$X$-Point Collapse and Saturation in the Nonlinear Tearing Mode Reconnection

We study the nonlinear evolution of the resistive tearing mode in slab geometry in two dimensions. We show that, in the strongly driven regime (large ${\Delta }^{\prime }$), a collapse of the $X$ point occurs once the island width exceeds a certain critical value $\sim 1/{\Delta }^{\prime }$. A current sheet is formed and the reconnection is exponential in time with a growth rate $\propto {\eta }^{1/2}$, where $\eta$ is the resistivity. If the aspect ratio of the current sheet is sufficiently large, the sheet can itself become tearing-mode unstable, giving rise to secondary islands, which then coalesce with the original island. The saturated state depends on the value of ${\Delta }^{\prime }$. For small ${\Delta }^{\prime }$, the saturation amplitude is $\propto {\Delta }^{\prime }$ and quantitatively agrees with the theoretical prediction. If ${\Delta }^{\prime }$ is large enough for the $X$-point collapse to have occurred, the saturation amplitude increases noticeably and becomes independent of ${\Delta }^{\prime }$.

Figure 1

Effective growth rate at the $X$ point ${\gamma }_{\mathrm{eff}}=d\ln \Psi /dt$ vs time for a strongly driven (large ${\Delta }^{\prime }$) tearing mode. FKR stands for Furth-Killeen-Rosenbluth, after the authors of Ref. 4.

Figure 2

Contours of $\psi$ at the beginning and end of stage III in Fig. 1. The boundaries of these plots are not the boundaries of the computational box.

Figure 3

The critical island width for collapse vs $\eta$ at fixed ${\Delta }^{\prime }=17.3,30.3$. Dashed lines are linear fits.

Figure 4

(a) Growth of the reconnected flux $\Psi$ during the SP stage for fixed ${\Delta }^{\prime }=17.3$ and various values of $\eta$. (b) Slopes of these lines vs $\eta$ during the exponential growth.

Figure 5

The exponential stage ($370<t<450$) of the run of Fig. 1: (a) maximum outflow velocity $v_{\mathrm{out}}$ vs the current-sheet length $L_{\mathrm{CS}}$; (b) $L_{\mathrm{CS}}$ vs $(\Psi -{\Psi }_{\mathrm{c}})/B_0$, where $B_0$ is defined as the maximum value of $B_y$ along the $x$ axis. These curves do not extrapolate to the origin because the full-width-half-maximum definition used for $L_{\mathrm{CS}}$ correctly reflects the growth of the current-sheet length but not its true length (thus, it formally gives $L_{\mathrm{CS}}>0$ for the $X$-point reconnection).

Figure 6

The current-sheet length $L_{\mathrm{CS}}$ and width ${\delta }_{\mathrm{CS}}$ (a) vs $\eta$ and (b) vs ${\Delta }^{\prime }$.

Figure 7

Contours of $\psi$ showing the current-sheet instability (a)–(c) and the subsequent nonlinear evolution of the secondary island (d)–(f) for a run with ${\Delta }^{\prime }=40.6,\eta =2.8\times {10}^{-4}$.

Figure 8

(a) Saturated amplitude ${\Psi }_{\mathrm{sat}}$ vs ${\Delta }^{\prime }$ for different values of $\eta$. The theoretical curves by POEM and White et al. [28] are also shown. The island width formula (3) has been used to convert $W_{\mathrm{sat}}$ calculated by these authors into ${\Psi }_{\mathrm{sat}}$. (b) ${\Psi }_{\mathrm{sat}}$ vs $\eta$ for ${\Delta }^{\prime }=8.2,17.3$. In both plots, open circles are the cases where $W_{\mathrm{sat}}$ exceeded the box size.