Principle of Minimum Energy in Magnetic Reconnection in a Self-organized Critical Model for Solar Flares

Abstract

Solar flares are an abrupt release of magnetic energy in the Sun's atmosphere due to reconnection of the coronal magnetic field. This occurs in response to turbulent flows at the photosphere which twist the coronal field. Similar to earthquakes, solar flares represent the behavior of a complex system, and expectedly their energy distribution follows a power law. We present a statistical model based on the principle of minimum energy in a coronal loop undergoing magnetic reconnection, which is described as an avalanche process. We show the distribution of peaks for the flaring events in this self-organized critical system is scale-free. The obtained power law index 1.84 ± 0.02 for the peaks is in good agreement with satellite observations of soft X-ray flares. The principle of minimum energy can be applied for general avalanche models to describe many other phenomena.

Synchronization of Small-scale Magnetic Features, Blinkers, and Coronal Bright Points

“…To obtain the power-law index (α), we applied a maximum-likelihood fitting for data without binning (Clauset et al 2009). This power-law behavior in the distribution tails can be due to a self-similar feature in the reconnection process that generates blinkers, analogous to what has been observed for flares and nanoflares as the system of self-organised criticality (Lin et al 1984;Crosby et al 1993;Krucker & Benz 1998;Parnell & Jupp 2000;Klimchuk et al 2009;Farhang et al 2018Farhang et al , 2019. We also observed that (Figure 12 right panel) the normalized maximum intensity of transients are correlated with the magnetic flux (the linear fits in log-log scale with slopes β about 1.31, 0.94, and 0.97 (red lines) in the scatter plots) for blinkers, ECBPs, and XCBPs, respectively.…”

天体物理中自组织临界现象的观测和模拟

“…统计发现, 雪崩的持续时间、 峰值强度和积分 能量在频数分布上表现为幂律形式. 此后, 又有一 系列基于不同的网格构建形式、能量驱动机制、临界 判定条件和雪崩分配规则的 CA 模型发展出来, 以 实现更加符合物理图像的太阳及天体爆发活动的 CA 模拟 [42,43,44,45,46,47,48,49,50] .…”

“…其中 ri 为(0, 1)之间的随机数，a = ∑ri, x 为自由参数, 由雪崩发生后系统能量达到最低的第一性原理所确 定 [49,50] , 也即 图11 Cluster卫星观测到的太阳风能谱 [125] .…”

Energy Balance in Avalanche Models for Solar Flares

“…In order to determine the value of x in Equation (8), which maximizes the amount of released energy, the first derivative of the energy difference between two consecutive driving steps, E n+1 − E n , is calculated. Considering the energy of the system as E = 1 2 A · J , and also assuming that the released energy during a redistribution depends on the nodal values of all 13 neighboring cells (Farhang et al 2018, Figure 1 therein), then x is determined by solving:…”