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Evolution equations for counterpropagating edge waves

Published online by Cambridge University Press:  26 April 2006

R. D. Pierce  and
E. Knobloch
R. D. Pierce
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA Present Address: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.
E. Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
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Abstract

Asymptotically exact evolution equations for counterpropagating shallow-water edge waves are derived. The structure of the equations depends only on the symmetries of the problem and on the fact that the group velocity of the edge waves is of order one. As a result the equations take the form of parametrically forced Davey–Stewartson equations with mean-field coupling. The calculations extend existing work on parametric excitation of edge waves by normally incident waves to arbitrary beach profiles with asymptotically constant depth, and include coupling to wave-generated mean longshore currents. Dissipation arises generically from radiation damping, but we also consider heuristically the effects of linear boundary-layer damping. Spatially modulated waves do not couple to the parametric forcing due to the non-locality of the evolution equations and are damped. Thus only spatially uniform wavetrains are expected as stable solutions. If linear dissipation is included the parametric coupling selects standing waves, but in the undamped case travelling wave states are possible. Both classes of solutions are examined for modulational instabilities, and stability conditions for the generic evolution equations are presented. However, modulational instability is found to be excluded in the shallow-water formulation through the effects of the mean flow. Explicit numerical results for two experimentally relevant beach profiles, exponentially decaying and piecewise linear, are presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 264 , 10 April 1994 , pp. 137 - 163
Copyright
© 1994 Cambridge University Press

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