The Vorticity Equation for a Homogeneous Fluid – Planetary waves

Chapter 6

Du 1 pfv

Dt x

Dv 1 pfu

Dt y

horizontal momentum equations

u(x,y,t)

The vorticity equation for a layer of homogeneous fluid of variable depth H(x,y,t)

H(x,y,t)

x

D v u u vf (f ) 0

Dt x y x y

z H z 0

H

0

u vw w dz

x y

Cross-differentiation

Continuity equation

Integrate with respect to height

0 u

Assume u and v to be functions of x and y only

DH u vH

Dt x y

2

D f 1 D (f ) DHf

Dt H H Dt H Dt

Use the identity

D v u u v Y Xf (f )

Dt x y x y x y

DH u vH

Dt x y

D f0

Dt H

The quantity (f + )/H is called the potential vorticity for a homogeneous fluid. It is conserved following a fluid column.

The atmosphere is a complex dynamical system which can support many different kinds of wave motion covering a wide range of time and space scales.

One of the most important wave types as far as the large-scale circulation of the atmosphere is concerned is the planetary wave, or Rossby wave.

These are prominent in hemispheric synoptic charts; either isobaric charts at mean sea level (msl) or upper level charts of the geopotential height of isobaric surfaces, e.g., 500 mb.

Planetary, or Rossby Waves

Southern hemisphere mean sea level isobaric analysis on a stereographic

projection illustrating the wavy nature of the flow in the zonal direction.

The 500 mb analysis corresponding to the previous one. Isopleths of the

500 mb surface are given in decametres.

Regional 500 mb analysis showing planetary-scale waves. Broken lines are isotachs in knots (2 kn = 1 ms1). Note the two “jet-streaks”, one just southwest of Western Australia, the other slightly east of Tasmania.

A planetary wave in its pure form is a type of inertial wave which owes its existence to the variation of the Coriolis parameter with latitude.

An inertial wave is one in which energy transfer is between the kinetic energy of relative motion and kinetic energy of absolute motion.

Such waves may be studied within the framework of the Cartesian equations described above by making the so-called "beta-plane" or "-plane" approximation.

Planetary Waves are Inertial Waves

This approximation regards f as a linear function of the north-south direction y, i.e., f = f0 + y, where is a positive constant, so that the effect of the variation of the Coriolis parameter with latitude is incorporated in the vorticity equation.

The beta-plane approximation allows us to study of the effects of varying f with latitude without the added complication of working in spherical geometry.

At the equator, f0 = 0 whereupon f = y. We call this an equatorial beta-plane approximation.

The beta-plane approximation

A simple description of the basic dynamics of a pure horizontally nondivergent planetary wave may be given by considering two-dimensional flow on a beta plane:

i.e., in a rectangular coordinate system with x pointing eastwards, y pointing northwards, and with f = f0 + y.

Horizontal nondivergence implies that H is a constant and therefore, in the absence of a body force, the vorticity equation reduces to

D

Dtf b g0

The absolute vorticity of each fluid column remains constant throughout the motion.

Non-divergent motions

A

y (north)

x (east)

B

C

D

cp

Northern hemisphere

Diagram illustrating the dynamics of a nondivergent Rossby wave

Each particle displacement conserves f +

Non-divergent Rossby waves

Two-dimensionality implies that there exists a streamfunction such that

u v andy x , 2

D

Dtf b g0 a PDE with as the sole

dependent variable.

For small amplitude motions, the equation can be linearized

t x

( ) 2 0

For motions independent of y (then u = y0), there exist travelling wave solutions of the form

Mathematical analysis

for a nontrivial solution :

sin ( )kx t

where k, and are constants.

Substitution of into

t x

( ) 2 0

( )( ) cos ( ) k k kx t2 0

( ) 0

k

This is the dispersion relation for the waves.

k is called the wavenumber and the frequency

The wavelength is = 2/k and the period T = 2/

The phase speed of the wave in the x-direction is cp = /k.

ckp

2

The phase speed of the wave in the x-direction is:

The perturbation northward velocity component is

v k kx t cos( )

This is exactly 90 deg. out of phase with .

