STELLARATOR BETA LIMITS WITH EXTENDED MHD MODELING USING NIMROD by Torrin A. Bechtel A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) at the UNIVERSITY OF WISCONSIN–MADISON 2021 Date of final oral examination: 4/14/21 The dissertation is approved by the following members of the Final Oral Committee: Chris C. Hegna, Professor, Engineering Physics Carl R. Sovinec, Professor, Engineering Physics David T. Anderson, Professor, Electrical Engineering Paul W. Terry, Professor, Physics © Copyright by Torrin A. Bechtel 2021 All Rights Reserved i abstract The nonlinear, extended MHD code NIMROD is employed to simulate self-consistent stellarator behavior at high beta. Finite anisotropic thermal conduction allows for sustained pressure gradients within regions of stochastic magnetic field. The con- figuration under investigation is an `=2, M=10 torsatron with vacuum rotational transform near unity. Finite-beta plasmas are generated from vacuum fields using a volumetric heating source and temperature dependent resistivity. With realis- tic parameters the configuration is unstable to interchange, which acts to limit the achievable beta. Simulations performed in a single field period domain do not ex- hibit a complete crash from the instability, but otherwise closely match theorized linear and nonlinear interchange behavior. In more dissipative regimes where in- stability is suppressed, steady-state solutions are obtained. A conventional equi- librium beta limit is observed due to pressure induced stochastic magnetic field formation. The parametric dependence of the equilibrium limit is examined in detail. Equilibrium results are compared with several reduced models for effec- tive collisional transport across stochastic magnetic fields and with the HINT code. Collisionality independent models suggest realistic parallel thermal conduction is below the Braginskii prediction, but require further development to allow appli- cation to stochastic fields. ii acknowledgments First and foremost, I would like to thank my advisors, Prof. Chris Hegna and Prof. Carl Sovinec, for their support and guidance during my PhD. They have always made themselves available when I needed assistance. It was an honor to learn from such brilliant physicists. I would also like to thank Dr. Jake King and the rest of the NIMROD team for all of their support behind the scenes. The perpetual maintenance and improvements to the code base were essential to my scientific progress. I am grateful to Dr. Mark Schlutt, Dr. Jonathan Hebert, and Dr. Nick Roberds who laid the foundation for this work and were instrumental in helping me get started. It was a pleasure be- ing able to learn the basics of stellarator design from Prof. Jim Hanson, Dr. Greg Hartwell, and the CTH team to whom I am highly appreciative. Additionally, I would like to thank all of the people who have offered useful insight and perspec- tives including Prof. Jim Callen, Prof. Eric Held, Dr. Jeong-Young Ji, my previous ERB officemates, former roommates, and graduate colleagues. Without the support of my friends and family, this accomplishment would not have been possible. I owe a special thanks to my mother, who has been preparing me for this even before I could walk, and to my father, who has aspired to keep an open mind. I am very thankful to the rest of my family who have always been there for me as well. In particular, I would like to thank Erin Middlemas for always believing in me. You have inspired me to be a better person in every way. Finally, I would like to thank my dear friend Dr. X for having my back through many highs and lows. My life wouldn’t be the same without you. This work could not have been performed with the immense computational resources provided by the high performance computing centers at the University of Wisconsin (CHTC) and NERSC (a DOE Office of Science user facility). I would like to thank all of the staff who keep these systems operating and accessible day and night. Both of the centers are funded through a broad tax base which is made possible by the public support of science. Thank you! iii contents Abstract i Contents iii 1 Introduction 1 I Background 3 2 High Beta Stability 4 2.1 Ideal MHD Stability ............................ 5 2.2 Resistive Interchange Theory ........................ 10 2.3 Simulated Stellarator Instabilities ...................... 13 2.4 High Beta Stellarator Experiments ..................... 19 3 Stochastic Field Transport 25 3.1 Ideal Equilibrium Limit ........................... 26 3.2 Non-Ideal Extensions ............................ 27 3.3 Realistic Equilibrium Beta Limit ...................... 28 3.4 Parallel Thermal Transport Models ..................... 39 3.5 Effective Transport In Stochastic Fields ................... 