9
Avalanches with reorganising grains
When grains are deposited on a sandpile, avalanches result. These have much in
common with many other varieties of avalanche; for example, snow or rocks, or
even the stress releases that result in earthquakes. The unifying phenomenon in all
these cases is that of a threshold instability: an overburden builds up, typically that
due to surface roughening, to the point where this threshold is crossed, and grains
are released in an avalanche. Avalanches can be classed in two main categories;
those that do not have intrinsic time or length scales, and those that do. Avalanches
relevant to granular media belong to the second category, and we shall discuss
their characteristics in depth. We will, however summarise some of the known
characteristics of the first category, referring readers who are interested in more
details to ref. [65] on the subject.
9.1 Avalanches type I – SOC
Bak, Tang and Wiesenfeld, in their now famous theory of self-organised criticality
(SOC), suggested [65] that extended systems were marginally stable, such that
the slightest overburdening would cause avalanching; a sandpile at its so-called
‘critical’ angle of repose was held to be paradigmatic of this. Although this turned
out to be, in retrospect, the wrong paradigm, the explorations that surrounded it in
fact greatly enriched the physics of granular avalanches. We touch briefly on the
important features of SOC here. Bak et al. claimed that such avalanches had no
intrinsic time or length scale; thus, avalanches of all sizes are equally probable in
the theory of SOC, with a power spectrum describable by a 1/ f power law. While
experiments [72, 74] have shown that typical avalanche traces are not so simply
described, we describe in the following paragraph the kind of physical scenario
that might lead to such statistics.
Granular Physics, ed. Anita Mehta. Published by Cambridge University Press.
C

A. Mehta 2007.
115

116 Avalanches with reorganising grains
Consider a sandpile whose surface is constrained to be smooth and elastic, on
which grains are deposited in the limit of low inertia; grains will land on the surface,
and possibly dislodge other surface grains, leading to a chain reaction where an
avalanche flows down the pile. Importantly, grains are not allowed to stick to the
surface of the pile, and cannot create roughness; the surface remains as smooth after
the avalanche as it was before. This mechanism is evidently stochastic; depending
on the first landing point of the deposited grain, and the local nature of the surface,
avalanches of variable sizes can be generated. Also, there is a minimal interaction
of the deposited grains with the surface; since the surface is constrained to stay
smooth, the implication is that as many grains leave the surface on average as are
deposited on it. No ‘bumps’ are allowed to form, as they are unsustainable by the
surface; equally, the deposited grains have low inertia, and do not ‘fluidise’ the
grains on the surface on impact. The surface remains essentially unchanged, so
that all pre-existing thresholds hold for avalanching; the angle of repose is close to
being unique.
In practice, sandpile surfaces are rough, angles of repose are non-unique, and
deposited grains interact strongly with the substrate. These give rise to the second
category of avalanches – Type II – specific to granular media which will be discussed
later in this chapter. To put this in context, we review below the standard results on
Type I avalanches.
9.1.1 Review of sandpile cellular automata – Type I
In general, lattice grain models, in which particles are simply represented by reg-
ular objects in discrete geometries, are powerful computational tools for studying
granular dynamics. Their discrete nature and geometrical parallelism are signif-
icant advantages; on the other hand, they require considerable interpretation and
analysis.
The development of lattice grain models follows from lattice gas models of fluid
flows [202, 203]. There, for a particular set of collision rules, the coarse-grained and
long-time behaviour of the lattice gas has been shown to have a precise mapping

onto the solutions of the Navier–Stokes equations for incompressible fluid flow. An
equivalent correspondence has not been made for the lattice grain methods, so that
a unique set of lattice based collision rules is not firmly established for granular
flow models. However, some models, such as that of [117], incorporate space filling
and inelastic interactions, acting as valuable indicators for good continuum models
of granular flow [7].
The most celebrated lattice grain model concerns the flow of grains down the
sloping surface of a sandpile [65]. The simplest nontrivial sandpile model consists
9.1 Avalanches typeI–SOC 117
of monodisperse, unit square grains stacked in columns on a one-dimensional base
of length L. The instantaneous state of the sandpile is described by a set of column
heights z
i
≥ 0, 1 ≤ i ≤ L. In turn, column heights can be used to define local slopes
s
i
such that
s
i
= z
i
− z
i−1
, i > 1, (9.1)
with s
1
= z
1
. At each timestep, one grain is added at column i for 1 ≤ i ≤ L such
that:

z
i
→ z
i+1
, s
i
→ s
i+1
, (9.2)
and, if i < L,
s
i+1
→ s
i+1
− 1. (9.3)
If, after the addition of a grain, the local slope s
i
exceeds some threshold slope
s
c
, then n
f
grains fall from the surface of column i onto columns below itself. In
local sandpile models, grains falling from column i land on column i − 1, but in
nonlocal models, falling grains may be distributed over all columns [1, i − 1], with
grains able to exit the sandpile from column 1. The number of grains falling at each
event, n
f
, may either be constant (as in the ‘limited’ model of Kadanoff et al. [77])
or determined dynamically (as in the ‘unlimited’ model of [77]).

Falling grains cause changes in several column heights (i and i − 1 in local
models), which could lead to the generation of supercritical slopes, s > s
c
, else-
where. More grains could now fall, leading to a chain reaction. When the sand-
pile returns to a state where all slopes are subcritical, the event chain for one
timestep is said to be complete. The number of falling events, n
s
, and the number
of grains which exit the pile, n
x
, are both measures of the size of an avalanche that
ensues.
It is known that the order in which the columns are updated is unimportant, so
that series or parallel updates are equally efficient. The evolution of the sandpile
model may be computed solely in terms of the discrete local slope variables, s
i
,
using an integer or bit representation, leading to a cellular automaton model [203].
Kadanoff et al. [77] have shown that the avalanche distribution function of models
of this type is typically multifractal. There are no special avalanche sizes and,
therefore, these manifest SOC. Nonlocal and/or unlimited models, different values
for n
f
and higher dimensionalities do not lead to any substantive change from SOC
behaviour, although they do lead to a change of universality class [77].
These simple models, however, fail to explain the dominance of large avalanches
in real sandpiles [72, 74]. In the next section we focus on these, and describe in
detail a cellular automaton model [75, 83, 168] which leads to their generation.
118 Avalanches with reorganising grains

9.2 Avalanches type II – granular avalanches
Avalanches are the signatures of instabilities on an evolving sandpile: spatiotempo-
ral roughness is alternately built up and smoothed away in the course of avalanche
flow. We present below an intuitive picture of avalanching in sandpiles, pointing
out that it could be relevant to other scenarios (e.g. granular flows along an inclined
plane [166], or sediment consolidation [204]).
As deposition occurs on a sandpile surface, clusters of grains grow unevenly at
different positions and roughness builds up until further deposition renders some
of these unstable. They then start ‘toppling’, so that grains from unstable clusters
flow down the sandpile, knocking off grains from other clusters. The net effect
of this is to ‘wipe off’ protrusions (where there is a surfeit of grains at a cluster)
and to ‘fill in’ dips, where the oncoming avalanche can disburse some of its grains.
Typically, small avalanches build up surface roughness, while large avalanches have
a smoothing effect; in the latter case, a rough precursor surface typically leads to
avalanche onset, and subsequently, an overall smoothing of the surface. This result
is rather robust, having been found independently using a variety of models, which
will be presented in this and succeeding chapters. In the next chapter, a coupled map
lattice model demonstrating stick–slip dynamics [22] will be discussed, while in
the following one, continuum equations coupling surface to bulk relaxation [95, 96]
will be presented.
In this chapter, we focus on a cellular-automaton model [75, 83, 168] of an evolv-
ing sandpile to look in more depth at the mechanisms by which a large avalanche
smooths the surface. This sandpile model is a ‘disordered’ and non-abelian version
of the basic Kadanoff cellular automaton [77]; in the present model grain ‘flip’ is
added to the grain flow which is already present in the Kadanoff model.
The disordered model sandpile
1
is built from rectangular lattice grains that have
aspect ratio a ≤ 1 arranged in columns i with 1 ≤ i ≤ L, where L is the system
size. Each grain is labelled by its column index i and by an orientational index 0

or 1, corresponding respectively to whether the grain rests on its larger or smaller
edge.
The dynamics of this model, which have been described at length elsewhere
[83, 168], are:
r
Grains are deposited on the sandpile with fixed probabilities of landing in the 0 or 1
position.
r
The incoming grains, as well as all the grains in the same column, can then ‘flip’ to
the other orientation stochastically (with probabilities which decrease exponentially with
1
Note that while the representation of disorder in this model is identical to that of the model [174] presented in
Chapter 7, its dynamics are entirely different.
9.2 Avalanches type II – granular avalanches 119
depth from the surface). This is a way of introducing a time-dependent disorder into the
problem.
r
Column heights are then computed as follows: the height of column i at time t, h(i, t),
can be expressed in terms of the instantaneous numbers of 0 and 1 grains, n
0
(i, t) and
n
1
(i, t) respectively:
h(i, t) = n
1
(i, t) + an
0
(i, t). (9.4)
r

