Part II: Waves in the Tropics- Theory and Observations

1. Part II: Waves in the Tropics- Theory and Observations Derivation of Rossby waves; mixed Rossby gravity waves and vertical propagation

2. Rossby Waves • Rossby waves result from the conservation of potential vorticity. • In the most simple case of relative and planetary vorticity conservation, displacing a parcel poleward in the northern hemisphere will lead to an increase in anticyclonic vorticity, the opposite for displacing it equatorward.

3. Rossby Waves • We will use the following QG Vorticity equation (not the simplest derivation for Rossby waves, nor the most complex). D/Dt 2ψ + β δψ/δx – (fo2/gh)(Dψ/Dt) = 0 ψ = -Uy + ψ’ h = H + h’ D/Dt 2ψ’ + β δψ’/δx – (fo2/gH)(1-h’/H)(Dψ/Dt) = 0

4. Rossby Waves D/Dt 2ψ’ + β δψ’/δx – (fo2/gH)(1-h’/H)(Dψ/Dt) = 0 Examine: D/Dt 2ψ’ = (δ/δt + (U+u’)δ/δx + v’ δ/δy) 2ψ’ Linearize: (δ/δt + U δ/δx) 2ψ’ In top equation, middle term with planetary vorticity does not need to be simplified, so we’ll turn to the green (stretching) term.

5. Rossby Waves (δ/δt + U δ/δx) 2ψ’ + β δψ’/δx – (fo2/gH)(1-h’/H)(Dψ/Dt) = 0 Examine: (fo2/gH)(1-h’/H)(D/Dt)ψ = (fo2/gH)(1-h’/H)(δ/δt + (U+u’)δ/δx + v’ δ/δy)(-Uy + ψ’) Linearizing: (fo2/gH){(1-h’/H)[(δ/δt + U δ/δx)ψ’- v’U]} =(fo2/gH)(δ/δt + U δ/δx)ψ’- (fo2U/gH) v’ v can be rewritten such that v = δψ/δx

6. Rossby Waves (δ/δt + U δ/δx) 2ψ’ + β δψ’/δx – (fo2/gH)(δ/δt + U δ/δx)ψ’+ (fo2/gH) U δψ’/δx = 0 Let fo2/gH = F (analogous to the Froude number). ψ’ = A exp{i(kx+ly-ωt)} Wave-like solutions yield: (-iω + Uik)(i2k2 + i2l2) + βik – F(-iω + Uik) + FUik = 0 (ω - Uk)(k2 + l2) + βk + Fω - FUk + FUk = 0 (ω - Uk)(k2 + l2 + F) + βk = 0 ω = Uk – βk/(k2 + l2 + F)

7. Rossby Waves ω = Uk – βk/(k2 + l2 + F) So, phase speed is: c = ω/k = U – β/ (k2 + l2 + F) That is, c – U = – β/(k2 + l2 + F) c – U is always < 0… waves always propagate westward relative the mean flow; small k,l propagate faster Cgx = δω/δk = U - β(k2 + l2 + F)-(βk)(2k)/(k2 + l2 + F) 2 Cgx – U = - β(l2 + F - k2)/(k2 + l2 + F) 2

8. N=1 Equatorial Rossby Wave In atmosphere, dry waves propagate to the west at 10-20 ms-1 For convectively coupled waves, propagation is 5-7 ms-1 in atmosphere, ~1 ms-1 in ocean

9. Mixed Rossby Gravity Waves • Propagation of moist mixed Rossby gravity waves is about 8-10 ms-1 in the atmosphere • Wavelength 1000-4000 km

10. Vertical Propagation of Rossby Waves Make stationary over topography to simplify terms. D/Dt( 2ψ + f + βy + [ (f/ρR) δ/δz ((ρR/N2) δψ/δz’)] ) = 0 ρR= air density ψ = -Uy + ψ’ N2 = g/θ δθ/δz’ (Brunt-Vaisala frequency) Zonal flow and N are made constant with height for ease. Linearize to obtain: δ/δt ψ* + u δ/δx ( 2ψ + [ (f2/ρR) δ/δz ((ρR/N2) δψ*/δz)] ) + β δψ*/δx = 0

11. Vertical Propagation of Rossby Waves ψ* = ψ(z) e (z’/2H)e {i(kx+ly)} After some substitution, we get: δ2ψ/δz2 + m2ψ = 0 …and after some more work, we find: m = +/- (N/f) [ β/U – (k2 + l2) – (f2/4N2H2) ] ½ = 0 We find m>0 (real solutions). If m is imaginary, we have evanescence (waves trapped and decay). For m < 0, cases would include: 1) u < 0 (easterly flow). Rossby waves must propagate westward, cannot be stationary over topography. 2) u > 0, with large k2 + l2, then wave speed too slow versus advection 3) u is very large

12. Vertical Wave Propagation Kelvin waves propagate better in easterly flow and dump westerly momentum mixed Rossby gravity propagate in westerly flow and dump easterly momentum

13. Equatorial Waves • Near/at equator, so let f ~ βy, such that: • x-mom: δu/δt – βyv = -g (δh/δx) • y-mom: δv/δt + βyu = -g (δh/δy) • cont: (δh/δt) = - H(δu/δx + δv/δy) • (ω/co)2 – k2 - βk/ω = (2n+1) β / co • If co = (gH)1/2 and n = -1, Kelvin wave solution • If n = 0, get mixed Rossby gravity waves • if k is large, more like e prop gravity waves, • k < 0: slow varying and w prop like Rossby waves • If n = 1 or greater, e prop high-freq gravity waves (k > 0), w prop equatorial Rossby waves or gravity waves (k < 0)

14. Important Waves in the Tropics

15. End Section • Continue with MJO, ENSO, and QBO