The Solar Memory from Hours to Decades

Waiting-time distributions allow us to distinguish at least three different types of dynamical systems, including (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 λ(t) ∝ t p with a polynomial function, which allows us to distinguish linear (p ≈ 1) from nonlinear (p 2) events. The power-law slopes α of the observed waiting times (with full solar cycle coverage) cover a range of α = 2.1–2.4, which agrees well with our prediction of α = 2.0 + 1/p = 2.3–2.6. 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 τG = τrise/(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 ± 0.1), partially occulted flare events (p = 1.8 ± 0.2), and the solar dynamo (p = 2.8 ± 0.5). The Sun exhibits memory on timescales of 2 hr to 3 days (for solar flare clustering), 6–23 days (for partially occulted flare events), and 1.5 month to 1 yr (for the rise time of the solar dynamo)