Self-similar resistive decay of a current sheet in a compressible plasma

Abstract

Self-similar solutions of the magnetogasynamic equations are derived which describe the resistive decay of a plane current sheet in a compressible plasma. Such current sheets are thought to provide the magnetic energy storage in solar flares. They also occur at the boundaries between regions containing different magnetic-field directions in interplanetary space, and at the interface between the solar wing and the earth' magnetic field. It is shown that the resistive decay of a current sheet in a compressible plasma must involve plasma motion. The convective effects associated with this motion are incorporated in the analysis; the inertia effects are not. The electrical and thermal conductivities are taken to be constant, but the analysis may easily be generalized to include realistic temperature and magnetic field dependences of these quantities. Radiative and viscous terms are not included. The ordinary differential equations resulting from the similarity hypothesis are solved numerically, yielding curves of the plasma density, temperature, and velocity, as well as of the magnetic and induced electric fields, as functions of the similarity variable. The non-dimensional groups of importance are: y, the ratio of specific heats at constant pressure and constant volume; Kx, the ratio of thermal to resistive diffusivity; β_∞, the ratio of plasma pressure to magnetic pressure at large distances from the current sheet. The first of these ratios is kept constant and equal to 5/3, corresponding to a monoatomic gas. The behaviour of the solution when the other two ratios are varied is investigated. The plasma velocity at large distances from the current sheet does not vanish in these solutions. It is always directed toward the sheet. However, when the diffusivity ratio K_∞ is small, plasma flow away from the centre of the sheet also occurs in two narrow regions, one on each side of the centre. As a result of the reversals in the flow direction, the density then displays a relative minimum at the centre of the sheet with two outward travelling maxima adjacent to it. The plasma temperature at the centre of the sheet becomes very large for small K_∞ and β_∞ The expansion of the sheet becomes explosive and inertia effects can no longer be neglected. The physical meaning of these results is discussed and directions for further research are outlined.