## 3. Recovering the differential equation behind the CA of LH## 3.1. Quiet evolution; the instability criterion; the unstable points and their nearest neighboursThe 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 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), or again 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.
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 ## 3.3. The consistent approachThe 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 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 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 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 |