Physics of Magnetically Confined Plasmas

Allen H. Boozer

Department of Applied Physics and Applied MathematicsColumbia University, New York, NY 10027

(Dated: September 1, 2003)

The physics of magnetically confined plasmas has had much of its development as part of the pro-gram to develop fusion energy and is an important element in the study of space and astrophysicalplasmas. Closely related areas of physics include Hamiltonian dynamics, kinetic theory, and fluidturbulence. A number of topics in physics have had their primary development through researchon magnetically confined plasmas. The physics that underlies the magnetic confinement of plas-mas is reviewed to make it more accessible to those beginning research on plasma confinementand for interested physicists.

Contents

I. Introduction 1

II. Fusion energy 2

III. Magnetic field lines 4A. Relation to Hamiltonian mechanics 4B. Methods of forming toroidal magnetic surfaces 9C. Evolution of magnetic field lines 12

IV. Confinement systems 15

V. Efficiency of magnetic confinement 17A. Equilibrium limits 17B. Stability limits 21

1. Current-driven instabilities 222. Resistive stability 243. Robustness of magnetic surfaces 25

C. Stability analyses 261. Expressions for δW 272. Pressure-driven ballooning modes 283. Minimization of δW 29

D. Interaction of plasmas with coils 301. Freedom of coil design 312. Plasma response to coil changes 32

VI. Radiation and transport 35A. Electromagnetic radiation 36B. Kinetic theory 37C. Landau damping and quasilinear diffusion 40D. Drift kinetic theory 42

1. Alfven’s guiding center velocity 432. Magnetic moment conservation 453. Drift Hamiltonian 464. Action invariant J 47

E. Particle trajectories and transport 481. Confinement of particle trajectories 492. Transport at low collisionality 513. Power required for maintaining fields 52

F. Microstability 53G. Gyrokinetic theory 57H. Alfven instabilities 60

1. Continuum Alfven wave damping 612. Weakly damped gap mode 623. Particle-Alfven wave interaction 63

VII. Plasma edge 64

Acknowledgments 65

A. General coordinate systems 65

References 67

I. INTRODUCTION

A plasma is a gas in which charged particles are ofsufficient importance for the gas to be a good electricalconductor. Ordinary matter becomes ionized and formsa plasma at temperatures above about 5000K, and mostof the visible matter in the universe is in the plasmastate. The high electrical conductivity implies currentscan flow in a plasma. These currents can interact withmagnetic fields to produce the forces that are needed forconfinement.

The physics of plasma confinement using magneticfields has been driven intellectually by the program todevelop fusion energy. Fusion has provided a focus forthe research, but much of the physics that has been de-veloped has far broader applications—most obviously tospace and astrophysical plasmas. In addition, the physi-cal insights and concepts are of intrinsic scientific impor-tance.

A number of topics in physics have had their primarydevelopment through research on plasma confinement.These include: (1) the relation between the field linesof divergence-free fields and Hamiltonian mechanics. (2)the constraints of magnetic helicity conservation on therapid evolution of magnetic fields. (3) collisionless relax-ation phenomena. (4) simplified kinetic equations thatare based on adiabatic invariants of classical mechanics.(5) the theory of small amplitude and short wavelengthturbulence, called micro-turbulence. (6) the experimen-tal observation and theoretical explanation of transportbarriers where plasma micro-turbulence is stabilized by astrong gradient in the plasma flow. (7) the rapid damp-ing of continuum shear Alfven waves, which propagateby twisting the magnetic field lines that are embeddedin a plasma. As well as, the existence of weakly dampeddiscrete shear Alfven modes that can be destabilized byparticles with a speed near the phase velocity of Alfvenwaves. One of the goals of this review is to make theseand other topics in the physics of magnetic confinement

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accessible to a broader audience.The physics of magnetic confinement is covered in

a number of books. Probably the most complete isTokamaks, (Wesson, 1997), which emphasizes the rela-tion between theory and experiment. (Miyamoto, 1997)gives the details of many of the classic derivations, TheTheory of Toroidally Confined Plasmas, (White, 2001),offers insights on a number of fundamental phenom-ena, and a standard textbook is Introduction to PlasmaPhysics, (Goldston and Rutherford, 1995). The Inter-national Thermonuclear Experimental Reactor (ITER)design team summarized the physics basis of tokamaksin a series of articles in the December 1999 issue of thejournal Nuclear Fusion. The first of these articles is anoverview (ITER Physics Basis Editors et al., 1999).

Despite the number of books and articles written onplasma confinement a review of the fundamental physicsthat includes derivations of major results has not existed.This review is designed to fit into that niche while beingaccessible to the general physics community.

Several important types of confined plasmas are notcovered in this review. Magnetically confined plasmasoccur in many space and astrophysical situations, andmuch of the physics is in common with laboratory plas-mas. However, this review does not provide sufficientcontext for understanding the breadth of applications tospace and astrophysical plasmas. Texts include BasicSpace Plasma Physics by (Baumjohann and Treumann,1996), Physics of Space Plasmas by (Parks, 1991), andPlasma Astrophysics by (Tajima and Shibata, 1997).(Bisnovatyi-Kogan and Lovelace, 2001) have written amajor review of the plasma issues associated with accre-tion disks, and (Ferrari, 1998) has reviewed extragalacticjets. Laboratory plasmas in which all of the magneticfield lines that are in the plasma intersect the chamberwalls are also not covered. Plasmas flow out of such sys-tems at a thermal speed or have a confinement time com-parable to a collision time.

Important plasma topics that are not covered arewaves, which are of particular importance for plasmaheating, and the methods of measurement of plasma pa-rameters, which is the topic of plasma diagnostics. Stan-dard books on waves in plasmas are (Brambilla, 1998;Stix, 1992; Swanson, 2003). Plasma diagnostics are basedon a broad range of physics principles. The standard textis Principles of Plasma Diagnostics (Hutchinson, 2002),and reviews have been given by (Gentle, 1995) and inan article, (ITER Physics Expert Group on Diagnosticset al., 1999), in the Nuclear Fusion series on ITER.

What is covered is the physics of the magnetic con-finement of plasmas that are flowing slowly compared tosonic speeds and at each point in the plasma are closethermodynamic equilibrium, which implies the confine-ment is long compared to collision times. Although thefundamental topic is not fusion, an understanding of thebasic features of magnetically confined fusion plasmas isuseful for placing research on plasma confinement in con-text. These features are explained in Section (II).

A review of the physics of magnetically confined plas-mas contains too much material to be absorbed at auniform level. Therefore, bullets are given near the be-ginning of most sections. Readers are encouraged toread just the bullets of the entire review before workingthrough the details of the sections. The flow of informa-tion in each section can be determined by reading the firstsentence of each paragraph. A result is generally givenbefore the derivation, so the derivations can be skipped.However, the derivations are sufficiently complete that areader should be able to reconstruct them using a tableof identities of vector calculus. The review is designed tobe read a various levels of detail, and that is the way itshould be read.

II. FUSION ENERGY

• The goal of research on fusion energy is a commer-cially viable source of energy. The primary researcheffort is on the fusion of two isotopes of hydrogen,deuterium and tritium, to form an isotope of he-lium, an alpha particle, and a neutron.

• Fundamental considerations imply that in a fu-sion power plant the particle distribution functionsmust be close to local Maxwellians with a tempera-ture of about 20kev and a density of approximately2× 1020nuclei/m3, and the plasma must have theshape of a torus with the minor radius of the torusa few meters in length.

• Technical limits on the magnitude of the magneticfield that can be used for confining fusion plasmasmake the efficiency of the utilization of the mag-netic field a central issue.

• Most of the freedom in the design of fusion plasmasis in the plasma shape, which means freedom in thedesign of the coils that surround the plasma. Aboutfifty shape parameters can be controlled, thoughonly about 10% of these are consistent with an ax-isymmetric torus.

Fusion is a potential source of energy, which is essen-tially unlimited in quantity and produces no greenhousegases. The fusion reaction that appears technically easi-est to harness is between two isotopes of hydrogen, deu-terium and tritium, with the product being ordinary he-lium (an alpha particle) and a neutron. Deuterium oc-curs naturally in water in sufficient abundance to be anessentially unlimited resource. Tritium has a half-life of12 years and must be produced through an interaction ofthe neutron with isotopes of lithium. Consequently, thefuel for fusion is deuterium and lithium. The waste prod-ucts of fusion energy production need not be radioactivein principle, but in practice some radioactive productswill be produced. The level, lifetime, and nature of theradioactive waste are dependent on the use of appropri-ate materials and the cleverness of the design. The fusion

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reaction is easily terminated, so a runaway reaction is nota safety concern. (Sheffield, 1994) has written a generalreview of fusion systems and (Baker et al., 1998) havediscussed fusion systems with a focus on the importantissue of the materials that would be used in fusion powerplants.

The development of fusion power is paced by bothphysics and engineering considerations. A magneticallyconfined plasma with an essentially self-sustaining fusionburn has not yet been produced, but this is thought tobe essentially an issue of the size of the experiments thathave been undertaken. A number of proposals have beenmade for a burning plasma experiment; the most ambi-tious currently under consideration is the InternationalTokamak Experimental Reactor (ITER) (Aymar, 2000).Many studies have claimed fusion power could have acompetitive cost, but the compatibility of the physicsand the engineering with energy at an acceptable costwill remain a primary issue until demonstration powerplants are built.

Fundamental considerations imply fusion power sys-tems will have a power output of order a gigawatt. First,the structure surrounding the plasma must have mini-mum thickness, about 1.5m, for the fusion neutrons toconvert lithium into tritium and heat as well as shieldthe external world from radiation. Second, until materiallimitations arise at a power density of several megawattsper square meter of chamber wall area, power becomescheaper the higher the power density. These two con-siderations plus the need for a toroidal plasma, whichis discussed in Section (III), imply that fusion energy ismost economical in units of approximately one gigawattof electrical power. A larger or smaller unit size wouldimply a larger or a smaller aspect ratio of the toroidalplasma.

The electrostatic repulsion of the deuterium and thetritium nuclei sets the energy scale at which fusion re-actions occur, which is a thermal energy of a few tensof kilovolts, which is a few hundred million Kelvin. Forthe plasma to maintain its burning state, the rate of pro-duction of high-energy alpha particles must be sufficientto offset the energy losses of the plasma through electro-magnetic radiation and thermal transport through diffu-sion, Section (VI). Since neutrons are electrically neutralthey have no significant interaction with the plasma orthe magnetic field. The maximum rate of energy produc-tion, at fixed plasma pressure, occurs at a temperature ofapproximately 20 kilovolts, or 2.3 × 108K. For a steadyburn at a temperature of 20 kilovolts, the number densityof nuclei times the energy confinement time must equal2× 1020 (nuclei/meters cubed) seconds. At that temper-ature the deuterium and tritium are completely ionizedand, therefore, form a plasma.

The required power density on the chamber walls foreconomic electric power determines the plasma density,about 2 × 1020 nuclei per cubic meter, and the plasmapressure, about ten atmospheres.

Plasmas of interest for magnetically confined fusion

are in a paradoxical collisionality regime. The rate forCoulomb collisions, which relax the particle distributionfunctions to Maxwellians, times the energy confinementtime is about 100 for the ions and 10,000 for the electrons.The rapidity of collisions implies the distribution func-tions must be close to Maxwellian. However the meanfree path of the thermal particles, about 10km, is enor-mous compared to the size of the plasma. This para-doxical collisionality regime places requirements on thequality of the collisionless particle trajectories in orderto have adequate confinement, Section (VI.E.1).

Three considerations set a minimal level for the mag-netic field strength, which is a few Tesla. (1) The mag-netic field pressure must be significantly larger than theplasma pressure to provide a stable force balance, Sec-tion (V). (2) A charged particle moves in a circle aboutmagnetic field lines, and the radius of this circle, the gyro-radius ρ, must be sufficiently small that the high-energyalpha particles remain in the plasma and heat it, Section(VI.E.1). (3) The thermal transport coefficients, whichare reduced by an increase in the magnetic field strength,must be sufficiently small to obtain the required energyconfinement time, Section (VI.F). Engineering consider-ations make it desirable to use the lowest magnetic fieldstrength that is consistent with the physics requirements.The technical limitations that arise for magnetic fieldslarger than about 10T make magnetic confinement fu-sion more difficult than if higher fields could be used. Ifhigher fields were available, power losses from electroncyclotron radiation, which set an upper limit on the elec-tron temperature, would be of increasing importance. Adiscussion of electron cyclotron losses is given in Sec-tion (VI.A) and by (Albajar, Bornatici, and Engelmann,2002).

In addition to fusion using magnetic fields to confinethe plasma, a large research program is being pursuedon inertial confinement fusion. The physics is describedthe book Inertial Confinement Fusion by (Lindl, 1998).Inertial confinement means the plasma is confined for atime of order a/Cs with Cs the speed of sound and a theplasma radius. Confinement is determined by the balancebetween the inertial and the pressure forces. Magneti-cally confined plasmas are in essentially a static balancebetween the magnetic and the pressure forces. The char-acteristic plasma density during the burn period of aninertially confined plasma is of order 1032particles/m3,which is about twelve orders of magnitude larger than thedensity of a magnetically confined fusion plasma. Thesize of the pellets that are imploded in inertial confine-ment have a millimeter scale, while magnetic fusion plas-mas have a scale of meters. However, the central plasmatemperatures are roughly the same in magnetically andinertially confined fusion plasmas in order to obtain anoptimal reaction rate.

The focus of much of the research on magnetic fusionduring the last decade has been on the understandingand the extrapolation of transport processes in confinedplasmas. Energy transport means the loss of energy by

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plasma processes rather than by electromagnetic radia-tion. Confined plasmas are usually, but not always, in astate in which microturbulence dominates the transportprocesses. Microturbulence, Section (VI.F), means thefluctuations have a wavenumber that is of order of theinverse of an ion gyroradius ρi and an amplitude of or-der the ion gyroradius to system size ρi/a < 10−2. Theextrapolation of microturbulent transport from existingto future experiments and to fusion power plants is es-sential for the scientific planning of fusion research andis an issue in the feasibility of fusion power.

The focus of innovation in fusion research is on ways toobtain more control over fusion plasmas and on improvedfusion systems. A plasma equilibrium is determined bythe shape of the plasma, the magnetic field strength, andthe profiles of the plasma current and pressure, Section(V.A). In a power plant, economics dictates that onlyabout 5% of the fusion power can be used for control.Consequently, the plasma current and pressure profilesare largely determined by internal plasma phenomena.The largest element of control that the designer has overthe plasma and its performance is on the plasma shape.Control of the plasma shape is actually control over thedesign of the coils that surround the plasma. As shownin Section (V.D.1), coil constraints limit the designer toabout fifty shape parameters of which only four (aspectratio, ellipticity, triangularity, and squareness) are con-sistent with an axisymmetric torus. The axisymmetrictokamak, Section (IV), is the most advanced plasma con-finement configuration and is the configuration of choicefor all major designs for experiments with a fusion burn.Careful consideration of the four axisymmetric shape pa-rameters is known to be essential for attractive tokamaks.However, tokamaks set more than 90% of the availableshape parameters to zero. The use of a larger set of shapeparameters allows the designer to sidestep issues in toka-mak design, such as current profile control, and is beingstudied under the topic of stellarator research, Section(IV).

III. MAGNETIC FIELD LINES

• Near-Maxwellian plasmas are in force balance be-tween the pressure and the magnetic forces, ~∇p =~j × ~B.

• The magnetic field lines confining a near-Maxwellian plasma must lie in the surfaces of con-stant pressure, ~B · ~∇p = 0. These surfaces can bespatially bounded only if they have the topologicalform of a torus. The pressure p(ψt) is a function ofψt, which is the flux of the toroidally directed mag-netic field that is enclosed by the pressure surfaces.

In plasmas of fusion interest, the ions and the electronsare in near-Maxwellian distributions, which implies theplasma has the pressure of an ideal gas, p = nT where Tis the temperature in energy units and n = ne+ni is the

sum of the number of electrons and ions per cubic meter.Plasma confinement implies a pressure gradient, and ~∇pis a force per unit volume. This force is balanced by theelectromagnetic force produced by the cross product ofthe current density in the plasma and the magnetic field,

~∇p = ~j × ~B. (1)

Equation (1) gives the force equilibrium of a near-Maxwellian plasma and places fundamental constraintson magnetic confinement systems.

A confined near-Maxwellian plasma is constrained tohave the topological form of a torus. Why is this? Equi-librium implies ~B · ~∇p = 0; a magnetic field line must liein a constant pressure surface through its entire length.For example, a constant pressure surface cannot be asphere and satisfy Equation (1). The magnetic field linesin a constant pressure surface resemble strands of hair.The hair on a topological sphere, such as a person’s head,always has a crown, a place where the hair spirals out of apoint. A crown is a point at which Equation (1) cannothold. A theorem of topology says a non-singular vec-tor field, ~B(~x), can be everywhere tangent to spatiallybounded function, namely p(~x), in only one shape, thetorus.

Since toroidal surfaces are central to the whole the-ory of plasma confinement, it is important to have amethod for describing them as a basis for terminology.The simplest description uses (R,ϕ,Z) cylindrical coor-dinates with spatial positions given by

~x(r, θ, ϕ) = R(r, θ, ϕ)R(ϕ) + Z(r, θ, ϕ)Z. (2)

Expressed in terms of the Cartesian unit vectors, theradial unit vector of cylindrical coordinates is R(ϕ) =x cosϕ+ y sinϕ and its derivative dR/dϕ is the unit vec-tor ϕ. Simple circular toroidal surfaces are given byR = Ro + r cos(θ) and Z = −r sin(θ). The constantRo is the major radius, r is the local minor radius, andε = r/Ro, which must be less than one, is the local in-verse aspect ratio. A function, such as ~x(r, θ, ϕ), whichdetermines the positions in space that are associated withthree quantities (r, θ, ϕ), defines an (r, θ, ϕ) coordinatesystem. Appendix (A) gives the theory of general co-ordinates that is required for understanding the review.The polar angle of cylindrical coordinates, ϕ, is also thetoroidal angle. The angle θ is called the poloidal an-gle, and r is a radial coordinate that labels the varioustoroidal surfaces.

A. Relation to Hamiltonian mechanics

• The field lines of a magnetic, ~B(~x), or any otherdivergence-free field, are the trajectories of a Hamil-tonian. (A short tutorial on Hamiltonians is givenjust after the bullets.) If ϕ(~x) is a toroidal an-gle, and ~B · ~∇ϕ 6= 0, then the magnetic field

5

lines are given by a one-and-a-half degree of free-dom Hamiltonian, which is the poloidal mag-netic flux ψp(ψt, θ, ϕ). The canonical momentumis the toroidal flux ψt, the canonical coordinateis a poloidal angle angle, θ, and the canonicaltime is the toroidal angle, ϕ, Figure (1). Thefull Hamiltonian system consists of the Hamilto-nian, ψp(ψt, θ, ϕ), and the transformation function,~x(ψt, θ, ϕ), which gives the spatial location of eachcanonical coordinate, (ψt, θ, ϕ), point.

• When the magnetic field lines lie in toroidal sur-faces, the canonical coordinates can be chosen sothe poloidal flux is a function of the toroidal fluxalone, ψp(ψt). The twist of the field lines, whichis called the rotational transform, is ι ≡ dψp/dψt.When the rotational transform is a rational numberι = n/m, the field lines close on themselves after npoloidal and m toroidal transits of the torus. Thecoordinates in which the poloidal flux is a functionof the toroidal flux alone are called magnetic coor-dinates and trivialize the solution of the magneticdifferential equation ~B · ~∇f = g, which arises fre-quently in plasma physics.

• If a magnetic field ~B0(~x) that forms perfect surfacesis perturbed by a field δ ~B, then the magnetic sur-faces can split to form islands, Figure (2), where therotational transform is a rational number, ι = n/m.If δ ~B · ~∇ψt/ ~B0 · ~∇ϕ =

∑bmn exp[i(nϕ−mθm)], the

width of the island that splits the surface ι = n/m

is proportional to√|bmn|. If islands from differ-

ent rational surfaces (ι different rational numbers)are sufficiently wide to overlap, the magnetic fieldlines in that region will come arbitrarily close toevery point in a volume of space rather than lyingon surfaces. Such field line trajectories are said tobe stochastic. Plasma confinement is destroyed inregions of stochastic field lines.

• The opening of an island is a singular process ina toroidal plasma. Let p0(~x) be any function thatsatisfies ~B0 · ~∇p0 = 0. Without an island dp0/dψt isgenerally non-zero near the resonant rational sur-face ι = n/m, but with an arbitrarily small islandonly the derivative dp0/dψh can be non-zero wherethe helical flux is defined by dψh ≡ (ι− n/m)dψt.

The nested toroidal surfaces of magnetic field lines arereminiscent of the torii formed by particle trajectories ofan integrable Hamiltonian H(p, x, t) that is periodic intime with T the period, H(p, x, t+T ) = H(p, x, t). SuchHamiltonians are said to have one-and-a-half degrees offreedom. An excellent reference for the Hamiltonian me-chanics that is used in this review is the book Regular andChaotic Dynamics, (Lichtenberg and Lieberman, 1992).

The Hamiltonian formulation of mechanics followsfrom Newton’s equations of motion, d~p/dt = −~∇V (~x, t)and d~x/dt = ~p/m. Let H(~p, ~x, t) = p2/2m+V (~x, t), then

Toroidal flux

Toroidal currentand

A constantor

surface

j

Poloidal flux

Poloidal currentandz

R

FIG. 1 Magnetic fluxes and currents are defined using thecross-sectional area for the toroidal flux ψt and the current Iand using the central hole of the torus for the poloidal flux ψp

and the current G. The poloidal angle is θ and the toroidalangle is ϕ. (R,ϕ,Z) are ordinary cylindrical coordinates.

Newton’s equations of motion are reproduced by the setof ordinary differential equations d~p/dt = −∂H/∂~x andd~x/dt = ∂H/∂~p. Any set of ordinary differential equa-tions of this form is said to be Hamiltonian with ~x thecanonical coordinate, ~p the canonical momentum, and tthe canonical time regardless of the functional form ofthe Hamiltonian, H(~p, ~x, t). If xi are the components ofthe canonical coordinate ~x and pi are the components ofthe canonical momentum, ~p, then Hamilton’s equationsare dxi/dt = ∂H/∂pi and dpi/dt = −∂H/∂xi. The su-perscripts number the components and are not powers.In Hamiltonian systems with one-and-a-half degrees offreedom ~x and ~p have only one component each.

Magnetic field lines are the trajectories of a one-and-a-half degree-of-freedom Hamiltonian, (Kerst, 1964),(Whiteman, 1977), (Boozer, 1983), (Cary and Littlejohn,1983). A magnetic field line moves through three space,(x, y, z), so the same number of coordinates are involvedas the (p, x, t) space of Hamiltonian mechanics. The onlyproblem is that Cartesian coordinates, (x, y, z), are notthe canonical coordinates for magnetic field lines. To ob-tain canonical coordinates, one must first write the mag-netic field in what is called the symplectic form, (Boozer,1983),

2π ~B = ~∇ψt × ~∇θ + ~∇ϕ× ~∇ψp. (3)

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with θ and ϕ arbitrary coordinates except ~B · ~∇ϕ 6= 0.In practice, ψt is generally the magnetic flux of thetoroidally directed field, θ is a poloidal, and ϕ is atoroidal angle, Figure (1). When ~B · ~∇ϕ 6= 0, pointsin three space can be described using (ψt, θ, ϕ) as coor-dinates, which means points in space are given as thefunction ~x(ψt, θ, ϕ). Using (R,ϕ,Z) cylindrical coordi-nates, Equation (2), points in space are defined by giv-ing R and Z as functions of the (ψt, θ, ϕ) coordinates,~x(ψt, θ, ϕ) = R(ψt, θ, ϕ)R(ϕ) + Z(ψt, θ, ϕ)Z.

The (ψt, θ, ϕ) coordinates are the canonical coordi-nates of the magnetic field lines. The mathematics ofgeneral coordinate systems is described in Appendix (A).The Jacobian, J , of the canonical coordinates is finite,1/J ≡ ~∇ψt · (~∇θ × ~∇ϕ) = 2π ~B · ~∇ϕ. In these coor-dinates magnetic field lines, which are the solutions tod~x/dτ = ~B(~x), are easily shown to have the canonicalform

dψtdϕ

= −∂ψp(ψt, θ, ϕ)∂θ

; (4)

dθ

dϕ=∂ψp(ψt, θ, ϕ)

∂ψt(5)

using Equations (A4) and (3). The canonical momen-tum is the toroidal magnetic flux, ψt =

∫~B · d~aϕ, which

is the magnetic field integrated over the cross section ofa constant ψt torus, Figure (1). The canonical coordi-nate is a poloidal angle, θ, and the canonical time is atoroidal angle, ϕ. The Hamiltonian is the poloidal fluxthat goes down through the hole in a constant ψp torus,ψp = −

∫~B · d~aθ, Figure (1). The field line Hamiltonian,

ψp, can be a function of clock time, t, in addition to ψt,θ, and ϕ, Section (III.C), but clock time is a parameter inthe Hamiltonian description and not one of the canonicalvariables.

The derivation of the symplectic representation of themagnetic field, Equation (3), clarifies the interpretation.Let (r, θ, ϕ) be an arbitrary set of well-behaved coordi-nates. For example, one can let the radial and verticalcoordinates of cylindrical coordinates, Equation (2), beR(r, θ, ϕ) = Ro + r cos θ and Z = −r sin θ, which makesr constant on circular toroidal surfaces. In three dimen-sions a vector can have only three independent compo-nents, so any vector can be written in the form

~A = ~∇g + ψt~∇(θ

2π

)− ψp~∇

( ϕ2π

). (6)

The functions of position, g, ψt, and ψp, represent thethree components of ~A. If ~A is interpreted as the vectorpotential of the magnetic field, then its curl gives thesymplectic representation of the magnetic field, Equation(3).

The theory of one-and-a-half degree-of-freedom Hamil-tonian systems (Lichtenberg and Lieberman, 1992) is thesame as that of magnetic field lines. A magnetic field line

has three fundamentally different types of trajectories.(1) It can close on itself after traversing the torusm timestoroidally, in the ϕ direction, and n times poloidally, inthe θ direction. (2) A field line can come arbitrarilyclose to every point on a toroidal surface as the num-ber of toroidal traversals goes to infinity. (3) A fieldline can come arbitrarily close to every point in a non-zero volume of space as the number of toroidal traversalsgoes to infinity. The first of these possibilities is topo-logically unstable; an arbitrarily small perturbation candestroy the closure of a set of field lines. For example,the closure of a pure toroidal field ~B = (µ0G/2πR)ϕcan be destroyed by an arbitrarily small field in thevertical direction, BZZ. The second possibility, eachfield line coming arbitrarily close to every point on atoroidal surface, is the one desired for magnetic confine-ment. The Kolmagorov-Arnold-Moser (KAM) Theorem,(Kolmogorov, 1954); (Arnold, 1963); (Moser, 1967), saysthis possibility is topologically stable over most of a vol-ume filled by field lines if the perturbation is sufficientlysmall. The statement that a magnetic field lies in toroidalsurfaces means the second possibility except on isolatedsurfaces of zero measure. The third possibility, magneticfield lines coming arbitrarily close to every point in a vol-ume, implies the absence of magnetic confinement in thatregion. Unfortunately this is the generic situation for amagnetic field. Consequently, magnetic confinement de-pends on the formation of very special magnetic fields,fields that have lines that lie on nested surfaces throughthe bulk of the plasma volume.

In regions of magnetic confinement of a near-Maxwellian plasma, the constraint ~B · ~∇p = 0 holds.This constraint greatly simplifies the field line trajecto-ries. In the language of Hamiltonian mechanics the pres-sure, p, is an isolating constant of the motion. Wheneveran isolating constant of the motion exists in a one-and-a-half degree-of-freedom Hamiltonian system, the canon-ical coordinates can be chosen so the Hamiltonian is afunction of only one canonical variable, the canonicalmomentum, rather than three. These canonical coor-dinates are called action-angle variables and are centralto Hamiltonian perturbation theory. For magnetic fieldlines, action-angle coordinates, are known as magneticcoordinates, (Hamada, 1962). Their existence is easilydemonstrated. Since ~B · ~∇p = 0, the magnetic fieldcan be written in arbitrary (p, θ, ϕ) coordinates as ~B =B1~∇p× ~∇θ+B2

~∇ϕ× ~∇p. The constraint that ~∇· ~B = 0implies ∂B1/∂ϕ = −∂B2/∂θ, which means the expansioncoefficients must have the forms 2πB1 = ψ′t(p)(1+∂λ/∂θ)and 2πB2 = ψ′p(p) − ψ′t(p)∂λ/∂ϕ. The magnetic field,therefore, has the form

2π ~B = ~∇ψt × ~∇θm + ~∇ϕ× ~∇ψp(ψt) (7)

where the magnetic poloidal angle θm ≡ θ + λ. In mag-netic coordinates (ψt, θm, ϕ), the field line Hamiltonian,which is the poloidal flux ψp, is a function of the toroidalflux, ψt, alone. The rotational transform ι(ψt), called

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iota, and its reciprocal, the safety factor q(ψt), are de-fined by

ι ≡ 1q≡ dψpdψt

. (8)

In general, the rotational transform is used in the stel-larator and the safety factor in the tokamak literature.We use ι to avoid confusion with the use of q for electricalcharge.

When magnetic coordinates exist, ψp(ψt), the field linetrajectories are said to be integrable and can be givenexplicitly in terms of the initial conditions: ψt = ψ0 andθm = θ0 + ι(ψ0)ϕ. The initial poloidal angle of the fieldlines α ≡ θ0/2π can be used as a coordinate in place ofθm. If this is done

~B = ~∇ψt × ~∇α, (9)

which is called the Clebsch representation. The theory ofthis representation predates (Clebsch, 1859), and in themathematics literature ψt and α are called Euler poten-tials. A history has been given by (Stern, 1970). Themagnetic field lines are orthogonal to both the toroidalflux, ψt, and to α for ~B · ~∇ψt = 0 and ~B · ~∇α = 0,so both are constants of the motion of the Hamiltonian.However, the constancy of α only prevents a field linefrom coming arbitrarily close to all points on a ψt sur-face if ι is the ratio of two integers, ι = n/m, in otherwords a rational number. When iota is a rational num-ber, the field lines close on themselves after m toroidalcircuits, which means in the ϕ direction, and n circuitsin the poloidal, which means θ direction. If ι is a rationalnumber, α is what is called in Hamiltonian mechanics anisolating invariant because it isolates the trajectory so itcan approach only a fraction of the spatial points it couldotherwise reach. However when ι is an irrational number,the trajectory of a single field line comes arbitrarily closeto every point on a constant ψt surface, and one says αis a non-isolating invariant.

The Clebsch representation of the magnetic field,Equation (9), exists in any region of space in which allfield lines pass through a ϕ = 0 plane. Let (r, α) beany set of coordinates in the ϕ = 0 plane, and let thetrajectory of a field line that passes through the point(r, α) be ~x(r, α, `) where ` is the distance along a fieldline from the ϕ = 0 plane. By construction both r

and α are constant along a field line, ~B · ~∇r = 0 and~B · ~∇α = 0, so ~B = f(r, α, `)~∇r × ~∇α. Since ~B is di-vergence free ∂f/∂` = 0. Define ψt so ∂ψt/∂r = f(r, α),then ~B = ~∇ψt × ~∇α.

Various choices are commonly made for the third co-ordinate of Clebsch coordinates. The most commonis the distance along the field lines `. The dual re-lations of general coordinates, Equation (A7), imply~B = ~∇ψt × ~∇α = (1/J )∂~x/∂` where J is the Jaco-bian of (ψt, α, `) coordinates. Since ` is the distancealong the lines, (∂~x/∂`)2 = (JB)2 = 1, which implies

the Jacobian of (ψt, α, `) coordinates is 1/B. Any vec-tor in three dimensions can be written in the covariantform, ~B = Bψt

~∇ψt + Bα~∇α+ B`~∇`. The orthogonalityrelation of general coordinates, Equation (A3), impliesB` ≡ ~B · ∂~x/∂` = B. That is,

~B = B~∇`+ Bψt~∇ψt + Bα~∇α. (10)

Another choice for the third coordinate is the magneticscalar potential φ, which is defined by the indefinite inte-gral φ(ψt, α, `) ≡

∫Bd` along each field line. Using this

coordinate, an arbitrary magnetic field can be written as

~B = ~∇φ+Bψt~∇ψt +Bα~∇α. (11)

The Jacobian of (ψt, α, φ) coordinates is 1/(~∇ψt × ~∇α) ·~∇φ = 1/B2.

Magnetic coordinates are central to the theory of mag-netically confined plasmas because they allow a simplesolution to a differential equation that frequently arises,the magnetic differential equation ~B · ~∇f = g, (Kruskaland Kulsrud, 1958; Newcomb, 1959). In magnetic co-ordinates, Equations (7) and (8), this equation has theform (

∂

∂ϕ+ ι(ψt)

∂

∂θm

)f =

g

~B · ~∇ϕ, (12)

which can be solved algebraically using the Fourier ex-pansion of g/ ~B · ~∇ϕ =

∑γmne

i(nϕ−mθm). The Fourierexpansion coefficients of the function f are fmn =−iγmn/(n− ιm).

A practical example of the use of the magnetic dif-ferential equation is the effect of a magnetic perturba-tion, δ ~B, on a field that has perfect surfaces ~B0. To findthe perturbed magnetic surfaces one solves for a func-tion p = p0 + δp such that ( ~B0 + δ ~B) · ~∇p = 0. Theunperturbed system has perfect surfaces so ~B0 · ~∇p0 = 0.Consequently the first order result is given by the mag-netic differential equation

~B0 · ~∇δp = −δ ~B · ~∇p0. (13)

For simplicity assume the Fourier expansion

δ ~B · ~∇ψt~B0 · ~∇ϕ

= bmn sin(nϕ−mθm), (14)

where ψt is the toroidal flux enclosed by a surface of con-stant p0. Then one obtains a finite solution near the reso-nant rational surface, ψt = ψmn on which ι(ψmn) = n/monly if the gradient of p0 vanishes there. More preciselyas ψt → ψmn, one must let dp0/dψt = (n− ιm)c0 with c0a constant. Then, δp = c0bmncos(nϕ −mθm), and nearthe rational surface p0(ψt) = p0(ψmn)−c0m(dι/dψt)(ψt−ψmn)2/2. The equation for the perturbed surfaces isp0(ψt) + δp = const. Using the identity cos(2x) = 1 −

8

separatrix

X-point

O-point 2d

FIG. 2 The solution to Equation (15) is plotted for an m=3magnetic island in the ϕ = 0 plane. The surface |s| = 1 isthe island separatrix, and the limit |s| → 0 gives the islandO-point. The half-width of the island is δ.

2 sin2(x), the equation for the perturbed surfaces nearthe resonant rational surface is

ψt − ψmn =s

|s|

√4bmnm dιdψt

s2 − sin2

(nϕ−mθm

2

).

(15)The constant s labels the surfaces of constant p, whichare the magnetic surfaces in the perturbed configuration.

Equation (15) for the perturbed magnetic surfaces hastwo topologically distinct regions, which can be studiedby holding ϕ fixed and varying θm, Figure (2). First,the surfaces that have a surface label |s| ≥ 1 cover thefull range of θm. The surfaces with s > 1 and s < −1are different sets of surfaces, which are distorted by theperturbation but not fundamentally changed. However,the surfaces that satisfy 1 > s > −1 cover only a limitedθm range and are said to form a magnetic island. Thetwo signs of s = ±|s| give two parts of the same surfacefor 1 > s > −1.

Each term bmn in a Fourier expansion of a magneticperturbation, Equation (14), produces an island if thereis a surface ψt = ψmn on which ι(ψmn) = n/m, which iscalled a resonant rational surface. The half-width of theisland in toroidal flux is

δmn ≡

√√√√∣∣∣∣∣ 4bmnm dιdψt

∣∣∣∣∣. (16)

Islands can also be calculated using the Hamiltonian forthe field lines. Equation (3) implies the function b ≡ δ ~B ·~∇ψt/ ~B0 · ~∇ϕ = −∂ψp/∂θ. Therefore, if ψp = ψp(ψt) +(ψp(ψt))mn cos(nϕ−mθM ), then bmn = −m(ψp)mn andEquation (16) can be rewritten as

δmn ≡

√√√√∣∣∣∣∣4(ψp)mndιdψt

∣∣∣∣∣. (17)

The Chirikov criterion, (Chirikov, 1979), says that ifthe half-widths of islands from different resonant rationalsurfaces become comparable to the separation betweenthese resonant surfaces, the magnetic field lines becomestochastic, which means a single field line comes arbitrar-ily close to every point in a finite volume of space.

A qualitative understanding of the formation of is-lands and the breakdown of magnetic surfaces can begained from a study of what is called the standardmap, (Chirikov, 1979). Given an initial poloidal an-gle θ0 and radial position Ψ0, the standard map as-sumes that after one toroidal circuit the poloidal an-gle of a field line becomes θ1 = θ0 + Ψ0 and the ra-dial coordinate becomes Ψ1 = Ψ0 + k sin(θ1). Whenk = 0, the radial coordinate is 2π times rotationaltransform, Ψ = 2πι(ψt). The parameter k is propor-tional to the perturbation. A trajectory of the stan-dard map is found by iteration from the original position(θ0,Ψ0). That is, the N th point along the trajectory isat θN = θN−1 + ΨN−1 and ΨN = ΨN−1 + k sin(θN ).The standard map has the essential property for mod-eling the field lines of a divergence free field, which is aunit Jacobian ∂(ΨN , θN )/∂(ΨN−1, θN−1) = 1. If a largecollection of trajectories are followed that were initiallyin a small area (δΨ0)(δθ0), then the area occupied by thetrajectories remains the same forever. For k 6= 0, islandsappear at Ψ = 2πn/m, where n and m are integers, withthe width of the islands scaling as

√k for small k. For

k < 0.9716 · · ·, (Greene, 1979) has shown that trajecto-ries of the standard map can cover only a limited range ofΨ. However for larger values of the perturbation k, sometrajectories cover an unlimited Ψ range, Figure (3). Suchtrajectories are said to be stochastic. The breakdownof the Ψ surfaces with increasing k is analogous to thebreakdown in the magnetic surfaces with an increasingperturbation.

In the presence of a perturbation that produces an ar-bitrarily small island, any function, p0(~x) that satisfies~B0 · ~∇p0 = 0 must have the form dp0/dψt = (n− ιm)c0.This constraint on p0 implies that the opening of an is-land is a singular process in a toroidal equilibrium. In theabsence of an island, a function such as the pressure canhave an non-zero derivative with respect to the toroidalflux, dp0/dψt, but an arbitrarily small island means thederivatives near the island can only be non-zero usingthe helical flux, dp0/dψh 6= 0, where the helical flux isdefined by dψh ≡ (ι − n/m)dψt. The singularities thatarise due to the modification of the pressure gradient bya small resonant perturbation are known as the Glassereffect (Glasser et. al., 1975). The singularities associatedwith the gradient in the electric potential can prevent asmall island from opening in a rotating plasma, Section(V.B.3).

9

FIG. 3 Iterates of the standard map are plotted for k = 0.3 in(a) and for k = 1.1 in (b). Islands arise near rational valuesof Ψ/2π for small values of k, but the trajectory followedby iterating the map remains bounded in Ψ space. For k >0.9716, trajectories exist that cover all values of Ψ. The largefuzzy region in (b) is a single stochastic trajectory found byone set of iterations. The periodicity of the standard map wasused to plot all iterates in the 0 ≤ θ/2π < 1 and 0 ≤ Ψ/2π < 1region.

B. Methods of forming toroidal magnetic surfaces

• A spatial region in which the magnetic field lineslie in toroidal surfaces requires either a net toroidalcurrent flowing within the region or helical shapingof the bounding toroidal surface.

To form magnetic surfaces, a tokamak, Figure (4), usesa net toroidal current in the plasma while the stellarator,Figures ( 5, 6, 7 ), uses helical shaping and in some casesa net toroidal current as well. Both of these plasma con-finement systems are discussed in Section IV. Ignoringthe topologically unstable case of all field lines closingon themselves, magnetic field lines form toroidal surfacesonly when there is both a toroidal and a poloidal mag-netic field. The toroidal magnetic field can be producedsimply by external coils, while the poloidal field is moredifficult.

A pure toroidal magnetic field is produced by a wirecarrying a current G along the Z axis of cylindrical co-

ToroidalFieldCoils

PoloidalFieldCoils

Plasma

Plasma Current

FIG. 4 A tokamak showing the axisymmetric plasma and thecoils necessary to support it. Magnetic surfaces exist in atokamak only when the toroidal plasma current, I, is non-zero.

FIG. 5 The coils and some magnetic field lines of the LargeHelical Device (LHD) are shown. The rotational transform inLHD is due to the wobble of the magnetic field lines producedby the helical coils.

FIG. 6 The W7-X stellarator is shown with its helicallyshaped plasma and the coils required to support it. Mag-netic surfaces exist in a stellarator even in the absence of aplasma.

10

FIG. 7 The surface of the proposed NCSX quasi-axisymmetric stellarator is shown with three toroidal circuitsof a magnetic field line. The rotational transform, ι, is justunder 2/3, so the line almost closes. Despite the strong helicalshaping, the particle drifts are similar to those of an axisym-metric tokamak. Much of the rotational transform in NCSXis due to the torsion of the magnetic axis.

ordinates,

~Bϕ =µ0G

2π~∇ϕ =

µ0G

2πϕ

R. (18)

An infinite straight wire is not a practical coil set, butexactly the same magnetic field is produced by any setof coils that surround the toroidal region of interest, runin a constant-ϕ plane, and are axisymmetric (no ϕ de-pendence). The toroidal field due to external coils isgiven by a multi-valued scalar potential, ~B = ~∇φ, thatobeys Laplace’s equation, ∇2φ = 0, which has the solu-tion φ = µ0Gϕ/2π. It is impossible to make the toroidalfield coils perfectly axisymmetric because that would pre-clude any direct access to the plasma. The effect of Nseparate toroidal field coils can be approximated by as-suming the coils have N vertical legs located at an outerradius Rc with the current returning along the z axis,R = 0. The magnetic potential becomes

φ =µ0G

2π

(ϕ+

(R

Rc

)N cosNϕN

), (19)

which gives a sinusoidal ripple in strength of the toroidalfield. In order to make the ripple sufficiently small, oneneeds the separation between the toroidal field coils tobe significantly less than the minor radius of the plasma,which means N must be somewhat larger than 2π/εcwith εc ≡ a/Rc the inverse aspect ratio of the toroidalsurface that forms the plasma edge at r = a. The phys-ical reasons that the toroidal ripple must be limited arediscussed in Section (VI.E) on particle trajectories andtransport.

