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

TL;DR

This paper reveals the geometric structure of the Craik-Leibovich equation as an Euler equation on a central extension of divergence-free vector fields, generalizes it to Riemannian manifolds, and proves a stability theorem for 2D steady flows.

Contribution

It demonstrates the geometric interpretation of the Craik-Leibovich equation and extends its applicability to Riemannian manifolds, along with a stability analysis.

Findings

Craik-Leibovich equation is an Euler equation on a central extension

Generalization to Riemannian manifolds with boundary

Stability theorem for 2D steady flows

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.