Reconciling Power-law Slopes in Solar Flare and Nanoflare Size Distributions

We unify the power laws of size distributions of solar flare and nanoflare energies. We present three models that predict the power-law slopes αE of flare energies defined in terms of the 2D and 3D fractal dimensions (DA, DV): (i) the spatiotemporal standard self-organized criticality model, defined by the power-law slope αE1 =1 + 2/(DV + 2) = (13/9) ≈ 1.44; (ii) the 2D thermal energy model, αE2 = 1 + 2/DA = (7/3) ≈ 2.33; and (iii) the 3D thermal energy model, αE3 = 1 + 2/DV = (9/5) ≈ 1.80. The theoretical predictions of energies are consistent with the observational values of these three groups, i.e., αE1 = 1.47 ± 0.07, αE2 = 2.38 ± 0.09, and αE3 = 1.80 ± 0.18. These results corroborate that the energy of nanoflares does not diverge at small energies, since (αE1 < 2) and (αE3 < 2), except for the 2D model (αE2 > 2). Thus, while this conclusion does not support nanoflare scenarios of coronal heating from a dimensionality point of view, magnetic reconnection processes with quasi-1D or quasi-2D current sheets cannot be ruled out.