On the intermediate long wave propagation for internal waves in the presence of currents

TL;DR

This paper develops a Hamiltonian-based model for internal wave propagation in the presence of currents, deriving an integrable Intermediate Long Wave Equation and its limits, which enhances understanding of wave dynamics in stratified fluids.

Contribution

It introduces a Hamiltonian formulation for internal waves with currents and derives an integrable ILWE, connecting it to Benjamin-Ono and KdV equations.

Findings

Derivation of an integrable ILWE from physical assumptions.

Identification of limits leading to Benjamin-Ono and KdV equations.

Model applicable to stratified fluids with deep lower layers.

Abstract

A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface approximation is assumed. The fluids are incompressible and inviscid. The model equations are obtained from the Hamiltonian formulation of the dynamics in the presence of a depth-varying current. It is shown that an appropriate scaling leads to the integrable Intermediate Long Wave Equation (ILWE). Two limits of the ILWE leading to the integrable Benjamin-Ono and KdV equations are presented as well.