Analytical Description of Steady Magnetic Reconnection in Hall Magnetohydrodynamics

Abstract

Steady magnetic reconnection in the framework of incompressible Hall magnetohydrodynamics is considered. The principal role of the Hall effect in the formation of the structure of the reconnecting current sheet is emphasized. Analytical expressions for the velocity and the magnetic field in the sheet are derived, based on the approximation of a weak two dimensionality of the planar components of the solution. The analytical solution illustrates key features of Hall magnetic reconnection, including the reconnection rate enhancement and the sheet thinning due to the Hall effect, the presence of a quadrupolar axial (out-of-the-plane) magnetic field that controls the geometry of the reconnecting planar magnetic field, and the dynamical coupling of the axial and planar components of the solution, with the coupling strength that is proportional to the ion skin depth. Scalings for the sheet thickness, width, and the reconnection inflow and outflow speeds in terms of the electric resistivity and the axial magnetic field are determined. Implications of the results for fast magnetic reconnection in a weakly collisional plasma of the solar corona are discussed.