Euler-Poincar\'{e} equations for anelastic fluid flows

TL;DR

This paper demonstrates that the ideal anelastic fluid equations can be derived as Euler-Poincaré equations, establishing a solid mathematical framework that preserves key physical conservation laws and introduces a mean flow model.

Contribution

It shows the anelastic equations follow from a constrained Hamilton's principle as Euler-Poincaré equations and introduces a mean flow model maintaining these properties.

Findings

Derivation of anelastic equations as Euler-Poincaré equations

Establishment of conservation laws and Hamiltonian structure

Introduction of a mean flow model preserving mathematical properties

Abstract

We show that the ideal (nondissipative) form of the dynamical equations for the Lipps-Hemler formulation of the anelastic fluid model follow as Euler-Poincar\'{e} equations, obtained from a constrained Hamilton's principle expressed in the Eulerian fluid description. This establishes the mathematical framework for the following properties of these anelastic equations: the Kelvin-Noether circulation theorem, conservation of potential vorticity on fluid parcels, and the Lie-Poisson Hamiltonian formulation possessing conserved Casimirs, conserved domain integrated energy and an associated variational principle satisfied by the equilibrium solutions. We then introduce a modified set of anelastic equations that represent the mean anelastic motion, averaged over subgrid scale rapid fluctuations, while preserving the mathematical properties of the Euler-Poincar\'{e} framework.