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Nonlinear dynamics of vorticity waves in the coastal zone

Published online by Cambridge University Press:  26 April 2006

Victor I. Shrira  and
Vyacheslav V. Voronovich
Victor I. Shrira
Affiliation:
P. P. Shirshov Institute of Oceanology Russian Academy of Sciences, 23 Krasikov str., 117218 Moscow, Russia
Vyacheslav V. Voronovich
Affiliation:
P. P. Shirshov Institute of Oceanology Russian Academy of Sciences, 23 Krasikov str., 117218 Moscow, Russia
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Abstract

Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 326 , 10 November 1996 , pp. 181 - 203
Copyright
© 1996 Cambridge University Press

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