Shear instability in spilling breakers

Abstract
High-speed photographs of a gently breaking water wave have shown the particular instability of the wave crest predicted by Longuet-Higgins et al. (J. Fluid Mech. 259, 333-344 (1994)) and Longuet-Higgins & Cleaver (J. Fluid Mech. 258, 115-129 (1994)), with a `bulge' on the forward face of the wave and the generation of parasitic capillaries ahead of the instability, emanating from the `toe' of the bulge. The photographs also show the unexpected occurence of longer (type 2) capillary waves above the toe of the bulge. In this paper it is shown that the type 2 capillaries are probably shear-flow instabilities arising from the vorticity shed by type 1 (parasitic) capillaries. At large amplitudes the initially linear shear instabilities must break up into discrete vortices. By formulating a simple model of the shear instabilities as vortex waves it is shown that the wavelength of the type 2 capillaries should be about 4 times the wavelength of the type 1 capillaries, a result in agreement with observation. When the flow separates at the toe of the bulge, the use of Prandtl's rule to estimate the strength of the vortices confirms the above relation. At larger lengthscales when surface tension is negligible, the argument leads to a simple rule for predicting the wavelength of the initial surface roughness of a spilling breaker.

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