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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.

A cylindrically symmetric, electrically driven, dissipative, energy-conserving magnetohydrodynamic equilibrium model is considered. The high-magneticfield Braginskii ion thermal conductivity perpendicular to the local magnetic field and the complete electron resistivity tensor are included in an energy equation and in Ohm's law. The expressions for the resistivity tensor and thermal conductivity depend on number density, temperature, and the poloidal and axial (z-component) magnetic field, which are functions of radius that are obtained as part of the equilibrium solution. The model has plasma-confining solutions, by which is meant solutions characterized by the separation of the plasma into two concentric regions separated by a transition region that is an internal boundary layer. The inner region is the region of confined plasma, and the outer region is the region of unconfined plasma. The inner region has average values of temperature, pressure, and axial and poloidal current densities that are orders of magnitude larger than in the outer region. The temperature, axial current density and pressure gradient vary rapidly by orders of magnitude in the transition region. The number density, thermal conductivity and Dreicer electric field have a global minimum in the transition region, while the Hall resistivity, Alfvén speed, normalized charge separation, Debye length, (ωλ)ion and the radial electric field have global maxima in the transition region. As a result of the Hall and electron-pressure-gradient effects, the transition region is an electric dipole layer in which the normalized charge separation is localized and in which the radial electric field can be large. The model has an intrinsic value of β, about 13·3%, which must be exceeded in order that a plasma-confining solution exist. The model has an intrinsic length scale that, for plasma-confining solutions, is a measure of the thickness of the boundary-layer transition region. If appropriate boundary conditions are given at R = 0 then the equilibrium is uniquely determined. If appropriate boundary conditions are given at any 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, which is almost homogeneous and cannot confine a plasma. Exact expressions for the two values of the axial current in the bifurcation are derived. If the boundary conditions are given at R = a then a solution exists if and only if the constant driving electric field exceeds a critical value. An exact expression for this critical electric field is derived. It is conjectured that the bifurcation is associated with an electric-field-driven transition in a real plasma, between states with different rotation rates, energy dissipation rates and confinement properties. Such a transition may serve as a relatively simple example of the L—H mode transition observed in tokamaks.

A cylindrically symmetric, electrically driven, dissipative, energy-conserving magnetohydrodynamic equilibrium model is considered. The high-magnetic-field Braginskii electron electrical resistivity η parallel to a constant axial magnetic field B and ion thermal conductivity ĸ perpendicular to B are included in an energy equation and in Ohm's law. The expressions for η and ĸ depend on number density and temperature, which are functions of radius that are obtained as part of the equilibrium solution. The model has plasma-confining solutions, by which are meant solutions characterized by the separation of the plasma into two regions separated by a relatively thin transition region that is an internal boundary layer across which temperature and current density vary rapidly. The inner region has a temperature, pressure and current density that are much larger than in the outer region. The number density and thermal conductivity attain their minimum values in the transition region. The model has an intrinsic value of β, about 6.6%, which must be exceeded in order that a plasma-confining solution exist. The model has an intrinsic length scale, which, for plasma-confining solutions, is a measure of the thickness of the transition region separating the inner and outer regions of plasma. A larger class of transport coefficients is modelled by artificially changing η and ĸ by changing the constant coefficients ηO and ĸO that occur in their expressions. Increasing ĸO transforms a state that does not exhibit confinement into one that does, improves the confinement in a state that already exhibits it, and leads to an increase in ĸ in the confined region of plasma. The improvement in confinement consists in a decrease in the thickness of the transition region. Decreasing ηO subject to certain constraints, also transforms a state that does not exhibit confinement into one that does, improves the confinement in a state that already exhibits it, and leads to a decrease in η in the confined region of plasma. Increasing ηO up to a critical point increases the current, temperature, and volume of the confined region of plasma, and causes the thickness of the transition region to increase. If ηO is increased beyond the critical point, a plasma-confining state cannot exist. In all cases it is found that an increase in ĸ and a decrease in η in the confined region of plasma are associated with an improvement in the confinement properties of the equilibrium state. If the pressure and temperature are given on the cylinder wall, the equilibrium bifurcates when the electric field decreases below a critical value. The equilibrium can bifurcate into a state that exhibits confinement and a state that does not.