We investigate physical scaling laws for magnetic energy dissipation in solar flares, in the framework of the Sweet-Parker model and the Petschek model. We find that the total dissipated magnetic energy $E_{diss}$ in a flare depends on the mean magnetic field component $B_f$ associated with the free energy $E_f$, the length scale $L$ of the magnetic area, the hydrostatic density scale height $\lambda$ of the solar corona, the Alfven Mach number $M_A=v_1/v_A$ (the ratio of the inflow speed $v_1$ to the Alfvenic outflow speed $v_A$), and the flare duration $\tau_f$, i.e., $E_{diss} = (1/4\pi) B_f^2\ L\ \lambda\ v_A\ M_A\ \tau_f$, where the Alfven speed depends on the nonpotential field strength $B_{np}$ and the mean electron density $n_e$ in the reconnection outflow. Using MDI/SDO and AIA/SDO observations and 3-D magnetic field solutions obtained with the vertical-current approximation nonlinear force-free field code (VCA-NLFFF) we measure all physical parameters necessary to test scaling laws, which represents a new method to measure Alfven Mach numbers $M_A$, the reconnection rate, and the total free energy dissipated in solar flares.

中文翻译：

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太阳耀斑的全球能量学。X. Petschek 重联率和 Alfvén 马赫磁重联流出数

我们在 Sweet-Parker 模型和 Petschek 模型的框架内研究了太阳耀斑中磁能耗散的物理标度定律。我们发现耀斑中总耗散的磁能 $E_{diss}$ 取决于与自由能 $E_f$ 相关的平均磁场分量 $B_f$、磁区的长度尺度 $L$、静水密度日冕标高 $\lambda$、阿尔文马赫数 $M_A=v_1/v_A$（流入速度 $v_1$ 与阿尔文流出速度 $v_A$ 的比值）和耀斑持续时间 $\tau_f$ ，即 $E_{diss} = (1/4\pi) B_f^2\ L\ \lambda\ v_A\ M_A\ \tau_f$，其中 Alfven 速度取决于非势场强度 $B_{np}$ 和重联流出中的平均电子密度 $n_e$。