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The Visualization of Quantum Turbulence in Superfluid 4He

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Title:
The Visualization of Quantum Turbulence in Superfluid 4He
Creator:
Marakov, Alexander
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (74 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
IHAS,GARY G
Committee Co-Chair:
STANTON,CHRISTOPHER JAY
Committee Members:
TANNER,DAVID B
BALACHANDAR,SIVARAMAKRISHNAN
Graduation Date:
5/2/2015

Subjects

Subjects / Keywords:
Counterflow ( jstor )
Excimers ( jstor )
Fluids ( jstor )
Heat flux ( jstor )
Helium ( jstor )
Ionization ( jstor )
Laser beams ( jstor )
Lasers ( jstor )
Turbulence ( jstor )
Velocity ( jstor )
Physics -- Dissertations, Academic -- UF
helium-4 -- lif -- superfluid -- turbulence
Genre:
Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
Physics thesis, Ph.D.

Notes

Abstract:
Past visualization experiments in superfluid 4He counterflow were severely limited to low heat fluxes to generate reliable data. Ambiguous coupling between micron-sized tracer particles and both components of the superfluid were a consequence of particle size and the scale of investigation. Thus, definite conclusions regarding the behavior of normal fluid flow and normal fluid turbulence could not be made. In light of this, we have developed a novel visualization technique based on the generation of a thin line of He2 excimer traces via femtosecond-laser field-ionization in order to investigate the motion of the normal fluid component of He-II in counterflow. Studying the drift and distortion of the tracer lines in laminar and turbulent flows, we discover a heretofore unpredicted flow profile in the transition from laminar to turbulent flow. Preliminary evidence is presented to suggest that distorted laminar region is related to the TI/TII transition in superfluid turbulence. Further, velocity PDFs and structure functions are calculated for fully turbulent normal fluid flow. We find no power law tails are present in the velocity PDFs and determine that the turbulent energy spectrum for steady state counterflow is given by E(k) is proportional to k-2 rather than the classical k-5/3 power law. ( en )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
General Note:
Adviser: IHAS,GARY G.
General Note:
Co-adviser: STANTON,CHRISTOPHER JAY.
Statement of Responsibility:
by Alexander Marakov.

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UFRGP
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Applicable rights reserved.
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LD1780 2015 ( lcc )

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THEVISUALIZATIONOFQUANTUMTURBULENCEINSUPERFLUID4HEByALEXANDERMARAKOVADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015AlexanderMarakov

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Tomyfather

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ACKNOWLEDGMENTS Foremost,Imustthankmyadvisor,Dr.GaryIhas,forhavingthefaithandcouragetoallowmetopursueaPh.D.projectfarremovedfromhisareasofexpertise.Garyhasalwaysbeenreadywithadviceorananecdote(oftenboth!)wheneverIranintotrouble(notuncommon)withcryogenicsorelectronics.And,perhapsmostimportantly,Garyallowedmethefreedomtomakemistakes,whichwereofteninstructive,butthenwouldappearjustintimetohelpmeavoidtotaldemoralizationanddisaster.TheworkpresentedhereinwasperformedattheNationalHighMagneticFieldLaboratoryinTallahasseeincollaborationwithJianGaoandunderthesupervisionofDr.WeiGuo.Thus,mythanksgoouttoDr.StevenvanSciverforpermittingmetobeamemberofhislabforthedurationoftheseexperiments.EnormousthankstoDr.Guonotonlyforhispatience,guidance,andadviceduringtheseexperiments,butalsoforteachingmealmosteverythingIknowaboutthepracticaluseoflasersandopticsinthelaboratoryduringtheearlydaysofthisexperiment.Enormousthankstomylabmate,JianGao,withoutwhomthelongnightsinlab,seeminglyendlessequipmentrepair,andunceasingbeamalignmentwouldhavebeenunbearableandimpossible.AlsothankstoErnestoBosque,JoeHurd,RamDhuleyandMarkVanderlaanfortheirfriendshipandtechnicaladviceduringmytimeattheCryolab.TherstattemptsatperformingtheseexperimentsoccurredintheReitzelabattheUniversityofFlorida.Thus,ImustthankDr.DavidReitzeforallowingustousehisfemtosecondlaser.Dr.VidyaRamanathandeservesmyendlessgratitudefortheextensiveadviceandassistanceshegavetomyself,Gary,andWeiwhenwewerecomingtogripswiththeprinciplesoffemtosecondlasersystems.ManythankstoDr.DanielMcKinseybothforhisadviceduringthecourseoftheseexperimentsandforloaningmuchoftheequipmentdescribedwithin.Thisexperimentwouldtrulyhavebeenimpossiblewithouthim. 4

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ThankstoDr.William`Joe'Vinenforthemanyfruitfuldiscussionsandexperimentalsuggestions.Consistently,IwouldndthattalkingtoJoehelpedmetourexperimentintothebroadercontextofquantumturbulencework.Additionally,IamgratefulforJoe'scommentsonthisdissertation,withoutwhichitwouldbeamuchpoorerdocument.Ofcourse,noneofthiswouldhavebeenpossiblewithoutnancialsupportfromtheUniversityofFlorida,theUFDepartmentofPhysicsandCenterforCondensedMatterSciences,theNationalScienceFoundation(DMR)]TJ /F1 11.955 Tf 9.3 0 Td[(1007974andDMR)]TJ /F1 11.955 Tf 9.29 0 Td[(1007937),FloridaStateUniversity,theNationalHighMagneticFieldLab,andtheUSDepartmentofEnergy(DE)]TJ /F1 11.955 Tf 9.3 0 Td[(FG02 96ER40952).Additionally,IamgratefulforfundingfromtheUFCollegeofLiberalArtsandSciencesandtheLT24GraduateStudentTravelFundforsupportingtraveltoInternationalMeetings.ThankstoJiheeYang,whoonmanyoccasionscoatedtransducersformeandalsoprovidedhelpfulfeedbackduringmypracticetalks.ThankstoBillMalphursandMarkLinkoftheUFmachineshop,whonodoubthaveseenenoughofmetolastalifetime.ThankstoPeteAxsonandDanEkdahloftheUFelectronicsshopandAndyPowellandEricStiersoftheNHMFLelectronicsshopwithoutwhoseconsiderableeortsIwouldbeburiedunderamountainofdeadinstrumentsandhalf-improvisedelectronicmonstrosities.ManythankstoGreggLabbeofUFCryogenicsserviceswithoutwhomIwouldn'tknowaheliumtransferifithitmeintheface.Greggalsotaughtmeagreatdealaboutvacuumsystems,leakcheckingandcryogenicplumbing.He,alongwithJayHorton,assembledtheheliuminfrastructureforourrstattemptsatthevisualizationexperiment.Specialthankstomyfriends,bothnewonesI'vemadehereinGainesvilleandtheoldonesscatteredacrosstheUnitedStates.Withoutyoulisteningtomyincessantgrumblingabout`lasers'and`mechanisms',Icertainlywouldhavequitalongtimeago.And,ofcourse,Imustthankmyfamily,whosesupportandcondenceinmehasneverwavered.Nodoubt,theyarequitereadyformetoendmylongexileinFlorida! 5

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Especially,Imustthankmygrandmother,whohasdoggedly(andsometimespointedly)kepttrackofmyprogresssincedayoneofgraduateschool.Shehasexperiencedmorethanherfairshareoftriumphanddespairinthelaboratory,andhasalwaysbeenreadytolendasympatheticearorawordofadvicethroughoutmygradschoolcareer.LastlyImustthankmyfather,towhomthisworkisdedicated,thoughhewillneverseeit.Myfathertaughtmethevalueofpatience,kindness,dedication,andagoodsenseofhumor.Withouthim,IwouldnotbethemanIamtoday. 6

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 11 1.1TurbulenceandLiquidHelium ......................... 11 1.1.1SecondSound .............................. 13 1.1.2QuantumTurbulence .......................... 16 1.1.3ThermalCounterow .......................... 17 1.2Visualization .................................. 20 1.2.1HeliumExcimers ............................ 22 1.2.2PenningIonization ........................... 24 2EXPERIMENTANDPROTOCOL ......................... 26 2.1ExperimentDesign ............................... 26 2.2Laser-eldIonizationinHelium ........................ 34 2.2.1IonizationThreshold .......................... 35 2.2.2Laser-inducedheating .......................... 37 2.2.3SecondSound .............................. 39 3DATAANALYSIS .................................. 43 3.1ImageProcessingAlgorithm .......................... 43 3.22ndSoundAttenuation ............................. 47 4STEADYSTATECOUNTERFLOW ........................ 49 4.1FullyTurbulentFlow .............................. 49 4.1.1TheNormalFluidVelocity ....................... 49 4.1.2TheSecondOrderTransverseStructureFunction .......... 53 4.2Laminar,DistortedLaminarFlowandtheTI/TIITransition ........ 55 5CONCLUSION .................................... 62 APPENDIX:VELOCITYPDFS ............................. 67 REFERENCES ....................................... 70 BIOGRAPHICALSKETCH ................................ 74 7

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LISTOFFIGURES Figure page 1-1FluidComponentFractions ............................. 13 1-22ndSoundVelocity .................................. 16 1-3EnergylevelsofHe2 ................................. 23 2-1SchematicofExperimentalChannel ......................... 29 2-2OpticalBenchLayout ................................ 31 2-3Measuredfsbeamprole ............................... 32 2-4SteadyStateCounterowTiming .......................... 33 2-5GeneratedExcimerLines ............................... 36 2-6IonizationLightIntensity .............................. 37 2-7IonizationThreshold ................................. 38 2-8LaserHeatingData .................................. 38 2-92ndSoundTransducer ................................ 39 2-102ndSoundWiring ................................... 40 2-112ndSoundFrequencySweep ............................. 42 3-1ImageBinning .................................... 45 3-2ImageAnalysis .................................... 46 3-32ndSoundResonanceAttenuation ......................... 48 4-1TurbulentExcimerLineDistortions ......................... 50 4-2MeanNormalFluidVelocity ............................. 51 4-3TurbulenceIntensity ................................. 52 4-4VelocityPDFs ..................................... 53 4-5SecondOrderTransverseStructureFunctions ................... 54 4-6DistortedLaminarFlow ............................... 56 4-7PoiseuilleFlowFit .................................. 57 4-8TI/TIITransition ................................... 59 8

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4-92ndSoundAttenuation ................................ 60 A-1180mW/cm2VelocityPDF .............................. 67 A-2200mW/cm2VelocityPDF .............................. 68 A-3243mW/cm2VelocityPDF .............................. 68 A-4275mW/cm2VelocityPDF .............................. 69 A-5350mW/cm2VelocityPDF .............................. 69 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEVISUALIZATIONOFQUANTUMTURBULENCEINSUPERFLUID4HEByAlexanderMarakovMay2015Chair:GaryG.IhasMajor:PhysicsPastvisualizationexperimentsinsuperuid4Hecounterowwereseverelylimitedtolowheatuxestogeneratereliabledata.Ambiguouscouplingbetweenmicron-sizedtracerparticlesandbothcomponentsofthesuperuidwereaconsequenceofparticlesizeandthescaleofinvestigation.Thus,deniteconclusionsregardingthebehaviorofnormaluidowandnormaluidturbulencecouldnotbemade.Inlightofthis,wehavedevelopedanovelvisualizationtechniquebasedonthegenerationofathinlineofHe2excimertracesviafemtosecond-lasereld-ionizationinordertoinvestigatethemotionofthenormaluidcomponentofHe-IIincounterow.Studyingthedriftanddistortionofthetracerlinesinlaminarandturbulentows,wediscoveraheretoforeunpredictedowproleinthetransitionfromlaminartoturbulentow.PreliminaryevidenceispresentedtosuggestthatdistortedlaminarregionisrelatedtotheTI/TIItransitioninsuperuidturbulence.Further,velocityPDFsandstructurefunctionsarecalculatedforfullyturbulentnormaluidow.WendnopowerlawtailsarepresentinthevelocityPDFsanddeterminethattheturbulentenergyspectrumforsteadystatecounterowisgivenbyE(k)/k)]TJ /F2 7.97 Tf 6.59 0 Td[(2ratherthantheclassicalk)]TJ /F2 7.97 Tf 6.58 0 Td[(5=3powerlaw. 10

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CHAPTER1INTRODUCTION 1.1TurbulenceandLiquidHeliumTurbulence,likegravity,isoneofthemostubiquitousandrelevantphysicalphenomenathatweexperienceonadailybasis.Itmanifestsconstantlyduringairandseatravel;itisvisibleinsinks,waterfalls,swimmingpoolsandoceans.Yetturbulenceremainsoneofthelastbastionsofunsolvedphysicsdueentirelytothenon-linearityintheNavier-Stokesequationswhichgovernuidow, @~u @t+~ur~u=rp+r2~u)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2 3r(r~u)+~g(1{1)whereisuiddensity,~uisvelocity,pispressure,isdynamicviscosityand~gistheaccelerationduetogravity(or,moregenerally,anoutsideforce).Thenon-linearityoftheNavier-Stokesequationcomesfromtheterm~ur~u.Itisthistermthatisresponsibleforthetransferofkineticenergydownlengthscales,alsoknownastheturbulentcascade.Indeed,whileturbulenceisdenedasachaotic,dissipativephenomenon,thedissipationinclassicaluidsoccursonlyatwhatiscalledtheKolmogorovlengthscale,denedas)]TJ /F2 7.97 Tf 6.59 0 Td[(1=43=4whereistheuxofturbulentkineticenergyperunitmassink-spaceandisthekinematicviscosity.AbovetheKolmogorovscale,inwhatiscalledtheinertialrange,energyissimplytransferredfromlargerlengthscalestosmallerones.Theinertialrangeisboundedabovebytheinjectionscaleatwhichenergyisinjectedintothesystem.Itisalsonaturaltowonderhowtheenergyinthesystemisdistributedacrosslengthscales.Thatistosay,whatistheenergyspectrum,E(k)oftheturbulence?Bydimensionalanalysis,KolmogorovfoundthatclassicalturbulenceshouldhaveenergyspectrumE(k)/k)]TJ /F2 7.97 Tf 6.59 0 Td[(5=3wherekisthewavevector.Strikingly,everyturbulentlyowinguideverinvestigatedexperimentallyhasexhibitedanenergyspectrumobeyingthepowerlawthatKolmogorovpredicted.Thuswhileexactanalyticsolutionsformotionofturbulentuidarecurrentlyoutofthereachofscience,wearestillabletodetermine 11

