The Effect of Helicity Dissipation on the Critical State of an Avalanche Model for Solar Flares

We perform numerical simulations using an avalanche model for solar flares, in which the effect of avalanche event duration is included and the lateral boundary condition can be either open or periodic. The results from the simulations show that self-organized criticality (SOC) exists under a variety of conditions of helicity dissipation, and they are similar in the sense that the indices of power-law distributions of avalanche sizes are the same, while the duration and waiting time distributions differ little. Of interest are: (1) if there exists helicity dissipation in the course of avalanches, SOC exists only when the amount of helicity dissipation per avalanche is small; (2) sudden reductions in the total helicity of the system do not severely disturb SOC, if the amount reduced is moderate; (3) the distribution of waiting times is shown to be exponential, in consistent with the Poisson statistics other avalanche models have predicted, and the exponent increases as the effect of event duration becomes larger. The self-similar structure of the avalanche time series is also examined. These results imply that, although helicity is dissipated in resistive diffusion of flares, the amount of dissipation should be small and the conservation law of helicity thus holds in an approximate sense. Also, the coronal mass ejections (CMEs), observed to take helicity out of the solar corona in an eruptive way, may disrupt the statistical equilibrium of the corona only temporarily.