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Eigenvalue bounds for compressible stratified magnetoshear flows varying in two transverse directions

Published online by Cambridge University Press:  17 June 2021

Kengo Deguchi Open the ORCID record for Kengo Deguchi [Opens in a new window]
Kengo Deguchi*
Affiliation:
School of Mathematics, Monash University, VIC3800, Australia
*
†Email address for correspondence: [email protected]
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Abstract

Three eigenvalue bounds are derived for the instability of ideal compressible stratified magnetohydrodynamic shear flows in which the base velocity, density and magnetic field vary in two directions. The first bound can be obtained by combining the Howard semicircle theorem with the energy principle of the Lagrangian displacement. Remarkably, no special conditions are needed to use this bound, and for some cases, we can establish the stability of the flow. The second and third bounds come out from a generalisation of the Miles–Howard theory and have some similarity to the semi-ellipse theorem by Kochar & Jain (J. Fluid Mech., vol. 91, 1979, p. 489) and the bound found by Cally (Astrophys. Fluid Dyn., vol. 31, 1983, p. 43), respectively. An important byproduct of this investigation is that the Miles–Howard stability condition holds only when there is no applied magnetic field and, in addition, the directions of the shear and the stratification are aligned everywhere.

JFM classification

Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A55
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© The Author(s), 2021. Published by Cambridge University Press

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