John R. de Bruyn*Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A
B. C. Lewis, M. D. Shattuck, and Harry L. SwinneyCenter for Nonlinear Dynamics and Department of Physics, University of Texas, Austin, Texas 78712
~Received 31 October 2000; published 28 March 2001!
Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscil-lation amplitude is suddenly increased from below the onset of pattern formation into the region where stripepatterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patternswith defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, andthe way in which they form and decay. Our results are compared with those for similar spiral patterns inRayleigh-Benard convection in fluids.
Granular materials appear simple: they consist of a lanumber of particles that interact solely through contforces. While the physics of a single inelastic collision btween two particles is conceptually straightforward, the clective dynamics of many particles, driven away from eqlibrium and undergoing many inelastic collisions,surprisingly rich and in many cases strongly nonintuitive.variety of interesting phenomena occurring in granular mterials has been studied recently, including convective floin granular layers, heaping instabilities, force chains, afront propagation@1,2#.
One of the most interesting types of collective behavseen in granular materials is the pattern formation that reswhen a layer of granular material is oscillated vertica@3–16#. Patterns of stripes, squares, and hexagons areserved, depending on the frequency and amplitude ofoscillation @6#. Localized structures called oscillons are alobserved@9#. The patterns seen in experiments have breproduced in event-driven molecular-dynamics simulatiof the oscillated granular system@13,14,17#.
A variety of theoretical models of granular pattern formtion have been introduced that capture the dynamics ofsystem to a greater or lesser degree@18–24#. This instabilityin an oscillating granular layer is analogous in many waysthe Faraday instability of an oscillated liquid layer, andanalysis similar to that for the Faraday instability has beperformed for the oscillated granular layer, using equatiderived from the kinetic theory of inelastic spheres@25,26#.Experiments and simulations have shown that the instaties of stripe patterns in vertically oscillated granular mateals are the same as those seen in fluid systems, suggethat a continuum description analogous to the Navier-Stoequation of fluid dynamics can be applied to vibrated gralar layers@14#.
Pattern formation in vibrated granular materials wasserved by Fauveet al. in a quasi-two-dimensional system
@3#, and by Thomaset al. @4# in a three-dimensional systemUmbanhowar, Melo, and Swinney carried out a detaistudy of the patterns which formed in a vertically vibratlayer of bronze spheres@6,9,12,16#. Their experimental sys-tem @10#, a modified version of which is used in the presework, consisted of an evacuated cell containing a layergranular material that was oscillated vertically by an electmagnetic shaker. The relevant experimental parametersthe amplitude of the sinusoidal accelerationG, measured inunits of g, the acceleration due to gravity, the frequencyvibration f, andN, the depth of the granular layer in units othe particle diameterd. For low G, the granular layer simplymoves up and down with the bottom plate of the cell. WhG becomes greater than 1, however, the layer leaves thetom plate for a portion of its cycle. The energy impartedthe layer by the bottom plate is dissipated in numerous clisions between particles, and between particles and thetom plate. The layer becomes somewhat dispersed whilfree fall, but overall it remains flat with uniform thicknesAs G is increased throughGc'2.8, there is a subcriticabifurcation at which a pattern of ‘‘hills’’ and ‘‘valleys’’ ap-pears in the layer. For lowf, the layer develops a pattern osquares, while for higherf, stripes appear@6,16#. WhenG isthen decreased, the pattern vanishes at a value ofG lowerthan Gc . The crossover from squares to stripes occurs af'0.35/(g/Nd)1/2 @13#. These patterns are subharmonic, this, the repeat time of the patterns is twice the period ofdriving. The stripe or square patterns remain stable up tG'4, at which point they lose stability to a hexagonal pattePeriod-2 flat layers and period-4 stripes and squares areserved at higherG. In the present paper we will be workinin the region in which period-2 stripe patterns occur; tpattern formation ‘‘phase diagram’’ as a whole is discussin Refs.@6,16#.
In this paper we present experimental results on sppatterns that form in a vertically oscillated granular layer inrange of driving amplitudes and frequencies. Large cfilling spirals, isolated smaller spirals, and a state of spdefect chaos@27# are all observed, depending on the expemental parameters. The existence of spiral patterns inbrated granular layers has been noted briefly elsewh
de BRUYN, LEWIS, SHATTUCK, AND SWINNEY PHYSICAL REVIEW E63 041305
FIG. 1. A schematic diagramof the experimental apparatus, described in detail in the text.
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@10,12#. Here we present a systematic survey of the rangparameters over which the spiral patterns exist, and distheir dynamics in some detail. Similar spiral patterns habeen studied in Rayleigh-Be´nard convection in fluids; Ref@28# is a recent review. Target and spiral patterns have abeen observed in experiments on the Faraday instability@29#.
The remainder of this paper is organized as follows. Stion II contains a description of our experimental setup atechniques. Our results are presented in Sec. III and arecussed in Sec. IV. Section V is a brief conclusion.
