A criterion for splitting of a reconnecting current sheet into MHD discontinuities

We consider the splitting of a reconnecting current sheet into MHD discontinuities, which is observed in many numerical simulations of the magnetic reconnection process, We suppose that the splitting takes place as a consequence of non-evolutionarity of the reconnecting current sheet as a one-dimensional discontinuity. This means that the problem of the time evolution of its small perturbations does not have a unique solution. Since a physical problem must always have a unique solution; a non-evolutionary discontinuity cannot exist in a real plasma, and splits into evolutionary discontinuities. However, this approach cannot be immediately applied to a current sheet, because the flow velocity inside the sheet is two-dimensional and it cannot be generally treated as a one-dimensional discontinuity. Solving the linear MHD equations inside and outside the sheet, we show that for large enough plasma conductivity, certain small perturbations interact with the sheet as with a discontinuity. On the basis of the non-evolutionarity criterion, with respect to these perturbations, we obtain a condition on the flow velocity at the sheet surface, under which the splitting takes place.