Jjokamaks A. E. COSTLEY B. LLOYD D. F. START THIRD EDITION JOHN WESSON with contributions from D. J. CAMPBELL J. W. CONNOR R. D. GILL J. HUGILL N. LASHMORE-DAVIES G. M. McCRACKEN H. R. WILSON and R. J. HASTIE A. HERRMANN G. F. MATTHEWS J. J. O’ROURKE B. J. D. TUBBING D. J. WARD CLARENDON PRESS - OXFORD 2004 UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP i i iversity of Oxford. Oxford University Press is a department of the University 9 / 1 furthers the University’s objective of excellence in tesearch, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dares Salaam Delhi Hong Kong Istanbul ; Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi SioPaulo Shanghai Taipe: Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © John Wesson, 1987, 1997, 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First edition 1987 Second edition 1997 Third edition 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford Universiry Press, oras expressly permitted by law, or under terms agreed with the appropriate prograph Tights ization. Enquiries concerning reproducti outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on ‘any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 8509227 10987654321 Typeset by Newgen Lmaging'S: i mn ystems (P) Ltd., Chennai, Indi Printed in Great Britain a on acid-free paper by The Bath Press, Avon Karachi Preface When I worked on toroidal devices in the early days of fusion research the plasma temperatures achieved were around 10eV and the confinement times were perhaps 100 microseconds. In the next thirty years there was steady progress and at the publication of the first edition of this book in 1987 the temperatures in large tokamaks were several keV and a confinement time of one second had been reached. By then the tokamak had become the predominant device in the attempt to achieve a useful power source from thermonuclear fusion. The accompanying increase in research activity and general interest in tokamaks led to the need for an introductory account of the subject and it was the aim of the first edition to provide such an introduction. In the subsequent decade up to publication of the second edition the subject was trans- formed again, There were now areas where the experimental behaviour could be understood in terms of accepted theory, which was encouraging. There had also been substantial research on large tokamaks leading to the long awaited achievement of significant amounts of fusion power, Inevitably this brought us face to face with the problems involved in design- ing and building a tokamak reactor. The aim of the second edition was to describe these advances, and it is perhaps a measure of the developments in the subject that the second edition was twice the size of the first. When the time came for a reprint the opportunity was taken of bringing the book up to date in this third edition. In the intervening period the emphasis has been on preparing the ground for an experimental reactor but there have also been significant advances in our understanding of the plasma behaviour, for example, the wider experience of internal transport barriers, the appreciation of the role of tearing modes driven by neoclassical effects, and insights from turbulence simulations. Despite the increasing complexity of the subject it is hoped that the book will still prove useful to those entering the subject, to specialists within tokamak research who wish to acquire knowledge of other areas in the subject, and to those outside tokamak research who would like to learn something of the principal concepts, methods, and problems involved. A further aim is to provide a handbook of equations, formulas, and data which the research worker frequently needs. T regard it as an honour to have worked with the distinguished physicists who are my co-authors. Their spirit of cooperation has made the endeavour a pleasure. Tam grateful to my wife Olive for her support during the time-consuming preparation of the manuscript. I would like to thank Caro! Simmons, Birgitta Croysdale, and Ingrid Farrelly who typed the eartier editions and Lynda Lee who has been unfailingly helpful in the preparation of this edition. 