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RAPID COMMUNICATIONS

PHYSICAL REVIEW E 88, 020201(R) (2013)

Spatially modulated kinks in shallow granular layers

J. E. Mac´

ıas, M. G. Clerc, C. Falc´

on,*and M. A. Garc´

ıa- ˜

Nustes

Departamento de F´

ısica, Facultad de Ciencias F´

ısicas y Matem´

aticas, Universidad de Chile, Casilla 487-3, Santiago, Chile

(Received 5 April 2013; revised manuscript received 30 June 2013; published 19 August 2013)

We report on the experimental observation of spatially modulated kinks in a shallow one-dimensional ﬂuidized

granular layer subjected to a periodic air ﬂow. We show the appearance of these solutions as the layer undergoes a

parametric instability. Due to the inherent ﬂuctuations of the granular layer, the kink proﬁle exhibits an effective

wavelength, a precursor, which modulates spatially the homogeneous states and drastically modiﬁes the kink

dynamics. We characterize the average and ﬂuctuating properties of this solution. Finally, we show that the

temporal evolution of these kinks is dominated by a hopping dynamics, related directly to the underlying spatial

structure.

DOI: 10.1103/PhysRevE.88.020201 PAC S num b er( s ) : 45.70.Qj, 05.45.−a, 47.20.Ky, 47.54.−r

Macroscopic systems under the inﬂuence of injection and

dissipation of quantities such as energy and momentum usually

exhibit coexistence of different states, which is termed mul-

tistability [1–3]. Heterogeneous initial conditions—usually

caused by the inherent ﬂuctuations—generate spatial domains

which are separated by their respective interfaces. These inter-

faces are known as fronts [2]. The evolution of these solutions

can be regarded as a particle-type one, i.e., they can be

characterized by a set of continuous parameters such as

the position, width, charge, and so forth. In the particular

case where fronts separate symmetric states, these front

solutions are termed kinks.Usually,thesetypesofstructures

have been studied in regimes where the symmetric states are

homogeneous ones [2]. Kinks have been a central element

in classical and quantum ﬁeld theory to understand the

dynamics and evolution of several physical systems [4]. In

parametrically driven systems this type of structure appears

through instabilities which lead to the emergence of symmetric

states which are out of phase by half the period of the

forcing [5]. A typical example of such systems are vertically

vibroﬂuidized two-dimensional granular layers where kink so-

lutions have naturally been observed (see references in Ref. [6]

therein). Although several studies have been performed in

two-dimensional ﬂuidized granular layers, only a handful of

studies on one-dimensional ﬂuidized granular layers where

kinks connecting homogeneous states have been reported

experimentally [7,8]andnumerically[9,10]. Furthermore,

to our knowledge, there is no observation of kink solutions

connecting spatially modulated states [11], which can strongly

inﬂuence their stability, bifurcation diagrams, and dynamical

properties.

The aim of this Rapid Communication is to report the

observation of kink solutions connecting spatially modulated

states. The system under study is a one-dimensional shallow

granular layer ﬂuidized by periodic air ﬂow (cf. Fig. 1). Air

ﬂows have already been used to study pattern formation in ﬂu-

idized granular layers in one- [12]andtwo-[13]dimensional

systems. Here, we show experimentally the emergence of kink

solutions as the layer undergoes a parametric instability, and

characterize its average and ﬂuctuating properties. Its proﬁle

*cfalcon@ing.uchile.cl

displays spatial oscillations on the homogeneous state induced

by intrinsic ﬂuctuations of the system. These oscillations

dictate the temporal evolution of the kink, which is a hopping

one, much similar to a Brownian particle in a periodic potential

[14,15].

Experimental setup.Theexperimentalsetupunderstudy

is displayed in Fig. 1.Acell(widthL=200 mm, height

H=200 mm, and depth D=3.5 mm) made out of two

large glass walls with a horizontally placed thick-band-like

sponge (6 mm thick, 200 mm wide, and 15 mm tall) acts

as a porous ﬂoor where approximately 25 000 monodisperse

bronze spheres (diameter a=350 µm) are deposited. In grain

diameter units, the granular layer is 570awide, 10adeep, and

5atall. Dis not changed in these experiments in order to

treat the interface dynamics as a quasi-one-dimensional one.

