A class of driven, dissipative, energy-conserving magnetohydrodynamic

equilibria with flow
Authors: GOODMAN MICHAEL L. 
Published: July 2002

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.