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    Plasma Confinement
    Plasma Confinement
    Plasma Confinement
    Ebook859 pages9 hours

    Plasma Confinement

    By R. D. Hazeltine and J. D. Meiss

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    Detailed and authoritative, this volume examines the essential physics underlying international research in magnetic confinement fusion. It offers readable, thorough accounts of the fundamental concepts behind methods of confining plasma at or near thermonuclear conditions.
    Starting with a review of fundamentals (including considerations of tensor calculus, Lagrangian and Hamiltonian mechanics, Maxwell-Lorentz equations, and charged particle motion), the text surveys four decades of controlled fusion research, with a focus on more recent concepts that have extended the understanding of confinement. Subjects include confined plasma equilibrium, kinetic description of a magnetized plasma, coulomb collisions, fluid description of magnetized plasma, stability of confined plasmas, collisional transport, and nonlinear processes.
    Designed for a one- or two-semester graduate-level course in plasma physics, it also represents a valuable reference for professional physicists in controlled fusion and related disciplines.
    LanguageEnglish
    PublisherDover Publications
    Release dateFeb 20, 2013
    ISBN9780486151038
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      Plasma Confinement - R. D. Hazeltine

      DOVER BOOKS ON PHYSICS

      METHODS OF QUANTUM FIELD THEORY IN STATISTICAL PHYSICS, A.A. Abrikosov et al. (63228-8)

      ELECTRODYNAMICS AND CLASSICAL THEORY OF FIELDS AND PARTICLES, A.O. Barut. (64038-8)

      DYNAMIC LIGHT SCATTERING: WITH APPLICATIONS TO CHEMISTRY, BIOLOGY, AND PHYSICS, Bruce J. Berne and Robert Pecora. (41155-9)

      QUANTUM THEORY, David Bohm. (65969-0)

      ATOMIC PHYSICS (8TH EDITION), Max Born. (65984-4)

      EINSTEIN’S THEORY OF RELATIVITY, Max Born. (60769-0)

      INTRODUCTION TO HAMILTONIAN OPTICS, H.A. Buchdahl. (67597-1)

      MATHEMATICS OF CLASSICAL AND QUANTUM PHYSICS, Frederick W Byron, Jr. and Robert W. Fuller. (67164-X)

      MECHANICS, J.P. Den Hartog. (60754-2)

      INVESTIGATIONS ON THE THEORY OF THE BROWNIAN MOVEMENT, Albert Einstein. (60304-0)

      THE PRINCIPLE OF RELATIVITY, Albert Einstein, et al. (60081-5)

      THE PHYSICS OF WAVES, William C. Elmore and Mark A. Heald. (64926-1)

      THERMODYNAMICS, Enrico Fermi. (60361-X)

      INTRODUCTION TO MODERN OPTICS, Grant R. Fowles. (65957-7)

      DIALOGUES CONCERNING Two NEW SCIENCES, Galileo Galilei. (60099-8)

      GROUP THEORY AND ITS APPLICATION TO PHYSICAL PROBLEMS, Morton Hamermesh. (66181-4)

      ELECTRONIC STRUCTURE AND THE PROPERTIES OF SOLIDS: THE PHYSICS OF THE CHEMICAL BOND, Walter A. Harrison. (66021-4)

      SOLID STATE THEORY. Walter A. Harrison. (63948-7)

      PHYSICAL PRINCIPLES OF THE QUANTUM THEORY, Werner Heisenberg. (60113-7)

      ATOMIC SPECTRA AND ATOMIC STRUCTURE, Gerhard Herzberg. (60115-3)

      AN INTRODUCTION TO STATISTICAL THERMODYNAMICS, Terrell L. Hill. (65242-4)

      EQUILIBRIUM STATISTICAL MECHANICS, E. Atlee Jackson (41185-0)

      OPTICS AND OPTICAL INSTRUMENTS: AN INTRODUCTION, B.K. Johnson. (60642-2)

      THEORETICAL SOLID STATE PHYSICS, VOLS. I & II, William Jones and Norman H. March. (65015-4, 65016-2)

      THEORETICAL PHYSICS, Georg Joos, with Ira M. Freeman. (65227-0)

      THE VARIATIONAL PRINCIPLES OF MECHANICS, Cornelius Lanczos. (65067-7)

