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Astron. Astrophys. 363, 1134-1144 (2000)

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1. Introduction

Cellular automata (CA) models for solar flares are successful in explaining solar flare statistics (peak flux, total flux, and duration distributions; Lu & Hamilton 1991 (hereafter LH91); Lu et al. 1993; Vlahos et al. 1995; Georgoulis & Vlahos 1996, 1998; Galsgaard 1996). They simplify strongly the details of the involved physical processes, and achieve in this way to model large volumes with complex field topologies and a large number of events. On the other hand, MHD simulations give insight into the details of the local processes, they are limited, however, to modeling relatively small fractions of active regions, due to the lack of computing power, yielding thus poor statistics and difficulties in comparing results to observations (e.g. Mikic et al. 1989; Strauss 1993; Longcope & Sudan 1994; Einaudi et al. 1996; Galsgaard & Nordlund 1996; Hendrix & Van Hoven 1996; Dmitruk & Gomez 1998; Galtier & Pouquet 1998; Georgoulis et al. 1998; Karpen et al. 1998; Einaudi & Velli 1999). The global MHD flare models are still in the state of rather qualitative flare scenarios.

The MHD and the CA approach to solar flares seem to have very little in common: The former are a set of partial differential equations, based on fluid-theory and Maxwell's equations, whereas the latter are a set of abstract evolution rules, based (in the case of solar flares) on the analogy to critical phenomena in (theoretical) sand-piles. The scope of this paper is to bridge the gap in-between these two approaches: the solar flare CA models are re-interpreted and extended so as (i) to make these models completely compatible with MHD and with Maxwell's equations, and so that (ii) all relevant MHD variables are made available (e.g. the current and the electric field, which so far were not available in CA models).

In an earlier paper (Isliker et al. 1998), we have analyzed the existing solar flare CA models for their soundness with MHD. We asked the question whether the fields in these CA models and the evolution rules can be interpreted in terms of MHD. It turned out that these models can indeed be interpreted as a particular way of implementing numerically the MHD equations. This fact is not trivial, since these models had been derived in quite close analogy to the sand-pile CA model of Bak et al. (1987 and 1988), with vague association of the model's variables with physical quantities. For instance, some authors (Lu et al. 1993) explicitly discuss the question whether their basic grid variable is the magnetic field or not, without reaching to a definite conclusion. Isliker et al. (1998) brought forth not only how the existing CA models are related to MHD and what simplifications are hidden, but also where they differ from or even violate the laws of MHD and Maxwell's equation. Important is the fact that though the existing CA models can be considered as a strongly simplified numerical solution of the (simplified) MHD equations, they do not represent the discretized MHD equations : the time-step and the spacing between two grid sites are not small (in a physical sense), but finite; they are a typical temporal and spatial scale of the diffusive processes involved (see Isliker et al. 1998).

From the point of view of MHD, the main short-comings of the existing CA models are (Isliker et al. 1998): (1) There is no control over consistency with Maxwell's equations. Interpreting, for instance, the vector-field in the CA models as the magnetic field leads to the problem that the gradient of the field ([FORMULA]) cannot be controlled. (2) Secondary quantities, such as currents, are not available, and they cannot be introduced in the straightforward way by replacing differential expressions by difference-expressions, since, as mentioned, the grid-size must be considered finite (see also App. B.1). This lack of knowing how to calculate derivatives made it also useless to interpret the primary vector-field in the CA models as the vector potential (to avoid the [FORMULA]-problem), since [FORMULA] could not be derived. The physical interpretation of these CA models remained so far problematic.

There are two basically different ways of developing CA models for flares further: (i) Either one considers CA models per se , tries to change the existing models further or invent new ones, with the only aim of adjusting them to reproduce still better the observations, i.e. one makes them a tool the results of which explain and maybe predict observed properties of flares. In this approach, one has not to care about possible inconsistencies with MHD or even Maxwell's equations, the various components of the model are purely instrumentalistic. (ii) On the other hand, one may care about the physical identification and interpretation of the various components of the model, not just of its results, and one may want the CA model to become consistent with the other approach to solar flares, namely MHD. In the approach (ii), some of the freedom one has in constructing CA models will possibly be reduced, since there are more `boundary conditions' to be fulfilled in the construction of the model: the observations must be reproduced and consistency with MHD has to be reached. (Trials to construct new CA models which are based on MHD and not on the sand-pile analogy were recently made by Einaudi & Velli 1999, MacPherson & MacKinnon 1999, Longcope & Noonan 2000, and Isliker et al. 2000a.)

Our aim is in-between these two alternatives: we construct a set-up which can be superimposed onto each classical solar flare CA model, and which makes the latter interpretable in a MHD-consistent way (by classical CA models we mean the models of LH91, Lu et al. 1993, Vlahos et al. 1995, Georgoulis & Vlahos 1996, 1998, Galsgaard 1996, and their modifications, which are based on the sand-pile analogy). The set-up thus specifies the physical interpretation of the grid-variables and allows the derivation of quantities such as currents etc. It does not interfere with the dynamics of the CA (unless wished): loading, redistributing (bursting), and the appearance of avalanches and self-organized criticality (SOC), if the latter are implied by the evolution rules, remain unchanged. The result is therefore still a CA model, with all the advantages of CA, namely that they are fast, that they model large spatial regions (and large events), and therewith that they yield good statistics. Since the set-up introduces all the relevant physical variables into the context of the CA models, it automatically leads to a better physical understanding of the CA models. It reveals which relevant plasma processes and in what form are actually implemented, and what the global flare scenario is the CA models imply. All this was more or less hidden so far in the abstract evolution rules. It leads also to the possibility to change the CA models (the rules) at the guide-line of MHD, if this should become desirable. Not least, the set-up opens a way for further comparison of the CA models to observations.

In Sect. 2, we introduce our set-up. Applying it to several CA models (Sect. 3), we will demonstrate the usefulness and some of the benefits such extended models (i.e. classical models extended with our set-up) provide over the classical CA models, and we will reveal basic physical features of the CA models. The potential of the extended models to explain more observational facts than the classical CA models is, among others, outlined in the conclusions (Sect. 4).

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Online publication: December 5, 2000