geophysical fluid dynamics

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Centre for Atmospheric and Oceanic Sciences INDIAN INSTITUTE OF SCIENCE BANGALORE 560 012 Lecture Notes on GEOPHYSICAL FLUID DYNAMICS Prof. Debasis Sengupta Workshop on PHYSICS OF THE ATMOSPHERES & OCEANS 1-12 July 2002

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Page 1: Geophysical Fluid Dynamics

Centre for Atmospheric and Oceanic Sciences

INDIAN INSTITUTE OF SCIENCE

BANGALORE 560 012

Lecture Notes on

GEOPHYSICAL FLUID DYNAMICS

Prof. Debasis Sengupta

Workshop on

PHYSICS OF THE ATMOSPHERES & OCEANS

1-12 July 2002

Page 2: Geophysical Fluid Dynamics

Geophysical Fluid Dynamics (GFD)

What is GFD ?GFD is the theory of large scale, low frequency flow of geophysical fluids (atmo-

sphere, ocean). These flows are strongly influenced by the earth’s rotation. The im-

portance of rotation is estimated from the Rossby number ε, which is the ratio of the

advective time scale L/U and the time scale of the earth’s rotation Ω−1. Large-scale,

low frequency flows have small Rossby number

ε =U

2ΩL≤ 1 (1)

where L and U are horizontal length and velocity scales of the flow and Ω angular

speed of earth rotation. For example an atmospheric flow with L = 1000 km and U = 10

m s−1 has a Rossby number of ε=0.07. An oceanic flow with L = 100 km and U = 1 ms−1

has ε= 0.07. Such flows have time to ”feel” the earth’s rotation, so they come within the

ambit of GFD. Waves on a beach, with L = 100 m and T = 10 sec, do not feel the earth’s

rotation.

The aim of GFD is to develop concepts that help to understand (and predict) geo-

physical flows. These concepts come from the study of idealized ”models” of geophys-

ical flow, which are constructed from the laws of physics with the help of simplifying

assumptions. The development of models is guided by the observed features of geo-

physical flows.

Equations of motionThe equations of motion in an inertial frame of reference are mass conservation (or

continuity) equation

∂ρ

∂t+ ~∇.(ρ~u) = 0 (2)

where ρ is density and ~u velocity, and Newton’s law (momentum conservation) for

a fluid

ρd~u

dt= −~∇p+ ρ~∇φ+ ~F (~u) (3)

1

Page 3: Geophysical Fluid Dynamics

where p is pressure, ρ ~∇ φ is a conservative body force and ~F (~u) is a nonconservative

fluid friction. In eqs. (2) and (3),

d

dt≡

∂t+ ~u.~∇

If the density is not constant, the momentum and continuity equations do not consti-

tute a closed dynamical system. The first law of thermodynamics plus the equation of

state for the fluid ρ = ρ(T ) must be considered to obtain an equation for ρ , where T is

temperature. 1

In a frame of reference rotating with angular velocity ~Ω,

~uI = ~u+ 2~Ω × ~r

where ~uI is the velocity as seen from an inertial frame of reference. and the equation of

motion (3) becomes

ρ

[d~u

dt+ 2~Ω × ~u

]= −~∇p+ ρ~∇φ+ ~F (4)

The observer in a uniformly rotating coordinate frame sees an extra acceleration, i.e.

the coriolis acceleration 2 ~Ω×~u. Spatial gradients are invariant under the transformation

to a rotating frame; the body force ρ~∇φ has an extra contribution from the centripetal

acceleration; the form of ~F determines if it is invariant or not (if ~F is Newtonian, i.e.

proportional to ∇2~u, it is invariant under transformation to a rotating frame).

The (non-dimensional) Rossby number can now be seen as the ratio between the

coriolis term and the relative acceleration term, i.e.

2~Ω × ~u = O(2ΩU),d~u

dt= O

(U2

L

),

giving

ε =U

2ΩL.

GFD deals with flows for which ε ≤ 1, i.e. the earth’s rotation is important for the

dynamics (the balance of forces). For small -ε flows, it is the coriolis acceleration that

mainly balances any applied force ~Fa (see Fig. 1). This implies that the relative ac-

celeration (or displacement) due to an applied force ~Fa must be at right angles to the1Apart from temperature, salinity or humidity also enter the equation of state for seawater or air.

Separate equations for conservation of salt or moisture are then required to close the system.

2

Page 4: Geophysical Fluid Dynamics

direction of ~Fa! This ”gyroscopic” property of fluids on a rotating earth is responsible

for the unique character of geophysical flows.

U

Fa

2Ω × U

Ω

Figure 1: Since coriolis acceleration balances the applied force ~Fa, the displacement (in unittime) ~U is to the right of ~Fa.