      A GUIDE TO FEYNMAN DIAGRAMS IN THE MANY-BODY PROBLEM, Richard D. Mattuck. (67047-3)

      MATTER AND MOTION, James Clerk Maxwell. (66895-9)

      FUNDAMENTAL FORMULAS OF PHYSICS, VOLS. I & II, Donald H. Menzel (ed.). (60595-7, 60596-5)

      OPTICKS, Sir Isaac Newton. (60205-2)

      THEORY OF RELATIVITY, W. Pauli. (64152-X)

      INTRODUCTION TO QUANTUM MECHANICS WITH APPLICATIONS TO CHEMISTRY, Linus Pauling and E. Bright Wilson, Jr. (64871-0)

      SURVEY OF PHYSICAL THEORY, Max Planck. (67867-9)

      THE PHILOSOPHY OF SPACE AND TIME, Hans Reichenbach. (60443-8)

      BOUNDARY AND EIGENVALUE PROBLEMS IN MATHEMATICAL PHYSICS, Hans Sagan. (66132-6)

      STATISTICAL THERMODYNAMICS, Erwin Schrodinger. (66101-6)

      PRINCIPLES OF ELECTRODYNAMICS, Melvin Schwartz. (65493-1)

      ELECTROMAGNETIC FIELD, Albert Shadowitz. (65660-8)

      PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS, S.L. Sobolev. (65964-X)

      PRINCIPLES OF STATISTICAL MECHANICS, Richard C. Tolman. (63896-0)

      LIGHT SCATTERING BY SMALL PARTICLES, H.C. van de Hulst. (64228-3)

      UNDERSTANDING THERMODYNAMICS, H.C. Van Ness. (63277-6)

      MATHEMATICAL ANALYSIS OF PHYSICAL PROBLEMS, Philip R. Wallace. (64676-9)

      STATISTICAL PHYSICS, Gregory H. Wannier. (65401-X)

      X-RAY DIFFRACTION, B.E. Warren. (66317-5)

      SPACE—TIME—MATTER, Hermann Weyl. (60267-2)

      Paperbound unless otherwise indicated. Available at your book dealer, online at www.doverpublications.com, or by writing to Dept. 23, Dover Publications, Inc., 31 East 2nd Street, Mineola, NY 11501. For current price information or for free catalogs (please indicate field of interest), write to Dover Publications or log on to www.doverpublications.com and see every Dover book in print. Each year Dover publishes over 500 books on fine art, music, crafts and needlework, antiques, languages, literature, children’s books, chess, cookery, nature, anthropology, science, mathematics, and other areas.

      Manufactured in the U.S.A.

      Copyright

      Copyright © 1992 by Addison-Wesley Publishing Company Copyright © 2003 by R. D. Hazeltine and J. D. Meiss All rights reserved.

      Bibliographical Note

      This Dover edition, first published in 2003, is a corrected republication of the work originally published as Volume 86 in the series Frontiers in Physics by Addison-Wesley Publishing Company, Redwood City, Calif., in 1992. A new Preface to the Dover Edition and a new Appendix have been added.

      International Standard Book Number: 0-486-43242-4

      9780486151038

      Manufactured in the United States of America

      Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

      Dedicated to the memory of Susannah Elizabeth Hazeltine

      Preface

      This book exposes the physical ideas and theoretical methods that have emerged from several decades of international research on plasma confinement. It attempts a relatively unified treatment of the numerous, rather diverse topics that affect the magnetic confinement of hot plasmas.

      Readers of the book are assumed to have some elementary acquaintance with plasma physics-to understand, for example, the significance of the Debye length and the gyroradius. However, the most important results are derived from first principles; they are intended to be comprehensible to anyone with a basic knowledge of classical physics. As a text, the book is intended for a graduate or advanced undergraduate course.

      The first two chapters concern introductory and review material. Chapters 3-6 form the core of the book (a synopsis may be found in Chapter 1—or gleaned from the table of contents). A one-semester course could cover these chapters, plus either Chapter 7, for a stability course, or Chapter 8, for a transport course. In either case, appropriate material from Chapter 9 could be included as time permits. A two-semester course would cover most of the book. Sections or subsections distinguished by an asterisk concern relatively advanced matters and may be omitted in a one-semester course.

