4. Summary and conclusions
We have introduced a new set-up for classical solar flare CA models which yields, among others, consistency with Maxwell's equations (e.g. divergence-free magnetic field), and availability of secondary variables such as currents and electric fields in accordance with MHD. Both are new for solar flare CA models. The set-up specifies the so far open physical interpretation of the CA models. This specification is to some extent unavoidably arbitrary, and it would definitely be interesting to see what alternative interpretations would yield - if they can be derived consistently. We can claim, however, that the interpretation we chose is reasonable, it is well-behaved in the sense that the derivatives of analytically prescribed vector-potentials are reproduced and that the abstract stress-measure of the CA models is related to the current, due to general properties of spline interpolation. The central problem which was to solve is how to calculate derivatives in a CA model, i.e. how to continue the primary grid-variable in-between the grid sites, since the notion of derivatives is alien in the context of CA models quite in general.
In this article, our main aim with the introduced set-up was to demonstrate that the set-up truly extends the classical CA models and makes them richer in the sense that they contain much more information, now. The main features we revealed about the CA models, extended with our set-up, are:
1. Large-scale organization of the vector-potential and the magnetic field: The field topology during SOC state is bound to characteristic large-scale structures which span the whole grid, very pronounced for the primary grid variable, the vector-potential, but also for the magnetic field. Bursts and flares are just slight disturbances propagating over the large-scale structures, which are always maintained, also in the largest events. The magnitude of the current, as a second order derivative of the primary field, does not show any obvious large-scale structure anymore, it reflects more or less only the random fluctuations of the large-scale organized magnetic field. It is worthwhile noting that the large-scale structure of the primary grid-variable is not an artificial result of our set-up, but a natural consequence of the SOC state in which the system finds itself. The appearance of large-scale structures for the primary grid variable was shown here for the first time. It may have been known to different authors, but it never has explicitly been shown: SOC models for flares are derived in analogy to sand-pile dynamics, and the paradigm of a pile reappears in the field topologies of the solar flare CA models.
2. Increased current at unstable grid-sites: Unstable sites are characterized by an enhanced current, which is reduced after a burst has taken place, as a result of which the current at a grid-site in the neighbourhood may be increased.
3. Availability of the electric field: The electric field is approximated with the resistive part of Ohm's law in its simple form, which can in general be expected to be a good approximation in coronal applications and where the interest is in current-dissipation events, e.g. in the case of solar flares.
4. Energy release in terms of Ohmic dissipation: We replaced the some-what ad hoc formula in the CA models to estimate the energy released in a burst with the expression for Ohmic dissipation in terms of the current. The distributions yielded in this way are very similar to the ones based on the ad hoc formula, so that the results of the CA models remain basically unchanged.
5. CA as models for current dissipations: As a consequence of point 2 and 4 in this list, and of the fact that there is an approximate linear relation between the current and the stress measure of the CA, we can conclude that the extended CA models can be considered as models for energy release through current dissipation.
Our set-up is to be contrasted to the recently suggested MHD-derived (not based on the sand-pile analogy) CA models of Einaudi & Velli (1999), MacPherson & MacKinnon (1999), Longcope and Noonan (2000), and Isliker et al. (2000a). They all suggest new evolution rules, derived from MHD, and all in different ways (they actually focus on different processes, namely the microscopic, macroscopic, and mesoscopic physics, respectively, in active regions). Our set-up, on the other hand, uses existing CA models, does not interfere (if not wished) with their evolution rules, does also not change their main results, as shown, but reinterprets them, extends them essentially, and makes them compatible with MHD.
The set-up we introduced allows different future applications and posing questions which could not be asked so far in the frame of CA models. In preparation is a study (Isliker et al. 2000b) to reveal in detail what physical flare scenario the extended CA models imply. We will address the questions: (1) how to interpret the small scale processes of the models (loading and bursting) in terms of MHD; (2) what the global flare-scenario implied by the models is; (3) whether the global magnetic field topology of the models can be considered to represent observed magnetic topologies in active regions; (4) what spatio-temporal evolution of the electric field during flares is yielded by the models.
A different future application we plan with CA models extended with our set-up is the introduction of particles into the models, with the aim to study thermal emission, particle acceleration, and non-thermal emission. This will allow a much deeper comparison of the CA models to observations than was possible so far, and this is actually the most important benefit of the set-up we introduced. Such comparisons will allow a new judgment of the adequateness or not of classical CA models (in their current form) to the problem of solar flares, beyond the three statistical distributions of the primarily released energy. Solar flare CA models which include particle acceleration would represent the first global and complete model for solar flares.
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000