global scale phenomena L ∼ a and the two approximations coincide.(Requirement (2.76b) for the hydrostatic approximation is derived in section 2.7.)

2.3 CARTESIAN APPROXIMATIONS: THE TANGENT PLANE

2.3.1 The f-plane

Although the rotation of the earth is central for many dynamical phenomena, thesphericity of the earth is not always so. This is especially true for phenomena on ascale somewhat smaller than global where the use of spherical coordinates becomesawkward, and it is more convenient to use a locally Cartesian representation of theequations. Referring to Fig. 2.4 we will define a plane tangent to the surface ofthe earth at a latitude ϑ0, and then use a Cartesian coordinate system (x,y, z)to describe motion on that plane. For small excursions on the plane, (x,y, z) ≈(aλ cosϑ0, a(ϑ−ϑ0), z). Consistently, the velocity is v = (u,v,w), so that u,v andw are the components of the velocity in the tangent plane. These are approximatelyin the east-west, north-south and vertical directions, respectively.

The momentum equations for flow in this plane are then

∂u∂t+ (v · ∇)u+ 2Ωyw − 2Ωzv = −

1ρ∂p∂x, (2.77a)

∂v∂t+ (v · ∇)v + 2Ωzu = −

1ρ∂p∂y, (2.77b)

∂w∂t

+ (v · ∇)w + 2(Ωxv −Ωyu) = −1ρ∂p∂z

− g, (2.77c)

2.3 Cartesian Approximations: The Tangent Plane 67

where the rotation vectorΩ = Ωxi+Ωy j+Ωzk andΩx = 0, Ωy = Ω cosϑ0 andΩz =Ω sinϑ0. If we make the traditional approximation, and so ignore the componentsof Ω not in the direction of the local vertical, then

DuDt

− f0v = −1ρ∂p∂x, (2.78a)

DvDt

+ f0u = −1ρ∂p∂y, (2.78b)

DwDt

= −1ρ∂p∂z

− ρg. (2.78c)

where f0 = 2Ωz sinϑ0. Defining the horizontal velocity vector u = (u,v,0), thefirst two equations may also be written as

DuDt

+ f0 × u = −1ρ∇zp, (2.79)

where Du/Dt = ∂u/∂t + v · ∇u, f0 = 2Ω sinϑ0k = f0k, and k is the directionperpendicular to the plane (it does not change its orientation with latitude). Theseequations are, evidently, exactly the same as the momentum equations in a systemin which the rotation vector is aligned with the local vertical, as illustrated in theright panel in Fig. 2.4. They will describe flow on the surface of a rotating sphere toa good approximation provided the flow is of limited latitudinal extent so that theeffects of sphericity are unimportant. This is known as the f-plane approximationsince the Coriolis parameter is a constant. We may in addition make the hydrostaticapproximation, in which case (2.78c) becomes the familiar ∂p/∂z = −ρg.

2.3.2 The beta-plane approximation

The magnitude of the vertical component of rotation varies with latitude, and thishas important dynamical consequences. We can approximate this effect by allow-ing the effective rotation vector to vary. Thus, noting that, for small variations inlatitude,

f = 2Ω sinϑ ≈ 2Ω sinϑ0 + 2Ω(ϑ − ϑ0) cosϑ0, (2.80)

then on the tangent plane we may mimic this by allowing the Coriolis parameter tovary as

f = f0 + βy , (2.81)

where f0 = 2Ω sinϑ0 and β = ∂f/∂y = (2Ω cosϑ0)/a. This important approxi-mation is known as the beta-plane, or β-plane, approximation; it captures the themost important dynamical effects of sphericity, without the complicating geomet-ric effects, which are not essential to describe many phenomena. The momentumequations (2.78a), (2.78b) and (2.78c) (or its hydrostatic counterpart) are unal-tered, save that f0 is replaced by f0 + βy to represent a varying Coriolis parameter.Thus, sphericity combined with rotation is dynamically equivalent to a differentiallyrotating system. For future reference, we write down the β-plane horizontal mo-mentum equations:

DuDt

+ f × u = −1ρ∇zp, (2.82)


where f = (f0 + βy)k. In component form this equation becomes

DuDt

− fv = −1ρ∂p∂x,

DvDt

+ fu = −1ρ∂p∂y, (2.83a,b)

The mass conservation, thermodynamic and hydrostatic equations in the β-planeapproximation are the same as the usual Cartesian, f -plane, forms of those equa-tions.

2.4 EQUATIONS FOR A STRATIFIED OCEAN: THE BOUSSINESQ APPROXIMATION

The density variations in the ocean are quite small compared to the mean density,and we may exploit this to derive somewhat simpler but still quite accurate equa-tions of motion. Let us first examine how much density does vary in the ocean.

2.4.1 Variation of density in the ocean

The variations of density in the ocean are due to three effects: the compressionof water by pressure (which we denote as ∆pρ), the thermal expansion of waterif its temperature changes (∆Tρ), and the haline contraction if its salinity changes(∆Sρ). How big are these? An appropriate equation of state to approximatelyevaluate these effects is the linear one

ρ = ρ0

[1− βT (T − T0)+ βS(S − S0)+

pρ0c2

s

], (2.84)

where βT ≈ 2 × 10−4 K−1, βS ≈ 10−3 psu−1 and cs ≈ 1500 m s−1 (see the table onpage 37). The three effects are then:

Pressure compressibility: We have ∆pρ ≈ ∆p/c2s ≈ ρ0gH/c2

s where H is the depthand the pressure change is quite accurately evaluated using the hydrostaticapproximation. Thus,

|∆pρ|ρ0

1 ifgHc2s 1, (2.85)

or if H c2s /g. The quantity c2

s /g ≈ 200 km is the density scale height ofthe ocean. Thus, the pressure at the bottom of the ocean (say H = 10 km inthe deep trenches), enormous as it is, is insufficient to compress the waterenough to make a significant change in its density. Changes in density dueto dynamical variations of pressure are small if the Mach number is small,and this is also the case.

Thermal expansion: We have ∆Tρ ≈ −βTρ0∆T and therefore

|∆Tρ|ρ0

1 if βT∆T 1. (2.86)

For ∆T = 20 K, βT∆T ≈ 4 × 10−3, and evidently we would require temper-ature differences of order β−1

T , or 5000 K to obtain order one variations indensity.

2.4 The Boussinesq Approximation 69

Saline contraction: We have ∆Sρ ≈ βSρ0∆S and therefore

|∆Sρ|ρ0

1 if βS∆S 1. (2.87)

As changes is salinity in the ocean rarely exceed 5 psu, for which βS∆S =5×10−3, the fractional change in the density of seawater is correspondinglyvery small.

Evidently, fractional density changes in the ocean are very small.

2.4.2 The Boussinesq equations

The Boussinesq equations are a set of equations that exploit the smallness of densityvariations in many liquids.4 To set notation we write

ρ = ρ0 + δρ(x,y, z, t) (2.88a)

= ρ0 + ρ(z)+ ρ′(x,y, z, t) (2.88b)

= ρ(z)+ ρ′(x,y, z, t), (2.88c)

where ρ0 is a constant and we assume that

|ρ|, |ρ′|, |δρ| ρ0. (2.89)

We need not assume that |ρ′| |ρ|, but this is often the case in the ocean. Toobtain the Boussinesq equations we will just use (2.88a), but (2.88c) will be usefulfor the anelastic equations considered later.

Associated with the reference density is a reference pressure that is defined tobe in hydrostatic balance with it. That is,

p = p0(z)+ δp(x,y, z, t) (2.90a)

= p(z)+ p′(x,y, z, t), (2.90b)

where |δp| p0, |p′| p and

dp0

dz≡ −gρ0,

dpdz

≡ −gρ. (2.91a,b)

Note that ∇zp = ∇zp′ = ∇zδp and that p0 ≈ p if |ρ| ρ0.

Momentum equations

To obtain the Boussinesq equations we use ρ = ρ0+δρ, and assume δρ/ρ0 is small.Without approximation, the momentum equation can be written as

(ρ0 + δρ)(

DvDt

+ 2Ω × v)= −∇δp − ∂p0

∂zk− g(ρ0 + δρ)k, (2.92)

and using (2.91a) this becomes, again without approximation,

(ρ0 + δρ)(

DvDt

+ 2Ω × v)= −∇δp − gδρk. (2.93)


If density variations are small this becomes

(DvDt

+ 2Ω × v)= −∇φ+ bk , (2.94)

where φ = δp/ρ0 and b = −gδρ/ρ0 is the buoyancy. Note that we should not anddo not neglect the term gδρ, for there is no reason to believe it to be small (δρ maybe small, but g is big). Eq. (2.94) is the momentum equation in the Boussinesq ap-proximation, and it is common to say that the Boussinesq approximation ignores allvariations of density of a fluid in the momentum equation, except when associatedwith the gravitational term.

For most large-scale motions in the ocean the deviation pressure and densityfields are also approximately in hydrostatic balance, and in that case the verticalcomponent of (2.94) becomes

∂φ∂z

= b. (2.95)

A condition for (2.95) to hold is that vertical accelerations are small compared togδρ/ρ0, and not compared to the acceleration due to gravity itself. For more discus-sion of this point, see section 2.7.

Mass Conservation

The unapproximated mass conservation equation is

DδρDt

+ (ρ0 + δρ)∇ · v = 0. (2.96)

Provided that time scales advectively — that is to say that D/Dt scales in the sameway as v · ∇ — then we may approximate this equation by

∇ · v = 0 , (2.97)

which is the same as that for a constant density fluid. This absolutely does not allowone to go back and use (2.96) to say that Dδρ/Dt = 0; the evolution of density isgiven by the thermodynamic equation in conjunction with an equation of state, andthis should not be confused with the mass conservation equation. Note also thatin eliminating the time-derivative of density we eliminate the possibility of soundwaves.

Thermodynamic equation and equation of state

The Boussinesq equations are closed by the addition of an equation of state, a ther-modynamic equation and, as appropriate, a salinity equation. Neglecting salinityfor the moment, a useful starting point is to write the thermodynamic equation,(1.116), as

DρDt

− 1c2s

DpDt

= Q(∂η/∂ρ)pT

≈ −Q(ρ0βTcp

)(2.98)

using (∂η/∂ρ)p = (∂η/∂T )p(∂T/∂ρ)p ≈ cp/(Tρ0βT ).


Given the expansions (2.88a) and (2.90a) this can be written as

DδρDt

− 1c2s

Dp0

Dt= −Q

(ρ0βTcp

), (2.99)

or, using (2.91a),DDt

(δρ + ρ0g

c2sz)= −Q

(ρ0βTcp

), (2.100)

as in (1.119). The severest approximation to this is to neglect the second term inbrackets, and noting that b = −gδρ/ρ0 we obtain

DbDt

= b , (2.101)

where b = gβT Q/cp. The momentum equation (2.94), mass continuity equation(2.97) and thermodynamic equation (2.101) then form a closed set, called the sim-ple Boussinesq equations.

A somewhat more accurate approach is to include the compressibility of thefluid that is due to the hydrostatic pressure. From (2.100), the potential density isgiven by δρpot = δρ + ρ0gz/c2

s , and so the potential buoyancy, that is the buoyancybased on potential density, is given by

bσ ≡ −gδρpot

ρ0= − g

ρ0

(δρ + ρ0gz

c2s

)= b − g z

Hρ, (2.102)

where Hρ = c2s /g. The thermodynamic equation, (2.100), may then be written

DbσDt

= bσ , (2.103)

where bσ = b. Buoyancy itself is obtained from bσ by the ‘equation of state’,b = bσ + gz/Hρ.

In many applications we may need to use a still more accurate equation of state.In that case (and see section 1.5.5) we replace (2.101) by the thermodynamic equa-tions

DθDt

= θ, DSDt

= S , (2.104a,b)

where θ is the potential temperature and S is salinity, along with an equation ofstate. The equation of state has the general form b = b(θ, S,p), but to be consis-tent with the level of approximation in the other Boussinesq equations we shouldreplace p by the hydrostatic pressure calculated with the reference density, that isby −ρ0gz, and the equation of state then takes the general form

b = b(θ, S, z) . (2.105)

An example of (2.105) is (1.174), taken with the definition of buoyancy b =−gδρ/ρ0. The closed set of equations (2.94), (2.97), (2.104) and (2.105) are the


Summary of Boussinesq Equations

The simple Boussinesq equations are, for an inviscid fluid:

Momentum equations:DvDt

+ f × v = −∇φ+ bk (B.1)

Mass conservation: ∇ · v = 0 (B.2)

Buoyancy equation:DbDt

= b (B.3)

A more general form replaces the buoyancy equation by:

Thermodynamic equation:DθDt

= θ (B.4)

Salinity equation:DSDt

= S (B.5)

Equation of state: b = b(θ, S,φ) (B.6)

Energy conservation is only assured if b = b(θ, S, z).

general Boussinesq equations. Using an accurate equation of state and the Boussi-nesq approximation is the procedure used in many comprehensive ocean generalcirculation models. The Boussinesq equations, which with the hydrostatic and tra-ditional approximations are often considered to be the oceanic primitive equations,are summarized in the shaded box above.

* Mean stratification and the buoyancy frequency

The processes that cause density to vary in the vertical often differ from those thatcause it to vary in the horizontal. For this reason it is sometimes useful to writeρ = ρ0 + ρ(z)+ ρ′(x,y, z, t) and define b(z) ≡ −gρ/ρ0 and b′ ≡ −gρ′/ρ0. Usingthe hydrostatic equation to evaluate pressure, the thermodynamic equation (2.98)becomes, to a good approximation,

Db′

Dt+N2w = 0, (2.106)

where D/Dt remains a three-dimensional operator and

N2(z) =(

dbdz

− g2

c2s

)= dbσ

dz, (2.107)

where bσ = b − gz/Hρ. The quantity N2 is a measure of the mean stratification ofthe fluid, and is equal to the vertical gradient of the mean potential buoyancy. N isknown as the buoyancy frequency, something we return to in section 2.9. Equations(2.106) and (2.107) also hold in the simple Boussinesq equations, but with c2

s = ∞.


2.4.3 Energetics of the Boussinesq system

In a uniform gravitational field but with no other forcing or dissipation, we writethe simple Boussinesq equations as

DvDt

+ 2Ω× v = bk−∇φ, ∇ · v = 0,DbDt

= 0. (2.108a,b,c)

From (2.108a) and (2.108b) the kinetic energy density evolution (c.f., section 1.9)is given by

12

Dv2

Dt= bw −∇ · (φv), (2.109)

where the constant reference density ρ0 is omitted. Let us now define the potentialΦ such that ∇Φ = −k (so Φ = −z) and so

DΦDt

= ∇ · (vΦ) = −w. (2.110)

Using this and (2.108c) gives

DDt(bΦ) = −wb. (2.111)

Adding this to (2.109) and expanding the material derivative gives

∂∂t

(12v2 + bΦ

)+∇ ·

[v(

12v2 + bΦ +φ

)]= 0. (2.112)

This constitutes an energy equation for the Boussinesq system, and may be com-pared to (1.189). (See also problem 2.12.) The energy density (divided by ρ0) isjust v2/2+bΦ. What does the second term represent? Its integral, multiplied by ρ0,is the potential energy of the flow minus that of the basic state, or

∫g(ρ − ρ0)z dz.

If there were a heating term on the right-hand side of (2.108c) this would directlyprovide a source of potential energy, rather than internal energy as in the compress-ible system. Because the fluid is incompressible, there is no conversion from kineticand potential energy into internal energy.

* Energetics with a general equation of state

Now consider the energetics of the general Boussinesq equations. Suppose first thatwe allow the equation of state to be a function of pressure; the equations are motionare then (2.108) except that (2.108c) is replaced by

DθDt

= 0,DSDt

= 0, b = b(θ, S,φ). (2.113a,b,c)

A little algebraic experimentation will reveal that no energy conservation law ofthe form (2.112) generally exists for this system! The problem arises because, byrequiring that the fluid be incompressible, we eliminate the proper conversion ofinternal energy to kinetic energy. However, if we use the consistent approximationb = b(θ, S, z), the system does conserve an energy, as we now show.5

Define the potential, Π, by the integral of b at constant potential temperatureand salinity

Π(θ, S, z) ≡ −∫b dz. (2.114)


Taking its material derivative gives

DΠDt

=(∂Π∂θ

)S,z

DθDt

+(∂Π∂S

)θ,z

DSDt

+(∂Π∂z

)θ,S

DzDt

= −bw, (2.115)

using (2.113a,b). Combining (2.115) and (2.109) gives

∂∂t

(12v2 +Π

)+∇ ·

[v(

12v2 +Π +φ

)]= 0. (2.116)

Thus, energetic consistency is maintained with an arbitrary equation of state, pro-vided the pressure is replaced by a function of z. If b is not an explicit function ofz in the equation of state, the conservation law is identical to (2.112).

2.5 EQUATIONS FOR A STRATIFIED ATMOSPHERE: THE ANELASTICAPPROXIMATION

2.5.1 Preliminaries

In the atmosphere the density varies significantly, especially in the vertical. How-ever deviations of both ρ and p from a statically balanced state are often quitesmall, and the relative vertical variation of potential temperature is also small. Wecan usefully exploit these observations to give a somewhat simplified set of equa-tions, useful both for theoretical and numerical analysis because sound waves areeliminated by way of an ‘anelastic’ approximation.6 To begin we set

ρ = ρ(z)+ δρ(x,y, z, t), p = p(z)+ δp(x,y, z, t), (2.117a,b)

where we assume that |δρ| |ρ| and we define p such that

∂p∂z

≡ −gρ(z). (2.118)

The notation is similar to that for the Boussinesq case except that, importantly, thedensity basic state is now a (given) function of vertical coordinate. As with theBoussinesq case, the idea is to ignore dynamic variations of density (i.e., of δρ)except where associated with gravity. First recall a couple of ideal gas relationshipsinvolving potential temperature, θ, and entropy s (divided by cp, so s ≡ logθ),namely

s ≡ logθ = logT − Rcp

logp = 1γ

logp − logρ, (2.119)

where γ = cp/cv , implying

δs = 1γδpp− δρρ≈ 1γδpp− δρρ. (2.120)

Further, if s ≡ γ−1 log p − log ρ then

dsdz

= 1γp

dpdz

− 1ρ

dρdz

= −gργp

− 1ρ

dρdz. (2.121)

In the atmosphere, the left-hand side is, typically, much smaller than either of thetwo terms on the right-hand side.

2.5 The Anelastic Approximation 75

2.5.2 The Momentum equation

The exact inviscid horizontal momentum equation is

(ρ + ρ′)DuDt

+ f × u = −∇zδp. (2.122)

Neglecting ρ′ where it appears with ρ leads to

DuDt

+ f × u = −∇zφ, (2.123)

whereφ = δp/ρ, and this is similar to the corresponding equation in the Boussinesqapproximation.

The vertical component of the inviscid momentum equation is, without approx-imation,

(ρ + δρ)DwDt

= −∂p∂z

− ∂δp∂z

− gρ − gδρ = −∂δp∂z

− gδρ. (2.124)

using (2.117). Neglecting δρ on the left-hand side we obtain

DwDt

= −1ρ∂δp∂z

− gδρρ= − ∂

∂z

(δpρ

)− δpρ2∂ρ∂z

− gδρρ. (2.125)

This is not a useful form for a gaseous atmosphere, since the variation of the meandensity cannot be ignored. However, we may eliminate δρ in favour of δs using(2.120) to give

DwDt

= gδs − ∂∂z

(δpρ

)− gγδpp− δpρ2∂ρ∂z, (2.126)

and using (2.121) gives

DwDt

= gδs − ∂∂z

(δpρ

)+ ds

dzδpρ. (2.127)

What have these manipulations gained us? Two things:(i) The gravitational term now involves δs rather than δρ which enables a more

direct connection with the thermodynamic equation.(ii) The potential temperature scale height (∼100 km) in the atmosphere is much

larger than the density scale height (∼10 km), and so the last term in (2.127)is small.

The second item thus suggests that we choose our reference state to be one of con-stant potential temperature (see also problem 2.17). The term ds/dz then vanishesand the vertical momentum equation becomes

DwDt

= gδs − ∂φ∂z

, (2.128)

where φ = δp/ρ, δs = δθ/θ and θ = θ0, a constant. If we define a buoyancy byba ≡ gδs = gδθ/θ, then (2.123) and (2.128) have the same form as the Boussinesqmomentum equations, but with a slightly different definitions of b.


2.5.3 Mass conservation

Using (2.117a) the mass conservation equation may be written, without approxi-mation, as

∂δρ∂t

+∇ · [(ρ + δρ)v] = 0. (2.129)

We neglect δρ where it appears with ρ in the divergence term. Further, the localtime derivative will be small if time itself is scaled advectively (i.e., T ∼ L/U andsound waves do not dominate), giving

∇ · u+ 1ρ∂∂z(ρw) = 0. (2.130)

It is here that the eponymous ‘anelastic approximation’ arises: the elastic com-pressibility of the fluid is neglected, and this serves to eliminate sound waves. Forreference, in spherical coordinates the equation is

1a cosϑ

∂u∂λ

+ 1a cosϑ

∂∂ϑ(v cosϑ)+ 1

ρ∂(wρ)∂z

= 0. (2.131)

In an ideal gas, the choice of constant potential temperature determines how thereference density ρ varies with height. In some circumstances it is convenient tolet ρ be a constant, ρ0 (effectively choosing a different equation of state), in whichcase the anelastic equations become identical with the Boussinesq equations, al-beit with the buoyancy interpreted in terms of potential temperature in the formerand density in the latter. Because of their similarity, the Boussinesq and anelas-tic approximations are sometimes referred to as a the strong and weak Boussinesqapproximations, respectively.

2.5.4 Thermodynamic equation

The thermodynamic equation for an ideal gas may be written

D lnθDt

= QTcp

. (2.132)

In the anelastic equations, θ = θ+δθ where θ is constant, and the thermodynamicequation is

DδsDt

= θTcp

Q. (2.133)

Summarizing, the complete set of anelastic equations, with rotation but with nodissipation or diabatic terms, is

DvDt

+ 2Ω× v = kba −∇φ

DbaDt

= 0

∇ · (ρv) = 0

, (2.134)

2.5 The Anelastic Approximation 77

where ba = gδs = gδθ/θ. The main difference between the anelastic and Boussi-nesq sets of equations is in the mass continuity equation, and when ρ = ρ0 =constant the two equation sets are formally identical. However, whereas the Boussi-nesq approximation is a very good one for ocean dynamics, the anelastic approxima-tion is much less so for large-scale atmosphere flow: the constancy of the referencepotential temperature state is then not a particularly good approximation and sothe deviations in density from its reference profile are not especially small, leadingto inaccuracies in the momentum equation. Nevertheless, the anelastic equationshave been used very productively in limited area ‘large-eddy-simulations’ where onedoes not wish to make the hydrostatic approximation but where sound waves areunimportant.7 The equations also provide a good jumping-off point for theoreticalstudies and the still simpler models that will be considered in the chapter 5.

2.5.5 * Energetics of the anelastic equations

Conservation of energy follows in much the same way as for the Boussinesq equa-tions, except that ρ enters. Take the dot product of (2.134a) with ρv to obtain

ρDDt

(12v2)= −∇ · (φρv)+ baρw. (2.135)

Now, define a potential Φ(z) such that ∇Φ = −k, and so

ρDΦDt

= −wρ. (2.136)

Combining this with the thermodynamic equation (2.134b) gives

ρD(baΦ)

Dt= −wbaρ. (2.137)

Adding this to (2.135) gives

ρDDt

(12v2 + baΦ

)= −∇ · (φρv), (2.138)

or, expanding the material derivative,

∂∂t

[ρ(

12v2 + baΦ

)]+∇ ·

[ρv

(12v2 + baΦ +φ

)]= 0. (2.139)

This equation has the form

∂E∂t+∇ ·

[v(E + ρφ)

]= 0, (2.140)

where E = ρ(v2/2 + baΦ) is the energy density of the flow. This is a consistentenergetic equation for the system, and when integrated over a closed domain thetotal energy is evidently conserved. The total energy density comprises the kineticenergy and a term ρbaΦ, which is analogous to the potential energy of a Boussinesqsystem. However, it is not exactly equal to that because ba is the bouyancy basedon potential temperature, not density; rather, the term combines contributions fromboth the internal energy and the potential energy.


2.6 CHANGING VERTICAL COORDINATE

Although using z as a vertical coordinate is a natural choice given our Cartesianworldview, it is not the only option, nor is it always the most useful one. Any vari-able that has a one-to-one correspondence with z in the vertical, so any variablethat varies monotonically with z, could be used; pressure and, perhaps surpris-ingly, entropy, are common choices. In the atmosphere pressure almost always fallsmonotonically with height, and using it instead of z provides a useful simplificationof the mass conservation and geostrophic relations, as well as a more direct con-nection with observations, which are often taken at fixed values of pressure. (In theocean pressure is almost the same as height, because density is almost constant.)Entropy seems an exotic vertical coordinate, but it is very useful in adiabatic flow,and we consider that in chapter 3.

2.6.1 Pressure coordinates

The primitive equations of motion for an ideal gas can be written,

DuDt

+ f × u = −1ρ∇p, ∂p

∂z= −ρg, (2.141a)

DθDt

= 0,DρDt

+ ρ∇ · v = 0, (2.141b)

where p = ρRT and θ = T(pR/p

)R/cp , and pR is the reference pressure. These arerespectively the horizontal momentum, hydrostatic, thermodynamic and mass con-tinuity equations. They can be put into a form similar to the Boussinesq equationsby transforming from Cartesian [i.e., (x,y, z)] to pressure coordinates, (x,y,p).The analog of the vertical velocity isω ≡ Dp/Dt, and the advective derivative itselfis given by

DDt

= ∂∂t+ u · ∇p +ω

∂∂p. (2.142)

The horizontal and time derivatives are taken at constant pressure. However, x andy are still purely horizontal coordinates, and u = ui+vj is still a strictly horizontalvelocity, perpendicular to the vertical (z) axis. The operator D/Dt is of course thesame in pressure or height coordinates because is simply the total derivative of someproperty of a fluid parcel. However, the individual terms comprising it in generaldiffer between height and pressure coordinates.

To obtain an expression for the pressure force, first consider a general verticalcoordinate, ξ. Then the chain rule gives(

∂∂x

)ξ=(∂∂x

)z+(∂z∂x

)ξ

∂∂z. (2.143)

Now let ξ = p and apply the relationship to p itself to give

0 =(∂p∂x

)z+(∂z∂x

)p

∂p∂z, (2.144)

which, using the hydrostatic relationship, gives(∂p∂x

)z= ρ

(∂Φ∂x

)p, (2.145)

2.6 Changing Vertical Coordinate 79

where Φ = gz is the geopotential. Thus, the horizontal pressure force in the mo-mentum equations is

1ρ∇zp = ∇pΦ, (2.146)

where the subscripts on the gradient operator indicate that the horizontal deriva-tives are taken at constant z or constant p. Also, from (2.141a), the hydrostaticequation is just

∂Φ∂p

= −α. (2.147)

Mass continuity

The mass conservation equation simplifies attractively in pressure coordinates, ifthe hydrostatic approximation is used. Recall that the mass conservation equationcan be derived from the Lagrangian form

DDtρδV = 0, (2.148)

where δV = δxδyδz is a volume element. But by the hydrostatic relationshipρδz = (1/g)δp and thus

DDt(δxδyδp) = 0. (2.149)

This is completely analogous to the expression for the Lagrangian conservation ofvolume in an incompressible fluid, (1.15). Thus, without further ado, we write themass conservation in pressure coordinates as

∇p · u+∂ω∂p

= 0, (2.150)

where the horizontal derivative is taken at constant pressure. The primitive equa-tions in pressure coordinates, equivalent to (2.141) in height co-ordinates, are thus:

DuDt

+ f × u = −∇pΦ,∂Φ∂p

= −α

DθDt

= 0, ∇p · u+∂ω∂p

= 0, (2.151)

where D/Dt is given by (2.142). The equation set is completed with the addition ofthe ideal gas equation and the definition of potential temperature. These equationsare isomorphic to the hydrostatic general Boussinesq equations (see shaded boxon page 72) with z ↔ −p, w ↔ −ω, φ ↔ Φ, b ↔ α, and an equation of stateb = b(θ, z)↔ α = α(θ,p). In an ideal gas, for example, α = −(θR/pR)(pR/p)1/γ .

The main practical difficulty with the pressure-coordinate equations is the lowerboundary condition. Using

w ≡ DzDt

= ∂z∂t+ u · ∇pz +ω

∂z∂p, (2.152)


and (2.147), the boundary condition of w = 0 at z = zs becomes

∂Φ∂t

+ u · ∇pΦ −αω = 0 (2.153)

at p(x,y, zs , t). In theoretical studies, it is common to assume that the lowerboundary is in fact a constant pressure surface and simply assume that ω = 0,or sometimes the condition ω = −α−1∂Φ/∂t is used. For realistic studies (withgeneral circulation models, say) the fact that the level z = 0 is not a coordinatesurface must be properly accounted for. For this reason, and especially if the lowerboundary is uneven because of the presence of topography, so-called sigma coor-dinates are sometimes used, in which the vertical coordinate is chosen so that thelower boundary is a coordinate surface. Sigma coordinates may use height itself asa measure of displacement (typical in oceanic applications) or use pressure (typicalin atmospheric applications). In the latter case the vertical coordinate is σ = p/pswhere ps(x,y, t) is the surface pressure. The difficulty of applying (2.153) is re-placed by a prognostic equation for the surface pressure, which is derived from themass conservation equation (problem 2.21).

2.6.2 Log-pressure coordinates

A variant of pressure coordinates is log-pressure coordinates, in which the verti-cal coordinate is Z = −H ln(p/pR) where pR is a reference pressure (say 1000 mb)and H a constant (for example the scale height RTs/g) so that Z has units of length.(Capital letters are conventionally used for some variables in log-pressure coordi-nates, and these are not to be confused with scaling parameters.) The ‘verticalvelocity’ for the system is now

W ≡ DZDt, (2.154)

and the advective derivative is now

DDt

≡ ∂∂t+ u · ∇p +W

∂∂Z. (2.155)

It is straightforward to show (problem 2.22) that the primitive equations of motionin these coordinates are:

DuDt

+ f × u = −∇ZΦ,∂Φ∂Z

= RTH, (2.156a)

DθDt

= 0,∂u∂x

+ ∂v∂y

+ ∂W∂Z

− WH= 0. (2.156b)

The last equation may be written∇Z·u+ρ−1R ∂(ρRW)/∂z = 0, where ρR = exp(−z/H),

so giving a form similar to the mass conservation equation in the anelastic equa-tions.

2.7 HYDROSTATIC BALANCE

In this section we consider one of the most fundamental balances in geophysicalfluid dynamics — hydrostatic balance, and in the next section we consider an-other fundamental balance, geostrophic balance. Neither hydrostasy (the state of

2.7 Hydrostatic Balance 81

geostrophic balance) nor geostrophy (the state of geostrophic balance) are exactlyrealized in the atmosphere or ocean, but their approximate satisfaction has pro-found consequences on the behaviour of the atmosphere and ocean. We first en-countered hydrostatic balance in section 1.3.4; we now look in more detail at theconditions required for it to hold.

2.7.1 Preliminaries

Consider the relative sizes of terms in (2.77c):

WT+ UWL

+ W2

H+ΩU ∼

∣∣∣∣∣1ρ∂p∂z

∣∣∣∣∣+ g. (2.157)

For most large-scale motion in the atmosphere and ocean the terms on the right-hand side are orders of magnitude larger than those on the left, and therefore mustbe approximately equal. Explicitly, suppose W ∼ 1 cm s−1, L ∼ 105 m, H ∼ 103 m,U ∼ 10 m s−1, T = L/U . Then by substituting into (2.157) it seems that the pressureterm is the only one which could balance the gravitational term, and we are led toapproximate (2.77c) by,

∂p∂z

= −ρg. (2.158)

This equation, which is a vertical momentum equation, is known as hydrostaticbalance.

However, (2.158) is not always a useful equation! Let us suppose that the den-sity is a constant, ρ0 . We can then write the pressure as

p(x,y, z, t) = p0(z)+ p′(x,y, z, t), (2.159)

where∂p0

∂z≡ −ρ0g. (2.160)

That is, p0 and ρ0 are in hydrostatic balance. The inviscid vertical momentumequation becomes, without approximation,

DwDt

= − 1ρ0

∂p′

∂z. (2.161)

Thus, for constant density fluids, the gravitational term has no dynamical effect:there is no buoyancy force, and the pressure term in the horizontal momentumequations can be replaced by p′. Hydrostatic balance, and in particular (2.160), iscertainly not an appropriate vertical momentum equation in this case. If the fluid isstratified, we should therefore subtract off the hydrostatic pressure associated withthe mean density before we can determine whether hydrostasy is a useful dynami-cal approximation, accurate enough to determine the horizontal pressure gradients.This is automatic in the Boussinesq equations, where the vertical momentum equa-tion is

DwDt

= −∂φ∂z

+ b, (2.162)

and the hydrostatic balance of the basic state is already subtracted out. In the moregeneral equation,

DwDt

= −1ρ∂p∂z

− g, (2.163)


we need to compare the advective term on the left-hand side with the pressurevariations arising from horizontal flow in order to determine whether hydrostasyis an appropriate vertical momentum equation. Nevertheless, if we simply needto determine the pressure for use in an equation of state then we simply needto compare the sizes of the dynamical terms in (2.77c) with g itself in order todetermine whether a hydrostatic approximation will suffice.

2.7.2 Scaling and the aspect ratio

In a Boussinesq fluid we write the horizontal and vertical momentum equations as

DuDt

+ f × u = −∇φ, DwDt

= −∂φ∂z

− b. (2.164a,b)

With f = 0, (2.164a) implies the scaling

φ ∼ U2. (2.165)

If we use mass conservation, ∇z · u+ ∂w/∂z = 0, to scale vertical velocity then

w ∼ W = HLU = αU, (2.166)

where α ≡ H/L is the aspect ratio. The advective terms in the vertical momentumequation all scale as

DwDt

∼ UWL

= U2HL2 . (2.167)

Using (2.165) and (2.167) the ratio of the advective term to the pressure gradientterm in the vertical momentum equations then scales as

|Dw/Dt||∂φ/∂z| ∼

U2H/L2

U2/H∼(HL

)2. (2.168)

Thus, the condition for hydrostasy, that |Dw/Dt|/|∂φ/∂z| 1, is:

α2 ≡(HL

)2 1 . (2.169)

The advective term in the vertical momentum may then be neglected. Thus, hy-drostasy is a small aspect ratio approximation.

We can obtain the same result more formally by nondimensionalizing the mo-mentum equations. Using uppercase symbols to denote scaling values we write

(x,y) = L(x, y), z = Hz, u = Uu, w = Ww = HULw,

t = T t = LUt, φ = Φφ = U2φ, b = Bb = U

2

Hb,

(2.170)

where the hatted variables are nondimensional and the scaling for w is suggested

2.7 Hydrostatic Balance 83

by the mass conservation equation, ∇z · u + ∂w/∂z = 0. Substituting (2.170) into(2.164) (with f = 0) gives us the nondimensional equations

DuDt

= −∇φ, α2 DwDt

= −∂φ∂z

− b, (2.171a,b)

where D/Dt = ∂/∂t + u∂/∂x + v∂/∂y + w∂/∂z and we use the convention thatwhen ∇ operates on nondimensional quantities the operator itself is nondimen-sional. From (2.171b) it is clear that hydrostatic balance pertains when α2 1.

2.7.3 * Effects of stratification on hydrostatic balance

To include the effects of stratification we need to involve the thermodynamic equa-tion, so let us first write down the complete set of non-rotating dimensional equa-tions:

DuDt

= −∇zφ,DwDt

= −∂φ∂z

+ b′, (2.172a,b)

Db′

Dt+wN2 = 0, ∇ · v = 0. (2.173a,b)

We have written, without approximation, b = b′(x,y, z, t) + b(z), with N2 =db/dz; this separation is useful because the horizontal and vertical buoyancy varia-tions may scale in different ways, and often N2 may be regarded as given. (We alsoredefine φ by subtracting off a static component in hydrostatic balance with b.) Wenondimensionalize (2.173) by first writing

(x,y) = L(x, y), z = Hz, u = Uu, w = Ww = εHULw,

t = T t = LUt, φ = U2φ, b′ = ∆bb = U

2

Hb′, N2 = N2N2,

(2.174)

where ε is, for the moment, undetermined, N is a representative, constant, valueof the buoyancy frequency and ∆b scales only the horizontal buoyancy variations.Substituting (2.174) into (2.172) and (2.173) gives

DuDt

= −∇zφ, εα2 DwDt

= −∂φ∂z

+ b′ (2.175a,b)

U2

N2H2

Db′

Dt+ εwN2 = 0, ∇ · u+ ε∂w

∂z= 0. (2.176a,b)

where now D/Dt = ∂/∂t+u·∇z+ε∂/∂z. To obtain a nontrivial balance in (2.176a)we choose ε = U2/(N2H2) ≡ Fr2, where Fr is the Froude number, a measure of thestratification of the flow. The vertical velocity then scales as

W = FrUHL

(2.177)

and if the flow is highly stratified the vertical velocity will be even smaller than apure aspect ratio scaling might suggest. (There must, therefore, be some cancella-tion in horizontal divergence in the mass continuity equation; that is, |∇z · u|


U/L.) With this choice of ε the nondimensional Boussinesq equations may be writ-ten:

DuDt

= −∇zφ, Fr2α2 DwDt

= −∂φ∂z

+ b′ (2.178a,b)

Db′

Dt+ wN2 = 0, ∇ · u+ Fr2 ∂w

∂z= 0. (2.179a,b)

The nondimensional parameters in the system are the aspect ratio and the Froudenumber (in addition to N, but by construction this is just an order one function ofz). From (2.178b) condition for hydrostatic balance to hold is evidently that

Fr2α2 1 , (2.180)

so generalizing the aspect ratio condition (2.169) to a stratified fluid. Because Fris a measure of stratification, (2.180) formalizes our intuitive expectation that themore stratified a fluid the more vertical motion is suppressed and therefore themore likely hydrostatic balance is to hold. Note also that (2.180) is equivalent toU2/(L2N2) 1.

Suppose we solve the hydrostatic equations; that is, we omit the advectivederivative in the vertical momentum equation, and by numerical integration weobtain u, w and b. This flow is the solution of the nonhydrostatic equations in thesmall aspect ratio limit. The solution never violates the scaling assumptions, even ifw seems large, because we can always rescale the variables in order that condition(2.180) is satisfied.

Why bother with any of this scaling? Why not just say that hydrostatic balanceholds when |Dw/Dt| |∂φ/∂z|? One reason is that we don’t have a good ideaof the value of w from direct measurements, and it may change significantly in dif-ferent oceanic and atmospheric parameter regimes. On the other hand the Froudenumber and the aspect ratio are familiar nondimensional parameters with a wideapplicability in other contexts, and which we can control in a laboratory settingor estimate in the ocean or atmosphere. Still, in scaling theory it is common thatascertaining which parameters are to be regarded as given and which should bederived is a choice, rather than being set a priori.

2.7.4 Hydrostasy in the ocean and atmosphere

Is the hydrostatic approximation in fact a good one in the ocean and atmosphere?

In the ocean

For the large scale ocean circulation, let N ∼ 10−2 s−1, U ∼ 0.1 m s−1 and H ∼ 1 km.Then Fr = U/(NH) ∼ 10−2 1. Thus, Fr2α2 1 even for unit aspect-ratiomotion. In fact, for larger scale flow the aspect ratio is also small; for basin-scaleflow L ∼ 106 m and Fr2α2 ∼ 0.012 × 0.0012 = 10−10 and hydrostatic balance is anextremely good approximation.

For intense convection, for example in the Labrador Sea, the hydrostatic approx-imation may be less appropriate, because the intense descending plumes may havean aspect ratio (H/L) of one or greater and the stratification is very weak. Thehydrostatic condition then often becomes the requirement that the Froude number

2.8 Geostrophic and Thermal Wind Balance 85

is small. Representative orders of magnitude are U ∼ W ∼ 0.1 m s−1, H ∼ 1 km andN ∼ 10−3 s−1–10−4 s−1. For these values Fr ranges between 0.1 and 1, and at theupper end of this range hydrostatic balance is violated.

In the atmosphere

Over much of the troposphere N ∼ 10−2 s−1 so that with U = 10 m s−1 and H = 1 kmwe find Fr ∼ 1. Hydrostasy is then maintained because the aspect ratio H/L is muchless than unity. For larger scale synoptic activity a larger vertical scale is appropriate,and with H = 10 km both the Froude number and the aspect ratio are much smallerthan one; indeed with L = 1000 km we find Fr2α2 ∼ 0.12 × 0.12 = 10−4 and theflow is hydrostatic to a very good approximation indeed. However, for smaller scaleatmospheric motion associated with fronts and, especially, convection, there can belittle expectation that hydrostatic balance will be a good approximation.

2.8 GEOSTROPHIC AND THERMAL WIND BALANCE

We now consider the dominant dynamical balance in the horizontal componentsof the momentum equation. In the horizontal plane (meaning along geopotentialsurfaces) we find that the Coriolis term is much larger than the advective terms andthe dominant balance is between it and the horizontal pressure force. This balanceis called geostrophic balance, and it occurs when the Rossby number is small, as wenow investigate.

2.8.1 The Rossby Number

The Rossby number characterizes the importance of rotation in a fluid.8 It is, essen-tially, the ratio of the magnitude of the relative acceleration to the Coriolis acceler-ation, and it is of fundamental importance in geophysical fluid dynamics. It arisesfrom a simple scaling of horizontal momentum equation, namely

∂u∂t+(v · ∇)u+ f × u = −1

ρ∇zp, (2.181a)

U2/L fU (2.181b)

where U is the approximate magnitude of the horizontal velocity and L is a typicallengthscale over which that velocity varies. (We assume that W/H Ü U/L, so thatvertical advection does not dominate the advection.) The ratio of the sizes of theadvective and Coriolis terms is defined to be the Rossby number,

Ro ≡ UfL

. (2.182)

If the Rossby number is small then rotation effects are important, and as the valuesin table 2.1 indicate this is the case for large-scale flow in both ocean and atmo-sphere.

Another intuitive way to think about the Rossby number is in terms of timescales.The Rossby number based on a timescale is

RoT ≡1fT, (2.183)


Variable Scaling Meaning Atmos. value Ocean valueSymbol

(x,y) L Horizontal length 106 m 105 mt T Timescale 1 day (105 s) 10 days (106 s)

(u,v) U Horizontal velocity 10 m s−1 0.1 m s−1

Ro Rossby number, U/fL 0.1 0.01

Table 2.1 Scales of large-scale flow in atmosphere and ocean. The choicesgiven are representative of large-scale eddying motion in both systems.

where T is a timescale associated with the dynamics at hand. If the timescale is anadvective one, meaning that T ∼ L/U , then this definition is equivalent to (2.182).Now, f = 2Ω sinϑ, where Ω is the angular velocity of the rotating frame and equalto 2π sinϑ/Tp where Tp is the period of rotation (24 hours). Thus,

RoT =Tp

4πT sinϑ= TiT, (2.184)

where Ti = 1/f is the ‘inertial timescale’, about three hours in midlatitudes. Thus,for phenomena with timescales much longer than this, such as the motion of theGulf Stream or a mid-latitude atmospheric weather system, the effects of the earth’srotation can be expected to be important, whereas a short-lived phenomena, suchas a cumulus cloud or tornado, may be oblivious to such rotation. The expressions(2.182) and (2.183) of course, just approximate measures of the importance ofrotation.

2.8.2 Geostrophic Balance

If the Rossby number is sufficiently small in (2.181a) then the rotation term willdominate the nonlinear advection term, and if the time period of the motion scalesadvectively then the rotation term also dominates the local time derivative. Theonly term which can then balance the rotation termis the pressure term, and there-fore we must have

f × u ≈ −1ρ∇zp, (2.185)

or, in Cartesian component form

fu ≈ −1ρ∂p∂y, fv ≈ 1

ρ∂p∂x. (2.186)

This balance is known as geostrophic balance, and its consequences are profound,giving geophysical fluid dynamics a special place in the broader field of fluid dy-namics. We define the geostrophic velocity by

fug ≡ −1ρ∂p∂y, fvg ≡

1ρ∂p∂x

, (2.187)


Fig. 2.5 Schematic of geostrophic flow with a positive value of the Coriolisparameter f . Flow is parallel to the lines of constant pressure (isobars). Cy-clonic flow is anticlockwise around a low pressure region and anticyclonicflow is clockwise around a high. If f were negative, as in the Southern hemi-sphere, (anti-)cyclonic flow would be (anti-)clockwise.

and for low Rossby number flow u ≈ ug and v ≈ vg. In spherical coordinates thegeostrophic velocity is

fug = −1ρa∂p∂ϑ, fvg =

1aρ cosϑ

∂p∂λ, (2.188)

where f = 2Ω sinϑ. Geostrophic balance has a number of immmediate ramifica-tions:

? Geostrophic flow is parallel to lines of constant pressure (isobars). If f > 0 theflow is anti-clockwise round a region of low pressure and clockwise around aregion of high pressure (see Fig. 2.5).

? If the Coriolis force is constant and if the density does not vary in the horizontalthe geostrophic flow is horizontally non-divergent and

∇z · ug =∂ug∂x

+ ∂vg∂y

= 0 . (2.189)

We may define the geostrophic streamfunction, ψ, by

ψ ≡ pf0ρ0

, (2.190)

whenceug = −

∂ψ∂y, vg =

∂ψ∂x. (2.191)

The vertical component of vorticity, ζ, is then given by

ζ = k · ∇× v = ∂v∂x

− ∂u∂y

= ∇2zψ. (2.192)


? If the Coriolis parameter is not constant, then cross-differentiating (2.187)gives, for constant density geostrophic flow,

vg∂f∂y

+ f∇z · ug = 0, (2.193)

which implies, using mass continuity,

βvg = f∂w∂z. (2.194)

where β ≡ ∂f/∂y = 2Ω cosϑ/a. This geostrophic vorticity balance is some-times known as Sverdrup balance, although that expression is better restrictedto the case when the vertical velocity results from external agents, and specif-ically a wind stress, as considered in chapter 14.

2.8.3 Taylor-Proudman effect

If β = 0, then (2.194) implies that the vertical velocity is not a function of height.In fact, in that case none of the components of velocity vary with height if densityis also constant. To show this, in the limit of zero Rossby number we first write thethree-dimensional momentum equation as

f0 × v = −∇φ−∇χ, (2.195)

where f0 = 2Ω = 2Ωk, φ = p/ρ0, and ∇χ represents other potential forces. Ifχ = gz then the vertical component of this equation represents hydrostatic balance,and the horizontal components represent hydrostatic balance. On taking the curlof this equation, the terms on the right-hand side vanish and the left-hand sidebecomes

(f0 · ∇)v − f0∇ · v − (v · ∇)f0 + v∇ · f0 = 0. (2.196)

But ∇ · v = 0 by mass conservation, and because f0 is constant both ∇ · f0 and(v · ∇)f0 vanish. Thus

(f0 · ∇)v = 0, (2.197)

which, since f0 = f0k, implies f0∂v/∂z = 0, and in particular we have

∂u∂z

= 0,∂v∂z

= 0,∂w∂z

= 0. (2.198)

A different presentation of this argument proceeds as follows. If the flow isexactly in geostrophic and hydrostatic balance then

v = 1f0

∂φ∂x, u = − 1

f0

∂φ∂y,

∂φ∂z

= −g. (2.199a,b,c)

Differentiating (2.199a,b) with respect to z, and using (2.199c) yields

∂v∂z

= −1f0

∂g∂x

= 0,∂u∂z

= 1f0

∂g∂y

= 0. (2.200)


Noting that the geostrophic veclocities are horizontally non-divergent (∇z · u = 0),and using mass continuity then gives ∂w/∂z = 0, as before.

If there is a solid horizontal boundary anywhere in the fluid, for example atthe surface, then w = 0 at that surface and thus w = 0 everywhere. Hence themotion occurs in planes that lie perpendicular to the axis of rotation, and the flowis effectively two-dimensional. This result is known as the Taylor-Proudman effect,namely that for constant density flow in geostrophic and hydrostatic balance thevertical derivatives of the horizontal and the vertical velocities are zero.9 At zeroRossby number, if the vertical velocity is zero somewhere in the flow, it is zeroeverywhere in that vertical column; furthermore, the horizontal flow has no verticalshear, and the fluid moves like a slab. The effects of rotation have provided astiffening of the fluid in the vertical.

In neither the atmosphere nor the ocean do we observe precisely such verticallycoherent flow, mainly because of the effects of stratification. However, it is typicalof geophysical fluid dynamics that the assumptions underlying a derivation are notfully satisfied, yet there are manifestations of it in real flow. Thus, one might havenaïvely expected, because ∂w/∂z = −∇z ·u, that the scales of the various variableswould be related by W/H ∼ U/L. However, if the flow is rapidly rotating we expectthat the horizontal flow will be in near geostrophic balance and therefore nearlydivergence free; thus ∇z · u U/L, and W HU/L.

2.8.4 Thermal wind balance

Thermal wind balance arises by combining the geostrophic and hydrostatic approx-imations, and this is most easily done in the context of the anelastic (or Boussinesq)equations, or in pressure coordinates. For the anelastic equations, geostrophic bal-ance may be written

− fvg = −∂φ∂x

= − 1a cosϑ

∂φ∂λ, fug = −

∂φ∂y

= −1a∂φ∂ϑ. (2.201a,b)

Combining these relations with hydrostatic balance, ∂φ/∂z = b, gives

−f ∂vg∂z

= − ∂b∂x

= − 1a cosλ

∂b∂λ

f∂ug∂z

= − ∂b∂y

= −1a∂b∂ϑ

. (2.202a,b)

These equations represent thermal wind balance, and the vertical derivative of thegeostrophic wind is the ‘thermal wind’. Eq. (2.202b) may be written in terms of thezonal angular momentum as

∂mg

∂z= − a

2Ω tanϑ∂b∂y, (2.203)

where mg = (ug + Ωa cosϑ)a cosϑ. Potentially more accurate than geostrophicbalance is the so-called cyclostrophic or gradient-wind balance, which retains a cen-trifugal term in the momentum equation. Thus, we omit only the material deriva-tive in the meridional momentum equation (2.50b) and obtain

2uΩ sinϑ + u2

atanϑ ≈ −∂φ

∂y= −1

a∂φ∂ϑ. (2.204)


Fig. 2.6 The mechanism of thermal wind. A cold fluid is denser than a warmfluid, so by hydrostasy the vertical pressure gradient is greater where thefluid is cold. Thus, the pressure gradients form as shown, where ‘higher’ and‘lower’ mean relative to the average at that height. The horizontal pressuregradients are balanced by the Coriolis force, producing (for f > 0) the hori-zontal winds shown (⊗ into the paper, and out of the paper). Only the windshear is given by the thermal wind.

For large-scale flow this only differs significantly from geostrophic balance veryclose to the equator. Combining cyclostrophic and hydrostatic balance gives a mod-ified thermal wind relation, and this takes a simple form when expressed in termsof angular momentum, namely

∂m2

∂z≈ −a

3 cos3 ϑsinϑ

∂b∂y. (2.205)

If the density or buoyancy is constant then there is no shear and (2.202) or(2.205) give the Taylor-Proudman result. But suppose that the temperature falls inthe polewards direction. Then thermal wind balance implies that the (eastwards)wind will increase with height — just as is observed in the atmosphere! In generala vertical shear of the horizontal wind is associated with a horizontal temperaturegradient, and this is one of the most simple and far-reaching effects in geophysicalfluid dynamics. The basic physical effect is illustrated in Fig. 2.6.

Pressure coordinates

In pressure coordinates geostrophic balance is just

f × ug = −∇pΦ, (2.206)

where Φ is the geopotential and ∇p is the gradient operator taken at constant pres-sure. If f if constant, it follows from (2.206) that the geostrophic wind is non-divergent on pressure surfaces. Taking the vertical derivative of (2.206) (that is,its derivative with respect to p) and using the hydrostatic equation, ∂Φ/∂p = −α,gives the thermal wind equation

f × ∂ug∂p

= ∇pα =Rp∇pT , (2.207)


where the last equality follows using the ideal gas equation and because the hori-zontal derivative is at constant pressure. In component form this is

− f ∂vg∂p

= Rp∂T∂x, f

∂ug∂p

= Rp∂T∂y. (2.208)

In log-pressure coordinates, with Z = −H ln(p/pR), thermal wind is

f × ∂ug∂Z

= −RH∇ZT . (2.209)

The physical meaning in all these cases is the same: a horizontal temperature gra-dient, or a temperature gradient along an isobaric surface, is accompanied by avertical shear of the horizontal wind.

2.8.5 * Effects of rotation on hydrostatic balance

Because rotation inhibits vertical motion, we might expect it to affect the require-ments for hydrostasy. The simplest setting in which to see this is the rotating Boussi-nesq equations, (2.164). Let us nondimensionalize these by writing

(x,y) = L(x, y), z = Hz, u = Uu, t = T t = ULt, f = f0f ,

w = βHUf0

w = βHULw, φ = Φφ = f0ULφ, b = Bb = f0uL

Hb,

(2.210)where β ≡ βL/f0. (If f is constant, then f is a unit vector in the vertical direction.)These relations are the same as (2.170), except for the scaling for w, which is sug-gested by (2.194), and the scaling for φ and b′, which are suggested by geostrophicand thermal wind balance.

Substituting into (2.164) we obtain the following nondimensional momentumequations:

RoDuDt

+ f × u = −∇φ, Ro βα2 DwDt

= −∂φ∂z

− b . (2.211a,b)

Here, D/Dt = ∂/∂t + u · ∇z + β∂/∂z and Ro = U/(f0L). There are two noteable as-pects to these equations. First and most obviously, when Ro 1, (2.211a) reducesto geostrophic balance, f × u = −∇φ. Second, the material derivative in (2.211b)is multiplied by three nondimensional parameters, and we can understand the ap-pearance of each as follows.

(i) The aspect ratio dependence (α2) arises in the same way as for non-rotatingflows — that is, because of the presence of w and z in the vertical momentumequation as opposed to (u,v) and (x,y) in the horizontal equations.

(ii) The Rossby number dependence (Ro) arises because in rotating flow the pres-sure gradient is balanced by the Coriolis force, which is Rossby number largerthan the advective terms.

(iii) The factor β arises because in rotating flow w is smaller than u by the β timesthe aspect ratio.


The factor Ro βα2 is very small for large-scale flow; the reader is invited to calculaterepresentative values. Evidently, a rapidly rotating fluid is more likely to be inhydrostatic balance than a non-rotating fluid, other conditions being equal. Thecombined effects of rotation and stratification are, not surprisingly, quite subtle andwe leave that topic for chapter 5.

2.9 STATIC INSTABILITY AND THE PARCEL METHOD

In this and the next couple of sections we consider how a fluid might oscillate if itwere perturbed away from a resting state. Our focus is on vertical displacements,and the restoring force is gravity, and we will neglect the effects of rotation, andindeed initially we will neglect horizontal motion entirely. Given that, the simplestand most direct way to approach the problem is to consider from first principlesthe pressure and gravitational forces on a dispaced parcel. To this end, consider afluid at rest in a constant gravitational field, and therefore in hydrostatic balance.Suppose that a small parcel of the fluid is adiabatically displaced upwards by thesmall distance δz, without altering the overall pressure field — that is, the fluidparcel instantly assumes the pressure of its environment. If after the displacementthe parcel is lighter than its environment, it will accelerate upwards, because theupward pressure gradient force is now greater downwards gravity force on theparcel — that is, the parcel is buoyant (a manifestation of Archimedes’ principle)and the fluid is statically unstable. If on the other hand the fluid parcel finds itselfheavier than its surroundings, the downward gravitational force will be greaterthan the upward pressure force and the fluid will sink back towards its originalposition and an oscillatory motion will develop. Such an equilibrium is staticallystable. Using such simple ‘parcel’ arguments we will now develop criteria for thestability of the environmental profile.

2.9.1 A simple special case: a density-conserving fluid

Consider first the simple case of an incompressible fluid in which the density ofthe displaced parcel is conserved, that is Dρ/Dt = 0 (and refer to Fig. 2.7 settingρθ = ρ). If the environmental profile is ρ(z) and the density of the parcel is ρthen a parcel displaced from a level z [where its density is ρ(z)] to a level z + δz[where the parcel’s density is still ρ(z)] will find that its density then differs fromits surroundings by the amount

δρ = ρ(z + δz)− ρ(z + δz) = ρ(z)− ρ(z + δz) = −∂ρ∂zδz. (2.212)

The parcel will be heavier than its surroundings, and therefore the parcel displace-ment will be stable, if ∂ρ/∂z < 0. Similarly, it will be unstable if ∂ρ/∂z > 0. Theupward force (per unit volume) on the displaced parcel is given by

F = −gδρ = g∂ρ∂zδz, (2.213)

and thus Newton’s second law implies that the motion of the parcel is determinedby

ρ(z)∂2δz∂t2

= g∂ρ∂zδz, (2.214)

2.9 Static Instability and the Parcel Method 93

Figure 2.7 A parcel is adiabaticallydisplaced upward from level z to z +δz. If the resulting density differ-ence, δρ, between the parcel and itsnew surroundings is positive the dis-placement is stable, and conversely.If ρ is the environmental values, andρθ is potential density, we see thatδρ = ρθ(z)− ρθ(z + δz)

or∂2δz∂t2

= gρ∂ρ∂zδz = −N2δz, (2.215)

where

N2 = −gρ∂ρ∂z

(2.216)

is the buoyancy frequency, or the Brunt-Väisälä frequency, for this problem. If N2 > 0then a parcel displaced upward is heavier than its surroundings, and thus experi-ences a restoring force; the density profile is said to be stable and N is the frequencyat which the fluid parcel oscillates. If N2 < 0, the density profile is unstable andthe parcel continues to ascend and convection ensues. In liquids it is often a goodapproximation to replace ρ by ρ0 in the demoninator of (2.216).

2.9.2 The general case: using potential density

More generally, in an adiabatic displacement it is potential density, ρθ, and notdensity itself that is materially conserved. Consider a parcel that is displaced adi-abatically a vertical distance from z to z + δz; the parcel preserves its potentialdensity, and let us use the pressure at level z + δz as the reference level. The insitu density of the parcel at z + δz, namely ρ(z + δz), is then equal to its potentialdensity ρθ(z + δz) and, because ρθ is conserved, this is equal to the potential den-sity of the environment at z, ρθ(z). The difference in in situ density between theparcel and the environment at z + δz, δρ, is thus equal to the difference betweenthe potential density of the environment at z and at z + δz. Putting this together(and see Fig. 2.7) we have

δρ = ρ(z + δz)− ρ(z + δz) = ρθ(z + δz)− ρθ(z + δz)= ρθ(z)− ρθ(z + δz) = ρθ(z)− ρθ(z + δz),

(2.217)

and therefore

δρ = −∂ρθ∂zδz, (2.218)


where the derivative on the right-hand side is the environmental gradient of poten-tial density. If the right-hand side is positive, the parcel is heavier than its surround-ings and the displacement is stable. Thus, the conditions for stability are:

Stability :∂ρθ∂z

< 0

Instability :∂ρθ∂z

> 0. (2.219a,b)

The equation of motion of the fluid parcel is

∂2δz∂t2

= gρ

(∂ρθ∂z

)δz = −N2δz, (2.220)

where, noting that ρ(z) = ρθ(z) to within O(δz),

N2 = − gρθ

(∂ρθ∂z

). (2.221)

This is a general expression for the buoyancy frequency, true in both liquids andgases. It is important to realize that the quantity ρθ is the locally-referenced potentialdensity of the environment, as will become more clear below.

An ideal gas

In the atmosphere potential density is related to potential temperature by ρθ =pR/(θR). Using this in (2.221) gives

N2 = gθ

(∂θ∂z

), (2.222)

where θ refers to the environmental profile of potential temperature. The referencevalue pR does not appear, and we are free to choose this value arbitrarily — thesurface pressure is a common choice. The conditions for stability, (2.219), thencorrespond to N2 > 0 for stability and N2 < 0 for instability. In the troposphere (thelowest several kilometers of the atmosphere) the average N is about 0.01 s−1, witha corresponding period, (2π/N), of about 10 minutes. In the stratosphere (whichlies above the troposphere) N2 is a few times higher than this.

