Magnetic reconnection via current sheets

A general picture of magnetic reconnection in the framework of 2-D incompressible resistive magnetohydrodynamic theory is presented. Numerical studies of (quasi-) steady-state driven reconnection reveal current sheet formation for Mach numbers M=u/v_A exceeding the Sweet-Parker reconnection rate M_SP=(η/Lv_A)^1/2. Since the thickness δ of the current sheet is found to be invariant to a change of the resistivity η, its length Δ increases rapidly with decreasing η or increasing M, which can be written in the form Δ∼(M/M_SP)⁴, so that Δ reaches the global system size L within a short range of the parameter M/M_SP. The results are rather insensitive to the particular choice of boundary conditions. Because of the presence of a current sheet, the overall reconnection process is quite slow. This picture essentially agrees with Syrovatsky's [Sov. Phys. JETP 33, 933 (1971)] theory and disproves Petschek's [AAS/NASA Symposium on the Physics of Solar Flares, (NASA, Washington, DC, 1964) p. 425] mechanism of fast magnetic reconnection. A theory of the solution in the external and in the diffusion region is developed and analytical expressions in agreement with the simulation results are obtained by means of a variational principle. For sufficiently long current sheets the tearing mode becomes unstable in spite of the stabilizing effect of the inhomogeneous flow. The tearing mode contributes to the overall reconnection process, but a general assessment of this effect in the asymptotic regime of almost vanishing η is difficult.