convection under rotation for prandtl numbers near 1: linear ...convection under rotation for...

PHYSICAL REVIEW E JUNE 1997VOLUME 55, NUMBER 6

Convection under rotation for Prandtl numbers near 1:Linear stability, wave-number selection, and pattern dynamics

Yuchou Hu,1,2 Robert E. Ecke,1 and Guenter Ahlers21Condensed-Matter and Thermal Physics Group, Center for Nonlinear Studies, Los Alamos National Laboratory,

Los Alamos, New Mexico 875452Department of Physics and Center for Nonlinear Sciences, University of California, Santa Barbara, California 93106

~Received 4 December 1996!

Rayleigh-Benard convection with rotation about a vertical axis was studied with the shadowgraph imagingmethod up to a dimensionless rotation rateV of 22. Most of the results are for a cylindrical convection cellwith a radius-to-height ratioG540 that contained CO2 at 33.1 bars with a Prandtl numbers50.93. Measure-ments of the critical Rayleigh numberRc and wave numberkc for 0,V,22 agree well with predictions basedon linear stability analysis. Above onset and with rotation, the average wave number and details of the patterndynamics were studied. ForV&5, the initial onset was to a pattern of straight or slightly curved rolls. For0.1&e[DT/DTc21&0.5 but below the onset of spiral-defect chaos, rotation withV&8 produced weakperturbations of nonrotating patterns. Typically, this gave an ‘‘S-shaped’’ distortion of the zero-rotation patternof straight or somewhat curved rolls. Rotation had a stronger effect on the source and motions of dislocationdefects. ForV.0 the defects were generated primarily at the wall, whereas forV50 they were nucleated inthe bulk via the skewed-varicose instability. Rotation picked a preferred direction of motion for the defectsonce they formed. Fore*0.5, recognizable spiral-defect chaos and the oscillatory instability were observed forV&12. ForV>8, domain growth and front propagation suggestive of the Ku¨ppers-Lortz instability wereobserved from onset up to ane value that increased withV. Increasinge at fixedV&12 enhanced dislocation-defect dynamics over Ku¨ppers-Lortz front propagation. Quantitative measurements of average pattern wavenumbers, correlation lengths, and spatially averaged roll curvature as functions ofe andV are presented. At afixedV*10, the average wave number had two distinct wave-number-selection regions with different slopesas a function ofe, one abovee'0.45 and the other near onset. The slope fore near onset reached a minimumat V512.1 and increased linearly for 12,V,20. @S1063-651X~97!03906-8#

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I. INTRODUCTION

Rayleigh-Benard convection with rotation is a simpllaboratory system incorporating essential forces that occunatural phenomena such as circulations in the atmospand ocean currents. It provides an ideal setting for the stof the interaction between thermally induced instabilities aCoriolis and centrifugal forces arising from rotation. Theretical analysis@1–4# of the problem has concentrated on tlimit where the centrifugal force is much less important thgravity and can be neglected. Experimentally this limit cbe well approximated by a combination of suitable physirotation frequenciesf , lateral system sizeL, and samplethicknessd for which the centrifugal acceleration (2p f )2Lis much smaller than the gravitational accelerationg. Underthe same conditions, the relevant dimensionless frequen

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which measures the strength of the Coriolis force, can coa wide range of interest (n is the kinematic viscosity!. Thesimple geometry of the problem, a thin horizontal layerfluid, combined with the easily controlled parametersheating from below and rotation about a vertical axis, mathis system experimentally attractive. Despite these featurelatively little experimental work has been done on this stem in the region close to the onset of convection. Mostthe previous work@5–18# was qualitative or semiquantita

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Linear stability analysis@1# predicts that convective motion in the form of rolls appears in the fluid when the temperature differenceDT across the layer exceeds a criticvalueDTc that depends on the rotation rate. The dimensiless distance from the onset of convection is measurede[DT/DTc(V)21. Rotation inhibits the onset of convection and decreases the wavelengthl of the convection rolls.The dimensionless control parameter proportional toDT isthe Rayleigh number

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wherea is the isobaric thermal expansion coefficient andkis the thermal diffusivity. The inhibition of convection brotation is illustrated in Fig. 1~a!, which shows the neutrastability curveRc(V) calculated from linear stability analysis @1#. The dimensionless wave number of the rollsk[2pd/l and its value atDTc is kc . The increase ofkcwith V is shown in Fig. 1~b!. For largeV, Rc and kc arepredicted to scale asV4/3 and V1/3, respectively@1#; forsmallV, as in the work presented here,Rc andkc should bequadratic inV because of the symmetry of the convectistate under reversal of the rotation direction. Previous expmental measurements of these quantities are sparse, pa

6928 © 1997 The American Physical Society

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55 6929CONVECTION UNDER ROTATION FOR PRANDTL . . .

larly for the pattern wave number because of the difficuof visualizing convection patterns near onset in a rotatsystem. Measurements ofRc as a function ofV for smallrotation rates (V&30) that agreed semiquantitatively wittheoretical predictions were reported by Rossby@6# andZhonget al. @16# for water, by Buhler and Oertel using ni-trogen gas@11#, and by Lucaset al. for liquid 4He @12#. Inthe experiments using water and forV,50 by Zhonget al.,the wave number ate'0.4 was found to agree with ththeoretical prediction forkc to within an experimental uncertainty of about 10%. This was the best determination ofkcprior to the results reported in this paper where pattecould be studied quantitatively fore*0.01. Sincekc as wellasRc can be calculated from an exact linear stability anasis, their accurate measurement is very desirable to provifirm starting point for comparison between experiment atheory. Thus, we also show in Fig. 1 the experimental dfor Rc andkc , which will be described in detail in Sec. IIIAs can be seen, the agreement between experimenttheory at thelinear level is excellent.

Above onset, the size of the stable region of infiniteextended straight-roll patterns in thek2e parameter spaceshrinks with increasingV @4#. As for zero rotation, the details of the stable region depend on the Prandtl nums[n/k of the fluid. In particular, above a critical rotatiorateVc(s), the region of stable straight rolls shrinks to zeand forV>Vc(s) the straight-roll pattern becomes unstabto the Kuppers-Lortz ~KL ! instability even at arbitrarilysmall e @2,3#. This instability causes otherwise steady covection rolls of a given orientation to become unstablerolls oriented at some angle advanced in the direction

FIG. 1. ~a! Rc and ~b! kc vs V. Solid lines are predictions olinear theory, and the data are forG540.

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rotation. Thus, the roll orientation changes in discrete stas rolls of one orientation grow to replace rolls in the ustable direction. The angle depends on bothV ands @3,4#and can vary between 30° and 70°. It is, however, close58° fors*1 andV*15 @4#. Because of the close proximitof the KL angle to 60°, theoretical models have been devoped that consist of equations for the amplitudes of thcoupled modes. The first such model, developed by Buand Heikes@9#, explained many qualitative features of earexperimental observations of the KL instability@8–10#. Itwas later extended to include spatial gradient terms byand Cross@19#. An important result of the latter work was tdemonstrate that a signature of the KL instability belowVcwas the propagation of fronts of one orientation into rdomains of a different orientation. This helped to explain texperimental observations of front propagation@16–18# andof an apparently lower value ofVc than theoretically pre-dicted @3#. Numerical solutions of generalized@20–23#Swift-Hohenberg ~GSH! equations @24,25# and of theNavier-Stokes equations in the Boussinesq approxima@26# have also been used to study the KL instability.

Although the KL instability is important in many of ouexperimental observations, its influence is often mixed wother effects. Thus, many of the experimental results arerectly comparable to theoretical models only quite closeonset and forV*Vc (Vc'13 for s50.93). Results in thatparameter range are reported in detail elsewhere@27,28#.Here we present measurements over a widee range of theeffects of rotation on pattern textures. These effects inclwave number selection and phenomena affecting patternnamics such as defect nucleation and propagation. Theinstability in this context comes in through the location ofstability boundary, which, for a given wave number, movto smallere with increasingV. We will divide our discus-sion into two parts: the first withV&8 is a description oftextures and dynamics where the KL instability is not appent. The second involves states forV*8 where the KL in-stability definitely plays a role. In the first it will be important to recall what is known theoretically about secondstability boundaries and pattern dynamics in both the nontating and rotating system.

For the nonrotating case, the theory for the stabilitystraight parallel rolls of infinite lateral extent, developed fa whole range of Prandtl numbers by Busse and Clever@29–31#, has been very useful in describing experimental obsvations of patterns. The assumption of uniform wave numin the theory leads to the expectation that the patterncomes unstable to a particular perturbation simultaneoueverywhere in space. On the other hand, many featurereal patterns that include wave-number gradients, roll curture, and defects are not considered by such theories. Hever, it was realized that, to a good approximation, the lowavelength instabilities of the laterally infinite system, suas the Eckhaus and skewed-varicose instabilities, will aoccur when thelocal wave number of a pattern crossesinstability boundary. In that case the instability leads to tformation of defects that travel away and relax the extremof local wave numbers back into the stable range. Thideas have evolved from early motivating experiments@32–37#, and by considering the infinite-system instabilitiesconjunction with phase equations@38–46# coupled to mean-

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6930 55YUCHOU HU, ROBERT E. ECKE, AND GUENTER AHLERS

drift flows that are relevant for low-s fluids @39,40#. In ad-dition to phenomena that can be explained to some exten‘‘finite-size’’ perturbations of concepts from infinite-systeinstabilities, there are states to which the theory of strairolls is not applicable at all. The state of spiral defect cha~SDC! discovered by Morriset al. @47# ~see, for instanceFig. 11 below! is a prime example. It consists of highlcurved rolls in the form of spirals and targets and existsregions ofk-e space where straight, parallel rolls are knowtheoretically to be stable@29–31,48,49#. These states posesignificant challenge to the existing theories of RayleigBenard convection.

