Fundamentals of Plasma Physics and Controlled Fusion

Fundamentals of Plasma Physics and Controlled Fusion

Kenro Miyamoto

Fundamentals of Plasma Physics and Controlled Fusion by Kenro Miyamoto

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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 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Plasma Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Velocity Space Distribution Function, Electron and Ion Temperatures . . . . . . . . . . . . . . . . . .7 Plasma Frequency, Debye Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cyclotron Frequency, Larmor Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Drift Velocity of Guiding Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Magnetic Moment, Mirror Con¯nement, Longitudinal Adiabatic Constant . . . . . . . . . . . . 12 Coulomb Collision Time, Fast Neutral Beam Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Runaway Electron, Dreicer Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Electric Resistivity, Ohmic Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Variety of Time and Space Scales in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Magnetic Con¯guration and Particle Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 3.2 3.3 3.4 3.5

Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Magnetic Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Equation of Motion of a Charged Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Particle Orbit in Axially Symmetric System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 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 E®ect of Longitudinal Electric Field on Banana Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4

Velocity Space Distribution Function and Boltzmann's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Phase Space and Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Boltzmann's Equation and Vlasov's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5

Plasma as Magnetohydrodynamic Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1 Magnetohydrodynamic Equations for Two Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Magnetogydrodynamic Equations for One Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.3 Simpli¯ed Magnetohydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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5.4 Magnetoacoustic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6

Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 6.2 6.3 6.4 6.5 6.6 6.7 7

Pressure Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Equilibrium Equation for Axially Symmetric and Translationally Symmetric Systems . 47 Tokamak Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Poloidal Field for Tokamak Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Upper Limit of Beta Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 P¯rsch-SchlÄ uter Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 Di®usion of Plasma, Con¯nement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.1 Collisional Di®usion (Classical Di®usion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 a Magnetohydrodynamic Treatment b A Particle Model 7.2 Neoclassical Di®usion of Electrons in Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3 Fluctuation Loss, Bohm Di®usion, and Stationary Convective Loss . . . . . . . . . . . . . . . . . . . 69 7.4 Loss by Magnetic Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8

Magnetohydrodynamic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.1 Interchange, Sausage and Kink Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 a Interchange Instability b Stability Criterion for Interchange Instability, Magnetic Well c Sausage Instability d Kink Instability 8.2 Formulation of Magnetohydrodynamic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 a Linearization of Magnetohydrodynamic Equations b Energy Principle 8.3 Instabilities of a Cylindrical Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 a Instabilities of Sharp-Boundary Con¯guration: Kruskal-Shafranov Condition b Instabilities of Di®use-Boundary Con¯gurations c Suydam's Criterion d Tokamak Con¯guration e Reversed Field Pinch 8.4 Hain-LÄ ust Magnetohydrodynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.5 Ballooning Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.6 ´i Mode due to Density and Temperature Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 9

Resistive Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.1 Tearing Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.2 Resistive Drift Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 10 10.1 10.2

10.3 10.4

Plasma as Medium of Electromagnetic Wave Propagation . . . . . . . . . . . . . . . . . . . 116 Dispersion Equation of Waves in a Cold Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Properties of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 a Polarization and Particle Motion b Cuto® and Resonance Waves in a Two-Components Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Various Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 a Alfv¶en Wave b Ion Cyclotron Wave and Fast Wave c Lower Hybrid Resonance d Upper Hybrid Resonance

Contents

10.5 11 11.1 11.2 11.3 11.4 12 12.1 12.2 12.3 12.4 12.5 12.6 13 13.1 13.2 13.3 13.4

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e Electron Cyclotron Wave Conditions for Electrostatic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Landau Damping and Cyclotron Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Landau Damping (Ampli¯cation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Transit-Time Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 Cyclotron Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Quasi-Linear Theory of Evolution in the Distribution Function . . . . . . . . . . . . . . . . . . . . . 136 Wave Propagation and Wave Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Dielectric Tensor of Hot Plasma, Wave Absorption and Heating . . . . . . . . . . . . . . . . . . . . 144 Wave Heating in Ion Cyclotron Range of Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Lower Hybrid Wave Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Electron Cyclotron Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Velocity Space Instabilities (Electrostatic Waves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157 Dispersion Equation of Electrostatic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Two Streams Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Electron Beam Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Harris Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

14 Instabilities driven by Energetic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 14.1 Fishbone Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 a Formulation b MHD Potential Energy c Kinetic Integral of Hot Component d Growth Rate of Fishbone Instability 14.2 Toroidal Alfv¶en Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 a Toroidicity Inuced Alfv¶en Eigenmode b Instabilities of TAE Driven by Energetic Particles c Various Alfv¶en Modes 15

Development of Fusion Researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

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Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

16.1 16.2

Tokamak Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 16.4 Beta Limit of Elongated Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 16.5 Impurity Control, Scrape-O® Layer and Divertor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 16.6 Con¯nement Scaling of L Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 16.7 H Mode and Improved Con¯nement Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 16.8 Noninductive Current Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 a Lower Hybrid Current Drive b Electron Cyclotron Current Drive c Neutral Beam Current Drive d Bootstrap Current 16.9 Neoclassical Tearing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 16.10 Resistive Wall Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

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a Growth Rate of Resistive Wall Mode b Feedback Stabilization of Resistive Wall Mode 16.11 Parameters of Tokamak Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 17 17.1

17.2

17.3

18 18.1 18.2

Non-Tokamak Con¯nement System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Reversed Field Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 a Reversed Field Pinch Con¯guration b MHD Relaxation c Con¯nement d Oscillating Field Current Drive Stellarator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 a Helical Field b Stellarator Devices c Neoclassical Di®usion in Helical Field d Con¯nement of Stellarator Plasmas Open End Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251 a Con¯nement Times in Mirrors and Cusps b Con¯nement Experiments with Mirrors c Instabilities in Mirror Systems d Tandem Mirrors Inertial Con¯nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Pellet Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Implosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Appendix A

Derivation of MHD Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

B Energy Integral of Axisymmetric Toroidal System . . . . . . . . . . . . . . . . . . . . . . . . . . . .271 B.1 Energy Integral in Illuminating Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 B.2 Energy Integral of Axisymmetric Toroidal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 B.3 Energy Integral of High n Ballooning Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 C Derivation of Dielectric Tensor in Hot Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 C.1 Formulation of Dispersion Relation in Hot Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 C.2 Solution of Linearized Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 C.3 Dielectric Tensor of Hot Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 C.4 Dielectric Tensor of bi-Maxwellian Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 C.5 Dispersion Relation of Electrostatic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 C.6 Dispersion Relation of Electrostatic Wave in Inhomogenous Plasma . . . . . . . . . . . . . . . . 287 Physical Constants, Plasma Parameters and Mathematical Formula . . . . . . . . . . . . . 291 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

Preface

v

Preface

Primary objective of this lecture note is to provide a basic text for the students to study plasma physics and controlled fusion researches. Secondary objective is to o®er a reference book describing analytical methods of plasma physics for the researchers. This was written based on lecture notes for a graduate course and an advanced undergraduate course those have been o®ered at Department of Physics, Faculty of Science, University of Tokyo. In ch.1 and 2, basic concept of plasma and its characteristics are explained. In ch.3, orbits of ion and electron are described in several magnetic ¯eld con¯gurations. Chapter 4 formulates Boltzmann equation of velocity space distribution function, which is the basic relation of plasma physics. From ch.5 to ch.9, plasmas are described as magnetohydrodynamic (MHD) °uid. MHD equation of motion (ch.5), equilibrium (ch.6) and di®usion and con¯nement time of plasma (ch.7) are described by the °uid model. Chapters 8 and 9 discuss problems of MHD instabilities whether a small perturbation will grow to disrupt the plasma or will damp to a stable state. The basic MHD equation of motion can be derived by taking an appropriate average of Boltzmann equation. This mathematical process is described in appendix A. The derivation of useful energy integral formula of axisymmetric toroidal system and the analysis of high n ballooning mode are described in appendix B. From ch.10 to ch.14, plasmas are treated by kinetic theory. This medium, in which waves and perturbations propagate, is generally inhomogeneous and anisotropic. It may absorb or even amplify the wave. Cold plasma model described in ch.10 is applicable when the thermal velocity of plasma particles is much smaller than the phase velocity of wave. Because of its simplicity, the dielectric tensor of cold plasma can be easily derived and the properties of various wave can be discussed in the case of cold plasma. If the refractive index becomes large and the phase velocity of the wave becomes comparable to the thermal velocity of the plasma particles, then the particles and the wave interact with each other. In ch.11, Landau damping, which is the most characteristic collective phenomenon of plasma, as well as cyclotron damping are described. Chapter 12 discusses wave heating (wave absorption) in hot plasma, in which the thermal velocity of particles is comparable to the wave phase velocity, by use of the dielectric tensor of hot plasma. In ch.13 the ampli¯cation of wave, that is, the growth of perturbation and instabilities, is described. Since long mathematical process is necessary for the derivation of dielectric tensor of hot plasma, its processes are described in appendix C. In ch.14 instabilities driven by energetc particles, that is, ¯shbone instability and toroidal Alfv¶en eigenmodes are described. In ch.15, con¯nement researches toward fusion grade plasmas are reviewed. During the last decade, tokamak experiments have made remarkable progresses. Now realistic designs of tokamak reactors have been actively pursued. In ch.16, research works of critical subjects on tokamak plasmas and reactors are explained. As non-tokamak con¯nement systems, reversed ¯eld pinch, stellarator, tandem mirror are described in ch.16. Elementary introduction of inertial con¯nement is added in ch.17. Readers may have impression that there is too much mathematics in this lecture note. However there is a reason for that. If a graduate student tries to read and understand, for examples, three of frequently cited short papers on the analysis of high n ballooning mode by Connor, Hastie, Taylor, ¯shbone instability by L.Chen, White, Rosenbluth, toroidal Alfven eigenmode by Betti, Freidberg without preparative knowledge, he must read and understand several tens of cited references and references of references. I would guess from my experience that he would be obliged to work hard for several months. It is one of motivation to write this lecture note to save his time to struggle with mathematical derivation so that he could spend more time to think physics and experimental results.

vi

This lecture note has been attempted to present the basic physics and analytical methods which are necessary for understanding and predicting plasma behavior and to provide the recent status of fusion researches for graduate and senior undergraduate students. I also hope that it will be a useful reference for scientists and engineers working in the relevant ¯elds. October 2000 Kenro Miyamoto Professor Emeritus

Unversity of Tokyo

[email protected]

The pdf ¯le of this lecture note is being prepared in Research Reports of http://www.nifs.ac.jp.

vii

viii

1.2 Charge Neutrality and Landau Damping

1

Ch.1 Nature of Plasma 1.1 Introduction As the temperature of a material is raised, its state changes from solid to liquid and then to gas. If the temperature is elevated further, an appreciable number of the gas atoms are ionized and become the high temperature gaseous state in which the charge numbers of ions and electrons are almost the same and charge neutrality is satis¯ed in a macroscopic scale. When the ions and electrons move collectively, these charged particles interact with coulomb force which is long range force and decays only in inverse square of the distance r between the charged particles. The resultant current °ows due to the motion of the charged particles and Lorentz interaction takes place. Therefore many charged particles interact with each other by long range forces and various collective movements occur in the gaseous state. The typical cases are many kinds of instabilities and wave phenomena. The word \plasma" is used in physics to designate the high temperature ionized gaseous state with charge neutrality and collective interaction between the charged particles and waves. When the temperature of a gas is T (K), the average velocity of the thermal motion, that is, thermal velocity vT is given by 2 mvT =2 = ∙T =2

(1:1)

where ∙ is Boltzmann constant ∙ = 1:380658(12) £ 10¡23 J/K and ∙T indicates the thermal energy. Therefore the unit of ∙T is Joule (J) in MKSA unit. In many ¯elds of physics, one electron volt (eV) is frequently used as a unit of energy. This is the energy necessary to move an electron, charge e = 1:60217733(49) £ 10¡19 Coulomb, against a potential di®erence of 1 volt: 1eV = 1:60217733(49) £ 10¡19 J: The temperature corresponding to the thermal energy of 1eV is 1.16£104 K(= e=∙). The ionization energy of hydrogen atom is 13.6 eV. Even if the thermal energy (average energy) of hydrogen gas is 1 eV, that is T » 104 K, small amount of electrons with energy higher than 13.6 eV exist and ionize the gas to a hydrogen plasma. Plasmas are found in nature in various forms (see ¯g.1.1). There exits the ionosphere in the heights of 70»500 km (density n » 1012 m¡3 ; ∙T » 0:2 eV). Solar wind is the plasma °ow originated from the sun with n » 106»7 m¡3 ; ∙T » 10 eV. Corona extends around the sun and the density is » 1014 m¡3 and the electron temperature is » 100 eV although these values depend on the di®erent positions. White dwarf, the ¯nal state of stellar evolution, has the electron density of 1035»36 m¡3 . Various plasma domains in the diagram of electron density n(m¡3 ) and electron temperature ∙T (eV) are shown in ¯g.1.1. Active researches in plasma physics have been motivated by the aim to create and con¯ne hot plasmas in fusion researches. Plasmas play important roles in the studies of pulsars radiating microwave or solar X ray sources observed in space physics and astrophysics. The other application of plasma physics is the study of the earth's environment in space. Practical applications of plasma physics are MHD (magnetohydrodynamic) energy conversion for electric power generation, ion rocket engines for space crafts, and plasma processing which attracts much attention recently. 1.2 Charge Neutrality and Landau Damping One of the fundamental property of plasma is the shielding of the electric potential applied to

2

1 Nature of Plasma

Fig.1.1 Various plasma domain in n - ∙T diagram.

the plasma. When a probe is inserted into a plasma and positive (negative) potential is applied, the probe attracts (repulses) electrons and the plasma tends to shield the electric disturbance. Let us estimate the shielding length. Assume that the ions are in uniform density (ni = n0 ) and there is small perturbation in electron density ne or potential Á. Since the electrons are in Boltzmann distribution usually, the electron density ne becomes ne = n0 exp(eÁ=∙Te ) ' n0 (1 + eÁ=∙Te ): Poisson's eqation is r(²0 E) = ¡²0 r2 Á = ½ = ¡e(ne ¡ n0 ) = ¡

E = ¡rÁ;

e 2 n0 Á ∙Te

and Á r Á= 2 ; ¸D 2

¸D =

µ

²0 ∙Te ne e2

¶1=2

3

= 7:45 £ 10

µ

1 ∙Te ne e

¶1=2

(m)

(1:2)

where ne is in m¡3 and ∙Te =e is in eV. When ne » 1020 cm¡3 ; ∙Te =e » 10keV, then ¸D » 75¹m. In spherically symmetric case, Laplacian r2 becomes r2 Á = (1=r)(@=@r)(r@Á=@r) and the solution is Á=

q exp(¡r=¸D ) : 4¼²0 r

It is clear from the foregoing formula that Coulomb potential q=4¼²0 r of point charge is shielded out to a distance ¸D . This distance ¸D is called the Debye length. When the plasma size is a and a À ¸D is satis¯ed, then plasma is considered neutral in charge. If a < ¸D in contrary, individual particle is not shielded electrostatically and this state is no longer plasma but an assembly of independent charged particles. The number of electrons included in the sphere of radius ¸D is called plasma parameter and is given by 3

n¸D =

µ

²0 ∙Te e e

¶3=2

1 1=2 ne

:

(1:3)

1.3 Fusion Core Plasma

3

When the density is increased while keeping the temperature constant, this value becomes small. If the plasma parameter is less than say »1, the concept of Debye shielding is not applicable since the continuity of charge density breaks down in the scale of Debye length. Plasmas in the region of n¸3 > 1 are called classical plasma or weakly coupled plasma, since the ratio of electron thermal energy ∙Te and coulomb energy between electrons Ecoulomb = e2 =4¼²0 d (d ' n¡1=3 is the average distance between electrons with the density n) is given by ∙Te Ecoulomb

= 4¼(n¸3D )2=3

(1:4)

and n¸3 > 1 means that coulomb energy is smaller than the thermal energy. The case of n¸3D < 1 is called strongly coupled plasma (see ¯g.1.1). Fermi energy of degenerated electron gas is given by ²F = (h2 =2me )(3¼ 2 n)2=3 . When the density becomes very high, it is possible to become ²F ¸ ∙Te . In this case quantum e®ect is more dominant than thermal e®ect. This case is called degenerated electron plasma. One of this example is the electron plasma in metal. Most of plasmas in experiments are classical weakly coupled plasma. The other fundamental process of plasma is collective phenomena of charged particles. Waves are associated with coherent motions of charged particles. When the phase velocity vph of wave or perturbation is much larger than the thermal velocity vT of charged particles, the wave propagates through the plasma media without damping or ampli¯cation. However when the refractive index N of plasma media becomes large and plasma becomes hot, the phase velocity vph = c=N (c is light velocity) of the wave and the thermal velocity vT become comparable (vph = c=N » vT ), then the exchange of energy between the wave and the thermal energy of plasma is possible. The existence of a damping mechanism of wave was found by L.D. Landau. The process of Landau damping involves a direct wave-particle interaction in collisionless plasma without necessity of randamizing collision. This process is fundamental mechanism in wave heatings of plasma (wave damping) and instabilities (inverse damping of perturbations). Landau damping will be described in ch.11, ch.12 and appendix B.

1.3 Fusion Core Plasma Progress in plasma physics has been motivated by how to realize fusion core plasma. Necessary condition for fusion core plasma is discussed in this section. Nuclear fusion reactions are the fused reactions of light nuclides to heavier one. When the sum of the masses of nuclides after a nuclear fusion is smaller than the sum before the reaction by ¢m, we call it mass defect. According to theory of relativity, amount of energy (¢m)c2 (c is light speed) is released by the nuclear fusion. Nuclear reactions of interest for fusion reactors are as follows (D; deuteron, T; triton, He3 ; helium3, Li; lithium): (1) (2) (3) (4) (5) (6)

D+D!T(1.01 MeV)+p(3.03 MeV) D+D! He3 (0.82 MeV)+n(2.45 MeV) T+D! He4 (3.52 MeV)+n(14.06 MeV) D+He3 ! He4 (3.67 MeV) +p(14.67 MeV) Li6 +n!T+He4 +4.8 MeV Li7 +n(2.5 MeV)!T+He4 +n

where p and n are proton (hydrogen ion) and neutron respectively (1 MV=106 eV). Since the energy released by chemical reaction of H2 + (1=2)O2 ! H2 O is 2.96 eV, fusion energy released is about million times as large as chemical one. A binding energy per nucleon is smaller in very light or very heavy nuclides and largest in the nuclides with atomic mass numbers around 60. Therefore, large amount of the energy can be released when the light nuclides are fused.

4

1 Nature of Plasma

Fig.1.2 (a) The dependence of fusion cross section ¾ on the kinetic energy E of colliding nucleous. ¾DD is the sum of the cross sections of D-D reactions (1) (2). 1 barn=10¡24 cm2 . (b) The dependence of fusion rate h¾vi on the ion temperature Ti .

Deuterium exists aboundantly in nature; for example, it comprises 0.015 atom percent of the hydrogen in sea water with the volume of about 1:35 £ 109 km3 . Although fusion energy was released in an explosive manner by the hydrogen bomb in 1951, controlled fusion is still in the stage of research development. Nuclear fusion reactions were found in 1920's. When proton or deuteron beams collide with target of light nuclide, beam loses its energy by the ionization or elastic collisions with target nuclides and the probability of nuclear fusion is negligible. Nuclear fusion researches have been most actively pursued by use of hot plasma. In fully ionized hydrogen, deuterium and tritium plasmas, the process of ionization does not occur. If the plasma is con¯ned in some speci¯ed region adiabatically, the average energy does not decrease by the processes of elastic collisions. Therefore if the very hot D-T plasmas or D-D plasmas are con¯ned, the ions have velocities large enough to overcome their mutual coulomb repulsion, so that collision and fusion take place. Let us consider the nuclear reaction that D collides with T. The e®ective cross section of T nucleous is denoted by ¾. This cross section is a function of the kinetic energy E of D. The cross section of D-T reaction at E = 100 keV is 5 £ 10¡24 cm2 . The cross sections ¾ of D-T, DD, D-He3 reaction versus the kinetic energy of colliding nucleous are shown in ¯g.1.2(a).1;2 The probability of fusion reaction per unit time in the case that a D ion with the velocity v collides with T ions with the density of nT is given by nT ¾v (we will discuss the collision probability in more details in sec.2.7). When a plasma is Maxwellian with the ion temperature of Ti , it is necessary to calculate the average value h¾vi of ¾v over the velocity space. The dependence of h¾vi on ion temperature Ti is shown in ¯g.1.2(b).3 A ¯tting equation of h¾vi of D-T reaction as a function of ∙T in unit of keV is4 h¾vi(m

¡3

µ

¶

20 3:7 £ 10¡18 )= exp ¡ ; 2=3 H(∙T ) £ (∙T ) (∙T )1=3

H(∙T ) ´

∙T 5:45 + 37 3 + ∙T (1 + ∙T =37:5)2:8 (1:5)

Figure 1.3 shows an example of electric power plant based on D-T fusion reactor. Fast neutrons produced in fusion core plasma penetrate the ¯rst wall and a lithium blanket surrounding the plasma moderates the fast neutrons, converting their kinetic energy to heat. Furthermore the lithium blanket breeds tritium due to reaction (5),(6). Lithium blanket gives up its heat to generate the steam by a heat exchanger; steam turbine generates electric power. A part of

1.3 Fusion Core Plasma

5

Fig.1.3 An electric power plant based on a D-T fusion reactor

the generated electric power is used to operate heating system of plasma to compensate the energy losses from the plasma to keep the plasma hot. The fusion output power must be larger than the necessary heating input power taking account the conversion e±ciency. Since the necessary heating input power is equal to the energy loss rate of fusion core plasma, good energy con¯nement of hot plasma is key issue. The thermal energy of plasma per unit volume is given by (3=2)n∙(Ti + Te ). This thermal energy is lost by thermal conduction and convective losses. The notation PL denotes these energy losses of the plasma per unit volume per unit time (power loss per unit volume). There is radiation loss R due to bremsstrahlung of electrons and impurity ion radiation in addition to PL . The total energy con¯nement time ¿E is de¯ned by ¿E ´

3n∙T (3=2)n∙(Te + Ti ) ' : PL + R PL + R

(1:6)

The necessary heating input power Pheat is equal to PL + R . In the case of D-T reaction, the sum of kinetic energies Q® = 3:52 MeV of ® particle (He4 ion) and Qn = 14:06 MeV of neutron is QNF =17.58 MeV per 1 reaction. Since the densities of D ions and T ions of equally mixed plasma are n=2 , number of D-T reaction per unit time per unit volume is (n=2)(n=2)h¾vi, so that fusion output power per unit volume PNF is given by PNF = (n=2)(n=2)h¾viQNF :

(1:7)

Denote the thermal-to-electric conversion e±ciency by ´el and heating e±ciency (ratio of the deposit power into the plasma to the electric input power of heating device) by ´heat . Then the condition of power generation is Pheat = PL + R =

3n∙T < (´el )(´heat )PNF ¿E

(1:8)

that is 3n∙T QNF 2 < (´heat )(´el ) n h¾vi; ¿E 4 12∙T n¿E > ´QNF h¾vi

(1:9)

where ´ is the product of two e±ciencies. The right-hand side of the last foregoing equation is the function of temperature T only. When ∙T = 104 eV and ´ » 0:3 (´el » 0:4; ´heat » 0:75), the necessary condition is n¿E > 1:7 £ 1020 m¡3 ¢ sec. The condition of D-T fusion plasma in the case of ´ » 0:3 is shown in ¯g.1.4. In reality the plasma is hot in the core and is cold in the edge.

6

1 Nature of Plasma

Fig.1.4 Condition of D-T fusion core plasma in n¿E - T diagram in the case of ´ = 0:3, critical condition (´ = 1) and ignition condition (´ = 0:2).

For the more accurate discussion, we must take account of the pro¯le e®ect of temperature and density and will be analyzed in sec.16.11. The condition Pheat = PNF is called break even condition. This corresponds to the case of ´ = 1 in the condition of fusion core plasma. The ratio of the fusion output power due to ® particles to the total is Q® =QNF = 0:2. Since ® particles are charged particles, ® particles can heat the plasma by coulomb collision (see sec.2.8). If the total kinetic energy (output energy) of ® particles contributes to heat the plasma, the condition Pheat = 0:2PNF can sustain the necessary high temperature of the plasma without heating from outside. This condition is called ignition condition, which corresponds the case of ´ = 0:2.

References 1. W. R. Arnold, J. A. Phillips, G. A. Sawyer, E. J. Stovall, Jr. and J. C. Tuck: Phys. Rev. 93, 483 (1954) 2. C. F. Wandel, T. Hesselberg Jensen and O. Kofoed-Hansen: Nucl. Instr. and Methods 4, 249 (1959) 3. J. L. Tuck: Nucl. Fusion 1, 201 (1961) 4. T. Takizuka and M. Yamagiwa: JAERI-M 87-066 (1987) Japan Atomic Energy Research Institute

2.1 Velocity Space Distribution Function, Electron and ¢ ¢ ¢

7

Ch.2 Plasma Characteristics 2.1 Velocity Space Distribution Function, Electron and Ion Temperatures Electrons as well as ions in a plasma move with various velocities. The number of electrons in a unit volume is the electron density ne and the number of electrons dne (vx ) with the x component of velocity between vx and vx + dvx is given by dne (vx ) = fe (vx )dvx : Then fe (vx ) is called electron's velocity space distribution function. When electrons are in thermally equilibrium state with the electron temperature Te , the velocity space distribution function becomes following Maxwell distribution: fe (vx ) = ne

µ

¯ 2¼

¶1=2

Ã

¯v 2 exp ¡ x 2

!

;

¯=

me : ∙Te

By the de¯nition the velocity space distribution function satis¯es following relation: Z

1

¡1

fe (vx )dvx = ne :

Maxwell distribution function in three dimensional velocity space is given by fe (vx ; vy ; vz ) = ne

µ

me 2¼∙Te

¶3=2

Ã

!

me (vx2 + vy2 + vz2 ) exp ¡ : 2∙Te

(2:1)

Ion distribution function is also de¯ned by the same way as the electron's case. The mean square of velocity vx2 is given by 2 = vT

1 n

Z

1

¡1

vx2 f (vx )dvx =

∙T : m

(2:2)

The pressure p is p = n∙T: Particle °ux in the x direction per unit area ¡+;x is given by ¡+;x =

Z

1

0

µ

∙T vx f (vx )dvx = n 2¼m

¶1=2

:

When an electron beam with the average velocity vb is injected into a plasma with a Maxwell distribution, the distribution function becomes humped pro¯le as is shown in ¯g.2.1(b). Following expression can be used for the modeling of the distribution function of a plasma with an electron beam: fe (vz ) = ne

µ

me 2¼∙Te

¶1=2

Ã

me vz2 exp ¡ 2∙Te

!

+ nb

µ

me 2¼∙Tb

¶1=2

Ã

me (vz ¡ vb )2 exp ¡ 2∙Tb

!

:

8

2 Plasma Characteristics

Fig.2.1 (a) Velocity space distribution function of Maxwellian with electron temperature Te . (b) velocity space distribution function of Maxwellian plasma with electron temterature Te and injected electron beam with the average velocity vb .

2.2 Plasma Frequency, Debye Length Let us consider the case where a small perturbation occurs in a uniform plasma and the electrons in the plasma move by the perturbation. It is assumed that ions do not move because the ion's mass is much more heavy than electron's. Due to the displacement of electrons, electric charges appear and an electric ¯eld is induced. The electric ¯eld is given by Poisson's equation: ²0 r ¢ E = ¡e(ne ¡ n0 ): Electrons are accelerated by the electric ¯eld: me

dv = ¡eE: dt

Due to the movement of electrons, the electron density changes: @ne + r ¢ (ne v) = 0: @t Denote ne ¡ n0 = n1 and assume jn1 j ¿ n0 , then we ¯nd ²0 r ¢ E = ¡en1 ;

me

@v = ¡eE; @t

@n1 + n0 r ¢ v = 0: @t

For simplicity the displacement is assumed only in the x direction and is sinusoidal: n1 (x; t) = n1 exp(ikx ¡ i!t): Time di®erential @=@t is replaced by ¡i! and @=@x is replaced by ik, then ik²0 E = ¡en1 ;

¡ i!me v = ¡eE;

¡ i!n1 = ¡ikn0 v

so that we ¯nd !2 =

n0 e2 : ² 0 me

(2:3)

This wave is called electron plasma wave or Langmuir wave and its frequency is called electron plasma frequency ¦e : ¦e =

Ã

ne e2 ² 0 me

!1=2

= 5:64 £ 10

11

µ

ne 1020

¶1=2

rad=sec:

2.2 Plasma Frequency, Debye Length

9

Fig.2.2 Larmor motion of charged particle in magnetic ¯eld

There is following relation between the plasma frequency and Debye length ¸D : ¸D ¦e =

µ

∙Te me

¶1=2

= vTe :

2.3 Cyclotron Frequency, Larmor Radius The equation of motion of charged particle with the mass m and the charge q in an electric and magnetic ¯eld E; B is given by m

dv = q(E + v £ B): dt

(2:4)

When the magnetic ¯eld is homogenous and is in the z direction and the electric ¯eld is zero, the equation of motion becomes v_ = (qB=m)(v £ b) (b = B=B) and vx = ¡v? sin(­t + ±); vy = v? cos(­t + ±); vz = vz0 ; ­=¡

qB : m

(2:5)

The solution of these equation is a spiral motion around the magnetic line of force with the angular velocity of ­ (see ¯g.2.2). This motion is called Larmor motion. The angular frequency ­ is called cyclotron (angular) frequency. Denote the radius of the orbit by ½­ , then the 2 centrifugal force is mv? =½­ and Lorentz force is qv? B. Since both forces must be balanced, we ¯nd mv? : (2:6) ½­ = jqjB This radius is called Larmor radius. The center of Larmor motion is called guiding center. Electron's Larmor motion is right-hand sence (­e > 0), and ion's Larmor motion is left-hand sence (­i < 0) (see ¯g.2.2). When B = 1 T, ∙T = 100 eV, the values of Larmor radius and cyclotron freqency are given in the following table:

10

2 Plasma Characteristics

Fig.2.3 Drift motion of guiding center in electric and gravitational ¯eld (conceptional drawing).

Fig.2.4 Radius of curvature of line of magnetic force B=1T, ∙T =100eV thermal velocity vT = (∙T =m)1=2 Larmor radius ½­ (angular) cyclotron frequency ­ cyclotron freqeuncy ­=2¼

electron 4:2 £ 106 m/s 23:8 ¹m 1:76 £ 1011 /s 28 GHz

  proton

9:8 £ 104 m/s 10.2 mm ¡9:58 £ 107 /s ¡15:2 MHz

2.4 Drift Velocity of Guiding Center When a uniform electric ¯eld E perpendicular to the uniform magnetic ¯eld is superposed, the equation of motion is reduced to m

du = q(u £ B) dt

by use of v = uE + u;

uE =

E£b : B

(2:7)

Therefore the motion of charged particle is superposition of Larmor motion and drift motion uE of its guiding center. The direction of guiding center drift by E is the same for both ion and electron (¯g.2.3). When a gravitational ¯eld g is superposed, the force is mg, which corresponds to qE in the case of electric ¯eld. Therefore the drift velocity of the guiding center due to the gravitation is given by ug =

m g£b (g £ b) = ¡ : qB ­

(2:8)

The directions of ion's drift and electron's drift due to the gravitation are opposite with each other and the drift velocity of ion guiding center is much larger than electron's one (see ¯g.2.3). When the magnetic and electric ¯elds change slowly and gradually in time and in space (j!=­j ¿ 1; ½­ =R ¿ 1), the formulas of drift velocity are valid as they are. However because of the curvature of ¯eld line of magnetic force, centrifugal force acts on the particle which runs along a ¯eld line with the velocity of vk . The acceleration of centrifugal force is g curv =

vk2 R

n

2.4 Drift Velocity of Guiding Center

11

where R is the radius of curvature of ¯eld line and n is the unit vector with the direction from the center of the curvature to the ¯eld line (¯g.2.4). Furthermore, as is described later, the resultant e®ect of Larmor motion in an inhomogeneous magnetic ¯eld is reduced to the acceleration of g rB = ¡

2 =2 v? rB: B

Therefore drift velocity of the guiding center due to inhomogenous curved magnetic ¯eld is given by the drift approximation as follows: 1 ug = ¡ ­

Ã

vk2

v 2 rB n¡ ? R 2 B

!

£ b:

(2:9)

The ¯rst term is called curvature drift and the second term is called rB drift. Since r£B = ¹0 j, the vector formula reduces @ 1 r(B ¢ B) = (b ¢ r)B + b £ (r £ B) = (Bb) + b £ ¹0 j 2B @l =

rp rp @B @b @B n ¡ ¹0 b+B = b ¡ B ¡ ¹0 : @l @l B @l R B

We used the following relation (see ¯g.2.4) n @b =¡ : @l R Then we have µ

rB rp n£b =¡ + ¹0 2 R B B

¶

£ b:

If rp is much smaller than rB 2 =(2¹0 ), we ¯nd ug = ¡

2 2 1 vk + v? =2 (n £ b): ­ R

The parallel motion along the magnetic ¯eld is given by m

2 dvk mv? =2 rk B = qEk + mgk ¡ dt B

where l is the length along the ¯eld line. Let us consider the e®ect of inhomogeneity of magnetic ¯eld on gyrating charged particle. The x component of Lorentz force F L = qv £ B perpendicular to the magnetic ¯eld (z direction) and the magnitude B of the magnetic ¯eld near the guiding center are FLx = qvy B = ¡jqjv? cos µB B = B0 +

@B @B ½­ cos µ + ½­ sin µ: @x @y

The time average of x component of Lorentz force is given by hFLx i = 12 (@B=@x)(¡jqj)v? ½­ and the y component is also given by the same way, and we ¯nd (see ¯g.2.5) hF L i? = ¡

2 =2 mv? r? B: B

12

2 Plasma Characteristics

Fig.2.5 Larmor motion in inhomogeneous magnetic ¯eld.

Next it is necessary to estimate the time average of z component of Lorentz force. The equation r ¢ B = 0 near the guiding center in ¯g.2.5 becomes Br =r + @Br =@r + @Bz =@z = 0 and we ¯nd @Br mv 2 =2 @B =¡ ? ; @r B @z since r is very small. Thus the necessary expression of g rB is derived. hFLz i = ¡hqvµ Br i = jqjv? ½­

2.5 Magnetic Moment, Mirror Con¯nement, Longitudinal Adiabatic Constant A current loop with the current I encircling the area S has the magnetic moment of ¹m = IS. Since the current and encircling area of gyrating Larmor motion are I = q­=2¼; S = ¼½2­ respectively, it has the magnetic moment of 2 q­ 2 mv? ¼½­ = : (2:10) 2¼ 2B This physical quantity is adiabatically invariant as is shown later in this section. When the magnetic ¯eld changes slowly, the magnetic moment is conserved. Therefore if B is increased, 2 mv? = ¹m B is also increased and the particles are heated. This kind of heating is called adiabatic heating. Let us consider a mirror ¯eld as is shown in ¯g.2.6, in which magnetic ¯eld is weak at the center and is strong at both ends of mirror ¯eld. For simplicity the electric ¯eld is assumed to be zero. Since Lorentz force is perpendicular to the velocity, the magnetic ¯eld does not contribute the change of kinetic energy and

¹m =

mvk2

2 mv? mv 2 = = E = const: 2 2 2 Since the magnetic moment is conserved, we ¯nd

+

µ

2 2 vk = § E ¡ v? m

¶1=2

µ

2 = § v ¡ ¹m B m 2

(2:11)

¶1=2

:

When the particle moves toward the open ends, the magnetic ¯eld becomes large and vk becomes small or even zero. Since the force along the parallel direction to the magnetic ¯eld is ¡¹m rk B, the both ends of the mirror ¯eld repulse charged particles as a mirror re°ects light. The ratio of magnitude of magnetic ¯eld at open end to the central value is called mirror ratio: RM =

BM : B0

Let us denote the parallel and perpendicular components of the velocity at the mirror center 2 at the position of maximum magnetic ¯eld BM is by vk0 and v?0 respectively. The value v? given by 2 = v?M

BM 2 v : B0 ?0

2.5 Magnetic Moment, Mirror Con¯nement, Longitudinal ¢ ¢ ¢

13

Fig.2.6 Mirror ¯eld and loss cone in vk - v? space.

If this value is larger than v 2 = v02 , this particle can not pass through the open end, so that the particle satisfying the following condition is re°ected and is trapped in the mirror ¯eld: µ

v?0 v0

¶2

>

B0 1 = : BM RM

(2:12)

Particles in the region where sin µ ´ v?0 =v0 satis¯es sin2 µ ∙

1 RM

are not trapped and the region is called loss cone in vk - v? space (see Fig.2.6). Let us check the invariance of ¹m in the presence of a slowly changing magnetic ¯eld (j@B=@tj ¿ j­Bj). Scalar product of v ? and the equation of motion is dv? d = mv ? ¢ dt dt

Ã

2 mv? 2

!

= q(v ? ¢ E ? ):

2 =2 During one period 2¼=j­j of Larmor motion, the change ¢W? of the kinetic energy W? = mv? is

¢W? = q H

Z

(v ? ¢ E ? )dt = q

I

E ? ¢ ds = q

Z

(r £ E ¢ n)dS

R

where ds is the closed line integral along Larmor orbit and dS is surface integral over the encircled area of Larmor orbit. Since r £ E = ¡@B=@t, ¢W? is Z

¢W? = ¡q

@B @B ¢ ndS = jqj¼½2­ : @t @t

The change of magnetic ¯eld ¢B during one period of Larmor motion is ¢B = (@B=@t)(2¼=j­j), we ¯nd ¢W? =

2 ¢B mv? ¢B = W? 2 B B

and ¹m =

W? = const: B

H

When a system is periodic in time, the action integral pdq, in terms of the canonical variables p; q, is an adiabatic invariant in general. The action integral of Larmor motion is J? = (¡m½­ ­)2¼½­ = ¡(4¼m=q)¹m . J? is called transversal adiabatic invariant. A particle trapped in a mirror ¯eld moves back and forth along the ¯eld line between both ends. The second action integral of this periodic motion Jk = m

I

vk dl

(2:13)

14

2 Plasma Characteristics

Fig.2.7 Probability of collision of a sphere a with spheres b.

is also another adiabatic invariant. Jk is called longitudinal adiabatic invariant. As one makes the mirror length l shorter, hvk i increases (for Jk = 2mhvk il is conserved), and the particles are accelerated. This phenomena is called Fermi acceleration. The line of magnetic force of mirror is convex toward outside. The particles trapped by the mirror are subjected to curvature drift and gradient B drift, so that the trapped particles move back and forth, while drifting in µ direction. The orbit (r; µ) of the crossing point at z = 0 plane of back and forth movement is given by Jk (r; µ; ¹m ; E) = const: 2.6 Coulomb Collision Time, Fast Neutral Beam Injection The motions of charged particles were analyzed in the previous section without considering the e®ects of collisions between particles. In this section, phenomena associated with Coulomb collisions will be discussed. Let us start from a simple model. Assume that a sphere with the radius a moves with the velocity v in the region where spheres with the radius b are ¯lled with the number density n (see ¯g.2.7). When the distance between the two particles becomes less than a + b, collision takes place. The cross section ¾ of this collision is ¾ = ¼(a + b)2 . Since the sphere a moves by the distance l = v±t during ±t, the probability of collision with the sphere b is nl¾ = n¾v±t since nl is the possible number of the sphere b, with which the sphere a within a unit area of incidence may collides, and nl¾ is the total cross section per unit area of incidence during the period of ±t. Therefore the inverse of collision time tcoll is (tcoll )¡1 = n¾v: In this simple case the cross section ¾ of the collision is independent of the velocity of the incident sphere a. However the cross section is dependent on the incident velocity in general. Let us consider strong Coulomb collision of an incident electron with ions (see ¯g.2.8) in which the electron is de°ected strongly after the collision. Such a collision can take place when

Fig.2.8 Coulomb collision of electron with ion.

2.6 Coulomb Collision Time, Fast Neutral Beam Injection

15

the magnitude of electrostatic potential of the electron at the closest distance b is the order of the kinetic energy of incident electron, that is, Ze2 me ve2 = : 4¼²0 b 2 The cross section of the strong Coulomb collision is ¾ = ¼b2 . The inverse of the collision time of the strong Coulomb collision is 1 tcoll

= ni ¾ve = ni ve ¼b2 =

ni ¼(Ze2 )2 ve Z 2 e4 ni = : (4¼²0 me ve2 =2)2 4¼²0 m2e ve3

Since Coulomb force is long range interaction, a test particle is de°ected by small angle even by a distant ¯eld particle, which the test particle does not become very close to. As is described in sec.1.2, the Coulomb ¯eld of a ¯eld particle is not shielded inside the Debye sphere with the radius of Debye length ¸D and there are many ¯eld particles inside the Debye sphere in the usual laboratory plasmas (weakly coupled plasmas). Accumulation of many collisions with small angle de°ection results in large e®ect. When the e®ect of the small angle de°ection is taken into account, the total Coulomb cross section increases by the factor of Coulomb logarithm ln ¤ ' ln

µ

2¸D b

¶

'

Z

¸D

b=2

1 dr ' 15 » 20: r

The time derivative of the momentum pk parallel to the incident direction of the electron is given by use of the collision time ¿ei k as follows:1;2 dpk pk ; =¡ dt ¿ei k 1 ¿ei k

=

Z 2 e4 ni ln ¤ 4¼²20 m2e ve3

(2:14)

where ¿ei k indicates the deceleration time of an electron by ions. When a test particle with the charge q, the mass m and the velocity v collides with the ¯eld ¤ particles with the charge q ¤ , the mass m¤ and the thermal velocity vT = (∙T ¤ =m¤ )1=2 in general, 1;2 the collision time of the test particle is given by 1 q 2 q ¤2 n¤ ln ¤ = = ¿k 4¼²20 mmr v 3

µ

qq ¤ n¤ ²0 m

¶2

ln ¤ 4¼(mr =m)v 3 n¤

(2:15)

¤ . mr is the reduced mass mr = mm¤ =(m + m¤ ). Taking the under the assumption of v > vT 2 average of (m=2)v = (3=2)∙T , 1=¿k becomes

1 q 2 q ¤2 n¤ ln ¤ = 1=2 : ¿k 3 12¼²20 (mr =m1=2 )(∙T )3=2

(2:16)

This collision time in the case of electron with ions is 1 ¿eik

=

Z 2 e4 ni ln ¤ 1=2

31=2 12¼²20 me (∙T )3=2

:

(2:17)

This collision time of electron with ions is within 20% of Spitzer's result4 1 ¿eik

Spitzer

=

Z 2 e4 ni ln ¤ 1=2

51:6¼ 1=2 ²20 me (∙Te )3=2

:

(2:18)

16

2 Plasma Characteristics

Fig.2.9 Elastic collision of test particle M and ¯eld particle m in laboratory system (a) and center-of-mass system (b).

When an ion with the charge Z and the mass mi collides with the same ions, the ion-ion collision time is given by Z 4 e4 ni ln ¤ 1 = : 1=2 ¿iik 31=2 6¼²20 mi (∙Ti )3=2

(2:19)

Electron-electron Coulomb collision time can be derived by substitution of mi ! me and Z ! 1 into the formula of ¿iik . 1 ¿eek

=

ne e4 ln ¤ 1=2

31=2 6¼²20 me (∙Te )3=2

:

(2:20)

However the case of ion to electron Coulomb collision is more complicated to treat because ¤ is no longer hold. Let us consider the case that a test particle with the assumption vi > vT the mass M and the velocity vs collides with a ¯eld particle with the mass m. In center-ofmass system where the center of mass is rest, the ¯eld particle m moves with the velocity of vc = ¡M vs =(M + m) and the test particle M moves with the velocity of vs ¡ vc = mvs =(M + m) (see ¯g.2.9). Since the total momentum and total kinetic energy of two particles are conserved in the process of elastic collision, the velocities of the test particle and the ¯eld particle do not change and two particles only de°ect their direction by the angle of µ in center-of -mass system. The velocity vf and scattering angle Á of the test particle after the collision in laboratory system are given by (see ¯g.2.9) vf2 = (vs ¡ vc )2 + vc 2 + 2(vs ¡ vc )vc cos µ = vs2 sin Á =

(M 2

(M 2 + 2M m cos µ + m2 ) ; (M + m)2

m sin µ : + 2M m cos µ + m2 )1=2

Denote the momentum and the kinetic energy of the test particle before and after the collision by ps ; Es ; and pf ; Ef respectively, then we ¯nd Ef ¡ Es ¢E 2M m ´ =¡ (1 ¡ cos µ): Es Es (M + m)2 When the average is taken by µ, we obtain the following relations in the case of m=M ¿ 1: h

¢E 2m i'¡ ; Es M

h

¢pk m i'¡ : ps M

From the foregoing discussion, the inverse of collision time 1=¿iek where a heavy ion collides with light electrons is about me =mi times the value of 1=¿eik and is given by1;2 1 ¿ie k

=

Z 2 e4 ne ln ¤ me : 3=2 mi (2¼)1=2 3¼²2 m1=2 e (∙Te ) 0

(2:21)

2.6 Coulomb Collision Time, Fast Neutral Beam Injection

17

When the parallel and perpendicular components of the momentum of a test particle are denoted by pk and p? respectively and the energy by E, there are following relations E=

p2k + p2? 2m

;

dpk dE dp2? ¡ 2pk = 2m : dt dt dt We de¯ne the velocity di®usion time ¿? in the perpendicular direction to the initial momentum and the energy relaxation time ¿ ² by p2 dp2? ´ ?; dt ¿? E dE ´¡ ² dt ¿ respectively. 1=¿? and 1=¿ ² are given by1 1 q2 q ¤2 n¤ ln ¤ q 2 q ¤2 n¤ ln ¤ = = ; ¿? 2¼²20 v(mv)2 2¼²20 m2 v 3

(2:22)

1 q 2 q ¤2 n¤ ln ¤ q 2 q ¤2 n¤ ln ¤ = = ¿² 4¼²20 m¤ v(mv 2 =2) 2¼²20 mm¤ v 3

(2:23)

¤ respectively under the assumption v > vT . In the case of electron to ion collision, we ¯nd

2 1 ' ; ¿ei? ¿eik

(2:24)

me 2 1 ' : ² ¿ei mi ¿ei k

(2:25)

In the case of electron to electron collision, and ion to ion collision, we ¯nd 1 ¿ee?

'

1 ¿eek

;

Ã

1 ¿eek

2 1 ' Z ¿eik

!

;

1 1 ' ² ¿ee ¿eek

(2:26)

(2:27)

and 1 ¿ii?

'

1 ; ¿ii k

(2:28)

1 1 ' ² ¿ii ¿ii k

(2:29)

respectively. In the case of ion to electron collision we have following relations:1 1 ¿ie?

'

Z 2 e4 ne ln ¤ 1=2 (2¼)3=2 ²20 me Ei (∙Te )1=2

me ; mi

(2:30)

18

2 Plasma Characteristics

1 ¿ie

²

'

Z 2 e4 ne ln ¤ 1=2 4¼²20 me (∙Te )3=2

4 me 1 me 2:77 ' ' 1=2 ¿iek mi ¿eik 3(2¼) mi

(2:31)

where Ei = (3=2)∙Ti is the kinetic energy of the ion. The inverse of collision time is called collisional frequency and is denoted by º. The mean free path is given by ¸ = 31=2 vT ¿ . High energy neutral particle beams can be injected into plasmas across strong magnetic ¯elds. The neutral particles are converted to high-energy ions by means of charge exchange with plasma ions or ionization. The high energy ions (mass mb , electric charge Zbe , energy Eb ) running through the plasma slow down by Coulomb collisions with the plasma ions (mi ; Zi ) and electrons (me ; ¡e) and the beam energy is thus transferred to the plasma. This method is called heating by neutral beam injection (NBI). The rate of change of the fast ion's energy, that is, the heating rate of plasma is Eb dEb Eb =¡ ² ¡ ² ; dt ¿bi ¿be 1 (Zb e)2 (Zi e)2 ln ¤ni = ² 3 ¿bi 2¼²20 mi mb vbi and3 dEb Z 2 e4 ln ¤ne =¡ b 2 dt 4¼²0 me vbi

à X me ni Z 2 i

4 + 1=2 3¼

mi ne

µ

me Eb mb ∙Te

¶3=2 !

(2:32)

when beam ion's velocity vb is much less (say 1/3) than the plasma electron thermal velocity and much larger (say 2 times) than the plasma ion thermal velocity. The ¯rst term in the righthand side is due to beam-ion collisions and the second term is due to beam-electron collisions respectively. A critical energy Ecr of the beam ion, at which the plasma ions and electrons are heated at equal rates is given by 2 mvcr = Ecr = 14:8∙Te Ab 2

Ã

1 X ni Zi2 ne Ai

!2=3

(2:33)

where Ab ; Ai are atomic weights of the injected ion and plasma ion respectively. When the energy of the injected ion is larger than Ecr , the contribution to the electron heating is dominant. The slowing down time of the ion beam is given by ¿slowdown =

Z

Eb

Ecr

Ã

¡dEb ¿² = be ln 1 + (dEb =dt) 1:5

µ

E Ecr

¶3=2 !

1 me Z 2 ne e4 ln ¤ = ; ² 1=2 2 1=2 3=2 ¿be m b (2¼) 3¼²0 me (∙Te )

;

(2:34)

² is the energy relaxation time of beam ion with electrons. where ¿be

2.7 Runaway Electron, Dreicer Field When a uniform electric ¯eld E is applied to a plasma, the motion of a test electron is me

dv 1 = ¡eE ¡ me v; dt ¿ee (v)

2.8 Electric Resistivity, Ohmic Heating

19

1 e4 ln ¤ = ne ¾v = : ¿ee 2¼²20 m2e v 3 The deceleration term decreases as v increases and its magnitude becomes smaller than the acceleration term j ¡ eEj at a critical value vcr . When v > vcr , the test particle is accelerated. The deceleration term becomes smaller and the velocity starts to increase without limit. Such an electron is called a runaway electron. The critical velocity is given by 2 me vcr e2 n ln ¤ = : 2e 4¼²20 E

(2:35)

The necessary electric ¯eld for a given electron velocity to be vcr is called Dreicer ¯eld. Taking ln ¤ = 20, we ¯nd 2 n me vcr = 5 £ 10¡16 : 2e E

(MKS units)

When n = 1019 m¡3 ; E = 1 V=m, electrons with energy larger than 5keV become runaway electrons. 2.8 Electric Resistivity, Ohmic Heating When an electric ¯eld less than Dreicer ¯eld is applied to a plasma, electrons are accelerated and are decelerated by collisions with ions to be an equilibrium state as follows: me (ve ¡ vi ) = ¡eE: ¿ei The current density j induced by the electric ¯eld becomes j = ¡ene (ve ¡ vi ) =

e2 ne ¿ei E: me

The speci¯c electric resistivity de¯ned by ´j = E is4 ´=

me ºeik (me )1=2 Ze2 ln ¤ = (∙Te )¡3=2 ne e2 51:6¼ 1=2 ²20

= 5:2 £ 10¡5 Z ln ¤

µ

∙Te e

¶¡3=2

(2:36)

: (­m)

The speci¯c resistivity of a plasma with Te = 1 keV; Z = 1 is ´ = 3:3 £ 10¡8 ­m and is slightly larger than the speci¯c resistivity of copper at 20± C, 1:8 £ 10¡8 ­m. When a current density of j is induced, the power ´j 2 per unit volume contributes to electron heating. This heating mechanism of electron is called Ohmic heating. 2.9 Variety of Time and Space Scales in Plasmas Various kinds of plasma characteristics have been described in this chapter. Characteristic time scales are a period of electron plasma frequency 2¼=¦e , an electron cyclotron period 2¼=­e , an ion cyclotron period 2¼=j­i j, electron to ion collision time ¿ei , ion to ion collision time ¿ii and electron-ion thermal energy relaxation time ¿ei² . Alf¶ ven velocity vA , which is a propagation 2 velocity of magnetic perturbation, is vA = B 2 =(2¹0 ½m ) (½m is mass density)(see chs.5,10). Alf¶ ven transit time ¿H = L=vA is a typical magnetohydrodynamic time scale, where L is a plasma size. In a medium with the speci¯c resistivity ´, electric ¯eld di®uses with the time scale of ¿R = ¹0 L2 =´ (see ch.5). This time scale is called resistive di®usion time.

20

2 Plasma Characteristics

Characteristic scales in length are Debye length ¸D , electron Larmor radius ½­e , ion Larmor radius ½­i , electron-ion collision mean free path ¸ei and a plasma size L. The relations between space and time scales are ¸D ¦e = vTe ; ½­e ­e = vTe , ½­i j­i j = vTi , 2 ¸ei =¿ei ' 31=2 vTe , ¸ii =¿ii ' 31=2 vTi , L=¿H = vA , where vTe ; vTi are the thermal velocities vTe = 2 ∙Te =me ; vTi = ∙Ti =mi . The drift velocity of guiding center is vdrift » ∙T =eBL = vT (½­ =L). Parameters of a typical D-T fusion plasma with ne = 1020 m¡3 ; ∙Te = ∙Ti = 10 keV; B = 5 T; L = 1 m are followings: ¸D = 74:5 ¹m 2¼=¦e = 11:1 ps (¦e =2¼ = 89:8 GHz) 2¼=­e = 7:1 ps (­e =2¼ = 140 GHz) ½­e = 47:6 ¹m ½­i = 2:88 mm 2¼=j­i j = 26 ns (j­i j=2¼ = 38 MHz) ¿ei = 0:34 ms ¸ei = 25 km ¿ii = 5:6 ms ¸ii = 9:5 km ¿ei² = 0:3 s ¿H = 0:13 ¹s ¿R = 1:2 £ 103 s. The ranges of scales in time and space extend to ¿R ¦e » 1014 ; ¸ei =¸D » 1:6 £ 108 and the wide range of scales suggests the variety and complexity of plasma phenomena. References 1. D. V. Sivukhin: Reviews of Plasma Physics 4 p.93 (ed. by M. A. Leontovich) Consultant Bureau, New York 1966 2. K. Miyamoto: Plasma Physics for Nuclear Fusion (Revised Edition) Chap.4 The MIT Press, Cambridge, Mass. 1989 3. T. H. Stix: Plasma Phys. 14, 367 (1972) 4. L. Spitzer, Jr.: Physics of Fully Ionized Gases, Interscience, New York 1962

3.1 Maxwell Equations

21

Ch.3 Magnetic Con¯guration and Particle Orbit In this chapter, the motion of individual charged particles in a more general magnetic ¯elds is studied in detail. There are a large number of charged particles in a plasma, thus movements do a®ect the magnetic ¯eld. But this e®ect is neglected here. 3.1 Maxwell Equations Let us denote the electric intensity, the magnetic induction, the electric displacement and the magnetic intensity by E, B, D, and H, respectively. When the charge density and current density are denoted by ½, and j, respectively, Maxwell equations are r£E+

@B = 0; @t

(3:1)

r£H ¡

@D = j; @t

(3:2)

r ¢ B = 0;

(3:3)

r ¢ D = ½:

(3:4)

½ and j satisfy the relation r¢j +

@½ = 0: @t

(3:5)

Eq.(3.2),(3.4) and (3.5) are consistent with each other due to the Maxwell displacement current @D=@t. From eq.(3.3) the vector B can be expressed by the rotation of the vector A: B = r £ A:

(3:6)

A is called vector potential. If eq.(3.6) is substituted into eq.(3.1), we obtain µ

@A r£ E+ @t

¶

= 0:

(3:7)

The quantity in parenthesis can be expressed by a scalar potential Á and E = ¡rÁ ¡

@A : @t

(3:8)

Since any other set of Á0 and A0 , A0 = A ¡ rÃ; Á0 = Á +

@Ã @t

(3:9) (3:10)

can also satisfy eqs.(3.6),(3.8) with an arbitrary Ã, Á0 and A0 are not uniquely determined. When the medium is uniform and isotropic, B and D are expressed by D = ²E;   B = ¹H:

22

3 Magnetic Con¯guration and Particle Orbit

² and ¹ are called dielectric constant and permeability respectively. The value of ²0 and ¹0 in vacuum are ²0 =

107 2 2 C ¢ s =kg ¢ m3 = 8:854 £ 10¡12 F=m 4¼c2

¹0 = 4¼ £ 10¡7 kg ¢ m=C2 = 1:257 £ 10¡6 H=m 1 = c2 ²0 ¹ 0 where c is the light speed in vacuum (C is Coulomb). Plasmas in magnetic ¯eld are anisotropic and ² and ¹ are generally in tensor form. In vacuum, eqs (3.2),(3.3) may be reduced to 1 @Á 1 @ 2A r = ¹0 j; + c2 @t c2 @t2 @A 1 r2 Á + r = ¡ ½: @t ²0 r£r£A+

(3:11) (3:12)

As Á and A have arbitrariness of à as shown in eqs.(3.9),(3.10), we impose the supplementary condition (Lorentz condition) r¢A+

1 @Á = 0: c @t

(3:13)

Then eqs.(3.11),(3.12) are reduced to the wave equations r2 Á ¡

1 @2Á 1 = ¡ ½; 2 2 c @t "0

(3:14)

r2 A ¡

1 @2A = ¡¹0 j: c2 @t2

(3:15)

In derivation of (3.15), a vector relation r £ (r £ a) ¡ r(r ¢ a) = ¡r2 a is used, which is valid only in (x; y; z) coordinates. The propagation velocity of electromagnetic ¯eld is 1=(¹0 ²0 )1=2 = c in vacuum. When the ¯elds do not change in time, the ¯eld equations reduce to E = ¡rÁ; r2 Á = ¡

B = r £ A;

1 ½; "0

r2 A = ¡¹j;

r ¢ A = 0;

r ¢ j = 0:

The scalar and vector potentials Á and A at an observation point P (given by the position vector r) are expressed in terms of the charge and current densities at the point Q (given by r 0 ) by (see ¯g.3.1) Á(r) =

1 4¼²0

¹0 A(r) = 4¼

Z

Z

½(r 0 ) 0 dr ; R j(r 0 ) 0 dr R

(3:16)

(3:17)

3.2 Magnetic Surface

23

Fig.3.1 Observation point P and the location Q of charge or current.

Fig.3.2 Magnetic surface à = const:, the normal rà and line of magnetic force.

where R ´ r ¡ r 0 , R = jRj and dr 0 ´ dx0 d0 dz 0 . Accordingly E and B are expressed by E=

1 4¼²0

¹0 B= 4¼

Z

Z

R ½dr 0 ; R3

(3:18)

j£R 0 dr : R3

(3:19)

When the current distribution is given by a current I °owing in closed loops C, magnetic intensity is described by Biot-Savart equation B I H= = ¹0 4¼

I

c

s£n ds R2

(3:20)

where s and n are the unit vectors in the directions of ds and R, respectively. 3.2 Magnetic Surface A magnetic line of force satis¯es the equations dx dy dz dl = = = Bx By Bz B

(3:21)

where l is the length along a magnetic line of force (dl)2 = (dx)2 + (dy)2 + (dz)2 . The magnetic surface Ã(r) = const: is such that all magnetic lines of force lie upon on that surface which satis¯es the condition (rÃ(r)) ¢ B = 0:

(3:22)

The vector rÃ(r) is normal to the magnetic surface and must be orthogonal to B (see ¯g.3.2). In terms of cylindrical coordinates (r; µ; z) the magnetic ¯eld B is given by Br =

@Aµ 1 @Az ¡ ; r @µ @z

Bµ =

@Az @Ar ¡ ; @z @r

Bz =

1 @ 1 @Ar (rAµ ) ¡ : r @r r @µ

(3:23)

In the case of axi-symmetric con¯guration (@=@µ = 0), Ã(r; z) = rAµ (r; z)

(3:24)

24

3 Magnetic Con¯guration and Particle Orbit

satis¯es the condition (3.22) of magnetic surface; Br @(rAµ )=@r+Bµ ¢ 0 + Bz @(rAµ )=@z = 0. The magnetic surface in the case of translational symmetry (@=@z = 0) is given by Ã(r; µ) = Az (r; µ)

(3:25)

and the magnetic surface in the case of helical symmetry, in which à is the function of r and µ ¡ ®z only, is given by Ã(r; µ ¡ ®z) = Az (r; µ ¡ ®z) + ®rAµ (r; µ ¡ ®z)

(3:26)

where ® is helical pitch parameter. 3.3 Equation of Motion of a Charged Particle The equation of motion of a particle with the mass m and the charge q in an electromagnetic ¯eld E; B is µ

¶

d2 r dr £B : m 2 =F =q E+ dt dt

(3:27)

Since Lorentz force of the second term in the right-hand side of eq.(3.27) is orthogonal to the velocity v, the scalar product of Lorentz force and v is zero. The kinetic energy is given by mv 2 mv0 2 ¡ =q 2 2

Z

t

t=t0

E ¢ vdt:

When the electric ¯eld is zero, the kinetic energy of charged particle is conserved. The x component of eq.(3.27) in the orthogonal coordinates (x; y; z) is written by md2 x=dt2 = q(Ex + (dy=dt)Bz ¡(dz=dt)By ), However the radial component of eq.(3.27) in the cylindrical coordinates (r; µ; z) is md2 r=dt2 6= q(Er + r(dµ=dt)Bz ¡ (dz=dt)Bµ ). This indicates that form of eq.(3.27) is not conserved by the coordinates transformation. When generalized coordinates qi (i = 1; 2; 3) are used, it is necessary to utilize the Lagrangian formulation. Lagrangian of a charged particle in the ¯eld with scalar and vector potentials Á; A is given by L(qi ; q_i ; t) =

mv 2 + qv ¢ A ¡ qÁ: 2

(3:28)

Lagrangians in the orthogonal and cylindrical coordinates are given by m 2 _ x + yA _ y + zA _ z ) ¡ qÁ; (x_ + y_ 2 + z_ 2 ) + q(xA 2

L(x; y; z; x; _ y; _ z; _ t) =

m _ 2 + z_ 2 ) + q(rA _ µ + zA _ z; _ r + r µA _ z ) ¡ qÁ L(r; µ; z; r; _ µ; _ t) = (r_ 2 + (r µ) 2 respectively. The equation of motion in Lagrangian formulation is d dt

µ

@L @ q_i

¶

¡

@L = 0: @qi

(3:29)

The substitution of (3.28) into (3.29) in the case of the orthogonal coordinates yields µ

@A @Á d ¡ (mv_ x + qAx ) ¡ q v ¢ dt @x @x µ

mÄ x=q ¡

@Ax ¡ @t

µ

¶

= 0; ¶

dx @ @A @Á dy @ dz @ ¡ Ax + v ¢ + + dt @x dt @y dt @z @x @x

¶

3.3 Equation of Motion of a Charged Particle

25

= q(E + v £ B)x ; and this equation is equivalent to eq.(3.27). Lagrangian equation of motion with respect to the _ 2 =r and the term of centrifugal force cylindrical coordinates is mÄ r = q(E + v £ B)r + m(r µ) appears. Canonical transformation is more general than the coordinates transformation. Hamiltonian equation of motion is conserved with respect to canonical transformation. In this formulation we introduce momentum coordinates (pi ), in addition to the space coordinates (qi ), de¯ned by pi ´

@L @ q_i

(3:30)

and treat pi as independent variables. Then we can express q_i as a function of (qj ; pj ; t) from eq.(3.30) as follows: q_i = q_i (qj ; pj ; t):

(3:31)

The Hamiltonian H(qi ; pi ; t) is given by H(qi ; pi ; t) ´ ¡L(qi ; q_i (qj ; pj ; t); t) +

X

pi q_i (qj ; pj ; t):

(3:32)

i

The x component of momentum px in the orthogonal coordinates and µ component pµ in the cylindrical coordinates are written as examples as follows: px = mx_ + qAx ;

x_ = (px ¡ qAx )=m;

pµ = mr 2 µ_ + qrAµ ;

µ_ = (pµ ¡ qrAµ )=(mr 2 ):

Hamiltonian in the orthogonal coordinates is H=

´ 1 ³ (px ¡qAx )2 +(py ¡qAy )2 +(pz ¡qAz )2 + qÁ(x; y; z; t)); 2m

and Hamiltonian in the cylindrical coordinates is Ã

!

1 (pµ ¡qrAµ )2 H= (pr ¡qAr )2 + +(pz ¡qAz )2 + qÁ(r; µ; z; t): 2m r2 The variation of Lagrangian L is given by ±L =

X µ @L i

@L ±qi + ± q_i @qi @ q_i

¶

=

X

(p_ i ±qi + pi ± q_i ) = ±

i

Ã

X

!

pi q_i +

i

X i

(p_ i ±qi ¡ q_i ±pi )

and ±(¡L +

X

pi q_i ) =

i

X i

(q_i ±pi ¡ p_ i ±qi );

±H(qi ; pi ; t) =

X i

(q_i ±pi ¡ p_ i ±qi ):

Accordingly Hamiltonian equation of motion is reduced to dqi @H ; = dt @pi

dpi @H : =¡ dt @qi

Equation (3.33) in the orthogonal coordinates is dx px ¡ qAx = ; dt m

@Á dpx q @A ¢ (p ¡ qA) ¡ q ; = dt m @x @x

(3:33)

26

3 Magnetic Con¯guration and Particle Orbit

m

dAx d2 x dpx ¡q = =q 2 dt dt dt

∙µ

v¢

¶

@Á @A ¡ ¡ @x @x

µ

@Ax + (v ¢ r)Ax @t

¶¸

= q(E + v £ B)x and it was shown that eq.(3.33) is equivalent to eq.(3.27). When H does not depend on t explicitly (when Á ; A do not depend on t), µ

dH(qi ; pi ) X @H dqi @H dpi = + dt @qi dt @pi dt i

¶

= 0;

H(qi ; pi ) = const:

(3:34)

is one integral of Hamiltonian equations. This integral expresses the conservation of energy. When the electromagnetic ¯eld is axially symmetric, pµ is constant due to @H=@µ = 0 as is seen in eq.(3.33) and pµ = mr 2 µ_ + qrAµ = const:

(3:35)

This indicates conservation of the angular momentum. In the case of translational symmetry (@=@z = 0), we have pz = mz_ + qAz = const:

(3:36)

3.4 Particle Orbit in Axially Symmetric System The coordinates (r¤ ; µ¤ ; z ¤ ) on a magnetic surface of an axially symmetric ¯eld satisfy à = r ¤ Aµ (r¤ ; z ¤ ) = cM : On the other hand the coordinates (r; µ; z) of a particle orbit are given by the conservation of the angular momentum (3.35) as follows; rAµ (r; z) +

m 2 _ pµ r µ= = const:: q q

If cM is chosen to be cM = pµ =q, the relation between the magnetic surface and the particle orbit is reduced to rAµ (r; z) ¡ r ¤ Aµ (r¤ ; z ¤ ) = ¡

m 2_ r µ: q

The distance ± (¯g.3.3) between the magnetic surface and the orbit is given by ± = (r ¡ r¤ )er + (z ¡ z ¤ )ez ; ± ¢ r(rAµ ) = ¡

m 2_ r µ: q

From the relations rBr = ¡@(rAµ )=@z; rBz = @(rAµ )=@r, we ¯nd [¡(z ¡ z ¤ )Br + (r ¡ r ¤ )Bz ] = ¡

m _ r µ: q

3.4 Particle Orbit in Axially Symmetric System

27

Fig.3.3 Magnetic surface (dotted line) and particle orbit (solid line).

Fig.3.4 The dotted lines are lines of magnetic force and the solid lines are particle orbit in cusp ¯eld.

This expression in the left-hand side is the µ component of the vector product of B p = (Br ; 0; Bz ) and ± = (r ¡ r¤ ; 0; z ¡ z¤). Then this is reduced to (B p £ ±)µ = ¡

m _ rµ: q

Denote the magnitude of poloidal component B p (component within (rz) plane) of B by Bp . _ and Then we ¯nd the relation ¡Bp ± = ¡(m=q)vµ (vµ = rµ) ±=

mvµ = ½­p : qBp

This value is equal to the Larmor radius corresponding to the magnetic ¯eld Bp and the tangential velocity vµ . If cM is chosen to be cM = (pµ ¡ mhrvµ i)=q (hrvµ i is the average of rvµ ), we ¯nd m ±= qBp

µ

¶

hrvµ i vµ ¡ : r

(3:37)

Let us consider a cusp ¯eld as a simple example of axi-symmetric system. Cusp ¯eld is given by Ar = 0; Br = ¡ar;

Aµ = arz; Bµ = 0;

Az = 0;

(3:38)

Bz = 2az:

(3:39)

From eq.(3.34) of energy conservation and eq.(3.35) of angular momentum conservation, we ¯nd pµ ¡ qazr; mr µ_ = r

28

3 Magnetic Con¯guration and Particle Orbit

Fig.3.5 Toroidal drift. µ

¶

m 2 (pµ ¡ qar 2 z)2 m = W = v02 : (r_ + z_ 2 ) + 2 2 2mr 2 These equations correspond to the motion of particle in a potential of X = (pµ ¡qar2 z)2 =(2mr2 ). When the electric ¯eld is zero, the kinetic energy of the particle is conserved, the region containing orbits of the particle with the energy of mv02 =2 is limited by (see ¯g.3.4) 1 X= 2m

µ

pµ ¡ qarz r

¶2

<

mv02 : 2

3.5 Drift of Guiding Center in Toroidal Field Let us consider the drift of guiding center of a charged particle in a simple toroidal ¯eld (Br = 0; B' = B0 R0 =R; Bz = 0) in terms of cylindrical coordinates (R; '; z). The ' component B' is called toroidal ¯eld and B' decreases in the form of 1=R outward. The magnetic lines of force are circles around z axis. The z axis is called the major axis of the torus. As was described in sec.2.4, the drift velocity of the guiding center is given by m v G = vk e' + qB' R

Ã

vk2

v2 + ? 2

!

ez :

Particles in this simple torus run fast in the toroidal direction and drift slowly in the z direction with the velocity of vdr

m = qB0 R0

Ã

vk2

v2 + ? 2

!

»

µ

¶

½­ v: R0

(3:40)

This drift is called toroidal drift. Ions and electrons drift in opposite direction along z axis. As a consequence of the resultant charge separation, an electric ¯eld E is induced and both ions and electrons drift outward by E £ B=B 2 drift. Consequently, a simple toroidal ¯eld cannot con¯ne a plasma (¯g.3.5), unless the separated charges are cancelled or short-circuited by an appropriate method. If lines of magnetic force connect the upper and lower regions as is shown in ¯g.3.6, the separated charges can be short-circuited, as the charged particles can move freely along the lines of force. If a current is induced in a toroidal plasma, the component of magnetic ¯eld around the magnetic axis (which is also called minor axis) is introduced as is shown in ¯g.3.6. This component B p is called poloidal magnetic ¯eld. The radius R of the magnetic axis is called major radius of torus and the radius a of the plasma cross section is called minor radius. Denote the radial coordinate in plasma cross section by r. When a line of magnetic force circles the major axis of torus and come back to cross the plane P, the cross point rotates around the minor axis O by an angle ¶ in P, there is following relation: r¶ Bp : = 2¼R B'

3.5 Drift of Guiding Center in Toroidal Field

29

Fig.3.6 The major axis A, the minor axis M of toroidal ¯eld and rotational transform angle ¶.

The angle ¶ is called rotational transform angle and is given by ¶ R Bp : = 2¼ r B'

(3:41)

A ´ R=a is called aspect ratio. 3.5a Guiding Center of Circulating Particles When a particle circulates torus with the velocity of vk , it takes T = 2¼R0 =vk . Accordingly the particle rotates around the minor axis with angular velocity of !=

¶vk ¶ = T 2¼R0

and drifts in z direction with the velocity of vdr . Introducing x = R ¡ R0 coordinate, the orbit of the guiding center of the particle is given by dz = !x + vdr : dt

dx = ¡!z; dt The solution is µ

x+

vdr !

¶2

+ z 2 = r2 :

If a rotational transform angle is introduced, the orbit becomes a closed circle and the center of orbit circle deviates from the center of magnetic surface by the amount of mvk 2¼ vdr ¢=¡ =¡ ! qB0 ¶ j¢j » ½­

µ

2¼ ¶

¶

Ã

v2 1 + ?2 2vk

!

;

(3:42)

where ½­ is Larmor radius. As is seen in ¯g.3.7, the sign of the deviation is ¢ < 0 for the case of vk > 0; q > 0 (ion) since vdr > 0; ! > 0 and the sign becomes ¢ > 0 for the case of vk < 0 (opposit to vk > 0) q > 0 (ion). 3.5b Guiding Center of Banana Particles In the case of jB' j À jBp j, the magnitude of toroidal ¯eld is nearly equal to B' and B=

µ

¶

B0 R0 r B0 ' B0 1 ¡ cos µ : = R 1 + (r=R) cos µ R0

30

3 Magnetic Con¯guration and Particle Orbit

Fig.3.7 Orbits (solid lines) of guiding center of circulating ions and electrons and magnetic surfaces (dotted lines).

Denote the length along magnetic line of force by l, and denote the projection of a location on the magnetic line of force to (R; z) plane by the coordinates (r; µ) as is shown in ¯g.3.8. Since the following relations rµ Bp ; = l B0

µ=

l Bp = ∙l r B0

holds, we ¯nd B = B0

µ

¶

r 1¡ cos(∙l) : R0

If vk (parallel component to magnetic ¯eld) is much smaller than v? component and satis¯es the condition; vk2

2 r v? >1¡ ; 2 v R

v2

<

r R

(3:43)

the particle is trapped outside in the weak region of magnetic ¯eld due to the mirror e®ect as is described in sec.2.5 (The mirror ratio is (1=R)=(1=(R + r))). This particle is called trapped 2 particle. Circulating particle without trapped is called untrapped particles. Since vk2 ¿ v? for the trapped particle, the r component of the toroidal drift vdr of trapped particle is given by r_ = vdr sin µ =

2 m v? sin µ: qB0 2R

The parallel motion of the guiding center is given by (see sec.2.4) dvk ¹m @B =¡ ; dt m @l v_ k = ¡ The solution is

d dt

Ã

¹m r v 2 Bp sin µ: ∙B0 sin ∙l = ¡ ? m R 2R B0

m r+ v qBp k

r ¡ r0 = ¡

!

= 0;

m v : qBp k

(3:44)

3.6 Orbit of Guiding Center and Magnetic Surface

31

Fig.3.8 (r; µ) coordinates

Fig.3.9 Banana orbit of ion

Here r = r0 indicates the radial coordinate of turning point by mirror e®ect. Since the orbit is of banana shape, the trapped particle is also called banana particle (see ¯g.3.9). The width of banana ¢b is given by ¢b =

mv vk B0 B0 m » vk » qBp qB0 v Bp Bp

µ

r R

¶1=2

½­ »

µ

R r

¶1=2 µ

¶

2¼ ½­ : ¶

(3:45)

3.6 Orbit of Guiding Center and Magnetic Surface The velocity of guiding center was derived in sec.2.4 as follows: 2 mvk2 =2 1 mv? (b £ rB) + (b £ (b ¢ r)B) v G = vk b + (E £ b) + B qB 2 qB 2

(3:46)

and 2 ¹m = mv? =(2B) = const:

When the electric ¯eld E is static and is expressed by E = ¡rÁ, the conservation of energy m 2 2 ) + qÁ = W (v + v? 2 k holds. Then vk is expressed by µ

2 vk = § m

¶1=2

(W ¡ qÁ ¡ ¹m B)1=2 :

Noting that vk is a function of the coodinates, we can write r £ (mvk b) = mvk r £ b + r(mvk ) £ b = mvk r £ b +

1 (¡qrÁ ¡ ¹m rB) £ b vk

(3:47)

32

and

3 Magnetic Con¯guration and Particle Orbit

2 mvk2 vk 1 =2 mv? r £ (mvk b) = r £ b + (E £ b) + (b £ rB): 2 qB qB B qB

Then eq.(3.46) for v G is reduced to v G = vk b +

Ã

= vk b +

!

mvk2 mvk2 vk r £ (mvk b) ¡ r£b + (b £ (b ¢ r)B) qB qB qB 2

mvk2 vk r £ (mvk b) ¡ (r £ b ¡ b £ (b ¢ r)b): qB qB

As the relation r(b ¢ b) = 2(b ¢ r)b + 2b £ (r £ b) = 0 ( (b ¢ b) = 1 ) holds (see appendix Mathematical Formula), the third term in right-hand side of the equation for v G becomes ( ) = (r £ b) ¡ (r £ b)? = (r £ b)k = (b ¢ (r £ b))b. Since r £ B = Br £ b + rB £ b = ¹0 j, we have b ¢ r £ b = ¹0 jk =B. The ratio of the third term to the ¯rst one, which are both parallel to the magnetic ¯eld, is usually small. If the third term can be neglected, eq.(3.46) for v G is reduced to µ

¶

mvk vk dr G = r£ A+ B : dt B qB

(3:48)

The orbit of guiding center is equal to the ¯eld line of magnetic ¯eld B ¤ = r £ A¤ with the vector potential A¤ ´ A +

mvk B: qB

By reason analogous to that in sec.3.2, the orbit surface of drift motion of guiding center is given by rA¤µ (r; z) = const:

(3:49)

in the case of axi-symmetric con¯guration.

3.7 E®ect of Longitudinal Electric Field on Banana Orbit In the tokamak con¯guration, a toroidal electric ¯eld is applied in order to induce the plasma current. The guiding center of a particle drifts by E £ B=B 2 , but the banana center moves in di®erent way. The toroidal electric ¯eld can be described by E' = ¡

@A' @t

in (R; '; z) coordinates. Since angular momentum is conserved, we can write R(mR'_ + qA' ) = const: Taking the average of foregoing equation over a Larmor period, and using the relation hR'i _ =

B' v B k

we ¯nd µ

R mvk

B' + qA' B

¶

= const:

(3:50)

3.7 E®ect of Longitudinal Electric Field on Banana Orbit

33

Fig.3.10 Coordinate system for explanation of Ware's pinch.

For particles in banana motion (vk ¿ v? ), vk becomes 0 at the turning points of the banana orbit. The displacement of a turning point (R; Z) per period ¢t is obtained from 0 = ¢(RA' (R; Z)) = ¢r

@ @ RA' + ¢t RA' @r @t

where r is the radial coordinate of the magnetic surface. The di®erentiations of RA' = const: with respect to ' and µ are zero, since RA' = const: is the magnetic surface. By means of the relation 1 1 @ (RA' ) = R @r R

µ

@R @(RA' ) @Z @(RA' ) + @r @R @r @Z

¶

= cos µBZ ¡ sin µBR = Bp ; we obtaine the drift velocity ¢r Eµ : = ¢t Bp

(3:51)

When the sign of Bp produced by the current induced by the electric ¯eld E' is taken account (see ¯g.3.10), the sign of ¢r=¢t is negative and the banana center moves inward. Since jBp j ¿ jB' j ' B, the drift velocity of banana center is (B=Bp )2 times as fast as the the drift velocity E' Bp =B 2 of guiding center of particle. This phenomena is called Ware's pinch.

4 Velocity Space Distribution Function ¢ ¢ ¢

34

Ch.4 Velocity Space Distribution Function and Boltzmann's Equation A plasma consists of many ions and electrons, but the individual behavior of each particle can hardly be observed. What can be observed instead are statistical averages. In order to describe the properties of a plasma, it is necessary to de¯ne a distribution function that indicates particle number density in the phase space whose ordinates are the particle positions and velocities. The distribution function is not necessarily stationary with respect to time. In sec.4.1, the equation governing the distribution function f (qi ; pi ; t) is derived by means of Liouville's theorem. Boltzmann's equation for the distribution function f (x; v; t) is formulated in sec.4.2. When the collision term is neglected, Boltzmann's equation is called Vlasov's equation. 4.1 Phase Space and Distribution Function A particle can be speci¯ed by its coordinates (x; y; z), velocity (vx ; vy ; vz ), and time t. More generally, the particle can be speci¯ed by canonical variables q1 ; q2 ; q3 ; p1 ; p2 ; p3 and t in phase space. When canonical variables are used, an in¯nitesimal volume in phase space ¢ = ±q1 ±q2 ±q3 ±p1 ±p2 ±p3 is conserved (Liouville's theorem). The motion of a particle in phase space is described by Hamilton's equations @H(qj ; pj ; t) dqi ; = dt @pi

dpi @H(qj ; pj ; t) : =¡ dt @qi

(4:1)

The variation over time of ¢ is given by d¢ = dt

µ

¶

d(±q1 ) d(±p1 ) ±p1 + ±q1 ±q2 ±p2 ±q3 ±p3 + ¢ ¢ ¢ ; dt dt µ

¶

=

@H d ±pi = ¡± dt @qi

¶

d @H ±qi = ± dt @pi µ

d¢ X = dt i

Ã

@2H ±qi ; @pi @qi =¡

@2H ±pi ; @qi @pi

@2H @2H ¡ @pi @qi @qi @pi

!

¢ = 0:

(4:2)

Let the number of particles in a small volume of phase space be ±N ±N = F (qi ; pi ; t)±q±p

(4:3)

where ±q = ±q1 ±q2 ±q3 ; ±p = ±p1 ±p2 ±p3 , and F (qi ; pi ; t) is the distribution function in phase space. If the particles move according to the equation of motion and are not scattered by collisions, the small volume in phase space is conserved. As the particle number ±N within the small phase space is conserved, the distribution function (F = ±N=¢) is also constant, i.e., µ

3 @F dqi dF @F X @F dpi = + + dt @t @q dt @p i i dt i=1

¶

µ

3 @H @F @H @F @F X ¡ = + @t @p @q @qi @pi i i i=1

¶

= 0:

(4:4)

4.2 Boltzmann's Equation and Vlasov's Equation

35

Fig.4.1 Movement of particles in phase space.

In the foregoing discussion we did not take collisions into account. If we denote the variation of F due to the collisions by (±F=±t)coll , eq.(4.4) becomes µ

3 @H @F @F X @H @F ¡ + @t @p @q @qi @pi i i i=1

¶

=

µ

±F ±t

¶

:

(4:5)

coll

4.2 Boltzmann's Equation and Vlasov's Equation Let us use the space and velocity-space coordinates x1 ; x2 ; x3 ; v1 ; v2 ; v3 instead of the canonical coordinates. The Hamiltonian is H=

1 (p ¡ qA)2 + qÁ; 2m

(4:6)

pi = mvi + qAi ;

(4:7)

qi = xi

(4:8)

dxi @H = vi ; = dt @pi

(4:9)

and

X (pk ¡ qAk ) @Ak @Á dpi @H ¡q = : =¡ q dt @xi m @xi @xi k

(4:10)

Consequently eq.(4.5) becomes Ã

3 3 3 X X @Á @F X @F @Ak ¡ vk +q vk + @t @xk @xi @xi i=1 k=1 i=k

!

@F = @pi

µ

±F ±t

¶

:

(4:11)

coll

By use of eqs.(4.7) (4.8), independent variables are transformed from (qi ; pi ; t) to (xj ; vj ; t) and 1 @vj (xk ; pk ; t) = ±ij ; @pi m @vj (xk ; pk ; t) q @Aj =¡ ; @xi m @xi @vj (xk ; pk ; t) q @Aj =¡ : @t m @t

4 Velocity Space Distribution Function ¢ ¢ ¢

36

We denote F (xi ; pi ; t) = F (xi ; pi (xj ; vj ; t); t) ´ f (xj ; vj ; t)=m3 . Then we have m3 F (xi ; pi ; t) = f (xj ; vj (xi ; pi ; t); t) and m3

m3

X @f @vj @ @ @f 1 F (xh ; ph ; t) = f (xj ; vj (xh ; ph ; t); t) = = ; @pi @pi @v @p @v j i im j X @f @vi @ @ @f F (xh ; ph ; t) = f (xi ; vi (xh ; ph ; t); t) = + @xk @xk @xk @vi @xk i

=

X @f @f + @xk @vi i

µ

¡q m

¶

@Ai @xk

@ @ @f X @f m F (xh ; ph ; t) = f (xi ; vi (xh ; ph ; t); t) = + @t @t @t @vi i 3

µ

¡q m

¶

@Ai : @t

Accordingly eq.(4.11) is reduced to @f X @f + @t @vi i +

µ

à X X i

k

¡q m

¶

@Ai X vk + @t k

@Á @Ak ¡ vk @xi @xi

!

Ã

X @f @f + @xk @vi i

q @f = m @vi

µ

±f ±t

¶

µ

¡q m

¶

@Ai @xk

!

;

coll

Ã

X @Ai X @Ai X @Ak @Á @f X @f ¡ ¡ ¡ vk + vk + vk + @t @xk @t @xk @xi @xi i k k k

!

q @f = m @vi

µ

±f ±t

¶

:

coll

Since the following relation is hold X

vk

k

X @Ai X @Ai @Ak = vk + (v £ (r £ A))i = vk + (v £ B)i : @xi @xk @xk k k

we have X q @f @f X @f vi + = + (E + v £ B)i @t @xi m @vi i i

µ

±f ±t

¶

:

(4:12)

coll

This equation is called Boltzmann's equation. The electric charge density ½ and the electric current j are expressed by ½=

X Z

q

f dv1 dv2 dv3 ;

(4:13)

vf dv1 dv2 dv3 :

(4:14)

i;e

j=

X Z

q

i;e

Accordingly Maxwell equations are given by r¢E =

Z 1 X q f dv; ²0

1 @E X r £ B = ²0 q + ¹0 @t r£E =¡

@B ; @t

(4:15) Z

vf dv;

(4:16)

(4:17)

References

37

r ¢ B = 0:

(4:18)

When the plasma is rare¯ed, the collision term (±f =±t)coll may be neglected. However, the interactions of the charged particles are still included through the internal electric and magnetic ¯eld which are calculated from the charge and current densities by means of Maxwell equations. The charge and current densities are expressed by the distribution functions for the electron and the ion. This equation is called collisionless Boltzmann's equation or Vlasov's equation. When Fokker-Planck collision term1 is adopted as the collision term of Boltzmann's equation, this equation is called Fokker-Planck equation (see sec.15.9).

  References 1. D. V. Sivukhin: Reviews of Plasma Physics 4 p.93 (ed. by M. A. Leontovich) Consultant Bureau, New York 1966 K. Miyamoto: Plasma Physics for Nuclear Fusion (Revised Edition) Chap.4 The MIT Press, Cambridge, Mass. 1989

38

5 Plasma as Magnetohydrodynamic Fluid

Ch.5 Plasma as Magnetohydrodynamic Fluid 5.1 Magnetohydrodynamic Equations for Two Fluids Plasmas can be described as magnetohydrodynamic two °uids of ions and electrons with mass densities ½mi , ½me , charge density ½, current density j, °ow velocities V i , V e , and pressures pi , pe . These physical quantities can be expressed by appropriate averages in velocity space by use of the velocity space distribution functions fi (r; v; t) of ions and electrons, which were introduced in ch.4. The number density of ion ni , the ion mass density ½m;i , and the ion °ow velocity V i (r; t) are expressed as follows: ni (r; t) =

Z

fi (r; v; t)dv;

(5:1)

½mi (r; t) = mi ni (r; t); R vfi (r; v; t)dv V (r; t) = R =

fi (r; v; t)dv

(5:2) 1 ni (r; t)

Z

vfi (r; v; t)dv:

(5:3)

We have the same expressions for electrons as those of ions. Since magnetohydrodynamics will treat average quantities in the velocity space, phenomena associated with the shape of the velocity space distribution function (ch.11) will be neglected. However the independent variables are r; t only and it is possible to analyze geometrically complicated con¯gurations. Equations of magnetohydrodynamics are followings: @ne + r ¢ (ne V e ) = 0; @t

(5:4)

@ni + r ¢ (ni V i ) = 0; @t

(5:5)

dV e = ¡rpe ¡ ene (E + V e £ B) + R; dt

(5:6)

dV i = ¡rpi + Zeni (E + V i £ B) ¡ R: dt

(5:7)

ne me

ni mi

Here R denotes the rate of momentum (density) change of the electron °uid by the collision with the ion °uid. The rate of momentum change of the ion °uid due to the collision with electron °uid is ¡R. The change of the number n(x; y; z; t)¢x¢y¢z of particles within the region of ¢x¢y¢z is the di®erence between the incident particle °ux n(x; y; z; t)Vx (x; y; z; t)¢y¢z into the surface A in ¯g.5.1 and outgoing particle °ux n(x + ¢x; y; z; t)Vx (x + ¢x; y; z; t)¢y¢z from the surface A0 , that is, (n(x; y; z; t)Vx (x; y; z; t) ¡ n(x + ¢x; y; z; t)Vx (x + ¢x; y; z; t))¢y¢z =¡

@(nVx ) ¢x¢y¢z: @x

5.1 Magnetohydrodynamic Equations for Two Fluids

39

Fig.5.1 Particle °ux and force due to pressure

When the particle °uxes of the other surfaces are taken into accout, we ¯nd (5.4), that is µ

¶

@n @(nVx ) @(nVy ) @(nVz ) ¢x¢y¢z: ¢x¢y¢z = ¡ + + @t @x @y @z The term ¡rp in eqs.(5.6),(5.7) is the force per unit volume of plasma due to the pressure p by the following reason. The force applied to the surface A in ¯g.5.1 is p(x; y; z; t)¢y¢z and the force on the surface A0 is ¡p(x + ¢x; y; z; t)¢y¢z. Therefore the sum of these two forces is (¡p(x + ¢x; y; z; t) + p(x; y; z; t))¢y¢z = ¡

@p ¢x¢y¢z @x

in the x direction. When the e®ects of the pressure on the other surfaces are taken account, the resultant force due to the pressure per unit volume is ¡

µ

¶

@p @p @p x ^+ y ^+ z ^ = ¡rp @x @y @z

where x ^; y ^; z ^ are the unit vector in x; y; z directions respectively. The second term in right-hand side of eqs.(5.6),(5.7) is Lorentz force per unit volume. The third term is the collision term of electron-ion collision as is mentioned in sec.2.8 and is given by R = ¡ne me (V e ¡ V i )ºei

(5:8)

where ºei is coulomb collision frequency of electron with ion. Let us consider the total time di®erential in the left-hand side of equation of motion. The °ow velocity V is a function of space coordinates r and time t. Then the acceleration of a small volume of °uid is given by @V (r; t) dV (r; t) = + dt @t

µ

¶

dr @V (r; t) ¢ r V (r; t) = + (V (r; t) ¢ r)V (r; t): dt @t

Therefore the equations of motion (5.6),(5.7) are reduced to µ

¶

@V e ne me + (V e ¢ r)V e = ¡rpe ¡ ene (E + V e £ B) + R µ @t ¶ @V i ni mi + (V i ¢ r)V i = ¡rpi + Zeni (E + V i £ B) ¡ R: @t

(5:9) (5:10)

Conservation of particle (5.4),(5.5), the equations of motion (5.9), (5.10) can be derived from Boltzmann equation (4.12). Integration of Boltzmann equation over velocity space yields eqs.(5.4), (5.5). Integration of Boltzmann equation multiplied by mv yields eqs. (5.9),(5.10). The process of the mathematical derivation is described in Appendix A.

