On the fractal dimension of small-scale magnetic structures in the Sun

We compare, by means of fractal analyses, the shapes of observed small-scale magnetic structures on the Sun with those of magnetic features resulting from numerical simulations of magnetoconvection. The observations were obtained with the “Gottingen” Fabry-Perot spectrometer at the Vacuum Tower Telescope at the Observatorio del Teide on Tenerife. Magnetograms with 0$\farcs$4–0$\farcs$5 spatial resolution were obtained from two-dimensional Stokes V polarimetry in the $\ion{Fe}{i}$ 6302.5 A line and by image reconstruction with speckle methods. The simulations of magnetoconvection was performed with the MURAM code. It solves the time-dependent MHD equations for a compressible, partly ionized plasma including radiative transfer in LTE. To determine the fractal dimensions the perimeter-area relation is used. We discuss the influence of seeing and noise in the fractal dimension D of the observed magnetograms. A dependence of D on the distance from disk center could not be found. The observations give $D=1.21\pm0.05$ for a pixel size corresponding to 0$\farcs$105, while for the numerical simulations $D=1.38\pm0.07$ for a pixel size of 20.83 km. If we use a yardstick adapted to the spatial resolution the observations give the dimension $D=1.41\pm0.05$ in close agreement with the simulations. This agreement is remarkable since the pixel sizes and spatial resolutions of the simulations and of the observations differ by a factor of 15. The finding supports the view of self-similarity of solar magnetic structures over a large range of scales. In addition, it demonstrates the realism of the simulations and suggests that all important physical processes are included. We discuss our results in comparison with other investigations.