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Title: Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation
Authors: Kamchatnov, A.M.
Kuo, Y.-H.
Lin, T.-C.
Horng, T.-L.
Gou, S.-C.
Clift, Richard
El, G.A.
Grimshaw, Roger H.J.
Keywords: Internal waves
Solitary waves
Topographic effects
Issue Date: 2013
Publisher: © Cambridge University Press
Citation: KAMCHATNOV, A.M. ... et al, 2013. Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation. Journal of Fluid Mechanics, 736, pp.495-531.
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 (eKdV), 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 modeled by the forced KdV 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.
Description: This paper was accepted for publication in the Journal of Fluid Mechanics and the definitive version is available at: http://dx.doi.org/10.1017/jfm.2013.556
Version: Accepted for publication
DOI: 10.1017/jfm.2013.556
URI: https://dspace.lboro.ac.uk/2134/14279
Publisher Link: http://dx.doi.org/10.1017/jfm.2013.556
ISSN: 0022-1120
Appears in Collections:Published Articles (Maths)

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