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manycharacteristicsofturbulentow.Onemightalsoconsiderwhetherauidwithverysmallorevenzeroviscositycouldbeturbulent.Asitturnsout,liquidhelium,whencooledbelow2:17K,exhibitssuperuidity,whichistosay,itsowbecomesirrotational.Althoughheliumwasrstliquiedin1908byKamerlinghOnnes[ 1 ],itwasnotuntil1938thatKapitza,AllenandMisenerindependentlydiscovereditssuperuidproperties[ 2 , 3 ].ThetwouidtheorythatdescribessuperuidheliumwasconceivedbyTisza[ 4 ]anddevelopedbyLandau[ 5 ]explicitlytoprovideatheoreticalexplanationfortheunusualbehaviorandpropertiesofsuperuidhelium(frequentlyreferredtoasheliumII).Forthesakeofbrevity,thisdocumentwillrestrictitselftoabriefdescriptionofthetwouidtheory,secondsound,andthermalcounterow.AnexcellentsurveyofearlyexperimentalresultsinheliumandthedevelopmentofliquidheliumtheorycanbefoundinRef.[ 6 ].Inshort,thetwouidtheoryofheliumIIpostulatesthatthesuperuidisactuallycomposedoftwo,non-interacting,intimatelymixedcomponents.Thesetwopartsaredubbedthesuperuidandthenormaluidwithdensitiessandn,respectively.Asexpected,thedensitiesofthecomponentsmustsumtothedensityofthebulkuid(s+n=).Additionally,eachcomponenthasitsownvelocityeld,~vsand~vn.Thesuperuidcomponentisdenedbyitsirrotationality(rvs=0),andwasfoundtohavenoviscosityandnoentropy,thusitdoesnotcarryanyheat.Indeed,theaptlynamednormaluidisconsideredtolargelyactasaclassicaluid,asithasalloftheentropyofthebulkliquidandalsoanon-zeroviscosity.Strictlyspeaking,thenormaluidisactuallyconsideredtobeagasofelementaryexcitations(phonons,rotons,maxons)andimpuritiesintheliquid,butthisisaconceptualclaricationwithlittlebearingonthephysicsofthetwouidmodel.Fig. 1-1 showsthesuperuidandnormaluidfractioninheliumIIasafunctionoftemperature[ 7 ].Themainbodyofworkdescribedinthefollowingchaptersisperformedatatemperatureof1:830Katwhichs n=2 3. 12

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Figure1-1: Thesuperuidfraction(s=)andnormaluidfraction(n=)asfunctionsoftemperature.Abovethetransition,intheheliumIstate,theentireuidmustbe`normal'(classical).Belowthetransitionthesuperuidfractionincreasesrapidly.Fortemperaturesbelow1:0K,theentireliquidisessentiallyinthesuperuidstate.[ 6 ] 1.1.1SecondSoundAdirectconsequenceofthetwouidmodelisthepossibilityforasecondmodeofwavepropagation.Regularsound,orrstsoundinheliumII,isdescribedasthecompressionandrarefactionoftheentireuidandsomustbeapressurewave.Inrstsound,boththenormalandsuperuidcomponentsmustmovetogetherinphase.However,sincevnandvsareindependentvelocityelds,thereisnoapriorireasonthattheuidcomponentsshouldalwaysmoveinphase.Insituationswherethecomponentsmoveoppositeeachother,oroutofphase,thereareregionswherethelocaldensityofnormaluidishigherthansuperuid(sotemperatureisgreaterthantheaveragetemperatureofthebulkliquid)andviceversa.Thissortofuidowcongurationresultsinawaveoftemperatureorentropy.Wecallthepropagationofatemperaturewavesecondsound(or2ndsound). 13

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Fortunately,wearenotlimitedtomerelyqualitativeargumentsfortheexistenceof2ndsound.Indeed,startingfromtheeighthydrodynamicequationsforasuperuid,itispossibletoderivetheexistenceoftwosoundmodes[ 6 ]. s@~vs @t=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(s rp+sSrT (1{2a)n@~vn @t=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(n rp)]TJ /F6 11.955 Tf 11.96 0 Td[(sSrT (1{2b)@ @t+r(s~vs+n~vn)=0 (1{2c)@ @t(S)+r(S~vn)=0 (1{2d)Here~vn;s,p,;s;nhavebeendenedaboveandS;Taretheentropyandtemperatureoftheuid,respectively.Thehydrodynamicequationslistedabovearenothingmorethanthefamiliarmomentum,massandentropyconservationequationsonemightencounterinaclassicaluidmechanicstext,butmodiedslightlytoaccountfortheindependentvelocityelds,~vsand~vnandtheentropy-lessnatureofthesuperuidcomponent.Weaimtoderivetwowavemodes,oneforpressureandoneforentropy,so( 1{2 a-d)canbemanipulatedtoeliminatevsandvntoyield,afterdroppingsecondorderterms: @2 @t2=r2p (1{3a)@2S @t2=s nS2r2T (1{3b)Wethenusetheequationsofstate: dp=@p @Sd+@p @SdS (1{4a)dT=@T @Sd+@T @SdS (1{4b) 14

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in( 1{3 a,b)toexpressTandpasfunctionsofandSinordertoobtainthefollowingwaveequations @2 @t2=@p @Sr2+@p @Sr2S (1{5a)@2S @t2=s nS2"@T @Sr2+@T @Sr2S# (1{5b)Ofcourse,solvingtheseequationsexplicitlyisnotnecessary.Instead,wemay`guess'planewavesolutions,( 1{6 ),forandSandchecktoseeifthesesolutionssatisfy( 1{5 a,b). =0+ei(kx)]TJ /F5 7.97 Tf 6.58 0 Td[(!t) (1{6a)S=S0+Sei(kx)]TJ /F5 7.97 Tf 6.59 0 Td[(!t)) (1{6b)WeuseandStodenotethatthewavesaresmalldeviationsfromtheequilibriumstate(0;S0),elsethelinearizationof( 1{2 )isinvalid.Substituting( 1{6 )into( 1{5 )anddeningthespeedofsoundasu=! kwenallyconcludethat "u u12)]TJ /F1 11.955 Tf 11.96 0 Td[(1#+@p @S@ @pSS=0 (1{7a)@T @SS@S @T+"u u22)]TJ /F1 11.955 Tf 11.96 0 Td[(1#S=0 (1{7b)whereu21=@p @S (1{7c)u22=s nS2@T @S (1{7d)Ifweletu=u1,thentosatisfy( 1{7 a,b)Smustvanish.Thuswhentherearedensityoscillations,entropyremainsconstantandwendthatwehavedescribedrstsound. 15

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Conversely,ifu=u2,thendensitymustremainconstantwhileentropy(ortemperature)oscillatesintheuidwhichisconsistentwiththequalitativedescriptionof2ndsoundgivenabove.Thespeedof2ndsoundispresentedinFig. 1-2 .Itisfoundtovaryslowlynear20m/sfrom1:0Ktoroughly1:8K,afterwhichitgrowsslightlyandplummetsto0m/satthe-point[ 8 ]. Figure1-2: Thevelocityof2ndsound,u2,asafunctionoftemperatureuptothelambdapoint.Thedashedlinesindicatethe2ndsoundvelocityfortheexperimentsconductedinthefollowingchapters.[ 9 ] 1.1.2QuantumTurbulenceCentraltothetwouidtheoryistherequirementthatthesuperuidbeirrotational[ 5 ].Thatistosay,r~vs=0.However,thenatureofturbulencerequiresthattherebesomesortofrotationpresent.Thisparadoxisresolvedbytheintroductionoflinedefectsintothesuperuid.Thesedefectschangethegeometryofthebulkuidfromasimplyconnectedregiontoamultiplyconnectedregion.Inamultiplyconnectedgeometry,theconditionr~vs=0issatised,butthecirculationequation( 1{8 )canstillyieldanon-zeroresult.Indeed,becausevs=rS,whereSisthephaseofthemacroscopicwavefunctionoftheBose-Einsteincondensateofthesuperuid,wendthatthecirculationisactuallyquantized[ 10 ]: 16

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I~vsd`=nh=m(1{8)Theintegrationisperformedalongapatharoundalinedefect,nisanintegerwindingnumber,hisPlank'sconstantandmisthemassofaheliumatom.Insuperuidhelium,nisrestrictedtobe1,sowedenethequantumofcirculationh=m.Wethenrenamethelinedefecttobeaquantizedvortexline,orifthelineterminatesuponitself,itistermedaquantizedvortexring.Inprinciple,quantizedvortexlinescanbenucleatedonlyabovesomecriticalowvelocity,whichshouldnotbeconfusedwiththeLandaucriticalvelocityof60m/satwhichelementaryexcitationsarecreatedintheuid[ 5 ].Inpractice,thecriticalvelocityforvortexlinecreationisfoundtobeseveralordersofmagnitudebelowtheLandaucriticalvelocity(see[ 11 { 14 ]forrelevantexperimentsoncriticalvelocities).Atowvelocitiesabovetheempiricalcriticalvelocity,avortexlinetangleissaidtoform.Itisthistangleanditsinteractionswiththenormaluidthroughmutualfrictionthatisdenedtobequantumturbulence.Thevortexlinetanglepossessesalloftheexpectedpropertiesofturbulence:itischaotic,ithassomesortofrotation,itisdiusive,anditisdissipative.Themutualfrictionforcebetweenthenormalandsuperuidcomponentsisthesourceofdissipationand,becauseoftheniterelativevelocitybetweentheuidsincounterow,thisdissipationisexpectedtooccuratmanylengthscales,possiblyallthewayuptotheinjectivescaleitself.Whatthen,canbesaidabouttheenergyspectrumofthiskindofquantumturbulence?Recently,thishasbecomeanexperimentallytractablequestion.Theremainderofthisdissertationisdevotedtodescribingtheexperimentbuilttoempiricallycharacterizethequantumturbulenceofcounterowingsuperuid4He. 1.1.3ThermalCounterowInsuperuidhelium,heatiscarriedentirelybythenormaluidandmassconservationrequiresthatthesuperuidcomponentowoppositetothenormaluid.Thus,applicationofanyheatuxtoavolumeofheliumIIwillcausecounterow.Thermal 17

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counterowturbulencehasnoknownclassicalanalogues,andthereexistavarietyofeectivetechniquestoprobevortexlinedensityintheuid.Thecounterowvelocityisonlyafunctionofappliedheatheatux,q,densityoftheliquid,,entropyperunitmass,s,andthetemperatureofthebath,T,andisexpressedinthefollowingway: vn=q sT(1{9)Therst,thoroughseriesofstudiesofcounterowturbulencewereperformedandpublishedbyVinen[ 11 , 15 { 17 ].Energydissipationincounterowturbulenceoccurslargelyintheinteractionsbetweenthevortexlinetangleandthenormaluid.Thisinteractioniscalledthemutualfriction[ 18 ].Theseexperimentswerethersttoshowthatthemutualfrictionbetweenthenormaluidandquantizedvortexlineswasproportionalonlytotherelativevelocitybetweentheuidcomponents(vs)]TJ /F6 11.955 Tf 12.21 0 Td[(vn).Vinen'sexperimentsperformedwithsteady-statecounterowdetermineanequationfor2ndsoundattenuationasafunctionofheatux: 0=C(W)]TJ /F6 11.955 Tf 11.95 0 Td[(W0)2(1{10)where0istheattenuation,Cisaconstantofproportionality,andW;W0areappliedandcriticalheatuxes,respectively.Theequationforlinedensity,L,asafunctionofattenuationis L=6 BA0 A)]TJ /F1 11.955 Tf 11.95 0 Td[(1(1{11)whereisresonancepeakwidth,istheconstantofcirculation,BisthemutualfrictionconstantandA0andAaretheinitialresonancepeakamplitudeandattenuatedamplitude,respectively.In( 1{11 ),)]TJ /F5 7.97 Tf 6.68 -4.87 Td[(A0 A)]TJ /F1 11.955 Tf 11.96 0 Td[(1=0.Combining( 1{10 )and( 1{11 ),expressingheatuxintermsofcounterowvelocityVandtakingthesquarerootwe 18

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areabletoproduceanequationfortheaveragespacingbetweenvortexlines,`(where`=L1=2)[ 19 ], `=(V)]TJ /F6 11.955 Tf 11.95 0 Td[(V0)(1{12)whereViscounterowvelocity,V0isthecriticalvelocityandisthecollectionofrelevantconstants.Practically,isusedasatparameterforcomparisonbetweendierentsteady-statecounterowexperiments.Intermsofexperimentalconsiderations,thebestsensitivityisachievedbyresonanceswithsmallfull-widthhalf-max(FWHM)valueandlargeamplitude.Experimentsinverysmall(m)]TJ /F6 11.955 Tf 12.48 0 Td[(mm)channelsservednotonlytofurtherconrmVinen'sinitialwork,butalsotounearthanotherfeatureofsuperuidturbulence:theTI/TIItransition.Therstevidence,largelyoverlooked,oftheexistenceofalessturbulentcounterowregime,laterdubbedtheTIstate,appearsinFig.3ofRef.[ 11 ].Subsequentexperiments[ 12 { 14 , 20 , 21 ]exploredandcharacterizedthetransitionfromTIturbulencetothemorefamiliarTIIturbulencedescribedby( 1{12 ).Insmallchannelsoflow-aspectratio(squareorcircularcross-section)theTI/TIIstatesarefoundtobereproduciblewithconsistentcriticalvelocitiesasafunctionofsize.Further,itwasfoundthatthefeaturesoftheTIandTIIsuperuidturbulencestatesareindependentofcharacteristictubesizeforcirculartubes.Itwouldnotbeunreasonabletoexpectthesametobetrueinchannelswithsquarecross-section,butcurrentlynoexperimentalevidenceexiststosupportsuchanexpectation.Itisnoteworthythatthereexistsasignicantbodyofworkrelatingtothedecayoftransientcounterowturbulence.Thedecayofcounterowturbulenceinawide(9mm)channel(at1:6K)exhibitsanumberofinterestingfeatures[ 22 ].First,itisclearthatthevorticityinthechanneldoesnotalwaysdecreasemonotonicallyafterheatuxisswitchedo(thoughthiswasseeninearlierwork).Indeed,thevorticityisobservedtoeitherattenoutforsometime(uptoseveralseconds!)orevenincrease,forminga`bump'inthedecay 19

