wave interactions in a shallow-water model

By using a Hamiltonian method, non-linear three-wave interaction in a class of systems related to the shallow water model is considered and a general coupling coefficient is presented. In the special case where two inertial waves and one Rossby wave interact resonantly, it is found that even a very small shear of the background velocity can be important in the interaction process. The stability of the system is considered by using a pseudo-energy method. Some implications for the dynamics of atmospheric flows are pointed out.