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26 3. 1 Rossby waves In the preceding section, we obtained for surface inertial-gravity waves a cut-off frequency of ω = f at which both the earth's rotation and gravity are important. Does this imply that there would be no waves falling into the frequency range ω < f? The answer is: yes or no. 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.

If we define β = ∂f/∂y (or β plane approximation)2 as the rate of change of f with latitude, and introduce the perturbation stream function based on Eq. (1c): u' = - ∂ψ , ∂y v' = ∂ψ , ∂x and ζ' = Δψ, then Eq. (2) becomes ( 2 ∂ ∂ ∂ψ + U ) Δψ + β = 0. 6 x 10-11 m1 s-1 at φ = 450, where a is the radius of the earth. The β-plane approximation is generally adequate when the wavelength of a Rossby wave is smaller than the earth's radius. 28 Substituting the following form of wave solution ψ = A e i(kx + ly - ωt) into (3) yields the frequency equation (ω - kU) (k2 +  2) + βk = 0, β ω = k(U - 2 ), (4) K where K2 = k2 +  2.

8a) implies that for a basic state in geostrophic balance, the free surface must slope to provide a pressure gradient force to balance the Coriolis force. Thus, the linearized continuity equation becomes ( or ∂H ∂ ∂ ∂u ' ∂v' + U )h' + v' + H( + ) = 0, ∂y ∂y ∂t ∂x ∂x ∂h' ∂u ' ∂v' + H( + ) = 0. ∂y ∂t ∂x (9) (9') Note that the geostrophic relation has been used to cancel U∂h'/∂x and v'∂H/∂y. We will treat H as constant except when taking derivatives, since the height difference caused by the sloping surface is very small, as compared to the depth of the fluid.