# Hamiltonian model for coupled surface and internal waves in the presence of currents

## TL;DR

This paper develops a Hamiltonian framework to analyze coupled surface and internal water waves influenced by depth-dependent currents, revealing that wave-current interactions are primarily affected by currents near the surface and interface.

## Contribution

It introduces a Hamiltonian model for coupled surface and internal waves with currents, providing new insights into wave-current interactions in stratified fluids.

## Key findings

- Wave-current interaction depends mainly on currents near the surface and interface.
- Hamiltonian equations of motion are derived for the system.
- Small amplitude and long-wave approximations are discussed.

## Abstract

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of 'wave' variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave-current interaction is influenced only by the current profile in the 'strips' adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.