A liquid ocean

No simple, accurate, analytic expression is available for computing static stabilityin the ocean. If the ocean had no salt, then the potential density referenced tothe surface would generally be a measure of the sign of stability of a fluid column,if not of the buoyancy frequency. However, in the presence of salinity, the surface-referenced potential density is not necessarily even a measure of the sign of stability,because the coefficients of compressibility βT and βS vary in different ways withpressure. To see this, suppose two neighbouring fluid elements at the surface havethe same potential density, but different salinities and temperatures. Displace them


both adiabatically to the deep ocean. Although their potential densities (referencedto the surface) are still equal, we can say little about their actual densities, andhence their stability relative to each other, without doing a detailed calculationbecause they will each have been compressed by different amounts. It is the profileof the locally-referenced potential density that determines the stability.

A sometimes-useful expression for stability arises by noting that in an adiabaticdisplacement

δρθ = δρ −1c2sδp = 0. (2.223)

If the fluid is hydrostatic δp = −ρgδz so that if a parcel is displaced adiabaticallyits density changes according to

(∂ρ∂z

)ρθ= −ρg

c2s. (2.224)

If a parcel is displaced a distance δz upwards then the density difference betweenit and its new surroundings is

δρ = −[(∂ρ∂z

)ρθ−(∂ρ∂z

)]δz =

[ρgc2s+(∂ρ∂z

)]δz, (2.225)

where the tilde again denotes the environmental field. It follows that the stratifica-tion is given by

N2 = −g[gc2s+ 1ρ

(∂ρ∂z

)]. (2.226)

This expression holds for both liquids and gases, and for ideal gases it is preciselythe same as (2.222) (problem 2.8). In liquids, a good approximation is to use areference value ρ0 for the undifferentiated density in the denominator, whence itbecomes equal to the Boussinesq expression (2.107). Typical values of N in theupper ocean where the density is changing most rapidly (i.e., in the pycnocline —‘pycno’ for density, ‘cline’ for changing) are about 0.01 s−1, falling to 0.001 s−1 inthe more homogeneous abyssal ocean. These correspond to periods of about 10and 100 minutes, respectively.

* Cabbeling

Cabbeling is an instability that arises because of the nonlinear equation of state ofseawater. From Fig. 1.3 we see that the contours are slightly convex, bowing up-ward, especially in the plot at sea level. Suppose we mix two parcels of water, eachwith the same density (σθ = 28, say), but with different initial values of tempera-ture and salinity. Then the resulting parcel of water will have a temperature and asalinity equal to the average of the two parcels, but its density will be higher thaneither of the two original parcels. In the appropriate circumstances such mixingmay thus lead to a convective instability; this may, for example, be an importantsource of ‘bottom water’ formation in the Weddell Sea, off Antarctica.10


2.9.3 Lapse rates in dry and moist atmospheres

A dry ideal gas

The negative of the rate of change of the temperature in the vertical is known asthe temperature lapse rate, or often just the lapse rate, and the lapse rate corre-sponding to ∂θ/∂z = 0 is called the dry adiabatic lapse rate and denoted Γd. Usingθ = T(p0/p)R/cp and ∂p/∂z = −ρg we find that the lapse rate and the potentialtemperature lapse rate are related by

∂T∂z

= Tθ∂θ∂z

− gcp, (2.227)

so that the dry adiabatic lapse rate is given by

Γd =gcp, (2.228)

as in fact we derived in (1.134). (We use the subscript d, for dry, to differentiateit from the moist lapse rate considered below.) The conditions for static stabilitycorresponding to (2.219) are thus:

Stability :∂θ∂z> 0; or −∂T

∂z< Γd

Instability :∂θ∂z< 0; or −∂T

∂z> Γd

, (2.229a,b)

where a tilde indicates that the values are those of the environment. The atmo-sphere is in fact generally stable by this criterion: the observed lapse rate, cor-responding to an observed buoyancy frequency of about 10−2 s−1, is often about7 K km−1, whereas a dry adiabatic lapse rate is about 10 K km−1. Why the discrep-ancy? One reason, particularly important in the tropics, is that the atmospherecontains water vapour.

* Effects of water vapour on the lapse rate of an ideal gas

The amount of water vapour that can be contained in a given volume is an in-creasing function of temperature (with the presence or otherwise of dry air in thatvolume being largely irrelevant). Thus, if a parcel of water vapour is cooled, it willeventually become saturated and water vapour will condense into liquid water. Ameasure of the amount of water vapour in a unit volume is its partial pressure,and the partial pressure of water vapour at saturation, es , is given by the Clausius-Clapeyron equation,

desdT

= LcesRvT 2 , (2.230)

where Lc is the latent heat of condensation or vapourization (per unit mass) andRv is the gas constant for water vapour. If a parcel rises adiabatically it will cool,and at some height (known as the ‘lifting condensation level’, a function of its initialtemperature and humidity only) the parcel will become saturated and any furtherascent will cause the water vapour to condense. The ensuing condensational heat-ing causes the parcel’s temperature, and buoyancy, to increase; the parcel thus rises


further, causing more water vapour to condense, and so on, and the consequence ofthis is that an environmental profile that is stable if the air is dry may be unstable ifsaturated. Let us now derive an expression for the lapse rate of a saturated parcelthat is ascending adiabatically apart from the affects of condensation.

Let w denote the mass of water vapour per unit mass of dry air, the mixingratio, and let ws be the saturation mixing ratio. (ws = αes/(p − es) ≈ αwes/pwhere αw = 0.62, the ratio of the mass of a water molecule to one of dry air.) Thediabatic heating associated with condensation is then given by

Qcond = −LcDwsDt

, (2.231)

so that the thermodynamic equation is

cpD lnθ

Dt= −Lc

TDwsDt

, (2.232)

or, in terms of p and and T

cpD lnT

Dt− RD lnP

Dt= −Lc

TDwsDt

. (2.233)

If these material derivatives are due to the parcel ascent then

d lnTdz

− Rcp

d lnpdz

= − LcTcp

dwsdz

, (2.234)

and using the hydrostatic relationship and the fact that ws is a function of T and pwe obtain

dTdz

+ gcp= −Lc

cp

[(∂ws∂T

)p

dTdz

−(∂ws∂p

)Tρg]. (2.235)

Solving for dT/dz, the lapse rate, Γs , of an ascending saturated parcel is given by

Γs = −dTdz

= gcp

1− ρLc(∂ws/∂p)T1+ (Lc/cp)(∂ws/∂T )p

≈ gcp

1+ Lcws/(RT)1+ L2

cws/(cpRT 2). (2.236)

where the last near-equality follows with use of the Clausius-Clapeyron relation.This (Γs) is variously called the pseudoadiabatic or moist adiabatic or saturated adia-batic lapse rate. Because g/cp is the dry adiabatic lapse rate Γd, Γs < Γd, and valuesof Γs are typically around 6 K km−1 in the lower atmosphere; however, dws/dT is anincreasing function of T so that Γs decreases with increasing temperature and canbe as low as 3.5 K km−1. For a saturated parcel, the stability conditions analogousto (2.229) are

Stability : −∂T∂z< Γs , (2.237a)

Instability : −∂T∂z> Γs . (2.237b)

where T is the environmental temperature. The observed environmental profilein convecting situations is often a combination of the dry adiabatic and moist adi-abatic profiles: an unsaturated parcel that is is unstable by the dry criterion will


rise and cool following a dry adiabat, Γd, until it becomes saturated at the liftingcondensation level, above which it will rise following a saturation adiabat, Γs . Suchconvection will proceed until the atmospheric column is stable and, especially inlow latitudes, the lapse rate of the atmosphere is largely determined by such con-vective processes.

* Equivalent potential temperature

Suppose that all the moisture in a parcel of air condenses, and that all the heatreleased goes into heating the parcel. The equivalent potential temperature, θeq is thepotential temperature that the parcel then achieves. We may obtain an approximateanalytic expression for it by noting that the first law of thermodynamics, dQ = Tdη,then implies, by definition of potential temperature,

− Lcdw = cpTd lnθ, (2.238)

where dw is the change in water vapour mixing ratio, so that a reduction of wvia condensation leads to heating. Integrating gives, by definition of equivalentpotential temperature,

−∫ 0

w

LcwcpT

dw =∫ θeq

θd lnθ, (2.239)

and so, if T and Lc are assumed constant,

θeq = θ exp(LcwcpT

). (2.240)

The equivalent potential temperature so defined is approximately conserved duringcondensation, the approximation arising going from (2.239) to (2.240). It is a use-ful expression for diagnostic purposes, and in constructing theories of convection,but it is not accurate enough to use as a prognostic variable in a putatively realis-tic numerical model. The ‘equivalent temperature’ may be defined in terms of theequivalent potential temperature by

Teq ≡ θeq

(ppR

)κ. (2.241)

2.10 GRAVITY WAVES

The parcel approach to oscillations and stability, while simple and direct, is divorcedfrom the fluid-dynamical equations of motion, making it hard to include other ef-fects such as rotation, or to explore the effects of possible differences between thehydrostatic and non-hydrostatic cases. To remedy this, we now use the equationsof motion to analyze the motion resulting from a small disturbance.

2.10.1 Gravity waves and convection in a Boussinesq fluid

Let us consider a Boussineq fluid, at rest, in which the buoyancy varies linearly withheight and the bouyancy frequency, N, is a constant. Linearizing the equations of

2.10 Gravity Waves 99

0 1 2 3 40

0.5

1

Scaled wavenumber (k/m)

Sca

led

Fre

quen

cy (ω

/N)

Figure 2.8 Scaled frequency,ω/N, plotted as a function ofscaled horizontal wavenumber,k/m, using the full dispersionrelation of (2.245) (solid line,asymptoting to unit value forlarge k/m) and with the hydro-static dispersion relation (2.249)(dashed line, tending to ∞ forlarge k/m).

motion about this basic state gives the linear momentum equations,

∂u′

∂t= −∂φ

′

∂x,

∂w′

∂t= −∂φ

′

∂z+ b′, (2.242a,b)

and mass continuity and thermodynamic equations,

∂u′

∂x+ ∂w

′

∂z= 0,

∂b′

∂t+w′N2 = 0, (2.243a,b)

where for simplicity we assume that the flow is a function only of x and z. A littlealgebra gives a single equation for w′,[(

∂2

∂x2 +∂2

∂z2

)∂2

∂t2+N2 ∂2

∂x2

]w′ = 0. (2.244)

Seeking solutions of the form w′ = Re W exp[i(kx+mz−ωt)] (where Re denotesthe real part) yields the dispersion relationship for gravity waves:

ω2 = k2N2

k2 +m2 . (2.245)

The frequency (see Fig. 2.8) is thus always less than N, approaching N for smallhorizontal scales, k m. If we neglect pressure perturbations, as in the parcelargument, then the two equations,

∂w′

∂t= b′, ∂b′

∂t+w′N2 = 0, (2.246)

form a closed set, and give ω2 = N2.If the basic state density increases with height then N2 < 0 and we expect this

state to be unstable. Indeed, the disturbance grows exponentially according toexp(σt) where

σ = iω = ±kN(k2 +m2)1/2

, (2.247)


where N2 = −N2. Most convective activity in the ocean and atmosphere is, ul-timately, related to an instability of this form, although of course there are manycomplicating issues — water vapour in the atmosphere, salt in the ocean, the effectsof rotation and so forth.

Hydrostatic gravity waves and convection

Let us now suppose that the fluid satisfies the hydrostatic Boussinesq equations.The linearized two-dimensional equations of motion become

∂u′

∂t= −∂φ

′

∂x, 0 = −∂φ

′

∂z+ b′, (2.248a)

∂u′

∂x+ ∂w

′

∂z= 0, (2.248b)

∂b′

∂t+w′N2 = 0, (2.248c)

being the horizontal and vertical momentum equations, mass continuity, and thethermodynamic equation respectively. Then a little algebra gives the dispersionrelation,

ω2 = k2N2

m2 . (2.249)

The frequency and, if N2 is negative the growth rate, is unbounded for as k/m →∞,and the hydrostatic approximation thus has quite unphysical behaviour for smallhorizontal scales (see also problem 2.10).11

2.11 * ACOUSTIC-GRAVITY WAVES IN AN IDEAL GAS

We now consider wave motion in a stratified, compressible fluid such as the earth’satmosphere. The complete problem is complicated and uninformative; we will spe-cialize to the case of an isothermal, stationary atmosphere and ignore the effectsof rotation and sphericity, but otherwise we will make few approximations. In thissection we will denote the unperturbed state with a subscript 0 and the perturbedstate with a prime (′); we will also omit many algebraic details. Because it is at rest,the basic state is in hydrostatic balance,

∂p0

∂z= −ρ0(z)g (2.250)

Ignoring variations in the y-direction for algebraic simplicity, the linearizedequations of motion are:

u-momentum: ρ0∂u′

∂t= −∂p

′

∂x(2.251a)

w-momentum: ρ0∂w′

∂t= −∂p

′

∂z− ρ′g (2.251b)

mass conservation:∂ρ′

∂t+w′ ∂ρ0

∂z= −ρ0

(∂u′

∂x+ ∂w

′

∂z

)(2.251c)

thermodynamic:∂θ′

∂t+w′ ∂θ0

∂z= 0 (2.251d)

2.11 * Acoustic-Gravity Waves in an Ideal Gas 101

equation of state:θ′

θ0+ ρ

′

ρ0= 1γp′

p0. (2.251e)

For an isothermal basic state we have p0 = ρ0RT0 where T0 is a constant, so thatρ0 = ρse−z/H and p0 = pse−z/H where H = RT0/g. Further, using θ = T(ps/p)κwhere κ = R/cp, we have that θ0 = T0eκz/H and so N2 = κg/H. It is also convenientto use (1.100) on page 22 to write the linear thermodynamic equation in the form

∂p′

∂t−w′p0

H= −γp0

(∂u′

∂x+ ∂w

′

∂z

). (2.251f)

Differentiating (2.251a) with respect to time and using (2.251f) leads to(∂2

∂t2− c2

s∂2

∂x2

)u′ = c2

s

(∂∂z

− 1γH

)∂∂xw′. (2.252a)

where c2s = (∂p/∂/ρ)η = γRT0 = γp0/ρ0 is the square of the speed of sound, and

γ = cp/cv = 1/(1− κ). Similarly, differentiating (2.251b) with respect to time andusing (2.251c) and (2.251f) leads to(

∂2

∂t2− c2

s

[∂2

∂z2 −1H∂∂z

])w′ = c2

s

(∂∂z

− κH

)∂u′

∂x, (2.252b)

Eqs. (2.252a) and (2.252b) combine to give, after some cancellation,

∂4w′

∂t4− c2

s∂2

∂t2

(∂2

∂x2 +∂2

∂z2 −1H∂∂z

)w′ − c2

sκgH∂2w′

∂x2 = 0. (2.253)

If we set w′ = W(x, z, t)ez/(2H), so that W = (ρ0/ρs)1/2w, then the term with thesingle z-derivative is eliminated, giving

∂4W∂t4

− c2s∂2

∂t2

(∂2

∂x2 +∂2

∂z2 −1

4H2

)W − c2

sκgH∂2w′

∂x2 = 0. (2.254)

Although superficially complicated, this equation has constant coefficients and wemay seek wavelike solutions of the form

W = Re Wei(kx+mz−ωt), (2.255)

where W is the complex wave amplitude. Using (2.255) in (2.254) leads to thedispersion relation for acoustic-gravity waves, namely

ω4 − c2sω2

(k2 +m2 + 1

4H2

)+ c2

sN2k2 = 0, (2.256)

with solution

ω2 = 12c2sK2

1±(

1− 4N2k2

c2sK4

)1/2 , (2.257)

where K2 = k2 +m2 + 1/(4H2). (The factor [1− 4N2k2/(c2sK4)] is always positive

— see problem 2.23.) For an isothermal, ideal-gas, atmosphere 4N2H2/c2s ≈ 0.8

and so this may be written

ω2

N2 ≈ 2.5K2

1±(

1− 0.8k2

K4

)1/2 , (2.258)

where K2 = k2 + m2 + 1/4, and (k, m) = (kH,mH).


−2 −1 0 1 20

1

2

3

4

5

6

Wavenumber, kH

Fre

quen

cy, ω

/N

m=1

m=1

m=0m=0

Lamb

ωa (acoustic)

ωg (gravity)

ωL (Lamb)

Fig. 2.9 Dispersion diagram for acoustic gravity waves in an isothermal at-mosphere, calculated using (2.258). The frequency is given in units of thethe buoyancy frequency N, and the wavenumbers are non-dimensionalizedby the inverse of the scale height, H. The solid curves indicate acousticwaves, whose frequency is always higher than that of the corresponding Lambwave at the same wavenumber (i.e. ck), and of the base acoustic frequency≈ 1.12N. The dashed curves indicate internal gravity waves, whose frequencyasymptotes to N at small horizontal scales.

2.11.1 Interpretation

Acoustic and gravity waves

There are two branches of roots in (2.257), corresponding to acoustic waves (usingthe plus sign in the dispersion relation) and internal gravity waves (using the minussign). These (and the Lamb wave, described below) are plotted in Fig. 2.9. If4N2k2/c2

sK4 1 then the two sets of waves are well separated. From (2.258) thisis satisfied when

4κγ(kH)2 ≈ 0.8(kH)2

[(kH)2 + (mH)2 + 1

4

]2; (2.259)

that is, when either mH 1 or kH 1. The two roots of the dispersion relationare then

ω2a ≈ c2

sK2 = c2s

(k2 +m2 + 1

4H2

)(2.260)

and

ω2g ≈

N2k2

k2 +m2 + 1/(4H2). (2.261)

2.11 * Acoustic-Gravity Waves in an Ideal Gas 103

corresponding to acoustic and gravity waves, respectively. The acoustic waves owetheir existence to the presence of compressibility in the fluid, and they have nocounterpart in the Boussinesq system. On the other hand, the internal gravity wavesare just modified forms of those found in the Boussinesq system, and if we take thelimit (kH,mH)→∞ then the gravity wave branch reduces toω2

g = N2k2/(k2+m2),which is the dispersion relationship for gravity waves in the Boussinesq approxima-tion. We may consider this to be the limit of infinite scale height or (equivalently)the case in which wavelengths of the internal waves are sufficiently small that thefluid is essentially incompressible.

Vertical structure

Recall that w′ = W(x, z, t)ez/(2H) and, by inspection of (2.252), u′ has the samevertical structure. That is,

w′ ∝ ez/(2H), u′ ∝ ez/(2H), (2.262)

and the amplitude of the velocity field of the internal waves increases with height.The pressure and density perturbation amplitudes fall off with height, varying like

p′ ∝ e−z/(2H), ρ′ ∝ e−z/(2H). (2.263)

The kinetic energy of the perturbation, ρ0(u′2+w′2) is constant with height, becauseρ0 = ρse−z/H .

Hydrostatic approximation and Lamb waves

Equations (2.252) also admit to a solution with w′ = 0. We then have(∂2

∂t2− c2

s∂2

∂x2

)u′ = 0 and

(∂∂z

− κH

)∂u′

∂x= 0, (2.264)

and these have solutions of the form

u′ = Re Ueκz/Hei(kx−ωt), ω = ck, (2.265)

where U is the wave amplitude. These are horizontally propagating sound waves,known as Lamb waves after the hydrodynamicist Horace Lamb. Their velocity per-turbation amplitude increases with height, but the pressure perturbation falls withheight; that is

u′ ∝ eκz/H ≈ e2z/(7H), p′ ∝ e(κ−1)z/H ≈ e−5z/(7H). (2.266)

Their kinetic energy density ρ0u′2 varies as

K.E.∝ e−z/H+2κz/H = e(2R−cp)z/(cpH)] = e(R−cv)z/(cpH) ≈ e−3z/(7H) (2.267)

for an ideal gas. (In a simple ideal gas, cv = nR/2 where n is the number ofexcited degrees of freedom, 5 for a diatomic molecule.) The kinetic energy densitythus falls away exponentially from the surface, and in this sense Lamb waves are anexample of edge waves or surface-trapped waves.

Consider now case in which we make the hydrostatic approximation ab initio,


Figure 2.10 An idealized boundary layer.The values of a field, such as velocity, U ,may vary rapidly in a boundary in order tosatisfy the boundary conditions at a rigidsurface. The parameter δ is a measure ofthe boundary layer thickness, andH is a typ-ical scale of variation away from the bound-ary.

but do not restrict the perturbation to have w′ = 0. The linearized equations areidentical to (2.251), except that (2.251b) is replaced by

∂p′

∂z= −ρ′g. (2.268)

The consequence of this is that first term (∂2w′/∂t2) in (2.252b) disappears, as dothe first two terms in (2.253) [the terms ∂4w′/∂t4 − c2(∂2/∂t2)(∂2w′/∂x2)]. It is asimple matter to show that the dispersion relation is then

ω2 = N2k2

m2 + 1/(4H2). (2.269)

These are long gravity waves, and may be compared with the corresponding Boussi-nesq result (2.249). Again, the frequency increases without bound as the horizontalwavelength diminishes. The Lamb wave, of course, still exists in the hydrostaticmodel, because (2.264) is still a valid solution. Thus, horizontally propagatingsound waves still exist in hydrostatic (primitive equation) models, but verticallypropagating sound waves do not — essentially because the term ∂w/∂t is absentfrom the vertical momentum equation.

2.12 THE EKMAN LAYER

In the final topic of this chapter, we return to geostrophic flow and consider theeffects of friction. The fluid fields in the interior of a domain are often set bydifferent physical processes than those occuring at a boundary, and consequentlyoften change rapidly in a thin boundary layer, as in Fig. 2.10. Such boundary layersnearly always involve one or both of viscosity and diffusion, because these appearin the terms of highest differential order in the equations of motion, and so areresponsible for the number and type of boundary conditions that the equations mustsatisfy — for example, the presence of molecular viscosity leads to the conditionthat the tangential flow (as well as the normal flow) must vanish at a rigid surface.

In many boundary layers in non-rotating flow the dominant balance in the mo-mentum equation is between the advective and viscous terms. In some contrast,

2.12 The Ekman Layer 105

in large-scale atmospheric and oceanic flow the effects of rotation are large, andthis results in a boundary layer, known as the Ekman layer, in which the dominantbalance is between Coriolis and frictional terms.12 Now, the direct effects of molec-ular viscosity and diffusion are nearly always negligible at distances more than afew millimeters away from a solid boundary, but it is inconceivable that the entireboundary layer between the free atmosphere (or ocean) and the surface is only afew millimeters thick. Rather, in practice a balance occurs between the Coriolisterms and the stress due to small-scale turbulent motion, and this gives rise to aboundary layer that has a typical depth of tens to hundreds of meters. Because thestress arises from the turbulence we cannot with confidence determine its preciseform; thus, we should try to determine what general properties Ekman layers mayhave that are independent of the precise form of the friction.

The atmospheric Ekman layer occurs near the ground, and the stress at theground itself is due to the surface wind (and its vertical variation). In the oceanthe main Ekman layer is near the surface, and the stress at ocean surface is largelydue to the presence of the overlying wind. There is also a weak Ekman layer atthe bottom of the ocean, analogous to the atmospheric Ekman layer. To analyze allthese layers, let us assume that:? The Ekman layer is Boussinesq. This is a very good assumption for the ocean,

and a reasonable one for the atmosphere if the boundary layer is not too deep.? The Ekman layer has a finite depth that is less than the total depth of the

fluid, this depth being given by the level at which the frictional stresses essen-tially vanish. Within the Ekman layer, frictional terms are important, whereasgeostrophic balance holds beyond it.

? The nonlinear and time dependent terms in the equations of motion are negli-gible, hydrostatic balance holds in the vertical, and buoyancy is constant, notvarying in the horizontal.

? As needed, we shall assume that friction can be parameterized by a viscousterm of the form ρ−1

0 ∂τ/∂z = A∂2u/∂z2, where A is constant and τ is thestress. In laboratory settings A may be the molecular viscosity, whereas inthe atmosphere and ocean it is a so-called eddy viscosity. (In turbulent flowsmomentum is transferred by the near-random motion of small parcels of fluidand, by analogy with the motion of molecules that produces a molecular vis-cosity, the associated stress is approximately represented, or parameterized,using a turbulent or eddy viscosity that may be orders of magnitude largerthan the molecular one.) In all cases it is the vertical derivative of the stressthat dominates.

2.12.1 Equations of motion and scaling

Frictional-geostrophic balance in the horizontal momentum equation is:

f × u = −∇zφ+∂ τ∂z. (2.270)


where τ ≡ τ/ρ0 is the kinematic stress (and τ is the stress itself), and f is allowedto vary with latitude. If we model the stress with an eddy viscosity (2.270) becomes

f × u = −∇zφ+A∂2u∂z2 . (2.271)

The vertical momentum equation is hydrostatic balance, ∂φ/∂z = b, and, becausebuoyancy is constant, we may without loss of generality write this as

∂φ∂z

= 0. (2.272)

The equation set is completed by the mass continuity equation ∇ · v = 0.

The Ekman number

We non-dimensionalize the equations by setting

(u,v) = U(u, v), (x,y) = L(x, y), f = f0f , z = Hz, φ = Φφ, (2.273)

where hatted variables are non-dimensional. H is a scaling for the height, and atthis stage we will suppose it to be some height scale in the free atmosphere orocean, not the height of the Ekman layer itself. Geostrophic balance suggests thatΦ = f0UL. Substituting (2.273) into (2.271) we obtain

f × u = −∇φ+ Ek∂2u∂z2 , (2.274)

where the parameter

Ek ≡(Af0H2

), (2.275)

is the Ekman number, and it determines the importance of frictional terms in thehorizontal momentum equation. If Ek 1 then the friction is small in the flowinterior where z = O(1). However, the friction term cannot necessarily be neglectedin the boundary layer because it is of the highest differential order in the equation,and so determines the boundary conditions; if Ek is small the vertical scales becomesmall and the second term on the right-hand side of (2.274) remains finite. The casewhen this term is simply omitted from the equation is therefore a singular limit,meaning that it differs from the case with Ek → 0. If Ek ≥ 1 friction is importanteverywhere, but it is usually the case that Ek is small for atmospheric and oceaniclarge-scale flow, and the interior flow is very nearly geostrophic. (In part this isbecause A itself is only large near a rigid surface where the presence of a shearcreates turbulence and a significant eddy viscosity.)

Momentum balance in the Ekman layer

For definiteness, suppose the fluid lies above a rigid surface at z = 0. Sufficientlyfar away from the boundary the velocity field is known, and we suppose this flowto be in geostrophic balance. We then write the velocity field and the pressure fieldas the sum of the interior geostrophic part, plus a boundary layer correction:

u = ug + uE , φ = φg + φE , (2.276)


where the Ekman layer corrections, denoted with a subscript E, are negligible awayfrom the boundary layer. Now, in the fluid interior we have, by hydrostatic balance,∂φg/∂z = 0. In the boundary layer we have still ∂φg/∂z = 0 so that, to satisfyhydrostasy, ∂φE/∂z = 0. But because φE vanishes away from the boundary wehave φE = 0 everywhere. This is an important result: there is no boundary layer inthe pressure field. Note that this is a much stronger result than saying that pressureis continuous, which is nearly always true in fluids; rather, it is a special result aboutEkman layers.

Using (2.276) with φE = 0, the dimensional horizontal momentum equation(2.270) becomes, in the Ekman layer,

f × uE =∂ τ∂z. (2.277)

The dominant force balance in the Ekman layer is thus between the Coriolis forceand the friction. We can determine the thickness of the Ekman layer if we modelthe stress with an eddy viscosity so that

f × uE = A∂2uE∂z2 , (2.278)

or, non-dimensionally,

f × uE = Ek∂2uE∂z2 . (2.279)

It is evident this equation can only be satisfied if z ≠ O(1), implying that H is not aproper scaling for z in the boundary layer. Rather, if the vertical scale in the Ekmanlayer is δ (meaning z ∼ δ) we must have δ ∼ Ek1/2. In dimensional terms thismeans the thickness of the Ekman layer is

δ = Hδ = HEk1/2 (2.280)

or

δ =(Af0

)1/2

. (2.281)

[This estimate also emerges directly from (2.278).] Note that (2.280) can be writ-ten as

Ek =(δH

)2. (2.282)

That is, the Ekman number is equal to the square of the ratio of the depth of theEkman layer to an interior depth scale of the fluid motion. In laboratory flowswhere A is the molecular viscosity we can thus estimate the Ekman layer thickness,and if we know the eddy viscosity of the ocean or atmosphere we can estimate thethickness of their respective Ekman layers. We can invert this argument and obtainan estimate of A if we know the Ekman layer depth. In the atmosphere, deviationsfrom geostrophic balance are very small in the atmosphere above 1 km, and usingthis gives A ≈ 102 m2 s−1. In the ocean Ekman depths are about 50 m or less, andeddy viscosities about 0.1 m2 s−1.


2.12.2 Integral properties of the Ekman layer

What can we deduce about the Ekman layer without specifying the detailed formof the frictional term? Using dimensional notation we recall frictional-geostrophicbalance,

f × u = −∇φ+ ∂ τ∂z, (2.283)

where τ is zero at the edge of the Ekman layer. In the Ekman layer itself we have

f × uE =∂ τ∂z. (2.284)

Consider either a top or bottom Ekman layer, and integrate over its thickness. From(2.284) we obtain

f ×ME = τt − τb, (2.285)

where

ME =∫

EkuE dz (2.286)

is the ageostrophic transport in the Ekman layer, and where τt and τb is the stressat the top and the bottom of the respective layer. The former (latter) will be zero ina bottom (top) Ekman layer. More explicitly, (2.285) may be written as:

Top Ekman Layer: ME = −1f

k× τt

Bottom Ekman Layer: ME =1f

k× τb. (2.287a,b)

The transport in the Ekman layer is thus at right-angles to the stress at the surface.This has a simple physical explanation: integrated over the depth of the Ekmanlayer the surface stress must be balanced by the Coriolis force, which in turn actsat right angles to the mass transport. This result is particularly useful in the ocean,where the stress at the upper surface is primarily due to the wind, and can beregarded as independent of the interior flow. If f is positive, as in the Northernhemisphere, then an Ekman transport is induced 90° to the right of the direction ofthe wind stress. This has innumerable important consequences — for example, ininducing coastal upwelling when, as is not uncommon, the wind blows parallel tothe coast. Upwelling off the coast of California is one example. In the atmosphere,however, the stress arises as a consequence of the interior flow, and we need toparameterize the stress in terms of the flow in order to calculate the surface stress.

Finally, we obtain an expression for the vertical velocity induced by an Ekmanlayer. The mass conservation equation is

∂u∂x

+ ∂v∂y

+ ∂w∂z

= 0. (2.288)

Integrating this over an Ekman layer gives

∇ ·Mt = −(wt −wb), (2.289)


where Mt is the total (Ekman plus geostrophic) transport in the Ekman layer,

Mt =∫

Ekudz =

∫Ek(ug + uE)dz ≡Mg +ME , (2.290)

and wt and wb are the vertical velocities at the top and bottom of the Ekman layer;the former (latter) is zero in a top (bottom) Ekman layer. From (2.285)

k× (Mt −Mg) =1f(τt − τb). (2.291)

Taking the curl of this (i.e., cross-differentiating) gives

∇ · (Mt −Mg) = curlz[(τt − τb)/f ], (2.292)

where the curlz operator on a vector A is defined by curlzA ≡ ∂xAy − ∂yAx. Using(2.289) we obtain, for bottom and top Ekman layers respectively,

wb = curlzτtf+∇ ·Mg, wt = curlz

τbf−∇ ·Mg , (2.293a,b)

where∇·Mg = −βMg/f is the divergence of the geostrophic transport in the Ekmanlayer, which is often small compared to the other terms in these equations. Thus,friction induces a vertical velocity at the edge of the Ekman layer, proportional to thecurl of the stress at the surface, and this is perhaps the most used result in Ekmanlayer theory. Numerical models sometimes do not have the vertical resolution toexplicitly resolve an Ekman layer, and (2.293) provides a means of parameterizingthe Ekman layer in terms of resolved or known fields. It is particularly useful for thetop Ekman layer in the ocean, where the stress can be regarded as a given functionof the overlying wind.

2.12.3 Explicit solutions. I: A bottom boundary layer

We now assume that the frictional terms can be parameterized as an eddy viscosityand calculate the explicit form of the solution in the boundary layer. The frictional-geostrophic balance may be written as

f × (u− ug) = A∂2u∂z2 , (2.294a)

where

f(ug, vg) =(−∂φ∂y,∂φ∂x

). (2.294b)

We continue to assume there are no horizontal gradients of temperature, so that,via thermal wind, ∂ug/∂z = ∂vg/∂z = 0.

Boundary conditions and solution

Appropriate boundary conditions for a bottom Ekman layer are:

At z = 0 : u = 0, v = 0 (the no slip condition) (2.295a)


Figure 2.11 The idealisedEkman layer solution inthe lower atmosphere,plotted as a hodograph ofthe wind components: thearrows show the velocityvectors at a particularheights, and the curvetraces out the continuousvariation of the velocity.The values on the curveare of the nondimen-sional variable z/d, whered = (2A/f)1/2, and vg ischosen to be zero.

As z →∞ : u = ug, v = vg (a geostrophic interior). (2.295b)

Let us seek solutions to (2.294a) of the form

u = ug +A0eαz, v = vg + B0eαz, (2.296)

where A0 and B0 are constants. Substituting into (2.294a) gives two homogeneousalgebraic equations

A0f − B0Aα2 = 0, −A0Aα2 − B0f = 0. (2.297a,b)

For non-trivial solutions the solvability condition α4 = −f 2/A2 must hold, fromwhich we find α = ±(1± i)

√f/2A. Using the boundary conditions we then obtain

the solution

u = ug − e−z/d[ug cos(z/d)+ vg sin(z/d)

](2.298a)

v = vg + e−z/d[ug sin(z/d)− vg cos(z/d)

], (2.298b)

where d =√

2A/f is, within a constant factor, the depth of the Ekman layer ob-tained from scaling considerations. The solution decays exponentially from thesurface with this e-folding scale, so that d is a good measure of the Ekman layerthickness. Note that the boundary layer correction depends on the interior flow,since the boundary layer serves to bring the flow to zero at the surface.

To illustrate the solution, suppose that the pressure force is directed in the y-direction (northward), so that the geostrophic current is eastward. Then the solu-tion, the now-famous Ekman spiral, is plotted in Fig. 2.11 and Fig. 2.12). The windfalls to zero at the surface, and its direction just above the surface is northeast-ward; that is, it is rotated by 45° to the left of its direction in the free atmosphere.Although this result is independent of the value of the frictional coefficients, it isdependent on the form of the friction chosen. The force balance in the Ekman layeris between the Coriolis force, the stress, and the pressure force. At the surface theCoriolis force is zero, and the balance is entirely between the northward pressureforce and the southwards stress force.


Figure 2.12 Solutions fora bottom Ekman layer witha given flow in the fluid in-terior (left), and for a topEkman layer with a givensurface stress (right), bothwith d = 1. On the leftwe have ug = 1, vg =0. On the right we haveug = vg = 0, τy = 0 and√

2τx/(fd) = 1.

−1 0 1 20

1

2

3

4

heig

ht

velocity

uv

−1 0 1−4

−3

−2

−1

0

dept

h

velocity

uv

Mass transport, force balance and vertical velocity

The cross-isobaric flow is given by (for vg = 0)

V =∫∞

0v dz =

∫∞0uge−z/d sin(z/d)dz = ugd

2. (2.299)

For positive f , this is to the left of the geostrophic flow — that is, down the pressuregradient. In the general case (vg ≠ 0) we obtain

V =∫∞

0(v − vg)dz = d

2(ug − vg). (2.300)

Similarly, the additional zonal transport produced by frictional effects are, for vg =0,

U =∫∞

0(u−ug)dz = −

∫∞0

e−z/d sin(z/d)dz = −ugd2, (2.301)

and in the general case

U =∫∞

0(u−ug)dz = −d

2(ug + vg). (2.302)

Thus, the total transport caused by frictional forces is

ME =d2

[−i(ug + vg)+ j(ug − vg)

]. (2.303)

The total stress at the bottom surface z = 0 induced by frictional forces is

τb = A∂u∂z

∣∣∣∣z=0

= Ad

[i(ug − vg)+ j(ug + vg)

], (2.304)

using the solution (2.298). Thus, using (2.303), (2.304) and d2 = 2A/f , we seethat the total frictionally induced transport in the Ekman layer is related to the


Figure 2.13 A bottom Ekman layer,generated from an eastwardsgeostrophic flow above it. An over-bar denotes a vertical integral overthe Ekman layer, so that −f ×uE isthe Coriolis force on the verticallyintegrated Ekman velocity. ME isthe frictionally induced boundarylayer transport, and τ is the stress.

stress at the surface by ME = (k × τb)/f , reprising the result of the more generalanalysis, (??). From (2.304), the stress is at an angle of 45° to the left of the velocityat the surface. (However, this result is not generally true for all forms of stress.)These properties are illustrated in Fig. 2.13.

The vertical velocity at the top of the Ekman layer, wE, is obtained using (2.303)and (2.304). If f if constant we obtain

wE = −∇ ·ME =1f0

curlzτb = Vx −Uy =d2ζg, (2.305)

where ζg is the vorticity of the geostrophic flow. Thus, the vertical velocity at thetop of the Ekman layer, which arises because of the frictionally-induced divergenceof the cross-isobaric flow in the Ekman layer, is proportional to the geostrophicvorticity in the free fluid and is proportional to the Ekman layer height

√2A/f0.

Another bottom boundary condition

In the analysis above we assumed a no slip condition at the surface, namely thatthe velocity tangential to the surface vanishes. This is certainly appropriate if Ais a molecular velocity, but in a turbulent flow, where A is interpreted as an eddyviscosity, the flow very close to the surface may be far from zero. Then, unless wewish to explicitly calculate the flow in an additional very thin viscous boundary layerthe no-slip condition may be inappropriate. An alternative, slightly more generalboundary condition is to suppose that the stress at the surface is given by

τ = ρ0Cu, (2.306)

where C is a constant. The surface boundary condition is then

A∂u∂z

= Cu. (2.307)

If C is infinite we recover the no-slip condition. If C = 0, we have instead a conditionof no stress at the surface, also known as a free slip condition. For intermediatevalues of C the boundary condition is known as a ‘mixed condition’. Evaluating thesolution in these cases is left as an exercise for the reader (problem 2.25).

2.12.4 Explicit solutions. II: The upper ocean


Figure 2.14 An idealized Ekmanspiral in a Southern Hemisphereocean, driven by an imposedwind-stress. A Northern Hemi-sphere spiral would be the re-flection of this about the verticalaxis. Such a clean spiral is rarelyobserved in the real ocean. Thenet transport is at right angles tothe wind, independent of the de-tailed form of the friction. Theangle of the surface flow is 45°to the wind only for a Newtonianviscosity.

Boundary conditions and solution

The wind provides a stress on the upper ocean, and the Ekman layer serves tocommunicate this to the oceanic interior. Appropriate boundary conditions are thus:

At z = 0 : A∂u∂z

= τx , A∂v∂z

= τy (a given surface stress) (2.308a)

As z → −∞ : u = ug, v = vg (a geostrophic interior) (2.308b)

where τ is the given (kinematic) wind stress at the surface. Solutions to (2.294a)with (2.308) are found by the same methods as before, and are

u = ug +√

2fd

ez/d[τx cos(z/d−π/4)− τy sin(z/d−π/4)

], (2.309)

and

v = vg +√

2fd

ez/d[τx sin(z/d−π/4)+ τy cos(z/d−π/4)

]. (2.310)

Note that the boundary layer correction depends only on the imposed surfacestress, and not the interior flow itself. This is a consequence of the type of bound-ary conditions chosen, for in the absence of an imposed stress the boundary layercorrection is zero — the interior flow already satisfies the gradient boundary condi-tion at the top surface. Similar to the bottom boundary layer the velocity vectors ofthe solution trace a diminishing spiral as they descend into the interior (Fig. 2.14,which is drawn for the Southern Hemisphere).

Mass flux, surface flow and vertical velocity

The mass flux induced by the surface stress is obtained by integrating (2.309) and(2.310) from the surface to −∞. We explicitly find

U =∫ 0

−∞(u−ug)dz = τ

y

f, V =

∫ 0

−∞(v − vg)dz = − τ

x

f, (2.311)


Fig. 2.15 Upper and lower Ekman layers. The upper Ekman layer in the oceanis primarily driven by an imposed wind stress, whereas the lower Ekman layerin the atmosphere or ocean largely results from the interaction of interiorgeostrophic velocityand a rigid lower surface. The upper part of figure showsthe vertical Ekman ‘pumping’ velocities that result from the given wind stress,and the lower part of the figure shows the Ekman pumping velocities giventhe interior geostrophic flow.

which indicates that the ageostrophic mass transport is perpendicular to the wind-stress, as noted previously from more general considerations.

Suppose that the surface wind is eastward. Then τy = 0 and the solutionsimmediately give

u(0)−ug = (τx/fd) cos(π/4), v(0)− vg = (τx/fd) sin(π/4). (2.312)

Therefore the magnitudes of the frictional flow in the x and y directions are equalto each other, and the flow is 45° to the right (for f > 0) of the wind. This resultis dependent on the form of the frictional parameterization chosen, but not on thesize of the viscosity.

At the edge of the Ekman layer the vertical velocity is given by (2.293), and so isproportional to the curl of the wind-stress. (The second term on the right-hand sideof (2.293) is the vertical velocity due to the divergence of the geostrophic flow, andis usually much smaller than the first term.) The production of a vertical velocity atthe edge of the Ekman layer is one of most important effects of the layer, especiallywith regard to the large-scale circulation, for it provides an efficient means wherebysurface fluxes are communicated to the interior flow (see Fig. 2.15).

2.12.5 Observations

Ekman layers are generally quite hard to observe, in either ocean or atmosphere,largely because of a signal-to-noise problem — the noise largely coming from iner-tial and gravity waves (section 2.10) and, especially in the atmosphere, the effects


of stratification and buoyancy-driven turbulence. As regards oceanography, fromabout 1980 onwards improved instruments have made it possible to observe thevector current with depth, and to average that current and correlate it with theoverlying wind, and a number of observations consistent with Ekman dynamicshave emerged.13 The main differences between observations and theory can beascribed to the effects of stratification (which causes a shallowing and flatteningof the spiral), and the interaction of the Ekman spiral with turbulence (and theinadequacy of the eddy-diffusivity parameterization). In spite of these differencesof detail, Ekman layer theory remains a remarkable and enduring foundation ofgeophysical fluid dynamics.

2.12.6 * Frictional parameterization

[Some readers will be reading these sections on Ekman layers after having beenintroduced to quasi-geostrophic theory; this section is for them. Other readers mayreturn to this section after reading chapter 5, or take (2.313) on faith.]

Suppose that the free atmosphere is described by the quasi-geostrophic vorticityequation,

DζgDt

= f0∂w∂z, (2.313)

where ζg is the geostrophic relative vorticity. Let us further model the atmosphereas a single homogeneous layer of thickness H lying above an Ekman layer of thick-ness d H. If the vertical velocity is negligible at the top of the layer (at z = H+d)the equation of motion becomes

DζgDt

= f0[w(H + d)−w(d)]H

= −f0d2H

ζg (2.314)

using (2.305). This equation shows that the Ekman layer acts as a linear drag onthe interior flow, with a drag coefficient r equal to f0d/2H and with associatedtimescale TEk given by

TEk =2Hf0d

= 2H√2f0A

. (2.315)

In the oceanic case the corresponding vorticity equation for the interior flow is

DζgDt

= 1H

curlzτs , (2.316)

where τs is the surface stress. The surface stress thus acts as if it were a body forceon the interior flow, and neither the Coriolis parameter nor the depth of the Ekmanlayer explicitly appear in this formula.

The Ekman layer is actually a very efficient way of communicating surfacestresses to the interior. To see this, suppose that eddy mixing were the sole mecha-nism of transferring stress from the surface to the fluid interior, and there were noEkman layer. Then the timescale of spindown of the fluid would be given by using

dζdt

= A∂2ζ∂z2 , (2.317)


implying a turbulent spindown time, Tturb of

Tturb ∼H2

A, (2.318)

where H is the depth over which we require a spin-down. This is much longer thanthe spin-down of a fluid that has an Ekman layer, for we have

Tturb

TEk= (H2/A)(2H/f0d)

= Hd 1, (2.319)

using d =√

2A/f0. The effects of friction are evidently enhanced because of thepresence of a secondary circulation confined to the Ekman layers (as in Fig. 2.15)in which the vertical scales are much smaller than those in the fluid interior andso where viscous effects become significant, and these frictional stresses are thencommunicated to the fluid interior via the induced vertical velocities at the edge ofthe Ekman layers.

Notes

1 The distinction between Coriolis force and acceleration has not always been made inthe literature. For a fluid in geostrophic balance, one might either say that there is abalance between the pressure force and the Coriolis force, with no net acceleration,or that the pressure force produces a Coriolis acceleration. The descriptions areequivalent, because of Newton’s second law, but should not be conflated.

The Coriolis forces is named after Gaspard Gustave de Coriolis (1792-1843), whointroduced the force in the context of rotating mechanical systems (Coriolis 1832,1835). See Persson (1998) for a historical account and interpretation.

2 Phillips (1973). See also Stommel and Moore (1989) and Gill (1982). (There are typo-graphic errors in the second term of each of Gill’s equations (4.12.11) and (4.12.12).)

3 Phillips (1966). See White (2003) for a review. In the early days of numerical mod-elling these equations were the most primitive — i.e., the least filtered — equationsthat could practically be integrated numerically. Associated with increasing com-puter power there is a tendency for comprehensive numerical models to use non-hydrostatic equations of motion that do not make the shallow-fluid or traditionalapproximations, and it is conceivable that the meaning of the word ‘primitive’ mayevolve to accomodate them.

4 The Boussinesq approximation is named for Boussinesq (1903), although similarapproximations were used earlier by Oberbeck (1879, 1888). Spiegel and Vero-nis (1960) give a physically based derivation for an ideal gas, and Mihaljan (1962)provides an asymptotic derivation of the equations. Mahrt (1986) discusses its ap-plicability in the atmosphere.

5 I thank W. R. Young for discussions on this point.

6 Various versions of anelastic equations exist — see Batchelor (1953a), Ogura andPhillips (1962), Gough (1969), Gilman and Glatzmaier (1981), Lipps and Hemler(1982) and Durran (1989) although not all have potential vorticity and energy con-servation laws (Bannon 1995, 1996; Scinocca and Shepherd 1992). The system wederive is most similar to that of Ogura and Phillips (1962) and unpublished notes by


J. S. A. Green. The connection between the Boussinesq and anelastic equations isdiscussed by, among others, Lilly (1996) and Ingersoll (2005).

7 A numerical model that includes sound waves must take very small timesteps inorder to maintain numerical stability, in particular to satisfy the CFL criterion. An al-ternative is to use an implicit time-stepping scheme that effectively lets the numericsdo the filtering of the sound waves, and this approach is favoured by many numericalmodellers. If we make the hydrostatic approximation then all sound waves exceptthose that propagate horizontally are eliminated, and there is little need, as regardsthe numerics, to also make the anelastic approximation.

8 It is named for C.-G. Rossby (see endnote on page 243) but was also used by Kibel(1940) and is sometimes called the Kibel or Rossby-Kibel number. The notion ofgeostrophic balance and so, implicitly, that of a small Rossby number, predateseither Rossby or Kibel.

9 After Taylor (1921b) and Proudman (1916). The Taylor-Proudman effect is some-times called the Taylor-Proudman ‘theorem’, but it is more usefully thought of as aphysical effect, with manifestations even when the conditions for its satisfaction arenot precisely met.

10 Foster (1972).

11 Many numerical models of the large-scale circulation in the atmosphere and oceando make the hydrostatic approximation. In these models convection must be param-eterized; otherwise, it would simply occur at the smallest scale available, namely thesize of the numerical grid, and this type of unphysical behaviour should be avoided.Of course in non-hydrostatic models convection must also be parameterized if thehorizontal resolution of the model is too coarse to properly resolve the convectivescales. See also problem 2.10.

12 After Ekman (1905). The problem was posed to Ekman, a student of Vilhelm Bjerk-nes, by Fridtjof Nansen, the polar explorer and statesman, who wanted to under-stand the motion of icebergs.

13 Davis et al. (1981), Price et al. (1987), Rudnick and Weller (1993).

Further Reading

Cushman-Roisin, B., 1994. An Introduction to Geophysical Fluid Dynamics.This book provides a compact introduction to a variety of topics in GFD.

Gill, A. E., 1982. Atmosphere-Ocean Dynamics.A rich book, especially strong on equatorial dynamics and gravity wave motion.

Holton, J. R., 1992. An Introduction to Dynamical Meteorology.A deservedly well-known textbook at the upper-division undergraduate/beginninggraduate level.

Pedlosky, J., 1987. Geophysical Fluid Dynamics.A primary reference, especially for flow at low Rossby number. Although the bookrequires some effort, there is a handsome pay-off for those who study it closely.

White (2002) provides a clear and thorough summary of the equations of motion formeteorology, including the non-hydrostatic and primitive equations.

Zdunkowski, W. and Bott, A., 2003. Dynamics of the Atmosphere: A Course in TheoreticalMeteorology.Concentrates on the equations of motion and the mathematical tools needed for afundamental understanding.


Problems

2.1 For an ideal gas in hydrostatic balance, show that:

(a) The integral of the potential plus internal energy from the surface to the top ofthe atmosphere (p = 0) is is equal to its enthalpy;

(b) dσ/dz = cp(T/θ)dθ/dz, where σ = I + pα+ Φ is the dry static energy;(c) The following expressions for the pressure gradient force are all equal (even

without hydrostatic balance):

− 1ρ∇p = −θ∇Π = − c

2s

ρθ∇(ρθ). (P2.1)

where Π = cpT/θ is the Exner function.(d) Show that item (a) also holds for a gas with an arbitrary equation of state, p =

p(ρ, T).

2.2 Show that, without approximation, the unforced, inviscid momentum equation maybe written in the forms

DvDt

= T∇η−∇(pα+ I) (P2.2)

and∂v∂t

+ω× v = T∇η−∇B (P2.3)

where ω = ∇ × v, η is the specific entropy (dη = cp d lnθ) and B = I + v2/2 + pαwhere I is the internal energy per unit mass.Hint: First show that T∇η = ∇I+p∇α, and note also the vector identity v×(∇×v) =12∇(v · v)− (v · ∇)v.

2.3 Consider two-dimensional fluid flow in a rotating frame of reference on the f−plane.Linearize the equations about a state of rest.

(a) Ignore the pressure term and determine the general solution to the resultingequations. Show that the speed of fluid parcels is constant. Show that the tra-jectory of the fluid parcels is a circle with radius |U|/f , where |U| is the fluidspeed.

(b) What is the period of oscillation of a fluid parcel?(c) If parcels travel in straight lines in inertial frames, why is the answer to (b) not

the same as the period of rotation of the frame of reference? [To answer this fullyyou need to understand the dynamics underlying inertial oscillations and inertiacircles. See Durran (1993), Egger (1999) and Phillips (2000).]

2.4 A fluid at rest evidently satisfies the hydrostatic relation, which says that the pressureat the surface is given by the weight of the fluid above it. Now consider a deepatmosphere on a spherical planet. A unit cross-sectional area at the planet’s surfacesupports a column of fluid whose cross-section increases with height, because thetotal area of the atmosphere increases with distance away from the center of theplanet. Is the pressure at the surface still given by the hydrostatic relation, or is itgreater than this because of the increased mass of fluid in the column? If it is stillgiven by the hydrostatic relation, then the pressure at the surface, integrated overthe entire area of the planet, is less than the total weight of the fluid; resolve thisparadox. But if the pressure at the surface is greater than that implied by hydrostaticbalance, explain how the hydrostatic relation fails.

2.5 By considering how the direction of the coordinate axes change with position [as inHolton (1992), for example] show geometrically that in spherical coordinates:

Di/Dt = u∂ i/∂x = (u/r cosϑ)(j sinϑ − k cosϑ), (P2.4)

Dj/Dt = u∂ j/∂x + v∂ j/∂y = −i(u/r) tanϑ − kv/a, (P2.5)


Dk/Dt = u∂k/∂x + v∂k/∂y = i(u/r)+ j(v/r). (P2.6)

Then, using (2.44a) show that (2.45) results.

2.6 At what latitude is the angle between the direction of Newtonian gravity (due solely tothe mass of the earth) and that of effective gravity (Newtonian gravity plus centrifugalterms) the largest? At what latitudes, if any, is this angle zero?

2.7 Write the momentum equations in true spherical coordinates, including the cen-trifugal and gravitational terms. Show that for reasonable values of the wind, thedominant balance in the meridional component of this equation involve a balancebetween centrifugal and pressure gradient terms. Can this balance be subtracted outof the equations in a sensible way, so leaving a useful horizontal momentum equa-tion that involves the Coriolis and acceleration terms? If so, obtain a closed set ofequations for the flow this way. Discuss the pros and cons of this approach versusthe geometric approximation discussed in section 2.2.1.

2.8 For an ideal gas show that the expressions (2.222) and (2.226) are equivalent.

2.9 Consider an ocean at rest with known vertical profiles of potential temperature andsalinity, θ(z) and S(z). Suppose we also know the equation of state in the formρ = ρ(θ, S,p). Obtain an expression for the buoyancy frequency. Check your expres-sion by substituting the equation of state for an ideal gas and recovering a knownexpression for the buoyancy frequency.

2.10 Convection and its parameterization

(a) Consider a Boussinesq system in which the vertical momentum equation is mod-ified by the parameter α to read

α2 DwDt

= −∂φ∂z

+ b, (P2.7)

and the other equations are unchanged. (If α = 0 the system is hydrostatic, andif α = 1 the system is the original one.) Linearise these equations about a state ofrest and of constant stratification (as in section 2.10.1) and obtain the dispersionrelation for the system, and plot it for various values of α, including 0 and 1.Show that, for α > 1 the system approaches its limiting frequency more rapidlythan with α = 1.

(b) Argue that if N2 < 0, convection in a system with α > 1 generally occurs ata larger scale than with α = 1. Show this explicitly by adding some diffusionor friction to the right-hand sides of the equations of motion and obtaining thedispersion relation. You may do this approximately.

2.11 (a) The geopotential height is the height of a given pressure level. Show that in an at-mosphere with a uniform lapse rate (i.e., dT/dz = Γ = constant) the geopotentialheight at a pressure p is given by

z = T0

Γ

1−(p0

p

)−RΓ /g (P2.8)

where T0 is the temperature at z = 0.(b) In an isothermal atmosphere, obtain an expression for the geopotential height as

function of pressure, and show that this is consistent with the expression (P2.8)in the appropriate limit.

2.12 Consider the simple Boussinesq equations, Dv/Dt = kb + ν∇2v, ∇ · v = 0, Db/Dt =Q + κ∇2b. Obtain an energy equation similar to (2.112) but now with the terms onthe right-hand side that represent viscous and diabatic effects. Over a closed volume,


show that the dissipation of kinetic energy is balanced by a buoyancy source. Showalso that, in a statistically steady state, the heating must occur at a lower level thanthe cooling if a kinetic-energy dissipating circulation is to be maintained.

2.13 Suppose a fluid is contained in a closed container, with insulating sidewalls, andheated from below and cooled from above. The heating and cooling are adjusted sothat there is no net energy flux into the fluid. Let us also suppose that any viscousdissipation of kinetic energy is returned as heating, so the total energy of the fluid isexactly constant. Suppose the fluid starts out at rest and at a uniform temperature,and the heating and cooling are then turned on. A very short time afterwards, thefluid is lighter at the bottom and heavier at the top; that is, its potential energyhas increased. Where has this energy come from? Discuss this paradox for both acompressible fluid (e.g., an ideal gas) and for a simple Boussinesq fluid.

2.14 Consider a rapidly rotating (i.e., in near geostrophic balance) Boussinesq fluid on thef -plane.

(a) Show that the pressure divided by the density scales as φ ∼ fUL.(b) Show that the horizontal divergence of the geostrophic wind vanishes. Thus,

argue that the scaling W ∼ UH/L is an overestimate for the magnitude of thevertical velocity. (Optional extra: obtain a scaling estimate for the magnitude ofvertical velocity in rapidly rotating flow.)

(c) Using these results, or otherwise, discuss whether hydrostatic balance is more orless likely to hold in a rotating flow that in non-rotating flow.

2.15 Estimate the size of the zonal wind 5 km above the surface in the midlatitude at-mosphere in summer and winter using (approximate) values for the meridional tem-perature gradient in the atmosphere. Also estimate the shear corresponding to thepole-equator temperature gradient in the ocean.

2.16 Using approximate but realistic values for the observed stratification, what is thebuoyancy period for (a) the mid-latitude troposphere, (b) the stratosphere, (c) theoceanic thermocline, (d) the oceanic abyss?

2.17 Consider a dry, hydrostatic, ideal-gas atmosphere whose lapse rate is one of constantpotential temperature. What is its vertical extent? That is, at what height does thedensity vanish? Is this a problem for the anelastic approximation discussed in thetext?

2.18 Show that for an ideal gas, the expressions (2.226), (2.221), (2.222) are all equiva-lent, and express N2 terms of the temperature lapse rate, ∂T/∂z.

2.19 Calculate a reasonably accurate, albeit approximate, expression for the buoyancyequation for seawater. (Derived from notes by R. deSzoeke)Solution (i): The buoyancy frequency is given by

N2 = −gρ

(∂ρpot

∂z

)env= gα

(∂αpot

∂z

)env= −g

2

α2

(∂αpot

∂p

)env

(P2.9)

where αpot = α(θ, S,pR) is the potential density, and pR a reference pressure. From(1.173)

αpot = α0

[1− α0

c20pR + βT (1+ γ∗pR)θ′ +

12β∗Tθ′2 − βS(S − S0)

]. (P2.10)

Using this and (P2.9) we obtain the buoyancy frequency,

N2 = −g2

α2α0

[βT

(1+ γpR +

β∗TβTθ)(

∂θ∂p

)env

− βS(∂S∂p

)env

], (P2.11)


although we must substitute local pressure for the reference pressure pR. (Why?)

Solution (ii): The sound speed is given by

c−2s = − 1

α2

(∂α∂p

)θ,S= 1α2

(α2

0

c20− γα1θ

)(P2.12)

and, using (P2.9) and (2.226) the square of the buoyancy frequency may be written

N2 = gα

(∂α∂z

)env− g

2

c2s= −g

2

α2

[(∂α∂p

)env

+ α2

c2s

](P2.13)

Using (1.173), (P2.12) and (P2.13) we recover (P2.11), although now with p explic-itly in place of pR.

2.20 Begin with the mass conservation in the height-coordinates, namely Dρ/Dt+ρ∇·v =0. Transform this into pressure coordinates using the chain rule (or otherwise) andderive the mass conservation equation in the form ∇p · u+ ∂ω/∂p = 0.

2.21 Starting with the primitive equations in pressure coordinates, derive the form of theprimitive equations of motion in sigma-pressure coordinates. In particular, show thatthe prognostic equation for surface pressure is,

∂ps∂t

+∇ · (psu)+ ps∂σ∂σ

= 0 (P2.14)

and that hydrostatic balance may be written ∂Φ/∂σ = −RT/σ .

2.22 Starting with the primitive equations in pressure coordinates, derive the form of theprimitive equations of motion in log-pressure coordinates in which Z = −H ln(p/pr )is the vertical coordinate. Here, H is a reference height (e.g., a scale height RTr/gwhere Tr is a typical or an average temperature) and pr is a reference pressure(e.g., 1000 mb). In particular, show that if the ‘vertical velocity’ is W = DZ/Dt thenW = −Hω/p and that

∂ω∂p

= − ∂∂p

(pWH

)= ∂W∂Z

− WH. (P2.15)

and obtain the mass conservation equation (2.156b). Show that this can be writtenin the form

∂u∂x

+ ∂v∂y

+ 1ρs∂∂Z(ρsW) = 0, (P2.16)

where ρs = ρr exp(−Z/H).2.23 (a) Prove that the argument of the square root in (2.257) is always positive.

Solution: The largest value or the argument occurs when m = 0 and k2 =1/(4H2). The argument is then 1 − 4H2N2/c2

s . But c2s = γRT0 = γgH and

N2 = gκ/H so that 4N2H2/c2s = 4κ/γ ≈ 0.8.

(b) This argument seems to depend on the parameters in the ideal gas equationof state. Is it more general than this? Is a natural system possible for which theargument is negative, and if so what physical interpretation could one ascribe tothe situation?

