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

Lockheed Martin Advanced Technology Center, Solar & Astrophysics Laboratory, Org. ADBS, Bldg. 252, 3251 Hanover St., Palo Alto, CA 94304, USA
e-mail: aschwanden@lmsal.com

Received: 10 October 2011
Accepted: 12 December 2011

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 ∝ 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 ≈ (1 + S)/2; and (iv) the occurrence frequency distributions N(x) ∝ x ^− α_x 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 ∝ L^D_S ∝ T^D_S/2 for the flux, P ∝ L^S ∝ T^S/2 for the peak energy dissipation rate, and E ∝ FT ∝ T^1 + D_S/2 for the total dissipated energy; the model predicts powerlaw distributions for all parameters, with the slopes α_T = (1 + S)/2, α_F = 1 + (S − 1)/D_S, α_P = 2 − 1/S, and α_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  ≈ 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 α_E = 1.40 − 1.67, and the peak energy distribution has a slope of α_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.