Measurements and Modification of Sheared Flows and Stability on the Compact Toroidal Hybrid Stellarator by MarkCianciosa AdissertationsubmittedtotheGraduateFacultyof AuburnUniversity inpartialfulfillmentofthe requirementsfortheDegreeof DoctorofPhilosophy Auburn,Alabama May6,2012 Keywords: Stellarator,Plasma Copyright2012byMarkCianciosa Approvedby EdwardE.Thomas,Jr.,Chair,ProfessorofPhysics StephenKnowlton,ProfessorofPhysics JamesHanson,ProfessorofPhysics StuartLoch,ProfessorofPhysics Abstract Sheared flows arising from spatially inhomogeneous, transverse electric fields are common phenomenafoundinspace,laboratory,andfusionplasmas. Theseflowsareasourceoffreeenergy that can drive or suppress instabilities. In space plasmas, numerous observations of electrostatic and electromagnetic instabilities at various scale lengths have been made. By contrast, in fusion plasmas, edge localized sheared flows provide a barrier against cross field particle transport and thepresenceoftheseflowsareassociatedwithenhancedconfinementregimes(H-mode). Under- standing how these flows provide enhanced confinement is of critical importance to current and futurefusionexperiments. Thisworkisanexperimentalinvestigationofshearedflowgeneration andthecorrespondingresponseoftheplasmainastellaratortypefusiondevice. ThisworkisperformedintheCompactToroidalHybrid(CTH)stellaratordevice. TheCTH stellaratorisafivefieldperiodcontinuouslywoundstellaratorrunwith 100mslongplasmas. Pri- mary plasma generation and heating is provided through Electron Cyclotron Resonance Heating (ECRH)withasecondaryOhmicheatingsystem. Flowexperimentsareperformedbymodifying the radial electric field by inserting a biased electrode past the last closed flux surface. Plasma parameters are measured using a Triple Probe. Plasma flows are measured using a Gundestrup Probe. Flows in CTH are studied by examining the effects that three dimensional geometries have on fluid flows. The interpretation of probe measurements in highly shaped fields is achieved by transforming laboratory space positions to magnetic flux coordinate space positions. Biasing ex- periments will modify the edge electric fields, measure the induced flows, demonstrate the role electricfieldsplayinducingflowsandmeasuretheenhancementordegradationofplasmastabil- ity in the presence of these flows. Instabilities that arise will be identified by examining various parameterscalestonarrowdownthevastspectrumofplasmainstabilities. ii Acknowledgments I would like to start by thanking mom and dad for putting up with me though out this long process. Payingmywaythoughundermyundergradeducationwasthegreatestgiftsomeonecan give. IwouldliketothankallthephysicsprofessorsatSUNYPotsdam. Yousetthebasesformy careertocome. AspecialthanksgoesouttoDr. WilliamE.Amatucci,Dr. GurudasGanguliand allthepeopleIworkedwithatNRL.ThoughyouIgotmystartinplasmaphysics,metmyfuture thesisadvisorandendedupatAuburnUniversity. InmytimeatAuburn,IneedtothankallthepeopleIhaveworkedwiththroughouttheyears. Everyone that I have worked with in the Plasma Sciences Laboratory and the Compact Toroidal Hybrid groups has made this work possible. Finally to my thesis committee for reading though this long document and making my defense seem less painless than I thought it was going to be. Finallyandmostimportantly, IthankDr. EdwardE.ThomasJr., mythesisadvisor. Noneofthis wouldbepossiblewithouttheresearchGrantawards,mentoringandfriendshipovertheyears. iii TableofContents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv ListofAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 MagneticConfinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 ToroidalDevices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 TheCTHDevice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 MissionofCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 FlowsinPlasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 OutlineofDissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 PlasmaFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 ParticleTrajectoryCode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 TransformationofCoordinateSystems . . . . . . . . . . . . . . . . . . . 20 2.2.2 Newton?sMethodInOptimization . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Runge-Kutta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.4 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 FluctuatingFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Kelvin-Helmholtz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 iv 3 ExperimentalDevice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1 TheCompactToroidalHybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 VacuumVessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 PlasmaHeating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 DataAcquisitionandAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.1 BiasingProbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 TripleProbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 GundestrupProbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 ProbeMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5.1 SimulatedDiagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.2 ExtrapolatingGlobalParameters . . . . . . . . . . . . . . . . . . . . . . . 70 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 EdgeBiasingExperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1.1 DataAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 PlasmaParametersDuringBiasing . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 IonFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.1 ComparisonWithTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.1 DrivingMechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.2 Wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4.3 Dispersionrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1 ComparisonsWithOtherExperiments . . . . . . . . . . . . . . . . . . . . . . . . 110 5.1.1 ComparisontoTokamakExperiments . . . . . . . . . . . . . . . . . . . . 110 5.1.2 ComparisontoStellaratorExperiments . . . . . . . . . . . . . . . . . . . 112 v 5.1.3 ComparisonwithLaboratoryExperiments . . . . . . . . . . . . . . . . . . 113 5.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2.1 ArgonPlasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2.2 HigherBiasVoltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.3 OhmicHeating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.4 MagneticShearandIslands . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.5 WavelengthMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A GeneralizedCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.1 BasisVectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.1.1 CovariantBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.1.2 ContravariantBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.1.3 MetricCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.1.4 VectorOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.1.5 DerivativesofCovariantandContravariantVectors . . . . . . . . . . . . . 124 BVMECCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.1 BasisVectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 C ComputerCodesUsed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 C.1 ShootingCode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 vi ListofFigures 1.1 Fusioncross-sectionsforvariousfusionreactions 1 . . . . . . . . . . . . . . . . . . . . 3 1.2 A torus. The blue arrow points in the toroidal direction. The red arrow points in the poloidaldirection. Thedrawingontherightshowsthemajorradius R,minorradius r andthedirectionsofincreasingtoroidal ?andpoloidal angle. . . . . . . . . . . . . 5 1.3 Schematic of the CTH device showing the Vacuum Vessel, Helical Coil Frame and variousCoilsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Various flux surfaces produced from the HF and TVF coils. Shaded regions shows magneticfieldresonancesforthevariousheatingsystems. . . . . . . . . . . . . . . . 11 2.1 DiagramofnestedfluxsurfacesproducedfromequilibriumreconstructionintheCTH device. Surfacecolorrepresents jBjbetween 0:75 T(Red)and 0:25 T(Blue). White linesrepresentmagneticfieldlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Plotsshowingthepathofconvergenceforbothgradientdescent(Red)andNewton?s method(Blue)forthefunction f (x;y) = x2 + 2y2. . . . . . . . . . . . . . . . . . . . 21 2.3 Singleparticletrajectorymagneticfieldonly. Red: jBj = 0:75 T,Green: jBj = 0:5 T, Blue: jBj = 0:25 T. Initial particle position s = 0:5;u = 0;v = . v0 = 1000ms ^z. Thefluxsurfaceplottedisthe s = 0:5 surface. . . . . . . . . . . . . . . . . . . . . . 25 2.4 Single particle trajectory with Es = 10. Red: jEj = 200Vm, Green: jEj = 112:5Vm, Blue: jEj = 25Vm. Initialparticleposition s = 0:5;u = 0;v = . v0 = 1000ms ^z. The fluxsurfaceplottedisthe s = 0:5 surface. . . . . . . . . . . . . . . . . . . . . . . . . 26 vii 2.5 Hierarchyofplasmainstabilitiesdrivenbyshearedflows 30 . . . . . . . . . . . . . . . 28 2.6 Plotoftheflowprofilefunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Plotofthefrequencyofthedispersionrelation. . . . . . . . . . . . . . . . . . . . . . 34 2.8 Plotofthegrowthrateofthedispersionrelation. . . . . . . . . . . . . . . . . . . . . 35 3.1 PhotographoftheCTHdevice. Thelargehorizontalportshownontheleftsideofthe photoisatatoroidalangleof ? = 180 . Therighthorizontalportisatatoroidalangle of ? = 252 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 ScaleddiagramoftheCTHvacuumvesselcross-sectionofthemajorandminorradii. 37 3.3 Various flux surfaces produced from the HF and TVF coils for a single field period. Symmetryplanesarelocatedatthe ? = 0 ,? = 36 and ? = 72 . Shadedareasshow themagneticfieldsresonatewiththeECRHheating(Green: 17:67 GHz,Blue: 14 GHz). 38 3.4 Diagram of various diagnostic and equipment locations on CTH. Blue marks the top portmountinglocations. Orangemarksthesideportmountinglocations. Bluemarks thebottomportmountinglocations. Diagnosticsrelevanttothisdissertationarehigh- lightedinbold. Diagnosticsmarked?Feed?arefeedthroughlocationsforthemagnetic diagnostics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Photographofbiasingprobetip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 CTHcross-sectionofthebiasingprobemountedat ? = 3:01 . Theprobeisdrawn atthefulltravelposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Circuitdiagramforthebiasingprobe. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.8 Rawvoltageandcurrentdataforthebiasingprobeforasingleshot. . . . . . . . . . . 46 viii 3.9 Photographoftripleprobetip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.10 CTHcross-sectionofthetripleprobemountedat 72 . Theprobeisdrawnatthefull travelposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.11 Photographofprobedrivesystemandbellows. . . . . . . . . . . . . . . . . . . . . . 52 3.12 Simplifiedtripleprobecircuitdiagram. . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.13 First op amp circuit for Triple Probe. This consists of two inverting amplifiers to measure potential differences, and a unity gain differential amplifier measuring the voltagedropacrossashuntresistortomeasurecurrent. Allopampsarepoweredbya 12 Vpowersupply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.14 Rawvoltageandcurrentdataforthetripleprobeforasingleshot. . . . . . . . . . . . 55 3.15 Analyzedtripleprobedataforasingleshot. . . . . . . . . . . . . . . . . . . . . . . . 56 3.16 Es calculated from triple probe p data in Flux surface space between the biasing probe(orange)andLCFS(verticaldashedline). . . . . . . . . . . . . . . . . . . . . . 57 3.17 PhotographoftheGundestrupprobewiththeAluminashieldpulledback. . . . . . . . 60 3.18 ScalediagramoftheassembledGundestrupprobetip. . . . . . . . . . . . . . . . . . 60 3.19 CTHcross-sectionoftheGundestrupprobemountedat 36 . Theprobeisdrawnatthe fulltravelposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.20 Scalediagramoverlayingthedeviationofacurvedtiptoaflattip. . . . . . . . . . . . 62 3.21 Op amp circuit diagram for a single Gundestrup probe tip. A complete Gundestrup probecircuitcontainssixidenticalcircuits. . . . . . . . . . . . . . . . . . . . . . . . 63 ix 3.22 Parallel and Perpendicular Mach numbers, 2 values, and the raw currents measured fromtheGundestrupprobeforasingleshot. . . . . . . . . . . . . . . . . . . . . . . . 65 3.23 Probe path for the CTH Gundestrup (Left) and Triple (Right) probes. The red box markstheregionwheretheprobetravelisperpendiculartothemagneticsurfaces(Re- gion1). Theblueboxmarkstheregionwheretheprobetravelisparalleltothemag- neticsurfaces(Region2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.24 Simulatedprobepositionsalongthemid-plane. . . . . . . . . . . . . . . . . . . . . . 68 3.25 Anarbitraryfluxsurfaceconstantquantityplottedasafunctionofbothmajorradius andfluxsurface sposition. Inlaboratoryspacefluxsurfaceconstantquantitiesdonot alignwithprobepositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.26 Plotcomparingthechangeinplasmapotential( p)tothechangeinfluxsurfacespace spositionasafunctionofposition. Thisshows p isafluxsurfaceconstantquantity. . 71 3.27 Starting from a potential profile (black (s) = 0 V, yellow (s) = 10 V), the flux surface space Es (red Es (s) = 10arb:) can be obtained. jEj(blue jEj = 0Vm, red = jEj = 240Vm)isobtainedbyconverting Es tolaboratoryspaceusingthecontravariant basisvectorsatany( s;u;v)position. Allcrosssectionsareplottedfrom s = 0:02 to s = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.28 a) Plot of Es measured at each probe position. b) Plot of jEjcalculated from Es. c) Difference between electric field calculated from Es quantity and the taking a finite differenceofdirectprobedataat ? = 18 . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1 AnexaggeratedviewoftheintersectionoftheGundestrupprobetip(blackbar)with thecurvedfluxsurface(redline). Themagneticfielddirection(purple)ispointinginto thepaperandtheorangelineshowsthedirectionoftheprobeshaft. TheGundestrup probeonlymeasurestheprojectionoftheflow(blueline)intheplaneoftheprobetips (dashedblueline). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 x 4.2 DataforexperimentAfromtimeinterval 1:62 s 1:64 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f),PlasmaPotential( p). b)Measuredandcalculatedvaluesof M?. c)Measured biasvoltageandcurrent. Theorangeshadedregionrepresentstheextentofthebiasing probe tip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthe lowerrightcornermarksthetimeintervalthatprofiledataisaveragedover. . . . . . . 80 4.3 DataforexperimentAfromtimeinterval 1:64 s 1:66 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f),PlasmaPotential( p). b)Measuredandcalculatedvaluesof M?. c)Measured biasvoltageandcurrent. Theorangeshadedregionrepresentstheextentofthebiasing probe tip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthe lowerrightcornermarksthetimeintervalthatprofiledataisaveragedover. . . . . . . 81 4.4 DataforexperimentAfromtimeinterval 1:66 s 1:68 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f),PlasmaPotential( p). b)Measuredandcalculatedvaluesof M?. c)Measured biasvoltageandcurrent. Theorangeshadedregionrepresentstheextentofthebiasing probe tip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthe lowerrightcornermarksthetimeintervalthatprofiledataisaveragedover. . . . . . . 82 4.5 Data for experiment B from time interval 1:625 s 1:65 s. a) From Top to Bottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Po- tential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. The orange shaded region represents the extent ofthebiasingprobetip. TheverticaldasheddottedlinemarkstheLCFS.Theshaded regioninthelowerrightcornermarksthetimeintervalthatprofiledataisaveragedover. 83 xi 4.6 Data for experiment B from time interval 1:65 s 1:675 s. a) From Top to Bottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Po- tential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. The orange shaded region represents the extent ofthebiasingprobetip. TheverticaldasheddottedlinemarkstheLCFS.Theshaded regioninthelowerrightcornermarksthetimeintervalthatprofiledataisaveragedover. 84 4.7 DataforexperimentBfromtimeinterval 1:675s 1:7 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f),PlasmaPotential( p). b)Measuredandcalculatedvaluesof M?. c)Measured biasvoltageandcurrent. Theorangeshadedregionrepresentstheextentofthebiasing probe tip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthe lowerrightcornermarksthetimeintervalthatprofiledataisaveragedover. . . . . . . 85 4.8 Plot comparing flux surface space electric field ( Es) with measurements of parallel andperpendicularMachnumberforexperimentA. . . . . . . . . . . . . . . . . . . . 87 4.9 Plot comparing flux surface space electric field ( Es) with measurements of parallel andperpendicularMachnumberforexperimentB. . . . . . . . . . . . . . . . . . . . 88 4.10 Rawvoltageandcurrentsignalsonalltripleprobe(TP),Gundestrupprobe(GP)and biasingprobe(BP)channelsforshotnumber11090836. . . . . . . . . . . . . . . . . 91 4.11 Plotoffluctuationspectrumfrom f measuredfromthetripleprobeunderpositivebias. 92 4.12 Plot of triple probe path(Blue line) in flux surface space. The red line draws a radial pathfromonepointinthetripleprobepath. Theorangeringshowsthepositionofthe biasingprobeandthedashed-dottedlineshowsthelocationoftheLCFS. . . . . . . . 93 xii 4.13 Plotcomparingmeasuredradialwavepowertopotential,density,andvelocitygradi- entsinfluxsurfacespaceunderapositivebias. Startingfromthetop,fluxsurfacespace electric field (Es) measured from the gradient in the plasma potential ( p), flux sur- facespacedensitygradient ( @ne@s ),fluxsurfacespaceshearfrequency (@M?@s ). Peak wavepowerismeasuredfromanFFTof Isat collectedoneachGundestrupprobetip labeledA-F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.14 Fluctuationsinthemeasuredin M? and M? flows. . . . . . . . . . . . . . . . . . . . 97 4.15 Plotcomparingtheparallelandperpendicularcomponentsofthezerothandfirstflow unitvectorstothepeakwavepowerasafunctionoffluxsurfacespace s coordinate. Peak wave power is measured from an FFT of Isat measured from each of the Gun- destrupprobetips. Peakwavepowercorrespondstotheregionwherezerothandfirst orderflowdirectionsareperpendiculartothemagneticfielddirection. . . . . . . . . . 99 4.16 a) Frequency ( !r) and Growth rate ( !i) for a classical Kelvin-Helmholtz mode. b) Normalized potential fluctuation amplitude for wavelength ( r) and imaginary parts ( i)fortheclassicalKelvin-Helmholtzmode( k = 30 m 1 and ! = 15320 + 6075i). . 104 4.17 a)Frequency( !r)andGrowthrate( !i)foradensitygradientmodifiedKelvin-Helmholtz mode. The presence of density gradients narrows the regions of instability growth anddecreasesthegrowthratecomparedtothepureKelvin-Helmholtzmode. b)Nor- malizedpotentialfluctuationamplitudeforwavelength( r)andimaginaryparts( i) for the density gradient modified Kelvin-Helmholtz mode ( k = 25 m 1 and ! = 13150 + 5050i). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1 Plot shows the relationship between Es and Te for all positive and negative biasing experimentsperformed. Radiallyoutward(positive)electricfieldscouldnotbegener- atedabove 13 eVelectrontemperatures. Belowthisthreshold,positiveelectricfields droveinstabilities. Instabilitiescouldnotbedrivenwithnegativeelectricfields. . . . . 109 xiii 5.2 FFTifthefluctuationsin f measurementsatdifferentappliedbiases. . . . . . . . . . 113 A.1 Coordinate curves and surfaces for the cylindrical coordinate system ( ; ;z). Red ? = const, Green = const, Blue z = const. The black lines show the coordinate curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 xiv ListofTables 3.1 CTHOperationalParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 TheCTHmdspluschannelsfortheBiasingProbe. . . . . . . . . . . . . . . . . . . . 45 3.3 TheCTHmdspluschannelsfortheTripleProbe. . . . . . . . . . . . . . . . . . . . . 54 3.4 TheCTHmdspluschannelsfortheGundestrupProbe. . . . . . . . . . . . . . . . . . 64 4.1 CTHrunparametersforbiasingexperimentA. . . . . . . . . . . . . . . . . . . . . . 76 4.2 CTHrunparametersforbiasingexperimentB. . . . . . . . . . . . . . . . . . . . . . 76 4.3 Variousparametersanddefinitionsusedforsolvingthedispersionrelationsforvarious instabilitymodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 B.1 ListofVMECOutputParameters. Parameterslabelledas?asymmetric?arepresentonly inoutputfilesallowingforup-downasymmetryofthemagneticfluxsurfaces. . . . . . 127 xv ListofAbbreviations f FloatingPotential RotationalTransform me ElectronMass mi IonMass ne ElectronDensity ElectricPotential p PlasmaPotential Bu Contravariantcomponentofmagneticfieldin eu direction. Bv Contravariantcomponentofmagneticfieldin ev direction. Bs Covariantcomponentofmagneticfieldin es direction. Bu Covariantcomponentofmagneticfieldin eu direction. Bv Covariantcomponentofmagneticfieldin ev direction. cs Acousticspeed. e Elementarycharge I Current J Jacobian kb Boltzmannconstant xvi M Ratioofspeedto cs. rL LarmorRadius Te ElectronTemperature Ti IonTemperature Z Ionizationnumber CTH CompactToroidalHybrid FFT FastFourierTransform LCFS LastClosedFluxSurface MHD Magnetohydrodynamic RK4 4th orderRunge-Kutta UHV UltraHighVacuum xvii Chapter1 Introduction Thestandardoflivinginthemodernwesternworldismadepossiblethroughthegeneration ofabundantenergymainlyintheformofelectricity. Electricityrightnowisgeneratedbyvarious non-renewable means such as coal, oil, natural gas and nuclear and renewable sources such as wind,solarandhydropower. Eachsystemhasitsstrengthsandweaknesses. Coal,oilandnatural gasrepresentrelativelycheapenergysourceswithamaturetechnologicalbasisthatenablespower plantsthataresimpletobuildandoperate,andallowcontinuousgenerationofelectricity. However, they come at a cost of environmental damage via the generation of harmful chemicals and the generationofgreenhousegasesduringnormaloperation. Moreover,thesesourcesallhaveafinite fuelsupplythatatcurrentconsumptionlevelscouldbedepletedwithinthenextcentury. The current generation of renewable energy technologies, particularly wind and solar, are clean forms of energy avoiding the generation of harmful chemicals and greenhouse gases. Fur- ther more, energy is extracted from resources with little danger of being depleted on short time scales. However these sources are often dependent on local climate and weather conditions such aswindpatternsandcloudcoverwhichmayvary. Thesetechnologiesarestillrelativelyimmature and expensive compared to fossil fuels. As such, these sources cannot be solely relied upon for continuousindustrialscaleproductionofelectricalpower. Therearetwoformsofenergyproductionwherethepropertiesofmatterattheatomiclevelcan beleveraged. Oneformisnuclearfissionwhereenergyisgeneratedfromthebreakupanddecayof heavyelements. Developedduringthe1950s,nuclearfissionpoweraccountsfor 19:6%ofpower generated in the United States today 2 . However, fears over safety and nuclear proliferation have resulted in a reluctance to rely on fission as a future energy source. The second form of nuclear 1 energy,fusion,isanactivelyresearchedenergysourcewiththegoalofsafecleanabundantenergy generation. 1.1 Fusion Fusion is a process where by lighter nuclei combine to form heavier nuclei. This fusion of particlesisaccompaniedbyareleaseofenergy. Inthesunnaturallyoccurringdeuterium-deuterium (D = H2) D + D ! T (1:01 MeV) + p(3:02 MeV) (1.1a) D + D ! He3 (0:82 MeV) + n(2:45 MeV) (1.1b) powersthesuninequalamounts. Inorderfortwoparticlestofusehowever,thenucleimustovercometheCoulombrepulsion. ClassicallytheCoulombbarrierisontheorderof 1 MeV. Howeverquantummechanicsallowsfor particlesoflowerincidentenergiestotunnelthroughthisbarrier. Figure 1.1 showvariousfusion crosssectionsforseveralfusionreactions 1 . Thefusioncross-sectionrepresentstheprobabilityof anincidentparticle,ofacertainenergy,toovercometheCoulombbarrier. Whiletherearemanypossiblefusionreactions,theparticularprocessofinterestforcontrolled fusionisthedeuterium-tritium(DT, T = H3)reactiondefinedas, D + T ! He4 (3:5 MeV) + n(14:1 MeV) (1.2) wherea 3:5 MeV particleanda 14:1 MeVneutronareproduced. Thisprocesshasthehighestfu- sioncross-sectionatlowincidentenergies. Howeveritstillrequiresplasmaswiththermalenergies intherangeof 10 keV. Thispresentsachallengetoafusionpowerplantasplasmasinthisrangecancausedamage toavesselcreatedtocontainit. Asidefromstarswhichusegravitytoconfinethehotplasma,there aretwotypesoffusionconfinementmethodologiesemployedtoday. Inertialconfinement,where 2 Atzeni: ? chap01 ?? 