Coupled continuum equations - the dynamics of sandpile surfaces

11 Coupled continuum equations: the dynamics ofsandpile surfaces 11.1 Introduction 11.1.1 Some general remarks The two previous chapters have dealt, in different guises, with the post-av

11 Coupled continuum equations: the dynamics of

sandpile surfaces

11.1 Introduction

11.1.1 Some general remarks

The two previous chapters have dealt, in different guises, with the post-avalanchesmoothing of a sandpile which is expected to happen in nature [213] It is clearwhat happens physically: an avalanche provides a means of shaving off roughnessfrom the surface of a sandpile by transferring grains from bumps to available voids[22, 69, 83], and thus leaves in its wake a smoother surface However, surprisinglylittle research has been done on this phenomenon so far, despite its ubiquity innature, ranging from snow to rock avalanches

In particular, what has not attracted enough attention in the literature is thequalitative difference between the situations which obtain when sandpiles exhibitintermittent and continuous avalanches [151] In this chapter we examine both thelatter situations, via coupled continuum equations [95, 96] of sandpile surfaces.These were originally envisaged [69] as the local version of coupled equations thathad been written down using global variables in [42]; subsequently, many versionswere introduced in the literature [214, 215] to model different situations The use

of these equations has also since been diversified into many areas, including rippleformation [216] and the propagation of sand dunes [217], about which we will havesomething to say at the end of this chapter

In order to discuss this, we introduce first the notion that granular dynamics is welldescribed by the competition between the dynamics of grains moving independently

of each other and that of their collective motion within clusters [69] A convenientway of representing this is via coupled continuum equations with a specific couplingbetween mobile grains ρ and clusters h on the surface of a sandpile [95] This

represents a formal outline of the most general situation of the coupling between

Granular Physics, ed Anita Mehta Published by Cambridge University Press.
C A Mehta 2007.

148

11.1 Introduction 149

surface and bulk in a sandpile; specific terms can now be modified to model specificscenarios In general, the complexity of sandpile dynamics leads us to equationswhich are coupled, nonlinear and noisy: these equations present challenges to thetheoretical physicist in more ways than the obvious ones to do with their detailedanalysis and/or their numerical solutions

11.1.2 Sand in rotating cylinders; a paradigm

A particular experimental paradigm that we choose to put the discussions in context

is that of sand in rotating cylinders [218] In the case when sand is rotated slowly in

a cylinder, intermittent avalanching is observed; thus sand accumulates in part ofthe cylinder to beyond its angle of repose [70] and is then released via an avalancheprocess across the slope This happens intermittently, since the rotation speed isless than the characteristic time between avalanches By contrast, when the rotationspeed exceeds the time between avalanches, we see continuous avalanching on thesandpile surface Though this phenomenon has been observed [70] and analysedphysically [151] in terms of avalanche statistics, we are not aware of measurementswhich measure the characteristics of the resulting surface in terms of its smoothness

or otherwise What we focus on here is precisely this aspect, and make predictionsfor future experiments

In the regime of intermittent avalanching, we expect that the interface will be theone defined by the ‘bare’ surface, i.e the one defined by the relatively immobileclusters across which grains flow intermittently This then implies that the rough-

ening characteristics of the h profile should be examined The simplest of the three

models we discuss in this chapter (an exactly solvable model referred to hereafter

as Case A) as well as the most complex one (referred to hereafter as Case C) treatthis situation, where we obtain in both cases an asymptotic smoothing behaviour in

h When on the other hand, there is continuous avalanching, the flowing grains

pro-vide an effective film across the bare surface and it is therefore the speciesρ which

should be analysed for spatial and temporal roughening In the model hereafterreferred to as Case B we look at this situation, and obtain the surprising result of agradual crossover between purely diffusive behaviour and hypersmooth behaviour

In particular, the analysis of Case C reveals the presence of hidden length scaleswhose existence was suspected analytically, but not demonstrated numerically inearlier work [95, 219]

The normal procedure for probing temporal and spatial roughening in interfaceproblems is to determine the asymptotic behaviour of the interfacial width withrespect to time and space, via the single Fourier transform Here only one of the

variables (x , t) is integrated over in Fourier space, and appropriate scaling

rela-tions are invoked to determine the critical exponents which govern this behaviour

150 Coupled continuum equations: sandpile surfaces

However, it turns out that this leads to ambiguities for those classes of problemswhere there is an absence of simple scaling, or to be more specific, where multiplelength scales exist [220] In such cases we demonstrate that the double Fourier trans-

form (where both time and space are integrated over) yields the correct answers.

