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Additional info for A short course on atmospheric and oceanic waves

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Averaging over a wave period, T, gives A2 ⎛kω + β/2⎞ = - 2 ⎜ ⎟ ⎝ ω ⎠ € 37 ⎛ β (k 2 − l 2 ) ⎞ A2(k2 + l2) ⎜ (k 2 + l 2 ) 2 ⎟ ⎜ ⎟ , = 4 ⎜⎜ 2βkl ⎟⎟ ⎝ (k 2 + l 2 ) 2 ⎠ in which the frequency equation (4) with U = 0 has been used. , € 2 2 ∂ω β (k − l − λi 2 ) cgx = = ; (21a) (K 2 + λi 2 ) 2 ∂k and cgy = € so that, ∂ω 2βkl = . 2 ∂ (K + λi 2 ) 2 (21b) ∂K ' + cg. ∇ Κ' = 0 (22) ∂€ t Therefore, we have shown that the Rossby wave energy propagates in the direction of and at the group velocity, cg.

Recall that in our previous model, the Coriolis parameter, f, was treated as a constant, namely, the earth curvature has been neglected. This assumption is valid only when the application range of latitude is small. In this chapter we shall show that if we allow f to vary with latitude when it is differentiated, we will find a type of low-frequency (ω < f) waves, called Rossby waves, that are of most importance for large-scale flows. Rossby waves exist in either a stratified fluid or a homogenous layer of fluid.

The scale at which the earth rotation becomes as important as gravity in determining the frequency of surface gravity waves is obtained by setting f2 = gHk2 in Eq. (23), which gives λ = k-1 = gH /f. The quantity, λ, which has unit of length, is called the (external) Rossby radius of deformation, which is of fundamental importance in the dynamics of rotating flows. 8 m s-2, H = 1 km and f = 10-4 s-1, so L ≈ 6300 km. It is evident that when waves are short compared with the Rossby radius, the pressure gradient or the gravity dominates the wave behavior, so the waves are non-dispersive, pure gravity, shallow water waves.

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