40

5 Plasma as Magnetohydrodynamic Fluid

5.2 Magnetohydrodynamic Equations for One Fluid Since the ion-to-electron mass ratio is mi =me = 1836A (A is atomic weight of the ion), the contribution of ions to the mass density of plasma is dominant. In many cases it is more convenient to reorganize the equations of motion for two °uids to the equation of motion for one °uid and Ohm's law. The total mass density of plasma ½m , the °ow velocity of plasma V , the electric charge density ½ and the current density j are de¯ned as follows: ½m = ne me + ni mi ; V =

(5:11)

ne me V e + ni mi V i ; ½m

(5:12)

½ = ¡ene + Zeni ;

(5:13)

j = ¡ene V e + Ze ni V i :

(5:14)

From eqs.(5.4),(5.5), it follows that @½m + r ¢ (½m V ) = 0; @t

(5:15)

@½ + r ¢ j = 0: @t

(5:16)

From eqs.(5.9) (5.10), we ¯nd ½m

@V + ne me (V e ¢ r)V e + ni mi (V i ¢ r)V i @t = ¡r(pe + pi ) + ½E + j £ B:

(5:17)

The charge neutrality of the plasma allows us to write ne ' Zni . Denote ¢ne = ne ¡ Zni , we have µ

½m = ni mi 1 + ½ = ¡e¢ne ;

¶

me Z ; mi

p = pi + pe ;

V =Vi+

me Z (V e ¡ V i ); mi

j = ¡ene (V e ¡ V i ):

Since me =mi ¿ 1, the second and third terms in left-hand side of eq.(5.17) can be written to be (V ¢ ¢)V . Since V e = V i ¡ j=ene ' V ¡ j=ene , eq.(5.9) reduces to µ

E+ V ¡

j ene

¶

£B+

me @V 1 R me @j ¡ rpe ¡ = 2 : ene ene e ne @t e @t

(5:18)

By use of the expression of speci¯c resistivity ´, (see sec.2.8) the collision term R is reduced to R = ne

µ

¶

me ºei (¡ene )(V e ¡ V i ) = ne e´j: ne e2

(5:19)

Equation (5.18) corresponds a generalized Ohm's law. Finally the equation of motion for one °uid model and a generalized Ohm's law are give by ½m

µ

@V + (V ¢ r)V @t

¶

= ¡rp + ½E + j £ B;

(5:20)

5.3 Simpli¯ed Magnetohydrodynamic Equations

µ

j E+ V ¡ ene

¶

£B+

41

1 me @j me @V ¡ ' 0: rpe ¡ ´j = 2 ene e ne @t e @t (j!=­e j ¿ 1)

(5:21)

The equation of continuity and Maxwell equations are @½m + r ¢ (½m V ) = 0; @t

(5:22)

@½ + r ¢ j = 0; @t

(5:23)

@B ; @t

(5:24)

r£E =¡

@D 1 r£B =j+ ; ¹0 @t

(5:25)

r ¢ D = ½;

(5:26)

r ¢ B = 0:

(5:27)

From eqs.(5.25),(5.24), it follows r £ r £ E = ¡¹0 @j=@t ¡ ¹0 ²0 @ 2 E=@t2 . A typical propagation velocity of magnetohydrodynamic wave or perturbation is Alfven velocity vA = B=(¹0 ½m )1=2 2 ¿ c2 as is described in sec.5.4 and is much smaller than light speed c and ! 2 =k2 » vA . Since 2 2 2 2 2 jr £(@B=@t)j = jr £r£Ej » k jEj, and ¹0 ²0 j@ E=@t j » ! jEj=c , the displacement current, @D=@t in (5.25) is negligible. Since the ratio of the ¯rst term (me =e)@j=@t in right-hand side of eq.(5.21) to the term (j £ B) in left-hand side is !=­e , the ¯rst term can be neglected, if j!=­e j ¿ 1. The second term (me =e)@V =@t in the right-hand side of eq.(5.21) is of the order of !=­e times as large as the term V £ B in the left-hand side. Therefore we may set the righthand side of eq.(5.21) nearly zero. When the term j £ B is eliminated by the use of eq.(5.20), we ¯nd E+V £B¡

¢ne mi dV 1 rpi ¡ ´j = E+ : en ne e dt

The ratio of (mi =e)dV =dt to V £ B is around j!=­i j, and ¢ne =ne ¿ 1. When j!=­i j ¿ 1, we ¯nd E+V £B¡

1 rpi = ´j: en

( j!=­i j ¿ 1)

(5:28)

5.3 Simpli¯ed Magnetohydrodynamic Equations When j!=­i j ¿ 1; j!=kj ¿ c, and the ion pressure term rpi can be neglected in Ohm's law, magnetohydrodynamic equations are simpli¯ed as follows: E + V £ B = ´j; ½m

µ

@V + (V ¢ r)V @t

(5:29) ¶

= ¡rp + j £ B;

(5:30)

42

5 Plasma as Magnetohydrodynamic Fluid

r £ B = ¹0 j; r£E =¡

(5:31)

@B ; @t

(5:32)

r ¢ B = 0;

(5:33)

@½m + (V ¢ r)½m + ½m r ¢ V = 0: @t

(5:34)

We may add the adiabatic equation as an equation of state; d (p½¡° m ) = 0; dt where the quantity ° is the ratio of speci¯c heats and ° = (2 + ±)=± (± is the number of degrees of freedom) is 5/3 in the three dimensional case ± = 3. Combined with eq.(5.34), the adiabatic equation becomes @p + (V ¢ r)p + °pr ¢ V = 0: @t

(5:35)

In stead of this relation, we may use the more simple relation of incompressibility r ¢ V = 0: if j(d½=dt)=½)j ¿ jr ¢ V j.

(5:36)

From eqs.(5.31),(5.32), the energy conservation law is given by

1 @ r ¢ (E £ B) + ¹0 @t

Ã

B2 2¹0

!

+ E ¢ j = 0:

(5:37)

From eq.(5.29), the third term in the right-hand side of eq.(5.37) becomes E ¢ j = ´j 2 + (j £ B) ¢ V :

(5:38)

By use of eqs.(5.30),(5.34), Lorentz term in eq.(5.38) is expressed by (j £ B) ¢ V =

@ ½m V 2 ½m V 2 ( )+r¢( V ) + V ¢ rp: @t 2 2

From eq.(5.35), it follows that ¡r ¢ (pV ) =

@p + (° ¡ 1)pr ¢ V @t

and @ V ¢ rp = @t

µ

p °¡1

¶

µ

¶

p +r¢ +p V: °¡1

Therefore the energy conservation law (5.37) is reduced to @ r ¢ (E £ H) + @t

Ã

½m V 2 p B2 + + 2 ° ¡ 1 2¹0

!

2

+ ´j + r ¢

Ã

!

½m V 2 p + + p V = 0: (5:39) 2 °¡1

5.4 Magnetoacoustic Wave

43

The substitution of (5.29) into (5.32) yields @B ´ = r £ (V £ B) ¡ ´r £ j = r £ (V £ B) + ¢B @t ¹0

(5:40)

@B ´ = ¡(V ¢ r)B ¡ B(r ¢ V ) + (B ¢ r)V + ¢B: @t ¹0

(5:41)

Here we used vector formula for r £ (V £ B) (see appendix) and r £ (r £ B) = ¡¢B (valid only in the case of orthogonal coordinates). The quantity ´=¹0 = ºm is called magnetic viscosity. The substitution of (5.31) into (5.30) yields Ã

dV B2 ½m = ¡r p + dt 2¹0

!

+

1 (B ¢ r)B: ¹0

(5:42)

The equation of motion (5.42) and the equation of magnetic di®usion (5.41) are fundamental equations of magnetohydrodynamics. Equation (5.33) r ¢ B = 0, equation of continuity (5.34) and equation of state (5.35) or (5.36) are additional equations. The ratio of the ¯rst term to the second term of the right-hand side in eq.(5.40), Rm , de¯ned by jr £ (V £ B)j V B=L ¹0 V L ´ Rm ¼ = 2 j¢B(´=¹0 )j (B=L )(´=¹0 ) ´

(5:43)

is called magnetic Reynolds number. The notation L is a typical plasma size. Magnetic Reynolds number is equal to the ratio of magnetic di®usion time ¿R = ¹0 L2 =´ to Alfven transit time ¿H = L=vA (it is assumed that v ¼ vA ), that is, Rm = ¿R =¿H . When Rm ¿ 1, the magnetic ¯eld in a plasma changes according to di®usion equation. When Rm À 1, it can be shown that the lines of magnetic force are frozen in the plasma. Let the magnetic °ux within the surface element ¢S be ¢©, and take the z axis in the B direction. Then ¢© is ¢© = B ¢ n¢S = B¢x¢y: As the boundary of ¢S moves, the rate of change of ¢S is @Vx d d (¢x) = (x + ¢x ¡ x) = Vx (x + ¢x) ¡ Vx (x) = ¢x; dt dt @x d (¢S) = dt

µ

@Vx @Vy + @x @y

¶

¢x¢y:

The rate of change of the °ux ¢© is dB d d (¢©) = ¢S + B (¢S) = dt dt dt

µ

dB + B(r ¢ V ) ¡ (B ¢ r)V dt

¶

z

¢S =

´ ¢Bz (¢S): ¹0

(5:44)

(see eq.(5.41)). When Rm ! 1; ´ ! 0, the rate of change of the °ux becomes zero, i.e., d(¢©)=dt ! 0. This means the magnetic °ux is frozen in the plasma. 5.4 Magnetoacoustic Wave As usual, we indicate zeroth-order quantities (in equilibrium state) by a subscript 0 and 1storder perturbation terms by a subscript 1, that is, ½m = ½m0 +½m1 ; p = p0 +p1 ; V = 0+V ; B = B 0 + B 1 . The case of ´ = 0 will be considered here. Then we ¯nd the 1st-order equations as follows: @½m1 + r ¢ (½m0 V ) = 0; @t

(5:45)

44

5 Plasma as Magnetohydrodynamic Fluid

½m0

@V + rp1 = j 0 £ B 1 + j 1 £ B 0 ; @t

(5:46)

@p1 + (V ¢ r)p0 + °p0 r ¢ V = 0; @t

(5:47)

@B 1 = r £ (V £ B 0 ): @t

(5:48)

If displacement of the plasma from the equilibrium position r 0 is denoted by »(r 0 ; t), it follows that »(r 0 ; t) = r ¡ r 0 ; V =

@» d» ¼ : dt @t

(5:49)

The substitution of (5.49) into eqs.(5.48),(5.45),(5.47) yields B 1 = r £ (» £ B 0 );

(5:50)

¹0 j 1 = r £ B 1 ;

(5:51)

½m1 = ¡r ¢ (½m0 »);

(5:52)

p1 = ¡» ¢ rp0 ¡ °p0 r ¢ »:

(5:53)

Then equation (5.46) is reduced to ½m0

1 @ 2» 1 = r(» ¢ rp0 + °p0 r ¢ ») + (r £ B 0 ) £ B 1 + (r £ B 1 ) £ B 0 : @t2 ¹0 ¹0

(5:54)

Let us consider the case where B 0 = const: p0 = const:, and the displacement is expressed by »(r; t) = » 1 exp i(k ¢ r ¡ !t), then eq.(5.54) is reduced to ¡½m0 !2 » 1 = ¡°p0 (k ¢ » 1 )k ¡ ¹¡1 0 (k £ (k £ (» 1 £ B 0 ))) £ B 0 :

(5:55)

Using the vector formula a £ (b £ c) = b (a ¢ c) ¡ c (a ¢ b), we can write eq.(5.55) as ³

´

³

´

(k ¢ B 0 )2 ¡ ¹0 ! 2 ½m0 » 1 + (B02 + ¹0 °p0 )k ¡ (k ¢ B 0 )B 0 (k ¢ » 1 ) ¡ (k ¢ B 0 )(B 0 ¢ » 1 )k = 0:

b ´ k=k; b ´ B 0 =B0 ; and the notations V ´ !=k, If the unit vectors of k; B 0 are denoted by k 2 ´ B2 2 b b vA 0 =(¹0 ½m0 ); ¯ ´ p0 =(B0 =2¹0 ); cos µ ´ (k ¢ b) are introduced, we ¯nd

(cos2 µ ¡

µ

¶

V2 °¯ b b ¢ » ) ¡ cos µ(b ¢ » )k b = 0: )» 1 + (1 + )k ¡ cos µb (k 1 1 2 2 vA

(5:56)

b and b, and the vector product of k b with eq.(5.56), yield The scalar product of eq.(5.56) with k

(1 +

°¯ V 2 b ¡ 2 )(k ¢ » 1 ) ¡ cos µ(b ¢ » 1 ) = 0; 2 vA

5.4 Magnetoacoustic Wave

45

2 °¯ b ¢ » ) ¡ V (b ¢ » ) = 0; cos µ(k 1 1 2 2 vA

(cos2 µ ¡

V2 b 2 )b ¢ (k £ » 1 ) = 0: vA

The solutions of these equations are magnetoacoustic wave. One solution is 2 V 2 = vA cos2 µ;

(» 1 ¢ k) = 0;

(» 1 ¢ B 0 ) = 0:

(5:57)

Since » 1 of this solution is orthogonal to k and B 0 , this is called torsional Alfv¶en wave (see sec.10.4). The other solutions are given by µ

V vA

¶4

¡ (1 +

µ

°¯ V ) 2 vA

¶2

+

°¯ cos2 µ = 0; 2

(5:58)

B 0 ¢ (k £ » 1 ) = 0: Since » 1 of these solutions are coplaner with k and B 0 , these solutions are compressional mode. If the velocity of sound is denoted by c2s = °p0 =½m0 , eq.(5.58) becomes 2 + c2s )V 2 + vA2 c2s cos2 µ = 0 V 4 + (vA

and Vf2 =

´ 1³ 2 2 2 2 vA + c2s ) + ((vA + c2s )2 ¡ 4vA cs cos2 µ)1=2 ; 2

(5:59)

Vs2 =

´ 1³ 2 2 2 2 vA + c2s ) ¡ ((vA + c2s )2 ¡ 4vA cs cos2 µ)1=2 : 2

(5:60)

The solution of eq.(5.59) is called compressional Alfv¶en wave (see sec.10.4) and the solution of eq.(5.60) is called magnetoacoustic slow wave. Characteristic velocity 2 vA =

B2 ¹0 ½m0

is called Alfv¶en velocity. The plasma with zero resistivity is frozen to the magnetic ¯eld. There is tension B 2 =2¹0 along the magnetic ¯eld line. As the plasma, of mass density ½m , sticks to the ¯eld lines, the magnetoacoustic waves can be considered as waves propagating along the strings of magnetic ¯eld lines (see sec.10.4).

46

6 Equilibrium

Ch.6 Equilibrium In order to maintain a hot plasma, we must con¯ne and keep it away from the vacuumcontainer wall. The most promising method for such con¯nement of a hot plasma is the use of appropriate strong magnetic ¯elds. An equilibrium condition must be satis¯ed for such magnetic con¯nement systems. 6.1 Pressure Equilibrium When a plasma is in the steady state, magnetohydrodynamic equation (5.30) yields the equilibrium equation rp = j £ B;

(6:1)

r £ B = ¹0 j;

(6:2)

r ¢ B = 0;

(6:3)

r ¢ j = 0:

(6:4)

and

From the equilibrium equation (6.1), it follows that B ¢ rp = 0;

(6:5)

j ¢ rp = 0:

(6:6)

Equation (6.5) indicates that B and rp are orthogonal, and the surfaces of constant pressure coincide with the magnetic surfaces. Equation (6.6) shows that the current-density vector j is everywhere parallel to the constant-pressure surfaces. Substitution of eq.(6.2) into eq.(6.1) yields Ã

B2 r p+ 2¹0

!

µ

¶

1 B @B=@l = (B ¢ r) = B2 ¡ n + b : ¹0 R B

(6:7)

The following vector relations were used here; B £ (r £ B) + (B ¢ r)B = r(B 2 =2);

(B ¢ r)B = B 2 [(b ¢ r)b + b((b ¢ r)B)=B]:

R is the radius of curvature of the line of magnetic force and n is the unit vector directed toward a point on the line of magnetic force from the center of curvature. l is the length along the ¯eld line. We ¯nd the right-hand side of eq.(6.7) can be neglected when the radius of curvature is much larger than the length over which the magnitude p changes appreciably, i.e., the size of the plasma, and the variation of B along the line of magnetic force is much smaller than the variation of B in the perpendicular direction. Then eq.(6.7) becomes p+

B2 B2 » 0; 2¹0 2¹0

6.2 Equilibrium Equation for Axially Symmetric and ¢ ¢ ¢

47

where B0 is the the value of the magnetic ¯eld at the plasma boundary (p=0). When the system is axially symmetric and @=@z = 0, eq.(6.7) exactly reduces to @ @r

Ã

B 2 + Bµ2 p+ z 2¹0

!

=¡

Bµ2 : r¹0

(6:8)

By the multipli¯cation of eq.(6.8) by r2 and the integration by parts we obtain Ã

B 2 + Bµ2 p+ z 2¹0

!

r=a

1 = ¼a2

Z

a

0

Ã

B2 p+ z 2¹0

!

2¼rdr

i.e., hpi +

hBz2 i B 2 (a) + Bµ2 (a) = pa + z : 2¹0 2¹0

(6:9)

h i is the volume average. As B 2 =2¹0 is the pressure of the magnetic ¯eld, eq.(6.9) is the equation of pressure equilibrium. The ratio of plasma pressure to the pressure of the external magnetic ¯eld B0 ¯´

p n(Te + Ti ) = B02 =2¹0 B02 =2¹0

(6:10)

is called the beta ratio. For a con¯ned plasma, ¯ is always smaller than 1, and is used as a ¯gure of merit of the con¯ning magnetic ¯eld. The fact that the internal magnetic ¯eld is smaller than the external ¯eld indicates the diamagnetism of the plasma. 6.2 Equilibrium Equation for Axially Symmetric and Translationally Symmetric Systems Let us use cylindrical coordinates (r; '; z) and denote the magnetic surface by Ã. The magnetic surface à in an axisymmetric system is given by (see (3.24)) à = rA' (r; z)

(6:11)

where (r; '; z) are cylindrical coordinates and the r and z components of the magnetic ¯eld are given by rBr = ¡

@Ã ; @z

rBz =

@Ã : @r

(6:12)

The relation B ¢ rp = 0 follows from the equilibrium equation and is expressed by ¡

@Ã @p @Ã @p + = 0: @z @r @r @z

Accordingly p is a functon of à only, i.e., p = p(Ã):

(6:13)

Similarly, from j ¢ rp = 0 and r £ B = ¹0 j , we may write ¡

@p @(rB' ) @p @(rB' ) + = 0: @r @z @z @r

This means that rB' is a function of à only and rB' =

¹0 I(Ã) : 2¼

(6:14)

48

6 Equilibrium

Fig.6.1 Magnetic surfaces à = rA' and I(Ã)

Equation (6.14) indicates that I(Ã) means the current °owing in the poloidal direction through the circular cross section within à = rA' (¯g.6.1). The r component of j £ B = rp leads to the equation on à : L(Ã) + ¹0 r 2 where L(Ã) ´

Ã

@p(Ã) ¹2 @I 2 (Ã) + 02 =0 @Ã 8¼ @Ã

@ 1 @ @2 r + 2 @r r @r @z

!

(6:15)

Ã:

This equation is called Grad-Shafranov equation. The current density is expressed in term of the function of the magnetic surface as jr =

¡1 @I(Ã) ; 2¼r @z

¡1 j' = ¹0 1 = ¹0 r

Ã

jz =

@ 1 @Ã 1 @ 2 Ã + @r r @r r @z 2

Ã

1 @I(Ã) ; 2¼r @r !

¹2 ¹0 r 2 p0 + 02 (I 2 )0 8¼

=¡

L(Ã) ¹0 r

!

or j=

I0 B + p0 re' ; 2¼

(6:16)

L(Ã) + ¹0 rj' = 0: The functions p(Ã) and I 2 (Ã) are arbitrary functions of Ã. Let us assume that p and I 2 are quadratic functions of à . The value Ãs at the plasma boundary can be chosen to be zero (Ãs = 0) without loss of generality. When the values at the boundary are p = ps ; I 2 = Is2 and the values at the magnetic axis are à = Ã0 ; p = p0 ; I 2 = I02 ; then p and I 2 are expressed by p(Ã) = ps + (p0 ¡ ps )

Ã2 ; Ã02

I 2 (Ã) = Is2 + (I02 ¡ Is2 )

Ã2 : Ã02

6.3 Tokamak Equilibrium

49

The equilibrium equation (6.15) is then reduced to L(Ã) + (®r2 + ¯)Ã = 0;

®=

2¹0 (p0 ¡ ps ) ; Ã02

¯=

¹20 (I02 ¡ Is2 ) : 4¼ 2 Ã02

Since Z

à (®r2 + ¯)ÃdV = 2¹0 r2

Z

1 ÃL(Ã)dV = r2

V

V

Z

S

Z

¹2 (p ¡ ps )dV + 02 4¼ V

1 Ãrà ¢ ndS ¡ r2

Z

V

Z

V

(I 2 ¡ Is2 ) dV; r2

1 (rÃ)2 dV = ¡ r2

Z

V

(Br2 + Bz2 )dV

eq.(6.15) of equilibrium equation is reduced to Z

(p ¡ ps )dV =

Z

´ 1 ³ 2 B'v ¡ B'2 + (Br2 + Bz2 ) dV: 2¹0

This is the equation of pressure balance under the assumption made on p(Ã) and I(Ã). The magnetic surface Ã, the magnetic ¯eld B and the pressure p in translationally symmetric system (@=@z = 0) are given by à = Az (r; µ);

Br =

1 @Ã ; r @µ

Bµ = ¡

@Ã ; @r

Bz =

¹0 I(Ã); 2¼

p = p(Ã): The equilibrium equation is reduced to µ

1 @ @Ã r r @r @r j=

¶

+

@p(Ã) 1 @ 2Ã ¹20 @I 2 (Ã) + ¹ + = 0; 0 r 2 @µ 2 @Ã 8¼ 2 @Ã

1 0 I B + p0 e z ; 2¼

¢Ã + ¹0 jz = 0: It is possible to drive the similar equilibrium equation in the case of helically symmetric system.

6.3 Tokamak Equilibrium1 The equilibrium equation for an axially symmetric system is given by eq.(6.15). The 2nd and 3rd terms of the left-hand side of the equation are zero outside the plasma region. Let us use toroidal coordinates (b; !; ') (¯g.6.2). The relations between these to cylindrical coordinates (r; '; z) are r=

R0 sinh b ; cosh b ¡ cos !

z=

R0 sin ! : cosh b ¡ cos !

50

6 Equilibrium

Fig.6.2 Toroidal coordinates.

The curves b = b0 are circles of radius a = R0 sinh b0 , centered at r = R0 coth b0 ; z = 0. The curves ! =const. are also circles. When the magnetic-surface function à is replaced by F , according to Ã=

F (b; !) ¡ cos !)1=2

21=2 (cosh b

the function F satis¯es @F @2F 1 @2F ¡ coth b + F =0 + 2 2 @b @b @! 4 outside the plasma region. When F is expanded as F = §gn (b) cos n!; the coe±cient gn satis¯es µ

¶

dgn 1 d2 gn ¡ n2 ¡ ¡ coth b gn = 0: 2 db db 4 There are two independent solutions: µ

n2 ¡

¶

1 d gn = sinh b Qn¡1=2 (cosh b); 4 db

µ

n2 ¡

¶

1 d fn = sinh b Pn¡1=2 (cosh b): 4 db

Pº (x) and Qº (x) are Legendre functions. If the ratio of the plasma radius to the major radius a=R0 is small, i.e., when eb0 À 1, then gn and fn are given by g0 = eb=2 ;

1 g1 = ¡ e¡b=2 ; 2

f0 =

2 b=2 e (b + ln 4 ¡ 2); ¼

f1 =

2 3b=2 : e 3¼

If we take terms up to cos !, F and à are F = c0 g0 + d0 f0 + 2(c1 g1 + d1 f1 ) cos !; Ã=

F ¼ e¡b=2 (1 + e¡b cos !)F: ¡ cos !)1=2

21=2 (cosh b

Use the coordinates ½; ! 0 shown in ¯g.6.3. These are related to the cylindrical and toroidal coordinates as follows: r = R0 + ½ cos ! 0 =

R0 sinh b cosh b ¡ cos !

z = ½ sin ! 0 =

R0 sin ! : cosh b ¡ cos !

6.3 Tokamak Equilibrium

51

Fig.6.3 The coordinates r; z and ½; ! 0

When b is large, the relations are ½ ¼ e¡b : 2R0

!0 = !;

Accordingly the magnetic surface à is expressed by 2 d0 (b + ln 4 ¡ 2) ¼ ∙µ ¶ µ ¶¸ 4 2 + c0 + d0 (b + ln 4 ¡ 2) e¡b + d1 eb ¡ c1 e¡b cos ! ¼ 3¼

à =c0 +

µ

=d00 ln

¶

8R ¡2 + ½

µ

µ

¶

¶

d00 8R h1 ¡1 ½+ ln + h2 ½ cos !: 2R ½ ½

In terms of Ã, the magnetic-¯eld components are given by @Ã ; @z @Ã ; rB½ = ¡ ½@! 0 rBr = ¡

@Ã ; @r @Ã = : @½

rBz = rB!0

From the relation ¡

¡¹0 Ip d00 = rB ! 0 ¼ R ; ½ 2¼½

the parameter d00 can be taken as d00 = ¹0 Ip R=2¼. Here Ip is the total plasma current in the ' direction. The expression of the magnetic surface is reduced to Ã=

µ

¶

µ

µ

¶

¶

¹ 0 Ip 8R 8R h1 ¹0 Ip R ¡2 + ¡1 ½+ ln ln + h2 ½ cos ! 0 2¼ ½ 4¼ ½ ½

(6:17)

where R0 has been replaced by R. In the case of a=R ¿ 1, the equation of pressure equilibrium (6.9) is hpi ¡ pa =

1 2 ((B'v )a + (Br2 + Bz2 )a ¡ hB'2 i): 2¹0

Here h i indicates the volume average and pa is the plasma pressure at the plasma boundary. The value of Br2 + Bz2 is equal to B!2 0 . The ratio of hpi to hB!2 0 i=2¹0 is called the poloidal beta ratio ¯p . When pa = 0, ¯p is ¯p = 1 +

2 ¡ hB 2 i B'v 2B'v ' ¼ 1 + 2 hB'v ¡ B' i: 2 B! 0 B! 0

(6:18)

52

6 Equilibrium

Fig.6.4 Diamagnetism (¯p > 1) and paramagnetism (¯p < 1)

Fig.6.5 Displacement of the plasma column. Ã0 (½0 ) = Ã0 (½) ¡ Ã00 (½)¢ cos !; ½0 = ½ ¡ ¢ cos !:

B' and B'v are the toroidal magnetic ¯elds in the plasma and the vacuum toroidal ¯elds respectively. When B' is smaller than B'v , the plasma is diamagnetic, ¯p > 1. When B' is larger than B'v , the plasma is paramagnetic, ¯p < 1. When the plasma current °ows along a line of magnetic force, the current produces the poloidal magnetic ¯eld B! 0 and a poloidal component of the plasma current appears and induces an additional toroidal magnetic ¯eld. This is the origin of the paramagnetism. When the function (6.17) is used, the magnetic ¯eld is given by B! 0 =

¡¹0 Ip 1 @Ã = + r @½ 2¼½

B½ = ¡

1 @Ã = r½ @!0

µ

µ

µ

h1 ¹0 Ip 8R 1 h2 ¡ 2 ln + 4¼R ½ R ½

µ

¶

µ

¶¶

¹ 0 Ip 8R 1 h1 ¡1 + ln h2 + 2 4¼R ½ R ½

cos ! 0 ;

¶¶

sin ! 0 :

9 > > > = > > > ;

(6:19)

The cross section of the magnetic surface is the form of Ã(½; ! 0 ) = Ã0 (½) + Ã1 cos ! 0 : When ¢ = ¡Ã1 =Ã00 is much smaller than ½, the cross section is a circle whose center is displaced by an amount (see ¯g.6.5) µ

¶

8R 2¼ ½2 ¡1 + ln (h1 + h2 ½2 ): ¢(½) = 2R ½ ¹ 0 Ip R Let us consider the parameters h1 and h2 . As will be shown in sec.6.4, the poloidal component B!0 of the magnetic ¯eld at the plasma surface (r = a) must be µ

B!0 (a; ! 0 ) = Ba 1 +

a ¤ cos ! 0 R

¶

(6:20)

6.3 Tokamak Equilibrium

53

at equilibrium. a is the plasma radius and ¤ = ¯p +

li ¡1 2

(6:21)

and ¯p is the poloidal beta ratio ¯p = and li is

R

li =

p

(6:22)

(Ba2 =2¹0 ) B!2 0 ½d½d! 0 : ¼a2 Ba2

(6:23)

The parameters h1 and h2 must be chosen to satisfy B½ = 0 and B!0 = Ba (1 + (a=R)¤ cos ! 0 ) at the plasma boundary, i.e., h1 =

µ

¶

¹0 Ip 2 1 ; a ¤+ 4¼ 2

h2 = ¡

µ

¶

8R ¹ 0 Ip 1 ln : +¤¡ 4¼ a 2

(6:24)

Accordingly à is given by µ

¶

¹ 0 Ip 8R ¹ 0 Ip R ¡2 ¡ ln Ã= 2¼ ½ 4¼

Ã

µ

½ 1 ln + ¤ + a 2

¶Ã

a2 1¡ 2 ½

!!

½ cos ! 0 :

(6:25)

The term h2 ½ cos ! 0 in à brings in the homogeneous vertical ¯eld Bz =

h2 ; R

which is to say that we must impose a vertical external ¯eld. When we write Ãe = h2 ½ cos ! 0 so that à is the sum of two terms, à = Ãp + Ãe , Ãe and Ãp are expressed by ¹ 0 Ip Ãe = ¡ 4¼

µ

¶

8R 1 ln ½ cos ! 0 +¤¡ a 2

µ

¶

¹ 0 Ip R 8R ¹0 Ip ¡2 + Ãp = ln 2¼ ½ 4¼

õ

¶

(6:26) µ

8R a2 1 ¡1 ½+ ln ¤+ ½ ½ 2

¶!

cos ! 0 :

(6:27)

These formulas show that a uniform magnetic ¯eld in the z direction, µ

¶

¹0 Ip 8R 1 ln ; +¤¡ B? = ¡ 4¼R a 2

(6:28)

must be applied in order to maintain a toroidal plasma in equilibrium (¯g.6.6). This vertical ¯eld weakens the inside poloidal ¯eld and strengthens the outside poloidal ¯eld. The amount of B? (eq.6.28) for the equilibrium can be derived more intuitively. The hoop force by which the current ring of a plasma tends to expand is given by ¯

@ Lp Ip2 ¯¯ 1 @Lp Fh = ¡ = I2 ; @R 2 ¯Lp Ip =const: 2 p @R

where Lp is the self-inductance of the current ring: µ

¶

8R li Lp = ¹0 R ln + ¡2 : a 2

54

6 Equilibrium

Fig.6.6 Poloidal magnetic ¯eld due to the combined plasma current and vertical magnetic ¯eld.

Fig.6.7 Equilibrium of forces acting on a toroidal plasma.

Accordingly, the hoop force is µ

¶

¹0 Ip2 8R li ln Fh = + ¡1 : 2 a 2 The outward force Fp exerted by the plasma pressure is (¯g.6.7) Fp = hpi¼a2 2¼: The inward (contractive) force FB1 due to the tension of the toroidal ¯eld inside the plasma is FB1 = ¡

hB'2 i 2 2 2¼ a 2¹0

and the outward force FB2 by the pressure due to the external magnetic ¯eld is 2 B'v 2¼ 2 a2 : 2¹0

FB2 =

The force FI acting on the plasma due to the vertical ¯eld B? is FI = Ip B? 2¼R: Balancing these forces gives µ

¶

Ã

2 B'v hB'2 i ¹0 Ip2 8R li ¡ ln + ¡ 1 + 2¼ 2 a2 hpi + 2 a 2 2¹0 2¹0

and the amount of B? necessary is B? =

µ

¶

¡¹0 Ip 1 8R li ln ; + ¡ 1 + ¯p ¡ 4¼R a 2 2

!

+ 2¼RIp B? = 0;

6.4 Poloidal Field for Tokamak Equilibrium

55

Fig.6.8 Volume element of a toroidal plasma.

where ¤ = ¯p + li =2 ¡ 1. Eq.(6.9) is used for the derivation. 6.4 Poloidal Field for Tokamak Equilibrium The plasma pressure and the magnetic stress tensor are given by2 Ã

B2 = p+ 2¹0

T®¯

!

±®¯ ¡

B® B¯ : ¹0

Let us consider a volume element bounded by (!; ! + d!); ('; ' + d'); and (0; a) as is shown in ¯g.6.8. Denote the unit vectors in the directions r; z; ' and ½; ! by er ; ez ; e' and e½ ; e! ; respectively. The relations between these are e½ = er cos ! + ez sin !;

@e! = ¡e½ ; @!

e! = ez cos ! ¡ er sin !;

@e½ = e! : @!

(Here ! is the same as ! 0 of sec.6.3). Let dS½ ; dS! ; dS' be surface-area elements with the normal vectors e½ ; e! ; e' : Then estimate the forces acting on the surfaces dS' ('); dS' ('+d') ; dS! ('); dS! (! + d!); and dS½ (a): The sum F ' of forces acting on dS' (') and dS' (' + d') is given by F ' = ¡d!d' = ¡d!d'

Z

0

Z

aµ

T''

0

a³

@e' @e! @e½ + T'! + T'½ @' @' @'

¶

½d½ ´

0 0 T'' (e! sin ! ¡ e½ cos !) ¡ T'! e' sin ! ½d½:

When the forces acting on dS! (!) and dS! (! + d!) are estimated, we must take into account the di®erences in e! ; T!® ; dS! = (R + ½ cos !) d½ d' at ! and ! + d!: The sum F ! of forces is Z

aµ

@ (e! (R + ½ cos !)) @! 0 @ @ +T!' (e' (R + ½ cos !)) + T!½ (e½ (R + ½ cos !)) @! @! ¶ @T!! @T!' @T!½ + Re! + Re' + Re½ d½ @! @! @!

F ! = ¡d!d'

µ

T!!

= d!d' Re½

Z

Ã

0

a

+e! sin ! Ã

¶

0 T!! d½

+d!d'e' sin !

Z

Z

"

µ

+ d!d' e½ cos !

0 T!! ½d½ 0 T!' ½d½

¡R ¡R

Z

Z

(1)

Z

!#

@T!! d½ @! (1)

!

@T!' d½ @!

0 T!! ½d½

+R

Z

¶

(1) T!! d½

56

6 Equilibrium Ã

+d!d' ¡e! R µ

= d!d' Re½ Ã

Z

a

0

Z

(1) T!½ d½

¶

¡ e½ R

Z

!

(1)

@T!½ d½ @!

0 T!! d½

Z

+d!d'e½ cos ! Ã

Z

+d!d'e! sin ! Ã

+d!d'e' sin !

Z

0 T!! ½d½

0 T!! ½d½

0 T!! ½d½

+R ¡R ¡R

Z Ã Z Ã Z

(1)

(1) T!!

@T!½ ¡ @!

(1) T!½

@T!! + @!

(1)

!

(1)

! !

!

d½

!

d½

@T!' d½ : @!

As B½ (a) = 0; the force F ½ acting on dS½ (a) is 0 (1) 0 2 Ra ¡ (T½½ Ra + T½½ a cos !)): F ½ = ¡e½ T½½ (R + a cos !)ad'd! = e½ (¡T½½

The equilibrium condition F ' + F ! + F ½ = 0 is reduced to Z

0 0 T!! d½ = T½½ (a)a;

@ @!

Z

Z Ã

cos !

Z

0 (T''

+

(1) T!' d½

2 sin ! = R (1)

(1) T!½

@T!! + @!

0 T!! )½d½

+R

!

Z

d½ =

Z Ã

(1) T!!

0 T!' ½d½;

sin ! R

(6:29)

Z ³ (1)

@T!½ ¡ @!

´

0 0 ¡ T'' T!! ½d½;

!

(6:30)

0 2 (1) d½ ¡ T½½ a cos ! ¡ T½½ Ra = 0:

(6:31)

From T (1) / sin !; cos !; it follows that @ 2 T (1) =@! 2 = ¡T (1) : So eq.(6.30) is Z Ã

(1)

!

@T!½ cos ! (1) ¡ T!! d½ = @! R

Z

0 0 ¡ T'' (T!! )½d½:

Using this relation, we can rewrite eq.(6.31) as (1) T½½ (a)

µ

a 2 0 = cos ! ¡T½½ (a) + 2 R a

Z

0

a

¶

0 T'' :½d½

:

(6:32)

T½½ and T'' are given by T½½

B'2 B!2 =p+ + ; 2¹0 2¹0

T''

B'2 B!2 ¡ =p+ : 2¹0 2¹0

(6:33)

From eq.(6.14), B' is B' =

µ

¶

µ

¶

¹0 I(Ã) ½ ½ ¹0 I(Ã) ・ ・ = B' (½) 1 ¡ cos ! +・ ・ ・ : 1 ¡ cos ! +・ = 2¼r 2¼R R R (1)

When B! (a) is written as B! (a) = Ba + B! , eqs.(6.33) and (6.34) yield the expression (1) T½½ (a) =

(1) 2 B'v (a) a Ba B! ¡ 2 cos !: ¹0 2¹0 R

(6:34)

6.5 Upper Limit of Beta Ratio

57 (1)

On the other hand, eqs.(6.9) and (6.32) give T½½ (a) as Ã

2 B'v (a) hB 2 i hB'2 i B2 a = cos ! ¡pa ¡ a ¡ + hpi + ! ¡ R 2¹0 2¹0 2¹0 2¹0

(1) T½½ (a)

a = cos ! R

Ã

2 (a) Ba2 B 2 B'v li + 2(hpi ¡ pa ) ¡ a ¡ 2¹0 ¹0 ¹0

!

!

where li is the normalized internal inductance of the plasma per unit length (the internal induc(1) tance Li of the plasma per unit length is given by ¹0 li =4¼). Accordingly, B! must be B!(1)

µ

¶

li 2¹0 (hpi ¡ pa ) a ¡1 : = Ba cos ! + R 2 Ba2

Ba is ! component of the magnetic ¯eld due to the plasma current Ip , i.e., Ba = ¡

¹ 0 Ip : 2¼a

When the poloidal ratio ¯p (recall that this is the ratio of the plasma pressure p to the magnetic (1) pressure due to Ba ) is used, B! is given by B!(1) =

µ

¶

li a Ba cos ! + ¯p ¡ 1 : R 2

(6:35)

6.5 Upper Limit of Beta Ratio In the previous subsection, the value of B! necessary for equilibrium was derived. derivation, (a=R)¤ < 1 was assumed, i.e., µ

li ¯p + 2

¶

<

In this

R : a

The vertical ¯eld B? for plasma equilibrium is given by µ

¶

a 8R 1 B? = Ba ln : +¤¡ 2R a 2 The direction of B? is opposite to that of B! produced by the plasma current inside the torus, so that the resultant poloidal ¯eld becomes zero at some points in inside region of the torus and a separatrix is formed. When the plasma pressure is increased and ¯p becomes large, the necessary amount of B? is increased and the separatrix shifts toward the plasma. For simplicity, let us consider a sharp-boundary model in which the plasma pressure is constant inside the plasma boundary, and in which the boundary encloses a plasma current Ip : Then the pressure-balance equation is 2 2 B'i B'v B!2 ¼p+ + : 2¹0 2¹0 2¹0

(6:36)

The ' components B'v ; B'i of the ¯eld outside and inside the plasma boundary are proportional 0 0 to 1=r; according to eq.(6.14). If the values of B'v ; B'i at r = R are denoted by B'v ; B'i respectively, eq.(6.36) may be written as 0 2 0 2 B!2 = 2¹0 p ¡ ((B'v ) ¡ (B'i ) )

µ

R r

¶2

:

58

6 Equilibrium

The upper limit of the plasma pressure is determined by the condition that the resultant poloidal ¯eld at r = rmin inside the torus is zero, 2¹0 pmax

2 rmin 0 2 0 2 = (B'v ) ¡ (B'i ) : R2

(6:37)

As r is expressed by r = R + a cos !; eq.(6.37) is reduced (with (rmin = R ¡ a)) to B!2

= 2¹0 pmax

Ã

r2 1 ¡ min r2

!

Here a=R ¿ 1 is assumed. From the relation beta ratio is ¯pc =

= 8¹0 pmax H

a ! cos2 : R 2

B! ad! = ¹0 Ip ; the upper limit ¯pc of the poloidal

R ¼2 R ¼ 0:5 : 16 a a

(6:38)

Thus the upper limit of ¯pc is half of the aspect ratio R=a in this simple model. When the rotational transform angle ¶ and the safety factor qs = 2¼=¶ are introduced, we ¯nd that B! a = B' R

µ

¶ 2¼

¶

a ; Rqs

=

so that p p ¼ 2 ¯= 2 B =2¹0 B! =2¹0

Ã

B! B'

!2

=

µ

a Rqs

¶2

¯p :

(6:39)

Accordingly, the upper limit of the beta ratio is ¯c =

0:5 a : qs2 R

(6:40)

6.6 P¯rsch-SchlÄ uter Current3 When the plasma pressure is isotropic, the current j in the plasma is given by eqs.(6.1) and (6.4) as j? =

b £ rp B

r ¢ j k = ¡r ¢ j ? = ¡r ¢ Then jk is r ¢ j k = ¡rp ¢

µµ

µ

¶

B £ rp = ¡rp ¢ r £ B2 ¶

1 ¹0 j r 2 £B + 2 B B

@jk (rB £ b) ; = 2rp ¢ @s B2

¶

= 2rp ¢

µ

¶

B : B2

rB £ B B3

(6:41) (6:42)

where s is length along a line of magnetic force. In the zeroth-order approximation, we can put B / 1=R; p = p(r); and @=@s = (@µ=@s)@=@µ = (¶=(2¼R))@=@µ; where ¶ is the rotational transform angle. When s increases by 2¼R, µ increases by ¶. Then eq.(6.42) is reduced to ¶ @jk @p 2 =¡ sin µ 2¼R @µ @r RB

6.6 P¯rsch-SchlÄ uter Current

59

Fig.6.9 P¯rsch-SchlÄ uter current jk in a toroidal plasma.

i.e., @jk 4¼ @p =¡ sin µ; @µ ¶B @r

jk =

4¼ @p cos µ: ¶B @r

(6:43)

This current is called the P¯rsch-SchlÄ uter current (¯g.6.9). These formulas are very important, and will be used to estimate the di®usion coe±cient of a toroidal plasma. The P¯rsch-SchlÄ uter current is due to the short circuiting, along magnetic-¯eld lines, of toroidal drift polarization charges. The resultant current is inversely proportional to ¶. Let us take the radial variation in plasma pressure p(r) and ¶ to be µ

p(r) = p0 1 ¡ ¶(r) = ¶0

µ ¶m ¶

r a

;

µ ¶2l¡4

r a

respectively; then jk is 4¼mp0 jk = ¡ B¶0 a

µ ¶m¡2l+3

r a

cos µ:

Let us estimate the magnetic ¯eld B ¯ produced by jk . As a=R is small, B ¯ is estimated from the corresponding linear con¯guration of ¯g.6.9. We utilize the coordinates (r; µ 0 ; ³) and put µ = ¡µ0 and jk ¼ j³ (¶ is assumed not to be large). Then the vector potential A¯ = (0; 0; A¯³ ) for B ¯ is given by 0

@A¯³

1

2 ¯ 1 @ A³ 1 @ @ A+ r = ¡¹0 j³ : r @r @r r 2 @µ0 2

When A¯³ (r; µ0 ) = A¯ (r) cos µ0 , and parameters s = m ¡ 2l + 3; ® = 4¼mp0 ¹0 =(B¶0 ) = m¯0 B=(¶0 =2¼) (¯0 is beta ratio) are used, we ¯nd 1 @ r @r

Ã

@A¯ r @r

!

A¯ ® ¡ 2 = r a

µ ¶s

r a

:

In the plasma region (r < a), the vector potential is A¯in

=

Ã

!

®r s+2 + ±r cos µ0 ((s + 2)2 ¡ 1)as+1

60

6 Equilibrium

and A¯out outside the plasma region (r > a) is A¯out =

° cos µ0 ; r

where ± and ° are constants. Since Br¯ ; Bµ¯0 must be continuous at the boundary r = a, the solution for B ¯ inside the plasma is ® Br¯ = ¡ (s + 2)2 ¡ 1 Bµ¯0

õ ¶ s+1 Ã

r a

µ ¶s+1

r ® =¡ (s + 2) 2 (s + 2) ¡ 1 a

and the solution outside is Br¯ = Bµ¯

!

s+3 ¡ sin µ0 ; 2

s+1 ® 2 (s + 2) ¡ 1 2

s+1 ¡® = (s + 2)2 ¡ 1 2

> > s+3 > ¡ cos µ0 > ; 2

µ ¶2

sin µ 0 ;

a R

cos µ0

a r

µ

!

9 > > > > =

¶2

(6:44)

(Br = r¡1 @A³ =@µ 0 ; Bµ0 = ¡@A³ =@r). As is clear from eq.(6.44), there is a homogeneous vertical-¯eld component Bz =

¡(m ¡ 2l + 6)m ¡(s + 3)® ¯ ¯ = B = ¡f (m; l) B: 2 2 2((s + 2) ¡ 1) 2((m ¡ 2l + 5) ¡ 1) (¶0 =2¼) (¶0 =2¼)

This ¯eld causes the magnetic surface to be displaced by the amount ¢. From eq.(3.42), ¢ is given by ¡2¼Bz ¯ ¢ (2¼)2 ¯ ¼ f (m; l) : = = f (m; l) R ¶(¢)B ¶(¢)¶0 (¶0 =2¼)2 f (m; l) is of the order of 1 and the condition ¢＜ a=2 gives the upper limit of the beta ratio: 1a ¯c ＜ 2R

µ

¶ 2¼

¶2

:

This critical value is the same as that for the tokamak. 6.7 Virial Theorem The equation of equilibrium j £ B = (r £ B) £ B = rp can be reduced to X @ i

@xi

Tik ¡

@p =0 @xk

(6:45)

where Tik =

1 1 (Bi Bk ¡ B 2 ±ik ): ¹0 2

(6:46)

This is called the magnetic stress tensor. From the relation (6.45), we have Z ÃÃ S

B2 p+ 2¹0

!

B(B ¢ n) n¡ ¹0

!

dS = 0

(6:47)

6.7 Virial Theorem

61

where n is the outward unit normal to the closed surface surrounding a volume V. Since the other relation X @

@xi

i

(xk (Tik ¡ p±ik )) = (Tkk ¡ p) + xk

X @ i

@xi

(Tik ¡ p±ik ) = (Tkk ¡ p)

holds,it follows that Z Ã V

B2 3p + 2¹0

!

dV =

Z ÃÃ

B2 p+ 2¹0

S

!

(B ¢ r)(B ¢ n) (r ¢ n) ¡ ¹0

!

dS:

(6:48)

This is called the virial theorem. When a plasma ¯lls a ¯nite region with p = 0 outside the region, and no solid conductor carries the current anywhere inside or outside the plasma, the magnitude of the magnetic ¯eld is the order of 1=r3 , so the surface integral approaches zero as the plasma surface approaches in¯nity (r ! 0). This contradicts that the volume integral of (6.48) is positive de¯nite. In other words, a plasma of the ¯nite extent cannot be in equilibrium unless there exist solid conductors to carry the current. Let us apply the virial theorem (6.48) and (6.47) to a volume element of an axisymmetric plasma bounded by a closed toroidal surface St formed by the rotation of an arbitrary shaped contour lt . We denote the unit normal and tangent of the contour lt by n and l respectively and a surface element of the transverse cross section by dS' . The volume and the surface element are related by dV = 2¼rdS' : The magnetic ¯eld B is expressed by B = B' e' + B p where B p is the poloidal ¯eld and B' is the magnitude of the toroidal ¯eld and e' is the unit vector in the ' direction. Let us notice two relations Z Z

®

r (r ¢ n)dSt = (® + 3) r® (er ¢ n)dSt = Z

Z

r

Z

r ® dV

(6:49)

r ¢ (r ® er )dV =

(®¡1)

dV = 2¼

Z

Z

1 @ ®+1 dV = (® + 1) r r @r

r® dS'

(6:50)

where er is the unit vector in the r direction. Applying (6.48) to the full torus surrounded by St , we get Z Ã

=

B'2 + Bp2 3p + 2¹0

Z ÃÃ

!

dV =

B 2 ¡ Bn2 p+ l 2¹0

!

Z ÃÃ

B'2 + Bp2 p+ 2¹0 !

!

Bn Bl (n ¢ r) ¡ (l ¢ r) ¹0

Bn (B ¢ r) (n ¢ r) ¡ ¹0

dSt +

Z

!

dSt

B' 2 (n ¢ r) dSt ; 2¹0

(6:51)

because of B p = Bl l + Bn n (see ¯g.6.10a). When the vacuum toroidal ¯eld (without plasma) is denoted by B'0 , this is given by B'0 = ¹0 I=(2¼r), where I is the total coil current generating the toroidal ¯eld. By use of (6.50), (6.51) reduces to4 Z Ã

2 Bp2 + B'2 ¡ B'0 3p + 2¹0

!

2¼rdS'

62

6 Equilibrium

Fig.6.10 Integral region of Virial theorem (a) (6.48) and (b) (6.47).

=

Z ÃÃ

B 2 ¡ Bn2 p+ l 2¹0

!

Bn Bl (n ¢ r) ¡ (l ¢ r) ¹0

!

dSt :

(6:52)

Applying eq.(6.47) to the sector region surrounded by ' = 0; ' = ¢' and St (see ¯g.6.10b) and taking its r component gives4 ¡¢' 2¼

Z Ã

Z Ã

B'2 B2 ¡ p+ 2¹0 ¹0

!

¢' dS' + 2¼

2 Bp2 ¡ B'2 + B'0 p+ 2¹0

!

dS' =

Z ÃÃ

Z ÃÃ

B2 p+ 2¹0

!

(B ¢ er )(B ¢ n) (n ¢ er ) ¡ ¹0

B 2 ¡ Bn2 p+ l 2¹0

!

! !

Bl Bn (n ¢ er ) ¡ (l ¢ er ) ¹0

dSt = 0

dSt = 0: (6:53)

Eqs. (6.52) and (6.53) are used to measure the poloidal beta ratio (6.18) and the internal plasma inductance per unit length (6.23) of arbitrary shaped axisymmetric toroidal plasma by use of magnetic probes surrounding the plasma.

References 1. V. S. Mukhovatov and V. D. Shafranov: Nucl. Fusion 11, 605 (1971) 2. V. D. Shafranov: Plasma Physics, J. of Nucl. Energy pt.C5, 251 (1963) 3. D. P¯rsch and A. SchlÄ uter: MPI/PA/7/62 Max-Planck Institut fÄ ur Physik und Astrophysik, MÄ unchen (1962) 4. V. D. Shafranov: Plasma Physics 13 757 (1971)

63

Ch.7 Di®usion of Plasma, Con¯nement Time Di®usion and con¯nement of plasmas are among the most important subjects in fusion research, with theoretical and experimental investigations being carried out concurrently. Although a general discussion of di®usion and con¯nement requires the consideration of the various instabilities (which will be studied in subsequent chapters), it is also important to consider simple but fundamental di®usion for the ideal stable cases. A typical example (sec.7.1) is classical di®usion, in which collisions between electrons and ions are dominant e®ect. The section 7.2 describe the neoclassical di®usion of toroidal plasmas con¯ned in tokamak, for both the rare-collisional and collisional regions. Sometimes the di®usion of an unstable plasma may be studied in a phenomenological way, without recourse to a detailed knowledge of instabilities. In this manner, di®usions caused by °uctuations in a plasma are explained in secs.7.3 and 7.4. The transport equation of particles is @ n(r; t) + r ¢ (n(r; t)V (r; t)) = 0 @t

(7:1)

provided processes of the ionization of neutrals and the recombination of ions are negligible (see ch.5.1). The particle °ux ¡ = nV is given by n(r; t)V (r; t) = ¡D(r; t)rn(r; t) in many cases, where D is di®usion coe±cient. (Additional terms may be necessary in more general cases.) Di®usion coe±cient D and particle con¯nement time ¿p are related by the di®usion equation of the plasma density n as follows: r ¢ (Drn(r; t)) =

@ n(r; t): @t

Substitution of n(r; t) = n(r) exp(¡t=¿p ) in di®usion equation yields r ¢ (Drn(r)) = ¡

1 n(r): ¿p

When D is constant and the plasma column is a cylinder of radius a, the di®usion equation is reduced to µ

1 @ @n r r @r @r

¶

+

1 n = 0: D¿p

The solution satisfying the boundary condition n(a) = 0 is n = n0 J 0

µ

¶

Ã

t 2:4r exp ¡ a ¿p

!

and the particle con¯nement time is ¿p =

a2 a2 = ; 2:42 D 5:8D

(7:2)

where J0 is the zeroth-order Bessel function. The relationship (7.2) between the particle con¯nement time ¿p and D holds generally, with only a slight modi¯cation of the numerical factor.

64

7 Di®usion of Plasma, Con¯nement Time

This formula is frequently used to obtain the di®usion coe±cient from the observed values of the plasma radius and particle con¯nement time. The equation of energy balance is given by eq.(A.19), which will be derived in appendix A, as follows: @ @t

µ

3 n∙T 2

¶

+r¢

µ

¶

X 3 @v i ¦ij : ∙T nv + r ¢ q = Q ¡ pr ¢ v ¡ 2 @x j ij

(7:3)

The ¯rst term in the right-hand side is the heat generation due to particle collisions per unit volume per unit time, the second term is the work done by pressure and the third term is viscous heating. The ¯rst term in the left-hand side is the time derivative of the thermal energy per unit volume, the second term is convective energy loss and the third term is conductive energy loss. Denoting the thermal conductivity by ∙T , the thermal °ux due to heat conduction may be expressed by q = ¡∙T r(∙T ): If the convective loss is neglected and the heat sources in the right-hand side of eq.(7.3) is zero, we ¯nd that @ @t

µ

3 n∙T 2

¶

¡ r ¢ ∙T r(∙T ) = 0:

In the case of n = const:, this equation reduces to @ @t

µ

3 ∙T 2

¶

=r¢

µ

¶

∙T r(∙T ) : n

When the thermal di®usion coe±cient ÂT is de¯ned by ÂT =

∙T ; n

the same equation on ∙T is obtained as eq.(7.1). In the case of ÂT = const:, the solution is ∙T = ∙T0 J0

µ

¶

µ

t 2:4 r exp ¡ a ¿E

¶

;

¿E =

a2 : 5:8(2=3)ÂT

(7:4)

The term ¿E is called energy con¯nement time. 7.1 Collisional Di®usion (Classical Di®usion) 7.1a Magnetohydrodynamic Treatment A magnetohydrodynamic treatment is applicable to di®usion phenomena when the electronto-ion collision frequency is large and the mean free path is shorter than the connection length of the inside regions of good curvature and the outside region of bad curvature of the torus; i.e., vTe 2¼R < ; » ºei ¶ 1 ¶ 1 ¶ ºei > » ºp ´ R 2¼ vTe = R 2¼

µ

∙Te me

¶1=2

where vTe is electron thermal velocity and ºei is electron to ion collision frequency. From Ohm's law (5.28) E+v£B¡

1 rpi = ´j; en

7.1 Collisional Di®usion (Classical Di®usion)

65

Fig.7.1 Electric ¯eld in a plasma con¯ned in a toroidal ¯eld. The symbols ­ and ¯ here show the direction of the P¯rsch-SchlÄ uter current.

the motion of plasma across the lines of magnetic force is expressed by 1 nv ? = B 1 B

=

µµ

¶

¶

me ºei rp ∙Ti rn £ b ¡ nE ¡ e e2 B 2

µµ

nE ¡

¶

¶

µ

¶

∙Ti Ti rn £ b ¡ (½­e )2 ºei 1 + rn e Te

(7:5)

where ½­e = vTe =­e ; vTe = (∙Te =me )1=2 and ´ = me ºei =e2 ne (see sec.2.8). If the ¯rst term in the right-hand side can be neglected, the particle di®usion coe±cient D is given by 2

D = (½­e ) ºei

µ

¶

Ti 1+ : Te

(7:6)

The classical di®usion coe±cient Dei is de¯ned by Dei ´ (½­e )2 ºei =

¯e ´k nTe = ; 2 ¾? B ¹0

(7:7)

where ¾? = ne e2 =(me ºei ); ´k = 1=2¾? . However the ¯rst term of the right-hand side of eq.(7.5) is not always negligible. In toroidal con¯guration, the charge separation due to the toroidal drift is not completely cancelled along the magnetic ¯eld lines due to the ¯nite resistivity and an electric ¯eld E arises (see ¯g.7.1). Therefore the E £ b term in eq.(7.5) contributes to the di®usion. Let us consider this term. From the equilibrium equation, the diamagnetic current j? =

b £ rp; B

¯

¯

¯ 1 @p ¯ ¯ j? = ¯¯ B @r ¯

°ows in the plasma. From r ¢ j = 0, we ¯nd r ¢ j k = ¡r ¢ j ? . By means of the equation B = B0 (1 ¡ (r=R) cos µ), jk may be written as (see eq.(6.43)) jk = 2

2¼ 1 @p cos µ: ¶ B0 @r

(7:8)

If the electric conductivity along the magnetic lines of force is ¾k , the parallel electric ¯eld is Ek = jk =¾k . As is clear from ¯g.7.1, the relation B0 Eµ ¼ Ek Bµ holds. From Bµ =B0 ¼ (r=R)(¶=2¼), the µ component of the electric ¯eld is given by B0 R 2¼ 1 2 R Eµ = Ek = jk = Bµ r ¶ ¾k ¾k r

µ

2¼ ¶

¶2

1 @p cos µ: B0 @r

(7:9)

66

7 Di®usion of Plasma, Con¯nement Time

Fig.7.2 Magnetic surface (dotted line) and drift surfaces (solid lines).

Accordingly eq.(7.5) is reduced to nVr = ¡n

µ

Eµ Ti ¡ (½­e )2 ºei 1 + B Te

Ã

µ

R 2¼ ¢2 =¡ r ¶

¶2

¶

@n @r µ

n∙Te r cos µ 1 + cos µ 2 R ¾ k B0

µ

n∙Te r + 1 + cos µ R ¾? B02

¶2 !

µ

£ 1+

Ti Te

¶

¶

@n : @r

Noting that the area of a surface element is dependent of µ, and taking the average of nVr over µ, we ¯nd that hnVr i =

1 2¼

Z

µ

2¼

nVr 1 +

0

µ

Ti n∙Te =¡ 1+ 2 Te ¾ ? B0

¶

r cos µ dµ R

¶Ã

2¾? 1+ ¾k

µ

2¼ ¶

¶2 !

@n : @r

(7:10)

Using the relation ¾? = ¾k =2, we obtain the di®usion coe±cient of a toroidal plasma: DP:S:

µ

nTe Ti = 1+ 2 ¾ ? B0 Te

¶Ã

1+

µ

2¼ ¶

¶2 !

:

(7:11)

This di®usion coe±cient is (1 + (2¼=¶)2 ) times as large as the di®usion coe±cient of eq.(7.2). This value is called P¯rsch-SchlÄ uter factor1 . When the rotational tranform angle ¶=2¼ is about 0.3, P¯rsch-SchlÄ uter factor is about 10. 7.1b A Particle Model The classical di®usion coe±cient of electrons Dei = (½­e )2 ºei is that for electrons which move in a random walk with a step length equal to the Larmor radius. Let us consider a toroidal plasma. For rotational transform angle ¶, the displacement ¢ of the electron drift surface from the magnetic surface is (see ¯g.7.2) ¢ ¼ §½­e

2¼ : ¶

(7:12)

7.2 Neoclassical Di®usion of Electrons in Tokamak

67

The § signs depend on that the direction of electron motion is parallel or antiparallel to the magnetic ¯eld (see sec.3.5). As an electron can be transferred from one drift surface to the other by collision, the step length across the magnetic ¯eld is ¢=

µ

¶

2¼ ½­e : ¶

(7:13)

Consequently, the di®usion coe±cient is given by 2

DP:S: = ¢ ºei =

µ

2¼ ¶

¶2

(½­e )2 ºei ;

(7:14)

thus the P¯rsch-SchlÄ uter factor has been reduced (j2¼=¶j À 1 is assumed). 7.2 Neoclassical Di®usion of Electrons in Tokamak The magnitude B of the magnetic ¯eld of a tokamak is given by B=

RB0 = B0 (1 ¡ ²t cos µ); R(1 + ²t cos µ)

(7:15)

²t =

r : R

(7:16)

where

Consequently, when the perpendicular component v? of a electron velocity is much larger than the parallel component vk , i.e., when µ

v? v

¶2

>

R ; R+r

that is, 1 v? > 1=2 ; vk ²t

(7:17)

the electron is trapped outside of the torus, where the magnetic ¯eld is weak. Such an electron drifts in a banana orbit (see ¯g.3.9). In order to complete a circuit of the banana orbit, the e®ective collision time ¿e® = 1=ºe® of the trapped electron must be longer than one period ¿b of banana orbit ¿b ¼

R vk

µ

2¼ ¶

¶

R

=

1=2

v? ²t

µ

¶

2¼ : ¶

(7:18)

The e®ective collision frequency ºe® of the trapped electron is the frequency in which the condition (7.17) of trapped electron is violated by collision. As the collision frequency ºei is the inverse of di®usion time required to change the directon of velocity by 1 radian, the e®ective collision frequency ºe® is given by ºe® =

1 ºei : ²t

(7:19)

Accordingly, if ºe® < 1=¿b , i.e., 3=2

v? ²t ºei < ºb ´ R

µ

¶ 2¼

¶

=

3=2 ²t

1 R

µ

¶ 2¼

¶µ

∙Te me

¶1=2

(7:20)

68

7 Di®usion of Plasma, Con¯nement Time

the trapped electron can travel the entire banana orbit. When the trapped electron collides, it can shift its position by an amount of the banana width (see sec.3.5(b)) ¢b =

mvk mv? vk B 1=2 R 2¼ ¼ ¼ ½­e ²t = eBp eB v? Bp r ¶ 1=2

As the number of trapped electrons is ²t electron contribution to di®usion is 1=2

1=2

¡3=2

µ

2¼ ¶

¶

2¼ ¡1=2 ²t ½­e : ¶

(7:21)

times the total number of electrons, the trapped-

DG:S: = ²t ¢2b ºe® = ²t = ²t

µ

¶2

µ

2¼ ¶

¶2

2 ²¡1 t (½­e )

1 ºei ²t

(½­e )2 ºei :

(7.22) ¡3=2

= (R=r)3=2 times as large as This di®usion coe±cient, introduced by Galeev-Sagdeev,2 is ²t the di®usion coe±cient for collisional case. This derivation is semi-quantitative discussion. The more rigorous discussion is given in ref.2. As was discussed in sec.7.1, MHD treatment is applicable if the electron to ion collision frequency is larger than the frequency ºp given by ºp =

1 ¶ 1 vTe = R 2¼ R

µ

¶ 2¼

¶µ

∙Te me

¶1=2

:

(7:23)

When the electron to ion collision frequency is smaller than the frequency 3=2

ºb = ² t ºp ;

(7:24)

the electron can complete a banana orbit. The di®usion coe±cients are witten by DP:S: =

µ

2¼ ¶

¶2

¡3=2

DG:S: = ²t

µ

(½­e )2 ºei ; 2¼ ¶

¶2

ºei > ºp ;

(½­e )2 ºei ;

(7.25) 3=2

ºei < ºb = ²t ºp :

(7.26)

If ºei is in the region ºb < ºei < ºp , it is not possible to treat the di®usion phenomena of electrons in this region by means of a simple model. In this region we must resort to the drift approximation of Vlasov's equation. The result is that the di®usion coe±cient is not sensitive to the collision frequency in this region and is given by2;3 Dp =

µ

2¼ ¶

¶2

(½­e )2 ºp ;

3=2

ºp > ºei > ºb = ²t ºp :

(7:27)

The dependence of the di®usion coe±cient on the collision frequency is shown in ¯g.7.3. The region ºei > ºp is called the MHD region or collisional region. The region ºp > ºei > ºb is the platau region or intermediate region; and the region ºei < ºb is called the banana region or rare collisional region. These di®usion processes are called neoclassical di®usion. There is an excellent review3 on neoclassical di®usion. The reason that the electron-electron collison frequency does not a®ect the electron's particle di®usion coe±cient is that the center-of-mass velocity does not change by the Coulomb collision. The neoclassical thermal di®usion coe±cient ÂTe is the same order as the particle di®usion coe±cient (ÂTe » De ). Although ion collision with the same ion species does not a®ect the ion's particle di®usion coe±cient, it does contribute thermal di®usion processes, if temperature

7.3 Fluctuation Loss, Bohm Di®usion, and Stationary ¢ ¢ ¢

69

Fig.7.3 Dependence of the di®usion coe±cient on collision frequency in a tokamak. ºp = (¶=2¼)vTe =R, 3=2 ºb = ²t ºp .

gradient exists. Even if the ions are the same species with each other, it is possible to distinguish hot ion (with high velocity) and cold ion. Accordingly the ion's thermal di®usion coe±cient in ¡3=2 (2¼=¶)2 ½2­i ºii , and ÂTi » (mi =me )1=2 Die (Die » Dei ). banana region is given by ÂTi » ²t Therefore ion's thermal di®usion coe±cient is about (mi =me )1=2 times as large as the ion's particle di®usion coe±cient. 7.3 Fluctuation Loss, Bohm Di®usion, and Stationary Convective Loss In the foregoing sections we have discussed di®usion due to binary collision and have derived the con¯nement times for such di®usion as an ideal case. However, a plasma will be, in many cases, more or less unstable, and °uctuations in the density and electric ¯eld will induce collective motions of particles and bring about anomalous losses. We will study such losses here. Assume the plasma density n(r; t) consists of the zeroth-order term n0 (r; t) and 1st-order perturbation terms n ~ k (r; t) = nk exp i(kr ¡ !k t) and n = n0 +

X

n ~k :

(7:28)

k

Since n and n0 are real, there are following relations: n ~ ¡k = (~ nk )¤ ;

n¡k = n¤k ;

!¡k = ¡!k¤ :

where ¤ denotes the complex conjugate. !k is generally complex and !k = !kr + i°k and !¡kr = ¡!kr ;

°¡k = °k :

The plasma is forced to move by perturbation. When the velocity is expressed by V (r; t) =

X

V~ k =

k

X k

V k exp i(k ¢ r ¡ !k t)

(7:29)

then V ¡k = V ¤k and the equation of continuity @n + r ¢ (nV ) = 0 @t may be written as µX ¶ X ~k @n0 X @ n ~ ~ 0 n0 V k + n ~ k V k = 0: + +r¢ @t @t k k k;k 0

When the ¯rst- and the second-order terms are separated, then X @n ~k

@t

+r¢

@n0 +r¢ @t

X

µX k;k 0

n0 V~ k = 0;

(7.30)

¶

(7.31)

n ~ k V~ k0

= 0:

70

7 Di®usion of Plasma, Con¯nement Time

Here we have assumed that the time derivative of n0 is second order. The time average of the product of eq.(7.30) and n ~ ¡k becomes )

°k jnk j2 + rn0 ¢ Re(nk V ¡k ) + n0 k ¢ Im(nk V ¡k ) = 0; !kr jnk j2 + rn0 ¢ Im(nk V ¡k ) ¡ n0 k ¢ Re(nk V ¡k ) = 0:

(7:32)

If the time average of eq.(7.31) is taken, we ¯nd that ³X ´ @n0 Re(nk V ¡k ) exp(2°k t) = 0: +r¢ @t

(7:33)

The di®usion equation is

@n0 = r ¢ (Drn0 ) @t and the outer particle °ux ¡ is ¡ = ¡Drn0 =

X k

Re(nk V ¡k ) exp 2°k t:

(7:34)

Equation (7.32) alone is not enough to determine the quantity rn0 ¢Re(nk V ¡k ) exp 2°k t. Denote ¯k = n0 k ¢ Im(nk V ¡k )= rn0 ¢ (Re(nk V ¡k )); then eq.(7.34) is reduced to P

2

Djrn0 j =

°k jnk j2 exp 2°k t 1 + ¯k

and D=

X

°k

k

j~ nk j2 1 : 2 jrn0 j 1 + ¯k

(7:35)

This is the anomalous di®usion coe±cient due to °uctuation loss. ~ k of the electric ¯eld is electrostatic and Let us consider the case in which the °uctuation E ~ can be expressed by a potential Ák . Then the perturbed electric ¯eld is expressed by ~ k = ¡rÁ~k = ¡ik ¢ Ák exp i(kr ¡ !k t): E ~ k £ B drift, i.e., The electric ¯eld results in an E ~ k £ B)=B 2 = ¡i(k £ b)Á~k =B V~ k = (E

(7:36)

where b = B=B. Equation (7.36) gives the perpendicular component of °uctuating motion. The substitution of eq.(7.36) into eq.(7.30) yields n ~ k = rn0 ¢

µ

b£k B

¶ ~ Ák

!k

:

(7:37)

In general rn0 and b are orthogonal. Take the z axis in the direction of b and the x axis in ^ , where ∙n is the inverse of the scale of the density the direction of ¡rn, i.e., let rn = ¡∙n n0 x ^ is the unit vector in the x direction. Then eq.(7.37) gives gradient and x n ~k ∙Te eÁ~k ∙n ky ~ ! ¤ eÁ~k Á k = ky ∙n = = k n0 B !k eB!k ∙Te !k ∙Te where ky the y (poloidal) component of the propagation vector k. The quantity !k¤ ´ ky ∙n

(∙Te ) : eB

7.3 Fluctuation Loss, Bohm Di®usion, and Stationary ¢ ¢ ¢

71

~ k and Á~k have the is called the drift frequency. If the frequency !k is real (i.e., if °k = 0), n same phase, and the °uctuation does not contribute to anomalous di®usion as is clear from ~ k and Á~k eq.(7.35). When °k > 0, so that ! is complex, there is a phase di®erence between n and the °uctuation in the electric ¯eld contributes to anomalous di®usion. (When °k < 0, the amplitude of the °uctuation is damped and does not contribute to di®usion.) Using the real parameters Ak ; ®k of !k = !kr +i°k = !k¤ Ak exp i®k (Ak > 0; ®k are both real), V k is expressed by ∙Te Á~k ∙Te n ~k V~ k = ¡i(k £ b) = ¡i(k £ b) Ak exp i®k : eB ∙Te eB n0 Then the di®usion coe±cient may be obtained from eq.(7.34) as follows: D=

¯ ¯2 ¯ nk ¯ Ak sin ®k ¯¯ ¯¯ exp 2°k t 2 ∙ n

X !¤ k n

k

=

0

à X ky k

! ¯ ¯2 ¯ nk ¯ ∙Te Ak sin ®k ¯¯ ¯¯ exp 2°k t : ∙n n0 eB

(7.38)

The anomalous di®usion coe±cient due to °uctuation loss increases with time (from eqs. (7.35) and (7.38)) and eventually the term with the maximum growth rate °k > 0 becomes dominant. However, the amplitude j~ nk j will saturate due to nonlinear e®ects; the saturated amplitude will be of the order of j~ nk j ¼ jrn0 j

∙n ¸ky ¼ n0 : 2¼ ky

Then eq.(7.35) becomes °k D= 2 ∙n

¯ ¯2 ¯n ¯ ¯ ~ k ¯ ¼ °k : ¯n ¯ ky2 0

(7:39)

When the nondimensional coe±cient inside the parentheses in eq.(7.38) is at its maximum of 1/16, we have the Bohm di®usion coe±cient DB =

1 ∙Te : 16 eB

(7:40)

It appears that eqs.(7.39) and (7.40) give the largest possible di®usion coe±cient. When the density and potential °uctuations n ~ k ; Á~k are measured, V~ k can be calculated, and the estimated outward particle °ux ¡ and di®usion coe±cient D can be compared to the values obtained by experiment. As the relation of n ~ k and Á~k is given by eq.(7.37), the phase di®erence will indicate whether !k is real (oscillatory mode) or °k > 0 (growing mode), so that this equation is very useful in interpreting experimental results. Next, let us consider stationary convective losses across the magnetic °ux. Even if °uctuations in the density and electric ¯eld are not observed at a ¯xed position, it is possible that the plasma can move across the magnetic ¯eld and continuously escape. When a stationary electric ¯eld exists and the equipotential surfaces do not coincide with the magnetic surfaces Á = const:, the E £ B drift is normal to the electric ¯eld E, which itself is normal to the equipotential surface. Consequently the plasma drifts along the equipotential surfaces (see ¯g.7.4) which cross the magnetic surfaces. The resultant loss is called stationary convective loss. The particle °ux is given by

72

7 Di®usion of Plasma, Con¯nement Time

Fig.7.4 Magnetic surface à = const: and electric-¯eld equipotential Á = const: The plasma moves along the equipotential surfaces by virtue of E £ B.

¡k = n 0

Ey : B

(7:41)

The losses due to di®usion by binary collision are proportional to B ¡2 ; but °uctuation or convective losses are proportional to B ¡1 . Even if the magnetic ¯eld is increased, the loss due to °uctuations does not decrease rapidly. 7.4 Loss by Magnetic Fluctuation When the magnetic ¯eld in a plasma °uctuates, the lines of magnetic force will wander radially. Denote the radial shift of the ¯eld line by ¢r and the radial component of magnetic °uctuation ±B by ±Br respectively. Then we ¯nd ¢r =

Z

L

br dl;

0

where br = ±Br =B and l is the length along the line of magnetic force. The ensemble average of (¢r)2 is given by 2

h(¢r) i = =

*Z

L

br dl

0

*Z

0

L

dl

Z

Z

L

0

br dl

0

+

=

*Z

L

dl

0

L¡l ¡l

ds br (l) br (l + s)

+

Z

L 0

0

0

+

dl br (l) br (l ) D E

¼ L b2r lcorr ;

where lcorr is lcorr =

DR 1

¡1 br (l) br (l + s) ds

hb2r i

E

:

If electrons run along the lines of magnetic force with the velocity vTe , the di®usion coe±cient De of electrons becomes4 h(¢r)2 i L 2 hb ilcorr = vTe lcorr De = = ¢t ¢t r

*µ

±Br B

¶2 +

:

We may take lcorr » R in the case of tokamak and lcorr » a in the case of reverse ¯eld pinch (RFP, refer sec.17.1).

References

73

References 1. D. P¯rsh and A. SchlÄ uter: MPI/PA/7/62, Max-Planck Institute fÄ ur Physik und Astrophysik MÄ unchen (1962) 2. A. A. Galeev and R. Z. Sagdeev: Sov. Phys. JETP 26, 233 (1968) 3. B. B. Kadomtsev and O. P. Pogutse: Nucl. Fusion 11, 67 (1971) 4. A. B. Rechester and M. N. Rosenbluth: Phys. Rev. Lett. 40, 38 (1978)

74

8 Magnetohydrodynamic Instabilities

Ch.8 Magnetohydrodynamic Instabilities The stability of plasmas in magnetic ¯elds is one of the primary research subjects in the area of controlled thermonuclear fusion and both theoretical and experimental investigations have been actively pursued. If a plasma is free from all possible instabilities and if the con¯nement is dominated by neoclassical di®usion in the banana region, then the energy con¯nement time ¿E is given by (3=2)a2 (3=2) ¼ ¿E ¼ 5:8ÂG:S: 5:8

µ

¶ 2¼

¶2

²

3=2

µ

a ½­i

¶2

1 ºii

where a is the plasma radius, ½­i is the ion Larmor radius, and ºii is the ion-ion collision frequency. For such an ideal case, a device of a reasonable size satis¯es ignition condition. (For example, with B = 5 T; a = 1 m; Ti = 20 keV, ¶=2¼ ¼ 1=3, and inverse aspect ratio ² = 0:2, the value of n¿E » 3:5 £ 1020 cm¡3 ¢ sec.) A plasma consists of many moving charged particles and has many magnetohydrodynamic degrees of freedom as well as degrees of freedom in velocity space. When a certain mode of perturbation grows, it enhances di®usion. Heating a plasma increases the kinetic energy of the charged particles but at the same time can induce °uctuations in the electric and magnetic ¯elds, which in turn augment anomalous di®usion. Therefore, it is very important to study whether any particular perturbed mode is stable (damping mode) or unstable (growing mode). In the stability analysis, it is assumed that the deviation from the equilibrium state is small so that a linearized approximation can be used. In this chapter we will consider instabilities that can be described by linearized magnetohydrodynamic equations. These instabilities are called the magnetohydrodynamic (MHD) instabilities or macroscopic instabilities. A small perturbation F (r; t) of the ¯rst order is expanded in terms of its Fourier components F (r; t) = F (r) exp(¡i!t)

! = !r + i!i

and each term can be treated independently in the linearized approximation. The dispersion equation is solved for ! and the stability of the perturbation depends on the sign of the imaginary part !i (unstable for !i > 0 and stable for !i < 0). When !r 6= 0, the perturbation is oscillatory and when !r = 0, it grows or damps monotonously. In the following sections, typical MHD instabilities are introduced. In sec.8.1, interchange instability and kink instability are explained in an intuitive manner. In sec.8.2 the magnetohydrodynamic equations are linearized and the boundary conditions are implemented. The stability criterion is deduced from the energy principle, eqs.(8.45) » (8.48). In sec.8.3, a cylindrical plasma is studied as an important example; and the associated energy integrals are derived. Furthermore, important stability conditions, the Kruskal-Shafranov limit, eq.(8.66), and the Suydam criterion, eq.(8.97) are described. Tokamak and reversed ¯eld pinch con¯gurations are taken approximately as cylindrical plasmas, and their stabilities are examined. In this chapter, only the most common and tractable magnetohydrodynamic instabilities are introduced; it should be understood that there are many other instabilities. General reviews of plasma instabilities may be found in ref.1. 8.1 Interchange, Sausage and Kink Instabilities Let us study simple examples of instabilities intuitively before discussing the general linear theory of instabilities.

8.1 Interchange, Sausage and Kink Instabilities

75

Fig.8.1 Ion and electron drifts and the resultant electric ¯eld for interchange instability.

8.1a Interchange Instability Let x = 0 be the boundary between plasma and vacuum and let the z axis be taken in the direction of the magnetic ¯eld B. The plasma region is x < 0 and the vacuum region is x > 0. It is assumed that the acceleration g is applied in the x direction (see ¯g.8.1). Ions and electrons drift in opposite directions to each other, due to the acceleration, with drift velocities v G;i =

M g£B ; e B2

v G;e = ¡

mg£B : e B2

Let us assume that, due to a perturbation, the boundary of the plasma is displaced from the surface x = 0 by the amount ±x = a(t) sin(ky y): The charge separation due to the opposite ion and electron drifts yields an electric ¯eld. The resultant E £ B drift enhances the original perturbation if the direction of the acceleration g is outward from the plasma. From ¯g.8.4b we see that this is the same as saying that the magnetic °ux originally inside but near the plasma boundary is displaced so that it is outside the boundary, while the °ux outside moves in to ¯ll the depression thus left in the boundary; because of this geometrical picture of the process, this type of instability has come to be called interchange instability. As the perturbed plasma boundary is in the form of °utes along the magnetic lines of force (¯g.8.4b), this instability is also called °ute instability. The drift due to the acceleration yields a surface charge on the plasma, of charge density ¾s = ¾(t) cos(ky y)±(x)

(8:1)

(see ¯g.8.1). The electrostatic potential Á of the induced electric ¯eld E = ¡rÁ is given by ²?

µ

¶

µ

¶

@2Á @Á @ 2 + @x ²? @x @y

= ¡¾s :

(8:2)

The boundary condition is ²0

µ

@Á @x

¶

@Á ¡ ²? @x +0

¡0

= ¡¾s ;

Á+0 = Á¡0 : Under the assumption ky > 0, the solution Á is Á=

¾(t) cos(ky y) exp(¡ky jxj): ky (²0 + ²? )

(8:3)

76

8 Magnetohydrodynamic Instabilities

The velocity of the boundary d(±x)=dt is equal to E £ B=B 2 at x = 0, with E found from the potential (8.3). The velocity is da(t) ¾(t) sin(ky y) = sin(ky y): dt (²0 + ²? )B

(8:4)

The charge °ux in the y direction is nejv G;i j =

½m g B

where ½m = nM . Accordingly the changing rate of charge density is ½m g d¾(t) d cos(ky y) = a(t) sin(ky y) dt B dy

(8:5)

d2 a ½m gky 2 = (² + ² )B 2 a: dt 0 ?

(8:6)

and

The solution is in the form a / exp °t; the growth rate ° is given by °=

µ

½m (²0 + ²? )B 2

¶1=2

(gky )1=2 :

(8:7)

In the low-frequency case (compared with the ion cyclotron frequency), the dielectric constant is given by ²? = ²0 (1 +

½m ) À ²0 B 2 ²0

(8:8)

as will be explained in ch.10. Accordingly the growth rate ° is2 ° = (gky )1=2 :

(8:9)

When the acceleration is outward, a perturbation with the propagation vector k normal to the magnetic ¯eld B is unstable; i.e., (k ¢ B) = 0:

(8:10)

However, if the acceleration is inward (g < 0), ° of eq.(8.9) is imaginary and the perturbation is oscillatory and stable. The origin of interchange instability is charge separation due to the acceleration. When the magnetic lines of force are curved, as is shown in ¯g.8.2, the charged particles are subjected to a centrifugal force. If the magnetic lines of force are convex outward (¯g.8.2a), this centrifugal acceleration induces interchange instability. If the lines are concave outward, the plasma is stable. Accordingly, the plasma is stable when the magnitude B of the magnetic ¯eld increases outward. In other words, if B is a minimum at the plasma region the plasma is stable. This is the minimum-B condition for stability. The drift motion of charged particles is expressed by Ã

!

2 =2) + vk2 (v? E£b b + £ g+ n + vk b vg = B ­ R

where n is the normal unit vector from the center of curvature to a point on a line of magnetic force. R is the radius of curvature of line of magnetic force. The equivalent acceleration is g=

2 =2) + vk2 (v?

R

n:

(8:11)

8.1 Interchange, Sausage and Kink Instabilities

77

Fig.8.2 Centrifugal force due to the curvature of magnetic-force lines.

Fig.8.3 Charge separation due to the di®erence in velocities of ions and electrons.

The growth rate becomes ° ¼ (a=R)1=2 (vT =a) in this case. Analysis of interchange instability based on the linearlized equation of motion (8.32) with the acceralation term is described in ref.1 For a perturbation with propagation vector k normal to the magnetic ¯eld B, i.e., (k ¢B) = 0, another mechanism of charge separation may cause the same type of instability. When a plasma rotates with the velocity vµ = Er =B due to an inward radial electric ¯eld (¯g.8.3), and if the rotation velocity of ions falls below that of electrons, the perturbation is unstable. Several possible mechanisms can retard ion rotation. The collision of ions and neutral particles delays the ion velocity and causes neutral drag instability. When the growth rate ° » (gky )1=2 is not very large and the ion Larmor radius ½i­ is large enough to satisfy (ky ½i­ )2 >

° j­i j

the perturbation is stabilized3 . When the ion Larmor radius becomes large, the average perturbation electric ¯eld felt by the ions is di®erent that felt by the electrons, and the E £ B=B 2 drift velocities of the ion and the electrons are di®erent. The charge separation thus induced has opposite phase from the charge separation due to acceleration and stabilizes the instabiltity. 8.1b Stability Criterion for Interchange Instability, Magnetic Well Let us assume that a magnetic line of force has \good" curvature at one place B and `bad" curvature at another place A (¯g.8.4). Then the directions of the centrifugal force at A and B are opposite, as is the charge separation. The charge can easily be short circuited along the magnetic lines of the force, so that the problem of stability has a di®erent aspect. Let us here consider perturbations in which the magnetic °ux of region 1 is interchanged with that of region 2 and the plasma in the region 2 is interchanged with the plasma in the region 1 (interchange perturbations, ¯g.8.4b). It is assumed that the plasma is low-beta so that the magnetic ¯eld is nearly identical to the vacuum ¯eld. Any deviation from the vacuum ¯eld is accompanied by an increase in the energy of the disturbed ¯eld. This is the consequence of Maxwell equation. It can be shown that the most dangerous perturbations are those which exchange equal magnetic °uxes, as follows.

78

8 Magnetohydrodynamic Instabilities

Fig.8.4 Charge separation in interchange instability. (a) The lower ¯gure shows the unstable part A and the stable part B along a magnetic line of force. The upper ¯gure shows the charge separation due to the acceleration along a °ute. (b) Cross section of the perturbed plasma.

The energy of the magnetic ¯eld inside a magnetic tube is Z

QM =

B2 dr = 2¹0

Z

dlS

B2 2¹0

(8:12)

where l is length taken along a line of magnetic force and S is the cross section of the magnetic tube. As the magnetic °ux © = B ¢ S is constant, the energy is QM

©2 = 2¹0

Z

dl : S

The change ±QM in the magnetic energy due to the interchange of the °uxes of regions 1 and 2 is ±QM =

1 2¹0

µµ

©21

Z

2

dl + ©22 S

Z

1

dl S

¶

µ

¡ ©21

Z

1

dl + ©22 S

Z

2

dl S

¶¶

:

(8:13)

If the exchanged °uxes ©1 and ©2 are the same, the energy change ±QM is zero, so that perturbations resulting in ©1 = ©2 are the most dangerous. The kinetic energy Qp of a plasma of volume V is Qp =

nT V pV = °¡1 °¡1

(8:14)

where ° is the speci¯c-heat ratio. As the perturbation is adiabatic, pV ° = const: is conserved during the interchange process. The change in the plasma energy is µ

¶

1 ±Qp = p0 V2 ¡ p1 V1 + p01 V1 ¡ p2 V2 : °¡1 2 where p02 is the pressure after interchange from the region V1 to V2 and p01 is the µ pressure ¶° after V1 0 interchange from the region V2 to V1 . Because of adiabaticity, we have p2 = p1 V2 , p01 = p2

µ

V2 V1

¶°

and ±Qp becomes ±Qp =

µ

µ

V1 1 p1 V2 °¡1

¶°

V2 ¡ p1 V1 + p2

µ

V2 V1

¶°

¶

V 1 ¡ p2 V 2 :

(8:15)

8.1 Interchange, Sausage and Kink Instabilities

79

Fig.8.5 Speci¯c volume of a toroidal ¯eld.

Setting p2 = p1 + ±p; V2 = V1 + ±V we can write ±Qp as ±Qp = ±p±V + °p

(±V)2 : V

(8:16)

Since the stability condition is ±Qp > 0, the su±cient condition is ±p±V > 0: Since the volume is Z

V=

dlS = ©

Z

dl B

the stability condition for interchange instability is written as ±p±

Z

dl > 0: B

Usually the pressure decreases outward (±p < 0), so that the stability condition is ±

Z

dl <0 B

(8:17)

in the outward direction4 . The integral is to be taken only over the plasma region. Let the volume inside a magnetic surface à be V and the magnetic °ux in the toroidal direction ' inside the magnetic surface à be ©. We de¯ne the speci¯c volume U by U=

dV : d©

(8:18)

If the unit vector of the magnetic ¯eld B is denoted by b and the normal unit vector of the in¯nitesimal cross-sectional area dS is denoted by n, then we have dV =

Z X i

(b ¢ n)i Si dl;

d© =

X i

(b ¢ n)i Bi dSi :

When the magnetic lines of force close upon a single circuit of the torus, the speci¯c volume U is ! I ÃX I X dl (b ¢ n)i dS i dl (b ¢ n)i Bi dS i Bi Xi U= = i X : (b ¢ n)i Bi dSi (b ¢ n)i Bi dSi i

i

80

8 Magnetohydrodynamic Instabilities

Fig.8.6 Sausage instability.

As the integral over l is carried out along a small tube of the magnetic ¯eld, independent of l (conservation of magnetic °ux). As constant, U is reduced to U=

I

I

X i

(b ¢ n)i dSi Bi is

dl=Bi on the same magnetic surface is

dl : B

When the lines of magnetic force close at N circuits, U is U=

1 N

Z

N

dl : B

(8:19)

When the lines of magnetic force are not closed, U is given by 1 N !1 N

U = lim

Z

N

dl : B

Therefore, U may be considered to be an average of 1=B. When U decreases outward, it means that the magnitude B of the magnetic ¯eld increases outward in an average sense, so that the plasma region is the so-called average minimum-B region. In other word, the stability condition for interchange instability is reduced to average minimum-B condition; dU d2 V < 0: = d© d©2

(8:20)

When the value of U on the magnetic axis and on the outermost magnetic surface are U0 and Ua respectively, we de¯ne a magnetic well depth ¡¢U=U as ¡

¢U U0 ¡ Ua : = U U0

(8:21)

8.1c Sausage Instability Let us consider a cylindrical plasma with a sharp boundary. Only a longitudinal magnetic ¯eld Bz , exists inside the plasma region and only an azimuthal ¯eld Hµ = Iz =2¼r due to the plasma current Iz exists outside the plasma region. We examine an azimuthally symmetric perturbation which constricts the plasma like a sausage (¯g.8.6). When the plasma radius a is changed by ±a, conservation of magnetic °ux and the current in the plasma yields ±Bz = ¡Bz

2±a ; a

8.1 Interchange, Sausage and Kink Instabilities

81

Fig.8.7 Kink instability.

