Capillary–gravity wave transport over spatially random drift

We derive transport equations for the propagation of water wave action in the presence of subsurface random flows. Using the Wigner distribution W(x, k, t) to represent the envelope of the wave amplitude at position x, time t contained in high frequency waves with wave vector k/ε (where ε is a small parameter compared to a characteristic distance of propagation), we describe surface wave transport over flows consisting of two length scales; one varying slowly on the wavelength scale, the other varying on a scale comparable to the wavelength. Both static underlying flows and time-varying underlying flows are considered. The spatially rapidly varying but weak surface flows augment the characteristic equations with scattering terms that are explicit functions of the correlations of the random surface currents. These scattering terms depend parametrically on the magnitudes and directions of the smoothly varying drift and are shown to give rise to a Doppler-coupled scattering mechanism. Conservation of wave action (CWA), typically derived for drift varying over long distances, is extended to systems with flow that varies on small length scales of order the surface wavelength. Our results provide a formal set of equations to analyze transport of surface wave action, intensity, energy, and wave scattering as a function of the smoothly varying drifts and the correlation functions of the random, highly oscillating surface flows. © 2002 Elsevier Science B.V. All rights reserved.