MHD ANALYSIS OF PETSCHEK-TYPE RECONNECTION IN NONUNIFORM FIELD AND FLOW GEOMETRIES

Analytical studies of reconnection have, for the most part, been confi ned to steady and uniform current sheet geometries. In contrast to the se implifications, natural phenomena associated with the presence of c urrent sheets indicate highly non-uniform structure and time-varying b ehaviour. Examples include the violent outbursts of energy on the Sun known as solar flares, and magnetospheric phenomena such as flux trans fer events, plasmoids, and amoral activity. Unlike the theoretical mod els, reconnection therefore occurs in a highly dynamic and structured plasma environment. In this article we review the mathematical tools a nd techniques which are available to formulate models capable of descr ibing the effects of reconnection in such situations. We confine atten tion to variants of the reconnection model first discussed by Petschek in the 1960s, in view of its successful application in predicting and interpreting phenomena in the terrestrial magnetosphere. The analysis of Petschek-type reconnection is based on the equations of ideal magn etohydrodynamics (MHD), which describe the large-scale behaviour of th e magnetic field and plasma flow outside the diffusion region, which w e determine as a localised part of the current sheet in which reconnec tion is initiated. The approach we adopt here is to transform the MHD equations into a Lagrangian or so-called 'frozen-in' coordinate system . In this coordinate system, the equation of motion transforms into a set of coupled nonlinear equations, in which the presence of inhomogen eous magnetic fields and/or plasma flows gives rise to a term similar to that which appears in the study of the ordinary string equation in a non-homogeneous medium. As demonstrated here, this approach not only clarifies and highlights the effects of such non-uniformities, it als o simplifies the solution of the original set of MHD equations. In par ticular, this is true for those types of problem in which the total pr essure can be considered as a known quantity from the outset. To illus trate the method we solve several 2D problems involving magnetic field and flow non-uniformities: reconnection in a stagnation-point how geo metry with antiparallel magnetic fields; reconnection in a Y-type magn etic field geometry with and without velocity shear across the current sheet; and reconnection in a force-free magnetic field geometry with field lines of the form xy = const. These case examples, chosen for th eir tractability, each incorporate some aspects of the field and flow geomtries encountered in solar-terrestrial applications, and they prov ide a starting point for further analytical as well as numerical studi es of reconnection.