On the Hamiltonian and Geometric structure of the Craik-Leibovich equation

Abstract: In this paper we show that the Craik-Leibovich (CL) equation in hydrodynamics is the Euler equation on the dual of a certain central extension of the Lie algebra of divergence-free vector fields. From this geometric viewpoint, one can give a generalization of CL equation on any Riemannian manifold with boundary. We also prove a stability theorem for 2-dimensional steady flows of the Craik-Leibovich equation.