II. EXPERIMENT
Our experimental apparatus is similar to that used in pvious work @6,9,10,12,14,16#. It is shown schematically inFig. 1. The cell containing the granular layer was shakvertically by an electromagnetic shaker. Considerable efwas taken to ensure that the cell was driven accuratelytically, and to eliminate vibrations transmitted to the cassembly by other than the shaker shaft. The shakermounted inside a housing which was filled with bags of leshot. This housing was supported by leveling screws reson an aluminum plate, which in turn sat on a granite sresting on the laboratory floor. The shaker was driven bsinusoidal voltage at a frequencyf produced by a functiongenerator and amplified by a power amplifier. The shadrove a hardened steel shaft which was connected by aible coupling to a square cross-section air bearing, whichused to minimize lateral and rotational motion of the drishaft. The housing of the air bearing was mounted via leving screws on an air-supported vibration-isolation tabthrough which a hole had been drilled to allow passage ofdrive column. A rigid aluminum baseplate was mountedtop of the air bearing, and the cell itself was bolted to tplate. An accelerometer mounted on the bottom of thewas monitored by a computer and used to measureG, theamplitude of the acceleration.
Several different sample cells were used. All were eva
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ated to a pressure of 0.1 Torr to eliminate the effects ofdrag. All cells were cylindrical with an aluminum bottomplate. The cell sidewall and top lid were made of PlexiglIn two of the cells the Plexiglas sidewall was coated withproprietary antistatic coating@30#. Without the coating, amonolayer of grains tended to stick to the sidewall up tofew grain diameters above the surface of the layer. Tmonolayer on the sidewall was not present in the cells wantistatic coating. The presence of these grains had a sigcant effect on the dynamics of the patterns, as discusbelow. The cell used for most of the work reported here ha flat bottom plate and a diameter of 14.6 cm. We also ua cell with antistatic walls and a flat bottom plate 13.9 cmdiameter. Another cell had a bottom plate which was flatto a diameter of 10.0 cm, then sloped upwards to the sidewith a 2° ‘‘beach;’’ the total cell diameter was 14.7 cm. Thfourth cell had antistatic walls and a steeper beach; its bwas flat to a diameter of 10.7 cm then sloped up at an anof 10° to a total diameter of 16.5 cm. The cells with a beawere used to reduce the effect of the sidewall on the oritation of the patterns. Finally, a cell with an inverted beahad the same dimensions as the cell with the 10° beachwith the opposite slope, so that the layer was deeper atperimeter of the cell than in the center.
The granular material consisted of spherical bronze pticles sieved between 150 and 180mm in diameter; we thustake the particle diameterd to be 165mm. Apart from siev-ing, the particles were used as received from the supp@31#. The depth of the granular layer was determined frothe volume of particles making up the layer and the cgeometry.
The patterns that formed in the granular layer were iaged using a digital video camera mounted above the cering of LEDs encircling the cell slightly above the level othe granular layer was strobed atf /2 to illuminate the pat-terns. As a result, higher regions of the granular layer willuminated and appear bright, while lower regions are
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shadow and appear dark. The phase of the strobe relativthe drive signal could be adjusted and was normally segive maximum visual contrast. The digital images westored on a personal computer and eventually written to CROM for storage and later analysis.
III. RESULTS
A. Cell-filling spirals and targets
1. General properties
In the frequency range of interest in this work, a subcrcal bifurcation to a stripe pattern occurs whenG is increasedslowly throughGc . The stripes are straight at onset, but asGis increased further in a cell with no beach, the stripes tenorient perpendicular to the cell wall. In a circular cell thboundary condition forces the stripes to become curved. Tleads to the presence of sidewall foci and bowed patteconsisting of two or more sidewall foci with curved stripwhich are generally perpendicular to the sidewall. In cewith a beach, the perpendicular orientation at the boundaless pronounced, but in both cases curvature of the strcauses spatial variations in the pattern wave number. Tcan lead to local instabilities in the pattern, resulting in tappearance of defects and time dependence@14#. As a resultthe steady-state pattern in this regime is predominastripelike, but with defects and a continual time dependen
In the experiments reported here,G was increased in asudden jump from below to above onset. The values oGgiven below are in all cases the final values attained afterjump. In many cases, this jump simply resulted in a strpattern with many defects, which annealed over time tpattern similar to those obtained for a quasistatic increasG, as described above. In a range of frequencies and ldepths, however, large cell-filling spiral or target pattedeveloped. Examples are shown in Fig. 2. In cells withbeach, these large spirals were observed in layers of dN515.4, 17.5, and 19.0, but not in runs withN513.1 orsmaller. The results in the cells with a beach are somewdifferent and will be discussed below.
The range of existence of the cell-filling spirals is showfor N515.4 by the triangles in Fig. 3. Very close toGc wesaw no spirals of any kind. Starting a few percent aboveGc ,however, there was a range off and G within which largespirals formed. For this particular depth, large spirals wseen between 29 and 35 Hz, and were most prevalent aand 32 Hz. Large spirals did not appear in every run pformed within their range of existence; often stripe pattewith defects formed instead, as described above and iltrated in Fig. 4. Typically, spirals formed more often at frquencies near the middle of their existence range, and wless common at other frequencies. In a somewhat larange of bothG and f, not shown in Fig. 3, we observecases in which the pattern near the edge of the cell initihad the circular symmetry of the cell, as in Fig. 7~b! below,but never developed into a cell-filling spiral. We also sasmall, isolated spirals within a mainly stripelike pattern ova much larger range, forG all the way up to the transition tohexagons.
