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

TL;DR

This paper develops a multiscale perturbation theory on principal G-bundles, showing that averaged equations for fast oscillations are Euler equations on central extensions of Lie algebras, with applications to hydrodynamics and ocean circulation models.

Contribution

It introduces a geometric framework linking multiscale perturbation theory to central extensions of Lie algebras, generalizing the Craik-Leibovich equation on Riemannian manifolds.

Findings

Averaged equations are Euler equations on central extensions of Lie algebras.

The difference between solutions of averaged and perturbed equations remains small over long times.

The framework provides adiabatic invariants for ocean circulation models.

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.