NONLINEAR-WAVE PROPAGATION ON A SPHERE - INTERACTION BETWEEN ROSSBY WAVES AND GRAVITY-WAVES - STABILITY OF THE ROSSBY WAVES

An analytical spectral model of the barotropic divergent equations on a sphere is developed using the potential-stream function formulation and the normal modes as basic functions. Explicit expressions of the c oefficients of nonlinear interaction are obtained in the asymptotic ca se of a slowly rotating sphere, i.e. when the normal modes can be expr essed as single spherical harmonics. Highly truncated versions of the model are used to illustrate some consequences of the interaction betw een the Rossby waves and the surface gravity waves. So, it is shown ho w the gravity waves can exchange energy through interaction with a Ros sby wave. A particular interaction between a Rossby wave and a gravity wave of zonal wavenumbers m and 2m, respectively, is discussed in gre ater detail and is shown to lead to a frequency shift of the Rossby wa ve. The stability of the Rossby waves to perturbations involving one o r two gravity waves is examined and it is shown that the corresponding critical amplitudes are unrealistically large. The decay of the Rossb y waves will probably be governed by the interactions inside Rossby wa ves triads.