Multiscale method, Central extensions and a generalized Craik-Leibovich equation

Abstract: In this paper we develop perturbation theory on the reduced space of a principal $G-$bundle. This theory uses a multiscale method and is related to vibrodynamics. For a fast oscillating motion with the symmetry Lie group $G$, we prove that the averaged equation (i.e. the equation describing the slow motion) is the Euler equation on the dual of a certain central extension of the corresponding Lie algebra $\mathfrak g$. As an application of this theory we study the Craik--Leibovich (CL) equation in hydrodynamics. We show that CL equation can be regarded as the Euler equation on the dual of an appropriate central extension of the Lie algebra of divergence-free vector fields. From this geometric point of view, one can give a generalization of CL equation on any Riemannian manifold with boundary. For accuracy of the averaged equation, we prove that the difference between the solution of the averaged equation and the solution of the perturbed equation remains small (of order $\epsilon$) over a very long time interval (of order $1/{\epsilon^2}$). Combining the geometric structure of the generalized CL equation and the averaging theorem, we present a large class of adiabatic invariants for the perturbation model of the Langmuir circulation in the ocean.