Plasma physics for controlled fusion second edition

Springer Series on Atomic, Optical, and Plasma Physics 92 Kenro Miyamoto Plasma Physics for Controlled Fusion Second Edition Springer Series on Atomic, Optical, and Plasma Physics Volume 92 Editor-in-chief Gordon W.F Drake, Windsor, Canada Series editors James Babb, Cambridge, USA Andre D Bandrauk, Sherbrooke, Canada Klaus Bartschat, Des Moines, USA Philip George Burke, Belfast, UK Robert N Compton, Knoxville, USA Tom Gallagher, Charlottesville, USA Charles J Joachain, Bruxelles, Belgium Peter Lambropoulos, Iraklion, Greece Gerd Leuchs, Erlangen, Germany Pierre Meystre, Tucson, USA The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire field of atoms and molecules and their interaction with electromagnetic radiation Books in the series provide a rich source of new ideas and techniques with wide applications in fields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the field, such as quantum computation and Bose-Einstein condensation The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the field More information about this series at http://www.springer.com/series/411 Kenro Miyamoto Plasma Physics for Controlled Fusion Second Edition 123 Kenro Miyamoto Tokyo Japan First edition published with the title: Plasma Physics and Controlled Nuclear Fusion ISSN 1615-5653 ISSN 2197-6791 (electronic) Springer Series on Atomic, Optical, and Plasma Physics ISBN 978-3-662-49780-7 ISBN 978-3-662-49781-4 (eBook) DOI 10.1007/978-3-662-49781-4 Library of Congress Control Number: 2016936992 © Springer-Verlag Berlin Heidelberg 2005, 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg Preface The worldwide effort to develop the fusion process as a new energy source has been going on for about a half century and has made remarkable progress Now construction stage of “International Tokamak Experimental Reactor”, called ITER, already started Primary objective of this textbook is to present a basic knowledge for the students to study plasma physics and controlled fusion researches and to provide the recent aspect of new results Chapter describes the basic concept of plasma and its characteristics The orbits of ion and electron are analyzed in various configurations of magnetic field in Chap From Chap to Chap 7, plasmas are treated as magnetohydrodynamic (MHD) fluid MHD equation of motion (Chap 3), equilibrium (Chap 4), and confinement of plasma in ideal cases (Chap 5) are described by the fluid model Chapters and 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 introduced in Appendix B From Chap to Chap 13, plasmas are treated by kinetic theory Boltzmann’s equation is introduced in Chap This equation is the starting point of the kinetic theory Plasmas, as mediums in which waves and perturbations propagate, are generally inhomogeneous and anisotropic It may absorb or even amplify the wave and perturbations Cold plasma model described in Chap 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 waves can be discussed in the case of cold plasma If the refractive index of plasma becomes large and the phase velocity of the wave becomes comparable to the thermal velocity of the plasma particles, then the particles and the waves interact with each other Chapter 10 describes Landau v vi Preface damping, which is the most important and characteristic collective phenomenon of plasma Waves in hot plasma, in which the wave phase velocity is comparable to the thermal velocity of particles, are analyzed by use of dielectric tensor of hot plasma Wave heating (wave absorption) in hot plasmas and current drives