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Vorticity generated by pure capillary waves

Capillary waves, like other surface waves on water, generate a rectified, or time-averaged, vorticity field extending beyond the oscillatory (Stokes) layer at the surface. This vorticity field ω is particularly interesting in relation to the parasitic capillary waves found on the forward slopes of steep gravity waves. Longuet-Higgins (1992) suggested that the rectified vorticity from the parasitic capillaries might contribute significantly to the vorticity observed beneath the crest of the gravity wave. The basic calculations by Longuet-Higgins (1992) were only of the horizontally averaged values of ω. Here we extend his theory by calculating, for pure capillary waves, the space variation of ω, to second order in the steepness of the capillary waves. Thus, the vorticity, and hence velocity, fields are calculated in the oscillatory Stokes layer and just beyond it, to the second order. Good agreement is found both with numerical simulations and with experimental measurements.