Fusion VOLUME 1 Magnetic Confinement PART В EDITED BY EDWARD TELLER Lawrence Livermore National Laboratory University of California Livermore, California 1981 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Toronto Sydney San Francisco COPYRIGHT © 1981, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX Library of Congress Cataloging in Publication Data Main entry under title: Fusion. Includes bibliographies and index. Contents, v. 1. Magnetic confinement (2 v.) 1. Nuclear fusion. I. Teller, Edward, Date. QC791.F87 621.48'A 80-69419 ISBN 0-12-685241-3 (v. 1, pt. B) AACR2 PRINTED IN THE UNITED STATES OF AMERICA 81 82 83 84 9 8 7 6 5 4 3 2 1 List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin. ROBERT W. CONN (193), Center for Plasma Physics and Fusion Engineering, School of Engineering and Applied Science, University of California, Los Angeles, California 90024 R. A. DANDL (79), Applied Microwave Plasma Concepts, Inc., Encinitas, California 92024 JOHN M. DAWSON (453), Department of Physics, University of California, Los Angeles, California 90024 G. E. GUEST (79), Applied Microwave Plasma Concepts, Inc., Encinitas, California 92024 W. B. KUNKEL (103), Department of Physics, Lawrence Berkeley Labora- tory, University of California, Berkeley, California 94720 R. W. MOIR (411), Lawrence Livermore National Laboratory, University of California, Livermore, California 94550 MIKLOS PORKOLAB (151), Department of Physics and Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 W. E. QUINN (1), Los Alamos National Laboratory, Los Alamos, New Mexico 87545 F. L. RIBE (39, 59), Department of Nuclear Engineering, University of Wash- ington, Seattle, Washington 98195 A. R. SHERWOOD (59), Los Alamos National Laboratory, Los Alamos, New Mexico 87545 R. E. SIEMON (1), Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ix Contents of Part A Introduction Edward Teller Tokamak Plasma Stability M. N. Rosenbluth and P. H. Rutherford The Tokamak H. P. Furth Stellarators J. L. Shohet Mirror Theory T. Ê. Fowler Experimental Base of Mirror-Confinement Physics R. F. Post The Reversed-Field Pinch D. A. Baker and W. E. Quinn INDEX xi FUSION, VOLUME 1, PART Β 8 Linear Magnetic Fusion Systems W. E. QUINN AND R. E. SIEMON Los Alamos National Laboratory Los Alamos, New Mexico I. Introduction 2 II. Heating Methods 5 A. Implosion Heating 5 Β. Laser Heating 7 C. Relativistic Electron-Beam (REB) Heating 9 D. Auxiliary Heating 11 III. Equilibrium and Stability 12 A. Rotational Instabilities 12 B. Curvature-Driven Instabilities 14 C. Tearing Modes 14 IV. Transport 15 A. Axial Particle Loss 16 B. Axial Heat Loss 16 C. Radial Particle Diffusion 17 D. Radial Heat Diffusion 18 V. Impurities 18 VI. Scaling Laws for LMF Devices 19 A. Particle End Loss 19 B. Electron Thermal Conduction 22 C. Radial Transport 24 D. Total Plasma Energy 25 VII. End-Stoppering Methods 26 A. Material End Plugs 26 B. Magnetic Field End-Stoppering Techniques 28 C. Reentrant End Plugs 29 VIII. Reactor Considerations 30 A. Linear Theta-Pinch Reactor (LTPR) 31 B. Laser-Heated Solenoid Reactor (LHSR) 31 C. Electron-Beam-Heated Solenoid Reactor (EBHSR) 32 D. Steady-State Solenoidal Fusion (SSF) Systems 32 References 33 / COPYRIGHT C 1981 BY ACADEMIC PRESS, INC. ALL RIGHTS OF REPRODUCTION IN ANY FORM RESERVED ISBN 0-12-685241-3 2 W. E. QUINN AND R. E. SIEMON I. Introduction Linear magnetic fusion (LMF) is a classification for systems having in common a linear, cylindrical geometry of plasma, blanket, and confinement system with transverse confinement primarily by a quasistatic magnetic field and open-ended field lines. Because of their relative simplicity, LMF system., provide an attractive approach to the development of fusion reactor systems. The basic advantages of LMF systems include MHD stability, high plasma beta, modest impurity problem, and simple fueling. In the case of the theta pinch, effective heating to fusion temperatures has been demonstrated. In addition, there are several heating methods available, and the reactor would have the potential of simple engineering and main- tenance for a modular system composed of identical, straight, cylindrical elements of modest size. The most serious problem of all LMF systems is the open field line geometry and the consequent length required to sustain the plasma for reactor burn times. Particle loss through open ends and classical heat loss by electron thermal conduction along the field present fundamental limitations on the particle and energy confinement times, implying reactor systems with lengths of tens of kilometers. 1 Such systems are found to produce an unacceptably large power and are unattractive unless their length is reduced by at least one order of magnitude. The length requirement is the single major disadvantage of LMF reactor systems and provides a strong impetus to find an appropriate end-stoppering arrangement. Several methods have been proposed, and in some cases demonstrated, for reducing system length by means of end stoppering. These include material end plugs, multiple mirrors, cusps, plasma injection, rf stoppering, electrostatic trapping, field reversal, and "reentrant" end plugs (elongated racetrack). (The "reentrant" end-plugged system consists of two long linear systems connected at the ends by semicircular sections to form a very elongated racetrack.) Both the field-reversal and reentrant systems have closed magnetic field lines to prevent both particle losses and heat loss by electron thermal conduction. The reactor embodiment for a system with closed field lines is hoped to be superior, but the