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

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.