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On vorticity waves propagating in a waveguide formed by two critical layers

Published online by Cambridge University Press:  15 June 2009

OLEG DERZHO  and
ROGER GRIMSHAW
OLEG DERZHO
Affiliation:
Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, NL A1B 3X7, Canada Institute of Thermophysics, Russian Academy of Sciences, Novosibirsk 630090, Russia
ROGER GRIMSHAW*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Leics LE11 3TU, UK
*
†E-mail address for correspondence: R.H.J.Grimshaw@lboro.ac.uk
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Abstract

A theoretical model for long vorticity waves propagating on a background shear flow is developed. The basic flow is assumed to be confined between two critical layers, respectively, located near the lower and upper rigid boundaries. In these critical layers even small disturbances will break, and eventually a thin zone of mixed fluid will appear. We derive a nonlinear evolution equation for the amplitude of a wave-like disturbance in this configuration, based on the assumption that the critical layers are replaced by thin recirculation zones attached to the lower and upper rigid boundaries, where the flow is very weak. The dispersive and time-evolution terms in this equation are typical for Korteweg–de Vries theory, but the nonlinear term is more complicated. It comprises nonlinearity associated with the shear across the waveguide, and the nonlinearity due to the flow over the recirculation zones. The coefficient of the quadratic nonlinear term may change sign, depending on the presence or otherwise of recirculation zones at the upper or lower boundary of the waveguide. We then seek steady travelling wave solutions, and show that there are no such steady solutions if the waveguide contains no density stratification. However, steady solutions including solitary waves and bores can exist if the fluid between the critical layers is weakly density stratified.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 629 , 25 June 2009 , pp. 161 - 171
Copyright
Copyright © Cambridge University Press 2009

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