1 must further thank Stuart Morris who produced most of the figures and Chad Heys who helped with the many new figures required for the present edition. 1 am also grateful to Graham O’Connor for his careful reading of the text and for the resulting corrections. Finally, I would like to dedicate this book to my friends and colleagues in the world- wide community of fusion physicists. They have set a splendid example of international collaboration for others to follow. England JOHN WESSON July 2003 Authorship L.A. WESSON J.A. WESSON 2.11 RJ. Hastie JA. WESSON 3.14 DE Start and B. Lloyd 3.W. CONNOR 4.1-4.5 ].A. Wesson 4.8 J.A. Wesson 4.11 B.D. Tubbing 4.13 L.A. Wesson 4.23-4.25 J.A, Wesson J.A. WESSON (5.1!-5.5) C.N. LASHMORE-DAVIES (5.6-5.10) LA. WESSON J.A. WESSON 14 H.R. WILSON G.M. McCRACKEN R.D. GILL 10.2 D.J. Ward 10.3 LI. O’Rourke 10.4 and 10.5 A.E, Costley 10:9 G.F. Matthews J.A. WESSON AND J, HUGILL D.J. CAMPBELL 12.6 A, Herrmann L.A. WESSON 13.5 D.J, Ward 1A. WESSON 14.13 J.W. Conner Acknowledgements The authors acknowledge the help of many colleagues and in particular as follows: Tokamak reactor—Roger Hancox and Terry Martin. Potato orbits—Bill Core and Per Helander. Current drive—Martin Cox and Martin O’Brien. Transport barriers—Barry Alper. Confinement—Ted Stringer and Geoff Cordey. Neutral beam heating—Andrew Bickley, Ron Hemsworth, Peter Massmann, and Emie Thompson. RF heating—Lars-Goran Eriksson, Jean Jacquinot, and Franz Sdldner. Neoclassical tearing modes—Richard Buttery and Tim Hender. TAE modes—Sergei Sharapov. Plasma surface interactions—Rainer Behrisch, Richard Pitts, and Peter Stangeby. Diagnostics—Wolfgang Engelhardt, Ian Hutchinson, and George Magyar. Tokamak experiments—Kar! Heinz Finken, Martin Greenwald, Otto Gruber, Jan Hutchinson, Louis Laurent, Niek Lopes ‘Cardozo, Kent McCormick, William Morris, Jerome Pamela, Chris Schiiller, Paul Smeulders, Alan Sykes, Paul Thomas, Pritz Wagner, Henri Weisen, Gerd Wolf and Hartinut Zohm. ITER—George Vayakis Contents Units and symbols 1 Mt 12 13 14 15 16 L7 18 19 1.10 2 22 2.2 2.3 24 2.5 2.6 27 2.8 29 2.10 211 2.12 Fusion Fusion and tokamaks Fusion reactions Thermonuclear fusion Power balance Ignition Tokamaks Tokamak reactor Fuel resources Tokamak economics Tokamak research Plasma physics Tokamak plasma Debye shielding Plasma frequency Larmor orbits Particle motion along B Particle drifts Adiabatic invariants Collisions Kinetic equations Fokker—Planck equation Gyro-averaged kinetic equations Fokker-Planck equation for a plasma Fokker-Planck coefficients for Maxwellian distributions Relaxation processes xiii 33 34 35 38 40 42 49 51 35 57 38 60 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 312 3.13 3.14 4l 42 43 Collision times Resistivity Runaway electrons Electromagnetism Fluid equations Magnetohydrodynamics Physics of plasma fluid Plasma diamagnetism Braginskii equations Plasma waves Landau damping Equilibrium Tokamak equilibrium Flux functions Grad—Shafranov equation Safety factor, q Beta Large aspect-ratio Shafranov shift Vacuum magnetic field Electric fields Particle orbits Particle trapping ‘Trapped particle orbits Plasma rotation Current drive Confinement Tokamak confinement Resistive plasma diffusion Diffusion in a cylinder 69 10 72 15 77 719 81 84 88 94 105 149 150 152 153 44 45 46 47 4.8 49 4.10 4.11 412 4,13 414 4.15 4,16 4.17 418 4.19 4.20 4.21 4.22 4.23 4.24 4.25 5.1 5.2 3.3 5.4 5.5 3.6 3.7 5.8 5.9 Contents Ptirsch-Schiiiter current Ptirsch-Schliiter diffusion Banana regime transport Plateau transport Ware pinch effect Bootstrap current Neoclassical resistivity Ripple transport Confinement modes and scaling expressions H-modes Intemal transport barriers Scaling laws Transport coefficients Fluctuations Turbulence-induced transport Radial electric field