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At higher f we observed spiral defect chaos — compltime-dependent patterns containing several small spiralsexisting with stripelike regions@12#. This state, which issimilar to that observed in Rayleigh-Be´nard convection@27,28# is discussed briefly in Sec. III B below.
As shown in Fig. 2, both right- and left-handed spirawere observed.~For a right-handed spiral, moving alongspiral arm in the counterclockwise sense takes one tocore of the spiral.! 59% of the spirals that formed after
FIG. 2. Examples of cell filling spiral and target patterns.~a! Atarget pattern;N517.5, f 532 Hz, andG52.94. ~b! A one-armed,right-handed spiral;N519.0, f 529 Hz, andG52.84. ~c! A three-armed, left-handed spiral;N515.4, f 532 Hz, andG53.15. ~d!An 11-armed, left-handed spiralN56.7, f 535 Hz, andG52.66.~a!–~c! were in a circular cell and~d! was recorded in the cell withthe 2° beach.
FIG. 3. A diagram showing the range of existence of spipatterns forN515.4 layers. The lower and upper dashed linesdicate the onset of the stripe pattern atGc and the transition fromstripes to hexagons, respectively. The triangles are points at wcell-filling spirals were observed, and the crosses indicate the sdefect chaos state. Dots indicate other points at which observawere made.
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jump in G were left handed and 41% were right handeSpirals with from zero arms~i.e., target patterns! up to ninearms were seen in the cells with no beach, and a short-lspiral with 11 arms, shown in Fig. 2~d!, was observed in acell with a beach. In some cases the number of arms chanas defects moved through the pattern. Normally the sparms extended all the way to the cell wall~except in the cellswith a beach, as discussed below!, but occasionally an armwas observed to end at a dislocation defect in the cell inrior. Figure 5 shows the relative probabilities for the formtion of n-armed spirals after a jump inG, for runs in theflat-bottomed cells.~Note that these data are only for runthat resulted in the formation of spirals.! Data for all threevalues ofN and many different values ofG and f are com-bined in this figure. Only one spiral withn.4 was seen inthese runs. One-armed spirals were most common, accoing for approximately half of all spirals observed at eadepth. No spirals withn.2 were observed in the deepelayer studied (N519.0), but otherwise the distribution ofnwas not strongly dependent onN over the limited range inwhich the large spirals were observed.
2. Spiral formation
Large spirals typically formed in one of two ways. Figu6 shows the formation of a target pattern following a jumpG from below to about 3.5% above onset. WhenG is in-creased above onset, a ring forms around the perimeter ocell. This develops into a pattern of concentric rings aspattern quickly propagates into the center of the cell fromedge. In the case shown, the end result was a defecttarget. A similar process can lead to the formationn-armed spirals if one or more defects form in the ring ptern as it develops. The circular cell boundary clearly plaan essential role in this process. The stripe nearest theorients parallel to the boundary, in contrast to the tendetowards perpendicular orientation observed whenG is in-creased slowly. Interestingly, in a few runs, a target pattformed first in the center of the cell, then propagated outwto the walls.
FIG. 4. An example of the type of pattern seen whenG isincreased slowly through onset, or a sufficiently long time aftejump in G. This pattern is time dependent, but its predominanstripelike character and the presence of sidewall foci and otherfects are robust features. HereN513.1, f 535 Hz, andG53.30.
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A somewhat less clean, but more typical, example offormation of a spiral pattern is shown in Fig. 7. Starting froa flat featureless layer,G is increased from below to 17%above onset. An imperfect circular ring near the cell wall cbe seen in Fig. 7~b!, but a disordered pattern simultaneousdevelops in the center of the cell. The pattern close towall retains its approximate circular symmetry as the pattin the interior of the cell gradually anneals and eventuaadopts the same symmetry. The end result is a spiral patThis process is substantially slower than that shown in F6, as indicated in the figure captions. If the pattern closethe cell wall develops too many defects or strays too far frcircular symmetry the spiral pattern will not form, and instead a stripe pattern with defects results.
3. Dynamics
In the cell with antistatic coating on the wall, the cefilling spirals rotated uniformly. In the uncoated cell rotatio
FIG. 5. Histogram of the probabilityP(n) for the formation ofn-armed spirals following a jump inG. A total of 154 cell-fillingspirals were observed at three layer depths (N515.4,17.5,19.0)for a range of values ofG and f.
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FIG. 6. A time sequence of images illustrating the formationa target pattern after a jump inG from below onset to 2.94. Thepattern develops first at the cell wall and propagates in to the ceThe remnants of a pattern left over from a previous run can be sfaintly in ~a!. The elapsed time between~a! and ~f! is 6.42 s.N517.5 andf 532 Hz.
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was also observed, but in this case the spirals often becpinned, apparently by the layer of particles adhering tocell wall due to static. As a result the rotation was not uform in these cells and in many cases was not present aFigure 8 is a sequence of images showing the rotationone-armed spiral in the coated cell. Like all spirals in cewith no beach, this one rotated in the direction that ‘‘wouit up.’’ This spiral was particularly long-lived and surviveintact for 720 s~or 21 000 oscillations of the plate! beforeeventually drifting off center and being destroyed by the pcesses described below. Its rotation period was apprmately 160 s.
