Nonlinear force-free modeling of the solar coronal magnetic field

The coronal magnetic field is an important quantity because the magnetic field dominates the structure of the solar corona. Unfortunately, direct measurements of coronal magnetic fields are usually not available. The photospheric magnetic field is measured routinely with vector magnetographs. These photospheric measurements are extrapolated into the solar corona. The extrapolated coronal magnetic field depends on assumptions regarding the coronal plasma, for example, force-freeness. Force-free means that all nonmagnetic forces like pressure gradients and gravity are neglected. This approach is well justified in the solar corona owing to the low plasma beta. One has to take care, however, about ambiguities, noise and nonmagnetic forces in the photosphere, where the magnetic field vector is measured. Here we review different numerical methods for a nonlinear force-free coronal magnetic field extrapolation: Grad-Rubin codes, upward integration method, MHD relaxation, optimization, and the boundary element approach. We briefly discuss the main features of the different methods and concentrate mainly on recently developed new codes.