Macroscopic Stability of Toroidally Confined Fusion Plasmas

Preface
Author

Chapter 1 ■ Magnetic confinement
1.1 Introduction
1.2 Thermonuclear fusion
1.3 Nuclear fusion reactions
1.4 The Lawson criterion
1.5 Fusion plasma parameters
1.6 Motion of a charged particle
1.7 The physics of magnetic confinement
1.8 Toroidal pinches
1.9 Macroscopic stability limits
1.10 Ideal and resistive instabilities

Chapter 2 ■ Plasma fluid theory
2.1 Introduction
2.2 Kinetic equation
2.3 Moments of distribution function
2.4 Moments of collision operator
2.5 Fluid equations
2.6 Fluid closure
2.7 Useful definitions
2.8 Braginskii equations
2.9 Normalization of Braginskii equations
2.10 MHD ordering
2.11 Drift ordering
2.12 Lack of collisional parallel confinement
2.13 Plasma turbulence
Chapter 3 ■ Ideal magnetohydrodynamics
3.1 Introduction
3.2 Magnetic pressure
3.3 Flux freezing
3.4 MHD waves

Chapter 4 ■ General ideal-MHD stability
4.1 Introduction
4.2 Ideal-MHD equations
4.3 Plasma equilibrium
4.4 Plasma boundary
4.5 Plasma equation of motion
4.6 Normal mode analysis
4.7 Perfectly conducting wall
4.8 Self-adjoint property of force operator
4.9 Reality of ω2
4.10 Orthogonality of normal modes
4.11 Variational principle
4.12 Energy principle
4.13 Vacuum region
4.14 Matching conditions at plasma-vacuum interface
4.15 Extended eigenmode problem
4.16 Extended self-adjoint property of force operator
4.17 Extended variational principle
4.18 Extended ideal-MHD energy principle
4.19 Alternative expression for δWp
4.20 Technical details
4.20.1 First useful result
4.20.2 Second useful result

Chapter 5 ■ General resistive wall mode stability
5.1 Introduction
5.2 Perturbed plasma equilibrium
5.3 Ideal-MHD energy principle
5.4 Perfect-wall and no-wall stability
5.5 Pressure balance boundary condition
5.6 Resistive wall physics
5.7 Timescale ordering
5.8 Variational principle
5.9 Minimization process
5.10 Resistive wall mode dispersion relation
5.11 Technical details
5.11.1 Self-adjoint property of F(ξ) in presence of resistive wall
5.11.2 Minimization of δW in presence of resistive wall
5.11.3 Useful results

Chapter 6 ■ Physics of outer region
6.1 Introduction
6.2 Axisymmetric plasma equilibrium
6.3 Grad-Shafranov equation
6.4 Tearing mode perturbation
6.5 Outer-region p.d.e.s
6.6 Primitive outer-region o.d.e.s
6.7 Outer-region o.d.e.s
6.8 Resonance condition
6.9 Symmetry properties of coupling matrices
6.10 Plasma perturbed potential energy
6.11 Toroidal electromagnetic torque
6.12 Behavior in vicinity of rational surface
6.13 Mercier indices
6.14 Asymptotic matching across rational surfaces
6.15 Magnetic island chains
6.16 Behavior close to magnetic axis
6.17 Behavior at plasma boundary
6.18 Tearing mode dispersion relation - I
6.19 Tearing mode solutions
6.20 Angular momentum conservation - I
6.21 Toroidal coordinates
6.22 Perturbed vacuum magnetic field
6.23 Matching across plasma boundary
6.24 Angular momentum conservation - II
6.25 No-wall matching condition
6.26 Perfect-wall matching condition
6.27 Tearing mode dispersion relation - II
6.28 Ideal solutions
6.29 Plasma perturbed potential energy matrix
6.30 Vacuum perturbed potential energy matrix
6.31 Total perturbed potential energy matrix
6.32 Resistive wall mode stability
6.33 Technical details
6.33.1 Nonorthogonal curvilinear coordinates
6.33.2 Derivation of outer-region p.d.e.s

Chapter 7 ■ Axisymmetric instabilities
7.1 Introduction
7.2 Axisymmetric perturbations
7.3 Axisymmetric ideal-MHD p.d.e.s
7.4 Axisymmetric ideal-MHD o.d.e.s
7.5 Properties of axisymmetric ideal-MHD o.d.e.s
7.6 Perturbed electric field
7.7 Toroidal electromagnetic torque
7.8 Electromagnetic energy flux
7.9 Perturbed plasma potential energy
7.10 Perturbed vacuum magnetic field
7.11 Matching across plasma boundary
7.12 Angular momentum conservation
7.13 Energy conservation
7.14 No-wall matching condition
7.15 Perfect-wall matching condition
7.16 Plasma perturnbed potential energy matrix
7.17 Vacuum perturbed potential energy matrix
7.18 Total perturbed potential energy matrix
7.19 Resistive wall mode stability
7.20 Technical details
7.20.1 Derivation of axisymmetric ideal-MHD p.d.e.s