2004/4/29 ? page 12 ? #12 12 1.3 Some important fusion reactions Table 1.2 Fusion reactions: cross sections at centre-of-mass ener gy of 10 k eV and 100 k eV , maximum cross-section ? max and location of the maximum epsilon1 max . V alues in parentheses are estimated theoretically; all others are measured data. Reaction ? (10 k eV) (barn) ? (100 k eV) (barn) ? max (barn) epsilon1 max (k eV) D + T ? ? + n 2.72 ? 10 ? 2 3.43 5.0 64 D + D ? T + p 2.81 ? 10 ? 4 3.3 ? 10 ? 2 0.096 1250 D + D ? 3 He + n 2.78 ? 10 ? 4 3.7 ? 10 ? 2 0.11 1750 T + T ? ? + 2n 7.90 ? 10 ? 4 3.4 ? 10 ? 2 0.16 1000 D + 3 He ? ? + p 2.2 ? 10 ? 7 0.1 0.9 250 p + 6 Li ? ? + 3 He 6 ? 10 ? 10 7 ? 10 ? 3 0.22 1500 p + 11 B ? 3 ? ( 4.6 ? 10 ? 17 ) 3 ? 10 ? 4 1.2 550 p + p ? D + e + + ? ( 3.6 ? 10 ? 26 )( 4.4 ? 10 ? 25 ) p + 12 C ? 13 N + ? ( 1.9 ? 10 ? 26 ) 2.0 ? 10 ? 10 1.0 ? 10 ? 4 400 12 C + 12 C (all branches) ( 5.0 ? 10 ? 103 ) Fig. 1.3 Fusion cross sections v ersus centre-of-mass ener gy for reactions of interest to controlled fusion ener gy . The curv e labelled DD represents the sum of the cross sections of the v arious branches of the reaction. Centre-of-mass kinetic ener gy (k eV) 1 10 100 1000 10,000 Fusion cross-se ction (bar n) 10 ? 5 10 ? 4 10 ? 3 10 ? 2 10 ? 1 10 1 DT D 3 He p 11 B DD Li 3 He TT T 3 He D 3 He TT 10 2 1.3.1 Main contr olled fusion fuels First, we consider the reactions between the hydrogen isotopes deuterium and tritium, which are most important for controlled fusion research. Due to Z = 1, these hydrogen reactions ha v e relati v ely small v alues of epsilon1 G and hence relati v ely lar ge tunnel penetrability . The y also ha v e a relati v ely lar ge S . Figure1.1: Fusioncross-sectionsforvariousfusionreactions 1 . 3 typically focused high power lasers are used to compress the fusion fuel, is outside the scope of thisdissertation. Thetypeoffusionconfinementrelatedtotheworkinthisdissertationismagnetic confinement,wheremagneticfieldsareusedtoholdtheplasma. 1.2 Magnetic Confinement AchargedparticleinamagneticfieldexperiencestheLorentzforcegivenby Fm = qv B (1.3) where Fm forceofthemagneticfield( B)onachargedparticle. qistheelectricchargeand visthe velocity of the chargedparticle. For a particle moving perpendicular to a magnetic field line, the Lorentzforceproducesaforcepullingtheparticletowardthatfieldline. Thiscausestheparticle toorbitaroundthemagneticfieldlinewithacharacteristicfrequencycalledthegyrofrequency ? = jqjBm (1.4) atacharacteristicradiusknownastheLarmorradius, L = mv?jqjB (1.5) where m is the mass of a charged particle and v? is the component of the particle velocity per- pendicular to the magnetic field vector B. In the absence of collisions, this orbital motion traps theparticleonthemagneticfieldline. Aninfinitelylongmagneticfieldwouldconfineaparticle indefinitely however, this configuration is impossible to build. Therefore magnetic confinement researchistaskedwithchallengeofdevelopinga?finite?magneticconfigurationthatcanconfine athermonuclearplasmawithminimallosses. 4 ? ? ? R r ? ? =0? = ? Figure 1.2: A torus. The blue arrow points in the toroidal direction. The red arrow points in the poloidal direction. The drawing on the right shows the major radius R, minor radius r and the directionsofincreasingtoroidal ?andpoloidal angle. 1.2.1 Toroidal Devices Inanattempttoproducean?infinitelylong?magneticfield,consideraconfigurationinwhich thefieldlinesarecurvedbackontothemselvesavoidingcontactwithavesselwall. Thisleadsto themostcommongeometryofmagneticconfinementdevices,thetorusisshowninFigure 1.2 . The geometryofatorusisdecriedbyamajorradiusmeasuredfromthecentralcoretothecenterofthe machine. The angle ? sweeps a direction the long way around the torus or the toroidal direction. In this dissertation, the angle ? is measured counterclockwise as seen from looking down on the topofthetorus. The minor radius ( r) is measured from the end of the major radius ( R). The angle sweeps adirectiontheshortwayaroundthetorusorthepoloidaldirection. Atanytoroidalcross-section, theangle ismeasuredcounterclockwise. Thatis, theoutboardsideisatanangleof = 0 with inboardsideat = . Thetopandbottomarelocatedat = 2 and = 3 2 respectively. Thecurvatureofthemagneticlinescausesalargermagneticfieldstrengthtobeproducedon theinboardside,wherethemagneticfieldlinesbecomecompressed,ascomparedtotheoutboard 5 side. Thisgradientinthemagneticfieldintroducesaparticledriftgivenby 3 vrB = v?rL2 B rBB2 (1.6) where rL is the Larmor radius defined by equation 1.5 . The is determined by the sign of the chargeofaparticle. Inadditionthiscurvaturecausesadriftdefinedby vR = mv 2 ? qB2 Rc B R2c (1.7) where Rc isaradiusofcurvature. Becausethesetwodriftsareadditive,particlesinapurelytoroidal magneticfieldwillacquireaverticaldriftandeventuallyintersectthevesselwall. Tonullifytheinherentverticaldriftofchargedparticlesinapurelytoroidalmagneticfield,a poloidalcomponentisaddedtothemagneticfield. Thiscomponentcreatesatwistingofthefield lines and produces a helical-toroidal trajectory of the magnetic field line. A consequence of this magneticgeometry,isthataparticletravelingalongafieldlinethatisdriftingupontheoutboard side, will begin to drift down as the field line curves down on the inboard side. Moreover, the additionofthepoloidalmagneticfieldleadstothegenerationofmagneticfluxsurfaces. Modern magnetic fusion devices vary in the production of magnetic fields from configura- tionswheremagneticfieldsareproducedentirelyfromexternalcoilstoconfigurationswherethe magneticfieldisgeneratedbyacombinationofexternalandinducedcurrentsintheplasmaitself. Thisdissertationwillfocusonmeasurementsperformedonastellarator,atoroidalfusionconfigu- rationinwhichthemagneticfieldisproducedentirelybyexternalcoils. However,anumberofthe phenomenaobservedintheworkhavealsobeenreportedintokamaks,acurrent-carryingtoroidal plasmaconfiguration. Becauseofdatafrombothtokamakandstellaratordeviceswillbediscussed inthiswork,abriefdescriptionofeachdeviceisgiven. 6 Tokamak The tokamak is a device producing a toroidally symmetric magnetic geometry with nested magnetic surfaces. In tokamaks, the poloidal component to the magnetic field is provided by a toroidallydrivenplasmacurrentinducedbyatransformer. Thedegreeof?twist?ofthemagnetic fieldinatokamakisparameterizedbythesafetyfactor q,theratioofthenumberoftimesamagnetic fieldlinetraversestoroidallyperthenumberoftimesthatamagneticfieldlinetraversedpoloidally. Inadditiontoprovidingapoloidaltwist,thedrivenplasmacurrenthastheaddedbenefitofheating theplasma. Thesimplicityofthedesignleadtorapidearlyadvancesintokamakperformance. By 1968 it was revealed at the Novosibirsk Conference that the Russian T-3 tokamak was achieving electron temperatures of 100 eV with energy confinement times of 2 4 ms, well ahead of its contemporarydevices 4 . Todaythetokamakisdominantdeviceforadvancedfusionconceptsand istheconfigurationthathasbeenchosenfortheITERproject. Thepresenceofplasmacurrent,whileallowingthetokamaktoachievehightemperaturesand densities, is also a tremendous reserve of free energy that can drive a wide range of magnetohy- drodynamic (MHD) instabilities. The most dangerous of these is an event known as a disruption in which there is a global reordering of the plasma and a complete loss of confinement. These disruptionscanhaveadetrimental, evendamagingeffectondevicesashotdenseplasmacrashes into the vessel wall. One focus of current tokamak research is on the detection and mitigation of disruptions. Typicalmethods ofdriving plasma currentsinvolve alargecentral transformer. By ramping upthecurrentinthetransformer,acurrentisinducedintheplasma. However,thislimitsthelength oftimeplasmacurrentcanbesustained. Thismakesthetokamakaninherentlypulseddeviceand todatetherearenooperatingsteadystatetokamaks. Stellarator In contrast to tokamaks, the magnetic fields in a stellarator are generated completely from external magnetic coils. The degree of ?twist? in stellarators is parameterized by the rotational 7 transform . Thisparameterisrelatedtothesafetyfactorintokamaksas = q 1. Amathematical definitionof willbeprovidedinAppendix B.1 . Instellarators,thereisnoneedtodriveahighplasmacurrent. Thismakesstellaratorsresistant tocurrentdriveninstabilitiesanddisruptions. Thelackofatransformertodriveaplasmacurrent means that the stellarator is inherently a steady state device. However, historically the stellarator sufferedpoorconfinementbroughtaboutbya?rippled?magneticfield. Alongafieldline,themag- neticfieldstrengthvariesonshortlengthscales. Thishastheeffectofproducingsmallmagnetic wellsthataparticlecanbecometrappedin. Ifaparticlebecomestrappedinthesemagneticwells thegradientandcurvaturedrifts(equations 1.6 and 1.7 )willcausetheparticlestoeventuallydrift tothewall. Modernstellaratorresearchisfocusedonmagnetictopologiesthatminimizethesemagnetic ripples. Inordertoachievethis,complexthree-dimensionalmagneticcoilsarerequired. However, the complexity of these coils needed to create the three-dimensional magnetic fields, presents a designchallengecomparedtothetokamak?ssimplerdesign. Inordertoachievegoodconfinement and nested stellarator magnetic surfaces, tight tolerances and minimization of external magnetic perturbations are necessary. This along with lack of a simple efficient heating mechanism has resulted in the stellarator not achieving the same level of performance as the tokamaks until the 1980?s 4 . Still today the highest performing stellarators still fall short of the highest performing tokamaks 5 . As of 2012, the largest stellarator currently in operation is the Large Helical Device (LHD) in Japan. LHD is a non-optimized configuration with superconducting coils. In Germany, the Wendelstine7-X(W7-X)stellaratoriscurrentlyunderconstruction 6 . W7-Xisanoptimizedstel- larator configuration with superconducting coils designed to study a fully optimized configura- tion,wheremagneticfieldshavebeencarefullydesignedtominimizeparticleloses. IntheUnited States,therearetwooperatingstellarators. TheHelicallySymmetriceXperiment(HSX) 7 located at University of Wisconsin in Madison, Wisconsin. HSX is designed to demonstrate and study quasi-helicalsymmetry,wherethetrappingofparticlesduetothestellaratorrippleisreduced. The 8 secondoperatingstellarator,andfocusofthisdissertation,istheCompactToroidalHybrid(CTH) locatedatAuburnUniversityinAuburn,Alabama. 1.3 The CTH Device TheCTHdeviceisafivefieldperiodcontinuouslywoundstellarator. The hybrid naturecomes fromtheinclusionoftoroidalfield(TF)coilsandohmicheating(OH)coils. Thepresenceofthese featuresallowtheCTHdevicetobeoperatedinaspectrumfromstellarator-likewithhighvacuum rotationaltransforms ,toamoretokamak-likelowvacuum . TheOHalsohastheaddedbenefit of providing plasma heating. CTH was designed to allow the superposition of three-dimensional magnetic fields and current driven poloidal fields, for the purposes of understanding the limits wherethetraditionalstabilityofthestellaratorbreaksdownasitbecomesmoretokamaklike. 1.3.1 Magnets CTH?s magnetic field is produced from seven sets of coils. Figure 1.3 shows a diagram of themajorcoilsets. TheHelicalField(HF)coilprovidesthemaintoroidalandpoloidalfieldsbut does not provide the entire vertical field magnitude required for equilibrium. The HF coil is a continuously wound helical coil using a l = 2, m = 5 winding law. This means that this coil wrapsthevacuumvesselfivetimespoloidallyandrejoinsitselfaftertraversingthevacuumvessel twicetoroidally. ThehelicalportionoftheHFcoilsisconnectedinseriestoasetofverticalcoils mountedaboveandbelowthehelicalwinding. TherestoftheverticalfieldisprovidedbytheTrim Vertical Field (TVF) coils. Figure 1.4 shows various nested flux surface cross-sections produced bytheHFandTVFcoils. TheOhmicHeating(OH)coilsareasetofinductivecoils,mountedinthecenterofthetorus, usedtodriveaplasmacurrent. Arapidlychangingfieldinthiscoilset,inducesatoroidalplasma current. Thiscurrentheatstheplasmaandcreatesapoloidalmagneticfieldslikeatokamak. Since fieldsproducesbythiscoilsetareconstantlychanging,magneticdiagnosticswillpickupitssignal. Tominimizethiseffect,thiscoilsetisdesignedinsuchawaythatitdoesnotprovidesignificant 9 Vacuum Vessel HF Coil TVF Coil OH Coil Helical Coil Frame TF Coil SVF Coil Figure 1.3: Schematic of the CTH device showing the Vacuum Vessel, Helical Coil Frame and variousCoilsets. fieldwithinthevacuumvessel. Thiscoilsetispoweredbyacapacitorbank. Forthisdissertation theOHsystemwasnotused. Allresultsarepresentedinplasmaswithoutplasmacurrent. The remaining coil sets are for varying the basic stellarator magnetic fields. The Toroidal Field(TF)coilsareasetoftencoilsthatproduceatoroidalfield. Onatokamak,thiscoilsetwould producethemainmagneticfield. Thesecoilsallowadjustmentfromstellaratorlikehigh toamore tokamaklikelow byaddingorsubtractingfromthetoroidalcomponentoftheHFcoils. TheShapingVerticalField(SVF)coilsprovideaquadrupolefieldforcontrollingthevertical elongationoftheplasmas. RadialField(RF)coilsshiftstheverticalpositionoftheplasma. Error Correction Coils (ECC) are a set of fifteen coils wrapped around horizontal and vertical ports. Thesecoilsprovidealocalradialfieldtocorrectforlocalasymmetriesinthemagneticfield. 10 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=0.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=9.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=18.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=27.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=36.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=45.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=54.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=63.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=72.00? R (m) Z (m) Figure 1.4: Various flux surfaces produced from the HF and TVF coils. Shaded regions shows magneticfieldresonancesforthevariousheatingsystems. 11 1.3.2 Mission of CTH For all toroidal fusion devices, there remains a number of outstanding physics issues to be addressedbeforeatruefusionpowerplantcanbebuilt. Amongthecriticalissuesaretheinteraction oftheplasmawiththesurroundingwalls,developmentofglobal,predictivemodelsofthebehaviors ofthermonuclear?burning?plasma,and,ofparticularrelevancetothisdissertation,thecontrolof particletransportandinstabilitiesintheplasma. MajorprojectscompletedandcurrentlyongoingonCTHinvolvemappingandreconstructing the equilibrium magnetic fields. Mapping of the magnetic field performed by Peterson et al. 8 , who developed a model of the magnetic field. It was show how the presence of magnetic errors canbeenhancedorreducedthroughtheuseofexternalcoils. Workmodelingthemagneticfields wasbuiltuponbyB.A.Stevensonforuseinperformingplasmaequilibriumreconstructionsfrom diagnostic measurements 9 . This dissertation builds upon this previous work for interpretation of probemeasurements. The ability to accurately model the magnetic field and reconstruct the plasma equilibrium, plays a vital role in the primary mission of CTH. The hybrid nature of CTH was produced to studytheeffectsofplasmadisruptionsinstellaratorswithsignificantplasmacurrent. Thegeneral shape and structure of CTH stellarator fields are varied to see the effects it has on current driven disruptions. The goal is to map out the boundaries where the stability of the stellarator design beginstobreakdownasplasmacurrentbecomesthedominanteffect. Thesestudiesareaimedat addressingthecentralquestionsoftheCTHresearchprogramondisruptioneffectsinstellarators. 1.4 Flows in Plasmas Some 30 years ago, studies performed on the ASDEX tokamak in Germany discovered an enhancedconfinementregimeofatoroidalplasma 10 . Thisregime,referredtoasthe?High?con- finement(H-Mode),wascharacterizedbyanincreaseindensityandtemperatureaccompaniedby a decrease in transport to the walls 11 . This regime exists as a common mode achievable on most largetokamaks 12 ? 16 andstellarators 17 ,18 . 12 Extensiveelectricfieldmeasurements 11 ,19 ofH-modeedgeelectricfields,showthegeneration ofaradiallyinward(negative)electricfieldfromtheambipolardiffusionofionsandelectronsinthe edgeplasma. ThisnegativeelectricfieldisconsistentacrossvariousfusiondevicesthatachieveH- mode. Thisnegativeelectricfieldisshearedoveradistanceof 3 cmintheplasmaedge. Sheared electricfieldsdriveashearedperpendicularflowintheplasma 20 ? 23 . Thisshearedflowiscredited withtheformationofaparticletransportbarrier 24 ? 26 andcanreducethelargescaleturbulence 27 ? 29 intheplasma. Ingeneral,shearedflowsarethoughttobestabilizinginfusionplasmas. By contrast, in space and laboratory plasmas, sheared flows are a source of free energy that can drive a wide spectrum of plasma instabilities 30 . The instability mode that can be generated variesbasedthedirectionofflowshearandonthescalesizesofthesystem. Specifically,thesize oftheionLarmorradius( i)tothesizeoftheshearlayer( L),playsanimportantconditiononthe instability generated. When i ? L, instabilities with frequencies greater than the ion cyclotron frequency ( ?ci) may be generated. When i L, instabilities with frequencies on the order the ioncyclotronfrequencymaybegenerated. When i ? L,instabilitieswithfrequenciesmuchless than the ion cyclotron frequency may be generated. Extensive experiments in the generation and suppressionofthesevariousinstabilitieshasbeenperformedextensivelyinlaboratoryplasmas 31 ,32 . Ingeneral,forspaceandlaboratoryplasmas,shearedflowsarethoughttobedestabilizing. While negative electric fields produce stabilizing flows in the edge of fusion plasmas, the generationofradiallyoutward(positive)electricfieldshavebeenobservedtoproducedestabilizing effects33 . Reports range fromthe generationofabifurcationof theplasma edge, tothecomplete loss in particle confinement. In order to achieve the goal of fusion as a viable power source, the optimaloperationconditionmustbeachieved. Assuch,extensiveresearchintooperationalmode thatshowadegradationinperformancearenotextensivelyexplored. WhilethisprojectisinabroadsenseconnectedtothestudyofH-Mode,itisnotaninvestiga- tionofenhancedconfinementregimes. Thisworkconsiderstheplasmaresponsetothegeneration of edge electric fields and is compared to instability regimes characterized by the size of the ion Larmorradiuscomparedtotheelectricfieldscalelength, andtheinstabilityfrequencycompared 13 tothesizeoftheioncyclotronfrequency. Furthermore,buildinguponthepreviousworkonequi- libriumreconstructioninCTH,thisprojectshowshowitispossibletouseamagneticfieldbased ?Flux?coordinatesystemtounderstanddiagnosticandflowmeasurementsinhighlyshapedthree- dimensional fields. As such, it extends the work on flows and edge biasing performed on the TEXTOR, CASTOR and T-10 tokamaks 33 , and Compact Auburn Torsatron (CAT) and the TJ-II stellarators 34 ,35 . 1.5 Outline of Dissertation InChapter 2 ,anoverviewoftheoreticalmodelsnecessarytointerpretexperimentalresultsof thisworkispresented. Section 2.1 discusseszerothorderflowsintheVMECcoordinatesystem. Sec- tion 2.2 presentsasimulationcodedevelopedtointerpretthecomplexparticleandfluidbehavior. Section 2.3 discussestheroleshearedflowshaveindrivingplasmainstabilities. In Chapter 3 a description and the theory of operation of all the experimental hardware is presented. Section 3.1 presentsanoverviewoftheCTHdevicewithkeyemphasisonkeysystems utilizedthroughoutthisdissertation. Thissectionalsoprovidesthetheoryanddesignofdiagnostic systemsusedforexperimentalmeasurements. Section 3.3.1 providesadescriptionofthebiasing probeusedtomodifytheedgeelectricfieldsandinduceaflowintheplasma. InChapter 4 ,theresultsofvariousedgebiasingexperimentswillbediscussed. Thegeneration ofedgeelectricfieldsthroughtheuseofedgebiasingwillbeverifiedintheCTHdevice. Plasma flowsparallelandperpendiculartothemagneticfieldswillbemeasured. Comparisonsofmeasured flowswillshowgoodagreementwiththeoreticalcalculationsofflows. Aninstabilitymodedriven bythepresenceofedgeflowswillbeidentified. 14 Chapter2 Theory Electricfieldsandpressuregradientstransversetothemagneticfieldcandriveplasmaflows. Inslabplasmageometriesthesedriftstaketheform 3 v = E BB B (2.1) v = rP BqnB B (2.2) for E B and diamagnetic drifts, respectively, where n is the plasma density. When electric fields and density gradients are spatially inhomogeneous, a nonuniform or sheared flow layer in theplasmaresults. However, a purely cartesian formulation of flows is complicated by the three-dimensional magneticfieldsproducedbystellarators. Figure 2.1 showsthethreedimensionalmagneticsurfaces producedbytheCTHdevice. Magneticfieldslines,showninwhite, wraparoundthetoruslying on surfaces of constant magnetic flux. The cross-sectional shape of these surfaces varies by the toroidalangle,asshowpreviouslyinFigure 1.4 . Electric field and pressure gradients, the primary drivers of perpendicular plasma flows, are assumedtopointinthedirectionperpendiculartothemagneticsurfacesassumingcertainplasma quantities are constant on a flux surface. The validity of these assumptions will be demonstrated inSection 3.5.2 . Forthissection,electricfieldwillbeassumedtopointinthedirectionnormalto amagneticsurface. InCartesiancoordinates, magneticandelectricfieldsarefullythreedimensional. Toreduce the dimensionality of these components, a coordinate system based around magnetic surfaces is employed. In this coordinate system, the magnetic field lines become straight lines on surfaces 15 Figure2.1: DiagramofnestedfluxsurfacesproducedfromequilibriumreconstructionintheCTH device. Surfacecolorrepresents jBjbetween 0:75 T(Red)and 0:25 T(Blue). Whitelinesrepresent magneticfieldlines. 16 formingconcentriccylinders. Electricfieldsandpressuregradientspointinasingleradialdirection normal to a magnetic surface. The resulting perpendicular flows are limited to these surfaces of constantmagneticfluxaswell. While this coordinate system reduces the dimensionality of parameters relevant to plasma flows, it comes at the cost of being no longer orthonormal. A formulation of the mathematical concepts necessary for working in generalized coordinates is provided in Appendix A. One such magneticcoordinatesystemisthecoordinatesystemusedbytheVariationalMomentsEquilibrium Code(VMEC)36 ,37 . OnCTH,VMECisusedbytheV3FIT38 codetoreconstructtheequilibriummag- neticsurfacesaftereachpulse. Thiscoordinatesystem,overviewedinAppendix B,isusedthrough outthisdissertation. 2.1 Plasma Flows AswasdiscussedinSection 1.4 ,thepresenceofplasmaflowsisanimportantcharacteristicof enhancedconfinementregimes. ForthestudiesperformedonCTH,thecontributionsoftheelectric fielddriven( E B)andpressuredriven(e.g.,diamagnetic, rP B)driftsareconsideredas themaindriversofplasmaflow. AsimpleCartesianmodelofthesedriftsfailstotakeintoaccount geometric corrections. Even simple cylindrical models require geometric corrections compared to Cartesian solutions. In order to account for geometric corrections to the plasma drifts, a fluid approachisemployedinthefluxcoordinateframe. Theionfluidmomentumequationincludingpressuregradientsis 39 ( @ @t +v r ) v = em i (E +v B) rPn i (2.3) Crossfieldtransportwillbeassumedtobeprimarilydiffusiveinnature,andthevelocityvectorof afluidelementisassumedtotaketheformof v = vu (s;u;v)eu + vv (s;u;v)ev (2.4) 17 Themagneticfield B isdefinedusingthecovariantbasisas B = Bu (s;u;v)eu + Bv (s;u;v)ev (2.5) Plasmapotentialandplasmapressureandassumedtobeflux surfaceconstantquantities. Taking thedotproductofvelocitywiththegradientoperatorbecomes v r = vu @@u + vu @@v (2.6) Asareminder,ingeneralizedcoordinates,basisvectorcanbefunctionoftheircoordinates. When taking the derivative of a vector quantity, the change in basis vector must also be accounted for. Substitutinginallcomponentsusingthegeneralizeddefinitionofthecrossproduct,themomentum equationbecomes gsuvu@v u @u + gsvv u@v v @u + (v u)2 @eu @u es + v uvv@ev @u es +gsuvv@v u @v + gsvv v@v v @v + v uvv@eu @v es + (v v)2 @ev @v es = em i [ @@s p + J (vuBv vvBu) ] 1n i @ @sP (2.7a) guuvu@v u @u + guvv u@v v @u + (v u)2 @eu @u eu + v uvv@ev @u eu +guuvv@v u @v + guvv v@v v @v + v uvv@eu @v eu + (v v)2 @ev @v eu = 0 (2.7b) guvvu@v u @u + gvvv u@v v @u + (v u)2 @eu @u ev + v uvv@ev @u ev +guvvv@v u @v + gvvv v@v v @v + v uvv@eu @v ev + (v v)2 @ev @v ev = 0 (2.7c) forthe es,eu and ev basisvectorsrespectively. InEquation 2.7a ,J istheJacobian, p istheplasma potential, P isthefluidpressure. Componentscontainingelementsoftheform @vui@uj representthe contributiontodriftduetomagneticfieldgradients. Componentscontainingelementsoftheform @eui @uj represent the contribution to the drift due to field line curvature. These elements represent changesinthedirectionofthebasisvectorsandcanbefoundinsimplercoordinatesystemssuch 18 as cylindrical coordinates. The remaining terms represent the combined E B and rP B drifts. However,thesecoupleddifferentialequationsaretoocomplextosolveanalytically. Ifweex- pandthemajorandminorradiiofourtoroidalsystemtoinfinity,thecoordinatesbecomeCartesian as the terms @eui@uj ! 