Equatorially trapped waves Shayne McGregor ARC Postdoctoral Fellow CCRC & ARC CSS, UNSW Climate Change Research Centre
Equatorially trapped waves The linear SWM Equatorial Kelvin waves Other equatorially trapped waves Mixed gravity- Rossby waves Interio- gravity waves Rossby waves Wave refleckon at the boundaries El Niño- Southern OscillaKon
The linear SWM (first baroclinic mode)
The linear SWM (first baroclinic mode) VerKcal mode example
The linear SWM (first baroclinic mode) East (x)!u!t "!yv = "g'!"!x North (y)!v!t "!yu = "g'!"!y ρ 1 H g'!!!t c"2 +!u!x +!v!y = 0 ρ 2 η
The linear SWM (first baroclinic mode) Reduced gravity Gravity wave speed g' =! 2!! 1! 1 g c = g'h g = 9.8 m s - 2 ρ diff 0.02 VerKcal mode example c generally gets smaller as mode increases
Beta plane Coriolis parameter: f = 2!sin(!) Eq β- plane approximakon: f =!y β = 2Ω/a (2.3 x 10-11 m - 1 s - 1 ) Ω = rotakon 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 10 4 1 0.5 0 0.5 1 1.5 Normal B plane 80 60 40 20 0 20 40 60 80 Latitude
The linear SWM (first baroclinic mode) East (x)!u!t "!yv = "g'!"!x North (y)!v!t "!yu = "g'!"!y ρ 1 H g'!!!t c"2 +!u!x +!v!y = 0 ρ 2 η Lord Kelvin (William Thompson) v=0
The Equatorial Kelvin wave v=0 SWM eqns reduce to!u!t = "g'!!!x!yu =!g' "" "y Wave solukon u = cg(x! ct)exp(!y 2 / 2R 2 eq )! = G(x! ct)exp(!y 2 / 2R 2 eq ) g'!!!t + c2!u!x = 0
The Equatorial Kelvin wave R(y) = c f (y) R eq = R y = R eq u = cg(x! ct)exp(!y 2 / 2R 2 eq ) 1 Meridional decay Equatorially trapped 0.8 R eq = c /! decay scale = (2c/B) 1/2 0.6 0.4 0.2 0 30 20 10 5 0 5 10 20 30
The equatorial Kelvin wave Coriolis PGF!yu =!g' "" "y Eastward flow direckon 1 0.8 0.6 0.4 0.2 PGF SSH perturbakon Coriolis Geostrophic balance 0 10 5 0 5 10 Coriolis PGF!yu =!g' "" "y Geostrophic balance Westward flow direckon 1 0.8 0.6 0.4 0.2 0 SSH perturbakon PGF Coriolis 10 5 0 5 10
The equatorial Kelvin wave 1 PropagaKon 0.5 Convergence 0 0.5 Divergence u du/dx 1 100 150 200 250 300 Longitude
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* Dispersion occurs when pure plane waves of different wavelengths have different propagakon velocikes, so that a wave packet of mixed wavelengths tends to spread out in space.
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 deformakon [(2c/β) 1/2 ]. 3. The flow along the boundary is in geostrophic balance with the pressure gradient perpendicular to the wall [!yu =!g' "" ]. "y 4. Propagate from west to east (boundary on the lep [right) in the SH (NH)] EffecKvely act to transfers info from the west side of an ocean basin to the east. 5. Have a speed, c = sqrt(g H) roughly 2-3 m s - 1 for the first ocean baroclinic mode (H=150m, g =5e - 2 m s - 2 ). [roughly 200 m s - 1 for the oceanic barotropic waves (H=5000m, g=9.8 m s - 2 )] [roughly 50 m s - 1 for the atmospheric baroclinic waves] So they allow the west Pacific to communicate with the east Pacific relakvely fast (~70 days) 6. Upwelling (downwelling) Kelvin waves are associated with eastward (westward) currents. 7. They are non- dispersive
Kelvin wave (modeled)
Kelvin wave (Obs)
Kelvin wave (Obs)
Irregular grid problems Course resolukon version of the MPI- ESM (along with many others) uses a curvlinear grid which shiped the north pole over Greenland. The south pole remains centrally located. a) b) 2 N 1 N EQ 1 S 2 S 80 N 2N 1N 0 1S 2S LaKtude where f 0 50E 150E 110W 10W 50 E 150 E 110 W 10 W 40 N EQ 40 S 80 S 50 E 150 E 110 W 10 W
Irregular grid problems
Irregular grid problems 30 Merdional velocity (m/s) on day 10 0.02 20 10 0.015 0.01 0.005 Latitude 0 10 20 0 0.005 0.01 0.015 30 100 150 200 250 300 Longitude 0.02
Irregular grid problems
Other equatorially trapped waves!u!t!v!t SWM equakons "!yv = "g'!"!x "!yu = "g'!"!y Zonally propagakng solukons of the form u =U(y)cos(kx!!t) v = V(y)sin(kx!!t) g'!!!t c"2 +!u!x +!v!y = 0 h = A(y)cos(kx!!t)
Other equatorially trapped waves SubsKtuKng the solukons funckons back into the SWM equakons yields:! 2 V!y + #! 2 " " 2 y 2 % " "k 2 c 2! " k 2 $ & (V = 0 ' Physics Schrödinger equakon Which can be shown to have solukons of the form: V(y) = H n! # " y R eq $ & % R eq = c /! & e'y 2 2 /2 R eq