are described in Chap 11 Non-inductive current drives combined with bootstrap current are essential in order to operate tokamak in steady state condition Instabilities driven by energetic particles (fishbone instability and toroidal Alfvén eigenmodes) are treated in Chap 12 In order to minimize the loss of alpha particle produced by fusion grade plasma, it is important to avoid the instabilities driven by energetic particles and alpha particles Chapter 13 discusses the plasma transport by turbulence Losses of plasmas with drift turbulence become Bohm type or gyro Bohm type depending on different magnetic configuration Analysis of confinement by computer simulations is greatly advanced Gyrokinetic particle model and full orbit particle model are introduced Furthermore it is confirmed recently that the zonal flow is generated in plasmas by drift turbulence Understanding of the zonal flow drive and damping has suggested several routes to improving confinement Those new topics are included in Chap 13 In Chap 14, confinement researches toward fusion plasmas are reviewed During the last two decades, tokamak experiments have made a remarkable progress Chapter 15 introduces research works of critical subjects on tokamak plasmas and the aims of ITER and its rationale are explained Chapter 16 explains reversed field pinch including PPCD (pulsed parallel current drive), and Chap 17 introduces the experimental results of advanced stellarator devices and several types of quasi-symmetric stellarator Boozer equation to formulate the drift motion of particles is explained in Appendix C Chapter 18 describes open-end systems including tandem mirrors Elementary introduction of inertial confinement including the fast ignition is added in Chap 19 Readers may have an impression that there is too much mathematics in this book However, it is one of motivation to write this text to save the time to struggle with the mathematical deduction of theoretical results so that students could spend more time to think physics of experimental results This textbook has been attempted to present the basic physics and analytical methods comprehensively 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 fields Tokyo, Japan Kenro Miyamoto Contents Nature of Plasma 1.1 Introduction 1.2 Charge Neutrality and Landau 1.3 Fusion Core Plasma References Damping 1 Orbit of Charged Particles in Various Magnetic Configuration 2.1 Orbit of Charged Particles 2.1.1 Cyclotron Motion 2.1.2 Drift Velocity of Guiding Center 2.1.3 Polarization Drift 2.1.4 Pondromotive Force 2.2 Scalar Potential and Vector Potential 2.3 Magnetic Mirror 2.4 Toroidal System 2.4.1 Magnetic Flux Function 2.4.2 Hamiltonian Equation of Motion 2.4.3 Particle Orbit in Axially Symmetric System 2.4.4 Drift of Guiding Center in Toroidal Field 2.4.5 Effect of Longitudinal Electric Field on Banana Orbit 2.4.6 Precession of Trapped Particle 2.4.7 Orbit of Guiding Center and Magnetic Surface 2.5 Coulomb Collision and Neutral Beam Injection 2.5.1 Coulomb Collision 2.5.2 Neutral Beam Injection 2.5.3 Resistivity, Runaway Electron, Dreicer Field 2.6 Variety of Time and Space Scales in Plasmas References 11 11 11 12 16 17 19 21 23 23 24 27 28 32 33 38 40 40 45 46 47 49 vii viii Contents Magnetohydrodynamics 3.1 Magnetohydrodynamic Equations for Two Fluids 3.2 Magnetohydrodynamic Equations for One Fluid 3.3 Simplified Magnetohydrodynamic Equations 3.4 Magnetoacoustic Wave 51 51 54 56 59 Equilibrium 4.1 Pressure Equilibrium 4.2 Grad–Shafranov Equilibrium Equation 4.3 Exact Solution of Grad–Shafranov Equation 4.4 Tokamak Equilibrium 4.5 