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Wave propagation in rotating shallow water in the presence of small-scale topography

Published online by Cambridge University Press:  28 July 2021

E.J. Goldsmith  and
J.G. Esler
E.J. Goldsmith*
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
J.G. Esler
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
*
†Email address for correspondence: [email protected]
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Abstract

The question of how finite-amplitude, small-scale topography affects small-amplitude motions in the ocean is addressed in the framework of the rotating shallow water equations. The extent to which the dispersion relations of Poincaré, Kelvin and Rossby waves are modified in the presence of topography is illuminated, using a range of numerical and analytical techniques based on the method of homogenisation. Both random and regular periodic arrays of topography are considered, with the special case of regular cylinders studied in detail, because this case allows for highly accurate analytical results. The results show that, for waves in a $\beta$-channel bounded by sidewalls, and for steep topographies outside of the quasi-geostrophic regime, topography acts to slow Poincaré waves slightly, Rossby waves are slowed significantly and Kelvin waves are accelerated for long waves and slowed for short waves, with the two regimes being separated by a narrow band of resonant wavelengths. The resonant band, which is due to the excitation of trapped topographic Rossby waves on each seamount, may affect any of the three wave types under the right conditions, and for physically reasonable results requires regularisation by Ekman friction. At larger topographic amplitudes, for cylindrical topography, a simple and accurate formula is given for the correction to the Rossby wave dispersion relation, which extends previous results for the quasi-geostrophic regime.

JFM classification

Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Topographic effects Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A24
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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