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

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Appendix A: temporal evolution of the CA

The temporal evolution of the CA models presented in this article is governed by the following rules:

  1. initializing

  2. loading

  3. scanning: create a list of the unstable sites; if there are none, return to loading (1)

  4. scanning and bursting: redistribute the fields at the unstable sites which are in the list created in the scannings 2 or 4

  5. 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 continuation

We mentioned in Sect. 2.2 that other possibilities for continuation of the vector-potential besides spline interpolation would be: a) continuation of [FORMULA] 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 [FORMULA] 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 [FORMULA]) 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 [FORMULA], are not the same, if the used interpolation is centered at [FORMULA], with the ones calculated with an interpolation centered at e.g. [FORMULA]. 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 n (with n the number of grid points in one direction), we reject immediately since it is notorious for its strong oscillations in-between grid-sites, mainly towards the edges of the grid. We tried the second candidate, trigonometric interpolation, in the form of discrete Fourier transform. Testing this by prescribing analytic functions for [FORMULA] and comparing the numerical derivatives with the analytic ones, it turned out that there arise problems with representing structures in [FORMULA] as large as the entire grid (the wave-number spectrum is too limited), and with structures as short as roughly the grid-spacing (different prescribed short structures are taken for the same).

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 [FORMULA] 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 [FORMULA] (Eq. (2)), which measures the stress at a site in the CA model, closely to [FORMULA], the Laplacian of [FORMULA] (see App. C). The latter is related to the current ([FORMULA]), 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 CA

We 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:

1. Consistency with the evolution rules: Isliker et al. (1998) have shown that the classical solar flare CA are not just the discretized form of a differential equations. Instead, they describe the time-evolution of a system by rules which express the direct transition from a given initial to a final state which is the asymptotic solution of a simple diffusion equation. The time-step corresponds therewith to the average time needed for smallest scale structures (structures as large as a neighbourhood) to diffuse, and the grid-size corresponds to the size of these smallest occurring structures. Assuming that the CA models were just discretized differential equations would lead to severe mathematical and physical contradictions and inconsistencies (continuity for [FORMULA] is violated (with [FORMULA] the grid-size), and negative diffusivities appear). Therewith, in order to be consistent with the evolution rules, which assume a finite grid-size, one cannot assume for the purpose of differentiating this same grid-size to be approximately infinitesimal.

2. Derivatives as difference expressions are not self-consistent: There are several equivalent ways to define numerical derivatives with the use of difference expressions: there are e.g. the backward difference [FORMULA], and the forward difference [FORMULA]. Both should give comparable values in a given application, else, in the context of differential equation integration, one would have to make the resolution higher. In the case of CA-models, we find that the two difference expressions yield values which differ substantially from each other: E.g. for an initial loading of the grid with independent random values for the [FORMULA]-field, the difference between the backward and the forward difference expression can be as large as the field itself. Such an initial condition would of course not make sense in the context of partial differential equations, in the context of CA, however, it is a reasonable starting configuration, and the evolution is unaffected by such an initial loading. Moreover, when the CA models we discuss in this article have reached the SOC state, then the differences between e.g. the backward- and forward-difference expressions can be as large as 400%. There is no way to reduce this discrepancy, since grid-refinement is principally impossible for CA: the evolution is governed by a set of rules, and making the grid spacing smaller by introducing new grid-points in-between the old ones would actually just mean to make the grid larger, since the evolution rules remain the same, there are no rules for half the grid-spacing.

Appendix C: relation of [FORMULA] to [FORMULA]

The stress measure of LH91, [FORMULA], can be related to continuous expressions by representing the values of [FORMULA] as Taylor-series expansions around [FORMULA], setting the spatial differences to [FORMULA]. It turns out that e.g.

[EQUATION]

and so on for the other two components. In general, it is therefore not adequate to consider [FORMULA] to be a good 4th order approximation to [FORMULA], since higher order corrections can be large, they depend on the way the vector potential is continued in-between grid-sites. If we had, for instance, chosen global polynomial interpolation instead of spline-interpolation, the higher order terms would not be negligible, above all towards the edges of the grid, since polynomial interpolation is known for introducing fluctuations near the edges of the grid. Consequently, [FORMULA] would be a bad approximation to [FORMULA]. In order [FORMULA] to be a good approximation to [FORMULA], interpolation with, for example, 3rd order polynomials would be an optimum choice ([FORMULA] would be an exact approximation to [FORMULA]). Thus, 3rd order polynomials would be the choice for local interpolation, which, however, is not applicable, since it introduces discontinuities in [FORMULA] and [FORMULA] (see App. B). The way out of the dilemma we suggested in this article is the use of cubic splines, which provide global interpolation with 3rd order polynomials, with [FORMULA] and [FORMULA] continuous, and only third order derivatives are discontinuous (this is the price of the compromise). For splines then, Eq. (C.1) writes as

[EQUATION]

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, [FORMULA] is a good approximation to [FORMULA].

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Online publication: December 5, 2000
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