Scale analysis of equations on a spherical earthEquation (4) takes the following form at a mid-latitude point on the earth’s surface,

with ~u = ui + vj + wk, where i, j, k are unit vectors in the eastward, northward and

vertically upward directions.

du

dt−uv tan θ

Re

+uw

Re

= −1

ρ

∂p

∂x+ 2Ωv sin θ − 2Ωw cos θ + Fx (5a)

dv

dt+u2 tan θ

Re

+vw

Re

= −1

ρ

∂p

∂y− 2Ωu sin θ + Fy (5b)

dw

dt−u2 + v2

Re

= −1

ρ

∂p

∂z− g − 2Ωu cos θ + Fz (5c)

where Re is earth radius, θ is latitude, g is gravitational acceleration and Fx, Fy, Fz, are

the components of fluid friction. For typical scales of atmospheric ”synoptic” flow, U

∼ 10 ms−1, W ∼ 1 cms −1, L ∼ 106 m, D ∼ 104 m (length scale in the vertical),∆P

ρ∼

103 m2 s−2 (horizontal pressure fluctuations),L

U∼ 105 s (advective time scale), the sizes

of other terms in the equations (5a-b) are small compared to the coriolis term and the

pressure gradient term .2, i.e. the dominant balance is

−fv ' −1

ρ

∂p

∂x, fu ' −

1

ρ

∂p

∂y(6)

2The friction terms can be important near boundaries.

3

Page 5: Geophysical Fluid Dynamics

where f = 2 Ω sin θ. Large-scale flows in the ocean also satisfy (6); such flows are called

”geostrophic”. Note that the local vertical component of coriolis force is the relevant

term in the dynamics. For a given horizontal pressure field, it is possible to calculate

the geostrophic flow ~Vg, which is

~Vg = k ×1

ρf~∇p. (7)

The geostrophic flow is everywhere normal to ~∇p, i.e. the flow is parallel to isobars

(lines of equal p). An atmosphere or ocean with variable density ρo(z) at rest must have

a standard pressure po(z) that depends only on the vertical coordinate z; po and ρo must

be in hydrostatic equilibrium, i.e.

1

ρo

∂po

∂z= −g (8)

In the presence of motion, the pressure and density fields can be written as

p(x, y, z, t) = po(z) + p′(x, y, z, t)

ρ(x, y, z, t) = ρo(z) + ρ′(x, y, z, t); (9)

for an atmosphere or ocean at rest, p′ and ρ′ are zero. Scale analysis of the vertical

momentum equation (5c) shows that ifρ′

ρo

1, then hydrostatic balance holds for the

perturbation pressure and density fields,

∂p′

∂z+ ρ′g = 0, (10)

i.e. vertical accelerations are negligible.

To obtain prediction equations it is necessary to retain the acceleration. The linear

equations are

∂u

∂t− fv = −

1

ρ

∂p

∂x,

∂v

∂t+ fu = −

1

ρ

∂p

∂y

4

Page 6: Geophysical Fluid Dynamics

VorticityVorticity is an important quantity in GFD. It is defined as

~ω = ~∇ × ~u (11)

From Stokes’ theorem,

∫ ∫A

~ω.ndA =

∮C

~u.d~r, (12)

where C is a contour enclosing the surface A whose normal at each point is the unit

vector n, and d~r is the infinitesimal vector tangent to the curve C at each point. In the

limit of small surface element δA (such that the mean vorticity normal to δA is well-

defined)

ωn =ucl

δA,

where uc is the mean circumferential velocity around the contour C of length L. If δA

is a small circle of radius r,

ωn = uc

2πr

πr2= 2

uc

r(13)

so that locally, the velocity is just twice the average angular velocity. The component of

~ω in the z-direction is

ωz =∂v

∂x−∂u

∂y,

illustrating that a unidirectional flow with shear (e.g. u = u(y) or v = v(x)) has non-

zero vorticity; curved trajectories are not required. Recall that a velocity vector ~u in a

rotating frame of reference appears as ~uI = (~u + 2~Ω × ~r) from an inertial frame, where

~r is the position vector of the fluid element. The absolute vorticity ~ωa (i.e. the vorticity

in an inertial frame) is

~ωa = ~∇ × (~u+ ~Ω × ~r) = ~w + 2~Ω (14)

where ~ω is the vorticity relative to the earth (relative vorticity) and 2~Ω is the planetary

vorticity.

5

Page 7: Geophysical Fluid Dynamics

At a latitude θ, the component of 2~Ω normal to the earth’s surface is f = 2Ω sin θ.

The size of the vertical component of relative vorticity ωz is O(U

L

). Therefore

ωz

f=

U

fL=

U

2Ω sin θL=

ε

sin θ(15)

Away from the equator sin θ ∼ O(1), so that small Rossby number flows have the prop-

erty that their relative vorticity is small compared to planetary vorticity.

Since vorticity is divergence - free by definition (~∇.~ω = ~∇.(~∇ × ~u) = 0), for an

arbitrary volume V ,

∫ ∫ ∫V

dV (~∇.~ω) = 0.

An equation for the rate of change of relative vorticityd~ω

dtcan be written by taking

the curl of the momentum equation (4). It is

d~ω

dt= (~ωa.~∇)~u− ~ωa(~∇.~u) +

~∇ρ× ~∇pρ2

+ ~∇ ×~F

ρ

The relative vorticity changes as a result of (a) vortex stretching, i.e. change in cross-

sectional area of vortex tubes. 3, due to non zero divergence in the plane normal to the

tube (e.g. if the tube is vertical,∂u

∂x+∂v

∂y6= 0), (b) Vortex tilting due to non-zero shear

u(z) or v(z), (c)non-zero ”baroclinic” vector ~∇ρ× ~∇p, and (d) friction.

A conservation ”law” for vorticityWe shall obtain a conservation law for vorticity in the context of a simple model

described by the ”shallow water equations”. This is not a general law like momentum

conservation (Newton’s second law), but a general principle of great power in GFD.

Consider a sheet of fluid of constant density ρ and mean depth D. We describe the

equations in a Cartesian coordinate system with its origin at latitude θ. We have seen

that the local vertical component of the earth’s rotation is relevant for small Rossby

number flows (eqs. 5 and the subsequent scale analysis). So we assume that the shallow

water system is rotating about the z - axis at Ω sin θ (therefore f = 2Ω sin θ). The surface

3A vortex tube is an imaginary volume whose sides are everywhere parallel to ~ω; it has two end faces

A and B. Since ~ω is divergence free, the flux of vorticity through the end faces is equal, i.e.∫∫

A~ω.nAdA

=∫∫

B~ω.nBdB.

6

Page 8: Geophysical Fluid Dynamics

of the fluid is at z = h(x, y, t), and the rigid bottom is not flat, but lies at z = hB(x, y). If

D and L are the vertical and horizontal scales of motion, shallow water theory requires

δ =D

L 1. (16)

For constant ρ, the equation of continuity (eq. (2)) reduces to the incompressibility

condition

~∇.~u = 0 (17)

Therefore there is no need to use thermodynamics to calculatedρ

dt; the shallow water

model is purely dynamical. Eq (17) gives

∂u

∂x+∂v

∂y+∂w

∂z= 0 (18)

The first two terms are of size O(U

L

), and it follows that the scale of the vertical

velocity W cannot be larger than O(UD

L

), otherwise eq (18) could not be balanced.

Therefore

W ≤ O(δU) (19)

The shallow water momentum equations are, with p(x, y, z, t) = po(z) + p′(x, y, z, t),

∂u

∂t+

[u∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

]− fv = −

1

ρ

∂p′

∂x(20a)

∂v

∂t+

[u∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

]+ fu = −

1

ρ

∂p′

∂y(20b)

∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z= −

1

ρ

∂p′

∂z, (20c)

where we have used the hydrostatic relation,

−1

ρ

∂po

∂z+ g = 0, i.e. po(z) = −ρgz

in deriving eq (20c).

Scale analysis of eqs (20) shows that the pressure fluctuation p′ is hydrostatic, i.e. at

depth z,

7

Page 9: Geophysical Fluid Dynamics

η(x,y,t)

hB(x,y)

H(x,y,t)h(x,y,t)

X

Z D

Figure 2: Geometry of the shallow water model.

p′(x, y, z, t) = ρg(h− z); (21)

Since the horizontal pressure gradient is independent of z, i.e.

∂p

∂x= ρg

∂h

∂x,

∂p

∂y= ρg

∂h

∂y,

the horizontal accelerations are independent of z. So we assume that initially the veloc-

ities u and v are z - independent, implying that they always remain z - independent.

The horizontal momentum equations are

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y− fv = −g

∂h

∂x(22a)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ fu = −g

∂h

∂y(22b)

If u and v are z - independent, eq (18) can be integrated in z to give

w(x, y, z, t) = −z(∂u

∂x+∂v

∂y

)+ w(x, y, z, t) (23)

where w(x, y, z, t) is a constant of integration. At the rigid surface z = hB(x, y), there can

be no normal flow (i.e. the flow must be parallel to the bottom)

w(x, y, hB, t) = u∂hB

∂x+ v

∂hB

∂y,

8

Page 10: Geophysical Fluid Dynamics

therefore the constant of integration is

w(x, y, t) = u∂hB

∂x+ v

∂hB

∂y+ hB

(∂u

∂x+∂v

∂y

)(24)

So that (23) gives w at any depth

w(x, y, z, t) = (hB − z)

(∂u

∂x+∂v

∂y

)+ hB

(∂u

∂x+∂v

∂y

)(25)

The kinematic boundary condition at the surface is

dh

dt= w =

∂h

∂t+ u

∂h

∂x+ v

∂h

∂yat z = h(x, y, t) (26)

When eqs. (26) and (25) are combined, we get

∂h

∂t+

∂x(h− hB)u +

∂y(h− hB)v = 0 (27)

Since the total depth of the fluid H (x, y, t) = h− hB, the equation of mass conservation

becomes

∂H

∂t+

∂x(uH) +

∂y(vH) = 0 (28)

or

dH

dt+H

(∂u

∂x+∂v

∂y

)= 0 (29)

i.e., horizontal divergence leads to decrease of H ; horizontal convergence leads to in-

crease of H .

Of the three components of vorticity,

wx =∂w

∂y= O

(W

L

)= O

(δU

L

),

wy = −∂w

∂x= O

(W

L

)= O

(δU

L

),

wz =∂v

∂x−∂u

∂y= O

(U

L

),

The vertical component is the dominant part. If h is eliminated from eq (15 a, b) by

differentiating (15a) w.r.t. y and (15b) w.r.t x, we obtain

9

Page 11: Geophysical Fluid Dynamics

dt≡∂ξ

∂t+ u

∂ξ

∂x+ v

∂ξ

∂y= −(ξ + f)

(∂u

∂x+∂v

∂y

)(30)

where ξ = wz. The physical content of (30) is that the convergence of absolute vorticity

(i.e. ξ + f ) tubes changes relative vorticity. Using eq. (29), we can write eq. (30) as

dt=

(ξ + f)

H

dH

dt, (31)

i.e. stretching of vortex tubes(dH

dt> 0

)increases relative vorticity

(dξ

dt> 0

)If f is

constant, (31) may be written

d

dt

(ξ + f

H

)= 0 (32)

The quantity πs =ξ + f

His called ”potential vorticity”, and is conserved following

the flow in shallow water theory.

Thus we have a model where the fluid motion is quasi-two-dimensional (recall that

u and v do not depend on z), and the vorticity is predominantly vertical. As we shall

see, this simple model is very useful indeed!

The Rossby waveThe Swedish meteorologist Carl-Gusstaf Rossby in 1939 introduced homogeneous

(constant ρ) fluid on the surface of a sphere, as the simplest model for the dynamics of

observed large-scale (or planetary) waves in the earth’s atmosphere.

Rossby considered the conservation of potential vorticity

πs =ξ + f

H

on a sphere, where f(= 2Ω sin θ) is a function of latitude. If a fluid parcel moves north

or south through a distance Y , f changes by

∆f =1

Re

∂f

∂θY =

Y

Re

2Ω cos θ (33)

Since ξ = O

(U

L

), ∆f will be of order ξ when

Y

Re

= O(ε tan θ),

10

Page 12: Geophysical Fluid Dynamics

where

ε =U

fL

Thus in mid-latitudes (tan θ ∼ O(1)), a small north-south displacement gives rise

to a sufficiently large change in f to be dynamically important. Introduce a Cartesian

coordinate system that is tangential to the earth’s surface (Fig. 3), such that f varies as

f = fo + βy, βy fo,

where fo is the coriolis frequency at the central latitude θo,

fo = 2Ω sin θ, β =2Ω

Re

cos θo (34)

If H = h − hB = D + η − hB, where η is the surface displacement due to the motion,

then for small η and hB

πs =fo + βy + ξ

D(1 + η

D− hB

D

)'

(fo + βy + ξ)(1 − η

D+ hB

D

)D

'fo +

(βy + fohB

D

)+ ξ − fo

ηD

D(35)

xy

Figure 3: The beta-plane.

11

Page 13: Geophysical Fluid Dynamics

We see from eq (35) that the β effect, i.e. the change of f with latitude due to cur-

vature of the earth, is equivalent to bottom relief (through thefohB

Dterm): both affect

the dynamics by changing the ambient potential vorticity. If hB = 0, the conserved

quantity is

πs =1

D

(fo + βy + ξ − fo

η

D

). (36)

The conservation law

(∂

∂t+ u

∂x+ v

∂y

) (fo + βy + ξ − fo

η

D

)= 0,

along with the geostrophic relations

v =g

fo

∂η

∂x, u = −

g

fo

∂η

∂y

and the relation

ξ =∂v

∂x−∂u

∂y=g

fo

(∂2η

∂x2+∂2η

∂y2

)≡

g

fo

∇2η, (37)

leads to the linear evolution equation

∂t

(∇2η −

f2o

gDη

)+ β

∂η

∂x= 0 (38)

where we have dropped all nonlinear terms (involving products of the dependent vari-

able and its derivatives). Notice that the surface displacement η is the stream function

of the geostrophic flow. Since we have used the geostrophic approximation to evaluate

the horizontal velocity, eq (38) represents the time evolution of the geostrophic vorticity.

It is an important equation called the ”quasi-geostrophic” potential vorticity equation.

Recall that the geostrophic equations cannot tell us anything about time evolution of

a flow; (38) is the simplest equation that can be used to study the time evolution of

relative vorticity in geophysical fluid flows.

