The Physics of Magnetic Fusion Reactors

The physics of magnetic fusion reactors

John Sheffield Oak Ridge National Laboratory, Qak Ridge, Tennessee 97891 During the past two decades there have been substantial advances in magnetic fusion research. On the experimental front, progress has been led by the mainline tokamaks, which have achieved reactor-level values of temperature and plasma pressure. Comparable progress, when allowance is made for their smaller programs, has been made in complementary configurations such as the stellarator, reversed-field pinch and field-reversed configuration. In this paper, the status of understanding of the physics of toroidal plasmas is reviewed. It is shown how the physics performance, constrained by technological and economic realities, determines the form of reference toroidal reactors. A comparative study of example reactors is not made, because the level of confidence in projections of their performance varies widely, rejecting the vastly different levels of support which each has received. Success with the tokamak has led to the initiation of the International Thermonuclear Experimental Reactor project. It is designed to produce 1500 MW of fusion power from a deuterium-tritium plasma for pulses of 1000 s or longer and to demonstrate the integration of the plasma and nuclear technologies needed for a demonstration reactor.

CONTENTS 12. Electron drift-wave instability 1050 13. Ion drift waves 1050 Symbols and Units 1016 14. Trapped ion modes 1050 I. Introduction 1017 15. Electromagnetic drift-Alfven modes 1050 II. Generic Reactor Characteristics 1019 16. Rippling modes and their relatives 1051 A. Representative reactor configuration 1019 17. MHD modes 1051 B. Nuclear fusion reactions and the fuel cycle 1020 D. Dimensionless plasma scaling 1051 C. Plasma power balance 1022 IV. Tokamak Reactors 1052 D. Generic reactor characteristics 1025 A. Introduction 1052 E. Beta and thermal diffusivity requirements 1028 B. Tokamak characteristics 1053 F. Alpha physics 1029 1. Coils 1053 G. Impurities, limiters, and divertors 1031 2. Startup 1054 1. Plasma-wall interactions. 1032 3. Plasma duration 1055 2. Limiters 1033 4. Bootstrap current 1055 3. Divertors 1035 C. Noninductive current drive 1056 H. Auxiliary heating 1038 D. Tokamak transport 1058 1. Neutral-beam injection 1039 1. The H mode 1060 2. Radio-frequency (RF) heating 1040 2. Dimensionless plasma scaling 1061 a. Absorption 1040 E. MHD limits 1061 b. Efficient coupling 1040 1. Resistive ballooning modes 1062 c. Electron cyclotron frequency range 1040 2. Beta limit 1062 d. Lower hybrid frequency range 1040 3. Second stable operation 1063 e. Ion cyclotron frequency range 1040 4. Resistive modes 1063 f. Adiabatic compression 1041 5. Disruptions 1063 I. Fueling 1041 F. Density limit 1064 1. Pellet injection fueling 1041 G. Alpha-particle effects 1064 2. Compact toroid plasma injection 1041 1. Destabilization of Alfven waves 1065 III. Taxonomy of Toroidal Systems 1042 2. Destabilization of ballooning modes 1066 A. Introduction 1042 3. Fishbone oscillations 1066 1. Axisymmetric systems 1042 H. Power and particle control 1066 2. Nonaxisymmetric systems 1043 I. Modeling of tokamak plasmas 1066 3. Strength and weaknesses 1043 J. Tokamak reactor options 1067 B. Magneto-hydrodynamics 1043 K. Conclusions 1069 1. Plasma equilibrium 1043 V. Stellarator Reactors 1069 2. Plasma stability 1045 A. Introduction 1069 3. Resistive modes 1045 B. Stellarator characteristics 1070 C. Transport 1045 C. Stellarator transport 1072 1. Diffusion 1046 D. MHD and density limits 1074 2. Drift 1046 E. Alpha-particle effects 1075 3. Thermal conductivity 1046 F. Power and particle control 1076 4. RFP thermal conductivity 1047 G. Computer modeling 1076 5. FRC thermal conductivity 1047 H. Stellarator reactor options 1076 6. Stellarator thermal conductivity 1047 I. Conclusions 1078

7. Ripple transport 1047 VI ~ Reversed-Field-Pinch Reactors 1078 8. Anomalous transport 1048 A. Introduction 1078 9. Weak turbulence 1048 B. RFP characteristics 1078 10. Strong turbulence 1048 1. Plasma startup 1078 11. Low-frequency microscopic modes 1050 2. Relaxed states 1080

Reviews of Modern Physics, Vol. 66, No. 3, July 1994 0034-6861 /94/66(3) /1 015(89)/31 3.90 1994 The American Physical Society 1015 1016 John Sheffield: Physics of magnetic fusion reactors

3. Equilibrium 1080 D. Startup 1087 C. Transport 1081 E. Adiabatic compression 1088 D. MHD limits 1082 F. Transport 1088 1. Infinitely conducting shell 1083 Cx. MHD limits 1089 2. Resistive shell 1083 1. Rotational instability 1090 E. Current drive 1083 2. Tilt instability 1090 F. Power and particle handling 1084 H. Alpha effects 1090 Cx. Computer modeling 1084 I. Power and particle control 1090 H. RFP reactors 1084 J. FRC reactor options 1090 I. Conclusions 1085 1. Pulsed reactor 1091 VII. Field-Reversed Configurations 1086 2. Conclusions 1091 A. Introduction 1086 Acknowledgments 1091 B. FRC characteristics 1086 References 1091 C. Equilibrium 1086

SYMBOLS AND UNITS

e =electron, i =ion, 0 =neutral, z =ion of charge z, b =beam, k =kilo, e.g., T,k =electron temperature in keV, m =mega or maximum, a =plasma edge, w =wall, P=toroidal, O=poloidal, r =radial, v =vertical, z =axial, ~~=parallel to magnetic field, J.=perpendicular to magnetic field, T =temperature or tritium, 0=deuterium, a=alpha particle, NC=neoclassical, nzo=density in units of 10 m TIo=temperature in units of 10 keV Superscripts: x =average (x ) =volume average, x =peak, or central value, x =fluctuating value. a (m) minor radius in the median Ib, (A) bootstrap current plane j (Am ) current density a (m) avel age minor 1ad1us J (Am ) current density A(m) area k =1.38X 10 (J K ')Boltzmann's constant atoIDic mass IluIIlbel k (m ') wave number b (m) minor radius in z direction Lo total magnetic helicity B (T) magnetic field l, I (m) length or gradient scale b,8„(T) magnetic-field swing in a length solenoid m (kg) mass D(m s ') diffusion coefficient poloidal mode number D~ (ms ') ambipolar di6'usion coeKcient MF, (kg) mass of the fusion island ' e =1.6X 10 (C) charge on an electron n(m ) number density E (Vm ') electric field n(m ) line-average density E„;, (eV) energy at which rate of toroidal mode number transfer to ions and p (Wm ) power density electrons is equal pb, (Wm ) breInsstrahlung radiation E,E„(J) fusion energy released in D-T power density fusion in alpha particle and p,„(Wm ) charge-exchange power neutron density Eb (MeV) neutral-beam energy p,, (Wm ) ion-electron transfer power fraction of alpha power lost density by GoIlductloIl ptR (Wm ) line radiation power density ratio of field in plasma to p, (Wm ) synchrotron radiation power maximum field on a coil density fE =N~Tk normalized potential p (Wm ) alpha power density f.s fraction of neutral-beam po (Wm ) ohmic power density shine-through nn (Wm 2) neutron Aux on first field-reversal parameter wall in an RFP p „(Wm ) average neutron Aux on first troyon factor for beta wall [a(m)B&(T)(P(%)]/I(MA) ] P (W) power I (A) current P„(W) neutron power

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P, (W) total thermal power v„(s ') electron-ion collision P (W) alpha power frequency P, (W) auxiliary power ratio of time to complete safety factor banana orbit to collision time at 95%%uo flux surface p (kgm ) density q& on Aux surface y p„p; (m) electron, ion gyroradius ratio of fusion power to plasma displacement power lost by plasma (harv)DT (m s ') rate coefficient for D-T fusion ratio of fusion power to total (cru), (m's ') rate coefficient for ionization recirculated power (ou)„ (m's ') rate coefficient for charge Q, (Jms ') energy Aux exchange r (m) minor radius of a torus rF. (s) energy con6nement time R (m) major radius of a torus r, (s) slowing down time for R„(m) solenoid radius energetic ions R~ (ohms) plasma resistivty p (&) potential s =rq'/q magnetic shear g(m s ') thermal diffusivity S Lundquist number magnetic-flux function S, (m s ') source rate of charge-Z ions co (rads ') angular frequency t (s) time co„, co„(rad s ') electron and ion cyclotron t~ (s) plasma current Qat-top time angular frequency T (eV) temperature coLH (rad s ) lower hybrid angular u (ms ') How velocity frequency U, (v) loop voltage ') u (ms speed I. INTRODUCTION v~ (ms ') Alfven speed vH (ms ') thermal speed During the past two decades there have been substan- V (ms ') plasma How speed tial advances across the board in magnetic fusion V (m) plasma volume research. In tokamaks, the main line of experimental 8' (J) plasma energy research in the world program, reactor-level plasma con- Z ionic charge ditions have been achieved (see Table I). Initial tests with g, Z n, deuterium-tritium fuel in the JET tokamak in 1991 led to Zeff the generation of 1.8 MW of fusion power (JET Team, ale 1992). More recently, the TFTR tokamak obtained over ratio of plasma pressure 6 MW of fusion power (TFTR Team, 1993). The charac- to magnetic pressure teristic time scale of plasma sustainment typical of the volume-average beta (generally 1970s was 1 s. Tokamak and stellarator plasma parame- given in %%uo) ters improved we11 above those attained in the 1970s have beta 8 poloidal now been sustained for tens of seconds. The development XR neoclassical resistance anomaly of noninductive current drive (Yoshikawa and ') Yamato, I (m s Aux 1966; Ohkawa, 1970; Fisch, 1980; Bevir and Gray, 1980) e=r/R inverse aspect ratio has opened the route to steady-state operation of the oth- ~h toroidal and helical 6eld ripple erwise pulsed devices. The small superconducting gQ efficiency of auxiliary plasma tokamak TRIAM-1M (S. Itoh et al. , 1991) has been power operated for one hour, at n =2X10' m, T,(0)=500 thermal-to-electrical eV. conversion efficiency Accompanying these advances in performance has Il fraction of power recirculated been substantial progress in our understanding of the g~~ (ohm m) parallel resistivity physics. The magnetohydrodynamics (MHD) behavior g„(ohm m) resistivity of plasmas is consistent with theoretical expectations, g,. =L„/Lz, and sophisticated computer models are able to describe 8 poloidal angle detailed MHD behavior. The theoretical predictions for 0 pinch parameter the plasma-driven bootstrap current and noninductive 4= 1/q rotational transform current drive have been con6rmed experimentally. Whi. le sc= b/a ellipticity the transport area exhibits the greatest gulf between A, (m) mean free path or gradient theory and experiment, neoclassical transport is observed scale length in some circumstances, and progress has been made in re- lnA, coulomb logarithm lating theory and experiment in the plasma edge of pa=4m X10 (Hm ') permeability of free space tokamaks, steBarators, and reversed-6eld pinches

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TABLE I. Key tokamak parameters: Achieved and required (Cordey et aI., 1992). Steady- Parameter state D-T (not simultaneous) 1981 1991 reactor Central ion temperature T;o (keV) 0.5 7 35 30 Central electron temperature T,o (keV) 1.5 3.5 15 30 Energy confinement time ~E (s) 0.007 0.05 1.4 3 Triple product n;o T;~~E (keV s m ') 1.5X 10" 5.5X 10" 9X10" 7X10 '

Plasma pressure P=2poXnkT/B X 100 0.1% 3% 11% S%%uo

(RFP's). The lack of a confirmed explanation of core damage, using simulated neutron spectra in fission reac- transport is compensated by burgeoning experimental in- tors, has led to the development of improved steels formation in tokamaks, stellarators, RFP's, and compact (Bloom, 1990). Tritium breeding tests in which small tori, which has allowed empirical confinement scalings to lithium blanket elements have been bombarded with neu- be developed and some theoretical models to be tested. trons are encouraging in showing a low retention for tri- Advances in technology supported many of the tium, i.e., the reactor tritium inventory may be small successes; see Table II. Multimegawatts of heating have (Johnson et a/. , 1989). While remote handling for fusion been provided efhciently by neutral-hydrogen beams and is in its infancy, some successes have been achieved in ap- by what is loosely called radio-frequency (rf) heating (fre- plications on the JET tokamak (Jones et a/. , 1991). quencies from 1 MHz to 100+ GHz). Fueling by the in- These advances support the recent studies of magnetic jection of high-speed frozen pellets of hydrogen has led to fusion reactors and have allowed a refinement of the un- improved plasma profiles and lower charge-exchange derstanding of reactor physics needs. losses (Milora, 1989). The ability to handle efFectively a In writing this review paper I have adopted the follow- large throughput of tritium has been demonstrated in the ing strategy. First, I shall summarize briefly the funda- tritium systems test assembly (Anderson and Bartlitt, mental building blocks that support the design of an 1989). The use of high-field copper magnets, pioneered efFective reactor: power density (cost of electricity), by MIT, has allowed high values of the plasma perfor- power balance, transport equations and transport models, mance parameter —nT~ —to be achieved in small facili- alpha-particle physics, MHD and beta limits, heating, fu- ties (Greenwald et a/. , 1984), and leads to the possibility eling, and power and particle control. These areas are of compact deuterium-tritium (D-T)-burning, high- presented as generically as possible though inevitably the fusion-power (high-Q) tokamaks (Coppi, 1977; Cohn far greater amount of research done for the tokamak un- et a/. , 1978). The ability to design for performance in derpins much of our knowledge, and this work is present- large superconducting coil systems has been demonstrated to illustrate particular reactor problems. ed in a number of facilities with coils at the 8-T level; see Second, I shall discuss the physics of four toroidal Table II. Improvements in superconducting materials reactor concepts: tokamak, stellarator, reversed-field and in manufacturing techniques permit 12-T coils to be Pinch (RFP), and a field-reversed theta-pinch con- considered for the next round of facilities, such as the In- figuration (FRC) as an example of a compact torus. The ternational Tokamak Experimental Reactor (ETER); see purpose is not to refer to all of the physics studies that ETER (Conceptual Design Teain, 1991). Still higher-field have been done, but to refer to those studies that have a coils are possible (Pourrahimi et a/. , 1987). particular bearing on reactor design. Various materials, including graphite and beryllium, Third, an accurate description of the observed trans- have been tested successfully as plasma facing com- port does not exist for any of the concepts. Reactor ponents (Gauster, 1990; Thomas et a/. , 1991). An in- designers must make do with a mixture of empirical scal- creased understanding of the efFects of 14-MeV neutron ing, neoclassical transport, and, in some cases, theoreti-

TABLE II. Technology achievements. Technology Facility Achievement Neutral beams TFTR 30 MW 0' at 110 keV RF JET 22 MW ICH at 23 to 57 MHz JT-60 8 MW LHH at 1.7 to 2.2 GHz T-15 3 MW ECH at 75 and 81 CxHz S/C coils LCTF MFTF-B S8-9 T Tore Supra with both NbT; and Nb3Sn T-15 Tritium handling TSTA 1 kg per day

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1019 cally based models. In order to illustrate how the physics 1989; Holdren et al., 1989; Bodin, 1990; Nuclear Fusion affects reactor scale and geometry, I have chosen to use Review, 1990; Krakowski et al., 1991; and Post et al., for each concept a popular transport scaling with a mul- 1991. tiplier. If readers feel that some other scaling is more ap- The symbols and units (SI) are listed at the beginning propriate, I hope they will find sufFicient information in of the article. In some places the units used differ be- this paper to permit its use. cause they are those used in the original paper. The orig- I have chosen not to make a comparative study of the inal units are retained to allow easier comparison with four concepts, because they are not at the same stage of the original paper. development. Since the early 1970s, the tokamak has been the major experimental vehicle for developing mag- II. GENERIC REACTOR CHARACTERISTICS netic fusion. The other concepts lag the tokamak in demonstrated plasma performance, because the experiments have been smaller and comparable only with A. Representative reactor configuration tokamaks of a decade or two ago. Nevertheless, in terms of relative performance, allowing for the difference in A cross section of a representative tokamak reactor is scale, a number of these concepts show sufficient promise shown in Fig. 1. The magnetic field of about 4—8 T is to warrant their longer-term consideration (She%eld, produced, typically, by two sets of coils, a toroidal set 1985). Success with the tokamak has led to the initiation producing a field 8& and a poloidal set producing a field of the engineering design phase of the International 8. The field lines twist around the toroida1 plasma, Thermonuclear Experimental Reactor (ITER) as a colla- forming nested Aux surfaces; see Fig. 2. Generally, in a borative venture of the European Community, Japan, the tokamak and stellarator 8&))8&, in a very-low-aspect- Russian Federation, and the United States under the ratio tokamak (R/a 51.5) and a pinch device B&)8&, auspices of the International Atomic Energy Agency. whereas in a field-reversed configuration B& «B&. ITER is designed to demonstrate controlled ignition and Linear systems such as the tandem mirror (Post, 1987, burn for 1000 seconds or longer in a deuterium-tritium 1988; Hershkowitz et al. , 1990) no longer receive as plasma, and will provide a demonstration and integration much attention in the world program as reactor candi- test of reactor technologies (ITER Conceptual Design dates, and will not be reviewed, though they are interest- Team, 1991;Rebut, 1993). The present schedule calls for ing as possible sources of 14-MeV neutrons for testing a construction start in 1998 and operation starting in purposes. In a 0-T-burning reactor, the average plasma 2005. ITER is a major element in the U.S. Magnetic temperature will be in the range 8 —25 keV Fusion Energy Program to develop a demonstration reac- (100000000—300000000 K), at a density of 10 —10 tor by about 2025. A comparison of the parameters of m ITER with those of present large tokamaks is given in The geometry of the Aux surfaces is characterized by Table III. the major radius of the toroidal plasma (R), the minor This review is built in great measure upon previous ex- radius (r), and the poloidal angle (8). The ilux surfaces cellent reviews of reactors and related topics, and the may be elliptical; the minor radius of the edge of the plas- reader is referred to them for a broader understanding of ma in the median plane is commonly denoted by (a), this interesting subject. The reviews include Rose, 1969; while the minor radius in the z (vertical) direction is Post and Ribe, 1974; Furth, 1975; Miyamoto, 1978; %'es- denoted by (b). The ellipticity is denoted by x=b/a (this. son, 1978; Rawls et al., 1979; Proceedings of the IEEE may vary with minor radius). In simplified treatments of 1981; Conn, 1981; Hagenson and Krakowski, 1981; this geometry, it is common practice to approximate the Shohet, 1981; Stacy, 1984; Liewer, 1985; Bodin et al., noncircular plasma by a circular plasma of equivalent 1986; Freidberg, 1987; Tuszewski, 1988; Grieger et aI., minor radius r (b /a ) ', where r is the radius in the medi-

TABLE III. Parameters of modern large tokamaks and ITER. Tokamak TFTR JET JT-60 ITER Major radius R(m) 2.45 3.10 3.40 7.75 Aspect ratio R/a 2.90 2.82 4.0 2.8 Ellipticity a 1.0 1.8 1.6 1.55 Current I(MA) 3.0 6.0 6.0 25 Field B& (T) 5.12 3.4 4.2 6.0 Divertor No Yes Yes Yes Coil type Cu CU Cu SC Fuel H, D,T H,D,T H,D D-T Pulse length (s) 2 20+ 20+ 1000 Estimated fusion power (MW)' —15 —50 —1500 'Projected fusion power. Obtained, JET 1992, 2 MW; TFTR 1993, 6 MW.

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POLOI DAL FIELD TOROIDAL FIELD

NKET D SHIELD FIG. 1. Cross section of a representative LOGICAL tokamak reactor {Najmabadi et al., 1991}. ELD LOIDAL ERTORS CUUM CT

gc a4aaat i iislaatthagala aa& l aais4cssli saalawilatstlia~tj 'f v f ~ t 1 I I ~ 4 fTTPT 7 1 ~ I I i ~ 1 f 4 1t 1 i I7 0 I I K I l 1 1 I 1 I I \ ~ I PRIMARY COOLANT SUPPLY l ( l I f l TIE l l f l 0 2 4 6 8 10 12 (METERS) an plane. In this approximation, the average minor ra- and beta (p), which is the ratio of the kinetic pressure dius of the plasma edge is (ab)' In re.ality the Aux sur- (n, kT, +n;kT;+Xn, kT, ) of the plasma divided by the faces may not be centered on the midpoint of the medium magnetic pressure, plane diameter, owing to the e8'ects of finite plasma pres- (pressure X l00) sure, causing an outward shift of the plasma (see Sec. (2.2) (&'/2po) III B, Fig. 26). In addition, the shape is often more like a D, and such shapes may be characterized by an addition- The development of reactor levels of rz and p has al parameter, the triangularity (5). dominated magnetic fusion research, and good progress Parameters of importance in characterizing the fusion has been made. However, much more is required for an plasma are the energy confinement time (rE), which attractive reactor than high rE and P. Equally critical equals the stored energy in the plasma (8') divided by are impurity minimization and control, low recirculating the heat (P) leaving the plasma (excluding the neutrons), power for plasma production and control, the development of radiation-resistant, low-activation materials, and (2.l) high reliability and Inaintainability. The magnet coils are protected from the nuclear radiation by a moderator of the neutrons (usually called the blanket) and by a shield which absorbs neutrons and gamma rays. In the case of a D-T plasma, the blanket contains lithium to breed tritium to replace that burned in fusion reactions. In most configurations the outer magnetic Aux surfaces are diverted onto targets, which absorb the heat, thus isolating most of the chamber wall from direct contact with the plasma. The first wall and the blanket and shield elements form a vacuum chamber. A coolant, liquid or gas, removes the heat deposited to the wall, blanket, shields, and divertor targets and trans- ports it to heat exchangers and generators which produce FLU SURFACES electricity.