physics of the more complicated configurations of magnetic field are not well enough understood to allow an evaluation at the pre- sent time. In LMF systems, the plasma is confined radially by an axial magnetic field. It is well established that plasma equilibrium exists in the straight field line configuration. Theory predicts the LMF plasma to have neutral stability in the absence of field curvature and plasma rotation. Finite Larmor radius (FLR) effects provide stabilization of m > 2 perturbations2-5 in the presence 8. LINEAR MAGNETIC FUSION SYSTEMS 3 of curvature or rotation that may result from field errors, boundary con- ditions, or from end-stoppering techniques. Unstable m = 1 motions are expected if field lines have unfavorable curvature or if the whole plasma column achieves a rotation which exceeds certain bounds. Experiments have demonstrated stable plasma confinement prior to propagation of end effects to the central plasma region. An m = 1 wobble instability, which saturates at low amplitude, is observed after an Alfvén transit time from the ends. The rotation which drives the instability is most likely associated with shorting of the radial electric fields and the boundary conditions at the ends. All LMF experimental results to date appear to be consistent with classical transport of particles and energy, in both the radial and axial directions, although the time scale imposed by axial losses limits the accuracy of radial loss measurements. Classical radial losses provide very favorable scaling to reactor parameters. When heating is performed in a theta pinch, i.e., by rapidly strengthening the magnetic field lines and thereby imploding the plasma which they carry, electron energy loss appears almost immediately as a result of thermal conduction parallel to the field lines. Subsequently the major energy loss becomes convective as a result of the end loss of particles. The theta pinch is an electrodeless discharge created by shock-implosion heating followed by adiabatic compression in an axial magnetic field generated by an azimuthal or theta current flowing in a solenoidal coil as illustrated in Fig. 1. It is to be distinguished from the Ζ pinch in which the magnetic field is perpendicular to that of a theta pinch. Results6-8 with material end plugs indicate that the free-streaming particle loss can be reduced significantly, leaving only the problem of energy loss by electron thermal conduction. Plasma confinement has also been demon- strated in linear systems at low beta using multiple magnetic mirrors. 9 The FIG. 1. Simplified schematic view of the theta pinch: (a) Implosion heating dynamic phase characterized by irreversible shock heating, (b) Adiabatic compression of the shock-heated plasma followed by quiescent phase and plasma confinement. 4 W. E. QUINN AND R. E. SIEMON study of end-stoppering techniques at high density (~ 10 16 cm"3) and high beta (~0.5 to 0.9) is in its infancy: no end-stoppering techniques have been tested at conditions at which they are expected to be the most effective; some have not been tested at all; and no experiment has used a combination of techniques. Somewhat related concepts are being worked on in the magnetic mirror program (cf. Chap. 6), but at lower plasma densities where ambipolar sheaths are effective in reducing the effects of electron thermal conduction. Plasma heating in LMF consists of a staged process beginning with rapid heating by a primary heater, which is followed by adiabatic compression. Three primary heaters have been studied: implosion heating by rapidly increasing the axial magnetic field, laser heating, and, to a lesser extent, relativistic electron-beam heating. Shock-implosion heating is a proven technique for achieving kilovolt ion temperatures and has been applied in many theta-pinch experiments. This technique heats the ions directly. Ex- periments have demonstrated effective coupling of laser and electron beams to plasma energy in plasma columns of mederate temperature (~ 100 eV). Laser heating occurs by inverse bremsstrahlung absorption and works best for higher-density plasmas (~10 17 cm"3). The energy-coupling process of e beams is anomalously high and appears to be effective for plasma densities in the 1016-cm~3 range. On the basis of classical collision processes, rela- tivistic electron beams (REB) should pass through a plasma without losing much energy, but anomalous effects arise through the coupling of the two- stream instability between the REB and the electrons in the plasma, resulting in a collective deposition of energy in the plasma. Both lasers and e beams heat the plasma electrons rather than the ions. Auxiliary heating processes may also be used to augment the primary heater and reduce its engineering requirements. The excessive length required for an LMF reactor to sustain the plasma density and energy against end loss for times required to achieve net energy breakeven presents a singular disadvantage that has limited LMF systems as a serious contender for a fusion power reactor. A solution to the axial confinement problem is crucial for the development of a viable reactor concept. Reactor technology requirements for LMF systems are likely to include high-field magnets, efficient energy transfer, and storage systems, and a large first-wall loading.10 The linear geometry is simple, and the magnets can be constructed in modular elements. Refueling can readily be performed during the interval between pulses, in the absence of a plasma. The linear theta-pinch device, which has been extensively studied, is typical of LMF systems. A review of the linear theta-pinch concept has been given by Freidberg et al.11 A summary of LMF topics discussed at an LMF 8. LINEAR MAGNETIC FUSION SYSTEMS 5 Workshop in 1977 gives a review of LMF physics and technology. 