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Emergence of spatiotemporal dislocation chains in drifting patterns

M. G. Clerc,

1,a)

C. Falc!

on,

1,b)

M. A. Garc!

ıa-~

Nustes,

2,c)

V. Odent,

1,d)

and I. Ortega

1,3

1

Departamento de F!ısica, Facultad de Ciencias F!ısicas y Matem!aticas, Universidad de Chile, Casilla,

487-3 Santiago, Chile

2

Instituto de Fisica, Pontiﬁcia Universidad Catlica de Valparaso Avenida Brasil, 2950 Valparaso, Chile

3

School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052,

Australia

(Received 30 December 2013; accepted 4 June 2014; published online 16 June 2014)

One-dimensional patterns subjected to counter-propagative ﬂows or speed jumps exhibit a rich and

complex spatiotemporal dynamics, which is characterized by the perpetual emergence of

spatiotemporal dislocation chains. Using a universal amplitude equation of drifting patterns, we

show that this behavior is a result of a combination of a phase instability and an advection process

caused by an inhomogeneous drift force. The emergence of spatiotemporal dislocation chains is

veriﬁed in numerical simulations on an optical feedback system with a non-uniform intensity pump.

Experimentally this phenomenon is also observed in a tilted quasi-one-dimensional ﬂuidized shallow

granular bed mechanically driven by a harmonic vertical vibration. V

C2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4883650]

Nature is ripe with patterns and structures of different

shapes and sized. These patterns naturally show imper-

fections where the pattern amplitude goes to zero, termed

defects, which can display a rich and complex spatiotem-

poral dynamics. In this article, we show that when one-

dimensional patterns are subjected to ﬂows that change

spatially their intensity, single defects or arrays of them

appear constantly over the observed pattern in space and

time. These defects are thus known as dislocations on the

space-time evolution of the pattern, as they are similar

the classical dislocations observed in solids or in the

stripe patterns of bi-dimensional systems out of equilib-

rium. We explain this phenomenon theoretically using a

universal model that describes how the amplitude of the

moving pattern creates such defects through a combina-

tion of an phase instability of the pattern interacting with

the inhomogeneous ﬂow. The explanation of the continu-

ous generation of defects in drifting patterns is veriﬁed

by simulating numerically the proposed model. This veri-

ﬁcation is also performed on a more complex model

describing the evolution of an optical feedback system

with a non-uniform intensity pump. Furthermore, we

observe experimentally the generation of spatiotemporal

dislocation chains on a quasi-one-dimensional ﬂuidized

shallow granular bed mechanically driven by a harmonic

vertical vibration, where the inhomogeneous ﬂow is

included by tilting the cell. In that sense, the results pre-

sented in this work are meaningful to the whole commu-

nity involved in understanding the way defects interact

within pattern forming systems and how their dynamics

inﬂuence the long-term evolution of the underlying

pattern.

I. INTRODUCTION

Non-equilibrium processes often lead to the formation

of spatially periodic structures arising from an homogeneous

state through the spontaneous breaking of symmetries pres-

ent in the system under study.

1–5

Pattern formation is gener-

ally observed by modifying a single parameter, usually

called bifurcation parameter, which controls the transition

from an homogeneous state to a patterned one as it surpasses

a certain threshold. In one-dimensional extended systems,

when one continues increasing this parameter above thresh-

old, the pattern can exhibit secondary instabilities, for exam-

ple, spatial and temporal period doubling, oscillatory,

Eckhaus, and parity breaking ones.

6

Secondary instabilities

are generically the cause of the transition from motionless to

propagative patterns. These transitions in pattern forming

systems have been studied in several physical contexts, such

as parametrically ampliﬁed surface waves in Newtonian

7

and non-Newtonian ﬂuids,

8

ﬂuidized granular beds,

9

binary

ﬂuid convection,

10

and nonlinear optics,

11

to mention a few.

This phenomenon can be produced by two mechanisms: (i)

spontaneous symmetry-breaking transitions, where the pat-

tern will choose spontaneously the direction of its propaga-

tion depending on initial conditions,

2,12

and (ii) induced

parity-breaking transitions, where stationary-to-propagating

pattern bifurcations arise when motionless patterns are

exposed to drift forces or spatial inhomogeneities.

