Nonlinear interactions of steady and oscillatory rolls in two dimensional langmuir circulations

Abstract

In this paper the linear and nonlinear stability of two dimensional Langmuir circulations in a fluid layer, stratified by vertical thermal and salinity gradients, is investigated. Stability curves for the onset of steady and oscillatory convection are determined and the point of intersection between these two curves is identified as a codimension two Hopf-zero bifurcation. The amplitude evolution equations describing this bifurcation point are then derived using the method of multiple scales and a numerical procedure for calculating the coefficients of the nonlinear terms is implemented. The results of unfolding this bifurcation are then presented for the case when the convection cells have a square cross-section.

Key Words:

• Linear stability
• multiple bifurcations
• O(2) symmetry
• multiple time scales
• modulated waves