A number of recent experiments and numerical simutions for V50 ands'1 have added considerably to ounderstanding of real patterns. For radius-to-height ratioGsimilar to those of the present work, experiments have pvided quantitative information about axisymmetric wavnumber selection and the focus instability@50#, nucleationrates of phase in the cores of sidewall foci@51#, the nucle-ation mechanisms of spiral defects@52# and the transition tospiral-defect chaos@53#, and the dynamics of textured paterns@54#. This work has provided an extensive characterition of pattern wave numbers, correlation lengths, roll curtures, and sidewall obliqueness. Computer simulationslow-Prandtl-number convection using both generalizSwift-Hohenberg models@55–57# and the Navier-Stokesequations in the Boussinesq approximation@48,58# have alsobeen extremely valuable. The experiments, simulations,theory reveal interesting features, many of which areunderstood. There are, however, some general conclusfor nonrotating, low-s convection in cylindrical containersFirst, in the absence of strong sidewall forcing, the onpattern is time independent and consists of straight rthroughout most of the cell with small cross-roll defects nthe part of the periphery where the sidewall is approximatparallel to the roll axis in the interior@see, for instance, Fig8~a! below#. With increasinge, the tendency for the rolls toterminate with their axes orthogonal to the circular sidewincreases@see Figs. 9~a!–9~c! below#. The resulting roll cur-vature in the cell interior leads to a wave-number distributwith a maximal wave-number increase near the cell cenTime dependence appears first ate'0.1 in the form of re-peated skewed-varicose dislocation-defect nucleationsthe cell center followed by defect motion and pattern rerangement@see, for instance, Fig. 9~a! below# @37,50#. Ex-periments@59,60# have demonstrated that mean drift drivby boundary-induced roll curvature is responsible for theserved time dependence. The appearance of SDC seemdepend onG in a complicated fashion@47,53,61–63#, but for20&G&60 ands.1 the transition occurs ate'0.5. Thepattern dynamics in the intermediate region 0.1,e,0.5 aredominated by the development of strong wall foci@see, forinstance, Figs. 9~b! and 9~c! below#, increasingly~with e)vigorous dislocation-defect motion, but little change in avage roll curvature or sidewall obliqueness@54#. These are themain features of pattern dynamics for nonrotating, lowsconvection. An important thing to keep in mind is that tmost obvious consequence of wave-number-relaxing inbilities is the formation of a small number of defects.

We now consider stability theory and numerical simutions for rotating convection and the limited number of e

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perimental observations of pattern dynamics for that caFor s'1, the KL and skewed-varicose~SV! instabilities athigh wave number and the Eckhaus~EK! instability at lowwave number are important secondary instabilities nearonset of convection@4,26#. Figure 2 shows the calculatestability boundaries for several values ofV for s51.0 @64#.As V is increased, the major features of these curves arestabilization of straight rolls against the skewed-varicosestability and their destabilization with respect to thKuppers-Lortz instability. The destabilization with respectthe KL instability leads to a shrinking straight-roll stabilitwindow asV is increased. At a critical valueVc(s), thisstability window disappears altogether and straight rollsno longer stable at anye. In contrast to the dramatic changin the SV boundary with increasing rotation, the low-wavnumber boundary is practically unaffected.

Another powerful approach to the study of pattern dynaics in Rayleigh-Be´nard convection, which augments thsecondary-stability calculations is the numerical simulatof model equations, in particular the Swift-Hohenberg~SH!equation@24,25# and its generalizations@46# to include meandrift @65#, non-Boussinesq effects@66#, and rotation about avertical axis@20–22#. The models that include rotation havbeen used to both confirm experimental observationsmake predictions regarding the behavior of patterns andfects under the influence of rotation. In relation to the expemental results presented here, a limitation of the modelshave included rotation is that they are effectively largesmodels since mean drift is ignored. Nevertheless, in systwith quite differents, the experimental observations@16–18# of sidewall-initiated front propagations caused by the Kinstability, the motions of sidewall defects, and the ‘‘S-shaped’’ distortion of the straight-roll pattern have beenproduced qualitatively@20,21#. These simulations have alssuggested the possible existence of rigid pattern rotacaused by defects@67# and the enhancement by rotationthe nonvariational property of dislocation glide@58,67#. Re-cently, a simulation of a GSH equation that included borotation and mean drift@23# showed a number of patterfeatures that we observe.

FIG. 2. Stability boundaries for straight parallel rolls with rspect to long-wavelength instabilities, EK~Eckhaus! and SV~skewed-varicose!, and to the Ku¨ppers-Lortz instability.s51.0.Values ofV are 0 ~—!, 5 ~- - -!, 10 ~– – –!, and 15~– - - –!.Boundaries for different instabilities are labeled on the plot withvalue ofV in parentheses.

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The dependence of the average wave number of theterns one as a function ofV and close to onset has not beinvestigated systematically prior to our work. There ahowever, several studies for highere that should be mentioned. Fore*3 and depending ons and V, Heikes ob-served hexagonal cellular patterns and cross-roll pattcomposed of two perpendicular sets of rolls in water ws between 3.8 and 14.5 in cells withG560 @8#. Heikes alsomeasured the wave numberk of the patterns frome of 0.5 toabout 10@8#. The wave number decreased with increase, in agreement with previous measurements by Rossbsilicone oil ats5100 @6# up to e'4. For e.4, k in watercontinued to decrease whereask in silicone oil increased.

In the following section, the apparatus is described. Nein Sec. III, we present the linear results for this systenamely, the critical Rayleigh numberRc and the criticalwave number kc as a function ofV. In Sec. IV, we giveresults that can be understood in the context of seconinstabilities and with insight developed from previous woon this system withV50 @27,53,54#. First we give a sum-mary of the different states in the parameter space ofe andV. Of particular interest here are the motions of defects,overall pattern textures, and the persistence of the spdefect-chaos state under rotation. We then discuss the ience of the skewed-varicose instability on the systemgive measurements of the characteristic angle assocwith it as a function ofV. A more detailed discussion of thmotion of defects forV.0 is then given, and their overaeffect on pattern rotation relative to the rotating frame ofexperiment is presented. In Sec. V quantitative results foraverage wave number, correlation length, roll curvature,sidewall obliqueness of the patterns are given as a funcof V and e. In Sec. VI, pattern dynamics for each of thregions identified in parameter space are presented inform of image sequences. We conclude in Sec. VII withsummary.

II. EXPERIMENTAL APPARATUS AND PROCEDURES

The apparatus is described thoroughly elsewhe@47,50,68# as are details of the cell construction and produres for determiningDTc @50,54#. The circular sidewallused in the majority of studies reported in this pap(G540) had a cross section shaped like the letterH andenclosed an active convective area measuring 86 mm inameter@69#. This H-shaped design minimized the sidewathickness seen by the fluid and reduced the effect of sideforcing @50#. Despite this design, two to three concentrolls were visible next to the sidewall whenDT was about1% below the value at which the center of the cell starconvecting. This, however, did not affect the determinatof DTc ~see Sec. III! significantly. The sidewall was sandwiched from the top by a 9.5-mm-thick optically flat saphire window and the bottom by an aluminum plate wmirror finish. The spacing between the window and pldetermined the layer heightd, but owing to anin situ adjust-ment of the uniformity ofd to within 0.002 mm, the absolutvalue of d could not be measured directly. We inferredindirectly through a comparison ofDTc atV50 with that ofa calibration cell with a known cell height operated at tsame temperature and pressure@70#, and through the as

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sumption that at onset the wave number of the straight-structure was the critical wave numberkc53.117. These twomethods gave the samed51.06 mm with an uncertainty o1%, and a radius-to-height ratioG of 40. The cell was filledwith CO2 at 33.1 bars, and the temperature of the bath tcooled the top plate was held constant within60.0002 °Cover periods of a week or longer near 33.70 °C. Under thconditions, the fluid had a Prandtl number of 0.93. The prsure was regulated by a temperature-controlled externallast volume to within 0.005%. The vertical thermal diffusiotime tv5d2/k was 4.5 sec, and the horizontal thermal diffsion time th5G2tv was 7200 sec. The measureDTc(V50) was 1.48760.004 °C. From Eq. ~2! andRc51708, we estimateDTc51.47 °C, which agrees with theexperimental value well within reasonable estimates of errin d and in the fluid properties@68# that we used.

We also made some measurements in a cell wd50.201 cm and a correspondingly smaller aspect raG523. For this case, the sidewall was made of Macormachinable ceramic. It was 0.200 cm thick and had‘‘spoiler tab’’ @50,71,72# that was intended to reduce thermsidewall forcing. In this cell, a thinner sapphire window~3.2mm! was used for the cell top. This resulted in a slight boing of the cell in the middle owing to a small pressure dferential between the sample and the water bath. Thuscell profile was approximately axisymmetric with maximuheight at the center. This caused the convection pattergrow in from the center where the effectivee was highestand had the consequence of reducing sidewall effectsonset, albeit at the price of a small spatial ramp ine. Bymeasuring the radial locationr where the pattern was firsnoticeable as a function ofDT, we estimated the local celthicknessd(r ). The result is shown in Fig. 3. As can be seed varied by 0.8% across the radius, leading to a total vation of e of 2.4%. For this cell the bath temperature was a33.70 °C, but the pressure was 16.55 bars. Using the medescribed in Sec. III, we measuredDTc to be 1.557 °C.Using this value as corresponding toe50, convection firststarted in the cell center ate520.015, and the cell wascompletely filled with convection fore>0.012. On the basisof the data in Fig. 3 we estimate that the value ofd corre-

FIG. 3. The thicknessd(r ) as a function of the radial positionr ~in units ofd) for the cell withG523.