9,11

In the

latter case, patterns are commonly deformed and advected,

which is usually related to the development of convective

instabilities.

2

The entire class of scenarios that can trigger the spatio-

temporal evolution of drifting patterns have not been identi-

ﬁed yet. Particularly, one expects a rich and complex

spatiotemporal dynamics. Experimental observations of drift-

ing patterns have been reported in particle-laden ﬂows inside

a partially ﬂuid ﬁlled, horizontal, rotating cylinder,

13

a free

electron laser

14

and a one-dimensional transverse Kerr-type

slice subjected to optical feedback.

15

In the former work, the

a)

Electronic mail: marcel@dﬁ.uchile.cl

b)

Electronic mail: cfalcon@ing.uchile.cl

c)

Electronic mail: mgarcianustes@ing.uchile.cl

d)

Electronic mail: vodent@ing.uchile.cl

1054-1500/2014/24(2)/023133/7/$30.00 V

C2014 AIP Publishing LLC24, 023133-1

CHAOS 24, 023133 (2014)

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proposed physical mechanism for this phenomenon is based

on nonuniformities of the control parameter, which induce an

Eckhaus instability.

15,16

The aim of this letter is to study and

explain the appearance of spatiotemporal dislocation chains

in drifting patterns. Despite of experimental reports, the rec-

ognition of the nature and origin of this universal phenom-

enon and the ingredients necessary for its existence are

absent. A dislocation chain is a defect line composed by

phase singularities. The distance between neighboring singu-

larities is of the same order of the underlying wavelength.

Indeed, this spatiotemporal dynamics can be understood as a

consequence of counter-propagative ﬂows, speed jumps, or

inhomogeneities in the parameters of drifting pattern systems.

That is, the inhomogeneous spatial coupling is responsible of

this phenomenon. Based on an amplitude equation describing

the evolution of a pattern, we identify the mechanism for the

emergence of spatiotemporal dislocation chains in the pres-

ence of a drift. This dynamical behavior is a combination of a

phase instability–local Eckhaus instability–and an advection

process caused by the inhomogeneous drift force. The appear-

ance and dynamics of dislocation chains are numerically veri-

ﬁed on a one-dimensional amplitude equation where the

ingredients described above are present, and also on a model

describing a transverse Kerr-type slice under to optical feed-

back illuminated by a non-uniform beam. Furthermore, the

phenomenon is observed experimentally in a tilted quasi-one-

dimensional ﬂuidized shallow granular bed mechanically

driven by harmonic vertical vibrations.

II. THEORETICAL DESCRIPTION OF DISLOCATION

CHAINS

Let us consider a one-dimensional extended system

described by the dimensionless partial differential equation

@t~

uðx;tÞ¼~

fð~

uðx;tÞ;@

x;fegÞ $ vðxÞ@x~

u;(1)

where ~

uðx;tÞis a vectorial ﬁeld that describes the system

under study, {x,t} respectively stand for the spatial and tem-

poral coordinates, ~

fis the vector ﬁeld, fegis a set of parame-

ters that characterizes the system under study, and v(x)

accounts for an inhomogeneous drift force.

In the case of zero-drift, i.e., v(x)¼0, we assume that

the system possesses a stationary state ~

u0that satisﬁes

~

fð~

uðx;tÞ;@

x;fegÞ ¼ 0, which exhibits a supercritical spatial

instability at a critical wavenumber k¼k

c

when one of the

parameters surpasses a certain threshold, say e

c

, generating a

stationary pattern.

Under suitable boundary conditions, when the pattern is

subjected to a small constant drift force ½vðxÞ¼vo6¼0&, it

remains motionless in a parameter region, i.e., there is a pin-

ning range.

11

Above a critical value of the drift force, the pat-

tern becomes propagative, which correspond to a regime of

absolute instability.

