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Linear stability of monopolar vortices over isolated topography

Published online by Cambridge University Press:  20 March 2023

Jeasson F. Gonzalez Open the ORCID record for Jeasson F. Gonzalez [Opens in a new window]  and
L. Zavala Sansón Open the ORCID record for L. Zavala Sansón [Opens in a new window]
Jeasson F. Gonzalez
Affiliation:
Departamento de Oceanografía Física, CICESE, Carretera Ensenada-Tijuana 3918, 22860, Ensenada, Mexico
L. Zavala Sansón*
Affiliation:
Departamento de Oceanografía Física, CICESE, Carretera Ensenada-Tijuana 3918, 22860, Ensenada, Mexico
*
†Email address for correspondence: lzavala@cicese.mx
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Abstract

The linear instability of circular vortices over isolated topography in a homogeneous and inviscid fluid is examined for the shallow-water and quasi-geostrophic models in the $f$-plane. The eigenvalue problem associated with azimuthal disturbances is derived for arbitrary axisymmetric topographies, either submarine mountains or valleys. Amended Rayleigh and Fjørtoft theorems with topographic effects are given for barotropic instability, obtaining necessary criteria for instability when the potential vorticity gradient is zero somewhere in the domain. The onset of centrifugal instability is also discussed by deriving the Rayleigh circulation theorem with topography. The barotropic instability theorems are applied to a wide family of nonlinear, quasi-geostrophic solutions of circular vortices over axisymmetric topographic features. Flow instability depends mainly on the vortex/topography configuration, as well as on the vortex size in comparison with the width of the topography. It is found that anticyclones/mountains and cyclones/valleys may be unstable. In contrast, cyclone/mountain and anticyclone/valley configurations are stable. These statements are validated with two numerical methods. First, the generalised eigenvalue problem is solved to obtain the wavenumber of the fastest-growing perturbations. Second, the evolution of the vortices is simulated numerically to detect the development of linear perturbations. The numerical results show that for unstable vortices over narrow topographies, the fastest growth rate corresponds to mode $1$, which subsequently forms asymmetric dipolar structures. Over wide topographies, the fastest perturbations are mainly modes $1$ and $2$, depending on the topographic features.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Topographic effects Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , A23
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Gonzalez and Zavala Sansón Supplementary Movie 1

Numerically calculated vorticity distribution of an anticyclone over a wide mountain. A wave number 2 instability is developed at the end of the simulation. The vortex core (mountain) is bounded by the black (magenta) circumference.
Download Gonzalez and Zavala Sansón Supplementary Movie 1(Video)
Video 10.4 MB

Gonzalez and Zavala Sansón Supplementary Movie 2

Numerically calculated vorticity distribution of a stable cyclone over a narrow mountain. The vortex core (mountain) is bounded by the black (magenta) circumference.
Download Gonzalez and Zavala Sansón Supplementary Movie 2(Video)
Video 8.2 MB