This point is illustrated by Case A, an exactly solvable model that we introduce;

we then use it to understand Case C, a nonlinear model where the analytical resultsare clearly only approximations to the truth

11.2 Review of scaling relations for interfacial roughening

In order to make some of these ideas more concrete, we now review some generalfacts about rough interfaces [221] Three critical exponents,α, β and z, characterise

the spatial and temporal scaling behaviour of a rough interface They are niently defined by considering the (connected) two-point correlation function ofthe heights,

in the whole long-distance scaling regime (x and t large) The scaling function F

is universal in the usual sense;α and z = α/β are respectively referred to as the

roughness exponent and the dynamical exponent of the problem In addition, we

have for the full structure factor which is the double Fourier transform S(k , ω),

S(k , ω) ∼ ω−1k −1−2α (ω/k z),

which gives in the limit of small k and ω,

(11.2)The scaling relations for the corresponding single Fourier transforms are

(11.3)

In particular, we note that the scaling relations for S(k , ω) (Eq (11.2)) always

involve the simultaneous presence of α and β, whereas those corresponding to

the double Fourier transforms, we need in each case information from the growing

11.3 Case A: the Edwards–Wilkinson equation with flow 151

as well as the saturated interface (the former being necessary forβ and the latter for α), whereas for the single Fourier transforms, we need only information from the

saturated interface for S(k , t = 0) and information from the growing interface for S(x = 0, ω) On the other hand, the information that we will get out of the double

Fourier transform will provide a more unambiguous picture in the case wheremultiple length scales are present, something which cannot easily be obtained inevery case with the single Fourier transform

In the sections to follow, we present, analyse and discuss the results of Cases A,

B and C respectively We then reflect on the unifying features of these models, andmake some educated guesses on the dynamical behaviour of real sandpile surfaces.Finally, we present as an example of the use of these equations, a study of thedynamics of aeolian sand ripples [216]

11.3 Case A: the Edwards–Wilkinson equation with flow

The first model involves a pair of linear coupled equations, where the equation

governing the evolution of clusters (‘stuck’ grains) h is closely related to the very

well-known Edwards–Wilkinson (EW) model [85] The equations are:

∂h(x, t)

∂ρ(x, t)

where the first of the equations describes the height h(x , t) of the sandpile surface

at (x , t) measured from some mean h, and is precisely the EW equation in the

presence of the flow term c ∇h The second equation describes the evolution of

flowing grains, whereρ(x, t) is the local density of such grains at any point (x, t).

As usual, the noiseη(x, t) is taken to be Gaussian, so that:

η(x, t)η(x , t) = 2δ(x − x)δ(t − t),

as over noise

11.3.1 Analysis of the decoupled equation in h

For the purposes of analysis, we focus on the first of the two coupled equations(Eq (11.4)) presented above,

∂h

∂t = D h∇2h + c∇h + η(x, t),

152 Coupled continuum equations: sandpile surfaces

noting that this equation is essentially decoupled from the second (This statement

is, however, not true in reverse, which has implications to be discussed later.) Wenote that this is entirely equivalent to the Edwards–Wilkinson equation [85] in a

frame moving with velocity c,

x = x + ct, t = t,

and would on these grounds expect to find only the well-known EW exponents

α = 0.5 and β = 0.25 [85] This would be verified by naive single Fourier transform

analysis of Eq (11.4), which yields these exponents via Eq (11.3)

Equation (11.4) can be solved exactly as follows The propagator G(k , ω) is

G h (k , ω) = (−iω + D h k2+ ikc)−1.

This can be used to evaluate the structure factor

S h (k , ω) = h(k, ω)h(k , ω)

which is the Fourier transform of the full correlation function S h (x − x , t − t)

defined by Eq (11.1) The solution for S h (k , ω) so obtained is:

11.3 Case A: the Edwards–Wilkinson equation with flow 153

1e –10

1 1e+10

k

(Case A) for the h–h correlation function, showing the crossover from high to

to decreasing grid sizes and increasing wavevector ranges The parameters used

appropriate parameters, to serve as a guide to the eye.