0. By ignoring magnetic field gradient and curvature effects, @vui@uj = 0 and @eui @uj = 0,equation 2.7 reducestoasimplifiedform. 0 = em i [ @@s p + J (vuBv vvBu) ] 1n i @ @sP (2.8) Thesecomponentsrepresentthe E B and rP B. Thesolutiontothisequationisthesame astheCartesiansolution. v = E BB B rPi Ben iB B (2.9) Thisdoesnotrepresentacompletesolutionbutisanappropriateapproximationforsystemswhere E Band rP Barelargecomparedtocurvatureandgradientdrifts. InSection 4.3 ,thiswill beshowntobeajustifiableassumptionforplasmasproducedinCTH. 2.2 Particle Trajectory Code As a need to understand the fluid and single particle motion of the plasma, a single particle code has been developed. The goal of this code is to understand the full motion of the particles incorporatingallguidingcenterdriftmotions. Withtheadventofhighperformancecomputingand highly parallel processors it is possible to examine large numbers of single particle motions in a reasonablecomputationaltime. ThesourcecodeisavailableontheCTHarchiveserver. ThesimplestmethodtoperformasingleparticlemotioncalculationisinCartesiancoordinates. Thetrajectorycodealgorithmfollowsthefollowingbasicsequenceateachtimestep. Theparticle positionisconvertedfrom( x;y;z)coordinatesto( s;u;v)coordinates. Themagneticfields,electric fieldsandanyotherparametersthatareafunctionof( s;u;v)arecomputed. Thoseparametersare usedtocalculatethetotalforceactingonaparticleinCartesiancoordinates. Usingthecomputed 19 force model, the particles position and velocity are modified, time is advanced and the algorithm repeats. It should be noted that since all forces acting on the particles are conservative, there are nocollisionaleffectstakenintoaccount. Thismeansthattimecanbeadvancedforwardaswellas reversed. 2.2.1 Transformation of Coordinate Systems This code and the dissertation as a whole, makes extensive use of two coordinate systems. A laboratory space cylindrical ( r;?;z) coordinate system and a flux surface space ( s;u;v) coor- dinate system. It is important to understand how positions and vectors are transformed between thetwospaces. Movingfromafluxsurfacespacepositiontolabspaceisperformedanalytically. Formulasforthetransformationofvectorquantitiesand R(s;u;v) and Z (s;u;v) areprovidedin Appendices Aand B.However,inorderconvertfrom( r;?;z)coordinatesto( s;u;v)coordinates, atwo-dimensionalrootfindingmustbeperformed. Thefluxsurfacepositionsof sand uarefound attheminimumofthefollowingfunction f (s;u) = (R(s;u) R0)2 + (Z (s;u) Z0)2 (2.10) where R(s;u) and Z (s;u) havethe v = ? coordinatefixed. R0 and Z0 arethelaboratoryframe coordinatesthatthe( s;u;v)coordinatesarebeingconvertedfrom. Onceincylindricalcoordinates, thepositionisconvertedtoCartesiancoordinatesusinganalyticalmeans. 2.2.2 Newton?s Method In Optimization Toconvertfrom( r;?;z)spaceto( s;u;v)spaceefficiently,equation 2.10 isminimizedusing Newton?s Method 40 . This method expands upon normal gradient descent methods by taking into account second derivatives to aid in faster convergence. Gradient descent methods work to find the minimum of a function by continuously moving parameters ?downhill?. That is, at each step the gradient of a function f(xi) is determined to find a direction of decreasing slope such that 20 -10 -5 0 5 10 -10 -5 0 5 10 x y Figure2.2: Plotsshowingthepathofconvergenceforbothgradientdescent(Red)andNewton?s method(Blue)forthefunction f (x;y) = x2 + 2y2. f(xi+1) < f(xi). Newton?s method refines this approach to find a ?steeper? path to follow for quicker convergence. Figure 2.2 shows the path taken by the gradient descent (Red Line) and Newton?s method (Blue Line) for the function f (x;y) = x2 + 2y2. Newton?s method takes one stepwhilethegradientdescenttookten. Startingataninitialpoint x0 thefunction 2.10 canbeminimizediterativelyuntilthefunction isbelowagiventhresholdvalue. Ateachiteration xi,thenextvalueisfoundby xi+1 = xi [Hf (xi)] 1rf (xi) (2.11) 21 where [Hf (x)] 1 istheinverseoftheHessianmatrix. TheHessianmatrixisamatrixofsecond derivativesoftheobjectivefunction. ForthissystemtheHessianmatrixbecomes Hf (xi) = 0 B@ @2 @s2f (s;u) @ @s @ @uf (s;u)@ @s @ @uf (s;u) @2 @u2f (s;u) 1 CA (2.12) where f (s;u) is the function to be minimized. The value of is initially chosen to be one, but subjecttotheWolfeconditions 40 whichmeansthatateachiteration, ischosensothat f (xi+1) < f (xi). Ifavalueof cannotbefoundtomeetthiscondition,thefunctionhasfailedtoconverge. In Figure 2.2 , = 1forNewton?smethodand = 0:2forthegradientdescenttoinsureconvergence. Once the function 2.10 has dropped close to machine precision, a good enough convergence has beenachieved. Newton?smethodrequiresthecomputationofboththefirstandsecondderivativesof R(s;u) and Z (s;u). Partial derivatives of R(s;u) and Z (s;u) with respect to u can be performed ana- lytically. However, partial derivatives with respect to the s coordinate must be interpolated. As such,acubicsplineinterpolation 40 waschosen. Thistypeofinterpolationprovidescontinuousand smooth function and first derivative and a continuous second derivative. Interpolated points are mirroredaboutthe s = 0 pointtoensurethatthefirstderivativesarezeroatthatpoint. 2.2.3 Runge-Kutta Theequationsofmotionoftheparticlescanbedefinedas, _v = F (x;v;t) (2.13) _x = v (2.14) where v isthevelocityand F (x;v;t) isthenetforceactingonaparticle. Tosolvethissystemof differentialequations,theparticlecodeusesafourthorderRunge-Kutta 40 (RK4)method. Ateach 22 timestep vi+1 = vi + 16 (k1 + 2(k2 +k3) +k4) (2.15) xi+1 = xi + 16 (l1 + 2(l2 +l3) +l4) (2.16) wherethecoefficientsaredefinedby k1 = 1mF (xi;vi;t)dt (2.17a) k2 = 1mF ( xi + 12l1;vi + 12k1;t + 12dt ) dt (2.17b) k3 = 1mF ( xi + 12l2;vi + 12k2;t + 12dt ) dt (2.17c) k4 = 1mF (xi +l3;vi +k3;t + dt)dt (2.17d) l1 = vidt (2.18a) l2 = ( vi + 12k1 ) dt (2.18b) l3 = ( vi + 12k2 ) dt (2.18c) l4 = (vi +k3)dt (2.18d) and mistheparticlemass. Thesecoefficientsactassubstepsateachtimestep dtandhelpcorrect forcomputationalerrors. Thetimestepisadaptedateachcalculationsteptobe 110 thegyroperiodof theparticle. Thisallowsaccuratemodelingofthefullgyromotionatallplaceswithinthemagnetic field. 2.2.4 Forces ThiscodehasbeenarchitectedusingthesamemodularforcemodelastheAuburnUniversity DEMONcode 41 . This modularity allows the creation of new forces without altering the underlying 23 solvers. The modularity is achieved by using object oriented programing techniques. A generic forceclassdefinesaninterfacethatallforcesconformto. TheRK4algorithm,actsonthisgeneric forceinterfacesothespecificsofaparticularforcearehidden. Each force is created as a subclass of the force class. Each specific force is responsible for adding its contribution to the total force acting on a particle for a particular RK4 sub step. Extra code associated with a particular force maybe add however that will have no affect or will be hiddenfromtheRK4solver. VariousforcesactingontheparticlesinCTHare,theLorentzforce contributionofthemagneticfieldandtheforcecontributionoftheelectricfield. Densitygradient effectscannotbemodeledbecause,diamagneticdriftsarenotguidingcenterdrifts. Magnetic Force Themagneticforceactingonachargedparticleisdefinedas Fm (v) = qv B (2.19) where q isthenetchargeofaparticle. Themagneticfieldvectorisobtainedataspecified( s;u;v) positionintheformofequation 2.5 andtransformedintothelaboratoryframe. Thisforceprovides theprimarymechanismofgyromotionoftheparticle. AsshowninSection 1.2.1 ,thetoroidalgeometryofthedevicecontributestoverticaldriftsof particles. InadditionthemagneticfieldstructureinCTHishighlynonuniform. Thehelicalwinding of the magnet coils leads to troughs and peaks in the magnetic field strength. These localized magnetic wells can trap particles as they mirror back and forth between the peaks. This trapping alongwithapreviouslymentionedparticledriftswillcausetheparticlestoeventuallydriftoutof thecoreplasma. InFigure 2.3 ,thetrajectoryofasingleparticlestartedfrom s = 0:5;u = 0;v = (mid-plane of a side port on the CTH device) is plotted. The particle is launched with an initial velocity of v = 1000^zms. The s = 0:5 fluxsurfaceisalsoplottedwiththecolorsrepresentingthemagnitude 24 Figure2.3: Singleparticletrajectorymagneticfieldonly. Red: jBj = 0:75 T,Green: jBj = 0:5 T, Blue: jBj = 0:25 T. Initial particle position s = 0:5;u = 0;v = . v0 = 1000ms ^z. The flux surfaceplottedisthe s = 0:5 surface. of the magnetic field strength jBj. Blue represents weak magnetic field strengths jBj = 0:25 T. Red represents strong magnetic field strengths jBj = 0:75 T with green in between. Figure 2.3 shows the particle trapped in one of the mirror fields of CTH. The curvature and gradient drifts causetheparticletoleavethe s = 0:5 surfaceandeventuallydriftbeyondtheLCFS. Electric Forces Atthebeginningofthischapter,plasmapotentialwasassumedtobeconstantonasurfaceof magneticflux. Usingthisassumption,electricfieldforcesaredefinedtobe FE = qEs (s)es = q @@s p (s)es (2.20) 25 Figure 2.4: Single particle trajectory with Es = 10. Red: jEj = 200Vm, Green: jEj = 112:5Vm, Blue: jEj = 25Vm. Initialparticleposition s = 0:5;u = 0;v = . v0 = 1000ms ^z. Thefluxsurface plottedisthe s = 0:5 surface. where q is the particle charge. Electric forces are defined either by defining a functional form of Es (s)directlyorbysplineinterpolating p profilesfromprobemeasurements. Whilepressuregra- dientswerealsoassumedtobeconstantonasurfaceofmagneticflux,driftsarisingfrompressure gradients cannot be simulated in a single particle code as these are fluid drifts and do not change theguidingcentermotion. InFigure 2.4 ,thetrajectoryofasingleparticlestartedfrom s = 0:5;u = 0;v = isplottedin thesamemannerasFigure 2.3 withtheadditionofauniform Eses. Againtheparticleislaunched with an initial velocity of v = 1000^zms. The s = 0:5 flux surface is also plotted with colors representing the jEj. Blue represents weak electric field strengths jEj = 25Vm. Red represents strongelectricfieldstrengths jEj = 200Vm withgreeninbetween. Figure 2.4 showsthattheparticle remains confined to a particular flux surface. Also the particle is no longer trapped in the mirror fields. Plasma parameter profile measurements may also be used to define the electric field. From measured profiles of p or f, data positions are transformed into ( s;u;v) coordinates. Again 26 assumingthatafluxsurfaceisanelectricequipotential,asplineinterpolationisusedtofitthedata pointsasafunctionof s. Theprofileisconstrainedtobesymmetricaboutthemagneticaxis. This insuresthattheelectricfielddropstozeroatthemagneticaxis. Electricfieldscanbetransformed backintothelaboratoryframeatanypositionwithinthelastclosedfluxsurface. Bymeasuringthe potentialprofilesatanyplace,theelectricfieldstructurecanbedeterminedeverywhere. Simulation Results The effect that electric fields and induced poloidal drifts is immediately apparent. In the puremagneticfieldonlysimulations,particlesremainedtrappedwithinthemagneticwells. These trappedparticleseventuallydriftoutoftheplasmabyacombinationofgradientandcurvaturedrifts (Equations 1.6 and 1.7 ). Under the influence of an electric field, the induced poloidal drift moves the particle to the inboard side of the magnetic field where the gradient and curvature drifts push the particle back ontoamagneticsurface. Asaresult,theparticleremainsconfinedwithintheplasma. Theelectric fieldalsoallowstheparticletoescapefromthemirrortrapsbetweenthehighfieldregions. These simulations can only provide a single particle view of the plasma. Collective fluid effectssuchasviscosity,diamagneticdriftsandplasmafluctuations,cannotbemodeledinasingle particle code. However this code is useful for providing insights into the complex motions of particlesinthethreedimensionalfields. 2.3 Fluctuating Flows When electric fields and pressure gradients are spatially inhomogeneous, a nonuniform or sheared flows can arise. In Section 1.4 , sheared flows were discussed as having different effects on the stability of the plasma depending on the plasma environment. In fusion plasmas, radially inward(negative)electricfieldsproduceshearedflowsthatreduceparticlelossesanddamplarge scaleplasmafluctuations. Inspaceplasmas,thepresenceofshearedflowsisasourceoffreeenergy thatcandriveawiderangeofplasmainstabilities. 27 GANGULI ET AL.' COUPLING OF MICROPROCESSES AND MACROPROCESSES 8877 V.LB I ?'s %H lectron Ion Hybrid Instability cor~ (OLH; kyL~ 1 Pi >L> Pe homogeneous Energy Density Driven Instability r ~ Qci; kyL> 1; L2 Pi elvin-Helmholtz Instability r << D'ci; kyL < 1; L ?? Pi.,J ? o?S > (Ope nmagnetized Plasma Limi (o r ~ 3' ~ COpe I vii B (os > (OI..H Streaming Instabilities (or~ (e)LH- Qe ) Pi >L> Pe (o s D-gi Current Driven Ion Cyclotron Instability cor~ D. ci' L? p i co s << D. ci D?Angelo Instability << ; L >> Pi (o r .O-ci Figure 2. A hierarchy of microinstabilities that can be triggered by velocity shear. Note only the instabilities investigated by us are listed. In principle, many other waves may be excited by velocity shear. ?, -? 12 mho, we see that the total power available is around 1.2 to 30 ergs cm -2 s-? which is orders of magnitude larger than necessary. Thus, even if a very small fraction of the total available energy can be dissipated by the instabilities leading to ion energization, then ion upwelling can easily be sustained. Now the question is whether the instability mech- anism we suggest is efiScient enough to accomplish this. We return to this point later. Since the waves discussed above are sustained by velocity shear and there is strong observational correlation of ion heating to velocity shear, it is highly probable that these waves play an important role in energizing the ions. A primary focus of this investigation is to explore, quantify, and establish this possibility. In the following we report our preliminary results. Our theory depends on an interplay between macroprocesses and microprocesses. We achieve this by coupling the outcome of a two-dimensional fluid code I ? ? with a 2?-d?mens?onal particle code. We find that the cou- pling of macroprocesses and microdynamics can explain a number of observed features such as low-altitude energiza- tion, formation of hot tails, and density morphology. 2. Model For the purpose of this study we assume an ideal iono- sphere (two species, no collision, etc.) but focus on to the important effects due to velocity shear. As low-frequency waves (such as the KH instability [Keskinen et al., 1988; Theilhaber and Birdsall, 1989]) evolve, they steepen and generate stressed regions with large shear frequency % self-consistently. We define the shear frequency, % = ]dV/dx]max "? V?/L, where V ? and L are the peak and the scale size of the flow velocity. It is a measure of the magnitude of velocity shear. Large to s is self-consistently generated by the density gradient [Romero et al., 1990]. It is found that as the density gradient scale size approaches an ion gyroradius, the self-consistent % can become compara- ble to the lower hybrid frequency [Romero et al., 1992b; Romero and Ganguli, 1993]. As % becomes large enough to resonate with various normal frequencies of the system, it can trigger high-frequency shear-driven waves as described below. Interestingly, in a recent laboratory experiment [Huang et al., 1992], it is shown that nonlinear evolution of the low-frequency Kelvin-Helmholtz mode can seed high- frequency noise. 2.1. Velocity Shear-Driven Microinstabilities We first summarize the various microinstabilities that can be excited by velocity shear. The important parameter for assessing the role of transverse velocity shear in exciting these instabilities is the shear frequency %. If % is greater than the gyrofrequency of the species j, flj, then that species becomes effectively unmagnetized. Also, the magnitude of the shear frequency determines the character of the waves. In general, we find that as the magnitude of the shear frequency gets close to various natural frequencies of the system it leads to instabilities around these frequencies. For example, if % < fl i, then both the ions and the electrons are magnetized and the resulting shear-driven waves oscillate around the ion cyclotron frequency (also referred to as the Figure2.5: Hierarchyofplasmainstabilitiesdrivenbyshearedflows 30 . 28 Studyingtheeffectsofshearedflowsinplasmasisamultifacetedproblem. Figure 2.5 ,taken from Ganguli et. al. 30 , shows a hierarchy of plasma instabilities that can be driven by sheared flows. Thishierarchyisdividedintotwomainbranches,shearinflowsparalleltomagneticfield linesandshearinflowsperpendiculartomagneticfieldlines. Thesetwobranchescanfurtherbe subdividedupbasedonthesizeofshearlayer. Anexhaustivestudyofallpossibleinstabilityregimesisbeyondthescopeofthisdissertation. However, asademonstrationoftheeffectsshearedflowsplayinthestabilityorinstabilityofthe plasma,aninstabilityinaregimerelevanttoCTHplasmaconditionswillbediscussed. ForCTH magnetic field strengths and ion species, the ion Larmor radius (Equation 1.5 ) is sub-millimeter sized. Measurements of plasma potential gradients in Section 4.2 , will show that CTH electric fieldareontheorderofafewcentimeters. ThisplacestheCTHrelevantconditionsatthebottom ofbothbranchesofFigure 2.5 ,where L ? i. In Sections 2.1 and 2.2.4 , electric fields were assumed to point normal to the magnetic sur- faces. In Section 2.1 , it was shown how this transverse electric field can drive a perpendicular flow. Assuming that electric fields are normal to a magnetic flux surface and that induced flows are perpendicular to a field line, it is assumed sheared flows arising in the CTH plasma will be perpendiculartomagneticfield. Derivationsofelectricfieldbasedontheconstancyofplasmapo- tentialonafluxsurfacetobediscussedinSection 3.5.2 verifiesthevalidityofthefirstassumption. Furthermore,measurementsofperpendicularflowwhichwillbepresentedinSection 4.3 showthe presenceofperpendicularshearintheCTHplasma. ExploringtheCTHrelevantscalessizesreducesthehierarchyofCTHrelevantplasmainsta- bilities to the Kelvin-Helmholtz region of Figure 2.5 . Therefore, to proceed, it will be assumed thatKelvin-HelmholtzinstabilitiesmayariseinCTHplasmas. Thedispersionrelationofthisin- stabilitywillbederivedandsolvedforvariousshearflowprofiles. Duetononlineareffectsofthe toroidal geometry, derivations of plasma instabilities will be carried out in Cartesian coordinates forsimplicity. 29 2.3.1 Kelvin-Helmholtz The derivation of the Kelvin-Helmholtz dispersion relation presented here follows the work presentedinGuzdaret. al. 57 . Thederivationwillbebeginbyassumingauniformmagneticfield pointinginthe B = B0^z direction. Thezerothorderelectricfieldisassumedtobeafunctionof x pointing in the E0 (x) = E0 (x) ^x direction. From the E-cross-B drift (Equation 2.1 ), a flow perpendiculartothemagneticfieldisproduced. V 0 (x) = E0 (x)B 0 ^y (2.21) Fluctuating quantities are assumed to take the form of a zeroth order component and a fluc- tuatingcomponent p = p0 + p1 (x)exp [i(ky !t)]. Parallelwavepropagationisassumedtobe largewavelengthandneglected( k? = 0). Zerothorderplasmadensitywillbeassumedtouniform andquasi-neutralsuchthat ne = ni = n. Withtheseassumptiontheionmomentumequationis ( @ @t +V i r ) V i = em i (E +V i B) (2.22) assumingthefollowingforms B = B0^z (2.23a) E = E0 (x) ^x r (x)exp [i(ky !t)] (2.23b) V = V0 (x) ^y +V 1 (x)exp [i(ky !t)] (2.23c) n = n0 + n1 (x)exp [i(ky !t)] (2.23d) SubstitutingthezerothordersolutionsandtheformgiveninEquation 2.23 intoEquation 2.22 andkeepingonlythefirstorderterms,thefirstorderionflowscanbesolvedforassuming ! ? ?ci. Vi1x (x) = ik 1 (x) ! ? @ @x 1 (x) B0 [ 1 + @@xV0(x)?ci ] (2.24a) 30 Vi1y (x) = [( 1 + @@xV0(x)?ci ) @ @x 1 (x) !? ?cik 1 (x) ] B0 [ 1 + @@xV0(x)?ci ] (2.24b) Intheseequations, !? istheDopplershiftedfrequencydefinedtobe !? = ! kV0 (x). Fortheelectrons,inertialtermsareignoredandrelevantequationsforthefluctuatingelectron driftaregivenby, Ve1x (x) = ik 1 (x)B 0 (2.25a) Ve1y (x) = 1B 0 @ @x 1 (x) (2.25b) Theseequationsarecombinedusingthedensitycontinuityequationforeachspecies . @n @t +r n V = 0 (2.26) Keepinguptoonlyfirstorderterms,solvingforthedensitygives i!?n 1 (x)n 0 = @@xV 1x (x) + ikV 1y (x) (2.27) Equations 2.24 and 2.25 are substituted into Equation 2.27 for each species. These equations are combinedassumingquasi-neutrality. Combiningtermsof 1 (x), andassumingthattheshearing rateissmallcomparedtotheioncyclotronfrequency, @@xV0 (x) ? ?ci,thedifferentialformofthe Kelvin-Helmholtzinstabilityisobtained. @2 @x2 1 (x) [ k2 k @2 @x2V0 (x) ! kV0 (x) ] 1 (x) = 0 (2.28) Toexploretheeffectsofplasmastabilityundertheeffectsofshearedflows,flowprofileswill beassumedtotakeafunctionalformof V0 (x) = V0 f (x L ) (2.29) 31 where Listhescalesizeoftheshearlayer. Definingthedimensionlessquantities, ~! = !kV0, ~x = xL and ~k = kL,Equation 2.28 becomes @2 @~x2 1 (~x) [ ~k2 @ 2 @~x2f (~x) ~! f (~x) ] 1 (~x) = 0 (2.30) Thisdispersionrelationcansolvedbymeansofanumericalshootingcodeforvariousflowprofile shapes. Shooting Code Considerthesecondorderdifferentialequationoftheform; @2 @x2 (x) A(!;k;x) @ @x (x) B (!;k;x) (x) = 0 (2.31) This differential equation is solved numerically by guessing a solution, at large values of x. The solutionisassumedtobeapproximatedatlargevaluesof xby; (x) = C1exp (kx) + C2exp ( kx) (2.32) Thenumerical solution ofthewave functionisstarted ataposition xlow suchthatthis positionis well outside the wave region. In this region, the + solution is asymptotically approaching zero whereas the solution approaches infinity making it unphysical. As such, the amplitude C2 is settobezero. Theremainingamplitude, C1, isarbitrary. Usingtheseassumptions, theboundary conditionsat xlow become (xlow) = 0:1 (2.33a) ? (xlow) = 0:1k (2.33b) At xhigh,thisisanunphysical,growingsolution. However,forcertainvaluesof !,thecoefficient ofthegrowingsolution C1 iszero. Theexponentialpartofthesolutioniseliminatedandtheroots 32 -10 -5 0 5 10 -1.0 -0.5 0.0 0.5 1.0 x f H x L ? -x 2 sech 2 HxL sechHxL tanh 2 HxL tanhHxL Figure2.6: Plotoftheflowprofilefunctions. oftheequationarefoundfrom; (xhigh) exp (kxhigh) = 0 (2.34) Therootsofthisequationarethefrequencyandcorrespondinggrowthrateoftheinstabilitymode. By solving this equation for ! at various values of k, the dispersion relation can be mapped out. The Mathematica codeimplementingtheshootingmethodisfoundinAppendix C.1 . Numerical Results Figure 2.6 showsthefunctionalformsthatwillbeusedtoexaminetheeffectsthatflowshear hasontheKelvin-Helmholtzinstability. Allfunctionalformsasymptoticallyapproachaconstant attheboundariestoensuretheshearlayerisisolatedinspace. The tanhformhastheprofileoftwo flowsmovingpasteachother. The tanh2 formhastheformofastationarylayerinabulkplasma flow. Bycontrast,the sech, sech2 and exp formshaveamovinglayerinastationarybulk. Since thecurvesofthesethreeprofilesaresimilar,itisexpectedthateachprofilewillproduceasimilar 33 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 kL Re J w kV 0 N ? -x 2 sech 2 HxL sechHxL tanh 2 HxL tanhHxL Figure2.7: Plotofthefrequencyofthedispersionrelation. dispersioncurve. Bycomparingthesethreesimilarprofilesitishopedtoexaminehowthewidth ofhearlayeraffectsthemodefrequencyandgrowthrate. Figures 2.7 and 2.8 show the frequency and growth rate dispersion relations. For all flow profiles, there is a region of instability growth. The shape and size of the shear layer determines the width of the instability region. As expected, the growth rate and frequency curves for the sech, sech2 and exp formstakeasimilarshape. Forthesethreeprofileforms,thewidertheshear layer, the smaller the instability growth. The flow profiles for the tanh2 and sech2 are inverted with respect to each other yet their growth rate curves are the exact same. This suggests that it isthescalesizeoftheshearlayerthatdeterminesthegrowthoftheKelvin-Helmholtzinstability. However,thefrequencycurves,oftwocases,arevastlydifferent. Forthe tanh2 profile,thereisa bulkflowwiththeshearproducedbyastationaryregioninthecenter. Forthisshearprofile,high frequenciesareassociatedwithlongwavelengths. Bycontrast,forthe sech2 profile,theoppositeis produces. Thebulkisstationaryandthewiththeshearproducedbyaflowingregioninthecenter. Forthisprofile,highfrequenciesareassociatedwithshortwavelengths. 