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plot.Secondly,foreachprobedheatux,thevorticitydecayeventuallyassumesaclassicalform.Thatistosay,thevorticitydecayobeysat)]TJ /F2 7.97 Tf 6.59 0 Td[(3=2powerlawinthelaterstagesofdecay.Inboththesteadystateandtransientcounterowcases,plentyofdataexistdescribingthebehaviorofthevortexlinetangleandsuperuidcomponent.However,manyseriousquestionsremain,notleastofwhichisthequestionofthenormaluidow.Inthetransitionstoturbulence,doesnormaluidowinstabilitycauseturbulenceinthesuperuid?Isnormaluidturbulencetotallyindependentofthesuperuidow?Whendoesthenormaluidbecometurbulent?Howdoesthedecayofnormaluidturbulencecomparetodecayofsuperuidvorticityinthechannel?Allofthesequestionsmustbeansweredtomovetowardacompletedescriptionofquantumturbulence.Thewayforwardabsolutelyrequiresdevelopmentofeectivevisualizationtechniquesinsuperuidhelium,thesubjectofthisdissertation. 1.2VisualizationAlthoughquantumturbulenceinsuperuid4Hehasbeenasubjectofstudyforwelloverhalfacentury,owvisualization,oneofthemostpowerfulandcommonexperimentaltechniquesinclassicalturbulence,hasonlyrecentlybeenimplementedinhelium-II.ThebestknownsuchexperimentswereParticleImageVelocimetryandParticleTrackingVelocimetry(PIV/PTV).Thebasicexperimentalschemerequiresthathelium-IIbeseededwithsmall(m)sphericalparticlesofpolystyrene[ 23 ]orhydrogen`snowballs'[ 24 , 25 ].ParticlepositionsaredetectedbyshiningalasersheetthroughtheexperimentalvolumeandcapturinglightreectedfromthetracerparticleswithaCCDcamera.PIV/PTVexperimentsdidyieldsomeinterestingdataaboutsuperuidhelium.ZhangandVanSciverdiscoveredvorticesupstreamofaxedcylinderinturbulentcounterowingheliumII[ 26 ].TherstvisualizationsofquantizedvortexlineswereachievedbyPackardetal.inrotatingcryostatsinwhichbothsinglevorticesandarraysofvorticeswereobserved[ 27 , 28 ].AnexperimentbyBewleyseemstoshowdecoratedquantizedvortexlineswith 20

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hydrogensnowballs,possiblyprovidingtherstimagesofquantizedvorticesincounterowingsuperuidhelium[ 24 ],butnoquantitativeproofisgiven.SimilarsubsequentexperimentsbyPaolettietal.showparticlevelocitystatisticsthatare,insomecases,consistentwiththeexpectedvelocityofthevortexlinetangleinthermalcounterow[ 29 ]Unfortunately,particlevelocimetrytechniquessuerfromambiguityathighheatux(highcounterowvelocity)duetotheinteractionsbetweenparticlesandquantizedvortices[ 29 , 30 ].PIVexperimentsshowedthat,inthermalcounterow,thetracervelocitydoesnotmatchthatofthenormaluid,aswasoriginallyexpected[ 31 ].SubsequentPTVexperimentsdemonstratedthatthetracersundergosuddenchangesofdirection,whichwereinterpretedtocorrespondtothetracerseitherbindingtoorbeingdislodgedfromthequantizedvortexlines[ 25 ].Whilesuchexperimentshaveyieldedawealthofinformationabouttheinteractionoftracerparticleswithvortexlines[ 32 , 33 ],and,atlowheatux,demonstratedtheexistenceoftwovelocityelds(vn&vs),comparativelylittlewaslearnedabouttheowprolesofthenormaluidandsuperuidcomponentsathighheatuxes.Somevisualizationworkhasbeendoneusingneutrontomographyof3Heseededhelium-II.Byusingatomsof3Heastracers,thistechniqueside-stepssomeoftheseriousissuesinherentinthePIV/PTVexperiments.Namely,above1K,the3Hemoleculesessentiallyactasnormaluid,sothereisnoambiguouscouplingbetweentracerandthesuperuid/normaluidcomponents.Additionally,3Hedoesnotspontaneouslyaggregateoradsorbontocellwalls,unliketheimpuritiesusedastracersinotherexperiments.Theonlysuchexperimentwaslimitedinitsspacialresolutionbythecomparativelylargesizeoftheneutronbeam(4mmFWHM)andinitstimeresolutionbytheneedtorasterthebeaminordertodevelopafullpictureoftheexperimentalvolume[ 34 ].Naturally,thisputsstrongrestrictionsontherangeofheatuxesthatcanbeinvestigatedinsuchasetup.Inprinciple,neutron-tomographyexperimentsofthissortcanbeimprovedbyreningtheexperimentaldesign,butitislikelythatsuchavisualizationschemewillremainmostsuitableforsteady-stateowmappingratherthanthestudyofdynamical 21

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processes.Giventhedicultiesencounteredinsuperuidowvisualizationthusfar,itisnaturaltoaskforatracerthatunambiguouslycouplestothenormaluidatT>1K,doesnotcontaminatethesuperuid,hasareasonablylonglifetimeanddoesnotrequireextremelyspecializedfacilitiessuchasbeamlinesatnationallaboratories.Onesuchcandidatetracerexists:theexcitedheliumdimer. 1.2.1HeliumExcimersTheexcitedheliumdimer,morecommonlycalledtheheliumexcimer(He2),hasanumberofagreeablequalitiesthatmakeitanidealtracerinhelium.Whilethespin-singletstateofHe2hasaradiativelifetimemeasuredinnanoseconds,thespin-tripletstatehasamuchlongerlifetime(13sinvacuum[ 35 ])duetoaforbiddenspin-ip.Additionally,incolddensegasorliquid,He2moleculesformsmallbubbles,withradiusabout7A[ 36 ],inthea(0)stateand12Ainthed(0)state,whichgivesthemanexceedinglyshortviscousrelaxationtime.Theviscousrelaxationtime,,isameasureofhowquicklytheparticle'smotionrelaxestothatoftheuidinwhichitislocatedandisgivenby[ 33 ] =2a2p 9n;=p+ 2(1{13)wherepisthedensityofthetracerparticle,isthedensityoftheuid,apistheradiusoftheparticle,andnistheviscosityofthenormaluid.Forsuperuidheliumat1:830Kandexcimerswithbubbleradius7A,theviscousrelaxationisonly4:4ps.Thus,thevelocityofexcimerswhichinteractwithvortexlinesrelaxestothatofthenormaluidinanextremelyshorttime.Inlightofthis,itisquitereasonabletoexpectthat,ontheprobingscaleofourexperiments,heliumexcimersareentirelyentrainedbythenormaluidattemperaturesabove1K.Thevisualizationofthesetracersisperformedentirelybylaser-induceduorescence(LIF).ExperimentsbyBenderskiietal.inbothliquidandcold,gaseousheliumindicatedthattheheliumexcimercanbemadetouoresceatseveraldierentwavelengths,themostdominantofwhichwas640nm.Acompletecharacterizationoflaser-induceduorescenceofHe2insuperuidheliumcanbefoundin 22

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WadeRellergert'sdissertation[ 37 ].WhatfollowswillbeabriefsummaryoftherelevantcharacteristicsandrequirementsofLIFexperimentsinsuperuidhelium.Fig 1-3 showsaschematicdrawingoftheenergylevelsofHe2inhelium-II. Figure1-3: EnergylevelsofHe2inliquidhelium.Thea3+u!c3+g!d3+utransitionispumpedbytwo905nmphotons.640nmuorescencefromthed3+u!b3gtransitionwillbecollectedbyanICCDcamera.Excimerstrappedinthelong-liveda(1)anda(2)vibrationalstatesarerepumpedbycwberlaserswithwavelengths1073nmand1099nm,respectively. Incoldheliumgas,theenergystatesaresuchthatthea3+u!c3+grequiresa919nmphotonandc3+g!d3+urequiresa1047nmphoton.Conveniently,inbulkliquid,theexcitationschemeforexcimersinbulksuperuidexploitsthebroadening[ 36 ]ofthec3+g!d3+utransitionsuchthatthea3+u!c3+gandc3+g!d3+utransitionscaneachbeexcitedwitha905nmphoton.About90%ofexcimersintheb3gquenchtothea(0)state.Thus,10%oftheexcimersenduptrappedineitherthea(1)ora(2)longlived(100ms)vibrationalstates.Withoutcyclingthetrappedexcimers,theuorescenceintensityfromthed3+u!b3gtransitionquicklydropstoasteadystatevalueof25%oftheinitialintensity.However,whencontinuouswave(cw)repumpinglaserbeamswithwavelengths1073and1099nmareintroduced,thesteadystateuorescenceintensityhas 23

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beenshowntobe85%oftheinitialvalue[ 37 ].Crucially,two-photonexcitationallowsforalargeseparationbetweenexcitationanduorescentwavelengths,sotheformercanbelteredoutfromthedetector. 1.2.2PenningIonizationPreviousexperimentshaveusedradioactivesources[ 37 ]andhighelectriceldsatthetipsofsharptungstenneedles[ 38 ]togeneratelargecloudsofexcimers.Furtherworkdemonstratedthattheexcimersfollowthenormaluid[ 38 ]andproducedevidenceofturbulenceathighheatux[ 39 ].However,duetolowexcimerdensity,theimagescapturedinthoseexperimentswereaveragesof40exposures,therebyaveragingoutimportantdetailsoftheturbulentow.Tostudythenerfeaturesofsuperuidturbulence,itisimportanttoobtainanexcimerdensitysuchthattheemitteduorescenceissucienttoallowforhighqualitysingle-shotimages.Onesuchmethodisexcimercreationbylaser-eldionization[ 40 ].Thebeamwaistandlaserpowercanbecontrolledtocreatelong,thinlinesofexcimerswithhighdensities(upto1013cm)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(3[ 41 ]).However,itisimportanttoconsiderthemechanismsbywhichexcimersmightbeeliminatedfromtheionizationvolume,therebylimitingthedensityofexcimersinanitevolume.Thedominantmechanismforexcimerremovalisduetocollisionsbetweenpairsofmolecules.ThisprocessiscommonlyreferredtoasPenningionizationorbimoleculardecay.Ingeneral,PenningionizationoccurswhenanexcitedmetastablemoleculeMcollideswithsomeothermoleculeoratomA,whichresultsinde-excitationofMandtheionizationofA[ 42 ]:M+A!M+A++e)]TJ /F1 11.955 Tf -267.94 -35.32 Td[(Ketoetal.[ 43 ]performedanexperimentalinvestigationofthePenningionizationofmetastable(a3+u)heliumexcimers.ThePenningionizationbetweentwoHe2moleculescanbedescribedas[ 41 ] He2+He2!He2+2He(1{14) 24

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whereHe2canbesaidtobeamorehighlyexcitedstatethanHe2.Further,theydeterminethat,forheliumexcimersinthea3+ustatenear1:830K,thedecayconstantforPenningionizationofHe2is410)]TJ /F2 7.97 Tf 6.58 0 Td[(10cm3/s.Thecorrespondingequationforexcimerdensity(intheabsenceofasourceterm)is[ 43 ] n(t)=n0 1+n0t(1{15)wheren(t)istheexcimerdensity,n0istheinitialexcimerdensity,isthePenningionizationdecayconstantandtisdecaytime.Thusitisclearthat,iftherateofexcimercreationisknown,thesteadystateconcentrationofexcimerscanbedeterminedasafunctionoflaserrepetitionrate(rep-rate)andlaserpowerperpulse.Specically,ifthenumberofexcimersisincreasedbyafactorof2mperpulse(wheremdependsonlaserpowerperpulse,describedforthisexperimentattheendofx 2.1 ),then,inthesteadystate,theexcimerdensitymustdecaybyafactorof2mbetweenpulses.Itthenfollowsthatthesteadystateexcimerdensity,nss,inatrainoflaserpulsesis nss=2m)]TJ /F1 11.955 Tf 11.95 0 Td[(1 trep(1{16)wheretrepisthetimebetweenlaserpulses.Sincetheintensityofuorescencewillbedirectlyproportionaltoexcimerdensityintheilluminatedvolume,thePenningionizationwillhaveasignicantimpactontheimagequalityandobservationtimescalesintheexperimentdescribedinCh. 2 . 25

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CHAPTER2EXPERIMENTANDPROTOCOLThischapterpresentsthedetailsofanexperimentalapparatusdesignedtoperformLIFexperimentsinsuperuid4He.x 2.1 describesthecomponentsoftheapparatusandtheircongurationsrelativetoeachother.Mostofthecomponentsarecommerciallyavailable,withtheexperimentalchannelbeingcustommade.Additionally,x 2.1 includesthedetailsofthelaserbeamprolecharacterizationanditsmanipulationbytherelevantopticstoachievethenecessaryenergyuxesrequiredtoproduceathinHe2.x 2.2.3 presentsthedetailsofthe2ndsoundapparatusforprobingvortexlinedensity,includingthedesignoftheoscillatingsuperleaktransducers,electronicssetupandconsiderationsrelatedtoresonancepeakselection.x 2.2.1 primarilyconcernspreliminaryexperimentsperformedtocharacterizetheactualperformanceofthefemtosecondlaserinheliumasafunctionofheliumdensity.Wendthatthinexcimerlinescanbecreatedinheliumgasnearstandardtemperatureandpressure(STP,T=273KandP=100kPa)throughmulti-photonionizationandthatalongerlinecanbecreatedinthesuperuidjustbelowthebreakdownthreshold.Finally,x 2.2.2 showsevidencethatthereisnoexcesslaserheatinginthesuperuidsincethereisnodeformationoftheexcimerlineunderzeroheatuxconditionsatlongdrifttimes. 2.1ExperimentDesignAschematicoftheexperimentalchannelisshowninFig. 2-1 .Wemountastainlesssteelchannelofsquarecross-sectionwithinnersidelength9:4mmandtotallength300mmatthebottomofatemperaturecontrolled,bath-pumpedheliumcryostat(customdesign,manufacturedbyAmericanMagnetics).MostofthesteelchannelcomponentsweremachinedattheNationalHighMagneticFieldLaboratoryinTallahassee,FL,whilethe2ndsoundcubeandtransducersweremachinedattheUniversityofFlorida,andthethin,sapphirewindowswerepurchasedfromThorlabs.Atthebottomofthechannel,weplaceaplanarheater(essentiallyathin-lmresistor,404)todrivethermalcounterow.The 26

