1. The Vorticity Equation for a Homogeneous Fluid – Planetary waves
Chapter 6
The Vorticity Equation for a Homogeneous Fluid – Planetary waves

2. The vorticity equation for a layer of homogeneous fluid of variable depth H(x,y,t)
u(x,y,t) x
horizontal momentum equations

3. Cross-differentiation
Continuity equation
Integrate with respect to height
Assume u and v to be functions of x and y only

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

5. Planetary, or Rossby Waves
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.

6. Southern hemisphere mean sea level isobaric analysis on a stereographic projection illustrating the wavy nature of the flow in the zonal direction.

7. The 500 mb analysis corresponding to the previous one
The 500 mb analysis corresponding to the previous one. Isopleths of the 500 mb surface are given in decametres.

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

9. Planetary Waves are Inertial Waves
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 "b-plane" approximation.

10. The beta-plane approximation
This approximation regards f as a linear function of the north-south direction y, i.e., f = f0 + by, where b 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 = by. We call this an equatorial beta-plane approximation.

11. Non-divergent motions
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 + by.
Horizontal nondivergence implies that H is a constant and therefore, in the absence of a body force, the vorticity equation reduces to
The absolute vorticity of each fluid column remains constant throughout the motion.

12. Non-divergent Rossby waves
y (north) B
Each particle displacement conserves f + z D C cp A
Suppose a parcel of fluid, originally at position A is displaced to position B. Now f + z is a constant during the displacement and, since f increases with y, z must decrease. Thus, an anticyclonic circulation is induced around B. As a result of this circulation, parcels such as C, to the east of B, will be swept southwards and parcels such as D, to the west of B, will be swept northwards. But the secondary circulations induced by the displacements of C and D will tend to sweep the parcel at B back towards its original position at A.
Clearly, the induced northward displacement of D represents a progression of the phase of the disturbance to the west.
This type of wave motion was first discovered and elucidated by C. G. Rossby and his collaborators in the late 1930's.
x (east)
Diagram illustrating the dynamics of a nondivergent Rossby wave
Northern hemisphere

13. Mathematical analysis
Two-dimensionality implies that there exists a streamfunction y such that
a PDE with y as the sole dependent variable.
For small amplitude motions, the equation can be linearized
For motions independent of y (then u = -yy º 0), there exist travelling wave solutions of the form

14. where k, w and are constants.
k is called the wavenumber and w the frequency
The wavelength is l = 2p/k and the period T = 2p/w
The phase speed of the wave in the x-direction is cp = w/k.
Substitution of y into
for a nontrivial solution :
This is the dispersion relation for the waves.

15. The phase speed of the wave in the x-direction is:
Since cp is a function of k (or l), the waves are called dispersive.
In this case, the longer waves travel faster than the shorter waves.
Note that b > 0 implies that cp < 0 and hence the waves travel towards the west, consistent with physical arguments.
The perturbation northward velocity component is
This is exactly 90 deg. out of phase with y.

16. A basic westerly airflow
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 u × s linearizes to U¶/¶x and
Travelling wave solutions of the type
exist as before:
The dispersion relation is: w = Uk - b/k

17. 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 = (b/U)
They move westwards for l > ls = 2p/ks and eastwards for l < ls

18. Planetary waves around the earth
If a planetary wave extends around the earth at latitude F, its wavelength cannot exceed the length of that latitude circle, 2pa cos F , a being the earth's radius.
For n waves around the latitude circle at 45° on which b = f/a, then:
with corresponding period T = 2p/|w| = 2pn/W.
Since W = 2p/(1 day), the period is simply n days.

19. Values of l, T and cp in the case U = 0 are listed in the table for values of n from 1 to 5.2 3 4 5
l 103 km
28.4
14.2
9.5
7.1
5.7
T days
cp m s-1
329
82.2
36.7
20.5
13.5
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.

20. The stationary wavelengths ls for various flow speeds U, calculated from the formula
are listed below for b as 1.6 x m -1s -1, appropriate to 45 deg. latitude.
U m s-1 20 40 60 80
l 103 km
7.0
9.0
12.0
14.0

21. Planetary waves in the atmosphere
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.

22. Inertia-gravity wavesf
undisturbed depth pa
p = pa + rg(H(1 + h) – z)
H(1 + h) H
z = 0
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.

23. Consider hydrostatic motions - then
independent of z
The fluid acceleration is independent of z.
If the velocities are initially independent of z, then they will remain so.

24. Linearized equations - no basic flow
Consider wave motions which are independent of y.
A solution exists of the form if

25. These algebraic equations for have solutions only if

26. The solution with w = 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
The phase speed of these is
The waves are dispersive

27. The existence of a positive and negative root for w (or cp) shows that these waves can propagate in either x-direction.
In the limit as k ® 0, w ® 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:

28. the Rossby radius of deformation for a homogeneous fluid
Note that the hydrostatic assumption restricts the validity of the whole analysis to long waves on shallow water; strictly for waves with wavelength l >> H.
Rotation is important if =>
the Rossby radius of deformation for a homogeneous fluid

29. In the deep ocean,
At 45° latitude, f = 10-4 s-1
and
Rotation effects are important only on the planetary scale.

30. Inclusion of the beta effect
Replace the v-momentum equation by the vorticity equation =>
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.

31. Again assume that ¶/¶y º 0 and consider travelling-wave solutions of the form:
These are consistent only if the determinant is zero

32. This is precisely the dispersion relation for inertia-gravity waves.
Now
expanding by the second row
A cubic equation for w with three real roots.
When w >> b/k, the two non-zero roots are given approximately by the formula
This is precisely the dispersion relation for inertia-gravity waves.
When w2 << gHk2, there is one root given approximately by

33. For wavelengths small compared with 2p times the Rossby length
reduces to the dispersion relation for nondivergent Rossby waves.
For longer wavelengths,
the effects of divergence due to variations in the free surface elevation become important, or even dominant.

34. 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!

35. To understand the effect of divergence on planetary waves, we introduce the meridional displacement x of a fluid parcel.
This is related to the meridional velocity component by v = Dx /Dt, or to a first approximation by
Substituting for v in and integrating with respect to time, assuming that all perturbation quantities vanish at t = 0, gives
This formula is equivalent (within a linear analysis) to the conservation of potential vorticity (see exercise 11.2).

36. In the absence of divergence,
holds for a fluid parcel which has x and z initially zero so that z = -bx.
The term f0h represents the increase in relative vorticity due to the stretching of planetary vorticity associated with horizontal convergence (positive h).

37. Phase diagram for Rossby waves1 HI LO
q = kx - wt p 2p -1
northern hemisphere case
The relative phases of the quantities v, x, h and z over one wavelength
The relative vorticity tendency due to stretching opposes that due to meridional motion and thereby reduces the restoring tendency of the induced velocity field.

38. Filtering
Equations:
If ¶/¶y º 0 as before, the vorticity equation reduces to an equation for h, and since z = ¶v/¶x =>
This has the solution , where

39. Dispersion relation for a divergent planetary wave
There is no other solution for w 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.

40. End of Chapter 6