±Bµ = ¡Bµ

±a : a

The longitudinal magnetic ¯eld inside the plasma acts against the perturbation, while the external azimuthal ¯eld destabilizes the perturbation. The di®erence ±pm in the magnetic pressures is ±pm = ¡

Bz2 2±a Bµ2 ±a + : ¹0 a ¹0 a

The plasma is stable if ±pm > 0 for ±a < 0, so that the stability condition is Bz2 >

Bµ2 : 2

(8:22)

This type of instability is called sausage instability. 8.1d Kink Instability Let us consider a perturbation that kinks the plasma column as shown in ¯g.8.7. The con¯guration of the plasma is the same as that in the previous subsection (sharp boundary, an internal longitudinal ¯eld, an external azimuthal ¯eld). Denote the characteristic length of the kink by ¸ and its radius of curvature by R. The longitudinal magnetic ¯eld acts as a restoring force on the plasma due to the longitudinal tension; the restoring force on the plasma region of length ¸ is Bz2 2 ¸ ¼a : 2¹0 R The azimuthal magnetic ¯eld becomes strong, at the inner (concave) side of the kink and destabilizes the plasma column. In order to estimate the destabilizing force, we consider a cylindrical lateral surface of radius ¸ around the plasma and two planes A and B which pass through the center of curvature (see ¯g.8.7). Let us compare the contributions of the magnetic pressure on the surfaces enclosing the kink. The contribution of the magnetic pressure on the cylindrical surface is negligible compared with those on the planes A and B. The contribution of the magnetic pressure on the planes A and B is Z

¸

a

¸ Bµ2 ¸ B 2 (a) 2 ¸ 2¼rdr £ ¼a ln £ : = µ 2¹0 2R 2¹0 a R

82

8 Magnetohydrodynamic Instabilities

Accordingly ¸ Bz2 > ln 2 a Bµ (a)

(8:23)

is the stability condition3 . However, the pressure balance p+

B2 Bz2 = µ 2¹0 2¹0

holds, so that perturbations of large ¸ are unstable. This type of instability is called kink instability. In this section, a cylindrical sharp-boundary plasma has been analyzed in an intuitive way. The stability of a cylindrical plasma column will be treated in sec.8.2 in more general and systematic ways. 8.2 Formulation of Magnetohydrodynamic Instabilities 8.2a Linearization of Magnetohydrodynamic Equations The stability problems of plasmas can be studied by analyzing in¯nitesimal perturbations of the equilibrium state. If the mass density, pressure, °ow velocity, and magnetic ¯eld are denoted by ½m ，p， V ; and B; respectively, the equation of motion, conservation of mass, Ohm's law, and the adiabatic relation are ½m

@V = ¡rp + j £ B; @t

E + V £ B = 0;

µ

@½m + r ¢ (½m V ) = 0; @t ¶

@ + V ¢ r (p½¡° m )=0 @t

respectively (° is the ratio of speci¯c heat). Maxwell's equations are then r£E =¡

@B ; @t

r £ B = ¹0 j;

r ¢ B = 0:

These are the magnetohydrodynamic equations of a plasma with zero speci¯c resistivity (see sec.5.2). The values of ½m , p, V , and B in the equilibrium state are ½m0 , p0 , V 0 = 0; and B 0 respectively. The ¯rst-order small quantities are ½m1 , p1 , V 1 = V , and B 1 . The zeroth-order equations are rp0 = j 0 £ B 0 ;

r £ B 0 = ¹0 j 0 ;

The ¯rst-order linearized equations are @½m1 + r ¢ (½m0 V ) = 0; @t ½m0

@V + rp1 = j 0 £ B 1 + j 1 £ B 0 ; @t

r ¢ B 0 = 0: (8:24)

(8:25)

@p1 + (V ¢ r)p0 + °p0 r ¢ V = 0; @t

(8:26)

@B1 = r £ (V £ B 0 ): @t

(8:27)

8.2 Formulation of Magnetohydrodynamic Instabilities

83

If displacement of the plasma from the equilibrium position r 0 is denoted by »(r 0 ; t), it follows that »(r 0 ; t) = r ¡ r 0 ;

V =

@» d» ¼ : dt @t

Equation (8.27) is reduced to @B 1 =r£ @t

µ

@» £ B0 @t

¶

and B 1 = r £ (» £ B 0 ): From ¹0 j = r £ B, it follows that ¹0 j 1 = r £ B 1 :

(8:28)

(8:29)

Equations (8.24) and (8.26) yield ½m1 = ¡r ¢ (½m0 »)

(8:30)

p1 = ¡» ¢ rp0 ¡ °p0 r ¢ »:

(8:31)

The substitution of these equations into eq.(8.25) gives ½m0

1 @2» 1 = r(» ¢ rp0 + °p0 r ¢ ») + (r £ B 0 ) £ B 1 + (r £ B 1 ) £ B 0 2 @t ¹0 ¹0 µ ¶ B0 ¢ B1 1 = ¡r p1 + + ((B 0 ¢ r)B 1 + (B 1 ¢ r)B 0 ): ¹0 ¹0

(8:32)

This is the linearized equation of motion in terms of ». Next let us consider the boundary conditions. Where the plasma contacts an ideal conductor, the tangential component of the electric ¯eld is zero, i.e., n £ E = 0: This is equivalent to n £ (» £ B 0 ) = 0, n being taken in the outward direction. The conditions (» ¢ n) = 0 and (B 1 ¢ n) = 0 must also be satis¯ed. At the boundary surface between plasma and vacuum, the total pressure must be continuous and 2 ¡ B2 2 ¡ B2 Bin Bex 0;in 0;ex p ¡ p0 + = 2¹0 2¹0

where B in , B 0;in give the internal magnetic ¯eld of the plasma and B ex , B 0;ex give the external ¯eld. The boundary condition is reduced to B 0;in ¢ (B 1;in + (» ¢ r)B 0;in ) ¹0 B 0;ex ¢ (B 1;ex + (» ¢ r)B 0;ex ) = ¹0

¡°p0 r ¢ » +

(8:33)

when B in (r); B ex (r) and p(r) are expanded in » = r ¡ r 0 (f (r) = f0 (r 0 ) + (» ¢ r)f0 (r) + f1 ). From Maxwell's equations, the boundary conditions are n0 ¢ (B 0;in ¡ B 0;ex ) = 0;

(8:34)

n0 £ (B 0;in ¡ B 0;ex ) = ¹0 K

(8:35)

84

8 Magnetohydrodynamic Instabilities

where K is the surface current. Ohm's law yields E in + V £ B 0;in = 0

(3:36)

in the plasma. As the electric ¯eld E ¤ in coordinates moving with the plasma is E ¤ = E+V £B 0 and the tangential component of the electric ¯eld E ¤ is continuous across the plasma boundary. The boundary condition can be written as E t + (V £ B 0;ex )t = 0

(8:37)

where the subscript t indicates the tangential component. Since the normal component of B is given by the tangential component of E by the relation r £ E = ¡@B=@t, eq.(8.37) is reduced to (n0 ¢ B 1;ex ) = n0 ¢ r £ (» £ B 0;ex ):

(8:38)

The electric ¯eld E ex and the magnetic ¯eld B ex in the external (vacuum) region can be expressed in terms of a vector potential: E ex = ¡

@A ; @t

B 1;ex = r £ A;

r ¢ A = 0:

If no current °ows in the vacuum region, A satis¯es r £ r £ A = 0:

(8:39)

Using the vector potential, we may express eq.(8.37) as µ

¶

@A n0 £ ¡ + V £ B 0;ex = 0: @t For n0 ¢ B 0;in = n ¢ B 0;ex = 0, the boundary condition is n0 £ A = ¡»n B 0;ex :

(8:40)

The boundary condition at the wall of an ideal conductor is n £ A = 0:

(8:41)

The stability problem now becomes one of solving eqs.(8.32) and (8.39) under the boundary conditions (8.33),(8.38),(8.40) and (8.41). When a normal mode »(r; t) = »(r) exp(¡i!t) is considered, the problem is reduced to the eigenvalue problem ½0 ! 2 » = F (»). If any eigenvalue is negative, the plasma is unstable; if all the eigenvalues are positive, the plasma is stable. 8.2b Energy Principle5 The eigenvalue problem is complicated and di±cult to solve in general. When we introduce a potential energy associated with the displacement », the stability problem can be simpli¯ed. The equation of motion has the form ½m0

@2» c ¢ »: = F (») = ¡K @t2

This equation can be integrated: 1 2

Z

½m0

µ

@» @t

¶2

dr +

1 2

Z

c » ¢ K»dr = const:

(8:42)

8.2 Formulation of Magnetohydrodynamic Instabilities

85

The kinetic energy T and the potential energy W are T ´

Z

1 2

½m0

µ

@» @t

¶2

dr;

W ´

1 2

Z

respectively. Accordingly if

c » ¢ K»dr =¡

1 2

Z

» ¢ F (»)dr

W >0 for all possible displacements, the system is stable. This is the stability criterion of the energy principle. W is called the energy integral. c is Hermite operator (self-adjoint operator).6;7 It is possible to prove that the operator K A displacement ´ and a vector potential Q are introduced which satisfy the same boundary conditions as » and A, i.e., n0 £ Q = ¡´n B 0;ex at the plasma-vacuum boundary and n0 £ Q = 0 at the conducting wall. By substitution of eq.(8.32), the integral in the plasma region Vin is seen to be Z

Vin

c ´ ¢ K»dr =

Z

Vin

µ

°p0 (r ¢ ´)(r ¢ ») + (r ¢ ´)(» ¢ rp0 ) +

1 (r £ (´ £ B 0 )) ¢ r £ (» £ B 0 ) ¹0

¶

¡

1 (´ £ (r £ B 0 )) ¢ r £ (» £ B 0 ) dr ¹0

+

Z

S

n0 ¢ ´

µ

¶

B 0;in ¢ r £ (» £ B 0;in ) ¡ °p0 (r ¢ ») ¡ (» ¢ rp0 ) dS: ¹0

(8:43)

Next let us consider the surface integral in eq.(8.43). Due to the boundary condition n0 £Q = ¡´n B 0;ex , we ¯nd that Z

S

´n B 0;ex ¢ B 1;ex dS = =¡ =

Z

Z

S

Vex

=

Z

Vex

Z

S

´n B 0;ex (r £ A)dS = ¡

n0 ¢ (Q £ (r £ A))dS =

Z

Z

S

Vex

(n0 £ Q) ¢ (r £ A)dS

r ¢ (Q £ (r £ A))dr

((r £ Q) ¢ (r £ A) ¡ Q ¢ r £ (r £ A)) dr (r £ Q) ¢ (r £ A)dr:

From the boundary condition (8.33), the di®erence between the foregoing surface integral and the surface integral in eq.(8.43) is reduced to Z

´n

µ

¶

B 0;in ¢ B 1;in ¡ B 0;ex ¢ B 1;ex ¡ °p0 (r ¢ ») ¡ (» ¢ r)p0 dS ¹0 µ

Z

¶

2 2 B0;in B0;ex ¡ ¡ p0 dS ´n (» ¢ r) = 2¹0 2¹0 S

=

Z

S

´n »n

µ

¶

2 2 B0;in @ B0;ex ¡ ¡ p0 dS @n 2¹0 2¹0

86

8 Magnetohydrodynamic Instabilities

2 2 =2¹0 ¡ B0;ex =2¹0 ) = 0 is used. The integral region Vex is where the relation n0 £ r(p0 + B0;in the region outside the plasma. Finally, the energy integral is reduced to

Z

Vin

c ´ ¢ K»dr =

Z

Vin

µ

°p0 (r ¢ ´)(r ¢ ») +

+(r¢´)(» ¢rp0 )¡

1 (r £ (´ £ B 0 )) ¢ r £ (» £ B 0 ) ¹0 ¶

1 1 (´ £(r£B 0 ))¢r£(» £B 0 ) dr + ¹0 ¹0

µ

Z

Z

(r£Q)¢(r£A)dr

Vex

¶

2 2 B0;in @ B0;ex ¡ ¡ p0 dS: + ´n »n @n 2¹0 2¹0 S

(8:44)

The energy integral W is divided into three parts WP , WS , and WV , the contributions of the plasma internal region Vin , the boundary region S ，and the external vacuum region Vex , i.e., 1 W = 2 Wp =

Z

c » ¢ K»dr = Wp + WS + WV ;

Vin

1 2

Z

Vin

µ

°p0 (r ¢ »)2 +

¡

(8:45)

1 (r £ (» £ B 0 ))2 + (r ¢ »)(» ¢ rp0 ) ¹0 ¶

1 (» £ (r £ B 0 )) ¢ r £ (» £ B 0 ) dr ¹0

1 = 2

Z

µ

B 21 ¡ p1 (r ¢ ») ¡ » ¢ (j 0 £ B 1 ) dr; ¹0

1 WS = 2

Z

»n2

@ @n

Vin

S

1 WV = 2¹0

Z

¶

Ã

2 2 B0;in B0;ex ¡ ¡ p0 dS; 2¹0 2¹0

!

(8:47)

Z

B 21 dr: 2¹0

(8:48)

2

Vex

(8:46)

(r £ A) dr =

Vex

The stability condition is W > 0 for all possible ». The frequency or growth rate of a perturbation can be obtained from the energy integral. When the perturbation varies as exp(¡i!t), the equation of motion is c !2 ½m0 » = K»:

(8:49)

The solution of the eigenvalue problem is the same as the solution based on the calculus of variations ±(!2 ) = 0, where 2

R

! = R

c » ¢ K»dr : ½m0 » 2 dr

(8:50)

c is a Hermitian operator, ! 2 is real. In the MHD analysis of an ideal plasma with zero As K resistivity, the perturbation either increases or decreases monotonically, or else the perturbed plasma oscillates with constant amplitude. The energy integral (8.46) can be further rearranged to the more illuminating form. The reduction of the form is described in appendix B. The energy integral of axisymmetiric toroidal system is also described in appendix B.

8.3 Instabilities of a Cylindrical Plasma

87

8.3 Instabilities of a Cylindrical Plasma 8.3a Instabilities of Sharp-Boundary Con¯guration: Kruskal-Shafranov Condition Let us consider a sharp-boundary plasma of radius a, with a longitudinal magnetic ¯eld B0z inside the boundary and a longitudinal magnetic ¯eld Bez and an azimuthal magnetic ¯eld Bµ = ¹0 I=(2¼r) outside. B0z and Bez are assumed to be constant. We can consider the displacement »(r) exp(imµ + ikz)

(8:51)

since any displacement may be expressed by a superposition of such modes. Since the term in r ¢ » in the energy integral is positive, incompressible perturbation is the most dangerous. We examine only the worst mode, r ¢ » = 0:

(8:52)

The perturbation of the magnetic ¯eld B 1 = r £ (» £ B 0 ) is B 1 = ikB0z »:

(8:53)

The equation of motion (8.32) becomes µ

¶

µ

2 k2 B0z B0 ¢ B1 ¡! ½m0 + » = ¡r p1 + ¹0 ¹0 2

As r ¢ » = 0, it follows that ¢p¤ = 0, i.e., Ã

Ã

d2 1 d m2 2 ¡ + k + r dr r2 dr2

!!

¶

´ ¡rp¤ :

(8:54)

p¤ (r) = 0:

(8:55)

The solution without singularity at r = 0 is given by the modi¯ed Bessel function Im (kr), so that p¤ (r) is p¤ (r) = p¤ (a)

Im (kr) : Im (ka)

Accodingly, we ¯nd

kp¤ (a) Im (ka) 0 »r (a) = 2 2 Im (ka): k B 0 ! 2 ½m0 ¡ ¹0

(8:56)

As the perturbation of the vacuum magnetic ¯eld B 1e satis¯es r £ B = 0 and r ¢ B = 0, B 1e is expressed by B 1e = rÃ. The scalar magnetic potential à satis¯es ¢Ã = 0 and à ! 0 as r ! 1. Then Ã=C

Km (kr) exp(imµ + ikz): Km (ka)

(8:57)

The boundary condition (8.33) is 1 1 p1 + B 0 ¢ B 1 = B e ¢ B 1e + (» ¢ r) ¹0 ¹0 =

à µ

B2 Be2 ¡ 0 ¡ p0 2¹0 2¹0 ¶

Bµ2 1 B e ¢ B 1e + (» ¢ r) : ¹0 2¹0

!

88

8 Magnetohydrodynamic Instabilities

Fig.8.8 Sharp-boundary plasma.

As Bµ / 1=r , p¤ (a) is given by p¤ (a) =

i m B2 (kBez + Bµ )C ¡ µ »r (a): ¹0 a ¹0 a

(8:58)

The boundary condition (8.38) is reduced to Ck

0 (ka) Km m = i(kBez + Bµ )»r (a): Km (ka) a

(8:59)

From eqs.(8.56), (8.58) and (8.59), the dispersion equation is 2 0 0 !2 Bµ2 (kBez + (m=a)Bµ )2 Im (ka) Km (ka) 1 Im (ka) B0z ¡ ¡ = : 0 2 2 k ¹0 ½m0 ¹0 ½m0 k Im (ka) Km (ka) ¹0 ½m0 (ka) Im (ka)

(8:60)

0 < 0). If the The 1st and 2nd terms represent the stabilizing e®ect of B0z and Bez (Km =Km propagation vector k is normal to the magnetic ¯eld, i.e., if

(k ¢ B e ) = kBez +

m Bµ = 0 a

the 2nd term (stabilizing term) of eq.(8.60) becomes zero, so that a °utelike perturbation is indicated. The 3rd term is the destabilizing term. Let us consider the m = 0 mode with Bez = 0. (i) The m = 0 Mode with Bez = 0 This con¯guration corresponds to that of the sausage instability described in sec.8.1c. Equation (8.60) reduces to B 2 k2 ! = 0z ¹0 ½m0 2

Ã

!

B 2 I00 (ka) 1 ¡ 2µ : B0z (ka)I0 (ka)

(8:61)

Since I00 (x)=xI0 (x) < 1=2, the stability condition is 2 B0z > Bµ2 =2:

(ii) The m = 1 Mode with Bez = 0 B 2 k2 ! 2 = 0z ¹0 ½m0

Ã

For the m = 1 mode with Bez = 0，eq.(8.60) is

B 2 1 I10 (ka) K1 (ka) 1 + 2µ B0z (ka) I1 (ka) K10 (ka)

!

:

(8:62)

8.3 Instabilities of a Cylindrical Plasma

89

For perturbations with long characteristic length, eq.(8.62) becomes B 2 k2 !2 = 0z ¹0 ½m0

Ã

1¡

µ

Bµ B0z

¶2

!

1 ln : ka

(8:63)

This dispersion equation corresponds to kink instability, which is unstable for the perturbation with long wavelength (refer (8.23)). When jBez j À jBµ j, the case jkaj ¿ 1 (iii) Instability in the Case of jBez j > jBµ j predominates. Expanding the modi¯ed Bessel function (m > 0 is assumed), we ¯nd µ

2 ¹0 ½m0 ! 2 = k2 B0z + kBez +

m Bµ a

¶2

¡

m 2 B : a2 µ

(8:64)

2 2 + Bez ) + (m=a)Bµ Bez ! 2 becomes minimum at @!=@k = 0, i.e., k(B0z 2 = 0. In this case, ! is 2 !min

Bµ2 = ¹0 ½m0 a2

Ã

2 m2 B0z 2 ¡m 2 Bez + B0z

!

=

µ

¶

Bµ2 1¡¯ ¡1 ; m m ¹0 ½m0 a2 2¡¯

(8:65)

where ¯ is the beta ratio. Accordingly, the plasma is unstable when 0 < m < (2 ¡ ¯)=(1 ¡ ¯). For a low-beta plasma only the modes m = 1 and m = 2 become unstable. However, if µ

Bµ Bz

¶2

< (ka)2

(8:66)

is satis¯ed the plasma is stable even for m = 1. Usually the length of the plasma is ¯nite so that k cannot be smaller than 2¼=L. Accordingly, when ¯ ¯ ¯ Bµ ¯ 2¼a ¯ ¯ ¯B ¯ < L z

the plasma is stable. This stability condition is called the Kruskal- Shafranov condition.8;9 When a cylindrical conducting wall of radius b surrounds the plasma, the scalar magnetic potential of the external magnetic ¯eld is µ

à = c1

¶

Km (kr) Im (kr) exp(imµ + ikz) + c2 Km (ka) Im (ka)

(8:570 )

instead of eq.(8.57). The boundary condition B1er = 0 at r = b yields c1 I 0 (kb)Km (ka) = ¡ m0 : c2 Km (kb)Im (ka) The dispersion equation becomes 2 0 (kBez + (m=a)Bµ )2 Im !2 (ka) B0z ¡ = 2 2 k ¹0 ½m0 ¹0 ½m0 k Im (ka)

£ ¡

µ

0 0 Km (ka)Im (kb) ¡ Im (ka)Km (kb) 0 0 0 0 Km (ka)Im (kb) ¡ Im (ka)Km (kb)

¶

0 Bµ2 1 Im (ka) : ¹0 ½m0 (ka) Im (ka)

Expanding the modi¯ed Bessel functions under the conditions ka ¿ 1, kb ¿ 1, we ¯nd 2 + ¹0 ½m0 ! 2 = k2 B0z

m 1 + (a=b)2m m (kBez + Bµ )2 ¡ 2 Bµ2 : 2m 1 ¡ (a=b) a a

90

8 Magnetohydrodynamic Instabilities

The closer the wall to the plasma boundary, the more e®ective is the wall stabilization. In toroidal systems, the propagation constant is k = n=R where n is an integer and R is the major radius of the torus. If the safety factor qa at the plasma boundary r = a qa =

aBez RBµ

(8:67)

is introduced, (k ¢ B) may be written as (k ¢ B) = (kBez +

m nBµ Bµ ) = a a

µ

qa +

¶

m : n

The Kruskal-Shafranov condition (8.66) of m = 1; n = ¡1 mode can then be expressed in terms of the safety factor as qa > 1:

(8:68)

This is the reason why qa is called the safety factor. 8.3b Instabilities of Di®use-Boundary Con¯gurations The sharp-boundary con¯guration treated in sec.8.3a is a special case; in most cases the plasma current decreases gradually at the boundary. Let us consider the case of a di®useboundary plasma whose parameters in the equilibrium state are p0 (r);

B 0 (r) = (0; Bµ (r); Bz (r)):

The perturbation » is assumed to be » = »(r) exp(imµ + ikz): The perturbation of the magnetic ¯eld B 1 = r £ (» £ B 0 ) is B1r = i(k ¢ B 0 )»r ;

(8.69)

B1µ = ikA ¡

(8.70)

B1z

(8.71)

d (»r Bµ ); dr µ ¶ imA 1 d =¡ + (r»r Bz ) r r dr

where m Bµ ; r A = »µ Bz ¡ »z Bµ = (» £ B 0 )r :

(k ¢ B 0 ) = kBz +

(8.72) (8.73)

Since the pressure terms °p0 (r ¢ »)2 + (r ¢ »)(» ¢ rp0 ) = (° ¡ 1)p0 (r ¢ »)2 + (r ¢ »)(r ¢ p0 ») in the energy integral are nonnegative, we examine the incompressible displacement r ¢ » = 0 again, i.e., im 1 d (r»r ) + »µ + ik»z = 0: r dr r

(8:74)

From this and eq.(8.73) for A, »µ and »z are expressed in terms of »r and A as i(k ¢ B)»µ = ikA ¡ ¡i(k ¢ B)»z =

Bµ d (r»r ); r dr

imA Bz d + (r»r ): r r dr

(8:75)

(8:76)

8.3 Instabilities of a Cylindrical Plasma

91

From ¹0 j 0 = r £ B 0 , it follows that dBz ; dr dBµ Bµ 1 d = + = (rBµ ): dr r r dr

¹0 j0µ = ¡

(8.77)

¹0 j0z

(8.78)

The terms of the energy integral are given by 1 4

Wp =

1 = 4 1 WS = 4

Z

Vin

µ

¶

1 jB 1 j2 ¡ » ¤ ¢ (j 0 £ B 1 ) dr ¹0

°p0 jr ¢ »j2 + (r ¢ » ¤ )(» ¢ rp0 ) +

Z µ

¶

1 ¡p1 (r ¢ ») + jB 1 j2 ¡ j 0 (B 1 £ » ¤ ) dr; ¹0

Z

µ

(8:79)

¶

2 2 B0;in @ B0;ex j»n j ¡ ¡ p0 dS; @n 2¹0 2¹0 S 2

1 WV = 4¹0

Z

Vex

(8:80)

jB 1 j2 dr:

(8:81)

»µ and »z can be eliminated by means of eqs.(8.75) and (8.76) and dBz =dr and dBµ =dr can be eliminated by means of eqs.(8.77) and (8.78) in eq.(8.79). Then Wp becomes 1 Wp = 4

Z

+ +

Vin

Ã

(k ¢ B)2 m2 j»r j2 + k2 + 2 ¹0 r ¯

µ

!

jAj2 ¹0

¶¯

¯ ¯ ¯ »r Bz d»r ¯¯2 ¯ + B z ¯ r dr ¯ ¶ ¶ ¤ µ

Bµ ¯¯2 1 ¯¯ d»r 1 ¡ B ¹ j + + » r 0 z µ ¹0 ¯ dr r ¯ ¹0 µ

µ

µ

Bµ d»r 2 Re ikA¤ Bµ + ¹0 j z ¡ ¹0 dr r µ

µ

+ 2Re »r¤ j0z ¡Bµ

»r ¹0 jz d»r ¡ + ikA dr 2

»r ¡

¶¶

imA r2

»r Bz + rBz

d»r dr

¶¶

dr:

The integrand of Wp is reduced to 1 ¹0

Ã

m2 k + 2 r 2

Ã

!

¯ ¯ ¯ ikBµ ((d»r =dr) ¡ »r =r) ¡ im(Bz =r)((d»r =dr) + (»r =r)) ¯¯2 ¯ £ ¯A + ¯ k2 + (m2 =r2 )

(k ¢ B)2 2jz Bµ ¡ + ¹0 r ¡

!

Bz2 ¹0

j»r j2 +

¯ ¯ ¯ d»r »r ¯¯2 Bµ2 ¯ + + ¯ dr r¯ ¹0

¯ ¯ ¯ d»r »r ¯¯2 ¯ ¡ ¯ dr r¯

jikBµ ((d»r =dr) ¡ (»r =r)) ¡ im(Bz =r)((d»r =dr) + (»r =r))j2 : ¹0 (k2 + (m2 =r 2 ))

Accordingly, the integrand is a minimum when A ´ »µ Bz ¡ »z Bµ =¡

i 2 k + (m2 =r2 )

µµ

kBµ ¡

m Bz r

¶

µ

d»r m ¡ kBµ + Bz dr r

¶

»r r

¶

:

Then Wp is reduced to Wp =

¼ 2¹0

Z aµ 0

µ

¶

¶

j(k ¢ B 0 )(d»r =dr) + h(»r =r)j2 2¹0 jz Bµ j»r j2 rdr + (k ¢ B 0 )2 ¡ 2 2 k + (m=r) r

(8:82)

92

8 Magnetohydrodynamic Instabilities

where h ´ kBz ¡

m Bµ : r

Let us next determine WS . From eq.(6.8), it follows that (d=dr)(p0 + (Bz2 + Bµ2 )=2¹0 ) = ¡Bµ2 =(r¹0 ). Bµ2 is continuous across the boundary r = a, so that d dr

Ã

B 2 + Bµ2 p0 + z 2¹0

!

d = dr

Ã

2 2 Bez + Beµ 2¹0

!

:

Accordingly we ¯nd WS = 0

(8:83)

as is clear from eq.(8.80). The expression for WV can be obtained when the quantities in eq.(8,82) for Wp are replaced as follows: j ! 0, Bz ! Bez = Bs (= const:), Bµ ! Beµ = Ba a=r, B1r = i(k ¢ B 0 )»r ! Be1r = i(k ¢ B e0 )´r . This replacement yields ¼ WV = 2¹0 +

Z b µµ a

m Ba a kBs + r r

¶2

j´r j2 ¶

j[kBs +(m=r)(Ba a=r)](d´r =dr)+[kBs ¡(m=r)(Ba a=r)]´r=rj2 rdr: k2 + (m=r)2

(8:84)

By partial integration, Wp is seen to be ¼ Wp = 2¹0

Z

a

0

Ã

r(k ¢ B 0 )2 k2 + (m=r)2

! ¯ ¯ 2 2 2 2 ¯ d»r ¯2 ¯ ¯ + gj»r j2 dr + ¼ k Bs ¡ (m=a) Ba j»r (a)j2 ¯ dr ¯ 2¹0 k2 + (m=a)2

d 2Bµ d(rBµ ) 1 (kBz ¡ (m=r)Bµ )2 ¡ g= + r(k ¢ B 0 )2 ¡ r k 2 + (m=r)2 r dr dr Using the notation ³ ´ rBe1r = ir(k ¢ B e0 )´r , we ¯nd that WV =

¼ 2¹0

Z bµ a

¯

¯

Ã

k2 Bz2 ¡ (m=r)2 Bµ2 k2 + (m=r)2

!

(8:85)

:

¶

¯ d³ ¯2 1 ¯ ¯ + 1 j³j2 dr: r(k2 + (m=r)2 ) ¯ dr ¯ r

(8:86)

(8:87)

The functions »r or ³ that will minimize Wp or WV are the solutions of Euler's equation: d dr

Ã

r(k ¢ B 0 )2 d»r k 2 + (m=r)2 dr

!

d dr

µ

1 d³ 2 2 r(k + (m=r) ) dr

¡ g»r = 0; ¶

1 ¡ ³ = 0; r

r ∙ a;

(8:88)

r>a:

(8:89)

There are two independent solutions, which tend to »r / r m¡1 , r ¡m¡1 as r ! 0. As »r is ¯nite at r = 0, the solution must satisfy the conditions r ! 0;

»r / rm¡1 ; µ

¶

r = a;

m ³(a) = ia kBs + Ba »r (a); a

r = b;

³(b) = 0:

8.3 Instabilities of a Cylindrical Plasma

93

Using the solution of eq.(8.89), we obtain ¯

¯

¯ d³ ¤ ¯b 1 ¼ ¯ ³ ¯ : WV = 2¹0 r(k2 + (m=r)2 ) ¯ dr ¯a

(8:90)

The solution of eq.(8.89) is

µ

¶

0 0 0 (kb) ¡ Km (kr)Im (kb) m I 0 (kr)Km r kBs + Ba »r (a): ³=i m 0 0 0 0 Im (ka)Km (kb) ¡ Km (ka)Im (kb) a

(8:91)

The stability problem is now reduced to one of examining the sign of Wp + WV . For this we use Z µ ¯

¯

9 > > > > > > > > =

¶

a ¯ d»r ¯2 ¼ ¯ + gj»r j2 dr + Wa ; = f ¯¯ 2¹0 0 dr ¯ ¼ k2 Bs2 ¡ (m=a)2 Ba2 j»r (a)j2 ; = 2¹0 k2 + (m=a)2 ¯ ¯ ¯ d³ ¤ ¯ ¡1 ¼ ¯ = ³ ¯ 2¹0 r(k2 + (m=a)2 ) ¯ dr ¯r=a

Wp Wa WV where

f=

r(kBz + (m=r)Bµ )2 ; k2 + (m=r)2

g=

1 (kBz ¡ (m=r)Bµ )2 m + r kBz + Bµ 2 2 r k + (m=r) r d 2Bµ d(rBµ ) ¡ ¡ r dr dr

Ã

(8:92)

> > > > > > > > ;

(8.93) µ

¶2

k2 Bz2 ¡ (m=r)2 Bµ2 k2 + (m=r)2

!

:

(8.94)

d (¹ p + B 2 =2) = ¡B 2 =r is used, eq.(8.94) of g is reduced to When the equation of equilibrium dr 0 µ g=

2 2 2 dp0 m 2k2 2 k + (m=r) ¡ (1=r) ¹ + ) + r(kB B 0 z µ k 2 + (m=r)2 dr r k2 + (m=r)2

+

(2k2 =r)(k2 Bz2 ¡ (m=r)2 Bµ2 ) : (k 2 + (m=r)2 )2

(8.95)

8.3c Suydam's Criterion The function f in the integrand of Wp in the previous section is always f ¸ 0, so that the term in f is a stabilizing term. The 1st and 2nd terms in eq.(8.94) for g are stabilizing terms, but the 3rd and 4th terms may contribute to the instabilities. When a singular point f / (k ¢ B 0 )2 = 0 of Euler's equation (8.88) is located at some point r = r0 within the plasma region, the contribution of the stabilizing term becomes small near r = r0 , so that a local mode near the singular point is dangerous. In terms of the notation r ¡ r0 = x; µ

2

f = ®x ;

¯

g = ¯;

dBz dBµ r0 kr + kBz + m ®= 2 2 2 dr dr k r0 + m

¶2

r=r0

2B 2 dp0 ¯¯ ¯ = 2µ ¹0 ; B0 dr ¯r=r0

rBµ2 Bz2 = B2

µ

¹ ~0 ¹ ~

¶2

r=r0

;

¹ ~´

Bµ : rBz

94

8 Magnetohydrodynamic Instabilities

Euler's equation is reduced to ®

d 2 d»r (x ) ¡ ¯»r = 0: dr dx

The solution is »r = c1 x¡n1 + c2 x¡n2

(8:96)

where n1 and n2 are given by n2 ¡ n ¡

¯ = 0; ®

ni =

1 § (1 + 4¯=®)1=2 : 2

When ® + 4¯ > 0, n1 and n2 are real. The relation n1 + n2 = 1 holds always. For n1 < n2 , we have the solution x¡n1 , called a small solution. When n is complex (n = ° § i±), »r is in the form exp((¡° ¨ i±) ln x) and »r is oscillatory. Let us consider a local mode »r , which is nonzero only in the neighborhood " around r = r0 and set r ¡ r0 = "t; »r (r) = »(t); »(1) = »(¡1) = 0: Then Wp becomes ¼ " Wp = 2¹0 Since Schwartz's inequality yields Z

1

¡1

2

0 2

t j» j dt

Z

1

¡1

Wp is

Z

1

¡1

¯ ¯2 ¶ ¯ ¯ 2 ®t ¯ ¯ + ¯j»j dt + O("2 ): dt 2 ¯ d» ¯

¯Z ¯ j»j dt ¸ ¯¯ 2

¼ 1 (® + 4¯) 2¹0 4

Wp >

µ

Z

1

¡1

1 ¡1

¯2 µ Z ¯ 1 ¯ t» » dt¯ = 2 0 ¤

1

¡1

2

j»j dt

¶2

j»j2 dt:

The stability condition is ® + 4¯ > 0, i.e., r 4

µ

¹ ~0 ¹ ~

¶2

+

2¹0 dp0 > 0: Bz2 dr

(8:97)

¹) is called shear parameter. Usually the 2nd term is negative, since, most often, dp0 =dr < r(~ ¹0 =~ ¹)2 represents the stabilizing e®ect of shear. This condition is called 0. The 1st term (~ ¹0 =~ 10 Suydam's criterion. This is a necessary condition for stability; but it is not always a su±cient condition, as Suydam's criterion is derived from consideration of local-mode behavior only. Newcomb derived the necessary and su±cient conditions for the stability of a cylindrical plasma. His twelve theorems are described in ref.11. 8.3d Tokamak Con¯guration In this case the longitudinal magnetic ¯eld Bs is much larger than the poloidal magnetic ¯eld Bµ . The plasma region is r ∙ a and the vacuum region is a ∙ r ∙ b and an ideal conducting wall is at r = b. It is assumed that ka ¿ 1; kb ¿ 1. The function ³ in eq.(8.90) for WV is µ

rm am bm (mBa + kaBs ) ¡ » (a) ³=i r 1 ¡ (a=b)2m bm r m bm

¶

(from eq.(8.91)), and WV becomes WV =

¼ (mBa + kaBs )2 2 »r (a)¸; 2¹0 m

¸´

1 + (a=b)2m : 1 ¡ (a=b)2m

8.3 Instabilities of a Cylindrical Plasma

95

Fig.8.9 The relation of the growth rate ° and nqa for kink instability (¡2W=(¼»a2 Ba2 =¹0 ) = ° 2 a2 (h½m0 i¹0 =Ba2 )). After V. D. Shafranov: Sov. Phys. Tech. Phys. 15, 175 (1970).

From the periodic condition for a torus, it follows that 2¼n = ¡2¼R k

(n is an integer)

so that (k ¢ B) is given by

µ

nqa m

a(k ¢ B) = mBa + kaBs = mBa 1 ¡

¶

in terms of the safety factor. The Wa term in eq.(8.92) is reduced to k

2

Bs2

¡

µ

m a

¶2

Ba2

µ

m = kBs + Ba a =

µ

nBa a

¶2 µµ

¶2

µ

m m ¡ 2 Ba kBs + Ba a a

nqa 1¡ m

¶2

µ

nqa ¡2 1¡ m

¶

¶¶

:

Accordingly, the energy integral becomes ¼ 2 2 B » (a) Wp + WV = 2¹0 a r ¼ + 2¹0

õ

Z µ µ

nqa 1¡ m

d»r f dr

¶2

+

¶2

g»r2

µ

nqa (1 + m¸) ¡ 2 1 ¡ m ¶

¶!

dr:

(8.98)

The 1st term of eq.(8.98) is negative when 1¡

2 nqa < < 1: 1 + m¸ m

(8:99)

The assumption nqa =m » 1 corresponds to ka » mBa =Bs . As Ba =Bs ¿ 1, this is consistent with the assumption ka ¿ 1. When m = 1, (m2 ¡ 1)=m2 in the 2nd term of eq.(8.95) for g is zero. The magnitude of g is of the order of k 2 r 2 , which is very small since kr ¿ 1. The term in f (d»r =dr)2 can be very small if »r is nearly constant. Accordingly the contribution of the integral term in eq.(8.98) is negligible. When m = 1 and a2 =b2 < nqa < 1, the energy integral becomes negative (W < 0). The mode m = 1 is unstable in the region speci¯ed by eq.(8.99) irrespective of the current distribution. The Kruskal-Shafranov condition for the mode m = 1 derived from the sharp-boundary con¯guration is also applicable to the di®use-boundary plasma. The growth rate ° 2 = ¡! 2 is ¡W Ba2 1 ° 'R = h½m0 i ¹0 a2 (½m0 j»j2 =2)dr 2

Ã

2(1 ¡ nqa )2 2(1 ¡ nqa ) ¡ 1 ¡ a2 =b2

!

;

(8:100)

96

8 Magnetohydrodynamic Instabilities

h½m0 i =

R

½m0 j»j2 2¼rdr : ¼a2 »r2 (a)

2 » (1 ¡ a2 =b2 )Ba2 =(¹0 h½ia2 ). When m 6= 1, (m2 ¡ 1)=m2 in the The maximum growth rate is °max 2nd term of eq.(8.95) for g is large, and g » 1. Accordingly, the contribution of the integral term to Wp must be checked. The region g < 0 is given by Â1 <  < Â2 , when  ´ ¡krBz =Bµ = nq(r) and

Â1;2

2k2 r2 2 k2 r2 § =m¡ 2 m(m ¡ 1) m(m2 ¡ 1)

Ã

m2 (m2 ¡ 1) ¹0 rp00 1¡ 2k 2 r 2 Bµ2

!1=2

:

(8:101)

Since kr ¿ 1, the region g < 0 is narrow and close to the singular point nq(r) = m and the contribution of the integral term to Wp can be neglected. Therefore if nqa =m is in the range given by eq.(8.99), the plasma is unstable due to the displacement »r (a) of the plasma boundary. When the current distribution is j(r) = j0 exp(¡∙2 r 2 =a2 ) and the conducting wall is at in¯nity (b = 1), ° 2 can be calculated from eq.(8.100), using the solution of Euler's equation; and the dependence of ° 2 on qa can be estimated. The result is shown in ¯g.8.9. When the value of nqa =m is outside the region given by eq.(8.99), the e®ect of the displacement of the plasma boundary is not great and the contribution of the integral term in Wp is dominant. However, the growth rate ° 2 is k2 r2 times as small as that given by eq.(8.100), as is clear from consideration of eq.(8.101).

8.3e Reversed Field Pinch12 The characteristics of the Reversed ¯eld pinch is that Ba and Bs are of the same order of magnitude, so that the approximation based upon ka ¿ 1 or Ba ¿ Bs can no longer be used. As is clear from the expression (8.82) for Wp , the plasma is stable if 2¹0 jz

Bµ 1 d Bµ d =2 2 (rBµ ) = 3 (rBµ )2 < 0 r r dr r dr

(8:102)

is satis¯ed everywhere. This is a su±cient condition; however, it can never be satis¯ed in real cases. When the expression (8.95) for g is rewritten in terms of P ´ rBz =Bµ (2¼P is the pitch of the magnetic-¯eld lines), we ¯nd g=

2(kr)2 ¹0 dp0 Bµ2 =r (kP + m) + m2 + (kr)2 dr (m2 + (kr)2 )2 ³

´

£ kP ((m2 + k2 r2 )2 ¡ (m2 ¡ k2 r2 )) + m((m2 + k2 r 2 )2 ¡ (m2 + 3k2 r2 )) : When m = 1, g becomes g=

2(kr)2 ¹0 dp0 (kr)2 Bµ2 =r (kP + 1)(kP (3 + k2 r2 ) + (k2 r 2 ¡ 1)): + 1 + (kr)2 dr (1 + (kr)2 )2

(8:103)

(8:104)

The 2nd term in eq.(8.104) is quadratic with respect to P and has the minimum value g(r) > 2¹0

4Bµ2 dp0 k2 r2 k2 r2 ¡ : dr 1 + k2 r 2 r (1 + k 2 r 2 )2 (3 + k2 r 2 )

The condition g(r) > 0 is reduced to r¹0 dp0 2 > 2 2 2 Bµ dr (1 + k r )(3 + k2 r 2 )

(8:105)

8.3 Instabilities of a Cylindrical Plasma

97

Fig.8.10 Dependence of pitch P (r) on r, and the region gj (P; r) < 0. The displacements »r (r) of the unstable modes are also shown. Parts a-d are for k < 0.

(dp0 =dr must be positive). Accordingly if the equilibrium solution is found to satisfy the condition (8.105) near the plasma center and also to satisfy the condition (8.102) at the plasma boundary, the positive contribution of the integral term may dominate the negative contribution from the plasma boundary and this equilibrium con¯guration may be stable. Let us consider the 2nd term of eq.(8.104): gj =

k2 rBµ2 (kP + 1)(kP (3 + k2 r 2 ) + (k2 r2 ¡ 1)): (1 + k2 r 2 )2

(8:106)

This term is positive when kP < ¡1 or

kP > (1 ¡ k2 r2 )=(3 + k2 r2 ):

(8:107)

The point kP = ¡1 is a singular point. The region gj < 0 is shown in the P; r diagrams of ¯g.8.10a-d for given k (< 0). rs is a singular point. Several typical examples of P (r) are shown in the ¯gure. It is clear that »r (r) shown in (b) and (c) makes W negative. The example (d) is the case where the longitudinal magnetic ¯eld Bz is reversed at r = r1 . If k (> 0) is chosen so that kP (0) < 1=3 and if the singular point rs satisfying kP (rs ) = ¡1 is smaller than r = b (i.e., rs does not lie on the conducting wall r = b), the plasma is unstable for the displacement »r (r) shown in ¯g.8.10d0 . The necessary condition for the stability of the reverse-¯eld con¯guration is ¡P (b) < 3P (0):

(8:108)

This means that Bµ cannot be very small compared with Bz and that the value of the reversed Bz at the wall cannot be too large. When m = 1, eq.(8.82) for Wp yields the su±cient condition of stability: 2¹0 jz

Bµ B2 < 2µ (1 + kP )2 : r r

(8:109)

98

8 Magnetohydrodynamic Instabilities

The most dangerous mode is k = ¡1=P (a). With the assumption Bµ > 0, the stability condition becomes 1 Bµ 2 r

¹0 jz <

µ

P (r) +1 P (a)

¡

¶2

:

(8:110)

Accordingly if the condition ¯ ¯ ¯ P (r) ¯ ¯ ¯ ¯ P (a) ¯ > 1

(8:111)

is satis¯ed at small r and jz is negative at r near the boundary, this con¯guration may be stable. Let us consider limitations on the beta ratio from the standpoint of stability. For this purpose the dangerous mode kP (a) = ¡1 is examined, using eq.(8.82) for Wp . The substitution 0 ∙ r ∙ a ¡ ";

»r (r) = »

»r (r) = 0

r ¸a+"

into eq.(8.82) yields ¼ 2 Wp = » 2¹0

Z

µ

dr d (krBz ¡ mBµ )2 ¡2Bµ (rBµ ) + (krBz + mBµ )2 + r dr m2 + k2 r2

a

0

!

:

When m = 1, then Wp is ¼ 2 » Wp = 2¹0

Z

µ

k2 r2 (krBz ¡ Bµ )2 dr d ¡2Bµ (rBµ ) + 2k2 r2 Bz2 + 2Bµ2 ¡ r dr 1 + k 2 r2

a

0

!

:

Since the last term in the integrand is always negative, the integration of the other three terms must be positive, i.e., µ

¼ 2 » ¡Bµ2 (a) + 2k2 2¹0

Z

0

a

rBz2 dr

¶

> 0:

Using kP (a) = ¡1 and the equilibrium equation (6.9), 2 a2

Z

a

0

Ã

B2 ¹0 p0 + z 2

!

Ã

B 2 + Bµ2 rdr = ¹0 p0 + z 2

!

r=a

we can convert the necessary condition for stability to 2

a

Bµ2 (a)

> 4¹0

Z

0

a

rp0 dr

i.e., ¯µ ´

Ã

2¹0 Bµ2

!

2¼ ¼a2

Z

p0 rdr < 1:

(8:112)

Next let us study the stability of the mode m = 0 in the reversed ¯eld pinch. It is assumed that Bz reverses at r = r1 . The substitution »r (r) = ¸r

0 ∙ r ∙ r1 ¡ ";

»r (r) = 0

r > r1 + "

into eq.(8.82) yields Wp =

¼ 2 ¸ 2¹0

Z

0

r1

µ

rdr 4Bz2 ¡ 2Bµ

¶

d (rBµ ) + k2 r 2 Bz2 : dr

8.4 Hain-LÄ ust Magnetohydrodynamic Equation

99

Using the equilibrium equation (6.8), we obtain the necessary condition for stability: 2 2 ¹¡1 0 r1 Bµ (r1 )

>8

Z

r1

0

rp0 dr ¡ 4r12 p0 (r1 ):

If p0 (r1 ) » 0, we have the condition 1 ¯µ < : 2

(8:113)

8.4 Hain-LÄ ust Magnetohydrodynamic Equation When the displacement » is denoted by »(r; µ; z; t) = »(r) exp i(mµ + kz ¡ !t) and the equilibrium magnetic ¯eld B 0 is expressed by B(r) = (0; Bµ (r); Bz (r)) the (r; µ; z) components of magnetohydrodynamic equation of motion are given by d ¡¹0 ½m ! »r = dr 2

Ã

µ

d ¹0 °p(r ¢ ») + B (r»r ) + iD(»µ Bz ¡ »z Bµ ) r dr

d ¡ F +r dr 2

¡¹0 ½m ! 2 »µ = i

¶

21

µ

Bµ r

¶2 !

»r ¡ 2ik

Bµ (»µ Bz ¡ »z Bµ ); r

(8:114)

1 d m Bµ Bz °¹0 p(r ¢ ») + iDBz (r»r ) + 2ik »r ¡ H 2 Bz (»µ Bz ¡ »z Bµ ); (8:115) r r dr r

¡¹0 ½m ! 2 »z = ik°¹0 p(r ¢ ») ¡ iDBµ

1 d B2 (r»r ) ¡ 2ik µ »r + H 2 Bµ (»µ Bz ¡ »z Bµ ) r dr r

(8:116)

where F =

m Bµ + kBz = (k ¢ B); r r¢» =

D=

m Bz ¡ kBµ ; r

H2 =

µ

m r

¶2

+ k2 ;

1 d im (r»r ) + »µ + ik»z : r dr r

When »µ , »z are eliminated by eqs.(8.115),(8.116), we ¯nd d dr

Ã

!

(¹0 ½m ! 2 ¡ F 2 ) 1 d (¹0 ½m ! 2 (°¹0 p + B 2 ) ¡ °¹0 pF 2 ) (r»r ) ¢ r dr

"

d + ¹0 ½m ! ¡ F ¡ 2Bµ dr 2

d +r dr

µ

2kBµ ¢r2

2

µ

µ

Bµ r

¶

¶

¡

4k2 Bµ2 (¹0 ½m ! 2 B 2 ¡ °¹0 pF 2 ) ¢ r2 ¶¸

m Bz ¡ kBµ (¹0 ½m ! 2 (°¹0 p + B 2 ) ¡ °¹0 pF 2 ) r

=0 where ¢ is ¢ = ¹20 ½2m ! 4 ¡ ¹0 ½m ! 2 H 2 (°¹0 p + B 2 ) + °¹0 pH 2 F 2 :

»r (8.117)

100

8 Magnetohydrodynamic Instabilities

Fig.8.11 Ballooning mode

This equation was derived by Hain-LÄ ust.13 The solution of eq.(8.117) gives »r (r) in the region of 0 < r < a. The equations for the vacuum region a < r < aw (aw is the radius of wall) are r £ B 1 = 0;

r ¢ B1 = 0

so that we ¯nd B 1 = rÃ;

4Ã = 0

and à = (bIm (kr) + cKm (kr)) exp(imµ + ikz); B1r =

¢ @Ã ¡ 0 0 (kr) exp(imµ + ikz): = bIm (kr) + cKm @r

(8:118)

B1r in the plasma region is given by

B1r = i(k ¢ B)»r = iF »r and the boundary conditions at r = a are B1r (a) = iF »r (a);

(8.119)

0 B1r (a) = i(F 0 »r (a) + F »r0 (a));

(8.120)

and the coe±cients b,c can be ¯xed. To deal with this equation as an eigenvalue problem, boundary conditions must be imposed on »r ; one is that »r / r m¡1 at r = 0, and the other is that the radial component of the perturbed magnetic ¯eld at the perfect conducting wall B1r (aw ) = 0. After ¯nding suitable ! 2 to satisfy these conditions, the growth rate ° 2 ´ ¡! 2 is obtained.14 8.5 Ballooning Instability In interchange instability, the parallel component kk = (k ¢ B)=B of the propagation vector is zero and an average minimum-B condition may stabilize such an instability. Suydam's condition and the local-mode stability condition of toroidal-system are involved in perturbations with kk = 0. In this section we will study perturbations where kk 6= 0 but jkk =k? j ¿ 1. Although the interchange instability is stabilized by an average minimum-B con¯guration, it is possible that the perturbation with kk 6= 0 can grow locally in the bad region of the average minimum-B ¯eld. This type of instability is called the ballooning mode (see ¯g.8.11).

8.5 Ballooning Instability

101

The energy integral ±W is given by ±W =

1 2¹0

Z

((r £ (» £ B 0 ))2 ¡ (» £ (r £ B 0 )) ¢ r £ (» £ B 0 ) +°¹0 p0 (r ¢ »)2 + ¹0 (r ¢ »)(» ¢ rp0 ))dr:

Let us consider the case that » can be expressed by

»=

B 0 £ rÁ ; B02

(8:121)

where Á is considered to be the time integral of the scalar electrostatic potential of the perturbed electric ¯eld. Because of » £ B 0 = r? Á the energy integral is reduced to 1 ±W = 2¹0

Z µ

2

(r £ r? Á) ¡

µ

¶

(B 0 £ r? Á) £ ¹0 j 0 r £ r? Á B02 ¶

2

+°¹0 p0 (r ¢ ») + ¹0 (r ¢ »)(» ¢ rp0 ) dr: r ¢ » is given by

µ

B 0 £ rÁ r¢» =r¢ B02

¶

= rÁ ¢ r £

µ

B0 B02

¶

= rÁ ¢

µµ

1 r 2 B

¶

¶

1 £ B + 2r £ B : B

The 2nd term in ( ) is negligible compared with the 1st term in the low beta case. By means of rp0 = j 0 £ B 0 , ±W is expressed by 1 ±W = 2¹0

Z

µ

¹0 rp0 ¢ (r? Á £ B 0 ) B 0 ¢ r £ r? Á (r £ r? Á) + B02 B02 2

µ

µ

¹0 (j 0 ¢ B 0 ) 1 ¡ r? Á ¢ r £ r? Á + °¹0 p0 r 2 B0 B02 µ

µ

1 ¹0 rp0 ¢ (B 0 £ r? Á) r + 2 B0 B02

¶

¶

¢ (B 0 £ r? Á)

¶

¶2

¶

¢ (B 0 £ r? Á) dr:

Let us use z coordinate as a length along a ¯eld line, r as radial coordinate of magnetic surfaces and µ as poloidal angle in the perpendicular direction to ¯eld lines. The r; µ; z components of rp0 , B, and rÁ are approximately given by rp0 = (p00 ; 0; 0);

B = (0; Bµ (r); B0 (1 ¡ rR¡1 c (z)));

rÁ = (@Á=@r ， @Á=r@µ; @Á=@z);

Á(r; µ; z) = Á(r; z)Re(exp imµ):

Rc (z) is the radius of curvature of the line of magnetic force: µ

¶

z 1 1 ¡w + cos 2¼ : = Rc (z) R0 L

102

8 Magnetohydrodynamic Instabilities

When Rc (z) < 0, the curvature is said to be good. If the con¯guration is average minimum-B, w and R0 must be 1 > w > 0 and R0 > 0. Since Bµ =B0 ; r=R0 ; r=L are all small quantities, we ¯nd µ

¶

@Á im r? Á = rÁ ¡ rk Á ¼ Re ; Á; 0 ; @r r r £ (r? Á) ¼ Re

Ã

!

¡im @Á @ 2 Á ; ;0 ; r @z @z@r

µ

¡im @Á B0 Á; B0 ; 0 B 0 £ r? Á ¼ Re r @r

¶

and ±W is reduced to 1 ±W = 2¹0

Z

m2 r2

õ

@Á(r; z) @z

¶2

!

¯ ¡ (Á(r; z))2 2¼rdrdz rp Rc (z)

where ¡p0 =p00 = rp and ¯ = p0 =(B02 =2¹0 ). The 2nd term contributes to stability in the region Rc (z) < 0 and contributes to instability in the region of Rc (z) > 0. Euler's equation is given by d2 Á ¯ + Á = 0: 2 dz rp Rc (z)

(8:122)

Rc is nearly equal to B=jrBj. Equation (8.122) is a Mathieu di®erential equation, whose eigenvalue is w = F (¯L2 =2¼ 2 rp R0 ): Since x ¿ 1;

F (x) = x=4;

F (x) = 1 ¡ x¡1=2 ;

x À 1;

we ¯nd the approximate relation ¯c »

4w 2¼ 2 rp R0 : (1 + 3w)(1 ¡ w)2 L2

Since w is of the order of rp =2R0 and the connection length is L ¼ 2¼R0 (2¼=¶) (¶ being the rotational transform angle), the critical beta ratio ¯c is ¯c »

µ

¶ 2¼

¶2 µ

rp R

¶

:

(8:123)

If ¯ is smaller than the critical beta ratio ¯c , then ±W > 0, and the plasma is stable. The stability condition for the ballooning mode in the shearless case is given by15 ¯ < ¯c : In the con¯guration with magnetic shear, more rigorous treatment is necessary. According to the analysis16;17;18 for ballooning modes with large toroidal mode number n À 1 and m¡ nq » 0 (see appendix C), the stable region in the shear parameter S and the measure of pressure gradient ® of ballooning mode is shown in Fig.8.12. The shear parameter S is de¯ned by S=

r dq q dr

8.5 Ballooning Instability

103

Fig.8.12 The maximum stable pressure gradient ® as a function of the shear parameter S of ballooning mode. The dotted line is the stability boundary obtained by imposing a more restricted boundary condition on the perturbation.16

where q is the safety factor (q ´ 2¼=¶: ¶ rotational transform angle) and the measure of pressure gradient ® is de¯ned by ®=¡

q2 R dp : B 2 =2¹0 dr

The straight-line approximation of the maximum pressure gradient in the range of large positive shear (S > 0:8) is ® » 0:6S as is shown in Fig.8.12. Since 1 1 ¯= 2 B0 =2¹0 ¼a2

Z

a

0

1 1 p2¼rdr = ¡ 2 B0 =2¹0 a2

Z

0

a

dp 2 r dr dr

the maximum ballooning stable beta is a ¯ = 0:6 R

µ

1 a3

Z

0

a

¶

1 dq 3 r dr : q 3 dr

Under an optimum q pro¯le, the maximum beta is given by19 ¯max » 0:28

a Rqa

(qa > 2)

(8:124)

where qa is the safety factor at the plasma boundary. In the derivation of (8.124) qa > 2; q0 = 1 are assumed. It must be noti¯ed that the ballooning mode is stable in the negative shear region of S, as is shown in ¯g.8.12. When the shear parameter S is negative (q(r) decreases outwardly), the outer lines of magnetic force rotate around the magnetic axis more quickly than inner ones. When the pressure increases, the tokamak plasma tends to expand in a direction of major radius (Shafranov shift). This must be counter balanced by strengthening the poloidal ¯eld on the outside of tokamak plasma. In the region of strong pressure gradient, the necessary poloidal ¯eld increases outwardly, so on outer magnetic surfaces the magnetic ¯eld lines rotate around the magnetic axis faster than on inner ones and the shear parameter becomes more negative18 . In reality the shear parameter in a tokamak is positive in usual operations. However the fact that the ballooning mode is stable in negative shear parameter region is very important to develope tokamak con¯guration stable against ballooning modes.

104

8 Magnetohydrodynamic Instabilities

Since 1 ¹0 r Bµ = = Rq B0 B0 2¼r

Z

r

j(r)2¼rdr

0

the pro¯le of safety factor q(r) is 1 R = q(r) 2B0

µ

¹0 ¼r2

Z

¶

r

0

j2¼rdr ´

¹0 R hj(r)ir : 2B0

Therefore a negative shear con¯guration can be realized by a hollow current pro¯le. The MHD stability of tokamak with hollow current pro¯les is analyzed in details in ref.20.

8.6 ´i Mode due to Density and Temperature Gradient Let us consider a plasma with the density gradient dn0 =dr, and the temperature gradient dTe0 =dr; dTi0 =dr in the magnetic ¯eld with the z direction. Assume that the ion's density becomes ni = ni0 + n ~ i by disturbance. The equation of continuity @ni + v i ¢ rni + ni r ¢ vi = 0 @t is reduced, by the linearization, to ¡i!~ ni + v~r

@n0 + n0 ikk v~k = 0: @r

(8:125)

It is assumed that the perturbation terms changes as exp i(kµ rµ + kk z ¡ !t) and kµ ; kk are the µ and z components of the propagation vector. When the perturbed electrostatic potential is ~ ~ the E £ B drift velocity is v~r = Eµ =B = ikµ Á=B. Since the electron density denoted by Á, follows Boltzmann distribution, we ¯nd n ~e eÁ~ = : n0 kTe

(8:126)

The parallel component of the equation of motion to the magnetic ¯eld ni mi

dvk = ¡rk pi ¡ enrk Á dt

is reduced, by the linearization, to ~ ¡i!ni mi v~k = ¡ikk (~ pi + en0 Á):

(8:127)

Similarly the adiabatic equation @ ¡5=3 ¡5=3 ) + v ¢ r(pi ni )=0 (pi ni @t is reduced to µ

p~i 5 n ~i ¡ ¡i! pi 3 n i

¶

0

1

dn0 dTi0 ikµ Á~ @ dr 2 dr A ¡ ¡ = 0: B Ti0 3 n0

(8:128)

¤ ¤ ; !T¤ ee and the ion drift frequency !ni ; !T¤ i by Let us de¯ne the electron drift frequencies !ne ¤ ´¡ !ne

kµ (∙Te ) dne ; eBne dr

¤ ´ !ni

kµ (∙Ti ) dni ; eBni dr

References

105

!T¤ e ´ ¡

kµ d(∙Te ) ; eB dr

!T¤ i ´

kµ d(∙Ti ) : eB dr

The ratio of the temperature gradient to the density gradient of electrons and ions is given by ´e ´

dTe =dr ne d ln Te ; = Te dne =dr d ln ne

´i ´

dTi =dr ni d ln Ti = Ti dni =dr d ln ni

respectively. There are following relations among these values; ¤ !ni =¡

Ti ¤ ! ; Te ne

¤ !T¤ e = ´e !ne ;

¤ !T¤ i = ´i !ni :

Then equations (8.125),(8.126),(8.127),(8.128) are reduced to v~k n ~i ! ¤ eÁ~ = + ne ; n0 !=kk ! ∙Te eÁ~ n ~e = ; n0 ∙Te µ

¶

v~k 1 p~i = eÁ~ + ; !=kk mi (!=kk )2 n0 µ

5 n p~i ~ ¡ pi0 3 n0

¶

µ

2 !¤ = ne ´i ¡ ! 3

¶

eÁ~ : ∙Te

~ e =n0 yields the dispersion equation21 Charge neutrality condition n ~ i =n0 = n !¤ 1 ¡ ne ¡ !

Ã

vTi !=kk

!2 µ

µ

¤ 2 Te 5 !ne + + ´i ¡ Ti 3 ! 3

¶¶

= 0:

2 (vTi = ∙Ti =mi ). The solution in the case of ! ¿ !n¤ e is 2

! =

2 ¡kk2 vTi

µ

¶

2 ´i ¡ : 3

The dispersion equation shows that this type of perturbation is unstable when ´i > 2=3. This mode is called ´i mode. When the propagation velocity j!=kk j becomes the order of the ion thermal velocity vTi , the interaction (Landau damping) between ions and wave (perturbation) becomes important as will be described in ch.11 and MHD treatment must be modi¯ed. When the value of ´i is not large, the kinetic treatment is necessary and the threshold of ´i becomes ´i;cr » 1:5. References 1. 2. 3. 4. 5. 6.

G. Bateman: MHD instabilities, The MIT Press, Cambridge Mass. 1978 M. Kruskal and M. Schwarzschield: Proc. Roy. Soc. A223, 348 (1954) M. N. Rosenbluth, N. A. Krall and N. Rostoker: Nucl. Fusion Suppl. Pt.1 p.143 (1962) M. N. Rosenbluth and C. L. Longmire: Annal. Physics 1, 120 (1957) I. B. Berstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud: Proc. Roy. Soc. A244, 17 (1958) B. B. Kadmotsev: Reviews of Plasma Physics 2, 153(ed. by M. A. Loentovich) Consultant Bureau, New York 1966 7. K. Miyamoto: Plasma Physics for Nuclear Fusion (revised edition) Chap.9, The MIT Press,

106

8 Magnetohydrodynamic Instabilities

Cambridge, Mass. 1988 8. M. D. Kruskal, J. L. Johnson, M. B. Gottlieb and L. M. Goldman: Phys. Fluids 1, 421 (1958) 9. V. D. Shafranov: Sov. Phys. JETP 6, 545 (1958) 10. B. R. Suydam: Proc. 2nd U. N. International Conf. on Peaceful Uses of Atomic Energy, Geneva, 31, 157 (1958) 11. W. A. Newcomb: Annal. Physics 10, 232 (1960) 12. D. C. Robinson: Plasma Phys. 13, 439 (1971) 13. K. Hain and R. LÄ ust: Z. Naturforsh. 13a, 936 (1958) 14. K. Matsuoka and K. Miyamoto: Jpn. J. Appl. Phys. 18, 817 (1979) 15. R. M. Kulsrud: Plasma Phys. and Controlled Nucl. Fusion Research,1, 127, 1966 (Conf. Proceedings, Culham in 1965 IAEA Vienna) 16. J. W. Connor, R. J. Hastie and J. B. Taylor: Phys. Rev. Lett. 40, 393 (1978) 17. J. W. Connor, R. J. Hastie and J. B. Taylor: Pro. Roy. Soc. A365, 1 (1979) 18. J. M. Green, M. S. Chance: Nucl. Fusion 21, 453 (1981) 19. J. A. Wesson, A. Sykes: Nucl. Fusion 25 85 (1985) 20. T. Ozeki, M. Azumi, S. Tokuda, S. Ishida: Nucl. Fusion 33, 1025 (1993) 21. B. B. Kadomtsev and O. P. Pogutse: Reviews of Plasma Physics 5,304 (ed. by M. A. Leontovich) Consultant Bureau, New York 1970

9 Resistive Instability

107

Ch.9 Resistive Instability In the preceding chapter we have discussed instabilities of plasmas with zero resistivity. In such a case the conducting plasma is frozen to the line of magnetic force. However, the resistivity of a plasma is not generally zero and the plasma may hence deviate from the magnetic line of force. Modes which are stable in the ideal case may in some instances become unstable if a ¯nite resistivity is introduced. Ohm's law is ´j = E + V £ B:

(9:1)

For simplicity we here assume that E is zero. The current density is j = V £ B=´ and the j £ B force is Fs = j £ B =

B(V ¢ B) ¡ V B 2 : ´

(9:2)

When ´ tends to zero, this force becomes in¯nite and prevents the deviation of the plasma from the line of magnetic force. When the magnitude B of magnetic ¯eld is small, this force does not become large, even if ´ is small, and the plasma can deviate from the line of magnetic force. When we consider a perturbation with the propagation vector k, only the parallel (to k) component of the zeroth-order magnetic ¯eld B a®ects the perturbation, as will be shown later. Even if shear exists, we can choose a propagation vector k perpendicular to the magnetic ¯eld B: (k ¢ B) = 0:

(9:3)

Accordingly, if there is any force F dr driving the perturbation, this driving force may easily exceed the force F s , which is very small for a perturbation where (k ¢ B) = 0, and the plasma becomes unstable. This type of instability is called resistive instability. 9.1 Tearing Instability Let us consider a slab model in which the zeroth-order magnetic ¯eld B 0 depends on only x and B is given as follows; B 0 = B0y (x)ey + B0z (x)ez :

(9:4)

From Ohm's law (9.1) we ¯nd @B ´ = ¡r £ E = r £ ((V £ B) ¡ ´j) = r £ (V £ B) + ¢B @t ¹0

(9:5)

where ´ is assumed to be constant. It is assumed that the plasma is incompressible. Since the growth rate of the resistive instability is small compared with the MHD characteristic rate (inverse of Alfven transit time) and the movement is slower than the sound velocity, the assumption of incompressibility is justi¯ed and it follows that r ¢ V = 0:

(9:6)

The magnetic ¯eld B always satis¯es r ¢ B = 0:

(9:7)

108

9 Resistive Instability

The equation of motion is ½m

dV 1 (r £ B) £ B ¡ rp = dt ¹0 1 = ¹0

Ã

rB 2 (B 0 ¢ r)B 1 + (B 1 ¢ r)B 0 ¡ 2

!

¡ rp:

(9:8)

Let us consider the perturbation expressed by f1 (r; t) = f1 (x) exp (i(ky y + kz z) + °t). Then eq.(9.5) reduces to °B1x

´ = i(k ¢ B)Vx + ¹0

Ã

!

@2 ¡ k2 B1x @x2

(9:9)

where k2 = ky2 + kz2 . The ¯rst term in the right-hand side of eq.(9.8) becomes (B 0 ¢ r)B 1 = i(k ¢ B 0 )B 1 . The rotation of eq.(9.8) is µ

µ

¶

¶

@ B0 : ¹0 ½m °r £ V = r £ i(k ¢ B 0 )B 1 + B1x @x

(9:10)

Equations (9.6),(9.7) reduce to @B1x + iky B1y + ikz B1z = 0; @x @Vx + iky Vy + ikz Vz = 0: @x

(9.11) (9.12)

Multiply ky and z component of eq.(9.10) and multiply kz and the y component and take the di®erence. Use the relations of eqs.(9.11) and (9.12); then we ¯nd ¹0 ½m °

Ã

!

Ã

!

@2 @2 2 ¡ k ¡ k2 B1x ¡ i(k ¢ B 0 )00 B1x V = i(k ¢ B ) x 0 @x2 @x2

(9:13)

where 0 is di®erentiation in x. Ohm's law and the equation of motion are reduced to eqs.(9.9) and (9.13). It must be noti¯ed that the zeroth-order magnetic ¯eld B 0 appears only in the form of (k ¢ B 0 ). When we introduce a function F (x) ´ (k ¢ B 0 )

(9:14)

the location of F (x) = 0 is the position where resistive instabilities are likely occurred. We choose this position to be x = 0 (see ¯g.9.1). F (x) is equal to (k ¢ B 0 ) ' (k ¢ B 0 )0 x near x = 0. As is clear from eqs.(9.9) and (9.13), B1x is an even function and Vx is an odd function near x = 0. The term j¢B1x j » j¹0 ky j1z j can be large only in the region jxj < ". Since the growth rate of resistive instability is much smaller than MHD growth rate, the left-hand side of the equation of motion (9.13) can be neglected in the region jxj > " and we have d2 B1x F 00 2 ¡ k B = B1x ; 1x dx2 F

jxj > ":

(9:15)

The solution in the region x > 0 is ¡kx

B1x = e

ÃZ

x

e ¡1

2k»

d»

Z

»

1

00

¡k´

(F =F )B1x e

!

d´ + A

9.1 Tearing Instability

109

Fig.9.1 Zeroth-order magnetic con¯guration and magnetic islands due to tearing instability. Pro¯les of B1x and Vx are also shown.

and the solution in the region x < 0 is B1x = ekx

ÃZ

x

e¡2k» d»

1

Z

»

1

!

(F 00 =F )B1x ek´ d´ + B :

0 0 (+") at x = +" and B1x (¡") at x = ¡" as follows; Let us de¯ne ¢0 as the di®erence between B1x

¢0 =

0 0 (+") ¡ B1x (¡") B1x : B1x (0)

(9:16)

Then the value of ¢0 obtained from the solutions in the region jxj > " is given by ¢0 = ¡2k ¡

1 B1x (0)

µZ

¡"

+

¡1

Z

"

1¶

exp(¡kjxj)(F 00 =F )B1x dx:

(9:17)

For a trial function of F (x) = Fs x=Ls (jxj < Ls );

F (x) = Fs x=jxj (x > jLs j)

we can solve eq.(9.15) and ¢0 is reduced to 0

¢ =

µ

2® Ls

¶

2 e¡2® + (1 ¡ 2®) ¼ ¡2® ¡ (1 ¡ 2®) e Ls

µ

1 ¡® ®

¶

Here ® ´ kLs was used and Ls is shear length de¯ned by Ls = (F=F 0 )x=0 . For more general cases of F (x), B1x (x) has logarithmic singularity at x = 0, since F 00 =F / 1=x generally. Referene 2 describes the method to avoid di±culties arising from the corresponding logarithmic singularity. Equations (9.9) and (9.13) in the region jxj < " reduce to µ

¶

°¹0 ¹0 @ 2 B1x ¡ k2 + B1x = ¡i F 0 xVx ; 2 @x ´ ´

(9:18)

110

9 Resistive Instability Ã

!

µ

¶

F 00 @ 2 Vx (F 0 )2 2 1 ¡ ¡ k2 + B1x : x Vx = i F 0 x 2 @x ½m ´° ½m ´ ¹0 ½m °

(9:19)

The value of ¢0 obtained from the solution in the region jxj < " is given from eq.(9.18) as follows; @B1x (+") @B1x (¡") ¡ @x @x

¢0 £ B1x (0) =

¹0 ´

=

Z

"

µµ

°+

¡"

¶

´ 2 k B1x ¡ iF 0 xVx ¹0

¶

dx:

(9:20)

The value ¢0 of eq.(9.20) must be equal to the value of ¢0 of eq.(9.17). This requirement gives the eigenvalue ° and the growth rate of this resistive instability can be obtained.1 However we try to reduce the growth rate in qualitative manner in this section. In the region jxj < ", it is possible to write ¢0 B1x @ 2 B1x » : @x2 " It is assumed that the three terms of eq.(9.9), namely the term of induced electric ¯eld (the left-hand side), the V £ B term (the ¯rst term in the right-hand side) and Ohm's term (the second term) are the same order: ´ ¢0 B1x ; ¹0 "

°B1x »

(9.21)

°B1x » iF 0 "Vx :

(9.22)

Then eq.(9.21) yields ´ ¢0 : ¹0 "

°»

(9:23)

Accordingly ¢0 > 0

(9:24)

is the condition of instability. In order to get the value of °, the evaluation of " is necessary. Equation (9.13) reduces to ¹0 ½m °

µ

¡Vx "2

¶

» iF 0 "

¢0 B1x : "

(9:25)

If the terms Vx ; B1x ; ° are eliminated by eqs.(9.21), (9.22) and (9.25), we ¯nd "5 » " » a

µ

´ ¹0 a2

õ

¿A ¿R

¶2

where the physical quantities ¿R =

¶2

¹0 a2 ; ´

(¢0 a)

½m ¹0 5 a ; (F 0 a)2

µ

B0 (¢ a) F 0 a2 0

¶2 !1=5

»S

¡2=5

0

1=5

(¢ a)

µ

B0 (k ¢ B 0 )0 a2

¶2=5

(9:26)

9.2 Resistive Drift Instability

¿A =

111

a B0 =(¹0 ½m )1=2

are the resistive di®usion time and Alfv¶en transit time respectively. A non-dimensional factor S = ¿R =¿A is magnetic Reynolds number and a is a typical plasma size. Accordingly the growth rate ° is given by ´ a 0 (¢0 a)4=5 °= a) = (¢ 3=5 2=5 ¹0 a2 " ¿R ¿A

Ã

(k ¢ B 0 )0 a2 B0

!2=5

(¢0 a)4=5 = S 3=5

Ã

(k ¢ B 0 )0 a2 B0

!2=5

1 : ¿A

(9:27)

Since this mode is likely break up the plasma into a set of magnetic islands as is shown in ¯g.9.1, this mode is called tearing instability.1 The foregoing discussion has been based on the slab model. Let us consider this mode in a toroidal plasma. The poloidal and the toroidal components of the propagation vector k are m=r and ¡n=R respectively. Accordingly there are correspondences of ky $ m=r; and kz $ ¡n=R, and n m n (k ¢ B 0 ) = Bµ ¡ Bz = Bµ r R r

µ

¶

m ¡q ; n

q´

r Bz : R Bµ

Therefore weak positions for tearing instability are given by (k ¢ B 0 ) = 0 and these are rational surfaces satisfying q(rs ) = m=n. The shear is given by ¡n dq (k ¢ B 0 ) = Bµ ; r dr 0

µ

rs (k ¢ B 0 )0 rs2 = ¡n B0 R

¶

q 0 rs : q

The tearing mode is closely related to the internal disruption in tokamak and plays important role as is described in sec.16.3. It has been assumed that the speci¯c resistivity ´ and the mass density ½m are uniform and there is no gravitation (acceleration) g = 0. If ´ depends on x, the resistive term in eq.(9.5) becomes r £ (´r £ B)=¹0 . When there is temperature gradient (´ 0 6= 0), rippling mode with short wavelength (kLs À 1) may appear in the smaller-resistivity-side (high-temperature-side) of x = 0 position. When there is gravitation, the term ½g is added to the equation of motion (9.8). If the direction of g is opposit to r½m (g is toward low-density-side), gravitational interchange mode may appear.1 9.2 Resistive Drift Instability A ¯nite density and temperature gradient always exists at a plasma boundary. Con¯gurations including a gradient may be unstable under certain conditions. Let us consider a slab model. The directon of the uniform magnetic ¯eld is taken in the z direction and B 0 = (0; 0; B0 ). The x axis is taken in the direction of the density gradient with the positive direction outward from the plasma. The pressure is p0 = p0 (x), (see ¯g.9.2). The zeroth-order plasma current is j 0 = (0; p00 =B0 ; 0) and we assume that the °ow velocity and the electric ¯eld are zero V 0 = 0; E 0 = 0 in the zeroth order. The °ow velocity due to classical di®usion is neglected here. Electron inertia and ion motion along magneitc lines of force are also neglected. The usual relations in this con¯guration are Mn

@V = j £ B ¡ rp; @t

E + V £ B = ´j +

1 (j £ B ¡ rp0 ) ; en

(9:28)

(9:29)

112

9 Resistive Instability

Fig.9.2 Slab model of resistive drift wave.

@n + r ¢ (nV ) = 0; @t

(9:30)

r¢j =0

(9:31)

where M is the ion mass. In this con¯guration, electrostatic perturbations are considered here. The 1st-order electric ¯eld E 1 is expressed by the electrostatic potential E 1 = ¡rÁ1 and the 1st-order magnetic ¯eld perturbation is zero B 1 = 0 (@B=@t = r £ E). The characteristics of electrostatic perturbation will be explained in ch.10 in detail. For simplicity the ion temperature is assumed to be zero Ti = 0. Let us consider the mode n1 = n1 (x) exp i(ky + kk z ¡ !t); Á1 = Á1 (x) exp i(ky + kk z ¡ !t): Equations (9.28),(9.29) reduce to ¡i!M n0 V 1 = j 1 £ B 0 ¡ ∙Te rn1 ;

(9:32)

j 1 £ B 0 ¡ ∙Te rn1 = en0 (¡rÁ1 + V 1 £ B 0 ¡ ´j 1 ):

(9:33)

Equations (7.32),(7.33) yields i!

µ

M e

¶

V 1 = rÁ1 ¡ V 1 £ B 0 + ´j 1 :

(9:34)

When ´ is small (ºei ¿ ­e ), the contribution of ´j can be neglected in eq.(9.34), i.e., we may write Vx = ¡ik

Á1 ; B0

Vy =

µ

! ­i

¶

µ

kÁ1 ; B0

Vz = ¡

­i !

¶

kk Á1 B0

­i is as usual, the ion cyclotron frequency (­i = ¡ZeB=M ). The wave frequency ! was assumed to be low (!=­i )2 ¿ 1. The x; y component of eq.(9.32) and the z component of eq.(9.33) are jx = ¡ik

∙Te n1 ; B0

jy =

µ

! @n1 ∙Te + kn0 @x B0 ­i

¶

eÁ1 ; B0

jz =

µ

¶

ikk n1 ¡ eÁ1 : ∙Te e´ n0

Since eq.(9.31) is jx0 + ikjy + ikk jz = 0; and eq.(9.30) is ¡i!n1 +n00 Vx +n0 ikVy + n0 ikk Vz = 0, we ¯nd kk2 ∙Te n1 e´

Ã

kk2 B

! ¡ ik en0 + ¡ n0 ´ ­i 2

!

Á1 = 0; B0

(9:35)

9.2 Resistive Drift Instability

113

Fig.9.3 Dependence of !=!e¤ = x + iz on y / kk =k for resistive drift instability.

n1 + n0

Ã

2 ¡k2 kk ­i n00 k + 2 + ­i ! n0 !

!

Á1 = 0: B0

(9:36)

The dispersion equation is given by the determinant of the coe±cients of eqs.(9.35),(9.36): µ

! ­i

¶2

µ

! ¡i ­i

¶µ

kk k

¶2

B0 n0 e´

Ã

2 ∙Te kk ∙Te k2 ¡ 1¡ eB0 ­i M !2

!

¡i

µ

kk k

¶2

B0 ∙Te k n00 = 0 (9:37) n0 e´ eB0 ­i n0

where ´ = me ºei =ne2 , B0 =(n0 e´) = ­e =ºei . The drift velocities v di , v de of ions and electrons due to the density gradient rn0 are given by v di =

¡(∙Ti rn0 =n0 ) £ b ¡∙Ti = eB0 eB0

v de =

(∙Te rn0 =n0 ) £ b ∙Te = eB0 eB0

µ

µ

¶

¡n00 ey ; n0 ¶

¡n00 ey : n0

The drift frequencies of ions and electrons are de¯ned by !i¤ ´ kvdi and !e¤ ´ kvde respectively. As n00 =n0 < 0, !e¤ > 0 and !i¤ = ¡(Ti =Te )!e¤ < 0. Since !e¤ = k(¡n00 =n0 )(∙Te =m­e ), the dispersion equation is reduced to µ

! !e¤

¶2

Ã

2 ∙Te kk ¡ i 1 + (k½­ ) ¡ M !2 2

!

­e ­i ºei !e¤

µ

kk k

¶2 µ

! !e¤

¶

+i

­e ­i ºei !e¤

µ

kk k

¶2

= 0:

(9:38)

½­ is the ion Larmor radius when the ions are assumed to have the electron temperature Te . Denote !=!e¤ = x+iz; and ¡(­e ­i =ºei !e¤ )(kk =k)2 = y 2 and assume (k½­ )2 ¡(∙Te =M )(kk2 =!2 ) ¿ 1. The dispersion equation is then (x + iz)2 + iy 2 (x + iz) ¡ iy 2 = 0:

(9:39)

The dependence of the two solutions x1 (y); z1 (y) and x2 (y); z2 (y) on y / (kk =k), is shown in ¯g.9.3. As z2 (y) < 0, the mode corresponding to x2 (y); z2 (y) is stable. This wave propagates in the direction of the ion drift. The solution x1 ; z1 > 0 propagates in the direction of the electron drift and it is unstable. If the value of (kk =k) is adjusted to be y ' 1:3, the z1 value becomes maximum to be z1 ¼ 0:25 and the growth rate is Im ! ¼ 0:25 !e¤ . If ´ is small, the wavelengh of the most unstable wave becomes long and the necessary number of collision to interrupt the electron motion along the magneitc line of force is maintained. If the lower limit of kk is ¯xed by an appropriate method, the growth rate is Im(!=!e¤ )

¼y

¡2

ºei !e¤ = ­e j­i j

Ã

k kk

!2

114

9 Resistive Instability

and the growth rate is proportional to ´ / ºei . This instability is called resistive drift instability or dissipative drift instability. If the ion's inertia term can be neglected, eq.(9.35) reduces to n1 =n0 = eÁ=∙Te and the dispersion equation becomes ! 2 ¡ !kvde ¡ kk2 Te =M = 0. The instability does not appear. This instability originates in the charge separation between electrons and ions due to ion inertia. The charge separation thus induced is neutralized by electrons motion along the lines of magnetic force. However, if the parallel motion of electrons is interrupted by collision,i.e.,resistivity, the charge separation grows and the wave becomes unstable.3;4 This instability is therefore also called collisional drift instability. The mechanism of charge separation between ions and electrons can be more easily understood if the equations of motion of the two components, ion and electron, are used. The equations of motions are 0 = ¡ikn1 ∙Te + ikÁ1 en0 ¡ en0 (V e £ B) ¡ 2n0 me ºei (V e ¡V i )k ; M

(9:40)

∙Ti n0 ¡ ikÁ1 en0 + en0 (V i £ B) + n0 me ºei (V e ¡ V i )k : (9:41) (¡i!)V i = ¡ikn1 2 Z

From these equations we ¯nd µ

¶

V e? =

¡i∙Te eÁ1 n1 ¡ ; (b £ k) eB n0 ∙Te

V ek =

¡ikk ∙Te me ºei

V i? =

¡M n1 ZeÁ1 i∙Ti + + (i!)(b £ V i? ); (b £ k) ZeB n0 ∙Ti BZe

(9:44)

V ik =

kk Zc2s n1 ! n0

(9:45)

µ

¶

(9:42)

eÁ1 n1 ¡ ; n0 ∙Te µ

(9:43) ¶ µ

¶

where it has been assumed that jV ek j À jV ik j and where cs ´ ∙Te =M has been used. The continuity equation @n=@t + r ¢ (nV ) = 0 for ions and electrons yield ¡i!n1 + r ¢

µ

b £ ik n0 Á1 + (b ¢ r)(n0 Vek ) = 0; B

¶

¡i!n1 + r ¢

µ

b £ ik M (¡i!) k2 ∙Ti n0 Á1 + B ZeB ZeB

¶

µ

¶ ikk2 c2s n1 ZeÁ1 + n0 + Zn1 = 0: n0 ∙Ti !

From the equations for electrons, it follows that !e¤ + ikk2 ∙Te =(me ºei ) µ eÁ1 ¶ n1 = n0 ! + ikk2 ∙Te =(me ºei ) ∙Te

(9:46)

and from the equations for ions, it follows that !i¤ + b! n1 = n0 !(1 + b) ¡ kk2 c2s Z=!

µ

¡ZeÁ1 ∙Ti

¶

(9:47)

where b = k2 (½­ )2 . Charge neutrality gives the dispersion equation, which is equivalent to eq.(9.38), !e¤ + ikk2 ∙Te =(me ºei ) ! + ikk2 ∙Te =(me ºei )

=

!e¤ ¡ (ZTe =Ti )b! : !(1 + b) ¡ kk2 c2s Z=!

(9:48)

References

115

Ion's motion perpendicular to the magnetic ¯eld has the term of y directon (the second term in the right-hand side of eq.(3.44)) due to ion's inertia in addition to the term of x direction. In collisionless case, eq.(9.38) becomes (1 + (k½­ )2 )! 2 ¡ !e¤ ! ¡ c2s kk2 = 0:

(9:49)

The instability does not appear in the collisionless case in the framework of MHD theory. However the instability may occur even in the collisionless case when it is analyzed by the kinetic theory (see app.B). This instability is called collisionless drift instability.

References 1. H. P. Furth and J. Killeen: Phys. Fluids 6, 459 (1963) 2. H. P. Furth, P. H. Rutherford and H. Selberg: Phys. Fluids 16, 1054 (1973) A. Pletzer and R. L. Dewar: J. Plasma Phys. 45, 427 (1991) 3. S. S. Moiseev and R. Z. Sagdeev: Sov. Phys. JETP 17, 515 (1963), Sov. Phys. Tech. Phys. 9, 196 (1964) 4. F. F. Chen: Phys. Fluids 8, 912 and 1323 (1965)

116

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

Ch.10 Plasma as Medium of Electromagnetic Wave Propagation A plasma is an ensemble of an enormous number of moving ions and electrons interacting with each other. In order to describe the behavior of such an ensemble, the distribution function was introduced in ch.4; and Boltzmann's and Vlasov's equations were derived with respect to the distribution function. A plasma viewed as an ensemble of a large number of particles has a large number of degrees of freedom; thus the mathematical description of plasma behavior is feasible only for simpli¯ed analytical models. In ch.5, statistical averages in velocity space, such as mass density, °ow velocity, pressure, etc., were introduced and the magnetohydrodynamic equations for these averages were derived. We have thus obtained a mathematical description of the magnetohydrodynamic °uid model; and we have studied the equilibrium conditions, stability problems, etc., for this model in chs. 6-9. Since the °uid model considers only average quantities in velocity space, it is not capable of describing instabilities or damping phenomena, in which the pro¯le of the distribution function plays a signi¯cant role. The phenomena which can be handled by means of the °uid model are of low frequency (less than the ion or electron cyclotron frequency); high-frequency phenomena are not describable in terms of it. In this chapter, we will focus on a model which allows us to study wave phenomena while retaining the essential features of plasma dynamics, at the same time maintaining relative simplicity in its mathematical form. Such a model is given by a homogeneous plasma of ions and electrons at 0K in a uniform magnetic ¯eld. In the unperturbed state, both the ions and electrons of this plasma are motionless. Any small deviation from the unperturbed state induces an electric ¯eld and a time-dependent component of the magnetic ¯eld, and consequently movements of ions and electrons are excited. The movements of the charged particles induce electric and magnetic ¯elds which are themselves consistent with the previously induced small perturbations. This is called the kinetic model of a cold plasma. We will use it in this chapter to derive the dispersion relation which characterizes wave phenomena in the cold plasma. Although this model assumes uniformity of the magnetic ¯eld, and the density and also the zero temperature, this cold plasma model is applicable for an nonuniform, warm plasma, if the typical length of variation of the magnetic ¯eld and the density is much larger than the wavelength and the phase velocity of wave is much larger than the thermal velocity of the particles. It is possible to consider that the plasma as a medium of electromagnetic wave propagation with a dielectric tensor K. This dielectric tensor K is a function of the magnetic ¯eld and the density which may change with the position. Accordingly plasmas are in general an nonuniform, anisotropic and dispersive medium. When the temperature of plasma is ¯nite and the thermal velocity of the particles is comparable to the phase velocity of propagating wave, the interaction of the particles and the wave becomes important. A typical interaction is Landau damping, which is explained in ch.11. The general mathematical analysis of the hot-plasma wave will be discussed in ch.12 and appendix C. The references 1»4 describe the plasma wave in more detail. 10.1 Dispersion Equation of Waves in a Cold Plasma In an unperturbed cold plasma, the particle density n and the magnetic ¯eld B 0 are both homogeneous in space and constant in time. The ions and electrons are motionless. Now assume that the ¯rst-order perturbation term exp i(k ¢ r ¡ !t) is applied. The ions and electrons are forced to move by the perturbed electric ¯eld E and the induced magnetic ¯eld

10.1 Dispersion Equation of Waves in a Cold Plasma

117

B 1 . Let us denote velocity by v k , where the su¯x k indicates the species of particle (electrons, or ions of various kinds). The current j due to the particle motion is given by j=

X

nk qk v k :

(10:1)

k

nk and qk are the density and charge of the kth species, respectively. The electric displacement D is D = ²0 E + P ; (10.2) @P = ¡i!P (10.3) @t where E is the electric intensity, P is the electric polarization, and ²0 is the dielectric constant of vacuum. Consequently D is expressed by i (10:4) D = ²0 E + j ´ ²0 K ¢ E: ! K is called dielectric tensor. The equation of motion of a single particle of the kth kind is dvk (10:5) mk = qk (E + vk £ B): dt Here B consists of B = B 0 + B 1 , where v k , E, B 1 are the ¯rst-order quantities. The linearized equation in these quantities is j=

¡i!mk v k = qk (E + v k £ B 0 ):

(10:6)

When the z axis is taken along the direction of B 0 , the solution is given by vk;x

¡iEx ­k ! Ey ­k2 ¡ = ; B0 ! 2 ¡ ­k2 B0 ! 2 ¡ ­k2

9 > > > > > > > > =

Ex ­k2 iEy ­k ! ¡ ; 2 2 > > B0 ! ¡ ­k B0 ! 2 ¡ ­k2 > > > > ¡iEz ­k > > ; vk;z = B0 ! where ­k is the cyclotron frequency of the charged particle of the kth kind: ¡qk B0 ­k = mk vk;y =

(10:7)

(10:8)

(­e > 0 for electrons and ­i < 0 for ions). The components of vk are the linear functions of E given by eq.(10.7); and j of eq.(10.1) and the electric displacement D of eq.(10.4) are also the linear function of E, so that the dielectric tensor is given by 2

where

32

3

K? ¡iK£ 0 Ex 4 5 4 K 0 iK Ey 5 K¢E = £ ? 0 0 Kk Ez K? ´ 1 ¡ K£ ´ ¡

k

X k

Kk ´ 1 ¡ ¦k2 ´

X

¦k2 ; ! 2 ¡ ­k2

!2

¦k2 ­k ; ¡ ­k2 !

