The Euler-Poincare Equations in Geophysical Fluid Dynamics

TL;DR

This paper develops a theoretical framework for Euler-Poincaré equations in geophysical fluid dynamics, deriving new variational principles and conservation laws applicable to standard GFD models.

Contribution

It introduces a variational derivation of Euler-Poincaré equations for GFD models and establishes associated Kelvin-Noether theorems and conservation laws.

Findings

Derived Euler-Poincaré equations for GFD models

Established Kelvin-Noether theorems for these models

Unified various GFD models within this theoretical framework

Abstract

Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: (1) Euler-Poincar\'e equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; (2) an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincar\'{e} form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws.