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

3. Applications of our set-up

3.1. Application to the CA model of Lu & Hamilton (1991)

Our first application is to the CA model of LH91 (see Sect. 2.1). The LH91 model has a fairly long transient phase and reaches finally a stationary state, the so-called SOC (self-organized criticality) state, in which spatially spreading series of bursts (avalanches) appear, alternating with quiet loading phases. The LH91 model gives basically three results concerning flare statistics, namely the distributions of total energy, peak-flux and durations, which are all power-laws with slopes that are in good agreement with the observations (Lu et al. 1993, Bromund et al. 1995).

Superimposing our MHD-frame onto the LH91 model such as it stands does not change anyone of the three results, since at this first stage we are not interfering with the dynamics (i.e. the evolution rules). The set-up allows, however, to address several questions in MHD language: Our main aim in the subsequent applications is to demonstrate that the set-up indeed yields a new and consistent interpretation of CA-models, to illustrate the behaviour of the secondary variables (currents, magnetic fields), and to reveal major features of them. (In the subsequent runs, we use a grid of size , as LH91 did to derive their main results.)

3.1.1. Global structures of the vector-fields

First, we turn to the question what the global fields (vector-potential, magnetic field, current) look like during the SOC state. Thereto, the temporal evolution of the model is stopped at an arbitrary time during SOC state (in a phase where there are no bursts, i.e. during loading), and the magnitude of the fields at a cut with fixed z-coordinate are shown as a function of the x- and y-coordinates in Fig. 1. obviously exhibits a large-scale organization over the whole grid, it forms a global convex surface (Fig. 1a). This convex surface has a slight random distortion over-lying, which visually cannot be discerned in Fig. 1a, but becomes visible in the plot of (Fig. 1b), the curl of , which still exhibits large-scale organization all over the grid, but is clearly wiggled. Finally, shows no left-overs of a large scale organization anymore, it reflects the random disturbances of the convexity of (Fig. 1c).

Fig. 1a-c. Surface and contour plots of the magnitudes of a the vector-potential (), b the magnetic field (), and c the current () as a function of x and y, for fixed.

The large-scale structures shown in Fig. 1 are always maintained during the SOC state, neither loading nor bursting (and avalanches) destroy them, they just `tremble' a little when such events occur. SOC state in the extended LH91 model thus implies large-scale organization of the vector-potential and the magnetic field, in the characteristic form of Fig. 1.

The large-scale organization of is not an artificial result of our superimposed set-up, but already inherent in the classical LH91 model: in the classical LH91 CA model, there is only one variable, the one we call here , whose values are not affected by the interpolation we perform since it is the primary grid variable, so that Fig. 1a is true also for the classical, non-extended LH91 model.

The large scale structure for the primary grid-variable is the result of a combined effect: The preferred directionality of the loading increments (see Sect. 2.1) tries to increase throughout the grid. The redistribution events, which already in Bak et al. (1987; 1988) were termed diffusive events, and which in Isliker et al. (1998) were analytically shown to represent local, one-time-step diffusion processes, smooth out any too strong spatial unevenness of , and they root the -field down to the zero level at the open boundaries. The result is the convex surface of Fig. 1a, blown-up from below through loading, tied to the zero-level at the edges, and forced to a maximum curvature which is limited by the local, threshold dependent diffusion events.

As the SOC state, so is the large-scale structure of independent of the concrete kind of loading, provided it fulfills the conditions that the loading increments exhibit a preferred directionality and are much smaller than the threshold (with symmetric loading, the SOC state is actually never reached, see LH91 and Lu et al. (1993)).

To make sure of the importance of the boundaries, we performed runs of the model with closed boundaries, and we found that neither a large-scale structure was developed in , nor the SOC state was reached.

3.1.2. Bursts

To illustrate the role of the current at unstable sites and during bursts, we plot in Fig. 2 the magnitude of the current before and after a typical burst: obviously, the current at the burst site has high intensity before the burst (Fig. 2a), and is relaxed after the burst (Fig. 2b). Inspecting a number of other bursts, we found that, generally, at sites where the LH91 instability criterion is fulfilled, the current is increased, too, and that bursts dissipate the currents. This is a first hint that classical CA models can be interpreted as models for energy release through current-dissipation.

