1112.4859 (Markus J. Aschwanden)
Markus J. Aschwanden
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.
View original:
http://arxiv.org/abs/1112.4859
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