Upper Limit of Beta Ratio 4.6 Pfirsch Schluter Current 4.7 Virial Theorem References 63 63 65 67 70 77 78 81 83 Confinement of Plasma (Ideal Cases) 5.1 Collisional Diffusion (Classical Diffusion) 5.1.1 Magnetohydrodynamic Treatment 5.1.2 A Particle Model 5.2 Neoclassical Diffusion of Electrons in Tokamak 5.3 Bootstrap Current References 85 87 87 90 91 93 96 Magnetohydrodynamic Instabilities 6.1 Interchange Instabilities 6.1.1 Interchange Instability 6.1.2 Stability Criterion for Interchange Instability 6.2 Formulation of Magnetohydrodynamic Instabilities 6.2.1 Linearization of Magnetohydrodynamic Equations 6.2.2 Rayleigh–Taylor (Interchange) Instability 6.3 Instabilities of Cylindrical Plasma with Sharp Boundary 6.4 Energy Principle 6.5 Instabilities of Diffuse Boundary Configurations 6.5.1 Energy Integral of Plasma with Diffuse Boundary 6.5.2 Suydam’s Criterion 6.5.3 Tokamak Configuration 6.6 Hain Lust Magnetohydrodynamic Equation 6.7 Ballooning Instability 6.8 ηi Mode Due to Density and Temperature Gradient References 97 98 98 102 105 105 109 110 115 118 118 123 124 126 128 133 135 Resistive Instabilities 7.1 Tearing Instability 7.2 Neoclassical Tearing Mode 7.3 Resistive Drift Instability 137 138 144 151 Contents ix 7.4 Resistive Wall Mode 155 References 160 Boltzmann’s Equation 8.1 Phase Space and Distribution Function 8.2 Boltzmann’s Equation and Vlasov’s Equation 8.3 Fokker–Planck Collision Term 8.4 Quasi Linear Theory of Evolution in Distribution Function References 163 163 164 167 171 173 Waves in Cold Plasmas 9.1 Dispersion Equation of Waves in Cold Plasma 9.2 Properties of Waves 9.2.1 Polarization and Particle Motion 9.2.2 Cutoff and Resonance 9.3 Waves in Two Components Plasma 9.4 Various Waves 9.4.1 Alfvén Wave 9.4.2 Ion Cyclotron Wave and Fast Wave 9.4.3 Lower Hybrid Resonance 9.4.4 Upper Hybrid Resonance 9.4.5 Electron Cyclotron Wave (Whistler Wave) 9.5 Conditions for Electrostatic Waves References 175 176 180 180 181 182 185 185 188 189 191 191 193 194 10 Waves in Hot Plasmas 10.1 Landau Damping and Cyclotron Damping 10.1.1 Landau Damping (Amplification) 10.1.2 Transit-Time Damping 10.1.3 Cyclotron Damping 10.2 Formulation of Dispersion Relation in Hot Plasma 10.3 Solution of Linearized Vlasov Equation 10.4 Dielectric Tensor of Hot Plasma 10.5 Dielectric Tensor of bi-Maxwellian Plasma 10.6 Plasma Dispersion Function 10.7 Dispersion Relation of Electrostatic Wave 10.8 Dispersion Relation of Electrostatic Wave in Inhomogeneous Plasma 10.9 Velocity Space Instabilities 10.9.1 Drift Instability (Collisionless) 10.9.2 Ion Temperature Gradient Instability 10.9.3 Various Velocity Space Instabilities References 195 195 195 199 200 202 205 207 210 212 215 217 222 222 223 223 223 478 Appendix C: Quasi-Symmetric Stellarators Table C.1 Vector calculus in general coordinates (1) a j ≡ ∂r/∂u j , ≡ ∇u i , V ≡ a1 · (a2 × a3 ) (2) dr = j ∂r/∂u j du j , · a j = δ ij a1 = V −1 (a2 × a3 ), a2 = V −1 (a3 × a1 ), a3 = V −1 (a1 × a2 ) 3 a1 = V (a × a ), a2 = V (a × a ), a3 = V (a1 × a2 ) −1 a · (a × a ) = V gi j ≡ · a j = g j i , g i j ≡ · a j = g j i F = i f i , f i ≡ F · (contravariant) i F = i fi a , f i ≡ F · (covariant) f j = i gj i f i, f i = j gi j f j (3) (4) (5) (6) (7) (8) (9) (10) (11) g ≡ |gi j | = V , dxdydz = g 1/2 du du du , g 1/2 = [∇u · (∇u × ∇u )]−1 (ds)2 = (dr)2 = i j gi j du i du j = i j g i j du i du j (12) (a × b)1 = g −1/2 (a2 b3 − a3 b2 ), (13) ∇φ = (14) ∇ × F= (15) ∇2φ ∂φ i ∂u i , ∇·F= ∂ f3 ∂u − = ∇ · (∇φ) = 1 g 1/2 ∂ f2 ∂u g 1/2 B = ∇ψ × ∇θ0 , ∂ i ∂u i g 1/2 a1 + i j (a × b)1 = g 1/2 (a b3 − a b2 ) ∂ f1 ∂u − (∂/∂u i ) (g 