By Derome J., Zhang D.L.
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Additional info for A short course on atmospheric and oceanic waves
Winds are in geostrophic balance at both sides of highs and lows, but out of balance at the interface of highs and lows. A vertical cross section of Kelvin waves would be the same as that for surface gravity waves (cf. Figs. 4). 45 Fig. 3 The equivalent depth As previously mentioned, the phase speed of shallow water waves is dependent on the depth, H, of the fluid if it is incompressible. When the concept is applied to the stratified atmosphere, one has to use the equivalent depth which is not exactly the thickness of a layered fluid.
Their ratio is € € ca2 c 2 K2 c a2 g 2γ 2 . c a2K 2 . 42 m2 = 4(γ - 1) . 11N); or b) ω2 < 2 k2g2(γ - 1) 2k = N ca2 K2 K2 (15b) in which case both sides are negative. Then Eq. (13) can be re-written as ω2 (ω2 - ωa2) - K2ω2 + k2N2 = 0 ca2 (13') Thus, we see that there are no solutions to (12) in the interval ωg to ωa in the above diagram. In other words, the possible waves in the (x, z) plane are separated into a highfrequency mode (to the right of ωa) and a low-frequency mode (to the left of ωg = N).
The eigenvalue for the vertical equation is (gH)-1, while the eigenvalue for the horizontal equations is the frequency ω. Unlike the shallow water equations in which H is the specified depth of a fluid, the value(s) of H in Eq. (5) must be determined first from the vertical equation, subject to the boundary conditions at the top and the bottom, and then H can be used in the horizontal equations to determine the frequency and other properties of waves. The vertical structure equation may yield a set of different equivalent depths which correspond to all possible vertical eigenvectors.