Short-Scale Waves on Wind-Driven Water (`Cat's Paws')

Wind-generated water waves are considered from the viewpoint of linear stability theory applied to the flow of one fluid, of density ρ_1 and viscosity μ_1, over another of density ρ_2 and viscosity μ_2. The velocity profiles in the fluids satisfy the full viscous interface conditions, but otherwise they can be of a general form. The short waves, which are found to arise at increased Reynolds numbers, travel almost at the interface velocity. Most attention is given to the range (μ_1/μ_2)^2 < (ρ_1/ρ_2) < 1 of viscosity and density ratios, which includes the wind-water combination. The short waves then are driven predominantly by a combination of the local shear stresses, the viscous forces and the surface tension, whereas gravity and the local curvature and other properties of the general velocity profiles play relatively little part. Although the corresponding main fluid motions near the interface are dominated by viscous dissipation, it so happens that the pressures induced by these viscous motions diminish at the interface and that allows inertial forces to exert a controlling influence there, bringing in the shear stresses above. The predictions agree fairly well with calculations for two-fluid systems at moderate Reynolds numbers, and the wave features seem to tie in with those observed in `cat's paws' on wind-driven stretches of water. For viscosity and density ratios outside of the range noted above, there is a change in structure that pulls in inertial forces more directly and suppresses the surface-tension effects.