2.24 Consider a wind stress imposed by a mesoscale cyclonic storm (in the atmosphere)given by

τ = −Ae−(r/λ)2(y i+ x j) (P2.17)

where r 2 = x2 + y2, and A and λ are constants. Also assume constant Coriolisgradient β = ∂f/∂y and constant ocean depth H. Find (a) the Ekman transport,(b) the vertical velocity wE(x,y, z) below the Ekman layer, (c) the northward velocityv(x,y, z) below the Ekman layer and (d) indicate how you would find the westwardvelocity u(x,y, z) below the Ekman layer.


2.25 In an atmospheric Ekman layer on the f -plane for a fluid with ρ = ρa = 1 let us writethe momentum equation as

f × u = −∇φ+ ∂τ∂z

(P2.18)

where τ = K∂u/∂z and K is a constant coefficient of viscosity. An independentformula for the stress at the ground is τ = Cu, where C is a constant. Assume thatin the free atmosphere the wind is geostrophic and zonal, with ug = Ui.(a) Find an expression for the wind vector at the ground. Discuss the limits C = 0

and C = ∞. Show that when C = 0 the frictionally-induced vertical velocity at thetop of the Ekman layer is zero.

(b) Find the vertically integrated horizontal mass flux caused by the boundary layer.(c) When the stress on the atmosphere is τ, the stress on the ocean beneath is

−τ. Determine the direction and strength of the surface current in terms ofthe surface wind, the oceanic Ekman depth and the ratio ρa/ρo, where ρo is thedensity of the seawater. How does the boundary-layer mass flux in the oceancompare to that in the atmosphere?

Partial solution: A useful trick in Ekman layer problems is to write the velocity asa complex number, u = u + iv. The Ekman layer equation, (2.294a), may then bewritten as

A∂2U∂z2 = if U, (P2.19)

where U = u− ug. The solution to this is

u− ug = (u(0)− ug)exp[− (1+ i)z

d

], (P2.20)

where the boundary condition of finiteness at infinity eliminates the exponentiallygrowing solution. The boundary condition at z = 0 is ∂ug/∂z = (C/K)ug whichgives (u(0)− ug)exp(iπ/4) = −Cd/(

√2K)u(0), and the rest of the solution follows.

2.26 The logarithmic boundary layerClose to ground rotational effects are unimportant and small-scale turbulence gener-ates a mixed layer. In this layer, assume that the stress is constant and that it can beparameterized by an eddy diffusivity the size of which is proportional to the distancefrom the surface. Show that the velocity then varies logarithmically with height.Solution: Write the stress as τ = ρ0u∗2 where the constant u∗ is called the frictionalvelocity. Using the eddy diffusivity hypothesis this stress is given by

τ = ρ0u∗2 = ρ0A∂u∂z

where A = u∗kz, (P2.21)

where k is von Karman’s (‘universal’) constant (approximately equal to 0.4). From(P2.21) we have ∂u/∂z = u∗/(Az) which integrates to give u = (u∗/k) ln(z/z0). Theparameter z0 is known as the roughness length, and is typically of order centimetersor a little larger, depending on the surface.

Another advantage of a mathematical statement is that it is so definitethat it might be definitely wrong. . . Some verbal statements have notthis merit.L. F. Richardson (1881–1953).

CHAPTER

THREE

Shallow Water Systems and IsentropicCoordinates

CONVENTIONALLY, ‘THE’ SHALLOW WATER EQUATIONS describe a thin layer of constantdensity fluid in hydrostatic balance, rotating or not, bounded from below bya rigid surface and from above by a free surface, above which we suppose is

another fluid of negligible inertia. Such a configuration can be generalized to mul-tiple layers of immiscible fluids lying one on top of each other, forming a ‘stackedshallow water’ system, and this class of systems is the main subject of this chapter.

The single-layer model is one of the simplest useful models in geophysical fluiddynamics, because it allows for a consideration of the effects of rotation in a simpleframework without with the complicating effects of stratification. By adding lay-ers we can then study the effects of stratification, and indeed the model with justtwo layers is not only a simple model of a stratified fluid, it is a surprisingly goodmodel of many phenomena in the ocean and atmosphere. Indeed, the models aremore than just pedagogical tools — we will find that there is a close physical andmathematical analogy between the shallow water equations and a description ofthe continuously stratified ocean or atmosphere written in isopycnal or isentropiccoordinates, with a meaning beyond a coincidental similarity in the equations. Webegin with the single-layer case.

3.1 DYNAMICS OF A SINGLE, SHALLOW LAYER

Shallow water dynamics apply, by definition, to a fluid layer of constant densityin which the horizontal scale of the flow is much greater than the layer depth.The fluid motion is then fully determined by the momentum and mass continuity

123

124 Chapter 3. Shallow Water Systems and Isentropic Coordinates

Fig. 3.1 A shallow water system. h is the thickness of a water column, H itsmean thickness, η the height of the free surface and ηb is the height of thelower, rigid, surface, above some arbitrary origin, typically chosen such thatthe average of ηb is zero. ∆η is the deviation free surface height, so we haveη = ηb + h = H +∆η.

equations, and because of the assumed small aspect ratio the the hydrostatic ap-proximation is well satisfied, and we invoke this from the outset. Consider, then,fluid in a container above which is another fluid of negligible density (and there-fore negligible inertia) relative to the fluid of interest, as illustrated in Fig. 3.1. Asusual, our notation is that v = ui + vj +wk is the three dimensional velocity andu = ui+vj is the horizontal velocity. h(x,y) is thickness of the liquid column, H isits mean height, and η is the height of the free surface. In a flat-bottomed containerη = h, whereas in general h = η − ηb, where ηb is the height of the floor of thecontainer.

3.1.1 Momentum equations

The vertical momentum equation is just the hydrostatic equation,

∂p∂z

= −ρg, (3.1)

and, because density is assumed constant, we may integrate this to

p(x,y, z) = −ρgz + po. (3.2)

At the top of the fluid, z = η, the pressure is determined by the weight of theoverlying fluid and this is assumed negligible. Thus, p = 0 at z = η giving

p(x,y, z) = ρg(η(x,y)− z). (3.3)

The consequence of this is that the horizontal gradient of pressure is independentof height. That is

∇zp = ρg∇zη, (3.4)

where∇z = i

∂∂x

+ j∂∂y

(3.5)

3.1 Dynamics of a Single, Shallow Layer 125

is the gradient operator at constant z. (In the rest of this chapter we will drop thesubscript z unless that causes ambiguity. The three-dimensional gradient operatorwill be denoted ∇3. We will also mostly use Cartesian coordinates, but the shal-low water equations may certainly be applied over a spherical planet — indeed,‘Laplace’s tidal equations’ are essentially the shallow water equations on a sphere.)The horizontal momentum equations therefore become

DuDt

= −1ρ∇p = −g∇η. (3.6)

The right-hand side of this equation is independent of the vertical coordinate z.Thus, if the flow is initially independent of z, it must stay so. (This z-independenceis unrelated to that arising from the rapid rotation necessary for the Taylor-Proudmaneffect.) The velocities u and v are functions only of x,y and t and the horizontalmomentum equation is therefore

DuDt

= ∂u∂t

+u∂u∂x

+ v ∂u∂y

= −g∇η. (3.7)

That the horizontal velocity is independent of z is a consequence of the hydrostaticequation, which ensures that the horizontal pressure gradient is independent ofheight. (Another starting point would be to take this independence of the horizontalmotion with height as the definition of shallow water flow. In real physical situationssuch independence does not hold exactly — for example, friction at the bottom mayinduce a vertical dependence of the flow in a boundary layer.) In the presence ofrotation (3.7) easily generalizes to

DuDt

+ f × u = −g∇η , (3.8)

where f = fk. Just as with the primitive equations, f may be constant or mayvary with latitude, so that on a spherical planet f = 2Ω sinϑ and on the β-planef = f0 + βy.

3.1.2 Mass continuity equation

From first principles

The mass contained in a fluid column of height h and cross-sectional area A is givenby∫A ρhdA (see Fig. 3.2). If there is a net flux of fluid across the column boundary

(by advection) then this must be balanced by a net increase in the mass in A, andtherefore a net increase in the height of the water column. The mass convergenceinto the column is given by

Fm = Mass flux in = −∫Sρu · dS, (3.9)

where S is the area of the vertical boundary of the column. The surface area of thecolumn is comprised of elements of area hnδl, where δl is a line element circum-scribing the column and n is a unit vector perpendicular to the boundary, pointingoutwards. Thus (3.9) becomes

Fm = −∮ρhu · ndl. (3.10)


Figure 3.2 The mass budget for acolumn of area A in a shallow watersystem. The fluid leaving the col-umn is

∮ρhu · ndl where n is the

unit vector normal to the bound-ary of the fluid column. There is anon-zero vertical velocity at the topof the column if the mass conver-gence into the column is non-zero.

Using the divergence theorem in two-dimensions, (3.10) simplifies to

Fm = −∫A∇ · (ρuh)dA, (3.11)

where the integral is over the cross-sectional area of the fluid column (looking downfrom above). This is balanced by the local increase in height of the water column,given by

Fm =ddt

∫ρ dV = d

dt

∫AρhdA =

∫Aρ∂h∂t

dA. (3.12)

Because ρ is constant, the balance between (3.11) and (3.12) leads to∫A

[∂h∂t

+∇ · (uh)]

dA = 0, (3.13)

and because the area is arbitrary the integrand itself must vanish, whence,

∂h∂t

+∇ · (uh) = 0 , (3.14)

or equivalently

DhDt

+ h∇ · u = 0 . (3.15)

This derivation holds whether or not the lower surface is flat. If it is, then h = η, andif not h = η− ηb. Eqs. (3.8) and (3.14) or (3.15) form a complete set, summarizedin the shaded box on the next page.

From the 3D mass conservation equation

Since the fluid is incompressible, the three-dimensional mass continuity equation isjust ∇ · v = 0. Writing this out in component form

∂w∂z

= −(∂u∂x

+ ∂v∂y

)= −∇ · u (3.16)

3.1 Dynamics of a Single, Shallow Layer 127

The Shallow Water Equations

For a single-layer fluid, and including the Coriolis term, the inviscid shallowwater equations are:Momentum:

DuDt

+ f × u = −g∇η. (SW.1)

Mass Continuity:

DhDt

+ h∇ · u = 0 or∂h∂t

+∇ · (hu) = 0. (SW.2)

where u is the horizontal velocity, h is the total fluid thickness, η is the heightof the upper free surface and ηb is the height of the lower surface (the bottomtopography). Thus, h(x,y, t) = η(x,y, t)− ηb(x,y). The material derivativeis

DDt

= ∂∂t+ u · ∇ = ∂

∂t+u ∂

∂x+ v ∂

∂y, (SW.3)

with the rightmost expresion holding in Cartesian coordinates.

Integrate this from the bottom of the fluid (z = ηb) to the top (z = η), noting thatthe right-hand side is independent of z, to give

w(η)−w(ηb) = −h∇ · u. (3.17)

At the top the vertical velocity is the matarial derivative of the position of a partic-ular fluid element. But the position of the fluid at the top is just η, and therefore(see Fig. 3.2)

w(η) = DηDt. (3.18a)

At the bottom of the fluid we have similarly

w(ηb) =DηbDt, (3.18b)

where, absent earthquakes and the like, ∂ηb/∂t = 0. Using (3.18a,b), (3.17) be-comes

DDt(η− ηb)+ h∇ · u = 0 (3.19)

or, as in (3.15),DhDt

+ h∇ · u = 0. (3.20)

3.1.3 A rigid lid

The case where the upper surface is held flat by the imposition of a rigid lid is some-times of interest. The ocean suggests one such example, for here the bathymetry


at the bottom of the ocean provides much larger variations in fluid thickness thando the small variations in the height of the ocean surface. Suppose then the up-per surface is at a contant height H then, from (3.14) with ∂h/∂t = 0 the massconservation equation becomes

∇h · (uhb) = 0. (3.21)

where hb = H − ηb Note that this allows us to define an incompressible mass-transport velocity, U ≡ hbu.

Although the upper surface is flat, the pressure there is no longer constant be-cause a force must be provided by the rigid lid to keep the surface flat. The hori-zontal momentum equation is

DuDt

= − 1ρ0∇plid, (3.22)

where plid is the pressure at the lid, and the complete equations of motion are then(3.21) and (3.22).1 If the lower surface is flat, the two-dimensional flow itself isdivergence-free, and the equations reduce to the two-dimensional incompressibleEuler equations.

3.1.4 Stretching and the vertical velocity

Because the horizontal velocity is depth independent, the vertical velocity plays norole in advection. However, w is certainly not zero for then the free surface wouldbe unable to move up or down, but because of the vertical independence of thehorizontal flow w does have a simple vertical structure; to determine this we writethe mass conservation equation as

∂w∂z

= −∇ · u (3.23)

and integrate upwards from the bottom to give

w = wb − (∇ · u)(z − ηb). (3.24)

Thus, the vertical velocity is a linear function of height. Eq. (3.24) can be written

DzDt

= DηbDt

− (∇ · u)(z − ηb), (3.25)

and at the upper surface w = Dη/Dt so that here we have

DηDt

= DηbDt

− (∇ · u)(η− ηb), (3.26)

Eliminating the divergence term from the last two equations gives

DDt(z − ηb) =

z − ηbη− ηb

DDt(η− ηb), (3.27)

which in turn givesDDt

(z − ηbη− ηb

)= D

Dt

(z − ηbh

)= 0. (3.28)

This means that the ratio of the height of a fluid parcel above the floor to the totaldepth of the column is fixed; that is, the fluid stretches uniformly in a column, andthis is kinematic property of the shallow water system.

3.2 Reduced Gravity Equations 129

Figure 3.3 The reduced gravity shallow wa-ter system. An active layer lies over a deep,more dense, quiescent layer. In a commonvariation the upper surface is held flat by arigid lid, and η0 = 0.

3.1.5 Analogy with compressible flow

The shallow water equations (3.8) and (3.14) are analogous to the compressiblegas dynamic equations in two dimensions, namely

DuDt

= −1ρ∇p (3.29)

and∂ρ∂t

+∇ · (uρ) = 0, (3.30)

along with an equation of state which we take to be p = f(ρ). The mass conser-vation equations (3.14) and (3.30) are identical, with the replacement ρ ↔ h. Ifp = Cργ , then (3.29) becomes

DuDt

= −1ρ

dpdρ∇ρ = −Cγργ−2∇ρ. (3.31)

If γ = 2 then the momentum equations (3.8) and (3.31) become equivalent, withρ ↔ h and Cγ ↔ g. In an ideal gas γ = cp/cv and values typically are in fact lessthan two (in air γ ≈ 7/5); however, if the equations are linearized, then the analogyis exact for all values of γ, for then (3.31) becomes ∂v′/∂t = −ρ−1

0 c2s∇ρ′ where

c2s = dp/dρ , and the linearized shallow water momentum equation is ∂u′/∂t =−H−1(gH)∇h′, so that ρ0 ↔ H and c2

s ↔ gH. The sound waves of a compressiblefluid are then analogous to shallow water waves, considered in section 3.7.

3.2 REDUCED GRAVITY EQUATIONS

Consider now a single shallow moving layer of fluid on top a deep, quiescent fluidlayer (Fig. 3.3), and beneath a fluid of negligible inertia. This configuration is oftenused a model of the upper ocean: the upper layer represents flow in perhaps theupper few hundred meters of the ocean, the lower layer the near-stagnant abyss. Ifwe turn the model upside-down we have a model, perhaps slightly less realistic, ofthe atmosphere: the lower layer represents motion in the troposphere above whichlies an inactive stratosphere. The equations of motion are virtually the same in bothcases.


3.2.1 Pressure gradient in the active layer

We’ll derive the equations for the oceanic case (active layer on top) in two cases,which differ slightly in the assumption made about the upper surface.

I Free upper surface

The pressure in the upper layer is given by integrating the hydrostatic equationdown from the upper surface. Thus, at a height z in the upper layer

p1(z) = gρ1(η0 − z), (3.32)

where η0 is the height of the upper surface. Hence, everywhere in the upper layer,

1ρ1∇p1 = −g∇η0, (3.33)

and the momentum equation is

DuDt

+ f × u = −g∇η0. (3.34)

In the lower layer the the pressure is also given by the weight of the fluid above it.Thus, at some level z in the lower layer,

p2(z) = ρ1g(η0 − η1)+ ρ2g(η1 − z). (3.35)

But if this layer is motionless the horizontal pressure gradient in it is zero andtherefore

ρ1gη0 = −ρ1g′η1 + constant, (3.36)

where g′ = g(ρ2−ρ1)/ρ1 is the reduced gravity. The momentum equation becomes

DuDt

+ f × u = g′∇η1. (3.37)

The equations are completed by the usual mass conservation equation,

DhDt

+ h∇ · u = 0, (3.38)

where h = η0 − η1. Because g g′, (3.36) shows that surface displacements aremuch smaller than the displacements at the interior interface. We see this in thereal ocean where the mean interior isopycnal displacements may be several tens ofmeters but variations in the mean height of ocean surface are of order centimeters.

II The rigid lid approximation

The smallness of the upper surface displacement suggests that we will make littleerror is we impose a rigid lid at the top of the fluid. Displacements are no longerallowed, but the lid will in general impart a pressure force to the fluid. Supposethat this is P(x,y, t), then the horizontal pressure gradient in the upper layer issimply

∇p1 = ∇P. (3.39)

3.3 Multi-Layer Shallow Water Equations 131

The pressure in the lower layer is again given by hydrostasy, and is

p2 = −ρ1gη1 + ρ2g(η1 − z)+ P= ρ1gh− ρ2g(h+ z)+ P,

(3.40)

so that∇p2 = −g(ρ2 − ρ1)∇h+∇P. (3.41)

Then if ∇p2 = 0 we haveg(ρ2 − ρ1)∇h = ∇P, (3.42)

and the momentum equation for the upper layer is just

DuDt

+ f × u = −g′1∇h. (3.43)

where g′ = g(ρ2 − ρ1)/ρ1. These equations differ from the usual shallow waterequations only in the use of a reduced gravity g′ in place of g itself. It is the densitydifference between the two layers that is important. Similarly, if we take a shallowwater system, with the moving layer on the bottom, and we suppose that overlyingit is a stationary fluid of finite density, then we would easily find that the fluidequations for the moving layer are the same as if the fluid on top had zero inertia,except that g would be replaced by an appropriate reduced gravity (problem 3.1).

3.3 MULTI-LAYER SHALLOW WATER EQUATIONS

We now consider the dynamics of multiple layers of fluid stacked on top of eachother. This is a crude representation of continuous stratification, but it turns outto be a powerful model of many geophysically interesting phenomena as well asbeing physically realizable in the laboratory. The pressure is continuous acrossthe interface, but the density jumps discontinuously and this allows the horizontalvelocity to have a corresponding discontinuity. The set up is illustrated in Fig. 3.4.

In each layer pressure is given by the hydrostatic approximation, and so any-where in the interior we can find the pressure by integrating down from the top.Thus, at a height z in the first layer we have

p1 = ρ1g(η0 − z), (3.44)

and in the second layer,

p2 = ρ1g(η0 − η1)+ ρ2g(η1 − z) = ρ1gη0 + ρ1g′1η1 − ρ2gz, (3.45)

where g′1 = g(ρ2 − ρ1)/ρ1, and so on. The term involving z is irrelevant for thedynamics, because only the horizontal derivative enters the equation of motion.Omitting this term, for the n’th layer the dynamical pressure is given by the sumfrom the top down:

pn = ρ1

n−1∑i=0

g′iηi, (3.46)


Figure 3.4 The multi-layershallow water system. Thelayers are numbered fromthe top down. The co-ordinates of the interfacesare denoted η, and thelayer thicknesses h, sothat hi = ηi − ηi−1.

where g′i = g(ρi+1 − ρi)/ρ1 (but g0 = g). The interface displacements may beexpressed in terms of the layer thicknesses by summing from the bottom up:

ηn = ηb +i=N∑i=n+1

hi. (3.47)

The momentum equation for each layer may then be written, in general,

DunDt

+ f × un = −1ρn∇pn, (3.48)

where the pressure is given by (3.46) and in terms of the layer depths using (3.48).If we make the Boussinesq approximation then ρn on the right-hand side of (3.48)is replaced by ρ1.

Finally, the mass conservation equation for each layer has the same form as thesingle-layer case, and is

DhnDt

+ hn∇ · un = 0. (3.49)

The two- and three-layer cases

The two-layer model is the simplest model to capture the effects of stratification.Evaluating the pressures using (3.46) and (3.47) we find:

p1 = ρ1gη0 = ρ1g(h1 + h2 + ηb) (3.50a)

p2 = ρ1[gη0 + g′1η1] = ρ1[g(h1 + h2 + ηb)+ g′1(h2 + ηb)

]. (3.50b)

The momentum equations for the two layers are then

Du1

Dt+ f × u1 = −g∇η0 = −g∇(h1 + h2 + ηb). (3.51a)

3.3 Multi-Layer Shallow Water Equations 133

Fig. 3.5 The two layer shallow water system. A fluid of density ρ1 lies over adenser fluid of density ρ2. In the reduced gravity case the lower layer may bearbitrarily thick and is assumed stationary and so has no horizontal pressuregradient. In the ‘rigid-lid’ approximation the top surface displacement isneglected, but there is then a non-zero pressure gradient induced by the lid.

and in the bottom layer

Du2

Dt+ f × u2 = −

ρ1

ρ2

(g∇η0 + g′1∇η1

)= −ρ1

ρ2

[g∇(ηb + h1 + h2)+ g′1∇(h2 + ηb)

].

(3.51b)

In the Boussinesq approximation ρ1/ρ2 is replaced by unity.In a three layer model the dynamical pressures are found to be

p1 = ρ1gh (3.52a)

p2 = ρ1[gh+ g′1(h2 + h3 + ηb)

](3.52b)

p3 = ρ1[gh+ g′1(h2 + h3 + ηb)+ g′2(h3 + ηb)

], (3.52c)

where h = η0 = ηb+h1+h2+h3 and g′2 = g(ρ3−ρ2)/ρ1. More layers can obviouslybe added in a systematic fashion.

3.3.1 Reduced-gravity multi-layer equation

As with a single active layer, we may envision multiple layers of fluid overlying adeeper stationary layer. This is a useful model of the stratified upper ocean overly-ing a nearly stationary and nearly unstratified abyss. Indeed we use such a modelto study the ‘ventilated thermocline’ in chapter 16 and a detailed treatment maybe found there. If we suppose there is a lid at the top, then the model is almostthe same as that of the previous section. However, now the horizontal pressuregradient in the lowest model layer is zero, and so we may obtain the pressures inall the active layers by integrating the hydrostatic equation upwards from this layer.


Fluid velocity, out of pageFluid velocity, into page

Coriolis force

Pressureforce

Free surface

Pressureforce

Coriolis force

Fig. 3.6 Geostrophic flow in a shallow water system, with a positive value ofthe Coriolis parameter f , as in the Northern hemisphere. The pressure forceis directed down the gradient of the height field, and this can be balancedby the Coriolis force if the fluid velocity is at right angles to it. If f werenegative, the geostrophic flow would be reversed.

Suppose we have N moving layers, then the reader may verify that the dynamicpressure in the n’th layer is given by

pn = −i=N∑i=nρ1g′iηi, (3.53)

where as before g′i = g(ρi+1 − ρi)/ρ1. If we have a lid at the top, so that η0 = 0,then the interface displacements are related to the layer thicknesses by

ηn = −i=n∑i=1

hi. (3.54)

From these expressions the momentum equation in each layer is easily constructed.

3.4 GEOSTROPHIC BALANCE AND THERMAL WIND

Geostrophic balance occurs in the shallow water equations, just as in the continu-ously stratifed equations, when the Rossby number U/fL is small and the Coriolisterm dominates the advective terms in the momentum equation. In the single-layershallow water equations the geostrophic flow is:

f × ug = −∇η. (3.55)

Thus, the geostrophic velocity is proportional to the slope of the surface, as sketchedin Fig. 3.6. (For the rest of this section, we will drop the subscript g, and take allvelocites to be geostrophic.)

In both the single-layer and multi-layer case, the slope of an interfacial surfaceis directly related to the difference in pressure gradient on either side and so, bygeostrophic balance, to the shear of the flow. This is the shallow water analog ofthe thermal wind relation. To obtain an expression for this, consider the interface,

3.5 Form Drag 135

Figure 3.7 Margules’ relation: us-ing hydrostasy, the difference inthe horizontal pressure gradientbetween the upper and the lowerlayer is given by −g′ρ1s wheres = tanφ = ∆z/∆y is the in-terface slope and g′ = (ρ2 −ρ1)/ρ1. Geostrophic balance thengives f∂y(u1 −u2) = g′s, which isa special case of (3.60).

η, between two layers labelled 1 and 2. The pressure in two layers is given by thehydrostatic relation and so,

p1 = A(x,y)− ρ1gz (at some z in layer 1) (3.56a)

p2 = A(x,y)− ρ1gη+ ρ2g(η− z)= A(x,y)+ ρ1g′1η− ρ2gz (at some z in layer 2) (3.56b)

where A(x,y) is the pressure where z = 0. (We don’t need to specifiy where thisis, except that it is in or at the top of the top layer). Thus we find

1ρ1∇(p1 − p2) = −g′1∇η. (3.57)

If the flow is geostrophically balanced and Boussinesq then, in each layer, the ve-locity obeys

fui =1ρ1

k×∇pi. (3.58)

Using (3.57) then givesf(u1 − u2) = −k× g′1∇η, (3.59)

or in generalf(un − un+1) = −k× g′n∇η. (3.60)

This is the thermal wind equation for the shallow water system. It applies at anyinterface, and it implies the shear is proportional to the interface slope, a result knownas the ‘Margules relation’ (Fig. 3.7).2

Suppose that we represent the atmosphere by two layers of fluid; a meridionallydecreasing temperature may then be represented by an interface that slopes upwardtoward the pole. Then, in either hemisphere, we have

u1 −u2 =g1

f∂η∂y

> 0, (3.61)

and the temperature gradient is associated with a positive shear. (See problem 3.2.)

3.5 FORM DRAG

When the interface between two layers varies with position — that is, when it iswavy — the layers exert a pressure force on each other. Similarly, if the bottom of


the fluid is not flat then the topography and the bottom layer will in general exertforces on each other. These kind of forces are known as form drag, and it is animportant means whereby momentum can be added to or extracted from a flow.3

Consider a layer confined between two interfaces, η1(x,y) and η2(x,y). Thenover some zonal interval L the average zonal pressure force on that fluid layer isgiven by

Fp = −1L

∫ x2

x1

∫ η1

η2

∂p∂x

dx dz. (3.62)

Integrating by parts first in z and then in x, and noting that by hydrostasy ∂p/∂zdoes not depend on horizontal position within the layer, we obtain

Fp = −1L

∫ x2

x1

[∂p∂xz]η1

η2

dx

= −η1∂p1

∂x+ η2

∂p2

∂x= +p1

∂η1

∂x− p2

∂η2

∂x,

(3.63)

where p1 is the pressure at η1, and similarly for p2, and to obtain the second linewe suppose that the integral is around a closed path, such as a circle of latitude,and the average is denoted with an overbar. These terms represent the transferof momentum from one layer to the next, and at a particular interface, i, we maydefine the form drag, τi, by

τi ≡ pi∂ηi∂x

= −ηi∂pi∂x. (3.64)

The form drag is a stress, and as the layer depth shrinks to zero its vertical deriva-tive, ∂τ/∂z, is the force on the fluid. It is a particularly important mechanism forthe vertical transfer of momentum and its ultimate removal in an eddying fluid,and it one of the the main mechanisms wherby the wind stress at the top of theocean is communicated to the ocean bottom. At the fluid bottom the form drag ispηbx, where ηb is the bottom topography, and this is proportional to the momen-tum exchange with the solid earth. This is a significant mechanism for the ultimateremoval of momentum in the ocean, especially in the Antactic Circumpolar Cur-rent where it is likely to be much larger than bottom (or Ekman) drag arising fromsmall scale turbulence and friction. In the two layer, flat-bottomed case the onlyform drag occurring is that at the interface, and the momentum transfer betweenthe layers is just p1∂η1/∂x or −η1∂p/∂x; then, the force on each layer due to theother is equal and opposite, as we would expect from momentum conservation.

For flows in geostrophic balance, the form drag is related to the meridional heatflux. The pressure gradient and velocity are related by ρfv′ = ∂p′/∂x and theinterfacial displacement is proportional to the temperature perturbation, b′ (in factone may show that η′ ≈ −b′/(∂b/∂z)). Thus −η′∂p′η/∂x ∝ v′b′, a correspondencethat will re-occur when we consider the Eliassen-Palm flux in chapter 7.

3.6 CONSERVATION PROPERTIES OF SHALLOW WATER SYSTEMS

There are two common types of conservation property in fluids: (i) material invari-ants and (ii) integral invariants. Material invariance occurs when a property (φsay) is conserved on each fluid element, and so obeys the equation Dφ/Dt = 0.

3.6 Conservation Properties of Shallow Water Systems 137

An integral invariant is one that is conserved following an integration over some,usually closed, volume; energy is an example.

3.6.1 A material invariant: potential vorticity

The vorticity of a fluid (considered at greater length in chapter 4), denoted ω, isdefined to be the curl of the velocity field, so that

ω ≡ ∇× v. (3.65)

Let us also define the shallow water vorticity, ω∗, as the curl of the horizontalvelocity, so that

ω∗ ≡ ∇× u (3.66)

and, because ∂u/∂z = ∂v/∂z = 0, only its vertical component is non-zero and

ω∗ = k(∂v∂x

− ∂u∂y

)≡ kζ. (3.67)

Using the vector identity

(u · ∇)u = 12∇(u · u)− u× (∇× u), (3.68)

we write the momentum equation, (3.8), as

∂u∂t

+ω∗ × u = −∇(gη+ 12u2). (3.69)

To obtain an evolution equation for the vorticity we take the curl of (3.69), andmake use of the vector identity

∇× (ω∗ × u) = (u · ∇)ω∗ − (ω∗ · ∇)u+ω∗∇ · u− u∇ ·ω∗

= (u · ∇)ω∗ +ω∗∇ · u, (3.70)

using the fact that ∇ ·ω∗ is the divergence of a curl and therefore zero, and (ω∗ ·∇)u = 0 because ω∗ is perpendicular to the surface in which u varies. The curl of(3.69) is then

∂ω∗

∂t+ (u · ∇)ω = −ω∗∇ · u, (3.71)

or∂ζ∂t

+ (u · ∇)ζ = −ζ∇ · u. (3.72)

where ζ = k ·ω∗. However, the mass conservation equation may be written as

− ζ∇ · u = ζh

DhDt. (3.73)

Thus, (3.72) becomesDζDt

= ζh

DhDt, (3.74)

which simplifies toDDt

(ζh

)= 0. (3.75)


The important quantity ζ/h, often denoted Q, is known as the potential vorticity,and (3.75) is known as the potential vorticity equation. We re-derive this conserva-tion law in a more general way in section 4.6

Because Q is conserved on parcels, then so is any function of Q; that is, F(Q)is a material invariant, where F is any function. To see this algebraically, multiply(3.75) by F ′(Q), the derivative of F with respect to Q, giving

F ′(Q)DQDt

= DDtF(Q) = 0. (3.76)

Since F is arbitrary there are an infinite number of material invariants correspond-ing to different choices of F .

Effects of rotation

In a rotating frame of reference, the shallow water momentum equation is

DuDt

+ f × u = −g∇η, (3.77)

where (as before) f = fk. This may be written in vector invariant form as

∂u∂t

+ (ω∗ + f )× u = −∇(gη+ 12u2), (3.78)

and taking the curl of this gives the vorticity equation

∂ζ∂t

+ (u · ∇)(ζ + f) = −(f + ζ)∇ · u. (3.79)

This is the same as the shallow water vorticity equation in a non-rotating frame,save that ζ is replaced by ζ + f , the reason for this being that f is the vorticitythat the fluid has by virtue of the background rotation. Thus, (3.79) is simply theequation of motion for the total or absolute vorticity, ωa =ω∗ + f = (ζ + f)k.

The potential vorticity equation in the rotating case follows, much as in non-rotating case, by combining (3.79) with the mass conservation equation, giving

DDt

(ζ + fh

)= 0 . (3.80)

That is, Q ≡ (ζ + f)/h, the potential vorticity in a rotating shallow system, is amaterial invariant.

Vorticity and circulation

Although vorticity itself is not a material invariant, its integral over a horizontalmaterial area is invariant. To demonstrate this in the non-rotating case, considerthe integral

C =∫Aζ dA =

∫AQhdA, (3.81)

over a surface A, the cross-sectional area of a column of height h (as in Fig. 3.2).Taking the material derivative of this gives

DCDt

=∫A

DQDthdA+

∫AQ

DDt(hdA). (3.82)

3.6 Conservation Properties of Shallow Water Systems 139

The first term is zero, by (3.74); the second term is just the derivative of the volumeof a column of fluid and it too is zero, by mass conservation. Thus,

DCDt

= DDt

∫Aζ dA = 0. (3.83)

Thus, the integral of the vorticity over a some cross-sectional area of the fluid is un-changing, although both the vorticity and area of the fluid may individually change.Using Stokes’s theorem, it may be written

DCDt

= DDt

∮u · dl, (3.84)

where the line integral is around the boundary of A. This is an example of Kelvin’scirculation theorem, which we shall meet again in more general form in chapter 4,where we also consider the rotating case.

A slight generalization of (3.83) is possible. Consider the integral I =∫F(Q)hdA

where again F is any differentiable function of its argument. It is clear that

DDt

∫AF(Q)hdA = 0. (3.85)

If the area of integration in (3.69) or (3.85) is the whole domain (enclosed by fric-tionless walls, for example) then it is clear that the integral of hF(Q) is a constant,including as a special case the integral of ζ.

3.6.2 Energy conservation — an integral invariant

Since we have made various simplifications in deriving the shallow water system, itis not self-evident that energy should be conserved, or indeed what form the energytakes. The kinetic energy density, that is the kinetic energy per unit area, is ρhu2/2.The potential energy density of the fluid is

PE =∫ h

0ρ0gz dz = 1

2ρ0gh2. (3.86)

The factor ρ0 appears in both kinetic and potential energies and, because it is aconstant, we will omit it.

Using the mass conservation equation (3.15) we obtain an equation for the evo-lution of potential energy density:

DDtgh2

2+ gh2∇ · u = 0 (3.87a)

or∂∂tgh2

2+∇ ·

(ugh2

2

)+ gh

2

2∇ · u = 0. (3.87b)

From the momentum and mass continuity equations we obtain an equation for theevolution of kinetic energy density, namely

DDthu2

2+ u

2h2∇ · u = −gu · ∇h

2

2(3.88a)


or∂∂thu2

2+∇ ·

(uhu2

2

)+ gu · ∇h

2

2= 0. (3.88b)

Adding (3.87b) and (3.88b) we obtain

∂∂t

12

(hu2 + gh2

)+∇ ·

[12u(gh2 + hu2 + gh2

)]= 0, (3.89)

or∂E∂t+∇ · F = 0. (3.90)

where E = KE + PE = (hu2 + gh2)/2 is the density of the total energy and F =u(hu2 + gh2 + gh2)/2 is the energy flux. If the fluid is confined to a domainbounded by rigid walls, on which the normal component of velocity vanishes, thenon integrating (3.89) over that area and using Gauss’s theorem, the total energy isseen to be conserved; that is

dEdt

= 12

ddt

∫A(hu2 + gh2)dA = 0. (3.91)

Such an energy principle also holds in the case with bottom topography. Note that,as we found in the case for a compressible fluid in chapter 2, the energy flux in(3.90) is not just the energy density multiplied by the velocity but it contains anadditional term guh2/2, and this represents the energy transfer occurring whenthe fluid does work against the pressure force (see problem 3.3).

3.7 SHALLOW WATER WAVES

Let us now look at the gravity waves that occur in shallow water. To isolate theessence of the phenomena, we will consider waves in a single fluid layer, with a flatbottom and a free upper surface, in which gravity provides the sole restoring force.

3.7.1 Non-rotating shallow water waves

Given a flat bottom the fluid thickness is equal to the free surface displacement (Fig.3.1), and we let

h(x,y, t) = H + h′(x,y, t) = H + η′(x,y, t), (3.92a)

u(x,y, t) = u′(x,y, t). (3.92b)

The mass conservation equation, (3.15), then becomes

∂η′

∂t+ (H + η′)∇ · u′ + u′ · ∇η′ = 0, (3.93)

and neglecting squares of small quantities this yields the linear equation

∂η′

∂t+H∇ · u′ = 0. (3.94)

3.7 Shallow Water Waves 141

Similarly, linearizing the momentum equation, (3.8) with f = 0, yields

∂u′

∂t= −g∇η′. (3.95)

Eliminating velocity by differentiating (3.94) with respect to time and taking thedivergence of (3.95) leads to

∂2η′

∂t2− gH∇2η′ = 0, (3.96)

which may be recognized as a wave equation. We can find the dispersion relation-ship for this by substituting the trial solution

η′ = Re η ei(k·x−ωt) (3.97)

where η is a complex constant, k = ik + jl is the horizontal wavenumber, andRe indicates that the real part of the solution should be taken. If for simplicity werestrict attention for the moment to the one-dimensional problem, with no variationin the y-direction, then substituting into (3.96) leads to the dispersion relationship

ω = ±ck, (3.98)

where c =√gH. That is, the wave speed is proportional to the square root of the

mean fluid depth and is independent of the wavenumber — that is, the waves aredispersionless. The general solution is a superposition of all such waves, with theamplitudes of each wave (or Fourier component) being determined by the Fourierdecomposition of the initial conditions.

Because the waves are dispersionless, the general solution can be written

η′(x, t) = 12[F(x − ct)+ F(x + ct)] , (3.99)

where F(x) is the height field at t = 0. From this, it is easy to see that the shape ofan initial disturbance is preserved as it propagates both to the right and to the leftat speed c. (See also problem 3.7.)

3.7.2 Rotating shallow water (Poincaré) waves

We now consider the effects of rotation. Linearizing the rotating, flat-bottomed f -plane shallow water equations [i.e., (SW.1) and (SW.2) on page 127] about a stateof rest we obtain

∂u′

∂t− f0v′ = −g

∂η′

∂x,

∂v′

∂t+ f0u′ = −g

∂η′

∂y,

∂η′

∂t+H

(∂u′

∂x+ ∂v

′

∂y

)= 0.

(3.100a,b,c)It is convenient to nondimensionalize these equations and we write

(x,y) = L(x, y), (u′, v′) = U(u,v), t = LUt, f0 =

f0

T, η′ = Hη. (3.101)


Eq. (3.100) then becomes

∂u∂t

− f0v = −c2 ∂η∂x,

∂v∂t

+ f0u = −c2 ∂η∂y,

∂η∂t+(∂u∂x

+ ∂v∂y

)= 0.

(3.102a,b,c)where c =

√gH/U is the nondimensional speed of nonrotating shallow-water waves.

(It is also the inverse of the Froude number U/√gH.) To obtain a dispersion rela-

tionship we let(u, v, η) = (u, v, η) ei(k·x−ωt), (3.103)

where k = ki + lj and ω is the nondimensional frequency, and substitute into(3.102), giving iω −f0 ic2k

f0 iω ic2lik il iω

uvη

= 0. (3.104)

This homogeneous equation has nontrivial solutions only if the determinant of thematrix vanishes. This condition gives

ω(ω2 − f 20 − c2K2) = 0. (3.105)

where K2 = k2 + l2. There are two classes of solution to (3.105). The first is simplyω = 0, time-independent flow corresponding to geostrophic balance in (3.100).(Because geostrophic balance gives a divergence-free velocity field for constantCoriolis parameter the equations are satisfied by a time-independent solution.) Thesecond set of solutions satisfies the dispersion relation

ω2 = f 20 + c2(k2 + l2). (3.106)

which in dimensional form is:

ω2 = f 20 + gH(k2 + l2) . (3.107)

The corresponding waves are known as Poincaré waves,4 and the dispersion rela-tionship is illustrated in Fig. 3.8. Note that the frequency is always greater than theCoriolis frequency f0. There are two interesting limits:

(i) The short waves limit: If

K2 f 20

gH, (3.108)

where K2 = k2 + l2, then the dispersion relationship reduces to that of thenonrotating case (3.98). This condition is equivalent to requiring that thewavelength be much shorter than the deformation radius, Ld. Specifically, ifl = 0 and λ = 2π/k is the wavelength, the condition is

λ2 L2d(2π)

2 (3.109)

The numerical factor of (2π)2 is more than an order of magnitude, so caremust be taken when deciding if the condition is satisfied in particular cases.Furthermore, the wavelength must still be longer than the depth of the fluid,else the shallow water condition is not met.

3.7 Shallow Water Waves 143

−3 −2 −1 0 1 2 30

1

2

3

Wavenumber (k × Ld)

Fre

quen

cy (ω

/f)Figure 3.8 Dispersionrelation for Poincaréwaves (solid) and non-rotating shallow waterwaves (dashed). Fre-quency is scaled by theCoriolis frequency f ,and wavenumber by theinverse deformation ra-dius

√gH/f . For small

wavenumbers the fre-quency is approximatelyf ; for high wavenumbersit asymptotes to that ofnonrotating waves.

(ii) The long wave limit: If

K2 f 20

gH, (3.110)

that is if the wavelength is much longer than the deformation radius Ld, thenthe dispersion relationship is

ω = f0. (3.111)

These are known as inertial oscillations. The equations of motion giving rise tothem are

∂u′

∂t− f0v′ = 0,

∂v′

∂t+ f0u′ = 0, (3.112)

which are equivalent to material equations for free particles in a rotatingframe, unconstrained by pressure forces, namely

d2xdt2

− f0v = 0,d2ydt2

+ f0u = 0. (3.113)

See also problem 3.9.

3.7.3 Kelvin waves

The Kelvin wave is a particular type of gravity wave that exists in the presence ofboth rotation and a lateral boundary. Suppose there is a solid boundary at y = 0;clearly harmonic solutions in the y-direction are not allowable, as these would notsatisfy the condition of no-normal flow at the boundary. Do any wavelike solutionsexist? The affirmative answer to this question was provided by Kelvin and theassociated waves are now eponymously known as Kelvin waves.5 We begin with thelinearized shallow water equations, namely

∂u′

∂t− f0v′ = −g

∂η′

∂x,

∂v′

∂t+ f0u′ = −g

∂η′

∂y,

∂η′

∂t+H

(∂u′

∂x+ ∂v

′

∂y

)= 0.

(3.114a,b,c)


The fact that v′ = 0 at x = 0 suggests that we look for a solution with v′ = 0everywhere, whence these equations become

∂u′

∂t= −g∂η

′

∂x, f0u′ = −g

∂η′

∂y,

∂η′

∂t+H∂u

′

∂x= 0. (3.115a,b,c)

Equations (3.115a) and (3.115c) lead to the standard wave equation

∂2u′

∂t2= c2 ∂2u′

∂x2 , (3.116)

where c =√gH, the usual wave speed of shallow water waves. The solution of

(3.116) isu′ = F1(x + ct,y)+ F2(x − ct,y), (3.117)

with corresponding surface displacement

η′ =√H/g

[−F1(x + ct,y)+ F2(x − ct,y)

]. (3.118)

The solution represents the superposition of two waves, one (F1) travelling in thenegative x-direction, and the other in the positive x-direction. To obtain the y-dependence of these functions we use (3.115b) which gives

∂F1

∂y= f0√

gHF1,

∂F2

∂y= − f0√

gHF2, (3.119)

with solutions

F1 = F(x + ct) ey/Ld F2 = G(x − ct) e−y/Ld , (3.120)

where Ld =√gH/f0 is the radius of deformation. The solution F1 grows exponen-

tially away from the wall, and so fails to satisfy the condition of boundedness atinfinity. It must be thus eliminated, leaving the general solution

u′ = e−y/LdG(x − ct), v′ = 0

η′ =√H/g e−y/LdG(x − ct)

. (3.121)

These are Kelvin waves, and they decay exponentially away from the boundary. Iff0 is positive, as in the Northern Hemisphere, the boundary is to the right of an ob-server moving with the wave. Given a constant Coriolis parameter, we could equallywell have obtained a solution on a meridional wall, in which case we would findthat the wave again moves such that the wall is to the right of the wave direction.(This is obvious once it is realized that f -plane dynamics are isotropic in x and y.).Thus, in the Northern Hemisphere the wave moves anticlockwise round a basin,and conversely in the Southern Hemisphere, and in both hemispheres the directionis cyclonic.

3.8 GEOSTROPHIC ADJUSTMENT

We noted in chapter 2 that the large-scale, extra-tropical circulation of the atmo-sphere is in near-geostrophic balance. Why is this? Why should the Rossby number

3.8 Geostrophic Adjustment 145

be small? Arguably, the magnitude of the velocity in the atmosphere and ocean isultimately given by the strength of the forcing, and so ultimately by the differentialheating between pole and equator (although even this argument is not satisfac-tory, since the forcing mainly determines the energy throughput, not directly theenergy itself, and the forcing is itself dependent on the atmosphere’s response). Buteven supposing that the velocity magnitudes are given, there is no a priori guaran-tee that the forcing or the dynamics will produce length-scales that are such thatthe Rossby number is small. However, there is in fact a powerful and ubiquitiousprocess whereby a fluid in an initially unbalanced state naturally evolves towarda state of geostrophic balance, namely geostrophic adjustment. This process occursquite generally in rotating fluids, stratified or not. To pose the problem in a simpleform we will consider the free evolution of a single shallow layer of fluid whose ini-tial state is manifestly unbalanced, and we will suppose that surface displacementsare small so that the evolution of the system is described by the linearized shallowequations of motion. These are

∂u∂t

+ f × u = −g∇η, ∂η∂t+H∇ · u = 0, (3.122a,b)

where η is the free surface displacement and H is the mean fluid depth, and weomit the primes on the linearized variables.

3.8.1 Non-rotating flow

We consider first the non-rotating problem set, with little loss of generality, in onedimension. We suppose that initially the fluid is at rest but with a simple disconti-nuity in the height field so that

η(x, t = 0) =

+η0 x < 0−η0 x > 0

(3.123)

and u(x, t = 0) = 0 everywhere. We can physically realize these initial conditionsby separating two fluid masses of different depths by a thin dividing wall, and thenquickly removing the wall. What is the subsequent evolution of the fluid? Thegeneral solution to the linear problem is given by (3.99) where the functional formis determined by the initial conditions so that here

F(x) = η(x, t = 0) = −η0sgn(x). (3.124)

Eq. (3.99) states that this initial pattern is propagated to the right and to the left.That is, two discontinuities in fluid height simply propagate to the right and left ata speed c =

√gH. Specifically, the solution is

η(x, t) = −12η0[sgn(x + ct)+ sgn(x − ct)] (3.125)

The initial conditions may be much more complex than a simple front, but, becausethe waves are dispersionless, the solution is still simply a sum of the translation ofthose initial conditions to the right and to the left at speed c. The velocity field inthis class of problem is obtained from

∂u∂t

= −g ∂η∂x, (3.126)


0

0 0

0

0

0

0

0

Perturbation height Velocity

t = 0

t = 1

t = 2

Fig. 3.9 The time development of an initial ‘top hat’ height disturbance, withzero initial velocity. Fronts propagate in both directions, and the velocity isnon-zero between fronts, but ultimately the velocity and height disturbanceare radiated away to infinity.

which gives, using (3.99),

u = − g2c[F(x + ct)− F(x − ct)]. (3.127)

Consider the case with initial conditions given by (3.123). At a given location,away from the initial disturbance, the fluid remains at rest and undisturbed untilthe front arrives. After the front has passed, the fluid surface is again undisturbedand the velocity is uniform and non zero. Specifically:

η =

−η0sgn(x)0

u =

0 |x| > ct(η0g/c) |x| < ct.

(3.128)

The solution with ‘top-hat’ initial conditions in the height field, and zero initialvelocity, is a superposition two discontinuities similar to (3.128) and is illustratedin Fig. 3.9. Two fronts propagate in either direction from each discontinuity and,in this case, the final velocity, as well as the fluid displacement, is zero after all thefronts have passed. That is, the disturbance is radiated completely away.

3.8.2 Rotating flow

Rotation makes a profound difference to the adjustment problem of the shallow wa-ter system, because a steady, adjusted, solution can exist with nonzero gradients inthe height field — the associated pressure gradients being balanced by the Coriolisforce — and potential vorticity conservation provides a powerful constraint on thefluid evolution.6 In a rotating shallow fluid that conservation is represented by

∂Q∂t

+ u · ∇Q = 0, (3.129)

where Q = (ζ + f)/h. In the linear case with constant Coriolis parameter (3.129)becomes

∂q∂t

= 0, q =(ζ − f0

ηH

). (3.130)


This equation may be obtained either from the linearized velocity and mass conser-vation equations, (3.122), or from (3.129) directly. In the latter case, we write

Q = ζ + f0

H + η ≈1H(ζ + f0)

(1− η

H

)≈ 1H

(f0 + ζ − f0

ηH

)= f0

H+ qH

(3.131)

having used f0 |ζ| and H |η|. The term f0/H is a constant and so dynami-cally unimportant, as is the H−1 factor multiplying q. Further, the advective termu · ∇Q becomes u · ∇q, and this is second order in perturbed quantities and so isneglected. Thus, making these approximations, (3.129) reduces to (3.130). Thepotential vorticity field is therefore fixed in space! Of course, this was also true inthe nonrotating case where the fluid is initially at rest. Then q = ζ = 0 and the fluidremains irrotational throughout the subsequent evolution of the flow. However, thisis rather a weak constraint on the subsequent evolution of the fluid; it does nothing,for example, to prevent the conversion of all the potential energy to kinetic energy.In the rotating case the potential vorticity is non-zero, and potential vorticity con-servation and geostrophic balance are all we need to infer the final steady state,assuming it exists, without solving for the details of the flow evolution, as we nowsee.

With an initial condition for the height field given by (3.123), the initial poten-tial vorticity is given by

q(x,y) =

−f0η0/H x < 0f0η0/H x > 0,

(3.132)

and this remains unchanged throughout the adjustment process. The final steadystate is then the solution of the equations

ζ − f0ηH= q(x,y), f0u = −g

∂η∂y, f0v = g

∂η∂x, (3.133a,b,c)

where ζ = ∂v/∂x−∂u/∂y . Because the Coriolis parameter is constant, the velocityfield is horizontally non-divergent and we may define a streamfunction ψ = gη/f0.Equations (3.133) then reduce to(

∇2 − 1L2d

)ψ = q(x,y), (3.134)

where Ld =√gH/f0 is known as the Rossby radius of deformation or often just the

‘deformation radius’ or the ‘Rossby radius’. It is a naturally occurring length-scalein problems involving both rotation and gravity, and arises in slightly different formin stratified fluids.

The initial conditions (3.132) admit of a nice analytic solution, for the flow willremain uniform in y, and (3.134) reduces to

∂2ψ∂x2 −

1L2dψ = −f0η0

Hsgn(x). (3.135)

We solve this separately for x > 0 and x < 0 and then match the solutions and their


−4 −2 0 2 4

−1

0

1

Fin

al H

eigh

t

−4 −2 0 2 4−1

−0.5

0

Vel

ocity

−4 −2 0 2 4

−1

0

1

Pot

. Vor

ticity

X/Ld

−4 −2 0 2 4

−1

0

1In

itial

Hei

ght

Fig. 3.10 Solutions of a linear geostrophic adjustment problem. Top panel:the initial height field, given by (3.123) with η0 = 1. Second panel: equi-librium (final) height field, η given by (3.136) and η = f0ψ/g. Third panel:equilibrium geostrophic velocity (normal to the gradient of height field), givenby (3.137). Bottom panel: potential vorticity, given by (3.132), and this doesnote evolve. The distance, x is non-dimensionalized by the deformation ra-dius Ld =

√gH/f0, and the velocity by η0(g/f0Ld). Changes to the initial

state occur only within O(Ld) of the initial discontinuity; and as x → ±∞ theinitial state is unaltered.

first derivatives at x = 0, imposing also the condition that the streamfunction decayto zero as x → ±∞. The solution is

ψ =

−(gη0/f0)(1− e−x/Ld) x > 0+(gη0/f0)(1− ex/Ld) x < 0.

(3.136)

The velocity field associated with this is obtained from (3.133b,c), and is

u = 0, v = − gη0

f0Lde−|x|/Ld . (3.137)

The velocity is perpendicular to the slope of the free surface, and a jet forms alongthe initial discontinuity, as illustrated in Fig. 3.10.

The important point of this problem is that the variations in the height and fieldare not radiated away to infinity, as in the non-rotating problem. Rather, potential


vorticity conservation constrains the influence of the adjustment to within a defor-mation radius (we see now why this name is appropriate) of the initial disturbance.This property is a general one in geostrophic adjusment — it also arises if the initialcondition consists of a velocity jump, as considered in problem 3.11.

3.8.3 * Energetics of adjustment

How much of the initial potential energy of the flow is lost to infinity by gravitywave radiation, and how much is converted to kinetic energy? The linear equations(3.122) lead to

12∂∂t(Hu2 + gη2)+ gH∇ · (uη) = 0, (3.138)

so that energy conservation holds in the form

E = 12

∫(Hu2 + gη2)dx,

dEdt

= 0, (3.139)

provided the integral of the divergence term vanishes, as it normally will in a closeddomain. The fluid has a non-zero potential energy, (1/2)

∫∞−∞ gη2 dx, if there are

variations in fluid height, and with the initial conditions (3.123) the initial potentialenergy is

PEI =∫∞

0gη2

0 dx. (3.140)

This is nominally infinite if the fluid has no boundaries, and the initial potentialenergy density is gη2

0/2 everywhere.In the non-rotating case, and with initial conditions (3.123), after the front

has passed, the potential energy density is zero and the kinetic energy density isHu2/2 = gη2

0/2, using (3.128) and c2 = gH. Thus, all the potential energy islocally converted to kinetic energy as the front passes, and eventually the kineticenergy is distributed uniformly along the line. In the case illustrated in Fig. 3.9,the potential energy and kinetic energy are both radiated away from the initialdisturbance. (Note that although we can superpose the solutions from differentinitial conditions, we cannot superpose their potential and kinetic energies.) Thegeneral point is that the evolution of the disturbance is not confined to its initiallocation.

In contrast, in the rotating case the conversion from potential to kinetic energyis largely confined to within a deformation radius of the initial disturbance, and atlocations far from the initial disturbance the initial state is essentially unaltered.The conservation of potential vorticity has prevented the complete conversion ofpotential energy to kinetic energy, a result that is not sensitive to the precise formof the initial conditions (see also problem 3.10).

In fact, in the rotating case, some of the initial potential energy is converted tokinetic energy, some remains as potential energy and some is lost to infinity; letus calculate these amounts. The final potential energy, after adjustment, is, using3.136,

PEF =12gη2

0

[∫∞0

(1− e−x/Ld

)2dx +

∫ 0

−∞

(1− ex/Ld

)2dx]

(3.141)


This is nominally infinite, but the change in potential energy is finite and is givenby

PEI − PEF = gη20

∫∞0(2 e−x/Ld − e−2x/Ld)dx = 3

2gη2

0Ld. (3.142)

The initial kinetic energy is zero, because the fluid is at rest, and its final value is,using (3.137),

KEF =12H∫u2 dx = H

(gη0

fLd

)2 ∫∞0

e−2x/Ld dx = gη20Ld2

. (3.143)

Thus one-third of the difference between the initial and final potential energies isconverted to kinetic energy, and this is trapped within a distance of order a defor-mation radius of the disturbance; the remainder, an amount gLdη2

0 is radiated awayto infinity. In any finite region surrounding the initial discontinuity the final energyis less than the initial energy.

3.8.4 * General initial conditions

Because of the linearity of the (linear) adjustment problem a spectral viewpoint isuseful, in which the fields are represented as the sum or integral of non-interactingFourier modes. For example, suppose that the height field of the initial disturbanceis a two-dimensional field given by

η(0) =∫∫ηk,l(0) ei(kx+ly) dkdl (3.144)

where the Fourier coefficients η(0)k,l are given, and the initial velocity field is zero.Then the initial (and final) potential vorticity field is given by

q = −f0

H

∫∫ηk,l(0) ei(kx+ly) dkdl. (3.145)

To obtain an expression for the final height and velocity fields, we express the po-tential vorticity field as

q =∫qk,l dkdl. (3.146)

The potential vorticity field does not evolve, and it is related to the initial heightfield by

qk,l = −f0

Hηk,l(0). (3.147)

In the final, geostrophically balanced, state, the potential vorticity is related to theheight field by

q = gf0∇2η− f0

Hη and qk,l =

(− gf0K2 − f0

H

)ηk,l, (3.148a,b)

where K2 = k2+ l2. Using (3.147) and (3.148), the Fourier components of the finalheight field satisfy (

− gf0K2 − f0

H

)ηk,l = −

f0

Hηk,l(0) (3.149)


or

ηk,l =ηk,l(0)K2L2

d + 1. (3.150)

In physical space the final height field is just the spectral integral of this, namely

η =∫∫ηk,l dkdl =

∫∫ηk,l(0)K2L2

d + 1dkdl. (3.151)

We see that at large scales (K2L2d 1) ηk,l is almost unchanged from its initial state;

the velocity field, which is then determined by geostrophic balance, thus adjusts tothe pre-existing height field. At large scales most of the energy in geostrophicallybalanced flow is potential energy; thus, it is energetically easier for the velocity tochange to come into balance with the height field than vice versa. At small scales,however, the final height field has much less variability than it did initially.

Conversely, at small scales the height field adjusts to the velocity field. To seethis, let us suppose that the initial conditions contain vorticity but have zero heightdisplacement. Specifically, if the initial vorticity is ∇2ψ(0), where ψ(0) is the initialstreamfunction, then it is straightforward to show that the final streamfunction isgiven by

ψ =∫∫ψk,l dkdl =

∫∫ K2L2dψk,l(0)

K2L2d + 1

dkdl. (3.152)

The final height field then obtained from this, via geostrophic balance, by η =(f0/g)ψ. Evidently, for small scales (K2L2

d 1) the streamfunction, and hencethe vortical component of the velocity field, are almost unaltered from their initialvalues. On the other hand, at large scales the final streamfunction has much lessvariability than it does initially, and so the height field will be largely governed bywhatever variation it (and not the velocity field) had initially. In general, the finalstate is a superposition of the states given by (3.151) and (3.152). The divergentcomponent of the initial velocity field does not affect the final state because it hasno potential vorticity, and so all of the associated energy is lost to infinity.

Finally, we remark that just as in the problem with a discontinuous initial heightprofile the change in total energy during adjustment is negative — this can be seenfrom the form of the integrals above, although we leave the specifics as a problemto the reader. That is, some of the initial potential and kinetic energy is lost toinfinity, but some is trapped by the potential vorticity constraint.

3.8.5 A variational perspective

In the non-rotating problem, all of the initial potential energy is eventually radiatedaway to infinity. In the rotating problem, the final state contains both potential andkinetic energy. Why is the energy not all radiated away to infinity? It is because po-tential vorticity conservation on parcels prevents all of the energy being dispersed.This suggests that it may be informative to think of the geostrophic adjustmentproblem as a variational problem: we seek to minimize the energy consistent withthe conservation of potential vorticity. We stay in the linear approximation in which,because the advection of potential vorticity is neglected, potential vorticity remainsconstant at each point.


The energy of the flow is given by the sum of potential and kinetic energies,namely

Energy =∫H(u2 + gη2)dA, (3.153)

(where dA ≡ dx dy) and the potential vorticity field is

q = ζ − f ηH. (3.154)

The problem is then to extremize the energy subject to potential vorticity conserva-tion. This is a constrained problem in the calculus of variations, sometimes calledan isoperimetric problem because of its origins in maximizing the area of a surfacefor a given perimeter.7 The mathematical problem is to extremize the integral

I =∫ H(u2 + v2)+ gη2 + λ(x,y)

[(vx −uy)− f0η/H

]dA. (3.155)

where λ(x,y) is a Lagrange multiplier, undetermined at this stage. It is a functionof space: if it were a constant, the integral would merely extremize energy subjectto a given integral of potential vorticity, and rearrangements of potential vorticity(which here we wish to disallow) would leave the integral unaltered.

As there are three independent variables there are three Euler-Lagrange equa-tions that must be solved in order to minimize I. These are

∂L∂h

− ∂∂x

∂L∂hx

− ∂∂y

∂L∂hy

= 0,

∂L∂u

− ∂∂x

∂L∂ux

− ∂∂y

∂L∂uy

= 0,∂L∂v

− ∂∂x

∂L∂vx

− ∂∂y

∂L∂vy

= 0,(3.156)

where L is the integrand on the right-hand side of (3.155). Substituting this into(3.156) gives, after a little algebra,

2gη− λf0 = 0, 2u+ ∂λ∂y

= 0, 2v − ∂λ∂x

= 0, (3.157)

and then eliminating λ gives the simple relationships

u = − gf0

∂η∂y, v = g

f0

∂η∂x, (3.158)

which are the equations of geostrophic balance. Thus, in the linear approximation,geostrophic balance is a minimum energy state for a given field of potential vorticity.

