Nonlinear interfacial waves in a constant-vorticity planar flow over variable depth

TL;DR

This paper derives an exact Lagrangian for internal waves in a two-fluid system with shear current and variable bottom, providing higher-order long-wave approximations in a simplified Hamiltonian framework.

Contribution

It introduces a compact exact Lagrangian formulation for internal waves with constant vorticity over variable depth, and develops higher-order asymptotic approximations.

Findings

Exact Lagrangian derived for internal waves with shear current.

Higher-order long-wave asymptotic approximations obtained.

Simplified Hamiltonian approach for different interface parametrizations.

Abstract

Exact Lagrangian in compact form is derived for planar internal waves in a two-fluid system with a relatively small density jump (the Boussinesq limit taking place in real oceanic conditions), in the presence of a background shear current of constant vorticity, and over arbitrary bottom profile. Long-wave asymptotic approximations of higher orders are derived from the exact Hamiltonian functional in a remarkably simple way, for two different parametrizations of the interface shape.