Planetary (Rossby) Waves

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 http://virga.sfsu.edu/scripts/nhemjet_archloop.html

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 fis 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.