In many pattern forming systems that exhibit traveling waves, sources and
sinks occur which separate patches of oppositely traveling waves. We show that
simple qualitative features of their dynamics can be compared to predictions
from coupled amplitude equations. In heated wire convection experiments, we
find a discrepancy between the observed multiplicity of sources and theoretical
predictions. The expression for the observed motion of sinks is incompatible
with any amplitude equation description.Comment: 4 pages, RevTeX, 3 figur

A. C. Newell

A. J. Simon

D. Bensimon

E. Knobloch

G. Faivre

H. Riecke

J. M. Vince

M. C. Cross

M. Dennin

M. Dubois

M. Rabaud

M. van Hecke and

Martin van Hecke

Roberto Alvarez

T. Bohr

W. Schöpf

W. van Saarloos

Wim van Saarloos

English

arXiv

Alvarez, R.

Hecke, M.L. van

Saarloos, W. van

NARCIS 

Sources and sinks separating domains of left- and right-traveling waves: experiment versus amplitude equations

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Hecke, M. van

Leiden University Scholary Publications

RAPID COMMUNICATIONSPHYSICAL REVIEW E AUGUST 1997VOLUME 56, NUMBER 2Sources and sinks separating domains of left- and right-traveling waves:Experiment versus amplitude equationsRoberto Alvarez,1 Martin van Hecke,2,* and Wim van Saarloos21Department of Physics, Drexel University, 32nd and Chestnut Streets, Philadelphia, Pennsylvania 191042Instituut-Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, the Netherlands~Received 18 November 1996!In many pattern forming systems that exhibit traveling waves, sources and sinks occur that separate patchesof oppositely traveling waves. We show that simple qualitative features of their dynamics can be compared topredictions from coupled amplitude equations. In heated wire convection experiments, we find a discrepancybetween the observed multiplicity of sources and theoretical predictions. The expression for the observedmotion of sinks is incompatible with any amplitude equation description.@S1063-651X~97!51308-0#PACS number~s!: 47.54.1r, 03.40.Kf, 47.20.Bp, 47.20.Kyhoosse-allyioeertiticni-ctthenpendeponpuapeledne-andey atlriseupsrtiveherebetal, intheands-ica-mSince its inception@1#, the amplitude equation approachas grown to become an important organizing principlethe theory of nonequilibrium pattern formation—it has nonly enabled us to uncover a number of general featurenear-threshold pattern dynamics, but it has also allowed uunderstand the influence of boundary conditions, defects,Many qualitativeandquantitativepredictions have been successfully confronted with experiments@2#. The most detailedand successful comparison with experiments has been mfor the type of systems for which the theory was originadeveloped, hydrodynamic systems that bifurcate to a statary periodic pattern. For traveling wave systems, the rangvalidity of the appropriate amplitude equation is, howevmuch more an open question, both because the theorederivation has been performed for only a few systems@3#,and because direct tests are difficult. Moreover, in praccomplications often arise due to the presence of additioimportant slow variables@4#. Nevertheless, when the ampltude equations include all the necessary terms and respethe proper symmetries, it is not unreasonable to hopethese equations still provide a goodqualitative descriptionoutside their proper range of validity; this has indeed befound to be the case in a number of systems~e.g., travelingwaves in binary mixtures and liquid crystals@2,4#!.It is the aim of this paper to point out that sources asinks that separate patches of traveling wave states canvide a clear way of testing the consistency of the experimtal observations withgeneric qualitativepredictions fromamplitude equations. Sources and sinks are distinguishewhether the group velocity points outwords or inwards—sFig. 1. They occur in a wide variety of systems where opsitely traveling waves suppress each other—in directiosolidification @5#, the printer instability@6#, eutectic growth@7#, as well as in convection@8,9#—but their properties haveremained largely unexplored. We illustrate the idea weforward with experiments on traveling waves occurring inliquid heated by a wire just below the surface@8–10#. In theparameter range we have been able to explore, the ex*Present address: The Niels Bohr Institute and CATS, Blegdavej 17, 2100 Copenhagen Ø, Denmark.561063-651X/97/56~2!