2

Increasing further v

o

, the system enters a

convective instability regime where the drift is large enough

to advect the patterned state away completely from the region

under study, returning the system to an homogeneous state.

17

The former scenario changes dramatically when we con-

sider non-uniformities in the drift force ½vðxÞ&. Physical

transport processes such as inhomogeneous diffusion or dis-

persion and inhomogeneous spatial coupling can lead to an

inhomogeneous drift force in the system. In such cases, the

drifting pattern can be deformed creating regularly isolated

or sequence of dislocations in the spatiotemporal diagram,

which corresponds to singularities in the phase or hole solu-

tions in the pattern envelope (see Figs. 1and 2).

Examples of non-uniform drift forces can be encoun-

tered in different ﬁelds ranging from biology to chemistry to

physics. For instance in physics, particle segregation in a

FIG. 1. Spatiotemporal dislocation chains in drifting patterns. (a) and (b)

are spatiotemporal diagrams of the phase uðx;tÞand (c) and (d) thei r re-

spective phase gradients @xugiven by model Eq. (3). Simulation parame-

ters are l¼0.4, dx ¼0.4, L¼400, X

0

¼0withj¼10$4(left panel) and

j¼3'10$4(right panel), respectively. Dislocation chains and isolated

defects are fra med by dashed circles and squares, respectively. X

c

corre-

sponds to critical position.

FIG. 2. Spatiotemporal dislocation chains in drifting patterns. (a) and (b) are

spatiotemporal diagrams of the phase uðx;tÞand (c) and (d) their respective

phase gradients @xugiven by model Eq. (3). Simulation parameters are

l¼0.4, dx ¼0.4, L¼400, X

0

¼80 with j¼10$4(left panel) and

j¼3'10$4(right panel), respectively. Dislocation chains and isolated

defects are framed by dashed circles and squares, respectively. X

c

corre-

sponds to critical position.

023133-2 Clerc et al. Chaos 24, 023133 (2014)

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laden ﬂow inside a horizontal, rotating cylinder exhibit band-

ing pattern formation.

13

The spatiotemporal dynamics shows

two counter propagative drifts. Due to this non-uniform drift

caused by the cylinder curvature, the pattern exhibits a con-

tinuous formation of defects. Another example is a liquid

crystal layer subject to an inhomogeneous optical pump.

15

In

this case, a Kerr medium slice is shined by a Gaussian laser

beam which through optical feedback generates an inhomo-

geneous ﬂow, locally advecting the pattern, creating disloca-

tions in the spatiotemporal diagram.

A. Unified description

To understand in a uniﬁed manner the previously dis-

cussed phenomenon, we study the amplitude evolution at the

onset of the spatial bifurcation assuming that the drift force

varies smoothly compared with the pattern wavelength.

Introducing the following ansatz for the critical propagative

mode:

~

uðx;tÞ¼l1=2AðX;TÞeiðkcx$v0tÞ~

ukþc:c:þh:o:t:; (2)

where A(X,T) is the envelope of the propagative pattern,

which is a slowly varying in time and space which scale as

X)l1=2x;T)lt, respectively, c.c. and h.o.t. stand for

complex conjugate and higher order terms, see Ref. 2.~

ukis

the marginal mode at e

c

with wavenumber k

c

. The corre-

sponding amplitude equation close to threshold is the

Ginzburg-Landau equation

@TA¼l0A$jAj2Aþ@XX A$i~

vðXÞA;(3)

where l0)ðe$ecÞ=land ~

vðXÞ)½vðXÞ$v0&kc=lis a spa-

tial function accounting for the effect of the inhomogeneous

drift force. Using polar ﬁelds representation of the amplitude,

A¼Reiu,oneintroducestwoscalarﬁeldsR(X,T)anduðX;TÞ,

the magnitude and phase of the amplitude, respectively.

B. Derivation of the amplitude equation in a simple

model

In order to illustrate the above procedure, let us consider

the following prototype model of pattern formation with

drifting force (supercritical drifting Swift-Hohenberg):

@tu¼lu$u3$@xx þq2

!"

