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

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 ϵ) over a very long time interval (of order 1/ϵ2). 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.