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Experimental Observation of Fluctuation-Driven Mean Magnetic Fields in the Madison Dynamo Experiment

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Experimental Observation of Fluctuation-Driven Mean Magnetic Fields in the Madison Dynamo Experiment EXPERIMENTAL OBSERVATION OF FLUCTUATION-DRIVEN MEAN MAGNETIC FIELDS IN THE MADISON DYNAMO EXPERIMENT by Erik J. Spence A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) at the UNIVERSITY OF WISCONSIN–MADISON 2006 c Copyright by Erik J. Spence 2006 All Rights Reserved i To my beautiful wife, Susanne. ii ACKNOWLEDGMENTS Many are the people that deserve thanks for the help they have given me over the years of my graduate career. Enumerating them all is a difficult endeavour and it is certain that not all will be thanked to the level they deserve. For this I apologize. Since this is inevitable it is my hope that this section will be the most incomplete part of this work. First and foremost, I express my thanks and gratitude to my advisor, Professor Cary Forest, for hiring me as a research assistant and bearing with me all these years. Obviously, this work would not exist without him. (since neither would the experiment!) This glaring point aside, his influence on my growth and maturation as a scientist has been noteworthy. Though I have failed him numerable times, he has been patient with me and encouraged me to reach the potential of which I am capable. His greatest lesson, “if it doesn’t look right it probably isn’t,” is a gift that will be put to great use in my future career as a scientist. My partner in research, Mark Nornberg, is a great man; I have nothing but praise for him. His kindness, patience, forbearance, piety, prudence, humour and dedication are inspirational. Many has been the time that I have been unjust to him, and he has repaid me with charity. He is truly a better laboratory partner than I deserve and I am a better man for him sharing his life with me. Mark’s contributions to this work are not limited to his superlative personality. His intelligence is significant, and his ability to learn exceeds my own. He has consistently helped me find the weaknesses in my work and to better my understanding of the science with which I have struggled. My eternal thanks to him for walking this journey with me. Another to whom I am very grateful is Roch Kendrick. His tireless dedication to the project has led to the construction of the leading experiment of its kind in the world, without which, of course, this thesis would never have happened. He is uncompromising on quality and is one of iii most pragmatic and adaptable people I have known. My thanks to him for all he has taught me about, but certainly not limited to, machining, plumbing, electrical wiring, metallurgy, chemistry, generators, motors, heating, cooling, insulating, and how to prepare a “killer” pork shoulder. While the previous three contributed by far the most academically to my effort to reach this point, many others have also had a hand in getting this thesis to completion. Craig Jacobson has worked extensively on the water model of the experiment, especially since it was relocated to the Tantalus facility. Adam Bayliss has helped me to develop my understanding of the physics of the experiment, both through his simulations and through our conversations. Carlos Parada, our latest team member, has been a great help in taking much of the data presented herein. Many others have helped with construction of the water and sodium versions of the experiment: Hal Canary, Michael Fix, Brian Grierson. Rob O’Connell, the first postdoctoral researcher for the experiment, did much of the development that brought the water into functioning form. Helping him in that effort was Jonathan Goldwin, who worked on the analysis of the 30 cm version of the water model, helping to develop the final impeller design. My thanks to all of them for their contribution to this work. But not all contributions to this work have been of an academic stripe. I have had the privilege of enjoying the company of many a fellow graduate student and friend on this journey, and I raise a glass to all of them in thanks: Stephie, Dan, Jim, Mike, Dave, Jodi, Steve, Bob, Olivia, Jenny, Hilarie, Yoyi, and many others. In particular I raise a cold one to my good friend Christian Ast, whose company I had the honour of sharing for many years as his roommate. His companionship and counsel continue to be a comfort to me, and I am sure that I would not have survived this adventure without him. Complementing the support of my friends has been the never-wavering support of my family. My nutty sisters, God bless them, continue to be a refreshing source of silliness and humour, even after all these years together. The support of my parents has been unquestionable, and their love never-ending. I will always be indebted to them. Since long before I