Entropy and Potential Vorticity in Dynamical Core ... and Potential... · simulated by a normalised...

THE UNIVERSITY OF READING

Entropy and Potential Vorticity in Dynamical

Core Atmosphere Models

Tim Woollings

A thesis submitted for the degree of Doctor of Philosophy

Department of Meteorology

October 2004

Declaration

I confirm that this is my own work and the use of all material from other sources has been

properly and fully acknowledged.

Tim Woollings

i

Abstract

Firstly, it is shown that the normal modes of atmospheric motion in isentropic coordinates

are best simulated using Charney-Phillips type vertical grids. Then, an isentropic model is

developed which uses a new technique to handle the intersection of model levels with the

ground, and which uses the PV conservation equation as a prognostic equation, so that the PV

evolution can be constrained to be closer to its evolution in the real atmosphere. Unfortunately,

the model is stable only for short periods. Nevertheless, the simulation of most of the life

cycle of an idealised baroclinic wave has been possible, and this is compared in detail with that

simulated by a normalised pressure coordinate model, the IGCM.

Some of the technical problems plaguing previous isentropic models have been overcome

here. For example, the surface flow is well represented, with strong fronts predicted which are

much more similar to those in a high resolution IGCM simulation than those the low resolution

IGCM predicts.

Different properties of the atmosphere are represented well in the two models, indicating

that both perspectives are valuable. Many of the synoptic differences are shown to result from

the contrasting small-scale dissipation mechanisms in the two models. The isentropic model

conserves entropy excellently, but it exhibits a drift in total energy that constitutes a more

serious error than that in entropy in the IGCM. This latter model shows a systematic increase

in global entropy, which is believed to be partly responsible for the cold bias of similar climate

models. The net increase of entropy is shown to be the residual effect of frequent changes

in the entropy of individual air parcels, so that the IGCM is seen to violate the Lagrangian

conservation law for entropy to a much greater extent than suggested by the global drift.

ii

Acknowledgements

I am deeply grateful to my supervisor, John Thuburn; without his enthusiastic support, guid-

ance and encouragement none of this would have happened. I am also very grateful to my thesis

committee, Brian Hoskins and Stephen Belcher, who have shown great interest and given in-

valuable advise. I have benefited greatly from interesting discussions with so many people both

in Reading, and elsewhere. A special mention must go to Mike Blackburn and John Methven

for their help with the IGCM, and to Manuel Pulido for his help with the isentropic model. I’d

also like to thank NERC for my funding.

Many thanks to my fellow PhD students past and present, and to all my friends in Reading,

Victoria and elsewhere, who have helped immensely and ensured that I’ve had a great few

years, and also to my parents for their ongoing support. Special thanks go to Helen for fantastic

proof-reading, and so much more.

iii

“The law that entropy always increases - the second law of thermodynamics - holds, I think,

the supreme position among the laws of nature. If someone points out that your pet theory

of the universe is in disagreement with Maxwell’s equations - then so much the worse for

Maxwell’s equations. If it is found to be contradicted by experiments - well, these experimen-

talists do bungle things sometimes. But if your theory is found to be against the second law

of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest

humiliation.”

Sir Arthur Stanley Eddington, in The Nature of the Physical World. Maxmillan, New York,

1948, p. 74.

iv

Contents

1 Introduction 1

1.1 The PV-θ Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Isentropic Modelling of the Atmosphere . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Advantages of the Isentropic Coordinate . . . . . . . . . . . . . . . . . 6

1.2.2 Disadvantages of the Isentropic Coordinate . . . . . . . . . . . . . . . 8

1.2.3 Pure Isentropic Models . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.4 Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.5 Isopycnal Ocean Models . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 PV as a Prognostic Variable in Models . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Vertical Grid Staggering in Isentropic Coordinates 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 The Equation Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 A Hydrostatic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 The Isentropic Model 50

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Boundary Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Controlling the Massless Region . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 The Advection Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1 PV Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.2 Flux Limiter for Density Advection . . . . . . . . . . . . . . . . . . . 59

Contents vi

3.4.3 Sub-gridscale Fit Near the Ground . . . . . . . . . . . . . . . . . . . . 60

3.4.4 Summary of the Mass Advection Step . . . . . . . . . . . . . . . . . . 62

3.5 Summary of the Model Timestep . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Basic Model Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6.1 Solid Body Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6.2 Perturbing the Massless Region . . . . . . . . . . . . . . . . . . . . . 66

3.6.3 Flow over a Hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.7 The Surface Divergence Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 The LC1 Baroclinic Life Cycle 79

4.1 Baroclinic Instability and a History of the LC1 Cycle . . . . . . . . . . . . . . 79

4.2 The IGCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 An Overview of the LC1 Cycle in the T42 IGCM . . . . . . . . . . . . . . . . 84

4.4 A High Resolution Version of the IGCM LC1 Cycle . . . . . . . . . . . . . . . 89

4.5 Initial Conditions for the Isentropic Model Simulation . . . . . . . . . . . . . . 92

4.6 Overview of the LC1 Cycle in the Isentropic Model . . . . . . . . . . . . . . . 94

4.7 The IGCM LC1 Cycle with∇4 Hyper-diffusion . . . . . . . . . . . . . . . . . 101

4.8 The Terminal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.9 Surface Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.10 Surface PV Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.11 Global Conservation Properties of the Two Models . . . . . . . . . . . . . . . 115

4.12 PV Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.13 Effect of the Relaxation Scheme on the LC1 Cycle . . . . . . . . . . . . . . . 123

4.13.1 The Surface Divergence Effect . . . . . . . . . . . . . . . . . . . . . . 123

4.13.2 Sensitivity to the Relaxation . . . . . . . . . . . . . . . . . . . . . . . 124

4.14 Smoothness of ∇2M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5 Entropy Production in the IGCM 130

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.1.1 Entropy and the Second Law . . . . . . . . . . . . . . . . . . . . . . . 131

5.1.2 The Entropy Budget of the Atmosphere . . . . . . . . . . . . . . . . . 132

Contents vii

5.1.3 Some Specific Motivations . . . . . . . . . . . . . . . . . . . . . . . . 134

5.2 Entropy Production in the LC1 Cycle . . . . . . . . . . . . . . . . . . . . . . 136

5.2.1 Magnitude and Sensitivity of the Source . . . . . . . . . . . . . . . . . 136

5.2.2 Location and Attribution of the Source . . . . . . . . . . . . . . . . . 140

5.2.3 Cross-Isentrope Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . 146

5.3 The Entropy Theory for the Cold Bias . . . . . . . . . . . . . . . . . . . . . . 148

5.3.1 Summary of Johnson (1997) . . . . . . . . . . . . . . . . . . . . . . . 148

5.3.2 Analysis of the IGCM Results . . . . . . . . . . . . . . . . . . . . . . 150

5.4 Alternative Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.5 A Comment on the MEP Theory . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 Conclusions 167

6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A The LC2 Cycle 174

References 180

Chapter 1

Introduction

The atmosphere is a three-dimensional fluid system capable of supporting three-dimensional

turbulence, and as such might be expected to exhibit even greater complexity than it actually

does. However, the combined effects of buoyancy stratification and rotation strongly constrain

the flow, acting, for example, to inhibit large-scale three-dimensional turbulence. The potential

complexity is further reduced because the atmosphere is known to satisfy several conservation

laws. Over timescales of the order of a few days, the large-scale motion of the atmosphere is

dominated by its adiabatic component (unless strong latent heating occurs), where an adiabatic

process is one in which an air parcel does not exchange heat with its surroundings. Under

adiabatic flow an air parcel will conserve its potential temperature θ. This quantity is intimately

linked to the stratification, which acts to inhibit vertical motion and thus inhibit mixing across

surfaces of constant θ. These are referred to as isentropic surfaces; the specific entropy is

proportional to ln θ, so a surface of constant potential temperature is also a surface of constant

entropy. These surfaces are nearly horizontal, sloping gradually with respect to the ground but

almost never overturning, only doing so under the action of very small scale three-dimensional

eddies. Since, above the boundary layer, θ almost always increases monotonically with height

it can be used as a vertical coordinate for the atmosphere; it is referred to as an isentropic

coordinate. The adiabatic motion of the atmosphere is then two-dimensional, with individual

air parcels only moving along isentropic surfaces and never across them.

Deeper understanding can be obtained by considering the potential vorticity or PV, another

scalar quantity which is conserved by a fluid parcel under adiabatic, frictionless flow. The es-

sential idea is that there is potential for changing the vorticity of the parcel by changing its

latitude, and hence the size of the effect of the Earth’s rotation, or by adiabatically changing

Introduction 2

the separation between isentropic surfaces, resulting in vortex stretching. The dominant adi-

abatic component of atmospheric flow can therefore be visualised by the evolution of PV on

isentropic surfaces, acting as a tracer for individual air masses. The PV, however, differs from

a passive tracer in that it is a dynamical quantity whose distribution actually determines the

subsequent evolution of a large, and dominant, component of the flow. A given PV anomaly on

an isentropic surface is said to ‘induce’ a wind field which in turn acts to redistribute the PV.

The apparent complexity of the atmosphere is thus greatly reduced by thinking in terms

of the evolution of PV on isentropic surfaces, often referred to as PV-θ or isentropic PV (IPV)

thinking. Use of the PV-θ concept has greatly improved understanding of atmospheric dynam-

ical processes from diagnostic studies, both in synoptic weather systems (e.g. Hoskins et al.

(1985)) and in the general circulation (e.g. Hoskins (1991)). Attempts to make use of this

in the field of numerical modelling by building models based on the PV-θ concept have en-

countered significant problems. To predict the evolution of the PV field from the field itself is

much simpler if the discrete model surfaces are isentropic surfaces. In the troposphere these

slope downwards from the poles towards the equator to the extent that all isentropic surfaces

which are below the tropopause at the poles intersect the Earth’s surface at some point before

the equator (see e.g. Figure 3.4). This implies intersections of model levels with the boundary

in an ‘isentropic’ numerical model, and it is technical difficulties in handling these intersec-

tions that has most hindered the development of weather and climate prediction models which

attempt to take advantage of the PV-θ concept.

The first aim of the work presented in this thesis is to build a numerical model which pre-

dicts PV on isentropic model levels using a new technique to handle the surface intersections,

and to perform a detailed comparison of this model with a more conventional one in the sim-

ulation of an idealised atmospheric disturbance. A further aim is to investigate entropy errors

in the conventional model; are these serious enough to make isentropic models an attractive

alternative, despite the technical difficulties? We begin with a very brief introduction to the

PV-θ concept, concentrating on those aspects which are of most relevance to this project.

1.1 The PV-θ Concept

Potential temperature θ is defined by

θ = T

(p

p0

)−κ, (1.1)

Introduction 3

in terms of the temperature T , pressure p and the constant κ = R/cp, where R is the gas

constant and cp is the specific heat at constant pressure. It is the temperature an air parcel

would have if it were to be transported adiabatically to the reference pressure p0. Potential

temperature is conserved following an adiabatic, frictionless flow, so that

Dθ

Dt= 0. (1.2)

By taking the gradient of (1.2), and using both this and mass conservation, it can be shown

from the vector vorticity equation that the Ertel PV

Q =1

ρζa · ∇θ (1.3)

is conserved following a fluid parcel. Here ρ is the density and the absolute vorticity is

ζa = 2Ω +∇× u, (1.4)

where u is the full three-dimensional velocity vector and Ω is the rotation vector of the Earth.

This conservation is remarkably general, and holds for any adiabatic, frictionless motion. If

we then make the shallow atmosphere approximation, the 2Ω cosφ terms, associated with the

local horizontal component of Ω, are neglected in order to retain conservation of energy and

angular momentum. The planetary term then reduces to fk, where f is the Coriolis parameter

2Ω sinφ in terms of the latitude φ, and k is the unit vector in the vertical. If hydrostatic balance

is then assumed, the horizontal components of the vorticity are also neglected. Even with these

approximations, the definition of PV is still complicated, containing the three-dimensional

product of a curl and a grad, but this simplifies significantly when expressed in isentropic

coordinates as

Q =f + k · ∇θ × v

σ, (1.5)

where

σ = −1

g

∂p

∂θ(1.6)

is termed the isentropic density. Note that here v is the horizontal velocity field. The conser-

vation of PV following an adiabatic, frictionless flow is expressed as

DQ

Dt= 0. (1.7)

Introduction 4

This section is intended as a brief review only, so proofs of these statements are not given

here. More details and proofs are given in the review by Hoskins et al. (1985), who emphasise

two main principles underlying the PV-θ concept. The first is this Lagrangian conservation

principle, and the second is that of invertibility.

Given the mass distribution in isentropic coordinate space and some balance relation be-

tween the mass and wind fields, and provided it is done globally with appropriate boundary

conditions, the PV field can be ‘inverted’ to give the balanced component of the flow. Differ-

ent balance relations can be used. For example, the simplest and least accurate is geostrophic

balance; the PV then contains all the dynamical information about the geostrophic flow. The

inversion corresponds to the mathematical inversion of an elliptic operator and therefore has a

smoothing effect. Discussions often centre around the effect of isolated PV anomalies. These

can be either positive or negative with respect to the background PV, and are generally a com-

bination of anomalies in static stability and vorticity. The wind field induced by a PV anomaly

will have smoother features than the anomaly itself and will also extend over a larger scale,

both horizontally and vertically. This latter feature is known as the scale effect. An analogy

has been drawn between the wind field induced by a PV anomaly and the field induced by an

electrical charge. This analogy was investigated by Bishop and Thorpe (1994, 1995), motivated

by the problem of how to attribute different components of a given wind field to the effects of

a number of different PV anomalies.

The picture given so far is of PV being advected in two-dimensional motion along isen-

tropic surfaces in exactly the same way as an inert, non-diffusive chemical tracer. While useful

to some extent, this analogy can be misleading, as shown by Haynes and McIntyre (1987,

1990). PV can be regarded as the mass mixing ratio of an imaginary ‘PV substance’ or PVS

(that is, a ratio describing the amount of PVS per unit mass which is, however, permitted to

take negative values). Isentropic surfaces act as if they are impermeable to PVS so that there

can be no net transport of PVS across any isentropic surface. PVS is exactly conserved away

from lateral boundaries, i.e. it cannot be created or destroyed in a layer bounded only by two

isentropic surfaces. PVS can only be created or destroyed at lateral boundaries, i.e. at the

Earth’s surface if such a layer intersects it. This theorem is exact in that it holds regardless of

whether frictional or diabatic forces act and regardless of whether hydrostatic balance is as-

sumed, e.g. even in regions of three-dimensional turbulence and associated diabatic mixing. In

such regions mass (and also chemical species) can be transported across isentropic surfaces but

these diabatic mass transports act to concentrate or dilute the PVS on either side of the isen-

Introduction 5

tropic surface rather than to transport PVS across it. Away from the boundary, PVS can only

be transported along isentropic surfaces or, more precisely, along the intersections of isentropic

surfaces and surfaces of constant Bernoulli function B = cpT + 12u2 + gz (Schar (1993)). This

statement holds even for diabatic, frictional flow when the mass flux is no longer along these

intersections.

The rapid formation of thin filaments of vorticity is a well-known feature of fluid dynam-

ics, and in the atmosphere it is the PV which is drawn out into thin filaments in this way. It is

therefore the potential enstrophy Q2, or the variance of the PV field, which is said to cascade

to small length scales (e.g. Salmon (1998)). Filaments of high and low PV on an isentropic

surface are frequently seen to wrap up together before ultimately mixing under the action of

diabatic processes such as heating or turbulence. In this process mass is invariably transported

across isentropic surfaces but PVS is not. As the filaments become narrower their dynamic

effect is reduced in line with the scale effect. When the fine-scale structure is integrated over,

destructive interference results in large cancellations between the effects of positive and neg-

ative PV anomalies. The circulation around an isentropic contour enclosing this fine-scale

structure represents all the dynamical effect of the PV features within it, and the details of the

fine-scale structure can be replaced by a uniform PV value within the contour. This is known

as coarse-grain PV, and it is this quantity which is generally derived from observed or averaged

fields. All of the theorems above can be shown to hold also for coarse-grain PV.

Discrete numerical models by their very nature have difficulty in simulating the potential

enstrophy cascade. Some form of scale-selective dissipation is invariably used to dissipate

potential enstrophy at the gridscale, where it would otherwise accumulate. From the discussion

above it appears that an ideal model would dissipate potential enstrophy by re-arranging PV

on isentropic surfaces only, while simultaneously allowing just the correct amount of mass to

move across these surfaces. In the vast majority of models, however, both PV and θ are merely

diagnostic variables and the cascade is halted by numerical diffusion acting on model variables

such as wind, vorticity or temperature. One of the central aims of this thesis is to improve

understanding of how well this represents the underlying PV-θ dynamics of the atmosphere.

One final development is of relevance. The potential temperature varies across the Earth’s

surface, and as a result the boundary conditions for PV dynamics are inhomogeneous, compli-

cating the application of the PV-θ concept to problems where boundary effects are significant.

In quasi-geostrophic theory a quantity can be derived, referred to as quasi-geostrophic PV or

QGPV, which is analogous to the PV in some ways but is not an approximation to it. One aspect

Introduction 6

of the relationship between the two is that the variation of the QGPV on a surface of constant

pressure or height is proportional to the variation of the Ertel PV on an isentropic surface. A

key difference is that while the Ertel PV is conserved following an air parcel, the QGPV is con-

served following the geostrophic flow so that, for example, there is no vertical advection term

in the conservation equation for QGPV. See Andrews et al. (1987), Section 3.8.3 or Hoskins

et al. (1985) for more details. Bretherton (1966) introduced the concept of boundary QGPV.

The underlying principle is that the dynamical effect of the varying surface θ field is identical

to that induced by replacing it with a constant θ but imposing a QGPV distribution confined

to an infinitesimally thin layer at the surface. When an isentropic surface hits the ground it is

envisioned to extend along just above the surface. Many such isentropic surfaces pile up with

negligible mass separating them and the lowest of these is exactly coincident with the ground.

The boundary condition is then one of constant θ, that of this lowermost surface, and so is

homogeneous. This theory has recently been extended to arbitrary, non-geostrophic flows by

Schneider et al. (2003). In the impermeability theorem of Haynes and McIntyre (1987, 1990),

creation and destruction of PV is permitted where an isentropic surface intersects the ground.

This can be simply interpreted as an exchange of boundary and interior PV.

There is a clear similarity between this view of PV-θ dynamics and the massless layer ap-

proach to handle the lower boundary problem employed in isentropic models. In this approach

an isentropic model level which intersects the ground is extended with negligible mass along

the surface (see Sections 1.2.3 and 3.2). Previous work in the field of isentropic modelling is

now reviewed, starting with a comprehensive list of reasons for using, and for not using, an

isentropic vertical coordinate.

1.2 Isentropic Modelling of the Atmosphere

1.2.1 Advantages of the Isentropic Coordinate

1. An air parcel conserves its potential temperature θ under adiabatic motion. As mentioned

by Bleck (1973), the only significant synoptic-scale process acting in the free atmosphere

which changes the potential temperature of a parcel on timescales of the order of a few

days is latent heating. This means that, except in regions with strong latent heating such

as fronts, isentropic surfaces in the free atmosphere act as material surfaces. When the

atmosphere is viewed in an isentropic framework, the large adiabatic component of its

Introduction 7

seemingly three-dimensional motion is therefore reduced to a stack of two-dimensional

motions. By using this coordinate in an atmospheric model the size of the vertical advec-

tion terms is greatly reduced, resulting in a reduction of the model’s truncation errors;

this was the motivation for the original isentropic model of Eliassen and Raustein (1968).

2. The coordinate is adaptive in nature, in that the model surfaces move close together in re-

gions of strong temperature gradients, which are likely to be of importance. Atmospheric

fronts are represented by the tight packing of isentropic levels between air masses of dif-

ferent temperatures. The front lies nearly parallel to the isentropic surfaces and so the

discontinuities seen in z or p space are much smoother features when viewed in θ space.

Shapiro (1975) quantified this by noting that on z and p surfaces which cut through a

front the variations of temperature and velocity occur on scales of order 100 km, but that

on isentropes contained within the front the variations of pressure and velocity occur

on scales of order 1000 km. There is potential, then, to simulate fronts accurately at a

significantly lower horizontal resolution using the isentropic coordinate.

3. In isentropic coordinates it is easy to satisfy the fourth of the integral constraints of the

continuous governing equations laid out by Arakawa and Lamb (1977), namely that the

integral over the entire mass of any function of θ is conserved under adiabatic processes.

In an isentropic model this reduces to the conservation of mass within each isentropic

level, which is relatively simple to enforce.

4. The conservation of the mass in every isentropic layer also ensures that the unavailable

part of the potential energy of the system is conserved, since this is given by the mass

distribution in θ space (see Section 4.1). The unavailable part is by far the largest pro-

portion of the total potential energy, and this in turn is far larger than the kinetic energy

so that the greater part of the total energy of the system is automatically conserved.

5. Hsu and Arakawa (1990) showed that in isentropic coordinates it is possible to construct

a vertical scheme that formally conserves both energy and angular momentum with-

out resorting to a Lorenz-type grid, which supports potentially damaging computational

modes (see Chapter 2).

6. The definition of Ertel’s PV (after making the hydrostatic and shallow atmosphere ap-

proximations) is simplest in isentropic coordinates (1.5), and does not involve the vertical

Introduction 8

derivative of the velocity. This should make it easier to conserve the PV in a vertically

discrete system.

7. In isentropic coordinates the horizontal pressure gradient force is given by −∇θM ,

where M is the Montgomery potential. This force is therefore represented by an irrota-

tional vector and so does not yield a horizontal circulation along any coordinate surface,

which should reduce the impact of the discretization error on the total circulation. In

other coordinate systems this force is generally expressed as the gradient of two oppos-

ing terms and so is a greater source of numerical error. As an aside it has recently been

suggested by Purser et al. (2004) that this force can be expressed as the gradient of one

‘modified Montgomery’ potential in a special hybrid σ − p coordinate framework.

8. It is believed that moist processes such as condensation heating could be simulated more

accurately in isentropic coordinates. As described by Konor and Arakawa (2000), this is

because these processes are highly dependent on the moisture transport, which is more

accurate in the absence of any truncation errors in vertical advection (see Johnson et al.

(1993, 2000, 2002)). Mahowald et al. (2002) showed that transport of chemical species

is improved when using isentropic coordinates for the same reason. Properties such as

water vapour and chemical species can exhibit large gradients in the vertical and are thus

especially susceptible to spurious numerical dispersion and truncation errors.

1.2.2 Disadvantages of the Isentropic Coordinate

1. The isentropic coordinate surfaces slope in the atmosphere and intersect the ground.

There are technical difficulties associated with representing these intersection points in

a discrete model, and furthermore in allowing them to be advected along the ground as

the surface θ distribution evolves. These intersections have always been the underlying

cause of the greatest problems encountered by isentropic models, in the guise of gravity

wave noise, instabilities or lack of conservation, and are still the source of the outstanding

problems today.

2. In the real atmosphere it is possible for unstable layers with ∂θ∂z< 0 to exist. This, how-

ever, is forbidden in isentropic models because the assumption of a monotonic variation

of θ with z must be made for it to be a valid coordinate choice. Without this assumption

it would be possible for z and p to become multivalued at a point in the model’s (x, y, θ)

Introduction 9

space, a problem referred to as the ‘folding’ of coordinate surfaces. The hybrid models

discussed in Section 1.2.4 are motivated partly by a desire to avoid this limitation.

3. The mass within an isentropic model layer can become very small, and indeed this is to

be expected given the adaptive nature of the coordinate. This has caused computational

problems in the past.

4. In isentropic coordinates some important regions, most notably the tropical troposphere,

are less well resolved in the vertical than the rest of the domain, due to the distribution

of θ throughout the atmosphere. Similarly there will be a lack of vertical resolution in

less stable regions near the surface, potentially making it harder to incorporate standard

boundary layer processes there.

5. If surface heating is included in an isentropic model then the coordinate surfaces may

become vertical near the ground. It therefore seems likely that a separate boundary layer

scheme would be necessary in a model more complete than the one developed here.

6. The pressure on an isentropic model level depends only on the temperature, so if numer-

ical errors introduce noise to the temperature profile, this can result in irregularly spaced

levels, potentially leading to further inaccuracies (see Webster et al. (1999)).

1.2.3 Pure Isentropic Models

Despite the clear advantages to be gained by diagnosing the state of the atmosphere in an isen-

tropic coordinate framework, the approach had only limited success, at least until the 1960s,

and so was used little by the meteorological community. As described by Bleck (1973), in

addition to the difficulties near the surface, this was at least partly due to a flaw in the adopted

procedure for calculating the Montgomery potential M on isentropes (which is analogous to

the geopotential Φ on pressure surfaces). The flaw was eventually discovered by Danielsen

(1959), but by this time pressure was firmly in place as the vertical coordinate of choice.

The first attempt to overcome the anticipated shortcomings of the isentropic coordinate

and create a prognostic, primitive equation isentropic model of the atmosphere was made by

Eliassen and Raustein (1968), and it was remarkably successful. Their model consisted of three

information surfaces: the ground and two isentropic surfaces, the lowest of which intersected

the ground. Horizontal velocities were predicted on all levels, as well as the pressure on the

Introduction 10

isentropic levels, and the surface potential temperature θs. A technical problem had to be over-

come, namely how to calculate horizontal differences at the point where the lowest isentropic

level intersected the ground and so terminated. To do this, imaginary values were created just

below the surface and used in the horizontal differencing. The pressure at these points was ex-

trapolated vertically by prescribing that the static stability in the imaginary underground region

was identical to that in the real air just above the surface.

The results were encouraging. A 72 hour test integration was completed successfully,

though a special non-linear diffusion was needed. Sharp fronts formed in the simulation so that

one of the key advantages of the isentropic coordinate already appeared to have been realised.

In a following paper, however, Eliassen and Raustein (1970) extended the model to include

five isentropic levels and noted that no such clear fronts developed. This they attributed to the

fact that the earlier model highly underestimated the static stability, due to its coarse vertical

resolution, and so frontogenetic circulations were exaggerated.

Results from the extended model, however, were still impressive, and the model performed

well enough to be used by Trevisan (1975), with a slightly modified underground extrapola-

tion, in a study to improve understanding of the influence of orography on cyclone formation.

Shapiro (1975) used the same model in his study of upper level frontogenesis, this time with

20 isentropic levels. The modified underground extrapolation technique used information from

both of the two lowest isentropic surfaces, giving improved results when several isentropic lev-

els intersected the surface in close proximity, and when the lowest isentropic level contained

only a small amount of mass. The model simulated upper level frontogenesis well, so that the

anticipated ability of the isentropic coordinate to adapt to give increased resolution in frontal

regions had indeed been realised.

By this time a predecessor of the isentropic model presented in this thesis had also been

developed. Bleck (1973) recognised not only the promise shown by the isentropic coordinate

but also its close association with the PV, and developed a balanced isentropic model with PV

as the only prognostic variable. It was Bleck (1973) that first described an improved extrap-

olation of the type used later by Trevisan (1975) and Shapiro (1975). The balanced model

performed well, successfully handling four cases of explosive cyclogenesis which ‘had consti-

tuted indisputable forecast failures’.

Later, however, Bleck (1974) abandoned the balanced model in favour of a primitive equa-

tion version which predicted the three variables u, v and p on isentropic surfaces. This version,

surprisingly, was significantly cheaper, and also ‘outperformed the potential vorticity model by

Introduction 11

a wide margin’. In this paper the first detailed discussion was given of an ongoing problem

which appears to have affected all the pure isentropic models mentioned so far. An undesirably

high level of gravity wave noise affected the flow near the surface, and Bleck showed that, as

suspected, the source of this noise was the points where the isentropic model levels intersected

the ground. A further, perhaps more intrinsic, problem with the underground extrapolation

models was that it proved hard to maintain conservation of integrated quantities such as energy

and angular momentum. These problems proved so significant that research focussed on the

possibility of coupling an upper level isentropic coordinate to a terrain-following one at the

surface to form a hybrid model. The next advance in the quest for a pure isentropic model was

not for a further 10 years.

Bleck (1984) identified the fundamental flaw holding back all of the previous models.

Each of these needed to predict the location of the intersection points by solving a prognostic

equation for θs on the regular horizontal grid. Inevitably this θs field would not always agree

exactly with the isentropic level thicknesses predicted by solving the continuity equation, and

imbalances and subsequent gravity wave noise would result. This is clear with hindsight and

in the light of recent results. As discussed in Section 4.9, there appears to be no limit to

the strength of a frontal gradient in adiabatic fluid dynamics, and isentropic models such as the

one described in this thesis, and that of Fulton and Schubert (1991), form fronts with incredibly

large temperature gradients. This happens as the mass between isentropic surfaces in the frontal

region becomes very small. Such gradients cannot be exhibited by the θs field, however, since

the gradient of θs is limited both by the horizontal grid scale and by any smoothing device

employed in the model, as had been noted by Bleck (1973). Disagreement between θs and the

distribution of isentropic levels was therefore inevitable at surface fronts, where large amounts

of noise in the pure isentropic models was reported by Bleck (1978).

Bleck (1984) then pioneered a new method recently enabled by a development in numeri-

cal advection techniques; that of Flux-Corrected Transport (FCT) schemes by Boris and Book

(1973) and Zalesak (1979). The idea behind FCT is to combine the effects of two different

schemes. The first is a low-accuracy scheme which has a diffusive effect but does not produce

unwanted ripples in the advected field; the second is higher-order accurate but prone to produce

ripples. The FCT combination has only a small diffusive effect and is free from ripples, i.e. no

new maxima or minima are produced. The details are similar to the flux-limiter described in

Section 3.4.2 so will not be given here.

The key benefit for isentropic models is that FCT allows the mass within an isentropic

Introduction 12

layer to go to zero anywhere within the domain without ever becoming negative (which would

constitute a new minimum). Bleck (1984) then developed the first prognostic model to use the

so-called massless layer idea first used by Lorenz (1955). An isentropic level which intersects

the ground is then envisioned to extend along just above the surface containing negligible

mass. The FCT algorithm ensures that there is no overshoot as the mass field goes to zero,

and so no negative mass is produced. The horizontal pressure gradient force (HPGF) terms

near the intersections (which previously depended on extrapolated underground values) are

now evaluated using values from the massless region, where the HPGF is formulated in Bleck

(1984) to be identical to the surface HPGF. Crucially, with the massless layer approach there is

no need for a predicted θs field; a method is given in this paper for obtaining it from the mass

distributions in the intersecting isentropic levels.

Bleck achieved good results with this method, and the most significant development in

pure isentropic modelling since, that of Hsu and Arakawa (1990), is based on his technique. In

this model considerable effort was put into the numerical schemes to ensure that the infinite PV

values implied in an isentropic layer of infinitesimal mass did not affect the above ground flow,

and also that dispersion errors were minimised at the intersection of a level with the ground,

where the gradient of the mass is discontinuous. The lower boundary condition is implemented

by defining M on the lowest massy level of a given column, i.e. the lowest level containing

mass. This is used to start the integration of the hydrostatic equation upwards from the surface,

so that θs is not needed and is never explicitly calculated. The same technique is used in the

model presented here, so further details can be found in Section 3.2. In a test run, the model

of Hsu and Arakawa (1990) simulated the evolution of a baroclinic wave with no significant

computational difficulties at the surface. The model is formulated with a strong emphasis on

conservation properties, indicating that use of the coordinate was becoming developed enough

for long-term integrations to be considered. The vertical scheme formally conserves both angu-

lar momentum and energy, and the horizontal momentum differencing conserves energy while

allowing the potential enstrophy to be dissipated at small scales, as suggested by Sadourny and

Basdevant (1985).

However, there were two, somewhat related, outstanding problems, the first of which is

mentioned at the end of Hsu and Arakawa (1990) and later in Arakawa et al. (1992). It was

found that the model had difficulty in representing sufficiently smooth M distributions, at all

levels, in points above the locations of the intersections. This is attributed to a lack of degrees

of freedom in the implied surface temperature distribution, since this is determined solely by

Introduction 13

the number of model levels intersecting the surface. The problem was studied in detail by

Randall et al. (2000) in a model combining the vertical scheme of Hsu and Arakawa (1990)

with a geodesic horizontal grid based on the Z grid of Randall (1994). M is affected by noise

throughout the domain to the extent that the geostrophic vorticity

ζg = ∇.(

1

f∇M

)(1.8)

is almost discontinuous at all points above an intersection, even in a test with a continuous

prescribed pressure distribution. The reason given is that, although θs is never explicitly calcu-

lated, the flow at all levels is affected by the discontinuous nature of the θs field implied by the

intersection with the ground of a number of levels of constant θ. The lower boundary condition

for the hydrostatic equation∂M

∂θ= Π(p) (1.9)

is therefore discontinuous, andM is affected at all levels during the integration of (1.9) up from

the surface. The solution proposed by both Arakawa and Randall is that the vertical resolution

should be matched to the horizontal resolution so that on average there is one isentropic level

intersecting the surface for every horizontal grid cell. Further discussion of this problem is

given in Section 4.14 where results from the isentropic model developed here are presented.

The second outstanding difficulty is attributed to essentially the same cause: the lack of

degrees of freedom in the implied θs field. Arakawa et al. (1992) studied fronts at mid and

upper levels in their model, as had Shapiro (1975) for the same reason, which was simply

that surface fronts were not well simulated by either model. However, while Shapiro’s model

failed because it explicitly predicted the θs field (as described above), Arakawa et al. (1992)

attributed the failure of their model to the fact that it did not. The θs field implied by the

intersecting of model levels effectively had too low a horizontal resolution to simulate a front,

they concluded, because the vertical resolution of 10 K level spacing used was too low. The

surface fronts predicted by the model described here are discussed in Section 4.9.

As a final comment in this section it should be noted that one of the major early concerns

over the isentropic coordinate was unfounded. It was anticipated that the coordinate surface

might overturn so that the pressure would become multi-valued at a given (x, y, θ) location.

This has never occurred in any of the isentropic models discussed here and the reason is laid

out clearly by Hsu and Arakawa (1990). By choosing θ as a vertical coordinate its variation

Introduction 14

with height is assumed to be monotonic. An isentropic model predicts the mass between, or

pressure difference across, two θ surfaces, and it is implicitly assumed by the monotonicity of

θ that this mass is distributed without folding of coordinate surfaces. Therefore an isentropic

model automatically adjusts any unstable layers with ∂θ∂z< 0 in the same way as the explicit

adjustment applied in any conventional model. Furthermore, as mentioned by Eliassen and

Raustein (1970), a dramatic folding of the isentropes is very unlikely as this ‘would normally

correspond to a significant increase in the available potential energy of the system’.

1.2.4 Hybrid Models

Significant success over the years has been achieved in the field of hybrid isentropic coordinate

modelling. The intention here is to avoid both the technical difficulties at the ground, and also

the restriction imposed by demanding that ∂θ∂z

> 0 near the surface, by using a generalised

vertical coordinate, which is typically terrain following near the surface and isentropic higher

up. Arakawa (2000) notes that in the presence of surface heating the movement of air parcels

near the ground is in fact closer to terrain-following surfaces than θ surfaces, and so a hybrid

coordinate may be even closer to a true material coordinate than an isentropic coordinate is.

Furthermore, the lack of resolution in the tropical troposphere is a fundamental drawback of the

pure isentropic model which has so far only been overcome by adopting a hybrid coordinate.

Bleck (1978) reviewed a selection of proposed methods of coupling a σ coordinate domain

to an isentropic one, concluding that surface noise was indeed significantly reduced in com-

parison with pure isentropic models. Note that in this section σ represents the conventional

terrain-following vertical coordinate p/ps rather than the isentropic density.

An early concern was that it would be hard to guarantee satisfaction of the various conser-

vation laws generally demanded of atmospheric models and yet still maintain full interaction

between the two domains. This concern, however, was dispelled by Uccellini et al. (1979) with

their careful formulation of a hybrid model in which isentropic levels in the free atmosphere in-

tersected a fixed number of σ levels near the surface. The University of Wisconsin hybrid θ−σmodel described by Zapotocny et al. (1994) is descended from that of Uccellini et al. (1979).

As hoped the hybrid model performs significantly better than a pure σ version in representing

the transport of trace constituents. This is an obvious benefit for the modelling of the important

stratosphere-troposphere exchanges of water vapour and chemical species, and it is for this

reason that Mahowald et al. (2002) chose to use a hybrid model for their study of stratospheric

Introduction 15

transport. Several chemical transport models (CTMs) employ isentropic or hybrid-isentropic

coordinates for this reason (e.g. Chipperfield (1999), McKenna et al. (2002)).

Zhu et al. (1992) formulated the first hybrid model (the HIGCM) to feature a smooth

transition between the σ and θ regions, thus eliminating the problem of noise generation due to

imbalances at the points where isentropic surfaces intersected the σ domain. As many levels as

possible were at constant θ, with at least one level at the surface of constant σ and some levels in

between on which θ and σ both varied. Early tests of the model were successful, although extra

vertical diffusion had to be included to avoid noisy temperature profiles and irregularly spaced

model levels. Thuburn (1993) performed further experiments, concluding that the anticipated

benefits of the isentropic coordinate ‘appeared to have been realised only in a small way’. The

vertical scheme was improved in Webster et al. (1999) so that the ‘ad-hoc’ vertical diffusion

was no longer necessary, and results improved. The problems with the vertical scheme were

attributed partly to the use of the Lorenz vertical grid which is now known to have drawbacks

(see Chapter 2). The same hybrid coordinate scheme was also implemented in a different GCM

by Zhu and Schneider (1997) who found significant improvements in the model’s simulation

of the stratosphere.

A fundamental disadvantage of the HIGCM scheme is that a large proportion of the tro-

posphere is not modelled isentropically. No isentropic surface which intersects the ground can

be used as a model level, and yet surfaces in the vicinity of 300 K which touch the surface in

the tropics also graze the tropopause near the poles. An alternative was proposed by Bleck and

Benjamin (1993) in which an individual coordinate surface is able to be isentropic in one re-

gion of (x, y) space and to follow a constant σ surface in another. Hence, a coordinate surface

can follow an intersecting isentropic surface until it is near the ground, and then is ‘nudged’ to

follow a σ surface instead. In this way a greater portion of the domain is modelled using the

advantageous isentropic coordinate. This scheme performed well enough to be the basis for

the RUC model of Benjamin et al. (2004) which is currently used operationally by NCEP.

Konor and Arakawa (1997) achieved a similar effect with their generalised vertical coor-

dinate η = F (θ, σ). An expression for F was chosen that was a function of σ only near the

surface, or where ∂θ∂z. 0, but quickly and smoothly became only a function of θ away from

these regions. The Charney-Phillips vertical staggering was used to avoid problems caused by

the Lorenz staggering as encountered by the HIGCM.