3.9 ISENTROPIC COORDINATES

We now return to the continuously stratified primitive equations, and consider theuse of potential density as a vertical coordinate. In practice this means using poten-tial temperature in the atmosphere or buoyancy (density) in the ocean; such coordi-nate systems are generically called isentropic coordinates, and sometimes isopycnalcoordinates if density is used. This may seem an odd thing to do but for adiabaticflow in particular the resulting equations of motion have an attractive form that aids

3.9 Isentropic Coordinates 153

the interpretation of large-scale flow. The thermodynamic equation then becomesa statement for the conservation of the mass of fluid with a given value of poten-tial density and, because the the flow of both the atmosphere and ocean is largelyalong isentropic surfaces, the momentum and vorticity equations have quasi-two-dimensional form.

The particular choice of vertical coordinate is determined by the form of thethermodynamic equation in the equation-set at hand; thus, if the thermodynamicequation is Dθ/Dt = θ, we transform the equations from (x,y, z) coordinates to(x,y, θ) coordinates. The material derivative in this coordinate system is

DDt

= ∂∂t+u

(∂∂x

)θ+ v

(∂∂y

)θ+ Dθ

Dt∂∂θ

= ∂∂t+ u · ∇θ + θ

∂∂θ,

(3.159)

where the last term on the right-hand side is zero for adiabatic flow, and the two-dimensional velocity u ≡ (u,v) is parallel to the isentropes.

3.9.1 A hydrostatic Boussinesq fluid

In the simple Boussinesq equations (see the table on page 72) the buoyancy isthe relevant thermodynamic variable. With hydrostatic balance the horizontal andvertical momentum equations are, in height coordinates,

DuDt

+ f × u = −∇φ, b = ∂φ∂z, (3.160)

where b is the buoyancy, the variable analogous to the potential temperature θ ofan ideal gas. The thermodynamic equation is

DbDt

= b, (3.161)

and because b = −gδρ/ρ0, isentropic coordinates are the same as isopycnal coor-dinates.

Using (2.143) the horizontal pressure gradient may be transformed to isentropiccoordinates(

∂φ∂x

)z=(∂φ∂x

)b−(∂z∂x

)b

∂φ∂z

=(∂φ∂x

)b− b

(∂z∂x

)b=(∂M∂x

)b, (3.162)

whereM ≡ φ− zb. (3.163)

Thus, the horizontal momentum equation becomes

DuDt

+ f × u = −∇bM. (3.164)

where the material derivative is given by (3.159), with b replacing θ. Using (3.163)the hydrostatic equation becomes

∂M∂b

= −z. (3.165)


The mass continuity equation may be derived by noting that the mass elementmay be written as

δm = ∂z∂bδbδxδy. (3.166)

The mass continuity equation, Dδm/Dt = 0, becomes

DDt∂z∂b

+ ∂z∂b∇3 · v = 0, (3.167)

where ∇3 ·v = ∇b ·u+ ∂b/∂b is the three dimensional derivative of the velocity inisentropic coordinates. Eq. (3.167) may thus be written

DσDt

+ σ∇b · u = −σ∂b∂b, (3.168)

where σ ≡ ∂z/∂b is a measure of the thickness between two isentropic surfaces.Equations (3.164), (3.165) and (3.168) comprise a closed set, with dependent vari-ables u, M and z in the space of independent variables x, y and b.

3.9.2 A hydrostatic ideal gas

Deriving the equations of motion for this system requires a little more work than inthe Boussinesq case but the idea is the same. For an ideal gas in hydrostatic balancewe have, using (1.112),

δθθ= δTT− κ δp

p= δTT+ δΦcpT

= 1cpT

δM, (3.169)

whereM ≡ cpT+Φ is the ‘Montgomery potential’, equal to the dry static energy. (Weuse some of the same symbols as in the Boussinesq case to facilitate comparison,but their meanings are slightly different.) From this

∂M∂θ

= Π, (3.170)

where Π ≡ cpT/θ = cp(p/pR)R/cp is the ‘Exner function’. Equation (3.170) repre-sents the hydrostatic relation in isentropic coordinates. Note also that M = θΠ + Φ.

To obtain the an appropriate form for the horizontal pressure gradient first notethat, in the usual height coordinates, it is given by

1ρ∇zp = θ∇zΠ, (3.171)

where Π = cpT/θ. Using (2.143) gives

θ∇zΠ = θ∇θΠ −θg∂Π∂z∇θΦ. (3.172)

Then, using the definition of Π and the hydrostatic approximation to help evaluatethe vertical derivative, we obtain

1ρ∇zp = cp∇θT +∇θΦ = ∇θM. (3.173)

3.10 Available Potential Energy 155

Thus, the horizontal momentum equation is

DvDt

+ f × u = −∇θM. (3.174)

As in the Boussinesq case the mass continuity equation may be derived by notingthat the mass element may be written as

δm = ∂p∂θδθδxδy. (3.175)

The mass continuity equation, Dδm/Dt = 0, becomes

DDt∂p∂θ

+ ∂p∂θ∇3 · v = 0 (3.176)

orDσDt

+ σ∇θ · u = −σ∂θ∂θ, (3.177)

where now σ ≡ ∂p/∂θ is a measure of the (pressure) thickness between two isen-tropic surfaces. Equations (3.170), (3.174) and (3.177) form a closed set, analo-gous to (3.165), (3.164) and (3.168).

3.9.3 Analogy to shallow water equations

The equations of motion in isentropic coordinates have an obvious analogy withthe shallow water equations, and we may think of the shallow water equations tobe a finite-difference representation of the primitive equations written in isentropiccoordinates, or think of the latter as the continuous limit of the shallow water equa-tions as the number of layers increases. For example, consider a two-isentropic-levelrepresentation of (3.170), (3.174) and (3.177), in which the lower boundary is anisentrope. A natural finite differencing gives

−M1 = −z0∆θ0 (3.178a)

M1 −M2 = −z1∆θ1, (3.178b)

and the momentum equations for each layer become

Du1

Dt+ f × u1 = −∆θ0∇z0 (3.179a)

Du2

Dt+ f × u2 = −∆θ0∇z0 −∆θ1∇z1. (3.179b)

Together with the mass continuity equation for each level these are just like thetwo-layer shallow water equations (3.51). This means that results that one mighteasily derive for the shallow water equations will often have a continuous analog.

3.10 AVAILABLE POTENTIAL ENERGY

We now revisit the issue of the internal and potential energy in stratified flow, mo-tivated by the following remarks. In adiabatic, inviscid flow the total amount of


Fig. 3.11 If a stably stratified initial state with sloping isentropes (left) isadiabatically re-arranged then the state of minimum potential energy hasflat isentropes, as on the right, but the amount of fluid contained betweeneach isentropic surface is unchanged. The difference between the potentialenergies of the two states is the available potential energy.

energy is conserved, and there are conversions between internal energy, potentialenergy and kinetic energy. In an ideal gas the potential energy and the internalenergy of a column extending throughout the atmosphere are in a constant ratio toeach other — their sum is called the total potential energy. In a simple Boussinesqfluid, energetic conversions involve only the potential and kinetic energy, and notthe internal energy. Yet, plainly, in neither a Boussinesq fluid nor an ideal gas canall the total potential energy in a fluid be converted to kinetic energy, for then allof the fluid would be adjacent to the ground and the fluid would have no thickness,which intuitively seems impossible. Given a state of the atmosphere or ocean, howmuch of its total potential energy is available for conversion to kinetic energy? Inparticular, because total energy is conserved only in adiabatic flow, we may usefullyask: how much potential energy is available for conversion to kinetic energy underan adiabatic re-arrangement of fluid parcels?

Suppose that at any given time the flow is stably stratified, but that the isen-tropes (or more generally the surfaces of constant potential density) are sloping,as in Fig. 3.11. The potential energy of the system would be reduced if the isen-tropes were flattened, for then heavier fluid would be moved to lower altitudes,with lighter fluid replacing it at higher altitudes. In an adiabatic re-arrangementthe amount of fluid between the isentropes would remain constant, and a statewith flat isentropes (meaning parallel to the geopotential surfaces) evidently con-stitutes a state of minimum total potential energy. The difference between the totalpotential energy of the fluid and the total potential energy after an adiabatic re-arrangement to a state in which the isentropic surfaces are flat is called the availablepotential energy, or APE.8


3.10.1 A Boussinesq fluid

The potential energy of a column of Boussinesq fluid of unit area is given by

P =∫ H

0bz dz =

∫ H0

b2

dz2. (3.180)

and the potential energy of the entire fluid is given by the horizontal integral of this.The minimum potential energy of the fluid arises after an adiabatic re-arrangementin which the isopycnals are flattened, and the resulting buoyancy is only a functionof z. The available potential energy is then the difference between the energy of theinitial state and of this minimum state, and to obtain an approximate expression forthis we first integrate (3.180) by parts to give

P = −∫ bm

0z2 db, (3.181)

where bm is the maximum value of b in the column. (We omit a constant of in-tegration that cancels when the state of minimum potential energy is subtracted.Alternatively, take the upper limit of the z-integral to be z = 0 and at the lowerlimit, at z = −H say, take b = 0.) The minimum potential energy state ariseswhen z is a function only of b, z = Z(b) say. Because mass is conserved in there-arrangement, Z is equal to the horizontally averaged value of z on a given isopy-cnal surface, z, and the surfaces z and b thus define each other completely. Theaverage available potential energy, per unit area, is then given by

APE =∫ bm

0(z2 − z2)db =

∫ bm0z′2 db, (3.182)

where z = z + z′; that is z′ is the height variation of an isopycnal surface. Theavailable potential energy is thus proportional to the integral of the variance of thealtitude of such a surface, and it is a positive-definite quantity. To obtain an expres-sion in z-coordinates, we express the height variations on an isopycnal surface interms of buoyancy variations on a constant-height surface by Taylor-expanding theheight about its value on the isopycnal surface. Referring to Fig. 3.12 this gives

z(b) = z + ∂z∂b

∣∣∣∣b=b

[b − b(z)] = z − ∂z∂b

∣∣∣∣b=b

b′ (3.183)

where b′ = b(z) − b is corresponding buoyancy perturbation on the z surface andb is the average value of b on the z surface. Furthermore, ∂z/∂b|z=b ≈ ∂z/∂b ≈(∂b/∂z)−1, and (3.183) thus becomes

z′ = z(b)− z ≈ −b′(∂z∂b

)≈ − b′

(∂b/∂z). (3.184)

where z′ = z(b) − z is the height perturbation of the isopycnal surface, from itsaverage value. Using (3.184) in (3.182) we obtain an expression for the APE perunit area, to wit

APE ≈∫ H

0

b′2

∂b/∂zdz . (3.185)


Fig. 3.12 An isopycnal surface, b = b, and the constant height surface, z = z.z is the height of the isopycnal surface after a re-arrangement to a minimumpotential energy state, equal to the average height of the isopycnal surface.The values of z on the isopycnal surface, and of b on the constant heightsurface, can be obtained by the Taylor expansions shown. For an ideal gas inpressure coordinates, replace z by p and b by θ.

The total APE of the fluid is the horizontal integral of this, and is thus proportionalto the variance of the buoyancy on a height surface. We emphasize that APE is notdefined for single column of fluid, for it depends on the variations of buoyancy overa horizontal surface. Note too that this derivation neglects the effects of topogra-phy; this, and the use of a basic state stratification, effectively restrict the use of(3.185) to a single ocean basin, and even for that the approximations used limit theaccuracy of the expressions.

3.10.2 An ideal gas

The expression for the APE for an ideal gas is obtained, mutatis mutandis, in thesame way as it was for a Boussinesq fluid and the trusting reader may skip directlyto (3.193). The internal energy of an ideal gas column of unit area is given by

I =∫∞

0cvTρ dz =

∫ ps0

cvgT dp, (3.186)

where ps is the surface pressure, and the corresponding potential energy is givenby

P =∫∞

0ρgz dz =

∫ ps0z dp =

∫∞0p dz =

∫ ps0

RgT dp. (3.187)

In (3.186) we use hydrostasy, and in (3.187) the equalities make successive use ofhydrostasy, an integration by parts, and hydrostasy and the ideal gas relation. Thus,the total potential energy is given by

TPE ≡ I + P = cpg

∫ ps0T dp. (3.188)


Using the ideal gas equation of state we can write this as

TPE = cpg

∫ ps0

(pps

)κθ dp = cpps

g(1+ κ)

∫∞0

(pps

)κ+1

dθ, (3.189)

after an integration by parts. (We omit a term proportional to pκ+1s θs that arises

in the integration by parts, because it plays no role in what follows.) The totalpotential energy of the entire fluid is equal to a horizontal integral of (3.189). Theminimum total potential energy arises when the pressure in (3.189) a functiononly of θ, p = P(θ), where by conservation of mass P is the average value of theoriginal pressure on the isentropic surface, P = p. The average available potentialenergy per unit area is then given by the difference between the initial state andthis minimum, namely

APE = cppsg(1+ κ)

∫∞0

( pps

)κ+1

−(pps

)κ+1 dθ, (3.190)

which is a positive definite quantity. A useful approximation to this is obtained byexpressing the right-hand side in terms of the variance of the potential temperatureon a pressure surface. We first use the binomial expansion to expand pκ+1 = (p +p′)κ+1. Neglecting third and higher order terms (3.190) becomes

APE = Rps2g

∫∞0

(pps

)κ+1 (p′

p

)2

dθ. (3.191)

The variable p′ = p(θ)− p is a pressure perturbation on an isentropic surface, andis related to the potential temperature perturbation on an isobaric surface by [c.f.,(3.184)]

p′ ≈ −θ′ ∂p∂θ

≈ − θ′

∂θ/∂p. (3.192)

where θ′ = θ(p)− θ(p) is the potential temperature perturbation on the p surface.Using (3.192) in (3.191) we finally obtain

APE = Rp−κs

2

∫ ps0pκ−1

(−g ∂θ∂p

)−1

θ′2 dp . (3.193)

The APE is thus proportional to the variance of the potential temperature on thepressure surface or, from (3.191), proportional to the variance of the pressure onan isentropic surface.

3.10.3 Use, interpretation, and the atmosphere and ocean

The potential energy of a fluid is reduced when the dynamics act to flatten theisentropes. Consider, for example, the earth’s atmosphere, with isentropes slopingupward toward the pole (Fig. 3.11 with the pole on the right). Flattening these isen-tropes amounts to a sinking of dense air and a rising of light air, and this reductionof potential energy leads to a corresponding production of kinetic energy. Thus, if


the dynamics is such as to reduce the temperature gradient between equator andpole by flattening the isentropes then APE is converted to KE by that proccess. Astastistically steady state is achieved because the heating from the sun continuallyacts to restore the horizontal temperature gradient between equator and pole, soreplenishing the pool of APE, and to this extent the large-scale atmospheric circula-tion acts like a heat engine.

It is a useful exercise to calculate the total potential energy of the atmosphereand ocean, the available potential energy and the kinetic energy (problem 3.15).One finds

TPE APE > KE (3.194)

with, very approximately, TPE ∼ 100 APE and APE ∼ 10 KE. The first inequalityshould not surprise us (for it was this that lead us to define APE in the first in-stance), but the second is not obvious (and in fact the ratio is larger in the ocean).It is related to the fact that the instabilities of the atmosphere and ocean occur ata scale smaller than the size of the domain, and are unable to release all the po-tential energy that might be available. Understanding this more fully is the topic ofchapters 6 and 9.

Notes

1 The algorithm to numerically solve these equations differs from that of the free-surface shallow water equations because the mass conservation equation can nolonger be stepped forward in time. Rather, an elliptic equation for plid must bederived by eliminating time derivatives from from (3.22) using (3.21), and this thensolved at each timestep.

2 After Margules (1903). Margules sought to relate the energy of fronts to their slope.In this same paper the notion of available potential energy arose.

3 The expression ‘form drag’ is also commonly used in aerodynamics, and the twousages are related. In aerodynamics, form drag is the force due to pressure differ-ence between the front and rear of an object, or any other ‘form’, moving througha fluid. Aerodynamic form drag may include frictional effects between the wind andthe surface itself, but this effect is omitted in most geophysical uses.

4 (Jules) Henri Poincaré (1854–1912) was a prodigious French mathematician, physi-cist and philosopher, regarded as one the greatest mathematicians living at the turnof the 20th century. He is remembered for his original work in algebra and topol-ogy, and in dynamical systems and celestial mechanics, obtaining many results inwhat would be called non-linear dynamics and chaos when these fields re-emergedsome 60 years later — the notion of ‘sensitive dependence on initial conditions’, forexample, is present in his work. He also obtained a number of the results of specialrelativity independently of Einstein, and worked on the theory of rotating fluids —hence the Poincaré waves of this chapter. He wrote extensively and successfully forthe general public on the meaning, importance and philosophy of science. Amongother things he discussed whether scientific knowledge was an arbitrary convention,a notion that remains discussed and controversial to this day. (His answer: ‘con-vention’, in part, yes; ‘arbitrary’, no.) He was a proponent of the role of intuitionin mathematical and scientific progress, and did not believe that mathematics couldever be wholly reduced to pure logic.


5 Thomson (1869). See Gill (1982) or Philander (1990) for more discussion.

6 As considered by Rossby (1938).

7 An introduction to variational problems may be found in Weinstock (1952) and anumber of other textbooks. Applications to many traditional problems in mechanicsare discussed by Lanczos (1970).

8 Margules (1903) introduced the concept of potential energy that is available forconversion to kinetic energy, Lorenz (1955) clarified its meaning and derived useful,approximate formulae for its computation. Shepherd (1993) showed that the APE isjust the non-kinetic part of the pseudo-energy, an interpretation that naturally leadsto a number of extensions of the concept. There are a host of other papers on thesubject, including that of Huang (1998) who looked at some of the limitations of theapproximate expressions in an oceanic context.

Further Reading

Gill, A. E., 1982. Atmosphere-Ocean Dynamics.This remains a good reference for geostrophic adjustment and gravity waves. Thetime-dependent geostrophic adjustment problem is discussed in section 7.3.

Problems

3.1 Derive the appropriate shallow water equations for a single moving layer of fluid ofdensity ρ1 above a rigid floor, and where above the moving fluid is a stationary fluidof density ρ0, where ρ0 < ρ1. Show that as (ρ0/ρ1) → 0 the usual shallow waterequations emerge.

3.2 (a) Model the atmosphere as two immiscible, ‘shallow water’ fluids of different den-sity stacked one above the other. Using reasonable values for the values of anyneeded physical parameters, estimate the displacement of the interfacial surfaceassociated with a pole–equator temperature gradient of 60 K.

(b) Similarly estimate an interfacial displacement in the ocean associated with a tem-perature gradient of 20 K over a distance of 4000 km. (This is a crude represen-tation of the main oceanic thermocline.)

3.3 For a shallow water fluid the energy equation, (3.90), has the form ∂E/∂t + ∇ ·(v(E + gh2/2) = 0. But for a compressible fluid, the corresponding energy equation,(1.190), has the form ∂E/∂t +∇ · (v(E +p) = 0. In a shallow water fluid, p ≠ gh2/2at a point so these equations are superficially different. Explain this and reconcile thetwo forms. (Hint: What is the average pressure in a fluid column?)

3.4 Can the shallow water equations for an incompressible fluid be derived by way ofan asymptotic expansion in the aspect ratio? If so, do it. That is, without assuminghydrostasy ab initio, expand the Euler equations with a free surface in small paramterequal to the ratio of the depth of the fluid to the horizontal scale of the motion, andso obtain the shallow water equations.

3.5 The inviscid shallow water equations, rotating or not, can support gravity waves ofarbitrarily short wavelengths. For sufficiently high wavenumber, the wavelength willbe shorter than the depth of the fluid. Is this consistent with an asymptotic nature ofthe shallow water equations? Discuss.

3.6 Show that the vertical velocity within a shallow-water system is given by

w = z − ηbh

DhDt

+ DηbDt. (P3.1)

Interpret this result, showing that it gives sensible answers at the top and bottom ofthe fluid layer.


3.7 What is the appropriate generalization of (3.99) to two-dimensions? Suppose that attime t = 0 the height field is given by a Gaussian distribution h′ = A e−r2/σ2

wherer 2 = x2 + y2. What is the subsequent evolution of this, in the linear approximation?Show that the distribution remains Gaussian, and that its width increases at speed√gH, where H is the mean depth of the fluid.

3.8 In an adiabatic shallow water fluid in a rotating reference frame show that the poten-tial vorticity conservation law is

DDtζ + fη− hb

= 0 (P3.2)

where η is the height of the free surface and hb is the height of the bottom topogra-phy, both referenced to the same flat surface.

(a) A cylindrical column of air at 30° latitude with radius 100 km expands horizontallyto twice its original radius. If the air is initially at rest, what is the mean tangentialvelocity at the perimeter after the expansion.

(b) An air column at 60° N with zero relative vorticity (ζ = 0) stretches from thesurface to the tropopause, which we assume is a rigid lid, at 10 km. The aircolumn moves zonally onto a plateau 2.5 km high. What is its relative vorticity?Suppose it then moves southward to 30° N. What is its vorticity? (Assume densityis constant.)

3.9 In the long-wave limit of Poincaré waves, fluid parcels behave as free-agents; that is,like free solid particles moving in a rotating frame unencumbered by pressure forces.Why then, is their frequency given by ω = f = 2Ω where Ω is the rotation rate ofthe coordinate system, and not by Ω itself? Do particles that are stationary or movein a straight line in the inertial frame of reference satisfy the dispersion relationshipfor Poincaré waves in this limit? Explain. (See also Durran 1993, Egger 1999, Phillips2000)

3.10 Linearize the f -plane shallow-water system about a state of rest. Suppose that thereis an initial disturbance that is given in the general form

η =∫∫ηk,l ei(kx+ly) dkdl (P3.3)

where η is the deviation surface height and the Fourier coefficients ηk,l are given, andthat the initial velocity is zero.

(a) Obtain the geopotential field at the completion of geostrophic adjustment, andshow that the deformation scale is a natural length scale in the problem.

(b) Show that the change in total energy during the adjustment is always less than orequal to zero. Neglect any initial divergence.N.B. Because the problem is linear, the Fourier modes do not interact.

3.11 Geostrophic adjustment of a velocity jumpConsider the evolution of the linearized f -plane shallow water equations in an infinitedomain. Suppose that initially the fluid surface is flat, the zonal velocity is zero andthe meridional velocity is given by

v(x) = v0sgn(x) (P3.4)

(a) Find the equilibrium height and velocity fields at t = ∞.(b) What are the initial and final kinetic and potential energies?

Partial Solution:The potential vorticty is q = ζ − f0η/H, so that the initial and final state is

q = 2v0δ(x). (P3.5)


(Why?) The final state streamfunction is thus given by(∂2/∂x2 − L−2

d

)ψ = q, with

solution ψ = ψ0 exp(x/Ld) and ψ = ψ0 exp(−x/Ld) for x < 0 and x > 0, whereψ0 = Ldv0 (why?), and η = f0ψ/g. The energy is E =

∫(Hv2 + gη2)/2 dx. The initial

KE is infinite, the initial PE is zero, and the final state has PE = KE = gLdη20/4 — that

is, the energy is equipartitioned between kinetic and potential.

3.12 In the shallow water equations show that, if the flow is approximately geostrophi-cally balanced, the energy at large scales is predominantly potential energy and thatenergy at small scales is predominantly kinetic energy. Define precisely what ‘largescale’ and ‘small scale’ mean in this context.

3.13 In the shallow-water geostrophic adjustment problem, show that at large scales thevelocity adjusts to the height field, and that at small scales the height field adjusts tothe velocity field.

3.14 Consider the problem of minimizing the full energy [i.e.,∫(hu2)+ gη2)dx], given

the potential vorticity field q(x,y) = (ζ + f)/h. Show that the balance relationsanalogous to (3.8.5) are uh = −∂(Bq−1)/∂y and vh = ∂(Bq−1)/∂x where B is theBernoulli function B = gη+ u2/2. Show that steady flow does not necessarily satisfythese equations. Discuss.

3.15 Using realistic values for temperature, velocity etc., calculate approximate values forthe total potential energy, the available potential energy, and the kinetic energy, ofeither a hemisphere in the atmosphere or an ocean basin.

All real fluid motions are rotational.Clifford Truesdell, The Kinematics of Vorticity, 1954.

CHAPTER

FOUR

Vorticity and Potential Vorticity

VORTICITY AND POTENTIAL VORTICITY both play a central role in geophysical fluiddynamics — indeed, we shall find that the large scale circulation of theocean and atmosphere is in large-part governed by the evolution of the lat-

ter. In this chapter we define and discuss these quantities and deduce some of theirdynamical properties. Along the way we will come across Kelvin’s circulation theo-rem, one of the most fundamental conservation laws in all of fluid mechanics, andwe will find that the conservation of potential vorticity is intimately tied to this.

4.1 VORTICITY AND CIRCULATION

4.1.1 Preliminaries

Vorticity, ω, is defined to be the curl of velocity and so is given by

ω ≡ ∇× v. (4.1)

Circulation, C, is defined to be the integral of velocity around a closed fluid loopand so is given by

C ≡∮v · dl =

∫Sω · dS (4.2)

where the second expression uses Stokes’ theorem, where S is any surface boundedby the loop. The circulation around some path is equal to the integral of the normalcomponent of vorticity over any surface bounded by that path. The circulation isnot a field like vorticity and velocity; rather, we think of the circulation around aparticular material line of finite length, and so its value generally depends on thepath chosen. If δS is an infinitesimal surface element whose normal points in the

165

166 Chapter 4. Vorticity and Potential Vorticity

direction of the unit vector n, then

n · (∇× v) = 1δS

∮δlv · dl (4.3)

where the line integral is around the infinitesimal area. Thus at a point the compo-nent of vorticity in the direction of n is proportional to the circulation around thesurrounding infinitesimal fluid element, divided by the elemental area bounded bythe path of the integral. A heuristic test for the presence of vorticity is to imagine asmall paddle wheel in the flow; the paddle wheel acts as a ‘circulation-meter’, androtates if the vorticity is non-zero.

4.1.2 Simple axisymmetric examples

Consider axisymmetric motion in two dimensions, so that the flow is confined toa plane. We use cylindrical coordinates (r ,φ, z) where z is the direction perpen-dicular to the plane, with velocity components (ur , uφ, uz). For axisymmetric flowuz = ur = 0 but uφ ≠ 0.

Rigid Body Motion

The velocity distribution is given by

uφ = Ωr (4.4)

where Ω is the angular velocity of the fluid. Associated with this is the vorticity

ω = ∇× v =ωzk, (4.5)

whereωz = 1

r∂∂r(ruφ) = 1

r∂∂r(r 2Ω) = 2Ω. (4.6)

The vorticity of a fluid in solid body rotation is thus twice the angular velocity ofthe fluid, and is pointed in a direction orthogonal to the plane of rotation.

The ‘vr ’ vortex

This vortex is so-called because the tangential velocity (historically denoted ‘v ’ inthis context) is such that the product vr is constant. In our notation we would have

uφ = Kr, (4.7)

where K is a constant determining the vortex strength. Evaluating the z-componentof vorticity gives

ωz = 1r∂∂r(ruφ) = 1

r∂∂r

(rKr

)= 0, (4.8)

except where r = 0, at which the expression is singular and the vorticity is infinite.Our paddle wheel rotates when placed at the vortex center, but, less obviously, doesnot if placed elsewhere.

The circulation around a circle that encloses the origin is given by

C =∮Krr dφ = 2πK. (4.9)

4.2 The Vorticity Equation 167

Figure 4.1 Evaluation of circulation in the axi-symmetric vr vortex. The circulation aroundpath A–B–C–D is zero. This result does not de-pend on the radii r1 or r2 or the angle φ, andthe circulation around any infinitesimal path notenclosing the origin is zero. Thus the vorticityis zero everywhere except at the origin.

This does not depend on the radius, and so it is true if the radius is infinitesimal.Since the vorticity is the circulation divided by the area, the vorticity at the originmust be infinite. Consider now an integration path that does not enclose the origin,for example the contour A–B–C–D–A in Fig. 4.1. Over the segments A–B and C–Dthe velocity is orthogonal to the contour, and so the contribution is zero. Over B–Cand D–A we have

CBC =Kr2θr2 = Kφ, CDA = −

Kr1θr1 = −Kφ. (4.10)

Adding these two expressions we see that net circulation around the contour CABCDA

is zero. If we shrink the integration path to an infinitesimal size then, within thepath, by Stokes’ theorem, the vorticity is zero. We can of course place the pathanywhere we wish, except surrounding the origin, and obtain this result. Thus thevorticity is everywhere zero, except at the origin.

4.2 THE VORTICITY EQUATION

Using the vector identity v×(∇×v) = ∇(v·v)/2−(v·∇)v, we write the momentumequation as

∂v∂t

+ω× v = −1ρ∇p − 1

2∇v2 + F, (4.11)

where F represents viscous or external forces. Taking the curl of (4.11) gives thevorticity equation

∂ω∂t

+∇× (ω× v) = 1ρ2 (∇ρ ×∇p)+∇× F. (4.12)

Now, the vector identity

∇× (a× b) = (b · ∇)a− (a · ∇)b+ a∇ · b− b∇ · a, (4.13)

implies that the second term on the left hand side of (4.12) may be written

∇× (ω× v) = (v · ∇)ω− (ω · ∇)v +ω∇ · v − v∇ ·ω. (4.14)


Because vorticity is the curl of velocity its divergence vanishes and so (4.12) be-comes

∂ω∂t

+ (v · ∇)ω = (ω · ∇)v −ω∇ · v + 1ρ2 (∇ρ ×∇p)+∇× F. (4.15)

The divergence term may be eliminated with the aid of the mass-conservation equa-tion to give

DωDt

= (ω · ∇)v + 1ρ3 (∇ρ ×∇p)+

1ρ∇× F . (4.16)

where ω ≡ω/ρ. We will set F = 0 in most of what follows.The third term on the right-hand side of (4.15), as well as the second term

on the right-hand side of (4.16), is variously called the baroclinic term, the non-homentropic term, or the solenoidal term. The solenoidal vector is defined by

S ≡ 1ρ2∇ρ ×∇p = −∇α×∇p (4.17)

A solenoid is a tube directed perpendicular to both ∇α and ∇p, with elements oflength proportional to ∇p × ∇α. If the isolines of p and α are parallel to eachother, then solenoids do not exist. This occurs when the density is a function onlyof pressure for then

∇ρ ×∇p = ∇ρ ×∇ρ dpdρ

= 0. (4.18)

The solenoidal vector may also be written

S = −∇η×∇T . (4.19)

This follows most easily by first writing the momentum equation in the form ∂v/∂t+ω× v = T∇η−∇B, and taking its curl (see problem 2.2). Evidently the solenoidalterm vanishes if: (i) isolines of pressure and density are parallel; (ii) isolines oftemperature and entropy are parallel; (ii) density or entropy or temperature orpressure are constant. A barotropic fluid has by definition p = p(ρ) and thereforeno solenoids. A baroclinic fluid is one for which ∇p is not parallel to ∇ρ. From(4.16) we see that the baroclinic term must be balanced by terms involving velocityor its tendency and therefore, in general, a baroclinic fluid is a moving fluid, even inthe presence of viscosity.

For a barotropic fluid the vorticity equation takes the simple form,

DωDt

= (ω · ∇)v. (4.20)

If the fluid is also incompressible, meaning that ∇ · v = 0, then we have the evensimpler form,

DωDt

= (ω · ∇)v. (4.21)

The terms on the right-hand side of (4.20) or (4.21) are conventionally divided into‘stretching’ and ‘tipping’ (or ‘tilting’) terms, and we return to these in section 4.3.1.

4.2 The Vorticity Equation 169

An integral conservation property

Consider a single Cartesian component in (4.15). Then, using superscripts to denotecomponents,

∂ωx

∂t= −v · ∇ωx −ωx∇ · v + (ω · ∇)vx + Sx

= −∇ · (vωx)+∇ · (ωvx)+ Sx ,(4.22)

where Sx is the (x-component of) the solenoidal term. Eq. (4.22) may be writtenas

∂ωx

∂t+∇ · (vωx −ωvx) = Sx , (4.23)

and this implies the Cartesian tensor form of the vorticity equation, namely

∂ωi∂t

+ ∂∂xj

(vjωi − viωj) = Si, (4.24)

with summation over repeated indices. The tendency of vorticity is given by thesolenidal term plus the divergence of a vector field, and thus if the former vanishesthe volume integrated vorticity can only be altered by boundary effects. In bothatmosphere and ocean the solenoidal term is important, but we will see in section4.5 that a useful conservation law for a scalar quantity can still be obtained.

4.2.1 Two-dimensional flow

In two-dimensional flow the fluid is confined to a surface, and independent of thethird dimension normal to that surface. Let us initially stay in a Cartesian geometry,and then two-dimensional flow is flow on a plane, and the velocity normal to theplane, and the rate of change of any quantity normal to that plane, are zero. Letthe normal direction be the z-direction and then the velocity in the plane, denotedby u, is

v = u = ui+ vj, w = 0. (4.25)

Only one component of vorticity non-zero and this is given by

ω = k(∂v∂x

− ∂u∂y

). (4.26)

That is, in two-dimensional flow the vorticity is perpendicular to the velocity. Welet ζ ≡ωz =ω ·k. Both the stretching and tilting terms vanish in two-dimensionalflow, and the two-dimensional vorticity equation becomes, for incompressible flow,

DζDt

= 0, (4.27)

where Dζ/Dt = ∂ζ/∂t + u · ∇ζ. That is, in two-dimensional flow vorticity is con-served following the fluid elements; each material parcel of fluid keeps its value ofvorticity even as it is being advected around. Furthermore, specification of the vor-ticity completely determines the flow field. To see this, we use the incompressibilitycondition to define a streamfunction ψ such that

u = −∂ψ∂y, v = ∂ψ

∂x, ζ = ∇2ψ. (4.28a,b,c)


Given the vorticity, the Poisson equation (4.28c) can be solved for the streamfunc-tion and the velocity fields obtained through (4.28a,b), and this process is called‘inverting the vorticity’.

Numerical integration of (4.27) is then a process of time-stepping plus inversion.The vorticity equation may then be written as an advection equation for vorticity,

∂ζ∂t

+ u · ∇ζ = 0 (4.29)

in conjunction with (4.28). The vorticity is stepped forward one time-step usinga finite-difference representation of (4.29), and the vorticity inverted to obtain avelocity using (4.28). (The notion that complete or nearly complete informationabout the flow may be obtained by inverting one field plays an important role ingeophysical fluid dynamics, as we will see later on.)

Two-dimensional flow is not restricted to a single, Cartesian plane, and we maycertainly envision two-dimensional flow on the surface of a sphere; in this case thevelocity normal to the spherical surface (the ‘vertical velocity’) vanishes, and theequations are naturally expressed in spherical coordinates. Nevertheless, vorticity(absolute vorticity if the sphere is rotating) is still conserved on parcels as theymove over the spherical surface.

4.3 VORTICITY AND CIRCULATION THEOREMS

4.3.1 The ‘frozen-in’ property of vorticity

Let us first consider some simple topological properties of the vorticity field and itsevolution.1 We define a vortex-line to be a line drawn through the fluid which is ev-erywhere in the direction of the local vorticity. This definition is analogous to that ofa streamline, which is everywhere in the direction of the local velocity. A vortex tubeis formed by the collection of vortex lines passing through a closed curve (Fig. 4.2.A material-line is just a line that connects material fluid elements. Suppose we drawa vortex line through the fluid; such a line obviously connects fluid elements andtherefore defines a co-incident material line. As the fluid moves the material linedeforms, and the vortex line also evolves in a manner determined by the equationsof motion. A remarkable property of vorticity is that, for an unforced and inviscidbarotropic fluid, the flow evolution is such that a vortex line remains co-incidentwith the same material line with which it was initially associated. Put another way,a vortex line always contains the same material elements — the vorticity is ‘frozen’or ‘glued’ to the material fluid.

To prove this we consider how an infinitesimal material line element δl evolves,δl being the infinitesimal material element connecting l with l + δl. The rate ofchange of δl following the flow is given by

DδlDt

= 1δt(δl(t + δt)− δl(t)

), (4.30)

which follows from the definition of the material derivative in the limit δt → 0.From the Taylor expansion of δl(t) and the definition of velocity it is also apparentthat

δl(t + δt) = l(t)+ δl(t)+ (v + δv)δt − (l(t)+ vδt) = δl + δvδt, (4.31)

4.3 Vorticity and Circulation Theorems 171

Figure 4.2 A vortex tube passingthrough a material sheet. The cir-culation is the integral of the veloc-ity around the boundary of A, andis equal to the integral of the nor-mal component of vorticity over A.

as illustrated in Fig. 4.3. Substituting into (4.30) gives

DδlDt

= δv (4.32)

But since δv = (δl · ∇)v we have that

DδlDt

= (δl · ∇)v (4.33)

Comparing this with (4.16), we see that vorticity evolves in the same way as a lineelement. To see what this means, at some initial time we can define an infinitesimalmaterial line element parallel to the vorticity at that location, that is,

δl(x, t = 0) = Aω(x, t = 0) (4.34)

where A is a constant. Then, for all subsequent times the magnitude of the vorticityof that fluid element, even as it moves to a new location x′, remains proportionalto the length of the fluid element at that point and is oriented in the same way; thatis ω(x′, t) = A−1δl(x′, t).

To see a similar result in a slightly different way note that a vortex line element

Fig. 4.3 Evolution of an infinitesimal material line δl from time t to time t+δt.It can be seen from the diagram that Dδl/Dt = δv.


is determined by the condition δl = Aω or, because A is just an arbitrary scalingfactor, ω× δl = 0. Now, for any line element we have that

DDt(ω× δl) = Dω

Dt× δl − Dδl

Dt×ω. (4.35)

We also have thatDδlDt

= δv = δl · ∇v (4.36a)

andDωDt

=ω · ∇v. (4.36b)

If the line element is initially a vortex line element then, at t = 0, δl = Aω and,using (4.36), the right hand side of (4.35) vanishes. Thus, the tendency ofω×δl iszero, and the vortex line continues to be a material line.

Stretching and tilting

The terms on the right-hand side of (4.20) or (4.21) may be interpreted in termsof ‘stretching’ and ‘tipping’ (or ‘tilting’). Consider a single Cartesian component of(4.21),

Dωx

Dt=ωx ∂u

∂x+ωy ∂u

∂y+ωz ∂u

∂z. (4.37)

The second and third terms of this are the tilting or tipping terms because theyinvolve changes in the orientation of the vorticity vector. They tell us that vorticityin x-direction may be generated from vorticity in the y- and z-directions if theadvection acts to tilt the material lines. Because vorticity is tied to these lines,vorticity oriented in one direction becomes oriented in another, as in Fig. 4.4.

The first term on the right-hand side of (4.37) is the stretching term, and itacts to intensify the x-component of vorticity if the velocity is increasing in thex-direction — that is, if the material lines are being stretched (Fig. 4.5). Thiseffect arises because a vortex line is tied to a material line, and therefore vorticityis amplified in proportion to the strretching of material line aligned with it. Thiseffect is important in tornadoes, to give one example. If the fluid is incompressiblestretching of a fluid mass in one direction must be accompanied by convergence inanother, and this leads to the conservation of circulation, as we now discuss.

4.3.2 Kelvin’s Circulation Theorem

Kelvin’s circulation theorem states that under certain circumstances the circulationaround a material fluid parcel is conserved; that is, the circulation is conserved ‘fol-lowing the flow’.2 The primary restrictions are that body forces are conservative(i.e., they are representable as potential forces, and therefore that the flow be in-viscid), and that the fluid is baroropic [i.e., p = p(ρ)]. Of these, the latter is themore restrictive for geophysical fluids. The circulation in the theorem is definedwith respect to an inertial frame of reference; specifically, the velocity in (4.41) isthe velocity relative to an inertial frame. To prove the theorem, we begin with theinviscid momentum equation,

DvDt

= −1ρ∇p −∇Φ, (4.38)


Figure 4.4 The tilting of vorticity. Supposethat vorticity, ω is initially directed horizon-tally, as in the lower figure, so that ωz, itsvertical component, is zero. The materiallines, and therefore the vortex lines also, aretilted by the postive vertical velocity W , socreating a vertically oriented vorticity. Thismechanism is important in creating verticalvorticity in the atmospheric boundary layer(and, one may show, the β-effect in large-scale flow).

Fig. 4.5 Stretching of material lines distorts the cylinder of fluid as shown.Vorticity is tied to material lines, and so is amplified in the direction of thestretching. However, because the volume of fluid is conserved, the end sur-faces shrink, the material lines through the cylinder ends converge, and theintegral of vorticity over a material surface (the circulation) remains constant,as discussed in section 4.3.2.


where ∇Φ represents the conservative body forces on the system. Applying thematerial derivative to the circulation, (4.2), gives

DCDt

= DDt

∮v · dr =

∮ (DvDt

· dr + v · dv)

=∮ [(

−1ρ∇p −∇Φ

)· dr + v · dv

]

=∮−1ρ∇p · dr

(4.39)

using (4.38) and D(δl)/Dt = δv. The line integration is over a closed, material,circuit. The second and third terms on the second line vanish separately, becausethey are exact differentials integrated around a closed loop. The term on the lastline vanishes if density is constant or, more generally, if pressure is a function ofdensity alone in which case ∇p is parallel to ∇ρ. To see this, note that∮

1ρ∇p · dr =

∫S∇×

(∇pρ

)· dA =

∫A

−∇ρ ×∇pρ2 · dA, (4.40)

using Stokes’s theorem where A is any surface bounded by the path of the lineintegral, and this evidently vanishes identically if p is a function of ρ alone. Thelast term is the integral of the solenoidal vector, and if it is zero (4.39) becomes

DDt

∮v · dr = 0 . (4.41)

This is Kelvin’s circulation theorem. In words, the circulation around a material loopis invariant for a barotropic fluid that is subject only to conservative forces. UsingStokes’ theorem, the circulation theorem may also be written

DDt

∫ω · dS = 0 . (4.42)

That is, the area-integral of the normal component of vorticity across any materialsurface is constant, under the same conditions. This form is both natural and useful,and it arises because of the way vorticity is tied to material fluid elements.

Stretching and circulation

Let us informally consider how vortex stretching and mass conservation work to-gether to give the circulation theorem. Let the fluid be incompressible so that thevolume of a fluid mass is constant, and consider a surface normal to a vortex tube,as in Fig. 4.5). Let the volume of a small material box around the surface be δV ,the length of the material lines be δl and the surface area be δA. Then

δV = δlδA. (4.43)

Because of the frozen-in property, vorticity passing through the surface is propor-tional to the length of the material lines. That is ω∝ δl, and

δV ∝ωδA. (4.44)


Figure 4.6 Solenoids and thecirculation theorem. Solenoidsare tubes perpendicularto both ∇α and ∇p, andthey have a non-zero cross-sectional area if isolines of αand p do not coincide. Therate of change of circulationover a material surface isgiven by the sum of all thesolenoidal areas crossing thesurface. If ∇α ×∇p = 0 thereare no solenoids.

The right-hand side is just the circulation around the surface. Now, if the cor-responding material tube is stretched δl increases, but the volume, δV , remainsconstant by mass conservation. Thus, the circulation given by the right-hand sideof (4.44) also remains constant. In other words, because of the frozen-in propertyvorticity is amplified by the stretching, but the vortex lines get closer together insuch a way that the product ωδS remains constant and circulation is conserved.

4.3.3 Baroclinic flow and the solenoidal term

In baroclinic flow, the circulation is not generally conserved. and from (4.39) wehave

DCDt

= −∮ ∇pρ· dl = −

∮dpρ, (4.45)

and this is called the baroclinic circulation theorem.3 Noting the fundamentalthermodynamic relation T dη = dI + p dα we have

αdp = d(pα)− T dη+ dI, (4.46)

so that the solenoidal term on the right-hand side of (4.45) may be written as

So ≡ −∮αdp =

∮T dη = −

∮ηdT = −R

∮T d logp, (4.47)

where the last equality holds only for an ideal gas. Using Stokes’s theorem, So canalso be written as

So = −∫A∇α×∇p · dA = −

∫A

(∂α∂T

)p∇T ×∇p · dA =

∫A∇T ×∇η · dA. (4.48)

The rate of change of the circulation across a surface depends on the existence ofthis solenoidal term (Fig. 4.6 and, for an example, problem 4.6).

However, even if the solenoidal vector is in general non-zero, circulation is con-served if the material path is in a surface of constant entropy, η, and if Dη/Dt = 0.In this case the solenoidal term vanishes and, because Dη/Dt = 0, entropy remainsconstant on that same material loop as it evolves. This result gives rise to the con-servation of potential vorticity, discussed in section 4.5


4.3.4 Circulation in a rotating frame

The absolute and relative velocities are related by va = vr + Ω × r so that in arotating frame the rate of change of circulation is given by

DDt

∮(vr +Ω× r) · dr =

∮ [(DvrDt

+Ω× vr)· dr + (vr +Ω× r) · dvr

]. (4.49)

But∮vr · dvr = 0 and, integrating by parts,∮

(Ω× r) · dvr =∮

d[(Ω× r) · vr ]− (Ω× dr) · vr

=∮

d[(Ω× r) · vr ]+ (Ω× vr ) · dr.

(4.50)

The first term is on the right-hand side is zero and so (4.49) becomes

DDt

∮(vr +Ω× r) · dr =

∮ (DvrDt

+ 2Ω× vr)· dr = −

∮dpρ, (4.51)

where the second equality uses the momentum equation. The term on the last linevanishes if the fluid is barotropic, and if so the circulation theorem is, unsurprisingly,

DDt

∮(vr +Ω× r) · dr = 0, or

DDt

∫(ωr + 2Ω) · dS = 0, (4.52a,b)

where the second equation uses Stokes’s theorem and we have used ∇× (Ω× r) =2Ω, and where ωr = ∇× vr is the relative vorticity.4

4.3.5 The circulation theorem for hydrostatic flow

Kelvin’s circulation theorem holds for hydrostatic flow, with a slightly differentform. For simplicity we restrict attention to the f -plane, and start with the hy-drostatic momentum equations,

DurDt

+ 2Ω× ur = −1ρ∇zp, 0 = −1

ρ∂p∂z

−∇Φ, (4.53a,b)

where Φ = gz is the gravitational potential and Ω = Ωk. The advecting field isthree-dimensional, and in particular we still have Dδr/Dt = δv = (δr · ∇)v. Thus,using (4.53) we have

DDt

∮(ur +Ω× r) · dr =

∮ [(DurDt

+Ω× vr)· dr + (ur +Ω× r) · dvr

]=∮ (

DurDt

+ 2Ω× ur)· dr

=∮ (

−1ρ∇p −∇Φ

)· dr, (4.54)

as with (4.51), having used Ω×vr = Ω×ur , and where the gradient operator ∇ isthree-dimensional. The last term on the right-hand side vanishes because it is the

4.4 Vorticity Equation in a Rotating Frame 177

integral of the gradient of a potential around a closed path. The first term vanishesif the fluid is barotropic, so that the circulation theorem is

DDt

∮(ur +Ω× r) · dr = 0, (4.55)

Using Stokes’s theorem we have the equivalent form

DDt

∫(ωhy + 2Ω) · dS = 0, (4.56)

where the subscript ‘hy’ denotes hydrostatic and, in Cartesian coordinates,

ωhy = ∇× u = −i∂v∂z

+ j∂u∂z

+ k(∂v∂x

− ∂u∂y

). (4.57)

4.4 VORTICITY EQUATION IN A ROTATING FRAME

Perhaps the easiest way to derive the vorticity equation appropriate for a rotatingreference frame is to begin with the momentum equation in the form

∂vr∂t

+ (2Ω+ωr )× vr = −1ρ∇p +∇

(Φ − 1

2v2r

), (4.58)

where the potential Φ contains the gravitational and centrifugal forces. Take thecurl of this and use the identity (4.13), which here implies

∇× [(2Ω+ωr )×vr ] = (2Ω+ωr )∇ ·vr + (vr ·∇)(2Ω+ωr )− [(2Ω+ωr ) ·∇]vr ,(4.59)

(noting that ∇ · (2Ω+ω) = 0), to give the vorticity equation

DωrDt

= [(2Ω+ωr ) · ∇]v − (2Ω+ωr )∇ · vr +1ρ2 (∇ρ ×∇p). (4.60)

Note that because Ω is a constant, Dωr/Dt = Dωa/Dt where ωa = 2Ω+ωr is theabsolute vorticity. The only difference between the vorticity equation in the rotatingand inertial frames of reference is in the presence of the solid-body vorticity 2Ω onthe right-hand side. The second term on the right-hand side may be folded in tothe material derivative using mass continuity, and after a little manipulation (4.60)becomes

DDt

(ωaρ

)= 1ρ(2Ω+ωr ) · ∇vr +

1ρ3 (∇ρ ×∇p). (4.61)

However, note that it is the absolute voricity,ωa, that now appears on the left-handside. If ρ is constant, ωa may be replaced by ωr .

4.4.1 The circulation theorem and vortex tilting

What are the implications of the circulation theorem on a rotating, spherical planet?Let us define relative circulation over some material loop as

Cr ≡∮vr · dl, (4.62)


Fig. 4.7 The projection of a material circuit on to the equatorial plane. Ifa fluid element moves poleward, keeping its orientation to the local verti-cal fixed (e.g., it stays horizontal) then the area of its projection on to theequatorial plane increases. If its total (absolute) circulation is to be main-tained, then the vertical component of the relative vorticity must diminish.That is,

∫A(ω + 2Ω) · dA =

∫A(ζ + f)dA = constant. Thus, the β term in

D(ζ + f)/Dt = Dζ/Dt + βv = 0 ultimately arises from the tilting of a parcelrelative to the axis of rotation as it moves meridionally.

and because vr = va − 2Ω× r we have

Cr = Ca −∫

2Ω · dS = Ca − 2ΩA⊥ (4.63)

where Ca is the total or absolute circulation and A⊥ is the area enclosed by theprojection of the material circuit onto the plane normal to the rotation vector; thatis, onto the equatorial plane (see Fig. 4.7). If the solenoidal term is zero then thecirculation theorem, (4.52), may be written as

DDt(Cr + 2ΩA⊥) = 0. (4.64)

This equation tells us that the relative circulation around a circuit will change ifthe orientation of the plane changes; that is, if the area of its projection on to theequatorial plane changes. In large scale dynamics the most common cause of this iswhen a fluid parcel changes its latitude. For example, consider the two-dimensionalflow of an infinitesimal, horizontal, homentropic fluid surface at a latitude ϑ witharea δS, so that the projection of its area on the equatorial plane is δS sinϑ. If thefluid surface moves, but remains horizontal, then directly from (4.64) the relativevorticity changes as

DζrDt

= −2ΩDDt

sinϑ = −vr2Ω cosϑa

= −βvr (4.65)

whereβ ≡ df

dy= 2Ωa

cosϑ, (4.66)

4.4 Vorticity Equation in a Rotating Frame 179

The means by which the relative vorticity of a parcel changes by virtue of its latitu-dinal displacement is known as the beta effect, or the β effect. It is a manifestation ofthe tilting term in the vorticity equation, and it is often the most important meansby which relative vorticity does change in large-scale flow. The β effect arises in thefull vorticity equation, as we now see.

4.4.2 The vertical component of the vorticity equation

In large-scale dynamics, the most important, although not the largest, component ofthe vorticity is often the vertical one, because this contains much of the informationabout the horizontal flow. We can obtain an explicit expression for its evolution bytaking the vertical component of (4.60), although care must be taken because theunit vectors (i, j, k) are functions of position (see problem 2.5.)

An alternative derivation begins with the horizontal momentum equations

∂u∂t

− v(ζ + f)+w∂u∂z

= −1ρ∂p∂x

− 12∂∂x(u2 + v2)+ Fx (4.67a)

∂v∂t

+u(ζ + f)+w∂v∂z

= −1ρ∂p∂y

− 12∂∂y(u2 + v2)+ Fy . (4.67b)

where in this section we again drop the subscript r on variables measured in therotating frame. Cross-differentiating gives, after a little algebra,

DDt(ζ + f) =− (ζ + f)

(∂u∂x

+ ∂v∂y

)+(∂u∂z∂w∂y

− ∂v∂z∂w∂x

)

+ 1ρ2

(∂ρ∂x∂p∂y

− ∂ρ∂y∂p∂x

)+(∂Fy

∂x− ∂F

x

∂y

).

(4.68)

We interpret the various terms as follows:

Dζ/Dt = ∂ζ/∂t +v ·∇ζ: The material derivative of the vertical component of thevorticity.

Df/Dt = v∂f/∂y = vβ: The β-effect. The vorticity is affected by the meridionalmotion of the fluid, so that, apart from the terms on the right-hand side,(ζ + f) is conserved on parcels. Because the Coriolis parameter changeswith latitude this is like saying that the system has differential rotation.This effect is precisely that due to the change in orientation of fluid surfaceswith latitude, as given above in section 4.4.1 and Fig. 4.7.

−(ζ+f)(∂u/∂x+∂v/∂y): The divergence term, which gives rise to vortex stretch-ing. In an incompressible fluid this may be written (ζ + f)∂w/∂z, so thatvorticity is amplified if the vertical velocity increases with height, so stretch-ing the material lines and the vorticity.

(∂u/∂z)(∂w/∂y) − (∂v/∂z)(∂w/∂x): The tilting term, whereby a vertical com-ponent of vorticity may be generated by a vertical velocity acting on a hor-izontal vorticity. See Fig. 4.4.

ρ−2 [(∂ρ/∂x)(∂p/∂y)− (∂ρ/∂y)(∂p/∂x)] = ρ−2J(ρ,p): The solenoidal term,also called the non-homentropic or baroclinic term, arising when isosur-faces of pressure and density are not parallel.


(∂Fy/∂x−∂Fx/∂y): The forcing and friction term. If the only contribution to thisis from molecular viscosity then this term is ν∇2ζ.

Two-dimensional and shallow water vorticity equations

In inviscid two-dimensional incompressible flow, all of the terms on the right-handside of (4.68) vanish and we have the simple equation

D(ζ + f)Dt

= 0, (4.69)

implying that the absolute vorticity, ζa ≡ ζ + f , is materially conserved. If f is aconstant, then (4.69) reduces to (4.29), and background rotation plays no role. Iff varies linearly with y, so that f = f0 + βy, then (4.69) becomes

∂ζ∂t

+ u · ∇ζ + βv = 0, (4.70)

which is known as the two-dimensional β-plane vorticity equation.For inviscid shallow water flow, we can show that (see chapter 3)

D(ζ + f)Dt

= −(ζ + f)(∂u∂x

+ ∂v∂y

). (4.71)

In this equation the vanishing of the tilting term is perhaps the only aspect which isperhaps not immediately apparent, but this succumbs to a little thought.

4.5 POTENTIAL VORTICITY CONSERVATION

Too much of a good thing is wonderful.

Mae West (1892–1990).

Although Kelvin’s circulation theorem is a general statement about vorticity con-servation, in its original form it is not always a practically useful statement for tworeasons. First, it is a not a statement about a field, such as vorticity itself. Sec-ond, it is not satisfied for baroclinic flow, such as is found in the atmosphere andocean. (Of course non-conservative forces such as viscosity also lead to circulationnon-conservation, but this applies to virtually all conservation laws and does not di-minish them.) It turns out that it is possible to derive a beautiful conservation lawthat overcomes both of these failings and one, furthermore, that is extraordinarilyuseful in geophysical fluid dynamics. This is the conservation of potential vorticityintroduced first by Rossby and then in a more general form by Ertel.5 The idea isthat we can use a scalar field that is being advected by the flow to keep track of,or to take care of, the evolution of fluid elements. For a baroclinic fluid this scalarfield must be chosen in a special way (it must be a function of the density and pres-sure alone), but there is no restriction to barotropic fluid. Then using the scalarevolution equation in conjunction with the vorticity equation gives us a scalar con-servation equation. In the next few subsections we derive the equation for potentialvorticity conservation in a number of superficially different ways.

4.5 Potential Vorticity Conservation 181

Fig. 4.8 An infinitesimal fluid element, bounded by two isosurfaces of theconserved tracer χ. As Dχ/Dt = 0, then Dδχ/Dt = 0.

4.5.1 PV conservation from the circulation theorem

Barotropic fluids

Let us begin with the simple case of a barotropic fluid. For an infinitesimal volumewe write Kelvin’s theorem as:

DDt[(ωa · n)δA] = 0 (4.72)

where n is a unit vector normal to an infinitesimal surface δA. Now consider avolume bounded by two isosurfaces of values χ and χ+δχ, where χ is any materiallyconserved tracer, so satisfying Dχ/Dt = 0, so that δA initially lies in an isosurfaceof χ (see Fig. 4.8). Since n = ∇χ/|∇χ| and the infinitesimal volume δV = δhδA,where δh is the separation between the two surfaces, we have

ωa · nδA =ωa ·∇χ|∇χ|

δVδh. (4.73)

Now, the separation between the two surfaces, δh may be obtained from

δχ = δx · ∇χ = δh|∇χ|, (4.74)

and using this in (4.72) we obtain

DDt

[(ωa · ∇χ)δV

δχ

]= 0. (4.75)

Now, as χ is conserved on material elements, then so is δχ, and it may be taken outof the differentiation. The mass of the volume element ρ δV is also conserved, sothat (4.75) becomes

ρδVδχ

DDt

(ωaρ· ∇χ

)= 0 (4.76)

orDDt

(ωa · ∇χ

)= 0 (4.77)

where ωa = ωa/ρ. Eq. (4.77) is a statement of potential vorticity conservationfor a barotropic fluid. The field χ may be chosen arbitrarily, provided that it bematerially conserved.


The general case

For a baroclinic fluid the above derivation fails simply because the statement ofthe conservation of circulation, (4.72) is not, in general, true: there are solenoidalterms on the right-hand side and from (4.45) and (4.47) we have

DDt[(ωa · n)δA] = S · nδA, S = −∇α×∇p = −∇η×∇T . (4.78a,b)

However, the right-hand side of (4.78a) may be annihilated by choosing the circuitaround which we evaluate the circulation to be such that the solenoidal term isidentically zero. Given the form of S, this occurs if the values of any of p,ρ, η, T areconstant on that circuit; that is, if χ = p,ρ, η or T . But the derivation also demandsthat χ be a materially conserved quantity, which usually restricts the choice of χ tobe η (or potential temperature), or to be ρ itself if the thermodynamic equation isDρ/Dt = 0. Thus, the conservation of potential vorticity for inviscid, adiabatic flowis

DDt

(ωa · ∇θ

)= 0 (4.79)

where Dθ/Dt = 0. For diabatic flow source terms appear on the right-hand side,and we derive these later on. A summary of this derivation provided Fig. 4.9.

4.5.2 PV conservation from the frozen-in property

In this section we show that potential vorticity conservation is a consequence ofthe frozen-in property of vorticity. This is not surprising, because the circulationtheorem itself has a similar origin. Thus, this derivation is not independent of thederivation in the previous section, just a minor re-expression of it. We first considerthe case in which the solenoidal term vanishes from the outset.

Barotropic fluids

If χ is a materially conserved tracer then the difference in χ between two infinitesi-mally close fluid elements is also conserved and

DDt(χ1 − χ2) =

DδχDt

= 0. (4.80)

But δχ = ∇χ · δl where δl is the infinitesimal vector connecting the two fluidelements. Thus

DDt(∇χ · δl) = 0 (4.81)

But since the line element and the vorticity (divided by density) obey the sameequation, we can replace the line element by vorticity (divided by density) in (4.81)to obtain again

DDt

(∇χ ·ωaρ

)= 0. (4.82)

That is, the potential vorticity, Q = (ωa ·∇χ) is a material invariant, where χ is anyscalar quantity that satisfies Dχ/Dt = 0.