Other equatorially trapped waves V(y) = H n! # " y R eq $ & % & e'y 2 2 /2 R eq H n are the Hermite polynomials of degree n, that take the form: H 0 =1, H 1 (ε)=2ε, H 2 (ε)=4ε 2-2, H 3 (ε)=8ε 3-12ε And the solukons only decay as y gets large when: Dispersion relakon "! 2 % $ # c 2! "k!! k 2 ' = & 2n +1 2 R eq
Other equatorially trapped waves The dispersion relakon provides frequencies, ω, as a funckon of wavenumber, k, for each mode. " $ #! 2 c 2! "k!! k 2 % ' = & 2n +1 2 R eq Westward Mixed Rossby- gravity Rossby InterKa- gravity Kelvin Eastward Eastward InterKa- gravity Westward
Mixed Rossby gravity waves " $ #! 2 c 2! "k!! k 2 % ' = & Set n=0 2n +1 2 R eq T eq = 1!c " (! + ck)!t eq! 1 % $! kr eq #!T ' = 0 eq &! 2 Convergence V!y + #! 2 " " 2 y 2 % " "k 2 c 2! " k 2 $ & (V = 0 ' Divergence
Dispersion InterKa- gravity Westward Mixed Rossby- gravity Kelvin Eastward Eastward Westward Source: Cushman- Roisen and Beckman 2009
Source: Wheeler 2002 " $ # Small at high frequencies! 2 % c 2! "k!! k 2 ' = & 2n +1 2 R eq!! ± InerKal gravity waves 2n +1 T eq 2 + g'hk 2, n "1 Divergence R eq Convergence
Source: Wheeler 2002 " $ # Small at high frequencies! 2 % c 2! "k!! k 2 InerKal gravity waves ' = & 2n +1 2 R eq!! ± 2n +1 + g'hk 2, n "1 2 R eq Convergence Divergence
Dispersion InterKa- gravity Westward Mixed Rossby- gravity Kelvin Eastward Eastward Westward Source: Cushman- Roisen and Beckman 2009
Source: Wheeler 2002 Equatorial Rossby waves " $ # Small at low frequencies! 2 % c 2! "k!! k 2 ' = & 2n +1 2 R eq!! " "R 2 eq (2n +1), n #1 Convergence Divergence
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
Dispersion InterKa- gravity Westward Mixed Rossby- gravity Kelvin Eastward Eastward Westward Source: Cushman- Roisen and Beckman 2009
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 (>=T eq, 2- days), InerKal gravity waves are not produced, leaving Kelvin, Rossby and mixed gravity- Rossby waves to do the adjustment. 4. If the perturbakon is quasi- symmetric about the equator, the mixed wave and even order Rossby waves are ruled out as solukons. 5. The Kelvin wave and short wave- length Rossby wave carry energy eastward. Rossby waves only propagate westward, meaning Rossby waves are dispersive for short wave lengths 6. Long wave length Rossby waves are largely non- dispersive as both energy and the wave propagate westward. 7. Kelvin waves travel roughly 3 Kmes faster (eastward) than the n=1 equatorially trapped Rossby waves do westward.
Equatorial Rossby wave (modeled)
Western boundary refleckons U r = " +$y #$y " 0 H u r!z!y U r! 2"y /!(y 0 2 # "y 2 ) Decreases as approx 1/y 0 2
Western boundary refleckons Kelvin wave h k = Aexp(!y 2 / 2R 2 eq ) u k = (g'/ c)h k Eq Kelvin wave zonal transport U k = +$ 0 " " u k!z!y = A[2c 3! / "] 1/2 #$ H
Western boundary refleckons Upper layer thickness ResulKng Eq Upper layer thickness Zonal current anomalies Eq Kelvin wave zonal transport U k = +$ 0 " " u k!z!y = A[2c 3! / "] 1/2 #$ H Kessler 1991
Eastern boundary refleckons Westward Rossby wave Westward InerKa- gravity wave Clark 1983
El Niño- Southern OscillaKon PC2 (dashed red)
El Niño- Southern OscillaKon Triggers
El Niño- Southern OscillaKon
El Niño- Southern OscillaKon
El Niño- Southern OscillaKon Why do the events end?
References Clarke, Allan J., 1983: The ReflecKon of Equatorial Waves from Oceanic Boundaries. J. Phys. Oceanogr., 13, 1193 1207. Cushman- Roisen, B. and M. Beckers, 2009: IntroducKon to Geophysical Fluid Dynamics. Kessler, W. S., 1991: Can reflected extra- equatorial Rossby waves drive ENSO? Journal of Physical Oceanography, 21, 444-452. Moore, D. W., 1968: Planetary- gravity waves in an equatorial Ocean, PhD thesis. Wheeler, M.C., 2002: Tropical meteorology: Equatorial waves. In: J. Holton, J. Curry, and J. Pyle (eds), Encyclopedia of Atmospheric Sciences. Academic Press, pages 2313-2325.
Rossby wave speed