      The citations in the text and at the end of each chapter list material, mostly books and review articles, that we think students might find helpful. A bibliography may be found at the end of the book. Parts of Chapter 6 draw on the unpublished thesis work of Chi-Tien Hsu and Xiang-Seng Lee; a part of Chapter 8 benefits from the unpublished thesis of M. Kotschenreuther. Our hope is that other contributions are at least indirectly cited, through secondary sources. There is no attempt to present a full bibliography of plasma confinement physics. We realize, with regret, that many excellent and important contributions are not mentioned.

      We are grateful to many friends whose encouragement, instruction and advice have benefitted the book. We especially wish to thank A. Y. Aydemir, D. E. Baldwin, Peter Catto, A. N. Kaufman, M. Kotschenreuther, W. H. Miner, Jr., P. J. Morrison, E. R. Solano, F. L. Waelbroeck, A. J. Wootton, X.-D. Zhang, E. Zweibel, Cheryl Hazeltine, Mary Sue Moore and Marie Pervil.

      Preface to the Dover Edition

      Preparing the Dover Edition has allowed us to correct a number of typographical errors in the previous edition of our book. We thank the many people, especially Frank Waelbroeck, Abinadab Dieter and Andrei Simakov, who helped us identify these errors. We have also taken this opportunity to enlarge the index and bibliography, to simplify parts of the argument, to add an appendix, and to make other improvements in the readability of the book.

      Table of Contents

      DOVER BOOKS ON PHYSICS

      Title Page

      Copyright Page

      Dedication

      Preface

      Preface to the Dover Edition

      Chapter 1 - Introduction

      Chapter 2 - Review of Fundamentals

      Chapter 3 - Confined Plasma Equilibrium

      Chapter 4 - Kinetic Description of a Magnetized Plasma

      Chapter 5 - Coulomb Collisions

      Chapter 6 - Fluid Description of Magnetized Plasma

      Chapter 7 - Stability of Confined Plasmas

      Chapter 8 - Collisional Transport

      Chapter 9 - Nonlinear Processes

      Appendix A - Useful Formulae

      Bibliography

      Index

      Chapter 1

      Introduction

      1.1 The Science of Plasma Confinement

      A sufficiently cold plasma can be confined by glass tubing; a sufficiently large plasma is confined gravitationally. This book is concerned with the confinement of terrestrial, very hot plasmas, which would be quenched by direct contact with ordinary container walls. Primarily of interest in application to controlled fusion research, such plasmas are confined magnetically: the magnetic field, itself ultimately confined by mechanical forces on the field coils, acts as an insulating and stress-bearing containment vessel. So our subject is more specifically magnetic confinement: the behavior of bounded, magnetized plasma.

      The international research program in magnetic plasma confinement, now some forty years old, stems almost entirely from interest in controlled fusion. It is therefore not surprising that the perspective and emphasis of that program pervades this book. Yet this is not a book about controlled fusion. Beyond a brief survey later in the present chapter, we rarely refer to the specific confinement criteria set by fusion, and make no attempt to cover the multitude of scientific, engineering and environmental issues involved in this potential energy source. Furthermore, mindful of the application of fusion-confinement ideas to such fields as astrophysics and solar physics, we have tried to adopt as broad and fundamental a viewpoint as possible. Thus we approach plasma confinement as a scientific issue with both extrinsic and intrinsic interest.

      Given the potent electrodynamics of an assemblage of charged particles, to what extent can they be kept away from, or at least insulated from, material walls? How do such fundamental physical processes as Coulomb collisions, fluid instability, and turbulence limit confinement? The attack on questions of this sort has yet to generate complete or final answers. But it has produced a body of science: a collection of observations, insights and methods that illuminate part of the physical world. This scientific domain is the subject of our book.

      1.2 Goals and Synopsis

      No book of reasonable size can attempt to cover the entire discipline of plasma confinement. Our coverage will be skimpy in several areas; some aspects of the subject are not treated at all. Our most important omission concerns experimental confinement physics. The design and operation of experimental confinement devices, the development of plasma diagnostic methods, and the vast technology of confinement, heating and plasma-current drive are areas of obvious critical importance to controlled fusion—and areas that have shown exciting progress in recent years. They deserve, in fact require, treatment in a separate volume.