B. Nuclear fusion reactions and the fuel cycle

The nuclear fusion reactions of greatest relevance to magnetic fusion, because they occur at the lowest temperatures, are listed in Table IV. The charged-particle power per unit volume released is shown in Fig. 3. The final number is the equivalent electrical energy imparted to the nuclei (kWh) per gram mass of the reacting nuclei. For comparison, the chemical reaction 2Hz+02 ~HzO+ HzO+0. 000006 MeV yields 0.0044 kWh/g. In the case of deuterium-tritium (D-T) fusion, because triti- FIG. 2. Nested flux surfaces in a toroidal system. um does not occur naturally it is necessary to breed triti-

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TABLE IV. Nuclear fusion reactions of greatest relevance to magnetic fusion. The final two quantities in each line refer to the total nuclear energy release in one reaction and to the energy release in the form of charged particles, respectively. Energy Total Charged equivalent particle Total energy Reaction {MeV) (MeV) (1 Wh/g) (1) D+T~ He(3.52 MeV)+n (14.06 MeV) 3.52 17.58 94,000 (2a) D+D —+'He(0. 82 MeV)+n (2.45 MeV) 0.82 3.27 22,000 (2b) D+D~T(1.01 MeV)+p (3.03 MeV) 4.04 4.04 27,000 (3) D+'He~ He (3.67 MeV)+p (14.67 MeV) 18.34 18.34 98,000

um by bombarding lithium with the fusion neutrons, ma temperatures T ~ 5 keV (fusion power exceeds brems- = strahlung radiation losses), and it will be the basis of Li+n He+T+4. 80 MeV, the discussion in this tern- (2.3) most of paper. The optimum Li+n = He+T+n —2.47 MeV . perature is about 15 keV. Deuterium is abundant in nature as about 1 part in The D-T-Li fuel cycle is the most attractive, because a 6500 in the hydrogen in water. It requires no fuel breed- self-sustaining plasma may be realized at the lowest plas- ing; however, a plasma temperature T ~20 keV is required by a self-sustaining deuterium plasma to overcome FED bremsstrahlung and synchrotron radiation. The op- 2 I I I IIII I I I I II& timum temperature is about 40 keV. It may be lowered to around 35 keV if the T and He fusion products are" re- a CAT DT ~ P Li circulated, a scenario referred to as "catalyzed D-D. At o CAT DD ~ p first sight the lower fraction of neutron energy produced b, CAT D He k ~He ~He per reaction appears attractive for reducing neutron ac- V CAT D Li v P ~Be tivation of the reactor structure. In fact, the number of neutrons per reaction, because of their lower energy, is not reduced enough to make a significant di6'erence. The )00 absence of a breeding blanket is an advantage. The D- He reaction is attractive because it produces no neutrons. However, to take advantage of this, it is necessary to run with a lean mixture of deuterium to &0" minimize D-D reactions. It also requires a high temperature, T~30 keV, for a self-sustaining plasma. A principal disadvantage is the lack of a source of He on earth. it is abundant on the moon, and 2 Apparently proposals $0 have been made to mine the moon to support earth-based

CL D- He fusion reactors (Kulcinski et al., 1989}. The charged-particle power per unit volume produced by fusion reactions is given by the product of the densities of the reacting ions, the rate parameter ( o u ), and the charged-particle power density released per reaction. For a D-T plasma, &04 p =8D 11T ( O' U )DTE (2.4)

where (cru ) is the product of the D-T reaction cross section o. and the relative velocity of the D and T nuclei, 5 )0 averaged over a Maxwellian velocity distribution. Cross &00 $02 sections and reaction rates are given in numerous reports T (keV) (e.g., Glasstone and Lovberg, 1960; Bosch and Hale, FIG. 3. Maximum charged-particle power density release vs 1992}. Formulas have been developed to approximate temperature for the principal fusion fuels in thermalized plas- the fusion power density for D-T plasmas in various tem- mas at n, .=10 m and n;/nJ=ZJ/Z;. Power output scales perature regions (Uckan et al., 1990). as the square of the electron density (McNally, 1982). For model density and temperature profiles

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2 T n =n0 1— T=T0 1— Q

t' (p ) =0.16f(a„,aT ) n20F (MWm ), (2.6) n~ T10C f(a„,az )=(1+a„)(1+2a„+3aT) /(1+2a„+2aT)', 3 T10 T10 ~10 + +10C T10C +10C 2 T10 T10C + T10 + 2T10C ~ ~10C 1.5 T10 =4 2+10C + T10 + 3 ~10C ~ 2 ~10C where

Tmc =[(1+a„)(1+2a„+3aT)]/[(1+a„+aT )(1+2a„+2aT)] . For reference profiles a„=0.5 and eT = 1.0

) =0.2S (n2oT, o) (MWm ) for (T) =7.S to 1S keV (p. n 2 05 nDT ' =0.25 (n2oT, o (MWm ) for (T) =1S to 22 keV, (2.7) n~ T10

where n 2=0(n, /10 m ) and T,o=(T/10 keV), nDT and loss and power input and loss. In general, we shall and n, are the densities of deuterium plus tritium and be discussing hydrogen plasmas (H, D, T); however, the electrons, respectively, and ( T ) is the density-weighted purity of such plasmas will vary owing to materials volume-averaged temperature. Note nDT & n, because of released from the chamber walls —impurities —and in impurities n, . the case of fusion plasmas owing to fusion products such In practice, pressure profiles in tokamaks and stellara- as He and He. Quasineutrality will hold in the plasma. tors are often more peaked, with u„&0.5 and aT-2, These impurities will be ionized and become multiply yielding somewhat higher fusion power for a given beta. charged (Z) 1) so that in general n, , ( n,, =n, + g, Zn„ In contrast, FRC profiles are somewhat Hatter. where Z is the charge state of each impurity ion. The There are two other useful quantities for characteriz- subscripts i and e refer, respectively, to the hydrogen ions ing a reactor. The neutron Aux leaving the plasma edge (combination of H, D, and T) and to electrons. A con- is venient measure of the impurity level is the parameter I'„ nDnr(cru )DYE„V Q, Z n, +n, p~n= (MWm ), (2.8) Z e6' (2.9) ~ W where E„(measured in megajoules) is the neutron ener- The particle balance in a basically single-ion-species plasma, setting n, = is given in a cylindrical (toroidal) sys- gy per reaction, V =2m Ra x is the plasma volume, x is ; n„ the plasma ellipticity, and 2 is the area of the chamber tem by wall. The neutron fluence is the time integral of the neu- Bn 1 8 Bn tron Aux falling on the wall. For a reactor we may expect =nn„(o U ),+— ra„+rV„n (m s ), p~„~ 2 MW m, and we require a wall that can survive neutron bombardment for a number of years, i.e., survive (2.10) a neutron fluence of, say, 15—25 MWyrm where (o.u ), is the ionization rate (Freeman and Jones, 1974) of the neutral particles that fuel the plasrnas; see C. Plasma power balance Sec. II.G, Fig. 17, where various rate coefticients are plotted. Note, different conventions are used for the sign The state of the plasma density and temperature is of V„' as used here a positive V„denotes an inward con- determined by the detailed balance of particle production vection of particles. Because the system acts to maintain

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1023 charge neutrality, the losses of electrons and ions are =30.9 —in(n T, ') represent the collisional transfer of equal and proceed at the rate of the better confined power from electrons to ions and vice versa, with n (m ) species. The ambipolar diffusion coe%cient Dz is used and T, (eV) (Braginskii, 1965). because the particle transport adjusts to maintain charge The line radiation is given by (Jensen et al. , 1977) neutrality via the self-consistent electric field. In piR= n, f(z)(Wm ) . (2.14) tokamak experiments it is observed that D z yn, —(0.2—0.3)y„where y, is the electron thermal diffusivity. The parameter is a radial convective ve- This is the dominant radiation term in present-day V, — locity. tokamaks (typically 20 40% of the power is radiated), The power balance may be written separately for each and it is particularly important at the plasma edge. As species. For the electrons, a simplified power balance is shown in Fig. 5, f (z) is a strongly increasing function of Z as the impurities become more massive. Consequently 8 —3 =—1 8 eBT, —3 Bn small amounts of heavy materials such as molybdenum neT, r n, y, + D&eT, +p& and tungsten can have a disproportionately large effect. Recombination radiation (p„) is generally less important +Jie JLR Jbr IS+Jr+I ae than line radiation. The bremsstrahlung power density (Rose and Clark, +p„(Wm ) . (2.11) 1961, p. 232) for the range of n(m ) and T, (eV) values For the ions appropriate to 0-T reactors is given approximately by

=1. Z T (Wm ) (2.15a) 3 1 8 3 Bn pb, 7X10 gn —neT; =— r ny; +—D~ eT; This represents the total emission at all wavelengths of the continuum from the free-free energy transitions of +p„.—p, +p„+p;+p„(W m ) . the (optically thin) plasma electrons. The Gaunt factor g (2.12) corrects for electron-electron collisions and relativistic eff'ects. For 1 keV & T, & 100 keV, g varies from 1.2 & & 1.1; see Ecker, 1972. Note that temperatures are in electron volts. In g given A formula for the average bremsstrahlung power den- addition there are similar equations for each impurity sity has been given by Uckan et al. (1990) for the model species. Generally, collisions between ion species main- density and temperature profiles in Eq. (2.5), tain T, —T, . The transport of impurities is, however, more complex and, as discussed below, depends on details of the plasma gradients. Impurities may move radially, relative to the background plasma, either in or out. The parameters y, and g; (m s ') are the electron and I I I I F ~ II( ~ ~ I 1 I III I I I l I I I ion thermal diffusivities, and the p (MWm ) are the power densities of the various mechanisms indicated. The overall power Bow is illustrated in Fig. 4: ' — 10 hei I ie 3+ei ne e(T, T) (Wm ), (213) Pl i ' where v„=2.92X10 Z,irnT, ' lnA, (s ') and lnA, E I

ADDiTIONAL ADD ITIONAL ~ 0M2 OHMIC HFATING C HEATING HEATING c III PLASMA ELECTRONS IONS INTERIOR

HEAT MASS MASS HEAT 33 CONDUCTION FLOW FLOW CONDUCTION )0

PLASMA EDGE ELECTRONS IONS

t CHARGE RADIATION MASS FLOW ~ ~ ~ I ~ l I I ~ ~ a a s a ~ ~ I EXCHANGE MASS 10 100 HEAT FLOW T (keV) WALL — li CONDUCTION LIMITER/DIVE RTOR FIG. 5. Line radiation factor f (z) as a function of electron I'"IG. 4. Power Sow in a typical toroidal plasma. temperature for representative impurities (Jensen et al., 1977).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1024 John Sheffield: Physics of magnetic fusion reactors

(1+a„) (1+a„+aT)' p =6 2X10 niinz T; (Wm ), T (20 keV, (2 19) (p, ) =1.6X10 Z, ( 1+2a„+0.52a T ) where T; (eV) is the assumed common temperature of the deuterons and tritons. X n AT tp/ (W m ) . (2.15b) The alpha particles are born with an energy of 3.5 For reference profiles u„=0.5 and o.T = 1.0 MeV and initially slow down mainly by collisions with electrons. At a critical energy E, the rate of loss to the =1.86X10 (Wm ) . (2.15c) „„ (pb, ) Z,sn2pTip/ ions becomes equal to that to the electrons, and at lower It is informative to study the ratio energies the loss to the ions predominates (Stix, 1972): (p„) 2/3 —0.074Z,s. T3/2 (2.15d) (eV) ( ) 10 Values of this ratio, for ( T ) =5.0 to 15 keV, are given in —0 1ATip (MeV) (2.20) Table V. For realistic levels it is impurity dificult to where A and A are the atomic weights of the slowing have a viable fusion reactor for an average temperature down particle and plasma ions, respectively, and Z. is the much less than 7.5 keV because the sum of bremsstrah- charge number of the plasma ions. The average fraction lung, line radiation, and direct alpha-particle 1osses G; of the total energy given up to the thermal ions of the leaves too little power to sustain a reasonable-size plas- plasma is ma. At low plasmas densities, low beta values, and high Crit 0~t G;= dy (2.21) temperatures, the sychrotron radiation (Spitzer, 1962, p. p ( l ++3/2) 152) emitted by the electrons from their magnetic orbits is an important factor in the plasma power balance For a pure Qfty-fifty, D-T plasma at a temperature of 20 (Trubnikov, 1979), keV, E„;,=660 keV and G;=0.20 for 10 keV, and G; =0.33 for 20 keV. p =1.3X10 T 8 [I+X ] It is convenient to describe the energy losses in terms of an confinement X(1—A) /a (Wm ), (2.16) energy time rE (s}, which is equal to the plasma stored energy divided by the total power loss where T, (eV} and X,„„=5.7X102a/RT, is a correc- from the plasma in equilibrium (P). tion factor for field inhomogeneity and Doppler broaden- For a toroidal plasma, major radius 8, minor radius in ing of the emission spectrum. A is the wall reffection the median plane a, and e11ipticity z, coef6cient. e(n, T, +n, T, ) The power input due to heating by a plasma (2.22) current —ohmic heating —is given by For a plasma sustained by the fusion alpha power, and 53X10 '/ (Wm p n7. y~Z, sj lnA, /T, ), & Tip (2Tl a Parabolic temPerature T)pc= pc Profile where ya —(1—1.95e +0.95e) ', e=r/R is a neoclas- (ar 1) and square-root parabolic density profile sical resistivity correction factor allowing for trapped- (aT =0.5), from Eq. (2.7) we have Tip& =0.75 and electron e6ects, and is the local plasma current density. j P =4.9X10 "(n T')'Ra K (W), (2.23) The losses due to charge exchange of the plasma ions on the background neutral particles are given by where nDT is the density of deuterium plus tritium ions. Substituting for P in Eq. (2.22), with T, =T;=T, leads =nn„(ou), „eT; (Wm ) (2.18) p, to a requirement for a self-sustaining pure D-T plasma when ( cr u ),„ is the rate coefficient. (n, =n;) The additional (or auxiliary) heating, i.e., other than s ( nDTT;) @r-—1.93X10 (m eVs) . (2.24) ohmic and alpha-particle heating, is represented for electrons and ions by p„and p„., respectively. For example, if (nDTT, )=1.5X10 p m . sX1(}keV', we The average alpha-particle power density for a D-T require &E = 1.3 s. plasma is given by Eq. (2.6). An approximate formula for For a D-T plasma contaminated by impurities and the local value is helium we must allow for the depletion of D-T fuel by multiplying by (n, /nDT ) . It is useful to derive a relationship between the global energy confinement time and the thermal difFusivity TABLE V. Values of the ratio (Pb, )/(P ) for Z,~=1.8 and y. nDT/ne=0. 8. In steady state Eq. (2.11)may be rewritten as

( Tio ) 0.50 0.75 1.0 1.25 1.50 1 8Te (Pb, )/(P ) 059 0.32 0.21 0.15 0.11 nieXr +neeX~ =f~~ ~ (2.25)

Rev. Mod. Phys. , Vol. 66, No. 3, July f 994 John Sheffield: Physics of magnetic fusion reactors 1025 where all of the nonconduction losses and auxiliary heat- generic considerations from those deriving from our ing sources have been wrapped up in the factor f . f is present understanding. On the one hand, advances can the product of two terms: f, =the fraction of alpha be expected in many areas, e.g., superconducting coils = and better optimized beta. On the other hand, generic power transferred to the plasma; f 2 1 f„—where f„ is the fraction of input power radiated. Sometimes quoted constraints such as classical (neoclassical) physics, neu- confinement times, derived experimentally, include radia- tron attenuation lengths in shield materials, and cross tion losses. In this case f„should be set equal to zero un- sections for tritium breeding and D-T fusion set ultimate less the radiation losses are expected to be different in the limits. Fortunately, there is a great deal of commonality reactor regime, e.g., if bremsstrahlung takes on a greater among the various reactor concepts (tokamak, RFP, stel- level of importance, when fI -—(pb, /(p ) is large [see larator, mirror). This fact has been used to generate the Eq. (21.5d)], or synchrotron radiation becomes large. If "generic toroidal reactor" (Shefield, 1986), which is a use- theoretical or experimentally based thermal diffusivities ful tool for identifying, in an approximate way, the com-" are used, then full account must be taken of line, brems- bined physics requirements for an "attractive reactor, strahlung, and synchrotron radiation. If auxiliary power i.e., one that might be competitive with fission. The stud- is applied, for example for current drive, then there is a ies have been extended to include consideration of envi- third factor 3—-1+f„where f, =p, /p . For exam- ronmental and safety aspects (Holdren et al., 1988, f 1989). A recent review has been ple, in a tokamak reactor one might have f I-—0.95, given by Holdren (1991). =0.85, allowing for bremsstrahlung losses at The generic reactor is a steady-state toroidal system f 2 with D-T which com- ( T ) = 15 keV and line radiation and, if auxiliary heating operating fuel, includes all of the ponents common to the various is applied with = 1.1, in total we have =0.89. types of fusion f 3 f reactor superconducting coils, a lithium breeding Integrating over the plasma volume yields — blanket, plasma heating and fueling systems, shielding, coil structure, power supplies, remote handling, build- aT aT. ings, generators, and cooling towers, as illustrated in Fig. —4m. Rr n, ey,- +n, ey, =f p r=a 6. In a small-aspect-ratio version the generic reactor ap- proximates a tokamak, at intermediate aspect ratios a (2.26) stellarator, and at large aspect ratios a tandem mirror. It where for a noncircular plasma we have used the nota- is a somewhat less accurate representation of systems tion a =(ab) (see Sec. II.A). Assuming a parabolic such as the RFP and field-reversed configuration (FRC), temperature profile and roughly Rat density profile, we in which the main field is produced by a plasma current. — n An example of coupled values of (P) and global thermal find that rn (BT/Br) l „=4( T ) . Consequently diff'usivity is shown for reference 1200-Mw(e) reactors when we equate Eqs. (2.26) and (2.22) with P =f P we gE in which the aspect ratio is varied (see Fig. 7). in this ex- find that ample the neutron flux to the first wall (p ) was in the 0.38ab range 5-6 MW and the maximum field on the XE m s (2.27) m, +E toroidal coil (8 ) was in the range of 8—12 T. A different optimization was used in the recent ARIES-1 where ~E represents the energy confinement in the face of study (Najmabadi et al., 1991), with p„=2.5 MWm conduction losses, and is a global value yE and 8 =21 T using more advanced superconducting (y, n, TI+y, n, T, & technology. Under these conditions a lower value of (P) 28 2. was permitted, =2% at R /a=4. 5. Thus we see that the (n, T, +n, T, & physics requirements for a magnetic fusion reactor are For example, if a=1.5 m, b=3, m, ~E=1.5 s, yE=1.14 m s

D. Generic reactor characteristics

l In recent years numerous studies have been made of l magnetic fusion reactors, including those of Badger, 1979; Hagenson et al., 1979; Baker, 1980; Hagenson and PUMPING l Krakowski, 1981; Logan et aI., 1983; Miller et a/. , 1983; l Najmabadi et al., 1990, 1991; Painter and Lyon, 1991; . FEEOWATER I and Seki et al., 1991. These studies, which involve esti- RECIRCULATING POWER HEAT TRANSPORT MAGNETS HEATING mates of the cost of the reactor and its operation, are BOP ANO CONOENSATION ANO MISC SYSTEM CRYO AUXILIARIES used to show how fusion might be competitive with other REACTOR energy sources, based on our present understanding of t t t physics and technology capabilities. In using these studies we must be careful to decouple the limitations set by FIG. 6. Principal components of a generic 0-T fusion reactor.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 't 026 John Sheffield: Physicsysics of magneticm fusion reactors

of the costs 6xed by the balance of plant1 and operations and maintenance.e. A ch ange from 10Q0 QOO tonnes to 20000 0.20 'n tonnes results is ononl y a 30% penalt y in COB. However 7 it shoulds be noted thaat at 10000 tonn e tji++ ~ O. ~ 0.16 -jP. is on t e high side of g or :l a ractiveness of thee usion reactor liies in or etter environvironmental and safet (P) Ot2 aey an a fission reactor. Whether the reac- I ' or needs to weigh 1ess th an 10000 toonnes or may w Clg h 0.08— 1.0 Q5 0.3 02 g (m ) more depends on factors such ass thee eaease of maintenance,

complexity, levele ofo powero recirculate d to the plasma 7 TANDEM MIRROR 0.04— /Bm — 8 d the importancce (costs benefit) assigned to envir OIl- TOKAMA K STELL A R ATOR p/B O. t Bp 0. (CENTER CELL) = y eatures. ~ The sim ple g CT RFP Bp /Bm EBS Bp/ Bm= 0.5 b/o 1.0 t' l I J us to understand the rel R 1VC (0 18 22 26 30 ysics and technolo ogy R/a ' p s in a simple modele, wwh'ich relates the ph ys- Coupled values aud se FIG. 7. Co of (P) s, of a g1VCIl IIla IlCt1IlC g 0 sustaining D-T toroid l l unct1onu of the aspectec rratio, n generic limitations for =3500 MW =100 = 8 to 12 and PF P, M(e),e, B T, e csessoson, 980 Sheffield,1,1985). This ~ 9, ~ e o a toroidal reactctor fusion island is ' ' shown in Fig. 9. Thee weightwei of the fusion is1an d is given intimately bound up witwith ththe capabilities of the technology. MFi=2~'R[(a+b)t+t t ' ]p„, (kg), (2.29) The cosost of electricity (COE, 10 mills=1 ' ' pp oximately linearlr y on theh size (we'ight) of the reactc where thehe average density of materia n is- wer output, as illustra 1'n ''n'lu '" 'h c scrape-o6' layeryer plasma-wallla gap) and ' ' ' reactor studies (Fiig.. 8).. From both th ac 01 '™)p studies and th e generic studies" it is c as R e scrape-off layer, blanket , maintenance hf""ofoO merit an "attractive"C 1Ve magnetic fuS1On ure, and coils. T a ro uce ~100 kate ac or is given by fusion island (coi1s, structure, blanket a ( iH e n an ross, 198S, . s P, =P [1+4(1+g)]+P, (W), (2.30) an be se in'n Fiig. 8, the rate of ch where is the blanket neutron ain.. We shall use th e f o .1 d eig ht is slow because of th e 1arge fraction g g ar re reactor stutudy (Baker et al. 198 3 o p ectric power is given by

P, =il, (1 —q„)[0.7+4.50]Pa (2.31) P, = 1228 MW(e) &P) =0.04-0.24 R/a = 2 - 20 100— = = Pt 4050 Mw 8 lYl =9T b/a ).6 MAINTENANCE GAP = & P. &00 MW(e) Bo/B~ 0.6 a~/a =&.2 RCON DUCT ING + CO IL S Pwn 6 MW. m BLAN

SHIELD E 50— UJ C)

F0 EL CYCLE SCRAPE -OFF OPERATIONS- MAINTENAANCE LAYER BALANCE OF PLANT

I I 10,000 20,000 50,000 R MF& (tonnes) aw

FIG. 8. Thee cost ofo electricity (COB) decrea e usion island for a generic reactor FIG 9 S'imphfied model of thee fusiusion island of a toroidal reac- (ShefBeld, 1986). tor.

Rev. Mod. Phys.~,, Vol.. 66, N o.. 3, July 1994 John SheffieId: Physics of magnetic fusion reactors 1027 where g, is the thermal-to-electric conversion efBciency the approximation leav- and we have approximated the useful thermal heat . 4X10-"("'T.' & the ) 0. to allow for incom- &—= ing plasma (P~+P, by 7P &P, , -0.35&8&% (2.37) plete recovery of the heat. The fraction of power recirculated to the reactor is g„. and (P, )-0.46(P)%. This is consistent with the im- A commonly used characteristic of a fusion reactor is purity depletion numbers given in Sec. II.G, for the ratio of the fusion power output to the power Q, Pz nHz ln, -0.05 to 0.10 and (P ) /(P) approaching 0.2 at injected in the plasma to maintain the plasma, 20 keV; see Table VI. Using the factor f to allow for P~ heat losses other than conduction and for auxiliary heat- P (2.32) ing, and substituting in Eq. (2.36), Eqs. (2.22), and (2.27) yields An engineering version, takes that ratio of to the Qz, Pz 3.6X10 P total recirculated power g„P, f (2.38) R&P&B' Defining g,o as the fraction of power recirculated to the balance of plant and coils and setting P, = 1.05', Equations (2.31) and (2.34)—(2.38) may now be solved for representative reactors to determine the coupled require- 0.95 0.95 'g — —'g «0+ CX'g qO+ (2.33) ments for (P )B and gz. To indicate the typical 'Qa '9E QQ T & demands placed on the physical performance, we shall where g, is the efficiency of use of power recirculated to consider the case in which t=2.5 m, g, =0.38, g„=0.15, the plasma. f~ =0.73, n&T /n, =0.8, b/a =2, there are parabolic tem- For a typical superconducting coil reactor producing perature and square-root parabolic density profiles, and —1 GWe, g,o—-0.07, g, =0.3—0.7, and g„related to the the figure of merit is 100 kWe/tonne. The results for Carnot eRciency, ranges from 0.35—0.45 (at most -0.6). four examples of net electric power (500, 1000, 1500, and For example, if g, =0.5 and g, =0.4, to obtain g, ~0.2, 2000 MWe) are shown as a function of minor radius (a) Qz ~ 5 we require Q ~ 37. and R /a in Figs. 11 and 12. A simplified power-How diagram for a reactor is shown For this simple model, yz, (P)B,p „,and P, /R vary in Fig. 10. with (a) independent of P, . Not surprisingly, the To make an "attractive" reactor we require (Sheflield, smaller-minor-radius reactors require a higher (P)B 1985) and power density (p„„)with lower yz than the larger- minor-radius versions. The physics requirements are 4.8 X 10 g, ( 1 —rI„)P ~ 100 (kWe/tonne) (2.34) eased for larger reactors, i.e., lower (13)B and higher yz R [(a+b)t+t2) are permitted. Study of Fig. 12 shows that the lower- devices more achievable at low where, allowing for impurities, Eq. (2.23) becomes power-density are readily where the requirements for are also relatively 2 R/a, yz less demanding. However, whether or not the 1ower- P~ =4.9X 10 (n, T, ) Ra a (W) . (2.35) ~e aspect-ratio regions are accessible depends on the mag-

From Eq. (2.2), substituting the details of the plasma composition, we have 800 — 8 ; 4X10 (n;T;+n, T, + n, T, +n E~) g, 700 — 7— g2 6oo — 6— (2.36) 500 CU 5 I- E 0 In a fusion plasma, n, & n, and the pressure of the ener- &o 400 4 —400 Al getic alpha particles (n E~) may be quite large. To al- E la LU low for this effect and impurity depletion, we shall use 300 300

200 — 2— 200 ~ Pt Peg Pe — qe 100 1 —100 — 0 ~ 0 0 3l Qr'rtept 0 a (m)

FICr. 11. Coupled values of &P &8, average neutron Ilux to the P jq grope Pt first wall p „,P, /R, and yE, shown as a function of minor radius (a), for a self-sustaining D-T plasma in the simplified FICx. 10. Simplified power-Aow diagram for a reactor. toroidal reactor model, operated at 100 kW{e) per tonne.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1028 John Sheffield: Physics of magnetic fusion reactors

l Pe (MW) larator, which require a strong, externally produced 2000 ~ 4.0— toroidal field, to access the smaller major radii (e.g., 4.0 —175 rn), because their superconducting coils need a thick 1500 —2.5 shield, and fz becomes very small. Proposals have been — —200 1000 2.0— CV I made for small-net-electricity-generating tokamaks using E 4.s I— — S E water-cooled copper coils at conventional aspect ratios, 1. ~O0 UJ 250 ~ but they suffer from high recirculating power or place ex- 5.0 —1.0 300 treme demands on the physics and technology (Wagner, s.s 6.0 —350 1981). One possible exception is the very-low-aspect- 400 6.5 —450 ratio (R/a 5 1.5) tokamak with no blanket, a negligible —500 550 shield in the bore, and a solid copper central toroidal conductor (Peng and Hicks, 1990). The RFP and FRC 0 I 0 4 have, respectively, a weak toroidal field and no conductor R/8 down the torus axis, and both have fields primarily de- FIG. 12. Coupled values of minor radius (a), p „, gE, and rived from plasma currents; theoretically, they can access (P&8 shown as a function of aspect ratio for four power levels, the smallest region. for the simplified toroidal reactor model, operated at 100 kW(e) per tonne.