12 An overview of end-stoppering techniques for open magnetic containment systems has been given by Hinrichs. 13 II. Heating Methods LMF systems employ at least two plasma heating techniques, a primary heater which produces a plasma of 1-2 keV, followed by adiabatic com- pression and possibly an auxiliary heater which increases the temperature to ignition. The primary heater usually delivers its energy on a microsecond time scale and requires expensive, high-quality energy storage, efficient transfer, and rapid conversion of electrical energy to plasma energy. The adiabatic compression or secondary heating usually occurs on a much slower time scale, typically in the millisecond range. The secondary heater generally requires considerably more energy than the primary heating, but the slower pulse time allows cheaper energy storage, such as fast-discharging homopolar generators. Additional auxiliary heating can be used to relax the engineering requirements of the primary heater. The primary plasma heating can be achieved by means of implosion heating in a theta pinch, or by absorption of energy from an axially directed laser or electron beam. Secondary heating can be achieved through adiabatic compression by a slowly rising magnetic field. Auxiliary heating could use magnetoacoustic heating, ICRH, Alfvén-wave heating, magnetic pumping, or neutral-beam injection. A. Implosion Heating Implosion or shock heating is a proven method for heating plasmas to thermonuclear conditions. It has the advantage of directly heating the ions, and multikilovolt ion temperatures have been achieved with plasma densities in the 10 14-1015 cm-3 range. Implosion heating requires a rapidly rising magnetic field (τ/4 ~ 0.5/isec,i? ~ lOkG), and therefore high-quality energy storage. In the theta pinch, the implosion phase is followed by slow magnetic compression. The implosion circuit is applied to a preionized plasma. During the initial phase of the implosion, some magnetic field becomes imbedded in the plasma. As the plasma current sheath becomes a few ion Larmor radii in thickness, the field diffusion slows. For low filling pressures (~ 3-10 mTorr), the density profile is approximately Gaussian and the peak value of beta is high but less than unity. Higher filling pressures 6 W. E. QUINN AND R. E. SIEMON implode to flat-topped density profiles with beta (ratio of plasma pressure to external magnetic field pressure) values approaching unity. The implosion heating phase of the theta pinch has been studied over a wide range of parameters and is understood both experimentally and theoretically. An international effort to understand collisionless shocks in the implosion heating process concentrated on low-density implosion phenomena.14-2 2 Experiments at higher densities with initial fill pressures of 5-10 mTorr, characteristic of reactor conditions, produced plasmas with ion energies of 1.6 keV.23 This temperature from implosion heating, when combined with a compression phase, represents a reasonable approximation to the heating requirements in a theta-pinch reactor. The implosion heating phase of the theta pinch is produced by a rapidly rising axial magnetic field which induces an azimuthal electric field in the outer plasma region. The electric field drives an azimuthal current in the cylindrical plasma sheath, which implodes toward the axis because of the increasing pressure of the axial magnetic field. Depending on the parameters of the experiment either a "bounce"24'25 or a "snowplow"26 model is used to describe the non- adiabatic heating process. The bounce model has been used to describe the implosion process in collisionless plasmas. In this model, the collisionless plasma ions are reflected from the rapidly inward moving plasma sheath at twice the imploding sheath velocity. The fast, radially imploding ions thermalize through nonadiabatic turbulent processes. The cold electrons are then heated by collisional equilibration with the ions. For higher-density, collisional plasmas the "snow plow" model describes the implosion process. As the plasma sheath is driven inward by the "magnetic piston," it sweeps up all the charged particles it encounters. The plasma ions are pushed ahead of the imploding sheath rather than being reflected since turn around is im- peded by collisions with other particles and by turbulence. The rate of momentum change of the plasma, balanced against the external magnetic pressure, then gives the inward velocity of the sheath as a function of time. In order to describe the broad sheaths, diffuse profiles, and magnetic field diffusion that are observed in the implosion heating phase, it has been necessary to invoke nonclassical effects, in particular, anomalous resistivity. Currently, numerical methods provide the best means of understanding the implosion heating phase. The hybrid code of Sgro and Nielson 27 uses kinetic ions and massless, fluid electrons; Ohm's law is supplemented with suitable anomalous resistivity coefficients. Simulations of experiments have demonstrated that this code, and others like it, 2 8 -0 3can accurately reproduce gross experimental features such as magnetic field, density and temperature profiles as well as more subtle effects such as neutron emissions, Faraday rotation, and plasma rotation. The staged theta-pinch experiment has successfully demonstrated the