2u$cðxÞ@xu;(4)

where u(x,t) is a scalar ﬁeld, lis the bifurcation parameter, q

is the pattern wavenumber, and caccounts for drift source of

the pattern. The Swift-Hohenberg model was introduced to

describe the onset of Rayleigh-Benard convection; however,

recent generalizations have been used intensively to account

for pattern formation in several systems.

2

Equation (4) under

the inﬂuence of a small drifting force ðcðxÞ*1Þdescribes a

spatial supercritical bifurcation. For l<0, the system

presents a stable uniform state u(x,t)¼0. At l¼0, the system

bifurcates, the uniform solution becomes unstable, giving rise

to pattern formation. For l>0, the pattern amplitude, at

wavenumber k

c

¼6q, grows as the square root of l.

To describe the dynamics of the pattern at the onset of

bifurcation ðl*1Þ, we introduce the ansatz

uðx;tÞ¼AðT;XÞ

ﬃﬃﬃ

3

peiqx þ

"

AðT;XÞ

ﬃﬃﬃ

3

pe$iqx þWðA;"

A;xÞ;(5)

where Aaccounts for the amplitude of the critical mode q,

which varies slowly in space and time (@XX A*@XA*1

and @TA*1) and WðA;"

A;xÞis a small correction function

including high order terms in Aand "

A. Introducing the above

ansatz in Eq. (4), linearizing in Wand considering the domi-

nant terms, we get

@XX þq2

!"

2W¼A3

3ﬃﬃﬃ

3

pei3qx þ$@TAþlA$jAj2A

$

þ4q2@XXA$iqcAgeiqx

ﬃﬃﬃ

3

pþc:c:

To solve the above equation, we must to impose the follow-

ing solvability conditions:

3

@TA¼lA$jAj2Aþ4q2@XX A$iqcðXÞA;(6)

and then

WðA;"

A;xÞ¼ A3

2633=2ei3qx þ

"

A3

2633=2e$i3qx:(7)

Then, one simultaneously determines the change of variable

[Eq. (5)] and the amplitude equation of the critical mode

[Eq. (6)]. Note that the Eq. (6) is valid considering the fol-

lowing scaling A+l1=2;@

T+l;@

X+l1=2and c+l.

Normalizing the spatial scale and the coefﬁcients in Eq. (6),

one obtains the amplitude Eq. (3).

C. Mechanism of spatiotemporal dislocations chain

Numerical simulations of amplitude Eq. (3) show the

formation of spatiotemporal dislocation chains (see Fig. 1).

We consider Eq. (3) with Neumann boundary conditions,

i.e., @XAð0;tÞ¼@XAðL;tÞ¼0, for any t, where Lis the sys-

tem size. Equation (3) is simulated with a 4th order Runge-

Kutta solver where the temporal step dt ¼0.02 and with a ﬁ-

nite difference solver with a spatial step dx ¼0.4.

For the sake of simplicity, we consider a linear ramp

forcing of the form vðXÞ¼jðX$X0Þðj*1Þ, where X

0

is

the point when ~

vðXÞ¼v0kc=l. The phase uðX;TÞ(Figs. 1(a)

and 1(b)) and the phase derivative @xuðX;TÞ(Figs. 1(c) and

1(d)) are shown for different values of jwith X

0

¼0. In this

case, only a left pointing drift appears. Successive appearan-

ces of phase instabilities are observed. For small values of j,

we observe the appearance of isolated dislocations at a dis-

tance X

c

from X

0

¼0. For larger values of j, formation of

dislocation chains is observed.

For a uniform drift (j¼0), the amplitude Eq. (3) has a

family of periodic solution ApðXÞ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

l0$p2

peipðX$X0Þ, par-

ametrized by a continuous parameter p<ﬃﬃﬃﬃﬃ

l0

p. This family

describes an homogenous pattern, which goes through a

023133-3 Clerc et al. Chaos 24, 023133 (2014)

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phase instability at p¼p

c

, where p2

c)l0=3, corresponding

to the well-known Eckhaus instability threshold.