ω a . ∇θω a

θ = θ 0

Mass: ρ δA δh = constant

Entropy: | ∇θ | δh = constant

δh

θ = θ 0 − δθ

| ∇θ |

δA

δA

Fig. 4.9 Geometry of potential vorticity conservation. The circulation equa-tion is D[(ωa · n)δA]/Dt = S · nδA where S ∝ ∇θ × ∇T . We choosen = ∇θ/|∇θ|, where θ is materially conserved, to annihilate the solenoidalterm on the right-hand side, and we note that δA = δV/δh, where δV isthe volume of the cylinder, and that δh = δθ/|∇θ|. The circulation isC ≡ ωa · nδA = ωa · (∇θ/|∇θ|)(δV/δh) = [ρ−1ωa · ∇θ](δM/δθ) whereδM = ρδV is the mass of the cylinder. As δM and δθ are materially con-served, so is the potential vorticity ρ−1ωa · ∇θ.

Baroclinic fluids

In baroclinic fluids we cannot casually substitute the vorticity for that of a lineelement in (4.81) because of the presence of the solenoidal term, and in any case alittle more care would not be amiss. From (4.81) we obtain

δl · D∇χDt

+∇χ · DδlDt

= 0 (4.83)

or, using (4.33),

δl · D∇χDt

+∇χ · [(δl · ∇)v] = 0. (4.84)

Now, let us choose δl to correspond to a vortex line, so that at the initial timeδl = εωa. (Note that in this case the association of δl with a vortex line can onlybe made instantaneously, and we cannot set Dδl/Dt ∝ Dωa/Dt.) Then,

ωa ·D∇χDt

+∇χ · [(ωa · ∇)v] = 0, (4.85)


or, using the vorticity equation (4.16),

ωa ·D∇χDt

+∇χ ·[

DωaDt

− 1ρ3∇ρ ×∇p

]= 0. (4.86)

This may be written

DDtωa · ∇χ =

1ρ3∇χ · (∇ρ ×∇p). (4.87)

The term on the right hand side is in general non-zero for an arbitrary choice ofscalar, but it will evidently vanish if ∇p,∇ρ and ∇χ are coplanar. If χ is anyfunction of p and ρ this will be satisfied, but χ must also be a materially conservedscalar. If, as for an ideal gas, ρ = ρ(η,p) (or η = η(p,ρ)) where η is the entropy(which is materially conserved), and if χ is a function of entropy η alone, then χsatisfies both conditions. Explicitly, the solenoidal term vanishes because

∇χ · (∇ρ ×∇p) = dχdη∇η ·

[(∂ρ∂p∇p + ∂ρ

∂η∇η

)×∇p

]= 0. (4.88)

Thus, provided χ satisfies the two conditions

DχDt

= 0 and χ = χ(p,ρ), (4.89)

then (4.87) becomesDDt

(ωa · ∇χρ

)= 0. (4.90)

The natural choice for χ is potential temperature, whence

DDt

(ωa · ∇θρ

)= 0 . (4.91)

The presence of a density term in the denominator is not necessary for incompress-ible flows (i.e., if ∇ · v = 0).

4.5.3 PV conservation — an algebraic derivation

Finally, we give a algebraic derivation of potential vorticity conservation. We willtake the opportunity to include frictional and diabatic processes, although thesemay also be included in the derivations above.6 We begin with the frictional vortic-ity equation in the form

DωaDt

= (ωa · ∇)v +1ρ3 (∇ρ ×∇p)+

1ρ(∇× F) . (4.92)

where F represents any nonconservative force term on the right-hand side of themomentum equation (i.e., Dv/Dt = −ρ−1∇p + F). We have also the equation forour materially conserved scalar χ,

DχDt

= χ (4.93)


where χ represents any sources and sinks of χ. Now

(ωa · ∇

) DχDt

= ωa ·D∇χDt

+[(ωa · ∇)v

]· ∇χ. (4.94)

which may be obtained just by expanding the left-hand side. Thus, using (4.93),

ωa ·D∇χDt

=(ωa · ∇

)χ −

[(ωa · ∇)v

]· ∇χ. (4.95)

Now take the dot product of (4.92) with ∇χ:

∇χ · DωaDt

= ∇χ · [(ωa ·∇)v]+∇χ ·[

1ρ3 (∇ρ×∇p)

]+∇χ ·

[1ρ(∇× F)

]. (4.96)

The sum of the last two equations yields

DDt

(ωa · ∇χ

)= ωa · ∇χ +∇χ ·

[1ρ3 (∇ρ ×∇p)

]+ ∇χρ· (∇× F). (4.97)

This equation reprises (4.87), but with the addition of frictional and diabatic terms.As before, the solenoidal term is annihilated if we choose χ = θ(p,ρ), so givingthe evolution equation for potential vorticity in the presence of forcing and diabaticterms, namely

DDt

(ωa · ∇θ

)= ωa · ∇θ +

∇θρ· (∇× F) . (4.98)

4.5.4 Effects of salinity and moisture

For seawater the equation of state may be written as

θ = θ(ρ,p, S) (4.99)

where θ is potential temperature and S is salinity. In the absence of diabatic terms(which include saline diffusion) potential temperature is a materially conservedquantity. However, because of the presence of salinity, θ cannot be used to annihi-late the solenoidal term; that is

∇θ · (∇ρ ×∇p) =(∂θ∂S

)p,ρ∇S · (∇ρ ×∇p) ≠ 0. (4.100)

Strictly speaking then, there is no potential vorticity conservation principle for sea-water. However, such a blunt statement rather overemphasizes the importance ofsalinity in the ocean, and the nonconservation of potential vorticity because of thiseffect is rather small.

In a moist atmosphere in which condensational heating occurs there is no ‘moistpotential vorticity’ that is generally materially conserved.7 We may choose to definea moist PV (Qe say) based on moist equivalent potential temperature but it doesnot always obey DQe/Dt = 0.


4.5.5 Effects of rotation, and summary remarks

In a rotating frame the potential vorticity conservation equation is obtained simplyby replacing ωa by ω + 2Ω, where Ω is the rotation rate of the rotating frame.The operator D/Dt is reference-frame invariant, and so may be evaluated using theusual formulae with velocities measured in the rotating frame.

In the above derivations, we have generally referred to the quantity ωa · ∇θ/ρas potential vorticity; however, it is clear that this form is not unique. If θ is amaterially conserved variable, then so is g(θ) where g is any function, so that ωa ·∇g(θ)/ρ is also a potential vorticity, although when such a non-standard definitionis used we qualify the expression ‘potential vorticity’ with some adjective.

The conservation of potential vorticity has profound consequences in fluid dy-namics, especially in a rotating, stratified fluid. The nonconservative terms areoften small, and large-scale flow in both the ocean and atmosphere is characterizedby conservation of potential vorticity. Such conservation is a very powerful con-straint on the flow, and indeed it turns out that potential vorticity is a much moreuseful quantity for baroclinic, or nonhomentropic fluids than for barotropic fluids,because the required use of a special conserved scalar imparts additional informa-tion. A large fraction of the remainder of this book explores, in one way or another,the consequences of potential vorticity conservation.

4.6 * POTENTIAL VORTICITY IN THE SHALLOW WATER SYSTEM

In chapter 3 we derived potential vorticity conservation by direct manipulation ofthe shallow water equations. We now show that shallow water potential vorticity isalso derivable from the conservation of circulation. Specifically, we will begin withthe three-dimensional form of Kelvin’s theorem, and then make the small aspectratio assumption (which is the key assumption underlying shallow water dynamics),and thereby recover shallow water potential vorticity conservation. In the followingtwo subsections we give two variants of such derivation (see also Fig. 4.10).

4.6.1 Using Kelvin’s theorem

We begin withDDt(ω3 · δS) = 0, (4.101)

where ω3 is the curl of the three-dimensional velocity and δS = nδS is an arbitraryinfinitesimal vector surface element, with n a unit vector pointing in the directionnormal to the surface. If we separate the vorticity and surface element into verticaland horizontal components we can write (4.101) as

DDt[(ζ + f)δA+ωh · δSh] = 0 (4.102)

where ωh and δSh are the horizontally-directed components of the vorticity andthe surface element, and δA = kδS is the area of a horizontal cross-section of afluid column. In Cartesian form the horizontal component of the vorticity is

ωh = i(∂w∂y

− ∂v∂z

)− j

(∂w∂x

− ∂v∂z

)= i∂w∂y

− j∂w∂x, (4.103)

4.6 * Potential Vorticity in the Shallow Water System 187

Fig. 4.10 The mass of a column of fluid, hA, is conserved in the shallow watersystem. Furthermore, the vorticity is tied to material lines so that ζA is also amaterial invariant, where ζ =ω · k is the vertical component of the vorticity.From this, ζ/hmust be materially conserved; that is D(ζ/h)/Dt = 0, which isthe conservation of potential vorticity in a shallow water system. In a rotatingsystem this generalizes to D[(ζ + f)/h]/Dt = 0.

where vertical derivatives of the horizontal velocity are zero by virtue of the natureof the shallow water system. Now, the vertical velocity in the shallow water sys-tem is smaller than the horizontal velocity by the order of the apect ratio — theratio of the fluid depth to the horizontal scale of motion. Furthermore, the size ofthe horizontally-directed surface element is also an aspect-ratio smaller than thevertically-directed component. That is,

|ωh| ∼ α|ζ| and |δSh| ∼ α|δA|, (4.104)

where α = H/L is the aspect ratio. Thus ωh · δSh is an aspect-number squaredsmaller than the term ζδA and in the small aspect ratio approximation should beneglected. Kelvin’s circulation theorem, (4.102) becomes

DDt[(ζ + f)δA] = 0 or

DDt

[(ζ + f)h

hδA]= 0, (4.105a,b)

where h is the depth of the fluid column. But hδA is the volume of the fluid column,and this is constant. Thus, (4.105b) gives, as in (3.80),

DDt

(ζ + fh

)= 0, (4.106)

where, because horizontal velocities are independent of the vertical coordinate, theadvection is purely horizontal.

4.6.2 Using an appropriate scalar field

In a constant density fluid we can write potential vorticity conservation as

DDt(ω3 · ∇χ) = 0, (4.107)


where χ is any materially-conserved scalar [c.f. (4.77) or (4.82)]. In the flat-bottomed shallow water system, a useful choice of scalar is the ratio z/h, where his the local thickness of the fluid column because, from (3.28),

DDt

(zh

)= 0, (4.108)

if (for simplicity) the fluid is flat-bottomed. With this choice of scalar, potentialvorticity conservation becomes

DDt

[ω · ∇

(zh

)]= 0, (4.109)

where ω and D/Dt are fully three dimensional. Expanding the dot product gives

DDt

[ζ + fh

− zh2ωh · ∇zh

]= 0. (4.110)

For an order-unity Rossby number, the ratio of the size of the two terms in thisequation is ∣∣ζ∣∣

|(z/h)ωh · ∇hh|∼ [U/L][WH/L2]

= ULWH

= α2 1. (4.111)

Thus, the second term in (4.110) is an aspect-ratio squared smaller than the firstand, upon its neglect, (4.106) is recovered.

4.7 POTENTIAL VORTICITY IN APPROXIMATE, STRATIFIED MODELS

If approximate models of stratified flow — Boussinesq, hydrostatic and so on — areto be useful then they should conserve an appropriate form potential vorticity, andwe consider a few such cases here.

4.7.1 The Boussinesq equations

A Boussinesq fluid is incompressible (that is, the volume of a fluid element is con-served, and ∇ · v = 0) and the equation for vorticity itself is isomorphic to thatfor a line element. However, the Boussinesq equations are not barotropic — ∇ρ isnot parallel to ∇p — and although the pressure gradient term ∇φ disappears ontaking its curl (or equivalently disappears on integration around a closed path) thebuoyancy term kb does not, and it is this that prevents Kelvin’s circulation theoremfrom holding. Specifically, the evolution of circulation in the Boussinesq equationsobeys

DDt[(ωa · n)δA] = (∇× bk) · nδA, (4.112)

where here, as in (4.72), n is a unit vector orthogonal to an infinitesimal surfaceelement of area δA. The right-hand side is annihilated if we choose n to be par-allel to ∇b, because ∇b · ∇ × (bk) = 0. In the simple Boussinesq equations thethermodynamic equation is

DbDt

= 0, (4.113)

4.7 Potential Vorticity in Approximate, Stratified Models 189

and potential vorticity conservation is therefore (with ωa =ω+ 2Ω)

DQDt

= 0, Q = (ω+ 2Ω) · ∇b. (4.114a,b)

Expanding (4.114b) in Cartesian coordinates with Ω = fk we obtain:

Q = (vx −uy)bz + (wy − vz)bx + (uz −wx)by + fbz. (4.115)

In the general Boussinesq equations b itself is not materially conserved. Wecannot expect to obtain a conservation law if salinity is present, but if the equationof state and the thermodynamic equation are:

b = b(θ, z), DθDt

= 0, (4.116)

then potential vorticity conservation follows, because taking n to be parallel to ∇θwill cause the right-hand side of (4.112) to vanish. That is,

∇θ · ∇× (bk) =(∂θ∂z∇z + ∂θ

∂b∇b

)· ∇× (bk) = 0. (4.117)

The materially conserved potential vorticity is thus

Q =ωa · ∇θ. (4.118)

Note that if the equation of state is b = b(θ,φ), where φ is the pressure, thenpotential vorticity is not conserved because then, in general, ∇φ · ∇× (bk) ≠ 0.

4.7.2 The hydrostatic equations

Making the hydrostatic approximation has no effect on the satisfaction of the circu-lation theorem. Thus, in a baroclinic hydrostatic fluid we have

DDt

∫(ωhy + 2Ω) · dS = −

∫∇α×∇p · dS (4.119)

where, from (4.57)ωhy = ∇×u = −ivz+juz+k(vx−uy), but the gradient operatorand material derivative are fully three-dimensional. Derivation of potential vorticityconservation then proceeds, as in section 4.5.1, by choosing the circuit over whichthe circulation is calculated to be such that the right-hand side vanishes; that is, tobe such that the solenoidal term is annihilated. Precisely as before, this occurs ifthe circuit is barotropic and without further ado we write

DQhy

Dt= D

Dt

((ωhy + 2Ω) · ∇θ

ρ

)= 0. (4.120)

Expanding this gives in Cartesian coordinates

Qhy =1ρ

[(vx −uy)θz − vzθx +uzθy + 2Ωθz

]. (4.121)

In spherical coordinates the hydrostatic approximation is usually accompanied bythe traditional approximation and the expanded expression for a conserved poten-tial vorticity is more complicated. It can still be derived from Kelvin’s theorem, butthis is left as an exercise for the reader (problem 4.4).


4.7.3 Potential Vorticity on isentropic surfaces

If we begin with the primitive equations in isentropic coordinates then potentialvorticity conservation follows quite simply. Cross differentiating the horizontal mo-mentum equations (3.164) gives the vorticity equation [c.f. (3.72)]

DDt(ζ + f)+ (ζ + f)∇θ · u = 0. (4.122)

where D/Dt = ∂/∂t + u · ∇θ. The thermodynamic equation is

DσDt

+ σ∇ · u = 0, (4.123)

where σ = ∂z/∂b (Boussinesq) or ∂p/∂θ (ideal gas) is the thickness of an isopycnallayer. Eliminating the divergence between (4.122) and (4.123) gives

DDt

(ζ + fσ

)= 0. (4.124)

The derivation, and the result, are precisely the same as with the shallow waterequations (sections 3.6.1 and 4.6).

A connection between isentropic and height coordinates

The hydrostatic potential vorticity written in height coordinates may be transformedinto a form that reveals its intimate connection with isentropic surfaces. Let us makethe Boussinesq approximation for which the potential vorticity is

Qhy = (vx −uy)bz − vzbx +uzby , (4.125)

where b is the buoyancy. We can write this as

Qhy = bz[(vx − vz

bxbz

)−(uy −uz

bybz

)]. (4.126)

But the terms in the inner brackets are just the horizontal velocity derivatives atconstant b. To see this, note that(

∂v∂x

)b=(∂v∂x

)z+ ∂v∂z

(∂z∂x

)b=(∂v∂x

)z− ∂v∂z

(∂b∂x

)z

/ ∂b∂z, (4.127)

with a similar expression for (∂u/∂y)b. (These relationships follow from standardrules of partial differentiation. Derivatives with respect to z are taken at constant xand y.) Thus, we obtain

Qhy =∂b∂z

[(∂v∂x

)b−(∂u∂y

)b

]= ∂b∂zζb. (4.128)

Thus, potential vorticity is simply the horizontal vorticity evaluated on a surface ofconstant buoyancy, multiplied by the vertical derivative of buoyancy, a measure ofstatic stability. An analogous derivation, with a similar result, proceeds for the idealgas equations, with potential temperature replacing buoyancy.

4.8 The Impermeability of Isentropes to Potential Vorticity 191

4.8 THE IMPERMEABILITY OF ISENTROPES TO POTENTIAL VORTICITY

A kinematical result is a result valid forever.

Clifford Truesdell, The Kinematics of Vorticity, 1954.

An interesting property of isentropic surfaces is that they are ‘impermeable’ to po-tential vorticity, meaning that the mass integral of potential vorticity (

∫Qρ dV) over

a volume bounded by an isentropic surface remains constant, even in the presenceof diabatic sources, provided the surfaces do not intersect a non-isentropic surfacelike the ground.8 This may seem surprising, especially because unlike most conser-vation laws the result does not require adiabatic flow, and for that reason it leads tointeresting interpretations of a number of phenomena. However, at the same timeimpermeability is a consequence of the definition of potential vorticity rather thanthe equations of motion, and in that sense is a kinematic property.

To derive the result we define s ≡ ρQ = ∇ · (θωa) and integrate over somevolume V to give

I =∫Vs dV =

∫V∇ · (θωa)dV =

∫Sθωa · dS, (4.129)

using the divergence theorem, where S is the surface surrounding the volume V . Ifthis is an isentropic surface then we have

I = θ∫Sωa · dS = θ

∫V∇ ·ωa dV = 0, (4.130)

again using the divergence theorem. That is, over a volume wholly enclosed bya single isentropic surface the integral of s vanishes. If the volume is bounded bymore than one isentropic surface neither of which intersect the surface, for exampleby concentric spheres of different radii as in Fig. 4.11a, the result still holds. Thequantity s is called ‘potential vorticity concentration’, or ‘PV concentration’. Theintegral of s over a volume is akin to the total amount of a conserved materialproperty, like salt content, and so may be called ‘PV substance’. That is, the PV con-centration is the amount of potential vorticity substance per unit volume (followingthe meaning for concentration introduced in section 1.2.3 on page 11) and

PV substance =∫s dV =

∫ρQdV. (4.131)

Suppose now that fluid volume is enclosed by an isentrope that intersects theground, as in Fig. 4.11b. Let A denote the isentropic surface, B denote the ground,θA the constant value of θ on the isentrope, and θB(x,y, t) the non-constant valueof θ on the ground. The integral of s over the volume is then

I =∫V∇ · (θωa)dV = θA

∫Aωa · dS +

∫BθBωa · dS

= θA∫A+B

ωa · dS +∫B(θB − θA)ωa · dS

=∫B(θB − θA)ωa · dS.

(4.132)

The first term on the second line vanishes after using the divergence theorem. Thus,the value of I, and so its rate of change, is a function only of an integral over the sur-face B, and the PV flux there must be calculated using the full equations of motion.


1

2

VEarth

B.x; y/

A

= == ==== == == ==== == ===

A D const.

B

V

B.x; y/

↓

(a) (b)

Fig. 4.11 (a) Two isentropic surfaces that do not intersect the ground. Theintegral of PV concentration over the volume between then, V , is zero, evenif there is heating and the contours move. (b) An isentropic surface, A, in-tersects the ground, B, so enclosing a volume V . The rate of change of PVconcentration over the volume is given by an integral over B.

However, we do not need to be concerned with a flux of PV concentration throughthe isentropic surface. Put another way, the PV substance in a volume can changeonly when isentropes enclosing the volume intersect a boundary such as the earth’ssurface.

4.8.1 Interpretation and application

Motion of the isentropic surface

How can the above results hold in the presence of heating? The isentropic surfacesmust move in such a way that the total amount of PV concentration containedbetween them nevertheless stays fixed, and we now demonstrate this explicitly.The potential vorticity equation may be written

∂Q∂t

+ v · ∇Q = SQ, (4.133)

where, from (4.98), SQ = (ωa/ρ) ·∇θ+∇θ · (∇×F)/ρ. Using mass continuity thismay be written as

∂s∂t+∇ · J = 0, (4.134)

where J ≡ ρvQ + N and ∇ · N = −ρSQ. Written this way, the quantity J/s is anotional velocity, vQ say, and s satisfies

∂s∂t+∇ · (vQs) = 0. (4.135)

That is, s evolves as if it were being fluxed by the velocity vQ. The concentration ofa chemical tracer χ (i.e., χ is the amount of tracer per unit volume) obeys a similarequation, to wit

∂χ∂t

+∇ · (vχ) = 0. (4.136)

However, whereas (4.136) implies that D(χ/ρ)/Dt = 0, (4.135) does not imply that∂Q/∂t + vQ · ∇Q = 0 because ∂ρ/∂t +∇ · (ρvQ) ≠ 0.

4.8 The Impermeability of Isentropes to Potential Vorticity 193

Now, the impermeability result tells us that there can be no notional velocityacross an isentropic surface. How can this be satisfied by the equations of motion?We write the right-hand side of (4.133) as

ρSQ = ∇ · (θωa + θ∇× F) = ∇ · (θωa + F ×∇θ). (4.137)

Thus, N = −θωa − F ×∇θ and we may write the J vector as

J = ρvQ− θωa − F ×∇θ = ρQ(v⊥ + v‖)− θω‖ − F ×∇θ, (4.138)

where, making use of the thermodynamic equation,

v‖ = v −v · ∇θ|∇θ|2 ∇θ, v⊥ = −

∂θ/∂t|∇θ|2 ∇θ, (4.139a)

ω‖ =ωa −ωa · ∇θ|∇θ|2 ∇θ =ωa −

Q|∇θ|2∇θ. (4.139b)

The subscripts ‘⊥’ and ‘‖’ denote components perpendicular and parallel to the localisentropic surface, and v⊥ is the velocity of the isentropic surface normal to itself.Eq. (4.138) may be verified by using (4.139) and Dθ/Dt = θ.

The ‘parallel’ terms in (4.138) are all vectors parallel to the local isentropic sur-face, and therefore do not lead to any flux of PV concentration across that surface.Furthermore, the term ρQv⊥ is ρQ multiplied by the normal velocity of the surface.That is to say, the notional velocity associated with the flux normal to the isentropicsurface is equal to the normal velocity of the isentropic surface itself, and so it tooprovides no flux of PV concentration across that surface (even through there maywell be a mass flux across the surface). Put simply, the isentropic surface alwaysmoves in such a way as to ensure that there is no flux of PV concentration across it.In our proof of the impermeability result in the previous section we used the factthat the potential vorticity multiplied by density is the divergence of something. Inthe demonstration above we used the fact that the terms forcing potential vorticityare the divergence of something.

* Dynamical choices of PV flux and a connection to Bernoullis’ theorem

If we add a non-divergent vector to the flux, J, then it has no effect on the evolutionof s. This gauge invariance means that the notional velocity, vQ = J/(ρQ) is sim-ilarly non-unique, although it does not mean that there are not dynamical choicesfor it that are more appropropriate in given circumstances. To explore this, let usobtain a general expression for J by starting with the definition of s, so that

∂s∂t= ∇θ · ∂ωa

∂t+ωa · ∇

∂θ∂t

= ∇θ · ∇× ∂v∂t

+∇ ·(ωa∂θ∂t

)= −∇ · J′,

(4.140)

whereJ′ = ∇θ × ∂v

∂t− ∂θ∂tωa +∇φ×∇χ. (4.141)

The last term in this expression is an arbitrary divergence-free vector. If we chooseφ = θ and χ = B, where B is the Bernoulli function given by B = I + v2/2 + p/ρwhere I is the internal energy per unit mass, then

J′ = ∇θ ×(∇B + ∂v

∂t

)−ωa(θ − v · ∇θ), (4.142)


having used the thermodynamic equation Dθ/Dt = θ. Now, the momentum equa-tion may be written, without approximation, in the form (see problems 2.1 and2.2)

∂v∂t

= −ωa × v + T∇η+ F −∇B, (4.143)

where η is the specific entropy (dη = cp d lnθ). Using (4.142) and (4.143) gives

J′ = ρQv − θωa +∇θ × F. (4.144)

which is the same as (4.138). Furthermore, using (4.141) for steady flow,

J = ∇θ ×∇B. (4.145)

That is, the flux of potential vorticity (in this gauge) is aligned with the intersectionof θ- and B-surfaces. For steady inviscid and adiabatic flow the Bernoulli functionis constant along streamlines; that is, surfaces of constant Bernoulli function arealigned with streamlines, and, because θ is materially conserved, streamlines areformed at intersecting θ- and B-surfaces, as in (1.193). In the presence of forcing,this property is replaced by (4.145), that the flux of PV concentration is along suchintersections.

This choice of gauge leading to (4.144) is physical in that it reduces to the trueadvective flux vρQ for unforced, adiabatic flow, but it is not a unique choice, normandated by the dynamics. Choosing χ = 0 leads to the flux

J1 = ρQv − θωa +∇θ × (F −∇B), (4.146)

and using (4.141) this vanishes for steady flow, a potentially useful property.

4.8.2 Summary Remarks

The impermeability result has a number of consequences, some obvious with hind-sight, and it also provides an interesting point of view and diagnostic tool.9 Here,we will just remark:? There can be no net transport of potential vorticity across an isentropic surface,

and the total amount of potential vorticity in a volume wholly enclosed byisentropic surfaces is zero.

? Thus, and trivially, the amount of potential vorticity contained between twoisentropes isolated from the earth’s surface in the northern hemisphere is thenegative of the corresponding amount in the southern hemisphere.

? Potential vorticity flux lines (i.e., lines everwhere parallel to J) can either closein on themselves or begin and end at boundaries (e.g., the ground, the oceansurface). However, J may change its character. Thus, for example, at thebase of the oceanic mixed layer J may change from from being a diabatic fluxabove to an adiabatic advective flux below. There may be a similar change incharacter at the atmospheric tropopause.

? The flux vector J is defined only to within the curl of a vector. Thus the vectorJ′ = J +∇×A, where A is an arbitrary vector, is as valid as is J in the abovederivations and diagnostics.


Notes

1 The frozen-in property — that vortex lines are material lines — was derived by Helm-holtz (1858) and is sometimes called Helmholtz’s theorem.

2 The theorem originates with Thomson (1869), who later became later Lord Kelvin.

3 Silberstein (1896) proved that that ‘the necessary and sufficient condition for thegeneration of vortical flow. . . influenced only by conservative forces . . . is that thesurface of constant pressure and surface of constant density. . . intersect’, as we de-rived in section 4.2, and this leads to (4.45). Bjerknes (1898a,b) explicitly put thisinto the form of a circulation theorem and applied it to problems of meteorolog-ical and oceanographic importance (see Thorpe et al. 2003), and the theorem issometimes called the Bjerknes theorem or the Bjerknes-Silberstein theorem. It isoccassionally stated as the evolution of circulation around a circuit is determined bythe number of solenoids passing through any surface bounded by that circuit, butthe meaning is that of (4.45).

Vilhelm Bjerknes (1862–1951) was a physicist and hydrodynamicist who in 1917moved to the University of Bergen as founding head of the Bergen Geophysical Insti-tute. Here he did what is probably his most influential work in meteorology, settingup and contributing to the ‘Bergen School of Meteorology’. Among other things heand his colleagues were the first to consider, as a practical proposition, the use ofnumerical methods — initial data in conjunction with the fluid equations of motion— to forecast the state of the atmosphere, based on earlier work describing howthat task might be done (Bjerknes 1904). Innaccurate initial velocity fields com-pounded with the shear complexity of the effort ultimately defeated them, but theeffort was continued (also unsuccessfully) by L. F. Richardson (Richardson 1922),before J. Charney, R. Fjortoft and J. Von Neumann eventually made what may be re-garded as the first successful numerical forecast (Charney et al. 1950). Their successcan be attributed to the used of a simplified, filtered, set of equations and the useof an electronic computer.

Vilhelm’s son, Jacob Bjerknes (1897-1975) was a leading player in the Bergen school.He was responsible for the now-famous frontal model of cyclones (Bjerknes 1919),and was one of the first to seriously discuss the role of cyclones in the generalcirculation of the atmosphere. In collaboration with Halvor Solberg and Tor Berg-eron the frontal model lead to a prescient picture of the life-cycle of extra-tropicalcyclones (see chapter 9), in which a wave grows initially on the polar front (akinto baroclinic instability with the meridional temperature gradient compressed to afront, but baroclinic instability theory was not yet developed), develops into a ma-ture cyclone, occludes and decays. In 1939 Bjerknes moved to the U.S. and, largelybecause of WWII, stayed, joining UCLA and heading its Dept. of Meteorology after itsformation in 1945. He developed an interest in air-sea interactions, and notably pro-posed the essential mechanism governing El Niño, a feedback between sea-surfacetemperatures and the strength of the trade winds (Bjerknes 1969). [See also Fried-man (1989), Cressman (1996), articles in Shapiro and Grønas (1999), and a memoirby Arnt Eliassen available fromhttp://www.nap.edu/readingroom/books/biomems/jbjerknes.html.]

4 The result (4.52) is sometimes attributed to Bjerknes (1902), although it was alsoknown to Poincaré (1893).

5 The first derivation of the PV conservation law was given for the shallow water equa-tions Rossby (1936), with a generalization to multiple layers in Rossby (1938). Inthe 1936 paper Rossby notes [his eq. (75)] that a fluid column satisfies f + ζ = cD


where c is a constant and D is the thickness of a fluid column; equivently, (f +ζ)/Dis a material invariant. In Rossby (1940) this was generalized slightly to an isentropiclayer, in which ζ is computed using horizontal derivates taken at constant density orpotential temperature. In this paper Rossby also introduces the expression ‘poten-tial vorticity’, as follows: ‘This quantity, which may be called the potential vorticity,represents the vorticity the air column would have it it were brought, isopycnallyor isentropically, to a standard latitude (f0) and stretched or shrunk vertically to astandard depth D0 or weight ∆0.’ (Rossby’s italics.) That is,

Potential Vorticity = ζ0 =(ζ + fD

)D0 − f0, (4.147)

which follows from his eq. (11), and this is the sense he uses it in that paper. How-ever, potential vorticity has come to mean the quantity (ζ + f)/D, which of coursedoes not have the dimensions of vorticity. We use it in this latter, now conven-tional, sense throughout this book. Ironically, quasi-geostrophic potential vorticityas usually defined does have the dimensions of vorticity.

The expression for potential vorticity in a continuously stratified fluid was given byErtel (1942a), and its relationship to circulation was given by Ertel (1942b). It is nowcommonly known as the Ertel potential vorticity. Interestingly, in Rossby (1940) wefind the Fermat-like comment ‘It is possible to derive corresponding results for an at-mosphere in which the potential temperature varies continuously with elevation. . . .The generalized treatment will be presented in another place.’(!) Opinions differ asto whether Rossby’s and Ertel’s derivations were independent: Charney (in Lindzenet al. 1990) suggests they were, and Cressman (1996) remarks that the origin ofthe concept of potential vorticity is a ‘delicate one that has aroused some passion inprivate correspondences’. In fact, Ertel visited MIT in autumn 1937 and presumablytalked to Rossby and became aware of his work. The likeliest scenario is that Ertelknew of Rossby’s shallow water theorems, and that he subsequently provided anindependent and significant generalization. Rossby and Ertel apparently remainedon good terms, but further collaboration was stymied by WWII. They later publisheda pair of short joint papers, one in German and the other in English, describingtheir conservation theorems (Ertel and Rossby 1949a,b). (I thank A. Persson and R.Samelson for some historical details on this.)

6 Truesdell (1951, 1954) and Obukhov (1962) were early explorers of the conse-quences of heating and friction on potential vorticity.

7 Schubert et al. (2001) provide more discussion of this topic. They derive a ‘moist PV’that, although not materially conserved, is an extension of the dry Ertel PV to moistatmospheres and has an impermeability result.

8 Haynes and McIntyre (1987, 1990). See also Danielsen (1990), Schär (1993) [whoobtained the result (4.145)], Bretherton and Schär (1993) and Davies-Jones (2003).

9 See for example McIntyre and Norton (1990) and Marshall and Nurser (1992). Thelatter use J vectors to study the creation and transport of potential vorticity in theoceanic thermocline.

Further Reading

Truesdell, C., 1954. The Kinematics of Vorticity.Written in Truesdell’s inimitable style, this book discusses many aspects of vorticityand has many historical references (as well as a generous definition of what consti-tutes a ‘kinematic’ result).

Batchelor, G. K., 1967. An Introduction to Fluid Dynamics.Contains an extensive discussion of vorticity and vortices.


Dutton, J. A., 1986. The Ceaseless Wind: An Introduction to the Theory of AtmosphericMotionContains derivations of the main vorticity theorems as well as more general materialon the dynamics of atmospheric flows.

Salmon, R., 1998. Lecture on Geophysical Fluid Dynamics.Chapter 4 contains a brief discussion of potential vorticity, and chapter 7 a longerdiscussion of Hamiltonian fluid dynamics, in which the particle relabeling symmetrythat gives rise to potential vorticity conservation is discussed.

Problems

4.1 For the vr vortex, choose a contour of arbitrary shape (e.g., a square) with segmentsneither parallel nor perpendicular to the radius, and not enclosing the origin. Showexplicitly that the circulation around the contour is zero. (This problem is a littleperverse.)

4.2 Vortex stretching and viscosity.Suppose there is an incompressible swirling flow given in cylindrical coordinates(r ,φ, z):

v = (vr , vφ, vz) = (−12αr,vφ, αz) (P4.1)

Show that this satisfies the mass conservation equation. Show too that vorticity isonly non-zero in the vertical direction. Show that the vertical component of the vor-ticity equation contains only the stretching term and that in steady state it is

− 12αr∂ζ∂r

= ζα+ ν 1r∂∂r

(r∂ζ∂r

). (P4.2)

Show that this may be integrated to ζ = ζ0 exp(−αr 2/4ν). Thus deduce that thereis a rotational core of thickness ro = 2 (ν/α)1/2, and that the radial velocity field isgiven by

vφ(r) = −1r

2ναζ0 exp

[−αr

2

4ν

]+ Ar

(P4.3)

where A = 2νζ0/α. What is the swirling velocity field? (From Batchelor 1967)

4.3 Beginning with the three-dimensional vorticity equation in a rotating frame of refer-ence [e.g., (4.60) or (4.61)], or otherwise, obtain an expression for the evolution ofthe vertical and radial coordinate of relative vorticity in Cartesian and spherical coor-dinates, respectively. Discuss the differences (if any) between the resulting equationsand (4.68). Show carefully how the β-term arises, and in particular that it may beinterpreted as arising from tilting term in the vorticity equation.

4.4 Making use of Kelvin’s circulation theorem obtain an expression for the potentialvorticity that is conserved following the flow (for an adiabatic and unforced fluid)and that is appropriate for the hydrostatic primitive equations on a spherical planet.Express this in terms of the components of the spherical coordinate system.

4.5 In pressure coordinates for hydrostatic flow on the f-plane, the horizontal momen-tum equation takes the form

DuDt

+ f × u = −∇φ (P4.4)

On taking the curl of this, there appears to be no baroclinic term. Show that Kelvin’scirculation theorem is nevertheless not in general satisfied, even for unforced, adi-abatic flow. By appropriately choosing a path on which to evaluate the circulationobtain an expression for potential vorticity, in this coordinate system, that is con-served following the flow. Hint: look at the hydrostatic equation.


4.6 Solenoids and sea-breezes.

A land-sea temperature con-trast of 20 K forces a sea breezein the surface "mixed layer"(potential temperature nearlyuniform with height), as illus-trated schematically. The layerextends through the lowest 10%of the mass of the atmosphere.

T = 300 Kx

z

L = 50 km

T = 280 K

(a) In the absence of dissipation and diffusion, at what rate does the circulationchange on a material circuit indicated? You may assume the horizonal flow isisobaric, and express your answer in m/s per hour.

(b) Suppose the sea breeze is equilibrated by a nonlinear surface drag of the form

dVdt

= −V2

LF(P4.5)

with LF = (3 m s−1)(3600 s). What is the steady speed of the horizontal wind inthe case L = 50 km?

(c) Suppose that the width of the circulation is determined by a horizontal thermaldiffusion of the form

DθDt

= κH∂2θ∂x2 (P4.6)

Provide an estimate of κH that is consistent with L = 50 km. Comment on whetheryou think the extent of real sea breezes is really determined this way.

This extreme generality whereby the equations of motion apply to theentire spectrum of possible motions — to sound waves as well as cy-clone waves — constitutes a serious defect of the equations from themeteorological point of view.Jule Charney, On the scale of atmospheric motions, 1948.

CHAPTER

FIVE

Simplified Equations for the Oceanand Atmosphere

LARGE-SCALE FLOW IN THE OCEAN AND ATMOSPHERE is characterized by an approximate

balance in the vertical between the pressure gradient and gravity (hydro-static balance), and in the horizontal between the pressure gradient and the

Coriolis force (geostrophic balance). In this chapter we exploit these balances tosimplify the Navier-Stokes equations and thereby obtain various sets of simplified‘geostrophic equations’. Depending on the precise nature of the assumptions wemake, we are led to the quasi-geostrophic system for horizontal scales similar tothat on which most synoptic activity takes place and, for very large-scale motion, tothe planetary-geostrophic set of equations. By eliminating unwanted or unimportantmodes of motion, in particular sound waves and gravity waves, and by building inthe important balances between flow fields, these filtered equation sets allow theinvestigator to better focus on a particular class of phenomenon and to potentiallyachieve a deeper understanding than might otherwise be possible.1

Simplifying the equations in this way relies first on scaling the equations. Theidea is that we choose the scales we wish to describe, typically either on some apriori basis or by using observations as a guide. We then attempt to derive a setof equations that is simpler than the original set but that consistently describesmotion of the chosen scale. An asymptotic method is one approach to this, for itsystematically tells us which terms we can drop and which we should keep. Thecombined approach — scaling plus asymptotics — has proven enormously useful,but it is useful to always remember two things: (i) that scaling is a choice; (ii) thatthe approach does not explain the existence of particular scales of motion, it justdescribes the motion that might occur on such scales. We have already employed

199

200 Chapter 5. Simplified Equations for Ocean and Atmosphere

this general approach in deriving the hydrostatic primitive equations, but now wego further.

5.1 GEOSTROPHIC SCALING

I have no satisfaction in formulas unless I feel their numerical magnitude.

William Thomson, Lord Kelvin (1824–1907).

5.1.1 Scaling in the Shallow Water Equations

Postponing the complications that come with stratification, we begin with the shal-low water equations. With the odd exception, we will denote the scales of vari-ables by capital letters; thus, if L is a typical length scale of the motion we wish todescribe, and U a typical velocity scale, and assuming the scales are horizontallyisotropic, we write

(x,y) ∼ L or (x,y) = O(L)(u,v) ∼ L or (u,v) = O(U).

(5.1)

and similarly for other variables. We may then nondimensionalize the variables bywriting

(x,y) = L(x, y), (u,v) = U(u, v) (5.2)

where the hatted variables are nondimensional and, by supposition, are O(1). Thevarious terms in the momentum equation then scale as:

∂u∂t

+ u · ∇u+ f × u = −g∇η (5.3a)

UT

U2

LfU g

HL

(5.3b)

where the∇ operator acts in the x, y plane andH is the amplitude of the variationsin the surface displacement. (We use η to denote the height of the free surface abovesome arbitrary reference level, as in Fig. 3.1. Thus, η = H +∆η, where ∆η denotesthe variation of η about its mean position.)

The ratio of the advective term to the rotational term in the momentum equa-tion (5.3) is (U2/L)

/(fU) = U/fL; this is the Rossby number, first encountered in

chapter 2.2 Using values typical of the large-scale circulation (e.g., from table 2.1)we find that Ro ≈ 0.1 for the atmosphere and Ro ≈ 0.01 for the ocean, small in bothcases. If we are interested in motion that has the advective timescale T = L/U thenwe scale time by L/U so that

t = LUt, (5.4)

and the local time derivative and the advective term then both scale as U2/L, andboth are order Rossby number smaller than the rotation term. Then, either theCoriolis term is the dominant term in the equation, in which case we have a stateof no motion with −fv = 0, or else the Coriolis force is balanced by the pressureforce, and the dominant balance is

− fv = −g ∂η∂x, (5.5)

5.1 Geostrophic Scaling 201

namely geostrophic balance, as encountered in chapter 2. If we make this nontrivialchoice, then the equation informs us that variations in η scale according to

∆η ∼H = fULg

(5.6)

We can also write H as

H = Rof 2L2

g= RoH

L2

L2d. (5.7)

where Ld =√gH/f is the deformation radius, and H is the mean depth of the fluid.

The variations in fluid height thus scale as

∆ηH

∼ RoL2

L2d, (5.8)

and the height of the fluid may be written as

η = H(

1+ RoL2

L2dη)

and ∆η = RoL2

L2dHη, (5.9)

where η is the O(1) nondimensional value of the surface height deviation.

Nondimensional momentum equation

If we use (5.9) to scale height variations, (5.2) to scale lengths and velocities, and(5.4) to scale time, then the momentum equation (5.3) becomes

Ro[∂ u∂t

+ (u · ∇)u]+ f × u = −∇η , (5.10)

where f = kf = kf/f0, where f0 is a representative value of the Coriolis param-eter. (If f is a constant, then f = 1, but it is useful to keep it in the equations toindicate the presence of Coriolis parameter. Also, where the operator ∇ operateson a nondimensional variable then the differentials are taken with respect to thenondimensional variables x, y.) All the variables in (5.10) will be supposed to beof order unity, and the Rossby number multiplying the local time derivative andthe advective terms indicates the smallness of those terms. By construction, thedominant balance in this equation is the geostrophic balance between the last twoterms.

Nondimensional mass continuity (height) equation

The (dimensional) mass continuity equation can be written

1H

DηDt

+(

1+ ∆ηH

)∇ · u = 0 (5.11)

Using (5.2), (5.4) and (5.9) this equation may be written

Ro(LLd

)2 DηDt

+[

1+ Ro(LLd

)2η]∇ · u = 0 . (5.12)


Equations (5.10) and (5.12) are the nondimensional versions of the full shallowwater equations of motion. Evidently, some terms in the equations of motion aresmall and may be eliminated with little loss of accuracy, and the way this is done willdepend on the size of the second nondimensional parameter, (L/Ld)2. We explorethis in sections 5.2 and 5.3.

Froude and Burger numbers

The Froude number may be generally defined as the ratio of a fluid particle speedto a wave speed. In a shallow-water system this gives

Fr ≡ U√gH

= Uf0Ld

= RoLLd. (5.13)

The Burger number3 is a useful measure of scale of motion of the fluid, relative tothe deformation radius, and may be defined by

Bu ≡(LdL

)2= gHf 2

0 L2=(Ro

Fr

)2. (5.14)

It is also useful to define the parameter F ≡ Bu−1, which is like the square of aFroude number but uses the rotational speed fL instead of U in the numerator.

5.1.2 Geostrophic scaling in the stratified equations

We now apply the same scaling ideas, mutatis mutandis, to the stratified primitiveequations. We use the hydrostatic anelastic equations, which we write as:

DuDt

+ f × u = −∇zφ, (5.15a)

∂φ∂z

= b, (5.15b)

DbDt

= 0, (5.15c)

∇ · (ρv) = 0. (5.15d)

where b is the buoyancy and ρ is a reference density profile. Anticipating that theaverage stratification may not scale in the same way as the deviation from it, let usseparate out the contribution of the advection of a reference stratification in (5.15c)by writing

b = b(z)+ b′(x,y, z, t). (5.16)

Then the thermodynamic equation becomes

Db′

Dt+N2w = 0, (5.17)

where N2 ≡ ∂b/∂z (and the advective derivative is still three-dimensional). Wethen letφ = φ(z)+φ′ where φ is hydrostatically balanced by b, and the hydrostaticequation becomes

∂φ′

∂z= b′. (5.18)

Equations (5.17) and (5.18) replace (5.15c) and (5.15b), and φ′ is used in (5.15a).

5.1 Geostrophic Scaling 203

Non-dimensional equations

We scale the basic variables by supposing that

(x,y) ∼ L, (u,v) ∼ U, t ∼ LU, z ∼ H, f ∼ f0, (5.19)

where the scaling variables (capitalized, except for f0) are chosen such that thenondimensional values have values of order unity. We presume that the scaleschosen are such that the Rossby number is small; that is Ro = U/(foL) 1. In themomentum equation the pressure term then balances the Coriolis force,

|f × u| ∼ |∇φ′| (5.20)

and so the pressure scales asφ′ ∼ Φ = foUL. (5.21)

Using the hydrostatic relation, (5.21) implies that the buoyancy scales as

b′ ∼ B = f0ULH

, (5.22)

and from this we obtain(∂b′/∂z)N2 ∼ Ro

L2

L2d, (5.23)

where Ld = NH/f0 is the deformation radius in the continuously stratified fluid,analogous to the quantity

√gH/f0 in the shallow water system, and we use the

same symbol, Ld, for both. In the continuously stratified system, if the scale ofmotion is the same as or smaller than the deformation radius, and the Rossby numberis small, then the variations in stratification are small. The choice of scale is the keydifference between the planetary geostrophic and quasi-geostrophic equations.

Finally, we will nodimensionalize the vertical velocity by using the mass conser-vation equation,

1ρ∂ρw∂z

= −(∂u∂x

+ ∂v∂y

), (5.24)

and we suppose that this implies

w ∼ W = UHL. (5.25)

This is a naïve scaling for rotating flow: if the Coriolis parameter is nearly constantthe geostrophic velocity is nearly horizontally non-divergent and the right-handside of (5.24) is small, and W UH/L. We might then estimate w by cross-differentiating geostrophic balance to obtain the linear geostrophic vorticity equa-tion and corresponding scaling:

βv ≈ f ∂w∂z, w ∼ W = βUH

f0. (5.26a,b)

However, rather than using (5.26b) from the outset, we will use (5.25) and letthe asymptotics guide us to a proper scaling in the fullness of time. Note that ifvariations in the Coriolis parameter are large and β ∼ f0/L, then (5.26b) is thesame as (5.25).


Given the scalings above [using (5.25) for w] we nondimensionalize by setting

(x, y) = L−1(x,y), z = H−1z, (u, v) = U−1(u,v), t = ULt,

w = LUH

w, f = f−10 f , φ = φ′

f0UL, b = H

f0ULb′.

(5.27)

where the hatted variables are nondimensional. The horizontal momentum andhydrostatic equations then become

RoDuDt

+ f × u = −∇φ, (5.28)

and∂φ∂z

= b. (5.29)

The non-dimensional mass conservation equation is simply

1ρ∇ · (ρv) =

(∂u∂x

+ ∂v∂y

+ 1ρ∂ρw∂z

)= 0. (5.30)

and the nondimensional thermodynamic equation is

f0ULH

UL

DbDt

+N2HULw = 0, (5.31)

or

RoDbDt

+(LdL

)2w = 0. (5.32)

The nondimensional primitive equations are summarized in the box on the facingpage.

5.2 THE PLANETARY GEOSTROPHIC EQUATIONS

We now use the low Rossby number scalings above to derive equation sets that aresimpler than the original, ‘primitive’, ones. The planetary geostrophic equations areprobably the simplest such set of equations, and we derive these equations first forthe shallow water equations, and then for the stratified primitive equations.

5.2.1 Using the shallow water equations

Informal derivation

The advection and time derivative terms in the momentum equation (5.10) are or-der Rossby number smaller than the Coriolis and pressure terms (the term in squarebrackets is multiplied by Ro), and therefore let us neglect them. The momentumequation straightforwardly becomes

f × u = −∇η. (5.33)

5.2 The Planetary Geostrophic Equations 205

Nondimensional Primitive Equations

Horizontal momentum: RoDuDt

+ f × u = −∇φ (NDPE.1)

Hydrostatic:∂φ∂z

= b (NDPE.2)

Mass continuity:(∂u∂x

+ ∂v∂y

+ 1ρ∂ρw∂z

)= 0 (NDPE.3)

Thermodynamic: RoDbDt

+(LdL

)2w = 0 (NDPE.4)

These equations are written for the anelastic equations. The Boussinesq equa-tions result if we take ρ = 1. The equations in pressure coordinates have avery similar form to the Boussinesq equations, but with a slight difference inhydrostatic equation.

The mass conservation equation (5.12), contains two nondimensional parame-ters, Ro = U/(f0L) (the Rossby number), and F = L/Ld (the ration of the lengthscale of the motion to the deformation scale) and we must make a choice as to therelationship of these two numbers. We will choose

FRo = O(1), (5.34)

which implies

L2 L2d or equivalently F 1, Bu 1. (5.35)

That is to say, we suppose that the scales of motion are much larger than the de-formation scale. Given this choice, all the terms in the mass conservation equation,(5.12), are of roughly the same size, and we retain them all. Thus, the shallowwater planetary geostrophic equations are the full mass continuity equation alongwith geostrophic balance and a geometric relationship between the height field andfluid thickness, and in dimensional form these are:

DhDt

+ h∇ · u = 0

f × u = −g∇η, η = h+ ηb. (5.36a,b)

We emphasize that the planetary geostrophic equations are only valid for scales ofmotion much larger than the deformation radius. The height variations are then aslarge as the mean height field itself; that is, using (5.8), ∆η/H = O(1).

Formal derivation

We assume that:


(i) The Rossby number is small. Ro = U/f0L 1.(ii) The scale of the motion is significantly larger than the deformation scale. That

is, (5.34) holds or

F = Bu−1 =(LLd

)2 1 (5.37)

and in particularFRo = O(1). (5.38)

(iii) Time scales advectively, so that T = L/U .The idea is now to expand the nondimensional variables velocity and height fieldsin an asymptotic series with Rossby number as the small parameter, substitute intothe equations of motion, and derive a simpler set of equations. It is a nearly trivialexercise in this instance, and so it illustrates well the methodology. The expansionsare

u = u0 + Rou1 + Ro2u2 + · · · (5.39a)

andη = η0 + Roη1 + Ro2η2 + · · · (5.39b)

Then substituting (5.39a) and (5.39b) into the momentum equation gives

Ro[∂ u0

∂t+ u0 · ∇u0 + f × u1

]+ f × u0 = −∇η0 − Ro

[∇0η1

]+O(Ro2) (5.40)

The Rossby number is now an asymptotic ordering parameter; thus, the sum of allthe terms at any particular order in Rossby number must vanish. At lowest orderwe obtain the simple expression

f × u0 = −∇η0. (5.41)

Note that although f0 is a representative value of f , we have made no assumptionsabout the constancy of f . In particular, f is allowed to vary by an order one amount,provided that it does not become so small that the Rossby number (U/f0L) is notsmall.

The appropriate height (mass conservation) equation is similarly obtained bysubstituting (5.39a) and (5.39b) into the shallow water mass conservation equa-tion. Because FRo = O(1) at lowest order we simply retain all the terms in theequation to give

FRo[∂η0

∂t+ u0 · ∇η0

]+[1+ FRoη

]∇ · u0 = 0. (5.42)

Equations (5.41) and (5.42) are a closed set, and constitute the nondimensionalplanetary geostrophic equations. The dimensional forms of these equations are just(5.36).

Variation of the Coriolis parameter

Suppose then that f is a constant (f0), or nearly so. Then, from the curl of (5.41),∇ · u0 = 0. This means that we can define a streamfunction for the flow and,

5.2 The Planetary Geostrophic Equations 207

from geostrophic balance, the height field is just that streamfunction. That is, indimensional form,

ψ = gf0η, u = −k×∇ψ, (5.43)

and (5.42) becomes, in dimensional form,

∂η∂t+ u · ∇η = 0, or

∂η∂t+ J(ψ,η) = 0. (5.44)

where J(a, b) ≡ axby−aybx. But since η∝ ψ the advective term is proportional toJ(ψ,ψ), which is zero. Thus, the flow does not evolve at this order. The planetarygeostrophic equations are uninteresting if the scale of the motion is such that theCoriolis parameter is not variable. On earth, the scale of motion on which thisparameter regime exists is rather limited, since the planetary geostrophic equationsrequire that the scale of motion is also larger than the deformation radius. In theearth’s atmosphere, any scale that is larger than the deformation radius will be suchthat the Coriolis parameter varies significantly over it, and we do not encounter thisparameter regime. On the other hand, in the earth’s ocean the deformation radiusis relatively small and there exists a small parameter regime that has scales largerthan the deformation radius but smaller than that on which the Coriolis parametervaries.4

Potential vorticity

The shallow water PG equations may be written as an evolution equation for an ap-proximated potential vorticity. A little manipulation reveals that (5.36) are equiva-lent to:

DQDt

= 0

Q = fh, f × u = −g∇η, η = h+ ηb

(5.45)

Thus, potential vorticity is a material invariant in the approximate equation set,just as it is in the full equations. The other variables — the free surface heightand the velocity — are diagnosed from it, a process known as potential vorticityinversion. In the planetary geostrophic approxmation, the inversion proceeds usingthe approximate form f/h rather than the full potential vorticity, (f+ζ)/h. (Strictlyspeaking, we do not approximate potential vorticity, because this is the evolvingvariable. Rather, we approximate the inversion relations from which we derive theheight and velocity fields.) The simplest way of all to derive the shallow water PGequations is to begin with the conservation of potential vorticity, and to note that atsmall Rossby number the expression (ζ+f)/h may be approximated by f/h. Then,noting in addition that the flow is geostrophic, (5.45) immediately emerges. Everyapproximate set of equations that we derive in this chapter may be expressed as theevolution of potential vorticity, with the other fields being obtained diagnosticallyfrom it.

5.2.2 The Planetary geostrophic equations for stratified flow


To explore the stratified system we will use the (inviscid and adiabatic) Boussinesqequations of motion with the hydrostatic approximation. The derivation carriesthrough easily enough using the anelastic or pressure-coordinate equations, but asthe PG equations have more oceanographic importance than atmospheric using theincompressible equations is quite appropriate.

Simplifying the equations

The nondimensional equations we begin with are (5.28)–(5.32). As in the shallowwater case we expand these in a series in Rossby number, so that:

u = u0 + εu1 + ε2u2 + · · · , b = b0 + εb1 + ε2b2 + · · · , (5.46)

and similarly for v, w and φ, where ε = Ro, the Rossby number. Substituting intothe nondimensional equations of motion (on page 205) and equating powers of εgives the lowest order momentum, hydrostatic, and mass conservation equations:

f × u0 = −∇φ0, (5.47a)

∂φ0

∂z= b0, (5.47b)

∇ · v0 = 0. (5.47c)

If we also assume that Ld/L = O(1), then the thermodynamic equation (5.32) be-comes (

LdL

)2w0 = 0. (5.48)

Of course we have neglected any diabatic terms in this equation, which wouldin general provide a non-zero right-hand side. Nevertheless, this is not a usefulequation, because the set of the equations we have derived, (5.47), can no longerevolve: all the time derivatives have been scaled away! Thus, although instructive,these equations are not very useful. If instead we assume that the scale of motionis much larger than the deformation scale then the other terms in the thermody-namic equation will become equally important. Thus, we suppose that Ld L2 or,more formally, that L2 = O(Ro−1)L2

d, and then all the terms in the thermodynamicequation are retained. A closed set of equations is then given by (5.47) and thethermodynamic equation (5.32).

Dimensional equations

Restoring the dimensions, dropping the asymptotic subscripts, and allowing for thepossibility of a source term, denoted S[b′], in the thermodynamic equation, theplanetary-geostrophic equations of motion are:

Db′

Dt+wN2 = S[b′]

f × u = −∇φ′

∂φ′

∂z= b′

∇ · v = 0

. (5.49)

5.3 The Shallow Water Quasi-Geostrophic Equations 209

The thermodynamic equation may also be written simply as

DbDt

= b (5.50)

where b now represents the total stratification. The relevant pressure, φ, is then thepressure that is in hydrostatic balance with b, so that geostrophic and hydrostaticbalance are most usefully written as

f × u = −∇φ, ∂φ∂z

= b. (5.51a,b)

Potential vorticity

Manipulation of (5.49) reveals that we can equivalently write the equations as anevolution equation for potential vorticity. Thus, the evolution equations may bewritten

DQDt

= Q

Q = f ∂b∂z

, (5.52)

where Q = f∂b/∂z, and the inversion — i.e., the diagnosis of velocity, pressure andbuoyancy — is carried out using the hydrostatic, geostrophic and mass conservationequations.

Applicability to the ocean and atmosphere

In the atmosphere a typical deformation radius NH/f is about 1,000 km. The con-straint that the scale of motion be much larger than the deformation radius is thusquite hard to satisfy, since one quickly runs out of room on a planet whose equator-to-pole distance is 10,000 km. Thus, only the largest planetary waves can satisfythe planetary-geostrophic scaling in the atmosphere and we should then also writethe equations in spherical coordinates.

In the ocean the deformation radius is about 100 km, so there is lots of room forthe planetary-geostrophic equations to hold, and indeed much of the theory of thelarge-scale structure of the ocean involves the planetary-geostrophic equations.

5.3 THE SHALLOW WATER QUASI-GEOSTROPHIC EQUATIONS

We now derive a set of geostrophic equations that is valid (unlike the PG equations)when the horizontal scale of motion is similar to that of the deformation radius.These equations are called the quasi-geostrophic equations, and are perhaps themost widely used set of equations for theoretical studies of the atmosphere andocean. The specific assumptions we make are:

(i) The Rossby number is small, so that the flow is in near-geostrophic balance.

(ii) The scale of the motion is not significantly larger than the deformation scale.Specifically, we shall require that

Ro(LLd

)2= O(Ro). (5.53)


For the shallow water equations, this assumption implies, using (5.9), thatthe variations in fluid depth are small compared to its total depth. For thecontinuously stratified system it implies, using (5.23), that the variations instratification are small compared to the background stratification.

(iii) Variations in the Coriolis parameter are small. That is, |βL| |f0| where L isthe length-scale of the motion.

(iv) Time scales advectively; that is, the scaling for time is given by T = L/U .

The second and third of these differ from the planetary geostrophic counterparts:we make the second assumption because we wish to explore a different parameterregime, and we then find that the third assumption is necessary to avoid a rathertrivial state (i.e., a leading order balance of βv = 0, see the discussion surrounding(5.77)). All of the assumptions are the same whether we consider the shallowwater equations or a continuously stratified flow, and in this section we considerthe former.

5.3.1 Single-layer shallow water quasi-geostrophic equations

The algorithm is, again, to expand the variables u, v, η in an asymptotic series withRossby number as the small parameter, substitute into the equations of motion, andderive a simpler set of equations. Thus we let

u = uo + Rou1 + Ro2u2 + · · · , v = vo + Rov1 + Ro2v2 + · · · (5.54a)

η = η0 + Roη1 + Ro2η2 · · · . (5.54b)

We will recognize the smallness of β compared to f0/L by letting β = βU/L2, whereβ is assumed to be a parameter of order unity. Then the expression f = f0 + βybecomes

f = f/f0 = f0 + Roβy. (5.55)

where f0 is the nondimensional value of f0; its value is unity, but it is helpful todenote it explicitly. Substitute (5.54) into the nondimensional momentum equation(5.10), and equate powers of Ro. At lowest order we obtain

f0u0 = −∂η0

∂y, f0v0 =

∂η0

∂x. (5.56)

Cross-differentiating gives∇ · u0 = 0, (5.57)

where, when ∇ operates on a nondimensional variable, the derivatives are takenwith respect to the nondimensional variables x and y. From (5.57) we see that thevelocity field is divergence-free, and that this arises from the momentum equationrather than the mass conservation equation.

