Wave mo(on in the tropics. Shayne McGregor Lecturer (Climate), EAE, Monash Associate Inves(gator (ARC CSS)

Transcription

1 Wave mo(on in the tropics Shayne McGregor Lecturer (Climate), EAE, Monash Associate Inves(gator (ARC CSS)

2 Equatorially trapped waves The linear SWM Equatorial Kelvin waves Other equatorially trapped waves Mixed gravity-rossby waves Interio-gravity waves Rossby waves Steady state (Gill) solu(ons

3 The linear Shallow Water Model (SWM) Pressure gradient terms Coriolis flow ρ 2 u t v t fv = g' η x fu = g' η y η t + H u x + v = 0 y H ρ 1 η East (x) North (y) Divergence/convergence

4 The linear SWM (baroclinic modes) Ver(cal mode examples Atmosphere Mode 0 is barotropic Higher modes are baroclinic Baroclinic mode 1 has surface flow which is opposite to that alox Ocean m=3 m=2 m=1 m=0

5 Simple ver(cal structure of this mode allows the upper level flow to mirror the lower level flow (e.g., Kleeman 1991). The first baroclinic mode

6 Beta plane Coriolis parameter: f = 2Ωsin(θ) Eq β-plane approxima(on: f = βy β = 2Ω/a (2.3 x m -1 s -1 ) Ω = rota(on rate of the earth (7.29 x 10-5 s -1 ) a = radius of the earth (6371km) y = distance from the equator 1.5 x Normal B plane Latitude

7 The linear SWM (first baroclinic mode) u t v t βyv = g' η x βyu = g' η y η t + H u x + v = 0 y H ρ 2 ρ 1 η East (x) North (y)

8 Non-dimensionalised linear SWM u t v t yv = η x yu = η y H ρ 2 η East (x) η t + u x + v y = 0 ρ 1 North (y) x* = Lx y* = λy u* = cu λ = c / B t* = L c t h* = Hh v* = (λ / L)cv c = g'h

9 The Equatorial Kelvin wave 1) 2) v=0 SWM eqns reduce to u t = η x yu = η y Looking for simple zonally propaga(ng solu(ons u = û(y)e i(kx σt) η = ˆη(y)e i(kx σt) 3) η t + u x = 0

10 The Equatorial Kelvin wave Looking for zonally propaga(ng solu(ons 1) 2) u t = η x yu = η y û = k σ ˆη yû = ˆη y Subs(tu(ng 1 into 3 σ ˆη k 2 σ ˆη = 0 σ 2 ˆη k 2 ˆη = 0 3) η t + u x = 0 σ ˆη kû = 0 (σ 2 k 2 ) ˆη = 0 4) σ = ±k

11 The Equatorial Kelvin wave Looking for zonally propaga(ng solu(ons 1) 2) u t = η x yu = η y û = k σ η yû = ˆη y Subs(tu(ng 1 into 2 ˆη y = y k σ ˆη Knowing 4) 3) η t + u x = 0 σ ˆη kû = 0 5) ˆη y = ±y ˆη

12 The Equatorial Kelvin wave 5) Meridional structure equa6on: ˆη(y) y = ±y ˆη(y) 4) σ = ±k σ = k Sign depends on 4 but must be a minus otherwise we will have exponen(al growth away from the equator Because of the above we have ˆη(y) y = y ˆη(y) Dispersion rela(on ˆη(y) = η 0 e (y2 /2) amplitude Pu9ng all of this back into 3: 3) û(y) = k σ ˆη(y) û(y) = ˆη(y) U and eta have the same meridional structure

13 The equatorial Kelvin wave A feature of a Kelvin wave is that it is non-dispersive, i.e., the phase speed of the wave crests is equal to the group speed of the wave energy for all frequencies. c=v*/k* Westward Mixed Rossbygravity Inter(o-gravity Kelvin Eastward Dispersion occurs when pure plane waves of different wavelengths have different propaga(on veloci(es, so that a wave packet of mixed wavelengths tends to spread out in space. Eastward Westward Source: Cushman-Roisen and Beckman 2009

14 The Equatorial Kelvin wave (meridional structure) η(y) = η 0 exp( y 2 / 2) η(y) = η 0 exp( y 2 β / 2c) Shown here =1 Dimensionalise by dividing y by: R eq = c / β Equatorially trapped Meridional decay Atmsophere (c = 50 m/s) Ocean (c = 2.5 m/s) Amplitude (m) Latitude ( o N)