      Thus, in keeping with the interests of its authors, this book primarily treats the theory of magnetic confinement. However, we have tried to make it more than a collection of theoretical methods. Instead we emphasize the relatively small set of physical ideas that lie at the heart of the confinement problem. Most of these ideas have a significance that is not only theoretically pervasive but also borne out, at least qualitatively, by experimental observation. In developing these concepts and demonstrating their application to various issues of confined plasma evolution, we attempt to display the scaffold of plasma confinement physics.

      Chapter 2 contains a review of certain ideas and methods that are used throughout the book. It is intended to indicate how much mathematical and physical background is assumed, and perhaps to jog memories. The treatment is very terse; students who find much of this material new should probably seek more detailed coverage elsewhere. Chapter 3, essentially a much more detailed and careful version of §1.4 below, concerns confined plasma equilibrium. The next three chapters concern how a system as complicated as a confined plasma is to be efficiently and convincingly described. The kinetic description is considered first: in Chapter 4 we derive and discuss the basic (drift- and gyro-) kinetic equations, and in Chapter 5, the collision operator. These treatments include applications of the methods discussed, so that, by the end of Chapter 5, key features of the confined plasma distribution function have been established. The fluid description of confined plasma is taken up in Chapter 6; special attention is given to the question of deriving a closed set of equations for the plasma and electromagnetic field. (Indeed, the closure issue, never trivial in plasma physics, is a theme of this book.) The results and methods of Chapters 4-6 are applied, in the three last three chapters, to the central confinement issues: linear stability, collisional transport and nonlinear evolution. The stability considerations of Chapter 7, including ideal and resistive MHD, as well as kinetic and nonzero gyroradius considerations, are organized around the shear-Alfvén law. The techniques for studying transport are introduced in Chapter 8. The emphasis here is on basic physical ideas—such as the importance of collisionless orbits, or the role of momentum conservation—rather than detailed results. Finally, Chapter 9 provides a brief introduction to some of the nonlinear processes, including magnetic island evolution, and quasilinear transport, which seem critically to affect confined plasma behavior.

      1.3 Confinement Demands of Controlled Fusion

      A nuclear reaction liberates energy when the mass of the reactant nuclei exceeds that of the reaction products. Stored nuclear energy is a function of atomic weight, with a broad minimum in the vicinity of the element iron and maxima at both ends. The variation is sharpest at the lower end, for the lightest elements; thus much more energy per nucleon is freed by hydrogen fusion than by fission of uranium.

      Hydrogen survives because of electrostatic repulsion: the repulsive Coulomb force between nuclei easily overrides nuclear attraction at atomic or molecular distances. The potency of this barrier results primarily from the infinite range of the Coulomb field, which impedes quantum tunneling. In fact tunneling becomes effective only after most of the Coulomb barrier has been classically penetrated, requiring large kinetic energy of the reactants. Thus fusion cross sections are negligibly small for kinetic energies less than several kilo-electron Volts (keV). For this reason, and because the density of fusion reactions depends on the product of the cross section with particle velocity, the interesting energy range for thermonuclear fusion is between ten and thirty keV.

      Fusion in our sun relies on the proton-proton interaction, whose cross section, adequate in a body of stellar dimensions, is too small for laboratory reactions. Terrestrial fusion research depends upon hydrogen isotopes that are much more reactive, if unfortunately less abundant. Table 1 lists the most important reactions. The deuterium-tritium (D-T) reaction is emphasized because of its relatively favorable cross section; D-T reactors would operate at a temperature close to 10 keV. (Note that we always measure temperature in units of energy.) Other reactions that appear feasible for energy production, such as D-D and D-He³, may have economic or environmental advantages but are more demanding with respect to confinement. For example, the D-D reaction is advantageous because of the relative abundance of deuterium (tritium is rare and would have to be bred from lithium) and because it provides a greater fraction of its energy in the form of charged particles, whose energy can be captured to sustain the reaction, or directly converted to output power. But a D-D reactor would apparently have to operate at temperatures exceeding 25 keV.