X ¦2 k k

nk qk2 : ² 0 mk

!2

;

(10:9)

(10.10) (10.11) (10.12) (10.13)

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

118

According to the Stix notation, the following quantities are introduced: R ´1¡ L´1¡

9 > > > > =

X ¦2 k k

! = K? + K£ ; 2 ! ! ¡ ­k

X ¦2 k k

> > > > ;

! = K? ¡ K£ : ! 2 ! + ­k

From Maxwell's equation r£E =¡

@B ; @t

r £ H = j + ²0

(10:14)

(10:15) @E @D = @t @t

(10:16)

it follows that k £ E = !B 1 ; k £ H 1 = ¡!²0 K ¢ E and k £ (k £ E) +

!2 K ¢ E = 0: c2

(10:17)

Let us de¯ne a dimensionless vector N´

kc !

(c is light velocity in vacuum). The absolute value N = jN j is the ratio of the light velocity to the phase velocity of the wave, i.e., N is the refractive index. Using N , we may write eq.(10.16) as N £ (N £ E) + K ¢ E = 0:

(10:18)

If the angle between N and B 0 is denoted by µ (¯g.10.1) and x axis is taken so that N lies in the z; x plane, then eq.(10.18) may be expressed by 2

32

3

K? ¡ N 2 cos2 µ ¡iK£ N 2 sin µ cos µ Ex 2 4 5 4 Ey 5 = 0: iK£ K? ¡ N 0 0 Kk ¡ N 2 sin2 µ N 2 sin µ cos µ Ez

(10:19)

For a nontrivial solution to exist, the determinant of the matrix must be zero, or AN 4 ¡ BN 2 + C = 0

(10.20),

A = K? sin2 µ + Kk cos2 µ;

(10.21)

2 2 ¡ K£ ) sin2 µ + Kk K? (1 + cos2 µ); B = (K?

(10.22)

2 2 ¡ K£ ) = Kk RL: C = Kk (K?

(10.23)

10.2 Properties of Waves

119

Fig.10.1 Propagation vector k and x; y; z coordinates

Equation (10.20) determines the relationship between the propagation vector k and the frequency !, and it is called dispersion equation. The solution of eq.(10.20) is N2 =

B § (B 2 ¡ 4AC)1=2 2A

2 2 ¡ K£ ) sin2 µ + Kk K? (1 + cos2 µ) = ((K? 2 2 2 ¡ K£ ¡ Kk K? )2 sin4 µ + 4Kk2 K£ §[(K? cos2 µ]1=2 )

³

£ 2(K? sin2 µ + Kk cos2 µ)

´¡1

:

(10.24)

When the wave propagates along the line of magnetic force (µ = 0), the dispersion equation (10.20) is 2 2 ¡ K£ )) = 0; Kk (N 4 ¡ 2K? N 2 + (K?

(10:25)

and the solutions are Kk = 0;

N 2 = K? + K£ = R;

N 2 = K? ¡ K£ = L:

(10:26)

For the wave propagating in the direction perpendicular to the magnetic ¯eld (µ = ¼=2), the dispersion equation and the solutions are given by 2 2 2 2 ¡ K£ ¡ K£ + Kk K? )N 2 + Kk (K? ) = 0; K? N 4 ¡ (K?

N2 =

2 ¡ K2 K? RL £ = ; K? K?

N 2 = Kk :

(10:27) (10:28)

10.2 Properties of Waves 10.2a Polarization and Particle Motion The dispersion relation for waves in a cold plasma was derived in the previous section. We consider here the electric ¯eld of the waves and the resultant particle motion. The y component of eq.(10.19) is iK£ Ex + (K? ¡ N 2 )Ey = 0; iEx N 2 ¡ K? = : Ey K£

(10.29)

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

120

The relation between the components of the particle velocity is

ivk;x vk;y

Ã

¡iEx ­k ! ¡ 2 i 2 2 Ey ! ¡ ­k ! ¡ ­k2 = ! Ex ­k ¡i 2 2 2 Ey ! ¡ ­k ! ¡ ­k2 =

!

(! + ­k )(N 2 ¡ L) + (! ¡ ­k )(N 2 ¡ R) : (! + ­k )(N 2 ¡ L) ¡ (! ¡ ­k )(N 2 ¡ R)

(10:30)

The wave satisfying N 2 = R at µ = 0 has iEx =Ey = 1 and the electric ¯eld is right-circularly polarized. In other word, the electric ¯eld rotates in the direction of the electron Larmor motion. The motion of ions and electrons is also right-circular motion. In the wave satisfying N 2 = L at µ ! 0, the relation iEx =Ey = ¡1 holds and the electric ¯eld is left-circularly polarized. The motion of ions and electrons is also left-circular motion. The waves with N 2 = R and N 2 = L as µ ! 0, are called the R wave and the L wave, respectively. The solution of the dispersion equation (10.25) at µ = 0 is Ã

jKk j 1 jR ¡ Lj R+L§ N = 2 Kk 2

!

(10:31)

so that R and L waves are exchanged when Kk changes sign. When K£ = R ¡ L changes sign, R and L waves are also exchanged. When µ = ¼=2, the electric ¯eld of the wave satisfying N 2 = Kk is Ex = Ey = 0, Ez 6= 0. For the wave sa¯sfying N 2 = RL=K? , the electric ¯eld satis¯es the relations iEx =Ey = ¡(R ¡ L)=(R + L) = ¡K£ =K? , Ez = 0. The waves with N 2 = Kk and N 2 = RL=K? as µ ! ¼=2 are called the ordinary wave (O) and the extraordinary wave (X), respectively. It should be pointed out that the electric ¯eld of the extraordinary wave at µ = ¼=2 is perpendicular to the magnetic ¯eld (Ez = 0) and the electric ¯eld of the ordinary wave at µ = ¼=2 is parallel to the magnetic ¯eld (Ex = Ey = 0). The dispersion relation (10.24) at µ = ¼=2 is 1 2 2 2 ¡ K£ ¡ Kk K? j) (K 2 ¡ K£ + Kk K? + jK? 2K? ? 1 = (RL + Kk K? § jRL ¡ Kk K? j) 2K?

N2 =

(10.32)

so that the ordinary wave and the extraordinary wave are exchanged at RL ¡ Kk K? = 0. Besides the classi¯cation into R and L waves, and O and X waves, there is another classi¯cation, namely, that of fast wave and slow wave, following the di®erence in the phase velocity. Since the term inside the square root of the equation N 2 = (B § (B 2 ¡ 4AC)1=2 )=2A is always positive, as is clear from eq.(10.24), the fast wave and slow wave do not exchange between µ = 0 and µ = ¼=2. 10.2b Cuto® and Resonance The refractive index (10.24) may become in¯nity or zero. When N 2 = 0, the wave is said to be cuto® ; at cuto® the phase velocity vph =

! c = k N

(10:33)

becomes in¯nity. As is clear from (10.20),(10.23), cuto® occurs when Kk = 0

R=0

L = 0:

(10:34)

10.3 Waves in a Two-Components Plasma

121

¯g.10.2 Wave propagation (a) near cuto® region and (b) near a resonance region

When N 2 = 1, the wave is said to be at resonance; here the phase velocity becomes zero. The wave will be absorbed by the plasma at resonance (see ch.11). The resonance condition is tan2 µ = ¡

Kk : K?

(10:35)

When µ = 0, the resonance condition is K? = (R + L)=2 ! §1. The condition R ! §1 is satis¯ed at ! = ­e , ­e being the electron cyclotron frequency. This is called electron cyclotron resonance. The condition L ! §1 holds when ! = j­i j, and this is called ion cyclotron resonance. When µ = ¼=2, K? = 0 is the resonance condition. This is called hybrid resonance. When waves approach a cuto® region, the wave path is curved according to Snell's refraction law and the waves are re°ected (¯g.10.2a). When waves approach a resonance region, the waves propagate perpendicularly toward the resonance region. The phase velocities tend to zero and the wave energy will be absorbed. 10.3 Waves in a Two-Components Plasma Let us consider a plasma which consists of electrons and of one kind of ion. Charge neutrality is ni Zi = ne :

(10:36)

A dimensionless parameter is introduced for convenience: ±=

¹0 (ni mi + ne me )c2 : B02

(10:37)

The quantity de¯ned by eq.(10.13), which was also introduced in sec.2.2, ¦e2 = ne e2 =(²0 me )

(10:38)

is called electron plasma frequency. Then we have the relations ¦e2 =¦i2 = mi =me À 1; ¦i2 + ¦e2 ¦2 = ± ¼ i2 : j­i j­e ­i

K? , K£ , Kk , and R, L are given by

¦2 ¦2 K? = 1 ¡ 2 i 2 ¡ 2 e 2 ; ! ¡ ­i ! ¡ ­e 2 ¦ 2 ­e ­i ¦ ¡ 2 e 2 K£ = ¡ 2 i 2 ; ! ¡ ­i ! ! ¡ ­e ! ¦2 ¦2 + ¦2 Kk = 1 ¡ e 2 i ' 1 ¡ 2e ! !

(10.39)

9 > > > > > > > > = > > > > > > > > ;

(10:40)

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

122

Fig.10.3 (a) Dispersion relations (! - ckk ) for R and L waves propagating parallel to the magnetic ¯eld (µ = 0). (b) Dispersion relations (! - ckk ) for O and X waves propagating perpendicular to the magnetic ¯eld (µ = ¼=2).

R =1¡

! 2 ¡ (­i + ­e )! + ­i ­e ¡ ¦e2 ¦e2 + ¦i2 ' ; (! ¡ ­i )(! ¡ ­e ) (! ¡ ­i )(! ¡ ­e )

(10:41)

L=1¡

! 2 + (­i + ­e )! + ­i ­e ¡ ¦e2 ¦e2 + ¦i2 ' : (! + ­i )(! + ­e ) (! + ­i )(! + ­e )

(10:42)

The dispersion relations for the waves propagating parallel to B 0 (µ = 0) are found by setting Kk = 0, N 2 = R, and N 2 = L. Then !2 = ¦e2 ;

(10:43)

!2 1 (! + j­i j)(! ¡ ­e ) (! ¡ ­i )(! ¡ ­e ) = = = c2 kk2 R ! 2 ¡ !­e + ­e ­i ¡ ¦e2 (! ¡ !R )(! + !L )

(10:44)

where !R ; !L are given by ­e !R = + 2

õ

­e !L = ¡ + 2

­e 2

õ

¶2

­e 2

+ ¦e2

¶2

+

!1=2

+ j­e ­i j

¦e2

> 0;

!1=2

+ j­e ­i j

> 0;

!2 1 (! ¡ j­i j)(! + ­e ) (! + ­i )(! + ­e ) = = = : c2 kk2 L !2 + !­e + ­e ­i ¡ ¦e2 (! ¡ !L )(! + !R )

(10:45)

(10:46)

(10:47)

Note that ­e > 0, ­i < 0 and !R > ­e . Plots of the dispersion relations ! ¡ ckk are shown in ¯g.10.3a. The dispersion relations for the waves propagating perpendicular to B 0 are found by 2 ¡ K2 setting N 2 = Kk (ordinary wave) and N 2 = (K? £ )=K? (extraordinary wave). Then !2 1 = = 2 2 Kk c k? !2 2 c2 k?

=

Ã

¦2 1 ¡ 2e !

!¡1

=1+

¦e2 2 ; c2 k?

(10:48)

K? K? 2(! 2 ¡­i2 )(! 2 ¡­e2 )¡¦e2 ((!+­i )(!+­e )+(!¡­i )(!¡­e )) = = 2 2) RL ¡ K£ 2(! 2 ¡ !L2 )(! 2 ¡ !R

2 K?

10.3 Waves in a Two-Components Plasma

123

Fig.10.4 The ! regions of R and L waves at µ = 0; O and X waves at µ = ¼=2; F and S waves; in the case (!L < ¦e < ­e ). The numbers on the right identify regions shown in the CMA diagram, ¯g.10.5.

Fig.10.5 CMA diagram of a two-component plasma. The surfaces of constant phase are drawn in each region. The dotted circles give the wave front in vacuum. The magnetic ¯eld is directed toward the top of the diagram.

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

124

=

! 4 ¡ (­i2 + ­e2 + ¦e2 )! 2 + ­i2 ­e2 ¡ ¦e2 ­i ­e : 2) (! 2 ¡ !L2 )(! 2 ¡ !R

(10:49)

Equation (10.48) is the dispersion equation of electron plasma wave (Langmuir wave). Let us de¯ne !UH and !LH by 2 ´ ­e2 + ¦e2 ; !UH

(10.50)

1 1 1 ´ 2 + : 2 2 j­i j­e !LH ­i + ¦i

(10.51)

!UH is called upper hybrid resonant frequency and !LH is called lower hybrid resonant frequency. Using these, we may write eq.(10.49) as 2 2 )(! 2 ¡ !UH ) !2 (! 2 ¡ !LH = : 2 2 2 2 2 2 c k? (! ¡ !L )(! ¡ !R )

(10:52)

2 We have !R > !UH > ¦e ; ­e and !LH < ­e j­i j; ­i2 +¦ 2 . Plots of the dispersion relation ! - ck? are shown in ¯g.10.3b. The gradient !=ck in ! - ck? diagram is the ratio of the phase velocity vph to c. The steeper the gradient, the greater the phase velocity. The regions (in terms of !) of R and L waves at µ = 0, and O and X waves at µ = ¼=2, and F and S waves are shown in ¯g.10.4, for the case of !L < ¦e < ­e . We explain here the CMA diagram (¯g.10.5), which was introduced by P. C. Clemmow and R. F. Mullaly and later modi¯ed by W. P. Allis3 . The quantities ­e2 =! 2 and (¦i2 + ¦e2 )=! 2 are plotted along the vertical and horizontal ordinates, respectively. The cuto® conditions R = 0 (! = !R ), L = 0 (! = !L ), Kk = 0 (! = ­e ) are shown by the dotted lines and the resonance conditions R = 1 (! = ­e ), L = 1 (! = ­i ), K? = 0 (! = ­LH ; ! = ­UH ) are shown by solid lines. The cuto® and the resonance contours form the boundaries of the di®erent regions. The boundary RL = Kk K? , at which O wave and X wave are exchanged, is also shown by broken and dotted line in ¯g.10.5. The surfaces of constant phase for R, L and O, X waves are shown for the di®erent regions. As the vertical and horizontal ordinates correspond to the magnitude of B and the density ne , one can easily assign waves to the corresponding regions simply by giving their frequencies !.

10.4 Various Waves 10.4a Alfv¶ en Wave When the frequency ! is smaller than the ion cyclotron frequency (! ¿ j­i j), the dielectric tensor K is expressed by 9 K? = 1 + ±; > > > = K£ = 0; (10:53) > > ¦e2 > ; Kk = 1 ¡ 2 ! where ± = ¹0 ni mi c2 =B02 . As ¦e2 =! 2 = (mi =me )(­i2 =! 2 )±, we ¯nd ¦e2 =!2 À ±. Assuming that ¦e2 =! 2 À 1, we have jKk j À jK? j; then A; B; C of eq.(10.20) are given by 9

¦2 > > > A ¼ ¡ 2e cos2 µ; > > ! > > = 2 ¦e 2 B ¼ ¡ 2 (1 + ±)(1 + cos µ);> > ! > > > > ¦e2 > 2 ; C ¼ ¡ 2 (1 + ±) !

(10:54)

10.4 Various Waves

125

and the dispersion relations are c2 !2 c2 = 2 = = 2 N k 1+±

B02 c2 2 ' ¹ ½ ; ¹0 ½m c 0 m 1+ B02

c2 !2 c2 = = cos2 µ N2 k2 1+±

(10.55)

(10.56)

(½m is the mass density). The wave satisfying this dispersion relation is called the Alfv¶en wave. We de¯ne the Alfv¶en velocity by 2 vA =

c2 = 1+±

c2 B02 2 ' ¹ ½ : ¹0 ½m c 0 m 1+ B02

(10:57)

Equations (10.55) and (10.56) correspond to modes appearing in region (13) of the CMA diagram. Substitution of eqs.(10.55) and (10.56) into (10.19) shows that Ez for either mode is Ez = 0; Ex = 0 for the mode (10.55) (R wave, F wave, X wave) and Ey = 0 for mode (10.56) (L wave, S wave). From eq.(10.6), we ¯nd for ! ¿ j­i j that E + vi £ B0 = 0

(10:58)

and v i = (E £ B 0 )=B 20 , so that v i of the mode (10.55) is ^ cos(kx x + kz z ¡ !t) vi ¼ x

(10:59)

and v i of the mode (10.56) is ^ cos(kx x + kz z ¡ !t) vi ¼ y

(10:60)

where x ^, y ^ are unit vectors along x and y axes, respectively. From these last equations, the fast mode (10.55) is called the compressional mode and the slow mode (10.56) is called the torsional or shear mode. The R wave (10.55) still exists, though it is deformed in the transition from region (13) to regions (11) and (8), but the L wave (10.56) disappears in these transitions. As is clear from eq.(10.58), the plasma is frozen to the magnetic ¯eld. There is tension B 2 =2¹0 along the magnetic-¯eld lines and the pressure B 2 =2¹0 exerted perpendicularly to the magnetic ¯eld. As the plasma, of mass density ½m , sticks to the ¯eld lines, the wave propagation speed in the direction of the ¯eld is B02 =(¹0 ½m ). 10.4b Ion Cyclotron Wave and Fast Wave Let us consider the case where the frequency ! is shifted from low frequency toward the ion cyclotron frequency and ¦e2 =! 2 À 1. The corresponding waves are located in regions (13) and (11) of the CMA diagram. When j!j ¿ ­e , ± À 1, and ¦e2 =!2 À 1, the values of K? , K£ and Kk are K? =

¡±­i2 ; ! 2 ¡ ­i2

K£ =

¡±!­i2 ; ! 2 ¡ ­i2

Kk = ¡

¦e2 : !2

(10:61)

Since ¦e2 =! 2 = (mi =me )(­i2 =! 2 )± À ±, the coe±cients A, B, C are 9

¦2 > > > A = ¡ 2e cos2 µ; > > ! > > > 2 2 = ¦e ±­i 2 B= 2 2 (1 + cos µ); ! ! ¡ ­i2 > > > > > 2 2 2 > ¦e ± ­i > > C= 2 2 : ; 2 ! ! ¡ ­i

(10:62)

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

126

2 = c2 =±) The dispersion equation becomes (¦i2 = ­i2 ±; vA

N 4 cos2 µ ¡ N 2

±­i2 ± 2 ­i2 2 (1 + cos µ) + = 0: ­i2 ¡ ! 2 ­i2 ¡ ! 2

(10:63)

2 =!2 , we may write eq.(10.63) as Setting N 2 cos2 µ = c2 kk2 =! 2 , and N 2 sin2 µ = c2 k? 2 2 c k?

=

! 4 ± 2 ­i2 ¡ ! 2 (2±­i2 kk2 c2 + kk4 c4 ) + ­i2 kk4 c4 ! 2 (±­i2 + kk2 c2 ) ¡ ­i2 kk2 c2

:

(10:64)

2 (!=vA kk )4 ¡ (!=vA kk )2 ¡ (!=­i )2 + 1 k? = kk2 (!=vA kk ) ¡ (1 ¡ ! 2 =­i2 )

=

((!=vA kk )2 ¡ (1 ¡ !=­i ))((!=vA kk )2 ¡ (1 + !=­i )) (!=vA kk )2 ¡ (1 ¡ ! 2 =­i2 )

(10:640 )

Therefore resonance occurs at !2 = ­i2 Ã

! vA kk

kk2 c2

kk2 c2 + ±­i2

!2

=1¡

µ

! ­i

= ­i2 ¶2

kk2 c2

(10:65)

kk2 c2 + ¦i2

(10:650 )

:

The dispersion equation (10.63) approaches N2 ¼

± ; 1 + cos2 µ

(10:66)

N 2 cos2 µ ¼ ±(1 + cos2 µ)

­i2 ­i2 ¡ ! 2

(10:67)

as j!j approaches j­i j. The mode (10.66) corresponds to the Alfv¶en compressional mode (fast wave) and is not a®ected by the ion cyclotron resonance. The dispersion relation (10.67) is that of the ion cyclotron wave, and can be expressed by 2

! =

­i2

Ã

¦2 ¦2 1 + 2i2 + 2 2 i 2 2 kk c kk c + k? c

!¡1

:

(10:68)

Note that here ! 2 is always less than ­i2 . The ions move in a left circular motion (i.e., in the direction of the ion Larmor motion) at ! ' j­i j (see eq.(10.30)). The mode (10.66) satis¯es iEx =Ey = 1, i.e., it is circularly polarized, with the electric ¯eld rotating opposite to the ion Larmor motion. The ion cyclotron wave satis¯es ! iEx à ¼¡ j­i j Ey

1 k2 1 + ?2 kk

!;

(10:69)

i.e., the electric ¯eld is elliptically polarized, rotating in the same direction as the ion Larmor motion.

10.4 Various Waves

127

Fig.10.6 Orbits of ions and electrons at lower hybrid resonance.

10.4c Lower Hybrid Resonance The frequency at lower hybrid resonance at µ = ¼=2 is given by 2 !2 = !LH ;

1 1 1 = 2 + ; 2 2 j­i j­e !LH ­i + ¦i

2 !LH ¦i2 + ­i2 = 2 : j­i j­e ¦i + j­i j­e + ­i2

(10:70)

When the density is high and ¦i2 À j­i j­e , it follows that !LH = (j­i j­e )1=2 . When ¦i2 ¿ 2 j­i j­e , then !LH = ¦i2 + ­i2 . At lower hybrid resonance, we have Ey = Ez = 0 and Ex 6= 0. When the density is high (that is ¦i2 > j­i j­e ), then j­i j ¿ !LH ¿ ­e and the analysis of the motions of ions and electrons becomes simple. From eq.(10.7), the velocity is given by vk;x =

i²k Ex !j­k j B0 ! 2 ¡ ­k2

(10:71)

and vk;x = dxk =dt = ¡i!xk yields xk =

¡²k Ex j­k j : B0 !2 ¡ ­k2

(10:72)

At !2 = j­i j­e , we ¯nd that xi ¼ ¡Ex =B0 ­e and xe ' ¡Ex =B0 ­e , or xi ¼ xe (see ¯g.10.6). Consequently, charge separation does not occur and the lower hybrid wave can exist. We have been discussing lower hybrid resonance at µ = ¼=2. Let us consider the case in which µ is slightly di®erent from µ = ¼=2. The resonance condition is obtained from eq.(10.24) as follows: K? sin2 µ + Kk cos2 µ = 0:

(10:73)

Using eqs.(10.46),(10.50) and (10.51), eq.(10.73) is reduced to Ã

2 2 (! 2 ¡ !LH )(! 2 ¡ !UH ) ¦e2 2 µ + 1 ¡ sin !2 (!2 ¡ ­i2 )(! 2 ¡ ­e2 )

!

cos2 µ = 0:

(10:74)

When µ is near ¼=2 and ! is not much di®erent from !LH , we ¯nd that 2 = ! 2 ¡ !LH

2 ¡ ­2 2 2 2 2 (!LH e )(!LH ¡ ­i ) ¦e ¡ !LH cos2 µ 2 ¡ !2 2 !LH ! UH LH

­2¦ 2 ¼ e2 e !UH

Ã

1¡

µ

­i !LH

¶2 ! Ã

1¡

µ

!LH ¦e

¶2 !

cos2 µ:

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

128

2 2 !LH = ­i2 ­e2 + ¦e2 j­i j­e , ! 2 is expressed by As !UH

2 6 6

mi + cos2 µ Zme 4

2 6 !LH 61

2

! =

Ã

1¡

µ

­i !LH µ

¶2 ! Ã

1¡

1+

j­i j­e ¦e2

µ

¶

!LH ¦e

¶2 ! 3

2 À 1, eq.(10.75) becomes When ¦e2 =j­i j­e ¼ ± = c2 =vA

µ

2 !2 = !LH 1+

¶

mi cos2 µ : Zme

7 7 7: 7 5

(10:75)

(10:76)

2 Even if µ is di®erent from ¼=2 only slight amount (Zme =mi )1=2 , ! 2 becomes ! 2 ¼ 2!LH , so that eq.(10.76) holds only in the region very near µ = ¼=2.

10.4d Upper Hybrid Resonance The upper hybrid resonant frequency !UH is given by 2 !UH = ¦e2 + ­e2 :

(10:77)

Since this frequency is much larger than j­i j, ion motion can be neglected. 10.4e Electron Cyclotron Wave (Whistler Wave) Let us consider high-frequency waves, so that ion motion can be neglected. When ! À j­i j, we ¯nd 9 ¦e2 > > > K? ¼ 1 ¡ 2 ; > > ! ¡ ­e2 > > > = 2 ¦e ­e (10:78) K£ ¼ ¡ 2 ; > > ! ¡ ­e2 ! > > > > > ¦e2 > ; Kk = 1 ¡ 2 : !

The solution of dispersion equation AN 4 ¡ BN 2 + C = 0 N2 =

B § (B 2 ¡ 4AC)1=2 2A

may be modi¯ed to N2 ¡ 1 = =

¡2(A ¡ B + C) 2A ¡ B § (B 2 ¡ 4AC)1=2 ¡2¦e2 (1 ¡ ¦e2 =! 2 ) ; 2! 2 (1 ¡ ¦e2 =! 2 ) ¡ ­e2 sin2 µ § ­e ¢

0

¢ = @­e2 sin4 µ + 4!2

Ã

¦2 1 ¡ 2e !

!2

11=2

cos2 µA

:

(10:79)

(10:80)

The ordinary wave and extraordinary wave will be obtained by taking the plus and minus sign, respectively, in eq.(10.79). In the case of ­e2

4

sin µ À 4!

2

Ã

¦2 1 ¡ 2e !

!2

cos2 µ

(10:81)

10.5 Conditions for Electrostatic Waves

129

we ¯nd N2 = N2 =

1 ¡ ¦e2 =! 2 ; 1 ¡ (¦e2 =! 2 ) cos2 µ

(1 ¡ ¦e2 =! 2 )2 ! 2 ¡ ­e2 sin2 µ : (1 ¡ ¦e2 =! 2 )! 2 ¡ ­e2 sin2 µ

(10.82) (10.83)

Equation (10.82) becomes N 2 = Kk = 1 ¡ ¦e2 =! 2 at µ » ¼=2 and does not depend on the magnitude of the magnetic ¯eld. This wave is used for density measurements by microwave interferometry. In the case of ­e2

4

sin µ ¿ 4!

2

Ã

¦2 1 ¡ 2e !

!2

cos2 µ

(10:84)

with the additional condition ­e2

2

sin µ ¿ 2!

2

Ã

¦2 1 ¡ 2e !

!

(10:85)

the dispersion relations become N2 = 1 ¡

¦e2 ; (! + ­e cos µ)!

(10.86)

N2 = 1 ¡

¦e2 : (! ¡ ­e cos µ)!

(10.87)

Equation (10.86) corresponds to the L wave, and eq.(10.87) to the R wave. R-wave resonance occurs near the electron cyclotron frequency. This wave can propagate in regions (7) and (8) of the CMA diagram, where the frequency is less than the plasma frequency. This wave is called the electron cyclotron wave. It must be noticed that the assumptions (10.84) and (10.85) are not satis¯ed near Kk = 1 ¡ ¦e2 =!2 ' 0. The electron cyclotron wave is also called the whistler wave. Electromagnetic disturbances initiated by lighting °ashes propagate through the ionosphere along magnetic-¯eld lines. The frequency of a lightning-induced whistler wave falls in the audio region, and its group velocity increases with frequency; so that this wave is perceived as a whistle of descending tone. This is why it is called the whistler wave. 10.5 Conditions for Electrostatic Waves When the electric ¯eld E can be expressed by an electrostatic potential E = ¡rÁ = ¡ikÁ

(10:88)

the resultant wave is called an electrostatic wave. The electric ¯eld E is always parallel to the propagation vector k, so that the electrostatic wave is longitudinal. The magnetic ¯eld B 1 of the electrostatic wave is always zero: B 1 = k £ E=! = 0:

(10:89)

Alf¶ ven waves are not electrostatic waves. We will here discuss the conditions for electrostatic waves. Since the dispersion relation is N £ (N £ E) + K ¢ E = 0

10 Plasma as Medium of Electromagnetic Wave ¢ ¢ ¢

130

the scalar product with N becomes N ¢ K ¢ (E k + E ? ) = 0: where E k and E ? are the components of the electric ¯eld parallel and perpendicular to k. If jE k j À jE ? j, the wave is electrostatic and the dispersion relation becomes N ¢ K ¢ N = 0:

(10:90)

Rewriting the dispersion relation as (N 2 ¡ K) ¢ E ? = K ¢ E k shows that jE k j À jE ? j holds when jN 2 j À jKij j

(10:91)

is satis¯ed for all Kij . The dispersion relation (10.90) for the electrostatic wave is then kx2 Kxx + 2kx kz Kxz + kz2 Kzz = 0:

(10:92)

The condition (10.91) for the electrostatic wave indicates that the phase velocity !=k = c=N of this wave is low. The Kij have already been given by eqs.(10.9-10.12) for cold plasmas, and the general formula for hot plasma will be discussed in chs.12-13. We have stated that magnetic ¯eld B 1 of the electrostatic wave is zero. Disturbances of the magnetic ¯eld propagate with the Alfv¶en velocity vA ' B02 =(¹0 ni mi ). If the phase velocity of the wave is much lower than vA , the disturbance of the magnetic ¯eld will be damped within a few cycles of the wave and the propagated magnetic-¯eld disturbance becomes zero. When the electron thermal velocity vTe is taken as a typical phase velocity for electrostatic waves, then the condition of vA > vTe reduced to 2me B02 = > 1; 2 ¯e mi ¹0 ni mi vTe ¯e <

2me : mi

This is a condition that a wave is electrostatic. At resonance the refractive index N becomes in¯nite. As the Kij are ¯nite for lower hybrid and upper hybrid resonance, the condition (10.91) is satis¯ed so that these hybrid waves are electrostatic. Since some of the Kij become in¯nite for the ion or electron cyclotron waves, these cyclotron waves are not always electrostatic.

References 1. T. H. Stix: The Theory of Plasma Waves, McGraw-Hill, New York 1962 2. T. H. Stix: Waves in Plasmas, American Institute of Physics, New York, 1992 3. W. P. Allis, S. J. Buchsbanm and A. Bers: Waves in AnisotropicPlasmas, The MIT Press, Cambrige Mass. 1963 4. G. Beke¯: Radiation Processes in Plasmas, John Willey and Son Inc. Gordon and Breach Science Publishers Inc. New York, 1961

11.1 Landau Damping (Ampli¯cation)

131

Ch.11 Landau Damping and Cyclotron Damping The existence of a damping mechanism by which plasma particles absorb wave energy even in a collisionless plasma was found by L.D.Landau, under the condition that the plasma is not cold and the velocity distribution is of ¯nite extent. Energy-exchange processes between particles and wave are possible even in a collisionless plasma and play important roles in plasma heating by waves (wave absorption) and in the mechanism of instabilities (wave ampli¯cation). These important processes will be explained in terms of simpli¯ed physical models in this chapter. In succeeding chs.12, 13, and app.C, these processes will be described systematically. In hot plasma models, pressure term and particle-wave interaction term appear in the dielectric tensor that are absent in the dielectric tensor for a cold plasma. 11.1 Landau Damping (Ampli¯cation) Let us assume that many particles drift with di®erent velocities in the direction of the lines of magnetic force. When an electrostatic wave (a longitudinal wave with k k E) propagates along the lines of magnetic force, there appears an interaction between the wave and a group of particles (see ¯g.11.1). Take the z axis in the direction of the magnetic ¯eld and denote the unit vector in this direction by z ^. Then the electric ¯eld and the velocity v = v^ z satisfy E=z ^E cos(kz ¡ !t);

(11.1)

m

(11.2)

dv = qE cos(kz ¡ !t): dt

The electric ¯eld E is a quantity of the 1st order. The zeroth-order solution of eq.(11.2) is z = v0 t + z0 and the 1st-order equation is m

dv1 = qE cos(kz0 + kv0 t ¡ !t): dt

(11:3)

The solution of eq.(11.3) for the initial condition v1 = 0 at t = 0 is v1 =

qE sin(kz0 + kv0 t ¡ !t) ¡ sin kz0 : m kv0 ¡ !

(11:4)

The kinetic energy of the particle becomes d mv 2 d d d = v mv = v1 mv1 + v0 mv2 + ¢ ¢ ¢ : dt 2 dt dt dt

(11:5)

From eqs.(11.2),(11.4), we have the relation m

d(v1 + v2 ) = qE cos(k(z0 + v0 t + z1 ) ¡ !t) dt = qE cos(kz0 + ®t) ¡ qE sin(kz0 + ®t)kz1 ;

z1 =

Z

0

t

v1 dt =

qE m

µ

¡ cos(kz0 + ®t) + cos kz0 t sin kz0 ¡ ®2 ®

¶

132

11 Landau Damping and Cyclotron Damping

Fig.11.1 Propagation of wave and motion of particles in the process of Landau Damping.

where ® ´ kv0 ¡ !: Using these, we may put eq.(11.5) into the form d mv 2 q2 E 2 = dt 2 m ¡

µ

kv0 q 2 E 2 m

µ

¶

sin(kz0 + ®t) ¡ sin kz0 cos(kz0 + ®t) ® ¶

¡ cos(kz0 + ®t) + cos kz0 t sin kz0 ¡ sin(kz0 + ®t): ®2 ®

The average of the foregoing quantitiy with respect to the initial position z0 is *

d mv 2 dt 2

+

= z0

q2 E 2 2m

µ

¶

¡! sin ®t !t cos ®t + t cos ®t + : 2 ® ®

(11:6)

When we take the velocity average of eq.(11.6) over v0 with the weighting factor i.e. distribution function f (v0 ) (® ´ kv0 ¡ !) f (v0 ) = f

µ

®+! k

¶

= g(®)

the rate of increase of the kinetic energy of the particles is obtained. The distribution function is normalized: Z

1

¡1

f (v0 ) dv0 =

1 k

Z

g(®) d® = 1:

The integral of the 2nd term of eq.(11.6) 1 k

Z

1 g(®)t cos ®t d® = k

Z

g

µ ¶

x cos x dx t

(11:7)

approaches zero as t ! 1. The integral of the 3rd term of eq. (11.6) becomes ! k

Z

g(®)t cos ®t ! d® = ® k

Z

µ ¶

t x cos x dx: g x t

(11:8)

The function g(®) can be considered to be the sum of an even and an odd function. The even function does not contribute to the integral. The contribution of the odd function approaches zero when t ! 1 if g(®) is continuous at ® = 0. Therefore, only the contribution of the 1st term in eq.(11.6) remains and we ¯nd *

d mv 2 dt 2

+

z0 ;v0

!q 2 E 2 =¡ P 2mk

Z

g(®) sin ®t d®: ®2

(11:9)

11.1 Landau Damping (Ampli¯cation)

133

Fig.11.2 (a) Landau damping and (b) Landau ampli¯cation.

P denotes Cauchy's principal value of integral. The main contribution to the integral comes from near ® = 0, so that g(®) may be expanded around ® = 0: g(®) = g(0) + ®g 0 (0) +

®2 00 g (0) + ¢ ¢ ¢ : 2

As sin ®t=®2 is an odd function, only the 2nd term of the foregoing equation contributes to the integral and we ¯nd for large t that *

d mv 2 dt 2

+

z0 ;v0

=¡

!q 2 E 2 2mjkj

¡¼q2 E 2 = 2mjkj

Z

1

¡1

g 0 (0) sin ®t d® ®

µ ¶µ

! k

@f (v0 ) @v0

¶

:

(11:10)

v0 =!=k

If the number of particles slightly slower than the phase velocity of the wave is larger than the number slightly faster, i.e., if v0 @f0 =@v0 < 0, the group of particles as a whole gains energy from the wave and the wave is damped. On the contrary, when v0 @f0 =@v0 > 0 at v0 = !=k, the particles gives their energy to the wave and the amplitude of the wave increases (¯g.11.2). This mechanism is called Landau damping1 or ampli¯cation. Experimental veri¯cation of Landau damping of waves in a collisionless plasma was demonstrated by J.M.Malemberg and C.B.Wharton2 in 1965, twenty years after Landau's prediction. The growth rate (11.10) of the kinetic energy of particles must be equal to the damping rate of wave energy. Therefore the growth rate ° of the amplitude of wave ¯eld is obtained by (° < 0 in the case of damping) ¿

n

d mv 2 dt 2

À

= ¡2°W

z0 v0

and the growth rate ° is given by ° ¼ = ! 2

µ

¦ !

¶2 µ

! jkj

¶µ

v0

@f (v0 ) @v0

R

¶

(11:11)

v0 =!=k

where ¦ 2 = nq 2 =²0 m; W ¼ 2²0 E 2 =4, f (v)dv = 1. There is a restriction on the applicability of linear Landau damping. When this phenomenon runs its course before the particle orbit deviates from the linear-approximation solution, the reductions leading to linear Landau damping are justi¯ed. The period of oscillation in the potential well of the electric ¯eld of the wave gives the time for the particle orbit to deviate from the linear approximation ( ! 2 » eEk=m from m! 2 x = eE). The period of oscillation is ¿osc

1 ¼ = !osc

µ

m ekE

¶1=2

:

134

11 Landau Damping and Cyclotron Damping

Consequently the condition for the applicability of linear Landau damping is that the Landau damping time 1=° is shorter than ¿osc or the collision time 1=ºcoll is shorter than ¿osc . j°¿osc j > 1;

(11.12)

jºcoll ¿osc j > 1:

(11.13)

On the other hand, it was assumed that particles are collisionless. The condition that the collision time 1=ºcoll is longer than ¸=vrms is necessary for the asymptotic approximation of the integral (11.9) as t ! 1, where ¸ is the wavelength of the wave and vrms is the spread in the velocity distribution; 2¼ 1 > : (11:14) ºcoll kvrms

11.2 Transit-Time Damping We have already described the properties of Alfv¶en waves in cold plasmas. There are compressional and torsional modes. The compressional mode becomes magnetosonic in hot plasmas, as is described in ch.5. In the low-frequency region, the magnetic moment ¹m is conserved and the equation of motion along the ¯eld lines is m

dvz @B1z = ¡¹m : dt @z

(11:15)

This equation is the same as that for Landau damping if ¡¹m and @B1z =@z are replaced by the electric charge and the electric ¯eld, respectively. The rate of change of the kinetic energy is derived similarly, and is *

d mv 2 dt 2

+

z0 ;v0

µ ¶µ

! ¼¹2 jkj = ¡ m jB1z j2 2m k

@f (v0 ) @v0

¶

:

(11:16)

v0 =!=k

This phenomena is called transit-time damping. 11.3 Cyclotron Damping The mechanism of cyclotron damping is di®erent from that of Landau damping. Here the electric ¯eld of the wave is perpendicular to the direction of the magnetic ¯eld and the particle drift and accelerates the particle perpendicularly to the drift direction. Let us consider a simple case in which the thermal energy of particles perpendicular to the magnetic ¯eld is zero and the velocity of particles parallel to the magnetic ¯eld B 0 = B0 z ^ is V . The equation of motion is m

@v @v ^ B0 + V z ^ £ B 1 ): + mV = q(E 1 + v £ z @t @z

(11:17)

^) = 0. B 1 is given by As our interest is in the perpendicular acceleration we assume (E 1 ¢ z B 1 = (k £ E)=!. With the de¯nitions v § = vx § ivy ; E § = Ex § iEy , the solution for the initial condition v = 0 at t = 0 is v§ =

­=

iqE § (! ¡ kV ) exp(ikz ¡ i!t) 1 ¡ exp(i!t ¡ ikV t § i­t) ; m! ! ¡ kV § ­ ¡qB0 : m

(11.18)

11.3 Cyclotron Damping

135

The macroscopic value of v ? is obtained by taking the average weighted by the distribution function f0 (V ) as follows: iq exp(ikz ¡ i!t) + ¡ ^); ((c +c )E ? + i(c+ ¡c¡ )E ? £ z 2m

hv ? i =

c§ = ®§ ¡ i¯ § ; §

® = ¯§ =

Z Z

1

(11:20)

dV

f0 (V )(1 ¡ kV =!)(1 ¡ cos(! ¡ kV § ­)t) ; ! ¡ kV § ­

(11:21)

dV

f0 (V )(1 ¡ kV =!) sin(! ¡ kV § ­)t : ! ¡ kV § ­

(11:22)

¡1 1

(11:19)

¡1

As t becomes large we ¯nd that ®§ ! P

Z

1

¡1

dV µ

f0 (V )(1 ¡ kV =!) ; ! ¡ kV § ­ ¶

¨¼­ !§­ ¯ ! : f0 !jkj k §

(11.23) (11.24)

When tÀ

2¼ kVrms

(11:25)

where Vrms = hV 2 i1=2 is the spread of the velocity distribution, the approximations (11.19)»(11.24) are justi¯ed. The absorption of the wave energy by the plasma particles is given by hRe(qE exp(ikz ¡ i!t))(Rehv ? i)iz =

q2 + (¯ jEx + iEy j2 + ¯ ¡ jEx ¡ iEy j2 ): 4m

(11.26)

Let us consider the case of electrons (­e > 0). As was described in sec.10.2, the wave N 2 = R propagating in the direction of magnetic ¯eld (µ = 0) satis¯es Ex + iEy = 0, so that the absorption power becomes Pe =

q2 ¡ ¯ jEx ¡ iEy j2 : 4m

When ! > 0, eq.(11.24) indicates ¯ ¡ > 0. In the case of ! < 0, ¯ ¡ is nearly zero since f0 ((! ¡ ­e )=k) ¿ 1. Let us consider the case of ions (¡­i > 0). Similarly we ¯nd Pi =

q2 + ¯ jEx + iEy j2 : 4m

When ! > 0, eq.(11.24) indicates ¯ + > 0. In the case of ! < 0, ¯ + is nearly zero, since f0 (! + ­i =k) ¿ 1. The cyclotron velocity Vc is de¯ned so that the Doppler shifted frequency (the frequency of wave that a particle running with the velocity V feels) is equal to the cyclotron frequency, that is, ! ¡ kVc § ­ = 0; µ

¶

! ­ Vc = 1§ : k !

136

11 Landau Damping and Cyclotron Damping

Accordingly particles absorb the wave energy when the absolute value of cyclotron velocity is smaller than the absolute value of phase velocity of the wave (§­=! < 0)(see eq.(11.24)). This phenomena is called cyclotron damping. Let us consider the change in the kinetic energy of the particles in the case of cyclotron damping. Then the equation of motion is m

dv ¡ q(v £ B 0 ) = qE ? + q(v £ B 1 ): dt

Since B 1 = (k £ E)=! and Ez = 0, we have m

dvz qkz = (v ? ¢ E ? ); dt ! µ

kz vz dv ? ¡ q(v ? £ B 0 ) = qE ? 1 ¡ m dt !

¶

so that µ

dv? kz vz mv ? ¢ = q(v ? ¢ E ? ) 1 ¡ dt !

¶

:

Then d dt 2 v?

Ã

mvz2 2 µ

!

kz vz d = ! ¡ kz vz dt

! + vz ¡ kz

¶2

Ã

2 mv? 2

!

;

= const:

In the analysis of cyclotron damping we assumed that vz = V is constant; the condition of the validity of linearized theory is3 2 j! ¡ k 3 kz2 q 2 E? z vz jt < 1: 2 2 24! m

We have discussed the case in which the perpendicular thermal energy is zero. When the perpendicular thermal energy is larger than the parallel thermal energy, so-called cyclotron instability may occur. The mutual interaction between particles and wave will be discussed again in chs.12 and 13 in relation to heating and instabilities. 11.4 Quasi-Linear Theory of Evolution in the Distribution Function It has been assumed that the perturbation is small and the zeroth-order terms do not change. Under these assumption, the linearized equations on the perturbations are analyzed. However if the perturbations grow, then the zeroth-order quantities may change and the growth rate of the perturbations may change due to the evolution of the zeroth order quantities. Finally the perturbations saturate (growth rate becomes zero) and shift to steady state. Let us consider a simple case of B = 0 and one dimensional electrostatic perturbation (B 1 = 0). Ions are uniformly distributed. Then the distribution function f (x; v; t) of electrons obeys the following Vlasov equation; e @f @f @f ¡ E +v = 0: @t @x m @v

(11:27)

Let the distribution function f be divided into two parts f (x; v; t) = f0 (v; t) + f1 (x; v; t)

(11:28)

11.4 Quasi-Linear Theory of Evolution ¢ ¢ ¢

137

where f0 is slowly changing zeroth order term and f1 is the oscillatory 1st order term. It is assumed that the time derivatives of f0 is the 2nd order term. When eq.(11.28) is substituted into eq.(11.27), the 1st and the 2nd terms satisfy following equations; @f1 @f1 e @f0 +v = E ; @t @x m @v @f0 e @f1 = E : @t m @v

(11:29) (11:30)

f1 and E may be expressed by Fourier integrals;

E(x; t) =

Z

1 (2¼)1=2

f1 (x; v; t) =

Z

1 (2¼)1=2

fk (v) exp(i(kx ¡ !(k)t))dk;

(11:31)

Ek exp(i(kx ¡ !(k)t))dk:

(11:32)

Since f1 and E are real, f¡k = fk¤ ; E¡k = Ek¤ ; !(¡k) = ¡! ¤ (k) ( !(k) = !r (k) + i°(k)). The substitution of eqs.(11.31) (11.32) into eq.(11.29) yields µ

e m

fk (v) =

¶

i @f0 Ek : !(k) ¡ kv @v

(11:33)

If eqs.(11.32) (11.33) are substituted into eq.(11.30), we ¯nd @f0 (v; t) = @t £

µ

e m

¶2

¿

@ 1 @v 2¼

Z

Ek0 exp(i(k0 x ¡ !(k 0 )t))dk0 À

i @f0 (v; t) Ek exp(i(kx ¡ !(k)t))dk : !(k) ¡ kv @v

(11:34)

Statistical average of eq.(11.34) (integration by x) is reduced to µ

¶

@f0 (v; t) @f0 (v; t) @ Dv (v) ; = @t @v @v Dv (v) =

µ

=

µ

e m

¶2 Z

1

¡1

e m

¶2 Z

1

¡1

(11:35)

ijEk j2 exp(2°(k)t) dk !r (k) ¡ kv + i°(k) °(k)jEk j2 exp(2°(k)t) dk: (!r (k) ¡ kv)2 + °(k)2

When j°(k)j ¿ j!r (k)j, the di®usion coe±cient in velocity space is Dv (v) =

µ

e m

¶2

=

µ

e m

¶2

¼

Z

jEk j2 exp(2°(k)t) ±(!r (k) ¡ kv)dk

¯ ¼ ¯ jEk j2 exp(2°(k)t)¯ : !=k=v jvj

From Poisson's equation and eq.(11.33), the dispersion equation can be derived: r¢E =¡

e ²0

Z

f1 dv;

(11:36)

138

11 Landau Damping and Cyclotron Damping Z

ikEk = ¡

e ²0

¦2 1 1+ e k n

Z µ

fk dv; 1 !(k) ¡ kv

¶

@f0 dv = 0: @v

(11:37)

Under the assumption of j°j ¿ j!r j(! = !r + i°), the solution of eq.(11.37) for ° is given by the same equation as eq.(11.11). Equation (11.35) is the di®usion equation in the velocity space. When the distribution function of electrons are given by the pro¯le shown in ¯g.11.2(b), in which the positive gradient of v @f =@v > 0 exists near v1 . Then waves with the phase velocity of !=k ¼ v1 grow due to Landau ampli¯cation and the amplitude of jEk j increases. The di®usion coe±cient Dv in velocity space becomes large and anomalous di®usion takes place in velocity space. The positive gradient of @f =@v near » v1 decreases and ¯nally the pro¯le of the distribution function becomes °at near v » v1 . Let us consider the other case. When a wave is externally exited (by antenna) in a plasma with Maxwellian distribution function as is shown in ¯g.11.2(a), di®usion coe±cient Dv at v = !=k is increased. The gradient of the distribution function near v = !=k becomes °at as will be seen in ¯g.16.18 of ch.16.

References 1. L. D. Landau: J. Phys. (USSR) 10, 45 (1946) 2. J. H. Malmberg, C. B. Wharton and W. E. Drummond: Plasma Phys.and Controlled Nucl. Fusion Research 1, 485 (1966) (Conf. Proceedings Culham in 1965, IAEA Vienna) 3. T. H. Stix: The Theory of Plasma Waves, McGraw Hill, New York 1962 T. H. Stix: Waves in Plasmas, American Institute of Physics, New York, 1992

12.1 Energy Flow

139

Ch.12 Wave Propagation and Wave Heating Wave heating in ion cyclotron range of frequency (ICRF), lower hybrid wave heating (LHH), electron cyclotron heating (ECH), and other heating processes are being studied actively. The power sources of high-frequency waves are generally easier to construct than the beam source of neutral beam injection (NBI). Although the heating mechanism of NBI can be well explained by the classical process of Coulomb collision (refer sec(2.6)), the physical processes of wave heating are complicated and the interactions of waves and plasmas have a lot of variety, so that various applications are possible depending on the development of wave heating. Waves are excited in the plasma by antennas or waveguides located outside the plasma (excitation of wave, antenna-plasma coupling). When the electric ¯eld of the excited wave is parallel to the con¯ning magnetic ¯eld of the plasma, the electron, which can move along the magnetic ¯eld, may cancel the electric ¯eld. However, if the frequency of the wave is larger than the plasma frequency the electron can not follow the change in the electric ¯eld, and the wave then propagates through the plasma. When the electric ¯eld of the excited wave is perpendicular to the magnetic ¯eld, the electrons move in the direction of E £ B (under the condition ! < ­e ) and thus they can not cancel the electric ¯eld. In this case the wave can propagate through the plasma even if the wave frequency is smaller than the plasma frequency. Excitation consists of pumping the high-frequency electromagnetic wave into plasma through the coupling system. If the strucure of the coupling system has the same periodicity as the eigenmode wave, the wave can be excited resonantly. The e±ciency of wave excitation is not high except such resonant excitation. Neutral beam injection and electron cyclotron heating can be launched in vacuum and propagate directly into the plasma without attenuation or interaction with the edge. Consequently the launching structures do not have to be in close proximity to the plasma and have advantage against thermal load and erosion by plasma. Excited waves may propagate and pass through the plasma center without damping (heating) in some cases and may refract and turn back to the external region without passing the plasma center or may be re°ected by the cuto® layer (see ¯g.12.1). The wave may be converted to the other type by the mode conversion (wave propagation). The waves propagating in the plasma are absorbed and damped at the locations where Landau damping and cyclotron damping occur and heat the plasma. Therefore it is necessary for heating the plasma center that the waves be able to propagate into the plasma center without absorption and that they be absorbed when they reach the plasma center (wave heating). 12.1 Energy Flow Energy transport and the propagation of waves in the plasma medium are very important in the wave heating of plasmas. The equation of energy °ow is derived by taking the di®erence between the scalar product of H and eq.(10.15) and the scalar product of E and eq.(10.16): r ¢ (E £ H) + E ¢

@D @B +H ¢ = 0: @t @t

(12:1)

P ´ E £ H is called Poynting vector and represents the energy °ow of electromagnetic ¯eld. This Poynting equation does not include the e®ect of electric resistivity by electron-ion collision. Plasmas are dispersive medium and the dielectric tensors are dependent on the propagation vector k and the frequency !. Denote the Fourier components of E(r; t) and D(r; t) by E ! (r; t)

140

12 Wave Propagation and Wave Heating

Fig.12.1 Passing through, refraction and re°ection, absorption near boundary, and absorption at center of plasma.

and D! (r; t), respectively. Then we ¯nd D! =

1 (2¼)2

E! =

1 (2¼)2

Z

Z

D(r; t) exp(¡i(k ¢ r ¡ !t)) dr dt; E(r; t) exp(¡i(k ¢ r ¡ !t)) dr dt:

There is following relation between them: D! (k; !) = ²0 K(k; !) ¢ E ! (k; !); and we have 1 D(r; t) = ²0 (2¼)2 1 E(r; t) = (2¼)2

Z

Z

K(k; !) ¢ E ! (k; !) exp(i(k ¢ r ¡ !t)) dk d!;

E ! (k; !) exp(i(k ¢ r ¡ !t)) dk d!:

From the formula of Fourier integral, following equations are derived: D(r; t) = ²0 c where K(r; t) is

c K(r; t) =

Z

c ¡ r 0 ; t ¡ t0 ) ¢ E(r 0 ; t0 ) dr 0 dt0 K(r

1 (2¼)4

Z

K(k; !) exp(¡i(k ¢ r ¡ !t)) dk d!:

Therefore analysis of general electromagnetic ¯elds in dispersive medium is not simple. However if the electric ¯eld consists of Fourier component in narrow region of k; ! and K changes slowly as k; ! change, then we can use the following relation: D(r; t) = ²0 K ¢ E(r; t): From now we will discuss this simple case. The relation between the magnetic induction B and the magnetic intensity H is B = ¹0 H; in plasmas. The quasi-periodic functions A; B may be expressed by µ

A = A0 exp ¡i µ

B = B0 exp ¡i

Z

t

¡1

Z

(!r + i!i )dt

0

t

¡1

(!r + i!i )dt

0

¶

¶

= A0 exp(¡iÁr + Ái ); = B0 exp(¡iÁr + Ái )

12.1 Energy Flow

141

where Ár and Ái are real. When the average of the multiplication of the real parts of A with the real part of B is denoted by AB, then AB is given by AB =

=

1 1 ¢ h(A0 exp(¡iÁr + Ái ) + A¤0 exp(iÁr + Ái )) £(B0 exp(¡iÁr + Ái ) + B0¤ exp(iÁr + Ái ))i 2 2 1 1 (A0 B0¤ + A¤0 B0 ) exp(2Ái ) = Re(AB ¤ ) : 4 2

(12:2)

The averaging of the Poynting equation becomes r¢P +

@W = 0; @t

(12:3)

1 P = Re(E 0 £ B ¤0 ) exp 2 2¹0 @W 1 = Re @t 2

µµ

B ¤ @B ¢ ¹0 @t

¶

Z

t

!i dt0 ;

¡1

¶

(12:4) µ

@ B¤ ¢ B 1 + ²0 E ¢ (K ¢ E) = Re ¡i! + ²0 (¡i!)E ¤ ¢ K ¢ E @t 2 ¹0 ¤

1 B ¢ B ¤ ²0 = !i + (!i Re(E ¤ ¢ K ¢ E) + !r Im(E ¤ ¢ K ¢ E)) : 2 ¹0 2

¶

(12:5)

From the relations E¤ ¢ K ¢ E =

X

E ¢ K ¤ ¢ E¤ = =

X i

Ei¤

i

Ei¤

Kij Ej ;

j

X i

X

Ei

X

X

¤ ¤ Kij Ej =

j

X j

Ej¤

X

T ¤ (Kji ) Ei

i

T ¤ (Kij ) Ej

j

we ¯nd Re(E ¤ ¢ K ¢ E) = E ¤ ¢

K + (K T )¤ ¢ E; 2

Im(E ¤ ¢ K ¢ E) = E ¤ ¢

(¡i)[K ¡ (K T )¤ ] ¢ E: 2

(K T )¤ is the complex conjugate of transpose matrix K T (lines and rows of components are T ¤ T ´ K exchanged) of K, i.e., Kij ji . When a matrix M and (M ) are equal with each other, this kind of matrix is called Hermite matrix. For the Hermite matrix, (E ¤ ¢ M ¢ E) is always real. The dielectric tensor may be decomposed to K(k; !) = K H (k; !) + iK I (k; !): As is described in sec.12.3, K H and K I are Hermite, when k; ! are real. It will be proved that the term iK I corresponds to Landau damping and cyclotron damping. When the imaginary part of ! is much smaller than the real part (! = !r + i!i , j!i j ¿ j!r j) we may write K(k; !r + i!i ) ¼ K H (k; !r ) + i!i

@ K H (k; !r ) + iK I (k; !r ): @!r

142

12 Wave Propagation and Wave Heating

Fig.12.2 F (x; t) and f (k) cos(kx ¡ w(k)t)

When the Hermite component of W (the term associated to K H in W ) is denoted by W0 , W0 is given by W0 =

µ

µ

¶

B ¤0 ¢ B 0 ²0 ¤ @ 1 ²0 + E 0 ¢ K H ¢ E 0 + E ¤0 ¢ !r KH ¢ E0 Re 2 2¹0 2 2 @!r µ

µ

¶

B ¤0 ¢ B 0 ²0 ¤ @ 1 = Re + E0 ¢ (!K H ) ¢ E 0 2 2¹0 2 @!

¶

¶

(12:6)

and eqs.(12.3),(12.5) yield 1 @W0 = ¡!r ²0 E ¤0 ¢ K I ¢ E 0 ¡ r ¢ P : @t 2

(12:7)

The 1st term in eq.(12.6) is the energy density of the magnetic ¯eld and the 2nd term is the energy density of electric ¯eld which includes the kinetic energy of coherent motion associated with the wave. Equation (12.6) gives the energy density of the wave in a dispersive media. The 1st term in the right-hand side of eq.(12.7) represents the Landau and cyclotron dampings and the 2nd term is the divergence of the °ow of wave energy. Let us consider the velocity of the wave packet F (r; t) =

Z

1

¡1

f (k) exp i(k ¢ r ¡ !(k)t)dk

(12:8)

when the dispersion equation ! = !(k) is given. If f (k) varies slowly, the position of the maximum of F (r; t) is the position of the stationary phase of @ (k ¢ r ¡ !(k)t) = 0: @ki

(i = x; y; z)

(see ¯g.12.2). Consequently the velocity of the maximum position is Ã

x @!(k) ; = t @kx

y @!(k) ; = t @ky

z @!(k) = t @kz

!

that is, vg =

Ã

@! @! @! ; ; @kx @ky @kz

!

:

(12:9)

12.2 Ray Tracing

143

This velocity is called group velocity and represents the velocity of energy °ow. 12.2 Ray Tracing When the wavelength of waves in the plasma is much less than the characteristic length (typically the minor radius a), the WKB approximation (geometrical optical approximation) can be applied. Let the dispersion relation be D(k; !; r; t) = 0. The direction of wave energy °ow is given by the group velocity v g = @!=@k ´ (@!=@kx ; @!=@ky ; @!=@kz ), so that the optical ray can be given by dr=dt = v g . Although the quantities (k; !) change according to the change of r, they always satisfy D = 0. Then the optical ray can be obtained by dr @D = ; ds @k

dk @D =¡ ; ds @r

(12:10)

dt @D =¡ ; ds @!

d! @D = : ds @t

(12:11)

Here s is a measure of the length along the optical ray. Along the optical ray the variation ±D becomes zero, ±D =

@D @D @D @D ¢ ±k + ¢ ±! + ¢ ±r + ¢ ±t = 0 @k @! @r @t

(12:12)

and D(k; !; r; t) = 0 is satis¯ed. Equations (12.10),(12.11) reduce to dr dr = dt ds

µ

dt ds

¶¡1

@D =¡ @k

µ

@D @!

¶¡1

=

µ

¶

@! = vg : @k r ;t=const:

Equation (12.10) has the same formula as the equation of motion with Hamiltonian D. When D does not depend on t explicitly, D = const: = 0 corresponds to the energy conservation law. If the plasma medium does not depend on z, kz =const. corresponds to the momentum conservation law and is the same as the Snell law, Nk = const:. When k = kr + iki is a solution of D = 0 for a given real ! and jki j ¿ jkr j is satis¯ed, we have @ReD(kr ; !) ¢ iki + iImD(kr ; !) = 0: D(kr + iki ; !) = ReD(kr ; !) + @kr Therefore this reduces to ReD(kr ; !) = 0; ki ¢

@ReD(kr ; !) = ¡ImD(kr ; !): @kr

(12:13)

Then the wave intensity I(r) becomes µ

I(r) = I(r 0 ) exp ¡2 Z

ki dr =

Z

Z r

r0

¶

ki dr ;

@D ki ¢ ds = ¡ @k

Z

ImD(kr ; !)ds = ¡

(12:14) Z

ImD(kr ; !) dl: j@D=@kj

(12:15)

where dl is the length along the optical ray. Therefore the wave absorption can be estimated from the eqs.(12.14) and (12.15) by tracing many optical rays. The geometrical optical approximation can provide the average wave intensity with a space resolution of, say, two or three times the wavelength.

144

12 Wave Propagation and Wave Heating

12.3 Dielectric Tensor of Hot Plasma, Wave Absorption and Heating In the process of wave absorption by hot plasma, Landau damping or cyclotron damping are most important as was described in ch.11. These damping processes are due to the interaction between the wave and so called resonant particles satisfying ! ¡ kz vz ¡ n­ = 0:

n = 0; §1; §2; ¢ ¢ ¢

In the coordinates running with the same velocity, the electric ¯eld is static (! = 0) or of cyclotron harmonic frequency (! = n­). The case of n = 0 corresponds to Landau damping and the case of n = 1 corresponds to electron cyclotron damping and the case of n = ¡1 corresponds to ion cyclotron damping (! > 0 is assumed). Although nonlinear or stochastic processes accompany wave heating in many cases, the experimental results of wave heating or absorption can ususally well described by linear or quasi-linear theories. The basis of the linear theory is the dispersion relation with the dielectric tensor K of ¯nite-temperature plasma. The absorbed power per unit volume of plasma P ab is given by the 1st term in the right-hand side of eq.(12.7): P

ab

= !r

µ

¶

²0 E ¤ ¢ K I ¢ E: 2

Since K H ; K I is Hermit matrix for real k; ! as will be shown later in this section, the absorbed power P ab is given by P

ab

= !r

µ

¶

²0 Re (E ¤ ¢ (¡i)K ¢ E)!=!r : 2

(12:16)

As is clear from the expression (12.19) of K, the absorbed power P ab reduces to P ab = !

²0 ³ jEx j2 ImKxx + jEy j2 ImKyy + jEz j2 ImKzz 2

+2Im(Ex¤ Ey )ReKxy + 2Im(Ey¤ Ez )ReKyz + 2Im(Ex¤ Ez )ReKxz ) :

(12:17)

Since eq.(10.3) gives j = ¡i!P = ¡i²0 !(K ¡ I) ¢ E, eq.(12.16) may be described by 1 P ab = Re(E ¤ ¢ j)!=!r : 2

(12:18)

The process to drive the dielectric tensor K of ¯nite-temperature plasma is described in appendix C. When the plasma is bi-Maxwellian; f0 (v? ; vz ) = n0 F? (v? )Fz (vz ); Ã

2 mv? m F? (v? ) = exp ¡ 2¼∙T? 2∙T?

Fz (vz ) =

µ

m 2¼∙Tz

¶1=2

Ã

!

;

m(vz ¡ V )2 exp ¡ 2∙Tz

!

the dielectric tensor K is given by K=I+

" X ¦ 2 Xµ i;e

!2

n

³

¶

i 1 ´ ³0 Z(³n ) ¡ 1 ¡ (1 + ³n Z(³n )) e¡b X n +2´02 ¸T L ; ¸T

(12:19)

12.3 Dielectric Tensor of Hot Plasma, Wave Absorption ¢ ¢ ¢

145

Fig.12.3 Real part Re Z and imaginary part Im Z of Z(x).

2

3

n2 In =b in(In0 ¡ In ) ¡(2¸T )1=2 ´n ®n In 6 7 X n = 4 ¡in(In0 ¡ In ) (n2 =b + 2b)In ¡ 2bIn0 i(2¸T )1=2 ´n ®(In0 ¡ In ) 5 ¡(2¸T )1=2 ´n ®n In ¡i(2¸T )1=2 ´n ®(In0 ¡ In ) 2¸T ´n2 In Z(³) ´

1 ¼ 1=2

Z

1

¡1

(12:20)

exp(¡¯ 2 ) d¯; ¯¡³

In (b) is the nth modi¯ed Bessel function

´n ´

! + n­ ; 21=2 kz vTz

³n ´

Tz ¸T ´ ; T? 2 ´ vTz

b´

∙Tz ; m

! ¡ kz V + n­ ; 21=2 kz vTz

µ

¶2

kx vT? ­

2 ´ vT?

;

® ´ b1=2 ;

∙T? : m

The components of L matrix are zero except Lzz = 1. When the plasma is isotropic Maxwellian (Tz = T? ) and V = 0, then ´n = ³n , and ¸T = 1, and eq.(12.19) reduces to K =I+

X ¦2 i;e

!2

"

1 X

¡b

³0 Z(³n )e X n +

n=¡1

2³02 L

#

:

(12:21)

Z(³) is called plasma dispersion function. When the imaginary part Im! of ! is smaller than the real part Re! in magnitude (jIm!j < jRe!j), the imaginary part of the plasma dispersion function is ImZ(³) = i

kz 1=2 ¼ exp(¡³ 2 ): jkz j

The real part Re Z(x) (x is real) is shown in ¯g.12.3. The real part of Z(x) is ReZ(x) = ¡2x(1 ¡ (2=3)x2 + ¢ ¢ ¢) in the case of x ¿ 1 (case of hot plasma) and ReZ(x) = ¡x¡1 (1 + (1=2)x¡2 + (3=4)x¡4 + ¢ ¢ ¢)

146

12 Wave Propagation and Wave Heating

in the case of x À 1 (case of cold plasma).1;2;3 The imaginary part of Z(³) represents the terms of Landau damping and cyclotron damping as is described later in this section. When T ! 0, that is, ³n ! §1, b ! 0, the dielectric tensor of hot plasma is reduced to the dielectric tensor (10.9)»(10.13) of cold plasma. In the case of b = (kx ½­ )2 ¿ 1 (½­ = vT? =­ is Larmor radius), it is possible to expand ¡b e X n by b using In (b) = =

µ ¶n X 1 b

2

l=0

µ ¶n Ã

b 2

1 l!(n + l)!

µ ¶2l

b 2

1 1 + n! 1!(n + 1)!

µ ¶2

b 2

1 + 2!(n + 2)!

µ ¶4

b 2

!

+ ¢¢¢ :

The expansion in b and the inclusion of terms up to the second harmonics in K give Kxx = 1 +

X µ ¦j ¶2

Kyy = 1 +

³0

!

j

X µ ¦j ¶2

³0

!

j

µ

µ

¶

1 b ¡ + ¢ ¢ ¢ + (Z2 + Z¡2 ) (Z1 + Z¡1 ) 2 2

µ

µ

Kxy = i

X µ ¦j ¶2

!

j

X µ ¦j ¶2

!

j

Kxz = 21=2

Kyz = ¡2

1=2

i

; j

¶ ¶

;

b2 +(³2 W2 + ³¡2 W¡2 ) + ¢¢¢ + ¢¢¢ 4

!

µ

µ

¶

b ¡ b2 + ¢ ¢ ¢ + ¢ ¢ ¢ 2

Ã

µ

³0 (Z1 ¡ Z¡1 )

!

!

µ

j

¶

µ

1=2

b

µ

³0 W0

¶

µ

¶

¶

1 b ¡ b + ¢ ¢ ¢ + (Z2 ¡ Z¡2 ) + ¢¢¢ + ¢¢¢ 2 2

b1=2 ³0 (W1 ¡ W¡1 )

X µ ¦j ¶ 2 j

!

³0 2³0 W0 (1 ¡ b + ¢ ¢ ¢) + (³1 W1 + ³¡1 W¡1 )(b + ¢ ¢ ¢)

X µ ¦j ¶2 j

!

b b2 ¡ + ¢¢¢ + ¢¢¢ 2 2

1 3b ¡ Z0 (2b + ¢ ¢ ¢) + (Z1 + Z¡1 ) + ¢¢¢ 2 2 + (Z2 + Z¡2 )

Kzz = 1 ¡

Ã

µ

µ

¶

µ

;

j

¶

¶

¶

(12:22)

1 b + ¢ ¢ ¢ + (W2 ¡ W¡2 ) + ¢¢¢ + ¢¢¢ 2 4 ¶

;

j

3 ¡1 + b + ¢ ¢ ¢ 2

µ

¶

µ

¶

1 b + (W1 + W¡1 ) + ¢ ¢ ¢ + (W2 ¡ W¡2 ) + ¢¢¢ + ¢¢¢ 2 4 Kyx = ¡Kxy ;

;

j

Kzx = Kxz ;

Kzy = ¡Kzy

where Z§n ´ Z(³§n );

Wn ´ ¡ (1 + ³n Z(³n )) ;

³n = (! + n­)=(21=2 kz (∙Tz =m)1=2 ):

When x À 1, ReW (x) is Re W (x) = (1=2)x¡2 (1 + (3=2)x¡2 + ¢ ¢ ¢):

j

12.3 Dielectric Tensor of Hot Plasma, Wave Absorption ¢ ¢ ¢

147

The absorbed power by Landau damping (including transit time damping) may be estimated by the terms associated with the imaginary part G0 of ³0 Z(³0 ) in eq.(12.22) of Kij : kz 1=2 ¼ ³0 exp(¡³02 ): jkz j

G0 ´ Im³0 Z(³0 ) = Since (ImKyy )0 =

µ

¦j !

¶2

2bG0 ;

(ImKzz )0 =

µ

¦j !

¶2

2³02 G0 ;

(ReKyz )0 =

µ

¦j !

¶2

21=2 b1=2 ³0 G0

the contribution of these terms to the absorption power (12.17) is P0ab

µ

¦j = 2! !

¶2

G0

µ

²0 2

¶µ

2

jEy j b +

jEz j2 ³02

+

Im(Ey¤ Ez )(2b)1=2 ³0

¶

:

(12:23)

The 1st term is of transit time damping and is equal to eq.(11.16). The 2nd term is of Landau damping and is equal to eq.(11.10). The 3rd one is the term of the interference of both. The absorption power due to cyclotron damping and the harmonic cyclotron damping is obtained by the contribution from the terms G§n ´ Im³0 Z§n =

kz 1=2 2 ¼ ³0 exp(¡³§n ) jkz j

and for the case b ¿ 1, (ImKxx )§n = (ImKyy )§n = (ImKzz )§n = (ReKxy )§n

µ

µ

¶2

¦j =¡ !

(ReKyz )§n = ¡ (ImKxz )§n

¦j !

µ

¦j !

µ

¦j =¡ !

®n = n2 (2 ¢ n!)¡1

µ

¦j !

¶2

G§n ®n ;

2 2³§n G§n b®n n¡2 ;

¶2

¶2

¶2

G§n (§®n ); (2b)1=2 ³§n G§n ®n n¡1 ; (2b)1=2 ³§n G§n (§®n )n¡1 ;

µ ¶n¡1

b 2

:

The contribution of these terms to the absorbed power (12.17) is ab =! P§n

µ

¦j !

¶2

Gn

µ

¶

²0 ®n jEx § iEy j2 : 2

Since ³n = (! + n­i )=(21=2 kz vTi ) = (! ¡ nj­i j)=(21=2 kz vTi )

(12:24)

148

12 Wave Propagation and Wave Heating

the term of +n is dominant for the ion cyclotron damping (! > 0), and the term of ¡n is dominant for electron cyclotron damping (! > 0), since ³¡n = (! ¡ n­e )=(21=2 kz vTe ): The relative ratio of E components can be estimated from the following equations: (Kxx ¡ Nk2 )Ex + Kxy Ey + (Kxz + N? Nk )Ez = 0; ¡Kxy Ex + (Kyy ¡ Nk2 ¡ N?2 )Ey + Kyz Ez = 0;

(12:25)

(Kxz + N? Nk )Ex ¡ Kyz Ey + (Kzz ¡ N?2 )Ez = 0: For cold plasmas, Kxx ! K? , Kyy ! K? , Kzz ! Kk , Kxy ! ¡iK£ , Kxz ! 0; Kyz ! 0 can be substituted into eq.(12.25), and the relative ratio is Ex : Ey : Ez = (K? ¡ N 2 ) £(Kk ¡ N?2 ) : ¡iKx (Kk ¡ N?2 ) : ¡Nk N? (K? ¡ N 2 ). In order to obtain the magnitude of the electric ¯eld, it is necessary to solve the Maxwell equation with the dielectric tensor of eq.(12.19). In this case the density, the temperature, and the magnetic ¯eld are functions of the coordinates. Therefore the simpli¯ed model must be used for analytical solutions; otherwise numerical calculations are necessary to derive the wave ¯eld. 12.4 Wave Heating in Ion Cyclotron Range of Frequency The dispersion relation of waves in the ion cyclotron range of frequency (ICRF) is given by eq.(10.64); Ã

Ã

N?2 ¡ 1¡ Nk2 = 2[1 ¡ (!=­i )2 ]

µ

! ­i

¶2 !

2Ã

2!2 + 2 2 §4 1¡ k? vA

µ

! ­i

¶2 !2

µ

! +4 ­i

¶2 µ

! k? vA

¶4

31=2 ! 5

:

The plus sign corresponds to the slow wave (L wave, ion cyclotron wave), and the minus sign corresponds to the fast wave (R wave, extraordinary wave). When 1 ¡ !2 =­i2 ¿ 2(!=k? vA )2 , the dispersion relation becomes kz2

=2

Ã

kz2 = ¡

!2 2 vA

!Ã

!2 1¡ 2 ­i

!¡1

2 k? !2 + 2: 2 2vA

;

(for slow wave)

(for fast wave)

2 > (¼=a)2 Since the externally excited waves have propagation vectors with 0 < kz2 < (1=a)2 , k? usually, there are constraints

!2 2 < 2 vA (1 ¡ !2 =­i2 )

µ ¶2

n20 a2 < 2:6 £ 10¡4

B 2 (1 ¡ ! 2 =­i2 ) A

for slow wave and !2 2 > 2vA

µ ¶2

¼ a

;

¼ a

;

12.4 Wave Heating in Ion Cyclotron Range of Frequency

n20 a2 > 0:5 £ 10¡2

149

(­i =!)2 A=Z 2

for the fast wave,4 where n20 is the ion density in 1020 m¡3 , a is the plasma radius in meters, and A is the atomic number. An ion cyclotron wave (slow wave) can be excited by a Stix coil1 and can propagate and heat ions in a low-density plasma. But it cannot propagate in a high-density plasma like that of a tokamak. The fast wave is an extraordinary wave in this frequency range and can be excited by a loop antenna, which generates a high-frequency electric ¯eld perpendicular to the magnetic ¯eld (see sec.10.2). The fast wave can propagate in a high-density plasma. The fast wave in a plasma with a single ion species has Ex + iEy = 0 at ! = j­i j in cold plasma approximation, so that it is not absorbed by the ion cyclotron damping. However, the electric ¯eld of the fast wave in a plasma with two ion species is Ex + iEy 6= 0, so that the fast wave can be absorbed; that is, the fast wave can heat the ions in this case. Let us consider the heating of a plasma with two ion species, M and m, by a fast wave. The masses, charge numbers, and densities of the M ion and m ion are denoted by mM , ZM , nM and mm , Zm , nm , respectively. When we use ´M ´

2 ZM nM ; ne

´m ´

2 Zm nm ne

we have ´M =ZM + ´m =Zm = 1 since ne = ZM nM + Zm nm . Since (¦e =!)2 À 1 in ICRF wave, the dispersion relation in the cold plasma approximation is given by eq.(10.2) as follows: N?2

=

(R ¡ Nk2 )(L ¡ Nk2 ) K? ¡ Nk2

¦2 R = ¡ 2i !

µ

¦i2 !2

µ

L=¡

¦2 K? = ¡ 2i ! ¦i2 ´

Ã

; ¶

´M ! ! (mM =mm )´m ! + ¡ ; ! + j­m j ! + j­M j j­M j=ZM ¶

´M ! ! (mM =mm )´m ! + + ; ! ¡ j­m j ! ¡ j­M j j­M j=ZM (mM =mm )´m !2 ´M ! 2 + 2 2 !2 ¡ ­m ! 2 ¡ ­M

!

;

ne e2 : ²0 mM

Therefore ion-ion hybrid resonance occurs at K? ¡ Nk2 = 0, that is, !2 2 ´m (mM =mm )! 2 ´M !2 ¼ ¡ + N ¼ 0; 2 2 ! 2 ¡ ­m !2 ¡ ­M ¦i2 k !2 ¼ !IH ´ ¹0 ´

mm ; mM

´M + ´m (¹2 =¹0 ) 2 ­m ; ´M + ´m =¹0 ¹´

­M mm ZM = : ­m mM Zm

Figure 12.4 shows the ion-ion hybrid resonance layer; K? ¡Nk2 = 0, the L cuto® layer; L¡Nk2 = 0, and R cuto® layer; R ¡ Nk2 = 0, of a tokamak plasma with the two ion species D+ (M ion) and

150

12 Wave Propagation and Wave Heating

Fig.12.4 L cuto® layer(L = Nk2 ), R cuto® layer (R = Nk2 ), and the ion-ion hybrid resonance layer (K? = Nk2 ) of ICRF wave in a tokamak with two ion components D+ , H+ . The shaded area is the 2 region of N? < 0.

H+ (m ion). Since the Kzz component of the dielectric tensor is much larger than the other component, even in a hot plasma, the dispersion relation of a hot plasma is5 Kxx ¡ Nk2 Kxy = 0: ¡Kxy Kyy ¡ Nk2 ¡ N?2

(12:26)

When we use the relation Kyy ´ Kxx + ¢Kyy , j¢Kyy j ¿ jKxx j, N?2 = ¼

2 (Kxx ¡ Nk2 )(Kxx + ¢Kyy ¡ Nk2 ) + Kxy

Kxx ¡ Nk2

(Kxx + iKxy ¡ Nk2 )(Kxx ¡ iKxy ¡ Nk2 ) Kxx ¡ Nk2

:

2 When ! 2 is near !IH , Kxx is given by

Kxx

¦2 = ¡ 2i !

Ã

mM ´M ! 2 ´m ³0 Z(³1 ) + 2 2 2mm ! ¡ ­M

!

:

The resonance condition is Kxx = Nk2 . The value of Z(³1 ) that appears in the dispersion equation is ¯nite and 0 > Z(³1 ) > ¡1:08. The condition 2 mm 1=2 vTi ´m ¸ ´cr ´ 2 Nk 1:08 mM c

Ã

2 ´M ! 2 2! + N k 2 ! 2 ¡ ­M ¦i2

!

is necessary to obtain the resonance condition. This point is di®erent from the cold plasma dispersion equation (note the di®erence between Kxx and K? ). It is deduced from the dispersion equation (12.26) that the mode conversion5 from the fast wave to the ion Berstein wave occurs at the resonance layer when ´m ¸ ´cr . When the L cuto® layer and the ion-ion hybrid resonance layer are close to each other, as shown in ¯g.12.4, the fast wave propagating from the outside torus penetrates the L cuto® layer partly by the tunneling e®ect and is converted to the ion Bernstein wave. The mode converted wave is absorbed by ion cyclotron damping or electron Landau damping. The theory of mode conversion is described in chapter 10 of ref.1. ICRF experiments related to this topic were carried out in TFR. When ´m < ´cr , K? = Nk2 cannot be satis¯ed and the ion-ion hybrid resonance layer disappears. In this case a fast wave excited by the loop antenna outside the torus can pass through the R cuto® region (because the width is small) and is re°ected by the L cuto® layer and bounced

12.5 Lower Hybrid Wave Heating

151

back and forth in the region surrounded by R = Nk2 and L = Nk2 . In this region, there is a layer satisfying ! = j­m j, and the minority m ions are heated by the fundamental ion cyclotron damping. The majority M ions are heated by the Coulomb collisions with m ions. If the mass of M ions is l times as large as the mass of m ions, the M ions are also heated by the lth harmonic ion cyclotron damping. This type of experiment was carried out in PLT with good heating e±ciency. This is called minority heating. The absorption power Pe0 due to electron Landau damping per unit volume is given by eq.(12.23), and it is important only in the case ³0 ∙ 1. In this case we have Ey =Ez ¼ Kzz =Kyz ¼ 2³02 =(21=2 b1=2 ³0 (¡i)) and Pe0 is6 Pe0

µ

¦e !²0 jEy j2 = 4 !

¶2 µ

k? vTe ­e

¶2

2 2³0e ¼ 1=2 exp(¡³0e ):

(12:27)

The absorption power Pin by the n-th harmonic ion cyclotron damping is given by eq.(12.24) as follows; µ

¦i !²0 jEx + iEy j2 Pin = 2 !

¶2 Ã

n2 2 £ n!

!µ ¶ n¡1

b 2

Ã

! (! ¡ nj­i j)2 £ 1=2 ¼ 1=2 exp ¡ 2(kz vTi )2 2 kz vTi

!

: (12:28)

The absorption power due to the second harmonic cyclotron damping is proportional to the beta value of the plasma. In order to evaluate the absorption power by eqs.(12.27) and (12.28), we need the spatial distributions of Ex and Ey . They can be calculated in the simple case of a sheet model.7 In the range of the higher harmonic ion cyclotron frequencies (! » 2­i ; 3­i ), the direct excitation of the ion Bernstein wave has been studied by an external antenna or waveguide, which generates a high-frequency electric ¯eld parallel to the magnetic ¯eld.8 12.5 Lower Hybrid Wave Heating Since j­i j ¿ ¦i in a tokamak plasma (ne ¸ 1013 ; cm¡3 ), the lower hybrid resonance frequency becomes 2 = !LH

¦i2 ¦i2 + ­i2 ¼ : 1 + ¦e2 =­e2 + Zme =mi 1 + ¦e2 =­e2

There are relations ­e À !LH À j­i j, ¦i2 =¦e2 = j­i j=­e . For a given frequency !, lower hybrid resonance ! = !LH occurs at the position where the electron density satis¯es the following condition: 2 ¦res ¦e2 (x) ´ p; = ­e2 ­e2

p=

!2 : ­e j­i j ¡ ! 2

When the dispersion equation (10.20) of cold plasma is solved about N?2 using N 2 = Nk2 + N?2 , we have 2Ã

f ¡ K2 + K K f f ¡ K2 + K K f K? K K? K ? ? k ? k ? £ £ §4 N?2 = 2K? 2K?

!2

31=2

Kk f2 )5 + (K 2 ¡ K ? K? £

;

f? = K? ¡ N 2 . The relations h(x) ´ ¦ 2 (x)=¦ 2 , K? = 1 ¡ h(x), K£ = ph(x)­e =!, where K e res k

2 2 =! 2 » O(mi =me ), ® ´ ¦res =(!­e ) » O(mi =me )1=2 and ¯¦ h À 1 Kk = 1 ¡ ¯¦ h(x), ¯¦ ´ ¦res reduce this to

N?2 (x)

µ

¶

h i1=2 ¯¦ h = Nk2 ¡ (1 ¡ h + ph) § (Nk2 ¡ (1 ¡ h + ph))2 ¡ 4(1 ¡ h)ph : 2(1 ¡ h)

(12:29)

152

12 Wave Propagation and Wave Heating

2 2 ¡ h(x) (= ¦e2 (x)=¦res Fig.12.5 Trace of lower hybrid wave in N? ) diagram for the case of 2 p = 0:353; Nkcr = 1 + p = 1:353. This corresponds to the case of H+ plasma in B = 3 T, and 2 =! 2 ) is f = !=2¼= 109 Hz. The electron density for the parameter ¯¦ = 7:06 £ 103 (= ¦res nres = 0:31 £ 1020 m¡3 .

The slow wave corresponds to the case of the plus sign in eq.(12.29). In order for the slow wave to propagate from the plasma edge with low density (h ¿ 1) to the plasma center with high 2 , h = 1), N? (x) must real. Therefore following condition density (¦e2 = ¦res Nk > (1 ¡ h)1=2 + (ph)1=2 is necessary. The right-hand side of the inequality has the maximum value (1 + p)1=2 in the range 0 < h < 1, so that the accessibility condition of the resonant region to the lower hybrid wave becomes 2 =1+p=1+ Nk2 > Nk;cr

2 ¦res : ­e2

(12:30)

If this condition is not satis¯ed, the externally excited slow wave propagates into the position where the square root term in eq.(12.29) becomes zero and transforms to the fast wave there. Then the fast wave returns to the low-density region. The slow wave that satis¯es the accessibility condition can approach the resonance region and N? can become large, so that the dispersion relation of hot plasma must be used to examine the behavior of this wave. Near the lower hybrid resonance region, the approximation of the electrostatic wave, eq.(13.1) or (C.36), is applicable. Since j­i j ¿ ! ¿ ­e , the terms of ion contribution and electron contribution are given by eqs.(13.3) and (13.4), respectively, that is, 1+

¦i2 mi ¦e2 me ¡b (1 + I e ³ Z(³ )) + (1 + ³Z(³)) = 0; 0 0 0 k 2 Te k2 Ti

where ³0 = !=(21=2 kz vTe ), and ³ = !=(21=2 kvTi ) ¼ !=(21=2 k? vTi ). Since I0 e¡b ¼ 1 ¡ b + (3=4)b2 , ³0 À 1, ³ À 1, 1 + ³Z(³) ¼ ¡(1=2)³ ¡2 ¡ (3=4)³ ¡4 , we have Ã

3¦i2 ∙Ti 3 ¦e2 ∙Te + ! 4 mi 4 ­e4 me

!

4 k?

Ã

¦2 ¦2 ¡ 1 + e2 ¡ 2i ­e !

Using the notations ½i = vTi =j­i j and Ã

j­e ­i j 1 Te ! 2 s2 ´ 3 + !2 4 Ti j­e ­i j

!

!

µ

2 k?

Ã

¦2 ¡ 1 ¡ 2e !

!

kz2 = 0:

¶

1 + p 1 Te p =3 ; + p 4 Ti 1 + p

(12:31)

12.5 Lower Hybrid Wave Heating

153

we have Ã

3¦i2 ∙Ti 3 ¦e2 ∙Te + ! 4 mi 4 ­e4 me

Ã

¦2 ¦2 1 + e2 ¡ 2i ­e !

!

=

!

=

2 2 s ¦i2 me vTi ! 2 mi ­i

1 1 ¡ h ¦i2 1 + p h !2

Then the dimensionless form of (12.31) is 1 ¡ h mi 1 (k? ½i ) ¡ (k? ½i )2 + h me (1 + p)s2 4

µ

mi me

¶2

1 (kz ½i )2 = 0: s2

(12:32)

This dispersion equation has two solutions. One corresponds to the slow wave in a cold plasma and the other to the plasma wave in a hot plasma. The slow wave transforms to the plasma wave at the location where eq.(12.31) or (12.32) has equal roots.9;10;11 The condition of zero discriminant is 1=h = 1 + 2kz ½i (1 + p)s and 2 p ¦e2 (x) ¦M:C: ´ = : ­e2 ­e2 1 + 2kz ½i (1 + p)s

Accordingly, the mode conversion occurs at the position satisfying !2 = ¦i2

Ã

!2 1¡ j­i j­e

!

à !2 11=2 p 0 2 Nk vTe 2 3 Ti ! 1 @ A + + c Te 4 ­i ­e

2 2 and the value of k? ½i at this position becomes 2 2 k? ½i jM:C: =

mi kz ½i : me s

If the electron temperature is high enough at the plasma center to satisfy vTe > (1=3)c=Nk , the wave is absorbed by electrons due to electron Landau damping. After the mode conversion, the value N? becomes large so that c=N? becomes comparable to the ion thermal velocity (c=N? » vTi ). Since ! À j­i j, the ion motion is not a®ected by the magnetic ¯eld within the time scale of ! ¡1 . Therefore the wave with phase velocity c=N is absorbed by ions due to ion Landau damping. When ions have velocity vi larger than c=N? (vi > c=N? ), the ions are accelerated or decelerated at each time satisfying vi cos(­i t) ¼ c=N? and are subjected to stochastic heating. The wave is excited by the array of waveguides, as shown in ¯g.12.6, with an appropriate phase di®erence to provide the necessary parallel index Nk = kz c=! = 2¼c=(¸z !). In the low-density region at the plasma boundary, the component of the electric ¯eld parallel to the magnetic ¯eld is larger for the slow wave than for the fast wave. Therefore the direction of wave-guides is arranged to excite the electric ¯eld parallel to the line of magnetic force. The coupling of waves to plasmas is discussed in detail in ref.12 and the experiments of LHH are reviewed in ref.13. For the current drive by lower hybrid wave, the accessibility condition (12.30) and c=Nk À vTe are necessary. If the electron temperature is high and ∙Te » 10 keV, then vTe =c is already » 1=7. Even if Nk is chosen to be small under the accessibility condition, eq.(12.30), the wave is subjected to absorbtion by electron damping in the outer part of the plasma, and it can not be expected that the wave can propagate into the central part of the plasma. When the value of Nk is chosen to be Nk » (1=3)(c=vTe ), electron heating can be expected and has been observed experimentally. Under the condition that the mode conversion can occur, ion heating can be expected. However, the experimental results are less clear than those for electron

154

12 Wave Propagation and Wave Heating

Fig.12.6 Array of waveguides to excite a lower hybrid wave (slow wave).

