The correlation of fractal structures in the photosphericand the coronal magnetic field

摘要

This work examines the relation between the fractal properties of the photospheric magnetic patterns and those of the corona.magnetic fields in solar active regions.Aims. We investigate whether there is any correlation between the fractal dimensions of the photospheric structures and the magneticdiscontinuities formed in the corona.Methods. To investigate the connection between the photospheric and coronal complexity, we used a nonlinear force-free extrapola-tion method that reconstructs the 3d magnetic fields using 2d observed vector magnetograms as boundary conditions. We then locatedthe magnetic discontinuities, which are considered as spatial proxies of reconnection-related instabilities. These discontinuities formwell-defined volumes, called here unstable volumes. We calculated the fractal dimensions of these unstable volumes and comparedthem to the fractal dimensions of the boundary vector magnetograms.Results. Our results show no correlation between the fractal dimensions of the observed 2d photospheric structures and the extrapo-lated unstable volumes in the corona, when nonlinear force-free extrapolation is used. This result is independent of efforts to (1) bringthe photospheric magnetic fields closer to a nonlinear force-free equilibrium and (2) omit the lower part of the modeled magnetic fieldvolume that is almost completely filled by unstable volumes. A significant correlation between the fractal dimensions of the photo-spheric and corona magnetic features is only observed at the zero level (lower limit) of approximation of a current-free (potential)magnetic field extrapolation.Conclusions. We conclude that the complicated transition from photospheric non-force-free fields to coronal force-free ones hampersany direct correlation between the fractal dimensions of the 2d photospheric patterns and their 3d counterparts in the corona at thenonlinear force-free limit, which can be considered as a second level of approximation in this study. Correspondingly, in the zero andfirst levels of approximation, namely, the potential and linear force-free extrapolation, respectively, we reveal a significant correlationbetween the fractal dimensions of the photospheric and coronal structures, which can be attributed to the lack of electric currents orto their purely field-aligned orientation.