E. Beta and thermal diffusivity requirements netic configuration and magnetic 6eld. Higher B lowers the requirement for (P) and lowers the experimental gz', It is informative to display the physics requirement in classically or neoclassically yE ~ and for a variety 8, ' the form of (P) /yz. First, we shall return to an impor- of anomalous transport mechanisms yE ~ 8 —B tant characteristic, the neutron Aux falling on the first though the multiplication factor is worse than the neo- wall, Eq. (2.8). This may be approximated as classical value. An important factor, therefore, is the field utilization factor fbi, which is the ratio of the magnetic field in the 0.8I'~ plasma (B) to the maximum magnetic field on the coils p (MWm ), (2.42) 4m (R/a)(a /a)(b/a)o 5a (B ):

B =f~B (2.39) where P~ is the fusion power in MW (P„=0.8 P~). For a tokamak, Inherent safety arguments, involving an analysis of ~w 6 a what could happen in a loss-of-blanket cooling accident, +- (2.40) a a R have been used by Logan (1988) to establish a maximum safe neutron Aux for current blanket designs. His recom- mended maximum value is (p„„),„-10MWm, and where a„/a is the ratio of the wall radius to the plasma more realistically may be S6 MW m . Economic stud- radius, and 5 is the distance from the wall to the toroidal ies, (Sheffield, 1986) suggest that it is necessary for the coil conductor on the median plane at the torus inside; 6 first wall and blanket to withstand a neutron Quence of includes the first wall, blanket, shield, coil structure, and 15—25 MW m before replacement. At an availability coil insulation. of 0.65, this requires surviving -4-7 years. Typical values are a /a = 1.1, 5= 1.5 m, e.g., for a =2 Consider now the plasma power balance for a self- and R/a=4, f~ =0.54. sustained D-T plasma, Eq. (2.24). Allowing for impuri- Typical values for other con6gurations are ties and losses other than conduction and some auxiliary heating (f ), the equation becomes

~ stellarator: =0.5(R /a 7), 2 1.7X10" ne (n, T, )rE= (m eVs), (2.43) a n RFP: = 1.0 —1.3(reactor), (2.41) where an impurity depletion leading to nDr/n, =0.73 FRC: =1.0 . has been assumed; see Sec. II.G. Using Eq. (2.37) and Eqs. (2.27), (2.35), (2.39), and (2.42) with (P, ) -0.46 (P) It is difficult for devices such as the tokamak and stel- yields the relationship for a self-sustaining plasma

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1029

scrape off neutron impurity geometry layer depletion

R/a &p) =1.9X10' %sm XE (b/ )0.5 f2g2 (2.44) field maximum fraction of fusion utilization coil field power to power Iadlai conduction

I where P„ is in MW and p „ is in MW m . Care must be For a 3.5-MeV alpha, a field of 5 T, and a minor radius of exercised in using Eq. (2.44) because there are, separately, 1 m, p /a=0. 054, showing that, to a close approxima- requirements on both &p) and yz. Using Eqs. (2.23) tion, the majority of alpha particles should be confined (with impurity depletion), (2.27), (2.43), and (2.44), we well away from the walls of the vacuum chambers. In find for the reference profiles, where It = b/a, that reality, in a toroidal system, incomplete cancellation of field gradient and curvature drifts between the top and 1 6X 10 2~'575P'7-5f' n s bottom halves of the torus can lead to a greater radial ex- XE 25 (m '), ( R /g )0 75( g /g )0 25(p )0 ne cursion of the alphas (banana orbits}. The requirement that 5 a, in order that alpha particles may be con6ned, (2.45) p may be used to establish a minimum current for a reactor when the radial excursion is normalized to the poloidal, rather than toroidal, magnetic field. The poloidal field /g)0. 25(a /a)0. 75(P )0.75 310(R n Be=l20f/2m. a; substituting in Eq. (2.47), we find I ~2.7 0.25 (%T ). )0.125PF MA for p Sa. In addition, alphas trapped in the poloidally varying (2.46) magnetic field, in externally produced field modulation As pointed out above in Sec. II.B, the fusion power varies (ripple), or in fields modulated by magnetohydrodynamic with density and temperature profiles and with the tem- (MHD) instabilities, can walk out of the plasma rapidly; perature range, for the same beta. A +20% change in see Fig. 13. Such losses lead to local heating and, poten- the constant in Eq. (2.35) leads to a +10% change in y, tially, to damage of the first wall (Tani et al., 1989). Rip- losses are discussed for the case in review and a +10%%uo change in the &P) 8 requirements. In the ple of ITER a sections below we shall see how various config- of Post et al. (1991; see also the discussion of ripple urations match up to these requirements. transport in Sec. III. C.7). Second, the energetic alphas, while they are small in F. Alpha physics number compared to the background plasma, constitute a signi6cant beta because of their high energy. The slow- In a D-T fusion reactor, energetic (3.5-MeV} alpha paring down for alphas (Stix, 1972}is given by ticles produced fusion reactions will provide the bulk ' 3/2 by 2.09 X 10 A T, of the plasma heating power. Modest auxiliary sources +sa ln 1+ (2.48) of power (P, ) may be used for control of the plasma, e.g., Z n, lnA, as drivers of plasma current. But for an economic reac- where A and Z are the atomic weight and charge number tor, we require, typically, that P, ~0. 1P . Consequently of the slowing down ion, E, is given by Eq. (2.20), and it is important to understand what effects the alphas „, A, is defined by Eq. (2.13). Classical alpha showing might have beyond providing bene6cial heat to the plas- down times are given in Table VI (Uckan et al., 1988}. ma. This subject is discussed in numerous review arti- The relative density of alphas, where is given by Eq. cles, including those of Kolesnichenko (1980), Lisak and p (2.19) or taken from a table of cross sections, is Wilhelmson (1987), Hively and Sigmar (1989), Sigmar (1989), Furth et al. (1990), and Bishop (1991). n a pa+sa (2.49) First, we are concerned about confinement of the al- n E n phas as they slow down. The ratio of the gyroradius of a fusion alpha to the plasma minor radius (a) in a magnetic The local beta of the energetic alphas, ignoring their ra- field (B) is given by dial excursions, is approximately (E )0.5 4n E =2.89 X 10 (2.47) p = (2.50} a aB

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1030 John Sheffield: Physics of magnetic fusion reactors

100.0 ( S (m s ') must not increase too rapidly as the plasma is heated, and the loss rate of alphas (ri ) must be slow 50.0 compared to the slowing down time [r, ; see Eq. (2.48)]:

E I 0 I N I B~, ' ) BS, 1 B(E, 1.5 1. -50. + (E +E 5) 0 S, Bt ~, Bt Bt (2.51) -100.0 I I --- I I I 150.0 200.0 250.0 300.0 350.0 400.0 The presence of loss regions in velocity space can modify R (cm) and destabilize the distribution function. Alphas are pro- 100.0 duced preferentially where the plasma is hottest ((o.v )zaT ) and can have strong radial, density, and pressure gradients. Their beta is finite, and their velocity v may exceed the local Alfven speed [v„=2.18X10' 8/( A;n;) (ms ')]. For a D-T plasma, E O 0 0. 5 N Ua "20 "DT =9.44 (2.52) Ug 8 pl~ -50.0- A11 of these factors represent sources of free energy which may drive instabilities. On the negative side, these -100.0 1 1 might lead to ejection of the alphas before they have 150.0 200.0 250.0 300.0 350.0 400.0 R (cm) transferred their heat to the plasma, and to increased losses. On the positive side, they might slow down the al- FIG. 13. Poloidal projection of (a) one trapped, and (b) one phas more rapidly, reducing the alpha component of passing alpha orbit in TFTR, perturbed by a stationary (m =2; beta; they might enhance heat transfer to the plasma ions n=1) mode, having peak amplitude at the inner dashed circle. [see Eq. (2.21) for the normal state of affairs], and the The wall is marked by the outer dashed circle, and the plasma large alpha orbits might stabilize some MHD modes. In edge by the solid circle (Mynick, 1992). addition, the rapidly lost alphas can generate radial electric fields in the plasma. Fourth, the slowed down alphas —helium ash —are a where for T =10—20 keV, the average energy during the ubiquitious and significant impurity in the plasma. Note slowing down thermal energies is = 1.4 MeV. to E that, in the absence of any other impurities, Values for (ov n and E are given )DT, /n„P /P, /E, o n DT =n, —2nH, ! The density of helium is given by in Table VII, where p is the plasma beta (Uckan et al., —3 1988). n Hg V]gcA7 Hc m Third, while the energetic alpha particles in a Maxwel- where is the helium confinement time and the pro- lian plasma are produced with an isotropic distribution ~H, duction rate of helium inside in the radius is function, they do not have a Maxwellian distribution; a (r) discussion of the conditions for the velocity-space stabili- P (r) the alpha-particle velocity distribution is given (2.53) ty of by 2m r Asc5eE Cordey et al. (1981). To prevent inversion of the distri- 0 bution function, i.e., Bf /Bv )0, the source rate Now

TABLE VI. Classical alpha slowing down time (local values).

(s) n/(10 m ) O.S 1.0 2.0 3.0 S.O 7.0 10.0

S 0.4 0.2 0.1 0.07 0.04 0.03 0.02 10 0.9 0.4S 0.23 0.1S 0.09 0.06S 0.04S 20 1.8 0.9 0.4S 0.31 0.18 0.13 0.09 30 2.7 1.3S 0.68 0.4S 0.27 0.19 0.13S 40 3.4 1.7 0.8S O.S7 0.34 0.24 0.17 SO 4.0 2.0 1.0 0.67 0.40 0.29 0.20

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors

TABLE VII. Fast alpha density and beta (local values) (Uckan et al., 1988).

T (OU &DT n /n, P.&l3 (keV) (m /s) (%)' (%)

5 1.35 X 10-" 0.01 0.73 0.30 10 1 13X10 0.1 4.2 0.34 20 4.31 X 10 0.8 19 0.39 30 6.65 X 10 1.8 31 0.41 40 793X10 2.7 34 0.41 50 8.54 X 10 3.45 34 0.40

6m r A~en, T variable magnetic ripple, to remove alphas if a tempera- rE(r) = (s), (2.54) starts. Consider the case in which P (r) ture excursion ~E=C, T "p ~ and p=& =C2T . The plasma power assuming only alpha heating. Consequently balance is given by n T +Bc V3nTT C',~T'~ =0.6 (2.55) V (3nT) =— + VC2 T'—Radlatlon n, E o r~(r) Bt (2.56) For example, for T=25 keV, if rH, ~12&@(r) we find nH, /ne ~0.05, i.e., a greater than 10% depletion of the If (1+x)~2(1—y), there will be no thermal runaway 0-T fuel. even if radiation losses are small (Engelmann, 1987). Provided helium confinement is comparable with hy- Empirical confinement scalings for tokamaks and stel- drogen confinement, ~H, =~ -3—5~E, there should not larators often have x=0 and y -0.5, so that the thermal be a problem. Unfortunately, the convection term V„ in runaway depends sensitively on details of the alpha Eq. (2.10) can be inward. A relative inward motion of power dependence on temperature, on radiation, and on impurities is predicted for some conditions (Braginskii, profiles. 1965; Hinton and Hazeltine, 1976), and it has been observed in, for example, tokamaks (Stacey et al. , 1985). Helium transport and exhaust are reviewed by Hogan and Hillis (1991). A further problem is that energetic al- G. Impurities, Iimiters, and divertors pha particles bombard the wall and generate impurities, some of which will penetrate the scrape-off layer and The control and minimization of impurities in a plas- enter the plasma. ma is a major concern for a fusion reactor. As we have The 0-T fusion reaction rate increases strongly with seen in the previous section, the helium ash created when temperature, as shown in Fig. 3 and Eq. (2.7), which have the fusion alpha particles slow down and thermalize p ~ T2 T' for ( T—) ~ 22 keV. Normally, ignition represents a minimum impurity constituent of the plas- occurs as a balance between alpha power and loss mecha- ma. Wall-generated impurities are an additional prob- nisms that have a weaker temperature dependence, such lem. Unfortunately, there is a strong tendency for highly con- as bremsstrahlung (T ) and conduction. Consequently, charged ions to diffuse from the wall region into the following ignition, the alpha power and temperature containment region (Taylor, 1974), and suppression of their tinue to rise until either (a) some other more strongly generation and transfer to the plasma are important con- temperature-dependent rnechanisrn, for example, syn- siderations for reactor design. Consider, for example, the chrotron radiation, sets in and reestablishes a power bal- reference ITER assumptions (Uckan et al. , 1990). ance or (b) the pressure (beta) limit is reached. In the thermal runaway lead to gross insta- latter case this may na bility of the plasma, as in the case of a tokamak disrup- helium = 5 and 10%, Z =2, tion (see Sec. IV. E). An example of the density and tem- n~ ' perature operating space for a tokamak, showing the ig- 2.6 nition region and beta limit, is given in Fig. 49 of Sec. IV. carbon 0.9+0.6 ZC=6, n~ n2p A review of burn control possibilities for tokamaks has been presented by Haney et al. (1990). np is possible that the may be self- oxygen =0.1%, Zp =8, While it system ne its behavior might involve temperature oscil- regulating, ' 2. 3 lations leading to a time-varying heat load on the first npc iron =005" +0 02 %, ZF, =26 . wall and divertor and stress-fatigue problems. This is not n2o a comfortable operating condition; therefore designers look for control methods, e.g., fuel density control and For ( n, ) =2 X 10 m, we find

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1032 John Sheffield: Physics of magnetic fusion reactors

Pl~ 1. Plasma-wall interactions. for =5%) =0.83, Z,~= 1.59, le ~e The complexity of the plasma-wall interactions is well f1~ nDT for = 10%, =0.73, Z,~=1.69 . illustrated by Fig. 14, which lists the wide range of ele- 7l~ ne mentary processes that can occur at the wall. There are numerous reviews of this topic, including those of The latter case has nearly a factor of two lower fusion McCracken and Stott (1979), Nygren et al. (1981},Lang- output than a pure D-T plasma would have. ley et al. (1984}, Post and Behrisch (1986), and Janev The impurities also radiate power from the plasma by (1991}.Sputtering is a major problem because after a lot increasing bremsstrahlung losses (2.15)] and line [Eq. by of material is removed a component must be replaced. (2.14)] and continuum radiation. High-Z impurities [Eq. Low sputtering rates are essential to ensure a reasonably are the worst offenders because they do not become fully long time between maintenance and replacement. The ionized, and minute quantities of, say, molybdenum and extent of sputtering can exceed that due to knock-on pro- tungsten can radiate substantial amounts of power. cesses, in which sputtered. particles receive enough ener- Low-Z impurities become fully ionized in the bulk of the from collisions with the incident particles to overcome plasma and radiate predominantly from the colder outer gy the surface binding energy, because of chemical interac- regions. This has led to the suggestion of their controlled tions in which molecules are found. For example, com- use to radiate heat from the plasma edge in a benign monly used wall, limiter, and target material —carbon— fashion to the walls, and to limit the conduction and con- interacts with hydrogen to form CH4. The sputtering vection load on limiters and divertors —the cold plasma yield for carbon bombarded by H, D, and He is given in mantle concept (gibson, 1978; Watkins et al., 1981). It Fig. 15, for varying carbon temperature. Radiation- is not clear whether this situation could be sustained enhanced sublimation increases the erosion rates for (Samm, 1993). carbon-based materials (Nygren et al., 1990}. Carbon A key issue is the transport of impurities in the plas- tlat, also suffers from the problem that it absorbs hydrogen, ma, both the helium ash and the wall-generated impuri- and in a D-T plasma this can lead to large trapped- ties that reach the plasma edge. Consider Eq. (2.10) for tritium inventories. Beryllium, a lower-Z material, has impurity transport with the source term turned off: been used successfully in JET (Thomas et al., 1990). Redeposition can off'set sputtering (Brooks, 1990), though rD, +rn, V, (2.57) unless the materials re- Bt r Br Br it is not an advantage redeposited tain the surface quality. Impurities, including the wall Let n, =n„exp( ); th—en if D, and V, are constant material itself, have higher sputtering yields than hydro- and aV, /D, is small, 0 PLASMA WALL 4(D~ —A V, ) Neglecting the convection term, we require

SOT ZS, PLASMA PLASMA BOUNDARY SURFACE LAYER SOLID REGION (AD SO R BATE) (WITH ADSORBED DDT NON WALL ATOMS) ELECTRONS

to prevent serious dilution of the D-T fuel. To ensure FUEL IONS low Z,z we need PHOTO NS BACK SCATTERING REF LECTION SOT S, FUEL NEUTRALS SECONDA R Y EMISSIO N ) gz2 ' PHOTOELECTRONIC EMISSION OPTICAL AND X-RAY EMISSION DDT, D. IMPU R IT I ES RE-COMBINATION SURFACE IONISATION ELECTRONS AD-AND DESORPTION where SOT and DDT are the source rate and diffusion CHEMICAL REACTIONS SPUTTERING FUEL IONS coefficient for the D-T fuel. THERMIONIC EMISSION THERMAL RADIATION However, if a V, is large compared to D„and positive, PHOTONS EVAPO RATION and in the convention used here positive V, means in- FUEL NEUTRALS ward convection, ~, can become negative and the impurities will accumulate (Braginskii, 1965; Hinton and Hazel- IMPURITIES tine, 1976; Hirshman and Sigmar, 1981). This phenomenon has been observed under some conditions in tokamaks; see the review by Stacy et al. (1985). Clearly, FIG. 14. Transition region (schematic) between plasma and impurity accumulation regions must be avoided, and this solid and list of elementary process occurring at the wall (Bick- fact places a restriction on operating regimes. erton, 1977).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffiead: Physics of magnetic fusion reactors 1033

10 gen, and this is another reason to suppress them. In e 3keV HE ~ 1keV 0 toroidal experiments, substantial reductions sn the a 1keVH C) amount of gas, mainly oxygen, desorbed from the walls E D have been achieved careful preparation of the sur- a5 by a faces, using a variety of cleaning techniques prior to UJ operation of the full plasma (Oren and Taylor, 1977. (3

CC Control of the plasma edge by a carefully contoured lim- LLI I- ~—— g~ iter or divertor is used to minimize impurity problems CL (f) CH - FORMATION and remove particles (Komarek et al., 1990). These op- 10 w ~ w w 4 tions are illustrated in Fig. 16.

I I I I ( I I I I ( 1 I I 300 500 100 1500 2000 TEMPERATURE (K) 2. Limiters

FICy. 15. Temperature dependence of the chemical sputtering4 + I.imiters have been used traditionally in toroidal de- yield of carbon with 1-keV H+ and D+, and 3-keV ions. H, vices to protect the vacuum chamber walls from plasma Methane is formed at about 900 K, but above 1300 K no hydro- bombardment. In recent years they have been shaped so carbon could be found (Langley et al., 1984). as to pump the gas formed by neutralization of the plasma —the pumped limiter [Kelley, 1973; Vershkov S imp I e L imi t e r and Mirnov, 1974; Schivel, 1977; see Fig. 16(b)]. The plasma intercepts the limiter over a small radia Confined distance the scrape-off layer (SQL) whose scale is Plasma — — determined by the ratio of the transport coefficients, radial and parallel to the magnetic field lines. The heat con- crapeof f ducted and convected from the main body of the plasma P I asrna streams along the field lines and is transferred by electrons and ions to the limiter. Most of the ions are neu- tralized at the limiter, either returning as dissociated molecules (energy -5 eV) or reflecting as neutrals. A cloud of neutrals forms around the limiter. Because the Pumped Limi ter (b) charge-exchange rate for hydrogen is greater than the ionization rate (see Fig. 17), the neutrals migrate into the plasma for considerable distances beyond the limiter. eutral Flow Typically, they have the local ion temperature. The mean free path for this diffusion has been calculated by Plasma Podesta and Engelmann (1973) as F low 0.5 ' 0.5 1 &ou ),„' eT, (m) n(& o&u, +&ou&, ) &ou), m, (2.59)

I I I Divertor 107 NIZATION—

Magnetic 10-8 Separatrix E O 09

CC Uj

I— 10-10 CC

10-11 l I I II 100 101 102 103 104 105 106 PARTlCI E ENERGY (eV) FIG. 16. Schematic illustration of power handling, particle exhaust, and impurity control options for a tokamak: (a} simple FICx. 17. Rate coe%cient for atomic hydrogen (Freeman and limiter; (b) pumped limiter; (c) poloidal divertor. Jones, 1974).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1034 John Sheffield: Physics of magnetic fusion reactors where, for T ~50 eV, n(ou), ~4X10 s ' and n = = n ( a U —3 X 10 s '. See also, Tendler and Agren n (0)exp (m ), T T(0)exp (eV) ), ~n (1982). AT & For T, 50 eV, the charge-exchange rate is sufficiently (2.65) greater than the ionization rate that most neutrals formed in the low-temperature region will march into the where A, „=(Drll) m and — higher-temperature region. Assuming T, T„ these ~n [(a+3) +4a(2y —3)] —(a+3) 50+eV neutrals can cause substantial sputtering. For (2.66) AT 2a carbon, the peak sputtering rate is for hydrogen with en- — ergies in the range 50 1000 eV (see Langley et al. , 1984, where a =g/D and y =1+y,. p. 77). The density and temperature at the point of the limiter A simple model for the SOL plasma density and tem- (y=0) are given by the requirement that all of the parti- perature is given by a slab model in which y represents cles and power that leave the plasma by cross-field trans- the radial direction and the the subscript ~~ denotes port Sow to the limiter: magnetic-field direction (Howe, 1981): ~ oo Pg HAT= A particle loss rate (s ')= (2.67) 3pl 7l 0 (2.60) jll Tp where 1V is the total number of particles in the =r — plasma. 8 BT 8 Bn 121 rz /(1 R„), where ~~ is the bulk-plasma particle + 3TD T( I+ ) (2.61) X confinement time and R„ is the recycling coefficient at the wall/limiter. Here D and are the diffusion and co~duction y=y, +y; ~ 2+nT coefficients, respectively; n =—n,,=n;, T=—T, = T;, and I' T= A dy thermal power (W) (2.68) y, = 1+0.25 ln(m, T, /m, T; ) is the enhancement of the electron energy Aux due to the surface sheath potential. where 3 =4m Ra. The plasma pa~t~~l~ I' and energy cruxes are given ll Qll by Integration yields Emmert et al. (1980) as PT SeT; no= m ), and To= . (1+ii) (eV) . 111=2inV;(m 's ') where V;= (ms '), n 2yeNT (2.69) (2.62) The fall of the heat Aux is given by 2T;P 1 1 In a toroidal system the Aux meets a toroidal limiter at (2.70) an angle determined by the field-line pitch Be/8& where =aB&/RB&. The particle fiux incident on a segment of q Ergodization of the magnetic field at the plasma edge, us- the toroidal limiter in the poloidal direction is given by ing local coils, increases the radial heat decay length and can lower the average power density on a limiter (Samain &e a et aI., 1984; DeGrassie et al., 1989; Monier-Gorbet et al., 1993). q Consider the following example for a pumped limiter: R=6 m, a=2 m, q=3, XT=5X10 s ', ET=120 MW, For a slab of between and Aux exact- plasma y y +dy this y =4, D=2 m s ', y=D, a=1, and q=1, in a D-T plas- balances ly the equivalent volume loss term in the density ma. To =375 eV; assume an average T = To/2, transport equation, V, =1.4X10 (ms )~ all 8.3X10 s, A, „=0.04m=AT k& =0.02 m, n0=2. 2X10' m 2I,2~Rdy = 2ma2mR dy, (2.63) I et the radial extent of the limiter be 0.08 m; then the II fraction of the heat intercepted on the front of the limiter fg =1—exp( D~/A, &)=0.98. The fra—ction of the parti- where the factor 2 in front of accounts for the two I, cle How that gets behind the limiter to a pumping region sides of the limiter: ~ /A, „ is q =(1—f&) "=0.14. &a71 2~Rq The neutral mean free path, assuming the peak tem- (s) . (2.64) I, y. perature and density in the SOL (if greater than -50 eV), is given by Eq. (2.59) as A,0=0.09 m. If the poloidal ex- For constant D and y and ignoring the T; dependence tent of the limiter is about 1 m, then, allowing that most of I the solutions are of the particle deposition will be towards the middle of II

Rev. Mod. Phys. , Vol. 66, No. 3, July 1 994 John Sheffield: Physics of magnetic fusion reactors 1035 the limiters, one can see that —10% of the neutrals may tor (-0.25 MW/m thermal flux at the plasma edge, be in a cloud extending beyond the limiter, i.e., can cause compared to 0.5 —1.0 MW/m expected for an attractive sputtering of the wall. A computation of neutral density reactor). contours for the INTOR/FED pumped limiters is shown in Fig. 18. 3. Divertors Of the neutrals that leave the limiter, part will contin- ue further into the eventually becoming ionized, plasma, It is not clear whether a limiter can be used in a reac- and a third, will return to the limiter and part, roughly tor plasm. a because of the problem of needing to remove wall. particles while preventing unacceptable erosion due to we assume one-third must return to the limiter and If the ions and neutrals. Certainly, if a cold plasma mantle the device is run for a with a carbon wall, year sputtering could be formed this might alleviate some of the prob- coefficient S, =0.1, the total number of carbon atoms re- ' lems. Of course it would also alleviate problems for the moved per year= —, X 5 X 10 XO. 1 X3600X8760=5.3 alternative solution —the magnetic divertor. X10 C/yr, or 1.0X10 kg, or a volume of 4 m . The Various designs of divertor have been proposed. They limiter area is 37.7 m, so the average depth of carbon re- all consist of an arrangement of coils producing magnetic Inoved is -0.1 m. Redeposition of carbon might reduce fields opposed to the confining fields, such that Aux lines the net erosion (Brooks, 1990). near the wall are diverted onto targets at a distance from The walls would be subject to a lower Aux and less ero- the plasma much greater than the SOL thickness; see sion. However, this example is operated at a much lower Fig. 19. For the tokamak, while a variety of approaches thermal power density than would be expected in a reac- has been tried, only the poloidal divertor remains as a serious consideration. However, in other devices, toroidal and local divertors are still being considered. The use of magnetic islands to create divertor action has been proposed by Karger and Lackner (1977). The divertor serves a number of purposes (see Harbor, 1981): (1) Act as a magnetic limiter, isolating the confined plasma from the wall. The fraction of particles that cross the separatrix and enter the divertor without first diffusing to the wall, the "unload eKciency, "is 'w 7/g = 1 exp (2.71)

where 8' is the width of the divertor throat. Again, if W/A, „~4 most of the particles go to the divertor region.