There are therefore three hybrid isentropic coordinate formulations in use in the research

community at this time. Current developments in isentropic modelling, of which there are

Introduction 16

many, are based on these hybrid methods. The Wisconsin model has evolved into the hybrid

θ−η model described by Schaack et al. (2004), which no longer has a discrete interface between

σ and θ domains. This model is currently undergoing testing in both weather forecasting

and climate scenarios with standard NCEP and NCAR physics packages incorporated, and

forms the dynamical core of the NASA/Wisconsin Regional Air Quality Modelling System

(RAQMS, Pierce et al. (2003)). A project is also under way to couple the Wisconsin θ −η model with the OMEGA model of Bacon et al. (2000) to produce a model with a finite-

difference formulation on isentropic surfaces above about 336 K and an adaptive, unstructured

grid formulation below. Version 3 of the Colorado State University model is under construction

as the Coupled Colorado State Model or CCoSM. The atmospheric component will feature the

hybrid coordinate of Konor and Arakawa (1997), a new explicit time-stepping scheme stable

for Courant numbers above one (Arakawa and Konor (2004), personal communication) and a

new ‘embedded’ planetary boundary layer comprising, by definition, the lowest nmodel levels.

Finally, there is the RUC model (Benjamin et al. (2004) based on Bleck and Benjamin (1993))

which is, at this time, the only hybrid-isentropic model in the world to be used operationally

for numerical weather prediction.

1.2.5 Isopycnal Ocean Models

While presenting his balanced atmospheric model, Bleck (1973) argued that skeptics of the

isentropic coordinate ‘should literally attempt to change their perspective and look at this prob-

lem in terms of an arbitrary boundary in a regularly spaced (x, y, θ) mesh’. This viewpoint is

even more natural in the ocean context where all models have to include lateral boundaries.

Furthermore, the interior of the ocean is dominated by adiabatic processes to a greater extent

than the atmosphere, making a material coordinate surface model even more attractive. The

presence of variable salinity S in the ocean results in a nonlinear equation of state, which gives

density as a function of T , p and S. The potential density is given by replacing T and p with θ

and the reference pressure p0 in this function, and it is this which is conserved by a fluid parcel

under adiabatic motion in the ocean, in the same way as θ is in the atmosphere.

Models with potential density as a vertical coordinate are termed isopycnal models and

show much similarity to isentropic atmosphere models. Development in the two disciplines

has, in fact, been closely related. For example, the isopycnal ocean model of Bleck and Boudra

(1986) and Bleck et al. (1992) is based on the massless layer technique introduced in the at-

Introduction 17

mosphere model of Bleck (1984), and the hybrid atmosphere model of Bleck and Benjamin

(1993) is based on the scheme introduced in the ocean model of Bleck and Boudra (1981).

It is interesting that, due to the extremely adiabatic nature of motion in the ocean interior,

and the small scale of the important eddies relative to typical resolutions, much work has been

done to ensure that the diabatic motion is not exaggerated in ocean models with non-isopycnal

coordinates. In particular, the conventional Laplacian-based diffusion operators acting on non-

isopycnal model surfaces clearly mix fluid across isopycnal surfaces, constituting spurious

diabatic motion. Common solutions are to rotate the diffusion operator to act parallel to isopy-

cnal surfaces (e.g. Redi (1982)), to diffuse isopycnal level thickness (Gent and McWilliams

(1990)), or some combination of the two (Griffies (1998)). The majority of modern ocean

models employ one or other of these techniques. In contrast, however, there have been few

attempts to stray from the standard∇q diffusion employed in most atmospheric models.

1.2.6 Summary

• There are both advantages and disadvantages to using an isentropic vertical coordinate.

However, it should be noted that in general the advantages spring from the fact that the

isentropic coordinate framework is fundamentally a natural one in which to view the

motion of the atmosphere; the disadvantages, on the other hand, are largely technical.

• Much success has been achieved with hybrid models, with only the region closest to the

surface not being modelled isentropically in modern schemes. It has been argued that

motion here is not isentropic in any case. Some benefit, however, could still be obtained

from the adaptive nature of the isentropic coordinate. This, after all, is the region where

the strongest atmospheric fronts are to be found.

• Pure isentropic models have also reached a certain maturity, with only two outstanding

problems described in the literature:

1. Noise in the Montgomery potential throughout the domain.

2. Inability to simulate surface fronts.

Both are attributed to the lack of degrees of freedom available to represent the surface

potential temperature.

Introduction 18

• Considerable effort has been made in the ocean modelling community to ensure that

parameterizations of the sub-gridscale flow do not result in spurious diabatic motions

in non-isopycnal models. Little or no effort however has been made to ensure this in

non-isentropic atmosphere models.

1.3 PV as a Prognostic Variable in Models

QGPV has frequently been used as a prognostic variable in quasi-geostrophic numerical models

of the atmosphere (e.g. Sadourny and Basdevant (1985)) but its counterpart, the Ertel PV of

equation (1.3), has rarely been used in non-geostrophic models and examples are few and far

between.

One such example is in the field of contour dynamics or contour advection. The evolution

of PV distributions, including the cascade to small scales, has been well simulated by these

techniques (e.g. Waugh and Plumb (1994)), in which imaginary particles are placed along a

contour of constant PV. These particles are then advected by a wind field, either prescribed from

observations or GCM runs or determined from the PV distribution, and their final positions de-

fine the new location of the contour. Recently, Mohebalhojeh and Dritschel (2004) performed

a trial study of the use of the contour-advective semi-Lagrangian algorithm of Dritschel and

Ambaum (1997) in a three-dimensional primitive equation model, and achieved encouraging

results.

Numerical shallow-water models have been developed which explicitly predict PV, for ex-

ample Temperton and Staniforth (1987) and Thuburn (1997) (from which the model described

here is ultimately descended). A different approach is taken by Lin and Rood (1997); while PV

is not a predicted variable in their model, a ‘reverse engineering’ procedure is used to ensure

that the PV is simulated better. The prognostic equations in u, v and thickness h are discretized

in such a way that the mass and absolute vorticity are transported consistently. The correlation

between the two fields is better preserved during transport, and so the shallow-water PV (ζa/h)

is transported more accurately.

Another shallow-water example is the model of Bates et al. (1995), which has recently

been developed into a full primitive equation model by Li et al. (2000). A variable termed

pseudo-PV is defined, and then used as a prognostic variable. Results are shown only for a

σ vertical coordinate. Future development is planned for a version built on the hybrid σ − θ

coordinate of Konor and Arakawa (1997), and the pseudo-PV has been constructed so as to

Introduction 19

reduce to the real PV in the isentropic part of the domain. The model developed in this thesis

can be viewed as an attempt to perform in a pure isentropic model what Li et al. (2000) are

attempting in a hybrid model.

The PV evolution is known to be central to the behaviour of the dominant, balanced pro-

cesses acting in the atmosphere, so it is perhaps surprising that so few attempts have been made

to improve model accuracy by directly predicting the PV. Two potential difficulties are, at least

partly, to blame for this. Firstly, if PV is predicted then the divergence must also be predicted in

any unbalanced model. The divergence equation is complicated, containing several terms other

than pure advection, and so it is difficult, and non-intuitive, to formulate numerical schemes for

it. Secondly, the PV field has to be ‘inverted’ in some way to find the winds at every timestep.

This adds to the computational expense of a model, and introduces potential for instabilities.

Robertson (1984) adapted the model of Hoskins and Simmons (1975) to predict PV on σ co-

ordinate levels. The inversion step was found to be very sensitive to the static stability, and the

model became unstable when ∂∂σ

ln θ became small. Robertson recommended changing to an

isentropic vertical coordinate to overcome this. As described above, this has its own technical

difficulties, which is certainly a further reason why attempts to formulate models predicting

PV have been limited.

1.4 Outline of the Thesis

The overall aim is to construct a global numerical model of the atmosphere solving the primi-

tive equations with PV as a prognostic variable, and with θ as a vertical coordinate throughout

the entire domain, and to compare this in detail with a conventional model. Is this approach

useful, and are the errors in entropy and PV in the conventional model significant?

Model development will be described in the first two chapters. Chapter 2 presents results

of an idealised study to determine which vertical scheme, out of many potential contenders,

best represents the linearised normal mode solutions of the continuous equations. Both hy-

drostatic and non-hydrostatic cases are considered. This work is part of a larger study across

three different vertical coordinates, published as Thuburn and Woollings (2005). The hydro-

static scheme recommended by this study is already used in the model of Gregory (1999) from

which the global model used here is constructed. This construction is described in Chapter 3,

which also includes some simple tests of the model and descriptions of known problems.

The remainder of the thesis is given over to a detailed comparison of this model with

Introduction 20

a conventional normalised pressure coordinate model (the Intermediate General Circulation

Model or IGCM). Note that from this point onwards the term ‘normalised pressure’ is used

for the quantity p/ps to avoid confusion with the isentropic density σ. Chapter 4 compares

an idealised baroclinic wave life cycle simulation performed by each model. Chapter 5 then

focusses on the IGCM life cycle and its representation of the entropy. This is motivated by two

controversial hypotheses which attribute large features of both real and modelled atmospheric

states to the effects of small terms in the global entropy budget. The extent to which filaments

of PV and gradients of θ are dissipated at small scales, as described in Section 1.1, is critical

to these theories.

Chapter 2

Vertical Grid Staggering in Isentropic

Coordinates

2.1 Introduction

When constructing a numerical model of the atmosphere there is freedom in the choice of

predicted variables used, and also how the variable set is arranged in space. Often it is desirable

to stagger one variable with respect to another, so that the two variables are not stored at exactly

the same points but at interleaving levels. In order to find the optimal scheme one has to

consider various combinations of variables and all the possible ways of staggering them, since

the best grid for one set of variables will not necessarily be the best for a different set.

There are many criteria which have to be considered when deciding which is the optimal

configuration. For example it is important that the scheme respects the conservation properties

of the continuous equation set, and that it couples well with the physical parameterisations used

in the model. Here we attempt to judge schemes by their ability to simulate normal mode so-

lutions of the continuous equations, and in particular to accurately represent their frequencies.

It is important to represent these modes accurately; the acoustic modes for correct simulation

of adjustment to hydrostatic balance, the gravity modes for correct simulation of adjustment to

geostrophic balance and the Rossby modes because of their meteorological significance.

This approach has been used previously to assess grids. For example, Randall (1994)

tested horizontal grids on their ability to simulate the dispersion relation of inertia-gravity

waves in the shallow-water equations. Fox-Rabinovitz (1994, 1996) judged vertically stag-

gered grids according to their representation of dispersion relations and group velocities. This

Vertical Grid Staggering in Isentropic Coordinates 22

work, however, is based on equations in a log(pressure) coordinate system and restricts atten-

tion to one particular choice of variables. Finally, Konor and Arakawa (2000) used a similar

technique to judge vertical discretizations of the moisture continuity equation in an isentropic

coordinate model.

The work presented here parallels studies by John Thuburn on vertical grids in a height

coordinate and a mass-based terrain-following coordinate. Results for all three studies are

given in Thuburn and Woollings (2005). Here we examine the basic atmospheric equations in

isentropic coordinates, which are linearised so that they support simple normal mode solutions

that can be found analytically. Each of the various cases of variables and grid staggering is then

solved numerically for the mode frequencies and these are compared with the analytic values.

Here we consider various choices of thermodynamic variables and vertical grid staggering.

There is also freedom of choice in the wind velocities and the horizontal grid staggering, but

this is assumed to be a separate problem independent of the vertical scheme.

2.2 The Equation Set

The normal mode analysis was performed on the non-hydrostatic compressible Euler equations

in isentropic coordinates. The usual shallow atmosphere approximation is made, resulting in

the following equation set, stated here with the friction and heating terms for completeness:

Dv

Dt= −fk× v−∇M + F, (2.1)

Dw

Dt= − 1

σ

∂p

∂θ− g, (2.2)

∂σ

∂t+∇ · (σv) +

∂

∂θ(σθ) = 0, (2.3)

Dθ

Dt=Q

π. (2.4)

Here DXDt

= X = ∂X∂t

+ u∂X∂x

+ v ∂X∂y

+ θ ∂X∂θ

, v and∇ are the horizontal components of the

velocity and the gradient operator, F is friction, Q is heating and π(p) is the Exner function.

These equations were generated following the procedure laid out by Kasahara (1974), where

the basic atmospheric equations are converted into various vertical coordinate systems. The

system here differs from the isentropic example in Kasahara’s work in that here we do not

make the hydrostatic assumption, and so we maintain the vertical velocity w (= DzDt

) as a


predicted variable. The small 2Ω cosφ Coriolis terms representing the horizontal component of

the Earth’s rotation Ω at latitude φ have been neglected, as usual, in order to retain conservation

of energy and angular momentum under the shallow atmosphere approximation. Initially we

will also neglect the latitudinal variation of the Coriolis parameter f = 2Ω sinφ, although

we include this later. The thermodynamic variables pressure p, density σ(= ρ ∂z∂θ

), height z,

Montgomery function M(= cpT + gz), total energy E(= cvT + gz) and the velocity variables

u and v are then small perturbations from this reference state, which is denoted by a subscript s

(e.g. ps, σs, zs, Ts). The equations are then linearised about an isothermal, hydrostatic reference

state, and now we neglect the diabatic heating and the effects of friction. With other subscripts

denoting partial derivatives, the simplest forms of the momentum equations are

∂u

∂t= fv − 1

σs(zsθpx − psθzx) = fv −Mx, (2.5)

∂v

∂t= −fu− 1

σs(zsθpy − psθzy) = −fu−My, (2.6)

∂w

∂t= − 1

σs(pθ + gσ) =

1

zsθ

(R

(psp0

)κp

ps−Mθ

), (2.7)

where p0 = 105 Nm−2. Note that two forms of the pressure gradient terms are given. These

will be referred to as the p-form and the M -form respectively. Each case studied has been

tested using both forms and that which gives the best results is shown here. It is anticipated

that if M is a predicted variable then the M -form will perform best. After all, one of the main

advantages of the isentropic coordinate is the fact that the horizontal pressure gradient force

can be expressed as the gradient of just one term, rather than two opposing ones. The five

possibilities considered here for the thermodynamic variables are predicted by the equations

∂p

∂t=

psκ− 1

(ux + vy +

wθzsθ

), (2.8)

∂σ

∂t= −σs(ux + vy), (2.9)

∂z

∂t= w, (2.10)

∂M

∂t=

R

κ− 1Ts

(ux + vy +

wθzsθ

)+ gw, (2.11)

∂E

∂t=

cvκ

κ− 1Ts

(ux + vy +

wθzsθ

)+ gw. (2.12)


The full equation set is formed by adding two of these thermodynamic equations to the three

momentum equations above. Here we have considered the combinations

σ, z p, z p, σ p,M p,E σ,M z,M

so there are seven thermodynamic cases. Any thermodynamic variables needed in the tendency

equations but not explicitly stored are calculated from the others using the linearised diagnostic

equations:pps

= σσs

+ TTs− zθ

zsθ, T

Ts= κ p

ps,

M = cpT + gz, E = cvT + gz.

(2.13)

Note that the temperature perturbation T on an isentrope is directly proportional to the pressure

perturbation p. These two variables are therefore considered equivalent and only the pressure is

used as a prognostic variable in this work. Note also that there are other possible combinations

of these thermodynamic variables, namely the pairing ofE with σ , z andM . In general M and

E were found to be interchangeable with negligible effect on the results, so only oneE scheme

is given as an example. The rigid boundary condition w = 0 was used in these experiments.

The non-hydrostatic isentropic equations have never been studied in depth and it is inter-

esting that z, the height of the isentrope, appears as a thermodynamic variable either explicitly

or contained in the Montgomery function M or the total energy E. In order to eliminate z in

the (p, σ) thermodynamic case the height z had to be found at each level by integrating up over

the levels beneath:

z(θ) =

∫ θ

θB

zsθ

(σ

σs+ (κ− 1)

p

ps

)dθ, (2.14)

where θB is the bottom isentrope of the model. Here z = 0 has been applied on the boundary

as a consequence of the rigid boundary condition. Similarly for the (σ,M) case, z had to be

found by integratingzθzsθ− g(1− κ)

RTsz =

σ

σs− (1− κ)

RTsM, (2.15)

which is obtained by eliminating p from (2.13).

All the coefficients in the equations (2.5) to (2.12) are independent of the variables x, y

and t and so the system has wave-like solutions of the form exp i(kx + ly − ωt) and these

are the normal modes that will be examined. Assuming solutions of this form the equations

become:

−iωu = fv − ik

σs(zsθp− psθz) = fv − ikM, (2.16)


−iωv = −fu− il

σs(zsθp− psθz) = −fu− ilM, (2.17)

−iωw = − 1

σs(pθ + gσ) =

1

zsθ

(R

(psp0

)κp

ps−Mθ

), (2.18)

−iωp =ps

κ− 1

(iku+ ilv +

wθzsθ

), (2.19)

−iωσ = −σs(iku+ ilv), (2.20)

−iωz = w, (2.21)

−iωM =cpκ

κ− 1Ts

(iku+ ilv +

wθzsθ

)+ gw, (2.22)

−iωE =cvκ

κ− 1Ts

(iku+ ilv +

wθzsθ

)+ gw. (2.23)

These equations assume a constant Coriolis parameter f so there is no β-effect in the

system, and therefore no Rossby restoring mechanism or time-dependent Rossby modes. These

modes, however, are of meteorological interest and so an attempt is now made to include a β-

effect in the equations in order to assess the ability of the various schemes to represent them.

If we simply replace the constant parameter f in equations (2.5) and (2.6) with the standard

β-plane approximation f = f0 + βy we no longer have constant coefficients, and so there

are no longer solutions of the form exp i(kx + ly − ωt). We begin by using this β-plane

approximation and taking the horizontal divergence and vertical component of the curl of the

horizontal equations (2.5) and (2.6) to give

∂δ

∂t= fξ − βu−∇2P, (2.24)

∂ξ

∂t= −fδ − βv, (2.25)

where δ = ux + vy, ξ = vx − uy and P is used to represent the pressure term 1σs

(zsθp− psθz)or M . With f and β now held constant, wave solutions proportional to exp i(kx+ ly−ωt) are

possible. Following White (2002), βu and βv need to be replaced by βudiv and βvrot (i.e. the

rotational and divergent components only) in order to retain energy conservation under this

approximation. The neglect of βurot and βvdiv are justifiable on the grounds of scale analysis.

For example, since β is of the same size as f/a (where a is the radius of the Earth), the βurot

term in (2.24) is much smaller than the fξ term because the length scale on which the velocity

varies is much smaller than the radius of the Earth. For solutions of this form equations (2.24)


and (2.25) become

−iω(iku+ ilv) = f(ikv − ilu)− βudiv − (k2 + l2)P, (2.26)

−iω(ikv − ilu) = −f(iku+ ilv)− βvrot. (2.27)

These equations are then rearranged to give the u and v tendencies:

∂u

∂t= −iωu = fv +

ikβ

k2 + l2u− ikP, (2.28)

∂v

∂t= −iωv = −fu+

ikβ

k2 + l2v − ilP. (2.29)

Here the identities ikurot+ ilvrot = 0 and ikvdiv− iludiv = 0 have been used; i.e. the rotational

component is non-divergent and the divergent component is non-rotational. Equations (2.28)

and (2.29) are the same as (2.16) and (2.17) but with the two linear β terms added and we use

these as the prognostic equations in the horizontal. We have maintained the linear normal mode

structure but also included a β-effect. Rotating the wave-vector (k, l) effectively rescales the

magnitude of the β-effect. With l = 0 this is a maximum, so here attention is restricted to this

case only.

Although it is not the full β-plane approximation, this device modifies the equations in

the same way as the full approximation by adjusting the horizontal velocity tendencies, so is

equivalent for the purposes of judging vertical grid staggerings. With these β terms included

the equations can now be used to examine realistic normal mode behaviour for Rossby modes

as well as for acoustic and gravity modes.

For each case the complete equation set consists of five prognostic equations in five pre-

dicted variables: the horizontal momentum equations with the β-effect (2.28, 2.29), the vertical

momentum equation (2.18) and then the appropriate pair of thermodynamic equations (2.19-

2.23). The rigid boundary condition w = 0 is imposed on both upper and lower boundaries.

2.3 Analytical Solution

If we restrict ourselves to an isothermal reference state (i.e. Ts = constant) then the dispersion

relation can be found analytically. The method used here for judging discrete vertical schemes

is to compare the normal mode frequencies they predict with those given by this analytical


solution. Since the mode frequencies are the same in isentropic coordinates as in height co-

ordinates, we can use the dispersion relation as derived in height coordinates in Thuburn and

Woollings (2005). An outline of this derivation is given here as this provides an insight into

the nature of the normal modes.

This derivation starts from the same basic equations but expressed in z coordinates with

predicted variables u, v, w, θ and p. In these coordinates the reference static stability is N 2s =

gθsz/θs, and with the assumption of an isothermal reference state both this quantity and the

reference sound speed squared, c2s = RTs/(1− κ), are constants. The variables u, v, w, and θ

are eliminated from the five equations to leave one equation in p:

(∂

∂z+ A

)(∂

∂z+B

)p+ Cp = 0, (2.30)

where the coefficients are

A = N2s

g, B = g

c2s,

C =

(1c2s

+(K2 + kβ

ω

) [(f − ilβ

K2

)2 −(ω + kβ

K2

)2]−1)

(ω2 −N2s ) .

(2.31)

The boundary condition w = 0 at z = 0 and z = D (the depth of the domain) becomes, after

elimination of the other variables,

(∂

∂z+B

)p = 0. (2.32)

We now look for solutions of the form p = p exp(− z2H

) where 1H

= (A+B) =(N2s

g+ g

c2s

).

Equations (2.30) and (2.32) become

(∂2

∂z2+(C − Γ2

))p = 0 (2.33)

with(∂

∂z+ Γ

)p = 0 on z = 0, D, (2.34)

where Γ = B−A2

= 12

(gc2s− N2

s

g

). These have solutions of the form

p = a exp(ikzz) + b exp(−ikzz), (2.35)


where

k2z = C − Γ2. (2.36)

Applying the boundary conditions (2.34) imposes the following restrictions on the values of

kz:

a(ikz + Γ) + b(−ikz + Γ) = 0, (2.37)

a(ikz + Γ) exp(ikzD) + b(−ikz + Γ) exp(−ikzD) = 0, (2.38)

and this pair of equations in a and b can only be solved if

(Γ2 + k2z)(exp(ikzD)− exp(−ikzD)) = 0. (2.39)

This condition has to be satisfied by any solutions of the form considered here, and any value of

the frequency ω that satisfies this condition for some vertical wave number kz is the frequency

of a normal mode of the system with that vertical wave number.

There are two possibilities. Firstly, we could have (Γ2 + k2z) = 0, and by (2.36) this is

equivalent to C = 0, i.e.

(1

c2s

[(f − ilβ

K2

)2

−(ω +

kβ

K2

)2]

+

(K2 +

kβ

ω

))(ω2 −N2

s

)= 0. (2.40)

Solutions with ω2 = N2s correspond to p ≡ 0 and are not physically interesting solutions, but

the other roots given by

ω

[(f − ilβ

K2

)2

−(ω +

kβ

K2

)2]

+ c2sK

2

(ω +

kβ

K2

)= 0 (2.41)

are physically significant and this is the dispersion relation for the external modes. This is a

cubic equation in ω, the three roots corresponding to two acoustic modes and one Rossby mode.

In the dispersion plots which follow, the external modes are those with zero wavenumber.

Secondly, equation (2.39) is also satisfied if (exp(ikzD) − exp(−ikzD)) = 0, which in

turn is satisfied if kz = mπ/D for some positive integer m. Substituting this expression for kz

and the definition of C (2.31) into equation (2.36) gives a quintic equation for ω. The roots of

this quintic are the internal mode frequencies, and for every positive integer m there are five

solutions which correspond to two acoustic modes, two inertia-gravity modes and one Rossby

mode.


The roots of these equations are calculated numerically using a standard iterative solver to

give the analytic values of the normal mode frequencies which the different discretizations are

tested against.

2.4 Numerical Solution

The values of the predicted variables are taken at either the full-levels of the model or at the

interleaving half-levels. For each case the equations are discretized by replacing any vertical

derivatives with a centred difference over the adjacent levels. For example, if the vertical

derivative of pressure pθ is needed on half-levels the difference is taken over the pressure values

from the full-levels immediately above and below the half-level, i.e. (pθ)k+1/2 =pk+1−pkθk+1−θk ,

where the full-levels k and k + 1 are separated by the half-level k + 1/2. A standard linear

interpolation is used to transfer variables between half-levels and full-levels when necessary, by

taking the average of its values on the adjacent levels (e.g. pk+1/2 = (pk +pk+1)/2). Intuitively

we would expect a good grid to be one which minimises the amount of averaging necessary, or

more specifically the averaging necessary in the largest and therefore most important terms in

the equations.

The boundary condition used is simply that the vertical velocity w is zero at both the top

and bottom boundaries of the model. In order to apply this condition it is necessary to store w

values at the boundaries, and for this reason both the boundaries are half-levels and w is always

stored on half-levels. The results shown here were generated on a grid of 21 half-levels with

20 interleaving full-levels. Each of the other four variables can either be stored on half-levels

(i.e. unstaggered with respect to w) or on full-levels (staggered with respect to w), so that

for every choice of thermodynamic variable pairing there are 16 possible choices of vertical

staggering. However, it is clear from the equations that any attempt to store the velocities

u and v on different levels will only add to the amount of averaging involved with no extra

advantage. With the additional constraint that u and v are stored on the same levels there are

eight staggering cases for each of the seven thermodynamic cases, making 56 different cases

to examine in total.

When the thermodynamic variables are p and σ it is necessary to evaluate the integral

given in equation (2.14) for the value of z. This is done by summing the value of the integrand

in the levels below, and a switch is built in to the program code to do this by summing either

half-level values or full-level values. Often it is necessary to add on a contribution from half


a level either at the top or bottom of the sum, or at both. For example, summing full-level

values up to full-level i will give the value of the integral at the next half-level up, so if z is

required on full-levels an extra contribution will have to be added to get the value of the integral

at full-level i + 1, and this contribution represents the final half a level of the domain of the

integral. These contributions were approximated by taking half of the relevant complete level

contribution, following ideas in Simmons and Burridge (1981).

For the (σ,M) cases z has to be obtained from equation (2.15). This equation is discretized

at full level k with (zθ)k =zk+1/2−zk−1/2

θk+1/2−θk−1/2and zk =

zk−1/2+zk+1/2

2, and then re-arranged to give

zk+1/2 from zk−1/2. It can then be integrated up from z = 0 on the lower boundary to give z at

all half-levels.

A further switch is built in to the program to prescribe whether the model layers are equally

spaced in θ or in height. In the latter case the coordinate levels are still isentropic surfaces but

their θ values are chosen so that the levels are approximately evenly spaced in z. This configu-

ration did give better results, with most of the cases capturing a few more modes correctly than

with the layers evenly spaced in θ, and all the results shown here use this coordinate system. It

is perhaps to be expected that this system gives better results as the solutions we are trying to

capture have p proportional to exp(ikzz), i.e. there is a wave-like variation with z rather than

θ.

Once the equations are discretized they can be solved to give the normal mode structures

and frequencies. A state vector s is formed of all the predicted variables at all levels, and then

the full system of equations can be written as

−iωs = As, (2.42)

where the elements on the left-hand side are the variable tendencies (since ∂s∂t

= −iωs for

normal modes) and those on the right-hand side are the forcings. The elements of the matrix A

are simply the coefficients of the linear prognostic equations.

This is then an eigenvalue problem, where the eigenvectors sj are the state vectors of the

normal modes and the eigenvalues are −iωj , where ωj are the corresponding normal mode

frequencies. The problem is solved for both mode structures and frequencies using a standard

NAG library routine. The frequencies can then be compared with the analytic values and the

structure of any mode with an unexpected frequency can be examined.

The following parameter values were chosen: domain depth = 10 km, β = 1.619 ×


10−11 s−1m−1 (corresponding to 45N), reference state temperature Ts = 250 K and horizontal

wavelength k = 1000 km.

2.5 Results

In general the accuracy of the vertical schemes was very varied. Many of the configurations

represented the dispersion relation extremely badly, with most of the modes either missing,

misrepresented or unstable. Others, however, gave very good results, with the acoustic and

gravity modes captured almost perfectly and just a slight deviation in the frequencies of the

Rossby modes.

As anticipated the schemes performed better using the M -form of the pressure gradient

terms if M was a prognostic variable. The results shown for all other cases used the p-form.

A numerical mode is classed as unstable if its frequency has a significant imaginary com-

ponent, specifically if |Im(ω)| ≥ 0.1 × |Re(ω)|. As the modal solutions are proportional to

exp(iωt) any significantly large imaginary part of ω will cause the solution to grow or decay

exponentially. Any configuration that has unstable modes cannot be recommended, simply be-

cause they are not physically realistic and are also likely to damage the conservation properties

of the model.

It is also unacceptable for a scheme not to capture one or more of the modes, as this shows

that it is unable to handle the equations accurately enough to represent these structures. As

an example, Figure 2.1 shows the dispersion plots for a staggered grid that has been unable to

correctly capture most of the gravity and Rossby modes, and those that are shown are mostly

unstable. The scheme shown in Figure 2.2 has performed slightly better but still has both

missing and unstable modes and therefore is not considered viable. Also note that for the

higher acoustic modes the variation of frequency with mode number has the wrong sign, which

will result in a simulated vertical group velocity in the wrong direction (e.g. Acheson (1990)).

Of the 56 configurations examined here there were only 12 with no missing or unstable

modes. These 12 cases are considered viable options for a vertical scheme, though there is still

a considerable hierarchy within this group. There is at least one viable configuration for all

choices of thermodynamic variable pair. These viable cases are laid out in Table 2.1 using the

notation (w..., ...) to indicate which variables are stored on half-levels with w and which are

staggered with respect to w.

Cases 42, 55, 43, 51 and 52 (e.g. Figures 2.3 and 2.4) simulate both the acoustic and grav-


Figure 2.1: Frequency ( s−1) plotted against vertical mode number for normal modes of aresting, isothermal, non-hydrostatic atmosphere. The three external modes are the points withvertical mode number 0. The other five sets of modes are, in order of ascending frequency,the internal Rossby, gravity and acoustic modes. Diamonds indicate analytical values. Crossesand asterisks indicate numerical values for the grid and variable configuration shown in theschematic below the dispersion plots. Asterisks indicate that the numerical mode is unstable.This particular vertical scheme has represented the normal modes extremely badly.


Figure 2.2: Dispersion curves, as in Figure 2.1, but for the configuration shown. Note thatthere are several modes missing from the numerical solution and that several of the Rossby andgravity modes that have been simulated are unstable.


Category Case ID Grid

1 42 (wz, uvM)M

55 (w, uvσM)Ma

2 44 (wz, uvp)39 (wp, uvM)Mb

43 (wz, uvσ)p

3 51 (w, uvpz)52 (w, uvpσ)53 (w, uvpM)54 (w, uvpE)56 (w, uvzM)

5 23 (wuv, pz)p

26 (wuv, pE)

Table 2.1: The configurations of the 12 viable cases arranged in the categories defined in thetext. Superscripts M and p indicate that the grid only qualifies for this category if the M - orp-form of the equations is used, respectively. Superscript a indicates that this scheme qualifiesfor category 1 only if one degree of freedom in σ is suppressed. Similarly, b indicates that theboundary values of the thermodynamic variables must be suppressed.

ity modes extremely well and slightly overestimate the last half of the Rossby modes. The

remaining seven viable cases each represent almost all the modes very well, only underesti-

mating the higher Rossby modes. Examples of these are given in Figures 2.6 and 2.5.

There are no obvious criteria which guarantee accurate representation of the modes, though

schemes which minimise the averaging in the largest terms of the equations generally perform

best. As in the companion coordinate studies, it appears that the Rossby modes are not well

represented unless the gravity modes are, and that these are not well represented unless the

acoustic modes are. This reflects the greater complexity of physics involved in Rossby waves

compared to gravity waves, and gravity waves compared to acoustic waves.

The dispersion plots enable us to eliminate configurations which have missing or unstable

modes, but not to identify those which have computational modes. These are modes of the

discrete system which do not correspond to any mode of the continuous system, and so have

no physical meaning. Computational modes are often two-grid oscillations in one or more of

the variables of the form ηk = (−1)k∆η for the general variable η, where ∆η is a constant

in the simplest case, but not in general. The interpolated half-level values ηk+1/2 etc. then

all vanish, and so modes like this can be invisible to the rest of the system. In this way they

are allowed to exist by the averagings involved, but additionally they can only occur if there

are extra degrees of freedom in the discrete system. Every discrete variable (e.g. ui, vi, pi)

corresponds to one row of the state vector and therefore one degree of freedom. For every


degree of freedom there is a solution to the matrix equation, so if there are more degrees of

freedom than there are resolvable modes of the continuous system it is possible that one of the

extra solutions is a computational mode. In order to find these modes it is necessary to compare

the number of eigenmodes with the number of degrees of freedom of the scheme, and then to

examine the structure of any modes which are not accounted for.

Modes such as these, although not directly interfering with the dynamics of a model, can

affect conservation properties and physical parameterisations. A well-known example of a

computational mode of this form is that which occurs when using the Lorenz grid in pressure

coordinates (see e.g.Arakawa and Moorthi (1988), who also discuss how the same extra degree

of freedom is indirectly responsible for the spurious amplification of short waves).

Some of the configurations contain computational modes which have zero frequency and

are just z, p and σ curves, resembling exponentials, with all other variables and tendencies

practically zero. This simply represents a time-independent shift in height of the isentropic

levels. For simplicity we have applied only the boundary condition w ≡ 0 on the boundaries,

whereas in operational use the additional condition that the height perturbation z is also zero

would be applied (as z must be constant to have w = 0). For each extra condition a degree

of freedom is removed and modes such as these are impossible. They are therefore ignored

in the results given here, although this can only be done in cases where z is stored on half-

levels so that this extra condition can be applied on the boundaries. Some configurations have

computational modes consisting of oscillations in z, p and σ which also have z 6= 0 on the

boundaries, and these are similarly ignored.

All of the other computational modes consist of a two-grid oscillation in two or more

of the variables. As an example of these modes consider case 23 (shown in Figure 2.6). On

examining the dispersion plots this configuration appears to give good results, with only the last

three Rossby mode frequencies significantly underestimated. However, this configuration has

three computational modes, and therefore cannot be recommended. One of the computational

modes is a two-grid oscillation in u, v, p and z which cancels in the discrete equations so that

all the tendencies are zero. The frequency of this mode is also zero to within numerical error

and so it does not appear on the dispersion plot. The other two computational modes are two-

grid oscillations in u, v and their tendencies, with all other variables and tendencies effectively

vanishing. They agree closely in both frequency and mode number with the mode number 18

gravity waves and so overlay these modes and appear as single points on the dispersion plots.

In the table the schemes are arranged in categories as in Thuburn and Woollings (2005)


which are outlined here:

• Category 1: Optimal. No computational modes and distinctly better dispersion rela-

tions than all other schemes.

• Category 2: Near-optimal. No computational modes and still reasonably good disper-

sion relations.

• Category 3: Single zero-frequency computational mode. Generally a two-grid oscil-

lation in one or both of the thermodynamic variables.

• Category 4: Inertial frequency decoupled modes. The only isentropic grid in this

category is (wuvz,M)M and since this does not qualify as a viable scheme this category

is not described here.

• Category 5: The rest.

Out of all the viable cases there are only five which can qualify for categories 1 and 2

(although they do all have one or two of the easily removable z modes described above). Ex-

amples are shown in Figures 2.3 and 2.5. These therefore are the configurations that can be

recommended as best representing the dispersion relations of the normal modes. Cases 42 and

55 clearly have the best numerical dispersion relations. These are practically identical so only

one is shown here (case 55 in Figure 2.4).

The recommendation of two out of these five grids is given under certain conditions. Case

55 supports a computational mode in which all prognostic variables are zero except for the

value of σ nearest the upper boundary. This is the boundary at which z is not defined as zero in

the integration of equation (2.15) and the computational mode corresponds to a non-zero value

of z there. This σ value has no effect on any of the other prognostic variables and is a redundant

degree of freedom. The computational mode is then trivially removed by dropping this degree

of freedom. Similarly, case 39 supports two trivial computational modes since the p values

stored on the two boundaries are never used in the equations; dropping these two degrees of

freedom removes the computational modes.

As mentioned in Section 2.4, when the variable pair is p and σ there is an option as to

whether the integral in equation 2.14 is evaluated by summing half-level values or full level

values. Of the two terms in the integrand it is the σ term which is dominant and the results

show that, in general, if σ is stored on half-levels it is best to sum over half-levels, and if it


Figure 2.3: Dispersion curves, as in Figure 2.1, but for the configuration shown. This is one ofthe five configurations recommended as best representing the normal mode frequencies. Thereare no unstable, missing or computational modes.


Figure 2.4: Dispersion curves, as in Figure 2.1, but for the configuration shown. This scheme ispotentially the best of all the configurations recommended here if a trivial computational modeis removed. Case 42 has an identical dispersion relation; these two schemes best represent thenormal mode frequencies out of all 56 schemes tested.


Figure 2.5: Dispersion curves, as in Figure 2.1, but for the configuration shown. This is anotherof the configurations recommended as best representing the normal mode frequencies. Thepattern of staggering in this case is identical to that in Figure 2.3, the only difference is that thiscase has p as a thermodynamic variable while the other has σ.


Figure 2.6: Dispersion curves, as in Figure 2.1, but for the configuration shown. From theseplots this would appear to be a good scheme, as only the last three Rossby modes are signifi-cantly wrong. However, this case has three computational modes which are invisible here andso this configuration cannot be recommended.


is stored on full-levels to sum over these. The only exception to this rule is the case with p

and σ on full-levels and u on half-levels. In this case it is best to sum over the half-levels

as the discrete integral then consists only of complete layer terms of thickness ∆θ, and it is

not necessary to add any approximated half-layer terms of thickness ∆θ/2 at either end of the

domain.

In fact, case 52 is the only viable configuration with p and σ as the thermodynamic vari-

ables. In this configuration σ is stored on full-levels and so the integral is evaluated by sum-

ming the full-level values. There is one computational mode, with p, σ, and therefore also the

implied z, resembling exponentials. On the upper boundary θT we have, by equation (2.14),

zt(θT ) =

∫ θT

θB

zsθ

(σtσs

+ (κ− 1)ptps

)dθ

=

∫ θT

θB

wθdθ using equations 2.8 and 2.9

= w(θT )− w(θB) = 0, (2.43)

i.e. z is constant on the upper boundary. This is satisfied for the computational mode because

its frequency is zero, so there is no time dependence. This mode, however, has a constant value

of z which is not zero, due to the exponential vertical structure. It could, then, theoretically be

removed by imposing boundary conditions such that z = 0. In practise this may be non-trivial

so the scheme is placed in category 3.