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channelrunsthroughastainlesssteelcubeequippedwiththreethin,sapphirewindows,twoofwhicharemountedonarms,removingthemfromthefocusedbeamstoavoidthepossibilityoflaserdamage.ThelowerinsetinFig. 2-1 showsaside-sectionviewofthevisualizationcube.Itisimportanttonotethethinslits(4mm)cutintothechannelratherthanthewindowsonewouldexpectatthisposition.Again,thisistoavoidlaserdamageduetothefocusingofthefsbeamatthispoint.Althoughtheenergyuenceofthefemtosecondlaserbeamforexperimentsonliquidheliumprovestobebelowthedamagethresholdofsapphire(between5)]TJ /F1 11.955 Tf 12.06 0 Td[(11J/cm2)[ 44 ],experimentsongaseousheliumathightemperatures,suchasthoseshowninSection 2.2.1 ,canbrushthedamageuencethreshold,causingsparkingandburnsonthewindow.Whilewindowsdamagedinthiswayareunlikelytoshatter,thedamagedareasdotendtosignicantlydistortthebeamproleofanylaserbeamsincidentonthespot.Adistortedbeamprole,especiallythatofthefslaser,leadstounpredictableexcimergenerationandionizationoftheliquidhelium.Justbelowthevisualizationcubeistheso-called"2ndsoundcube",onwhichcanbemountedtwooscillatingsuperleaktransducers.Onetransducerwillgenerate2ndsoundresonancesandtheotherwillactasadetector.Theaveragedvortexlinedensityintheliquidcanbecalculatedfromtheattenuationoftheresonanceamplitude(x 1.1.3 ).Seesection 2.2.3 foramoredetaileddescriptionofthe2ndsoundexperimentaltechnique.Finally,thetemperatureoftheliquidheliumintheexperimentisdeterminedbythesaturatedvaporpressurecurveofhelium.Apressuretransducercontrollinganelectronicvalvekeepsthetemperatureofthebathconstanttowithin1mK.Thethermometerisafourwirecernoxresistor,placedatthetopofthechannel,whichiscalibratedfrom300Kto4:2Kandfrom2:17Ktonearly1:5K.Thefemtosecondlasersystem(SpectraPhysics)consistsofaMaiTaifemtosecondlasersource,whichisfedintoaSpitreAceregenerativeamplier(pumpedbyanEmpower20Wgreenlaser).TheoutputfromtheSpitreAceisastable,nearlyGaussian(M2=1.04;M2valuescloseto1indicatethatthebeamproleisveryclose 27

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toGaussian[ 45 ])beamwithpulselength35fsandaninitialmaximumrepetitionrate1kHz.Mostoftheexperimentsperformedinsection 2.2.1 usearep-rateof1kHz.However,theresultsdetailedinchapter4areobtainedwiththeSpitreAceupgradedtooperateatamaximumrep-rateof5kHz.TheimaginglaserisanEKSPLANd:YAGlaserwithxedwavelengthof905nm,whichcanbeoperatedateither1kHzor500Hz.Wechose500Hz.Theimagingbeamisfocusedbytwocylindricallensestohavearoughlyrectangularshapewithwidth1:5mmandheight4mm.Inordertosaturatethea3+u!d3+utransitionintheilluminatedvolume,theimaginglasermustdeliver2:5mJ/cm2perpulse[ 37 ].Twoberdiodecontinuouswave(cw)lasersarerequiredtoliberateexcimerstrappedinthea(1)anda(2)vibrationalstates.Theyoperatewithwavelengths1073nmand1099nm,respectively,andarealignedontotheimaginglaserbeforepassingthroughthesamecylindricallenses.Afterfocusing,thewidthofthecwlasersiscomparabletothatoftheimagingbeam,but,duetothelargeinitialbeamsize,thefocusedheightisconsiderablygreater.Bothcwlasersdeliverapproximately13W/cm2atthefocalplane,someofwhichisinterceptedbyeitherthechannelbodyorthethermalshielding.Tominimizetheheatloadfromthecwlasers,weensurethattheyareonlyonforashorttime:17msatmost.Theeect,orlackthereof,ofthisonexcimerlinequalityisdiscussedinsection 2.2.2 .Acustommadeintensiedcharge-coupleddevice(ICCD)camera(PI-MAXIIfromPrincetonInstruments)isplacedfacingthefrontofthevisualizationcube.Althoughthe640nmemissionpeakofHe2isdominant,thelightfromtheexcimerlineisstillexpectedtobequitedim.Thus,mostofthemodicationstothecameraarecenteredaroundlenscoatingsandhardwareupgradesthatimprovethequantumeciencyofthedetectorfortherelevantwavelength.Alenswithfocallength15cmisinstalledonthecryostat'snitrogenshieldtoensurethatthechanneltakesupthemajorityofthecamera'seldofview.Asmentionedinx 1.2.1 ,thattheuorescencewavelengthwearedetectingissignicantlysmallerthantheimagingandcwlaserwavelengthsisanadvantageous 28

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consequenceoftheLIFschemeweareusing(seeFig. 1-3 ).Welteroutunwantedlightbyinstallinga640nmbandpasslterwithafull-widthhalf-max(FWHM)bandwidthof20nmonthecamerawithoutworryingaboutadditionallightbeingcreatedbyhigherenergyphotonsfromtheexcitationlasers. Figure2-1: Aschematicdiagramofthestainlesssteelexperimentalchannel.Thechannelhassquarecross-sectionwithsidelength9:4mmandtotallength30cm.Itisboltedtoabath-pumpedcryostatandthereforesharesitsliquidwiththatofthecryostatitself.Athermometerisplacednearthetopofthechannelandisusedinconjunctionwithanelectronicvalvetoregulatethetemperatureofthebath.The`visualizationcube'hastwoprotrudingarmssealedwiththinsapphirewindows.Thefsandimagingbeamsarealignedtobecollinearonthecenterofthevisualizationcubearmssothatthebeamscanpassunobstructedthroughtheexperimentalvolume.TheICCDcameraisplacedfacingthethirdwindowinthevisualizationcube(facingintothepage).Justbelowthevisualizationcubeisthe2ndsoundcubeonwhicharemountedtwooscillatingsuperleaktransducers,whichgenerateanddetect2ndsoundwaves.Theattenuationofthesewavesprobestheaveragevortexlinedensityintheexperimentalvolume.Theresistiveheater(404)atthebottomoftheowchannelgeneratesthermalcounterow.Theinsetshowsasketchofthefslaserbeamprole(1/e2width)asitpassesthroughtheexperimentalvolume. TheupperinsetinFig. 2-1 showsasketchofthefsbeamproleasitpassesthroughthechannel.Naturally,thefocusinglensisplacedsuchthatthebeamwaistisinthe 29

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centerofthechannel.Sinceitispossibleforthechanneltoshiftduringcool-down,thefsbeamfocusinglensismountedonanopticalrailsothatthebeamwaistcanalwaysbeplacedmid-channel.Additionally,wechosethefocallengthtogiveaRayleighrangehalfthelengthofthechannel.ForaGaussianbeam,the1=e2beamwidth,!(thisisp 2 p ln(2)FWHM),asafunctionofdistancefromthefocalplane,z,isgivenby !(z)=!0s 1+z !202(2{1)where!0isthediameterofthebeamwaistandisthewavelengthoflight.TheRayleighrange,whichisthedistancefromthefocalplaneatwhichthecross-sectionalareaofthebeamdoubles,isdenedtobezR=!20 .Giventhisconvenientdenition,wehaveawayofchoosingtheopticsandlaserpowersuchthatthereissucientenergyux(foundtobeI=51013W/cm2atthefocalplane[ 41 ])throughoutthewidthofthechanneltocreateathinlineofexcimermolecules.Indeed,notonlymusttheenergyuxbesucient,butalsoitmustnot,atanypoint,exceedtheionizationbreakdownthresholdofliquidhelium(discussedinsection 2.2.1 ).Fig. 2-3 showsthemeasuredfsbeamproleand1/e2widthofthebeamproleasafunctionofdistancefromthefocalplane.Thedatashowthat,whenpassedthroughalenswithfocallength75cm,thebeamproleisanearlyperfectGaussianwithM2=1:04andRayleighrange5:5mm.Further,passageofthebeamsthroughtheexperimentiscontrolledbytwoshutters.Therst,anelectro-mechanicalsolenoidshutter(Newport,model76993ElectronicFastShutter),ispairedwiththefslaser.Ithasminimumopentimeofabout8mswithacharacteristicopen/closingtimeof4ms.Atarep-rateof1kHzorbelow,thisprovidesreasonablygoodcontrolofthefslaserpulsespassingthroughthechannel.Atmuchhigherrep-rates,suchas5kHz,theshutterisunabletoprovideanythingbetterthancoarsecontrolofthefslaser.Inprinciple,withtwosuchshutters,itispossibletoselectexactlyhowmanylaserpulseswillbepassedthroughthechannel,evenat5kHz.Aboveacertainnumberoffslaserpulses,theintensityoftheexcimerlineuorescenceiscappedbythe 30

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steadystateexcimerdensity,sosuchrigorouscontrolisnotnecessary.Inlightofthis,wechoseaoneshuttersystemforboththefsandimaginglasersinordertosimplifytheexperimentasmuchaspossible.Forcontroloftheimagingandberlasersweuseaslower,mechanicalshutter(Thorlabs,SH05opticalbeamshutter)withminimumopentime17msandcharacteristicopen/closetimeof8ms.Asindicatedabove,theimaginglaserisoperatedatarep-rateof500Hz,soevensuchaslowshuttercanprovideadequatecontrolofthe905nmpulses.Fig. 2-2 providesaschematicrepresentationoftheopticalbenchlayoutwithallrelevantopticalcomponents. Figure2-2: Aschematicdrawingoftheopticalbenchforthisexperiment.Theimaginglasers(905nm,1073nmand1099nm)arealignedontoeachotherusingbeamsplittersandthenpassedthroughtwocylindricallensestoobtainthenecessaryprole.Mostcritically,the905nmlasermustbefocusedsothatitis1mmwideand4mmtallsothatitmaypassthroughtheslitsinthechannelwallsandalsopumptheexcimerlinewhenitmovesduringexperiments.TheSH05mechanicalshuttercontrolstheentryoftheimaginglasersintothechannel.Thefemtosecondlaserpoweriscontrolledbyhalf-waveplate(/2plate)andathinlmpolarizer.TheSpitreAcealwaysoutputs4Woflaserpower,thevastmajorityofwhichissentintothebeamdump.TheremainderofthefslaserpoweristhensentthroughthefastNewport76993shutter,throughthe75cmfocusinglensandalignedwiththeimagingbeamswithanotherthin-lmpolarizer.Thenaltwomirrorsallowthecollinearbeamstobecarefullyalignedthroughtheslitsinthechannel. 31

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Figure2-3: (a)Measuredcross-sectionintensityofthefemtosecondlaserbeamatthefocalplane.TheproleisanearlyperfectGaussianbeamwithM2=1:04.b)Themeasured1/e2widthofthefemtosecondlaserbeamasafunctionofpositionzalongthedirectionofpropagation.ThedataarettoEq. 2{1 (redline),whichgivesaRayleighRangeof5.5mm.Theerrorbarsaresmallerthanthedatapoints. TheexperimentaltimingisshowninFig. 2-4 .Theheaterisactivatedatthespeciedpowerforatleast15spriortoopeningthefslasershuttertoensurethatthermalcounterowhasfullydevelopedbeforedatacollection.Subsequently,thefsshutterisopened,usuallyfortheminimumpossibletimeat5kHzsinceevenasfewas10pulsesareexpectedtobesucienttoreachasteady-statedensityofheliumexcimers.Afterexcimerlinecreationwewaitsometimedenotedastdrifttoallowtheexcimerstobecarriedwiththenormaluid.Forexperimentsatlowheatux,tdriftcanbeaslargeas900ms(atwhichpointimageaveragingisrequired).However,inordertoextractthestructurefunctionfromsingle-shotimages,tdriftmustn'texceedtheeddyturnovertime,sincetheturnovertimegivesthemaximumlifetimeoftheeddy.ForowswitheddysizeRandvelocityincrementu(R)acrosstheeddy,theeddyturnovertime,,isgivenby=R p hu(r)2i[ 46 ].Fromourempiricalvelocityuctuationdata,owdrivenbyaheatuxof300mW/cm2hasturnovertimeofabout51msatthesmallestscales(200m).Whilefor225mW/cm2theturnovertimeincreasestoabout100msatthesamescale.Inpractice,tdriftmustbechosentobelargeenoughsothatdistortionsoftheexcimerlinecanberesolved,butlessthantheeddyturnovertimesothateddypropertiescanstillbeprobed. 32

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Afterthedrifttimehaselapsed,weopentheimagingshuttertopassthroughboththe905nmpumppulsesandthe1073and1099nmCWberlaserbeamsforvibrationallevelcycling.Empirically,wendthatimagequalitydoesnotnoticeablyimprovewhenaveragingmorethan6imagingpulses.TheICCDcameramicro-channelplates(MCP)aresettotriggerforeach905nmpulse,therebycollectingonlyuorescencelightandminimizingpollutionofthesignalbyoutsidelightsources.Asetofsiximagingpulsesiscollectedbythecameraandthentheregistersareshiftedandread,deningthatsetofsixasasingleshot.Atthemaximumachievablecounterowvelocities,2cm/s,themovementofthenormaluidbetweenimagingpulsesisontheorderof40m,signicantlysmallerthanthewidthoftheexcimerlineatitsnarrowestpoint.Therefore,smearingoftheimageduetomovementbetweenMCPtriggersisnotexpectedtobeameasurableeect. Figure2-4: Schematicdiagramofexperimenttimingforallsteadystatecounterowvisualizationexperiments.Initially,theresistiveheateratthebottomoftheexperimentalchannelisturnedonforatleast15stoensurefullyformedcounterow.Thefslaseristhenallowedtopassthroughthechanneltocreatethemaximumachievabledensityofexcimers.Sometimetdriftisallowedtoelapsebetweenexcimercreationandimagingpulses.Siximagingpulsesarefoundtocreatethebestuorescencesignal.Morepulsesdonotproduceanoticeableimprovementinimagequality.tdriftcanrangebetween25)]TJ /F1 11.955 Tf 11.96 0 Td[(900msdependingonthemagnitudeoftheheatux. Thenatureoftheexcimercreationmechanism,controlledelectronavalancheionization(see 2.2 ),necessarilyputsanupperboundontheheatuxesatwhichqualityuorescenceimagescanbecaptured.Benderskiietal.determinedthateachfslaserpulsemultipliesthenumberofexcimersinagivenvolumeofLHebyafactorof2m. 33

