Asymptotic Models for Internal Waves

TL;DR

This paper systematically derives and analyzes asymptotic models for internal wave propagation at the interface of two immiscible fluids, establishing their consistency with the full Euler equations across various scaling regimes.

Contribution

It introduces a unified framework for deriving classical and new asymptotic models for internal waves, including rigorous consistency proofs.

Findings

Derivation of a large class of asymptotic models for internal waves.

Identification of new model systems not previously documented.

Validation of models' consistency with full Euler equations.

Abstract

We derived here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on $\R^d$, $d=1,2$, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators with respect to suitable small parameters that depend variously on the amplitude, wave-lengths and depth ratio of the two layers. We rigorously derive classical models and also some model systems that appear to be new. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.