/1306~4!/$10.00ftoftotc.den-of,calealallatndro--bye-altri-mental properties of sources and sinks we observe areincon-sistentwith the behavior predicted by the standard coupamplitude equations for the near-threshold behavior of odimensional traveling waves@2#,~] t1s0]x!AR5«~11 ic0!AR1~11 ic1!]x2AR2~12 ic3!uARu2AR2g2~12 ic2!uALu2AR ,~1a!~] t2s0]x!AL5«~11 ic0!AL1~11 ic1!]x2AL2~12 ic3!uALu2AL2g2~12 ic2!uARu2AL .~1b!In these equations, we have used suitable units of spacetime, andAR is the amplitude of the right traveling mode2 i (vct2kcx) and AL the one of the left traveling modee2 i (vct1kcx). Furthermore« is the control parameter whichmeasures the distance from the threshold of the instabilit«50, and the parametersc02c3 are related to the linear (c0,c1) and nonlinear (c2, c3) dispersion of the waves. It is welknown that, strictly speaking, the above equations only aas the lowest order amplitude equations if the linear grovelocity s0 is of order«1/2 @11#; nevertheless, the equationare frequently also used whens0 is finite at threshold, undethe assumption that they still provide an adequate qualitadescription@12#.When the coupling parameterg2 in Eq. ~1! is larger than1, the left- and right-traveling waves suppress each other@2#,and the system evolves to a state consisting of patches weitherAL or AR is zero. Within such a patch, asingleampli-tude equation suffices, and the group velocity term canremoved by a Galilean transformation. Many experimenand theoretical studies have focused on this situationwhich the theory seems to give a reasonable account ofqualitative observations@2#. Here, we wish to study thesources and sinks that connect these patches of left-right-traveling waves~see Fig. 1!. These coherent structureinvolve bothamplitudesAR andAL ; therefore the group velocity terms cannot be removed, and they probe the applbility of Eq. ~1! to real patterns with finite group velocitys0.s-R1306 © 1997 The American Physical Societygbknththsejuysthrvisnlaeiohinaintoouthce.ner-edbletheig-ingne.eryandglyer-taketosur-beorgeandbothinrcesve,eeandaofapa-nge-re-thehen inaatameiogavathoc,kseoereavvinlet.fre-withandRAPID COMMUNICATIONS56 R1307SOURCES AND SINKS SEPARATING DOMAINS OF . . .In our experiments, we have explored the ran0.25&«&0.5—we cannot make accurate measurementslow «'0.25 since the convection then becomes too weamake careful measurements. Although we therefore canstudy the limit «→0 systematically, which in principle isnecessary to study the validity of~1!, we do believe that weeffectively probe small amplitude convective patterns:flow is weak, and there are no systematic trends as« ilowered. Moreover, the relative frequency difference ofwaves is so small (1/25 to 1/50) and the pictures of@8,9#show that in this range sources are wide objects; both obvations indicate that an adiabatic approximation appearstified.Our experimental setup, shown in Fig. 2, is a simple stem based on an electrically heated wire immersed belowsurface of a fluid@8–10#. Beyond a critical heating poweQc , traveling periodic modulations appear at the surfacea forward Hopf bifurcation@8,9#. The apparatus is similar tothe one used in@8,9# but the design of the Plexiglas cellsomewhat different and larger~5531536 cm!, so that thesides are further away from the wire. Both top to bottom alateral views are possible in our setup; in particular, theeral view proved especially useful for recording the timseries signals. A tungsten wire with a diameter of 0.1 mmheated by means of an electrical current and immersed inin the middle of the cell and parallel to the longest side. Tviscosity of the GE SF 96 silicone oil is 0.5~1 S51cm2/S!. Both the voltage across the wire and the heatcurrent were continuously monitored to check that their vues did not change during the measurements. Four spr~two on each end of the cell! provide the necessary tensionkeep the wire parallel to the surface of the fluid throughthe cell. A pair of micrometers attached to the ends ofFIG. 1. Definition of sources and sinks.