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sponding to the radial onset location forDT5DTc isd50.201 cm. With Eq.~2! and the known fluid propertie@68# this yieldsDTc51.54 °C, in very good agreement witthe experiment. For this cell,tv was 6.7 sec ands was 0.83.It follows that a particular value ofV is reached at a dimensioned rotation ratef , which is smaller than in the larger-Gcell by a factor of about 1.68~the ratio of the values otv /s). Since the centrifugal acceleration is proportionalf 2, it was smaller by a factor of about 2.81 at a givenV inthe thicker cell. This cell gave essentially equivalent resuwherever a comparison with the larger cell could be maexcept very close to onset as described below.

Convection in both of the cells was well approximatedthe Boussinesq approximation. The parameterP defined byBusse@73# that describes the size of departures fromBoussinesq approximation is 0.17 and 0.10 for theG540and 23 cells, respectively. Non-Boussinesq effects in a slar system but with smaller layer depth have been studcarefully @74#.

The cell was placed inside a pressure vessel thatmounted on a turntable with the electrical connections routhrough a slip-ring assembly concentric with the rotatiaxis. A schematic illustration of the arrangement of the mjor components of the apparatus is shown in Fig. 4. A mcrostepping controller drove a stepping motor that rotatedturntable through a belt-and-pulley arrangement with orevolution of 360° completed in 50 000 microsteps. Therotation rate achievable with this setup was limited byfriction and stepping-motor power. For theG540 cell, 1 Hzwas equivalent toV530. The effect of the centrifugal forcat the highest rotation rate in the experiments of 0.7 Hz wsmall, with (2p f )2Gd/g50.09. The rotation direction isspecified as seen when viewed from above and is counclockwise ~positive V) for all the results reported in thipaper.

The convection patterns were observed by the shad

FIG. 4. Schematic diagram of the apparatus.

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graph visualization method with the necessary optics ctained inside a cylindrical tube mounted on top of the prsure vessel. The tube rotated with the cell, so observatiopatterns was in the rotating frame of reference. The shadgraph images were captured and digitized as 8-bit grey-sccoded images. In this paper, the contrast-enhanced imshow black regions corresponding to hot fluid~upflow! andwhite regions corresponding to cold fluid~downflow!. Forthe work discussed in the present paper, approxima200 000 images were analyzed.

The complex spatial and temporal dynamics of the pterns made quantitative analysis challenging. Here we uthe same algorithms presented in Ref.@54# to characterizethese dynamical patterns. To describe the spatial scales opatterns, the first two moments^k& and j22 were obtainedfrom the structure factorS(k), which was calculated bysquaring the modulus of the Fourier transform F~k! @47,54#.In the computation of the Fourier transform, the Hanniwindow described elsewhere@54# was used. Averages oS(k) were computed over image sequences taken at fiV and e, with each sequence containing between 128 a256 images equally spaced in time. The time interval at ee was adjusted so that the images were uncorrelated exfor e&0.1 where the time scale of the dynamics was velong. The physical interpretation of^k& is that of the averagepattern wave number but that ofj is much less obvious. Thewidth of the distribution depends on the roll curvature, tnumber of defects, wave-number gradients, and the sizdomains~a spatial region with roughly uniform pattern texture! in a complicated manner@75,76#. It is, however, asimple and well-defined measure of spatial order disorand as such we use it here.

Measures of the pattern texture are the roll curvaturesidewall obliqueness. These were extracted from each imby locally fitting the roll to a parabola in the interior andstraight line near the wall, respectively@54#. Unfortunately,these real-space techniques were not suitable for cellulagions with coexisting rolls and defects or for regions neKL fronts. Thus, they were primarily used to characteripatterns at small rotation rates,V<8, where cellular regionsappeared infrequently, and KL fronts did not occupy a snificant fraction of the total area. The roll curvatureg mea-sures locally the amount of angular change per unit lenalong the rolls, and the spatially and temporally averagcurvaturegs,t is the average over all the images in the sasequence. The similarly averaged sidewall obliquenessbs,tis based on roll orientations in the area adjacent to the swall. It provides a measure of the average deviation frperpendicular termination of the roll axes at the sidewall.

III. LINEAR RESULTS FOR DTc„V… AND Kc„V…

Heat-transport measurements and simultaneous shagraph visualization allow for the experimental determinatiof Rc andkc . The critical temperature difference is obtainefrom Nusselt-number data. The Nusselt numberN is definedas the ratio of total heat conducted by the fluid in the covecting state to that conducted only by thermal diffusion.example forG540 atV515.4 of typical results forN isshown in Fig. 5. There is a small region of rounding f20.01&e&0.01 (1.865 °C&DT&1.895 °C!. In principle,

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rounding at some level would be expected from large-scflows induced by the centrifugal acceleration. In our expements, however, the rounding was of the same size aV includingV50. Thus, we attribute it to experimental imperfections in the convection cell and conclude that centrgal effects onN were much less than revealed by the dataFig. 5.

Away from the rounded region we fit the function

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to the data forN in the convecting region. In the fit,DTc ,S1 , andS2 were adjusted. From the results forDTc we com-putedRc using the fluid properties evaluated at the meantemperature. Figure 1~a! shows the results forRc(V) as wellas the prediction based on linear stability analysis@1#. Theagreement between the data and the prediction is excelA different way to illustrate this agreement and to showquadratic dependence ofRc on V is to plot Rc(V)/Rc(0)againstV2 as in Fig. 6~a!, where results for both cells argiven. The dashed line is drawn with the initial slope of ttheoretical results, and illustrates that noticeable deviatifrom the quadratic relationship occur forV*10.

We determinedkc for rotating patterns from Fourier transforms of shadowgraph images. As discussed in Sec. II,wave number at onset forV50 is assumed to bekc53.117. For nonzeroV we determinedkc from an ex-trapolation of the average wave number^k& as a function ofe for e&0.1 ~see Fig. 30 below!. In Fig. 1~b! the experimen-tal results for theG540 cell are seen to agree well with thpredictions from linear stability analysis. Again one cdemonstrate the quadratic scaling for smallV by plottingkc(V)/kc(0) versusV

2 as is done in Fig. 6~b! for both cells.

IV. NONLINEAR STABILITY

The behavior of patterns as a function of rotation depeon many influences and is extremely complicated in detFrom previous experiments without rotation on large-G cellsand fors.1 @27,37,51,53,54#, and from a large amount otheoretical and numerical modeling@46#, we know that the

FIG. 5. Nusselt number vsDT for the determination ofDTc .V515.4 andG540. The solid lines are least-squares fits to the dbelow and aboveDTc outside the rounded region.

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dynamics of patterns forV50 is governed by a complexinteraction of long-wavelength instabilities~Eckhaus,skewed-varicose, etc.!, large-scale vertical-vorticity mode~mean drift!, and interactions with the convection-cell laterboundaries. A complete picture has not been establispartly because of the difficulty in directly measuring thmean-drift field. To this we add rotation. Clearly, we cannachieve a full characterization of pattern dynamics in roting convection. What we will describe are phenomenawhich a reasonable picture could be developed based oncepts from the nonrotating problem or that were seen mtimes and were somehow generic to the convecting statparticular parameter values. In reporting quantitative msures of patterns, we nondimensionalize length scalesd and time scales withtv .

One general feature of the rotating system is that theeral boundaries become more important than for the nontating case. Previous experimental work@16# and subsequennumerical models@20# showed that even for small rotationdefects could be nucleated at and travel along the sidew~at higher rotation, the wall has such a dominant effect thanew wall state becomes unstable at a smallerR thanRc forthe bulk state@77#!. In the present work we also observmany sidewall defect nucleations and defect motions, whwe describe in detail. These wall-generated defects mcomparisons with theories problematic since most treatmconsider either a laterally infinite system, or a large-asperatio limit where sidewalls should have only a small infl

a

FIG. 6. ~a! Rc(V)/Rc(0) and ~b! kc(V)/kc(0) vs V2. Solidlines are predictions of linear theory and the dashed lines areinitial slopes of the theoretical curves. The data are forG540(d) andG523 ~1!.

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ence. Theoretical analysis will have to include realistic siwall interactions because experimentally sidewall effecannot be avoided merely by increasing lateral system sHowever, we will make some comparisons between datatheG540 cell and theG523 cell that had a slight axisymmetric ramp ofe as described in Sec. II. The ramp causconvection rolls to form in the interior of the cell before theformed near the walls. This isolated the bulk state from siwall influences for a small range ofe and had a simplifyingeffect on the pattern dynamics over that range.

Another difficulty in analyzing our experimental resultsthat the theoretical and numerical modeling of patternnamics in rotating convection is not nearly as extensivefor nonrotating convection although there has been a fluof recent activity@20,21,58,67#. For example, whereas thstability boundaries for nonrotating convection are knowndetail for arbitrarys owing to the large body of work byBusse and co-workers@73#, the only published work on theeffects of rotation@4# has almost no data fors51. Althoughsome recent work@26# is beginning to fill that gap, a complete characterization of secondary stability boundariesfunction of rotation is not available. Nevertheless, as the ccept of secondary stability has proven so valuable for nontating convection, we apply it here as best we can.

