Evolution of Relative Magnetic Helicity: Method of Computation and Its Application to a Simulated Solar Corona above an Active Region

For a better understanding of solar magnetic field evolution it is appropriate to evaluate the magnetic helicity based on observations and to compare it with numerical simulation results. We have developed a method for calculating the vector potential of a magnetic field given in a finite volume; the method requires the magnetic flux to be balanced on all the side boundaries of the considered volume. Our method uses a fast Laplace/Poisson solver to obtain the vector potentials for a given magnetic field and for the corresponding potential (current-free) field. This allows an efficient calculation of the relative magnetic helicity in a finite 3D volume. We tested our approach on a theoretical model (Low and Lou, Astrophys. J. 352, 343, 1990) and also applied our method to the magnetic field above active region NOAA 8210 obtained by a photospheric-data-driven MHD model. We found that the amount of accumulated relative magnetic helicity coincides well with the relative helicity inflow through the boundaries in the ideal and non-ideal cases. The temporal evolution of relative magnetic helicity is consistent with that of magnetic energy. The maximum value of normalized helicity, H (m)/I broken vertical bar(2)=0.0298, is reached just before a drastic energy release by magnetic reconnection. This value is close to the corresponding value inferred from the formula that connects the magnetic flux and the accumulated magnetic helicity based on the observations of solar active regions.