Hall Magnetic Reconnection Rate

Two-dimensional Hall magnetohydrodynamic simulations are used to determine the magnetic reconnection rate in the Hall limit. The simulations are run until a steady state is achieved for four initial current sheet thicknesses: $L=1,5,10$, and $20c/{\omega }_{pi}$, where $c/{\omega }_{pi}$ is the ion inertial length. It is found that the asymptotic (i.e., time independent) state of the system is nearly independent of the initial current sheet width. Specifically, the Hall reconnection rate is weakly dependent on the initial current layer width and is $\partial \Phi /\partial t\lesssim 0.1V_{A0}{B}_0$, where $\Phi$ the reconnected flux, and $V_{A0}$ and $B_0$ are the Alfvén velocity and magnetic field strength in the upstream region. Moreover, this rate appears to be independent of the scale length on which the electron “frozen-in” condition is broken (as long as it is $<c/{\omega }_{pi}$) and of the system size.