~a! Illustration ofsources and sinks as coherent structures in terms of the behavthe amplitudesAL and AR of the adjacent left- and right-travelinwaves. A source is defined as a coherent structure at which wwith total group velocitys pointing outward are generated, andsink is one between waves with group velocitys pointing inwards.~b! Illustration of the kinematics of sinks and sources in terms ofproperties of the adjacent waves, for the case that the group vels has the same sign as the phase velocity~as in the experimentwhere s'vph/3!. In this case, the definition of sources and singiven under~a! agrees with the intuitive notion that the waves travaway from a source and into a sink. The thin lines indicate linesconstant phase of the traveling waves. In accord with our expmental observations, illustrated in Fig. 3~b!, we have drawn a caswith two stationary and symmetric sources, each generating wwith different wave numbers and frequencies, and one sink moaccording to the phase matching rule, Eq.~2!. According to thisrule, every constant phase line coming in at the source from thematches up with a constant phase line coming in from the righee-tooteer-s--eiadt-silegl-gstecell enables us to carefully adjust the wire-surface distanA shadowgraph technique is used to record the signal geated by the waves. The cell is illuminated with unpolarizwhite light. Two photodetectors can be placed at adjustapositions along the wire, and the light signal captured bydetectors is sent to a digital oscilloscope. This temporal snal has a very local character as a result of the diverggeometry from the light source towards the acquisition plaIt therefore allows us to measure the local frequency vaccurately, even though the relation between the signalthe surface modulations is quite nonlinear due to the stroninhomogeneous temperature distribution in the direction ppendicular to the wire. Since a single measurement mayseveral hours due to the long times needed for the fluidreach a steady state condition, the temperature of therounding was carefully controlled.The picture of the typical sequence of events candrawn as follows@8,9#. Having chosen an adequate depth fthe wire, domains of left- and right-traveling waves emerafter the powerQ is turned on ~our control parameter«[Q/Qc21). These patches are separated by sinkssources. Sources in our experiment send out waves tosides, while sinks have oppositely traveling waves comingfrom both sides. Once transients have died out, the soustay at some fixed position while the sinks generically moeither towards a source~in which case they usually annihilateach other! or a boundary~thus also disappearing from thscene!. A typical example is shown in Fig. 3~b!. The timethat a simple state, say one with two or three sourcessinks, remains in the cell is quite arbitrary; in the endsource usually is the longest living object@9#. A space-timeplot of a source solution in this regime is shown in Fig. 2@9#, while sideways snapshots of regions of the cell withsource and a sink are shown in Figs. 3~c! and 3~d!.Our main experimental observationsconcerning the dy-namics of sources and sinks are the following:~i! The rela-tive motion of sinks and sources is independent of their seration, and so there does not appear to be a long-ranteraction between them.~ii ! Sources always have zero velocity, vso50, and are symmetric: the wave number and fquency of the outcoming left-traveling mode are alwayssame as those of the outcoming right-traveling mode. Tdata that illustrate the stationarity of the source are showFig. 3~b!, while the fact that the waves emerging fromsource are symmetric is illustrated by the photodetector dof Fig. 3~a!. In this figure, the frequencies of the signals frothe detectorsD1 andD2 at both sides of the left source arr ofeseitylfi-esgftFIG. 2. Schematic drawing of the experimental setup. Thequencies of the traveling waves can be probed at two positionsphotodetectors. The distance between the surface of the liquidthe wire was varied from 1 to 3 mm.-nersone-reualThisenestheonceinEq.e-am-achle-u-ers,.ofgcedecriti-fse-urboafreevinithethnkRAPID COMMUNICATIONSR1308 56ALVAREZ, van HECKE, AND van SAARLOOSexactly the same, and so are those of theD3 andD4 signalstaken around the other source.