Since cp is a function of k (or ), the waves are called dispersive.

In this case, the longer waves travel faster than the shorter waves.

Note that > 0 implies that cp < 0 and hence the waves travel towards the west, consistent with physical arguments.

Suppose now there is a basic westerly airflow (U, 0), U being a constant.

Remember westerly means from the west; it is sometimes called a zonal flow.

Then ulinearizes to U/x andD

Dtf b g0

t

Ux x

LNMOQP 2 0

Travelling wave solutions of the type exist as before:

sin ( )kx t

The dispersion relation is: = Uk /k

A basic westerly airflow

Thus the waves are simply advected with the basic zonal flow.

The above equation is known as Rossby's formula.

It shows that waves propagate westwards relative to the air at a speed proportional to the square of the wavelength.

In particular, waves are stationary when ks = (/U)

They move westwards for > s = 2/ks and eastwards for < s

c Ukp

2

If a planetary wave extends around the earth at latitude , its wavelength cannot exceed the length of that latitude circle, 2a cos , a being the earth's radius.

2 a n/

/ /2 1 n s

with corresponding period T = 2/|| = 2n/

Since = 2/(1 day), the period is simply n days.

Planetary waves around the earth

For n waves around the latitude circle at 45° on which = f/a, then:

n 1 2 3 4 5

103 km 28.4 14.2 9.5 7.1 5.7

T days 1 2 3 4 5

cp m s-1 329 82.2 36.7 20.5 13.5

Values of , T and cp in the case U = 0 are listed in the table for values of n from 1 to 5.

For n equal to 1 or 2, cp is unrealistically large. => This results from the assumption of nondivergence, which is poor for the ultra-long waves.

The stationary wavelengths s for various flow speeds U, calculated from the formula

are listed below for as 1.6 x 10-11 m -1s -1, appropriate to 45 deg. latitude.

s U2 /a f

U m s-1 20 40 60 80

103 km 7.0 9.0 12.0 14.0

Hemispheric upper air charts such as those at 500 mb show patterns with between about 2 and 6 waves.

These waves are considered to be essentially planetary waves forced by three principal mechanisms:

orographic forcing resulting from a basic westerly air stream impinging on mountain ranges such as the Rockies and Andes;

thermal forcing due to longitudinal heating differences associated with the distribution of oceans and continents: and

nonlinear interaction with smaller scale disturbances such as extra-tropical cyclones.

Planetary waves in the atmosphere

Shallow water model configuration

In pure inertial wave motion, horizontal pressure gradients are zero.

Consider now waves in a layer of rotating fluid with a free surface where horizontal pressure gradients are associated with free surface displacements.

f

H(1 + ) H

pa

z = 0

p = pa + g(H(1 + ) – z)

undisturbed depth

Inertia-gravity waves

Consider hydrostatic motions - then

ap(x, y,z, t) p g[H{1 (x, y, t)} z]

1

h hp gH

independent of z

The fluid acceleration is independent of z.

If the velocities are initially independent of z, then they will remain so.

Linearized equations - no basic flow

ufv gH

t xv

fu gHt y

u v

x y t

Consider wave motions which are independent of y.

A solution exists of the form u u kx t

v v kx t

kx t

cos ( ),

sin ( ),

cos ( ),

, tanu v and cons ts

if

,

,

,

u fv gHk

fu v

ku

0

0

0

These algebraic equations for have solutions only if

, u v and

f gHk

f

k

f gHk0

0

03 2 2( )

0 2 2 2or f gHk

The solution with= 0 corresponds with the steady solution (/t = 0) of the equations and represents a steady current in strict geostrophic balance in which

The other two solutions correspond with so-called inertia-gravity waves, with the dispersion relation

gHv

f x

The phase speed of these is

2 2 2 f gHk

c k gH f kp / [ / ]2 2

The waves are dispersive

The existence of a positive and negative root for (or cp) shows that these waves can propagate in either x-direction.

In the limit as k0, f and the x-dependence of the motion drops out.

Then the motion corresponds with the pure inertial wave discussed earlier.