34 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 kL Im J w kV 0 N ? -x 2 sech 2 HxL sechHxL tanh 2 HxL tanhHxL Figure2.8: Plotofthegrowthrateofthedispersionrelation. Thisdemonstratessomeoftheeffectsthatshearedflowshaveonthestabilityoftheplasma. However,thisisonlyasmallcornerinthevastregionofplasmainstabilities. Othersheardrivenin- stabilitiesmayhavedifferentresponsestothevariousscalelengthsandshearprofiles. Furthermore non-lineareffectofthethreedimensionalplasmashape,maychangetheresponseoftheplasma. 35 Chapter3 ExperimentalDevice Thischapterwillprovideanoverviewofthehardwareanddiagnosticsusedinthisdissertation. AnoverviewoftheCTHdevicedesignisprovided. Thetheory,designandoperationofallmajor diagnosticsystemsusedintheprojectwillbediscussed. Figure3.1: PhotographoftheCTHdevice. Thelargehorizontalportshownontheleftsideofthe photoisatatoroidalangleof ? = 180 . Therighthorizontalportisatatoroidalangleof ? = 252 . 36 VECTORWORKS EDUCATIONAL VERSION VECTORWORKS EDUCATIONAL VERSION R a Figure3.2: ScaleddiagramoftheCTHvacuumvesselcross-sectionofthemajorandminorradii. 3.1 The Compact Toroidal Hybrid Experimentspresentedinthisdissertation,areperformedinthepurelystellaratorconfiguration ofCTH;i.e.,withoutthepresenceofdrivenplasmacurrents. Asaresult,thisprojectmadeuseofa subsetoftheCTHcoilsetdiscussedinSection 1.3.1 . Specifically,onlytheHelicalField(HF)and TrimVerticalField(TVF)coilsetsareused; theShapingVerticalField(SVF),RadialField(RF) andErrorCorrectionCoils(ECC)werenotusedforthiswork. Foreachexperimentdiscussed,the currentsettingfortheactivecoilswillbegiven. 3.1.1 Vacuum Vessel TheCTHvacuumvesselisatoruswithacircularcross-section. CTHhasmajorradiusof R = 0:75 m and minor radius of a = 0:26 m. Figure 3.2 shows a scaled diagram of the CTH vacuum vessel cross-section. A feature of the magnetic field structure is the symmetry planes. These are toroidalcross-sectionwherethemagneticfluxsurfacesare vertically(up-down)symmetric. Port placementonCTHiscenteredonthesesymmetryplanes. Therearefive, 18?-diameterConFlatflangesmountedhorizontallyontheoutboardsideat 36 andrepeatingevery 72 aroundthemachine. Aboveandbelowthehorizontalports, areapairof 412?ConFlatportsmountedata 45 anglewithrespecttothehorizontal. Fivepairsof 10?ConFlat 37 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=0.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=9.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=18.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=27.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=36.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=45.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=54.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=63.00? R (m) Z (m) 0.490.600.700.800.901.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=72.00? R (m) Z (m) Figure 3.3: Various flux surfaces produced from the HF and TVF coils for a single field period. Symmetryplanesarelocatedatthe ? = 0 ,? = 36 and ? = 72 . Shadedareasshowthemagnetic fieldsresonatewiththeECRHheating(Green: 17:67 GHz,Blue: 14 GHz). 38 VECTORWORKS EDUCATIONAL VERSION VECTORWORKS EDUCATIONAL VERSION Biasing Probe ECRH MicroWave (17.65) GHz Bolometer SXR Camera 4 T r i p l e Pr o b e H ? d e t e ct o r G u n d e s tr u p Pr o b e EC R H Mi cro W a ve (1 4 G H z a n d 1 7 . 6 5 G H z) SXR Sp e ct ro me t e r Feed G a s Pu f e r I n t e rf e ro me t e r Feed Ti Gettering Hall Probe Camera Visible Spectrometer Paddle SXR C a me r a 1 -3 V a cu u m Pu mp s Fe ed C a rb o n L i mi t e r Feed D i sch a rg e C l e a n i n g Arm Feed Feed Figure3.4: DiagramofvariousdiagnosticandequipmentlocationsonCTH.Bluemarksthetopport mountinglocations. Orangemarksthesideportmountinglocations. Bluemarksthebottomport mountinglocations. Diagnosticsrelevanttothisdissertationarehighlightedinbold. Diagnostics marked?Feed?arefeedthroughlocationsforthemagneticdiagnostics. 39 Vacuum Vessel MajorRadius R0 = 0:75 m MinorRadius a = 0:26 m Plasma Parameters ElectronTemperature y Te 5 20 eV ElectronDensity y ne 1018 m 3 IonCyclotronFrequency fci 10 MHz IonPlasmaFrequency fpi 150 MHz IonLarmorRadius y i 0:2 mm IonSpecies Hydrogen PlasmaBetay 10 5 MagneticFieldStrength jBj 0:75 T Table3.1: CTHOperationalParameters y ECRHonly verticalportsaremounted,onthetopandbottomofthevacuumvessel,atamajorradiusof 0:71 m at0 andrepeatingevery ?? = 72 . Figure 3.4 showsthemountinglocationsofvariousdiagnostic systemsandvacuuminfrastructureonCTH.TheCTHvacuumvesselisacontinuouslyconducting shellmadeofInconel 625 alloywithalowtoroidalresistance. Forthiswork,since,plasmaswere operatedwithoutinducedplasmacurrent,vacuumvesselcurrentsandinducedplasmacurrentwill beassumedtobenegligible. 3.1.2 Plasma Heating The plasma is generated using three Electron Cyclotron Resonance Heating (ECRH) mi- crowavesourcesprovidingupto 10 kWeach. Twosourcesprovideheatingpowerat 14 GHzres- onantwith 0:5 Tmagneticfields. Onesourceprovidespowerat 17:67 GHzresonantwith 0:64 T. Figure 3.3 showstheresonantfieldsatvariouscross-sectionsaroundCTH.Thegreenareasrepre- sentthe 17:67 GHzresonanceandtheblueareasrepresentthe 14 GHzresonance. ECRHplasmas haveamaximumcutoffdensity 4 1018 m 3. Table 3.1 showsvariousplasmaparametersunder ECRHheating. 40 3.2 Data Acquisition and Analysis CTHdata is acquired on one of twosystems. The slower speed (8 kilo-samplesper second) SCXI and the higher speed (100 kilo-samples per second) DTAC systems. All of the raw data from the diagnostics is stored in anmdsplus42 data base, a standard fusion data storage system developed at MIT and used through out the fusion community. The raw data is stored in 16 bits andiscoveredtomVbyaconversionfactorof 0:305185. CTHrunparametersarealsostoredsuch astimingfortheECRHsuppliesandthetiminginformationforeachofthedataacquisitionboards. Inthisdissertation,ECRHdrivetimesareusedtocropacquireddatadowntoatimeintervalwhen theplasmaispresentintheCTHdevice. Forthemajorityoftheexperimentsdescribedinthiswork,measurementsaremadebymoving the diagnostic probes, typically in 1 cm intervals, across the plasmas, as described in Chapter 2 . Eachpositioncorrespondstoasingleplasmadischargeor?shot?. ShotsonCTHarerepeatedwith arepetitionrateofapproximateoneshotevery 5 6 min. Afulldatasettypicallyconsistsofabout 16spatialpositions. Duringasingleshot,atimeseriesofvoltageandcurrentmeasurementsaremade. Depending ontheparticularexperimentbeingperformed,thetimeseriesdataisdividedintosubsectionsand averaged. In this dissertation, time intervals chosen are typically specified by the ECRH drive or a chosen biasing interval. Error in measurements is accounted for by calculating the standard deviation 43 ofthedatawithinthechosentimeinterval. 2 = 1n n? i (xi ) (3.1) Here isthemeanofthecollecteddata. Aftereachshot,theprobeismovedandthisprocedureis repeateduntilthedesireddepthisreached. 41 3.3 Limiters The edge of the CTH plasma is defined by a various limiters throughout the vacuum vessel. Theselimitersdefinetheboundarybetweentheopenfieldlinesandthescrapeofflayerknownas thelastclosedfluxsurface(LCFS).Theotherfunctiontheyserveistoprotectthevacuumvesseland preventdamagetoanyequipmentinsidethevacuumvessel. Eachlimiteriselectricallyconnected tothevacuumvessel. Duringnormaloperation,thevacuumvesselisgrounded. CTHhasfourpermanentlimiters. Twolimitershaveaminorradiusof 0:26 m. Oneismadeof molybdenummountedat ? = 144 whiletheotherismadeofstainlesssteelmountedat ? = 184 . Two molybdenum limiter blocks with a minor radius of 0:245 m are mounted at ? = 300 and ? = 358 respectively. Amoveablecarbonlimitermountedatthe ? = 216 canbeusedtoadjust thesizeoftheplasma. Forthisdissertationonlythefixedlimitersareused. TheLCFSisdetermined from theV3FITequilibrium by calculating where the innermost flux surface intersects any of the limiters. 3.3.1 Biasing Probe Inordertostudytheeffectsofedgeelectricfieldsandplasmarotationintheedge,itisneces- sarytocontroltheinternalelectricfieldsoftheplasma. Atypicalwayofmodifyingelectricfields involveseitherdrawingcurrentfromorinjectingcurrentintotheplasma. TheCTHedgeelectric fields are modified by inserting a biased electrode past the last closed flux surface and drawing currentwhenabiasisapplied. Design Thebiasingtipisconstructedofa 34?? diameter, 1?? long 316alloystainlesssteelcylinderwelded toa 38?? diametershaft. AcylindricalAluminaceramictubeplacedovertheshaft,insuresthatthe probe tip is electrically isolated past the LCFS. The tip is coupled to a 12?? diameter probe shaft mounted on a 10 cm vacuum positioner. A photograph of the assembled probe tip is shown in 42 Figure3.5: Photographofbiasingprobetip. Figure 3.5 . The probe is mounted on a vertical port just off the symmetry plane at ? = 3:01 . Figure 3.6 showsacross-sectionaldiagramofthemountedprobe. Whennotinuse, theentireprobeassemblyisretractedfromtheplasmaandremainsbehind the limiter. When operational, the conducting part of the biasing probe is extended past the last closedfluxsurface. TheinsulatingceramicportionremainsbehindtheCTHlimiters. Thisinsures thatplasmadoesnotbecomeelectricallyshortedouttothelimiter. Earlierversionsofthebiasing probe, thatused asingleconductingrod, wereunsuccessfulatmodifyingtheedgeelectricfields. Theentireplasmawouldfloatupordowntothebiasappliedwhilepotentialprofileswouldremain unaltered. Bias Circuitry The entire biasing probe assembly is biased with respect to ground using a 100 V, 10 A Kepcobipolarpowersupplywithabuiltinfunctiongenerator. Figure 3.7 showsacircuitdiagram of the biasing setup. The probe is electrically isolated from the vacuum vessel using a ceramic 43 VECTORWRKS EDUCATIONL VERSION VECTORWRKS EDUCATIONL VERSION Limiters Ceramic Break Vacuum Positioner Vertical Port LCFS Vacuum Vessel Probe Tip Figure3.6: CTHcross-sectionofthebiasingprobemountedat ? = 3:01 . Theprobeisdrawn atthefulltravelposition. . Kepco A ProbeTip V Figure3.7: Circuitdiagramforthebiasingprobe. 44 Channel Name PlateDesignation ACQ1962:INPUT_47EDPB1 MeasuredBiasVoltage ACQ1962:INPUT_49EDPB3 MeasuredBiasCurrent Table3.2: TheCTHmdspluschannelsfortheBiasingProbe. break. Theactualvoltageappliedandthecurrentdrawnbytheprobearemeasuredduringtheshot. Currentisdeterminedfroma 10 VoutputsignalfromtheKepco. Thevoltageisdeterminedusing afractionalgaindifferentialamplifiercircuit. Intypicalbiasingexperiments,theinitialbiasisheldat 0 Vwithrespecttoground. Duringthe courseofthe 100 mslongshot,apre-programmedsetofvoltages,intherangeof 100 V,canbe appliedtotheprobe. Analysisoftheexperimentaldataisoftenperformedasacomparisonbetween the 0 Vandnon-zerobiasingregions. Atypicalbiasingschemewillstartwitha 0 Vbiasfor 25 ms. During this time period the initial plasma is still forming. As a result, the plasma parameters are generally unstable and make a poor choice for the 0 V reference. The bias is then increased to 100 Vforthenext 25 ms,anddecreasedbackdown 0 Vfor 25 ms. Unliketheinitial 0 Vbias,in this time period, the plasma conditions remain relatively steady. Finally the bias is decreased to 100 V. AtypicalbiasingwaveformisshowninFigure 3.8 . Data Analysis Raw measured voltage and current data is collected throughout the entire CTH shot on the DTAC system. Table 3.2 shows themdsplusdatabase entries for the Biasing Probe. Raw data is convertedtovoltageandcurrentsby Vreal = Raw0:3051851000mV V 36:66 V+ 0:61V (3.2a) Ireal = Raw0:3051851000mV V 1:3AV (3.2b) 45 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t (A) 120.00 -120.00 0.00 60.00 -60.00 L i mi t e r Bi a s (V) Figure3.8: Rawvoltageandcurrentdataforthebiasingprobeforasingleshot. Itisunnecessarytozerooutdataonthisprobebecausevoltageoffsetsofthecircuitaremeasureda prioriandtakenintoaccountinEquation 3.2a . Figure 3.8 showsprocessedbiasingprobedatafor atypicalbiasingprocess. 46 3.4 Diagnostics CTHhasanextensivearrayofplasmadiagnostics. Figure 3.4 showsthemountinglocations ofvariousplasmadiagnostics. ThemajorityofCTHdiagnosticsaremagneticdiagnostics. There areawidevarietyofmagneticpickupcoilslocatedthroughouttheinsideandoutsideofthevacuum vessel. These diagnostics detect fluctuations in the magnetic field particularly magnetohydrody- namicinstabilitiesinthepresenceofplasmacurrent. AHalleffectprobeallowsin-situmeasure- mentsofthenon-fluctuatingmagneticfield. Allofthemagneticdiagnosticsystemsarediscussed extensively in work by Stevenson 9 . The line averaged density is measured through the use of an interferometer. CTHalsocontainsvarioushardandsoftX-raydiagnostics. Howeverintheabsence ofplasmacurrent,andthelowvalueofplasma 10 5,theusefulnessofthesediagnosticsisdi- minished. Thisdissertationwillfocusonmeasurementsprimarilyfromtwo,in-situprobesystems thatwillmeasuretheplasmaparametersanddetectflowsintheplasma. BecausethisworkreliesonprobemeasurementstakenatdifferentlocationswithintheCTH vacuum vessel, it is essential to have a good reconstruction of the evolution of the magnetic sur- facesduringeachshot. ThisisaccomplishedbyusingtheV3FITequilibriumreconstructioncode 38 . V3FITusesthethree-dimensionalequilibriumcodeVMECtoreconstructtheplasmaequilibrium. The calculatedequilibriumiscomparedtoplasmadiagnosticsandinputparametersareadjusted. This procedure is repeated until the equilibrium reconstruction and plasma diagnostics agree. Recon- structedequilibriaareplottedtoshowthesize,shapeandpositionoftheplasma. Thisdissertation makesextensiveuseofreconstructedequilibriumattainedfromV3FITtofacilitatecomparisonof probemeasurementsindifferentareasoftheplasma. Adescriptionofhowthiscomparisonismade isgiveninSection 3.5 . Aspartofthisresearchproject,IwasresponsibleforinterfacingtheCTH dataacquisitionsystemwiththeV3FITcode. Thiswasachievedbybuildingaclientserversystem that has been successfully operating for over three years. A copy of the server source code and LabVIEWclientinterfaceisavailableontheCTHarchiveserver. 47 3.4.1 Triple Probe Generalplasmaparametersareobtainedbymeansoftheinstantaneoustripleprobebasedupon theChenandSekiguchidesign 39 . AtripleprobeisathreetippedLangmuirprobethatusesafixed bias. Thenatureofthefixedbiasallowsasimplifiedanalysis,andafasttimeresponse. Inessence, this Langmuir probe technique allows real time in-situ measurements of electron temperature Te, electron density ne and floating potential f, and an estimation of plasma potential p. Electric fieldscanbeestimatedbytakingthegradientofeitherthe f or p profiles. Theory The current flowing through any biased Langmuir probe can be written as the sum of the electronandioncurrentasfollows: 39 In = Ieexp [ e kbTe ( n p) ] Isat (3.3) where kb is Boltzmann?s constant, Te is the electron temperature and p is the plasma space po- tentialorplasmapotential. Thetotalcurrentflowingthroughaprobetip nisthecombinedcurrent contribution from the electrons Ie, scaled by the probe tip bias potential n, and the contribution from the ion saturation current Isat. Unbiased, each probe tip, when isolated from ground, will chargetothefloatingpotential f andnonetcurrentwillbecollectedbytheprobe. Thetripleprobeisarrangedsuchthatonetip(probetip 2)remainselectricallyfloating. There- for,Equation 3.3 asappliedtoprobetip 2 becomes Ieexp ( ek bTe p ) = Isatexp ( ek bTe f ) (3.4) A fixed bias is applied between probe tips 1 and 3 such that probe tip 1 is based positively with respecttoprobetip 3. Probetip 1willreachsomepotentialabove f,labeled 1,andstartsdrawing current. Sinceprobetip 1isbiasedwithrespecttoprobetip 3,thecurrententeringprobetip 1must 48 equalthecurrentexitingprobetip 3. Probetip 3willacquireapotentiallowerthan f,labeled 3. Thekeycriterionoftheinstantaneoustripleprobeisthatthepotentialappliedbetweenprobetips 1and 3issufficientlylargetoforcetip 3toremaininionsaturation. CombiningEquations 3.3 and 3.4 ,thecurrentflowingthroughprobetips 1 and 3 respectivelyis I1 = I = Isatexp [ e kbTe ( 1 f) ] Isat (3.5) I3 = I = Isat (3.6) where I = I1 = I3. Havingobtainedexpressionforthecurrentflowingthrougheachprobetipthevariousplasma parameterscanbesolvedfor. CombiningEquations 3.5 and 3.6 ,Te canbesolvedfor. kbTe e = 1 f ln (2) (3.7) Itshouldbenotedthatthisexpressionisonlyvalidwhenbiasbetweenprobetips 1 and 3 issuffi- cientlylargethatprobetip 3 reachesionsaturation. PlacingEquation 3.7 into 3.5 andcombining itwithEquation 3.6 ,theelectrondensitycanbesolvedfor: Isat = ene ?k bTe mi (3.8) ne = IeA ? m i kbTe exp (1 2 ) (3.9) Here,I,isthenetcurrentflowingthroughprobetips 1and 3,Aistheprobetiparea,and mi isthe ionmass. Combiningalloftheseresults,Equation 3.4 canbeusedtoestimate p. p = kbTe2e [ ln ( m i 2 me ) + 1 ] + f (3.10) 49 Figure3.9: Photographoftripleprobetip. Design Theprobetipisconstructedfromthree, 1 mmdiametertungstenwiresinsertedthroughafour bore alumina ceramic tube. A photograph of the probe tip is shown in Figure 3.9 . The ceramic tubeislongenoughsuchthattheprobetipscanreachtothemid-planewithanymetalpiecesleft in the shadow of the limiter. Each probe tip is exposed 2 3 mm. triple probe theory assumes all probe tip have equal areas so each tip is exposed the same length. The ceramic tip is coupled to a 12?? diameter 316 alloy stainless steel probe shaft ending to an expanded 112?? tube with 234?? ConFlat vacuum flanges. Three Kapton coated ultra high vacuum (UHV) wires are connected to thetungstenwiresandfeedthroughtheprobeshaft. Wiresconnectedtotips 1 and 3 areatwisted pairtoavoidinducedcurrentpickup. AllprobewiresareconnectedtoathreeBNCvacuumfeed through. Thetripleprobeismountedonthecenterofatopverticalportat ? = 72 . Figure 3.10 shows across-sectionaldiagramofthemountedprobe. A 1? extensionholdsaweldedbellowsandprobe driveawayfromthevacuumvesseltoavoidmagneticperturbationsbytheelectricsteppermotor. Aweldedbellows,witha 34?? innerdiameteranda 14?? throw,ismountedtoa 20?? travelprobedrive. Asteppermotorandfinethreadedwormdriveallowspreciseprobepositioningwithin 0:005 mm. The flange that mounts the probe shaft to the bellows is rotatable allowing the probe tips to be adjusted so that no tips are shadowed in the toroidal direction. A photograph of the assembled probedrivesystemwithprobemountedisshowninFigure 3.11 . Whenfullyretracted, theprobe 50 VECTORWRKS EDUCATIONAL VERSION VECTORWRKS EDUCATIONAL VERSION LCFS Vacuum Vessel Vertical Port Bellows Limiters Probe Tip Figure3.10: CTHcross-sectionofthetripleprobemountedat 72 . Theprobeisdrawnatthefull travelposition. 51 Figure3.11: Photographofprobedrivesystemandbellows. . VVf P2 V P1 A Bias P3 Figure3.12: Simplifiedtripleprobecircuitdiagram. is contained within a 316 alloy stainless steel shell to prevent contamination of the tips during dischargecleaningandtitaniumgettering. Measurement Circuitry In its simplest form, all of the triple probe parameters can be obtained from three measure- ments. Figure 3.12 shows a simple circuit diagram for the triple probe. The potential difference 52 . InvertingAmplifier InvertingAmplifier DifferentialAmplifier + V 1M? 100k? + V 1M? 100k? + V 1M? 1M?1M? 100? 1M? P3 200V P1 P2 741 741 741 Figure 3.13: First op amp circuit for Triple Probe. This consists of two inverting amplifiers to measure potential differences, and a unity gain differential amplifier measuring the voltage drop acrossashuntresistortomeasurecurrent. Allopampsarepoweredbya 12 Vpowersupply. betweenprobetip P2 andgroundmeasuresthefloatingpotential f directly. Thepotentialdiffer- encemeasuredbetweenprobestips P1 and P2 isrelatedtotheelectrontemperature Te byEquation 3.7 . Theelectronplasmadensity ne,isdeterminedfromthecurrent( I)flowingthroughprobetips P1 and P3 using Equation 3.9 . It is important to note that this circuit does not provide a current pathtoground. Allthreetipsmustremainelectricallyfloatinginorderforthetripleprobeanalysis tobeperformed. Topreventacurrentpathtogroundandtoprovideascaledandbuffered 10 Vsignalforthe DTACcards,twoopampcircuitsarebuilt. Figure 3.13 showsthecircuitdiagramforthefirstofthe probecircuits. The f andthepotentialbetween P1 andground( 1)aremeasuredbymeansofa 53 Channel Name Description ACQ1962:INPUT_44MKPB1 1 ACQ1962:INPUT_45MKPB2 f ACQ1962:INPUT_46MKPB3 Isat Table3.3: TheCTHmdspluschannelsfortheTripleProbe. fractionalgaininvertingamplifier 44 . Resistorvaluesarechosentoprovideadividebytenscaling. Thepotentialdifferencebetween P2 and P1 ( f 1)issubtractedinsoftware. Currentflowing through the circuit is measured by unity gain differential amplifier 44 across a shunt resistor. A 100 ?,75 ?or 50 ?shuntisuseddependingonplasmaconditions. Differentialamplifierresistors arechosentobe 1 M? 1%resistors. Allopampsarepoweredbya 12 Vpowersupplyandare generic 741 models. Whilethetripleprobewasabletomakemeasurementsofplasmaparameters during positive biasing, during negative biasing, the circuit shown in Figure 3.13 suffered from commonmoderejection 44 problems. Data Analysis RawpotentialandcurrentdataiscollectedthroughouttheentireCTHdatashotontheDTAC systems. Table 3.3 showsatableofthemdsplusdatabaseentriesfortheTripleProbe. Rawdatais convertedtovoltagesandcurrentsby Vreal = Raw 0:305185 Gain1000mV V (3.11a) Ireal = Raw 0:305185 1 10 7 A A 1000mVV R (3.11b) where Gain isthevalueofthesignaldividerand R isthevalueoftheshuntresistor. Channels 1 and 2areconvertedusingEquation 3.11a . Channel 3isconvertedusingEquation 3.11b . Sincethe probetipsareelectricallyfloating,itcannotbepredeterminedwhatvalueschannels 1and 2should have when there is no plasma present. However the probe should not be drawing current when thereisnoplasma. Intheabsenceofshortcircuits,anycurrentoffsetsinthesignal,whenthereis 54 1.7001.600 1.650 1.675 1.6871.6621.625 1.6371.612 t (s) 4.5e+04 -1.7e+04 1.4e+04 3.0e+04 -1.4e+03 I ( ? A) 93.4 -27.6 32.9 63.1 2.6 ? f (V) 98.1 -21.3 38.4 68.2 8.5 ? 1 (V) Figure3.14: Rawvoltageandcurrentdataforthetripleprobeforasingleshot. no plasma, should be the result potential offsets in the measurement circuits. The initial value of thecurrentsignalismeasuredandusedtooffsetthetotalsignalinsoftware. Figure 3.14 showstherawvoltageandcurrentdataforasingleshot. The f and 1 areused tocalculate Te ateachdatapointusingEquation 3.7 . Datapointswhere f ismeasuredasahigher potentialthan 1 calculateasanegativetemperature. Thiscanoccurwhenthedifferencebetween f and 1 are within the noise signal or when plasma parameters are rapidly changing such as duringthetransitionbetweenbiasingregions. Thisoccursrelativelyinfrequentlyintheregionsof interestofthisdissertation. Assuchthesedatapointsareconsideredinvalidandremovedfromthe analysis. 55 1.7001.600 1.650 1.675 1.6871.6621.625 1.6371.612 t (s) 131.82 -0.70 65.56 98.69 32.43 ? p (V) 93.39 -27.63 32.88 63.14 2.63 ? f (V) 1.5e+12 0.0e+00 7.5e+11 1.1e+12 3.8e+11 n e (cm -3 ) 24.98 4.23 14.61 19.79 9.42 T e (e V) Figure3.15: Analyzedtripleprobedataforasingleshot. Once Te is calculated, it is combined into Equation 3.9 along with the measured current to measure ne. Current should only ever flow in one direction. Areas that display negative current are indicative of local electron density being too small for the circuit to detect. Finally electron temperatureandfloatingpotentialarecombinedtocalculatetheplasmapotentialateachdatapoint usingEquation 3.10 . Figure 3.15 showstheanalyzedtimedata. Profilesofplasmaparametersare acquiredbythemethodoutlinedinSection 3.2 . From the triple probe data, electric fields and density gradients can be calculated. Electric fields can be calculated from the average p in one of two ways. The first way involves taking a finite difference of the s coordinate in ( s;u;v) space using triple probe positions. The second 56 1.000.40 0.70 0.85 0.930.960.890.770.810.740.55 0.620.660.590.470.510.44 s (arb.) 200.00 -200.00 0.00 100.00 150.00 50.00 -100.00 -50.00 -150.00 E s (a rb . ) Figure 3.16: Es calculated from triple probe p data in Flux surface space between the biasing probe(orange)andLCFS(verticaldashedline). methodinvolvestakingasplineinterpolation 40 influxsurfacespaceassumingthatdataisaxisym- metric. In other words, the spline is interpolated across the entire plasma column in flux surface space to ensure that first derivative is zero at s = 0. The electric field is then calculated taking thefirstderivativeoftheresultingsplinefunction. Ineithercase, Es iscalculatedfromEquation 3.23 . Thevectoristransformedbackintolabspace( r;?;z)tofind E. Electronpressuregradients are calculated from Te and ne in the same manner. For ion pressure gradients, quasi-neutrality is invoked, ne = ni,andtheiontemperatureisassumedtobecold Ti 1 eV. Figure 3.16 showsa measured Es profileinCTH. 