~iii ! While sources are stationary and symmetric, they are not unique: each source seout waves with a well-defined frequency and wave numbbut different sources send out different waves — compae.g., the two sources of Fig. 3~a!: the frequency of the signalFIG. 3. Experimental results.~a! Example of the signals fromthe photodetectors placed at various positions in between soand sink solutions, as indicated. Typical signal amplitudes are aa factor 8 above the noise level. The absolute value of each signarbitrary, especially the frequency is relevant. Note that thequencies of the two waves sent out by each separate sourcexactly the same, but that the two sources send out different waAs a result, the sink is sandwiched between different incomwaves.~b! Example of traces of the position of sinks and sourcesthe experiment, as drawn schematically in Fig. 1~b!. ~c! and ~d!Snapshots of regions of the experimental cell with a source~c! anda sink~d!, taken from a sideways video image of the cell. Notesymmetry of the pattern around the source, which is roughly inmiddle of ~c!, and the asymmetry of the pattern around the siwhich is just right of center in~d!.dsr,e,of D1 andD2 is different from those ofD3 andD4 . We takethis as evidence that in these experiments at least aparameter family of sources exists.~iv! As Fig. 3~b! illus-trates, sinks typically move. Moreover, most of our sinks afound to move in such a way that the incoming phasesmatchat the sink@13,14#: in the frame traveling with the sink, thefrequencies of the waves coming in from both sides are eqand no phase difference builds up across these sinks.was already illustrated in Fig. 1~b!. If we write the two ap-propriate incoming modes ase2 i (vRt2 ikRx) ande2 i (vLt1 ikLx),then the velocityvsimatch of such a sink is simplyvsimatch5vR2vLkR1kL. ~2!This relation implies that when a sink is sandwiched betwetwo sources, it moves away from the source with the largfrequency and its velocity is completely determined by tproperties of the adjacent sources.We now confront these results witht eoretical predic-tions. Since the pattern occurs via a forward Hopf bifurcati@8,9#, the generic amplitude equations are given by Eqs.~1!.We takeg2.1 since traveling wave states occur, and sinthe group velocity is about a third of the phase velocitythis experiment, we allow the linear group velocitys0 to beof order 1. Note that for our analysis,we do not needknowl-edge of the values of the other parameters occurring in~1!.Property ~i!, the absence of long-range interactions btween sources and sinks, is consistent with the fact thatplitudes at both sides of source and sink solutions approtheir asymptotic value exponentially fast, as in the singmode equation@Eq. ~1a!# with AL50 @15,16#.To compare with observations~ii ! and~iii !, we have ana-lyzed the generic existence and multiplicity ofsourcesolu-tions of Eq.~1! with an extension of previous counting argments @15# for solutions of the form AR5e2 iv0tÂR(x2vsot) andAL5e2 iv0tÂL(x1vsot). Our analysis@17# showsthat independent of the specific values of the parametsource solutions of Eq.~1! genericallycome in discrete setsIn particular, one typically expects there to be only auniquesymmetric source solution withvso50, and numerical simu-lations of Eq.~1! confirm this. This is in clear contradictionwith the experiments, where we find a continuous familythem.We now turn tosinks, which according to~iv! move in theexperiments with a velocity~2!. Can the phase matchinproperty of the sinks underlying this equation be reproduat all in an amplitude approach based on Eq.~1!? The answeris no. To see this, note thatvsimatch is according to Eq.~2!given in terms of thetotal frequenciesvR , vL and wavenumberskR , kL of the incoming modes. In an amplitudexpansion, these are written as an expansion about theircal values, e.g.,vR5vc1vAR wherevAR is the frequency ofthe amplitudeAR of the right-traveling mode, etc. In terms othese, the experimentally observed velocity of phamatching sinks becomesceutl is-aree .gne,devintdt.e-dtheansts ofor-ta-hets.gRAPID COMMUNICATIONS56 R1309SOURCES AND SINKS SEPARATING DOMAINS OF . . .vsimatch5vAR2vAL2kc1kAR1kAL, ~3!which underscores once more the fact that this velocitypends on both the fast and the slow spatial scales. Howeas an amplitude description is based on an adiabatic elimtion of the fast scales, the amplitude equations~1! do notinvolve the parametersvc and kc associated with the fasscales. So, although families of moving sink solutionsexist for Eq.