An asymptotic theory for the turbulent flow over a progressive water wave

The turbulent flow over a progressive water wave is studied using an eddy viscosity model. The governing equations are treated asymptotically for the case ε [Lt ] 1, where ε is the square root of a characteristic drag coefficient. A calculation of the phase shift between the wave-induced pressure perturbation and the surface elevation shows that the phase shift is induced by a term in the gradient of the Reynolds stress. Growth rates are determined, and are shown to agree well with observations for the most rapidly amplifying waves. However, the present model and previous turbulence calculations are found to provide significantly lower growth rates than those measured by Snyder et al. (1981) for waves with phase velocities comparable to the wind speed.