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Theyshowthattheexponentmisdeterminedbytheratioofincidentenergyperpulseofthelasertotheenergyperpulserequiredtodoublethenumberofions.Thatistosay,m=W=W2whereW2=13J/pulse[ 41 ].Wendthatat60J/pulsewearejustunderthebreakdownthresholdofliquidheliumatT=1:830K(seex 2.2.1 ).Basedonthesevalues,m=4:62inourcase.Betweenpulses,theexcimersundergoPenningionizationdecay(seex 1.2.2 andRefs.[ 41 , 42 , 47 ])untilthenextpulsecomesalong.Ofcourse,attheexcimersaturationdensity,theexcimerdensitydecaysbyexactlythefactor2mbetweenpulses.Inthespecialcaseofcounterow,notonlyistheexcimerdensitydecreasingduetoPenningionization,butalsoduetothemotionofthenormaluidremovingthemfromthevolumeofheliumthatisilluminatedbythelaserpulsesandintroducingnew,un-ionizedheliumatoms.Atlargeenoughowvelocities,itispossiblethatthesteadystatedensityreachedatzeroheatuxissimplyunachievable.Empirically,wendthatatafslaser(rep-rate)of1kHz,theheatuxatwhichimagequalityistotallycompromisedisabout100mW/cm2(6:44mm/s).Athigherrep-rates,weexpectthisheatuxtoincreaseand,indeed,wendthatat5kHzwecanobtainhighqualityimagesuptoheatuxescloseto(andsometimesexceeding)400mW/cm2(2:58cm/s).Largelyforthisreasonwechoosetoperformsteadystatecounterowvisualizationexperimentswithafslaserrep-rateof5kHz. 2.2Laser-eldIonizationinHeliumCreatingausableexcimerlineinliquidorcoldgaseousheliumrequiresanunderstandingofhowtheionizationofatomscanoccur.Inthemulti-photonregime,anatomisionizedwhenitsoutermostelectrongainsenoughenergyfromincomingphotonsthatthebindingenergyisexceeded.Forhelium,theionizationenergyis24:6eV.Ifexcimersweretobecreatedinthisway,eachheliumatomwouldhavetobestruckbyabout16photonswith=800nm.Alternatively,inthetunnelingregime,theelectriceldsuppressesanddistortsthecoulombpotential,allowingtheelectrontosimplytunnelfromboundstatestothecontinuum. 34

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ThetransitionbetweenthetworegimesisgivenbytheKeldyshadiabaticityparameter[ 48 ]: =Uion UP1=2=Uion e2F20=2me!21=2(2{2)whereUionistheionizationpotentialoftheatom,andUpistheponderomotivepotentialofanelectroninalaserelectriceldgivenbyF=F0cos(!t).When1(highfrequency,loweldstrength),multi-photonionizationisdominant,butwhen1(lowfrequency,higheld),tunnelingionizationdominates.Foratomichelium,thetunnelingregimeisonlyreachedatlaserintensitiesof41014W/cm2,wellabovethemaximumenergyuxachievedduringthecourseoftheexperiments.Conversely,themulti-photonregimeisreachedbelow41010W/cm2,wellbelowtheenergyuxesinthisexperiment.However,forheliumexcimers,theionizationpotentialislow,4:26eV,sothetunnelingionizationregimecanbereachedaboveintensitiesof1012W/cm2.Thus,werequiresomepopulationofHe2toalreadybepresentinuidinordertoallowtheelectronavalanchetooccur.Possibly,thisseedpopulationcanbegeneratedbymulti-photonionizationfromtheinitialfemtosecondpulse. 2.2.1IonizationThresholdWeperformedexperimentstodeterminetheionizationthresholdofheliuminordertoconrmthatourexperimentfunctionsinthewaywewouldexpectbasedonthetheoryoflaser-eldionization.Theionizationthresholdexperimentsweredonewiththefslaserrep-ratesetto1kHz.Sincenouorescencelightisrequired,theexperimentalprotocolissimplerthanthedescriptioninx 2.1 .Wesetexposuretimetobe500msandcollectlightateachfslaserpulse.Eachimageisanaverageof500exposures.Fig 2-5 showsionizationlinesingaseousandsuperuidhelium.Ataglance,itisclearthatwecancreatealong,thinexcimerlineinsuperuidheliumaslongasthelaserenergyremainsbelowthebreakdownthreshold.Abovethethreshold,excimerclustersform,compromisingthequalityofthetracerlineandmakingvisualizationexperimentsinfeasible. 35

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Figure2-5: a)Anaveragedimageofanionizationlineinthemultiphotonregime(gaseoushelium,T=300K,P=760T).b)Theionizationlineinliquidhelium,justbelowthebreakdownthreshold(T=1:830K).c)Clustersofionizationforminliquidheliumwhenthefemtosecondlaserenergyuxexceedsthebreakdownthreshold. Wecollectedsuchimagedataacrossawiderangeoffslaserpowers;fromwellbelowtheionizationthresholdtowellabove.Wecollectedintegratedintensitydatainaregionofinterestdenedasasmallareaaroundtheionizationline,spanningthewidthofthechannel.Anexampleofsuchdataforgaseousheliumat300KandnearlyatmosphericpressureispresentedinFig. 2-6 .Whenplottedonalog-logscale,itisclearthatthedatafollowtwodierentpowerlawsandthusindicatethattherearetwodierentlightintensityregimes.Wetaketheionizationthresholdtobethe`kink'whichmarksthetransitionbetweenregimes,whichisdeterminedbytheintersectionoftheexponentialts(redlines).TheionizationthresholdscalculatedinthiswayareshowninFig. 2-7 asafunctionofheliumdensity(relativetothedensityofliquidhelium,LHe).Blacksquaresindicatedatatakeninthewarmgasphase.Theessentiallyconstantnatureoftheionizationthresholdexhibitedinthisdensityregionischaracteristicofthemulti-photonionizationregime[ 49 ].Theredcirclerepresentstheionizationthresholdincoldgas(13K),andthebluetriangle 36

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Figure2-6: Arepresentativedatasetoftheionizationlightintensity(inarbitraryunits)inheliumgasispresentedasafunctionofthefemtosecondlaserenergyux.Thereareclearlytwoseparatelightintensitytrendsinthedatashownbyexponentialts(redlines).Thetransitionfromtheslowlyrisingtorapidlyrisingregionsistakentobetheionizationthreshold.Eachdatapointisanaverageoflightintensityfromtenimages.Theerrorbarsindicatethelightintensityuctuationsforeachenergyux. isionizationthresholdinsuperuidhelium(1:830K).ThattheionizationthresholdfallsbynearlyanorderofmagnitudeasdensityapproachesLHesuggeststhatincoldgasandliquidhelium,theionizationiscausedbyacontrolledelectron-avalanchecascade.Further,thisobservedionizationregimechangeisconsistentwiththeworkofBenderskiietal.[ 41 ]. 2.2.2Laser-inducedheatingAnotherissuepertinenttotheLIFvisualizationtechniqueisthepossibleheatingoftheuidduetothefemtosecondandimaginglasers.Fluidheatingwould,ofcourse,causethermalcounterowastheheatisdissipated.Inanextremecase,itiseasytoimaginethatthelasersintroducesuchlargeheatloadsthattheexcimerlineisimmediatelydistortedandnomeasurementscanbeperformed.Fig. 2-8 (a)-(c)showsthattheexcimerlinebroadens,butdoesnotdistort,overthecourseof50ms.Wendthatthebroadeningisconsistentwiththeprocessofmolecular 37

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Figure2-7: Ionizationthresholdofheliumasafunctionofrelativedensity.Theblacksquaresareionizationthresholdsinwarmgas,theredcircleisionizationthresholdincoldgas(13K)andthebluetriangleistheionizationthresholdinheliumII(1:830K).Inthewarmgas,oratdensitiesLHe,theionizationthresholdremainslargelyunchanged,whichsuggestsionizationisoccurringinthemultiphotonregime.Asdensityapproachesthatofliquidhelium,theionizationthresholddropsbyafactorof6. Figure2-8: Singleshotimagesoftheexcimerlineunder0mW/cm2heatux.Theimageistaken(a)immediatelyafterthefslaserpassesthroughthechannel;(b)25msaftertheexcimerlinecreation.Thelineisdimmerandmorediuse,butisatthesamepositionasin(a),withnodistortions;(c)50msafterexcimerlinecreation.Stilldimmer,butwithnorelativemovementofanyportionsoftheline. diusion,thusweconcludethattheeectoflaserheatingwillplaynoroleinsubsequentexperiments. 38

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2.2.3SecondSoundAlthoughmostofthisdissertationisconcernedwiththevisualizationofnormal-uidturbulence,itisneverthelessinterestingandimportanttocoupleinformationaboutthenormaluidwiththebehaviorofthesuperuidunderthesameconditions.Tothisend,secondsoundresonancesinthechannelareexcited(anddetected)byoscillatingsuperleaktransducers[ 50 ].Fig. 2-9 showsadiagramofthetransducerdesign.Theporesizeofthemembraneis0:2mandthechannel-facingsideofthemembraneiscoatedwith200-500Aofgold. Figure2-9: Across-sectioneddiagramofthe2ndsoundtransducer(designedbyWeiGuowiththeassistanceofGaryIhas)tobemountedbelowthevisualizationcube.Thegold-coated(200to500A)microporous(0:2m)membraneisstretchedtautontheplasticmembraneholder.Whenthebrasselectrodeisbiasedwithavoltagebetween50)]TJ /F1 11.955 Tf 10.78 0 Td[(150V,themembraneispulledtowardtheelectrode.Oscillationsofthemembranearethenproducedbyanoscillatingdrivevoltagefromafunctiongenerator.Suchatransducercanfunctionasanemitterorreceiverof2ndsoundwaves.ThisdesigndeviatesfromtheoriginaldesignpresentedinRef.[ 50 ]inthatwehaveaddedaplasticbasesupportedbyspringwasher,allsetintoastainlesssteelangethatistobesealedtothechannelwithanindiumo-ring.Theadvantageofsuchadesignisthat,inprinciple,itisentirelyreusable. Fig. 2-10 demonstratestheelectronicssetupforthe2ndSoundexperiments.Bothemittingandreceivingtransducersarebiasedat50Vwitha6Vpeak-to-peakdrive 39

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voltageappliedtotheemitterbythefunctiongenerator.CapacitorsareaddedtoblockDCsignalsfromthelow-levelelectronics.ThedetectedwaveformissenttoaStanfordResearchSystems830lock-inamplier,whichthenampliesthesignalanddisplaysitviaLabView. Figure2-10: Adiagramshowingthecongurationofthe2ndsoundgeneration/detectionelectronics.Eachtransducer(labelled`rcvr'forreceiverand`xmitter'fortransmitter)isgivenaDCbiasbetween50-150V.Themoresensitiveelectronics,namelytheStanfordResearchSystemsmodel830lock-inamplierandalsothefunctiongeneratorareprotectedfromunexpectedvoltagespikesfromthevoltagessourcesbytwo200nFcapacitors.Thefunctiongeneratordrivesthetransmittingtransducerwithasinusoidalvoltagewaveformwithpeak-to-peakvoltageof2Vinordertoexcite2ndsoundresonances.Thefunctiongeneratoris,ofcourse,alsoprovidingthereferencesignalusedbythelock-intodetecttheoscillatedvoltagefromthereceivingtransducer.ThedetectedresonanceisthenqueriedbyaLabViewprogramandwrittentoatextlestoredonthedataacquisitioncomputer. The2ndsoundattenuationexperimentsareperformedinthefollowingway.First,thefunctiongeneratorissettosweepalargerangeoffrequencies,fromthefundamentalfrequencyofthechanneltoabout40kHztondaresonancepeakwiththelargestqualityfactor(Q)andgoodsymmetry,whichcorrespondstolargestsensitivitytovortexlinedensity.Fig. 2-11 showsaportionofthefrequencysweepfrom15kHzto30kHz.Wendthat,generally,resonancepeakswiththelargestQ(600)arefoundbetween19-26kHz. 40

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Onceacandidatepeakisidentied,wesetthefrequencygeneratortomakeconstantsweepsofwidth600Hzcenteredaroundtheresonancefrequencyofthepeak.Thechosenwidthmustincludeaatregionbetweenresonancepeaks,sothatpeakamplitudecanbereliablycalculatedwhileatthesametimeminimizingthelengthofdatacollection.Duetothetemperaturedependenceofthe2ndSoundvelocity,temperatureuctuationsalterthefundamentalresonancefrequencyofthechannel.Althoughthisfrequencyshiftmaybesmallforthefundamental,theshiftismultipliedbytheharmonicnumberforhigherresonances.Thusforourexperimentsnear20kHz,frequencyshiftsinthefundamentalcanhaveexaggeratedresults.Duetoexcellenttemperaturecontrolinthecryostat,peakshiftsduetotemperatureuctuationsareheldtonomorethan4Hzat20kHzatT=1:830K.Dataisonlycollectedinthiswayaftertheheaterhasbeenactivefor60sormore,toensurefullydevelopedcounterowinthechannel.Betweeneachdataset,wecollecteda`baseline'measurementatzeroheatuxtocheckforremnantvorticityandcontrolforanomalousbaselineshift. 41

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Figure2-11: Asweepof2ndsoundresonancesfrom15kHzto30kHz.Wendthat,althoughtheQmaybequitehighcloseto30kHz,ashoulderbeginstoappearinthepeaks.Additionally,peaksatsuchhighfrequenciesexhibitevergreaterfrequencyshiftsduetotemperatureuctuations.WendthatthebestcompromisesbetweenresonancefrequencyandQfactorarethosebetween19-26kHz. 42