All the results shown here are for modes with a horizontal wavelength of 1000 km in a

10 km deep atmosphere. To demonstrate that our conclusions are robust and can be used in a

range of applications, the same test was run with these parameter values altered. In order to

demonstrate the application to features of different horizontal scales the wavelength was set at

100 km and at 4000 km, and for stratospheric applications the vertical domain was extended

to 50 km and then to 80 km. In all these cases the results proved to be robust, in that the five

recommended configurations in categories 1 and 2 remained the best and no other configura-

tions became viable. The dispersion plots for all cases were simply stretched or compressed

versions of their originals.


2.6 A Hydrostatic System

In the sections above, the non-hydrostatic equations in isentropic coordinates were exam-

ined for direct comparison with the studies in other coordinate systems. However, the non-

hydrostatic isentropic equations are currently not widely used as most applications assume

hydrostatic balance. For the linearised system in isentropic coordinates this is simply the as-

sumption that the w tendency ∂w∂t

can be neglected in the vertical momentum equation (2.7).

This results in a smaller system with only three predicted variables, since the vertical momen-

tum equation becomes a diagnostic linking p and σ. To compare vertical grid staggerings for

hydrostatic models the experiment was repeated using the following equation set in the three

variables u, v and σ:

∂u

∂t=∂M

∂x, (2.44)

∂v

∂t=∂M

∂y, (2.45)

∂σ

∂t= −σs

(∂u

∂x+∂v

∂y

), (2.46)

whereM(= cpT+gz) is the Montgomery potential as before. The diagnostic equation obtained

from the vertical momentum equation is

σ∂ps∂θ

= σs∂p

∂θ(2.47)

so that, since it is the vertical derivative of p that is equated to σ, a grid with p on half levels,

for example, is equivalent to a similar grid but with σ on full levels.

Two different vertical schemes were examined, one with the horizontal velocities u and v

staggered with respect to the density σ, and one without. The unstaggered grid is equivalent

to a grid predicting the pressure p staggered with respect to the horizontal velocities u and v.

This scheme is used in the models of UCLA (Konor and Arakawa (1997), Hsu and Arakawa

(1990)), Colorado State University (Randall et al. (2000)), The University of Wisconsin (Pierce

et al. (1991), Uccellini et al. (1979)) and in the stratosphere model of Gregory (1999), and is of-

ten referred to as the Charney-Phillips grid in isentropic coordinates. This grid has recognised

advantages when using pressure coordinates. In both Fox-Rabinovitz (1996) and the compan-

ion study in height coordinates (Thuburn and Woollings (2005)), the Charney-Phillips θ-p grid


is found to have the best simulated dispersion relation, and the grid is also recommended by

Arakawa and Moorthi (1988) for its lack of computational modes. At least one of the hybrid

coordinate models above (Konor and Arakawa (1997)) uses this grid for its advantages in the

region of normalised pressure coordinates (p/ps) near the ground.

An alternative, used by Zhu et al. (1992) following Simmons and Burridge (1981), is the

Lorenz grid in isentropic coordinates. This model predicts u, v and temperature T on the same

levels. Due to the equivalence of T and p mentioned in Section 2.2 this can be compared with

grids predicting p on the same levels as the velocities, i.e. with σ staggered from them. This

therefore corresponds to the staggered grid in our hydrostatic test.

In the original experiment a rigid boundary condition (i.e. w = 0) was applied at both

the top and bottom boundaries. This is harder to apply in the current system where w does not

appear in the prognostic equations. In order to rewrite this boundary condition as a condition on

σ and p, the non-hydrostatic equations for σ and p (equations (2.19) and (2.20)) were combined

to eliminate the horizontal velocities, and then this equation was integrated over the entire

domain. The boundary conditions w = 0 at the limits of integration were then applied, and the

result is an integral constraint on the values of p and σ throughout the entire domain:

(1− κ)

∫

D

∂zs∂θ

p

psdθ =

∫

D

∂zs∂θ

σ

σsdθ. (2.48)

The procedure used in this study to evaluate the tendencies ∂u∂t, ∂v∂t

and ∂σ∂t

is as follows.

Define a relative pressure p′ by p = pT + p′, where pT is the pressure at the top of the domain.

At first pT is set to zero. Then the relative pressure p′ at all other levels is calculated from the

σ values by integrating equation (2.47) down from the top of the model. In both of the vertical

schemes σ is stored on full levels, so p is then naturally obtained on half-levels. Equation (2.48)

becomes an expression for pT :

(1− κ)

∫

D

1

ps

∂zs∂θ

dθ pT =

∫

D

∂zs∂θ

(σ

σs− p′

ps

)dθ (2.49)

in which both integrals can be evaluated. This gives pT , and the full pressures are given by

p = pT + p′.

In order to calculate the Montgomery potential M from the pressure we use the diagnostic

relation∂M

∂θ= Π = cp

(p

p0

)κ(2.50)


which, after linearising about the reference state, becomes

∂M

∂θ= κ

∂Ms

∂θ

p

ps. (2.51)

M is defined on the bottom boundary (denoted by a subscript B) by MB = RTspBps

and then

can be found at all levels by integrating up following equation (2.51). We have p stored on

half-levels and so we evaluate this integral by summing these half-level values, naturally giv-

ing M on full-levels. Up to this point the two schemes tested are identical. For the unstaggered

(Charney-Phillips) grid the calculation is now complete as M is needed on full-levels to evalu-

ate the horizontal tendencies. In the staggered (Lorenz) grid however the horizontal velocities

are stored on half-levels so the M values need to be interpolated from the full-levels first.

The Lorenz grid appears, therefore, to have two extra sources of error that do not arise

when using the Charney-Phillips grid: the M field has to be interpolated in order to evaluate

the horizontal equations, and the u and v fields will have to be interpolated to evaluate the

σ equation. We would therefore expect the Charney-Phillips grid to perform better than the

Lorenz grid in our experiment.

Dispersion plots for the two schemes are shown in Figures 2.7 and 2.8. The Charney-

Phillips grid (Figure 2.8) does indeed simulate the normal mode frequencies better than the

Lorenz grid (Figure 2.7), which significantly underestimates the frequency of the higher Rossby

modes. Both of the schemes are viable, however, in that all the modes are captured and are

stable. (Note that as there are only three degrees of freedom in the continuous equation set

there are only three sets of modes; the eastward and westward gravity modes and the Rossby

mode.)

Furthermore, as the Lorenz grid has the velocities u and v on half-levels there are two

extra degrees of freedom in the model (there are 62 in the discrete equation set but only 60

resolvable modes). These permit the existence of two computational modes, both of which are

two-grid waves in u and v. On the dispersion plot these overlay the gravity waves with mode

numbers 14 eastward and 18 westward. The Charney-Phillips grid, in contrast, introduces no

extra degrees of freedom into the system, and there are no computational modes. On these

grounds we conclude that, for a hydrostatic system, the Charney-Phillips grid is preferable to

the Lorenz grid for the simulation of vertical normal modes in isentropic coordinates.


Figure 2.7: Dispersion curves, as in Figure 2.1, but for the configuration shown, which isreferred to here as the Lorenz grid for a hydrostatic system in isentropic coordinates. There aretwo computational modes whose frequencies appear exactly overlaying the eastward gravitymode number 14 and westward gravity mode number 18.


Figure 2.8: Dispersion curves, as in Figure 2.1, but for the configuration shown, which isreferred to here as the Charney-Phillips grid for a hydrostatic system in isentropic coordinates.There are no computational modes and all the frequencies are simulated well, so this grid isrecommended over the Lorenz grid.


2.7 Discussion

In this study we have identified two optimal schemes for the accurate simulation of the disper-

sion properties of the continuous non-hydrostatic system, though one of these (case 55) comes

with a proviso. Further to this, the three schemes in category 2 can be recommended for hav-

ing relatively good dispersion properties and supporting no damaging computational modes.

The remaining seven schemes listed in the table all support one or more potentially damaging

computational modes.

All five of the category 1 and 2 configurations represent the acoustic and gravity modes

extremely well. The only deviations from the analytic values are in the higher Rossby modes.

It could be argued that, if a background horizontal wind is added to the system, the observed

frequencies would be dominated by the Doppler shift, and so these errors would not be sig-

nificant. However, it could also be argued that the vertical group velocity of these higher

Rossby modes would be misrepresented, and this could affect the simulation of features such

as the Quasi-Biennial Oscillation, which is thought to be partly driven by equatorially trapped

Rossby-gravity modes. This issue deserves further consideration. Cases 42 and 55 (Figure 2.4)

represent all the modes very well, even these higher Rossby modes, and are therefore recom-

mended as the optimal non-hydrostatic schemes for normal mode representation in isentropic

coordinates. The mode frequencies predicted by these schemes are as accurate as those pre-

dicted by the best configuration in other coordinate systems tested in Thuburn and Woollings

(2005).

On examination of the underlying equations it is apparent that the category 1 and 2 schemes

are indeed sensible choices of grid staggering for the variables given. When z is one of the ther-

modynamic variables it seems sensible to store it on the same levels as w because of the close

relationship between the two. Both the viable (p, z) and (σ, z) grids have this layout. When

either p or σ are variables they appear staggered with respect to w, which is convenient because

often their derivatives appear in the tendency equation for w, or vice-versa. Finally p and σ

appear on the same levels as the horizontal velocities, and in the tendency equations for u and

v it is always p and σ that are required rather than their vertical derivatives.

The non-hydrostatic equations studied here are interesting in that they naturally predict z as

a prognostic variable. Several ways were found to eliminate z from the equations. With p and σ

as prognostic variables the height z is calculated at every timestep from the diagnostic integral

in equation (2.14), and one of the resulting schemes qualifies for category 3 as described in


section 2.5. Another method is to combine z with the temperature T in either the Montgomery

potential M or the total energy E, but none of the resulting schemes are recommended. In

both cases very promising dispersion plots (similar to that in Figure 2.5) are obtained if all

variables are staggered with respect to w. However, these configurations each contain one

computational mode of zero frequency consisting of two-grid waves in the thermodynamic

variables. If these computational modes could be removed somehow then the two schemes

would be valid options.

Elimination of z using equation (2.15) gives the optimal scheme (w, uvσM)M (case 55)

with its proviso that the redundant degree of freedom in σ be removed. This would be easy

in the linear normal mode calculations presented here but might not be in a fully nonlinear

model. Furthermore, the need to solve the nonlinear, nonlocal equation (2.15) at each timestep

would add to the cost of such a scheme. In spite of these potential drawbacks this scheme holds

significant promise, as follows.

It should be stressed that the ability of a vertical scheme to simulate the dispersion relation

is only one of several criteria which should be considered when choosing such a scheme. Other

features are desirable, such as the ability to model advection, to support any physical parame-

terisations and to satisfy conservation laws. One potential benefit of isentropic coordinates is

that, in the absence of diabatic processes, air parcels move along coordinate surfaces and so it

is easier to achieve certain conservation properties in models, such as conservation of mass and

of potential vorticity. The potential offered by case 55 is that it might, in addition, enable other

conservation properties to be satisfied easily, since its mass variable σ is stored at the same lo-

cations as its wind velocities. This feature makes it significantly easier to conserve properties

such as energy and angular momentum which involve a product of density and velocity. When

using height or pressure coordinates it has been difficult to find a scheme which gives both

good dispersion properties and good conservation properties; this scheme has the potential to

do both.

Of the two hydrostatic schemes assessed in Section 2.6 the Charney-Phillips grid (Fig-

ure 2.8) clearly simulates the normal modes best, a reassuring result as this scheme is widely

used in existing isentropic models. Again, the mass and wind variables are stored at the same

levels, with the potential to easily satisfy conservation laws.


2.8 Summary

In this chapter a wide range of vertical grid schemes in isentropic coordinates have been as-

sessed on their ability to represent the normal modes of the linearised system, in particular the

dispersion relation. The principal findings are outlined below.

• Some general guidelines for constructing vertical grids have emerged. In general, schemes

which minimise the averaging required in the dominant terms of the equations will per-

form well. The number of degrees of freedom in the discretized system is an accurate

guide to the existence of computational modes.

• The non-hydrostatic equations in isentropic coordinates are most readily formulated with

the height z as a prognostic thermodynamic variable. All successful schemes without z

as a prognostic variable have to derive it using a non-local diagnostic equation.

• For the non-hydrostatic system the scheme (wz, uvM)M is optimal for representation of

the mode frequencies and supports no computational modes. The scheme (w, uvσM)Ma

is also optimal under a minor proviso, and has the potential to offer both good dispersion

and conservation properties. The isentropic coordinate is the only one in which such a

scheme has been found. Both of these schemes have dispersion properties as good as

any scheme examined in other coordinate systems.

• Three further non-hydrostatic schemes are identified which support no computational

modes and have dispersion relations only slightly worse than the two optimal schemes.

• For the hydrostatic system the Charney-Phillips grid (−, uvσ) has the best dispersion

properties and also supports no computational modes. This scheme is used in the isen-

tropic model developed in the following chapter.

Chapter 3

The Isentropic Model

3.1 Introduction

The isentropic model used here is an extension to the ground of the stratosphere model of

Gregory (1999), itself a multi-level extension of the shallow-water model of Thuburn (1997).

The model solves the hydrostatic primitive equations on a full global domain and, for the sim-

ulations presented in this thesis, is adiabatic so that DθDt

= 0. The vertical levels are labelled

with the generalised vertical coordinate η which is identically equal to the potential tempera-

ture θ = T (p/p0)−κ at all points above the ground. Each level retains its η coordinate label

after it has intersected the ground, though this should no longer be thought of as the potential

temperature.

The three prognostic variables are the potential vorticity Q, the divergence δ and the isen-

tropic density σ, defined by

Q = f+ξσ, σ dθ = ρ dz, δ = ux + vy (3.1)

where f is the Coriolis parameter 2Ω sinφ, ξ is the relative vorticity vx − uy and ρ is the

density in height (z) coordinates. In this framework the primitive equations are particularly

simple, with PV and σ evolving purely by advection:

σt = −∇.(σv), (3.2)

(σQ)t = −∇. [σQv + k×X] , (3.3)

The Isentropic Model 51

δt = −∇.[σQk× v +∇

(M +

1

2v2

)−X

], (3.4)

where v is the horizontal velocity, k is the unit vertical vector, M = cpT + gz is the Mont-

gomery potential and X represents Rayleigh friction at upper levels. The friction is used to

reduce reflection of gravity waves from the upper boundary; it is optional and is not needed

with the relatively low model top used in the simulations in this thesis.

By formulating the model with (3.3) as a prognostic equation it is hoped that the La-

grangian conservation of PV will be improved. Note, however, that it is σQ rather than Q

which is predicted by this equation, so that the model does not attempt to formally conserve

the PV of an air parcel. The correlation between σQ and Q can change during advection, be-

cause in each case the advective flux is dependent on an upstream interpolation of the respective

field (see Section 3.4). In fact, the PV field is smoothed because of this upstream interpolation,

but it should be remembered that, in any case, perfect Lagrangian conservation is not possible

in any discrete Eulerian model.

Finite difference versions of these equations are solved on a hexagonal-icosahedral hori-

zontal grid, an example of which is shown in Figure 3.1. The grid is nested so that a dense grid

is formed by sub-dividing the elements of a sparse grid, and then projecting the new elements

onto the sphere. For example, a regular hexagon is comprised of six equilateral triangles. Each

of these triangles is sub-divided into four by connecting the central points of its edges, and

new hexagons, each a quarter of the area of the original, can be formed. The first grid in the

hierarchy is the icosahedron; sub-dividing this results in a grid with a pentagon at the centre

of each of the original pentagons, surrounded by five hexagons. Subsequent grids all have four

times as many hexagons as the previous, but still exactly 12 pentagons.

As a result of the projection onto the sphere, the grid cells are not regular hexagons or

pentagons. This introduces error in the horizontal differential operators when the straight line

between the centres of two adjacent cells is not perpendicular to the edge between them. To

overcome this the grid is ‘tweaked’ using the method of Heikes and Randall (1995) to ensure

that all operators converge uniformly to their continuous counterparts in the limit of infinite

resolution. The model can still suffer from small amplitude noise because truncation errors

invariably reflect the structure of the grid. This is most noticeable at the pentagons, whose

surface area is slightly smaller than any of the hexagons. This gives a wavenumber 5 shape to

the noise due to the pentagonal shape of the original icosahedron. Recently an optimisation has

been proposed by Tomita et al. (2002) to ensure a more homogeneous grid; this optimisation


has not been implemented here. The magnitude of the noise is small, and the only reason for

concern is the possible triggering of spurious Rossby and gravity waves of wavenumber 5 (see

Appendix A).

In the horizontal, the three prognostic variables are stored at box centres and the winds

at box edges in components normal and tangential to the edge. This is equivalent to the Z

grid recommended by Randall (1994) for simulating the dispersion relation in a geostrophic

adjustment problem better than any of the staggered A-E grids of Arakawa and Lamb (1977).

The vertical grid is of the Charney-Phillips type recommended in Chapter 2, with just the

pressure staggered with respect to the prognostic variables.

At every timestep the wind fields need to be obtained from Q and δ. The streamfunction

ψ and velocity potential φ are found by solving the Poisson equations:

∇2ψ = σQ− f, ∇2φ = δ. (3.5)

These are solved efficiently by a multigrid method. This first solves the equations on the

coarsest grid in the hierarchy, and then uses this solution as a first guess for the solution on the

second grid, and so on. The rotational and divergent wind fields are given by:

v = vrot + vdiv = k×∇ψ +∇φ. (3.6)

The divergence field is stepped using a predictor-corrector method whereby a dummy diver-

gence step gives a new vdiv to use in the final divergence update (for more details see Gregory

(1999)). A ∇4 hyper-diffusion is applied to the divergence field with a damping timescale of

12 hours on grid scale features. No explicit diffusion is applied to either PV or density as these

are updated with a slightly diffusive advection scheme.

3.2 Boundary Formulation

The greatest problem in isentropic modelling is in handling the intersection of model levels

with the ground. One technique for dealing with this is to extend an intersecting level along

the ground as a so-called ‘massless layer’. This approach has proved successful, for example

in the model of Hsu and Arakawa (1990) (see Chapter 1). The isentropic density is given by

(3.1) and the hydrostatic law as σ = − 1g∂p∂θ

. In an isentropic model this represents both the


Figure 3.1: The fourth buckyball grid in the hierarchy. The simulations presented here wereproduced on the fifth grid, which has roughly four times the number of cells.

mass contained in a grid cell and the static stability. In a massless layer there is, by definition,

no mass, and yet the infinite PV values implied by this present difficulties in a discrete model

predicting PV.

To avoid such difficulties the model developed here uses two different density variables,

which are identically equal in all cells which are entirely above the ground, but behave differ-

ently when a level intersects the surface. The aim is to separate the two quantities represented

by the isentropic density; one of the densities ensures mass is treated correctly at the surface,

while the other gives a finite value to the static stability in a massless layer so that Q is well

defined. These densities can be defined using the generalised vertical coordinate η by

σ = −1

g

∂p

∂η, σ = −1

g

∂p

∂θ, (3.7)

where the derivative in the definition of σ is assumed to exist in the massless region where

δp, δθ → 0.

The first of the two, denoted σ , is used to keep track of the mass in the model. The value

of σ goes to zero when a model level hits the ground, as in the massless layer approach, since

δp vanishes but δη does not. The mass in a cell of area A and depth δη is therefore given by

σAδη.

The second density, denoted σ, is used to relate the PV to the absolute vorticity ζ (= f + ξ)

in (3.1). In order that Q is finite for non-zero ζ , σ has a non-zero value after a level has

intersected the surface. This is intended to represent the limit of the static stability as both δp

and δθ tend to zero, and is in line with the ideas of Eliassen and Raustein (1968) who extended

their model levels underground by choosing values such that the static stability just below


U U

Figure 3.2: The application of Kelvin’s Theorem to the massless region. The circulation arounda closed region U on a surface is given by the area integral over the region of the component ofvorticity normal to the surface, and is independent of time for certain flows. On the right, theregion U is the massless region of an isentropic level that intersects the ground in the tropics;vorticity values in this region define the circulation along a θ contour on the Earth’s surface.See also Hoskins (1991).

the surface was equal to the static stability just above it. These σ values in the ‘massless’ or

‘boundary’ region are used practically for several purposes:

1. To aid in accurate advection of mass (σ ) at the ground (see Section 3.4.3).

2. To evaluate horizontal finite differences near the ground, for example the vorticity term

−∇. (σQk× v) in the divergence equation (3.4) and the horizontal advection of Q in

the PV equation (3.3).

3. To define sensible PV values in the massless region which then yield the correct surface

flow under solution of the Poisson equation (3.5). This can be thought of as an applica-

tion of Kelvin’s Circulation Theorem (see e.g. Acheson (1990)), whereby defining the

vorticity at every point in a massless region U of a given model level defines the circula-

tion around the boundary ∂U , in this case the contour where the level meets the ground.

See Figure 3.2.

For an arbitrary flow, initial σ values in the massless region are created by extrapolating

horizontally along the intersecting level from the above ground values. Both densities then

evolve according to the mass continuity equation (3.2), with the distribution of σ defining the

location of the ground relative to the isentropic levels.

The representation of the ground is critical in the calculation of the Montgomery poten-

tial M , the Laplacian of which gives the horizontal pressure gradient force in the divergence


i

^

ss iΜ := Φ + π η

ii−1M i−1/2

i+1/2

^

η = θ

i+1M

η

sΦ = Φ

i+1η

i−1/2η

ηi

i−1η

i+1/2

i

σ = σ

σ = 0^

+ π ∆ η= M

= M − π ∆ η

pi−1/2

i+1/2

i+1/2p

i−1/2

iσ

Figure 3.3: A schematic of the vertical scheme in the model showing the intersection with theground. The equation on the right definingMi applied at the lowest massy cell of each column;the arrow indicates which cell this is for the furthest right column shown.

equation (3.4). This is given by integrating the hydrostatic equation

∂M

∂θ= π(p) := cp

(p

p0

)κ(3.8)

from the surface where

Ms = Φs + πsθs. (3.9)

Here κ = R/cp, Φ = gz is the geopotential, a subscript s represents a value at the Earth’s

surface and π is termed the Exner function. The procedure for calculating M from the mass

field is now described and is sketched in Figure 3.3.

The pressure p is calculated on half-levels by integrating ∂p∂η

= −gσ down from the model

top where p = 0; similarly, p is calculated by integrating ∂p∂η

= −gσ. The surface pressure ps

is given by p on the lowest model half-level, and this in turn gives πs. In any given column the

lowest massy cell is defined as the lowest cell containing a non-zero σ. Labelling this by its

level i (counting down from the top of the domain), the Montgomery potential in this cell Mi

is defined using a finite difference approximation to the hydrostatic equation (3.8):

Mi = Ms + πs(ηi − θs)

= Φs + πsηi with substitution from (3.9). (3.10)

This definition of Mi is identical to that used in the models of Hsu and Arakawa (1990) and

Randall et al. (2000), and has the advantageous property that θs is not needed to calculate

Mi. From here M is calculated at all levels j in the column above using the finite difference


0 10 20 30 40 50 60 70 80 90

0

1

2

3

4

5

6

7

8

9

10

x 104

Latitude

Pre

ssur

e (P

a)

Location of model levels

Figure 3.4: Isentropic level locations on a pressure axis for the initial conditions of the baro-clinic wave life cycle experiments of Chapter 4.

approximation to the hydrostatic equation (3.8),

Mj−1 −Mj

ηj−1 − ηj= πj− 1

2, (3.11)

and at all levels j in the column below using the pressure derived from the second density σ:

Mj−1 −Mj

ηj−1 − ηj= πj− 1

2. (3.12)

The vertical levels have been chosen to ensure, as far as possible, a similar vertical resolu-

tion to the 15 level version of the Reading IGCM used in later comparisons. There are 21 full

levels of which typically 19 are fully above ground at the poles and 11 at the equator. The level

values were calculated by demanding an exponential change in spacing ∆η with roughly a 6 K

spacing for most of the troposphere and a top half-level height of 800 K, giving half-levels

every 6 K from 244 K through to 334 K and then at 341 K, 350 K, 366 K, 402 K, 501 K and

800 K. Typical locations of the levels in pressure coordinates are shown in Figure 3.4.

In summary, when an isentropic level intersects the ground it continues along the surface

as a massless layer, with σ = 0. A second density variable σ represents the limit of the static

stability as the ground is approached, and does not vanish. In the massless region σ , Q and δ


evolve according to (3.2–3.4), and σ according to

σt = −∇.(σv). (3.13)

3.3 Controlling the Massless Region

The vertical scheme has been designed to avoid contamination of the real atmospheric flow by

the artificial massless region. As shown in Figure 3.3, information only propagates downward

below ground so that the massless values of p and M in a column do not affect the above

ground region of the column. This, however, has an unwanted side-effect since the lack of

vertical feedback means that the massless region suffers from an unusual kind of instability.

Suppose a small positive density perturbation is applied in a grid cell above ground, for

example by the action of a small convergence. This results in a positive perturbation to the pres-

sure field at all levels below, and so to the surface pressure ps. By equation (3.10) this causes

a positive perturbation to Mi, and so to M at all levels above ground through the integration

upward of the hydrostatic equation (3.11). The grid cell originally perturbed is then forced by

a positive divergence tendency −∇2M which acts to reduce the original density perturbation.

This feedback loop is simply the buoyancy effect which comprises the gravity wave feedback

mechanism. In the unphysical massless region, however, this feedback loop does not exist and

so any perturbations applied here will simply accumulate. Perturbations grow polynomially

rather than exponentially, but nevertheless the region is unstable.

In order to control this instability, σ in the massless region is continually relaxed back

towards its initial value σin, and the winds u and v are relaxed towards zero, without changing

the surface fields. The relaxation uses a forward timestep for stability and is given by

σ → σ+τFσin1+τF

, where F (σ, σ) =(σ−σσ

)2

l−1. (3.14)

Here τ is the ratio of the model timestep to the relaxation timescale, and F ensures a smooth

transition at the edge of the massless region, since the strength of the relaxation increases if the

fraction of the cell above (i.e. at level l − 1) that is filled with mass decreases. This relaxation

needs to be strong to control the instability; for the simulations presented in this thesis τ = 12.

Given that such a strong relaxation has to be taken to control the instability, and that the

massless region is, by definition, unphysical in nature, one should ask whether these measures


are justified.

The σ values at the edge of the massless region are used in various ways, for example,

to evaluate horizontal finite differences at the ground such as the vorticity term ∇. (σQk× v)

in the divergence equation (3.4). They are also used to improve the accuracy of the advection

scheme near the ground by providing a means to quantify the proportion of a grid box which

lies above ground, and so define exactly where, within the grid box, the ground lies. This is

explained in Section 3.4.3.

The fields at ‘lower’ massless levels, however, are used solely in the solution of the Poisson

equations (3.5). To obtain the winds u and v above ground in an intersecting level these need

to be solved in that level. This is done either by imposing boundary conditions at the ground

∂U or by creating values in the empty region U to reproduce the correct surface winds under

solution of (3.5). Only with this second approach can an efficient multigrid method be used.

The relaxation can then be justified because, unphysical though it is, by choosing to relax the

winds rather than the PV we ensure that the surface fields are unchanged. This, coupled with

the fact that Q and δ are unchanged above ground, means that the relaxation does not affect the

above ground region through the solution of (3.5).

3.4 The Advection Scheme

In the absence of friction the PV and densities evolve solely by advection. This alone means

that the advection scheme is a key part of the model, but in addition it is the advection scheme

that handles many of the technical difficulties introduced by the surface intersection. The

underlying scheme is that of the original shallow-water model of Thuburn (1997) with the two-

dimensional flux-limiter of Thuburn (1996); the only modifications aim to represent the flow

near the ground more accurately.

Note the importance of the surface θ distribution as a part of the PV-θ view; dynamical

features are often represented by perturbations of both internal PV and surface θ. By using this

scheme, with the same winds, to advect both PV and isentropic density (which defines the sur-

face θ) we improve consistency in the evolution of the internal PV and surface θ distributions,

which should improve the representation of dynamical features.


3.4.1 PV Advection

The PV is advected using the original scheme, which is both shape-preserving and conserva-

tive. In order to calculate the flow across a cell edge an interfacial value of the PV at the edge

needs to be defined. A two dimensional quadratic function is fitted through the PV values in

an upstream-biased neighbourhood of the edge, and this quadratic is integrated to calculate the

amount of PV which would be swept across the edge during the timestep. The constant value

which reproduces this PV flux over the timestep forms a first guess at the interfacial PV value.

This first guess is then adjusted by the flux-limiter by applying bounds to the edge value to

ensure that no new maxima or minima are created in the PV distribution; the scheme is thus

shape-preserving. For more details see Thuburn (1996).

It is worth mentioning here that it is the quadratic upstream interpolation which makes

the scheme slightly diffusive; the PV distribution is smoothed slightly by using a quadratic fit

to calculate the flux across the cell edge. This implicit diffusion is also present in the density

advection.

3.4.2 Flux Limiter for Density Advection

Both densities σ and σ are advected by the model winds. When a level hits the ground, σ drops

suddenly to zero so the horizontal gradient of σ is discontinuous, and this sharp edge should

be preserved by the advection scheme. The first concern is that no negative mass is produced:

σ should drop to zero (actually an infinitesimally small positive value) but not below. This is

ensured by a modification of the flux-limiter as follows.

Convergence and divergence during the flow evolution will act to create new maxima and

minima in the density distribution so, with this in mind, the outflow bounds imposed by the

flux-limiter to ensure shape-preservation are removed. Instead, an outflow bound is applied

to simply limit the flow of mass from a grid box so that the scheme does not remove more

mass from the box than it contains. In the rest of this subsection a suitable value for this

outflow bound is derived using notation from Thuburn (1996) and following the suggestions of

Eckhnel Kleine (personal communication).

Let qk denote the density (either σ or σ ) in grid cell k and qj the interfacial density at edge

j of the cell. (Note that when used with q,ˆdenotes an edge value rather than any connection

with σ.) The aim is to define a maximum outflow bound for cell k, denoted(q

(out)k

)max

, and


to apply this bound to the interfacial values so that

qj ≤(q

(out)k

)max

(3.15)

for all edges j of cell k. Then the mass flow out of cell k during the timestep qout is given by

qout =∑

j

cj qj, (3.16)

where cj is the outflow courant number for edge j given by

cj = max

(0,vnj γj∆t

Ak

)(3.17)

and vnj is the velocity in the outwards normal direction, γj is the length of edge j, ∆t is the

time increment and Ak is the area of cell k. On applying the new outflow bound to each qj

equation (3.16) becomes

qout ≤∑

j

cj

(q

(out)k

)max

=(q

(out)k

)max

∑

j

cj. (3.18)

The scheme will not remove more mass from cell k than it contains so long as

qout ≤ qk (3.19)

and so, by comparison with (3.18), the new outflow bound can be defined as:

(q

(out)k

)max

=qk∑j cj

. (3.20)

This formally prevents the production of negative mass; both densities are indeed strictly posi-

tive in all the simulations in this thesis.

3.4.3 Sub-gridscale Fit Near the Ground

As mentioned earlier it is important to preserve the shape of the sharp change in σ as a level

intersects the ground, yet this feature is bound to be smoothed by the diffusive effect of the


x

x0

B C D E

1

Gradient m

u

A

Figure 3.5: A schematic of the σ profile (shaded) in a level intersecting the ground. The methodused to construct the sub-grid fit for the sharp feature in box B is given in the text.

quadratic fit described in Section 3.4.1. In order to prevent this the advection scheme has

been modified to detect the intersection with the ground and to replace the quadratic fit with a

piecewise linear fit if necessary.

This process takes advantage of one of the benefits of the two-density formulation. The

ratio σ/σ gives the fraction of the relevant grid cell which lies above ground, and so it is easy

to clearly define when the cell is fully above ground, i.e. full with respect to σ . Currently the

tolerance is set at 0.1% so that

σ/σ < 0.001 ⇒ the cell is empty,

0.001 ≤ σ/σ ≤ 0.999 ⇒ the cell is partially filled,

0.999 < σ/σ ⇒ the cell is full.

(3.21)

Consider the advection of the shaded σ profile across the edge AB in the one dimensional

example in Figure 3.5. Box A is empty but box B is partially filled so that when the sub-

gridscale distribution of the σ field is constructed as shown the sharp feature lies within box

B. The original scheme would fit a quadratic through the values of σ in boxes A, B and C

to represent this distribution, and the area under this quadratic, integrated between appropriate

limits, would be the mass transferred to box A during the timestep. Clearly the quadratic fit is

not an accurate representation of the sub-gridscale distribution of the mass in box B. Instead

the advection scheme uses a piecewise linear fit with the σ values in boxes B and C defining

the gradient m of the linear fit and the intercept x0, which is then the horizontal extent of mass

within box B. If the wind advects by a distance less than x0 then there is no flux into box A

and so the interfacial value at edge AB will be set to zero. However, if it advects by a distance

x1 as shown, the heavily shaded region of mass should flow into box A. To ensure this the

interfacial σ value at edge AB is set to Muγ∆t

where M is the mass in the shaded region, γ is the

length of edge AB and ∆t is the timestep.


0 10 20

0

0.2

0.4

0.6

0.8

1

No Flux−Limiter and No Subgrid FIT

0 10 20

0

0.2

0.4

0.6

0.8

1

Flux−Limiter but No Subgrid FIT

0 10 20

0

0.2

0.4

0.6

0.8

1

Flux−Limiter and Subgrid FIT

Figure 3.6: Results of a 1D test of the sub-grid fit. In each case the dotted line is the initialcondition of a sharp density profile and the solid line is the profile after 400 advection steps.Not all of the 40 cell domain is shown.

The same sub-gridscale fit would also be used to calculate the interfacial value at edge

BC if the wind were reversed to flow from left to right. There is another sharp feature in the

sub-gridscale σ distribution in box E and this is handled in exactly the same way by applying

the above procedure to the variable σ − σ.

Figure 3.6 shows the results of a test of this sub-grid fit using a one-dimensional version

of the advection scheme. A prescribed density profile was advected by a constant wind U = 1

and courant number C = U ∆t∆X

= 12

on a periodic domain of 40 cells of width ∆X = 140

,

where all values are non-dimensional. The sharp initial density profile is dotted, and the solid

lines show the density after 400 advection timesteps when the profile has been advected five

times around the domain. The flux limiter succeeds in preventing the generation of negative

mass, but the sharp edges are diffused by the advection scheme. With the piecewise linear fit,

however, these edges are maintained. This scheme has been applied on the two dimensional

hexagonal grid in an approximate manner as follows. To calculate the interfacial value at a cell

edge, the first and second cells upwind of the edge are identified and used as boxes B and C in

the above routine.

3.4.4 Summary of the Mass Advection Step

The full advection scheme used to advance the two density fields σ and σ by one timestep is as

follows:

1. First guess interfacial σ values are calculated using quadratic upstream interpolation.

2. These first guess values are adjusted to lie within inflow bounds for the downwind box,


which are the maximum and minimum values in an upstream biased neighbourhood of

this downwind box.

3. The outflow bound(q

(out)k

)max

is applied to the interfacial σ values as in Section 3.4.2

to ensure that no negative mass is generated.

4. First guess interfacial σ values are calculated for each edge as follows:

• If the first upstream box contains a sharp feature as in boxes B and E in Figure 3.5

then the piecewise linear fit is used.

• If, by (3.21), the first upstream box is full then the σ interfacial value is set to equal

the σ interfacial value.

• If, by (3.21), the first upstream box is empty then the σ interfacial value is set equal

to the (small) σ value in this first upstream box (see Section 3.7).

• If none of the previous three criteria are satisfied then the interfacial σ value is

found using the quadratic upstream interpolation.

5. Inflow bounds are applied to the interfacial σ values as for σ .

6. The new outflow bound is applied to the interfacial σ values as for σ .

7. The interfacial σ values are stored and used for the mass flux in the PV advection step.

8. Both σ and σ are advected using the interfacial values.

3.5 Summary of the Model Timestep

To advance the prognostic variables by one timestep (15 minutes for the horizontal resolution

used here) the following procedure is applied.

1. The densities σ and σ are advected as in Section 3.4.4.

2. The PV is advected as in Section 3.4.1.

3. The Poisson equations are solved using a multigrid method to obtain wind fields from ξ

and δ.

4. The Montgomery potential M is calculated from the mass distribution as in Section 3.2.


5. The δ field is time-stepped using a predictor-corrector method and applying a ∇4 hyper-

diffusion.

6. The Poisson equation is solved once more to obtain the new divergent wind.

7. Relaxation is applied to σ and winds as in Section 3.3, only to grid cells which are defined

as empty by equation (3.21).

8. The PV is re-calculated from the new σ and wind distributions.

3.6 Basic Model Tests

In order to test the model, and in particular the surface formulation, some simulations of sim-

ple flow patterns were performed. The focus of these tests was to assess the accuracy of the

numerical surface formulation rather than of the global-scale flow produced.

3.6.1 Solid Body Rotation

The model was initialised in a state of isothermal solid body rotation, which is an exact steady-

state solution of the primitive equations (3.2)-(3.4) defined by

T = T0, u = U0 cosφ, v = 0, (3.22)

where T0 = 220 K is the constant temperature and U0 = 50 ms−1 is the zonal wind at the

equator. This is a steady state if the divergence tendency vanishes. Substitution of (3.22) into

(3.4) shows that this is satisfied providing

M(θ, φ) = f(θ) +1

2(2Ωa+ U0)U0 cos2 φ, (3.23)

where the function f(θ) is given by the hydrostatic law. The initial state is constructed as

follows:

1. Pressure on half levels and σ on full levels are defined as

pj+ 12

= p0

(T0

ηj+ 12

) 1κ

, σj = −1

g

δpjδηj

. (3.24)


2. The wind fields are then defined by

Qj =2(Ω + U0

a

)sinφ

σj, δj = 0. (3.25)

3. To satisfy (3.23) the Montgomery potential at the bottom of the model domain is defined

as

M0 =1

2(2Ωa+ U0)U0 cos2 φ+ α, (3.26)

where α = 1.9× 105 m2 s−2 is a suitable constant which determines the mean η value of

the z = 0 surface.

4. The discrete hydrostatic equation (3.12) is then integrated upwards to give the Mont-

gomery potential Mj on all model levels.

