Langmuir circulation that has been disturbed by a perturbed source in a horizontal Couette flow with a vertical density gradient (i.e., the effective Rayleigh number R) and horizontal Couette flow is investigated. The evolution of instability developing in the presence of a vertical density gradient influenced by the disturbance at various depths and the horizontal Couette flow is considered near the onset of convection under a moderate rate of shear. We use velocity as the basic variable and solve the pressure Poisson equation in terms of the associated Green function. Growth competition between the longitudinal vortices (Lv) and the transverse vortices (Tv), whose axes are respectively in the direction parallel to and perpendicular to the Couette flow, is investigated by the weakly nonlinear analysis of coupled-mode equations. The results show that the Tv mode is characterized in some range of the effective Rayleigh number, and that the stability is dominated by the Lv mode in the system.
