Three-dimensional Structures of Magnetostatic Atmospheres. VII. Magnetic Flux Surfaces and Boundary Conditions

We treat a long-standing difficulty with the boundary conditions for the governing equations derived in the third paper of this series describing magnetic fields in equilibrium in a three-dimensional atmosphere. The force-free magnetic fields are special cases of this magnetostatic model. Among the flux surfaces of these magnetic fields, those of constant twist parameter, the so-called α-surfaces, have a basic topological property previously pointed out by Aly for force-free fields. This property dictates a natural set of boundary conditions for the field equations and can be used to reduce the vector field equations to a novel form, illustrated here with analytical examples of three-dimensional force-free fields. The physical implications of this theoretical development for solar coronal magnetic fields are discussed.