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Non-ideal instabilities in sinusoidal shear flows with a streamwise magnetic field

Published online by Cambridge University Press:  06 October 2022

A.E. Fraser Open the ORCID record for A.E. Fraser [Opens in a new window] ,
I.G. Cresswell Open the ORCID record for I.G. Cresswell [Opens in a new window]  and
P. Garaud Open the ORCID record for P. Garaud [Opens in a new window]
A.E. Fraser*
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California Santa Cruz, Santa Cruz, CA 95064, USA
I.G. Cresswell
Affiliation:
Department of Astrophysical and Planetary Sciences & LASP, University of Colorado, Boulder, CO 80309, USA
P. Garaud
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California Santa Cruz, Santa Cruz, CA 95064, USA
*
†Email address for correspondence: adr.fraser@gmail.com
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Abstract

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number ${{{Pm}}} < 1$. We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.

JFM classification

Instability: Shear-flow instability MHD and Electrohydrodynamics: Plasmas
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 949 , 25 October 2022 , A43
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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