Long wave expansions for water waves over random topography

Abstract

In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process β(x, ω) whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. This is a question of homogenization theory in the scaling regime for the Boussinesq and KdV equations.The analysis is performed from the point of view of perturbation theory for Hamiltonian PDEs with a small parameter, in the context of which we perform a careful analysis of the distributional convergence of stationary mixing random processes. We show in particular that the problem does not fully homogenize, and that the random effects are as important as dispersive and nonlinear phenomena in the scaling regime that is studied. Our principal result is the derivation of effective equations for surface water waves in the long wave small amplitude regime, and a consistency analysis of these equations, which are not necessarily Hamiltonian PDEs. In this analysis we compute the effects of random modulation of solutions, and give an explicit expression for the scattered component of the solution due to waves interacting with the random bottom. We show that the resulting influence of the random topography is expressed in terms of a canonical process, which is equivalent to a white noise through Donsker's invariance principle, with one free parameter being the variance of the random process β. This work is a reappraisal of the paper by Rosales & Papanicolaou [24] and its extension to general stationary mixing processes.2000 Mathematics Subject Classification. 76B15, 35Q53, 76M50, 60F17.

Numerical simulation of three-dimensional nonlinear water waves

“…This formulation of the water wave problem is convenient for the numerical simulation of three-dimensional nonlinear waves as presented in this paper, as well as in a number of other settings. These include numerical and asymptotic studies of surface waves over a rough bottom [15,32,7], and waves at the interface between immiscible fluids [14]. It is also understood that overturning waves (i.e.…”

“…We assume periodic boundary conditions in x and use a pseudospectral method [8] for spatial discretization of the DNO as well as the equations of motion (6), (7). This is a natural choice for the computation of G since each term in its Taylor series expansion (11) consists of concatenations of Fourier multipliers with powers of g. More specifically, both functions g and n are expanded in truncated Fourier series…”

“…To this aim, we first separate the linear and nonlinear parts in (6) and (7). Defining u ¼ ðg; nÞ > , the equations of motion can be written in the form…”

Asymptotics of surface waves over random bathymetry

“…In this paper we present recent results concerning the asymptotic description of surface waves in the long wave regime over a variable bottom, extending and elaborating upon [3]. The underlying assumption is that the bottom topography is modeled by a random process varying on short spatial scales.…”

Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves

“…These systems are readily computed using pseudospectral methods. Some examples of applications of this approach are [15], [16], [18], [21], [30].…”