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.

Two flux surfaces pressed into contact

The fields (a) ${\bf B}_0$ and (b) ${\bf B}_{\tau} ({\bf r})$ described in the text. The sketch (c) of two flux surfaces pressed into contact through a hole punched into a third intervening flux surface, discussed in Section II.D.
Fig. 2 (a) The Lundquist $B_{\varphi}(R) = {\rm J}_1(\alpha_