There are two influences of rotation on secondary insbility: the shift of stability boundaries present without rottion and the introduction of new rotation-induced instabties. In Fig. 2, the long-wavelength instability boundarithat are present also without rotation are shown for sevrotation rates. There is not much effect of rotation onlow-wave-number boundaries forV&15; for V520 theEckhaus boundary at smalle&0.3 is replaced by the crossroll instability. On the high-wave-number side the effectsrotation are more dramatic since the SV boundary is shito higher wave number asV increases. This suggests that tSV instability should be less relevant to the dynamicshigherV. Another result of this analysis is that the angle fthe SV instability decreases with rotation@26#.

The rotation-induced Ku¨ppers-Lortz instability emergeasV is increased and is usually associated with the critrotation rateVc . But just as the skewed-varicose instabilihas a big influence on pattern dynamics before theboundary is actually crossed by the mean wave numbercause of pattern textures with roll curvature and wanumber gradients, one might expect that the KL instabiwould have a similar effect forV,Vc . In Fig. 2, the shrink-ing domain of stability in wave-number space is showndifferentV. ForV,4, the long-wavelength instabilities arimportant, and the KL instability should play little part in thpattern dynamics. ForV.4 the KL-limited region is smallerthan the long-wavelength-limited one and KL dynamshould dominate. We will revisit this issue later and plandiscuss it as well in another paper focusing on the KL dnamics@28#. One would expect from this theoretical analysthat, even for modestV, KL dynamics would be preferentially observed relative to other dynamics related to lonwavelength straight-roll instabilities. It may be difficulhowever, to distinguish the signature of the KL instabiliwhen it occurs locally, from that of other instabilities. Befodelving into specific signatures of secondary instabilitiesis useful to give an overall representation of the differe

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pattern textures that one sees in different regions of pareter space.

A. The big picture

The parameter space shown in Fig. 7 can be roughlyvided into six regions based on the types of patterns thatobserved. The determinations of the boundaries of thosegions are approximate because the coexistence of behaattributable to different instabilities was observed over laparts. Some of the data indicate boundaries between regof linear variation withe of the average pattern wave numb@see Sec. V and Fig. 30~b! below# that is roughly correlatedwith some of the overall pattern dynamics we observe.region I, for V&3 and e&0.1, the pattern consisted ostraight or gently curved rolls that were time independenthe onset of convection~Fig. 8!. These straight rolls werebounded on roughly opposite sides by a short region ocross-roll grain boundary~not to be confused with the crossroll instability!. At slightly higherV, the issue of time de-pendence near onset was complicated by the nucleatio

FIG. 7. Phase diagram in thee-V parameter space forG540ands50.93. The division into regions was done by visual obsvation of the patterns, in some cases supplemented by quantitmeasurements. Symbols representing these determinations ar(L) and SDC (n). Solid lines denote known boundaries wheredashed lines are interpolations or suggestions of boundaries. Ational data for the boundariese1 ande2 for the linear dependence othe average wave number one ~see Fig. 30 below! are shown ascrosses and plusses, with dashed lines to guide the eye.

FIG. 8. Representative patterns for~a! V50, e50.04 and~b!V54.4, e50.04 illustrating straight or slightly curved convectiorolls in region I. For~b!, the overall pattern rotation is counterclockwise, and the short cross rolls travel in the clockwise direction.

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defects at the sidewalls and the interaction of sidewall foing with such nucleations. Images for theG523 cell inwhich much more data were taken for smalle helps to elu-cidate this behavior, but we postpone this discussion ulater. In region II, the patterns were similar to the nonrotatpatterns and had an ‘‘S-shaped’’ ~SS! distortion of thestraight rolls. Representative samples of images in regioare shown in Figs. 9 and 10. The SS distortion appears ta generic response of straight rolls to rotation, as obseralso in experiments at larger Prandtl numbers@17,18# andnoted in GSH simulations@20# for which s is effectivelyinfinite. The degree of distortion increases withV as illus-trated in Figs. 9~d! and 10~a! and 10~d!. An observation re-lated to the SS distortion is that the large sidewall fociopposing sides of the cell were not as symmetrical asones atV50 @Fig. 9~b!#. At V58.8 @Fig. 10~e!#, the asym-metry of the sidewall foci became very pronounced. Onside clockwise from the foci, rolls ended nearly perpendilar to the sidewall whereas on the other side rolls weremost parallel to the sidewall. Since the direction of rotatiwas counterclockwise, it was the side opposite to the rotasense that contained rolls ending perpendicular to the w

The other general aspect seen in region II was the nuation and motions of dislocation defects. The defects wnucleated both in the bulk and near the sidewall and wsometimes identifiable as arising from either the Eckhauskewed-varicose instabilities. An example of a SV nucleapair of defects is illustrated in Fig. 9~a!. There are manyinstances, however, especially at higherV, when the mecha-nism for defect nucleation was not clear, and the only ththat could be determined was whether it was a wave numincreasing or decreasing nucleation. Further, it is not c

FIG. 9. Representative patterns from region II for~a! V50,e50.12, ~b! V50, e50.25, ~c! V50, e50.44, ~d! V53.9,e50.13, ~e! V53.9, e50.29, and~f! V53.9, e50.44. The en-closed region in~a! identifies a SV defect pair.

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whether the concept of laterally infinite long-wavelengthstability applies in the presence of other defects or nearwalls. Thus, we will sometimes refer to specific nucleatimechanisms and at other times to a general defect nucleaof unknown origin. The other type of defect that appearinfrequently fore values that bounded region II from abovwere small spiral defects@Figs. 10~c! and 10~f!#. The appear-ance of these spirals indicates the border with region III.

Spiral-defect chaos, which was observed fore*0.55 atV50 @53,54#, was the distinguishing feature of region IIThe SDC onset forG540 decreased with increasingV, con-sistent with results for a slightly larger cell withG552@63,78#. Patterns illustrative of the SDC state with and witout rotation are shown in Fig. 11. The trends with increasrotation are that the spirals become smaller, that they becpredominantly counterclockwise in their winding sen@78,63# in alignment with the external rotation direction, anthat the spirals become more multiarmed. SpiralsV.8.8 were more angular in appearance@Figs. 11~e! and11~f!#. At V512.1, the pattern contained mostly small targdefects~a locally axisymmetric pattern of concentric rolls!rather than spirals, and spirals were seldom larger than 4d indiameter. Similar patterns full of target defects were oserved at higherV. As there were no longer recognizabspirals, the ‘‘existence of spirals’’ criterion indicates thSDC did not exist forV*13 and this criterion rather arbitrarily marks the boundary between regions III and VI. Othe other hand certain average properties such as the liscaling of ^k& with e described below seemed to continusmoothly across both regions, indicating that the state wmany spiral defects atV50 was transformed continuousl

FIG. 10. Representative patterns from region II for~a!V56.2, e50.15, ~b! V56.2, e50.25, ~c! V56.2, e50.35, ~d!V58.8, e50.16, ~e! V58.8, e50.25, and~f! V58.8, e50.35.Patterns in~c! and ~f! have small spiral defects that indicate ththese parameter values are quite close to the SDC onset.

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into a disordered state without spirals for largeV. The dif-ficulty arises in the label of ‘‘spiral-defect chaos,’’ which bdefinition must include spirals. In any event, the overall pterns in region VI had features reminiscent of the SDC sas shown in Fig. 12. Whether one labels this SDC or nosomewhat a matter of preference.

At intermediateV in region IV, the time dependence neonset consisted of a mixture of KL fronts and dislocatidefects~Fig. 13!. At higher e, the KL instability was sup-pressed, and dislocation defects became dominant in reII. For 8&V&10.5 and near onset, the dynamics werecombination of dislocation-defect motion and KL-fropropagation with the fronts being nucleated near the swall. The KL fronts were initiated by cross rolls at the sidwall and grew into the unstable central region of the celprocess similar to observations by Zhonget al. @16# in aG510 ands56.4 system. Three such fronts are visiblethe images in Fig. 13 with the distinguishing features bethat the fronts are curved outward in their propagation dirtion and that only a small number of fronts are present atsame time. The coexistence of isolated defects and KL frocan be seen in Fig. 13~b!. Another feature not shown herethat KL fronts could disintegrate into isolated defectsdifferent parameter values. Fronts appearing spontaneoaway from the wall were the distinguishing new featureregion V in Fig. 14 forV*10.5. This value is somewhat lesthan the theoretically predicted KL instability thresholdVc513 for s50.93 @3,79#. The result of front nucleationaway from the wall was that the average size of a ‘‘patc~region occupied by neighboring rolls of similar orientatio!was smaller compared to those in region IV where the whcell often appeared as a single patch. The number of pat

FIG. 11. Representative patterns from region III for~a! V50,e50.69, ~b! V53.9, e50.60, ~c! V53.9, e50.81, ~d! V58.8,e50.40, ~e! V58.8, e50.59, and~f! V58.8, e50.82. The arrowin ~b! points to a target defect.

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and dislocation defects grew asV increased. ForV.15, thenumber of patches became quite large, and parts of theterns where two sets of differently oriented rolls coexistappeared cellular, as seen in Figs. 14~c!–14~f!. The cellularregions were not fixed in space as both their size and shevolved via KL fronts, causing different regions to appecellular at different times. The basic characteristicspatches and cellular regions remained the same up tohighest measured value ofV (V520).

At very high e'3, the oscillatory instability was observed forV&12.1 ~data at largerV were not gathered asuch highe values!. The instability caused the appearancetraveling waves that propagated along the rolls@37,54#, andtheir dynamics were similar to observations atV50 with thewaves on rolls that ended at the sidewall traveling towardwall @54#.

