The Solar Memory From Hours to Decades

Abstract: 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).