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CHAPTER3DATAANALYSISThischapterpresentsthedataanalysisalgorithmsandconsiderationsforthedatacollectedinboththeLIFand2ndsoundattenuationportionsoftheexperimentdescribedinCh. 2 .Therstportion,x 3.1 ,dealswiththesubtletiesofanalyzingLIFimagesinareal,experimentalenvironment.Confoundingfactorssuchaslightreectionfromthechannelwalls,darkcurrentsandtheeectoftheuidvelocityonexcimerdensityarediscussed.Then,analgorithmthatcanamelioratetheeectsoftheaforementioneddicultieswhilecalculatingvelocitystatisticsisbrieysketchedout.Section 3.2 describesthestatisticsperformedontheattenuated2ndsoundresonancepeakstocalculateavortexlinedensityfromtheavailabledata. 3.1ImageProcessingAlgorithmAreliableandwell-consideredschemeforimageprocessingiscriticalforexperimentsofthisnature.Therawdatacomesintheformofa1024x1025matrixofintegers.TheICCDcameraitselfhasaresolutionof1024x1024,butanextracolumnisappendedtonumbertherows.Thefollowingchapterdescribesthesequenceoftheimageprocessingalgorithm:binning,Gaussiantting,comparisontobaselineandgenerationoftherelevantvalues(velocityproles,structurefunctions,velocityPDFs,etc).ThealgorithmdescribedinthischapterwasdevelopedentirelybyJianGaoasapartofhisPhDworkunderthesupervisionofDr.WeiGuooftheFloridaStateUniversityMechanicalEngineeringDepartmentandNationalHighMagneticFieldLaboratory.Imageprocessingstartswithbinningoftheimage.Sincethechannelllsapproximately80%oftheeldofview(FOV)oftheCCD(about800pixels),generallytherst100andlast100pixels(px)canbediscarded,sincetheydonotcontainanyexcimers.Theremainingpixelswillbedividedinto32pxbinsandthepixelsaresummedacrosstheirrowsineachbin.Fig. 3-1 (a)showsasampleimage,withoneexaggeratedbindesignatedbythedashedyellowlines.Fig. 3-1 (b)showsthesubsequentstep,inwhichtheintensity 43

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ofeachbinmustbettedtoaGaussianinordertodeterminethepositionoftheexcimerlinesegmentineachbin.Whentheimageinquestionisofanexcimerlineatnearly0sdrifttimeand0mW/cm2heatux,itisacomparativelysimpletasktondtheexcimerlinewithaGaussiant.However,mostimageswillbetakenatmanytensofms,withheatuxesaslargeas400mW/cm2,whichcanintroduceseveralconfoundingfactors.Forinstance,atlargedrifttimes,Penning-Ionizationrapidlydecreasestheexcimerdensity(seex 1.2.2 andRef.[ 43 ]),whichisdemonstratedbythesignicantlyattenuateduorescencelightintensity.Similarly,atlargeheat-uxes,thesaturationdensityofmoleculesmightbedecreased,whichalsowilldecreaseuorescencelightintensity(seediscussionofexcimercreationneartheendofx 2.1 ).Further,darkcurrents,whicharepresentinallCCDcameras,andreectionsoftheuorescencelightfromthebackofthechannel,maycombinewiththeaforementionedeectstocreateimageswherethelocationofthelinemaynotbeobvious,especiallytoacomputer.Itispossibletoempiricallydeterminethemeanlightintensityemittedbytheexcimerlineunderthemostfavorableconditions(0sdrifttime,0mW/cm2heatux).Withthisvalueinmind,itisthenpossibletoeliminatepixelscontainingcountsthataresignicantlylargerthanthemean.Additionally,particularlybrightpixelsthatarefarfromtheexperimentalregion(generally,theexcimerlinedoesnotdriftmorethanafewhundredpixels)canalsobeeliminated.Subsequently,thebinscanbesummedacrossthecolumnstofurtherincreasethesignal-to-noise.TheGaussiantoneachbinthendeterminestheexcimerlinelocationandlinewidthinthebin.IfthelightintensityinacertainsectionoftheimageistoolowtoproduceareliableGaussiant,thatsectioncansimplybeomitted.Aslongastherearenottoomanysuchsections,thestatisticsoftheimageanalysisdonotsuersignicantly.Every10imagesarematchedwithacorrespondingbaselineimage(takenat0msdrift,0mW/cm2heatux)inordertohedgeagainstanygradualbaselineshifts.Eachsectioniseitherassignedasabaselinevalue,ifitcomesfromabaselineimage,orit 44

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iscomparedtotherelevantbaselinesectiontodetermineadisplacement.Inthiswaywecancalculateupto6000velocitiesfromasetof200images.Expressionsformeanvelocity,umean,velocityuctuations,u,andturbulentintensityIaregivenbyumean=1 NNXn=1unun=un)]TJ /F6 11.955 Tf 11.95 0 Td[(umeanI=( u2)1 2 umeanwhereunisthevelocityinbinn,Nisthetotalnumberofbinsorlinesegments,unisthevelocityuctuationinbinn,andthebaron u2indicatesanaverageoverNlinesegments.TheresultsofthesecalculationsareplottedinCh. 4 . Figure3-1: Binningandttingofthecaptureimage.(a)Showsabaselineimage(200mW/cm2,0msdrift)withadistortedlineoverlay(200mW/cm2,40msdrift).Thedistortedlinehasbeenosetupwardtoprovideanexaggeratedcomparisontothebaseline.Thedashedverticalyellowlinesindicatethewidthofabin,alsohighlyexaggeratedforclarity.Thepositionofthedistortedlinesegmentfoundinthebiniscomparedtothepositionofthebaselinesegmentlocatedinthebin.Thedisplacementisdividedbytdrifttocalculatethevelocityofthelinesegment.(b)ArepresentativesampleoftheGaussiantsofthelightintensityineach32pxbin.ThecenteroftheGaussianistakentobethepositionofthelinesegment(convertedfrompixelstomm)whiletheFWHMistakentobethewidthoftheline. Thestructurefunctionisacalculationofthevelocityincrementbetweenanytwopointsintheuid.Inthetransversecase,wechoosetwodierentpointslocatedontheexcimerline.Inthelongitudinalcase,wewouldcalculatethevelocitydierencebetweenthesameexcimerlinesegmentontwodierentexcimerlinesseparatedbysomedistancer.Weareinterestedspecicallyinthesecondorderstructurefunctionsprimarilybecause, 45

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undertherightconditions,theexponentofthesecondorderstructurefunctioncanbesimplyrelatedtotheexponentoftheenergyspectrum.Specically,iftheenergyspectrumhasformE(k)/k)]TJ /F5 7.97 Tf 6.58 0 Td[(pwhere1
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corner.Sinceturbulenceisarandomphenomenon,wecanimmediatelyconcludethatodd-order,transversestructurefunctionsmustbebeidenticallyzero.Inlightofthis,wecalculateonlythesecond-order,transversestructurefunction.Bymovingthereferencepoint(usuallyatthecenteroftheexcimerline)todierentpositionsalongtheexcimerline,wecancheckwhethertheturbulenceishomogeneous.Ofcourse,thestatisticalweightingforpointswithaseparation,r,closetothechannelwidthismuchsmallerthanforpointsonthelinewithsmallseparation. 3.22ndSoundAttenuationThe2ndsoundexperimentsareperformedasdescribedinthepreviouschapter.Wecollectdatapointsatarateof10pointspersecondoverthecourseof62s.Eachsuchscanoveraresonancepeakhaswidth600Hztoensurepeakandbaselinearecaptured.600HzismuchwiderthantheFWHMoftheresonancepeak,evenwhenitisstronglyattenuatedbythepresenceofvortexlines.Eachexperimentalrunsweepsfrequenciesoverthesamepeakabout20times.Zeroheatuxdataiscollectedbetweeneachnon-zeroheatuxtoavoidtheeectsofpossiblebaselineshifting.Theresonancepeakandleft(orright)baselineiscollectedbyaprogramthatiteratesthrougheachdatale.Forheatuxesinexcessof60mW/cm2,itissucienttoaveragepeakheightstondtheattenuationofthe2ndsoundsignal.However,atheatuxes<60mW/cm2theattenuationissmallenoughthatasamplesizeof20producesextremelylargeerrorbars.Insuchcases,eachnon-zeroheatuxamplitudeissubtractedfromeachzeroheatuxbaselineamplitude.Analyzingthedatainthiswayallowsforbetterstatisticsintheverylowheatuxregimes.Fig. 3-3 comparesanunattenuated2ndsoundresonancepeak(collectedat0mW/cm2)withanattenuatedpeakcollectedat300mW/cm2heatux.Recallthatthevortexlinedensityasafunctionofattenuationwasgivenbyequation( 1{11 ):L=6 BA0 A)]TJ /F1 11.955 Tf 11.95 0 Td[(1 47

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Figure3-3: Acomparisonoftheunattenuated2ndsoundresonanceat19:3kHztothesameresonancecollectedundersteadystatecounterowconditionsgeneratedbyaheatuxof300mW/cm2.TheblackarrowmarkedbyA0indicatestheunattenuatedresonanceamplitude,whiletheredarrowmarkedbyAindicatestheattenuatedamplitudeattheaforementionedheatux,anddenotestheFWHMoftheunattenuatedpeak.Themeasuredattenuationinthiscaseisabout33%.Additionally,itisclearthatthethewaveformisnotsignicantlyaectedbythevorticityinthechannel,norshouldweexpectanysignicantbaselineshiftsintheexperiment. whereAandA0aretheattenuatedandunattenuatedamplitudes,respectively.istheFWHMoftheunattenuatedpeak,isthequantumofcirculation(~=m4)andBisthemutualfrictioncoecient.Theunattenuatedamplitude(A0)andattenuatedamplitude(A)arebothshowninthegureusingtherightsideoftheresonanceasabaseline. 48

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CHAPTER4STEADYSTATECOUNTERFLOWThischapterwillpresentanddiscussthedatacollectedoverthecourseoftheexperiment.Theresultsthatfollowcanbefoundinanextremelyabbreviatedforminreference[ 51 ].Sectionx 4.1 consistslargelyofimagestakeninfullyturbulentcounterow.x 4.1.1 presentsvelocityprobabilityfunctionsandthemeanvelocityasafunctionofheatuxandtheturbulenceintensityofthefullyturbulentnormaluid.x 4.1.2 presentsstructurefunctionscomputedacrossawiderangeofheatuxesandtheircorrespondingenergyspectrapowerlaws.Itwillbeshownthattheenergyspectrumhasanon-classicalpowerlaw,apparentlyindicativeofquantumturbulence.Finally,x 4.2 presentstheexcimerlinedistortioninwhatwetermthedistortedlaminarregionandpresentssomepreliminaryevidencethatthisdistortionisrelatedtotheTI/TII(seex 1.1.3 )transitioninthesuperuidcomponent. 4.1FullyTurbulentFlowFig. 4-1 showsthedistortionsoftheexcimerlineinfullyturbulentowconditions.Fig. 4-1 (a)showsthebaselineimageunder0mW/cm2heatuxand0msdrifttime.Fig. 4-1 (b)-(d)showtherandomdistortionsoftheexcimerlineunderfully-turbulentthermalcounterowconditions.Indeed,suchconditionswillbethefocusofthissection. 4.1.1TheNormalFluidVelocityFig. 4-2 showsthemeannormaluidvelocity,umeanasafunctionofheatux.Theowvelocity,u,iscalculatedasdescribedinCh. 3 .Eachopencircledatumistheaverageofuovertheexcimerlineandapproximately200single-shotimagesundertherelevantcounterowconditions.Therst5datapointsareobtainedinthelaminar(openredtriangle)anddistortedlaminar(opensquare)regions.Thesolidblacklineisthenormaluidvelocitygivenbyequation( 1{9 )[ 52 ]vn=q sT 49

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Figure4-1: Imagesoftheexcimerlineunderavarietyofsteady-statecounterowconditions.(a)Abaselineimageat200mW/cm2and0msdrifttime.Thelineismeasuredtobe110mwideand9:5mmlong.(b)-(d)Single-shotimageoftheexcimerlinedistortionataheat-uxof200mW/cm2and40msdrifttime.Weexpectthisheatuxtobermlyinthefully-turbulentregime,andtherandomdistortionsofthelinesupportthisexpectation. whereqistheheatux,vnisthenormaluidvelocity,isthedensityoftheuid,sistheentropyperunitmassandTisthetemperatureoftheuid.Theerrorbarsinthefullyturbulentregimearetheaverages (u2)1=2=h (u)]TJ /F6 11.955 Tf 11.96 0 Td[(umean)2i1=2,whichrepresentthemagnitudeofthevelocityuctuations.Asexpected,theybecomemoreextremeasheatuxincreases.SinceallofthedatainFig. 4-2 liealongthelinedescribedby( 1{9 ),weconsiderthisconrmationthattheheliumexcimersarewell-coupledtothenormaluid.Withthevelocitydataextractedfromtheimages,wecanalsocalculatetheturbulenceintensityoftheow,givenby (u2)1=2 umean.Again,eachdatumiscalculatedfromanaverageoverthelengthoftheexcimerlinein200images.Fromthedata,itiscleartheturbulenceintensityismoreorlessindependentofheatuxupto350mW/cm2, 50

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Figure4-2: Normaluidvelocityasafunctionofheatux(_q).Theopenredtrianglesdenotevelocitiesobtainedduringfullylaminarow,theopenblacksquaresarevelocitiesobtainedinthedistortedtransitionregimeandopenbluecirclesarevelocitiesinthefullyturbulentow.Thesolidlineisaplotof( 1{9 )andtheerrorbarsindicatethemagnitudeofvelocityuctuationsintheow.Thedataclearlyshowthattheexcimersarewell-coupledtothenormaluid[ 51 ]. beyondwhichitbecomesimpossibletoproducehighqualitysingle-shotimagesofthedistortedexcimerline.Inclassicaluids,theturbulenceintensityinsimplegeometries,suchasachannelwithsquarecross-section,isconsistentlyfoundtorangebetween1%and5%.Notably,theturbulenceintensityweobserveinsuperuidheliumisnearly30%,signicantlyhigherthaninaclassicaluid.Althoughthisisareal,reproducibleeect,asyettherehavebeennocluesastowhyexactlythisshouldbethecase.ItmaybeinterestingtonotethatthecalculatedturbulenceintensitiesplottedinFig. 4-3 assumethattherelevantmeanvelocityisthevelocityofthenormaluid,vn.However,iftherelevantmeanvelocityisthecounterowvelocity,vn)]TJ /F6 11.955 Tf 11.69 0 Td[(vs,thenwendthattheturbulenceintensitydropsto20%.Arepresentativesampleofthevelocityprobabilitydensityfunctions(PDFs)ispresentedinFig. 4-4 .Ineachinstance,theredlinerepresentsatofthedatatoaGaussiandistribution.Itisapparentthatthevelocitydistributionisverycloseto 51