Fig.12.7 The locations of electron cyclotron resonance (! = ­e ), upperhybrid resonance (! = !LH ) and R cut o® (! = !R ) in case of ­e0 > ¦e0 , where ­e0 and ¦e0 are electron cyclotron resonance frequency and plasma frequency at the plasma center respectively (left ¯gure). The right ¯gure is the CMA diagram near electron cyclotron frequency region.

heating. 12.6 Electron Cyclotron Heating The dispersion relation of waves in the electron cyclotron range of frequency in a cold plasma is given by eq.(10.79). The plus and minus signs in eq.(10.79) correspond to ordinary and extraordinary waves, respectively. The ordinary wave can propagate only when !2 > ¦e2 as is clear from eq.(10.86) (in the case of µ = ¼=2). This wave can be excited by an array of waveguides, like that used for lower hybrid waves (¯g. 12.6), which emits an electric ¯eld parallel to the magnetic ¯eld. The phase of each waveguide is selected to provide the appropriate value of the parallel index Nk = kz c=!= 2¼c=(!¸z ). The dispersion relation of the extraordinary wave is given by eq.(10.87). When µ = ¼=2, it is 2 2 > !2 > !L2 ; !LH . As is seen from the CMA given by eq.(10.52). It is necessary to satisfy !UH diagram of ¯g.10.5, the extraordinary wave can access the plasma center from the high magnetic ¯eld side (see ¯g.12.7) but can not access from the low ¯eld side because of ! = !R cuto®. The extraordinary wave can be excited by the waveguide, which emits an electric ¯eld perpendicular to the magnetic ¯eld (see sec.10.2a). The ion's contribution to the dielectric tensor is negligible. When relations b ¿ 1, ³0 À 1 are satis¯ed for electron, the dielectric tensor of a hot plasma is Kxx = Kyy = 1 + X³0 Z¡1 =2; Kxy = ¡iX³0 Z¡1 =2;

Kzz = 1 ¡ X + N?2 Âzz ;

Kxz = N? Âxz ;

Kyz = iN? Âyz ;

12.6 Electron Cyclotron Heating

155

Âxz ¼ Âyz ¼ 2¡1=2 XY ¡1 Âzz ¼ XY ¡2 X´

¦e2 ; !2

µ

vT c

¶2

Y ´

vT ³0 (1 + ³¡1 Z¡1 ); c

³0 ³¡1 (1 + ³¡1 Z¡1 );

­e ; !

³¡1 =

! ¡ ­e ; 21=2 kz vT

N? =

k? c : !

The Maxwell equation is (Kxx ¡ Nk2 )Ex + Kxy Ey + N? (Nk + Âxz )Ez = 0; ¡Kxy Ex + (Kyy ¡ Nk2 ¡ N?2 )Ey + iN? Âyz Ez = 0; N? (Nk + Âxz )Ex ¡ iN? Âyz Ey + (1 ¡ X ¡ N?2 (1 ¡ Âzz ))Ez = 0: The solution is iN 2 Âxz (Nk + Âxz ) + Kxy (1 ¡ X ¡ N?2 (1 ¡ Âzz )) Ex =¡ ? ; Ez N? (iÂxz (Kxx ¡ Nk2 ) + Kxy (Nk + Âxz )) N?2 (Nk + Âxz )2 ¡ (Kxx ¡ Nk2 )(1 ¡ X ¡ N?2 (1 ¡ Âzz )) Ey =¡ : Ez N? (iÂxz (Kxx ¡ Nk2 ) + Kxy (Nk + Âxz )) The absorption power P¡1 per unit volume is given by eq.(12.24) as follows: P¡1

Ã

(! ¡ ­e )2 ¼ 1=2 = !X³0 exp ¡ 2 2 2kz2 vTe

!

²0 jEx ¡ iEy j2 : 2

When ! = ­e , then ³¡1 = 0, Z¡1 = i¼ 1=2 , Kxx = 1+ih, Kxy = h, Âyz = Âxz = 21=2 X(vTe =c)³0 = X=(2Nk ), Âzz = 0, h ´ ¼ 1=2 ³0 X=2. Therefore the dielectric tensor K becomes 2

3

1 + ih h N? Âxz 4 ¡h 1 + ih iN? Âxz 5 : K= N? Âxz ¡iN? Âxz 1 ¡ X For the ordinary wave (O wave), we have iN?2 (O)Nk (Nk + Âxz ) ¡ i(1 ¡ Nk2 )(1 ¡ X ¡ N?2 (O)) Ex ¡ iEy = : Ez N? (O)(Nk h + iÂxz (1 ¡ Nk2 )) When Nk ¿ 1 and the incident angle is nearly perpendicular, eq.(10.82) gives 1 ¡ X ¡ N?2 (O) = (1 ¡ X)Nk2 . Since Âxz = X=2Nk , Âxz À Nk . Therefore the foregoing equation reduces to iN? (O)Nk Âxz Ex ¡ iEy = : Ez Nk h + iÂxz For extraordinary wave (X wave), we have 2 (X)) iN?2 (X)Nk (Nk + Âxz )¡ i(1¡ Nk2 )(1¡ X ¡ N? Ex ¡ iEy =¡ 2 : 2 2 2 Ey N? (X)(Nk + Âxz ) ¡ (Kxx ¡ Nk )(1 ¡ X ¡ N? (X))

156

12 Wave Propagation and Wave Heating

When Nk ¿ 1 and ! = ­e , eq.(10.83) gives 1 ¡ X ¡ N?2 (X) ¼ ¡1 + Nk2 . Since Â2xz = (2¼)¡1=2 (vTe =cNk )Xh ¿ h, the foregoing equation reduces to

¡(1 + N?2 (X)Nk (Nk + Âxz )) ¡1 Ex ¡ iEy » = : 2 2 Ey h h ¡ i(1 + N? (X)(Nk + Âxz ) ) The absorption power per unit volume at ! = ­e is 2

P¡1 (O) ¼

2 2

hN? (O)Nk Âxz !²0 2 jEz j2 exp(¡³¡1 ) 2 (Nk h)2 + Â2xz

!²0 1 ¼ jEz j2 2 (2¼)1=2

µ

¦e !

¶2 Ã

vTe cNk

!

N?2 (O)Nk2

Nk2 + (vTe =c)2 (2=¼)

for ordinary wave and µ ¶1=2 µ

1 2 !²0 !²0 jEy j2 = jEy j2 2 P¡1 (X) » 2 h 2 ¼

¦e !

¶¡2 µ

¶

Nk vTe : c

for extraordinary wave.14;15 1=2 1=2 Since P (O) / ne Te =Nk , P (X) / Nk Te =ne , the ordinary wave is absorbed more in the case of higher density and perpendicular incidence, but the extraordinary wave has opposite tendency. The experiments of electron cyclotron heating have been carried out by T-10, ISX-B, JFT-2, D-IIID, and so on, and the good heating e±ciency of ECH has been demonstrated.

References 1. T. H. Stix: The Theory of Plasma Waves, McGraw-Hill, New York 1962 T. H. Stix: Waves in Plasmas, American Institute of Physics, New York, 1992 2. B. D. Fried and S. D. Conte: The Plasma Dispersion Function, Academic Press, New York 1961 3. K. Miyamoto: Plasma Physics for Nuclear Fusion (Revised Edition), Chap.11 The MIT Press, Cambridge, Mass. 1989 4. M. Porkolab: Fusion (ed. by E. Teller) Vol.1, Part B, p.151, Academic Press, New York (1981) 5. J. E. Scharer, B. D. McVey and T. K. Mau: Nucl. Fusion 17, 297 (1977) 6. T. H. Stix: Nucl. Fusion 15, 737 (1975) 7. A. Fukuyama, S.Nishiyama, K. Itoh and S.I. Itoh: Nucl. Fusion 23 1005 (1983) 8. M. Ono, T.Watari, R. Ando, J.Fujita et al.: Phys. Rev. Lett. 54, 2339 (1985) 9. T. H. Stix: Phys. Rev. Lett. 15, 878 (1965) 10. V. M. Glagolev: Plasma Phys. 14, 301,315(1972) 11. M. Brambilla: Plasma Phys. 18, 669 (1976) 12. S. Bernabei, M. A. Heald, W. M. Hooke, R. W. Motley, F.J. Paoloni, M. Brambilla and W.D. Getty : Nucl. Fusion 17, 929 (1977) 13. S. Takamura: Fundamentals of Plasma Heating, Nagoya Univ. Press, 1986 (in Japanese) 14. I. Fidone, G. Granata and G. Ramponi: Phys. Fluids 21, 645 (1978) 15. A. G. Litvak, G. V. Permitin, E. V. Suvorov, and A. A. Frajman: Nucl. Fusion 17, 659 (1977)

13.1 Dispersion Equation of Electrostatic Wave

157

Ch.13 Velocity Space Instabilities (Electrostatic Waves) Besides the magnetohydrodynamic instabilities discussed in ch.8, there is another type of instabilities, caused by deviations of the velocity space distribution function from the stable Maxwell form. Instabilities which are dependent on the shape of the velocity distribution function are called velocity space instabilities or microscopic instabilities. However the distinction between microscopic and macroscopic or MHD instabilities is not always clear, and sometimes an instability belongs to both. 13.1 Dispersion Equation of Electrostatic Wave In this chapter, the characteristics of the perturbation of electrostatic wave is described. In this case the electric ¯eld can be expressed by E = ¡rÁ = ¡ikÁ. The dispersion equation of electrostatic wave is give by (sec.10.5) kx2 Kxx + 2kx kz Kxz + kz2 Kzz = 0: The process in derivation of dispersion equation of hot plasma will be described in details in appendix B. When the zeroth-order distribution function is expressed by f0 (v? ; vz ) = n0 F? (v? )Fz (vz ); Ã

2 mv? m exp ¡ F? (v? ) = 2¼∙T? 2∙T?

Fz (vz ) =

µ

m 2n∙T?

¶1=2

Ã

!

;

mvz2 exp ¡ 2∙Tz

!

the dispersion equation is given by eq.(C.36) as follows kx2 + kz2 +

X i;e

m ¦2 ∙Tz

Ã

1+

1 µ X

n=¡1

¶

Tz (¡n­) 1+ ³n Z(³n )In (b)e¡b T? !n

!

=0

(13:1)

where ³n ´

!n ; 1=2 2 kz vTz

b = (kx vT? =­)2 ;

!n ´ ! ¡ kz V + n­; 2 vTz = ∙Tz =m;

2 vT? = ∙T? =m;

In (b) is n-th modi¯ed Bessel function Z(³) is plasma dispersion function. When the frequency of the wave is much higher than cyclotron frequency (j!j À j­j), or b ! 0 (T? ! 0; kx ! 0), then we ¯nd ³n ! ³0 ; n­=!n ! 0; In ! 0 (n 6= 0); I0 ! 1, so that the dispersion equation is reduced to X m kx2 + kz2 + ¦2 (1 + ³0 Z(³0 )) = 0 (j!j À j­j): (13:2) ∙Tz i;e The dispersion equation in the case of B = 0 is given by k2 +

X i;e

¦2

m (1 + ³Z(³)) = 0: ∙T

µ

³=

! ¡ kV ; B=0 21=2 kvT

¶

(13:3)

158

13 Velocity Space Instability (Electrostatic Wave)

When the frequency of wave is much lower than cyclotron frequency (j!j ¿ j­j), then we ¯nd P ³n ! 1 (n 6= 0); ³n Zn ! ¡1 and In (b)e¡b = 1. kx2

+ kz2

+

X i;e

m ¦ ∙Tz 2

µ

I0 e

´¶ Tz ³ ¡b (1 + ³0 Z(³0 )) + 1 ¡ I0 e = 0: T?

¡b

(j!j ¿ j­j) (13:4)

When the frequency of wave is much higher than cyclotron frequency or the magnetic ¯eld is very small, the dispersion equations (13.2),(13.3) are reduced to

kx2 + kz2 +

X

¦2

Partial integration of eq.(13.5) gives

0

m B Bkz ∙Tz ∙Tz @ m

kx2 + kz2 X 2 = ¦ kz2

Z

Z

1

@(f =n0 ) C @vz dvz C A = 0: ! ¡ kz vz

(f =n0 ) dvz : (! ¡ kz vz )2

(13:5)

13.2 Two Streams Instability The interaction between beam and plasma is important. Let us consider an excited wave in the case where the j particles drift with the velocity Vj and the spread of the velocity is zero. The distribution function is given by fi (vz ) = nj ±(vz ¡ Vj ): The dispersion equation of the wave propagating in the direction of the magnetic ¯eld (kx = 0) is 1=

X j

¦j2 : (! ¡ kVj )2

In the special case ¦12 = ¦22 (n21 q12 =m1 = n22 q22 =m2 ), the dispersion equation is quadratic: (! ¡ kV¹ ) = 2

¦t2

Ã

1 + 2x2 § (1 + 8x2 )1=2 2

!

where ¦t2 = ¦12 + ¦22 ;

x=

k(V1 ¡ V2 ) ; 2¦t

V1 + V2 V¹ = : 2

For the negative sign, the dispersion equation is (! ¡ kV¹ )2 = ¦t2 (¡x2 + x3 + ¢ ¢ ¢)

(13:6)

and the wave is unstable when x < 1, or k2 (V1 ¡ V2 )2 < 4¦t2 : The energy to excite this instability comes from the zeroth-order kinetic energy of beam motion. When some disturbance occurs in the beam motion, charged particles may be bunched and the electric ¯eld is induced. If this electric ¯eld acts to amplify the bunching, the disturbance grows. This instability is called two-streams instability.

13.4 Harris Instability

159

13.3 Electron Beam Instability Let us consider the interaction of a weak beam of velocity V0 with a plasma which consists of cold ions and hot electrons. The dispersion equation (13.5) of a electrostatic wave with kx = 0 (Ex = Ey = 0; Ez 6= 0; B 1 = 0) is given by Kzz = Kk ¡

¦b2 ¦i2 ¡¦e2 = 1¡ (! ¡ kV0 )2 !2

Z

1

¡1

¦b2 fe (vz )=n0 ¡ dv = 0: z (! ¡ kvz )2 (! ¡ kV0 )2

(13:7)

For the limit of weak beam (¦b2 ! 0), the dispersion equation is reduced to Kk (!; k) ¼ 0, if ! 6= kV0 . The dispersion equation including the e®ect of weak beam must be in the form of ! ¡ kV0 = ±! (k): (±! (k) ¿ kV0 ) Using ±!2 we reduce eq.(13.7) to ¦b2 = Kk (! = kV0 ; k) + ±!2

µ

@Kk @!

¶

±! :

!=kV0

If ! = kV0 does not satisfy Kk = 0, Kk 6= 0 holds and the 2nd term in right-hand side of the foregoing equation can be neglected: ¦b2 = Kk (! = kV0 ; k): ±!2 The expression for Kk (! = kV0 ; k) is Kk (!r ) = KR (!r ) + iKI (!r ): The KI term is of the Laudau damping (see sec.12.3). When the condition ! = kV0 is in a region where Landau damping is ine®ective, then jKI j ¿ jKR j and the dispersion equation is reduced to ¦b2 = KR : (! ¡ kV0 )2

(13:8)

Therefore if the condition KR < 0

(13:9)

is satis¯ed, ±! is imaginary and the wave is unstable. When the dielectric constant is negative, electric charges are likely to be bunched and we can predict the occurrence of this instability. If ! = kV0 is in a region where Landau damping is e®ective, the condition of instability is that the wave energy density W0 in a dispersive medium (eq.12.6) is negative, because the absolute value of W0 increases if @W0 =@t is negative; @ @W0 = @t @t

µ

²0 ¤ @ E (!Kzz )Ez 2 z @!

¶

=¡

!r ²0 Ez¤ KI Ez < 0: 2

When energy is lost from the wave by Landau damping, the amplitude of wave increases because the wave energy density is negative. Readers may refer to ref.1 for more detailed analysis of beam-plasma interaction. 13.4 Harris Instability When a plasma is con¯ned in a mirror ¯eld, the particles in the loss cone ((v? =v)2 < 1=Rm Rm being mirror ratio) escape, so that anisotropy appears in the velocity-space distribution function.

160

13 Velocity Space Instability (Electrostatic Wave)

The temperature perpendicular to the magnetic ¯eld in the plasma heated by ion or electron cyclotron range of frequency is higher than the parallel one. Let us consider the case where the distribution function is bi-Maxwellian. It is assumed that the density and temperature are uniform and there is no °ow of particles (V = 0). In this case the dispersion equation (13.1) is kx2 + kz2 +

X

¦j2 mj

j

³l =

µ

µ

1

¶

1 X 1 1 1 (¡l­) + Il (b)e¡b + ³l Z(³l )A = 0; ∙Tz ∙T ∙T ! + l­ z ? l=¡1

(13:10)

j

! + l­ : (2∙Tz =m)1=2 kz

We denote the real and imaginary parts of the right-hand side of eq.(13.10) for real ! = !r by K(!r ) = Kr (!r ) + iKi (!r ). When the solution of eq.(13.10) is !r + i°, i.e., for K(!r + i°) = 0, !r and ° are given by Kr (!r ) = 0; ° = ¡Ki (!r )(@Kr (!r )=@!r )¡1 when j°j ¿ j!r j and Taylor expansion is used. Accordingly we ¯nd kx2

+

kz2

+

X

¦j2 mj

j

Ã

!

µ ¶ 1 X l­ 1 1 ¡b ! + l­ ¡ + I e Zr (³l ) l ∙Tz ∙Tz ∙T? (2∙Tz =m)1=2 kz l=¡1

= 0; j

(13:11) 0

1 X X 1 1 ° = ¡ ¼ 1=2 ¦j2 mj £ @ Il e¡b A (2∙Tz =m)1=2 jkz j j l=¡1

A=

X j

¦j2 mj

1 X

l=¡1

"

¡b

Il e

Ã

µ

¶

¶

l­ ! + l­ ¡ exp(¡³l2 ) ∙Tz ∙T?

Zr (³l )=∙Tz 1 + 1=2 (2∙Tz =m)kz2 (2∙Tz =m) kz

Ã

!

;

!#

l­ ! + l­ ¡ Zr0 (³l ) ∙Tz ∙T?

(13:12)

j

:

j

Zr (³l ) is real part of Z(³l ) and Zr0 (³l ) is the derivatives by ³l . Let us assume that the electron is cold (be ¼ 0, j³0e j À 1) and the ion is hot. Then the contribution of the electron term is dominant in eq.(13.11) and the ion term can be neglected. Equation (13.11) becomes k2 ¡ ¦e2

kz2 = 0: !r2

(13:13)

i.e., !r = §¦e

kz : k

The substitution of !r into eq.(13.12) yields 0

1

µ ¶ µ ¶ 1 X ¼ 1=2 X 2 m 1 Tz ¡b @ £ °= ¦j Il e !r ¡(!r + l­) + l­ exp(¡³l2 )A : 2k2 j ∙Tz j (2∙Tz =m)1=2 jkz j T ? l=¡1 j j

(13:14)

exp(¡³l2 ) has meaningful value only near !r + l­i = !r ¡ lj­i j ¼ 0. The ¯rst term in the braket of eq.(13.14), ¡(!r ¡ lj­i j)j can be destabilizing term and the 2nd term ¡(Tz =T? )lj­i j is stabilizing term. Accordingly the necessary conditions for instability (° > 0) are !r » lj­i j;

!r < lj­i j;

Tz 1 l< ; T? 2

References

161

that is, !r = ¦e

kz < lj­i j; k

(13.15)

T? > 2l: Tz

(13.16)

When the density increases to the point that ¦e approaches j­i j, then plasma oscillation couples to ion Larmor motion, causing the instability. When the density increases further, an oblique Langmuir wave couples with an ion cyclotron harmonic wave lj­i j and an instability with !r = ¦e kz =k » lj­i j is induced. As is clear from eq.(13.16), the degree of anisotropy must be larger for the instability in the region of higher frequency (l becomes large). In summary, the instability with ion cyclotron harmonic frequencies appear one after another in a cold-electron plasma under the anisotropic condition (13.16) when the electron density satis¯es me ne » l2 Z 2 mi

Ã

B2 ¹0 mi c2

!

k2 : kz2

(l = 1; 2; 3; ¢ ¢ ¢)

This instability is called Harris instability.2;3 Velocity space instabilities in simple cases of homogeneous bi-Maxwellian plasma were described. The distribution function of a plasma con¯ned in a mirror ¯eld is zero for loss cone region (v? =v)2 < 1=RM (RM is mirror ratio). The instability associated with this is called loss-cone instability.4 In general plasmas are hot and dense in the center and are cold and low density. The instabilities driven by temperature gradient and density gradient are called drift instability. The electrostatic drift instability5;7 of inhomogeneous plasma can be analyzed by the more general dispersion equation described in app.C. In toroidal ¯eld, trapped particles always exist in the outside where the magnetic ¯eld is weak. The instabilities induced by the trapped particles is called trapped particle instability.6;7

References 1. R. J. Briggs: Electron-Stream Interaction with Plasma, The MIT Press, Cambridge, Mass. 1964 2. E. G. Harris: Phys. Rev. Lett. 2, 34 (1959) 3. E. G. Harris: Physics of Hot Plasma, p.145 (ed. by B. J. Rye and J. B. Taylar) Oliver & Boyd, Edinburgh (1970) 4. M. N. Rosenbluth and R. F. Post: Phys. Fluids 8, 547 (1965) 5. N. A. Krall and M. N. Rosenbluth: Phys. Fluids 8, 1488 (1965) 6. B. B. Kadomtsev and O. P. Pogutse: Nucl. Fusion 11, 67 (1971) 7. K. Miyamoto: Plasma Physics for Nuclear Fusion (revised edition) Chap.12, The MIT Press, Cambridge, Mass. 1989

162

14 Instabilities Driven by Energetic Particles

Ch.14 Instabilities Driven by Energetic Particles Sustained ignition of thermonuclear plasma depends on heating by highly energetic alpha particles produced from fusion reactions. Excess loss of the energetic particles may be caused by ¯shbone instability and toroidal Alfv¶en eigenmodes. Such losses can not only reduce the alpha particle heating e±ciency, but also lead to excess heat loading and damage to plasma-facing components. These problems have been studied in experiments and analyzed theoretically. In this chapter basic aspects of theories on collective instabilities by energetic particles are described. 14.1 Fishbone Instability Fishbone oscillations were ¯rst observed in PDX experiments with nearly perpendicular neutral beam injection. The poloidal magnetic ¯eld °uctuations associated with this instabilities have a characteristic skeletal signature on the Mirnov coils, that has suggested the name of ¯shbone oscillations. Particle bursts corresponding to loss of energetic beam ions are correlated with ¯shbone events, reducing the beam heating e±ciency. The structure of the mode was identi¯ed as m=1, n=1 internal kink mode, with a precursor oscillation frequency close to the thermal ion diamagnetic frequency as well as the fast ion magnetic toroidal precessional frequency. 14.1a Formulation Theoretical analysis of ¯shbone instability is described mainly according to L. Chen, White and Rosenbluth1 . Core plasma is treated by the ideal MHD analysis and the hot component is treated by gyrokinetic description. The ¯rst order equation of displacement » is (refer eq.(8.25)) ½m ° 2

d» = j £ ±B + ±j £ B ¡ r±pc ¡ r±ph : dt

(14:1)

where ±pc is the ¯rst order pressure disturbance of core plasma r±pc = ¡» ¢ rpc ¡ °s pr ¢ ». ±ph is the ¯rst order pressure disturbance of hot component. The following ideal MHD relations hold: ±E ? = i!» £ B; By multiplying

R

±E k = 0;

±B = r £ (» £ B);

±j = r±B:

dr» ¤ on (14.1) and assuming a ¯xed conducting boundary, we have ±WMHD + ±WK + ±I = 0

(14:2)

where ±I =

°2 2

Z

½m j»j2 dr Z

(14:3)

1 » ¢ r±ph dr (14:4) ±WK = 2 and ±WMHD is the potential energy of core plasma associated with the displacement », which was discussed in sec.8.2b and is given by (8.79). ±WK is the contribution from hot component.

14.1 Fishbone Instability

163

14.1b MHD potential Energy s from Let us consider the MHD term of ±WMHD , which consists of the contribution of ±WMHD ext singular region near rational surface and the contribution ±WMHD from the external region. ext External contribution ±WMHD of cylindrical circular plasma is already given by (8.92)

! ¯ Z aà ¯ ext ±WMHDcycl ¯ d»r ¯2 ¼ 2 ¯ + gj»r j dr f ¯¯ = 2¼R 2¹0 0 dr ¯

(14:5)

where f and g are given by (8.93) and (8.95). When r=R ¿ 1 is assumed, f and g of (-m, n) mode are µ

n r3 1 ¡ f = 2 Bz2 R q m r g = Bz2 R

õ

n 1 ¡ q m

¶2 Ã

¶2 Ã

1¡

n2 r2 (m ¡ 1) + 2 R 2

µ

!µ

nr mR

¶2 !

nr 1¡( mR

¶2

+2

Ã

1 ¡ q2

µ

n m

¶2 ! µ

n m

¶µ

r R

¶2 !

where q(r) ´ (rBz =RBµ (r)) is the safety factor. Let us consider the m=1 perturbation with the singular radius r = rs (q(rs ) = m=n). In this case the displacement is »r =const. for 0 < r < rs ext and »r = 0 for rs < r < a (refer sec.8.3b). Then ±WMHDcycl is reduced to2 µ

ext 2 ±WMHDcycl rs ¼Bµs j»s j2 = 2¼R 2¹0 R

¶2 µ

Z

¡¯p ¡

0

1

3

½

µ

¶

¶

1 1 + ¡ 3 d½ 2 q q

(14:6)

2 =2¹0 ) and Bµs ´ (rs =Rq(rs ))Bz is the poloidal ¯eld at r = rs . where ½ = r=rs , ¯p ´ hpis =(Bµs The pressure hpis is de¯ned by

hpis = ¡

Z

0

rs

µ

r rs

¶2

dp 1 dr = 2 dr rs

Z

rs

0

(p ¡ ps )2rdr:

(14:7)

ext =2¼R per unit length of toroidal plasma with circular cross MHD potential energy ±WMHDtor 3 section is given by

µ

ext ±WMHDtor 1 = 1¡ 2 2¼R n

¶

ext 2 ±WMHDcycl ¼Bµs ^ T; j»s j± W + 2¼R 2¹0

µ

^ T = ¼ rs ±W R

¶2

µ

¶

13 2 ¡ ¯ps 3(1 ¡ q0 ) : 144 (14:8)

ext ^ T. =2¼R is reduced to only the term of ± W In the case of m=1 and n=1, ±WMHDtor Let us consider the contribution from singular region. In this case we must solve the displacement »r in singular region near rational surface. The equation of motion in singular surface was treated in sec.9.1 of tearing instability. From (9.13) and (9.9), we have (in the limit of x ¿ 1)

¹0 ½m ° 2

@ 2 »r @ 2 B1r = iF 2 @x @x2

°B1r = iF °»r +

´ @2 B1r ¹0 rs2 @x2

(14:9)

(14:10)

where F ´ (k¢B) =

¯

dq ¯¯ ¡m n Bµ r ¡ rs Bµ n dq Bµ ns x; x ´ ; s ´ rs : Bµ + Bz = (¡m+nq) = ¢r = r R r r dr rs rs dr ¯rs

164

14 Instabilities Driven by Energetic Particles

By following normalizations ô

iB1r rs ; Bµ;s sn

rs ; 2 (Bµ =¹0 ½m )1=2

¿Aµ ´

¿R ´

¹0 rs2 ; ´

SR ´

¿R ; ¿Aµ

we have °

2

µ

¿Aµ ns

¶2

»r00 = xà 00 ;

à = ¡x»r +

1 Ã 00 : °¿Aµ SR

(14:11)

In the limit of SR ! 1, (14.11) yields õ

¿Aµ ° ns

¶2

+x

2

!

»r00 + 2x»r0 = 0

and the solution is4 »r0

µ

(»0 =¼)(°¿Aµ =ns)¡1 = 2 ; x =(°¿Aµ =ns)2 + 1

¶

»0 x : »r (x) = »1 ¡ tan¡1 ¼ °¿Aµ =(ns)

(14:12)

Since the external solution of m=1 is »r = »s as x ! ¡1 and »r = 0 as x ! 1, the matching conditions to external solution yield »1 = »s =2 and »0 = »s . s from singular region is The term ±WMHD s ±WMHD ¼ = 2¼R 2¹0

Z

rs +¢

rs ¡¢

r 3 (k ¢ B)2

µ

@»r @r

¶2

dr =

2 ¼ Bµs sn°¿Aµ : 2¹0 2¼

(14:13)

Eq.(14.13) is the expression in the case of cylindrical plasma. For the toroidal plasma ¿Aµ is replaced by 31=2 rs =(Bµ2 =¹0 ½), where 31=2 is the standard toroidal factor (1 + 2q 2 )1=2 (ref.6). Therefore the total sum of MHD contributions of m=1, n=1 is (°¿Aµ ¿ 1 is assumed.) ±WMHD

µ

B2 2 ^ T + °¿Aµ s + ¼° 2 ¿Aµ + ±I = 2¼R µs j»s j2 ± W 2¹0 2 µ

¶

¶

B2 ^ T + °¿Aµ s : ¼ 2¼R µs j»s j2 ± W 2¹0 2

(14:14)

14.1c Kinetic Integral of Hot Component The perturbed distribution of hot ion component ±Fh is given by gyrokinetic equation in the case of low beta and zero gyro-radius approximation as follow5 e @ ±Fh ´ ±Á F0h + ±Hh ; m @E

µ

¶

@ e ^ dh ) ±Hh = i Q±Á vk ¡ i(! ¡ ! @l m

(14:15)

where v2 E´ ; 2 v dh ´

Ã

v2 ¹´ ?; 2B

v2 vk2 + ? 2

!

eB !c ´ ; m

m (b £ ∙); eB

µ

¶

@ Q´ ! +! ^ ¤h F0h ; @E ! ^ ¤h ´ ¡i!c

! ^ dh ´ ¡iv dh ¢ r;

¡m 1 @ b £ rF0h ¢r¼ : F0h eBr F0h @r

!dh j is diamagnetic drift frequency of hot ion. ∙ = (b¢r)b v dh is the magnetic drift velocity and j^

14.1 Fishbone Instability

165

is the vector toward the center of curvature of magnetic ¯eld line and the magnitude is R (refer sec.2.4). ±Á is scaler potential and related by r±Á = ¡i!» £ B. When we set ±Hh = ¡

1 e Q±Á + ±Gh !m

(14:16)

we have µ

¶

1 e @±Hh 1 e ! ^ dh e = i(! ¡ ! ^ dh ) ±Gh ¡ Q±Á + i ±Á = i(! ¡ ! ^ dh )±Gh + i ±Á: vk @l !m !m ! m H

H

Taking the average A¹ ´ (A=vk )dl= dl=vk of both side of the foregoing equation yields ±Gh = ¡

! ^ dh ±Á e 1 Q m !¡! ^ dh !

(14:17)

and 2 2 2 2 ! ^ dh ±Á i m(vk + v? =2) 1 m(vk + v? =2) =¡ (b £ ∙) ¢ r±Á = ¡ (b £ ∙) ¢ !(» £ B) ! ! eB ! eB

=

2 m(vk2 + v? =2)

e

(∙ ¢ ») = ¡

m m 2 J v = ¡ J2E e e

where 2 2 =2 (cos µ»r + sin µ»µ ) =2 e¡iµ »r vk2 + v? vk2 + v? ¡1 2 2 » J ´ 2 (vk + v? =2)(∙ ¢ ») ¼ : v v2 R v2 R

(14:18)

One notes that frequencies !; !dh are much smaller than the hot ion transit and bounce frequencies vk =R; ²1=2 v=qR. For untrapped particles (±Ghu ) and trapped particles (±Ght ), we have ±Ghu ¼ 0;

±Ght ¼ 2QE

J¹ : !¡! ^ dh

(14:19)

The perturbed pressure tensor due to hot ion component is ±Ph = ¡» ? ¢ r(P? I + (Pk ¡ P? )bb) + ±P? I + (±Pk ¡ ±P? )bb

(14:20)

where ±P? =

Z

±Pk =

Z

2 mv? ±Fh 2¼v? dv? dvk 2

mvk2 ±Fh 2¼v? dv? dvk :

The ¯rst term of the right-hand side of (14.20) has similar form to the pressure term of core plasma. Since beta of hot ion component ¯h is much smaller than ¯c of core plasma, the ¯rst 2 term in (14.20) can be neglected. Since E = v 2 =2; ¹ = v? =2B and ® ´ ¹=E are de¯ned, we have vk2 = 2E(1 ¡ ®B);

2 v? = 2B®E;

2¼v? dv? dvk = 22 ¼

BE E 1=2 dEd® = 23=2 ¼B d®dE: vk (1 ¡ ®B)1=2

166

14 Instabilities Driven by Energetic Particles

Then the perturbed pressure of hot ion component is reduced to Z

±P? = 23=2 ¼B 5=2

=2

±Pk = 2

3=2

mB

¼B

E 1=2 d®dE m®BE±Fh (1 ¡ ®B)1=2

Z

1=2

d®(1 ¡ ®B)

¡1 Bmax

Z

= 25=2 mB

B ¡1

Z

E

dEE 3=2

0

®B ±Fh 2(1 ¡ ®B)

(14:21)

E 1=2 d ®dE m2E(1 ¡ ®B)±Fh (1 ¡ ®B)1=2

Z

B ¡1

d®(1 ¡ ®B)1=2

¡1 Bmax

Z

E 0

dEE 3=2 ±Fh :

(14:22)

The divergence of the second pressure term of (14.20) is (refer appendix A) (r±Ph )¯ =

X @±P?

@x®

®

=

±®¯ +

X @(±Pk ¡ ±P? )

@x®

®

b® b¯ + (±Pk ¡ ±P? )

X

(b® b¯ )

®

@±P? + b¯ (b ¢ r)(±Pk ¡ ±P? ) + (±Pk ¡ ±P? )((b ¢ r)b¯ + b¯ (r ¢ b)) @x¯

(r±Ph )? = r? ±P? + (±Pk ¡ ±P? )(b ¢ r)b = r? ±P? + (±Pk ¡ ±P? )∙

(14:210 )

(r±Ph )k = rk ±P? + (b ¢ r)(±Pk ¡ ±P? ) + (±Pk ¡ ±P? )r ¢ b:

(14:220 )

The kinetic integral ±WK is 1 ±WK = 2 1 =¡ 2

Z

3=2

= ¡2

Z

1 ¢ r±P dr = 2

» ¤?

Z

» ¤? ¢ (r? ±P? + (±Pk ¡ ±P? )∙)dr

r ¢ » ¤? ±P? ¡ (±Pk ¡ ±P? )» ¤? ¢ ∙)dr ¼m

Z

drB

Z

E 3=2 d®dE (1 ¡ ®B)1=2

Ã

2

r¢

» ¤?

2

Since r ¢ » ? + 2(» ¢ ∙) ¼ 0 (refer (B.7)), the term of ( ) in the integrand is 2 =2 ¤ vk2 + v? 1 (» ? ¢ ∙) ¼ ¡ (» ¤? ¢ ∙); ( )=¡ 2 v 2 ±WK is reduced to ±WK 23=2 ¼ =¡ mh 2¼R 2¼R ¼ mh

Z

= ¡27=2 ¼ 2 mh

Z

5=2 2

= ¡2

0

Z

drB Z

1 dr r 2¼ rs

dr r

Z

Z

d®dE

dµB

Z

¹ J¹¤ QE J2 E 3=2 1=2 (1 ¡ ®B) !¡! ^ dh

d®dE

(1+r=R)

(1¡r=R)

d(®B)

¹ J¹¤ QE J2 E 3=2 (1 ¡ ®B)1=2 ! ¡ ! ^ dh

Z

!

2 vk ¡ v? =2 v? ¡ (» ¤? ¢ ∙) ±Fh : 2 2v v2

dEKb E 5=2

J¹¤ QE J¹ !¡! ^ dh

14.1 Fishbone Instability

¼ 23=2 ¼ 2 mh

j»j2 R2

Z

167

rs

dr r

0

where Kb =

I

Z

d(®B)

Z

B2 K22 Q ^ K; ´ µs j» s j2 ± W Kb ! 2¹0 ^ dh ¡ !

dEE 5=2

dµ 1 = 2¼ (1 ¡ ®B)1=2

I

dµ v ; 2¼ vk

K2 =

I

(14:23)

dµ cos µ : 2¼ (1 ¡ ®B)1=2

Therefore dispersion relation (14.3) is reduced to ¡i! ^ T + ±W ^ K = 0; + ±W !A

(14:24)

where !A ´ (¿A s=2)¡1 and ° is relaced by ¡i!. 14.1d Growth Rate of Fishbone Instability 2 Let us assume a model distribution for slowing down hot ions with the initial velocity vmx =

Emx Fh0 = c0

±(® ¡ ®0 ) ; E 3=2

(E < Emx ):

(14:25)

Then the pressure ph and the density nh of hot ions are ph =

Z

23=2 ¼B

E 1=2 d®dE(mE)Fh0 = c0 (1 ¡ ®B)1=2

ph = c0 23=2 ¼BmKb Emx ;

nh =

Z

Emx

23=2 ¼B

Tc

c0 =

Z

Emx

23=2 ¼B

0

m dE (1 ¡ ®0 B)1=2

ph ; 23=2 ¼BmKb Emx

(14:26) (14:27)

ln(Emx =Tc ) E 1=2 d®dEFh0 = c0 23=2 ¼BmKb Emx = ph : 1=2 Emx (1 ¡ ®B)

(14:28)

The kinetic integral is Z

1 r2 ±WK = s2 j» s j2 2 2¼R R rs

0

1 r2 B 2 j» j2 = s2 R 2¹0 s rs2

Z

1 r2 B 2 j» s j2 2 = s2 R 2¹0 rs

Z

rs

0

0

Ã

rs

rs

dr r

Z

dE2

3=2 2

K2 1 dr r¼ 22 Kb Emx

¼ mBE

Z

Emx

dE

0

µ

5=2

Ã

¡(3=2)¯h ¡ 2(@¯h =@r)R(mE=2eBRr!) mE=(2eBRr!) ¡ 1

K2 ! ¡(3=2)¯h dr r¼ 22 Kb !dh;mx µ

!

K22 ¡(3=2)!c0 E ¡5=2 ¡ (@c0 =@r)(m=eBr)E ¡3=2 Kb mE=(2eBRr) ¡ !

Z

ymx

0

!dh;mx ! ¼ K22 B2 = µs j» s j2 (¡3=2)h¯h i ln 1¡ 2 2¹0 2 Kb !dh;mx !

¶

@¯h dy ¡2 R y¡1 @r Ã

Z

ydy y¡1

¶ µ

!dh;mx @¯h ! iR 1 + ¡ 2h ln 1¡ @r !dh;mx !

¶!!

:

(14:29) As the second term of h(@¯h =@r)iR is dominant, the dispersion relation is reduced to ¡i­

µ

µ

¶¶

2 !dh;mx ^ T + ¼ K2 h¡ @¯h iR 1 + ­ ln 1¡ 1 + ±W !A @r ­ Kb2

= 0:

(14:30)

168

14 Instabilities Driven by Energetic Particles

Fig.14.1 Toroidal precession of banana orbit of trapped ions

where ­´

! !dh;mx

!dh;mx ´

;

2 mvmx =2 ; eBRr

¯h ´

ph 2 B =2¹

: 0

^ T = 0. Then (14.30) is Let us consider the case of ± W µ

¶

1 ¡i®h ­ + ­ ln 1¡ + 1 = 0; ­

(14:31)

where !dh;mx ®h ´ !A

Ã

!¡1

K 2 @¯h iR ¼ 22 h¡ Kb @r

:

Under the assumption (1 ¡ 1=­r ) < 0 and j­i j ¿ j­r j, (14.31) is reduced to µ

¡i®h (­r + i­i ) + (­r + i­i ) ln

µ

¶

¶

1 ­i ¡ 1 + ¼i ¡ i + 1 = 0: ­r (1=­r ¡ 1)­r2

(14:32)

From the real and imaginary parts of (14.32), we have ­i =

¼ ¡ ®h ­r ¡ ln(1=­r ¡ 1) + (1 ¡ (1=­r ))

(14:33)

­r =

1 ¡ (¼ ¡ ®h )­i : ¡ ln(1=­r ¡ 1)

(14:34)

In the case of marginally unstable state ¼ = ®h , that is, ­i = 0; ­r is given by 1 ; ­r = ¡ ln(1=­r ¡ 1)

!

µ

1 1 1 ­r = 1 + tanh = 1 + exp(¡1=­r ) 2 2­r

¶

and ­r ¼ 0:75. For the excitation of ¯shbone instability, the necessary condition is ­i > 0, that is, ®h < ¼ and h¡

@¯h rs !dh;mx 1 Kb2 irs > : @r R !A ¼ 2 K22

There is a threshold for hj@¯h =@rjirs for the instability. Banana orbits of trapped ions drift in toroidal direction as is shown in ¯g.14.1. The toroidal precession velocity and frequency are¤ vÁ =

2 =2 mv? ; eBr

!Á =

2 =2 mv? : eBRr

(14:35)

14.2 Toroidal Alfv¶en Eigenmode

169

Therefore !dh;mx is equal to the toroidal precession frequency of trapped ions with the initial (maximum) velocity. It seems that the ¯shbone instability is due to an interaction between energetic particles and m=1, n=1 MHD perturbation. The interaction of resonant type is characterized by Landau damping. The resonance is between the toroidal wave velocity of instability and the toroidal precession of trapped energetic particles. 2 =2eBR), so that the poloidal dis(* Note: The toroidal vertical drift velocity is vd = (mv? placement of particles between bounces is r±µ » vd ¿d , ¿d being the bounce period. Since dÁ=dµ = q along the magnetic line of ¯eld, the associated toroidal displacement between bounces is RdÁ = (Rqvd ¿d =r); q = 1. Thus toroidal precession velocity is given by (14.35).) 14.2 Toroidal Alfv¶ en Eigenmode Alfv¶en waves in homogeneous magnetic ¯eld in in¯nite plasma have been analyzed in sec.5.4. Shear Alfv¶en wave, fast and slow magnetosonic waves appear. In the case of incompressible plasma (r ¢ » = 0 or ratio of speci¯c heat ° ! 1), only the shear Alfv¶en wave can exists. In the case of cylindrical plasma in the axisymmetric magnetic ¯eld, the displacement of MHD perturbation »(r; µ; z) = »(r) exp i(¡mµ + kz ¡ !t) is given by Hain-LÄ ust equations (8.114-117) as was discussed in sec.8.4. In the case of incompressible plasma, Hain-LÄ ust equation (8.117) is reduced to [In sec.8.4 perturbation is assumed to be »(r) exp i(+mµ + kz ¡ !t)] d dr

Ã

F 2 ¡ ¹0 ½m ! 2 m2 =r 2 + k 2

!

µ

1 d d (r»r ) + ¡(F 2 ¡ ¹0 ½m !2 ) + 2Bµ r dr dr

4k 2 Bµ2 F 2 d + 2 2 2 + 2r 2 2 2 r (m =r + k )(F ¡ ¹0 ½m ! ) dr

µ

µ

Bµ r

(m=r)F Bµ 2 r (m2 =r 2 + k 2 )

¶

¶!

»r = 0

(14:36)

where F = (k ¢ B) =

µ

¶

¡m n Bz Bµ (r) + Bz (r) = r R R

µ

¶

m n¡ ; q(r)

q(r) =

R Bz : r Bµ

2 2 ´ B2 The position at which F 2 ¡ ¹0 ½m ! 2 = 0 ! ! 2 = k2k vA ; vA =¹0 ½m holds is singular radius. 7 It was shown by Hasegawa and L.Chen that at this singular radius (resonant layer) shear Alfv¶en wave is mode converted to the kinetic Alfv¶en wave and absorbed by Landau damping. Therefore Alfv¶en wave is stable in the cylindrical plasma. Alfv¶en waves were also treated in sec.10.4a, 10.4b by cold plasma model. The dispersion relation in homogenous in¯nite plasma is given by (10.640 ) showing that Alfv¶en resonance occurs 2 at ! 2 ¼ kk2 vA and cuts o® of compressional Alfv¶en wave and shear Alfv¶en wave occur at ! 2 = 2 2 (1 + !=­i ) and ! 2 = kk2 vA (1 ¡ !=­i ) respectively. kk2 vA

14.2a Toroidicity Induced Alfv¶ en Eigenmode Let us consider shear Alfv¶en waves in toroidal plasma and the perturbation of (-m, n) mode given by Á(r; µ; z; t) = Á(r) exp i(¡mµ + n where R is major radius of torus and kk is µ

z ¡ !t) R

¶

k¢B m 1 kk = n¡ : = B R q(r) The resonant conditions of m and m+1 modes in linear cylindrical plasma are !2 2 2 ¡ kkm = 0 vA

(14:37)

170

14 Instabilities Driven by Energetic Particles

Fig.14.2 The Alfv¶en resonance frequency ! of toroidally coupled m and m+1 modes.

!2 2 2 ¡ kkm+1 = 0: vA However wave of m mode can couple with m§1 in toroidal plasma since the magnitude of toroidal ¯eld changes as Bz = Bz0 (1 ¡ (r=R) cos µ), as will be shown in this section later. Then the resonant condition of m and m+1 modes in toroidal plasma becomes ¯ ¯ ¯ ¯ ¯ ¯

!2 2 vA

2 ¡ kkm 2

®² v!2

A

2

®² v!2 !2 2 vA

A

2 ¡ kkm+1

¯ ¯ ¯ ¯=0 ¯ ¯

where ² = r=R and ® is a constant with order of 1. Then the solutions are 2 !§ 2 vA

=

³

2 2 2 ¡ k2 2 2 2 2 § (kkm kkm + kkm+1 km+1 ) + 4®² kkm kkm+1

2(1 ¡ ®2 ²2 )

´1=2

:

(14:38)

2 2 = kkm+1 , the The resonant condition (14.38) is plotted in ¯g.14.2. At the radius satisfying kkm di®erence of !§ becomes minimum and the radius is given by

µ

1 m n¡ R q(r)

¶

µ

¶

1 m+1 =¡ n¡ ; R q(r)

q(r0 ) =

m + 1=2 ; n

kkm = ¡kkm+1 =

1 : (14:39) 2q(r0 )R

q(r0 ) = 1:5 for the case of m=1 and n=1. Therefore Alfv¶en resonance does not exist in the frequency gap !¡ < ! < !+ . The continuum Alfv¶en waves correspond to the excitation of shear Alfv¶en waves on a given 2 2 vA (r) and such a resonance leads °ux surface where the mode frequency is resonant ! 2 = kkm wave damping. However frequencies excited within the spectral gaps are not resonant with the continuum and hence will not damp in the gap region. This allows a discrete eigen-frequency of toroidicty-induced Alfv¶en eigenmode or toroidal Alfv¶en eigenmode (TAE) to be established. This TAE can easily be destabilized by the kinetic e®ect of energetic particles. The equations of TAE will be described according to Berk, Van Dam, Guo, Lindberg8 . The equations of the ¯rst order perturbations are r ¢ j 1 = 0;

½

dv 1 = (j £ B)1 ; dt

(14:40)

14.2 Toroidal Alfv¶en Eigenmode

E 1 = rÁ1 ¡

171

@A1 ; @t

B 1 = r £ A1 :

(14:41)

For ideal, low ¯ MHD waves, we have following relations: B k1 = 0;

E k = 0;

A1 = Ak1 b

(14:42)

E1 £ b : B

(14:43)

so that i!Ak1 = b ¢ rÁ1 ;

v1 =

From (14.40), we have r ¢ j ?1 + r ¢ (jk1 b) = 0;

(14:44)

and ¡i!½(v 1 £ b) = (j ?1 £ B) £ b + (j £ B 1 ) £ b;

j ?1 = ¡

jk i!½ E + B ?1 : ?1 B2 B

(14:45)

Equations (14.41-43) yield B ?1 = r £ (Ak1 b) = r

jk1 = b ¢ j 1 =

=

µ

µ

Ak1 B

¶

£B+

µ

Ak1 ¡i b ¢ rÁ1 r£B ¼ r B ! B µ

µ

¶

£

i r ¢ B 2 r? !¹0 B

µ

¡i (B ¢ r)Á1 1 b ¢ r £ B ?1 = b ¢ r £ B 2 r? ¹0 !¹0 B2 ¶

µ

µ

B ¢ rÁ1 i B b ¢ 2 r ¢ B 2 r? !¹0 B B2

¶¶

=

¶

µ

£B

B B2

(14:46)

¶

B ¢ rÁ1 B2

¶¶

:

(14:47)

Then (14.44-47) yield Ã

!

µ

¶

µ

¶

jk jk1 ! 1 r? Á1 + r r¢ i B ?1 + r B =0 2 ¹0 vA B B Ã

!

µ

¶

µ

¶

µ

µ

µ

jk !2 (B ¢ r)Á1 1 B ¢ rÁ1 ¢B £r r ¢ B 2 r? ¢ r¢ 2 r? Á1 +¹0 r +(B ¢r) 2 2 vA B B B B2

¶¶¶

= 0:

(14:48) When (R; '; Z) and (r; µ; ³) coordinate are introduced by, R = R0 + r cos µ;

Z = r sin µ;

'=¡

³ R

172

14 Instabilities Driven by Energetic Particles

Fig.14.3 Lefthand-side ¯gure: The toroidal shear Alfv¶en resonance frequencies ­ that corresponds to (n=1, m=1) and (n=1. m=2), q(r) = 1 + (r=a)2 , a=R = 0:25, ­ ´ !=(vA (0)=R0 ). Righthand-side ¯gure: The structure of the global mode amplitude as a function of radius.

and following notations are used X

Á1 (r; µ; ³; t) =

m

Ám (r) exp i(¡mµ + n' ¡ !t);

µ

¶

i m kkm Ám = n¡ ; R0 q(r)

Em ´

(b ¢ r)Ám =

µ

i m n¡ R0 q(r)

¶

= ¡ikkm Ám ;

Ám ; R

(14.48) is reduced to8 d dr

Ã

r

3

Ã

!2 2 2 ¡ kkm vA

d + dr

Ã

r

3

µ

!

dEm dr

¶2

! vA

2r R0

!

d + r Em dr

µ

dEm+1 dEm¡1 + dr dr

2

µ

! vA

¶2

Ã

!

!2 2 ¡ (m ¡ 1) 2 ¡ kkm rEm vA 2

¶!

= 0:

(14:49)

As is seen in ¯g.14.3, mode structure has a sharp transition of m=1 and m=2 components at the gap location. Therefore m and m+1 modes near gap location reduces Ã

!2 2 2 ¡ kkm vA

!

Ã

!2 2 2 ¡ kkm+1 vA

dEm 2r + dr R0 !

µ

! vA

dEm+1 2r + dr R0

¶2 µ

dEm+1 ¼0 dr

! vA

¶2

dEm ¼ 0; dr

so that toroidal shear Alfv¶en¯ resonance frequency is given by ¯ ¶ µ ¯ ¯ ¯ ¯ ¯ ¯ ¯

!2 2 vA

2²

2 ¡ kkm

³

! vA

´2

µ

!2 2 vA

³

´2

¯ ¯ ¯ ¶ ¯ = 0: ¯ 2 ¯ ¡ kkm+1 ¯

2²

! vA

(14:50)

When Shafranov shift is included in the coordinates of (R; '; Z) and (r; µ; ³), coupling constant becomes 2.5² instead of 2².8 The energy integral from (14.49) without coupling term of m§1 modes is reduced to following equation by use of partial integral: G(!; Em ) ´ P

Z

0

a

dr r

ÃÃ

r

2

µ

dEm dr

¶2

2

+ (m ¡

2 1)Em

!Ã

!

!2 2 2 d 1 ¡ kkm ¡ ! 2 rEm 2 2 dr vA vA

!

14.2 Toroidal Alfv¶en Eigenmode

173

= Em (rs¡ )Cm (rs¡ ) ¡ Em (rs+ )Cm (rs+ ) where Cm (r) =

Ã

!

!2 dEm 2 ¡ kkm r2 ; 2 dr vA

(14:51)

Em (a) = 0:

2 The radius r = rs is singular at which (! 2 =vA )2 ¡kkm = 0 and P is principal value of the integral. From this formulation, it is possible to estimate the damping rate of TAE and is given by8

±! = ¡i¼ !

sign(! )C (r )2

0 m s ¯ µ ¶¯ : ¯ ¯ @ !2 @G 2 3 ¯ ¯ rs ¯ @r v 2 ¡ kkm ¯ !0 @!0 A

(14:52)

Since !0 @G=@!0 > 0; Im(±!) < 0. This is called continuum damping. 14.2b Instability of TAE Driven by Energetic Particles Dynamics of energetic particles must be treated by kinetic theory. Basic equations will be described according to Betti and Freidberg9 . @fj qj (E + v £ B) ¢ rv fj = 0; + v ¢ fj + @t mj

(14:53)

@nj + r ¢ (nj uj ) = 0; @t

(14:54)

mj

@ (nj uj ) + r ¢ Pj = qj nj (E + juj £ B); @t

Pj = mj

Z

(14:55)

vvfj dv;

(14:56)

B 1 = r £ (» ? £ B);

(14:57)

¹0 j 1 = rB 1 = r £ r £ (» ? £ B);

j 1 £ B + j £ B1 =

X j

((14:58))

(rP1j ¡ i!mj (n1j uj + nj u1j )) ¼

X³ j

´

rP1j ¡ ½! 2 » ?j :

(14:59)

Fj is equilibrium distribution function of axisymmetric torus. Fj ("; p' ) is assumed to be a function of constants of motion " and p' , where "=

mj 2 v + qj Á; 2

p' = mj Rv' + qj Ã;

à = RA'

(14:60)

174

14 Instabilities Driven by Energetic Particles

RBZ =

@Ã ; @R

RBR = ¡

@Ã ; @Z

@f1j qj qj (v £ B) ¢ rv f1j = ¡ (E + v £ B 1 ) ¢ rv Fj + v ¢ f1j + @t mj mj

(14:61)

and ^ rv F j = '

@p' @Fj @Fj @Fj @Fj ^ jR + (rv ") + mj v = 'm : @v' @p' @" @p' @"

(14:62)

The solution is obtained by integral along the particle orbit (refer appendix sec.C.2) f1j = ¡

qj mj

Z

t ¡1

(E + v £ B 1 ) ¢ rv Fj dt0 :

(14:63)

It is assumed that the perturbations are in the form of Q1 = Q1 (R; Z) exp i(n' ¡ !t): The second term mj v(@Fj =@") of the right-hand side of (14.62) contributes to the integral ¡

qj mj

Z

t

¡1

(E + v £ B 1 ) ¢ mj v

@Fj @Fj 0 dt = ¡qj @" @"

Z

t

¡1

E ¢ vdt0 :

The contribution from the ¯rst term mj R(@Fj =@p' ) is qj ¡ mj

ÃZ

t

@Fj 0 (E ' mj R dt + @p ¡1 '

@Fj @p'

µZ

@Fj = ¡qj @p'

µZ

@Fj = ¡qj @p'

µZ

= ¡qj

t

¡1

t

¡1

t

¡1

E ' Rdt0 +

Z

Z

t

@Fj 0 mj R(b £ B 1 )' dt @p ¡1 '

t

¡1

Rv £

1 @(E' R) 0 dt + ¡i! @t

Z

1 d(E' R) 0 dt + ¡i! dt

Z

µ

t

¡1

t

¡1

¶

(r £ E)' dt0 ¡i! µ

!

¶ ¶

n 1 (v ¢ E) ¡ (v ¢ r)(E' R) dt0 ! i!

¶

¶

n (v ¢ E)dt0 : !

The solution is qj f1j = ¡ !

Ã

Ã

@Fj @Fj @Fj i RE' + ! +n @p' @" @p'

!Z

t

¡1

(E ¢ v)dt

0

!

:

Since E1k = 0;

¡ i!» ? =

E? £ B ; B2

E ? = i!(» ? £ B);

RE' = i!(» ? £ B)' R = i!(»?R BZ ¡ »?Z BR )R = ¡i!(» ¢ rÃ);

(14:64)

14.2 Toroidal Alfv¶en Eigenmode

175

E ¢ v = i!(»? £ B) ¢ v = ¡i!» ? ¢ (v £ B) = ¡i!» ? ¢ dv mj mj = ¡i! » ? ¢ = ¡i! qj dt qj

µ

d» d(»? ¢ v) ¡v¢ ? dt dt

¶

!µ

Z

mj dv qj dt

;

f1j becomes Ã

@Fj @Fj @Fj f1j = ¡qj (» ¢ rÃ) + imj ! +n @p' @" @p'

t

d» »? ¢ v ¡ v ¢ ? dt0 dt ¡1

@Fj @Fj + imj (! ¡ !¤j ) (» ¢ v ¡ sj ) @Ã @" ?

= ¡qj

¶

(14:65)

where sj ´

Z

t

¡1

v¢

d» ? 0 dt ; dt

!¤j ´ ¡

n@Fj =@p' : @Fj =@"

sj is reduced to s j = mj

Z

t

¡1

Ã

2 v? r ¢ »? + 2

Ã

!

!

2 v? ¡ vk2 » ¢ ∙ dt0 2

(14:66)

as will be shown in the end of this subsection. The perturbed pressure tensor is P1j =

Z

mj vvdv = P1?j I + (P1kj ¡ P1?j )bb

(14:67)

and rP1j is given by eqs.(14.210 ) and (14.220 ). Then the equation of motion is ¡½! 2 » ? = F ? (» ? ) + iD? (»? );

(14:68)

F ? (» ? ) = j 1 £ B + j £ B 1 + r(» ? ¢ rP1 );

(14:69)

D ? (» ? ) = mj

Z Ã

Ã

2 v2 v? r? + vk2 ¡ ? 2 2

! !

∙ (! ¡ !¤j )

@Fj sj dv: @"

(14:70)

F ? (»? ) is the ideal MHD force operator for incompressible displacement. D ? (» ? ) contains the contribution of energetic particles. Eqs. (14.68-70) describe the low frequency, ¯nite wave number stability of energetic particle-Alfv¶en waves in axisymmetric torus. The energy integral of (14.68) consists of plasma kinetic energy normalization KM , ideal MHD perpendicular potential energy ±WMHD and the kinetic contribution to the energy integral ±WK : !2 KM = ±WMHD + ±WK : where KM =

1 2

Z

½j» ? j2 dr;

±WMHD = ¡

1 2

Z

» ¤? F ? (» ? )dr;

(14:71)

176

14 Instabilities Driven by Energetic Particles

±WK = ¡

i 2

Z

» ¤? D? (» ? )dr:

After a simple integration by parts, ±WK can be written as iX ±WK = 2 j

Z

(! ¡ !¤j )

@Fj ds¤j sj dvdr; @" dt

(14:72)

since ds¤j = mj dt

Ã

Ã

2 v? r? ¢ » ¤ + 2

!

!

2 v? ¡ vk » ? ¢ ∙ : 2

On the other hand ds¤j =dt is given by ds¤j = i!¤ s¤j + Ds¤j ; dt

D ´ (v ¢ r) +

qj (v £ B) ¢ rv : mj

With use of the notation sj ´ aj + icj (aj and cj are real), we have sj

ds¤j 1 = i! ¤ jsj j2 + i(cj Daj ¡ aj Dcj ) + D(a2j + c2j ): dt 2

Contribution of the last term to the integral (14.72) by drdv is zero, since Fj and !¤j are functions of the constants of motion " and p' and ±WK =

1X 2 j

Z

(! ¡ !¤j )

@Fj (i!i jsj j2 + Rj )dvdr; @"

Rj = cj Daj ¡ aj Dcj ¡ !r jsj j2 : The desired expression for the growth rate is obtained by setting the real and imaginary parts of (14.71) to zero: !r2 =

±WMHD + O(¯): KM

(14:73)

O(¯) is the contribution of Rj term. In the limit of !i ¿ !r , the imaginary part yields !i ¼

WK ; KM

0

WK ´ lim @ !i !0

Z 1 X

4!r

j

(! ¡ !¤j )

1

@Fj !i jsj j2 dvdr A : @"

Let us estimate (14.74). Since r ¢ » ? + 2» ? ¢ ∙ ¼ 0 (refer (B.7)), sj is sj = ¡mj

Z

t

¡1

µ

vk2

v? + 2

¶

0

(∙ ¢ » ? )dt = mj

Z

t

¡1

µ

vk2

v? + 2

where »R = »r cos µ ¡ »µ sin µ = »r

eiµ + e¡iµ eiµ ¡ e¡iµ ¡ »µ : 2 2i

»r and »µ are (r ¢ » = (1=r)(@(r»r )=@r) ¡ i(m=r)»µ ¼ 0) »r =

X m

»m (r)e¡imµ ;

»µ = ¡i

X (r»m (r))0 m

m

e¡imµ :

¶

»R 0 dt R

(14:74)

14.2 Toroidal Alfv¶en Eigenmode

177

Since the leading-order guiding center of orbits of energetic particles are given by r(t0 ) = r(t);

vk Bµ 0 (t ¡ t) + µ(t); rB'

µ(t0 ) =

'(t0 ) =

vk 0 (t ¡ t) + '(t) r

perturbations along the orbit become Ã

!

nvk mBµ ¡ !)(t0 ¡ t) exp i(¡mµ(t) + n'(t) ¡ !t) exp i(¡mµ(t ) + n'(t ) ¡ !t ) = exp i(¡ vk + rB' R 0

0

0

¡

¢

= exp ¡i(! ¡ !m )(t0 ¡ t) exp i(¡mµ(t) + n'(t) ¡ !t) where

µ

¶

vk m n¡ : !m = R q and sj =

2 =2) 1 X mj (vk2 + v?

R

2

£ Ã

mj v2 =i vk2 + ? 2R 2

!

m

Z

0

1

(»m¡1 + »m+1 ¡ i»µ(m¡1) + i»µ(m+1) ) exp i(¡mµ + n' ¡ !t)

exp(¡i(! ¡ !m )t00 dt00

Xµ m

»m¡1 + »m+1 ¡ i

(r»m¡1 )0 (r»m+1 )0 +i (m ¡ 1) (m + 1)

¶

exp i(¡imµ + n' ¡ !t) : (14:75) (! ¡ !m )

It is assumed that perturbation consists primarily of two toroidally coupled harmonics » m and » m+1 and all other harmonics are essentially zero. Strong coupling occurs in a narrow region of thickness » ²a localized about the surface r = r0 corresponding to q(r0 ) = (2m + 1)=2n = q0 . 0 terms dominate in (14.75). Substituting these results The mode localization implies that »m§1 into the expression for sj and maintaining only these terms which do not average to zero in µ leads to the following expression for jsj j2 : m2j r02 jsj j = 4R 2

Ã

vk2

v2 + ? 2

!2 Ã

0 j»(m+1) j2

0 j2 j»m + (m + 1)2 m2

!µ

1 1 + 2 j! ¡ !m j j! ¡ !m¡1 j2

¶

since !m+1 = ¡!m and !m+2 = ¡!m¡1 . KM is given by KM

r2 ½2 = 0 0 2

Z Ã

0 0 j2 j»m+1 j2 j»m + m2 (m + 1)2

!

dr:

(14:76)

Using the relations !r ¼ kk vA , kk = 1=(2q0 R), q0 = (2m + 1=2n), we obtain the following expression for growth rate: Ã ! Ã !µ ¶ 2 2 X ¹0 m2j q02 Z @Fj !i v? !i n @Fj !i 2 = lim vk + !r + dv: + j! ¡ !m j2 j! ¡ !m¡1 j2 kk vA !i !0 j 2B 2 2 @" qj @Ã

(14:77)

178

14 Instabilities Driven by Energetic Particles

With use of the formula X 2¼ !i = kk vA j

2

¹0 m2j Rq03 2B 2

R1

¡1

²=(x2 + ²2 )dx = ¼, short calculation of integral by vk yields

Z Ã

vk2 +

2 v?

2

!2 Ã

!r

@Fj n @Fj + @" qj @Ã

!

¯ ¯ ¯ v? dv? ¯¯ ¯

+ vk =vA

¯ ¯ ¯ ¯ ¯ ¯ ¯

:

vk =vA =3

(14:78)

(Note !m = vk =(2q0 R), !m¡1 = 3vk =(2q0 R), !r = vA =2q0 R.) Equation (14.78) gives the TAE growth rate for arbitrary distribution function Fj ("; Ã). The second term of (14.78) is due to sideband resonance. The growth rate can be easily evaluated for a Maxwellian distribution Fj = nj

Ã

mj 2¼Tj

!3=2

Ã

mj v 2 exp 2¼Tj

!

:

Here nj = nj (Ã) and Tj = Tj (Ã). Some straightforward calculation leads to Ã

!i kk vA

!

= j

Ã

¡q02 ¯j

GT mj

T T Hmj + ´j Jmj ¡ nq0 ±j 1 + ´j

!

(14:79)

where ¯j =

nj T j ; B 2 =2¹0

±j = ¡rLpj

dpj =dr ; pj

mj vTj ; qj Bp

rLpj ´

´j ´

d ln Tj : d ln nj

T T Each of these quantities is evaluated at r = r0 . The functions GT mj , Hmj and Jmj are functions of single parameter ¸j ´ vA =vTj and are given by 2

T T GT mj = gmj (¸j ) + gmj (¸j =3);

T gmj (¸j ) = (¼ 1=2 =2)¸j (1 + 2¸2j + 2¸4j )e¡¸j

T T Hmj = hT mj (¸j ) + hmj (¸j =3);

1=2 hT =2)(1 + 2¸2j + 2¸4j )e¡¸j mj (¸j ) = (¼

T T T Jmj = jmj (¸j ) + jmj (¸j =3);

2

(14:80)

2

T jmj (¸j ) = (¼ 1=2 =2)(3=2 + 2¸2j + ¸4j + 2¸6j )e¡¸j

For the alpha particles it is more reasonable to assume a slowing down distribution F® =

A ; 2 (v + v02 )3=2

m® v®2 = 3:5MeV) 2

( 0 < v < v® ;

A and v0 are related to the density and pressure as follows: A¼

n® ; 4¼ ln(v® =v0 )

p® ¼

n® m® v®2 =2 ; 3 ln(v® =v0 )

mj v02 ¼ Tj : 2

After another straightforward calculation we obtain an analogous expression for the alpha particle contribution to the growth rate: Ã

!i kk vA

!

®

³

T = ¡q02 ¯® GT s® ¡ nq0 ±® Hs®

´

(14:82)

14.2 Toroidal Alfv¶en Eigenmode

179

where ¯® =

p® ; 2 B =2mu0

dp® =dr 2 ±® = ¡ rL® ; 3 p®

rLp® =

m® v® : q ® Bp

T The functions GT s® and Hs® are functions of the parameter ¸® ´ vA =v® and are given by T T GT s® = gs (¸® ) + gs (¸® =3);

gsT (¸® ) = (3¼=16)¸® (3 + 4¸® ¡ 6¸2® ¡ ¸4® )H(1 ¡ ¸® )

T HsT = hT s (¸® ) + hs (¸® =3);

2 3 4 hT s® (¸® ) = (3¼=16)(1 + 6¸® ¡ 4¸® ¡ 3¸® )H(1 ¡ ¸® ) (14:83)

H(1 ¡ ¸® ) is the Heaviside step function (H(x) = 1 for x > 0, H(x) = 0 for x < 0). The ¯nal form of the growth rate is obtained by combining the contributions of ions and electrons of core plasma and ® particles: ³ ´ !i T T T = ¡q02 ¯i GT (14:84) mi + ¯e Gme + ¯® (Gs® ¡ nq0 ±® Hs® ) kk vA where ¯i , ¯e and ¯® are ¯j ´ nj Tj =B 2 =2¹0 of ions and electrons of core plasma and ® particles. The contributions of ions and electron of core plasma are Landau damping. The marginal condition for excitation of TAE is ¯® >

¯i GT i (¸i ) ; T ¡ GT nq0 ±® Hs® s®

±® >

Gs® : T nq0 Hs®

(14:85)

Derivation of (14.66) is described as follow:

v¢

X X d» X d»i X @»i X @»i ¡i!vi »i + vi vi vi (v ¢ r)»i = vi vj = = + dt dt @t xj i i i i ij

^? ) v = vk b + v? cos(­t)^ e? ¡ v? sin(­t)(b £ e

2 2 ^? )(b £ e ^? ) vv = vk2 bb + v? cos(­t)2 + v? sin(­t)2 (b £ e

2 2 2 2 ^? e ^? + (b £ e ^? )(b £ e ^? )) = (vk2 + v? = (vk2 + v? =2)bb + v? =2)(bb + e =2)bb + v? =2)I;

v¢

X @»i d» 2 2 =2)r ¢ » + (vk2 ¡ v? =2) bi bj ; = ¡i!vk »k + (v? dt @x j ij X ij

Ã

@»i @bi bi bj + bj »i @xj @xj

!

=

X ij

bj

@ (»i bi ) @xj

180

14 Instabilities Driven by Energetic Particles

Fig.14.4 Representative shear Alfv¶en frequency continuum curves as function of minor radius r. Horizontal lines indicating the approximate radial location and mode width for toroidal Alfv¶en eigenmode (TAE), kinetic TAE mode (KTAE), core-localized TAE mode (CLM), ellipticity Alfv¶en eigenmode (EAE), noncicular triangularity Alfv¶en eigenmode (NAE), and energetic particle continuum mode (EPM).

X ij

v¢

bi bj

@»i = ¡» ¢ (b ¢ r)b + (b ¢ r)(» ¢ b) = ¡∙ ¢ » + (b ¢ r)»k ; @xj

@»k d» 2 2 2 =2)r ¢ » + (v? =2 ¡ vk2 )∙ ¢ » ? ¡ i!vk »k + (vk2 ¡ v? =2) = (v? : dt @l

Since j»k j ¿ j» ? j, we obtain v¢

d» 2 2 =2)r ¢ » + (v? =2 ¡ vk2 )∙ ¢ » ? + a1 e¡i­t + ¢ ¢ ¢ : = (v? dt

The third term is rapidly oscillating term and the contribution to (14.66) is small. 14.2c Various Alfv¶ en Modes In the previous subsection we discussed the excitation of weakly damped TAE by superAlfv¶enic energetic particles. There are various Alfv¶en Modes. In high-temperature plasmas, non-ideal e®ects such as ¯nite Larmor radius of core plasma become important in gap region and cause the Alfv¶en continuum to split into a series of kinetic Alfv¶en eigenmodes at closely spaced frequencies above the ideal TAE frequency. In central region of the plasma, a low-shear version of TAE can arise, called the core-localized mode. Noncicular shaping of the plasma poloidal cross section creates other gaps in the Alfv¶en continuum, at high frequency. Ellipticity creats a gap, at about twice of TAE frequency, within which exist ellipticity-induced Alfv¶en eigenmodes and similarly for triangularity-induced Alfv¶en eigenmodes at about three times the TAE frequency. The ideal and kinetic TAE's are "cavity" mode, whose frequencies are determined by the bulk plasma. In addition, a "beam mode" can arise that is not a natural eigenmode of plasma

References

181

but is supported by the presence of a population of energetic particles and also destabilized by them. This so-called energetic particle mode, which can also exit outside the TAE gaps, has a frequencies related to the toroidal precession frequency and poloidal transit/bounce frequency of the fast ions. The schematic in ¯g.14.4 illustrates these various modes. Close interaction between theory and experiment has led many new discoveries on Alfv¶en eigenmodes in toroidal plasma. A great deal of theoretical work have been carried out on energetic particle drive and competing damping mechanism, such as continuum and radiative damping, ion Landau damping for both thermal and fast ions, electron damping and trapped electron collisional damping. For modes with low to moderate toroidal mode numbers n, typically continuum damping and ion Landau damping are dominant, where as high n modes, trapped collisional damping and radiative damping are strong stabilizing mechanism. There are excellent reviews on toroidal Alfv¶en eigenmode10 .

References 1. Liu Chen and R. B. White and M. N. Rosenbluth: Phys. Rev. Lett. 52, 1122 (1984) Y. Z. Zhang, H. L. Berk S. M. Mahajan: Nucl. Fusion 29, 848 (1989) 2. V. D. Shafranov: Sov. Phys. Tech. Phys. 15, 175 (1970) 3. M. N. Bussac, R. Pella, D. Edery and J. L. Soule: Phys. Rev. Lett. 35, 1638 (1975) 4. G. Ara, B.Basu, B.Coppi, G. Laval, M. N. Rosenbluth and B. V. Waddell: Annals of Phys. 112, 443 (1978) 5. P. J. Catto, W. M. Tang and D. E. Baldwin: Plasma Phys. 23, 639 (1981) 6. B. N. Kuvshinov, A. B. Mikhailovskii and E. G. Tatarinov: Sov. J. Plasma Phys. 14, 239, (1988) 7. A. Hasegawa and Liu Chen: Phys. Fluids: 19, 1924 (1976) 8. H. L. Berk, J. W. Van Dam, Z. Guo and D.M. Lindberg: Phys. Fluids B4, 1806 (1992) 9. R. Betti and J. P. Freidberg: Phys. Fluids B4, 1465 (1992) 10. ITER Physics Basis: Nucl. Fusion 39, No.12, p2495 (1999) E. J. Strait, W. W. Heidbrik, A. D.Turnbull, M.S. Chu, H. H. Duong: Nucl. Fusion 33, 1849 (1993) King-Lap Wong: Plasma Phys. Control. Fusion 41 R1 (1999) A. Fukuyama and T. Ozeki: J. Plasma Fusion Res. 75, 537 (1999) (in Japanese)

182

15 Development of Fusion Researches

Ch.15 Development of Fusion Researches The major research e®ort in the area of controlled nuclear fusion is focused on the con¯nement of hot plasmas by means of strong magnetic ¯elds. The magnetic con¯nements are classi¯ed to toroidal and open end con¯gurations. Con¯nement in a linear mirror ¯eld (sec.17.3) may have advantages over toroidal con¯nement with respect to stability and anomalous di®usion across the magnetic ¯eld. However, the end loss due to particles leaving along magnetic lines of force is determined solely by di®usion in the velocity space; that is, the con¯nement time cannot be improved by increasing the intensity of the magnetic ¯eld or the plasma size. It is necessary to ¯nd ways to suppress the end loss. Toroidal magnetic con¯nements have no open end. In the simple toroidal ¯eld, ions and electrons drift in opposite directions due to the gradient of the magnetic ¯eld. This gradient B drift causes the charge separation that induces the electric ¯eld E directed parallel to the major axis of the torus. The subsequent E £ B drift tends to carry the plasma ring outward. In order to reduce the E £ B drift, it is necessary to connect the upper and lower parts of the plasma by magnetic lines of force and to short-circuit the separated charges along these ¯eld lines. Accordingly, a poloidal component of the magnetic ¯eld is essential to the equilibrium of toroidal plasmas, and toroidal devices may be classi¯ed according to the method used to generate the poloidal ¯eld. The tokamak (ch.16) and the reversed ¯eld (17.1) pinch devices use the plasma current along the toroid, whereas the toroidal stellarator (sec.17.2) has helical conductors or equivalent winding outside the plasma that produce appropriate rotational transform angles. Besides the study of magnetic con¯nement systems, inertial con¯nement approaches are being actively investigated. If a very dense and hot plasma could be produced within a very short time, it might be possible to complete the nuclear fusion reaction before the plasma starts to expand. An extreme example is a hydrogen bomb. This type of con¯nement is called inertial con¯nement. In laboratory experiments, high-power laser beams or particle beams are focused onto small solid deuterium and tritium targets, thereby producing very dense, hot plasma within a short time. Because of the development of the technologies of high-power energy drivers, the approaches along this line have some foundation in reality. Inertial con¯nement will be discussed brie°y in ch.18. The various kinds of approaches that are actively investigated in controlled thermonuclear fusion are classi¯ed as follows:

Magnetic con¯nement

8 > > > > > > > > > > Toroidal > > > > system > > > > <

8 8 > Axially < Tokamak > > > > Reversed ¯eld pinch symmetric > > : > > Spheromak < 8 > > > Axially < Stellarator system > > > > Heliac asymmetric > > : :

> > > Bumpy torus > > > > > > 8 > > > Open end < Mirror; Tandem mirror > > > > Field Reversal Con¯guration system > > : :

Cusp

½ Inertial Laser con¯nement Ion beam; Electron beam

15 Development of Fusion Researches

183

From Secrecy to International Collaboration Basic research into controlled thermonuclear fusion probably began right after World War II in the United States, the Soviet Union, and the United Kingdom in strict secrecy. There are on record many speculations about research into controlled thermonuclear fusion even in the 1940s. The United States program, called Project Sherwood, has been described in detail by Bishop.1 Bishop states that Z pinch experiments for linear and toroidal con¯gurations at the Los Alamos Scienti¯c Laboratory were carried out in an attempt to overcome sausage and kink instabilities. The astrophysicist L. Spitzer, Jr., started the ¯gure-eight toroidal stellarator project at Princeton University in 1951. At the Lawrence Livermore National Laboratory, mirror con¯nement experiments were conducted. At the Atomic Energy Research Establishment in Harwell, United Kingdom, the Zeta experiment was started2 and at the I.V. Kurchatov Institute of Atomic Energy in the Soviet Union, experiments on a mirror called Ogra and on tokamaks were carried out.3 The ¯rst United Nations International Conference on the Peaceful Uses of Atomic Energy was held in Geneva in 1955. Although this conference was concerned with peaceful applications of nuclear ¯ssion, the chairman, H.J. Bhabha, hazarded the prediction that ways of controlling fusion energy that would render it industrially usable would be found in less than two decades. However, as we have seen, the research into controlled nuclear fusion encountered serious and unexpected di±culties. It was soon recognized that the realization of a practical fusion reactor was a long way o® and that basic research on plasma physics and the international exchange of scienti¯c information were absolutely necessary. From around that time articles on controlled nuclear fusion started appearing regularly in academic journals. Lawson's paper on the conditions for fusion was published in January 1957,4 and several important theories on MHD instabilities had by that time begun to appear.5;6 Experimental results of the Zeta7 (Zero Energy Thermonuclear Assembly) and Stellarator8 projects were made public in January 1958. In the fusion sessions of the second United Nations International Conference on the Peaceful Uses of Atomic Energy, held in Geneva, September 1-13, 1958, 9;10 many results of research that had proceeded in secrecy were revealed. L.A.Artsimovich expressed his impression of this conference as \something that might be called a display of ideas." The second UN conference marks that start of open rather than secret international cooperation and competition in fusion research. In Japan controlled fusion research started in Japan Atomic Energy Institute (JAERI) under the ministry of science and technology and in Institute of Plasma Physics, Nagoya University under the ministry of education and culture in early 1960's.11 The First International Conference on Plasma Physics and Controlled Nuclear Fusion Research was held in Salzburg in 1961 under the auspices of the International Atomic Energy Agency (IAEA). At the Salzburg conference12 the big projects were fully discussed. Some of there were Zeta, Alpha, Stellarator C, Ogra, and DCX. Theta pinch experiments (Scylla, Thetatron, etc.) appeared to be more popular than linear pinches. The papers on the large scale experimental projects such as Zeta or Stellarator C all reported struggles with various instabilities. L.A. Artsimovich said in the summary on the experimental results: \Our original beliefs that the doors into the desired regions of ultra-high temperature would open smoothly...have proved as unfounded as the sinner's hope of entering Paradise without passing through Purgatory." The importance of the PR-2 experiments of M.S. Io®e and others was soon widely recognized (vol.3, p.1045). These experiments demonstrated that the plasma con¯ned in a minimum B con¯guration is MHD stable. The Second International Conference on Plasma Physics and Controlled Nuclear Fusion Research was held at Culham in 1965.13 The stabilizing e®ect of minimum B con¯gurations was con¯rmed by many experiments. An absolute minimum B ¯eld cannot be realized in a toroidal con¯guration. Instead of this, the average minimum B concept was introduced (vol.1, pp.103,145). Ohkawa and others succeeded in con¯ning plasmas for much longer than the Bohm time with

184

15 Development of Fusion Researches

toroidal multipole con¯gurations (vol.2, p.531) and demonstrated the e®ectiveness of the average minimum B con¯guration. Artsimovich and others reported on a series of tokamak experiments (T-5, vol.2, p.577; T-3, p.595; T-2, p.629; TM-2, p.647; TM-1, p.659). Further experiments with Zeta and Stellarator C were also reported. However, the con¯nement times for these big devices were only of the order of the Bohm time, and painful examinations of loss mechanisms had to be carried out. Theta pinch experiments were still the most actively pursued. The ion temperatures produced by means of theta pinches were several hundred eV to several keV, and con¯nement times were limited only by end losses. One of the important goals of the theta pinch experiments had thus been attained, and it was a turning point from linear theta pinch to toroidal pinch experiments. In this conference, the e®ectiveness of minimum B, average minimum B, and shear con¯gurations was thus con¯rmed. Many MHD instabilities were seen to be well understood experimentally as well as theoretically. Methods of stabilizing against MHD instabilities seemed to be becoming gradually clearer. The importance of velocity-space instabilities due to the non-Maxwellian distribution function of the con¯ned plasma was recognized. There had been and were subsequently to be reports on loss-cone instabilities,17 Harris instability18 (1959), drift instabilities19 (1963, 1965),etc. The experiment by J.M.Malmberg and C.B.Wharton (vol.1,p.485) was the ¯rst experimental veri¯cation of Landau damping. L.Spitzer,Jr., concluded in his summary talk at Culham that \most of the serious obstacles have been overcome, sometimes after years of e®ort by a great number of scientists. We can be sure that there will be many obstacles ahead but we have good reason to hope that these will be surmounted by the cooperative e®orts of scientists in many nations." Artsimovich Era The Third International Conference14 was held in 1968 at Novosibirsk. The most remarkable topic in this conference was the report that Tokamak T-3 (vol.1, p.157) had con¯ned a plasma up to 30 times the Bohm time (several milliseconds) at an electron temperature of 1 keV. In Zeta experiments a quiescent period was found during a discharge and MHD stability of the magnetic ¯eld con¯guration of the quiescent period was discussed. This was the last report of Zeta and HBTX succeeded this reversed ¯eld pinch experiment. Stellarator C (vol.1, pp.479, 495) was still con¯ning plasmas only to several times the Bohm time at electron temperatures of only several tens to a hundred eV. This was the last report on Stellarator C; this machine was converted into the ST tokamak before the next conference (Madison 1971). However, various aspects of stellarator research were still pursued. The magnetic coil systems of Clasp (vol.1, p.465) were constructed accurately, and the con¯nement of high-energy electrons were examined using the ¯ decay of tritium. It was demonstrated experimentally that the electrons ran around the torus more than 107 times and that the stellarator ¯eld had good charge-particle con¯nement properties. In WII the con¯nement of the barium plasma was tested, and resonant loss was observed when the magnetic surface was rational. Di®usion in a barium plasma in nonrational cases was classical. In 2X(vol.2, p.225) a deuterium plasma was con¯ned up to an ion temperature of 6-8 keV at a density of n < 5 £ 1013 cm¡3 for up to ¿ = 0:2 ms. Laser plasmas appeared at this conference. At the Novosibirsk conference toroidal con¯nement appeared to have the best overall prospects, and the mainstream of research shifted toward toroidal con¯nement. L.A.Artsimovich concluded this conference, saying; \We have rid ourselves of the gloomy spectre of the enormous losses embodied in Bohm's formula and have opened the way for further increases in plasma temperature leading to the physical thermonuclear level." The Tokamak results were seen to be epoch making if the estimates of the electron temperature were accurate. R.S.Pease, the director of the Culham Laboratory, and L.A.Artsimovich agreed the visit of British team of researchers to Kurchatov to measure the electron temperature of the T-3 plasma by laser scattering methods. The measurements supported the previous estimates

15 Development of Fusion Researches

185

by the tokamak group.20 The experimental results of T-3 had a strong impact on the next phase of nuclear fusion research in various nations. At the Princeton Plasma Physics Laboratory, Stellarator C was converted to the ST tokamak device; newly built were ORMAK at Oak Ridge National Laboratory, TFR at the Center for Nuclear Research, Fontaney aux Rose, Cleo at the Culham Laboratory, Pulsator at the Max Planck Institute for Plasma Physics, and JFT-2 at the Japan Atomic Energy Research Institute. The Fourth International Conference was held in Madison, Wisconsin, in 1971.15 The main interest at Madison was naturally focused on the tokamak experiments. In T-4(vol.1, p.443), the electron temperature approached 3 keV at a con¯nement time around 10 ms. The ions were heated to around 600 eV by collision with the electrons. ST(vol.1, pp.451, 465) produced similar results. Trek to Large Tokamaks (since around oil crisis) Since then the IAEA conference has been held every two years; Tokyo in 1974,16 Berchtesgarden in 1976, Innsburg in 1978, Brussels in 1980, Baltimore in 1982, London in 1984, Kyoto in 1986, Nice in 1988, Washington D.C. in 1990, WÄ urzburg in 1992, Seville in 1994, Montreal in 1996, Yokohama in 1998, Sorrento in 2000 ¢ ¢ ¢. Tokamak research has made steady progress as the mainstream of magnetic con¯nement. Pease stated in his summary talk of the IAEA conference at Berchtesgarden in 1976 that \one can see the surprisingly steady progress that has been maintained. Furthermore, looked at logarithmically, we have now covered the greater part of the total distance. What remains is di±cult, but the di±culties are ¯nite and can be summed up by saying that we do not yet have an adequate understanding or control of cross-¯eld electron thermal conduction." After the tokamaks of the ¯rst generation (T-4, T-6, ST, ORMAK, Alcator A, C. TFR, Pulsator, DITE, FT, JFT-2, JFT-2a, JIPP T-II, etc.), second generation tokamaks (T-10, PLT, PDX, ISX-B, Doublet III, ASDEX, etc.) began appearing around 1976. The energy con¯nement time of ohmically heated plasmas was approximately described by the Alcator scaling law (¿E / na2 ). The value of n¿E reached 2 £ 1013 cm¡3 s in Alcator A in 1976. Heating experiments of neutral beam injection (NBI) in PLT achieved the ion temperature of 7keV in 1978, and the e®ective wave heating in an ion cyclotron range of frequency was demonstrated in TFR and PLT around 1980. The average ¯ value of 4.6 % was realized in the Doublet III non-circular tokamak (∙ = 1:4) in 1982 using 3.3 MW NBI. Noninductive drives for plasma current have been pursued. Current drive by the tangential injection of a neutral beam was proposed by Ohkawa in 1970 and was demonstrated in DITE experimentally in 1980. Current drive by a lower hybrid wave was proposed by Fisch in 1978 and demonstrated in JFT-2 in 1980 and in Versator 2, PLT, Alcator C, JIPP T-II, Wega, T-7 and so on. Ramp-up experiments of plasma current from 0 were succeeded by WT-2 and PLT in 1984. TRIAM-1M with superconducting toroidal coils sustained the plasma current of Ip = 22 kA; (ne ¼ 2 £ 1018 m3 ) during 70 minutes by LHW in 1990. The suppression of impurity ions by a divertor was demonstrated in JFT-2a (DIVA) in 1978 and was investigated by ASDEX and Doublet III in detail (1982). At that time the energy con¯nement time had deteriorated compared with the ohmic heating case as the heating power of NBI was increased (according to the Kaye-Goldston scaling law). However, the improved mode (named H mode) of the con¯nement time, increased by about 2 times compared with the ordinary mode (L mode), was found in the divertor con¯guration of ASDEX in 1982. The H mode was also observed in Doublet III, PDX, JFT-2M, and DIII-D. Thus much progress had been made to solve many critical issues of tokamaks. Based on these achievements, experiments of third-generation large tokamaks started, with TFTR (United States) in the end of 1982, JET (European Community) in 1983 and JT-60 (Japan) in 1985. Originally these large tokamaks are planned in early 1970s. TFTR achieved nDT (0)¿E » 1:2 £ 1019 m¡3 ¢ s, ∙Ti (0) = 44 keV by supershot (H mode-like). JET achieved

186

15 Development of Fusion Researches

Fig.14.1 Development of con¯nement experiments in n ¹ e ¿E - ∙Ti (0) (¹ ne is the line average electron density, ¿E is the energy con¯nement time ¿E ´ W=(Ptot ¡ dW=dt ¡ Lthr ), Ti (0) is the ion temperature). tokamak(²), stellarator (4), RFP(±), tandem mirror, mirror, theta pinch( closed triangle). Q = 1 is the critical condition. W ; total energy of plasma, Ptot ; total heating power, Lthr ; shine through of neutral beam heating.

nD (0)¿E » 3:2 £ 1019 m¡3 ¢ s, ∙Ti (0) = 18:6 keV by H mode with divertor con¯guration. JT-60 drove a plasma current of 1.7 MA (¹ ne = 0:3 £ 1013 cm¡3 ) by lower hybrid wave (PRF = 1:2 MW) in 1986 and upgraded to JT60U in 199121 . JT60U achieved nD (0)¿E » 3:4 £ 1019 m¡3 ¢ s, ∙Ti (0) = 45 keV by high ¯p H mode. A high performance con¯nement mode with negative magnetic shear was demonstrated in TFTR, DIII-D, JT60U, JET, Tore Supra22 , T10. JET performed a preliminary tritium injection experiment23 (nT =(nD + nT ) ' 0.11) in 1991 and the production of 1.7 MW (Q » 0:11) of fusion power using 15MW of NBI. Extensive deuterium-tritium experiment was carried out on TFTR in 1994.24 Fusion power of 9.3 MW (Q » 0:27) was obtained using 34 MW of NBI in supershot (Ip =2.5 MA). JET set records of DT fusion output of 16.1MW (Q » 0:62) using 25.7 MW of input power (22.3MW NBI + 3.1MW ICRF)25 in 1998. A pumped divertors were installed in JET, JT60U, DIIID, ASDEX-U and others in attempt to suppress impurity ions and the heat load on divertor plate. Now these large tokamaks are aiming at the scienti¯c demonstration of required conditions of critical issues ( plasma transport, steady operation, divertor and impurity control and so on) of fusion reactors. Based on the development of tokamak researches, the design activities of tokamak reactors have been carried out. The International Tokamak Reactor (INTOR)26 (1979-1987) and The International Thermonuclear Experimental Reactor (ITER)27 (1988-2001) are collaborative efforts among Euratom, Japan, The United States of America and Russian Federation under the auspices of IAEA. The status of ITER in 200028 is described in sec.16.11. Alternative Aproaches Potentail theoretical advantages of spherical tokamak was outlined by Peng and Strickler29 , in which the aspect rato A = R=a of standard tokamak is substantially reduces toward unity.