The production of the poloidal, θ directed, magneticfield is more difficult. Of course a poloidal field can beproduced by a coil that encircles the Z axis at a radiusRo and carries a current I. Close to the coil, r/Ro 1,the poloidal field is Bθ = µ0I/2πr, and the poloidal fluxobeys ∂ψp/∂r = 2πRoBθ(r). The toroidal magnetic flux

is approximately ψt = πr2Bϕ, so the rotational trans-form, ι ≡ dψp

dψt, is approximated by

ι ≈ RoBθrBϕ

. (20)

A circular coil produces magnetic surfaces, but this con-figuration by itself is not suitable for confining fusionplasmas. Connections are needed between the coil andthe outside world so the heat produced by the slowingof the fusion neutrons within the coil structure can beremoved. Such connections would have have to passthrough the hot fusing plasma and would destroy its con-finement.

Although a circular coil cannot be used to producethe poloidal magnetic field, much the same effect can beproduced by a current in the plasma that is parallel tothe magnetic field, which is called the net plasma cur-rent ~jnet = k(~x) ~B/µ0. This method was suggested by I.Tamm and A. Sakharov in the Soviet Union in the early1950’s and is used to form the poloidal field in tokamaks.If the plasma pressure gradient is negligibly small, thenet current is the total current. Such equilibria are calledforce free and obey the equation

~∇× ~B = k(~x) ~B. (21)

Taking the divergence of both sides of this equation, onefinds that k(~x) is constrained to be constant along themagnetic field,

~B · ~∇k = 0. (22)

This implies a gradient in k ≡ µ0j||/B, with j|| ≡ ~j · ~B/B,can only occur in regions of good magnetic surfaces wherek(ψt) is a function of the enclosed toroidal flux. If themagnetic surfaces are essentially circular then the rota-tional transform is given by Equation (20) with Bθ =µ0I/2πr and I the current enclosed by a surface of ra-dius r. Usually k can be approximated by a constant forsufficiently small r, the rotational transform in that re-gion is ι = Rok/2. For radii r larger than the currentchannel the rotational transform, ι ∝ 1/r2.

Stellarators (Spitzer, 1958) have magnetic surfaces anda non-zero rotational transform even in the absence of aplasma. Stellarator magnetic surfaces must have helicalshaping. The existence of magnetic surfaces without anyenclosed currents is demonstrated starting with the ax-isymmetric magnetic surfaces formed by a coil encirclingthe Z axis at a radius Ro, which is designed so the ro-tational transform ι is an integer Np in the center of theregion of interest. If this system is modified by a mag-netic perturbation in which bm=1,n=Np

is non-zero, anisland is formed, Equation (16). Since the magnetic sur-faces inside an island do not cover the full range of theta,Figure (2), room exists for connections and supports forthe coil. The magnetic surfaces inside the island fill avolume and have no current on them. The integer Np is

11

the number of periods of this type of stellarator, whichis called a Heliac, (Boozer et al., 1983).

The poloidal field in a stellarator can be produced intwo ways in the absence of a plasma. The first wayuses helical wobbles of the magnetic field lines that aredriven by a helical currents in coils (Spitzer, 1958). Therotational transform in the Large Helical Device, (Ya-mada et al., 2001), is produced entirely by the field linewobble produced by helical coils, Figure (5). The mag-netic field that is produced by the helical wobbles acts,when averaged over the wobbles, as if there were a force-free current that fills the space around the coils that be-comes exponentially small with increasing distance fromthe coils. To understand this, consider a weak heli-cal perturbation to a spatially constant magnetic field,~B/B0 = z + ~∇φ with φ(x, y, z) = ∆(x) cos(kzz − kyy) inCartesian coordinates. Since ∇2φ must be zero, the dis-tance ∆(x) ∝ exp(kx) with k2 = k2

y + k2z . The magnetic

field lines are given by dx/dz = Bx/Bz ' k∆ cos(· · ·) orx ' x0 + (k∆/kz) sin(· · ·), and dy/dz = By/Bz. That is,dy/dz = ky∆ sin(· · ·)/1−kz∆ sin(· · ·), which when ex-panded to keep the important second order terms in ∆ isdy/dz ' ky∆ sin(· · ·)+(kyk2/kz)+ (kykz)∆2 sin2(· · ·).The magnetic field lines have an oscillation in the y di-rection, δy = (ky/kz)∆ sin(· · ·), but also a systematicdrift in that direction, which is equivalent to a y-directedmagnetic field

〈By〉B0

=12kykz

(1 +

k2z

k2

)(k∆)2. (23)

The second order terms in the dx/dz equation give nosystematic drift. Consequently, field lines wobble in andout of a constant-x surface but have a drift in the y-direction. The rotational transform produced per periodof the helix by this drift is ιp ≡ (kz/ky)〈By〉/Bz, or

ιp =12

(1 +

k2z

k2

)(k∆)2. (24)

There is an upper limit on the transform per period,ιp <∼ 0.2, since k∆ must be small compared to one. Theeffective force-free current that fills the space around thehelical coil is µ0

~jeff = 2k〈By〉z.The second way to produce a poloidal field without

a plasma is through torsion in the magnetic axis. Themagnetic axis is the field line at ψt = 0 and is the axisof the poloidal angle. The torsion τ of a curve measuresthe extent to which the curve fails to lie flat on a plane.The rotational transform due to torsion is

ι = 〈τ〉 L2π

(25)

with 〈τ〉 the torsion of the magnetic axis averaged alongits length L. Torsion was used in the early 1950’s toproduce the poloidal field in Spitzer’s Figure-8 stellara-tors, (Spitzer, 1958). The optimization of the particletrajectories in as stellarator, Section (VI.E.1), requiresthe use of torsion. Indeed, torsion gives a major part

of the rotational transform in both the Wendelstein-7X(W7-X), Figure (6), (Beidler et al., 1990), and the Na-tional Compact Stellarator Expermiment (NCSX), Fig-ure (7), (Zarnstorff et al., 2001), stellarator designs. Thederivation of the relation between the torsion and therotational transform uses the Frenet formulae (Math-ews and Walker, 1964) to analyze the behavior of fieldlines near the magnetic axis, ~x0(`) with ` the distancealong the axis. The derivative of a curve is its tan-gent, b0 ≡ d~x0/d`, which is a unit vector along the mag-netic field line that forms the axis. The derivative ofthe tangent is db0/d` = κκ with κκ the curvature ofthe axis. The derivative of the curvature unit vector isdκ/d` = −(κb0 + τ τ) with τ τ the torsion. The derivativeof the torsion unit vector is dτ/d` = τ κ. The Frenet unitvectors are mutually orthogonal and satisfy b0 × κ = τ .(Mercier, 1964) used the Frenet unit vectors to estab-lish a coordinate system near an arbitrary magnetic axis.We will simplify his analysis by making the magneticsurfaces circular near the axis, which excludes rotationaltransform from helical wobbles. That is we choose coor-dinates ~x(r, θ, `) = ~x0(`) + r cos θκ + r sin θτ . Using themethods of general coordinates given in Appedix (A), onefinds ~∇r = cos θκ+ sin θτ , ~∇θ = (− sin θκ+ cos θτ)/r +τ b0/(1− κr cos θ), and ~∇` = b0/(1− κr cos θ). The mag-netic field in the absence of plasma currents has the form~B = (µ0G/2π)~∇ϕp with G a constant and with the po-tential ϕp, which can be interpreted as a toroidal angle,obeying ∇2ϕp ≡ ~∇ · ~∇ϕp = 0. If the magnetic strengthis constant along the magnetic axis, then for r → 0 thetoroidal angle ϕp = 2π(`+κτr3 sin θ+ · · ·)/L with L thelength of the axis. For r → 0, the magnetic field lineequations are dθ/dϕp = ~B · ~∇θ/ ~B · ~∇ϕp = τL/2π anddr/dϕp = ~B · ~∇r/ ~B · ~∇ϕp = 0. Consequently, the rota-tional transform near the axis is ι = 〈τ〉L/2π with theaxis average of the torsion 〈τ〉 ≡

∮τd`/

∮d`.

The magnetic configuration due to the coils alone iscalled the vacuum configuration. It is in principle easyto design vacuum configurations that have magnetic sur-faces of any desired shape, but unless the surfaces havehelical shaping the rotational transform will be zero. Todesign a vacuum configuration, first assume there is anaxisymmetric toroidal field, ~Bϕ = µ0Gϕ/2πR, whichproduces the toroidal magnetic flux. One then definesthe desired shape of an outermost magnetic surface using(R,ϕ,Z) cylindrical coordinates, ~xs = Rs(θ, ϕ)R(ϕ) +Zs(θ, ϕ)Z, by giving the appropriate Fourier series rep-resentation of Rs and Zs. The unit normal to this surfaceis n ∝ ∂ ~xs

∂θ ×∂ ~xs

∂ϕ . Coils just outside the plasma can inprinciple produce any desired normal magnetic field onthe surface ~xs. If the coils are designed so they produce anormal field equal but opposite in sign to n · ~Bϕ, then thesurface ~xs will be a magnetic surface since the magneticfield lines do not cross it, n · ~B = 0. When vacuum mag-netic fields produce one magnetic surface, the volumeenclosed by that surface is generally dominated by re-

12

gions of good surfaces rather than islands and stochasticregions. The practical limitations on designing magneticfield configurations are discussed in Section (V.D).

C. Evolution of magnetic field lines

• Clock time is a parameter in the Hamiltonian de-scription of evolving magnetic field lines. Foran evolving magnetic field, the Hamiltonian,ψp(ψt, θ, ϕ, t), as well as the coordinate transfor-mation function, ~x(ψt, θ, ϕ, t), depend on time aswell as the canonical coordinates.

• A magnetic field evolves ideally (without a topologychange) if the field line Hamiltonian can be madetime independent by an appropriate choice of thecoordinate transformation function ~x(ψt, θ, ϕ, t).This can always be done in a sufficiently localizedspatial region. The mathematical condition for anideal evolution is that a function s(~x, t) exist suchthat ~B · ~∇s = − ~E · ~B, where the electric field is de-termined by Faraday’s law, ∂ ~B/∂t = −~∇× ~E. Mag-netic field lines evolve ideally under more generalconditions than the tying together of the plasmaand the field.

• The self-entanglement of magnetic field lines ismeasured by the magnetic helicity, K ≡

∫~A· ~Bd3x.

A spiky current profile causes a rapid dissipation ofenergy relative to magnetic helicity. If the evolu-tion of a magnetic field is rapid, then it must be atconstant helicity

Ideally a fusion plasma would be in a steady state, butthe theory of the evolution of magnetic fields is importantfor finding the conditions for: (1) establishing the fieldconfiguration, (2) maintaining the magnetic fields, and(3) finding the conditions under which rapid changes inthe magnetic field can occur. The evolution of the mag-netic field means the evolution of the magnetic field lines.The equations for the evolution of a magnetic field linesare also of interest for the evolution of other intrinsicallydivergence-free fields such as the vorticity field of fluidmechanics.

The topological properties of the magnetic field linesare independent of time if the canonical coordinates(ψt, θ, ϕ) can evolve so the field line Hamiltonian doesnot change, (∂ψp/∂t)c = 0. The subscript c means thepartial derivative is performed at a fixed point in canon-ical coordinates. The conclusion follows from two state-ments. First, the trajectories of magnetic field lines incanonical coordinates are determined by the field lineHamiltonian ψp(ψt, θ, ϕ, t) alone, Equations (4) and (5).Second, topological properties are unchanged by a contin-uous temporal variation in a coordinate transformationfunction, ~x(ψt, θ, ϕ, t). The topological properties of fieldlines are determined by the field line Hamiltonian alone,

and the topological properties cannot change unless thefield line Hamiltonian changes.

The evolution equation for the magnetic field lineHamiltonian is obtained from the evolution of the vectorpotential in canonical coordinates. Using only the the-ory of general coordinates, Appendix (A), one can write(Boozer, 1992) the time derivative of an arbitrary vec-tor ~A = ψt~∇(θ/2π) − ψp~∇(ϕ/2π) + ~∇g in a coordinatesystem defined by the function ~x(ψt, θ, ϕ, t) as, Equation(A19),(

∂ ~A

∂t

)~x

= −(∂ψp∂t

)c

~∇ ϕ

2π+ ~u× ~B + ~∇s, (26)

where we have let ~B ≡ ~∇× ~A and introduced the velocityof a fixed point in canonical coordinates

~u ≡ ∂~x(ψt, θ, ϕ, t)∂t

. (27)

The subscript ~x means a fixed spatial point, and the sub-script c means a fixed point in the canonical (ψt, θ, ϕ)coordinates. The function s = (∂g/∂t)c− ~A ·~u, Equation(A20), is the freedom of infinitesimal canonical transfor-mations, and in the language of Hamiltonian mechan-ics, s is the generating function for infinitesimal canon-ical transformations. Even when the vector potential isindependent of time,

(∂ ~A/∂t

)~x

= 0, the canonical co-ordinates can change, ~u 6= 0, using the freedom of thefunction s.

All of the topological properties of the field lines are in-dependent of time if canonical coordinates exist in which(∂ψp/∂t)c = 0. Equation (26) implies such coordinatesexist if and if only a function s(~x, t) exists such that

~B · ~∇s = ~B · (∂ ~A/∂t)~x. (28)

A magnetic field that is embedded in a perfectly con-ducting fluid evolves conserving the topology of its fieldlines. A Lorentz transformation in the non-relativisticlimit implies the electric field in a conductor moving withvelocity ~v is ~Emov = ~E + ~v × ~B. In simple conductor~Emov =

↔η ·~j with

↔η the resistivity, and Ohm’s law be-

comes

~E + ~v × ~B =↔η ·~j. (29)

The electric field is related to the vector and scalar poten-tials by ~E = −∂A/∂t − ~∇Φ, so Ohm’s law is equivalentto (

∂ ~A

∂t

)~x

= ~v × ~B − ~∇Φ−↔η ·~j, (30)

which with Equation (26) implies(∂ψp∂t

)c

~∇ ϕ

2π= (~u− ~v)× ~B + ~∇(s+ Φ)+

↔η ·~j. (31)

13

If ~B · (↔η ·~j) = 0, then Equation (31) says (∂ψp/∂t)c = 0

with the choice s+ Φ = 0.In an ideal fluid the topology of the field lines of the

vorticity, ~ω ≡ ~∇ × ~v, are conserved. The equivalent ofOhm’s law for the vorticity is the Navier-Stokes equation,

∂~v

∂t+ ~v · ~∇~v = −~∇w(p)− ν ~∇× ~ω. (32)

This form of the equation assumes the density ρ is afunction of the pressure alone with dw(p) ≡ dp/ρ(p). Thekinematic viscosity is ν. The term ~v · ~∇~v = −~v × ω +~∇(v2/2), so the Navier-Stokes equation can be written ina form analogous to Equation (30),

∂~v

∂t= ~v × ~ω − ~∇

(w +

12v2

)− ν ~∇× ~ω. (33)

The Hamiltonian of the vorticity field lines can be madetime independent if the viscosity ν = 0.

A magnetic field evolution obeys two distinct conserva-tion laws when the resistivity vanishes,

↔η= 0: (1) the ty-

ing together of the toroidal and poloidal magnetic fluxes,and (2) the tying of the magnetic field lines to the fluid,~u = ~v. A distinction between these two laws is rarelymade in the literature, but the first holds under moregeneral conditions. In plasmas the resistivity tensor isdiagonal but with distinct values along, η||, and across,η⊥, the magnetic field. In a quiescent plasma, η⊥ is abouttwice η||, but the micro-turbulence that is present in mostconfined plasmas enhances η⊥ by several orders of mag-nitude while leaving η|| essentially unchanged, Section(VI.F). The slippage of the poloidal flux relative to thetoroidal flux, which changes the magnetic field line topol-ogy, is determined by the parallel resistivity, η||, alone.The flow of the plasma relative to the magnetic field lines,~v−~u, depends on the perpendicular resistivity, η⊥, and ingeneral on the parallel resistivity as well. The componentof ~u parallel to the magnetic field does not appear in theequation for the evolution of the vector potential, Equa-tion (26), and can be chosen arbitrarily. When

↔η= 0, the

choice ~u · ~B = ~v · ~B simplifies equations, so this choice ismade.

The distinction between the two conservation laws formagnetic evolution is even clearer in a Type II super-conductor with a melted flux lattice (Huebener, 1979).The technically important superconductors are Type II,which means the magnetic field can penetrate into thematerial, which lowers the magnetic field energy, but themagnetic field lines must lie in narrow flux tubes. Theseflux tubes can form a lattice, which holds the tubes inplace, and in the technically important superconductorsthe flux lattice is rigid. When the lattice is rigid, thesuperconductor has zero resistivity. However, the fluxlattice can melt, which allows the flux tubes to movethrough the superconductor with a velocity, ~u − ~v, pro-portional to the applied force. This motion is equivalentto a perpendicular resistivity, η⊥. Therefore, a Type II

superconductor with a melted flux lattice has a tensorresistivity with η|| = 0 but η⊥ non-zero. The first con-servation law for magnetic evolution, which is the tyingtogether of the two fluxes, holds rigorously in a super-conductor since η|| = 0. However, if the flux lattice ofa superconductor melts the perpendicular resistivity be-comes large, and the second conservation law, the tyingof the field lines to the superconductor, is not even ap-proximately valid.

The evolution of a given magnetic field ~B(~x, t) is calledideal if its evolution is consistent with the field beingembedded in a perfectly conducting fluid. This is alwaystrue in a sufficiently small spatial region even if the field isin a vacuum or a resistive fluid, (Boozer, 2002). To provethis, note that given ~B(~x, t) one can construct the vectorpotential ~A(~x, t) up to the gradient of an arbitrary gaugefunction g(~x, t). The freedom of gauge means that thereis no constraint on the function s in the general evolutionequation for ~A. A function s(~x, t) always exists locallythat satisfies the magnetic differential equation ~B · ~∇s =~B · (∂ ~A/∂t)~x, Equation (28). Although s(~x, t) alwaysexists in a sufficiently small spatial region, a single-valueds(~x, t) may not exist everywhere along a magnetic fieldline.

There are several ways a solution s(~x, t) to the equationthat guarantees an ideal evolution, ~B · ~∇s = ~B ·(∂ ~A/∂t)~x,may fail to exist. First, the loop voltage V on a magneticsurface may not be zero. The loop voltage on a magneticsurface is defined as

V (ψt, t) ≡ − ∂

∂ψt

∫~B ·

(∂ ~A

∂t

)~x

d3x, (34)

where the volume integral is performed over the regionenclosed by the constant ψt surface. If the surface loopvoltage is non-zero, then Gauss’ theorem implies that nosolution s(~x, t) exists. In magnetic coordinates d3x =dψtdθmdϕ/(2π ~B · ~∇ϕ), and one can easily show usingEquation (26) that the surface loop voltage is

V =∂ψp(ψt, t)

∂t. (35)

Since ∂ ~A/∂t = −( ~E + ~∇Φ) and ~B · ~∇Φ = ~∇ · (Φ ~B),the surface loop voltage can be written as V (ψt, t) =∫

( ~E· ~B/ ~B·~∇ϕ)dθdϕ. Although the field line Hamiltonianevolves when the surface loop voltage is non-zero, themagnetic surfaces need not split into islands.

A solution s(~x, t) to the equation ~B · ~∇s = ~B ·(∂ ~A/∂t)~xcan also fail to exist along individual field lines and leadto a rapid breakup of magnetic surfaces. The loop volt-age of a closed magnetic field line is V` ≡ −

∮( ~B/B) ·

(∂ ~A/∂t)d` where ` is the distance along the field line.If V` is non-zero then s(~x, t) cannot exist on that fieldline. In a magnetic configuration with magnetic surfaces,closed field lines occur only at rational surfaces whereι = n/m. If the loop voltage along the lines of a rational

14

surface differ, then the magnetic Hamiltonian will de-velop a term that goes as ei(nϕ−mθm), and an island willopen. If V` is the same on each field line of a rationalsurface, then the surface loop voltage V is V` divided bythe number of toroidal circuits a field line makes beforeclosing on itself.

The important phenomenon of fast reconnection arisesif the surface loop voltage V is negligible, and the vio-lation of the constraint equation ~B · ~∇s = ~B · (∂ ~A/∂t)~xoccurs only in a region of infinitesimal volume about therational surfaces, ι = n/m. In this case, the currentsthat arise in a highly conducting fluid, η → 0, to preservethe topology are surface currents, which means they aredelta functions—non-zero only on the surface but infinitethere. Such currents lead to rapid dissipation and thesplitting of the rational surface to form an island. Per-turbations can open islands on a very short time scalecompared to a global resistive time, τη ≡ (µ0/η)a2, of aplasma of radius a. The actual time scale in a plasmais subtle question, a question on which recent progresshas been made. See (Rogers et al., 2001) and reviews by(Bhattacharjee et al., 2001) and by (Priest and Forbes,2000).

In astrophysical and space plasmas, as well as in manyfluid problems, the magnetic or vorticity lines enter andleave the volume of interest. In these problems the fieldline Hamiltonian may be forced to change if the inte-gral

∫( ~B/B) · (∂ ~A/∂t)d` is non-zero when calculated be-

tween boundary interceptions but with a jump in s(~x, t)inconsistent with the boundary conditions. In systems inwhich field lines enter and leave the volume of interest,another possibility arises if neighboring field lines sepa-rate from each other exponentially with distance alongthe lines. Such separations occur, as in all one-and-a-half degree of freedom Hamiltonian systems, in regionsof stochastic field lines. In this case a solution s(~x, t)may exist but have poor analytic properties due to ex-ponentially large gradients across the field lines. Nev-ertheless, in systems in which the magnetic field linesenter and leave the volume of interest, it is more difficultthan in systems with the two periodic directions to focusthe breaking of the conservation of the Hamiltonian intoregions of infinitesimal spatial extent. Such focusing isrequired for fast reconnection (Boozer, 2002).

The rapidity of the evolution of magnetic fields islimited by the properties of magnetic helicity (Woltjer,1958). Magnetic helicity is defined as

K(t) ≡∫

~A · ~Bd3x. (36)

In a system with perfect magnetic surfaces, the helic-ity is K =

∫(ψtι − ψp)dψt with ι(ψt, t) ≡ ∂ψp/∂ψt the

rotational transform. The helicity is a measure of thetopological entanglement of the magnetic field lines.

The importance of magnetic helicity comes from itsrate of dissipation, 2

∫~B · ~Ed3x. If the current density

in a plasma has a large spatial variation, as it does inturbulent situations, the rate of loss of magnetic energy,

∫~j · ~Ed3x, is rapid in comparison to the rate of helic-

ity dissipation. It is relatively easy to change the mag-netic energy quickly compared to the resistive time scale,τη ≡ (µ0/η)a2, of a plasma of radius a. However, ifthe magnetic energy changes rapidly compared to τη, thechange must be at an essentially constant helicity.

The time derivative of the helicity can be placed in aconvenient form using ~E = −∂ ~A/∂t− ~∇Φ,

dK

dt= −2

∫~B · ~Ed3x+ Ses + Sin. (37)

The volumetric term, 2∫~B · ~Ed3x, is the helicity dissi-

pation. The two surface terms are the external sourcesof helicity: the electrostatic source

Ses ≡ −2∮

Φ ~B · d~a, (38)

and the inductive source

Sin ≡ −∮

~A× ∂ ~A

∂t· d~a. (39)

If the bounding surface is a perfect conductor, the ex-ternal sources of helicity vanish. When the boundingsurface is a magnetic surface, the inductive source isSin = ψtdψp/dt − ψpdψt/dt. The poloidal and toroidalloop voltages are defined as Vp ≡

∮~E · (∂~x/∂θm)dθm and

Vt ≡∮~E · (∂~x/∂ϕ)dϕ, so Sin = Vtψt + Vpψp, the loop

voltages times the fluxes. The toroidal loop voltage isequal to the surface loop voltage, V , if the poloidal loopvoltage, Vp = −∂ψt/∂t, is zero.

The magnetic energy, WB =∫

(B2/2µ0)d3x, has thetime derivative dWB/dt = −

∫~j · ~Ed3x. Using the

standard Ohm’s law, Equation (29) with↔η having dis-

tinct parallel and perpendicular components, −dWB/dtis the sum of two terms: the dissipative loss of energy∫

(η||j2|| + η⊥j2⊥)d3x and a non-dissipative transfer of en-

ergy between the magnetic field and the fluid in whichit is embeded,

∫~v · (~j × ~B)d3x. Retaining only the

dissipative term, −dWB/dt ≥∫η||j

2||d

3x. In contrast,

the volumetric dissipation of helicity is 2∫~B · ~Ed3x =

2∫η||~j · ~Bd3x.

The relative rates of dissipation of energy and helicity,

−dWB

dt≥ 1∫

η||B2d3x

(12dK

dt

)2

, (40)

can be demonstrated (Berger, 1984) using the Schwarzinequality. This inequality says the average of the squareof a function is at least as large as the square of theaverage of the function. The appropriate average forthe relative dissipation rates is the η average, whichis defined as 〈f〉η ≡

∫(η||B2fd3x)/

∫(η||B2d3x). The

rate of magnetic energy dissipation satisfies −dWB/dt ≥(∫η||B

2d3x)⟨(j||/B)2

⟩η

while the rate of helicity dissi-pation satisfies dK/dt = 2(

∫η||B

2d3x)⟨j||/B

⟩η. The

15

Schwarz inequality,⟨(j||/B)2

⟩η≥⟨j||/B

⟩2η, then implies

Equation (40). The greater the spatial variation in j||/B,the stronger the Schwarz inequality becomes. In Section(V.B.1), we will find that unstable, or more generally tur-bulent, plasmas develop very spiky, Dirac delta functionlike, spatial distributions of j||/B. In such situations, thedissipation of the magnetic energy is extremely rapid incomparison to the dissipation of helicity.

Magnetic energy can be dissipated rapidly comparedto helicity, so highly unstable plasmas evolve to mini-mize their energy for fixed helicity. As Woltjer showed,this evolution relaxes the current density to the form~j = (k/µ0) ~B, where k is a spatial constant. The prop-erties of the state of minimum energy with fixed helicityare found by varying ~A in

∫(B2/µ0 − k ~A · ~B)d3x. The

constant k, which is called a Lagrange multiplier, is cho-sen to make the helicity after the minimization equal toits initial value. The terms δ ~B = ~∇× δ ~A are integratedby parts, and the boundary terms

∮~B × δ ~A · d~a can be

ignored if the boundary is a perfect conductor. One finds~∇× ~B = k ~B.

The physical importance of helicity conservation wasdemonstrated by (Taylor, 1974). He showed that tur-bulent periods in the reversed field pinch (RFP) plasmaconfinement device (Prager, 1999) lead to flattened cur-rent profiles with a more quiescent plasma. Since Taylor’simportant work, the relaxation of the current profile toform a more quiescent plasma has been known as a Tay-lor relaxation. Taylor’s work demonstrated that helicityconservation should be a central element of any theoryof rapidly evolving magnetic fields.

Spiky current profiles cause a rapid loss of magneticenergy,

∫~j · ~Ed3x. On the time scale of energy dis-

sipation, the magnetic helicity can be transported butnot dissipated. The absence of helicity dissipation im-plies 2 ~E · ~B must be the divergence of a flux, the helic-ity flux ~Fh with 2 ~E · ~B = ~∇ · ~Fh. Energy dissipation,which in the Schwarz inequality argument takes placethrough the parallel current

∫(j||/B) ~B · ~Ed3x, implies

−∫~Fh · ~∇(j||/B)d3x must be positive. This condition is

satisfied if ~Fh = −λh~∇(j||/B). The positive coefficientλh is called the hyper-resistivity. The gradient of j||/B isa source of free energy, which can drive instabilities muchas the pressure gradient can, Section (V.B.1).

If a system has a large source of free energy other thanthe magnetic field energy, the helicity flux, ~Fh, can haveterms that add energy to the magnetic field. Such termsare needed in a magnetic dynamo, where magnetic fieldsare generated by taking energy from another source, suchas a fluid flow. Taylor relaxation in the reversed fieldpinch demonstrates the existence of a dissipative termin the helicity flux, such as −λh~∇(j||/B), but not a termthat can add energy to the magnetic field, which is neededfor a dynamo. The role of helicity and its conservation inlimiting the forms of magnetic dynamo theories remainscontroversial.

The external sources of helicity, the electrostatic Sesand the inductive Sin, are important for creating theplasma currents that are needed for magnetic surfacesin axisymmetric devices. The inductive source is usuallya toroidal loop voltage Vt supplied by varying the mag-netic flux in a solenoid that goes through the central holeof the torus while an essentially constant toroidal flux issupplied by toroidal field coils. If the loop voltage Vt issmall, the plasma is relatively quiescent, and this is themode of operation of the tokamak. In the reversed fieldpinch, the loop voltage is made sufficiently large that theplasma has periods of turbulence, which relax the cur-rent profile to form transiently quiescent states. Electro-static helicity injection, Ses, requires magnetic field linesthat penetrate conducting plates, or electrodes, outsideof the plasma. A given magnetic field line goes from aplate held at one voltage to a plate held at a differentvoltage with the voltage difference, Φh, driving a currentalong the magnetic field line. If Φh is small the magneticfield structure is affected only slightly by the presenceof the voltage. When Φh is large, the currents drivenby the electric potential difference produce a field thatis large compared to the field Bh that penetrates theplates. When Φh is large, unstable plasma states can becreated that undergo a Taylor relaxation to plasmas thatcan have closed magnetic surfaces (Jarboe et al., 1983;Raman et al., 2003).

IV. CONFINEMENT SYSTEMS

• The two most prominent magnetic configurationsfor confining plasmas are tokamaks and stellara-tors. An ideal tokamak is axisymmetric and uses anet toroidal current to produce magnetic surfaces.The tokamak has the most extensive data base ofall magnetic confinement systems and has been thebasis for a number of proposals for experimentsin which deuterium and tritium are burned underconditions similar to those of a fusion power plant.Stellarators use helical shaping to produce at leastpart of the rotational transform.

• The primary control a machine designer has overthe performance of a confinement system is theshape of the outer-most plasma surface. Aboutfifty properties of the plasma shape can be con-trolled but only about four of these are consistentwith axisymmetry: aspect ratio, ellipticity, trian-gularity, and squareness.

In the world research effort on magnetically confinedplasmas, the tokamak, Figure (4), and the stellarator,Figures ( 5, 6, 7 ), are by far the largest programs. Toka-mak plasmas are axisymmetric and carry a net toroidalcurrent in order to form the magnetic surfaces. Stellara-tor plasmas have the form of a torus with helical shap-ing, which produces some or all of the rotational trans-form ι. Of the two the tokamak has the more extensive

16

FIG. 8 The cross section of the NSTX spherical torus (ST) isshown (Peng, 2000). The plasma cross-sectional shape resem-bles that of all modern tokamaks and ST devices. The fieldlines that go from the bottom of the plasma to the wall forma divertor. Divertors are discussed in Section (VII).

data base and has been the basis for a number of pro-posals for experiments in which deuterium and tritiumare burned under conditions similar to those of a fusionpower plant. In addition, the tokamak has a variant, thespherical torus or spherical tokamak, which is called anST, Figure (8). The ST is like a tokamak at a very tightaspect ratio, εa ≡ a/Ro much closer to unity, which hasphysics advantages (Peng, 2000; Sykes, 2001) as well as asmaller unit size for fusion systems. Tokamaks have de-sirable physics properties only at a tight aspect ratio, butstellarators with desirable physics properties tend to beeasier to design the larger the aspect ratio, Ro/a. The he-lical shaping of the stellarator allows one to design aroundcertain issues of the tokamak and ST. Many of these is-sues are associated with the maintenance and stability ofthe net plasma current, issues that can be avoided in thestellarator.

In addition to the tokamak and stellarator, many othermagnetic configurations are being actively pursued in theworld fusion program, though at a much lower financiallevel. (Sheffield, 1994) reviewed many of these configu-rations. The most prominent are the reversed field pinch(RFP) (Ortolani and Schnack, 1993; Prager, 1999), thespheromak (Bellan, 2000), and the magnetic dipole (Gar-nier, Kesner, and Mauel, 1999; Kesner et al., 2001).

The largest tokamaks have been the Joint EuropeanTorus (JET), which is at Culham, England, (Keilhackeret al., 2001, 1999); the Tokamak Fusion Test Reactor,which was at Princeton, NJ, (Hawryluk et al., 1998); andthe Japanese Tokamak JT-60U, (Kamada et al., 1999).

The largest tokamak that is operating in the U.S. is theDIII-D tokamak at General Atomics, which is investi-gating ways of making tokamak plasmas that are moresuited for high pressure, steady-state operation (Chanet al., 2000; Petty et al., 2000). Both the JET and theTFTR tokamaks produced power above the 10 megawattlevel by fusing deuterium and tritium (Hawryluk, 1998).However as expected from the time of their design in theearly 1970’s, both tokamaks required a high level of exter-nally injected power, comparable to or greater than thefusion power, in order to maintain the required plasmatemperature, and neither was, therefore, what is meantby a burning plasma experiment.

The largest stellarator (Yamada et al., 2001) is theJapanese Large Helical Device (LHD), Figue (5. A stel-larator of comparable scale Wendelstein-7X (W7-X) isbeing constructed (Beidler et al., 1990) in Greifswald,Germany. The W7-X stellarator, Figure (6), has es-sentially no net current and an unusually small currentflowing parallel to the magnetic field lines. This impliesthe shape of the magnetic surfaces is similar with andwithout plasma. The W7-X design has many innova-tive features of engineering and physics, which have beenstudied in the smaller W7-AS stellarator (McCormicket al., 2003) at Garching, Germany. The National Com-pact Stellarator Experiment (NCSX), Figure (7), whichis planned for construction at the Princeton PlasmasPhysics Laboratory, (Zarnstorff et al., 2001) is quasi-axisymmetric, Section (VI.E.1). Quasi-axisymmetry im-plies the particle orbits closely resemble those of a toka-mak even though a large fraction of the rotational trans-form comes from helical shaping. Quasi-axisymmtricstellarators have the feature that they can be designedwith an arbitrarily small level of helical shaping, so thereis a continuous connection between quasi-axisymmetricstellarator designs and tokamaks. The potential impor-tance is that quasisymmetry allows a minimal modifica-tion of the tokamak while giving the freedom to designaround physics issues that could potentialy affect the useof tokamak for practical fusion power.

The primary control that a machine designer has onthe plasma properties is the shape of the plasma. Asshown is Section (V.A), a plasma equilibrium is speci-fied (Bauer, Betancourt, and Garabedian, 1984) by giv-ing its toroidal flux content, which is the ψt enclosedby the plasma surface; the rotational transform profileι(ψt); the pressure profile p(ψt); and the plasma shape~xs = Rs(θ, ϕ)R(ϕ) + Zs(θ, ϕ)Z. The optimal shape forthe plasma surface in both tokamaks and stellarators isfar from a simple torus with a circular cross section, Rs =Ro+a cos θ and Zs = −a sin θ. Since a tokamak design isby definition axisymmetric, both Rs and Zs can dependonly on θ. A frequently used expression for the shape ofa tokamak plasma is Rs(θ) = Ro + a cos(θ+ ∆ sin θ) andZs(θ) = −κa sin θ. The parameter κ is the elongation orellipticity, and ∆ is the triangularity. Stellarators have amuch larger design space since Rs and Zs can depend onboth the poloidal angle θ and the toroidal angle ϕ.

17

The number of plasma shape parameters that can becontrolled by coils at a sufficient distance from the plasmafor a fusion power system is limited to about four for atokamak and about ten times that number for the stel-larator, Section (V.D.1). The number of free shape pa-rameters for the tokamak is sufficiently small that theyall have names: aspect ratio, ellipticity, triangularity,and squareness, though squareness is not defined by anagreed-upon shape function. The performance of toka-maks is markedly improved by a careful choice of theseparameters relative to a circular cross section. The num-ber of free parameters that are available to stellaratordesigners is so large that extensive runs with optimiza-tion codes are needed to choose design points. Many ofthese optimization codes and techniques were developedby Jurgen Nuhrenberg and co-workers as part of W7-Xdesign effort (Nuhrenberg et al., 1995).

V. EFFICIENCY OF MAGNETIC CONFINEMENT

• The cost of fusion power depends on the efficiencywith which the magnetic field can be utilized andon the fraction of the power output that must beused to maintain the plasma and the magnetic field,which is called the recirculating power fraction.

• The efficiency of magnetic field utilization dependson (1) the ratio of the pressure of the plasma tothat of the magnetic field, β ≡ 2µ0p/B

2, and (2)the ratio of the magnetic field at the plasma tothat at the coils. The plasma pressure, or β, canbe limited by equilibrium, stability, or transportissues.

• The power required to maintain the net current ofa tokamak places significant constraints on the de-sign.

The cost of fusion power depends on the efficiency ofthe confinement. Engineers tend to emphasize the ra-tio of the power output of a fusion system to its mass.But the more physics oriented efficiency measures are (1)the field strength on the coils needed to confine a givenplasma and (2) the ratio of the power output to the powerrequired to sustain the current in the plasma and coilsor more generally the recirculating power fraction. Thepower required to sustain the plasma current is discussedin Section (VI.E.3).

The magnetic field at the coils that is needed to confinea given plasma depends on two numbers. The first is theplasma beta, which is the ratio of the plasma pressure tothe magnetic field pressure,

β ≡ 2µ0p

B2, (41)

and the second is the ratio of the magnetic field strengthon the coils to that in the plasma. The limits on the

plasma beta due to equilibrium considerations are dis-cussed in Section (V.A), and limits from the existence ofunstable perturbations of large spatial scale are discussedin Sections (V.B) and (V.C). The considerations that setthe ratio of the magnetic field strength on the plasma tothat on the coils are discussed in Section (V.D). Section(VI) will consider limitations on plasma beta, and hencethe magnetic field strength, due to transport.

A. Equilibrium limits

• A plasma equilibrium is determined by the shapeof the outermost plasma surface, the toroidal fluxenclosed by that surface, and the profiles of theplasma pressure, p(ψt), and rotational transform,ι(ψt) ≡ dψp/dψt. The rotational transform of amagnetic field is the average number of poloidaltransits of the torus a field line makes per toroidaltransit. In tokamaks, ι is generally less than unity.

• The plasma β ≡ 2µ0p/B2 is limited by the distor-

tions to the plasma shape caused by the currentdensity j|| parallel to the magnetic field that arisesto make ~∇ ·~j = 0. That is, ~B · ~∇(j||/B) = −~∇ ·~j⊥with ~j⊥ given by the force exerted by the plasma,~∇p = ~j × ~B. These distortions can be impor-tant when beta is just a few percent though muchhigher betas can be achieved by careful plasma de-sign. In asymmetric plasmas, these distortions ofthe plasma shape can cause the breakup of the mag-netic surfaces that are needed for plasma confine-ment.

• A magnetic field that lies in surfaces, ~B · ~∇p = 0,has a simple contravariant representation, Equa-tion (7), which means a representation that usescross products of pairs of coordinate gradients. Amagnetic field associated with a plasma equilib-rium simultaneously has a simple covariant repre-sentation, Equation (56) or Equation (57), whichmeans a representation that uses coordinate gradi-ents. Coordinates in which the magnetic field hasboth a simple cotravariant and a simple covariantrepresentation simplify the analysis of plasmas.

If at each point in a stationary plasma the plasma isclose to thermodynamic equilibrium, then the force ex-erted by a stationary plasma is accurately approximatedby its pressure gradient, −~∇p. The equilibrium betweenthe pressure and electromagnetic forces is ~∇p = ~j × ~B,which is known as the equilibrium equation for a plasma.

A toroidal plasma equilibrium is specified by giving(1) the shape of the magnetic surface that bounds theplasma, ~xs(θ, ϕ) with θ and ϕ arbitrary poloidal andtoroidal angles, (2) the total toroidal magnetic flux in theplasma, (3) the pressure profile p(ψt), and (4) rotational

18

transform profile ι(ψt) (Bauer, Betancourt, and Garabe-dian, 1984). This result is proven by showing equilib-ria are extrema of the energy in a toroidal region thatis bounded by a fixed surface ~xs(θ, ϕ) when the ideal,which means dissipationless, plasma constraints are ob-served. The energy of the plasma plus the magnetic fieldis

W =∫

plasma

(B2

2µ0+

p

γ − 1

)d3x, (42)

where γ is the adiabatic index. The variation thatis carried out is in the shape of the magnetic sur-faces ~x(ψt, θ, ϕ) with the shape of the outermost plasmasurface ~xs(θ, ϕ) held fixed. That is, ~x(ψt, θ, ϕ) →~x(ψt, θ, ϕ)+ ~ξ where ~ξ is the displacement of the surfaceswith the displacement ~ξ zero on the plasma surface.

Before variations in the magnetic field and pressure canbe discussed, the relation between the total and the par-tial time derivative must be explained. In fluid mechanicsthe total time derivative dg/dt gives the rate of changeof any function g(~x, t) in the frame of reference of themoving fluid while the partial time derivative (∂g/∂t)~xgives the time derivative at a fixed spatial point. Thepartial derivative (∂g/∂t)~x is usually written as ∂g/∂t.The rate of change of the position vector ~x is the flowvelocity of the fluid, ~v ≡ d~x/dt. The chain rule impliesdg/dt = ∂g/∂t+(d~x/dt)· ~∇g, so the total, time derivativeis

dg

dt≡ ∂g

∂t+ ~v · ~∇g. (43)

In an ideal fluid the total time derivative of the canon-ical coordinates is zero, for they are carried with thefluid, though their partial time derivatives are non-zeroin a moving plasma. This is proven using the the-ory of general coordinates, Appendix (A), which implieseach canonical coordinate of the transformation function~x(ψt, θ, ϕ, t) obeys

dψtdt

= (~v − ~u) · ~∇ψt (44)

with ~u ≡ (∂~x/∂t)c, Equation (27). In an ideal plasma,~u = ~v, so the canonical coordinates satisfy dψt/dt = 0.