      Table 1.1: Fusion reactions with relatively large cross-sections (Rose and Clark, 1961)

      Some explanation of the term thermal, or thermonuclear fusion is appropriate here. One can imagine penetrating the Coulomb barrier by means of energetic, oppositely directed particle beams at low temperature. Indeed, non-thermal schemes of this general sort have been proposed and remain objects of research. However, especially because the Coulomb cross section for elastic scattering always greatly exceeds fusion cross sections, most research attention has been devoted to a population of reactants whose velocity distribution approximates a Maxwellian: a thermal population, with temperature close to the optimal reaction energy. In this book we study thermal plasmas exclusively. On the other hand we treat seriously both the process of relaxation to a Maxwellian as well as the non- Maxwellian components of the steady-state distribution; the latter are critically important.

      Power Balance and the Lawson Criterion

      Fusion confinement requirements are conveniently quantified in terms of energy and particle confinement times. The energy confinement time, τE, is operationally defined in terms of a steady-state plasma. One measures the plasma energy content, U, as well as the rate at which energy must be added to sustain the steady state. Unavoidable energy loses due to radiation losses (see below) are subtracted from the latter to provide a net rate of energy input, Win. Then we have

      τE ≡ U/Win .

      The particle confinement time, τp, is defined analogously, in terms of particle content and replacement rate.

      In a steady-state, self-sustained fusion plasma, the energy input would come from fusion reactions. We denote the power input by fusion reactions by Fin. Note that Fin measures the rate at which fusion energy is deposited in the plasma; it is smaller than the rate of fusion energy production, because only the kinetic energy of charged reaction products, such as alpha particles, can be contained. Neutron energy is too quickly lost to the walls of the containing vessel to contribute to Fin. We express the radiation losses by Rb, where the subscript refers to bremstrahlung radiation. Cyclotron radiation losses are conventionally omitted because their longer wavelength allows at least partial reflection at the plasma boundary. Thus Win = Fin - Rb and fusion plasma power balance is expressed as

      (1.1)

      It was noted by Lawson that, while the energy content is proportional to plasma density, n, both quantities on the left-hand side of (1.1) vary with the square of the density (Rose and Clark, 1961). That Fin is proportional to n² follows immediately from the binary nature of fusion reactions; for Rb it is a consequence of the role of ions in accelerating electrons to produce bremstrahlung—and the fact that plasma quasineutrality makes the electron and ion densities vary together. It follows that the density enters (1.1) only through the combination nτE, the Lawson parameter.

      The critical value of the Lawson parameter, (nτE)c, satisfies (1.1). It is evident that smaller values of nτE correspond to energy losses too severe for self-sustained fusion; larger values could presumably be cured by artificially enhanced losses. Thus the Lawson criterion for self-sustained, or ignited fusion states that the product of density and confinement time must equal or exceed (nτE)c.

      Using the known fusion-reaction, energy-deposition and bremstrahlung rates, it is possible to evaluate (nτE)c. The result is sensitive to the particular fusion reaction considered, as well as to various plasma parameters (such as impurity content) that are not easy to predict, so that a simple, general specification is difficult. Nonetheless most estimates are in rough agreement; for the important case of D-T reactions, one finds

      (1.2)

      Confinement Q

      Another measure of confinement quality, alternative to nτE, realistically allows for nuclear energy production that is short of ignition, or self-sustained fusion. Thus a certain fusion device might produce excess energy only in the presence of continuous energy input, the deposited fusion energy being insufficient to maintain thermonuclear temperature. If the ratio (fusion power produced)/(operating power supplied) is sufficiently large, the device is a useful fusion reactor. Sometimes this ratio is called "engineering Q."

      A more commonly used ratio is simplified by including in the denominator only the plasma heating power that is required to sustain nuclear reactions. This ratio is denoted by

      Q ≡ (fusion power produced)/(heating power supplied) .

      Here the numerator is some multiple, α > 1, of the deposited power Fin:

      fusion power produced = αFin .

      For the case of D-T fusion, where 80% of the fusion energy is carried off by 14 MeV neutrons, α ≈ 5.

      The denominator of Q is a fraction, γ < 1, of the total externally supplied power, Pin, where Pin includes heating power as well as the power needed to magnetize the plasma, drive plasma currents required for equilibrium and so on:

      (1.3)

      The value of γ is very sensitive to reactor concept and design; it could in principle approach unity, but might be as small as 10%. One object of magnetized plasma research is to find confinement schemes consistent with γ ≈ 1.

