Solutions and symmetries of force-free magnetic fields

Abstract

New analytical results concerning force-free magnetic fields are presented. A number of examples of exact solutions for two-dimensional nonlinear force-free fields described by the Liouville equation are shown. These include classical solutions, such as, the Gold–Hoyle field and the force-free Harris sheet as special cases. The connection between these solutions and the Lie point symmetries of the Liouville equation is illustrated. Lie point symmetries of the equation describing force-free magnetic fields in helical symmetry in cylindrical geometry are also investigated and an infinitesimal generator that, in the vicinity of the cylinder axis, makes it possible to transform purely radially dependent solutions into helically symmetric solutions, is found. Finally we point out the existence of a formal analogy between the equations for the vector potential components of a class of force-free fields and the equations of motion of a charged particle in a magnetic field. This analogy makes it possible to transfer known results from the theory of the motion of a charged particle, into the context of force-free magnetic fields. Explicit examples of such application are given.