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

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6. Conclusion

We have shown that the evolution rules of the CA model of LH can be interpreted in continuous time and space as a PDE of diffusive type - acting in a localized region, and subdued to an instability criterion -, together with a source term. Since the diffusive part of this equation has a trivial solution after the diffusive time has elapsed, the CA rules of LH implement directly this solution, namely flattened magnetic field in a bounded region.

The main assumptions on solar flares we have in this way revealed for the LH model are: (A) Every instability which possibly occurs has the same properties, namely the same characteristic length and the same diffusion time. (B) The diffusivity is an average quantity, without any spatial variations inside the unstable volume, modeling thus the complex dynamics of magnetic reconnection and diffusion in a simplified way. (C) The convective term is replaced by a simple random function, so the dynamics of the turbulent active region (plasma motion) is modeled in a strongly simplified way.

Despite of these limitations, the benefit of running a CA model for solar flares is that in this way an understanding of the global statistical properties of flares can be achieved. For instance, it provides an interpretation of the observed fragmented nature of the energy release process (see e.g. Vilmer and Trottet 1997, and references therein) as an avalanche-like phenomenon (chain-reactions of localized bursts). Such insights are far out of reach for a pure MHD-treatment of the problem: MHD and CA cannot replace each other, but they must be combined.

Our considerations allow extraction of more information from the so far existing CA models for solar flares, and to interpret some aspects in a more physical way:

  1. Units: Since in Sect. 4 we have established the connection between the CA of Lu and Hamilton and a PDE, and in Sect. 5 the connection of this PDE to the induction equation has been shown, we can associate real physical units with the length scale [FORMULA] (the grid size in the CA) and the diffusive time [FORMULA] (the iteration time step in the CA). From this the units of the diffusivity, the magnetic field, the threshold, and the released energy can be derived. To introduce physical units in solar flare CA was not feasible, so far.

  2. Diffusivity: Having real units (point 1), the values given in the literature for the diffusivity [FORMULA] can be used and discussed, e.g. the question can be addressed whether the values needed in the CA to be in a self-organized critical (SOC) state are physically reasonable or not, compared to anomalous diffusivity.

  3. Released energy: Lu and Hamilton assume the amount of released energy to be [FORMULA] (Eqs. 7, 10, and 37). In the frame of MHD, an expression for the released energy should be in terms of the currents, or else it can be estimated by the difference in [FORMULA] before and after a burst (flare), as given in Eq. (43).

  4. Energy balance: If the previous point is clarified, then the emitted energy can be calculated (in physical units), and it can be compared to the injected energy, analyzing thus the energy balance of CA models.

Furthermore, on the basis of the given discussion, concrete suggestions of how CA-rules can be modified to include more physical insight (MHD) can be made, for instance:

  1. The driver: Lu and Hamilton had replaced the convective term [FORMULA] in the induction equation (Eq. 39) by a simple random function ([FORMULA] in Eq. 33). To use the convective term would mean to do full MHD, since also the momentum equation would have to be included, and so a treating of the whole problem with a CA would become as complex as to integrate the full PDEs (as e.g. Einaudi et al. 1996 have done). The question is whether this random loading term [FORMULA] of LH could be replaced by a description of the convective motion which is still simplified, but which catches more of the physical picture we have on the convective motion in the corona, in such a way, however, that it still is possible to reduce the problem to a CA. A possible set-up would be that, instead of random loading everywhere, the random loading is only from below (the photosphere), and thereafter the magnetic field is shuffled from site to site, e.g. through a term [FORMULA], with [FORMULA] a random variable, varying from site to site and with a distribution function taken from Kolmogoroff 's theory of turbulence.

  2. Instability criterion: LH use [FORMULA] as a critical quantity for the onset of an instability. A different approach would be to consider the slope of the magnetic field in some direction [FORMULA]: [FORMULA], or the current [FORMULA]. Related to this question is a discussion of the magnitude and nature of the threshold (in physical units).

  3. Released energy: An improved way of formulating the amount of released energy has been given as point 3, above.

We believe that the connection of MHD with CA models, for which we have given here a first example, will help to improve the global modeling of active regions, since insights accumulated in MHD may give guidelines to improve CA rules. Moreover, it is a step towards the combination of MHD and CA models, i.e. towards a model which incorporates the micro-physical as well as the global aspects of flares.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998