5. In order to define the location of the ground in the η domain the σ distribution needs to

be defined so that, in the lowest massy level i of each column, the boundary condition

(3.10) is satisfied. For each column, level i is identified as the highest level for which the

half-level pressure pi+ 12

exceeds the value of the surface pressure ps given by (3.10) as

π(ps) =Mi − Φs

ηi, (3.27)

where Φs = gzs = 0 here for a flat surface.

6. This level i is then the lowest massy level in this column, and is partially filled by

σi = −1

g

pi− 12− ps

ηi− 12− ηi+ 1

2

, (3.28)

with σ = σ in all cells in the column above and σ = 0 in all cells below.

It should be noted that this steady-state solution is rather unphysical; the atmosphere is

isothermal and on the surface (z = 0) the potential temperature increases from equator to pole.

It is nonetheless a useful analytical solution with which to quantify model errors.

The zonal and meridional wind fields at 383 K after 20 days are shown in Figure 3.7 for

comparison with an identical test of the original stratosphere model (described in Section 2.5.3

of Gregory (1999)). The errors are larger in the full atmosphere model; the maximum merid-

ional wind speed is 1.5 ms−1 compared to 0.3 ms−1 in Gregory (1999), and large density errors


0 100 200 300

−80

−60

−40

−20

0

20

40

60

80

Longitude

Latit

ude

Zonal wind

45

30

15

45

30

15

0 100 200 300

−80

−60

−40

−20

0

20

40

60

80

Longitude

Latit

ude

Meridional wind

Figure 3.7: Zonal and meridional wind on the 383 K surface on day 20 of the solid bodyrotation simulation. The contour interval is 5 ms−1 for zonal wind and 0.4 ms−1 for meridionalwind.

are present with a drift of up to 5% from the initial distribution. These density errors are prob-

ably the cause of the wind errors since they act to push the flow away from the steady-state

solution, and are discussed in Section 3.7.

3.6.2 Perturbing the Massless Region

In order to test that the real atmospheric flow is not contaminated by the massless region and

its relaxation, the model was initialised in solid body rotation but with a large perturbation

applied to the wind fields in the massless region. A wavenumber 5 perturbation centred on the

mid-latitudes was applied to the wind field in order to force the model at the same wavenumber

as the grid noise (see section 3.1). The perturbation was given by

ξ′ = −5v0

asin2 φ cos 5λ F (σ, σ),

δ′ = 2v0

asin 5λ sinφ cos 2φ F (σ, σ),

where F (σ, σ) =

(σ − σσ

)2

l−1

. (3.29)

Here λ is the longitude, a is the radius of the Earth and the resulting wind fields are shown

in Figure 3.8. This construction ensures a large meridional component as to force flow on

model levels normal to their intersection with the ground; a value of v0 = 75 ms−1 was chosen,


0 100 200 300

−80

−60

−40

−20

0

20

40

60

80

Longitude

Latit

ude

Zonal Wind

50

35

15

50

35

20

0 100 200 300

−80

−60

−40

−20

0

20

40

60

80

Longitude

Latit

ude

Meridional Wind

Figure 3.8: Initial wind fields in the massless region from Section 3.6.2. The contour intervalis 5 ms−1 with the zero contour dotted and negative contours dashed.

resulting in a maximum perturbation meridional wind speed of 20 ms−1. As in the relaxation

equation (3.14), the factor F ensures a smooth growth of the perturbation just below the surface.

Provided σ and σ are zonally symmetric, which is certainly true during the relaxation period,

the integral over longitude of the equivalent vorticity perturbation vanishes so that there is no

net surface circulation implied by this perturbation.

In the resulting simulation the perturbation is rapidly removed by the model relaxation.

The global energy and angular momentum decrease significantly during this time as shown in

0 2 4 6 8 10 12 14 16 18 2097

97.5

98

98.5

99

99.5

100

100.5Global Energy and Angular Momentum

Time (days)

Drif

t (%

)

Figure 3.9: Global integrals of total energy and angular momentum for a 20 day solid body ro-tation run with perturbed flow in the massless region. Energy values are marked by circles andmomentum values by crosses, with dashed lines for an integral taken over the whole domainand solid lines for an integral taken over the above ground region only. All quantities are givenas a percentage of their initial value.


Figure 3.9 but the above ground values remain constant. Errors in the flow field above ground

are of the same size as for unperturbed solid body rotation; there is no adverse effect on the

above ground flow due to the perturbation of the massless region.

3.6.3 Flow over a Hill

In more complicated atmospheric flows the surface θ distribution would be expected to vary in

both space and time. The aim of this test was to verify that the intersecting isentropic levels

are able to move freely across the ground in response to the evolving flow.

A hill of maximum height 2 km was introduced, centred at φ0 = 45N, λ0 = 90E, and

defined as a surface geopotential distribution:

Φb =ghw2

a2 [(λ− λ0)2 + (φ− φ0)2] + w2, (3.30)

where a is the radius of the Earth, h is the height of the hill and w = 1000 km is a width scale.

The model was initialised in solid body rotation with a flat surface, and the Φb perturbation

added gradually to allow the flow to evolve; i.e. the hill grows during the initial hours of the

simulation. If the growth is rapid, for example, occurring in the first hour of integration only,

then a large gravity wave is produced as would be expected (see Figure 3.10). To reach a steady

state the hill was instead grown slowly over the first 24 hours, and the model run for a further

5 days after this.

An air parcel is able to pass over an obstacle of height h provided the Richardson number

Ri = N2h2

u2 < 1, where u and N are the upstream velocity and Brunt-Vaisala frequency

respectively (see e.g. Gill (1982), p. 293). Using this, and that N 2 = gθdθdz

, we estimate the

maximum elevation obtainable by the parcel as

hmax ≈u√gθdθdz

≈ U0 cosφ√g

300 K6 K

500 m

≈ 1750 m. (3.31)

Only at the very centre of the hill does the height exceed this, so by this estimate most of the

air approaching the hill does have enough energy to flow over it. The final location of the


0 100 200 300

−50

0

50

Longitude

Latit

ude

Divergence after 4 hours

0 100 200 300

−50

0

50

Longitude

Latit

ude


0 100 200 300

−50

0

50

Longitude

Latit

ude


0 100 200 300

−50

0

50

LongitudeLa

titud

e


Figure 3.10: The divergence on the 301 K model level shows a gravity wave propagating awayfrom the rapidly-grown hill, and later converging on the opposite side of the globe. The contourinterval is 2× 10−6 s−1 with negative contours dashed.

0 10 20 30 40 50 60 70 80 900

1000

2000

3000

4000Location of isentropic levels after 120 hours

Latitude

Hei

ght (

m)

0 20 40 60 80 100 120 140 160 1800

500

1000

1500

2000

2500

3000

3500

Longitude

Hei

ght (

m)

Figure 3.11: The lower levels (spaced every 6 K) are shown as a function of height z after 5days of the flow over a hill simulation, with the initial level locations shown as dotted lines.The flow is from west to east i.e. from left to right in the lower panel and out of the page in theupper panel.


40 60 80 100 120 140 160 18020

25

30

35

40

45

50

55

60

65

70

Longitude

Latit

ude

Flow and height at 265K after 5 days

500

1000

1500

Figure 3.12: Wind vectors on the 265 K model level and height of the level in solid contours.The dashed line marks the location where this model level intersects the ground.

isentropic levels over the hill is shown in Figure 3.11. The model levels rise up over the hill

indicating that the general flow is indeed over the hill as predicted.

Figure 3.12 shows the wind and height of the 265 K level at day 5. Isentropic levels are

compressed in the region above the hill resulting in an anticyclonic anomaly over the hill due

to conservation of Q, as given by (3.1). On the downslope side of the hill a parcel acquires a

southward velocity under the influence of the eastern edge of the anticyclone and accelerates

through a region of low pressure on the lee of the hill. The parcel’s planetary vorticity f

decreases and its momentum carries it south past the point where f = Qσ so that, to satisfy PV

conservation, its relative vorticity ξ increases, turning the flow northwards cyclonically. The

parcel oscillates northward and southward under this vorticity-induced restoration mechanism,

forming a Rossby wave train which is shown in Figure 3.13. The wave is similar to that

predicted by Grose and Hoskins (1979) for flow over an isolated mountain of similar size.

Although it is reassuring to see that the surface topography forces the atmosphere in a

reasonable manner, the aim here is to test the ability of the intersecting levels to move with the

flow. Figure 3.12 shows the location of the 265 K surface intersection as a dashed line which

lies just to the north of the hill under this solid body rotation background flow. It is clear that

the intersection has indeed adjusted to the flow as the meridional meanders of individual air


0 50 100 150 200 250 300 350

−50

0

50

Meridional wind after 24 hours

0 50 100 150 200 250 300 350

−50

0

50


0 50 100 150 200 250 300 350

−50

0

50


Figure 3.13: The meridional wind field shows the propagation of a Rossby wave train behindthe hill. Contours are drawn every 5 ms−1, with the zero contour dotted and negative contoursdashed.

parcels moving on isentropes imply an oscillation of surface θ with longitude.

3.7 The Surface Divergence Effect

As mentioned in Section 3.3 there is no vertical contamination of the above ground flow by

the artificial massless region. However, there is evidence of a small horizontal effect and this

is the cause of the density errors in the solid body rotation experiment. Figure 3.14 shows the

location of the lowest model levels after 20 days of solid body rotation, and demonstrates that

there is a tendency for mass to collect over the equator, especially noticeable in the lowest level

but also present above. There is a σ error of up to 5% at 383 K, compared to less than 0.1%

at 1240 K in Gregory (1999), and it is likely that this density error is responsible for the larger

wind errors noted in section 3.6.1. The total global angular momentum of the above ground

region increases by almost 0.2% over 20 days due to a 5% increase in the angular momentum

in level 20, the lowest massy level in Figure 3.14.

The development of this distribution is shown in Figure 3.15. In the early stages a negative

σ anomaly develops only in the first empty grid cell encountered when the level intersects the


−80 −60 −40 −20 0 20 40 60 800

500

1000

1500

2000

2500

Latitude

Hei

ght(

m)

Height of isentropic model levels

Figure 3.14: The intersecting isentropic levels on a height axis for the solid body rotationinitial conditions (dotted contours) and at day 20 (solid contours), showing the migration ofmass towards the equator.

ground, as defined by equation (3.21). This anomaly grows with time and increases in extent

so that partially filled cells equatorward of the empty cell are affected. Mass of both varieties

(σ and σ) migrates from the region of intersection towards the equator where it collects form-

ing the unphysical spike seen in Figure 3.14. The underlying cause of this clearly erroneous

feature is a persistent anomaly in the divergence field at the intersection, itself a product of the

relaxation.

The divergence tendency is given by equation (3.4) which is reprinted here with the friction

term omitted

δt = −∇.[σQk× v +∇

(M +

1

2v2

)]. (3.32)

The tendency is therefore the sum of three terms which are generally nearly in balance: the first

is the vorticity term, referred to here as the Q term, the second is the horizontal pressure gra-

dient force or the M term, and the third is the kinetic energy or the E term. Figure 3.16 shows

the balance between these three terms and the resulting tendency at the initial conditions, and

after one hour of integration (i.e. four timesteps). The large positive anomaly in the divergence

tendency at the surface intersection persists throughout the run and it is the resulting divergence

which is removing mass from this region.

From Figure 3.16 it is clear that it is the Q term in equation (3.32) that is causing this

divergence tendency. The grid cell in which the anomaly starts is just in the massless region.

According to equation (3.21) the cell is empty but its first equatorward neighbour is not. This

means that the wind at the poleward edge is relaxed towards zero, but that the wind on the


−50 0 500

50

100

150

200

250ICs

−50 0 500

50

100

150

200

250After 2 hours

−50 0 500

50

100

150

200

250After 4 hours

−50 0 500

50

100

150

200

250After 6 hours

−50 0 500

50

100

150

200

250After 5 days

−50 0 500

50

100

150

200

250After 10 days

−50 0 500

50

100

150

200

250After 15 days

−50 0 500

50

100

150

200

250After 20 days

Figure 3.15: Snapshots of the latitudinal distributions of σ (solid lines) and σ (dotted lines) inthe lowest model level which contains any mass. This shows the migration of mass towards theequator in this solid body rotation run. The units are kg m−2 K−1.


−100 −50 0 50 100

−2

−1

0

1

2

x 10−6 Sum of terms

−100 −50 0 50 100

−2

−1

0

1

2

x 10−6 E term

−100 −50 0 50 100

−2

−1

0

1

2

x 10−6 M term

−100 −50 0 50 100

−2

−1

0

1

2

x 10−6 Q term

Figure 3.16: Divergence tendency contributions from the various terms in equation (3.32) asfunctions of latitude for the lowest massy level, in s−2. The first panel is the sum of the otherthree and is therefore the divergence tendency δt. Values are shown for the initial state as dottedlines, and after 1 hour as solid lines.

equatorward edge is not. The wind is westerly everywhere so the relaxation induces a cyclonic

relative vorticity perturbation in the grid cell, and therefore a cyclonic PV perturbation. The

relaxation also results in a large wind shear ∂u∂y

which is negative in the northern hemisphere,

and it is this which causes the divergence anomaly. The Q term is given by

Q term = −∇. (σQk× v)

= −∇. (σQui) if v = ui

= − ∂

∂y(uσQ) . (3.33)

While there are Q and σ perturbations in the first massless cell, the wind u has a variation

which is both larger relative to its mean value and monotonic with latitude, and so it is this

factor which dominates the Q term, resulting in the large positive divergence tendency seen.

The cyclonic PV perturbation induced by the relaxation is enhanced by the negative σ anomaly

resulting from the divergence. These effects are most noticeable in the lower levels because

these levels intersect the surface closer to the equator where the wind, and therefore also the


−100 −80 −60 −40 −20 0 20 40 60 80 100−3

−2

−1

0

1

2

3

Latitude

PV

(P

VU

)

−100 −80 −60 −40 −20 0 20 40 60 80 100−1

−0.5

0

0.5

1x 10

−6

Latitude

Mas

s−w

eigh

ted

PV

(s−1

)

Figure 3.17: Zonal mean PV (top) and above ground mass-weighted PV (σQ) (bottom) in thelowest massy level for the initial conditions (dotted line), after 2 days (dashed line) and after20 days (solid line). Note the large change in above ground mass-weighted PV between days2 and 20 due to the horizontal extension of the PV anomaly.

wind shear, is stronger.

Both the size and extent of the PV and σ anomalies increase steadily with time. As de-

scribed in Section 3.3, there is no buoyancy feedback mechanism in the massless region to

restore the σ perturbation. The horizontal extent of the anomalies increases so that σ and σ de-

crease in the partially filled region as mass is advected away in response to the large divergence

just below the ground. The PV anomaly also spreads into the partially filled region, as shown in

Figure 3.17, and runtime diagnostics show that it is the advection step which causes this. The

flux-limiter in the advection scheme prevents the generation of new PV maxima, but once the

maximum just below the surface is created by the relaxation step this PV anomaly is spread into

the partially filled region by the advection. The divergence anomaly forces a PV flux normal to

the surface intersection, and the quadratic upstream interpolation in the advection scheme acts

to diffuse the below ground PV anomaly into the partially filled region.

The PV anomaly induced in the partially filled region is a significant feature of the above

ground flow. The PV anomalies are large, as can be seen in Figure 3.17, and of opposing sign

in the two hemispheres. In any level the integral of the mass-weighted PV over all cells j


(∑j σQ δA δθ

)should be zero since the flow is symmetric about the equator. In general this

is observed in the model with

∑σQ∑ |σQ| = O(10−6) = O(round-off error) (3.34)

in levels which do not intersect the ground. However in the lowest massy level this figure is

O(10−3); this error is larger than pure round-off error.

There is one final manifestation of the divergence anomaly at the surface. Suppose an

amount of σ mass leaks into the massless region that is too small for the cells to be counted as

partially filled, i.e.

0 <σ

σ< 0.001. (3.35)

This mass will then be advected by the wind, the divergent part of which is forced as in the first

panel of Figure 3.16 to give divergence at the surface intersection and convergence towards

the poles. This convergence is also due to the relaxation: at the initial conditions the σ field

is in balance with the wind in solid body rotation, but with the winds relaxed to zero the

Q term in (3.32) is too small to balance the M term. Acting under this divergent field the

mass migrates away from the surface and collects at the poles. This development is shown

in Figure 3.18, and it is clear that the σ values are small. After 20 days these values are still

small but they are larger than 0.001 σ, so that the lowest of these ‘outcropped’ levels at the

pole does count as the lowest massy level for some columns in the calculation of M described

in Section 3.2. This is concerning in its own right but to date no problems have arisen as a

result. The original σ leak could perhaps be prevented in future by altering the algorithm in

step 4 of the advection summary in Section 3.4.4 so that, if the first upstream box is empty

(σ < 0.001σ), the interfacial value is set to zero rather than to this small σ value. This small

modification was not made for the simulations in this thesis.

To summarise, there is a contamination of the above ground flow by the massless values

because the relaxation causes a strong horizontal wind shear which in turn induces a divergence

anomaly at the surface. The contamination is stronger in the solid body rotation flow studied

here than might be expected in more realistic flows due to the strong and persistent surface

wind field. Nevertheless, this process is monitored closely in the simulations presented later.


0 20 40 60 800

1

2

3

4

Latitude

Hei

ght(

m)

Day 5

0 20 40 60 800

1

2

3

4

LatitudeH

eigh

t(m

)

Day 15

0 20 40 60 800

0.01

0.02

0.03

0.04

0.05

0.06

Latitude

Hei

ght(

m)

6 hours

0 20 40 60 800

0.01

0.02

0.03

0.04

0.05

0.06

Latitude

Hei

ght(

m)

Day 1

Figure 3.18: Meridional sections of the height of intersecting level surfaces to show the devel-opment of the outcropping at the pole. Even at day 5 the σ mass in the outcropped region isso small that the cells are considered empty. By day 15, however, these cells are consideredpartially filled and the lowest is counted as the lowest massy cell for columns over the pole,and so used to define the surface Montgomery potential by (3.10).

3.8 Summary

The isentropic stratosphere model of Gregory (1999) has been extended to include a surface

formulation. Two densities are used: σ goes to zero at the ground as in the massless layer

approach, while σ has non-zero values used to compute ξ from Q and also to aid in smooth

evaluation of terms near the ground. This massless region is unstable and a relaxation term is

used to control this. The advection scheme has been modified in two ways: the flux-limiter

simply prevents the production of negative mass, and a piecewise linear sub-grid fit more ac-

curately advects the sharp edge in the σ field at the ground.

The model is able to simulate a 20 day period of solid body rotation, albeit with larger

errors than the original stratosphere model. Large perturbations to the massless region do not

affect the above ground flow, and the model is able to realistically simulate the flow over a

large hill. The larger errors seen in the solid body rotation simulation are due to a horizontal

contamination of the above ground flow by surface divergence anomalies induced by the relax-


ation term. This effect is considerably smaller in the simulations in following chapters, where

the surface winds are smaller and less persistent. The contamination needs to be monitored

closely and should be reduced in future development of the model.

Chapter 4

The LC1 Baroclinic Life Cycle

4.1 Baroclinic Instability and a History of the LC1 Cycle

The atmosphere of the Earth is a baroclinic fluid, defined as a fluid for which the density at a

given location is dependent on the temperature as well as the pressure. The temperature is then

able to vary on a surface of constant pressure, and the atmosphere can develop a temperature

gradient on the planetary surface in response to the differential solar heating. This implies

a distribution of potential temperature θ in which isentropes high above the ground at the

poles slope down to intersect the surface at warmer regions towards the equator. Air of a high

potential temperature towards the equator is then lower, when viewed in height or pressure

coordinates, than relatively colder air towards the poles, and so there is potential energy in the

basic state which is available for conversion to kinetic energy.

It is useful to define here this notion of available potential energy. Air of a high potential

temperature is essentially more buoyant than air of a lower potential temperature. In an at-

mosphere at its state of lowest possible potential energy, more buoyant air must lie above less

buoyant air, and therefore in this state surfaces of constant potential temperature and pressure

are both uniformly horizontal, i.e. they coincide with surfaces of constant height. There is still

a large amount of potential energy in this state but there is no way to adiabatically extract it

from the system. This is therefore referred to as the unavailable part of the potential energy and

that part gained by adiabatically re-arranging the mass within isentropic layers is the available

part. For a more rigorous and detailed description see e.g. Holton (1992), Section 8.3.

This basic state is unstable to wave-like perturbations which grow into the patterns of

cyclonic and anticyclonic weather systems commonly seen in the mid-latitudes. These are

The LC1 Baroclinic Life Cycle 80

essentially synoptic-scale eddies which act to advect warm air upwards and towards the poles

and cold air downwards and towards the equator. In this way heat is transported poleward

and upwards, contributing significantly to the work of the atmosphere as a heat engine. From

this viewpoint the atmosphere is seen as an engine transporting heat from the tropics, where

incoming solar radiation dominates the outgoing radiation and so the Earth is heated, to the

extratropics, where the outgoing radiation dominates and the Earth is cooled (see e.g. Curry

and Webster (1999)).

The wave acts to flatten the isentropes in the mid-latitudes, so they lie closer to surfaces of

constant pressure, as available potential energy is extracted from the mean flow and converted

into eddy kinetic energy. The wave therefore modifies the original basic state into a more

stable one. Surface temperature contours deform during the wave growth, and the mean merid-

ional surface temperature gradient in the mid-latitudes temporarily decreases. The background

state is therefore more stable and the wave decays. The atmosphere, however, is continually

forced by relative heating in the tropics and cooling towards the poles, enforcing a permanent

temperature gradient, and new systems frequently grow during the decay of their predecessors.

Instabilities of this kind which can only exist in a baroclinic fluid are referred to as baro-

clinic instabilities. They can only be realised with a temperature gradient on pressure surfaces,

and a corresponding vertical structure in the wind field in accordance with thermal wind bal-

ance. Many, more detailed, introductions are given in the standard textbooks, for example

Holton (1992).

A key aim of atmospheric science over the past century has been to improve our under-

standing of this type of atmospheric flow by modelling the growth of wave-like perturbations in

baroclinically unstable flows. Early models were analytic, starting with the pioneering work of

Charney (1947) and Eady (1949). These models represented very idealised situations in which

the zonal mean is independent of latitude and the flow is assumed to be quasi-geostrophic and

takes place on an f or β plane. Such models greatly improved our understanding of baroclinic

instability in the atmosphere. However, to study the more realistic case of solutions to the

primitive equations in spherical geometry demands the use of numerical models. The life cy-

cles studied in this thesis are just some out of perhaps hundreds of numerical wave simulations

which have been presented in the literature; they have, however, evolved from some of the

earliest examples of such simulations.

Hoskins and Simmons (1975) presented a multi-layer model of the atmosphere which

solved the primitive equations on the sphere by representing the prognostic variables as trun-


cated spherical harmonics. They demonstrated the use of their model by simulating the growth

of a baroclinic wave on a zonal flow of differential solid body rotation with velocity increas-

ing with height. Simmons and Hoskins (1976) went on to study the waves which developed

on background states consisting of the same solid body rotation but with the addition of two

different mid-latitude westerly jets. Using an initial value technique they obtained the growth

rates, phase speeds and structures of the most unstable linear modes at various zonal wave-

lengths by perturbing the zonal flow and taking the perturbation at the time when its growth

became exponential to be the normal mode. The waves were in general considered too shallow,

and to remedy this the vertical resolution was increased, and a more realistic representation of

the zonal mean structure in the tropopause region was added in Simmons and Hoskins (1977).

Though improved, the linear solutions still failed to represent the observed patterns and

eddy strengths at upper levels, which prompted a study of the nonlinear development of waves

from the same initial conditions. Simmons and Hoskins (1978) left behind the linear initial

value technique and used their fully nonlinear spectral model to generate the first complete

baroclinic wave life cycles. The initial zonal flow was perturbed by a small amplitude distur-

bance in the shape of their previously calculated normal modes and the model was integrated

in time. It was found that the growth rate at lower levels decreased more rapidly than at upper

levels, and so upper level amplitudes did indeed become larger relative to those at the surface

than they had in the linear modes. Barotropic processes were found to bring about a decay of

the wave at later stages and so a well defined life cycle existed.

Waves of zonal wavenumber 6 were found to give deeper disturbances than the waves

of maximum growth rate (wavenumbers 7 and 8), which tended to be shallow in nature and

therefore did not extract as much energy from the basic state. Simmons and Hoskins (1980)

studied various wavenumber 6 life cycles in which the initial conditions differed only in a

barotropic component. One of these featured an added cyclonic horizontal shear, originally

added to enhance warm frontogenesis as suggested by Hoskins and West (1979). The life

cycle, however, displayed strikingly different energetics and was referred to as an ‘anomalous’

case to distinguish it from the ‘basic’ cycles.

A version of the ‘basic’ cycle was studied for evidence of frontal cyclogenesis in Thorn-

croft and Hoskins (1990), where a detailed description of the low level evolution of the wave

is given. Later Thorncroft et al. (1993) re-christened the basic and anomalous cycles as LC1

and LC2 respectively, and extensively analysed the differences between the two. They identi-

fied two extreme types of behaviour which they termed cyclonic and anticyclonic according to


whether air turns in the same direction as the Earth’s rotation or in the opposite, respectively. A

key difference between the cycles is that, while the LC1 cycle exhibits both types of behaviour,

the LC2 cycle is dominated by the cyclonic type. The two cycles are characterised by very

different patterns of eddy kinetic energy activity in the later stages and also by very different

synoptic structures and evolutions.

By this stage a picture of the general evolution of such life cycles was developing which

comprised four, more or less distinct phases as described by Thorncroft et al. (1993):

1. Exponential growth of a normal mode perturbation as in the linear theory of baroclinic

instability.

2. Saturation at low levels as nonlinear processes such as the formation of fronts and the

occlusion process slow and eventually stop the growth of the lower wave.

3. Vertical propagation of a Rossby wave, triggered by the low level saturation, which car-

ries wave activity upwards into the jet region in the upper troposphere / lower strato-

sphere. This replaces the baroclinically unstable growth as the dominant cause of eddy

kinetic energy increase.

4. A similar saturation event now occurs at upper levels where nonlinear processes again

act to slow and eventually stop the growth of the wave, which then decays mostly under

the action of barotropic processes.

This is widely referred to as the ‘saturation-propagation-saturation’, or SPS, picture of a

general baroclinic disturbance.

Both life cycles have been used in several important studies over the years, of which the

most relevant to this thesis is Methven (1996). The LC2 cycle is described in Appendix A,

while the rest of this chapter compares the evolution of the LC1 cycle in the isentropic model

described in Chapter 3 with its evolution in a model with a normalised pressure coordinate.

This second model, the Intermediate General Circulation Model or IGCM, has developed from

the original spectral model described by Hoskins and Simmons (1975).

4.2 The IGCM

The model referred to here as the IGCM is more formally named IGCM1 to distinguish it

from later versions which incorporate some representation of the various physical processes at


work. Excepting some technical alterations this version of the model has changed little from

the original version of Hoskins and Simmons (1975), which therefore remains the principal

source for a more detailed description of the model.

The IGCM is a spectral model in which the variables are represented in terms of truncated

series of spherical harmonics on a set of discrete levels of normalised pressure p/ps. Here ps

is the surface pressure, which is itself predicted by an integration of the continuity equation.

The quantity p/ps would normally be referred to as the σ coordinate, a term which is avoided

here purely to avoid confusion with the isentropic density σ. For the basic life cycles presented

here the resolution is given by a triangular truncation of T42 in the horizontal and 15 levels in

the vertical, at normalised pressure values of 0.0185, 0.0596, 0.1057, 0.1520, 0.1968, 0.2407,

0.2865, 0.3382, 0.4002, 0.4765, 0.5687, 0.6740, 0.7844, 0.8871 and 0.9673.

The hydrostatic primitive equations in vorticity, divergence, temperature and surface pres-

sure are integrated using a semi-implicit timestepping scheme. The IGCM is a spectral trans-

form model, meaning that the nonlinear terms are evaluated in grid space and then transformed

back to spectral space in order to compute the spectral tendencies. The most significant alter-

ation from the original model is in the vertical scheme. The original mass and energy conserv-

ing scheme has been replaced by one based on that of Simmons and Burridge (1981) so that the

angular momentum is also conserved by the vertical differencing. By restricting the spherical

harmonics used to only every sixth zonal harmonic a six-fold symmetry in longitude is implied

which, when coupled with an assumed symmetry across the equator, means that calculations

need only be performed for one twelfth of the global domain.

This version of the model is dry and the only diabatic process is a horizontal ∇6 hyper-

diffusion term acting on all three prognostic variables. For the basic T42 simulation this dif-

fusion acts to remove features of the shortest retained scale with an e-folding timescale of 4

hours. This diffusion is necessary to avoid the accumulation of energy, and thus noise, at the

grid scale during the cascade of flow features to small scales, which is characteristic of fluid

motion. By acting along normalised pressure surfaces this diffusion mixes air across isentropic

surfaces. This is in direct contrast to the isentropic model in which the mass in each isentropic

level is exactly conserved, and is one of the key processes which gives rise to differences in the

life cycles predicted by the two models.


Figure 4.1: A reproduction of Figure 3 of Thorncroft et al. (1993) showing the zonal meanstate of the LC1 and LC2 cycles (left and right, respectively) at days 0 and 7 (top and bottom).Contours show the potential temperature every 5 K and the zonal wind every 5ms−1. Theheavy contour is the 2PV U surface, and the shading indicates the eddy kinetic energy maxima.

4.3 An Overview of the LC1 Cycle in the T42 IGCM

The zonal mean initial conditions have been likened to a western ocean basin wintertime sit-

uation. The initial distributions of potential temperature and zonal wind for the LC1 cycle

are shown in the top left panel of Figure 4.1. The meridional surface temperature gradient at

mid-latitudes is clear, indicating that this region is strongly baroclinic. The isentropes here

slope relative to the normalised pressure surfaces so there is a large amount of potential energy

available for conversion to eddy kinetic energy. The thermal wind balance relation equates the

meridional temperature gradient to the vertical shear of the zonal wind. In agreement with this

the initial conditions display vertical wind shear in the region of sloping isentropes, so that the

zonal mean wind distribution takes the form of a jet which reaches a maximum of 47 ms−1 at

roughly 45N in the vicinity of the tropopause, and drops to almost zero at the surface. To seed

the wave, the surface pressure is perturbed by up to 1 mb in the shape of the most unstable


zonal wavenumber six normal mode.

In the initial period of exponential growth the wave more or less retains the shape of

the normal mode perturbation. Fields of all flow variables vary approximately sinusoidally

in longitude and the wave grows much as predicted by the linear normal mode theory, with

the eddy activity concentrated at low levels. The growth in EKE during this stage is due

to baroclinic conversion from available potential energy as in the standard baroclinic growth

theory. Thorncroft et al. (1993), however, showed that this period of exponential growth lasts

only for the first four days of the cycle, and that at the end of this first stage the EKE is less than

a fifth of its maximum value. The EKE evolution in both models is shown later in Figure 4.6.

By day 6 the flow at low levels has departed significantly from the sinusoidal shape held

during the initial growth period, as it undergoes the nonlinear saturation event which comprises

the second phase of the SPS picture of the evolution. The low level temperature distribution

is shown in Figure 4.2 for days 6 to 9. The flow features associated with the nonlinear satura-

tion processes are clearly visible by day 6, and are reminiscent of features observed in mature

cyclone systems. A sector of warm air has moved northward between two regions of colder,

polar air, and strong surface fronts have developed between the air masses. The temperature

gradients are strongest in the cold front which trails out from a deep low pressure region. The

warm sector and associated warm front display a NE-SW tilt, comprising what Thorncroft

et al. (1993) refer to as a bent-back baroclinic zone of inverse meridional temperature gradient.

The warm sector decays during the following days in a manner resembling observed occlu-

sion events. The observed events however are associated with a lifting of the warm sector air

whereas in the simulated life cycle the dominant processes acting are horizontal advection and

diffusion. By day 6 the tip of the warm sector has turned cyclonically to the west, forming a

patch of warm air that is later pinched off into the centre of the low in a feature that has been

termed a seclusion. Both of these features are of a relatively small scale and are thus greatly

affected by the model’s diffusion. Thorncroft and Hoskins (1990) studied the low level evolu-

tion in great detail, concentrating on the possibility of a frontal cyclogenesis event occurring

on the cold front.

By day 8 the low level temperature gradients are weak in the mid-latitudes, indicating

that the baroclinicity in the region has largely been destroyed. The low level saturation event

has stopped the growth of the wave at low levels where it is now in barotropic decay. The

upwards propagating Rossby wave of the third phase of the SPS picture is triggered by this first

saturation event, and Thorncroft et al. (1993) presented the two phases as occurring more or


Figure 4.2: Temperature on the level p/ps = 0.9673 in the LC1 cycle simulated by the IGCMat T42 resolution. Contours are drawn every 5 K with the warmest (i.e. southernmost) contourline at 295 K .


less concurrently. Eliassen-Palm cross-sections as developed by Edmon et al. (1980) showed

that, above the surface, the dynamics of the life cycle are dominated by the vertical Rossby

wave propagation from days 5 to 7. The EKE growth after day 4 greatly exceeds the growth

up to that point, and the diagnosis of Thorncroft et al. (1993) demonstrated that this growth is

associated, not with baroclinic instability, but with the propagation of Rossby wave activity up

into the strong winds of the upper tropospheric jet.

The evolution of the upper level wave is illustrated in Figure 4.3 by the PV distribution

on the 330 K isentropic surface. The upper wave is clearly still growing on day 7 when the

lower wave has saturated and is in decay. Thorncroft et al. (1993) identified two contrasting

types of behaviour which they termed ‘cyclonic’ and ‘anticyclonic’, and stressed that a key

characteristic of the LC1 cycle is that it exhibits both of these, as is frequently observed in real

systems. Cyclonic wrap-up of air occurs on the poleward side of the westerly jet where there

is strong cyclonic shear of the zonal wind; air on the equatorward side of the jet, in contrast,

wraps up anticyclonically under the influence of the anticyclonic shear there. This anticyclonic

feature is clearly visible from day 7 in the PV field shown in Figure 4.3, but on the poleward

side of the jet the field is dominated by large positive PV anomalies which, while indicative

of cyclonic vorticity, do not exhibit the same process of wrap-up as is seen to the south of

the jet. Thorncroft et al. (1993) chose to illustrate the upper flow by plotting the distribution

of potential temperature on the surface where PV = 2 PVU (= 2 × 10−6 m2Ks−1kg−1), an

approximation to the extratropical tropopause. These fields, shown in their Figure 7, do indeed

show a cyclonic wrapping-up of air polewards of the jet in the location of the positive PV

anomalies in Figure 4.3. This feature is discussed further in Section 4.4. Figure 4.5 of that

section shows a higher resolution version of the life cycle in which the cyclonic wrap-up is

clearly visible from day 6.

The tongues of high PV air in Figure 4.3 which extend south across the jet into the region

of anticyclonic shear are generally referred to as troughs, or as tropopause folds. They mark

regions where stratospheric air with high PV has intruded down into the troposphere. During

days 8 and 9 these troughs fold westward in the anticyclonic shear and are thinned under the

influence of the large scale PV field as described in Thorncroft et al. (1993), Section 6(c). PV

contours are irreversibly deformed as a region of high PV air is cut off from the southern end of

the trough, which weakens as it thins. This ‘equatorward Rossby wave breaking’ event heralds

the decay of the upper level wave in the final phase of the SPS picture. The SW-NE tilt of the

trough means that air moving north towards the jet along the eastern edge of the trough has


Figure 4.3: Distribution of PV interpolated to the 330 K isentropic surface during the IGCMsimulation of the LC1 cycle at T42 resolution. Contours are drawn every 0.5 PVU. Note theincrease of the PV maximum; the highest contour is at 7 PVU on day 6 and 8 PVU on day 9.


a large westerly velocity component. Momentum is therefore transported from the wave into

the jet signalling the barotropic conversion of EKE into zonal basic state kinetic energy. The

period from day 7 to day 9 is the main period of this barotropic conversion during which EKE

is lost at roughly the same rate as it was generated in days 4 to 7.

The reduction of EKE has stopped by day 10 and the wave has left the atmosphere in a

much more stable state. The horizontal temperature gradient at mid-latitudes has been greatly

reduced, and the zonal mean wind has a much larger barotropic component than the initial

conditions so that the strong westerly winds of the jet extend low down towards the surface.

A strong surface front is still in existence equatorward of the jet, indicating that significant

baroclinicity is still present. The late stages of the cycle show some evidence of a wave-like

disturbance forming in this baroclinic zone but this does not develop into a significant feature.

4.4 A High Resolution Version of the IGCM LC1 Cycle

The life cycles predicted by both the T42 IGCM and the isentropic model (resolution roughly

equivalent to T50) will be compared with a much higher resolution version of the IGCM cy-

cle. This is the high resolution simulation performed by Methven (1996) with 30 levels in

the vertical and a horizontal resolution of T341. Much fine detail is observed in this simula-

tion, with many features reminiscent of those seen in the similar high-resolution life cycles of

Polavarapu and Peltier (1990). Despite differences in the flow at the smallest scales the large

scale evolution of the life cycle is as seen in the T42 cycle.

The low level temperature distribution in the T341 cycle is shown in Figure 4.4. On

comparison with the T42 fields in Figure 4.2, the most striking difference is in the strength

of the surface fronts; clearly in these cycles the horizontal temperature gradient is limited by

the model resolution. The warm sector pushing poleward during days 6 and 7 is of smaller

area in the high resolution cycle, while the cold regions on either side are wider and generally

colder. By day 8 these cold regions have been cut off from the polar reservoir where the air

originated, but in contrast to the T42 cycle they remain large in area and extend equatorward

up to the very edge of the frontal region. This demonstrates the extent to which air masses of

differing temperatures are mixed by the diffusion acting on temperature in the T42 life cycle.

Figure 4.5 shows the PV field on the 330 K isentropic surface in the T341 simulation for

comparison with Figure 4.3. The initial conditions for the life cycle have a maximum PV value

attained not at the pole but at roughly 60N. In this figure, the cyclonic wrap-up poleward of the


Figure 4.4: Temperature on the lowest model level from the T341 LC1 cycle of the IGCM.Colours are as in Figure 4.2 with contours again every 5 K from 295 K.


Figure 4.5: PV on the 330 K isentropic surface from the T341 LC1 cycle of the IGCM. Con-tours are drawn every 0.5 PVU and colours are identical to Figure 4.3.


jet is clearly seen as filaments of this maximum PV air wrap up with filaments of lower PV air

from the surrounding regions. In the T42 cycle this is inaccurately represented as an increase

in the value of the PV maximum. The troughs of high PV air which extend equatorward

demonstrate the same qualitative behaviour as at low resolution but in an exaggerated manner.