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Figure4-3: Turbulenceintensityplottedasafunctionofheatux,_q.Asexpected,theintensityisapproximatelyconstantacrosstheentirerangeofusableheatuxes.Note,however,thatthemagnitudeoftheturbulenceintensityinsuperuid4He(30%)ismuchhigherthaninaclassicaluid,whichis5%.Althoughthiseectisrealandreproducible,itisunclearwhytheturbulenceintensityforheliumIIshouldbesomuchhigherthaninclassicaluids[ 51 ]. Gaussianforallheatuxes.(TheplotsinFig. 4-4 arereproducedsomewhatlargerintheAppendixalongwithadditionalvelocityPDFplotsatheatuxesinbetweenthoseofFig. 4-4 ).Furthermore,thepowerlawtailsreportedintheexperimentsofRef[ 53 ]andattributedtovortexlinereconnectionsarenotpresentinourdata.Thisdoesnotnecessarilysuggestthatthesetailsdonotexist.Basedontheviscousrelaxationtime(seeEq 1{13 ),wecanbeverysurethattheexcimersarewellcoupledtothenormaluid,notvortexlines.However,thenormaluidnearthelocationsofvortexlinereconnectionscan,briey,haveaveryhighvelocityduetothemutualfriction,whichthencontributestotheformationofpower-lawtails.Onshortprobingtimescales,thesehighvelocitiesshouldbeobservable.However,atthecomparativelylongtimescales(10ms)inthisexperiment,theexcimershavetimetomoveawayfromthesehighvelocityregionsandexhibitanormalvelocity. 52

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Figure4-4: ThevelocityPDFsareshownforsteadystatethermalcounterowwithheatuxes(a)180mW/cm2,(b)243mW/cm2and(c)350mW/cm2.EachPDFisshownwithaGaussiant(redline),whichmatchesthedatacloselyineachcase.ThecalculatedmeanowvelocityandRMSvelocityuctuationsaredisplayed.Themeanvelocityvaluesagreewellwith( 1{9 ). 4.1.2TheSecondOrderTransverseStructureFunctionWithknowledgeoftheexcimerlinesegmentvelocities,wecalculatethesecond-order,transversestructurefunctionasdescribedinx 3.1 .Fig. 4-5 presentsS?2asafunctionofhorizontaldistance,r,ontheexcimerline.Wendthat,infullyturbulent,steadystatethermalcounterow,S?2/rn,wheren=10:05[ 51 ]independentofheatuxforr<2mm.Additionally,forrsmallerthanthesizeofthechannel,wendnodependenceofthecalculatedstructurefunctionsonthereferencecoordinate,R,whichsuggeststhatthenormaluidturbulenceishomogeneousattheselengthscales.Whilethecalculatedstructurefunctionforlargeralsoappearstobeindependentofthelocationof 53

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thereferencepoint,thestatisticalweightingforthesevaluesismuchsmaller,sinceonlyoneortwovaluesofS?2canbeextractedatr1cm.Thus,wecannotbecertainofthehomogeneityoftheturbulenceatlargescales(r1cm).Thebehaviorofthe2ndorder,transversestructurefunctioninsteadystatecounterowstronglysuggeststhatthecharacterofcounterowturbulenceisnon-classicalatallprobedlengthscales. Figure4-5: Aplotofthetransverse,second-orderstructurefunctionat150mW/cm2(opengreencircles),225mW/cm2(openredtriangles)and300mW/cm2(openbluesquares).Theblackdottedlineover-layoneachofthedatasetsisaplotofr1.Thepowerlawofthesecond-order,transversestructurefunctionisthencalculatedtobeS?2r1,whichcorrespondstoanenergyspectrumpowerlawofE(k)k)]TJ /F2 7.97 Tf 6.58 0 Td[(2[ 51 ]. Inclassicalturbulence,farfromboundaries,weexpectthatthestructurefunctionoftheturbulencewillleveloastheseparationdistancerexceedsthesizeoftheenergycontainingeddiesandthevelocitiesbecomeuncorrelated.ExperimentsusingtheRELIEFtaggingmethodinasupersonicjetofairndthatjetowexhibitsfeaturesofhomogeneousturbulence.Thatistosay,thesecondordertransversestructurefunctionhasformS?2(r)/r2=3,whichcorrespondstoenergyspectrumE(k)/k)]TJ /F2 7.97 Tf 6.59 0 Td[(5=3[ 54 ].Bothoftheseexponentialformscorrespondtoclassical,Kolmogorovturbulence.Althoughsuchexperimentshavebeendoneforchannelow(usingthearelatedmoleculartaggingtechnique:PHANTOMM),theregionofinterestwassmallandfarfromboundaries[ 55 ]. 54

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Thebehaviorofthestructurefunctioninclassicalturbulentchannelownearthewallsisstillnotobvious.However,numericalevidencesuggeststhatthevalueshouldincreaseduetoincreaseinthemeanshear[ 56 ].SincethestructurefunctionhasformS?2/rn,wherenis1,weareabletoimmediatelydeducethepowerlawoftheenergyspectrum:E(k)/k)]TJ /F2 7.97 Tf 6.58 0 Td[(2(seethestructurefunctiondiscussioninx 3.1 ).Indeed,itispossibletodeterminethefullformoftheenergyspectrumfromthestructurefunctiondata,buttheprocessisnon-obviousandnon-trivial.Knowingthepoweroftheenergyspectruminthecascaderegionwassucientfortheseexperiments,butthereiscurrentworkbeingperformedtodeterminetheexactenergyspectrumanditsconsequencesforcounterowturbulence[ 57 ].Aclassical,E(k)/k)]TJ /F2 7.97 Tf 6.58 0 Td[(5=3,(Kolmogorov)powerlawwouldimplythatbothuidsmoveasone,classicaluid.Indeed,thisoccursafterthecounterowisallowedtodecayforsucienttime(someevidenceofthedecayingthermalcounterowstructurefunctionsisprovidedinRef.[ 51 ]).TheexperimentalsetupdescribedinCh. 2 canbeusedtoprobethebehaviorofthenormaluidindecayingcounterow,butadetaileddiscussionofsuchexperimentsisoutsidethescopeofthisdissertation.Thatweobserveanon-Kolmogorovenergyspectruminsteadystatecounterowsuggeststhattheturbulenceisindeednon-classical.Insuchnon-classicalturbulence,energyisexpectedtobedissipatedinawiderangeoftheavailablelengthscalesduetothemutualfrictionbetweennormalandsuperuidcomponents. 4.2Laminar,DistortedLaminarFlowandtheTI/TIITransitionFig. 4-6 showsthedeformationoftheexcimerlinestartinginthelaminarowregimeandprogressingthroughwhatwehavetermedthedistortedlaminarowregime.Eachimageaftertheinitial,undistortedbaselineisanine-shotaveragewhichbothemphasizestheconsistencyofthecentralparabolicstructureandimprovesthesignaltonoiseratiointheimageatdrifttimes100ms.Theatteningoftheedgesoftheexcimerlinebeginsnear55mW/cm2,butispresentedhereathigherheatuxestounambiguouslydisplaythephenomenon. 55

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Figure4-6: Thedistortedlaminarregioninsteadystatecounterow.(a)Theexcimerlinebeforetheheaterisactivated.b)Anineshotaverageofthelineat10mW/cm2and900msdrifttime.Fromtheparabolicdistortionoftheline,weconcludethattheowislaminar.c)Anine-shotaverageoftheexcimerlinedistortionat62mW/cm2and150msdrifttime.Notethatthecenterofthelineisdistortedinawaythatsuggestslaminarow,buttheedgesoftheexcimerlinehavebeguntoattenout.d)Anine-shotaverageoftheexcimerlinedistortion75mW/cm2at100msdrifttime.Atthispoint,thereisstillsomelaminardistortioninthecenteroftheexcimerline,butmuchofthelinehasbecomeattenedout. AlthoughFig 4-6 (b)certainlyappearstobelaminar,itispreferabletoprovidesomequantitativeevidencetothateect.Fig. 4-7 showsanaveragedimageoftheexcimerlineunderowconditionsweexpecttobelaminar(_q=20mW/cm2,un=1:29mm/s)andthecorrespondingtoftheexcimerlinevelocitytoaparabolicvelocityprole.Wendthattheempiricallydeterminedmeanvelocityisconsistentwiththenormaluidvelocitycalculatedfrom( 1{9 )and,further,thatthevelocityatthecenteroftheexcimerlineistwicethatofthemean,whichisconsistentwithPoiseuilleow.Havingconrmedthatthermalcounterowproducedbylowheatuxes(<50mW/cm2,<3:23mm/s)resultsinalaminarnormaluidowprole,wenowturnourattention 56

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Figure4-7: Aquantitativeanalysisoflowheatuxnormaluidow.(a)Showsa9-imageaverageofthedeformationoftheexcimerlineatT=1:830Kandaheatuxof20mW/cm2.Thedistortionoftheexcimerlineappearstobelaminar.(b)Showsaparabolictoftheexcimerlinevelocity(determinedfrom(a))whichyieldstheexpectednormaluidvelocity. tothedistortionsinthelaminarproleencounteredbetween50)]TJ /F1 11.955 Tf 12.53 0 Td[(80mW/cm2(3:23)]TJ /F1 11.955 Tf -438.44 -23.91 Td[(5:15mm/s).Theatteningoftheexcimerlineproceedstoincreaseuntil,ataheatuxofroughly80mW/cm2(5:15mm/s),thecentrallaminarregiondisappearsentirely.Theatportionsofthelineinthedistortedlaminarregimeareexpectedtoactuallyberandomdistortionsduetointeractionsbetweennormaluidandthevortexlinetangle,butwhichareaveragedouttoaatlineduetotheaveragingofcapturedimages.Above80mW/cm2(5:15mm/s),itbecomespossibletotakesingle-shotimagesoftheexcimerlineandrandomdistortionsareobserved,conrmingthetransitiontofullturbulence.Fairlyrecentnumericalstudies[ 58 { 61 ]ndthat,underPoiseuilleowinasimple,squarechannelgeometry,vortexlinedensityinitiallyaccumulatesnearthewalls.Inlightofthesesimulations,wethenpositthattheatteningoftheexcimerlineiscausedprimarilybymutualfrictioninteractionsbetweenthenormaluidandvortexlinesnearthewallsofthechannel.Itispossibletoexcludesmall-scalenormaluidturbulenceasthecauseoftheexcimerlineattening.Recentanalysisofsingle-shotimagestakeninthelaminar(40mW/cm2),distortedlaminar(70mW/cm2)andfullyturbulentows 57

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(150mW/cm2)withaxeddrifttimeof100msindicatethatthewidthofthelineincreasesby<5%fromlaminartodistortedlaminarows,butwidenssignicantly(14%)infullyturbulentow[ 62 ].Sincethelinewidthinthedistortedlaminarowisconsistentwiththeexpectedwidthduetonon-turbulentdiusion,small-scalenormaluidturbulenceisruledoutasacauseofthelineatteningshowninFig. 4-6 .Ifthedistortedlaminarregioniscausedbyvortexlinebuildupnearthewallsofthechannel,thentheincreasedaveragevorticityinthechannelshouldbedetectableina2ndsoundattenuationexperiment.Furthermore,somenumericalsimulations[ 59 ]predictthatthetransitiontofullnormaluidturbulenceshouldcorrespondtoanabruptincreaseinvortexlinedensity.Similarly,earliernumericalworkperformedbyMelotteandBarenghi[ 63 ]suggeststhatthetransitionfromTIsuperuidturbulencetoTIIsuperuidturbulenceisintimatelylinkedtoinstabilitiesinthenormaluid.Nowthatthebehaviorofboththesuperuidandnormaluidcomponentscanbereliablyprobed,weseektotestthesepredictionsexperimentally.TheworkofToughandMartin[ 14 ]isreproducedbelowinFig. 4-8 todemonstratethecharacteristicsoftheTI/TIItransition.AlthoughthedatapresentedinFig. 4-8 (a)-(c)arecollectedatsignicantlydierenttemperatures,theydemonstratethattheTI/TIItransitionoccurswithreproduciblecharacteristics.Asexpected,theinitiallinedensityiszerountiltherstcriticalvelocity,vc1,isreached,afterwhichthelinedensityincreasesslowlyuntilthesecondcriticalvelocity,vc2isreached,atwhichtimethelinedensityjumpsupandthensmoothlysettlesintothebehaviorrstmeasuredbyVinen[ 15 ].CalculatingvortexlinedensityfrommeasurementsofthetemperaturegradientnecessarilyrequiredthatallexperimentsthatdemonstratedaTI/TIItransitioninsuperuidheliumwereperformedinchannelsofwidthatleastanorderofmagnitudesmallerthantheonedescribedinx 2.1 .Thecharacteristicsizeofourchannelissuchthatthetemperaturegradientinthechannelwouldbebelowtheresolutionoftheavailable 58

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Figure4-8: FiguresreproducedfromRef[ 14 ]plottingadimensionlessquantity,L1=20dasafunctionofcounterowvelocity,vns.(a),(b)and(c)showessentiallythesamebehaviorofthevortexlinedensity,butattemperaturesof1:5K,1:6K,and1:7Krespectively,inachannelofcharacteristicsize1mm.Ascounterowvelocityincreasesfromzero,thelinedensityisinitiallyzero,beginstoriseattherstcriticalvelocity,vc1andtendstoincreaseabruptlyatthesecondcriticalvelocityvc2.ReprintedFig.10withpermissionfromK.P.Martin,J.T.Tough,Phys.Rev.B27(1983), http://dx.doi.org/10.1103/PhysRevB.27.2788 .Copyright2015bytheAmericanPhysicalSociety. thermometers.Conveniently,inlargechannels,thevortexlinedensitycanbeprobedbymeasuringtheattenuationofstanding2ndsoundwaves,thedetailsofwhicharedescribedinx 2.2.3 .Fig. 4-9 (a)showsthesquarerootofthevortexlinedensity,L1=2,asafunctionofheatux,fromthelaminarthroughthefullyturbulentregime.Theredlinerepresents 59

