Modelling intermediate internal waves with currents and variable bottom

TL;DR

This paper develops a Hamiltonian-based model for internal interfacial waves considering currents and variable bottom topography, deriving integrable equations like ILWE, Benjamin-Ono, and KdV as special cases, with higher-order extensions.

Contribution

It introduces a comprehensive Hamiltonian framework for internal waves with variable bottom and currents, deriving a variable-coefficient ILWE and related integrable equations.

Findings

Derivation of ILWE with variable coefficients for non-flat bottom.

Reduction to Benjamin-Ono and KdV equations in specific limits.

Development of higher-order ILWE models.

Abstract

A model for internal interfacial waves between two layers of fluid in the presence of current and variable bottom is studied in the flat-surface approximation. Fluids are assumed to be incompressible and inviscid. Another assumption is that the upper layer is considerably deeper with a lower density than the lower layer. The fluid dynamics is presented in Hamiltonian form with appropriate Dirichlet-Neumann operators for the two fluid domains, and the depth-dependent current is taken into account. The well known integrable Intermediate Long Wave Equation (ILWE) is derived as an asymptotic internal waves model in the case of flat bottom. For a non-flat bottom the ILWE is with variable coefficients. Two limits of the ILWE lead to the integrable Benjamin-Ono and Korteweg-de Vries equations. Higher-order ILWE is obtained as well.