Fractal Analysis of Michelson Doppler Imager Magnetograms: A Contribution to the Study of the Formation of Solar Active Regions

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Abstract
In this paper a fractal analysis of active regions observed by the Michelson Doppler Imager (MDI) instrument on the SOHO spacecraft is performed. The purpose of such an analysis is to study the shape of these magnetic regions via their fractal dimension, which characterizes their complexity, and to deduce some information on the processes that led to their formation. Both full disk and high-resolution MDI magnetograms are used and the fractal dimensions using the perimeter/area relation (d1) and the linear size/area relation (d2) are calculated. A statistical analysis of the estimate of a fractal dimension is performed, as well as that of the influence of noise on it. The fractal dimension for a range of sizes is calculated. The fractal dimension d1 is found to increase with the area of the active regions (from 1.48 for supergranular size structures to 1.68 for the largest structures). The fractal dimension d2 also increases with the area of the active regions (from 1.78 to 1.94). The fractal dimension d1 decreases with the magnetic threshold for moderate sizes and increases for the largest structures. The high-resolution results match those of full disk magnetograms when the images are degraded. This fractal analysis is performed in relation to the size distribution properties: the two analyses provide complementary information. Then some models of active regions very similar to those of Wentzel & Seiden are introduced to interpret the observations in terms of a percolation process at the bottom of the convective zone and diffusion at the surface. These models lead to structures that are more complex than the observed structures.