      Evidently the ignited state corresponds to Q = ∞. Finite Q-values have scientific interest or even, if sufficiently large, power-production relevance. For example, a Q value of five or more in a D-T plasma yields fusion heating comparable to external heating; such a plasma, although far from economic application, might reveal much about plasma ignition physics.

      Most fusion reactors designs presume ignition, although, as we have observed, this is not economically essential. On the other hand, the confinement quality required by high Q differs little from what is required for ignition: large values of the Lawson parameter, approaching (nτE)c, are required to make Q much bigger than unity.

      Summary

      Fusion makes two demands on the quality of confinement: the temperature must exceed some critical temperature for significant fusion yield; and the product of density and confinement time must approach, or, for ignition, exceed the critical Lawson value. The critical temperature is close to ten keV (or, for reactions other than D-T, a few tens of keV), while the critical value of the Lawson parameter is several times 10¹⁴ sec /cm³.

      It is possible to satisfy both demands without any plasma confinement in the usual sense. For inertial confinement fusion, as it is called, τE is measured in microseconds and fixed essentially by inertia of the reactants; the density is correspondingly large. A controlled energy release is made possible by limiting the total reactant mass for each, essentially explosive, fusion cycle. The fusion devices considered in this book allow for steady-state, or nearly steady-state operation. Then densities are limited by the maximum containable plasma pressure—ultimately fixed by mechanical stresses on the confining magnetic field coils. A typical reactor might have n ≈ 10¹⁴/cm³, with confinement times measured in seconds. Experimental densities and confinement times approach these figures.

      1.4 Magnetized Plasma Confinement

      Nonequilibrium Physics

      A charged particle subject to magnetic force executes a helical orbit about an axis parallel to the local direction of the magnetic field; one says that the particle gyrates about the field line, while moving uniformly along it. We suppose at first that the field is uniform, so each field line is straight. By increasing the strength of the field, we can make the particle’s Larmor radius or gyroradius indefinitely small, thereby restricting the particle orbit to the neighborhood of a single line. In so far as the field line avoids hitting boundaries, the particle is confined.

      An obvious weakness of this scheme is its two-dimensional character: confinement in the direction of the field is not addressed. But before considering the third dimension, we note a more fundamental problem: consistency with classical thermodynamics. Since the equilibrium distribution of charged particles depends only on particle energy, it is unaffected by magnetic fields. Thus thermodynamic equilibrium is inconsistent with magnetic confinement.

      For concreteness, consider the canonical two-dimensional confinement structure: an infinite circular cylinder (solenoid) with radius a and uniform, axial magnetic field. If the field is strong enough to make all gyroradii small compared to a, then single particle motion will be effectively axial. For magnetic confinement however, we must require in addition that field lines within a gyroradius of the wall are unpopulated. Thus the confined distribution must have a density gradient, with maximum near the cylindrical axis and vanishing density at the radius a. It is this gradient that contradicts thermodynamic equilibrium. Evidently any initial gradient is worn away by, inter alia, the random Coulomb interactions of discrete charges.

      Single-particle confinement is straightforward, even in three dimensions, because it involves only external forces. Plasma confinement, on the other hand, invokes electromagnetic interactions. At the very least, binary Coulomb collisions will relax density and temperature gradients; often collective interactions—plasma instability and turbulence—produce the same effect much more rapidly. The nonequilibrium character of magnetized plasma confinement has a controlling influence on confinement physics.

      Toroidicity

      Confinement in the third dimension is addressed, in different ways, by two broad classes of devices: open confinement systems and closed confinement systems.

      In an open system, the confinement vessel is roughly cylindrical in shape, and the field is essentially axial. Field lines intersect the material walls at either end of the vessel. Thus particles moving in the direction of the cylindrical axis are not confined in the usual sense, and a strategy is needed to plug the ends of the cylinder, or otherwise ameliorate the effects of end loss. The most common technique relies on the conservation of particle energy and magnetic moment, implying that particles with sufficient perpendicular energy will be reflected by positive gradients in the field magnitude. Hence confinement in the third dimension can be addressed by strong field regions, magnetic mirrors, at each end of the cylinder. The burden of mirror confinement is to insure that particles approaching a plug have enough perpendicular energy to be reflected by it.

