3. Recovering the differential equation behind the CA of LH
3.1. Quiet evolution; the instability criterion; the unstable points and their nearest neighbours
The expressions used in LH involve differences of the magnetic fields in space and time, and it is natural to interpret the difference-expressions as discretized differential-expressions. We start with considering the control quantity (Eq. 2), considering the x-coordinate only:
We now explicitly introduce , the distance between adjacent grid sites (LH assume ), and multiply the equation with , whence
The r.h.s. obviously is - besides a factor - the discretization of , done in a standard way (see e.g. Gerald and Wheatley 1989). Treating the other coordinates analogously, we can identify
The time evolution of the process is the following: In the non-critical state, the evolution is, according to Eq. (1),
or, introducing the time step , which was assumed to be always 1 in LH, we multiply with
so that for , we find
The singularity on the r.h.s. is removed by interpreting as , where is the field injected per unit time, and saying that Lu and Hamilton consider the field accumulated over the time step . We get finally the quiet evolution
The instability criterion (Eq. 3) turns into
(by first multiplying Eq. (3) with , and then inserting Eq. 10). Again, we have to interpret as a 'cumulative threshold', in the sense that , so that the criterion is
If the instability criterion is met, then according to Eq. (4) we have for the point where the instability occurs, inserting Eq. (10) for ,
or by dividing by the time-step
and for , , we find
where we have introduced the diffusion constant . Again, a singularity seems to appear on the r.h.s. We can absorb it by assuming for some , i.e. we interpret Lu and Hamilton as having used a 'cumulative' diffusion coefficient , which, for , adopts the value . Whence we have
with the diffusion coefficient
It remains to consider the nearest neighbours of the points where the instability criterion is fulfilled. They evolve according to a different rule than the center point, and we have to check whether the different rules are compatible. The nearest neighbours evolve according to Eq. (5), so that, by using Eq. (10),
and for , , we find
with a dummy diffusion constant , or absorbing again the seeming singularity by introducing so that ,
with the diffusion coefficient
3.2. Problems and inconsistencies
In what we have recovered so far, a site which becomes unstable has a temporal evolution (Eqs. (20) and (21)) which is different from the one of the nearest neighbour sites (Eqs. (25) and (26)): the diffusion coefficients are different. If we consider the evolution law on the grid as a discretized partial differential equation (PDE), then we may choose the grid size arbitrarily. If we let it go to zero, then the central point and its nearest neighbours approach each other until they coincide. For physical reasons, we must demand continuity of the fields, which can be achieved only if the coefficients in the PDE are continuous, too, i.e. the evolution laws for the central point and its nearest neighbours must coincide for . However, from Eqs. (21) and (26) it is clear that this condition is not satisfied, the diffusion coefficients are different.
The way out of this dilemma is realizing that the grid size in the CA of LH is not an arbitrary quantity, but it is physically meaningful, it is a property of the considered system, some kind of a characteristic size (termed in the following). Whence, the preliminary interpretation we give so far to the CA dynamics in the LH model is that, in the burst mode, the system evolves according to a diffusion equation (Eq. 20), with a diffusivity which is given by Eq. (21) at the point where the instability criterion is met, drops to a value given by Eq. (26) for points which are one characteristic length away from the central point, and finally settles to 0 for points which are further than two characteristic lengths away (in Fig. 1 a sketch of vs. x (1D) is given).
B. Two physical and a mathematical problem
The interpretation given so far is still not satisfying, for physical as well as mathematical reasons: (i) The diffusivity in the neighbourhood of an unstable point is negative (Eq. 26). This cannot be motivated with reasonable physical arguments, and mathematically, such a diffusion equation has exploding solutions. (ii) Both diffusion coefficients, the central and the peripheral one, carry still an element which is reminiscent of the particular grid chosen, namely the numerical factors 1/7 in Eq. (21) and 1/42 in Eq. (26). Following the argumentation in LH, these factors in a different grid would have to be in Eq. (21) and in Eq. (26), where n is the number of nearest neighbours (e.g. in a grid with hexagonal prismas one has ). Obviously, the particular n appears still in the continuous formulation of the process, which is contradictory. Letting gives useless results for the central point (Eq. 21): if we replace the 7 in Eq. (21) with and try to absorb the singularity which appears for , then, since we already identified with , we would have to introduce , for some given (any characteristic time of the process). Therewith, however, the time-step in the CA would depend on the number of nearest neighbours, which is in contradiction with all the CA models. (iii) Having the diffusivity varying from a positive value to a negative value is like having two different processes acting, but the two processes are actually connected: the neighbourhood of an unstable point receives exactly the field which is lost by the central point.
