Steady state magnetic reconnection at high magnetic Reynolds number: A boundary layer analysis

Abstract

A boundary layer analysis of two‐dimensional current sheets appearing in magnetic reconnection is presented in the framework of resistive incompressible two‐dimensional magnetohydrodynamics. The coordinate normal to the current sheet is rescaled by a power of the resistivity in order to obtain inner limit equations which satisfy the symmetry conditions and represent magnetic annihilation to lowest order. They can be matched to the outer expansion of the incompressible resistive MHD equations with a domain of validity smaller than the current sheet length. The magnetic vector potential and the stream function are expanded in powers of the inverse magnetic Reynolds number Rm. For the inner limit equations, four cases are found depending on the boundary conditions in the overlap region. These cases correspond to different scalings of the velocity and the magnetic field which are equivalent to different prescribed boundary conditions of the magnetic field and plasma velocity. For the case of the magnetic field dominating over the flow, inner and outer solutions are obtained up to the second order in the expansion parameter and are matched in the intermediate region. Scaling laws for the width of the current sheet are derived: It decreases for increasing magnetic Reynolds number. The characteristic length for growth of the magnetic field component normal to the current sheet increases with magnetic Reynolds number. A Sweet‐Parker scaling is found for the external reconnection rate.