Fundamentals of Plasma Physics and Controlled Fusion Kenro Miyamoto (Received Sep. 18, 2000) NIFS-PROC-48 Oct. 2000 Fundamentals of Plasma Physics and Controlled Fusion by Kenro Miyamoto i Fundamentals of Plasma Physics and Controlled Fusion Kenro Miyamoto Contents Preface 1 Nature of Plasma ....................................................................1 1.1 Introduction .......................................................................... 1 1.2 Charge Neutrality and Landau Damping ..............................................1 1.3 Fusion Core Plasma ...................................................................3 2 Plasma Characteristics ..............................................................7 2.1 Velocity Space Distribution Function, Electron and Ion Temperatures ..................7 2.2 Plasma Frequency, Debye Length ......................................................8 2.3 Cyclotron Frequency, Larmor Radius ..................................................9 2.4 Drift Velocity of Guiding Center ..................................................... 10 2.5 Magnetic Moment, Mirror Confinement, Longitudinal Adiabatic Constant ............12 2.6 Coulomb Collision Time, Fast Neutral Beam Injection ................................14 2.7 Runaway Electron, Dreicer Field .....................................................18 2.8 Electric Resistivity, Ohmic Heating ...................................................19 2.9 Variety of Time and Space Scales in Plasmas .........................................19 3 Magnetic Configuration and Particle Orbit ......................................21 3.1 Maxwell Equations ...................................................................21 3.2 Magnetic Surface .....................................................................23 3.3 Equation of Motion of a Charged Particle ............................................24 3.4 Particle Orbit in Axially Symmetric System ..........................................26 3.5 Drift of Guiding Center in Toroidal Field .............................................28 a Guiding Center of Circulating Particles b Guiding Center of Banana Particles 3.6 Orbit of Guiding Center and Magnetic Surface .......................................31 3.7 Effect of Longitudinal Electric Field on Banana Orbit ................................32 4 Velocity Space Distribution Function and Boltzmann’s Equation .............................................................35 4.1 Phase Space and Distribution Function ...............................................35 4.2 Boltzmann’s Equation and Vlasov’s Equation ........................................ 36 5 Plasma as MHD Fluid ..............................................................39 5.1 Magnetohydrodynamic Equations for Two Fluids .....................................39 5.2 Magnetogydrodynamic Equations for One Fluid ......................................41 5.3 Simplified Magnetohydrodynamic Equations ..........................................42 5.4 Magnetoacoustic Wave ...............................................................44 6 Equilibrium ..........................................................................47 6.1 Pressure Equilibrium .................................................................47 6.2 Equilibrium Equation for Axially Symmetric and Translationally Symmetric Systems . 48 ii Contents 6.3 Tokamak Equilibrium ................................................................50 6.4 Poloidal Field for Tokamak Equilibrium ..............................................55 6.5 Upper Limit of Beta Ratio ...........................................................58 6.6 Pfirsch-Schl¨uter Current ..............................................................59 6.7 Virial Theorem .......................................................................61 7 Diffusion of Plasma, Confinement Time ..........................................65 7.1 Collisional Diffusion (Classical Diffusion) .............................................66 a Magnetohydrodynamic Treatment b A Particle Model 7.2 Neoclassical Diffusion of Electrons in Tokamak .......................................69 7.3 Fluctuation Loss, Bohm Diffusion, and Stationary Convective Loss ...................71 7.4 Loss by Magnetic Fluctuation ........................................................75 8 Magnetohydrodynamic Instabilities ...............................................77 8.1 Interchange, Sausage and Kink Instabilities ...........................................77 a Interchange Instability b Stability Criterion for Interchange Instability, Magnetic Well c Sausage Instability d Kink Instability 8.2 Formulation of Magnetohydrodynamic Instabilities ...................................85 a Linearization of Magnetohydrodynamic Equations b Energy Principle 8.3 Instabilities of a Cylindrical Plasma ..................................................90 a Instabilities of Sharp-Boundary Configuration: Kruskal-Shafranov Condition b Instabilities of Diffuse-Boundary Configurations c Suydam’s Criterion d Tokamak Configuration e Reversed Field Pinch 8.4 Hain-L¨ust Magnetohydrodynamic Equation .........................................102 8.5 Ballooning Instability ...............................................................103 8.6 ηi Mode due to Density and Temperature Gradient .................................. 107 9 Resistive Instability ............................................................... 111 9.1 Tearing Instability ..................................................................111 9.2 Resistive Drift Instability ...........................................................115 10 Plasma as Medium of Waves ....................................................121 10.1 Dispersion Equation of Waves in a Cold Plasma ....................................121 10.2 Properties of Waves ................................................................124 a Polarization and Particle Motion b Cutoff and Resonance 10.3 Waves in a Two-Components Plasma .............................................. 126 10.4 Various Waves .....................................................................129 a Alfv´en Wave b Ion Cyclotron Wave and Fast Wave c Lower Hybrid Resonance d Upper Hybrid Resonance e Electron Cyclotron Wave 10.5 Conditions for Electrostatic Waves .................................................134 11 Landau Damping and Cyclotron Damping .....................................137 11.1 Landau Damping (Amplification) ..................................................137 11.2 Transit-Time Damping .............................................................140 11.3 Cyclotron Damping ................................................................140 11.4 Quasi-Linear Theory of Evolution in the Distribution Function .....................142 12 Wave Propagation and Wave Heating ..........................................145 12.1 Energy Flow .......................................................................145 Contents iii 12.2 Ray Tracing .......................................................................148 12.3 Dielectric Tensor of Hot Plasma, Wave Absorption and Heating ....................149 12.4 Wave Heating in Ion Cyclotron Range of Frequency ................................154 12.5 Lower Hybrid Wave Heating .......................................................157 12.6 Electron Cyclotron Heating ........................................................159 13 Velocity Space Instabilities (Electrostatic Waves) .............................163 13.1 Dispersion Equation of Electrostatic Wave .........................................163 13.2 Two Streams Instability ........................................................... 164 13.3 Electron Beam Instability ..........................................................165 13.4 Harris Instability ..................................................................165 14 Instabilities driven by Energetic Particles ......................................169 14.1 Fishbone Instability ................................................................169 a Formulation b MHD Potential Energy c Kinetic Integral of Hot Component d Growth Rate of Fishbone Instability 14.2 Toroidal Alfv´en Eigenmodes .......................................................176 a Toroidicity Inuced Alfv´en Eigenmode b Instabilities of TAE Driven by Energetic Particles c Various Alfv´en Modes 15 Development of Fusion Researches .............................................189 16 Tokamak ...........................................................................197 16.1 Tokamak Devices ..................................................................197 16.2 Equilibrium ........................................................................201 a Case with Conducting Shell b Case without Conducting Shell c Equilibrium Beta Limit of Tokamaks with Elongated Plasma Cross Section 16.3 MHD Stability and Density Limit ..................................................203 16.4 Beta Limit of Elongated Plasma ...................................................205 16.5 Impurity Control, Scrape-Off Layer and Divertor ...................................206 16.6 Confinement Scaling of L Mode ....................................................210 16.7 H Mode and Improved Confinement Modes ........................................212 16.8 Noninductive Current Drive ........................................................217 a Lower Hybrid Current Drive b Electron Cyclotron Current Drive c Neutral Beam Current Drive d Bootstrap Current 16.9