The Astrophysical Journal, 632:1168–1175, 2005 October 20 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.
SELF-CONSISTENT MAGNETOHYDRODYNAMIC MODELING OF CURRENT SHEET STRUCTURE AND HEATING USING REALISTIC DESCRIPTIONS OF TRANSPORT PROCESSES Michael L. Goodman Institute for Scientific Research, Inc., 2500 Fairmont Avenue, Suite 734, Fairmont, WV 26555-2720;
[email protected] Received 2005 April 6; accepted 2005 June 29
ABSTRACT A magnetohydrodynamic (MHD) model of an electron-ion, collision-dominated plasma that includes the electrical conductivity and thermoelectric tensors in Ohm’s law is used to generate current sheet solutions in parameter ranges that correspond to those of the solar transition region and lower corona. The model contains a prescribed sheared magnetic field with a characteristic length scale L. The characteristic sheet width is 2L, but it is found that the temperature has transition region or coronal values only within a diffusion region (DR) with a width several orders of magnitude smaller than 2L. The heating rate per unit mass and flow speed in the DR are orders of magnitude larger, and the density is orders of magnitude smaller than in the surrounding plasma. The heating rate per unit volume is a maximum in the DR and falls off steadily outside the DR. The Joule heating rate and current density each consist of a conduction component driven by the center-of-mass electric field and a thermoelectric component driven by the temperature gradient. It is found that these components largely cancel, leading to a total heating rate and current density orders of magnitude smaller than either of their components. This suggests that thermoelectric current drive is important in determining current sheet structure. The center-of-mass electric field that provides the energy to maintain the plasma in a steady state is almost entirely the convection electric field. The electron magnetization Me is the product of the electron cyclotron frequency and the electron-ion collision time. Nonzero values of Me cause the conductivity and thermoelectric tensors to be anisotropic. It is found that the large values of Me that occur in the DR increase the heating rates per unit volume and mass by several orders of magnitude and can change the sign of the heating rate per unit mass from negative to positive, corresponding to a change from a cooling process to a heating process. This suggests that electron magnetization, and hence anisotropic transport, is a major factor in current sheet heating. Subject headinggs: MHD — stars: coronae — Sun: corona — Sun: magnetic fields — Sun: transition region
1. INTRODUCTION
out most of the solar corona, the solar wind, the Earth’s magnetosphere, and in the core regions of fusion plasma devices such as tokamaks. Only classical transport is considered here. The parameter ranges of density, temperature, and magnetic field strength used in the examples considered here are characteristic of the transition region and lower corona where collision-dominated transport is likely to be a significant part of the total transport (Goodman 1998). The representation of transport processes in an MHD model must take the form of transport coefficients that are functions of the MHD variables of temperature, number density, and magnetic field. Such a representation exists in closed form for classical transport in an electron-ion plasma (Braginskii 1965; Chapman & Cowling 1970; Mitchner & Kruger 1973; Balescu 1988). The use of MHD models that include classical transport processes to model current sheets and magnetic reconnection can make important contributions to understanding these basic plasma physics phenomena. The solutions presented here include a selfconsistent calculation of the effects of the exact electrical conductivity and thermoelectric tensors for a collision-dominated, electron-ion plasma. Current sheet solutions of this type do not appear to have been considered previously in the literature. Contrary to the case of classical transport, there is no single theory of anomalous transport. This is because the turbulent electric and magnetic fields responsible for this transport can be generated by many types of plasma waves and instabilities. The corresponding transport is in general different for different generation mechanisms. The relevant generation mechanisms for each situation of interest must be known in order to determine
Current sheet solutions to magnetohydrodynamic (MHD) models that include a self-consistent description of anisotropic and inhomogeneous transport processes are necessary for determining the role of current sheets and the associated process of magnetic reconnection in coronal heating, flares, and coronal mass ejections, and are of general interest for understanding current sheets and magnetic reconnection as basic plasma physics phenomena. A simple steady state model that generates a class of such solutions is presented here. The model makes detailed predictions of the structure of the current sheet diffusion region (DR), in which transport processes are most important. The model is intended to be the first in a series of successively more realistic MHD models that predict the structure of the DR, in particular the heating rates per unit mass and volume, in the presence of realistic anisotropic and inhomogeneous transport processes. An extensive literature exists that describes properties of current sheets in the MHD approximation. Most of this work uses models based on ideal MHD, which by definition does not contain effects of transport processes, or uses models that contain the simplest descriptions of transport processes, typically constant scalar representations of the electrical and thermal conductivities and viscosity, and the Hall term in Ohm’s law. Some of this literature explores the role of anomalous transport, also called collisionless transport, in models of current sheets. Anomalous transport, due to wave-particle scattering by microturbulent electric and magnetic fields, as opposed to classical transport (also called collision-dominated transport, due to particle-particle collisions), is dominant in plasmas such as those that exist through1168
CURRENT SHEET MODEL the corresponding transport. It is not clear which generation mechanisms primarily determine anomalous transport in the transition region and lower corona. Authoritative and comprehensive descriptions of the development and state of the art of current sheet theory, the related process of magnetic reconnection, and applications to space and fusion plasmas, including discussions of anomalous transport, are given by Parker (1979, 1994), Priest & Forbes (2000), and Biskamp (1997, 2000, 2003). An extensive review of plasma turbulence in general is given by Krommes (2001). Reviews that discuss anomalous transport in plasmas are given by Horton (1999, 1997), Horton & Ichikawa (1996), O’Neil & Coroniti (1999), Ottaviani et al. (1997), Krommes (1997), Robinson (1997), and Kadomstev (1996). Ichimaru (1975, 1996) derives a collisionless anisotropic conductivity due to microturbulence in the DR of a current sheet. 2. BASIC EQUATIONS
Equation of state:
Maxwell’s equations: J¼
J
