Example of exponentially enhanced magnetic reconnection driven by a spatially-bounded and laminar ideal flow

Abstract

In laboratory and natural plasmas of practical interest, the spatial scale $\Delta_d$ at which magnetic field lines lose distinguishability differs enormously from the scale $a$ of magnetic reconnection across the field lines. In the solar corona, plasma resistivity gives $a/\Delta_d\sim10^{12}$, which is the magnetic Reynold number $R_m$. The traditional resolution of the paradox of disparate scales is for the current density $j$ associated with the reconnecting field $B_{rec}$ to be concentrated by a factor of $R_m$ by the ideal evolution, so $ j\sim B_{rec}/\mu_0\Delta_d$. A second resolution is for the ideal evolution to increase the ratio of the maximum to minimum separation between pairs of arbitrarily chosen magnetic field lines, $\Delta_{max}/\Delta_{min}$, when calculated at various points in time. Reconnection becomes inevitable where $\Delta_{max}/\Delta_{min}\sim R_m$. A simple model of the solar corona will be used for a numerical illustration that the natural rate of increase in time is linear for the current density but exponential for $\Delta_{max}/\Delta_{min}$. Reconnection occurs on a time scale and with a current density enhanced by only $\ln(a/\Delta_d)$ from the ideal evolution time and from the current density $B_{rec}/\mu_0a$. In both resolutions, once a sufficiently wide region, $\Delta_r$, has undergone reconnection, the magnetic field loses static force balance and evolves on an Alfv\'enic time scale. The Alfv\'enic evolution is intrinsically ideal but expands the region in which $\Delta_{max}/\Delta_{min}$ is large.