They have thinned more and are sheared further to the west by the mean flow. Air of a higher

PV value is cut off from the end of the trough; this air has clearly been advected from the

region of the PV maximum at 60N. Lying south of the zonal jet this high PV anomaly remains

roughly stationary as a small, intense cyclone event which, by day 9, can be seen to induce a

cyclonic turning in the remnants of the trough.

4.5 Initial Conditions for the Isentropic Model Simulation

In order to perform a simulation of the LC1 life cycle using the isentropic model developed

in Chapter 3, the initial conditions from the IGCM run were interpolated onto the isentropic

model grid. This initial state comprises both the zonal mean background state and the normal

mode surface pressure perturbation. The following algorithm was used to transform the data

from an initial output of the T42 IGCM:

1. The IGCM output is first processed using the Tradv program written by John Methven to

give the meridional wind, zonal wind and temperature on normalised pressure surfaces,

and the surface pressure, all on the model’s latitude-longitude spectral transform grid.

The IGCM predicts vorticity and divergence on normalised pressure surfaces, so it is

important to interpolate the winds onto the isentropic model levels and then re-calculate

the vorticity and divergence, since these derivatives should be taken along the isentropic

surfaces.

2. Due to the symmetry assumptions of the IGCM cycle (see Section 4.2), this data only

covers one twelfth of the globe. This patch is simply copied to cover the sphere, assuming

symmetry about the equator.

3. The pressure on normalised pressure levels is readily obtained using the surface pressure,

and is then interpolated onto the isentropic model half levels ηh by assuming that ln p is

a linear function of ln η. This gives the pressure ph associated with the isentropic model

density σ .


4. A second pressure ph is defined by ph = min(ph, ps), which will be associated with the

second isentropic model density σ . Values of the two pressures in the first η level below

the IGCM domain are chosen by extrapolating from the IGCM data assuming again that

ln p is a linear function of ln η. These pressure values are used to give density values

in grid boxes which intersect the ground. At this stage the pressure ph is given dummy

values in the massless region; σ values here are chosen directly later.

5. The two densities σ and σ are then calculated from the pressures according to

σ = −1

g

∆ph∆ηh

σ = −1

g

∆ph∆ηh

. (4.1)

The definition of ph then gives σ = 0 in the massless region. Massless cells are given

a σ value by simply copying σ horizontally at constant longitude along a level from

the intersection with the ground. This definition of massless σ is unique because of the

imposed symmetry about the equator, and is designed so that σ represents the limit of

−1g∂p∂θ

within the level as it becomes massless.

6. Zonal and meridional velocities u and v are interpolated onto η levels in the same manner

as the pressure. In the region outside the IGCM domain, i.e. the surface intersection and

below, the winds are calculated from the mass field by assuming thermal wind balance.

This is given in η coordinates by eliminating the Montgomery potential M from the

hydrostatic equation∂M

∂η= π(p) (4.2)

and the equation of geostrophic balance

−∇ηM = fk× v (4.3)

to give

f∂v

∂η=cpη

∂T

∂x, −f ∂u

∂η=cpη

∂T

∂y. (4.4)

7. The PV and divergence are then calculated on η levels using centred differences.

8. At this point the prognostic variables Q, δ, σ and σ have been obtained on η levels but

still as fields on a regular horizontal grid. The final step then is to interpolate the fields

horizontally onto the hexagonal grid.


The initial conditions generated by this technique proved to be smooth and balanced

enough for a successful simulation of the LC1 cycle. This was not the case for the LC2 cycle,

however, for which the intermediate latitude-longitude grid had to be refined in order to reduce

imbalances in the interpolated fields. This is described in Appendix A.

4.6 Overview of the LC1 Cycle in the Isentropic Model

The isentropic model of Chapter 3 was run from the initial conditions as constructed above.

The model parameters were identical to those used in the simulations shown in Chapter 3, and

are summarised briefly here. Horizontal grid number 5 was used, which is roughly equivalent

to a spectral truncation of T50. There were 21 full levels in the vertical, of which on average

15 were above the surface. The relaxation in the massless region was exactly as described

in Section 3.3, with a timescale of 30 minutes, and the only explicit diffusion was a hyper-

diffusion acting on the divergence field, with a timescale of 12 hours on grid-scale features.

Despite the strong relaxation employed in the massless region, an instability developed,

and the model crashed during the tenth day of the life cycle. The instability is limited to the

massless region and appeared only a few hours before the end of the integration; there was no

effect on the above ground flow up to and including day 9. This is examined in more detail in

Section 4.8.

Figure 4.6 shows the evolution of the global average EKE for the LC1 cycle in the isen-

tropic model and for the T42 IGCM. The isentropic model has completed much of the decay

phase of the life cycle by the time it crashes, indicating that the terminal instability is not likely

to occur in response to forcing by any above ground feature of the wave evolution.

In general, the EKE growth is slower in the isentropic model than in the IGCM, and the

maximum attained is lower. The growth stage, however, has a longer duration and the de-

cay rate is also slower than in the IGCM. This could perhaps indicate that over the cycle the

two waves exert similar amounts of work in terms of transporting heat polewards, though this

cannot be decisively inferred from this data.

The lower panel of Figure 4.7 shows the zonal mean structure after 7 days of integration

of the isentropic model. Isentropes at mid-latitudes are closer to horizontal as available po-

tential energy has been removed from the basic state by the wave, and baroclinicity has been

destroyed. In line with thermal wind balance the zonal mean jet extends down towards the

surface with a reduction of vertical wind shear in the mid-latitudes. In general the jet by day 7


0 5 10 150

2

4

6

8

10

12x 10

5

Day

Eddy Kinetic Energy (Jm−2)

Figure 4.6: Eddy Kinetic Energy (EKE) averaged over the Earth’s surface during the evolutionof the LC1 cycle. The isentropic model is shown as a solid line, and the IGCM dashed.

is slightly weaker and broader than that predicted by the T42 IGCM.

Figure 4.1 shows the same picture for the IGCM life cycle. One of the clearest differences

is in the distribution of isentropes intersecting the surface. The IGCM zonal mean shows a

surface cold region which is isolated at mid-latitudes by warm air on its poleward side, while

the isentropic model does not show this warm region, at least in the zonal mean. Most of

the isentropes below 300 K in the isentropic model cycle intersect the surface close to 20N,

indicating that the surface front is stronger than in the IGCM cycle.

Further understanding of these differences is obtained by examining the low level temper-

ature distribution. Figure 4.8 shows the evolution of the temperature predicted by the isentropic

model interpolated onto the p/ps = 0.967 surface for comparison with the IGCM equivalents

shown in Figures 4.2 and 4.4. Note that to plot all the fields from the isentropic model which

are shown in this thesis the data was first interpolated from the hexagonal grid onto a regular

latitude-longitude grid. The general behaviour is the same as in the IGCM cycle; warm sectors

extending poleward turn cyclonically while the cold air moving equatorward turns anticycloni-

cally. There are, however, some differences.

The warm sectors in the isentropic model life cycle appear wider than their counterparts

in the IGCM. In the IGCM at both low and high resolution these warm sectors decay to leave


0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initial Conditions

p/p s

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Day 7

Latitude

p/p s

Figure 4.7: Zonal mean sections on a normalised pressure axis for the LC1 cycle simulatedby the isentropic model. The black lines are isentropes corresponding to the isentropic modelhalf levels, which are spaced every 6 K from 298 K (dotted) to 341 K (dashed). The slopingred line is the 2 PVU surface, and the other coloured contours show the zonal wind field at5 ms−1 intervals, with the zero contour dotted and negative contours dashed. All fields arezonal means.


Figure 4.8: Temperature from the isentropic model LC1 cycle interpolated onto the surfacep/ps = 0.967 for comparison with Figures 4.2 and 4.4. Contours are drawn every 5 K from255 K to 290 K.


cut-off regions of warm air near 30N. The cold regions equatorward are cut-off in the same

way. In the isentropic cycle, however, while warm cut-offs are visible by day 9, these fea-

tures are significantly weaker than those in the IGCM cycles. It is supposed that, in the T42

simulation at least, this cutting-off process occurring in the IGCM is largely attributable to the

action of the model diffusion on the thin filament of warm air linking the high-latitude warm

sector to the equatorward warm reservoir from which it originated (see Methven (1996)). At

T341 resolution the warm filaments visible on day 6 are very thin and are bounded by large

temperature gradients on either side, so it is probable that the diffusion has the same effect

here. The diffusion essentially transports air across isentropes, a process which cannot occur

in the isentropic model.

By day 9 the temperature field in Figure 4.8 appears disorganised in that the sixfold sym-

metry in longitude has been lost. This is perhaps to be expected given that the wave at this time

is well into the decay phase and the basic state is significantly more stable than at earlier times.

The flow is not, therefore, dominated by the strong, wavenumber 6 normal mode to the extent

that it was before. When comparing the output of the two models it should also be remembered

that the IGCM cycle is, by definition, symmetric in longitude (see Section 4.2).

In Figure 4.8 the low-level front separating the warm and cold regions does not appear to

be significantly stronger than in the T42 IGCM cycle. However, this is not the case, and this

misleading impression serves to illustrate that care must be taken when interpreting the output

from the isentropic model. Figure 4.8 was created by interpolating the temperature from the

isentropic model onto a normalised pressure surface assuming that ln η is a linear function of

ln p. In the isentropic model, however, the strength of a front is represented by the distribution

of model levels which can become infinitesimally close to each other. By simply interpolating

onto a regular grid on a normalised pressure surface the temperature gradient is limited by the

horizontal grid scale and the advantage of the isentropic coordinate is lost. The details shown

in Figure 4.8 in regions of strong temperature gradients should not, therefore, be trusted.

An alternative is to view the surface potential temperature distribution by plotting the inter-

section of each model half-level with the surface as a contour line drawn where p = (1− δ)psfor some small constant δ. Results are shown in Figure 4.9 for days 7 and 9. The temperature

gradient at the front is indeed increased when compared with that in Figure 4.8. The distribu-

tion to the north of the front is much the same, however, indicating that the interpolation used

to produce Figure 4.8 is valid where isentropes are closer to horizontal. The surface fronts are

studied in more detail in Section 4.9.


Figure 4.9: Surface θ from the isentropic model, obtained by plotting the line where p =0.995ps for all intersecting model half-levels.

The upper level wave is illustrated in Figure 4.10 by the distribution of PV on the 331 K

isentropic level for comparison with Figures 4.3 and 4.5. The cyclonic wrap-up poleward of the

jet is visible here, which is an improvement over the T42 cycle. As discussed in Section 4.3, the

T42 cycle in this region suffers from a spurious increase of the PV maximum near 30N. The

opposite effect occurs in the isentropic life cycle where the maximum is spuriously reduced. In

the isentropic model the only process affecting the PV above the surface is the advection. It is

therefore certain that this spurious reduction of the PV maximum is due to the diffusion which

is implicit within the advection scheme, arising because a quadratic upstream interpolation

is used to determine the interfacial PV value. Thuburn (1995) tested the advection scheme

used in the isentropic model, estimating that the diffusive effect is roughly equivalent to a

∇4 diffusion operator. Furthermore, an effective resolution was calculated as the length scale

at which ‘spreading by advection balances the generation of small scales by straining’, and

the fact that this is similar to the grid scale was taken to imply that the cascade of potential

enstrophy to small scales, and its subsequent dissipation, is well represented by the scheme.

The troughs of high PV air which turn anticyclonically on the equatorward side of the jet

are broader and weaker in the isentropic model, though they do extend to the same latitude as

in the IGCM cycles. On day 9 the anticyclonic shear of the troughs is weaker than in either of

the IGCM cycles. It is most likely that the implicit diffusion is also responsible for the broad

nature of the troughs in the isentropic model. The high PV air extends south from the polar

reservoir through the region of strong westerly wind in the jet centre. The jet advects the high


Figure 4.10: PV on the 331 K level of the isentropic model during the LC1 cycle. Contoursare drawn every 0.5 PVU from 0.5 PVU to 6 PVU, and the colours are identical to Figures 4.3and 4.5.


PV trough in the direction perpendicular to its orientation; the trough, therefore, is bound to be

broadened by the implicit diffusion. The reduction in the PV maximum discussed above is due

to the implicit diffusion and is a significant feature of the PV evolution. It must, therefore, be

concluded that the trough is also significantly affected by the implicit diffusion.

As both the troughs and the ridges in between are eroded into each other, the edge of the

polar reservoir of high PV air is essentially broadened. The PV gradient is weakened which

agrees with the weaker and broader zonal jet seen in the isentropic life cycle. A weaker, broader

jet has less cyclonic shear to the north and less anticyclonic shear to the south, corresponding

to a reduction in the cross-jet vorticity gradient. It is within the anticyclonic shear south of

the jet that the troughs turn dramatically in the IGCM life cycles. The effect of the weakened

shear in the isentropic model compared to the IGCM is to reduce the anticyclonic turning of

the troughs, as seen.

A picture thus emerges of an underlying difference between the life cycles predicted by

the two models. In the isentropic model the PV is diffused on isentropes resulting in the

weaker, broader nature of features such as the jet and the troughs. In the IGCM, however,

PV maxima are increased, resulting in stronger PV gradients and correspondingly stronger,

narrower features. Further evidence to support this theory is given in the following section.

This explains more of the differences between the zonal mean states of the life cycles, as

predicted by the two models (Figures 4.1 and 4.7). Both zonal means show a front of strong θ

gradients near 60N in the lower troposphere, with low-level easterlies to the north. However,

in the isentropic model these features are again weaker and broader than in the IGCM. This is

consistent with the erosion of the strip of high PV at this latitude in the isentropic model life

cycle (seen clearly at 331 K in Figure 4.10), which results in weaker wind shear. In the T341

IGCM life cycle (Figure 4.5) the strip of high PV values has clearly been well maintained.

Neither of the two models exactly reproduces the Lagrangian conservation property of the

PV. The truly adiabatic evolution of the life cycle must lie somewhere between the two. A

comparison of the relative error in the conservation of PV is given in Section 4.12.

4.7 The IGCM LC1 Cycle with∇4 Hyper-diffusion

Thuburn (1995) estimated the diffusive effect of the isentropic model’s advection scheme to

be roughly equivalent to a ∇4 hyper-diffusion. To support this, and to demonstrate that the

difference in dissipation explains most of the differences between the two models, the T42


0 5 10 150

2

4

6

8

10

12x 10

5

Day


IGCM ∇6

IGCM ∇4

Isentropic model

Figure 4.11: Eddy kinetic energy evolution, as in Figure 4.6, but now for all three simulations.

IGCM life cycle was repeated with the hyper-diffusion changed from∇6 to∇4 (while keeping

the diffusion timescale fixed at 4 hours).

Figure 4.11 shows the EKE evolution of the LC1 cycle, as in Figure 4.6, but with the

IGCM∇4 run added. The EKE evolution in the isentropic model is indeed much closer to that

of the ∇4 run, both in the timing, and magnitude, of the maximum. Figure 4.12 shows the PV

field at 330 K for days 8 and 9 of the IGCM ∇4 life cycle. Here the troughs of high PV air are

broader than in the ∇6 simulation (Figure 4.3), they do not shear to the west as dramatically,

and they show no sign of being cut-off from the polar reservoir. This behaviour is also closer to

that of the isentropic model than it is to the ∇6 run. Note that the spuriously large PV maxima

near 60N are still a major feature of the∇4 life cycle.

The life cycle predicted by the isentropic model is, therefore, very similar to that predicted

by the IGCM with ∇4 hyper-diffusion, which is in agreement with the estimate of Thuburn

(1995). However, this does not mean that the diffusive effect of the advection scheme is identi-

cal to a∇4 hyper-diffusion. For example, if the velocity is zero then there will be no advection,

and hence no diffusion. The ∇6 version of the IGCM life cycle is the standard, as studied by

Thorncroft et al. (1993), and so it is this one with which the isentropic model is compared in

the following sections.


Figure 4.12: PV on the 330 K isentropic surface, as in Figure 4.3, but for the IGCM ∇4 lifecycle. Only days 8 and 9 are shown.

4.8 The Terminal Instability

The isentropic model cycle crashed after just over 9.5 days of integration due to an instability

in a massless grid cell, where a large gridpoint PV spike rapidly developed. The location of

this cell is shown in Figure 4.13 a few timesteps before the model crashed. In the horizontal the

cell is near to the point where the warm and cold fronts meet. This is most likely a coincidence,

but the proximity of the front in general is relevant.

The vertical cross-section in the second panel of the figure shows that the cell lies entirely

within the massless region. The cell has, in fact, been empty with respect to σ for the entirety

of the run, and the cell directly above was also empty until near the end of the cycle. Between

days 8.5 and 9 some mass was advected into this upper cell from the north as the front moved

southward (signalling the spread of the surface cold region from the north). The smoothing

function F of equation (3.14) then decreased in the lower cell, resulting in a weaker relaxation

there which enabled the instability to grow in the manner described in Section 3.3.

It can therefore be concluded that the model crashed due to the unstable nature of the

massless region, and not in any direct response to a development of the interior flow. Further-

more, the interior flow cannot have been affected in any way at least until after day 9 when the

gridpoint PV spike began to grow. In fact, the surface θ and mass fields shown in Figure 4.13

appear unaffected by the instability even at this late stage. This is as might be expected since


0 20 40 60 80 100

0

10

20

30

40

50

60

Longitude

Latit

ude

0 9 18 27 36 45 54 63 72 81 90

250

256

262

268

274

280

Latitude

Isen

trop

ic M

odel

Hal

f−Le

vels

Figure 4.13: The location of the unstable grid cell in the isentropic LC1 cycle plotted on ahorizontal plane (left) and a meridional section (right). The flow is represented in the horizontalplane by the surface θ distribution. In each case the cell is marked by selected PV contourswhich are coloured. In the right-hand plot the solid black contours show the locations of theIGCM p/ps levels to indicate the mass field and the dashed black line indicates the location ofthe surface, plotted as the p = (1− 10−5)ps contour.

the model is, after all, formulated to prevent contamination of the above ground flow by the

artificial massless region.

Note that the precise timing and location of the instability is sensitive to the choice of

machine and of compiler, indicating that there is some sensitivity to the truncation errors. In

a run performed on a different machine, with a different compiler, the instability occurred at

roughly the same time but in a grid cell at the surface, rather than in an entirely massless cell

as in the run described in this chapter.

The instability could be addressed in future development by attempting a smoother, or

more gradual implementation of relaxation with depth. Preliminary attempts to do this have

been made by applying a gradual shift from relaxation at depth, to a slow evolution or a dif-

fusion near the surface. While the resulting schemes have been as successful as the standard

relaxation, there has not been a significant improvement in stability. This has not been pur-

sued further; the intention instead is to redesign the surface formulation so that relaxation is

unnecessary (see Chapter 6).

4.9 Surface Fronts

There has been significant interest within the atmospheric science community in the question

of whether the strength of atmospheric fronts has any limitation in the theory of inviscid fluid

dynamics. Hoskins and Bretherton (1972) recognised that the quasi-geostrophic equations are


invalid for flows with gradients in temperature and velocity comparable to those observed in

fronts. Accelerations tangential to the front were considered important, and they developed a

semi-geostrophic (SG) model in which geostrophic balance is assumed only in the cross-front

direction. In this model Hoskins and Bretherton showed that any true discontinuity in θ can-

not form first in the interior of the fluid, but must do so at the boundary. The model was then

solved analytically for a variety of forcing mechanisms, showing that there is indeed a tendency

to form discontinuities on the boundary in finite time. Cullen and Purser (1984) extended this

model past the formation of a frontal discontinuity by assuming that the Lagrangian conserva-

tion law form of the SG equations remains valid for the majority of fluid parcels. In these SG

solutions a discontinuity formed first on the boundary, but then extended into the interior of the

fluid.

Shapiro et al. (1985) reported on the observation of an atmospheric front over Boulder,

Colorado in which the majority of the temperature gradient occurred over a horizontal scale

of just 200 m. Motivated in part by this, Gall et al. (1987) simulated deformation frontoge-

nesis in a high-resolution, non-hydrostatic primitive equation model with no explicit diffu-

sion (although they do acknowledge the presence of implicit smoothing within their difference

schemes, and an explicit smoothing near the boundaries to reduce wave reflections). The min-

imum horizontal scale of their fronts was never less than a few kilometres, and vertical resolu-

tion was shown to be the limiting factor in the frontal gradient; no natural limiting process was

found. For further investigation, see also the study of Snyder et al. (1993).

Garner (1989) used a fully Lagrangian numerical model of the 2D hydrostatic primitive

equations to simulate a similar deformation front. The grid elements moved to follow the

flow while conserving both their volume and PV exactly, except for time-differencing errors.

A frontal singularity in finite time was demonstrated by the co-location of two model grid-

points on the boundary, after which the model becomes invalid. Garner’s simulations therefore

support the theory of Gall et al. (1987) that the strength of frontal gradients has no inviscid

limitation.

In a similar study, Fulton and Schubert (1991) took advantage of the Lagrangian na-

ture of the isentropic vertical coordinate to model surface frontogenesis, combining it with

a geostrophic coordinate in the horizontal in a SG model. This model also produced a true dis-

continuity in finite time as the transformation from geostrophic to physical space broke down

and θ became multi-valued at some point on the surface.

The difference in frontal strength shown by the IGCM in Figures 4.2 and 4.4 is in agree-


60 70 80 90 100

0.4

0.5

0.6

0.7

0.8

0.9

1

Isentropic Model

p/p s

Longitude60 70 80 90 100

T42 IGCM

Longitude

60 70 80 90 100

0.4

0.5

0.6

0.7

0.8

0.9

1

T213 IGCM

p/p s

Figure 4.14: Cross sections of the surface fronts at 35N on day 7.5 in the isentropic model, andin the IGCM at T42 and T213 resolution. Isentropic surfaces coincident with the half-levels ofthe isentropic model are shown.

ment with Methven (1996) that the frontal gradient in the LC1 cycle is limited by spectral trun-

cation and the model’s hyper-diffusion. It is therefore of interest to examine the frontal gradient

in the isentropic model in more detail. Figures 4.8 and 4.9 highlight that care must be taken in

interpreting the output of the isentropic model. In the rest of this section the fronts are shown

by plotting the pressure and height of the model half-levels. The only interpolation needed is

to translate pressure from the hexagonal horizontal grid onto a regular latitude-longitude grid.

Figure 4.14 compares the fronts predicted by the isentropic model with those predicted by the

IGCM at both high and low resolutions. The fronts in the isentropic model are indeed much

more similar to those in a T213 simulation than those in the T42 life cycle are. At the cold front

in particular a very strong surface θ gradient occurs, as several isentropic model levels intersect

the surface very close together. It is clear that one of the main potential benefits of an isentropic

approach has been realised; a sharp front has been well represented by extreme deformation

of the model levels, which are squeezed tightly together along the front. The question, then, is

how tightly?

Figure 4.15 shows close-ups of the surface front at days 6 and 8. All of the isentropic levels

become massless when they intersect the surface; a frontal discontinuity is predicted when two


0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

p/p s

Latitude

Day 8

27 31.5 36 40.5 45 49.5 54 58.50

200

400

600

Hei

ght (

m)

Latitude

Day 8

0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

p/p s

Day 6

27 31.5 36 40.5 45 49.5 54 58.50

200

400

600

Hei

ght (

m)

Day 6

Figure 4.15: Meridional sections showing the locations of the model half-levels at 22.5E. Theleft-hand panels show the whole northern hemisphere, plotting the normalised pressure on eachmodel half-level. The right-hand panels focus on the region in the vicinity of the front, this timeshowing the height above the surface of each half-level. In all panels the 298 K half-level isdotted, and the spacing is 6 K for all the half-levels below this.

or more levels become massless at exactly the same point. The latitude of each of the regular

grid cells is shown as an increment on the x axis of the right-hand panels. It is then clear from

these right-hand panels that by day 8 all of the levels which comprise the front are intersecting

the surface in the same horizontal grid cell. However, each cell is roughly 500 km wide, so to

look closer we need to examine the implied sub-gridscale mass distribution.

On each isentropic level the intersection with the ground is defined by the σ distribution.

The sharp σ feature at the surface is advected using a linear sub-grid fit as described in Sec-

tion 3.4.3. When viewing the output from the model the same sub-grid fit should be used to

identify the distribution of mass within the intersecting grid cell. This technique simply redis-

tributes the mass in the intersecting cell and its non-empty neighbour to lie under a straight

line, while conserving the mass within each cell. For more details see Section 3.4.3.

The results obtained by applying the sub-grid fit to the frontal region in the model output

are shown in Figure 4.16. It appears that, at this longitude, a true discontinuity in surface θ

has developed at the front by day 7.5. Not only do multiple levels intersect within the same


0

100

200 Day 4.5

Height in metres of model half−levels

0

100

200 Day 5.5

0

100

200 Day 6.5

0

100

200 Day 7.5

0

100

200 Day 8.5

20 25 30 35 40 45 50 550

100

200 Day 9.5

Latitude

Figure 4.16: The height of intersecting model half-levels at 22.5E given by using the linearsub-grid fit to reconstruct the distribution of mass within the intersecting grid cell and its non-empty neighbour. The location of each half-level is only plotted for the two grid cells in whichthe linear fit is valid.


0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000

4500

Day

dz (

m)

Minimum Vertical Spacing

0 2 4 6 8 10−2

0

2

4

6

8

10

12

14

16

Day

dx (

Deg

rees

of L

atitu

de)

Minimum Horizontal Spacing

Figure 4.17: Two different measures of the strength of the surface front. The left-hand panelshows the global minimum vertical spacing between selected model half-levels, with the pro-viso that both are at least 10 m above the ground. The right-hand panel shows the minimumhorizontal spacing between the intersection points of the same levels using the sub-gridscalefit. In both panels the upper line is the spacing between the 268 K and 292 K half-levels andthe lower line is the spacing between the 280 K and 292 K half-levels. These ranges boundfour and two full model levels respectively.

horizontal grid cell at the front, but in the reconstruction of the sub-grid mass distribution the

levels intersect at almost exactly the same point on the surface.

Further evidence that a discontinuity is predicted, and an impression of its time evolution,

is given by Figure 4.17. The left-hand panel shows the minimum vertical spacing attained

between two half-levels while both are at least 10 m above the surface. The sub-grid fit is

not used to produce this plot. The spacing decreases rapidly towards zero as the front forms,

reaching minima of just 1.2 m and 7 cm for the two half-level pairs shown.

An alternative approach to quantifying the strength of the front is to calculate the horizon-

tal spacing in latitude of the intersection points of the sub-grid representations of various levels.

Results are shown in the second panel of Figure 4.17 for the same two pairs of half-levels. In

both cases the minimum horizontal spacing exhibited at some point along the front becomes

negative in finite time. This means that in the sub-grid reconstruction two isentropic surfaces

have intersected just above the surface. However, it should be remembered that the sub-grid fit

is only a linear approximation to the mass field near the front. Compared to a horizontal grid

scale of 4.5 the overlap indicated by the negative spacing is small, and certainly within the

margin of error that should be expected of the linear fit. This figure is strongly suggestive of


0 2 4 6 8 10−6

−4

−2

0

2

4x 10

−6

Day

Divergence (s−1)

0 2 4 6 8 10260

280

300

320

340

360

Day

Density σ (kg m−2 K−1)

Figure 4.18: Divergence and model density σ at the surface front for each of the four modellevels bounded by the 268 K and 292 K half-levels. At each longitude the fields are taken inthe intersecting grid cell for each level, and the zonal mean of these values is calculated.

an actual singularity in finite time but should not be taken as proof of the propagation of this

singularity into the interior of the fluid.

The first panel of Figure 4.18 shows the divergence at the front on the four appropriate

model levels. The intention here is to show that the development of the frontal discontinuity

is not simply the consequence of a spurious divergence caused by the problem described in

Section 3.7. The figure clearly demonstrates that this is not the case; there even appears to be a

slight convergence along model levels during the formation of the front. The frontal divergence

values are an order of magnitude smaller than the surface divergence observed in the solid body

rotation simulation, and are of the same size as features in the divergent field above the surface.

In general, although the surface divergence problem has affected the LC1 cycle the effect is

small. This is discussed in more detail in Section 4.13.1.

The strength of frontal gradients is often measured by the ratio of the absolute vorticity

to the planetary vorticity f . Cullen and Purser (1984) reported vorticities of up to 12f in

both their SG and primitive equation simulations, a value near the maximum allowed within

the validity of their SG equations. In their baroclinic life cycle simulations, Polavarapu and

Peltier (1990) noted maximum vorticities of 6f in a run at 50 km horizontal resolution, and

11f in one at 30 km. At the fronts, isentropic surfaces slope relatively steeply to the surface,

so that a velocity gradient taken on an isentropic surface will be different from one taken on a

constant height or pressure surface. It is this latter vorticity, rather than the isentropic vorticity

k · ∇θ × v, which should be compared with the values given above. To calculate this in the

isentropic model we would have to take the surface wind either side of the front and divide this


by the distance across the front. However, this distance vanishes because a frontal discontinuity

is predicted. The wind fields could be interpolated onto a normalised pressure surface, and the

vorticity calculated by finite differencing, but the result would depend strongly on the method

of interpolation and the resolution used on the pressure surface. This has not been attempted

here.

The isentropic model clearly demonstrates a frontal discontinuity as the mass σ between

two isentropic surfaces vanishes. However the zonal mean evolution of the second density

variable σ (shown in the final panel of Figure 4.18) decreases during frontogenesis, but does

not approach zero. To determine if this is realistic we have to consider what σ represents. The

surface fronts are formed by the intersection of model levels with the ground, as occurs in the

cell labelled B in Figure 3.5. To the right of the projected intersection point (i.e. for x > x0)

the static stability will be very large, as model half-levels squeeze close together. However, to

the left (x < x0) the level is massless, and the static stability just above the surface will be

small, as seen in Figure 4.15. In massless regions σ represents the limit of ∂p∂θ

just above the

surface, and will be large here. The σ value in the cell is an average over the whole cell, and

hence an average of these very different values, and as such, it should not vanish.

The isentropic model simulation therefore supports the hypothesis that there is no limita-

tion to the frontal gradient within the inviscid primitive equations. It should be noted however

that in the real atmosphere gradients will be limited by small-scale physical processes and var-

ious forms of instability. For example, Hoskins and Bretherton (1972) suggested that since the

Richardson number can be approximated by f/ζ it must become small in strong frontal regions,

thus satisfying the necessary condition for Kelvin-Helmholtz shear instabilities to occur.

4.10 Surface PV Anomaly

Zonal sections of the PV and θ distributions in the two models are shown in Figure 4.19, taken

after 7.5 days of integration when both waves are practically at the end of the growth phase.

One of the most noticeable differences is the large PV anomaly at the surface fronts predicted

by the IGCM. Real cyclone systems have been observed in which large PV anomalies were

apparent at low levels (e.g. Hoskins and Berrisford (1988) and Wernli et al. (2002)). These

anomalies, however, are considered to be products of latent heating and so are clearly distinct

from those in the IGCM cycle.

PV anomalies at surface fronts are a common feature of dry ‘adiabatic’ simulations, and


60 80 100 120 140 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Longitude

p/p s

Isentropic Model at 40N

40 60 80 100 120 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Longitude

IGCM at 40N

Figure 4.19: Zonal sections at 40N of PV and θ on day 7.5 for the isentropic model (left) andthe T42 IGCM (right). Solid contours are the PV every 0.5 PVU up to 5 PVU, and the dottedcontours are the θ surfaces coincident with the isentropic model half-levels.

have been studied by several authors. The anomalies are of particular interest in simulations

of two-dimensional Eady Waves. These two-dimensional waves are unable to develop the

meridional temperature structure which is fundamental in bringing about the equilibration or

halt in the growth of three-dimensional waves. Nakamura and Held (1989) attribute the wave

equilibration in this case to an effective increase in stability linked with the PV anomaly in the

frontal region.

The source of the anomaly was identified as follows, using the concept of boundary PV

first introduced by Bretherton (1966). Isentropic layers which intersect the boundary are con-

sidered to extend along the surface as layers of infinitesimal thickness in the same way as in

the isentropic model described here. The inhomogeneous boundary condition of a varying θ

on the surface is then replaced by a homogeneous condition of constant θ, with the variation

represented instead by variations in PV along the nearly massless layers. During the occlusion

of the fronts, fluid is diabatically transported into the massless layers by diffusive mixing. The

massless layers are thus lifted from the surface, intruding boundary PV into the interior to form

the anomaly in interior PV seen in the simulations. This is in line with the theorem of Haynes

and McIntyre (1987) which allows creation or destruction of PV by diabatic processes at the

intersection of an isentropic layer with the boundary. These diabatic processes, such as the

diffusive mixing, can then be thought of as exchanging interior and boundary PV along the

isentropic layer, while the sum of the two types of PV in the layer is conserved. For a recent


extension of the boundary PV concept see Schneider et al. (2003).

Later, Garner et al. (1992) modelled an Eady wave which equilibrated without explicit

diffusion by replacing the rigid boundaries with semi-infinite regions of high PV. The equi-

libration was then attributed to a vortex roll-up of this boundary PV as an intrusion into the

interior. In the limit that the value of the boundary PV is infinitely larger than that of the in-

terior PV the boundary represents a rigid boundary and the PV intrusion becomes infinitely

thin. The role of diffusion was then viewed as spreading this PV intrusion onto the grid scale,

effectively giving resolution to frontal discontinuities as PV structures. Cooper et al. (1992)

studied various aspects of the surface PV production, concluding that significant amounts of

interior PV are produced but also that the occlusion itself is not necessary for the production of

interior PV, which only requires a change in the mean surface θ.

In three dimensions the equilibration of the wave is largely due to barotropic processes

as described by Simmons and Hoskins (1980). Of particular importance is the reduction in

baroclinicity and increase of static stability associated with the reduction of the meridional

temperature gradient by the wave itself. The PV anomalies, however, are still common features

and are thought by some to have an important effect. Ziemianski and Thorpe (2000) used the

primitive equation baroclinic wave simulation of Rotunno et al. (1994) to perform an attribution

analysis of the flow induced by a frontal PV anomaly. They found that up to 10% of the

vertical velocity acting to lower the tropopause fold could be attributed to the PV anomaly,

hypothesising that this resulted in a stronger coupling of the upper and lower waves. This, they

claimed, would counter the direct stabilising effect of the anomaly described by Nakamura

and Held (1989) in the Eady wave. Ziemianski and Thorpe (2002) went on to demonstrate a

simulation of the wave by a nonlinear balance model in which the PV was exactly conserved.

Frontal temperature gradients were stronger in the balance model, and no surface PV anomalies

were produced. In contrast, frontal PV anomalies were produced in a primitive equation model

by day 7. After this time the wave growth rate in the nonlinear balance model decreased but

that in the primitive equation model did not, adding weight to their hypothesis that the PV

anomaly acts to phase-lock the upper and lower wave together, and thus to allow continued

growth.

Methven (1996) studied the surface PV anomalies in the IGCM LC1 cycle, attributing

their generation to scale limitation of the surface front, of which the dominant cause is the

diffusion on temperature. This effectively transports mass diabatically into the massless layers

and so extrudes boundary PV into the interior in the manner described above. The mass-


weighted domain-averaged PV increases as a result. In the isentropic model life cycle, however,

there is nothing which limits the horizontal scale of the surface front. The so-called occlusion

event in which the surface warm sector decays takes place solely due to horizontal advection

along isentropes, and there is no frontal PV anomaly produced. The total PV contained in the

intersecting isentropic model levels does change in a manner similar to that seen in the surface

divergence problem of Section 3.7, but the mechanism is entirely different and the effect is

small (see Section 4.13.1).

The zonal sections of Figure 4.19 show that the troughs, or tropopause folds, in the IGCM

cycle do indeed penetrate further into the troposphere than their equivalents in the isentropic

cycle, on first impressions supporting the hypothesis of Ziemianski and Thorpe (2002). How-

ever, the folds in the isentropic model are broader, a consequence of the diffusion on PV which

is implicit in the advection scheme (see Section 4.6).

It is therefore interesting to compare the growth rates of the waves, as implied by the EKE

evolutions of Figure 4.6, in search of evidence to support the hypothesis that the surface PV

anomalies act to prolong the growth stage. The growth rate in the isentropic wave does indeed

begin to decrease slightly before that in the IGCM wave, but the equilibration itself occurs

later. The EKE comparison is therefore inconclusive. In Figure 4.19 there is also no apparent

difference between the two models in terms of the simulated phase difference between the

upper and lower waves. Upper and lower level fields have also been compared in more detail

and there is no significant difference in the phase differences predicted by the two models.

It must be concluded, then, that there is no evidence here to support the theory of Ziemian-

ski and Thorpe (2002) that the PV anomalies are prolonging the growth stage of the life cycle.

This, however, does not disprove the theory. The waves predicted by the two models have dif-

ferent behaviour in fundamental quantities such as the growth rate of EKE; the two waves are

by no means identical, even in the absence of the frontal PV anomaly predicted by the IGCM.

A further possibility is suggested now. Following Garner et al. (1992), the diffusive mixing

of boundary and interior PV in the IGCM is viewed as acting ‘to give resolution to frontal

discontinuities as PV structures’. The surface PV anomaly in the IGCM and the apparent

surface θ discontinuity in the isentropic model can therefore be viewed as two different, but

equivalent, representations of the same dynamical feature. It is hypothesised, then, that the

effect of this feature on the surrounding wind fields, and thus the large scale flow, is similar in

both representations. At the ‘occlusion’ the IGCM predicts a positive PV anomaly, while the

isentropic model instead has a warm surface θ anomaly between the stronger cold and warm


fronts. These two anomalies are indeed expected to have similar dynamical influence according

to PV-θ theory (e.g. Hoskins et al. (1985)).

Both model simulations could be considered unrealistic representations of a truly adia-

batic front in the primitive equations for various reasons. However, the front predicted by the

isentropic model is much more similar to a high resolution simulation than that predicted by

the T42 IGCM is. Furthermore, since the IGCM representation involves the production of both

mass-weighted interior PV and entropy, it is surely a worse approximation to a truly adiabatic

frontal evolution than that predicted by the isentropic model, in which the entropy is exactly

conserved and the change in PV is significantly smaller. These, and other, conserved quantities

are now discussed in detail.

4.11 Global Conservation Properties of the Two Models

The real atmosphere has an infinite number of conserved quantities of which only a limited

number can be conserved in a numerical model. The two models compared here have been

formulated with very different conservation properties in mind. In addition, the IGCM uses

a vertical scheme derived from that of Simmons and Burridge (1981) which conserves both

energy and angular momentum, although neither are formally conserved by the horizontal

spectral scheme or by the time differencing. The isentropic model, on the other hand, exactly

conserves the mass, mi say, in each isentropic layer of constant θi, and therefore also the

entropy since this is given bymicp ln θi. The mass-weighted PV in levels which do not intersect

the ground is conserved, but in intersecting levels this quantity varies due to the mixing of

boundary and internal PV by the diffusion implicit in the advection scheme.

