Statistical Properties of the Energy Release in Emerging and Evolving Active Regions

The formation and evolution of active regions are inherently complex phenomena. Magnetic fields generated at the base of the convection zone follow a chaotic evolution before reaching the solar surface. In this article, we use a two-dimensional probabilistic cellular automaton to model the statistical properties of the magnetic patterns formed on the solar surface and to estimate the magnetic energy released in the interaction of opposite polarities. We assume that newly emerged magnetic flux tubes stimulate the emergence of new magnetic flux in their neighborhood. The flux tubes move randomly on the surface of the Sun, and they cancel and release their magnetic energy when they collide with magnetic flux of opposite polarity, or diffuse into the "empty" photosphere. We assume that cancellation of magnetic flux in collisions causes "flares" and determine the released energy as the difference in the square of the magnetic field flux (E ~ B2). The statistics of the simulated flares follow a power-law distribution in energy, f(E) ~ E-a, where a = 2.2 ± 0.1. The size distribution function of the simulated active regions exhibits a power-law behavior with index k ≈ 1.93 ± 0.08, and the fractal dimension of the magnetized areas on the simulated solar surface is close to DF ~ 1.42 ± 0.12. Both quantities, DF and k, are inside the range of the observed values.