Asymptotic solution for field-line reconnexion. Compressible case of Petschek's model†

For the solution of Petschek's problem of field-line reconnexion, a new method is elaborated which is based on the introduction of a special co-ordinate system in which the streamlines and the magnetic lines of force become co-ordinates simultaneously. We have constructed the zero-order and the first-order approximation (for small Alfvén Mach numbers) for the solution of Petschek's problem in the steady-state, compressible, two-dimensional symmetric case. It is shown that the density across the slow shock wave increases by a factor

and the pressure by

(β = 8πρ₀/B²₀, γ being the adiabatic exponent), and the plasma accelerates up to the Alfvén velocity. On the bases of the results obtained and of the analysis of numerical experiments on the reconnexion problem we draw the conclusion that during the initial phase of the process there develops a current sheet as described by Syrovatskii and that simultaneously there is a development of the tearing mode instability whose nonlinear phase creates the condition for the reconnexion process in the sense of Petschek.