Hamiltonian reductions, scalings, and effective wave models in stratified fluids

TL;DR

This paper uses Hamiltonian reduction techniques to derive and analyze effective wave models in stratified fluids, connecting various asymptotic models through scaling limits and providing a canonical Hamiltonian framework.

Contribution

It introduces a Hamiltonian reduction approach to derive effective wave models in stratified fluids, including the SGN and CC equations, with a focus on their canonical structures and scaling limits.

Findings

Derived the stratified effective model via Hamiltonian reduction.

Recovered the SGN equations through a double scaling limit.

Established the Hamiltonian structure of the CC equations.

Abstract

We apply Poisson reduction techniques to describe asymptotic fully nonlinear models of fluid wave motion in the Hamiltonian setting. We start by considering Zakharov and Benjamin Hamiltonian settings for a stably stratified $2D$ Euler fluid. We use a Marsden-Ratiu reduction scheme for sharply stratified fluids to obtain a canonical formulation of the stratified effective model in one space variable. The long-wave Serre-Green Naghdi (SGN) equations is then recovered by means of a suitable double scaling limit in the Hamiltonian function. We also consider the opposite double-scaling limit, which leads to a local model in the "large-lower layer" regime. Furthermore, applying the previous results on the canonical structure of the SGN equations, we provide the Miyata-Choi Camassa (CC) equations for fully non-linear waves in sharply stratified fluids with a natural Hamiltonian structure. We also study the reduced Hamiltonian system obtained taking the natural constraints of the CC equations into account. To this end, we perform a Dirac-type reduction on a suitable constrained submanifold of fluid field configurations.