The mass conservation equation is also, at lowest order, ∇ · u0 = 0, and at nextorder we have

F∂η0

∂t+ Fu0 · ∇η0 +∇ · u1 = 0 (5.58)


This equation is not closed, because the evolution of the zeroth order term involvesevaluation of a first order quantity. For closure, we go to next order in the momen-tum equation,

∂ u0

∂t+ (u0 · ∇)u0 + βyk× u0 − f0k× u1 = −∇η1, (5.59)

and take its curl to give the vorticity equation:

∂ζ0

∂t+ (u0 · ∇)(ζ0 + βy) = −f0∇ · u1. (5.60)

The term on the right-hand side is the vortex stretching term. Only vortex stretchingby the background or planetary vorticity is present, because the vortex stretchingby the relative vorticity is a factor Rossby number smaller. Eq. (5.60) is also notclosed; however, we may use (5.58) to eliminate the divergence term to give

∂ζ0

∂t+ (u0 · ∇)(ζ0 + βy) = f0

(F∂η0

∂t+ Fu · ∇η0

), (5.61)

or∂∂t(ζ0 − f0Fη0)+ (u0 · ∇)(ζ0 + βy − Fη0) = 0. (5.62)

The final step is to note that the lowest order vorticity and height fields arerelated through geostrophic balance, so that using (5.56) we can write

u0 = −∂ψ0

∂y, v0 =

∂ψ0

∂x, ζ0 = ∇2ψ0, (5.63)

where ψ0 = η0/f0 is the streamfunction. Eq. (5.62) can thus be written,

∂∂t(∇2ψ0 − f 2

0 Fψ0)+ (u0 · ∇)(ζ0 + βy − f 20 Fψ0) = 0, (5.64)

orD0

Dt(∇2ψ0 + βy − f 2

0 Fψ0) = 0, (5.65)

where the subscript ‘0’ on the material derivative indicates that the lowest order ve-locity, the geostrophic velocity, is the advecting velocity. Restoring the dimensions,(5.65) becomes

DDt(∇2ψ+ βy − 1

L2dψ) = 0 , (5.66)

where ψ = (g/f0)η, L2d = gH/f 2

0 , and the advective derivative is

DDt

= ∂∂t+ug

∂∂x

+ vg∂∂y

= ∂∂t− ∂ψ∂y

∂∂x

+ ∂ψ∂x

∂∂y

= ∂∂t+ J(ψ, ·). (5.67)

Another form of (5.66) is

DDt(ζ + βy − f0

Hη) = 0, (5.68)


with ζ = (g/f0)∇2η. Equations (5.66) and (5.68) are forms of the shallow-waterquasi-geostrophic potential vorticity equation. The quantity

q ≡ ζ + βy − f0

Hη = ∇2ψ+ βy − 1

L2dψ (5.69)

is the shallow water quasi-geostrophic potential vorticity.

Connection to shallow water potential vorticity

The quantity q given by (5.69) is an approximation (except for dynamically unim-portant constant additive and multiplicative factors) to the shallow water potentialvorticity. To see the truth of this statement, begin with the expression for the shal-low water potential vorticity,

Q = f + ζh

. (5.70)

Now let h = H(1 + η′/H), where η′ is the perturbation of the free-surface height,and assume that η′/H is small to obtain

Q = f + ζH(1+ η′/H) ≈

1H(f + ζ)

(1− η

′

H

)≈ 1H

(f0 + βy + ζ − f0

η′

H

). (5.71)

Because f0/H is a constant it has no effect in the evolution equation, and the quan-tity given by

q = βy + ζ − f0η′

H(5.72)

is materially conserved. Using geostrophic balance we have ζ = ∇2ψ and η′ =f0ψ/g so that (5.72) is identical to (5.69). [Note that only the variation in η areimportant in (5.68) or (5.69).]

The approximations needed to go from (5.70) to (5.69) are the same as thoseused in our earlier, more long-winded, derivation of the quasi-geostrophic equa-tions. That is, we assumed that f itself is nearly constant, and that f0 is muchlarger than ζ, equivalent to a low Rossby number assumption. It was also nec-essary to assume that H η′ to enable the expansion of the height field which,using assumption (ii) on page 209, is equivalent to requiring that scale of motionnot be significantly larger than the deformation scale. The derivation is completedby noting that the advection of the potential vorticity should be by the geostrophicvelocity alone, and we recover (5.66) or (5.68).

Two interesting limits

There are two interesting limits to the quasi-geostrophic potential vorticity equa-tion:

(i) Motion on scales much smaller than the deformation radius.That is, L Ld and thus Bu 1 or F 1. Then (5.65) becomes

∂ζ∂t

+ uψ · ∇ζ = 0 or∂ζ∂t

+ J(ψ,ζ) = 0, (5.73)

where ζ = ∇2ψ and J(ψ,ζ) = ψxζy −ψyζx. Thus, the motion obeys the two-dimensional vorticity equation. Physically, on small length scales the deviationsin the height field are very small and may be neglected.


(ii) Motion on scales much larger than the deformation radius.Although scales are not allowed to become so large that Ro(L/Ld)2 is of orderunity, we may, a posteriori, still have L Ld, whence the potential vorticityequation becomes

∂η∂t+ uψ · ∇η = 0 or

∂η∂t+ J(ψ,η) = 0. (5.74)

However, because ψ = gη/f0, the Jacobian term vanishes. Thus, one is leftwith a trivial equation that implies there is no advective evolution of the heightfield. There is nothing wrong with our reasoning; the mathematics have indeedpointed out a limit interesting in its uninterestingness. From a physical pointof view, however, such (lack of) motion is likely to be rare, because on suchlarge scales the Coriolis parameter varies considerably, and we are led to theplanetary geostrophic equations.

In practice, often the most severe restriction of quasi-geostrophy is that variationsin layer thickness are small: what does this have to do with geostrophy? If we scaleη assuming geostrophic balance then η ∼ fUL/g and η/H ∼ Ro(L/Ld)2. Thus, ifRo is to remain small, η/H can only be order one if (L/Ld)2 1. That is, the heightvariations must occur on a large scale, or we are led to a scaling inconsistency. Putanother way, if there are order-one height variations over a length-scale less than ororder of the deformation scale, the Rossby number will not be small. Large height vari-ations are allowed if the scale of motion is large, but this contingency is describedby the planetary geostrophic equations.

Another flow regime

Although perhaps of little terrestrial interest, we can imagine a regime in which theCoriolis parameter varies fully, but the scale of motion remains no larger than thedeformation radius. This parameter regime is not quasigeostrophic, but it gives aninteresting result. Because η′/H ∼ Ro(L/Ld)2 deviations of the height field are atleast order Rossby number smaller than the reference height and |η′| H. Thedominant balance in height equation is then

H∇ · u = 0, (5.75)

presuming that time still scales advectively. This zero horizontal divergence mustremain consistent with geostrophic balance

f × u = −g∇η, (5.76)

where now f is a fully variable Coriolis parameter. Taking the curl of (i.e., cross-differentiating) (5.76) gives

βv + f∇ · u = 0, (5.77)

whence, using (5.75), v = 0, and the flow is purely zonal. Although not at all usefulas an evolution equation, this illustrates the constraining effect that differentialrotation has on meridional velocity. This effect may be the cause of the banded,highly zonal flow on some the the giant planets, and we will revisit this issue in ourdiscussion of geostrophic turbulence.


Fig. 5.1 A quasi-geostrophic fluid system consisting of two immiscible fluidsof different density. The quantities η′ are the interface displacemens fromthe resting basic state, denoted with dashed lines, with ηb being the bottomtopography.

5.3.2 Two-layer and multi-layer quasi-geostrophic systems

Just as for the one-layer case, the multi-layer shallow water equations simplify toa corresponding quasi-geostrophic system in appropriate circumstances. The as-sumptions are virtually same as before, although we assume that the variation inthe thickness of each layer is small compared to its mean thickness. The basic fluidsystem for a two-layer case is sketched in Fig. 5.1 (and see also Fig. 3.5), and forthe multi-layer case in Fig. 5.2.

Let us proceed directly from the potential vorticity equation for each layer. Wewill also stay in dimensional variables, foregoing a strict asymptotic approach forthe sake of informality and insight, and use the Boussinesq approximation. For eachlayer the potential vorticity equation is just

DQiDt

= 0, Qi =ζi + fhi

. (5.78)

Let hi = Hi + h′i where |h′i| << Hi. The potential vorticity then becomes

Qi ≈1Hi(ζi + f)

(1−

h′iHi

)— variations in layer thickness are small

(5.79a)

≈ 1Hi

(f + ζi − f

h′iHi

)— the Rossby number is small (5.79b)

≈ 1Hi

(f + ζi − f0

h′iHi

)— variations in Coriolis parameter are small

(5.79c)


Now, because Q appears in the equations only as an advected quantity, it is only thevariations in Coriolis parameter that are important in the the first term on the right-hand side of (5.79c), and given this all three terms are of the same approximatemagnitude. Then, because mean layer thicknesses are constant, we can define thequasi-geostrophic potential vorticity in each layer by

qi =(βy + ζi − f0

h′iHi

), (5.80)

and this will evolve according to Dqi/Dt = 0, where the advective derivative is bythe geostrophic wind. As in the one-layer case, quasi-geostrophic potential vorticityhas different dimensions from the full shallow water potential vorticity.

Two-layer model

To obtain a closed set of equations we must obtain an advecting field from the po-tential vorticity. We use geostrophic balance to do this, and neglecting the advectivederivative in (3.51) gives

f0 × u1 = −g∇η0 = −g∇(h′1 + h′2 + ηb), (5.81a)

f0 × u2 = −g∇η0 − g′∇η1 = −g∇(h′1 + h′2 + ηb)− g′∇(h2 + ηb), (5.81b)

where g′ = (ρ2 − ρ1)/ρ1 and ηb is the height of any bottom topography, and,because variations in the Coriolis parameter are presumptively small, we use a con-stant value of f (i.e., f0) on the left-hand side. For each layer there is therefore astreamfunction, given by

ψ1 =gf0(h′1 + h′2 + ηb), ψ2 =

gf0(h′1 + h′2 + ηb)+

g′

f0(h′2 + ηb), (5.82a,b)

and these two equations may be manipulated to give

h′1 =f0

g′(ψ1 −ψ2)+

f0

gψ1, h′2 =

f0

g′(ψ2 −ψ1)− ηb. (5.83a,b)

We note as an aside that the interface displacements are given by

η′0 =f0

gψ1, η′1 =

f0

g′(ψ2 −ψ1). (5.84a,b)

Using (5.80) and (5.83) the quasi-geostrophic potential vorticity for each layerbecomes

q1 = βy +∇2ψ1 +f 2

0

g′H1(ψ2 −ψ1)+

f 20

gH1ψ1

q2 = βy +∇2ψ2 +f 2

0

g′H2(ψ1 −ψ2)+ f0

ηbH2

. (5.85a,b)

In the rigid-lid approximation the last term in (5.85a) is neglected. The poten-tial vorticity in each layer is just advected by the geostrophic velocity, so that theevolution equation for each layer is just

∂qi∂t

+ J(ψi, qi) = 0, i = 1,2. (5.86)


Fig. 5.2 A multi-layer quasi-geostrophic fluid system. Layers are numberedfrom the top down, i denotes a general interior layer and N denotes thebottom layer.

* Multi-layer model

A multi-layer quasi-geostrophic model may be constructed by a straightforward ex-tension of the above two-layer procedure (see Fig. 5.2). The quasi-geostrophicpotential vorticity for each layer is still given by (5.80). The pressure field in eachlayer can be expressed in terms of the thickness of each layer using (3.46) and(3.47) on page 132, and by geostrophic balance the pressure is proportional to thestreamfunction, ψi, for each layer. Carrying out these steps we obtain, after a littlealgebra, the following expression for the quasi-geostrophic potential vorticity of aninterior layer, in the Boussinesq approximation:

qi = βy +∇2ψi +f 2

0

Hi

(ψi−1 −ψig′i−1

− ψi −ψi+1

g′i

), (5.87)

and for the top and bottom layers,

q1 = βy +∇2ψ1 +f 2

0

H1

(ψ2 −ψ1

g′1

)+ f 2

0

gH2ψ1, (5.88a)

qN = βy +∇2ψN +f 2

0

HN

(ψN−1 −ψNg′N−1

)+ f0

HNηb. (5.88b)

In these equations Hi is the basic-state thickness of the i’th layer, and g′i = g(ρi+1−ρi)/ρ1. In each layer the evolution equation is (5.86), now for i = 1 · · ·N. Thedisplacements of each interface are given, similarly to (5.84), by

η′0 =f0

gψ1, η′i =

f0

g′i(ψi+1 −ψi). (5.89a,b)

5.4 The Continuously Stratified Quasi-Geostrophic System 217

5.3.3 †Non-asymptotic and intermediate models

The form of the derivation of the previous section suggests that we might be able toimprove on the accuracy and range of applicability of the quasi-geostrophic equa-tions, whilst still filtering gravity waves. For example, a seemingly improved set ofgeostrophic evolution equations might be

∂qi∂t

+ ui · ∇qi = 0, (5.90)

withqi =

f + ζihi

, ζi =∂vi∂x

− ∂ui∂y, (5.91a,b)

and with the velocities given by geostrophic balance, and therefore a function of thelayer depths. Thus, the vorticity, height, and velocity fields may all be inverted frompotential vorticity. Note that the inversion does not involve the linearization of po-tential vorticity about a resting state [compare (5.91a) with (5.80)], and we mightalso choose to keep the full variation of the Coriolis parameter in (5.81). Thus, themodel consisting of (5.90) and (5.91) contains both the planetary geostrophic andquasi-geostrophic equations. However, the informality of the derivation hides thefact that this is not an asymptotically consistent set of equations: it mixes asymp-totic orders in the same equation, and good conservation properties are not assured.The set above does not, in fact, exactly conserve energy. Models that are eithermore accurate or more general than the quasi-geostrophic or planetary geostrophicequations yet that still filter gravity waves are called ‘intermediate models’.5

A model that is derived asymptotically will, in general, maintain the conser-vation properties of the original set. To see this, albeit in a rather abstract way,suppose that the original equations (e.g., the primitive equations) may be writtenin non-dimensional form, as

∂φ∂t

= F(φ, ε) (5.92)

where φ is a set of variables, F is some operator, and ε is a small parameter, like theRossby number. Suppose also that this set of equations has various invariants (suchas energy and potential vorticity) that hold for any value of ε. The asymptotically-derived lowest order model (such as quasi-geostrophy) is simply a version of thisequation set valid in the limit ε = 0, and therefore it will preserve the invariantsof the original set. These invariants may seem to have a different form in thesimplified set: for example, in deriving the hydrostatic primitive equations from theNavier-Stokes equations the small parameter is the aspect ratio, and this multipliesthe vertical velocity. Thus, in the limit of zero aspect ratio, and therefore in theprimitive equations, the conserved kinetic energy has contributions only from thehorizontal velocity. In other cases, some invariants may be reduced to trivialities inthe simplified set. On the other hand, there is nothing to preclude new invariantsemerging that hold only in the limit ε = 0, and enstrophy (considered later in thischapter) is one example.6

5.4 THE CONTINUOUSLY STRATIFIED QUASI-GEOSTROPHIC SYSTEM

We now consider the quasi-geostrophic equations for the continuously stratifiedhydrostatic system. The primitive equations of motion are given by (5.15), and


we extract the mean stratification so that the thermodynamic equation is givenby (5.17). We also stay on the β-plane for simplicity. Readers who wish for abriefer, more informal derivation may peruse the box on page 224; however, it isimportant to realize that there is a systematic asymptotic derivation of the quasi-geostrophic equations, for it is this that ensures that the resulting equations havegood conservation properties, as explained in section 5.3.3.

5.4.1 Scaling and assumptions

The scaling assumptions we make are just those we made for the shallow watersystem on page 209, with a deformation radius now given by Ld = NH/f0. Thenondimensionalization and scaling is initially precisely that of section 5.1.2, and sowe obtain the following non-dimensional equations:

Horizontal momentum: RoDuDt

+ f × u = −∇zφ, (5.93)

Hydrostatic:∂φ∂z

= b, (5.94)

Mass continuity:∂u∂x

+ ∂v∂y

+ 1ρ∂ρw∂z

= 0, (5.95)

Thermodynamic: RoDbDt

+(LdL

)2w = 0. (5.96)

In Cartesian coordinates we may express the Coriolis parameter as

f = f0 + βy k (5.97)

where f0 = f0 k. The variation of the Coriolis parameter is assumed small (this is akey difference between the quasi-geostrophic system and the planetary geostrophicsystem), and in particular we shall assume that βy is approximately the size of therelative vorticity, and so much smaller than f0 itself.7 Thus,

βy ∼ UL, β ∼ U

L2 , (5.98)

and so we define an O(1) non-dimensional beta parameter by

β = βL2

U= βL

Rof0. (5.99)

From this it follows that if f = f0 + βy, the corresponding nondimensional versionis

f = f0 + Ro βy. (5.100)

where f = f/f0 and f0 = f0/f0 = 1.

5.4.2 Asymptotics

We now expand the nondimensional dependent variables in an asymptotic series inRossby number, and write

u = u0+Rou1+· · · , φ = φ0+Roφ1+· · · , b = b0+Rob1+· · · . (5.101)


Substituting these into the equations of motion, the lowest order momentum equa-tion is simply geostrophic balance,

f0 × u0 = −∇φ0 (5.102)

with a constant value of the Coriolis parameter. (For the rest of this chapter we dropthe subscript z from the ∇ operator.) From (5.102) it is evident that

∇ · u0 = 0. (5.103)

Thus, the horizontal flow is, to leading order, non-divergent; this is a consequenceof geostrophic balance, and is not a mass conservation equation. Using (5.103) inthe mass conservation equation, (5.95), gives

∂∂z(ρw0) = 0, (5.104)

which implies that if w0 is zero somewhere (e.g., at a solid surface) then w0 iszero everywhere (essentially the Taylor-Proudman effect). A physical way of sayingthis is that the scaling estimate W = UH/L is an overestimate of the size of thevertical velocity, because even though ∂w/∂z ≈ −∇ · u, the horizontal divergenceof the geostrophic flow is small if f is nearly constant and |∇ · u| U/L. Wemight have anticipated this from the outset, and scaled w differently, perhaps usingthe geostrophic vorticity balance estimate, w ∼ βUH/f0 = RoUH/L as the scalingfactor for w, but there is no a priori guarantee that this would be correct.

At next order the momentum equation is

D0u0

Dt+ βyk× u0 + f × u1 = −∇φ1, (5.105)

where D0/Dt = ∂/∂t + (u0 · ∇), and the next order mass conservation equation is

∇z · (ρu1)+∂∂z(ρw1) = 0. (5.106)

From (5.96), the lowest order thermodynamic equation is just(LdL

)2w0 = 0 (5.107)

provided that, as we have assumed, the scales of motion are not sufficiently largethat Ro(L/Ld)2 = O(1). (This is a key difference between quasi-geostrophy andplanetary geostrophy.) At next order we obain an evolution equation for the buoy-ancy, and this is

D0b0

Dt+ w1

(LdL

)2= 0. (5.108)

The potential vorticity equation

To obtain a single evolution equation for lowest order quantities we eliminate w1

between the thermodynamic and momentum equations. Cross differentiating thefirst order momentum equation (5.105) gives the vorticity equation,

∂ζ0

∂t+ (u0 · ∇)ζ0 + v0β = −f0∇z · u1. (5.109)


(In dimensional terms, the divergence on the right-hand side is small, but is multi-plied by the large term f0, and their product is the same order as the terms on theleft-hand side.) Using the mass conservation equation (5.106), (5.109) becomes

D0

Dt(ζ0 + f ) =

f0

ρ∂∂z(w1ρ) (5.110)

Combining (5.110) and (5.108) gives

D0

Dt(ζ0 + f ) = −

f0

ρ∂∂z

(D0

Dt(Fρb0)

)(5.111)

where F ≡ (L/Ld)2. The right-hand side of this equation is

∂∂z

(D0b0

Dt

)= D0

Dt

(∂b0

∂z

)+ ∂ u0

∂z· ∇b0. (5.112)

The second term on the right-hand side vanishes identically using the thermal windequation

k× ∂ u0

∂z= − 1

f0∇b0, (5.113)

and so (5.111) becomes

D0

Dt

[ζo + f +

f0

ρ∂∂z

(ρFb0

)]= 0, (5.114)

or, after using the hydrostatic equation,

D0

Dt

[ζ0 + f +

f0

ρ∂∂z

(ρF∂φ0

∂z

)]= 0. (5.115)

Since the lowest-order horizontal velocity is divergence-free, we can define astreamfunction ψ such that

u0 = −∂ψ∂y, v0 =

∂ψ∂x

(5.116)

where also, using (5.102), φ0 = f0ψ. The vorticity is then given by ζ0 = ∇2ψ and(5.115) becomes a single equation in a single unknown, to wit

D0

Dt

∇2ψ+ βy + f20

ρ∂∂z

(ρF∂ψ∂z

) = 0 , (5.117)

where the material derivative is evaluated using u0 = k × ∇ψ. This is the nondi-mensional form of the quasi-geostrophic potential vorticity equation, one of themost important equations in dynamical meteorology and oceanography. In derivingit we have reduced the Navier Stokes equations, which are six coupled nonlinearpartial differential equations in six unknowns (u,v,w,T ,p, ρ) to a single (albeitnonlinear) first-order partial differential equation in a single unknown.8


Dimensional equations

The dimensional version of the quasi-geostrophic potential vorticity equation maybe written,

DqDt

= 0,

q = ∇2ψ+ f + f20

ρ∂∂z

(ρN2∂ψ∂z

) . (5.118a,b)

where only the variable part of f (e.g., βy) is relevant in the second term on theright-hand side of the expression for q. The quantity q is known as the quasi-geostrophic potential vorticity. It is analogous to the exact (Ertel) potential vorticity(see section 5.5 for more about this), and it is conserved when advected by thehorizontal geostrophic flow. All the other dynamical variables may be obtainedfrom potential vorticity as follows:

(i) Streamfunction, using (5.118b).(ii) Velocity: u = k×∇ψ [≡ ∇⊥ψ = −∇× (kψ)].(iii) Relative vorticity: ζ = ∇2ψ .(iv) Perturbation pressure: φ = f0ψ.(v) Perturbation buoyancy: b′ = f0∂ψ/∂z.

The length-scale Ld = NH/f0, emerges naturally from the QG dynamics. Itis the scale at which buoyancy and relative vorticity effects contribute equally tothe potential vorticity, and is called the deformation radius; it is analogous to thequantity

√gH/f0 arising in shallow water theory. In the upper ocean, with N ≈

10−2 s−1, H ≈ 103 m, and f0 ≈ 10−4 s−1, then Ld ≈ 100 km. At high latitudes theocean is much less stratified and f is somewhat larger, and the deformation radiusmay be as little as 30 km (see Fig. 9.11 on 407, where the deformation radius isdefined slightly differently). In the atmosphere, with N ≈ 10−2 s−1, H ≈ 104 m,then Ld ≈ 1000 km. It is this order of magnitude difference in the deformationscales that accounts for a great deal of the quantitative difference in the dynamicsof the ocean and atmosphere. If we take the limit Ld → ∞ then the stratified quasi-geostrophic equations reduce to

DqDt

= 0, q = ∇2ψ+ f (5.119)

This is the two-dimensional vorticity equation, identical to (4.69). The high stratifi-cation of this limit has suppressed all vertical motion, and variations in the flowbecome confined to the horizontal plane. Finally, we note that it is typical inquasi-geostrophic applications to omit the prime on the buoyancy perturbations,and write b = f0∂ψ/∂z; however, we will keep the prime in this chapter.

5.4.3 Buoyancy advection at the surface

The solution of the elliptic equation in (5.118) requires vertical boundary conditionson ψ at the ground and at the top of the atmosphere, and these are given by use ofthe thermodynamic equation. For a flat, slippery, rigid surface the vertical velocity


is zero so that the thermodynamic equation may be written

Db′

Dt= 0, b′ = f0

∂ψ∂z. (5.120)

We apply this at the ground and at the tropopause, treating the latter as a lid on thelower atmosphere. In the presence of friction and topography the vertical velocityis not zero, but is given by

w = r∇2ψ+ u · ∇ηb (5.121)

where the first term represents Ekman friction (with the constant r proportionalto the thickness of the Ekman layer) and the second term represents topographicforcing. The boundary condition becomes

∂∂t

(∂ψ∂z

)+ u · ∇

(∂ψ∂z

+N2ηb)+N2r∇2ψ = 0, (5.122)

where all the fields are evaluated at z = 0 and z = H, the height of the lid. Thus, thequasi-geostrophic system is characterized by the horizontal advection of potentialvorticity in the interior and the advection of buoyancy at the boundary. Instead ofa lid at the top, then in a compressible fluid like the atmosphere we may supposethat all disturbances tend to zero as z →∞.

* A potential vorticity sheet at the boundary

Rather than regarding buoyancy advection as providing the boundary condition, itis sometimes useful to think of there being a very thin sheet of potential vorticityjust above the ground and another just below the lid, specifically with a verticaldistribution proportional to δ(z − ε) or δ(z − H + ε). The boundary condition(5.120) or (5.122) can be replaced by this, along with the condition that there areno variations of buoyancy at the boundary and ∂ψ/∂z = 0 at z = 0 and z = H.9

To see this, we first note that the differential of a step function is a delta function.Thus, a discontinuity in ∂ψ/∂z at a level z = z1 is equivalent to a delta function inpotential vorticity there:

q(z1) =[f 2

0

N2∂ψ∂z

]z1+

z1−δ(z − z1). (5.123)

Now, suppose that the lower boundary condition, given by (5.120), has some arbi-trary distribution of buoyancy on it. We can replace this condition by the simplercondition ∂ψ/∂z = 0 at z = 0, provided we also add to our definition of potentialvorticity a term given by (5.123) with z1 = ε. This term is then advected by the hori-zontal flow, as are the other contributions. A buoyancy source at the boundary mustsimilarly be treated as a sheet of potential vorticity source in the interior. Any flowwith buoyancy variations over a horizontal boundary is thus equivalent to a flowwith uniform buoyancy at the boundary, but with a spike in potential vorticity adja-cent to the boundary. This approach brings notational and conceptual advantages,in that now everything is expressed in terms of potential vorticity and its advection.However, in practice there may be less to be gained, because the boundary termsmust still be included in any particular calculation that is to be performed.


5.4.4 Quasi-geostrophy in pressure coordinates

The derivation of the quasi-geostrophic system in pressure coordinates is very sim-ilar to that in height coordinates, with the main difference coming at the bound-aries, and we give only the results. The starting point is the primitive equations inpressure coordinates, (2.151). In pressure coordinates quasi-geostrophic potentialvorticity is found to be

q = f +∇2ψ+ ∂∂p

(f 2

0

S2∂ψ∂p

), (5.124)

where ψ = Φ/f0 is the streamfunction and Φ the geopotential, and

S2 ≡ −Rp

(ppR

)κdθdp

= − 1ρθ

dθdp

(5.125)

where θ is a reference profile, a function of pressure only. In log-pressure coordi-nates, with Z = −H lnp, the potential vorticity may be written

q = f +∇2ψ+ 1ρ∗

∂∂Z

(ρ∗f 2

0

N2Z

∂ψ∂Z

), (5.126)

where

N2Z = S2

(pH

)2= −

(RH

)(ppR

)κdθdZ

(5.127)

is the buoyancy frequency and ρ∗ = exp(−z/H). Temperature and potential tem-perature are related to the streamfunction by

T = −f0pR∂ψ∂p

= Hf0

R∂ψ∂Z, (5.128a)

θ = −(pRp

)κ (f0pR

)∂ψ∂p

=(pRp

)κ (Hf0

R

)∂ψ∂Z. (5.128b)

In pressure or log-pressure co-ordinates, potential vorticity is advected along iso-baric surfaces, analogous to the horizontal advection in height co-ordinates.

The surface boundary condition again is derived from the thermodynamic equa-tion. This is, in log-pressure coordinates,

DDt

(∂ψ∂Z

)+ N

2Zf0W = 0. (5.129)

where W = DZ/Dt. This is not the real vertical velocity, w, but it is related to it by

w = f0

g∂ψ∂t

+ RTgHW. (5.130)

Thus, choosing H = RT(0)/g, we have, at Z = 0,

∂∂t

(∂ψ∂Z

− N2Zgψ)+ u · ∇ψ = −N

2

f0w, (5.131)

wherew = u · ∇ηb + r∇2ψ. (5.132)

This differs from the expression in height coordinates only by the second term inthe local time derivative. In applications where accuracy is not the main issue thesimpler boundary condition D(∂Zψ)/Dt = 0 is sometimes used.


Informal Derivation of Stratified QG Equations

We will use the Boussinesq equations, but similar derivations could be given for theanelastic equations or pressure coordinates. The first ingredient is the vertical com-ponent of the vorticity equation, (4.68); in the Boussinesq version (or the pressurecoordinate or anelastic versions) there is no baroclinic term and we have:

D3

Dt(ζ + f) = −(ζ + f)

(∂u∂x

+ ∂v∂y

)+(∂u∂z∂w∂y

− ∂v∂z∂w∂x

). (QG.1)

We now apply the assumptions on page 209. The advection and the vorticity onthe left-hand side are geostrophic, but we keep the horizontal divergence (which issmall) on the right-hand side where it is multiplied by the big term f . Furthermore,because f is nearly constant we replace it with f0 except where it is differentiated.The second term (tilting) on the right-hand side is smaller than the advection termson the left-hand side by the ratio [UW/(HL)]

/[U2/L2] = [W/H]

/[U/L] 1, be-

cause w is small (∂w/∂z equals the divergence of the ageostrophic velocity). Wetherefore neglect it, and given all this (QG.1) becomes

DgDt(ζg + f) = −f0

(∂u∂x

+ ∂v∂y

)= f0

∂w∂z, (QG.2)

where the second equality uses mass continuity and Dg/Dt = ∂/∂t + ug · ∇.

The second ingredient is the three-dimensional thermodynamic equation,

D3b/Dt = 0. (QG.3)

The stratification is assumed nearly constant, so we write b = b(z) + b′(x,y, z, t),where b is the basic state buoyancy. Furthermore, becausew is small it only advectsthe basic state, and with N2 = ∂b/∂z (QG.3) becomes

Dgb′/Dt +wN2 = 0. (QG.4)

Hydrostatic and geostrophic wind balance enable us to write the geostrophic veloc-ity, vorticity, and buoyancy in terms of streamfunction ψ [= p/(f0ρ0)]:

ug = k×∇ψ, ζg = ∇2ψ, b′ = f0∂ψ/∂z. (QG.5)

The quasi-geostrophic potential vorticity equation is obtained by eliminating wbetween (QG.2) and (QG.4), and this gives

DgqDt

= 0, q = ζg + f +∂∂z

(f0b′

N2

). (QG.6)

This equation is the Boussinesq version of (5.118), and using (QG.5) it may beexpressed entirely in terms of the streamfunction, with Dg ·/Dt = ∂/∂t + J(ψ, ·).The vertical boundary conditions, at z = 0 and z = H say, are given by (QG.4) withw = 0, with straightforward generalizations if topography or friction are present.


Fig. 5.3 The two-level quasi-geostrophic system with a flat bottom and rigidlid at which w = 0.

5.4.5 The two-level quasi-geostrophic system

The quasi-geostrophic system has, in general, continuous variation in the vertical(and horizontal, of course). By finite-differencing the continuous equations we canobtain a multi-level model, and a crude but important special case of this is thetwo-level model, which allows only two-degrees of freedom in the vertical. To ob-tain the equations of motion one way to proceed is to take a crude finite differenceof the continuous relation between potential vorticity and streamfunction given in(5.118b). In the Boussinesq case (or in pressure coordinates, with a slight reinter-pretation of the meaning of the symbols) the continuous expression for potentialvorticity is

q = ζ + f + ∂∂z

(f0b′

N2

), (5.133)

where b′ = f0∂ψ/∂z. In the case with a flat bottom and rigid lid at the top (andincorporating topography is an easy extension) the boundary condition of w = 0 issatisfied by D∂zψ/Dt = 0 at the top and bottom. An obvious finite differencing of(5.133) in the vertical (see Fig. 5.3) then gives

q1 = ζ1 + f +2f 2

0

N2H1H(ψ2 −ψ1), q2 = ζ2 + f +

2f 20

N2H2H(ψ1 −ψ2). (5.134)

In atmospheric problems it is common to choose H1 = H2, whereas in oceanicproblems we might choose to have a thinner upper layer, representing the flowabove the main thermocline. Note that the boundary conditions of w = 0 at thetop and bottom are already taken care of (5.134): they are incorporated into thedefinition of the potential vorticity — a finite-difference analog of the delta-functionconstruction of section 5.4.3. At each level the potential vorticity is advected by thestreamfunction so that the evolution equation for each level is:

DqiDt

= ∂qi∂t

+ ui · qi =∂qi∂t

+ J(ψi, qi) = 0, i = 1,2. (5.135)


Models with more than two levels can be easily constructed by extending the finite-differencing procedure.

Connection to the layered system

The two-level expressions, (5.134), have an obvious similarity with the two-layerexpressions, (5.85). Noting that N2 = ∂b/∂z and that b = −gδρ/ρ0 it is natural tolet

N2 = − gρ0

ρ1 − ρ2

H/2= g′

H/2. (5.136)

With this identification we find that (5.134) becomes

q1 = ζ1 + f +f 2

0

g′H1(ψ2 −ψ1), q2 = ζ2 + f +

f 20

g′H2(ψ1 −ψ2). (5.137)

These expressions are identical with (5.85) in the flat-bottomed, rigid lid case.Similarly, a multi-layered system with n layers is equivalent to a finite-differencerepresentation with n levels. It should be said, though, that in the pantheon ofquasi-geostrophic models the two-level and two-layer models hold distinguishedplaces.

5.5 * QUASI-GEOSTROPHY AND ERTEL POTENTIAL VORTICITY

When using the shallow water equations, quasi-geostrophic theory could be natu-rally developed beginning with the expression for potential vorticity. Is such an ap-proach possible for the stratified primitive equations? The answer is yes, althoughthe algebra is more complicated, as we see.

5.5.1 * Using height coordinates

Noting the general expression, (4.121), for potential vorticity in a hydrostatic fluid,potential vorticity in the Boussinesq hydrostatic equations is given by

Q =[(vx −uy)bz − vzbx +uzby + fbz

], (5.138)

where the x,y, z subscripts denote derivatives. Without approximation, we writethe stratification as b = b(z)+ b′(x,y, z, t), and (5.138) becomes

Q = [f0N2]+ [(βy + ζ)N2 + f0b′z]+ [(βy + ζ)b′z − (vzb′x −uzb′y)], (5.139)

where, under quasi-geostrophic scaling, the terms in square brackets are in decreas-ing order of size. Neglecting the third term, and taking the velocity and buoyancyfields to be in geostrophic and thermal wind balance, we can write the potentialvorticity as Q ≈ Q+Q′, where Q = f0N2 and

Q′ = (βy + ζ)N2 + f0b′z = (βy +∇2ψ)N2 + f 20∂ψ∂z. (5.140)

The potential vorticity evolution equation is then

DQ′

Dt+w∂Q

∂z= 0. (5.141)

5.5 * Quasi-geostrophy and Ertel Potential Vorticity 227

The vertical advection is important only in advecting the basic state potential vor-ticity Q. Thus, after dividing by N2, (5.141) becomes

∂q∗∂t

+ ug · ∇q∗ +wN2∂q∂z

= 0, (5.142)

where

q∗ = (βy + ζ)+f0

N2b′z. (5.143)

This is the approximation to the (perturbation) Ertel potential vorticity in the quasi-geostrophic limit. However, it is not the same as the expression for the quasi-geo-strophic potential vorticity, (5.118b) and, furthermore, (5.142) involves a verti-cal advection. (Thus, we might refer to the expression in (5.118) as the ‘quasi-geostrophic pseudo-potential vorticity’, but the prefix ‘quasi-geostrophic’ alone nor-mally suffices.) We can derive (5.118) by eliminating w between (5.142) and thequasi-geostrophic thermodynamic equation ∂b′/∂t + ug · ∇b′ +w∂b/∂z = 0.

5.5.2 Using isentropic coordinates

An illuminating and somewhat simpler path from Ertel potential vorticity to thequasi-geostrophic equations goes by way of isentropic coordinates.10 We begin withthe isentropic expression for Ertel potential vorticity for an ideal gas,

Q = f + ζσ

(5.144)

where σ = −∂p/∂θ is the thickness density (which we will just call the thick-ness), and in adiabatic flow potential vorticity is advected along isopycnals. Wenow employ quasi-geostrophic scaling to derive an approximate equation set fromthis. First assume that variations in thickness are small compared to the referencestate, so that

σ = σ (θ)+ σ ′, |σ ′| |σ |. (5.145)

and similarly for pressure and density. Assuming also that the variations in Coriolisparameter are small, (5.144) becomes

Q ≈[f0

σ

]+[

1σ(ζ + βy)− f0

σσ ′

σ

]. (5.146)

We now use geostrophic and hydrostatic balance to express the terms on the right-hand side in terms of a single variable, noting that the first term does not vary alongisentropic surfaces. Hydrostatic balance is

∂M∂θ

= Π (5.147)

where M = cpT + gz and Π = cp(p/pR)κ . Writing M = M(θ) + M′ and Π =Π(θ) + Π′, where M and Π are hydrostatically balanced reference profiles, weobtain

∂M′

∂θ= Π′ ≈ dΠ

dpp′ = 1

θρp′ (5.148)


where the last equality follows using the equation of state for an ideal gas and ρ isa reference profile. The perturbation thickness field may then be written as

σ ′ = − ∂∂θ

(ρθ∂M′

∂θ

). (5.149)

Geostrophic balance is f0 × u = −∇θM′ where the velocity, and the horizontalderivatives, are along isentropic surfaces. This enables us to define a flow stream-function by

ψ ≡ M′

f0. (5.150)

and we can then write all the variables in terms of ψ:

u = −(∂ψ∂y

)θ, v =

(∂ψ∂x

)θ,

ζ = ∇2θψ, σ ′ = f0

∂∂θ

(ρθ∂M′

∂θ

).

(5.151)

Using (5.146) (5.150) and (5.151), the quasi-geostrophic system in isentropic co-ordinates may be written

DqDt

= 0

q = f +∇2θψ+

f 20

σ∂∂θ

(ρθ∂ψ∂θ

) . (5.152a,b)

where the advection of potential vorticity is by the geostrophically balanced flow,along isentropes. The variable q is an approximation to the second term in squarebrackets in (5.146), multiplied by σ ,

Projection back to physical-space coordinates

We can recover the height or pressure coordinate quasi-geostrophic systems by pro-jecting (5.152) onto the appropriate coordinate. This is straightforward because,by assumption, the isentropes in a quasi-geostrophic system are nearly flat. Recallthat [c.f., (2.143)] a transformation between vertical coordinates may be effectedby

∂∂x

∣∣∣∣θ= ∂∂x

∣∣∣∣p+ ∂p∂x

∣∣∣∣∣θ

∂∂p, (5.153)

but the second term is O(Ro) smaller than the first because, under quasi-geostrophicscaling, isentropic slopes are small. Thus∇2

θψ in (5.152b) may be replaced by∇2pψ

or ∇2zψ. The vortex stretching term in (5.152) becomes, in pressure coordinates,

f 20

σ∂∂θ

(ρθ∂ψ∂θ

)≈ f

20

σdpdθ

∂∂p

(ρθ

dpdθ∂ψ∂p

)= ∂∂p

(f 2

0

S2∂ψ∂p

)(5.154)

where S2 is given by (5.125). The expression for the quasi-geostrophic poten-tial vorticity in isentropic coordinates is thus approximately equal to the quasi-geostrophic potential vorticity in pressure coordinates. This near-equality holds

5.6 * Energetics of Quasi-Geostrophy 229

because the isentropic expression, (5.152b), does not contain a component propor-tional to the mean stratification: the second square-bracketed term on the right-hand side (5.146) is the only dynamcally relevant one, and its evolution alongisentropes is mirrored by the evolution along isobaric surfaces of quasi-geostrophicpotential vorticity in pressure coordinates.

5.6 * ENERGETICS OF QUASI-GEOSTROPHY

If the quasi-geostrophic set of equations is to represent a real fluid system in aphysically meaningful way, then it should have a consistent set of energetics. Inparticular, total energy should be conserved, and there should be analogs of kineticand potential energy and conversion between the two. We now show that suchenergetic properties do hold, using the Boussinesq set as an example.

Let us write the governing equations as a potential vorticity equation in theinterior,

DDt

[∇2ψ+ ∂

∂z

(f 2

0

N2∂ψ∂z

)]+ β∂ψ

∂x= 0, 0 < z < 1, (5.155)

and buoyancy advection at the boundary,

DDt

(∂ψ∂z

)= 0, z = 0,1. (5.156)

For lateral boundary conditions we may assume that ψ = constant, or imposeperiodic conditions. If we multiply (5.155) by −ψ and integrate over the domain,using the boundary conditions, we easily find

dEdt

= 0, E = 12

∫V

[(∇ψ)2 + f

20

N2

(∂ψ∂z

)2]dV. (5.157a,b)

The term involving β makes no direct contribution to the energy budget. Eq.(5.157) is the fundamental energy equation for quasi-geostrophic motion, and itstates that in the absence of viscous or diabatic terms the total energy is conserved.The two terms in (5.157b) can be identified as the kinetic and available potentialenergy of the flow, where

KE = 12

∫V(∇ψ)2 dV, APE = 1

2

∫V

f 20

N2

(∂ψ∂z

)2dV. (5.158a,b)

The available potential energy may also be written as

APE = 12

∫V

HL2d

(∂ψ∂z

)2dV, (5.159)

where Ld is the deformation radius NH/f0 and we may choose H such that z ∼H. At some scale L the ratio of the kinetic energy to the potential energy is thus,roughly,

KEAPE

∼L2dL2 . (5.160)

For scales much larger than Ld the potential energy dominates the kinetic energy,and contrariwise.


5.6.1 Conversion between APE and KE

Let us return to the vorticity and thermodynamic equations,

DζDt

= f ∂w∂z

(5.161)

where ζ = ∇2ψ, andDb′

Dt+N2w = 0 (5.162)

where b′ = f0∂ψ/∂z. From (5.161) we form a kinetic energy equation namely

12

ddt

∫V(∇ψ)2 dV = −

∫Vf0∂w∂zψdV =

∫Vf0w

∂ψ∂z

dV. (5.163)

From (5.162) we form a potential energy equation, namely

ddt

12

∫V

f 20

N2

(∂ψ∂z

)2

dV = −∫Vf0w

∂ψ∂z

dV. (5.164)

Thus, the conversion from APE to KE is represented by

ddt

KE = − ddt

APE =∫vf0w

∂ψ∂z

dV. (5.165)

Because the buoyancy is proportional to ∂ψ/∂z, when warm fluid rises there isa correlation between w and ∂ψ/∂z and available potential energy is convertedto kinetic energy. Whether such a phenomenon occurs depends of course on thedynamics of the flow; however, such a conversion is in fact a common feature ofgeophysical flows, and in particular of baroclinic instability, as we see in chapter 6.

5.6.2 Energetics of two-layer flows

Two-layer or two-level flows are an important special case. For layers of equalthickness let us write the evolution equations as

DDt

(∇2ψ1 −

12k2d(ψ1 −ψ2)

)+ β∂ψ1

∂x= 0 (5.166a)

DDt

(∇2ψ2 +

12k2d(ψ1 −ψ2)

)+ β∂ψ2

∂x= 0 (5.166b)

where k2d/2 = (2f0/NH)2. On multiplying these two equations by −ψ1 and −ψ2

respectively and integrating over the horizontal domain, the advective term in thematerial derivatives and the beta term all vanish, and we obtain

ddt

∫A

[12(∇ψ1)2 +

12k2dψ1(ψ1 −ψ2)

]dA = 0, (5.167a)

ddt

∫A

[12(∇ψ2)2 −

12k2dψ2(ψ1 −ψ2)

]dA = 0. (5.167b)

Adding these gives

ddt

∫A

[12(∇ψ1)2 +

12(∇ψ2)2 + k2

dψ2]

dA = 0. (5.168)

This is the energy conservation statement for the two layer model. The first twoterms represent the kinetic energy and the last term the available potential energy.

5.6 * Energetics of Quasi-Geostrophy 231

Energy in the baroclinic and barotropic modes

A useful partitioning of the energy is between the energy in the barotropic and baro-clinic modes. The barotropic streamfunction, ψ is the vertically averaged stream-function and the baroclinic mode is the difference between the streamfunctions inthe two layers. That is, for equal layer thicknesses,

ψ ≡ 12(ψ1 +ψ2), τ ≡ 1

2(ψ1 −ψ2) (5.169)

Substituting (5.169) into (5.168) reveals that

ddt

∫A

[(∇ψ)2 + (∇τ)2 + k2

dτ2]

dx = 0 (5.170)

The energy density in the barotropic mode is thus just (∇ψ)2, and that in the baro-clinic mode is (∇τ)2 + k2

dτ2. This partitioning will prove particularly useful when

we consider baroclinic turbulence in chapter 9.

5.6.3 Enstrophy conservation

Potential vorticity is advected only by the horizontal flow, and thus it is materiallyconserved on the horizontal surface at every height and

DqDt

= ∂q∂t+ u · ∇q = 0. (5.171)

Furthermore, the advecting flow is divergence-free so that u · ∇q = ∇ · (uq). Thus,on multiplying (5.171) by q and integrating over a horizontal domain, A, usingeither no-normal flow or periodic boundary conditions, we straightforwardly obtain

dZdt

= 0, Z = 12

∫Aq2 dA. (5.172)

The quantity Z is known as the enstrophy, and this is conserved at each height aswell as, naturally, over the entire volume.

The enstrophy is just one of an infinity of invariants in quasi-geostrophic flow.Because the potential vorticity of a fluid element is conserved, any function of thepotential vorticity must be a material invariant and we can immediately write

DDtF(q) = 0. (5.173)

To verify that this is true, simply note that (5.173) implies that (dF/dq)Dq/Dt = 0,which is true by virtue of (5.171). (However, by virtue of the material advection, thefunction F(q) need not be differentiable in order for (5.173) to hold.) Each of thematerial invariants corresponding to different choices of F(q) has a correspondingintegral invariant; that is

ddt

∫AF(q)dA = 0. (5.174)

The enstrophy invariant corresponds to choosing F(q) = q2; it plays a particularlyimportant role because, like energy, it is a quadratic invariant, and its presence pro-foundly alters the behaviour of two-dimensional and quasi-geostrophic flow com-pared to three-dimensional flow (see section 8.3).


5.7 ROSSBY WAVES

In the final topic of this chapter we consider wave motion in a quasi-geostrophicsystem. (A brief introduction to wave kinematics is given in the appendix to thischapter.) Although we consider the closely related topics of hydrodynamic instabil-ity and wave–mean flow interaction in Part II, Rossby waves are such a fundamentalpart of geophysical fluid dynamics, and intimately tied to quasi-geostrophic dynam-ics, that they find a natural place in this chapter.

5.7.1 Waves in a single layer

Consider flow of a single homogeneous layer on a flat-bottomed β-plane. The un-forced, inviscid equation of motion is

DDt(ζ + f −ψ/L2

d) = 0, (5.175)

where ζ = ∇2ψ is the vorticity and ψ the streamfunction.

Infinite deformation radius

If the scale of motion is much less than the deformation scale then the β-plane theequation of motion is governed by

DDt(ζ + βy) = 0. (5.176)

Expanding the material derivative gives

∂ζ∂t

+ u · ∇ζ + βv = 0 or∂ζ∂t

+ J(ψ,ζ)+ β∂ψ∂x

= 0. (5.177)

We now linearize this equation — that is, we suppose that the flow consistsof a time-independent component (the ‘basic state’) plus a perturbation, with theperturbation being small compared to the mean flow. Such a mean flow must satisfythe time independent equation of motion, and purely zonal flow will do this. Forsimplicity we choose a flow with no meridional dependence and let

ψ = Ψ +ψ′(x,y, t) (5.178)

where Ψ = −Uy and |ψ′| |Ψ |. (The symbol U represents the zonal flow of thebasic state, not a magnitude for scaling purposes.) Substitute (5.178) into (5.177)and neglect the nonlinear terms involving products of ψ′ to give

∂ζ′

∂t+J(Ψ , ζ′)+β∂ψ

′

∂x= 0 or

∂∂t∇2ψ′+U ∂∇

2ψ′

∂x+β∂ψ

′

∂x= 0. (5.179a,b)

Solutions to this equation may be found in the form of a plane wave,

ψ′ = Re ψ ei(kx+ly−ωt), (5.180)

where Re indicates the real part of the function (and this will sometimes be omittedif no ambiguity is so-caused). Solutions of the form (5.180) are valid in a domainwith doubly-periodic boundary conditions; solutions in a channel can be obtained

5.7 Rossby Waves 233

using a meridional variation of sin ly, with no essential changes to the dynamics.The amplitude of the oscillation is given by ψ and the phase by kx+ly−ωt, wherek and l are the x- and y-wavenumbers and ω is the frequency of the oscillation.

Substituting (5.180) into (5.179) yields

[(−ω+Uk)(K2)+ βk]ψ = 0, (5.181)

where K2 = k2 + l2. For nontrivial solutions this implies

ω = Uk− βkK2 . (5.182)

This is the dispersion relation for Rossby waves. The phase speed, cp, and groupvelocity, cg, in the x-direction are

cxp ≡ωk= U − β

K2 , cxg ≡∂ω∂k

= U + β(k2 − l2)

(k2 + l2)2 . (5.183a,b)

The velocity U provides a uniform translation, and doppler shifts the frequency.The phase speed in the absence of a mean flow westwards, with waves of longerwavelengths travelling more quickly, and the eastward current speed required tohold the waves of a particular wavenumber stationary (i.e., cxp = 0) is U = β/K2.We discuss the meaning of the group velocity in section .

Finite deformation radius

For finite deformation radius the basic state Ψ = −Uy is still a solution of theoriginal equations of motion, but the potential vorticity corresponding to this stateis Q = Uy/L2

d + βy and its gradient is ∇Q = (β+ U/L2d)j. The linearized equation

of motion is thus,(∂∂t+U ∂

∂x

)(∇2ψ′ −ψ′/L2

d)+ (β+U/L2d)∂ψ′

∂x= 0. (5.184)

Substituting ψ′ = ψ ei(kx+ly−ωt) we obtain the dispersion relation,

ω = k(UK2 − β)

K2 + 1/L2d= Uk− k

β+U/L2d

K2 + 1/L2d. (5.185)

The corresponding x-components of phase speed and group velocity are

cxp = U −β+Uk2

d

K2 + k2d= UK

2 − βK2 + k2

d, cxg = U +

(β+Uk2d)(k

2 − l2 − k2d)

(k2 + l2 + k2d)2

, (5.186a,b)

where kd = 1/Ld. The uniform velocity field now no longer provides just a simpleDoppler shift of the frequency, nor a uniform addition to the phase speed. From(5.186a) the waves are stationary when K2 = K2

s ≡ β/U; that is, the current speedrequired to hold waves of a particular wavenumber stationary is U = β/K2. How-ever, this is not simply the magnitude of the phase speed of waves of that wavenum-ber in the absence of a current — this is given by

cxp = −β

K2s + k2

d= − U

1+ k2d/K

2s. (5.187)


Fig. 5.4 The mechanism of a two-dimensional (x-y) Rossby wave. An initialdisturbance displaces a material line at constant latitude (the straight hor-izontal line) to the solid line marked η(t = 0). Conservation of potentialvorticity, βy + ζ, leads to the production of relative vorticity, as shown fortwo parcels. The associated velocity field (arrows on the circles) then advectsthe fluid parcels, and the material line evolves into the dashed line. Thephase of the wave has propagated westwards.

Why is there a difference? It is because the current does not just provide a uni-form translation, but, if Ld is non-zero, also modifies the basic potential vorticitygradient. The basic state height field η0 is sloping, that is η0 = −(f0/g)Ψy, andthe ambient potential vorticity field increases with y, that is Q = (β + U/L2

d)y.Thus, the basic state defines a preferred frame of reference, and the problem is notGalilean invariant.11

The mechanism of Rossby waves

The fundamental mechanism underlying Rossby waves is easily understood. Con-sider a material line of stationary fluid parcels along a line of constant latitude,and suppose that some disturbance causes their displacement to the line markedη(t = 0) in Fig. 5.4. In the displacement, the potential vorticty of the fluid parcelsis conserved, and in the simplest case of barotropic flow on the β-plane the potentialvorticity is the absolute vorticity, βy +ζ. Thus, in either hemisphere, a northwardsdisplacement leads to the production of negative relative vorticity and a southwardsdisplacement leads to production of positive relative vorticity. The relative vorticitygives rise to a velocity field which in turn advects the parcels in material line in themanner shown, and the material line propagates eastward.

In more complicated situations, such as flow in two layers, considered below, orin a continuously stratified fluid, the mechanism is essentially the same: a displacedfluid parcel conserves its potential vorticity, and in the presence of a potential vor-ticity gradient in the basic state, the displacement leads to the production of relativevorticity and an associated velocity field. The velocity field then further displacesthe fluid parcels, leading to the formation of a Rossby wave. The vital ingredient isa basic state potential vorticity gradient, such as that provided by the change of theCoriolis parameter with latitude.

5.7 Rossby Waves 235

5.7.2 Rossby waves in two layers

Now consider the dynamics of the two-layer model, linearized about a state of rest.The two (coupled) linear equations describing the motion in each layer are

∂∂t

[∇2ψ′1 + F1(ψ′2 −ψ′1)

]+ β∂ψ

′1

∂x= 0, (5.188a)

∂∂t

[∇2ψ′2 + F2(ψ′1 −ψ′2)

]+ β∂ψ

′2

∂x= 0, (5.188b)

where F1 = f 20 /g′H1 and F2 = f 2

0 /g′H2. By inspection these may be transformedinto two uncoupled equations: one equation is obtained by multiplying the first byF2 and the second by F1 and adding, and the other is obtained as the difference ofthe two equations. Then, defining

ψ = F1ψ′2 + F2ψ′1F1 + F2

, τ = 12(ψ′1 −ψ′2), (5.189a,b)

(think ‘τ for temperature’), (5.188) become

∂∂t∇2ψ+ β∂ψ

∂x= 0, (5.190a)

∂∂t

[(∇2 − k2

d)τ]+ β∂τ

∂x= 0. (5.190b)

where now kd = (F1 + F2)1/2. The internal radius of deformation for this problemis the inverse of this, namely

Ld = k−1d = 1

f0

(g′H1H2

H1 +H2

)1/2. (5.191)

The variables ψ and τ are the normal coordinates for the two layer model; theyoscillate independently of each other, and the solution in physical space is justtheir superposition. [For the continuous equations the analogous eigenfunctionsare given by solutions of ∂z[(f

20 /N2)∂zφ] = λ2φ, where eigenvalue, λ, is inversely

proportional to the deformation radius.] The equation for ψ is identical to thatof the single-layer, rigid-lid model, namely (5.179) with U = 0, and its dispersionrelation is just

ω = −βkK2 . (5.192)

The barotropic mode corresponds to synchronous, depth-independent, motion inthe two layers with a flat interface — the displacement of the interface is given by2f0τ/g′ and so proportional to the amplitude of the baroclinic mode. The disper-sion relation for the baroclinic mode is

ω = − βkK2 + k2

d. (5.193)

The mass transport associated with this mode is identically zero, since from (5.189)we have

ψ1 = ψ+2F1τF1 + F2

, ψ2 = ψ−2F2τF1 + F2

, (5.194a,b)


Fig. 5.5 Left: The dispersion relation for barotropic (ωt , solid line) and baro-clinic (ωc , dashed line) Rossby waves in the two-layer model, calculated using(5.192) and (5.193) with ky = 0, plotted for both positive and negative zonalwavenumbers and frequencies. The wavenumber is nondimensionalised bykd, and the frequency is non-dimensionalized by β/kd. Right: the corre-sponding zonal group and phase velocities, cg = ∂ω/∂kx and cp = ω/kx,with superscript ‘t’ or ‘c’ for the barotropic or baroclinic mode. The velocitiesare non-dimensionalized by β/k2

d.

and this impliesH1ψ1 +H2ψ2 = (H1 +H2)ψ. (5.195)

The left-hand side is proportional to the total mass transport, which is evidentlyassociated with the barotropic mode.

The dispersion relation and associated group and phase veloicties are plottedin Fig. 5.5. The x-component of phase speed, ω/k, is negative (westward) forboth baroclinic and barotropic Rossby waves. The group velocity of the barotropicwaves is always positive (eastward), but the group velocity of long baroclinic wavesmay be negative (westward). For very short waves, k2 k2

d, the baroclinic andbarotropic velocities coincide and their phase and group velocities are equal andopposite. With a deformation radius of 50 km, typical for the mid-laititude ocean,then a nondimensional frequency of unity in the figure corresponds to a dimen-sional frequency of 5 × 10−7 s−1 or a period of about 100 days. In an atmospherewith a deformation radius of 1000 km a non-dimensional frequency of unity corre-sponds to 1 × 10−5 s−1 or a period of about 7 days. Nondimensional velocities ofunity correspond to respective dimensional velocities of about 0.25 m s−1 (ocean)and 10 m s−1 (atmosphere).

The deformation radius only affects the baroclinic mode. For scales much smallerthan the deformation radius, K2 k2

d, we see from (5.190b) that the baroclinicmode obeys the same equation as the barotropic mode so that

∂∂t∇2τ + β∂τ

∂x= 0. (5.196)

Using this and (5.190a) implies that

∂∂t∇2ψi + β

∂ψi∂x

= 0, i = 1,2. (5.197)

5.8 * Rossby Waves in Stratified Quasi-Geostrophic Flow 237

That is to say, the two layers themselves are uncoupled from each other. At theother extreme, for very long baroclinic waves the relative vorticity is unimportant.

5.8 * ROSSBY WAVES IN STRATIFIED QUASI-GEOSTROPHIC FLOW

5.8.1 Setting up the problem

Let us now consider the dynamics of linear waves in stratified quasi-geostophicflow on a β-plane, with a resting basic state. (In chapter 13 we explore the roleof Rossby waves in a more realistic setting.) The interior flow is governed by thepotential vorticity equation, (5.118), and linearizing this about a state of rest gives

∂∂t

[∇2ψ′ + 1

ρ(z)∂∂z

(ρ(z)F(z)

∂ψ′

∂z

)]+ β∂ψ

′

∂x= 0, (5.198)

where ρ is the density profile of the basic state, and F(z) = f 20 /N2. (F is the square

of the inverse Prandtl ratio, N/f0.) In the Boussinesq approximation ρ = ρ0, aconstant. The vertical boundary conditions are determined by the thermodynamicequation, (5.120). If the boundaries are flat, rigid, slippery surfaces then w =0 at the boundaries and if there is no surface buoyancy gradient the linearizedthermodynamic equation is

∂∂t

(∂ψ′

∂z

)= 0. (5.199)

We apply this at the ground and, with somewhat less justification, at the tropopause— the higher static stability of the stratosphere inhibits vertical motion. If theground is not flat or if friction provides a vertical velocity by way of an Ekman layerthe boundary condition must be correspondingly modified, but we will stay withthe simplest case here and apply (5.199) at z = 0 and z = H.

5.8.2 Wave motion

As in the single-layer case, we seek solutions of the form

ψ′ = Re ψ(z) ei(kx+ly−ωt) (5.200)

where ψ(z) will determine the vertical structure of the waves. The case of a sphereis more complicated but introduces no truly new physical phenomena.

Substituting (5.200) into (5.198) gives

ω[−K2ψ(z)+ 1

ρ∂∂z

(ρF(z)

∂ψ∂z

)]− βkψ(z) = 0. (5.201)

Now, if ψ satisfies1ρ∂∂z

(ρF(z)

∂ψ∂z

)= −Γ ψ, (5.202)

where Γ is a constant, then the equation of motion becomes

−ω[K2 + Γ

]ψ− βkψ = 0, (5.203)


and the dispersion relation follows, namely

ω = − βkK2 + Γ . (5.204)

Equation (5.202) constitutes an eigenvalue problem for the vertical structure; theboundary conditions, derived from (5.199), are ∂ψ/∂z = 0 at z = 0 and z = H.The resulting eigenvalues, Γ are proportional to the inverse of the squares of thedeformation radii for the problem and the eigenfunctions are the vertical structurefunctions.

A simple example

Consider the case in which F(z) and ρ are constant, and in which the domain isconfined between two rigid surfaces at z = 0 and z = H. Then the eigenvalueproblem for the vertical structure is

F∂2ψ∂z2 = −Γ ψ (5.205a)

with boundary conditions of

∂ψ∂z

= 0, at z = 0,H. (5.205b)

There is a sequence of solutions to this, namely

ψn(z) = cos(nπz/H), n = 1,2 . . . (5.206)

with corresponding eigenvalues

Γn = n2 Fπ2

H2 = (nπ)2(f0

NH

)2, n = 1,2 . . . . (5.207)

Eq. (5.207) may be used to define the deformation radii for this problem, namely

Ln ≡1√Γ n

= NHnπf0

. (5.208)

The first deformation radius is the same as the expression obtained by dimensionalanalysis, namely NH/f , except for a factor of π. (Definitions of the deformationradii both with and without the factor of π are common in the literature, andneither is obviously more correct. In the latter case, the first deformation radiusin a problem with uniform stratification is given by NH/f , equal to π/

√Γ1.) In

addition to these baroclinic modes, the case with n = 0, that is with ψ = 1, is alsoa solution of (5.205) for any F(z).