A variation in the magnetic field, δ ~B, in an idealplasma is related to a sufficiently small plasma displace-ment, ~ξ, by

δ ~B = ~∇× (~ξ × ~B). (45)

This equation follows from the Ohm’s law, Equation(29), with zero resistivity, ~E + ~v × ~B = 0, plus Fara-day’s law, ∂ ~B/∂t = −~∇ × ~E, and ~v = ∂~ξ/∂t. Actually~v ≡ d~x/dt = ∂~ξ/∂t + ~v · ~∇~ξ, but for a sufficiently smalldisplacement, ~ξ → 0, one has ~v = ∂~ξ/∂t. The variationin the magnetic energy is determined using a vector iden-tity to write ~B · δ ~B = (~ξ× ~B) · ~∇× ~B+ ~∇·(~ξ× ~B)× ~B.

The divergence term makes no contribution because theplasma surface has a fixed shape, which means ~ξ = 0on the bounding surface. As shown in Section (III.C)the rotational transform retains its initial profile, ι(ψt),through all variations in the plasma shape when the re-sistivity is zero.

A pressure variation in an ideal plasma con-serves the entropy per particle, which implies(d ln p/dt)/(d lnn/dt) = γ. The plasma number densityobeys the continuity equation, ∂n/∂t + ~∇ · (n~v) = 0,so dp/dt = −γp~∇ · ~v. This means the change in thepressure at a fixed point in space, δp ≡ (∂p/∂t)~xδt, is

δp = −~ξ · ~∇p− γp~∇ · ~ξ. (46)

The variation in the pressure can be rewritten in a formthat makes that part of the energy variation obvious δp =(γ−1)~ξ · ~∇p− ~∇·(γp~ξ). The (γ−1) term in δp cancels thesimilar factor in W and the divergence term in δp makesno contribution to the change in the energy since thechange in the shape, ~ξ, is zero on the plasma boundary.

Using the expressions for the variation in magnetic fieldenergy and the variation in the pressure in response to asmall plasma displacement, ~ξ, one finds that

δW =∫

plasma

(~∇p−~j × ~B) · ~ξd3x, (47)

which means W has equilibria, ~∇p = ~j × ~B as its ex-trema. The minimization of the energyW gives the shape~x(ψt, θ, ϕ) of all of the magnetic surfaces in the plasma.VMEC is the best known code for finding equilibria byminimizing the energy (Hirshman, van Rij, and Merkel,1986). It should be noted that the adiabatic index, γ,does not appear in the equilibrium equation, so γ can bechosen arbitrarily. The choice γ = 0 is frequently madebecause with this choice dp/dt = 0. When dp/dt = 0,the pressure profile p(ψt) is unchanged by variations ~ξ inthe shape of the surfaces.

An important theorem of equilibrium theory is thatgiven an arbitrary set of magnetic surfaces, ~x(ψt, θ, ϕ),and an arbitrary rotational transform profile, ι(ψt), themagnetic force ~fB ≡ ~j× ~B can always be written as ~fB =fψt

~∇ψt with fψt(ψt, θ, ϕ) a known function. The angles θ

and ϕ are arbitrary poloidal and toroidal angles, but ψt isthe toroidal magnetic flux enclosed by a magnetic surface.If ~x(ψt, θ, ϕ) and ι(ψt) are consistent with an equilibrium,the function fψt

depends on ψt alone, and the equilibriumpressure profile, p(ψt), is given by dp/dψt = fψt

. Toprove this theorem, note that if ~∇ψt× ~fB ≡ ( ~B · ~∇ψt)~j−(~j · ~∇ψt) ~B is zero, then ~fB = fψt

~∇ψt. Since ~B · ~∇ψt = 0by the definition of magnetic surfaces, the result is provenif the constraint ~j · ~∇ψt = 0 can be imposed. The mostgeneral divergence-free field, ~∇ · ~B = 0, that has therequired surfaces, ~B · ~∇ψt = 0, and rotational transform

19

profile, ι(ψt), is

~B =(

1 +∂λ

∂θ

) ~∇ψt × ~∇θ2π

+(ι− ∂λ

∂ϕ

) ~∇ϕ× ~∇ψt2π

,

(48)which is proven using the argument that led to Equa-tion (7). The constraint µ0

~j · ~∇ψt = (~∇ × ~B) · ~∇ψt =~∇ · ( ~B × ~∇ψt) = 0 determines λ as the solution to aPoisson-like equation that has derivatives only in the θand ϕ coordinates. Periodicity in θ and ϕ implies λ hasa definite expression, which can be determined magneticsurface by magnetic surface. The function λ is the dif-ference between the arbitrary and the magnetic poloidalangles, θm = θ + λ and determines how the field lineswind through the magnetic surfaces. Once λ is deter-mined, one has a definite expression for ~B, which givesthe magnetic force ~fB . The calculation of the curl ofthe magnetic field is complicated but is explained in Ap-pendix (A).

One can always minimize the energy W over any setof shape functions ~x(ψt, θ, ϕ), so it may first appear thatequilibria always exist. This is not true for two reasons.First, the set of shape functions may not be rich enoughto find the extremum, which means the equilibrium con-straint dp/dψt = (~j × ~B) · ∂~x/∂ψt may not be satisfied.That is, fψt

may not be a function of ψt alone. Second,if one considers an arbitrarily rich set of shape functionsthe current ~j may become singular near rational surfaces.

Equilibria with perfect magnetic surfaces tend to havesingular currents parallel to the magnetic field lines,j|| ≡ ~j · ~B/B, at rational surfaces. Rational surfacesare defined by the rotational transform being the ratioof two integers, ι = n/m, and are surfaces on which themagnetic field lines close on themselves after m toroidaland n poloidal circuits of the torus. The equation forvariation of the parallel current determines much of thetheory of force balance of plasmas embedded in a mag-netic field and follows from ~∇·~j = ~B·~∇(j||/B)+~∇·~j⊥ = 0with the perpendicular current determined by the forcethe magnetic field exerts on the plasma, ~fB ≡ ~j × ~B.The equation for variation of the parallel current alongthe field lines is

~B · ~∇j||

B= ~∇ ·

(~fB × ~B

B2

), (49)

which is a magnetic differential equation, Equation (12),for the parallel current. In a plasma equilibrium ~fB =~∇p. Solutions to the magnetic differential equation forj||/B are the sum of (1) a special solution, which variesover the magnetic surface and is given by any solutionto the magnetic differential equation, and (2) a homo-geneous solution in which j||/B is constant along fieldlines, ~B · ~∇(j||/B)h = 0. Both the special and the ho-mogeneous solution can be singular at rational surfaces.The singularity of the special solution, (j||/B)s = jPS/Bwith jPS called the Pfirsch-Schluter current (Pfirsch and

Schluter, 1962), can arise if the pressure gradient, dp/dψtis non-zero at a rational surface. This singularity is dis-cussed more extensively below. The homogeneous solu-tion, (j||/B)h = jnet/B with jnet called the net current,has the form

µ0jnetB

= k(ψt) +∑m,n

kmnei(nϕ−mθm)δ(ψt − ψmn) (50)

with ψmn defined by ι(ψmn) = n/m and δ(ψt−ψmn) theDirac delta function. Each non-zero constant kmn impliesa surface current on a rational surface, which producesa normal magnetic field that cancels the resonant partof the normal field due to external perturbations andprevents the formation of an island.

The precise relation between the special solution forj||/B of Equation(49), the Pfrisch-Schluter current, andthe pressure gradient depends on having a covariant rep-resentation, Equation (55), of the magnetic field in mag-netic coordinates in addition to the contravariant repre-sentation,

2π ~B = ~∇ψt × ~∇θm + ι(ψt)~∇ϕ× ~∇ψt, (51)

which follows from Equation (7) and ι ≡ dψp/dψt.The covariant representation of the magnetic field,

Equation (55), follows from the contravariant represen-tation of the current, ~j = ~∇× ~B/µ0, Equation (52). Thederivation of the contravariant representation of the mag-netic field, Equation (7), depended on two conditions: (1)~∇ · ~B = 0 and (2) that a non-constant function ψt existswith ~B · ~∇ψt = 0. These two conditions hold for thecurrent associated with a plasma equilibrium since it isdivergence free and in an equilibrium satisfies ~j · ~∇ψt = 0,so the current associated with an equilibrium must havethe contravariant representation

~j = −∂Gtot∂ψt

~∇ϕ× ~∇ψt2π

+∂Itot∂ψt

~∇ψt × ~∇θm2π

, (52)

where the radial derivatives of the total poloidal and totaltoroidal current are

∂Gtot∂ψt

=dG

dψt+∂ν

∂ϕ(53)

and

∂Itot∂ψt

=dI

dψt+

∂ν

∂θm. (54)

The function G(ψt) =∫~j ·d~aθm

is the poloidal current inthe region exterior to a constant ψt surface, which is thecurrent through the hole in the torus, Figure (1), andI(ψt) =

∫~j · d~aϕ is the toroidal current in the region

interior to a constant ψt surface, which is the currentthrough a cross section of the torus, Figure (1). Thecontravariant representation of the current implies themagnetic field has the covariant repesentation

~B =µ0

2π

G(ψt)~∇ϕ+ I(ψt)~∇θm − ν ~∇ψt + ~∇F

. (55)

20

We will find that the function ν(~x) is determined by theequilibrium equation. The function F (~x) is determinedby the constraint that ~∇ · ~B = 0 and the boundary con-ditions on the magnetic field.

The contravariant representation of the magnetic field,Equation (51), is unchanged if one defines new poloidaland toroidal angles by θm = θn+ιω and ϕ = ϕn+ω whereω is any well-behaved function of position. If these formsare substituted into the covariant representation of themagnetic field, Equation (55), one obtains an equation ofthe same form but with ν replaced by νn ≡ ν+(dG/dψt+ιdI/dψt)ω and with F replaced by Fn ≡ F + (G+ ιI)ω.There are two obvious choices for ω. The first makesνn = 0, and the resulting coordinates are called Hamadacoordinates, (ψt, θH , ϕH), (Hamada, 1962), in which

~B =µ0

2π

G(ψt)~∇ϕH + I(ψt)~∇θH + ~∇F

. (56)

The second makes Fn = 0, and the resulting co-ordinates are called Boozer coordinates,(ψt, θB , ϕB),(Boozer, 1981), in which

~B =µ0

2π

G(ψt)~∇ϕB + I(ψt)~∇θB + β∗~∇ψt

. (57)

The cross product between the contravariant represen-tation of the current, Equation (52), and of the magneticfield, Equation (51), must equal the pressure gradient.This equality gives an equation for ν, Equation (62), andan equation for the average equilibrium on a magneticsurface, Equation (61). Using arbitrary magnetic coordi-nates, (ψt, θm, ϕ),

~j × ~B = −~B · ~∇ϕ

2π

(∂Gtot∂ψt

+ ι∂Itot∂ψt

)~∇ψt. (58)

Since ~∇p = (dp/dψt)~∇ψt,

∂Gtot∂ψt

+ ι∂Itot∂ψt

= − 2π~B · ~∇ϕ

dp

dψt. (59)

The contravariant form for the magnetic field, Equation(51), implies that the Jacobian of magnetic coordinates,1/J = (~∇ψt × ~∇θm) · ~∇ϕ, is

J =1

2π ~B · ~∇ϕ(60)

with the volume enclosed by the magnetic surfacesV (ψt) =

∫J dψtdθmdϕ. If Equation (59) is averaged

over the poloidal and toroidal angles one obtains the av-erage equilibrium equation (Kruskal and Kulsrud, 1958),

dG

dψt+ ι

dI

dψt= − dV

dψt

dp

dψt, (61)

which is a coordinate independent equation. The func-tion ν satisfies a magnetic differential equation(

∂

∂ϕ+ ι

∂

∂θm

)ν =

(dV

dψt− (2π)2J

)dp

dψt. (62)

The function ν and hence the current are singular ata rational surface ι = n/m unless either the resonantFourier term of 2πJ = 1/ ~B · ~∇ϕ is zero or the pressuregradient, dp/dψt, is zero.

Since Equation (62) for ν holds in any set of mag-netic coordinates, it must hold in Hamada coordinates, inwhich ν = 0. This means the Jacobian of Hamada coor-dinates is independent of the angles, (2π)2JH = dV/dψt,but also that the coordinate transformation to Hamadacoordinates is singular if the function ν is singular whenone uses the cylindrical angle ϕ as the toroidal angle.

The Jacobian of Boozer coordinates can be obtained bydotting the co- and contra-variant representations of ~Btogether, JB = µ0(G+ ιI)/(2πB)2. The parallel currentis particularly simple in these coordinates,

j||

B=G dIdψt

− I dGdψt

G+ ιI+G ∂ν∂θB

− I ∂ν∂ϕG+ ιI

. (63)

The first term on the right hand side is jnet/B =k(ψt)/µ0 with jnet the net current, and the second termis jPS/B with jPS the Pfirsch-Schluter current. Equa-tion (63) for j||/B is obtained from the dot product ofEquations (52) and (57).

In axisymmetric equilibria the expression for thePfirsch-Schluter current can be simplified to

jPSB

=1ι

G

G+ ιI

(dV

dψt− 2π~B · ~∇ϕ

)dp

dψt(64)

with dV/dψt =∮dθm/ ~B · ~∇ϕ. This equation for jPS

follows from Equation (63) with ∂ν/∂ϕ = 0 and Equation(62), which gives an expression for ∂ν/∂θB . Equation(64) is written in a form that is valid in any coordinatesin which ϕ is the toroidal angle and ψt is the toroidalflux.

Even in simple axisymmetric equilibria, the Pfirsch-Schluter current, jPS , provides a significant limitationon the plasma pressure. At large aspect ratio, Ro/r 1,the vacuum toroidal field is dominant, G ιI, and1/ ~B · ~∇ϕ ≈ R2/(RoBo). The Pfirsch-Schluter current,Equation (64), is in the toroidal direction and approxi-mated by

jPS ≈ −2(R−Ro)ιBo

1r

dp

dr(65)

with R − Ro = r cos θ. If the pressure is assumed to de-pend on the minor radius r as p = p0(1 − r2/a2), onefinds the Pfirsch-Schluter current produces a magneticfield in the vertical or Z direction, ∂δBZ/∂R = −µ0jPS ,which at the plasma edge, (R − Ro)2 = a2 is δBZ ≈−2µ0p0/(ιBo). This vertical field opposes the poloidalfield due to the plasma current, Bθ ≈ εaιBo, on the in-board side with εa ≡ a/Ro. When the vertical field be-comes sufficiently strong to cancel the poloidal field onthe inboard side, the equilibrium has reached its pressurelimit. In other words, equilibria must have a volume av-eraged beta, 〈β〉 ≡ 2µ0〈p〉/B2

o with 〈p〉 = p0/2, which

21

satisfies

〈β〉 <∼12εaι

2. (66)

Stability limits the average rotational transform in atokamak to ι ≈ 1/2 and the inverse aspect ratio isεa ≈ 1/3, so 〈β〉 <∼ 4% when the magnetic surfaces arecircular. Actual tokamaks with additional shaping haveachieved 〈β〉 greater than 10%, and ST plasmas haveachieved 〈β〉 approaching 50%. However, the poor scal-ing of the beta limit with aspect ratio limits tokamaksto having a tight aspect ratio. The weakening of thepoloidal field on the inboard side and the strengthen-ing on the outboard side causes the magnetic surfacesto shift outward, called the Shafranov shift, so the sameflux passes between the plasma axis and the inboard andoutboard edges.

The equilibrium of an axisymmetric plasma can alsobe found using the Grad-Shafranov equation (Grad andRubin, 1959; Lst and Schluter, 1957; Shafranov, 1958).Using (R,ϕ,Z) cylindrical coordinates, an arbitrary ax-isymmetric, divergence-free field can be written in amixed covariant–contravariant representation as 2π ~B =µ0G~∇ϕ+ ~∇ϕ× ~∇ψp, which follows from ~∇· (~∇ϕ) = 0 incylindrical coordinates. The curl of this mixed represen-tation gives 2π~j = ~∇G× ~∇ϕ+ ϕ(∆∗ψp)/(µ0R) where

∆∗ψp ≡ R∂

∂R

(1R

∂ψp∂R

)+∂2ψp∂Z2

(67)

is called the Grad-Shafranov operator. From ~j × ~B =(dp/dψp)~∇ψp one finds G is a function of ψp alone and

∆∗ψp = −µ20G

dG

dψp− µ0(2πR)2

dp

dψp, (68)

which is the Grad-Shafranov equation.Equation (49) for the variation in the parallel current

also has important implications for non-toroidal plasmasembedded in magnetic fields. For example, the photo-sphere of the sun becomes sufficiently tenuous with alti-tude that the forces that the magnetic field exerts on theplasma must be small, ~fB ≡ ~j× ~B → 0. In this situation,which is called a force-free equilibrium, the parallel cur-rent satisfies the conservation law that j||/B is constantalong magnetic field lines. The constancy of j||/B impliesthat stars, like the sun, with outer convective zones and amagnetic field must have a corona, a region in the uppersolar atmosphere of very high energy electrons comparedto the energy that would be expected from thermal diffu-sion (Boozer, 1999). The reason is the current j = envs isthe product of the electron charge, number density, andstreaming velocity vs of the electrons relative to the ions.If the streaming velocity exceeds the electron thermal ve-locity ve ≡

√Te/me, the electrons will break away from

a Maxwellian distribution and reach whatever energy isnecessary to carry the current. This is the phenomenonof runaway electrons (Dreicer, 1960) and (Connor and

Hastie, 1975), which arises from the collision frequencydropping as one over the speed of a particle cubed, Equa-tion (133), when the speed is larger than the thermal ve-locity. In the absence of a corona in a star like the sun,the density of the atmosphere drops exponentially withheight, n(z) ∝ exp(−z/h), due to gravitational force bal-ance, ~∇p = ρ~g, with a scale height h ≈ 100km. Althoughthe fractional ionization increases with altitude, the to-tal number of electrons that are potentially available tocarry the current rapidly drops. The magnetic field abovethe sun has structures with a scale of 104km, or about102 e-folds of the atmospheric density. Since j||/B is ap-proximately constant due to the rapid drop in the atmo-spheric pressure, the streaming parameter vs/ve wouldincrease by a factor of roughly e100 ≈ 1043 without acorona. This is of course impossible without the stream-ing parameter becoming greater than one and the elec-trons running away. At the bottom of the photosphere,magnetic fields are observed to have variations of 0.1Tover 103km, so local current densities have to be greaterthan j ≈ 102A/m2. The current that could be carried bythe full electron density (neutral plus ionized) moving atthe electron thermal speed is of order 1010A/m2, whichis only eight orders of magnitude larger. Actually the lo-cal current density on magnetic field lines emerging fromthe sun should have a value orders of magnitude largerthan the observed current density of 102A/m2, which bythe observation process involves a spatial average. Thereason a large current density should be expected is thatwhen field lines are churned around in a conducting fluid,as they are in the outer solar convective zone, they de-velop strong parallel currents with very short correlationdistances (Thiffeault and Boozer, 2003). These currentsflow all along the field lines, Equation (49), relaxing totheir equilibrium values via shear Alfven waves, Section(VI.H). Runaway electrons are not generally consideredas an explanation for the actual solar corona. Whetherthe currents along the magnetic field lines that penetratethe solar surface have sufficient strength and shortnessof correlation length to cause the actual solar corona arequestions that are not easily answered observationally.

B. Stability limits

• A magnetically confined plasma cannot be in ther-modynamic equilibrium, so a potential for instabil-ity always exists. Instabilities can be driven by thepressure gradient or the net current in the plasma.

• The pressure gradient is destabilizing when the cen-ter of curvature of the magnetic field lines, ~κ ≡ b·~∇bwith b ≡ ~B/B, is in the direction of higher pressure(bad curvature) and stabilizing when the curvaturehas the opposite sign (good curvature).

• Stability calculations can be done by preserving theconstraints of an ideal, dissipationless, plasma evo-lution, which prevent islands from opening but give

22

singular currents, or by keeping smooth current dis-tributions, which allow islands to open. Throughlinear order in the perturbation theory, there is nodistinction in these two types of analyses exceptnear rational surfaces ι = n/m. However, the idealanalysis always predicts greater stability.

A magnetically confined plasma always has the ther-modynamic free energy that is required for instability.The maximum entropy state of the charged particles thatform a plasma is a Gibbs distribution, exp(−H/T ), butthe particle energy H = 1

2mv2 +qΦ is independent of the

magnetic field, ~B(~x). A plasma that is carrying a currentcannot be in a Gibbs distribution and, therefore, alwayshas thermodynamic free energy. The stability of current-carrying plasmas depends on the existence of constraintson the plasma motion that prevent the reduction in thefree energy.

The free energy that is available for instabilities allowstoroidal plasmas to spontaneously break their symmetryby kinking. This kinking is driven by the gradients inthe parallel current distribution, k ≡ µ0j||/B, and in thepressure. Both drives limit the achievable plasma beta(ITER Physics Expert Group on Disruptions, PlasmaControl, and MHD et al., 1999). A book by (Freidberg,1987) discusses plasma stability under the assumptionsof ideal magnetohydrodynamics (MHD), which means inthe absence of dissipative effects.

Instabilities driven by the gradient in the parallel cur-rent distribution, d(j||/B)/dψt, limit the magnitude ofthe poloidal magnetic field or more precisely the rota-tional transform, and both equilibrium and stability lim-its involve the strength of the poloidal magnetic field.

The pressure limit due to plasma stability for a toka-mak plasma is approximated with remarkable success,even for highly shaped plasmas, by the Troyon limit(Troyon and Gruber, 1985). The volume-averaged β ≡2µ0p/B

2 can be no greater than

〈β〉 =βn10

µ0

4πI

aB, (69)

where I is the net toroidal current and a is the half widthof the plasma in the Z = 0 plane. The Troyon coefficient,also called beta normal, is a number, which is βn ≈ 3 insomewhat optimized plasmas but can be a factor of twolarger. The results of tokamak experiments are oftenexpressed by giving the βn that was achieved.

The stability of a plasma to a pressure gradient islargely determined by the relative orientation of the pres-sure gradient, ~∇p, and the curvature of the magnetic fieldlines ~κ ≡ b · ~∇b where b ≡ ~B/B is a unit vector alongthe magnetic field. For example, the curvature of a cir-cular line b = ϕ(ϕ) of (R,ϕ,Z) cylindrical coordnatesis ~κ = −R/R. The pressure gradient directly entersthe energy change due to a displacement ~ξ of an equi-librium plasma through −

∫(~ξ · ~∇p)(~ξ · ~κ)d3x, Equation

(91). Good curvature means ~κ·~∇p < 0 and bad curvature

means ~κ · ~∇p > 0. Good curvature is stabilizing becausethat geometry compresses plasma motion down the pres-sure gradient. Unfortunately, topology implies the fieldline curvature must be bad somewhere on a toroidal mag-netic surface, and pressure driven instabilities localize inthese places. Localization is stabilizing because it impliesa bending of the magnetic field. If the bad curvature re-gions are sufficiently localized, the pressure gradient doesnot destabilize the equilibrium.

If the plasma is assumed to be ideal, which meanszero resistivity, then the evolution equations for the mag-netic field imply the rotational transform ι is a time-independent function of ψt. In the presence of a per-turbation, this assumption generally leads to a singularnet plasma current, µ0

~jnet = k ~B with ~B · ~∇k = 0. Onecan avoid a singular net current by assuming that thedistribution of net current k(ψt) is independent of timeinstead of ι(ψt). The assumption of a fixed net currentis a resistive stability analysis.

Remarkably, linear perturbation theory makes no dis-tinction between an ideal, ι(ψt) independent of time, anda resistive analysis, k(ψt) independent of time, except atresonant rational surfaces, ι = n/m. In an ideal analy-sis a singular current arises at resonant rational surfacesand in a resistive analysis an island opens. Away fromresonant rational surfaces, the change in the rotationaltransform scales as the amplitude of the magnetic per-turbation squared and, therefore, cannot enter a linearstability analysis. To see this consider the perturbed fieldline Hamiltonian ψp = ψp(ψt) + ψ1 cos(nϕ −mθ). Onecan solve for the magnetic field lines to second order inthe perturbation ψ1 and find that the rotational trans-form is changed by an amount

∆ι(ψt) =m

4

d

dψt

(1

n− ιm

dψ21

dψt

)+

mι′′ψ21

(n− ιm)2

,

(70)where ι′′ ≡ d2ι/dψ2

t . The perturbation ψ1 is linear in theperturbation amplitude for a sufficiently small perturba-tion, so ∆ι is quadratic in the perturbation amplitudewhere n− ιm is non-zero.

1. Current-driven instabilities

• The strongest effect of the net plasma currenton stability is near the rational surfaces, ι =n/m. Defining a helical flux by dψh ≡ (ι −n/m)dψt, a gradient in the net current is stabi-lizing if d(j||/B)/dψh is positive and destabilizingif d(j||/B)/dψh is negative.

• Only low poloidal mode number, m, perturbationscan be driven unstable by the net plasma current.These instabilities are called kinks.

• A current density proportional to a Dirac deltafunction generally arises on the rational surfaces,ι = n/m, in the presence of perturbations that

23

conserve the constraints of an ideal, dissipation-less, plasma. The amplitude of these delta-functioncurrents is proportional to the jump in resonantFourier term of the radial displacement of theplasma,

[~ξ · ~∇ψt

]mn

.

The stability properties of a force-free equilibrium,~∇× ~B = k(ψt) ~B, illustrates the effect of the distributionof net plasma current k(ψt) on plasma stability. The fun-damental results are that the net current affects stabilityprimarily through the gradient dk/dψt. Near a rationalsurface ι = n/m the gradient of the net current distribu-tion enters through the quantity

m

n− ιm

dk

dψt, (71)

which is destabilizing when positive and stabilizing whennegative. In other words, plasma stability is extremelysensitive to the gradient in the current distribution k ≡µ0j||/B near rational surfaces, but the effect on stabilityof the gradient is opposite on the the two sides of therational surface. For example in a tokamak, a low orderrational surface just outside the plasma is very destabi-lizing because dk/dψt is negative near the plasma edgewhile ι is positive but becoming smaller dι/dψt < 0. Thedestabilizing effect of the current gradient is obtained onthe plasma side of the rational surface, but there is nocurrent to provide a stabilizing effect on the opposite sideof the rational surface. This instability, which is called anexternal kink, is particularly pronounced when q = 1/ιis just above an integer at the plasma edge.

The stability of force-free equilibria is very sensitive tothe gradient of the current distribution k due to the prop-erties of the magnetic differential equation ~B · ~∇k = 0,which must hold for both the perturbed and the un-perturbed equilibrium. In the perturbed equilibrium,k = k0(ψt)+δk, and ( ~B0+δ ~B)·~∇k = 0. The unperturbedequilibrium is ~∇ × ~B0 = k0(ψt) ~B0, and the magneticperturbation is given by ~∇ × δ ~B = δk ~B0 + k0δ ~B where~B · ~∇δk = −δ ~B · ~∇k0. That is, δk is given by a magneticdifferential equation and is proportional to dk0/dψt. Theperturbed current distribution, δk, can become very largenear a resonant rational surface, ∝ 1/(n−ιm), because ofthe properties of the magnetic differential equation. Thelargeness of δk is what allows unstable perturbations toexist.

Instabilities due to the gradient in the net currentare driven by the parallel part of the perturbed vec-tor potential, δ ~A|| ≡ (b · δ ~A)b with b ≡ ~B/B. Thisis proven by noting that the three independent com-ponents of any vector, including the perturbed vec-tor potential, can be written at each spatial point asδ ~A = (δAB/B) ~B + δAψ ~∇ψt + ~∇δg. If one chooses agauge in which ~B · ~∇δg = 0, then δAB = δA||, andδ ~B · ~∇k0(ψt) = (~∇ × δ ~A||) · ~∇k0. This representationof δ ~A must be used with care. Given a change in the

poloidal flux, δψp = −∮δ ~A · (∂~x/∂ϕ)dϕ, the change in

the toroidal flux, δψt =∮δ ~A · (∂~x/∂θ)dθ, is constrained

if δg is a single-valued function of θ and ϕ. One cansatisfy the constraint ~B · ~∇δg = 0 and have arbitrarychanges in the poloidal and toroidal fluxes if δg has theform δg = (θ − ιϕ)∆(ψt).

The stability of force-free equilibria can be studiedwith greater simplicity when the aspect ratio is very largeRo/r → ∞. In a large aspect ratio torus with circularmagnetic surfaces, one can let ϕ = z/Ro, replace ϕ by z,and describe the magnetic field using (r, θ, z) cylindricalcoordinates, ~B = B0

(z + ι(r) r

Roθ)

where the rotationaltransform ι is assumed to be of order unity. The toroidalflux is ψt = πB0r

2. The important part of the perturbedvector potential is the parallel component, δ ~A = δA||z,so δ ~B = ~∇δA||× z. If the parallel component of the vec-tor potential is written as δA|| = A||(r, t) sin(nφ −mθ),then

δk =RorB0

m

n− ιm

dk0

drδA||, (72)

and the equation ~∇× δ ~B = δk ~B0 is

1r

d

dr

(rdA||

dr

)−m2A||

r2= −Ro

r

m

n− ιm

dk0

drA||. (73)

The rotational transform and the current distribution arerelated by

1r

dr2(ι− ιv)dr

= Rok0(r) (74)

where ιv(r) is the rotational transform of the vacuum, orcurrent-free, magnetic field. In a tokamak, ιv is zero, butit is non-zero in a stellarator.

Externally driven currents are required to perturb theplasma, but it is mathematically simpler to view theseas lying just outside the plasma. To understand theneed for externally driven currents, note that solutionsto Equation (73) should be well-behaved as r → 0, whichleaves only one free boundary condition. The solution toEquation(73) outside of the plasma, r > a, is A|| ∝ 1/rmwith the assumption that there are no perturbing cur-rents away from the plasma boundary. The matchingcondition at the plasma boundary at r = a is that A||be continuous. However, the radial derivative of A|| cannot also be made continuous, for that would be a secondboundary condition. The jump in the derivative, whichis denoted by [∂A||/∂r], gives the current that must beexternally driven at the edge of the plasma to producethe magnetic perturbation.

The stability of a magnetic perturbation to a plasmais determined by the sign of the power that is needed todrive the current on the plasma surface that supportsthe perturbation. Positive power means stability andnegative power instability. The surface current that is

24

required to support the perturbation is

δ~j = −δ(r − a)z

µ0

[∂δA||

∂r

], (75)

where it is assumed that there are no currents outsideof the plasma surface, δA|| ∝ (a/r)m for r > a. Thisfollows from Ampere’s law, ∇2δA|| = −µ0δjz. The powerper unit length in z that is needed to drive the magneticperturbation is

pd = −πaµ0

∂A||

∂t

[∂A||

∂r

], (76)

where the quantities are to be evaluated at r = a.To prove this, note that the power per unit volumegoing into the electromagnetic fields is − ~E · δ~j. Theelectric field associated with the magnetic perturbationδ ~B = ~∇ × (δA||z) is obtained using Faraday’s Law~E = −z∂δA||/∂t, which implies Equation (76). In theabsence of a plasma A|| = A(t)(r/a)m for r < a andA|| = A(t)(a/r)m for r > a, which requires a powerpd = (mπ/µ0)dA2/dt per unit length, which is positiveas one would expect. To make the power pd negativeand obtain an instability, (∂2A||/∂r

2)/A|| must be neg-ative in some part of the plasma. The condition that(∂2A||/∂r

2)/A|| be negative implies the stability proper-ties of the current driven modes given in the discussionof Equation (71) and that only perturbations with lowvalues of m can be unstable.

The restriction of current driven modes to low modenumbers comes from the stabilizing effect of the termm2A||/r

m in Equation (73), which tends to make(∂2A||/∂r

2)/A|| positive. Although current driven in-stabilities must be low m, pressure driven perturbationshave comparable stability properties at all m. The dom-inant drive for pressure driven instabilities is the Pfirsch-Schluter current that arises in the perturbed plasmastate. The destabilization of short wavelength pertur-bations by the pressure gradient is discussed in Section(V.C.2) on ballooning modes.

In a tokamak the vacuum transform ιv vanishes, anda very simple solution to Equations (73) and (74) ex-ists for any k0(r) when m = 1. This solution is δA|| =(ι(r)− n) rA sin(nφ−θ) where A is a constant. Define rnby ι(rn) = n. In an ideal stability analysis, the generalm = 1 solution for r < rn can be matched to an externalsolution A|| = 0 for r > rn. The matching gives a strongsurface current, µ0jz = nδ(r− rn)(d ln ι/d ln r)rn

A. Thism = 1 perturbation is called the the sawtooth insta-bility, which in a simple analysis has zero growth ratewithout plasma dissipation and an infinite growth ratewith dissipation. The rotational transform in the cen-ter of a tokamak whose current is maintained by a loopvoltage tends to rise until it exceeds unity. After thecentral transform exceeds unity, strong m = 1, n = 1kink-like relaxation oscillations occur that maintain the

central transform just above, but close to, unity. Theserelaxation oscillations are called sawteeth due their ap-pearance on electron temperature diagnostics where theywere first observed. Reviews on the m = 1 mode havebeen written by (Migliuolo, 1993) and (Hastie, 1997).

The location of the resonant rational surface, ι(rmn) =n/m, is a special radial position in the differential equa-tion for A||, Equation (73). In ideal theory an island mustbe avoided, which means δ ~B·r = −(m/r)A|| cos(nϕ−mθ)must vanish there. The surface current that arises on theresonant surface in an ideal analysis to prevent an islandfrom opening is given by Equation (75) with a replacedby rmn. The relaxation of this current dissipates energyand allows an island to open.

A jump in the resonant component of a radial displace-ment, ~ξ·~∇ψt, is the signature of a singular surface currenton a rational surface ι = n/m in an ideal perturbationanalysis. The demonstration is easiest in a large aspectratio torus. The perturbed parallel component of thevector potential δA|| can be replaced by the radial com-ponent of the displacement,

ξr = − RorB0

m

n− ιmδA||. (77)

The radial component means ~ξ · r, where r ≡ ~∇r. Therelation between ξr and δA|| is proven using δBr = ~B0 ·~∇ξr, which follows from δ ~B = ~∇× (~ξ × ~B0), and δBr =θ·~∇δA||, which follows from δ ~B = ~∇δA||×z. The currentflowing on a rational surface r = rmn, Equation (75),can then be written in terms of the jump in the radialdisplacement,

δ~j = − z

µ0

dι

dr

rB0

Roδ(r − rmn)[ξr]. (78)

Much of the theory of instabilities in a cylinder, in-cluding the existence of jumps in the radial displacement,was developed by (Newcomb, 1960). Alan Glasser in un-published work during the 1990’s extended Newcomb’sresults to general axisymmetric equilibria. Glasser’swork can be trivially generalized so it applies to arbi-trary scalar pressure plasma equilibria (Nuhrenberg andBoozer, 2003).

2. Resistive stability

• The stability of a plasma with non-zero resisitiv-ity can be tested by considering the perturbationdriven by a delta function current on a rationalsurface. If this current increases when power is re-moved,

∫~j · ~Ed3x > 0, then the plasma is unstable

to the formation of an island with a width that in-creases linearly in time. Such instabilities are calledtearing modes.

• The stability of a plasma to perturbations that aresufficiently slow for all current perturbations to re-lax can be tested by making the distribution of the

25

force-free current a time independent function ofthe toroidal magnetic flux, ψt.

In the presence of a perturbation, a resonant sur-face current naturally arises on each resonant surface,which means a current density that is proportional toδ(ι − n/m) cos(nϕ −mθ). The relaxation of this singu-lar current density due to the plasma resistivity leads toan island and potentially to a resistive instability. Thepaper that sparked research on resisitive instabilities is(Furth, Killeen, and Rosenbluth, 1963).

Magnetic islands are produced by the resonant partof perturbed parallel component of the vector potential.That is,

δ ~B · ~∇ψt~B · ~∇ϕ

= µ0G(ψt)∂δA||/B

∂θ− µ0I(ψt)

∂δA||/B

∂ϕ(79)

with (ψt, θ, ϕ) magnetic coordinates coordinates in which~B has a simple covariant representation, Equation (57).A resonant Fourier term in δ ~B · ~∇ψt/ ~B · ~∇ϕ, Equation(14), is related to the half width of an island by Equation(16). The derivation of Equation (79) is straightforwardusing the representation for the perturbed vector poten-tial δ ~A = (δAB/B) ~B + δAψ ~∇ψt + ~∇δg with the gaugeδg chosen so δA|| = δAB and using Equation (57) for ~B

in δ ~A|| = (δA||/B) ~B. One can easily show that a changein the gauge does not affect the resonant components ofEquation (79).

The rate of growth of an island can be calculated usinga (Rutherford, 1973) analysis. The component of Ohm’slaw along the resonant magnetic field line, ι = n/m, isE|| = −∂δA||/∂t = ηδj||. As the island grows the force-free current near the rational surface flows on the mag-netic surfaces of the island, so the width of the currentchannel, δR, is proportional to the width of the magneticisland. For a slowly growing island, one can find the cor-rect current distribution by solving the induction equa-tion (Boozer, 1984a). Ampere’s law, µ0δj|| = −∇2δA||implies the width, δR, of a narrow current channel canbe defined by δj|| = −2∆′δA||/(µ0δR) with

∆′ ≡ 1A||

[∂A||

∂r

]. (80)

The width of the island and the current channel are pro-

portional to√|A|||, so δ2R ∝ |A|||, and

dδRdt

=η

µ0∆′. (81)

If ∆′ > 0, the width of the island grows until the currentsingularity disappears, ∆′ → 0.

In the limit of a resistive analysis with no singular cur-rents, both A|| and ∂A||/∂r must be continuous. The sin-gularity of the right hand side of the differential equationfor A||, Equation (73), is sufficiently weak that one can

integrate straight through, even numerically, without realdifficulty, which makes both A|| and ∂A||/∂r continuous.Said more precisely, the solution to f ′′ = fx/(x2 + δ2)is well-behaved in the limit as δ → 0. Because the re-laxation of the singular currents that arise in an idealanalysis always takes energy from the field, a perturba-tion always takes more energy to drive, and is thereforemore stable, in an ideal than in a resistive analysis.

3. Robustness of magnetic surfaces

• Plasma rotation prevents an arbitrarily small ex-ternal perturbation from opening a magnetic islandinside a plasma.

• The pressure driven net parallel current, which iscalled the bootstrap current, can make an other-wise stable plasma unstable to the opening of anisland. This instability is called the neoclassicaltearing mode.

An arbitrarily small external magnetic perturbationcannot force an island to open in a rotating plasma (Fitz-patrick and Hender, 1991). To understand why, we mustfirst understand the equations for plasma rotation . Theforce balance equation of an ion species is mini~v · ~∇~v +~∇pi = qni( ~E + ~v × ~B). If the plasma is rotating slowlycompared to the thermal speed,

√T/mi, but rapidly

compared to the ion diamagnetic speed, |~∇pi/qniB|, ionforce balance is ~E+~v× ~B = 0 with the electric field givenby a potential, ~E = −~∇Φ. The magnitude of the dia-magnetic speed is |~∇pi/qnB| ≈ (ρi/a)

√T/mi. The ion

gyroradius is ρi = (mi/qB)√T/mi with ρi/a ≈ 1/500

under fusion conditions. The equation ~v× ~B = ~∇Φ, plusthe requirement that a steady flow be divergence free,lead to two magnetic differential equations, ~B · ~∇Φ = 0and ~B · ~∇(v||/B) = ~∇ · (~∇Φ × ~B/B2). These equationsact as constraints on the two independent directions ofplasma rotation, poloidal and toroidal. Plasma rotationis determined by two functions that depend on only ψt.These functions are Φ(ψt) and the solution to the ho-mogeneous equation ~B · ~∇(v||/B) = 0. The equation~B · ~∇Φ = 0 holds under very general conditions becauseof the ease with which electrons flow along the magneticfield lines. If the magnetic surfaces are good and therotation is slow compared to the ion thermal speed, theelectric potential is accurately given by a function of thetoroidal flux alone, Φ(ψt).

Even a small stationary island forces one component ofthe plasma rotation to be zero; only rotation parallel tothe resonant field lines is allowed. On the outermost sur-face of an island, the separatrix, Figure (2), the magneticdifferential equation ~B · ~∇Φ = 0 forces dΦ/dψt to be zero.This follows from the general result discussed at the endof Section (III.A) that in the vicinity of an arbitrarilysmall island any function Φ that satisfies ~B · ~∇Φ = 0

26

can only have a non-zero derivative relative to the helicalflux, which is defined by dψh ≡ (ι − n/m)dψt. Withoutthe island Φ can be an arbitrary function of ψt.

The zeroing of a component of rotation on a surface,which is required if an island is to exist, requires a torque.If an external magnetic perturbation is not strong enoughto transport this torque to the current that produces theperturbation, δ ~Bx, then an island cannot open. The Zcomponent of that torque, Tz, is associated with toroidalrotation. From the plasma perspective, any externalmagnetic perturbation in a tokamak can be viewed asbeing created by a surface current that flows in a thinaxisymmetric surface that surrounds the plasma. Torquebalance between the plasma and the perturbing externalcurrent allows an exact calculation of the total electro-magnetic torque within the plasma,

∫~x×(~j× ~B)d3x, that

the perturbation exerts by carrying out an integral overthe external current-carrying surface. The Z componentof the torque is

Tz =1µ0

∮surf

R[δ ~Bx] · ϕδ ~Bi · d~a. (82)

δ ~Bi is the field produced by currents internal to theplasma. δ ~Bx is the external magnetic perturbation,and [δ ~Bx] is its jump across the current-carrying layerthat represents the external current that produces δ ~Bx.Equation (82) is derived by noting that the surface cur-rent that flows in the current-carrying layer is δ ~Js =n × [δ ~Bx]/µ0, where n is the normal to the surface,and the force per unit area in the torodial direction is(δ ~Js × δ ~Bi) · ϕ. The torque that can be exerted by thecurrent-carrying layer has an upper limit. Since δ ~Bi fora stable plasma is proportional to the external perturba-tion δ ~Bx, Section (V.D.2), the maximum torque scalesas (δBx)2. If the torque that is required to introduce anisland is greater than this limit, then the island cannotopen. Equation (82) also contains the essence of the phe-nomenon of the torque on a plasma due to a magneticfield error becoming much larger as a plasma approachesmarginal stability. This is because the ratio δBi/δBx be-comes large as marginal stability is approached, Section(V.D.2).

