University:
Massachusetts Institute of Technology
Course:
12.802 | Wave Motions in the Ocean and Atmosphere
Academic year:

2024
Views:

452

Pages:

9
Author:

Origbo2ol

Rotating Shallow-Water Waves We now consider the effects of rotation and boundaries on fluids obeying (in the horizontal) the linearized shallow-water equations a a +gHVu=0 where u and V are horizontal vectors/ operators. Plane waves For the simplest case, we take all fields proportional to exp(skx-wt) to find -f - 9Hrk gHil ak TA =0 which implies τωω - 12 - 9H(2+2)] = 0 which has three roots, w0 and w² = f²+gHk2 which is the generalization of the long gravity wave dispersion relation. In the presence of rotation, the waves become dispersive, with k f2 These can be simplified by using the deformation radius √gH/f as a length scale 7=√1+kRa Cg kR VgHV1+kRa Note that the shallow water equations will only be applicable for √9/H kH=kR<<1 kR <<9TH ~500 f for a 4000m deep ocean. Rotating shallow water waves 6 omega/f c/sqrt(gH) 5 4- 3 2- 0 0 2 3 5 Dispersion relation for rotating plane waves The w0 root is non-trivial; to see this, let us look at the equations in vorticity/ divergence form. If ( =(x)=- a ay and +SD=0 D-FC-2 Ot at +9HD=0 D = Vu, then Eliminating D from the first and third equation gives gH 90 Ot The (linearized) potential vorticity q = (-) is conserved. This equation implies either the frequency is zero and the potential vorticity is not, or vice-versa. The zero- frequency waves correspond to D=0, (+V/f and are geostrophically balanced. When f varies, these turn into Rossby waves. If we recast the divergence equation in terms of q f2 --- 9H and use the conservation of mass equation, we find +fgHq+f² = 9HV2 For the gravity waves with no PV signal, +f²=gHV2 and we recover the dispersion relation above. (1) (2) Adjustment If we consider the initial value problem, we can specify the three fields, or, alterna- tively, we can specify q(x), (x,0) 4(x), and (x, 0) = Φερ(x). Since q remains unchanged, we can split the pressure up into the geostrophic part and the gravity wave part 6 = (x) + (x, t) f gh 02 Φω (x, 0) = φο(x) - g(x) Φω + 12 = 9HV² θε Φω (x, 0) = Φερ (κ) The equation for the geostrophic pressure shows that the influence of a localized potential vorticity anomaly spreads out over a scale Ra√gH/f called the "deformation radius." L.c., if q = 4od(x) (independent of y), the geostophic pressure is qo Ra 2 exp(-/Ra)

Rotating Shallow-Water Waves

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