Speed-up of Langmuir instability at large scales when the wave scattering is significant

We analytically consider the problem of the Langmuir instability. The instability describes emergence of transverse modulation of horizontal current with vertical shear, when it is superimposed on surface wave co-directional with the flow. Compared to the Craik–Leibovich scheme, we account for the scattering of surface wave on the modulated current, which is significant if the unperturbed wave possesses sufficient spatial coherence. We show that the modulation of the Stokes drift, caused by the interference of the scattered wave with the unperturbed one, leads to an acceleration of the instability development. This acceleration becomes substantial when the modulation period is large enough, so the dispersion law for the scattered wave is approximately satisfied. Specifically, the frequency correction caused by the change in the wavenumber due the modulation should be compensated by the frequency change imposed by the sheared flow. We trace how a partial loss of the wave coherence leads to a reduction in the acceleration. Our results are in qualitative agreement with the available data from wave-resolving numerical simulations.