3.3. The consistent approach
The conclusion so far is that the derived PDEs (Eqs. 20 and 25) cannot be the continuous version of the CA in its bursting phase. To find the correct continuous formulation for the CA of LH, one has to reconsider the CA rules (Sect. 2) for the case where an instability occurs: The central point of the unstable region evolves according to Eq. (4), and the neighbour-sites according to Eq. (5). This evolution is characterized by the fact that after a time step , we have (Eq. 6): the site is no more unstable, the fields are flattened, or, in continuous language, since (Eq. 10), we have .
Every burst in the CA is such that after a time step the magnetic field has diffused through a distance and is completely relaxed. This implies that has also a physical meaning, it is a relaxation time of the ongoing process, and is the characteristic size of the system, as also stated above. This means that the discrete evolution laws (Eqs. 4 and 5) should not be interpreted as the equivalent to differential equations (as done in Eqs. 20 and 25), though this is formally possible, but they describe the solution to a continuous equation which has to be found from the following general properties of the ongoing process:
1) After the relaxation time , the magnetic field around an unstable site is almost homogeneously distributed over the sphere with radius the characteristic-size of the system. Thereby, homogeneity is measured through :
2) The process is not interacting with the region outside the sphere around the unstable point, the field in this sphere is conserved, neither in- or outflow of field takes place:
We are looking for a differential equation which fulfills these two conditions: generally spoken, we need to determine in or such that the above properties are recovered. Of course, there is an infinite number of solutions for , and we will look for the simplest one (higher order terms cannot be motivated by the few properties wanted). To find , we first note that the quantity which drives the process obviously is , since if this quantity is not zero, then the process acts, until this Laplacian vanishes. Now assume that is negative in some region (considering the first component of , the others are analogous), then is a convex function in this region, and it has to decrease in order that the convexity disappears (and thereby goes to 0). Analogously, has to increase where is positive. Whence, if we assume that is proportional to we have a simple equation with the wished property, and analogously for the other components, so that, in vector form,
is the wanted equation, with some constant , which can be interpreted as a diffusivity, and for which we must have
in order that the system has relaxed in a volume after a time .
The second wanted property (Eq. 28) yields the boundary condition: We have (for one of the components)
where we have first inserted the differential equation (Eq. 29) and then used Gauss' theorem ( is a unit vector normal to the surface of the sphere ). We therefore conclude that the boundary conditions are
for all times t during which the burst process is acting. (From this derivation it is clear that the form of the boundary conditions which describe no interaction with the outside region depend on the differential equations and cannot be stated in general, irrespective of what equation should have this property.)
Going from the uncovered continuous equation (Eq. 29) back to the CA, it is now clear that running the CA of LH corresponds to solving the diffusion equation (Eqs. 29 with 32) in a particular way: time and space are discretized by using a grid-size which is the characteristic length and a time step which is the diffusion time of the system. If the sphere contains n neighbour points in a discretization, then, since we know that after the fields in this sphere are flattened, we may just redistribute the fields by using a dilution factor . Exactly this is implemented in the evolution rules for central point (Eq. 4), the nearest neighbours (Eq. 5), and in that grid sites further away are not influenced. In this sense, the CA solves the diffusion equation: its trivial solution (flattening of the magnetic field in the sphere after time ) is the basis of the evolution law of the central point and its nearest neighbours. We note, however, that these evolution rules are not unique, they just must fulfill that (in continuous language) is 0 after one time-step. An alternatively possible rule would for instance be the complete equi-distribution of the fields (see also the remark in the next section).
This way of proceeding simplifies greatly the solution of the diffusion equation (Eq. 29), it has the disadvantage, however, that nothing is known on the dynamics on time scales shorter than or length scales shorter than . If this evolution would be of interest, then the CA frame would not help anymore, and a usual PDE integration scheme would have to be used. On the other hand, the advantage of this approach is that one can run a simplified model without implementing the unknown details of the process, and can monitor therewith the global evolution of a spatially extended complex system, given that it consists of many localized, randomly triggered diffusion events.
© European Southern Observatory (ESO) 1998
Online publication: June 26, 1998