2

To

describe the dynamics in the inhomogeneous media, Eq. (3),

we promote the parameter p to a function of space, which at

dominant order is quadratic function of X. Thus, we propose

the ansatz

ApðZ¼X$X0Þ, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

l0$pþj

6Z2

%&

2

seipþj

6Z2

ðÞ

Z;(8)

which corresponds to a pattern with increasing modulus and

phase as a function of the spatial coordinate. The Eckhaus

instability is characterized by the local deformation of phase

gradients, which generates phase singularities and, respec-

tively, hole solutions on the envelope modulus. Later, the

pattern modiﬁes locally its wavenumber, diffusing it within

itself and ﬁnally materializing a stable pattern state. From

expression (8), using the Eckhaus instability criterium, the

inhomogeneous pattern state becomes unstable at a critical

position

XcðpÞ¼sgnðjÞﬃﬃﬃﬃﬃﬃ

6

jjj

sﬃﬃﬃﬃﬃ

l0

3

r$jpj

"#

1=2

:(9)

Notice that the critical distance is a function of the wave-

number of the pattern that is parametrized by p. Therefore,

these phase singularities generated at this position are

advected as a consequence of the drift force, restarting the

process. Hence, the system exhibits the perpetual creation of

spatiotemporal dislocations as a result of the above process.

As in the case of the Eckhaus instability, depending on initial

conditions, it is possible to generate several phase singular-

ities on different locations of the pattern, which diffuse the

stable wavelength. Then, when j$1is larger than the size of

phase singularities ðj>ﬃﬃﬃ

l

pÞ, the system presents the forma-

tion of dislocation chains. If we consider counter propagative

ﬂows at a conﬂuence point X

0

ðX06¼0Þ, a similar behavior

composed by two opposite propagative patterns is observed

in the region of conﬂuence (cf. Fig. 2). For j>0ðj<0Þ,

the defects annihilate (create) successively. For small j, we

observe the annihilation (creation) of defects only in the con-

ﬂuence region. For greater values of j, the pattern starts to

destabilize at a distance X

c

from X

0

, creating new defects at

each side. We like to emphasize that the former dynamical

behavior was equally observed in our numerical simulations

with periodical boundary conditions.

III. OPTICAL SPATIOTEMPORAL DISLOCATION

CHAINS

As an example of the former dynamics in a physical sys-

tem, we have conducted numerical simulations of a one-

dimensional transverse Kerr-type slice with optical feedback

shined by a non-uniform laser beam.

15

This system is com-

posed of a nematic liquid crystal sample (Kerr media repre-

sented in Fig. 3(a) by LC) and a mirror, which is illuminated

with a non-uniform beam, which crosses ﬁrst the liquid

crystal layer, then reﬂected on the mirror and re-crosses the

liquid crystal layer. Figure 3(a) depicts a schematic sketch of

the transverse Kerr-type slice with optical feedback. The

numerically simulated nonlinear medium is assumed to be a

50 lm thick layer of E7 nematic liquid crystal with homeo-

tropically anchored. The laser beam is chosen as monomode

frequency source, k

0

¼532 nm and unidimensional (1D) fol-

lowing the xaxis (see Fig. 3(a)), which can be produced by

cylindrical lenses for instance. The laser ﬁeld proﬁle is repre-

sented by a stationary linear ramp function, as F(x)¼F

0

(1 –

x/2L), where jF0j2is the maximum laser intensity and

L¼1.6 mm is the system size. This kind of proﬁle is feasible

experimentally with a Spatial Light Modulator. B(x,t) is the

backward ﬁeld which crosses the slide, which is reﬂected by

the mirror M. This ﬁeld is directly reinjected itself onto the

slide from the back. Notice that this ﬁeld is depend on space

and time because its phase is modiﬁed by the liquid crystal

refractive index when it crosses the sample. Ris the mirror

intensity reﬂectivity, which is positioned at a distance dof

the liquid crystal sample. The modiﬁcation of the refractive

index dn(x,t) induced by light with respect to the unperturbed

refractive index n

0

is a satisfactory order parameter to

describe the dynamics of this system. Thus, we will concern

ourselves with nðx;tÞ¼n0þdnðx;tÞ, where n(x,t) is the

effective refractive index of the Kerr-type slice and dn*n0.