Thin rods of plexiglass were introduced vertically between the

glass walls effectively shortening Lto study the dynamics of

areducedgranularlayer,whichwewillexplainbelow.

The excitation system of the granular layer is similar to the

one described in [12], where a periodic air ﬂow is generated

by an air compressor (Indura Hurac´

an 1520) and regulated by

an electromechanical proportional valve (Teknocraft 203319),

aprecisioncontrolregulator(Controlair100),andanairlung.

The valve aperture is set by a variable voltage signal controlled

by the ﬁrst output of a two-channel function generator (RIGOL

DG1022) through a power ampliﬁer (NF model HFA4011).

Asymmetricaltriangularsignalwithfrequencyfoand a

nonzero offset is used to generate through the air ﬂow a

time-modulated controllable pressure signal, as shown in the

inset of Fig. 1(a). Pressure oscillations are measured 50 cm

before the ﬂow enters the cell with a dynamic pressure sensor

(PCB 106B) and a signal conditioner (PCB 480C02). We have

checked experimentally the linearity between the peak voltage

delivered by the function generator and the peak pressure

ﬂuctuations Poat the forcing frequency fo. Hence, the control

parameters are foand Po.Wehavealsocheckedthattheextra

pressure drop due to the motion and ﬂuidization of the granular

layer is negligible with respect to the one measured on the

unloaded cell.

Images of the granular bed motion are acquired with a

CCD camera over a 100 s time window in a 1080 ×200 px

spatial window (0.19 mm/px sensitivity in the horizontal

direction and 0.18 mm/px in the vertical direction). For each

020201-1

1539-3755/2013/88(2)/020201(4) ©2013 American Physical Society

RAPID COMMUNICATIONS

J. E. MAC´

IAS et al. PHYSICAL REVIEW E 88, 020201(R) (2013)

y(x)

x

Camera

PC

2-channel

Function

Generator

Power

Amplifier

Air

Compressor

+

Air

Lung

Dynamic

Pressure

Sensor

(a)

y(

x

)

x

Ca

mer

a

2-

ch

ann

e

l

Fu

n

c

ti

on

Ge

ne

ra

to

r

Po

wer

A

m

p

lifier

A

ir

C

om

p

resso

r

+

Air

L

un

g

Dynami

c

P

r

essu

r

e

S

ensor

(a)

y(x)

x

Proportional

Valve

(c)

(b)

0 0.1

0

0.4

0.8

t [s]

(a)

0.2 0.3 0.4 0.5

P(t) [kPa]

1 cm

1 cm

FIG. 1. (Color online) (a) Experimental setup. The inset depicts a

typical temporal trace of the pressure ﬂuctuations. (b) Typical image

of the excited granular layer. The dashed white line corresponds to

the numerically calculated granular interface y(x). (c) Granular

surface kink on shallow granular layers.

experimental conﬁguration, two image sequences are taken.

The ﬁrst one, acquired at high frame rate (100 fps), is used to

study the typical oscillation frequencies of the granular layer.

The second one, set at the subharmonic frequency fo/2using

the second output of the function generator as a trigger, is used

to ensure a stroboscopic view of the oscillating layer. The

granular interface y(x,t) is tracked for every point in space x

at each time tusing a simple threshold intensity algorithm (see

Ref. [12]formoreinformation),asshowninFig.1.Todothis,

white light is sent through a diffusing screen from behind the

granular layer as images are taken from the front, enhancing

contrast and thus surface tracking algorithms. Figure 1shows

asnapshotofthegranularlayerandakinksolutionusingthe

above mentioned tracking algorithm.