shear and transport Candidate modes Turbulence simulations, critical gradients, and temperature pedestals Impurity transport Experimental discoveries Radiation losses Impurity radiation Heating Heating Ohmic heating Neutral beam injection Neutral beam heating Neutral beam production Radio frequency heating Physics of radio frequency heating Ion cyclotron resonance heating Lower hybrid resonance heatin g 155 158 159 167 169 172 174 175 180 185 189 191 195 198 202 210 213 217 219 223 227 229 237 238 240 243 246 253 258 261 270 286 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 71 12 73 14 1S 76 17 18 79 7.10 WY 712 743 TA4 TAS Electron cyclotron resonance heating Mhd stability Mhd stability Stability theory Growth rates Energy principle Tokamak instabilities Large aspect-ratio tokamak Kink instability Tearing modes Tearing stability Internal kink Resistive m = 1 modes Localized modes Ballooning modes Ballooning stability Axisymmetric modes B limit Instabilities Instabilities Magnetic islands Tearing modes Mirnov instabilities Current penetration Sawtooth oscillations Dismptions Causes of disruptions Physics of disruptions Mode locking Error field instability Vertical instability Ergodicity Fishbone instability Toroidal Alfvén eigenmodes 290 303 304 306 307 309 311 312 313 318 324 329 332 336 337 340 342 343 351 352 354 356 362 364 365 374 376 382 390 394 396 397 399 402 7.16 VAT 78 8.t 8.2 8.3 8.4 8.5 91 9.2 93 9.4 95 9.6 97 9.8 9.9 9.10 OAL 10 10.1 MARFEs ELMs Operationat overview Microinstabilities Microinstabilities Electron drift wave Passing particle instabilities Trapped particle instabilities Micro-tearing modes Plasma-surface interactions Plasma—surface interactions The plasma sheath The scrape-off layer Recycling Atomic and molecular processes Wall conditioning Sputtering Arcing Limiters Divertors Heat flux, evaporation, and heat transfer The behaviour of tritium Diagnostics Tokamak diagnostics 10.2 Magnetic measurements 10.3. Interferometry 10.4 Reflectometry 10.5 Measurement of electron temperature 406 409 4 417 418 420 422 430 435 443 446 449 453 457 462 466 473 415 AT] 489 492 497 498 500 507 Su 514 10.6 10,7 10.8 10.9 10.10 10.11 11 Wl 112 143 14 115 11.6 17 11.8 1L9 11.10 tay 11.12 11.13 1144 1145 11.16 thal? 1L18 11.19 11.20 11.21 Contents lon temperature and the ion distribution function Radiation from plasmas Total radiation Measurements Langmuir probes Measurements of fluctuations Determination of the q-profile Tokamak axperiments Tokamak experiments T-3 ST JFT-2 Alcator A, Alcator C, and Alcator C-Mod TFR DITE PLT T-10 ISX FT and FT Upgrade Doublet-II ASDEX TEXT TEXTOR Tore Supra COMPASS RTP START, MAST, and NSTX TCV Tokamak parameters 12 Large Tokamaks 121 12.2 12.3 Large Tokamaks TFTR JET xi 522 332 34t 551 556 561 562 563 564 566 567 569 570 572 572 573 574 576 576 379 580 583 584 586 589 590 593 594 597 617 xii Contents 12.4 JT-60/T-60U 12.5 DU-D (2.6 ASDEX Upgrade 13 The future 13.1 Status 13.2 Strategy 13.3 Reactor requirements 134 ITER 13.5 Prospects 14 Appendix 14.1 Vector relations 14.2 Differential operators 645 665 687 705 106 707 708 mM N18 721 722 722 14.3 14.4 14.5 14.6 147 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 14.16 Index Units—conversions Physical constants Coulomb logarithm Collision times Lengths Frequencies Velocities Resistivity Chang-Hinton formula for x; Bootstrap current Confinement scaling relations Plasma shape Formulae Symbols 14 725 725 729 731 733 734 735 737 738 740 741 142 743 745 Units and symbols The system of units used is m.k.s. Following the convention generally accepted in the subject, temperatures are written either in joules or in electron-volts (or keV), Thus in place of conventional kT° (where k is Boltzmann's constant and T° is in degrees Kelvin) we write T (joules), so that T° = 7 (joules)/1.381 x 10-23, The temperature in electron- volts is defined by the potential difference in volts through which an electron must fall to acquire an energy 7, that