Atms 4320 / 7320 Lab 11

1. Atms 4320 / 7320 Lab 11 Anthony Lupo

2. The Rossby Wave Equation • Rossby Waves (Holton p 165 – 8, Pedlosky Ch. 3) • Long waves that show up on a 500 hPa chart are typically referred to as “Rossby Waves”. A Rossby Wave is an absolute vorticity conserving motion which owes it’s existence to the variation of the Coriolis Force w/ latitude (Beta plane – if we assume change in coriolis is constant).

3. The Rossby Wave Equation • These waves are important for synoptic-scale and large-scale processes. • We’ll see that they are not really a “Rossby Wave” in the Fluid Dynamic sense, since a pure Rossby Wave is a westward propagating wave (derived from the Barotropic Rossby wave equation). • This is as opposed to a Kelvin wave which is an Eastward Propagating wave (especially in the tropics).

4. The Rossby Wave Equation • What are the waves at 500 hPa? They are mixed wave (Rossby-Gravity, or an Inertia -Gravity wave) Pressure gradient force balances with Coriolis force! • Start with the Barotropic Vorticity Equation in Pressure Coordinates:

5. The Rossby Wave Equation • Vorticity Equation (2-D): • Apply the assumption of inviscid, barotropic (non-divergent), 2-D flow (no vertical motions!). • (Essentially a Taylor Fluid –or- “Shallow Water Body”)

6. The Rossby Wave Equation • Now Expand:

7. The Rossby Wave Equation • Now use perturbation method (“linearize”): • and assume geostrophic balance and use stream function:

8. The Rossby Wave Equation • Now assume a wavelike solution for the streamfunction: • apply all these and plug into (1) to get:

9. The Rossby Wave Equation • Then,

10. The Rossby Wave Equation • where C is a dispersive relationship, which is a function of the Zonal windspeed and the zonal and meridional wave number. Thus, the Rossby Phase speed (Cp) is slower than the mean flow. • Also assume (and an example):

11. The Rossby Wave Equation • Here it is:

12. The Rossby Wave Equation • Example: • if Lx = 10,000 km  ?? m • U = 20 m/s • fo = 1 x 10-4 (s-1) • bo = 1 x 10-11 (s-1 m-1)

13. The Rossby Wave Equation • Example cont: • Cp = -5.4 m/s (westward moving wave) • If Lx = 4,000 km • Cp = 15.9 m/s (eastward moving wave)

14. The Rossby Wave Equation • You can calculate a stationary zonal wind speed required to get a stationary wave (or a stationary wave length)

15. The Rossby Wave Equation • If U = 10 m/s and Beta are the same as used above, we get: • Lx = 6350 km (or about the wave length of a typical block?)

16. The Rossby Wave Equation • Also, the 2-D Rossby wave equation is called the Rossby - Haurwitz Equation (Haurwitz modification)

17. The Rossby Wave Equation • and

18. The Rossby Wave Equation • Rossby waves are important in tropical regions. • Also Kelvin waves are important in that region. Kelvin waves are eastward propagating and “trapped” waves, which means they do not propagate vertically. • Kelvin waves (In a pure barotropic fluid):

19. The Rossby Wave Equation • In tropics, if Cp > 0, then likely a Kelvin wave! • These are important in tropical circulations like the 30 - 50 day oscillation, and El Nino.

20. The Rossby Wave Equation • The End !