Fig. 2a and b. The magnitude of the current (, contour-plot, with the ticks pointing `downhill') as a function of x and y at a zoomed cut through the grid, before a and after b a burst, which occurs in the middle of the plot, at .

After the burst, at the neighbouring site , the intensity of the current is increased, and indeed the presented burst gives rise to subsequent bursts, it is one event during an avalanche.

The magnetic field at the bursting site is reshaped, in a way which is difficult to interpret when using only the magnitude of it () for visualization. May-be field line plots would help visualization, but we leave this for a future study.

3.1.3. Energy release and Ohmic dissipation

We now turn to the question what relation the energy release formula of LH91 (Eq. 6) has to the respective MHD relations: In parallel to using the formula of LH91, we determine the released energy in the following ways, closer to MHD: First, we assume it to be proportional to (with the diffusivity at unstable sites, see Sect. 2.2), which we linearly interpolate between the two states before and after the burst. This is done in two ways, (i) summing over the local neighbourhood,

and (ii) without summing, but just taking into account the current at the central point,

and finally, we monitor the change in magnetic energy due to a burst using the difference in magnetic energy in the local neighbourhood,

(In Eqs. (10), (11), (12), we assume and for the grid-spacing and the time-step , since, according to Isliker et al. (1998), in the classical CA models both values are not specified and set to one.)

The corresponding distributions of total energy and peak-flux are shown in Fig. 3, together with the distributions yielded by the energy-release formula of LH91, Eq. (6) (the duration distribution remains the same as in the classical LH91 model, namely a power-law, and is not shown). Obviously, the four ways of defining the released energy give basically similar results, with larger deviations only at the low and high energy ends (note that the energy in Fig. 3 is in arbitrary units). Using the formula of Ohmic dissipation does thus not change the results of the classical LH91 model.

Fig. 3a and b. The distribution of the total energy (a ), and of the peak-flux (b ), for different ways of measuring the released energy in a burst: (solid); (dotted); as the difference in before and after the burst (dashed); (dash-dotted). (The distributions are normalized probability distributions, the last two were shifted in both directions for viewing them together with the first two.)

With an estimate of the numerical value of the anomalous resistivity and of the typical size of a diffusive region or the typical diffusive time, it would be possible to introduce physical units. We did not undertake this, since all three parameters are still known only with large observational and theoretical uncertainties.

3.1.4. The relation of to

From the similarity of the distributions of the extended model with the ones of the classical LH91 model (Fig. 3), and from Fig. 2, where it was seen that an instability is accompanied by an enhanced current, we are led to ask directly for the relation of to , which we plot as a function of each other in Fig. 4. Obviously, the two quantities are related to each other: above , the current is an approximate linear function of the stress, around the current is zero, and below there is again an approximate linear relationship, with negative slope, however (above the current is actually preferably along , whereas below it is preferably along , i.e. is an approximatly linear function of in the whole range, it merely changes its directivity at with respect to ). In Appendix C, we show analytically why with our set-up a more or less close relation between and has to be expected.

Fig. 4. Plot of the magnitude of the current vs. the LH91 stress measure , using the corresponding values in the whole grid at a time fixed in a loading phase in the SOC state, together with the values at bursting sites during an avalanche.

Of particular interest in Fig. 4 is that if is above the threshold , then is also reaching high values: obviously, large values of imply large values of . This confirms the statement made above: The extended CA models can be considered as models for energy release through current dissipation. It also explains why the energy distributions remain very similar when the LH91 formula for the amount of energy released in a burst (, Eq. (6)) is replaced by Ohmic dissipation (, Sect. 3.1.3): bursts occur only for large stresses , where is also large and an approximate linear function of , so that the distributions of and can be expected to be the same in shape.

