Flux Coordinates and Magnetic Field Structure - University of ...

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VIII Preface The book is divided into three parts. In Part I we establish the foundation on which the remainder of the text is built. It comprises a first chapter on basics concerning vectors, differential geometry and curvilinear coordinates (Chap. 2), a second chapter deals with tensors (Chap. 3) and the last chapter of Part I (Chap. 4) deals with the basic concepts of the magnetic-field geometry in toroidal and open-ended devices. Chapters 2 and 4 in particular should be considered as prerequisites for the rest. Part I1 treats the various distinct flux-coordinate systems. In Chaps. 5 through 9, we discuss Clebsch coordinates, the generic straight-field-line coordinate system, the Boozer toroidal flux coordinates, Hamada coordinates and the form of flux coordinates in axisymmetric tokamaks. In addition, we treat two coordinate systems that superficially resemble flux coordinates, but which are not flux coordinates: symmetry coordinates in a tokamak and so-called canonical coordinates. In Part I11 of this monograph, we consider some topics that deserve atten- tion, but which were tangential to the main issues in the previous chapters. Here, we treat "proper" toroidal coordinates (Chap. lo), the dynamic equilibrium of an ideal tokamak plasma (Chap. 1 I), the relationship between the proper length of a field line j dl/B and the specific volume of a flux surface dV/dYto, (Chap. 12), the transformation properties of vector and tensor components (Chap. 13), and finally two alternative derivations for the divergence of a vector (Chap. 14). There are a number of people who have been helpful in getting this monograph into its present shape We are most grateful to Mrs. Sarah Cohen, for whom it must have been a nightmare to type the various formulae with their variety of superscripts and subscripts. She retired soon after having finished this manuscript; we hope that we are not to blame for this decision. We also would like to thank Springer-Verlag and its Physics Editor, Dr. H.-U. Daniel, for the professional handling of our request for the publication of this work. We also appreciate that we had the opportunity to update the manuscript with regard to recently (1988, 1989) published scientific literature Also, we wish to express our gratitude to the referees for their useful comments. One of us (W. D'H) wishes to thank Dr. J. A. Tataronis for numerous discus- sions on the subject of magnetic-field structure This work was supported by the US Department of Energy through Grants No. ~~-FG02-87ER53216, No. DE-FG02-85ER53201 and No. DE- FG02-86ER53218, and in Part by the Wisconsin Alumni Research Foundation. We hope that the reader who works through all the steps given will reach a level where hdshe can consider co- and contravariant representations to be quite simple. Whether we have succeeded in fulfilling this expectation must be judged by those patient and persistent enough to have worked through the book. Garching and Madison, November 1990 W D. D 'haeseleer W N. G. Hitchon J D. Callen J L. Shohet Contents 1. Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I Fundamental Concepts 2. Vector Algebra and Analysis in Curvilinear Coordinates . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Reciprocal Sets of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Transformation to Curvilinear Coordinates . . . . . . . . . 2.3.2 Tiingent-Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Reciprocal-Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Covariant and Contravariant Components of a Vector 2.4 Covariant and Contravariant "Vectors" . . . . . . . . . . . . . . . . . . 2.5 Vector Relationships in Curvilinear Coordinates . . . . . . . . . . . 2.5.1 The Metric Coefficients gii and gii . . . . . . . . . . . . . . . . 2.5.2 The Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) The Jacobian of the Curvilinear Coordinate System . . . . . . . . . . . b) The Relationship Between J and g . . . . . . . . . . . . . . 2.5.3 The Dot and Cross-Products in Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . a) Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b) Cross Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 The Differential Elements dl(i), dS(i), d3R . . . . . . . . a) The Differential Arc Length dl(i) . . . . . . . . . . . . . . . b) The Differential Area Element dS(i) . . . . . . . . . . . . c) The Differential Volume Element d3R . . . . . . . . . . . 2.6 Vector Differentiation in Curvilinear Coordinates . . . . . . . . . 2.6.1 The Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . a) Differentiation of Vector Components and Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . b) The Covariant Derivative in nrms of the Metric Coefficients . . . . . . . . . . . . . . . . . . . . . c) The Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . 2.6.2 The Del Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) Definition of the Del Operator . . . . . . . . . . . . . . . . .
