Visco-resistive plasmoid instability in Sweet-Parker current sheets

The linear analysis by Loureiro et al. [1] is generalized to investigate the plasmoid instability in visco-resistive Sweet-Parker sheets [2]. We cover both the linear and nonlinear growth of the plasmoids. The linear growth rate and the wavenumber scale as $S^{1/4} (1+P_m)^{-5/8}$ and $S^{3/8} (1+P_m)^{-3/16}$ with respect to the Lundquist number $S$ and the magnetic Prandtl number $P_m$. The growth of the plasmoids slows down from an exponential growth to an algebraic growth when they enter into the nonlinear regime. The time-scale of the nonlinear growth of the plasmoids is found to be $\tau_{NL} \sim S^{-3/16} (1+P_m)^{19/32} \tau_{A,L}$. We also discuss how the plasmoid instability can enable fast magnetic reconnection [3] in visco-resistive plasmas [4]. In this regime, the global reconnection rate is shown to be $\left\langle {{\left. {d\psi /dt} \right|}_X} \right\rangle \approx 0.01 v_{A,u} B_u (1+P_m)^{-1/2}$ [2]. [1] N.F. Loureiro, A.A. Schekochihin and S.C. Cowley, Phys. Plasmas 14, 100703 (2007). [2] L. Comisso and D. Grasso, Phys. Plasmas 23, 032111 (2016) [3] A. Bhattacharjee, Y.-M. Huang, H. Yang and B. Rogers, Phys. Plasmas 16, 112102 (2009). [4] L. Comisso, D. Grasso and F.L. Waelbroeck, Phys. Plasmas 22, 042109 (2015).