Complementary mild-slope equations in a two-layer fluid

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. By using a streamfunction formulation instead of a velocity potential one, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact linear theory compared to other MS-type equations. The main goal of this work is to extend the CMSE model for solving two-layer flow with a free-surface. In order to allow for an exact reference, an analytical solution for a two-layer fluid over a sloping plane beach is derived. This analytical solution is used for validating the results of the approximated MS-type models. It is found that the two-layer CMSE model performs better than the potential based one. In addition, the new model is used for investigating the scattering of linear surface water waves and interfacial ones over variable bathymetry.

Graphical Abstract
► A streamfunction (SF) formulation models better bottom BCs in mild-slope models.
► A linear SF mild-slope equation for two-layer flow with a free-surface is derived.
► An analytical solution for a two-layer fluid over a sloping plane beach is derived.
► This analytical solution is used for validating the results of MS-type models.
► Scattering of surface and interfacial waves over variable bottom is investigated.