Plasmoid and Kelvin-Helmholtz instabilities in Sweet-Parker current sheets

A two-dimensional (2D) linear theory of the instability of Sweet-Parker (SP) current sheets is developed in the framework of Reduced MHD. A local analysis is performed taking into account the dependence of a generic equilibrium profile on the outflow coordinate. The plasmoid instability [Loureiro et al, Phys. Plasmas 14, 100703 (2007)] is recovered, i.e., current sheets are unstable to the formation of a large-wave-number chain of plasmoids (k(max)L(CS) similar to S-3/8, where k(max) is the wave number of fastest growing mode, S = L-CS V-A/eta is the Lundquist number, L-CS is the length of the sheet, V-A is the Alfven speed, and eta is the plasma resistivity), which grows super Alfvenically fast (gamma(max)iota(A) similar to S-1/4, where gamma(max) is the maximum growth rate, and iota(A)= L-CS/V-A). For typical background profiles, the growth rate and the wave number are found to increase in the outflow direction. This is due to the presence of another mode, the Kelvin-Helmholtz (KH) instability, which is triggered at the periphery of the layer, where the outflow velocity exceeds the Alfven speed associated with the upstream magnetic field. The KH instability grows even faster than the plasmoid instability, gamma(max)iota(A) similar to k(max) L-CS similar to S-1/2. The effect of viscosity (nu) on the plasmoid instability is also addressed. In the limit of large magnetic Prandtl numbers, Pm = nu/eta, it is found that gamma(max) similar to (SPm-5/8)-Pm-1/4 and k(max)L(CS) similar to (SPm-3/16)-Pm-3/8, leading to the prediction that the critical Lundquist number for plasmoid instability in the Pm >> 1 regime is S-crit similar to 10(4)Pm(1/2). These results are verified via direct numerical simulation of the linearized equations, using a new, analytical 2D SP equilibrium solution. DOI: 10.1103/PhysRevE.87.013102