This is illustrated in Fig 11.1, while representative graphs for S h (k , ω = 0) and

S h (k = 0, ω) are presented in Figs 11.2 and 11.3 respectively.

It is obvious from Eq (11.6) that S h (k , ω) does not show simple scaling More

with k0= c/D h, andω0 = c2/D h, we see that there are two limiting cases:

The first of these contains no surprises, being the normal EW fixed point [85],while the second represents a new ‘smoothing’ fixed point

We now explain this smoothing fixed point via a simple physical picture Thecompetition between the two terms in Eq (11.4) determines the nature of the fixedpoint observed: when the diffusive term dominates the flow term, the canonical EW

154 Coupled continuum equations: sandpile surfaces

(11.4) (Case A) for the h–h correlation function The different markers in the figure

fixed point is obtained, in the limit of large wavevectors k On the contrary, when the

flow term predominates, the effect of diffusion is suppressed by that of a travellingwave whose net result is to penalise large slopes; this leads to the smoothing fixed

point obtained in the case of small wavevectors k We emphasise, however, that

this is a toy model of smoothing, which will be used to illuminate the discussion ofmodels B and C below

11.3.2 Some caveats

We realise from the above that the interface h is smoothed because of the action of

the flow term which penalises the sustenance of finite gradients∇h in Eq (11.4).

However, Eq (11.4) is effectively decoupled from Eq (11.5), while Eq (11.5) ismanifestly coupled to Eq (11.4) In order for the coupled Eqs (11.4) to qualify as

a valid model of sandpile dynamics, we would need to ensure that no instabilities

are generated in either of these by the coupling term c∇h.

In this spirit, we look first at the value of ρ averaged over the sandpile, as a

function of time (Fig 11.4a) We observe that the incursions of ρ into

nega-tive values are limited to relanega-tively small values, suggesting that the addition of

a constant background ofρ exceeding this negative value would render the

cou-pled system meaningful, at least to a first approximation In order to ensure that

11.3 Case A: the Edwards–Wilkinson equation with flow 155

(a) Variation ofρ(t) with time t Here ρ(t) is the average over the sandpile

surface of 100 sample configurations.

0.1

1 10

Time (t)

(b) The root mean square widthρrms(t) =ρ2 

− ρ2 ) 1/2against time t over 100

at the minimum and maximum value of ρ at any point in the pile over a large range

of times (Fig 11.4c); this appears to be bounded by a modest (negative) value of

156 Coupled continuum equations: sandpile surfaces

−1 0 1

Time (t)

(c) The variation ofρmax(t) andρmin(t) with time t.

Fig 11.4 (cont.)

‘bare’ρ Our conclusions are thus that the fluctuations in ρ saturate at

computation-ally accessible times and that the negativity of the fluctuations inρ can always be

handled by starting with a constantρ0, a constant ‘background’ of flowing grains,which is more positive than the largest negative fluctuation

Physically, then, the above implies that, at least in the presence of a constantlarge densityρ0of flowing grains, it is possible to induce the level of smoothingcorresponding to the fixed pointα = β = 0 This model is thus one of the simplest

possible ways in which one can obtain a representation of the smoothing of the

‘bare surface’ that is frequently observed in experiments on real sandpiles afterintermittent avalanche propagation [213]

11.4 Case B: when moving grains abound

These model equations, first presented in [95], involve a simple coupling between

the species h and ρ, where the transfer between the species occurs only in the

presence of the flowing grains and is therefore relevant to the regime of continuous

avalanching when the duration of the avalanches is large compared to the time

between them The equations are:

11.4 Case B: when moving grains abound 157

where the termsη h (x , t) and η ρ (x , t) represent Gaussian white noise as usual:

η h (x , t)η h (x , t) = 2

h δ(x − x )δ(t − t),

η ρ (x , t)η ρ (x , t) = 2

ρ δ(x − x)δ(t − t),

and the· · · stands for average over space as well as noise

A simple physical picture of the coupling or ‘transfer’ term T (h , ρ) between h

to regions of the interface which are at less than the critical slope, and vice versa,

provided that the local density of flowing grains is always nonzero This form of

interaction becomes zero in the absence of a finite density of flowing grains ρ

(when the equations become decoupled) and is thus the simplest form appropriate

to the situation of continuous avalanching in sandpiles We analyse in the following

the profiles of h and ρ consequent on this form.

