Journal of Physics of Plasmas—E. N. Parker Special Topic: B. C. Low
The two-plate initial boundary-value problem of Parker is reviewed, treating the relaxation of a 3D magnetic field prescribed with an arbitrary topology to a terminal force-free field in a cold, viscous, electrically perfect fluid-conductor. Anchored by their foot-points at the perfectly conducting rigid plates, the relaxing field preserves its topology. The Parker Magnetostatic Theorem states that for most prescribed field topologies, the terminal field must embed current sheets. The elements of this Theorem are examined to relate this initial boundary-value problem to (i) the variational problem for a force-free field of a given topology and (ii) the direct construction of a force-free field in terms of its pair of Euler flux functions. Insights are presented on the Theorem as the compelling basis of the Parker theory of solar coronal heating.
The uniform magnetic field in (a) is given a twisted topology in its lines of force by a continuous displacement of the magnetic foot-points at the boundary plates z = 0, L, continuing the displacement smoothly into the interior to create the deformed continuous field (b) that is then anchored to the plates treated as rigid electrically perfect conductor. The twisted field so created is then allow to push its way by its Lorentz force in the idealized, cold, perfectly-conducting viscous fluid between the plates, to evolve and attain a force-free equilibrium with a vanishing Lorentz force everywhere. The MHD of such a system constrains the field to preserve its topology, in consequence of which the relaxed force-free equilibrium generally must contain infinitesimally-thin sheets of electric current, according to the Parker Magnetostatic Theorem, depicted graphically in (c) in terms of two magnetic flux subsystems coming into contact by expelling a third subsystem out of the way. In a real plasma of high conductivity, this effect is still present, except that the thin current sheets cannot reach zero thickness only to dissipate at the otherwise negligible resistivity. This article examines and understands the elements of the Theorem with a mathematically and physically completeness, as a fundamental explanation of the observed solar corona at its million-degree temperature.