The Tearing Mode Instability of Thin Current Sheets: The Transition to Fast Reconnection in the Presence of Viscosity

Abstract

This paper studies the growth rate of reconnection instabilities in thin current sheets in the presence of both resistivity and viscosity. In a previous paper, Pucci and Velli (2014), it was argued that at sufficiently high Lundquist number S it is impossible to form current sheets with aspect ratios L/a which scale as L/a ∼ S α with α > 1/3 because the growth rate of the tearing mode would then diverge in the ideal limit S → ∞. Here we extend their analysis to include the effects of viscosity, (always present in numerical simulations along with resistivity) and which may play a role in the solar corona and other astrophysical environments. A finite Prandtl number allows current sheets to reach larger aspect ratios before becoming rapidly unstable in pile-up type regimes. Scalings with Lundquist and Prandtl numbers are discussed as well as the transition to kinetic reconnection.

Resistive Magnetohydrodynamics Simulations of the Ideal Tearing Mode

“…Tenerani et al (2015) have considered the effects of viscosity on the "ideal" tearing scenario, showing that the main result is unchanged at Prandtl numbers of the order of unity, while greater critical aspect ratios, even close to the SP limit, are possible in more viscous regimes. On the other hand, for typical conditions of the solar corona, the smaller scales arising in nonlinear evolution require the inclusion of further kinetic effects, such as the Hall term and electron pressure and inertial terms, in Ohmʼs law.…”

Phase diagrams of forced magnetic reconnection in Taylor’s model

“…Therefore, forΨ c < Ψ W /3 an increase in the amplitude perturbationΨ 0 drives the system directly from scenario (1) to scenario (3). This situation is depicted in figure 3(b), where it is clearly shown that the increase of the magnetic Prandtl number has the effect of making possible or extending the domain of existence of scenario (2), as recently pointed out in Tenerani et al (2015) and Comisso et al (2015).…”

“…In addition to the works discussed above, which adopt a magnetohydrodynamic (MHD) description of the plasma, we emphasize that many other efforts have been devoted to investigating the Taylor problem assuming MHD, two-fluid and kinetic descriptions (see Wang & Bhattacharjee 1992b;Ma et al 1996;Avinash et al 1998;Rem & Schep 1998;Valori et al 2000;Fitzpatrick 2003;Fitzpatrick 2004aFitzpatrick ,b, 2008Fitzpatrick et al 2003;Cole & Fitzpatrick 2004;Bian & Vekstein 2005;Birn et al 2005;Vekstein & Bian 2006;Birn & Hesse 2007;Hosseinpour & Vekstein 2008;Gordovskyy et al 2010a,b;Lazzaro & Comisso 2011;Dewar et al 2013;Hosseinpour 2013). Indeed, the Taylor problem has important applications besides being interesting from the point of view of basic physics.…”

Boosting magnetic reconnection by viscosity and thermal conduction

“…The condition Pr m ) 1 and Pr ( 1 can be expected in actual plasma environments such as the solar atmosphere and the interstellar medium. 12 The result indicates the importance of viscosity and heat transfer for the reconnection against the conventional resistive MHD model in which the Prandtl numbers are not explicitly defined.…”

“…Resistive MHD assumes this number to be zero, meaning that the vortex scale is negligibly small compared with the current scale. However, it is not necessarily true in actual plasma environments; 12 the number can be much larger than unity in a classical Spitzer model for hot tenuous plasmas. 13 Numerical simulations have demonstrated that it affects the nonlinear evolution of MHD phenomena such as small-scale turbulence and dynamo.…”