The Solar Memory from Hours to Decades

Waiting time distributions allow us to distinguish at least three different types of dynamical systems, such as (i) linear random processes (with no memory); (ii) nonlinear, avalanche-type, nonstationary Poisson processes (with memory during the exponential growth of the avalanche rise time); and (iii) chaotic systems in the state of a nonlinear limit cycle (with memory during the oscillatory phase). We describe the temporal evolution of the flare rate $\lambda(t) \propto t^p$ with a polynomial function, which allows us to distinguish linear ($p \approx 1$) from nonlinear ($p \gapprox 2$) events. The power law slopes $\alpha$ of observed waiting times (with full solar cycle coverage) cover a range of $\alpha=2.1-2.4$, which agrees well with our prediction of $\alpha = 2.0+1/p = 2.3-2.5$. The memory time can also be defined with the time evolution of the logistic equation, for which we find a relationship between the nonlinear growth time $\tau_G = \tau_{rise}/(4p)$ and the nonlinearity index $p$. We find a nonlinear evolution for most events, in particular for the clustering of solar flares ($p=2.2\pm0.1$), partially occulted flare events ($p=1.8\pm0.2$), and the solar dynamo ($p=2.8\pm0.5$). The Sun exhibits memory on time scales of $\lapprox$2 hours to 3 days (for solar flare clustering), 6 to 23 days (for partially occulted flare events), and 1.5 month to 1 year (for the rise time of the solar dynamo).