A Fractal Analysis of Magnetograms Within Active Regions

The magnetograms of 16 active regions, observed during June–November 2015, were obtained from the Joint Science Operations Center (JSOC) from the Solar Dynamics Observatory (SDO) using the Helioseismic and Magnetic Imager (HMI) and used to determine the fractal dimensions of various magnetic field groups. The fields were divided into strong (positive and negative) and weak (positive and negative) groupings. The area-perimeter method was used to determine the fractal dimensions for the umbral and penumbral regions of the magnetograms. The fractal dimensions were found to be 1.79±0.49\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1.79 \pm 0.49$\end{document} and 1.96±0.29\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1.96 \pm 0.29$\end{document} for strong and weak magnetic fields, respectively. When compared to the umbral and penumbral fractal dimensions determined by Rajkumar, Haque and Hrudey (Sol. Phys.292, 170, 2017) using white light images, an inverse relationship was found, with the umbra showing lower fractal dimensions than the penumbra for the magnetic field groups compared to the intensity images. This directly implies that there is greater complexity in the penumbral magnetic field groupings than the umbral region. This has implications for models that constrain how the magnetic field groups are contained in the cooler umbral regions.