On the spacing of Langmuir circulation in strong shear

Abstract

The inviscid instability of $O(\epsilon)$ two-dimensional free-surface gravity waves propagating along an $O(1)$ parallel shear flow is considered. The modes of instability involve spanwise-periodic longitudinal vortices resembling oceanic Langmuir circulation. Here, not only are wave-induced mean effects important but also wave modulation, caused by velocity anomalies which develop in the streamwise direction. The former are described by a generalized Lagrangian-mean formulation and the latter by a modified Rayleigh equation. Since both effects are essential, the instability may be called ‘generalized’ Craik–Leibovich (CLg). Of specific interest is whether spanwise distortion of the wave field, both at the free surface and in the interior, acts to enhance or inhibit instability to longitudinal vortices. Also of interest is whether the instability gives rise to a preferred spacing for the vortices and whether that spacing concurs well or poorly with experiment. The layer depth is varied from much less than the e-folding depth of the $O(\epsilon)$ wave motion to infinity. Relative to an identical shear flow with rigid though wavy top boundary, it is found, inter alia, that wave modulation acts in concert with the free surface, at some wavenumbers, to increase the maximum growth rate of the instability. Indeed, two preferred spanwise spacings occur, one which gives rise to longitudinal vortices through a convective oscillatory bifurcation and a second, at higher wavenumber and growth rate, through a stationary bifurcation. The preferred spacings set by the stationary bifurcation concur well with those observed in laboratory experiments, with the implication that the instability acting in the experiments is very likely to be CLg.