Capillary rollers and bores

At any free surface at which the tangential stress τ ns vanishes there must be a surface vorticity ω = −2κq, where κ is the curvature and q the tangential velocity. In a surface wave on water, this condition produces a (Stokes) boundary layer with thickness of order δ = (2ν/σ) ½, where ν is the kinemetic viscosity and σ the radian frequency of the wave. To first order in the wave steepness parameter ak, the vorticity remains within the boundary layer, but at second order some escapes through the Stokes layer. The mean vorticity at the outer edge of the Stokes layer is of order 2 (ak) 2 σ, twice the mean vorticity generated at the free surface. These results are applied to steep capillary waves, particularly the parasitic capillaries often seen on the forward face of short gravity waves. Because of the high value of σ for the capillaries, the vorticity they generate is much larger than that generated by the gravity wave itself. Hence the capillaries contribute significantly to the vortex (roller) often found at the crest of short gravity waves, when capillaries are present. It is argued that the crest roller and the capillaries form a cooperative system, a ‘capillary roller’ in which each supports the other, with the aid of surface tension and viscosity. Energy is supplied by the gravity wave. A capillary roller is one instance of a more general phenomenon: a ‘capillary bore’, which is a noticeable feature of many disturbed water surfaces.