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Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

Published online by Cambridge University Press:  06 September 2018

Shota Suzuki ,
Makoto Hirota  and
Yuji Hattori Open the ORCID record for Yuji Hattori [Opens in a new window]
Shota Suzuki
Affiliation:
Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
Makoto Hirota
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai 980–8577, Japan
Yuji Hattori*
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai 980–8577, Japan
*
†Email address for correspondence: [email protected]
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Abstract

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.

JFM classification

Instability: Parametric instability Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 854 , 10 November 2018 , pp. 293 - 323
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Copyright
© 2018 Cambridge University Press 

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