Hence, dn(x,t) satisﬁes

18,19

s@dn

@t¼$dnþl2

d

@2dn

@x2þjFðxÞj2þjBðx;tÞj2;(10)

with sis the relaxation time of the liquid crystal molecules,

and l

d

is the diffusion length. The explicit form of B(x,t) is

given by

18,19

FIG. 3. Spatiotemporal dislocation chains in 1D optical Feedback with linear

transverse pumping. (a) Schematic sketch of the Kerr-type slice with optical

feedback. M is the mirror, LC is the liquid crystal, and F and B are the forward

and the backward ﬁelds. (b) Drifting pattern with long dislocation chains. (c)

Local phase gradient. F

0

¼1.1, d¼5mm, k

0

¼532 nm, vl¼1, l

d

¼10 lm,

and s¼2.23s. Fx

ðÞ¼F01$x

2L

!"

, with L¼512. Iðx;tÞ¼jBðx;tÞj2.

023133-4 Clerc et al. Chaos 24, 023133 (2014)

93.126.137.222 On: Wed, 18 Jun 2014 19:03:46

Bðx;tÞ¼ ﬃﬃﬃ

R

pe

idk0

2p@xx ðeivldnðx;tÞFx

ðÞ

Þ;

where k

0

is the laser wavelength, vis the coefﬁcient of the

Kerr nonlinearity, and lis the liquid crystal sample thickness.

We assume that free space propagation length is much bigger

that the liquid crystal sample thickness, d-l, to neglect the

diffraction in the nonlinear medium. When F

0

exceeds a crit-

ical value F0c)ðk2

cþ1Þ=2Rvsinðk2

cdk0=2pÞ, the refractive

index exhibits a spatial modulation instability with wave-

number kc¼p=ðdk0Þ.

19

Based on the amplitude equation

method, close to the spatial instability, we can introduce

the ansatz nðx;tÞ¼Aeikcx=ﬃﬃﬃ

q

pþh:o:t, where q¼RF0;c

v2½3vsinðrk2

cÞ$vsinð3rk2

cÞþ2a1vðcosðrk2

cÞ$cosð3rk2

cÞÞ& in

Eq. (10), and we get at dominant order the amplitude Eq. (3)

with

l0¼2RðF0$F0;cÞvsinðrk2

cÞ

F0;c

;

~

vðxÞ¼2RF0;cvkcsinðrk2

cÞ

2Lx:

(11)

Therefore, we expect that this system exhibits spatiotemporal

dislocation chains. To verify the above predictions, we con-

ducted numerical simulations of model (10) with a variable

step of RungeKutta order 8 solver (dop853)

20

and with peri-

odical boundary conditions. We perform the spatial deriva-

tives using the Fourier space (FFTW3 Library) with a spatial

step dx ¼0.27 l

d

. Figure 3displays a numerical simulation of

Eq. (10) considering the linear forward propagation proﬁle

F(x). Such proﬁle generates a non-uniform drift in one direc-

tion, as we can see in Fig. 3(b) where the intensity has been

displayed as a function of time. Close to the left border, the

formation of sequences of dislocation chains can be

observed. As seen in Fig. 3(c), these sequences correspond

to phase singularities. Thus, the optical feedback system

veriﬁes the appearance of spatiotemporal dislocation chains

when this is subjected to an inhomogeneous drifting force as

predicted by amplitude Eq. (3).