DIVE RTOR CURRENT ERTED LD LINE DIVERTED FLUX BUNDLE CURRE LOOP

SEPARATRIX

TOROIDAL DIVERTOR BUNDLE DIVERTOR

RRENT PROJECTION OF A DIVERTED FIELD LINE PLASMA PROJECTION OF A DIVERTED FIELD LINE PLASMA

Limiter SEPARATRIX

SINGLE-NULL DOUBLE-NULL FIG. 18. Contour plot of the log of the density of the D origi- POLOIDAL DIVERTOR POLOIDAL DIVERTOR nating at the INTOR/FED pumped limiter at 6.6 s into the base discharge case, showing the D population concentrated in FIG. 19. Types of divertor incorporated in toroidal systems the limiter region (Heifetz et al., 1983). (Rawls et al., 1979).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors

(2) Ionize incoming impurities from the wall in the HIGH BE TA PLASMA SOI. and sweep them into the divertor. The probability of ionization of an impurity in crossing the scrape-o6' lay- D Transpor t er is Modeling Edge Floe — (oU), ~-Limi ter g; =1 exp (n, A, „), (2.72) Physics where ( o v ), is the ionization rate for the impurity and UI is the impurity radial velocity. If n., A, „)4UI/( o;U, ), q, =1, e g., for oxygen at 1 eV, v1=25X10 ms it ' Irnpur y (au), =1X10 m s '. n, =lX10' m, 5=0.04 m, Transpor t the condition is met easily. 8 Atomic Physics Inlpur ity (3) Transport the heat to the divertor chambers rather Mo lee u le than to the first wall. Transpor t (4) Minimize the neutral content of the SOL and the Spu t ter ing al main body of the plasma. tion (S) Be reliable and maintainable. (6) Allow operation with a low T and high n at the tar- To Pump get plate, even in some conditions in which there is a !3iverted high T and low n at the edge of the main plasma. P la sma Divertor Analyses of divertors have led to the identification of Gas three regimes of operations (Post, Heifetz, and Petravic, 1982); a tokamak example is given in Table VIII. FIG. 20. Types of processes involved in a divertor for a high- In the first regime there is very little recycling, the beta plasma (Post et al., 1982). temperatures are high, and the densities are low. The neutrals can easily escape down the pump or back to the main plasma without being ionized. The neutral pressure is low. length and the width of the divertor. The electron tem- The second regime is characterized by a medium level perature is ~30—40 eV and the density is high —10 of recycling. The neutral mean free path is shorter than m . The neutral pressure is high, ~ 10 Torr. Radia- the length of the divertor channel, but greater than the tion and charge exchange can be important. Such opera- width of the channel, so that neutrals can escape to the tion was achieved first in the D-III expanded boundary pump but not to the main plasma. Radiation from hy- divertor (Shimada et al., 1981;Ohyabu et al. , 1982). drogen and charge exchange does not transfer much of A representative divertor geometry is shown in Fig. 20, the heat Aux to the divertor walls before the plasma which also illustrates the large number of physical pro- reaches the neutralizer plate. Sputtering yields are high cesses thai must be taken into account. in both regimes because ion energies exceed sputtering It is necessary to treat ions and electrons Aowing both thresholds for realistic materials. across the magnetic field and along the field to a target The third regime is characterized by large recycling plate, where they combine and reenter the SOL as neu- rates. Each ion typically recycles 10 or more times be- trals. Neutrals atoms and molecules transport through fore being pumped or reaching the main plasma. The the plasma, reInecting from the walls and undergoing neutral mean free paths are short compared to both the charge exchange and ionizing collisions. Impurities are

TABLE VIII. Approximate operating regimes for a divertor of length L and width w, with INTOR level heat and particles cruxes (Post et aI., 1982). Neutral recycling Low Medium High

A,o, neutral Incan L )Ao&w b, a &AD free path + I plate~I throat R —1 1~A 10 R ~10 T, (eV) 200—1000 30—200 &30 Edge plasma density 10"-10"m-' 10" m ' 10 m Importance of hydrogen Negligible —10 10—90% radiation estimate — — Neutral pressure (Torr) 10 6 10 7 10 —10 &10 ' T, gradient Small —10% Large

Rev. Mod. Phys. , Yol. 66, No. 3, July 1994 John Sheffieid: Physics of magnetic fusion reactors 1037 produced from the walls and targets by sputtering, arc- adequate description. The appropriate plasma equations ing, vaporization, chemical erosion, desorption, etc. for the base plasma, to be used in estimating the impurity Zero-dimensional, one-dimensional, and two-dim. ensional generation and shielding and impurity removals are models have been used. The last are necessary for an (Petravic et al., 1985)

8 8 dn ) =S„+ Di Bx (nuit By By

B 2 B B [n(mu~~+T;+T,. )]=S~+= Di nmv~~ X By By (2.73) 5 1 dT; Q(piTe) (j 5 mu (jyi dT; nv— nm— — = — — (( i T; u(( y((; u(( +SE;+ DL T;+ +pi; +Q„,

B 5 BT, B(iiT, ) g 5 gpi — e &Ile +SE,+ Di T, +y 2" II B~

where n =n, =n;, v is the ion parallel velocity and m is private flux region of the divertor (it decreases nonlinear- the ion mass. Q„ is the electron-ion Coulomb energy- ly with increasing null-to-strike-point length A, measured transfer term [see Eq. (2.13)] and S„,S~,SE;,SE, are the in the poloidal plane). The geometric parameter ionization (particle) source, momentum source, and ion 6 =sinQ(Bs/8, ) «1 allows for spreading of power and electron energy sources arising from the plasma in- over a toroidally symmetric divertor plate, inclined at an teraction with neutrals. The use of a Quid model is angle Q relative to the magnetic Qux surface. justi6ed for most useful conditions, since I., m &)A,;, but The key points illustrated by these equations are (1) An not at the sheath in front of the divertor target, which is increase in P causes a nonlinear increase in pD and TD. handled with boundary conditions. At the sheath the (2) Tz increases very strongly as n, is reduced. (3} TD parallel velocity is estimated to be the sound speed, can be reduced by increasing A. '" ' v = [(T +T ) Im ] While these results have been obtained for a tokamak, To obtain solutions it is customary to specify values for they have wider generic application. Two-dimensional the Qux nu~ and the electron and ion heat Quxes at the simulations of the scrape-off-layer support the validity of upstream boundary x =0. Neutral source terms are com- the analytic expression, for temperature (Ueda et al., puted using a separate neutral-particle transport code, 1989;Itoh and Itoh, 1990). for example, the Monte Carlo code DEGAs (Heifetz et al., 1982), or with simple diffusion models. While simplified analytic expressions have been obtained for SOL and divertor parameters, it is necessary to calibrate their va- solve full lidity using a computer to the set of equations. 10" —104 300 I An example solution for the international thermonuclear I 275 CASE ON A. 1 experimental reactor (ITER}is given in Fig. 21. 250 a5 for the unit area CL Simpli6ed expressions peak power per - E 225 CL on the divertor plate pD and the adjacent plasma temper- 200 UJ I—~ CC I— ature TD have been given for ITER by Harrison and 175 (f) M Co p20 LLJ Hotston (1990). The expressions are useful for indicating 150 1 CC Cl CL trends: 125 OC OC OC I—CC CC 7/9 100 P CC CC QC 75 CL CL p~ Z,~ G~G +nonpertinent terms, LLI UJ 50 03 CO (2.74) P2 10/9 25 I I OC 1P19 1P3 n, Z ff GA+nonpertinent terms . 2.5 5.0 7.5 XJ. FIELD LINE LENGTH, x ((T})

Here is the alpha power and additional heating P FIG. 21. Variation of the plasma parameters on the separatrix (current-drive) power fiowing across the scrape-off layer, with position on the separatrix. The temperatures fall strongly yj is the cross-6eld thermal diffusivity, n, is the plasma and density rises as the Seld lines approach the divertor in such density in the stagnation zone of the SOL, Gz (1 is a a way that the pressure remains nearly constant (Cohen et al., geometric parameter accounting for energy loss in the 1990).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1038 John Sheffield: Physics of magnetic fusion reactors

H. Auxiliary heating plasma volume, using model profiles for the density and The requirements for heating to ignition are obtained temperature (ShefBeld, 1986; Uckan et al., 1990). For by solving the time-dependent power-balance equations startup conditions in a 0-T reactor, the synchrotron ra- (2.11) and (2.12). A computer code, for example, wHIsT diation term is smaller than the bremsstrahlung term and (Houlberg, 1982) or BALDUR (Singer et al., 1986), is may be dropped. The line radiation and charge-exchange needed to account properly for the details of the radial terms are important at the plasma edge but play only a dependences of ions, electrons, impurities, neutrals, and small role in the plasma center, for a clean dense evolving magnetic field, let alone MHD efFects. For the plasma, and are dropped. Assuming n =no, T, = T; purpose of providing a simple solution, a rough answer = To(1 r l—a ), and taking the temperature gradient at may be obtained by integrating the equations over the r /a =0.8, leads to the following power-balance equation:

W= —0. 16Rn i9Tk(y, +y;)—1. 1 X 10 RabZ, Irn i9Tk' Bt conduction losses

0.08y ~ Z,ffRI — nDT +1 9~10 4g b 2 Tk +P, (MW) abT," Ple ohmic heating auxiliary heating (2.75)

where n»(10' m ), Tk (keV), I (MA), and average ties, such as the tokamak with toroidal fields —5 T, values have been used for /=1. 1, lnA=19. The alpha Ohmic heating is insufhcient to raise the temperature to a power formula (p ~ T, ) is a best fit for a parabolic level where alpha heating can take over. The Ohmic temperature profile for 2 5 T;k ~ 10 keV. The factor temperature ( Tz) is obtained primarily by a balance of (nDT /n, ) allows for fuel depletion owing to wall- Ohmic heating and conduction losses. Setting y, +y; generated impurities and helium ash. For most tokamak =yE(Tk/10), we find conditions the various terms have the form shown in Fig. ''"" " '05(10)'XR Z effI'm 22; the conduction term is not shown. For neoclassical keV . (2.76) losses y ~ T, and as we shall see in later discussion of QAn (9+E various devices, theoretical mechanisms for anomalous For example, let 5=0, =1.5, Z,ff=2, I =20 MA, transport have a wide variety of dependences on temper- yz a=2, 6=4, n = 10, and =2 m s '; then Tk -—1.7 ature. For configurations having moderate current densi- )9 yE keV. A second temperature of interest is that at which the alpha power exceeds the bremsstrahlung losses,

p keV .

For Z,Ir=2, nDT/n, =0.73, T~bz —4.6 keV. This value is less than the value for a Bat temperature profile because of gain in alpha power from the higher central temperatures. The addition of conduction losses raises the temperature at which the alpha power takes over (ignition). The gap from the Ohmic temperature must be bridged with auxiliary power. In a stellarator with no Ohmic heating, the auxiliary power Inust also be used to produce the initial plasma. / TO IGNITION For very high current-density configurations, high-field tokamaks, the RFP's, it is theoretically possible for the Ohmic power to provide all the heating and lead to ignition, the dashed line in Fig. 22. Assuming that FICx. 22. Auxiliary power requirements for tokamaks, showing = a characteristic variation with plasma temperature owing to the 8 8 /Bt constant, the maximum auxiliary power occurs temperature dependences of ohmic, bremsstrahlung, and alpha where BP, /8 T=0. This is given, approximately, by power. balancing the rate of change of' alpha heating against the

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1039 rate of change of conduction losses, provided 5 ~ 1, Wilson, 1981; Bornatici et QI., 1983; Colestock, 1984; — Cairns, 1991;Stix, 1992), and adiabatic compression (Bol 340(1+5)y,' 2 1/(1.5 5) et al., 1975). mK (keV) . (2.77) (10) abn» nDr

For ~~~~pl~, if 5=0, yE=1, nND/n, =0.73, a=2, b=4, 1. Neutral-beam injection n 19 10 T k —3.2 keV. The minimum value of P, occurs when BiR'/dt=0. For the above example and This heating technique involves injecting an intense beam of energetic atoms, generally hydrogen or deuteri- y~ = 1.5, Zeff 2, I~ =20, R =7, um, into the plasma. To date, energies up to 150 keV conduction Ohmic have been used. At such energies it is possible to form — — the beams by neutralization of energetic positive ions. P, =72+22 15 21=58 (MW) . For the higher energies envisaged for ITER and rectors ~ bremsstrahlung alpha ( 1 MeV), positive-ion neutralization is inefficient and negative ions must be used. Because the atoms are neu- Ignition occurs at about 7 keV. tral, they can cross the outer regions of the confining To raise the plasma to ignition in a finite length of time magnetic field and reach the plasma. The atoms requires additional heating. Assuming a rise time of 30 s penetrate the plasma until they are either ionized or to 7.5 keV for the example above requires BR'/Bt =2.5 charge exchange on plasma ions. In most applications a MW. So the total auxiliary power needs to be about 80 small percentage of the beam passes through the plasma. MW. For Ohmic ignition the minimum heating power (Avoidance of damage to the chamber wall sets a limit on occurs at T,o, where P =Po, and this represents the the maximum beam energy. ) The energetic ions pro- highest hurdle to cross, assuming conduction losses, like duced are trapped by the magnetic field and orbit within bremsstrahlung losses, vary slowly with temperature: the plasma, slowing down by collisions that heat the plasma. Some energetic ions may su6'er collisions that transfer them into uncontained orbits as they slow down. 0.5 In practice, for most practical conditions losses are 5„~0.25Z0. 25 I aQ 4' f R eff keV . (2.78) 5 10%. The fraction of beam shine-through (Uckan et al. , for hydrogen-beam energies 5 is The ignition requirement is that 1990), of -0. MeV/amu,

0. 16y~ Z,ffI )0 16Rn $9+ET Q b-exp[ —(n20db)(0. 795/Eb) . (2.79) ahT.'~5 f ]

+ 1. 1 X 10 RabZ, ffn»T, ~ Here Eb (MEV) is the beam energy, n20 (10 m ) is A variety of methods have been used successfully to the average density along the beam path, and d& heat toroidal plasmas (see Fig. 23) —neutral-beam injec- =2RO[(1+a/Ro) (R„„s/Ro) —]'~, where R„„/Ro is tion (Menon, 1981), radio-frequency heating (Hwang and the cosine of the angle between the beam ions and the toroidal field at R0. The ions slow down initially on the electrons RADIO FREQUENCY energetic HEAT IN G and, after they reach a critical energy (E, ), on the ions. OSCILLATOR OSCILLATOR The formulas for the critical energy and for the fraction C0 I L WAVEGUI DE of energy transferred to the ions are in Eqs. (2.20) and OHMIC (2.21). The first demonstration of substantial ion heating in a toroidal device was on PLT (Stodiek et al., 1981), where the central ion temperature was raised to 7.5 keV (the electron temperature to 3.5 keV) with 3 MW of power. Since that time the ion temperature has reached

ING over 30 keV in TFTR (Bell et al., 1989), and the electron A temperature 13 keV in JET (Keilhacker et al., 1991), ATOMS IONIZED SSION AND TRAPPED with up to 25 MW of power. For deuterium injection into a 0-T plasma, the energetic ions can fuse, thereby

ENERGETIC enhancing the alpha power in the plasma (Jassby, 1976). ATO While the gain is too small to make an economic power NEUTRAL INJE reactor, it is of interest in making volume neutron HEATING ACCELERATOR sources for fusion nuclear testing. Neutral-beam injec- FIT+. 23. An illustration of the main heating systems that have tion may also be used to drive a plasma current; see Sec. been used on tokamaks. IV.B.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1040 John Sheffield: Physics of magnetic fusion reactors

2. Radio-frequency (RF) heating turn may be transferred to the plasma in a way that drives a plasma current. This technique involves driving oscillating currents in frequencies for which the plasma im- the plasma at b. Efficient coupling pedance is high. In fusion, the term RF heating is used in the from 25 kKz to loosely for systems operating range Efficient coupling of the power requires that the in- consists a oscilla- 250 GHz. Each system of high-power cident wave penetrate the plasma and be absorbed. The with feed cables or waveguides to a launching tor going behavior of the waves will depend upon the radial profiles which are close structure made of coils or waveguides to for the plasma density and magnetic field, and to a lesser the plasma. extent on the temperature. At very high frequencies [ro))co~, (electron plasma frequency) and co„(electron cyclotron frequency)], the waves are transmitted with a. Absorption only a minute amount of power being scattered by the plasma. At lower frequencies (ro ~ co, and co„) the For good absorption, by Landau damping or transit- waves may be rejected or absorbed, depending on the time magnetic pumping, it is necessary for the phase ve- plasma profiles and launcher geometry. In an unmagnet- locity of the propagating wave to match the velocity (v~ ) ized plasma, the waves will propagate to the point at of either the electrons (q =e) or the ions (q =i) Fo.r cy- which the density has risen to a level where clotron resonant interactions the requirement is co~, =(n, /m, to)' =w. At this point, called a "cutoII", (w —mw, ~)/k ~v~ where m is an integer and w, is the the waves will be rejected. The situation is more compli- cyclotron frequency. The thermalization of power by the cated in a plasma containing a magnetic field. particles resonant with the wave may be by collisions, Landau damping, or cyclotron damping or, at very high c. Electron cyclotron frequency range power densities, by nonlinear interactions resulting from turbulence generated by the incident wave. In a magnetized plasma, the behavior is modified and The attractions of RF heating are that, in principle, cutofF may occur where ro, =co or where (co~, —co ) the power may be preferentially deposited in the resonant =co„co, depending upon the mode of the waves. Propa- region, which may be selected; either electrons or ions gation occurs provided the density remains below a value may be heated; and, as discussed in Sec. IV.B, momen- corresponding to cutoff,

1 1+ extraordinary mode (E -8),

24X10' (m ) X (2.80) n,„=1. 10 GHz 1 ordinary mode (E ~~8),

I where w and ~=2m. /1, are the incident-wave frequency The frequency is sufficiently high that waveguides may be and wave number, respectively, and I is the cyclotron used for launching it, e.g., n =10' —10 m, B-4—10 harmonic number. Power is coupled to harmonics of the T, in a deuterium plasma. The launching structure must electron cyclotron frequency, generally to the first or be placed close enough to the plasma that the plasma second harmonic. Because these are high frequencies, density near the launcher is above the cutofF level for the ' w„=7—14X 10" (rad s ) (110—120 GHz) and A, &(a, it waves. The wave is then evanescent to the cutoff region is necessary to satisfy the cutoff condition. The develop- and propagates beyond it. For a simple launching ment of the gyrotron in Russia opened the way for the geometry (e.g., single waveguide), the wave would be deployinent of high power ( -MWs). A program in T-10 rejected. Access to the resonance zone then requires the (Parail and Alikaev, 1984; Alikaev and Parail, 1991) incident wave to have a finite wave number parallel to demonstrated both heating —the ability to modify MHD the magnetic field. This may be provided by using a set activity with up to 2.2 MW of power at 86 GHz —and of waveguides with the wave phase alternated (Brambilla, current drive (Lohr et al. , 1991). Significant heating has 1976). Experiments show that the coupling varies as a been obtained in DIII-D (Luce et al., 1991), with 1.4 function of phase angle, in agreement with the Brambilla MW at 60 GHz. An excellent review has been given by theory. Substantial heating and efficient current drive Bornatici et al. (1993). have been demonstrated on JT-60 with 4.5 MW of power (Nagami et al. , 1991). d. Lower hybrid frequency range e. ion cyclotron frequency range The lower hybrid frequency is mLH where Two modes of the wave are possible: a "slow wave, " 1 (2.81) which behaves in a manner similar to the lower hybrid 2 WLH case and requires a complex launcher structure to avoid

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1041 reAection; and a "fast wave, " which is evanescent be- wall bombardment by energetic neutra1s and to wall ero- tween the launcher and a cutoff region in the plasma, but sion. In addition, the sources at the plasma edge gen- can propagate beyond the cutoff without attenuation. erates a relatively Qat density profile unless the particle Decay of the curves in the evanescent region has the pinch term is large. form exp( —2v~~b, r), where b r is the launcher-to-cutoff region separation. Provided the scale of the launcher is 1. Pellet injection fueling large enough ( ~4m b, r), most of the wave power will propagate into the plasma. The injection of high-speed frozen pellets of hydrogen The ion cyclotron frequency is to„=eB/m; (rads '). isotopes has been used to overcome these problems For a deuterium plasma and a field in the range 4 to 8 T, (Milora, 1981, 1989; Combs, 1993). The model of pellet co„=1.9—3.8X 10 (rad s '). There are a variety of reso- ablation which best matches the experimental measure- nances to which power may be coupled: (1) the harmon- ment of pellet penetration is the neutral-gas shielding ics of the ion cyclotron frequency, and (2) when two ion mode (Parks, Turnbull, and Foster, 1977). In this model, species are present, the hybrid frequency. the plasma heat Aux ablates molecular hydrogen from the Most of the applications to date have been directed at pellet surface. The cloud of hydrogen gas shields the pel- heating a minority H or He component in a deuterium let from the hot plasma and in effect increases the energy plasma. The region of power deposition depends upon required to remove a molecular from the surface from the fraction i/=nH/nD or n& /nD. For r/-0. 05 the the sublimation energy of 0.01 to -2 eV. The pellet power is absorbed at the minority cyclotron frequency, suffers little charge exchange, alleviating the wall bom- providing a very energetic minority component; this bardment problem. The rate of reduction of the pellet component then slows down in a manner similar to the radius (r~) is energetic ions in neutral injection, heating both electrons 6 dr 3.1X10 &/3~] 64r 2/3 —$/3 and ions. For large i/ ( ~0.1), near the ion hybrid reso- Be ~e r& m (2.82) nant layer, there is a competition between cyclotron at B damping on the minority species and conversion to Bern- where n, (m ) and T, (eV) are the local plasma parame- stein waves, which are absorbed subsequently by Landau ters, B, =3. 1 X 10 m is the molecular number density damping, again leading to both electron and ion heating. of solid deuterium, and m is the molecular mass. For Power has also been successfully coupled at the second T, = (0)(1 r/a) and n—=n, (0)(1 r/a) the pe—llet harmonic frequency. While experiments to data have T, , lifetime, for penetration to the plasma axis, is used loop antennae, a recent development, the folded waveguide, makes the waveguide possible for ' ' approach r=58X10 n s Be(0)n Te(0)' r ( ) (2.83) larger or higher-field devices (Owens, 1986). Ion cyclotron heating has been applied successfully to a wide num- The speed required to reach the axis is r/a The. ber of experiments. For example, ion and electron tem- response of the plasma density to the injection of a pellet peratures of 10 keV or more have been obtained in JET in the JET tokamak is shown in Fig. 48. (Tubbing et a/. , 1991). For use in a reactor, the pellet size should be small enough not to overperturb the plasma, e.g., —10% of the plasma particle content. For example, if R=7 m, a=2 f. Adiabatic compression m, b=4 In, B =2X10 m, N=4. 4X10 atoms. A pellet radius of about 5.5 min would yield a 10% density Major radius adiabatic compression was first demon- increase. The time to penetrate to the center assuming strated on ATC (Bol, 1975), and has been explored fur- T,(0)=20 keV, n, (0)=3X10 m would be 61 ps, re- ther on TFTR (Tait et al., 1985). It is not favored in quiring a pellet speed of 33 km/s. The present maximum tokamak reactors because of the constraints it imposes speed is 3 km/s without a sabot to protect the pellet dur- on the coils. However, it is of interest to FRCs; see Sec. ing acceleration. Speeds of 5 —10+km/s are expected for VII.E. systems using sabots. It seems unlikely that speeds cap- able of reaching the axis of large reactors will be I. Fueling developed. Nevertheless this technique can certainly lead to penetration to one-third to half of the radius (50 to In the earliest experiments, the plasma chamber was 75 % of the plasma volume) and in any circumstances to evacuated and then filled with the working gas to a densi- much reduced charge-exchange problems. ty somewhat less than that required for the plasma. Ion- ization of the gas fill plus gas evolved from the walls led 2. Compact toroid plasma injection to the final plasma density. The plasma density varied in time as the plasma came into equilibrium with the walls. An alternative approach to fueling using the injection Today, to control the density, a gas input with feedback of compact toroid plasmas has been proposed (Perkins from the line density is used. As discussed in Sec. II.Cx, et al. , 1987). Both plasma guns (Jarboe et a/. , 1980), and charge exchange of the gas on the plasma ions can lead to field-reversed pinch devices (Hoffmann, 1992) have been