Experimental results.Wehaveconductedexperimentsin

the parameter space of peak pressures Poranging from 100 Pa

to 10 kPa and excitation frequencies foranging from 5 to

20 Hz. We have concentrated our studies in the frequency

range fo∈[12.5,14.5] Hz, as the phenomenology is quite

reproducible and less input pressure is needed. We have

restricted our experimental cell, shortening Lto 5 cm, in order

to study the dynamics of the homogeneous state, preventing

the appearance of kinks which form for larger widths.

As we increase Pofor a ﬁxed excitation frequency fo,

the granular bed displays small surface ﬂuctuations (less

than a diameter) of the upper layer of grains. This motion

is enhanced as Poincreases, lifting the complete layer over

aperiodofthepressureﬂuctuations.Foracriticalvalue

of Po=Pc

o,theﬂatoscillatinglayerbecomesunstableto

small perturbations through a parametric instability, displaying

subharmonic oscillations at fo/2. Therefore, the granular layer

presents an effective parametric resonance as a consequence

of the forcing [16]: The periodic air ﬂow is responsible

for inducing the oscillatory behavior of the layer and its

respective parametric resonance. This subharmonic response

can be observed by measuring the space averaged motion

12.5 13 13.5 14 14.5

2000

2100

2200

2300

2400

2500

Frequency (Hz)

P

o (Pa)

10 10

10

10

10

F (Hz)

PSD (mm2/Hz)

fo

100101

10

10

10

F (Hz)

PSD (mm

2

/Hz)

fo

fo/2

500 1000 1500 2000 2500 3000 3500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

P

o

(Pa)

<A >

e (cm)

0

−

0.6

−

0.4

−

0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

ε=(P

o

−P

o

c

)/P

o

c

<A>

~

(b)

(a)

FIG. 2. (Color online) (a) Parametric instability curve. The continuous line shows the experimentally computed phase line Pc

oas a function

of fo.ForPo<P

c

oonly harmonic oscillations of the ﬂat layer are present in the reduced cell (L=5cm).ForPo>P

c

osubharmonic

oscillations are dominant, arising from a parametric instability. Lower inset: power spectral density (PSD) of the ﬂat layer harmonic oscillations

at fo=14 Hz for Po=1520 ±20 Pa. Upper inset: PSD of the dominant subharmonic oscillations of the ﬂat layer at fo/2=7Hzfor

Po=3840 ±20 Pa. (b) Bifurcation diagram for the subharmonic amplitude "Ae#vs Pofor fo=14 Hz. Dashed line is the theoretical prediction

for "Ae#=α1/2(Po−Pc

o)1/2where α=33.64 ±1.16×cm2/kPa is a calibration factor and Pc

o=2247 ±39 Pa. The continuous line is

a theoretical ﬁt of "Ae#=α{(Po−Pc

o)+[(Po−Pc

o)2+2˜η]1/2/2}1/2where ˜η=9264 ±2913 Pa2is the noise intensity. Inset: Normalized

amplitude "˜

A#="Ae#/α√Pc

ovs #=(Po−Pc

o)/P c

o. The dashed line follows the prediction ˜

A=#1/2. The continuous line is a theoretical ﬁt

of "˜

A#={[#+(#2+2η)1/2]/2}1/2,whereηis the normalized noise intensity.

020201-2

RAPID COMMUNICATIONS

SPATIALLY MODULATED KINKS IN SHALLOW GRANULAR ... PHYSICAL REVIEW E 88, 020201(R) (2013)

of homogeneous granular interface y(x,t), that is, Y(t)=

L−1!L

0y(x,t)dx as a function of time t.ForsmallPothe

power spectral density of Y(t) displays a peak at foshowing

the harmonic character of the oscillation (cf. Fig. 2,lower

inset). As Pois increased, a subharmonic oscillation appears

[cf. Fig. 2(a),upperinset].Foreachexcitationfrequency

there is a transition from harmonic to subharmonic dominant

oscillations of the ﬂat layer as Posurpasses a critical value Pc

o,

which is displayed by the continuous line in Fig. 2(a).This

transition is found to be smooth and supercritical in nature

for all fo∈[12.5,14.5] Hz. For the sake of simplicity in what

follows fowill be ﬁxed at 14 Hz. It must be noticed that

increasing Hfrom 5ato 10awill change this dynamical state,

as a patterned state appears [12].

