We can summarize the result of the previous section in the claim that the CA of Lu and Hamilton (see Sect. 2) is equivalent to the following continuous system:
Given is a magnetic field in 3D-space. The initial condition is a random distribution. The field evolves according to
with the diffusion coefficient
and the boundary condition
where denotes the sphere of radius around , and its surface. is the diffusive time and the characteristic length of the system. The source-function is asymmetric in its values and random in space and time, with , a time scale much larger than , and spatial correlations which decay over a length smaller than . An example of such a process is a Poisson process in time and space, with mean time-interval between two shots and mean spacing between two shots .
Eqs. (33), (34) and (35) describe a system in which a field quantity is randomly increased, until it reaches a threshold. This turns on a fast diffusion process which takes place over a volume of size and acts during a time (it does not stop when the critical quantity falls below the threshold, but only when the critical quantity has reached a value 0). It is thus a localized, threshold-dependent, fast diffusion process, completely disconnected from the surrounding region. The restructuring of the magnetic field may eventually cause that in an adjacent volume the instability criterion is met, and so on, so that an avalanche-like event may occur. Lu (1995) has shown that continuous systems with localized diffusion events can show avalanche behaviour. The critical point is that in a CA the region which is unstable completes its diffusive process before the neighbour sites may eventually start to become unstable, whereas in continuous systems, neighbour regions may become unstable at any time instant during the primary instability, causing effects which are difficult to predict, in general.
In 1D, the diffusive part of the equation can be solved analytically. If an instability starts at and with an initial distribution of the fields in (with ), then we have
where in , and outside it is defined in such a way that it is an even function with respect to and (which implies that is a particular choice of a periodic continuation). As an illustration, we plot in Fig. 2 for the times (), with the initial condition , , , , , and consequently : The field goes asymptotically to a flat state, which has effectively been reached already after the time ( in Fig. 2). We note that in 1D, a straight line with any slope would also be an asymptotic solution to Eq. (33). However, the boundary condition (Eq. 35) demands that the field has zero slope at the two edges, which introduces convex regions of , and which drive then the diffusion equation again until a flat distribution of the fields is reached.
Of course, the equation could also be solved numerically on a spatial grid. The essential point of the CA of LH is that the equation is not discretized to solve it, but directly its solution after the relaxation time is implemented in the form of CA evolution-rules - which is feasible since this solution can be expected to be trivial, namely a flat distribution of the fields after the diffusive time, as illustrated above through the analytical example. If one chooses a grid-size
and a time-step
and makes the identifications (for Eq. 1), and (for Eq. 3), then the CA rules in Sect. 2, since they correspond to flattening of the field ( is 0 after one time step), are a solution of the diffusion equation after the diffusive time has elapsed (leaving though some rest-fluctuations in the field, since the latter is only flattened and not exactly equi-distributed, as we would demand from the analytical example given above).
Note that if one wants to have information on the process on time scales smaller than or spatial scales smaller than then the CA of LH is useless. One would have to construct a different CA, with different rules, which would be nothing else than an integration scheme of the PDE Eq. (33). But due to its simplicity, the CA allows to study the statistics of the large scale events, such as the possible occurrence of avalanches, and this is its true benefit.
© European Southern Observatory (ESO) 1998
Online publication: June 26, 1998