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THE ROLE OF MHD TURBULENCE IN MAGNETIC SELF-EXCITATION: A STUDY OF THE MADISON DYNAMO EXPERIMENT by Mark D. Nornberg 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 Mark D. Nornberg 2006 All Rights Reserved i For my parents who supported me throughout college and for my wife who supported me throughout graduate school. The rest of my life I dedicate to my daughter Margaret. ii ACKNOWLEDGMENTS I would like to thank my adviser Cary Forest for his guidance and support in the completion of this dissertation. His high expectations and persistence helped drive the work presented in this thesis. I am indebted to him for the many opportunities he provided me to connect with the world-wide dynamo community. I would also like to thank Roch Kendrick for leading the design, construction, and operation of the experiment. He taught me how to do science using nothing but duct tape, Sharpies, and Scotch-Brite. He also raised my appreciation for the artistry of engineer- ing. My thanks also go to the many undergraduate students who assisted in the construction of the experiment, especially Craig Jacobson who performed graduate-level work. My research partner, Erik Spence, deserves particular thanks for his tireless efforts in modeling the experiment. His persnickety emendations were especially appreciated as we entered the publi- cation stage of the experiment. The conversations during our morning commute to the lab will be sorely missed. I never imagined forging such a strong friendship with a colleague, and I hope our families remain close despite great distance. I would also like to thank Jim, Mike, and Julie for both moral support and epicurean sustenance. I would never have endured the struggle of graduate school without the steadfast support of my family. I will always be grateful for the education I received through the sacrifices of my parents. My beautiful wife Julianne has been my comfort and encouragement throughout my graduate career. She checked my discouragement with unwavering optimism and provided the hope that is needed in accomplishing such an endeavor. iii It seems an almost incredible thing to me that an invention of the human intellect and the structure of the universe coincide. — Benedict XVI iv TABLE OF CONTENTS Page List of tables .......................................... vi List of figures .......................................... vii Nomenclature ......................................... x Abstract ............................................. xi 1 Introduction ........................................ 1 1.1 A brief history of the dynamo problem ........................ 1 1.2 Dynamo experiments ................................. 3 1.3 Motivation for the experiment ............................ 6 1.4 Outline of the thesis .................................. 8 2 The kinematic dynamo problem ............................. 10 2.1 The magnetic induction equation ........................... 10 2.2 The Bullard and Gellman formalism ......................... 12 2.3 The kinematic eigenvalue problem .......................... 15 2.4 Simple roll flow dynamos ............................... 17 2.5 The stretch-twist-fold mechanism .......................... 21 2.6 Comparison with simulation ............................. 24 2.7 Anticipated effects of turbulence ........................... 27 2.7.1 Intermittent excitations ............................ 27 2.7.2 Mean-field dynamo .............................. 27 2.7.3 Small-scale dynamo ............................. 30 3 Description of the sodium experiment .......................... 32 3.1 Industrial automation ................................. 36 3.2 Holding tank ..................................... 36 3.3 Experimental vessel .................................. 38 3.4 Expansion tank .................................... 41 3.5 Melt station ...................................... 41 3.6 Pneumatic transfer system .............................. 42 v Page 3.7 Magnetic sensors and data acquisition ........................ 43 3.8 External field coils .................................. 46 3.9 Safety equipment ................................... 47 3.9.1 Protective clothing and gear ......................... 47 3.9.2 Scrubber system ............................... 47 3.9.3 Fume hood for small sodium fires and material tests ............ 48 3.9.4 Sodium cleaning station ........................... 48 4 Hydrodynamic experiments to model the flow ..................... 50 4.1 Description of the water experiment ......................... 50 4.2 Laser Doppler velocimetry .............................. 55 4.3 Model of the mean flow ................................ 56 4.4 Velocity fluctuations ................................. 