is T(eV) = T(joules)/e where e is the electronic charge. Thus T(eV) = T(joules)/1.602 x 10-8, Whenever the temperature given is in electron-volts this is explicitly stated. To avoid continual redefinition of frequently used symbols a list of such symbols is given in Section 14,16. Fusion vd Fusion and tokamaks 1.1 Fusion and tokamaks If a nucleus of deuterium fuses with a nucleus of tritium, an o-particle is produced and a neutron released. The nuclear rearrangement results in a reduction in total] mass and a consequent release of energy in the form of the kinetic energy of the reaction products. The energy released is 17.6MeV per reaction. In macroscopic terms, just ! kg of this fuel would release 108 kWh of energy and would provide the requirements of a 1 GW (electrical) power station for a day. Deuterium is a plentiful resource but tritium does not occur naturally. It should, however, be possible to use the neutrons released in the fusion reaction to breed tritium from lithium, of which there are large reserves. In order to induce the fusion of nuclei of deuterium and tritium it is necessary to overcome the mutual repulsion due to their positive charges, and as a result the cross-section for fusion is small at low energies. How- ever, the cross-section increases with energy, reaching a maximum at 100 keV, and a positive energy balance is possible if the fuel particles can be made to react before they lose their energy. To achieve this the particles must retain their energy and remain in the reacting region fora sufficient time. More precisely the product of this time and the density of Teacting particles must be sufficiently large. The simple schemes of firing a beam of particles into a solid target or through another beam fail to satisfy this criterion, In the first case the Particles lose their energy too rapidly and in the second the density is too low, The most promising method of supplying the energy is to heat the deuterium-tritium fuel to a sufficiently high temperature that the thermal velocities of the nuclei are high enough to produce the required reac- tions. Fusion brought about in this way is called thermonuclear fusion. The optimum temperature is not as high as that corresponding to the energy of maximum cross-section because the Fequired reactions occur in the high energy tail of the Maxwellian distribution of heated particles, The necessary temperature is around 10 keV, that is about 100 million Although the required temperature, densi all been obtained in tokamaks, they have not been achieved in the same Fig. 1.1.1 Ina reactor the product nz of ion density and energy confinement time, and the temperature, 7, must hoth be in the right range. Taking peak values, the required fitg is 2-5 x 10°? m—?s and the temperature range is around 10-20keV. The required value of the product ize? is approximately 5 x 107! m—? skev (Section 5.1). The figure shows the progress in improving this product, leading to the verge of reactor conditions. 1.2 Fusion reactions 3 10226 SS. [REACTOR 102! Ol 10? O° ATET 19!9b ° -3 ry) 1018 1017 1016 1015+ ° 1014/0 ! i L | ! L ! i 1955 1960 1965 1970 1975 1980 1985 1990 1995 T ° ° T ° plasma. However, the progress toward this goal has been remarkable, and a thermonuclear power of more than sixty percent of the input power has been produced. A further step is to achieve ignition where, as with fossil fuels, the burning process becomes self-sustaining without further applied heating. The progress toward ignition can be measured by means of a single parameter. The form of the dependence of the fusion cross- section on energy fortuitously allows the requirement for ignition to be expressed approximately by AtaT > 5x 107! m-35KeV where fi and T are the peak ion density and temperature in the plasma and tg is the energy confinement time. The improvement in the achieved value of this parameter over the years is shown in Fig. 1.1.1. It is now believed that a tokamak can be built which would produce ignition. However, the design of such a reactor raises a wide range of questions. A commercial reactor even more so. Present research is aimed at answering these questions and this book gives an introductory account of our knowledge of the underlying physics. 