Within this work, a numerical method was used to calculate the magnetic field above solar active regions based on photospheric vector magnetic field measurements. Since it is the magnetic field which is the major influence controlling the structure of the outer solar atmosphere, our method is based on the force-free assumption, i. e. the adoption that the coronal currents are coaligned with the magnetic field. We extrapolated the measured photospheric field vector into the chromosphere and corona in order to derive the 3D magnetic field structure in these atmospheric layers. Potential (current-free) and nonlinear force-free field models were used to calculate the coronal magnetic field, where the latter represents the currently most sophisticated and most realistic approximation to the true coronal magnetic field.
By the analysis of the coronal magnetic field above two solar active regions we demonstrated that nonlinear force-free field extrapolations are a useful tool to investigate its evolution before and after solar eruptions, like flares and coronal mass ejections. We were able to calculate the magnetic energy content, the free magnetic energy and the distribution of the energy density. This allowed us to show that magnetic energy is slowly build up before the eruptions and that the amount of stored magnetic energy indicates the magnitude of the associated explosive events. Besides localizing the height and exact position in the solar atmosphere where the excess energy was stored prior to and released during the eruptions, we found evidence for a proposed coronal implosion scenario which states that the disposal of free magnetic energy in a flaring region leads to a decrease of the magnetic pressure.
As a consequence of solar eruptions, part of the coronal magnetic energy is transformed into other forms of energy such as kinetic energy or heat. Therefore, an alternative measure to quantify the topological properties of the magnetic field in form of the magnetic helicity is favored due to its conservation properties. To evaluate the magnetic helicity content of the solar corona we need to calculate the magnetic vector potential and a newly developed method to compute it is presented.