Two-layer sharply stratified Euler fluids in three dimensions: a Hamiltonian setting

TL;DR

This paper develops a Hamiltonian framework for three-dimensional two-layer Euler fluids, deriving effective 2D models like the KBK-B and KP equations through reduction techniques.

Contribution

It introduces a Hamiltonian reduction method to derive 2D models from 3D Euler fluids, connecting them to well-known integrable equations.

Findings

Derived a Hamiltonian structure for the 2D interface model.

Connected the weakly nonlinear limit to the KBK-B model.

Showed unidirectionalization leads to the KP equation.

Abstract

Three-dimensional two-layer incompressible Euler fluids are studied from a Hamiltonian perspective. A natural Hamiltonian structure for the effective 2D model described by the interface-value of the field variables is obtained by means of a Hamiltonian reduction process from the 3D Poisson structure. The problem of expressing the fluid's energy in terms of the reduced variables is considered, and it is shown that in the weakly non linear approximation our procedure gives rise to a so-called 2D Kaup-Broer-Kupershmidt Boussinesq (KBK-B) model with ``critical" parameters. A model weakly dependent on one of the two horizontal directions is also discussed, whose unidirectionalization turns out to be the well-known Kadomtsev-Petviashvili (KP) equation. A Dirac-type reduction process of the Hamiltonian structure of the KBK-B model yields a natural Hamiltonian structure for KP qua 2+1-dimensional model.