How does the torque that an island induces prevent theisland from opening? For simplicity assume the plasmapressure is zero, then force balance is ~fvis = δ~j× ~B where~fvis is the viscous force exerted by the plasma. The par-allel current obeys ~B ·~∇(δj||/B) = −~∇·( ~B× ~fvis/B2). Toprevent an external magnetic perturbation, such that ofEquation (14), from opening an island, the parallel cur-rent must have the form δj||/B = kr(ψt) cos(nϕ−mθm)so it can cancel the perturbing field on the rational sur-face. Without the viscous force this current can onlyflow on the rational surface ι = n/m, which meanskr(ψt) must be a delta function. With a viscous force,(n − ιm)kr(ψt) is given by the resonant Fourier compo-nent of ~∇· ( ~B× ~fvis/B

2). Since (n− ιm) vanishes at the

rational surface, even a small viscous force can spread outand also maintain the current that is required to preventan island from opening.

If a plasma is initially confined on perfect magnetic sur-faces, then it is a complicated question whether these sur-faces will break due to either instabilities or small exter-nal perturbations. Rotational shielding of islands, whichwe discussed, is only one example of several importanteffects. In the modern literature, these effects are pri-marily discussed under the topic of neoclassical tearingmodes; see for example (Rosenberg et al., 2002).

A neoclassical tearing mode causes a magnetic islandto open through the formation of a strong gradient inthe net plasma current near a rational surface. An islandremoves the pressure gradient near its resonant rationalsurface, Section (III.A), and zeros the bootstrap current,which is a net parallel current due to the pressure gradi-ent, Equation (219). The zeroing of the bootstrap currentin the vicinity of the rational surface produces a largegradient d(j||/B)/dψh, which is of the same sign on bothsides of the rational surface, where dψh ≡ (ι− n/m)dψt.Depending on the relative signs of the bootstrap currentand dι/dψt, this effect either strongly favors, or mitigatesagainst, the formation of an island, Section (V.B.1). Ina tokamak with ι peaked on axis, the formation of anisland is favored.

Magnetic surface quality does seem to have some hys-teresis: good surfaces tend to stay good and surfaces splitby islands tend to stay split by islands. The preservationof magnetic surfaces is an important design and opera-tional consideration in both tokamaks and stellarators.

In tokamaks instability can lead to a catastrophicloss of surfaces, which is called a disruption (ITERPhysics Expert Group on Disruptions, Plasma Control,and MHD et al., 1999). Disruption avoidance is criticalto the success of magnetic fusion. Stellarators are farmore immune to disruptions. There are two reasons forthis. First, the free energy in a stellarator equilibriumtends to be smaller because of the contribution of thevacuum magnetic field to the rotational transform. Sec-ond, the vacuum field in a stellarator strongly centersthe plasma in its surrounding chamber. In a tokamakthe plasma is centered in its chamber by a vertical, Z-directed, magnetic field. The toroidal current that pro-vides the poloidal field in a tokamak produces a self forcein the R direction, which is called the hoop stress, thatmust be balanced by the force of interaction with a ver-tical field. Changes in the plasma equilibrium cause achange in the plasma position, which may result in theplasma striking the chamber walls with a fast loss of theplasma energy and current. Such an event in a powerplant could cause severe damage to the walls.

C. Stability analyses

• Instability of an equilibrium can be demonstratedby finding a plasma displacement ~ξ(~x) that re-

27

duces the energy, W , of the magnetic field plus theplasma, Equation (42).

• The two components of the plasma displacementthat lie in a magnetic surface can be expressed interms of the radial displacement, ξψ ≡ ~ξ · ~∇ψt, fordisplacements that minimize the energy W .

• The change in the energy, δW , produced by aplasma displacement is a quadratic operator on ξψfor perturbations that minimize the energy W .

The theory of long wavelength plasma stability isclosely related to the theory of equilibria. Equilibria canbe found by minimizing the energy, Equation (47), whileimposing either torodial symmetry for tokamaks or peri-odicity for stellarators, ~x(ψt, θ, ϕ + 2π/Np) = ~x(ψt, θ, ϕ)with Np the number of periods. These equilibria are un-stable if the energy can be lowered by a more generalshape function ~x(ψt, θ, ϕ). Mathematically, equilibria areextrema and stable equilibria are minima of the energy,(Lundquist, 1951) and (Bernstein et al., 1958). The longwavelength instabilities found by minimizing the energyare called magnetohydrodynamic (MHD) instabilities indistinction to the microinstabilities that have a wave-length comparable to an ion gyroradius.

1. Expressions for δW

If ~xe(ψt, θ, ϕ) gives the shape of the magnetic surfacesin an equilibrium plasma, the stability of that equilib-rium can be tested by considering the shape function~x(ψt, θ, ϕ) = ~xe(ψt, θ, ϕ) + ~ξ with ~ξ a small perturbingdisplacement. The displacement causes a change in themagnetic field, Equation (45), and a change in the pres-sure, Equation (46). The change in the energy, Equation(42), due to a small change from equilibrium, Equation(47), is quadratic in the displacement and given by

δW =12

∫plasma

(~∇δp− δ~j × ~B −~j × δ ~B) · ~ξd3x (83)

with δ~j ≡ ~∇×δ ~B/µ0. The integral is over the volume oc-cupied by the plasma, for outside the plasma δp, δ~j, and~j are zero. The factor of two arises from the force opera-tor ~F [~ξ] ≡ ~∇δp−δ~j× ~B−~j×δ ~B being linearly dependenton the displacement ~ξ. Partial integrations demonstratethat the force operator has the important property of be-ing self-adjoint,

∫~ξ1 · ~F [ξ2]d3x =

∫~ξ2 · ~F [ξ1]d3x where ~ξ1

and ~ξ2 are two arbitrary displacements (Bernstein, 1983;Bernstein et al., 1958). This implies that if the displace-ment is expanded in a set of vectors, ~ξ =

∑cj~uj then

δW =∑c∗iWijcj where Wij is a Hermitian matrix.

As shown below, Equation (83) for δW can be writtenas the sum of a vacuum energy and a plasma energy(Bernstein et al., 1958),

δW = δWv + δWp. (84)

The vacuum energy is

Wv ≡∫ext

(δB)2

2µ0d3x. (85)

with the integral performed over the region exterior tothe plasma. The plasma contribution is

δWp =∫

plasma

(wξ⊥ +

12γp(~∇ · ~ξ)2

)d3x (86)

where

wξ⊥ ≡(δ ~B)2

2µ0+

12(~∇· ~ξ⊥)(~ξ⊥ · ~∇p)−

12~ξ⊥ · (~j× δ ~B) (87)

has no dependence on the parallel part of the displace-ment, ~ξ|| ≡ (b · ~ξ)b, where b ≡ ~B/B. Equation (86)implies that to minimize δW the parallel component ofthe displacement should be chosen so ~∇ · ~ξ = 0.

The derivation of Equations (84)–(87) is given in thisparagraph, which can be skipped. The derivation followsfrom noting that the term (δ~j× ~B) · ~ξ in δW can be inte-grated over all of space since δ~j = 0 outside of the plasma.One finds that

∫(δ~j× ~B) ·~ξd3x = −

∫(δB)2d3x/µ0 where

the last volume integral is over all of space. Using Equa-tion (46),

~ξ · ~∇δp = −~ξ · ~∇(~ξ · ~∇p)− ~∇· (γp~ξ~∇·~ξ)+γp(~∇·~ξ)2. (88)

Using Equation (45) one can easily show δ ~B · ~∇p =~B · ~∇(~ξ · ~∇p), which implies ~ξ|| ·(~j×δ ~B) = −~ξ|| · ~∇(~ξ · ~∇p).The integral over the plasma volume of a pure divergencevanishes because the plasma pressure and pressure gra-dient vanish at the plasma edge.

Equation (87) for wξ⊥ can be rewritten (Furth et. al.,1965) as

wξ⊥ = w+ + w±, (89)

where

w+ ≡ (δB)2⊥2µ0

+B2

2µo(~∇ · ~ξ⊥ + ~ξ⊥ · κ)2 (90)

is always positive but the term

w± ≡ −12j||

B(~ξ⊥ × ~B) · δ ~B⊥ − (~ξ⊥ · ~κ)(~ξ⊥ · ~∇p) (91)

has an indefinite sign.The derivation of Equations (89)—(91), which is given

in this paragraph and can be skipped, follows from anexpression for the parallel component of the perturbedmagnetic field,

~B · δ ~B = −(~∇ · ~ξ⊥ + 2~ξ⊥ · ~κ)B2 + µ0~ξ⊥ · ~∇p. (92)

The derivation of this identity uses the vector identity~B · ~∇ × (~ξ⊥ × ~B) = −~ξ⊥ · (~∇ × ~B) × ~B − ~∇ · (B2~ξ⊥)

28

plus Ampere’s law, ~∇× ~B = µ0~j, and force balance ~∇p =

~j × ~B. Force balance plus Ampere’s law imply ~∇⊥(B2 +2µ0p) = 2B2~κ where

~κ ≡ b · ~∇b (93)

is the curvature of the field lines, b ≡ ~B/B. Assemblingresults one obtains Equation (92). The identity of Equa-tion (92) can be written in a second form, which is

(δ ~B||)2

2µ0= ∆2 B

2

2µ0+ δB||

~ξ⊥ · ~∇p2B

− 12(~ξ⊥ · ~∇p)∆, (94)

where

∆ ≡ ~∇ · ~ξ⊥ + 2~ξ⊥ · ~κ. (95)

The last term in Equation (87) for wξ⊥ can be rewrittenusing the fact that the vector triple product in three di-mensions is non-zero only when one of the vectors comesfrom each of the three independent directions. Conse-quently, ~ξ⊥ · (~j× δ ~B) = ~ξ⊥ · (~j||× δ ~B⊥)+ ~ξ⊥ · (~j⊥× δ ~B||),which can be rewritten as ~ξ⊥ · (~j × δ ~B) = (j||/B)δ ~B⊥ ·(~ξ⊥ × ~B) + (δB||/B)~ξ⊥ · ~∇p.

A minimization of δW is really a minimization of aquadratic functional of ξψ ≡ ~ξ · ~∇ψt that involves radialderivatives only up to the first order. If the perturbingdisplacement is written as

~ξ = ξψ∂~x

∂ψ+ η

∂~x

∂θ+ µ~B, (96)

then the only place the coefficient µ(ψ, θ, ϕ) enters δWis in the positive definite term involving (~∇ · ~ξ)2. A min-imization of δW makes ~∇ · ~ξ = 0, which is a magneticdifferential equation for µ in terms of the η and ξψ coef-ficients.

The η coefficient of ~ξ can be expressed in terms ofξψ for perturbations that minimize δW . A perturbationthat minimizes δW must give a perturbed equilibriumand in a perturbed equilbrium the current, ~j+~∇×δ ~B/µ0,must be orthogonal to the toroidal flux in that equilib-rium. At a given spatial point the perturbation causesthe toroidal flux to go from ψt to ψt − ~ξ⊥ · ~∇ψt, whichfollows from dψt/dt = ∂ψt/∂t + (∂~ξ/∂t) · ~∇ψt and theconstraint that dψt/dt = 0 in an ideal plasma. Seethe discussion of Equation (44). Therefore, δW min-imizing perturbations obey the constraint δ~j · ~∇ψt =~j · ~∇(~ξ⊥ · ~∇ψt), where δ~j ≡ ~∇× δ ~B/µ0. Now ~j · ~∇(~ξ⊥ ·~∇ψt) = ~∇ · [(~ξ⊥ · ~∇ψt)~j]. The normal to the surfaceis n ≡ ~∇ψt/|~∇ψt|, so one can write ~j = n × (~j × n)and (~ξ⊥ · ~∇ψt)~j = ~∇ψt × [(~j × n)~ξ⊥ · n]. This implies~j · ~∇(~ξ⊥ · ~∇ψt) = −~∇ψt · ~∇× [(~j × n)~ξ⊥ · n]. Letting

~C ≡ δ ~B + (~ξ⊥ · n)(µ0~j × n) (97)

one finds that

(~∇× ~C) · ~∇ψt = ~∇ · (~C × ~∇ψt) = 0 (98)

is the condition for the current in the perturbed equilib-rium to lie in the perturbed flux surfaces. This constraintcan be imposed ψt surface by ψt surface since the onlyderivatives that arise are in θ and ϕ. The two constraints~∇ · ~ξ = 0 and Equation (98) determine both µ(ψt, θ, ϕ)and η(ψt, θ, ϕ) in terms of ξψ(ψt, θ, ϕ).

The constraint that (~∇× ~C) · ~∇ψt = 0 must vanish fora δW minimizing perturbation can be obtained directlyfrom the energy principle if wξ⊥ is written as

wξ⊥ =~C2

2µ0− wd, (99)

where wd ≡ (~ξ · n)2(~j × n) · ( ~B · ∇)n (Bernstein, 1983;Bernstein et al., 1958). For a divergence-free perturba-tion the only place the coefficient η enters the energyis through ~C, and there it enters only through δ ~B. Thecontravariant representation of the magnetic field in mag-netic coordinates, Equation (51), plus the dual relationsimply (∂~x/∂θ) × ~B = ~∇ψt/2π. Consequently, the partof ~C that depends on η is ~Cη = ~∇ × (η~∇ψt)/2π. Theminimization of

∫C2d3x through the variation δη gives

2∫~C · ~∇ × (δη~∇ψt)/2πd3x. An integration by parts

then gives the desired result.

2. Pressure-driven ballooning modes

• Perturbations of arbitrarily large wavenumber canbe destabilized by a pressure gradient that is in thesame direction as the field line curvature. Theseinstabilities are called ballooning modes since theyballoon out, have their largest amplitude, at theleast stable place on a constant-pressure surface.

• Plasma instabilities that have a high wavenumberhave a short wavelength in the constant pressuresurfaces across the magnetic field lines but a longwavelength along the magnetic field lines. For bal-looning modes this anisotropy arises to minimizethe stabilizing effect of field line bending. Formicroinstabilities, Section (VI.F), the anisotropyarises to avoid stabilization by Landau damping,Section (VI.C).

The expression for δW is greatly simplified when theperturbations that are considered have a short wave-length across the magnetic field lines, such instabilitiesare called ballooning modes (Connor, Hastie, and Taylor,1978; Todd et al., 1977). Despite their short wavelengthacross the magnetic field lines, the least stable pertur-bations of this type have a long wavelength along themagnetic field. These disparate scales can be efficientlyrepresented using a concept from geometric optics, theeikonal S(α), which satisfies ~B · ~∇S = 0. Let

~ξ⊥ = ~Ξcos (S(α)) , (100)

29

where α is a Clebsch coordinate, Equation (9). For bal-looning mode calculations it is conventional to choose theClebsch coordinates so ~B = ~∇α × ~∇ψp where ψp is thepoloidal magnetic flux and

2πα = ϕ− q(ψp)θm − θ0((ψp) (101)

with q(ψp) ≡ 1/ι(ψp) ≡ dψt/dψp the safety factor. Theeikonal S could be a function of both α and ψp, butthis provides no generality beyond that in the functionθ0(ψp). The eikonal can be written as

S = 2πNα (102)

where N is the toroidal mode number of the perturba-tion. The short wavelength limit means N → ∞. Theexpression that one obtains is

δW

(2πN)2=∫ (~∇α)2

2µ0

(~B · ~∇FB

)2

− κdp

dψpF 2

d3x,

(103)where the relevant component of the field line curvatureis

κ ≡ ~κ · ~B × ~∇αB2

. (104)

The perpendicular displacement of an unstable perturba-tion must have the form of Equation (107), which definesthe function F .

The derivation of the short wavelength form for δW isgiven in this paragraph and can be skipped for those notinterested in the mathematical details. Using Equation(100) for the perpendicular displacement, derivatives ofthe displacement enter δW , Equations (89–91), through

~∇ · ~ξ⊥ = cos(S)~∇ · ~Ξ− sin(S)~Ξ · ~∇S (105)

and

δB⊥ = cos(S)~∇× (Ξ× ~B)⊥. (106)

Note that δB = cos(S)~∇× (Ξ× ~B)− sin(S)~∇S× (~ξ⊥×~B), but the term ~∇S× (~ξ⊥× ~B) = −(~ξ⊥ · ~∇S) ~B doesnot contribute to the magnetic perturbation perpendic-ular to the field lines. When these expressions are in-serted into Equations (89–91), the terms proportional to~∇S and hence N appear only in a positive definite termthat is proportional to (~∇ · ~ξ⊥ + ~ξ · κ)2, consequently in-stability is possible in the limit as N → ∞ only if onechooses ~ξ⊥ · ~∇S = 0. By choosing a 1/N correction to~ξ⊥ appropriately, one can eliminate the positive definiteterm (~∇ · ~ξ⊥ + ~ξ · κ)2 from δW altogether since no otherterm is affected in the lowest non-trivial order by the 1/Ncorrection. The perpendicular displacement for unstableshort wavelength modes must have the form

~ξ⊥ =~B × ~∇SB2

F (107)

since it must be perpendicular to both the magnetic fieldand ~∇S. One then finds ~ξ⊥ × ~B = F ~∇S and δ ~B =~∇F × ~∇S. The parallel current term in Equation (91)vanishes because (~ξ⊥× ~B) ·δ ~B⊥ = F ~∇S ·(~∇F× ~∇S) = 0.As discussed earlier, the parallel current can not be asource of instability for short wavelength perturbations.The perpendicular part of the magnetic perturbation isδ ~B⊥ = (b · ~∇F )(b × ~∇S) with b ≡ ~B/B. Putting thepieces together one obtains Equation (103).

The form for δW for short wavelength perturbationsimplies unstable modes must have a long wavelengthalong the magnetic field lines in order to minimize the( ~B · ~∇F )2 term. If one can choose θ0 in α so κdp/dψtis positive, then δW predicts instability if the variationof F along the field lines is not taken into account. The( ~B · ~∇F )2 term can stabilize the mode in places whereκdp/dψt is positive, but the mode amplitude is naturallylarger there. That is, the plasma displacement balloonsout at the locations where κdp/dψt is positive.

If the ballooning experession for δW is extremized, oneobtains the differential equation

∂

∂θ

(~∇α)2

µ0

~B · ~∇θB2

∂F

∂θ

+

2κ~B · ~∇θ

dp

dψpF = 0, (108)

where ~B · ~∇ = ( ~B · ~∇θ)∂/∂θ and the Jacobian of (ψp, θ, α)coordinates is 1/ ~B · ~∇θ. Equation (108) predicts insta-bility if a solution F crosses F = 0 twice. To under-stand why this prescription works, note that if one ofthe F ’s in the F 2 term of Equation (103) is replacedby the F from Equation (108), one finds after an in-tegration by parts that δW = 0. Now add a term∫

∆sF2dψpdθdα/2 to both sides of the ballooning ex-

pression for δW/(2πN)2, Equation (103), with ∆s aconstant. Extemizing δWm/(2πN)2 ≡ δW/(2πN)2 +∫

∆sF2dψpdθdα/2, one finds Equation (108) is modified

by the zero on the right hand side becoming ∆sF . Solvethis modified equation. If the solution has two placeswhere F = 0 when ∆s > 0, replace an F in the mod-ified equation, δWm/(2πN)2, with the F from the dif-ferential equation. One then finds δWm/(2πN)2 = 0, soδW/(2πN)2 = −

∫∆sF

2dψpdθdα/2, which is negative.That is, one has found a perturbation that reduces theenergy, so instability is predicted.

3. Minimization of δW

Given any positive definite normalization of the dis-placement, ‖ξ2‖, a plasma is unstable if one can find adisplacement such that

λ ≡ δW

‖ξ2‖(109)

is negative. The sign of λ is independent of the normal-ization. However, different normalizations provide differ-ent types of information. Three normalizations will be

30

discussed, the kinetic energy norm, the ξψ norm, and thesurface norm.

The conventional normalization of the displacementis the kinetic energy norm, ‖ξ2‖k ≡

∫ρξ2d3x/2, which

makes√

(−λ) the growth rate of the instability. Thisnorm is poorly behaved when the plasma is stable be-cause then a minimization of λ can mean a maximiza-tion of ‖ξ2‖k. As we have seen, the coefficient µ of thedisplacement, Equation (96), is determined by a mag-netic differential equation, which means the Fourier com-ponent of µ that resonates with a rational surface canbe made arbitrary large in that neighborhood with lit-tle change in δW . That is, one can always find a con-tinuum of modes that make λ → 0 for an essentiallyfixed δW . The existence of this continuum (Grad, 1973),(Goedbloed, 1998) makes it difficult to find the points ofmarginal stability. Even in the unstable region, the dis-placement that minimizes δW is usually not the pertur-bation that minimizes λ. Consequently, a minimizationof λ using displacements that are divergence free and sat-isfy (~∇× ~C) · ~∇ψt = 0 does not give the correct growthrate for unstable modes.

The singularity of the displacement, ~ξ, near rationalsurfaces is particularly important for the theory of per-turbations that are rotating relative to the plasma. Asingular plasma displacement rotating through a plasmagives an infinite correction to the energy. This singular-ity can resolved (Betti and Freidberg, 1995), but at thecost of an imaginary contribution to the energy, whichrepresents the torque between the perturbation and theplasma. Though not noted by (Betti and Freidberg,1995), the resolution of the singularity also changes thereal part of the energy (Boozer, 2003), which implies thecritical value of the plasma parameters that are requiredfor the stability of the perturbation are very sensitive torotation.

A second choice for the normalization of the displace-ment is similar to the kinetic energy except only thecomponent ξψ is retained, ‖ξ2‖ψ ≡

∫ρ(ξψ)2d3x/2. The

quantity λ has no special interpretation, but the contin-uum problem in finding points of marginal stability iseliminated.

A third choice of normalization uses ξψ but only itsvalue on the plasma surface, ‖ξ2‖s =

∮(ξψ)2wdθdϕ with

w(θ, ϕ) > 0 an arbitrary weight function. This nor-malization has pathological features for internal plasmamodes, which have a displacement that reduces the en-ergy while keeping the plasma boundary fixed. Forthese modes λ → −∞. However if the minimizing λis bounded, the displacements of the plasma surface, theξψ’s, that are associated with a spectrum of λ’s give themodified equilibria of the plasma in the presence of ex-ternal perturbations.

An instability with a negative but bounded λ with afixed perturbation of the surface shape, the ‖ξ2‖s norm,is called an external mode. External modes would bestabilized if a perfect conductor were close enough tothe plasma surface. A magnetic perturbation cannot

penetrate a perfect conductor, so the component of themagnetic field line displacement ~ξ that is normal to theconductor must vanish. Actual plasmas are surroundedby chamber walls, which are conductors. An instabilitythat would be stabilized if these conducting structureswere perfectly conducting is called a wall mode. Wallmodes can grow on the resistive time scale of the con-ducting structures, which is many orders of magnitudeslower than the growth rate determined by plasma iner-tia. Due to their slow growth, resistive wall modes canbe stabilized by plasma rotation and by feedback, Section(V.D.2). The stabilization of wall modes is considered animportant topic for tokamak and ST plasmas that havethe profile of net current that is required for steady-stateoperation.

When one uses a fixed perturbation of the boundary,the ‖ξ2‖s norm, the Fourier coefficient of ξψ in magneticcoordinates can jump at each resonant rational surface,ι = n/m, and the magnitude of the jump gives the magni-tude of the surface current on the rational surface. Oncethe expansion coefficients η and µ of the displacement,Equation (96), have been eliminated from δW in favorof ξψ, radial derivatives of the displacement arise only inthe form ∂b/∂ψt where

b ≡ δ ~B · ~∇ψt~B · ~∇ϕ

. (110)

Equation (45), δ ~B = ~∇ × (~ξ × ~B0), implies that b =(∂/∂ϕ + ι∂/∂θm)ξψ. If one allows the resonant Fouriercoefficient of ξψmn to change by a fixed amount over a dis-tance δψt about a rational surface ι = n/m, the resonantFourier coefficient of ∂b/∂ψt has a well-defined limit oneach side of the rational surface as δψt → 0. One canshow that the jump in the resonant coefficient of ξψ orin ∂b/∂ψt is proportional to the surface current on therational surface (Nuhrenberg and Boozer, 2003). For alarge aspect ratio torus, this was shown in the deriva-tion of Equation (78). The amplitude of the jumps inξψ at rational surfaces gives a measure of the width ofthe islands that would open in the absence of a singularcurrent. For a large aspect ratio torus with circular sur-faces, ψt ∝ r2 with r the minor radius, one can easilyshow that the half-width of the island that would arisein the absence of the singular current is

(δr)mn =

√∣∣∣∣2rmnm[(ξr)mn]

∣∣∣∣. (111)

One can use an ideal δW method to assess and im-prove the quality of equilibria and to find the requiredperturbations to the plasma surface to remove islands(Nuhrenberg and Boozer, 2003).

D. Interaction of plasmas with coils

• Between the plasma and the surrounding coils, themagnetic field has the form ~B = ~∇φ. Since ~∇· ~B =

31

0, the interaction of a plasma with currents outsidethe plasma must be through solutions to Laplace’sequation, ∇2φ = 0, which greatly constrains theform of the interaction.

• Practical coils can control only a certain number,Nf , features of the plasma shape. For axisymmet-ric systems Nf ≈ 4, but for stellarators Nf ≈ 50.

• Important long-wavelength instabilities of toka-maks would be stabilized if the surrounding cham-ber walls were perfectly conducting. Such instabil-ities are called resistive wall modes since the insta-bilities can grow on a time scale determined by theresistivity of the chamber walls. Their slow growthpermits feedback stabilization. The interaction ofthe plasma with the perturbing currents in wallsand in feedback coils can be calculated by coupledcircuit equations. The plasma circuit elements havea determinate response to changes in the other cir-cuit elements.

The external coils that are required for the magneticconfinement of a plasma have two distinct functions.First, they provide the required toroidal magnetic flux,ψt. Second, they ensure ~B · n is zero on the plasma sur-face. These two functions are separated in tokamak coildesign: the toroidal field coils provide the flux and thepoloidal field coils ensure ~B · n = 0, Figure (4). However,in stellarators these functions are often coupled in a sin-gle coil set, Figure (6). Nevertheless in discussions of theefficiency and practicality of stellarator coils, it is oftenuseful to separate the two functions.

Critical issues in coil design are the ratio of the mag-netic field at the coils to that on the plasma, the com-plexity and the forces associated with the required coilcurrents, and the production of a field with the requiredsymmetry while providing for ports, for plasma access,and for discrete coils.

A point that is often emphasized in the magnetic fusionliterature is that, if all else is equal, the power producedby a fusion plant scales as the square of β ≡ 2µ0p/B

2.A point less frequently made is that, under the same as-sumptions as those that give the β2 scaling, the powerproduction scales as the fourth power of the ratio of themagnetic field on the plasma to that on the coils. If theplasma β is limited, then the optimal plasma temperatureis the one that maximizes the fusion power production ata fixed plasma pressure, p = nT . The power from the fu-sion of deuterium and tritium scales as pDT = n2fDT (T )with n the plasma number density and fDT a functionof the plasma temperature. The maximum of pDT atfixed pressure occurs at d(fDT /T 2)/dT = 0. This con-dition implies that near the optimal temperature, whichis about 20kev, the fusion power density pDT is propor-tional to the pressure squared or equivalently to β2.

Between the coils and the plasma, the magnetic fieldis curl free,

~B = ~∇φ (112)

as well as divergence free, so

∇2φ = 0. (113)

The interactions of the coils with the plasma must oc-cur through solutions to Laplace’s equation, which placesstrong constraints on the form of this interaction. Muchof the theory of coil design, such as the work of (Merkel,1987, 1988) on the coil design for the W7-X stellarator,is an application of the theory of Laplace’s equation.

1. Freedom of coil design

The choice of plasma shape is the major determinantof the quality of a confined plasma. Unfortunately, thetheory of Laplacians says that a general plasma shapecannot be supported by distant coils. What can be done,through the design of coils, is to enforce a certain numberof conditions on the plasma shape. This number, whichis the number of degrees of freedom in the coil design Nf ,is in practice about four for tokamaks and about fifty forstellarators.

The theory of Laplace’s equation implies that is ismathematically impossible for a distant coil set to pro-vide a generic normal magnetic field, ~B · n, on the plasmasurface, even when ~B · n is a smooth analytic function.This statement is proven by giving a generic normal fieldthat cannot be produced by distant coils. Consider thecylindrical problem in which the radial magnetic fieldthat must be produced by coils, Br(θ) =

∑Bm cos(mθ),

is given on the surface r = a. Between the coils andthe plasma the magnetic field obeys Equations (112)and (113), so φ =

∑ aBm

m

(ra

)m cos(mθ). In general amaximum value of r/a exists for which this Fourier se-ries converges, which means there is a maximum valueof r/a for which a solution exists. To find this maxi-mum, consider the covergence properties of the Fouriercoefficients Bm =

∫Br(θ)cos(mθ)dθ/π. If Br is an an-

alytic function of θ this integral can be performed us-ing the method of residues of complex analysis. If θpis the distance of the nearest pole of Br(θ) from thereal axis, then as m → ∞ the Fourier coefficients havethe form Bm = Bc exp(−mθp). In other words, theFourier coefficients of a generic function Br(θ) decay ex-ponentially as m → ∞. Now consider the coefficientsBm (r/a)m. As m → ∞ these coefficients are propor-tional to exp[m(ln(r/a)− θp)], which means they divergeexponentially with m if r/a > exp(θp).

Even when distant coils can in principle produce therequired magnetic field on the plasma surface, practi-cal coils may not exist. Laplace’s equation implies thatthe intrinsic difficulty of distant coils is exponentiallydependent on the wavenumber k of the distribution ofnormal magnetic field that they are producing on theplasma surface. The number of independent distribu-tions of magnetic field that can be produced by practicalcoils is comparable to the number of Fourier componentsof the magnetic potential φ, Equation (112), that have

32

a wavenumber smaller than some critical value, km. Ina large aspect ratio circular tokamak, k = m/a and theratio of magnetic field on the coils at r = b to that on theplasma at r = a is (b/a)(m−1). For a practical separationbetween the coils and the plasma in a fusion power plant,b/a ≈ 1.7, the ratio of the field at the coil to that on theplasma is five to one for m = 4. Practical tokamak coilscan control about four properties of the plasma, crudelythe poloidal Fourier harmonics up to Mm = 4, so toka-mak coil designs have four degrees of freedom, Nf = 4.For stellarators the wavenumber of a magnetic field dis-tribution is k '

√(m/a)2 + (n/Ro)2 where a is the minor

and Ro is the major radius of the torus. To preserve theNp-fold periodicity of the stellarator the toroidal modenumber of a field distribution must satisfy n = nhNpwhere nh is an integer. Summing over the possible com-binations of m and nh that satisfy k ≤Mm/a, one findsthat the number of degrees of freedom of stellarator coilsis

Nf ≈ 1.8M2m

εp, (114)

where the inverse aspect ratio per period of the stellaratorεp ≡ Npa/Ro. Typically εp ≈ 1/2, so with Mm = 4,stellarator coils have about fifty degrees of freedom.

Efficient coils for stellarators should exist since thenumber degrees of freedom in stellarator coil design,Nf ≈ 50, is larger than the number of shape param-eters, Ns, that are needed to achieve reasonably opti-mized plasma shapes. The COILOPT code (Strickler,Berry, and Hirshman, 2002) tackles this problem directlyby varying parameters that define the coils until a com-bined optimum is found for the coils and the plasma.However, the large number of degrees of freedom and thenon-linearity of the optimization process means the de-sign of stellarator coils for optimal plasmas is extremelysubtle, and it is important to understand the choices thatare made during this optimization.

The fundamental choices in the optimization of a stel-larator are the choice of the plasma shape and the choiceof the coils. The optimized plasma shapes, ~xs(θ, ϕ), arefound using a set of Ns parameters, (Nuhrenberg et al.,1995) and (Neilson et. al., 2000). For example, the shapeparameters could be the Fourier coefficients of Rs(θ, ϕ)and Zs(θ, ϕ) of a representation of the surface in (R,ϕ,Z)cylindrical coordinates. The optimization process forthe plasma shape determines Ns shape parameters, soplasma optimization can only determine Ns constraintson the coils. Practical limits exist on the number of shapeparameters that can be determined. These limits arisenot only from the increasing difficulty of optimizationwith additional parameters, but also from the numericalaccuracy of the codes that determine the physics prop-erties associated with a given plasma shape. The con-straints on the coils should be chosen to ensure the Nsknown shape parameters have their desired values butwith no constraints imposed for fitting unknown shapeparameters. This can be done by studying the relation

between changes in the plasma shape and small changesin the fields due to the coils, which is discussed in Section(V.D.2).

2. Plasma response to coil changes

The relation between perturbations to the plasmashape and small changes in the magnetic field due to thecoils is important for: stellarator coil design, the elimina-tion of islands due magnetic field errors, and the designof feedback systems for resistive wall modes in tokamaks.

A small change in the plasma shape is equivalent to anormal displacement ~ξ · n of the plasma surface and hasa simple and unique relation to the normal componentof the perturbed magnetic field on the original plasmasurface δ ~B · n. To prove this, we first note that a tangen-tial displacement, which means ~ξ · n = 0 but ~ξ 6= 0 with~xs(θ, ϕ) → ~xs(θ, ϕ)+~ξ(θ, ϕ), does not change the plasmashape but does change the (θ, ϕ) coordinate system thatdescribes that shape. In an ideal plasma δ ~B = ~∇(~ξ× ~B),Equation (45), and δ ~B·~∇ψt = ~B·~∇(~ξ·~∇ψt), which relates~ξ · n and δ ~B · n since the normal to the plasma surface isn = ~∇ψt/

∣∣∣~∇ψt∣∣∣. Actually the equation δ ~B = ~∇×(~ξ× ~B)holds whether or not the magnetic field is embedded ina perfectly conducting plasma provided the rotationaltransform on the surface is an irrational number. Thereason is δ ~B = ~∇ × δ ~A. But, any vector, includingthe perturbed vector potential, can be written in theform δ ~A = ~ξ × ~B + ~∇δg provided a function δg(~x)exists that satisfies the magnetic differential equation~B · ~∇δg = ~B · δ ~A. This equation for δg is solvable whenthe rotational transform ι is an irrational number. Thegeneral validity of δ ~B = ~∇×(~ξ× ~B) on irrational surfacesis another way of stating that the rotational transform isnot changed in linear order by a perturbation ~ξ exceptnear a rational surface, Equation (70).

Functions of θ and ϕ, such as the magnetic field per-turbation normal to the original plasma surface δ ~B · n,are conveniently expressed using ortho-normal functions.A set of dimensionless functions fi(θ, ϕ) on the plasmasurface are ortho-normal if∮

fif∗j wda = δij (115)

where w > 0 is an arbitrary weight function and da is thearea element of the plasma surface. The normal compo-nent of the magnetic perturbation on the original plasmasurface is expanded in the fi’s in the form,

δ ~B · n = w∑

Φjf∗j . (116)

The expansion coefficients have units of flux

Φi =∮fiδ ~B · nda. (117)

33

The derivation of theNs constraints on stellarator coilsthat are required to fit Ns parameters in the plasmashape illustrates a response analysis. A small changein each of the Ns shape parameters produces a smallchange in the plasma shape, which is a normal displace-ment ~ξs · n of the plasma surface, and determines a smallchange in the normal magnetic field δ ~Bs · n. The Nsfunctions βs(θ, ϕ) ≡ δ ~Bs · n can be transformed into aset of not more than Ns ortho-normal functions fs(θ, ϕ)by a Gram-Schmidt process such that any of the δ ~Bs · ncan be expanded in terms of the fs’s, Equation (116).The constraints on the coils

∮fs ~B · nda = 0 then en-

sure that through linear order in the field error that theNs plasma shape parameters are reproduced but with noconstraints on the coils from unknown shape parameters.Good stellarator coils exist if the Ns field distributionsthat are needed are in the solution space of the Nf fielddistributions that efficient stellartor coils can produce.

In a number of problems involving coils, it is neces-sary to know the change in the normal magnetic field onthe plasma surface due to the coils, δ ~Bx · n, that is re-quired to produce a given normal magnetic field on theplasma surface, δ ~B · n. The difference between δ ~B · n andδ ~Bx · n is the normal field produced by the perturbedplasma current δ ~Bi · n. The decomposition of δ ~B · ninto a part produced by coil currents and a part pro-duce by internal plasma currents can be carried out ifthere is enough information to solve Laplace’s equationfor the magnetic potential φ between the plasma and theconducting structures. Two boundary conditions are re-quired. In a cylinder, a solution to Laplace’s equation hasthe form φ =

φi(ar

)m + φx(ra

)m cos(mθ) and bound-ary conditions must determine both φi, which gives theeffect of currents inside the plasma, and φx, which givesthe effect of currents outside the plasma.

The response of the plasma to perturbations deter-mines the relation between the perturbed normal fielddue to external current δ ~Bx · n and the normal field dueto the internal perturbed plasma currents, δ ~Bi · n. Sinceonly two boundary conditions are required to determinea unique solution to Laplace’s equation, any two inde-pendent pieces of information will suffice. One exampleis the relation between the perturbed normal and tangen-tial magnetic field components. This information is con-tained in δ ~B at the location of the unperturbed plasmasurface. Since δ ~B is continuous at the plasma surface,but δ ~B = ~∇φ outside the plasma, the vector δ ~B givestwo functions of information on the plasma surface, φand n · ~∇φ = n · δ ~B, and not three as one might firstassume.

Given a plasma model, a linear relation exists at thelocation of the orginal plasma surface between a smallchange in the normal component of the magnetic fieldproduced by external currents δ ~Bx · n and the perturbednormal field δ ~B · n. This linear relation is convenientlyexpressed using matrices. The fluxes that are the ex-

pansion coefficients of the perturbed normal field on theplasma surface form a matrix vector ~Φ ≡

∮~f(δ ~B · n)da.

The matrix vector ~f has the functions fi(θ, ϕ) as its com-ponents. Analogously, the fluxes that are the expansioncoefficients of the normal field on the plasma surface dueto perturbed coil currents form a matrix vector ~Φx. Thelinear relation between δ ~Bx · n and δ ~B · n determines thepermeability matrix,

~Φ =↔P ·~Φx. (118)

The permeability matrix↔P is a property of the plasma

and for an ideal plasma can be determined using a δWstability code. Once

↔P is known, it is straight forward

to uniquely relate any small perturbation in the plasmashape to the change in the normal magnetic field onplasma surface that is produced by the coils.

Tokamak and stellarator plasmas are particularly sen-sitive to magnetic perturbations that resonate with low-order rational surfaces and cause islands to open. Thecoils can be modified to eliminate the islands in oneplasma state (Hudson et al., 2002), though error correc-tion or trim coils may be needed to eliminate islandsover a broad range of plasma states. The issue of islandswould not exist in a tokamak with perfect axisymme-try, but it does exist even for an ideal stellarator if thenumber of periods is small Np <∼ 6. As the plasma equi-librium changes, a magnetic field that was non-resonantcan develop resonant components. However, there is arelatively small number, Nr, of low order rational sur-faces that are consistent with the periodicity of a stel-larator. This means that any set of trim coils that obeysthe periodicity of the stellarator and has Nr sets of leadscan control the islands provided the matrix between thefields produced by the trim coils and the resonant fieldsis non-singular. In practice tokamaks are not preciselyaxisymmetric and stellarators are not precisely periodic,so additional coils may be needed to correct field errorsthat break these symmetries.

The information that is required to determine whethera given set of trim coils can control the islands of a plasmaequilibrium can be found using an ideal δW stabilitycode. As discussed in Section (V.C.3), perturbations atthe edge of an ideal plasma, ~ξ · n, can cause a jump in~ξ · ~∇ψt at the rational surfaces. These jumps imply is-lands would open in a resistive plasma with the islandwidth proportional to the square root of the jump. Ifthere are Nr sets of integers n/m that give the rationalnumbers associated with these jumps, then there are Nrspecific forms for ~ξ · n on the plasma surface that areproportional to the jumps but otherwise have the small-est possible amplitude,

∮(~ξ · n)2wda. All other small

displacements of the plasma edge cause no jumps and,therefore, no islands. These Nr specific forms for ~ξ · n onthe plasma surface give Nr distributions of normal mag-netic field, δ ~Br · n = n · ~∇φr and have Nr associated per-turbed tangential fields on the plasma surface, which give

34

the magnetic scalar potentials φr there. From the φr andthe n · ~∇φr, one can construct the Nr ortho-normal func-tions fr(θ, ϕ) that give the externally produced fluxes∮frδ ~Bx · nda that control the islands. These fluxes are

linearly related to the currents in the trim coils. If thereare Nt independent sets of trim coils, which means inde-pendent currents, It, then the fluxes required for islandcontrol obey

∮frδ ~Bx · nda =

∑tMrtIt with the mu-

tual inductance matrix Mrt a Nr × Nt matrix. If thismatrix has Nr non-zero eigenvalues, the trim coils cancontrol the island causing resonances, though practical-ity requires that none of theNr eigenvalues be excessivelysmall.

In tokamaks important kink-like instabilities, the resis-tive wall modes, can be stabilized by conducting struc-tures, such as the chamber wall that surrounds theplasma, Section (V.C.3). These instabilities grow on theresistive time scale of the wall and can be feedback sta-bilized using external coils. The permeability matrix, P

↔,

contains all of the plasma information that is needed forcalculating the stability and the feedback of wall modes.The circuit representation of the plasma that is associ-ated with the permeability matrix has been used to de-sign and interpret feedback systems for a number of ma-jor tokamak experiments (Bialek et. al., 2001). Priorto the development of the circuit representation of re-sistive wall modes, (Lazarus, Lister, and Neilson, 1990)developed a closely related theory for the feedback of thevertical instability of tokamaks. The vertical instabil-ity causes a Z motion of the plasma and limits the de-gree to which tokamak plasmas can be shaped to achievehigher beta operation. A number of groups have de-veloped methods for studying feedback of resistive wallmodes starting with the early work of (Bishop, 1989). Re-cent discussions of feedback techniques have been givenby (Fitzpatrick, 2001), (Chance et al., 2002), (Bondesonet al., 2002), and (Boozer, 2003).

