Nonlinearly coupled Rossby-type and inertio-gravity waves in self-gravitating systems

A dispersion relation is derived for the azimuthal propagation of inertio-g ravity and gravitational Rossby waves in a nonuniform self-gravitating syst em. The instability condition for the azimuthally propagating perturbations and its possible application to the formation of galaxy disk spirals is di scussed. In the region where the system is stable, the nonlinear coupling b etween inertio-gravity and gravitational Rossby waves is considered, accoun ting for the action of the low frequency force associated with the inertio- gravity waves. It is shown that this interaction is governed by a pair of e quations, which can be useful for studying the modulational instability of a constant amplitude inertio-gravity wave as well as the dynamics of nonlin early coupled inertio-gravity and gravitating Rossby waves.