Equation (38) or its variants (i.e. equations closely related to it) have been very

successful in explaining a number of observed features of large scale flows in the at-

mosphere and ocean. Examples include the evolution of mid latitude weather patterns

(leading to the first successful weather forecasts), and the existence of strong poleward

western boundary currents in the ocean such as the Gulf Stream, the Kuroshio and the

12

Page 14: Geophysical Fluid Dynamics

Agulhas current. The reason for the success of the quasi-geostrophic potential vorticity

equation is that it contains the essence of vorticity evolution, including the planetary

wave or Rossby wave.

Plane wave solutions of eq. (38)

η ∼ Re exp(i(kx+ ly − σt))

must satisfy the dispersion relation

σ =−βk

k2 + l2 + F, (39)

where F =f 2

o

gD; k and l are wavenumbers in the eastward and northward direction, σ

is frequency. For positive σ, k has to be negative.

The Rossby wave has peculiar properties. The phase speed cx =σ

kis always west-

ward, whereas group velocity Cg =

(∂σ

∂ki,∂σ

∂lj

)can have either eastward or westward

component. Unlike familiar waves such as sound or light waves, the frequency in-

creases with increasing wavelength! Further, as the following calculations show, the

Rossby wave has an upper cutoff in frequency.

The frequency has a maximum (σ is a positive quantity by definition) for a given

value of l, when

∂σ

∂k=

(k2 + l2 + F )β − βk(2k)

(k2 + l2 + F )2= 0,

i.e., at

k = − | (l2 + F )12 |,

So that σmax =β

2(l2 + F )12

. Over all l and k, the Rossby wave cannot have fre-

quency higher than

σ =β

2F12

(40)

i.e. it is a low-frequency wave.

Further interesting properties of the Rossby waves can be studied conveniently with

the help of a dispersion diagram. Equation (39) can be put in the form, with F = 0 for

simplicity,

13

Page 15: Geophysical Fluid Dynamics

(k −

β

)2

+ l2 =β2

4σ2,

which is the equation for a circle of radiusβ

2σon the (k, l) plane, with its centre at(

− β

2σ, 0

)(Fig. 4). Wavevectors satisfying the Rossby wave dispersion relation (39)

have their origin at (0, 0) and their tip on the circle. For a given value of l, k can have two

values −k1, and −k2; the wavevector ~K1 with the smaller absolute value of k(= −k1)

represents a wave with longer wavelength than the wavevector ~K2with k = −k2 (see

fig. 4). Waves with wavevectors lying on the right (left) half of the circle are the long

(short) Rossby waves.

k

l

l = loCg1

Cg2

−k1 −k2 (−β/2σ,0)

σ = constant

K1

K2

Figure 4: Dispersion diagram of the Rossby wave.

Consider a Rossby wave with frequency σ + ∆σ (∆σ > 0) on the k, l plane. Its

dispersion curve is a circle with slightly smaller radius, and center at(

−β2(σ + ∆σ)

, 0

).

Now recall that the group velocity of a wave (which is the velocity of energy propaga-

tion) is given by

~cg =

(∂σ

∂ki,∂σ

∂lj

). (41)

This is nothing but the gradient of σ in the (k, l) plane. Since the lines of constant σ are

circles, the gradient must be in the radial direction. As the gradient vector is directed

from low values to high values, the group velocity of a Rossby waves must be directed

towards the center of the circle. Therefore the group velocity of the long Rossby wave

14

Page 16: Geophysical Fluid Dynamics

(with wavevector ~K1) ~cg1 (see Fig. 4) has a westward component, whereas that of the

short Rossby wave (wavevector ~K2) ~cg2 has an eastward component. It is easy to show

that long Rossby waves carry energy westward much faster than short Rossby waves

carry energy eastward (see the expression for∂σ

∂kon the previous page).

This last property of the Rossby wave provides a simple explanation for the ob-

served ”western intensification” in the ocean. The alongshore (i.e. parallel to the coast-

line) currents near the western boundary of all major ocean basins (both in the northern

and southern hemisphere) are strong compared to currents at eastern boundaries or in

the open ocean away from coasts.

Assume that low frequency fluctuations of surface winds generate a broad spectrum

of Rossby waves (i.e. over a range of frequencies) everywhere in the ocean. At all

frequencies, both long and short Rossby waves will exist. The long Rossby waves will

rapidly carry energy to the west. One would expect that the eastern region of the ocean

will not have energetic flows (i.e. the kinetic energy associated with low frequency

motions will be small). Further, the energy will eventually be carried to the western

boundary region by long Rossby waves. Therefore, the western boundary region will

be a region of energetic flows. Let us do a simple calculation.

Let the governing equation for Rossby waves be

∂t∇2ψ + β

∂ψ

∂x= 0, (42)

which is the same as eq. (38) with F = 0 for simplicity (our results are valid in the

presence of nonzero F as well). 4; ψ is the streamfunction such that

u =−∂ψ∂y

, v =∂ψ

∂x;

ψ is proportional to h or to pressure (see the geostrophic relations). Now consider a

north-south oriented wall (representing a coastline) at x = 0 on the western side of an

ocean basin (Fig. 5). Let a long Rossby wave be incident on this wall (i.e. the incident

wave carries energy towards the wall): It is

ψi = Aiexp i(kix+ liy − σit)4Eq. (38) represents the conservation of relative vorticity due to north-south displacement (the β term)

and due to vortex stretching/shrinking associated with surface displacement (the F term).

