Vorticity in the

Ocean

Principles of

Oceanography

Introduction

Definitions of Vorticity

Conservation of Vorticity

Influence of Vorticity

Vorticity and Ekman pumping

Consequence

Definitions of vorticity

Vorticity is analagous to angular momentum.

Vorticity is a conserved quantity (Conservation of Vorticity)

Two types of vorticity

Planetary Vorticity f

Rotation imparted by Earth

Relative Vorticity z

Due to currents in the ocean

2D flow assumption

Order z << f

Planetary Vorticity f and Relative

Vorticity z Planetary Vorticity: Every object on earth has a vorticity given

to it by the rotation of the earth (except an object on the

equator). This vorticity is dependent on latitude.

Relative Vorticity: The ocean and rotate at exactly the same

rate as earth. They haveatmosphere do not some rotation

relative to earth due to currents and winds. Relative vorticity ζ is

the vorticity due to currents in the ocean. Mathematically it is:

ky

u

x

vj

x

w

z

ui

z

v

y

w

wvuzyx

kji

u

ζ is usually much smaller than f, and it is greatest at the edge of fast currents such as the Gulf Stream. To obtain some understanding of the size of ζ, consider the edge of the Gulf Stream off Cape Hatteras where the velocity decreases by 1 m/s in 100 km at the boundary. The curl of the current is approximately (1m/s)/(100 km) = 0.14 cycles/day = 1 cycle/week. Hence even this large relative vorticity is still almost seven times smaller than f. A more typical values of relative vorticity, such as the vorticity of eddies, is a cycle per month.

Absolute Vorticity

The sum of the planetary and relative vorticity

Equations of Motion for frictionless flow

Derivation of Vorticity equation

xFfvx

p

Dt

Du

1

yFfuy

p

Dt

Dv

1

0

0

fvx

p

y

uv

x

uu

t

u

Dt

Du

1)()(

fuy

p

y

vv

x

vu

t

v

Dt

Dv

1)()(

y

vfv

y

f

yx

p

y

uv

y

u

y

v

yx

uu

x

u

y

u

y

u

t

2

2

22 1)()(

x

ufu

x

f

yx

p

yx

vv

y

v

x

v

x

vu

x

v

x

u

x

v

t

22

2

2 1)()(

vy

fu

x

f

y

v

x

uf

y

u

x

v

y

v

y

u

x

v

x

u

y

u

x

v

yv

y

u

x

v

xu

y

u

x

v

t

0

v

y

fu

x

f

y

v

x

uf

y

v

x

u

yv

xu

tz

zzz

0

v

y

f

y

v

x

uf

Dt

Dz

z

0

0 0

0

Dt

Df

y

v

x

uf

Dt

Dz

z

Potential Vorticity

As a conserved quantity potential vorticity is a valuable tool in

studying ocean dynamics. The potential vorticity is defined as the product of the absolute vorticity and the stratification.

P is conserved along a fluid trajectory

Barotropic, geostrophic flow in an ocean with depth H(x, y, t)

Fig 1.1

Integrate the continuity equation

Vertical velocity

Conservation of Vorticity

Conservation of z in a spinning ocean

Transfer of momentum between two bodies

Friction is essential

Air-sea boundary Ekman layer transfer momentum

Sea-bottom boundary Ekman layer transfer momentum

Sides of subsea mountains friction pressure differences form drag

In the vast interior of the ocean frictionless vorticity is conserved

Conservation of the angular momentum

The angular momentum of any isolated spinning body is conserved

Transfer of angular momentum between two bodies

Need not be in physical contact; gravitational forces can transfer momentum between bodies in space

Ocean Surface

Ocean bottom

A parcel of water moves east

(constant latitude) in N.Hemis.

As the parcel hits the bump, H

decreases. We know that (f +

ξ)/H=Constant. So if H decreases,

(f + ξ) must decrease. If f

decreases, the parcel moves

equatorward. If ξ decreases the

parcel spins clockwise.

An example of

conservation of

vorticity when H

doesn’t stay constant

H

Bump in bottom

H

What happens when the

parcel leaves the bump?

North

South

A parcel of water moves east

(constant latitude) in N.Hemis.

As the parcel hits the bump, H

decreases. We know that (f + ξ

)/H=Constant. So if H decreases, (f

+ ξ ) must decrease. If f decreases,

the parcel moves equatorward. If

ξ decreases the parcel spins

clockwise. Or a combination.

An example of

conservation of

vorticity when

H doesn’t stay

constant

Bump in bottom

H

H

From ABOVE

Parcel Moves Equatorward

Conservation of Potential

Vorticity

The conservation of potential vorticity couples changes in depth,

relative vorticity, and changes in latitude. All three interact:

Changes in the depth H of the flow causes changes in the

relative vorticity. The concept is analogous with the way figure

skaters decreases their spin by extending their arms and legs.

The action increases their moment of inertia and decreases

their rate of spin

Changes in latitude require a corresponding change in ζ. As a

column of water moves equatorward, f decreases, and ζ must

increase.

Influence of Vorticity

The concept of conservation of potential vorticity has

far reaching consequences, and its application to

fluid flow in the ocean gives a deeper understanding

of ocean currents.

f >> z f / H = constant

The flow in an ocean of constant depth be zonal

Depth is not constant, but in general, currents tend to be

east-west rather than north south

Wind makes small changes in z, leading to a small

meridional component to the flow

Figure 11.3

Influence of Vorticity

Barotropic flows are diverted by seafloor features

Figure 1.4: a flow encounters a subsea ridge, Friction along the sides of sub-sea mountains leads to pressure differences on either side of the mountain which causes another kind of drag called form drag.

