Fast magnetic field line reconnection

Petschek (1964) has given a qualitative model for fast magnetic field line reconnection, at speeds up to a significant fraction of the Alfven speed. It is supposed that an electrically conducting fluid is permeated by an almost uniform magnetic field which reverses direction across a plane of symmetry parallel to the field lines. An almost uniform stream flows steadily towards the plane of symmetry and is maintained by pressure forces. Magnetic field line reconnection occurs at the origin inside a small central diffusion region. The reconnected magnetic field is swept away rapidly in two thin jets aligned with the plane of symmetry. The inflow and outflow regions are separated by discontinuities at which the tangential components of the magnetic field and fluid velocity suffer abrupt changes. Sonnerup (1970) and Yeh & Axford (1970), on the other hand, have given alternative solutions for the incompressible case which include a second set of discontinuities. Their solutions are of similarity type, valid over some length scale which is much less than the overall distance between the magnetic field sources but is much greater than the size of the central diffusion region. The second set of discontinuities is, however, unacceptable for an astrophysical plasma, since they need to be generated at corners in the flow rather than at the central diffusion region. This paper presents other solutions for the incompressible case, which are locally self-similar, without discontinuities or singular behaviour at a second set of discontinuities. The solutions are valid everywhere outside the central diffusion region when the inflow Alfven Mach number M1 (see (2.3) below) is much less than unity and are valid at large distances from the diffusion region when M1 = 0(1). The analysis has been summarized by Priest & Soward (1976). It puts Petschek’s mechanism on a sound mathematical basis and shows that the discontinuities are not in general straight but curve away from the incoming flows. Our estimate of the maximum reconnection rate Me,max (see (10.9) below) depends weakly on the value of the magnetic Reynolds number Rm,e (see (10.7) below). It decreases from 0.2 when Rm,e> = 10 to 0.03 when Rm,e = 106.