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The plasmoid instability has revolutionized our understanding of magneticreconnection in astrophysical environments. By preventing the formation ofhighly elongated reconnection layers, it is crucial in enabling the rapidenergy conversion rates that are characteristic of many astrophysicalphenomena. Most of the previous studies have focused on Sweet-Parker currentsheets, which, however, are unattainable in typical astrophysical systems.Here, we derive a general set of scaling laws for the plasmoid instability inresistive and visco-resistive current sheets that evolve over time. Our methodrelies on a principle of least time that enables us to determine the propertiesof the reconnecting current sheet (aspect ratio and elapsed time) and theplasmoid instability (growth rate, wavenumber, inner layer width) at the end ofthe linear phase. After this phase the reconnecting current sheet is disruptedand fast reconnection can occur. The scaling laws of the plasmoid instabilityare \emph{not} simple power laws, and depend on the Lundquist number ($S$), themagnetic Prandtl number ($P_m$), the noise of the system ($\psi_0$), thecharacteristic rate of current sheet evolution ($1/\tau$), as well as thethinning process. We also demonstrate that previous scalings are inapplicableto the vast majority of the astrophysical systems. We explore the implicationsof the new scaling relations in astrophysical systems such as the solar coronaand the interstellar medium. In both these systems, we show that our scalinglaws yield values for the growth rate, wavenumber, and aspect ratio that aremuch smaller than the Sweet-Parker based scalings.

Plasmoid Instability in Forming Current Sheets