Two-Layer Models for Nonlinear Internal Wave Interaction

We present a hierarchy of two-layer models for the propagation across ocean basins of nonlinear internal waves (NLIWs) and for their interactions. The models are derived using the Euler-Poincaré framework, and include a two layer system, derived from the principle that at leading order the pressure in the top layer is hydrostatic, but nonhydrostatic pressure is present in the bottom layer, which avoids the ill-posedness under shear flow found in the two layer Green-Naghdi equations. We compare the stability conditions and traveling wave solutions from the various models. The most reduced model in the hierarchy, the two component Camassa-Holm (CH2) equation, may be modified by regularization of the potential energy term to produce a system with conservation equations which admit solutions with peaked solitons in both the velocity and layer thickness. The filament solutions of this equation in two dimensions can be used to model efficiently the propagation and interaction of NLIWs using Lagrangian particle methods. We present some preliminary numerical results.