Topographic steering:

H ζ f the flow is turned toward the equator

Topographic blocking

If the change in depth is sufficiently large, no change in latitude will be sufficient to conserve potential vorticity, and the flow will be unable to cross the ridge

Fig.1.4

Influence of Vorticity An alternate explanation for the existence

of western boundary currents Figure 12.5

Wind blow negative z

Eastern boundary southward flow f providing positive z conservation

Western boundary northward flow f providing negative z conservation?!!

A strong source of positive vorticity is provided by the current shear in the western boundary current as the current rubs against the coast causing the northward velocity to go to zero at the coast

Convergence/Divergence This idea is nothing more then the piling up or moving

of water away from a region.

Conservation of VOLUME: (du/dx+dv/dy+dw/dz=0)

Rearranging... du/dx + dv/dy = -dw/dz

If water comes into the box (du/dx + dv/dy)>0 there is

a velocity out of the box: dw/dz < 0 DOWNWARD

Vorticity and Ekman Pumping

First consider flow in a fluid with constant rotation

Secondly, how vorticity constrains the flow of a fluid

with rotation that varies with latitude

Fluid dynamics on the f Plane: the Taylor-Proudman

Theorem

f-plane, constant rotation f = f0

Slowly varying flow in a homogeneous (constant density ρ0), rotating, inviscid fluid

Geostrophic equations

Continuity equation

Vorticity and Ekman Pumping

Taking the z derivative and using gives;

Similarly, for the u-component of velocity

The vertical derivative of the horizontal velocity field

must be zero

Because w = 0 at the sea surface and at the sea floor,

if the bottom is level, there can be no vertical velocity

on an f–plane.

Vorticity and Ekman Pumping (Fluid dynamics on the f Plane: the Taylor-Proudman

Theorem)

Implication

Rotation greatly stiffens the flow

Cannot expand or contract in the vertical

direction

As rigid as a steel bar

Geostrophic flow cannot go over a seamount,

it must go around it

w(z=0) = 0 + w(z=H) = 0 + w/z = 0 w(z) = 0

Vorticity and Ekman Pumping (Fluid

Dynamics on the Beta Plane)

Ekman pumping

b-plane

Consider, f = f0 + b y

G: geostrophic flow

Using the continuity equation, and recalling that β y ≪ f0

Thus the variation of Coriolis force with latitude allows

vertical velocity gradients in the geostrophic interior of the ocean, and the vertical velocity leads to north-south currents.

This explains why Sverdrup and Stommel both needed to

do their calculations on a β-plane.

In the central region of the jet, some streamlines that are not purely horizontal are also

visible. They correspond to beads trapped in the bottom or top Ekman layers and

show net fuid exchange from source to sink.

Vorticity and Ekman Pumping Winds at the sea surface drive Ekman transports to the right of the

wind in this northern hemisphere example (bold arrows in shaded Ekman layer). The converging Ekman transports driven by the trades and westerlies drives a downward geostrophic flow just below the Ekman layer (bold vertical arrows), leading to downward bowing constant density surfaces i. Geostrophic currents associated with the

warm water are shown by bold arrows.

Vorticity and Ekman Pumping

An example of how winds

produce geostrophic currents running upwind. Ekman transports due to winds in the north Pacific (Left) lead to Ekman pumping (Center), which sets up north-south

pressure gradients in the upper ocean. The pressure gradients are balanced by the Coriolis force due to east-west geostrophic currents (Right). Horizontal lines

indicate regions where the curl of the zonal wind stress changes sign. AK: Alaskan Current, NEC: North Equatorial Current, NECC: North Equatorial Counter Current

Ocean Surface

Mixed Layer

Ocean bottom

A parcel of water moves into an

area of downwelling. It

becomes shorter (and fatter).

f/H must be

conserved!

We know that (f + ξ)/H= Constant. So if H

decreases, (f + ξ ) must decrease. I gave

examples before that either f or ξ could

change. But in this process; it is f that

decreases. f can only decrease by the parcel

moving equatorward.

With DOWNWELLING, the

vertical velocity is downward.

This pushes on the column of

water, making it shorter (and

fatter). What happens when a

column of water gets short and

fat (Vorticity must be

conserved).

H H

Ekman Convergence

Ekman transport creates convergence and

divergence of upper waters.

Convergence

Convergence

Divergence

Divergence

Divergence

Sea Surface Height and Mean Geostrophic Ocean Circulation

Important concepts

Vorticity strongly constrains ocean dynamics.

Vorticity due to Earth's rotation is much greater

than other sources of vorticity.

Taylor and Proudman showed that vertical velocity is impossible in a uniformly rotating flow.

The ocean is rigid in the direction parallel to the

rotation axis. Hence Ekman pumping requires

that planetary vorticity vary with latitude. This explains why Sverdrup and Stommel found that

realistic oceanic circulation, which is driven by Ekman pumping, requires that f vary with latitude

Important concepts (cont.)

The curl of the wind stress adds relative vorticity to

central gyres of each ocean basin. For steady state

circulation in the gyre, the ocean must lose vorticity in

western boundary currents.

Positive wind stress curl leads to divergent flow in the

Ekman layer. The ocean's interior geostrophic

circulation adjusts through a northward mass transport.

Conservation of absolute vorticity in an ocean with

constant density leads to the conservation of potential

vorticity. Thus changes in depth in an ocean of

constant density requires changes of latitude of the

current.