Typical spirals survived much shorter times, usually lethan one rotation period. As a result it was not possiblestudy the rotation rate as a function of the experimentalrameters. In general, the spiral core and the outer tips ofarms rotate at similar rates as long as the core is close tocenter of the cell. The motion of the core is a combinationrotation and translation~see Sec. III A 4!, and as the coretranslates off center, its rotation rate increases. Meanwhthe rotation rate of the outer defect at the cell wall canmain steady, decrease slowly, or simply go to zero, depeing on the presence or absence of nearby wavelenchanging instabilities of the pattern. As a result, the rotatrate of the spiral is a complicated function of both time aposition in the cell.
FIG. 7. A time sequence of images showing the formation ospiral pattern after a jump inG from below onset to 3.35. In thiscase a disordered pattern appears throughout the cell, with a ncircular ring at the wall. The pattern in the interior of the ceanneals to form a spiral. The time elapsed between~a! and ~f! is39.1 s. HereN515.4,f 532 Hz.
FIG. 8. A time sequence of images showing the rotation oone-armed spiral. The elapsed time is 139 s, slightly less thanrotation period. Note the rotation of the spiral core as well as thathe cell wall. The spiral rotates in the direction which winds it uHereN519.0, f 529 Hz, andG52.84.
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Interesting dynamics occurred in the cores of the sppatterns. Examples are shown in Figs. 9 and 10. Figurshows the core of a three-armed spiral. The three boundfects that form the spiral core interact, and the ‘‘bridgejoining two of the three arms in Fig. 9~a! decays, then reappears between another pair of arms. This behavior repwith a period that varied somewhat, but on average w6.161.6 s, much faster than the rotation period of the spitself. The core of a four-armed spiral is shown in Fig. 10.this case the period of the core dynamics was more unifoit was 4.860.5 s over the course of this run, but appearedbecome larger and more erratic as the core drifted off-ce~see below!. Similar dynamics were observed in the coresspirals with differentn. The core dynamics illustrated inFigs. 9 and 10 is the same as that observed in cell-fillspirals in Rayleigh-Be´nard convection@32–34#, as discussedbelow.
The core of a target pattern also exhibits interestingnamics, as illustrated in Fig. 11. The phase of the patterthe core changes continuously, and the core of the taemits circular rings which propagate outwards. A simiprocess is also seen in Rayleigh-Be´nard convection@35#. Inour experiments this process is slow when the target cornear the center of the cell — the period is approximatelys in Fig. 11 — but as the core drifts off center as describ
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FIG. 9. A time sequence of images showing one period ofcore dynamics of a three-armed spiral atN515.4, f 532 Hz, andG53.15. The times of the images are~a! 0 s, ~b! 1.83 s,~c! 2.20 s,~d! 2.89 s,~e! 3.58 s,~f! 5.47 s, and~g! 6.17 s.
FIG. 10. A time sequence of images showing one period ofcore dynamics of a four-armed spiral atN515.4, f 531 Hz, andG53.23. The times of the images are~a! 0 s, ~b! 1.10 s,~c! 1.82 s,~d! 2.14 s,~e! 2.86 s,~f! 3.64 s,~g! 4.35 s, and~h! 4.68 s.
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below the period decreases by a factor of three. In onethe opposite behavior was observed; that is, circular ripropagated in towards the core of the target where they wabsorbed.
Instabilities of granular stripe patterns have been dcussed in Ref.@14#. Local skew-varicose and crossroll instbilities play an important role in the time dependencethese patterns. Crossrolls in particular are important inbreakup of spiral patterns, as discussed below. We haveobserved the oscillatory instability at low frequencies in delayers. Figure 12 shows an example of an oscillatory insbility observed on a spiral pattern in a flat-bottomed circucell. The same instability has been observed on strastripes under similar conditions@see Fig. 7~a! of Ref. @12##.
4. Breakup
All of the cell-filling spirals were ultimately unstable ithe range of experimental parameters we studied. They tcally survived for 1 or 2 min before breaking up. As 1 mcorresponds to roughly 2000 oscillations of the plate,1000 periods of the subharmonic pattern, these lifetimesin fact quite long in terms of the characteristic time scalewhich energy is injected into the system.
The breakup of the spiral pattern starts with the corethe spiral slowly drifting off center. As a result, the wav
FIG. 11. Time sequence showing dynamics in the center otarget. The times of the images are~a! 0 s,~b! 15.8 s,~c! 31.1 s,~d!41.5 s,~e! 51.3 s,~f! 61.11 s,~g! 73.3 s, and~h! 81.5 s.N519.0,f 529 Hz, andG52.63.
FIG. 12. A one-armed spiral showing an oscillatory instabiliThis image was taken in a flat-bottomed cell withN525, f522 Hz, andG53.30.