The energy that is required to drive resistive wallmodes comes from the plasma, so a representation of theeffects of the perturbed plasma current on the externalcircuits is critical. Since these effects must be propa-gated through a solution to Laplace’s equation, the ef-fects can be represented by the magnetic field normal tothe plasma surface that is produced by the perturbedcurrents inside the plasma, δ ~Bi · n. This means the ef-fects of the perturbed plasma current can be representedby a surface current flowing on the plasma surface thatproduces the same normal field at the location of the un-perturbed plasma surface as the perturbed plasma cur-rent.

A surface current can be represented as a matrix vectorwith elements that are discrete currents Ji. The normalfield on the surface due to these currents is linear in thecurrents, so it can be represented using Equation (117) asa magnetic flux matrix vector ~ΦJ = L

↔p· ~J . The matrix L

↔p

is called the plasma inductance and depends only on thegeometric shape and the size of the surface. The surface

current that gives the same normal field on the unper-turbed plasma surface as the perturbed plasma currentis

~Ip =↔L−1

p ·(↔P −

↔1 ) · ~Φx. (119)

The representation of a surface current by a matrix vector~J is demonstrated by noting that the current flowing ona surface ψt = ψs has the form

~js = δ(ψt − ψs)~∇κ× ~∇ψt. (120)

This is the most general vector that is divergence free,lies in the surface ~js · ~∇ψt = 0, and is zero except in thesurface ψt = ψs. The current potential κ(θ, ϕ) can beexpanded in the orthogonal functions, κ =

∑J∗i fi(θ, ϕ).

Each element Ji of the matrix vector ~J has units of Am-peres and can be viewed as a current in a circuit. Ac-tually, the general expression for the current potentialκ(θ, ϕ) is the sum of the single-valued current potential∑J∗i fi plus two non-single-valued terms. One of these

terms is −θIs/(2π), which gives the net toroidal currentIs in the surface, and the other is ϕGs/(2π), which givesthe net poloidal current Gs in the surface. The currentsIs and Gs are not usually retained in analyses of wallmodes.

When the plasma is ideal, the permeability matrix,↔P has real eigenvalues that are customarily written as−1/si, (Boozer, 2003). The stability coefficients si areproportional to the energy required to drive the pertur-bation with a positive si implying a negative energy. Anunstable wall mode arises if the matrix

↔P has a nega-

tive eigenvalue, which means a positive si. The relationbetween P

↔and the energy is demonstrated by consider-

ing the power required to drive a perturbing current ~Jon a surface infinitesimally outside of the plasma. Thatpower is Pd = −

∫~js · δ ~Ed3x. Using Equation (120),

Pd = −∮κ(~∇× δ ~E) · d~a. Since ∂δ ~B/∂t = −~∇× δ ~E, the

power is Pd = ~J† · d~Φ/dt. The flux and the current areproportional to each other, ~Φ =

↔P ·

↔Lp · ~J , so the energy

δW required to drive the perturbation, Pd = dδW/dt,has the explicitly real form δW = 1

4 ( ~J† · ~Φ + ~Φ† · ~J).

Since the inductance matrix↔Lp is a positive Hermitian

matrix, the energy δW is negative only if the permeabil-ity,

↔P , has a negative eigenvalue.

The circuit equations for wall modes are particularlysimple in the case of primary importance, a single modepassing through marginal stability, which means one sta-bility coefficient s passing through zero. Because of thesingularity of the permeability matrix at s = 0, onlythe marginal mode is important, so the matrix vectorscan be approximated by a single matrix element, the el-ement that represents the marginally stable perturba-tion. For this problem, Φ = −Φx/s, Φx = MpwIw,Φw = LwIw +MwpIp, and dΦw/dt = −RwIw. Φw is themagnetic flux that penetrates the wall, Iw is the current

35

in the wall, Lw is the wall inductance, Mpw = Mwp is themutual inductance between the wall and the plasma, andRw is the wall resistivity. Simple algebra demonstratesthat the flux through the wall is

Φw = Lw

1− 1 + s

s

MpwMwp

LwLp

Iw. (121)

When the effective inductance of the wall, Φw/Iw, is neg-ative the wall mode grows with a growth rate that is pro-portional to Rw. By adding sensors and actively drivencoils to the circuit equations, one can stabilize the wallmode by feedback.

In a rotating tokamak plasma, the eigenvalues of thepermeability matrix (Boozer, 2003) are complex numbers1/(−si + iαi). The quantity αi gives the toroidal torquebetween the plasma and the mode i. The toroidal torqueexerted by an axisymmetric surface that carries a surfacecurrent ~js is given by τϕ = −

∫(~js × δ ~B) · (∂~x/∂ϕ)d3x,

which implies τϕ = iN2 ( ~J† · ~Φ− ~Φ† · ~J) when the ϕ depen-dence of fi is written as exp(−iNϕ). When an externalperturbation is applied to a rotating tokamak plasma,the perturbation on the plasma surface can be amplifiedand have toroidal phase shift. The amplification and thephase shift are given by si and αi (Boozer, 2003). Plasmarotation can stabilize the resistive wall mode by draggingthe mode toroidally at too rapid a rate for it to penetratethe wall.

VI. RADIATION AND TRANSPORT

• A fusion burn of deuterium and tritium requiresthat the characteristic time for loss of energy fromthe plasma satisfy τE ≈ 4×1021/niT , where τE is inseconds, ni is the number of deuterium-tritium ionsper cubic meter, and T is the plasma temperaturein kilovolts. The characteristic energy confinementtime of a plasma with few impurities and a tem-perature of about 20kev is predominately due tothermal energy transport by plasma processes, notelectromagnetic radiation.

• Energy transport determines the minimum size andhence cost of an experiment to study plasmas thatburn deuterium and tritium. For a power plant,the level of transport must be consistent with therequired system parameters, Section (II).

• The energy confinement time τE of proposed ex-periments is estimated using empirical power-lawscalings, which have the fundamental assumptionthat no critical values are crossed of the parame-ters that enter the power law.

The scaling of the energy confinement time of tori-oidally confined plasmas is a primary issue in the feasi-bility and cost of a burning plasma experiment (ITERPhysics Expert Groups on Confinement and Transport

et al., 1999). By definition a burning plasma experimentburns deuterium and tritium while requiring little ex-ternal power. The loss of energy from a fusing plasmaat 20kev with few impurities is dominated by plasmaprocesses, called energy transport, not electromagneticradiation. The focus on proposals for burning plasmaexperiments has meant that energy transport has been amajor topic for plasma research.

Energy transport is also an issue for the feasibility offusion power, but differences exist between the transportissues for fusion power and for a burning plasma experi-ment. The size of a burning plasma experiment is essen-tially determined by the magnitude of the energy trans-port, the smaller the transport the smaller and cheaperthe required experiment. However, the basic size of apower plant is determined by issues that have little todo with plasma transport coefficients. The critical is-sues in a power plant are whether the energy transportis consistent with what is required to maintain a steadyburn—neither too large nor too small—and whether theparticle transport is sufficiently large to remove alphaparticle ash. In addition, the feasibility of steady-statetokamak power plants is dependent upon the natural pro-files that the temperature and density and current takein a fusion plasma. The performance of stellarator powerplants is far less dependent on the issue of profiles.

Energy transport can provide an upper limit on theplasma beta, β ≡ 2µ0p/B

2, because transport for givenplasma conditions is generally larger the smaller the mag-netic field.

Predictions of the energy confinement times of ex-periments are usually based on empirical scaling rela-tions (ITER Physics Expert Groups on Confinement andTransport et al., 1999). Empirical scaling relations gen-erally assume the mathematical form of a power law,

τE(x1, x2, · · ·) = a0xa11 x

a22 · · · , (122)

so each parameter of the set (x1, x2, · · ·) enters multi-plicatively. The constants (a0, a1, · · ·) are chosen toobtain the best fit to the data. Mathematics impliespower law scaling is a precise description if and only ifthe parameters (x1, x2, · · ·) have no characteristic valuesor scale. A function is independent of the scale, s, iff(x/s) = f(1/s)f(x), which implies f(x) ∝ xa. All otherfunctions have a scale, and the properties of the functiondepend on the size of x relative to s. For example, thescale of sinx is 2π. The constants of a power law, (a0,a1, · · ·), are easily fit to data by a linear regression of thelogarithm of τE against the logarithms of the parameters.

The most commonly used scaling relations are powerlaws based on experimental parameters such as inputpower, plasma current, and plasma size. Less used butmore scientifically appealing scaling relations are basedon dimensionless parameters (Connor, 1984). The di-mensionless energy confinement time is the confinementtime times the cyclotron frequency of deuterium, Ω ≡eB/md. Important dimensionless parameters are the gy-roradius of deuterium ρ ≡ (

√Tmd)/(eB) divided by the

36

plasma radius, which is called ρ∗; the number of bouncesbetween collisions a deuterium ion makes when trappedin the variation in the magnetic field strength on a mag-netic surface, a ratio called 1/ν∗; and β = 〈2µ0p/B

2〉. Interms of these parameters

ΩτE = a0ρaρ∗ ν

aν∗ βaβ . (123)

A. Electromagnetic radiation

• Although electromagnetic radiation is usually asubdominant energy loss mechanism for deuterium-tritium (DT) fusion systems, electromagnetic radi-ation limits the feasibility of non-DT fusion fuelsand the tolerable level of impurities in all fusionsystems.

• The most important types of radiative losses frommagnetically confined plasmas are bremsstrahlung,cyclotron, and atomic.

Electromagnetic radiation is a subdominant energyloss mechanism in a deuterium-tritium (DT) fusionplasma operating at 20kev with a low level of impuri-ties. However, electromagnetic radiation limits the toler-able level of impurities, the range of temperatures, andthe types fuel (Nevins, 1998) for which fusion energy isfeasible. Three types of radiative losses are important:bremsstrahlung, cyclotron, and atomic.

Electromagnetic radiation arises when the current den-sity ~j has both a non-zero curl and time derivative. Thetime derivative of the current density of a single particleis ~j(~x) = q~aδ(~x − ~xp) where ~xp(t) is the position of theparticle, ~a = d2~xp/dt

2 is its acceleration, and δ(· · ·) isthe Dirac delta function. The rate a non-relativistic par-ticle of charge q loses energy H through electromagneticradiation is given by the Larmor formula,

dH

dt= − q2a2

6πε0c3, (124)

which is derived in the standard electrodynamics texts.Bremsstrahlung, which in German means radiation

due to deceleration, arises from the electrostatic scat-tering of one charged particle by another. The powerloss due to bremsstrahlung is dominated by the scatter-ing of electrons by ions. The power loss per unit volumedue to scattering by ions of charge Ze with a numberdensity nz is proportional to Z2nzne

√Te. The typical

photon emitted in bremsstrahlung has an energy equalto the electron temperature. In plasmas of fusion in-terest there is negligible reabsorption of bremsstrahlungradiation because the total radiated power is far be-low the blackbody level. An accurate derivation ofbremsstrahlung is given by (Karzas and Latter, 1961).The basic dependences and power loss of bremsstrahlungcan be understood starting with the expression for theacceleration of one charged particle by another, ~a1 =

(q1q2/4πε0m1)(~x1 − ~x2)/|~x1 − ~x2|3. The maximum ac-celeration occurs occurs at the point of closest approachof the two particles b ≡ min(|~x1 − ~x2|). The characteris-tic frequency of the Bremsstrahlung radiation is ω = v/bwith v the particle velocity, which implies the character-istic wavenumber satisfies kb = v/c. For non-relativisticparticles v/c << 1, the wavelength of the radiation islong compared to the distance between the radiating par-ticles. The time derivative of the current density of twoparticles is ~j(~x) = q1~a1δ(~x − ~x1) + q2~a2δ(~x − ~x2). Sincethe wavelength the radiation is long compared to the dis-tance between the radiating particles, one can ignore thatdistance and let ~j(~x) = ( q1m1

− q2m2

)m1~a1δ(~x − ~x1) us-ing Newton’s third law to write m2~a2 = −m1~a1. Theexpression for ~j implies radiation from electron-electronscattering vanishes in the non-relativistic limit, and thatbremsstrahlung is primarily a result of the electron-ionscattering. The change in energy of an electron withspeed v that passes by an ion that has a charge Ze at adistance b is δH ≈ (dH/dt)(b/v) where dH/dt is given bythe Larmor formula, Equation (124), with the accelera-tion a ≈ (Ze2/4πε0me)/b2. The bremsstrahlung powerpb emitted per unit volume is given by the number of elec-trons per unit volume times the number of ions the elec-trons pass per unit time. The number of ions of chargeZ that the electrons pass per unit time at a range be-tween b and b+db is nzv2πbdb, so pb ≈ nzne

∫δHv2πbdb.

This integral is proportional to 1/bmin with bmin the dis-tance of closest possible approach. In classical mechan-ics the distance of closest possible approach is given by(Ze2/4πε0)/bmin = mev

2/2, but for electrons that haveenergies of importance for fusion systems this distanceis much smaller than the limit set by quantum mechan-ics bminmev ≈ h. Quantum effects determine the closestpossible approach when v/c > 2Zα, where the fine struc-ture constant α ≡ e2/(4πε0ch) ≈ 1/137. The typical en-ergy of an emitted photon is hω ≈ hv/bmin ≈ mev

2 ≈ T .The basic dependences of the emitted power, Z2nzne

√Te

are obtained by assembling the results.Cyclotron radiation results from the acceleration of

electrons with velocity ~v by the magnetic field, ~a =−(e/me)~v× ~B. The characteristic frequency of the asso-ciated motion is the electron cyclotron frequency Ωe ≡eB/me, and this is the frequency at which cyclotron ra-diation is emitted by an electron. A straight forwardapplication of the Larmor formula, Equation (124), givesthe power emitted per unit volume due to the cyclotronmotion of Maxwellian electons, pc = ω2

peΩ2eT/(3πc

3).Since

⟨mv2

⊥/2⟩

is the energy in two independents com-ponents of velocity, and the energy in three indepen-dent components is 3

2T , one has⟨mv2

⊥/2/⟩

= T . Theplasma frequency, ωpe =

√e2ne/(meε0), which is the

frequency with which the electrons would oscillate if dis-placed enmasse from the ions. Thermodynamics impliesthe emission of electromagnetic waves due to the cy-clotron motion can not exceed the energy flux leavinga blackbody in the relevant frequency range. The en-

37

ergy flux leaving a blackbody at frequencies less thanω is Fbb = ω3T/(12π2c2) provided the energy per pho-ton is far less than the temperature, hω << T . Thisis the Rayleigh-Jeans limit, which is the relevant limitwhen ω ∼ Ωe. To be consistent with thermodynamics,the plasma must reabsorb the radiated power within adistance La ≡ 4πFbb/pc = (Ωe/ωpe)(c/ωpe), which isa fraction of a millimeter in a fusion plasma. Due torelativistic effects, electrons also emit radiation at har-monics of the cyclotron frequency with the emission ateach higher harmonic reduced by a factor T/(mec

2) com-pared to the previous harmonic. The power emitted by aregion is comparable to the black body flux Fbb up to afrequency ω that is set by the first cyclotron harmonic forwhich the power can leave the plasma without strong re-absorption. Accurate calculations the power losses due tocyclotron emission are complicated. (Albajar, Bornatici,and Engelmann, 2002) give recent results and historicalreferences.

Atomic radiation arises from electrons switchingatomic levels and from the capture of free electrons intobound states. Significant atomic radiation only arisesif the electron temperature is low enough for the atomor ion to have bound electrons. The atomic radiationfrom an element with a number density nz has the frompz = nenzRz(T ) with Rz(T ) a function that is large atelectron temperatures at which ionization occurs. Thefunction Rz(T ) for a single element can have large varia-tions with multiple peaks. Under fusion conditionsRz(T )is negligible except for elements with high atomic num-bers, such as iron. Functions Rz(T ) have been given by(Post et al., 1977).

At a given impurity level, the power losses due tobremsstrahlung and atomic radiation depend quadrati-cally on the plasma density as does the nuclear powerinput, so the nuclear power input minus the radiativelosses due to bremsstrahlung and atomic radiation canbe viewed as an effective power input, peff = n2feff (T ).The power loss due to cyclotron emission and absorp-tion is more complicated and in some features resemblesa diffusive transport process.

Fusion power is feasible only at low impurity levels.In addition to the enhancement of electromagnetic radi-ation, the ionization of impurities adds many electrons tothe plasma. These electrons exert a pressure and trans-port energy but are unrelated to the production of fusionpower.

B. Kinetic theory

• Kinetic theory is needed to calculate transport co-efficients such as particle diffusivities, thermal con-ductivities, and bootstrap currents.

• Collisions in plasmas change the trajectories of par-ticles diffusively. This is in contrast to collisions inordinary gases which produce large abrupt changesin the particle trajectories.

• Plasma confinement forces the distribution func-tions of the particles to deviate from localMaxwellians. The rate of entropy production thatis required to hold the distribution functions awayfrom local Maxwellians allows a simple calculationof transport coefficients in the low collisionalitylimit.

Transport calculations are carried out using kinetictheory. The fundamental quantity in kinetic theory isthe distribution function f(~x, ~p, t), which describes theevolution of a large group of identical particles. In aplasma, there is a distribution function for the electronsand for each of the ion species. The distribution func-tion is the density of particles in phase, or (~x, ~p), space.In other words, the number of particles in a region ofmomentum, ~p, and ordinary space ~x is the integral ofthe distribution function over that region,

∫fd3pd3x. A

more detailed treatment of kinetic theory than that givenhere can be found in (Helander and Sigmar, 2002) and inmost plasma textbooks.

The evolution of the distribution function is given bythe kinetic equation

df

dt= C(f) (125)

where C(f) is the collision operator. The total timederivative of kinetic theory is an extension of the con-cept of the total time derivative of fluid mechanics, Equa-tion (43), which operates on functions g(~x, t) in ordinaryspace, to functions f(~x, ~p, t) in phase space,

df

dt=∂f

∂t+d~x

dt· ∂f∂~x

+d~p

dt· ∂f∂~p

(126)

where ∂f/∂~x ≡ ~∇f . Equation (126) can also be writtenas

df

dt=∂f

∂t+

∂

∂~x·(d~x

dtf

)+

∂

∂~p·(d~p

dtf

). (127)

If ~p is the canonical momentum of Hamiltonian mechan-ics, Equation (127) follows from (126) because of Hamil-ton’s equations d~x/dt = ∂H/∂~p and d~p/dt = −∂H/∂~x.In the presence of a magnetic field, the canonical momen-tum is ~p = m~v+q ~A, Equation (190). If ~p ≡ m~v, Equation(127) follows because ~x = ~p/m and ~p = q( ~E+ ~p× ~B/m),so ∂

∂~x · ~x = 0 and the momentum space divergence of ~p iszero, ∂

∂~p · ~p = 0.Without collisions the kinetic equation is called the

Vlasov equation, df/dt = 0. The Vlasov equation is eas-ily understood. Given the distribution function at t = t0,the distribution function at t = t0 + δt is obtained by ad-vancing each particle in the distribution along its tra-jectory, ~x = ~x0 + (d~x/dt)δt and ~p = ~p0 + (d~p/dt)δt.If f(~x0, ~p0) is the given distribution, then for an in-finitesimal interval of time, δt = t − t0, the distributionfunction is f(~x− (d~x/dt)δt, ~p− (d~p/dt)δt), which implies

38

∂f/∂t = −(d~x/dt) ·∂f/∂~x− (d~p/dt) ·∂f/∂~p. The Vlasovequation is a hyperbolic, partial differential equation thathas one characteristic. The characteristic is the trajec-tory of a particle.

The collision operator is the momentum space diver-gence of a flux ~F of particles through phase, (~x, ~p), space,

C(f) = − ∂

∂~p· ~F . (128)

Collisions are caused by the graininess of the plasma, aneffect that is not directly described by the distributionfunction f(~x, ~p).

The graininess of the plasma leads to complicated elec-tric fields that scatter particles and cause the collisionalflux ~F . The electric field at position ~x due a charge q at~x1 is

~E(~x, ~x1) = − q

ε0

∫i~k

k2ei~k·(~x−~x1)

d3k

(2π)3, (129)

which is demonstrated by checking that ~∇ · ~E =(q/ε0)δ(~x − ~x1), ~∇ × ~E = 0, and ~E(~x → ∞, ~x1) = 0.Note the Dirac delta function has the representation

δ(~x− ~x1) =∫ei~k·(~x−~x1)

d3k

(2π)3. (130)

The electric field that is obtained from a uniform densityn of charges, 〈 ~E〉 ≡

∫~E(~x, ~x1)nd3x1 is zero. However,

the root–mean–square electric field from a uniform den-sity of charge is not zero,

〈E2〉 ≡∫

~E(~x, ~x1) · ~E(~x, ~x1)nd3x1, (131)

and can be written as

〈E2〉 = n

(q

ε0

)2 ∫ 1k2

d3k

(2π)3. (132)

This electric field causes accelerations ∆~a = (q/m) ~E ofthe particles, which for a fast moving particle persistsfor a time ∆t = 1/kv with v the speed of the particle.That is, a particle has essentially random changes in itsvelocity, ∆~v = (∆~a)(∆t), separated by time intervals oforder ∆t. These random changes scatter the velocityat an average rate ν = 〈(∆v)2/∆t〉/v2. That is, ν =〈(q/m)2E2∆t〉/v2, which can be written as

ν ≈ nq4

m2ε20

1v3

∫1k3

d3k

(2π)3. (133)

The integral over wavenumbers is logarithmically diver-gent without a cutoff at large and at small wavenumbers,

lnΛ ≡∫

d3k

4πk2= ln

(kmaxkmin

). (134)

The cutoff at large wavenumbers, kmax is given by thedistance of closest possible approach of two charged par-ticles, bmin = 1/kmax. As shown in the discussion of

bremsstrahlung, Section (VI.A), bmin is determined byquantum effects at fusion temperatures, bmin = h/(mv).The cutoff at small wavenumbers is given by the De-bye length, kmin = 1/λD. The Debye length is λD ≡√ε0T/(nq2). The Coulomb logarithm ln Λ ≈ 17 in labo-

ratory plasmas.The Debye length, λD ≡

√ε0T/(nq2), is the shield-

ing distance for the electrostatic potential of a charge.What is meant by this? Let Φ be the potential due to acharge Q that is placed in a plasma. The electric force−qn~∇Φ on a species with charge q and number density nis balanced by the pressure force of that species ~∇(nT ).In equilibrium the temperature is constant, so the equi-librium density is n = n∞ exp(−qΦ/T ). The Poissonequation for the potential is then

∇2Φ = −Qε0δ(~x)− qn∞

ε0

(e−

qΦT − 1

). (135)

Far from the charge qΦ/T << 1, and Poisson’s equationbecomes ∇2Φ = Φ/λ2

D. Consequently, for r >> λD theelectric potential of a charge in a plasma is

Φ =Q

ε0

exp(− rλD

)r

. (136)

Conservation laws place three conditions on the colli-sion operator, C(f), and the thermodynamic law of en-tropy increase places a fourth condition. The discussionof these conditions is simpler using phase space coordi-nates in which the momentum is ~p = m~v rather thanthe canonical momentum, ~p = m~v + q ~A, of Hamiltonianmechanics, so these are the phase space coordinates thatwill be used. The three conservation laws are particle,

n(~x) ≡∫fd3p; (137)

momentum,

nm~u(~x) ≡∫~pfd3p; (138)

and energy,

ε(~x) ≡∫

p2

2mfd3p, (139)

conservation. The entropy per unit volume is

s ≡ −∫f ln(f)d3p. (140)

This definition of the kinetic entropy is shown below togive results that are in agreement with the thermody-namic entropy.

The Maxwellian distribution, fM , has the maximumentropy per unit volume, s, with a fixed number of par-ticles per unit volume, n, momentum per unit volume,

39

nm~u, and energy per unit volume, ε = 12nmu

2 + 32nT ,

which defines the temperature T ,

fM (~x, ~p) =n

(2πT/m)3/2exp

(− (~p−m~u)2

2mT

). (141)

That is, one extremizes s+ λ1n+~λ2 ·nm~u+ λ3ε by con-sidering variations in the distribution function δf whiletreating the λ’s as constants. When the λ’s, which arecalled Lagrange multipliers, are chosen to obtain the cor-rect density, momentum, and energy, the Maxwellian isobtained. To be consistent with thermodynamics, a col-lision operator must cause the entropy to increase exceptwhen the distribution function is a local Maxwellian.

A collision operator, C(f) = − ∂∂~p · F , that is simple

but obeys the four conditions has the phase space flux

~F = −↔ν

2·

(~p−m~u)f +mT∂f

∂~p

. (142)

The collision frequency↔ν in this simple collision operator

is momentum independent. The correct collision opera-tor is considerably more complicated and in particularthe collision frequency

↔ν is dependent on the energy of

the particles being scattered, Equation (133).The simplified collision operator of Equation (142) is

useful for illustrating the diffusive nature of collisions inplasmas. A distribution function that develops compli-cated structures in momentum space can be smoothedarbitrarily quickly. More precisely, if the distributionfunction changes over a range of momenta ∆p then thatchange is smoothed out at a rate of order ν(p/∆p)2.

The change in the entropy density s(~x, t) at each pointin a plasma is due to a flux of entropy, ~Fs, and the cre-ation of entropy by collisions, sc. A differentiation of theentropy density s(~x, t), Equation (140), with the use ofdf/dt = C(f) and Equation (127), implies

∂s

∂t+ ~∇ · ~Fs = sc. (143)

The flux of entropy is ~Fs ≡ −∫~v(ln f)fd3p where ~v =

~p/m, and the rate of entropy production by collisions is

sc ≡ −∫

ln(f)C(f)d3p. (144)

The definition of entropy per unit volume s(~x, t), Equa-tion (140), is consistent with the thermodynamic entropyif and only if it satisfies three conditions. (1) The entropy,∫sd3x, must be additive. That is, entropy in a region

that consists of two parts must be the sum of the twoentropies, a condition that s satisfies. (2) The entropy ofan isolated system cannot decrease. This condition im-plies sc ≥ 0. If the distribution function is written inthe form f = fM exp(f) with fm is defined so its n(~x),~u(~x), and ε(~x) are the same as those of f , then Equa-tion (142) for simplified collisional flux gives a positive

entropy production as long as↔ν is a positive matrix,

sc =mT

2

∫∂f

∂~p· ↔ν ·∂f

∂~pfd3p. (145)

(3) The change in the energy due to a transfer of heatis TδS. By taking the time derivative of the energy perunit volume ε(~x, t) ≡

∫(p2/2m)fd3p, one can show, us-

ing Equation (127) and df/dt = C(f) to eliminate ∂f/∂t,that the energy flux is the sum of two parts: a part pro-portional to the fluid velocity ~u and a part equal to theheat flux ~Fh ≡

∫12m(~v − ~u)3fd3p where ~v = ~p/m. The

entropy flux ~Fs is also the sum of two parts when theplasma is near thermodynamic equilibrium, which meansf = fM exp(f) with |f | << 1. One part of the entropyflux is proportional to the fluid velocity ~u and the otherpart is ~Fh/T . In a continuum system the temperature iswell-defined only near themodynamic equilibrium, and inthat limit the third condition on the entropy is satisfied.

Thermodynamics gives a simple but rigorous expres-sion for the heating power P required to maintain a sta-tionary, ~u = 0, steady-state plasma,

P (ψt) = T (ψt)∫scd

3x (146)

where the volume integral is over a region enclosed bya flux surface ψt, and P (ψt) is the total heating powerin that region. To prove this equation, note the powerper unit volume that must be added to balance the heatflux is εh = ~∇· ~Fh while the entropy created by collisionsmust satisfy sc = ~∇ · ( ~Fh/T ).

Equation (146) coupled with Equation (145) im-plies the distribution functions must be close to localMaxwellians on each pressure surface to obtain the con-finement needed for fusion. The required energy confine-ment time in a fusion power plant is about 102 ion col-lision times and 104 electron collision times. The main-tenance of near Maxwellians requires the trajectories ofcharged particles remain close to constant pressure sur-faces during the time between collisions.

Thermodynamics relates the collisional entropy pro-duction sc and the plasma transport in an even morecomplete form. Given sc, one can read off the trans-port coefficients. For simplicity, assume that at eachpoint in the plasma the distribution function is well-approximated by a stationary Maxwellian, so the flowvelocity ~u = 0. The textbook thermodynamic equa-tion, dU = TdS − pdV + µdN , relates thermodynamicquantities of an entire system. Plasma studies use theenergy ε ≡ U/V , the entropy s ≡ S/V , and the num-ber of particles n ≡ N/V per unit volume. Substitutingthese definitions into the thermodynamic equation yields(dε−Tds−µdn)V = −(ε−Ts+p−µn)dV . The thermo-dynamic properties of a plasma are independent of theplasma volume, V , so both sides of this equation mustbe zero. The chemical potential µ can be evaluated for aMaxwellian using µ = (ε− Ts+ p)/n, and one finds

µ

T= c0 + ln

( n

T 3/2

)(147)

40

with c0 a constant. Consequently a stationaryMaxwellian has the form

fM = cM exp(µ−mv2/2

T

)(148)

where cM is a constant. The thermodynamic equationdε = Tds + µdn implies that the time rate of change ofthe entropy density is ∂s/∂t = (1/T )∂ε/∂t−(µ/T )∂n/∂t.Now ∂n/∂t = −~∇ · ~Γ with ~Γ the diffusive particle flux,and ∂ε/∂t = −~∇ · ~Fh with ~Fh the heat flux. Therefore,

∂s

∂t= −~Γ · ~∇µ

T+ ~Fh · ~∇

1T− ~∇ ·

(1T~Fh −

µ

T~Γ). (149)

The divergence term on the right hand side of Equa-tion (149) is the divergence of the entropy flux, but theother terms are due to the irreversible production of en-tropy, which means the production by collisions. Sincethe chemical potential and the temperature depend onlyon the ψt coordinate, the collisional entropy productionis

sc = −Γdµ/T

dψt+ Fh

d1/Tdψt

(150)

with Γ ≡ ~Γ · ~∇ψt and Fh ≡ ~Fh · ~∇ψt. The quantities−d(µ/T )/dψt and d(1/T )/dψt are called thermodynamicforces, and Fh and Γ are the conjugate fluxes. Near ther-modynamic equilibrium, the fluxes are proportional tothe forces, Γ = −Dnd(µ/T )/dψt − Dcd(1/T )/dψt andFh = Dcd(µ/T )/dψt +DT d(1/T )/dψt. The cross terms,the terms proportional to Dc, have the same coefficientdue the symmetry of the collision operator, or more gen-erally due to Onsager symmetry (Onsager, 1931). Con-sequently,

sc = Dn

(dµ/T

dψt

)2

+ 2Dcdµ/T

dψt

d1/Tdψt

+DT

(d1/Tdψt

)2

.

(151)By calculating sc, one can obtain all three independenttransport coefficients.

Much freedom exists in the choice of variables ξj thatare used to describe the distribution function. The lefthand side of kinetic equation df/dt = C(f) becomesdf/dt = ∂f/∂t+

∑j(∂f/∂ξ

j)(dξj/dt). All that is neededis knowledge of how each of the variables changes alongthe trajectory of a particle, dξj/dt. For example, f is asolution of the collisionless kinetic equation df/dt = 0,which is called the Vlasov equation, if and only if itsvariables are constants of the motion of the particles,dξj/dt = 0. In a system that is independent of time,f(H) is a solution to the Vlasov equation where H is theHamiltonian or energy of a particle.

The calculation of density perturbations using kinetictheory is subtle when the unperturbed distribution func-tion is given as a function of a constant of the motionsuch as the Hamiltonian, f0(H). Consider a collisionlessplasma that is perturbed by a change in the electric po-tential, δΦ. The distribution function in the presence of

the perturbation is f = f0(H) + δf . The density per-turbation that is caused by the perturbation δΦ is not∫δfd3v, but δn =

∫δfd3v +

∫qδΦ(df0/dH)d3v. That

is, the total change in the distribution function is

∆f ≡ δf + qδΦdf0dH

. (152)

To understand the reason, suppose the plasma is per-turbed by an electric potential δΦ = Φ sin(kx − ωt).The kinetic equation of a collisionless plasma, df/dt = 0,can be solved in two ways, and a subscript of v or Hwill be placed on δf to indicate which set of variables,(x, v, t) or (x,H, t), is used to find the solution. First, us-ing variables (x, v, t) one can write dδfv/dt = ∂δfv/∂t+v∂δfv/∂x = −(qE/m)df0/dv since dv/dt = qE/m. Thisequation implies

δfv =k

kv − ω

qδΦm

df0dv. (153)

Second, using variables (x,H, t) one can write dδfH/dt =∂δfH/∂t + v∂δfH/∂x = −(df0/dH)dH/dt where thechange in the energy of a particle is dH/dt = q∂δΦ/∂t.The solution is

δfH =ω

kv − ωqδΦ

df0dH

. (154)

These two solutions are not equal even after making thesubstitution df0/dv = (df0/dH)mv. Indeed, δfv = δfH+qδΦdf0/dH. The resolution of this paradox is that theunperturbed Hamiltonian H0 goes to H = H0 + qδΦ.A first order Taylor expansion implies f0(H) = f0(H0)+qδΦdf0/dH0. The term qδΦdf0/dH is called the adiabaticpart of the plasma response.

C. Landau damping and quasilinear diffusion

• A collisionless, but unidirectional, transfer of en-ergy can occur between a wave and the particlesthat form a plasma. The sign of this transfer, whichis called Landau damping, is determined by the signof the velocity derivative of the distribution func-tion at the place where the particle velocity equalsthe phase velocity, ω/k, of the wave.

• If the energy density of waves has a continuous de-pendence on their phase velocity, ω/k, the wavescause a collisionless diffusion of the particles whosephase velocities resonate with the phase velocitiesof the waves. This is called quasilinear diffusion.

In fusion, and many other plasmas of interest, the col-lision frequency is small compared to the time it takesa charged particle to cross a distance comparable to theplasma size. Since collisional effects are weak, it is nat-ural to study the Vlasov equation, which is df/dt = 0.This hyperbolic partial differential equation has the par-ticle trajectories as its single characteristic. In other

41

words, the distribution function is carried along by theparticle motion, so any function of the constants of mo-tion is a solution to the Vlasov equation.

Since the Vlasov equation, df/dt = 0, contains onlyinformation about the particle trajectories, which aretime reversible, it is natural to assume that solutionsto the Vlasov equation are themselves time reversible.This turns out to be false in practice, with the sense oftime determined by the nature of the initial conditions.Solutions to Vlasov equation can evolve from smooth ini-tial conditions into functions with arbitrarily complicatedstructures. Plasma observations have finite resolution,and the averaging that is implicit in the observation pro-cess leads to irreversibility. These time irreversible col-lisionless effects are important in plasma physics for thedamping of externally driven waves, which gives plasmaheating, and for amplifying plasma waves, which givesinstabilities. The two collisionless phenomena that willbe discussed are Landau damping (Landau, 1946) andquasilinear diffusion (Drummond and Pines, 1962; Vede-nov, Velikhov, and Sagdeev, 1961).

Landau damping is a collisionless, but unidirectional,transfer of energy between an electromagnetic wave anda plasma. Landau damping is central to the theoryof plasma heating by waves and the destabilization ofelectro-magnetic perturbations by non-Maxwellian dis-tributions of the plasma species.

To understand Landau damping, assume that at t = 0the distribution function of a plasma species is indepen-dent of position, f0(v), but a weak electric field is intro-duced that for t > 0 has the form

E = Ek cos(kx− ωt). (155)

The problem has only one spatial coordinate x and onevelocity coordinate, which is vx but will be denoted by vfor simplicity. The perturbed Vlasov equation, df/dt = 0with f = f0 + δf , is

∂δf

∂t+ v

∂δf

∂x+

q

mE∂f0∂v

= 0. (156)

The solution for the perturbed distribution function δfthat obeys the initial condition that δf = 0 is

δf =qEkm

∂f0∂v

sin(kx− ωt)− sin(kx− kvt)ω − kv

, (157)

which is non-singular though growing at v = ω/k,

(δf)v= ωk

= −∂f0∂v

qE

mt. (158)

Particles moving at the phase velocity of the perturba-tion, v = ω/k, see a time independent electric field andare accelerated by it. A spatial average is defined by thelimit as L goes to infintiy of 〈g(x)〉 ≡ 1

2L

∫ +L

−L gdx. Thespatially averaged power going to the plasma is

〈P 〉 =∫v〈qEδf〉dv. (159)

Using the trigonometric identity sin(kx−kvt) = sin(kx−ωt) cos[(ω − kv)t] + cos(kx− ωt) sin[(ω − kv)t], one has

〈qEδf〉 = −q2E2

k

2m∂f0∂v

sin[(ω − kv)t]ω − kv

. (160)

When ωt→∞, the velocity integral that must be per-formed to obtain the power, Equations (159) and (160),has the form∫F (v)

sin[(ω − kv)t]ω − kv

dv =∫F

[ω

k

(1 +

ξ

ωt

)]sin ξξ

dξ

|k|,

(161)where ξ ≡ (kv−ω)t. The integral of sin(ξ)/ξ from minusto plus infinity is π, so

limωt→∞

∫F (v)

sin[(ω − kv)t]ω − kv

dv = πF (ω/k)|k|

. (162)

The spatially averaged power is then

〈P 〉 = −π q2E2

k

2m |k|

(v∂f0∂v

)v= ω

k

. (163)

Power is transfered to the plasma (positive power) if thereare fewer particles at high energy than low and to theelectric perturbation (an incipient instability) if there aremore particles at high energy than low.

One can simplify some of the analysis of Landau damp-ing by writing form δf = f exp[i(kx− ωt)]. The criticalstep in Landau damping is then the determination of theimaginary part of what appears to be a real integral, theLandau integral,(∫

f ′(v)ω − kv

dv

)imag

= −i π|k|f ′(ωk

). (164)

Landau damping is closely related to a second phe-nomenon, quasilinear diffusion, which is important forunderstanding the interaction of waves with plasmasand the effect of short wavelength perturbations on theplasma. If one identifies the distribution function f0 ofthe Landau damping discussion with the spatially aver-aged distribution function, f0 ≡ 〈f〉, then a spatial av-erage of the Vlasov equation, df/dt = 0, Equation (127)with p = mv, gives

∂f0∂t

= − ∂

∂v

⟨ qmEδf

⟩. (165)

Equation (160) gives an expression for 〈qEδf〉, which isproportional to ∂f0/∂v. Therefore

∂f0∂t

=∂

∂vD(v)

∂f0∂v

(166)

where the velocity diffusion coefficient

D =q2E2

k

2m2

sin[(ω − kv)t]ω − kv

. (167)

42

This expression for the velocity space diffusion is not use-ful unless there is a spectrum of perturbations with dif-ferent k’s. The spatially averaged energy density of asingle wave is (ε0/2)〈E2〉 = (ε0/4)E2

k. If∫E(ω/k)dk is

the energy per unit volume of a spectrum of waves, onecan integrate Equation (167) over k using the integrationformula of Equation (162) to obtain

D =2πq2

ε0m2

E(v)|v|

. (168)

The quantity ω(k)/k is the phase velocity of the elec-tric perturbation, so a perturbation diffuses particles thathave a velocity equal to the phase velocity.

Quasilinear diffusion is closely related to the stochas-ticity of magnetic field lines that was discussed in Section(III.A). In principle the equations of mechanics are re-versible. However when stochastic, trajectories that havean infinitesimal initial separation increase that separationexponentially with distance along the trajectories. If thetrajectories are resolved to finite accuracy, then the in-formation is quickly lost that would be needed to launchthe time-reversed trajectories.

The spatial averaging that is involved in deriving thequasilinear diffusion coefficient, destroys information andallows an entropy-like quantity s0 ≡ −

∫f0 ln f0d3p to

increase. However, s0 is not the entropy, nor even thespatial average of the entropy, s ≡ −

∫f ln fd3p. The

spatially averaged entropy 〈s〉 cannot change if f is a so-lution to the Vlasov equation, df/dt = 0, Equation (143).Let f = f0 exp(f), then 〈exp(f)〉 = 1 since 〈f〉 = f0. As-suming f is small, the relation between 〈s〉 and s0 is〈s〉 = s0 − 1

2

∫〈f2〉f0d3p. Consequently, s0 must increase

if the magnitude of f does to avoid a change in 〈s〉. Actu-ally, the complicated velocity-space structure that f de-velops, coupled with the diffusive nature of plasma col-lisions, means that f may not reach a large amplitudeeven when the fractional change in the volume-averagedentropy is of order unity.

Quasilinear diffusion can represent the heating of aplasma by a fluctuating electric field even in steady state.Of course for a steady state in the presence of heating,there must be a cooling term in the kinetic equation, suchas ∂f0/∂t = ∂(νcvf0)/∂v where 2νc(v) is the cooling rate.For this cooling term, the volume-averaged distributionfunction relaxes to ∂ ln f0/∂(v2/2) = −νc/D, which hasthe shape of a Maxwellian with an effective temperatureTeff/m ≡ D/νc in velocity regions where D/νc is inde-pendent of velocity.

D. Drift kinetic theory

• The single characteristic of the Vlasov operator,df/dt, Equation (126), is the trajectories of the par-ticles. Consequently one can use approximations tothe particle trajectories to simplify kinetic calcula-tions.

• Charged particles make a circular gyration aboutmagnetic field lines. When the radius of gyration issmall compared to the scale of the spatial variationof the electric and the magnetic fields, the magneticmoment, µ = mv2

⊥/2B, is a constant of the motion.

• When the gyration radius is small, the particle tra-jectory can be accurately approximated by trackingthe center of the circle, ~xg about which the parti-cle gyrates, the guiding center. The guiding centerdrifts at a velocity, ~vg, given by Equation (180).The equation of motion is first order, d~xg/dt = ~vg,and not the usual second order equation, md2~x/dt2

equal to a force.

• The drift velocity of the gyro-centers is also givenby a Hamiltonian, Equation (194), which has onlyfour canonical variables (θ, ϕ, pθ, pϕ), Equations(198) and (199), instead of the six canonical vari-ables of the full trajectories. The form of the Hamil-tonian for the drift motion demonstrates that thevariation in the magnetic field strength on the mag-netic surfaces is the primary determinant of theconfinement properties of the particle trajectories.

