Landslides, Rockfalls and Sandpiles - Environmental Modelling: Finding Simplicity in Complexity - page 260

Figure 16.4 Evolution of a sandpile in the simple sandpile model, starting from a flat surface. The last avalanches that have occurred
are marked grey. For convenience, an additional frame of inactive sites with zero height is plotted around the boundary of the
domain.

unstable. As a consequence, the avalanche-size distribu-
tion shown in Figure 16.5 is rather trivial; it just reflects
the distance of the sites to the boundary. On an l

10 − 1

64 x 64

l grid,
the largest avalanches involve l 2 sites, but these can only
be initiated at the top of the pyramid. Conversely, the
smallest avalanches (size one) can be initiated everywhere
at the boundary. So the observed avalanche-size distribu-
tion is rather a geometric property of the state the model
approaches by self-organization than a dynamic property
of the model.
In order to improve the model we should first find
out the deeper reason for the behaviour of the model.
It is an awkward combination of relaxation rule and
threshold of instability: when a site becomes unstable as
the result of adding a grain, this grain topples, so that
the site returns to its original height. Thus, only the site
where the grain has toppled to can be destabilized as
a consequence; the other neighbourhood sites are not
affected. Thus, each avalanche always involves only one
grain that topples towards the boundary or comes to rest
before it reaches the boundary. This result is independent
of the actual shape of the surface and is in stark contrast to
the BTW model where, in principle, all four neighbours
may become unstable.
This problem can only be solved by allowing that
more than one grain topples when one grain is added

×

256 x 256

10 − 2

1024 x 1024

10 − 3

10 − 4

10 0 10 1 10 2 10 3

Event size

Figure 16.5 Probability density of the avalanches in the simple
sandpile model on grids of different sizes.

the surface reveals the reason for this behaviour. The
surface evolves towards a regular pyramid with com-
pletely smooth faces. For each site, the height difference
towards its lowest neighbour is one. Therefore, each addi-
tional grain immediately becomes unstable. After it has
moved one site downslope, this site becomes unstable,
and so on. Thus, each grain that is randomly added top-
ples downslope until it passes the boundary of the lattice;
it can neither stop somewhere nor make any other grain