Nonlinear force‐free magnetic fields

Astrophysical magnetic fields in a low beta plasma can evolve in a quasi‐steady state for extended periods of time which are much longer than the characteristic dynamical time scale. The evolution proceeds with the electric current flowing largely parallel to the magnetic field, so that as a first approximation the Lorentz force may be set to zero everywhere. This paper reviews in a unified picture the results of various theoretical investigations of this process. These investigations seek to uncover those circumstances under which extreme conditions develop to make the assumption of a quasi‐steady state untenable, and the field must pass into a fast dynamical phase at this point. The astrophysical interest is in explaining how a solar magnetic field would occasionally break into a flare or other eruptions. Two types of mechanisms have been treated. In the first the force‐free field is in a medium of infinite electrical conductivity. The field evolves in response to slowly changing boundary conditions brought about by photospheric motions in the solar active region. In the second mechanism the medium has a small finite electrical conductivity, and the field evolves in response to the slow Ohmic dissipation of the electric current. The field evolution in both cases is nonlinear, posing difficult mathematical problems. This review emphasizes exact mathematical treatment, outlining the progress the theory has made from isolated discoveries of particular solutions to general theorems upon which broad physical conclusions can be based.