## 1. IntroductionCellular 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 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, 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 () 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 -problem), since 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 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
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). © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |