Asymptotically Reduced Equations for Weakly Three-Dimensional Langmuir Turbulence

Ocean observations, numerical simulations and theoretical considerations all suggest that Langmuir circulation (LC) is characterized by counter-rotating vortical structures elongated in the wind direction. We identify the source of this downwind invariance by exploring the Craik--Leibovich (CL) equations modeling LC in the physically-relevant large $Re_s$ limit; here, the Stokes Reynolds number $Re_s$ is based on the Stokes drift velocity of the surface waves and on the depth of the mixed layer. Inspection of the CL equations reveals that the CL vortex force dominates the flow physics when $Re_s\gg 1$, and vortices aligned with the wind direction are preferred. Using multiscale asymptotics, we leverage this limit to derive a reduced set of PDEs governing weakly 3D Langmuir turbulence. Linear and secondary (i.e. nonlinear) stability studies show that the reduced equations economically capture the key 3D instabilities.