3.4.2 Gundestrup Probe Inordertostudytheeffectsofedgeflowsonplasmastability,alocalizedmeasurementofthe plasmaflowisrequired. AGundestrupprobeisamultidirectionalvariantofaMachprobe. Mach Probesareusedtomeasureplasmaflows. AMachprobeconsistsoftwotips,onefacesupstream and the other faces downstream. Since the upstream tip leaves a wake downstream, the two tips collectunequalionsaturationcurrents. Theratiooftheupstreamtothedownstreamcurrentswill beshowntoberelatedtotheMachnumberoftheplasmaflow. Here,theMachnumber, M = vcs, is defined as the ratio of the plasma flow velocity, v, to the plasma sound speed, cs. The sound 57 speedisdefinedas 45 : cs = ?Zk bTe + kbTi mi (3.12) where Z istheionizationstateoftheplasma. FortheseexperimentsonCTH,theplasmaismade fromhydrogenandtherefore Z = 1. TheGundestrupprobecanresolvenotonlyspeedbutalsothegeneraldirectionofplasmaflow inatwodimensionalplane. Shearedflowscanbestudiedbymovingtheprobetodifferentspatial locationsandmeasuringchangesinspeedand/ordirection. Theprimaryobjectiveofthisworkis to measure the changes in flow induced by modifying the edge electric field over the course of a plasmashot. Theory Strong electric fields or sharp density gradients, such as the edges of stellarators and toka- maks, induce a poloidal flow perpendicular to a magnetic field line. In order to understand the Mach probe theory in these conditions it is necessary to look beyond the diffusion as a primary mechanism. A purely convective treatment is more appropriate. From both isothermal fluid and kineticsolutions 45 ,46 theratioofionfluxtoopposingplatesinaplasmaisgivenforsubsonicflows by R = IupI down = exp [( M? M?cot ) Mc ] (3.13) where R is the ratio of the upstream to downstream collected currents. The angle between the magnetic field vector and the tangent of a plate surface is and M? and M? are the parallel and perpendicular Mach numbers respectively. For cold ion plasmas, the calibration factor, Mc 12, willbethevalueassumedinthisdissertation 45 . To extend this formulation from a single pair of plates to a Gundestrup Probe, the values of M? and M? aresolvedforusinga 2 minimizationtechnique. 2 = ?[ RM (M?;M?; ) RE]2 (3.14) 58 The modeled current ratio RM is taken from the right hand side of Equation 3.13 . To aid the minimizationalgorithmandpreventnumericalinstabilities,Equations 3.13 and 3.14 arecombined andrewrittenintotheequivalentform 2 = ? n (M ?sin n M?cos n Mcsin nln Rn )2 (3.15) andsummedovereachplatepair. Thevalue isdeterminedby = probe B + 2 (3.16) where probe is the angle between the plate normal and the horizontal. The angle, B, is the an- gle between the magnetic field vector and the horizontal in the plane of the probe. This angle is determinedby B = tan 1 B ^? ^n B ^? (3.17) where ^?istheunitvectorinthetoroidaldirectionand ^nistheunitvectorpointinginthedirection oftheprobeshaft. Design The Gundestrup probe in this work consists of six tips biased into ion saturation Isat. Each probetiphasacorrespondingtipmountedontheoppositesideoftheprobearrangedinsuchaway thatitscollectionareaonlyfacesoneareaoftheplasmaandshadowedfromplasmaflowsonthe opposite side. Six identical probe tips are mounted on an AX05 grade boron nitride core. Each probetipiscreatedfromasingle 1?diameter, 2?long, 316alloystainlesssteelrod. A 34?diameter stepiscut 18 ofaninchdownfromthetop. A 12?diameterboreisdrilledthroughthecenter. This pieceisthencutlengthwiseevery 60 producingsixidenticalprobetipswithanelectricalgapthe widthofasawbladebetweenthetips. 59 Figure3.17: PhotographoftheGundestrupprobewiththeAluminashieldpulledback. VECTORWORKS EDUCATIONAL VERSION VECTORWORKS EDUCATIONAL VERSION Alumina Shield Boron Nitride CoreProbe Tip Plate Mounting Screws Figure3.18: ScalediagramoftheassembledGundestrupprobetip. 60 Theboronnitridecoreisturneddownfroma 1?diameterstockbillet. A 18?long, 1?diameter capisleftinplaceatoneendwiththerestofcoreturneddowntoa 12?diameter. Thiscapconfines signalpickuptojustthepoloidalandtoroidaldirections. A 1?longsectionisturneddowntomate with the similar probe shaft used for the Triple Probe. The six probe tips screwed into the boron nitridecoreusing 4 40 14?inchflatheadscrews. Thisscrewsizesetstheconstraintonthenumber ofpossibleprobetipsforagivenprobediameter. SixkaptoninsulatedUHVwiresarespotwelded toanotchcutoutofthebottomoftheprobetips. A 1?OD, 34?IDAluminatubeisplacedoverthe probetipssothatonlya 18?sectionoftheprobetipisleftexposed. Totalexposedareaforeachtip is A 84:5 mm2. AphotographoftheassembledprobetipwiththeAluminashieldpulledback isshowninFigure 3.17 . Two 1?longslotsaremilledintooppositesidesoftheprobeshaft. Thisallowsventingof theprobeshaftandaholetofeedthroughwires. Thesixwiresarerundownthelengthoftheprobe shafttoasixpinfeedthrough. Theboronnitridecoreisheldinplacewithasetscrew. A 1?long, 1? diametercoupler 316 alloystainlesssteelcouplerholdthealuminatubepressfitagainsttheboron nitridecoreandprobetips. Thereisa 12?longsectionturneddowntoa 34?diametertomatewith thecenterofthealuminatube. Fourholesdrilledlengthwiseallowventingandthecollarisheldin placewithtwosetscrews. TheGundestrupprobeismountedon 36 horizontalport. Figure 3.19 showsacross-sectional diagram of the mounted probe. The center of this port lies along a symmetry plane. The probe ismountedabovethemidplaneata = 14:04 angleabovethemid-planesuchthat, whenfully extended, theprobetipreachesthecenterofthevacuumvessel. LiketheTripleProbe, theprobe shaft is mounted onto a bellows. The bellows are mounted to a 1? long extension from plasma chamberinthesamemannerastheTripleProbe. A 412?to 234?zerolengthreducingflangeadapts theprobesystemtotheCTHport. Insidethechamber,a 112?diameter 316alloystainlesssteeltube withaflappeddoorpreventscontaminationoftheelectrodesduringdischargecleaning. Ateflon collar around the probe shaft inside the 1? long extension, prevents the probe from sagging. The 61 VECTORWRKS EDUCATIONAL VERSION VECTORWRKS EDUCATIONAL VERSION LCFS Vacuum Vessel Horizontal Port Bellows Limiters Probe Tip Figure 3.19: CTH cross-section of the Gundestrup probe mounted at 36 . The probe is drawn at thefulltravelposition. VECTORWORKS EDUCATIONAL VERSION VECTORWORKS EDUCATIONAL VERSION Figure3.20: Scalediagramoverlayingthedeviationofacurvedtiptoaflattip. bellows are mounted to a 20? stepper motor driven probe drive in the same manner as the Triple Probe. As built, each probe plate has a curved surface. The Gundestrup probe theory presented in thisSectionassumesflatplates. Toquantifytheapplicabilityofthistheorytothe?asbuilt?probe, theratioofprobeareasandtheratioofthesurfacenormalswillbeexamined. Figure 3.20 shows 62 . + V ProbeTip 50? 200V AD629 Figure 3.21: Op amp circuit diagram for a single Gundestrup probe tip. A complete Gundestrup probecircuitcontainssixidenticalcircuits. ascalediagramoverlappingaflatplatetoanasbuiltcurvedplate. Takingtheratioofaflatplate surfaceareatoacurvedplatesurfacearea wr wr 3 = 3 95% (3.18) where w is the probe plate width and r is the probe radius. On a curved probe, the normal unit vector will deviate from the direction of the normal on a flat plate. Taking the component of the curvedplatenormalinthedirectionofflatplatenormalatthepointofhighestdeviationis ^nc ^nf = cos 6 87% (3.19) where ^nc and ^nf arethe unit vectors in the normal direction of the curves and flat plates respec- tively. TheseresultssuggestthatmodelingtheasbuiltplatesasflatisreasonableforthisGunde- strupProbe. 63 Channel Name PlateDesignation ACQ1962:INPUT_77Probe1 A ACQ1962:INPUT_78Probe2 B ACQ1962:INPUT_79Probe3 C ACQ1962:INPUT_91Probe4 D ACQ1962:INPUT_81Probe5 E ACQ1962:INPUT_82Probe6 F Table3.4: TheCTHmdspluschannelsfortheGundestrupProbe. Measurement Circuitry CurrentdrawnfromeachGundestrupprobeplateismeasuredthroughaunitygaindifferential amplifier circuit across a shunt resistor. Figure 3.21 shows a diagram of a measurement circuit for a single tip. Each probe circuit contains six identical circuits. Each probe tip is biased 200 V negativelywithrespecttogroundtoplaceeachprobetipintoionsaturation. Becauseeachcircuit isbiasedto 200 V,normalopampscannotbeused. TheAnalogDevicesAD629isusedtoavoid problemswithcommonmoderejection 44 . ForconditionsinCTH,itisfoundthatashuntresistor of 50 ? isworksbestforkeepingsignalswithina 10 Vrange. Allopampsarepoweredwitha 12 Vpowersupplywith 0:1 Ffilteringcapacitors. Data Analysis TherawmeasuredcurrentdataiscollectedthroughouttheentireCTHdatashotontheDTAC system. Table 3.4 shows a table of themdsplusdatabase entries for the Gundestrup Probe. Raw dataisconvertedtocurrentusing Ireal = Raw 0:305185 1 10 7 A A 1000mVV R (3.20) where R isthevalueoftheshuntresistor. Formeasurementspresentedinthisdissertation, shunt resistorsof 50 ? areused. SincetheGundestrupprobeisbasedonratiosofthecollectedcurrents, it is not strictly necessary to convert the raw signal to the actual current. None of the probe tips 64 1.7001.600 1.650 1.6751.6871.6621.6251.6371.612 t (s) -9.5e+03 -1.1e+05 -5.7e+04 -3.3e+04 -8.1e+04 I sa t ( ? A) A B C D E F 1.00 -1.00 0.00 0.50 -0.50 ? 2 0.45 -0.45 0.00 0.22 -0.22 Ma ch N u mb e r M ? M ? Figure3.22: ParallelandPerpendicularMachnumbers, 2 values,andtherawcurrentsmeasured fromtheGundestrupprobeforasingleshot. should be drawing current when there is no plasma present. In the absence of short circuits, any currentoffsetsinthesignalwhenthereisnoplasmashouldbetheresultofpotentialoffsetsinthe measurementcircuits. Theinitialvalueofthecurrentsignalismeasuredandusedtooffsetthetotal signalinsoftwareforeachchannel. Figure 3.22 showstherawcurrentdataforasingleshot,themeasuredparallel( M?)andper- pendicular( M?)Machnumbersandthe 2 forsignifyinghowclosetotheexactvaluefitteddata reached. Theratiosofcurrentscollectedonchannels A : D, B : E and C : F areusedtofind M? and M? by minimizing Equation 3.15 for each data point. For each shot, aV3FITreconstruction 65 is performed from values averaged over the shot length. The magnetic field vector at each Gun- destrup probe position is used to calculate the pitch angle between the magnetic field vector and the Gundestrup probe plate normal vector. Profiles of M? and M? are obtained from the method outlinedinSection 3.2 . 3.5 Probe Measurements To compare probe measurements from different areas of the plasma, a common framework must be employed. In fusion plasmas with closed magnetic surfaces, various plasma parameters areassumedtobeconstantoveramagneticfluxsurface. Thismakesthe scoordinate,asdiscussed in appendix B, a natural value for normalizing probe positions. Each probe position is converted fromlabspace( r;?;z)positiontoVMECfluxsurfacespace( s;u;v)positionusinganaverageV3FIT reconstructed equilibrium for each time interval using the method discussed in Section 2.2.1 . In thisdissertationallprofiledataispresentedasafunctionoftheVMECcoordinate sposition. The coils necessary to create stellarator magnetic field topologies, place constraints on the shapeofvacuumvesselsandtheplacementofdiagnosticaccessports. This,alongwithspacecon- straintsandcompetitionforportaccesswithotherdiagnosticandvacuuminfrastructure,canlead tomountingplasmadiagnosticsinlocationswhereinterpretationofmeasurementsischallenging. Toillustratethis, themeasurementpositionsoftwoprobesystemsareoverlaidontopofthe CTH magnetic surfaces in Figure 3.23 . At the toroidal angle of 36 , the Gundestrup probe is mounted on a horizontal port pitched at a downward angle of 14 . At the toroidal angle of 72 , thetripleprobeismountedonaverticalportoffsetfromthecenterofthevacuumvesselby 4 cm. Both probes have a maximum travel extent of 26 cm. However, because of their mounting loca- tions,movementdirectionsanddifferencesintheshapeofthemagneticfluxsurfaces,theseprobes sampledifferentregionsoftheplasma. Whenchangingthepositionsofbothprobesbyanequaldistanceinlaboratoryspace,anun- equal number of flux surfaces are traversed. Consider the triple probe path of the right side of Figure 3.23 ,forthefirstportionofprobetraveldefinedfrom z = 0:26 mto z = 0:15 m(Region1, 66 0.490.60 0.70 0.80 0.90 1.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=36.00? R (m) Z (m ) 0.490.60 0.70 0.80 0.90 1.01 -0.26 -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.26 Phi=72.00? R (m) Z (m ) Figure 3.23: Probe path for the CTH Gundestrup (Left) and Triple (Right) probes. The red box markstheregionwheretheprobetravelisperpendiculartothemagneticsurfaces(Region1). The blueboxmarkstheregionwheretheprobetravelisparalleltothemagneticsurfaces(Region2). redbox),theprobemovesmainlyperpendiculartothemagneticsurfaces. However,forthesecond portion of probe travel defined by z = 0:1 m to z = 0 m (Region 2, blue box), the probe moves mainly parallel to the magnetic surfaces. That is, a 1 cm change in probe position in Region 1 crossesagreaternumberofmagneticsurfacesthana 1 cmchangeinpositionofRegion2. More- over,spatialfeaturesinthemeasuredprofilesbythesetwoprobeswillnotlineupcorrectlywhen plottedsolelyasafunctionofprobeposition. Moreover, when trying to determine vector quantities such as the electric field in laboratory space, only the component of that quantity in the direction of the diagnostic travel can be deter- minedfromonedimensionalmeasurements. OnCTH,theonlypossiblelocationwhereadiagnostic could be mounted in a position allowing for diagnostic travel perpendicular to the magnetic flux surfaces is at the ? = 36 symmetry plane along the mid-plane and subsequent symmetry planes ever ?? = 72 . The lack of poloidal or toroidal symmetry in the stellarator magnetic surfaces, meansthatmeasurementsobtainedfromaregionwherethedirectionofprobetravelisorthogonal tothemagneticsurfaces,areonlyvalidforthatspecificlocationandatmatchinglocationsineach 67 0.490.600.700.800.901.01-0.26 -0.20-0.15 -0.10-0.05 -0.000.05 0.100.15 0.200.26 Phi=0.00? R (m) Z (m) 0.490.600.700.800.901.01-0.26 -0.20-0.15 -0.10-0.05 -0.000.05 0.100.15 0.200.26 Phi=18.00? R (m) Z (m) 0.490.600.700.800.901.01-0.26 -0.20-0.15 -0.10-0.05 -0.000.05 0.100.15 0.200.26 Phi=36.00? R (m) Z (m) 0.490.600.700.800.901.01-0.26 -0.20-0.15 -0.10-0.05 -0.000.05 0.100.15 0.200.26 Phi=54.00? R (m) Z (m) Figure3.24: Simulatedprobepositionsalongthemid-plane. fieldperiod. Atdifferentlocationswithintheplasma,themagneticsurfacesexpandandcompress alteringtheelectricfieldstructure. 3.5.1 Simulated Diagnostics Thepositionofanin-situmeasurementcanbetransformedintofluxsurfacespace. Tocompare plasmaprofiles,itisnaturaltousethemostradial-likefluxcoordinate sasmanyplasmaparameters canbeassumedtobeconstantalongamagneticfieldline. Todemonstratetheuseoftransforming 68 1.010.75 0.88 0.94 0.980.910.81 0.850.78 Major Radius (m) 10.0 0.0 5.0 7.5 2.5 A (a rb . ) ?=0? ?=18? ?=36? ?=54? 1.000.00 0.50 0.75 0.880.620.25 0.380.12 s 10.0 0.0 5.0 7.5 2.5 A (a rb . ) ?=0? ?=18? ?=36? ?=54? Figure3.25: Anarbitraryfluxsurfaceconstantquantityplottedasafunctionofbothmajorradius andfluxsurface s position. Inlaboratoryspacefluxsurfaceconstantquantitiesdonotalignwith probepositions. laboratoryspace( r;?;z)measurementpositionstofluxsurfacespace( s;u;v)spacefordataanaly- sis,Figure 3.24 showsfourmeasurementpathsallutilizingthesame (r;z) coordinatesatdifferent toroidalangles. Thesetoroidalcross-sectionsarechosenforuniformspacingwithinafieldperiod anddonotrepresentrealdiagnosticpositions. Asimulatedfluxsurfaceconstantquantitytakestheformof; A(s) = 8 >< >: A0 (1 jsj) jsj < 1 0 jsj > 1 (3.21) 69 Figure 3.25 shows a plot of the simulated flux surface constant quantity in both laboratory space (r;?;z)andfluxsurfacespace( s;u;v)coordinatesforatypicalECRHpulseonCTHforeachof thefourprobepaths. Thelaboratoryspace( r;?;z)positionofeachprobepositionisconvertedinto fluxsurfacespace( s;u;v)positionbythemethodoutlinedinSection 2.2.1 usingsplineinterpolated VMECquantities. Since the quantity is constant on a flux surface, when plotted as function of s, all probes measurethesameprofileeventhoughthefluxsurfaceprobe spositionsdonotalign. Whenplotted asafunctionofmeasurementpositionhowever,measuredprofilesshowdeviations. Thelocation of the last closed flux surface ( s = 1) in laboratory space, shown where the quantity A becomes zero, does not align. The toroidal angles of ( ? = 18 and ? = 52 ) are antisymmetric about the mid-planeoneithersideofthesymmetryplanesof( ? = 0 and ? = 36 ). Theprobepositionsof (? = 18 and ? = 54 )representaspecialcasewherelaboratoryspacemeasurementsalign. 3.5.2 Extrapolating Global Parameters Infusiondevices,nestedfluxsurfacesaredefinedassurfacesofconstantpressurewherethe magneticpressurebalancestheplasmapressure. Figure 2.1 showsthenestedmagneticsurfacesin theCTHdevice. Itisfurtherassumedthatafluxsurfaceisasurfaceofconstantelectricpotential. Thisimpliesthattheplasmapressure P andplasmapotential p areonlyfunctionsofthe scoordi- nate. Whilethisisclearlytrueforthepressure(becauseitistheinherentassumptionforcomputing theequilibrium),itremainstobeverifiedfortheelectricpotential. Atestofthisassumptioncanedonebyconsideringthepathtakenbythetripleprobe. Asnoted inRegion2ofFigure 3.24 ,themotionofthetripleprobeismostlyparalleltothemagneticsurfaces. This means that the u position of the probe is changing, while the s position nearly constant. If plasmapotentialisafluxsurfaceconstantquality,thetripleprobeshouldmeasurethesamevalue regardlessof uposition. Figure 3.26 , shows a plot of the plasma potential as a function of the real space position of the probe. The last four positions correspond to the movement of the probe at approximately a 70 0.260.05 0.16 0.210.10 z (m) 140.00 35.00 87.50 113.75 61.25 ? p (V) 1.00 0.00 0.50 0.75 0.25 s (a rb . ) Figure3.26: Plotcomparingthechangeinplasmapotential( p)tothechangeinfluxsurfacespace spositionasafunctionofposition. Thisshows p isafluxsurfaceconstantquantity. 71 constantvalueof s. Inthisregion,thevariationof p? p? = 1:2%,while s?s? = 10:9%. Fromthis,itis concludedthatmodelingthefluxsurfacesasequipotentialsurfacesisareasonableapproximation. This implies that the plasma pressure P and the plasma potential p are only functions of the s coordinate. Thissectionwillfocusonderivingtheelectricfield. However,allmethodsdiscussed applyequallytogradientsintheplasmafluidpressure. Thegeneralizedgradientoperatorisdefinedtobe r = @ @uiei (3.22) usingthecontravariantbasisvectors. Theelectricfieldcannowbedefinedas E = r p (s) = es @@s (s) = Es (s)es (3.23) Thecontravariantbasisvectoroftheelectricfieldandrelatedtothecovariantbasisvectorsofthe magneticfield(Equation 2.5 )by ei ej = ji (3.24) Thismeansthat Epointsinadirectionnormaltoamagneticsurfaceandisorthogonalto B. Since p is constant on a flux surface, the covariant electric field components ( Es) is also constant on a flux surface. As a consequence, by measuring a profile of p at any arbitrary position in any arbitrarydirection,thetotal E canbedeterminedeverywherewithinthe s extentofthemeasured profile. Derivativesoffluxsurfaceconstantquantitiescanbeobtainedbytakingafinitedifference in sorthroughacubicsplineinterpolation. Bothmethodswillbeusedthroughoutthisdissertation. Themethodusedwillbeidentifiedforeachmeasurementpresented. Asasimpleexampleofcalculatingtheelectricfield,consideramodeledelectricpotentialof the form in Equation 3.21 . From the definition of the electric field (Equation 3.23 ), the value of @ @s (s) isalsoonlyafunctionofthe scoordinateandthusconstantonafluxsurface. Figure 3.27 showstheprogressionfromafluxsurfacespacepotentialtoalaboratoryspaceelectricfield. Asa 72 ?=0? ?=18? ?=36? ?=54? F l u x Sp a ce L a b o ra t o ry Sp a ce (s) E s (s) | ~ E| Figure3.27: Startingfromapotentialprofile(black (s) = 0 V,yellow (s) = 10 V),theflux surfacespace Es (red Es (s) = 10arb:)canbeobtained. jEj(blue jEj = 0Vm,red= jEj = 240Vm)is obtainedbyconverting Es tolaboratoryspaceusingthecontravariantbasisvectorsatany( s;u;v) position. Allcrosssectionsareplottedfrom s = 0:02 to s = 1. reminder, the flux surface space vector components do not carry the same units as the laboratory spacecounterparts. Figure 3.28 showsvariousplotsofelectricfieldquantitiesmeasuredattheprobepositionsin Fig 3.24 for a potential of the form of Equation 3.21 . Figure 3.28 a shows the magnitude of the gradientofpotentialinfluxsurfacespace( Es). Thequantity Es representsafluxsurfaceconstant quantity. In Figure 3.28 b Es is transformed back into laboratory space providing the complete electricfieldvectorateachprobeposition. Figure 3.28 cshowsthedifferencebetweencalculating the total electric field from the flux surface constant Es quantity and measuring the electric field directlyfromfinitedifferenceinprobepositionatthe ? = 18 probepath. Thedeviationinthese twomethodsarisesfromthefactthatprobetravelisnotinthedirectionoftheelectricfield. 73 1.000.00 0.50 0.750.880.620.250.380.12 s 20.0 0.0 10.0 15.0 5.0 E s (a rb . ) ?=0? ?=18? ?=36? ?=54? a) 1.010.75 0.88 0.940.980.910.810.850.78 Major Radius (m) 240.0 0.0 120.0 180.0 60.0 | E| ( V / m ) ?=0? ?=18? ?=36? ?=54? b) 0.870.75 0.81 0.840.78 Major Radius (m) 240.0 0.0 120.0 180.0 210.0 150.0 60.0 90.0 30.0 | E| ( V / m ) Flux Space Lab Space c) Figure3.28: a)Plotof Es measuredateachprobeposition. b)Plotof jEjcalculatedfrom Es. c) Differencebetweenelectricfieldcalculatedfrom Es quantityandthetakingafinitedifferenceof directprobedataat ? = 18 . Directprobemeasurementscanonlymeasurethecomponentoftheelectricfieldinthedirec- tion of the probe travel. However, by transforming potential profiles into flux surface space, the flux surface constant quantity Es can be obtained. As a consequence, by measuring a potential profile anywhere, the full electric field vector can now be determined everywhere in the plasma. Figure 3.27 showsthemagnitudeoftheelectricfieldatthetoroidalcross-sectionsshowninFigure 3.24 . The red shaded areas shown represent regions of strong electric field. The blue shaded re- gionsrepresentareasofweakelectricfield. Onrightside(CTHoutboard),themagneticsurfaces becomehighlycompressedandresultantelectricfieldsbecomestrong. Towardthemagneticaxis, thesurfacesexpandandelectricfieldsbecomeweak. Themethodsdevelopedherewillbeusedfordeterminationofthefullelectricfieldandpres- suregradientvectors. Ithasbeenshownhowthesevectorsareorthogonaltothemagneticfields. Thepresenceofelectricfieldsandpressuregradientstransversetoamagneticfieldwillinducea flow perpendicular to both. Understanding the electric fields and pressure gradients is important forunderstandingplasmaflow. 74 Chapter4 Results To study the effect of driven plasma rotation on plasma stability, edge biasing experiments have been performed on the CTH device. The goals of edge biasing experiments are to: modify theedgeelectricfield, measureaninducedperpendicularflow, demonstratetheroleelectricfield playsinducingflowsandmeasuretheenhancementorsuppressionofplasmainstabilitiesassociated withtheseflows. ThischapterwilldiscussvariousedgebiasexperimentsperformedinECRHonly plasmasintheCTHdevice. 4.1 Edge Biasing Experiments AlledgebiasingexperimentsareperformedbyinsertingtheBiasingprobedescribedinSection 3.3.1 past the last closed flux surface. Various biasing schemes are employed over the course of a shot. In each case, when the bias voltage is set to 0 V, this condition will be considered the referenceorbaselineagainstwhichallotherbiasingconditionsarecomparedto. Foreachshot,dataiscollectedfromthetripleprobeandGundestrupProbe. Betweenshots,the probesaremovedin 1 cmstepsintotheplasmaasdescribedinSections 3.4.1 and 3.4.2 . Thetriple probe data is used to determine the plasma parameters. Electric fields will be measured from the p bytakingthegradientinfluxsurfacespaceasdescribedinSection A.1.4 andtransformedback intothelaboratoryframe. Thepressuregradientisdeterminedfrom Te and ne inthesamemanner. ParallelandperpendicularflowsaremeasuredbyscanningtheGundestrupProbe,asdescribedin Section 3.4.2 . Two main experiments are described in this study. The first, Experiment A, is a high input heatingpowerruninwhichallthreeECRHpowersuppliesareusedtogeneratetheplasma. The second,ExperimentB,isalowerpowerruninwhichonlyoneoftheECRHpowersuppliesisused. 75 ShotNumbers 11081821 11081837 HFCurrent 4500 A TVFCurrent 750 A RF1Power 4 kW RF2Power 4 kW RF3Power 6 kW Table4.1: CTHrunparametersforbiasingexperimentA. ShotNumbers 11090828 11090832;11090834 11090850 HFCurrent 4600 A TVFCurrent 650 A RF2Power 6 kW Table4.2: CTHrunparametersforbiasingexperimentB. TheoveralloperatingparametersforeachofthesecasesaregivenisTable 4.1 ,forExperimentA, Table 4.2 ,forExperimentB. 4.1.1 Data Analysis Each shot is divided into a number of time slices, typically 4 or 5. At each time slice, the magnet coil currents are averaged over this time interval andV3FITis run to produce awoutfile for that shot number and time interval.V3FITis configured to calculate the plasma equilibrium on 100 flux surface s positions. This is done to minimize errors in interpolated values between calculatedfluxsurfaces. Appendix Bdetailsthevariousquantitiesthatcanbecalculatedfromthe V3FITreconstruction. Thereconstructionisusedtoobtainamodelofthemagneticfieldstructure. This model is used to transform probe positions using the procedures outlined in Chapter 3 and Appendices Aand B,andtodetermineflowsintheplasma. Probe Data TripleprobeandGundestrupprobedataareanalyzedandaveragedforeachshotnumberand timeslice. ThemethodoffindingtimesliceaverageddataerroranalysisisoutlinedinSection 3.2 . TripleprobeandGundestrupanalysesareoutlinedinSections 3.4.1 and 3.4.2 respectively. 