A comparison of conserved quantities over the two LC1 life cycles is shown in Figure 4.20.

Mass conservation is shown in the top two panels and is extremely good in both models; the

column average mass is 104 kg so the error is of order 10−4 % for both models, i.e. of the same

size as the expected round-off error.

The energy conservation properties in the second two panels are particularly interesting.

The difference in kinetic energy between the two models is mostly accounted for by the differ-

ence in EKE shown in Figure 4.6. The potential energy in the isentropic model decreases by

a larger amount than in the IGCM despite the fact that less kinetic energy has been generated

by conversion from available potential energy. This error in the isentropic model’s potential

energy must be due to an error in the available part of the potential energy since, in line with


0 5 10 15−2

−1

0

1

x 106

Ene

rgy

(Jm

−2)

0 5 10 15−2

−1

0

1

x 106

5 10 15−10

−5

0

x 10−3

Mas

s (K

g m

−2)

IGCM

5 10 15−10

−5

0

x 10−3 Isentropic Model

0 5 10 15−8

−6

−4

−2

0

2x 10

10

Ang

. Mom

entu

m (

Kg

s−2)

0 5 10 15−8

−6

−4

−2

0

2x 10

10

0 5 10 150

200

400

600

Ent

ropy

(JK

−1m

−2)

Day0 5 10 15

−40

−20

0

Day

Figure 4.20: Global conservation of quantities from the IGCM (left-hand panels) and the isen-tropic model (right-hand panels). All quantities are integrated over the finite difference gridof the respective model, then averaged over the Earth’s surface and the change from the initialvalue is plotted. In the energy plots the dashed line is the total kinetic energy, the dotted lineis the potential plus internal energy (given by cpT for a hydrostatic system) and the solid lineis the sum of these, i.e. the total energy. Note that the y axis scaling is different for the twolowermost panels.


the discussion of Section 4.1, the unavailable part is that part which cannot be extracted by an

adiabatic re-arrangement of mass within an isentropic layer. The unavailable part is therefore

defined by the mass within each isentropic layer, and this is exactly conserved. The error in the

total energy of the isentropic model is thus the imbalance between the EKE generated and the

available potential energy from which it is converted.

The error in the total energy in the IGCM is known to be simply due to the non-closure

of the energy cycle (see Blackburn (1983)); kinetic energy which is lost during the diffusion

process is not put back into the system as heat. With this in mind it is suggested that the

same process is occurring in the isentropic model. As has been shown, the PV is diffused

significantly which broadens and weakens features in the velocity distribution such as the jet.

This smoothing of the velocity field will lead to a reduction in the kinetic energy as the variance

in the velocity field is reduced, and the kinetic energy lost is again not put back into the system

as heat.

As shown later (Section 4.12) there is a general loss of interior PV to the massless region,

also attributable to the diffusive nature of the advection scheme. Velocities near the surface are

reduced by this effect. This can be seen by noting that the sum of mass-weighted interior PV

within an intersecting isentropic layer is equal to the circulation around the circuit bounding

the mass within the layer, i.e. the circuit marking the intersection of the level with the surface

(Hoskins (1991)). If the sum of mass-weighted interior PV is reduced then so is the circulation,

and thus the component of surface wind lying parallel to this circuit. The loss of PV could

therefore play a part in the loss of total energy by the isentropic model.

The conservation of total energy, and also of angular momentum (shown in the third set of

panels), is worse in the isentropic model than in the IGCM. This should probably be expected

given that the isentropic model makes no attempt to formally conserve these quantities, whereas

the IGCM does. The isentropic model errors, while larger than those in the IGCM, are certainly

of a similar scale. To put the errors in perspective the drift in angular momentum corresponds

to 0.3% of the total, and that in energy just 0.05%.

The potential temperature θ satisfies a Lagrangian conservation law(DθDt

= 0)

under adi-

abatic motion, so that not only is θ conserved for any given fluid parcel, but any function of θ

is as well. The entropy s = cp ln θ is just one of the infinite number of functions of θ which

are conserved by the continuous primitive equations, although it does have a particular phys-

ical significance. The entropy, and in particular one potential consequence of not conserving

it in models, is discussed at length in Chapter 5. Here the conservation of entropy is simply


Model Entropy error Energy error Imbalance

T42 IGCM 550 -2000 2550θ Model -50 -5000 4950

Table 4.1: Errors in the second law of thermodynamics by expressing energy errors as theimplied entropy error using equation (4.6). Changes in entropy and energy were taken over thelife cycles and values of T = 250 K and ∆t = 10 days were assumed. Units are Jm−2K−1.

compared for the two models.

The final two panels show the change in entropy over the course of the two life cycles.

While the IGCM shows a systematic increase in global average entropy the change in the en-

tropy of the isentropic model is simply a reflection of the error in global mass due to rounding

error. This time the errors in the two models do appear to have significantly different scales.

However it should be asked whether the reduction in entropy error obtained by using the isen-

tropic model is significant given the increase in the error in the energy.

An obvious way to relate the energy and entropy errors is through the second law of ther-

modynamics, given bydS

dt=Q

T(4.5)

in the absence of friction. Over a fixed time period ∆t this reduces to an equation

∆S∆E =∆E

T(4.6)

for the entropy error ∆S∆E equivalent to an energy error ∆E. Since both models are suppos-

edly adiabatic, both sides of equation (4.5) should be zero for each model. The relative errors

can therefore be compared if they are viewed as errors in the representation of the second law

in each of the models. Table 4.1 compares the errors in entropy and energy, both expressed as

entropies using equation (4.6), and also the imbalance in the second law which is implied. The

imbalance can be interpreted as the entropy error which would be implied if the energy had in

fact been conserved exactly. A temperature of T = 250 K was chosen as a suitable average

over the global domain. The error in the second law is seen to be greatest in the isentropic

model, which exhibits both the largest individual error and the largest imbalance. From this

point of view the energy error in the isentropic model is more significant than the entropy error

in the IGCM.


4.12 PV Conservation

In the continuous primitive equations the PV satisfies a Lagrangian conservation law in the

same way as the potential temperature. This conservation theorem was one of the motivations

for the construction of the isentropic model described here, along with the recognition that the

PV is a fundamental quantity that completely determines the balanced part of the flow (see

Section 1.1). It is therefore hoped that this model will conserve its PV better than the IGCM,

and hence that some improvement in the representation of the dynamics might be obtained.

It should be noted that there is a key difference in the nature of the conservation properties

of PV and θ. The mass-weighted integral of any function of θ over an isentropic layer is a

robust invariant, because the stratification implied by the θ distribution itself acts to inhibit

mixing across isentropes, i.e. to inhibit any non-conservative processes. No such robustness

exists for a similar integral of any function of PV over an isentropic layer. It is expected that

the potential enstrophy Q2 will cascade to small scales as thin filaments of PV are produced,

and will ultimately be dissipated in any realistic flow. In this way a given PV distribution

will evolve adiabatically in such a way that some violation of Lagrangian PV conservation is

inevitable.

The variation of PV in the two models is illustrated in Figure 4.21. The IGCM simulation

assumes symmetry about the equator and so the PV is positive everywhere. The isentropic

model, however, simulates the whole global domain, and so the total mass-weighted PV is zero.

While errors can be examined by studying the departure from zero, this gives no impression

of the size of errors relative to the absolute PV values. To overcome this it is the global mass-

weighted integral of |PV | which is plotted here. This is, after all, a function of the PV and so

should be conserved. As expected, the isentropic model conserves this better than the IGCM

does, but it does still show a slight drift.

The lower plots show the same change in |PV | but this time broken down into the con-

tributions from each of the isentropic layers corresponding to model levels in the isentropic

model. For the IGCM some interpolation is required in order to do this. For each column of

the transform grid, the mass-weighted PV in each isentropic layer is calculated as follows. The

pressure at the two θ values bounding the layer is found by interpolating from the IGCM levels

assuming a linear relation between ln θ and ln p. This gives the pressure corresponding to a

given θ value from the pressures at the model levels below (p1 say) and above (p2 < p1), and


0 5 10 15−0.01

−0.005

0

0.005

0.01

0.015

0.02|PV|

0 5 10 15−0.5

0

0.5

1

1.5PV levelwise (IGCM)

0 5 10 15−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

PV levelwise (θ Model)

Figure 4.21: PV conservation diagnostics from the LC1 cycle simulated by the two models.All quantities are plotted as the change from their initial value, normalised by the initial valueso that, e.g. a value of 0.5 indicates a 50% increase. The top panel shows global mass-weighted|PV | for the IGCM (dashed line) and isentropic model (solid line). The following panels showthe change in total PV within individual isentropic layers which correspond to the isentropicmodel levels. Levels which intersect the surface are shown dashed.


their respective θ values, as

p(θ) = p1−α1 pα2 where α =

ln θθ1

ln θ2θ1

. (4.7)

The mass-weighted PV in the layer is then given by integrating between the bounds of the

layer, labelled θb and θt say:

−1

g

∫ p(θt)

p(θb)

Q dp, (4.8)

where Q is diagnosed at the IGCM level, or levels, in the range (p(θb), p(θt)). Some error is

expected because of the need for interpolation; in Section 5.2.2 this is shown to be small for a

similar application.

The change in |PV | in the isentropic model is seen to be almost entirely due to the change

which occurs in the intersecting levels. There is, in fact, a small decrease in PV in the upper

levels due to implicit diffusive mixing of positive and negative PV across the equator; this,

however, is tiny compared to the change within the intersecting levels.

Larger variations are seen in the IGCM, and there is a clear difference between the errors

in the two models. While the isentropic model loses PV through the lower boundary, the IGCM

gains PV there through the limiting of the surface frontal gradients described in Section 4.10.

This time there is a significant drift in non-intersecting levels, of order 10% in the worst case.

The IGCM thus fails to respect the PV conservation theorem of Haynes and McIntyre (1987)

which holds for the primitive equations. This is a result not only of the non-conservation of the

PV of a fluid parcel but also of the non-conservation of its potential temperature.

The increase in maximum PV values in the IGCM cycle (noted in Section 4.4) is an exam-

ple of this lack of Lagrangian conservation. As described in Section 4.6, the implicit diffusion

in the isentropic model acts instead to erode these PV maxima. Both of these changes are

spurious; to estimate the effects on the life cycle, their magnitudes are compared. Figure 4.22

shows the evolution of the maximum PV value in a selection of isentropic levels. The IGCM

error is always larger than that in the isentropic model. The erosion of PV maxima in the isen-

tropic model is most obvious in the first panel, but the error is still significantly less than that

in the IGCM. Note that in all four cases the PV maximum corresponds to a significant feature.

For example, the increase in the IGCM PV maximum in the intersecting levels is due to the

frontal anomalies (Section 4.10). At 331 K the largest PV values occur at the local maxima

near 60N , which are seen to spuriously increase in the IGCM in Figure 4.3. It is uncertain


0 5 10 150.95

1

1.05

1.1

1.15383K Level

0 5 10 150.8

1

1.2

1.4

1.6331K Level

0 5 10 150.5

1

1.5

2

2.5

3277K Level

Day0 5 10 15

0

1

2

3

4271K Level

Day

Figure 4.22: Variation with time of the maximum interior PV value in a selection of isentropiclevels from the IGCM (dashed line) and isentropic model (solid line). For the isentropic modelinterior PV was defined as the PV in any cell which has σ/σ > 0.1. Values are normalised bythe initial value for that level. The lower panels show levels which intersect the surface.

however what the effect is of these latter increased maxima in the IGCM. Comparison with the

high resolution run (Figure 4.5) suggests that the flow in the vicinity of one of these anomalies

is largely unchanged, so the effect is small. The anticyclonic shear of the high-PV trough is

weaker at low resolution, which would be consistent with a spuriously large cyclonic influence

from the 60N anomaly. This, however, could also be due to the resolution limits on the width

of the trough, resulting in a spurious amplification of its influence as a dynamical feature, in

accordance with the scale effect.

In the intersecting levels the isentropic model can show spurious increases similar to that

seen in the IGCM, but they are significantly smaller and generally more transient. The sections

shown in Figure 4.19 show no large surface PV anomalies as seen in the IGCM. As in the IGCM

these increases are due to the mixing of boundary and interior PV due to diffusion. Here the

diffusion implicit in the advection scheme (as described in Section 3.4.1) acts to diffuse PV

across the interface along isentropic levels. The increase occurs at roughly the same time as

that in the IGCM, since this is the time when strong gradients are formed at the boundary

and the isentropic layers move most dramatically across the surface, indicating a significant

velocity normal to the intersection. Note that only the value of the PV maximum is shown in


Figure 4.22; on average interior PV is lost from these levels (Figure 4.21). Also note that these

spurious increases are limited to only those grid cells which actually intersect the boundary; if

only cells with σ/σ > 0.99 are considered then the PV maximum is practically unchanged.

The average loss of interior PV in almost all the intersecting levels in the isentropic model

run is due to the implicit diffusion in the advection step. This loss of PV is less than the error

in the PV contained within isentropic levels by the IGCM. The fact that PV is diffused into the

boundary rather than out of it is probably due to the relaxation which, by relaxing the winds to

zero, acts as a sink of vorticity.

The mixing of internal and boundary PV by diffusion is also responsible for the spreading

into the interior of the boundary anomalies caused by relaxation in the surface divergence prob-

lem described in Section 3.7. The magnitude of this problem, and the effect of the relaxation

in general, will now be assessed for the LC1 simulation.

4.13 Effect of the Relaxation Scheme on the LC1 Cycle

4.13.1 The Surface Divergence Effect

The fundamental signature of this problem is the large value of divergence at the intersection

of a model level with the ground, which can be up to an order of magnitude larger than other

features in the interior divergence field, as seen in Figure 3.16 for the solid body rotation

experiment. A careful study of the divergence field in the LC1 cycle (not shown) reveals no

such anomalies; any effect is therefore of a size no greater than the real divergent processes.

There is also no evidence of the outcropping seen as a by-product of the problem in the solid

body rotation (SBR) simulation.

The evolution of PV in the intersecting levels for the LC1 cycle compared to the SBR

simulation is summarised in Figure 4.23. The surface divergence problem in the SBR run is

mostly confined to the lowest massy level, and it is this level which shows by far the largest

error. In the LC1 cycle all the intersecting levels show a general reduction in both the total PV

and its variance, apart from the lowest massy level which shows a small increase. This increase

is almost certainly a sign of the surface divergence effect and its size shows that the effect is

much smaller than in SBR.

The effect of the surface divergence problem in LC1 is small because the surface wind

field is weaker, and less organised, than in SBR. From the lower panel of Figure 4.7 it is clear


0 5 10 15 20−0.2

0

0.2

0.4

0.6SBR

|PV

|

0 2 4 6 8 10−0.2

0

0.2

0.4

0.6LC1

0 5 10 15 200.5

1

1.5

2

2.5

Var

(PV

)

Day0 2 4 6 8 10

0.5

1

1.5

2

2.5

Day

Figure 4.23: Comparison of the PV change in the intersecting model levels between the solidbody rotation simulation (left) and the LC1 cycle (right). Upper panels show the total mass-weighted |PV | and lower panels show the mass-weighted variance of |PV |, i.e. the sum of thesquares of the deviation of |PV | in each cell from its mean value for that level. Quantities areagain normalised by their initial values, and the lowest massy level is shown dashed.

that, even when the jet has extended down to the surface, it does so to the north of the frontal

region where the levels are intersecting the ground. From this picture it looks possible that the

easterly wind maximum of flow along the front could have caused a spurious divergence which

would act to increase the frontal gradient. This, however, was disproved in Section 4.9.

The general loss of interior PV into the boundary in the LC1 cycle is not seen in the SBR

experiment. This is because in SBR the flow is largely parallel to the intersections, which

therefore move little over the surface. While this error is larger in LC1, it is significantly

smaller than the error in the SBR experiment which is caused by the surface divergence effect.

4.13.2 Sensitivity to the Relaxation

The relaxation of fields in the massless region back towards their initial values (see Section 3.3)

is necessary to improve stability, but is certainly one of the less attractive features of this isen-

tropic model. All possible care has been taken to ensure that the massless region and the

relaxation scheme do not adversely affect the flow in the interior. In the vertical scheme infor-

mation only propagates downwards in the massless region, and the surface divergence effect of


Figure 4.24: Eddy kinetic energy in Jm−2 for versions of the isentropic model LC1 cyclewhich differ only in the relaxation timescale as shown. The T42 IGCM value is shown forcomparison.

Section 3.7 has been shown to be small.

It is hard to imagine, however, that the above ground flow is entirely unaffected by the

relaxation process, especially given the implicit mixing of boundary and interior PV described

in Sections 4.12 and 4.13.1 above. To investigate the sensitivity of the LC1 simulation to the

relaxation the cycle was run several times with different relaxation timescales. The eddy kinetic

energy of each of these runs is shown in Figure 4.24. There is significant sensitivity to the

relaxation timescale, though the differences are smaller than that between the isentropic model

and IGCM runs, with less EKE is produced than in the IGCM for all relaxation timescales. The

effect of increasing the strength of the relaxation is very systematic; reducing the timescale

results in a simulation which is stable for longer, and in a wave with a larger EKE peak which

is attained slightly later. Note that all the runs are similar until day 7, i.e. until after the collapse

of the surface fronts.

The reason why there is such a systematic dependence on the timescale is unknown. As

mentioned above, one mechanism by which the relaxation could affect the interior flow is

through the PV advection. Changing the timescale will affect massless PV values near to cells

which are at least partially above ground. These ‘massy’ cells will feel the effect through the

implicitly diffusive PV advection step.


Note that the EKE in the IGCM cycle is very sensitive to the value of certain model param-

eters, so certainly doesn’t represent an absolute truth. MacVean (1983) showed that the EKE

maximum of a baroclinic wave in the IGCM could be increased by a factor of 2.5 by changing

the timescale, and scale-selectivity, of the hyper-diffusion terms. The sensitivity of the isen-

tropic model to the relaxation is an order of magnitude smaller than this. This sensitivity is

therefore interesting, but is not too alarming given the qualitative similarity between all four

runs. Apart from significant differences in the length of simulation achievable there is little to

choose between the various timescales; it is the 30 minute run which is described in the rest of

this chapter.

4.14 Smoothness of∇2M

Hsu and Arakawa (1990) and Arakawa et al. (1992) reported a problem in their isentropic

model as follows. The surface potential temperature distribution θs is determined by the lo-

cation of the model level intersections, and its resolution is given by the vertical spacing of

the isentropic levels. The horizontal pressure gradient force term is given by ∇2M , where the

Montgomery potential M is obtained by integrating the hydrostatic equation (3.8) up from the

surface as described in Section 3.2. With θs insufficiently smooth they found that ∇2M , and

thus also the geostrophic wind, became almost discontinuous at all levels above an intersection

point.

Randall et al. (2000) studied the effect further by prescribing an analytic pressure and θs

distribution (that of Held and Suarez (1994)), and then calculating the geostrophic vorticity

∇.(

1f∇θM

)using both the continuous equations, and also the discrete equations of Hsu and

Arakawa (1990). Significant computational noise was evident in the discrete vorticity and

attributed to the lack of smoothness in θs. The conclusion was that vertical resolution in an

isentropic model should be chosen so that the horizontal resolution of θs as defined by the

intersection of model levels matches the horizontal resolution of the model.

Figure 4.25 shows meridional sections of ∇2M during the LC1 cycle simulated by the

isentropic model described here. No gridscale noise is seen in ∇2M in connection with the

intersection of model levels with the surface. Figure 4.7 shows six isentropic levels intersect-

ing the surface in about 30 of latitude in the initial conditions. In this simulation the fifth

horizontal grid was used, which has roughly seven cells in this latitude band. The horizontal

and vertical resolutions are therefore fairly closely matched. If the horizontal resolution is in-


0 10 20 30 40 50 60 70 80 90

260

280

300

320

340

360

380

Pot

entia

l Tem

pera

ture

∇2M − Initial Conditions

0 10 20 30 40 50 60 70 80 90

260

280

300

320

340

360

380

Latitude

Pot

entia

l Tem

pera

ture

∇2M − After 5 days

Figure 4.25: Meridional sections at 0E of the Laplacian of the Montgomery potential. Con-tours are drawn every 0.5× 10−9 s−2, with negative contours dashed. The lowermost contourline marks the location of the surface.

0 10 20 30 40 50 60 70 80 90

260

280

300

320

340

360

380

Latitude

Pot

entia

l Tem

pera

ture

Laplacian of M − Initial Conditions

Figure 4.26: As Figure 4.25 but at higher horizontal resolution (grid 6). Here the noise in∇2Mis clearly visible.


creased then the two are no longer well matched; Figure 4.26 shows the initial conditions of

the LC1 cycle at grid number six, i.e. with approximately four times the number of cells in

the horizontal but with the vertical resolution unchanged. Here noise is clearly visible so it is

concluded that θs is not resolved smoothly enough, and therefore this problem does still exist

in the isentropic model described here. In the grid number five simulation discussed in the

rest of this chapter the resolutions are matched, indicating that this is indeed a solution to the

problem.

It is worth noting that the noise in the high resolution version is very damaging. The noise

seen in Figure 4.26 extends down into the unstable massless region in the initial conditions and

causes the model to crash almost immediately.

4.15 Summary

In this chapter the simulation of a baroclinic wave life cycle by the isentropic model of Chap-

ter 3 has been compared with its simulation by a standard normalised pressure coordinate

model, the IGCM. The principal findings are as follows:

• An underlying difference between the two models is that, while the PV distribution is

slightly diffused by the isentropic model, it is intensified by the IGCM. This contrasting

behaviour is largely responsible for the synoptic differences observed between the two

waves. The isentropic model life cycle is, in fact, much more similar to the IGCM version

with ∇4 hyper-diffusion than to the standard ∇6 version. The error in PV maxima is

significantly smaller in the isentropic model, but certain features of the T341 IGCM

wave, such as the thin, heavily sheared troughs, are better represented in the ∇6 T42

IGCM wave than by the isentropic model.

• The isentropic model predicts an actual frontal discontinuity in surface potential temper-

ature, while in the IGCM the frontal gradient is limited by hyper-diffusion on tempera-

ture and (to a lesser extent) the spectral truncation. In contrast, the surface front in the

IGCM is characterised by a weaker temperature gradient and a large PV anomaly. Both

model simulations can be considered as approximations to a truly adiabatic front in the

primitive equations. Since the IGCM simulation involves the generation of entropy, and

a much larger change in the total interior mass-weighted PV, it is a worse approximation

than the isentropic model simulation. It is the front predicted by the isentropic model


which shows most similarity to that in a high resolution version of the IGCM life cycle.

• The ability of the two models to satisfy the various conservation laws of the continuous

equations contrasts markedly. For example, the relative error in global mass-weighted

PV in the IGCM is roughly the same size as the relative error in angular momentum

in the isentropic model. The IGCM strikingly violates the impermeability theorem of

Haynes and McIntyre (1987), which is almost exactly satisfied in the isentropic model

in all non-intersecting levels. While the entropy conservation is better by an order of

magnitude in the isentropic model, the energy error it exhibits means that the total error

in representing the second law of thermodynamics is larger than in the IGCM. Improved

energy conservation is clearly desirable in the isentropic model.

• The problem of noise in∇2M found previously in pure isentropic models is successfully

overcome here by matching the horizontal and vertical resolutions.

• The instability which caused the isentropic model to crash is purely a feature of the

massless boundary region, and was not forced by any feature of the interior flow. The

EKE comparison shows that the life cycle is largely complete when the model crashes, so

a full comparison of the two cycles is possible. There is some sensitivity to the timescale

chosen in the relaxation scheme which is used to control the instability; this, however,

is much less than the sensitivity of the IGCM to parameters in its hyper-diffusion terms.

The instability, though, clearly needs to be addressed if the model is to be used routinely.

• The surface divergence problem of Section 3.7 has little effect on the life cycle predicted

by the isentropic model. The mixing of boundary and interior PV by the diffusive advec-

tion scheme is more significant, but is smaller than the error shown in mass-weighted PV

in the same levels by the IGCM. The same mechanism, however, is likely responsible for

part of the energy loss seen in the isentropic model.

Chapter 5

Entropy Production in the IGCM

5.1 Introduction

The most fundamental difference between the two models compared in the previous chapter

is in the representation of the entropy. The isentropic model is constructed on the principle of

Lagrangian conservation of entropy under adiabatic, frictionless flow. While the IGCM is also

constructed from a continuous equation set which satisfies this constraint, the discretized sys-

tem does not. As shown in Figure 4.20, the isentropic model exhibits changes in global entropy

which are attributable entirely to truncation errors in the conservation of mass; the IGCM, in

contrast, shows a systematic increase in global entropy which is an order of magnitude larger.

In Section 4.11 it was shown that for this particular isentropic model the drift in total en-

ergy constitutes a larger error in modelling the thermodynamics of the system than the entropy

drift in the IGCM does. However, it is important to understand these changes in the total en-

tropy in the IGCM simulation because the vast majority of weather and climate predictions are

currently made using similar, non-isentropic models. Relatively little is known about the en-

tropy budgets of such models and how well these represent the real atmosphere. This chapter

is dedicated, therefore, to studying the entropy change during the IGCM LC1 cycle, and its

relation to both observed and modelled entropy budgets of the atmosphere. We begin with an

introduction to entropy principles and some further, more specific, motivations.

Entropy Production in the IGCM 131

5.1.1 Entropy and the Second Law

Specific entropy s is a thermodynamic state variable defined by

ds =dE

T(5.1)

for the change in entropy from an initial state to one differing by energy dE. Note that strictly

the change between the two states must be reversible in this definition (see e.g. Curry and

Webster (1999) for more detail). Dividing by the temperature T has converted the inexact

differential dE into the exact differential ds, meaning that while the change in energy between

the two states depends on the history of the thermodynamic state throughout the transition, the

change in entropy depends only on the initial and final states (e.g. Dutton (1976)). Using the

first law of thermodynamics and the ideal gas law, (5.1) can be written

ds = cp d(lnT )− R d(ln p), (5.2)

leading to the expression

s− s0 = cp ln

(θ

θ0

)(5.3)

for the entropy in terms of the potential temperature θ, where a subscript 0 indicates the initial

state. An isentropic surface is therefore a surface of both constant s and constant θ.

Entropy can be formally related to the notion of order in a thermodynamic system using

probability theory. A state with a high degree of order or information is less probable, and

has lower entropy, than a more disordered state. For example, a room in which all the air

molecules are crowded into one corner has a lower entropy than an identical room with the

molecules spread evenly throughout. In the same way, a volume of air consisting of parcels of

differing θ is more ordered than if all parcels had identical θ. Any mixing of the air parcels

will increase the total entropy (e.g. Johnson (1997), Appendix B). The entropy also has an

interpretation as the availability of energy; an increase in entropy is associated with a change

of energy from a more to a less usable form.

The second law of thermodynamics has been stated in several equivalent ways. Here we

use

Tds

dt= q + f (5.4)

for an open system which can exchange heat with its surroundings. Here q is the flow of heat


into the system and f is friction internal to the system. Since f ≥ 0, this implies that an isolated

system for which q = 0 can never move to a state of lower entropy. The entropy of an open

system such as the atmosphere can decrease through a flow of heat to its surroundings, but the

total entropy of the system and its environment, in this case the universe, can never decrease.

It is important, therefore, to differentiate between the entropy of the system (ssys) and the total

entropy (stot) of system and environment.

A process is termed reversible if the system is in a state of thermodynamic equilibrium

throughout the process; the process can then be reversed such that both the system and its en-

vironment return to their initial states. The second law demands that dstot = 0 for a reversible

process, otherwise the total entropy will be decreased during the reversal. A process is irre-

versible if dstot > 0 so that the system can still be restored to its original state but not both the

system and the environment can be restored. It is important that these definitions apply to the

total entropy; an open system such as the atmosphere is, for example, able to have reversible

sources and sinks of entropy which simply correspond to fluxes of heat across its boundary.

From (5.1) it follows that stot is unchanged when heat is transferred from one element of a

system to another only if the two are at identical temperatures; the transfer is then reversible.

Any exchange of heat between two elements of differing temperature is irreversible.

5.1.2 The Entropy Budget of the Atmosphere

Over timescales of a few days the behaviour of an air parcel is dominated by reversible trans-

port, leading to the beneficial quasi-Lagrangian nature of the isentropic coordinate. A plethora

of highly irreversible processes do, however, act within the atmosphere, for example the pre-

cipitation out of water vapour, and the dissipation of kinetic energy in the turbulent planetary

boundary layer. Despite this, the total entropy of the atmosphere is approximately constant on

timescales of the order of a few years or longer. The continual production of entropy within

the atmosphere is balanced by a net loss to space by radiation as follows. The motion of the

atmosphere is driven by net heating where it is already hot (in the tropics) and cooling where it

is already cool (the extratropics). The average global incoming and outgoing energy fluxes are

equal, but the entropy removed by long-wave radiation is roughly 20 times the amount brought

in by incoming short-wave radiation (Peixoto et al. (1991)). This is easily seen from (5.1)

since long-wave radiation is emitted from the Earth at temperatures of order 300 K, while the

emission of short-wave radiation from the sun occurs at temperatures close to 6000 K. A given


amount of energy is carried by far fewer photons in solar than in terrestrial radiation and so the

energy is more ordered and the radiation is said to be of high quality, meaning low entropy. The

balance between internal generation of entropy and the net radiative loss to space is generally

referred to as the entropy balance of the atmosphere.

Peixoto et al. (1991) made the first detailed attempt at estimating the entropy budget of

the atmosphere. They estimated the entropy production by irreversible processes internal to

the atmosphere by considering each process individually. The dominant contributions were

from latent heating and absorption of solar radiation, with all other contributions one or two

orders of magnitude smaller. This internal generation was then compared to their estimate

of the net flux of entropy to space. The balance appeared to be represented reasonably well,

with a difference of 17 mWm−2K−1 between the two terms, each of roughly 600 mWm−2K−1.

The leading order terms in the entropy budget seemed to be well understood. This impression

however proved misleading.

Goody (2000) revisited the entropy budget and re-interpreted the data of Peixoto et al.

(1991) by applying two new principles. Firstly, it is the budget of the whole climate system

which is of interest, not that of the atmosphere in isolation. Large radiative fluxes between

the surface and atmosphere are unimportant for the entropy of the climate system and are thus

removed from the balance equation. The second principle is a consequence of a subtlety in

the interaction between the fluid and the radiation field, described earlier by Goody and Abdou

(1996). As noted above, the transfer of energy from one medium to another is reversible if

and only if the temperatures of the two media are identical. In the absorption of radiation by a

fluid the radiation temperature of the photons and the kinetic temperature of the molecules are

generally different, strikingly so, in fact, in the absorption of solar radiation by the atmosphere.

A large irreversible source of entropy would therefore be expected. The subtlety lies in the tra-

ditional fluid dynamical assumption that the fluid is in thermodynamic equilibrium. Absorbed

photons are thus instantaneously thermalized to the fluid temperature so that the transfer of heat

to the fluid occurs at the same temperature, and so constitutes a reversible source of entropy

for the fluid. There is a significant irreversible entropy source due to the thermalization but

this is a source in the entropy budget of the radiation field, not the fluid system. It is therefore

only the reversible component of the radiative entropy flux which is of relevance to the climate

system, and it is this term which is equal and opposite to the internal entropy sources of the

system. The irreversible component largely balances the net outward flux of radiative entropy;

both terms are components of the radiative entropy field rather than that of the climate system


and so are removed in the budget of Goody (2000). This viewpoint is not intuitively obvious,

but could be considered equivalent to treating radiation purely as a diabatic heating term which

transfers energy to the atmosphere.

After Goody’s re-arrangement of the data the difference is striking; while Peixoto et al.

(1991) show am imbalance of -17 mWm−2K−1 between sources and sinks of 589 mWm−2K−1

and -606 mWm−2K−1, the re-arranged budget shows the same imbalance but this time between

sources and sinks of 50 mWm−2K−1 and -33 mWm−2K−1. The imbalance in the entropy

budget is now of the same size as individual source terms, and Goody’s conclusion is that

‘atmospheric entropy inventories are not yet established to a desirable degree of accuracy’.

5.1.3 Some Specific Motivations

Clearly there is much still to be understood concerning exchanges of entropy within the climate

system and its environment. This is of practical as well as academic interest. The real system

obeys strict physical laws and exhibits a delicate balance between a multitude of entropy fluxes.

There is obvious potential for the entropy budget to be used as an additional constraint on

numerical models and their parameterizations, in an attempt to make the simulated climate

more realistic (e.g. Gohar (2002)). Interest in climate entropy is further stimulated by two

intriguing and controversial hypotheses.

Paltridge (1975) presented a simple, thermodynamical multi-box model of the atmosphere

and ocean system containing only parameterized dynamics, and showed that the observed mean

meridional distribution of temperature and cloud cover could be predicted simply by maximis-

ing the entropy produced by poleward heat transport. This work gave birth to the Maximum

Entropy Production or MEP hypothesis, namely that the climate system continually adjusts it-

self to the state in which this entropy generation term is maximised, the global dynamics acting

as ‘something of a passive variable’ in the words of Paltridge.

Several studies have provided supporting evidence for the MEP hypothesis; we only give

a few recent examples. Lorenz et al. (2001) used the MEP principle to realistically predict

the zonal mean temperatures of other planetary bodies in a simple thermodynamical model.

Shimokawa and Ozawa (2001, 2002) applied perturbations to an oceanic GCM to induce tran-

sitions between multiple steady states of the thermohaline circulation. In their experiments all

the transitions led to a final state with a higher rate of entropy production by heat transport than

the initial state, and they took this as evidence that ‘a nonlinear system is likely to move to a


state with maximum entropy production by perturbation’.

There is, however, widespread skepticism over the hypothesis, mostly because all the ev-

idence is empirical in nature and no physical explanation has been agreed upon for why the

system might seek such a state. Recently Ozawa et al. (2003) gave a review of the theory,

including attempts to provide a physical basis. MEP supporters are also hailing the work of

Dewar (2003) as an answer to the objections; information theory is used to show that, of all

possible states, that of MEP is statistically the most probable. Despite this, whether or not the

MEP hypothesis is correct still remains to be proven indisputably. It has certainly motivated

many studies of the entropy of the climate system. The fact that it has neither been proved

or disproved in the past 30 years testifies that much remains to be understood of the entropy

exchange processes at work, and of their importance.

A direct motivation for the work of this chapter is the second hypothesis, that of Johnson

(1997), that errors in entropy are to blame for what he refers to as the ‘general coldness of

climate models’. Boer et al. (1992) compared the current climates predicted by 14 atmospheric

GCMs with observations. All 14 suffered a common deficiency; not only were the simulated

climates too cold in a global average, but all showed remarkably similar zonal-mean distribu-

tions of the cold bias, leading to classification of the problem as ‘systematic’ and ‘tenacious’.

By far the largest biases (up to 20 K) were in the polar upper troposphere and lower strato-

sphere (UTLS) region, hence this is often referred to as the cold pole problem. Smaller, but

equally systematic, biases occurred in the ‘nonpolar’ lower troposphere. The same systematic

bias was seen in the AMIP intercomparison (Gates et al. (1999)), although the magnitude was

slightly reduced. It was noted that ‘the cause of this error has remained elusive’. The GRIPS

intercomparison of middle atmosphere climate models (Pawson et al. (2000)) recorded a cold

pole bias in the range 3 to 15 K in almost all of the models. This bias is still the largest and

most systematic error in the numerical simulations of the current climate.

The theory put forward by Johnson (1997) to explain this is as follows. Climate models all

contain spurious, aphysical sources of entropy from processes such as excessive numerical dis-

persion/diffusion and inadequacies of parameterizations. These models are tuned to simulate

a climate state without drift, so the total entropy of the system must be constant in a long time

average. In order to maintain this driftless state there must then be a corresponding entropy

sink to offset the spurious sources, and the cold bias develops to provide for this sink. The ba-

sic mechanism is clear from equation (5.1); more entropy is lost to space if the same amount of

energy is radiated at a lower temperature. Johnson used entropy balance arguments to explain


the distribution of the bias, and estimated that it would take an error of just 4% in the entropy

source (or 2 mWm−2K−1, Goody (2000)) to give a cold bias of 10 K. Johnson’s theory is far

from being unanimously accepted, partly since all the evidence is somewhat circumstantial,

but it has attracted widespread interest and has further motivated investigation of the entropy

budgets of both real and modelled atmospheres.

Explicit numerical diffusion is cited in the theory as being one of the major spurious en-

tropy sources, since any mixing of air parcels of differing θ will increase the entropy of the sys-

tem. Implicit dispersion from numerical transport schemes is also a spurious entropy source,

as shown by Egger (1999). The magnitude of these two sources in atmospheric simulations has

never been studied in depth. In the dry, adiabatic life cycle studied in the previous chapter these

are the only two ways that entropy can be changed, and they must therefore be responsible for

the systematic increase of global entropy predicted by the IGCM. This idealised simulation of

typical mid-latitude synoptic dynamics presents a good opportunity to quantify and study the

entropy production by these processes in isolation, i.e. in the absence of physical parameterisa-

tions such as heating, and the large real entropy sources and sinks they involve. In the following

sections the entropy sources in the IGCM LC1 life cycle are studied in detail, identifying their

locations and causes, and investigating their sensitivity to changes of model resolution and dif-

fusivity. Key questions are: is the size of the sources in agreement with Johnson’s theory, and

what physical processes, and hence real entropy sources, are being represented by the model

diffusion?