Chapter 8 ■ Physics of inner region
8.1 Introduction
8.2 Drift-MHD fluid equations
8.3 Simplified drift-MHD fluid equations
8.4 Normalization scheme
8.5 Reduction process
8.6 Reduced drift-MHD fluid equations
8.7 Linearized reduced drift-MHD equations
8.8 Generalized linearized reduced drift-MHD equations
8.9 Resonant layer equations
8.10 Asymptotic matching
8.11 Fourier transformation
8.12 Constant-ψ limit
8.13 Constant-ψ linear resonant response regimes
8.14 Nonconstant-ψ limit
8.15 Nonconstant-ψ linear resonant response regimes
8.16 General layer equations
8.17 Riccati matrix differential equation
8.18 Small argument behavior
8.19 Large argument behavior
8.20 Method of solution
8.21 Example calculation

Chapter 9 ■ Ideal stability of cylindrical plasma equilibria
9.1 Cylindrical model
9.2 Cylindrical plasma equilibrium
9.3 Plasma ideal potential energy
9.4 Surface ideal potential energy
9.5 Boundary conditions
9.6 Vacuum ideal potential energy
9.7 Minimization of potential energy
9.8 Vacuum solution
9.9 Total ideal potential energy
9.10 Alternative vacuum solution
9.11 Behavior close to magnetic axis
9.12 Behavior close to rational surface
9.13 Mercier stability criterion
9.14 Internal mode stability
9.15 External mode stability
9.16 Non-resonant mode stability
9.17 Resistive wall mode stability
9.18 Perturbed poloidal magnetic flux
9.19 The reversed-field pinch
9.20 The low-β tokamak
9.21 Internal kink modes
9.22 m = 1 internal kink modes
9.23 External modes
9.24 External kink modes

Chapter 10 ■ Tearing modes in reversed-field pinches
10.1 Introduction
10.2 Plasma equilibrium
10.3 Perturbed plasma equilibrium
10.4 Perturbed quantities
10.5 Eigenmode equation
10.6 Stepped pressure equilibrium
10.7 Simplified linear eigenmode equation
10.8 Tearing mode stability
10.9 Nonlinear mode coupling
10.10 Regularization at rational surfaces
10.11 Validity of quasilinear approach
10.12 Phase locking of m = 0 modes
10.13 Phase locking of m = 1 modes
10.14 Experimental results
10.15 Technical details 27
10.15.1 Derivation of nonlinear eigenmode equation
10.15.2 Nonlinear coupling coefficients
10.15.3 Integrated electromagnetic torques
10.15.4 Localized electromagnetic torques

Chapter 11 ■ Tearing modes in low-pressure tokamaks
11.1 Introduction
11.2 Plasma equilibrium
11.3 Perturbed plasma equilibrium
11.4 Perturbed quantities
11.5 Linear eigenmode equation
11.6 Tearing mode stability
11.7 Nonlinear evolution of tearing modes

Chapter 12 ■ Vertical stability of tokamaks
12.1 Introduction
12.2 Aspect-ratio expanded equilibrium
12.3 Vertical instabilities
12.4 Plasma perturbed potential energy matrix
12.5 Vacuum perturbed potential energy matrix
12.6 Total perturbed potential energy matrix
12.7 Resistive wall mode stability
12.8 Vertical stability of tokamaks
12.9 Technical details
12.9.1 Coordinate transformation
12.9.2 Metric elements
12.9.3 Expansion of inverse Grad-Shafranov equation
12.9.4 Edge boundary condition
12.9.5 Calculation of coupling coefficients

Chapter 13 ■ General stability of high-pressure tokamaks
13.1 Introduction
13.2 Aspect-ratio expanded equilibrium
13.3 Outer-region o.d.e.s
13.4 Determination of ideal stability
13.5 Determination of resistive stability
13.6 Ideal stability of n = 1 internal kink mode
13.7 Resistive stability of n = 1 internal kink mode
13.8 Ideal stability of n = 1 external kink mode
13.9 Resistive stability of n = 1 external kink mode
13.10 Technical details
13.10.1 Intuitive form of δWp
13.10.2 Calculation of coupling coefficients
13.10.3 Resistive layer model

Bibliography
Index

Biography

Richard Fitzpatrick is a Professor of Physics at the University of Texas at Austin, where he has been a faculty member since 1994. He is a member of the Royal Astronomical Society, a fellow of the American Physical Society, and the author of Maxwell’s Equations and the Principles of Electromagnetism (2008), An Introduction to Celestial Mechanics (2012), Oscillations and Waves: An Introduction (2013). Plasma Physics: An Introduction (2014), Quantum Mechanics (2015), Theoretical Fluid Mechanics (2017), Oscillations and Waves: An Introduction, 2nd Edition (2019), Thermodynamics and Statistical Mechanics (2020), and Newtonian Dynamics: An Introduction (2022). He earned a Master’s degree in physics from the University of Cambridge and a DPhil in astronomy from the University of Sussex.