References

187

Predicted advantages include a naturally high elongation (∙s » 2), high toroidal beta and tokamak like con¯nement. These predictions have been veri¯ed experimentally, in particular by START device30 at Culham (R=a ¼ 0:3=0:28 = 1:31; Ip ¼ 0:25MA; Bt ¼ 0:15T). The toroidal beta reached 40% and observed con¯nement times follow similar scaling of standard tokamaks. Spherical tokamak (ST) experiments were also conducted by Globus-M (Io®e), Pegasus (Madison), TST (Tokyo), TR-3 (Tokyo). The next generation ST projects MAST (Culham) and NSTX (Princeton) started experiments in 1999 » 2000. Non-tokamak con¯nement systems have been investigated intensively to catch up with the achievements of tokamaks. The stellarator program proceeded from small-scale experiments (Wendelstein IIb, Clasp, Uragan-1, L-1, JIPP-I, Heliotron D) to middle-scale experiments (Wendelstein VIIA, Cleo, Uragan-2, L-2, JIPP T-II, Heliotron E). The plasmas with Te » Ti ¼ several hundred eV to 1 keV, ne » several 1013 cm¡3 were sustained by NBI heating without an ohmic heating current, and the possibility of steady-state operation of stellarators was demonstrated by WVIIA and Heliotron E. Scaling of con¯nement time of currentless plasma was studied in Heliotron E, CHS, ATF and WVII AS. Large helical device LHD started experiments in 1998 and advanced stellarator WVII-X is under construction. The reversed ¯eld pinch (RFP) con¯guration was found in the stable quiescent period of Zeta discharge just before the shutdown in 1968. J.B. Taylor pointed out that RFP con¯guration is the minimum energy state under the constraint of the conservation of magnetic helicity in 1974 (see sec.17.1). RFP experiments have been conducted in HBTX-1B, ETA-BETA 2, TPE-1RM, TPE-1RM15, TPE-1RM20, ZT-40M, OHTE, REPUTE-1, STP-3M, MST. An average ¯ of 1015% was realized. ZT-40M demonstrated that RFP con¯guration can be sustained by relaxation phenomena (the so-called dynamo e®ect) as long as the plasma current is sustained (1982). The next step projects RFX and TPE-RX are proceeding. Spheromak con¯gurations have been studied by S-1, CTX, and CTCC-1, and ¯eld reversed con¯gurations have been studied by FRX, TRX, LSX, NUCTE and PIACE. In mirror research, 2XIIB con¯ned a plasma with an ion temperature of 13 keV and n¿E £ 1011 cm¡3 s in 1976. However, the suppression of the end loss is absolutely necessary. The concept of a tandem mirror, in which end losses are suppressed by electrostatic potential, was proposed in 1976-1977 (sec 17.3). TMX, TMX-U and GAMMA 10 are typical tandem mirror projects. The bumpy torus is the toroidal linkage of many mirrors to avoid end loss and this method was pursued in EBT and NBT. Inertial con¯nement research has made great advances in the implosion experiment by using a Nd glass laser as the energy driver. Gekko XII (30 kJ, 1 ns, 12 beams), Nova (100 kJ, 1 ns, 10 beams), Omega X(4 kJ, 1 ns, 24 bemas), and Octal (2 kJ, 1 ns, 8 beams) investigated implosion using laser light of ¸ = 1:06 ¹m and its higher harmonics ¸ = 0:53 ¹m and 0.35 ¹m. It was shown that a short wavelength is favorable because of the better absorption and less preheating of the core. A high-density plasma, 200 » 600 times as dense as the solid state, was produced by laser implosion (1990). Based on Nova results, Lawrence Livermore National Laboratory is in preparation on National Ignition Facility (NIF31;32 ) (1.8MJ, 20ns, 0.35¹m, 192 beams, Nd glass laser system). Nuclear fusion research has been making steady progress through international collaboration and competition. A summary of the progress of magnetic con¯nement is given in ¯g.14.1 of the n ¹ e ¿E - Ti (0) diagram. TFTR demonstrated Q » 0:27 DT experiments and JET demonstrated Q » 0:62 DT experiments. JET and JT60U achieved equivalent break-even condition by D-D plasma, that is, the extrapolated D-T fusion power output would be the same as the heating input power (Qequiv =1). References 1. A. S. Bishop: Project Sherwood, Addison Wesley, Reading Mass. (1958)

188

2. 3. 4. 5. 6.

15 Development of Fusion Researches

R. S. Pease: Plasma Phys. and Controlled Fusion 28, 397 (1986) L. A. Artsimovich: Sov. Phys. Uspekhi 91, 117 (1967) J. D. Lawson: Proc. Phys. Soc. 70B, 6 (1957) M. D. Kruskal and M. Schwarzschild: Proc. Roy. Soc. A223, 348 (1954) M. N. Rosenbluth and C. L. Longmire: Ann. Phys. 1, 120 (1957) I. B. Berstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud: Proc. Roy. Soc. A244, 17 (1958) 7. Nature 181, No. 4604, p. 217, Jan. 25 (1958) 8. L. Spitzer, Jr.: Phys. Fluids 1, 253 (1958) 9. Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy in Geneva Sep. 1{13 (1958), Vol.31, Theoretical and Experimental Aspects of Controlled Nuclear Fusion, Vol.32, Controlled Fusion Devices, United Nations Publication, Geneva (1958) 10. The Second Geneva Series on the Peaceful Uses of Atomic Energy (editor of the series by J. G. Beckerley, Nuclear Fusion) D. Van Nostrand Co. Inc., New York (1960) 11. S.Hayakawa and K.Kimura: Kakuyugo Kenkyu 57, 201, 271 (1987) in Japanese 12. Plasma Physics and Controlled Nuclear Fusion Research (Conference Proceeding, Salzburg, Sep. 4{9 1961) Nucl. Fusion Suppl. (1962) (Translation of Russian Papers: U.S. Atomic Energy Commission, Division of Technical Information O±ce of Tech. Service, Depart. of Commerce, Washington D.C.(1963)) 13. ibid: (Conference Proceedings, Culham, Sep. 6{10 1965) International Atomic Energy Agency, ( Vienna (1966) Translation of Russian Paper:U.S. Atomic Energy Commission, Division of Technical Information Oak Ridge, Tenn. (1966)) 14. ibid: (Conference Proceedings, Novosibirsk, Aug.1{7 1968) International Atomic Energy Agency, Vienna (1969)(Translation of Russian Paper: Nucl. Fusion Suppl. (1969)) 15. ibid: (Conference Proceedings, Madison, June 17{23 1971) International Atomic Energy Agency, Vienna(1971) (Translation of Russian Paper: Nucl. Fusion Suppl.(1972)) 16. ibid: (Conference Proceedings, Tokyo. Nov. 11{15 1974) International Atomic Energy Agency, Vienna (1975) 17. M. N. Rosenbluth and R. F. Post: Phys. Fluids 8, 547 (1965) 18. E. G. Harris: Phys. Rev. Lett. 2, 34 (1959) 19. A. B. Mikhailovskii and L. I. Rudakov: Sov. Phys. JETP 17, 621 (1963) N. A. Krall and M. N. Rosenbluth: Phys. Fluids 8, 1488 (1965) 20. M. J. Forrest, N. J. Peacock, D. C. Robinson, V. V. Sannikov and P. D. Wilcock: Culham Report CLM{R 107, July (1970) 21. JT60U Team: Plasma Physics and Controlled Nucl. Fusion Research 1, 31, (1995) (Conference Proceedings, Seville in 1994) IAEA, Vienna 22. O1-2, O1-6, O1-3, A5-5, O2-2 in 16th IAEA Fusion Energy Conference (Montreal in 1996) 1, (1997) IAEA Vienna 23. JET Team: Nucl. Fusion 32, 187 (1992) 24. TFTR Team: Plasma Physics and Controlled Nucl. Fusion Research 1, 11, (1995) (Conference Proceedings, Seville in 1994) IAEA, Vienna 25.JET Team: 17th IAEA Fusion Energy Conference (Yokohama in 1998) 1, 29 (1999) IAEA Vienna 26. INTOR Team: Nucl. Fusion 23, 1513, (1983) 27. ITER PhysicsBasis: Nucl. Fusion 39 No.12 pp2137-2638 (1999) 28. ITER-FEAT: Technical Basis for the ITER-FEAT Outline Design (Dec 1999) IAEA Vienna 29. Y-K M. Peng, D. J. Stricker: Nucl.Fusion 26, 769 (1986) 30. A. Sykes: 17th IEA Fusion Energy Conference (Yokohama in 1998) 1, 129 (1999) IAEA Vienna 31. J. D. Lindl, M. M. Marinak: 16th IEA Fusion Energy Conference (Monreal in 1996) 3, 43 (1997) IAEA Vienna 32. J. D. Lindl: Inertial Con¯nement Fusion AIP Press, Springer 1998

16.1 Tokamak Devices

189

Ch.16 Tokamak The word \tokamak" is said to be a contraction of the Russian words for current (ток), vessel (камер), magnet (магнит), and coil (катущка). Tokamaks are axisymmetric, with the plasma current itself giving rise to the poloidal ¯eld essential to the equilibrium of toroidal plasmas. In a tokamak the toroidal ¯eld used to stabilize against MHD instabilities, is strong enough to satisfy the Kruskal-Shafranov condition. This characteristics is quite di®erent from that of reversed ¯eld pinch, with its relatively weak toroidal ¯eld. There are excellent reviews and textbooks of tokamak experiments,1;2 equilibrium,3 and diagnostics.4;5 16.1 Tokamak Devices The structure of the devices of large tokamaks JET, JT60U and TFTR are shown in ¯gs.16.1, 16.2 and 16.3 as typical examples. The toroidal ¯eld coils, equilibrium ¯eld coils (also called the poloidal ¯eld coils, which produce the vertical ¯eld and shaping ¯eld), ohmic heating coils (the primary windings of the current transformer), and vacuum vessel can be seen in the ¯gures. Sometimes \poloidal ¯eld coils" means both the equilibrium ¯eld coils and the ohmic heating coils. By raising the current of the primary windings of the current transformer (ohmic heating coils), a current is induced in the plasma, which acts as the secondary winding. In the JET device, the current transformer is of the iron core type. The air-core type of current tranformer is utilized in JT60U and TFTR. The vacuum vessel is usually made of thin stainless steel or inconel so that it has enough electric resistance in the toroidal direction. Therefore the voltage induced by the primary windings can penetrate it. The thin vacuum vessel is called the linear. Before starting an experiment, the liner is outgassed by baking at a temperature of 150-400 ℃ for a long time under high vacuum. Table 16.1 Parameters of tokamaks. R; a; b in m, Bt in T, and Ip in MA. T-4 T-10 PLT TFTR JET JT60U

R 1.0 1.5 1.32 2.48 2.96 3.4

a(£b) 0.17 0.39 0.4 0.85 1.25(£2.1) 1.1(£1.4)

R=a 5.9 3.8 3.3 2.9 2.4 3.1

Bt 5.0 5.0 3.2 5.2 3.45 4.2

Ip 0.3 0.65 0.5 2.5   7 6

Remarks

compact noncircular JT60 upgraded

Furthermore, before running an experiment, a plasma is run with a weak toroidal ¯eld in order to discharge-clean the wall of the liner. Inside the liner there is a diaphragm made of tungsten, molybdenum, or graphite that limits the plasma size and minimizes the interaction of the plasma with the wall. This diaphragm is called a limiter. Recently a divertor con¯guration was introduced instead of the limiter. In this case the magnetic surface, including the separatrix point, determines the plasma boundary (see sec.16.5). A conducting shell surrounds the plasma outside the liner and is used to maintain the positional equilibrium or to stabilize MHD instabilities during the skin time scale. The magnitude of the vertical ¯eld is feedback controlled to keep the plasma at the center of the liner always. Many improvements have been made in tokamak devices over the years. Accuracy of the magnetic ¯eld is also important to improve the plasma performance in tokamak and other toroidal devices. The parameters of typical tokamak devices are listed in table 16.1. Measurements by magnetic probes are a simple and useful way to monitor plasma behavior. Loop voltage VL and the plasma current Ip can be measured by the magnetic loop and Rogowsky coil, respectively.4 Then the electron temperature can be estimated by the Spitzer formula from

190

16 Tokamak

Fig.16.1 Artist's drawing of JET (Joint European torus), JET Joint Undertaking, Abingdon, Oxfordshire, England. The toroidal ¯eld coils (TFC) are arranged around the vacuum vessel(VV). The outer poloidal ¯eld coils (Outer PFC, equilibrium ¯eld coils) and inner poloidal ¯eld coils (Inner PFC, ohmic heating coils) are wound in the toroidal direction outside the toroidal ¯eld coils(TFC). JET uses an ion-core current transformer (TC). The mechanical structures (MS) support the toroidal ¯eld coils against the large amount of torque due to the equilibrium ¯eld. Reprinted with permission from JET Joint Undertaking.

16.1 Tokamak Devices

Fig.16.2 A birdview of JT60U, Japan Atomic Energy Research Institute.

191

192

16 Tokamak

Fig.16.3 A birdview of TFTR( Tokamak Fusion Test Reactor), Plasma Physics Laboratory, Princeton University.

16.1 Tokamak Devices

193

Fig.16.4 (a) Locations of magnetic probes around plasma (¢ shown in the ¯gure is minus) (b) an array of soft X ray solid-state detectors. Each detector's main contribution to a signal comes from the emission at the peak temperature along the line of sight of the detector. the °uctuation of the electron temperature at this point can be detected.

the resistivity of the plasma, which can be evaluated using VL and Ip . From eq.(6.18), the poloidal beta ratio ¯p is given by ¯p = 1 +

2B' hB'V ¡ B' i B!2

(16:1)

where jB'V ¡ B' j ¿ jB' j and B! = ¹0 Ip =2¼a. Since the diamagnetic °ux ±© is ±© = ¼a2 hB'V ¡ B' i

we have ¯p =

p 8¼B' = 1 + 2 2 ±©: B!2 =2¹0 ¹ 0 Ip

(16:2)

Therefore measurement of the diamagnetic °ux ±© yields ¯p and the plasma pressure. Magnetic probes g1 ; g2 located around the plasma, as shown in ¯g.16.4a, can be used to determine the plasma position. Since the necessary magnitude of the vertical ¯eld for the equilibrium B? is related to the quantity ¤ = ¯p + li =2 , the value of ¤ can be estimated from B? (li is the normalized internal inductance). The °uctuations in the soft X-ray (bremsstrahlung) signal follow the °uctuations in electron temperature. The °uctuations occur at the rational surfaces ( qs (r) = 1; 2; :::). The mode number and the direction of the propagation can be estimated by arrays of solid-state detectors, as shown in ¯g.16.4b. When the positions of the rational surfaces can be measured, the radial current pro¯le can be estimated for use in studies of MHD stability.

16.2 Equilibrium The solution of the Grad-Shafranov equation (6.15) for the equilibrium gives the magnetic surface function ª = rA' (A' is the ' component of the vector potential). Then the magnetic ¯eld B is described by @Ã ; @z and the current density j is rBr = ¡

j=

rBz =

@Ã @r

I0 B + rp0 e' 2¼

where 0 means @=@Ã. When the plasma cross section is circular, the magnetic surface Ã(½; !) outside the plasma is given by eq.(6.25) as follows: ¹ 0 Ip Ã(½; !) = 2¼

µ

¶

µ

µ

¹ 0 Ip ½ 1 8R ¡2 ¡ ln ln + ¤ + ½ a 4¼ 2

¶Ã

a2 1¡ 2 ½

!¶

½ cos !:

(16:3)

194

16 Tokamak

Fig.16.5 Positions of plasma boundary and shell

The plasma boundary is given by ½ = a, that is, by Ã(½; !) =

µ

¶

¹ 0 Ip R 8R ¡2 : ln a 2¼

(16:4)

16.2a Case with Conducting Shell When a conducting shell of radius b surrounds the plasma, the magnetic surface à must be constant at the conducting shell. Therefore the location of the shell is given by µ

¶

¹ 0 Ip R 8R Ã(½; !) = ¡2 : ln b 2¼ (In practice, the position of the shell is ¯xed, and the plasma settles down to the appropriate equilibrium position; the important point is their relative position). When the magnetic surface is expressed by Ã(½; !) = Ã0 (½) + Ã1 cos ! the magnetic surface is a circle with the shifted center by the amount of ¢ = ¡Ã1 =Ã00 . Therefore the plasma center is displaced from the center of the shell by an amount ¢0 given by (see ¯g. 16.5. ½0 = ½ ¡ ¢ cos !; ©0 (½0 ) = ©0 (½) ¡ (@©0 =@½)¢ cos !) Ã

µ

b2 b 1 ¢0 (b) = ¡ ln + ¤ + a 2R 2

¶Ã

a2 1¡ 2 b

!!

:

(¢0 < 0 means that the plasma center is outside the center of the shell.) 16.2b Case without Conducting Shell If the vertical ¯eld B? is uniform in space, the equilibrium is neutral with regard to changes to plasma position. When the lines of the vertical ¯eld are curved, as shown in ¯g.16.6, the plasma position is stable with regard to up and down motion. The z component Fz of the magnetic force applied to a plasma current ring with mass M is Fz = ¡2¼RIp BR : From the relation (@BR =@z) ¡ (@Bz =@R) = 0;

µ

R @Bz @BR M zÄ = ¡2¼RIp z = 2¼Ip Bz ¡ @z Bz @R

¶

z:

As Ip Bz < 0, the stability condition for decay index n is n´¡

R @Bz > 0: Bz @R

(16:5)

16.2 Equilibrium

195

Fig.16.6 Vertical ¯eld for plasma equilibrium

The horizontal component FR of the magnetic force is M

d2 (¢R) = FR = 2¼RIp (Bz ¡ B? )¢R: dt2

The amount of B? necessary for plasma equilibrium (see eq.(6.28)) is B? =

µ

¶

¡¹0 Ip 1 8R ; ln +¤¡ a 4¼R 2

¤=

li + ¯p ¡ 1: 2

When the plasma is ideally conductive, the magnetic °ux inside the plasma ring is conserved and @ (Lp Ip ) + 2¼RB? = 0: @R Here the self-inductance is Lp = ¹0 R(ln(8R=a) + li =2 ¡ 2): Therefore the equation of motion is M

µ

¶

3 d2 (¢R) ¡ n ¢R = 2¼Ip B? dt2 2

under the assumption ln(8R=a) À 1. Then the stability condition for horizontal movement is 3 > n: 2

(16:6)

16.2c Equilibrium Beta Limit of Tokamaks with Elongated Plasma Cross Sections The poloidal beta limit of a circular tokamak is given by ¯p = 0:5R=a, as was given by eq.(6.38). The same poloidal beta limit is derived by similar consideration for the elongated tokamak with horizontal radius a and vertical radius b. When the length of circumference along the poloidal direction is denoted by 2¼aK for the elongated plasma and the average of poloidal ¹p =Bt = aK=(RqI ), ¹p = ¹0 Ip =(2¼aK), the ratio of the poloidal and toroidal ¯eld is B ¯eld is B 2 1=2 where K is approximately given by K = [(1 + (b=a) )=2] . Therefore the beta limit of an elongated tokamak is ¯ = ¯p (aK=RqI )2 = 0:5K 2 a=(RqI2 ) and is K 2 times as large as that of circular one. In order to make the plasma cross section elongated, the decay index n of the vertical ¯eld must be negative, and the elongated plasma is positionally unstable in the updown motion. Therefore feedback control of the variable horizontal ¯eld is necessary to keep the plasma position vertically stable.6 16.3 MHD Stability and Density Limit A possible MHD instability in the low-beta tokamak is kink modes, which were treated in sec.8.3. Kink modes can be stabilized by tailoring the current pro¯le and by appropriate choice of the safety factor qa . When the plasma pressure is increased, the beta value is limited by the ballooning modes (sec.8.5). This instability is a mode localized in the bad curvature region

196

16 Tokamak

Fig.16.7 Magnetic islands of m = 1; m = 3=2; m = 2 appears at q(r)=1,3/2,2.

Fig.16.8 The hot core in the center is expelled by the reconnection of magnetic surfaces

driven by a pressure gradient. The beta limit of ballooning mode is given by ¯max » 0:28(a=Rqa ) of eq.(8.124). The ¯ limit by kink and ballooning modes depends on the radial pro¯le of the plasma current (shear) and the shape of the plasma cross section. The limit of the average beta, ¯c = hpi=(B 2 =2¹0 ), of the optimized condition is derived by MHD simulation codes to be ¯c (%) = ¯N Ip (MA)=a(m)Bt (T) (¯N = 2 » 3:5):7;8 ¯max of eq.(8.124) is consistent with the result of MHD simulation. Even if a plasma is ideally MHD stable, tearing modes can be unstable for a ¯nite resistive plasma. When ¢0 is positive at the rational surfaces (see sec.9.1) in which the safety factor q(r) is rational q(r) = 1; 3=2; 2, tearing modes grow and magnetic islands are formed, as shown in ¯g.16.7. When the pro¯le of the plasma current is peaked, then the safety factor at the center becomes q(0) < 1 and the tearing mode with m = 1; n = 1 grows at the rational surface q(r) = 1, and the hot core of the plasma is pushed out when the reconnection of magnetic surfaces occurs and the current pro¯le is °attened (¯g.16.8). The thermal energy in the central hot core is lost in this way.9;2 Since the electron temperature in the central part is higher than in the outer region and the resistance in the central part is smaller, the current pro¯le is peaked again and the same process is repeated. This type of phenomenon is called internal disruption or minor disruption . The stable operational region of a tokamak with plasma current Ip and density ne is limited.

16.4 Beta Limit of Elongated Plasma

197

With Greenward normalized density or Greenward-Hugill-Murakami parameter, de¯ned by NGHM ´

n20 Ip (MA)=¼a(m)2

(16:7)

an empirical scaling NGHM < 1

(16:8)

is hold for most of tokamak experiments10 , where n20 is the electron density in the unit of 1020 m¡3 . NGHM is expressed by the other form (refer the relations in sec.16.4) NGHM =

n20 0:628 qI : K 2 Bt (T)=R(m)

(16:70 )

The upper limit of the electron density depends critically on the plasma wall interaction and tends to increase as the heating power increases, although the scaling NGHM < 1 does not re°ect the power dependence. When hydrogen ice pellets are injected into a plasma for fueling from high ¯eld side of ASDEX-U with advanced divertor11 , NGHM becomes up to » 1.5. Therefore there is possibility to increase NGHM furthermore. The safety factor qa at the plasma boundary is qa > 3 in most cases. Beyond the stable region (NGHM < 1; 1=qa < 1=2 » 1=3), strong instability, called disruptive instability, occurs in usual operations. Negative spikes appear in the loop voltage due to the rapid expansion of the current channel (°attened current pro¯le), that is, the rapid reduction of the internal inductance. The thermal energy of the plasma is lost suddenly. The electron temperature drops rapidly, and the plasma resistance increases. A positive pulse appears in the loop voltage. Then the plasma discharge is terminated rapidly. In some cases, the time scale of disruption is much faster than the time scale (9.27) predicted by the resistive tearing mode. For possible mechanisms of the disruptive instability, overlapping of the magnetic islands of m = 2=n = 1 (q(r) = 2) and m = 3=n = 2 (q(r) = 1:5) or the reconnection of m = 2=n = 1; m = 1=n = 1 magnetic islands are being discussed. Reviews of the MHD instabilities of tokamak plasmas and plasma transport are given in ref.12-15. 16.4 Beta Limit of Elongated Plasma The output power density of nuclear fusion is proportional to n2 h¾vi: Since h¾vi is proportional to Ti2 in the region near Ti » 10 keV, the fusion output power is proportional to the square of plasma pressure p = n∙T . Therefore the higher the beta ratio ¯ = p=(B 2 =2¹0 ), the more economical the possible fusion reactor. The average beta of h¯i » 3% was realized by NBI experiments in ISX-B, JET-2, and PLT. All these tokamaks have a circular plasma cross section. The theoretical upper limit of the average beta ¯c of an elongated tokamak plasma due to kink and ballooning instability is7;8 ¯c (%) ' ¯N Ip (MA)=a(m)Bt (T)

(16:9)

¯N is called Troyon factor or normalized beta (¯N = 2 » 3:5). When the following de¯nitions ¹p ´ ¹0 Ip ; B 2¼aK

qI ´ K

a Bt ¹p RB

(16:10)

are used, the critical beta is reduced to ¯c (%) = 5¯N K 2

a RqI

(16:90 )

where 2¼Ka is the the length of circumference of the plasma boundary and K is approximately given by K 2 ' (1 + ∙2s )=2

198

16 Tokamak

Fig.16.9 The observed beta versus I=aB for DIII-D. Various ¯ limit calculations are summarized in the curves with di®erent assumptions on the location of a conducting wall (rw =a). (After DIII-D team: Plasma Phys. Controlled Nuc. Fusion Research 1, 69, (1991) IAEA. ref.17)

and ∙s is the ratio of the vertical radius b to the horizontal radius a. The safety factor qà at a magnetic surface à is given by qà =

1 2¼

I

1 = 2¼dÃ

Bt 1 dl = RBp 2¼dà I

Bt dsdl =

I

Bt

dà dl RBp

1 d© 2¼ dÃ

where © is the toroidal °ux through the magnetic surface Ã. It must be noti¯ed that qI is di®erent from qà in the ¯nite aspect ratio. As the approximate formula of the e®ective safety factor at the plasma boundary, qe®

a2 B 1 + ∙2s = (¹0 =2¼)RI 2

Ã

1+²

2

Ã

¹2 ¤ 1+ 2

!!

³

£ 1:24 ¡ 0:54∙s + 0:3(∙2s + ± 2 ) + 0:13±

´

(16:11)

¹ ´ is used (including the divertor con¯guration (sec.16.5))16 . The notations are ² = a=R, ¤ ¯p + li =2 (see (6.21)) and ± = ¢=a is triangularity of the plasma shape (see ¯g.16.10). In non-circular tokamak DIII-D, h¯i=11% was realized in 199017 , in which a=0.45 m,Bt =0.75 T, Ip =1.29 MA, Ip =aBt =3.1 MA/Tm, g »3.6, ∙s =2.35 and R=1.43m. Fig.16.9 shows the experimental data of DIII-D on the observed beta versus Ip =aBt . 16.5 Impurity Control, Scrape-O® Layer and Divertor Radiation loss power Pbrems by bremsstrahlung due to electron collision with ion per unit volume is Pbrems = 1:5 £ 10¡38 Ze® n2e (∙Te =e)1=2 :

(W=m3 )

The loss time due to bremsstrahlung de¯ned by ¿brems = (3=2)ne ∙Te =Pbrems is ¿brems

1 = 0:16 Ze® n20

µ

∙Te e

¶1=2

(sec)

16.5 Impurity Control, Scrape-O® Layer and Divertor

199

Fig.16.10 Divertor con¯guration using separatrix S of the magnetic surface (lefthand side). De¯nition of the triangularity ± = ¢=a (righthand side)

where n20 is in units of 1020 m¡3 ∙Te =e is in unit of eV . When ne » 1020 m¡3 ; and ∙Te » 10 keV, then we have ¿brems » 8=Ze® (s). Therefore if the radiation losses such as bremsstrahlung, recombination radiation and line spectre emission are enhanced much by impurity ions, fusion core plasma can not be realized even by the radiation losses only. When the temperature of the plasma increases, the ions from the plasma hit the walls of the vacuum vessel and impurity ions are sputtered. When the sputtered impurities penetrate the plasma, the impurities are highly ionized and yield a large amount of radiation loss, which causes radiation cooling of the plasma. Therefore impurity control is one of the most important subjects in fusion research. The light impurities, such as C and O, can be removed by baking and discharge-cleaning of the vacuum vessel. The sputtering of heavy atoms (Fe, etc.) of the wall material itself can be avoided by covering the metal wall by carbon tiles. Furthermore a divertor, as shown in ¯g.16.10, is very e®ective to reduce the plasma-wall interaction. Plasmas in Scrap-o® layer °ow at the velocity of sound along the lines of magnetic force just outside the separatrix S into the neutralized plates, where the plasmas are neutralized. Even if the material of the neutralized plates is sputtered, the atoms are ionized within the divertor regions near the neutralized plates. Since the thermal velocity of the heavy ions is much smaller than the °ow velocity of the plasma (which is the same as the thermal velocity of hydrogen ions), they are unlikely to °ow back into the main plasma. In the divertor region the electron temperature of the plasma becomes low because of impurity radiation cooling. Because of pressure equilibrium along the lines of magnetic force, the density in the divertor region near the neutralized plates becomes high. Therefore the velocity of ions from the plasma into the neutralized plates is collisionally damped and sputtering is suppressed. A decrease in the impurity radiation in the main plasma can be observed by using a divertor con¯guration. However the scrape o® layer of divertor is not broad and most of the total energy loss is concentrated to the narrow region of the target divertor plate. The severe heat load to the divertor plate is one of the most critical issues for a reactor design. Physical processes in scrapeo® layer and divertor region are actively investigated experimentally and theoretically18 . Let us consider the thermal transport in scrape-o® layer. It is assumed that the thermal transport parallel to the magnetic line of force is dominated by classical electron thermal conduction and the thermal transport perpendicular to the magnetic ¯eld is anomalous thermal di®usion. We use a slab model as is shown in ¯g.16.11 and omit Boltzmann constant in front of temperature. Then we have rqk + rq? + Qrad = 0

(16:12) 7=2

qk = ¡∙c

@Te @Te 2 @Te = ¡∙0 Te5=2 = ¡ ∙0 @s @s @s 7

(16:13)

200

16 Tokamak

Fig.16.11 Con¯guration of scrape-o® layer (SOL) and divertor. The coordinate of the slab model (right-hand side) µ

q? = ¡n Âe?

@Te @Ti + Âi? @r @r

¶

@n 3 ¡ D(Te + Ti ) @r 2

(16:14)

∙c » n¸2ei ºei = 2:8 £ 103 m¡1 W (eV)¡7=2 Te5=2 (eV)5=2 : Here qk and q? are heat °uxes in the directions of parallel and perpendicular to the magnetic ¯eld and Qrad is radiation loss. ∙c is heat conductivity and Âe? ; Âi? are thermal difusion coe±cients and D is di®usion coe±cient of particles. The stagnation point of heat °ow is set as s = 0 and the X point of separatrix and divertor plate are set as s = Lx and s = LD respectively. Then the boundary condition at s = 0 and s = LD are qk0 = 0

(16:15)

1 qkD = °TD nD dD + mi u2D nD uD + »nD uD 2 = nD MD cs ((° + MD2 )TD + »)

(16:16)

where uD is °ow velocity of plasma at the divertor plate and MD is Mach number MD = uD =cs . ° ¼ 7 is sheath energy transfer coe±cient and » ¼ 20 » 27eV is ionization energy. The sound 1=2 velocity is cs = c~s TD , c~s = 0:98(2=Ai )1=2 104 ms¡1 (eV)¡1=2 , Ai being ion atomic mass. The ¯rst and the second terms of eq.(16.16) are the power °ux into the sheath and the third term is power consumed within the recycling process. The equations of particles and momentum along the magnetic lines of force are @(nu) = Si ¡ Scx;r ¡ r? (nu? ) ¼ Si ¡ Scx;r @s mnu

@p @u ¡ muSm =¡ @s @s

(16:17)

(16:18)

where Sm = nn0 h¾vim is the loss of momentum of plasma °ow by collision with neutrals, Si = nn0 h¾vii is the ionization term and Scx;r = nn0 h¾vicx;r is energy loss by charge exchange and radiation recombination. Eqs.(16.17) and (16.18) reduce to @(nmu2 + p) = ¡mu(Sm + Scx;r ) + muSi @s

(16:19)

16.5 Impurity Control, Scrape-O® Layer and Divertor

201

The °ow velocities at s = 0 and s = LD are u0 = 0 and uD = MD cs ; MD ¼ 1 respectively. Eqs.(16.12),(16.13) and the boundary conditions (16.15),(16.16) reduce to 2∙0 @ 2 7=2 T = r? q? + Qrad 7 @s2 e 2∙0 7=2 7=2 (Te (s) ¡ TeD ) = 7

Z

s

(16:20)

ds0

LD

Z

s0 0

(r? q? + Qrad )ds00 :

(16:21)

When r? q? =const. Qrad = 0 in 0 < s < Lx and r? q = 0, Qrad =const. in Lx < s < LD , we have 2∙0 7=2 7=2 (Te (s)¡TeD ) = 0:5(¡r? q? )(2Lx LD ¡L2x ¡s2 )+0:5Qrad (LD ¡Lx )2 7

(0 < s < Lx ):

When radiation term is negligible, Te0 ´ Te (0) becomes 7=2 Te0

=

7=2 TeD

7 + 4∙0

µ

¶

2LD ¡ 1 (¡r? q? )L2x : Lx

If TeD < 0:5Te0 and LD ¡ Lx ¿ Lx , we have Te0

Ã

(¡r? q? )L2x ¼ 1:17 ∙0

!2=7

= 1:17

Ã

q? L2x ∙0 ¸q

!2=7

(16:22)

where 1=¸q ´ ¡r? q? =q? . When the scale lengths of gradients of temperature and density are ¸T and ¸n respectively (T (r) = T exp(¡r=¸T ); n(r) = n exp(¡r=¸n )) and Âi? ¿ Âe? andD » Âe? are assumed, eq.(16.14) becomes q? =

nÂe?

Te ¸T

µ

Ti ¸T 3 1 + (1 + ) Te ¸n 2

¶

:

(16:23)

Therefore if Âe is known as a function Âe (Te ; n; B), ¸T is given as ¸T (Te ; n; B; q? ). Let us consider the relations between ns ; Tes ; Tis at stagnation point s = 0 and nD ; TD at divertor plate s = LD . The momentum °ux at divertor region decreases due to collision with neutrals, charge exchange and ionization and becomes smaller than that at stagnation point. fp =

2(1 + n2D )nD TD < 1: ns (Tes + Tis )

(16:24)

The power °ux to divertor plate is reduced by radiation loss from the power °ux q? Lx into scrape-o® layer through the separatrix with length of Lx Z

0

1

qk dr = (1 ¡ frad )q? Lx

(16:25)

where frad is the fraction of radiation loss. Eqs.(16.25) and(16.16) reduce 1=2 MD nD c~s TD

Ã

» (° + MD2 )TD + 3=2¸T + 1=¸n 1=(2¸T ) + 1=¸n

!

= (1 ¡ frad )q? Lx

that is (1 ¡ frad )q? Lx =

Tes + Tis c~s fp ¸T ns G(TD ) 1:5 + ¸T =¸n 2

(16:26)

202

16 Tokamak

Fig.16.12 Dependence of G(TD )(eV)1=2 on TD (eV)

Ã

»¹ MD 1 1=2 G(TD ) ´ M (° + )T 1 + D D ° + MD TD 1 + MD2

!

:

(16:27)

The curve of G(TD ) as the function of TD is shown in ¯g.16.12 and G(TD ) has a minimum at ¹ TD = »=(° + MD2 ). In the case of MD ¼ 1; ° ¼ 7; » = 24eV, G(TD ) is GD =

1=2 4TD

µ

¶

4:5 : 1+ TD 1=2

G(TD ) is roughly proportional to TD when TD > 15eV in this case. Since Tes depends on ns ¡2=7 through ¸q as is seen in eq.(16.22), the dependence of Tes on ns is very weak. From (16.26) and (16.24), we have roughly following relations TD / n¡2 s

nD / n3s

(16:28)

and the density nD at divertor increases nonlinearly with the density ns of upstream scrape-o® layer. When the upstream density ns increases while keeping the left-hand side of eq.(16.26) constant, the solution TD of (16.26) can not exists beyond a threshold density, since G(TD ) has the minimum value (¯g.16.12). This is related to the phenomenon of detached plasma above a threshold of upstream density18 . The heat load ÁD of divertor normal to the magnetic °ux surface is given by µ

¸T a (1 ¡ frad )Psep ÁD ¼ = (1 ¡ frad )¼K q? 1:5 + ¸T ¸n 2¼R2¸ÁD

¶

Bµ BµD

(16:29)

where Psep is the total power °ux across the separatrix surface and ¸ÁD is the radial width of heat °ux at divertor plate Bµ 1 Psep = 2¼aK2¼Rq? ¸ÁD = ¸T : 1:5 + ¸T =¸n BµD The term Bµ =BµD = 2 » 3 is the ratio of separations of magnetic °ux surfaces at stagnation point and divertor plate. If the divertor plate is inclined with angle ® against the magnetic °ux surface, the heat load of the inclined divertor plate becomes sin ® times as small as that of the divertor normal to magnetic °ux surface. 16.6 Con¯nement Scaling of L Mode The energy °ow of ions and electrons inside the plasma is schematically shown in ¯g.16.13. Denote the heating power into the electrons per unit volume by Phe and the radiation loss and

16.6 Con¯nement Scaling of L Mode

203

Fig.16.13 Energy °ow of ions and electrons in a plasma. Bold arrows, thermal conduction (Â). Light arrows, convective loss (D). Dashed arrow, radiation loss (R). Dot-dashed arrows, charge exchange loss (CX).

the energy relaxation of electrons with ions by R and Pei , respectively; then the time derivative of the electron thermal energy per unit volume is given by d dt

µ

¶

µ

¶

@∙Te @ne 3 3 1 @ : r Âe ne ∙Te = Phe ¡ R ¡ Pei + + De ∙Te r @r @r @r 2 2

where Âe is the electron thermal conductivity and De is the electron di®usion coe±cient. Concerning the ions, the same relation is derived, but instead of the radiation loss the charge exchange loss Lex of ions with neutrals must be taken into account, and then d dt

µ

¶

µ

¶

@∙Ti @ni 3 3 1 @ : r Âi ni ∙Ti = Phi ¡ Lcx + Pei + + Di ∙Ti r @r @r @r 2 2

The experimental results of heating by ohmic one and neutral beam injection can be explained by classical processes. The e±ciency of wave heating can be estimated fairly accurately by theoretical analysis. The radiation and the charge exchange loss are classical processes. In order to evaluate the energy balance of the plasma experimentally, it is necessary to measure the fundamental quantities ne (r; t); Ti (r; t); Te (r; t), and others.4 According to the many experimental results, the energy relaxation between ions and electrons is classical, and the observed ion thermal conductivities in some cases are around 2 » 3 times the neoclassical thermal conductivity; Âi;nc = ni f (q; ")q2 (½­i )2 ºii : ¡3=2

(f = 1 in the P¯rsch-SchlÄ uter region and f = ²t in the egion) and the observed ion thermal conductivities in some other cases are anomalous. The electron thermal conduction estimated by the experimental results is always anomalous and is much larger than the neoclassical one (more than one order of magnitude larger). In most cases the energy con¯nement time of the plasma is determined mostly by electron thermal conduction loss. The energy con¯nement times ¿E is de¯ned by ¿E ´

R

(3=2)(ne ∙Te + ni ∙Ti )dV : Pin

in steady case. The energy con¯nement time ¿OH of an ohmically heated plasma is well described by Alcator (neo-Alcator) scaling as follows (units are 1020 m¡3 , m): ¿OH (s) = 0:103q 0:5 n ¹ e20 a1:04 R2:04 :

204

16 Tokamak

Fig.16.14 Comparison of con¯nement scaling ¿EITER¡P with experimental data of energy con¯nement time ¿EEXP of L mode. (After Yushimanov et al: Nucl. Fusion 30, 1999, (1990). ref.20)

However, the linearity of ¿OH on the average electron density n ¹ e deviates in the high-density region ne > 2:5£ 1020 m¡3 and ¿OH tends to saturate. When the plasma is heated by high-power NBI or wave heating, the energy con¯nement time degrades as the heating power increases. Kaye and Goldston examined many experimental results of NBI heated plasma and derived so-called Kaye-Goldston scaling on the energy con¯nement time,19 that is, 2 2 ¿E = (1=¿OH + 1=¿AUX )¡1=2 ¡0:5 ¡0:37 1:75 ¿AUX (s) = 0:037∙0:5 R s Ip Ptot a

where units are MA, MW, m and ∙s is the elongation ratio of noncircularity and Ptot is the total heating power in MW. ITER team assembled data from larger and more recent experiments. Analysis of the data base of L mode experiments (see next section) led to the proposal of following ITER-P scaling20 0:2 ¿EITER¡P (s) = 0:048Ip0:85 R1:2 a0:3 n ¹ 0:1 (Ai ∙s =P )1=2 20 B

(16:31)

where unite are MA, m, T, MW and the unit of n ¹ 20 is 1020 m¡3 . P is the heating power corrected for radiation PR (P = Ptot ¡ PR ). A comparison of con¯nement scaling ¿EITER¡P with the experimental data of L mode is presented in ¯g.16.14. For burning plasmas, heating power is roughly equal to ® particle fusion output power P® ¼ 0:04n2DT20 T 2 Aa3 ∙s (MW, 1020 m¡3 , keV, m) at around T » 10keV (refer sec.16.11). It is interesting to note that nDT T ¿E depends mainly only on the product of AIp in cases of Goldstone and L mode scalings.

16.7 H Mode and Improved Con¯nement Modes An improved con¯nement state \H mode " was found in the ASDEX21;22 experiments with divertor con¯guration. When the NBI heating power is larger than a threshold value in the divertor con¯guration, the D® line of deuterium (atom °ux) in the edge region of the deuterium plasma decreases suddenly (time scale of 100 ¹s) during discharge, and recycling of deuterium

16.7 H Mode and Improved Con¯nement Modes

205

Fig.16.15 Plots of various edge plasma pro¯les at times spanning the L-H transition in DIIID. (a) Er pro¯le, (b) Pro¯les of the ion temperature measured by CVII charge exchange recombination spectroscopy, (c)(d) Pro¯les of electron temperature and electron density measured by Thomson scattering. (After Doyle et al: Plasma Phys. Controlled Nuc. Fusion Research 1, 235, (1991) IAEA. ref.24)

atoms near the boundary decreases. At the same time there is a marked change in the edge radial electric ¯eld Er (toward negative). Furthermore the electron density and the thermal energy density increase and the energy con¯nement time of NBI heated plasma is improved by a factor of about 2. H mode was observed in PDX, JFT-2, DIII-D, JET, JT60U and so on. The con¯nement state following Kaye-Goldston scaling is called the \L mode". In the H mode, the gradients of electron temperature and the electron density become steep just at the inside of the plasma boundary determined by the separatrix. In the spontaneous H mode, and Er becomes more negative (inward) (see ¯g.16.15).23;24 The ion orbit loss near the plasma edge was pointed out and analyzed as a possible cause of the change of radial electric ¯eld on L-H transition25;26 . The radial electric ¯eld causes plasma rotation with the velocity of vµ = ¡Er =B in the poloidal direction and with the velocity vÁ = ¡(Er =B)(Bµ =B) in the toroidal direction. If the gradient of Er exists, sheared poloidal rotation and sheared toroidal rotation are generated. The importance of sheared °ow for supression of edge turbulence and for improved con¯nement was pointed out in ref.27. Let us consider the following °uid model µ

¶

@ ~ ) ¢ r + Ld »~ = s~ + (v 0 + v @t

where »~ is the °uctuating ¯eld. v 0 is taken to be the equilibrium E £ B °ow. s~ represents a driving source of the turbulence and Ld is an operator responsible for dissipation of turbulence. ~ »(2)i ~ ~ at a point 1 and »(2) ~ at The mutual correlation function h»(1) of the °uctuating ¯eld »(1) 28 a point 2 is given by µ

¶

@ @ @ @ ~ »(2)i ~ D(r+ ; y¡ ) ¡ =T + Ld h»(1) + (vµ0 ¡ vµ =r+ )r+ @t @y¡ @r+ @r+

(16:32)

206

16 Tokamak

where D is radial di®usion coe±cient of turbulence and T is the driving term and r+ = (r1 +r2 )=2, µ¡ = µ1 ¡µ2 , y¡ = r+ µ¡ . The decorrelation time ¿d in the poloidal direction is the time in which the relative poloidal displacement between point 1 and point 2 due to sheared °ow becomes the ¡1 space correlation length of the turbulence k0k , that is, k0k ±y » 1; ±y = vµ0 (¢r)¿d ; ¿d =

1

: vµ0 ¢rk0k

The decorrelation rate !s in the poloidal direction is !s =

1 = (¢rk0k )vµ0 : ¿d

When ¢r is the radial correlation length of the turbulence, the radial decorrelation rate ¢!t is given by ¢!t =

D : (¢r)2

Since there is strong mutual interaction between radial and poloidal decorrelation processes, the decorrelation rate 1=¿corr becomes a hybrid of two decorrelation rates, that is, 1 ¿corr

= (!s2 ¢!t )1=3 =

µ

!s ¢!t

¶2=3

¢!t :

(16:33)

The decorrelation rate 1=¿corr becomes (!s =¢!t )2=3 times as large as ¢!t ; ¢!t is the decorrelation rate of the turbulence in the case of shearless °ow. Since the saturation level of °uctuating ¯eld »~ is ~ 2 » T £ ¿corr j»j the saturation level of °uctuating ¯eld is reduce to µ ¶ µ ¶2=3 ~2 j»j ¢!t 2=3 1 1 ; » » 2 ~ !s (dvµ =dr)t0 (k0y ¢r)2 j»0 j 2 t¡1 0 ´ hk0y iD

where j»~0 j is the level in the case of shearless °ow. The e®ect of sheared °ow on the saturated resistive pressure gradient driven turbulence is shown in ¯g.16.16. The coupling between poloidal and radial decorrelation in shearing °uctuation is evident in this ¯gure. Since the thermal ~ 2 , the thermal di®usion is reduced, that is, thermal di®usion coe±cient is proportional to j»j barrier near the plasma edge is formed. Active theoretical studies on H mode physics are being carried out. In addition to the standard H mode as observed in ASDEX and others, the other types of improved con¯nement modes have been observed. In the TFTR experiment,29 outgassing of deuterium from the wall and the carbon limiter located on the inner (high-¯eld) side of the vacuum torus was extensively carried out before the experiments. Then balanced neutral beam injections of co-injection (beam direction parallel to the plasma current) and counterinjection (beam direction opposite to that of co-injection) were applied to the deuterium plasma, and

16.7 H Mode and Improved Con¯nement Modes

207

Fig.16.16 Snapshot of equidensity contour for shearless (top) and strongly-sheared (bottom) °ows. (After Bigrali et al: Plasma Phys. Controlled Nuc. Fusion Reseach 2, 191, (1991) IAEA. ref.27)

an improved con¯nement \supershot" was obsered. In supershot, the electron density pro¯le is strongly peaked (ne (0)=hne i= 2:5 » 3). In DIII-D experiment, VH mode30 was observed, in which the region of strong radial electric ¯eld was expanded from the plasma edge to the plasma interior (r=a » 0:6) and ¿E =¿EITER¡P becomes 3.6. In JT60U experiment, high beta-poloidal H mode31 was observed, in which ¯p was high (1.2»1.6) and the density pro¯le was peaked (ne (0)=hne i= 2:1 » 2:4). Furthermore the edge thermal barrier of H mode was formed. Hinton et al 32 pointed out the peaked pressure and density pro¯les induce the gradient of the radial electric ¯eld. From the radial component of the equation of motion (5.7) of ion °uid or (5.28), we have Er ' Bp ut ¡ Bt up +

1 dpi : eni dr

(16:34)

Di®erentiation of Er by r is 1 dni dpi dEr »¡ 2 dr eni dr dr since the contribution from the other terms is small in usual experimental condition of H mode. Recently a high performance mode of negative magnetic shear con¯guration is demonstrated in DIII-D, TFTR, JT60U, JET and Tore Supra (ref.33). As described in sec.8.5, ballooning mode is stable in the negative shear region S=

r dq < 0: q dr

(16:35)

An example of radial pro¯les of temperature, density and q pro¯le of JT60U is shown in ¯g.16.17. By combination of the central heating and the magnetic negative shear, the steep gradients in temperature and density appear at around the q minimum point. This internal transport barrier is formed probably by the e®ects of the negative magnetic shear and E £ B °ow shear.

208

16 Tokamak

Fig.16.17 Radial pro¯les of ion and electron temperatures and density and q pro¯les in the negative magnetic shear con¯guration of JT60U.

As a measure of high performance of improved con¯nement mode, the ratio, HL factor, of observed energy con¯nement time ¿EEXP to ITER-P scaling ¿EITER¡P is widely used. HL ´

¿EEXP ¿EITER¡P

(16:36)

Observed HL factors are in the range of 2»3. ITER H mode database working group assembled standard experimental data of H mode from ASDEX, ASDEX-U, DIII-D, JET, JFT-2M, PDX, PBX and Alcator C-Mod and so on. Results of regression analysis of H mode experiments led to the following thermal energy con¯nement time34 IPB98y2 0:58 0:78 ¿E;th ∙ = 0:0562Ip0:98 Bt0:15 P ¡0:69 Mi0:19 R1:97 n ¹ 0:41 e19 ²

(16:37)

where units of sec, MA, T, MW, amu, m, 1019 m¡3 are used and the total heating power corrected for shine through of NBI heating, orbit loss and charge exchange loss, less the time derivative of stored energy. This scaling is used when edge-localized-modes (ELM) exist. The heating power threshold scaling PLH , which de¯nes the boundary of H mode operating window, is 1:00 0:81 a PLH = 2:84Mi¡1 Bt0:82 n ¹ 0:58 e20 R

(16:38)

In most of experiments of hot plasmas, neutral beam injections are used to heat the plasma. Combined with improved con¯nement mode operations, such as H mode, supershot and high ¯p mode in large tokamaks, fusion grade plasmas are produced by neutral beam injection. The plasma parameters of typical shots of JET35 , JT60U31 and TFTR29 are listed in table16.2. In the present neutral beam source, the positive hydrogen ions are accelerated and then passed through the cell ¯lled with neutral hydrogen gas, where ions are converted to a fast neutral beam by charge exchange (attachment of electron). However, the conversion ratio of positive hydrogen ions to neutral becomes small when the ion energy is larger than 100 keV (2.5% at 200 keV of H+ ). On the other hand, the conversion ratio of negative hydrogen ions (H¡ ) to neutral

16.8 Noninductive Current Drive

209

Table 16.2 Plasma parameters of large tokamaks JET35 , JT60U31 and TFTR29 . ni (0)¿Etot Ti (0) is fusion triple product. ∙s is the ratio of vertical radius to horizontal radius. q's are the e®ective safety factors near plasma boundary with di®erent de¯nitions. q95 is the safety factor at 95% °ux surface. qe® and q ¤ are de¯ned in (16.11) and ref.29 respectively. qI is the factor de¯ned in sec.16.4. ENB is a particle energy of neutral beam injection.

Ip (MA) Bt (T) R=a(m/m) ∙s q's qI ne (0)(1019 m¡3 ) ne (0)=hne i ni (0)(1019 m¡3 ) Te (0)(keV) Te (0)=hTe i Ti (keV) Wdia (MJ) dWdia =dt(MJ/s) Ze® ¯p ¯t (%) g(Troyon factor) PNB (MW) ENB (keV) ¿Etot = W=Ptot (s) H = ¿Etot =¿EITER¡P ni (0)¿Etot Ti (0)(1020 keVm¡3 s) nT (0)=(nT (0) + nD (0)) Pfusion (MW)

JET ELM free No.26087 3.1 2.8 3.15/1.05 1.6 q95 =3.8 2.8 5.1 1.45 4.1 10.5 1.87 18.6 11.6 6.0 1.8 0.83 2.2 2.1 14.9 135, 78 0.78 » 3.0 5.9 0 -

JT60U ELMy No.E21140 2.2 4.4 3.05/0.72 1.7 qe® =4.6 3.0 7.5 2.4 5.5 10 30 7.5 2.2 1.2 »1.3 »1.9 24.8 95 0.3 » 2.1 5 0 -

TFTR supershot 2.5 5.1 »2.48/0.82 1 q ¤ =3.2 2.8 8.5 6.3 11.5 44 6.5 7.5 2.2 »1.1 »1.2 2 33.7 110 0.2 » 2.0 5.5 0.5 9.3

(stripping of electron) does not decrease in the high energy range (»60%); a neutral beam source with a negative ion source is being developed as a high-e±ciency source. Wave heating is another method of plasma heating and was described in detail in ch.12. The similar heating e±ciency of wave heating in ICRF (ion cyclotron range of frequency) to that of NBI was observed in PLT. In the ICRF experiments of JET, the parameters ∙Ti (0) = 5:4 keV; ∙Te (0) = 5:6 keV; ne (0) = 3:7 £ 1013 cm¡3 ; ¿E » 0:3 s were obtained by PICRF = 7 MW: 16.8 Noninductive Current Drive As long as the plasma current is driven by electromagnetic induction of the current transformer in a tokamak device, the discharge is a necessarily pulsed operation with ¯nite duration. If the plasma current can be driven in a noninductive way, a steady-state tokamak reactor is possible in principle. Current drive by neutral beam injection has been proposed by Ohkawa,36 and current drive by traveling wave has been proposed by Wort.37 The momenta of particles injected by NBI or of traveling waves are transferred to the charged particles of the plasma, and the resultant charged particle °ow produces the plasma current. Current drive by NBI was demonstrated by DITE, TFTR, etc. Current drive by a lower hybrid wave (LHW), proposed by Fisch,38 was demonstrated by JFT-2, JIPPT-II, WT-2, PLT, Alcator C, Versator 2, T-7, Wega, JT-60 and so on. Current drive by electron cyclotron wave40 was demonstrated by Cleo, T-10, WT-3, Compass-D, DIII-D, TCV and so on. 16.8a Lower Hybrid Current Drive The theory of current drive by waves is described here according to Fisch and Karney.38 When a wave is traveling along the line of magnetic force, the velocity distribution function near the

210

16 Tokamak

phase velocity of the wave is °attened by the di®usion in velocity space. Denote the di®usion coe±cient in velocity space by the wave by Drf ; then the Fokker-Planck equation is given by39 @f + v ¢ rr f + @t

µ

F m

¶

@ ¢ rv f = @vz

µ

@f Drf @vz

¶

+

µ

±f ±t

¶

(16:39) F:P:

where (±f =±t)F:P: is Fokker-Planck collision term µ

±f ±t

¶

F:P:

=¡

Xµ 1 @

v 2 @v

i;e

Jv = ¡Dk

(v 2 Jv ) +

¶

1 @ (sin µJµ ) ; v sin µ @µ

(16:40)

@f 1 @f : + Af; Jµ = ¡D? @v v @µ

(16:41)

¤ of ¯eld particles When the velocity v of a test particle is larger than the thermal velocity vT ¤ (v > vT ), the di®usion tensor in velocity space Dk ; D? and the coe±cient of dynamic friction A are reduced to

v ¤2 º0 Dk = T 2 A = ¡Dk

µ

¤ vT v

¶3

;

D? =

¤2 º v ¤ vT 0 T ; 2 2v

m v ¤2 m¤ vT

¤ where vT and º0 are ¤2 vT

T¤ = ¤; m

º0 =

µ

qq ¤ ²0

¶2

n¤ ln ¤ ln ¤ : = ¦ ¤4 ¤3 ¤3 n¤ 2 2¼vT m 2¼vT

¤ ; q ¤ ; n¤ where ¦ ¤2 ´ qq ¤ n¤ =(²0 m). (v; µ; Ã) are spherical coordinates in velocity space. vT are the thermal velocity, charge, and density of ¯eld particles, respectively, and v; q; n are quantities of test particles. Let us consider the electron distribution function in a homogeneous case in space without external force (F = 0). Collision terms of electron-electron and electronion (charge number = Z) are taken into account. When dimensionless quantities ¿ = º0e t; u = ¤ ; w ¤ ; D(w) ¤2 º v=vTe = vz =vTe = Drf =vTe 0e are introduced, the Fokker-Planck equation reduces to

µ

@f @f @ D(w) = @¿ @w @w

¶

+

1 @ 2u2 @u

µ

1 @f +f u @u

¶

+

µ

@f 1+Z 1 @ sin µ 3 @µ 4u sin µ @µ

¶

:

When Cartesian coordinates in velocity space (vx ; vy ; vz ) = (v1 ; v2 ; v3 ) are used in stead of spherical coordinates in velocity space, the Fokker-Planck collision term in Cartesian coordinates ¤ is assumed): is given as follows (v > vT ¤ Ai = ¡D0 vT

m vi m¤ v 3

¤ D0 vT Dij = 2 v3

Ã

2

X

Dij

Ji = Ai f ¡ D0 ´

(16:42) !

v ¤2 (v ±ij ¡ vi vj ) + T2 (3vi vj ¡ v 2 ±ij ) v

j

@f @vj

¤2 º vT (qq ¤ )2 n¤ ln ¤ 0 ´ ¤ 2 4¼²20 m2 vT

(16:43)

(16:44)

(16:45)

16.8 Noninductive Current Drive

211

Fig.16.18 Distribution function f (vk ) of electrons is °atten in the region from v1 = c=N1 to v2 = c=N2 due to the interaction with the lower hybrid wave whose spectra of parallel index Nk ranges from N1 to N2 . µ

±f ±t

¶

F:P:

= ¡rv ¢ J :

Ai is the coe±cient of dynamic friction and Dij is the component of di®usion tensor. Let us assume that the distribution function of the perpendicular velocities vx ; vy to the line of magnetic force is Maxwellian. Then the one-dimensional Fokker-Planck equation on the distriR ¤ can be deduced by (vx ; vy ) bution function F (w) = f dvx dvy of parallel velocity w = vz =vTe integration: ZZ µ

=

ZZ

±f ±t

¶

dvx dvy =

F:P:

0

@ @ ¡Az f + @vz

ZZ

X

(¡rv ¢ J ) dvx dvy

Dzj

j

1

@f A dvx dvy : @vj

When jvz j À jvx j; jvy j; the approximation v ¼ jvz j can be used. The resultant one-dimensional Fokker-Planck equation on F (w) is µ

@F @F @ D(w) = @¿ @w @w

¶

µ

Z + 1+ 2

¶

@ @w

µ

¶

1 @ 1 + 2 F (w) 3 w @w w

and the steady-state solution is F (w) = C exp

Z

w

1+

¡wdw

w3 D(w)=(1

+ Z=2)

;

and F (w) is shown in ¯g.16.18 schematically (when D(w) = 0, this solution is Maxwellian). F (w) is asymmetric with respect to w = 0, so that the current is induced. The induced current density J is ¤ J = envTe j

where j =

R

wF (w)dw, and j¼

w1 + w2 F (w1 )(w2 ¡ w1 ): 2

(16:46)

212

16 Tokamak

On the other hand, this current tends to dissipate by Coulomb collision. Dissipated energy must be supplied by the input energy from the wave in order to sustain the current. Necessary input power Pd is

Pd = ¡ =

Z

nmv 2 2

¤2 nmvTe º0

Z

µ

±f ±t

¶

dv =

F:P:

Z

µ

w2 @ @F D(w) @w 2 @w

nmv 2 @ 2 @vz ¶

µ

Drf

@f @vz

¶

dv

¤2 º0 p d dw = nmvTe

where pd is given by use of the steady-state solution of F (w), under the assumption of w 3 D(w) À 1, as follows: µ

Z pd = 1 + 2

¶

F (w1 ) ln

µ

w2 w1

¶

µ

Z ¼ 1+ 2

¶

F (w1 )

w2 ¡ w1 : w1

and j 2 2 1:5 w : = pd 1 + 0:5Zi 3

(16:47)

More accurately, this ratio is38 j 1:12 = 1:7w2 : pd 1 + 0:12Zi

(16:470 )

The ratio of the current density J and the necessary input power Pd per unit volume to sustain the current is given by ¤ j envTe ∙TkeV 2 J 1:12 hw i = = 0:16 Pd nTe º0 pd n19 1 + 0:12Zi

Ã

A=m2 W=m3

!

(16:48)

where ∙TkeV is the electron temperature in 1 keV units and n19 is the electron density in 1019 m¡3 units. The ratio of total driven current ICD to LHCD power WLH is R

ICD J2¼rdr 1 R = WLH 2¼R Pd 2¼rdr

T is and the current drive e±ciency of LHCD ´LH T ´LH

Rn19 ICD ´ = WLH

R

µ

´LH (r)Pd (r)2¼rdr A R 1019 Pd (r)2¼rdr W m2

¶

where ´LH (r) is local current drive e±ciency given by

µ

Rn19 J(r) 1:12 A ´LH (r) = 1019 = 0:026(∙Te )keV hw 2 i W m2 2¼RPd (r) 1 + 0:12Zi

¶

(16:49)

(R is the major radius in meter). The square average hw 2 i of the ratio of the phase velocity (in the direction of magnetic ¯eld) of traveling waves to the electron thermal velocity is of the order of 20 » 50. In the experiment (1994) of JT60U, a plasma current of Ip = 3 MA was driven by a lower hybrid wave with WLH = 4.8 MW when n = 1:2 £ 1019 m¡3 , h∙Te i » 2 keV, R=3.5 m and Bt =4 T (´LH » 2:6). These results are consistent with the theoretical results. The necessary power of current drive is proportional to the density, and the current cannot be driven beyond a threshold density in the case of lower hybrid current drive because of accessibility

16.8 Noninductive Current Drive

213

Fig.16.19 The displacement in velocity space of small number, ±f , of electrons from coordinates to be subscripted 1 to those to be subscripted 2.

condition (refer 12.5). Other possible methods, such as drive in cyclotron range of frequencies (refer 16.8b), fast wave and by neutral beam injection (refer sec.16.8c) are also being studied. A ramp-up experiment of the plasma current from zero was carried out ¯rst by WT-2 and PLT and others by the application of a lower hybrid wave to the target plasma produced by electron cyclotron heating and other types of heating. When the plasma current is ramped up in the low-density plasma and the density is increased after the plasma current reaches a speci¯ed value, all the available magnetic °ux of the current transformer can be used only for sustaining the plasma current, so that the discharge duration can be increased several times. 16.8b Electron Cyclotron Current Drive Electron Cyclotron Current Drive (ECCD) relies on the generation of an asymmetric resistivity due to the selective heating of electrons moving in a particular toroidal direction. N. J. Fisch and A. H. Boozer40 proposed that the collisionality of plasma is somehow altered so that, for example, electrons moving the left collide less frequently with ions than do electrons moving to the right. There would result a net electric current with electrons moving, on average, to the left and ions moving to the right. Consider the displacement in velocity space of small number, ±f , of electrons from coordinates to be subscripted 1 to those to be subscripted 2 as is shown in¯g.16.19. The energy expended to produce this displacement is given by ¢E = (E2 ¡ E1 )±f where Ei is the kinetic energy associated with velocity-space location i. Electrons at di®erent coordinates will lose their momentum parallel to the magnetic ¯eld, which is in the z direction, at a rate º1 , but now lose it at a rate º2 . The z-directed current density is then given by j(t) = ¡e±f (vz2 exp(¡º2 t) ¡ vz1 exp(¡º1 t)):

(16:50)

Consider the time-smoothed current J over a time interval ±t which is large compared with both 1=º1 and 1=º2 so that J=

1 ¢t

Z

0

¢t

j(t)dt = ¡

e±f ¢t

µ

¶

vz2 vz1 : ¡ º2 º1

Therefore the necessary input power density Pd to induce the current density is Pd =

¢E E2 ¡ E1 ±f: = ¢t ¢t

214

16 Tokamak

The ratio of J=Pd becomes J vz2 =º2 ¡ vz1 =º1 s ¢ r(vz =º) ) ¡e = ¡e Pd E2 ¡ E1 s ¢ rE

(16:51)

where s is the unit vector in the direction of the displacement in velocity space. Let us estimate º of eq.(16.51). The deceleration rate of momentum of a test electron by collision with electrons and ions is expressed by (refer (2.14),(2.20)) µ

p Zi p dp ¡ =¡ 1+ =¡ ¿eek ¿eik dt 2

¶

º0 p u3

where º0 =

Ã

e2 ne ² 0 me

!2

ln ¤ 3 ; 2¼ne vTe

u´

v : vTe

vTe = (∙Te =me )1=2 is electron thermal velocity. Therefore we have dp = ¡ºM p; dt

ºM ´ (2 + Zi )

º0 : 2u3

² (refer (2.27)) In order to estimate du=dt, we must use the energy relaxation time ¿ee

E dE =¡ ² ; ¿ee dt

E=

me 2 2 u vTe 2

that is º0 u du = ¡ ² = ¡ 3 u: dt 2¿ee 2u Each term in eq.(16.50) of j(t) must be modi¯ed as follow: j(t) = j0 exp(¡

Z

ºM dt) = j0

µ

u(t) u0

¶2+Zi

(16:500 )

because of ¡

Z

ºM dt = ¡

Z

ºM

dt du = (2 + Zi ) du

Z

u(t) du : = (2 + Zi ) ln u u0

Then the integral of j(t) of (16:500 ) reduces to Z

0

1

j(t)dt = j0

Z

0

u0

µ

u(t) u0

¶2+Zi

j0 2u30 dt : du = º0 5 + Zi du

Accordingly º in (16.51) is º = º0

5 + Zi 2u3

(16:52)

and J ene vTe j ; = Pd ne Te º0 pd

j 4 s ¢ r(u3 w) ´ pd 5 + Zi s ¢ ru2

16.8 Noninductive Current Drive

215

where w ´ vz =vTe . In the case of ECCD we have j=pd ¼ 6wu=(5 + Zi ) and J ene vTe h6wui (∙Te )keV h6wui = = 0:096 Pd ne Te º0 5 + Zi n19 5 + Zi

(16:53)

The ratio of driven current ICD to ECCD power WEC is R

J2¼rdr ICD 1 R = WEC 2¼R Pd 2¼rdr

T of ECCD is and the current drive e±ciency ´EC T ´EC

Rn19 ICD ´ = WCD

R

´EC (r)Pd (r)2¼rdr R Pd 2¼rdr

where ´EC (r) is local current drive e±ciency given by ´EC (r) =

Rn19 J(r) h6wui : = 0:015(∙Te )keV 2¼RPd 5 + Zi

(16:54)

16.8c Neutral Beam Current Drive When a fast neutral beam is injected into a plasma, it changes to a fast ion beam by charge 2 =2 exchange or ionization processes. When the fast ions have higher energy than Ecr = mb vcr given by (2.33), they are decelerated mainly by electrons in the plasma and the fast ions with E < Ecr are decelerated mainly by ions in the plasma. The distribution function of the ion beam can be obtained by solving the Fokker-Planck equations. The Fokker-Planck collision term eq.(16.40) of the fast ions with E À Ecr is dominated by the dynamic friction term in eq.(16.41) due to electrons. The dynamic friction term of electrons on the fast ion in the case ¤ is given by39 of v < vT A=¡

v ² : 2¿be

Then the Fokker-Planck equation is reduced to @fb @ + @t @v

Ã

¡vfb ² 2¿be

!

= Á±(v ¡ vb )

(16:55)

² is the energy relaxation time of beam ions and where vb is the initial injection velocity and ¿be electrons as described by (2.34). The right-hand side is the source term of beam ions. The steady-state solution of the Fokker-Planck equation is

fb / 1=v: However, the dynamic friction term due to ions or the di®usion term dominates the collision term in the region of v < vcr . Therefore the approximate distribution function of the ion beam 3 is given by fb / v 2 =(v 3 + vcr ), that is, fb (v) =

v2 nb 3 ln(1 + (vb =vcr )3 )1=3 v 3 + vcr

fb (v) = 0:

(v ∙ vb )

(16:56)

(v > vb )

(16:560 )

216

16 Tokamak

The necessary ion injection rate Á per unit time per unit volume to keep the steady-state condition of the beam is derived by substitution of the solved fb (v) into the Fokker-Planck equation nb (1 + (vcr =vb )3 )¡1 Á= ² 2¿be (ln(1 + (vb =vcr )3 ))1=3 and necessary power is Pb =

mb vb2 nb mb vb2 Á¼ ² : 2 4 ln(vb =vcr )¿be

(16:57)

The average velocity of the decelerating ion beam is v¹b = vb (ln(vb =vcr ))¡1 :

(16:58)

Then the current density J driven by the fast ion's beam consists of terms due to fast ions and bulk ions and electrons of the plasma: J = Zi eni v¹i + Zb enb v¹b ¡ ene v¹e ne = Zi ni + Zb nb ; where v¹i and v¹e are the average velocities of ions with density ni and electrons with density ne , respectively. The electrons of the plasma receive momentum by collision with fast ions and lose it by collision with plasma ions, that is me n e

ve d¹ vb ¡ v¹e )ºebk + me ne (¹ vi ¡ v¹e )ºeik = 0 = me ne (¹ dt

so that ve = Zb2 nb v¹b + Zi2 ni v¹i : (Zi2 ni + Zb2 nb )¹ Since nb ¿ ni , ne v¹e =

Zb2 nb v¹b + Zi ni v¹i Zi

so that36 µ

¶

Zb J = 1¡ Zb enb v¹b : Zi

(16:59)

The driven current density consists of the fast ion beam term Zb enb v¹b and the term of dragged electrons by the fast ion beam, ¡Zb2 enb v¹b =Zi . Then the ratio of J=Pd becomes µ

² Zb enb v¹b Zb J ) 2eZb (2¿be = (1 ¡ Zb =Zi ) = 1¡ ² Pd mb nb vb v¹b =4¿be mb vb Zi

¶

:

(16:60)

When the charge number of beam ions is equal to that of the plasma ions, that is, when Zb = Zi , the current density becomes zero for linear (cylindrical) plasmas. For toroidal plasmas, the motion of circulating electrons is disturbed by collision with the trapped electrons (banana electrons), and the term of the dragged electrons is reduced. Thus J=Pd becomes41 µ

¶

² J Zb ) 2eZb (2¿be = 1¡ (1 ¡ G(Ze® ; ²)) Pd mb vb Zi

(16:600 )

16.8 Noninductive Current Drive

217

µ

G(Ze® ; ²) = 1:55 +

¶

µ

¶

0:85 1=2 1:55 ² ¡ 0:2 + ² Ze® Ze®

(16:61)

where ² is inverse aspect ratio. When the e®ect of pitch angle of ionized beam is taken into account, the factor » ´ vk =v = Rtang =Rion must be multiplied to eq.(16:600 ), where Rtang is the minimum value of R along the neutral beam path and Rion is R of ionization position. The driving e±ciency calculated by the bounce average Fokker Planck equation41 becomes µ

¶

µ

¶

² Zb J ) 2eZb (2¿be = 1¡ (1 ¡ G(Ze® ; ²)) »0 Fnc xb J0 (xb ; y) Pd mb vb Zi

and ² J Zb ) 2eZb (2¿be = 1¡ (1 ¡ G(Ze® ; ²)) »0 Fnc J0 (xb ; y) Pd mb vcr Zi

(16:62)

where xb ´

vb ; vcr

J0 (x; y) =

y = 0:8

Ze® Ab

x2 x3 + (1:39 + 0:61y 0:7 )x2 + (4 + 3y)

and Fnc = 1 ¡ b²¾ is the correction factor41 . Finally we have J Pd

µ

Am W

¶

=

µ

¶

Zb 15:8(∙Te )keV »0 1¡ (1 ¡ G) (1 ¡ b²¾ )J0 (xb ; y) Zb ne19 Zi

(16:63)

The local current drive e±ciency ´NB of NBCD is ´NB

Rne19 J ´ Pd

µ

A 10 Wm2 19

µ

= 2:52(∙Te )keV »0 1 ¡

¶ ¶

Zb (1 ¡ G) (1 ¡ b²¾ )J0 (xb ; y): Zi

(16:64)

When Zb = 1; Ze® = 1:5; Ab = 2; x2b = 4, then ((1 ¡ b²¾ )J0 ) » 0:2: When h²i » 0:15; then ´NB » 0:29(∙Te )keV (1019 )A/Wm2 ): The current drive by NBI is demonstrated by the experiments of DITE, TFTR JT60U and JET. When the application of a current drive to the fusion grade plasma with ne » 1020 m¡3 is considered, the necessary input power for any current drive of full plasma current occupies a considerable amount of the fusion output. Therefore substantial part of plasma current must be driven by bootstrap current as is described in the next section. 16.8d Bootstrap Current It was predicted theoretically that radial di®usion induces a current in the toroidal direction and the current can be large in banana region42¡45 .Later this current called 'bootstrap current' had been well con¯rmed experimentally. This is an important process which can provide the means to sustain the plasma current in tokamak in steady state. As was described in sec.7.2, electrons in collisionless region ºei < ºb make complete circuit of the banana orbit. When density gradient exists, there is a di®erence in particles number on neighboring orbit passing through a point A, as is shown in ¯g.16.20. The di®erence is (dnt =dr)¢b , ¢b being the width of the banana orbit. As the component of velocity parallel to

218

16 Tokamak

Fig.16.20 Banana orbits of trapped electrons those induce the bootstrap current

magnetic ¯eld is vk = ²1=2 vT , the current density due to the trapped electrons with the density nt is jbanana

µ

¶

dnt 1 dp ¢b = ¡²3=2 : = ¡(evk ) Bp dr dr

The untrapped electrons start to drift in the same direction as the trapped electrons due to the collisions between them and the drift becomes steady state due to the collisions with ions. The drift velocity Vuntrap of untrapped electrons in steady state is given by me Vuntrap ºei =

µ

jbanana ºee me ² ¡ene

¶

where ºee =² is e®ective collision frequency between trapped and untrapped electrons. The current density due to the drift velocity Vuntrap is jboot ¼ ¡²1=2

1 dp : Bp dr

(16:65)

This current is called 'bootstrap current'. When the average poloidal beta is ¯p = hpi=(Bp2 =2¹0 ) is used, the ratio of the total bootstrap current Ib to the plasma current Ip to form Bp is given by µ

Ib a »c Ip R

¶1=2

¯p

(16:66)

where c » 0:3 is constant. This value can be near 1 if ¯p is high (¯p » R=a) and the pressure pro¯le is peaked. Experiments on bootstrap current were carried out in TFTR, JT60U and JET. 70 %» 80 % of Ip =1 MA was bootstrap driven in high ¯p operation. As the bootstrap current pro¯le is hollow, it can produce negative magnetic shear q pro¯le, which is stable against ballooning. MHD stability of hollow current pro¯le is analyzed in details in ref.46.

16.9 Neoclassical Tearing Mode Much attention has been focused on tokamak operational pressure limit imposed by nonideal MHD instabilities, such as the e®ects of bootstrap current driven magnetic islands. At high ¯p (poloidal beta) and low collisionality, the pressure gradient in the plasma gives rise to

16.9 Neoclassical Tearing Mode

219

a bootstrap current (refer sec.16.8d). If an island develops, the pressure within the island tends to °atten out, thereby removing the drive for the bootstrap current. This gives rise to a helical 'hole' in the bootstrap current, which increases the size of the island (refer ¯g.16.23). Tearing instability was treated in slab model in sec.9.1. The zero order magnetic ¯eld B 0 depends on only x and is given by B 0 = B0y (x)ey + B0z ez , jB0y (x)j ¿ jB0z j, B0z =const.. The basic equations are µ

¶

@v ½ + (v ¢ r)v = ¡rp + j £ B @t

¡E = v £ B ¡ ´j =

¡

@A ; @t

A = (0; 0; ¡Ã)

(16:67)

Bx = ¡

@Ã @Ã ; By = @y @x

@Ã = (vx By ¡ vy Bx ) ¡ ´jz = (v ¢ r)Ã ¡ ´jz @t

(16:68)

(16:680 )

r2 Ã = ¹0 jz

(16:69)

Since v=

E£B = B2

µ

¶

µ

¶

Ex Ey 1 @Á 1 @Á ;¡ ;0 = ¡ ; ;0 ; B0z B0z B0z @y B0z @x

it is possible to introduce a stream function ' such as vx = ¡

@' ; @y

vy =

@' : @x

Furthermore z component of vorticity wz = (rv)z is introduced, then wz = r2 '. Rotation of (16.67) yieds ½

@wz + (v ¢ r)wz = (r £ (j £ B))z = (B ¢ r)jz ¡ (j ¢ r)Bz = (B ¢ r)jz @t

(16:70)

The relations r ¢ B = 0, r ¢ j = 0 were used here. The 0th order °ux function Ã0 and the ¯rst order perturbation Ã~ are 0 Ã0 (x) = B0y

~ t) = B1x (t) cos ky; Ã(y; k

x2 ; 2

0 x; B0z ) B 0 = (0; B0y

B1x (t) Ã~A (t) ´ k

B 1 = (B1x (t)) sin ky; 0; 0);

2 ~ t) = B 0 x + B1x (t) cos ky à = Ã0 (x) + Ã(y; 0y k 2

(16:71)

x = 0 is the location of singular layer. The separatrix of islands is given by 0 x B0y

2

B1x (t) B1x (t) ; cos ky = + k k 2

xs = 2

Ã

B1x 0 kB0y

!1=2

;

220

16 Tokamak

Fig.16.21 Tearing mode structure in the singular layer

and the full width w of the island is w=4

Ã

B1x 0 kB0y

!1=2

=4

Ã

Ã~A (t) 0 B0y

!1=2

:

(16:72)

The perturbation B1x (t) sin ky growing with the growth rate ° induces a current j1z = E1z =´ = 0 x °B1x =´k, which provides the x direction linear force f1x = ¡j1z B0y indicated on ¯g.16.22. These drive the °ow pattern of narrow vortices. Moving away from the resistive singular layer, 0 x). For incomthe induced electric ¯eld produces a °ow vx = ¡Ez =By = ¡°B1x cos ky=(kB0y pressible °ow (in strong equilibrium ¯eld B0z ), this requires a strongly sheared °ow vy (x) over the layer x » xT , that is the narrow vortex pattern shown in ¯g.16.21 and we have vy xT » vx =k;

0 vy » vx =kxT » °B1x =(k 2 B0y xT ):

That this shear °ow be driven against inertia by the torque produced by the linear forces requires

xT j1z B0y = °½vy =k;

0 B0y = B0y xT ;

!

x4T =

°B1x °½ °½´ = 0 0 0 )2 : 2 j1z kB0y k B0y (kB0y

since j1z = Ez =´ = °B1x =´k. Thus determined width of perturbation is47 xT =

(°½´)1=4 : 0 )1=2 (kB0y

(16:73)

This is consistent with the results (9.26) and (9.27) obtained by linear theory of tearing instability, which were described in sec.9.1 (Notation ² was used in sec.9.1 instead of xT ). Rutherford showed that the growth of the mode is drastically slowed and perturbation grows only linearly in time, when non-linear e®ects are taken into account47 . The vortex °ow will 2 =(´k 2 B 0 x2 ). induce the second order y-independent eddy current ±j1z = ¡vy B1x =´ » °B1x 0y p The y-direction third order nonlinear forces ±fy » ±jz B1x indicated on ¯g.16.22 provide a torque opposing vortex °ow and decelerate vy °ow.

16.9 Neoclassical Tearing Mode

221

Fig.16.22 Nonlinear forces decelerating vy °ow in tearing mode.

We restrict ourselves to the case where the inertia may be neglected in eq.(16.70). @Ã @jz @Ã @jz + = 0; @y @x @x @y

(B ¢ r)jz = ¡

!

jz = jz (Ã):

Eq.(16.680 ) yields @Ã0 @ Ã~ 0 x + ´j1z ; + = ¡vx B0y @t @t

@Ã0 = ´j0z ; @t

!

@ Ã~ @' 0 x + ´j1z ¡ ´j0z = ¡ B0y @t @y

(16:6800 )

We may eliminate ' from (16.6800 ) by dividing by x and averaging over y at constant Ã. From (16.71), x is given by x=

Ã

2 ~ 0 (Ã ¡ Ã) B0y

!1=2

=

Ã

2 0 B0y

!

1=2 Ã~A (W ¡ cos ky)1=2 ;

W ´

à Ã~A

(16:74)

and *

1 ~ 1=2 (Ã ¡ Ã)

+

(´j1z (Ã) ¡ ´j0z (Ã)) =

1 j1z (Ã) = j0z (Ã) + ´

*

~ @ Ã=@t ~ 1=2 (Ã ¡ Ã)

+

D

*

~ t)=@t @ Ã(y; ~ t))1=2 (Ã ¡ Ã(y;

~ ¡1=2 (Ã ¡ Ã)

+

;

E¡1

(16:75)

where hf i ´

k 2¼

Z

2¼=k

f dy: 0

For the outer solution we require the discontinuity in the logarithmic derivatives across singularity. We must match the logarithmic discontinuity from the solution within the singular layer to that of outer solution: 0

¢ ´

Ã

¯

¯

@ Ã~A ¯¯ @ Ã~A ¯¯ ¯ ¡ ¯ @x ¯+0 @x ¯¡0

!

¯

¯+0 @ 1 = ln Ã~A ¯¯ : @x Ã~A ¡0

2 ¼¹ j ~ We utilize r2 Ã~ = ¹0 j1z and @ 2 Ã=@x 0 1z and

¿

¢ Ã~A = 2¹0 cos ky 0

Z

1

¡1

À

j1z dx ;

dx =

Ã

1 0 B0y

!1=2

dà : ~ 1=2 (à ¡ Ã)

(16:76)

222

16 Tokamak

Fig.16.23 The coordinates in slab model and the coordinates in toroidal plasma. The coordinates (x; y; z) correspond radial coordinate r ¡ rs , poloidal coordinate rµ and toroidal coordinate R' in toroidal plasma respectively. The arrows in the island indicate the direction of magnetic ¯eld Bp ¡ (nr=mR)Bt (refer eq.(16.79)).

Insertion of (16.75) into (16.76) yields ¹0 ¢ Ã~A = 2 ´ 0

Z

4¹0 = 0 )1=2 ´(2B0y

1

¡1

Z

*

1

~ @ Ã=@t ~ 1=2 (Ã ¡ Ã)

dÃ

Ãmin

*

+

D

~ ¡1=2 (Ã ¡ Ã)

@ Ã~A cos ky=@t ~ 1=2 (Ã ¡ Ã)

+

E¡1

D

cos ky

~ (Ã ¡ Ã)

¡1=2

Ã

E¡1

1 0 B0y *

!1=2

dà ~ 1=2 (à ¡ Ã)

cos ky ~ 1=2 (Ã ¡ Ã)

+

:

Since Z

dÃ

*

cos ky ~ 1=2 (Ã ¡ Ã)

we obtain

+2

1 D E = ~ ¡1=2 (Ã ¡ Ã)

¢0 Ã~A =

Z ¿

cos ky (W ¡ cos ky)1=2

À2

dW Ã~A 1=2 ® ´ AÃ~A ; ¡1=2 (W ¡ cos ky)

­

4¹0 A @ Ã~A ~1=2 ÃA 0 )1=2 @t ´(2B0y

and 0 1=2 ) @ ~1=2 ´(2B0y ÃA = ¢0 Ã~A : @t 8¹0 A

Taking note (16.72), the time variation of the island width is reduced to 1 dw = 1=2 ´¹0 ¢0 ¼ ´¹0 ¢0 ; dt 2 A

¿R

d w = ¢0 rs : dt rs

Let us consider a toroidal plasma as is shown in ¯g.16.23. The magnetic ¯eld Bp ¡

nr Bt = nR

µ

1 1 ¡ q(r) qs

¶

r Bt ; R

(qs =

m ) n

(16:77)

16.9 Neoclassical Tearing Mode

223

corresponds B0y in slab model near singular radius. The coordinates (x; y; z) in slab model correspond radial coordinate r¡rs , poloidal coordinate rµ and toroidal coordinate R' in toroidal plasma respectively (see ¯g.16.23). The °ux function is Ã(x; y) =

Z

r¡rs

0

µ

1 1 ¡ q(r) qs

¶

r B1x Bt dr + cos ky R k

(16:78)

and the magnetic ¯eld is given by B1x = ¡

@Ã = B1x sin ky; @y

B0y =

@Ã = @x

µ

1 1 ¡ q(r) qs

¶

r q0 0 x Bt = ¡ Bp x = B0y R q

(16:79)

Eq.(16.78) is reduced to 0 Ã(x; y) = B0y

x2 B1x cos ky + k 2

(16:780 )

b induces the change in °ux function ±Ãb and elecric ¯eld The change of bootstrap current ±j1z Ez .

Ez =

@Ãb b : = ´±j1z @t

b is Discontinuity of logarithmic derivative due to ±j1z

¢0b

1 = ~ ÃA

where

Ã

¯

b¯ @ Ã~A ¯ ¯ @r ¯r

s

¯

b¯ @ Ã~A ¯ ¡ ¯ ¯ @r + r

s¡

!

=

1 ~ ÃA

Z

rs +

rs ¡

b ¹0 ±j1z dr

0 w 2 B0y B1x Ã~A = = k 16

so that ¢0b =

16 0 2 w B0y

Z

rs +

rs ¡

b ¹0 ±j1z dr:

b is given by (refer Because of °attening of pressure pro¯le due to the formation of island, ±j1z (16.65))

b ±j1z

Ã

1=2

²s dp =0¡ ¡ Bp dr

!

1=2

=

²s dp Bp dr

(16:80)

This is called by helical hole of bootstrap current. Thus discontinuity of logarithmic derivative b is reduced to due to ±j1z 16¹0 ¢0b rs = 2 0 w B0y

Ã

1=2

²s dp Bp dr

!

rs =

rs

Lq p 8rs ²1=2 ; s 2 w Bp =2¹0 Lp

Bp q0 0 B0y = ¡ Bp ´ ¡ ; q Lq

p dp ´¡ : Lp dr

Then the time variation of island's width is given by ¿R

Lq rs d w ; = ¢0 rs + a²1=2 s ¯p Lp w dt rs

a » 8:

(16:81)

The ¯rst term of right-hand side of (16.81) is Rutherford term and the second is the destabilizing term of bootstrap current. This is the equation of neoclassical tearing mode. When the transport

224

16 Tokamak

Fig.16.24 The curve of (16.810 ). wth is the threshold width of island for the onset of neoclassical tearing mode and wsat is the saturated width.

across the island is taken into account, a reduction in the bootstrap current takes place. Then the term of bootstrap current is modi¯ed to ¿R

Lq rs w d w : = ¢0 rs + a²1=2 s ¯p Lp w 2 + wc2 dt rs

(16:810 )

where wc is the e®ect of transport across the island parameterizing the magnitude of the contribution of the Â? =Âk model48 and being given by the relation wc = 1:8rs

µ

8RLq rs2 n

¶1=2 Ã

Â? Âk

!1=4

:

Figure 16.24 shows the curve of (16.810 ). When the e®ect of wc is included, there is a threshold wth for the onset of neoclassical tearing mode. When w becomes large, destabilizing term of bootstap current becomes weak and the island width is saturated. It is possible to control neoclassical tearing mode by local current drive in rational (singular) surface49 . 16.10 Resistive Wall Mode MHD kink instabilities in tokamak are of major importance because they have a beta limit7;8 . In the absence of a conducting wall, the results obtained in ref.7 with a variety of current and pressure pro¯le show that this beta limit is of form ¯=(Ip =aB) ´ ¯N < 2:8 (refer (16.9)). However the external kink can be stabilized at higher value of ¯N by including a closely ¯tting conducting wall. The situation is complicated by the existence of resistive wall in the case when an ideal MHD instability is stabilized by perfect conducting wall, but unstable if the wall is removed. In this situation, there is a resistive wall mode that grows on the resistive time of wall. Furthermore it is an interesting issue whether this resistive wall mode is stabilized by plasma rotation or not. In this section the analysis of resistive mode is described. 16.10a Growth Rate of Resistive Wall Mode Basic equations of motion in slab model are already given by (9.9) and (9.13), that is, B1x = i(k ¢ B)»x ¹0 ° 2 r½m r»x = i(k ¢ B)rB1x ¡ i(k ¢ B)00 B1x

16.10 Resistive Wall Mode

225

Fig.16.25 Upper ¯gure: Pro¯les of mass density ½(r), current density j(r) and q pro¯le q(r). The plasma radius is r = a and the wall is located at r = d. Lower ¯gure: Pro¯le of °ux function Ã(r) in the case of conducting wall.

For cylindrical coordinates, the same mathematical process as for the slab model leads to (rB1r ) = iF (r»r )

(16:82)

00

¹0 ° 2 F r½m r(ir»r ) = ¡r2 (rB1r ) + (rB1r ) F F

(16:83)

where F = (k ¢ B) =

µ

¡m n Bµ + Bz r R

¶

Bz = R

µ

m n¡ q(r)

¶

=

Bµ (nq(r) ¡ m): r

The °ux function à = Az (r; µ) = Az (r) exp(¡mµ), z component of vector potential, is introduced and then B1r =

1 @Az m = ¡i Ã; r @µ r

Bz = ¡

@Az @Ã =¡ : @r @r

(16.83) is reduced to ¹0 ° 2 Ã m¹0 j 0 Ã; r½m r? = ¡r2 Ã ¡ F F F r

00

F ¼¡

m¹0 j r

0

(16:830 )

Firstly a step function model of cylindrical plasma presented by Finn50 is used for analysis. The mass density and plasma current pro¯les are °at within the plasma r < a as is shown in ¯g.16.25, that is, j(r) = j0 ;

½(r) = ½0 ;

q(r) = q;

for r < a;

and j(r) = 0; ½(r) = 0; q(r) = q(r) for r < a. Then (16.830 ) for r < a yields Ã

¹0 ½0 ° 2 1+ F2

!

2

r à = 0;

Ã(r) = Ã0

µ ¶m

r a

Ã(a¡ )0 m = Ã(a) a

;

(16:8300 )

and (16.830 ) for r > a yields 2

r à = 0;

Ã(a) Ã(r) = 1¡®

õ ¶ ¡m

r a

¡®

µ ¶m !

r a

:

226

16 Tokamak

Fig.16.26 Stable region in nq ¡ m-d=a-diagram.

Fig.16.27 Pro¯le of Ã(r) in the case of resistive wall at r = d.

