Part II: Waves in the Tropics- Theory and Observations Derivation of gravity and Kelvin waves

Background: Occur in response to density displacement on fluid interface or within a fluid Restoring force is gravity A Kelvin wave is a type of gravity wave. 2 types of Kelvin waves: a) coastal b) equatorial Gravity and Kelvin Waves

Hydrostatic external gravity waves (such as those occurring on the ocean-atmosphere interference) Compressibility effects can be ignored p = p(z) and dw/dt in z-momentum equation is ignored p(z) = -ρ o gz External Gravity Waves

Use the following equations to describe motion… x-momentum: δu/δt + u δu/δx = -(1/ρ o ) (δp’/δx) (8) z-momentum: 0 = -(1/ρ o ) (δp/δx) – g (9) Continuity (compressibility effects ignored): δu/δx + δw/δz = 0 (10) Shallow water form: large scale, where x and y >> z Vertical motions << horizontal. 2-D plane for simplification. Neglect f in this case, so –fv drops out. External Gravity Waves

Write in shallow water form by vertically integrating (10): § h 0 (δu/δx + δw/δz) dz = 0 (10) = δu/δx § h 0 dz + w| h 0 = 0 w(0) = 0 So, we end up with w(h) for right hand term, such that: w(h) = δh/δt + u δh/δx For thoroughness, for w(h) comes from w(h) = Dh/Dt = δh/δt + u δh/δx + v δh/δy + w δh/δz Left- hand term simply becomes h δu/δx External Gravity Waves

Eliminate the z-dependence for shallow water eqns: Simplify (10): δh/δt + δ/δx (hu) = 0 (10*) Note that p’(x,t) = ρ o gh(x,t). Plug into (8): δu/δt + u δu/δx = -g (δh/δx) (8*) External Gravity Waves

Linearize: u = u + u’ h = h + h’ Remember: derivative of constants = 0 and product of two perturbations is very small. We end up with: δh'/δt + u δh’/δx + h δu’/δx = 0 (11) δu'/δt + u δu’/δx = -g (δh’/δx) (12) We are going to attempt to find solutions now. We have two equations in terms of u and h. External Gravity Waves

Eliminate u’, that is, (δ/δt + u δ/δx) : Rearrange equations first… (δ/δt + u δ/δx ) h’+ h δu’/δx = 0 (11) (δ/δt + u δ/δx) u’ + g (δh’/δx) = 0 (12) Multiply (11) by (δ/δt + u δ/δx): (δ/δt + u δ/δx) 2 h’ + h δ/δx (δ/δt + u δ/δx)u’ = 0 (11*) (δ/δt + u δ/δx)u’ = -g δh’/δx from (12) Plug the above line into (11*) and get: (δ/δt + u δ/δx) 2 h’ – gh δ/δx (δh’/δx) = 0 (13) External Gravity Waves

Tidy up right term a bit to get: (δ/δt + u δ/δx) 2 h’ – gh (δ 2 h’/δx 2 ) = 0 (13) Solutions of the form… δ 2 h’ / δt 2 - C 2 δ 2 h’/δx 2 = 0 Plug in wave-like solution: h’ = Ae (ik(x-ct)) (-ikc) 2 + (uik) 2 –gh(ik) 2 = 0 c 2 = u 2 + gh c = u +/- (gh) 1/2 External Gravity Waves

Phase speed (also, can be written as ω/k): c = u +/- (gh) 1/2 Angular frequency: ω = ck = k(u +/- (gh) 1/2 ) k = zonal wavenumber = 1/λ x (where λ x = zonal wavelength) Group velocity: dω/dk = (1)(u +/- (gh) 1/2 ) + k*0 = u +/- (gh) 1/2 Group velocity is the same phase speed, so it’s a nondispersive wave. Gravity waves can propagate either east or west in this case. External Gravity Waves

If internal gravity wave (within atmosphere or ocean, much slower): c ~ (g’h) 1/2 where g’ = g ((ρ o -ρ 1 )/ρ o ) where g’ = “reduced gravity” Difference in densities within the same fluid is much less, which considerably slows down the wave. Gravity Waves

