Capillary–gravity wave transport over spatially random drift

Abstract

We derive transport equations for the propagation of water wave action in the presence of a static, spatially random surface drift. Using the Wigner distribution W(x, k, t) to represent the envelope of the wave amplitude at position x contained in waves with wavevector k, we describe surface wave transport over static flows consisting of two length scales; one varying smoothly on the wavelength scale, the other varying on a scale comparable to the wavelength. 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. The Doppler interaction in the presence of slowly varying drift modifies the scattering processes and provides a mechanism for coupling long wavelengths with short wavelengths. Conservation of wave action (CWA), typically derived for slowly varying drift, is extended to systems with rapidly varying flow. At yet larger propagation distances, we derive from the transport equations, an equation for wave energy diffusion. The associated diffusion constant is also expressed in terms of the surface flow correlations. Our results provide a formal set of equations to analyse transport of surface wave action, intensity, energy, and wave scattering as a function of the slowly varying drifts and the correlation functions of the random, highly oscillatory surface flows.

Wave modelling – The state of the art

“…In the form given by Magne and Ardhuin (2006), it also includes interactions between two waves and one Fourier component of the current or surface elevation that arises from the adjustment of a mean current to the topography. This scattering theory over the current fluctuations should be consistent with the theory of Bal and Chou (2002) for the scattering of gravity-capillary waves over depth-uniform and irrotational current fluctuations.…”

Scattering of surface gravity waves by bottom topography with a current

“…The geometry of the resonant wavenumbers is modified in that case, with with incident and reflected waves having the same absolute frequency, but different wavenumber magnitudes if incident and reflected waves propagate at different angles relative to the current direction. Kirby (1988) also considered the short scale fluctuations of the current, due to the sinusoidal bottom, that may be interpreted as a separate scattering mechanism, and generalized further to any irrotational current fluctuations, leading to results similar to those obtained for gravity-capillary waves by Bal & Chou (2002).…”

“…The geometry of the resonant wavenumbers is modified in that case, with incident and reflected waves having the same absolute frequency, but different wavenumber magnitudes. Kirby (1988) also considered the short-scale fluctuations of the current, due to the sinusoidal bottom, that may be interpreted as an extra source of scattering, identical to the scattering of gravity and gravity-capillary waves over irrotational current fluctuations studied by Watson & West (1975) and Bal & Chou (2002). It should be noted that a more general theory for deep-water waves over any current was given by Rayevskiy (1983) and Fabrikant & Raevsky (1994).…”

Kinetic Modeling of Multiple Scattering of Acoustic Waves in Randomly Heterogeneous Flows

“…This corresponds to the so-called weak coupling limit as defined in the dedicated literature (see, for example, [21]), whereby an explicit separation of scales can be invoked. The analysis developed in [1,3,4,5,9,10,20,28,48,49] is based on the use of a Wigner transform of the wave field, the high-frequency, nonnegative limit of which characterizes its angularly resolved energy density. It can be made mathematically rigorous as in [1,25,38], ignoring, however, the influence of random inhomogeneities, except for some particular situations [19,39].…”

“…The developments of [20] rely on the same mathematical tools, but they consider time-independent ambient quantities and the first-order linearized Euler equations rather than the second-order convected wave equation addressed below. The developments of [4] consider a time-dependent flow velocity as done here. Last, the analysis in [9] considers a constant mean velocity of the ambient flow and acoustic waves in a forward-scattering regime of propagation.…”