Hamiltonian dynamics of wave mean-flow interaction

Darryl D. Holm, Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory

At all wavenumbers, there is a separation in time scales between internal waves and the mean flow of an inviscid stratified rotating fluid. We discuss new equations for wave, mean-flow interaction, obtained by applying asymptotic expansions and phase averaging in Hamilton's principle for the Euler dynamics of such a fluid. By construction, these equations possess a Kelvin circulation theorem, conserve a potential vorticity, and are Lie-Poisson Hamiltonian dynamical systems in the Eulerian variables. We will also discuss the relation of these results to Charney-Drazin non-acceleration theorems, Whitham averaging and Lagrangian mean flow equations such as those of Craik and Lebovich.