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Contents
Contents
Tearing Mode Dynamics in Tokamak Plasmas
Richard Fitzpatrick
Contents
Preface
Introduction
Introduction
Thermonuclear Fusion
Nuclear Fusion Reactions
Lawson Criterion
Fusion Plasma Parameters
Particle Balance
Energy Balance
Linear Pinches
Toroidal Pinches
Tokamaks
Tearing Modes
Tearing Mode Rotation
Error-Field Penetration
Neoclassical Tearing Modes
Tearing Modes in Toroidal Plasmas
Bibliography
Plasma Fluid Theory
Introduction
Kinetic Theory
Fluid Theory
Fundamental Quantities
Fluid Closure Schemes
Classical Closure Scheme
Trapped and Passing Particles
Neoclassical Closure Scheme
Drift and Transport Orderings
Toroidal Plasma Equilibrium
Lowest-Order Flows
Flux-Surface Average Operator
Chew-Goldberger-Low Forms
Parallel Flows
Useful Results
Friction Force Densities
Parallel Viscous Force Densities
Determination of Ion Flows
Determination of Electron Flows
Parallel Current Density
Neoclassical Transport
Perpendicular Closure Scheme
Parallel Closure Scheme
Derivation of Neoclassical Fluid Equations
Normalization of Neoclassical Fluid Equations
Discussion
Bibliography
Cylindrical Tearing Mode Theory
Introduction
Cylindrical Tokamak Equilibrium
Magnetic Field and Current Density Perturbations
Density and Temperature Perturbations
Fluid Continuity
Velocity Perturbation
Cylindrical Tearing Mode Equation
Solution in Presence of Perfectly Conducting Wall
Solution in Presence of Resistive Wall
Resistive Wall Physics
Resistive Layer Physics
Solution in Presence of External Magnetic Field-Coil
Electromagnetic Torques
Plasma Angular Equations of Motion
Solution of Plasma Angular Equations of Motion
Modification of Rotation Frequency
Tearing Mode Evolution Equations
Bibliography
Reduced Resonant Response Model
Introduction
Drift-MHD Fluid Equations
Normalization Scheme
Reduction Process
Reduced Drift-MHD Model
Bibliography
Linear Resonant Response Model
Introduction
Reduced Drift-MHD Model
Plasma Equilibrium
Linearized Reduced Drift-MHD Equations
Resonant Layer Equations
Asymptotic Matching
Fourier Transformation
Constant-
Limit
Constant-
Linear Resonant Response Regimes
Nonconstant-
Limit
Nonconstant-
Linear Resonant Response Regimes
Linear Resonant Response Regimes
Response Regimes in Tokamak Fusion Reactors
Numerical Solution of Resonant Layer Equations
Plasma Rotation
Magnetic Reconnection
Bibliography
Linear Tearing Mode Stability
Introduction
Linear Dispersion Relation
Determination of Linear Growth-Rates
Linear Growth-Rate Regimes
Resonant Layer Thickness
Numerical Solution of Resonant Layer Equations
Bibliography
Error-Field Penetration in Tokamak Plasmas
Introduction
Asymptotic Matching
Resonant Layer Response
Torque Balance
Error-Field Penetration
Numerical Solution of Layer Equations
Bibliography
Nonlinear Resonant Response Model
Introduction
Reduced Drift-MHD Model
Rescaled Reduced Drift-MHD Model
Ordering Scheme
Lowest-Order Solution
Flux-Surface Average Operator
Fluid Velocities
Need for Higher-Order Solution
Higher-Order Solution
Asymptotic Matching
Evaluation of Integrals
Bibliography
Nonlinear Tearing Mode Stability
Introduction
Rutherford Island Width Evolution Equation
Composite Linear/Nonlinear Model
Saturated Island Width
Island Rotation Frequency
Bibliography
Rotation Braking in Tokamak Plasmas
Introduction
Rotation Braking by a Thin Conducting Wall
Rotation Braking by a Thick Conducting Wall
Improved Torque Balance Model
Bibliography
Nonlinear Neoclassical Resonant Response Model
Introduction
Neoclassical Drift-MHD Equations
Reduced Neoclasssical Drift-MHD Model
Magnetic Field-Line Curvature
Rescaled Reduced Neoclassical Drift-MHD Model
Ordering Scheme
Zeroth-Order Solution
Higher-Order Solution
Asymptotic Matching
Evaluation of Integrals
Bibliography
Neoclassical Tearing Modes
Introduction
Isolated Magnetic Island Chain
Island Rotation Frequency
Generalized Rutherford Equation
Stabilization via RF-Driven Current
Bibliography
Mode Locking in Tokamak Plasmas
Introduction
Asymptotic Matching
Rutherford Island Width Evolution Equation
Island Phase Evolution Equations
Analytic Solution of Phase Evolution Equations
Numerical Solution of Phase Evolution Equations
Locked Magnetic Island Chains
Island Width Evolution
Bibliography
Toroidal Tearing Modes
Introduction
Coordinate Systems
Useful Identities
Equilibrium Magnetic Field
Equilibrium Plasma Current Density
Grad-Shafranov Equation
Perturbed Magnetic Field
Perturbed Current Density
Electromagnetic Torques
Magnetic Island Chains
Inductance Matrix
Calculation of Inductance Matrix
Toroidal Tearing Mode Dispersion Relation
Example Tokamak Discharge
Linear Calculation
Nonlinear Calculation
Effect of Electromagnetic Torques
Neoclassical Tearing Modes
Bibliography
Neoclassical Theory
Introduction
Plasma Species
Collisionality Parameters
Friction Force Matrices
Neoclassical Viscosity Matrices
Parallel Force and Heat Balance
Neoclassical Fluid Velocities
Glasser-Greene-Johnson Parameters
Bibliography
About this document ...
Limit
Linear Resonant Response Regimes
Limit
Linear Resonant Response Regimes