IV. GRANULAR DISLOCATION CHAINS

The phenomenon theoretically explained above was also

observed experimentally on a simple ﬂuidized granular sys-

tem, which presents drifting patterns. The experimental setup

is depicted in Fig. 4(a). A container (60 '40 '7mm

3

) made

out of two 5 mm thick plexiglas walls with a aluminum

frame between them holds in the space between the walls

4.0 g of monodisperse brass spheres of diameter D¼150 lm,

creating a granular layer 1.7 mm in depth. In units of grain

diameters, the granular layer is approximately 400 Dwide,

47 Din thickness, and 12 Din depth. The cell is mounted on

an electromagnetic vibration exciter, driven by a frequency

synthesizer (FS), via a power ampliﬁer (Amp), providing a

vertical sinusoidal acceleration (horizontal acceleration less

than 1% of the vertical one). The sinusoidal gravity modula-

tion gef f ðtÞ¼aex cosð2pfextÞis measured by a piezoelectric

accelerometer (Acc) and a charge ampliﬁer, where a

ex

is pro-

portional to the applied tension with a 1.0 Vs

2

/m sensitivity

and f

ex

is the excitation frequency. A biaxial tilt sensor

driven by a 12 V power supply is positioned solidary on top

of the cell in order to measure the inclination of the cell with

respect to the axis of gravity in the x–yplane with a sensi-

tivity of 100 mV/8. This inclination is represented by the

angle /(cf. Fig. 4). In this experimental conﬁguration, /is

monitored by measuring the x-axis voltage difference. The

variations of the off-plane inclination angle on the x-axis are

also monitored to ensure that only in-plane movements of

the cell are allowed. The control parameters are the forcing

frequency f

ex

, the acceleration amplitude a

ex

, and the

in-plane inclination angle /. Images were acquired at 20 fps

over 100 s using a CCD camera over a 120 '840 px window

(6.5 '10

$3

cm/px sensitivity in the horizontal direction and

6.3 '10

$3

cm/px in the vertical direction). Using a simple

tracking scheme,

21

the granular pattern interface y(x,t) is

computed (as shown in Fig. 4(b)), where xcorresponds to

the spatial coordinate along the x-axis and tto the temporal

one. Using this data, the local envelope, phase gradient, and

velocity are obtained. In this work, we focused solely on the

characterization of the spatiotemporal dynamics of the sub-

harmonic standing waves formed on top of the quasi-one-di-

mensional ﬂuidized shallow granular bed appearing through

a supercritical parametric instability

22,23

as a certain acceler-

ation threshold of the container is surpassed. To do this,

the excitation parameters are ﬁxed at f

ex

¼40 Hz, and

a

ex

¼58 m/s

2

, which is +20% larger than the critical acceler-

ation for subharmonic patterns at f

ex

.

Dislocation chains appear locally in the spatiotemporal

diagram of the pattern evolution,

24

as it is depicted in Fig. 5.

These dislocations, appearing isolated or in groups, can be

clearly pinpointed as we compute the phase gradient of pat-

tern @xu(cf. Fig. 5(b)). In our experimental setup, due to the

horizontal inclination angle of the cell

9

and the intrinsic

heaping of the system arising from air-grain interactions,

25

inhomogeneous drift force appears (cf. Fig. 5), which can be

measured by the local time-averaged phase speed of the pat-

tern (cf. Fig. 5(c)). The local averaged phase speed is

obtained by two independent methods: the Hilbert transform

FIG. 4. Granular drifting patterns. (a) Schematic representation of the exper-

imental setup. (b) Typical snapshot of the granular pattern. The continuous

white line corresponds to the numerically calculated granular interface.

023133-5 Clerc et al. Chaos 24, 023133 (2014)

93.126.137.222 On: Wed, 18 Jun 2014 19:03:46

algorithm

26

and the rigid solid method.

27

For very small in-

clination angle /, the dominant mechanism for the genera-

tion of drift forces are inhomogeneities created by the air-

grain interactions such as heaping (speed jumps, see Fig.

