On Townsend's rapid-distortion model of the turbulent-wind-wave problem

Abstract

Townsend's (1980) model of wind-to-wave energy transfer, which is based on a putative interpolation between an inner, viscoelastic approximation and an outer, rapid-distortion approximation and predicts an energy transfer that is substantially larger (by as much as a factor of three) than that predicted by Miles's (1957) quasi-laminar model, is revisited. It is shown that Townsend's interpolation effectively imposes a rapid-distortion approximation throughout the flow, rather than only in the outer domain, and that his asymptotic (far above the surface) solution implicitly omits one of the two admissible, linearly independent solutions of his perturbation equations. These flaws are repaired, and Townsend's dissipation function is modified to render the transport equation for the perturbation energy of the same form as those for the perturbation Reynolds stresses. The resulting wind-to-wave energy transfer is close to that predicted by Townsend's (1972) viscoelastic model and other models that incorporate the perturbation Reynolds stresses, but somewhat smaller than that predicted by the quasi-laminar model. We conclude that Townsend's (1980) predictions, although closer to observation than those of other models, rest on flawed analysis and numerical error.