HOPF BIFURCATIONS IN LANGMUIR CIRCULATIONS

The equations governing two-dimensional Langmuir circulations in a continuously stratified layer of fluid possess O(2) symmetry when laterally periodic boundary conditions are applied. In one limit, which is among those treated here, the mathematical problem is strictly analogous to a double-diffusive problem. Steady, oscillatory and multiple oscillatory states are all possible. The method of multiple scales is used to obtain evolution equations for the amplitudes of oscillatory convection when a single wavenumber is destabilized, and when two wavenumbers are simultaneously destabilized. In the process, this paper provides the first example of the derivation of amplitude equations for a double-Hopf bifurcation with symmetry in a fluid mechanical problem. Parallel numerical simulations of the full partial differential equations are carried out, and quantitative comparisons are made between the two methods, both when periodic boundary conditions are enforced, and when the more restrictive flux-free side-wall boundary conditions are enforced.