Plasmoid Instability in Evolving Current Sheets and Onset of Fast Reconnection

The scaling of the plasmoid instability maximum linear growth rate with respect to the Lundquist number S in a Sweet-Parker current sheet, γ_max ∼ S^1/4, indicates that at high S, the current sheet will break apart before it approaches the Sweet-Parker width. Therefore, a proper description for the onset of the plasmoid instability must incorporate the evolving process of the current sheet. We carry out a series of two-dimensional simulations and develop diagnostics to separate fluctuations from an evolving background. It is found that the fluctuation amplitude starts to grow only when the linear growth rate is sufficiently high (γ_max τ_A > O(1)) to overcome advection loss and the stretching effect due to the outflow. The linear growth rate continues to rise until the sizes of plasmoids become comparable to the inner layer width of the tearing mode. At this point, the current sheet is disrupted and the instability enters the early nonlinear regime. The growth rate suddenly decreases, but the reconnection rate starts to rise rapidly, indicating that current sheet disruption triggers the onset of fast reconnection. We identify important timescales of the instability development, as well as scalings for the linear growth rate, current sheet width, and dominant wavenumber at disruption. These scalings depend not only on the Lundquist number, but also on the noise amplitude. A phenomenological model that reproduces scalings from simulation results is proposed. The model incorporates the effect of reconnection outflow, which is crucial for yielding a critical Lundquist number S_c below which disruption does not occur. The critical Lundquist number S_c is not a constant value, but has a weak dependence on the noise amplitude.