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Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

Published online by Cambridge University Press:  08 November 2013

A. M. Kamchatnov ,
Y.-H. Kuo ,
T.-C. Lin ,
T.-L. Horng ,
S.-C. Gou ,
R. Clift ,
G. A. El  and
R. H. J. Grimshaw
A. M. Kamchatnov
Affiliation:
Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 142190, Russia
Y.-H. Kuo
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan
T.-C. Lin
Affiliation:
Institute of Applied Mathematical Sciences, National Center of Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan
T.-L. Horng
Affiliation:
Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan
S.-C. Gou
Affiliation:
Department of Physics, National Changhua University of Education, Changhua 50058, Taiwan
R. Clift
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
G. A. El*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
R. H. J. Grimshaw
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
*
†Email address for correspondence: G.El@lboro.ac.uk
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Abstract

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg–de Vries, or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modelled by the forced Korteweg–de Vries equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including kinks, rarefaction waves, classical undular bores, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 495 - 531
Copyright
©2013 Cambridge University Press 

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