5.2 Entropy Production in the LC1 Cycle

5.2.1 Magnitude and Sensitivity of the Source

Figure 4.20 of the previous chapter shows a change in global entropy during the IGCM LC1

cycle of roughly 600 mWm−2K−1. However, as noted in Section 4.11, and described in Black-

burn (1983), there is a slight drift in the total energy during the life cycle since kinetic energy

lost by diffusion is not put back into the model as heat. This constitutes a flux of energy, and

therefore entropy, out of the system. The entropy generation during the life cycle occurs by

the implicit or explicit mixing of air masses of different potential temperatures, and is in part

masked by the energy drift. This should be taken into account in evaluating the entropy genera-

tion; after all, in a complete climate model simulation there would hopefully be negligible drift


0 1 2 3 4 5 6 7 8 90

1

2

3

4

x 10−3

Rat

e of

Ent

ropy

Cha

nge

(WK

−1m

−2)

Timescale (days−1)

0 5 10 150

500

1000

1500

2000

2500

3000

Day

Cum

ulat

ive

Ent

ropy

Cha

nge

(JK

−1m

−2) Effect of varying diffusivity (decay time on shortest wavelength)

3 Hours4 Hours6 Hours12 Hours1 Day2 DaysNo Diffusion

Figure 5.1: Entropy production in the IGCM LC1 cycle at T42 resolution as a function ofdiffusivity. The upper panel shows the total entropy change, corrected for the energy drift asdescribed in the text. The lower panel shows the rate of production averaged over the final 10days of the life cycle. Here the dotted line is uncorrected, the dashed line the correction and thesolid line the corrected total. Errorbars on the total are calculated using extremal temperaturesin the correction term.

in the total energy of the system. Here we first calculate the total entropy as the mass integral

of cp ln θ, and then correct for the energy drift by adding ∆ETa

, simply an estimate of the entropy

lost in the energy drift using a representative average temperature of the atmosphere Ta.

The results are shown in Figure 5.1 for various LC1 simulations which differ only in the

strength of the IGCM’s hyper-diffusion. The top panel shows the total entropy change after

correction for the energy drift using a value of 250 K for Ta. The lower panel plots the average

rate of entropy generation over the last 10 days of the cycle as a function of diffusivity. Here the

effect of the correction is evident: the dotted line is the uncorrected rate, the dashed line is that

due to the correction term and the solid line is the corrected total rate of entropy production.


The errorbars on the total indicate the effect of varying Ta in the correction term; the lower limit

results from choosing 280 K and the upper from choosing 200 K. In this life cycle most of the

atmosphere, at least up to 100 mb, lies between these two extremes, and 250 K is a suitable

mean value. The figure shows that the choice of Ta has relatively little effect on the total rate

of generation.

The intention of Figure 5.1 is to show the effect on the entropy generation of varying the

diffusivity. The IGCM, in common with almost all numerical models, contains diffusion terms

in the discrete equations which have no analogues in the continuous equations, and are there

to perform several duties, for example to soak up dispersion errors and Gibbs oscillations,

and as an arguably crude representation of the sub-gridscale flow. In these experiments a

hyper-diffusion term K∇6 acts on each of the prognostic variables vorticity ζ , divergence δ

and temperature T . The hyper-diffusion on T is the only explicit process in these experiments

which mixes air across isentropic surfaces and hence generates entropy. The constantK defines

the strength of the diffusion which is referred to here, following MacVean (1983), by the e-

folding decay time of features at the shortest wavelength (here that of wavenumber 42).

There is little change in the entropy in the period before day 5, during the exponential

growth stage of the life cycle when the flow is unrealistically zonal in nature. After day 5

the highest rate of entropy production is seen, coinciding with the low level saturation stage

described in Section 4.1. Of central importance here are the collapse of the surface fronts and

the occlusion-like decay of the warm sector (Section 4.3). The hyper-diffusion is acting in both

cases to reduce temperature gradients, effectively mixing air of different potential temperatures

and hence creating entropy. The life cycle was also run with no hyper-diffusion in the model,

resulting in little entropy production as shown by the dashed line in the upper panel, and the

zero entry in the lower. Considerable noise is evident in the fields predicted in this run, and this

is even visible in the global entropy sum. Structures at the gridscale become unrealistic as the

cascade of potential enstrophy to small scales is artificially truncated; avoiding this is, after all,

one of the main aims of the hyper-diffusion. That entropy production is significantly reduced

in this run serves to illustrate that the diffusion scheme is responsible for most of the entropy

source.

The key result of this experiment is that varying the diffusivity has relatively little effect

on the entropy source. This holds even when it is varied over a range as large as that used

here, which certainly includes most values used in practise by models (4-6 hours is common

for models at this resolution). The time averaged rate of generation in the lower panel clearly


0 5 10 150

2

4

6

8

10

12x 10

5 Effect of varying horizontal resolution

EK

E (

Jm−2

)

T42T63T106T213

0 5 10 150

500

1000

1500

2000

2500

3000

Day

Cum

ulat

ive

Ent

ropy

Cha

nge

(JK

−1m

−2)

T42T63T106T213

Figure 5.2: The effect of varying the horizontal resolution. The upper panel shows the EddyKinetic Energy evolution and indicates that the synoptic evolution is significantly different afterabout day 8. The total corrected entropy change is shown in the lower panel.

bears testament to this, changing by less than 1 mWm−2K−1, i.e. under a third of its value,

when the diffusivity is increased by a factor of 16. Furthermore, the upper panel shows that the

rate of production is almost identical for all diffusivities until after about day 6, i.e. after the

fronts have collapsed to their smallest scale (e.g. Figure 4.2). The entropy production clearly

does not asymptote to the case of no diffusion as the diffusivity is reduced.

In a similar experiment the horizontal resolution was varied while keeping the diffusivity

fixed, with a decay timescale of four hours on features of the shortest resolved scale. The

results are shown in Figure 5.2. Changing the horizontal resolution has a large effect on the

synoptic evolution. At higher resolutions a secondary wave development grows from the debris

of the first, which is still baroclinically unstable (as described by MacVean (1983)). The EKE

evolution in the upper panel indicates that the four life cycles are very similar until about day 8

when this secondary wave develops. This synoptic difference means that the entropy changes


shown in the lower panel should not be compared after this time. Again, the rate of production

during frontal collapse is remarkably similar for all four runs. The only difference before day

8 is in the timing of the large entropy source corresponding to the low level saturation, which

begins slightly earlier and lasts longer at lower resolution. As a result, more entropy has been

generated by the end of this phase, but again the difference is surprisingly small. For example,

the entropy source during the first wave is reduced only slightly by doubling the resolution

from T106 to T213.

The results so far serve to motivate further investigation. The rate of entropy generation

is certainly of a similar size to the 2 mWm−2K−1 predicted by Johnson to yield a 10 K cold

bias. Furthermore, this source is shown to be relatively insensitive to the choice of horizontal

resolution or diffusivity. If Johnson’s theory is correct and explicit diffusion is a spurious

entropy source, then the problem cannot be solved by going to higher resolution or weaker

diffusion. The entropy generation is therefore investigated further, firstly demonstrating two

points which have so far only been suggested: that the entropy production is attributable in the

greater part to the hyper-diffusion, and that this largely occurs at the surface.

5.2.2 Location and Attribution of the Source

The IGCM timestep is split into an adiabatic step and a diabatic one. Here the diabatic step

consists only of the hyper-diffusion term, and hence simulations such as this are often referred

to as adiabatic, though this is obviously not strictly correct. The entropy changes shown above

were diagnosed by a simple global mass integral of entropy s = cp ln θ, with θ in turn diagnosed

from the model variables T , ps and σ = p/ps. This gives the total entropy change but does

not distinguish between the two anticipated sources: the hyper-diffusion and the numerical

transport errors. Now the entropy source due to the diffusion is calculated using a different

technique and the results compared. In this technique the temperature tendencies from the

diabatic step are used to calculate the heat addition, and thus the entropy source due to the

diffusion only. This source is given as a rate of generation sdi by the global mass integral ofcp∆T diT∆t

, where ∆T di is the temperature change over a diabatic step of time ∆t.

The results are shown in Figure 5.3. Here the red line is sdi and the black lines are the

total s given by differentiating the corrected entropy change of Figure 5.1 in time. The close

similarity between the two measures supports the use of the simple energy drift correction in

s, since this correction is not needed in calculating sdi from local heat additions.


0 5 10 15−2

0

2

4

6

8

10x 10

−3

Day

Rat

e of

cha

nge

of e

ntro

py (

Wm

−2K

−1)

Figure 5.3: Total corrected rate of entropy production s in black. The solid line correspondsto a temperature of 250 K for the energy drift correction, and the dashed lines to the extremaltemperatures of 200 K and 280 K. The source from diffusion only (sdi) is shown in red; thisterm does not need correction for the energy drift. The data was diagnosed from an LC1 cycleat T42 horizontal resolution with 4 hour∇6 hyper-diffusion.

It is clear that the diffusion is responsible for the vast majority of the entropy source up

until about day 7, i.e. until after the frontal collapse and much of the low level saturation event

is complete. The diffusion certainly plays a large part in the decay of the surface warm sector

by mixing air of different temperatures. A simple estimate using Figure 4.2 puts the scale

of the warm sector at roughly twice the smallest resolvable scale and, bearing in mind the

strength of gradients here, this means that the diffusion will have a large effect. In addition to

this, Methven (1996) showed that it is the diffusion on temperature that limits the scale of the

surface fronts, doing so just above the smallest resolvable scale.

Though the diffusive contribution dominates the entropy source during the low level satu-

ration, it is interesting that there are other periods when it does not, namely around days 4 and

8 to 9. Numerical transport errors are presumably responsible for the entropy source at these

times, and the effect is not negligible. It looks, for example, as if the second, slightly smaller

peak in total s (around days 8 and 9) is attributable to numerical transport errors rather than

explicit diffusion. Near the end of the cycle there are periods where sdi exceeds s, implying

a sink of entropy elsewhere. One possible explanation is that Gibbs oscillations occurring as


Figure 5.4: A snapshot on day 7.5 of the entropy source sdi due to explicit numerical diffusion,in Wm−2K−1 on the lowest model level of the IGCM. The solid contours show temperatureevery 5 K down from 295 K.

a side-effect of the spectral technique could constitute a small spurious sink of entropy, acting

to add a small amount of order over the equivalent continuous field, and so to increase the

variance of θ.

In general though, Figure 5.3 shows that the diffusion is the dominant source of entropy

in the LC1 cycle. Since sdi is determined locally, the distribution of the source throughout

the domain can be analysed. Figure 5.4 shows a snapshot of the horizontal distribution of sdi

on the lowest model level. The effect of the diffusion is seen everywhere in this picture, but

the largest sources are in the vicinity of the fronts and the warm sector. The diffusion acts to

reduce temperature gradients by heating cool regions and cooling hot regions, resulting in local

entropy sources and sinks, respectively. For example, there is an entropy sink along the warm

sector with sources either side, reflecting the contribution of the diffusion to the occlusion-like

decay of the sector. The diffusion essentially transports heat, and thus entropy, from one region

to another, effectively moving mass across isentropic surfaces. Adjacent sources and sinks


−8 −6 −4 −2 0 2 4 6

10 20 30 40 50 60 70 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Latitude

σ=p/

p s

(∂ s / ∂ t)diffusion

on day 6.5 (Wm−2K−1 per unit ∆σ)

0 5000 10000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Entropy change (Jm−2K−1 per unit ∆σ)

Horizontally integrated source

Figure 5.5: The left panel shows the source sdi in a meridional section through the warm sectorat day 6.5. The solid contours are isentropic surfaces drawn every 5 K. The right hand panelshows the entropy change due to diffusion as a vertical profile, i.e. sdi integrated over the whole15 day life cycle and horizontally over the globe for each model level.

could be viewed as fluxes of entropy from one region to another. Note that the sources shown

in Figure 5.4 are in Wm−2K−1 so that the local sources and sinks are three orders of magnitude

larger than the global averages shown in the previous section. The global entropy drift is thus

seen as the small residual left after many cancellations between much larger individual sources

and sinks.

A similar picture is seen in the vertical, an example snapshot of which is shown in Fig-

ure 5.5. As expected, the largest entropy fluxes are at the surface since this is where the

strongest temperature gradients occur. These fluxes do, however, extend a significant way

up from the surface, and there is also evidence of weaker fluxes higher up, indicating the effect

of diffusion at the tropopause level, for example at tropopause folds and upper level fronts.

This picture is just a snapshot; the overall effect of the diffusion is seen in the horizontally and

time integrated source shown as a vertical profile on the right. Diffusion at the tropopause level

is seen to be only a small contribution to the total entropy change. Surface effects dominate,

but these extend higher in the atmosphere than might be expected, with approximately a third

of the entropy produced outside of the lowest two levels, i.e. above p/ps = 0.84. This implies

that a significant amount of entropy is produced outside the region below about 1 km, which is


−6 −4 −2 0 2

x 105

240

260

280

300

320

340

360

380

400

θ (K

)

∆ s (JK−1m−2)

−5000 0 5000240

260

280

300

320

340

360

380

400

∆ sdiff

(JK−1m−2)

Figure 5.6: Horizontally and time integrated vertical profiles of s and sdi as in the right panelof Figure 5.5, but shown here as functions of θ. The left panel is thus the total entropy changeand the right panel is the entropy change due to the hyper-diffusion.

generally represented by a boundary layer scheme in comprehensive models. Numerical dif-

fusion would, therefore, be a significant source of entropy, even if the surface processes were

modelled perfectly by such a scheme.

Figure 5.5 shows an overall source which is positive at all normalised pressure levels.

The entropy source is due to effective mixing of air across isentropic surfaces, and so looks

significantly different when viewed in isentropic coordinates. Figure 5.6 shows the vertical

profiles of the entropy sources as a function of θ. The calculation of these inevitably involved

some interpolation from the model’s normalised pressure levels. This was performed as in

the isentropic PV analysis of Section 4.12 by dividing the atmosphere into a set of ‘isentropic

layers’, or regions between two isentropic surfaces, roughly coincident with the vertical levels

of the isentropic model. The entropy within each isentropic level was evaluated by calculating

the pressure at each of the bounding θ values, then using these pressures as bounding values

in a mass integral. The pressure at a given θ bound was calculated by interpolating from the

model levels assuming a linear variation of ln θ with ln p. Some error is to be expected from the

interpolation procedure. To show that this is small, the sum was repeated several times with the

isentropic levels shifted by 1 K each time (the level spacing is 6 K for most of the troposphere)

and each of the resulting profiles is plotted in a different colour in Figure 5.6. The general

structure is the same for all the profiles, which is taken to imply that this structure is robust.


The right hand panel shows that the diffusion results in a local entropy sink in the isentropic

levels around 300 K and a source of similar size in levels below. This simply corresponds to

a flow of mass downwards in isentropic space. The diffusion is limiting surface temperature

gradients by transporting mass into isentropic layers whose thickness is tending to become very

small. The figure shows that the net transport is downwards everywhere, from levels above

about 290 K to those below. These lower levels are those which are involved in the surface

fronts (e.g. Figure 4.15), so it is not surprising that mass is transported into these layers. It is

interesting, though, that all this mass comes from the levels above rather than those below.

The total entropy change is shown on the left and here some other entropy fluxes are

visible, for example from the region around 270 K to its surroundings. An important point to

note is that the scales of the two horizontal axes are different; local sources and sinks in the

total entropy are two orders of magnitude larger than can be explained by the action of the

diffusion. Despite this, Figure 5.3 reminds us that the overall entropy source, given as a small

residual between these local sources and sinks, is largely attributable to the diffusion. This

can be understood as follows. Numerical errors in temperature advection, for example, will

effectively move air parcels across isentropic surfaces. If an air parcel moves to a higher θ the

entropy of the whole system increases slightly, while if it moves to a lower θ the total entropy

decreases. If there is no preferred direction for the temperature error then many such local

sources and sinks will cancel with little effect on the total entropy. The diffusion, on the other

hand, has a systematic effect of heating cool air and cooling warm air. This systematic mixing

across isentropic surfaces acts to bring the temperature of neighbouring parcels closer together,

and is always a positive source of entropy.

Note that the residual effect of the numerical transport errors is also positive, shown as

the difference between red and black lines in Figure 5.3, and is doubtless due to the implicit

entropy generation by numerical schemes, as studied by Egger (1999).

These large local sources and sinks show that the dynamics of the IGCM clearly violate the

principle of Lagrangian conservation of entropy to a greater extent than indicated by the overall

entropy changes. This is in agreement with the error analyses of Johnson et al. (2000, 2002),

who demonstrated large errors in the Lagrangian conservation of moist potential temperature

under moist adiabatic conditions in the NCAR Community Climate Models.


−100 −80 −60 −40 −20 0 20200

250

300

350

400

450

500

Cross−isentrope mass transport (kg m−2)

θ (K

)

Figure 5.7: Total cross-isentrope mass transport (positive upwards) over the 15 day LC1 cycle.First the mass change of each isentropic level was calculated by interpolating from model levelsas in Section 4.12 and Figure 5.6, and then this was integrated down from the top of the domainto any given isentropic surface to give the mass transport across that surface.

5.2.3 Cross-Isentrope Mass Flow

In a perfectly adiabatic simulation there should be no transport across isentropic surfaces. In

the IGCM LC1 cycle, however, there is; to quantify this, Figure 5.7 shows the vertical profile

of net cross-isentrope mass flow accumulated over the 15 day life cycle. The largest mass flux

is of almost 100 kg m−2 through an isentropic surface just under 300 K, an amount equal to

roughly 1% of the total column mass of the atmosphere.

Thuburn (1993) advocated the use of accumulated cross-isentrope mass flow as a diagnos-

tic of the diabatic circulation showing, for example, localised fluxes across the 358 K surface

of 1-4 kg m−2day−1 associated with regions of tropical convection. From Figure 5.7 the global

average mass flux at this level of roughly 10 kg m−2 represents a flux of the same order (given

that the active part of the cycle lasts roughly 10 days). The effects of diffusion and numerical

transport errors in the IGCM are therefore significant. This serves to illustrate one of the major

advantages of the isentropic coordinate. The diabatic motion is generally of great importance

to the behaviour of the atmosphere but is much smaller than the adiabatic component, and so

is easily contaminated by small transport errors in the adiabatic flow (e.g. in vertical advection

terms in non-isentropic models).

Another example is in the field of stratosphere-troposphere exchange, which is of par-


ticular interest for the transport of trace chemical species between the two regions. Schoeberl

(2004), following Appenzeller et al. (1996), estimated the net mass flux across the extratropical

tropopause by considering the mass of the lowermost stratosphere, i.e. the region bounded by

the tropopause and the 380 K surface. Subtracting the mass flux across the 380 K surface from

the rate of change of mass of the lowermost stratosphere gave a downwards flux across the

Northern Hemisphere tropopause of roughly 1010 kg s−1 in two assimilated data sets. Analysis

of the output from a GCM gave a slightly lower value of 0.7 × 1010 kg s−1. Figure 5.7 shows

a flux of order 10 kg m−2 across isentropic surfaces near 380 K over the life cycle. When as-

sumed to occur over 10 days, and multiplying by the surface area of the Northern Hemisphere

extratropics (poleward of 30) this represents a flux of 1.7× 109 kg s−1. This is clearly a crude

estimate of the extratropical flux but, since most of the dynamical activity in the life cycle

occurs in the extratropics, it is more likely to be an underestimate than an overestimate. The

diabatic mass flux due to diffusion and numerical errors in the tropopause region of the IGCM

is thus seen to be roughly one sixth of the observed flux, and so counts as a significant term.

This is unlikely to be realistic. It is also of a similar size to the discrepancy between Schoeberl’s

(2004) estimates from GCM and assimilated data. It should be noted that the life cycle is a very

idealised simulation; this should not therefore be taken to suggest that this error is the cause of

the discrepancy, but simply that errors of this size are to be expected in non-isentropic models.

It seems likely that an unrealistically weak Brewer-Dobson circulation (see Section 5.4) could

also be contributing.

By far the strongest dynamical feature in the tropopause region during the life cycle is the

folding of the tropopause, so it would be reasonable to assume that a significant amount of

the cross-isentrope mass flux is associated with this feature. This has not been proven here,

but if true would have interesting consequences. The stratosphere-troposphere exchange due

to tropopause folding is primarily downwards, i.e. from the stratosphere to the troposphere

(e.g. Holton et al. (1995)). In the life cycle, however, the diffusion and numerical errors have

had mixed effects, acting to transport mass upwards across isentropic surfaces above 350 K

and downwards in the region just below. This is clearly not a rigorous or robust result, merely

an indication that further investigation might be worthwhile. In general, it is believed that too

much mass is transported downwards in stratosphere to troposphere transport events simulated

by GCMs due to excessive numerical diffusion (Stohl et al. (2003a)). Models therefore overes-

timate the ozone flux from the stratosphere to the troposphere and, furthermore, stratospheric

ozone reaches the surface more often than expected, perhaps indicating that there is too much


mixing across isentropic surfaces.

Finally, it should be noted that the cross-isentrope flow due to diffusion and numerical

errors are presumably rather shallow vertical motions in which a parcel’s potential temperature

does not change by a large amount. In the stratosphere-troposphere exchange problem it is the

deep exchange events which bring air parcels of radically different chemical compositions into

contact, and are thus of greatest importance (Stohl et al. (2003b)).

5.3 The Entropy Theory for the Cold Bias

Are the entropy sources shown in these experiments consistent with the coldness theory of

Johnson (1997) introduced in Section 5.1.3; i.e. is there any evidence here to support or con-

tradict the theory? We begin with a brief summary of the arguments made by Johnson.

5.3.1 Summary of Johnson (1997)

Models do contain spurious, aphysical sources of entropy. In his Appendix B, Johnson demon-

strates positive definite aphysical entropy production from numerical diffusion/dispersion since

any ‘mixing of energy’, be it implicit or explicit, will increase the total entropy of the system.

In his Section 9 he asserts that the primary source stems from the action of this numerical mix-

ing during the long-range transport of energy, both dry and moist, and of potential temperature

θ. As described by Egger (1999), the total entropy S of a discrete system of grid cells i, each

of volume δV , is given by

S =∑

i

(cp ρ δV ln θi)

= cp ρ δV ln

(∏

i

θi

), (5.5)

where here the density ρ is assumed constant for simplicity. Hence, in a discrete system the

entropy is essentially the product of all gridpoint values of θ, and is thus a moment of high

order. Inevitably the variance of θ will decrease as a result of the implicit or explicit numerical

mixing, and the total entropy given by (5.5) will increase. (Note that while most numerical

schemes will mix unrealistically, some can un-mix, for example the ‘VL2’ scheme studied by

Thuburn and McIntyre (1997), though this is obviously unrealistic as well.)

Johnson defines the following entropy sources: s is the total entropy source in a model,


and se

is the true source which the model is aiming to represent. (He uses a wavy underbar

wherever a variable applies to the true atmospheric state.) The spurious positive definite source

is labelled sa. Departures of heat addition ∆Q, and of temperature ∆T , from the true climate

state are defined by

Q = Qe

+ ∆Q and T = Te

+ ∆T, (5.6)

where Q and T represent the model’s heat flux and temperature, respectively. Departure from

the true entropy source is expected in the model as a result of these heat and temperature

departures; this entropy source is labelled ∆s (∆Q,∆T ). Johnson then writes the relation

between the entropy sources as

〈s〉 =⟨se

⟩+ 〈∆s (∆Q,∆T )〉+ 〈sa〉 , (5.7)

where 〈〉 indicates a temporal and areal mass-weighted average value over an incremental isen-

tropic layer. This is his equation (23). If both real and modelled climate states are assumed to

be without drift then

〈∆s (∆Q,∆T )〉 = −〈sa〉 < 0, (5.8)

i.e. a departure entropy sink develops to counter the spurious source. The theory is that this

happens through a departure ∆T of the model temperature field from that of the real climate.

Note that in any case the presence of an aphysical source sa precludes the unbiased representa-

tion of the true state, since there must be a departure in either the heat addition or temperature

fields to allow for ∆s.

By considering the dry entropy balance, Johnson derives his equation (38):

〈sa〉 ∼=1⟨Te

⟩

〈s?(∆T )?〉 − 〈∆T 〉⟨

Te

⟩⟨s?Te?⟩− 〈∆Q〉+

1⟨Te

⟩⟨

(∆Q)?Te?⟩ (5.9)

showing that the aphysical source can be balanced by a combination of the four terms on the

right. Here ? represents a deviation from the average, i.e.

f ? = f − 〈f〉 . (5.10)

Note that some assumptions have been made in this derivation, namely that ∆T is small com-

pared to 〈T 〉, and that the locations of the spurious source and the temperature departure are


uncorrelated, so that the covariance⟨s?a

(∆TTe

)?⟩is negligible. Johnson then assumes that

‘the emphasis given by climate modelers to replicating the atmospheric energy balance leads

to successfully modeled estimates of heat addition’, and sets ∆Q to zero. This leaves only the

first two terms on the right-hand side of (5.9) to balance the aphysical source.

Consider the second term. The circulation of the atmosphere is maintained by radiative

heating where it is hot and cooling where it is cool, so the covariance⟨s?Te?⟩

must be positive.

For this term to balance the positive aphysical source therefore requires 〈∆T 〉 to be negative,

i.e. there must be a mean cold bias of this infinitesimal isentropic layer. Similarly, for the first

term to offset the aphysical source requires a particular distribution of the cold bias; that is

coldest where the net entropy flux is out of the layer, and less cold where it is inwards.

The cold bias shown by Boer et al. (1992) has a clear distribution, with the severest bias

in the extratropics at high altitude and a weaker, but still systematic, cold bias in the tropical

troposphere. Johnson notes the striking similarity between these regions and that bounded by

the 300 and 325 K isentropic surfaces, which roughly fills the troposphere from the ground up

to 500 mb in the tropics, and sits in the band 300 to 200 mb in the polar regions. He also notes

that this region contains both areas of heating where the atmosphere is already hot and cooling

where it is already cool, and is therefore the region where the covariance in the second term is

likely to be largest. Furthermore, the distribution of the bias is such that it is coldest where the

net entropy flux is to space in the polar high altitudes, and less cold in the tropical troposphere

where the net flux is inwards; the covariance in the first term will therefore be positive. In

incremental isentropic layers within this region, he claims, both terms contribute to offsetting

the spurious, aphysical entropy source, as the model develops a cold bias which becomes more

pronounced towards the poles. Note, however, that only the second term on the right of (5.9)

was considered in Johnson’s estimate of a 4% error in the entropy source being required to

cause a 10 K mean cold bias in an isentropic layer.

5.3.2 Analysis of the IGCM Results

The global entropy source shown in Section 5.2.1 is of a similar size to Johnson’s prediction.

Section 5.2.2 shows that the entropy is produced largely at the surface in the extratropics in

connection with the low-level evolution of the wave. The source, therefore, is outside the 300-

325 K region identified by Johnson, and so would not appear directly in the entropy balance

equation (5.9) for infinitesimal isentropic layers in this region. This, however, does not mean


that the theory does not apply (Johnson (2004), personal communication). If a model is able

to develop additional entropy sinks to balance aphysical sources, it is clearly able to support

additional fluxes of entropy between different isentropic levels. (After all, the results of Sec-

tion 5.2.2 show that non-isentropic models do not conserve the entropy between two isentropic

surfaces, even over a short period of time.) It is within the limits of the theory for entropy to

be spuriously produced in one isentropic level and radiated to space in another; this simply

implies a spurious flux of entropy between the two.

As noted in Section 5.2.2, the difference between the red and black lines in Figure 5.3

represents a source of entropy which occurs during the adiabatic timestep of the model. This,

therefore, is without doubt a spurious source. Entropy should be conserved under adiabatic

flow, but will be spuriously increased by numerical schemes as described by Egger (1999).

In the IGCM this would apply, for example, to the nonlinear advection terms calculated in

gridspace. Gregory and West (2002) showed that numerical advection schemes can have a large

diffusive effect when the wind field is oscillating. The vertical wind will oscillate in a model

such as the IGCM during the vertical propagation of planetary waves, as isentropic surfaces

are displaced relative to isobaric surfaces. This could partly explain this systematic entropy

source from the adiabatic step, since this occurs at a similar time as the propagation phase of

the saturation-propagation-saturation picture of the life cycle. Averaged over the ‘active’ latter

10 days of the cycle, the source during the adiabatic step is of the order of 0.4 mWm−2K−1,

which is smaller than Johnson’s estimate.

Including the diabatic step (i.e. the effect of the diffusion), however, the overall entropy

source is of the right size to give a bias of 10 K according to Johnson’s theory. The diffusion

is a source of entropy, but it must now be asked whether it is a spurious source; i.e. what real

processes does the diffusion represent, and how well does this entropy source compare to any

real sources?

The hyper-diffusion in the IGCM is one, very common, type of the scale-selective dissi-

pation which almost all numerical models include for various reasons, for example:

• To clean up small-scale noise introduced by various components of the model, e.g. physical

parameterizations, dispersion and truncation errors introduced by discretization of the

dynamical equations, and Gibbs oscillations introduced in the use of the spectral method.

• To help suppress any nonlinear instabilities.

• To maintain realistic flow features by dissipating potential enstrophy near the gridscale


as it cascades to small scales.

• As a crude representation of the Reynolds stress terms, and hence of the sub-gridscale

flow.

Some of these are now discussed in detail.

Numerical diffusion can be interpreted as a representation of the Reynolds stress terms in

the large-scale equations derived by Reynolds averaging. As described by Kalnay (2003), for

example, variables such as velocity u and a tracer mixing ratio q can be expressed as the sum

of resolved (i.e. cell average) and unresolved components so that, for example,

u = u+ u′, where u′ = 0. (5.11)

By substituting these expansions into the continuous prognostic equations for u and q, alterna-

tive equations are formed which predict u and q, i.e. the large-scale or resolved variables. The

only dependence on the unresolved variables is in nonlinear terms such as − ∂∂x

(ρ u′q′) which

represent transport by the unresolved eddies. Diffusion, and similarly hyper-diffusion, essen-

tially parameterizes these eddy fluxes in terms of the large-scale variables by, for example,

−ρ u′q′ = K∂q

∂x. (5.12)

This is a first order closure for the Reynolds stress terms and is generally considered to be

crude. There is no a priori reason for closing the large-scale equations in this way, and there

can be significant sensitivity to the choice of the eddy diffusivity K (e.g. MacVean (1983),

Stephenson (1994)). It is widely believed that diffusion mixes the model atmosphere more

than the unresolved eddies would. The real atmosphere shows no clear separation of scales

at the resolution limit of models; the necessity to artificially truncate this range of scales and

simultaneously avoid noise at the gridscale inevitably leads to flow features near the gridscale

which are smoother than their counterparts in the real atmosphere.

The stretching out of vortex filaments to ever smaller horizontal scales is a familiar feature

in fluid dynamics. This occurs under the action of the large-scale, adiabatic deformation field as

seen, for example, in the contour advection study of Waugh and Plumb (1994). Charney (1971)

showed that in the atmosphere rotation and stratification apply a ‘geostrophic constraint’ that

acts to inhibit strong vortex stretching, and thus also to inhibit the cascade of energy to small

scales which is seen in three-dimensional turbulence. In the atmosphere it is the potential


enstrophy (Q2), or variance of the PV field, which cascades to small scales rather than the

energy (e.g. Salmon (1998)). In the absence of a dissipation scheme this variance will collect

at the model’s smallest resolvable scale resulting in noisy and unrealistic small-scale features.

This is often referred to as ‘spectral blocking’.

In the real atmosphere quasi-horizontal chaotic advection along isentropic surfaces acts

to stir the air creating thin filaments of varying PV which are not mixed (e.g. Pierrehumbert

and Yang (1993)). Actual mixing, and hence dissipation of potential enstrophy, occurs under

the action of radiative heating, and in intermittent encounters with localised patches of three-

dimensional turbulence (e.g. Haynes and Ward (1993)). So it is, therefore, diabatic processes

which ultimately dissipate the potential enstrophy, and real entropy sources are to be expected.

In the troposphere, three-dimensional turbulence is encountered in the atmospheric boundary

layer, in connection with convective events or in strong frontal zones where Kelvin-Helmholtz

instability is expected. As shown by Figure 5.5, most of the entropy source from diffusion

occurs near the surface in response to strong horizontal temperature gradients. It could then

be argued that the diffusion gives a good representation of these processes, which act to mix

across isentropic surfaces, and so reduce temperature gradients at or near the surface.

In the IGCM LC1 cycle the scale limitation of the surface fronts described by Methven

(1996) clearly represents a large part of the entropy source from diffusion. The formation of

fronts is driven by the large-scale deformation field (e.g. Hoskins and Bretherton (1972)). Dif-

fusion limits the frontal scale by transporting mass across isentropic surfaces, and Figure 5.2

shows that the entropy source sdi during the frontal collapse is remarkably similar at all reso-

lutions, i.e. regardless of the scale at which the diffusion is operating. Furthermore, Figure 5.1

shows that the entropy source is also largely independent of the eddy diffusivity.

This is in agreement with the findings of Roberts and Marshall (1998), who studied spuri-

ous potential temperature tendencies resulting from diffusion on temperature in ocean models.

They found that the diapycnal velocities(DθDt/∂θ∂z

)due to diffusion on temperature were inti-

mately related to the dissipation of vorticity gradients in the cascade of potential enstrophy to

the gridscale. In numerical experiments, the spurious diapycnal transfers were relatively in-

sensitive to changes in the model resolution, indicating that the same diapycnal transfer was

needed to dissipate the vorticity gradients no matter which scale it occurred on. Compare the

evolution of the low-level temperature field in the IGCM at T42 and T341 resolutions (Fig-

ures 4.2 and 4.4). In both cases a warm sector extends towards the pole, collapses to near the

gridscale and is eroded by the hyper-diffusion. The only difference between the two is the scale


at which this occurs.

The argument of Roberts and Marshall also holds for atmosphere models. As in the orig-

inal argument, thermal wind balance must be assumed, expressed here on pressure surfaces

by

− fRp∂v

∂p=

(p

p0

)κ∂θ

∂x,

f

Rp∂u

∂p=

(p

p0

)κ∂θ

∂y, (5.13)

where R is the gas constant and f the Coriolis parameter (assumed constant). A ∇4 hyper-

diffusion acting on temperature on constant pressure model levels can then be expressed as

∇4 T = ∇4

(θ

(p

p0

)κ)

= ∇2

(f

Rp

(− ∂2v

∂p ∂x+

∂2u

∂p ∂y

))using (5.13)

= − fRp∂

∂p

(∇2ξg

). (5.14)

Hyper-diffusion on temperature in pressure coordinate models is therefore seen to dissipate

gradients of geostrophic vorticity ξg.

The following picture thus emerges. Entropy is produced by the limitation of frontal tem-

perature gradients by diffusion in the IGCM. The size of this source is determined not by the

details of the scale-limiting process, however, but by the rate of frontal collapse, itself deter-

mined by the deformation field of the large-scale flow. It is to be hoped that the large-scale flow

will be represented well, so that the rate of collapse will be very similar in both real and mod-

elled atmospheres. This suggests that, on sufficiently long timescales, the entropy source will

also be very similar, even though in the real atmosphere it is turbulent and radiative processes

which act to limit the frontal scale.

It is proposed, therefore, that the entropy source due to the action of diffusion to limit the

frontal gradients is a good representation of the real entropy source involved in limiting such

gradients in the continuous atmosphere. The IGCM has no mechanism other than diffusion or

spectral blocking to limit the gradient. What, then, is the part played by the diffusion in the

entropy budget of a full GCM?

Goody (2000) analysed the entropy budget of the GISS GCM and compared it to his

analytical estimations of the entropy sources of the atmosphere. The numerical scale-selective

dissipation of this model is in the form of a binomial filter which was found to dissipate energy

at an average rate of 0.684 Wm−2. Goody combined this with the dry convection term since,


though it has no direct physical meaning, it ‘most resembles the action of a turbulent viscosity’.

A temperature of 260 K was assumed for the dry convection, and using the same value gives an

entropy source of 2.6 mWm−2K−1 for the action of the filter. This is in close agreement with

the source from diffusion in the IGCM which is shown in Section 5.2.1. The closeness of this

agreement may, of course, be a coincidence, but the size of the source does indicate that the

diffusion still plays a large part in dissipating enstrophy during the cascade, even in this, more

comprehensive model.

Goody quantifies the GISS entropy sources from parameterized dissipative processes such

as dry turbulence and boundary layer stresses, which would be expected to play a part in limit-

ing frontal gradients. The total dissipative source compares well with his theoretical estimate,

but given the uncertainties emphasised by Goody in the atmospheric budget, it cannot be in-

ferred from this that these sources in the model are correct. It is surprising that the source due

to the binomial filter is so large even with radiative and convective parameterizations acting in

the model. Here, however, it is hypothesised that the overall source due to truncation of the

potential enstrophy cascade, by a combination of parameterizations and the binomial filter, is

realistic. If potential enstrophy is dissipated by the filter then there will be weaker temperature

gradients and presumably less mixing will be done by the parameterizations. In the extreme

case, if the fluid is well mixed it simply cannot be mixed any more.

Hence it is argued here that, even in a full GCM, the action of diffusion to limit frontal

gradients is not a spurious source of entropy, since the entropy generation is determined by the

deformation field of the large-scale flow. This is not to say that there is no spurious source due

to diffusion. Here we have focussed on the surface fronts; Figure 5.5 shows that while these

have a dominant contribution, the diffusion does also act to increase entropy elsewhere. At the

surface fronts significant sub-gridscale features would be expected, and so this crude closure

of the large-scale equations is perhaps acceptable. It could well be less so, however, in other

regions where the sub-gridscale ‘eddies’ are expected to be weaker, for example in the vicinity

of stratosphere-troposphere exchange events where the diffusion is generally considered to do

too much mixing (Section 5.2.3). As emphasised by Salmon (1998), one of the fundamental

concerns over this closure is that there is no dependence of the eddy diffusivity on the nature

of the flow.

As another example, consider the blocking anticyclone shown in Figure 11 of Hoskins

et al. (1985), which is reproduced in Figure 5.8. A large pool of low PV air from the subtropics

gets cut off in the high PV air to the north, where it remains for four days, with little change in


Figure 5.8: Reproduction of Figure 11 of Hoskins et al. (1985), showing PV on the 330 Kisentropic surface from analyses for the period 30 September - 7 October 1982. The contourinterval is 1 PVU with the region between 1 and 2 PVU shaded.

magnitude, before being re-absorbed into the low PV region to the south. In a model simulation

of this event scale-selective dissipation was needed to reduce noise; numerical diffusion thus

has some effect on such features. In contrast to features such as the surface fronts, here the

potential enstrophy is not cascading to small scales in this region, so the argument presented

above does not apply. It is not clear in situations such as this that the entropy source will be

independent of the scale, or the rate at which the dissipation acts. It is possible that the entropy

produced by diffusion will be spuriously large in such events. This is, of course, entirely

speculative.

In summary this study has shown that there are spurious, aphysical entropy sources during

the life cycle due to numerical errors in the adiabatic transport terms. It also seems likely that

a part of the entropy generated by diffusion constitutes a spurious source. The dominant part

of this source is that due to the limiting of the surface frontal gradients. It is proposed that

this dominant part is of a realistic size, however, despite the unrealistic nature of the actual

scale-limiting mechanism.