76 ? b ?n p ~v ~v ? Figure4.1: Anexaggeratedviewof theintersection ofthe Gundestrup probe tip(blackbar) with thecurvedfluxsurface(redline). Themagneticfielddirection(purple)ispointingintothepaper and the orange line shows the direction of the probe shaft. The Gundestrup probe only measures theprojectionoftheflow(blueline)intheplaneoftheprobetips(dashedblueline). Theerrorinthefluxsurfacespaceprobeposition( s),isestimatedbyconvertingthemaximum extentsoftheprobetipsfromlaboratoryspacetofluxsurfacespace. Forthetripleprobe,theprobe tipis 3 mmlong. Themaximumandminimum s valuesarecalculatedbytakingthe 1:5 mm oftheprobepositionandconvertingthatintofluxsurfacespace. Thecompletemeasurementtipof theGundestrupprobeisa 1?diametercylinder 18?inlength. Toestimatetheerror,eightpositions ontheinwardandoutwardfaceofthetiparesampledandconvertedintofluxsurfacespace. The maximumandminimum sextentinfluxsurfacespaceofthesesampledpositionsareusedtoesti- matetheresultingspatialerror. SincetheGundestrupprobetipisphysicallybiggerthanthetriple probetip,itisexpectedthatGundestrupprobepositionerrorwillbegreaterthanthecorresponding tripleprobeerror. Electricfieldandpressuregradientsaredeterminedfromthetripleprobeusingtheprocedure outlinedinSection 3.4.1 . Theresultingprofilesvaluesofthefluxsurfacespaceelectricfield( Es) 77 andpressuregradient( rP)areinterpolatedtotheGundestrupprobe spositions. Thecontributions to the plasma flows by the E-cross-B ( E B) and diamagnetic ( rP B) drift are estimated fromEquation 2.9 . However,theseflowscannotbedirectlycomparedtoGundestrupprobemea- surements. ThecomponentofthetotaldriftrepresentingtheperpendicularflowintheplaneoftheGun- derstrupProbeisestimatedusing v? = v ^b ^np (4.1) where ^bistheunitvectorinthedirectionofthemagneticfieldand ^np istheunitvectorpointing inthedirectionoftheGundestrupprobeshaft. ThelocalsoundspeediscalculatedfromEquation 3.12 ,ateachGundestrupprobe sposition,usingeitheralinearinterpolationorsplineinterpolation of Te and assuming a Ti value of 1 eV. Since Ti is on the order of 10% of Te it is expected that, errors in Ti estimation will produce errors in cs on the order of 10%. Normalizing the perpendicularflowtothesoundspeedallowsadirectcomparisonto M?. Biasing Probe and LCFS Positions Inordertointerprettheprofilemeasurements,thepositionsoftheLCFSandthebiasingprobe must be converted in the flux surface s coordinate as well. Under normalV3FIToperation the limiterpositionsareprovidedasafittinginputparameter. Thismakesthe s = 1surfacetheLCFS. However to allow probe measurements to extend into the scrape off layer, the CTH limiter input isartificiallyexpandedbeyondthewavelengthlimiters. InECRHonlyplasmas,thelowvaluesof plasma makesitreasonabletoassumethatvacuumsurfacesandplasmasurfacesarethesame. Tointerprettheprobepositions,itbecomesnecessarytodeterminethelocationofthephysical limiters which determine the location of the LCFS. This is achieved by performing a search for wheretheinnermostfluxsurfaceintersectsanyofthewavelengthlimiters. Thissearchisperformed foreachtimesliceofallCTHshots. Thesepositionsareaveragedtogether. Thestandarddeviation iscalculatedtoaccountforanyerrorcausedbyshiftsintheplasmapositionbetweenshots. 78 Thebiasingprobepositionisdirectlytransformedintofluxsurfacespace. Thetransformation is performed for each probe position for each time slice. The bottom center ( r = 0:712 m;? = 3:07 ;z = 0:165 m)and 1?(r = 0:712 m;? = 3:07 ;z = 0:187 m)abovethatpositionofthe biasprobetipareusedasthelabspacepositionsofthebiasingprobe. Thetransformed spositions areaveragedtogether. Errorsinthepositionarisingfromshiftsintheplasmapositionshottoshot areaccountedbycalculatingthestandarddeviationofthebiasingprobe spositions. 4.2 Plasma Parameters During Biasing Inthissection,themeasurementsfromthetwoexperimentalconfigurationsdescribedintables 4.1 (Experiment A) and 4.2 (Experiment B) are presented. Figures 4.2 to Figures 4.4 present the results from experiment A. Figures 4.5 to Figures 4.7 present the results from experiment B. In each figure, three plots are given. As discussed throughout this dissertation, it is essential to be abletomakeadirectcomparisonbetweenthedifferentsystems. Assuch,allprobespatialdatais presentedasafunctionoftheVMECfluxcoordinate s. Plot (a) in each figure summarizes the measurements from the triple probe and Gundestrup Probe. The top plot presents the parallel ( M?) and perpendicular ( M?) Mach number measured fromtheGundestrupProbe. Theremainingfourplotsareelectrontemperature( Te),electronden- sity ( ne), floating potential ( f) and plasma potential ( p) measured from the triple probe. It is importanttonotethatinfluxsurfacespace,thetripleprobeandGundestrupprobereachdifferent depthsintotheplasma. Ineachplot, theprojectedextentofthebiasingprobeinfluxcoordinates isshownasanorangestrip. TheLCFSisshownasaverticaldashedline. Plot(b)ineachfigure,comparesthemeasured M? tothetheoreticalcalculationoftheplasma drift. Electric fields and pressure gradients are measured from triple probe data. This limits the- oretical flow calculation to the depth of the triple probe. In each plot, the projected extent of the biasing probe in flux coordinates is again shown as an orange strip and the LCFS is shown as a verticaldashedline. 79 Forplot(c),ineachofthefollowingfigures,showsthetimeevolutionoftheappliedbiasing voltage and resulting current measured off the biasing probe. The experimental data is analyzed during the three time intervals when the probe is at 100 V, 0 V and 100 V. The shaded area represents the time interval used for each figure. It is noted that during the initial 0 V biasing phase, the plasma is still forming in the CTH vacuum vessel. As a result plasma conditions are somewhatunstable. Thesecond 0 Vbiasingphaseisusedasthecomparisoncaseforthesestudies asdescribedinSection 3.3.1 . 1.000.00 0.50 0.750.25 s 165.0 -80.0 42.5 103.8 -18.8 ? p (V) 95.0 -115.0 -10.0 42.5 -62.5 ? f (V) 2.0e+12 0.0e+00 1.0e+12 1.5e+12 5.0e+11 n e (cm -3 ) 45.0 0.0 22.5 33.8 11.2 T e (e V) 0.50 -0.55 -0.03 0.24 -0.29 Ma ch N u mb e r M ? M ? 1.000.00 0.50 0.750.25 s 0.50 -0.50 0.00 0.25 0.38 0.12 -0.25 -0.12 -0.38 M ? Measured Theory 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t 100.00 -100.00 0.00 50.00 -50.00 L i mi t e r Bi a s (V) a) b) c) Figure4.2: DataforexperimentAfromtimeinterval 1:62 s 1:64 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. Theorangeshadedregionrepresentstheextentofthebiasingprobetip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthelowerrightcornermarksthetimeintervalthatprofile dataisaveragedover. 80 1.000.00 0.50 0.750.25 s 165.0 -80.0 42.5 103.8 -18.8 ? p (V) 95.0 -115.0 -10.0 42.5 -62.5 ? f (V) 2.0e+12 0.0e+00 1.0e+12 1.5e+12 5.0e+11 n e (cm -3 ) 45.0 0.0 22.5 33.8 11.2 T e (e V) 0.50 -0.55 -0.03 0.24 -0.29 Ma ch N u mb e r M ? M ? 1.000.00 0.50 0.750.25 s 0.50 -0.50 0.00 0.25 0.38 0.12 -0.25 -0.12 -0.38 M ? Measured Theory 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t 100.00 -100.00 0.00 50.00 -50.00 L i mi t e r Bi a s (V) a) b) c) Figure4.3: DataforexperimentAfromtimeinterval 1:64 s 1:66 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. Theorangeshadedregionrepresentstheextentofthebiasingprobetip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthelowerrightcornermarksthetimeintervalthatprofile dataisaveragedover. 81 1.000.00 0.50 0.750.25 s 165.0 -80.0 42.5 103.8 -18.8 ? p (V) 95.0 -115.0 -10.0 42.5 -62.5 ? f (V) 2.0e+12 0.0e+00 1.0e+12 1.5e+12 5.0e+11 n e (cm -3 ) 45.0 0.0 22.5 33.8 11.2 T e (e V) 0.50 -0.55 -0.03 0.24 -0.29 Ma ch N u mb e r M ? M ? 1.000.00 0.50 0.750.25 s 0.50 -0.50 0.00 0.25 0.38 0.12 -0.25 -0.12 -0.38 M ? Measured Theory 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t 100.00 -100.00 0.00 50.00 -50.00 L i mi t e r Bi a s (V) a) b) c) Figure4.4: DataforexperimentAfromtimeinterval 1:66 s 1:68 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. Theorangeshadedregionrepresentstheextentofthebiasingprobetip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthelowerrightcornermarksthetimeintervalthatprofile dataisaveragedover. 82 1.000.00 0.50 0.750.25 s 140.0 -40.0 50.0 95.0 5.0 ? p (V) 95.0 -60.0 17.5 56.2 -21.2 ? f (V) 2.0e+12 0.0e+00 1.0e+12 1.5e+12 5.0e+11 n e (cm -3 ) 35.0 0.0 17.5 26.2 8.8 T e (e V) 0.45 -0.40 0.03 0.24 -0.19 Ma ch N u mb e r M ? M ? 1.000.00 0.50 0.750.25 s 0.50 -0.50 0.00 0.25 0.38 0.12 -0.25 -0.12 -0.38 M ? Measured Theory 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t 100.00 -100.00 0.00 50.00 -50.00 L i mi t e r Bi a s (V) a) b) c) Figure 4.5: Data for experiment B from time interval 1:625 s 1:65 s. a) From Top to Bottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. The orange shaded region represents the extent of the biasing probe tip. The vertical dashed dotted line marks the LCFS. The shaded region in the lower right corner marks the time intervalthatprofiledataisaveragedover. 83 1.000.00 0.50 0.750.25 s 140.0 -40.0 50.0 95.0 5.0 ? p (V) 95.0 -60.0 17.5 56.2 -21.2 ? f (V) 2.0e+12 0.0e+00 1.0e+12 1.5e+12 5.0e+11 n e (cm -3 ) 35.0 0.0 17.5 26.2 8.8 T e (e V) 0.45 -0.40 0.03 0.24 -0.19 Ma ch N u mb e r M ? M ? 1.000.00 0.50 0.750.25 s 0.50 -0.50 0.00 0.25 0.38 0.12 -0.25 -0.12 -0.38 M ? Measured Theory 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t 100.00 -100.00 0.00 50.00 -50.00 L i mi t e r Bi a s (V) a) b) c) Figure 4.6: Data for experiment B from time interval 1:65 s 1:675 s. a) From Top to Bottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. The orange shaded region represents the extent of the biasing probe tip. The vertical dashed dotted line marks the LCFS. The shaded region in the lower right corner marks the time intervalthatprofiledataisaveragedover. 84 1.000.00 0.50 0.750.25 s 140.0 -40.0 50.0 95.0 5.0 ? p (V) 95.0 -60.0 17.5 56.2 -21.2 ? f (V) 2.0e+12 0.0e+00 1.0e+12 1.5e+12 5.0e+11 n e (cm -3 ) 35.0 0.0 17.5 26.2 8.8 T e (e V) 0.45 -0.40 0.03 0.24 -0.19 Ma ch N u mb e r M ? M ? 1.000.00 0.50 0.750.25 s 0.50 -0.50 0.00 0.25 0.38 0.12 -0.25 -0.12 -0.38 M ? Measured Theory 1.7001.600 1.650 1.6751.625 Time (s) 10.00 -10.00 0.00 5.00 -5.00 L i mi t e r C u rre n t 100.00 -100.00 0.00 50.00 -50.00 L i mi t e r Bi a s (V) a) b) c) Figure4.7: DataforexperimentBfromtimeinterval 1:675s 1:7 s. a)FromToptoBottom: Mach Number ( M), Electron Temperature ( Te), Electron Density ( ne), Floating Potential ( f), Plasma Potential ( p). b) Measured and calculated values of M?. c) Measured bias voltage and current. Theorangeshadedregionrepresentstheextentofthebiasingprobetip. Theverticaldasheddotted linemarkstheLCFS.Theshadedregioninthelowerrightcornermarksthetimeintervalthatprofile dataisaveragedover. 85 Electric fields are established by gradients in the plasma potential ( p). For positive biases, figures 4.2 aand 4.5 a,experimentAwasnotabletoestablishanelectricfield. Theplasmapotential p profiles(a)bottomplot)showaflatprofile. Bycontrast,experimentBdoesgenerateasignif- icant electric field. For 0 V biases, it is expected that there should be no electric fields since the biasingprobeisfixedatthesamepotentialastheCTHvacuumvessel. p profilesremainflatfor both experiments as seen in figures 4.3 a and 4.6 a. Negative biases, figures 4.4 a and 4.7 a, show theoppositeeffectthatpositivebiasesshow. InExperimentA,negativebiasesproducearadially inward(negative)electricfield. InExperimentB,negativebiasescouldnotproduceasignificant electricfield. One fundamental difference between experiments A and B is the amount of ECRH heating powerused. InExperimentB,theelectrontemperature(a;4thplotfromthebottom)islower. On average, electron temperatures in the core of the plasma were Te 15 eV for Experiment A and Te 10 eVforExperimentB.Whilethisisnotunexpectedduetothereducedinoutheatingpower, results show that lower electron temperatures play a pivotal role in ability to produce an electric fielddirection. InthehighelectrontemperaturesofexperimentA,radiallyinwardornegativeelec- tricfieldswereproducedundernegativebiasing. Significantradiallyoutwardorpositiveelectric fields could not be produced by positive biasing. By contrast in the lower electron temperatures ofexperimentB,theoppositeoccurs. Apositiveelectricfieldwasproducesfrompositivebiasing. However,asignificantnegativeelectricfieldcouldnotbeproducedfromnegativebiasing. Inei- ther case, induced gradients in p are localized between the biasing limiter tip and the scrape off layer. 4.3 Ion Flows FlowsinCTHshouldarisefromtwosources,intrinsicflowspresentintheplasmaandflows arising from edge biasing. The Gundestrup Probe, introduced in Section 3.4.2 , can measure the presence of flows both parallel and perpendicular to the magnetic field lines. It is expected that electricfieldsinducedfromedgebiasingwillmodifytheperpendicularflows. 86 1.000.40 0.70 0.85 0.930.770.55 0.620.47 s 300.00 -300.00 0.00 150.00 -150.00 E s (. a rb ) 0.25 -0.30 -0.03 0.11 -0.16 M ? 0.50 -0.60 -0.05 0.22 -0.32 M ? 100V 0V -100V Figure4.8: Plotcomparingfluxsurfacespaceelectricfield( Es)withmeasurementsofparalleland perpendicularMachnumberforexperimentA. AsshowninSection 4.2 ,the 0 Vcaseofbothpresentedexperimentsdoesnotshowthepres- enceofasignificantedgeelectricfield. Thisprovidesagoodbaselinefromwhichtocomparethe extenttowhichtheapplicationofapositiveornegativebiasleadstothegenerationofanelectric field. Furthermore,itwillbedeterminedwhetherthepresenceofthiselectricfieldleadstoamodi- ficationoftheperpendicularflowintheplasmathatislocalizedtotheregionwherethegradientin theplasmapotentialisgreatest. Casesthatshowthegenerationofedgeelectricfields,shouldshow a modification of the perpendicular flow localized to the gradient region of the plasma potential p. 87 1.000.40 0.70 0.85 0.930.770.55 0.620.47 s 300.00 -300.00 0.00 150.00 -150.00 E s (. a rb ) 0.25 -0.30 -0.03 0.11 -0.16 M ? 0.50 -0.60 -0.05 0.22 -0.32 M ? 100V 0V -100V Figure4.9: Plotcomparingfluxsurfacespaceelectricfield( Es)withmeasurementsofparalleland perpendicularMachnumberforexperimentB. Figure 4.8 comparesthefluxsurfacespaceelectricfield( Es)comparedtothemeasuredpar- allel and perpendicular Mach numbers for experiment A. For the 0 V case (dotted red line, open boxes), theperpendicularMachnumberisnearzerocorrespondingtoanearlyzeroelectricfield. Plasma flows in this case are dominated by parallel flows. By comparison to the positive bias case(solidblackline,opencircles),theparallelflownowreversesdirection. Perpendicularflows showanegativeflowinthescrapeofflayerhowever,perpendicularflowsdonotcorrespondwell tomeasuredelectricfields. Forthenegativebiascase(dashedblueline, opentriangles), showsa significant negative perpendicular flows corresponding to a negative electric field. Parallel flow remainsunalteredfrom 0 Vcase. 88 Figure 4.9 comparesthefluxsurfacespaceelectricfield( Es)comparedtothemeasuredpar- allel and perpendicular Mach numbers for experiment B. For the 0 V case (dotted red line, open boxes), theperpendicularMachnumberisnearzero. Plasmaflowsinthiscasearedominatedby parallelflows. Incomparisontothepositivebiascase(solidblackline,opencircles),theparallel flownowreversesdirection. Perpendicularflowsshowasignificantpositiveflowcorresponding to a positive electric field. The negative bias case (dashed blue line, open triangles) shows par- allel flows mainly unaltered from 0 V case. The perpendicular flow profile oscillates about zero correspondingtoanelectricfieldthatroughlyfollowsthesamepattern. In both experiments positive biases show a significant change in the parallel Mach number comparedtotheirrespective 0 Vcases. Theperpendicularflowsfor 0 Vcases,where p profiles areflatandcorrespondingelectricfieldsarenearzero,showmostlynoperpendicularflow. Parallel flowsremainunalteredfromthe 0 Vbiasesinboth 100 Vbiascases. Thecaseswherethereare largegradientsinthe p profiles,perpendicularflowsvelocitiesshowasignificantdeviationfrom the 0 Vcase. Perpendicularflowdirectioncorrespondswellwiththedirectionofthefluxsurface spaceelectricfields. 4.3.1 Comparison With Theory In order to interpret the ion flow measurements, the two mechanisms that contribute to the flow are considered. The first of these is the drift that arises in the plasma from the application of the electric field; i.e. the E-cross-B drift. The second of these arises from the gradient in the plasma pressure which is the diamagnetic drift. Both of these effects were discussed in detail in Section 2.1 . These combined drifts are solutions to the fluid equations when ignoring magnetic fieldgradientandcurvaturedrifts. TheGundestrupprobemeasurestheioncomponentofthefluid flow. Onlythepressuregradientsintheionfluidimposeadriftontheions. Duetothecoldions, diamagneticdrifteffectsaresmallcomparedtoE-cross-Bdrifts( 10%). Asaresult,itisexpected thatplasmaflowisdominatedbythe E B drift. 89 In figures 4.2 b, 4.3 b and 4.4 b and 4.5 b, 4.6 b and 4.7 b, total calculated drift is overlaid on top of measured perpendicular flows (right column, top plot). For cases of 0 V biases and cases wherethereisasignificantgradientinthe p profiles,experimentalandcalculateddatashowgood agreement. The negative bias case of experiment B shows similar trends. In regions where the flowisnegative,theelectricfieldgoesnegative. Inregionswheretheflowispositive,theelectric fieldsispositive. HoweverthemagnitudesofelectricfieldsproduceanE-cross-Bdriftinexcess ofthemeasuredperpendicularflow. Thiserrormayarisefromthelinearinterpolationsandfinite differencemethodsusedtocalculatetheelectricfields. Onlyinthepositivebiascaseofexperiment A,doboththeE-cross-Bdriftspeedanddirectiondisagreewithmeasuredperpendicularflows. 4.4 Instabilities Oneofthekeyfeaturesthathasoftenbeenassociatedwithflowingplasmasinfusiondevices has been the suppression of plasma instabilities 47 ? 51 . This is in contrast to wide ranging studies in laboratory and space plasma environments in which plasma flows are often the source of free energy that gives rise to instabilities 32 ,52 ? 55 . It is well-known that fusion plasmas are often in a vastlydifferentrangeofparametersfromtypicallaboratoryandspaceplasmaenvironments. This isanimportantcontributortothedifferencesinplasmaresponse. However,fortheexperimentsperformedontheCTHdevice,theplasmasformedaregenerally have a low electron temperature ( Te 10 eV) and moderate electron density ( ne 1018 m 3). These conditions are comparable to many laboratory experiments. Therefore, this work reports thatduringthepositivebiasofexperimentB( t = 1:625 s : 1:65 s),thereisthegenerationofalow frequencyoscillation. Figure 4.10 ,showstheoscillationonalltripleprobe,Gundestrupprobeand biasing probe raw signals. All signals are filtered with a bandpass pass filter between 1 kHz and 3 kHz. In further investigations of this phenomena, all probe signals will be cropped and filtered inthesamemannerforspectralanalysis. Figure 4.11 showsafastFouriertransformation(FFT)of the f asafunctionoftheprobeposition. Thismodehasafrequencyof f 2 kHz. 90 0.1000.000 0.050 0.0750.0870.0620.0250.0370.012 t (s) 2.50e+04 -1.20e+05 -4.75e+04 -1.12e+04 -8.38e+04 I ( ? A) GP A GP B GP C GP D GP E GP F TP 10.00 -10.00 0.00 5.00 -5.00 I (A) BP 120.00 -120.00 0.00 60.00 -60.00 ? (V) TP 1 TP 2 BP Figure4.10: Rawvoltageandcurrentsignalsonalltripleprobe(TP),Gundestrupprobe(GP)and biasingprobe(BP)channelsforshotnumber11090836. There is a broad spectrum of instabilities in the presence of sheared flows. However large categoriesofinstabilitiescanbeeliminatedfromexaminationofvariousscalesizes. Immediately, theinvestigationislimitedtoelectrostaticbranchesonlyarisingfromthelowvalueof . Ganguliet. al.30 providesahierarchyofelectrostaticinstabilitiesassociatedwiththepresenceofflowsparallel andperpendiculartomagneticfieldlines. This,however,isnotanexhaustivelistofinstabilitiesbut does provide guidance in narrowing down the broad spectrum of instabilities. From Gundestrup probemeasurements,thisinstabilityappearswhenthereisasignificantperpendicularflowinduced andparallelflowissuppressed. Asaresult,investigationsofthisinstabilitywillfocusontransverse flow-driveninstabilities. 91 6.000.00 3.00 4.50 5.253.751.50 2.250.75 f (kHz) 0.26 0.05 0.16 0.21 0.23 0.25 0.22 0.18 0.19 0.17 0.10 0.13 0.14 0.12 0.08 0.09 0.06 Po si t i o n (m) 100.00 0.00 50.00 75.00 87.50 93.75 81.25 62.50 68.75 56.25 25.00 37.50 43.75 31.25 12.50 18.75 6.25 Amp l i t u d e (. a rb ) Figure 4.11: Plot of fluctuation spectrum from f measured from the triple probe under positive bias. This instability is further parameterized by the ratio of the mode frequency compared to the ioncyclotronfrequency. IntheCTHdevice,theioncyclotronfrequencyis fci 10 MHzandthe observed instability frequency is fl 2 kHz where ! = 2 f 12:6 kHz such that !r ? ?ci. However,withthepresenceofanelectricfieldandinducedperpendicularflow,theobservedfre- quencyinthelaboratoryframeislikelytobeDopplershifted. Formeasuredflows,andreasonable wavelengths, it is anticipated that the mode frequency will satisfy the condition !r ? ?ci. As a result,theinvestigationofpossibleinstabilitieswillbelimitedtolowfrequencymodes. This regime is also characterized by the scale length of the potential gradient size to the ion Larmorradius( L ? i). ForthepositivebiasofexperimentB,thedensityandpotentialgradients, 92 Figure4.12: Plotoftripleprobepath(Blueline)influxsurfacespace. Theredlinedrawsaradial path from one point in the triple probe path. The orange ring shows the position of the biasing probeandthedashed-dottedlineshowsthelocationoftheLCFS. arelocatedbetweentheLCFSandtheBiasingprobetip L 3 cm. FromTable 3.1 , i 0:2 mm, theconditionof L ? i isalsoapplicable. Inthisregime,somecommonpossiblelowfrequencymodesinclude,Kelvin-Helmholtz 30 ,56 ? 58 , Rayleigh-Taylor or interchange modes 57 ,58 , drift waves 30 ,58 ,59 and ion acoustic waves 30 ,59 . Some ofthesemodesrequirethepresenceofshearedflowsasadrivingmechanismwhileotherscanbe stabilized or destabilized by the presence of sheared flows. To identify the possible instability, characteristics of the mode will be compared with experimental data. If possible, the dispersion relationwillbesolvedforrelevantCTHparameterstodetermineifthatinstabilityislikelytooccur. 93 4.4.1 Driving Mechanism To investigate the radial structure of the observed instability, the velocity, density and elec- tric field gradients will be compared to the peak wave power in flux surface space at each of the Gundestrup probe positions. Peak wave power is obtained by means of taking the FFT of each probe signal and spatially locating the peak of the spectrum. Radial structure of the instability is examined by means of a cross spectral analysis. The fluctuating floating potential measurement of the triple probe is cross-correlated to the fluctuation measured on the fixed biasing probe. At eachtripleprobelocation,thecrosscorrelationmagnitudeandphasedifferencebetweenthetriple and biasing probes are obtained. The radial phase difference is obtained by locating the phase at themaximumofthecross-correlationamplitude. Thephaseatthefirstprobepositionissettobe zero. Theradialphasestructureisaccumulatedfromthephasedifferenceateachprobeposition. It should be noted that this does not represent the true radial structure of the plasma because, as Figure 4.12 shows,thetripleprobepathisnotradialinfluxsurfacespace. Toinvestigatepossibledrivingmechanisms,profilesof ne, p and M? aresplineinterpolated influxsurfacespace. Thederivativeofeachprofileistakenwithrespectto s fromthecalculated piecewise spline functions to construct the flux surface gradient of these quantities. The results ofthiscalculationarethreekeyquantities. Thefirstofthese, thefluxsurface spaceelectricfield (Es),computedfromthegradientoftheplasmapotential( p). Thedensitygradientinfluxsurface space is computed from @ne@s . Finally the flux surface space shear frequency is competed from !s = @@sM?. These quantities are not the same as their laboratory space counterparts, however they are instructive for comparing with the spatial profile of the wave power with these various quantities. Figure 4.13 showsacomparisonof(frombottomtotop)thewavepowerforeachGundestrup probe tip, the radial phase difference of the triple probe f cross correlated to the biasing probe potential, !s, @ne@s and Es. Wave power is peaked between the biasing probe tip and the LCFS. Sincewavepowerdisappearsinboardofthebiasingprobe,furtherinvestigationswillconcentrate ontheregionfromthebiasingprobetothescrapeofflayer. 