Monte-Carlo Simulation of Solar Active-Region Energy

A Monte-Carlo approach to solving a stochastic jump transition model for active-region energy (Wheatland and Glukhov, Astrophys. J. 494, 1998; Wheatland, Astrophys. J. 679, 2008) is described. The new method numerically solves the stochastic differential equation describing the model, rather than the equivalent master equation. This has the advantages of allowing more efficient numerical solution, the modelling of time-dependent situations, and investigation of details of event statistics. The Monte-Carlo approach is illustrated by application to a Gaussian test case, and to the class of flare-like models presented in Wheatland (2008), which are steady-state models with constant rates of energy supply, and power-law distributed jump transition rates. These models have two free parameters: an index ($\delta$), which defines the dependence of the jump transition rates on active-region energy, and a non-dimensional ratio ($\overline{r})$ of total flaring rate to rate of energy supply. For $\overline{r}\ll 1$ the non-dimensional mean energy $<\overline{E}>$ of the active-region satisfies $<\overline{E}> \gg 1$, resulting in a power-law distribution of flare events over many decades in energy. The Monte-Carlo method is used to explore the behavior of the waiting-time distributions for the flare-like models. The models with $\delta\neq 0$ are found to have waiting times which depart significantly from simple Poisson behavior when $<\overline{E}> \gg 1$. The original model from Wheatland and Glukhov (1998), with $\delta=0$ (no dependence of transition rates on active-region energy), is identified as being most consistent with observed flare statistics.