Implosion of a uniform current sheet in a low-beta plasma

Using ideal, one-dimensional MHD equations, numerical and analytic solutions are presented which describe the nonlinear behaviour of an imploding current sheet in a low-fl plasma. Initially the current density is uniformly distributed in asheet of finite thickness, and the Lorentz force tending to pinch the plasma together is unopposed by any fluid pressure force. As the implosion develops the current density in the sheet is concentrated into a thin layer at the centre of the sheet, and both the current density and the current in this layer become infinite in a finite time if β = 0. At the moment this occurs, fast-mode shocks are produced which propagate outward from the centre of the current sheet, and as the shocks move away an infinitely thin current sheet is left behind. Although the solutions are related to electric discharges, they are also closely related to a problem posed by Dungey concerning the evolution of a uniformly distributed current in the vicinity of an X-type magnetic neutral line. The implications of these solutions for Dungey's problemare discussed.