Hamiltonian model for coupled surface and internal waves over currents and uneven bottom

TL;DR

This paper develops a Hamiltonian model for coupled surface and internal water waves over currents and uneven bottom topography, deriving a variable-coefficient KdV equation to describe internal wave dynamics.

Contribution

It introduces a Hamiltonian framework for coupled internal and surface water waves over complex bottom topography, deriving a new variable-coefficient KdV equation.

Findings

Hamiltonian formulation of water wave interactions

Derivation of a variable-coefficient KdV equation

Framework for analyzing internal waves over uneven bottoms

Abstract

A Hamiltonian model for the propagation of internal water waves interacting with surface waves, a current and an uneven bottom is examined. Using the so-called Dirichlet-Neumann operators, the water wave system is expressed in the Hamiltonian form, and thus the motions of the internal waves and surface waves are determined by the Hamiltonian formulation. Choosing an appropriate scaling of the variables and employing the Hamiltonian perturbation theory from Hamiltonian formulation of the dynamics, we derive a KdV-type equation with variable coefficients depending on the bottom topography to describe the internal waves.