62 4.5 Kinematic growth rate of the magnetic field ..................... 62 4.6 Kolmogorov turbulence ................................ 66 4.7 Measurements of the velocity spectrum ....................... 70 4.8 Turbulent conductivity ................................ 75 5 Comparison of measurements of the induced field with predictions ......... 80 5.1 The predicted mean induced magnetic field ..................... 80 5.2 Measurements of the magnetic field ......................... 81 5.3 Reconstruction of the mean magnetic field ...................... 85 5.4 Magnetic field amplification and feedback ...................... 87 5.5 The magnetic spectrum due to turbulence ...................... 94 5.6 Measurements of the magnetic spectrum ....................... 98 5.7 Influence of the back reaction on turbulence ..................... 103 6 Observation of an intermittently excited magnetic field ................ 104 7 Summary and discussion ................................. 112 References ........................................... 115 APPENDICES Appendix A: Calculating the Mode Energy ....................... 128 vi LIST OF TABLES Table Page 2.1 Optimized flow parameters ................................ 20 3.1 Experiment parameters .................................. 35 4.1 Turbulence characteristics ................................ 68 6.1 Results from conditional averaging ........................... 109 vii LIST OF FIGURES Figure Page 1.1 Schematic of the sodium experiment ........................... 7 2.1 Dudley and James t2s2 flow ............................... 18 2.2 Growth rate plot of the Dudley and James t2s2 flow .................. 19 2.3 Stretch-twist-fold mechanism .............................. 23 2.4 Structure of the excited magnetic field .......................... 25 2.5 Growth rates for different flows ............................. 26 3.1 Photo of the Madison Dynamo Experiment ....................... 33 3.2 Sodium conductivity ................................... 34 3.3 Automation software display ............................... 37 3.4 Motor power ....................................... 40 3.5 Photo of magnetic sensors ................................ 44 4.1 Photo of the water experiment .............................. 51 4.2 Photo of the impellers .................................. 52 4.3 Schematic of the water experiment ............................ 57 4.4 LDV time series measurements ............................. 59 4.5 Measurements of the mean flow ............................. 60 4.6 Reconstruction of the mean flow ............................. 61 viii Figure Page 4.7 PDF of velocity fluctuations ............................... 63 4.8 Contour plot of velocity fluctuation levels ........................ 64 4.9 Eigenmode growth rate .................................. 65 4.10 Particle arrival time .................................... 71 4.11 Sample and hold interpolation .............................. 72 4.12 Velocity power spectra from LDV ............................ 74 4.13 Velocity correlation functions .............................. 76 4.14 Turbulent conductivity .................................. 78 4.15 Spatial distribution of turbulent conductivity ...................... 79 5.1 Time series measurements of the magnetic field ..................... 83 5.2 Internal magnetic field measurements .......................... 84 5.3 Spherical harmonic modes of the field induced by the flow ............... 88 5.4 Reconstruction of the magnetic field induced by the flow ................ 89 5.5 Predicted gain and growth rate of induced transverse dipole field ............ 92 5.6 Gain and orientation of induced transverse dipole field ................. 93 5.7 Magnetic power spectrum from simulation ....................... 96 5.8 Magnetic power spectra measurements ......................... 99 5.9 Magnetic spectrum varying applied field strength .................... 101 5.10 Dependence of dissipation scale on Rm ......................... 102 6.1 PDF of magnetic field fluctuations ............................ 106 6.2 Surface magnetic field during burst ........................... 107 6.3 Time series of transverse dipole field .......................... 108 ix Appendix Figure Page 6.4 Conditionally averaged magnetic bursts ......................... 108 6.5 Energy PDFs of transverse dipole field .......................... 110 x Nomenclature MHD Magnetohydrodynamics LDV Laser Doppler Velocimetry VKS Von Karm´ an´ Sodium Experiment PDF Probability Distribution Function PSD Power Spectral Density ACF Autocorrelation Function xi Abstract Determining the onset conditions for magnetic field growth in magnetohydrodynamics is fun- damental to understanding how astrophysical dynamos such as the
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