15

Page 17: Geophysical Fluid Dynamics

On encountering the wall, it is reflected as

ψr = Arexp i(krx+ lry − σrt)

The total stream function describing the motion in the ocean 0 < x <∞ is

ψ = ψi + ψr

ψi

ψr

XX=0

Figure 5: Reflection of a long Rossby wave ψi incident on a western boundary.

The condition of no normal flow must be satisfied at the wall, i.e.,

u |x=0=−∂ψ∂y

|x=0= −Ailiexp i(liy − σit)−Arlrexp i(lry − σrt) = 0 (43)

This relation cannot be satisfied at all y and all t unless li = lr and σi = σr. This

implies that the frequency and y - wavenumber of the incident and reflected waves are

identical. Since both waves have to satisfy the dispersion relation for the Rossby wave,

and the reflected wave has to carry energy away from the wall (i.e. ~cg has to have an

eastward x - component), the only possibility is that ψi is the long Rossby wave while ψr

is the short Rossby wave. Therefore the wavelength of the Rossby wave is changed on

reflection. The boundary condition also implies that Ar = −Ai, i.e. there is a 180phase

change on reflection.

The x-wavenumber of the incident wave (the long wave) is

ki =−β2σ

+

√β2

4σ2− l2

while that of the reflected wave (the short wave) is

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Page 18: Geophysical Fluid Dynamics

kr =−β2σ

√β2

4σ2− l2

The ratio of alongshore speeds (north-south speeds) at the wall associated with the

reflected and incident waves is

| vr || vi |

=|∂ψr

∂x|

|∂ψi

∂x|

∣∣∣∣∣∣∣∣x=0

=kr

ki

> 1,

i.e. the reflected wave has larger v velocity. Since the group speed of the Rossby

wave is

| ~Cg |=β

k2 + l2,

energy in the short wave travels much slower than in the long wave. Any fluid friction

(which we have not considered explicitly) will therefore damp the short wave before it

has moved very far away from the wall. The net result of the reflection process is that

the western boundary region of the ocean has stronger (more energetic) flow than the

open ocean, i.e. ”western intensification”. It can be shown that energy is conserved in

the reflection process, i.e.

< Er >| ~Cgr |=< Ei >| ~Cgi |

across any line parallel to the wall, where < E > is energy density and < E >| ~Cg | is

the energy flux due to the Rossby wave.

At the eastern boundary, the incident short Rossby wave is reflected as a long Rossby

wave that rapidly carries energy to the west. Therefore eastern boundary regions in

mid-latitudes do not have energetic flows. An analysis similar to the one above shows

that there is no wavelength change on reflection from northern or southern boundaries.

In summary, the increase of the coriolis parameter with latitude (i.e. the β-effect)

is responsible for the existence of the Rossby wave. It has some peculiar properties,

including anisotropy, giving rise to westward phase and group propagation of large

scale, low frequency motion in the atmosphere and ocean.

We left out the surface displacement Fη term for simplicity in our analysis of Rossby

wave reflection. However, it can be a very important term for long waves. Consider a

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Page 19: Geophysical Fluid Dynamics

very long Rossby wave with purely westward wave vector, such that k2 f 2o

gD. The

dispersion relation (eq. (39)) can be approximated by

σ = −βk

f2o

gD

(44)

the phase speedσ

kfor long Rossby waves goes as f−2

o . In regions relatively close to the

equator, i.e. in the tropics, β varies little with latitude since it goes as cos θ. However,

fo goes as sin θ, which is proportional to θ for moderate values of θ. Equation (44) then

predicts that if a disturbance along the eastern boundary of the ocean generates long

Rossby waves, the phase front of these waves will have a characteristic1

θ2shape : the

wave will travel fast at low latitudes, and its speed will drop off with increasing latitude

on either side of the equator (Fig. 6). Such long Rossby wave phase fronts are often seen

the ocean.

θ = 20οS

θ = 20οN

eastern boundary

east

Figure 6: Phase front of long Rossby waves generated simultaneously all along the easternboundary.

A final point can only be mentioned here (we do not offer a proof): stratification

(i.e. the increase of density ρ(z) with depth z in the ocean, or ρ(p) with pressure p in the

atmosphere) makes the Rossby wave travel much slower than suggested by eqs. (39)

or (43).In the presence of stratification (the real atmosphere and ocean are generally

stably stratified), the F term in the denominator of eq. (38) or (43) isf 2

o

gDe

rather than

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Page 20: Geophysical Fluid Dynamics

f 2o

gD; where the ”equivalent depth” De is much smaller than D, the geometric scale in

the vertical direction. In other words, the effective vertical scale of motion is small in

the presence of stratification. Therefore for a given k, the Rossby wave has smaller fre-

quency and propagation speeds than suggested by shallow water theory (with constant

ρ).