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length of the pattern becomes spatially nonuniform, wheventually leads to the development of local instabilitiesthe wavelength is forced outside of the range in which striare stable@14,36#. Examples of this are shown in Figs. 1and 14. Figure 13 shows a target pattern. The center oftarget drifts towards the wall at the bottom left of the figurThe stripes become compressed at the wall at the bottomand are destroyed by the process illustrated in Fig. 15described below. The wavelength of the pattern increawhere stripes end at the cell wall as well as on the upstreside of the core. Crossroll instabilities can occur at eitherthese locations when the local wavelength becomeslarge. In Figs. 13~d! and 13~e! crossrolls have appearewhere stripes meet the cell walls, as well as at the upright, where the wavelength has been stretched by the drithe pattern. These regions become disordered, and inspace of a few seconds the disorder spreads into the re
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FIG. 13. A time sequence of pictures showing the drift off ceter and breakup of a target: crossrolls form at the walls wherelocal wavelength is large. The times of the images are~a! 0 s, ~b!102 s, ~c! 126 s, ~d! 132 s, ~e! 138 s, and~f! 143 s.N519.0, f529 Hz, andG52.63.
FIG. 14. A time sequence of pictures showing the drift off ceter and breakup of a spiral: crossrolls form in the interior of tspiral pattern on the ‘‘upstream’’ side, again where wavelengthlarge. The times of the images are~a! 0 s, ~b! 4.2 s,~c! 8.5 s,~d!10.7 s, ~e! 12.7 s, and~f! 14.7 s. N519.0, f 528 Hz, andG52.79.
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the pattern. In Fig. 13~f! the center of the original target harun into the wall and turned into a sidewall focus. Whremains is a disordered stripe pattern with no sign oforiginal circular symmetry remaining. This process is vesimilar to that shown in Fig. 21 of Ref.@35# for Rayleigh-Benard convection. Figure 14 shows a similar scenario,with crossrolls developing in the interior of the pattern on tupstream side of the core in Fig. 14~c!. These crossrolls anneal to form two dislocation defects, which distort the spipattern and eventually lead to its destruction both at the wand in the core, as seen in Fig. 14~f!. In all cases the endresult of the breakup is a time-dependent stripe pattern wdefects similar to that shown in Fig. 4.
As the core of a granular spiral drifts off center, the oermost arm downstream of the core gets pushed closer towall. When the arm gets too close to the wall, an instabioccurs, as illustrated in Fig. 15. The portion of the spiral aclosest to the wall distorts@Fig. 15~b!# and moves right up tothe wall @Fig. 15~c!#, where a pair of defects form. Thesdefects then move apart from each other around the cell@Figs. 15~d! and 15~e!#. This process appears to be due tolocal Eckhaus instability@37# involving the cell wall. Onestripe is eliminated locally and the local wavelength iscreased, as can be seen by comparing Figs. 15~a! and 15~f!.Typically this process repeats several times as the spiraldrifts towards the wall.
5. Wave number evolution
The pattern wave numberk changes as the spirals formand evolve in the manner described above. As in Ref.@14#,we determined the local wave number using the methodtroduced in Ref.@36#. An image of the central region of thpattern is band-pass filtered around the fundamental spfrequency andk is determined as a function of position btaking derivatives of the filtered image@36#. The wave num-ber was then spatially averaged. The spatially averaged wnumber^k& from one run is shown in Fig. 16. In this paticular run, G was increased suddenly from below onsetG53.15 at timet50. The high wave numberk&;6 ob-
FIG. 15. Time sequence of pictures showing the local desttion of a stripe at the wall. The times of the images are~a! 0 s, ~b!7.3 s, ~c! 8.1 s, ~d! 10.1 s, ~e! 12.1 s, ~f! 14.0 s. N517.5, f531 Hz, andG53.45.
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served at small times is that found by our algorithm for tfeatureless granular layer. The pattern starts at the celland propagates inwards as illustrated in Fig. 6, and as it dso the calculated value of^k& decreases steadily. During thtime the granular layer is partly patterned and partly flat, athe value of^k& we calculate is not characteristic of thpattern itself. However, the pattern fills the cell at appromately the time of the first inset image shown in Fig.~which shows the central region of a circular cell!, and be-yond this time^k& is in fact the mean wave number of thpattern. After the pattern fills the cell,^k& continues to de-crease steadily as high-spatial-frequency features smout. By the time of the second inset image in Fig. 16, at51.3 s, the pattern is essentially a spiral with severalfects. From the second to the third inset (t55.0 s) the de-crease of k& continues, but more slowly. The topology othe pattern changes little but the pattern becomes sharpeless noisy. Just before the fourth inset att538.1 s, the finaldefects anneal out and we are left with a target patternthis point the wave number of the pattern is at its minimuIn the fourth inset this target has started to drift off center,described above. This drift is accompanied by anincreasein^k&. In the last inset (t5259 s), the target pattern has brokeup completely and we are left with a stripe pattern with dfects. The mean wave number of the final stripe patternroughly 10% higher than that of the target pattern whpreceded it.
The evolution of^k& described above and shown in Fi16 is typical of that observed in several runs. Much the sabehavior was observed independent of whether a spiraltern formed or not, although it was not possible to identifyconsistent functional form for the time dependence of^k&@38# from our data. In particular, the final increase in wanumber also occurred in runs where no spiral pattern formThis increase appears to be associated with the formatio
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FIG. 16. The mean wave number of the pattern as a functiontime for a run withN517.5, G53.15, and f 531 Hz. Time isplotted on a logarithmic scale for display purposes;t50 corre-sponds to the time at whichG was suddenly increased from beloonset. The five inset images show the central region of the pattethe data points indicated by large circles.
