Vorticity and waves: geometry of phase-space and the problem of normal variables

Abstract

The problem of finding Hamiltonian variables free of constraints imposed by an infinity of conservation laws is studied for two wave-vortex systems: non-linear Rossby waves and non-linear internal gravity waves in the Boussinesq approximation. The structure of their phase-spaces is displayed and a perturbative relation between old and new variables is established. The canonical variables of Zakharov and Piterbarg for Rossby waves arise as a particular case of this construction. Their domain of validity is further specified and the role of flow topology in establishing of a Hamiltonian picture is discussed.