In Fig. 6, we did not draw the phase front of the Rossby wave within a few de-

grees north and south of the equator. This is because fo goes to zero at the equator,

and the dynamics of the equatorial region is somewhat different from that of higher

latitudes. We shall not discuss equatorial dynamics here, but mention a few impor-

tant facts. A model similar to the mid-latitude shallow water model can be constructed

for the equatorial region. It can be shown using this equatorial beta plane model that

the equator acts like a waveguide, i.e. there exist several classes of waves which can

propagate along the equator, with amplitude falling sharply away from the equatorial

region. These waves can have simple or complex structure in the y -direction depend-

ing on a meridional mode number. The eastward moving equatorial Kelvin wave is

the simplest of these waves. There is also a family of equatorial Rossby waves, which

are low-frequency waves, just like their off-equatorial counterparts. In addition we

mention the Yanai wave (or mixed Rossby-gravity wave) which can have high or low

frequency, and the high frequency intertia-gravity waves. All these waves can be seen

in the equatorial atmosphere and ocean. Many of them are important players in climate

variability. For example, the oceanic Kelvin wave plays a key role in interannual climate

variability through its influence on El Nino. The theoretical prediction in 1966 of the

existence, structure and propagation characteristics of equatorially trapped waves by

the Japanese scientist Taroh Matsuno is one of the great achievements of Geophysical

Fluid Dynamics. As better observations continue to be made in the tropics from in situ

and space - based sensors, the importance of equatorially trapped waves in climate is

becoming clearer.

Wind stress and forced motionSo far we have discussed free evolution, i.e. the evolution of a geophysical fluid in

the absence of ”external” forcing. An important forcing, both for the atmosphere and

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Page 21: Geophysical Fluid Dynamics

ocean, arises from the action of surface stress. Stress arises from the near-surface tur-

bulent transport of momentum in the atmosphere or ocean, or between the atmosphere

and ocean. If we use the momentum equations to study large scale flows, the stress

term in these equations represents the influence of small scale (turbulent) motion on

these flows. We do not discuss the form of the stress in the near-surface atmosphere or

ocean, but assume that a such a stress exists and study its consequences using simple

models.

Since the vertical scale of the boundary layers in the lower atmosphere (1 km) and

upper ocean (10 m-100 m) is small compared to the horizontal scale on which the

stresses vary (100-1000 km), it is the vertical gradient of the horizontal stress that enters

the dynamics. Let the x and y components of the stress ~τ be denoted by τx and τy. Then

the shallow water equations with forcing due to ~τ can be written as

∂u

∂t− fv = −

1

ρ

∂p

∂x+

1

ρ

∂τx

∂z

∂v

∂t+ fu = −

1

ρ

∂p

∂y+

1

ρ

∂τy

∂z(45)

The solution to the linear system can be written as u = up + uE, v = vp + vE , where the

pressure gradient driven flows up and vp satisfy

∂up

∂t− fvp = −

1

ρ

∂p

∂x,

∂vp

∂t+ fup = −

1

ρ

∂p

∂y(46)

and the stress driven flows uE and vE satisfy

∂uE

∂t− fvE =

1

ρ

∂τx

∂z,

∂vE

∂t+ fuE =

1

ρ

∂τy

∂z(47)

In steady state up, vp represent the geostrophic flow. The flow driven by stress uE , vE

is called the Ekman velocity (after the Norwegian explorer-scientist V. Walfrid Ekman).

This flow is confined to the shallow layer of fluid over which the stress acts, called the

Ekman layer. If τx and τy are zero outside the Ekman layer, integration of eq. (47) across

the layer gives

ρ

(∂UE

∂t− fVE

)= −τxs, ρ

(∂VE

∂t+ fUE

)= −τys (48)

if the boundary is below, and

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Page 22: Geophysical Fluid Dynamics

ρ

(∂UE

∂t− fVE

)= τxs, ρ

(∂VE

∂t+ fUE

)= τys (49)

if the boundary is above, (see Fig. 7). Here (UE, VE) =∫

(uE, vE)dz is the Ekman volume

transport, and τxs and τys are the values of τx and τy at the surface. The Ekman mass

transport in the atmosphere and ocean add to zero (as can be seen by adding eqs. (48)

and (49)). The volume transport in the atmosphere is about 1000 times that in the ocean,

because the density of air is about 1000 times smaller than water. In steady state, the Ek-

man transport is directed at right angles to the surface stress, illustrating the gyroscopic

property of geophysical fluids. The stress is entirely balanced by coriolis force (in the

absence of pressure gradients) in steady state. For a typical value of surface stress, 0.1

Newton per meter square, the Ekman mass transport in mid-latitudes (f ∼ 10−4s−1) in

the lower atmosphere or upper ocean is about 1000 kgm−1s−1. The Ekman formulas

ρVE =τxs

f, ρUE =

τys

f(50)

are remarkable because transport does not depend on the details of the turbulence that

generates the stress. Experimental verification of eq. (50) is difficult, because it is not

possible to measure accurately the pressure gradient term, which is generally non-zero

in the atmosphere and ocean.