41 3.6 Stochastic Field Analysis in Stellarators . 48 4 NIMROD Modeling 52 4.1 Available Physics .............................. 52 4.2 Numerical Methods ............................. 54 4.3 Stellarator Simulations ........................... 56 4.4 Previous Stellarator Applications ...................... 58 4.5 Ongoing Development ........................... 63 iv II Setup 65 5 Vacuum Configuration 66 5.1 Improvements to CTH ........................... 66 5.2 New Configuration Design ......................... 68 6 Simulation Methods 76 6.1 Model .................................... 76 6.2 Initial Condition and Sources ........................ 80 6.3 Vacuum Initialization ............................ 82 6.4 Plasma Density Redistribution ....................... 89 6.5 Resolution and Convergence ........................ 91 6.6 Low Resolution Simulations ........................ 95 7 Heat Confinement Analysis 103 7.1 Flux Surface Identification . 103 7.2 Plasma Beta Calculation . 108 7.3 Effective Diffusivity in Simulation . 110 7.4 Magnetic Fieldline Diffusion . 111 7.5 Kolmogorov Length Scale Computation . 114 III Results 119 8 Interchange Instability 120 8.1 Simulation Characteristics . 120 8.2 Theory Based Analysis . 134 9 Equilibrium Transport 143 9.1 Beta Limit Dependencies . 144 9.2 Fieldline Proxies ............................... 153 9.3 Low Collisionality Extensions . 173 v 10 Comparison With HINT 182 11 Conclusion 187 11.1 Summary of Results ............................. 187 11.2 Outstanding Questions . 188 11.3 Future Directions .............................. 190 IV Appendices 192 A Rotational Transform Calculations 193 B Vacuum Field Prescription Comparison 197 B.1 Comparison of Different Initializations . 197 B.2 New Coilset For External Comparison . 202 References 208 1 1 introduction A confined plasma with both high temperature and density is essential to self sus- tained nuclear fusion; also known as burning fusion. While sustained burning fusion is currently only possible within massive solar objects, its creation within laboratories could be used for energy production, neutron breeding and improved understanding of astrophysical systems. The biggest challenge to creating fusion in a lab is the confinement of a plasma which is sufficiently hot, but has a man- ageable size. Many different confinement schemes have been proposed, each with their own benefits and downsides. With the primary goal of long term, steady state operation and minimal risk from disruptive instabilities, the stellarator magnetic confinement is a natural choice. The defining feature of a stellarator is that the magnetic field required to sta- bly confine a plasma is provided entirely by external coils. Plasma current is often minimized by design in these configurations. As a result, current driven instabili- ties are effectively eliminated and no inductive systems are required. The primary trade-off is the loss of a direct symmetry in the magnetic field. The axisymmetry of tokamaks and other magnetic configurations both simplifies analysis and reduces losses from plasma drifts. Some quasi-symmetry properties can be recovered with careful design in stellarators, but the fully 3D nature makes the potential operating space vastly more complex. Confining plasma is the first step to achieving burning fusion, but the required temperature and density can only be achieved if the confinement is efficient enough. This is frequently parameterized for magnetic confinement by the plasma beta. Beta is the ratio of thermal or internal pressure, p, to the pressure exerted by the ex- 2 2 ternally applied magnetic fields, B =2µ0. It therefore takes the form β = 2µ0p=B , which is dimensionless and generally expressed as a percentage. Maximizing beta is a common goal for all magnetic confinement schemes. In tokamaks, pressure and current driven instabilities take control of the plasma at high beta and are re- sponsible the Troyon beta limit. The lack of large current in stellarators means that the Troyon limit is not applicable and different mechanisms are believed to 2 be responsible for the beta limit. Theoretical examination of stellarator beta limits, however, have only been performed with highly simplified plasma models. The intention of this work is to advance the understanding of high beta stellarators by analyzing simulations with more complete physics. In the first part of this document, we outline the present understanding of stel- larator behavior at high beta. Chapter 2 considers instabilities which have been observed in stellarators from theory, simulations, and experiments. In chapter 3 we discuss the transport effects produced by changes in stellarator equilibrium and how they can lead to a soft beta limit. We introduce the NIMROD