The scaling of embedded collisionless reconnection

The scaling of the reconnection rate is examined in situations in which the equilibrium current supporting a reversed magnetic field has a spatial scale length that is much greater than all nonmagnetohydrodynamic (non-MHD) kinetic scales. In this case, denoted as embedded reconnection, the narrow non-MHD region around the x-line where dissipation is important is embedded inside of a much larger equilibrium current sheet. In this system, the magnetic field just upstream of this non-MHD region, $B_d,$ changes significantly during the reconnection process. This wide equilibrium current sheet is contrasted with the very thin equilibrium current sheets of width ${\mathrm{c/\omega }}_{\mathrm{pi}}$ used in previous simulations to establish the importance of the Hall term in Ohm’s law in allowing fast reconnection in large scale collisionless systems. In the present study we lay out a procedure for determining $B_d$ directly from simulation data and use this value to renormalize the reconnection rate using Sweet–Parker-like scaling arguments. Using two-dimensional two-fluid simulations, we find that the time evolution of the reconnection process can be broken into two phases: A developmental phase that is quite long and strongly dependent on system size and presumably the dissipation mechanisms, and a fast asymptotic phase in which the flow velocity into the x-line is on the order of 0.1 of the Alfvén speed based on $B_d.$ The reconnection rate during the asymptotic phase is independent of system size and the majority of island growth and flux reconnection occurs during this phase. The time to reconnect a significant amount of magnetic flux is roughly consistent with solar flare timescales.