To characterize the transition pressure Pc

o,wefollowthe

scheme proposed in [17]. We compute the bifurcation diagram

of the envelope Aeof subharmonic oscillations of Y(t),

Aecos (πfot), as Pois increased. From the temporal trace of

the layer oscillations the harmonic part is ﬁltered out and the

amplitude of the remaining subharmonic oscillations is com-

puted using a Hilbert transform algorithm [18]. The bifurcation

diagram is shown in Fig. 2(b),wherethetemporalaverageof

Ae,#Ae$, is plotted versus Po.Theerrorbarscorrespondto

the standard deviation of the values of the envelope σA=

"A2

e−#Ae$2.Thesmoothbifurcationcurvecanbedescribed

with a simple model which takes into account noise in a su-

percritical transition [17]. Thus, we can compute the threshold

value of Pc

ofor each excitation frequency fo, and the intensity

of the noise ηof the layer ﬂuctuations following the expression

#Ae$=α#[(Po−Pc

o)+"(Po−Pc

o)2+2˜η]/2, where ˜ηis

the noise intensity and αis a calibration factor. For every

foin our experiments, all bifurcation curves follow the above

expression.

The spatial structure of the granular layer was also studied

to characterize the stationary states as the layer oscillates. For

Po<P

c

o,theharmonicallyoscillatingﬂatlayerdisplaysno

typical spatial scale. For Po>P

c

o, ﬂuctuations of the ﬂat layer

display a characteristic wavelength and frequency sporadically

[see Fig. 1(b)], disappearing randomly with a typical lifetime,

which is known as a precursor [19]. This phenomenon is a

consequence of the balance between energy injection, caused

by internal ﬂuctuations of the granular layer, and the local

dissipation of the slowest decaying spatial mode of the uniform

steady state of the layer interface. In our experimental setup,

the typical wavelength λof the precursor is typically ∼2cm,

which is of the order of 60a.Wehavecheckedthatλis

independent of the periodicity or the position of the air inlets.

No discernible change is observed for our experimental control

parameters. λcan be understood as the typical wavelength of a

secondary spatial instability which occurs for larger pressures

than the ones reported here. It must be noticed that this

type of supercritical noisy bifurcation has also been observed

in vibroﬂuidized granular layers, although the analysis of

the transition was performed via spectral properties of the

ﬂuctuations [20].

Now, we will concern ourselves with kinks appearing

through the above described transition in the extended cell

for L=20 cm. Maintaining foat 14 Hz and increasing

ε=(Po−Pc

o)/P c

oabove the transition, the subharmonic

motion described above allows the system to exhibit bistability

between two states which are out of phase and, thus, a spatial

connection between them. More precisely, there is a height

jump as we go from left to right through a ﬁnite region of the

layer where this shift occurs. This means that, at any given

instant, on one side of the region the granular layer is moving

upwards and on the other side it is moving downwards.

Akinkcanappearinanypointoftheexperimentalcell

spontaneously. By choosing the phase mismatch between the

triggering signal and the layer oscillation, we can image the

kink when the separation between the in-phase and out-of-

phase parts of the oscillating granular layer is at its maximum.

Ave rag ing o ver a ll th e com put e d int erf ace s in an i mag e s e-

quence, we calculate the averaged front, its height d,andwidth

'for different ε,asshowninFig.3.Here,2dcorresponds to

the distance between the in- and out-of-phase states, measured

at its maximum separation. 'is computed as the average width

of the spatial derivative of the kink solution. The error bars

correspond to the standard deviation of dand '.Wecansee

that dgrows linearly with εand 'is roughly constant at 0.7 cm

(independent of ε). Note that the computed kink displays

a spatial modulation on both connected states, as discussed

above. Thus, its typical wavelength is again λ(cf. Fig. 3).

