Abstract

The Craik-Leibovich Equation (CLE) describes the Langmuir circulations in the upper layer of the oceans, lakes, etc. In this lecture we consider CLE and the related notions of drifts and pseudo-diffusion. In our approach, CLE describes general vortex dynamics of oscillating flows, not only the Langmuir circulations. A number of new results on CLE are presented. An important elements of our presentation is the systematic deriving of boundary conditions, which represents a more difficult tusk than the deriving of averaged equations. We also consider a 'linearized’ version of CLE , which is different from that obtained by a straightforward linearization of an averaged equation. A possible tree-timing procedure for the generalisation of CLE is proposed. The effects of viscosity and density stratification has been additionally re-examined. We also discuss two generalisations of CLE . First one is a Magneto-Hydro-Dynamic version of CLE . It may have relation to the MHD dy namo and to the forming of various flows and phenomena in stars, galaxies etc. Second generalisation deals with a version of CLE for compressible fluid, where similar to CLE equations appears due to oscillations caused be acoustic waves.

Mathematically, our consideration is based on a unified viewpoint. We use the two-timing method, Eulerian averaging, and the concept of distinguished limit. Such a consideration emphasises the generality, simplicity, and the rigour in all our derivations. We do not accept any additional assumption and suggestions except the presence of small parameters and the applicability of rigorous asymptotic procedure. Our approach allows us to obtain the classical results much simpler than it has been done before, and hence a noticeable progress in the obtaining of new results can be achieved.

Key Words: Craik-Leibovich Equation, Lamgmuir Circulations, Two-Timing Method, Distinguished Limits, Drift, Pseudo-Diffusion.