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focus defects at the cell sidewall@28#, which evidently selecta higher wave number than that preferred in their absen
B. Spiral defect chaos
As shown in Fig. 3, a spiral defect chaos state is obserfor frequencies higher than those for which cell-filling spiraoccur. In this region the pattern that develops after a jumpG consists of several small spirals along with stripelikegions and other defects. These patterns are time depenAs with the cell-filling spirals discussed above, the spidefect chaos is a transient and the patterns eventually evto a stripe pattern with defects.
A snapshot of the spiral defect chaos state is shownFig. 17. This figure shows several small sidewall foci arouthe perimeter of the cell, and a collection of six small zeand one-armed spirals in the interior. Most of the small srals seen in this state had either zero or one arm, butnumber of spirals and their handedness varied from runrun.
C. Cells with a beach
The beach geometry used in some experiments modthe boundary conditions at the cell perimeter. For sufficienthin layers, the patterns formed in the center of the cell adid not extend all the way to the cell wall. For layers wiN*2 at the edge, the patterns filled the entire cell, buttendency for the stripes to orient perpendicular to the wwas reduced by the beach, particularly for the thinner lay
In the cell with the shallow(2°) beach, no large spiralwere seen for the values ofN which produced large spirals ithe flat-bottomed cells. On the other hand, large spirals wobserved in this cell with thinner layers. An example ofspiral with 11 arms observed withN56.7 in the center ofthis cell is shown in Fig. 2~d!. In this case the layer is only1.8 particles deep at the cell wall and the pattern does nothe cell. This spiral lasted in the form shown for only abo10 s. Crossrolls continually developed over much of its coas can be seen in Fig. 2~d!, and led to changes in the numb
FIG. 17. A snapshot of the spiral defect chaos state in abottomed cell atN515.4, f 540 Hz, andG53.13. This pattern istime dependent and eventually evolves to a stripe pattern withfects as in Fig. 4.
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of arms on a time scale of a few seconds. In the cell withsteeper (10°) beach, large spirals were seen at a depN517.5 forf from 22 Hz to 35 Hz, and atN511 for f in therange 30–35 Hz, but none were observed forN57.
The lifetimes of spirals in the cells with the 2° and 10beaches were on the order of only 10 s, substantially shothan in the flat-bottomed cells. The spirals decayed by ming off-center towards the cell walls, and then evolving todisordered stripe pattern. Typically, however, a new laspiral would thenspontaneouslyreform and the proceswould repeat. In the cell with the inverted beach, stripalways formed perpendicular to the sidewall and cell-fillispirals were not observed.
A two-armed spiral in the cell with the 2° beach is showin Fig. 18. Examination of the core of this spiral in Fig18~b!–18~d! shows that it rotates in theunwindingsense, thatis, in the opposite direction to those in the flat-bottomcells. This reverse rotation was always observed in the cwith a beach. In the flat-bottomed cell the rotation of tspiral was accompanied by the motion of the ends ofspiral arms around the cell perimeter, as in Fig. 8, but hthe spiral arms do not reach the cell wall. The outer regioof the spiral appear not to change over the rotation periodthe spiral shown in Fig. 18; in particular, the arm which diaway at approximately the 3 o’clock position in Fig. 18~a!continues to do so, independent of the orientation of theral core.
IV. DISCUSSION
We observe large spirals developing whenG is increasedsuddenly from below to aboveGc . For flat-bottomed circular
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FIG. 18. A time sequence of images showing the rotation otwo-armed spiral in the cell with the 2° beach. HereN56.7 in thecenter of the cell,f 535 Hz, andG52.81. Image~a! shows theentire cell at time 0 s.~b!–~d! show the core of the spiral at time9.8, 19.5, and 29.3 s. The spiral does not extend all the way tocell wall because of the beach. Note that the spiral rotates inunwinding direction, in contrast to the case in the flat-bottomedshown in Fig. 8.
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SPIRAL PATTERNS IN OSCILLATED GRANULAR LAYERS PHYSICAL REVIEW E63 041305
cells, spirals appear in deep layers (N*13), in a regionwhere stripes are the stable pattern whenG is increased quasistatically. These cell-filling spirals are simply stripes thare wound up, and whether stripes or spirals are obsedepends strongly on the effect of the boundary conditionthe cell wall.
Smaller spirals within a larger stripelike pattern formthe same frequency range over essentially the whole rangG, all the way up to the transition to hexagons.
For all experimental conditions we have studied, the cfilling spirals are transients. They form either by propagatof a circularly symmetric pattern in from the sidewall or bthe annealing of the disordered central region of the patin the presence of a circularly symmetric pattern at thewall. During this stage of the pattern’s evolution, its mewave number k& decreases, approaching a constant valuethe defects anneal. The spirals survive for of order 10oscillations of the bottom plate. The core of the spiral thdrifts slowly off center, leading to distortions of the patternwave number field and eventually to the disruption of tpattern by the crossroll instability in regions where the lopattern wavelength is large. Once the spiral symmetry isrupted, the pattern fairly quickly evolves to a time-dependstripe or bowed pattern with defects, similar to those otained in quasistatic experiments. The mean wave num^k& increases during this process as sidewall foci form athe pattern becomes stripelike.