      The vast majority of plasma confinement devices are closed. In this case the field lines are bent, through appropriate coil geometry, in such a way that each is effectively endless. A field line without ends can be hoop-shaped, closing on itself, or it can fill some two- or three-dimensional region of space without self-intersections. In any case it can be shown that the magnetization of the entire plasma requires the boundary of the confinement region to be topologically equivalent to a torus: a closed confinement system is necessarily toroidal.

      Figure 1.1: Simplest features of a torus.

      A simple torus is depicted in Figure 1.1. The toroidal axis is vertical by convention; it is encircled by the magnetic axis, a single toroidal field line that generally locates the peak of the plasma density profile. The magnetic axis exemplifies a closed toroidal curve; any such curve can be loosely identified with the toroidal direction. Similarly, closed poloidal curves, encircling the magnetic axis, indicate the local poloidal direction. One says that toroidal curves traverse the torus the long way, and poloidal curves, the short way. Such statements reflect the essence of a torus: the existence of two topologically distinct round trips.

      The torus in Figure 1.1 is axisymmetric—symmetric with respect to rotation about the major axis. Of course the same geometric concepts also apply to an asymmetric torus, such as the generic torus depicted in Figure 1.2.

      Figure 1.2: Features of a generic torus.

      Corresponding to the two closed curves are two radii: the major radius, R, measuring distance from the toroidal axis, and the minor radius, r, measuring distance from the magnetic axis. The major radius of the magnetic axis is denoted by R0, while the minor radius of the confinement region is denoted by a. The ratio R0/a, characterizing the fatness of the torus, is the toroidal aspect ratio.

      A consequence of toroidicity is that the confining field cannot be uniform. For example, the magnitude of a purely toroidal, axisymmetric magnetic field, formed by bending a solenoid upon itself, varies inversely with major radius. Field variation and curvature are responsible for most of the difficulties in confining thermonuclear plasma.

      Nonuniformity affects even single-particle confinement: the guiding-center drifts associated with non-uniform fields allow charged particles to stray from the field lines. In the simplest case of closed toroidal field lines, the drift is vertical (orthogonal to B and to ∇B), and in opposite directions for oppositely charged species. The electric field resulting from such uncompensated drifts would quickly sweep any confined plasma outward in major radius, so that confinement based on strictly toroidal, axisymmetric field is not possible.

      However, vertical drifts can be rendered innocuous. By imposing strong poloidal motion on the guiding center trajectories, one controls the net excursion away from the magnetic axis. Such compensation is considered explicitly in Chapter 3. The required poloidal motion can result from gradients in the magnitude of purely toroidal field lines, but more commonly it is provided by field lines that have poloidal as well as toroidal components. Notice that in this case the lines are roughly helical, a typical line winding around a toroidal surface much like copper wire is wound on a ferromagnetic core. A toroidal surface that is covered in this way by lines of magnetic field is called a magnetic surface, or flux surface.

      Thus toroidicity poses no insurmountable threat to single-particle confinement; the problem again stems from particle interactions. First, classical dissipation, resulting from Coulomb collisions, is enhanced by particle drifts, even when collisionless drifts are perfectly compensated. This issue is a major theme of Chapter 8. More severely, line-curvature threatens plasma confinement by its potentially unstable interaction with plasma pressure gradients. Indeed, the most dangerous electromagnetic instabilities usually involve curvature of the field. This topic is taken up in Chapter 7.

      1.5 Character of Toroidal Confinement

      Confinement Parameters

      We have noted that the boundary surface of any closed magnetic confinement system is necessarily toroidal. Since the field lines are not to intercept material walls they must lie on this surface. It follows that the magnetic field on the surface can always be expressed as

      (1.4)

      where Bp and BT are the poloidal and toroidal components respectively. This decomposition is intended to be descriptive; more rigorous definitions of Bp and BT will be found in Chapter 3. As we have noted, (1.4) suggests the picture of a single, helical field line, winding endlessly around the toroidal surface.

      Systems have been envisioned in which all confinement forces are provided only at the boundary, the plasma pressure being nearly uniform inside. Much more commonly however, confinement is distributed in minor radius, in the sense that a smooth pressure gradient is supported. Then (1.4) applies not only to the plasma boundary, but to a sequence of nested tori, each bounding and confining some fraction of the plasma. The innermost such torus is evidently a single toroidal field line, the magnetic axis. Every other toroidal surface has the crucial property of being nowhere penetrated by magnetic field lines. The lines lie everywhere in the surfaces, which are therefore called magnetic surfaces, or flux surfaces. It should be pointed out that this picture of nested toroidal flux surfaces, however basic to confinement theory, is considerably idealized; we consider its relation to real confining fields in Chapter 3.

