Hall magnetohydrodynamic reconnection in the plasmoid unstable regime

A set of reduced Hall magnetohydrodynamic (MHD) equations are used to evaluate the stability of large aspect ratio current sheets to the formation of plasmoids (secondary islands). Reconnection is driven by resistivity in this analysis, which occurs at the resistive skin depth $dη≡SL-1/2LνA/γ$⁠, where S_L is the Lundquist number, L, the length of the current sheet, ν_A, the Alfvén speed, and γ, the growth rate. Modifications to a recent resistive MHD analysis [N. F. Loureiro et al., Phys. Plasmas 14, 100703 (2007)] arise when collisions are sufficiently weak that d_η is shorter than the ion skin depth d_i ≡ c/ω_pi. Secondary islands grow faster in this Hall MHD regime: the maximum growth rate scales as $(di/L)6/13SL7/13νA/L$ and the number of plasmoids as $(di/L)1/13SL11/26$⁠, compared to $SL1/4νA/L$ and S^3/8, respectively, in resistive MHD.