THE STATISTICAL FLARE

Solar and stellar flares are interpreted so far as an instability of a large scale magnetic neutral sheet. In this article, however, we assu me that the active region is highly inhomogeneous: a large number of m agnetic loops are simultaneously present interacting and randomly form ing discontinuities in many independent points in space. These magneti c discontinuities release energy and force weaker discontinuities in t heir neighbourhood to release energy as well. This complex dynamical s ystem releases constantly energy in the form of small and large scale explosions. Clustering of many discontinuities in the same area has th e effect of larger scale explosions (flares). This type of flare with spatiotemporal fragmentation and clustering in small and large scale s tructures will be called here the statistical flare. The statistical f lare is simulated using avalanche models originally introduced by Bak et al. (1988). Avalanche models applied so far to solar flares (Lu & H amilton 1991) were isotropic (the field was distributed equally to the closest neighbours of an unstable point). These models simulate relat ively large events (microflares and flares). Here we introduce a more refined isotropic avalanche model as well as an anisotropic avalanche model (energy is distributed only among the unstable point and those n eighbours that develop gradients higher than a critical value). The an isotropic model simulates better the smaller events (nanoflares): in c ontrast to the well-known results of the isotropic model (a power law with index similar to -1.8 in the peak-luminosity distribution), the a nisotropic model produces a much steeper power law with index similar to -3.5. Finally, we introduce a mixed model (a combination of isotrop ic and anisotropic models) which gives rise to two distinct power-law regions in the peak-luminosity distribution, one with index similar to -3.5 accounting for the small events, and one with index similar to - 1.8 accounting for large events. This last model therefore explains co ronal heating as well as flaring. The three models introduced in this paper show length-scale invariant behaviour. Model-dependent memory ef fects are detected in the peak-luminosity time series produced by thes e models.