Resonant Langmuir-circulation–internal-wave interaction. Part 2. Langmuir circulation instability

A long-wavelength weakly nonlinear analysis is used to investigate the possibility for resonant energy exchange between low-mode internal waves and counter-rotating roll vortices known as Langmuir circulation. The analysis is based on a two-layer ocean model in which the Langmuir circulation is confined to the upper layer and counter-propagating internal waves travel along the sharp thermocline normal to the axes of the vortices. An asymptotically consistent description of the slow-time behaviour is obtained by making a WKBJ approximation to treat the comparatively high-frequency internal-wave reflections identified in Part 1. When the vortices and waves are modelled as linearly neutral modes, the resulting dynamics take the form of nonlinear oscillations. The theory suggests that Langmuir cells may transiently lose stability to standing internal-wave disturbances whose nodes are aligned with the cell downwelling zones. An exact solution of the Langmuir-circulation–standing-wave interaction is used to gain insight into the nonlinear instability mechanism. As in Part 1, the modification of the linear internal-wave dynamics by the Craik–Leibovich ‘vortex force’ is found to be crucial to the interaction.