When a conducting wall is located at r = d, Ã(d) = 0 must be satis¯ed and ® = (a=d)2m . Then à 0 (a+ ) m 1 + (a=d)2m : =¡ Ã(a) a 1 ¡ (a=d)2m

(16:83000 )

At the plasma boundary, (16.830 ) yields Ã

¹0 ½0 ° 2 Ã 0 (a+ ) ¡ 1+ 2 Ã(a) Bµ (nq ¡ m)2 =a2

!

à 0 (a¡ ) m m¹0 2 ; j0 = = Ã(a) Fa a (nq ¡ m)

(16:84)

since ¹0 j0 = 2Bµ =a for °at current pro¯le. The growth rate °c (d) of the MHD perturbation in the case of conducting wall at r = d is reduced from (16.8300 , 83000 ) and (16.84) as follow: 2 °c (d)2 ¿Aµ

µ

¶

(nq ¡ m) : = ¡2(nq ¡ m) 1 + 1 ¡ (a=d)2m

(16:85)

The stable region in d=a-q diagram is shown in ¯g.(16.26). When a thin resistive wall is located at r = d instead of conducting wall, the external solution of à is modi¯ed and is given by (see ¯g.(16.27)) Ã(r) = Ã(d)(r=d)¡m ; (r > d);

Ã(r) =

´ Ã(d) ³ (r=d)¡m ¡ ®res (r=d)2m ; (d > r > a):(16:86) 1 ¡ ®res

When the wall current and wall speci¯c resistivity are denoted by jw and ´w , there are following relations: ° jw = Ez =´w = ¡ Ã: r2 Ã = ¡¹0 jw ; ´w Then the discontinuity of logarithmic derivative at r = d is R

R

Ã(d+ )0 ¡ Ã(d¡ )0 °¿w ¹0 jw dr ¹0 °=´w Ãdr ¹0 °±w ; = =¡ = = Ã(d) Ã Ã ´w d

¿w ´ ¹0 d±w ´w

16.10 Resistive Wall Mode

227

where ±w is the wall thickness and we obtain Ã(d¡ )0 m °res ¿w : =¡ ¡ Ã(d) d d Thus ®res in (16.86) is given by ®res =

°res ¿w =(2m) : 1 + °res ¿w =(2m)

We have already à 0 (a+ )=Ã(a) from (16.84) as follow: à 0 (a+ ) m = Ã(a) a

Ã

2 ¿2 °res 2 Aµ 1+ + (nq ¡ m)2 (nq ¡ m)

!

:

On the other hand à 0 (a+ )=Ã(a) is also given by (16.86) as follow: à 0 (a+ ) m 1 + ®res (a=d)2m : =¡ Ã(a) a 1 ¡ ®res (a=d)2m

(16:860 )

Therefore the growth rate of the mode in the resistive wall is given by 2 °res (d)2 ¿Aµ

µ

¶

(nq ¡ m) : = ¡2(nq ¡ m) 1 + 1 ¡ ®res (a=d)2m

(16:87)

Since R 1 1 1 ; = + 1 ¡ ®res (a=d)2m 1 + R 1 + R 1 ¡ (a=d)2m

R = (1 ¡ (a=d)2m )

°res (d)¿w 2m

(16.87) is reduced to 2 °res (d)2 ¿Aµ =

°c2 (1) + R°c2 (d) : 1+R

(16:88)

Let us consider the case where the mode is stable with the conducting wall at r = d and is unstable without the wall, that is, °c (d) < 0 and °c (1) > 0. Then the growth rate of the mode with thin resistive wall at r = d is (under the assumption of °res (d)2 ¿ °c2 (d); °c2 (1)) ° 2 (1) ; R = ¡ c2 °c (d)

2m °res (d)¿w = 1 ¡ (a=d)2m

Ã

!

° 2 (1) : ¡ c2 °c (d)

(16:89)

Therefore the growth rate is order of inverse resistive wall time constant. For d ! a; °res (d)¿w ! ¡2m(1 + nq ¡ m)=(nq ¡ m) is remains ¯nite. This mode is called resistive wall mode (RWM). When wall position d approaches to critical one dcr , where ideal MHD mode becomes unstable even with conducting wall °c (dcr ) = 0, the growth rate of RWM becomes in¯nity as is seen in (16.89) and connects to ideal MHD mode. When plasmas rotate rigidly and perturbations also rotate without slip in the plasmas, the e®ect of rigid rotation is included by adding a Doppler shift µ

n vz ¡ m!µ ° ! ° + ik ¢ v = ° + i R

¶

= ° + !rot on the left¡hand side of (16:88);

but not on the right-hand side of (16.88). Fig.16.28 shows the dependence of the growth rate °res (d) on the resistive wall position d=a while ­ is ¯xed for the case of !rot ¿A = 0:5, ¿A¡1 ´ ¡1 B=(a(¹0 ½)1=2 ) = (B=Bµ )¿Aµ , R=a = 5; q0 = 1:05, m=2, n=1, ¿A =¿w = 5 £ 10¡4 . When d/a increases beyond dcr =a, plasma becomes unstable in ideal MHD time scale. For d=a approaching

228

16 Tokamak

Fig.16.28 The growth rate °res (d) versus resistive wall position d=a in the case of !rot = 0:5. °res (d) ¡1 ; dcr = 2:115. (R=a = 5; q0 = 1:05, m=2, and !rot are in unit of ¿A¡1 = B=(a(¹0 ½)1=2 ) = (B=Bµ )¿Aµ ¡4 n=1, ¿A =¿w = 5 £ 10 ). (Afer ref.50).

near unity, there is an enhancement in the growth rate due to the inductance factor (1¡(a=d)2m ). Due to the fact the e®ective °ux decay time becomes shorter, the resistive wall behaves as if the wall is more resistive. There is an initial increase in the growth rate with !rot , after which the growth rate decreases, but does not go to zero as !rot ! 1. Ward and Bondeson51 analyzed the full toroidal plasma with resistive wall by numerical code. Numerical analysis indicates that there are two modes. One is the mode that has zero frequency in the frame of plasma and perturbation hardly penetrates the resistive wall, the "plasma mode". In other word resistive wall behaves as if the wall is ideal when !rot À ¿w¡1 . The other one is the mode that the perturbation rotates slowly with resistive wall, the "resistive wall mode". In other word perturbation rotates with respect to the plasma. The two modes are in°uenced in opposite way by the wall distance. The plasma mode is destabilized as the wall is moved farther from the plasma, while the resistive wall mode is stabilized. Therefore there can be a ¯nite window for the wall position such that both modes are stable (see ¯g.16.29). They suggest that important aspects of stabilization mechanism are that inertial e®ects become important near the resonant layers in the plasma where the rotation speed exceeds the local Alfv¶en frequency kk vA and that coupling to sound waves contributes to the stabilization. These numerical results can be interpreted by analytical consideration. In the case of resistive wall at r = d, à 0 (a+ )=Ã(a) is already given by (16.860 ). Formally à 0 (a¡ )=Ã(a) can be expressed by à 0 (a¡ ) m = ¡ (1 + Z) Ã(a) a Z should be calculated by solving à for a given plasma model. Then the dispersion relation is given by (1 + Z) =

0 0 ) + °res (a=d)2m (1 + °res 1 + ®res (a=d)2m ; = 0 ) ¡ ° 0 (a=d)2m 1 ¡ ®res (a=d)2m (1 + °res res

0 °res =

°res ¿w 2m

and the growth rate is given by ´ °res ¿w ³ Z ; 1 ¡ (a=d)2m = w¡Z 2m

w´

2(a=d)2m 2 : = 1 ¡ (a=d)2m (d=a)2m ¡ 1

(16:90)

In the case when plasma does no rotate, Z is real Z = x and positive and the resistive wall mode is unstable for w > x or equivalently for (1 + 2=x) > (d=a)2m ;

a < d < dideal = a(1 + 2=x)1=2m :

16.10 Resistive Wall Mode

229

Fig.16.29 The growth rate °res and slip frequency ¢!slip = !rot ¡ !res of resistive wall mode and the growth rate °ideal of plasma mode versus resistive wall radius d=a for n=1. !rot = 0:06 is unit of ¡1 ¿A¡1 = B=(a(¹0 ½)1=2 ) = (B=Bµ )¿Aµ . (Afer ref.51).

As the wall radius approaches to dideal (w ! x), °res tends to increase to in¯nity and the resistive wall mode connects to ideal MHD instability, which is ideally unstable for d > dideal . When plasma rotates, the logarithmic derivative has non zero imaginary part z = x + yi and the growth rate is ³

´

°res ¿w 2m 1 ¡ (a=d)2m =

wx ¡ x2 ¡ y2 + iwy : (w ¡ x)2 + y 2

(16:900 )

This eliminates the zero in the demominator of (16.900 ) and °res remains ¯nite and complex for all wall distances. The resistive wall mode does not connect to the ideal instability. When wx < (x2 + y2 ), the resistive wall mode becomes stable. This condition becomes d > dres ;

dres ´ (1 + 2x=(x2 + y 2 ))1=2m :

These results are consistent with numerical results. 16.10b Feedback Stabilization of Resistive Wall Mode Feedback stabilization of resistive wall mode is discussed according to ref.52 by Okabayashi, Pomphrey and Hatcher. We begin with the eigenmode equation used to determine the stability of a large aspect ratio tokamak with low beta53 . µ

Ã

¶

d dF 2 d (°¿A2 + F 2 )r (r»r ) ¡ m2 (°¿A2 + F 2 ) + r dr dr dr µ

m n F = ¡ Bµ + Bz r R

¶

=

!

»r = 0

(16:91)

Bµ (nq ¡ m): r

This formula can be derived from (14.36) under the assumption ² = r=R ¿ 1. In the vacuum, the perturbation of magnetic ¯eld B 1 = rÁ is the the solution of r2 Á = 0;

¡

¢

Á = A (r=bw )¡m + ®w (r=bw )m exp(¡imµ + nz=R):

(16:92)

230

16 Tokamak

A plasma-vacuum boundary condition at plasma edge r = a is of the form of53 µ

aà 0 (a+ ) 2m 1 d(r»r ) ¡ (° ¿ + f ) = f2 »r dr Ã(a) f 2 2

2

¶

(16:93)

where f = nq¡m and à is the °ux function of external perturbation B 1ex . The °ux function à = Ã(r) exp(¡imµ) (z component of vector potential Az ) in the vacuum is given by à = ¡rB1r =m. This formula can be derived from the plasma-vacuum boundary condition (8.33) and (8.38). The boundary condition (8.38) n¢B 1ex = n¢r£(»£B) becomes @Á=@r = r¡1 @(»r Bµ =@µ+@(»r Bz )=@z and determines A in (16.92) as follow: A = ¡i

(m=a) ((a=bw

»r F ¡m ¡ )

®w (a=bw )m )

:

The constant ®w is to be determined by the boundary condition on the wall at r = bw . The boundary condition (8.33) becomes Bµ B1µin + Bz B1zin = Bµ B1µex + Bz B1zex where B 1in = r £ (» £ B) is given by (8.69-71) and B 1ex is given by (16.92). » is given by HainLÄ ust equation (8.114-116), in which low ¯ and incompressivility are assumed. Two boundary conditions yield eq.(16.93). The boundary condition (16.93) is used to provide a circuit equation for the plasma by de¯ning ¯0 ´ (1=»r )(d(r»r )=dr)ja . In principle, ¯0 would be determined by eq.(16.91) and (16.93) selfconsistently. The °ux function 2¼RÃ(a+ ) is the perturbed poloidal °ux of (m, n) mode in the vacuum region (B1µ = ¡(@Ã=@r). 2¼RÃ(a+ ) consists of contributions from the perturbed current I1 , resistive wall current I2 and circuit current I3 corresponding to active feedback coil for (m, n) mode, that is, 2¼RÃ(a+ ) = L1 I1 + M12 I2 + M13 I3

0 0 I2 + M13 I3 : 2¼RÃ0 (a+ ) = L01 I1 + M12

Therefore eq.(16.93) is reduced to µ

2 2

2

(° ¿ + f )¯0 ¡ f µ

2 2

2

0 2 aL1

L1

+ (° ¿ + f )¯0 ¡ f

¶

µ

2 2

2

+ 2m L1 I1 + (° ¿ + f )¯0 ¡ f

0 2 aM13

M13

0 2 aM12

M12

¶

+ 2m M12 I2

¶

+ 2m M13 I3 = 0:

For circuit corresponding to resistive wall, the °ux function at resistive wall (r = rw ) is given by 2¼R

@Ã(rw ) @I2 @I1 @I3 = L2 + M21 + M23 @t @t @t @t

2¼R

@Ã(rw ) = ¡2¼R´w jw = ¡R2 I2 @t

and

where I2 ´ 2¼rw ±w jw =(2m);

R2 ´ 2m

´w 2¼R : 2¼rw ±w

16.11 Parameters of Tokamak Reactors

231

For the circuit corresponding the active feedback control, a voltage term to drive the feedback current must be included and M31

@I1 @I2 @I3 + M32 + M31 + R3 I3 = V3 : @t @t @t

The form of the feedback voltage V3 should be applied to minimize Ã(a+ ) by use of appropriate sensors of the perturbations.

16.11 Parameters of Tokamak Reactors Although there are many parameters to specify a tokamak device, there are also many relations and constrains between them54 . If the plasma radius a, toroidal ¯eld Bt and the rato Q of fusion output power to auxiliary heating power are speci¯ed, the other parameters of tokamak are determined by use of scaling laws of electron density, beta, energy con¯nement time and burning condition, when cylindrical safety factor qI (or e®ective q value qe® ), the elongation ratio ∙s , triangularity ± of plasma cross section are given. By the de¯nition of qI , we have qI ´

Ka Bt 5K 2 aBt ; = R Bp AIp

Bp =

¹ 0 Ip Ip = 2¼Ka 5Ka

the plasma current is Ip =

5K 2 aBt AqI

where K 2 = (1 + ∙2s )=2 (Ip in MA, Bt in T and a in m). Aspect ratio A = R=a will be given as a function of a anf Bt in (16.990 ) later. The e®ective safety factor qe® at plasma boundary is given by16 qe® = qI

Ã

1 + ¤¹2 =2 1+ A2

!

(1:24 ¡ 0:54∙s + 0:3(∙2s + ± 2 ) + 0:13±)

where ¤¹ = ¯p + li =2. The volume average electron density n20 in unit of 1020 m¡3 is n20 = NG

Ip ¼a2

(16:94)

where NG is Greenward normalized density. The beta ratio of thermal plasma ¯th ´

hpi

Bt2 =2mu0

= 0:0403(1 + fDT + fHe + fI )

hn20 T i Bt2

is expressed by ¯th = 0:01¯N

Ip aBt

(16:95)

where ¯N is normalized beta. The notations fDT , fHe and fI are the ratios of fuel DT, He and impurity density to electron density respectively and the unit of T is keV. hXi means volume average of X. Thermal energy of plasma W is W =

B2 3 ¯th t V = 0:5968¯th Bt2 V 2 2¹0

232

16 Tokamak

where W is in unit of MJ and plasma volume V is in unit of m¡3 . Plasma shape with elongation ratio ∙s and triangularity ± is given by R = R0 + a cos(µ + ± sin µ) z = a∙s sin µ Plasma volume V is given by V ¼ 2¼ 2 a2 R∙s fshape where fshape is a correction factor due to triangularity fshape

±4 a ± ¡ =1¡ + 8 192 4R

Ã

±3 ±¡ 3

!

:

We utilize the thermal energy con¯nement scaling of IPB98y234 1:97 1:39 0:78 ¡0:69 ¿E = 0:0562 £ 100:41 Hy2 I0:93 Bt0:15 M 0:19 n0:41 A ∙s P 20 a

(16:96)

where M (= 2:5) is average ion mass unit and P is loss power in MW by transport and is equal to necessary absorbed heating power subtracted by radiation loss power Prad . The total ® particle fusion output power P® is P® =

Q 2 hn h¾viv iV 4 DT

where Q® = 3:515MeV. h¾viv is a function of T and a ¯tting equation for h¾viv is given in (1.5). Since the fusion rate ¾v near T =10keV is approximated by 2 h¾viv ¼ 1:1 £ 10¡24 TkeV (m3 =s)

the folowing £ ratio is introduced: £(hT i) ´

hn2DT h¾viv i : 1:1 £ 10¡24 hn2DT T 2 i

£ is a function of average temperature hT i in keV and the pro¯les of density and temperature and has a peak of around 1 near hT i ¼ 8 » 10keV. The curves of £ versus hT i in cases of n(½) = hni (1 ¡ ½2 )®n =(1 + ®n ), T (½) = hT i(1 ¡ ½2 )®T =(1 + ®T ) is shown in ¯g.16.3055 . Then P® is reduced to P® = 0:9551

2 fprof fDT ¯ 2 B 4 £V (1 + fDT + fHe + fI )2 th t

(16:97)

where fprof ´ hn2 T 2 i=hnT i2 ¼ (®n + ®T + 1)2 =(2®n + 2®T + 1) is the pro¯le e®ect of temperature and density. When absorbed auxiliary heating power is denoted by Paux and heating e±ciency of ® heating is f® , the total heating power is f® P® + Paux . When the fraction of radiation loss power to the total heating power is fR , the heating power to sustain buring plasma is given by P = (1 ¡ fR )(f® P® + Paux ): . When Q ratio is de¯ned by the ratio of total fusion output power Pn +P® = 5P® (Pn is neutron output power) to absorbed auxiliary heating power Paux , Q is Q=

5P® Paux

16.11 Parameters of Tokamak Reactors

233

Fig.16.30 £ is function of average temperature hT i in cases with pro¯le parameters (®T = 1:0; ®n = 0:0), (®T = 2:0; ®n = 0:0), (®T = 1:0; ®n = 0:5) and (®T = 2:0; ®n = 0:3)

Then P is reduced to µ

P = (1 ¡ fR ) 1 +

¶

5 P® : Q

Therefore burning condition µ

W 5 = (1 ¡ fR ) f® + ¿E Q

¶

(16:98)

and (16.96) reduce55 a=

1:64 3:22fshape

Ã

(1 + fDT + fHe + fI )2 2 £ (1 ¡ fR )(f® + 5=Q)fprof fDT

!0:738

0:905 q 2:29 ¯N I 2:36 0:976 Hy2 NG ∙0:214 K 4:57 s

A0:619 : Bt1:74

(16:99)

Therefore the inverse aspect ratio A is given as a function of (16:990 )

A = c1:616 a1:616 Bt2:81

where c is coe±cient of A0:619 =Bt1:74 in (16.99). When the distance of plasma separatrix and the conductor of toroidal ¯eld coil is ¢ and the maximum ¯eld of toroidal ¯eld coil is Bmx (see ¯g.16.31), there is a constraint of µ

R¡a¡¢ ¢ Bt = =1¡ 1+ Bmx R a

¶

1 A

and 1¡

Bt 2 1 < < A Bmx A

under the assumption a > ¢ > 0. By speci¯cation of ¢ and Bmx , Bt is a function of a. The ratio » of the °ux swing ¢© of ohmic heating coil and the °ux of plasma ring Lp Ip is given by »´

¢© 5Bmx ((ROH + dOH )2 + 0:5d2OH ) = 1=2 L p Ip (ln(8A=∙s ) + li ¡ 2)RIp

234

16 Tokamak

Fig.16.31 Geometry of plasma, toroidal ¯eld coil and central solenoid of current transformer in tokamak

Table 16.3-a Speci¯ed design parameters. Speci¯ed value of ¯N is the normalized beta of thermal plasma which does not include the contribution of energetic ion components a 2.0

Bt 5.3

A 3.1

qI 2.22

∙s 1.7

± 0.35

fR 0.3

f® 0.95

¯N 1.63

NG 0.85

fDT 0.82

fHe 0.04

qe® 3.38

£ 0.99

fI 0.02

Hy2 1.0

®T 1.0

®n 0.1

Table 16.3-b Reduced parameters Q 10.2

R 6.2

Ip 15.0

¿E 3.8

n20 1.01

hT i 8.1

Pn 324

P® 81

Paux 40

Prad 35

Table 16.4 Parameters of ITER-FEAT．¿Etr is the energy con¯nement time corrected for radiation loss. Q=10. ∙s is the ratio of vertical radius to horizontal radius. q95 is the safety factor at 95% °ux surface. The maximum ¯eld of toroidal ¯eld coils is Bmax =11.8T. The number of toroidal ¯eld coils is 18. Single null divertor con¯guration. 1 turn loop voltage is Vloop = 89mV. Inductive pulse °at-top under Q = 10 condition is several hundreds seconds. Pfus is the total fusion output power. NG is de¯ned in eq.(16.7). fR is the fraction of radiation loss and fDT , fBe , fHe , fHe are fractions of DT, Be, He, Ar densities to the electron density. Ip Bt R a R=a ∙s hne i h0:5 £ (Te + Ti )i Wthermal Wfast ¿Etr Pfus (P® ) Paux Prad

15MA 5.3T 6.2m 2.0m 3.1 1.7 1:01 £ 1020 m¡3 8.5keV 325MJ 25MJ 3.7s 410MW (82MW) 41MW 48MW

Ze® fDT fHe fBe fAr fR ¯t ¯p ¯N (normalized beta) NG Hy2 = ¿Etr =¿EIPB98y2 q95 qI li

1.65 » 82% 4.1% 2% 0.12% 0.39 2.5% 0.67 1.77 0.85 1.0 3.0 2.22 0.86

16.11 Parameters of Tokamak Reactors

Fig.16.32 Toroidal cross section of ITER-FEAT outline design in 2000.

235

236

16 Tokamak

where ROH = R = (a + ¢ + dTF ) + ds + dOH , dTF and dOH being the thickness of TF and OH coil conductors and ds being the separation of TF and OH coil conductors (refer ¯g.16.31). The average current densities jTF , jOH of TF and OH coil conductors in MA/m2 =A/m2 are jTF =

1 2:5 Bmx ¼ dTF 1 ¡ 0:5dTF =(R ¡ a ¡ ¢)

jOH =

2:5 Bmx : ¼ dOH

When parameters a, Bt , A are speci¯ed instead of a, Bt , Q, then Q value and other parameters can be evaluated and are shown in Table 16.3 The conceptual design of tokamak reactors has been actively pursued according to the development of tokamak experimental research. INTOR (International Tokamak Reactor)56 and ITER (International Thermonuclear Experimental Reactor)57;58 are representative of international activity in this ¯eld. ITER aims58 achievement of extended burn in inductively driven plasmas with Q » 10 and aims at demonstrating steady state operation using non-inductive drive with Q » 5. The main parameters of ITER in 2000 are given in table 16.4. A cross section of ITER-FEAT outline design in 2000 is shown in ¯g.16.32.

References 1. L. A. Artsimovich: Nucl. Fusion 12, 215 (1972) H. P. Furth: Nucl. Fusion 15, 487 (1975) 2. J. Wesson: Tokamaks, Clarendon Press, Oxford 1997 ITER Physics Basis: Nucl. Fusion 39, No.12, 2138-2638 (1999) 3. V. S. Mukhovatov and V. D. Shafranov: Nucl. Fusion 11, 605 (1971) 4. Equip TFR: Nucl. Fusion 18, 647 (1978) 5. J. She±eld: Plasma Scattering of Electromagnetic Radiation, Academic Press, New York(1975) 6. For example Y. Nagayama, Y. Ohki and K. Miyamoto: Nucl. Fusion 23, 1447 (1983) 7. F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi: Plasma Phys. and Controlled Fusion 26, 209 (1984) 8. A. Sykes, M. F. Turner and S. Patel: Proc. 11th European Conference on Controlled Fusion and Plasma Physics, Aachen Part II, 363 (1983) T. Tuda, M. Azumi, K. Itoh, G. Kurita, T. Takeda et al.: Plasma Physics and Controlled Nuclear Fusion Research 2, 173 (1985) (Conference Proceedings, London in 1984, IAEA Vienna) 9. B. B. Kadomtsev: Sov. J. Plasma Phys. 1, 389 (1975) 10. M. Greenwald, J. L. Terry, S, M. Wolfe, S. Ejima, M. G. Bell, S. M. Kaye, G. H. Nelson: Nucl. Fusion 28, 2199, (1988) 11. ASDEX-U Team: IAEA Fusion Energy Conference, O1-5 (Montreal in 1990) IAEA, Vienna 12. J. A. Wesson: Nucl. Fusion 18, 87 (1978) 13. B. B. Kadomtsev and O. P. Pogutse: Nucl. Fusion 11, 67 (1971) 14. J. W. Connor and H. R. Wilson: Plasma Phys. Control. Fusion 36, 719 (1994) 15. F. Wagner and U. Stroth: Plasma Phys. Control. Fusion 35, 1321 (1993) 16. T. N. Todd: Proceedings of 4th International Symposium on Heating in Toroidal Plasmas, Rome, Italy, Vol.1, p21,(1984) International School of Plasma Physics, Varenna Y. Kamada, K. Ushigusa, O. Naito, Y. Neyatani, T. Ozeki et al: Nucl. Fusion 34, 1605 1994 17. DIII-D team: Plasma Physics and Controlled Nuclear Fusion Research 1, 69, 1991 (Conference Proceedings, Washington D. C. in 1990) IAEA, Vienna 18. K. Borrass: Nucl. Fusion: 31, 1035, (1991) K. Borrass, R. Farengo, G. C. Vlases: Nucl. Fusion 36, 1389, (1996) B. LaBombard, J. A. Goetz, I. Hutchinson, D. Jablonski, J. Kesner et al: Nucl. Materials 241-243, 149 (1997) 19. R. J. Goldston: Plasma Physics and Controlled Fusion 26, 87, (1984) S. M. Kaye: Phys. Fluids 28, 2327 (1985) 20. P. N. Yushimanov, T. Takizuka, K. S. Riedel, D. J. W. F. Kardaun, J. G. Cordey, S. M. Kaye, D. E. Post: Nucl. Fusion 30, 1999, (1990) N. A. Uckan, P. N. Yushimanov, T. Takizuka, K. Borras, J. D. Callen, et al: Plasma Physics and

References

21.

22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.

237

Controlled Nuclear Fusion Research 3, 307 (1991) (Conference Proceedings, Washington D. C. in 1990) IAEA, Vienna F. Wagner, G. Becker, K. Behringer et al: Phys. Rev.Lett.49, 1408 (1982) F. Wagner, G. Becker, K. Behringer, D. Campbell, M. Keilhacker et al.: Plasma Physics and Controlled Nuclear Fusion Research 1, 43, (1983) (Conference Proceedings, Baltimore in 1982) IAEA, Vienna ASDEX Team: Nucl. Fusion 29, 1959 (1989) R. J. Groebner: Phys. Fluids B5, 2343 (1993) E. J. Doyle, C. L. Rettig, K. H. Burrell, P. Gohil, R. J. Groebner et al: Plasma Physics and Controlled Nucl. Fusion Research 1, 235 (1992) (Conference Proceedings, WÄ urzburg in 1992) IAEA, Vienna S. I. Itoh and K. Itoh: Phys. Rev. Lett. 63, 2369 (1988) K. C. Shaing and E. C. Crume: Phy. Rev. Lett. 63, 2369 (1989) H. Bigrali, D. H. Diamond, Y.-B. Kim, B. A. Carreras, V. E.Lynch, F. L. Hinton et al: Plasma Physics an Controlled Nucl. Fusion Research 2, 191, (1991) (Conference Proceedings, Washington D.C. in 1990) IAEA Vienna T. H. Dupree: Phys. Fluids: 15, 334 (1972) T. Boutros-Ghali and T. H. Dupree: Phys. Fluids: 24, 1839 (1981) TFTR Team: Plasma Physics and Controlled Nucl. Fusion Research 1, 11, (1995) (Conference Proceedings, Seville in 1994) IAEA Vienna ibid. 1, 9, (1991) (Conference Proceedings, Washington D.C. in 1990) T. S. Taylor, T. H. Osborne, K. H. Burrel et al: Plasma Physics and Controlled Nuclear Fusion Research 1, 167, (1992) (Conference Proceedings, WÄ urzburg in 1992) IAEA, Vienna JT60U Team: ibid, 1, 31, (1995) (Conference Proceedings, Seville in 1994) IAEA Vienna F. L. Hinton and G. M. Staebner: Phys. Fluids B5, 1281 (1993) O1-6, O1-2, O1-3, A5-5, O2-2 in IAEA Fusion Energy Conference (Montreal in 1996) IAEA, Vienna ITER Physics Basis, Chapter 2 in Nucl. Fusion 39, No.12 (1999) JET Team: Nucl. Fusion 32, 187 (1992) T. Ohkawa: Nucl. Fusion 10, 185 (1970) D. J. H. Wort: Plasma Phys. 13, 258 (1971) N. J. Fisch: Phys. Rev. Lett. 41, 873 (1978) C. F. F. Karney and N. J. Fisch: Phys. Fluids 22, 1817 (1979) D. V. Sivukhin: Reviews of Plasma Physics 4, 93 (ed. by M. A. Leontobich) Consultants Bureau New York 1966 N. J. Fisch and A. H. Boozer: Phys. Rev. Lett 45, 720 (1980) D. F. H. Start, J. G. Cordey and E. M. Jones: Plasma Phys. 22, 303 (1980) R. J. Bickerton, J. W. Connor and J. B. Taylor: Nature Physical Science 229, 110(1971) A. A. Galeev: Sov. Phys. JETP 32, 752 (1971) M. N. Rosenbluth, R. D. Hazeltine and F. L. Hinton: Phys.Fluids 15, 116 (1972) D. J. Sigmar: Nucl. Fusion 13, 17 (1973) T. Ozeki, M. Azumi, S. Tokuda, S. Ishida: Nucl. Fusion 33, 1025, (1993) P. H. Rutherford: Phys. Fluids 16, 1903, (1973) R. Fitzpatrick: Phys. Plasmas 2, 825, (1995) D. A. Gates, B. Lloyd, A. W. Morris, G. McArdle, M. R. O'Brien et.al.: Nucl. Fusion 37, 1593, (1997) J. M. Finn: Phys. Plasmas 2, 198, (1995) D. J. Ward, A. Bondeson: Phys. Plasmas: 2, 1570, (1995) M. Okabayashi, N. Pomphrey, R. E. Hatcher: Nucl. Fusion 38, 1607, (1998) J. A. Wesson: Nucl. Fusion 18, 87, (1978) ITER Team: Plasma Physics and Controlled Nuclear Fusion Research (Conf. Proceedings, Washington D.C., 1990) 3, 413, (1991) IAEA, Vienna K. Miyamoto: Jour. Plasma Fusion Research 76, 166 (2000) INTOR Team: Nucl Fusion 23,1513 (1983) ITER Team: 16th IAEA Fusion Energy Conference ( Montreal in 1996 ) O1-1, F1 » F5, IAEA, Vienna ITER Team: 18th IAEA Fusion Energy Conference ( Sorrento in 2000 ) OV/1, ITER/1-6, IAEA, Vienna ITER Team: Technical Basis for th ITER-FEAT Outline Design Dec. 1999

238

17 Non-Tokamak Con¯nement System

Ch.17 Non-Tokamak Con¯nement System 17.1 Reversed Field Pinch 17.1a Reversed Field Pinch Con¯guration Reversed ¯eld pinch (RFP) is an axisymmetric toroidal ¯eld used as a tokamak. The magnetic ¯eld con¯guration is composed of the poloidal ¯eld Bp produced by the toroidal component of the plasma current and the toroidal ¯eld Bt produced by the external toroidal ¯eld coil and the poloidal component of the plasma current. The particle orbit loss is as small as that of tokamak. However, RFP and tokamaks have quite di®erent characteristics. In RFP, the magnitudes of the poloidal ¯eld Bp and the toroidal ¯eld Bt are comparable and the safety factor qs (r) =

r Bz (r) R Bµ (r)

is much less than 1 (qs (0) » a=(R£); £ » 1:6): The radial pro¯le of the toroidal ¯eld is shown in ¯g.17.1. The direction of the boundary toroidal ¯eld is reversed with respect to the direction of the on-axis ¯eld, and the magnetic shear is strong. Therefore high-beta (h¯i = 10 » 20%) plasmas can be con¯ned in an MHD stable way. Since the plasma current can be larger than the Kruskal-Shafranov limit (q < 1), there is a possibility of reaching the ignition condition by ohmic heating only (although it depends on the con¯nement scaling). RFP started in an early phase of nuclear fusion research. A stable quiescent phase of discharge was found in Zeta in Harwell in 1968.1 The con¯guration of the magnetic ¯eld in the quiescent phase was the reversed ¯eld pinch con¯guration, as shown in ¯g.17.1. The electron temperature, the energy con¯nement time, and the average beta of Zeta were ∙Te = 100 » 150eV; ¿E = 2ms; h¯i » 10% at the time of IAEA conference at Novosibirsk. However, the epoch-making result of tokamak T-3 (∙Te = 1keV; ¿E = several ms; ¯ » 0:2 %) was also presented in the same conference, and Zeta was shut down because of the better con¯nement characteristics in tokamaks. On the other hand, RFP can con¯ne higher beta plasma and has been actively investigated to improve the con¯nement characteristics (ZT-40 M, OHTE, HBTX1-B, TPE1RM 20, MST and RFX, TPE-RX)2¡5 . The important issues of RFP are con¯nement scaling and impurity control in the high-temperature region. 17.1b MHD Relaxation Even if the plasma is initially MHD unstable in the formation phase, it has been observed in RFP experiments that the plasma turns out to be a stable RFP con¯guration irrespective of the initial condition. J.B.Taylor pointed out in 1974 that RFP con¯guration is a minimum energy state by relaxation processes under certain constraints.6 Let us introduce a physical quantity \magnetic helicity" for the study of this subject. By use of scalar and vector potentials Á; A of electric and magnetic ¯eld E; B, the magnetic helicity K is de¯ned by the integral of the scalar product A ¢ B over the volume V surrounded by a magnetic surface K=

Z

V

A ¢ Bdr

(17:1)

where dr ´ dx dy dz. Since E = ¡rÁ ¡

@A ; @t

B =r£A

17.1 Reversed Field Pinch

239

we ¯nd from Maxwell equations that7 @B @ @A ¢B+A¢ (A ¢ B) = = (¡E ¡ rÁ) ¢ B ¡ A ¢ (r £ E) @t @t @t = ¡E ¢ B ¡ r ¢ (ÁB) + r ¢ (A £ E) ¡ E ¢ (r £ A) = ¡r ¢ (ÁB + E £ A) ¡ 2(E ¢ B): When the plasma is surrounded by a perfect conductive wall, then the conditions (B ¢ n) = 0; E £ n = 0 are hold (n is unit outward vector normal to the wall), so that we ¯nd @ @K = @t @t

Z

V

A ¢ Bdr = ¡2

Z

V

E ¢ Bdr:

(17:2)

The right-hand side in eq.(17.1) is the loss term of the magnetic helicity. When the Ohm's law E + v £ B = ´j is applicable, the loss term is reduced to @K = ¡2 @t

Z

V

´j ¢ Bdr:

(17:3)

When ´ = 0, the magnetic helicity is conserved; in other words, if a plasma is perfectly conductive, K integral over the volume surrounded by arbitrary closed magnetic surfaces is constant. However, if there is small resistivity in the plasma, the local reconnections of the lines of magnetic force are possible and the plasma can relax to a more stable state and the magnetic helicity may change locally. But J. B. Taylor postulates that the global magnetic helicity KT integrated over the whole region of the plasma changes much more slowly. It is assumed that KT is constant within the time scale of relaxation processes. Under the constraint of KT invariant ±KT =

Z

B ¢ ±Adr +

Z

±B ¢ Adr = 2

Z

B ¢ ±Adr = 0

the condition of minimum energy of magnetic ¯eld Z

(2¹0 )¡1 ± (B ¢ B) dr = ¹¡1 0

Z

Z

B ¢ r £ ±Adr = ¹¡1 (r £ B) ¢ ±Adr 0

can be obtained by the method of undetermined multipliers, and we have r £ B ¡ ¸B = 0:

(17:4)

This solution is the minimum energy state in the force-free or pressureless plasma (j £ B = rp = 0; jkB). The axisymmetric solution in cylindrical coordinates is Br = 0;

Bµ = B0 J1 (¸r);

Bz = B0 J0 (¸r)

(17:5)

and is called a Besssel function model. The pro¯les of Bµ (r) and Bz (r) are shown in ¯g.17.1(a). In the region ¸r > 2:405, the toroidal ¯eld Bz is reversed. The pinch parameter £ and the ¯eld reversal ratio F are used commonly to characterize the RFP magnetic ¯eld as follows: £=

(¹0 =2)Ip a Bµ (a) =R ; hBz i Bz 2¼r dr

F =

Bz (a) hBz i

(17:6)

240

17 Non-Tokamak Con¯nement System

Fig.17.1 (a) Toroidal ¯eld Bz (r) and poloidal ¯eld Bµ (r) of RFP. The radial pro¯les of the Bessel function model (BFM) and the modi¯ed Bessel function model (MBFM) are shown. (b) F -£ curve

where hBz i is the volume average of the toroidal ¯eld. The values of F and £ for the Bessel function model are £=

¸a ; 2

F =

£J0 (2£) J1 (2£)

(17:60 )

and the F -£ curve is plotted in ¯g.17.1(b). The quantity ¸=

¹0 j ¢ B (r £ B) ¢ B = = const: 2 B B2

is constant in Taylor model. The observed RFP ¯elds in experiments deviate from the Bessel function model due to the ¯nite beta e®ect and the imperfect relaxation state. The ¸ value is no longer constant in the outer region of plasma and tends to 0 in the boundary. The solution of r £ B ¡ ¸B = 0 with ¸(r) is called modi¯ed Bessel Function model (MBFM). The stability condition of the local MHD mode8 is 1 4

µ

qs0 qs

¶2

+

2¹0 p0 (1 ¡ qs2 ) > 0: rBz2

(17:7)

This formula indicates that the strong shear can stabilize the RFP plasma in the p0 (r) < 0 region but that the °at pressure pro¯le, p0 (r)∼0, is preferable in the central region of weak shear. When qs2 < 1, the local MHD mode is unstable near qs0 = 0 (pitch minimum). When the e®ect of ¯nite resistivity of a plasma is taken into account, it is expected by the classical process of magnetic dissipation that the RFP con¯guration can be sustained only during the period of ¿cl = ¹0 ¾a2 , where ¾ is the speci¯c conductivity. However, ZT-40M experiments9 demonstarated that RFP discharge was sustained more than three times (» 20 ms) as long as ¿cl . This is clear evidence that the regeneration process of the toroidal °ux exists during the relaxation process, which is consumed by classical magnetic dissipation, so that that the RFP con¯guration can be sustained as long as the plasma current is sustained. When there are °uctuations in plasmas, the magnetic ¯eld B in the plasma, for example, is ~ of the time average hBit and the °uctuation term B. ~ The expressed by the sum B = hBit + B time average of Ohm's law ´j = E + v £ B is reduced to ~ t h´jit = hEit + hvit £ hBit + h~ v £ Bi

(17:8)

~ t appears due to °uctuations. Since where h it denotes the time average. NewR term h~ v £ Bi the time average of the toroidal °ux ©z = Bz dS within the plasma cross section is constant

17.1 Reversed Field Pinch

241 H

during qasi-stationary state, the time average of the electric ¯eld in µ direction is 0 ( Eµ dl = ¡d©z =dt = 0 ) and hv r it = 0. Steady state RFP plasmas require the following condition ~ µ it : h´jµ it = h(~ v £ B)

(17:9)

In other words, resistive dissipation is compensated by the e®ective electric ¯eld due to the °uctuations. This process is called MHD dynamo mechanism. Much research 10;11;12 on the relaxation process has been carried out. When electron mean free path is very long, local relation such as Ohm's law may not be applicable. In stead of MHD dynamo theory, kinetic dynamo theory was proposed,13;14 in which anomalous transport of electron momentum across magnetic surfaces plays essential role to sustain RFP con¯guration. 17.1c Con¯nement The energy con¯nement time ¿E in an ohmically heated plasma can be obtained by energy balance equation (3=2)hn∙(Te + Ti )iv 2¼R¼a2 = Vz Ip ¿E where Vz is the loop voltage and Ip is the plasma current. The notation h iv means the volume average. Using the de¯nition of the poloidal beta ¯µ ´

hn∙(Te + Ti )iv 8¼ 2 a2 hn∙(Te + Ti )iv = Bµ2 =2¹0 ¹0 Ip2

the energy con¯nement time is given by ¿E =

Ip 3¹0 R¯µ : 8 Vz

(17:10)

Therefore the scalings of ¯µ and Vz are necessary for the scaling of ¿E . In order to apply a loop voltage on RFP plasma, a cut in the toroidal direction is necessary in the shell conductor surrounding the plasma. In this case, the contribution of surface integral must be added in eq.(17.1) of magnetic helicity as follows: @K = ¡2 @t

Z

E ¢ B dr ¡

Z

(ÁB + E £ A) ¢ n dS:

The induced electric ¯eld in the (conductive) shell surface is zero and is concentrated between the both edge of shell cut. The surface integral consists of the contribution 2Vz ©z from the shell cut and the contribution from the other part of surface S¡ , that is, @K = ¡2 @t

Z

´j ¢ B dr + 2Vz ©z ¡

Z

S¡

(ÁB + E £ A) ¢ n dS

(17:11)

where ©z is the volume average of toroidal magnetic °ux ©z = ¼a2 hBz iv . In quasi-steady state, the time average h@K=@tit is zero. Then the time average of eq.(17.2) yields Vz =

=

R

h´j ¢ Bit dr + (1=2)

2¼R ´0 Ip ³ + VB ; ¼a2

R

S¡ hÁB

h©z it

+ E £ Ait ¢ n dS

242

17 Non-Tokamak Con¯nement System

VB =

2¼R hh(ÁB + E £ A) ¢ nit iS¡ hhBz it iv a

where h iS¡ is the average in the surface region S¡ . The notation ³ is a non-dimensional factor determined by the radial pro¯les of speci¯c resistivity and magnetic ¯eld as follows: ³´

g ¢f hh´jit ¢ hBit iv + hh(´j) hh´j ¢ Bit iv Bit iv = : ´0 hhjz it iv hhBz it iv ´0 hhjz it iv hhBz it iv

Here ´0 is the speci¯c resistivity at the plasma center. When the term of °uctuation is negligible, the value ³ of modi¯ed Bessel function model is ³ » 10, but the value is generally ³ > 10 due to °uctuation. The value of VB is 0 when whole plasma boundary is conductive shell. In reality, plasma boundary is liner or protecting material for the liner. Lines of magnetic force can cross the wall by the magnetic °uctuation or shift of plasma position (B ¢ n 6= 0; E 6= 0). Then the term VB has a ¯nite value. The substitution of Vz into the equation of energy con¯nement time ¿E gives µµ ¶ ¶ ´0 8 VB =2¼R 1 = 2³ + : ¿E 3¯µ ¹0 a2 aBµ (a) When plasmas become hot, the resistive term becomes small and the term of °uctuation and contribution of VB are no longer negligible. The experimental scaling in the region of Ip < 0:5 MA, are Ip =¼a2 hniv = (1 » 5) £ 10¡14 A ¢ m; ¯µ » 0:1; (∙Te (0))keV » Ip (MA). 17.1d Oscillating Field Current Drive RFP plasmas tend to be modi¯ed Bessel function model due to non-linear phenomena of MHD relaxaton. Oscillating ¯eld current drive (OFCD) was proposed15 for sustaining the plasma current and preliminary experiments have been done16 . If terms Vz and © of the second term in the right-hand side of magnetic helicity balance equation(17.11) are modulated as Vz (t) = V~z cos !t; ©z (t) = ©z0 + ©~z cos !t, a direct current component V~z ©~z in the product of 2Vz ¢ ©z appears and compensates the resistive loss of the magnetic helicity. The period of the oscillating ¯eld must be longer than the characteristic time of relaxation and must be shorter than magnetic di®usion time. The disturbing e®ect of the oscillating ¯eld to RFP plasma must be evaluated furthermore. 17.2 Stellarator A stellarator ¯eld can provide a steady-state magnetohydrodynamic equilibrium con¯guration of plasma only by the external ¯eld produced by the coils outside the plasma. The rotational transform, which is necessary to con¯ne the toroidal plasma, is formed by the external coils so that the helical system has the merit of steady-state con¯nement. Although Stellarator C17 was rebuilt as the ST tokamak in 1969, at the Princeton Plasma Physics Laboratory, con¯nement experiments by Wendelstein 7A, 7AS, Heliotoron-E, and ATF are being carried out, since there is a merit of steady-state con¯nement, without current-driven instabilities. Large helical device LHD started experiments in 1998 and advanced stellarator WVII-X is under construction. 17.2a Helical Field Let us consider a magnetic ¯eld of helical symmetry. By means of cylindrical coordinates (r; µ; z), we can express the ¯eld in terms of (r; ' ´ µ ¡ ±®z), where ® > 0; ± = §1. A magnetic ¯eld in a current-free region (j = 0) can be expressed by a scalar potential ÁB , satisfying ¢ÁB = 0, and we can write ÁB = B0 z + ' ´ µ ¡ ±®z:

1 1X bl Il (l®r) sin(l'); ® l=1

(17:12)

17.2 Stellarator

243

Fig.17.2 Current of helical coils

The ¯eld components (Br ; Bµ ; Bz ) of B = rÁB are given by Br =

1 X

lbl Il0 (l®r) sin(l');

(17:13)

l=1

Bµ =

¶ 1 µ X 1

®r

l=1

Bz = B0 ¡ ±

lbl Il (l®r) cos(l');

1 X

(17:14)

lbl Il (l®r) cos(l'):

(17:15)

l=1

The vector potential corresponding to this ¯eld has components Ar = ¡

1 ± X bl Il (l®r) sin(l'); ®2 r l=1

1 B0 ± X bl I 0 (l®r) cos(l'); r¡ 2 ® l=1 l

Aµ =

Az = 0: Using these, we can write Br = ¡

@Aµ ; @z

Bµ =

@Ar ; @z

Bz =

1 @(rAµ ) 1 @Ar ¡ : r @r r @µ

The magnetic surface à = Az + ±®rAµ = ±®rAµ = const: is given by Ã(r; ') = B0

1 X ±®r 2 ¡r bl Il0 (l®r) cos(l') = const: 2 l=1

(17:16)

Such a helically symmetric ¯eld can be produced by a helical current distribution as is shown in ¯g.17.2. Let the magnetic °uxes in z and µ directions inside the magnetic surface be denoted by © and X (X is the integral over the pitch along z, i.e., over 2¼=®); then these may be expressed by ©=

Z

0

2¼

Z

0

r(')

Bz (r; ')r dr dµ;

244

17 Non-Tokamak Con¯nement System

X=

Z

0

2¼=® Z r(') 0

Bµ (r; ') dr dz =

1 ®

Z

2¼

0

Z

r(')

0

Bµ (r; ') dr dµ:

Since ®rBz ¡ ±Bµ = ®@(rAµ )=@r = ±@Ã=@r, we ¯nd that © ¡ ±X = 2¼Ã=±®: Let us consider only one harmonic component of the ¯eld. The scalar potential and the magnetic surface are expressed by ÁB = B0 z +

b Il (l®r) sin(lµ ¡ ±l®z); ®

µ

¶

2±(®r)b 0 B0 B0 Ã= (®r)2 ¡ Il (l®r) cos(lµ ¡ ±l®z) = (®r0 )2 : 2±® B0 2±® The singular points (rs ; µs ) in the z = 0 plane are given by @Ã = 0; @r

@Ã = 0: @µ

Since the modi¯ed Bessel function Il (x) satis¯es Il00 (x)

Ã

1 l2 + Il0 (x) ¡ 1 + 2 x x

!

Il = 0;

the singular points are given by sin(lµs ) = 0; µ

µ

¶

¶

±bl 1 1+ ll (l®rs ) cos(lµs ) = 0 B0 (®rs )2

®r 1 ¡ or

µs = 2¼(j ¡ 1)=l; µ

= 2¼ j ¡

±b=B0 > 0;

¶

1 =l; 2

±b=B0 < 0;

j = 1; ¢ ¢ ¢ ; l;

¯ ¯ ¯ ±bl ¯ 1 ¯ ¯ ¯ B ¯ = (1 + (®r )¡2 )I (l®r ) : 0 s s l

The magnetic surfaces for l = 1，l = 2，l = 3 are shown in ¯g.14.13. The magnetic surface which passes through the hyperbolic singular point is called separatrices. When x ¿ 1, the modi¯ed Bessel function is 1 Il (x) ¼ l!

µ ¶l

x 2

:

The magnetic surfaces in the region ®r ¿ 1 is expressed by (®r)2 ¡

±b(l=2)l¡1 (®r)l sin l(µ ¡ ±®z) = const: B0 (l ¡ 1)!

The magnitude B is µ

B B0

¶2

±lb = 1 ¡ 2 Il cos(l') + B0

µ

lb B0

¶2 µ

Il2

µ

¶

¶

1 1+ cos2 (l') + (Il0 )2 sin2 (l') : (®r)2

17.2 Stellarator

245

Fig.17.3 Magnetic surfaces, showing separatrix points and separatrices, of the helical ¯eld.

The magnitude B at the separatrix (rs ; µs ) is µ

B B0

¶2

=1¡

(®r)2 1 + (®r)2

and B at the point (rs ; µs + ¼=l) is µ

B B0

¶2

=1+

(®r)2 : 1 + (®r)2

Therefore the magnitude B is small at the separatrix points. Let us estimate the rotational transform angle ¶. As the line of magnetic force is expressed by dr rdµ dz = = Br Bµ Bz the rotational transform angle is given by r¶ = 2¼R

¿

rdµ dz

À

=

¿

Bµ Bz

À

=

¿

À

(1=®r)lbIl (l®r) cos l(µ ¡ ±z) : B0 ¡ lbIl (l®r) cos l(µ ¡ ±z)

Here r and µ are the values on the lineH of magnetic R force and are functions of z and h i denots the average over z. In a vaccum ¯eld, Bµ dl = (r £ B) ¢dS = 0 is holds, so that the rotational transform angle is 0 in the ¯rst order of b=B0 . However the ¯rst order components of Bµ and Bz resonate to yield the resultant second order rotational transform angle. The average method gives the formula of the rotational transform angle18;19 µ

¶ b =± 2¼ B

¶2

l3 2

µ

d dx

µ

Il Il0 x

¶¶

x=l®r

R : r

(17:17)

By use of the expansion Il (x) =

µ ¶l µ

x 2

1 1 1 + x2 + x4 + ¢ ¢ ¢ l! (l + 1)! 2!(l + 2)!

¶

we ¯nd µ

b ¶ =± 2¼ B

¶2 µ

1 2l l!

¶2

³

´

l5 (l ¡ 1)®R (l®r)2(l¡2) + : : : :

(l ¸ 2)

(17:18)

An example of the analysis of toroidal helical ¯eld is given in the ref.20. 17.2b Stellarator Devices Familiar helical ¯elds are of pole number l = 2 or l = 3. The three dimensional magnetic axis system of Heliac has l=1 component. When the ratio of the minor radius ah of a helical coil to

246

17 Non-Tokamak Con¯nement System

Fig.17.4 Cross-sectional views of helical coils in the l = 2 case. (a) standard stellarator. (b) heliotron/torsatron.

Fig.17.5 (a) Arrangement of elliptical coils used to produce an l = 2 linear helical ¯eld. (b) twisted toroidal coils that produce the l = 2 toroidal helical ¯eld.

the helical pitch length R=m (R is the major radius and m is the number of ¯eld periods) is much less than 1, that is, mah =R ¿ 1, the rotaional transform angle is ¶2 (r) = const: for l = 2 and ¶3 (r) = ¶(r=a)2 for l = 3. In this case the shear is small for the l = 2 con¯guration, and ¶3 (r) is very small in the central region for the l = 3 con¯guration. However, if mah =R » 1, then ¶2 (r) = ¶0 + ¶2 (r=z)2 + ¢ ¢ ¢, so that the shear can be large even when l = 2. The arrangement of coils in the l = 2 case is shown in ¯g.17.4. Figure 17.4 (a) is the standard type of stellarator,21;22 and ¯g.17.4(b) is a heliotron/torsatron type.23;24 Usually helical ¯elds are produced by the toroidal ¯eld coils and the helical coils. In the heliotron/torsatron con¯guration the current directions of the helical coils are the same so that the toroidal ¯eld and the helical ¯eld can be produced by the helical coils alone.27;28 Therefore if the pitch is properly chosen, closed magnetic surfaces can be formed even without toroidal ¯eld coils.29;30 The typical devices of this type are Heliotron E, ATF and LHD. The device of LHD is shown in ¯g.17.6(a). When elliptical coils are arranged as shown in ¯g.17.5(a), an l=2 helical ¯eld can be obtained.25 The currents produced by the twisted toroidal coil system shown in ¯g.17.5(b) can simulate the currents of toroidal ¯eld coils and the helical coils taken together.26 The typical devices of this modular coil type are Wendelstein 7AS and 7X. Modular coil system of Wendelstein 7X is shown in ¯g.17.6(b). For linear helical ¯elds the magnetic surface ª = rAµ exists due to its helical symmetry. However, the existence of magnetic surfaces in toroidal helical ¯elds has not yet been proven in the strict mathematical sense. According to numerical calculations, the magnetic surfaces exist in the central region near the magnetic axis, but in the outer region the lines of magnetic force behave ergodically and the magnetic surfaces are destroyed. Although the helical coils have a relatively complicated structure, the lines of magnetic force can be traced by computer, and the design of helical ¯eld devices becomes less elaborate. The e®ect of the geometrical error to the helical ¯eld can be estimated, and accurate coil windings are possible with numerically controlled devices (¢l=R < 0:05 » 0:1%).

17.2 Stellarator

247

Fig.17.6 (a) Schematic view of the LHD device in Toki (R=3.9m, a » 0:6, B=3T). (b) Modular coil system and a magnetic surface of the optimized stellerator Wendelstein 7-X in Greifswald (R=5.5m, a=0.55m, B=3T).

248

17 Non-Tokamak Con¯nement System

Fig.17.7 Variation of the magnitude B along the length l of line of magnetic force.

Fig.17.8 Orbit of helical banana ion trapped in helical ripple.

17.2c Neoclassical Di®usion in Helical Field For the analysis of classical di®usion due to coulomb collision, the study of the orbit of charged particles is necessary. In a helical ¯eld or even in a tokamak toroidal ¯eld produced by ¯nite number of coils, there is an asymmetric inhomogenous term in the magnitude B of magnetic ¯eld B ¼ 1 ¡ ²h cos(lµ ¡ m') ¡ ²t cos µ (17:19) B0 in addition to the toroidal term ¡²t cos µ. The variation of B along lines of magnetic force is shown in ¯g.17.7. Particles trapped by the inhomogeneous ¯eld of helical ripples drift across the magnetic surfaces and contribute to the particle di®usion in addition to the banana particles as was discussed in tokamak. The curvature of line of magnetic force near the helically trapped region is convex toward outward and is denoted by Rh . Helically trapped particles drift in 2 poloidal direction (µ direction) due to rB drift with the velocity of vh ¼ mv? =(qBRh ) (see ¯g.17.8). The angular velocity of poloidal rotation is wh = vh =r ¼ (r=Rh )(kT =qBr2 ):

(17:20)

In the case of linear helical ¯eld (²t = 0), helically trapped particles rotate along the magnetic surface. However in the case of toroidal helical ¯eld, the toroidal drift is superposed and the toroidal drift velocity is vv = kT =(qBR) in the vertical direction (see sec.3.5). When the e®ective collision time (ºe® )¡1 = (º=²h )¡1 is shorter than one period (!h )¡1 of poloidal rotation, the deviation of orbit of helical banana from the magnetic surface is ¢h1 = vv

²h kT 1 = ²h : º qBR º

Then the coe±cient of particle di®usion becomes31 Dh1 »

1=2 ²h ¢2h1 ºe®

=

3=2 ²h

µ

kT qBR

¶2

µ

1 kT 1 3=2 = ²2t ²h º qBr2 º

¶µ

kT qB

¶

:

17.2 Stellarator

249

Fig.17.9 Dependence of the neoclassical di®usion coe±cient of helical ¯eld on collision frequency. 3=2 ºp = (¶=2¼)vTe =R, ºb = ²t ºp , !h = ²h ∙Te =(qBr 2 ).

Since Rh » r=²h , the other expression is µ

1=2

Dh1 » °h ²h ²2t

!h º

¶µ

¶

kT ; qB

(º=² > !h )

(17:21)

where °h is a coe±cient with the order of O(1) (¯g.17.9). When the e®ective collision time (ºe® )¡1 is longer than (!h )¡1 , the deviation ¢h2 of the orbit and the magnetic surface is ¢h2 ¼ vv =!h ¼

Rh ²t r » r; R ²h

and the Dh2 becomes (¯g.17.9) 1=2

Dh2 ¼ ²h ¢2h2 ºe® =

³

²t ²h

´2

1 2 1=2 r º: ²h

(º=²h < !h )

When a particle is barely trapped in a local helical mirror, the particle moves very slowly near the re°ection point where the magnetic ¯eld is locally maximum and the ¯eld line is concave to outward. The e®ective curvature, which the particle feels in time average, becomes negative (concave). The orbit of the trapped particle in this case becomes so called superbanana 31 . However this theoretical treatment is based on the assumption of the longitudinal adiabatic invariant Jk =const. along the orbit of helically trapped particle. The adiabatic invariance is applicable when the poloidal rotation angle, during the one period of back and forth motion in the helical local mirror, is small. As the one period of back and forth motion of barely trapped particles becomes long, the adiabatic invariance may not be applicable in this case. The orbit trace by numerical calculations shows that the superbanana does not appear32 in the realistic case of ²h » ²t . If a particle orbit crosses the wall, the particle is lost. This is called orbit loss. A loss region in velocity space appears due to orbit loss in some case.33 When a radial electric ¯eld appears, the angular frequency of the poloidal drift rotation becomes !h + !E (!E = Er =B0 ), the orbit is a®ected by the radial electric ¯eld. The thermal di®usion coe±cient Âh1 due to helically trapped particles in the region of º=²h > !h is given by 3=2

Âh1 » °T ²2t ²h

µ

kT qBr

¶2

1 : º

(°T » 50)

(17:22)

Since º / T ¡1:5 , it means Âh1 / T 3:5 . This may suggest the serious problem that the thermal conduction loss becomes large in hot plasma.31;34;35;36

250

17 Non-Tokamak Con¯nement System

Fig.17.10 Equidensity contours of plasmas con¯ned in JIPP I-b stellarator (l = 2) with the rotational transform angles of ¶=2¼ = 1=2;1=3 and 0:56:38

17.2d Con¯nement of Stellarator System37;38;39 After Stellarator C, the basic experiments were carried out in small but accurate stellarator devices (Clasp, Proto Cleo, Wendelstein IIb, JIPP I, Heliotron D, L1, Uragan 1). Alkali plasmas, or afterglow plasmas produced by wave heating or gun injection, were con¯ned quiescently. The e®ect of shear on the stability and con¯nement scaling were investigated. The l=2 stellarators with long helical pitch, such as Wendelstein IIa or JIPP I-b, have nearly constant rotational transform angles and the shears are small. When the transform angle is rational, ¶=2¼ = n=m, a line of magnetic force comes back to the initial position after m turns of the torus and is closed. If electric charges are localized in some place, they cannot be dispersed uniformly within the magnetic surface in the case of rational surfaces. A resistive drift wave or resistive MHD instabilities are likely to be excited, and convective loss is also possible. The enhanced loss is observed in the rational case (¯g.17.10). This is called resonant loss. Resonant loss can be reduced by the introduction of shear. Medium-scale stellarator devices (Wendelstein VIIA, Cleo, JIPP T-II, Heliotron-E, L2, Uragan 2, Uragan 3) have been constructed. The con¯nement time of the ohmically heated plasmas (Te < 1 keV) is smilar to that of tokamaks with the same scale. When the rotational transform angle is larger than ¶h =2¼ > 0:14, the major disruption observed in tokamaks is suppressed (W VIIA, JIPP T-II). NBI heating or wave heatings, which were developed in tokamaks, have been applied to plasma production in helical devices. In Wendelstein VIIA, a target plasma was produced by ohmic heating; then the target plasma was sustained by NBI heating while the plasma current was gradually decreased, and ¯nally a high-temperature plasma with ∙Ti » several hundred eV, ne » several 1013 cm¡3 was con¯ned without plasma current (1982). In Heliotron-E, a target plasma was produced by electron cyclotron resonance heating with ∙Te » 800 eV, ne » 0:5 £ 1013 cm¡3 , and the target plasma was heated by NBI heating with 1.8 MW to the plasma with ∙Ti » 1keV; ne = 2 £ 1013 cm¡3 (1984). The average beta, h¯i » 2%, was obtained in the case of B = 0:94 T and NBI power PNB » 1 MW: These experimental results demonstrate the possibility of steady-state con¯nement by stellarator con¯gurations. Experimental scaling laws of energy con¯nement time are presented from Heliotron-E group40 as follows; 0:84 ¡0:58 P ¿ELHD = 0:17a2:0 R0:75 n0:69 20 B

(17:23)

17.3 Open End Systems

251

where the unit of n20 is 1020 m¡3 . W7AS group presented W7AS con¯nement scaling41 of ¿EW7AS

=

0:73 ¡0:54 0:115A0:74 a2:95 n0:5 P 19 B

µ

¶ 2¼

¶0:43

(17:24)

¶0:4

(17:25)

The scaling law of the international stellarator database41 is ¿EISS95

= 0:079a

2:21

0:83 ¡0:59 R0:65 n0:51 P 19 B

µ

¶ 2¼

where the unit of n19 is 1019 m¡3 and ¶=2¼ is the value at r = (2=3)a. Units are s, m, T, MW. A high con¯nement NBI discharge was observed in WVII-AS with the improved factor of »2 against W7AS scaling in 199842 . Large helical device (LHD)43 with superconductor helical coils started experiments in 1998 (refer ¯g.17.6(a)) and advanced stellarator Wendelstein 7-X44 with superconductor modular coils is under construction (refer ¯g.17.6(b)). 17.3 Open End Systems Open end magnetic ¯eld systems are of a simpler con¯guration than toroidal systems. The attainment of absolute minimum-B con¯gurations is possible with mirror systems, whereas only average minimum-B con¯gurations can be realized in toroidal systems. Although absolute minimum-B con¯gurations are MHD stable, the velocity distribution of the plasma becomes non-Maxwellian due to end losses, and the plasma will be prone to velocity-space instabilities. The most critical issue of open-end systems is the supression of end loss. The end plug of the mirror due to electrostatic potential has been studied by tandem mirrors. The bumpy torus, which is a toroidal linkage of many mirrors, was used in other trials to avoid end loss. A cusp ¯eld is another type of open-end system. It is rotationally symmetric and absolute minimum-B. However, the magneitc ¯eld becomes zero at the center, so the magnetic momentum is no longer invariant and the end loss from the line and point cusps are severe. 17.3a Con¯nement Times in Mirrors and Cusps Particles are trapped in a mirror ¯eld when the velocity components v? and vk , perpendicular and parallel to the magnetic ¯eld, satisfy the condition 2 b0 v? > ; 2 v bL

2 (v 2 = v? + vk2 )

where b0 and bL are the magnitudes of the magnetic ¯eld at the center and at the end, respectively. Denoting the mirror ratio bL =b0 by RM , so that 1 b0 = = sin2 ®L bL RM the trapping condition is reduced to v? > sin ®L : v When the particles enter the loss cone, they escape immediately, so that the con¯nement time is determined by the velocity space di®usion to the loss cone. The particle con¯nement time ¿p of a mirror ¯eld is essentially determined by the ion-ion collision time ¿ii as follow:45 ¿p ¼ ¿ii ln RM :

(17:26)

Even if the mirror ratio RM is increased, the con¯nement time is increased only as fast as ln RM . The time is independent of the magnitude of the magnetic ¯eld and the plasma size. If the

252

17 Non-Tokamak Con¯nement System

Fig.17.11 Line cusp width ±L and point cusp radius ±p :

density is n ¼ 1020 m¡3 , the ion temperature ∙Ti ¼ 100keV, the atomic mass number A = 2, and ion charge Z = 1, the ion-ion collision time is ¿ii = 0:3s, so that n¿p = 0:3 £ 1020 m¡3 s. This value is not enough for a fusion reactor. It is necessary to increase the e±ciency of recovery of the kinetic energy of charged particles escaping through the ends or to suppress the end loss. Next let us consider cusp losses. Since the magnetic ¯eld is zero at the cusp center 0, the magnetic moment is not conserved. Therefore the end losses from the line cusp and the point cusp (¯g.17.11) are determined by the hole size and average ion velocity v¹i . The °ux of particles from the line cusp of width ±L and radius R is given by 1 FL = nL v¹i 2¼R ¢ ±L ; 4

v¹i =

µ

8 ∙Ti ¼ M

¶1=2

where nL is the particle density at the line cusp and Ti and M are the temperature and mass of the ions. The °ux of particles from two point cusps of radius ±p is46 1 Fp = np v¹i ¼±p2 £ 2: 4 The observed value of ±L is about that of the ion Larmor radius. As the lines of magnetic force are given by rAµ = ar2 z = const: we have ±p2 R ¼ R2

±L ; 2

±p2 =

R±L 2

so that Fp ¼ FL =2: The total end-loss °ux is µ

np 1 F ¼ v¹i 2¼R±L nL + 4 2

¶

and the volume V of the cusp ¯eld is (¯g.17.11) µ

V = ¼R2 ±L ln

¶

2R +1 : ±L

(17:27)

The con¯nement time of cusp is now given by ¿=

µ

¶

R 4 2R n nV ¼ ln +1 F v¹i ±L 3 nL + np =2

(17:27)

17.3 Open End Systems

253

Fig.17.12 Minimum-B mirror ¯eld. (a) Coil system with mirror coil (A) and stabilizing coils (Io®e bars) (B). (b) Magnitude of the magnetic ¯eld in a simple mirror. (c) Magnitude of the magnetic ¯eld in the minimum-B mirror ¯eld. (d) The shape of a quadrupole minimum-B (¯shtail).

where n is the average density. When R = 10 m; B = 10T; ∙Ti = 20keV; A = 2; and ±L = ½iB , then ¿ ¼ 0:1 ms, which is 10-20 times the transit time R=¹ vi . Even if the density is n ¼ 1022 m¡3 , the product of n¿ is of the order of 1018 m¡3 s, or 10¡2 times smaller than fusion plasma condition. 17.3b Con¯nement Experiments with Mirrors One of the most important mirror con¯nement experiments was done by Io®e and his colleagues, who demonstrated that the minimum-B mirror con¯guration is quite stable.47 Addition of stabilizing coils as shown in ¯g.17.12(a) results in a mirror ¯eld with the minimum-B property (¯g.17.12(c)). De¯ne the wall mirror ratio as the ratio of the magnitude of the magnetic ¯eld at the radial boundary to its magnitude at the center. When the wall mirror ratio is increased, °ute instability disappears and the con¯nement time becomes long. In the PR-5 device, a plasma of density 109 » 1010 cm¡3 and ion temperature 3 » 4 keV has been con¯ned for 0.1 s.48 However, when the density is larger than 1010 cm¡3 (¦i > j­i j), the plasma su®ers from loss-cone instability. One of the most promising mirror con¯nement results was brought about in the 2X experiments.49 Here a plasmoid of mean energy of 2.5 keV produced by a plasma gun is injected into a quadrupole mirror and then compressed. The initial density of the trapped plasma is n ¼ 3 £ 1013 cm¡3 , and n¿ ¼ 1010 cm¡3 s. The magnitude of the magnetic ¯eld is 1.3 T and ¯ ¼ 5%. The average energy of the ions is 6-8 keV, and the electron temperature is about 200 eV. Since n¿ for an ideal case (classical coulomb collision time) is n¿ ¼ 3 £ 1010 cm¡3 s, the results obtained are 1=3 » 1=15 that of the ideal. In the 2XIIB experiment,50 microscopic instability is suppressed by adding a small stream of warm plasma to smooth out the loss cone. A 15-19 keV, 260 A neutral beam is injected, and the resulting plasma, of 13 keV ion temperature, is con¯ned to n¿ ¼ 1011 cm¡3 s (n ¼ 4 £ 1013 cm¡3 ). 17.3c Instabilities in Mirror Systems Instabilities of mirror systems are reviewed in refs.51-53. MHD instability can be suppressed by the minimum-B con¯guration. However, the particles in the loss-cone region are not con¯ned, and the non-Maxwellian distribution gives rise to electrostatic perturbations at the ion cyclotron frequency and its harmonics, which scatter the particles into the loss cone region and so enhance the end loss. It can be shown that instabilities are induced when the cyclotron wave couples with other modes, such as the electron plasma wave or the drift wave. (i) Instability in the Low-Density Region (¦e ¼ lj­i j)

Let us consider the low-density

254

17 Non-Tokamak Con¯nement System

case. When the plasma electron frequency ¦e reaches the ion cyclotron frequency j­i j, there is an interaction between the ion Larmor motion and the electron Langmuir oscillation, and Harris instability occurs53 (sec.13.4). When the density increases further, the oblique Langmuir wave with ! = (kk =k? )¦e couples with the harmonics lj­i j of the ion cyclotron wave. The condition ! ¼ k? v?i is necessary for the ions to excite the instability e®ectively; and, if ! > 3kk vke , then the Landau damping due to electrons is ine®ective. Thus excitation will occur when ! ¼ lj­i j ¼ ¦e kk =k ¼ k? v?i > 3kk vke where kk ; k? are the parallel and perpendicular components of the propagation vector and vk ; v? are the parallel and perpendicular components of the velocity. Therefore the instability condition is ¦e > lj­i j

Ã

1+

2 9vke 2 v?i

!1=2

µ

9M Te ¼ 1+ mTi

¶1=2

:

(17:28)

Let L be the length of the device. As the relation ∙k > 2¼=L holds, we have55;56 L 6¼ vke > : ½­i l v?i Harris instability has been studied in detail experimentally.57 (ii) Instability in the High Density (¦i > j­i j) When the density increases further, so that ¦i is larger than j­i j (while ­e > ¦i still holds), loss-cone instability with !r ¼ !i ¼ ¦i will occur.58 This is a convective mode and the length of the device must be less than a critical length, given by Lcrit = 20A½­i

Ã

!1=2

­e2 +1 ¦e2

(17:29)

for stability. Here A is of the order of 5 (A ¼ 1 for complete re°ection and A ¼ 10 for no re°ection at the open end). Therefore the instability condition is L > 100½iB . Loss-cone instability can occur for a homogeneous plasma. When there is a density gradient, this type of instability couples with the drift wave, and drift cyclotron loss-cone instability may occur.57 When the characteristic length of the density gradient is comparable to the radial dimension Rp of the plasma, the instability condition of this mode is 0

Ã

¦i ¦2 1 + e2 Rp < ½­i @ j­i j ­e

!¡1=2 14=3 A :

(17:30)

(iii) Mirror Instability When the beta ratio becomes large, the anisotropy of plasma pressure induces electromagnetic-mode mirror instabilities. (Note that the Harris and loss-cone instabilities are electrostatic.) The instability condition is Ã

!

T?i ¡ 1 2¯ > 1: Tki

(17:31)

To avoid the instabilities described in (i)-(iii), the following stability conditions must be met: L < 100 » 200½­i ;

(17:32)

17.3 Open End Systems

255

Fig.17.13 Negative mass instability.

Rp > 200½­i ; 1 ¯< 2

Ã

T?i ¡1 Tki

(17:33) !¡1

¼ 0:3 » 0:5:

(17:44)

(iv) Negative Mass Instability Let us assume that charged particles in Larmor motion are uniformly distributed at ¯rst and then a small perturbation occurs, so that positive charge (for example) accumulates at the region A shown in ¯g.17.13. The electric ¯eld decelerates ions in the region to the right of A, and their kinetic energy ² corresponding to the velocity component perpendicular to the magnetic ¯eld is decreased. Ions to the left of A are accelerated, and their ² is increased. When the rotational frequency ! depends on ² through the relation d!=d² < 0, the frequency ! of the ions in the region to the right of A is increased and the ions approach A despite the deceleration. These ions behave as if they had negative mass. This situation is exactly the same for the ions in the left-hand region. Therefore the charge accumulates at A, and the perturbation is unstable. This type of instability is called negative mass instability.59 The condition d!=d² < 0 is satis¯ed when the magnitude of the magnetic ¯eld decreases radially. Thus, as expected, simple mirrors, where B decreases radially and the Larmor radius is large, have been reported to exhibit negative mass instability.60 The PR-5 device is of minimum-B con¯guration, so that the magnitude of the magnetic ¯eld increases radially. However, here another type of negative mass instability is observed.61 When the perpendicular energy ² decreases, ions can enter the mirror region more deeply, so that the ion cyclotron frequency is increased. Thus the condition d!=d² < 0 is satis¯ed even in PR-5. (v) Instability in Hot Electron Plasmas A hot electron plasma can be produced by electron cyclotron resonant heating in mirror ¯elds. The temperature of the hot component is raised up to the range of several keV to several hundred keV, and the density range is 1010 » 1011 cm¡3 . The electromagnetic whistler instability61;62 is excited by anistropy of the velocity distribution function (T? > Tk ). This whistler instability of hot electron plasmas has been observed experimentally.63

17.3d Tandem Mirrors The input and output energy balance of a classical mirror reactor is quite marginal even in the ideal con¯nement condition. Therefore the suppression of the end loss is the critical issue for realistic mirror reactors. This section describes the research on the end plug by the use of electrostatic potential. In a simple mirror case, the ion con¯nement time is of the order of ion-ion collision time, and the electron con¯nement time is of the order of electron-electron or electron-ion collision times,(¿ee » ¿ei ). Since ¿ee ¿ ¿ii , the plasma is likely to be ion rich, and the plasma potential in the mirror tends to be positive. When the two mirrors are arranged in the ends of th central mirror in tandem, it is expected that the plasma potentials in the plug

256

17 Non-Tokamak Con¯nement System

Fig.17.14 The magnitudes of magnetic ¯eld B(z), electrostatic potential Á(z), and density n(z) along the z axis (mirror axis) of a tandem mirror.

Fig.17.15 Loss-cone regions of an ion (q = Ze) and electron (¡e) for positive electrostatic potential Ác > 0 2and the mirror 2ratio Rm . (v?c =v)2 = (eÁc =(Rm ¡ 1))=(me v 2 =2); in the electron case. (vkc =v) = ZeÁc =(mi v =2) in the ion case.

mirrors (plug cell) at both ends become positive with respect to the potential of the central mirror (central cell). This con¯guration is called a tandem mirror.64;65 The loss-cone region in velocity space of the tandem mirror, as shown in ¯g.17.14, is given by µ

v? v

¶2

µ

1 qÁc < 1¡ Rm mv 2 =2

¶

(17:35)

where q is the charge of the particle and Rm is the mirror ratio. The loss-cone regions of electrons and ions are shown in ¯g.17.15 in the positive case of the potential Ác . Solving the Fokker-Planck equation, Pastukhov66 derived the ion con¯nement time of a tandem mirror with positive potential as ¿PAST

µ

¶

µ

eÁc eÁc = ¿ii g(Rm ) exp ∙Tic ∙Tic

¶

(17:36)

g(Rm ) = ¼ 1=2 (2Rm + 1)(4Rm )¡1 ln(4Rm + 2) where Tic is the ion temperature of the central cell and Ác is the potential di®erence of the plug cells in both ends with respect to the central cell. Denote the electron densities of the central cell and the plug cell by nc and np respectively; then the Boltzmann relation np = nc exp(eÁc =∙Te ) gives µ

np ∙Te Ác = ln e nc

¶

:

(17:37)

17.3 Open End Systems

257

Fig.17.16 The magnitudes of magnetic ¯eld B(z), electrostatic potential Á(z), density n(z), and electron temperature Te (z) at the thermal barrier. C.M., central mirror. T.B., thermal barrier. P, plug mirror.

By application of neutral beam injection into the plug cell, it is possible to increase the density in the plug cell to be larger than that of the central cell. When Rm » 10 and eÁc =∙Tic » 2:5; ¿PAST » 100¿ii , so that the theoretical con¯nement time of the tandem mirror is longer than that of the simple mirror. In this type of tandem mirror it is necessary to increase the density np in the plug cell in order to increase the plug potential Ác , and the necessary power of the neutral beam injection becomes large. Since the plug potential Ác is proportional to the electron temperature Te , an increase in Te also increases Ác . If the electrons in the central cell and the electrons in the plug cells can be thermally isolated, the electrons in the plug cells only can be heated, so that e±cient potential plugging may be expected. For this purpose a thermal barrier67 is introduced between the central cell and the plug cell, as shown in ¯g.17.16. When a potential dip is formed in the thermal barrier in an appropriate way, the electrons in the plug cell and the central cell are thermally isolated. Since the electrons in the central cell are considered to be Maxwellian with temperature Tec , we have the relation nc = nb exp

µ

¶

eÁb : ∙Tec

(17:38)

The electrons in the plug cell are modi¯ed Maxwellian,68 and the following relation holds: np = nb exp

Ã

e(Áb + Ác ) ∙Tep

where º » 0:5. These relations reduce to Ã

np ∙Tep Ác = ln e nb

Ã

Tec Tep

!

£

!º !

µ

¡

Tep Tec

¶º

(17:39)

µ

∙Tec nc ln e nb

¶

258

17 Non-Tokamak Con¯nement System

∙Tep = ln e Áb =

à µ

np nc

nc ∙Tec ln e nb

à ¶

Tec Tep

:

!º !

+

µ

¶

µ

¶

Tep ∙Tec nc ¡1 ; ln Tec e nb

(17:40)

(17:41)

Therefore, if the electron temperature Tep in the plug cell is increased, Ác can be increased without the condition np > nc ; thus e±cient potential plugging may be expected. Experiments with tandem mirrors have been carried out in TMX-U, GAMMA-10 and others. References 1. D. C. Robinson and R. E. King: Plasma Physics and Controlled Nuclear Fusion Research 1, 263 (1969) (Conference Proceedings, Novosibirsk in 1968) IAEA Vienna 2. H. A. B. Bodin and A. A. Newton: Nucl. Fusion 20, 1255 (1980) 3. H. A. B. Bodin: Plasma Phys. and Controlled Fusion 29, 1297 (1987) 4. MST Team: Plasma Physics and Controlled Nuclear Fusion Research 2, 519 (1991) (Conference Proceedings, Washington D. C. in 1990) IAEA Vienna TPE-1RM20 Team: 19th Fusion Energy Conference 2, 95 (1997) (Conference Proceedings, Montreal in 1996) IAEA Vienna 5. EX4/3(RFX), EX4/4(TPE-RX): 17th Fusion Energy Conference 1 367, 375 (1999) (Conference Proceedings, Yokohama in 1998 IAEA Vienna 6. J. B. Taylor: Phys. Rev. Lett. 33, 1139 (1974) 7. T. H. Jensen and M. S. Chu: Phys. Fluids 27, 2881 (1984) 8. V. D. Shafranov and E. I. Yurchenko: Sov. Phys. JETP 26, 682 (1968) 9. D. A. Backer, M. D. Bausman, C. J. Buchenauer, L. C. Burkhardt, G. Chandler, J. N. Dimarco et al.: Plasma Physics and Controlled Nuclear Fusion Research 1, 587 (1983) (Conference Proceeding, Baltimore in 1982) IAEA Vienna 10. D. D. Schnack, E. J. Caramana and R. A. Nebel: Phys. Fluids 28, 321 (1985) 11. K. Kusano and T. Sato: Nucl. Fusion 26, 1051 (1986) 12. K. Miyamoto: Plasma Phys. and Controlled Fusion 30, 1493 (1988) 13. A. R. Jacobson and R. W. Moses: Phys. Rev. A29 3335 (1984) 14. R. W. Moses, K. F. Schoenberg and D. A. Baker: Phys. Fluids 31 3152 (1988) 15. M. K. Bevir and J. W. Gray: Proc of Reversed Field Pinch Theory Workshop (LANL Los Alamos, 1981) Report No{8944{C, p.176 M. K. Bevir and C. G. Gimblett: Phys. Fluids 28, 1826 (1985) 16. K. F. Schoenberg, J. C. Ingraham, C. P. Munson, P.G. Weber et al: Phys. Fluids 31, 2285 (1988) 17. L. Spitzer, Jr.: Phys. Fluids 1, 253 (1958) 18. A. I. Morozov and L. S. Solov¶ev: Reviews of Plasma Physics 2, 1 (ed. by M. A. Leontovich) Consultants Bureau, New York 1966 19. K. Miyamoto: Plasma Physics for Nuclear Fusion (revised edition) Chap.2 The MIT Press 1989 20. K. Nagasaki, K. Itoh, M. Wakatani and A. Iiyoshi: J.Phys. Soc. Japan 57, 2000 (1988) 21. W VIIA Team: Plasma Physics and Controlled Nuclear Fusion Research 2, 241 (1983) (Conference Proceedings, Baltimore in 1982) IAEA Vienna 22. E. D. Andryukhina, G. M. Batanov, M. S. Berezhetshij, M. A. Blokh, G. S. Vorosov et al.: Plasma Physics and Reseach Controlled Nuclear Fusion 2, 409 (1985) (Conference Proceedings, London in 1984) IAEA Vienna 23. K. Uo, A. Iiyoshi, T. Obiki, O. Motojima, S. Morimoto, M. Wakatani et al.: Plasma Physics and Controlled Nuclear Fusion Research 2, 383 (1985) (Conference Proceedings, London in 1984) IAEA Vienna 24. L. Garcia, B. A. Carreras and J. H. Harris: Nucl. Fusion 24, 115 (1984) 25. Yu. N. Petrenlco and A. P. Popyyadukhin: The 3rd International Symposium on Toroidal Plasma Con¯ne ments, D 8 March 1973 Garching 26. H. Wobig and S. Rehker: Proceedings of the 7th Symposium on Fusion Technology, Grenoble 345 (1972) S. Rehker and H. Wobig: Proceedings of the 6th European Conference on Controlled Fusion and Plasma Physics p.117 in Moscow(1973), IPP 2/215 Max Planck Inst. of Plasma Phys. (1973) 27. C. Gourdon, D. Marty, E. K. Maschke and J. P. Dumont: Plasma Physics and Controlled Nuclear Fusion Research 1, 847 (1969) (Conference Proceedings, Novosibirsk in 1968) IAEA Vienna 28. K. Uo: Plasma Phys. 13, 243 (1971) 29. A. Mohri: J. Phys. Soc. Japan 28, 1549 (1970)

References

30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

259

C. Gourdon, D. Marty, E. K. Maschke and J. Touche: Nucl. Fusion 11, 161 (1971) B. B. Kadomtsev and O. P. Pogutse: Nucl. Fusion 11, 67 (1971) J. A. Derr and J. L. Shohet: Phys. Rev. Lett. 44, 1730 (1979) M. Wakatani, S. Kodama, M. Nakasuga and K. Hanatani: Nucl. Fusion 21,175 (1981) J. W. Connor and R. J. Hastie: Phys. Fluids 17, 114 (1974) L. M. Kovrizhnykh: Nucl. Fusion 24, 851 (1984) D. E. Hastings, W. A. Houlberg and K.C.Shaing: Nucl. Fusion 25, 445 (1985) K. M. Young: Plasma Phys. 16, 119 (1974) K. Miyamoto: Nucl. Fusion 18, 243 (1978) B. A. Carreras, G. Griegen, J. H. Harris, J. L. Johnson, J. F. Lyon, O. Motojima, F. Rau, H. Renner, J. A. Rome, K. Uo, M. Wakatani and H. Wobig: Nucl. Fusion 28, 1613 (1988) 40. S. Sudo, Y. Takeiri, Z. Zushi, F. Sano, K. Itoh, K. Kondo and A. Iiyoshi: Nucl. Fusion 30, 11 (1990) 41. U. Stroth, M. Murakami, R. A. Dory, H. Yamada, S. Okamura, F. Sano, T. Obiki: Nucl. Fusion 36, 1063, (1996) 42. U. Stroth et al: Plasma Phys. Control. Fusion 40, 1551 (1998) 43. A. Iiyoshi, A. Komori, A. Ejiri, M. Emoto, Funaba et al: Nucl Fusion 39, 1245 (1999) O. Motojima, N. Ohyabu, A. Komori, N. Noda, K. Yamazaki et al: Plasma Phys. Control. Fusion 38, A77, (1996) 44. H. Wobig: Plasma Phys. Control. Fusion 41, A159, (1999) 45. D. V. Sivukhim: Reviews of Plasma Physics 4, 93 (ed. by M. A. Leontovich) Consultants Bureau, New York 1966 46. I. J. Spalding: Nucl. Fusion 8, 161 (1968) I. J. Spalding: Advances in Plasma Physics 4, 79 (ed. by A. Simon and W. B. Thompson), Interscience, New York (1971) 47. Yu. V. Gott, M. S. Io®e and V. G. Tel'kovskii: Nucl. Fusion Suppl. Pt.3, 1045 (1962) (Conference Proceedings, Salzburg in 1961) IAEA Vienna Yu. T. Baiborodov, M. S. Io®e, V. M. Petrov and R. I. Sobolev: Sov. Atomic Energy 14, 459 (1963) 48. Yu. T. Baiborodov, M. S. Io®e, B. I. Kanaev, R. I. Sobolev and E. E. Yushimanov: Plasma Physics and Controlled Nuclear Fusion Research 2, 697 (1971) (Conference Proceedings, Madison in 1971) IAEA Vienna 49. F. H. Coensgen, W. F. Cummins, V. A. Finlayson, W. E. Nexsen, Jr. and T. C. Simonen: ibid. 2, 721 (1971) 50. F. H. Coensgen, W. F. Cummins, B. G. Logan, A. W. Halvik, W. E. Nexsen, T. C. Simonen, B. W. Stallard and W. C. Turne: Phys. Rev. Lett. 35, 1501(1975) 51. ed. by T. K. Fowler: Nucl. Fusion 9, 3 (1969) 52. M. S. Io®e and B. B. Kadomotsev: Sov. Phys. Uspekhi 13, 225 (1970) 53. R. F. Post: Nucl. Fusion 27, 1579 (1987) 54. E. G. Harris: Phys. Rev. Lett. 2,34 (1959) 55. R. A. Dory, G. E. Guest, E. G. Harris: Phys. Rev. Lett. 14, 131 (1965) 56. G. E. Guest and R. A. Dory: Phys. Fluids 8, 1853 (1965) 57. J. Gordey, G. Kuo-Petravic, E. Murphy, M. Petravie, D. Sweetman and E. Thompson: Plasma Physics and Controlled Nuclear Fusion Research 2, 267 (1969) (Conference Proceedings, Novosibirsk in 1968) IAEA Vienna 58. R. F. Post and M. N. Rosenbluth: Phys. of Fluids 9, 730 (1966) 59. H. Postman, H. Dunlap, R. Dory, G. Haste and R. Young: Phys. Rev. Lett. 16, 265 (1966) 60. B. B. Kadomtsev and O. P. Pogutse: Plasma Physics and Controlled Nuclear Fusion Research 2, 125 (1969) (Conference Proceedings, Novosibirsk in 1968) IAEA Vienna 61. R. Z. Sagdeev and V. D. Shafranov: Sov. Phys. JETP 12, 130 (1961) 62. J. Sharer and A. Trivelpiece: Phys. Fluids 10, 591 (1967) 63. H. Ikegami, H. Ikezi, T. Kawamura, H. Momota, K. Takayama and Y. Terashima: Plasma Physics and Controlled Nuclear Fusion Research 2, 423 (1969) (Conference Proceedings, Novosibirsk in 1968) IAEA Vienna 64. G. I. Dimov, V. V. Zakaidakov and M. E. Kishinevskii: Sov. J. Plasma Phys. 2, 326 (1976) 65. T. K. Fowler and B. G. Logan: Comments Plasma Phys. Contorolled Fusion Res. 2,167 (1977) 66. V. P. Pastukhov: Nucl. Fusion 14, 3 (1974) 67. D. E. Baldwin and B. G. Logan: Phys. Rev. Lett. 43, 1318 (1979) 68. R. H. Cohen, I. B. Bernstein, J. J. Dorning and G. Rowland: Nucl. Fusion 20, 1421 (1980)

260

18 Inertial Con¯nement

Ch.18 Inertial Con¯nement The characteristic of inertial con¯nement is that the extremely high-density plasma is produced by means of an intense energy driver, such as a laser or particle beam, within a short period so that the fusion reactions can occur before the plasma starts to expand. Magnetic con¯nement play no part in this process, which has come to be called inertial con¯nement. For fusion conditions to be reached by inertial con¯nement, a small solid deuterium-tritium pellet must be compressed to a particle density 103 » 104 times that of the solid pellet particle density ns = 4:5 £ 1022 cm¡3 . One cannot expect the laser light pressure or the momentum carried by the particle beam to compress the solid pellet: they are too small. A more feasible method of compression involves irradiating the pellet from all side, as shown in ¯g.18.1. The plasma is produced on the surface of the pellet and is heated instantaneously. The plasma expands immediately. The reaction of the outward plasma jet accelerates and compresses the inner pellet inward like a spherical rocket. This process is called implosion. The study of implosion processes is one of the most important current issues, and theoretical and experimental research is being carried out intensively. 18.1 Pellet Gain1;2 Let the pellet gain ´G be the ratio of the output nuclear fusion energy ENF to the input driver energy EL delivered to the pellet. The heating e±ciency ´h of the incident driver is de¯ned by the conversion ratio of the driver energy EL to the thermal energy of the compressed pellet core. Denote the density and the volume of the compressed core plasma by n and V , respectively, and assume that Te = Ti = T : the following relation can be obtained: 3n∙T V = ´h EL :

(18:1)

The densities nD , nT of the duetrium and tritium are decreased by the D-T fusion reaction (nD = nT = n=2) and 1 dnD = ¡nT h¾vi; nD dt n(t) = n0

1 : 1 + n0 h¾vit=2

When the plasma is con¯ned during the time ¿ , the fuel-burn ratio ´b is given by ´b ´

n0 ¡ n(¿ ) n0 h¾vi¿ =2 = n0 1 + n0 h¾vi¿ =2

(18:2)

and the fusion output energy ENF is ENF = ´b nV ®

QNF : 2

(18:3)

The coe±cient ® is an enhancement factor due to the fusion reaction occuring in the surrounding plasma of the core, which are heated by ® particles released from the hot compressed core. Then the pellet gain ´G is reduced to µ

¶

ENF QNF ´G ´ = ´h ® ´b : EL 6∙T

(18:4)

18.1 Pellet Gain

261

Fig.18.1 Conceptual drawing of implosion. (a) Irradiation by laser or particle beam from all side. (b) Expansion of plasma from the pellet surface and the implosion due to the reaction of the outward plasma jet.

Fig.18.2 Energy °ow diagram of an inertial con¯nement reactor.

The ratio ´NF ´

QNF =2 ® 3∙T

(18:5)

is the thermonuclear fusion gain, since one ion and one electron are of energy 3T before fusion and QNF =2 is the fusion output energy per ion. As QNF = 17:6 MeV, then ´NF ¼ 293®=(∙T )10 , where (∙T )10 is the temparature in 10 keV. Equations (18.4) (18.5) yield the pellet gain ´G = ´h ´b ´NF :

(18:6)

Consider the energy balance of a possible inertial fusion reactor. The conversion e±ciency of the thermal-to-electric energy is about ´el » 0:4, and the conversion e±ciency of the electric energy to output energy of the driver is denoted by ´L . Then the condition ´el ´L ´G > 1 is at least necessary to obtain usable net energy from the reactor (see ¯g.18.2) (When ´L » 0:05; ´el » 0:4 are assumed, ´G > 50 is necessary). Therefore we ¯nd ´G = ´h

µ

QNF ® 6∙T

¶µ

nh¾vi¿ 2 + nh¾vi¿

¶

>

1 ´el ´L

(18:7)

that is, n¿ >

1 12∙T : ´el (´L ´h )®QNF h¾vi [1 ¡ (´el ´L ´h ´NF )¡1 ]

(18:8)

Here the following condition is necessary; ´el ´L ´h ´NF > 1:

(18:9)

When ´el » 0:4; ´L » 0:05; ´h » 0:1; ∙T » 10 keV, then ´el ´L ´h ´NF » 6®=∙T (keV) » 0:6® > 1 is a necessary condition.

262

18 Inertial Con¯nement

The con¯nement time ¿ is the characteristic expansion time and is expressed by r ; 3cs

¿¼

c2s =

5 p 10 ∙T = 3 ½m 3 mi

(18:10)

where cs is the sound velocity. Since the volume V of the core is V =

4¼r3 3

eq.(18.1) reduces to EL =

4¼ n∙T r 3 : ´h

(18:11)

Equations (18.2) and (18.10) yield ¿=

´b 2 ; (1 ¡ ´b ) nh¾vi µ

∙T r = 5:5 mi

¶1=2

´b0 ´

´b ; 1 ¡ ´b

¿:

(18:12)

(18:13)

When ∙T = 10 keV, and the plasma density is expressed in units of solid density ns = 4:5 £ 1028 m¡3 , then h¾vi = 1:3£10¡22 m3 s¡1 ; mi = 2:5mp (mp is proton mass) and eqs.(18.12),(18.13), (18.11) yield ¿ = 0:34 £ 10¡6 ´b0 r=

1:1´b0

µ

µ

¶

ns ; n

(s)

¶

ns ; n

EL = 1:2 £ 1015

(´b0 )3 ´h

(m) µ

ns n

¶2

(18:14)

: (J)

When the plasma is compressed to 103 times the solid density n = 103 ns ; and ´b0 » 0:1 is assumed, we have ¿ = 34 ps;

r = 0:11 mm;

EL = 12 MJ

and n¿ » 1:5 £ 1021 m¡3 ¢ s. The necessary condition (18.7) is ® > 19 in this case. Equation (18.14) is equivalent to r½m » 2g=cm2 when the mass density ½m = 2:5 mp n of the compressed core is used (mp is the proton mass). The critical issue for an inertial fusion reactor are how extremely high-density plasmas can be produced by implosion and how the enhancement factor of fusion output in the surrounding plasma heated by ® particles released from the compressed core can be analyzed. Optimum design of fuel pellet structures and materials also important. Technological issues of energy drivers are increase of the e±ciency of laser drivers and improvement of the focusing of electrons, light ions, and heavy ion beams.

18.2 Implosion

263

Fig.18.3 Pellet structure. A:ablator, P:pusher, D-T:solid D{T fuel, V:vacuum.

18.2 Implosion A typical structure of a pellet is shown in ¯g.18.3. Outside the spherical shell of deuteriumtritium fuel, there is a pusher cell, which plays the role of a piston during the compression; an ablator cell with low-Z material surrounds the pusher cell and the fuel. The heating e±ciency ´h is the conversion ratio of the driver energy to the thermal energy of the compressed core fuel. The heating e±ciency depends on the interaction of the driver energy with the ablator, the transport process of the particles, and the energy and motion of the plasma °uid. The driver energy is absorbed on the surface of the ablator, and the plasma is produced and heated. Then the plasma expands and the inner deuterium-tritium fuel shell is accelerated inward by the reaction of the outward plasma jet. The implosion takes place at the center. Therefore the heating e±ciency ´h is the product of three terms, that is, the absorption ratio ´ab of the driver energy by the ablator, the conversion ratio ´hydro of the absorbed driver energy to the kinetic energy of the hydrodynamic °uid motion, and the conversion ratio ´T of the kinetic energy of the hydrodynamic motion to the thermal energy of compressed core: ´h = ´ab ´hydro ´T : The internal energy of the solid deuterium-tritium fuel per unit volume is given by the product of Fermi energy "F = (¹h2 =2me )(3¼ 2 n)2=3 and the density with a factor 3/5(¹h = h=2¼ is Planck's constant, me is the electron mass). The internal energy of solid deuterium-tritium per unit mass w0 can be estimated to be w0 = 1:0 £ 108 J/Kg. If the preheating occurs before the compression starts, the initial internal energy is increased to ®p w0 , and then the solid deuterium-tritium fuel is compressed adiabatically. By use of the equation of state for an ideal gas, the internal energy w after the compression is

w = ®p w0

µ

½ ½0

¶2=3

where ½0 and ½ are the mass densities before and after the compression. If the preheating is well suppressed and ®p is of the order of 3, the internal energy per unit mass after 1000 £ compression is w » 3 £ 1010 J/Kg. This value w corresponds to the kinetic energy of unit mass with velocity v » 2:5 £ 105 m/s (w = v 2 =2). Therefore, if the spherical fuel shell is accelerated to this velocity and if the kinetic energy is converted with good e±ciency ´T into the thermal energy of the fuel core at the center, then compression with 1000 times mass density of the solid deuterium-tritium is possible. When the pellet is irradiated from all sides by the energy driver, the plasma expands with velocity u from the surface of the ablator. Then the spherical shell with mass M is accelerated inward by the reaction with the ablation pressure Pa . The inward velocity v of the spherical

264

18 Inertial Con¯nement

shell can be analyzed by the rocket model with an outward plasma jet,3;4 that is, d(M v) dM ¢ u = SPa : =¡ dt dt

(18:15)

where S is the surface area of the shell. When the average mass density and the thickness of the spherical shell are denoted by ½ and ¢, respectively, the mass M is M = ½S¢. Usually the outward velocity of the expanding plasma u is much larger than the inward velocity of the spherical shell v, and u is almost constant. The change of sum of the kinetic energies of the plasma jet and the spherical shell is equal to the absorbed power of the energy driver as follows: ´ab IL S =

d dt

µ

¶

µ

dM 1 1 ¡ M v2 + 2 2 dt

¶

u2

(18:16)

where IL is the input power per unit area of the energy driver. From eqs.(18.15) and (18.16), the absorbed energy Ea is reduced to Ea =

Z

1 ´ab ID S dt ¼ (¢M )u2 2

(18:17)

where the approximations u À v; and u=const. are used. The quantity ¢M is the absolute value of the change of mass of the spherical shell. The pressure Pa is estimated from eqs.(18.15) and (18.16) as follows: µ

dM u ¡ Pa = S dt

¶

1 ¼ 2´ab IL : u

(18:18)

Then the conversion ratio ´hydro of the absorbed energy to the kinetic energy of the spherical shell is ´hydro

1 M0 ¡ ¢M = (M0 ¡ ¢M )v 2 = 2Ea ¢M

µ ¶2

v u

:

Since the rocket equation (18.15) implies v=u = ln((M0 ¡ ¢M )=M0 ), the conversion ratio ´hydro is ´hydro =

µ

¶µ

M0 ¡1 ¢M

µ

¢M ln 1 ¡ M0

¶¶2

¼

¢M M0

(18:19)

where ¢M=M0 ¿ 1 is a assumed. The ¯nal inward velocity of the accelerated spherical shell must still be larger than v » 3 £ 105 m=s. The necessary ablation pressure Pa can be obtained from eq.(18.15) with relation S = 4¼r2 ; and the approximation M ¼ M0 ，Pa ¼ const: as follows: dv Pa 4¼Pa 2 r = r2 ; = dt M0 ½0 r02 ¢0

v=¡

dr : dt

Integration of v ¢ dv=dt gives ¢0 3 Pa = ½0 v 2 2 r0

(18:20)

where ½0 ; r0 ; and ¢0 are the mass density, the radius, and the thickness of the spherical shell at the initial condition, respectively. When r0 =¢0 = 30 and ½0 = 1g=cm3 ; the necessary ablation pressure is Pa = 4:5 £ 1012 Newton=m2 = 45 Mbar (1 atm = 1.013 bar) in order to achieve the velocity v = 3 £ 105 m=s. Therefore the necessary energy °ux intensity of driver IL is ´ab IL =

Pa u : 2

(18:21)

18.2 Implosion

265

Fig.18.4 Structure of hohlraum target.

For the evaluation of the velocity u of the expanding plasma, the interaction of the driver energy and the ablator cell must be taken into account. In this section the case of the laser driver is described. Let the sound velocity of the plasma at the ablator surface be cs and the mass density be ½c . The energy extracted by the plasma jet from the ablator surface per unit time is 4½c c3s and this must be equal to the absorbed energy power ´ab ID . The plasma density is around the cut o® density corresponding to the laser light frquency (wavelength), that is u » 4cs ; ´ab IL » 4mDT nc c3s where mDT = 2:5 £ 1:67 £ 10¡27 kg is the average mass of deuterium and tritium and the cuto® density is nc = 1:1 £ 1027 =¸2 (¹m) m¡3 (¸ is the wavelength of laser in units of ¹m). From eq.(18.21) we have µ

(´ab IL )14 Pa = 13 ¸(¹m)

¶2=3

; (Mbar)

(18:22)

where (´ab IL )14 is the value in 1014 W/cm2 . This scaling is consistent with the experimental results in the range 1 < (´ab IL )14 < 10. Most implosion research is carried out by the laser driver. The observed absorption rate ´ab tends to decrease according to the increase of laser light intensity IL . The absorption rate is measured for a Nd glass laser with wavelength 1.06 ¹m (red), second harmonic 0.53 ¹m (green), and third harmonic 0.35 ¹m (blue). The absorption is better for shorter wavelengths, and it is ´ab ¼ 0:9 » 0:8 in the case of ¸ = 0:35 ¹m in the rage of IL = 1014 » 1015 W=cm2 . The conversion ratio ´hydro determined by the experiments is 0.1»0.15. The conversion ratio ´T is expected to be ´ T ¼ 0:5. The necessary ablation pressure of Pa = 45 Mbar has been realized. In order to compress the fuel to extremely high density, it is necesary to avoid the preheating of the inner pellet during the implosion process, since the pressure of the inner part of pellet must be kept as low as possible before the compression. When laser light with a long wavelength (CO2 laser ¸ = 10:6 ¹m) is used, the high-energy electrons are produced by the laser-plasma interaction, and the high-energy electrons penetrate into the inner part of the pellet and preheat it. However, the production of high-energy electrons is much lower in short-wavelength experiments. The implosion process just described is for directly irradiated pellets. The other case is indirectly irradiated pellets. The outer cylindrical case surrounds the fuel pellet, as shown in ¯g.18.4. The inner surface of the outer cylindrical case is irradiated by the laser light, and the laser energy is converted to X-ray energy and plasma energy. The converted X-rays and the plasma particles irradiate the inner fuel pellet and implosion occurs. The X-ray and plasma energy are con¯ned between the outer cylindrical case and the inner fuel pellet and are used for the implosion e®ectively. This type of pellet is called a hohlraum target5;6 .

266

18 Inertial Con¯nement

Recent activities in inertial con¯nement fusion including NIF (National Ignition Facility) are well described in ref.7. References 1. K. A. Brueckner and S. Jorna: Rev. of Modern Phys. 46, 325 (1974) 2. J. L. Emmett, J. Nuckolls and L. Wood: Scienti¯c American 230, No.6, p.2 (1974) 3. R. Decoste, S. E. Bodner, B. H. Ripin, E. A. McLean, S. P. Obenshain and C. M. Armstrong: Phys. Rev. Lett. 42, 1673 (1979) 4. K.Mima: Kakuyugo Kenkyu 51, 400 (1984) in Japanese 5. J. H. Nuckolls: Physics Today 35, Sept. 25 (1982) 6. J. Lindl: Phys. Plasma 2, 2933, (1995) 7. J. Lindl: Inertial Con¯nement Fusion Springer/AIP Press 1998

A Derivation of MHD Equations of Motion

267

App.A Derivation of MHD Equations of Motion When the distribution function is obtained from Boltzmann's equation introduced in ch.4 F @f + v ¢ rr f + rv f = @t m

µ

±f ±t

¶

(A:1) coll

average quantities by velocity space v such as mass density, current density, charge density, °ow velocity and pressure can be estimated as the functions of space coordinates r and time t. The relations between these average variables can be reduced by multiplication of a function g(r; v; t) by Boltzmann's equation and integration over the velocity space. When g = 1; g = 2 mv; g = m 2 v , equations in terms of the density, the momentum, and energy, respectively, can be obtained. Averages of g weighted by the distribution function are denoted by hgi, i.e., hg(r; t)i =

R

Since density is expressed by n(r; t) =

Z

g(r; v; t) f (r; v; t) dv R : f (r; v; t) dv

f (r; v; t) dv

we ¯nd that n(r; t)hg(r; t)i = By integrating by parts, we obtained Z Z

(A:2)

(A:3) Z

g(r; v; t) f (r; v; t) dv: ¿

À

@ @f @ g dv = (nhgi) ¡ n g ; @t @t @t ¿

(A:4)

À

@f @ @(nhvi gi) ¡n gvi dv = (vi g) ; @xi @xi @xi ¿ À Z @ Fi @f n g dv = ¡ gFi : m @vi m @vi For the force F = qE + q(v £ B), the relation @Fi =0 @vi holds, and

¿

Z

@g Fi @f n g dv = ¡ Fi m @vi m @vi Consequently we ¯nd, in terms of averages, that ¿

À

:

À

n @g @ + rr ¢ (nhgvi) ¡ nhrr ¢ (gv)i ¡ hF ¢ rv gi (nhgi) ¡ n @t @t m Z µ ¶ ±f dv: = g ±t coll

(A:5)

Equation (A.5), with g = 1, is @n + rr ¢ (nhvi) = @t

Z µ

±f ±t

¶

coll

dv:

(A:6)

268

A Derivation of MHD Equations of Motion

This is the equation of continuity. When the e®ects of ionization and recombination are neglected, collision term is zero. Equation (A.5), with g = mv, yields the equation of motion: X @ @ (nmhvj vi) ¡ nhF i = (mnhvi) ¡ @t @xj j

Z

mv

µ

±f ±t

¶

dv:

(A:7)

coll

Let us de¯ne the velocity of random motion v r by v ´ hvi + v r By de¯nition of v r , its average is zero: hv r i = 0: Since hvi vj i = hvi ihvj i + hvri vrj i; X @ j

@xj

(nhvi vj i) =

X @

@xj

j

=n

X j

(nhvi ihvj i) +

hvj i

X @ j

@xj

(nhvri vrj i)

X @ X @ @ hvi i + hvi i (nhvj i) + (nhvri vrj i) : @xj @xj @xj i j

(A:8)

Multiplication of the equation of continuity (A.6) by mhvi gives Z µ

X @ @ @ m (nhvi) = mn hvi ¡ mhvi (nhvj i) + mhvi @t @t @xj j

±f ±t

¶

dv:

(A:9)

coll

De¯ne the pressure tensor by mnhvri vrj i ´ Pij

(A:10)

then the equation of motion is reduced from (A.7),(A.8),(A.9) as follows: µ

¶

X @ @ Pij + + hvir hvi = nhF i ¡ mn @t @xj j

Z

mvr

µ

±f ±t

¶

dv:

(A:11)

coll

When the distribution function is isotropic, the pressure tensor is 2 i±ij = nm Pij = nmhvri

hvr2 i ±ij = p±ij ; 3

and p = n∙T for a Maxwell distribution. (For an anisotropic plasma, the pressure tensor is given by Pij = P? ±ij + (Pk ¡ P? )bb, where b is the unit vector of magnetic ¯eld B.) We introduce the tensor ¦ij = nmhvri vrj ¡ (hvr2 i=3)±ij i; then the pressure tensor may be expressed by Pij = p±ij + ¦ij :

(A:12)

A Derivation of MHD Equations of Motion

269

A nonzero ¦ij indicates anisotropic inhomogeneous properties of the distribution function. When the collision term is introduced by R=

Z

mvr

µ

¶

±f ±t

dv

(A:13)

coll

the equation of motion is expressed by µ

¶

X @¦ij @ mn + R: + hvir hvi = n hF i ¡ rp ¡ @t @xj j

(A:14)

Here R measures the change of momentum due to collisions with particles of other kind. When g = mv 2 =2, then (v £ B) ¢ rv v 2 = 0. The energy- transport equation is reduced to µ

¶

µ

¶

Z

@ nm 2 n hv i +rr mhv 2 i = qnE ¢ hvi + @t 2 2

mv 2 2

µ

±f ±t

¶

dv:

coll

(A.15) The 2nd term in left-hand side of (A.15) is modi¯ed to nm 2 nm 3 hv i = (hvi)2 + p 2 2 2 hv 2 i =(hvi)2 hvi + 2h(hvi ¢ v r )ihvi + hvr2 ihvi

+ hvi2 hv r i + 2h(hvi ¢ v r )v r i + hvr2 v r i:

The 2nd and 4th terms are zero, and the 5th term is X i

hvi ihvri vrj i = =

and

µ

X i

hvi i

Pij 1 X hvi i(p±ij + ¦ij ) = nm nm i

1 1 X hvi i¦ij phvi + nm nm i ¶

X ¦ij p hvi + 2 hvi i hv i = hvi + 5 + hvr2 vr i: nm nm i 2

2

Consequently (A15) is reduced to @ @t

µ

Ã

!

¶ µ ¶ X nm 2 3 nm 2 5 hvi + p +r¢ hvi + p v+r¢ ¦ij hvi i +r¢q = qnE¢hvi+R¢hvi+Q; (A:16) 2 2 2 2 i

q(r; t) =

Z

m 2 v v r f dv; 2 r

Q(r; t) =

Z

mvr2 2

µ

±f ±t

¶

(A:17)

dv:

(A:18)

coll

Q is the heat generation by collision and q is the energy-°ux density due to random motion. The scalar product of (A.14) and hvi is µ

¶

X @¦ij hvi2 @ hvi i = qnE ¢ hvi + R ¢ hvi mn + hvi ¢ r + hvi ¢ rp + @t 2 @xj i;j

270

A Derivation of MHD Equations of Motion

and the equation of continuity (A.6) gives µ

¶

m 2 @n hvi + r ¢ (nhvi) = 0 2 @t then (A.16) is reduced to @ @t

µ

¶

nm 2 hvi + r ¢ 2

µ

¶

X @¦ij nm 2 hvi hvi + hvi ¢ rp + hvi i = qnE ¢ hvi + R ¢ hvi: 2 @xj i;j

Subtraction of this equation and (A.16) gives µ

¶

X 3 @p 3 @ hvi i + r¢q = Q: ¦ij + r¢ phvi + pr ¢ hvi + 2 @t 2 @xj i;j

(A:19)

The relation p = n∙T and the equation of continuity (A.6) yield 3 @(n∙T ) +r¢ 2 @t ³

´

³

µ

¶

3 3 d∙T n∙T hvi = n : 2 2 dt ´

By setting s ´ ln (∙T )3=2 =n = ln p3=2 =n5=2 , we may write (A.19) as ds ∙T n = ∙T dt

µ

¶

X @(ns) @hvi i ¦ij + Q: + r ¢ (nshv3i) = ¡r ¢ q ¡ @t @xj i;j

(A:20)

B.1 Energy Integral in Illuminating Form

271

App.B Energy Integral of Axisymmetric Toroidal System B.1 Energy Integral in Illuminating Form The energy integral W =

1 2

Z µ

°p(r ¢ »)2 +

V

1 2

Z µ V

¶

1 (» £ (r £ B)) ¢ r £ (» £ B) dr ¹0

¡ =

1 (r £ (» £ B))2 + (r ¢ »)(» ¢ rp) ¹0

¶

B 21 + °p(r ¢ »)2 + (r ¢ »)(» ¢ rp) ¡ » ¢ (j £ B 1 ) dr ¹0

(B:1)

was derived in (8.46) of ch.8. This expression can be further rearranged to the more illuminating form1;2 of 1 W = 2

Z µ

°p(r ¢ »)2 + ¡

¹0 (» ¢ rp) 2 1 1 jB 1? j2 + jB 1k ¡ B j ¹0 ¹0 B2 ¶

(j ¢ B) (» £ B) ¢ B 1 ¡ 2(» ¢ rp)(» ¢ ∙) dr: B2

(B:2)

The ¯rst term of the integrand of (B.2) is the term of sonic wave. The second and the third terms are of Alf¶ ven wave. The fourth term is of kink mode. The last one is the term of ballooning mode. ∙ is the vector of curvature of ¯eld line. The rearrangement from (B.1) to (B.2) is described in the followings. When » is expressed by the sum of the parallel component »k b and the perpendicular component » ? to the magnetic ¯eld B = Bb » = »k b + » ? the last two terms of (B.1) are reduced to (r ¢ »)(» ¢ rp) + (j £ ») ¢ B 1 = (» ¢ rp)r ¢ (»k b) + (» ¢ rp)r ¢ » ? + »k (j £ b) ¢ r £ (» £ B) + (j £ » ? ) ¢ B 1 = (» ¢ rp)r ¢ (»k b) +

»k rp ¢ r £ (» £ B) + (» ¢ rp)r ¢ » ? + (j £ » ? ) ¢ B 1 B

= (» ¢ rp)(B ¢ r)(

»k »k ) + r ¢ ((» £ B) £ rp) + (» ¢ rp)r ¢ »? + (j £ »? ) ¢ B 1 B B

= (» ¢ rp)(B ¢ r)(

»k »k ) + r ¢ ((» ¢ rp)B) + (» ¢ rp)r ¢ » ? + (j £ » ? ) ¢ B 1 B B

=r¢

µ

¶

»k (» ¢ rp)B + (» ¢ rp)r ¢ » ? + (j £ » ? ) ¢ B 1 : B

(B:3)

272

B Energy Integral of Axisymmetric Toroidal System

The current density j can be expressed by the sum of parallel and perpendicular components to the magnetic ¯eld as follows: B £ rp B2

j = ¾B + where ¾=

j ¢B : B2

The last term of (B.3) is (j £ » ? ) ¢ B 1 = ¾(B £ » ? ) ¢ B 1 ¡

(» ? ¢ rp) B ¢ B1 B2

and r ¢ » ? in the second term of (B.3) is µ

¶

B B B r ¢ »? = r ¢ £ (» £ B) = (» £ B) ¢ r £ 2 ¡ 2 ¢ r £ (» £ B) 2 B B B

= (» £ B) ¢

=¡

r£B rB B ¡ 2(» £ B) ¢ 3 £ B ¡ 2 r £ (» £ B) 2 B B B

(» ¢ ¹0 rp) B B £ rB ¡ 2 ¢ B1: + 2(» £ B) ¢ 2 3 B B B

(B:4)

Then the energy integral (B.2) is reduced to 1 W = 2

Z µ

°p(r ¢ ») +

V

B ¢ B1 B 21 ¹0 (» ¢ rp)2 ¡ ¡ (» ¢ rp) 2 ¹0 B B2 ¶

B ¢ B1 B £ rB ¡(» ? ¢ rp) + ¾(B £ » ? ) ¢ B 1 + 2(» ¢ rp)(» £ B) ¢ dr 2 B B3 1 = 2

Z µ V

1 °p(r ¢ ») + ¹0

¡2(» ¢ rp)

µ

¯ ¯2 ¯ ¹20 (» ¢ rp)2 ¯¯ (j ¢ B) ¯ B¯ ¡ (» ? £ B) ¢ B 1 ¯B 1 ¡ ¯ ¯ B2 B2

B £ rB ¹0 (» ¢ rp) ¡ (» £ B) ¢ 2 B B3

¶¶

dr:

The last ballooning term can be expressed as ¡2(» ¢ rp)(» ¢ ∙) by introducing a vector ∙ ∙´

´ 1 ³ ¹0 rp (B £ rB) £ B 2 £B = B £ r(B + 2¹ p) + ; 0 4 2B B2 B3

(B:5)

since (» ¢ ∙) =

¹0 (» ¢ rp) (» £ B) ¢ (B £ rB) ¹0 (» ¢ rp) » ¢ (B £ rB) £ B ¡ + = : (B:6) 2 3 B B B2 B3

B.2 Energy Integral of Axisymmetric Toroidal System

273

¯g.B1 Orthogonal coordinate system (Ã; Â; ') and eà ; e ; e' are unit vectors of Ã; Â; ' directions respectively.

(B.4) and (B.5) reduce r ¢ » ? + 2(» ? ¢ ∙) =

¹0 (» ? ¢ rp) B ¢ B 1 ¡ B2 B2

(B:7)

From (6.7) of ch.6 we have r(2¹0 p + B 2 ) = 2(B ¢ r)B: Then it becomes clear that ∙ is equal to the vector of curvature as follows: µ

¶

1 1 ∙ = (b £ (b ¢ r)(Bb)) £ b = b £ ((b ¢ r)b + b (b ¢ r)B) £ b B B n = ((b ¢ r)b)? = ¡ R where R is the radius of curvature and n is the unit vector from the center of curvature to the point on the line of magnetic force (see ¯g. 2.4 of ch.2). B.2 Energy Integral of Axisymmetric Toroidal System In any axisymmetric toroidal system, the energy integral may be reduced to more convenient form. The axisymmetric magnetic ¯eld is expressed as ^ I(Ã) ¹0 I(Ã) ^ I(Ã) ´ e' + B e R 2¼ where ' is the angle around the axis of torus and à is the °ux function de¯ned by B=

(B:8)

à = ¡RA'

(B:9)

where R is the distance from the axis of symmetry and A' is the ' component of the vector potential of the magnetic ¯eld. B is the poloidal component of the magnetic ¯eld. e' and e are the unit vectors with the directions of toroidal angle and poloidal angle respectively (see ¯g.B1). The R and Z components of the magnetic ¯eld are given by @à @à RBZ = ¡ : @Z @R We can introduce an orthogonal coordinate system (Ã; Â; '), where Ã=const. are the magnetic surfaces and Â; ' are poloidal and toroidal angles respectively. The metric for these coordinates is RBR =

2

ds =

Ã

dà RBÂ

!2

+ (JBÂ dÂ)2 + (Rd')2

(B:10)

274

B Energy Integral of Axisymmetric Toroidal System

where the volume element is dV = J(Ã)dÃdÂd'. A ¯eld line is de¯ned by Ã=const. and by ^ I(Ã) B' Rd' = = JB d B RB that is ^ d' J(Ã)I(Ã) ´ q^(Ã; Â) = d R2 and the toroidal safety factor is given by I

1 q(Ã) = 2¼

^ J(Ã)I(Ã) dÂ: R2

The energy integral of axisymmetric toroidal system is given by3 1 2

W =

=

Z µ V

Z µ V

¶

jB 1 j2 + °pj(r ¢ »)j2 + (r ¢ » ¤ )(» ¢ rp) ¡ » ¤ ¢ (j £ B 1 ) dr ¹0

2 2 1 B kk 1 R2 2 jXj + 2¹0 BÂ2 R2 2¹0 J 2

BÂ2 + 2¹0

¯ µ ¶ ¯2 ¯ @U JX 0 ¯¯ ¯ ¡I ¯ ¯ ¯ @ R2 ¯

¯ ¯2 ¯ ¯2 ¯ ¯ ¹0 j' ¯¯ 1 ¯¯ 1 ¯ 0 0 ¯ X + + iBk Y + inU °p (JX) ¯inU + X ¡ ¯ k ¯ ¯ ¯ RBÂ2 ¯ 2 J

¡KXX

¤

¶

J dÃdÂd'

(B:11)

The derivation of (B.11) is described in the followings. The notations in (B.11) are explained successively. In a general orthogonal coordinate system (u1 ; u2 ; u3 ) with the metric of ds2 = h21 (du1 )2 + h22 (du2 )2 + h23 (du3 )2 g1=2 = h1 h2 h3 operators of gradient, divergence and rotation of a vector F = F1 e1 + F2 e2 + F3 e3 (ej are unit vectors) are expressed by rÁ =

X 1 @Á

r¢F =

hj @uj 1

g 1=2

µ

1 r£F = h2 h3

ej ¶

@ @ @ (h2 h3 F1 ) + 2 (h3 h1 F1 ) + 3 (h1 h2 F3 ) 1 @u @u @u µ

¶

@ @ (h3 F3 ) ¡ 3 (h2 F2 ) e1 2 @u @u

1 h3 h1

µ

@ @ (h1 F1 ) ¡ 1 (h3 F3 ) e2 3 @u @u

1 + h1 h2

µ

@ @ (h2 F2 ) ¡ 2 (h1 F1 ) e3 : 1 @u @u

+

¶ ¶

B.2 Energy Integral of Axisymmetric Toroidal System

275

In the coordinate system (Ã; Â; '), (» ¢ rp) is reduced to (» ¢ rp) = »Ã RBÂ

@p = »Ã RB p0 @Ã

The prime on p means the di®erentiation by Ã. From (6.15),(6.16) in ch.6, we have ¡j' = Rp0 + p0 = ¡

I^I^0 ¹0 R

(B:12)

j' I^I^0 ¡ R ¹0 R 2

Note that à de¯ned by (B.9) in this appendix is ¡RA' , while à in (6.15),(6.16) of ch.6 is RA' . r ¢ » is expressed as 1 r¢» = J

Ã

@ @ (JB R»Ã ) + @à @Â

Ã

»Â BÂ

!

@ + @'

µ

J»' R

¶!

:

It is convenient to introduce X ´ RB »Ã Y ´

»Â BÂ

U´

1 »' I^ ¡ 2 »Â : (B »' ¡ B' »Â ) = RB R R BÂ

»Ã =

X RBÂ

Then

»Â = B Y »' = RU +

I^ Y R

and Ã

I^I^0 j' » ¢ rp = Xp = X ¡ ¡ R ¹0 R 2 0

!

:

(B:13)

We analyze an individual Fourier mode » = »(Ã; Â) exp(in'). Then (Bikk )Y ´

Ã

!

1 @ 1 @ BÂ Y = + B' JBÂ @Â R @'

Ã

!

I^ 1 @ + 2 in Y J @Â R

I^ 1 @ Y = (Bikk )Y ¡ in 2 Y: J @Â R Since r¢» =

1 (JX)0 + iBkk Y + inU J

(B:14)

276

B Energy Integral of Axisymmetric Toroidal System

(rp)(r ¢ » ¤ ) is

Ã

I^I^0 j' (» ¢ rp)(r ¢ » ) = X ¡ ¡ R ¹0 R 2 ¤

!µ

¶

1 (JX ¤ )0 ¡ iBkk Y ¤ ¡ inU ¤ : J

(B:15)

Let us derive the expression of B 1 B 1 = r £ (» £ B) (» £ B)à = »Â B' ¡ »' B (» £ B) = ¡»Ã B' (» £ B)' = »Ã B B1Ã

1 = JBÂ R

µ

µ

@X @ JB' + X @Â @' R

µ

@X @Ã

¶

J I^ X R2

!

B1Â = ¡BÂ inU + B1'

R = J

Ã

@ ¡ @Ã

Ã

@U + @Â

!

¶¶

=

1 BÂ R

iBkk X

:

Each component of the current density is ¹0 jà = 0 ¹0 j = ¡B ¹0 j' =

@ (RB' ) = ¡BÂ I^0 @Ã

R @ (J BÂ2 ) J @Ã

(B:16)

and R @U ¤ R » ¡ (B 1 £ » )Â = J @Â Ã J ¤

(B 1 £ » ¤ )' =

µ

IJ X R2

¶0

»Ã¤ ¡

iBkk X» ¤ BÂ R '

iBkk X» ¤ + (inU + X 0 )B »Ã¤ : B R Â

Then » ¤ ¢ (j £ B 1 ) is » ¤ ¢ (j £ B 1 ) = j ¢ (B 1 £ » ¤ ) I^0 = ¡BÂ ¹0

Ã

R @U ¤ R » ¡ J @Â Ã J

µ

IJ X R2

¶0

!

iBkk »Ã¤ ¡ X» ¤ + j' B R '

Ã

iBkk X» ¤ + (inU + X 0 )B »Ã¤ B R Â

!

à ! µ ¶0 iBkk I^0 I^ ¤ I^0 I^0 @U ¤ IJ 1 1 ¤ ¤ ¤ ¡ = X X X Y j' + (RU + Y ) + X +inU X ¤ j' +X 0 X ¤ j' 2 R ¹0 R ¹0 J R ¹0 J @ R R

B.2 Energy Integral of Axisymmetric Toroidal System

277

and (r ¢ » ¤ )(» ¢ rp) ¡ (» ¤ £ j) ¢ B 1 Ã

j' I^I^0 = ¡ ¡ R ¹0 R2 +in

j' (inXU ¤ ¡ inX ¤ U ) R

¤

Ã

I^I^0 J 0 J 0 j' I^I^0 R0 I^02 ¡ ¡ 2 ¡ + 2 J R ¹0 R2 ¹0 R2 R ¹0 R2 J

!

(B:17)

¯ ¯2 µ ¶ ¯ @U JX 0 JX ^0 ¯¯ ¯ ¡I ¡ 2I¯ ¯ ¯ @ ¯ R2 R

¯ µ ¶ ¯2 µ ¶ ¯ @U I^0 JX 0 ¯¯ @U ¤ @U ¤ ¯ ¡I ¡ + X X ¯ ¯ ¯ @ R2 ¯ ¹0 J @ @Â

R2 = ¹0 J 2 +

(XX ¤0 + X 0 X ¤ ) +

^0 ^0 I^ 1 @U ¤ I^0 ¤ I ¤I XU + iBk XU + X k R2 ¹0 ¹0 J @Â ¹0

+XX

jB1' j2 R2 = ¹0 ¹0 J 2

!

µ ¶ I^I^0 I^I^0 J 0 I^02 2R0 0 ¤ ¤0 ¡ (X X + X X) + 2 J XX ¤ + XX ¤ 2 2 3 ¹0 R ¹0 J R R ¹0 R 2

jB1Â j2 BÂ2 jinU + X 0 j2 = ¹0 ¹0 BÂ2 = ¹0

¯ ¯2 ¯ ¹0 j'2 ¹0 j' ¯¯ j' ¯ 0 ¤ ¤ 0 ¤ ¤0 j' ¡ ¡ inU X + (inU X X) X + X ) XX ¤ : + (X ¯inU + X ¡ ¯ ¯ RBÂ2 ¯ R R R2 BÂ2

Finally the energy integral of axisymmetric toroidal system becomes W =

=

1 2

Z µ V

Z µ V

¶

jB 1 j2 + °pj(r ¢ »)j2 + (r ¢ » ¤ )(» ¢ rp) ¡ » ¤ ¢ (j £ B 1 ) dr ¹0

2 2 1 B kk 1 R2 2 jXj + 2¹0 BÂ2 R2 2¹0 J 2

BÂ2 + 2¹0

¯ µ ¶ ¯2 ¯ @U JX 0 ¯¯ ¯ ¡I ¯ ¯ ¯ @ R2 ¯

¯ ¯2 ¯ ¯2 ¶ ¯ ¯ ¯ ¹ j 1 ¯¯ 1 0 ' ¯ ¯ 0 0 ¤ ¯ X ¯ + °p ¯ (J X) + iBkk Y + inU ¯ ¡ KXX JdÃdÂd' ¯inU + X ¡ ¯ RBÂ2 ¯ 2 J

(B:18)

where j' I^I^0 R0 + K´ 2 ¹0 R R 2R

Ã

J0 ¹0 j' + J RBÂ2

I^I^0 R0 j' = + ¹0 R2 R R

Ã

!

J 0 BÂ0 + J BÂ

!

;

278

B Energy Integral of Axisymmetric Toroidal System

since (refer (B.13)) ¹0 j' =

R 0 2 (J BÂ + J2BÂ BÂ0 ): J

B.3 Energy Integral of High n Ballooning Modes The energy integral (B.18) was used for stability analysis of high n modes and ballooning mode3;4 . The ¯rst step in minimization of ±W is to select Y so that the last second positive term in (B.18) vanishes (r¢» = 0). The second step is minimization with respect to U . When we concern ballooning modes, the perturbations with large toroidal mode number n and jm ¡ q^nj ¿ n are important (see sec.8.5). After long mathematical calculation, the energy integral with O(1=n) accuracy is derived as follows3 ; ¼ W = ¹0

Z

dÃdÂ

Ã

2J¹0 p0 ¡ B2

+

¯2 R2 BÂ2 JB 2 ¯¯ ¯ k X + ¯ ¯ k R2 BÂ2 JB 2

Ã

¯ ¯2 ¯1 @ ¯ ¯ ¯ ¯ n @Ã (JBkk X)¯

@ B2 iI^ @ jXj (¹0 p + )¡ @Ã 2 JB 2 @Â 2

Ã

B2 2

¶

X¤ 1 JBkk (X¾ 0 ) ¡ (P ¤ JBkk Q + c:c:) n n

!

X ¤ @X n @Â

!

(B:19)

where P = X¾ ¡

BÂ2 I @ (JBkk X) ºB 2 n @Ã

Q=

I^2 X¹0 p0 1 @ + (JBkk X) 2 2 2 2 B º R B n @Ã

¾=

^ 0 p0 I¹ + I^0 B2

iJkk B =

@ + in^ q; @Â

q^(Ã; Â) =

^ IJ : R2

±W must be minimized with respect to all periodic functions X subject to an appropriate normalization ¼

Z

JdÃd½m

Ã

jXj2 + R2 BÂ2

µ

RBÂ B

¯ ! ¶2 ¯ ¯ 1 @X ¯2 ¯ ¯ = const: ¯ n @Ã ¯

(B:20)

where ½m is mass density and (B.20) corresponds to the total kinetic energy of perpendicular component to ¯eld line. The Euler equation for the minimizing function X(Ã; Â) can be deduced form (B.19) and (B.20). As X(Ã; Â) is periodic in Â, it can be expanded in Fourier series X(Ã; Â) =

X m

Xm (Ã) exp(imÂ)

References

279

A continuous function Xs (Ã) of s, which is equal to Xm (Ã) at s equal to integer m, can be constructed and can be expressed by Fourier integral Xs (Ã) =

Z

1

^ X(Ã; y) exp(isy)dy:

¡1

Then X(Ã; Â) is reduced to X(Ã; Â) =

X

exp(¡imÂ)

1

^ X(Ã; y) exp(imy)dy:

(B:21)

¡1

m

Since

Z

X 1 X exp(¡im(Â ¡ y)) = ±(y ¡ Â + 2¼N ) 2¼ m N

^ (±(x) is ± function), the relation of X(Ã; Â) and X(Ã; y) is X(Ã; Â) =

X N

^ 2¼ X(Ã; Â ¡ 2¼N ):

(B:22)

X(Ã; Â) is expressed by an in¯nite sum of quasi-modes. ^ The Euler equation for X(Ã; Â) is converted into an identical equation for X(Ã; y) but with ^ ^ X in the in¯nite domain of y and free of periodicity requirement. Let us consider X(Ã; y) with the form of ^ X(Ã; y) = F (Ã; y) exp(¡in

Z

y

q^dy)

(B:23)

y0

in which the amplitude F (Ã; y) reminds a more slowly varying function as n ! 1. Then ^ y) = iJkk B X(Ã;

µ

=

¶

@ ^ + in^ q X(Ã; y) @y

@F (Ã; y) exp(¡in @y

Z

y

q^dy):

y0

^ The leading term of the Euler equation for X(Ã; y) is reduced to3 0

0

1 @ @ 1 @1 + J @y JR2 BÂ2 Ã

2¹0 p0 @ + B 2 @Ã

Ã

0

Ã

1

B2 ¹0 p + 2

!2 (Ã; y0 ) @ + 1+ R2 BÂ2

1

!2 R2 BÂ2 Z y 0 @F0 A q^ dy A B @y y0

Ã

!

! ¶ ^ 0 p0 µZ y I¹ 1 @B 2 0 ¡ q^ dy F0 B4 J @y y0 1

!2 R2 BÂ2 Z y 0 q^ dy A F0 = 0: B y0

By use of this Euler equation, stability of ballooning mode is analyzed4 (see sec.8.5).

References 1. 2. 3. 4.

J. M. Greene and J. L. Johnson: Plasma Phys. 10, 729 (1968) G. Bateman: MHD Instabilities, The MIT Press, 1978 J. W. Connor, R. J. Hastie and J. B. Talor: Proc. Roy. Soc. A365 1, (1979) J. W. Connor, R. J. Hastie and J. B. Talor: Phys. Rev.Lett. 40, 396 (1978)

(B:24)

280

C Derivation of Dielectric Tensor in Hot Plasma

App.C Derivation of Dielectric Tensor in Hot Plasma C.1 Formulation of Dispersion Relation in Hot Plasma In ch.10 dispersion relation of cold plasma was derived. In the unperturbed state, both the electrons and ions are motionless in cold plasma. However, in hot plasma, the electron and ions move along spiral trajectories even in the unperturbed state. The motion of charged particles in a uniform magnetic ¯eld B 0 = B0^ z may be described by dr 0 = v0 ; dt0

dv 0 q = v0 £ B0 : dt0 m

(C:1)

Assuming that r 0 = r, v0 = v = (v? cos µ; v? sin µ; vz ) at t0 = t, the solution of eqs.(C.1) is obtained as follows: vx0 (t0 ) = v? cos(µ + ­(t0 ¡ t)); vy0 (t0 ) = v? sin(µ + ­(t0 ¡ t));

(C:2)

vz0 (t0 ) = vz ; v? (sin(µ + ­(t0 ¡ t)) ¡ sin µ); ­ v? y0 (t0 ) = y ¡ (cos(µ + ­(t0 ¡ t)) ¡ cos µ); ­ x0 (t0 ) = x +

(C:3)

z 0 (t0 ) = z + vz (t0 ¡ t) where ­ = ¡qB0 =m and vx = v? cos µ; vy = v? sin µ. The analysis of the behavior due to a perturbation of this system must be based on Boltzmann's equation. The distribution function fk (r; v; t) of kth kind of particles is given by qk @fk (E + v £ B) ¢ rv fk = 0; + v ¢ rr fk + @t mk r¢E =

Z 1 X qk vfk dv; ²0 k

@E X 1 r £ B = ²0 qk + ¹0 @t k r£E =¡

@B ; @t

r ¢ B = 0:

(C:4) (C:5)

Z

vfk dv;

(C:6) (C:7) (C:8)

As usual, we indicate zeroth order quantities (the unperturbed state) by a subscript 0 and the 1st order perturbation terms by a subscript 1. The 1st order terms are expressed in the form of exp i(k ¢ r ¡ !t). Using fk = fk0 (r; v) + fk1 ;

(C:9)

C.2 Solution of Linearized Vlasov Equation

281

B = B0 + B1;

(C:10)

E = 0 + E1

(C:11)

we can linearize eqs.(C.4)»(C.8) as follows: v ¢ rr fk0 + X

qk

k

Z

qk (v £ B 0 ) ¢ rv fk0 = 0; mk

(C:12)

fk0 dv = 0;

X 1 r £ B0 = qk ¹0 k

(C:13) Z

vfk0 dv = j 0 ;

(C:14)

@fk1 qk qk (v £ B 0 ) ¢ rv fk1 = ¡ (E 1 + v £ B 1 ) ¢ rv fk0 ; + v ¢ rr fk1 + @t mk mk Z 1 X ik ¢ E 1 = qk fk1 dv; ²0 k

(C:15) (C:16)

Ã

!

Z i X 1 k £ B 1 = ¡! ²0 E 1 + qk vfk1 dv ; ¹0 ! k

(C:17)

B1 =

(C:18)

1 (k £ E 1 ): !

The right-hand side of (C.15) is a linear equation in E 1 as is clear from (C.18), so that fk1 is given as a linear function in E 1 . The electric tensor of the hot plasma is de¯ned by K(D = ²0 K ¢ E) is given by Z i i X qk vfk1 dv ´ K ¢ E 1 : (C:19) j = E1 + E1 + ²! ²0 ! k The linear relation of E 1 is derived from eqs.(C.17) (C.18): k £ (k £ E 1 ) +

!2 K ¢ E1 = 0 c2

(C:20)

and the dispersion relation is obtained by equating the determinant of the coe±cient matrix of the linear equation to zero as is eqs.(10.19),(10.20). Consequently if fk1 can be solved from (C.15), then, K can be obtained. As for cold plasmas, the properties of waves in hot plasmas can be studied by the dispersion relation of hot plasma. C.2 Solution of Linearized Vlasov Equation When the right-hand side of (C.15) is time-integrated along the particle orbit (C.2),(C.3) in the unperturbed state, we ¯nd fk1 (r; v; t) = ¡

qk mk

Z

t

¡1

µ

E 1 (r 0 (t0 ); t0 ) +

¶

1 0 0 v (t ) £ (k £ E 1 (r 0 (t0 ); t0 )) ¢ r0v fk0 (r 0 (t0 ); v 0 (t0 ))dt0 ! (C:21)

Substitution of (C.21) into (C.15) yields µ

¶

qk 1 E 1 + v £ (k £ E 1 ) ¢ rv fk0 mk ! ¶ Z t µ qk @ qk ¡ (v £ B 0 ) ¢ rv [integrand]dt0 + v ¢ rr + mk ¡1 @t mk

¡

282

C Derivation of Dielectric Tensor in Hot Plasma

=¡

qk (E 1 + v £ B 1 ) ¢ rv fk0 : mk

(C:22)

Therefore if it is proven that the 2nd term of the left-hand side of eq.(C.22) is zero, eq.(C.21) is con¯rmed to be the solution of (C.15). When the variables (r; v; t) are changed to (r 0 ; v 0 ; t0 ) by use of (C.2),(C.3), the di®erential operators in the 2nd term of left-hand side of (C.22) are reduced to @ @r 0 @v 0 @t0 @ 0 ¢ r ¢ r0v + + = r @t @t @t0 @t @t @(t0 ¡ t) = @t

µ

= ¡v 0 ¢ r0r ¡

@r 0 @v 0 0 ¢ r ¢ r0v + r @(t0 ¡ t) @(t0 ¡ t)

¶

qk 0 (v £ B 0 ) ¢ r0v ; mk

v ¢ rr = v ¢ r0r ; @ @r 0 @v0 ¢ r0r + ¢ r0v = @vx @vx @vx µ

@ @ 1 = sin ­(t0 ¡ t) 0 + [¡ cos ­(t0 ¡ t) + 1] 0 ­ @x @y Ã

@ @ + cos ­(t ¡ t) 0 + sin ­(t0 ¡ t) 0 @vx @vy 0

!

µ

;

@ @ @ 1 = (cos ­(t0 ¡ t) ¡ 1) 0 + sin ­(t0 ¡ t) 0 @vy ­ @x @y Ã

@ @ + ¡ sin ­(t0 ¡ t) 0 + cos ­(t0 ¡ t) 0 @vx @vy Ã

q @ @ ¡ vx (v £ B 0 ) ¢ rv = ¡­ vy m @vx @vy µ

@ @ @ @ = vx0 + vy0 0 ¡ vx 0 + vy 0 0 @x @y @x @y = (v 0 ¡ v) ¢ r0r +

¶

¶

!

¶

;

! Ã

@ @ ¡ ­ vy0 0 ¡ vx0 0 @vx @vy

!

q 0 (v £ B 0 ) ¢ r0v : m

Therefore the 2nd term of left-hand side of (C.22) is zero. Since the 1st order terms vary as exp(¡i!t), the integral (C.21) converges when the imaginary part of ! is positive. When the imaginary part of ! is negative, the solution can be given by analytic continuation from the region of the positive imaginary part. C.3 Dielectric Tensor of Hot Plasma The zeroth-order distribution function f0 must satis¯y eq.(C.12), or f0 (r; v) = f (v? ; vz );

2 v? = vx2 + vy2 :

Let us consider E 1 (r 0 ; t0 ) = E exp i(k ¢ r 0 ¡ !t0 ):

C.3 Dielectric Tensor of Hot Plasma

283

The z axis is taken along B 0 direction and x axis is taken in the plane spanned by B 0 and the propagation vector k, so that y component of the propagation vector is zero (ky = 0), that is: ^ + kz z^: k = kx x Then (C.21) is reduced to f1 (r; v; t) = ¡

q exp i(kx x + kz z ¡ !t) m

µ

Z

t

1

µ³

1¡

k k ¢ v0 ´ E + (v 0 ¢ E) ! ! ¶

¶

¢ r0v f0

kx v? kx v? £ exp i sin(µ + ­(t0 ¡ t)) ¡ i sin µ + i(kz vz ¡ !)(t0 ¡ t) dt0 : ! ­ We introduce ¿ = t0 ¡ t and use following formulas of Bessel function: 1 X

exp(ia sin µ) =

Jm (a) exp imµ;

m=¡1

J¡m (a) = (¡1)m Jm (a); µ

exp

¶

1 X

=

1 X

m=¡1 n=¡1

Jm exp(¡imµ)Jn exp (in(µ + ­¿ )) exp i(kz vz ¡ !)¿:

Since µ³

k k ¢ v0 ´ 1¡ E + (v 0 ¢ E) ! !

¶

¢ r0v f0

µ ¶ @f0 ³ kx vx0 ´ kz @f0 = 1¡ Ez + (vx0 Ex + vy0 Ey ) + @vz ! ! @v?

=

µ

@f0 ³ kz vz ´ @f0 kz v? 1¡ + @v? ! @vz ! ³ @f k v 0 x z

+

@v? !

¡

¶µ

Ã

³

vy0 ´ kx vx0 kz vz0 ´³ vx0 1¡ Ex + Ey + vz Ez ! v? v? ! v?

!

¶

Ex i(µ+­¿ ) Ey i(µ+­¿ ) ¡ e¡i(µ+­¿ ) ) + e¡i(µ+­¿ ) ) + (e (e 2 2i

@f0 kx v? ´ Ez i(µ+­¿ ) @f0 + e¡i(µ+­¿ ) ) + Ez (e @vz ! 2 @vz

we ¯nd µ

³J ´ ³J ´ X q n¡1 + Jn+1 n¡1 ¡ Jn+1 U Ex ¡ iU Ey f1 (r; v; t) = ¡ exp i(kx x + kz z ¡ !t) m 2 2 mn ¶

³

Jm (a) exp(¡i(m ¡ n)µ) Jn¡1 + Jn+1 @f0 ´ + W Jn Ez ¢ + 2 @vz i(kz vz ¡ ! + n­)

where µ

U = 1¡ W = a=

kz vz !

¶

@f0 kz v? @f0 + ; @v? ! @vz

(C:23)

kx v? @f0 kx vz @f0 ¡ ; ! @v? ! @vz

(C:24)

¡qB ; m

(C:25)

kx v? ; ­

­=

284

C Derivation of Dielectric Tensor in Hot Plasma

and

nJn (a) Jn¡1 (a) ¡ Jn+1 (a) d Jn¡1 (a) + Jn+1 (a) = ; = Jn (a): 2 a 2 da Since f1 is obtained, the dielectric tensor K of hot plasma is reduced from (C.19) to Z i X (K ¡ I) ¢ E = qj vfj1 dv: ²0 ! j

(C:26)

Since vx = v? cos µ, vy = v? sin µ, vz = vz , only the terms of ei(m¡n)µ = e§iµ in fj1 can contribute to x, y components of the integral (C.26) and only the term of ei(m¡n)µ = 1 in fj1 can contribute to z component of the integral (C.26) and we ¯nd: K =I¡

Sjn

2

X ¦j2 j

!nj0

1 Z X

dv

n=¡1

v? (n Jan )2 U 6 = 4 iv? Jn0 (n Jan )U vz Jn (n Jan )U

Sjn ; kz vz ¡ ! + n­j

(C:27) 3

@f0 ¡iv? (n Jan )Jn0 v? (n Jan )Jn ( @v + na W ) z 7 @f0 v? (Jn0 )2 U iv? Jn0 Jn ( @v + na W ) 5 z @f0 ¡iv? Jn Jn0 U vz Jn2 ( @v + na W ) z

where ¦j2

nj qj2 = : ²0 mj

When we use the relations @f0 + vz U ¡ v? ( @v z

n­ kx v? W )

kz vz ¡ ! + n­

1 X

Jn2

n=¡1

= 1;

1 X

=¡

Jn Jn0

vz @f0 v? @f0 + ; ! @v? ! @vz 1 X

= 0;

n=¡1

nJn2 = 0

n=¡1

and replace n by ¡n, then eq.(C.27) is reduced to K=I¡

1 Z X ¦j2 X j

!

n=¡1

Tjn

(Jn = (¡1)n Jn )

¡1 X ¦j2 v? Uj n¡1 j0 Tjn dv ¡ L kz vz ¡ ! ¡ n­j !2 j

Ã

1 1+ nj0

2

Z

!

vz2 @fj0 dv ; v? @v? 3

2 v 2 (n Jn )(n Jn ) iv? (n Jan )Jn0 ¡v? vz (n Jan )Jn 6 ? 2 a 0 Jan 7 2 = 4 ¡iv? Jn (n a ) v? Jn0 Jn0 iv? vz Jn0 Jn 5 ¡v? vz Jn (n Jan ) ¡iv? vz Jn Jn0 vz2 Jn Jn

where all the components of matrix L are zero except Lzz = 1. From the relations µ

¶

Uj @fj0 @fj0 1 @fj0 1 ¡n­j =¡ + + kz v? ; kz vz ¡ ! ¡ n­j ! @v? !(kz vz ¡ ! ¡ n­j ) @v? @vz 1 X

1 (Jn0 )2 = ; 2 n=¡1

1 X n2 Jn2 (a)

n=¡1

a2

=

1 2

another expression of the dielectric tensor is obtained: Ã

!

µ

¶

X ¦j2Z ¦j2 ¡n­j @fj0 Tjn @fj0 1 +kz dv: K= 1¡ 2 I ¡ 2 ! ! kz vz ¡ ! ¡ n­j v? @v? @vz nj0 j;n

(C:28)

C.4 Dielectric Tensor of bi-Maxwellian Plasma

285

Using N´

k c !

the dispersion relation (C.20) is given by (Kxx ¡ Nk2 )Ex + Kxy Ey + (Kxz + N? Nk )Ez = 0; Kyx Ex + (Kyy ¡ N 2 )Ey + Kyz Ez = 0; (Kzx + N? Nk )Ex + Kzy Ey + (Kzz ¡ N?2 )Ez = 0 where Nk is z component of N (parallel to B ) and N? is x component of N (perpendicular to B ). The dispersion relation is given by equating the determinent of the coe±cient matrix to zero. C.4 Dielectric Tensor of bi-Maxwellian Plasma When the zeroth-order distribution function is bi-Maxwellian, f0 (v? ; vz ) = n0 F? (v? )Fz (vz ); Ã

2 mv? m F? (v? ) = exp ¡ 2¼∙T? 2∙T?

Fz (vz ) = we ¯nd

µ

¡

µ

m 2¼∙Tz

n­j @f0 @f0 +kz v? @v? @vz

¶1=2

¶

(C:29) !

;

Ã

m(vz ¡ V )2 exp ¡ 2∙Tz

(C:30) !

µ

(C:31) ¶

kz (vz ¡ V ) 1 n­j ¡ =m F? (v? )Fz (vz ): n0 ∙T? ∙Tz

Integration over vz can be done by use of plasma dispersion function Z(³). Plasma dispersion function Z(³) is de¯ned by: Z(³) ´

1 ¼ 1=2

Z

1

¡1

exp(¡¯ 2 ) d¯: ¯¡³

Using following relations Z

1

¡1

Z

1

¡1

Z

1

¡1

Z

1

¡1

Fz 1 dvz = ³n Z(³n ); kz (vz ¡ V ) ¡ !n !n kz (vz ¡ V )Fz dvz = 1 + ³n Z(³n ); kz (vz ¡ V ) ¡ !n (kz (vz ¡ V ))2 Fz dvz = !n (1 + ³n Z(³n )); kz (vz ¡ V ) ¡ !n (kz (vz ¡ V ))3 Fz k2 (∙Tz ) dvz = z + !n2 (1 + ³n Z(³n )); kz (vz ¡ V ) ¡ !n m

!n ´ ! ¡ kz V + n­;

(C:32)

286

C Derivation of Dielectric Tensor in Hot Plasma

³n ´ Z

1

0

! ¡ kz V + n­ ; kz (2∙Tz =m)1=2 ³

Jn2 (b1=2 x) exp ¡

1 X

x2 ´ xdx = ®In (®b)e¡b® ; 2® 1 X

In (b) = eb ;

n=¡1

1 X

nIn (b) = 0;

n=¡1

n2 In (b) = beb

n=¡1

(where In (x) is nth modi¯ed Bessel function) the formula for the dielectric tensor of a biMaxwellian plasma is obtained as follows: K=I+

X ¦ 2 µX µ i;e

!2

n

2

¶

¶

3 ¡(2¸T )1=2 ´n ®n In in(In0 ¡ In ) (n2 =b + 2b)In ¡ 2bIn0 i(2¸T )1=2 ´n ®(In0 ¡ In ) 5 1=2 0 ¡i(2¸T ) ´n ®(In ¡ In ) 2¸T ´n2 In

! + n­ ; 21=2 kz vT z

® ´ b1=2 ;

¶

1 ³0 Z(³n ) ¡ 1 ¡ (1 + ³n Z(³n )) e¡b X n + 2´02 ¸T L ; ¸T

n2 In =b 4 Xn = ¡in(In0 ¡ In ) ¡(2¸T )1=2 ´n ®n In

´n ´

µ

2 ´ vTz

¸T ´ ∙Tz ; m

Tz ; T?

b´

2 ´ vT?

µ

kx vT? ­

¶2

(C:33)

(C:34)

;

∙T? ; m

L matrix components are Lzz = 1 and all others are 0. C.5 Dispersion Relation of Electrostatic Wave When the electric ¯eld E of waves is expressed by electrostatic potential Á: E = ¡rÁ the waves is called electrostatic wave. In this section the dispersion relation of electrostatic wave in hot plasma is described. Since @B 1 =@t = r £ E and B 1 = k £ E=! = 0 the dispersion relation is reduced from (10.92) to kx2 Kxx + 2kx kz Kxz + kz2 Kzz = 0:

(C:35)

When K given by (C.33) is substituted in (C.35), we ¯nd: kx2 + kz2 +

X ¦2 ∙ i;e

!2

kz2 2´02 ¸T +

1 µ 2 X n In

n=¡1

b

kx2 ¡ (2¸T )1=2 ´n

µ

µ

n

In 2kx kz + 2¸T ´n2 In kz2 b1=2 ¶

¶

¸

¶

1 £ ´0 Z(³n ) ¡ 1 ¡ (1 + ³n Z(³n )) e¡b = 0 ¸T where !n ´ ! ¡ kz V + n­;

1 X

n=¡1

In (b) = eb ;

C.6 Dispersion Relation of Electrostatic Wave in ¢ ¢ ¢

³n =

!n ; 21=2 kz vTz

´n =

! + n­ ; 21=2 kz vTz

b=

¸T =

287

Tz ; T?

³ k v ´2 x T?

­

and kx2

+

kz2

+

X ¦ 2 µ m!2 i;e

kx2

+

kz2

!2

∙T?

1 X m! 2

+

n=¡1

µ

∙T?

In

³

¶

³

´ 1 ´ ³0 Z(³n ) ¡ 1 ¡ (1 + ³n Z(³n )) e¡b = 0; ¸T

µ

¶

¶

1 X m Tz ³ ¡n­ ´ + ¦ 1+ 1+ ³n Z(³n )In e¡b = 0: ∙T T ! z n ? n=¡1 i;e

X

2

(C:36)

C.6 Dispersion Relation of Electrostatic Wave in Inhomogenous Plasma Equation (C.36) is the dispersion relation of electrostatic wave in a homogenenous bi-Mawellian plasma. When the density and temperature of the zeroth-order state change in the direction of y, we must resort to (C.5),(C.21) and @B 1 = ¡r £ E 1 = 0; @t

E 1 = ¡rÁ1 ; ¡r2 Á1 = fk1

qk = mk

Z 1 X qk fk1 dv; ²0 k

Z

t

¡1

(C:37)

r0r Á1 (r 0 ; t0 ) ¢ r0v fk0 (r 0 ; v 0 )dt0 :

(C:38)

The zeroth-order distribution function fk0 must satisfy eq.(C.12), or vy

@f0 @ @ ¡ ­(vy ¡ vx )f0 = 0: @y @vx @vy

2 , ¯ = (vz ¡ V )2 , ° = y + vx =­. The solution of the equation for particle motion are ® = v? Consequently f0 (®; ¯; °) satis¯es (C.12) and we adopt the following zeroth order distribution function

f0

µ

2 v? ; (vz

vx ¡ V ) ;y + ­ 2

¶

= n0

Ã

Ã

v2 (vz ¡ V )2 1 + ¡² + ±? ? + ± z 2 2 2vT? 2v?z

£

Ã

1 2 2¼vT?

!Ã

1 2 2¼vTz

!1=2

Ã

!µ

vx y+ ­

¶!

v2 (vz ¡ V )2 ¡ exp ¡ ? 2 2 2vT? 2vTz

The density gradient and temperature gradient of this distribution function are ±z 1 dn0 = ¡² + ±? + ; n0 dy 2 1 dT? = ±? ; T? dy

!

: (C:39)

288

C Derivation of Dielectric Tensor in Hot Plasma

1 dTz = ±z : Tz dy Let us consider the following perturbation: Á1 (r; t) = Á1 (y) exp(ikx x + ikz z ¡ i!t): Then the integrand becomes r0r Á1 ¢ r0v f0 = (v 0 ¢ r0r Á1 )2

µ

¶

@f0 @f0 ikx @f0 @f0 + 2ikz (vz0 ¡ V ) 0 ¡ 2ikz vz0 + Á1 : @®0 @¯ @®0 ­ @° 0

Using @Á1 dÁ1 = + (v 0 ¢ r0r Á1 ) = ¡i!Á1 + v 0 ¢ r0r Á1 ; 0 dt @t0 Z

t

0

v ¢

r0r Á1 dt0

®0 = ®;

= Á1 + i!

¯ 0 = ¯;

Z

t

Á1 dt0 ;

°0 = °

we ¯nd that f1 =

µ

µ

@f0 @f0 ikx @f0 q @f0 @f0 ¡ 2ikz vz 2 Á1 + 2i! + 2ikz (vz ¡ V ) + m @® @® @¯ @® ­ @° £

Z

t

0

¡1

0

0

0

Á1 (y ) exp(ikx x + ikz z ¡ i!t )dt

0

¶

¶

(C:40)

and Z

t

Á(r 0 ; t0 )dt0 =

¡1

Z

t

¡1

Á1 (y 0 ) exp(ikx x0 + ikz z 0 ¡ i!t0 )dt0 µ

= Á1 (y) exp(ikx x + ikz z ¡ i!t) exp ¡i £

Z

t

exp

¡1

µ

kx v? sin µ ­

¶

¶

ikx v? sin(µ + ­¿ ) + i(kz vz ¡ !)¿ d¿: ­

(C:41)

Using the expansion exp(ia sin µ) =

1 X

Jm (a) exp imµ;

m=¡1

J¡m (a) = (¡1)m Jm (a) we write the integral as Z

t

¡1

Á1 (r 0 ; t0 )

= Á1 (r; t)

1 X

i (Jn2 (a) + Jn (a)Jn¡1 (a) exp iµ + Jn (a)Jn+1 (a) exp(¡iµ) + ¢ ¢ ¢) ¡ n­ ! ¡ k v z z n=¡1 (C:42)

C.6 Dispersion Relation of Electrostatic Wave in ¢ ¢ ¢

289

where a = kx v? =­. Substitution of the foregoing equation into eq.(C.40) gives "

#

µ ¶ @f0 ikx @f0 X (Jn2 (a) + ¢ ¢ ¢) q @f0 @f0 ¡ ¡ V ) f1 = Á1 2 2(! ¡ kz vz ) : (C:43) + 2kz (vz + m @® @® @¯ ­ @° ! ¡ kz vz ¡ n­

When this expression for f1 is substituted into eq.(C.37), we ¯nd the dispersion relation of electrostatic wave in more general inhomogenous plasma as follows: Ã

kx2

+

kz2

@2 ¡ 2 @x

!

Á1 = Á1

µ

X qj2

² 0 mj

j

Z Z Z "

2

@f0 @®

@f0 ikx @f0 @f0 ¡ 2(! ¡ kz vz ) + 2kz (vz ¡ V ) + @® @¯ ­ @° £

1 X Jn2 + Jn Jn¡1 exp iµ + Jn Jn+1 exp(¡iµ)

! ¡ kz vz ¡ n­

n=¡1

¶

#

+ ¢ ¢ ¢ dµdv? dv? dvz :

(C:44)

j

For j(kx2 + ky2 )Á1 j À j@ 2 Á1 =@x2 j, eq.(C.44) is reduced to (kx2 + kz2 ) ¡

X

¦j2

j

1 n0j

Z Z Z

[ ]j dµdv? dv? dvz = 0:

By the same way as sec.C.3, this dispersion relation is reduced to kx2

+ kz2

+

X

¦j2

j

+

1 X

¡b

In (b)e

n=¡1

"

1 2 vTz

ÃÃ

1 n­ 1 ¡ 2 2 vTz vT? !n

!

"

µ

¶

n n­ ³n Z(³n ) ¡ 2 (² + ±? ¡ fn (b)±? ) 1 + ³n Z(³n ) !n vT? kx 1

±z ¡ 2 "

1 n ±z + 2 (² + ±z ¡ fn (b)±? )(1 + ³n Z(³n )) ¡ 2 vTz kx

Ã

"

Ã

!2 1 + 2 n2 (1 + ³n Z(³n )) kz vTz #!#

kx ³n ±z !n + (² ¡ fn (b)±? ) Z(³n ) ¡ 2 (1 + ³n Z(³n )) ­ !n 2 kz2 vTz

!#

n­!n 1 + 2 2 (1 + ³n Z(³n )) kz vTz

= 0:

!#

(C:45)

j

Here we set y = 0 and used the following relations: !n = ! ¡ kz V + n­; Z

1

¡1

Jn2 (b1=2 x) exp

Ã

x2 ¡ 2

!

¢

x2 xdx = fn (b)In (b)e¡b ; 2

fn (b) ´ (1 ¡ b) + bIn0 (b)=In (b): When ² = ±? = ±z = 0, this dispersion relation becomes (C.36). In the case of low frequency ! ¿ j­j, we have ³n À 1 (n 6= 0), ³n Z(³n ) ! ¡1 (n 6= 0) and 1 + ³n Z(³n ) ! ¡(1=2)³n¡2 (n 6= 0). Then (C.45) reduces to

290

C Derivation of Dielectric Tensor in Hot Plasma kx2

+

kz2

+

X

¦j2

j

Ã

1 2 vTz

+ I0 (b)e

¡b

Ã

1 1 2 (1 + ³0 Z(³0 )) ¡ v 2 vTz T?

kx kx + (² ¡ f0 (b)±? )³0 Z(³0 ) ¡ ±z ³ 2 (1 + ³0 Z(³0 )) ­!0 ­!0 0

¶!

= 0:

(C:46)

j

The relation 1 X

In (b)e¡b = 1

¡1

was used. When vT? = vTz = vT , ±? = ±z = 0, V = 0, we have familiar dispersion relation of drift wave due to the density gradient as follows, kx2

+

kz2

+

X j

¦j2

Ã

1 ¡b 2 + I0 (b)e vT

Ã

1 kx ³0 Z(³0 ) + ²³0 Z(³0 ) 2 ­!0 vT

!!

= 0:

(C:47)

j

We can usually assume be = 0 for electrons. (C.47) reduces to 0=

(kx2

v2 + kz2 ) Te2 ¦e

µ

!¤ + 1 + ³e Z(³e ) 1 ¡ e !

where !e¤ =

2 ¡kx ²vTe ¡kx ²Te = ­e eB

!i¤ =

2 ¡kx ²vTi kx ²Ti = : ­i ZeB

¶

µ

µ

ZTe !¤ + 1 + I0 (b)e¡b ³0 Z(³0 ) 1 ¡ i Ti !

¶¶

;

(C:48)

291

Physical Constants, Plasma Parameters and Mathematical Formula 2:99792458 £ 108 m=s (de¯nition)

c (speed of light in vacuum)

8:8541878 £ 10¡12 F=m 1:25663706 £ 10¡6 H=m (= 4¼ £ 10¡7 )

²0 (dielectric constant of vacuum) ¹0 (permeability of vacuum) h (Planck0 s constant) ∙ (Boltzmann0 s constant)

6:6260755(40) £ 10¡34 Js 1:380658(12) £ 10¡23 J=K

A (Avogadro0 s number) e (charge of electron)

6:0221367(36) £ 1023 =mol(760 torr;0± C; 22:4l) 1:60217733(49) £ 10¡19 C 1:60217733(49) £ 10¡19 J 1:6726231(10) £ 10¡27 kg

1 electron volt (eV) mp (mass of proton)

9:1093897(54) £ 10¡31 kg 11; 604 K=V

me (mass of electron) e=∙ mp =me (mp =me )1=2

1836 42:9

me c2

0:5110 MeV

Units are MKS, ∙T =e in eV, ln ¤ = 20 101=2 = 3:16 ¶1=2 ³ n ´1=2 ne e 2 e = 5:64 £ 1011 ; me ² 0 1020 eB ­e = = 1:76 £ 1011 B; me Z ZeB ¡­i = = 9:58 £ 107 B; mi A ¦e =

µ

³ n ´1=2 ¦e e ; = 8:98 £ 1010 2¼ 1020 ­e = 2:80 £ 1010 B; 2¼ ¡­i Z = 1:52 £ 107 B; 2¼ A

¶¡3=2 ∙Te ni = = 2:9 £ 10 Z ; ºeik = 20 1=2 2 1=2 3=2 ¿eik e 10 51:6¼ ²0 me (∙Te ) ¶¡3=2 µ 4 ∙Ti 1 ni Z 4 e4 ln ¤ ni 9 Z ºiik = = = 0:18 £ 10 ; 1=2 1=2 ¿iik e 1020 A 31=2 6¼²20 mi (∙Ti )3=2 µ ¶¡3=2 2 ∙Te Z 2 ne e4 ln ¤ me ne ² 6Z = = 6:35 £ 10 ; ºie A e 1020 (2¼)1=2 3¼²20 me 1=2 (∙Te )3=2 mi ni Z 2 e4 ln ¤

1

¸D =

µ

²0 ∙T ne e 2

¶1=2

= 7:45 £ 10

9

¡7

µ

∙Te e

¶1=2µ

2

µ

ne 1020

¶¡1=2

;

¶1=2 ∙Te 1 ; e B µ ¶1=2 A∙Ti 1 ¡4 1 ½­i = 1:02 £ 10 Z e B ¶2µ ¶1=2 µ ¶¡1 µ ne 3∙Te 1 ¡4 ∙Te ¸ei = = 2:5 £ 10 ; me ºeik e 1020 ¶2µ ¶1=2 µ ¶¡1 µ ∙Ti ni 3∙Ti 1 ¡4 1 = 0:94 £ 10 ; ¸ii = mi ºiik Z4 e 1020

½­e = 2:38 £ 10¡6

µ

Physical Constants, Plasma Parameters and Mathematical ¢ ¢ ¢

292

¶1=2 B2 B = 2:18 £ 106 ; ¹0 n i m i (Ani =1020 )1=2 ¶1=2 ¶1=2 µ µ ∙Te ∙Te vTe = = 4:19 £ 105 ; me e ¶1=2 ¶1=2 µ µ ∙Ti 3 ∙Ti = 9:79 £ 10 ; vTi = mi Ae vA =

µ

¶¡3=2 µ 1=2 Ze2 me ln ¤ ∙Te ¡5 ´k = = 5:2 £ 10 Z ln ¤ ; e 51:6¼ 1=2 ²20 (∙Te )3=2 µ ¶¡1=2 ¶µ n ∙Te me ∙Te ¡2 Z ; Dcl = 2 2 ºei? = 3:3 £ 10 e B B 1020 e 1 ∙Te DB = ; 16 eB µ ¶¡1 ³ v ´2 m ne ²0 B 2 ∙Te 2 A i 2 = = = = 0:097B ; me n e c me me c2 ¯e 1020 ¶3=2 µ µ ¶¡1=2 4¼ ∙Te ne ; N¸ ´ ne ¸3D = 1:73 £ 102 3 e 1020 ¶µ µ ¶ n ∙T n∙T ¡5 1 ; ¯= = 4:03 £ 10 (B 2 =2¹0 ) B2 e 1020 µ ¶2 vTe mi = ¯e ; vA 2me µ ¶2 vTi 1 = ¯i ; vA 2 ¶2 µ ¶2 µ ¸D vA me n e = ; c ½­e mi n i ¶ µ 2¶µ B ¹0 a aB(∙Te =e)3=2 ¿R = 2:6 £ 103 = ; S number ´ 1=2 ¿H ´ ¹0 (ni mi ) a ZA1=2 (n=1020 )1=2 DB 1 ­e = ; Dcl 16 ºei? ¦e 51:6¼ 1=2 = ne ¸3D : ºeik ln ¤Z µ

­e ¦e

¶2

Physical Constants, Plasma Parameters and Mathematical ¢ ¢ ¢ a¢(b £ c) = b¢(c £ a) = c¢(a £ b);

293

cylindrical coordinates (r; µ; z)

a £ (b £ c) = (a ¢ c)b ¡ (a ¢ b)c;

ds2 = dr 2 + r 2 dµ 2 + dz 2 ;

(a £ b) ¢ (c £ d) = a ¢ b £ (c £ d) = a ¢ ((b ¢ d)c ¡ (b ¢ c)d) = (a ¢ c)(b ¢ d) ¡ (a ¢ d)(b ¢ c); r ¢ (Áa) = Ár ¢ a + (a ¢ r)Á; r £ (Áa) = rÁ £ a + Ár £ a; r(a ¢ b) = (a ¢ r)b + (b ¢ r)a +a £ (r £ b) + b £ (r £ a); r ¢ (a £ b) = b ¢ r £ a ¡ a ¢ r £ b; r £ (a £ b) = a(r ¢ b) ¡ b(r ¢ a) +(b ¢ r)a ¡ (a ¢ r)b;

rà =

@Ã 1 @Ã @Ã i1 + i2 + i3 ; @r r @µ @z 1 @ 1 @F2 @F3 (rF1 ) + + ; r @r r @µ @z

r¢F =

¶ µ 1 @F3 @F2 ¡ i1 r£F = r @µ @z ¶ µ ¶ µ 1 @ @F1 @F3 1 @F1 ¡ i+ (rF2 ) ¡ i3 ; + @z @r r @r r @µ µ ¶ @Ã 1 @2Ã @2Ã 1 @ r + 2 2 + r Ã= ; r @r @r r @µ @z 2 2

spherical coordinates (r; µ; Á)

r £ r £ a = r(r ¢ a) ¡ r2 a (x; y; z coordinates only); r £ rÁ = 0;

rà =

r ¢ (r £ a) = 0; r = xi + yj + zk; r £ r = 0;

Z

rÁ ¢ dV =

Z

Z

r ¢ adV =

Z

Z

r £ adV =

Z

n £ rÁda =

I

V

V

Z

S

Z

S

Ánda;

S

a ¢ nda;

S

S

r £ a ¢ nda =

@Ã 1 @Ã 1 @Ã i1 + i2 + i3 ; @r r @µ r sin µ @Á

r¢F =

r ¢ r = 3;

V

ds2 = dr 2 + r 2 dµ2 + r2 sin2 µdÁ2 ;

n £ ada;

Áds;

C

I

C

a ¢ ds:

+

1 @ 2 (r F1 ) r 2 @r

1 @ 1 @F3 (sin µF2 ) + ; r sin µ @µ r sin µ @Á

µ ¶ @ @F2 1 (sin µF3 )¡ i1 r£F= r sin µ @µ @Á 1 + r

µ

¶ 1 @F1 @ ¡ (rF3 ) i2 sin µ @Á @r

1 + r

µ

@ @F1 (rF2 ) ¡ @r @µ

r2 Ã = +

1 @ r2 @r

¶

i3 ;

µ ¶ µ ¶ 1 @Ã @Ã @ r2 + 2 sin µ @r r sin µ @µ @µ

@2Ã 1 : r 2 sin2 µ @Á2

294

Index

Index (Numbers are of chapters, sections and subsections ) Absolute minimum-B ¯eld, 8.1a, 17.3b Accessibility of lower hybrid wave, 12.5 Adiabatic heating, 2.5 Adiabatic invariant, 2.5 Alfv¶en eigenmodes, 14.2 Alfv¶en velocity, 5.4, 10.4a Alfv¶en wave 5.4, 10.4a compressional mode, 5.4, 10.4a torsional mode, 5.4, 10.4a Aspect ratio, 3.5 Average minimum-B ¯eld, 8.1b Axial symmetry, 3.2, 3.4, 6.2 Ballooning instability, 8.5, 16.4 B.3 Banana, 3.5b orbit, 3.5b, 7.2 width, 3.5b, 7.2 Banana region, 7.2 Bernstein wave, 12.4 Bessel function model, 17.1b Beta ratio, 6.1 upper limit, 6.5, 6.6, 15.4 Bi-Maxwellian, 12.3 Biot-Savart equation, 3.1 Bohm di®usion, 7.3 Boltzmann's equation, 4.2 Bootstrap current, 16.8d Break even condition, 1.3 Bremsstrahlung, 1.3 Canonical variables, 3.3, 4.1 Carbon tiles, 16.5 Charge exchange, 2.6 Charge separation, 3.5, 7.1a Circular polarization, 10.2a Circulating particle, 3.5a Classical di®usion, 7.1a CMA diagram, 10.3 Cold plasma, 10.1 Collisonal drift instability, 9.2 Collisional frequency, 2.6 Collisional region, 7.2 Collision time, 6.2 Conducting shell, 15.2a, 16.10 Conductive energy loss, 7.0 Con¯nement time of cusp, 17.3a of mirror, 17.3a of stellarator, 17.2d of tandem mirror, 17.3d of tokamak, 16.6, 16.7 Connection length, 7.1a Convective loss, 7.3, 17.2d Coulomb collision, 2.6 Coulomb logarithm, 2.6 Cross section, 2.6 of Coulomb collision, 2.6 of nuclear fusion, 1.3 Cusp ¯eld, 17.3a, 3.4 Cuto®, 10.2b Cyclotron damping, 11.3, 12.3 Cyclotron frequency, 2.3, 10.1 Cyclotron resonance, 10.4b Debye length, 1.2 Debye shielding, 1.2 Degenerated electron plasma, 1.1 Deceleration time, 2.6 Diamagnetism, 6.1, 6.3 Dielectric constant, 3.1

Dielectric tensor of cold plasma, 10.1 of hot plasma, 12.3, C.4 Di®usion, 7.1, 7.2, 7.3, 7.4 Di®usion coe±cient, 7.1, 7.2, 7.3, 7.4 Di®usion tensor in velocity space, 16.8a Dispersion relation of cold plasma, 10.1 of electrostatic wave, 13.1, 13.4, C.5, C.6 of hot plasma, 12.3, 12.4, 12.5, 12.6 Disruptive instability, 16.3 Dissipative drift instability, 9.2 Distribution function in phase space, 4.1 Divertor, 16.5 Dragged electron, 16.8c Dreicer ¯eld, 2.7 Drift approximation, 2.4 Drift frequency of ion and electron, 8.6 Drift instability, 9.2, C.6 Drift velocity of guiding center, 2.4 curvature drift, 2.4 E cross B drift, 2.4 gradient B drift, 2.4 Dynamic friction, coe±cient of, 16.8a E®ective collision frequency, 7.2 Electric displacement, 3.1 Electric intensity, 3.1 Electric polaization, 10.1 Electric resistivity, 2.8 Electron beam instability, 13.3 Electron cyclotron current drive, 16.8b Electron cyclotron heating, 12.6 Electron cyclotron resonance, 10.3 Electron cyclotron wave, 10.4e Electron plasma frequency, 2.2, 10.1 Electron plasma wave, 2.2 Electrostatic wave, 10.5, 12.5, C.5, C.6 Elliptic coil, 16.2b Elongated plasma, 16.4 Energy con¯nement time, 7.0, 16.6, 16.7, 17.2d, 17.3a,17.3d Energy density of wave in dispersive medium, 12.1 Energy integral, 8.2b, B.1, B.2, B.3 Energy principle, 8.2b Energy relaxation time, 2.6 Energy-transport equation, 7.0, 15.6, A Equation of continuity, 5.1 Equation of motion, 5.2, 5.2 Equilibrium condition in tokamak, 6.1, 6.2, 6.3, 16.2 Equilibrium equation in axially symmetric system, 6.2 Eta i (´i ) mode, 8.6 Euler's equation, 8.3 Excitation of waves, 12.0 Extraordinary wave, 10.2a Fast wave, 10.2a Fermi acceleration, 2.5 Field particle, 2.6 Fishbone instability, 14.1 Fluctuation loss, 7.3 Flute instability, 8.1a Fokker-Planck collision term, 4.1, 4.2 Fokker-Planck equation, 4.2 Fusion core plasma, 1.3 Fusion reactor, 1.3, 16.11 Galeev-Sagdeev di®usion coe±cient, 7.2 Gradient-B drift, 2.4 Grad-Shafranov equation, 6.2 Greenward normalized density, 16.3 Greenward-Hugill-Murakami parameter, 16.3

Index Group velocity, 12.1 Guiding center, 2.4 Hamiltonian, 3.3 Harris instability, 13.4 Heating additional in tokamak plasma, 16.7 by neutral beam injection, 2.6, 16.7 Helical coil, 17.2a Helical symmetry, 17.2a Helical system, See Stellarator Heliotron/torsatron, 17.2b Hermite matrix, 12.1 Hermite operator, 8.2b H mode, 16.7 Hohlraum target, 18.2 Hoop force, 6.3 Hybrid resonance, 10.3, 12.5 Ignition condition, 1.3 Implosion, 18.2 Inertial con¯nement, 18.1 Interchange instability, 8.1a Intermediate region, 7.2 Internal (minor) disruption, 16.3 INTOR, 16.11 Io®e bar, 17.3b Ion cyclotron resonance, 10.2b Ion cyclotron range of frequency (ICRF), 12.4 Ion cyclotron wave, 10.4b ITER, 16.11 Kaye and Goldston scaling, 16.6 Kink instability, 8.1d, 8.3a(iii) Kruskal-Shafranov limit, 8.3a(iii) Lagrangian formulation, 3.3 Landau ampli¯cation, 11.1 Landau damping, 11.1, 12.3 Langmuir wave, 2.2 Larmor radius, 2.3 Laplacian, 1.2 Laser plasma. See Inertial con¯nement, 18 Linearized equaion of magnetrohydrodynamics, 8.2 Line of magnetic force, 3.2 Liouville's theorem, 4.1 Lithium blanket, 1.3 L mode, 16.6 Longitudinal adiabatic invariant, 2.5 Loss cone angle, 2.5, 17.3d instability, 17.3c(ii) Loss time. See Con¯nement time Lower hybrid resonant frequency, 10.4c Lower hybrid current drive, 16.8a Lower hybrid wave heating, 12.5 L wave, 10.2a Macroscopic instability, 8.0 Magnetic axis, 3.5 Magnetic con¯nement, 14 Magnetic di®usion, 5.3 Magnetic °uctuation, 7.4 Magnetic helicity, 17.1b Magnetic induction, 3.1 Magnetic intensity, 3.1 Magnetic moment, 2.5 Magnetic probe, 16.1 Magnetic Reynolds number, 5.3 Magnetic surface, 3.2 Magnetic viscosity, 5.3 Magnetic well depth, 8.1b Magnetohydrodynamic (MHD) equations of one °uid, 5.2

295 of two °uids, 5.1 Magnetohydrodynamic instability, 8.0 instability in tokamak, 16.3 region, 7.2 Magnetoacoustic waves, 5.4 Major axis, 3.5 Major radius, 3.5 Mass defect, 1.3 Maxwell distribution function, 2.1 Maxwell equations, 3.1 Mean free path, 6.1 Microscopic instability, 13.0 Minimum-B ¯eld absolute, 8.1a, 17.3b, average, 8.1b stability condition for °ute, 8.1b Minor axis, 3.5 Minor (internal) disruption, 16.3 Minor radius, 3.5 Mirror ¯eld condition of mirror con¯nement 2.5, 17.3a Instabilities in, 17.3c Mirror instability, 17.3c(iii) Mode conversion, 12.0, 12.5 Negative ion source, 16.7 Negative dielectric constant, 13.3 Negative energy density, 13.3 Negative mass instability, 17.3c(iv) Negative shear, 8.5, 16.7, 16.9d Neoclassical di®usion in helical ¯eld, 17.2c of tokamak, 7.2 Neoclassical tearing mode, 16.9 Neutral beam injection (NBI), 2.6, 16.7 Neutral beam current drive, 2.6, 16.8c Noncircular cross section, 16.4 Ohmic heating, 2.8, 5.3 Ohm's law, 2.8 Open-end magnetic ¯eld, 17.3 Orbit surface, 3.6 Ordinary wave, 10.2a Oscillating ¯eld current drive, 17.1d Paramagnetism, 6.3 Particle con¯nement time. See also Con¯nement time Particle orbit and mgnetic surface, 3.6 Pastukhov's con¯nement time, 17.3d Pellet gain, 18.1 Permeability, 3.1 P¯rsch-SchlÄ uter current, 6.6 P¯rsch-SchlÄ uter factor, 7.1a Pitch minimum, 17.1b Plasma dispersion function, 12.3 Plasma parameter, 1.2 Plateau region, 7.2 Poisson's equation, 3.2 Polarization, 10.2 Poloidal beta ratio, 6.3 Poloidal ¯eld, 3.5 Poynting equation, 12.1 Poynting vector, 12.1 Pressure tensor, A Quasi-linear theory of distribution function, 11.4 Radiation loss, 1.3, 16.6 Ray tracing, 12.2 Relaxation process, 17.1b Resistive drift instability, 9.2 Resistive instability, 9.1 Resistive wall mode, 16.10

296 Resonance, 10.2b Resonant excitation, 12.0 Reversed ¯eld pinch (RFP), 17.1 con¯gurations, 17.1a relaxation, 17.1b Reversed shear con¯guration, See Negatve shear Rotational transform angle 3.5 of helical ¯eld, 17.2a of tokamak, 3.5, 16.4 Runaway electron, 2.7 R wave, 10.2 Safety factor, 8.3a(iii), 16.4 Sausage instability, 8.1c Scalar potential, 3.1 Scrape o® layer, 16.5 Self-inductance of plasma current ring, 6.3 Separatrix, 16.5, 17.2 Sharfranov shift, 8.5 Shear, 8.3c Sheared °ow, 16.7 Spherical tokamak, 15 Slowing down time of ion beam, 2.6 Slow wave, 10.2a Small solution, 8.3c Speci¯c resistivity, 2.8 Speci¯c volume, 8.1b Sputtering, 16.5 Stability of di®use-boundary current con¯guration, 8.3a(iii) Stability of local mode in torus, 17.1b Stellarator, con¯nement experiments in, 17.2d devices, 17.2b ¯eld, 17.1a neoclassical di®usion in, 17.1a rotational transform angle of, 17.1a upper limit of beta ratio of, 6.6 Stix coil, 12.4 Strongly coupled plasma, 1.2 Superbanana, 17.2c Supershot, 16.7 Suydam's criterion, 8.3c Tandem mirror, 17.3d Torsatron/heliotron, 17.2b Tearing instability, 9.1 Test particle, 2.6 Thermal conductivity, 7.0, 16.5 Thermal barrier, 17.3d Thermal energy of plasma, 1.3, 16.11 Thermal °ux, 7.0 Tokamak 16 conducting shell, 16.2a con¯nement scaling, 16.6, 16.7 devices, 16.1 equilibrium condition, 6.2, 16.2 impurity control, 16.5 MHD stability, 16.3 neoclassical di®usion in, 7.2 poloidal ¯eld in, 6.4 reactor, 16.11 Toroidal Alfv¶en eigenmode, 14.2 Toroidicity induced Alfv¶en eigenmode, 14.2 Toroidal coordinates, 6.3 Toroidal drift, 3.5 Transit-time damping, 11.2, 12.3 Translational symmetry, 3.3 Transversal adiabatic invariant, 2.5 Trapped particle, 3.5b see banana Trapped particle instability, 13.4 Triangularity of plasma cross section, 16.4, 16.11 Troyon factor, 16.4 Two stream instability, 13.2

Index Untrapped particles, 3.5a Upper hybrid resonant frequency, 10.4d Vector potential, 3.1 Velocity di®usion time, 2.6 Velocity space distribution function, 2.1 Velocity space instability, 13.0 VH mode, 16.7 Virial theorem, 6.7 Vlasov's equation, 4.2 Wall mirror ratio, 16.3b Ware pinch, 3.7 Wave absorption, 12.2 propagation, 12.0, 12.2 Wave heating electron cyclotron, 12.6 in ion cyclotron range, 12.4 lower hybrid, 12.5 Whistler instability, 17.3c(v) Whistler wave, 10.4e