Important in deep tropics, such as deep moist convection in the tropical Pacific, with periods of a couple days. Very large gravity waves can be generated from other features, such as swell from an earthquake that would later become a tsunami. When convectively coupled (i.e., moist convection in atmosphere), wave speed tends to slow. Gravity Waves

Kelvin Waves- Trapped along Coast

Kelvin Waves- Trapped along Coast

Near/at equator, so let f ~ βy, such that: x-mom: δu'/δt – βyv’ = -(1/ρ o ) (δp’/δx) y-mom: δv'/δt + βyu’ = -(1/ρ o ) (δp’/δy) cont: (1/ρ o ) (δp’/δt) + gh(δu’/δx + δv’/δy) = 0 Assume perturbation cross-velocity (perturbation meridional velocity) is small, so v’ drops out. Equatorially Trapped Kelvin Waves

If v’ is very small, we get: x-mom: δu'/δt = -(1/ρ o ) (δp’/δx) y-mom: βyu’ = -(1/ρ o ) (δp’/δy) cont: (1/ρ o ) (δp’/δt) + gh(δu’/δx) = 0 Let ϕ = = -(1/ρ o ) δp’ And let: u’ = u exp{i(kx-ωt)} v’ = v exp{i(kx-ωt)} ϕ ’ = ϕ exp{i(kx-ωt)} Equatorially Trapped Kelvin Waves

Plug in wave-like solutions to obtain… x-mom: -(iω)u = - (ik) ϕ y-mom (keep from earlier): βyu = - (δ ϕ /δy) cont: (-iω) ϕ + gh(iku) = 0 From top: ϕ = ( ω /k)u Plug in to continuity: -i ω 2 /ku + gh(iku) = 0 Multiply by k, divide by i and divide by u: gh(k 2 ) - ω 2 = 0 Equatorially Trapped Kelvin Waves

Rearrange: ω 2 = gh(k 2 ) (ω/k) = c x (phase speed) c x 2 = gh Like shallow-water gravity waves derived earlier. (δω/δk) = c gx (group velocity) c x 2 = c gx 2 (nondispersive wave) Observed dry Kelvin wave speed ~30-60 ms -1, while moist is ~12-25 ms -1 Equatorially Trapped Kelvin Waves

Go back to: x-mom: -iωu = - ik ϕ y-mom (keep from earlier): βyu = - (δ ϕ /δy) From top: ϕ = ( ω /k)u or ϕ = cu Plug in to y-mom: βyu = -c(δu/δy) Integrate to find: u = u o exp(-βy 2 /2c) Equatorially Trapped Kelvin Waves

u = u o exp(-βy 2 /2c) Perturbation zonal velocity at equator = u o If solutions exist at equator, then they decay away from equator and phase speed must be positive (c > 0). That is, waves propagate to the east. Equatorially Trapped Kelvin Waves

Therefore, from slides (21) and (15), we know that an equatorially trapped Kelvin wave would propagate to the east in an ocean basin and then, after hitting a coastline, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. The wave will decay exponentially away from the equator and away from the coast. Equatorially Trapped Kelvin Waves

Kelvin Waves

- Wheeler et al. (2000) -UL Kelvin wave structure -1 st order baroclinic wave -Convectively coupled, where moist convection slows down wave propagation Kelvin Waves

- UL OLR anomalies indicate that deep convection trails upper level high as wave propagates eastwards. Deep convection likewise trails LL low. -Moist convection slows down wave propagation by inducing pressure falls west of the low and pressure rises west of the high. Kelvin Waves

Examples include MJO and ENSO influences, which often feature convectively coupled Kelvin waves in equatorial Pacific. Vertical propagation into stratosphere of Kelvin or gravity waves (as well as Rossby waves) also influences wind fields higher up. Localized impacts, too, such as affecting weather along a mountain range, which can serve as a topographic barrier, like a coastline or continental shelf margin. Kelvin Waves

End First Presentation of Section 2