X Contents b) Gradient ..................................... c) Divergence ................................... d) Curl ......................................... 2.7 The Parallel and Perpendicular Components of a Vector .... 2.8 A Summary of Vector Related Identities .................. 3 . Tensorial Objects ........................................... 3.1 Introduction ........................................... 3.2 The Concept of Tensors; A Pragmatic Approach ........... 3.3 Dot Product. Double Dot Product. Contraction ............ 3.4 The Relationship Between Covariant. Contravariant and Mixed Components ................................ 3.5 Special Tensors ........................................ 3.5.1 The Kronecker Delta and the Metric Tensor ......... 3.5.2 Levi-Civita and Christoffel Symbols ................ a) Levi-Civita Symbols ........................... b) Christoffel Symbols ............................ 3.6 Tensor and Dyadic Identities ............................ 3.7 Suggestions for Further Reading ......................... 4 . Magnetic-Field-Structure-Related Concepts ..................... 4.1 The Equation of a Magnetic-Field Line ................... 4.2 The Frozen-Flux Theorem ............................... 4.3 The Magnetic Field-Line Curvature ...................... 4.4 Magnetic Pressure and Magnetic Tension .................. 4.5 Magnetic Surfaces ...................................... a) Toroidal Systems .................................... b) Open-Ended Systems ................................ 4.6 Curvilinear Coordinate Systems in Confinement Systems with "Simple" Magnetic Surfaces ............................. 4.6.1 Toroidal Systems ................................. a) "The Cylindrical-Toroidal" or "Elementary" Toroidal System ............................... b) Generalized Cylindrical-Toroidal Coordinates; Flux Coordinates .............................. c) An Illustrative Example ........................ d) (Q. /3. I) Coordinates in Toroidal Systems ......... 4.6.2 Open-Ended Systems ............................. 4.7 Magnetic-Surface Labeling .............................. 4.7.1 Toroidal Systems ................................. 4.7.2 Open-Ended Systems ............................. 4.8 The Rotational lkansform in Toroidal Systems ............. 4.9 The Flux-Surface Average ............................... 4.9.1 Toroidal Systems ................................. 4.9.2 Open-Ended Systems ............................. 4.9.3 Properties of the Flux-Surface Average .............. 4.10 The Magnetic Differential Equation ...................... Part I1 Flux Coordinates Contents XI 5 . The Clebsch-Type Coordinate Systems ........................ 100 5.1 Stream Functions ..................................... 100 5.2 Generic Clebsch Coordinates ........................... 100 5.3 Relationship to the Contra- and Covariant Formalism ..... 103 5.4 Boozer-Grad Coordinates 107 .............................. . .................................. ......................... ............... ............ ............... 6 Toroidal Flux Coordinates 116 6.1 Straight Field-Line Coordinates 116 6.2 Symmetry Flux Coordinates in a Tokamak 122 6.3 Interlude: Non-Flux Coordinates in Tokamaks 126 6.4 Straight Current-Density-Line Coordinates 129 6.5 6.6 Covariant B Components and Their Relationship to the Boozer-Grad Form .............................. 131 6.5.1 The Vector Potential ............................. 131 6.5.2 Covariant B Components ........................ 133 6.5.3 Relationship to the Boozer-Grad Form ............. 140 Boozer's Toroidal Flux Coordinates ..................... 142 6.7 Ideal-MHD-Equilibrium Conditions for Toroidally Confined Plasmas ........................ 146 6.8 Hamada Coordinates 150 .................................. 7 . Conversion from Clebsch Coordinates to Toroidal Flux Coordinates ............................... 156 7.1 The Generic Clebsch Coordinate System (Q. v. I) .......... 156 7.2 Boozer-Grad Coordinates (Q. v. X) 157 ...................... 8 . Establishment of the Flux-Coordinate 'hnsformation; A Summary .............................................. 160 8.1 Toroidal Systems ...................................... 160 8.2 Open-Ended Systems .................................. 164 9 . Canonical Coordinates or "Generalized Magnetic Coordinates" . 165 9.1 Flux Coordinates Versus Canonical Coordinates ........... 165 9.2 On the Existence of Flux Surfaces. Revisited ............. 165 ..................................... 9.3 Flux Coordinates 166 9.4 Canonical Coordinates; The Field-Line Hamiltonian ....... 170 9.5 Practical Evaluation of the Field-Line Harniltonian ........ 174 Part 111 Selected Topics . .............................. .......................................... 10 "Proper" Toroidal Coordinates 179 10.1 Introduction 179 10.2 Bipolar Coordinates: Intuitive Considerations ... ; ......... 179
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Subject Index Abraham 54 Absolute d
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