1/2 f i ) ∂ f3 ∂u a2 + ∂ f2 ∂u − ∂ f1 ∂u a3 g 1/2 g i j (∂φ/∂u j ) θ0 = θ − q −1 ζ, q −1 = dψp , dψ (C.42) ˜ B = ∇χ + β∇ψ, ψ ≡ ψt , χ = μ0 It θ + μ0 Ip ζ, β ≡ μ0 It θ + μ0 Ip + w (C.43) The term b · ∇ × b in (C.40) is expressed by b·∇ ×b= 1 B · ∇ × B = ∇χ · (∇β × ∇ψ) = (∇ψ × ∇χ) · ∇β B2 B B Equations (C.42), (C.43) reduce to (∇ψ × ∇θ0 ) · ∇χ = B The substitution of u = ψ, u = θ0 , and u = χ into (1), (4), and (5) in Table C.1 reduces (∇ψ × ∇χ) = −B (dr/dθ0 ) Therefore, we obtain ∇ × (ρ B) · ∇χ = ρ (∇β × ∇ψ) · ∇χ + (∇ρ × B) · ∇χ = ρ (∇ψ × ∇χ) · ∇β + β(∇ψ × ∇χ) · ∇ρ ∂β ∂ρ = −B ρ − B2β ∂θ0 ∂θ0 and χ˙ = v · ∇χ = v B (B + ∇ × (ρ B)) · ∇χ − ρ (∂β/∂θ0 ) Appendix C: Quasi-Symmetric Stellarators =v B 1− 479 β(∂ρ /∂θ0 ) − β(∂ρ /∂θ0 ) (C.44) (∂ρ /∂θ0 ) means differentiation by θ0 while keeping ψ, χ, and Hamiltonian H0 constant Hamiltonian is (μ is magnetic moment) H0 ≡ e 2 μ B ρ + B + Φ, 2m e (C.45) where B = B(ψ, θ0 , χ) and Φ = Φ(ψ, θ0 , χ) and ρ = (m/e)(v /B) We define θ0c ≡ θ0 − β(ψ, θ0 , χ)ρ (C.46) There are the following relations: ∂θ0 ∂ρ = β+ρ θ0c − ∂θ0 ∂χ θ0c ∂β ∂θ0 ∂θ0 ∂ρ e ∂ρ B ρ m ∂θ0 = , θ0c = H0 θ0c = = θ0c β , − ρ (∂β/∂θ0 ) e ∂ B2 μ ∂B ∂Φ ρ + + , m ∂θ0 e ∂θ0 ∂θ0 ρ ∂β/∂χ , − ρ (∂β/∂θ0 ) ∂θ0 ∂ψ ∂θ0 ∂ρ ∂θ0 = , ∂θ0c − ρ (∂β/∂θ0 ) ρ ∂β/∂ψ − ρ (∂β/∂θ0 ) Let us change the independent variable θ0 to θ0c = θ0 − βρ and H (ρ , ψ, θ0c , χ) ≡ H0 (ρ , ψ, θ0c + βρ , χ), that is, H (ρ , ψ, θ0c , χ) = B = B(ψ, θ0c + βρ , χ), e 2 μ B ρ + B + Φ, 2m e (C.47) Φ = Φ(ψ, θ0c + βρ , χ) Then we have ∂H ∂ H0 ∂ H0 ∂θ0 e ∂ H0 ∂θ0 B ρ + = + = ∂ρ θ0c,ψ,χ ∂ρ ∂θ0 ∂ρ θ0c m ∂θ0 ∂ρ θ0c = e B ρ + m =v B− e ∂ B2 μ ∂B ∂Φ ρ + + 2m ∂θ0 e ∂θ0 ∂θ0 ∂ρ v B ∂θ0 × β − ρ (∂β/∂θ0 ) β(∂ρ /θ0 ) β =v B 1− − ρ (∂β/∂θ0 ) − ρ (∂β/∂θ0 ) 480 Appendix C: Quasi-Symmetric Stellarators Finally we reduce desirous result χ˙ = ∂ H/∂ρ |θ0 c By similar way to reduce χ, ˙ the equation of drift motion is expressed in Hamilton’s canonical form [4] χ˙ = ∂H , ∂ρ ∂H , θ˙0c = ∂ψ ∂H , ∂χ (C.48) ∂H ψ˙ = − ∂θ0c (C.49) ρ˙ = − The canonical transformation to new coordinates (θ0c , χ, ψ, ρ ) → (θc , ζ, Pθ , Pζ ) is given by the following generating function: F(ψ, ρ , θc , ζ) = μ0 ρ (g(ψ)ζ + I (ψ)θc ) + ψθc − ζψ/q, that is, θ0c = ∂F ζ = θc − ∂ψ q χ= + μ0 ρ (ζg + θc I ), (C.50) (C.51) ∂F = μ0 (gζ + I θc ), ∂ρ (C.52) ∂F = μ0 ρ I + ψ, ∂θc (C.53) ∂F = μ0 ρ g + ψ/q ∂ζ (C.54) Pθ = Pζ = New Hamilton is [5] H (θc , ζ, Pθ , Pζ ) = where ρ = e 2 μ B ρ (Pθ , Pζ , ψ) + B + Φ, 2m e Pθ /q − Pζ , μ0 (I /q − g) θ0 = θc − ζ/q + ρ δ, ψ= Pθ g − Pζ I , I /q − g δ = μ0 (ζg + θ I ) + β (B, Φ) = (B, Φ)(ψ, θ0 , χ) = (B, Φ)(ψ, θc − ζ/q + δρ , χ) (C.55) (C.56) (C.57) (C.58) Appendix C: Quasi-Symmetric Stellarators 481 Then Hamilton’s equation in new coordinates is ∂H , θ˙c = ∂ Pθ ∂H P˙θ = − , ∂θc (C.59) ∂H , ζ˙ = ∂ Pζ ∂H P˙ζ = − ∂ζ (C.60) The substitution of θc by θ gives formally H (θ, ζ, Pθ , Pζ ) = e 2 μ B ρ (Pθ , Pζ , ψ) + B + Φ 2m e (C.61) ∂H , θ˙ = ∂ Pθ ∂H P˙θ = − ∂θ (C.62) ∂H , ζ˙ = ∂ Pζ ∂H P˙ζ = − ∂ζ (C.63) The distinction between the solutions of (C.59), (C.60) and (C.62), (C.63) is the order of gyroradius (θ = θc + ρ δ) Therefore, the nonresonant difference between them is negligible [6] In the currentless case (β = δ = 0), they are identical References S Hamada, Hydromagnetic equilibria and their proper coordinates Nucl Fusion 2, 23 (1962) A.H Boozer, Guiding center drift equations Phys Fluids 23, 904 (1980) N Nakajima, J Todoroki, M Okamoto, On relation between Hamada and Boozer magnetic coordinates systems J Plasma Fusion Res 68, 395 (1992) R.B White, A.H Boozer, R Hay, Drift hamiltonian in magnetic coordinates Phys Fluids 25, 575 (1982) R.B White, M.S Chance, Hamiltonian guiding center drift orbit calculation for plasmas of arbitrary cross section Phys Fluids 27, 2455 (1984) A.H Boozer, Time-dependent drift hamiltonian Phys Fluids 27, 2441 (1984) Appendix D Physical Constants, Plasma Parameters and Mathematical Formula c (speed of light in vacuum) 2.99792458 × 108 m/s (definition) −12 F/m (dielectric constant of vacuum) 8.8541878 × 10 1.25663706 × 10−6 H/m (=4π × 10−7 ) μ0 (permeability of vacuum) h (Planck’s constant) 6.6260755(40) × 10−34 Js κ (Boltzmann’s constant) 1.380658(12) × 10−23 J/K A (Avogadro’snumber) 6.0221367(36) × 1023 /mol (760 torr,0◦ C, 22.4l) e (charge of electron) 1.60217733(49) × 10−19 C electron volt (eV) 1.60217733(49) × 10−19 J 1.6726231(10) × 10−27 kg m p (mass of proton) 9.1093897(54) × 10−31 kg m e (mass of electron) e/κ 11,604 K/V 1836 m p /m e 42.9 (m p /m e )1/2 0.5110 MeV m e c2 Units are MKS, T /e in eV, ln Λ = 20, 101/2 = 3.16 Πe = Ωe = n e e2 me eB = me Z eB −Ωi = m i 1/2 = 5.64 × 1011 n e 1/2 Πe , 2π 1020 1.76 × 1011 B, = 9.58 × 107 ZA B, = 8.98 × 1010 Ωe = 2.80 × 1010 B 2π −Ωi = 1.52 × 107 ZA B 2π −3/2 Te νei = n i Z e4 ln Λ = = 3.9 × 10 Z 1/2 3/2 τei 31/2 12π 20 m e Te νii = n i Z e4 ln Λ Z = = 0.18 × 10 1/2 3/2 τii A1/2 31/2 6π 20 m i Ti νi.e = Z n e e4 ln Λ n e 1/2 1020 me 6Z = 6.35 × 10 3/2 A (2π)1/2 3π 20 m e 1/2 Te m i ni 1020 e Ti e Te e −3/2 −3/2 © Springer-Verlag Berlin Heidelberg 2016 K Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic, Optical, and Plasma Physics 92, DOI 10.1007/978-3-662-49781-4 ni 1020 ne 1020 483 484 Appendix D: Physical Constants, Plasma Parameters and Mathematical Formula 1/2 0T n e e2 λD = 1/2 ρΩe = m e 2Te e2 B ρΩi = Am i 2Ti Z e2 B 1/2 λei = 3Te me λii = 3Ti mi vA = B2 μ0 n i m i vTe = Te me vTi = Ti mi 1/2 = 3.37 × 10−6 1/2 = 1.44 × 10−4 1/2 ne 1020 Te e 1/2 = 2.18 × 106 1/2 Te e 1/2 Ti Ae 1/2 = 9.79 × 103 Z e2 m e ln Λ 3/2 51.6π 1/2 20 Te Te e n Z m e Te νei⊥ = 3.3 × 10−2 2 e B B 1020 Te DB = 16 eB = Nλ ≡ β= vTe vA 0B m ene = vA c 3/2 T nT = 4.03 × 10−5 (B /2μ0 ) B e = Snumber ≡ mi βe , 2m e ni 1020 −1 −3/2 −1/2 ne mi Te = = 0.097B me m e c2 βe 1020 Te 4π n e λ3D = 1.73 × 102 e −1 Te e = 5.2 × 10−5 Z ln Λ Dcl = Ωe Πe B B (An i /1020 )1/2 = 4.19 × 105 1/2 1/2 ne 1020 1 Ti = 0.94 × 10−4 νii Z e 1/2 −1/2 B ATi e Z Te = 2.5 × 10−4 νei e 1/2 η = Te e = 7.45 × 10−7 vTi vA = βi , ne 1020 −1 −1/2 n 1020 vA c = λD ρΩe m ene m ini B a BTe /e3/2 μ0 a τR = 2.6 × 10 = τH η μ0 (n i m i )1/2 a Z A1/2 (n/1020 )1/2 DB Ωe = , Dcl 16 νei⊥ Πe 51.6π 1/2 n e λ3D = νei ln ΛZ Appendix D: Physical Constants, Plasma Parameters and Mathematical Formula a·(b × c) = b·(c × a) = c·(a × b) a × (b × c) = (a · c)b − (a · b)c (a × b) · (c × d) = a · b × (c × d) = a · ((b · d)c − (b · c)d) = (a · c)(b · d) − (a · d)(b · c) ∇ · (φa) = φ∇ · a + (a · ∇)φ ∇ × (φa) = ∇φ × a + φ∇ × a ∇(a · b) = (a · ∇)b + (b · ∇)a + a × (∇ × b) + b × (∇ × a) ∇ · (a × b) = b · ∇ × a − a · ∇ × b ∇ × (a × b) = a(∇ · b) − b(∇ · a) + (b · ∇)a − (a · ∇)b ∇ × ∇ × a = ∇(∇ · a) − ∇ a ∇ × ∇φ = ∇ · (∇ × a) = r = x i + y j + zk ∇ · r = 3, ∇×r =0 ∇φ · dV = V φnda S ∇ · adV = V a · nda S (x, y, z coordinates only) 485 486 Appendix D: Physical Constants, Plasma Parameters and Mathematical Formula ∇ × adV = V n × ada S n × ∇φda = S φds C ∇ × a · nda = S a · ds C Cylindrical Coordinates (r, θ, z) ds = dr + r dθ2 + dz ∂ψ ∂ψ ∂ψ i1 + i2 + i3 ∂r r ∂θ ∂z ∇ψ = ∇·F = ∂ F3 ∂ ∂ F2 (r F1 ) + + r ∂r r ∂θ ∂z ∇×F= ∇2ψ = ∂ F2 ∂ F3 ∂ F1 ∂ F3 − − i1 + r ∂θ ∂z ∂z ∂r i2 + ∂ ∂ F1 (r F2 ) − r ∂r r ∂θ ∂ψ ∂2ψ ∂2ψ ∂ r + 2 + r ∂r ∂r r ∂θ ∂z Spherical Coordinates (r, θ, φ) ds = dr + r