In the absence of rotation (i.e., f = 0) the theory reduces to that for small amplitude surface gravity waves on shallow water.

Such waves are non-dispersive:

2 2 2 f gHk2

p 2

fc gH

k k

c gHp

Note that the hydrostatic assumption restricts the validity of the whole analysis to long waves on shallow water; strictly for waves with wavelength >> H.

Rotation is important if =>

f k gH2 2/ ~

2 21 2( ) //gH f LR

the Rossby radius of deformation for a homogeneous fluid

2 2 2 f gHk

In the deep ocean, 3H ~ 4 km 4 10 m

At 45° latitude, f = 104 s1

3 1/ 2 4 7 42 (10 4 10 ) /10 1.3 10 m ~ 10 km

Rotation effects are important only on the planetary scale.

gH ms 200 1and .

Replace the v-momentum equation by the vorticity equation =>

0

0

uf v gH

t x

v ft t

u v0

t x y

v u

x y

where

Assume that f can be approximated by its value f0 at a particular latitude, except when differentiated with respect to y in the vorticity equation. This is justified provided that meridional particle displacements are small.

Inclusion of the beta effect

Again assume that /y0 and consider travelling-wave solutions of the form:

u u kx t

v v kx t

kx t

cos ( ),

sin ( ),

cos ( ),

,

( ) ,

,

u f v gHk

k v f

ku

0

0

0

0

0

These are consistent only if the determinant is zero

f gHk

k f

k

gHk k f k0

02 2

020

0

0( ) ( )( )

Nowexpanding by the second row

A cubic equation for with three real roots.

When >> /k, the two non-zero roots are given approximately by the formula

2 202 gHk f

This is precisely the dispersion relation for inertia-gravity waves. When 2 << gHk2, there is one root given approximately by

2 20k /(k f / gH)

For wavelengths small compared with 2 times the Rossby length 2 0 gH / f

reduces to the dispersion relation for nondivergent Rossby waves.

k f0 0( )gH

the effects of divergence due to variations in the free surface elevation become important, or even dominant.

22 0

k

fk

gH

For longer wavelengths,

20

kgH~

f

For extremely long waves

These are nondispersive.

The importance of divergence effects on the ultra-long waves explains why the calculated phase speeds for planetary waves of global wave-numbers-1 and -2 were unrealistically large.

The latter were obtained from the phase speed formula for nondivergent waves.

In the atmosphere, divergence effects are not associated with the displacement of a free surface!

To understand the effect of divergence on planetary waves, we introduce the meridional displacement of a fluid parcel.

This is related to the meridional velocity component by v = D /Dt, or to a first approximation by

vt

Substituting for v in and integrating with respect to time, assuming that all perturbation quantities vanish at t = 0, gives

t tv f 0

f0

This formula is equivalent (within a linear analysis) to the conservation of potential vorticity (see exercise 11.2).

f f f 0 0

holds for a fluid parcel which hasandinitially zero so that = .

In the absence of divergence,

The term f0 represents the increase in relative vorticity due to the stretching of planetary vorticity associated with horizontal convergence (positive ).

1

1

0

/| | , /| | /| |

v v/| |

HI LO = kx t

The relative phases of the quantities v, , and over one wavelength

northern hemisphere case

The relative vorticity tendency due to stretching opposes that due to meridional motion and thereby reduces the restoring tendency of the induced velocity field.

Phase diagram for Rossby waves

This has the solution , where

Equations:

0

gHv

f x

If /y0 as before, the vorticity equation reduces to an equation for , and since= v/x =>

2

020 0

gH gHf 0

t f x f x

cos ( )kx t

Filtering

0

0

uf v gH

t x

v ft t

u v0

t x y

k k f gH/ ( / )202

There is no other solution for as there was before.

In other words, making the geostrophic approximation when calculating v has filtered out in the inertia-gravity wave modes from the equation set, leaving only the low frequency planetary wave mode.

This is not too surprising since the inertia-gravity waves, by their very essence, are not geostrophically-balanced motions.

Dispersion relation for a divergent planetary wave

End of Chapter 6