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 dislod

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

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

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

si such that

si = z i − z i−1, i > 1, (9.1)

with s1= z1 At each timestep, one grain is added at column i for 1 ≤ i ≤ L such

that:

and, if i < L,

If, after the addition of a grain, the local slope s i exceeds some threshold slope

sc, then nf 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, nf, 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 > sc, 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, ns, and the number

of grains which exit the pile, nx, 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 nfand 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

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 sandpile1is 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.

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, n0(i , t) and

n1(i , t) respectively:

h(i , t) = n1(i , t) + an0(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 crossover ≡ L z

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, whereβ is the temporal roughening exponent This regime is

appropri-ate for all times less than the crossover time tcrossover≡ 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 ofα, 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

The mean slopes(t) defines expected column heights, hav(i , t), according to

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) − hav(i , t) = h(i, t) − is(t). (9.8) The mean slope must therefore satisfy

since instantaneous height deviations about it vanish; thus2

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 giveW, 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 functionC( 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 quantitys (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 z0 =

2, z = 20 and a = 0.7, unless otherwise stated.

in Fig 9.1(b), the marked ‘second peak’ [83] in the avalanche size distribution is composed of such large avalanches

The large avalanche highlighted in Fig 9.1(a) is pictured in Fig 9.1(c) The initiation site is marked by an arrow and the outline of the surface before and after the avalanche (corresponding to about 5 per cent of the mass of the sandpile) shaded in black We note that this is an example of an ‘uphill’ avalanche described

in Chapter 5 [165, 166]; the avalanche propagates downwards, of course, but it

destabilises particles above its point of initiation The inset shows the relative motion

of the surface during this event; the signatures of smoothing by large avalanches are already evident, as the precursor state in the inset is much rougher than the final state

In Fig 9.1(d), the grain-by-grain picture of the aftermath pile is superposed on the precursor pile, which is shown in shadow An examination of the aftermath pile and the precursor pile [83] shows that the propagation of the avalanche across the upper half of the pile has left only a very few disordered sites in its wake (i.e the majority of the remaining sites are of type 0) whereas the lower half (which was undisturbed by the avalanche) still contains many disordered, i.e type 1 sites

in the boundary layer This suggests that larger avalanches rid surface layers of their disorder-induced roughness, a fact that is borne out by more quantitative investigations below

Detailed studies have revealed [83] that the very largest avalanches, which are system-spanning, remove virtually all disordered sites from the surface layer; one

is then left with a normal ‘ordered’ sandpile, where the avalanches have their usual scaling form for as long as it takes for a layer of disorder to build up When the disordered layer reaches a critical size, another large event is unleashed; this

is the underlying reason for the quasiperiodic form of the time series shown in Fig 9.1(a) This picture is confirmed by a totally different model, the coupled-map lattice model of a reorganising sandpile [22], to be discussed in the next chapter

The sequence of Figs 9.1 a–d is shown also for an ordered pile – Fig 9.2 a–d and a small disordered pile – Fig 9.3 a–d The small disordered pile has a mass– time series (Fig 9.3a) that is midway between the scale-invariance of the ordered pile (Fig 9.2a) and the quasiperiodicity of the large disordered pile (Fig 9.1a) The avalanche size distribution of the small disordered pile (Fig 9.3b) is likewise intermediate between that of the ordered pile (which shows the scale invariance

observed by Kadanoff et al [77]) and the two-peaked distribution characteristic of

the disordered pile [75, 83] This suggests that a crossover length must exist, after which the fully non-invariant scale characteristics of real sandpiles would begin to manifest; interestingly, such a crossover length in the mass time series has indeed been observed in experiment [74]

51000

52000

53000

54000

55000

56000

80 90 100 110 120

Time (k)

(a)

(a) A time series of the mass The vertical line

indicates the position in this series of the large

avalanche illustrated in c, d.

−8

−6

−4

−2

(b)

(b) A log–log plot of the event size distribution.

0

50

100

150

200

250

Position

(c)

(c) An illustration of a large wedge shaped avalanche;

a lighter aftermath pile has been superposed onto the

dark precursor pile, and an arrow shows the point

at which the event was initiated The inset shows

the relative positions of the two surfaces and their

relationship to a pile that has a smooth slope.

(d)

(d) A detailed picture of the internal structure in the aftermath of a large avalanche event The individual grains

of the aftermath pile (for columns 1 −

128 of the sandpile with L= 256) are superposed on the gray outline of the precursor pile.

Fig 9.1 Statistics for a model sandpile (L= 256) with a structurally disordered surface layer

45500

46000

46500

20 30 40 50 60

Time (k)

(a)

(a) A time series of the mass The vertical line

indicates the position in this series of the large

avalanche illustrated in c, d.

−8

−6

−4

−2

(b)

(b) A log–log plot of the event size distribution.

0

50

100

150

200

250

300

350

Position

(c)

(c) An illustration of a large wedge shaped

avalanche; a lighter aftermath pile has been

super-posed onto the dark precursor pile, and an arrow

shows the point at which the event was initiated The

inset shows the relative positions of the two surfaces

and their relationship to a pile that has a smooth

slope.

(d)

(d) A detailed picture of the inter-nal structure in the aftermath of a large avalanche event The individ-ual grains of the aftermath pile (for columns 1 − 128 of the sandpile

with L= 256) are superposed on the gray outline of the precursor pile.

Fig 9.2 Statistics for a model sandpile (L= 256) without structural disorder.

(a) A time series of the mass The vertical line

indicates the position in this series of the large

avalanche illustrated in c, d.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

log 10 (N)

−8

−6

−4

−2

(b)

(b) A log–log plot of the event size distribu-tion.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

Position

(c)

(c) An illustration of a large wedge shaped

avalanche; a lighter aftermath pile has been

superposed onto the dark precursor pile, and an

arrow shows the point at which the event was

ini-tiated The inset shows the relative positions of

the two surfaces and their relationship to a pile

that has a smooth slope.

(d)

(d) A detailed picture of the internal struc-ture in the aftermath of a large avalanche event The individual grains of the after-math pile (for columns 1 − 128 of the

sandpile with L= 256) are superposed on the gray outline of the precursor pile.

Fig 9.3 Statistics for a small model sandpile (L= 64) with a structurally disor-dered surface layer

In both small and large disordered piles, we see evidence of large ‘uphill’ avalanches which shave off a thick boundary layer containing large numbers of disordered sites, and leave behind a largely ordered pile (see Figs 9.1 c–d and Figs 9.3 c–d) By contrast, the ordered pile loses typically two commensurate