dθ2 + r sin2 θdφ2 ∇ψ = ∂ψ ∂ψ ∂ψ i1+ i2+ i3 ∂r r ∂θ r sin θ ∂φ ∇·F= ∂ ∂ ∂ F3 (r F1 ) + (sin θF2 ) + r ∂r r sin θ ∂θ r sin θ ∂φ i3 Appendix D: Physical Constants, Plasma Parameters and Mathematical Formula ∇×F= ∂ ∂ F2 (sin θF3 )− i1 + r sin θ ∂θ ∂φ r + ∇2ψ = r ∂ ∂ F1 (r F2 ) − ∂r ∂θ ∂ r ∂r r2 ∂ψ ∂r + ∂ F1 ∂ − (r F3 ) i sin θ ∂φ ∂r i3 ∂ ∂ψ sin θ r sin θ ∂θ ∂θ + ∂2ψ r sin2 θ ∂φ2 487 Curriculum Vitae in Sentence of Kenro Miyamoto Dr Kenro Miyamoto is Professor Emeritus, University of Tokyo He received his diploma in physics (1955) from University of Tokyo and his Ph.D in 1961 from University of Rochester He engaged the constructions and experiments of stellarator and tokamak in the Institute of Plasma Physics, Nagoya University during 1964–1979 His main interest was confinement physics of stellarator and tokamak He moved to Department of Physics, Faculty of Science, University of Tokyo in 1979 He started reversed field pinch (RFP) experiment and studied the relaxation and reconnection phenomena of RFP plasma (1979–1992) He was a member of working group Phase 2A (1981–1983) of INTOR (International Tokamak Reactor) and a member of Tokamak Physics Expert Group of ITER (International Tokamak Experimental Reactor) during 1999–2002 Literary work Plasma Physics for Nuclear Fusion The MIT Press 1980 Plasma Physics for Nuclear Fusion (Revised Ed.) The MIT Press 1989 Plasma Physics and Controlled Nuclear Fusion Springer 2005 © Springer-Verlag Berlin Heidelberg 2016 K Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic, Optical, and Plasma Physics 92, DOI 10.1007/978-3-662-49781-4 489 Index A Accessibility of lower hybrid wave, 242 Adiabatic heating, 16 invariant, 15 Alfven wave, 187 compressional mode, 187 shear mode (torsional mode), 187 Aspect ratio, 30, 69 Average minimum B, 105 Axial symmetry, 26 B Ballooning mode, 128 Banana orbit, 33 region, 93 width, 32 Bernstein wave, 217 Bessel function model, 390 Beta ratio, 65, 78, 80, 348 Bi-Maxwellian, 210 Biot-Savart equation, 21 Bohm diffusion, 288 Boltzmann’s equation, 163 Bootstrap current, 93, 367 Boozer coordinates, see magnetic coordinates Boozer equation of drift motion, 27, 481 Break even condition, Bremsstrahlung, 8, 351 Burning condition, 374 C Carbon tile, 351 Charge exchange, 45, 365 Charge separation, 28 Chirp pulse amplification, 450 Circular polarization, 180 Circulating particle, 30 Classical diffusion, 88 CMA diagram, 184 Cold plasma, 175 Collision time, 41 Collisional drift instability, see resistive drift wave Collisional region, 93 Connection length, 87 Convective loss, 291, 414 Core gain, 440 Coulomb collision, 40 logarithm, 41 Cross section of fusion, Current drive electron cyclotron-(ECCD), 252 lower hybrid-(LHCD), 248 neutral beam-(NBCD), 256 oscillating field current drive, 401 Cusp, 424 Cutoff, 181 Cyclotron damping, 235, 236 frequency, 11, 177 resonance, 181 D Debye length, shielding, Deceleration time, 41 © Springer-Verlag Berlin Heidelberg 2016 K Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic, Optical, and Plasma Physics 92, DOI 10.1007/978-3-662-49781-4 491 492 Degenerated electron plasma, Detached plasma, 355, 363 Diamagnetism, 65 Dielectric constant, 18, 20 Dielectric tensor of cold plsama, 177 of hot plasma, 211 Diffusion coefficient, 85, 90 coefficient due to fluctuation loss, 287 tensor in velocity space, 168, 249 Dimensional analysis of energy confinement, 292 Dimit shift, 302 Dispersion relation of cold plasma, 179 of drift wave, 221 of electrostatic wave, 215, 220 Disruptive instability, 348 Dissipative drift wave, see resistive drift wave Distributuion function