Plasmoid instability in the semi-collisional regime

We investigate analytically and numerically the semi-collisional regime of the plasmoid instability, defined by the inequality $δ_{SP} \gg ρ_s \gg δ_{in}$, where $δ_{SP}$ is the width of a Sweet-Parker current sheet, $ρ_S$ is the ion sound Larmor radius, and $δ_{in}$ is width of boundary layer that arises in the plasmoid instability analysis. Theoretically, this regime is predicted to exist if the Lundquist number $S$ and the length of the current sheet $L$ are such that $(L/ρ_S)^{14/9} < S <(L/ρ_S)^2$ (for a sinusoidal-like magnetic configuration; for a Harris-type sheet the lower bound is replaced with $(L/ρ_S)^{8/5}$). These bounds are validated numerically by means of simulations using a reduced gyrokinetic model (Zocco & Schekochihin, $\it{Physics~of~Plasmas}$, ${\bf 18}$, 2011) conducted with the code $\tt{Viriato}$. Importantly, this regime is conjectured to allow for plasmoid formation at relatively low, experimentally accessible, values of the Lundquist number. Our simulations obtain plasmoid instability at values of $S$ as low as $\sim 250$. The simulations do not prescribe a Sweet-Parker sheet; rather, one is formed self-consistenly during the nonlinear evolution of the initial tearing mode configuration. This proves that this regime of the plasmoid instability is realizable, at least at the relatively low values of the Lundquist number that are accessible to current dedicated experiments.