Oceanic Ekman mass transport τs/f

Atmospheric Ekman mass transport τs/f

Surface stress τsZ

Figure 7: The Ekman transport in the upper ocean and lower atmosphere are at right angles tothe surface stress.

The surface stress varies from place to place, and this implies that the (horizontal)

Ekman transport is not constant. If the Ekman transport has non-zero horizontal diver-

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Page 23: Geophysical Fluid Dynamics

gence, then fluid must be ”sucked” vertically into or out of the Ekman (or boundary)

layer.

The magnitude of vertical velocity wE just outside the Ekman layer that results from

convergence or divergence of horizontal Ekman transport can be estimated by integrat-

ing the continuity equation. For constant density,

∂u

∂x+∂v

∂y+∂w

∂z= 0; (51)

integrating in z across the Ekman layer, and assuming that w = 0 at the surface gives

ρwE =∂

∂x

(τys

f

)−

∂y

(τxs

f

)(52)

where we have used eqs. (48) and (49). Usually the stress varies over smaller scales

than f , and the ”Ekman pumping” velocity is given by

wE =1

ρ(curl~τs) (53)

where curl~τs =∂τys

∂x−∂τxs

∂y.

Ekman pumping

Surface

Atmosphere boundary layer

Ocean boundary layer

Ekman pumping

Isotherms

Figure 8: Ekman pumping velocity wE is upward in both ocean and atmosphere if curl~τs > 0

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Page 24: Geophysical Fluid Dynamics

The sign of wE is the same in the ocean and atmosphere (see Fig. 8), with wE be-

ing 1000 times larger in the atmosphere than in the ocean. Upwelling (upward Ekman

pumping) in the upper ocean below the Ekman layer causes the lines of constant tem-

perature (isotherms) to dome upward; there is upward motion above the atmospheric

boundary layer, which can promote convection. Since f appears in the denominator

of eq. (53), the Ekman pumping velocities are larger near the equator than in mid-

latitudes. At the equator, of course, eqs. (50) or (53) are not valid because f is zero.

Thus not only does surface stress directly drive lateral motion in the Ekman layer, it

also gives rise to three dimensional motion (via continuity) in the region outside the

turbulent boundary layer.

Finally, we obtain a simple relation between the wind field and the ocean transport

including the effect of Ekman pumping. Consider the linear momentum equations for

the ocean on the β plane,

∂u

∂t− fv = −

1

ρ

∂p

∂x,

∂v

∂t+ fu = −

1

ρ

∂p

∂y;

These equations have the same form as equation (45). Note that they are valid (at a

given depth z) for a stratified ocean as well. By cross differentiation, we can form the

vorticity equation

∂ξ

∂t+ f

(∂u

∂x+∂v

∂y

)+ βv = 0. (54)

For an incompressible fluid,

∂ξ

∂t+ βv = f

∂w

∂z(55)

which represents the vorticity balance for a fluid column moving northward in the

planetary vorticity field, in the presence of vortex stretching due to change in w with

depth. For steady flow,

βv = f∂w

∂z. (56)

The interpretation is that if the horizontal velocity field is convergent, the r.h.s is posi-

tive; since vortex tubes are (almost) vertical for small Rossby number flows, stretching

23

Page 25: Geophysical Fluid Dynamics

of vortex tubes increases the absolute vorticity. Eq. (56) says that this is only possible if

the fluid column moves north. What happens in the presence of turbulent stress?

We have mentioned that stress is important in a shallow (∼ O (100 m)) Ekman layer

in the upper ocean. Assume that at some level −H in the deep ocean (∼ O (1000 m)),

the vertical motion is zero (or negligible). Consider the vertical velocity at the base of

the Ekman layer to act at the ocean surface. Integration of Eq. (56) in z from −H to the

surface then gives

βV =1

ρ

(∂τys

∂x−∂τxs

∂y

)=

1

ρcurl~τs (57)

where V =∫ 0

−Hvdz, and we have used eq. (53) for wE

Equation (57) is called the Sverdrup relation after the Norwegian oceanographer

Harald Sverdrup. It relates the curl of surface wind stress to northward transport in

the entire upper ocean influenced by wind (through Ekman pumping). It is another

form of the steady vorticity balance, and has proved useful for understanding aspects

of large scale wind driven ocean circulation. We note that like the geostrophic relation,

the Ekman formula and the shallow water theory, it is based on drastic assumptions;

its popularity is due to its apparent simplicity.

ReferencesAdrian E. Gill (1982): Atmosphere-ocean Dynamics. Academic Press, London, 662pp.

Joseph Pedlosky (1987): Geophysical Fluid Dynamics, Second Edition. Springer-Verlag,

New York, 710pp.

24