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

JOURNAL OF GEOMETRY AND PHYSICS

卷:116

Article

Yang, Cheng^1,2 

[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China

[2] Univ Toronto, Dept Math, Toronto, ON M5S 2E4, Canada

关键词: Multiscale method;    Central extensions;    Craik-Leibovich equation;    Averaging theorem;    Adiabatic invariants;   

DOI  :  10.1016/j.geomphys.2017.02.004

来源: Elsevier

【 摘 要 】

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 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. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free