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Linear stability analysis for flows over sinusoidal bottom topography

Published online by Cambridge University Press:  28 January 2021

Jack Davies Open the ORCID record for Jack Davies [Opens in a new window] ,
Hemant Khatri Open the ORCID record for Hemant Khatri [Opens in a new window]  and
Pavel Berloff Open the ORCID record for Pavel Berloff [Opens in a new window]
Jack Davies*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Hemant Khatri
Affiliation:
Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ08540, USA
Pavel Berloff
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK Institute of Numerical Mathematics of the Russian Academy of Sciences, 119333Moscow, Russia
*
†Email address for correspondence: jack.davies18@imperial.ac.uk
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Abstract

This is an ocean motivated study which investigates the impacts of sinusoidal bottom topography on baroclinic instability of zonal vertically sheared flows in the two-layer quasigeostrophic model. The corresponding linear stability problem is solved by assuming Fourier-mode solutions in both the zonal and meridional directions. In the presence of variable topographic features, the Fourier modes become coupled due to phase shifts in the wavevectors. The spectral discretisation method used in this work retains the primary relationship between different Fourier modes; thus, the linear stability eigenproblem can be solved for any periodic topography. Moreover, this method does not need any additional assumptions, such as considering small-amplitude or large-scale bottom irregularities, as in some previous studies. In this work, the eigenproblem is solved for a range of topographic amplitudes and wavenumbers, and their effects on the growth rates and shapes of the most unstable eigenmodes are discussed. In general, both the zonal and meridional variations in topography tend to suppress the baroclinic instability. However, it is found that only meridionally varying topography affects the magnitudes of the fastest growth rates. In this instance, unstable modes appear to form two clusters well separated in the zonal wavenumber axis and growth rate maxima occur at two distinct zonal wavenumbers. Dependencies of the characteristics of these clusters on the values of topography amplitude and ridge width are reviewed. Finally, doubly periodic numerical simulations are used to verify the results from the linear stability analysis.

JFM classification

Geophysical and Geological Flows: Topographic effects Geophysical and Geological Flows: Quasi-geostrophic flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A33
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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