Further increasing the number of images used in the average

values of d,',andλdoes not affect the computed values.

1.2 1.6 2.0 2.4 2.8 3.2

0.0

0.2

0.4

0.6

0.8

1.0

d, ∆ (cm)

ε

0 2 4 6 8 10 12 14

0.4

0.8

1.2

1.6

2.0

x (cm)

y (cm)

2d

∆

λ

1 cm

1 cm

FIG. 3. (Color online) Top: Typical image of a granular kink

at Po=8038 ±20 Pa (ε=2.42 ±0.01). The dashed line is the

numerical interface detection. Middle: Granular kink averaged over

1000 frames. dstands for the granular kink height with respect to

the middle plane and 'stands for the typical core size of the kink.

Bottom: Granular kink height d(×) and typical core size '(◦)asa

function of ε. Error bars stand for the standard deviation for dand '

for each value of ε.

020201-3

RAPID COMMUNICATIONS

J. E. MAC´

IAS et al. PHYSICAL REVIEW E 88, 020201(R) (2013)

0 500 1000 1500 2000 2500 3000 3500 4000 4500

7

8

9

10

11

12

13

14

t (s

)

x (cm)

0

500 1000

1

50

0

2

000 2500 300

0

3

500

4

00

0

4

500

7

8

9

10

11

12

13

1

4

t (s

)

s

s

x

(

cm

)

y(x,t) (cm)

t (s) x (cm)

0200 400 600 800 1000 0510 15 20

2

1

0

2

1.5

1

0.5

0

-0.5

(a)

(b)

o

FIG. 4. (Color online) Motion of the granular kink. (a) Image

sequence of 20 min of Po=8038 ±20 Pa (ε=2.42 ±0.01).

(b) Temporal trace of the core of the kink xo(t) as a function of time.

The long term dynamics of the spatially modulated kink

are dictated by its structure and inherent ﬂuctuations. A

typical image sequence of the kink motion acquired over

long time periods (∼104periods of oscillation) is depicted

in Fig. 4(a), where the complete structure shifts its position in

the experimental cell through discrete jumps. This motion is

tracked in time by following the kink position, xo(t), which

is the position in space where the spatial derivative of the

kink reaches its maximum. The typical distance between

these jumps is λ[cf. Fig. 4(b)]andtheyoccuratrandom

times either to the left or the right of the cell. Although

the kink displays these jumps, the temporal average of xo(t),

"xo#,doesnotchangeintheexperimentalobservationtime.

Hence, the dynamics of xo(t)canbeunderstoodasarandom

motion (where ﬂuctuations come from the inherent noise of

the granular layer) within a periodical potential (arising from

the spatial structure of the precursor) [14]. It can be foreseen

that in the case of the existence of a small asymmetry in the

system (for instance, tilting the cell) the evolution of the kink

could resemble that of a Brownian-type motor [15,21], but this

is only an extrapolation of the previous dynamics and needs

experimental conﬁrmation.

In summary, we have studied the stability properties and

bifurcation diagrams of kink solutions in a shallow one-

dimensional ﬂuidized granular layer subjected to a periodic

air ﬂow. The inherent noise of the system simultaneously

induces ﬂuctuations on the kink position, and sustains an

effective pattern over the extended homogeneous state. These

ingredients combined allow us to ﬁgure out the long time

dynamics of the kink solution as a Brownian-type motor. A

deeper understanding on the existence, properties, dynamics,

and interaction of kinks is still lacking. Theoretical and

experimental work in this direction is in progress.

Acknowledgments.Theauthorswouldliketothankan

anonymous referee for his constructive criticism towards

our work. We acknowledge ﬁnancial support from ANR-

CONICYT 39 and ACT127. M.G.C., C.F., and M.A.G.-N.

are grateful for the ﬁnancial support of FONDECYT projects

1120320, 1130354, and 3110024, respectively.

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