Using (5.204) and (5.207) the dispersion relation becomes

ω = − βkK2 + (nπ)2(f0/NH)2

, n = 0,1,2 . . . (5.209)

and, of course, the horizontal wavenumbers k and l are also quantized in a finite


domain. The dynamics of the barotropic mode are independent of height and in-dependent of the stratification of the basic state, and so these Rossby waves areidentical with the Rossby waves in a homogeneous fluid contained between two flatrigid surfaces. The structure of the baroclinic modes, which in general dependson the structure of the stratification, becomes increasingly complex as the verticalwavenumber n increases. This increasing complexity naturally leads to a certaindelicacy, making it rare that they can be unambiguosly identified in nature. Theeigenproblem for a realistic atmospheric profile is further complicated because ofthe lack of a rigid lid at the top of the atmosphere.12

APPENDIX: WAVE KINEMATICS, GROUP VELOCITY AND PHASE SPEED

This appendix provides a brief and informal look at wave kinematics and the mean-ing of phase speed and group velocity.13

5.A.1 Kinematics and definitions

A wave may be defined as a disturbance that satisfies a dispersion relation. Tosee what this means, suppose a disturbance, ψ(x, t) (where ψ might be velocity,streamfunction, pressure, etc), satisfies some equation

L(ψ) = 0, (5.210)

where L is a linear operator, typically a polynomial in time and space derivatives;an example is ∂∇2 · /∂t + β∂ · /∂x . If (5.210) has constant coefficients (if β isconstant in our example) then solutions may often be found as a superposition ofplane waves, each of which satisfy

ψ = Re ψ eiθ(x,t) = Re ψ ei(k·x−ωt). (5.211)

where ψ is a constant, θ is the phase, k is the vector wavenumber (kx , ky , kz), andω is the wave frequency. [We also often write the wave vector as k = (k, l,m).]The frequency and wavevector are related by the dispersion relation:

ω = Ω(k), (5.212)

where Ω(k) [meaning Ω(k, l,m)] is some function determined by the form of L.Unless it is necessary to explicitly distinguish the function Ω from the frequency ωwe will often write ω = ω(k). Two examples of dispersion relations are (2.245)and (5.182).

If the medium in which the waves are progating is inhomogeneous, then (5.210)will probably not have constant coefficients (for example, βmay vary meridionally).Nevertheless, if the medium is slowly varying, wave solutions may often still befound — although we do not prove it here — with the the general form

ψ = Re ψ(x, t) eiθ(x,t), (5.213)

where ψ(x, t) [meaning ψ(x,y, z, t)] varies slowly compared to the variation ofthe phase, θ. The frequency and wavenumber are then defined by

k ≡ ∇θ, ω ≡ −∂θ∂t, (5.214)


which in turn implies the formal relation between k and ω:

∂k∂t+∇ω = 0. (5.215)

Even if the medium is inhomogeneous, we may still have a local dispersion relationbetween frequency and wavevector,

ω = Ω(k,x, t). (5.216)

5.A.2 Wave propagation

Phase speed

First consider the propagation of monochromatic plane waves. Given (5.211) awave will propagate in the direction of k (Fig. 5.6). At a given instant and locationwe can align our coordinate axis along this direction, and we may write k·x = Kx∗where x∗ increases in the direction of k and K2 = |k|2 is the magnitude of thewavenumber. With this, we can write (5.211) as

ψ = Re ψ ei(Kx∗−ωt) = Re ψ eiK(x∗−ct). (5.217)

where c = ω/K. From this equation it is evident that the phase of the wave propa-gates at the speed c in the direction of k, and we define the phase speed:

cp ≡ωK. (5.218)

The wavelength of the wave, λ, is the distance between two wavecrests — thatis, the distance between two locations along the line of travel whose phase differsby 2π — and evidently this is given by

λ = 2πK. (5.219)

In (for simplicity) a two-dimensional wave, and referring to Fig. 5.6a, the wave-length and wave vectors in the x- and y-directions are given by,

λx = λcosφ

, λy = λsinφ

, kx = K cosφ, ky = K sinφ. (5.220)

In general, lines of constant phase intersect both the coordinate axes and propagatealong them. The speed of propagation along these axes is given by

cxp = cplx

l= cp

cosφ= cp

Kkx

= ωkx, cyp = cp

ly

l= cp

sinφ= cp

Kky

= ωky, (5.221)

using (5.220). The speed of phase propagation along any one of the axis is ingeneral larger than the phase speed in the primary direction of the wave. Thephase speeds are clearly not components of a vector: for example, cxp ≠ cp cosφ.To summarize, the phase speed and its components are given by

cp =ωK, cxp =

ωkx, cyp =

ωky

. (5.222)


Fig. 5.6 The propagation of a two-dimensional wave. (a) Two lines of con-stant phase (e.g., two wavecrests) at a time t1. The wave is propagating inthe direction k with wavelength λ. (b) The same line of constant phase at twosuccessive times. The phase speed is the speed of advancement of the wave-crest in the direction of travel, and so cp = l/(t2 − t1). The phase speed inthe x-direction is the speed of propagation of the wavecrest along the x-axis,and so cxp = lx/(t2 − t1) = cp/ cosφ.

Group velocity

Let us consider how the wavevector and frequency might change with position andtime. Using (5.216) in (5.215) gives

DcgkDt

≡ ∂k∂t+ cg · ∇k = −∇Ω (5.223)

where

cg ≡∂Ω∂k

≡(∂Ω∂k,∂Ω∂l,∂Ω∂m

), (5.224)

is the group velocity, sometimes written as cg = ∇kω or, in subscript notation, ascgi = ∂Ω/∂ki. The group velocity is a vector. If the frequency is spatially constantthe wavevector is evidently propagated at the group velocity.

The frequency is, in general, a function of space, wavenumber and time andfrom the dispersion relation, (5.216), its variation is governed by

∂ω∂t

= ∂Ω∂t

+ ∂Ω∂k∂k∂t

= ∂Ω∂t

+ ∂Ω∂k∇ω (5.225)

using (5.215). Thus, using the definition of group velocity, we have

DcgωDt

≡ ∂ω∂t

+ cg · ∇ω = ∂Ω∂t. (5.226)

If the dispersion relation is not an function of time, the frequency propagates atthe group velocity. We may define a ray as the trajectory traced by the group ve-locity, and if the frequency is not an explicit function of space or time, then bothwavevector and frequency are constant along a ray.


5.A.3 Meaning of group velocity

In the discussion above we introduced the group velocity in a kinematic and ratherformal way. What is the group velocity, physically?

Information and energy travel clearly cannot travel at the phase speed, for as thedirection of propagation of the phase line tends to a direction parallel to the y-axis,the phase speed in the x-direction tends to infinity! Rather, they travel at the groupvelocity, and to see this we consider the superposition of plane waves, noting that amonochromatic plane wave already fills space uniformly. We will restrict attentionto waves propagating in one direction, but the argument may be extended to twoor three dimensions.

Superposition of two waves

Consider the linear superposition of two waves. Limiting attention to the one-dimensional case for simplicity, consider a disturbance represented by

ψ = Re ψ(ei(k1x−ω1t) + ei(k2x−ω2t)). (5.227)

Let us further suppose that the two waves have similar wavenumbers and frequency,and in particular that k1 = k + ∆k and k2 = k − ∆k, and ω1 = ω + ∆ω andω2 =ω−∆ω. With this, (5.227) becomes

ψ = Re ψ ei(kx−ωt)[ei(∆kx−∆ωt) + e−i(∆kx−∆ωt)]

= 2Re ψ ei(kx−ωt) cos(∆kx −∆ωt).(5.228)

The resulting disturbance, illustrated in Fig. 5.7 has two aspects: a rapidly vary-ing component, with wavenumber k and frequency ω, and a more slowly vary-ing envelope, with wavenumber ∆k and frequency ∆ω. The envelope modulatesthe fast oscillation, and moves with the velocity ∆ω/∆k; in the limit ∆k → 0 and∆ω → 0 this is the group velocity, cg = ∂ω/∂k. It evidently differs from the phasespeed,ω/k, when the latter depends on the the wavenumber. The energy in the dis-turbance must move at the group velocity — note that node of the envelope movesat the speed of the envelope and no energy can cross the node. These concepts gen-eralize to more than one dimension, and if the waveumber is the three-dimensionalvector k = (k, l,m) then the three-dimensional envelope propagates at the groupvelocity given by (5.224).

* Superposition of many waves

Now consider a generalization of the above arguments. We will suppose that thedisturbance is a wave packet of the form

ψ = A(x, t) ei(kx−ωt) (5.229)

where A(x, t), like the envelope in Fig. 5.7, modulates the amplitude of the waveon a scale much longer than that of the wavelength 2π/k. We assume that packetis confined to a finite region of space, and that it contains a superposition of Fouriermodes with wavenumbers near to k. That is, at some fixed time, t = 0 say,

A(x) =∫ +∆k−∆k

A(k′) eik′x dk′ (5.230)


Fig. 5.7 Superposition of two sinusoidal waves with wavenumbers k and k+δk, producing a wave (solid line) that is modulated by slowly varying waveenvelope or wave packet (dashed line). The envelope moves at the groupvelocity, cg = ∂ω/∂k and the phase of the wave moves at the group speedcp =ω/k.

where ∆k is small. Each of the wavenumber components oscillates at a frequencygiven by the dispersion relation at hand, so that the evolution of the packet is givenby

A(x, t) =∫ +∆k−∆k

A(k′) ei(k′x−ω′t) dk′ ≈∫ +∆k−∆k

A(k′) eik′(x−t∂ω/∂k) dk′ (5.231)

where we have used the fact that the wavenumber range is small, so that ω′ ≈k′(∂ω/∂k), where the derivative is evaluated at the central wavenumber k. Thus,comparing (5.230) and (5.231), we see that

A(x, t) = A(x − cgt) (5.232)

where cg = ∂ω/∂k. Thus, the packet moves at the group velocity.

Notes

1 The phrase ‘quasi-geostrophic’ seems to have been introduced by Durst and Sutcliffe(1938) and the concept used in Sutcliffe’s development theory of baroclinic sys-tems (Sutcliffe 1939, 1947). The first systematic derivation of the quasi-geostrophicequations based on scaling theory was given by Charney (1948). The planetary geo-strophic equations were used by Robinson and Stommel (1959) and Welander (1959)in studies of the thermocline (and were first known as the ‘thermocline equations’),and were put in the context of other approximate equation sets by Phillips (1963).

2 Carl-Gustav Rossby (1898-1957) played a dominant role in the development of dy-namical meteorology in the early and middle parts of the 20th century, and his workpermeates all aspects of dynamical meteorology today. Perhaps the most funda-mental non-dimensional number in rotating fluid dynamics, the Rossby number, isnamed for him, as is the perhaps the most fundamental wave, the Rossby wave.He also discovered the conservation of potential vorticity (later generalised by Ertel)and contributed important ideas to atmospheric turbulence and the theory of air


masses. Swedish born, he studied first with V. Bjerknes before taking a position inStockholm in 1922 with the Swedish Meteorological Hydrologic Service and receivinga ‘Licentiat’ from the University of Stockholm in 1925. Shortly thereafter he movedto the United States, joining the Government Weather Bureau, a precursor of NOAA’sNational Weather Service. In 1928 he moved to MIT, playing an important role indeveloping the meteorology department there, while still maintaining connectionswith the Weather Bureau. In 1940 he moved to the University of Chicago, wherehe similarly helped develop meteorology there. In 1947 he became director of thenewly-formed Institute of Meteorology in Stockholm, and subsequently divided histime between there and the United States. Thus, as well as his scientific contribu-tions, he played a very influential role in the institutional development of the field.

3 Burger (1958)

4 This is the so-called ‘frontal geostrophic’ regime (Cushman-Roisin 1994).

5 Numerical integrations of the potential vorticity equation using (5.91), and per-forming the inversion without linearizing potential vorticity, do in fact indicate im-proved accuracy over either the quasi-geostrophic or planetary geostrophic equa-tions (Mundt et al. 1997). In a similar vein, McIntyre and Norton (2000) show howuseful potential vorticity inversion can be, and Allen et al. (1990a,b) demonstratethe high accuracy of certain intermediate models. Certainly, asymptotic correctnessshould not be the only criterion used in constructing a filtered model, because theparameter range in which the model is useful may be too limited. Note that thereis a difference between extending the parameter range in which a filtered model isuseful, as in the inversion of (5.91), and going to higher asymptotic order accuracyin a given parameter regime, as in Allen (1993) and Warn et al. (1995). Using Hamil-tonian mechanics it is possible to derive equations that span different asymptoticregimes, and that also have good conservation properties (Salmon 1983, Allen et al.2002).

6 I thank T. Warn for a conversation on this matter.

7 There is a difference between the dynamical demands of the quasi-geostrophic sys-tem in requiring β to be small, and the geometric demands of the Cartesian ge-ometry. On earth the two demands are similar in practice. But without dynamicalinconsistency we may imagine a Cartesian system in which βy ∼ f , and indeedthis is common in idealized, planetary geostrophic, models of the large-scale oceancirculation.

8 Atmospheric and oceanic sciences are sometimes thought of as not being ‘beautiful’in the same way as are some branches of theoretical physics. Yet surely quasi-geostrophic theory, and the quasi-geostrophic potential vorticity equation, are quitebeautiful, both for their austerity of description and richness of behaviour.

9 Bretherton (1966). Schneider et al. (2003) look at the non QG extension. The equiva-lence between boundary conditions and delta-function sources is a common featureof elliptic problems, and is analogous to the generation of electromagnetic fieldsby point charges. It is sometimes exploited in the numerical solution of ellipticequations, both as a simple way to include non-homogeneous boundary conditionsand, using the so-called capacitance matrix method, to solve problems in irregulardomains (e.g., Hockney 1970, Pares-Sierra and Vallis 1989).

10 Charney and Stern (1962). See also Vallis (1996).

11 This non-Doppler effect also arises quite generally, even in models in height coordi-nates. See White (1977) and problem 5.9.


12 See Chapman and Lindzen (1970).

13 For a review of waves and group velocity, see Lighthill (1965).

Further Reading

Majda, A., 2003. Introduction to PDEs and waves for the atmosphere and ocean.Provides a compact, somewhat mathematical introduction to various equation setsand their properties, including quasi-geostrophy.

Problems

5.1 In the derivation of the quasi-geostrophic equations, geostrophic balance leads to thelowest order velocity being divergence-free — that is, ∇z · u0 = 0. It seems that thiscan also be obtained from the mass conservation equation at lowest order. Is thisa coincidence? Suppose that the Coriolis parameter varied, and that the momentumequation yielded ∇2 · u0 ≠ 0. Would there be an inconsistency?

5.2 In the planetary geostrophic approximation, obtain an evolution equation and cor-responding inversion conditions that conserves potential vorticity and that is accu-rate to one higher order in Rossby number than the usual shallow water planetarygeostrophic equations.

5.3 Consider the flat-bottomed shallow water potential vorticity equation in the form

DDtζ + fh

= 0 (P5.1)

(a) Suppose that deviations of the height field are small compared to the mean heightfield, and that the Rossby number is small (so |ζ| f ). Further consider flowon a β-plane such that f = f0 + βy where |βy| f0. Show that the evolutionequation becomes

DDt

(ζ + βy − f0η

H

)= 0 (P5.2)

where h = H + η and |η| H. Using geostrophic balance in the form f0u =−g∂η/∂y , f0v = g∂η/∂x , obtain an expression for ζ in terms of η.

(b) Linearize (P5.2) about a state of rest, and show that the resulting system sup-ports two-dimensional Rossby waves that are similar to those of the usual two-dimensional barotropic system. Discuss the limits in which the wavelength ismuch shorter or much longer than the deformation radius.

(c) Linearize (P5.2) about a geostrophically balanced state that is translating uni-formly eastwards. Note that this means that:

u = U +u′ η = η(y)+ η′ (P5.3)

where η(y) is in geostrophic balance with U . Obtain an expression for the formof η(y).

(d) Obtain the dispersion relation for Rossby waves in this system. Show that theirspeed is different from that obtained by adding a constant U to the speed ofRossby waves in part (b), and discuss why this should be so. (That is, why is theproblem not Galilean invariant?)

5.4 Obtain solutions to the two-layer Rossby wave problem by seeking solutions of theform

ψ1 = Re ψ1 ei(kxx+kyy−ωt),

ψ1 = Re ψ2 ei(kxx+kyy−ωt).(P5.4)


Substitute (P5.4) directly into (5.188) to obtain the dispersion relation, and showthat the ensuing two roots correspond to the baroclinic and barotropic modes. Showthat the baroclinic mode has no net (vertically integrated) transport associated withit, and that the motion of the barotropic is independent of depth.

5.5 (Not difficult, but messy.) Obtain the vertical normal modes and the dispersionrelationship of the two-layer quasi-geostrophic problem with a free surface, for whichthe equations of motion linearized about a state of rest are

∂∂t

[∇2ψ1 + F1(ψ2 −ψ1)

]+ β∂ψ1

∂x= 0 (P5.5a)

∂∂t

[∇2ψ2 + F2(ψ1 −ψ2)− Fextψ1

]+ β∂ψ2

∂x= 0. (P5.5b)

where Fext = f0/(gH2).

5.6 Given the baroclinic dispersion relation

ω = − βkxk2x + k2

d, (P5.6)

for what value of kx is the x-component of group velocity the largest (i.e., the mostpositive), and what is the corresponding value of the group velocity?

5.7 Beginning with the vorticity and thermodynamic equations for a two layer model, ob-tain an expression for the conversion between available potential energy and kineticenergy in the two-layer model. Show that these expressions are consistent with theconservation of total energy as expressed by (5.168). Show also that the expressionmight be considered to be a simple finite-difference approximation to (5.165).

5.8 The vertical normal modes are the eigenfunctions of

1ρ∂∂z

(ρN2

∂Ψ∂z

)= −ΓΨ (P5.7)

along with boundary conditions on Ψ(z). Numerically obtain the vertical normalmodes for some or all of the following profiles, or others of your choice.

(a) ρ = 1, N2(z) = 1, Ψz = 0 at z = 0,1. (This is profile ‘b’ of Fig. 9.10. An analyticsolution is possible.)

(b) ρ = 1 and an N2(z) profile corresponding to a density profile similar to profile‘a’ of Fig. 9.10 (e.g., an exponential), and Ψz = 0 at z = 0,1.

(c) An isothermal atmosphere, with Ψz = 0 at z = 0,1. (Similar to (i), except that ρvaries with height.)

(d) An isothermal atmosphere, now assuming that ψ → 0 as z →∞.(e) An fluid with N2 = 1 for 0 < z < 0.5 and N2 = 4 for 0.5 < z < 1, with continuous

b, and with Ψz = 0 at z = 0,1.

In the atmospheric cases it is easiest to do the problem first with ρ = 1 (the Boussi-nesq case) and then extend the problem (and the code) to the compressible case.Then remove the upper boundary to larger and larger values of z.

5.9 Show that the non-Doppler effect arises using geometric height as the vertical co-ordinates, using the modified quasi-geostrophic set of White (1977). In particular,obtain the dispersion relation for stratified quasi-geostrophic flow with a resting ba-sic state. Then obtain the dispersion relation for the equations linearized about auniformly translating state, paying attention to the lower boundary condition, andnote the conditions under which the waves are stationary. Discuss.


5.10 Derive the quasi-geostrophic potential vorticity equation in isopycnal coordinates fora Boussinesq fluid. Show that the isopycnal expression for potential vorticity is ap-proximately equal to the corresponding expression in height coordinates, carefullystating any assumptions that may be necessary to show this.

5.11 (a) Obtain the dispersion relationship for free Rossby waves for the single-layerquasi-geostrophic potential vorticity equation with linear drag.

(b) Obtain the dispersion relation for free Rossby waves in the linearized two-layerpotential vorticity equation with linear drag in the lowest layer.

(c) Obtain the dispersion relation for free waves in the continuously stratifiedquasi-geostrophic equations, with the effects of linear drag appearing in the ther-modynamic equation for the lower boundary condition. That is, the boundarycondition at z = 0 is ∂t(∂zψ) + N2w = 0 where w = αζ where α is a constant.You may make the Boussinesq approximation and assume N2 is constant if youlike.

Part II

INSTABILITY, WAVE–MEAN FLOW

INTERACTION AND TURBULENCE

Ceci n’est pas une pipe.René Magritte (1898–1967), title of painting.

CHAPTER

SIX

Barotropic and Baroclinic Instability

WHAT HYDRODYNAMIC STATES ACTUALLY OCCUR IN NATURE? Any flow must clearly bea solution of the equations of motion, and there are, in fact, many steadysolutions to the equations of motion — a purely zonal flow, for exam-

ple. However, steady solutions do not abound in nature because, in order to persist,they must be stable to those small perturbations that inevitably arise. Indeed, all thesteady solutions that are known for the large-scale flow in the earth’s atmosphereand ocean have been found to be unstable.

There are a myriad forms of hydrodynamic instability, but our focus in this chap-ter is on barotropic and baroclinic instability. The latter is at the heart of the large-and mesoscale motion in the atmosphere and ocean — it gives rise to atmosphericweather systems, for example. Barotropic instability is important to us for two rea-sons. First, it is important in its own right as an instability mechanisms for jets andvortices, and is a driving mechanism in both two- and three-dimensional turbulence.Second, many problems in barotropic and baroclinic instability are formally and dy-namically similar, so that the solutions and insight we obtain in the often simplerproblems in barotropic instability are often useful in the baroclinic problem.

6.1 KELVIN-HELMHOLTZ INSTABILITY

To introduce the issue, we will first consider, rather informally, perhaps the simplestphysically interesting instance of a fluid-dynamical instability — that of a constant-density flow with a shear perpendicular to the fluid’s mean velocity, this being anexample of a Kelvin-Helmholtz instability.1 Let us consider two fluid masses of equaldensity, with a common interface at y = 0, moving with velocities −U and +Uin the x-direction respectively (Fig. 6.1). There is no variation in the basic flowin the z-direction (normal to the page), and we will assume this is also true for

251

252 Chapter 6. Barotropic and Baroclinic Instability

Figure 6.1 A simple basic state givingrise to shear-flow instability. The veloc-ity profile is discontinuous, the densityuniform.

→

→

y = 0

y

x

the instability (these restrictions are not essential). This flow is clearly a solutionof the the Euler equations. What happens if the flow is perturbed slightly? If theperturbation is initially small then even if it grows we can, for small times after theonset of instability, neglect the nonlinear interactions in the governing equationsbecause these are the squares of small quantities. The equations determining theevolution of the initial perturbation are then the Euler equations linearized aboutthe steady solution. Thus, denoting perturbation quantities with a prime and basicstate variables with capital letters, for y > 0 the perturbation satisfies

∂u′

∂t+U ∂u

′

∂x= −∇p′, ∇ · u′ = 0 (6.1a,b)

and a similar equation hold for y < 0, but with U replaced by −U . Given periodicboundary conditions in the x-direction, we may seek solutions of the form

φ′(x,y, t) = Re∑kφk(y) exp[ik(x − ct)], (6.2)

where φ is any field variable (e.g., pressure or velocity), and Re denotes that onlythe real part should be taken. (Typically we use tildes over variables to denoteFourier-like modes, and we will often omit the marker ‘Re’.) Because (6.1a) is lin-ear, the Fourier modes do not interact and we may confine attention to just one.Taking the divergence of (6.1a), the left-hand side vanishes and the pressure satis-fies Laplace’s equation

∇2p′ = 0 (6.3)

This has solutions in the form

p′ =

Re p1 eikx−ky eσt y > 0,Re p2 eikx+ky eσt y < 0,

(6.4)

where, anticipating the possibility of growing solutions, we write σ = −ikc. In gen-eral the growth-rate σ is complex: if it has a positive real component, the amplitude

6.2 Instability of Parallel Shear Flow 253

of the perturbation will grow and there is an instability; if σ has a non-zero imagi-nary component, then there will be oscillatory motion, and there may of course beboth oscillatory motion and an instability. To obtain the dispersion relationship, weconsider the y-component of (6.1a), namely (for y > 0)

∂v′1∂t

+U ∂v′1

∂x= −∂p

′1

∂y(6.5)

Substituting a solution of the form v′1 = v1 exp(ikx + σt) yields, with (6.4),

(σ + ikU)v1 = kp1. (6.6)

But the velocity normal to the interface is, at the interface, nothing but the rate ofchange of the position of interface itself. That is, at y = +0

v1 =∂η′

∂t+U ∂η

′

∂x, (6.7)

orv1 = (σ + ikU)η. (6.8)

where η′ is the displacement of the interface from its equilibrium position. Usingthis in (6.6) gives

(σ + ikU)2η = kp1. (6.9)

The above few equations pertain to motion on the y > 0 side of the interface.Similar reasoning on the other side gives (at y = −0)

(σ − ikU)2η = −kp2. (6.10)

But at the interface p1 = p2 (because pressure must be continuous). The dispersionrelationship then emerges from (6.9) and (6.10), giving

σ 2 = k2U2. (6.11)

This equation has two roots, one of which is positive. Thus, the amplitude of theperturbation grows exponentially, like eσt, and the flow is unstable. The instabilityitself can be seen in the natural world when billow clouds appear wrapped upinto spirals: the clouds are acting as tracers of fluid flow, and the billows are amanifestation of the instability at finite amplitude, as in Fig. 6.6.

6.2 INSTABILITY OF PARALLEL SHEAR FLOW

We now consider a little more systematically the instability of parallel shear flows,such as are illustrated in Fig. 6.2.2 This is a classic problem in hydrodynamic stabil-ity theory, and there are two particular reasons for our own interest:

(i) The instability is an example of barotropic instability, which abounds in theocean and atmosphere. Loosely, barotropic instability arises when a flow isunstable by virtue of its shear, with gravitational and buoyancy effects beingsecondary.


(ii) The instability is in many ways analagous to baroclinic instability, which is themain instability giving rise to weather systems in the atmosphere and similarphenomena in the ocean.

Let us will restrict attention to two dimensional, incompressible flow; this illustratesthe physical mechanisms in the most transparent way, in part because it allows forthe introduction of a streamfunction and the automatic satisfaction of the masscontinuity equation. In fact, for parallel two-dimensional shear flows the mostunstable disturbances are two-dimensional ones.3

The vorticity equation for incompressible two-dimensional flow is just

DζDt

= 0. (6.12)

We suppose the basic state to be a parallel flow in the x-direction that may vary iny-direction. That is

u = U(y)i. (6.13)

The linearized vorticity equation is then

∂ζ′

∂t+U ∂ζ

′

∂x+ v′ ∂Z

∂y= 0 (6.14)

where Z = −Uy . Because the mass continuity equation has the simple form

∂u′

∂x+ ∂v

′

∂y= 0, (6.15)

we may introduce a streamfunction ψ such that u′ = −∂ψ′/∂y , v′ = ∂ψ′/∂x andζ′ = ∇2ψ′. The linear vorticity equation is then

∂∇2ψ′

∂t+U ∂∇

2ψ′

∂x+ ∂Z∂y∂ψ′

∂x= 0. (6.16)

The coefficients of the x-derivatives are not themselves functions of x. Thus we mayseek solutions that are harmonic functions (sines and cosines) in the x-direction,but the y-dependence must remain arbitrary at this point and we write

ψ′ = Re ψ(y) eik(x−ct). (6.17)

The solution is a superposition of all wavenumbers, but since the problem is linearthe waves do not interact and it suffices to consider them separately. If c is purelyreal then c is the phase speed of the wave; if c has a positive imaginary componentthen the wave will grow exponentially is thus unstable.

From (6.17) we have

u′ = u(y) eik(x−ct) = −ψy eik(x−ct), (6.18a)

v′ = v(y) eik(x−ct) = ikψ eik(x−ct), (6.18b)

ζ′ = ζ(y) eik(x−ct) = (−k2ψ+ ψyy) eik(x−ct), (6.18c)

where the y subscript denotes a derivative. Using (6.18) in (6.14) gives

(U − c)(ψyy − k2ψ)−Uyyψ = 0 , (6.19)


Fig. 6.2 Left: example of smooth velocity profile — both the velocity andvorticity are continuous. Right: example piecewise continuous profile — thevelocity and vorticity may have finite discontinuities.

sometimes known as Rayleigh’s equation.4 It is the linear vorticity equation fordisturbances to parallel shear flow, and in the presence of a β-effect it generalizesslightly to

(U − c)(ψyy − k2ψ)+ (β−Uyy)ψ = 0 . (6.20)

6.2.1 Piecewise linear flows

Although Rayleigh’s equation is linear and has a simple form, it is nevertheless quitedifficult to analytically solve for an arbitrary smoothly varying profile. It is simplerto consider piecewise linear flows, in which Uy is a constant over some interval, withU or Uy changing abruptly to another value at a line of discontinuity, as illustratedin Fig. 6.2. The curvature, Uyy is accounted for through the satisfaction of matchingconditions, analogous to boundary conditions, at the lines of discontinuity (as insection 6.1), and solutions in each interval are then exponential functions.

Jump or Matching conditions

The idea, then, is to solve the linearized vorticity equation separately in the contin-uous intervals in which vorticity is constant, matching the solution with that in theadjacent regions. The matching conditions arise from two physical conditions:

(i) That normal stress should be continuous across the interface. For an inviscidfluid this implies that pressure be continuous.

(ii) That the normal velocity of the fluid on either side of the interface should beconsistent with the motion of the interface itself.

Let us consider the implications of these two conditions.

(i) Continuity of pressure:The linearized momentum equation in the direction along the interface is:

∂u′

∂t+U ∂u

′

∂x+ v′ ∂U

∂y= −∂p

′

∂x. (6.21)


For normal modes, u′ = −ψy eik(x−ct) and v′ = ikψ eik(x−ct) and (6.21) be-comes

ik(U − c)ψy − ikψUy = −ikp. (6.22)

Because pressure is continuous across the interface we have the first matchingor jump condition,

∆[(U − c)ψy − ψUy] = 0 (6.23)

where the operator ∆ denotes the difference in the values of the argument (insquare brackets) across the interface. That is, the quantity (U − c)ψy − ψUy iscontinuous.

We can obtain this condition directly from Rayleigh’s equation, (6.20), writtenin the form

[(U − c)ψy −Uyψ]y + [β− k2(U − c)]ψ = 0. (6.24)

Integrating across the interface gives (6.23).

(ii) Material interface condition:At the interface, the normal velocity v is given by the kinematic condition

v = DηDt

(6.25)

where η is the interface displacement. The linear version of (6.25) is

∂η′

∂t+U ∂η

′

∂x= ∂ψ

′

∂x. (6.26)

If the fluid itself is continuous then this equation must hold at either side of theinterface, giving two equations and their normal mode counterparts, namely,

∂η′

∂t+U1

∂η′

∂x= ∂ψ

′1

∂x−→ (U1 − c)η = ψ1, (6.27)

∂η′

∂t+U2

∂η′

∂x= ∂ψ

′2

∂x−→ (U2 − c)η = ψ2. (6.28)

Material continuity at the interface thus gives the second jump condition:

∆[ψU − c

]= 0 . (6.29)

That is, ψ/(U − c) is continuous at the interface. Note that if U is continuousacross the interface the condition becomes one of continuity of the normalvelocity.


Figure 6.3 Velocity profile of a stable jet. Although thevorticity is discontinuous, a small perturbation gives riseonly to edge waves centered at y = 0.

6.2.2 Kevin-Helmholtz instability, revisited

We now use Rayleigh’s equation and the jump conditions to consider the situationillustrated in Fig. 6.1; that is, vorticity is everywhere zero except in a thin sheet aty = 0. On either side of the interface, Rayleigh’s equation is simply

(U − c)(∂yyψi − k2ψi) = 0 i = 1,2 (6.30)

or, assuming that U ≠ c, ψyy − k2ψ = 0. (This is just Laplace’s equation, comingfrom ∇2ψ′ = ζ′, with ζ′ = 0 everywhere except at the interface.) Solutions of thisthat decay away on either side of the interface are

y > 0 : ψ1 = Ψ1 e−ky , (6.31a)

y < 0 : ψ2 = Ψ2 eky , (6.31b)

where Ψ1 and Ψ2 are constants. The boundary condition (6.23) gives

(U1 − c)(−k)Ψ1 = (U2 − c)(k)Ψ2, (6.32)

and (6.29) givesΨ1

(U1 − c)= Ψ2

(U2 − c). (6.33)

The last two equations combine to give (U1 − c)2 = −(U2 − c)2, which, supposingthat U = U1 = −U2 gives c2 = −U2. Thus, since U is purely real, c = ±iU , and thedisturbance grows exponentially as exp(kU1t), just as we obtained in section 6.1.All wavelengths are unstable, and indeed the shorter the wavelength the greater theinstability. In reality, viscosity will damp the smallest waves, but at the same timethe presence of viscosity would also mean that the initial profile is not an exact,steady solution of the equations of motion.

6.2.3 Edge Waves

We now consider a case sketched in Fig. 6.3 in which the velocity is continuous,but the vorticity is discontinuous. Since on either side of the interface Uyy = 0,


Rayleigh’s equation is just

(U − c)(ψyy − k2ψ) = 0. (6.34)

Provided c ≠ U this has solutions,

ψ =

Φ1 e−ky y > 0Φ2 eky y < 0.

(6.35)

The value of c is found by applying the jump conditions (6.23) and (6.29) at y = 0.Using (6.35) these give

−k(U0 − c)Φ1 − Φ1U1y = k(U0 − c)Φ2 − Φ2U2y (6.36a)

Φ1 = Φ2 (6.36b)

where U1 and U2 are the values of U at either side of the interface, and both areequal to U0 at the interface. After a line of algebra these equations give

c = U0 +∂yU1 − ∂yU2

2k. (6.37)

This is the dispersion relationship for edge waves that propagate along the interface aspeed equal to the sum of the fluid speed and a factor proportional to the differencein the vorticity between the two layers. No matter what the shear is on either sideof the interface, the phase speed is purely real and there is no instability. Eq. (6.37)is imperfectly analogous to the Rossby wave dispersion relation c = U0 − β/K2,and reflects a similarity in the physics — β is a planetary vorticity gradient, whichin (6.37) is collapsed to a front and represented by the difference U1y − U2y =−(Z1 − Z2), where Z1 and Z2 are the basic-state vorticities on either side of theinterface.

6.2.4 Interacting edge waves producing instability

Now we consider a slightly more complicated case in which edge waves may inter-act giving rise, as we shall see, to an instability. The physical situation is illustratedin Fig. 6.4. We consider the simplest case, that of a shear layer (which we denoteregion 2) sandwiched between two semi-infinite layers, regions 1 and 3, as in theleft panel of the figure. Thus, the basic state is:

y > a : U = U1 = U0 (a constant), (6.38a)

−a < y < a : U = U2 =U0

ay, (6.38b)

y < −a : U = U3 = −U0. (6.38c)

We assume a solution of Rayleigh’s equation of the form:

y > a : ψ1 = A e−k(y−a), (6.39a)

−a < y < a : ψ2 = B e−k(y−a) + C ek(y+a), (6.39b)

y < −a : ψ3 = D ek(y+a). (6.39c)


to 8

................ ................................. y=a

.......................... .......................y=-a

to - 8

y = b

y = -b

Fig. 6.4 Barotropically unstable velocity profiles. In the simplest case, on theleft, a region of shear is sandwiched between two infinite regions of constantvelocity. The edge waves at y = ±a interact to produce an instability. If a =0, then the situation corresponds to that of Fig. 6.1, giving Kelvin-Helmholtzinstability. In the case on the right, the flow is bounded at y = ±b. Itmay be shown that the flow is still unstable, provided that b is sufficientlylarger than a. If b = a (plane Couette flow) the flow is stable to infinitesimaldisturbances.

Applying the jump conditions (6.23) and (6.29) at the interfaces at y = a andy = −a gives the following relations between the coefficients:

A[(U0 − c)k] = B[(U0 − c)k+

U0

a

]+ C e2ka

[U0

a− (U0 − c)k

], (6.40a)

A = B + C e2ka, (6.40b)

D[(U0 + c)k] = B e2ka[−(U0 + c)k+

U0

a

]+ C

[U0

a+ (U0 + c)k

], (6.40c)

D = B e2ka + C. (6.40d)

These are a set of four homogeneous equations, with the unknown parametersA, B, C and D, which may be written in the form of a matrix equation,k(U0 − c) −k(U0 − c)−U0/a e2ka[k(U0 − c)− (U0/a)] 0

1 −1 − e2ka 00 e2ka[k(U0 + c)− (U0/a)] −k(U0 + c)− (U0/a) k(U0 + c)0 e2ka 1 −1

ABCD

= 0.

(6.41)

For non-trivial solutions the determinant of the matrix must be zero, and solvingthe ensuing equation gives the dispersion relationship5

c2 =(U0

2ka

)2 [(1− 2ka)2 − e−4ka

], (6.42)

and this is plotted in Fig. 6.5. The flow is unstable for sufficiently long wavelengths,


0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

Wavenumber

Wav

e S

peed

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Wavenumber

Gro

wth

Rat

e

Fig. 6.5 Left: Growth rate (σ = kci) calculated from (6.42) with c nondi-mensionalized by U0 and k nondimensionalized by 1/a (equivalent to settinga = U0 = 1). Right: Real (cr , dashed) and imaginary (ci, solid) wave speeds.The flow is unstable for k < 0.63, with the maximum instability occurring atk = 0.39.

for then the right-hand side of (6.42) is negative. The critical wavenumber belowwhich instability occurs is found by solving (1−2ka)2 = e−4ka, which gives instabil-ity for ka < 0.63293. A numerical solution of the initial value problem is illustratedin Fig. 6.6 and Fig. 6.7.6 Here, the initial perturbation is small and random, con-taining components at all wavenumbers. All the modes in the unstable range growexponentially, and the pattern is soon dominated by the mode that grows fastest —a horizontal wavenumber three in this problem. Eventually, the perturbation growssufficiently that the linear equations are no longer valid and, as is seen in the sec-ond column of Fig. 6.6, vortices form and pinch off. Eventually, the vortices interactand the flow develops into two-dimensional turbulence, considered in chapter 8.

The mechanism of the instability — an informal view

[A similar mechanism is discussed in section 6.7, and the reader may wish to readthe two descriptions in tandem.] We have seen that an edge wave in isolation is sta-ble, the instability arising when two edge waves have sufficient cross-stream extentthat they can interact with each other. This occurs for sufficiently long wavelengthsbecause the cross-stream decay scale is proportional to the along-stream wavelength— hence the high-wavenumber cut-off. To transparently see the mechanism of theinstability, let us first suppose that the interfaces are, in fact, sufficiently far awaythat the edge waves at each interface do not interact. Using (6.37) the edge wavesat y = −a and y = +a have dispersion relationships

c+a = U0 −U0/a2k

, c−a = −U0 +U0/a2k

(6.43a,b)

If the two waves are to interact these phase speeds must be equal, giving thecondition

c = 0, k = 1/(2a). (6.44a,b)

That is, the waves are stationary, and their wavelength is proportional to the sepa-ration of the two edges. In fact, (6.44) approximately characterizes the conditions


Fig. 6.6 A sequence of plots of the vorticity, at equal time intervals, from a numericalsolution of the nonlinear vorticity equation (6.12), with initial conditions as in Fig. 6.4 witha = 0.1, plus a very small random perturbation. Time increases first down the left column,then down the right column. The solution is obtained in a rectangular (4× 1) domain, withperiodic conditions in the x-direction and slippery walls at y = (0,1). The maximum linearinstability occurs for a wavelength of 1.57, which for a domain of length 4 corresponds to awavenumber of 2.55. Since the periodic domain quantizes the allowable wavenumbers, themaximum instability is at wavenumber 3, and this is what emerges. Only in the first two orthree frames is the linear approximation valid.

Fig. 6.7 Total streamfunction (top panel) and perturbation streamfunctionfrom the same numerical calculation as in Fig. 6.6, at a time corresponding tothe second frame. Positive values (a clockwise circulation) are solid lines, andnegative values are dashed. The perturbation pattern leans into the shear,and grows exponentially in place.


at the critical wavenumber k = 0.63/a (see Fig. 6.5). In the region of the shear thetwo waves have the form

ψ+a = Re ψ+a(t) ek(y−a) eiφ eikx , ψ−a = Re ψ−a(t) e−k(y+a) eikx (6.45a,b)

where φ is the phase shift between the waves; in the case of pure edge waves wehave ψ±a = A±a e−ikct where we may take A±a to be real.

Now consider how the wave generated at y = −a might affect the wave aty = +a and vice versa. The contribution of ψ−a to acceleration of ψ+a is givenby applying the x-momentum equation, (6.21), at either side of the interface aty = +a, and similarly for the acceleration at y = −a. Thus we take the kinematicsolutions, (6.45), and use them in a dynamical equation, the momentum equation,to calculate the ensuing acceleration. We obtain

∂u′+a∂t

≈ −v′−a(+a)∂U∂y,

∂u′−a∂t

≈ −v′+a(−a)∂U∂y, (6.46a,b)

at y = +a and y = −a respectively, omitting the terms that give the neutral edgewaves. Here v′−a(+a) denotes the value of v at y = +a due to the edge wavegenerated at −a. If the spatial dependence of the waves is given by (6.45) thisgives, at y = ±a,

− k eiφ ∂ψ+a∂t

≈ −ikψ−a∂U∂y, k

∂ψ−a∂t

≈ −ik eiφψ+a∂U∂y, (6.47a,b)

If ψ+a and ψ−a have the appropriate phase with respect to each other, then the twoedge waves can feed back on each other. In particular, from (6.47) we see that thesystem is unstable when φ = π/2, for then (6.47) gives

∂2ψ+a∂t2

≈ ψ+a(∂U∂y

)2

(6.48)

In this case, the wave at y = +a lags the wave at y = −a. That is, the perturbationis unstable when it tilts into the shear, and this is seen in the full solution, Fig. 6.7.

6.3 NECESSARY CONDITIONS FOR INSTABILITY

6.3.1 Rayleigh’s criterion

For simple profiles it may be possible to calculate, or even intuit, the instabilityproperties, but for continuous profiles of U(y) this is often impossible and it wouldbe nice to have some general guidelines as to when a profile might be unstable. Tothis end, we will derive a couple of necessary conditions for instability, or sufficientconditions for stability, that will at least tell us if a flow might be unstable.

We first write Rayleigh’s equation, (6.20), as

ψyy − k2ψ+ β−UyyU − c ψ = 0. (6.49)

Multiply by ψ∗ (the complex conjugate of ψ) and integrate over the domain ofinterest. After integrating the first term by parts, this gives∫ y2

y1

∣∣∣∣∣∂ψ∂y∣∣∣∣∣

2

+ k2|ψ|2 dy −

∫ y2

y1

β−UyyU − c |ψ|2 dy = 0, (6.50)

6.3 Necessary Conditions for Instability 263

assuming that ψ vanishes at the boundaries. (The limits to the integral may beinfinite, in which case it is assumed that ψ decays to zero as |y| approaches ∞.)The only variable in this expression that is complex is c, and thus the first integralis real. The imaginary component of the second integral is

ci∫ β−Uyy|U − c|2 |ψ|

2 dy = 0. (6.51)

Thus, either ci vanishes or the integral does. For there to be an instability, ci must benonzero and because the eigenvalues of Rayleigh’s equation come in pairs (becauseit is a second order ODE), and for each decaying mode (negative ci) there is acorresponding growing mode (positive ci). Therefore:

A necessary condition for instability is that the expression

β−Uyy

change sign somewhere in the domain.

Equivalently, a sufficient criterion for stability is that β − Uyy not vanish in thedomain interior. This condition is known as Rayleigh’s inflection-point criterion, orwhen β ≠ 0, the Rayleigh-Kuo inflection point criterion.7

An alternate, more general, derivation

Consider again the vorticity equation, linearized about a parallel shear flow [c.f.,(6.14) with a β term],

∂ζ∂t

+U ∂ζ∂x

+ v(∂Z∂y

+ β)= 0, (6.52)

(dropping the primes on the perturbation quantities). Multiply by ζ and divide byβ+Zy to obtain

∂∂t

(ζ2

β+Zy

)+ Uβ+Zy

∂ζ2

∂x+ vζ = 0, (6.53)

and then integrate with respect to x to give

∂∂t

∫ (ζ2

β+Zy

)dx = −

∫vζ dx. (6.54)

Now, using ∇ · u = 0,

vζ = − ∂∂y(uv)+ 1

2∂∂x(u2 + v2). (6.55)

That is, the flux of vorticity is the divergence of some quantity. Its integral thereforevanishes provided there are no contributions from the boundary, and integrating(6.54) with respect to y gives

ddt

∫ (ζ2

β+Zy

)dx dy = 0. (6.56)


If there is to be an instability ζ must grow, but the integral is identically zero.These two conditions can only be simultaneously satisfied if β+Zy , or equivalentlyβ−Uyy , is zero somewhere in the domain.

This derivation shows that the inflection-point criterion applies even if distur-bances are not of normal-mode form. The quantity ζ2/(β+Zy) is an example of awave-activity density — a wave activity being a conserved quantity, quadratic in theamplitude of the wave. Such quantities play an important role in instabilities, andwe consider then further in chapter 7.

6.3.2 Fjørtoft’s criterion

Another necessary condition for instability was obtained by Fjørtoft.8 In this sectionwe will derive the condition for normal mode disturbances, and provide a moregeneral derivation in section 7.7. From the real part of (6.50) we find∫ y2

y1

(β−Uyy)(U − cr )|U − c|2 |ψ|

2 dy =∫ y2

y1

∣∣∣∣∣∂ψ∂y∣∣∣∣∣

2

+ k2|ψ|2 dy > 0. (6.57)

Now, from (6.51), we know that for an instability we must have∫ y2

y1

β−Uyy|U − c|2 |ψ|

2 dy = 0. (6.58)

Using this and (6.57) it is clear that, for an instability,∫ y2

y1

(β−Uyy)(U −Us)|U − c|2 |ψ|

2 dy > 0 (6.59)

where Us is any real constant. It is most useful to choose this constant to be thevalue of U(y) at which β−Uyy vanishes. This leads directly to the criterion:

A necessary condition for instability is that the expression

(β−Uyy)(U −Us)

where Us is the value of U(y) at which β − Uyy vanishes, be positivesomewhere in the domain.

This is equivalent to saying that the magnitude of the vorticity must have an ex-tremum inside the domain, and not at the boundary or at infinity, as can be seenby perusing Fig. 6.8. Why choose Us in the manner we did? Suppose we choseUs to have a very large negative or large positive value, so that U − Us is of onesign everywhere. Then (6.59) just implies that β − Uyy must be negative some-where and must be positive somewhere, which is already known from Rayleigh’scriterion. The most stringent criterion is obtained by choosing Us to be the valueof U(y) at which β − Uyy vanishes. Both Fjørtoft’s and Rayleigh’s criteria are nec-essary conditions for instability, and examples may be constructed which do satisfytheir criterion, yet which are stable to infinitesimal perturbations. Note that the β-effect can stabilize the middle two profiles of Fig. 6.8, because if it is large enoughβ−Uyy will be one-signed. However, the β-effect can destabilize a westward pointjet, U(y) = −(1 − |y|) (the negative of the jet in Fig. 6.3), because β − Uyy isnegative at y = 0 and positive elsewhere. An eastward point jet is stable, with orwithout β.

6.4 Baroclinic Instability 265

0 1−1

0

1

−5 0 5−1

0

1

0 1−1

0

1

−5 0 5−1

0

1

−1 0 1−1

0

1

−5 0 5−1

0

1

0 0.5 1−1

0

1

U−5 0 5

−1

0

1

d2U/dy2

Fig. 6.8 Example parallel velocity profiles (left column) and their secondderivatives (right column). From the top: Poiseuille flow (u = 1−y2); a Gaus-sian jet; a sinusoidal profile; a polynomial profile. By Rayleigh’s criterion,the top profile is stable, whereas the lower three are potentially unstable.However, the bottom profile is stable by Fjørtoft’s criterion (and note thatthe vorticity maxima are at the boundaries). If the β-effect were present andlarge enough it would stabilize the middle two profiles.

6.4 BAROCLINIC INSTABILITY

Baroclinic instability is a hydrodynamic instability that occurs in stably stratified,rotating fluids, and it is ubiquitous in the planetary atmospheres and oceans. Itgives rise to weather, and thus is perhaps the form of hydrodynamic instability thatmost affects the human condition.

6.4.1 A physical picture

We will first draw a picture of baroclinic instability as a form of ‘sloping convection’in which the fluid, although statically stable, is able to release available potentialenergy when parcels move along a sloping path. To this end, let us first ask: whatis the basic state that is baroclinically unstable? In a stably stratified fluid potentialdensity decreases with height; we can also easily imagine a state in which the basicstate temperature decreases, and the potential density increases, polewards. (We


Fig. 6.9 A steady basic state giving rise to baroclinic instability. Potentialdensity decreasing upwards and equatorwards, and the associated horizontalpressure gradient is balanced by the Coriolis force. Parcel ‘A’ is heavier than‘C’, and so statically stable, but it is lighter than ‘B’. Hence, if ‘A’ and ‘B’ areinterchanged there is a release of potential energy.

will couch most of our discussion in terms of the Boussinesq equations, and hence-forth drop the qualifier ‘potential’ from density.) Can we construct a steady solutionfrom these two conditions? The answer is yes, provided the fluid is also rotating;rotation is necessary because the meridional temperature gradient generally im-plies a meridional pressure gradient; there is nothing to balance this in the absenceof rotation, and a fluid parcel would therefore accelerate. In a rotating fluid thispressure gradient can be balanced by the Coriolis force and a steady solution main-tained even in the absence of viscosity. Consider a stably-stratified Boussinesq fluidin geostrophic and hydrostatic balance on an f -plane, with buoyancy decreasinguniformly polewards. Then fu = −∂φ/∂y and ∂φ/∂z = b, where b = −gρ′/ρ0

is the buoyancy. These together give the thermal wind relation, ∂u/∂z = ∂b/∂y .If there is no variation of these fields in the zonal direction, then, for any variationof b with y, this is a steady solution to the primitive equations of motion, withv = w = 0.

The density structure corresponding to a uniform increase of density in themeridional direction is illustrated in Fig. 6.9. Is this structure stable to pertur-bations? The answer is no, although the perturbations must be a little special.Suppose the particle at ‘A’ is displaced upwards; then, since the fluid is (by assump-tion) stably stratified it will be denser than its surroundings and hence experiencea restoring force, and similarly if displaced downwards. Suppose, however, we in-terchange the two parcels at positions ‘A’ and ‘B’. Parcel ‘A’ finds itself surroundedby parcels of higher density that itself, and it is therefore buoyant; it is also higherthan where it started. Parcel ‘B’ is negatively buoyant, and at a lower altitude thanwhere is started. Thus, overall, the centre of gravity of the fluid has been lowered,and so its overall potential energy lowered. This loss in potential energy of the basicstate must be accompanied by a gain in kinetic energy of the perturbation. Thus,the perturbation amplifies and converts potential energy to kinetic energy.

6.4 Baroclinic Instability 267

The loss of potential energy is easily calculated. Since

PE =∫ρgdz (6.60)

the change in potential energy due to the interchange is

∆PE = g(ρAzA + ρBzB − ρAzB − ρBzB)= g(zA − zB)(ρA − ρB) = g∆ρ∆z (6.61)

If both ρB > ρA and zB > zA then the initial potential energy is larger than thefinal, energy is released and the state is unstable. If the slope of the isopycnals isφ [so that φ = −(∂yρ)/(∂zρ)] and the slope of the displacements is α, then for adisplacement of horizontal distance L the change in potential energy is given by

∆PE = g∆ρ∆z = g(L∂ρ∂y

+ Lα∂ρ∂z

)αL = gL2α

∂ρ∂y

(1− α

φ

), (6.62)

if α andφ are small. If 0 < α < φ then energy is released by the perturbation, and itis maximized when α = φ/2. For the atmosphere the actual slope of the isothermsis about 10−3, so that the slope and potential parcel trajectories are indeed shallow.

Although intuitively appealing, the thermodynamic arguments presented in thissection pay no attention to satisfying the dynamical constraints of the equations ofmotion, and we now turn our attention to that,

6.4.2 Linearized quasi-geostrophic equations

To explore the dynamics of baroclinic instability we use the quasi-geostrophic equa-tions, specifically a potential vorticity equation for the fluid interior and a buoyancyor temperature equation at two vertical boundaries, one representing the groundand the other the tropopause — the boundary between the troposphere and strato-sphere at about 10 km. (The tropopause is not a true rigid surface, but the higherstatic stability of the stratosphere does inhibit vertical motion. We return to this insection 6.9.) For a Boussinesq fluid the equations are

∂q∂t+ u · ∇q = 0, 0 < z < H,

q = ∇2ψ+ βy + ∂∂z

(F∂ψ∂z

),

(6.63)

where F = f 20 /N2, and the buoyancy equation with w = 0,

∂b∂t+ u · ∇b = 0, z = 0,H,

b = f0∂ψ∂z.

(6.64)

A solution of these equations is a purely zonal flow, u = U(y, z)i with a correspond-ing temperature field given by thermal wind balance. The potential vorticity of thisbasic state is

Q = βy −Uy +∂∂zF∂Ψ∂z

= βy + Ψyy +∂∂zF∂Ψ∂z

(6.65)


where Ψ is the streamfunction of the basic state, related to U by U = −∂Ψ/∂y .Linearizing (6.63) about this zonal flow gives the potential vorticity equation forthe interior,

∂q′

∂t+U ∂q

′

∂x+ v′ ∂Q

∂y= 0, 0 < z < H (6.66)

where q′ = ∇2ψ′+∂z(F∂zψ′

)and v′ = ∂ψ′/∂x . Similarly, the linearized buoyancy

equation is∂b′

∂t+U ∂b

′

∂x+ v′ ∂B

∂y= 0, z = 0,H, (6.67)

where b′ = f0∂ψ′/∂z and ∂B/∂y = ∂y(f0∂zΨ) = −f0∂U/∂z.Just as for the barotropic problem, a standard way to of proceeding is to seek

normal-mode solutions of these equations. Since the coefficients of the equationsare functions of y and z, but not of x, we seek solutions of the form

ψ′(x,y, z, t) = Re ψ(y, z) eik(x−ct), (6.68)

and similarly for the derived quantities u′, v′ and q′. In particular

q = ψyy +∂∂zF∂ψ∂z

− k2ψ. (6.69)

Substituting (6.68) and (6.69) into (6.66) into (6.67) gives

(U − c)(ψyy + (Fψz)z − k2ψ

)+Qyψ = 0 0 < z < H, (6.70a)

(U − c)ψz −Uzψ = 0 z = 0,H. (6.70b)

These equations are analogous to Rayleigh’s equations for parallel shear flow, andemphasize the similarity between baroclinic instability and that of a parallel shearflow.

6.4.3 Necessary conditions for baroclinic instability

Necessary conditions for instability may be obtained following a procedure analo-gous to that used for parallel shear flows. First, integrating by parts, we note that∫ y2

y1

ψ∗ψyy dy =[ψ∗ψy

]y2

y1−∫ y2

y1

|ψy |2 dy. (6.71)

If the integral is performed between two quiescent latitudes, or the domain is achannel with ψ = 0 at the boundaries, then the first term on the right-hand sidevanishes. Similarly,∫ H

0ψ∗(Fψz)z dz =

[Fψ∗ψz

]H0−∫ H

0F|ψz|2 dz

=[FUz|ψ|2(U − c)

]H0−∫ H

0F|ψz|2 dz, (6.72)

6.5 The Eady Problem 269

using (6.70b). Now, multiply (6.70a) by ψ∗ and integrate over y and z, and use(6.71) and (6.72) to obtain∫ H

0

∫ y2

y1

|ψy |2 + F|ψz|2 + k2|ψ|2]dy dz

−∫ y2

y1

∫ H

0

QyU − c |ψ|

2 dz +[FUz|ψ|2U − c

]H0

dy = 0.(6.73)

The term on the first line is purely real. The term on the second line is complex,and its imaginary component is given by

− ci

∫ y2

y1

∫ H

0

Qy|U − c|2 |ψ|

2 dz +[FUz|ψ|2|U − c|2

]H0

dy = 0. (6.74)

If there is to be instability ci must be non-zero, and the integrand must thereforevanish. This gives the Charney-Stern-Pedlosky (CSP) necessary condition for insta-bility, namely that one of the following criteria be satisfied:9

(i) Qy changes sign in the interior.(ii) Qy is the opposite sign to Uz at the upper boundary, z = H.(iii) Qy is the same sign as Uz at the lower boundary, z = 0.(iv) Uz is the same sign at the upper and lower boundaries, a condition which

differs from (ii) or (iii) if Qy = 0.In the earth’s atmosphere, Qy is often dominated by β, and is positive every-where, as, frequently, is the shear. The instability criterion is then normally satisfiedthrough (iii): that is, both Qy and Uz(0) are positive. A more general derivationthat does not rely on normal mode disturbances is given in section 7.7.2.

6.5 THE EADY PROBLEM

We now proceed to explicitly calculate the stability properties of a particular con-figuration that has become known as the Eady problem. This was one of the firsttwo mathematical descriptions of baroclinic instability, the other being the Charneyproblem.10 The two were formulated independently, each being the (largely unsu-pervised) Ph.D. thesis of its respective author, and although the Charney problem isin some respects more complete (for example in allowing a β-effect) the Eady prob-lem displays the instability in a more transparent form. The Charney problem in itsentirety is also quite mathematically opaque,11 and for these reasons we will firstconsider the Eady problem. The β-effect can be incorporated relatively simply inthe two-layer model (the ‘Phillips problem’) considered in the next section, and insection 6.9.1 we look at some aspects of the Charney problem approximately. Theseproblems were all initially envisioned as models for instabilities in the atmosphere,but the process of baroclinic instability is also ubiquitous in the ocean. To begin, letus make the following simplifying assumptions:

(i) The motion is on the f -plane (β = 0). This assumption, although not particu-larly realistic, greatly simplifies the analysis.

(ii) The fluid is uniformly stratified; that is, N2 is a constant. This is a decentapproximation for the atmosphere below the tropopause, but less so for the


ocean where the stratification is quite non-uniform, being much larger in theupper ocean.

(iii) The basic state has uniform shear; that is, U0(z) = Λz = Uz/H where Λ isthe (constant) shear and U is the zonal velocity at z = H where H the domaindepth. Again, this profile is more appropriate for the atmosphere than theocean — below the thermocline the ocean is relatively quiescent and the shearsmall.

(iv) The motion is contained between two rigid, flat horizontal surfaces. In theatmosphere this corresponds to the ground and a ‘lid’ at a constant-heighttropopause.

Assumptions (ii)–(iv) are rather inappropriate for the ocean, and will preclude usfrom drawing any quantitative conclusions about that system from our analysis.The most restrictive assumption vis-a-vis the atmosphere is (i).

6.5.1 The linearized problem

With a basic state streamfunction of Ψ = −Λzy, the basic state potential vorticity,Q, is

Q = ∇2Ψ + H2

L2d

∂∂z

(∂Ψ∂z

)= 0. (6.75)

The fact that Q = 0 makes the Eady problem a special case, albeit an illuminatingone. The linearized potential vorticity equation is

(∂∂t+Λz ∂

∂x

)(∇2ψ′ + H

2

L2d

∂2ψ′

∂z2

)= 0 (6.76)

This equation has no x-dependent coefficients and in a periodic channel we mayseek solutions in the form (6.68), namely ψ′(x,y, z, t) = Re ψ(y, z) eik(x−ct). Sub-stituting this into (6.76) yields

(Λz − c)(∂2ψ∂y2 +

H2

L2d

∂2ψ∂z2 − k

2ψ)= 0, (6.77)

which is (6.70a) applied to the Eady problem.

Boundary Conditions

There are two sets of boundary conditions to satisfy, the vertical boundary condi-tions at z = 0 and z = 1 and the lateral boundary conditions. In the horizontalplane we may either consider the flow to be confined to a channel, periodic in xand confined between two meridional walls, or, with a slightly greater degree ofidealization but with little change to the essential dynamics, suppose that domainis doubly-periodic. Either case is dealt with easily enough by the choice of geometricbasis function; we will choose a channel of width L and impose ψ = 0 at y = +L/2and y = −L/2 and, to satisfy this, seek solutions of the form Ψ = Φ(z) sin ly or,using (6.68)

ψ′(x,y, z, t) = ReΦ(z) sin ly eik(x−ct). (6.78)

where l = nπ/L where n is a positive integer.


