A Statistical Fractal-Diffusive Avalanche Model of a Slowly-Driven Self-Organized Criticality System

Abstract

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. 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 $L\propto T^{1/2}$; (ii) The instantaneous energy dissipation rate $f(t)$ occupies a fractal volume with dimension $D_S$, which predicts the relationships $F=f(t=T)\propto L^{D_S}\propto T^{D_S/2}$, $P\propto L^S\propto T^{S/2}$ for the peak energy dissipation rate, and $E\propto FT\propto T^{1+D_S/2}$ for the total dissipated energy; (iii) The mean fractal dimension of avalanches in Euclidean space $S=1,2,3$ is $D_S\approx (1+S)/2$; and (iv) The occurrence frequency distributions $N(x)\propto x^{-{\alpha }_x}$ based on spatially uniform probabilities in a SOC system are given by $N(L)\propto L^{-S}$, which 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 ${\alpha }_E=1+(S-1)/(D_S+2)$. We test the predicted fractal dimensions, occurrence frequency distributions, and correlations with numerical simulations of cellular automaton models in three dimensions $S=1,2,3$ and find satisfactory agreement within $\approx 10$. 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.

9 Figures, 1 movie available at authors website, http://www.lmsal.com/~aschwand/eprints/cellular_automaton_movie.jpeg

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