      We next introduce, qualitatively, the two key parameters describing a toroidal confinement device: the safety factor, and the plasma beta.

      The safety factor, denoted by q, measures the field line pitch. A field line will, upon a single circuit of the magnetic axis (poloidal circuit), perform q circuits of the toroidal axis (toroidal circuits). Thus small q corresponds to a tightly wound helix; the limiting case of infinite q describes a purely toroidal line. An approximate, useful formula can be inferred from consideration of the simple cylindrical torus (as in Figure 1.1):

      where qc is called the cylindrical safety factor.

      The more elaborate definition of Chapter 3 shows that the safety factor has a fixed value on each toroidal surface. But it obviously varies within the confining volume: not all field lines have the same pitch. Indeed, the variation of q from one surface to the next, called magnetic shear, has crucial effects on plasma stability. In particular, rational values of q, corresponding to surfaces with closed field lines, are especially vulnerable to electromagnetic perturbation. In the present chapter we associate a q-value with some confinement geometry, globally, by referring implicitly to an average or typical value.

      The parameter beta measures the efficiency of magnetic confinement, in terms of the ratio

      (1.5)

      Like q, this quantity will vary throughout the confinement region, one should strictly distinguish maximum beta, average beta, and so on; for simplicity we will ignore such distinctions here.

      Economical fusion reactors require high beta: the numerator roughly measures fusion yield, while the denominator measures input energy. Experimental devices have beta-values ranging from less than one percent to perhaps 30%.

      Plasma diamagnetism, the depression of magnetic fields by plasma pressure, increases with beta. Thus higher beta devices have stronger diamagnetic currents—currents that usually play a critical role in the confinement mechanism. For this reason high beta devices tend to be relatively self-confining, in the sense of relying less on external coils to produce the desired field. However, the relation between beta and plasma current is easily exaggerated. The point is that diamagnetic current, while indeed measured by beta, is always perpendicular to the local field; plasma current parallel to B is unrelated to diamagnetism or beta. Thus large parallel currents can exist even at zero beta, and certain low-beta devices make critical use of plasma current.

      Confinement Systems

      We next briefly survey some representative toroidal devices. Our discussion is emphatically incomplete: its purpose is simply to impart some sense of the variety and nature of actual confinement geometries.

      A convenient initial characterization involves the safety-factor. When q is infinite (Bp = 0), each line is a hoop encircling the toroidal axis. Equilibrium is possible in this case only if the lines are asymmetric: the hoops cannot be simple circles. Devices with this geometry are called bumpy tori, or, more generally, closed field-line devices. The opposite limit, with vanishing toroidal field, does not globally describe an operating machine, although q vanishes locally in certain devices discussed below.

      The most common confinement geometries have both poloidal and toroidal field components, with values of q within an order of magnitude of unity. In this case only exceptional field lines close on themselves; the typical field line eventually covers its toroidal surface ergodically (i.e., comes arbitrarily close to every point on the surface). There are two broad categories of such devices, which we shall call self-excited and externally excited.

      Self-excited devices tend to have to low q,

      q < 1 ,

      and relatively large beta:

      β ∼ 10 .

      Their characteristic feature is the reliance on self-generated magnetic fields—plasma currents—to provide the primary confining force. (We will find in Chapter 3 that part of the confining force must always be provided external coils.) The smaller q-values reflect the fact that poloidal fields are most easily generated by toroidal plasma current, rather than external field coils. Examples are the stabilized Z-pinch, the spheromak, and various compact toroid geometries. The first of these has received by far the greatest attention; its most popular manifestation is the reversed-field pinch (RFP). In the RFP, plasma currents parallel to the toroidal minor axis not only generate the poloidal field, but also diamagnetically alter the toroidal field, allowing BT to change sign near the plasma boundary. (Such field reversal improves fluid stability.) In fact all three of these devices usually have small toroidal fields—small or vanishing q—in the boundary region.

      The most common versions of all three concepts are axisymmetric. The spheromak, as well as most compact toroid devices, has nearly unit aspect ratio. Z-pinch devices have aspect

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