1.2 Fusion reactions By far the most promising fusion reaction is that in which the nuclei of deuterium and tritium fuse to produce an alpha particle with the release 4 1.2 Fusion reactions of a neutron, that is :D? + it? > He* + gn! I | 3.5 MeV + 14.1 MeV = 17.6MeV where the energies given are the kinetic energies of the reaction products. energy balance follows from the overall mass deficit 5m in the The mass- reaction D + T (2 — 0.000 994)rp (3 — 0,006 284)mp _ a + n (4 — 0.027 404)mp (1 + 0.001 378), where my is the mass of the proton ( 1.6726 x 10-27 kg). The mass deficit is 0.018 75mp, and so the energy released is & = 5m .c? = 0.018 75m,c? = 2.818 x 107" joules = 17.59 Mev. The reaction is induced in collisions between the particles, and the cross- section for the reaction is therefore of fundamental importance. The cross-section at low impact energies is small because of the Coulomb barrier which prevents the nuclei from approaching to. within nuclear dimensions as is required for fusion to take place. The potential is illustrated in Fig, 1.2.1. Because of quantum mechanical tunneling, D-T fusion occurs at ener- gies somewhat less than that required to overcome the Coulomb barrier. The cross-section for the reaction-is given in Fig. 1.2.2 and it is seen that the maximum cross-section occurs at just over 100 keV. e ! An bm Potential energy Tm Nuclear separation Fig. 1.2.1 Potential 6 ener, ‘i of nuclear separation, gy as a function Fig. 1.2.2 Cross-sections for the reactions DT, D~D and D-He?. The two D_D reactions have similar cross-sections, the graph gives their sum. 1.3 Thermonuciear fusion 5 TTT Tey TTT rn 1077 10° pol ad 10” a (m’) 10° 10" TOT Ty yt 4 rol 4 ul 1 vil 10°? pe dl 10 100 1000 Deuteron energy (keV) -_ The reason the D-T reaction is preferred to other reactions is clear from Fig. 1.2.2 where the cross-sections for D? + D? > He? +n! + 3.27MeV Dp? +p? > T 4 H! +4.03 MeV D? + He? > He* + H! + 18.3MeV are also shown. It is seen that these cross-sections are considerably less than that for D-T except at impractically high energies. 1.3 Thermonucleer fusion Calculation of the reaction rate in a hot D-T plasma requires an integration over the distribution functions of both species. The rate of reaction per unit volume between particles of one species with a velocity v, and particles of the other species with velocity v2 is a(v')v! fi(v1) fo(v2) where v' = 0) — 02 and f; and f2 are the distribution functions. \ 3 Thermonucles’ fusion if the distributions are Maxwellian, ane mj \ ogy ME, soop=ni (see) OP OT the total reaction rate per unit volume aa [foo noone Pv, dv2 may be written 2 (mya)? _ my timo (v Lm =m) R= nine Ty exp oF + mam 2 x o(v'yu' exp (-5) Bu &BV p. bemg the reduced mass. The integral over Vis (2 F/(m + m))?/? so that 3/2 (2 R= arnyna( ==) | ov’? exp (=) dv’. 1.3.1 The cross-sections measured in laboratory experiments are usually given Gn terms of the energy of the bombarding particle, say type |. at is 72 s= amv nie so that eqn 1.3.1 may be more conveniently written ‘gy li? 3/2 1 R= () mn (ey! ( we o(e)e exp ( -—— 2 nr =) me (e) Pp miT de. 13 tthe cress section o(e) for D-T reactions given in Section 1.2 is sub- stated into the integra of eqn 1,3.1, the reaction rate R = nam (ov) is i ov) is given in Fig, 1.3.1. For a given i i menmum rate is achieved for na = m ra given ion density Ne t temperatures of interest the n| , i ¢ 1 uclear reactions come predominant! from the tail of the distribution. This is illustrated in Fig. P32 where the Fig. 1.3.1 (ov) for D-T reactions as a function of plasma temperature. Fig. 1.3.2 Graph of the imegrand of eqn 1.3.2 and of its two factors o(€) and € eXp(—ye/myT) against the normalized energy €/T for a D-T plasma at T= 10kev. 1.3. Thermonucleer fusion 7 107! 