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

In plasmas of practical interest the spatial scale $\Delta_d$ at which magnetic field lines lose distinguishability differs greatly from the scale $a$ of magnetic reconnection (MR) across the field lines. In the solar corona, plasma resistivity gives $a/\Delta_{d}\sim 10^{12}$ which is the magnetic Reynold number $R_m$. The scale paradox is typically resolved by assuming the current density $j$ of the reconnecting field $B_{rec}$ concentrates by a factor of $R_{m}$ ideally, $j\sim B_{rec}/\mu_0\Delta_{d}$. A second resolution is for the ideal evolution to increase the ratio of the maximum to minimum separation between pairs of neighboring magnetic field lines $\Delta_{max}/\Delta_{min}$. MR is inevitable where $\Delta_{max}/\Delta_{min}\sim R_m$. A simple model of the solar corona is used to numerically illustrate that $j$ increases linearly in time while $\Delta_{max}/\Delta_{min}$ increases exponentially. MR occurs on a time scale and with a current density enhanced by only $\ln(a/\Delta_d)$ from the ideal evolution time and current density $B_{rec}/\mu_0 a$. In both resolutions, once a sufficiently wide region has reconnected, the magnetic field loses static force balance and evolves on an Alfvénic time scale, expanding the region in which $\Delta_{max}/\Delta_{min}$ is large.