• Particles with a sufficiently small ratio of their ve-locity parallel to the magnetic field, v||, to their to-tal velocity, v, are trapped between maxima of themagnetic field strength along a magnetic field line.These trapped particles drift from one field line toanother conserving their action, J =

∮mv||d`.

The Vlasov operator df/dt in the kinetic equation isdetermined by the trajectories of the particles, and thedetermination of these trajectories is a major difficultyin solving the kinetic equation. Charged particles in amagnetic field move in circles with a gyration frequencyΩ ≡ qB/m, Figure (9), which makes a direct calculationinefficient when the circles are small as they are in a fu-sion plasma. In addition to being inefficient, a direct cal-culation obscures fundamental properties of the trajecto-ries. For example, the confinement properties of particletrajectories are essentially determined by the variationof the magnetic field strength on the constant pressuresurfaces.

This section derives the asymptotic expressions for theparticle trajectories, (Alfven, 1940), and the associatedkinetic theory that hold when the circles that are madeby charged particle orbits in a magnetic field are smallcompared to the scale of field variations. The basic re-sults are: (1) the magnetic moment,

µ =mv2

⊥2B

, (169)

is a constant of the motion with v⊥ the velocity compo-nents perpendicular to ~B. The invariance properties ofµ, which is an adiabatic invariant are discussed in Sec-tion (VI.D.2). (2) The center of the circle, ~xg, aboutwith a particle gyrates, Figure (9), drifts at a velocity

43

xxxx

xxxx

a)

b)

c)

FIG. 9 A charged particle gyrates in a circle about straightmagnetic field line while moving along the line, (a). Theinstantaneous center of this circle is the guiding center ~xg,and the difference between the position of the particle ~x andthe guiding center is the vector gyroradius ~ρ, (b). A uniformelectric field perpendicular to the magnetic field causes theparticle velocity, and hence its gyroradius, to be slightly largeron one side of its gyroorbit than the other causing a driftacross the magnetic field lines, (c).

~vg = d~xg/dt. This velocity is called the guiding centeror the drift velocity, and d~xg/dt = ~vg gives a first orderordinary differential equation for the trajectories. Theguiding center velocity is given by Equation (180) as wellas by a drift Hamiltonian, Equation (194). The Hamilto-nian of the drift motion, Section (VI.D.3), has only fourcanonical variables (θ, ϕ, pθ, pϕ), which are the poloidaland toroidal angles of Boozer coordinates, Equation (57)and their conjugate momenta, Equations (198) and (199),(Boozer, 1984b). This Hamiltonian formulation was uti-lized in a code by (White and Chance, 1984). Their codehas been the basis of numerous investigations of phenom-ena that depend on the particle drifts. In addition to thecanonical formulation of the drift equations, which is em-phasized here, (Littlejohn, 1981) has given an importantnon-canonical, though Hamiltonian, treatment based onLie theory.

The velocity-space coordinates of the guiding centermotion that simplify the kinetic equation, df/dt = C(f),are the HamiltonianH, which is the energy, and the mag-netic moment µ, where H = mv2

||/2 + µB + qΦ. Thephase angle ϑ of the circular gyromotion of the particles

is averaged over to obtain the drift equations. The ve-locity space volume element in cylindrical velocity spacecoordinates is d3v = v⊥dv⊥dϑdv||. Using (H,µ, ϑ, ~x) ascoordinates, dH = mv||dv||, so

d3v =B

m2

∑±

dHdµdϑ

|v|||, (170)

where the∑± means that the sum is taken over the

sign of velocity along the magnetic field, v||. In driftkinetic theory, one can integrate over the gyrophase ϑ,which is equivalent to replacing dϑ in Equation (170) by2π. The drift kinetic equation is obtained by calculatingdf(H,µ, ~x, t)/dt and setting d~x/dt = ~vg, the drift velocityof the guiding centers.

df

dt=∂f

∂t+ ~vg · ~∇f +

dH

dt

∂f

∂H= C(f), (171)

where the energy exchange between the particles and theelectromagnetic fields, dH/dt, is given by Equation (195).

The use of the drift kinetic theory to calculate flowsrequires care. The current density is not q

∫~vgfd

3v asone would naively expect but

~j = q

∫~vgfd

3v − ~∇×(b

∫µfd3v

). (172)

The second term is called the magnetization current andarises, just as a magnetization current in a solid, fromthe magnetic moments of the particles that constitutethe medium (Jackson, 1999), Figure (10). The actualmagnetic moment produced by the circular gyromotionis

~m = −µb. (173)

The definition of the magnetic moment of a current dis-tribution is ~m ≡ 1

2

∫~x×~jd3x, which for a charged particle

moving in a gyro-orbit is ~m = (q/2)~ρ×~v with ~ρ the vectorgyroradius, Equation (176).

1. Alfven’s guiding center velocity

(Alfven, 1940) derived an expression for the guidingcenter or drift velocity that follows from the expressionfor the trajectories of a charged particle in given electricand magnetic fields,

md~v

dt= q( ~E + ~v × ~B). (174)

The position of the particle is given by d~x/dt = ~v. Ina uniform magnetic field with no electric field, a particlemoves in a circle at the frequency

Ω ≡ qB

m(175)

44

n

Flowvelocity

FIG. 10 The magnetic field is out of the figure. In the pres-ence of a density gradient, the number of particles moving inone direction perpendicular to both the magnetic field anddensity gradient differs from the number of particles movingin the opposite direction. This leads to a divergence-free flowof the particles in the ~B × ~∇n direction even if all the par-ticles are moving in perfect circles. This flow is called themagnetization drift.

with the circle having a radius, Figure (9),

~ρ =b× ~v

Ω(176)

with b ≡ ~B/| ~B|. The center of the circle ~xg, which iscalled the guiding center, moves with a velocity v|| ≡ b ·~vparallel to the magnetic field. The position of the chargedparticle is, Figure (9),

~x = ~xg + ~ρ. (177)

The addition of a constant electric field that is perpen-dicular to the magnetic field causes the guiding centersto drift in the ~E × ~B direction, Figure (9). Let

~vE×B ≡~E × ~B

B2, (178)

which is independent of position and time if ~E and ~B

are. If ~E · ~B = 0, Equation (174) can be written asm ddt (v − ~vE×B) = q(v − ~vE×B) × ~B. The motion of a

charged particle in a constant electric that is perpendic-ular to the field lines is identical to the motion withoutthe electric field in a frame of reference moving with theE ×B velocity ~vE×B .

The properties of the trajectories become far morecomplicated when the particles move in electric and mag-netic fields that depend on position and time. However,Equation (177) can be viewed as defining the guiding cen-ter of a particle in arbitrary magnetic and electric fields,and the exact motion of the guiding center,

d~xgdt

= v||b+~E × b

B+ ~v × d

dt

(b

Ω

), (179)

is obtained using Equations (174), (176), and (177).

When the spatial variation of the electric and magneticfields is on a scale much longer than the gyroradius |~ρ|,particles can be tracked by following the velocity of theguiding center averaged over the rapid gyromotion, ~vg ≡〈d~xg/dt〉. The expression for the guiding center velocityis (Alfven, 1940)

~vg = v||b+b

Ω×

(v2||~κ+ v||

∂b

∂t+v2⊥2

~∇BB

)+~E × ~B

B2,

(180)where the curvature of the magnetic field lines is

~κ ≡ b · ~∇b (181)

with b ≡ ~B/| ~B|. The curvature of a pure toroidal field~B = (µ0G/2πR)ϕ is ~κ = −R/R since dϕ/dϕ = −R.The term involving ∂b/∂t in the guiding center velocityis usually dropped since it is small if the time scale Tover which b varies is long compared to the characteristictime for a particle to cross the system R/v.

In time independent magnetic and electric fields,Alven’s expression for the guiding center velocity canbe written in an alternative form, which simplifies thethe theory of particle confinement. One can easily show(Morozov and Solov’ev, 1966) that

~vg =v||

B~∇×

(~A+ ρ|| ~B

)(182)

has the same drift across the magnetic field as Equation(180), The parallel gyroradius is defined by

ρ||(H,µ, ~x) ≡v||

Ω(183)

with the energy H = 12mv

2|| + µB + qΦ and the mag-

netic moment µ treated as constants. That is, the curlof v|| is calculated holding the energy H and the mag-netic moment µ constant. The trajectories are obtainedby solving first order equations for the guiding centerposition, d~xg/dt = ~vg. The only difficulty in the integra-tion is at turning points of the parallel motion where v||passes through zero and changes its sign. This difficultycan be removed by the Hamiltonian formulation, Section(VI.D.3).

The terms in Alfven’s expression for the drift of theguiding center can be understood by analogy to the E×Bdrift, Equation (178). If a force ~F is applied to a chargedparticle gyrating in a magnetic field, then the same ar-gument that led to the E×B drift yields a drift velocity~vF×B ≡ ~F × ~B/(qB2). If the magnetic field lines have anon-zero curvature ~κ, the centrifugal force, ~Fc = −mv2

||~κ,gives the curvature drift. If the magnetic field strengthvaries, the quantity µB acts like a potential energy givingthe ∇B force, ~F∇B = −µ~∇B, and an associated drift.

The only part of the derivation of the guiding centervelocity, Equation (180), that is not obvious is the part

45

of ~vg perpendicular to ~B that follows from the last termin Equation (179). This part of ~vg is given by

b×

~v × d

dt

(b

Ω

)= ~vb· d

dt

(b

Ω

)−v||

d

dt

(b

Ω

). (184)

One can let d/dt = ∂/∂t+v||b · ~∇+~v⊥ · ~∇⊥ for b ≡ ~B/| ~B|and Ω ≡ qB/m have no velocity dependence. Since b isa unit vector b · db/dt = 0, and

b×

~v × d

dt

(b

Ω

)= ~v⊥

d

dt

(1Ω

)−v||

Ωdb

dt. (185)

The direction of the velocity perpendicular to the mag-netic field changes over a gyroperiod, so 〈~v⊥〉 = 0 inlowest order, and⟨

~v⊥d

dt

1Ω

⟩=v2⊥2~∇⊥

1Ω. (186)

The gyrophase average⟨v||

Ωdb

dt

⟩=v2||

Ωb · ~∇b+

v||

Ω∂b

∂t. (187)

2. Magnetic moment conservation

The integration of the guiding center velocity, Equa-tion (180), to obtain ~xg is greatly simplified by the ex-istence of an adiabatic invariant, the magnetic moment,which has the approximate form µ = mv2

⊥/2B.An important general principle of Hamiltonian me-

chanics, which is rarely taught in mechanics courses, isthat if the parameters that define a periodic motion of aparticle change sufficiently slowly, then the action of theperiodic motion is conserved. If ~pΩ are the componentsof the canonical momentum of the particle Hamiltonianthat are oscillating and ~xΩ are the conjugate canonicalcoordinates, then the action is given by an integrationover the periodic motion,

∮~pΩ · (d~xΩ/dt)dt. Such invari-

ants are called adiabatic invariants, and the magneticmoment µ is an example. The conservation propertiesof adiabatic invariants is a complicated subject (Kruskal,1962), which is closely related to the subject of magneticsurface existence, Section (III.A). The basic result is thatif the parameters of a Hamiltonian are changed on a timescale T, which is slow compared to the frequency Ω of aperiodic motion, then the action of that motion is con-served with exponential accuracy. That is, the variationof the action is of order exp(−ΩT ).

Canonical momenta are defined using the Lagrangianformulation of the equations of motion. The Lagrangianof a particle in electric and magnetic fields is

L(~x, ~x) =m

2~x

2+ q~x · ~A− qΦ (188)

where ~x ≡ d~x/dt, ~B = ~∇ × ~A and ~E = −∂ ~A/∂t − ~∇Φ.The canonical momenta are defined by,

~p ≡ ∂L

∂~x. (189)

The time derivatives of the canonical momenta are ~p =∂L/∂~x.

The Lagrangian gives the standard equations of mo-tion, Equation (174), in Cartesian coordinates. In thesecoordinates, the canonical momentum, ~p = ∂L/∂~v, is

~p = m~v + q ~A, (190)

where ~v ≡ ~x. The gradient of the Lagrangian is ∂L/∂~x =q~∇(~v · ~A) − q~∇Φ. A vector identity implies ~∇(~v · ~A) =~v × (~∇× ~A) + ~v · ~∇ ~A, where ~v is not differentiated sinceit is an independent coordinate in a Lagrangian analysis.Now, d~p/dt = md~v/dt+ q∂ ~A/∂t+ ~v · ~∇ ~A, so ~p = ∂L/∂~x

is equivalent to md~v/dt = q( ~E + ~v × ~B).The beauty of the Lagrangian approach is the ease

of finding the equations of motion in arbitrary (ψt, θ, ϕ)coordinates, ~x(ψt, θ, ϕ, t). If L is independent of a coor-dinate ϕ, as it is in axisymmetric plasmas, then pϕ is aconstant of the motion, pϕ = ∂L/∂ϕ = 0. The ϕ compo-nent of the canonical momentum is

pϕ = m~v · ∂~x∂ϕ

+ q ~A · ∂~x∂ϕ

(191)

since pϕ ≡ ∂L/∂ϕ and d~x/dt = ∂~x/∂t+(∂~x/∂ψt)ψt+· · ·,so ∂(d~x/dt)/∂ϕ = ∂~x/∂ϕ. The poloidal flux is ψp =−2π ~A ·(∂~x/∂ϕ) using Equation (6) for ~A. The constancyof pϕ ensures the confinement of particles of sufficientlysmall velocity. Particles cannot cross the field lines bya distance greater than their gyroradius in the poloidalfield, ~Bp ≡ ~∇(ϕ/2π)× ~∇ψp, alone.

The definition of the adiabatic invariant that is knownas the magnetic moment is the integral

µ ≡ q

m

12π

∮~p⊥ ·

d~x⊥dt

dt (192)

over a gyroperiod where ~p is the canonical momentum,~p ≡ m~v + q ~A. The quantity mµ/q is the action of thegyromotion.

The approximate expression for the adiabatic invari-ant, µ = mv2

⊥/2B, which is correct to lowest order in thegyroradius to system size, obtained from Equation (192)using

∮~v⊥ · d~x⊥dt dt = (2π/Ω)v2

⊥ and∮~A · (d~x⊥/dt)dt =

−∮~A · d~x⊥. The minus sign arises because a positive

charge moves about its circular orbit in a clockwise di-rection while the convention for a line integral is counterclockwise. The integral

∮~A · d~x⊥ =

∫~B · d~a = πB|~ρ|2.

The approximate expression for the magnetic moment,µ = mv2

⊥/2B, follows.The magnetic moment is proportional to the magnetic

flux enclosed by the circular gyromotion, so magnetic

46

moment conservation can be viewed as flux conservation.The conservation of the magnetic moment is also equiv-alent to a fixed Landau level of the quantum theory ofparticle motion in a magnetic field. That is, the adiabaticinvariance of µ follows from the adiabatic approximationof quantum mechanics. The lowest order conservation ofµ can also be demonstrated by differentiating the expres-sion for µ of Equation (169) with respect to time.

Just as magnetic surfaces can only be broken by res-onant perturbations, the invariant µ can only be bro-ken if there is a resonance between the time variation ofthe gyromotion and the gyromotion itself. The Fouriertransform of an analytic function

∫f(t) exp(−iΩt)dt ∼

exp(−ΩT ) where T is the distance of the closest pole off(t) from the real axis, Section (V.D.1). T is the charac-teristic time scale for variations. It is sometimes said themagnetic moment is conserved to all orders in ε ≡ 1/Ωt.What is meant is that if the function exp(−1/ε) is Taylorexpanded in ε about ε = 0, then every term in the Taylorseries is zero. The function exp(−1/ε) is the most im-portant example in physics of a function that is not zerobut has a Taylor series that is indentically zero. It servesas a warning that an expansion in a small parametercan be subtle. A complicated variation in the magneticand electric fields across the field lines is irrelevant to µconservation if the total time derivatives of these fields,d/dt = ∂/∂t + v||b · ~∇, are small compared to the gy-rofrequency Ω. The irrelevance of variations across themagnetic field to the conservation of µ is important forthe validity of gyrokinetic theory, which is discussed inSection (VI.G).

The constancy of the magnetic moment implies thatthe number of independent variables in a guiding centercalculation is four instead of the six required for the fullparticle motion. The four variables can be taken to bethe three components of the guiding center position andthe energy. The energy, or Hamiltonian, of a chargedparticle is

H ≡ 12mv2 + qΦ. (193)

Magnetic moment conservation implies the energy canalso be written as

H =12mv2

|| + µB + qΦ. (194)

As shown below, the gyrophase averaged change in theenergy is ⟨

dH

dt

⟩=

∂

∂t(µB + qΦ)− v||

∂A||

∂t, (195)

which can be integrated along with ~xg to obtain the en-ergy H. The parallel velocity is given by the energy andmagnetic moment,

v|| = ±√

2m(H − µB − qΦ). (196)

Equation(195) for the gyrophrase averaged energychange can be derived by writing the Hamiltonian in its

canonical variables H(~p, ~x, t) = (~p−q ~A)2/(2m)+qΦ withthe particle velocity ~v = (~p − q ~A)/m. Hamilton’s equa-tions imply that dH/dt = ∂H/∂t, so

dH

dt= q

(∂Φ∂t

− ~v · ∂~A

∂t

). (197)

The only subtle term in the derivation of Equation(195)is⟨q~v⊥ · ∂ ~A/∂t

⟩. This term is calculated using the same

technique as that for∮~A⊥ · (d~x⊥/dt)dt in the magnetic

moment. That is,⟨~v⊥ · ∂ ~A/∂t

⟩= −(Ω/2)(∂B/∂t)|~ρ|2.

3. Drift Hamiltonian

When the scale of the spatial and temporal variation ofthe magnetic and electric fields is long compared to thegyroradius and the gyrofrequency of a particle, the mo-tion of the particle can be tracked using the drift Hamil-tonian, which is the Hamiltonian for the guiding centermotion. This Hamiltonian is the energy, Equation (194),H(pθ, pϕ, θ, ϕ, t) = 1

2mv2|| + µB + qΦ . The canonical

momenta of the drift Hamiltonian are

pθ =µ0I

2πBmv|| +

q

2πψt, (198)

and

pϕ =µ0G

2πBmv|| −

q

2πψp. (199)

The (ψt, θ, ϕ) coordinate system is the Boozer coordinatesystem, Equation (57), with the subscripts omitted tosimplify the notation.

To obtain the drift Hamiltonian and its canonical coor-dinates, we start with a Lagrangian for the guiding centermotion that was given by (Taylor, 1964). To lowest or-der in in gyroradius to system size, the trajectories of theTaylor Lagrangian,

LT (~x, ~x) =12m(~x · b)2 + q~x · ~A− (µB + qΦ) (200)

with ~v = ~x and b ≡ ~B/B, agree with those given byAlfven’s expression for the guiding center motion, Equa-tion (180). The validity of the Taylor Lagrangian isdemonstrated by an explicit calculation. The canoni-cal momentum ~p ≡ ∂LT /∂~x is ~p = mv||b + q ~A. Theenergy or Hamiltonian is H ≡ ~p · ~x − LT , which givesEquation (194). The time derivative of the canonicalmomentum is d~p/dt = mv||b + mv||db/dt + qd ~A/dt, butdb/dt ' ∂b/∂t + v||~κ with ~κ ≡ b · ~∇b, and d ~A/dt =∂ ~A/∂t+ ~v · ~∇ ~A. The equations of motion in Lagrangiandynamics are d~p/dt = ∂LT /∂~x. The gradient of the firstterm in LT is zero, ~∇(~v · b) ' 0. This follows from~∇(~v · b) = ~v × (~∇ × b) + ~v · ~∇b, but b × (~∇ × b) = −~κ

47

and ~v · ~∇b ' v||~κ. Using ~∇(~v · ~A) = ~v × ~B + ~v · ~∇ ~A, onehas ~∇LT ' q~v× ~B+ q~v · ~∇ ~A− ~∇(µB+ qΦ). Putting thepieces together mv||b + mv||(∂b/∂t + v||~κ) + q∂ ~A/∂t 'q~v × ~B − ~∇(µB + qΦ). The component of the equa-tions of motion that is along the magnetic field is thenmv|| = µb · ~∇B+ qE||, which is consistent with Equation(195) for the gyrophase averaged energy change. Thecomponents of the velocity that are perpendicular to themagnetic field agree with Equation (180) for ~vg.

Given a Lagrangian, it is usually trivial to obtain aHamiltonian description of trajectories. However, a sub-tlety exists in obtaining a Hamiltonian description ofguiding center motion from the Taylor Lagrangian. Thecanonical momenta plus the coordinates of the TaylorLagrangian depend on only four independent variables,which can be taken to be (~x, v||). Consequently, theHamiltonian can have only four variables, two canonicalcoordinates and two canonical momenta.

The canonical coordinates of the drift Hamiltonianare closely related to the Boozer magnetic coordinates,Equation (57), so one needs to transform the Taylor La-grangian into these coordinates. The expression is

LT =m

2

(µ0Gφ+ Iθ

2πB

)2

+ qψtθ − ψpϕ

2π− µB − qΦm,

(201)where Φm ≡ Φ + s. The function s is determined us-ing Equation (26) and gives the effect of the motion ofthe magnetic coordinate system. Usually the distinctionbetween Φm and Φ is negligible, and so it will not be re-tained. The most difficult and subtle point in the trans-formation of the Taylor Lagrangian into Boozer coordi-nates is the parallel velocity, ~v · ~B = ~B ·d~x(ψt, θ, ϕ, t)/dt,which gives v|| = µ0

2πB

(G(ψt)ϕ+ I(ψt)θ

). To prove this

use the chain rule to write d~x/dt = ∂~x/∂t+(∂~x/∂ψt)ψt+· · ·. The term ~B · ∂~x/∂t = ~B · ~u, Equation (27), can betaken to be zero, because the flow of the canonical coordi-nates along the field lines can always be chosen to be zero.The time derivatives θ ' (v||/B) ~B · ~∇θ and ϕ are largerby the gyroradius to the system size than the derivativeψt, so one can let ~v · ~B = ( ~B · ∂~x/∂θ)θ + ( ~B · ∂~x/∂ϕ)ϕ.The transformation of ~x· ~A = ~A·~u+(ψt/2π)θ−(ψp/2π)ϕ,while ~A·~u = −s with the choice of gauge g = 0, Equation(A20). Assembling results one obtains Equation (201).

Given the Taylor Lagragian in magnetic coordinates,Equation (201), the determination of the canonical mo-menta and the drift Hamiltonian are straight forward.The canonical momenta are pθ ≡ ∂L/∂θ, which givesEquation (198 and pϕ ≡ ∂L/∂ϕ, which gives Equation(199).

It is also useful to have the Hamiltonian for the guid-ing center motion in Clebsch coordinates (ψ, α, φ). Inthese coordinates the magnetic field has the contravari-ant representation ~B = ~∇ × (ψ~∇α), Equation (9), andthe covariant representation ~B = ~∇φ+Bα~∇α+Bψ ~∇ψ,

Equation (11). Dotting the two representations together,one finds that the inverse of the coordinate Jacobian is~B · ~∇φ = B2. The Taylor Lagrangian is transformed intoClebsch coordinates by noting that ~v· ~B = ~B·(∂~x/∂φ)φ =φ and ~v · ~A = (d~x/dt) · ~A = ψα. The canonical momentaare

pα = qψ (202)

and

pφ = mv||/B = qρ||. (203)

The Hamiltonian is the energy, Equation (194),

H(pα, pφ, α, φ) =B2

2mp2φ + µB + qΦ. (204)

The comparison of the trajectories given by the driftHamiltonian in Clebsch coordinates with those obtainedfrom Equation (182) is instructive. Equation (182) im-plies

dψ

dt= ~vg · ~∇ψ = v||B

(∂ρ||

∂α−∂Bαρ||

∂φ

). (205)

The first of these two terms is reproduced by the Hamil-tonian formulation, but the second is not. The changein ψ as the particle moves along its trajectory due tothe term −v||B∂(Bαρ||)/∂φ = v||∂(Bαρ||)/∂` is given byδψ = −Bαρ||. This follows from dφ = Bd`, with ` thedistance along ~B, and dt = d`/v||. Bα has dimensions ofthe magnetic field times a length, which means a length oforder a, the scale of the plasma. Consequently, the max-imal deviation of ψ along a trajectory is approximatelyδψ/ψ ≈ ρ||/a, which can be viewed as a redefinition ofthe guiding center position, not a systematic drift. Thelocation of the guiding center is a question of definitionto within a distance of order a gyroradius. That arbi-trariness can be used to simplify the equations for theguiding center motion.

4. Action invariant J

The conservation of the magnetic moment µ causesthe parallel velocity of a particle to pass through zeroand change sign at a point where H = µB + qΦ. Par-ticles that are trapped between two such points of highmagnetic field or electric potential are called trapped par-ticles, Figure (11). They have a periodic oscillation be-tween the points and hence an adiabatic invariant, whichis customarily written as

J =∮mv||d`. (206)

J is called the action invariant (Northrop, 1963). In Cleb-sch coordinates (ψ, α, φ), Equations (9) and (11), theaction is J(H,µ, ψ, α) ≡

∮pφdφ =

∮mv||d`, Equation

48

Passing

Trapped

B

H

FIG. 11 A particle is trapped or passing depending on itsenergy H relative to the energy µBmax. Bmax is the maxi-mum of the field strength along a magnetic field line, and µ isthe adiabatically conserved magnetic moment. The distancealong the line is `. The electric potential Φ is ignored forsimplicity.

(203). which means J is an action in the standard senseof Hamiltonian mechanics.

The derivatives of the action, J(H,µ, ψ, α) give im-portant information about the long-term trajectories oftrapped particles. The derivative with respect to the en-ergy gives the bounce time, τb the time to go form oneturning point to the other. Equation (196) for the paral-lel velocity implies ∂v||/∂H = 1/mv||, so

∂J

∂H=∮

d`

v||= 2τb. (207)

Writing J = q∮ρ||dφ makes the ψ and α derivatives

easier to evaluate. Equation (205) implies

∂ρ||

∂α=~vg · ~∇ψv||B

+∂(Bαρ||)∂φ

. (208)

Therefore,

∂J

∂α= q

∮~vg · ~∇ψv||

d` = 2q∆ψ, (209)

where ∆ψ is the change in ψ going from one turning pointto the next. Similarly,

∂J

∂ψ= −2q∆α. (210)

The long-term motion of a trapped particle consists of aradial drift,

dψ

dt=

1q

∂J/∂α

∂J/∂H, (211)

and a precession,

dα

dt= −1

q

∂J/∂ψ

∂J/∂H. (212)

E. Particle trajectories and transport

• Particles that have a sufficiently large ratio of thevelocity parallel to the magnetic field, v||, to thetotal velocity, v, can move all along the field lines,Figure (11), and are generally well-confined whenmagnetic surfaces exist. Such particles are calledpassing particles.

• Particles with a small ratio of v||/v are trapped be-tween maxima of the field strength, Figure (11),and are well confined only when stringent condi-tions are met on the variation of the magnetic fieldstrength in the magnetic surfaces. The trappedparticles can be well confined if the field strengthdepends on only one angle in the magnetic surface,as is the case in axisymmetry, or if the magneticfield strength is the same at all minima of the fieldstrength in a magnetic surface.

• The electric potential in a confined plasma, witha temperature T , has the characteristic magnitude|Φ| ≈ T/e. The reason is that one species is gener-ally more poorly confined, so that species preferen-tially leaves the plasma until its pressure gradientis balanced by the electric field

∣∣∣~∇p∣∣∣ = ∣∣∣en~E∣∣∣. Onlya tiny fraction of the particles are lost while settingup this electric field, which is called the ambipolarfield, so the plasma is approximately quasi-neutral.

• A pressure gradient drives a net current along themagnetic field lines. This current is called the boot-strap current.

In a confined fusion plasma, particles can move a dis-tance more than a thousand times the size of the plasmabetween collisions. Plasma confinement for times longcompared to the collision time requires that the trajecto-ries of all particles that form a Maxwellian distributionstay close to the constant pressure surfaces. In addition,the alpha particles that are produced by the fusion reac-tion must remained confined as they slow from their birthenergy of 3.5 Mev and heat the plasma (ITER PhysicsExpert Group on Energetic Particles, Heating and Cur-rent Drive et al., 1999a). Two questions need to be ad-dressed: (1) Do the trajectories of high energy particlesremain in the plasma? (2) What effect does the strayingof particles that have an energy near the thermal energyfrom the pressure surfaces have on the transport coeffi-cients?

Before discussing the motion of particles and the as-sociated transport phenomena, it is useful to have anestimate of the radial electric field in a plasma. Thecharacteristic change in the electric potential across aconfined plasma is |q∆Φ/T | of order unity. This po-tential difference is associated with only a small netcharge density qn∆ with n∆/n << 1. That is a con-fined plasma is generally quasi-neutral. The reason forthe potential difference, ∆Φ is that one species, ions or

49

electrons, is more poorly confined than the other. Themore poorly confined species leaves the plasma until theelectric potential becomes sufficiently strong to provideconfinement for that species d ln p/dψt = −qndΦ/dψt. Ifthe temperature were constant, the density and poten-tial would be related by n ∝ exp(−qΦ/T ). Even a sub-stantial density drop is associated with a modest changein qΦ/T . The net charge density, qn∆, associated withthis potential is given by Gauss’ law, ∇2Φ = −qn∆/ε0,which implies qn∆ ≈ ε0∆Φ/a2 with a the minor radiusof the plasma. The fractional charge imbalance can bewritten as n∆/n ≈ (λD/a)2 where the Debye length isλD ≡

√ε0T/q2n. In a fusion plasma, the Debye length

is of order a tenth of a millimeter, so the charge imbalancebetween electrons and ions n∆/n is extremely small, andthe plasma is said to be quasi-neutral. The force exertedon the overall plasma qn∆

~E is also a factor of (λD/a)2

smaller than the pressure force ~∇p.A related concept to quasi-neutrality is ambipolarity,

which means the electrons and the ions diffuse at thesame rate so there is no radial current. In steady statesituations, plasma transport is generally ambipolar forotherwise the plasma would lose either all of the ions orall of the electrons.

1. Confinement of particle trajectories

The confinement of individual particles is strongly de-pendent on whether a particle is trapped or passing, Fig-ure (11). If Bmax(ψt) is the maximum magnetic fieldstrength on the magnetic surface that contains toroidalflux ψt, then a particle is trapped if µBmax ≥ H−qΦ(ψt).If a particle is not trapped then it is said to be passing.For passing particles the parallel velocity retains a givensign between collisions, v|| > 0 or v|| < 0, and is neverzero. For a barely trapped particle, the conservation of µimplies µBmax = µBmin+mv2

||/2 where v|| is the parallelvelocity at the location of the minimum of the magneticfield strength. If one defines 2ε ≡ (Bmax −Bmin)/Bmax,then at the magnetic field minimum v||/v =

√2ε for a

barely trapped particle. In a circular cross section toka-mak, ε ' r/Ro.

The confinement of passing particles is rarely a prob-lem unless magnetic surfaces are lost. The confinementof passing particles is easily derived using the form forthe guiding center velocity of Equation (182). Since theparallel velocity never vanishes, the motion of a passingparticle is along an effective magnetic field ~B∗(H,µ, ~x) ≡~B + ~∇ × (ρ|| ~B) where the energy H and the magneticmoment µ are constants of the motion. The field ~B∗ isdivergence free and differs from the magnetic field ~B onlyby a small term that is proportional to the gyroradius tothe system size. If ~B has magnetic surfaces, then thesurface quality of the ~B∗ field can be investigated usingthe methods of Section (III.A).

The confinement of trapped particles in toroidal plas-

mas is more difficult than passing particles, and con-strains the design of confinement systems. In principle,the guiding center drifts can carry a trapped particle outof the plasma after is has travelled a distance of order(R/ρ)a where R is the radius of curvature of the fieldlines, ρ is the gyroradius, and a is the minor radius ofthe plasma. This distance is too short for confining fu-sion plasmas. The time constant that is associated withunbounded drift motion is τ ≈ aR/(ρv), which is of orderthe time for Bohm diffusion, Equation (228).

Trapped particles are well-confined if the magneticfield strength along the magnetic field lines satisfies pe-riodicity, B(`) = B(` + L) where ` is the distance alonga field line and L is a constant along that line. Mag-netic configurations that satisfy this constraint are calledquasi-symmetric. If the field strength is given in Boozercoordinates, Equation (57), then periodicity implies thefield strength can depend on the poloidal and toroidalangles only though the linear combination θh ≡ θ+Nhϕwhere Nh is an integer. That is the field strength hasthe form B(ψt, θh). The helical canonical momentum,ph ≡ ~p · (∂~x/∂ϕ)θh

, of the Taylor Lagrangian, Equation(201) is conserved,

ph = µ0G−NhI

2πBmv|| − q

ψp +Nhψt2π

. (213)

The expression for ph is obtained using (∂~x/∂ϕ)θh=

(∂~x/∂ϕ)θ − Nh∂~x/∂θ, so ph = pϕ − Nhpθ. The conser-vation of ph means the excursions that trapped particlesmake from the magnetic surfaces are proportional to thethe gyroradius.

The axisymmetric tokamak satisfies the condition ofquasi-symmetry with Nh = 0. But the condition ofquasi-symmetry can also be satisfied to high accuracy instellarators in which the magnetic surfaces are not sym-metric, in ϕ. The Quasi-Helically Symmetric stellarator(QHS) at the University of Wisconsin, the first operat-ing quasi-symmetric stellarator, has four periods, Np = 4and Nh = 4 (Talmadge et al., 2001). The NCSX stel-larator, which is to be constructed at Princeton (Zarn-storff et al., 2001), is quasi-axisymmetric, Nh = 0, justas a tokamak, but has three periods, Np = 3, Figure(7). (Garren and Boozer, 1991) have shown that quasi-symmetry cannot be precisely achieved except in perfectaxisymmetry. However, the required breaking of quasi-symmetry, which is of order the local inverse aspect ratiocubed, (r/R)3, can be very small.

A different method than quasi-symmetry for obtainingtrapped particle confinement can be derived starting withthe requirement that a deeply trapped particle must re-main close to a flux surface. The condition for good con-finement of the deeply trapped particles is that all theminima of the field strength Bmin along each field lineoccur at essentially the same value of B, (Mynick et al.,1982). Field minima satisfy ∂B/∂` = 0 and ∂2B/∂`2 > 0with ` the distance along a magnetic field line. Deeplytrapped particles have (v||/v⊥)2 1 even when they arenear the minimum of the field strength and have esen-

50

tially zero action, J =∮mv||d` = 0, Equation (206).

Since the action is conserved, they must remain deeplytrapped throughout their drift motion. That is, theymust remain at a minimum of the field strength. Theguiding center drift, Equation(180), for a deeply trappedparticle is ~vg = ( ~B/qB2) × ~∇(µB + qΦ) so the particlesdrift on surfaces of constant µB + qΦ. Since the po-tential is a function of the toroidal flux, Φ(ψt), Section(V.B.3), the deeply trapped particles stay on a flux sur-face if ∂Bmin/∂α = 0. The derivative ∂Bmin/∂α is zeroat minimia if all minimia of the magnetic field on a ψtsurface have the same value.

If deeply trapped particles are confined to the magneticsurfaces, then it is relatively easy to shape the variationin magnetic field strength along the field lines to confinemost of the trapped particles. However, particles near thetrapped passing boundary tend to have bad orbits unlessall field maxima are at the same field strength, (Caryand Shasharina, 1997). The condition of all field max-ima being at the same field strength is not as importantbecause a radial variation in the average field strength

B0(ψt) ≡√∫

B2J dθdϕ/∫J dθdϕ and the electric po-

tential Φ(ψt) cause particles to drift in a way that con-verts particles near the trapped-passing boundary intoeither trapped or passing particles. In addition collisionsin a plasma are a diffusive phenomenon, and thermal par-ticles near the trapped-passing boundary switch rapidlybetween the two types of orbits, which means they slowlydiffuse rather than rapidly drift out of the confinementregion.

The W7-X stellarator is designed (Nuhrenberg et al.,1995) to make the minima of the field strength on a mag-netic surface have the same value. This is accomplishedby the plasma having a pentagonal shape when viewedalong the Z axis, Figure (6). Each magnetic surface hasfive straight sections where the field strength is low andfive high curvature corners where the field strength ishigh. It is easy to design all field minima to have thesame field strength since they occur in straight sections.The maxima of the field strength occur in the cornersof the pentagonal shape. The fractional variation in thefield strength at the maxima is approximately 2δR/R,where δR is the half width of the plasma along the ma-jor radius at the corners, and δR is very narrow in W7-X,δR/R ≈ 1/20. Pressure balance in W7-X approximatelysatisfies p(ψt) + B2

0(ψt)/2µ0 constant, so at the designbeta value, 〈2µ0p/B

2〉 = 5% the radial variation in B0

is sufficient to confine most particles near the trappedpassing boundary. The principle used to confine parti-cle orbits in W7-X has been given many names: linkedmirrors, quasi-isodynamic, quasi-omnigenious, and mostrecently quasi-poloidal symmetry. Quasi-poloidal sym-metry is broken by the strong curvature in the cornersin first order in the inverse aspect ratio r/R while quasi-symmetry in its more traditional usage is broken in thirdorder, (r/R)3.

A tokamak with too few toroidal field coils is an ex-

ample of the bad particle confinement that occurs whenthe minima of the field strength on a magnetic surfacehave differing values. Even when the field minima createdby the space between the individual coils are shallow,∂Bmin/∂α is large and the particles trapped in these min-ima drift out of the device on a relatively short time scaleaR/(ρv). The fractional variation in the magnetic fielddue to toroidal asymmetry is called the toroidal ripple,δ. The approximate expression for the field strength isB(ψt, θ, ϕ) = B01−(r/Ro) cos θ+δ cos(Nϕ), Equation(19). Unless the toroidal ripple satisfies δ (r/Ro)(ι/N)it causes a large number of secondary minima at varyingvalues of the field strength.

When the minima of the magnetic field on a pressuresurface are not all at the same field strength, the colli-sionless drift trajectories generally cross a large fractionof the pressure surfaces. From the conservation of action,Section (VI.D.4), one has

dψtdα

= − ∂J/∂α

∂J/∂ψt. (214)

Since α is the Clebsch angle with a characteristic rangeof unity, the radial excursion of particles ∆ψt is approx-imately (∂J/∂α)/(∂J/∂ψt) and can be reduced by en-hancing the precession, which is the change in the Cleb-sch angle per bounce ∆α = −(∂J/∂ψt)/2q, Equation(210). The precession of the deeply trapped particlesis proportional to ∂(µB + qΦ)/∂ψt, which for thermalparticles is generally dominated by the radial variationof the potential, ∂Φ/∂ψt with either sign of the radialelecric field, ~E = −(dΦ/dψt)~∇ψt enhancing the confine-ment of drift orbits. However, for super-thermal parti-cles a precession zero, or resonance, can occur in which∂(µB+qΦ)/∂ψt = 0. Frequently, ∂B/∂ψt < 0 for deeplytrapped particles, which means they are in a region ofbad field line curvature. In this case, an electric field thattends to push a charge species out q(∂Φ/∂ψt) < 0 cannothave precession resonance and provides better confine-ment of the individual trajectories than an electric fieldthat pulls that species in. For very high energy particles,such as fusion α particles, µ = (mv2

⊥/2)/B is sufficientlylarge that only the term µ∂B/∂ψt in the precession isimportant.

The action, J , is only an adiabatic invariant and isnot conserved if the integrand of Equation (206) varieson the time scale of the bounce motion τb. Even in theabsence of collisions, the action invariant can be brokenin two ways. First, the drift motion of a particle cantake it to a place where the local minimum in which it istrapped no longer confines particles with the action thatthat particle has. Each region in which a particle can betrapped has a maximum value of the action, Jmax thatit can confine. As the particle drifts from one field lineto another, the maximum action varies, if the action Jexceeds Jmax the particle escapes from the region whereit has been trapped. A particle can also be captured bylocal minima if it drifts so Jmax is increasing.

51

The second way the action, J , is broken by the par-ticle drift motion is if a particle drifts a sufficient dis-tance in a full bounce that the magnetic field near aturning point changes significantly from one bounce tothe next. This is particularly important for finding theeffect on trapped alpha particles of toroidal ripple intokamaks and can determine the required limitation onripple. The change in ψt per full bounce of a particleis 2∆ψt(ψt, α) = ∂J/∂α. If the change in α during afull bounce, which is −2∂J/∂ψt is sufficiently large, thesign of 2∆ψt(ψt, α) is essentially random with each fullbounce and the particle diffuses collisionlessly with a dif-fusion rate that is appoximately (∆ψt)2/τb, which is pro-portional to the square of the ripple amplitude, δ2. Thiseffect (Goldston et. al., 1981) can occur at arbitrarilysmall ripple, even ripple that is too small to cause sec-ondary minima δ (r/Ro)(ι/N), if the precession rate ofthe alphas is sufficiently large in the axisymmetric field.

The breaking of the conservation of the action J due tothe particle drift motion can give trapped particle trajec-tories a complexity that can only be studied by numeri-cally integrating the guiding center equations of motion.

2. Transport at low collisionality

The deviation of trapped particle trajectories from thepressure surfaces leads to enhanced transport and to anet parallel current that is proportional to the pressuregradient, the bootstrap current. The transport phenom-ena associated with the deviation of the particle drifttrajectories from the pressure surfaces are known as neo-classical transport.

Usually the transport rates for particles and energy intokamak plasmas are much larger than their neoclassicalvalues because of micro-turbulence, Section (VI.F), sothe neoclassical transport theory is not as important asone might think. However, the bootstrap current is im-portant for steady-state tokamaks and can be a complica-tion in the design of stellarators. The theory of neoclassi-cal transport in tokamaks has been reviewed by (Hintonand Hazeltine, 1976), and a book on the theory of colli-sional transport in toroidal plasmas has been written by(Helander and Sigmar, 2002).