took her as my wife, Susanne has been a continuous source of support during the many trials, failures and eventual successes of this project. I could never possibly earn iv the love that she has shown me, but offer it she does, each and every day. The choice of dedication of this work is an easy one to make. One further expression of thanks is necessary. Throughout my graduate career the presence of Jesus the Christ has been a source of strength and perseverance for me like none other. The Holy Spirit has been generous with guidance and wisdom and peace, and because of this the thesis that you are reading exists. Thus, if there is any good in this work then let it be for the glory of God. v ËÓÐi DeÓ GÐÓÖia vi TABLE OF CONTENTS Page LIST OF TABLES ....................................... ix LIST OF FIGURES ...................................... x NOMENCLATURE ......................................xiii ABSTRACT ..........................................xiv 1 Introduction ........................................ 1 1.1 MHDandDynamoExperiments . .. 4 1.2 MotivationfortheMadisonDynamoExperiment . ......... 8 1.3 TheRoleofTurbulence. .. .. .. .. .. .. .. .. ... 11 1.4 ThesisOutline................................... .. 13 2 Experimental Apparatus ................................. 15 2.1 WaterModel ..................................... 16 2.1.1 DescriptionoftheWaterApparatus . ..... 16 2.1.2 LaserDopplerVelocimetry. ... 21 2.1.3 WaterDataRuns ............................... 25 2.2 SodiumExperiment................................ .. 26 2.2.1 DescriptionoftheSodiumApparatus . ..... 26 2.2.2 HeatingandCoolingtheExperiment. ..... 30 2.2.3 LaboratorySodiumSafety . .. 34 2.2.4 FillingtheSphere.............................. 37 2.2.5 MagneticFieldsandDataAcquisition . ...... 38 2.2.6 SodiumDataRuns .............................. 43 3 Determination of the Mean Velocity Field ........................ 45 3.1 LDVMeasurements................................. 46 3.2 FittingtheVelocityField . ...... 47 3.2.1 SphericalHarmonicExpansion. .... 49 3.2.2 VelocityFieldFits ............................. 50 vii Page 4 Predicting the Induced Field ............................... 57 4.1 MagneticInductionEquation . ...... 57 4.1.1 InteractionTerms.............................. 59 4.1.2 GauntandElsasserIntegrals . .... 60 4.1.3 MagneticBoundaryConditions . ... 61 4.2 CodingtheProblem................................ .. 62 4.2.1 RadialProfilesandOperators . ... 62 4.2.2 CalculatingtheInducedField . .... 63 4.2.3 TestingtheCode ............................... 63 4.3 DeterminingtheAppliedField . ...... 65 4.4 PredictedMagneticFields . ..... 67 5 Measurements of the Induced Magnetic Field ..................... 71 5.1 MeasuringInducedMagneticFields . ....... 72 5.1.1 ExternalMagneticFields. ... 73 5.1.2 InternalMagneticFields . ... 76 5.2 FittingInternalMagneticFields . ........ 78 5.2.1 FittingInternalPoloidalFields . ....... 80 5.2.2 FittingInternalToroidalFields . ....... 81 6 Evidence for Fluctuation-Driven Currents ....................... 87 6.1 PoloidalInducedFields. ..... 88 6.1.1 ExternalPoloidalFields . ... 88 6.1.2 InternalPoloidalFields. .... 91 6.1.3 InducedDipoleMoment . 91 6.2 ToroidalInducedFields. ..... 95 6.3 Fluctuation-InducedFields . ....... 96 7 Discussion and Summary .................................102 7.1 ComparisonwithSimulation . .102 7.2 FormsoftheTurbulentEMF . 103 7.3 ImplicationsfortheDynamo . .105 7.4 Summary .......................................106 viii Page LIST OF REFERENCES ...................................108 APPENDICES AppendixA: HallProbeInputVoltageDrift. ........114 AppendixB: FINDFrequencyShifting . .118 AppendixC: MagneticFieldBoundaryConditions . .........121 AppendixD: Advectionand DiffusionMatrixOperators . ...........123 Appendix E: Axisymmetric Velocity and Magnetic Fields Cannot Induce Dipole Mo- ments....................................131 Appendix F: Calculation of the Magnetic Field due to the EMF . ...........135 ix LIST OF TABLES Table Page 1.1 ExperimentalSpecifications . ........ 11 2.1 Parameter values used to calculate the radial LDV measurementposition . 24 3.1 ParametervaluesusedwiththeFINDsoftware . ........... 45 5.1 Inducedexternalpoloidalharmonicstatistics . ............... 77 6.1 Energy in the measured and predicted induced magnetic field.............. 88 A.1 Circuitboardelementvalues . ........114 A.2 Probe circuit board voltages as a function of magnetic fieldstrength . .116 B.1 LDVlaserlightcharacteristics . ..........118 B.2 Minimum shift frequency values needed by the FIND software.............119 x LIST OF FIGURES Figure Page 1.1 SchematicoftheRigadynamoexperiment . ......... 6 1.2 SchematicoftheKarlsruhedynamoexperiment . ............ 7 1.3 Dudley and James’ t2s2 flow............................... 10 1.4 Growth rate versus Rm forDJflow ........................... 11 2.1 Kinematic
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