in phase space, 164 Divertor, 352 Dragged electron, 256 Dreicer field, 46 Drift approximation, 14 frequency, 153, 221 instability, 155, 222, 223 velocity, 153 velocity of guiding center, 48 Dynamic friction, 168, 249 Dynamo, 396 E Edge harmonic oscillation (EHO), 363 Edge localized mode, 369 Effective collision frequency, 91 Electric displacement, 19 intensity, 19 Electron cyclotron heating (ECH), 247 cyclotron wave, 191 drift frequency, 287 plasma frequency, plasma wave, temperature gradient mode, 322 Electrostatic wave, 193 Electrostatic wave (Langmuir wave), 217 Elliptcal coil, 409 ELM, see edge localized mode Index Elongated tokamak, 348 Energy integral, 115 principle, 115 relaxation time, 44 Energy confinement time of H mode tokamak, 298 of H mode tokamak, 364 of L mode tokamak, 358 of RFP, 399 of stellarator, 415 Energy density of wave in dispersive media, 229 Energy integral of axisymmetric torus, 15, 469 of ballooning mode, 17, 471 of illuminating form, 7, 461 Equation of continuity, 2, 456 energy transport, 5, 459 motion, 3, 53, 457 Equilibrium, 63 Eta i mode, 135, 223 ETG, see electron temperature gradient mode Euler’s equation, 121 Excitation of wave, 225 Extraordinary wave, 180 F Fast ignition, 18, 450 Fast wave, 181 Fermi acceleration, 22 Fermi energy, 442 Field particle, 41 Field reversed configuration, 306 Fish bone instability, 259 Fokker–Planck collision term, 168 Fokker–Plank equation, 167 Fuel-burn ratio, 440 Full orbit particle model, 303 Fusion reactor, G Galeev–Sagdeev diffusion, 92 GAM, see geodesic acoustic mode Geodesic acoustic mode, 320 Grad–Shafranov equation, 66 Solovev solution, 68 Solovev-Weening solution, 69 Gradient B drift, 14 Greenwald density, 347 Index Group velocity, 230 Guiding center, 12 Gyro-Bohm diffusion, 290 Gyrokinetic particle model, 299, 301 H Hamada coordinates, see natural coordinates Hamiltonian formulation, 25 Harris instability, 223, 427 Hasegawa–Mima equation, 308 Helical coil, 405 Helical symmetry, 403 Heliotron/torsatron, 408 Hermite matrix, 229 Hermite operator, 115 H mode, 359 Hohlraum target, 452 I Ignition condition, Implosion, 439, 445 Inertial confinement, 439 Interchange instability, 104, 109 Internal disruption, 347 INTOR, 380 Ioffe bar, 426 Ion cyclotron range of frequency heating (ICRF), 237 Ion temperature gradient mode, see eta i mode Ion-ion hybrid resonance, 238 Isobar model, 442 ITER, 380 K Kadomtsev’s constraint, 296 Kink instability, 113 Kruskal-Shafranov condition, 114 L Lagrange formulation, 24 Landau amplification, 198 collision integral, 170 damping, 198, 243 Langmuir wave, Larmar radius, 12 Laser plasma, 439 Line of magnetic force, 23 Linearized equation of MHD, 105 493 Liouville’s theorem, 163 Lithium blanket, L mode, 358 Longitudinal adiabatic invariant, 22 Loss cone, 22 instability, 427 Loss-cone instability, 223 region of tandem mirror, 429 Lower hybrid frequency, 184 heating (LHH), 244 resonance, 189 L wave, 180 M Macroscopic instability, 98 Magnetic axis, 28 coordinates, 23, 477 diffusion, 58 fluctuation, 291 flux function, 24 helicity, 390 induction, 19 intensity, 19 moment, 15 Reynolds number, 58, 142 viscosity, 57 well depth, 105 Magnetohydrodynamic(MHD) equation of motion, 55 Major disruption, 348 Major radius, 29 Maxwell equation, 19 Mean free path, 48 Metal wall, 351 Microinstability, 222 Minimum B condition, 101 Minor disruption, see internal disruption Minor radius, 29 Mirror, 21 Mirror instability, 428 Mode conversion, 243 N Natural coordinates, 22, 476 Negative ion source, 365 mass instability, 428 shear, 133, 363 Neoclassical diffusion of stellarator, 