The vertical boundary conditions are that w = 0 at z = 0 and z = H. We followthe procedure of section 6.4.2 and from (6.67) we obtain(

∂∂t+Λz ∂

∂x

)∂ψ′

∂z−Λ∂ψ

′

∂x= 0, at z = 0,H. (6.79)

Solutions

Substituting (6.78) into (6.77) gives the interior potential vorticity equation

(Λz − c)[H2

L2d

∂2Φ∂z2 − (k

2 + l2)Φ]= 0, (6.80)

and substituting (6.78) into (6.79) gives, at z = 0 and z = H,

cdΦdz

+ΛΦ = 0 and (c −ΛH)dΦdz

+ΛΦ = 0. (6.81a,b)

These are equivalent to (6.70b) applied to the Eady problem. If Λz ≠ c then (6.80)becomes12

H2 d2Φdz2 − µ

2Φ = 0, (6.82)

where µ2 = L2d(k

2+ l2). The nondimensional parameter µ is a horizontal wavenum-ber, scaled by the inverse of the Rossby radius of deformation. Solutions of (6.82)are

Φ(z) = A coshµz + B sinhµz, (6.83)

where z = z/H; thus, µ determines the vertical structure of the solution. Theboundary conditions (6.81) are satisfied if

A[ΛH]+ B [µc] = 0,A [(c −ΛH)µ sinhµ +ΛH coshµ]+ B [(c −ΛH)µ coshµ +ΛH sinhµ] = 0.

(6.84)

Equations (6.84) are two coupled homogeneous equations in the two unknowns Aand B. Non-trivial solutions will only exist if the determinant of their coefficients(the terms in square brackets) vanishes, and this leads to

c2 −Uc +U2(µ−1 cothµ − µ−2) = 0, (6.85)

where U ≡ ΛH and cothµ = coshµ/ sinhµ. The solution of (6.85) is

c = U2± Uµ

[(µ2− coth

µ2

)(µ2− tanh

µ2

)]1/2. (6.86)

The waves, being proportional to exp(−ikct), will grow exponentially if c has animaginary part. Since µ/2 > tanh(µ/2) for all µ, for an instability we require that

µ2< coth

µ2, (6.87)

which is satisfied when µ < µc where µc = 2.399. The growth rates of the in-stabilities themselves are given by the imaginary part of (6.86), multiplied by thex-wavenumber. That is

σ = kci = kUµ

[(coth

µ2− µ

2

)(µ2− tanh

µ2

)]1/2. (6.88)


These solutions suggest a natural nondimensionalization: scale length by Ld,height by H and time by Ld/U . The Eady growth rate is the inverse of the timescaling, and is defined by

σE ≡ΛHLd

= ULd. (6.89)

Its inverse, the Eady timescale, may also be written as

TE =LdU= NHf0U

= 1Frf0

=√

Rif0, (6.90)

where Fr = U/(NH) and Ri = N2/Λ2 are the Froude and Richardson numbers forthis problem.

From (6.88) we may determine that the maximum growth rate occurs whenµ = µm = 1.61, with associated (nondimensional) growth rate of kci/σE = 0.31,and phase speed cr/U = 0.5. Note that for any given x-wavenumber, the mostunstable wavenumber has l = 0, so that Ldk = µ. The unstable x-wavenumbersand corresponding wavelengths occur for

k < kc =µcLd

= 2.4Ld, λ > λc =

2πLdµc

= 2.6Ld. (6.91a,b)

The wavenumber and wavelength at which the instability is greatest are:

km =1.6Ld, λm =

2πLdµm

= 3.9Ld. (6.92a,b)

These properties are illustrated in the left panels of Fig. 6.10 and in Fig. 6.11.Given c, we may use (6.84) to determine the vertical structure of the Eady wave

and this is, to within an arbitrary constant factor,

Φ(z) = coshµz − Uµc

sinhµz =[

coshµz − Ucr sinhµzµ|c2| + iUci sinhµz

µ|c2|

]. (6.93)

The wave therefore has a phase, θ(z), given by

θ(z) = tan−1

(Uci sinhµz

µ|c2| coshµz −Ucr sinhµz

). (6.94)

The phase and amplitude of the Eady waves are plotted in the right panels of Fig.6.10, and their overall structure in Fig. 6.12, where we see the unstable wave tiltinginto the shear.

6.5.2 Atmospheric and oceanic parameters

To get a qualitative sense of the nature of the instability we choose some typicalparameters, as follows.

For the atmosphere

Let us choose

H ∼ 10 km, U ∼ 10 m s−1, N ∼ 10−2 s−1. (6.95)


0 1 2 3 40

0.1

0.2

0.3

0.4G

row

th R

ate,

kC

i

Zonal Wavenumber

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Wav

e sp

eeds

, Cr a

nd C

i

Zonal Wavenumber

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Phase

Hei

ght

0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Hei

ght

Amplitude

Cr

Ci

(a) (c)

(b) (d)

Fig. 6.10 Solution of the Eady problem, in non-dimensional units. (a) Growth rate, kci, ofthe most unstable Eady modes (i.e., those with the gravest meridional scale) as a functionof scaled wavenumber µ, from (6.88) with Λ = H = 1. (b) The real (solid) and imaginary(dashed) wave speeds of those modes, as a function of horizontal wavenumber. (c) Thephase of the single most unstable mode as a function of height. (d) The amplitude of thatmode as a function of height. To obtain dimensional values, multiply the growth rate byΛH/Ld and the wavenumber by 1/Ld.

0 1 2 30

1

2

3

0.3

0.2

0.1

x-wavenumber

y-wavenumber

Figure 6.11 Contours of growthrate, σ , in the Eady problem,in the k–l plane using (6.88),nondimensionalized as in Fig.6.10. The growth rate peaksnear the deformation scale, andfor any given zonal wavenumberthe most unstable wavenumberis that with the gravest merid-ional scale.


Fig. 6.12 Left column: Vertical structure of the most unstable Eady mode.Top: contours of streamfunction. Middle: temperature, ∂ψ/∂z. Bottom:meridional velocity, ∂ψ/∂x . Negative contours are dashed, and two completewavelengths are present in the horizontal. Poleward flowing (positive v) airis generally warmer than equatorward flowing air. Right column: Same, butnow for a wave just beyond the short-wave cut-off.

We then obtain:

Deformation Radius: Ld =NHf

≈ 10−2 104

10−4 ≈ 1000 km, (6.96)

Scale of maximum instability: Lmax ≈ 3.9Ld ≈ 4000 km, (6.97)

Growth Rate: σ ≈ 0.3ULd

≈ 0.3× 10106 s−1 ≈ 0.26 day−1. (6.98)

For the ocean

For the main thermocline in the ocean let us choose

H ∼ 1 km U ≈ 0.1 m s−1 N ∼ 10−2 s−1. (6.99)

We then obtain:

Deformation Radius: Ld =NHf

≈ 10−2 100010−4 = 100 km, (6.100)

Scale of maximum instability: Lmax ≈ 3.9Ld ≈ 400 km, (6.101)

Growth Rate: σ ≈ 0.3ULd

≈ 0.3× 0.1105 s−1 ≈ 0.026 day−1.

(6.102)

6.6 Two-Layer Baroclinic Instability 275

Summary of Results from the Eady Problem

? The length-scales of the instability are characterized by the deformationscale. The most unstable scale has a wavelength about four times thedeformation radius Ld, where Ld = NH/f0.

? The growth rate of the instability is approximately

σE ∼ULd

= Λf0

N. (1)

That is, it is proportional to the shear, and scaled by the Prandtl ratiof0/N. The value σE is known as the Eady growth rate.

? The most unstable waves for a given zonal scale are those with the gravestmeridional scale.

? There is a short-wave cutoff beyond which (i.e., at higher wavenumberthan) there is no instability. This occurs near the deformation radius.

? The instability relies on an interaction between waves at the upper andlower boundaries. If either boundary is removed, the instability dies. Thispoint is be considered further in section 6.7.

In the ocean, the Eady problem is not quantitatively applicable because of the non-unifomity of the stratification and non-zonality of the flow. Nevertheless, the aboveestimates give a qualitative sense of the scale and growth rate of the instabilityrelative to the corresponding values in the atmosphere. A summary of the mainpoints of the Eady problem is given in the shaded box on the next page.

6.6 TWO-LAYER BAROCLINIC INSTABILITY

The eigenfunctions displaying the largest growth rates in the Eady problem have arelatively simple vertical structure. This suggests that an even simpler mathematicalmodel of baroclinic instability might be constructed in which the vertical structure isa priori restricted to a very simple form, namely the two-layer QG model of sections5.3.2 and 5.4.5. One notable advantage over the Eady model is that it is possible toinclude the β–effect in a simple way.

6.6.1 Posing the problem

For two layers or two levels of equal thickness, we write the potential vorticityequations in the dimensional form,

DDt

[ζi + βy +

k2d

2(ψj −ψi)

]= 0, i = 1,2, j = 3− i, (6.103)


where, using two-level notation for definiteness,

k2d

2=(

2f0

NH

)2→ kd =

√8Ld, (6.104)

where H is the total depth of the domain, as in the Eady problem. The basic statewe choose is:

Ψ1 = −U1y, Ψ2 = −U2y = +U1y. (6.105)

It is possible to choose U2 = −U1 without loss of generality because there is notopography and the system is Galilean invariant. The basic basic state potentialvorticity gradient is then given by

Q1 = βy + k2dUy, Q2 = βy − k2

dUy (6.106)

where U = U1. (Note that this differs by a constant factor from the U in the Eadyproblem.) Even in the absence of β there is a non-zero potential vorticity gradient.Why should this be different from the Eady problem? — after all, the shear is uni-form in both problems. The difference arises from the vertical boundary conditions.In the standard layered formulation the temperature gradient at the boundary isabsorbed into the definition of the potential vorticity in the interior. This resultsin a nonzero interior potential vorticity gradient at the two levels adjacent to theboundary (the only layers in the two-layer problem), but isothermal boundary con-ditions D/Dt(∂ψ/∂z) = 0. In the Eady problem we have a zero interior gradientof potential vorticity but a temperature gradient at the boundary. The two formu-lations are physically equivalent — a finite-difference example of the Brethertonboundary layer.

The linearized potential vorticity equation is, for each layer,

∂q′i∂t

+Ui∂q′i∂x

+ v′i∂Qi∂y

= 0, i = 1,2 (6.107)

or, more explicitly,

[∂∂t+U ∂

∂x

][∇2ψ′1 +

k2d

2(ψ′2 −ψ′1)

]+ ∂ψ

′1

∂x(β+ k2

dU) = 0, (6.108a)

[∂∂t−U ∂

∂x

][∇2ψ′2 +

k2d

2(ψ′1 −ψ′2)

]+ ∂ψ

′2

∂x(β− k2

dU) = 0. (6.108b)

For simplicity we will set the problem in a square, doubly-periodic domain, and soseek solutions in the form,

ψ′i = Re ψi ei(kx+ly−ωt) = Re ψi eik(x−ct) eily , i = 1,2. (6.109)

Here, k and l are the x- and y-wavenumbers, and (k, l) = (2π/L)(m,n) where Lis the size of the domain andm and n are integers. The constant ψi is the complexamplitude.


6.6.2 The solution

Substituting (6.109) into (6.108) we obtain

[ik(U − c)][−K2ψ1 + kd2(ψ2 − ψ1)/2

]+ ikψ1(β+ k2

dU) = 0, (6.110a)

[−ik(U + c)][−K2ψ2 + kd2(ψ1 − ψ2)/2

]+ ikψ2(β− kd2U) = 0, (6.110b)

where K2 = k2 + l2. Re-arranging these two equations gives[(U − c)(kd2/2+K2)− (β+ k2

dU)]ψ1 −

[k2d(U − c)/2

]ψ2 = 0, (6.111a)

−[k2d(U + c)/2

]ψ1 +

[(U + c)(kd2/2+K2)+ (β− kd2U)

]ψ2 = 0. (6.111b)

These equations are of the form

[A]ψ1 + [B]ψ2 = 0, [C]ψ1 + [D]ψ2 = 0, (6.112)

and for nontrivial solutions the determinant of coefficients must be zero; that isAD − BC = 0. This gives a quadratic equation in c and solving this we obtain

c = − βK2 + k2

d

1+ kd2

2K2 ±k2d

2K2

1+ 4K4(K4 − kd4)k4βk

4d

1/2 , (6.113)

where K4 = (k2 + l2)2 and kβ =√β/U (its inverse is known as the Kuo scale).

We may nondimensionalize this equation using the deformation radius Ld as thelength scale and the shear velocity U as the velocity scale.13 Then, denoting non-dimensional parameters with hats, we have

k = kLd, c = c U, t = Ld

Ut (6.114)

and the nondimensional form of (6.113) is just

c = −k2β

K2 + k2d

1+k2d

2K2±k2d

2K2

1+4K4(K4 − k4

d)

k4βk

4d

1/2 , (6.115)

where kβ = kβLd and kd =√

8, as in (6.104). The nondimensional parameter

γ = 14k2β =

βL2d

4U, (6.116)

is often useful as a measure of the importance of β; it is proportional to the squareof the ratio of the deformation radius to the Kuo scale

√U/β. (It is also the two

layer version of the ‘Charney-Green number’ considered more in section 6.9.1.) Letus look at two special cases first, before considering the general solution to theseequations.

I. Zero shear, non-zero β


Figure 6.13 Growth rates formodels with varying numbersof vertical layers, all with β = 0and the same uniform stratifi-cation and shear. The dashedline is the solution to the con-tinuous (Eady) problem, andthe solid lines are results ob-tained using two, four andeight layers. The two and fourlayer results are labelled, andthe eight layer result is almostcoincident with the dashedline. 0 1 2 3 4

0

0.1

0.2

0.3

0.4

Gro

wth

Rat

e

Wavenumber

2

4

If there is no shear (i.e., U = 0) then (6.111a) and (6.111b) are identical andtwo roots of the equation give the purely real phase speeds c

c = − βK2 , c = − β

K2 + k2d

(6.117)

The first of these is the dispersion relationship for Rossby waves in a purelybarotropic flow, and corresponds to the eigenfunction ψ1 = ψ2. The secondsolution corresponds to the baroclinic eigenfunction ψ1 + ψ2 = 0.

II. Zero β, non-zero shearIf β = 0, then (6.111) yields, after a little algebra,

c = ±U(K2 − k2

d

K2 + k2d

)1/2

(6.118)

or, defining the growth rate σ by σ = −iω,

σ = Uk(k2d −K2

K2 + k2d

)1/2

(6.119)

These expressions are very similar to those in the Eady problem. Indeed, aswe increase the number of layers (using a numerical method to perform thecalculation) the growth rate converges to that of the Eady problem (Fig. 6.13).We note that:

? There is an instability for all values of U .

? There is a high-wavenumber cut-off, at a scale proportional to the radiusof deformation. For the two-layer model, if K > kd = 2.82/Ld there isno growth. For the Eady problem, the high wavenumber cut-off occurs at2.4/Ld.

? There is no low wavenumber cut-off.


0 1 2 3 40

0.1

0.2

0.3

Gro

wth

Rat

e

Wavenumber0 1 2 3 4

-0.4

-0.2

0

0.2

Wav

e sp

eeds

Wavenumber

a b

c

a

b

c

Fig. 6.14 Growth rates and wave speeds for the two-layer baroclinic instabilityproblem, from (6.115), with three (nondimensional) values of β: a, γ = 0(kβ = 0); b, γ = 0.5 (kβ =

√2); c, γ = 1 (kβ = 2). As β increases, so does

the low-wavenumber cut-off to instability, but the high-wavenumber cut-offis little changed. (The solutions are obtained from (6.115), with kd =

√8 and

U1 = −U2 = 1/4.)

? For any given k, the highest growth rate occurs for l = 0. In the two-layermodel, from (6.119), for l = 0 the maximum growth rate occurs whenk = 0.634kd = 1.79/Ld. For the Eady problem, the maximum growth rateoccurs at 1.61/Ld.

Solution in the general case: non-zero shear and non-zero β

Using (6.115), the growth rate and wave speeds as function of wavenumber areplotted in Fig. 6.14. We observe that there still appears to be a high wavenumbercut-off and, for β = 0, there is a low-wavenumber cut-off. A little analysis elucidatesthese features.

The neutral curve:For instability, there must be an imaginary component to the phase speedin (6.115). That is, we require

k4βk

4d + 4K4(K4 − k4

d) < 0. (6.120)

This is a quadratic equation in K4 for the value of K, Kc say, at which thegrowth rate is zero. Solving, we find

K4c =

12k4d

(1±

√1− k4

β/k4d

), (6.121)

and this is plotted in Fig. 6.15. From (6.120) useful approximate expres-sions can be obtained for the critical shear as a function of wavenumberin the limits of small K and K ≈ kd, and these are left as exercises for thereader.


Fig. 6.15 Contours of growth rate in the two-layer baroclinic instability prob-lem. The dashed line is the neutral stability curve obtained from (6.121),and the other curves are contours of growth rates obtained from (6.115).Outside of the dashed line, the flow is stable. The wavenumber is scaled by1/Ld (i.e., by kd/

√8) and growth rates are scaled by the inverse of the Eady

timescale (i.e., by U/Ld). Thus, for Ld = 1000 km and U = 10 m s−1, a nondi-mensional growth rate of 0.25 corresponds to a dimensional growth rate of0.25× 10−5 s−1 = 0.216 day−1.

Minimum shear for instability:From (6.120), instability arises when β2k4

d/U2 < 4K4(k4

d − K4). The maxi-mum value of the right-hand side of this expression arises when K4 = k4

d/2;thus, instability arises only when

β2k4d

U2 < 4k4d

2k4d

2(6.122)

or

Us >2βk2d

(6.123)

where Us = U1 − U2 = 2U . In terms of the deformation radius itself theminimum shear for instability is

Us >14βL2

d. (6.124)

Fig. 6.16 sketches how this might vary with latitude in the atmosphereand ocean. (In (6.124), the shear is the difference in the velocity betweenlevel 1 and level 2, whereas the deformation radius, NH/f0, is based onthe total height of the fluid. If we were to use half the depth of thefluid in the definition of the deformation radius, the factor of 4 woulddisappear.) If the shear is just this critical value, the instability occurs at


0 20 40 60 800

10

20

30

40

Latitude

Min

imum

She

ar

0 20 40 60 800

0.1

0.2

0.3

0.4

Latitude

Atmosphere Ocean

Fig. 6.16 The minimimum shear (Us = U1−U2, in m/s) required for baroclinicinstability in a two-layer model, calculated using (6.124), i.e. Us = βL2

d/4where β = 2Ωa−1 cosϑ and Ld = NH/f , where f = 2Ω sinϑ. The left paneluses H = 10 km and N = 10−2 s−1, the right panel uses parameters represen-tative of the main thermocline, H = 1 km and N = 10−2 s−1. The results arenot quantitatively accurate, but the implications that the minimum shear ismuch less for the ocean, and that in both atmosphere and ocean the shearincreases rapidly at low latitudes, are robust.

k = 2−1/4kd = 0.84kd = 2.37/Ld. As the shear increases, the wavenumberat which the growth rate is maximum decreases slightly (see Fig. 6.15), andfor a sufficiently large shear the β-effect is negligible and the wavenumberof maximum instability is, as we saw earlier, 0.634kd or 1.79/LdNote the relationship of the minimum shear to the basic state potentialvorticity gradient in the respective layers. In the upper and lower layersthe potential vorticity gradients are given by, respectively,

∂Q1

∂y= β+ kd2U,

∂Q2

∂y= β− kd2U (6.125a,b)

Thus, the requirement for instability is exactly that which causes the po-tential vorticity gradient to change sign somewhere in the domain, in thiscase becoming negative in the lower layer. This is an example of the gen-eral rule that potential vorticity (suitably generalized to include the surfaceboundary conditions) must change sign somewhere in order for there to bean instability.

High-wavenumber cut-off:Instability can only arise when, from (6.120),

4K4(k4d −K4) > k4

βk4d, (6.126)

so that a necessary condition for instability is

k2d > K

2. (6.127)

Thus, waves shorter than the deformation radius are always stable, no mat-ter what the value of β. We also see from Fig. 6.14 and Fig. 6.15 that


the high wavenumber cut-off in fact varies little with β if kd kβ. Notethat the critical shear required for instability approaches infinity as K ap-proaches kd.

Low-wavenumber cut-off:Suppose that k kd. Then (6.120) simplifies to k4

β < 4K4. That is, forinstability we require

K2 >12k2β =

β2U. (6.128)

Thus, using (6.127) and (6.128) the unstable waves lie approximately inthe interval β/(

√2U) < k < kd.

6.7 AN INFORMAL VIEW OF THE MECHANISM OF BAROCLINIC INSTABILITY

In this section we take a more intuitive look at baroclinic instability, trying to under-stand the mechanism without treating the problem in full generality or exactness.We will do this by way of a semi-kinematic argument that shows how the waves ineach layer of a two-layer model, or the waves on the top and bottom boundaries inthe Eady model, can constructively interact to produce a growing instability. It iskinematic in the sense that we initially treat the waves independently, and only sub-sequently allow them to interact — but it is this dynamical interaction that gives theinstability. We first revisit the two-layer model and simplify it to its bare essentials.

6.7.1 The two-layer model

A simple dynamical model

We first re-derive the instability ab initio from the equations of motion written interms of the baroclinic streamfunction τ and the barotropic streamfunctionψwhere

τ ≡ 12(ψ1 −ψ2), ψ ≡ 1

2(ψ1 +ψ2). (6.129)

We linearize about a sheared basic state of with zero barotropic velocity and withβ = 0. Thus, with ψ = 0+ψ′ and τ = −Uy +τ′ the linearized equations of motion,equivalent to (6.108) with β = 0, are

∂∂t∇2ψ′ = −U ∂

∂x∇2τ′, (6.130a)

∂∂t(∇2 − k2

d)τ′ = −U ∂

∂x(∇2 + k2

d)ψ′. (6.130b)

Seeking solutions of the form (ψ′, τ′) = Re (ψ, τ) exp[ik(x − ct)] gives

cψ−Uτ = 0, (6.131a)

c(K2 + k2d)τ −U(K2 − k2

d)ψ = 0. (6.131b)

These equations have nontrivial solutions if the determinant of the matrix of coeffi-cients is zero, giving the quadratic equation c2(K2+k2

d)−U2(K2−k2d) = 0. Solving

6.7 An Informal View of the Mechanism of Baroclinic Instability 283

this gives, reprising (6.118),

c = ±U(K2 − k2

d

K2 + k2d

)1/2

. (6.132)

Instabilities occur for K2 < k2d, for which c = ici; that is, the wave speed is

purely imaginary. From (6.131) unstable modes have

τ = iciUψ = eiπ/2 ci

Uψ. (6.133)

That is, τ lags ψ by 90° for a growing wave (ci > 0). Similarly, τ leads ψ by 90° fora decaying wave. Now, the temperature is proportional to τ, and in the two-levelmodel is advected by the vertically averaged perturbation meridional velocity, vsay (with Fourier amplitude v), where v = ∂ψ/∂x . Thus, for growing or decayingwaves,

v = τ kUci

(6.134)

and the meridional velocity is exactly in phase with the temperature for growingmodes, and is out of phase with the temperature for decaying modes. That is,for unstable modes, polewards flow is correlated with high temperatures, and fordecaying modes polewards flow is correlated with low temperatures. For neutralwaves, τ = cr ψ/U and so v = ikτU/cr and the meridional velocity and tempera-ture are π/2 out of phase. Thus, to summarize:? Growing waves transport heat (or buoyancy) polewards.? Decaying waves transport heat equatorward.? Neutral waves do not transport heat.

Further simplifications to the two-layer model

First consider (6.130) for waves much larger than the deformation radius, K2 k2d;

we obtain∂∂t∇2ψ = −U ∂

∂x∇2τ,

∂∂tτ = U ∂

∂xψ. (6.135a,b)

for which we obtain [either directly or from (6.132)] c = ±iU; that is, the flow isunstable. To see the mechanism, suppose that the initial perturbation is barotropicand sinusoidal in x, with no y variation. Polewards flowing fluid (i.e., ∂ψ/∂x > 0)will, by (6.135b), generate a positive τ, and the baroclinic flow will be out of phasewith the barotropic flow. Then, by (6.135a), the advection of τ by the mean shearproduces growth of ψ that is in phase with the original disturbance. Contrast thiscase with that for very small disturbances, for which K2 k2

d and (6.130) becomes

∂∂t∇2ψ = −U ∂

∂x∇2τ,

∂∂t∇2τ = −U ∂

∂x∇2ψ, (6.136a,b)

or, in terms of the equations for each layer,

∂∂t∇2ψ1 = −U

∂∂x∇2ψ1,

∂∂t∇2ψ2 = +U

∂∂x∇2ψ2. (6.137a,b)

That is, the layers are completely decoupled and no instability can arise. Motivatedby this, consider waves that propagate independently in each layer on the potential


vorticity gradient caused by β (if non-zero) and shear. Thus in (6.108) we keep thepotential vorticity gradients but neglect k2

d where it appears alongside ∇2 and find[∂∂t+U ∂

∂x

]∇2ψ′1 +

∂ψ′1∂x

∂Q1

∂y= 0, (6.138a)[

∂∂t−U ∂

∂x

]∇2ψ′2 +

∂ψ′2∂x

∂Q2

∂y= 0. (6.138b)

where ∂Q1/∂y = β + k2dU and ∂Q2/∂y = β − k2

dU . Seeking solutions of the form(6.109), the phase speeds of the associated waves are

c1 = U −∂yQ1

K2 , c2 = −U −∂yQ2

K2 . (6.139a,b)

In the upper layer the phase speed is a combination of an eastward advection anda fast westward wave propagation due to a strong potential vorticity gradient. Inthe lower layer the phase speed is a combination of a westward advection anda slow eastward wave propagation due to the weak potential vorticity gradient.The two phase speeds are, in general, not equal, but they would need to be so ifthey are to combine to cause an instability. From (6.139) this occurs when K2 =k2d and c1 = c2 = −β/k2

d. These conditions are just those occuring at the high-wavenumber cut-off to instability in the two-level model. At higher wavenumbers,the waves are unable to synchronize, whereas at lower wavenumbers they maybecome inextricably coupled.

Let us suppose that the phase of the wave in the upper layer lags that (i.e.,is westward of) that in the lower layer, as illustrated in the top panel Fig. 6.17.The lower panel shows the temperature field, τ = (ψ1 − ψ2)/2, and the averagemeridional velocity, v = ∂x(ψ1 +ψ2)/2. In this configuration, the temperaturefield is in phase with the meridional velocity, meaning that warm fluid is advectedpolewards. Now, let us allow the waves in the two layers to interact by adding onedynamical equation, the thermodynamic equation, which in its simplest form is

∂τ∂t

= −v ∂τ∂y

= vU, (6.140)

where τ is proportional basic state temperature field. The temperature field, τ,grows in proportion to v, which is proportional to τ if the waves tilt westward withheight, and an instability results. This dynamical mechanism is just that which iscompactly described by (6.135). It is a straightforward matter to show that if thestreamfunction tilts eastward with height, v is out of phase with τ and the wavesdecay.

6.7.2 Interacting edge waves in the Eady problem

A very similar description applies to the Eady problem. As in the two-layer case, firstconsider the case in which the bottom and top surfaces are essentially uncoupled.Instead of solutions of (6.82) that have the structure (6.83) (which satisfies bothboundary conditions) consider solutions that separately satisfy the bottom and topboundary conditions and that decay into the interior. These are, including the x-dependence,

ψB = ReAB eik(x−cT t) e−µz/H , ψT = ReAT eiφ eik(x−cBt) eµ(z−H)/H . (6.141a,b)

6.7 An Informal View of the Mechanism of Baroclinic Instability 285

Figure 6.17 Baroclinicallyunstable waves in a twolayer model. The stream-function is shown in thetop panel, ψ1 for the toplayer and ψ2 for the bot-tom layer. Given the west-ward tilt shown, the tem-perature, τ, and merid-ional velocity, v, (bottompanel) are in phase, andthe instability grows.

1

0

1

ψ2

ψ1

0 0.2 0.4 0.6 0.8 1

0

1V

τ

X

-

1-

for the bottom and top surfaces respectively, and φ is the phase shift, with AB andAT being real constants. The boundary conditions (6.81) then determine the phasespeeds of the two systems and we find

cB =ΛHµ, cT = ΛH

(1− 1

µ

). (6.142a,b)

These are the phase speeds of edge waves in the Eady problem; they are real and ingeneral they are unequal. It must therefore be the interaction of the waves on theupper and lower boundaries that is necessary for instability, because the unstablewave has but a single phase speed. This interaction can occur when their phasespeeds are equal and from (6.142) this occurs when µ = 2, giving

k = 2Ld

and c = ΛH2

(6.143a,b)

This phase speed is just that of the flow at mid-level, and at the critical wavenumberin the full Eady problem [kc = 2.4/Ld, from (6.91)] the phase speed is purelyreal and equal to that of (6.143b) — see Fig. 6.10. Thus, (6.143) approximatelycharacterizes the critical wavenumber in the full problem.

To turn this kinematic description into a dynamical instability, suppose that thetwo rigid surfaces are close enough so that the waves can interact, but still farenough so that their structure is approximately given by (6.141). (Note that if µis too large, the waves decay rapidly away from the edges and will not interect.)Specifically, let the buoyancy perturbation at a given boundary be advected by thetotal meridional velocity perturbation, including that arising from the perturbationat the other boundary, so that at the top and bottom boundaries

∂b′T∂t

= −(v′B + v′T )∂bT∂y

,∂b′B∂t

= −(v′B + v′T )∂bB∂y

(6.144)

The waves will reinforce each other if v′T is in phase with b′B at the lower boundary,and if v′B is in phase with b′T at the upper boundary. Now, using (6.141), the velocity


Figure 6.18 Interactingedge waves in the Eadymodel. The upper panelshows waves on the topsurface, and the lowerpanel waves on the bot-tom. If the streamfunctiontilts westward with height,then the temperature onthe top (bottom) is corre-lated with the meridionalvelocity on the bottom(top), the waves canreinforce each other. Seealso Fig. 6.12.

1

0

1 ψTv

TbT

0 0.2 0.4 0.6 0.8 1

1

0

1 ψB

vB

bB

X

-

-

and buoyancy associated with the edge waves are given by, omitting the harmonicx-dependence,

bB = −kNAB e−µz/H , bT = kNAT eiφ eµ(z−H)/H , (6.145a)

vB = ikAB e−µz/H , vT = ikAT eiφ eµ(z−H)/H . (6.145b)

The fields bT and vT , and bB and vB , will will be positively correlated if 0 < φ < π,and will be exactly in phase if φ = π/2, and this case is illustrated in Fig. 6.18. Justas in the two-layer case, this phase corresponds to a westward tilt with height, andit is this, in conjunction with geostrophic and hydrostatic balance, that allows warmfluid to move poleward and available potential energy to be released. From (6.144),the perturbation will grow and an instability will result. The analogy betweenbaroclinic instability and barotropic instability should be evident from the similarityof this description and that of section 6.2.4, with z in the baroclinic problem playingthe role of y in the barotropic problem, and b the role of v. However, the analogyis not perfect because the boundary condition that w = 0 does not have an exactcorrespondence in the barotropic problem. Also, the nonlinear development of thebaroclinic problem, discussed in chapter 9, is generally three-dimensional.

6.8 * THE ENERGETICS OF LINEAR BAROCLINIC INSTABILITY

In baroclinic instability, warm parcels move poleward and cold parcels move equa-torward. This motion draws on the available potential energy of the mean state,because warm light parcels move upward, and cold dense parcels downward andthe height of the mean center of gravity of the fluid falls, and the loss of potentialenergy is converted to kinetic energy of the perturbation. However, because the in-stability is growing, the energy of the perturbation is of course not conserved, andboth the kinetic energy and the available potential energy of the perturbation willgrow. However, we still expect a conversion of potential energy to kinetic, and thepurpose of this section is to demonstrate that explicitly. For simplicity, we restrictattention to the flat-bottomed two-level model with β = 0.

As in section 5.6, the energy may be partitioned into kinetic energy and available

6.8 * The Energetics of Linear Baroclinic Instability 287

potential energy. In a three-dimensional quasi-geostrophic flow the kinetic energyis given by, in general,

KE = 12

∫(∇ψ)2 dV (6.146)

which, in the case of the two-layer model becomes

KE = 12

∫(∇ψ1)2 + (∇ψ2)2 dA =

∫(∇ψ)2 + (∇τ)2 dA. (6.147)

Restricting attention to a single Fourier mode this becomes

KE = k2ψ2 + k2τ2. (6.148)

The available potential energy in the continuous case is given by

APE = 12

∫ (f0

N

)2 (∂ψ∂z

)2dV. (6.149)

For a single Fourier mode in a two layer model this becomes

APE = k2dτ

2. (6.150)

Now, the nonlinear vorticity equations for each level is

∂∂t∇2ψ1 + J(ψ1,∇2ψ1) = −2

f0wH

(6.151a)

∂∂t∇2ψ2 + J(ψ2,∇2ψ2) = 2

f0wH

(6.151b)

where w is the vertical velocity between the levels. (These equations are the two-level analogs of the continuous vorticity equation, with the right-hand sides beingfinite difference versions of f0∂w/∂z.) Multiplying the two equations of (6.151) by−ψ1 and −ψ2, respectively, and adding we readily find

ddt

KE = 4f0

H

∫wτ dA. (6.152a)

For a single Fourier mode this becomes

ddt

KE = Re4f0

Hwτ∗, (6.152b)

where w = w exp[i(kx − ct)]+ c.c., and the asterisk denotes complex conjugacy.The continuous thermodynamic equation is

DbDt

+wN2 = 0. (6.153)

Using b = f0∂ψ/∂z and finite-differencing [with ∂ψ/∂z → (ψ1 − ψ2)/(H/2) =

4τ/H], we obtain the two-level thermodynamic equation:

∂τ∂t

+ J(ψ,τ)+ wN2H

4f0= 0. (6.154)


The change of available potential energy is obtained from this by multiplying byk2dτ and integrating, giving∫ (

12

ddtk2dτ

2 + τw 2f0

H

)dA = 0 (6.155)

orddt

APE = −4f0

H

∫wτ dA (6.156a)

or, for a single Fourier mode,

ddt

APE = −Re4f0

Hw τ∗. (6.156b)

From (6.152) and (6.156) it is clear that in the nonlinear equations the sum of thekinetic energy and the available potential energy is conserved.

We now specialize by obtaining w from the linear baroclinic instability problem.Using this in (6.152) and (6.156) will give us the conversion between kinetic en-ergy and potential energy in the growing baroclinic wave. It is important to realizethat the total energy of the disturbance will not be conserved — both the poten-tial and kinetic energy are growing, exponentially in this problem, because theyare extracting energy from the mean state. To calculate w we use the linearizedthermodynamic equation. From (6.154) this is

∂τ∂t

−U ∂ψ∂x

+ HwN2

4f0= 0, (6.157)

omitting the primes on perturbation quantities. For a single Fourier mode, this gives

HN2

4f0w = ik(cτ +Uψ). (6.158)

But, from (6.131), cψ = Uτ in two-layer f -plane baroclinic instability and so

HN2

4f0w = ikcτ

(1+ U

2

c2

)= ikcτ

(2K2

K2 − k2d

). (6.159)

using (6.132). For stable waves, K2 > k2d and c = cr and in that case the vertical

velocity is π/2 out of phase with the temperature, and there is no conversion ofAPE to KE. For unstable waves c = ici and K2 < k2

d, and the vertical velocity is inphase with the temperature. That is, warm air is rising and so there is a conversionof APE to KE To see this more formally, recall that the conversion from APE to KE isgiven by 4wτ∗f0/H. Thus, using (6.159),

ddt(APE → KE) = Re 2ikck2

d

(2K2

K2 − k2d

)τ2, (6.160)

using also the definition of kd given in (6.104). If the wave is growing, then K2 < k2d

and c = ici and the right-hand side is real and positive. For neutral waves, If c = crthe right-hand side of (6.160) is pure imaginary, and so the conversion is zero. Thiscompletes our demonstration that baroclinic instability converts potential energyinto kinetic energy.

6.9 * Beta, Shear and Stratification in a Continuous Model 289

6.9 * BETA, SHEAR AND STRATIFICATION IN A CONTINUOUS MODEL

The two-layer model of section 6.6 indicates that β has a number of important ef-fects of β on baroclinic instability. Do these carry over to the continuously stratifiedcase? The answer by-and-large is yes, but with some important qualifications thatgenerally concern weak or shallow instabilities. In particular, we will find that thereis no short-wave cut-off in the continuous model with non-zero beta, and that theinstability determines its own depth scale. We will illustrate these properties firstby way of scaling arguments, and then by way of numerical calculations.14

6.9.1 Scaling arguments for growth rates, scales and depth

With finite density scale height and non-zero β, the quasi-geostrophic potentialvorticity equation, linearized about a mean zonal velocity U(z), is(

∂∂t+U ∂u

∂x

)q′ + ∂ψ

′

∂x∂Q∂y

= 0, (6.161)

where

q′ = ∇2ψ′ + f20

ρ∂∂z

(ρN2∂ψ′

∂z

), (6.162)

∂Q∂y

= β− f20

ρ∂∂z

(ρN2∂U∂z

), (6.163)

and ρ is a specified density profile. If we assume that U = Λz where Λ is constantand that N is constant, and let H−1

ρ = −ρ−1∂ρ/∂z, then

∂Q∂y

= β+ f 20Λ

N2Hρ= β(1+α) (6.164)

where

α =(f 2

0ΛβN2Hρ

), (6.165)

The boundary conditions on (6.161) are(∂∂t+U ∂

∂x

)∂ψ′

∂z− ∂ψ

′

∂x∂U∂z

= 0, at z = 0 (6.166)

and that ψ → 0 as z → ∞. The problem we have defined essentially constitutes theCharney problem. We can reduce this to the Eady problem by setting β = 0 andHρ = ∞, and providing a lid some finite height above the ground.

As in the Eady problem, we seek solutions of the form

ψ = Re ψ(z) ei(kx+ly−kct), (6.167)

and substituting into (6.161) gives(f 2

0

H2ρN2

)(H2ρ

d2ψdz2 −Hρ

dψdz

)−(K2 − β+Λf

20 /(N2Hρ)

Λz − c

)ψ = 0. (6.168)


The Boussinesq version of this expression for a fluid contained between two hori-zontal surfaces is obtained by letting Hρ = ∞, giving(

f 20

N2

)d2ψdz2 −

(K2 − β

Λz − c

)ψ = 0. (6.169)

It seems natural to nondimensionalize (6.168) using:

z = Hρz, c = ΛHρ c, K =(f0

NHρ

)K, (6.170)

whence it becomesd2ψdz2 −

dψdz

−(K2 − γ + 1

z − c

)ψ = 0 (6.171)

where

γ = α−1 = βN2Hρf 2

0Λ=βL2

dHρΛ

= Hρh. (6.172)

where h ≡ Λf 20 /(βN2). The non-dimensional parameter γ is known as the Charney-

Green number.15 The Boussinesq version, (6.169), may be non-dimensionalizedusing HD in place of Hρ, where HD is the depth of the fluid between two rigidsurfaces. In that case

d2ψdz2 −

(K2 − γ

z − c

)ψ = 0, (6.173)

where here the non-dimensional variables are scaled with HD.Now, suppose that γ is large, for example if β or the static stability are large

or the shear is weak. Eq. (6.171) admits of no non-trivial balance, suggesting thatwe rescale the variables using h instead of Hρ as the vertical scale in (6.170). Therescaled version of (6.171) is then

d2ψdz2 −

1γ

dψdz

−(K2 − 1+ γ−1

z − c

)ψ = 0, (6.174)

or, approximately,d2ψdz2 −

(K2 − 1

z − c

)ψ = 0. (6.175)

This is exactly the same equation as results from a similar rescaling of the Boussi-nesq system, (6.173), as we might have expected because now the dynamical ver-tical scale, h, is much smaller than the scale height Hρ (or HD) and the systemis essentially Boussinesq. Thus, noting that (6.175) has the same nondimesionalform as (6.173) save that γ is replaced by unity, and that (6.175) with γ = 1 mustproduce the same scales and growth rates as in the Eady problem, we may deducethat:

(i) The wavelength of the instability is O(Nh/f0).(ii) The growth rate of the instability is O(Kc) = O(f0Λ/N).(iii) The vertical scale of the instability is O(h) = O(f 2

0Λ/(βN2)).These are the same as for the Eady problem, except with the dynamical height h re-placing the geometric or scale height HD. Effectively, the dynamics has determined


its own vertical scale, h, that is much less than the scale height or geometric height,producing ‘shallow modes’.

In the limit γ 1 (strong shear, weak β), the Boussinesq and compressibleproblems differ. The Boussinesq problem reduces to the Eady problem, consideredpreviously, whereas (6.171) becomes, approximately,

d2ψdz2 −

dψdz

−(K2 − 1

z − c

)ψ = 0, (6.176)

and in this limit the appropriate vertical scale is the density scale heightHρ. BecauseHρ h these are ‘deep modes’, occupying the entire vertical extent of the domain.

The scale h does not arise in the two-level model, but there is a connectionbetween it and the critical shear for instability in the two-level model. The conditionγ 1, or h H, may be written as

HΛ β(NHf0

)2

. (6.177)

Compare this with the necessary condition for instability in a two-level model,(6.124), namely

(U1 −U2) > β(NH∆f0

)2

(6.178)

where H∆ is the vertical distance between the two levels. Thus, essentially the samecondition governs the onset of instability in the two-level model as governs the pro-duction of deep modes in the continuous model. This correspondence is a naturalone, because in the two-level model all modes are ‘deep’, and the model fails (asit should) to capture the shallow modes of the continuous system. For similar rea-sons, there is a high-wavenumber cut-off in the two-level model: in the continuousmodel these modes are shallow and so cannot be captured by two-level dynamics.Somewhat counter-intuitively, for these modes the β-effect must be important, eventhough the modes have small horizontal scale: when β = 0 the instability arises viaan interaction between edge waves at the top and bottom of the domain, whereasthe shallow instability arises via an interaction of the edge waves at the surface withRossby waves just above the surface.

6.9.2 Some numerical calculations

Adding β to the Eady model

Our first step is add the β-effect to the Eady problem.16 That is, we suppose aBoussinesq fluid with uniform stratification, that the shear is zonal and constant,and that the entire problem is sandwiched between two rigid surfaces. Growthrates and phase speeds of such an instability calculation are illustrated in Fig. 6.19and the vertical structure is shown in Fig. 6.20. As in the two-layer problem, thereis a low-wavenumber cut-off to the main instability, although there is now an ad-ditional weak instability at very large-scales. These so-called Green modes have nocounterpart in the two-layer model — they are deep, slowly growing modes thatwill be dominated by faster growing modes in most real situations. (Also, the fact


0 1 2 3 40

0.1

0.2

0.3

0.4G

row

th R

ate

Wavenumber0 1 2 3 4

-0.2

0

0.2

0.4

0.6

Wav

e sp

eeds

Wavenumber

Fig. 6.19 Growth rates and wave speeds for the two-layer (solid) and con-tinuous (dashed) models, with the same values of the Charney-Green num-ber, γ, and uniform shear and stratification. (In the two-layer case γ =βL2

d/[2(U1 −U2)] = 0.5, and in the continuous case γ = βL2d/(HΛ) = 0.5.) In

the continuous case only the wave speed associated with the unstable modeis shown. In the two-layer case there are two real wave speeds which coa-lesce in the unstable region. The two-layer model has an abrupt short-waveand long-wave cut-off, whereas the growth rate of the continuous model tailsoff gradually at small wavelengths, and has a weak instability (the ‘Greenmodes’) at large wavelengths.

that the Green modes have a scale much larger than the deformation scale sug-gests that a degree of caution in the accuracy of the quasi-geostrophic calculationis warranted.) At high wavenumbers is no cut-off to the instability in the continu-ous problem in the case of non-zero beta; the high-wavenumber modes are shallowand unstable via an interaction between edge waves at the lower boundary andRossby waves in the lower atmosphere, and so have no counterpart in either thethe two-layer problem (where the modes are deep) or the Eady problem (whichhas no Rossby waves).

Effects of nonuniform shear and stratification

If the shear or stratification is non-uniform an analytic treatment is, even in prob-lems without β, usually impossible and the resulting equations must be solvednumerically. However, if we restrict attention to a discontinuity in the shear orthe stratification, then resulting problem is very similar to the problem with rigidboundaries, and this property provides much of the justification for using the Eadyproblem to model instabilities in the earth’s atmosphere: in the troposphere thestratification is (approximately) constant, and the rapid increase in stratification inthe stratosphere can be approximated by a lid at tropopause. Heuristically, we cansee this from the form of the thermodynamic equation, namely

DbDt

+N2w = 0. (6.179)

If N2 is high this suggests w will be small, and a lid is the limiting case of this.The oceanic problem is rather more involved, because although both the stratifica-tion and shear are concentrated in the upper ocean, they vary relatively smoothly;


0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Amplitude

Hei

ght

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Phase

Fig. 6.20 Vertical structure of the most unstable modes in a continuouslystratified instability calculation with β = 0 (dashed lines, the Eady problem)and β ≠ 0 (solid lines), as in Fig. 6.19. The effect of beta is to depress theheight of maximum amplitude of the instability.

furthermore, the shear is high where the stratification is high, and the two haveopposing effects.

To go one step further, consider the Boussinesq potential vorticity equation, lin-earized about a zonally uniform state Ψ(y, z), with a rigid surface at z = 0. Thenormal-mode evolution equations are similar to (6.70), namely

(U − c)[∂2

∂y2 − k2 + ∂

∂z

(F∂∂z

)]ψ+ ∂Q

∂yψ = 0, z > 0, (6.180a)

(U − c)∂ψ∂z

− ∂U∂zψ = 0, at z = 0. (6.180b)

where ∂yQ = β − ∂yyU − ∂z(F∂zU). Now suppose that there is a discontinuity inthe shear and/or the stratification in the interior of the fluid, at some level z = zc .

Integrating (6.180a) across the discontinuity, noting that ψ is continuous in z,gives

(U − c)[F∂ψ∂z

]zc+zc−

− ψ(y, zc)[F∂U∂z

]zc+zc−

= 0. (6.181)

which has similar form to (6.180b). This construction is evocative of the equiva-lence of a delta-function sheet of potential vorticity at a rigid boundary, except thatnow a discontinuity in the potential vorticity in the interior has a similarity with arigid boundary.

We can illustrate the effects of an interior discontinuity that crudely representsthe tropopause by numerically solving the linear eigenvalue problem. For simplicity,we pose the problem on the f -plane, in a horizontally doubly-periodic domain,with no horizontal variation of shear, and between two horizontal rigid lids. Theeigenvalue problem is defined by (6.70), and the numerical procedure then solvesfor the complex eigenvalue c and eigenfunction ψ(z); various results are illustratedin Fig. 6.21. To parse this rather complex figure, first look at the solid curves in allthe panels. These arise when the problem is solved with a uniform shear and auniform stratification, with a lid at z = 0 and z = 1, so simply giving the Eady


0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Gro

wth

Rat

es

Wavenumber1 1.05

0

0.5

1

1.5

2

A

a

b

Hei

ght

Potential Temp.0 1 2

0

0.5

1

1.5

2

Velocity

D

c d

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

Amplitude

Hei

ght

0 0.5 1 1.5 20

0.5

1

1.5

2

Phase

Hei

ght

A+Da+cb+db+c

A+Da+cb+db+c

A+Da+cb+db+c

(a) (b)

(c) (d)

Fig. 6.21 The effect of a stratosphere on baroclinic instability. (a) the given profiles of shearand stratification; (b) The growth rate of the instabilities; (c) amplitude of the most unstablemode as a function of height; (d) phase of the most unstable mode. The instability problemis numerically solved with various profiles of stratification and shear. In each profile, inthe idealized troposphere (z < 1) the shear and stratification are uniform and the samein each case. We consider four idealized stratospheres (z ≥ 1): 1, A lid at z = 1, i.e., nostratosphere, so Eady problem itself (profiles A+D, solid lines); 2, Stratospheric stratificationsame as the troposphere, but zero shear (profiles a+c, dashed); 3, Stratospheric shear sameas troposphere, but stratification (N2) four times the tropospheric value (b+d, dot-dashed);4, Zero shear and high stratification in the stratosphere (b+c, dotted). In the tropospherethe amplitude and structure of the instability is similar in all cases, illustrating the similarityof a rigid-lid and abrupt changes in shear or stratification. Either a high stratification or alow shear (or both) will result in weak stratospheric instability.

problem. The familiar growth rates and vertical structure of the solution are givenby the solid curves in panels (b), (c) and (d), and these are just the same as inFig. 6.10. The various dotted and dashed curves show the results when the lidat z = 1 is replaced by stratosphere stretching from 1 < z < 2 either with highstratification, zero shear, or both, an in all of these cases the stratosphere acts inthe same qualitative way as a rigid lid. The vertical structure of the solution in thetroposphere in all cases is quite similar, and the amplitude decays rapidly above theidealized tropopause, consistent with the almost uniform phase of the disturbanceillustrated in panel (d) — recall that a tilting of the disturbance with height is


0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

Gro

wth

Rat

es

Wavenumber

b+da+db+ca+c

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

Amplitude

Dep

th

b+da+db+ca+c

0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

Phase

Dep

th

b+da+db+ca+c

1 1.005 1.01−1

−0.5

0

Dep

th

Density

a

b

0 0.5 1Velocity

c

d

(a) (b)

(c) (d)

Fig. 6.22 The baroclinic instability in an idealized ocean, with four different profiles of shearor stratification. The panels are: (a) The profiles of velocity and density (and so N2) used; (b)the growth rates of the various cases; (c) the vertical structure of the amplitude of the mostunstable models; (d) the phase in the vertical of the most unstable modes. The instabilityis numerically calculated with four combinations of shear and stratification: 1, Uniformstratification and shear i.e. the Eady problem, (profiles b+d, solid lines). 2, Uniform shear,upper-ocean enhanced stratification (a+d, dashed); 3, Uniform stratification, upper oceanenhanced shear (b+c, dot-dashed); 4, Both stratification and shear enhanced in upper ocean(a+c, dotted). Case 2 (a+d, dashed) is really more like to an atmosphere with a stratosphere(see Fig. 6.21), and the amplitude of the disturbance falls off, rather unrealistically, in theupper ocean. Case 4 (a+c, dotted) is the most oceanically relevant.

necessary for instability. It is these properties that make the Eady problem, or moregenerally any baroclinic instability problem that is posed between two rigid lids, ofmore general applicability to the earth’s atmosphere than might be first thought:the high stratification above the tropopause and consequent decay of the instabilityis mimicked by the imposition of a rigid lid. (Of course, the β effect is still absentin the Eady problem.)

In the ocean, the stratification is highest in the upper ocean where the shearis also strongest, and numerical calculations of the structure and growth rate ofidealized profiles illustrated in Fig. 6.22. The solid curve shows the Eady problem,and the various dashed curves show the phase speeds, growth rates and phasewith combinations of the profiles illustrated in panel (a). Much of the ocean is


characterized by having both a higher shear and a higher stratification in the upper1 km or so, and this case is the one with the dotted line in Fig. 6.22. In this case theamplitude of the instability is also largely confined to the upper ocean, and unlikethe Eady problem it does not arise through the interaction of edge waves at the topand bottom: the potential vorticity changes sign because of the interior variationsdue to the nonuniform shear, mainly in the upper ocean. Consistently, the phase ofthe baroclinic waves is nearly constant in the lower ocean in the two cases in whichthe shear is confined to the upper ocean. The ocean itself is still more complicated,because the most unstable regions near intense western boundary currents are oftenalso barotropically unstable, and the mean flow itself may be meridionally directed.Nevertheless, the result that linear baroclinic instability is primarily an upper oceanphenomonom is quite robust.17 However, we will find in chaper 9 that the nonlinearevolution of baroclinic instability leads to eddies throughout the water column.

Notes

1 Thomson (1871), Helmholtz (1868). The more general case, considered by Thomson(later Lord Kelvin), allows the fluid’s density to vary.

2 See Drazin and Reid (1981) or Chandrasekhar (1961) for more detail.

3 This is Squire’s theorem, which states that for every three-dimensional disturbanceto a plane-parallel flow there corresponds a more unstable two-dimensional one.This means there is no need to consider three dimensional effects to determinewhether such a flow is unstable.

4 Rayleigh, Lord (1880).

5 First obtained by Rayleigh, Lord (1894).

6 The solution of Fig. 6.6 is obtained with a gridpoint code with 400 × 400 equallyspaced gridpoints. This kind of problem is also well suited to contour dynamicsapproach, as in Dritschel (1989).

7 Rayleigh, Lord (1880) and, for the case with β, l. Kuo (1949).

8 Fjørtoft (1950).

9 Charney and Stern (1962), Pedlosky (1964).

10 Eady (1949), Charney (1947). Eric Eady (1915–1966) is best remembered today asthe author of the iconic ‘Eady model’ of baroclinic instability, which describes thefundamental hydrodynamic instability mechanism that gives rise to weather systems.After an undergraduate education in mathematics he joined the U. K. MeteorologicalOffice in 1937, becoming a forecaster and upper air analyst, in which capacity heserved throughout the war. In 1946 he joined the Department of Mathematics atImperial College, presenting his Ph.D. thesis in 1948 on ‘The theory of developmentin dynamical meteorology’, subsequently summarized in Tellus (Eady 1949). Thiswork, masterly in its combination of austerity and relevance, provides a mathemat-ical description of the essential aspects of cyclone development that stands to thisday as a canonical model in the field. It also includes, rather obliquely, a derivationof the stratified quasi-geostrophic equations, albeit in a special form. The impactof the work was immediate and it led to visits to Bergen (in 1947 with J. Bjerknes),Stockholm (in 1952 with C.-G. Rossby) and Princeton (in 1953 with J. von Neumann


and Charney). Eady followed his baroclinic instability work with prescient discus-sions of the general circulation of the atmosphere (Eady 1950, Eady and Sawyer1951, Eady 1954). A perfectionist who sought to understand it all, Eady’s subse-quent published output was small and he later turned his attention to fundamentalproblems in other areas of fluid mechanics, the dynamics of the sun and the earth’sinterior, and biochemistry. He finally took his own life. There is little publishedabout him, save for the obituary by Charnock et al. (1966).

Jule Charney (1917–1981) played a defining role in dynamical meteorology in thesecond half of the 20th century. He made seminal contributions in many areas in-cluding: the theory of baroclinic instability (Charney 1947); a systematic scaling the-ory for large-scale atmospheric motions and the derivation of the quasi-geostrophicequations (Charney 1948); a theory of stationary waves in the atmosphere (Char-ney and Eliassen 1949); the demonstration of the feasibility of numerical weatherforecasts (Charney et al. 1950); planetary wave propagation into the stratosphere(Charney and Drazin 1961); a criterion for baroclinic instability (Charney and Stern1962); a theory for hurricane growth (Charney and Eliassen 1964); and the conceptof geostrophic turbulence (Charney 1971). His Ph.D. is from UCLA in 1946 and this,entitled ‘Dynamics of long waves in a baroclinic westerly current’, became his well-known 1947 paper. After this he spent a year at Chicago and another at Oslo, and in1948 joined the Institute of Advanced Study in Princeton where he stayed until 1956(and where Eady visited for a while). He spent most of his subsequent career at MIT,interspersed with many visits to Europe, especially Norway. For a more completepicture of Charney, see Lindzen et al. (1990) and a brief biography by N. Phillipsavailable at http://www.nap.edu/readingroom/books/biomems/jcharney.html.

11 At least I find it so. My treatment of the Eady problem draws from unpublished notesby J. S. A. Green, as well as Eady (1949) itself.

12 If c is purely real (and so the waves are neutral), then there exists the possibility thatΛz − c = 0, and the equation for Φ is

d2Φdz2 − µ

2Φ = Cδ(z − zc), zc = c/Λ. (6.182)

where C is a constant. Because zc is continuous in the interval [0,1] so is c, andthese solutions have a continuous spectrum of eigenvalues. The associated eigen-functions provide formal completeness to the normal modes, enabling any functionto be represented as their superposition.

13 Our nondimensionalization of the two-layer system is such as to be in correpondencewith that for the continuous system. Thus we choose H to be the total depth of thedomain. This choice produces growth rates and wavenumbers that are equivalent tothose in the Eady problem.

14 Green (1960) and Branscome (1983). Lindzen and Farrell (1980) also provide anapproximate calculation of growth rates in the Charney problem.

15 After Charney (1947), in whose problem it appears, and Green (1960), who appreci-ated its importance.

16 Our numerical procedure is to assume a wavelike solution in the horizontal of theform ψ exp[i(kx + ly −ωt)], and to finite difference the equations in the vertical.The resulting eigenvalue equations are solved by standard matrix methods, for eachhorizontal wavenumber. See Smith and Vallis (1998).

17 Gill et al. (1974) and Robinson and McWilliams (1974) were among the first to lookat baroclinic instability in the ocean.


Further Reading

Drazin P. and Reid, W. H., 1981. Hydrodynamic Stability.A standard text on hydrodynamic instability theory. It discusses nearly all the classiccases in a straightforward and clear fashion. It includes a more extensive discussionof the linear instability of parallel shear flow than is contained here, although thetreatment of baroclinic instability is rather brief.

Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability.A classic text discussing many forms of instability but not, alas, baroclinic instability.

Pierrehumbert and Swanson (1995) review many aspects of baroclinic instability.

Problems

6.1 Derive the jump condition (6.29) without directly considering the motion of the in-terface. In particular, from the momentum equation along the interface show that

∂∂y

(ψU − c

)= − p

(U − c)2 (P6.1)

and show that (6.29) follows. Be explicit about the conditions under which the right-hand side vanishes when integrated across the interface. (For help see Drazin andReid 1981).

6.2 By applying the matching conditions (6.23) and (6.29) at y = ±a to Rayleigh’s equa-tion, explicitly derive the dispersion relationship (6.42).

6.3 Show that for very long waves, or as the shear layer becomes thinner, the growth rategiven by (6.42) reduces to that of Kelvin-Helmholz instability of a vortex sheet.

6.4 Obtain the stability properties of the triangular jet, with a basic state velocity givenby

U(y) =

0 for z ≥ 11− |y| for − 1 ≤ y ≤ 10 for z ≤ −1

(P6.2)

In particular, obtain the eigenfunctions and eigenvalues of the problem, and showthat each eigenfunction is either even or odd. Perturbations with even ψ′ are knownas ‘sinuous modes’ and those with odd ψ′ are ‘varicose modes’. Show that sinuouswaves are unstable for sufficiently long wavelengths in the z-direction, but that allvaricose modes are stable.

6.5 Consider the incompressible piecewise linear shear flow below:

u =

AL+ B(y − L), y ≥ LAy, −L ≤ y ≤ L−AL+ B(y + L), y ≤ −L,

(P6.3)

The flow is two-dimensional, and A and B are constants with B > 0.

(a) Find the two normal mode frequencies as a function of zonal wavenumber k.(b) Find the stability boundaries in terms of k and A and provide a physical interpre-

tation. If A = B is the flow stable or unstable? Why?


(If the algebra defeats you, explain carefully the method for doing the problem.)

6.6 Show numerically or analytically that, in the Eady problem:

(a) Instability occurs for µ < 2.399.(b) The wavenumber at which the instability is greatest is µ = 1.61.(c) The nondimensional growth rate at that wavenumber is 0.31.

6.7 Consider the vertical modes of continuously stratified problems:

(a) When solving the continuous form of the eigenvalue stability problem (as in theEady problem, for example) the differential equation typically seems to have justone pair of eigenvalues. However, if the equation is solved on a vertical gridwith N levels, the resulting difference equation has N roots. Does this mean thatN − 2 roots are spurious, and if so how might the ‘correct’ eigenvalues be iden-tified? Alternatively, are there corresponding additional roots in the continuouslystratified problem?

(b) The N-level problem is equivalent to a physically realizable N-layer system, inwhich there are presumably N physically meaningful eigenvalues. As N becomeslarge, with the density differences and thicknesses of each layer chosen to be-come smaller in a consistent way, the equations describing the layered systempresumably converge to those describing the continuous system, yet there are Neigenvalues in the former. What is the physical nature of the eigenvalues in thelayered system, and how do they relate to those of the continuous system?

6.8 Show, using the two-layer model (or otherwise) that the presence of β reduces theefficiency of baroclinic instability. For example, show that it makes the meridionalvelocity slightly out of phase with the temperature.

6.9 Consider the baroclinic instability problem with a discontinuity in the stratification,but a uniform shear. For example, suppose the shear is uniform for z ∈ (0,1) withan abrupt change in stratification at z = 0.5. How does the amplitude of the insta-bility vary on either side of the discontinuity? Your answer may be an analytical or anumerical calculation, or both.

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