Singular magnetic equilibria in the solar x-ray corona

Interlaced magnetic field line topologies are expected in the active x-ray solar coronal arches. In the presence of uniform fluid pressure the equilibrium involves magnetic stresses in equilibrium among themselves, i.e. the divergence of the magnetic stress tensor Mij is zero, ∂Mij/∂xj = 0. In the presence of an ideal infinitely conducting fluid the requirement reduces to the familiar force-free field equilibrium equation, ∇ × B = αB. This paper reviews and comments on the formal analytical demonstration that the equilibrium of almost all interlaced field line topologies involves surfaces of tangential discontinuity (TD). Consider, then, the static equilibrium of a randomly interlaced field extending in the z-direction through an infinitely conducting fluid throughout the space 0 < z < L between the end plates z = 0 and z = L. With the field anchored at both ends it is obvious that a stable equilibrium exists for all interlacing topologies. The equilibrium equation can be reduced to the form of the familiar 2D time-dependent vorticity equation. The solutions to the vorticity equation represent a topological set of measure zero compared with the set of all interlacing field line topologies. Lacking the special topology of the vorticity solutions, it follows that the equilibria of almost all interlaced topologies are described by the so-called weak solutions, containing TDs (current sheets). The spaces between the TDs are filled with a continuous field satisfying the vorticity equation. A brief discussion of the mathematical difficulties in constructing illustrative examples of weak solutions is assisted by recent effort at numerical simulation. In the real world resistive dissipation at incipient TDs appears to be the principal heat source for the solar x-ray corona. (paper)