A New Approach for the Calculation of Non-linear Force Free Fields

The magnetic field is the major ordering factor for the low-beta plasma in the solar corona. As measurements of the magnetic field in the corona itself are not: (yet) possible the only way of inferring the coronal field structure is by extrapolation of the magnetic field measured at sub-coronal level. The appropriate lowest order equilibrium equations for the corona are those of non-linear force free magnetic fields. The fundamental problems associated with the solution of these equations are their non-linearity which prevents the use of analytical methods and that from a mathematical point of view the whole problem is ill-posed.

In the present contribution a new idea for the numerical calculation of non-linear force free fields is presented. It is based on the application of Galerkin's method to the force free equations written in tensorial form with the magnetic stress tensor as basic quantity. In its present form the approach imposes the solenoidal condition for the magnetic field by the method of Lagrangean multipliers. The implementation of boundary conditions and possible ways of a numerical realisation will be discussed.