Rev. Mod. Phys. , YoI. 66, No. 3, July )994 John Sheffield: Physics of magnetic fusion reactors suggested for the production of dense, high-beta net result of these motions is a displacement of the parti- plasmoids. Calculations of the energetics (Gouge et al., cle orbit from a Aux surface, and some complicated orbits 1988; Hoffman, 1992), suggest that the higher-beta, FRC which afFect transport. An additional vertical field is re- approach may be favored. quired to counteract the tendency of the plasma and plasma current to expand in major radius and provide an equilibrium. Such a current and vertical field preserve III. TAXONOMY OF TOROIDAL SYSTEMS the axisymmetry of the simple torus. The field lines circle the torus in such a way that their intersections on a poloidal plane form a set of nested magnetic surfaces, A. Introduction g= f ds Bz=constant, as illustrated in Fig. 2. The number times field line circles the torus in the The majority of work in magnetic fusion, directed to- of a toroidal direction for one transit in the poloidal direction wards power reactor development, is concentrated on the be relatively small number of closed magnetic configurations is called the safety factor q. The safety factor may over the the shown in Fig. 24. Work on open systems, such as the written in terms of an integral projection of radius linear pinch and mirror, has declined because of the surfaces on a poloidal plane, where the major coordinate is denoted by R: difhculty of blocking the end losses efIIiciently. Closed magnetic systems avoid the end losses by bending the field back on itself to form a torus, with average toroidal (3.1) field 8&. However, the curvature of the field, introduced by the bending, leads to a new rapid loss of plasma be- Here g and 8 are the angular variables the long and short cause charged particles of opposite sign drift apart paral- way around the torus. For a large-aspect-ratio circular lel to the axis of symmetry. The ensuing electric field plasma, drives the plasma outward radially, and there is no equilibrium. There are a number of ways of overcoming this r8~ problem. g— (3.2)

1. Axisymmetric systems Note that situations in which a field line closes on itself after an integer number of toroidal (n) and poloidal (m) The addition of a current, driven along the toroidal transits (q =n/m) can be unstable to perturbations. The field, causes the total field to twist owing to the poloidal Kruskal-Shafranov stability limit addresses the case in field (Bz) of the current (see Fig. 27 below), and the which q, =l at the plasma edge, for which case violent charged particles tend to follow the helical field lines. instability can occur. Consequently a particle that starts drifting up will soon A variety of systems use a plasma current to provide follow the helical field line down, and vice versa, thus equilibrium. They are distinguished by the range of q, in preventing the charges from separating. However, the which they operate to avoid crippling instabilities:

Coil Tokamak roId Mac aB q, ~3 typically, 8&- &(Bp, Stellarator Rq,

— Tokamak Increasing I+, j@ I+/~a' ~ Conventional low and high noncircularity ~ =b /a ~ 2. ~ except 8 /a (2, High Field B(4 ~ Spherical Torus Increasing Dominance ~ Second Stability of PF Reversed-Field Pinch (RFP) Higher Beta RFP

Increased OH since Compact Toroids typically R/a -6, B~ ~&~. B@ —B9 in Plasma Spheromak B+ —0 Outside Plasma Reduced Coil Fields I Field-Reserved Configuration (FRC)

I I FRC — I Be 0 Everywhere

I I L b Pulsed Plasma ——1, —large (e.g. , —6), DZP a Q

for reactor candidates. radial profiles FICz. 24. Representative toroidal systems with nested fIIux sur- 8&»B& Typical faces; stellarator, tokamak, reversed-field pinch, spheromak, of B& and 8 for representative configurations are shown field-reversed configuration, and dense Z pinch. in Fig. 25.

Rev. Mod. Phys. , Yol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1043

2. Nonaxisymmetric systems (3.3) An alterative approach is to produce the twist in the field lines by twisting the external coils —the stellarator. Such systems also have magnetic-Aux surfaces, but intro- As shown in Sec. V (Fig. 50), stellarators inhabit a wider duce a nonaxisymmetry into the field which modifies the range of 4-space than do tokamaks, because of their particle orbits and increases the role of the electric field greater ability to vary the form of their poloidal field. In in transport. general, the radial dependence of the safety factor is op- In stellarators it is commonplace to use the rotational posite to that in a tokamak [which rises center to edge, transform (4) rather than q to characterize the flux sur- q (0}& q (a)]. Stellarators can also be created with a heli- faces, cal axis. Representative configurations include

stellarator l =2, 3 alternate pairs of coils carry opposing currents, torsatron-heliotron l =2, 2 coils carry currents in the same direction, heliac I =1, helical axis.

In all cases the helical coils, vertical-field coils, and which, in certain conditions, can lead to rapid plasma poloidal-field coils can be modularized as nonplanar ejection —disruption —notably at the highest plasma toroidal coils. pressures and densities. In addition, a mechanism must be found to sustain it or the plasma will be pulsed. These 3. Strength and weaknesses two problems constrain the optimization of plasma per- The strength of the axisymmetric systems lies in the formance. use of a plasma current which provides relatively good The strength of the stellarators is their external control coupled values of confinement and beta. The mainline of the magnetic field. It allows planned optimization of tokamak has exhibited the best confinement, with low the plasma performance, in which the inherent steady- thermal di6'usivities and, in large devices such as JET, state nature of the field does not constrain the optimiza- energy confinement times of more than 1 s. The current tion. However, in the present configurations the coil is also a weakness because it is a source of free energy, geometry is more complex than that of the axisymmetric systems. Also, while thermal difFusivities in the present modest-scale facilities are comparable with those in similar-scale tokamaks, there is the concern that at lower collisionality the non-axisymmetric eFect will degrade confinement. An appealing concept, which has not yet received significant attention in reactor design, because of the lack of an experimental and theoretical basis, is to combine stellarator and. axisymmetric current-carrying features in

0 0 a single hybrid. One potential advantage is the suppres- 0 r 0 r sion of disruptions in tokamaks by the application of an - RFP FINITE TOKAMAK I ~0(1 r /a ) (AT P) helium demon- P=Oq =3, R/a=3 externally applied transform, as has been strated experimentally (Fussman et al. , 1979; Robinson, 1980}.

B. Magneto-hydrodynamics

On the macroscopic scale, the toroidal plasma may be treated as a conducting Quid; the behavior of this Quid and of the confining magnetic fields is described by the magnetohydrodynamic (MHD) equations:

0 0 Bp 0 a 0 I' +V pv=0, p=gmn, TORSATRON Bt ) R/a 7 t 0 3+ 0 65(/ )2 FRC (AT FINITE P) HELIOTRON (3.4) HELIAS R/a = 10, t = 0.7 conservation of mass, ATI3=0 a FICx. 25. +v-P ~=0, adiabatic law, (3.5) Typical radial profiles of B& (average B&) and Be for Bt pp representative toroidal systems.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1044 John Sheffield: Physics of magnetic fusion reactors

Bv Pof a 8R 1 +v.Vv =—Vp+ XB, Be(a, 9)= 1 ——ln ——cos8 . (3.10) p Bt j 2+a R a 2 (3.6) conservation of momentum, The plasma pressure is constant on a Aux surface; however, the magnetic pressure is nonuniform, being stronger Ohm's law, (3.7) E+v XB=gj, on the inside (g=n ) than on the outside (8=0). Conse- quently a vertical field B, must be provided to establish VXE= (3.8a) Bt an equilibrium. In early tokamaks this was provided by image currents in a conducting shell; nowadays it is pro- T.B=0, (3.8b) vided by coils. Shafranov (1966) showed that the re- V' X8=ppj, Maxwell's equations . (3.8c) quired field is In "ideal " the plasma is assumed to be perfectly PP 8R MHD, ——+p+— (3.11) conducting and the resistivity (g) in Eq. (3.7) is set to 4mR a 2 zero; the magnetic field is frozen into the plasma. In resistive MHD, the resistivity allows field lines to reor- where I; is the internal inductance per unit length of the ganize. plasma, typically I; 5 1. The poloidal beta is the ratio of the plasma pressure to the poloidal magnetic pressure, 1. Plasma equilibrium

In a stationary system with no plasma flow, Eqs. (3.6) 2po fpdV and (3.8) may be combined to give v(a', )„„„' (3.12) B2 V p+' (8 V)B. (3.9) where (, is the flux-surface average of the poloidal 2pp Po Be )s„„ field at the plasma edge. In the large-aspect-ratio approximation r/R « 1, ana- The equilibrium in an RFP is also provided by a con- lytic solutions may be obtained for the equilibria ducting shell and, on long time scales, by fields produced (Shafranov, 1966). The flux surfaces, assumed to be cir- by external coils. The equilibrium in an FRC is provided cles, have their centers displaced (6) from the magnetic by a conducting shell. axis (r=0); see Fig. 26. Analytical equilibria for ellipti- There is no exact, simple solution to the equilibrium of cal surfaces have been calculated by Freidberg (1982). If the toroidal stellarator, unlike the case for axisymmetric the poloidal Aux is due entirely to a plasma current, the systems. Approximate solutions may be found analyti- poloidal field at the plasma boundary (r =a) is cally by expansion of the magnetic-field properties about the magnetic axis. More exact numerical solutions may be found by looking for an energy minimum of the sys- tern when the plasma profile is perturbed. Such solutions TOTAL POLOIDAL FIELD must be found at the required beta because the IMAGE configuration evolves owing to plasma-driven currents, as &LZ J&Z CURRENT the pressure is raised. The Pfirsch-Schliiter current is given by 2 1 ()p PLASMA Jy — cosO . (3.13) & Bo dr SHELL PLASM It yields a vertical field B„which causes an outward shift of the magnetic axis (b ), given by APPLIED 8„-FIELD B„ JIL Z i/Z (3.14) S(&)&o An approximate equilibrium limit to beta in a stellarator may be defined by requiring the shift to be less than the minor radius (typically b =a/2 is used),

(A) P& —(j) X100% . (3.15) R FICi. 26. Plasma equilibrium may be provided by a conducting shell {a), or by external coils only {b). In either case, the re- In fact, higher beta can be achieved than this simple quired inward force {along A ) is due to the Lorentz force from model by careful adjustment of the configuration to plasma currents Rowing across the vertical field S„. reduce the current (Carreras et al. , 1983; Nuhrenberg

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1045 and Zille, 1986}. Since the equilibrium in a stellarator is and plasma currents represent significant sources of free established by the constant external fields, feedback sta- energy. Good reviews of this topic have been given by bilization is not required. Changes in the external fields, Bateman (1978), Wesson (1978), Schmidt (1979), and e.g., vertical field or quadrupole field, may be used to op- Freidberg (1982, 1987). timize the configuration for equilibrium, stability, and The energy principle may be used to determine which transport as the beta is raised (Carreras et al. , 1984). ideal MHD modes are linearly unstable (Bernstein et al. , 1958). A perturbation g is applied to the plasma equilib- 2. Plasma stability rium and an equation is derived for the change in energy The previous section discussed conditions for a of the plasma and magnetic field, 5 8' =5 8' &„, toroidal plasma to be in equilibrium. An important ques- +58,„&„,+58'„„„„.The major instabilities may be tion is whether this equilibrium is stable, for the plasma discussed in terms of

4 Vpo & = —d'x — V. I' ~i i.,m. +pa +ypo I J 2 f pp Po Bo,&o

Slow, shear fast, magnetoacoustic acoustic wave Alfven wave compressional wave

I e stabilizing terms

Jo &o + BOX)' —2(g' Vpo)(g. K) . (3.16) g22 B,

Kink free energy interchange or ballooning free energy

destabilizing terms

with Bi=VX(/Xylo) and where K=b.Vb is the curva- The magnetic field and plasma Quid are no longer tied ture vector of the equilibrium field and b=BO/go. The together, and the magnetic field can diff'use out of the sys- subscripts 0 and 1 refer to the equilibrium and perturbed tem. There is less restriction on plasma collective fields. An equilibrium is stable only if 5W+0 for all al- motions, particularly near rational surfaces, and this lowed displacements (g'}. The first term represents the leads to a modification of the ideal MHD instabilities. In energy required to bend the magnetic-field lines. The addition the field lines can, in e6ect, break and rejoin to second term corresponds to the energy needed to form a lower-energy state (Furth, Killeen, and Rosen- compress the magnetic field, while the third term bluth, 1963). The reconnected fields form islands, which represents the energy needed to compress the plasma. extend radially and lead to enhanced transport. The latter two terms are sometimes referred to as current-driven and pressure-driven destabilizing terms (Freidberg, 1982). C. Transport

Broadly speaking there are three ways in which the 3. Resistive modes charged particles that constitute a plasma can cross a magnetic field: (a) by virtue of collisions causing them to Combining Ohm's law, Eq. (3. and Maxwell's equa- 7}, difFuse across the field; (b) because of drift motion arising tions, Eq. (3.8), leads to an equation for the evolution of from gradients and curvature of the magnetic field; and magnetic perturbations, (c) because of electric fields. Such transport is termed 8 classical. When it includes the more complicated orbits =VX(vXB}+ Vi8 (3.17) Bt Po to be found in a toroidal configuration such as the tokamak or stellarator, it is called neoclassical. Owing to instabilities within the plasma, the transport may be in- motion of the plasma fluid slippage of the magnetic field creased above the classical or neoclassical level. In either with the magnetic field relative to the plasma the microscopic or the macroscopic domain, this may

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Ph ysics of magneticic fusion reactors

& COLLISIONAL occur because nontherermal Auctuatina ing electric field S RI'C ! ! (PF I RSC H —SC H LUTE R I ~ COL LISIONLESS "PLATEAU" set u or bbecauseu currrents are set u thRt locR11 Ic REGIME p up REGiME REGIME

ges to a y with ease. rme "anomalous. "

1. Diffusion

The characteristicc"c~~ 'cc diIu1 usion Auxilx ls glveI1 e if%sion coeKci t D — v ' coe o tx~t~~e particle.. Innt h'1scasc Axx particle from a fi ld ision frequenc . F FIG.. 28. Schematicic illustrationillu of th e dependencece ofo the neo- 1 t magnetic 6cld thc gyIQIRd ius of thec g x-p, ca If%sion coeKc1 o 1sj.on fre uen etween thhe dl~ere regimes occurrin'ng at 2 —Uth /gR. (3.18) orbit exceeds the For elcctron-1on coll 1s1ons v 1s given1v by Eq. (2.13). ere ra ius in the poloidal field (8s) is g o 2. Drift ' TThe fraction oof ccharargedRl ed particles exexeciltlIigxe bRI1R11R ol'- r . The banana diffusion In nonuniformform Inagnetic fie o a char ged e simple classical is s onl the basic h e1'ical motion not y of owever, for thisis term to be im o rifts arising from gra-ra- dcents0 an" curvature of thcthe magnetic field. one may systcms, thc Il Q c 1cal motion ' ' e 1 ms o the parameter nto g oroi al ' ad d a d f out in mayor rad ius. If R R ' (3.19) parallel to thee 6elde, its orbit V, pro]ected on the R- p Rccd c1I'clc Particles I' o 1on IIlay bc rcfIIccted b which i's the ratio of thee timetim to complete a t is conventional to defin ' Il 1Il c collls1'onal ' ' s ic radial1 excursion of R bRIlaIlR - c uter regiine, where bananana CQ'ects are unimportant aroun, the colhslonless or banan I'C p ateau re im ana CA'ects laay apeak role. A scschematic illus- p ' i usion coefticientcient is given in ig. 28.

PAR

~ Th errnal PARTICLE GUIDING——CENTER ORBIT 3. conduct ivity

zh The precedinng d'1SCUSS1on was centerede on particle Ex 8 82 OJECTION PARTICLE ORBIT VERTICAL DRIFT in terms of th e collision

FLUX URFACE

PROJECTION 0OF PART CENTER ORBIT on and Hazeltine (1976 has ) and Chang VERTiCAL DRIFT C i regard to imppurity and BY MOTION ALO NG HELICAL FIELD — as ect- . s 0okamak with T,e =T.T; Rlld FIG. 27. Trarapped-particlee- orb'ti s in a tokamak. , n ~T ", a simple

Rev. Mod. Phys. , Vol.ol. 66,66 No.N . 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1047 formula for the radially averaged thermal diffusivity may prove the particle orbits, and reduce transport as shown be derived for a reactor-grade D-T plasma (Sheffield, in Fig. 51 below. 1985}, The modifications to the ion thermal transport have 3/2 been calculated by Shaing (1984) for a torsatron and/or R — , (P& heliotron. The average compared to a tokamak is g,iv, -0.04 q 2 (m s ), (3.20) y;&, Tk given by where a =x'~ a and T„ is the temperature in kilovolts. 3.2X 103 0' ~ ~ m s Equating Eqs. (3.20) and (2.44) yields an expression for Xi' (XiNc }tok 6+ 2 2 & T, & (R /a)'f g the minimum fusion power that can be sustained against neoclassical ion losses, assuming that electron losses are (3.23) similar: where fz=p/Tk and p is the radial potential. For a (R/a)' (a /a)p carefully tailored magnetic Geld, such as in a helias I'F ~ 1.3X10, „ (b/a)5/3f LB' f configuration, the transport is similar to that with —2 —3 (see 51 Sec. 2 fF Fig. of V). For example, if —2 n, ( ) = 15 keV, R /a =7— and =2— we find X g (MW) . (3.21) Tk 10, fF 3, (XiXC )stell For example, for R /a=4, b/a=2, a~/a=1. 1, f~ =0.5, (XiNC )tok B =12 T, p „=4 MWm, f =08, nDT/n, =08, q=2, (T&)=15 keV, we find I'+~450 MW. Raising Tk while reducing the density provides more margin; howev- 7. Ripple transport er, a balance must be struck with the need to meet other objectives, such as achieving low impurity levels. A cross Even in nominally axisymmetric systems there can be check against MHD limits is needed to obtain consistent inhomogeneities owing to the use of discrete coils. In a values of T, n, and P. tokamak, the toroidal field ripple, resulting from the use of a finite number of toroidal coils, can lead to local trapping of a small fraction of particles, proportional to the 4. RFP thermal conductivity square root of the modulation. An excellent review of the subject has been given by Kovrizhnykh (1984). Neoclassical transport in a RFP is similar to that in a Increased losses of particles and heat occur both due to tokamak. However, the higher value of B&/Bo (much the ripple trapped-particle difFusion and due to asym- lower q, -0.2, compared to ~ 3 in a tokamak) leads to metry induced in the reAection points of toroidally trapped particles, owing to the toroidal field ripple's improved orbits, which reduce g;iv, (Boozer, 1983). causing incomplete compensation of the vertical drift. For large ripple, the losses can be severe, and 5. FRC thermal conductivity tokamaks are generally designed to have a toroidal field ripple e&=(B,„B;„)l(B,„—+B;„}~1 or 2% at the For this device, the classical thermal diffusivity is simi- plasma edge and ~ 0.1% on axis, to avoid problems. lar to that of a cylindrical system and A simple empirical formula has been derived to relate ') the major radius in the median plane to the ripple (5&), g;=0.02 3/2 (m s . (3.22) ) number of toroidal coils (X, ), and major and minor radii of the circular coil set, R, and a„respectively (Core, 1974; Sheffield and Gibson, 1975): 6. Stellarator thermal conductivity 2.6 sin/(, Rs =(R, +a, ) 1 —ln 1.1N, , (3.24) As in a tokamak, there are toroidally circulating parti- 26'y A „smk, p cles and particles trapped in the magnetic mirrors of the high-field region on the inside of the torus. In addition, where Ao=m[(R, —a, )/(R, +a, )], A, =Ao A„, and A„ is there are particles trapped in the helical field, which are the ratio of the coil cross section on the outside of the rejected from its local mirror regions. In contrast to ax- torus to the cross section in the bore of the toroidal as- isymmetric systems, there is an increasing particle trans- sembly. There are some limited expenmental data that port as collisionality decreases (see Fig. 51 of Sec. V). support the theoretical models of heat and particle loss The large level of transport would make it very dificult (Scott et al. , 1985). One unpleasant effect of the ripple to design an economic reactor were it not for the ability can be to concentrate an energetic particle Aux on the to tailor the magnetic geometry (Nuhrenberg and Zille, walls, causing damage. 1986, 1988; Mynick, Chu, and Boozer, 1982). Alterna- Direct loss of energetic alpha particles and alpha tively, large radial electric fields may induce rotation, im- power is a concern for devices such as ITER and reac-

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1048 John Sheffield: Physics of magnetic fusion reactors tors, both because of the loss of heating and because of where the latter term is a quasilinear expression valid for the potential for damage to the walls, as discussed in Sec. collisionless plasmas (Wooton, 1991). II.F. Recent studies of energetic-ion losses have been made in TFTR (Boivin et al. , 1990; Zweben et al. , 9. Weak turbulence 1983), and in JT-60U (Nishitani et a/. , 1993). In comparing theory and experiment for energetic-particle losses In weak-turbulence theory it is assumed that the electric-field in JT-60U it was necessary to allow for decorrelation time of fluctuations is short enough that effects and Alpha-particle losses in (Itoh Sanuki, 1991). they do not affect each other directly. In quasilinear ITER are discussed by Post et al. (1991). For represen- theory, contributions are truncated at first order, tative plasma profiles a variety of computations indicate = o+ ef„and a modification to the background a power loss of 1% for full-current operation in ITER. f f 5 comes from terms ( ). Saturation of the Auctuations The results are sensitive to the profile. f, P& occurs when the background equilibrium is modified sufficiently to limit the fluctuation level, e.g., flattening of 8. Anomalous transport a gradient. Dimensional analysis and/or mixing-length theory leads to a diffusion coefficient Unfortunately, the observed transport in all of the devices described above is anomalously higher, particularly D=h w (3.27) for the electrons, to the extent that, instead of the expect- where 5 is the mixing length and ~ is the decorrelation ed y, «y,. in the axisymmetric systems, y, ~g; and time. It is common practice to take 5-1/k~, the width — — g, (1 10)Xy,~,. Therefore ion neoclassical losses of the mode, and ~-1/y, where y is the linear growth (tokamak case) and electron neoclassical losses (stellara- rate of the mode. There are very few models solved tor case) can be used only as lower limits to set a analytically. In the case of resistive pressure-gradient- minimum reactor size for a given power and field. driven turbulence it was found analytically that Anomalously high transport in plasmas results from in- D =A y/k~, where the multiplier A depends logarith- stabilities that are driven by sources of free energy in the mically on the collisional dissipation. At the next level of plasma, e.g., pressure, density, temperature, resistivity, theory, nonlinear wave-particle and wave-wave interac- etc. gradients, which are inherent to the plasma tions are included, as discussed by Tsytovich (1970). configuration. Various reviews of plasma instabilities have been given, including those of Kadomtsev and Po- gutse (1970, 1971), Mikhailovskii (1974), Wesson (1978), Tang (1978), Rosenbluth and Rutherford (1981), Liewer )O. Strong turbulence (1985), Ross (1987), Romanelli (1989), Callen (1991), and Carreras (1992). While much of this discussion has been In strong turbulence the various modes are strongly in- in relation to tokamaks, nevertheless, the underlying teracting, and the level of each mode depends directly on mechanisms have wider application to toroidal systems. interactions with the other modes, rather than on a There has been only limited success in identifying partic- modification of the background conditions or weak inter- ular mechanisms as being credible candidates to explain mode interactions. Solutions require either the use of a transport in certain regions of toroidal devices. I am in- computer or the incorporation of strong, averaged effects cluding a brief review of this subject in the hope that, of the modes on each other —renormalization theory (see during the useful life of this paper, greater progress will for example, Kadomtsev, 1965). be made and it will be helpful to the reader. Substantial efforts are underway to relate theory and In a turbulent plasma the particle flux given in Eq. experiment (Ross, 1987; Callen, 1990,1991), but it is a (2.10), I =D„Bn/Br + V„n, must now be modified to in- difficult task because of the plethora of terms in the clude additional terms resulting from correlations of the transport equations. There is even debate on the correct various fluctuations. For example, when n =(n(r, r)) form for the transport equations to be used (Ross, 1990). +n, where ( ) denotes a time average and P=(P)+P, In practice, in predicting reactor performance it is com- there is an additional term involving (E(r, t)P(r', t')). monplace to use empirical values for D, V, and y, based Allowing also for Inagnetic fluctuations, we find upon experimental data, in combination with the neoclassical values. On occasion it is possible to replace or sup- &E,n & &Jiib„) (3.25) plement the empirical models with theoretical models for B~ eBp anomalous transport. Some progress has been made in characterizing the where B& is the toroidal field, and j~~ and b„are, respectively, the fluctuating parallel current and radial magnet- turbulence in the edge of the plasma by comparing fluc- ic fields. tuations in tokamaks, stellarators, and RFP's with vari- Additional fluxes in the power balances are ous theories (Wootton, 1991).It has been shown that the fluctuations in n, P, and T are sufficient to account for &ET& 3 &Em& b„ na —+ kT +~AU, —, (3.26) the anomalous edge transport in tokamaks and stellara- 2 Bo 2 Bo Bo tors at moderate temperatures ( —100 eV), while b enters

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fUsion reactors 1049

into the losses for RFP's. As for our understanding of turbulence in the core plasma, there has been less progress, owing to the greater difficulty of making fluctuation and correlation measurements. A wide range of instabilities remain candidates to explain the anomalous losses. Various models are classified in Table IX (Liewer, 1985). In resistive MHD, magnetic-field lines reconnect to form a lower-energy state (Furth, Killeen, and Rosen- bluth, 1963). The reconnected fields form islands in the magnetic field, as seen in a field-line plot in a poloidal plane; the islands extend the radial distance which electrons can transverse without a collision (V„=v~~Ã„/&). The instabilities start on rational surfaces at different radii, forming islands as indicated in Fig. 29. If the islands grow large enough, they may overlap, leading eventually to an ergodization of the field. This ergodization may be seen in the fuzziness of the field lines at intermediate radii in Figs. 29(c) and 29(d). Mode overlap is also shown in Fig. 42 of Sec. IV. As tangling of the field lines increases, the electron radial transport is enhanced. It is generally assumed that, unless tangling is extreme, the ions, with their greater natural excursions, are less FIG. 29. Field-line plots during a crash (simple sawtooth). affected. The average squared radial displacement Ar in Note the development and healing of stochastic regions (Ay- moving along a field line a distance s is given by demir et a/. , 1989). ((hr) ) =2sD, where

TABLE IX. Classification of some modes of turbulent transport, Liewer (198S).

