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

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      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 \(E \propto F T \propto T^{1+D_S/2}\) 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.

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      Self-organized criticality: An explanation of the 1/fnoise

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        Self-organized criticality

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          Self-organized criticality

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            Author and article information

            Journal
            20 December 2011
            1112.4859 10.1051/0004-6361/201118237

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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

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