Abstract
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 ∝ T1/2; (ii) the energy dissipation rate f(t) occupies a fractal volume with dimension DS; (iii) the mean fractal dimension of avalanches in Euclidean space S = 1,2,3 is DS ≈ (1 + S)/2; and (iv) the occurrence frequency distributions N(x) ∝ x – αx based on spatially uniform probabilities in a SOC system are given by N(L) ∝ 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 ∝ LDS ∝ TDS/2 for the flux, P ∝ LS ∝ TS/2 for the peak energy dissipation rate, and E ∝ FT ∝ T1 + DS/2 for the total dissipated energy; the model predicts powerlaw distributions for all parameters, with the slopes αT = (1 + S)/2, αF = 1 + (S – 1)/DS, αP = 2 – 1/S, and αE = 1 + (S – 1)/(DS + 2). The cellular automaton simulations reproduce the predicted fractal dimensions, occurrence frequency distributions, and correlations within a satisfactory agreement within ≈ 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 αE = 1.40 – 1.67, and the peak energy distribution has a slope of αP = 1.67 (for any fractal dimension DS = 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.
Author
Aschwanden, M. J.
Journal
Astronomy & Astrophysics
Paper Publication Date
March 2012
Paper Type
Astrostatistics