The details of the time independence near onset inG540 cell for small rotation rates is complicated by sidew

FIG. 12. Representative patterns from region VI for~a!V512.1,e50.37,~b! V512.1,e50.54,~c! V515.4,e50.30,~d!V515.4, e50.62, ~e! V519.8, e50.43, and ~f! V519.8,e50.60.

FIG. 13. Representative patterns from region IV showing wanucleated KL fronts. ~a! V56.2, e50.07 and ~b! V58.8,e50.06. Notice the localized cross rolls in~a! and ~b!.

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effects fore&0.03. Sidewall forcing in this cell induced several concentric rolls that interacted with the mostly straigroll pattern in the cell interior. In the presence of rotatiothis interaction would often nucleate dislocation defects tproduced some time dependence. Similarly, in regionclose to onset, KL fronts would begin from sidewall-defestructures. One is tempted to speculate that in the absenwalls, the interior structure would be time independeSome support for this conjecture is obtained from expments using the cell withG523. In that cell there was aslight radial variation ofe ~see Sec. II! that caused the convection pattern to form first at the center where the effecte was 2.4% higher than near the wall. This had the conquence of reducing sidewall effects near onset, albeit atprice of a ramp in e. Some representative images fG523 are shown in Fig. 15. In Fig. 15~e!, a time-independent pattern very close to onset (e50.01) and at thefairly large V511.5 is shown with SS rolls in the interioand no visible convection near the sidewalls. With thisduced sidewall influence the phase diagram at lowe is modi-fied as shown in Fig. 16. In Fig. 15~f!, another pattern veryclose to onset withe50.001 andV512.0 shows interior KLfronts, again with no rolls near the sidewall. Note that,contrast to the continuous transition from sidewall KL fronto interior generation of KL boundaries seen forG540, theonset of the KL instability as a function ofV for G523 ismuch sharper and in quite good agreement with the theoical prediction ofVc512 for s50.83. From these resultsis apparent that defects nucleated near the sidewall habig impact on the overall pattern dynamics near onset. Sowhat above onset, however, the patterns observed in

FIG. 14. Representative patterns from region V for~a!V512.1,e50.06,~b! V512.1,e50.20,~c! V515.4,e50.05,~d!V515.4, e50.20, ~e! V519.8, e50.05, and ~f! V519.8,e50.18.

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smaller aspect ratio~Fig. 15! are quite similar to those foG540.

B. Skewed-varicose instability

The SV instability dominates low-s convection and leadsto wave-number selection through the nucleation of dislo

FIG. 15. Representative patterns atG523 for ~a! V54,e50.07, ~b! V54, e50.10, ~c! V58, e50.10, ~d! V58,e50.13, ~e! V511.5, e50.01, ~f! V512, e50.001, ~g! V512,e50.04, and~h! V514, e50.03.

FIG. 16. Phase diagram in thee-V plane for G523 ands50.83. The symbols indicate actual points at which the instabties appeared whene was increased: KL (d) and SV (L). Thedashed line indicates an approximate boundary for the instabili

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tion pairs. Mean drift plays a crucial role in the pattern dnamics that result. For our system, the SV instability is oserved frequently forV50 but becomes less obvious asV isincreased. Nevertheless, rotation has a definite effect ontails of the SV instability, and one observes SV defect-pnucleations throughout region II. Here we consider charterizations of the SV instability with and without rotationwhich includes the rate of dislocation-defect separation,distortion area, and the SV angle.

In the theory of this instability@40,73#, there is an angleassociated with the small-amplitude, long-wavelength perbation that has componentsqx andqy where the wave vectoof the straight rolls is in thex direction. ForV50 the anglearctan(qx /qy) is about 45° ate'0.1 and decreases slowlwith increasinge. In the presence of rotation, the angleexpected to decrease asV is increased at constante @64#. Inexperiments, the manifestation of the SV instability is a pof dislocation defects nucleated along a line that is orienat an angle with respect to the local straight-roll wave-vecdirection. A comparison can be made between the theoreprediction and experimental results at different rotation raby defining an angle associated with SV defect-pair nucations as illustrated in Fig. 17:u is the angle between thlocal wave vector and the line along which the roll pinchimanifests itself. In the early stages of the defect pair nuation the roll-pinching line is one of small amplitude~noticethe fuzzy region in Fig. 17! whereas later the line connecthe two dislocations. Taking an average over a small numof events fore'0.15 yields the SV angle as a functionV ~Fig. 18!. Also included in the plot as a solid curve is thcalculated angle fore50.1. Becauseu also depends one,the variation ofe between the differentV values might beimportant. Quantitatively, however, thee variation of uyields a shift that is well within the plotted error bars andsuch we ignore it. The rate of decrease of the SV angle wincreasingV seen in the data agrees quantitatively with tcalculation. The magnitude of the angle is systematica

FIG. 17. Representative patterns of the skewed-varicose inbility for ~a! V50, e50.12, ~b! V53.9, e50.19, ~c! V56.2,e50.14, and~d! same as~c! but 6.7tv later.

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lower by about 7°. This could arise from comparing tresults of a calculation valid for infinitesimal amplitude anlarge spatial extent to an experimental observation wlarge-amplitude variations on a small spatial scale. An intesting aspect of the SV instability is that, without rotatiothe dislocation pairs had either a positive or negative anof equal magnitude with respect to the rolls with apparenequal probability. Rotation removed this degeneracy andlected a particular sign of the angle that depended on theof V. This phenomenon is related to the influence of rotaton the motion of isolated defects, which is discussed innext section. Finally we note that the images of Croque~Fig. 22 of Ref.@37#! for V50, s50.69, ande50.085 sug-gest an angle of about 90°, where the defects move apaa nearly pure climbing motion. This behavior would be epected for the Eckhaus instability.

In the absence of rotation, SV-nucleated defects ocalmost exclusively near the center of the convection cell.characterized such events away from sidewalls in two waFirst we took the difference between two images, one shing an SV event like Figs. 19~a! and 19~d!, and another with-out the SV distortion taken 9tv earlier. The difference im-ages corresponding to Figs. 19~a! and 19~d! are shown inFigs. 19~b! and 19~e!. The area that is influenced by distotions from the SV event is readily determined. It is showna function of time as solid circles in Fig. 20, where it canseen to grow essentially linearly with time. The second chacterization was a measurement of the distance betweentwo defects generated by the SV event as a function of tiThis was done by demodulating the Fourier transform soto remove the mode with the dominant wave vector froboth the image with the defect pair and the reference imtaken earlier when there was no distortion from the Sevent. The demodulated transforms still showed considerstructure due to the overall roll curvature. Nevertheless,ratio of the moduli of these two demodulated transforgave a sharp determination of the defect-core locationshown by Figs. 19~c! and 19~f!. The distance between thcores is easily measured, and is also plotted in Fig. 20.

With rotation, dislocation nucleations were concentraat the sidewall and became much less frequent in the inteIn Fig. 21, we show a time sequence of images that ill

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FIG. 18. The skewed-varicose instability angleu vs V for enear 0.15 (V50, e50.11; V50, e50.19; V50, e50.13;V50, e50.16). The data (d) were measured as shown in Fig. 1and the solid line is the theoretical prediction fore50.1.

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trates the nucleation of a dislocation defect near the wala complicated process that increases the local wave numAlso shown starting on the far right-hand edge is the appance of an extra roll via an Eckhaus-like instability thatcreases the local wave number.

C. Pattern rotation and dislocation-defect motion

In variational systems, dislocation defects move primaby climbing along the roll direction as opposed to glidin

FIG. 19. Images~a! and ~d! are forV50, e50.11. They areseparated in time by 1.8tv . The differences between a referenimage taken 9tv before~a! and images~a! and~d! are shown in~b!and~e!, respectively. Complex demodulation of images~a! and~d!normalized by the complex-demodulated reference imageshown in~c! and ~f!, respectively.

FIG. 20. Roll-distortion area and dislocation-defect separaas a function of time forV50, e50.11. The left axis correspondto data for the distortion area (d) and the right axis to the distancbetween dislocation defects (n).

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along the wave-vector direction@46#. In nonvariational sys-tems such as low-s convection and/or rotating convectiongliding motion can occur. A recent observation@58,67# inlarge-s GSH and Boussinesq-convection simulations oftating convection was that dislocation motion in rotaticonvection for larges occurred mostly by glide motion, tharotation picked a preferential direction for that motion, athat the defects provided a mechanism for overall pattrotation. In our system, dislocation defects were observemany of the regions inV-e parameter space. They are, however, clearly delineated from other dynamics only in regiII. Much of our work on dislocation-defect motion in thiregion will be presented elsewhere@80# but here we includea summary of the observed behavior. We address two toraised in simulations, namely, the net rotation of the mewave-vector direction and the selection of a unique propation direction for isolated defects.

One very noticeable feature of the SS patterns in regiois that their mean wave-vector direction rotates. Simulatioyielded a rotation rate of the overall pattern that was proptional toV and suggested that gliding defects were respsible for this rigid rotation@67#. We measured the patterrotation rate for several values ofV. Our results for thepattern rotation ratev in units of milliradians pertv areplotted versuse in Fig. 22. ~We chose to use the verticadiffusion timetv for the time scale although it is not obviouthat it is relevant.! By interpolating these data, the variatioof v as a function ofV is obtained as shown in Fig. 23. Acorresponding calculation based on numerical simulatiwould be interesting for comparison.