It turns out that a singularity discovered by Edwards [222] three decades ago inthe context of fluid turbulence is present in models with a particular form of the

transfer term T ; the above is one example, while another example is the model due

to Bouchaud et al (BCRE) [214], where

T = −ν∇h − µρ(∇h) and the noise is present only in the equation of motion for h This singularity, the so-

called infrared divergence, largely controls the dynamics and produces unexpectedexponents

11.4.1 Numerical analysis

We focus now on the numerical results for Case B The coupled equations in thissection and the following one were numerically integrated using the method of finitedifferences Grids in time and space were kept [96] as fine-grained as computationalconstraints allowed so that the grid size in space x was chosen to be in the

range (0.1, 0.5), whereas that in time was in the ranget (0.001, 0.005) Thus the

instabilities associated with the discretisation of nonlinear continuum equationswere avoided and convergence was checked by keepingt small enough such

that the quantities under investigation were independent of further discretisation.These results were also checked for finite size effects In the calculations of this

section D h = D ρ = 1.0 and µ = 1, with the results being averaged over several

independent configurations The exponentsα and β and the corresponding error

bars were calculated from the slopes of the fitted straight lines, −(1 + 2β) and

−(1 + 2α) respectively.

158 Coupled continuum equations: sandpile surfaces

0 2 4 6 8 10

On discretising Eqs (11.7)–(11.9), the divergences that were previously observed

in [95] were found once again These have since become a field of study in their ownright [223] These divergences are a direct representation of the infrared divergencementioned above, and we follow here a parallel course to [95] in regulating thesevia an explicit regulator, replacing the functionµρ∇h by the following:

= µρ(∇h) for − 1 ≤ µρ(∇h) ≤ 1,

The Fourier transform S h (k , t = 0) (Fig 11.5) is consistent with a spatial

roughen-ing exponentα h ∼ 0.501 ± 0.007 via the observation of

S h (k , t = 0) ∼ k −2.03±0.014 ,

and the Fourier transform S h (x = 0, ω) (Fig 11.6) is consistent with a temporal

roughening exponentβ h ∼ 0.465 ± 0.008 via the observation of

S h (x = 0, ω) ∼ ω −1.93±0.017

Hence the value z h ∼ 1.07 is obtained.

The full structure factor S h (k , ω) has been calculated at two different k points

and Fig 11.7 displays the results The solid and dashed lines in Fig 11.7 are plots

for k = 0.1 and k = 0.2 with 0= 0.4 and 0.5 respectively The spatial structure

11.4 Case B: when moving grains abound 159

0 2 4 6 8 10 12

ln(w)

Data Best fit

B obtained from Eqs (11.7)–(11.9) The best fit shown in the figure has a slope of

0.01 0.1 1 10 100 1000

w

line 1 k1 line 2 k2

estimates (computed in Ref [96]) of the solutions, meant as a guide to the eye.

factor S h (k , ω = 0) shows a power-law behaviour (Fig 11.8) given by

S h (k , ω = 0) ∼ k −3.40±.029 ,

and the temporal structure factor S h (k = 0, ω) shows a power-law behaviour

(Fig 11.9) given by

S h (k = 0, ω) ∼ ω −1.91±.017

160 Coupled continuum equations: sandpile surfaces

from Eqs (11.7)–(11.9) (Case B) The best fit displayed in the figure has a slope

11.4 Case B: when moving grains abound 161

0 2 4 6 8 10

behaviour

S ρ (x = 0, ω) ∼ ω −1.81±0.017

While the range of wavevectors in Fig.11.10 over which crossover in S ρ (k , t = 0)

is observed was restricted by the computational constraints [96], the form of thecrossover appears conclusive Checks (with fewer averages) over larger systemsizes revealed the same trend

11.4.2 Homing in on the physics: a discussion of smoothing in Case B

We focus in this section on the physics of the equations and the results In theregime of continuous avalanching in sandpiles, the major dynamical mechanism isthat of mobile grainsρ flowing into voids in the h landscape as well as the converse

process of unstable clusters (a surfeit of∇h above some critical value) becoming