Hamiltonian Approach to Internal Wave-Current Interactions in a Two-Media Fluid with a Rigid Lid

TL;DR

This paper models the interaction between internal waves and currents in a two-layer fluid system using a Hamiltonian framework, revealing that wave-current dynamics are primarily affected by the current profile near the internal wave interface.

Contribution

It introduces a Hamiltonian formulation for internal wave-current interactions in a two-media fluid with a rigid lid, considering non-zero vorticities and depth-dependent currents.

Findings

Wave-current interaction depends mainly on the current profile near the internal wave.

The Hamiltonian approach captures the influence of currents on internal wave dynamics.

The model accounts for non-zero constant vorticities in both media.

Abstract

We examine a two-media 2-dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface with wind generated surface waves but considered bounded above by a lid by an assumption that surface waves have negligible amplitude. An internal wave driven by gravity which propagates in the positive $x$-direction acts as a free common interface between the media. The current is such that it is zero at the flatbed but a negative constant, due to an assumption that surface winds blow in the negative $x$-direction, at the lid. We are concerned with the layers adjacent to the internal wave in which there exists a depth dependent current for which there is a greater underlying than overlying current. Both media are considered incompressible and having non-zero constant vorticities. The governing equations are written in canonical Hamiltonian form in terms of the variables, associated to the wave (in a presence of a constant current). The resultant equations of motion show that wave-current interaction is influenced only by the current profile in the 'strip' adjacent to the internal wave.