To understand neoclassical transport, suppose the typ-ical deviation of a trapped particle from a pressure sur-face is a distance (∆r)t. In a low collisionality limit,the distribution function must be consistent with theVlasov equation, df/dt = 0. Since the overall confine-ment is long compared to a collision time, the distributionfunction must also be close to a local Maxwellian fM ∝n(r)e−H/T (r). The deviation from a local Maxwellian isf = (∆r)t∂ ln(fM )/∂r. If this is inserted into Equation(145) for the entropy production, ignoring the tempera-ture gradient for simplicity, one finds that

sc =ν√2ε

(∆r)2t

(d ln(n)dr

)2

(215)

where the fraction of trapped particles is√

2ε. For circu-lar magnetic surfaces ε = r/Ro, the inverse aspect ratio.Actually there are two factors of

√2ε in this equation

for the entropy production. The first factor is an en-hanced effective collision rate ν/(2ε) because the colli-sion operator is diffusive and particles need to be scat-tered through velocity space by a distance of only

√2εv

to move all the way across the trapped particle part ofvelocity space and become passing particles. The sec-ond factor of

√2ε comes from the trapped particles be-

ing the only particles that have a large deviation fromthe pressure surfaces, Section (VI.E.1). The rate of en-tropy production is also (d ln(n)/dr)2 times the diffusioncoefficient, Equation (151), so the diffusion coefficient isD ≈ (ν/

√2ε)(∆r)2t .

For quasi-symmetric confinement systems, the devia-tion of the trapped particle trajectories from a constantpressure surface can be calculated using the conservationof the helical canonical momentum ph, Equation (213).The deviation of a particle from a magnetic surface de-pends on the variation in its parallel velocity v||, whichis maximized by the barely trapped particles. For barelytrapped particle, v||/v = ±

√2ε. Consequently a barely

trapped particle deviates by an amount

∆ψbt =√

2εµ0G−NhI

ι+Nhρ (216)

on either side of the flux surface on which its turningpoints are located where ρ = mv/qB is the gyroradius.At large aspect ratio, ψt = πBr2, I << G = 2πRoB/µ0,and the deviation of a barely trapped particle is

(∆r)bt =

√2ε

ρ

ι+Nh. (217)

This deviation of trapped particles is called a banana or-bit because of its shape when projected on a constant ϕplane. The diffusion, D ≈ (ν/

√2ε)(∆r)2t , is then approx-

imately

D ≈ ν

ε3/2

(ρ

ι+Nh

)2

, (218)

which is called the neoclassical diffusion coefficient(Galeev and Sagdeev, 1968). In tokamaks and quasi-axisymmetric stellarators, Nh = 0, but Nh is non-zero inquasi-helically symmetric stellarators (Talmadge et al.,2001).

As shown by (Kovrizhnikh, 1969), the statement thatEquation (218) gives the neoclassical diffusion in a quasi-symmetric system is misleading. This equation is approx-imately correct for the diffusion of heat, but the diffusionof particles in a quasi-symmetric system can only arisefrom unlike particle collisions. The reason is the mo-mentum conserving properties of the collision operator.Three expressions are important: the collisional entropyproduction, Equation (144), the relation between the col-lisional entropy production and the transport coefficients,

52

Equation(151), and f = (∆ψt)∂ ln(fM )/∂ψt. The devia-tion of a particle from a magnetic surface, ∆ψt, is calcu-lated using ph conservation, Equation (213), which for adensity gradient implies f ∝ (dn/dψt)mv||. The momen-tum conservation properties of the collision operator thenmake the entropy production zero, sc = 0, so no parti-cle transport occurs. For a temperature gradient, f hasadditional factors of the velocity, so the entropy produc-tion, sc, is non-zero, and the diffusion of heat is approx-imated by Equation (218). Particle transport does notvanish in quasi-symmetric systems because ion and elec-trons can exchange momentum. For ions this rate of mo-mentum exchange is a factor of approximately

√me/mi

smaller than the rate of ion-ion collisions, though for elec-trons the rates of momentum exchange with like and un-like particles are comparable. Momentum conservationmeans the transport of ions and electrons must be at thesame rate, to lowest non-trivial order in gyroradius tosystem size, independent of the radial electric field. Thisphenomenon is called intrinsic ambipolarity.

In systems that are not quasi-symmetric the constraintof ambipolarity sets the radial electric field. In quasi-symmetric systems, the radial electric field, and hencethe rate of toroidal rotation, are only weakly affected byneoclassical transport.

One neoclassical effect that is very important forsteady state tokamaks is the bootstrap current (Bick-erton et. al., 1971). The conservation of ph, Equation(213), implies that the deviation of a trapped particletrajectory from pressure surface has one sign, say pos-itive (∆r)t > 0, when the parallel velocity is positiveand the deviation is negative when the parallel veloc-ity is negative. In the presence of a density gradient,dn/dr < 0, this implies that at a fixed radius there aremore barely trapped particles with a positive than witha negative parallel velocity. The passing particles thathave a parallel velocity that is positive interact througha diffusive collision operator with passing particles with anegative parallel velocity only through the trapped par-ticles. This and the fact that there are more trappedparticles moving in the positive direction along the mag-netic field lines than in the negative direction leads to anexcess of passing particles moving in the positive direc-tion, Figure (12). The excess of passing particles movingin the positive direction produces a net parallel currentjb ≈ qvT (∆r)tdn/dr, where vT ≡

√T/m is the thermal

speed. That is

jb ≈1√εB

T

ι+Nh

dn

dr. (219)

The calculation of transport coefficients in non-quasi-symmetric configurations is complicated. Transport co-efficients as well as the effect of collisions on the particletrajectories can be determined using the Monte Carloequivalent to the collision operator (Boozer and Kuo-Petravic, 1981). The Monte Carlo collision operator rep-resents the effects of collisions during a time step by achange that has a random component in the velocity co-

-1 1

f

v || /v

FIG. 12 In the presence of a density gradient, the numberof trapped particles with a parallel velocity greater than zerodiffers from the number with a parallel velocity less than zero.This variation in density is transmitted to the passing parti-cles by diffusive property of collisions in a plasma. A particleis trapped if −

√2ε < v||/v <

√2ε at the minimum of the

magnetic field strength along a magnetic field line with ε thevariation in the field strength. Otherwise the particle is pass-ing.

ordinates of each particle. The simplest Monte Carlocollision operator represents the Lorentz collision opera-tor

C(f) =ν

2∂

∂λ(1− λ2)

∂f

∂λ, (220)

where λ ≡ v||/v is the pitch of the particle relative to themagnetic field. The Monte Carlo equivalent is

λn = (1− ντ)λo ±√

(1− λ2o)ντ , (221)

where λn and λo are the new and the old values of thepitch with the change caused by collisions during a timeinterval τ . The symbol ± means the sign is chosen atrandom. Since the particle trajectories are the charac-teristics of the operator df/dt, the inclusion of effects ofcollisions by Monte Carlo methods means one can findthe solution f to the equation df/dt − C(f) = g by themethod of characteristics. This is the basis of δf MonteCarlo studies of transport, (Sasinowski and Boozer, 1995)and (Lin et. al., 1995), in which one calculates transportcoefficients by using Monte Carlo methods to determinethe deviations from a local Maxwellians of the particledistribution functions. An alternative to the Monte Carlocodes for numerical evaluation of transport coefficients isthe DKES code (van Rij and Hirshman, 1989) with solvesan approximate form of the drift kinetic equation by avariational principle.

3. Power required for maintaining fields

• The net current in a tokamak can be maintainedin steady state using externally produced radio fre-quency waves. However, the required power is un-acceptably large for practical fusion power if morethan about a third of the total current is wavedriven.

53

• Tokamaks can be designed so more than two thirdsof the net plasma current is maintained by thebootstrap current, which is the net current drivenby the pressure gradient. The requirement of alarge bootstrap current makes the feasibility of asteady-state tokamaks more sensitive to the pres-sure profile given by plasma transport processesthan is a steady-state stellarator.

An important issue for steady-state tokamaks is thepower that is required for maintaining the plasma current(Fisch, 1987; ITER Physics Expert Group on EnergeticParticles, Heating and Current Drive et al., 1999b). Thetraditional way to maintain the current in a tokamak isto have a solenoid within the central hole of the torus.A change in the magnetic flux in this solenoid producesa loop voltage, V = ∂ψp/∂t, Equation (35), which drivesthe current. A loop voltage can only be maintained tran-siently, so a different method must be adopted for long-pulse or steady-state tokamaks.

The most developed method of steady-state currentdrive uses waves to maintain a distribution of electrons inwhich more electrons are moving in the direction requiredfor the current than in the opposite direction. Unfortu-nately, too much power is required to maintain the fulltokamak current. However, the total current need notbe externally driven because a pressure gradient drivesa net toroidal current, the bootstrap current, Equation(219). From the point of view of non-equilibrium ther-modynamics, the bootstrap current arises from a crossterm in the transport matrix. The bootstrap current canprovide most of the current in a tokamak, but the mag-nitude and profile of this current is dependent of plasmapressure and density profiles. Since plasma profiles aredifficult to control in the absence of external input power,the requirement of a large bootstrap current places anadditional uncertainty on the feasibility of steady-statetokamaks that does not exist for steady-state stellara-tors.

A physics constraint on the minimum power needed forcurrent drive is simply derived (Boozer, 1988). The mostefficient steady-state current drive has the current carriedby high energy, mildly relativistic, electrons that form atail on the background Maxwellian. The current densityis j = entc where nt is the number of the tail electronsper unit volume and c is the speed of light. The powerper unit volume that is required to maintain this currentcannot be smaller than the power required to maintainthe tail, pd = (γ−1)ntmec

2ν(γ) where γ ≡ 1/√

1− v2/c2

and ν(γ) is the slowing down rate of electrons with kineticenergy (γ−1)mec

2. The power per unit volume pd that isrequired to drive a current of density j determines an ef-fective electric field for power dissipation, Ed ≡ pd/j. Forelectrons at the most efficient energy, which is γ ≈ 2, theeffective electric field for current drive is approximately

Ed = Er where

Er ≡e ln(Λ)

4πε0(c/ωpe)2'(

0.087V olts

meter

)(n

1020 1m3

),

(222)the background electron density is n, the electron plasmafrequency is ωpe ≡

√ne2/ε0me, and ln(Λ) ≈ 17 is the

Coulomb logarithm, Equation (134). Equation (222) isobtained within a numerical factor if one uses the colli-sion frequency ν of Equation (133) in the calculation ofEd ≡ pd/j with v = c.

The power required to maintain a current using sub-relativistic electrons can be calculated by similar argu-ments, the current density is j = entv, and the powerper unit volume is pd = (ntmev

2/2))ν(v). The rate ofslowing of high energy particles ν(v) is proportional to1/v3, Equation (133), so

Ed ≈ Erc2

v2. (223)

An important point is that, unlike the electric field asso-ciated with resistive dissipation, Ed is independent of thecurrent being driven. In other words, the power is pro-portional to the driven current instead of current squaredas it is when the current is dissipated by resistivity. Thetotal power to maintain the current is obtained by mul-tiplying Ed by 2πR, where R is the average major radiuson a magnetic surface, to obtain the effective loop volt-age for power dissipation, Vd ≡ 2πREd. The power isPd =

∫Vd(ψt)(dI/dψt)dψt where I(ψt) is the net toroidal

current inside a magnetic surface that contains toroidalflux ψt. The loop voltage that gives the power requiredfor current drive, Vd, can be tens of volts, which makesthe power requirements for driving the total current un-acceptable.

F. Microstability

• Confined plasmas are generally unstable to pertur-bations that have a wavelength across the magneticfield that is comparable to, or smaller than, theion gyroradius, ρi, but with a wavelength along thefield lines that is comparable to the overall systemsize. For near-Maxwellian plasmas, the character-istic growth rates and frequencies of microinstabil-ities are Cs/a where Cs ≡

√(Te + Ti)/mi is called

the sound speed and a is the plasma radius.

• Microinstabilities lead to micro-turbulence with anassociated particle diffusion of order the gyro-Bohmrate, Dg = ρ2

iCs/a. The amplitude of the fluctua-tions in micro-turbulence is small, roughly the iongyroradius to system size, δn/n ≈ ρi/a.

• The most prominent microinstabilities in the mod-ern literature are the ion and the electron temper-ature gradient modes. They are called the ITG orηi mode and the ETG or ηe mode.

54

• The logarithmic temperature gradients, d lnT/drcan have critical values at which the transportgreatly increases. Plasma temperature gradientsmay remain close to these critical values, whichmakes the temperature throughout the plasma pro-portional to the temperature near the plasma edge.

Even when a plasma is stable to perturbations thathave a wavelength comparable to the plasma size, theplasma may be unstable to perturbations with wave-lengths comparable to or smaller than the gyroradius ofthe ions, ρi. Such instabilities are called microinstabili-ties. Microinstabilities do not cause a sudden loss of theplasma equilibrium but can greatly enhance the plasmatransport across the magnetic field lines. Long wave-length instabilities, such as the instabilities discussed inSection (V), can be so catastrophic in their effect thatthe only issue of interest is whether they are stable ornot. With microinstabilities the primary issue is thenon-linear, or saturated, state, which is generally turbu-lent. In other words, the primary issue is the transportcaused by the micro-turbulence. Unlike turbulence in or-dinary fluids, which is associated with large fluctuations,the fluctuations associated with plasma micro-turbulenceare small, δn/n ≈ ρi/a ∼ 1/500. (Horton, 1999) has re-viewed the theory of microinstabilities associated withdrift waves, which is the type of microinstability of mostrelevance to toroidal plasmas. (Yoshizawa, Itoh, Itoh,and Yokoi, 2001) have reviewed the theory of turbulencein fluids and plasma with an emphasis on plasma micro-turbulence.

Although collisions play an important role in some mi-croinstabilities, generally the issue is whether the Vlasovequation df/dt = 0 is consistent with the growth of elec-tromagnetic perturbations. Gardner’s theorem gives acondition under which the Vlasov-Maxwell equations canhave no unstable solutions. Unfortunately this conditionis violated in current-carrying plasmas, such as magnet-ically confined plasmas. (Gardner, 1963) noted that thedistribution function in the Vlasov equation is the den-sity of a conserved fluid in phase, (~x, ~p), space. If f(~x, ~p)has a form such that the plasma energy is increased bythe interchange of any packets of this fluid, then no en-ergy can be removed from the plasma to support electro-magnetic fluctuations, and the system must be stable. Ifthe distribution function of a single plasma species in thedirection r is

F (u,~v0) ≡∫δ[u− r · (~v − ~v0)]f(~v)d3v, (224)

Garner’s theorem says the Vlasov-Maxwell equationshave no unstable solutions if a ~v0 exists such thatu∂F/∂u ≤ 0 for all species, for all values of u, and for alldirections r. Distribution functions can be far from localMaxwellians and satisfy Gardner’s condition for stabil-ity. Unfortunately Gardner’s condition is not satisfiedfor a current-carrying plasma, so microinstabilities arean issue in magnetic confinement.

(Rosenbluth and Rutherford, 1981) have argued thatonly low frequency microinstabilities are energeticallypossible when the source of their free energy is the pres-sure gradient. By low frequency they mean no higherthan approximately Cs/a where Cs ≡

√Ts/mi, Ts is

the sum of the electron and ion temperatures, a is theplasma radius, and mi is the ion mass. The energy perunit volume that is associated with the oscillations of aninstability is roughly nmi(ω∆)2 with ω the frequency and∆ the spatial scale of the microinstability. The maximumenergy that the oscillation can tap from the pressure gra-dient is approximately nTs(∆/a)2 with a the plasma ra-dius. The first order term nTs∆/a vanishes because theaverage motion of the plasma 〈∆〉 is zero in an oscilla-tion. The instability is not energetically favored unlessnmi(ω∆)2 ≤ nTs(∆/a)2, or |ω| ≤ Cs/a.

A heuristic model clarifies some of the properties ofthe plasma transport caused by the turbulence associatedwith fully developed microinstabilities. The perturbedelectric potential δΦ of a fully developed microinstabilityhas a scale across the magnetic field lines in the magneticsurface of 1/k⊥ and ∆ in the radial direction. The vari-ation of the potential δΦ along the magnetic field lines isvery weak, k||/k⊥ 1, in order to avoid Landau damp-ing Section (VI.C). The variation in the electric potentialcauses a drift velocity ~E × ~B/B2. The magnitude of theradial component of this velocity is δvr ≈ k⊥δΦ/B. Letτc be the correlation time of the microturbulence, whichmeans the time scale over which significant changes inthe pattern of the perturbed potential δΦ occur. Dur-ing each correlation time, particles in the plasma takea radial step δr ≈ δvrτc with equal probability of thestep being inwards or outwards, assuming δr <∼ ∆. Therandom steps cause diffusion with the diffusion coefficientD ≈ δ2r/τc, which can be rewritten as D ≈ k2

⊥(δΦ/B)2τc.The correlation time of micro-turbulence has differ-

ent values in two limits. If the micro-turbulence is ex-tremely weak the correlation time is determined by thegrowth rate of the underlying microinstability, and thetheory is essentially that of quasilinear diffusion, Section(VI.C). However, when the micro-turbulence is fully de-veloped the correlation time is determined by the tur-bulence itself changing the potential δΦ. In this strongturbulence limit, the correlation time is determined byhow long it takes particles to diffuse across the poten-tial contours, τc ≈ 1/(k2

⊥Ds). The diffusion coefficientDs is the coefficient for diffusion in the magnetic sur-faces across the field lines. In this direction, the E × Bvelocity is δvs ≈ (δΦ/∆)/B, so the diffusion in the mag-netic surfaces is related to diffusion across the surfacesby Ds ≈ D/(k⊥∆)2. The correlation time is τc ≈ ∆2/D,and the radial step is δr ≈ ∆. Inserting this expressionfor τc into D ≈ k2

⊥(δΦ/B)2τc, one finds

D ≈ k⊥∆∣∣∣∣δΦB

∣∣∣∣ . (225)

In strong micro-turbulence the particle diffusion is linearin the perturbation amplitude rather than quadratic as

55

it is in quasilinear diffusion.The perturbation amplitude that is reached in a micro-

turbulent plasmas is bounded by∣∣∣~∇δn∣∣∣ ≈ ∣∣∣~∇n∣∣∣. That

is, the micro-turbulence cannot create steeper gradientsthan the ambient gradient while transferring energy fromthe gradients. If one of the species is responding adia-batically, which means eδΦ/T = δn/n, the bound on thefluctuation amplitude is

eδΦT

=δn

n≈ 1k⊥a

(226)

where 1/a ≡ dln(n)/dr is the scale of the ambient densitygradient. The radial diffusion coefficient is then

D ≈ ∆a

T

eB, (227)

where T/(eB) = ρiCs, which is the ion gyroradius timesthe speed of sound.

The transport caused by microturbulence is dependenton the radial extent ∆ of the constant δΦ contours. Thereare two extreme assumptions about ∆. The more pes-simistic is that ∆ is proportional to the size of the plasma,∆ ∝ a. This assumption makes the diffusion proportionalto the Bohm diffusion coefficient, which in the modernliterature is usually defined as

DB ≡T

eB, (228)

although David Bohm’s unpublished work, which firstgave this coefficient, included an unjustified factor of oneover sixteen. The confinement time given by Bohm dif-fusion, τB ≈ a2/DB is similar to the time it takes par-ticles to drift out of a torus due to unconfined particletrajectories τd = a/vg where a is the plasma radius andvg ≈ (ρi/a)Cs is the guiding center drift velocity. Themore optimistic assumption is that ∆ is of order the iongyroradius ρi. The assumption that ∆ ≈ ρi gives thegyro-Bohm rate,

Dg ≡ρia

T

eB=ρ2iCsa

. (229)

The Bohm time is far too short for a fusion powerplant. The gyro-Bohm confinement time, τg = a2/Dg,is marginal,

τg ≈ 7msa3B2

(T/10kev)3/2, (230)

where the plasma radius is in meters and the magneticfield is in Tesla.

What sets the radial scale ∆ of the perturbed poten-tial δΦ? First, consider a microinstability, such as theion temperature gradient (ITG) instability discussed be-low, in which the electrons have an adiabatic response,which means δn/n = eΦ/T , but the ions behave non-adiabatically. In linear theory, ∆ can be much larger than

the ion gyroradius. However, the long radial contours ofthe potential are broken up in the non-linear microtur-bulent state by what are known as zonal flows (Diamondand Kim, 1991). These flows come from the E×B drift inthe part of potential perturbation that has a non-zero av-erage over the magnetic surface, δΦ(ψt, t) ≡ 〈δΦ〉. Thesurface-averaged fluctuation in the potential, δΦ(ψt, t),cannot be damped by electrons flowing along the fieldlines, unlike the rest of the variation in the potential,δΦ − δΦ. Indeed, the adiabatic response of the elec-trons, if written correctly, is δn/n = e(δΦ− δΦ)/T . Thedrive for the zonal flows is the surface-averaged inertialforce of the fluctuating E × B velocity. The left-handside, or inertial part, of the Navier-Stokes equation isρ(∂~v/∂t+~v · ~∇~v) = ∂(ρ~v)/∂t+ ~∇· (ρ~v~v) where the conti-nuity equation ∂ρ/∂t+ ~∇· (ρ~v) = 0 was used to place theinertial terms in the second form. The force that drivesthe zonal flows is the average over the magnetic surfacesof⟨~∇ · (ρ~v~v)

⟩, where ~v = ( ~B × ~∇δΦ)/B2 is the E × B

velocity of the plasma in the turbulence. The tensor ρ~v~vis known as the Reynolds stress tensor, so zonal flows aredriven by the Reynolds stresses. When the microturbu-lence is strong, the zonal flows are sufficiently robust tomake ∆ ≈ 1/k⊥. When the ions are the non-adiabaticspecies, the perpendicular wavenumber k⊥ tends to becomparable to the ion gyroradius, ρi. The reason is thatwhen k⊥ 1/ρi the instability grows faster the largerk⊥. However, when k⊥ 1/ρi the ions respond adiabat-ically to changes in the potential by moving across thefield lines, Section (VI.G), which removes the drive forthe instability. The typical fluctuation amplitude, Equa-tion (226), is of order the ion gyroradius to system size,ρi/a.

Microinstabilities, such as the electron temperaturegradient (ETG) mode that is discussed below, also ex-ist in which the ions respond adiabatically but electronsnon-adiabatically. The ions behave adiabatically whenthe wavenumber of the perturbations satisfies k⊥ρi 1,so the ions are free to cross the field lines in responseto changes in the potential. For this case zonal flowsare not important because the adiabatic ion response,δn/n ≈ −eδΦ/T , responds to the full variation in theelectric potential and not just the part that varies on themagnetic surfaces as is the case with adiabatic electrons.The radial extent ∆ of the constant-potential contours isstill limited by the ion gyroradius ρi because otherewisethe ions cannot respond adiabatically although the per-pendicular wavenumber for the ETG mode is compara-ble to the electron gyroradius, k⊥ρe ≈ 1. Because of thegreat anisotropy, k⊥∆ ≈ ρi/ρe, ETG micro-turbulencecan produce transport comparable to the ITG micro-turbulence with zonal flows (Dorland et al., 2000; Jenkoet al., 2000). For both the diffusion is comparable tothe gyro-Bohm rate. However, it should be noted thata number of experiments see a reduction in the diffusionwith ion mass rather than the increase that would be ex-pected from gyro-Bohm diffusion. This effect, which is

56

discussed by (Bessenrodt-Weberpals et al., 1993) is notunderstood theoretically.

An important feature of recent experiments has beenthe observation of transport barriers, which are narrowregions in which transport is greatly reduced. This topichas been recently reviewed (Terry, 2000; Wolf, 2003). De-spite the narrowness of the transport barriers, the re-duction in transport is sufficiently great to significantlyenhance the overall plasma confinement. The theoreticalexplanation, (Biglari et al., 1990), is a stabilization of themicroturbulence by a strong radial gradient, or shear, inthe E × B flow in the magnetic surfaces. The shearingrate of the flow is γs ≡

∣∣∣d[( ~E × ~B)/B2]/dr∣∣∣. If the shear-

ing rate, γs, is greater than the growth rate of a microin-stability, the instability is stabilized by being torn apartby the sheared flow and is stabilized. Since the jump inthe potential across a shear layer cannot be greater thanroughly ∆Φ ≈ Ts/e and the typical growth rate is Cs/a,one finds the width of a shear layer is roughly δs ≈

√aρi.

Neoclassical diffusion ion a tokamak, Equation (218) di-vided by the gyro-Bohm diffusion, Equation (229) is

Dg

Dnc≈ ι2ε3/2

ν

Csa, (231)

which is roughly 102 in a fusion plasma. A simple esti-mate of the temperature drop that one can obtain acrossa transport barrier, ∆T/T , is (δs/a)(Dg/Dnc). Actually(δs/a)(Dg/Dnc) is generally greater than unity under fu-sion conditions, so one can obtain a large temperaturedrop across a transport barrier. The difference betweenzonal flows and transport barriers is that zonal flows havea radial scale and a time variation determined by themicroturbulence. Transport barriers are quasi-static fea-tures with the strong variation in the electric potentialrelated to the strong variation in the pressure throughthe tendency in plasmas for

∣∣∣~∇p∣∣∣ ≈ ∣∣∣en~E∣∣∣ for one of thetwo species.

The diffusion coefficients that have been discussed aremore properly considered transport coefficients for heatthan for particles. The reason is that if either species re-sponds adiabatically there can be no particle transport.Radial particle transport averaged over a magnetic sur-face is Γr = 〈(n+ δn)δvr〉. For an adiabatic responseδn/n = ±eδΦ/T while vr = r · ( ~B × ~∇δΦ)/B2, so δnand vr are out of phase, which means their average overthe surface is zero. Although individual particles of thenon-adiabatic species diffuse, so heat can be transported,the electric field arranges itself so the net particle flux iszero is order to preserve quasi-neutrality, qni ' ene.

Neither species responds fully adiabatically in fully de-veloped micro-turbulence, so micro-turbulence generallyleads to particle transport across the magnetic field lines,which is effectively an enhancement of the perpendic-ular component of the resistivity tensor η⊥, Equation(29). The component of the resistivity along the mag-netic field, η|| is, however, rarely enhanced by micro-turbulence. There are two reasons. First, the parallel

component of the fluctuating electric field δE|| = −ik||δΦis small compared to the electric field across the magneticfield lines because |k||/k⊥| << 1, so little scattering ofthe parallel motion of the particles occurs. Second, ifany group of ions and electrons can diffuse rapidly acrossthe magnetic field lines, then η⊥ is enhanced, but if anygroup of electrons can flow freely along the magnetic fieldlines, then η|| remains close to its quiescent plasma value.

Our discussion of micro-turbulence has assumed onlythe electric potential is perturbed. Micro-turbulence cou-ples to shear Alfven modes, Section (VI.H), when theAlfven frequency, ωA ≡ vAι/Ro, is comparable to thatof microinstabilities, Cs/a. Since the Alfven velocity isvA ≡

√B2/µ0min, the coupling occurs when the plasma

pressure satisfies β ≡ 2µ0p/B2 > (aι/Ro)2. Alfven cou-

pling means that the component of the vector poten-tial parallel to the magnetic field, δA||, is perturbed. IfδA||/B is Fourier decomposed in magnetic coordinatesthat have a simple covariant form, Equation (57), thenthe magnetic surfaces are broken to form an island un-less each Fourier component (δA||/B)mn vanishes at itsrational surface, ι = n/m, Equation (79). It is unclearwhether the resonant components of (δA||/B)mn are non-zero under standard plasma conditions, but if they werenon-zero they would produce a qualitative change inthe micro-turbulence due to electron transport along thestochastic magnetic field lines. On the other hand if theFourier components (δA||/B)mn exactly vanish at theirresonant rational surfaces, then the effect of the couplingof the micro-turbulence to the shear Alfven modes wouldbe to wobble the magnetic surfaces, which makes cal-culations more difficult but causes no qualitatively newphysical effects.

The microinstability that has received the most atten-tion in recent years is the ion temperature gradient (ITG)instability, which is also called the ηi mode,

ηi ≡d ln(T )d ln(n)

, (232)

where T (ψt) is the ion temperature and n(ψt) is the iondensity. The ITG mode, which is discussed in the reviewby (Horton, 1999), appears to be responsible for the en-hanced transport of heat by ions in tokamak plasmas andcan be viewed as a kinetic version of sound waves thatare destabilized by the ion temperature gradient.

The simplest version of the ITG instability (Kadomt-sev and Pogutse, 1970) occurs in a uniform magnetic field~B = Bz with the plasma having temperature and densitygradients in the x direction. The fluctuations depend ony, z, and time as exp[i(kyy+kzz−ωt)] and only weakly onx. In a fluid model, the ions obey the continuity equation,∂n/∂t+ ~∇· (n~v) = 0 with the perpendicular componentsof the velocity given by ~v⊥ = ~E × ~B/B2. Using a tildeto denote perturbed quantities, the continuity equationimplies

ωn

n+kyB

d lnndx

Φ = kz v||. (233)

57

The parallel component of the velocity is given by forcebalance along the field line, nmidv||/dt = −∂p/∂z −en∂Φ/∂z, or

ωminv|| = kz(p+ enΦ). (234)

The ions are assumed to respond adiabatically, whichmeans is p/n5/3 is carried with the flow, ∂(p/n5/3)/∂t+~v · ~∇(p/n5/3) = 0, and

p

p=

53n

n− kyωB

(ηi − 2/3)d lnndx

Φ. (235)

The electrons are of sufficiently low mass that they canmove rapidly along the magnetic field line and remain inthermodynamic equilibrium,

n

n=eΦTe, (236)

which is called an adiabatic response. In giving the elec-tron response we have assumed that the perturbation hasa long spatial scale compared to the Debye length, whichimplies the electron and ion densities are essentially equalin both the unperturbed and the perturbed states. Inother words, the perturbed plasma is quasi-neutral. As-sembling results one finds

1− ω∗eω

=(kzω

)2T

mi

TeT

+53

+ω∗eω

(ηi −

23

),

(237)where the electron drift frequency is

ω∗e ≡ −kyTeeB

d lnndx

. (238)

If frequency of the perturbation, ω, is high compared tothe drift frequency ω∗e, then Equation (237) gives soundwaves

ω2 =Te + 5

3T

mik2z . (239)

However when the frequency of the perturbation is lowcompared to the drift frequency, ω ω∗e, the frequencyof the perturbation is

ω2 =T

mi

(23− ηi

)k2z , (240)

which has an exponentially growing ITG mode for ηi >2/3. In a plasma such as a fusion plasma in which colli-sions are weak, neither the sound wave nor the ITG modeare treated realistically in this analysis unless ω/k|| >>√T/mi, because otherwise strong ion Landau damping,

Section (VI.C), implies only decaying solutions exist. Inother words the analysis is only realistic for sound wavesif T/Te 1 and for the ITG mode if ηi 1. To theextent the electrons respond adiabatically, the ηi modecauses ion heat transport but no electron transport and,because of quasi-neutrality, no particle transport.

In addition to ignoring kinetic effects, two other impor-tant determinants of ITG stability were ignored. Theseare shear and the ~B × ~∇B drift. Magnetic shear, whichmeans the magnetic field lines change direction from onepressure surface to another, is stabilizing. Magnetic sheareffects will be considered in the discussion of Alfven in-stabilities, Section (VI.H.2). The ~B × ~∇B drift destabi-lizes the ITG drift in regions of bad magnetic field linecurvature, which means the center of curvature is on thehigher pressure side of the field lines. This destabilizationof the ITG mode is closely related to the destabilizationof ballooning modes by bad curvature, Section (V.C.2).The version of the ITG mode in which curvature effectsdominate is called the toroidal branch, and the versionin which curvature effects are subdominant is called theslab branch.

An important feature of ITG turbulence is that it be-comes strong only when a critical gradient is reached.In the simplest version of the theory, the critical gradi-ent is a critical value of ηi. When curvature effects areretained, the turbulence can become strong when the ra-dius of curvature R times 1/LT ≡ |d lnT/dr| exceeds acritical value. The R/LT critical gradient is counter in-tuitive since the weaker the curvature the easier it is forcurvature to destabilize the mode. However, the growthrate of the mode and the maximum rate of transport be-come small when R is large. In many experiments, inparticular those with a high ion temperature, the plasmais thought to operate just above the critical gradient.In this situation, the logarithmic ion temperature gradi-ent, d lnT/dr, is given by the critical gradient with themagnitude of the heat flux determined by conditions atthe plasma edge. That is, the ion temperature through-out the plasma is proportional to the ion temperaturenear the plasma edge. A similar phenomenon occursin the earth’s atmosphere where the temperature pro-file is determined by the critical gradient for convection,d ln(p)/d ln(n) ≥ γ, with the magnitude of the heat fluxdetermined by the boundaries of the convection zone.

A second instability, the electron temperature gradi-ent (ETG) mode, has very similar physics, only the roleof the electrons and ions is reversed. The ETG modehas wavenumbers comparable to the electron gyroradius,ρe, on this scale the ions have only weak magnetic ef-fects, ρi ≈ 60ρe, and the ions respond adiabatically,δn/n = −eΦ/T . The ETG mode may be responsiblefor the enhancement of the electron heat transport aboveits neoclassical value (Dorland et al., 2000; Jenko et al.,2000), and the combination of the ITG and the ETGmodes may give the enhanced particle transport that isobserved in experiments.

G. Gyrokinetic theory

• The simplification of the particle trajectories andkinetic theory that occurs when the gyroradius ρ issmall compared to the system size can be extended

58

to include perturbations of the magnetic and elec-tric fields that have arbitrarily large wavenumbersperpendicular to the magnetic field, k⊥ρ arbitrary.This approximate kinetic theory is called gyroki-netic theory.

Computational studies of microinstabilities, which areon the scale of the gyroradius of one of the species, arecarried out using the gyrokinetic equations. This andother computational studies of plasmas have been dis-cussed by (Tang, 2002). The gyrokinetic equations arethe kinetic equations but with the particle velocity repre-sented by the gyrokinetic drift velocity. The derivation ofthese equations was developed by (Rutherford and Frie-man, 1968), (Taylor and Hastie, 1968), and (Antonsenand Lane, 1980).

The gyrokinetic drift velocity consists of two parts.The first part is the guiding center drift of the parti-cles in the large scale electric and magnetic fields. Thisis just the ordinary guiding center drift, ~vg, which wasdiscussed in Section (VI.D). The second part, δ~vg, isthe modification of the drift of the guiding center by thesmall scale perturbations. The perturbations that arisein microturbulence are small, of order gyroradius to sys-tem size ρ/a, but the drifts that the perturbations causecan be of order of other guiding center drifts due to thestrong spatial gradients, of order 1/ρ.

Despite having a wavelength across the magnetic fieldlines that is comparable to the gyroradius of one ofthe species, the perturbations that arise in a microtur-bulent plasma have a wavelength along the magneticfield comparable to the plasma size. The long parallelwavelength arises to minimize Landau damping, Section(VI.C). This complicated spatial structure can be ac-commodated mathematically by writing perturbed quan-tities, such as the electric potential, in eikonal form

δΦ = Φ(~x, t)eiS(ψt,α) (241)

just a ballooning modes are. The eikonal S(ψt, α) de-pends on the two Clebsch coordinates, where ~B = ~∇ψt×~∇α, Equation (9), and represents the rapid variation ofδΦ across the magnetic field lines. Φ varies on a muchlonger spatial scale, which involves the plasma size, andwith a slow time scale, of order the thermal sound speeddivided by the plasma radius Cs/a. Although

~k⊥ ≡ ~∇S (242)

is very large, k⊥ρ ≈ 1, the spatial variation of ~k⊥ itselfis much longer involving the overall scale of the plasma.

Three electromagnetic quantities affect the particledrifts due to their rapid spatial variation: the perturbedelectric potential, δΦ = Φ exp(iS), the perturbed parallelcomponent of the vector potential, δA|| = A|| exp(iS),and the perturbed parallel component of the magneticfield, δB|| = B|| exp(iS). As shown below, their effect onthe drifts is given by

δ~vg ≡⟨e−iSg

d~xgdt

⟩= − i

~k⊥ × b

Bχ (243)

with the generalized potential

χ =(Φ− v||A||

)J0(k⊥ρ) +

µB||

q

2J1(k⊥ρ)k⊥ρ

(244)

defined using the time derivative of the particle energyor Hamiltonian,

q∂χ

∂t≡⟨e−iSg

dH

dt

⟩. (245)

Sg means the eikonal is evaluated at the guiding centerposition, and 〈· · ·〉 means an average over the gyrophaseϑ. J0 and J1 are the zeroth and the first cylindrical Besselfunctions, µ ≡ mv2

⊥/2B is the magnetic moment, whichis an adiabatic invariant, and b ≡ ~B/B. The time deriva-tive of the guiding center position, ~xg, Equation (179),depends on the electric and magnetic fields at the trueposition of the particle, ~x rather than at the location ofthe guiding center, ~xg. The difference between the trueposition ~x and the guiding center ~xg is the vector gyrora-dius ~ρ = ~x− ~xg, Equation (176). The difference betweenthe eikonal evaluated at the guiding center position andthe actual position of a particle is Sg − S = −~k⊥ · ~ρ.

The Bessel functions that appear in the gyrokineticequations measure the importance of the interaction ofa charged particle with the magnetic field. As k⊥ρ → 0the interaction is strong, and the Bessel functions havethe limits J0(k⊥ρ) → 1 and 2J1((k⊥ρ)/(k⊥ρ) → 1. Fork⊥ρ → ∞, the interaction becomes weak but at a slowrate, 1/

√k⊥ρ. As z ≡ k⊥ρ → ∞, the Bessel functions

have the asymptotic forms Jn(z) =√

(2/πz) cos[z− (n+1/2)(π/2)].

The distribution function of gyrokinetic theory isevaluated at the guiding center position of the parti-cles ~xg rather than their actual position ~x and hasthe form f(H,µ, ~xg, t) = f0(H,~xg) + δf where δf =f(H,µ, ~xg, t) exp(iSg) and f0 is the equilibrium distri-bution function.

The gyrokinetic equation for the amplitiude of the per-turbed distribution function, δf = f exp(iSg), is(

∂

∂t+ ~vg · ~∇+ i~k⊥ · ~vg

)f +Q+ FI = CL(f) (246)

where the inhomogeneous term is

FI ≡ δ~vg · ~∇f0 + q∂χ

∂t

∂f0∂H

(247)

with f0 the distribution function for the unperturbedequilibrium. CL(f) is a linearized collision operator, andthe quantity Q is a quadratic non-linearity. The domi-nant non-linearity in gyrokinetic theory is generally takento be the (d~xg/dt) · ~∇δf non-linearity for which Q hasthe form

Q =b ·⟨⟨e−iSg ~∇(χeiSg )× ~∇(f eiSg )

⟩⟩B

. (248)

59

The notation 〈〈· · ·〉〉means an average over the gyrophaseand a projection of the expression using the orthogonal-ity of the solutions to the linear gyrokinetic equation.The function χ exp(iSg) comes from converting the elec-tric and magnetic fields that appear in d~xg/dt from theparticle position to the guiding center position plus anaverage over the gyrophase. The gyrophase average isdiscussed below in the derivations of δ~vg and χ.

The non-linear part of the gyrokinetic equation Q re-quires a definition of orthogonality among the solutionsto the linear gyrokinetic equation, which means solutionsthat have different eikonals S. For example, in axisym-metric systems different solutions in eikonal form canhave different toroidal mode numbers, as discussed inSection (V.C.2) on ballooning modes. The gradient ofthe eikonal is the effective perpendicular wavenumber.As is well known from Fourier analysis, perturbationswith different wavenumbers are orthogonal. The defini-tion of orthogonality and the evaluation of the requiredintegrals to obtain Q is the major subtlety of study ofmicro-turbulence with the gyrokinetic equation. An ap-proximate method, which is relatively easy to follow, hasbeen given by (Beer, Cowley, and Hammett, 1995).

The linear part of gyrokinetic equation follows fromEquations (243, 245) and the kinetic equation df/dt =C(f), where df/dt = df0/dt + dδf/dt. The time deriva-tive of the perturbed distribution function is dδf/dt =(df/dt)eiSg + ifeiSgdSg/dt, and the time derivativeof the equilibrium distribution function is df0/dt =(df0/dH)dH/dt + (d~xg/dt) · ~∇f0. The linear gyroki-netic equation is obtained by multiplying both sides ofthe kinetic equation by e−iSg and then averaging overthe gyrophase, ϑ. The rapidly varying part of δf isin the eikonal, so f is slowly varying, and df/dt =∂f/∂t + ~vg · ~∇f . The gyrophase average 〈dSg/dt〉 =〈~vg · ~∇Sg〉 = ~k⊥ · ~vg.

To have a complete set of equations, relations areneeded between the perturbed distribution function fand the perturbed electric potential Φ, parallel compo-nent of the vector potential A||, and parallel componentof the magnetic field B||. While taking the averages, theperpendicular velocity of the particle will be written as

~v⊥ = v⊥cosϑk⊥ × b+ sinϑk⊥). (249)

The perturbed electric potential is given by Pois-son’s equation, which can be approximated as k2

⊥δΦ =(q/ε0)δn, or

k2⊥Φ =

q

ε0n. (250)

The perturbed density of a species is δn =∫

∆fd3vwhere ∆f ≡ qδΦ(∂f0/∂H) + δf , Equation (152). Sinceδf = f exp(iSg) but δn = n exp(iS) with Sg − S =−k⊥ρ cosϑ, the perturbed density is

n = qΦ∫∂f0∂H

d3v +∫J0(k⊥ρ)fd3v (251)

where we used the integral expression for the zeroth orderBessel function

J0(z) =12π

∫ π

−πe−iz cosϑdϑ (252)

to carry out the integration over the gyrophase ϑ, whichis one of the integration variables in d3v, Equation (170).

The perturbed parallel component of the vector poten-tial can be calculated using Ampere’s law, ~∇× ~∇× δ ~A =µ0δ~j, or

k2⊥A|| = µ0j||. (253)

The perturbed current denstiy

j|| = q

∫v||J0(k⊥ρ)fd3x, (254)

where we have assumed df0/dH is symmetric in v||.The equation for B|| is given by force balance across

the field lines, which for short wavelength perturbationshas the form k · ~∇(p⊥ +B2/2µ0) = 0. This follows fromµ0~j × ~B = ~B · ~∇ ~B − ~∇B2/2 since ~B · ~∇ ~B = B2b · ~∇b +

b ~B · ~∇B is negligible when k⊥ >> k||. The perturbed

perpendicular pressure is p⊥ =⟨∫

m(~v · k⊥)2∆fd3v⟩,

which means

p⊥ =(qΦ∫µ∂f0∂H

d3v +∫µf

2J1(k⊥ρ)k⊥ρ

d3v

)B (255)

where µ ≡ mv2⊥/2B is the magnetic moment and the first

order Bessel function obeys

J1(z)z

=12π

∫ π

−πe−iz cos θ sin2 θdθ. (256)

The adiabatic term in p, which is the term proportionalto Φ, is equal and opposite for the electrons and the ionsprovided the Debye length satisfies k⊥λD 1, so eδne =qδni. Force balance is p⊥ +BB||/µ0 = const., so

B|| = −µ0

∫µf

2J1(k⊥ρ)k⊥ρ

d3v. (257)

The remainder of the section gives the derivations ofEquation (244) for χ, which is defined by q∂χ/∂t ≡〈e−iSgdH/dt〉, and δ~vg ≡ 〈e−iSgd~xg/dt〉, Equation (243).

