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

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2. Introduction of the set-up

The set-up we propose can be superimposed onto solar flare CA models which use a 3-D grid and a basic 3-D vector grid-variable, say [FORMULA]. The corresponding set of evolution rules is not changed. (With a few modifications, the set-up can also be superimposed onto CA models which use a scalar field in a planar grid, which our set-up necessarily interprets as slab geometry, as will become clear later.)

We introduce our model on the example of the solar flare CA model of LH91, which we summarize here in order to make the subsequent presentation more concrete:

2.1. Summary of the CA model of LH91

In the LH91 model, to each grid-site [FORMULA] of a 3-D cubic grid a 3-D vector [FORMULA] is assigned. Initially, [FORMULA] is set to [FORMULA], everywhere. The system is then loaded with the repeated dropping of increments at randomly chosen sites [FORMULA] (one per time-step)

[EQUATION]

where [FORMULA] has all its components as random numbers uniformly distributed in [FORMULA].

After each loading event, the system is checked for whether the local `stress', defined as

[EQUATION]

where the sum goes over the six nearest neighbours of the central point [FORMULA], exceeds a threshold [FORMULA], i.e. whether

[EQUATION]

where [FORMULA] is used. If this is the case, the field in the neighbourhood of the critical site is redistributed according to

[EQUATION]

for the central point, and

[EQUATION]

for its six nearest neighbours. In such a redistribution event (burst), energy amounting to

[EQUATION]

is assumed to be released. The grid is scanned again and again to search for second, third etc. generation bursts, until the system is nowhere critical anymore and returns to the loading phase (the details we apply concerning the temporal evolution of the model are given in App. A; they are not explicitly stated in LH91). The field outside the grid is held constant and assumed to be zero.

2.2. Our set-up

We turn now to introducing our set-up, starting with a specification: We interpret the vector [FORMULA] at the grid sites [FORMULA] to denote the local vector-field, [FORMULA]. Note that this was not specified in the classical CA models. Lu et al. (1993) for instance discuss this point: it might also have been thought of as a mean local field, i.e. the average over an elementary cell in the grid.

Guided by the idea that we want to assure [FORMULA] for the magnetic field [FORMULA], which is most easily achieved by having the vector-potential [FORMULA] as the primary variable and letting [FORMULA] be the corresponding derivative of [FORMULA] ([FORMULA]), we furthermore assume that the grid variable [FORMULA] of the CA model is identical with the vector-potential.

The remaining and actually most basic problem then is to find an adequate way to calculate derivatives in the grid. In general, CA models assume that the grid-spacing is finite, which also holds for the CA model of LH91 (as shown in detail by Isliker et al. 1998), so that the most straightforward way of replacing differential expressions with difference expressions is not adequate (see the detailed discussion in App. B.1, below; Vassiliadis et al. (1998) suggested to interpret CA models as the straightforwardly discretized (simplified) MHD equations, which we find problematic for the reasons given in App. B.1, and we therefore do not follow this approach, here).

Consequently, one has to find a way of continuing the vector-field into the space in-between the grid-sites, which will allow to calculate derivatives. There is, of course, an infinite number of possibilities to do so, and the problem cannot have a unique solution. Adequate possibilities definitely include: a) continuation of [FORMULA] with the help of an equation (e.g. demanding the resulting [FORMULA]-field to be potential or force-free); b) interpolation, either locally (in a neighbourhood), or globally (through the whole grid). Trying several methods, we concluded that 3-D cubic spline interpolation is particularly adequate to the problem since it has remarkable advantages over other methods (e.g. it does not introduce oscillations in-between grid-sites, which would strongly influence the values of the derivatives, and it well reproduces the derivatives of analytically prescribed primary fields). The process of evaluating different continuation methods we went through, as well as the comparison of spline interpolation to other continuation-methods are described in App. B.

The 3-D interpolation is performed as three subsequent 1-D interpolations in the three spatial directions (Press et al. 1992). For the 1-D splines, we assume natural boundaries (the second derivatives are zero at the boundaries). Moreover, since in the CA model of LH91 it is assumed that around the grid there is a zero field which is held constant (see Sect. 2.1), we enlarge the grid by one grid point in all directions to include this constant zero-layer explicitly, using it however only for the interpolation. In the interpolation, the derivatives at the grid-points are immediately given by the analytically differentiated interpolating polynomials.

With the help of this interpolation, the magnetic field [FORMULA] and the current [FORMULA] are calculated as derivatives of [FORMULA], according to the MHD prescription:

[EQUATION]

[EQUATION]

To determine the electric field [FORMULA], we make the assumption that under coronal conditions the MHD approach is in general valid, and that [FORMULA] is reasonably well approximated by Ohm's law in its simple form, [FORMULA], with [FORMULA] the diffusivity and [FORMULA] the fluid velocity. Since the classical CA models use no velocity-field, our set-up can yield only the resistive part,

[EQUATION]

In applications such as to solar flares, where the interest is in current dissipation events, i.e. in events where [FORMULA] and [FORMULA] are strongly increased, Eq. (9) can be expected to be a good approximation to the electric field. Theoretically, the convective term in Ohm's law would in general yield just a low-intensity, background electric field.

Eq. (9) needs to be supplemented with a specification of the diffusivity [FORMULA]: Isliker et al. (1998) have shown that in the classical CA models the diffusivity adopts the values [FORMULA] at the unstable (bursting) sites, and [FORMULA] everywhere else. This specifies Eq. (9) completely.

Remark 1: It is worthwhile noting that, since spline interpolation has the property to be the least curved of all twice differentiable interpolating functions, the grid-size is a typical smallest-possible length-scale of field structures, or, if we would think the CA to model MHD turbulence, it is something like an average smallest possible eddy-size.

Remark 2: With our set-up, field lines are made available: the interpolation was introduced to define derivatives at the grid-sites, but it can as well be used to determine the vector-potential, and therewith the magnetic field and the current, in between the grid sites. However, it is important to note that this field in between the grid sites is not used for the time-evolution of the model, it merely allows to visualize the evolution in the standard way through field-lines, if wished so (second order derivatives in-between the grid-sites, if needed, would have to be done numerically, since else their numerical values would depend on the order in which the three 1-D interpolations are done).

Remark 3: In this paper, we do not change the rules of the classical CA models to which we apply our set-up - except for the definition of energy release (Sect. 3.1.3). Our aim here is to show the usefulness of the set-up and to give some results and reveal some aspects by extending published, classical CA models. While the redistribution rules have in detail been shown to represent diffusion events and fit nicely into an MHD scenario (Isliker et al. 1998), the loading process is strongly simplified and poorly follows a reasonable flare scenario: For instance, the loading process acts independently everywhere in the simulation box, whereas according to a realistic flare scenario (see e.g. Parker 1993) disturbances should appear independently only on one boundary (the photosphere, due to random foot-point motions or newly appearing flux), and propagate then into the interior of the simulation box along the magnetic field lines.

To translate such a realistic loading scenario into the language of CA models has not been undertaken, so-far. We just note that it would be quite straightforward to introduce a velocity field into the CA models: e.g. Isliker et al. (2000a) propose a CA model which uses a velocity field for the loading phase, but this model does not fall into the category of classical CA models since it does not follow the sand-pile analogy and uses different, MHD based, evolution rules. We leave the problem of introducing a velocity field and a more physical loading process into the classical CA models for a future study. In Isliker et al. 2000b, we will - among others - analyze in details what this simplified loading process physically represents.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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