Visco-resistive plasmoid instability

The plasmoid instability in visco-resistive current sheets is analyzed in both the linear and nonlinear regimes. The linear growth rate and the wavenumber are found to scale as S1/4(1+Pm)-5/8 and S3/8(1+Pm)-3/16 with respect to the Lundquist number S and the magnetic Prandtl number Pm. Furthermore, the linear layer width is shown to scale as S-1/8(1+Pm)1/16. The growth of the plasmoids slows down from an exponential growth to an algebraic growth when they enter into the nonlinear regime. In particular, the time-scale of the nonlinear growth of the plasmoids is found to be tNL~S-3/16(1+Pm)19/32tA,L. The nonlinear growth of the plasmoids is radically different from the linear one, and it is shown to be essential to understand the global current sheet disruption. It is also discussed how the plasmoid instability enables fast magnetic reconnection in visco-resistive plasmas. In particular, it is shown that the recursive plasmoid formation can trigger a collisionless reconnection regime if S?Lcs(?clk)-1(1+Pm)1/2, where Lcs is the half-length of the global current sheet and lk is the relevant kinetic length scale. On the other hand, if the current sheet remains in the collisional regime, the global (time-averaged) reconnection rate is shown to be ?d?/dt|X?~?cvA,uBu(1+Pm)-1/2, where ?c is the critical inverse aspect ratio of the current sheet, while vA,u and Bu are the Alfvén speed and the magnetic field upstream of the global reconnection layer.