Flux Coordinates and Magnetic Field Structure

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.