part ii: waves in the tropics- theory and observations derivation of gravity and kelvin waves

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

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.


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


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*)


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.


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)


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


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.


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

Gravity Waves

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)}


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


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


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)


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.


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.


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

End First Presentation of Section 2

part ii: waves in the tropics- theory and observations derivation of gravity and kelvin waves

Documents