This book is devoted to mathematical modeling of tokamak plasma. Since the appearance in 1982 of the first edition (in Russian), a considerable amount of experimental and theoretical material on tokamak research has been accumu lated. The new-generation devices, viz. , TFTR, JET and JT-60 were put into operation. The first experiments on these units have confirmed the correctness of the basic physical concepts underlying their construction. Experiments on plasma heating with the help of neutral beams and high-frequency (HF) waves on previous generation devices made it possible to obtain high-P plasmas. The number of "medium-size" tokamaks in operation has increased. New experi mental results and advances in the theory have led to more complicated and perfected models of high-temperature plasma. Rapid progress in computer hardware and software has played an important role in the further development of mathematical modeling. While preparing the English edition of the book, we have revised the text considerably. Several new models which have undergone significant advance ment in recent years are described. A section devoted to models of RF (radio frequency) current drive has been added to Chap. 2. The reduced magneto hydrodynamic (MHD) equations for high-P plasma are now considered in detail in Chap. 3. Chapter 4 contains the latest results on anomalous thermal conductivity, diffusion coefficient and pinching. Two new sections are added to Chap. 5.
Table of Contents1. Controlled Fusion and Numerical Simulation.- 1.1 Controlled Fusion.- 1.1.1 The Lawson Criterion.- 1.1.2 Magnetic and Inertial Confinement of Plasma.- 1.1.3 The Role of Numerical Simulation in Fusion Research.- 1.2 Tokamaks.- 1.2.1 Design and Principle of Operation.- 1.2.2 The Current Status of Tokamak Research.- 1.2.3 Mathematical Models of Plasma in Tokamak Devices.- 1.3 Motion of Charged Particles in Tokamaks.- 1.3.1 Drift Equation of Motion.- 1.3.2 Tokamak Magnetic Field with a Circular Cross Section.- 1.3.3 The Motion of Charged Particles in the Tokamak Magnetic Field.- 2. Simulation of Kinetic Processes Involving Coulomb Interaction.- 2.1 Operator of Coulomb Collisions.- 2.1.1 Coulomb Collision Operator.- 2.1.2 Properties of Coulomb Operator.- 2.1.3 Coulomb Collision Operator for Axisymmetric Velocity Distributions.- 2.1.4 Coulomb Collision Operator for Isotropic Velocity Distribution of ? Particles.- 2.2 Cauchy Problem. Characteristic Relaxation Times.- 2.2.1 Cauchy Problem.- 2.2.2 Collisions Between Particles of the Same Species. The Simplest Relaxation Time.- 2.2.3 Relaxation of Relative Motion of Electrons and Ions.- 2.2.4 Energy Exchange and Temperature Equalization in Nonisothermal Plasma.- 2.2.5 Qualitative Description of the Behavior of the Cauchy Problem Solution for Two-Component Plasma.- 2.3 Linear Problem on the Interaction Between Fast Ions and Maxwellian Plasma.- 2.3.1 Mathematical Formulation.- 2.3.2 Isotropic Problem.- 2.3.3 Two-Dimensional Problem.- 2.3.4 Difference Scheme for the Solution of a Linear Kinetic Equation.- 2.4 Electric Field Effects.- 2.4.1 Critical Electric Field.- 2.4.2 Runaway Electrons.- 2.4.3 Effective Electric Field Acting on Ions.- 2.4.4 Interaction of Fast Ions with Plasma in the Presence of an Electric Field.- 2.5 The a).- b) Internal Modes (rs < a).- c) Dissipative Modes (? ? ?).- d) Effect of Toroidicity on the Helical Mode Stability (R/a ? ?).- e) Local Stability Criteria (m ? 1, n ? 1).- 3.3.4 Numerical Solution of Stability Problems.- a) Time Evolution of Solutions of System (3.3.1).- b) Investigation of the Potential Energy Sign.- c) Minimization of Functional (3.3.13) to Obtain the Natural Frequency Spectrum.- d) Examples.- 3.4 Nonlinear Problems.- 3.4.1 Nonlinear Evolution of External Modes.- 3.4.2 Evolution of the Internal Mode m/n =1/1 and Reconnection of Magnetic Surfaces.- 3.4.3 Nonlinear Evolution of Modes m ? 2 and Growth of Islands.- 3.4.4 Helical Modes with Two Resonant Surfaces (“Double Tearing Modes”).- 3.4.5 Interaction Between Helical Modes.- 4. Transport Models.- 4.1 Physical Grounds of Transport Models.- 4.1.1 Basic Equations.- 4.1.2 Neoclassical Fluxes of Particles and Energy.- 4.2 Development of the Transport Model.- 4.2.1 Model of Classical Energy Balance.- 4.2.2 Anomalous Thermal Conductivity of Electrons.- 4.2.3 Particle Flux.- 4.2.4 Model for Neutrals.- 4.2.5 The Effect of Magnetic Field Rippling.- 4.2.6 Compression of Plasma by a Magnetic Field.- a) Compression Along the Minor Radius.- b) Compression Along the Major Radius.- 4.3 Impurities.- 4.3.1 Influx of Impurities into Plasma.- 4.3.2 Basic System of Equations.- 4.3.3 Atomic Processes.- 4.3.4 Particle Fluxes.- 4.3.5 Approximate Solutions of System (4.3.1).- 4.3.6 Comparison with Experiments.- 4.3.7 Radiation of Impurities in Energy Balance Models.- 4.4 Numerical Solution of Systems (4.1.1, 3.1).- 4.4.1 Linear Implicit Scheme.- 4.4.2 Nonlinear Implicit Scheme.- 4.4.3 Gear’s General Methods for Stiff Systems.- 4.5 Appendix.- 5. Hybrid Models.- 5.1 Models of Plasma Heating by High-Energy Neutral Injection.- 5.1.1 Ionization and Capture of Energetic Neutrals.- a) Narrow-Beam Model.- b) Wide-Beam Model.- 5.1.2 Simple Model of Energy Balance with Neutral Injection.- 5.1.3 Hybrid Model of Energy Balance with Neutral Injection.- 5.1.4 Effect of Multiple Charge Exchange on Energy Transfer from Fast Ions to Bulk Plasma Particles.- 5.2 Effect of MHD Mixing on Energy and Particle Balance.- 5.2.1 Experimental Data on Mixing.- 5.2.2 Structure of the Hybrid Model.- 5.2.3 The Kadomtsev Model of Internal Mixing for Tearing Mode m/n = 1/1.- 5.2.4 Mixing for Nonmonotonic Current Profile.- 5.2.5 Electric Field in Mixing.- 5.2.6 Properties of the Hybrid Model and Its Application to the Experiment Description.- 5.3 Kinetic Convective Transport of Ions in Longitudinal Magnetic Field Ripples.- 5.3.1 Diffusive and Convective Transport.- 5.3.2 Basic Equations.- 5.3.3 Solution of Systems (5.3.5-10) and (5.3.18-21).- 5.3.4 Numerical Solution of Systems (5.3.5-10), (5.3.28-32).- References.