Flux Coordinates and Magnetic Field Structure

Flux Coordinates and Magnetic Field Structure gives a systematic and rigorous presentation of the mathematical framework and principles underlying the description of magnetically confined fusion plasmas. The various specific types of coordinate system are dealt with in detail.

1. Introduction and Overview.- 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 Tangent-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 gij and gij.- 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 Terms of the Metric Coefficients.- c) The Christoffel Symbols.- 2.6.2 The Del Operator.- a) Definition of the Del Operator.- 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 Sýstems 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) (?, ?, l) 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 Transform 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.- II Flux Coordinates.- 5. The Clebsch-Type Coordinate Systems.- 5.1 Stream Functions.- 5.2 Generic Clebsch Coordinates.- 5.3 Relationship to the Contra- and Covariant Formalism.- 5.4 Boozer-Grad Coordinates.- 6. Toroidal Flux Coordinates.- 6.1 Straight Field-Line Coordinates.- 6.2 Symmetry Flux Coordinates in a Tokamak.- 6.3 Interlude: Non-Flux Coordinates in Tokamaks.- 6.4 Straight Current-Density-Line Coordinates.- 6.5 Covariant B Components and Their Relationship to the Boozer-Grad Form.- 6.5.1 The Vector Potential.- 6.5.2 Covariant B Components.- 6.5.3 Relationship to the Boozer-Grad Form.- 6.6 Boozer’s Toroidal Flux Coordinates.- 6.7 Ideal-MHD-Equilibrium Conditions for Toroidally Confined Plasmas.- 6.8 Hamada Coordinates.- 7. Conversion from Clebsch Coordinates to Toroidal Flux Coordinates.- 7.1 The Generic Clebsch Coordinate System (?, v, l).- 7.2 Boozer-Grad Coordinates (?, v, ?).- 8. Establishment of the Flux-Coordinate Transformation; A Summary.- 8.1 Toroidal Systems.- 8.2 Open-Ended Systems.- 9. Canonical Coordinates or “Generalized Magnetic Coordinates”.- 9.1 Flux Coordinates Versus Canonical Coordinates.- 9.2 On the Existence of Flux Surfaces, Revisited.- 9.3 Flux Coordinates.- 9.4 Canonical Coordinates; The Field-Line Hamiltonian.- 9.5 Practical Evaluation of the Field-Line Hamiltonian.- III Selected Topics.- 10. “Proper” Toroidal Coordinates.- 10.1 Introduction.- 10.2 Bipolar Coordinates: Intuitive Considerations.- 10.3 The Relationship of 3-D Spherical Coordinates to 2-D Polar Coordinates.- 10.4 “Proper” Toroidal Coordinates as a 3-D Version of Bipolar Coordinates.- 10.5 The Bipolar Coordinate System: A Detailed Analysis.- 11. The Dynamic Equilibrium of an Ideal Tokamak Plasma.- 11.1 Introduction.- 11.2 A Useful Identity in Tokamak Geometry.- 11.3 The Existence of an Electric Field in a Tokamak Plasma with Infinite Conductivity.- 11.4 Motion of the Plasma and the Flux Surfaces.- 11.4.1 Plasma Motion due to E×B Drift.- 11.4.2 Motion of Flux Surfaces Defined by the Poloidal Disk Flux ?pold.- 11.4.3 Motion of Flux Surfaces Defined by the Poloidal Ribbon Flux ?polr.- 11.4.4 Motion of Flux Surfaces Defined by the Toroidal Flux ?tor.- 11.5 Constancy of the Rotational Transform Flux Function.- 11.6 Remarks on the Evolution of a Finite-Resistivity Plasma.- 11.7 The Relationship Between Poloidal Disk and Ribbon Fluxes.- 12. The Relationship Between ?dl/B and dV/d?tor.- 13. Transformation Properties of Vector and Tensor Components.- 13.1 Transformation of the Basis Vectors.- 13.2 Transformation of Vector Components.- 13.3 Transformation of Tensor Components.- 13.4 Transformation of Components of Special Tensors and Symbols.- 13.4.1 The Kronecker Delta.- 13.4.2 Metric Coefficients.- 13.4.3 Levi-Civita Symbols.- 13.4.4 Christoffel Symbols.- 14. Alternative Derivations of the Divergence Formula.- 14.1 A Straightforward Derivation of the Divergence Formula.- 14.2 Divergence-Formula Derivation Employing Christoffel Symbols.- References.