3.2. Application to loading with power-law increments

Georgoulis and Vlahos (1996, 1998) introduced power-law distributed increments for the loading. The main result of such a way of driving the system is that the power-law indices of the energy-distributions depend on the power-law index of the distribution of the loading increments, explaining thus the observed variability of the indices through the variability of the intensity of the driving. We generalize their way of power-law loading, which is for a scalar primary field, to a vector field in the following way: The anisotropic directivity of the loading increment is kept (see Sect. 2.1), but is now distributed according to

with and , the power-law index, a free parameter. Simulations were performed for and . Interested in global features implied by the CA model, our concern here is the structure of the magnetic field. It turns out that the magnetic field exhibits still a large scale organization, which is very similar to the one of the -field of the (extended with our set-up) LH91 model (Fig. 1b): for , the respective plots are visually indiscernible, and for the overall shape is still roughly the same, it merely seems slightly more distorted. Thus, though the statistical results depend on , the strength and variability of the loading, the structure of the magnetic field remains approximately the same as in the case of the extended model of LH91. Large-scale organization (in the characteristic form of Fig. 1) must consequently be considered as an inherent property of SOC state, through the mechanism explained in Sect. 3.1.1.

3.3. Application to anisotropic bursts

Vlahos et al. (1995) introduced anisotropic bursts for solar flare CA models, which lead only to small events, but yield a steep distribution at small energies, predicting thus a significant over-abundance of small events with a significant contribution to coronal heating. We have first to generalize the anisotropic evolution rules, which are again for a scalar primary field, to the case of a primary vector field. A natural generalization would be to apply the anisotropic rules to the absolute magnitude of , but it turns out that this causes the algorithm to get trapped in infinite loops (two neighbouring grid-sites trigger each other mutually for ever). The same holds if we apply the anisotropic rules to the absolute magnitudes of the three components of independently. We finally applied the anisotropic rules to the three components of directly, not using absolute magnitudes, as also Vlahos et al. (1995) did not use absolute magnitudes, and this turned out to lead to a stationary asymptotic state: The anisotropic stress in the x-component is thus defined as

where stands for one of the six nearest neighbours. The instability criterion is

and the redistribution rules become

for the central point and

for those nearest neighbours which fulfill the instability criterion (Eq. 15), where the primed sum is over those neighbours for which Eq. (15) holds. The rules for are completely analogous (so that actually there are 18 possibilities to exceed the threshold (Eq. 15) at a given site). The released energy is assumed to amount to

We performed a run where only the anisotropic burst-rules were applied, in order to isolate their effect, although the anisotropic burst-rules are used always together with the isotropic ones by Vlahos et al. (1995), since alone they cannot explain the complete energy distributions of flares. In Fig. 5, the magnitude of the magnetic field at a cut through the grid is shown (fixed z), for an arbitrary time (in the loading phase) during the asymptotic stationary state of the model. Clearly, there is no overall large scale structure anymore, except that the magnetic field along the boundaries is increased. The magnetic field topology is thus nearer to the concept of a random, relatively unstructured magnetic field than the magnetic field topology yielded by the isotropic models in SOC state.

Fig. 5. Surface and contour plot of the magnitude of the the magnetic field () as a function of x and y at a cut through the grid, for the case of anisotropic redistribution rules.

The anisotropic burst rules do not yield large-scale structures, as they are, when used alone, also not able to lead the system to SOC state: this is obvious from the energy distributions they yield, which are much smaller in extent than the ones given by the isotropic rules (see Vlahos et al. 1995), and confirmed by the result of Lu et al. (1993) that isotropy of the redistribution rules - at least on the average - is a prerequisite to reach SOC state, at all. The anisotropic bursts occur independently in all directions and are in this way not able to organize the field in a neighbourhood systematically, and, as a consequence, also not in the entire grid.

The inquiry of the relation of the energy release formula Eq. (18), which is different from the isotropic formula (Eq. 6), to MHD based formulae we leave for a future study. We just note that the distributions the anisotropic model in our vector-field version yields are at lower energies, smaller in extent, and steeper than the ones of the isotropic models.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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