10” (ov) (m's") 107 10% T(keV) _ bE cera) oe exp(- # £ ~p integrand of eqn 1.3.2 is plotted against ¢/T together with the two factors o(e) and € exp(—ye/mgT) for a D-T plasma ata temperature of 10 keV. Experiments are more usually carried out using deuterium rather than a deuterium-tritium mixture. A graph of (ov) for deuterium is given in Fig. 1.3.3 together with that for D-He’. In the temperature range 5-20 keV the ratio of (av) for D-T to that of deuterium is around 80. 1.4 Power balance Fig, 1.3.3 (ov) for D-D (total) and D-He’ reactions as a function of plasma temperature. The values are much smaller than those for D-T which are included for comparison. (ov) (m?s') TikeV) | 1.4 Power belance Thermonnclear power The thermonuclear power per unit volume in a D-T plasma is Pty = Nan (ov)&, 1.4.1 where ng and are the deuterium and tritium densities, (o v) is the rate given in Fig. 1.3.1 and 6 is the energy released per reaction. The total ion density is n=natnm, so eqn 1.4.1 can be written Pt, = naln — na)(ov)€. 1.4 Power belence 9 For a given n this power is maximized by ng = in that is equal deuterium and tritium densities. For this optimum mixture the thermonuclear power density is Pte = 4x7 (ov) 6. 142 Energy loss In a tokamak there is a continuous loss of energy from the plasma which has to be replenished by plasma heating. The average energy of plasma particles at a temperature T is 37, comprised of 47 per degree of free- dom. Since there is an equal number of electrons and ions, the plasma energy per unit volume is 3n7 The total energy in the plasma is therefore W= | 3nT d°x =3nTV, 1.43 where the bar represents the average value, and V is the plasma volume. The rate of energy loss, Py,, is characterized by an energy confinement time defined by the relation R= We 1.44 TE In present tokamaks the thermonuclear power is usually small and in steady state the energy loss is balanced by externally supplied heating. Thus if the power supplied is Py, Py = Pt 1.4.5 and eqns 1.4.4 and 1.4.5 give aul -¥. This expression provides a means of determining tz from experimentally known quantities. a-particle heating The thermonuclear power given by eqn 1.4.2 consists of two parts. Four fifths of the reaction energy is earried by the neutrons and the remainder, &y, is carried by the e-particles. The neutrons leave the plasma with- out interaction but the a-particles, being charged, are confined by the magnetic field. The o:-particles then transfer their 3.5 MeV energy to the 1.5 Ignition plasma through collisions. ‘Thus the w-particle heating per unit volume is 14.6 Pa = tna v) 8a and the total o-particle heating is Py = | Po ax = gn? (ov) GeV. 147 Power balance In the overall power balance the power loss is balanced by the externally supplied power plus the w-particle power. That is Put Po = P, and using eqns 1.4.3, 1.4.4, and 1.4.7 this balance is given by —— 3nT Put 4n?(ov)€aV = —-V. 1.4.8 The implications of this equation are described in the next section. 1.5 Ignition Ignition condition As a D-T plasma is heated to thermonuclear conditions the a-particle heating provides an increasing fraction of the total heating. When adequate confinement conditions are provided, a point is reached where the plasma temperature can be maintained against the energy losses solely by -particle heating. The applied heating can then be removed and'the plasma temperature is sustained by internal heating. By analogy with the burning of fossil fuels this event is called ignition. The power balance is described by eqn 1.4.8 and, taking constant density and temperature for simplicity, this can be written 3nT = 12 Py = Ce —jn (on6.) Vv. 1.5.1 Equation 1.5.1 provides the condition for ignition, the requirement for the plasma burn to be self-sustaining being (ou) Bs 1.5.2 The right-hand side of inequality 1.5.2 is a function of temperature only and a graph of the temperature dependence of the required value of nt¢