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

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

Dec 2011

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2011agufmsh51c2018a&link_type=abstract

American Geophysical Union, Fall Meeting 2011, abstract #SH51C-2018

Astronomy and Astrophysics

Astrophysics

[4499] Nonlinear Geophysics / General Or Miscellaneous, [7519] Solar Physics, Astrophysics, And Astronomy / Flares

Scientific paper

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 slowly-driven self-organized criticality (SOC). 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 instantaneous energy dissipation rate P occupies a fractal volume with dimension DS, which predicts the relationships P ∝ LDS ∝ TDS/2 and the total dissipated energy E ∝ P T ∝ T1+DS/2; (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-α x based on spatially uniform probabilities in a SOC system are given by N(L) ∝ L-S, which predicts powerlaw distributions for all parameters, with the slopes α T=(1+S)/2, α P=1+(S-1)/D_S, and α 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 ≈ 10%. 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 (for any fractal dimension) 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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