On driven, dissipative, energy-conserving magnetohydrodynamic equilibria

Abstract

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.