1. Planetary (Rossby) Waves
References:
1 “Atmospheric Physics” Andrews, Chapter 5 (5.6)
2 “Introduction to Dynamic Meteorology”, Holton, Chapter 7 (7.7)

2. General remarks Large horizontal scales (planetary scales);
The rotation of the planet is important.
Typical sources: large topography like continents; land-sea thermal contrast;
These waves are important for large scale meteorological processes (middle latitude cyclones).
Typical zonal wavenumber (how many wavelengths in a latitudinal circle) for the Earth jet stream is 4-5.

3. The jet stream and Rossby waves
Also see the animations at

4. Saturn’s north polar hexagon

5. Schematic representation
Consider a purely zonal non-rotational
eastward (westerly) wind.
The wind is deflected northward:
To conserve absolute vorticity the wind needs to acquire negative relative vorticity. The wind turns in counterclockwise direction and swings towards the original latitude. W
f is increasing
xr<0
xr>0

6. Returning Force
When the wind turns south of the original latitude then f decreases and the wind will generate positive xr in order to conserve xa. The wind will turn counterclockwise towards the original latitude.
The fluid parcel oscillates around its equilibrium latitude.
This oscillating motion propagates to the west.
This westward propagating field of positive and negative relative vorticity constitutes the Rossby wave.
What happens if the wind is originally coming from the east and goes west (easterly, westward zonal wind)?

7. Now the math…
Start with the quasi-geostrophic potential vorticity equation (QGPV) for a zonal wind u=(U,0,0)
Superimpose a perturbation so that:

8. Linearised Equation: The linearised QGPV equation is:
Substitute above a wavelike solution for y’
We obtain a dispersion equation:

9. Understanding the dispersion relation
For a wave to propagate upward in the atmosphere m must be real!
Obviously if there is no b-effect (the planetary vorticity does not vary with latitude) there is no vertically propagating Rossby wave!
Rossby waves move westward with respect to the background wind U.

10. More …
The strongest wind in which a wave with a given horizontal wavelength can propagate vertically is:
For stationary waves (horizontal phase velocity c=0) can propagate only in eastward zonal winds (U>0) that are not too strong (U<Uc)!
Waves with large horizontal wavelengths (k and l very small) have the best chance to survive since Uc is largest.
Rossby waves are highly dispersive. The phase velocity strongly depends on the wave wavelengths.
For upward propagating waves (m>0), the surfaces of constant phase slope westward with height.