Plasmoid instability in double current sheets

The linear behavior of plasmoid instability in double current sheet configurations, namely, double plasmoid mode (DPM), is analytically and numerically investigated within the framework of a reduced magnetohydrodynamic model. Analytical analysis shows that if the separation of double current sheets is sufficiently small [$κxs≪κ2/9SL1/3$], the growth rate of DPMs scales as $κ2/3SL0$ in the non-constant-$ψ$ regime, where $κ=kLCS/2$ is the wave vector measured by the half length of the system $LCS/2$⁠, $2xs$ is the separation between two resonant surfaces, and $SL=LCSVA/2η$ is Lundquist number with $VA$ and $η$ being Alfven velocity and resistivity, respectively. If the separation is very large [$κxs≫κ2/9SL1/3$], the growth rate scales as $κ−2/5SL2/5$ in the constant-$ψ$ regime. Furthermore, it is also analytically found that the maximum wave number scales as $xs−9/7SL3/7$ at the transition position between these two regimes, and the corresponding maximum growth rate scales as $xs−6/7SL2/7$ there. The analytically predicted scalings are verified in some limits through direct numerical calculations.