 07228 Anne de Bouard,Walter Craig, Oliver D\'{\i}azEspinosa, Philippe Guyenne,Catherine Sulem
 Long wave expansions for water waves over random topography
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Oct 1, 07

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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 $\beta(x, \omega)$ 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 $\beta$.
This work is a reappraisal of the paper by Rosales \& Papanicolaou~\cite{RP83}
and its extension to general stationary mixing processes
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