Finally, grains fall to the next column down the sandpile (maintaining their orientation
as they do so) if the height difference exceeds a specified threshold as in the Kadanoff
model [77]. At this point, avalanching occurs.
The presence of the flipping mechanism – ‘annealed disorder’ – leads, for large
enough system sizes, to a preferred size of large avalanches [75], while in the
absence of disorder, scale-invariant avalanche statistics are observed. Below, the
evolving state of the sandpile surface is correlated with the onset and propagation
of avalanches.
9.2.1 Dynamical scaling for sandpile cellular automata
It is customary in the study of generalised surfaces to examine the widths generated
by kinetic roughening [169], and then establish properties related to dynamical scal-
ing. This procedure can be generalised to include the kinetic roughening of sandpile
cellular automata. The hypothesis of dynamical scaling for sandpile surfaces [83]
reads, in terms of the surface width W of the sandpile:
W (t) ∼ t
β
, t  t
crossover
≡ L
z
; (9.5)
W (L) ∼ L
α
, L → ∞. (9.6)
Thus, to start with, roughening occurs at the CA sandpile surface in a time-
dependent way; after an initial transient, the width scales asymptotically with time
t as t
β
, where β is the temporal roughening exponent. This regime is appropri-
ate for all times less than the crossover time t

crossover
≡ L
z
, where z = α/β is
the dynamical exponent and L the system size. After the surface has saturated,
i.e. its width no longer grows with time, the spatial roughening characteristics of
the mature interface can be measured in terms of α, an exponent characterising the
dependence of the width on L.
The surface width W (t) for a sandpile automaton is defined in terms of the mean-
squared deviations from a suitably defined mean surface; the instantaneous mean
surface of a sandpile automaton is thus defined as the surface about which the sum
of column height fluctuations vanishes. Clearly, in an evolving surface, this must be
a function of time; hence all quantities in the following analysis will be presumed
to be instantaneous.
120 Avalanches with reorganising grains
The mean slope s(t) defines expected column heights, h
av
(i, t), according to
h
av
(i, t) = is(t), (9.7)
where it is assumed that column 1 is at the bottom of the pile. Column height
deviations are defined by
dh(i, t) = h(i, t) − h
av
(i, t) = h(i, t) − is(t). (9.8)
The mean slope must therefore satisfy

i
[h(i, t) − is(t)] = 0 (9.9)

since instantaneous height deviations about it vanish; thus
2
s(t)=2
i
[h(i, t)]/L(L + 1) (9.10)
The instantaneous width of the surface of a sandpile automaton, W (t), can be
defined as:
W (t) =


i
[dh(i, t)
2
]/L, (9.11)
which can in turn be averaged over several realisations to give W , the average
surface width in the steady state.
Another quantity of interest is the height–height correlation function, C( j, t);
this is defined by
C( j, t) =dh(i, t)dh(i + j, t)/dh(i, t)
2
, (9.12)
where the mean values are evaluated over all pairs of surface sites separated by j
lattice spacings:
dh(i, t)dh(i + j, t)=
i
(dh(i, t)dh(i + j, t))/(L − j ) (9.13)
for 0 ≤ j < L. This function is symmetric and can be averaged over several reali-
sations to give the time-averaged correlation function C( j).
9.2.2 Qualitative effects of avalanching on surfaces
Figure 9.1(a) shows a time series for the mass of a large (L = 256) evolving dis-

ordered sandpile automaton.
3
The series has a typical quasiperiodicity [74]. The
vertical line in Fig. 9.1(a) denotes the position of a particular ‘large’ event, while
2
Note that this slope is distinct from the quantity s

(t)=h(L , t)/L that is obtained from the average of all the
local slopes s(i, t) = h(i, t ) − h(i − 1, t), about which slope fluctuations would vanish on average.
3
Throughout this chapter we refer to disordered sandpiles described in reference [168] with parameters z
0
=
2, z
1
= 20 and a = 0.7, unless otherwise stated.