A set of fully nonlinear equations for surface and internal gravity waves

On an interface between upper and lower fluid layers of different densities, internal gravity waves occur and propagate. In this study a set of fully nonlinear equations for a multi-layer fluid system was obtained. The velocity potential in each layer was expressed as a series in terms of a given set of vertical distribution functions. Then the Lagrangian was integrated vertically and the variational principle was applied to yield a set of time-dependent, horizontally two-dimensional, fully nonlinear equations. The application of this model is free from any limitation concerning the layer's relative thickness or the wave frequency. Numerical simulations were carried out for various two-layer fluid systems. It was shown that internal waves change their profiles in shoaling and disintegrate over the submerged breakwater.