Nonlinear internal waves over variable topography

A nonlinear two-layered fluid system on a variable bottom with freely moving upper surface and abrupt density change at the interface is considered. The two stably stratified layers with constant densities are assumed to be immiscible. Starting from the continuity and the mometum equations of an incompressible inviscid fluid for each layer and associated boundary and transition conditions at the free upper surface, the interface and the bottom surface, vertically averaged equations are deduced that incorporate nonlinear advection and dispersion. The approximate equations are deduced by scaling the governing equations accordingly and by using a limit analysis for small aspect ratios (shallowness) and small amplitude disturbances. The limiting equations extend known model equations and include finite amplitude nonlinearities, dispersion and variable topography. In a second step viscous effects and wind forcing are also incorporated.