Stability of the Rossby-Haurwitz wave

A spectral method is used to integrate the primitive equations for the motion on a sphere of a shallow layer of fluid with a free surface. A simple Rossby-Haurwitz wave of zonal wavenumber 4 is found to change its form little over 24 days, whilst one of wavenumber 8 breaks down completely in 5 days. Inspired by the integrations, the barotropic stability of a wave to a perturbation composed of another wave and a zonal flow is considered. The theory suggests that waves of zonal wavenumber greater than 5 may be unstable because the feedback of the perturbation on itself is positive. Those of wavenumber 5 and less are stable, the feedback being negative. The theory and the general barotropic instability of Rossby waves is tested by integrations of the non-divergent barotropic model, in which the wave is perturbed by a zonal flow. Some agreement with theory is found. Waves of zonal wavenumber 8 rapidly break down to produce jets in the same direction as the perturbation zonal flow. The stability of these jets is also examined.

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