# The Statistical Flare

## Abstract

Solar and stellar flares are interpreted so far as an instability of
a large scale magnetic neutral sheet. In this article, however, we
assume that the active region is highly inhomogeneous: a large number
of magnetic loops are simultaneously present interacting and randomly
forming discontinuities in many independent points in space. These
magnetic discontinuities release energy and force weaker discontinuities
in their neighbourhood to release energy as well. This complex dynamical
system releases constantly energy in the form of small and large scale
explosions. Clustering of many discontinuities in the same area has the
effect of larger scale explosions (flares). This type of flare with
spatiotemporal fragmentation and clustering in small and large scale
structures will be called here the * statistical flare *.
The statistical flare is simulated using avalanche models originally
introduced by Bak, Tang and Wiesenfeld (1988). Avalanche models applied so far
to solar flares (Lu and Hamilton 1991) were isotropic (the field was distributed
equally to the closest neighbours of an unstable point). These models simulate
realtively 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 neighbours
that develop gradients higher than a critical value). The anisotropic model
simulates better the smaller events (nanoflares): in contrast to the
well-known results of the isotropic model (a power law with index -1.8
in the peak-luminosity distribution), the anisotropic model produces a much
steeper power law with index -3.5. Finally, we introduce a mixed
model (a combination of isotropic and anisotropic models) which gives rise
to two distinct power-law regions in the peak-luminosity distribution, one
with index -3.5 accounting for the small events, and one with
index -1.8 accounting for large events. This last model therefore
explains coronal heating as well as flaring.
The three models introduced in this paper show
length-scale invariant
behaviour. Model-dependent memory effects are detected
in the peak-luminosity time series produced by these models.

Subject Headings: Sun: activity - Sun: corona - Sun:flare-Sun: magnetic fields
- Stars: activity - Stars: flare

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