Formation of Plasmoid Chains in Magnetic Reconnection

A detailed numerical study of magnetic reconnection in resistive MHD for very large, previously inaccessible, Lundquist numbers (${10}^4\le S\le {10}^8$) is reported. Large-aspect-ratio Sweet-Parker current sheets are shown to be unstable to super-Alfvénically fast formation of plasmoid (magnetic-island) chains. The plasmoid number scales as $S^{3/8}$ and the instability growth rate in the linear stage as $S^{1/4}$, in agreement with the theory by Loureiro et al. [Phys. Plasmas 14, 100703 (2007)]. In the nonlinear regime, plasmoids continue to grow faster than they are ejected and completely disrupt the reconnection layer. These results suggest that high-Lundquist-number reconnection is inherently time-dependent and hence call for a substantial revision of the standard Sweet-Parker quasistationary picture for $S>{10}^4$.

Figure 1

Contour plots of the current density showing the time evolution of an SP current sheet for $S={10}^8$. The times shown are, from top to bottom, $t=0.20{\tau }_A$, $t=0.40{\tau }_A$, $t=0.45{\tau }_A$ and $t=0.50{\tau }_A$. The domain shown is $-{\delta }_{\mathrm{SP}}\le x\le {\delta }_{\mathrm{SP}}$ (inflow direction, vertical), and $-0.12L\le y\le 0.12L$ (outflow direction, horizontal), where ${\delta }_{\mathrm{SP}}\simeq {10}^{-4}$ is the SP layer width and $L=1$ is the (half-)length of the current sheet (see text; only the central half of the simulation box is shown).

Figure 2

Current density for $S={10}^4$, $S={10}^5$, $S={10}^6$ and $S={10}^7$. $S={10}^8$ is shown in Fig. 1.

Figure 3

Growth rate of the central plasmoid vs $S$. The theoretical slope $S^{1/4}$ [26] is shown for reference.

Figure 4

Time evolution of the half-width of the plasmoid closest to the center of the sheet for $S={10}^8$. Half-widths of the inner layer, ${\delta }_{\mathrm{inner}}$ [26], and of the SP sheet, ${\delta }_{\mathrm{SP}}=L_x$, are shown for reference.

Figure 5

$O$ points position vs time, $S={10}^8$. Dotted lines mark the times at which the current sheet is shown in Fig. 1.

Figure 6

The maximum number of plasmoids in the central part of the sheet, $-0.12L\le y\le 0.12L$, vs $S$. The theoretical slope from linear theory, $S^{3/8}$ [26], is shown for reference.