15 The equatorial Kelvin wave Coriolis PGF yu = η y Eastward flow direc(on PGF High pressure perturba(on Coriolis Geostrophic balance Coriolis PGF yu = η y Geostrophic balance Westward flow direc(on High pressure perturba(on PGF Coriolis

16 The equatorial Kelvin wave u = η = G(x ct)e (y2 /2) 1 Propaga(on 0.5 Convergence Divergence u du/dx Longitude With a propaga(on speed of 50 m/s a dry Kelvin wave will cover this distance in around 5 days and make it around the globe in around 9 days

17 The equatorial Kelvin wave Convec(vely-coupled atmospheric Kelvin waves have a typical period of 6-7 days when measured at a fixed point and much slower phase speeds of m/s.

18 The equatorial Kelvin wave u = η = G(x ct)e (y2 /2)

19 The equatorial Kelvin wave What do we know about equatorial Kelvin waves? They: 1. No flow normal to the boundary (equator) [v=0]. 2. Have maximum amplitude on the equator and their decay away from the equator follows the equatorial radius of deforma(on [(c/β) 1/2 ]. 3. The flow along the boundary/equator is in geostrophic balance with the pressure gradient perpendicular to the wall [ βyu = g' η ]. y 4. Relies on the change of sign of coriolis at the equator makes it func(on like a barrier. 5. Have a speed, c = sqrt(g H) roughly 2-3 m s -1 for the first ocean baroclinic mode. roughly 50 m s -1 for the first atmospheric baroclinic mode. 6. Propagate from west to east (boundary on the lex [right) in the SH (NH)] Ocean - effec(vely act to transfers info from the west side of an ocean basin to the east. Atmos - allow an equatorial signal to propagate around the globe in a liple over a week. 7. They are non-dispersive

20 Other equatorially trapped waves 1. u t SWM equa(ons βyv = φ x Zonally propaga(ng solu(ons of the form u = û(y)e i(kx σt) 2. v t βyu = φ y v = ˆv(y)e i(kx σt) 3. φ u + g'h t x + v y = 0 φ = ˆφ(y)e i(kx σt)

21 Other equatorially trapped waves iωû βyˆv = ik ˆφ iω ˆv βyû = ˆφ y iω ˆφ + c 2 ikû + ˆv y 1.1 û = k ˆφ βyˆv ω ( β 2 y 2 ω 2 ) ˆv 2.1 = ikβyφ iω ˆφ y = ( ω 2 + ghk 2 ) ˆφ = gh iω ˆv y + βyˆv

22 Other equatorially trapped waves Subs(tu(ng 3.1 into 2.1 yields: 2 ˆv y + ω 2 β 2 y 2 k 2 c 2 ω β k 2 ˆv = 0 Physics Schrödinger equa(on Which can be shown to have solu(ons of the form: ˆv(y) = H n y R eq e y 2 2 /2 R eq R eq = c / β

23 Other equatorially trapped waves ˆv(y) = H n y R eq H n are the Hermite polynomials of degree n, that take the form: H 0 =1, H 1 (y)=2y, H 2 (y)=4y 2-2, H 3 (y)=8y 3-12ε e y 2 2 /2 R eq 0.8 Hermite Functions Amplitude th 1st 2nd 3rd 4th Latitude ( o N)

24 Other equatorially trapped waves The dispersion rela(on provides frequencies, ω, as a func(on of wavenumber, k, for each mode. gh β c k 2 ω β k 2 = 2n +1 ω 2 Allows two main types of solu(on ω -> 0 (low frequencies), we get geostrophic flow (Kelvin and Rossby waves). ω >> 0 (high frequencies), we get inerca gravity waves (equatorial gravity waves) There is also the n=0 mixed Rossby-gravity wave.

25 Mixed Rossby gravity waves " $ # ω 2 c 2 βk ω k 2 % ' = & Set n=0 2n +1 2 R eq ω = kc 1 2 ± 1 4β 1+ 2 k 2 c o N 26 o N 13 o N 2 Convergence V y + # ω 2 β 2 y 2 % 2 $ c 2 βk ω k 2 & (V = 0 ' 13 o S 26 o S 39 o S Divergence

26 Dispersion Inter(a-gravity Westward Mixed Rossbygravity Kelvin Eastward Eastward Westward Source: Cushman-Roisen and Beckman 2009

