Numerical investigation of Rossby waves for nonlinear shallow‐water equations on the sphere

Although Rossby modes of the nonlinear shallow-water system on the rotating sphere are suspected to be close to their counterpart for the non-divergent barotropic vorticity equation, little is known about their properties. A numerical procedure is employed to approximate some of these modes for which an analytical expression is known in the non-divergent case. For stationary modes, the procedure simply consists in iteratively applying a time filter in a forecast initialised with the non-divergent mode. For zonally propagating modes, a change of frame allows a similar method to be used with only very minor changes to the forecast model. For stable modes, the structure may be approached with high precision by extending the length of the time filter used in the numerical procedure. The structure of the wind divergence field of Rossby modes is then revealed. The modes identified by this mean undergo instabilities of a similar nature to those for the barotropic vorticity equation system. The stability domain and growth rates are explored. Some avenues for improving test cases based on Rossby modes for shallow-water systems are finally discussed.