RADIAL LOCALIZED MODES kl.a» 1 GLOBAL MODES kl-a - 1 SCAlE SIZE GENERALLY DIFFUSIVE TRANSPORT GENERALLY NQN-DIFFUSIVE TRANSPORT SOURCE OF ENERGY Vj

MODE AND Vo INSTABILITY DRIFT NAVE INSTABILITIES

ION IDEAL AND

TEMPERATURE R ESI STIVE GRADIENT INTERCHANGE

TRAPPED CQLLIS IONLESS COLL l 8 IO MAL MICROTEARING RIPPLING IDEAL AND KINK AND ELECTRON UNIVERSAL RESISTIVE TEARING BALLOONING

BASIC TWO-FLUID OR KINETIC EQUATION = DESCR IPTIQN IDEAL OR RESISTIVE MHD EQUATION

Rev. , Mod. Phys. , Vol. 66, No. 3, July 1994 1050 John Sheffield: Physics of magnetic fusion reactors

D = dz(b„(0)b„(z)) =L, b', . (3.28) 11. Low-frequency microscopic modes I0 Here b„=B„/Bo, bc= ( ~b„(0)~ ), and Lo is the parallel These modes have a frequency ro«co„(r0„=2X10 autocorrelation length of the radial magnetic fluctua- radss ' in a D-T plasma at 5 T). In one type of mode, tions. In the collisionless regime, A, , ))L,k, drift instabilities, plasma inhornogeneity leads to amplification of ion sound waves into two branches related to the drifts of electrons and ions, respectively. In the Xe Veam (3.29) case of drift modes, the density gradient provides the drift that produces the wave part of the instability. How- In the collisional regime, A, &(L,k, where I.k is the , ever, some other phenomenon is needed to break the adi- characteristic length over which two neighboring field abaticity and lead to transport. For example, trapped lines diverge, Rechester and Rosenbluth (1978) found electrons can provide this function and introduce a drive RR through the electron temperature gradient. These modes VD (3.30) are electrostatic at low in that no involvement of the k p, magnetic field is required. As p is raised (e.g., p& l%%uo), A simple formula for the enhanced losses has been given the fluctuation of'the pressure causes magnetic-field 6uc- by Callen et al. (1979), tuations, which can enhance transport. p3Q For p& m, /m;, the shear Alfven. mode can be driven rE, ~rE, 1 —4g (3.31) unstable, leading to microtearing, kinetic ballooning, and s drift-tearing modes. These modes are intrinsically elec- where r, and 5, are the radial position and width of each trornagnetic. island. Islands may also be caused by field errors, stemming 12. Electron drift-wave instability from an inadequately designed coil system, e.g., in current leads. In stellarators, it is possible to measure the For these modes, ki-p, —=c, /co„wher cthe sound islands directly using electron beams and detecting speed c, =(eT, /m, )'~ . The most unstable modes have screens. An example of calculated Aux surfaces for the kii «ki. The phase velocity satisfies the relationship ATF stellarator in the presence of field errors, in good u; «co/kii (u„where co=co, =keT, /eBL„, where the agreement with experimental measurements, is shown in density-gradient scale length is L„= n/(—dn/dr). The Fig. 30. Increased transport in the outer plasma region modes are electrostatic provided co ((k~ V~, where V~ is was observed, leading to peaked profiles, when the field the Alfven speed, V„=B/(pcnm; )'~ . The potential errors were uncompensated. fluctuation is eP/T, =n/n. There are three classes of instability, dependent on electron collisionality: (a) Collisional or dissipative drift instabilities, 0.4 v„(qR /u, ) & 1, temperature-gradient driven. (b) Collisionless instabilities, ( r /R ) ' (v„(qR /u, ) ( 1, curvature driven. When a polarization drift is included, the universal drift wave is obtained. 4 ~ e' r+ ~ ~ gf ~~ (c) Trapped-electron modes for v„=(r/R)' v, (qR/ 0.2— ~ V s'P~ u~ ) ( 1.

', *:. ' = 1/3 ISLAND . 13. ion drift waves E j OUTER FIELD "LINES ERGODIC 0— I::: in direc- g These modes propagate ihe ion diamagnetic LU tion; their phase velocity is ro/kii-u; «u, . The ion temperature-gradient mode is unstable when r/;=L„/ I ~ 1. This mode can also give rise to electron thermal r ~ ~ ~o+ ~~ee T; -4.2— 0 and particle transport. &= 1/2 ISLAND t

~ Qo ~ ~ 14. Trapped ion modes C3' —0.4 These occur when both electrons and ions are trapped. '1 .7 2.1 MAJOR RADIUS (m) 15. Electromagnetic drift-Alfven modes FIG. 30. Calculated Aux surface islands formed when a perturbation of 8, =68 G on axis is applied [6 kA in the perturbation These occur when p&m, /m, . Their poloidal mode coil for a 1 T helical field (Colchin et al., 1992)]. number is large (m ))1).

Rev. Mod. Phys. , Yol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1051

16. Rippling modes and their relatives date empirical formulas (Connor and Taylor, 1977). An extension of the use of invariance properties to determine These are electron drift waves that are obtained when formulas for plasma transport was made by Connor and the resistivity (g„)is large (typically at the edge). Similar Taylor (1984). They determined confinement scalings for modes may also be driven by impurity and ionization transport governed by the "reduced" resistive MHD gradients. equations for tokamaks and RFP's. The dimensionless parameters for a fusion plasma are 17. MHD modes R/a, a, 5, q, v„p/a, P, 2;, Z,s., n, /n;, T, /TI, P/T, V/v, I„v,h /c, ND, and A, „/a To make an analysis correct, There are two types of MHD mode with generally low the source terms for particles, momentum, and power poloidal model number and co/k~~ &&v„which can lead must scale appropriately to maintain similar solutions for to transport. the governing equations (Sheffield, 1990}. Kink modes are driven by the free energy in the po- The particle source term scales as nT'/ /a. If pellet loidal magnetic 6eld of the plasma current. injection is used, and a divertor suppresses neutral gas fu- Interchange modes are driven by unfavorable curva- eling, A, „/a may be neglected. The penetration distance ture in the magnetic field, VBVp&0. The modes are sta- for a pellet A~ is given by Eq. (2.82) and the pellet speed bilized by favorable curvature and by magnetic shear (v~ }. To keep A~/a constant and have the same source ~ ' dq /dr %0. term requires v B ', the pellet radius r B,and the pellet repetition rate f ~B, when we compare to D. Dimensionless plasma scaling plasmas at constant v„p, and p/a. The momentum source term R,„, scales as nT/a. Because of the limited progress in relating plasma The power density source (p) scales as gj (the ohmic theory to experiment, it is a common practice to use power density). In order to maintain the same local empirical confinement scalings based upon the wealth of values for T, /T, , it is necessary to maintain the same experimental confinement data. A concern with the scaled power separately, for electrons and ions: empirical approach is the lack of a theoretical basis for — l(p. t), +w' p, d l the prediction; what if some new physical mechanism . . =const. , 'N 'W should intercede between the limits of present operation (3.32) and the region of operations While reliance on — projected l(p..t }; p,.f = existing data can never, short of a full theoretical model, nj' const. guarantee performance in a new regime (i.e., different dimensionless parameters), there is a complementary ap- In practice, as pointed out by Kadomtsev, not all of the proach that can at least establish the performance of the dimensionless parameters need be held constant. The new plasma under some conditions. This approach, de- number of particles in a Debye sphere ND is not impor- scribed by Buckingham (1914) and Bridgeman (1937), tant unless, for example, V/v, I, is very large or we are uses dimensionless analysis to scale from one experiment concerned about the divertor sheath. The potential func- to another. Its use in scaling plasma and Quid data was tion PIT is related to other dimensionless parameters if proposed by Bickerton and London (1958, for pinch ex- V.E=O and the displacement current is negligible, i.e., periments), Beiser and Raab (1962), Kadomtsev (1975), phenomena such as lower hybrid or c/w, modes are and Bickerton (1989). It has been applied in one form or unimportant. The normalized fiow velocity V/v, &, another to the tokamak, stellarator, and RFP. n, /n;, and T, /T; are also invariant if the other parame- The set of equations that describe a physical system- ters are invariant and the source terms are scaled ap- in fusion, a reasonable picture is given by Maxwell's propriately. Relativistic effects may normally be ignored. equations, the Fokker-Planck equation, and the The change in the distribution functions when alpha multifiuid equations (Braginskii, 1965)—may be written heating is involved may lead to differences; Bickerton in terms of dimensionless parameters. A set of scaling (1989) showed that to achieve similarity among a family laws may be established in which the dimensionless com- of burning plasma devices we should adjust the isotopic binations of the physical parameters remain constant mix. Assuming that alpha power may be treated like when the quantities are changed in scale. A corollary to another power source, the key dimensionless parameters this approach is the use of dimensionless vali- scaling to for fixed R /a, ~, 5, q, A;, and Z,& are I qRn the collisionality v ~ T2 g 1/2T1/2 the normalized gyroradius + ~ a aB (3. nT 33) the beta ~ P g2

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y 3/2p and the normalized power density —-2 ~ gj BR2 Combining these parameters, we find in comparing two devices that similar conditions will hold if ' O. S 0.4 1.6 0.6 Q2 T2 B2 n, B2 I'2 2 ai B2 Ti Bi n, B, I'i BI 0.2 ' 0.2 (3.34) Bi (nT&)2 B2 I2 &a2 1 ' ' 82 (nTr)i Bi I, &Bi B2

I The thermal diffusivity may be written in the general able time, an economical reactor will be at least 1 GW(e) form y-Dz(p/a) F.(v, t3, R /a, q, . . . ), where Dz in output and more likely -2 GW(e) (Rebut et al. , —T/B is the Bohm diffusion coefficient, which is charac- 1990). This large size may be the best route, not just for teristic of processes having large scale lengths. For a =0, tokamaks, because of the potential need to operate at the losses have a Bohm-like scahng. For a=1, the moderate power density ( S4 MMm neutron fiux) in characteristic scale of the loss processes is a gyroradius, order not to overstress the system. The alternative and the losses have a so-called gyro-Bohm scaling. route —higher power density, smaller size, ~ 1 GW(e) —requires either very high coil fields (15—20 T) or much better physics performance, that is, higher ((P))/gz coupled with optimum divertor conditions IV. TOKAMAK REACTORS and low recirculating power. ITER is planned to provide a demonstration of a con- A. Introduction trolled, burning D-T plasma for times of 1000 s or more, at a reactor level of fusion power ( —1500 MW) (Rebut, Numerous design studies have been made of tokamak 1993). ITER will also demonstrate and integrate fusion reactors, including those of Badger et al. (1979), Baker reactor technologies. Reference parameters are given in et al. (1980), Hancox et al. (1981), Cooke et al. (1989), Table X. The Tokamak Physics Experiment (TPX; Najmabadi et al. (1991), and Seki et al. (1991). The pa- Schmidt et al. , 1993; and Thomassen et al. , 1993) is be- rameters are listed in Table X. These studies and those ing designed in the U.S. program to help optimize a of other configurations have been useful in refining our tokamak reactor by studying steady-state, advanced understanding of what is needed to make an "attractive (improved-transport, higher-beta, lower-recirculating- reactor. " Based on present achievements, with a margin power) modes of operation. TPX parameters are given in allowed to ensure divertor and wall survival for a reason- Table XI.

TABLE X. Parameters of recent tokamak reactor designs.

Reactor 1 ARIES 1 SSTR ARIES II Reactor Cooke et al., Najmabadi et al., Seki et al., Najmabadi et al., reference 1989 1991 1991 1992

Major radius R (m) 7.07 6.52 7.00 5.5 Aspect ratio R/a 3.49 4.5 4.0 4.0 Ellipticity ~ 2.0 1.6 1.8 1.4 Current I (MA) 22.4 12.0 12.0 5.5 Field S& (T) 6.44 13.0 9.0 7.5 Field on coil 8 (T) 21.0 16.5 $16 Con6nement factor CH S2 3

Troyon factor g(%%u& Tm/MA) 3.0 3.2 3.0 6.7 Current drive Neutral beam Fast wave Neutral beam ICRF Bootstrap fraction 0.3 0.53 0.75 0.95 Burn time (s) 10' p„„(MWm ) 3.1 2.8 5.0 (peak) 3.5 Divertor Double null Double null Single null Single null Total thermal power MW (th) 3710 Net electric power MW (e) 1080

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TABLE XI. Representative modern tokamak parameters. Tore Asdex Tokamak DIII-D TFTR JET Supra U T-15 JT-60U Triam U TPX Major radius R (m) 1.65 2.45 3.10 2.25 1.65 2.43 3.40 0.80 2.25 Aspect ratio R/a 2.75 2.90 2.82 3.75 3.3 3.5 4.0 5.0 4.5 Ellipticity ~ -2.0 1.0 1.8 1.0 1.6 1.0 1.6 1.5 2.0 Current I (MA) 2.0 3.0 6.0 2.0 2.0 1.4(2.0) 6.0 0.1 —+0.5 2.0 Field Bp (T) 2.0 5.2 3.4 4.5 4.0 3.5(5.0) 4.2 8.0 4.0 Pulse length (s) 10+ 2 20+ 5 600 10+ 1.5+ 20+ 100~ cc Divertor Yes No Yes No Yes No Yes Yes Yes Coil type' CU CQ CU S/C Cu S/C CU S/C S/C 'Cu =copper; S/C =superconducting.

B. Tokamak characteristics generated impurities from entering the plasma (see Sec. II.G). The torus walls, blanket, and shield must have There are numerous reviews of tokamak research, for sufhcient electrical resistance to avoid drawing large example, those of Artsimovich (1972), Furth (1975), currents during plasma current initiation, but must also Rawls et al. (1979), Sheffield (1981), Kadomtsev et al. act as a metal shell to provide image currents, which will (1990), and Post et al. (1991). The parameters of some stabilize some MHD modes. representative modern tokamaks are listed in Table XI. in tokamak area is well illustrated the Success the by plot 1. Coils of progress in achieving good confinement N;SET; vs T; in Fig. 31. The drawing of the ARIES-1 tokamak reactor To date, most tokamaks have used water-cooled or in Fig. 32 illustrates the basic features of a modern liquid-nitrogen-cooled copper coils, though a few super- tokamak. The plasma is produced in a toroidal vacuum conducting coil tokamaks are operating, namely, Tore chamber (torus) of elliptical cross section, which, in a Supra, T-15, and TRIAM (Komarek et al. , 1990). Ac- reactor, will be made of a low-activation material. cording to present knowledge of the physics capabilities, While tests are being made of a pumped limiter, recent superconducting coils seem to be essential for tokamak engineering test reactors and reactor design use a po- reactors with R /a ~ 3, to provide an acceptably low re- loidal divertor to remove helium ash and limit wall- circulating power. As a consequence, the blanket shield thickness on the inner bore of the plasma must be 6-1—1.5 m thick to limit radiation damage to the coil insulation and conductor and overheating of the coils. Reactor ', :::::" Since the dominant field in a tokamak is provided the 100 ,Orldltloll8 j:;:;:::, by YEAR toroidal field, this sets a limit on the field ratio, Eq. (2.40), :::.::'' lgnitiorI, ':, aw I 5 a CD I fs= 1— +- 10 ~-':::::::-::./!::.':-:'::.'::. a a R T "JET .:: —1991 To/' ~ ~ E I =4 ED TFTR For example, let (a„/a)=1.1, Pz+3500 MW, p OJ /JET ~ I C) ~ then a=3.38 ' and we ~ I MWm, b/a=2; (R/a) m, :: . I '::::::::;,,:f.pg:.":Q:.:'.::::.::::::::::Q'DT-0.1 1 I find for 5=1.5 m: I-D':::::i:::::Diil I I— D I UJ I-D I I R/a 2.5, 3, 3.5, 4, 4.5, 5, Reactor relevant I Conditions —19SO e ~ ASDEX Hot )on Mode I 0.28, 0.38, 0.45, 0.50, 0.55, 0.58 . ~~ 0.10 ALC-A/ I fs ASDEX'~ I O PLT I CC / 0 ! The decrease in useful field affects the gains in beta of go- ( T10 ~ I ing to a lower aspect ratio, and drives tokamak reactor O / ~ I I designers to higher aspect ratios than those used in most ~ 0.01 TFR, I LL experiments, which do not need a shield. —1970 I An alternative solution (Peng and Hicks, 1990) is to T3 I / 0 D—T Exp I dispense with the blanket and shield in the torus bore, JG92.12 I I 1965 and use a solid copper central toroidal conductor and an 0.1 1 10 100 aspect ratio R/a 5 1.5. This approach has the potential CENTRAL ION TEMPERATURE Tl (keV) to achieve a high-mass power density, but at the expense FICs. 31. Performance of tokamaks, JET Team, 1992. of a higher recirculating power and regular replacement

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors

STATIC ROTATING SYSTEM SYSTEM

RECTIFIERS =-

I l POLOIDAL TOROIDAL COILS COILS I

I I EQUILIBRIUM MAGNETISING COILS COIL I

FIG. 32. Drawing of representative tokamak showing the principal features. ADD ITIONAL HEATING L ~+~~ PLASMA ' ING

OIDAL RTORS

UM DUCT

P RIMARY

D IAG NOSTI CS ADJUST DATA ANALYS IS DATA ACQUISITION OP ERATING AND M ONITO R ING COND ITIONS INTERP RETATION CONTROL SYSTEM

of the central coil limb. the plasma it is common practice to work in Aux coordi- The magnetic field has, in essence, three components: nates in order to accommodate the efFects of the chang- a strong toroidal field, poloidal fields produced by the ing Aux geometry. plasma current, and a set of poloidal coils. An approximate formula for Pe is The poloidal coils perform four functions: (1) a 0.375a [I+a (1+25 —1.25 )](nq (T solenoid on the inner bore of the plasma produces the o+kTk) plasma current by transformer action; (2) a second set of [I(MA) ] coils provides a (mainly) vertical field 8„,which provides (4.1) some volt-seconds for driving the plasma current, interacts with the plasma current, and balances the out- where A. =i/a is the plasma ellipticity and 5 is a measure ward expansion of the plasma and plasma current (see of the triangularity of the plasma fIux surfaces. Sec. III.B). In addition (3) the coils shape the plasma and With allowance for plasma shaping and toroidal provide stability against a bulk vertical shift of the plas- e6'ects, the safety factor at the 95% Aux surface is ma curving the field lines. They may also be used (4) by 2 5a B [I+A. (1+25 —1.25 )](1.17—0.65m) to produce a field opposing the poloidal field of the plas- ~~5 = IR(1—e ) ma current, forming a separatrix and diverting Aux to a divertor region. In practice, in optimized designs, the (4.2) smallest number of coils is used to provide all of the where @=a /R, a, R, (m), B (T), and I (MA). above functions in a time-dependent way, by careful pro- For a system with a poloidal divertor, q&, ~ ao, and to gramming of the current in each coil. define plasma characteristics it is normal practice to use The fields of the plasma current and coil set produce a q9&, the safety factor at the Aux surface, which includes set of nested Aux surfaces. As the plasma pressure is 95% of the toroidal flux in the plasma. raised, the poloidal beta [(Ps), Eq. (3.12)] increases and the plasma attempts to shift outward in major radius. 2. Startup An increase of the vertical field is used to hold the position of the outer Aux surface, but the center of the plasma The first step is to evacuate the torus to —10 torr. moves out, as shown in Fig. 26. In computer models of To ensure a low residual impurity gas level, the wall ma-

Rev. Mod. Phys. , Yol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1 055 terials will be run at elevated temperatures, e.g., The plasma resistivity is a result of collisions between 300—500'C. The torus is then prefilled to —10 torr the electrons and ions in the plasma. The resistivity with 0-T fuel, which will lead to an initial density of parallel to the magnetic field is given by Book (1990, p. ~ 10' rn of D-T plasma. The toroidal field is already 37) as established. A voltage (U&) is then induced around the 5.2X 10 lnA, torus transformer action, leading to breakdown. y~Z, ff by "Ill ohm m (4.5) Preionization by electron cyclotron heating, and auxili- T ary heating in general, may be used to lower the loop ' where =(1—1.95&r/R +0.95r/R) is an increase voltage and reduce the usage of transfer volt-seconds in yz in resistivity owing to the effects of electrons trapped by raising the plasma current (Gilgenbach et al. , 1980; the increasing field on the inner side of the torus. A, is Lloyd, 1991). The breakdown voltage depends on the (2.13) and (2.9). We can calcu- electron loss channels, toroidal drift, transverse diffusion, given by Eq. Z,s by Eq. electron tempera- recombination and attachment, and poloidal stray mag- late the plasma resistance for a given constant and lnA, and an netic fields (Bi). If the stray magnetic field dominates ture profile. Assuming Z,ff average at r =a /2); for R /a -2.5 —4 an empir- (Bi ) 10 B&), the requirement on loop voltage is y~ [(say, ical fit is yz —-4. 3 —0.6R /a (Uckan et al., 1988)],we find ~ B~ U& 1.3X10 2mRp/In 530pa (V), gZ ff lnA (4.6) Bj RP (Ohins) Q KTe0 where p is measured in torr, and R and a are measured in meters. With electron cyclotron heating assistance, a for a parabolic temperature profile and a noncircular toroidal electric field E&-—0.3 V/m is sufficient to initiate plasma, a=b/a. Combining Eqs. (4.6) and (2.8) yields breakdown and start the plasma current. the maximum Aat-top current duration, Care must be exercised during current ramp-up to al- [1.3$„—( —1.5L I))a tr T,~~ low current penetration into ihe conducting plasma; oth- (4.7) erwise a hollow current profile may develop, which is un- 2.6X 10 Ry~Z, ff lnA, I stable to the double tearing-mode instability (Furth, Rutherford, and Selberg, 1973). In a reactor the main For present-day experiments this time is relatively concerns are waste of volt-seconds (for a pulsed reactor), short ( 860 s). However, for a large reactor operating at wall bombardment during instability, leading to damage high temperature, e.g., T,0=20 keV, it can be —1 hour. and impurity generation, and the need to establish a Consider the following case: R = 8 m, a =2.0 m, ~=2 m, route to the desired current profile to support an opti- R„=3.4 m& y~ =1.9& Z,ff=1.8 m& lnA, =17& I=17 MA mized plasma. (q95 =3.3) a, /a =3, Bo =6.5 T, B„=20T, 1.3$„=950 V s., 1.5L~I = 350, we find t~ =6 600 (s). 3. Plasma duration Optimization to a higher aspect ratio would allow even more room for the solenoid. Pulsed operation is not If transformer volt-seconds were the only source of desirable, because of thermal stresses caused by the cy- current drive, the pulse length of the tokamak would be cling and the need for heat reservoirs to maintain the limited. Denoting by tz the time to raise the current, blanket and wall temperature, as well as electricity out- and by t~ the maximum duration of the plasma at peak put. Nevertheless, it is a tradeoff issue to be balanced current, we find that the flux (volt-seconds) required to against the cost and complexity of providing steady-state support the current is given by operation with noninductive current drive. One natural t~+ feature of the tokamak can help to alleviate this U, dt =p =L I+ R Idt+R ItF (Vs), problem —the bootstrap current. J0 J0 (4.4) where L =poR [1n(a, /a )+0.25] (H) is the plasma induc- 4. Bootstrap current tance and R~ (0) its resistance, while a, and a are the average radii of the poloidal coils and plasma, respective- This current appears in the neoclassical theory of ly. In tokamaks with unassisted startup, the second term transport through the equation for conservation of is typically -0.5 I I, which is the Aux used up in over- momentum (Galeev and Sagdeev, 1968; Bickerton et al. , corning losses during startup. In assisted startup, some 1971; Galeev, 1971). It is driven by the radial pressure of this fiux may be saved (Lloyd, 1991). gradient and depends on details of the particle orbits, The available flux (volt-seconds) is typically 1.3 times notably the fraction of trapped particles [f, ~ (R/R) ]. that given by the central solenoid (P„=mR„bB„), A "seed" current must still be driven on axis where there where R „is the radius of the solenoid and b,B„(T)is the is no pressure gradient. A theory applicable for all col- full swing in the solenoid. The extra 30% comes from lisionality regimes has been given by Hirshrnan and Sig- other poloidal coils, whose Aux links the plasma. mar (1981), showing the detailed dependence of the