In the cells with a beach, large spirals were observedthinner layers, for which the boundary conditions allowthe stripes to orient parallel to the sidewall. These spirwere also transients, but in this case spirals would breakthen spontaneously reform, in contrast to what happenthe flat-bottomed cells.
Cell-filling spirals and target patterns have been obserin experiments on the Faraday instability with high-viscosfluids @29#. They formed whenG was increased suddenlthrough the pattern onset, as in the present case, and pfor the duration of the experiment.
Similar large spirals have also been observed in RayleBenard convection experiments@28,32,33,39–41#. InRayleigh-Benard convection the flow pattern is driven bytemperature difference applied across a fluid layer. Spiratarget patterns are observed in circular cells in the preseof thermal forcing by the cell sidewalls@35,42#. Static side-wall forcing can occur as a result of lateral temperature gdients near the walls, while transient forcing occurs whentemperature difference is suddenly changed. Convectionalign perpendicular to the sidewalls in the absence of swall forcing, and the effect of the forcing is to cause the roto align parallel to the walls.
Target patterns (n50 spirals! in convection have beenstudied experimentally by Huet al. @35#, as well as by sev-eral other groups@32,33,43–46#. In the presence of well-characterized static sidewall forcing, Huet al. @35# foundthat target patterns were stable at onset and up to a cevalue of the reduced control parameter,e. Above that valueof e, the targets become unstable to the so-called focusstability @45,47#, which causes the core of the target to droff center. Beyond this instability the behavior of the targ
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varies depending on experimental conditions. Under soconditions, the core of the target continues to drift off centeventually reaching the wall. Under other conditions, tcore periodically emits traveling convection rolls@35#. Thedistortion of the pattern due to the drift of the target core clead to local wave number changing instabilities, which crestabilize the core@44,45#. In other cases, a periodic oscilation of the position of the core is observed@33#.
Although we have not observed stable target patterns,have seen in the granular system some similarities to tapatterns in convection. In particular, the first stage indecay of an initially symmetric target pattern is the driftinoff center of the core, as in Fig. 13. The emission of travelstripes from the target core was also observed, as showFig. 11. We also observed the absorbtion of stripes bycore; to our knowledge this process has not been observeconvection experiments. In convection the drift of the coand the emission of stripes are both accompanied by a lscale mean flow in the fluid layer@35,46,48#; presumably asimilar large scale granular flow occurs in the granular stem. We do not see the target pattern restabilizing as a reof wave number adjustments away from the core; in our clocal instabilities tend to destroy the circular symmetry of tpattern.
Cell-filling spirals with n.0 have been studied in convection experiments by several groups@27,32,33,39,40#.Again it is found that large spirals develop in the presencesidewall forcing, and are stable at lowe. The spirals rotate inthe winding-up direction, and then spiral arms end atndislocation defects in the interior of the cell; outside of thedefects the pattern consists of concentric rolls@32,33,39,40#.Above a certain value ofe the spiral core moves off centeThis drift causes local variations in the wave number ofpattern, and eventually the pattern becomes unstable toskew varicose instability where it is most compressed. Tleads to the appearance of defects which migrate throughpattern, leading eventually to a spiral pattern with a differen, a bowed pattern, or spiral defect chaos. Large spiralsconvection have been studied theoretically by several gro@49–51#.
There are both similarities and differences betweenbehavior of large spirals in the granular system and thobserved in Rayleigh-Be´nard convection. In the absencepinning at the walls, the granular spirals in the flat-bottomcells rotate in the winding-up sense. In the cells with a beahowever, they rotate in the opposite sense. In convectionspiral arms end at defects in the interior of the cell, but ingranular case the arms almost always ended at the cell wIn both cases the breakup of a spiral starts with its cmoving off-center and distorting the wave number fieldthe pattern. In convection, the skew varicose instabilityvelops where the wave number is large. In the granular cin contrast, we usually observe the crossroll instabilityregions where the local wave number is too small, as welthe wall-mediated instability shown in Fig. 15.
The fast dynamics observed in the spiral cores in ourperiments and in convection appears to be the same.example, our Figs. 9 and 10 can be compared directly w
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Figs. 9 and 11 of Ref.@32#. Core dynamics in large spiralhas been studied theoretically in Ref.@52#, and simulationsdone for the convection system@28# suggest that the cordynamics in that case results from a mean flow in the spcore which couples the tips of the spiral arms.
In convection, spirals are stabilized by sidewall forcinand sidewall forcing must also play a role in the granucase. The role of the forcing, as in convection, is to orientstripes parallel to the cell wall, and so impose the circugeometry of the cell on the pattern formation process. Twas illustrated in Figs. 6 and 7. In the granular system, hoever, the sidewall forcing in the flat-bottomed cells appeto be transient and the spirals eventually decay to patternwhich stripes preferentially orient perpendicular to the siwall. In this case, the~transient! forcing mechanism is strongenough to produce spirals only in deep layers. In the cwith a beach, the forcing is continuous, since it is causedthe slope of the cell bottom plate, and produces spiralsthinner layers. The fact that large spirals continuously dieand reform in the cells with a beach suggests that althothe forcing is not transient in this case, it is not stroenough to completely stabilize that spirals.
