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Intrinsic Plasma Rotation and Reynolds Stress at the Plasma

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Intrinsic Plasma Rotation and Reynolds Stress at the Plasma Measurements of Reynolds Stress and its Contribution to the Momentum Balance in the HSX Stellarator by Robert S. Wilcox A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) at the University of Wisconsin–Madison 2014 Date of final oral examination: 12/2/2014 The dissertation is approved by the following members of the Final Oral Committee: David T. Anderson, Professor, Electrical Engineering Amy E. Wendt, Professor, Electrical Engineering Chris C. Hegna, Professor, Engineering Physics Paul W. Terry, Professor, Physics Joseph N. Talmadge, Associate Scientist, Electrical Engineering c Copyright by Robert S. Wilcox 2014 All Rights Reserved i ABSTRACT In a magnetic configuration that has been sufficiently optimized for quasi-symmetry, the neoclas- sical transport and viscosity can be small enough that other terms can compete in the momentum balance to determine the plasma rotation and radial electric field. The Reynolds stress generated by plasma turbulence is identified as the most likely candidate for non-neoclassical flow drive in the HSX stellarator. Using multi-tipped Langmuir probes in the edge of HSX in the quasi-helically symmetric (QHS) configuration, the radial electric field and parallel flows are found to deviate from the values calculated by the neoclassical transport code PENTA using the ambipolarity con- straint in the absence of externally injected momentum. The local Reynolds stress in the parallel and perpendicular directions on a surface is also measured using the fluctuating components of floating potential and ion saturation current measurements. Although plasma turbulence enters the momentum balance as the flux surface averaged radial gradient of the Reynolds stress, the locally measured quantity implies a significant contribution to the momentum balance. If extrapolated to a flux surface average, this locally measured Reynolds stress gradient is calculated to result in a flow drive many times larger than the observed flows. Probe measurements made at two locations on the device in regions with different magnetic geometry indicate very different, but consistently large Reynolds stress drive terms. The large variation of the local Reynolds stress on a flux sur- face suggests that a small number of measurement locations is insufficient to properly sample the flux surface averaged quantity. Contrary to expectations, measurements in configurations with the quasi-symmetry intentionally degraded deviate more from the neoclassically calculated velocity profiles than those in the QHS configuration. Measured density fluctuations and the Reynolds stress are reduced in these cases, indicating that additional terms may be important in the momen- tum balance. ii For my parents, Betty and Steven Wilcox iii ACKNOWLEDGMENTS This work could not have been done without the help and support of many people, both within and beyond the HSX group. First and foremost, I would like to thank my advisor, David Ander- son, for the guidance and support throughout the course of my studies, starting many years ago when he gave an inspiring lecture in my undergraduate electromagnetics course about magnetic confinement and fusion energy. His unique combination of scientific aptitude, real-world knowl- edge and enthusiasm are invaluable assets to his students and the university, as well as to the world fusion community. Discussions with Chris Hegna and Paul Terry were vitally important to gaining a deeper understanding of the transport mechanisms at play in this work. In addition to Professors Anderson, Terry and Hegna, I would like to thank Professor Amy Wendt and Joe Talmadge for serving on my thesis committee. Joe Talmadge provided scientific guidance through the course of this work, and most often asks the probing questions that need to be investigated when linking experiments to theory. Simon Anderson painstakingly maintained many of the systems on HSX and aided greatly in day-to-day operations and on-the-spot data analysis in order to make adjustments to experiments during run days. Konstantin Likin, Chuanbao Deng, Santhosh Kumar, Kan Zhai and Paul Probert all provided crucially important expertise into the diagnostic systems of HSX that they helped design and build, and they kept everything up and running as experiments were performed. Mike Frankowski passed away during my time at HSX, but I remember him fondly for the conversations we had and the help that he provided in the design and construction of the Langmuir probe system. He is dearly missed. During the course of my research, I traveled to Madrid and worked alongside Carlos Hidalgo, Boudewijn van Milligen and Arturo Alonso, and published a paper based in part on this collab- oration. I learned much from this exchange and I am very appreciative of their guidance and hospitality. iv My time at HSX was made much more