http://arxiv.org/abs/2011.11822
In laboratory and natural plasmas of practical interest, the smallest spatial scale $\Delta_d$ over which magnetic field lines are distinguishable on the time scale set by an ideal evolution 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 standard resolution of the paradox of disparate time scales is for the current density $j$ associated with the reconnecting field $B_{rec}$ to be concentrated by the ideal evolution, so $ j\sim B_{rec}/\mu_0\Delta_d$, an amplification by a factor $R_m$. A second resolution is for the ideal evolution to increase the ratio of the maximum to minimum separation between two arbitrarily chosen magnetic field lines, $\Delta_{max}/\Delta_{min}$, when calculated at various points in time. Reconnection becomes inevitable when $\Delta_{max}/\Delta_{min}\sim R_m$. As demonstrated using a simple model of the solar corona, 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 of the paradox of disparate time scales, once a sufficient region has undergone reconnection, the magnetic field loses force balance and evolves ideally on an Alfvén transit time. This ideal evolution generally expands the region in which $\Delta_{max}/\Delta_{min}$ is large.
A. Boozer and T. Elder
Wed, 25 Nov 2020
6/65
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