Based on the area fractal dimension D2 of solar flares measured in Paper I, we carry out modeling of the three-dimensional (3D) flare volume here and derive an analytical relation between the volume fractal scaling V(L) ∝ LD3 and the area fractal scaling A(L) ∝ LD2. The 3D volume model captures a flare arcade with a variable number of flare loops; its fractal structure is not isotropic, but consists of aligned one-dimensional substructures. The geometry of the arcade model has three free parameters and makes some simplifying assumptions, such as semicircular loops, east-west orientation, location near the equator, and no magnetic shear. The analytical model predicts the scaling of the area filling factor qA(nloop) and volumetric filling factor qV(nloop) as a function of the number of loops nloop, and allows one to predict the volume filling factor qV(qA) and volume fractal dimension D3(D2) from the observationally measured parameters qA and D2. We also corroborate the analytical model with numerical simulations. We apply this fractal model to the 20 flares analyzed in Paper I and find maximum volume filling factors with a median range of qV ≈ 0.03–0.08 (assuming solid filling for loop widths of 1 Mm). The fractal nature of the flare volume has important consequences for correcting electron densities determined from flare volume emission measures and density-dependent physical quantities, such as the thermal energy or radiative cooling time. The fractal scaling has also far-reaching consequences for frequency distributions and scaling laws of solar and stellar flares.
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