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1056 John Sheffield: Physics of magnetic fusion reactors current on the temperature and density gradients. The existence of the current has been inferred in tokamaks by x = 1.75, 5 = 0.50 analysis of contributions to the at po- ~ = Po(~ —V) loop voltage high A=2 loidal beta (Zarnstorff et a/ .1988; Challis et a/. 1989; ioh=ig(~ -V ) , , =3. Kikuchi et a/. , 1990; Navratil et a/. , 1991), but for times q. 0qo less than a skin time. Studies for ITER (Fujisawa, 1990a, 1990b) show that the bootstrap current depends on the current profile, owing to changes in the areas of the vari- 1.0— , = 0.95') ous Aux surfaces. A simple empirical formula valid for 1.0&q&(95)/q&(0)&5. 0, which may be used to give a rough global estimate of the fraction of bootstrap current, is UNST 0.5 I.3 0.5— TABLE IbS = CbS (4.8) where

q~(95) q~(95) I = 1.32 —0.235 +0.0185 2 Cb, 0 0 qg q~ qo and be represented (4.1). Pe may by Eq. FICx. 34. Values of eP~ and qo required for stable operation at For cases with high bootstrap current, where the de- high bootstrap current, for representative aspect ratio (Ramos, tailed distribution of current can affect MHD stability, it 1991). is necessary to use the complete description given by Hirshman and Sigmar (1981)to properly account for the effects of the temperature and density gradients. range of 5 —10, i.e., lower current than normal, to obtain The recent results from TFTR (Navratil et a/. , 1991) high In addition this would facilitate operating with support the idea that a valuable mode of operation for a Ps. q&(0) ~ 2 (Ramos, 1991), which has been predicted to al- reactor might be at relatively high q&(95), e.g., q in the low access to a second region of stability and higher beta (Coppi et a/. , 1979; Mercier, 1979). Recent results from D-III D, which show relatively improved transport as q& is raised (Jackson et a/. , 1991) are encouraging for this mode of operation. The question is what self-consistent optimized level of beta, transport, and bootstrap current can be achieved (without impurity problems). Examples of model profiles for pressure and current, which lead to very high bootstrap current ()90%) in the second stable region, have been developed by Ramos (1991), but without a self-consistent transport model. An example of the required profiles is given in Fig. 33. Note the zero bootstrap current on axis (r/A ~0), which means that some sort of noninductive current drive will be required to complete and control a desirable profile. In Fig. 34, values of @Pe are given for stable operation as a function of q (0). Advanced modes of operation, such as those described above, are being studied in present-day, 1.00 moderate-pulse-length tokamaks and will be tested in 10 steady-state operation in TPX (Schmidt et a/. , 1993; 0.98 Thomassen et a/. , 1993).

RBy 0.96 C. Noninductive current drive

0.94 Numerous techniques have been proposed and tested for driving a plasma current noninductively; for recent FIG. 33. Reference plasma parameter profiles, in Aux coordi- reviews see Uckan (1985), Fisch (1987), Post et a/. (1991), nates, for high-bootstrap-current operation (Ramos, 1991). and Moreau et a/. (1991). These techniques rely on

Rev. Mod. Phys. , Yol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1057 transferring axial momentum to one of the charge species along the beam path in the plasma. n20 is the line aver- in the plasma in a way that leads to a net drift asymmetry age density, in the parallel velocity distribution function, without producing a compensating e6'ect on the other charge species. 0.5 A current-drive figure of merit is commonly used to T10 lnA J =23.7+in. AbEb characterize the various techniques, (A +Ab) "20 PcD(10 A/ W} n 20Ic R I I c (4.9) To obtain adequate penetration for reactor plasmas, i.e., In neutral-beam driue, the ions formed when a neutral to provide current drive on axis, requires MeV-level neu- beam is ionized the by plasma constitute a current. In a tral beams. In ITER, a=2 m, n —10 m, 1.3-MeV uniform, pure hydrogen plasma, this current is exactly deuterium beams are optimum to provide an on-axis compensated by an electron shielding current, created as current drive, but have only a small percentage of shine- electrons gain momentum from the beam ions and them- ~ through on the far wall, b 0.05. selves slow down on the background In f, plasma. a Radio-frequency (rf) power may be injected so as to tokamak, the presence of trapped electrons and impuri- transfer axial momentum to one of the charge species. ties affects the momentum transfer, and complete com- One scheme, that of Fisch (1980), relies on the Landau pensation no longer occurs (Ohkawa, 1970; Cordey, damping of high-phase-velocity waves launched in one Jones et al., 1979; Hirshman, 1980}: direction parallel to the magnetic 6eld. There are three Zb main approaches to current drive: JcD =Jb 1 — [1—6 (Zese)] (A m ) (4.10) (i) High-speed waves with phase velocities that are Zeff several times the electron thermal speed. where the subscript b refers to beam ions, and @=r /R (ii) Low-speed waves with subthermal phase speeds, A model for neutral-beam current drive incorporating which have the most momentum per unit energy —for approximate corrections —(r /R ) to the trapped-electron example, the fast wave [i.e., compressional Alfven wave contribution has been developed by Mikkelsen and Singer (CAW}] at low frequencies and the magnetosonic wave (1983) for a tokamak plasma, taking into account profile above the ion cyclotron frequency. e6'ects: (iii) Selective heating, which creates an anisotropic particle distribution and indirectly drives current. Examples ' are ion and electron cyclotron heating. ICD=Ib 1 — [1—(1.55+0.85/Z, s)c e8' A normalized efficiency factor J/P may be used to compare current-drive schemes, as shown in Fig. 35 (Uckan, 1985), +(0.22+1.55Z,a)e] '(A) r J=neV d (Am ), (4.13) e :Ib ZbE (Zgff y Zb p ) (4.11) which is a reasonable approximation in the banana regime of collisionality for e 0.2 and 1.0 & Z,z & 3.0. For deuteron beam energies —1 MeV, the ef6ciency for neutral-beam drive is given approximately by

R tang 1 ND (5AbdT10)(1 fsb } R )p2 &CL X [5J(x,y)]I' (Z,ff, Zb F), . (4.12) where the following notations are used: F=a/2R0 J(x,y)=x /[4+3y+x (x+1.39+0.61y )]; x = (Eb /Bbd E, ) ', y =Z2/3, and E, is gimme~ by Eq. (2.20). R „„s/R0 is the pitch angle (cosine of the angle of the beam ions to the B field). &po f»-—exp[ (n20db)[0. 77—5/Eb (MeV)] ] is the fraction of the beam that penetrates the plasma and reaches lp the far wall, for deuteron beams at energies —1 MeV. Uo Z2 = [QJ. (1nAJnjz~ /Ab AJnJZ /A = 12Z, ]/[QJ ] s/ FIG. 35. Normalized current-drive efBciency factor vs 5 Ab when n ~ 0.9. DT/n, Uo =(co —leo, ) /k v, with co„ the cyclotron frequency for — 08 and — are ~~ Abd 0. Sbd 1.0 Stting factors. waves, and Uo= Vb/v„ for particle injection at Vb (Uckan, db-—2R0[(1+a/R0) —(R„„s/R0) ] is the distance 1985).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1058 John Sheffield: Physics of magnetic fusion reactors where V„& is the relative velocity of the ion and electron fast-wave current drive on the electrons. It has the ad- species. vantage of higher efficiency and avoids ion cyclotron res- The power to maintain the current is onances within the plasma. It concentrates current drive near the plasma center, neatly complementing the I' =mnvV„) (Wm ) . bootstrap current. Optimized computer figures of merit The normalized values are have P 2 &Te) 102o nev, rnenve vp 7 FWCD (2+Z,a ) (4.14) 10 keV m W J ~e~o 1 (4.16) P mv Up where v, is the electron thermal speed, vp=ne lnA, / depending on the sophistication of the calculation, with a (2m.mom, v, ), v is the collision frequency, Uo = /U, and V„, multiplier 2 0.5 or the ITER conceptual design vo/v is a function of Uo. Various rf heating schemes are (Batchelor et al. , 1990). Further refinements are needed discussed in Sec. II.G. to include, more fully, toroidal effects and relativistic For lower hybrid current drive the figure of merit is efFects, which can limit efFiciency. given Ehst and Karney by (1990), Electron cyclotron current drive is mainly considered )'LH= xo—F(»Z.n)G «x), (4.15) for plasma initiation and for detailed profile control to stabilize MHD modes. The drive relies on creating an where yp is the "Fisch-Boozer" efFiciency, I =1.5 —1.9 is asymmetric plasma resistivity by preferentially heating a temperature correction term, and 6-0.7 is a correc- the electrons that are moving in one direction along the tion to allow for trapped electrons: magnetic field so that they collide with the ions less than those moving the other way (Fisch and Boozer, 1980; 2@1e 6'pc 4 Cohen, 1987). The efficiency may be comparable with e lnA, (5+Z,s ) that of other techniques (see Fig. 35), but the generally

2 higher cost of electron cyclotron systems compared to 1 1 + ln )]max radio-frequency systems has restricted their considera- 2 2 2 de- +[[min +]]max +J]min tion to that of meeting high leverage needs such as tailed profile control. where X~~,„and %~I,„are minimum- and maximum- Oscillating geld current driue involves applying audio- wave parallel refractive indices in the local lower- frequency oscillating voltages to the toroidal and poloidal hybrid-wave spectrum. The minimum value is set by the circuits in the appropriate phase (5=m/2) to drive a dc accessibility limit, toroidal current in the plasma (Bevir and Cxray, 1980). This technique has been considered primarily in relation W e W e w + 1+ to RFPs and is discussed in Sec. VI.E. /f, acc Wce wee J. w a function mainly of n/8 . The maximum penetration penetra- value is set by Landau damping, which limits D. Tokamak transport tion,

Lt maxTe max ~~, An excellent review of tokamak and stellarator trans- (for ITER conditions 60—90 keV). The two restrictions port has been given by Wagner and Stroh (1993). The lead in effect to be a beta limit for lower hybrid penetra- neoclassical theory of tokamak transport is mentioned in tion. To obtain maximum penetration requires a narrow Sec. III.C. A complete calculation of the neoclassical spectrum, e.g., hN~~/X~~, „~0.1. Assuming a reactor transport coefficients given by Hinton and Hazeltine has conditions similar to or more restrictive than ITER, (1976) has been updated by Hirshman and Sigmar (1981) this limits lower-hybrid-wave current drive to low- and Change and Hinton (1982,1986) with regard to im- density startup and in the full-density plasma to heating purity and aspect-ratio effects: and current drive in the outer part of the plasma, ~ r/a 0.6. 3/2 Ion cyclotron waves can drive current by damping fast R 2Te Ale 2 2 (m's (4.17) waves on either electrons or ions and at a variety of tem- 0 Ps~es ", peratures, at low frequency (below the tritium ion cyclotron resonance), between the second harmonics of tritium and deuterium, and above the third deuterium harmonic. where p, =1.02X10 (T, A;) «/B (m), v„ is given by The optimum scenario is calculated to be low-frequency Eq. (2.13), A; is the atomic mass, and

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 3 059

— *'i kz = [ [0.66(1+2.54a„,)+( l. 88e'i 1.54e)(1+2.29a„,) ]/[1+ 1.03p, +0.31@,*)] I eff

+[1.17[1+4.38a„,(1+0.3a«) j(1+1.17a„,)]p; e /[1+0.74@,*e ]] Zeff

Here e=r/R, a„,=n; /n„and This may be combined with Eq. (2.27), the power-balance equation (2. and Ohmic power (4. ~ 22), Pn=R&I, Eq. 6), p,'=[1+2.54a«]v„qR (T, /T;) /(&2e U, ) . to eliminate T„yielding an approximate formula for = ~E The second term in k2 is the Pfirsch-Schluter contribu- (assuming T; /T, constant, and ignoring additional tion. For reactor conditions, the first term dominates ex- terms)

cept, possibly, in the scrape-off layer (T, -0.1 keV). o(- — 2.4 0.2 1. —0.3 ~E n R a q 2g (4.20) Theoretically, the neoclassical electron losses are much less [(m, jm, )'~ less]. As pointed out in Sec. III.C, While the agreement is tantalizingly good, the experi- neoclassical ion losses are usually small, but they should mental evidence remains equivocal as to the cause of core not be neglected at large (R /a), large (P), and low tem- transport (Callen, 1991; Wooton, 1991). Nevertheless, perature; see Eq. (3.20). the analysis shows that even diffusive transport does not If neoclassical losses represented the only mechanism, have to scale as a, and the explicit dependence on field then, provided impurity levels were low, it would be may be weak, even though gE ~ 8 straightforward to provide tokamak reactor-grade plas- Following the application of substantial auxiliary heat- mas at modest fusion power levels, i.e., a few hundred ing (P, ))Pn },it was discovered on ISX-B (Swain, 1981) megawatts. Unfortunately, transport is much higher, that confinement degraded with increasing power but in- owing to instabilities that generate Quctuating electric creased with current, rz ~ (I /P ). Again, there and magnetic fields. Anomalous transport mechanisms is a similarity to the collisionless trapped-electron are also discussed in Sec. III C. The transport in tokamaks is often discussed in terms of three radial regions —the edge, where atomic processes are impor- ppp - P [kW] tant; the center, q ~ 1, where sawtooth MHD activity dominates; and the high-gradient region in between, which provides most of the confinement. This is illustrated in Fig. 36. To date, while some progress has been made in characterizing edge transport (Wootton, 1991),there is no clear connection between theory and experiment that would allow the accurate projection of reactor plasma parame- 400 ters. As a consequence, reliance is placed on empirical models of transport obtained by regression analysis of data from a wide range of experiments (Reidel, 1990). Early empirical formulas were derived for Ohmically heated plasmas (Cohn et al. , 1976; Pfeiffer and Waltz, 1979). The most striking dependences of the formulas are (a} a strong dependence on density, (b) no explicit dependence on field (though the maximum density attain- 200 able increases with field), and (c) a greater than quadratic dependence on the plasma scale. The so-called neo- Alcator formula (Parker et al. , 1985) is commonly used:

roH 0 07nzoR aq, (s) (4.18) Variants of this formula add dependences on A; and ~. It is interesting that this formula is very similar to what cm] 0 would be expected if the dissipative trapped-electron ll 0 )0 (DTE} mode dominated the plasma (Perkins, 1984). Al- electron losses dominate in an Ohmically lowing that FIG. 36. Power balance for a TFR-600 plasma. In the inserts heated plasma, 3/2 are shown the experimental results for the dominant physical T'~ mechanism in each of the three plasma regions: sawtooth ac- yDTE= $ / ${) /[8 L n ln(A, )Z,s ] . R tivity, microturbulence, and spectral radiation (Equipe TFR, (4.19) 1980).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1060 John Sheffie|d: Physics of magnetic fusion reactors mode scaling (Perkins, 1984), which yields and allows a jump to a different solution for transport. rE ~ a ' R n 8'/I' . Subsequently, as data be- The effect of changing the radial current has been seen came available from other experiments, formulas involv- (Taylor, 1989; Weynants and Taylor, 1990), in experi- ing the plasma size were developed (Goldston, 1984; ments in which an electrode inserted in the plasma was Kaye and Goldston, 1985; Rebut et al., 1987; Odajima used to induce a current (Fig. 37). Various models have and Shimomura, 1988). A model based upon the role of a been developed to show how turbulence may be reduced critical electron temperature gradient has been developed by gradients and curvature in the poloidal rotation by Rebut, Lallia, and Watkins (1989; see also Rebut (Biglari, Diamond, and Terry, 1990; Leboeuf et al., et al., 1991). More recently the ITER group has 1991); and how a shear can be generated by turbulence developed formulas (Yushmanov et al., 1990). There are (Carreras, Garcia, and Lynch, 1991; Diamond and Kim, two types of ITER formulas for operation in the so-called 1991). Flow generation and turbulence reduction are low (L) mode of tokamak operation (CH=1): first, a closely interrelated. As L-to-H transition model, in- power series, tegrating self-consistently both effects, has been presented by Diamond et al. (1991). Detailed examinations +TER-P 1.2 0. 3 —0. 1 of —() ()48C I0.85R & 0 ~2O the edge parameter which have been made in DIII-D Xa 8' A /P (s) (4.21) (Groebner et al., 1990; Burrell et al., 1991) and in JFT- 2M (Ida et al., 1990) support some of the postulates of and second, an offset-linear type of scaling, which rejects these models. the view that there is an Ohmic scaling plus an incremen- In most experiments the transition from L to H mode tal auxiliary heating scaling: leads to a substantial increase in the plasma edge density and to a relatively Aat density profile. The temperature ITER-OL — IO. 1.6g0. 0.6~0 280. 0.. of the edge also rises, but the temperature retains its nor- () ()64C SR Q 6pPl 2pK 35' 2/P mally peaked profile. These features are well represented =— +0.04CIII 'R 'a a 2 (s) (4.22) by the model profiles, T=T0(1 r /a ) and n n0(1 r la—) ",a„=0—0.5. where the units are (MA), R line- I a, (m), n20(10 m, A condition must be set for the LH transition to occur. average density), P (MW) and is the atomic 8 (T), A; Empirical formulas for the condition have been given mass number. These formulas are valid for q» ~3. For (Ryter, 1993;Kardaun, 1993), lower safety factors the confinement is usually worse, owing to MHD activity. I'„,R (0.04 —0.07)n208 (T)2 (MW), (4.23) Interestingly, for very low aspect ratio, the where A (m is the surface area of the plasma. confinement in the Ohmic regime is better than neo- ) Empirical scalings for H-mode operation are at an ear- Alcator scaling would predict and closer to the more lier development (Christiansen et al. The favorable Rebut-Lallia-Watkins scaling (Hender, 1993). stage of , 1991). quality of confinement depends upon the level of activity of edge localized modes (ELMs), in which the H mode 1. The Hrnode crashes on a transient basis (Wagner et al. , 1982). For

As discovered in the ASDEX tokamak (Wagner et al., 1982; Asdex Team, 1989), the confinement can be improved in tokamaks with a divertor —the high or H B-r = 0.25 T mode. A typical value for the H-mode factor (CH) is 2.0 lp —25 kA CH-—2 for q&, ~3. Similar confinement improvements ne =2x10 cm have been obtained in plasmas with a limiter, e.g., "su- pershots" on TFTR, C~ —-3, when careful conditioning is used to reduce deuterium recycling from the limiter (Hawryluk, 1987). Recently a very high (VH) mode has been seen in diverted discharges in DIII-D, when wall conditioning with boron was used, C~=3.5 at —4 q&, 0.5 (Jackson et al. , 1991). A possible explanation for the LH transition lies in the effect of a radially varying poloidal rotation on the tur- I I bulence, which causes anomalous transport. It is postu- -100 -200 -300 lated that a bifurcated solution for transport exists with E, (vicm) two different, coupled levels of rotation and turbulence. FIG. 37. The radial particle current in CCT varies as a func- Itoh and Itoh (1988, 1989, 1990, 1991) and Shaing et al. tion of the applied electric field {poloidal rotation). The current (1988, 1989, 1990; Shaing, 1991) proposed that a change is related to the poloidal damping force. This "runaway"-like in ion orbit losses at the edge leads to a radial current damping can support bifurcation in the poloidal rotation and in and a j, XB& force, which changes the poloidal rotation the corresponding radial electric field (Taylor et al., 1989).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1061

ELM-free data an empirical fit is given by Post et a/. 10 I I I I (1991),

o R' ' tT rg (ELM-free)=0. 064I a 20 1.0 X KO'35B0'15 g 0'5/pO'5 (S) (4.24) exp When ELM activity is high, 7., =0.757; (ELM-free). (T sec) Most theoretical work has concentrated on explaining 0.1 the anomalous transport in terms of drift-wave-type instabilities, with frequencies below the ion cyclotron frequencies and ~0.5, i.e. short wavelengths. Howev- 01 0. I I k~p, , I I I I I I I I I I I I er, recent experimental data, e.g., those of Efthimion 0.01 0.1 1.0 5.0 et a/. , 1991, show the largest fluctuations occurring at B g (T sec) lower values, kep, 50.4. Also, the frequencies ((50 L kHz), when allowance is made for the Doppler shift com- FIG. 38. Plot of B~E" vs an empirical scaling law for a variety ing from plasmas Bows, are much less than those expect- of tokamaks operating in the H mode, showing a reproducible ed from drift waves. Possible candidates for these low- and systematic enhancement of con6nement over L-mode frequency, long-radial-correlation-length modes are in- operation (Goldston et al., 1992). stabilities such as the trapped-ion modes and neoclassical MHD tearing inodes (Callen, 1991). Ion temperature- gradient modes, which are important when I.„/Lz; ~1, ITER, of the empirical scalings should give confidence in were believed to be an important contributor to such the ability to predict reactor performance. phenomena as the saturation of confinement at high densities in Ohmically heated plasmas and I.-mode E. MHD limits confinement degradation. However, recent calculations, which incorporate more realistic kinetic effects, predict a Good reviews of this topic have been given by Bate- lower level of transport and suggest they may have im- man (1978), Wesson (1978), Schmidt (1979), and Freid- portance only in the plasma center (Callen, 1991). berg (1982,1987; see also Sec. III.B). The dominant modes in a tokamak are (a) kink instabilities, (b) balloon- 2. Dimensionless plasma scaling ing instabilities in which a pressure-gradient drive causes interchange of plasma and field in the region of poor The scaling relations derived in Sec. III.D may now be magnetic-field curvature loca1ized on the outside of the instabili- applied to the empirical formulas to see if they satisfy plasma (Fig. 39); and (c) low-order resistive kink CC '. ~ ties, which lead to magnetic island formation. 7 z B We find for Eq. (4.18) AH B,suggesting Whether the equilibria are stable can be checked that a slightly weaker dependence on density n might by be appropriate. For Eq. (4.21), 7. R ~B 97; for Eq. — — E (4 24) PTER-OLIX. B 1.11+B 0.78. d f E (4 24) (ELM-free) ~B', in fair agreement with the requirements. In the Connor and Taylor (1977) analysis, the ex- l ~ ponents of the parameters are related for particular plasma models. Evidence is growing that tokamak transport is Bohm- like in character rather than gyro-Bohm-like (Deboo et a/. , 1991; Christiansen et a/. , 1992; Waltz et a/. , 1990; Scott et al. , 1993). In fact, when ITER-P scaling is expressed in terms of a diffusivity, il1Il l' ' r

v* ' = ~ constant T, /B (P ) (4.25) p r r r Il I ' r p 0 p (Scott et al. 1993). I , ~ l ~ r I \ ~ ~ A plot of experimental ~EB is given in Fig. 38 for the I ITER, I;mode scaling (Cxoldston, et al. , 1992). JET data can be used as benchmarks for both an ignition experi- 0 ment and ITER; at reduced operating parameters corre- spond to equal dim ensionless parameters (Sheffield, FIG. 39. Projection of the displacement vector of the n=3 1990). It is expected that a power reactor will not be fixed boundary (b/a=1) mode for a P=9% case with R/a=3. 5 much larger than ITER; therefore benchmarking, in and 1 & q & 3.0 (Todd, 1979).