The other very visible effect of rotation on the motionsdislocations throughout the cell but especially noticeablethe interior was that their direction of propagation was detmined by the direction of rotation. In Fig. 24, a patternshown with several dislocations and the paths of thoselocations. For this case, the rotation vectorVW is out of~clockwise! the plane of the image. We define a vectorbW ,which is parallel locally to the convection rolls. The diretion of bW is along the extra roll of the dislocation towards thsingularity as illustrated in Fig 24. Another way to descrithe dislocation is by its topological charge

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FIG. 21. Formation of defects at the side wall fore50.13 andV53.9. The images are 40tv apart. Two defects traveling in opposite directions merged between~g! and ~h!.

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n51/2prdefect(cw)dQ, whereQ is the phase of the compleconvection roll amplitude with wave numberk. We use thevectorbW instead of deriving a direction fromn because theformer is clearer and yields an unambiguous direction. Tdirection of the glide-velocity vector~the component of ve-locity along the localkW ) is then given by the direction oVW 3bW . We did not see any dislocations that violated thpropagation direction for the rotation rates we investigatV.3.9. Overall the motion of defects was complicated wa variety of speeds and relative ratios of climb and glide tdepended on the local pattern texture and the distance odislocations from the sidewalls. A systematic study of teffects of rotation on dislocation motion in straight-roll paterns similar to those done without rotation@81# would beextremely valuable.

V. CHARACTERIZATIONS OF AVERAGE QUANTITIES

The pattern textures and their spatiotemporal dynamcan be characterized to some extent by averages over mimages of suitably defined parameters. The propertieswe extracted from the images by either Fourier techniquespecialized algorithms as discussed in Ref.@54# are the av-erage wave numberk&, the correlation lengthj, the roll

FIG. 22. Pattern rotation frequencyv vs e for V53.9 ~circles!,V56.6 ~triangles!, andV58.8 ~squares!. The solid lines are guidesto the eye.

FIG. 23. v vs V interpolated from data in Fig. 22. Values oe are given in the figure. The solid and dashed lines are guidethe eye. The dashed line indicates less certainty in the interpola

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curvatureg, and the sidewall obliquenessb. The roll curva-ture and sidewall obliqueness are useful for smallV&8 ~re-gions I, II, and III! but yield little information about cellularpatterns at higherV ~regions IV, V, and VI!. The correlationlength provides some insight into the intermediate regioand the transition to the KL state. Finally, the average wanumber is useful for highV where several regimes of lineadependence one were observed. We consider these quanties in order of their utility for increasingV.

A. Roll curvature and sidewall obliqueness

The average curvatures are plotted in Fig. 25~a!. ForV50, the main features of the data are an increase ofgs,t ataboute50.1, which reflects the appearance of sidewall fothe relatively flat region 0.1,e,0.5, and the large increasbeyond e'0.55, which is related to the onset of SD@53,54#. The data with rotation do not show an increasesmalle but do have the flat dependence at intermediatee andthe rise that indicates the SDC state. Another way to consthese data is to construct contours ofgs,t as a function ofV at fixed e, determined from the data in Fig. 25~a! andshown in Fig. 26. The accumulation of defects near the swall contributed significantly togs,t atV50 and resulted invalues comparable to those atV58.8 fore&0.4. These side-wall defect clusters were much smaller and appearedoften forV.0. On the other hand, the increase ings,t for4,V,9 and 0.1&e&0.4 was caused by the SS distortioton.

FIG. 24. Trajectory of some defects atV56.2 ande50.13. Thebox in the image is expanded above to show details of the

vectors describing the directions of the dislocationb, of rotation

about a vertical axisV, and of the glide direction of the dislocatio

V3b. Also shown is the decomposition of the dislocation veloc

vectorvW into glide and climb components. Note that by definitiothe glide direction is along the local wave-vector direction.

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of the rolls taking on the ‘‘S’’ shape as seen in Figs. 9 an10. The depressed onset for SDC as a function ofV wasresponsible for the largergs,t at higherV for 0.4,e,1.Whereas the SDC onset wase'0.55 for zero rotation, itdecreased to almost 0.5 forV53.9 and 0.4 forV56.2. Theslopes ofgs,t as a function ofe after the onset of SDC werapproximately independent ofV as seen in Fig. 25, evethough the size of the spirals decreased for largerV. This issurprising at first glance because, averaged over a singleral, the curvature would be larger for a small spiral thana large one and, additionally, small spirals could be pac

FIG. 25. ~a! Average curvaturegs,t and ~b! average sidewallobliquenessbs,t vs e for V 5 0 (h), 3.9 (3), 6.2 (d), and 8.8~1!.

FIG. 26. Average curvaturegs,t vs V for the values ofe indi-cated in the figure interpolated from data in Fig. 25~a!. Solid linesare guides to the eye.

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more densely than large ones. These factors were ofhowever, by the straighter rolls between the spirals andmore angular appearance of the spirals at larger rotationso the average curvature at the same distance from theonset remained approximately independent ofV.

The sidewall obliqueness data are presented in Fig. 25~b!.Unlike the curvature, the obliqueness atV50 was alwayssmaller than that at the samee under rotation. Below theonset of SDC, this is so because the rolls of the SS patend less perpendicular to the wall forV.0. Above the on-set, the smallerbs,t was due to a decrease of the sizesidewall foci with increasingV. For V58.8 ande&0.2,bs,t was smaller than atV56.2. The appearance of KLfronts at the sidewall caused rolls to be more perpendicto the sidewall, sobs,t was smaller. At largere where KLfronts were rarely observed,bs,t became larger again because of the increased curvature of the ‘‘S’’-shaped patterns.

B. Correlation length

The average correlation lengthj calculated from thestructure factor of pattern sequences has been shown tovide a simple, well-defined measure of the spatial orderthese sequences@27,47,54,61#. Previously, we used thecorrelation-length analysis to determine spatial scaling inKuppers-Lortz regime~region V! at smalle,0.2 @27#. Herewe apply it also to aid in the characterization of pattern orin regions II and III where weak rotation effects are seen.Fig. 27,j is shown as a function ofe. Generallyj decreaseswith e with the exception of the bumps in the rang0.2,e,0.5 forV5 8.8 and 10.5. For higher and lowerV,j decreases monotonically withe. The bumps are well cor-related with the development of the SS distortion seenregion II of Fig. 7. More insight is obtained by plotting contours ofj as a function ofV, as shown in Fig. 28 where thdata were interpolated from data in Figs. 27 and 29. At fix0.2&e&0.5 and forV,10,j initially increases withV con-sistent with an increased ordering resulting from the devopment of theS shape. AboveV'12 for that samee range,j falls quickly, indicating the transition to the cellular regimgenerated by the Ku¨ppers-Lortz instability.

ForV50, the dependence ofj on e was first studied byMorris et al. @47# who showed that a power-law scaling o

FIG. 27. Correlation lengthj vs e for the values ofV indicatedin the figure. The data were smoothed slightly.

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the formj}e2x with x521/2 was consistent with their datover roughly a decade ine between 0.1 and 1. Our measurments @27# for V.0 showed an exponent value betwe0.20 and 0.25 in region V. In Fig. 29, we show datalog-log scales for three values ofV and for values ofe thatspan regions V and VI. The data for the different rotatirates fall roughly on parallel curves in each of two distinregimes. Fore&0.1 they show again the small exponent rported earlier. However, for 0.3&e&1 the slopes given bythe data are consistent with an effective exponent of ab3/4. This latter result is larger than theV50 exponent@47#but smaller than the exponents of between 1 and 3/2 sgested recently by GSH simulations fors52 andV522 and28 @23#. The two separate effective scaling ranges aretriguing, but their origin is completely unclear. On the baof the general structure of Ginzburg-Landau equationsexponent value of 1/2 might have been expected closonset. Fore of order 1, we do not know of any prediction

FIG. 28. Correlation lengthj vsV for values ofe as indicated.Data were interpolated from data at fixedV. Solid lines are guidesto the eye.

FIG. 29. Log-log plot of correlation lengthj vs e forV514.3 (s), V516.5 (n) andV519.8 (1). For comparison,the solid lines indicate power-law behavior with exponents ofand 3/4.

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C. Average wave number

The average wave number^k& is a reflection of the dif-ferent wave-number-selection processes that competecomplicated textured pattern. Only in very ordered pattesuch as an axisymmetric one of concentric rolls is the wanumber-selection process unique. In previous studies withrotation, we considered the wave-number selection for asymmetric patterns@50# and the more complex situation fotextured patterns@54#. In textured patterns there is a distrbution of wave numbers that can be partially characteriby ^k&, with ^k& still uniquely selected by the prevailintextures. One interesting feature of the selection at smaefor G540 was that the wave number initially increased bfore turning back towards low wave number fore*0.1. Wenow consider the effects of rotation on the behavior ofaverage wave number. The wave numberskc(V) at onsetdetermined by linear extrapolation of^k& for small e toe50 were presented in Sec. III.

The variation of k& with e is shown in Fig. 30~a! for V50, 3.9, 6.2, and 8.8. The feature of increasing wave numat smalle is maintained in the presence of rotation exceptV58.8 where there is an initial decrease followede'0.1 by an increase ink& and finally a decreasing wavnumber fore*0.2. ForV50, 3.9, and 6.2, the variation o^k& is about the same fore&0.30, at which point the data foV56.2 break off and move more quickly to lowerk. Simi-larly, at e'0.45 the data for 3.9 move sharply to lowerk.This trend towards lower wave number is consistent withappearance of the SDC state and is also reflected in thecurvature@Fig. 25~a!#. At higher e and after the jump tolowerk, the slope of k& becomes steeper, indicative of weldeveloped SDC@54#. The behavior forV58.8 is probablycaused by KL dynamics in region IV beginning to influenwave-number selection for smalle, followed by dislocationprocesses winning out for a small interval ofe, and finallythe SDC state coming in a little higher. Multiple influencare clearly at work here near the borders of several ofregions of parameter space shown in Fig. 7.

