## Appendix A: temporal evolution of the CAThe temporal evolution of the CA models presented in this article is governed by the following rules: -
initializing -
loading -
scanning: create a list of the unstable sites; if there are none, return to loading (1) -
scanning and bursting: redistribute the fields at the unstable sites which are in the list created in the scannings 2 or 4 -
scanning: create a list of the unstable sites. If there are any, go to bursting (3), else return to loading (1)
The extra scannings 2 and 4 are needed for causality: if a site becomes unstable through a burst in the neighbourhood, then it should be redistributed in the subsequent scan, and not in the same as the primary unstable site. The same is true for the scanning 4, since in the next bursting phase (if any) only those sites should burst who had become unstable through a burst in their neighbourhood during the foregoing time-step. As a time-step is considered one scanning of the grid, point 3. The released energy per time-step is the sum of all the energy released by bursts in this time-step (a burst is considered a single redistribution event in 3). We term a flare or avalanche the loop 3,4, from the occurring of the first burst in 3 until the activity has died out and one returns via the scanning 4 to loading (1). The duration of the flare is the number of time-steps it lasted, the total flare energy is the sum of all the energies released in the duration of the flare, and the peak flux or peak energy is the maximum of the energies of all the time-steps of the flare. ## Appendix B: why spline interpolation is particularly adequate: comparison to other methods of continuationWe mentioned in Sect. 2.2 that other possibilities for continuation of the vector-potential besides spline interpolation would be: a) continuation of with the help of an equation; b) other kinds of interpolation, either locally (in a neighbourhood), or globally (through the whole grid). Possibility a) implies that an equation has to be solved in each time-step (after each loading and after each burst), in the worst case numerically, with open boundary condition and the given at the grid-sites. This computational effort might slow down the algorithm of the model considerably (and bring it near to the computational effort of MHD equation integration). Besides that, the problem is what equation to use: to make the magnetic field always a potential field (i.e. using a corresponding equation for the vector-potential ) implies that, from the point of view of MHD, at all times a very `well-behaved' magnetic field resides in the CA, with no tendency towards instabilities, which makes it difficult to understand why bursts should occur at all, since critical quantities such as currents do not become excited. A better candidate could be expected to be force-freeness, except that, possibly, one may be confronted with incompatibility of the boundary conditions with the vector-potential values given at the grid-sites, i.e. existence-problems for solutions eventually arise. Though definitely possibility a) cannot be ruled out on solid grounds, we found it more promising to proceed with possibility b), interpolation. A guide-line for choosing a particular interpolation method is the reasonable demand that the interpolation should not introduce wild oscillations in-between grid-sites, for we want to assure that the derivatives at the grid sites, which are very sensitive to such oscillations, are not `random' values solely due to the interpolation, but that they reflect more or less directly the change of the primary grid-variable from grid-site to grid-site. This calls for interpolating functions which are as little curved as possible. The easiest and fastest way of interpolating would be to perform local interpolations around a point and its nearest neighbours (e.g. using low-order polynomials or trigonometric functions of different degrees). This interpolation leads, however, to ambiguities for the derivatives: the derivatives, say at a point , are not the same, if the used interpolation is centered at , with the ones calculated with an interpolation centered at e.g. . In this sense, local interpolation is not self-consistent, the derivatives at a grid-site depend on where the used interpolation is centered. Finally, we are left with global interpolation through the whole
grid. Among the candidates are, besides more exotic interpolating
functions, polynomials of degree equal to the grid size, trigonometric
functions (also in the form of Fourier-transforms), low-order smooth
polynomials (e.g. splines). The first candidate, polynomials of a high
degree Trying cubic spline-interpolation, we found that it does not suffer from the problems stated for the other types of interpolation: neither does it introduce wild oscillations, unmotivated by the values at the grid-sites, nor does spline interpolation have problems with describing large or small scale structures (if a functional form of is prescribed, then the analytic derivatives and the derivatives yielded by the interpolation give very close values, in general). Moreover, based on results of Sect. 3, App. C, and Isliker et al. (1998), there is another reason why spline-interpolation is particularly adequate to our problem: It relates the quantity (Eq. (2)), which measures the stress at a site in the CA model, closely to , the Laplacian of (see App. C). The latter is related to the current (), which, from the point of view of MHD, can be considered as a measure of stress in the magnetic field configuration. If this relation would not hold, then the redistribution rules (Eqs. (4) and (5)) of the CA would not be interpretable as the diffusion process revealed by Isliker et al. (1998), and the instability criterion (Eq. 3) would not be so closely related to the current (see Sect. 3 and App. C). ## B.1. why in particular differencing is not adequate to calculate derivatives in a CAWe had rejected above (Sect. 1, Sect. 2.2) the use of difference expressions to calculate derivatives, stating that differencing is not in the spirit of CA models quite in general, since the nature of CA is truly discrete. We think it worthwhile to make this argument more concrete and to show what problems arise if differencing were used:
## Appendix C: relation of toThe stress measure of LH91, , can be related to continuous expressions by representing the values of as Taylor-series expansions around , setting the spatial differences to . It turns out that e.g. and so on for the other two components. due to the discontinuities in the 3rd order derivatives (the superscripts + and - refer to the right and left derivative, respectively). Thus, in case where the third order right and left derivatives are not too different, is a good approximation to . © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |