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Resonant coupling of mode-1 and mode-2 internal waves by topography

Published online by Cambridge University Press:  02 December 2020

Zihua Liu Open the ORCID record for Zihua Liu [Opens in a new window] ,
Roger Grimshaw Open the ORCID record for Roger Grimshaw [Opens in a new window]  and
Edward Johnson Open the ORCID record for Edward Johnson [Opens in a new window]
Zihua Liu*
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Roger Grimshaw
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Edward Johnson
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider the resonant coupling of mode-1 and mode-2 internal solitary waves by topography. Mode-2 waves are generated by a mode-1 wave encountering variable topography, modelled by a coupled Korteweg–de Vries (KdV) system. Three cases, namely $(A)$ weak resonant coupling, $(B)$ moderate resonant coupling and $(C)$ strong resonant coupling, are examined in detail using a three-layer density-stratified fluid system with different stratification and topographic settings. The strength of the resonant coupling is determined by the range of values taken by the ratio of linear long-wave phase speeds ($c_2/c_1$, where $c_1$ is the mode-1 speed and $c_2$ the mode-2 speed) while the waves are above the slope. In case $A$ the range is from $0.42$ (ocean edge) to $0.48$ (shelf edge), in case $B$ from $0.58$ (ocean) to $0.72$ (shelf) and in case $C$ from $0.44$ (ocean) to $0.92$ (shelf). The feedback from mode-2 to mode-1 is estimated by comparing the coupled KdV system with a KdV model. In case $A$, a small-amplitude convex mode-2 wave is generated by a depression mode-1 wave and the feedback on the mode-1 wave is negligible. In case $B$, a concave mode-2 wave of comparable amplitude to that of the depression incident mode-1 wave is formed; strong feedback enhances the polarity change process of the mode-1 wave. In case $C$, a large-amplitude concave mode-2 wave is produced by an elevation mode-1 wave; strong feedback suppresses the fission of the mode-1 wave. Simulations for a wider range of topographic slopes and three-layer stratifications are then classified in terms of these responses.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Topographic effects Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A2
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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