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Water waves over a variable bottom: a non-local formulation and conformal mappings

Published online by Cambridge University Press:  24 February 2012

A. S. Fokas  and
A. Nachbin
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK Research Center of Mathematics of the Academy of Athens, Athens, 11527, Greece
A. Nachbin*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina, 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil
*
†Email address for correspondence: nachbin@impa.br
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Abstract

In Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, pp. 313–343) a novel formulation was proposed for water waves in three space dimensions. In the flat-bottom case, this formulation consists of the Bernoulli equation, as well as of a non-local equation. The variable-bottom case, which now involves two non-local equations, was outlined but not explored in the above paper. Here, the variable-bottom formulation is addressed in more detail. First, it is shown that in the weakly nonlinear, weakly dispersive regime, the above system of three equations can be reduced to a system of two equations. Second, by combining the novel non-local formulation of the above authors with conformal mappings, it is shown that in the two-dimensional case, it is possible to obtain a system of two equations without any asymptotic approximations. Furthermore, for the weakly nonlinear, weakly dispersive regime, the nonlinear equations are simpler than the equations obtained without conformal mappings, since they contain lower order derivatives for the terms involving the bottom variable.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 695 , 25 March 2012 , pp. 288 - 309
Copyright
Copyright © Cambridge University Press 2012

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