A statistical fractal-diffusive avalanche model of a slowly-driven self-organized criticality system

Aims. We develop a statistical analytical model that predicts the occurrence frequency distributions and parameter correlations of avalanches in nonlinear dissipative systems in the state of a slowly-driven self-organized criticality (SOC) system. Methods. This model, called the fractal-diffusive SOC model, is based on the following four assumptions: (i) the avalanche size L grows as a diffusive random walk with time T, following L proportional to T-1/2; (ii) the energy dissipation rate f (t) occupies a fractal volume with dimension D-S; (iii) the mean fractal dimension of avalanches in Euclidean space S = 1, 2, 3 is D-S approximate to (1+ S)/2; and (iv) the occurrence frequency distributions N(x). x(-ax) based on spatially uniform probabilities in a SOC system are given by N(L) proportional to L-S, with S being the Eudlidean dimension. We perform cellular automaton simulations in three dimensions (S = 1, 2, 3) to test the theoretical model. Results. The analytical model predicts the following statistical correlations: F proportional to L-DS. T-DS/2 for the flux, P proportional to L-S proportional to T-S/2 for the peak energy dissipation rate, and E proportional to FT proportional to T1+DS/2 for the total dissipated energy; the model predicts powerlaw distributions for all parameters, with the slopes alpha(T) = (1+ S)/2, alpha(F) = 1+(S - 1)/D-S, alpha(P) = 2-1/S, and alpha(E) = 1+(S - 1)/(D-S + 2). The cellular automaton simulations reproduce the predicted fractal dimensions, occurrence frequency distributions, and correlations within a satisfactory agreement within approximate to 10% in all three dimensions. Conclusions. One profound prediction of this universal SOC model is that the energy distribution has a powerlaw slope in the range of alpha(E) = 1.40-1.67, and the peak energy distribution has a slope of alpha(P) = 1.67 (for any fractal dimension D-S = 1, ... , 3 in Euclidean space S = 3), and thus predicts that the bulk energy is always contained in the largest events, which rules out significant nanoflare heating in the case of solar flares.