94 1.000.40 0.70 0.850.930.770.550.620.47 s (.arb) 1.50e+08 0.00e+00 7.50e+07 1.12e+08 3.75e+07 Amp l i t u d e (. a rb ) A B C D E F 185.00 -32.15 76.43 130.71 22.14 Ph a se Dif f (D e g . ) 2.50 -2.50 0.00 1.25 -1.25 ? s (. a rb ) 1.50e+13 -5.00e+12 5.00e+12 1.00e+13 -4.30e+04 - ? n (. a rb ) 300.00 -200.00 50.00 175.00 -75.00 E s (. a rb ) Figure4.13: Plotcomparingmeasuredradialwavepowertopotential,density,andvelocitygradi- ents in flux surface space under a positive bias. Starting from the top, flux surface space electric field (Es)measuredfromthegradientintheplasmapotential ( p),fluxsurfacespacedensitygra- dient ( @ne@s ), fluxsurfacespaceshearfrequency (@M?@s ). Peakwavepowerismeasuredfroman FFTof Isat collectedoneachGundestrupprobetiplabeledA-F. 95 First, we must rule out the possibility of this fluctuation coming from probe measurement circuity. Ifthisoscillationwaspresentonthecommongroundofallprobemeasurementcircuits, itwouldbeexpectedthatallprobemeasurementfluctuationswouldbeinphase. Radialprofilesof thephasechangeofthe f showaradialstructureofthephase. Furthermore,thechangeinphase is correlated to the peak in the fluctuation power. This provides good evidence that this is most likelyaplasmaeffect. The perpendicular velocity shear, i.e., shear frequency, !s, has a local extrema at the edge of the biasing probe and at the LCFS as shown in Figure 4.13 . Profiles of perpendicular Mach number (Figure 4.5 ), show the induced flow is localized between the biasing probe tip and the LCFS. Perpendicular flows inboard of the biasing probe and in the scrape off layer are generally quitesmall. Peakwavepowerisnotstronglycorrelatedwiththepeaksintheshearfrequency. This isindicativethatvelocityshearmaybeanunlikelydrivingmechanismofthisinstability. Thepeak ofthewavepower,howeveriscorrelatedwiththemaximaindensitygradientsandelectricfields. ThismaybeindicativeofadriftwaveorRayleigh-Taylorinstability 58 orsomeintermediatemode in-between. 4.4.2 Wavenumber Thenextstepinidentifying,possiblemodes,istheidentificationofthewavenumber k. Un- fortunately,duetolackofdiagnostics,atrue kmeasurementcannotbedirectlyobtained. However, insightsintothewavenumbermaybeobtainedfromanexaminationofthefirstordervelocityfluc- tuations. Figure 4.14 shows the fluctuations in M? and M? for a single data shot. At each data pointofmeasuredsaturationcurrent, M? and M? areobtainedasafunctionoftime. Byaveragingoverthisregion,thezerothorderflowscanbecalculatedasdescribedinSection 4.3 . The radial plasma flow is assumed to be diffusive in nature and small compared to induced parallel and perpendicular flows. In order to understand zeroth order flow direction, the zeroth 96 1.6501.625 1.637 1.644 1.6471.6411.631 1.6341.628 t (s) 0.15 -0.21 -0.03 0.06 0.11 0.02 -0.12 -0.07 -0.16 M ? 0.33 -0.08 0.12 0.23 0.28 0.17 0.02 0.07 -0.03 M ? Figure4.14: Fluctuationsinthemeasuredin M? and M? flows. orderflowisseparatedintoitsunitvectorcomponentsusing; ^m0? = M?( M2? + M2? )1 2 (4.2a) ^m0? = M?( M2? + M2? )1 2 (4.2b) For first order velocity fluctuations, M? and M? are filtered with the same band pass filter usedinearlierinthissection. Powerspectrafortheparallelandperpendicularflowsaremeasured fromthetimeseriesofGundestrupprobedata. FirstorderFourieramplitudesarelocatedfromthe 97 peakinthepowerspectra. Fourieramplitudesarenormalizedtofirstorderunitvectorsusing ^m1? = M ? ?( M?2? + M?2? )1 2 (4.3a) ^m1? = M ? ?( M?2? + M?2? )1 2 (4.3b) where M? representstheFouriertransformedamplitudes. Figure 4.15 shows the measured flow direction of zeroth and first order Mach numbers nor- malized to unit vectors. In general, zeroth order flows are mostly perpendicular to the magnetic fielddirection. Thesameholdstrueforthefirstordervelocityfluctuationsaswell. Atthepeakof thewavepowerfortheA,BandFcollectingplates,thefirstordervelocityfluctuationarenearly completely perpendicular. This indicated that the parallel wavenumber k? is small or nearly zero indicatingeitherlongwavelengthsornopropagationalongthefieldlines. 4.4.3 Dispersion relations Todetermineaninstabilitymoderequiresameasurementofthedispersionrelation. Normally, thisisachievedbymeasuringthe k atvariousmodefrequencies. Atypicalwayofmeasuringthe wavenumberofanelectrostaticmodeisbymeasuringthephasedifferencebetweentwoprobetips. However, an estimate of the expected wavelength is required a-priori to insure proper probe tip spacing. If the probe tips are spaced too far apart, multiple wavelengths can occur between the tips. Ifthetipsarespacedtwoclosetogether,therelativephaseshiftmeasuredcanbetoosmallto detect. ForflowstudiesonCTH,itwasnotknownwhattheeffectsofedgebiasingwouldproduce. A diagnosticmeasurementof kcouldnotbeperformeduntilapossiblewavemodeisidentified. Itis possiblehowevertoidentifypossiblewavenumbersthroughmodelingknownplasmainstabilities constrained by experimental observations. When possible, the dispersion relation will be solved 98 1.000.40 0.70 0.85 0.930.770.55 0.620.47 s (.arb) 1.00 -1.00 0.00 0.50 -0.50 Unit V e ct o r (. a rb ) Pa ra l l e l Zeroth First 1.00 -1.00 0.00 0.50 -0.50 Unit V e ct o r (. a rb ) Pe rp e n d i cu l a r Zeroth First 1.50e+08 0.00e+00 7.50e+07 1.12e+08 3.75e+07 Amp l i t u d e (. a rb ) A B C D E F Figure4.15: Plotcomparingtheparallelandperpendicularcomponentsofthezerothandfirstflow unit vectors to the peak wave power as a function of flux surface space s coordinate. Peak wave power is measured from an FFT of Isat measured from each of the Gundestrup probe tips. Peak wavepowercorrespondstotheregionwherezerothandfirstorderflowdirectionsareperpendicular tothemagneticfielddirection. 99 directlyor kwillbeestimatedfromamodelofthedispersionrelation. Thecalculatedvaluesof k aresubjecttotwolimits. 1. WavenumbersmustcorrespondtowavelengthsthatfitwithintheCTHvacuumvessel. 2. Wavenumbersmustcorrespondtowavelengthsthatarelargerthanthesizeofaprobemea- surementtip. For the first limit, if it is assumed that the vacuum vessel acts like a waveguide, the poloidal or toroidalcircumferenceofthechamberwouldconstrainthemaximumwavelength. Forthesecond limit, if the wavelength was smaller than the probe tip, the probe would not be able to measure a fluctuation because the that fluctuation would average out over the length of the probe. Using theselimitsandthedispersionrelationcalculations,thewavelengthwillbedeterminedfromthe k valuecorrespondingtothemeasuredmodefrequency. Possibleinstabilitymodeswillbeaccepted orrejectedbasedonvaluesof k thatfallwithintheseconstraints. For the ion acoustic and drift wave modes the models for frequency and growth rate pro- videdbySwanson 59 willbeused. FortheKelvin-Helmholtzwecansolvethedispersionrelation directly 30 ,57 usingashootingmethoddescribedinsection 2.3.1 . Forthedispersionrelationincor- porating all the features measured in the CTH plasma, we will solve the wave function provided by Guzdar et. al. 57 using the same shooting method. Parameters and values that will be used are defined in Table 4.3 . For simplicity, the plasma will be modeled using a slab geometry. Density andpotentialprofileswillbemodeledas n0 (x) = 7:45 1017 m 3tanh (x L ) + 7:55 1017 m 3 (4.4) ?0 (x) = 17 Vtanh (x L ) + 90 V (4.5) toqualitativelymatchtheprofilesmeasuredinCTH.Intheslabgeometry Bisassumedtopointin positive ^z. Tomatchthedirectionofgradientsrelativeto B withrespecttoCTH,theplasmacore isassumedtobeatpositive xandthescrapeofflayerisassumedtobeatnegative x. 100 Name Symbol Definition Charge e 1:6 10 19c PermittivityofFreeSpace ?0 8:9 10 12 Fm 1 MagneticField B 0:5T^z ScaleLength L 0:03 m IonMass mi 1:7 10 27 kg ElectronMass me 9:1 10 31 kg IonTemperature Ti 1 eV ElectronTemperature Te 10 eV IonThermalVelocity vi ? 3:2 10 19Ti mi ElectronThermalVelocity ve ? 3:2 10 19Te me IonAcousticSpeed cs ? 1:6 10 19Te mi Density n0 (x) Equation 4.4 Potential ?0 (x) Equation 4.5 E B Drift v0 (x) ? 0 (x) B ElectronDebyeLength De (x) ? ?01:6 10 19Te n0 (x)e2 IonCyclotronFrequency ?ci eBm i Table4.3: Variousparametersanddefinitionsusedforsolvingthedispersionrelationsforvarious instabilitymodes. Ion Acoustic Fortheionacousticwave,thefrequencyandgrowthratearedefinedtobe !2r = k 2c2 s 1 + k2 2De [ 1 + 3TiT e (1 + k2 2 De )] (4.6) and !i !r = p 3 ir 1 + 3 2ir ( exp ( 2ir)+ TiviT eve ) (4.7) where ri = !rkvi. The growth rate for the ion acoustic wave is negative ( !r < 0) implying this mode is damped. Without a driving mechanism external to the plasma, this mode will not exist. 101 This condition does not immediately rule out this mode as an external driving source is possible. Theionacousticmodedoesnottakeintoaccountthepossibilityofflow. ForapplicationtoCTH, it will be assumed that if this mode exists, that it is a rotating mode whose frequency !r is given by 2 fl + kv0 (x). Solvingforbothpositiveandnegativebranchesforthewavelengthmeasuredfrequency. The positivebranchgivesa k = 0:37 m 1 convertedtowavelengthis = 17 m. Thenegativebranch givesa k = 0:35 m 1 convertedtowavelengthis = 18 m. Withaminorradiusof 0:26 mthese wavelengthsaretoobigtofitintheCTHvacuumvesselradiallyandpoloidally. TheCTHmajor radius R0 = 0:75 mgivingthecenterofthevacuumvesselacircumferenceof 4:71 m. Ifthewave was traveling purely toroidally, the CTH vacuum vessel would be too small to contain this wave mode. However,thereisthepossibilityofthewavetravelingalongafieldline. Thelengthofafield linecanbelongerthanthecircumferenceofthevacuumvessel. Toestimatethesmallervalueof k thatcanfittheCTHmagneticfield,define k? = n R0 where nisthetoroidalmodenumberand isthe rotationaltransform. TypicalECRHplasmasinCTHareperformedinplasmaswhere isbetween = 0:22 and = 0:18. Thisplacestheboundariesof k inCTHtobebetween k0:22 = 0:3 m 1 and k0:18 = 0:24 1 which correspond to wavelength of 0:22 = 21 m and 0:18 = 26 m respectively. While these wavelengths are able to fit into the machine, estimates of ^k show wave propagation is mostly perpendicular to the magnetic field line. Evidence presented shows that the measured instabilityismostlikelynotanionacousticmode. Drift Waves Forthedriftwave,thefrequencyandgrowthratearedefinedtobe !r = ! e 1 i1 + k2 ? 2s (4.8) and !i = p ! e ( k2 ? 2 s 1 + k2? 2s )2( 1 + 2TiT e ) (4.9) 102 where the drift frequency defined to be ! e (x) = k?1:6 10 19TeeB n?0(x)n0(x), s = cs!ci and i = k2?v2i2?ci . Similarly to the ion acoustic wave treatment, for application to CTH, it will be assumed that if this mode exists, that it is a rotating mode and !r is replaced by 2 fl + kv0 (x). For fi the calculated k? is4 m 1 whichcorrespondstoawavelengthof 1:5 m. CTHhasamaximumpoloidal circumferenceof 1:63 m. Whilethiswavelengthisontheedgeoffittingwithinthevacuumvessel, thismodeshouldnotbedismissedentirelybecauseEquations 4.8 and 4.9 werederivedassuming anon-rotatingdriftwave. Thegrowthrateforthedriftwave(Equation 4.9 )ispositive. Thismeansthatthisisanaturally occurring instability in the presence of a density gradient. The estimated growth rate for CTH conditions and the k? = 4 m 1 is on the order of !i 10 8. This growth rate estimate shows a veryslowlygrowingmode. Therefore,whilethedriftwaveisapossiblecandidate,theslowgrowth ratemakeitanunlikelycandidate. Kelvin-Helmholtz In the appropriate limits, the generalized dispersion relations presented in Ganguli, et. al. 30 andGuzdar,et. al. 57 reducetothewellknownKelvin-Helmholtzdispersionrelation. @2 @x2 (x) k 2 y + kyv??0 (x) ! kyv0 (x) (x) = 0 (4.10) For this dispersion relation we can solve this directly using the shooting method presented at the beginning of this section. Figure 4.16 a shows the frequency and growth rate as a function of wavenumber. The growth rate is peaked at about k = 30 m 1 corresponding to a wavelength of = 0:21 m. Thefrequencyassociatedwiththispeakis 15320rads . Figure 4.16 bshowsaplotof the real and imaginary parts potential fluctuation amplitude. This is peaked at about x = 0, this roughlythesamelocationasthepeakinthevelocityprofileandisconsistentwithmeasurements of the location of peak wave amplitude (Figure 4.13 ). At x = 0, the Doppler shifted frequency 103 0 10 20 30 40 50 60 0 10 20 30 40 km -1 w kH z w i w r -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 xm F H x L ? F H 0 L F i F r a) b) Figure 4.16: a) Frequency ( !r) and Growth rate ( !i) for a classical Kelvin-Helmholtz mode. b) Normalized potential fluctuation amplitude for wavelength ( r) and imaginary parts ( i) for the classicalKelvin-Helmholtzmode( k = 30 m 1 and ! = 15320 + 6075i). inthelaboratoryframebecomes f = 3 kHz. Thisfrequencyiswithinareasonablerangeofthe measuredfrequencyshowninFigure 4.11 . 104 Density Gradient Modified Kelvin-Helmholtz DerivingthegeneralizeddispersionrelationforadensitygradientmodifiedKelvin-Helmholtz begins by neglecting parallel wave propagation ( k? = 0), assuming ! ? ?ci and the ion-neutral collision frequency i ? ?ci. The last assumption states that the ions are magnetized. Ignoring gravityandion-neutralcollisions,thegeneraldispersionrelationbecomes 57 @2 @x2 (x) + p(!;k;x) @ @x (x) + q(!;k;x) (x) = 0 (4.11) where p(!;k;x) = @@x ln n0 (x) (4.12) and q(!;k;x) = k2 + kv0 (x)! kv 0 (x) [ 1 v0 (x) @2 @x2v0 (x) + @ @x ln v0 (x) @ @x ln n0 (x) ] (4.13) For this dispersion relation we can solve this directly using the shooting method presented at the beginning of this section. Figure 4.17 a shows the frequency and growth rate as a function of wavenumber. The growth rate is peaked at about k = 26 m 1 corresponding to a wavelength of = 0:24 m. Thefrequencyassociatedwiththispeakis 13150rads . Figure 4.17 bshowsaplotof therealandimaginarypartspotentialfluctuationamplitude. Thisispeakedatabout x = 0,thisis roughlythesamelocationasthepeakinthevelocityprofileandisconsistentwithmeasurements of the location of peak wave amplitude (Figure 4.13 ). At x = 0, the Doppler shifted frequency in the laboratory frame becomes f = 2:6 kHz. This frequency is within a reasonable range of themeasuredfrequencyshowninFigure 4.11 . ComparedtothepureKelvin-Helmholtzmode,the presenceofthedensitygradientnarrowstheinstabilityregionanddecreasesthegrowthrate. 105 0 10 20 30 40 50 0 5 10 15 20 25 30 35 km -1 w kH z w i w r -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.5 0.0 0.5 1.0 xm F H x L ? F H 0 L F i F r a) b) Figure 4.17: a) Frequency ( !r) and Growth rate ( !i) for a density gradient modified Kelvin- Helmholtzmode. Thepresenceofdensitygradientsnarrowstheregionsofinstabilitygrowthand decreases the growth rate compared to the pure Kelvin-Helmholtz mode. b) Normalized poten- tial fluctuation amplitude for wavelength ( r) and imaginary parts ( i) for the density gradient modifiedKelvin-Helmholtzmode( k = 25 m 1 and ! = 13150 + 5050i). 106 Chapter5 Conclusions The study of flowing plasmas is a topic that has long been of interest to the basic, space andfusionplasmaresearchcommunities. Dependinguponthelocalplasmaparameters,numerous observations have shown that these flows can have both stabilizing and destabilizing effects on plasmas. Intheparticularareaoffusionenergyresearch,strongplasmaflowsintheplasmaedge areknowntosuppresslargescaleturbulence,reducetheradialtransportofparticlesandgenerally provide a stabilizing influence leading to improved energy confinement in fusion devices. These effectsarestronglybeneficialtothelong-termdevelopmentoffusionasaviableenergysource. Theworkpresentedinthisdissertationsummarizesaseriesofexperimentsthatseektounder- standtheroleofdrivenplasmaflowsonthestabilityofastellaratorplasmas. Whilemanystudies of driven plasma flows in fusion devices are focused on the transition to enhanced confinement regimes(so-called,?H-modes?),itisreiteratedthatthisisnotthefocusofthiswork. Instead,this investigation has sought to gain a more fundamental understanding of the mechanisms that can allowsubstantialflowstobeestablishedintheplasma,characterizingthoseflows,andmeasuring theresponseoftheplasmatothoseflows. This work is performed using the Compact Toroidal Hybrid (CTH) stellarator device oper- atedwithECRHgeneratedplasmasatlowtomoderateplasmadensities( ne 1018 m 3)andlow electrontemperatures( Te 15 eV). Themoderateplasmaconditionstheseexperimentswereper- formed in allow the use in-situ probe diagnostic systems. The flow in the plasma is established byusingabiasedelectrodetogenerateanelectricfieldthatisperpendiculartothemagneticflux surfaces. As a result of the application of the electric field, quasi-poloidal flows are established in the plasma. Experimental studies were performed to compare the behavior of the plasma in thepresenceandabsenceofthedrivenflows. Aseriesofinsitudiagnosticprobes, specificallya 107 tripleprobeandaGundestrupprobe,weredeveloped. Thetripleprobewasusedtomake?instan- taneous? measurements of the plasma parameters, i.e., the electron temperature, electron density, floatingpotentialandplasmapotential. TheGundestrupprobewasusedtomake?instantaneous? measurementsoftheflowintheplasma. Comparison of probe systems in the three dimensionally shaped magnetic surfaces of CTH, wasmadepossiblethroughtheuseoftheequilibriumreconstructioncodeVMEC.Thedirect,detailed modelofthemagneticfieldprovidebyVMEC,wascriticalforinterpretationofprobemeasurements andplasmaflowsintheCTHdevice. Assumptionsthatcertainquantitiessuchasplasmapotential and plasma pressure, are constant on a magnetic flux surface, allow localized measurements to diagnosetheglobalstructureoftheplasma. Furthermore,asaresultoftheplasmageneratedwith lowvaluesofbeta( 10 5), thevacuummagneticfluxsurfaceswereassumedtobeunaltered by the presence of the plasma. As a result, the equilibrium reconstruction could be extended out pastthenormaledgeoftheplasmaintothescrapeofflayer. Figure 5.1 showsanoverviewofallbiasingexperimentsperformed. Electricfieldsarecalcu- latedbetweentheLCFSandtheBiasingProbe. Electrontemperatureistakenfromthe z = 0:1m triple probe position during the zero bias time interval. The experiments performed showed that radiallyoutward(positive)electricfieldsweregeneratedusingapositivebiasvoltagebutonlyin plasmas with low electron temperatures ( Te 13 eV). In plasmas with higher electron tempera- tures( Te > 13 eV),positivebiasvoltageswouldnotgenerateapositiveelectricfield. Bycontrast, theoppositeistrueforradiallyinward(negative)electricfields. Negativeelectricfieldswerebest generatedwhen( Te 13 eV). In the cases where the electric field is formed, it was noted that this electric field layer was formed between the biased electrode and the limiter making the scale-length of the electric field significantlylargerthantheionLarmorradius. Inparticular,thisfactisimportanttocharacterizing the instability that is formed in the plasma. Only in the case of the low temperature plasma with aradiallyoutwardelectricfieldwasaplasmainstabilityobserved. Undernegativeelectricfields, thepresenceofacoherentplasmainstabilitywasnotfound. 108 0 5 10 15 20 25 -800 -600 -400 -200 0 200 400 T e (e V) E s (.arb) -Bias -Bias Instability +Bias +Bias Instability Figure 5.1: Plot shows the relationship between Es and Te for all positive and negative biasing experiments performed. Radially outward (positive) electric fields could not be generated above 13 eV electron temperatures. Below this threshold, positive electric fields drove instabilities. In- stabilitiescouldnotbedrivenwithnegativeelectricfields. Investigations of possible instability modes leads to the Kelvin-Helmholtz mode as a likely candidate. This mode is driven by the shear in the perpendicular velocity. This mode typically appearsinaregionwheretheiongyroradius i ? Lthescalesizeofshearlayerandthemodefre- quency ! ? ?ci ismuchlessthantheiongyrofrequency,consistentwithCTHplasmaconditions. Whenincludingthepresenceofthedensitygradient,theregionofinstabilitygrowthnarrowsand growthrateisreducedcomparedtothepureKelvin-Helmholtzmode. However,duetothelackof ameasurementofthewavenumber k,atrueidentificationofthewavemodecannotbeperformed. ItisnotedthattheoperatingconditionsfortheCTHdeviceforthisworkaresimilartoconditions found in basic laboratory type plasma experiments, so perhaps the observation of instabilities of theKelvin-Helmholtztypearenotunexpected. 109 5.1 Comparisons With Other Experiments Experimentsondrivenflowsinplasmashavebeenperformedextensivelyinlaboratoryplas- mas31 ,32 and in fusion devices 33 ,35 ,60 . In many of these devices, the generation of a strong edge plasma flows and the associated transition to higher confinement regimes, is often driven by the presence of a radially inward (negative) electric field 11 ,17 ,19 ,48 . In these cases, the edge electric fieldsareself-generatedfromtheambipolardiffusionofionsandelectrons. Theworkpresentedin thisdissertationismoretypicalofedgebiasingstudiesthathavebeenperformedontheTEXTOR, CASTOR, T-10 and ISTTOK tokamak experiments 33 and on the TJ-II stellarator experiment 35 ,60 andCompactAuburnTorsatron(CAT) 34 . 5.1.1 Comparison to Tokamak Experiments BiasingexperimentsonTEXTORsoughttoinvestigatetherolebetweenE-cross-Bshearand transport barrier formation. In TEXTOR, a mushroom shaped carbon limiter was inserted past the LCFS and biased to a maximum of 600 V. As the bias was increased, the edge electric field increased until a bifurcation occurred. The edge electric field jumps to a high level and density gradient is generated. A low frequency ringing of the plasma is developed. This is similar to results shown on CTH. By contrast, the ringing on TEXTOR was assumed to be caused by the biasingsupply. Inthiswork,extensivemeasurementsshowthattheoscillationonCTHhasaradial structureintheplasmaedge. Onbothdevices,underpositivebiases,astrongedgeelectricfieldis generatedalongwithastrongdensitygradientandlowfrequencyoccultationinduced. Biasing experiments on CASTOR examined the sought to measure the edge E-cross-B flow and the structure of turbulence. On CASTOR instead of biasing past the LCFS, the biasing lim- iter was just touching the edge of the plasma and not extending into it. Flows were measured by a Gundestrup probe and compared to calculation of E-cross-B flows made from electric field measurements. The radial structure of the edge plasma flow show good agreement with E-cross- B flows with diamagnetic drifts included as well. Measurements of the turbulent structure show 110 a wave like mode propagating poloidally. This is similar to results obtained on CTH however, measuredelectricfieldsonCASTORareradiallyinwardornegativeelectricfield. BiasingexperimentsonT-10achievedH-modebyapplyingapositivebiastotheplasmaedge. Inabroadrangeofdensityandplasmacurrentconditions,negativebiaseshadnoobservableeffect on plasma parameters. This is consistent with the results of biasing experiments on CTH. Mea- surementsofelectricfieldproducedshowedastrongshearelectricfield. Attheedgeofthebiasing electrode,astrongradiallyoutward(positive)electricfieldimmediatelyfollowedbyaradiallyin- ward(negative)electricfieldintheregionbetweenthelimiterandthebiasingelectrode. Itshould be noted also be noted that in CTH, there was a broadband suppression of low frequency fluctu- ationsduringthisphaseoftheexperiment. Thepresenceoftheshearedelectricfieldssuppressed broadbandfluctuationsbetween 3 30 kHz. AnimportantdifferencebetweenthesebiasingexperimentsandCTHisthateachoftheafore- mentionedstudieswereperformedintokamaks. Assuch,eachoftheseexperimentswereareper- formedinplasmaswithohmicheatingcurrentandwithanaxisymmetricplasma. Typicaldensities undertheseconditionsare ne 1019 m 3 areafactorof 10largerandcoreplasmatemperaturesare Te 1 keVareafactorof 100largerthandensitiesandtemperaturesonCTHunderECRHheating. As such absolute electric fields were determined from measured values of the f. Comparisons ofperpendicularflowsandE-cross-Bflowsmeasuredfromprofilesof f havepooragreementas comparedtoE-cross-Bflowsmeasuredfrom p. ForconditionsonCTH,profilesof f areapoor proxyforelectricfieldmeasurements. InTEXTOR,T-10andCTH,thebiasingelectrodeisinsertedpasttheLCFSwhereasonCAS- TOR, the bias was applied to the scrape off layer only. Experiments on ISTTOK compared both biasingregimes. ItwasshownthatwhenthebiasingelectrodewasinsertedpasttheLCFSasper- formedonTEXTOR,T-10andCTH,thatpositivebiasedproducedthebestcorrelationbetweenen- hancementofconfinementandshearedE-cross-Bflows. Whenthebiaswasappliedtothescrape off layer only, it was found that negative biases decreased particle losses. While absolute com- parison between these biasing experiments and CTH is not possible due to differences in plasma 111 conditions,qualitatively,resultsobtainedonCTHshowqualitativeagreementwithpreviousresults obtainedontokamakedgebiasingexperiments. 