Oscillatory and competing instabilities in a nonlinear model for Langmuir circulations

Abstract

A five-mode truncation of the full partial differential equations which describe Langmuir circulations is studied. Instabilities of different kinds are possible depending on the relative strengths of a stabilizing vertical density stratification and a destabilizing vertical Stokes drift gradient. For sufficiently small buoyancy, steady convection obtains, whereas for larger values of the buoyancy oscillatory convection obtains. In addition the competition can create multiple instabilities. Techniques from dynamical systems theory are used to derive equations which describe nonlinear oscillatory convection and the multiple bifurcation, whose character has been explored in the literature. The center manifold and normal form transformations are explained in a direct and elementary manner in order to be clear to readers encountering these methods for the first time. The problem treated may also be interpreted as one of double diffusion, but with boundary conditions not usually considered. The results are discussed in terms of the physics of Langmuir circulations.