First, we need the approximate relation between thevector potential and the magnetic field that holds whenk⊥a >> 1 with a the plasma radius. The magnetic fieldis δ ~B = ~∇ × δ ~A where δ ~A = ~A exp(iS). Dotting thisexpression with the unperturbed magnetic field ~B, onefinds that ~B · δ ~B = ~∇ · (δ ~A× ~B) + δ ~A · ~∇× ~B. The onlyterm on the right hand side that is large is the first termthrough its dependence on ~k⊥ ≡ ~∇S,

δB|| = −(i~k⊥ × b) · δ ~A⊥. (258)

60

The components of the perturbed magnetic field that areperpendicular to the unperturbed field are given by ~B ×δ ~B = ~∇( ~B · δ ~A)− δ ~A× ~∇× ~B− δ ~A · ~∇ ~B− ~B · ~∇δ ~A. Thelarge term is the first term, which gives

b× δ ~B = i~k⊥δA||. (259)

Second, we drive the expression for χ, Equation (244),from its definition, Equation (245). The exact equa-tion for the change in the energy of the particles isdH/dt = q(∂Φ/∂t − ~v · ∂ ~A/∂t), Equation (197). Therequired gyrophase average 〈e−iSgdH/dt〉 is the sum oftwo terms. The first term is

q

(∂Φ∂t

− v||∂A||

∂t

)⟨ei(S−Sg)

⟩. (260)

The gyrophase average 〈exp[i(S − Sg)]〉 can be writtenas 〈exp(ik⊥ρ cosϑ)〉 = J0(k⊥ρ). The second term is

−⟨~v⊥ ·

∂

∂t

(δ ~A⊥e

−iSg

)⟩=µ

q

∂B||

∂t

2J1(k⊥ρ)k⊥ρ

, (261)

which is derived using Equations (249) and (258), ~v⊥ ·δ ~A⊥ = v⊥k⊥ · δ ~A⊥ sinϑ + i(v⊥/k⊥)δB|| cosϑ. The gy-rophase average 〈sinϑ exp[i(S − Sg)]〉 is zero because

12π

∫ π

−πeiz cosϑ sinϑdϑ = 0, (262)

and the gyrophase average 〈cosϑ exp[i(S − Sg)]〉 is

12π

∫ π

−πeiz cosϑ cosϑdϑ = iJ1(z), (263)

where z = k⊥ρ and ρ = mv⊥/qB. Assembling the partsone obtains 〈e−iSgdH/dt〉 = q∂χ/∂t with χ given byEquation (244).

Third, we calculate the part of δ~vg, Equation (243),that is across the magnetic field. The time derivative ofthe guiding center position is given by Equation (179).Because of the slowness of the time variation, the per-turbed electric field is given by the gradient of the po-tential,

δ ~E = −i~k⊥δΦ. (264)

The contribution of the E ×B drift to δ~vg is⟨e−iSg

δ ~E × ~B

B2

⟩= − i

~k⊥ × b

BJ0(k⊥ρ)Φ (265)

where we used 〈exp[i(S − Sg)]〉 = J0(k⊥ρ). The effect ofthe perturbation on the term

δ

⟨e−iSg~v⊥ × b

d

dt

1Ω

⟩= −

iB||

ΩB

⟨ei(S−Sg)~v⊥ × b

dS

dt

⟩(266)

The large term is dS/dt = ~v⊥ · ~∇S = ~k⊥ · ~v⊥, whichmeans dS/dt = k⊥v⊥ sinϑ. Equation (249) implies~v⊥ × b = v⊥(k⊥ × b) sinϑ − v⊥k⊥ cosϑ. The gyrophaseintegral that involves the second term in ~v⊥ × b is zero,and the gyrophase integral that involves the first term iscalculated using Equation (256),⟨

e−iSg~v⊥ × bd

dteiS⟩

= −i(~k⊥ × b)v2⊥J1(k⊥ρ)k⊥ρ

. (267)

Fourth, we calculate the part of δ~vg, Equation (243),that is along the perturbed magnetic field. Motion alongthe magnetic field is given by v||b+ v||δb with the changein the direction of the magnetic field, δb = δ ~B⊥/B.Equation (259) implies

δ⟨e−iSgv||δb

⟩=i~k⊥ × b

Bv||A||J0(k⊥ρ). (268)

Putting the pieces together, one finds the effect of theperturbations on the drift Equation (243) with the gen-eralized potential χ given by Equation (244).

H. Alfven instabilities

• Shear Alfven waves are a twisting of the magneticfield lines. These waves have a continuous spec-trum of possible frequencies but are generally heav-ily damped. The damping occurs if there is suffi-cient field line shear, which means variation in thethe direction of the magnetic field lines, or suffi-cient variation in the Alfven velocity, B/

√µ0ρ with

ρ the plasma mass density, transverse to the mag-netic field.

• If the Alfven velocity varies along the magneticfield lines and the magnetic field lines have shear,then the continuous spectrum of Alfven waves hasgaps and weakly damped discrete (sharp frequency)shear Alfven modes can exist in these gaps.

• Energy is transfered between shear Alfven modesand particles with a resonant velocity, ω/k||, alongthe field lines. A destabilizing transfer of energyfrom the particles to the waves occurs if the dia-magnetic drift frequency of the interacting parti-cles, ω∗ of Equation (289), is larger than the fre-quency ω of the Alfven mode; otherwise the trans-fer is stabilizing. Only particles, such as fusion al-phas, that have a poloidal gyroradius comparableto their density gradient have a destabilizing inter-action.

If a magnetic field is twisted and released, the twisttravels down the magnetic field lines at the Alfven veloc-ity,

vA ≡

√B2

µ0ρ, (269)

61

and is called a shear Alfven wave. Most shear Alfvenperturbations are strongly damped, but certain pertur-bations called gap modes have weak natural damping.The weakly damped gap modes are susceptible to beingdriven to high amplitude by interactions with particlesthat are moving along the field lines with a velocity com-parable to the Alfven velocity. This instability of thegap modes goes under the name of the toroidal Alfveneigenmode (TAE) instability. The theory of short wave-length TAE modes was developed by (Cheng et al., 1985),and the theory of the more important low mode numberTAE modes by (Cheng and Chance, 1986). The concernis that alpha particles produced by the fusion reactionwill drive gap modes to a high amplitude causing a lossof the alphas from the plasma before they can transfertheir energy to the bulk plasma (ITER Physics ExpertGroup on Energetic Particles, Heating and Current Driveet al., 1999a). Only at a low plasma β ≡ 2µ0p/B

2 is theAlfven velocity above the speed vα at which fusion al-phas are produced, β ' 1.8%(T/20kev)(vα/vA)2 with Tthe plasma temperature. (Wong, 1999) has reviewed therelation between the experiments and theory for TAEmodes. The properties of shear Alfven modes that arecentral to the theory of TAE instabilities will be discussedin this section.

1. Continuum Alfven wave damping

Shear Alfven waves have a continuous spectrum that isheavily damped (Tataronis and Grossmann, 1973). Thephysics is illustrated by a model in which the equilibriummagnetic field is uniform and in the z direction, ~B = Bzbut the plasma mass density, ρ depends on x. Forcebalance is ρ(x)∂~v/∂t = δ~j× ~B with µ0δ~j = ~∇× δ ~B. Theperturbation to the magnetic field is given by the idealOhm’s law, which means ∂δ ~B/∂t = ~∇ × (~v × ~B). Thetwo components of velocity that enter the calculation arewritten as vx = vx(x) exp[i(kyy + k||z − ωt)] and vy =vy(x) exp[i(kyy+k||z−ωt)]. The wave number ky is in thesurface of constant Alfven velocity, and the wavenumberk|| is along the magnetic field.

If the model equations for Alfven waves are analyzed,keeping all three components of δ ~B, one finds(

ω2 − k2||v

2A(x)

)vx + v2

A

d2vxdx2

+ ikyv2A

dvydx

= 0 (270)

and(ω2 − (k2

|| + k2y)v

2A(x)

)vy + ikyv

2A

dvxdx

= 0. (271)

These equations, which are more general than the deriva-tion given here, can be combined into a single equationfor vx,

d

dx

(ω2 − k2

||v2A

ω2 − (k2|| + k2

y)v2A

dvxdx

)+ω2 − k2

||v2A

v2A

vx = 0. (272)

When the Alfven velocity is independent of position, thisequation has two types of solutions. One type of solu-tion is the compressional Alfven mode with v2

Ad2vx/dx

2+ω2 − (k2

|| + k2y)v

2Avx = 0, which is a propagating wave

if k2x ≡ (ω2/v2

A) − (k2y + k2

||) ≥ 0 and evanescent (expo-nential dependence on x) otherwise. The other solutionis the shear Alfven wave with ω2 = k2

||v2A, which has

an arbitrary dependence on x. When the Alfven velocitydepends on position, these two types of solutions are cou-pled, and the point where v2

A(x) = ω2/k2|| is a singular

point, a point at which energy is absorbed.Equation (272) has a singular point, which will be

denoted by x = 0, where v2A(x) = ω2/k2

||. At thesingular point, the velocity vy has a 1/x singularity,which means there is an infinite amount of energy inthe vicinity of the singular point, an unphysical result.This singularity in the energy means an arbitrarily largeamount of energy can accumulate. This build-up of en-ergy can be represented by letting the frequency be com-plex, ω = ω0 + iγ. Assume γ is small, let x = 0 be thepoint where k2

||v2A = ω2

0 , and let

δ ≡ 2ω0γ

k2||dv

2A/dx

(273)

evaluated at x = 0. Then near its singular pointEquation(272) can be approximated as

d

dx(x− iδ)

dvxdx

= (x− iδ) k2y vx, (274)

while Equation (271) implies ky vy = idvx/dx.To calculate the power per unit area flowing into the

singular point of Equation (272), we study the two in-dependent solutions of Equation (274), which holds inthe vicinity of the singular point. These solutions arethe cylindrical zeroth order modified Bessel functionsof the first, I0(ξ), and the second kind, K0(ξ), whereξ ≡ ky(x− iδ). As ξ → 0, I0 → 1 and K0 → − ln ξ. Theregular, or I0, part of the solution is undamped, but thesingular, or K0, part is. Focusing on the singular part ofthe solution, let

vx = VsK0(ξ) (275)

where Vs is a constant, then near ξ = 0, vy = −iVs/ξ.The average power per unit y− z area that goes into thevelocity singularity is P = Re

∫ρv∗y(∂vy/∂t)dx/2

. We

used the easily proven result that if a function f(y) is thereal part of fc exp(ikyy) and another function g(y) hasthe same form, then y-average of f(y)g(y) isRef∗c gc/2.Since ∂vy/∂t = −iωvy, P = (γ/2)

∫ρv∗y vydx. The inte-

gral that must be performed has the form∫ ∞

−∞

dx

x2 + δ2=

π

|δ|, (276)

so P = (π/2)(γ/|δ|)ρ0V2s /k

2y where ρ0 = ρ(x = 0). In

other words, the power per unit area going into the sigu-

62

larity is

P =π

2ω0ρ0V

2s

k2y

∣∣∣∣d ln vA(x)dx

∣∣∣∣ . (277)

2. Weakly damped gap mode

The existence of weakly damped gap modes requiresboth a shear in the magnetic field lines across the linesand a variation in the magnetic field strength along themagnetic field lines. An equation for the shear Alfvenmode, which contains both shear and variation in theAlfven velocity, is relatively simple when the perturba-tion is localized to a magnetic field line. This occurs whenthe perpendicular wave number, k⊥ is large in compar-ison to the wavenumber along the magnetic field linesk||. The assumption k⊥ k|| leads to the eikonal ap-proximation as in discussion of ballooning modes, Section(V.C.2), and gyrokinetics, Section (VI.G).

All the perturbed quantities in a shear Alfven wave canbe expressed in terms of the perturbation to the electricpotential δΦ = Φ exp(iS), where S(α) = 2πmα is theeikonal andm is a large integer. The perpendicular wave-length is k⊥ ≡ ~∇S. The Clebsch angle α = (θm− ιϕ)/2πis defined so ~B = ~∇ψt× ~∇α, Equation (9). The equationfor the perturbed electric potential is

∂

∂ζ

(k2⊥B

∂δΦ∂ζ

)= −k

2⊥B

Ω(ζ)2δΦ, (278)

where ζ is a dimesionless coordinate along the magneticfield line. The distance along a line is ` = (L/2π)ζ withL a characteristic distance. Ω is a normalized frequency,

Ω(ζ)2 ≡ ω2

(L

2πvA(ζ)

)2

(279)

with ω the frequency of the perturbation, Φ ∝ exp(−iωt).The characteristic form is Ω2(ζ) = Ω2

0(1 + 2ε cos ζ). In alarge aspect ratio tokamak, ε = r/Ro gives the variationin the magnetic field strength, and L = 2πRo/ι is theperiodicity length of the field strength. The period ofthe magnetic field strength in a tokamak is θm = 2π, butalong a field line θm = ιϕ with d` = Rodϕ. The frequencyωA ≡ (2π/L)vA is known as the Alfven frequency, so Ω =ω/ωA. For a large aspect ratio tokamak, ωA = vAι/Ro.

A magnetic field has global shear if dι/dψt is non-zero.When the field sheared the characteristic dependence ofk2⊥/B on ζ is

k2⊥B

∝ (1 + s2ζ2) (280)

with s ∝ dι/dψt. This follows from 2π~∇α = ~∇θm −ι~∇ϕ − ϕ(dι/dψt)~∇ψt with ϕ ∝ ζ. For a large aspectratio tokamak, ζ = ιϕ and s = d ln ι/d ln r.

This paragraph contains the derivation of Equation(278) for δΦ and can be skipped. The derivation starts

with the ideal Ohm’s law, δ ~E + ~v × ~B. The mag-netic perturbation of a shear Alfven wave is perpendic-ular to the main field, and when k⊥ → ∞ Equation(259) shows that δ ~B⊥ = i~k⊥ × bδA|| with b ≡ ~B/B.Therefore, δ ~E = −b∂δA||∂t − ~∇δΦ, which implies ~v =( ~B × i~k⊥)δΦ/B2 and ∂δA||/∂t = −( ~B · ~∇δΦ)/B. Forcebalance ρ∂~v/∂t = δ~j× ~B implies δ~j⊥ = ( ~B×ρ∂~v/∂t)/B2

and that ~∇ · δ~j⊥ = k2⊥ρ(∂δΦ/∂t)/B

2. Since the cur-rent is divergence free, ~B · ~∇(j||/B) = −~∇ · δ~j⊥. There-fore, ~B · ~∇

(j||/B

)= −k2

⊥(ρ/B2)∂δΦ/∂t. Ampere’s lawgives k2

⊥δA|| = µ0j||, and ∂2δΦ/∂t2 = −ω2δΦ. Let~B · ~∇ = (2πB/L)∂/∂ζ with L a characteristic lengthalong the field line. Ω2 is defined by Equation (279).Combining results, one obtains Equation (278).

The energy density of a shear Alfven wave,

w =ρk2⊥

4B2

(δΦ∗δΦ +

1Ω2

∂δΦ∗

∂ζ

∂δΦ∂ζ

), (281)

is the sum of the kinetic energy ρ~v∗ ·~v/4 and the magneticenergy δ ~B∗ · δ ~B/4µ0. The factors of four, where two isexpected, come from the use of complex arithmetic asdiscussed just above Equation (276).

A shear in the magnetic field lines, s 6= 0, causesa rapid loss of energy from Alfven waves if the fieldstrength is constant, ε = 0. Consider the Fourier trans-form of Equation (278) with k2

⊥/B replaced using Equa-tion (280). That is, let δΦ =

∫φ exp(iκ||ζ)dκ||, which

implies dδΦ/dζ → iκ||φ and ζδΦ → −idφ/dκ||. Thenwhen the variation of the Alfven speed along the fieldlines is ignored, Ω2 = const., Equation (278) implies

s2d

dκ||

(Ω2 − κ2

||

) dφ

dκ||=(Ω2 − κ2

||

)φ. (282)

This equation is singular at the place where κ|| = Ω,which is the equation for a shear Alfven wave in the ab-sence of shear, s = 0. The energy density in κ|| space,wκ, is defined so

∫wκdκ|| =

∫wdζ, and is given by

wκ =π

2ρ

(k⊥B

)2

0

(1 +

κ2||

Ω2

)(φ∗φ+ s2

dφ∗

dκ||

dφ

dκ||

)(283)

with the subscript naught on (k⊥/B)20 implying evalu-ation at ζ = 0. Near the singular point κ|| = Ω, thedominant component in the energy density wκ scales asthe square of sdφ/dκ||, and an infinite amount of energyis located near this point in κ|| space. These equationsand the resolution of their singularity exactly follow thederivation that led to Equation (277) for the dissipationof Alfven wave energy. The singular point, κ|| = Ω,of Equation (282) represents the transfer of energy tothe region ζ → ∞, which in steady-state has infiniteenergy content, while the singular point k||vA = ω ofEquation(272) represents the transfer of energy to an ar-bitrarily small region in x, which means to kx →∞.

63

0 20 40 60 80 100 120 140-0.4

-0.2

0

0.2

0.4

0.6

0.8

1u

s z

FIG. 13 The solution to Equation (285) is plotted for threevalues of the eigenvalue. Ω0 = 0.42 (red) gives a solution inthe continuum, Ω0 = 0.4775 (green) gives a singular solution,and Ω0 = 0.487 (blue) gives a discrete mode. The parameterss = 0.5 and ε = 0.2 are the same for all three solutions.

The existence of gaps in the spectrum of Alfen wavesand the existence of weakly damped modes in those gapsis easier to demonstrate if the perturbed quantities areexpressed in terms of u ≡ (ik⊥/

√B)δΦ, which places the

energy density in a non-singular form,

w =ρ

4B

(|u|2 +

1Ω2

∣∣∣∣dudζ − sζ

1 + s2ζ2su

∣∣∣∣2). (284)

The equation for u has the characteristic form

∂2u

∂ζ2− s2

(1 + s2ζ2)2u = −(1 + 2ε cos ζ)Ω2

0u, (285)

Figure (13). The equation obtained from transformingEquation (278) is ∂2u/∂ζ2 − σ(ζ)u+ Ω2(ζ)u = 0, whereσ(ζ) ≡ (

√B/k⊥)∂2(k⊥/

√B)/∂ζ2. Using k⊥/

√B ∝√

1 + s2ζ2 and Ω2(ζ) = Ω20(1 + 2ε cos ζ), one obtains the

characteristic equation for u. If Ω2 = const., then thesolution is u → exp(iΩζ) for either s → 0 or sζ → ∞.This means as ζ →∞ the Fourier transform of the shearAlven wave satisfies κ|| → Ω. Since the amplitude of uis constant as ζ → ∞, the solution u → exp(iΩζ) hasinfinite energy.

Now consider the effect of a variation in the Alfvenspeed along the magnetic field lines. The characteristicequation for u, Equation (285), reduces to the Mathieuequation (Bender and Orszag, 1978),

d2u

dx2+ (1 + 2ε cosx)Ω2

0u(x) = 0, (286)

for either zero shear, s = 0, or for |sζ| 1. The Math-ieu equation has two types of solutions depending on the

value of the eigenvalue Ω20. For certain ranges of Ω2

0, theregular regions, the equation yields oscillatory solutions,and for others, the singular, or gap, regions, there areexponentially growing solutions, Figure (13). The small-est Ω2

0 > 0 region that is singular occurs in the vicinityof Ω2

0 = 1/4 for |ε| << 1, which means ω = ωA/2 withωA ≡ vAι/Ro the Alfven frequency. More precisely, thesingular, or gap, region is Ω2

0 = 14 (1±ε) for |ε| << 1. The

regular ranges of Ω20 give continuum shear Alfven waves.

These solutions extend over the full range of ζ and, there-fore, require an infinite energy to drive. The singular re-gions, or gaps, are values of Ω2

0 for which physical shearAlfven waves do not exist. The energy in a singular so-lution is in the region |ζ| → ∞, since the amplitude of uincreases exponentially with |ζ|.

The characteristic equation for u, Equation (285), alsohas spatially bounded solutions, which means the solu-tion is concentrated in the region where |sζ| is small,Figure (13). These solutions are the gap modes becausethey occur in the singular gap regions of the Mathieuequation and have little intrinsic damping. They are themodes of interest for Alfven instabilities. Since they havefinite energy, they can be driven unstable by their in-teractions with high energy particles. The existence ofdiscrete modes in the gaps of a frequency spectrum isa well-known phenomenon in condensed matter physics,for a simple discussion see (Allen et al., 2000).

3. Particle-Alfven wave interaction

The interaction of the particles that form the plasmawith the shear Alfven wave is given by the kinetic equa-tion. This interaction is weaker than one might first ex-pect for modes, such as the discrete or gap mode, thathave no parallel electric field. When E|| = 0, particles inresonance with a shear Alfven wave, v|| = ω/k||, do notexchange energy with the wave unless there is a crossfield drift, vd, in the equilibrium magnetic field.

To simplify the analysis of the interaction of particleswith a shear Alfven wave, the perturbed electric potentialwill be assumed to have the form

δΦ = Φei(k||z+kyy−ωt). (287)

The distribution function f of the species that has a res-onant interaction with the Alven wave can be written asthe sum of an equilibrium distribution and a perturba-tion, f = fr(H,x) + δf and obeys the Vlasov equationdf/dt = 0. The time-averaged power to the Alfven waveper unit volume will be shown to be

pw =π

2k2yv

2d

|k|||

(ω∗ω− 1) q2Tr

Φ2fr

(ω

k||

), (288)

where vd is the guiding center drift velocity of particlesin the y direction in the unperturbed magnetic field,

ω∗ ≡kyTrqB

∂ ln fr∂x

(289)

64

is the diamagnetic drift frequency, fr(vz) ≡∫frdvxdvy,

and Tr ≡ −∂ ln fr/∂H is the effective temperature of theinteracting particles, which have a Hamiltoian H.

The power to the shear Alfven wave, Equation (288),has a destabilizing sign only if ω∗ is greater than ω, whichis equivalent to the requirement that the gyroradius ofthe interacting particles in the poloidal field alone, ρθ, besufficiently large to be comparable to their density gradi-ent. The condition on the poloidal gyroradius is derivedby first noting that the gap mode has a typical frequencyω = ωA/2 with the Alfven frequency ωA ≡ vAι/Ro. Forthe interacting species Tr ∼ mrv

2A, and ky ' m/r with

m the poloidal mode number. Defining the poloidal gry-roradius ρθ ≡ (Ro/ιr)ρ, one finds the condition ω∗ > ωimplies ρθd lnnr/dr >∼ 1/m. Only particles, such as fu-sion alphas, that have a poloidal gyroradius comparableto their density gradient can destabilize Alfven modes.Extremely high m-number Alfven modes are not of suchgreat concern because of the small radial extent of theperturbations that they produce.

To derive the power going to an Alfven wave, both thetime derivative of the unperturbed and the perturbedparts of the distribution function are needed. The timederivative of the unperturbed part of the distributionfunction fr is given by

dfrdt

=dxgdt

∂fr∂x

+dH

dt

∂fr∂H

, (290)

which depends on dH/dt = ∂H/∂t and the x componentof the guiding center velocity, dxg/dt. The time rate ofchange of the Hamiltonian is

∂H

∂t= q

(∂δΦ∂t

− v||∂δA||

∂t

)= q

(1−

k||v||

ω

)∂δΦ∂t(291)

where we used δE|| = −∂δΦ/∂z − ∂δA||/∂t = 0, whichmeans ωδA|| = k||δΦ. The relation between δA|| and δΦalso gives the guiding center velocity in the x direction,

dxgdt

=δEyB

+ v||δBxB

= −(

1−k||v||

ω

)1B

∂δΦ∂y

, (292)

and that the time derivative

dfrdt

= i(ω − k||v||)(1− ω∗

ω

) qδΦTr

fr, (293)

where we have written ∂ ln fr/∂H = −1/Tr and used thediamagnetic drift frequecy, Equation (289).

The time derivative of the perturbed part of the distri-bution function δf is given by dδf/dt = ∂δf/∂t+~vg ·~∇δf ,which can be written as dδf/dt = −i(ω−k||v||−kyvd)δf ,where vd is the drift of particles in the y direction in theunperturbed magnetic field. The perturbed distributionfunction is given by dδf/dt = −dfr/dt, or

δf =(1− ω∗

ω

)(1 +

kyvdω − k||v|| − kyvd

)qδΦTr

fr. (294)

The time-averaged power per unit volume going to theAlfven wave is pw = 〈−δ~j · δ ~E〉. The perturbed electricfield is δ ~E = −ikyδΦy, and the part of δjy =

∫vdδfd

3vthat is in phase is the imaginary part, which is given bythe Landau integral, Equation (164),

iδjiy = −i π|k|||

qkyv2d

(1− ω∗

ω

) qδΦTr

fr

(ω

k||

), (295)

where fr(vz) ≡∫frdvxdvy. The time-averaged power

per unit volume to the Alfven wave is pw = 〈−δ~j · δ ~E〉 =〈δ~j⊥ · ~∇δΦ〉, which implies pw = ky j

iyΦ/2. Consequently,

the power to the wave per unit volume is given by Equa-tion (288).

VII. PLASMA EDGE

• Control of the plasma edge is important to (a)maintain plasma purity, (b) limit the maximumheat flux to the chamber walls, and (c) achieve ahigh edge temperature if the plasma transport isassociated with a critical temperature gradient.

• The plasma edge can be defined by a solid object,called a limiter, or by a separatrix between themagnetic field lines that lie on toroidal surfacesand open magnetic field lines that intercept thechamber walls. Most modern plasma confinementdevices define the edge with a separatrix. Thismethod of defining the plasma edge is called a di-vertor.

The edge is the interface between the hot plasma andthe walls. This interface must remove impurities, such asthe helium ash, and the waste heat. Ideally the wasteheat is removed by electromagnetic radiation becausethat avoids hot spots on the walls. In addition the edgeis more important to plasma confinement than one mightexpect because transport often appears to obey a criticalgradient theory, which means the temperature through-out the plasma is proportional to the edge temperature.

In modern plasma confinement devices the plasma edgeis generally defined by the separatrix between magneticfield lines that lie on toroidal magnetic surface and openmagnetic field lines that go into specially designed regionson the chamber walls, Figure (8). This method of defin-ing the plasma edge is called a divertor (ITER PhysicsExpert Group on Divertor Modelling and Database et al.,1999; Loarte, 2001).

Simple features of divertors can be understood from afluid model, ρ~v · ~∇~v + ~∇p = ~j × ~B with ~∇ · (ρ~v) = 0and ρ = mn the mass density of the plasma. The plasmaflow is rapid along the magnetic field. Let ρ~v = Γb+ρ~v⊥,then

~B · ~∇(

ΓB

)= S, (296)

65

where the source S ≡ −~∇ · (ρ~v⊥). Equation (296) givesthe flux of plasma Γ along a magnetic field line. Sincethe flow is essentially parallel to the magnetic field, theparallel component of force balance is Γb · ~∇(Γ/ρ) + b ·~∇p = 0, which can be written as

Γ(`)d(Γ/ρ)d`

+dp

d`= 0, (297)

where d/d` ≡ b · ~∇. If the density ρ(`) is used as theindependent variable,

dΓ2

dρ= −2ρ

(C2s −

Γ2

ρ2

), (298)

where C2s ≡ dp/dρ. That is, the effective sound speed

is C2s ≡ (dp/d`)/(dρ/d`). Equation (298) is well known

from the theory of nozzles in fluid mechanics and saysthe density drops as the flux Γ increases when the flowstarts with zero speed, Γ/ρ = 0. The maximum fluxthat can be obtained is Γ = ρCs, which implies a flow atthe speed of sound Cs. If the divertor chamber exerts asufficiently small back pressure, then the plasma flow willreach the sonic rate as it flows down the field lines. Inother other limit in which the back pressure pb keeps theflow slow compared to the sound speed, the flux reachesΓ2 = 2ρ0(p0 − pb) with p0 the pressure where Γ = 0.Divertors have been operated in both the high and thelow back-pressure limit. The advantage of a high backpressure is that more of the energy moving down thedivertor channel can be radiated away primarily throughatomic radiation, Section (VI.A), which reduces the peakheat loads. The advantage of a sonic flow is that theplasma is efficiently swept into the divertor chamber.

An important issue is the ability of materials to with-stand the high energy fluxes is the energy of the im-pinging particles. The energy of the impinging parti-cles is greatly modified by the sheath potential betweena plasma and a material surface. The flow of a plasmainto a wall is roughly the speed of sound. However, theelectrons would naturally flow into the wall at the elec-tron thermal speed, which is a factor

√mi/me faster.

To preserve the quasineutrality of the plasma, a jumpin electrostatic potential occurs on a Debye length scalebetween a plasma and a wall with the potential holdingback the electrons. This means

√Te/me exp(e∆Φ) ≈ Cs,

or ∆Φ/eTe ≈ ln(√mi/me) ≈ 4. The ions impinge on the

wall not only with their thermal energy but also with thekinetic energy obtained from the sheath potential q∆Φ.

The required divertor flux is Γ = nπa2/(τpδ) wheren is the average density in the plasma, τp is the confine-ment time for particles, and δ is the width of the divertoroutflow channel.

Stellarators use an island chain about a rational sur-face, ι equal to a rational number, at the plasma surfaceto divert the plasma into the divertor chamber (Renneret al., 2002). This idea is analogous to that used in toka-maks where the separatrix between field lines that encir-cle the plasma and those that go the divertor chamber is

at the ι = 0 rational surface, which is the only rationalsurface on which an island can form in axisymmetry.

The variation in the electrostatic potential across thedivertor region has important implications for the overallconfinement of the plasma. The ions can sense the pres-ence of open magnetic field lines once they are within abanana orbit width of the plasma edge. Because of theirhigher thermal speed electrons tend to leave as soon asthey cross on to open field lines while the ions penetratedeeper into the open field line region. Consequently it isnatural for a strong shear in the E×B flow to occur nearthe plasma edge, which can stabilize the microtubulence.A region of greatly reduced transport at the plasma edgeis called a high, or H, mode of confinement and was thefirst transport barrier observed (Wagner et. al., 1982).

Acknowledgments

The author would like to thank the Reviews of ModernPhysics for the invitation to write such a broad review,and in particular Robert Siemann of the Stanford Lin-ear Accelerator, who extended the invitation and madevery useful comments on an early draft. Many colleaguesand friends have answered questions, read drafts, andmade helpful suggestions. Three physicists, who workwith me at Columbia, deserve special thanks: DmitryMaslovsky, who drew or placed in appropriate form allof the figures, David Lazanja, and Ronald Schmitt. Fig-ure (5) was provided by the National Institute for FusionScience in Japan, Figure (6) by the Max-Planck-Institutfur Plasmaphysik in Germany, and Figures (7) and (8)by the Princeton Plasma Physics Laboratory in theUnited States. This work was supported by the grantsDE-FG02-95ER54333 and DE-FG02-03ER5496 from theU.S. Department of Energy.

APPENDIX A: General coordinate systems

Clever choices of coordinates are an important part ofclassical physics. Any three quantities, which are con-ventionally denoted by (x1, x2, x3), can be used as coor-dinates if they are well behaved functions of the Carte-sian coordinates and if positions in Cartesian coordinatesare well behaved functions of (x1, x2, x3). The super-scripts on the x’s number the coordinates and are notpowers. The position in space associated with each co-ordinate point is defined by the transformation function~x(x1, x2, x3). The transformation function can be givenin Cartesian coordinates as

~x = x(x1, x2, x3)x+y(x1, x2, x3)y+z(x1, x2, x3)z. (A1)

For example, cylindrical coordinates (R,ϕ,Z) are definedby ~x(R,ϕ,Z) = R cosϕx+R sinϕy + Zz.

Most people find it surprising that a coordinate trans-formation is defined by giving ~x(x1, x2, x3) rather that

66

xi(~x) with the index i going from 1 to 3. The rea-son ~x(x1, x2, x3) is given is that we want to convertfunctions of position, like the temperature T (~x), intofunctions of (x1, x2, x3). Given ~x(x1, x2, x3) one hasT (x1, x2, x3) = T (~x(x1, x2, x3)).

The quantities (x1, x2, x3) are valid coordinates only ifthe Cartesian coordinates (x, y, z) are well behaved func-tions of (x1, x2, x3). This implies the Jacobian

J ≡ ∂~x

∂x1·(∂~x

∂x2× ∂~x

∂x3

)(A2)

cannot be infinite. The condition that the quantities(x1, x2, x3) be well behaved functions of the Cartesiancoordinates implies that the Jacobian cannot be zero.

The gradients of the three coordinates, ~∇xi with theindex i = 1, 2, 3, and the three tangent vectors, ∂~x/∂xiare related by the orthogonality relation,

~∇xi · ∂~x∂xj

= δij . (A3)

This relation, which is fundamental to the whole theoryof general coordinates can be proven using the chain rule.First consider each coordinate to be a function of theCartesian coordinates, xi(x, y, z), so ~∇xi = ∂xi/∂~x =(∂xi/∂x)x + · · ·. The derivatives of the transforma-tion equations, called tangent vectors, are ∂~x/∂xj =(∂x/∂xj)x + (∂y/∂xj)y + (∂z/∂xj)z. Take a dot prod-uct between the gradient of a coordinate and a tan-gent vector, and one finds using the chain rule that~∇xi · ∂~x/∂xj = ∂xi/∂xj . If i = 1 and j = 2, one has~∇x1 · ∂~x/∂x2 = (∂x1/∂x2)x1,x2 , which is zero. In words,the derivative of x1 with respect to x2 while holding x1

constant is zero. If one lets i = 1 and j = 1, one has~∇x1 · ∂~x/∂x1 = (∂x1/∂x1)x2,x3 , which is one.

The use of general coordinates is illustrated by thederivation of the equations for field lines. Field linesare defined by d~x/dτ = ~B(~x) where τ is a parameterthat defines positions along the line. We want the tra-jectory of the field lines in general coordinates, whichmeans we want the functions xi(τ). Now d~x/dτ =∑

(∂~x/∂xi)dxi/dτ . Dotting both sides of the definingequation for field lines by the various coordinate gradi-ents and using the orthogonality relation, one finds that

dxi

dτ= ~B · ~∇xi. (A4)

If ~B · ~∇x3 is non-zero, one can use x3 as the parame-ter that defines positions along the line. The chain ruleimplies dx1/dx3 = ~B · ~∇x1/ ~B · ~∇x3.

The orthogonality relation allows one to write the gra-dients of the coordinates in terms of the tangent vectors,for example

~∇x1 =1J

∂~x

∂x2× ∂~x

∂x3. (A5)

This relation is called a dual relation and is important inpractical calculations for otherwise the calculation of a

coordinate gradient would require a function inversion of~x(x1, x2, x3) to obtain x1 as a function of the Cartesiancoordinates (x, y, z). A vector in three dimensions canbe expanded using any three independent vectors, so

~∇x1 = a1∂~x

∂x2× ∂~x

∂x3+a2

∂~x

∂x3× ∂~x

∂x1+a3

∂~x

∂x1× ∂~x

∂x2. (A6)

Dotting this equation with ∂~x/∂x1, one finds that a1 =1/J . Dotting the equation with ∂~x/∂x2, one finds a2 =0. Similarly, one finds a3 = 0.

Dual relations also exist that give the tangent vectorsin terms of the gradients, for example

∂~x

∂x1= J ~∇x2 × ~∇x3. (A7)

These dual relations are derived in an analogous manner.Given a transformation function ~x(x1, x2, x3) one can

expand any vector using the tangent vectors ∂~x/∂xi asbasis vectors. This expansion, ~B =

∑Bi(∂~x/∂xi), is

called the contravariant representation of the vector ~B.The orthogonality relation implies the expansion coeffi-cients are given by Bi = ~B · ~∇xi.

An arbitrary vector can also be expanded using thegradients of the coordinates as the expansion vectors,~B =

∑Bi~∇xi, which is called the covariant represen-

tation. The expansion coefficients are given by Bi =~B · ∂~x/∂xi. The dot product of two vectors is given by~A · ~B =

∑AiB

i =∑AiBi.

The covariant and contravariant representation of vec-tors have distinct roles in both differential and integralvector calculus. First consider the gradient of a scalar,

~∇f =∑ ∂f

∂xi~∇xi, (A8)

which is a covariant vector.Next consider the curl of a vector. This is easy for a

vector written in the covariant representation, which isthe expansion using the coordinate gradients,

~∇× ~B =∑

~∇Bi × ~∇xi. (A9)

One can expand ~∇Bi in terms of the coordinate gradientsand use the dual relations to show that

~∇× ~B =1J∑

εijk∂iBj∂~x

∂xk(A10)

where

∂i ≡∂

∂xi(A11)

and εijk is the contravariant fully antisymmetric tensor.This tensor is defined by ε1,2,3 = 1 with the sign changingif the order of two adjacent indices are changed. Forexample ε2,1,3 = −1. If two indices are identical, thefully antisymmetric tensor is zero, ε1,1,3 = 0. The curl ofa covariant vector is a contravariant vector.

67

The divergence can be calculated using the contravari-ant vector. One uses the dual relations to write

~B =J2

∑εijkB

i~∇xj × ~∇xk (A12)

where the components of the covariant antisymmetrictensor, εijk have identical values to the components ofthe contravariant tensor εijk. The divergence is simplebecause the divergence of cross gradients is zero, and~∇ · (f~v) = ~v · ~∇f + f ~∇ · ~v. Writing the Jacobian as1/J = ~∇x1 · (~∇x2 × ~∇x3), which can be proven usingthe dual relations. One finds

~∇ · ~B =1J∑ ∂JBi

∂xi. (A13)

The curl can be taken only if a vector is in covariantform and the divergence can be taken only if a vectoris in contravariant form. How can one convert a vectorfrom one form to another? The covariant componentsare given by Bi = ~B · (∂~x/∂xi), which if ~B is known incontravariant form yields Bi =

∑(∂~x/∂xi) · (∂~x/∂xj)Bj .

The metric tensor is defined as

gij ≡∂~x

∂xi· ∂~x∂xj

, (A14)

so Bi =∑gijB

j .The name metric tensor comes from its role in deter-

mining distances between two coordinate points. Thevector between two adjacent coordinate points is δ~x =∑

(∂~x/∂xi)δxi. The square of the distance is (δ~x)2 =∑gijδx

iδxj .A vector in covariant form can be rewritten in con-

travariant form using Bi =∑gijBj where gij ≡ ~∇xi ·

~∇xj . Using the dual relations one can show that gij isthe matrix inverse of gij .

There are three types of integrals: line, area, andvolume. A line integral is performed along a curve,which means one of the three coordinates is varied whiletwo are held constant. Let us assume the first coordi-nate is varied, then d~x = (∂~x/∂x1)dx1 and

∫~B · d~x =∫

~B ·(∂~x/∂x1)dx1, which can also be written as∫~B ·d~x =∫

B1dx1. In other words, a line integral is an ordinary

integral of a covariant coefficient.The only difficulty in area, or surface, integrals is be-

coming comfortable with the definition of the area ele-ment. A surface is defined by holding one coordinateconstant, say x1, and varying the other two. The areaelement is then

d~a1 ≡ ∂~x

∂x2× ∂~x

∂x3dx2dx3 = ~∇x1J dx2dx3, (A15)

where a dual relation was used to obtain the second form.An area integral is then the double integral

∫B · d~a =∫

B1J dx2dx3, where B1 = ~B · ~∇x1 is the contravariantcoefficient.

The volume element is obtained by dotting the areaelement with the distance across the surface, d3x ≡

(∂~x/∂x1)dx1 · d~a1, which can be written as d3x =J dx1dx2dx3. In other words, the integral of a functionf over a volume is the triple integral

∫fJ dx1dx2dx3.

The expression for the time derivative of a general co-variant vector, namely the vector potential, is importantfor the theory of the evolution of magnetic fields. Let~A = ψt~∇(θ/2π)−ψp~∇(ϕ/2π) + ~∇g. The time derivativeat a fixed spatial point ~x is(∂ ~A

∂t

)~x

=(∂ψt∂t

)~x

~∇ θ

2π−(∂θ/2π∂t

)~x

~∇ψt + · · ·+ ~∇s.

(A16)where the · · · stands for terms involving ψp and ϕ, whichare of the same form as those for ψt and θ. The quantity

s ≡(∂g

∂t

)~x

+ ψt

(∂θ/2π∂t

)~x

− ψp

(∂ϕ/2π∂t

)~x

. (A17)

Time derivatives holding the coordinates (ψt, θ, ϕ) fixed,which are denoted by a subscript c, are related to timederivatives holding the spatial position ~x fixed by thechain rule,(

∂f

∂t

)c

=(∂f

∂t

)~x

+(∂~x

∂t

)c

· ~∇f. (A18)

The velocity of the (ψt, θ, ϕ) coordinates through spaceis u ≡ (∂~x/∂t)c, so the time derivative of the coordinateshas the form (∂ψt/∂t)~x = −~u · ~∇ψt while the derivativeof ψp has the form (∂ψp/∂t)~x = (∂ψp/∂t)c − ~u · ~∇ψp.Let ~B ≡ ~∇ × ~A, then ~u × ~B = (~u · ~∇(θ/2π))~∇ψt − (~u ·~∇ψt)~∇(θ/2π) + · · ·, and the time derivative of ~A can bewritten(

∂ ~A

∂t

)~x

= −(∂ψp∂t

)c

~∇ ϕ

2π+ ~u× ~B + ~∇s. (A19)

The expressions ~u · ~∇θ = −(∂θ/∂t)~x and ~u · ~∇ϕ =−(∂ϕ/∂t)~x plus Equation (A17) imply

~A · ~u = −s+ (∂g/∂t)~x. (A20)

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