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Astron. Astrophys. 335, 1085-1092 (1998)

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1. Introduction

Cellular Automata (CA) models have been used to model solar flares (Lu and Hamilton 1991, hereafter LH; Lu et al. 1993; Vlahos et al. 1995; Georgoulis and Vlahos 1996), and they have been successful in reproducing several statistical properties of the latter, such as peak-flux distributions, total-flux distributions and duration distributions, as derived from HXR observations (Dennis 1985; Dennis 1988; Vilmer 1993). CA mimic the temporal evolution of the magnetic field on a spatial grid. They usually have a random loading function during their quiet evolution phase (build-up of magnetic field in the active region), changing however to a bursting type of evolution if a certain local threshold-criterion is fulfilled. The magnetic fields are then locally relaxed, loosely modeling an assumed magnetic reconnection process in this way. Chain reactions (avalanches) of such elementary bursts are interpreted as energy release events in flares.

In general, CA have been developed to model complex systems, i.e. systems which consist of a large number of interacting subsystems. The essence of the CA approach to such systems is to assume that the global dynamics, if described statistically, are not sensitive to the details of the elementary processes, the system has the property that most local information gets lost if viewed globally. The 'classical' approach to complex systems, on the other hand, is analytical: from a precise description of the elementary processes - in the optimum case involving the fundamental laws of physics, i.e. differential equations - one tries to understand a process globally. Both approaches have drawbacks and advantages. The CA approach does not explain what happens locally or over short time intervals, but it allows to understand the statistics of the global behaviour. The analytical approach may reveal insights into the local processes, but coupling this understanding to a global description is practically not feasible, mainly due to the large number of (in astrophysics even unobserved) boundary conditions. In this sense, the two approaches can be considered as complementary, and a description of a complex system should ideally combine them.

Concerning the problem of solar flares, such a combination is still missing. In this article, we try to make a first step towards this direction, starting from the point where the two approaches touch, namely at the scale where the local micro-physics can be summarized into simple CA evolution rules.

The analytical (micro-physical) theory of the processes in solar active regions is kinetic plasma physics, or, with some idealizations, MHD, i.e. a set of partial differential equations. The task is to establish a connection between the solar flare CA rules and the (local) MHD equations. The way we choose to do so is first to see how the CA rules are related to differential equations, and then to compare these equations to the relevant ones of MHD. We note that Lu (1995) made a general discussion of whether there exist continuous driven dissipative systems which show analogous features as CA models for solar flares, namely avalanches. He found a general type of such continuous systems, however, they were not derived from a given CA model, but constructed in order to mimic the general statistical properties of CA dynamics. Our approach, on the other hand, starts from a particular (solar flare) CA and derives the continuous system which exactly corresponds to this CA, establishing thus a translation scheme to go from a given CA to a partial differential equation (and vice-versa), which in turn is very general.

We will concentrate this inquiry on the model of LH, since it was the first CA model suggested for an application to flares, and the later developed CA models, though being improved in details, still have the essential features of this first model.

We will first review the CA model of LH (Sect. 2), the starting point of our discussion. In Sect. 3 we will recover the differential equation behind this CA, giving the derivation in enough details so that the way of proceeding may be applied to other CA models. The result, the explicit form of this equation, will be stated in Sect. 4. In Sect. 5, we discuss the relation of this equation to the MHD equations relevant to solar flares, unveiling thus the nature of the process, the assumptions, and the simplifications which are hidden behind the CA model of LH. In the conclusion (Sect. 6) we finally make some suggestions on how the results of the existing CA models can be interpreted more thoroughly by introducing physical units, and how improved (in the sense of more physically motivated) CA models could be constructed (in a parallel paper the idea of deriving CA rules from MHD will be pursued (Vassiliadis et al. 1998, as a continuation of Vassiliadis et al. 1996)).

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998