Geometry of shallow-water dynamics with thermodynamics

TL;DR

This paper explores the geometric and Hamiltonian structure of the IL0PE model, a rotating shallow-water system with variable buoyancy, revealing its relation to classical rigid body dynamics and providing a detailed proof of its Poisson bracket properties.

Contribution

It demonstrates the geometric structure of the IL0PE model and connects it to Euler--Poincaré equations and Hamiltonian systems, extending the understanding of shallow-water models with thermodynamics.

Findings

The IL0PE model has a Lie--Poisson Hamiltonian structure.

The model equations can be derived from Morrison and Greene's system.

An explicit proof of the Jacobi identity for the Poisson bracket is provided.

Abstract

We review the geometric structure of the IL$^0$PE model, a rotating shallow-water model with variable buoyancy, thus sometimes called ``thermal'' shallow-water model. We start by discussing the Euler--Poincar\'e equations for rigid body dynamics and the generalized Hamiltonian structure of the system. We then reveal similar geometric structure for the IL$^0$PE. We show, in particular, that the model equations and its (Lie--Poisson) Hamiltonian structure can be deduced from Morrison and Greene's (1980) system upon ignoring the magnetic field ($\vec{\mathrm B} = 0$) and setting $U(\rho,s) = \frac{1}{2}\rho s$, where $\rho$ is mass density and $s$ is entropy per unit mass. These variables play the role of layer thickness ($h$) and buoyancy ($\t$) in the IL$^0$PE, respectively. Included in an appendix is an explicit proof of the Jacobi identity satisfied by the Poisson bracket of the system.