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Water waves over a random bottom

Published online by Cambridge University Press:  02 November 2009

Department of Mathematics, McMaster University, Hamilton, ON L8S 4K1, Canada
Department of Mathematical Sciences, University of Delaware, Newark DE 19716, USA
Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
Email address for correspondence:


This paper gives a new derivation and an analysis of long-wave model equations for the dynamics of the free surface of a body of water which has random bathymetry. This is a problem of hydrodynamical significance to coastal regions and to global-scale propagation of tsunamis, for which there may be imperfect knowledge of the detailed topography of the bottom. The surface motion is assumed to be in a long-wavelength dynamical regime, while the bottom of the fluid region is given by a stationary random process whose realizations vary over short length scales and are decorrelated on the longer principal length scale of the surface waves. Our basic conclusions are that coherent solutions propagating over a random bottom maintain basic properties of their structure over long distances, but however, the effect of the random bottom introduces uncertainty in the location of the solution profile and modifies the amplitude by random factors. It also gives rise to a random scattered component of the solution, but this does not result in the dispersion of the principal component of the solution, at least over length and time scales considered in this regime. We illustrate these results with numerical simulations.

The mathematical question is one of homogenization theory in the long-wave scaling regime, for which our work is a reappraisal of the paper of Rosales & Papanicolaou (Stud. Appl. Math., vol. 68, 1983, pp. 89–102). In particular, we derive appropriate Boussinesq and Korteweg–deVries type equations with random coefficients which describe the free-surface evolution in this regime. The derivation is performed from the point of view of perturbation theory for Hamiltonian partial differential equations with a small parameter, with a subsequent analysis of the random effects in the resulting solutions. In the analysis, we highlight the distinction between the effective equations for a fixed typical realization, for which there are coherent solitary-wave solutions, and their ensemble average, which may exhibit diffusive effects. Our results extend the prior analysis to the case of non-zero variance σ2β > 0, and furthermore the analysis identifies the canonical limit random process as a white noise with covariance σβ2δ(XX′) and quantifies the variations in phase and amplitude of the principal and scattered components of solutions. We find that the random topography can give rise to an additional linear term in the KdV limit equations, which depends upon a skew property of the random process and whose sign affects the stability of solutions. Finally we generalize this analysis to the case in which the bottom has large-scale deterministic variations on which are superposed random fluctuations with slowly varying statistical properties.

Copyright © Cambridge University Press 2009

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