part ii: waves in the tropics- theory and observations derivation of gravity and kelvin waves
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Background: Occur in response to density displacement on fluid interface or within a
fluidRestoring 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) = -ρogz
External Gravity Waves
Use the following equations to describe motion…
x-momentum: δu/δt + u δu/δx = -(1/ρo) (δp’/δx) (8)
z-momentum: 0 = -(1/ρo) (δp/δx) – g (9)
Continuity (compressibility effects ignored): δu/δx + δw/δz = 0 (10)
Shallow water form: large scale, where x and y >> zVertical 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 §h0 dz + w|h
0 = 0
w(0) = 0So, we end up with w(h) for right hand term, such that:
w(h) = δh/δt + u δh/δxFor thoroughness, for w(h) comes from
w(h) = Dh/Dt = δh/δt + u δh/δx + v δh/δy + w δh/δzLeft- 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) = ρogh(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 (δ2h’/δx2) = 0 (13)
Solutions of the form…δ2h’ / δt2 - C2 δ2h’/δx2 = 0 Plug in wave-like solution:
h’ = Ae(ik(x-ct))
(-ikc)2 + (uik)2 –gh(ik) 2 = 0c2 = u2 + 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/λx (where λx = 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 ((ρo-ρ1)/ρo) 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
http://faculty.washington.edu/luanne/pages/ocean420/notes/kelvin.pdf
Kelvin Waves- Trapped along Coast
http://faculty.washington.edu/luanne/pages/ocean420/notes/kelvin.pdf
Near/at equator, so let f ~ βy, such that:x-mom: δu'/δt – βyv’ = -(1/ρo) (δp’/δx)
y-mom: δv'/δt + βyu’ = -(1/ρo) (δp’/δy)
cont: (1/ρo) (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/ρo) (δp’/δx)
y-mom: βyu’ = -(1/ρo) (δp’/δy)
cont: (1/ρo) (δp’/δt) + gh(δu’/δx) = 0
Let ϕ = = -(1/ρo) 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 = -(δϕ/δy)
cont: (-iω)ϕ + gh(iku) = 0
From top: ϕ = (ω/k)u
Plug in to continuity:
-iω2/ku + gh(iku) = 0Multiply by k, divide by i and divide by u:
gh(k2) - ω2 = 0
Equatorially Trapped Kelvin Waves
Rearrange:ω2 = gh(k2)
(ω/k) = cx (phase speed)
cx2 = gh
Like shallow-water gravity waves derived earlier.(δω/δk) = cgx (group velocity)
cx2 = cgx
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 = -ikϕ
y-mom (keep from earlier): βyu = -(δϕ/δy)
From top: ϕ = (ω/k)u or ϕ = cu
Plug in to y-mom:
βyu = -c(δu/δy)Integrate to find:
u = uo exp(-βy2/2c)
Equatorially Trapped Kelvin Waves
u = uo exp(-βy2/2c)
Perturbation zonal velocity at equator = uo 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
-Wheeler et al. (2000)
-UL Kelvin wave structure
-1st orderbaroclinic 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
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