Interoccurrence Times in the Bak-Tang-Wiesenfeld Sandpile Model: A Comparison with the Observed Statistics of Solar Flares

A sequence of bursts observed in an intermittent time series may be caused by a single avalanche, even though these bursts appear as distinct events when noise and/or instrument resolution impose a detection threshold. In the Bak-Tang-Wiesenfeld sandpile, the statistics of quiet times between bursts switches from Poissonian to scale invariant on raising the threshold for detecting instantaneous activity, since each zero-threshold avalanche breaks into a hierarchy of correlated bursts. Calibrating the model with the time resolution of GOES data, qualitative agreement with the interoccurrence time statistics of solar flares at different intensity thresholds is found.

Figure 1

Time series of the instantaneous number of topplings, $n(t)$, in the two dimensional BTW with $L=2048$ and $\Delta T=100$. Dark (red online) areas and the black line represent the actual signal and the signal with some additive noise, respectively. Shaded regions indicate a detection thresholds $n_c=10$ (yellow, dashed line) and $n_c=100$ (blue, dot-dashed line); any signal below $n_c$ is considered undetectable. Intervals between a rise of the signal above $n_c$ and its next fall below constitute measured durations $t_d$ of events, followed by a “quiet time” $t_q$ until the next rise. The “waiting time” between two consecutive rises is $t_w=t_d+t_q$.

Figure 2

Distribution of quiet times $t_q$ in BTW for different thresholds $n_c$ for model A and model B with $\Delta T=100$ (groups of curves are offset vertically). In model B the distribution changes from approximately exponential at $n_c=0$ to a power law for increasing $n_c$. The straight reference lines decay as $t_d^{-5/3}$, suggesting ${\gamma }_q^{\mathrm{BTW}}=1.67(5)$ in Eq. (1).

Figure 3

Rescaled distribution of quiet times in BTW with $L=2048$, $\Delta T=100$ for different thresholds $n_c$ and for the solar flare data corresponding to “all” at different thresholds $I$ in Fig. 3(b) of Ref. 33. The quiet times $t_q$ have been divided by the average quiet time $⟨t_q⟩$ at each threshold, and the distributions rescaled to preserve normalization. BTW and flare data are similar in the intermediate regime. The former are shifted up by one unit on the log scale.

Figure 4

The distribution of event durations for BTW model B with $L=2048$, $\Delta T=100$ and for flare data corresponding to “minimum” at different thresholds $I$ in Fig. 4 of Ref. 33. One parallel update step in BTW has been calibrated to the time resolution of the GOES data, or 1 min. Although the critical exponents are different, the overall distributions are roughly comparable. The inset shows that, for increasing $n_c$, the behavior changes from that in Refs. 41, 42 towards a plateau indicating an asymptotic exponent $\approx 5/3$ for BTW at large thresholds [$n_c$: 0 (●), 15 (+), 255 (⋄), and 1023 (▽)]. Similar behavior is found for model A (not shown).