411 494 of tokamak, 91 Neoclassical tearing mode, 368 Neutral beam injection (NBI), 45 of negative ion source, 365 Normalized beta, 348 O Ohmic heating, 47 Open end magnetic field, 423 Orbit surface, 40 Ordinary wave, 180 Oscillating field current drive, 401 P Paramagnetism, 73 Particle confinement time of cusp, 425 of mirror, 424 of tandem mirror, 429 Pastukhov’s confinement time, 429 Pellet, 444 Pellet gain, 439, 441 Permiability, 20 Pfirsch-Schluter current, 79 factor, 90 Plasma dispersion function, 212 parameter, Plateau region, 93 Poisson’s equation, 3, 300, 306 Polarization current, 17 Polarization drift, 16, 17 Poloidal beta, 69, 74 Poloidal magnetic field, 29 Ponderomotive force, 18, 452 Poynting equation, 228 Poynting vector, 226 PPCD, see pulsed parallel current drive Precession of helical banana, 410 of ion banana in tokamak, 36 Preheating, 445 Pressure tensor, 3, 457 Pulsed parallel current drive, 399 Q QH mode, 363 Quasi axisymmetric stellarator, 418 helically symmetric stellarator, 418 Index isodynamic stellarator, 419 linear theory of distribution function, 172 omnigenous stellarator, 419 poloidally symmetric stellarator, 419 symmetric stellarator, 417 R Radiation loss, 8, 351 Ray tracing, 231 Rayleigh-Taylar instability, see interchange instability Rayleigh-Taylor instability, 448 Reconnection, 393 Relaxation, 393 Resistive drift wave, 154 instability, 137 wall mode, 159, 368 Resonant magnetic perturbation, 363 Reversed field pinch (RFP), 389 Reversed shear, see negative shear Richtmyer-Meshkov instability, 449 Rosnebluth potential, 171 Rotational transform angle, 29 of stellarator, 407 of tokamak, 350 Runaway electron, 46 Rutherford term, 150 R wave, 180 S Safety factor, 115, 350 Sausage instability, 113 Scalar potential, 19 Scrape-off layer, 352 Self-inductance of the current ring, 75 Separatrix, 353, 406 Shafranov shift, 132 Shear parameter, 124 Sheared flow, 361 Slow wave, 181 Specific resistivity, 47 Specific volume, 104 Spherical tokamak, 382 Sputtering, 351 ST, see spherical tokamak Stability of local mode, 392 Stellarator, 403 Stix coil, 237 Strongly coupled plasma, Superbanana, 412 Superparticle, 304 Index Supershot, 362 Suydam’s criterion, 124 T Tandem mirror, 429 Tearing instability, 142 Test particle, 41 Thermal barrier, 431 conductivity, 86 diffusion coefficient, 87 energy of plasma, 371 Tokamak, 337 Toroidal Alfven eigenmode, 270 coordinates, 70 drift, 28 Transit time damping, 200, 235 Trapped particle instability, 223 Trapped particle, see banana Troyon factor, see normalized beta Twisted coil, 409 U Untrapped particle, see circulating particle Upper hybrid 495 frequency, 184 resonance, 191 V Vector potential, 19 Velocity space distribution function, 163 Virial theorem, 81 Vlasov’s equation, 163 W Wall mirror ratio, 426 Ware’s pinch, 33 Wave heating of electron cyclotron frequency, 247 of ion cyclotron range of frequency (ICRF), 241 of lower hybrid frequency, 243 Weakly coupled plasma, Whistler instability, 428 wave, 191 Z Zonal flow, 316 ... field More information about this series at http://www.springer.com/series/411 Kenro Miyamoto Plasma Physics for Controlled Fusion Second Edition 123 Kenro Miyamoto Tokyo Japan First edition published... centrifugal force is mv⊥ /ρΩ and Lorentz force is qv⊥ B © Springer-Verlag Berlin Heidelberg 2016 K Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic, Optical, and Plasma Physics. .. Therefore the unit of κT is Joule (J) in SI unit In many fields © Springer-Verlag Berlin Heidelberg 2016 K Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic, Optical, and Plasma