Nonlinear steady states to Langmuir circulation in shallow layers: an asymptotic study

摘要

The nonlinear steady states of perturbation equations describing the instability of wavy shear flows to counter-rotating vortical structures aligned with the flow in shallow water layers is considered. The structures are described by the Craik-Leibovich equations and are known as Langmuir circulation; they arise through an instability that requires the presence of shear U' and differential drift D' of the same sign provided a threshold Rayleigh number is exceeded. Of specific interest here is the aspect ratio of the Langmuir circulation and how that ratio is affected by nonlinearities when the layer is shallow, as in coastal waters and estuaries. For context it is known from observation that the aspect ratio (width of two cells to depth) is two to three in deep water, whereas in shallow waters it can range up to ten. Accordingly, while the ratio for deep water is well predicted by linear theory, the ratio for shallow water is not, which explains why nonlinearities are of interest. Present always, but of key importance in effecting a preferred spacing with nonzero wavenumber l in shallow water Langmuir circulation, is an extra stress induced by the perturbed motion at the free surface. This stress is reflected in the boundary conditions. Moreover, since it is most influential in the limit l -> 0, it is instructive to expose that influence through a small-l asymptotic approximation. It is found that nonlinearities ensure supercritical stability and that the critical wavenumber at onset to instability is significantly less than its linear counterpart. This results from a subcritical bifurcation due to symmetry breaking of the governing equations. The precise level of reduction is affected by the distributions of U' and D', but herein it ranges from 50 to 70%. This means that nonlinearities act to effect aspect ratios up to twice those given by linear theory and which are in good agreement with observation in shallow coastal waters. Finally, the extra stress at the free surface is found to play no role in the ratio of linear to nonlinear spacing but, as in the linear case, acts to ensure the spanwise wavenumber at onset to instability is nonzero.