Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves

A train of uniform two-dimensional gravity waves in deep water is known to be unstable to certain sideband disturbances. If the time of propagation is sufficiently long for the fourth-order terms to be important, the sidebands may grow at unequal rates, resulting in a downward shift of peak frequency. But this shift is only a temporary phase of a recurrent evolution process. Recent work by us (Hara & Mei 1991) has shown that wind and dissipation can help maintain this downshift at large time. In this paper we examine a similar two-dimensional problem for capillary-gravity waves. The basic flow in air and water is assumed to be steady, horizontally uniform and turbulent; the wave-induced flow in both media is assumed to be laminar. Evolution equations are deduced with wind and dissipation included in such a way that their influence is comparable to the asymmetric spectral evolution. After finding the initial growth rates of unstable sidebands, the nonlinear development of modulational instability is examined by integrating the evolution equations numerically. Computed results show that persistent downshift of frequency can happen for relatively long waves, but upshift occurs for very short waves.