enjoyable by the company of my fellow graduate students. Jeremy Lore passed his knowledge of Thomson scattering to me before leaving, and his research and career advice were always helpful. Alexis Briesemeister’s CHERS measurements were an important piece of evidence in support of my work, and she always provided much-needed enthusiasm around the office. Chris Clark has an incredibly wide range of knowledge, and observ- ing his thorough, systematic approach to problems has helped me immensely as a scientist. John Schmitt always came to work with an enviable combination of an excellent work ethic and positive attitude. I benefited greatly from Enrico Chlechowitz’s unique German perspective on whatever topic was at hand, whether it was physics, politics, or food and drink. My time at HSX over- lapped significantly with Gavin Weir and Carson Cook. We shared many good times and fruitful discussions at conferences and around the office. Laurie Stephey perfectly embodies the hopeful optimism that brings most of us into the field of fusion, and it is encouraging to know that there are people like her in the world. There have been many nights after work that I enjoyed drinks and “talking shop” at the Library and elsewhere with Adrian Akerson, Jason Smoniewski, Carlos Ruiz, and Tom Dobbins, as well. I would also like to thank my parents, Steve and Betty, who have been very supportive of my pursuits, as well as my sisters, Annie and Julie. My family has always provided a solid grounding and a support structure that I found more necessary than ever as I worked through graduate school. My girlfriend Peggy has been patient with my long hours in the lab, and she has made the process of finishing this dissertation much more tolerable than it might have otherwise been. Finally, I would like to acknowledge the financial support of the United States Department of Energy, who provided the funding over the course of this work. v TABLE OF CONTENTS Page ABSTRACT .......................................... i ACKNOWLEDGMENTS ................................... iii LIST OF FIGURES ......................................viii 1 Introduction ........................................ 1 1.1 Need for fusion energy . 1 1.2 Magnetic confinement of plasma . 5 1.2.1 Tokamaks . 6 1.2.2 Conventional stellarators . 8 1.2.3 Optimized stellarators . 11 1.3 HSX . 14 1.3.1 Quasi-helical symmetry and predicted effects . 15 2 Momentum balance and intrinsic flows ......................... 23 2.1 Rotation and Er determination in the absence of external momentum sources . 26 2.1.1 Er determination in unoptimized stellarators . 26 2.2 Neoclassical transport and radial electric field calculations . 28 2.2.1 VMEC . 29 2.2.2 DKES . 30 2.2.3 PENTA . 36 2.3 Momentum balance . 38 2.3.1 Reynolds stress . 40 2.3.2 Maxwell/kinetic stress . 43 2.3.3 Ion orbit loss . 45 2.3.4 Charge exchange drag force . 46 2.3.5 ECRH-driven electron flux . 48 2.3.6 Turbulent parallel flow acceleration . 50 2.4 Rotation in configurations approaching quasi-symmetry . 50 2.4.1 Magnetic configurations in HSX . 53 vi Page 2.4.2 Effective ripple . 55 2.4.3 Observations of intrinsic rotation in tokamaks . 58 2.5 Predictions of limits of non-neoclassical Er determination in HSX . 60 2.5.1 Helander and Simakov estimates . 60 2.5.2 Calculations of closeness to quasi-symmetry by Calvo et. al. 63 2.5.3 Comparison of non-ambipolar neoclassical particle flux to total experi- mental particle flux . 67 2.6 Previous comparisons of measured flows in HSX to neoclassical calculations . 69 2.6.1 Flow damping . 69 2.6.2 CHERS intrinsic flow measurements . 70 2.6.3 Probe floating potential . 72 2.7 Measurements of Reynolds stress in other devices . 73 3 Experimental setup .................................... 85 3.1 Langmuir probes . 85 3.1.1 Floating potential . 86 3.1.2 Ion saturation current . 91 3.1.3 Mach probes . 92 3.2 Reynolds stress probe design . 94 3.3 Probe positioning and magnetic geometry . 97 3.3.1 Probe alignment to the magnetic field . 99 3.3.2 Radial positioning of probes . 100 3.3.3 Magnetic geometry at probe locations . 103 3.4 Probe perturbation tests . 107 4 Measurements of Er and Vjj and comparisons with neoclassical modeling . 113 4.1 Neoclassical ambipolarity calculations . 113 4.1.1 Inputs to PENTA . 114 4.1.2 Calculated neoclassical particle fluxes and ambipolar solution for Er . 116 4.1.3 Sensitivity studies . 118 4.2 Raw Langmuir probe data and synthesis . 120 4.3 Radial electric field measurements . 123 4.3.1 Measurement of Te from fluctuating signals . 124 4.3.2 Er profile and comparison to neoclassical calculations . 128 4.4 Parallel flow measurements . 131 4.4.1 Pfirsch-Schluter¨ component of the parallel ion flow . 132 4.4.2 Total measured Vjj profiles and comparison to neoclassical calculations in QHS . 134 vii Appendix Page 4.5 Er and Vjj measurements in other configurations . 136 4.5.1 Er and Vjj measurements in the Mirror configuration . 137 4.5.2 Er and Vjj measurements in the Flip-1-4 configuration .
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