Modeling solar force-free magnetic fields

A class of nonlinear force-free magnetic fields is presented, described in terms of the solutions to a second-order, nonlinear ordinary differential equation. These magnetic fields are three-dimensional, filling the infinite half-space above a plane where the lines of force are anchored. They model the magnetic fields of the sun over active regions with a striking geometric realism. The total energy and the free energy associated with the electric current are finite and can be calculated directly from the magnetic field at the plane boundary using the virial theorem. In the study of solar magnetic fields with data from vector magnetographs, there is a long-standing interest in devising algorithms to extrapolate for the force-free magnetic field in a given domain from prescribed field values at the boundary. The closed-form magnetic fields of this paper open up an opportunity for testing the reliability and accuracy of algorithms that claim the capability of performing this extrapolation. The extrapolation procedure as an ill-posed mathematical problem is discussed.