The behavior of k& at higherV is quite different fromthat forV&9. Figure 30~b! shows^k& as a function ofe forV510.5,15.4, and 19.8. Common to all three data setsthe two linear sections, one near the onset and the othee*0.5. The straight lines in Fig. 30~b! are fits to the twolinear sections, and they highlight the differences amongplots, namely, the rounding at the transitional region frolow to high e. The origin of the linear sections and throunding is unclear, but their properties show a systemdependence onV. At V510.5, ^k& deviated from its linearbehavior near onset fore'0.15; it resumed its linear dependence one at e'0.45 with a different slope. Over a largrange ofe, the values of k& were larger than extrapolationbased on either of the two linear regions. This roundingthe transition between the two sections decreased asV in-creased. AtV519.8, there was essentially no rounding, a^k& varied linearly withe but with two different slopes, onefor e&0.36 and another fore*0.36. In Fig. 30~b!, the quan-tities e1 ande2 are the points at whichk& deviated from thelinear fits for small and largee, respectively. The changes oe1 and e2 with V are summarized in Fig. 31~a!. Whereas4

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e2 did not vary in a simple way,e1 increased linearly withV.

The values of k& at the end points of the linear regione1 ande2, are denoted ask1 andk2, respectively, and illus-trated in Fig. 30~b!. As can be seen in Fig. 31~b!, the valuesof k2 were all smaller thank1, and the difference betweek1 and k2 decreased with increasingV. The slopesd^k&/de from the fits, plotted in Fig. 32, are negative, rflecting the wave-number decrease withe. The slope at largee did not vary much withV, but near onset it varied considerably.

The problem of wave-number selection for concentpatterns andV.0 was also studied, but in aG541 cell witha ‘‘spoiler tab.’’ Details of the construction and thconcentric-pattern dynamics atV50 are contained in Ref@50#. As shown in Fig. 33~a!, the pattern consisted of axsymmetric rolls coexisting with an annular cross-roll staconfined near the sidewall. AtV50, this state was the preferred pattern above onset and remained stable fore<0.08.Under rotation, the wave number changed, as seen in a cparison between Figs. 33~a! and 33~b!, but the pattern stillremained stable fore<0.08 andV<9.5. ForV.9.5, thepattern could not be maintained at anye as the cross rolls

FIG. 30. Plots of k& vs e for ~a! V50 (d),3.9 (h), 6.2 ~1!,and 8.8 (3), and ~b! V510.5 (d),15.4 (h), and 19.8~1!. Thesolid and dashed straight lines are fits to data at low and higerespectively. The definitions ofk1 , k2 , e1, ande2 are illustratedfor V510.5.

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grew into the axisymmetric rolls in a manner similarsidewall-initiated KL fronts. The destruction of the concetric pattern left a textured pattern similar to the ones ingion IV as in Fig. 13. The experiments were conductedrampingV at constante50.025 from 0 to 9.5 and then bacdown to 0 in steps of 0.25. The measured wave numbepresented in Fig. 34 together with the theoretical curvekc for comparison. The length of the cross rolls changedaccommodate smooth increases or decreases of the wlength of the axisymmetric rolls. The large changes in^k&came from a reversal in the fluid flow direction at the umlicus that accompanied a creation or destruction of one rHysteresis in the selected wave number as a function oV

FIG. 31. ~a! Plots ofe1 (s), e2 (d) and ~b! k1 (s), k2 (d).

FIG. 32. Plots ofd^k&/de vs V, near onset (s) and for highe (d).

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occurred atV'4 andV'7. This is similar to the behavioof concentric patterns in water without rotation@82# wherehysteresis was observed ase changed. The hysteresis hereunusual as k& remained close to 3.2 over a wider rangeV during ramping up than ramping down as seen in Fig.The cause of this behavior is unclear.

VI. PATTERN DYNAMICS

The textures and averaged properties of patterns presein previous sections are the result of complicated patternnamics. In this section, we present pattern sequencesillustrate the dynamics in the regions labeled in tparameter-space diagram of Fig. 7. The richness of thenamics needs to be condensed for presentation here anhave no doubt omitted or missed altogether interesting pcesses that might be important. Nevertheless, the sequepresented do reflect the most obvious characteristics ofseparate regions.

A. Region II: S-shaped patterns and dislocation motion

The dynamics of patterns in region II involved primaridislocation-defect motion within the generallyS-shaped pat-tern. The defect nucleation could be divided into bulk nucation, identifiable with the SV instability, and sidewall nuclations of dislocations. AtV50, the roll pinching typical ofSV usually took place near the central region of the cell,under rotation, more of the roll pinchings occurred near siwall foci. This is illustrated by the time series of imag

FIG. 33. Concentric patterns in aG541 cell at e'0.03: ~a!V50 and~b! V56.25.

FIG. 34. The mean wave number^k& of concentric patterns ae50.025 for increasing (s) and decreasingV (d). The solid lineshowskc from linear stability analysis for comparison.

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taken 40tv apart ate50.13 forV53.9 shown in Fig. 35. Apair of defects was generated when roll pinching occurnear a wall defect at the lower left side of Fig. 35~a!. One ofthe defects glided across the rolls while climbing alongrolls toward the central region of the cell. It eventualmerged with the cross-roll patch at the right side of the cThe other traveled in the opposite direction along the siwall and merged with the left cross-roll patch. The motionthe defects was mainly across the rolls~gliding! with lessmovement along the roll axes~climbing!. This is seen in thepath of the right-traveling defects, which moved acrossmost the entire width of the cell while climbing less than hthe cell width. This is similar to results from a numericsimulation fors5` where the defect motion was found tbe mostly gliding rather than climbing@83#. In Fig. 35~a!, thelower-left sidewall-defect structure near which the ropinching occurred was pinned and remained immobthroughout this sample image sequence, but at other tisuch structures could move and merge with the cross-patch. Figure 35~e! contains an infrequent SV defect-pacreation away from the sidewall, an event resembling occrences atV50. While the defects were formed and subsquently propagated toward the cross rolls, the underlystraight-roll pattern gradually rotated in the direction of rtation. This change in roll orientation was smooth and dnot involve discrete large jumps as would be expected frthe KL instability.

At higherV in region II, the rolls were much more curveand theS shape was accentuated. Many defects were presimultaneously but, unlike atV53.9, defects did not accumulate near the sidewall as in Fig. 9. This time dependewas captured in the image sequence, Fig. 36, takene50.23 andV58.8, spaced 20tv apart. Whereas the nucle

FIG. 35. Time sequence showing defect motion and the skewvaricose instability ate50.13 andV53.9 in region II. The circle in~e! encloses a pair of skewed-varicose-generated defects. Theages are 40tv apart.

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ation of defects took only about 3–5tv , the defect move-ment was slow enough that individual defects can belowed in this sequence. The defects traveled accordingtheir topological charge as discussed earlier. The orientaof the underlying roll pattern still advanced slowly in thdirection of rotation, as expected from similar observatioat smallerV. Infrequent KL fronts were also initiated bdefects at the sidewall. They rarely propagated acrosswhole cell and only affected a portion of the pattern.

B. Regions III and VI: Spiral-defect chaos

The existence of spiral and target defects was characistic of region III. At low V53.9, target defects first appeared neare50.45. They sometimes evolved into spiraand the number and sizes of the spirals fluctuated in tisimilar to the case forV50. At e'0.7 spirals were alwayspresent, but most spirals with diameter larger than 728dwere two-armed. Smaller spirals generally evolved from tget defects or dislocations and were usually one-armed.dynamics were quite fast with continuous creations andstructions of spirals, but the images in Fig. 37 taken 40tvapart ate50.81 captured the evolution of several structuin detail. The arrow in Fig. 37~b! points to a three-ring targethat grew from a small spiral and then became a spiral agin Fig. 37~c!. The spiral then merged with surrounding sprals to form a large two-armed spiral, about 14d in diameter,in Fig. 37~d!. This large spiral was subsequently destroywhen it moved left toward the wall several time steps latThe change of a target defect into a small spiral is alsoible in these images as seen in a comparison between37~a! and Fig. 37~b! of the defect indicated by the arrow iFig. 37~a!.

At e'0.3 sidewall-defect clusters nucleated small tardefects, which stayed within 10d and 12d of the sidewall,

FIG. 36. Time sequence showing defect motion ate'0.23 andV58.8 in region II. The images are 20tv apart.

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and ate'0.4, these target defects sometimes evolved ismall spirals. The patterns then resembled those ate'0.5andV50; an image sequence taken ate50.40 illustratingthe creation and annihilation of small spirals is shown in F38. At even highere, recognizable spirals appeared mooften, and the average number of spirals increased. Laspirals appeared infrequently, as seen in Fig. 39, which isimage sequence ate50.59 spaced 6.7tv apart. Most spiralsmeasured 628d in diameter and were smaller than typicspirals at smallerV. The spirals were more angular as illutrated by the large double-armed spiral indicated by therow in Fig. 39~d!. The average lifetime of a spiral was quishort, about 20tv . The short duration is typified by the smaspiral, indicated in Fig. 39~a!, which was nucleated and destroyed in less than 20tv . Away from the spirals, the rollsjoined in a more angular fashion. This rather angular featof the rolls was reminiscent of the discrete orientatichange caused by the KL instability despite the almost coplete absence of KL fronts fore*0.2. A final interestingphenomenon seen in some images in this region, sevwhite cells in a row, is shown by the arrows in Figs. 39~f!and 40~a!. This pattern may be an indication of local crosroll instability.

