W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 1991 Completely bootstrapped tokamak Richard Henry Weening College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Plasma and Beam Physics Commons Recommended Citation Weening, Richard Henry, "Completely bootstrapped tokamak" (1991). Dissertations, Theses, and Masters Projects. Paper 1539623812. https://dx.doi.org/doi:10.21220/s2-2nck-2c81 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 ------------------------~ Order Number 9219102 Completely bootstrapped tokamak Weening, Richard Henry, Ph.D. The College of William and Mary, 1991 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 --------------------------- COMPLETELY BOOTSTRAPPED TOKAMAK A Dissertation Presented to The Faculty of the Department of Physics The College of William and Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Richard H. Weening 1991 This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Richard H. Weening Approved, November 1991 & A - . __________ Allen H. Boozer - l l . 6 ^ — Roy L. Champion Alkesh Punjabi Department of Mathematics Hangpfon University i"A V ^/ 'h / ^------- ' George T. Rublein Department of Mathematics Eugene R. T^py ______________ SlA^U. ______________L George M. Vahala ii It is not your obligation to complete the work of perfecting the world, but you are not free to desist from it either. - RABB! TARFON, Ethics of the Fathers ^ But all things very clear are as difficult as rare. - BARUCH SPINOZA, ETHIC God is inexorable in offering His gifts. He only gave me the stubbornness o f a mule. No! - He also gave me a keen sense of smell. - ALBERT EINSTEIN iii TABLE OF CONTENTS PREFACE....................................................................................................Vi ACKNOWLEDGEMENTS......................................................................... Viii LIST OF TABLES.....................................................................................xi LIST OF FIGURES..................................................................................xii ABSTRACT...............................................................................................xiv CHAPTER I. INTRODUCTION Section I-A. Thermonuclear Fusion ............................................2 Section 1-B. Magnetic Confinement .............................................8 Section I-C. Tokamaks .................................................................. 16 Section I-D. Tokamak Current ...................................................29 CHAPTER II. GENERALIZED OHM’S LAW Section II-A. MHD Theory .............................................................35 Section I I-B. Kinetic Theory ............................ 44 Section II-C. Fluid Equations ................................. 51 CHAPTER III. MAGNETIC FIELDS Section 11 I-A. Canonical Coordinates .......................................59 Section III-B. Magnetic Coordinates ........................................ 65 Section III-C. Magnetic Diffusion Equation ............................71 CHAPTER IV. TEARING MODES Section IV-A. Ideal MHD ...............................................................81 Section IV-B. Magnetic Topology .............................................. 91 iv Section IV-C. Tokamak Plasma Stability .............................I ll Section IV-D. Magnetic Island Growth .................................126 CHAPTER V. MEAN-FIELD THEORY Section V-A. Magnetic Helicity ...............................................133 Section V-B. Mean-field Ohm's Law ......................................144 Section V-C. Current Viscosity ..............................................152 CHAPTER VI. BOOTSTRAP EFFECT Section VI-A. Neo-classical Transport ................................ 162 Section VI-B. Current Amplification ....................................169 CHAPTER VII. NUMERICAL METHODS Section VII-A. Scaled Model Equations .................................178 Section VII-B. Poloidal Flux Evolution ..................................182 Section VII-C. Delta Prime Analysis ....................................198 CHAPTER VIII. INDUCTIVELY DRIVEN TOKAMAK Section VIII-A. Tokamak Temperature Profile ..................207 Section VIII-B. MHD-Stable Regime of the Tokamak . .218 CHAPTER IX. COMPLETELY BOOTSTRAPPED TOKAMAK Section IX-A. Tokamak Pressure Profile .............................237 Sec. IX-B. Self-maintenance of the Tokamak Current . .245 CHAPTER X. CONCLUSION Section X-A. Plasma Theory and Experiment ....................256 BIBLIOGRAPHY.....................................................................................261 VITA.......................................................................................................265 v PREFACE This is a dissertation about the leading fusion plasma confinement configuration, the tokamak. The problem we examine here is a fundamental one: How can one efficiently maintain the tokamak toroidal current in the steady-state? As will be seen later, this question is of great practical importance if one considers using the tokamak as a fusion reactor. The theoretical tools used to analyze our problem are fairly sophisticated ones. That is, we employ a new theoretical approach based on a magnetic helicity conserving mean-field Ohm’s law. Unfortunately, helicity is a very abstract topological concept. Nevertheless, an attempt has been made throughout the work to motivate theoretical arguments by physical rather than abstract considerations. The main accomplishment of this work is that, for the first time, a self-consistent mean-field helicity transport model for the tokamak has been successfully implemented on the computer. Two kinds of numerical simulations are accomplished: (1) simulations of inductively driven tokamaks and (2) simulations of completely bootstrapped tokamaks. The inductively driven tokamak simulations are compared with experimental results from the present generation of tokamak devices. The completely bootstrapped tokamak simulations are of great interest, since they suggest that an intrinsic tokamak steady-state can be obtained from the bootstrap current amplification effect alone. The organization of the dissertation is as follows. First, the Introduction has been written so that it is accessible to all readers and places our research within the context of the magnetic fusion program. Chapters II, III, and IV contain materials of a preliminary or background nature which describe the plasma Ohm's law, magnetohydrodynamics (MHD), representations of magnetic fields, tearing modes, and some tokamak physics. Reading these preliminary chapters will be vi particularly worthwhile for those uninitiated in plasma theory or those who want an in-depth view of our work. The reader who already possesses some knowledge of plasma physics may wish to move directly to chapters V and VI. which emphasize our mean-field theoretical approach. Chapter VII describes the numerical methods we have developed for the computer simulations which implement our mean-field theory. The results of the simulations are given in two chapters. Chapter VIII gives the results of simulations involving inductively driven tokamaks. Chapter IX analyzes the possiblity of the self-maintenance of the tokamak current by performing simulations of completely bootstrapped tokamaks. Finally, some concluding remarks are given in chapter X. vii ACKNOWLEDGEMENTS I would like to acknowledge and thank some of the many wonderful people who have helped make my graduate school career at William and Mary so enjoyable. First, I want to thank former Physics