INTERNAL WAVES COUPLED TO SURFACE GRAVITY WAVES IN THREE DIMENSIONS

We consider the nonlinear interaction of internal waves and surface waves in a three-dimensional fluid composed of two distinct layers. Using Hamiltonian perturbation theory, we show that long internal waves are modeled by the KPII equation and generate a resonant interaction with modulated surface waves at resonant wavenumbers. The surface wave envelope is described by a linear Schrodinger equation in two space dimensions with a potential given by the internal wave. We review the two-dimensional case where an analysis of the model equations, in analogy with radiative absorption in the semi-classical limit, provides an explanation of characteristic features observed on the sea surface due to the presence of an internal wave. In the three-dimensional case, for an internal wave in the form of an oblique line soliton, it is possible to relax the resonance condition to one admitting families of carrier frequencies. We also discuss open problems related to more general KP internal waves.