The State of Self-organized Criticality of the Sun during the Last Three Solar Cycles. II. Theoretical Model

The observed power-law distributions of solar-flare parameters can be interpreted in terms of a nonlinear dissipative system in a state of self-organized criticality (SOC). We present a universal analytical model of an SOC process that is governed by three conditions: i) a multiplicative or exponential growth phase, ii) a randomly interrupted termination of the growth phase, and iii) a linear decay phase. This basic concept approximately reproduces the observed frequency distributions. We generalize it to a randomized exponential growth model, which also includes a (log-normal) distribution of threshold energies before the instability starts, as well as randomized decay times, which can reproduce both the observed occurrence-frequency distributions and the scatter of correlated parameters more realistically. With this analytical model we can efficiently perform Monte-Carlo simulations of frequency distributions and parameter correlations of SOC processes, which are simpler and faster than the iterative simulations of cellular automaton models. Solar-cycle modulations of the power-law slopes of flare-frequency distributions can be used to diagnose the thresholds and growth rates of magnetic instabilities responsible for solar flares.