<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi></mml:math>-Point Collapse and Saturation in the Nonlinear Tearing Mode Reconnection

Abstract

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 ${\ensuremath{\Delta}}^{\ensuremath{'}}$), a collapse of the $X$ point occurs once the island width exceeds a certain critical value $\ensuremath{\sim}1/{\ensuremath{\Delta}}^{\ensuremath{'}}$. A current sheet is formed and the reconnection is exponential in time with a growth rate $\ensuremath{\propto}{\ensuremath{\eta}}^{1/2}$, where $\ensuremath{\eta}$ 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 ${\ensuremath{\Delta}}^{\ensuremath{'}}$. For small ${\ensuremath{\Delta}}^{\ensuremath{'}}$, the saturation amplitude is $\ensuremath{\propto}{\ensuremath{\Delta}}^{\ensuremath{'}}$ and quantitatively agrees with the theoretical prediction. If ${\ensuremath{\Delta}}^{\ensuremath{'}}$ is large enough for the $X$-point collapse to have occurred, the saturation amplitude increases noticeably and becomes independent of ${\ensuremath{\Delta}}^{\ensuremath{'}}$.