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attoL1=2=(Vns)]TJ /F6 11.955 Tf 13.11 0 Td[(Vc)andreturnsavalueof162s/cm2,whichisconsistentwithprevious2ndsoundmeasurementsinthefullyturbulentTIIregion.Fig. 4-9 (b)presentsacloseupofseveralexperimentsconnedtoheatuxeswhichcorrespondtothelaminaranddistortedlaminarregionsinthenormaluidow.Atheatuxesbelowabout55mW/cm2,theuncertaintyonthemeasurementsisequivalenttothemeasuredvalues,butthedistributionofL1=2valuesaround0cm)]TJ /F2 7.97 Tf 6.59 0 Td[(1suggeststhatthisisbelowtherstcriticalvelocity.Thenextregion,from55mW/cm2to80mW/cm2beginswithasharpjumpinL1=2,andsubsequentlydemonstratesasmall,positiveslopethatincreasesuntilitobtainsthevalueforobtainedinthefullyturbulentregime(80mW/cm2). Figure4-9: Squarerootofthevortexlinedensityasafunctionof_Q.(a)TheredlineisatinthefullyturbulentregiontotheequationL1=2=(Vns)]TJ /F6 11.955 Tf 12.38 0 Td[(Vc).Thettedvalueofis162s/cm2,whichisconsistentwithpreviousworkonsuperuidturbulence.(b)focusesonthelowerheatuxregion,indicatingthatthedataarerepeatable,butthelargeerrorbarsbelow50mW/cm2makeitimpossibletodrawconclusionsabouttheexistenceandsizeoftheTIregion.NotethaterrorbarsthatextendbelowzeroarenotmeanttosuggestthatthevalueofL1=2candropbelowzero.[ 51 ]. AcomparisonwithFig. 4-8 suggeststhatthesuperuidenterstheTIIturbulencestateat55mW/cm2and,duetothelargeerrorbars,theTIregion,ifitexists,cannotbeseen.Itappearsthatanincreasingvortexlinedensity,andthusincreasingmutualfrictionbetweenthenormalandsuperuidcomponents,contributestonormaluidinstabilityratherthantheotherwayaroundassuggestedbyMelotteandBarenghi[ 63 ].Further,it 60

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isclearthatthereisnosharpjumpinvortexlinedensityatthepointwherethenormaluidbecomesfullyturbulent.Thisdisagreementwithnumericalsimulationsisperhapsnotverysurprising.Todate,nosimulationsofsuperuidowareself-consistent.Thatistosay,allsimulationsofsuperuidownormallyenforceasinglelaminarorturbulentowproleinthenormaluid,whichisnotpermittedtoevolveasafunctionoftime.However,veryrecently,YuiandTsubotahaveperformedsimulationswhich,thoughtheyarenotself-consistent,allowforatteningofthePoiseuilleprole.Theirresultspresentfurtherevidencethattail-attenedowor,aswecallit,distortedlaminarow,isaowstatewithcharacteristicsbetweenthoseoffullylaminarowandfullyturbulentow[ 64 ].Giventheselatestadvances,itisreasonabletoexpectthat,withguidancefromexperimentalresults,futureself-consistentsimulationscanreproduceknownphenomenaandprovidefurtherinsightintoquantumturbulence. 61

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CHAPTER5CONCLUSIONTheneedforaneective,reliabletracerforthevisualizationofturbulenceinsuperuidowshasdriventhedevelopmentofanexperimentalapparatuswhichusestheexcitedheliumdimer(He2)asatracerinthenormaluid,andexploitstheavailableelectronicenergylevelsofthismoleculetoperformlaser-induceduorescencevisualization.Webuilduponpreviousworkdetailingthedevelopmentofheliumexcimersastracers[ 35 , 37 , 38 ]tocreateasystemthatgeneratesandimagesathinlineofheliumexcimersinbulkliquidheliumIIow.Conveniently,thevastmajorityoftheexperimentalapparatuscomponentsarecommerciallyavailable,theonlycustommachinedcomponentsbeingtheexperimentalchannelitselfandthe2ndsoundtransducers.Wepresentaninitialinvestigationoftheionizationthresholdof4Heasafunctionofdensitytocharacterizethewidthoftheexcimerlineanddeterminetheoptimaloperationparametersoftheapparatus.Wendthationizationlightcanbeproducedevenatlowheliumdensities(i.e.atSTP)whenionizationoccursentirelybythemultiphotonmechanism.Inthesuperuid,wendthattheexcimerlinehaswidth100mjustunderthebreakdownthreshold(60J/pulse).Abovethebreakdownthreshold,weseediuseclustersofexcimersform,butwhichareunusableforthevisualizationexperimentstofollow.TheLIFexperimentsinsteadystatecounterowperformedat1:830Kspannedtherangeofheatuxesfrom0)]TJ /F1 11.955 Tf 11.96 0 Td[(350mW/cm,2andrevealedanumberofinterestingcharacteristicsofnormaluidow.Thenormaluiddoeshaveafullylaminarregimebelow50mW/cm2(vn3:26mm/s).Unexpectedly,thelaminarproleoftheexcimerlinebegantodisplayaattening(inaveragedimages)ofthelineneartheexperimentalchanneledgesastheheatuxroseabove55mW/cm2untiltheentirelinewasattenedabove80mW/cm2(vn5:15mm/s.Somenumericalsimulations[ 58 , 59 , 61 ]indicatethatthereisapreferentialbuildupofvorticesnearthewallsofthechannel,whichmaysuggestthatlineatteninginthedistortedlaminarregionisduetothenormaluidinteractionwiththesevortices.Very 62

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recentexperimentalresultsruleoutsmall-scaleturbulenceinthenormaluidasacauseoftheattening,lendingmorecredencetotheaforementionednumericalstudies.Thesesimulationsalsopredictthatthevortexlinedensityjumpssharplywhenthenormaluidtransitionstofullturbulence.However,ourexperimentsprobingthevortexlinedensityinthechannelusingthemethodof2ndsoundattenuationndthatthevortexlinedensityincreasessmoothlyfromthestartofthedistortedlaminarregionthroughthetransitiontofullturbulenceandbeyond.Furthermore,anattempttoexperimentallyinvestigatethenumericalworkofMelotteandBarenghi[ 63 ]regardingtherelationshipbetweennormaluidinstabilityandthestatesofsuperuidturbulence(TIandTII)metwithextremelylimitedsuccess.WhileitisclearthataTIIturbulencestateexistsabove80mW/cm2,coincidentwithafullyturbulentnormaluid,theerrorbarsonthe2ndsoundmeasurementatheatuxesbelow80mW/cm2quicklybecomecomparabletothemagnitudeoftheattenuation,makingdenitestatementsregardingtheTIstateandtheTI/TIItransitionimpossible.DisagreementwithsomeofthepredictionsofRefs.[ 59 , 61 ]likelystemsfromlimitationsofthesimulationsthemselves,whicharenotself-consistent.Thesesimulationsdeneaowproleforthenormaluid,eitherlaminarorturbulent,butdonotallowittoevolvewhilecalculatingthetheevolutionofthevortextangleinthesuperuid.Self-consistentsimulationsrequireaknowledgeoftheboundaryconditionsforsuperuidowinthepresenceofvortexlinesthatcanbepinnedtochannelsurfaces.Ourwork,especiallyinthedistortedlaminarregion,canprovideimportantempiricalevidenceagainstwhichsimulationscanbecheckedandcorrectboundaryconditionsselected.Theover-archinggoalofthiswork,tocapturehighquality,single-shotimagesofnormaluidow,isrealizedinheatuxesabove80mW/cm2(subsequentrenementstotheapparatusextendedthistothedistortedlaminarregionaswell).CalculatingaveragevelocitiesfortheexcimertracersshowsthatthenormaluidvelocityagreeswellwiththevelocitypredictedbyEq. 1{9 ,whichsuggestsstrongcouplingbetweenthenormal 63

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uidandtheexcimertracers.Thenormaluidturbulenceintensity,ameasureoftheroot-mean-squarevelocityofthevelocityuctuationsdividedbymeanowvelocity,showsnodependenceonheatux,buthasvalue30%ratherthantheclassicalvalueof(1%)]TJ /F1 11.955 Tf 12.51 0 Td[(5%)foundinsimpleowgeometries.ThecalculatedvelocityprobabilitydensityfunctionofthenormaluidowisttedwellbyaGaussianfunctionanddisplaysnoneofthepower-lawtailsobservedinRef.[ 53 ].Thedatadonotprovideevidenceagainstpower-lawtailsinthevelocityPDF;insteadtheobservedGaussianPDFislikelyaconsequenceofthecomparativelylargetimescaleofobservationintheexperiment.Thesecond-order,transversestructurefunctionsofnormaluidowarecalculatedforthersttimeeverandconclusivelyshowthattheenergyspectrumofnormaluidturbulenceisdecidedlynon-classical.Thecalculatedenergyspectrumpowerlaw,E(k)/k)]TJ /F2 7.97 Tf 6.58 -.01 Td[(2,isshowntobeindependentofheatuxuptothelimitationsoftheexperiment(350mW/cm2,vn=2:25cm/s).Thesenovelcontributionstothebodyofquantumturbulencenotonlyprovideimportantguidanceforfutureexperimentation,butalsoprovidecriticalnewinformation,suchasboundaryconditionsandvelocityproles,forthedevelopingtheoryofquantumturbulenceanditsnumericalsimulations.Theseexperimentssetthestageforthecreationofself-consistentsimulationsofthermalcounterowandalsosuggestsomeobviousavenuesoffurtherexperimentalwork.SubsequentexperimentsmustbeperformedacrossabroadrangeoftemperaturesinheliumIIandtheymustprobebothsteadystatecounterowanddecayingcounterow,aswellasthevortexlinedensity.Suchworkwouldbenetfromimprovedsensitivityin2ndsoundattenuationmeasurementsatlowheatuxes,perhapsbyusingatechniquepioneeredbyVinen[ 11 ]formeasuringthecharacteristictimeofvortexlinebuildupinthermalcounterow.Additionally,therearetwonaturalextensionsofthecurrentLIFworkbeingperformedonsuperuidhelium.Therstistheinvestigationandcharacterizationofthenormaluidproleinturbulencegeneratedbyatowedgrid.Some2ndsoundattenuationexperimentsforgridturbulenceinheliumIIhave 64

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alreadybeenperformed,initiallybyStalpetal.[ 65 ].Morerecently,theseexperimentshavebeenextendedtoawidervarietyofgridtransparencies[ 66 ]andcurrentexperimentsarecharacterizingthenatureofthesuperuidturbulenceinmuchlargerchannelsasafunctionofgridspeedandgridtransparency[ 67 ].Theseexperimentslackonlyimportantinformationaboutthenormaluidow,whichtheLIFvisualizationtechniqueisuniquelysuitedtoprovide.Secondly,attemperatureswellbelow1K,itisexpectedthattheheliumexcimerscanbecometrappedbyvortexlines.Thusatlowtemperatures,theLIFtechniquecanbeusedtovisualizethebehaviorofthevortexlinesthemselves.Agreatdealofimportantgroundworkonthetrappingdiameterhasalreadybeendone,andshownthatnotonlyisthetrappingdiameternearly100timestheradiusofanexcimer,butalsothatexcimerscanbecarriedthroughthesuperuidbythevortices[ 68 ].Thatathinlineofexcimerscanbecreatedinheliumgas(x 2.2.1 )presentsanumberofinterestingpossibilitiesforclassicalturbulenceinvestigationathighReynoldsandRayleighnumbers(denotedasReandRa,respectively).Comparedtomorecommontestuids,suchasSF6andcompressedair,theviscosity(andthereforeRe)ofcryogenicheliumisbotheasilytunable[ 69 ]andspansseveralordersofmagnitude.Cryogenicheliumhasbeenusedincompact(L1cm),highReynoldsnumber(107)[ 70 ]owexperimentswhichstandinstarkcontrasttotraditionalhighReexperimentsthatrequireenormous,andenormouslyexpensive,specializedfacilities[ 71 ].Indeed,sincetheLIFtechniqueforowvisualizationinheliumisapplicableinallphasesoftheuid,includinggaseousheliumatitscriticalpoint,itisparticularlysuitedtothestudyoflargecirculationsandstructureswhicharecharacterizedbyhighRayleighandReynoldsnumbers[ 72 ],butweremadeinfeasibleduetothelackofeectivevisualizationandvelocimetrytechniquesincryogenichelium.Insummary,wehavepresentedthedetailsofanexperimentalapparatusthatcangenerateathinlineofexcimersinsuperuidheliumandsubsequentlycapturehighquality,single-shotimagesofthemolecularlinedistortioninthecounterowingsuperuid. 65

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Imageanalysisofcounterowat1:830Krevealsnovelstructureinthetransitionofthenormaluidfromlaminartoturbulentow.Weproducetherstevercharacterizationofnormaluidturbulenceandndthattheenergyspectrumhasadecidedlynon-classicalpowerlawofE(k)/k)]TJ /F2 7.97 Tf 6.58 0 Td[(2acrossallprobedheatuxes.Theseexperiments(andtheirextensionacrossawidertemperaturerangeandintodecayingcounterow)provideanimportantcontributionnotonlytoourunderstandingofquantumturbulence,butalsoarepoisedtobecomeavaluablenewtechniqueinthestudyofclassicalhighReynoldsandRayleighnumberows. 66

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APPENDIX:VELOCITYPDFS ThevelocityPDFspresentedinthemainbodyinthetextarereproducedhere,butlargerandsupplementedwithPDFplotscalculatedatintermediateheatuxestoreinforcetheassertionthatthevelocityPDFsareGaussianacrossallprobedheatuxes.Foreachplot,theredlineindicatesaGaussiantandtheheatuxisindicatedintheupperleft-handcorner. FigureA-1: Thenormaluidprobabilitydensityfunctionforaheatuxof180mW/cm2(vn=1:16cm/s). 67

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FigureA-2: Thenormaluidprobabilitydensityfunctionforaheatuxof200mW/cm2(vn=1:29cm/s). FigureA-3: Thenormaluidprobabilitydensityfunctionforaheatuxof243mW/cm2(vn=1:57cm/s). 68

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FigureA-4: Thenormaluidprobabilitydensityfunctionforaheatuxof275mW/cm2(vn=1:77cm/s). FigureA-5: Thenormaluidprobabilitydensityfunctionforaheatuxof350mW/cm2(vn=2:25cm/s). 69

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BIOGRAPHICALSKETCH AlexanderMarakovwasborninMoscowin1988.In1991,hisparentsmadetheeminentlysensibledecisiontoabandontheUSSRinfavoroftheAmericanDream.Atage10,Alexmadethelife-changingdecisiontobecomeaphysicist.In2010hereceivedhisB.Sc.inPhysicsfromCarnegieMellonUniversity.HebeganhisgraduatestudiesattheUniversityofFloridaDepartmentofPhysicsinAugust2010,receivingaMaster'sdegreein2013and,nallygraduatingwithhisPh.D.in2015. 74