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Development of tearing instability in a current sheet forming by sheared incompressible flow

Published online by Cambridge University Press:  21 February 2018

Elizabeth A. Tolman ,
Nuno F. Loureiro  and
Dmitri A. Uzdensky
Elizabeth A. Tolman*
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Nuno F. Loureiro
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Dmitri A. Uzdensky
Affiliation:
Center for Integrated Plasma Studies, University of Colorado, Boulder, CO 80309, USA Institute for Advanced Study, Princeton, NJ 08540, USA
*
†Email address for correspondence: [email protected]
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Abstract

Sweet–Parker current sheets in high Lundquist number plasmas are unstable to tearing, suggesting they will not form in physical systems. Understanding magnetic reconnection thus requires study of the stability of a current sheet as it forms. Formation can occur due to sheared, sub-Alfvénic incompressible flows which narrow the sheet. Standard tearing theory (Furth et al. Phys. Fluids, vol. 6 (4), 1963, pp. 459–484, Rutherford, Phys. Fluids, vol. 16 (11), 1973, pp. 1903–1908, Coppi et al. Fizika Plazmy, vol. 2, 1976, pp. 961–966) is not immediately applicable to such forming sheets for two reasons: first, because the flow introduces terms not present in the standard calculation; second, because the changing equilibrium introduces time dependence to terms which are constant in the standard calculation, complicating the formulation of an eigenvalue problem. This paper adapts standard tearing mode analysis to confront these challenges. In an initial phase when any perturbations are primarily governed by ideal magnetohydrodynamics, a coordinate transformation reveals that the flow compresses and stretches perturbations. A multiple scale formulation describes how linear tearing mode theory (Furth et al. Phys. Fluids, vol. 6 (4), 1963, pp. 459–484, Coppi et al. Fizika Plazmy, vol. 2, 1976, pp. 961–966) can be applied to an equilibrium changing under flow, showing that the flow affects the separable exponential growth only implicitly, by making the standard scalings time dependent. In the nonlinear Rutherford stage, the coordinate transformation shows that standard theory can be adapted by adding to the stationary rates time dependence and an additional term due to the strengthening equilibrium magnetic field. Overall, this understanding supports the use of flow-free scalings with slight modifications to study tearing in a forming sheet.

Keywords

astrophysical plasmasplasma flowsplasma instabilities
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 1 , February 2018 , 905840115
Copyright
© Cambridge University Press 2018 

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