For progressively higherV in region III, spirals becomesmaller and less frequent. The last remnants of discernspiral activity was forV512.1. In Fig. 40, fore50.37 andV512.1, a time sequence of images shows several sspirals and some clustered curved defects. Also noticeare lines of cells that are similar to the cross-roll nucleatioseen at highers andV50 @84#. These cross rolls sometimedisappeared without affecting the texture, much as typifiby the ones indicated by the left arrow in Fig. 40~a!. Some-

FIG. 37. Time sequence of spiral-defect chaos ate'0.81 andV53.9 in region III. The images are 40tv apart. Arrows in~a! and~b! point to target defects that evolve into spiral defects.

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times spirals would arise from these cross rolls as shownthe right arrow in Fig. 40.

At e'0.5, where SDC was observed at lowerV, the pat-terns consisted of many target defects and some small spmeasuring 425d in diameter. No spiral larger than 6d indiameter was seen. A time series of images spaced 6tv

FIG. 38. Time sequence of spiral-defect chaos ate'0.40 andV58.8 in region III. The images are 13.3tv apart.

FIG. 39. Time sequence of spiral-defect chaos ate'0.59 andV58.8 in region III. The images are 6.7tv apart. Arrows point to~a! single-arm spiral,~d! double-arm spiral, and~f! localized crossrolls.

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apart, taken ate50.55, is shown in Fig. 41. The dynamicwas faster than captured by this sequence, but typicaltures can be seen in the figure. The target defects were qmobile and occasionally annihilated each other by collisi

For higherV.12.1, the dynamics were complicated, cosisting of many defects and cellular patches, and are impsible to represent clearly with still images. Clearly identi

FIG. 40. Time sequence ate'0.37 andV512.1 in region VI.The images are 6.7tv apart. The arrows point to two cross roregions.

FIG. 41. Time sequence ate'0.55 andV512.1 in region VI.The images are 6.7tv apart.

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able spirals were only observed rarely, but small higcurved defects were present at all times.

C. Region IV: Kuppers-Lortz fronts

The Kuppers-Lortz instability was not observed foV&5 up toe'1 where spiral-defect chaos dominated. Tfirst appearance of the KL instability was atV56.2 ande50.07 where the main time dependence was associwith dislocation motion. ByV58.8, KL fronts occurred neathe onset of convection and were responsible for much ofdynamics, as seen in Fig. 42, which was taken ate50.06.Three KL fronts, labeledA, B, andC, are visible in Fig.42~a!, and dislocation defects are seen in the region wnearly horizontal rolls. The horizontal rolls were unstabwith respect to frontsA andB while the rolls marking frontC were unstable to the horizontal rolls. As a result, froA andB propagated away from the wall and grew into thorizontal rolls, but frontC moved toward the wall. In Fig42~b! front C had reached the wall, and frontB had mergedsmoothly into the horizontal rolls. This smooth merging ocurred frequently in KL front propagation forV&12; itsmain feature involved the growing rolls linking up with thdecaying rolls, eventually forming smoothly curved continous rolls. A line of defects separated the growing fromunstable rolls, as indicated by the arrow in Fig. 42~c!, and thedefects traveled left toward the wall leaving behind smootcurved rolls as in Fig. 42~d!. The same merging phenomenoalso occurred at the end of large orientational changeillustrated by frontA, which was more well defined thafront B. The angles between the growing rolls behind froA and the unstable rolls varied little from one end of t

FIG. 42. Time sequence of Ku¨ppers-Lortz fronts mixed withdislocation defects ate'0.06 andV58.8 in region IV. The imagesare 40tv apart. Arrows in~a! point to different KL fronts. Thearrow in ~c! shows the dissipated remains of KL frontB indicatedin ~a!.

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front to the other, as seen in Fig. 42~b!. In Fig. 42~c!, the leftside of frontA had disappeared, and the two sets of rolls hmerged into curved rolls as sharp angles between thesets of rolls disappeared. The remaining defects at the inface propagated toward the right, leaving behind slighcurved rolls, as seen in Fig. 42~d! and Fig. 42~e!. Even withthis smoothing process, the overall roll orientation schanged by discrete steps, but the changes varied betw30° and 70°. This is fundamentally different from thsmooth angular changes observed at largere, which werecaused by repeated dislocation-defect nucleations. BefrontA had traversed the whole cell, another front was nucated near the wall at the upper right side, and its progresshown in Fig. 42~e! and Fig. 42~f!.

D. Region V: Kuppers-Lortz dynamics

For a spatially uniform straight-roll pattern at the onsetconvection, theoretical calculations predicted the appearaof the KL instability at the critical rotation rateVc'13 forour system@79#. The inclusion of the spatial dependencethe roll amplitude showed that KL fronts could propagateV,Vc provided such fronts exist@19#. The existence ofgrain boundaries near the sidewall resulted in the appearof KL front propagation forV*6, substantially belowVc .At V512.1 ande50.11 the KL fronts initiated by the grainboundaries remained quite dominant in the dynamics, butpattern was always composed of several domains of slarly oriented rolls with different roll orientations from patcto patch as in Fig. 43. In addition to fronts nucleated atsidewall, occasionally new fronts appeared spontaneousthe interior of the cell. One such front is indicated by tarrow in Fig. 43. This spontaneous formation of new fronwas not observed forV&11. More details on region V arepresented elsewhere@27,28#.

VII. CONCLUSIONS

The experimental results presented in this paper fall iseveral categories:~1! the linear results for the onset Rayleigh number and the pattern wave number that agree qtitatively with theoretical predictions;~2! qualitative behav-ior of pattern dynamics that was consistent with theoretiexpectations and numerical simulations;~3! new phenomenaboth qualitative and quantitative, for which no theory hbeen developed so far. The qualitative behavior of patteincluded observations about the form and frequency ofknown long-wavelength instabilities. The skewed-varicoinstability was chief among these and was responsiblemuch of the dynamics for smalle andV. The form of the SVinstability under rotation was studied, and the SV angle afunction of V was determined. Dislocation defects and tS-shaped pattern dominated this region in which slow ovall pattern rotations were observed. The precession frequcies for this slow rotation were determined. Key to undstanding the results near onset was the importcontribution of the sidewall, which served to nucleate bodislocation defects or Ku¨ppers-Lortz fronts, depending oV. This was explicitly illustrated by the different patterdynamics near onset in cells with differentG that had differ-ences in spatial homogeneity. The other striking feature

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smallV ande was the strong polarizing effect of rotation othe direction of dislocation motion and the significant ehancement of glide motions for these defects.

A well-studied state forV50 is the spiral-defect-chaostate. Under rotation, this state dominates for highere butundergoes a continual metamorphosis asV is increased untilthere are no spirals at all but a state of highly curved roreminiscent of the SDC state. The first appearance of spas a function ofe is also affected by rotation with smalleonset values ofe for larger V. This combination of SDCdynamics and rotation is complex and may be hard totangle as it includes mean-drift, local defect structures,

FIG. 43. Time sequence of Ku¨ppers-Lortz-unstable domainmixed with dislocation defects ate'0.11 andV512.1 in region V.The images are 8.9tv apart. Arrow in~c! points to a region where anew domain nucleated away from the boundary can be seen in~d!.

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Kuppers-Lortz dynamics. It is certainly a complicated eample of spatiotemporal chaos in two-dimensional patter

Several quantitative characterizations of wave-numberlection and spatial correlations are not addressed by thebut may be amenable to theoretical analysis. The first isscaling of the correlation length as a function ofe, whichdivides itself naturally into two regions, one near onset wa scaling exponent of about 1/4 and another at intermede where the exponent is close to 3/4. The other very intesting quantitative feature of the data is the linear depdence of the average wave number near onset near and athe predicted criticalVc for the KL instability. Whereas theaverage wave number for smallerV fluctuates on long timescales, the faster dynamics and more cellular structureduced by the KL instability produces an average wave nuber that is very steady in time and has a very linear depdence one.

In summary, we have illustrated some of the richness texists for rotating layers of fluid heated from below. Whave observed local signatures of skewed-varicose, Eckhand cross roll instabilities as well as the global Ku¨ppers-Lortz instability and the state of spiral-defect chaos. A though theoretical study of the stability boundaries as a fution of Prandtl number and rotation rate would be vevaluable. Numerical simulations of these states is anoarea that is just beginning and for which a wealth of coparisons could be made. We hope that our work will stimlate such investigations.

ACKNOWLEDGMENTS

We gratefully acknowledge support from a UC/LANINCOR grant and from the U.S. Department of Energy. Wthank Ning Li, Mike Cross, and Werner Pesch for usediscussions, Eberhard Bodenschatz, John de Bruyn, DavCannell, Stephen Morris, and Steve Trainoff for their conbutions to the design of the apparatus, and Tom CluneEdgar Knobloch for providing us with data on the linestability boundary and on the Ku¨ppers-Lortz critical rotationrate.

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ica

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@69# The cell and the CO2 gas pressure are the same as for Re@53# and @54#. The slightly different fluid properties in thispaper are based on updated CO2 data given in Ref.@68#.

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1704 ~1995!.@79# The first calculation was reported by G. Ku¨ppers in Ref.@3#.

The value quoted in this paper for our particular Prandtl nuber is from T. Clune and E. Knobloch~private communica-tions!.

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convection under rotation for prandtl numbers near 1: linear ...convection under rotation for...

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