Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach

TL;DR

This paper develops a Hamiltonian framework for two-layer stratified flows beyond the Boussinesq approximation, revealing integrability properties and deriving solutions for the resulting deformed nonlinear equations.

Contribution

It introduces a Hamiltonian reduction approach to derive and analyze non-Boussinesq two-layer flow equations, demonstrating their integrability and constructing explicit solutions.

Findings

The deformed system retains an infinite number of conserved quantities.

Riemann invariants for the deformed equations are explicitly constructed.

Some local solutions are obtained using hodograph-like methods.

Abstract

The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids' inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, and thence, in a non-trivial way, to the dispersionless non-linear Schr\"odinger equation. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, it is shown that at first order the deformed system possesses an infinite sequence of constants of the motion, thus casting this system within the framework of completely integrable equations. The Riemann invariants of the deformed model are then constructed, and some local solutions found by hodograph-like formulae for completely integrable systems are obtained.