5(d)). For />0:5., contrarily, the horizontal inclination

leads the dynamics (counter ﬂows, see Fig. 5(g)). For inter-

mediate angles, both mechanisms contribute to generate

inhomogeneities in the speed proﬁle. It must be stressed that

without inhomogeneous drift force, no defects are observed

in the stationary pattern. Figure 5shows, respectively, the

stroboscopic spatiotemporal diagram of y(x,t) acquired at

f

ex

/2, /¼0:02.(top) and /¼0:15.(bottom). We observe

that for the counter-propagative ﬂow (cf. bottom panel

Fig. 5) or speed jumps (cf. top panel Fig. 5), the spatiotempo-

ral diagrams of ﬂuidized shallow granular bed show the

emergence of dislocations. These dislocations, which are sin-

gularities in the local phase of the pattern, appear in the

region where the speed of the pattern presents substantial

spatial variations (see Fig. 5(g)). Hence, in these regions, the

drifting pattern manifests a phase instability that leads to the

emergence of phase singularities. As we have mentioned,

analogous spatiotemporal diagrams have been reported in

particle-laden ﬂows inside a partially ﬂuid ﬁlled, horizontal,

rotating cylinder,

13

a free electron laser,

14

and a one-

dimensional transverse Kerr-type slice subjected to optical

feedback.

15

Therefore, spatiotemporal dislocation chains are

a robust phenomenon displayed by pattern forming systems

in the presence of inhomogeneous forcing.

To complete our analysis, we perform a numerical and

experimental veriﬁcation of the theoretical prediction for X

c

,

Formula (9). As seen in Fig. 6(left panel), numerical simula-

tions of the amplitude Eq. (3) exhibit a power law of the

form Xc¼að1=jÞbwith a¼3.25 y b¼0.43 60.05 close to

the expected critical exponent b¼0.5 of expression (9). The

experimental veriﬁcation in drifting granular patterns shows

a deviation of the predicted behavior, exhibiting a power law

Xc¼að1=jÞbwith a¼23.29 y b¼0.28 60.1 [see Fig. 6

(right panel)]. Such deviation can be due to the fact that the

inhomogeneous drift does not follow a linear proﬁle in space

[Figs. 5(d) and 5(g)]. Notwithstanding, we can approximate

in some limit this proﬁle to a linear ramp close to the conﬂu-

ence region X

0

¼0. An additional effect that can modify this

scale-invariant behavior is defect interaction that is not

included in the above approach.

V. CONCLUSION

We have shown that the complex spatiotemporal dy-

namics exhibited by inhomogeneous drifting patterns can be

ﬁgured out as a perpetual phase singularity production result-

ing from phase instabilities (coming from a local Eckhaus

instability) and pattern propagation induced by an inhomoge-

neous drift force. Thus, inhomogeneities in spatial coupling

are the origin of this phenomenon. Hence, this rich and com-

plex spatiotemporal dynamics can be understood as a combi-

nation of simple phenomena. The experimental ﬁndings of

the spatiotemporal dynamics of dislocation chains are in

quite good agreement with our theoretical general descrip-

tion valid for a broad class of physical systems. The study of

the effect of ﬂuctuations (noise) on the spatiotemporal dy-

namics of phase singularities and their interactions is a work

in progress.

FIG. 5. Experimental dislocation chains. (a) Drifting pattern with long dislocation chains (dashed ovals) for /¼0:07.. (b) Drifting pattern for /¼0:02.. (c)

Local phase gradient. Dashed squares represent isolated defects and dashed circles dislocation chains. (d) Local velocity averaged over the entire image

sequence. The continuous line corresponds to the velocity computed using a rigid solid approximation (see Ref. 27) and the dashed line using a Hilbert trans-

form. (e) Drifting pattern for /¼0:15.. (f) Local phase gradient. Dashed squares represent isolated defects and dashed circles dislocation chains. (g) Local ve-

locity averaged over the entire image sequence. The continuous line corresponds to the velocity computed using a rigid solid approximation (see Ref. 27) and

the dashed line using a Hilbert transform.

FIG. 6. Plot of X

c

vs j$1obtained by numerical simulations (left) and direct

experimental measurements (right). Continuous lines are numerical ﬁts.

023133-6 Clerc et al. Chaos 24, 023133 (2014)

93.126.137.222 On: Wed, 18 Jun 2014 19:03:46

ACKNOWLEDGMENTS

The authors acknowledge ﬁnancial support by the ANR-

CONICYT 39, “Colors”. M.G.C., M.A.G-N., C.F., and V.O.

thank for the ﬁnancial support of FONDECYT projects

1120320, 3110024, 1130354, and 3130382, respectively.

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