The Topological Nature of Boundary Value Problems for Force‐Free Magnetic Fields

Abstract

The difficulties of constructing a three-dimensional, continuous force-free magnetic field in the solar corona are investigated through a boundary value problem posed for the unbounded domain external to a unit sphere. The normal field component Bn and the boundary value αb of the twist function α on the unit sphere, combined with the demand for a vanishing field at infinity, do constitute sufficient conditions for determining a solution if it exists, but Bn and αb cannot be prescribed independently. An exhaustive classification of the admissible (Bn,αb)-pairs is developed, using the topological properties of the α flux surfaces implied by their footprints described by the constant-αb curves on the unit sphere. The incompatibilities arising from boundary conditions contradicting the field equations are distinguished from the interesting one of (Bn,αb) being, in principle, admissible but requiring a weak solution describing a force-free field containing inevitable magnetic tangential discontinuities. This particular incompatibility relates the boundary value problem to the Parker theory of spontaneous current sheets in magnetic fields embedded in electrically perfectly conducting fluids. Our investigation strategically skirts around some important but formidable mathematical problems to arrive at physically definite conclusions and insights on the construction of force-free fields, both in the practical task of modeling coronal magnetic fields with magnetopolarimetric data and in the basic understanding of the Parker theory. Two specific demonstrations of (Bn,αb) are given to illustrate circumstances under which a continuous solution to the boundary value may or may not exist.