Hamiltonian aspects of 3-layer stratified fluids

TL;DR

This paper analyzes the Hamiltonian structure of 3-layer stratified ideal fluids, exploring their properties, limits, and symmetries, and extends some results to the n-layer case with implications for momentum conservation.

Contribution

It introduces a Hamiltonian formulation for the 3-layer fluid model, generalizes to n-layers, and discusses implications for momentum conservation and special solutions.

Findings

The long-wave limit forms a system without Riemann invariants.

The Hamiltonian structure is derived from the full 2D equations.

Pressure imbalances affect momentum conservation in layered fluids.

Abstract

The theory of 3-layer density stratified ideal fluids is examined with a view towards its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Virissimo and Milewski are also discussed.