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

Analytical studies of reconnection have, for the most part, been confined to steady and uniform current sheet geometries. In contrast to these implifications, natural phenomena associated with the presence of current sheets indicate highly non-uniform structure and time-varying behaviour. Examples include the violent outbursts of energy on the Sun known as solar flares, and magnetospheric phenomena such as flux transfer events, plasmoids, and auroral activity. Unlike the theoretical models, reconnection therefore occurs in a highly dynamic and structured plasma environment. In this article we review the mathematical tools and techniques which are available to formulate models capable of describing the effects of reconnection in such situations. We confine attention 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 magnetohydrodynamics (MHD), which describe the large-scale behaviour of the magnetic field and plasma flow outside the diffusion region, which we determine as a localised part of the current sheet in which reconnection 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 inhomogeneous 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 also simplifies the solution of the original set of MHD equations. In particular, this is true for those types of problem in which the total pressure can be considered as a known quantity from the outset. To illustrate the method, we solve several 2D problems involving magnetic field and flow non-uniformities: reconnection in a stagnation-point flow geometry with antiparallel magnetic fields; reconnection in a Y-type magnetic 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 their tractability, each incorporate some aspects of the field and flow geomtries encountered in solar-terrestrial applications, and they provide a starting point for further analytical as well as numerical studies of reconnection.