The stability of planetary waves on an infinite beta?plane

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Abstract
The stability of a planetary wave on an infinite ß‐plane is examined. There are two parameters in the problem, (a) ? = Uκ2/β where U is the velocity amplitude of the planetary wave and κ its wavenumber, and (b) the direction of the wavenumber of the planetary wave. For large M, the problem reduces to the Rayleigh problem for a sinusoidal velocity distribution. The growth rate of the most unstable wave is 0.27 Uκ. 38 % of the energy lost by this wave is transferred to wavenumber 0.59 κ and 61 % is transferred to wavenumber 1.16 κ. As ? decreases and β effects become more important, there are fewer unstable waves, but unstable waves can always be found. For small M, the disturbance comprises two waves which form a resonantly interacting triad with the primary wave. Some geophysical applications are discussed. It is shown, for instance, that for strong currents like the Gulf Stream, this type of instability is much more important than inertial instability. Also, observations indicate that the dominant eddies in the energy spectra of both ocean and atmosphere have ? of order unity. But they also have length scales comparable with the radius of deformation, indicating the importance of more physical processes than those considered here.

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