The source of forcing in our experiments is probablyradially oriented convective flow of the granular maternear the side wall. In the cells with a beach, such a flarises due to the cell bottom geometry, with a radial lenscale determined by the length of the beach. Since thiscess depends only on the cell geometry, it will be continuas long as the cell is oscillating. In the flat-bottomed cethe convection is driven by friction between the granuparticles and the sidewall — particles next to the sidewexperience a drag force different from those in the bulkthe layer. This effect is strongest when the amplitude ofoscillation is suddenly changed, then dies out as the partinear the wall adjust. The radial extent of this flow is detmined by the appropriate length scale for dissipation witthe layer.
Knight et al. have studied convection driven by sidewafriction in vertically-shaken granular systems@53#. Theyfound that the direction of flow at the sidewalls dependedthe wall geometry as well as on the friction coefficient.particular, in a container with a vertical cylindrical wall theobserved downflow next to the wall and upflow in the ccenter, while in a conical cell — with a wall slope analogoto that of the cell bottom in our cells with a beach — thobserved upflow at the wall and downflow in the centDownflow at sidewalls with friction has also been observin numerical simulations.
The fact that spirals rotate in the winding-up senseflat-bottomed cells but in the unwinding sense in cells witbeach suggests that the convective flows near the boundmay be in the opposite directions in the two cases. Dirobservation, however, indicates that in both the flat-bottomcells and in the cell with the 10° beach there is a flowparticles radially outwards along the top surface ofgranular layer. Thus while this radial flow certainly affecthe boundary conditions of the layer and influences themation of spirals, the details remain unclear.
The spiral defect chaos state observed at high frequen
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~Fig. 17! is very similar in appearance to that studiedRayleigh-Benard convection@27,28,51,54–59#. Spiral defectchaos exists over a large range off and G in our granularsystem, but is always a transient that anneals to a stripetern with defects as in Fig. 4@12#. A similar transient spiral-defect chaos state has been observed in convection exments in a fluid with Prandtl numbers;4 @57#, while forlower values ofs this state was stable in convection@27#,and for highers it does not appear@54#.
In convection, the Prandtl numbers, which is the ratio ofthe fluid’s viscosity to its thermal diffusivity, plays a key rolin the formation of spiral patterns, since it is easier for meflows to produce curvature in the convection pattern in fluwith small s. Coupling between mean flows and curvatuoccurs in other systems as well@60#. It has been demonstrated in simulations@14# that mean flows couple to thpattern in oscillated granular layers in the same way asfluids. The existence of large spirals, the observed tendefor small spirals to form even outside of the range in whicell-filling spirals are observed, and the fact that curvstripes and sidewall foci are common in steady-state patteall suggest that the coupling between mean flows andpattern is relatively strong in the range of experimentalrameters that we have studied.
V. CONCLUSIONS
We have observed large, cell-filling spirals in verticaloscillated granular layers. These transient patterns exista range of parameters within the region where stripe patteare stable. The formation of large spirals depends stronglythe boundary conditions at the cell sidewall. In cells withbeach~for which the depth decreases with radial position inring near the cell wall!, unstable spirals that rotate in thunwinding sense form and decay continuously. In flbottomed cells, spirals form after a jump inG from below toabove onset. They rotate in the winding-up sense and deover time to stripe patterns with defects. Stripes normaorient perpendicular to the cell wall in the granular systebut spirals form when the boundary conditions at the cwall are modified by convective flow of the granular matrial, driven by the bottom plate geometry in the case of cewith a beach, or by friction at the wall for the flat-bottomecells.
Although some differences have been noted above,properties and dynamics of these large spirals are genestrikingly similar to those seen in Rayleigh-Be´nard convec-tion experiments. The role of sidewall forcing is also tsame. Such similar behavior in such physically different stems illustrates the universality of the pattern formation pcess.
Patterns with strong curvature, spiral defect chaos, andoscillatory instability are all observed at low Prandtl numbin fluid convection, where coupling between curvature in tpattern and mean flows is strong. Our observations sugthat a similar coupling plays an important role in the dynaics of our granular system.
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SPIRAL PATTERNS IN OSCILLATED GRANULAR LAYERS PHYSICAL REVIEW E63 041305
Several other results from this work warrant more detaiexperimental or numerical study. These include the sidewforcing and the mechanism by which it influences the pattformation; the evolution of the pattern and in particular ofwave number after its initial formation; the dynamics of tspiral defect chaos state; and the development and influof mean flows in the granular layer on the pattern anddynamics.
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ACKNOWLEDGMENTS
We are grateful to C. Bizon, D. Goldman, W. D. McComick, B. B. Plapp, J. B. Swift, and S. W. Morris for helpfudiscussions. This research was supported by the EngineeResearch Program of the U.S. Department of Energy Ofof Basic Energy Sciences, and the Natural Sciences andgineering Research Council of Canada.
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