27 Source: Wheeler 2002 Iner(al gravity waves " $ # Small at high frequencies ω 2 c βk 2 ω k % 2 ' = & 2n +1 2 R eq ω 2 ig ± ((2n +1)βc + k 2 c 2 ) 39 o N 26 o N Divergence 13 o N 13 o S 26 o S 39 o S Convergence

28 Source: Wheeler 2002 " $ # Small at high frequencies ω 2 c βk 2 ω k % 2 Iner(al gravity waves ' = & 2n +1 2 R eq ω 2 ig ± ((2n +1)βc + k 2 c 2 ) 39 o N 26 o N Convergence 13 o N 13 o S 26 o S 39 o S Divergence

29 Dispersion Westward Inter(o-gravity Eastward Mixed Rossbygravity Kelvin Eastward Westward Source: Cushman-Roisen and Beckman 2009

30 Equatorial Rossby waves " $ # Small at low frequencies ω 2 c βk 2 ω k % 2 ' = & 2n +1 2 R eq ω rossby Bk k 2 + (2n +1)β / c 39 o N 26 o N Convergence 13 o N 13 o S 26 o S 39 o S Divergence Source: Wheeler 2002

31 Source: Wheeler 2002 Equatorial Rossby waves Equatorial Rossby wave speed: For long waves largely non-dispersive: Phase velocity= ω/k = -c/(2n+1) So: n=1 0.9m/s n=2 0.55m/s n=3 0.4m/s 39 o N 26 o N 13 o N 13 o S 26 o S 39 o S

32 Dispersion Inter(a-gravity Westward Mixed Rossbygravity Rossby waves Kelvin Eastward Eastward Westward Source: Cushman-Roisen and Beckman 2009

33 Other Equatorially trapped waves What do we know about equatorially trapped waves? They: 1. They are how the tropical ocean/atmosphere adjusts when perturbed. 2. Waves of an even (odd) order are asymmetric (symmetric) about the equator. 3. For long periods, Iner(al gravity waves are not produced, leaving Kelvin, Rossby and mixed gravity-rossby waves to do the adjustment. 4. The Kelvin wave and short wave-length Rossby are the only low frequency waves that carry energy eastward. Rossby waves only propagate westward, meaning Rossby waves are dispersive for short wave lengths 5. Long wave length Rossby waves are largely non-dispersive as both energy and the wave propagate westward. 6. Kelvin waves travel roughly 3 (mes faster (eastward) than the n=1 equatorially trapped Rossby waves does westward.

34 Steady state (Gill) solu(ons u t v t βyv = φ x βyu = φ y φ u + g'h t x + v y = Q Hea(ng rate Looking for steady state solu(ons we add Rayleigh fric(on and Newtonian cooling. We assume fric(on has the same value as the cooling (ε), then dropping the (me deriva(ve (steady state) the solu(ons become: Ver(cal velocity of the geopoten(al surface is given by: w = φ t +Q

35 Steady state (Gill) solu(ons εu βyv = φ x εv βyu = φ y εφ + g'h u x + v = Q y Hea(ng rate Looking for steady state solu(ons we add Rayleigh fric(on and Newtonian cooling. We assume fric(on has the same value as the cooling (ε), then dropping the (me deriva(ve (steady state) the solu(ons become: Ver(cal velocity of the geopoten(al surface is given by: w = φ t +Q

36 Steady state (Gill) solu(ons Heat source (symmetric) Contours = pressure Rossby waves Kelvin wave Rela(onship to Walker circula(on

37

38 Westerly wind events Bursts of low-level westerly winds are oxen observed along the Equator. These bursts oxen result in (or are caused by) the forma(on of tropical cyclone pairs or twins located at approximately the same longitude and the same distance on opposite sides of the Equator. The forma(on of twin tropical cyclones typically occurs in the western Pacific and Indian Ocean during the months of December through May. GOES-9 12 UTC 2 Dec micron image

39 Steady state (Gill) solu(ons Heat source (~asymmetric) Cross equatorial flow Contours = pressure

40 SE Asian monsoon

41 References Clarke, Allan J., 1983: The Reflec(on of Equatorial Waves from Oceanic Boundaries. J. Phys. Oceanogr., 13, Cushman-Roisen, B. and M. Beckers, 2009: Introduc(on to Geophysical Fluid Dynamics. Wheeler, M.C., 2002: Tropical meteorology: Equatorial waves. In: J. Holton, J. Curry, and J. Pyle (eds), Encyclopedia of Atmospheric Sciences. Academic Press, pages