5.1.2 Comparison to Stellarator Experiments The forerunner to CTH was the smaller CAT device. Edge biasing experiments on CAT used a heated filament to inject current into the plasma. Typical plasma densities on CAT were ne 1016 m 3, a factor of 100 lower than CTH. Plasma temperatures were comparable to CTH conditions under ECRH at Te 5 20 eV. Rotation experiments on CAT focused on a neg- ative biases and the generation of radially outward (negative) electric fields. Measurements of plasmaflowvelocitywereperformedwitharotatableMachprobe. Comparisonsofpoloidalflow measurementsandE-Cross-Bflowcalculationsinplasmaswithalowneutralpressureshowgood agreementconsistentwithpreviousexperimentsandmeasurementsperformedonCTH.Therota- tion experiments on CAT show a consistent increase in plasma density and temperature when in thepresenceofedgebiasing. ExperimentsonCTHdidnotshowasignificantchangeinelectron temperatureduringbiasing. TJ-II is a stellarator with a plasma cross-section that has a ?C? shape. Edge biasing experi- mentsperformedonTJ-IIillustratetheabilityofpositiveandnegativeelectricfieldstomodifythe plasma parameters. Under negative biasing, there is a reduction of board band turbulent fluctua- tionsobserved. ExperimentsonCTHqualitativelyshowassuppressionofbroadbandfluctuations duringpositive biasing. Figure 5.2 showsthechangein potentialfluctuationonCTHforvarious biases. This is opposite of plasma response seen on TJ-II. However, plasma conditions on these devicesdiffer. Electrontemperaturesanddensityareafactorof 10 higherinTJ-IIthanthevalues obtainedintheCTHplasma. Bothmachinesuseabiasof Bias 100 V. OnefundamentaldifferencebetweenexperimentsperformedonCATandTJ-IIistheaddition of equilibrium reconstruction for probe measurements. On CAT, all probe measurements were performedonthemid-planeatequalfieldperiods. Measurementsonbothmachinesdonotaccount fortheshapednatureofthemagneticfieldswheninterpretingdata. ThisismadenecessaryonCTH, 112 75.0 -75.0 0.0 37.5 56.2 65.6 46.9 18.8 28.1 9.4 -37.5 -18.8 -9.4 -28.1 -56.2 -46.9 -65.6 50.00.0 25.0 37.5 43.731.212.5 18.76.2 f (kHz) Bi a s (V) 5.0 0.0 2.5 3.8 4.4 4.7 4.1 3.1 3.4 2.8 1.2 1.9 2.2 1.6 0.6 0.9 0.3 L o g (Amp l i t u d e ) Figure5.2: FFTifthefluctuationsin f measurementsatdifferentappliedbiases. due to the mounting position of probes. It has been shown in section 3.5.2 that measurements of global electric fields and density measurements are made possible through the use of theVMEC coordinatesystemprovidedfromequilibriumreconstruction. 5.1.3 Comparison with Laboratory Experiments Extensive work performed on linear machines shows the role that sheared flows play in the drivingandsuppressionofplasmainstabilities. Ingeneral,laboratoryexperimentstypicallyhave densities in the range of ne 1016 m 3 and temperature of about Te 5 eV. Magnetic fields are generally in the range of B 0:1 T to 0:01 T. Under these conditions, fundamental plasma frequencies,suchastheioncyclotronfrequency,areontheorderof ?ci 10 kHzandionLarmor 113 radiusisontheorderof i 1 cmfortypicalargonplasmas. Bycontrast,onCTHunderECRH heating in hydrogen plasmas, Densities are ne 1018 m 3. Temperatures can be produced in a comparablerange. Magneticfieldsareanorderofmagnitudehigher,making ?ci 10 MHzand i 0:1 mm. ExperimentsontheAuburnLinearExperimentforInstabilitystudies,showthatmeasurements ofanelectrostaticioncyclotroninstabilityinthepresenceofflowsheardrivenbyaradiallyinward (negative)electricfield. Measurementsofmodefrequencyshowthemodefrequencyontheorder of the ion cyclotron frequency. Biasing experiments performed on the Space Physics Simulation Chamber(SPSC)showthatashearednegativeradialelectricfieldprofiledrivesanelectrostaticion cyclotronwaveviatheInhomogeneousEnergyDensityDrivenInstability(IEDDI)mechanism. In boththeseexperiments,electricfieldsareinducedintheplasmabybiasingaringelectrode. Electric fieldscale-lengthsgeneratedareontheorderof L 2 5 cmwhichisroughlythesizeoftheion gyroradius. By contrast, observed modes in CTH exist in a region where mode frequencies are muchlessthan ?ci andelectricfieldscale-lengthsaremuchlargerthantheiongyroradius. Assuch directcomparisonofCTHfluctuationandtypicallaboratoryexperimentscannotbeperformed. 5.2 Future Work There are a number of unanswered questions and experimental regimes not covered in this dissertation. Due to limitations of experimental hardware and operational conditions, not all ex- perimentalopportunitiescanbeexploredatthemoment. However,therearesomepossibleavenues beyondthescopeofthisdissertation. 5.2.1 Argon Plasmas InCTH,loweringtheioncyclotronfrequencybydroppingthemagneticfieldstrengthisnot possibleduetotheuseofelectroncyclotronresonanceheatingastheprimarymethodofgenerating the plasma. Assuch, certain magneticfieldstrengthson theorderof B 0:5 Tarenecessaryto make use of the first harmonic ECRH. However, the ion cyclotron frequency can be lowered by 114 increasingthemassoftheions. InCTH,hydrogenplasmashave ?ci 10 MHz. Bychangingthe ionmasstoargon,theioncyclotronfrequencycanbereducedto ?ci 200 kHz. TheionLarmor radius increases to i 1 mm. If a similar electric field structure could be produced in argon plasmas, as hydrogen plasmas, it maybe possible to drive an instability closer to the conditions seeninlaboratoryexperimentsorsomeintermediatemode. 5.2.2 Higher Bias Voltages PositiveelectricfieldsonCTHwereonlycapableofproducingpositiveelectricfieldsincold plasmas. Duetolimitationsofthebiasingpowersupply,biasingexperimentsarelimitedto 100 V. Theplasmacouldonlyachieveuptosubsonicspeeds M 0:1 0:2. Inexperimentsperformed onTEXTOR,abiasof 600 Vwasabletoproducepositiveelectricfieldsinaplasmawithmuch highertemperaturethanCTHconditions. Byincreasingthebiasingvoltage,itmaysimplifyedge biasingexperimentsbyexpandingtherangewherepositiveedgeelectricfieldsareproduced. 5.2.3 Ohmic Heating CTH has the unique design where the device can be operated in stellarator like condition or near tokamak like conditions. Due to the fragility of diagnostics build for this dissertation it was discoveredearly,thatoperationinplasmaswithdrivenplasmacurrentswasnotpossible. Ifanew, morerobustdiagnosticcanbebuild,CTHhastheuniqueopportunitytobridgethegapbetweenthe earlybiasingexperimentsontokamakdeviceandedgebiasingexperimentsonstellaratordevices. 5.2.4 Magnetic Shear and Islands If the conditions of electric field generation can be achieved consistently or edge rotational transform can be altered over the course of a single shot, then the effects of magnetic shear and itsroleinenhancingorsuppressingedgeplasmaflowsandconfinementcanbestudied. TheCTH device has a number of toroidal field coils that can alter the toroidal field and change the edge rotational transform. If the coils currents are varied enough, a rational surface can be created in 115 the edge. Measurements of plasma flows and confinement in the presence of magnetic islands is ofspecificinterestinstellaratorsdevices. However,itshouldbenotedthatthediagnosticanalysis techniqueusingVMEC,developedinchapter 2 ,isinvalidinthepresenceofmagneticislandssince VMECassume nested magnetic surfaces. Future numerical studies on CTH using theNIMRODcode maybeabletostudytheeffectsofmagneticislands. 5.2.5 Wavelength Measurements Asdiscussedinsection 4.4 atrueidentificationofthewavemodecannotbeidentifiedwithout a measurement of the wavenumber. It has been stated in this dissertation that a density gradient modifiedKelvin-Helmholtzmodeisalikelycandidatefortheidentificationoftheinstabilitymode. Estimationofwavepropagationdirectionandstructuresuggestthatthisinstabilityisarealplasma effectandnotapowersupplyeffect. Byrecreatingtheconditionstodrivetheinstabilityobserved againalongwithameasurementofthewavenumber,itmaybepossibletofurtherconfirmorreject theexistenceofthiswavemode. Furtheritmaybepossibletoidentifythestructureoftheturbulence likeexperimentsperformedontheCASTORtokamak. 116 Bibliography 1 S.AtzeniandJ.Meyer-ter-Vehn, ThePhysicsofInertialFusion (OxfordUniversityPress,2004). 2 R. Repice, R. Adler, J. Berry, J. Micheals, R. Schmitt, et al. , ?Annual energy review 2009,? Tech.Rep.DOE/EIA-0384(2009)(U.S.EnergyInformationAdministration,2010). 3 F.F.Chen, IntroductiontoPlasmaPhysicsandControlledFusion ,2nded.(PlenumPress,1990). 4 C.M.BraamsandP.E.Stott, Nuclear Fusion, Half a Century of Magnetic Confinement Fusion Research (IOPPublishingLtd,2002). 5 U.Stroth, PlasmaPhysicsandControlledFusion 40 ,9(1998) . 6 H.-S. Bosch, A. Werner, R. Konig, R. Stadler, et al. , Fusion Engineering, 2009. SOFE 2009. 23rd IEEE/NPSS Symposium on , ,1 (2009) . 7 F. S. B. Anderson, A. F. Almagri, D. T. Anderson, P. G. Matthews, J. N. Talmadge, and J. L. Shohet,FusionTechnol. 27 ,273(1995). 8 J.Peterson,G.Hartwell,S.Knowlton,J.Hanson,R.Kelly, andC.Montgomery, FusionEnergy 26 ,145(2007) . 9 B.A.Stevenson, 3D Reconstruction of Plasma Equilibrium using Magnetic Diagnostics on the Compact Toroidal Hybrid ,Ph.D.thesis ,AuburnUniversity(2011). 10 F. Wagner, G. Becker, K. Behringer, D. Campbell, A. Eberhagen, et al. , Phys. Rev. Lett. 49 , 1408(1982) . 11 F.Wagner, PlasmaPhysicsandControlledFusion 49 ,B1(2007) . 12 M.Greenwald,R.Boivin,F.Bombarda,P.Bonoli, et al. ,NuclearFusion 37 ,793(1997) . 13 B.Tubbing,B.Balet,D.Bartlett,C.Challis, et al. ,NuclearFusion 31 ,839(1991) . 14 K. H. Burrell, S. L. Allen, G. Bramson, N. H. Brooks, et al. , Plasma Physics and Controlled Fusion 31 ,1649(1989) . 15 S.Tsuji,K.Ushigusa,Y.Ikeda,T.Imai, et al. ,Phys.Rev.Lett. 64 ,1023(1990) . 16 R.J.Hawryluk,S.Batha,W.Blanchard,M.Beer, et al. ,PhysicsofPlasmas 5 ,1577(1998) . 17 V.Erckmann, F.Wagner,J.Baldzuhn, R.Brakel, R.Burhenn, et al. ,Phys.Rev.Lett. 70 ,2086 (1993) . 18 M. Shigeru, M. Tomohiro, T. Kenji, et al. , Journal of Plasma and Fusion Research 80 , 279 (2004) . 117 19 R.M.McDermott,B.Lipschultz,J.W.Hughes,P.J.Catto, etal. ,PhysicsofPlasmas 16 ,056103 (2009) . 20 H.Biglari,P.H.Diamond, andP.W.Terry, PhysicsofFluidsB:PlasmaPhysics 2 ,1(1990) . 21 Z.Lin,T.S.Hahm,W.W.Lee,W.M.Tang, andR.B.White, Science 281 ,1835(1998) . 22 T.S.Hahm, PlasmaPhysicsandControlledFusion 44 ,A87(2002) . 23 K.ItohandS.-I.Itoh, PlasmaPhysicsandControlledFusion 38 ,1(1996) . 24 P.H.Diamond,S.-I.Itoh,K.Itoh, andT.S.Hahm, PlasmaPhysicsandControlledFusion 47 , R35(2005) . 25 K.Itoh,S.-I.Itoh,P.H.Diamond,T.S.Hahm, et al. ,PhysicsofPlasmas 13 ,055502(2006) . 26 Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and P. H. Diamond, Phys. Rev. Lett. 83 , 3645 (1999) . 27 C.Hidalgo,M.A.Pedrosa,L.Garc?a, andA.Ware, Phys.Rev.E 70 ,067402(2004) . 28 M.A.Pedrosa,C.Hidalgo,E.Calder?n,T.Estrada, etal. ,PlasmaPhysicsandControlledFusion 47 ,777(2005) . 29 T.Estrada,T.Happel,L.Eliseev,D.L?pez-Bruna, et al. ,PlasmaPhysicsandControlledFusion 51 ,124015(2009) . 30 G.Ganguli,M.J.Keskinen,H.Romero,H.R, et al. ,J.Geophys.Res. 99 ,8873(1994) . 31 W. E. Amatucci, D. N. Walker, G. Ganguli, J. A. Antoniades, et al. , Phys. Rev. Lett. 77 , 1978 (1996) . 32 E. E. Thomas Jr., J. D. Jackson, E. A. Wallace, and G. Ganguli, Physics of Plasmas 10 , 1191 (2003) . 33 G.V.Oost,J.Ad?ek,V.Antoni,P.Balan, et al. ,PlasmaPhysicsandControlledFusion 45 ,621 (2003) . 34 E. E. Thomas Jr., An Investigation of Helically Trapped Ion Orbits in the Compact Auburn Torsatron ,Ph.D.thesis,AuburnUniversity(1996). 35 C. Hidalgo, M. A. Pedrosa, N. Dreval, K. J. McCarthy, et al. , Plasma Physics and Controlled Fusion 46 ,287(2004) . 36 S.P.HirshmanandJ.C.Whitson, PhysicsofFluids 26 ,3553(1983) . 37 S.Hirshman,W.vanRij, andP.Merkel, ComputerPhysicsCommunications 43 ,143 (1986) . 38 J.D.Hanson,S.P.Hirshman,S.F.Knowlton,L.L.Lao,E.A.Lazarus, andJ.M.Shields, Nucl. Fusion 49 ,075031(2009) . 118 39 S.ChenandT.Sekiguchi, JournalofAppliedPhysics 36 ,2363 (1965) . 40 W.H.Press,S.A.Teukolsky,W.T.Vetterling, andB.P.Flannery, Numerical Recipes: The Art of Scientific Computing ,3rded.(CambridgeUniversityPress,2007). 41 R.A.Jefferson,M.Cianciosa, andE.Thomas, PhysicsofPlasmas 17 ,113704(2010) . 42 J.A.Stillerman,T.W.Fredian,K.Klare, andG.Manduchi, ReviewofScientificInstruments 68 ,939 (1997) . 43 D.S.Sivia, Data Analysis: A Bayesian Tutorial ,2nded.(OxfordUniversityPress,2007). 44 B.K.Jones, Electronics for Experimentation and Research (Prentice-HallInternational,1986). 45 L.PatacchiniandI.H.Hutchinson, Phys.Rev.E 80 ,036403(2009) . 46 I.H.Hutchinson, Phys.Rev.Lett. 101 ,035004(2008) . 47 C.P.Ritz,H.Lin,T.L.Rhodes, andA.J.Wootton, Phys.Rev.Lett. 65 ,2543(1990) . 48 E.J.Doyle,R.J.Groebner,K.H.Burrell,P.Gohil, et al. ,PhysicsofFluidsB:PlasmaPhysics 3 ,2300(1991) . 49 T.S.HahmandK.H.Burrell, PhysicsofPlasmas 2 ,1648(1995) . 50 R.J.Groebner,K.H.Burrell, andR.P.Seraydarian, Phys.Rev.Lett. 64 ,3015(1990) . 51 E.Mazzucato,S.H.Batha,M.Beer,M.Bell, et al. ,Phys.Rev.Lett. 77 ,3145(1996) . 52 M. E. Koepke, W. E. Amatucci, J. J. Carroll, and T. E. Sheridan, Phys. Rev. Lett. 72 , 3355 (1994) . 53 M.E.Koepke,W.E.Amatucci,J.J.C.III,V.Gavrishchaka, andG.Ganguli, PhysicsofPlasmas 2 ,2523(1995) . 54 W. E. Amatucci, D. N. Walker, G. Ganguli, J. A. Antoniades, et al. , Phys. Rev. Lett. 77 , 1978 (1996) . 55 W. E. Amatucci, D. N. Walker, G. Ganguli, D. Duncan, et al. , J. Geophys. Res. 103 , 11,711 (1998) . 56 V.Gavrishchaka,M.E.Koepke, andG.Ganguli, PhysicsofPlasmas 3 ,3091(1996) . 57 P. N. Guzdar, P. Satyamarayama, J. D. Huba, and S. L. Ossakow, Geophys. Res. Lett. 9 , 547 (1982) . 58 F.Brochard,E.Gravier, andG.Bonhomme, PhysicsofPlasmas 12 ,062104(2005) . 59 D.G.Swanson, Plasma Waves ,2nded.(IOPPublishing,2003). 60 C.Hidalgo,C.Alejaldre,A.Alonso,J.Alonso, et al. ,NuclearFusion 45 ,S266(2005) . 61 W. D. D?haeseleer, W. N. G. Hitchon, J. D. Callen, and J. L. Shohet, Flux Coordinates and Magnetic Field Structure (Spriner-Verlag,1991). 119 Appendices 120 AppendixA GeneralizedCoordinates Abriefoverviewforthedevelopmentofgeneralizedcoordinatesrelevanttothisdissertation, isprovidedfromthefirstchapterofthebook?FluxCoordinatesandMagneticFieldStructure? 61 . In this section, a set a vectors defining the directions of our coordinate system is developed in a generalized manner. These direction vectors will be allowed to be non-orthonormal. As a result, definitionsofcommonvectoroperationsneedtoberedefined. A.1 Basis Vectors Apointinthreedimensionalspacecanbespecifiedasavectorparameterizedbythreeinde- pendentparametersorcoordinates, ui. R(u1;u2;u3) (A.1) Byholdingonecoordinateconstantandallowingtheothertwotovary,surfacesofconstant ui can bemappedout. Varyingonecoordinatewhileholdingtheothertwoconstant,mapsoutcoordinate curves. A coordinate curve along the direction of ui is located at the intersection of coordinate surfacesdefinedby uj and uk. Figure A.1 showsthecoordinatesurfacesandcurvesforthefamiliar cylindrical coordinate system. The direction tangental to a coordinate curve produces covariant basis vectors. The direction normal to a coordinate surface produces contravariant basis vectors. InCartesiancoordinates,thesedirectionsarethesameandthereisnodistinctionbetweencovariant andcontravariantvectors. ThiswillnotbethecasefortheVMEC-coordinatesystem. A.1.1 Covariant Basis The vectors tangential to the coordinate curve (covariant basis vectors) are determined by takingthederivativeofequation A.1 withrespecttoeachcoordinate. @R @ui = ei (A.2) This set of vectors forms the covariant basis set of a coordinate system. It should be noted that unlikethefamiliarcartesiancoordinatesystem,thebasisvectors ei arenotconstrainedtohaveany specialpropertiessuchasbeingorthonormalorevenunit-less. A.1.2 Contravariant Basis The vectors normal to the coordinate surface (contravariant basis vectors) are defined from thecovariantbasissetas ei = 1J (ej ek) (A.3) 121 Figure A.1: Coordinate curves and surfaces for the cylindrical coordinate system ( ; ;z). Red ? = const,Green = const,Blue z = const. Theblacklinesshowthecoordinatecurves. 122 forcyclicpermutationsof i,j and k. J istheJacobianandisdefinedtobe; J = ei ej ek (A.4) Thissetofvectorsformsthecontravariantbasissetofacoordinatesystem. Againlikethecovariant basis,thissetofcoordinatesmightalsonotbeorthonormalorunit-less. Inthemorefamiliarcarte- siancoordinatesystem,thecovariantandcontravariantbasissetsareequaltoeachother. Thusno distinctionismadebetweenthetwo. However,inthegeneralizedcase,thedifferencesbetweenba- sissetsbecomeimportant. Becausethesebasissetsformreciprocalsets,thefollowingrelationship betweenbasissetsholds; ei ej = ji (A.5) A.1.3 Metric Coefficients Avector Dmaybedefinedineitherbasissetas D = { (D e1)e 1 + (D e2)e 2 + (D e3)e 3 = D1e1 + D2e2 + D3e3 (D e1)e1 + (D e2)e2 + (D e3)e3 = D1e1 + D2e2 + D3e3 (A.6) where Di and Di arethecontravariantandcovariantcomponentsrespectivelyofthevector D. To facilitate conversion between covariant and contravariant vectors, metric coefficients are defined as gij = ei ej (A.7a) gij = ei ej (A.7b) forthecovariantandcontravariantcases,respectively. Whenacoordinate ei or ei isorthogonalto ej or ej respectively,theassociatedmetriccoefficientiszero. Wecanconvertthecovariantcom- ponentsofavector D tocontravariantcomponents,andviceversa,usingthemetriccoefficients. Di = gijDj (A.8a) Di = gijDj (A.8b) In this notation there is an implied summation over repeated indices. In the same manner, the conversionbetweenbasisvectorsis ei = gijej (A.9a) ei = gijej (A.9b) for the covariant and contravariant bases vectors respectively. A curvilinear coordinate system vectormaybetransformedbackintoavectorintheoriginalcoordinatesystemby Dxi = GTDui (A.10) where GT G = 0 @ e1 ^x1 e1 ^x2 e1 ^x3 e2 ^x1 e2 ^x2 e2 ^x3 e3 ^x1 e3 ^x2 e3 ^x3 1 A (Covariant) (A.11a) 123 G = 0 @ e1 ^x1 e1 ^x2 e1 ^x3 e2 ^x1 e2 ^x2 e2 ^x3 e3 ^x1 e3 ^x2 e3 ^x3 1 A (Contravariant) (A.11b) is the transposed matrix of the covariant or contravariant basis vectors depending on whether the Dui isdefinedusingthecovariantorcontravariantbasisvectorsrespectively. A.1.4 Vector Operators Fromequations A.5 and A.7 ,therearefourpossiblevariationsonthedotproductoperator. A B = 8 >>< >>: AiBj ij AiBj ji gijAiBj gijAiBj (A.12) Again there is an implied summation over repeated indices. For orthogonal systems, the cross componentsinthelasttwo-formsarezero. Unlikethedotproduct,thereareonlytwonaturalformsofthecrossproduct, A B = 8 >>< >>: J ? k (AiBj AjBi)ek 1 J ? k (AiBj AjBi)ek (A.13) forcyclicpermutationsof i,j and k. Thegradientoperatorisdefinedas r = @ @uiei (A.14) againwithanimpliedsummationoverrepeatedindices. A.1.5 Derivatives of Covariant and Contravariant Vectors TheMHDequationscontaintermoftheform (A r)Awheretheresultingdifferentialop- erator acts on each component of a vector A. Unlike the cartesian coordinate system, covariant and contravariant basis vectors can be functions of their coordinates. When taking the derivative of vector one also must account for changes in the basis vectors as well. For a vector W, the derivativemaybetakenintwoways @W @uk = 8 >>< >>: (@W @uk )j ej (@W @uk ) j ej (A.15) 124 intermsofthecovariantandcontravariantbasisvectors. Thecontravariantcomponentscannow beexpandedintwoways. (@W @uk )j = 8 >< >: @Wi @uk j i + W i @ei @uk e j @Wi @uk g ij + Wi @e i @uk e j (A.16a) (@W @uk ) j = 8> < >: @Wi @uk i j + Wi @ei @uk ej@Wi @uk gij + W i @ei @uk ej (A.16b) 125 AppendixB VMECCoordinates VMECis a three-dimensional MHD equilibrium solver that uses a steepest-descent moment method 36 assuming nested magnetic surfaces. For a given set of input parameters (coil currents, pressure profile, current profile, etc.)VMECwill construct the resulting plasma equilibrium. This equilibriumissolvedinafluxcoordinatesystemparameterizedbycoordinates s, u,and v. The s coordinateisanormalizedcoordinatelabelingamagneticsurface. Thecoordinate sisnormalized insuchawaythat s = 0 isthemagneticaxisand s = 1 istheLCFS.Thecoordinate u represents a poloidal like angle. The coordinate v represents the toroidal angle and is the same as ? in the cylindricalcoordinatesystem. The output ofVMECis a file containing various parameters that can be used to describe the magneticfieldstructureatanypointinsidetheLCFSoftheCTHdevice. Apartiallistofimportant availableparametersareprovidedinTable B.1 . Eachvalueisrepresentedasalinearcombination ofmodesof uand v. Avalue Amaybecalculatedby 36 ; A(s;u;v) = ? mn Amnc (s)cos (mu nv) + ? mn Amns (s)sin (mu nv) (B.1) Thecoefficients Amnc (s)and Amns (s)aredefinedonagridofdiscrete svalues. Valuesaredefined oneitherafullgridstartingat s = 0goingto s = 1atregularintervalsorahalfgridon spositions betweenthefullgridpoints. AnyvalueinTable B.1 maybeobtainedatany uand v positionfora given svalue. However,becauseofthediscretenatureofvaluesinthe scoordinate,aninterpolation mustbeemployedforvaluesof sthatfallbetweengridpoints. 126 ValueNameDescription Gridspacing Rc (s)rmncRadiuscoscomponents Fullgrid Rs (s)rmnsRadiussincomponents y Fullgrid Zc (s)zmncHeightabovethemid-planecoscomponents y Fullgrid Zs (s)zmnsHeightabovethemid-planesincomponents Fullgrid Buc (s)bsupumncContravariantmagneticfieldcomponent Halfgrid in eu coscomponents Bus (s)bsupumnsContravariantmagneticfieldcomponent Halfgrid in eu sincomponents y Bvc (s)bsupvmncContravariantmagneticfieldcomponent Halfgrid in ev coscomponents Bvs (s)bsupvmnsContravariantmagneticfieldcomponent Halfgrid in ev sincomponents y Bsc (s)bsubsmncCovariantmagneticfieldcomponent Halfgrid in es coscomponents y Bss (s)bsubsmnsCovariantmagneticfieldcomponent Halfgrid in es sincomponents Buc (s)bsubumncCovariantmagneticfieldcomponent Halfgrid in eu coscomponents Bus (s)bsubumnsCovariantmagneticfieldcomponent Halfgrid in eu sincomponents y Bvc (s)bsubvmncCovariantmagneticfieldcomponent Halfgrid in ev coscomponents Bvs (s)bsubvmnsCovariantmagneticfieldcomponent Halfgrid in ev sincomponents y TableB.1: ListofVMECOutputParameters. Parameterslabelledas?asymmetric?are present only in output files allowing for up-down asymmetry of the magnetic flux surfaces. y AsymmetricOnly 127 B.1 Basis Vectors Apositionaroundthetorusisdefinedincylindricalcoordinates,parameterizedbyfluxcoor- dinates s,uand v,as R(s;u;v) = R(s;u;v) ^r + Z (s;u;v) ^z (B.2) where R(s;u;v)and Z (s;u;v)havetheformofequation B.1 . Whenapplying A.2 ,thederivatives withrespectto uand v canbetakenanalytically. However,sincethecoefficientsareonlydefined ondiscrete spositions,aninterpolationorfinitedifferencemustbetakenforthesefunctions. Lastly since v isequivalentto ?, @^r @v = ^? (B.3) ThecovariantbasisvectorsforVMECcoordinatesystemare; es = [? mn @ @sRmnc (s)cos (mu nv) + ? mn @ @sRmns (s)sin (mu nv) ] ^r + [? mn @ @sZmnc (s)cos (mu nv) + ? mn @ @sZmns (s)sin (mu nv) ] ^z (B.4a) eu = [ ? mn Rmnc (s)msin (mu nv) + ? mn Rmns (s)mcos (mu nv) ] ^r [? mn Zmnc (s)msin (mu nv) ? mn Zmns (s)mcos (mu nv) ] ^z (B.4b) ev = [? mn Rmnc (s)nsin (mu nv) ? mn Rmns (s)ncos (mu nv) ] ^r + [? mn Rmnc (s)cos (mu nv) + ? mn Rmns (s)sin (mu nv) ] ^? + [? mn Zmnc (s)nsin (mu nv) ? mn Zmns (s)ncos (mu nv) ] ^z (B.4c) ThecontravariantbasisvectorsfortheVMECcoordinatesystemareobtainedthroughequation A.3 . Magnetic fields, in this coordinate system, can be defined as either using covariant or con- travariantbasisvectors. B = Bueu + Bvev (B.5a) B = Bses + Bueu + Bvev (B.5b) When defined using equation B.5a , the field lines have no es component. Bu represents a quasi poloidal component and Bv represents a quasi toroidal component. The magnetic field lines run alongsurfacesofconstant s. Thus,thesurfacesdefinedbyaconstant scoordinaterepresentanested fluxsurfaceinVMEC.However,itshouldbenotedthatthe es basisvectorneednotbeorthogonalto B andneednotrepresenta?radial?direction. Fromthisdefinitionwecanmeasureanimportant 128 fluxsurfaceconstantquantityinstellarators. = B u Bv (B.6) representsthemeasureofthetwistofthefieldline. 129 AppendixC ComputerCodesUsed 130 131 C.1 Shooting Code Kelvin-Helmholtz Shooting Code CTH Introduction Attempting to solve for a dispersion relation using CTH relivant parameters. The equation we are tying to solve is equation 6 in Guzdar, P. N., P. Satyanarayana, J. D. Huba, and S. L. Ossakow (1982), Influence of velocity shear on the Rayleigh-Taylor instability, Geophys. Res. Lett., 9(5), 547?550. To simplify this we start by assuming g=0 and ignore, ion-neutral colsitions. Initialization ? Constants In[1]:=b0=0.5; l=0.03; n=10; me=9.1094*^-31; ? Fuctions ? Density and Potential profiles. In[5]:=f0=17*Tanh@??lD+90&; Plot@f0@xD,8x,-n*l,n*l<,PlotRange?AllD Out[6]= -0.3 -0.2 -0.1 0.1 0.2 0.3 80 85 90 95 100 105 ? Drift velocity In[7]:=ve0=f0'@?D?b0&; In[8]:=k=?2^2-H?2*ve0''@?3DL?H?1-?2*ve0@?3DL&; 132 ? Dopler Shifted Frequency In[9]:=w1=?1-?2*ve0@?3D&; Shooting Code Set up the functions for the shooting code. 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