Asymptotics of surface waves over random bathymetry

This paper addresses the propagation of free surface water waves over a variable seabed in the long wavelength scaling regime. We consider the situation in which the bathymetry is given by a stationary random process which has a correlation length substantially shorter than the wavelength of the principal surface wave components. An asymptotic description shows that the water waves problem is modeled by an effective system of equations that is related to the KdV, however in a reference frame given in terms of random characteristic coordinates, and in addition with a random amplitude modulation and random scattered component. The resulting random processes are strongly correlated and have canonical limits due to the Donsker invariance principle. Our analysis is based on the Hamiltonian description of water waves and long wave perturbation theory and a new criterion for asymptotic expansions of partial differential equations with rapidly varying coefficients. In this paper we give a detailed analysis of the transformation to random characteristic coordinates and the asymptotic form of the resulting transformed partial differential equations. A companion paper (de Bouard, A., Craig, W., Díaz-Espinosa, O., Guyenne, P., Sulem, C., Long wave expansions for water waves over random topography, Nonlinearity 21 (2008), 2143–2178) analyses in detail the asymptotic behavior of the resulting expression for solutions, and their consistency with the derivation of the effective model equations.