A class of driven, dissipative, energy-conserving magnetohydrodynamic equilibria with flow

Abstract

The classical transport coefficients provide an accurate description of transport processes in collision-dominated plasmas. These transport coefficients are used in a cylindrically symmetric, electrically driven, steady-state magnetohydrodynamic (MHD) model with flow and an energy equation to study the effects of transport processes on MHD equilibria. The transport coefficients, which are functions of number density, temperature and magnetic field strength, are computed self-consistently as functions of radius R. The model has plasma-confining solutions characterized by the existence of an inner region of plasma with values of temperature, pressure and current density that are orders of magnitude larger than in the surrounding, outer region of plasma that extends outward to the boundary of the cylinder at R=a. The inner and outer regions are separated by a boundary layer that is an electric-dipole layer in which the relative charge separation is localized, and in which the radial electric field, temperature, pressure and axial current density vary rapidly. By analogy with laboratory fusion plasmas in confinement devices, the plasma in the inner region is confined plasma, and the plasma in the outer region is unconfined plasma. The solutions studied demonstrate that the thermoelectric current density, driven by the temperature gradient, can make the main contribution to the current density, and that the thermoelectric component of the electron heat flux, driven by an effective electric field, can make a large contribution to the total heat flux. These solutions also demonstrate that the electron pressure gradient and Hall terms in Ohm's law can make dominant contributions to the radial electric field. These results indicate that the common practice of neglecting thermoelectric effects and the Hall and electron pressure-gradient terms in Ohm's law is not always justified, and can lead to large errors. The model has three, intrinsic, universal values of β at which qualitative changes in the solutions occur. These values are universal in that they only depend on the ion charge number and the electron-to-ion mass ratio. The first such value of β (about 3.2% for a hydrogen plasma), when crossed, signals a change in sign of the radial gradient of the number density, and must be exceeded in order that a plasma-confining solution exist for a plasma with no flow. The second such value of β (about 10.4% for a hydrogen plasma), when crossed, signals a change in sign of the poloidal current density. Some of the solutions presented exhibit this current reversal. The third such value of β is about 2.67 for a hydrogen plasma. When β is greater than or equal to this value, the thermoelectric, effective electric-field-driven component of the electron heat flux cancels 50% or more of the temperature-gradient-driven ion heat flux. If appropriate boundary conditions are given on the axis R=0 of the cylinder, the equilibrium is uniquely determined. Analytical evidence is presented that, together with earlier work, strongly suggests that if appropriate boundary conditions are enforced at the outer boundary R=a then the equilibrium exhibits a bifurcation into two states, one of which exhibits plasma confinement and carries a larger axial current than the other state, which is close to global thermodynamic equilibrium, and so is not plasma-confining. Exact expressions for the two values of the axial current in the bifurcation are presented. Whether or not a bifurcation can occur is determined by the values of a critical electric field determined by the boundary conditions at R=a, and the constant driving electric field, which is specified. An exact expression for the critical electric field is presented. Although the ranges of the physical quantities computed by the model are a subset of those describing fusion plasmas in tokamaks, the model may be applied to any two-component, electron–ion, collision-dominated plasma for which the ion cyclotron frequency is much larger than the ion–ion Coulomb collision frequency, such as the plasma in magnetic flux tubes in the solar interior, photosphere, lower transition region, and possibly the upper transition region and lower corona.