Stochastic Reconnection for Large Magnetic Prandtl Numbers

We consider stochastic magnetic reconnection in high-beta plasmas with large magnetic Prandtl numbers, Pr-m > 1. For large Prm, field line stochasticity is suppressed at very small scales, impeding diffusion. In addition, viscosity suppresses very small-scale differential motions and therefore also the local reconnection. Here we consider the effect of high magnetic Prandtl numbers on the global reconnection rate in a turbulent medium and provide a diffusion equation for the magnetic field lines considering both resistive and viscous dissipation. We find that the width of the outflow region is unaffected unless Prm is exponentially larger than the Reynolds number Re. The ejection velocity of matter from the reconnection region is also unaffected by viscosity unless Re similar to 1. By these criteria the reconnection rate in typical astrophysical systems is almost independent of viscosity. This remains true for reconnection in quiet environments where current sheet instabilities drive reconnection. However, if Prm. >. 1, viscosity can suppress small-scale reconnection events near and below the Kolmogorov or viscous damping scale. This will produce a threshold for the suppression of large-scale reconnection by viscosity when Prm > Re. In any case, for Pr-m > 1 this leads to a flattening of the magnetic fluctuation power spectrum, so that its spectral index is similar to-4/3 for length scales between the viscous dissipation scale and eddies larger by roughly Pr-m(3/2). Current numerical simulations are insensitive to this effect. We suggest that the dependence of reconnection on viscosity in these simulations may be due to insufficient resolution for the turbulent inertial range rather than a guide to the large Re limit.