~1!, there is no mechanism in these equationssingle out the velocity~3! as the selected velocity of sinksIn summary, our results demonstrate thatgeneric proper-tiesof sources and sinks in traveling wave systems, like thmultiplicity and dynamics, allow a simple yet powerful comparison between experiments and amplitude equationHS.et-er,a-ooire-scriptions. For the heated wire convection experiment inrange 0.25&«&0.5 our experiments are inconsistent withamplitude description. Although a final conclusion muawait further study of the«→0 limit, our results point to twoimportant issues. First of all, they question the soundnesusing Eqs.~1! for systems with finite group velocitys0. Sec-ondly, they provide a clear example of the possible imptance of nonadiabatic effects~coupling of the slow and fasscales@18#! in sinks, even though the two are widely seprated, as the relative frequency modulationDv/vc ~which iscomparable to the ratio of the typical sink velocity and tphase velocity! can be as small as 1/50 in our experimenR.A. is grateful to N. Kwasnjuk for help in constructinthe experimental cell.nged touseallyn ahibitatch.thede. A@1# A. C. Newell and J. A. Whitehead, J. Fluid Mech.38, 279~1969!.@2# M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65, 851~1993!.@3# See, e.g., W. Scho¨pf and W. Zimmerman, Phys. Rev. E47,1739~1993! for the derivation for binary fluids, and E. Y. Kuoand M. C. Cross,ibid. 47, R2245~1993! and M. van Heckeand W. van Saarloos, Phys. Rev. E55, R1259~1997! for thederivation for rotating convection cells.@4# For an example in the context of binary mixtures, seeRiecke, Phys. Rev. Lett.68, 301 ~1992!; of chemical waves,see M. Ipsen~unpublished!; of convection in liquid crystals,see M. Dennin, M. Treiber, L. Kramer, G. Ahlers, and D.Cannell, Phys. Rev. Lett.76, 319 ~1995!.@5# A. J. Simon, J. Bechhoefer, and A. Libchaber, Phys. Rev. L61, 2574~1988!.@6# M. Rabaud, S. Michelland, and Y. Couder, Phys. Rev. Lett.64,184 ~1990!; L. Pang and J. R. de Bruyn,ibid. 70, 1791~1993!.@7# G. Faivre and J. Mergy, Phys. Rev. A46, 963 ~1992!.@8# M. Dubois, F. Daviaud, O. Ronsin, and P. Berge´, Physica D61, 140 ~1992!.@9# J. M. Vince and M. Dubois, Europhys. Lett.20, 505 ~1992!.@10# R. Alvarez~unpublished!..t.@11# E. Knobloch and J. de Luca, Nonlinearity3, 575 ~1990!; C.Martel and J. M. Vega,ibid. 9, 1129~1996!.@12# H. Riecke, Europhys. Lett.11, 213 ~1990!.@13# Note, however, that Duboiset al. @8# show an example of aweakly oscillating sink close to a source, without matchiphases. It is not stated at which value of« this occurs, orwhether it is a transient phenomenon.@14# When the phases match across a sink, one might be temptsay that no phase slips occur there. We prefer to avoid tothe words phase slip in this context, however, as they usurefer to events at which the amplitude goes through zero isingle amplitude equation@15#. In the case of two coupledmodes, each separate amplitude does not need to exphase-slip events even if the phases across a sink do not mThe relation between phase slips of the amplitudes and ofpatterns is quite different for single and double amplituequations.@15# W. van Saarloos and P. C. Hohenberg, Physica D56, 303~1992!; 9, 209~E! ~1993!.@16# T. Bohr, G. Huber, and E. Ott, Europhys. Lett.33, 589~1996!.@17# M. van Hecke, C. Storm, and W. van Saarloos~unpublished!.@18# D. Bensimon, B. I. Shraiman, and V. Croquette, Phys. Rev38, 5461~1988!.

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Sources and sinks separating domains of left- and right-traveling waves: Experiment versus amplitude equations

Alvarez, R. Rossi

van Hecke, M.

van Saarloos, W.

Copenhagen University Research Information System

Sources and Sinks Separating Domains of Left- and Right-Traveling Waves: Experiment versus Amplitude Equations

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Sources and sinks separating domains of left- and right-traveling waves:
  Experiment versus amplitude equations

Sources and sinks separating domains of left- and right-traveling waves: Experiment versus amplitude equations

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