By Derome J., Zhang D.L.
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Additional info for A short course on atmospheric and oceanic waves
8) has an additional set of wave solutions for the isothermal atmosphere that are not of the form (7b). These waves have no vertical velocity and they propagate horizontally with the speed of acoustic waves. This type of waves was first discovered by Laplace (1778) in the study of tidal oscillations, and later examined in detail by Lamb (1910); so it is now known as the Lamb wave. With w = 0, we can write (8a), (8c) and (8d) in matrix form as ⎡−ω ⎢ −k ⎢⎣ 0 0 −δ cω −c 20 ω k ⎤ ⎡ uˆ ⎤ 0 ⎥ ⎢αˆ ⎥ = 0.
This is because these waves have w' = 0, and they are therefore in hydrostatic balance and not affected by the approximation. In obtaining (25) we have not used Eq. (8b). We need to find the vertical dependence of the wave solutions in (7b). Assuming w' = 0, δc = 1 and ω = k ca, Eqs. (8a)-(8c) can be written in matrix form as ⎡−ω ⎢ 0 ⎢⎣ −k 0 −g −ω k ⎤ ⎡ uˆ ⎤ −1 im − (2H ) ⎥ ⎢αˆ ⎥ = 0.
This perturbation vorticity will induce a north-south velocity field, which moves the fluid parcels southward (northward) west of the vorticity maximum (minimum). Therefore, the fluid parcels will oscillate about their equilibrium latitude, and the wave pattern will propagate to the west. Clearly, in this case, the north-south gradient of absolute vorticity provides the restoring mechanism for Rossby waves, just as a positive vertical gradient of potential temperature (θ) resists vertical displacement and provides the restoring force for vertical oscillation (see Fig.
A short course on atmospheric and oceanic waves by Derome J., Zhang D.L.