5.4 Alternative Theories

Johnson’s theory is currently not widely accepted. Alternative explanations for the cold bias

have been suggested, and several studies published demonstrating a reduction in the bias. These

are now briefly reviewed. In the original intercomparison of Boer et al. (1992) it was suggested

that the simplest explanation would be a minor error in albedo or emissivity due to, for example,

the treatment of clouds, or of trace gases and aerosols in radiation schemes. Consistent with this

the temperature structure was ‘much more sensitive to changes in physical parameterizations

than to resolution’. Later, Boer (2000) commented that ‘the error is diminishing, but still exists,

in recent models’, although it was unclear which aspects of model development had reduced

the bias; improved resolution and treatment of clouds or radiation were given as possibilities.

Clearly there is much uncertainty in the representation of clouds and cloud-radiation inter-

actions in climate models. Improvements have been achieved by developing parameterizations

of these effects, but never to the extent that the cold bias is fully corrected in either size or dis-

tribution. For example Gu et al. (2003) tested a new cloud-radiation parameterization scheme

in the UCLA GCM which corrected, to some extent, the cold bias in the troposphere of the

original model. The lower stratosphere, however, became even colder under the new scheme,

leading the authors to comment that ‘the general coldness of climate models is not simply a

radiation problem’.

One feature of climate models which is clearly improving is the resolution, both in the

horizontal and the vertical. The effects of this, however, appear to be mixed. Hamilton et al.

(1999) achieved a significant reduction in the cold pole bias by increasing the horizontal res-

olution in the GFDL SKYHI model, which extends with good vertical resolution up into the

mesosphere. They note that the bias reflects an extratropical stratospheric circulation that is

too close to radiative equilibrium. The middle atmosphere is driven away from radiative equi-

librium by the Brewer-Dobson circulation, a global-scale, meridional overturning circulation

with large-scale ascent across the tropopause in the tropics and descent nearer the poles (see

e.g. Holton et al. (1995)). As air parcels are pushed downwards near the poles they are warmed

adiabatically above the radiative temperature. In the eyes of Hamilton et al. (1999) and many

others the cold pole bias is therefore a symptom of an unrealistically weak Brewer-Dobson

circulation in climate models. This circulation is driven by momentum deposited in the extrat-

ropical stratosphere by the breaking of Rossby and gravity waves; this is the downward control

mechanism of Haynes et al. (1991). An unrealistically weak circulation is thus, in turn, a symp-


tom of weak wave fluxes. Increasing the resolution means that shorter waves are resolved and

the wave fluxes are stronger, leading to a stronger Brewer-Dobson circulation and a warmer

polar stratosphere.

At first sight the results of Brankovic and Gregory (2001) appear to contradict this. In

experiments with the ECMWF model an increase in horizontal resolution reduces the cold bias

throughout the troposphere but not in the extratropical UTLS region. A closer look shows

that the two results are, however, consistent. In the SKYHI model used by Hamilton et al.

(1999) there is no parameterization of sub-gridscale gravity wave effects; any such effects must

come from resolved gravity waves, so the wave forcing of the circulation would be expected

to increase with resolution. An increase in temperature with resolution is seen everywhere

above a few hundred millibars, but by far the largest warming (up to 40 K) is in the region

of 10 to 0.1 mb, i.e. the upper stratosphere and lower mesosphere. In the polar UTLS region

originally identified as cold by Boer et al. (1992) (about 200 mb) there is a warming with

increased resolution, but this is slight, especially above N90 (i.e. 90 grid rows between pole and

equator). Brankovic and Gregory (2001) focus on this UTLS region and so the lack of warming

with increased resolution is consistent. Furthermore, the ECMWF model used in their study

contains a gravity wave drag parameterization, and so it is not obvious that a large increase

in wave forcing would result from an increase in resolution. These parameterizations aim to

represent the momentum deposited during the dissipation above the tropopause of unresolved

gravity waves which are triggered almost exclusively in the troposphere, mainly by orography

but also by dynamical events such as convection. An example is the scheme of McFarlane

(1987) who also shows that, while much of the wave dissipation is in the upper stratosphere

and mesosphere, there is an effect on the extratropical UTLS region with temperatures rising

by up to 7 K.

Clearly the strength of the wave driving in the stratosphere has an effect on the polar

temperatures. A common opinion in the modelling community is that the cold bias is due

to the omission or underestimation of some element of gravity wave forcing. Most of the

middle atmosphere climate models in the intercomparison of Pawson et al. (2000) contain some

kind of gravity wave drag parameterization and yet still show significant cold biases. Among

these models two main types of systematic error in 100 mb temperatures were identified, and

different causes suggested for each. Many models were too cold at all latitudes, yet showed a

realistic annual temperature cycle; here an error in radiation was suggested. Some models, on

the other hand, had a realistic annual cycle but were too warm in the Tropics and cold at high


latitudes, suggesting that the residual circulation described above was too weak, i.e. with too

little downwelling at high latitudes and upwelling at low latitudes. This raises an interesting

point: if an unrealistically weak meridional overturning circulation is to blame, why are models

not systematically too warm in the tropical stratosphere due to underestimated upwelling? The

answer to this, however, could be that the descent near the poles is more concentrated than the

ascent in the tropics, which occurs over a larger horizontal area.

Williamson and Olson (1994) noted a reduction in polar tropopause cold bias in changing

from Eulerian to semi-Lagrangian model dynamics. This was confirmed by Chen and Bates

(1996), who achieved a reduction of the polar cold bias by roughly half everywhere above

200 mb, showing this was due to more efficient poleward heat transport in the polar regions.

This, in turn, is likely to be a result of increased wave-driving of the stratospheric circulation.

Finally, Zhu and Schneider (1997) achieved a reduction in the cold pole bias in the third

generation ECHAM GCM by switching from a hybrid σ − p to a σ − θ vertical coordinate.

Warming over the poles in the 100-300 mb region had also been noted by Thuburn (1993) in a

similar experiment. Zhu and Schneider show the temperature error of the original model which

appears to contain two distinct cold biases; one of up to 9 K centred at 200 mb in the polar

region of both hemispheres, and a second centred at 30 mb only in the winter hemisphere (9 K

in the Southern Hemisphere and up to 27 K in the Northern Hemisphere). Both of these biases

are reduced in the σ − θ version, consistent with differences in eddy heat-flux convergence

between the two models. Changes in Eliassen-Palm fluxes are observed in the same regions,

indicating a change in wave forcing of the zonal mean.

Webster et al. (1999) and Schaack et al. (2004) have also achieved reductions in the cold

bias by using hybrid-isentropic coordinates. Webster et al. noted that errors in the vertical

advection of temperature and moisture are reduced in both isentropic models, and in those em-

ploying semi-Lagrangian advection schemes, and suggested that this could be one reason for

the reduced bias in these models. Increased wave-driving contributed significantly to the re-

duction, especially in the winter hemisphere. In the hybrid-isentropic model this was attributed

to two factors. Firstly, model diffusion acts along isentropic surfaces in the stratosphere, and

therefore is expected to be a better representation of the isentropic mixing of PV which occurs

in wave-breaking events in the real atmosphere (Haynes and McIntyre (1987, 1990)). Secondly,

the vertical propagation of planetary waves involves large displacements of isentropic surfaces

from isobaric surfaces, so that with a pressure coordinate there are large vertical mass fluxes,

with associated truncation errors, that do not occur with an isentropic coordinate. Schaack et al.


(2004) comment that the improvements in advection, and in wave-driving, in hybrid isentropic

models ‘are consistent with Johnson’s (1997) theoretical study’, though no further explanation

of this is given.

5.5 A Comment on the MEP Theory

The work presented here was not performed with the Maximum Entropy Production, or MEP,

theory in mind. It is, however, of relevance. According to the MEP theory it is the entropy

produced by poleward heat transport in the climate system which is at a maximum. The size of

this entropy production in the atmosphere can be estimated using a very simple model. Suppose

the atmosphere can be represented as a warm reservoir of temperature T+ (the tropics) and a

cool reservoir at T− (the polar regions), with a poleward heat flux F between the two. Using

(5.1) the total entropy source is then

s =F

T−− F

T+

=T+ − T−T+T−

F > 0. (5.15)

Assuming values of T+ = 295 K, T− = 275 K and F = 5 PW this gives an entropy production

rate of 4 mWm−2K−1. This is obviously a very rough value, but indicates the order of magni-

tude of the source; it is certainly a small number compared to terms in the atmospheric entropy

budget (Peixoto et al. (1991), Goody (2000)).

It is the large-scale circulation which transports heat towards the poles, but this is, in

principle, a reversible process. Entropy is only produced by the mixing of energy, and it is

the small-scale dissipation processes such as turbulence which actually mix the warm tropical

and cool extratropical air masses, and hence generate the entropy to which the MEP theory

applies (Ozawa et al. (2003)). The dry dissipation processes, however, are also precisely those

represented by the diffusion in the IGCM LC1 cycle.

In investigating the MEP theory, Kleidon et al. (2003) used a simple, dry GCM to perform

experiments similar to those described in this chapter. The only physical parameterizations

included were Rayleigh friction to represent the turbulent boundary layer and Newtonian heat-

ing to represent the effect of radiation by relaxing temperatures towards a radiative-convective

equilibrium state. In 10 year simulations the total entropy source was determined by comput-

ing the change in heat content of each grid cell and dividing by the temperature. The size of


the source was seen to be independent of horizontal resolution above T42, having converged

to 2.5 mWm−2K−1. In these experiments the heating, surface friction and numerical diffusion

all contribute to dissipating potential enstrophy and generating entropy, yet the overall source

is remarkably similar to that seen in Figure 5.1 for the source during the LC1 cycle (roughly

3 mWm−2K−1).

It is perhaps a coincidence that these values are so similar, particularly given the short

duration of the IGCM runs. However, here it is argued that this is not the case. The generation

of entropy in both experiments is by the mixing of air masses with different potential temper-

atures, and is a direct consequence of the dissipation of potential enstrophy to halt the cascade

to small scales. The similarity of the two values could then be taken to support the hypothe-

sis that the size of the source is determined not by the details of the dissipation mechanism,

here strikingly different in the two experiments, but by the rate at which the potential enstro-

phy is cascading, itself determined by the characteristics of the large-scale flow. It could be

argued that the large-scale flows are significantly different in the two experiments, one driven

by parameterized heating and the other evolving from an idealised initial condition. The same

dominant dynamics act in each, however; the evolution of synoptic scale eddies in response

to baroclinic instability. It is entirely possible, if not anticipated, that both flows will generate

small scales in PV at the same rate.

This comparison is, of course, entirely circumstantial evidence for the hypothesis that the

entropy source is determined by the large-scale flow. It could, in fact, be considered weak evi-

dence given the idealistic nature of the LC1 cycle, and its limited duration. Longer simulations

in the absence of any heating parameterization would, however, be unrealistic. Clearly further

investigation is warranted.

It is a widely held view that there is no physical basis for the climate system to choose to

be at a state of MEP, and the fact that it appears to be close to this is merely a coincidence.

Here it is suggested that the rate of entropy production is determined by the large-scale flow,

and hence by the degree of baroclinic instability. It is much easier to imagine that there is some

critical level of stability at which the stabilising effect of the baroclinic eddies balances the

destabilising effect of the uneven solar heating. The atmosphere is thus continually adjusting

towards this state, and the entropy source is indirectly determined by this through the strength

of the eddy field, rather than itself determining the nature of the flow.

As an example, consider again the two box model introduced at the start of this section.

In slightly more complex models (e.g. Lorenz et al. (2001), Kleidon and Lorenz (2004)) the


temperatures are allowed to vary, and the heat flux is expressed as a diffusive flux by

F = k(T+ − T−), (5.16)

where k is considered an effective heat diffusivity and is also allowed to vary. The heat flux

F is fixed by energy conservation. The parameter k measures how efficient the atmosphere

is at transporting heat, i.e. how quickly heat is transported for a given temperature gradient

(T+ − T−). The mechanism inducing heat transport is baroclinic instability, so k would be

expected to decrease if the temperature gradient decreases. This system now shows potential

for feedback: if k is increased then heat is transported more efficiently and the equator to pole

temperature gradient is reduced, in turn reducing k. There is therefore a stable state at some

value of k; the system will act to correct any perturbation of k away from this state. MEP

supporters claim that this state is the one in which the entropy production given by (5.15) is

a maximum, though clearly this depends on the exact nature of the dependence of k on the

temperature gradient.

Now suppose the system is at the critical stability state described above, and is perturbed

so it is slightly more unstable. This would correspond to an increase of APE in the continuous

system, or an increased gradient (T+ − T−) in the two box model. Stronger eddies would

form, effectively increasing k and reducing the temperature gradient again by more efficient

heat transport. There is therefore a clear physical argument for the system to seek this critical

stability state. This may correspond to the state of maximum entropy production via APE

arguments (Ozawa et al. (2003)) but if so this is a symptom of the system’s behaviour rather

than the cause.

5.6 Discussion

It should be noted that the experiments presented in this chapter represent very idealised situ-

ations. Any conclusions drawn are valid only to the extent that the flows modelled are repre-

sentative of both real flows and those simulated by more comprehensive models. Furthermore,

several of the ideas presented here are clearly hypotheses rather than proven theories.

In addition it is important to remember that the effect of diffusion on model simulations is

not limited to its characteristics as an entropy source. MacVean (1983) also used the IGCM to

study the effect of varying diffusion on idealised baroclinic waves. The linear growth rate of


the normal modes was found to be very sensitive; without diffusion, the growth rate monoton-

ically increased with wavenumber so that waves of unrealistically high wavenumber (e.g. 20)

would be more unstable than the commonly observed events of wavenumber 6 to 9. Adding

sufficiently scale-selective diffusion had the beneficial effect of damping the high wavenum-

bers while leaving the more realistic, lower wavenumber disturbances unchanged. The details

of the diffusion, therefore, greatly affected the growth of high wavenumber secondary waves

after the main development, resulting in wildly different energetics as seen in Figure 5.2.

Later, Stephenson (1994) studied the sensitivity of simulations by a more realistic GCM to

the horizontal diffusion. Significant effects were observed, for example that ‘weakened diffu-

sion results in weakened equator-pole temperature gradients, weakened baroclinicity and much

weaker barotropic energy conversions’. In some ways, then, the details in representation of the

small-scale processes clearly do profoundly affect the larger scales.

While entropy is generated by hyper-diffusion on temperature in the IGCM LC1 cycle, it

is argued here that the majority of this does not constitute a spurious entropy source in climate

models. The combination of diffusion and physical parameterizations acts to dissipate PV

gradients and to mix energy. The amount of mixing needed is determined by the large-scale

flow and is not overestimated; if air masses of different potential temperature have been mixed

by diffusion then there is simply nothing remaining to be mixed by the parameterizations and

the entropy will not be further increased. If it is true that, in this context at least, the effect of

small-scale processes is determined by the large-scale flow, this provides further justification

for predicting the flow evolution from a discretized equation set, providing perhaps that some

minimum resolution is used so that the large-scale deformation field is accurately simulated.

This is not to say that there is no spurious entropy generated by the diffusion, only that

the dominant contribution from the limiting of surface frontal gradients is not spurious, since

these gradients are limited by diabatic processes in the real atmosphere. Diffusion, however,

also acts on features which are not cascading to small scales in the way that fronts and PV

filaments are. Here the above arguments do not apply; the amount of mixing performed by the

diffusion would be expected to be dependent on the details of the scheme, and is likely to be

an overestimate. Entropy generated in this way would constitute a spurious source.

Entropy is also produced in the life cycle by numerical schemes representing the adiabatic

dynamics. This is considered to be a spurious source, though it is five times smaller than that

predicted by Johnson (1997) to yield a 10 K mean cold bias in the 300-325 K region. This

does not contradict Johnson’s theory. Large spurious entropy sources are expected from the


representation of moist processes (Johnson et al. (2000, 2002)), which are not included here.

If the dry dynamics alone generated entropy at 2 mWm−2K−1 then the total source would be

larger and the predicted bias would significantly exceed that actually seen in climate models.

It is worth noting that the entropy exchanges associated with moist processes are clearly the

most uncertain (Goody (2000)). Any subtlety in whether, for example, the entropy generated

by dry dynamics during a model occlusion event is realistic will be swamped by uncertainty in

the moist effects.

It appears that, in general, the cold bias has reduced since the original study of Boer et al.

(1992), but has not disappeared. It is clear that the representation of gravity waves plays a part;

any underestimation of the gravity wave forcing will result in a weakened Brewer-Dobson

circulation, less downwelling and therefore reduced adiabatic heating over the poles. This

cannot explain all of the bias, however. The effect of increasing gravity wave forcing is seen

mainly in the upper stratosphere and lower mesosphere, while the tropopause region is also too

cold in models. Furthermore, if an unrealistically weak Brewer-Dobson circulation is to blame

then the tropical tropopause region would show a systematic warm bias. There does appear

to be a mean warm bias here (see Figure 8 of Gates et al. (1999)), but also a large ensemble

standard deviation, indicating a large spread between models. The fact that the warm bias is

not as systematic as the cold bias could be because the polar downwelling is more concentrated

than the tropical upwelling. However, both Pawson et al. (2000) and Shepherd (2002) conclude

that the stratospheric circulation is too weak in some models, but not in others.

Entropy errors of the size required by Johnson’s theory certainly exist (Goody (2000))

and the theory does provide an explanation for the cold bias in the 300-325 K region, i.e. the

region where little warming is expected from improved representation of gravity wave effects.

Perhaps the greatest problem with Johnson’s theory is the neglect of the ∆Q terms in (5.9). If

the temperature of an air parcel decreases then the energy that it loses by longwave radiation

will also decrease. It is then hard to imagine departures of the temperature structure from that

of the real atmosphere, without also experiencing some departure of the energy exchanges.

This is hinted at in Goody’s final comment that ‘there may be compensating changes in energy

sources’. Despite this, Johnson’s entropy theory is attractive, and seems likely to account for

part of the observed cold bias. Aphysical sources must be offset by additional sinks and, given

the strong constraints on energy exchanges, a systematic bias in temperatures seems to be an

obvious mechanism to provide for this. (Another candidate could be a systematic bias in the

hydrological cycle; moist effects enter Johnson’s theory only through the heat addition Q.)


Hamilton et al. (1999), Zhu and Schneider (1997) and Webster et al. (1999) all show two,

seemingly distinct, regions of bias, one centred at roughly 200 mb and another higher up. Both

are reduced in the switch to a hybrid isentropic vertical coordinate in the latter two studies.

Here, it is hypothesised that the reduction in spurious entropy sources associated with the

isentropic coordinate has reduced the cold bias in the 300-325 K region, in line with Johnson

(1997). The improved eddy heat fluxes resulting from more accurate planetary wave dynamics

would account for the reduction in bias centred on 30 mb in the winter hemisphere, where the

large-scale circulation has a strong warming effect.

5.7 Summary

• The entropy source during the IGCM LC1 cycle averages approximately 3 mWm−2K−1,

and is remarkably insensitive to changes in horizontal resolution and diffusivity.

• The effect of the model’s hyper-diffusion dominates the source, and this component is,

in turn, dominated by the entropy generation near the surface. A source of roughly

0.4 mWm−2K−1 is generated in the evaluation of the adiabatic terms in the prognostic

equations; this is certainly spurious.

• It is hypothesised here that much of the entropy generated by the hyper-diffusion does

represent a real, physical source. The diffusion generates entropy as it limits the scale

of surface fronts, and as it dissipates potential enstrophy in the cascade to small scales.

The entropy source is determined by the rate at which these small scales are generated,

and this is determined by the large-scale deformation field rather than the dissipation

mechanism.

• The diffusion clearly acts on features which are not cascading to small scales. In this case

the above argument does not apply. The resulting entropy source is expected to depend

on the formulation of the diffusion scheme, and is likely to be overestimated. Some part

of the entropy source due to diffusion is, therefore, likely to be spurious.

• The spurious entropy sources demonstrated in the IGCM are of a size which would result

in a cold bias of up to a few Kelvin, according to Johnson’s theory. It does appear

probable that this mechanism accounts for a part of the bias seen in current climate

models.


• The Lagrangian conservation of entropy in the IGCM is much worse than the net source

suggests. The global increase in entropy is the residual effect of frequent increases and

decreases in the entropy of individual fluid parcels.

Chapter 6

Conclusions

The general conclusions of this thesis are that:

• There are errors in the representation of the entropy balance in conventional atmosphere

models, and these are significant.

• Isentropic models do show improved accuracy in certain areas, such as Lagrangian trans-

port and conservation, and are therefore a useful alternative to conventional atmosphere

models.

• Some of the potential benefits expected from using PV as a prognostic variable have also

been realised.

On the first of these, the most basic argument is simply that the real system obeys physical laws

such as the second law of thermodynamics, and the Lagrangian conservation of entropy under

adiabatic motion. Models which do not obey these laws are therefore unphysical in some way.

On the second, it is important that the atmosphere is modelled from a range of perspec-

tives, and using a range of different techniques. Isentropic models offer the trivial satisfaction

of Lagrangian conservation laws which are not satisfied in other models. Several hybrid isen-

tropic models are at an advanced stage of development, but technical difficulties do still exist

at the Earth’s surface in pure isentropic models, including the one developed here. These mod-

els are still useful, however, as the only tools currently available that are able to predict the

completely adiabatic evolution of a primitive equation atmosphere. The interplay between adi-

abatic and diabatic processes in the atmosphere is not fully understood, and these models have

the potential to provide valuable insight.

Conclusions 168

Before discussing these conclusions in more detail, we briefly review some specific results

of this work. More detailed summaries can be found at the end of each chapter.

6.1 Summary of Results

• Charney-Phillips type vertical grids best represent the normal modes of both hydrostatic

and non-hydrostatic atmospheres in isentropic coordinates. The isentropic coordinate

is the only one out of three coordinates tested which has the potential to easily achieve

good conservation and dispersion properties in the same grid.

• An isentropic model is presented which uses a new variation of the massless layer tech-

nique to handle the intersection of model levels with the ground. Unfortunately the

massless region is unstable, and a relaxation method is employed to control this. Simple

flows are modelled successfully, and some known problems analysed, in Chapter 3. The

model is not stable enough to simulate a whole 15 day baroclinic wave life cycle, but

does successfully simulate the growth and most of the decay phase.

• The isentropic model life cycle is much more similar to an IGCM version with∇4 hyper-

diffusion than it is to the standard∇6 version. This is in agreement with estimates of the

implicit diffusion in the advection scheme by Thuburn (1995). There is, therefore, a

slight mixing of interior and boundary PV during advection. While PV is conserved bet-

ter in the isentropic model, it is the∇6 IGCM version of the cycle which most resembles

a high resolution simulation, because the scale of PV features is limited in the isentropic

model as it is in the ∇4 IGCM.

• Despite this, some of the expected benefits of using PV as a prognostic variable have

been realised. For example, there is no spurious amplification of PV maxima, as is seen

in the IGCM life cycle, and there is no spurious generation of potential enstrophy.

• Lagrangian conservation of entropy is significantly better in the isentropic model. How-

ever, the model exhibits a total energy drift which is more serious than the error in total

entropy shown by the IGCM. This energy drift may be partly caused by the implicit

mixing of interior and boundary PV.

• As described in Section 1.2.3, recent pure isentropic models have suffered from two

outstanding technical problems. The first of these, i.e. lack of smoothness in ∇2M , is

Conclusions 169

avoided here because the horizontal and vertical resolutions are well matched. If this

is not the case, this problem still exists. The second, that of inability to model surface

fronts, appears to have been overcome. This model predicts a discontinuity in surface

potential temperature, and so supports the hypothesis that frontal discontinuities are a

feature of purely inviscid atmospheres. The model forms fronts without entropy produc-

tion, and with only slight mixing of interior and boundary PV, so these are considered

better representations of truly adiabatic fronts than those in the IGCM. As hoped, the

isentropic model is able to simulate realistic fronts at much lower resolution than the

IGCM.

• There is a spurious entropy source in the IGCM due to numerical transport errors; this

source averages 0.4 mWm−2K−1 over the lifecycle.

• Entropy is generated by numerical diffusion at a rate of approximately 2.5 mWm−2K−1.

This varies only slightly with resolution and diffusivity, and is dominated by the con-

tribution from the lowest two levels. It is argued in Chapter 5 that the majority of this

source does represent a real, physical entropy source, because it is diabatic processes

which ultimately limit both the potential enstrophy cascade, and the gradient of surface

fronts. The rate of cascade, and therefore the entropy source, is determined by the defor-

mation field of the large-scale flow, rather than the details of the dissipation mechanism.

Some fraction of this entropy source from diffusion is spurious, however, since model

diffusion does not only act on features which are cascading to small scales.

• No evidence has been found to disprove the theory of Johnson (1997) that spurious en-

tropy production in climate models is the cause of their systematic cold bias. On the

contrary, the spurious entropy source seen in the IGCM life cycle is of the size required

by the theory to cause a bias of at least a few Kelvin, according to Johnson’s estimate. It

appears likely that this is one factor responsible for the general coldness of climate mod-

els, with an unrealistically weak wave-driven stratospheric circulation also contributing.

• The IGCM, in any case, does not respect the Lagrangian conservation laws of entropy

and PV. This would undoubtedly lead to errors in the transport of moisture and chemical

species in a more complete model.

Conclusions 170

6.2 Discussion

The instability of the isentropic model clearly needs addressing. The definition of PV and

static stability values throughout the massless region appears to be justified, since the flow near

the surface is realistic. However, evolution according to the primitive equations in this region

is less justifiable, especially since a strong relaxation needs to be applied. Future work will

aim to determine these values in the massless region diagnostically using data from just above

the surface. This would eliminate the need for a relaxation scheme, preventing the surface

divergence effect of Sections 3.7 and 4.13.1, and hopefully reducing the mixing of internal and

boundary PV by reducing PV gradients. This, in turn, should reduce the energy drift.

In general the model has shown promise, especially in its Lagrangian conservation prop-

erties. Despite this, however, some synoptic PV features simulated by the ∇6 IGCM are more

similar to features in the high resolution IGCM run than those predicted by the isentropic model

are. This is because the scale limitation in the isentropic model acts like a ∇4 operator, and

at this resolution features such as the high PV troughs are affected. It is hoped that at higher

resolution this scale limitation will act to dissipate potential enstrophy at a more realistic scale,

with less effect on features such as the troughs. Unfortunately, higher resolution runs with the

current model have been unsuccessful due to the instability of the massless region.

The surface flow is well simulated by the isentropic model, which is a significant improve-

ment over similar previous models. Arakawa et al. (1992) describe how the surface evolution in

their model is inhibited by the lack of degrees of freedom in the surface potential temperature.

Baroclinic waves are too weak because the low-level wave does not evolve correctly (Konor

(2004), personal communication). Their isentropic model levels slump excessively and pile up

along the surface, as seen in Hsu and Arakawa (1990), so that the fronts where they intersect

the surface lie too far towards the equator. This is believed to be partly a result of the model

adjusting itself to smooth the ∇2M distribution. This does not happen in the isentropic model

developed here; the fronts it predicts are no further towards the equator than those in the IGCM

(compare Figures 4.2 and 4.8), and the EKE generated is almost identical to that shown by the

∇4 IGCM. The sub-gridscale fit of Section 3.4.3 is key in the accurate representation of the

surface potential temperature distribution.

One potential objection to the modelling philosophy used here is based on the scale effect

of PV. An isolated PV anomaly corresponds to wind and temperature fields that have a larger

scale, in both the horizontal and vertical, than the anomaly itself. The anomaly is therefore

Conclusions 171

harder to resolve on a discrete grid than its associated flow field. This is perhaps the reason

why the PV distribution predicted by the isentropic model in Figure 4.10 still appears slightly

more diffused than that predicted by the∇4 IGCM in Figure 4.12. However, there is very little

difference between the two simulations in either the synoptic scale features, or the eddy kinetic

energy evolution, so this effect appears to be only minor, at least in the simulation studied here.

Another objection is that the ‘inversion’ step needed to calculate the winds from the vor-

ticity and divergence, while cheap in a spectral model such as the IGCM, is expensive in a grid-

point model. Multigrid methods, as used in the isentropic model, have significantly reduced

this computational burden, but not removed it. The horizontal scheme offers many benefits: it

allows the balanced flow to evolve only by the action of a carefully constructed conservative

advection scheme on PV and density, it allows gravity modes to be damped separately if need

be, and it effectively gives the vorticity and divergence on the unstaggered Z grid, which sim-

ulates geostrophic adjustment better than any of the well-known A-E grids (Randall (1994)).

(Note that it may be possible to achieve the benefits of the Z grid in a model predicting momen-

tum directly, and so not requiring an inversion step (Ringler and Randall (2002a,b)), though

this does require the filtering of computational modes.) In future development of the isentropic

model, it will be important to establish whether these potential benefits have been realised, and

the expense justified.

Spurious entropy production occurs in the IGCM life cycle by two mechanisms, as argued

in Chapter 5. Firstly, numerical transport errors during the evaluation of adiabatic terms in the

primitive equations result in a net production of entropy. This positive source is the residual of

many errors, both positive and negative, in the Lagrangian transport of heat. While the overall

source is small, local errors are large; the potential temperature of an air parcel is certainly not

conserved by the IGCM. This supports the conclusion of Johnson et al. (2000, 2002) that non-

isentropic Eulerian atmosphere models are very bad at Lagrangian conservation. The second

mechanism is by explicit numerical diffusion. Here we claim that some of the entropy produced

by hyper-diffusion in the IGCM constitutes a spurious entropy source.

Both of these spurious sources can be removed by using an isentropic vertical coordinate,

as demonstrated by the model developed here. In this model, potential enstrophy is dissipated

on isentropic levels with no mass exchange between levels. Real entropy sources could then

be carefully parameterized as diabatic mass fluxes. It should be remembered, however, that

the energy drift shown by this isentropic model constitutes a larger error in the second law of

thermodynamics than the entropy drift in the IGCM. Given the ongoing technical problems

Conclusions 172

with pure isentropic models such as this, it is wise to also consider some alternatives.

In hybrid models not all of the atmosphere is modelled isentropically, but a large propor-

tion is. In the RUC model of Benjamin et al. (2004), 70-80% of the atmosphere is modelled

with a purely isentropic coordinate (Benjamin (2004), personal communication). In the Uni-

versity of Wisconsin model this figure is 85% (Schaack et al. (2004)). Furthermore, these

models both demonstrate exceptionally good Lagrangian entropy conservation in the tests laid

out by Johnson et al. (2000, 2002). The use of a non-isentropic coordinate in the region near

the lower boundary does not appear to damage these Lagrangian conservation properties sig-

nificantly. These coordinates are, therefore, a good solution to the problem of how to represent

entropy in an atmosphere model. These hybrid models currently employ standard physical

parameterizations which were designed, and tuned, for models based on other coordinate sys-

tems. The community is now looking to design parameterization schemes specifically to take

advantage of the benefits of the isentropic coordinate, for example, the direct link between

diabatic heating and vertical velocity.

It might be possible to reduce errors in the transport of entropy in non-isentropic Eulerian

models by developing more accurate transport schemes, but this would be very challenging.

There is, however, an option for reducing the second spurious entropy source, that from nu-

merical diffusion. The ocean is dominated by adiabatic processes to an even greater extent

than the atmosphere, so much attention has been focussed on efforts to reduce spurious dia-

batic motion in ocean models. Gent and McWilliams (1990), and subsequently Griffies (1998),

parameterize the sub-gridscale flow by effectively diffusing isopycnal layer thickness, i.e. by

moving fluid along isopycnal surfaces rather than across them. It is hoped that such a scheme

would improve the accuracy of atmosphere models, reducing both the spurious entropy source,

and the flow of moisture and chemical species across isentropic surfaces unless explicitly pa-

rameterized. Some benefit is expected even in comprehensive models which incorporate a

separate boundary layer scheme, since Figure 5.5 shows that roughly a third of the entropy

generation by the hyper-diffusion occurs above the boundary layer. Numerical diffusion is

generally believed to transport too much mass across isentropic surfaces in the tropopause re-

gion (Stohl et al. (2003a)); it is hoped that more accurate transport could be achieved if some

kind of adiabatic diffusion operator is used. Work is currently in progress to implement diffu-

sion schemes in the IGCM which are based on that of Gent and McWilliams (1990), and on

the more scale-selective scheme of Roberts and Marshall (1998).

The atmosphere is strongly constrained by the Lagrangian conservation of entropy and

Conclusions 173

PV, and exhibits a delicate balance between numerous entropy sources and sinks. If numerical

models can be made to reproduce these features, their realism can only be improved.

Appendix A

The LC2 Cycle

The second of the two life cycles studied by Thorncroft et al. (1993) is the LC2 cycle, whose

initial conditions differ from those of the LC1 cycle only by an enhanced cyclonic horizontal

shear (see Section 4.1 and Figure 4.1). A simulation of this life cycle was also attempted using

the isentropic model, but suffered from imbalances in the initial conditions. The initial state

was created in the same way as in Section 4.5 for the LC1 cycle; LC2 however proved more

sensitive to errors in the interpolation procedure. Imbalances were introduced because the

mass field was interpolated to the centre of each hexagonal or pentagonal cell, while the wind

fields were interpolated onto the cell edges. The imbalances were successfully removed using

an adjustment technique, but this significantly changed the zonal mean initial conditions. The

purpose of this appendix is to document this adjustment technique, and the effect on the mean

state, and to show that a closer fit to the IGCM LC2 cycle can be obtained by using a higher

resolution intermediate grid during the interpolation of initial conditions.

If no attempt is made to balance the initial state, then the simulation is unsuccessful. The

imbalances result in small-scale noise, and in the initial forcing of gravity waves. The noise is

largest at the pentagons, and therefore has a wavenumber 5 shape which seeds a wavenumber

5 wave in the same way as the surface pressure perturbation seeds a wavenumber 6 wave. The

resulting evolution shows a combination of waves of these two wavenumbers.

A first attempt to remove the noise was made by enforcing a period of adjustment, or bal-

ancing, following Thuburn (1995). Firstly, the normal mode perturbation was removed from

the PV field by subtracting off the difference from the zonal mean. This perturbation was then

stored in order to be re-applied later. The model was run with the density and PV fields ad-

vected only by the divergent wind, and with the timescale for hyper-diffusion on divergence

The LC2 Cycle 175

0 20 40 60 80 100 120

−80

−60

−40

−20

0

20

40

60

80

Latit

ude

Zonal Mean Divergence

0 20 40 60 80 100 120289

301

313

325

337

358

451

Time (hours)

θ (K

)

Figure A.1: Evolution of the zonal mean divergence as a function of latitude at 345 K (top),and as a function of potential temperature at 45N (bottom), during adjustment. Contours aredrawn every 2× 10−6s−1.

The LC2 Cycle 176

reduced by a factor of 10. The intention was to allow the system to adjust towards a balanced

state by emitting gravity waves. These waves would be damped by the additional divergence

damping, with the balanced dynamics essentially frozen. Note that this approach is only pos-

sible because this model is formulated with PV and divergence as prognostic variables.

Figure A.1 shows that gravity waves are indeed emitted from the region near 60N, where

the imbalance is greatest, and propagate away while being damped by the hyper-diffusion.

After 96 hours of balancing, the normal mode PV perturbation was added back on, and the

model integrated as usual from these ‘adjusted’ initial conditions. Figure A.2 compares the

subsequent PV evolution with that predicted by the IGCM. The isentropic model does predict

the correct general behaviour, with cut-off cyclones dominating in the extratropics. The inten-

sity of these cyclones, however, is much weaker than it is in the IGCM simulation, and the

disagreement between the two models is greater than that seen in the LC1 cycle.

The cause of these differences is obvious when the adjusted initial conditions are examined

in detail. Figure A.3 shows the zonal mean initial conditions before, and after, the adjustment

process. This process has significantly affected the state, as is seen most clearly in the low

level winds. After adjustment the zonal mean wind near the surface reaches 20 ms−1, a value

that is not attained until over 600 mb in the original state. The adjustment has also resulted in

a uniformity of wind on isentropic model levels in the mid-latitude troposphere, so that in this

figure the wind contours lie more parallel to the isentropes. The life cycle is very sensitive to the

zonal mean initial state, as described by Thorncroft et al. (1993); it is, after all, only changes in

the zonal mean wind shear that distinguish the LC2 cycle from LC1. The changes in the initial

state shown in Figure A.3 result in a wave which is significantly different, as is to be expected,

and so this simulation cannot be directly compared to the IGCM LC2 cycle. While adjustment

techniques such as this are clearly successful in bringing the state back towards balance, this

example shows that care must be taken since the mean state can be altered significantly.

A second attempt was then made to remove the imbalances. As described in Section 4.5,

the spectral fields from the IGCM are first evaluated on a latitude-longitude grid, and then

interpolated onto the hexagonal-icosahedral grid. The spectral coefficients define the initial

conditions exactly at a given point on the sphere, so using a higher resolution latitude-longitude

grid will result in smaller interpolation errors, and in more accurate fields on the hexagonal

grid. The resolution of this intermediate grid was doubled, and the LC2 cycle run from the

resulting initial conditions with no adjustment. Figure A.4 shows the eddy kinetic energy

of this life cycle, compared with that predicted by the IGCM, and by the isentropic model

The LC2 Cycle 177

Figure A.2: PV on the 330 K isentropic surface for the LC2 cycle in the isentropic model (left)and the IGCM (right). Contours and colour are as in Figure 4.3.

The LC2 Cycle 178

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pre−adjustment

p/p s

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Post−adjustment

Latitude

p/p s

Figure A.3: Zonal mean sections, as in Figure 4.7, but for the LC2 initial conditions before, andafter, the adjustment process. The black lines are isentropes corresponding to the isentropicmodel half levels, which are spaced every 6K from 298K (dotted) to 341K (dashed). Thesloping red line is the 2 PVU surface, and the other coloured contours show the zonal windfield at 5 ms−1 intervals, with the zero contour dotted and negative contours dashed. All fieldsare zonal means.

The LC2 Cycle 179

0 5 10 150

5

10

15x 10

5

Day


IGCMAdjustedUnadjusted

Figure A.4: Eddy kinetic energy evolution for the LC2 simulations. Two runs are shown fromthe isentropic model, demonstrating the two methods for creating balanced initial conditions.The ‘adjusted’ run only used the adjustment technique, while the ‘unadjusted’ run only usedthe refined intermediate grid.

cycle using the adjustment technique. The life cycle that evolves from the adjusted initial

conditions has dramatically weaker EKE than the IGCM cycle, a result that is consistent with

the weaker cyclones seen in Figure A.2. By using the refined intermediate grid, rather than

the adjustment technique, an EKE evolution is obtained which is much closer to that predicted

by the IGCM. It is anticipated that further refinement of the intermediate grid would lead to

even closer agreement. This has not been pursued here, however, because the isentropic model

proved unstable in this life cycle, as in LC1. Addressing this deficiency is considered a higher

priority than further corrections to the initial conditions in this test case.

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