Rev. Mod. Phys. , Vol. 66, No. 3, Juty 1994 1062 John Sheffield: Physics of magnetic fusion reactors

linearizing Eqs. (3.4)—(3.8) and then subjecting the fluid The prime denotes difFerentiation with respect to g. For elements to small perturbations of the form the normal case of p'&0, both terms are stabilizing if q& 1. g„=g'(r)exp[i(m8+nP ~t) j, (4.26) The addition of resistivity modifies the ideal criterion in a narrow around the resonant surfaces, where m and n are integers (0,1,2,3, . . . ). Modes for because, region it allows the field lines to break and rejoin. The modified which the imaginary part of m =~+i y is positive will criterion is grow; these modes are linearly unstable. They may be 3 stabilized by nonlinear eFects when the plasma and field reach a new but time- or space-varying equilibrium. Various modes are observed in toroidal plasma, even when, on a long time scale, the system is in a stationary & 0 . (4.28) state. When modes are not stabilized, the plasma can move into the wall —the "major disruption" —or can The resistivity has removed the large shear stabilizing suffer relaxation oscillations with periodic enhancements term. Nevertheless, stability is still achieved for q &0 if of transport —tokamak "sawtooth" oscillations. A sta- q&1. bility diagram for modes having a variety of toroidal (n) Some modes are localized in the parallel direction with and poloidal (I) mode numbers and difFerent current a finite wavelength on the outside of the toroidal plasma, where the magnetic curvature is unfavorable —these are distributions is shown in Fig. 40. In the ideal MHD cal- " culations with a vacuum region surrounding the plasma, the so-called "ballooning modes. Analytic expressions these bulk modes would appear as perturbations to the for stability against ideal ballooning modes have been plasma surface. When finite resistivity is included in the given by Pogutse and Yurchenko for circular (1979) and calculation, the sharp plasma-vacuum boundary becomes noncircular (Pogutse, Chudin, and Yurchenko, 1980) blurred and other instabilities can be found inside the plasmas, plasma, for instance, the resistive kink modes termed — 1 2 2~Po, 2 7 5 2 1 "tearing modes" (Furth, Killeen, and Rosenbluth, 1963), —s + p' 1 —q 1 — 1 ——s exp 2 2 4e 6 Isl which dominate the low-P MHD activity in the plasma. jp = = In tokamaks, the (m 1, n 1) mode occurs when q & 1 Shear magnetic well or hill inside the plasma and induces sawtooth oscillations; see '2 Fig. 29. (4.29) Large m-number modes are localized near the resonant g2 ~2 surface, q =m jn, while small m numbers may have a greater radial extent. The stability criterion for radially ballooning drive localized modes was derived Mercier (1960). For a by where large-aspect-ratio tokamak (Shafranov and Yurchenko, 1968), p' s= —r,q',' e= —. 2 R P'8 2,q' q — — y p'(q 1)+ )0 . (4.27) This formula illustrates the stabilizing influence of 8po q magnetic shear (the first term) and of a magnetic well (the second term). The third term shows the stabilizing MODE NUMBERS (I) efFects of negative shear (s (0). 1111 1 2 1323 1 2 3 4567 2 3 4 5678 8765 4 7 3857 -- V=O 3. Resistive ballooning modes

Resistivity modifies the stability properties of balloon- j PROFILES ing modes (shear is ineffective) and lowers the stability O threshold. These modes can be important at high beta Ctf and low temperatures ( 51 keV) where the resistivity is higher. They may be important in the edge region of tokamaks and stellarator reactors, where they can enhance transport.

8765 4 3 2. Beta limit q (a) and which FICi. 40. Stability diagram for ideal kink and internal modes. For optimized pressure q(r) profiles, just max- Stability to internal modes requires q(0) & 1; to m=1 kinks re- avoid ideal ballooning and kink instabilities, the quires q(a) & 1; and to m ) 1 kink modes requires a minimum imum value average beta is (Bernard et a/. , 1983; Sykes peaking of the current pro61e (lesson, 1978). et a/. , 1983; Troyen et a/. , 1984)

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors

12 culated to self-stabilize is often referred to as a second Dilly-D stable region. Use of such terminology does not neces- sarily imply that the whole plasma is stable nor that some 10 other mode is stable in some other part of the plasma. For example, in a tokamak, modification of the magnetic field in the outer region might lead to self-stabilization of ballooning modes there, while the sawtooth mode was still active in the plasma interior. Direct access to the re- ~O 6 gion is theoretically possible for highly D shaped or even indented "kidney bean" tokamak plasma cross sections, as in the PBX-M tokamak, or by raising q&(0) to -2. Kink modes, however, are an obstable to self- stabilization. Calculations with optimized profiles predict g up to 5.5, stable to both classes of modes (Ramos, 1991). Operation in the second stable region at high bootstrap current fractions is an interesting mode for reactors; see 0 0 0.5 1.5 2 2.5 3.5 Fig. 34.

I/aB (MA/m/T) Ideal modes can be driven unstable by non-Maxwellian particle distribution functions, as in the "fishbone" insta- FIG. 41. The stable operating space for each indicated bility driven by perpendicular neutral-beam injection tokamak lies below the corresponding curve. A pressure-driven (Chen et al , 1984).or in the case of the predicted alpha- limit, often accompanied by ballooning-type modes with low particle instabilities (Spong et al. , 1985, 1987). mode numbers, is encountered at about Pr=3.5 in all these high-beta experiments. The vertical line to the right in each is 4. Resistive modes approximately I/aB, the "kink" limit at q, =2 for that device (Callen et al., 1992}. The high electrical conductivity of tokamak reactor plasmas ( T, ~ 10 keV) leads to a high Lundquist number S—magnetic difFusion time/poloidal Alfven I 8 : ~ (p, )=g %, g=2. to 4.4, (4.30) time=(por jr/)j(Roqjv„e) 10, where v~a=2 aBpo 18. X 10' Be/(2 n; ) (ms '). As a consequence, the where I (MA), a (m), B (T), and (P, ) =2po modes in a tokamak are generally constrained to a nar- (pressure X 100)/B, %, B, is the total field, and B&0 is the row region around the mode rational surface (q =m/n) vacuum toroidal field. The coefficient g is a weak func- The modes grow relatively slowly ( —10 —10 tion of 5 and ~. A plot of performance in a number of seconds) compared to ideal modes, and saturate when all tokarnaks, Fig. 41, shows that g=3.5 is typical of the of the accessible free energy has been accommodated in beta level achieved. Generally, the limit is set by a com- rearranging the local magnetic field, creating an island. bination of ballooning and kink modes. Qnly the low mode numbers are unstable, i.e., At low aspect ratio (R /a 5 2), paramagnetic efFects be- I/n =1/1, 3/2, 2/1, 3/1, . . . . These modes account come important and the toroidal field becomes larger for the macroscopic MHD phenomena observed in than the vacuum value. In addition, the poloidal field tokamaks. The 1/1 mode leads to a relaxation oscillation may be large (pe~ B&) as R /a drops to ~ 1.5. The total and the characteristic sawtooth behavior of ihe plasma field B,=(Be+B&)' -2B& at R/a —1.2. Calculations center when qo 5 1 (see Fig. 36). This mode flattens the of MHD stability show similar values of the Troyon central temperature profile, reducing the potential for coefFicient to values calculated for conventional aspect fusion reactions at fixed beta. The mode also decreases ratios (Bespoludennov et al. , 1986; Hender, 1993). confinement below the normal empirical value if the q= 1 surface becomes large, generally a problem when q& (3. It may be stabilized by careful control of the current 3. Second stable operation profile to hold q(0) ) 1, for example by using ICRH (see Phillips et al 1992). . For high enough pressure the distortion of the magnet- , The other modes are normally present and lead to is- ic geometry by the plasma moves the low (absolute) shear land formation with some modest degradation of regions to places with more favorable curvature and per- confinement [see Eq. (3.18)). rnits higher values of beta. This self-stabilization effect was first calculated for ballooning modes in tokamaks by Mercier (1979) and Coppi et al. (1979). Subsequently the 5. Disruptions efFect was also seen in calculation of interchange (Mer- cier) modes in stellarators (Carreras et al. , 1983; Rapid loss of energy and plasma current can occur in a Shafranov, 1983). The region in which some mode is cal- tokamak as a result of MHD instabilities, usually involv-

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1064 John Sheffield: Physics of magnetic fusion reactors ing the M=2, N=1 mode. A computation showing Troyon limit to ensure minimal disruption. Two ap- growth and overlap of the 2/1 and 3/2 modes leading to proaches to disruption prevention are being considered: instability is shown in Fig. 42. Conditions that are (a) Using current drive to control the current profile and vulnerable to disruptions, because they can have unstable suppress mode growth, and (b) using an externally ap- profiles, include (a) plasmas cooled at the edge by impuri- plied transform (X~0.15), which has been shown to ties near the density limit; (b) low-q operation, i.e., 2; prevent disruptions in Pulsator (Fussman et al. , 1979) (c) operation near the beta limit; and (d) transient and in TOSCA (Robinson, 1980). conditions —startup, current ramp-down, strong In summary, the limits set by MHD activity, in terms sawtooth activity, strong ELM activity, changes in of the Troyon factor g, are estimated to be roughly (a) con6gurations, and termination of additional heating. A g 2.5 if profiles are not controlled, in order to minimize review of disruptions is given by Post et al (19.91). A disruptions; (b) g =3.5 in the first stable region if there is slow rise in radiation or in MHD activity (e.g., the 2/1 some profile control; (c) g 55.5 when on-axis q(0) is mode) can be a precursor to a disruption. The disruption raised to -2, and with higher q&(a) to retain sufficient then leads to an energy quench, in which the plasma loses magnetic shear and put some of the plasma in a second from 50%%uo to 80% of its energy directly to plasma facing stable region. components or by radiation, in a time typically in the range of several hundred microseconds to a few mil- F. Density limit liseconds. The decrease in pressure causes the plasma to move inward in major radius. Following this phase the There is a maximum density at which the plasma be- current usually Aattens and rises, owing to the lower in- comes disruptive. %ith increasing density the plasma ductance. The current then quenches in a time as fast as edge cools, leading to a shrinking of the inductively 5 to 15 milliseconds. Most of the magnetic energy (over driven current channel. In turn the m/n=2/I tearing 75%) is lost radiation. The plasma usually moves in- by mode is destabilized when q&=2 is reached. A formula ward and may also suffer a vertical displacement. Large for the density limit was given first by Murakami (1976) currents can appear in plasma facing components, and for Ohmically heated plasmas, very energetic electrons ( ~ 50 MeV) may be generated. In present-day experiments, a few percent of disrup- n ~1.5 (10 m ), tions cannot be attributed to an identified cause (Post Rq,„) et al. , 1991). Disruptions remain an obstacle to the development of an acceptable tokamak reactor. Reactor are often restricted less than the designs to 2.' 5a B 2 q,„,= (1+i' ) .

For auxiliary heated plasma the multiplier was found 0.8 to be -2.5. Subsequently, Cxreenwald et al. (1988) showed that for noncircular plasmas the limit was higher 0.7 2 / than given by the Murakami formula, and — I(MA) (1()2O 3) 0.6 (4.31) 5/ sr[a (m)] A second limit may occur owing to loss of thermal 0.5 equilibrium in front of the divertor plate (Borrass, 1989). An between O analytic model provides a relationship the & 0.4 separatrix density and the divertor temperature, in terms of the heat Aux across the separatrix, the connection length, and the impurity radiation loss in the scrape-off layer. 0.2

G. Alpha-particle effects

0 The general effects of alpha particles in fusion plasmas 0.5 ).0 2.0 are discussed in Sec. II.F; see also Furth et al. (1990) and ( tx&0~) /~, Sigmar (1989). There are also specific collective efFects which be in tokamak when the FIG. 42. Time evolution of the magnetic island width for the may important reactors parallel velocity v approaches the Alfven speed. A (m =2; n = 1) and ( m =5; n =3}modes in a multiple-helicity cal- ~~ culation, leading to a disruptive condition (Carreras et al., summary of collective alpha instabilities is given in Table 1980). XII.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1065

TABLE XII. Collective alpha instabilities. Frequencies evaluated for TFTR (Zweben 1988). Symbols: ~d =alpha precession frequency; co& =Alfven frequency; co, =alpha cyclotron frequency', m~ =alpha diamagnetic frequency; S =alpha source rate. Frequency Physical Important Possible Possible Instability (kHz) mechanism(s) parameters effects diagnostic(s)

Alpha-driven & 0.1 Centra1 electron Modification of Gyrotron alpha sawteeth heating by q(r) profile; scattering, soft

fishb P (0) expulsion of alphas- x-ray emission Ph„,(0) alphas from sawtooth crash center Alpha-driven —10 '-10 Resonance Expulsion of Escaping alpha one of alpha p (o} trapped alphas detectors, soft precession and pth from center x-ray emission internal m=1 COd /CO g mode Alpha- —10-10 Resonance of Gradients Reduction of Microwave driven drift alphas with of p and beta limit; scattering wave or m »1 modes n /n, change of escaping alpha ballooning plasma detectors transport Alpha-driven —10 -10 Passing alphas Anomalous loss Microwave Alfven with u & V& v /V& of passing scattering waves (e.g., excite Alfven COg ~/CO g alphas; external B TAE mode) modes vp electron heating Alpha loss- —10 -10 Velocity space Anomalous loss Ion cyclotron cone-driven instability near TF ripple of trapped emission Alfven trapped/passing n /n, alphas; ion escaping alpha waves boundary ~ca heating detectors Alpha «10 -10 Bump-on-tail Anomalous Ion cyclotron population- instability due alpha slowing emission inversion- to fast alpha down; ion escaping alpha driven turn-on heating detectors Alfven wave

1. Destabilization of Alfvdn waves p threshold caused by balancing the alpha pressure- gradient drive against a variety of damping mechanism. The existence of toroidal Alfven modes has been demon- A variety of shear Alfven waves may be destablized by strated on TFTR (Wong et al. , 1991) and DIII-D (Heid- alpha populations. The radially localized kinetic Alfven brink et al. , 1991)by using neutral beams at low magnet- waves (KAW) can be excited both by trapped energetic ic fields. The fact that the observed p thresholds gen- and particles (Mikhailovskii, 1975) by passing particles, erally exceed those of the simple analytic theory by an or- through inverse Landau damping with to =k v (Rosen- i ~~ der of magnitude has focused attention on various damp- bluth and Rutherford, 1975). The global Alfven eigen- ing mechanisms for the toroidal Alfven eigenmode, such mode (GAE) is a radially extended, low-n, low-m mode as continuum damping (Rosenbluth et al. , 1992), ion occurring below the continuum [to ((k~~vz );„];it has Landau damping (Betti and Freidberg, 1992), Landau been shown to be destabilized by alphas (Li, Mahajan, damping on trapped collisional electrons (Gorelenkov Alfven re- and Ross, 1987}. The global eigenmode has and Sharapov, 1992), and finite-orbit effects (Berk et al. , cently been excited with neutral beams in the Wendel- 1992). stein 7-AS stellarator (Jaenicke et a/. , 1993}and may be Assessing the potential importance of these modes will an issue in the low-shear central regions of tokamaks, require (a) a better understanding of linear stability boun- where the alpha pressure gradients are large. When daries (including all relevant damping effects) and (b) toroidal geometry is included, spectral gaps occur in the nonlinear studies to determine the saturated amplitudes shear Alfven continuum, leading to the existence of the and their impact on the a particles. Some of the ap- toroidal Alfven eigenmodes (TAE) (Kieras and Tataronis proaches being used to address these questions are 1982; Cheng et al. , 1984; Fu and Van Dam, 1989). The gyrofiuid models (Spong et al. , 1992), hybrid fluid- conditions for the instability to occur are v Uz and particle models (Gang et al. , 1992; Park et al. , 1992), mv~p~lrL~)0. 5vz/Roq, evaluated at the radial loca- and Monte Carlo following of alpha-particle orbits in an tion of the gap, where I. is the radial scale length of externally imposed toroidal Alfven eigenmode structure n (r}. In addition, the toroidal Alfven eigenmode has a (Hsu and Sigmar, 1992).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1066 John Sheffield: Physics of magnetic fusion reactors

2. Destabilization of ballooning modes co, /co& &0 or destabilizing in the opposite case. The former limit was apparently achieved using anisotropic Ballooning modes may be destabilized by trapped and minority ICRF tails in JET, where sawtooth stabilization circulating alpha particles, with mode frequencies either was demonstrated (Campbell et a/. , 1988). However, in in the toroidal Alfven eigenmode range (co—=U~/2qR; the case of isotropic alphas, both signs of cod will be con- Spong et a/. , 1990; Rewoldt, 1988) or in the ion diamag- tained in the pitch-angle distribution, and it is not ex- netic frequency range (co-=co~;,„/2; Spong et a/. , 1985, pected to be possible to maintain either the stabilizing 1987; Biglari and Chen, 1991. The underlying destabili- (co~ /cod &0) or purely nonresonant limits (co&(cod ) zation mechanism is inverse Landau damping through over the entire tokamak radius (Cheng, 1990). transit, drift, or bounce resonances, which then couple ei- Some experimental data on the alpha-induced modes ther with a toroidal Alfven eigenmode (TAE) or a kinetic should come from TFTR and JET. However, ITER will ballooning mode (KBM). The calculated efFect is to probably be the first device to have a fully reactor-grade lower the background plasma beta limit (see Fig. 43; plasma in which alpha heating dominates. Spong et a/. , 1990) and increase the radial difFusion of the particles (Rewoldt, 1988). An important ques- alpha H. Power and particle control tion of future devices will be which mode (TAE or KBM) is dominant in the interest. operational regime of The The generic issues and solutions for power and particle toroidal Alfven eigenmode growth rate is expected to be control are discussed in Sec. II.Cx. For a tokamak it lowered enhanced continuum near the (due to damping) seems that a poloidal divertor (Figs. 19 and 20), with beta limit, while the kinetic ballooning mode per- Troyon high recycling of the neutrals (Table VI) is the best solu- sists in this regime Zonca, and 1992). (Biglari, Chen, tion for handling the heat leaving the plasma edge, for blocking wall-generated impurities, for pumping, and for 3. Fishbone oscillations minimizing divertor plate erosion. The operation of the divertor may be improved by radiating power in the Trapped alphas with drift frequencies resonant with divertor region, provided this is controllable. The most the m =n= 1 internal kink mode (co =—codh) will lower the extensive studies of tokamak reactor-relevant divertor beta threshold for this instability, leading to "fishbone" configurations have been made for ITER (Post et a/. , oscillations (Coppi et a/. , 1990; White et a/. , 1990). This 1991). Tests in existing facilities, their upgrades, some instability leads to ejection of fast particles in bursts. new steady-state facilities, and ultimately ITER will be Nonresonant alphas (co ((co& ) can be stabilizing if required to validate predictions, optimize divertor designs, and determine the best reactor plasma operating conditions. 12 External sources of power and particles are important for establishing plasma profiles and thereby controlling

l,' the plasma behavior. In a reactor, for example, produc- I ing 3500 MeV of fusion power [-GW(e)], the alpha WITHOUT power is 700 MW. It is not likely that a6ordable external sources of heating can have impact on the temperature PARTICLES g profile. However, power may be used to control the current profile with noninductive current drive and to afFect low-order MHD activity (Zakharov, 1989). Fueling is discussed in Sec. II.I. While pellet injection

o~0 to the axis of a tokamak reactor may be dificult to 6 achieve, penetration well into the plasma may be used to WITH aQect the edge density profile. ALPHA PARTICLE

I. Modeling of tokamak plasmas

The system of equations that describes the tokamak core, edge, divertor region, and walls is far too complex to solve, even with a computer, except by splitting up the problem and simplifying the calculations for each region. I 0 l I Nevertheless very good progress has been made on mod- 0 0.2 04 0.6 0.8 1.0 eling experiments, and there is some ability to predict performance (McNamara, 1981; Killeen, 1992). FIG. 43. ITER stability contours (at y=0.02co& ) for an elonga- Numerous programs exist which model the plasma in cir- tion of b /a=2. 0 (Spong et al., 1990). cular and noncircular geometries and follow its time his-

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffie Id:'. ph Vslcs of magnetic fusio

JET SHOT No. 16211

15 1 I (b) ns~(10 m ) 10-

5" T (keV 5

FIG. 44. A comparison of computeuted and ex- 0 0 perimenta l data for pellet injection in JE 42 43 44 42 43 s owing (a) the central electron and ion tem- TIME (s) TIME (s) peratures; (b) thee central electron density; (cc) th eevovilving electron temperature pro e, an (c) (d) the evolving density profile ( ou 1990). ns(1

9.27 1.45 44

0 0 184 4 4 4~0 «o R t't. ~) +(cm)

r 1982; Pfeiffer et al. , 1980; Singer et al. , f a fast-ion distri'h utionion, (d) model rf heating; (e model ' 1986). These codes solve the multi ui equa gas and pellet fueling; trackrac neutral hydrogen as it g current rofiles, and inc u e su ' neutral injection; (1) inc u e u tions and fusion product heating; (c fo ow e ev all i include the efFects of divertor action and mo e

peellete injection in JET is shown in Fig. . e used to map t e power cocontours for operation over T, n space as iillustustrated in Fig. 45 0'h""d s of the scrape-off layer andan divertoriver (Heifetz et al. , 1982; Petravic et a ., 1987).

CD CU J. Tokamak reactor options Q7 MW/ m2 508 MW ON POWER) studies have been made of tokamak reac- rs for example, those of Badger et al. l. (1989), Najmabadi et al. Seki et al. (1991). Only recently, however,

l l 15 20 25 30 port model; see Sec. IV.D. A systematic study of princi- Te (ke Y) P P'o now being undertaken in th e Advanced Reactoror Innovationn and Evaluation Study FIG.. 45.. ITER pplasma operating contoursntours, POPCONS) in n-T ' 1' ARIES-1 design uses the presentnt bestes physicsy base, withh space foror ITER H-mode scaling (H=2.0}. Contours denote ' ' ~ ~ ~ advanced technology and eneng'ineering. . The',design cn-n- auxiliary neutral-beam powerower requirere uired too maintain an equilibri- po ce. P =0 is the true ignition oun a to a requirement for a 21-T toroidal coi. 1'gh d'ff O eration constraints of the beta limit, ensi y im', ly pp o quired neutron flux are shown (Post et al. 1991). design (Seki et al. , 1991) has a 16-T coil. The

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fosion reactors design relieves the pressure on the technology and en- quirements for +E and (B) for an ignited plasma [Eqs. gineering by utilizing the potential benefits of advances in (2.4S) and (2.46)]. These requirements are now equated tokamak physics, such as may be achieved with better to the empirical formulas for rE, Eq. (4.21), and (P), Eq. current profile control —that is, second stable operation (4.30). Using Eq. (2.42) for the average neutron flux at (higher g) —and better edge control and enhanced the wall and Eq. (4.2) for the plasma current, we find the confinement (higher CH ). requirements for fusion power Pz, toroidal magnetic field In Sec. II.E a simple model was used to define the re- (8), and minor radius (a) of a tokamak reactor:

i06( — )0. 1061r0947q i. i 2)22ig 1.79 16X 1()4(aaw /a)0.a pan & qyg i( I a 2/R (MW), (4.32) i li n 034' 17f 7(n /n )i 62(R )0277[1+ 2(1+2g2 1 2g2)]i. I 17 () 65

46(a ja)0.36(P )0.36 0.36 0 2if 0 21 /a)0. 66(1 2/R 2)0.72( 0 426 N wn fa 'i a (R H )0.30~0 . 723 (T) (4.33) (n /n 18[ 1+ ~2( 1+2/2 1 2/3) ]0 36( 1. 1 7 0.65a /R)0 36g0 0. 142P~ (m) . (4.34) (R /a )0 5( a /a )0 5lr0 25(p )0 5

The examples below illustrate how these formulas may be Whether such parameters are possible depends on wheth- used to define various reactor options. er the actual transport leads to stable profiles and wheth- (1) ITER-like plasma parameters for an Engineering er impurity accumulation is avoided. Reactor (ITER). a„/a = 1.1, J7 „=1.0 (MW m ), Note that the average neoclassical ion thermal a=1.6, q&, =3.0, R/a=2. 8, g=2.0, n20=1.2, A, =2.5, difFusivity, Eq. (3.20), is about half of the permitted level. f =0.9, nDTjn, =0.55, 5=0.4, CH=2. These values (4) Better-confinement reactor. The better the yield R=7.91 m, a=2.83 m, $=6.2 T, PF-—1540 MW. confinement, the lower the unit power is allowed to be, The tradeoff' issues for the ITER design are discussed by Eq. (4.32). However, everything else being equal, the Post and Uckan (1992), who show the high leverage of field will be higher for improved confinement as can be using higher ellipticity. seen in Eq. (4.33). Improved confinement is still advanta- (2) Aries-1-like reactor (see Table X). a /a = l.1, geous in providing a margin that allows more power to be radiated: less is p „=2.6 MWm, v=1.8, q&, =4.5, R/a=4. 5, g=3.2, power required to support conduction n20=1.S, A; =2.5, f =0.8, nDT/n, =0.73, 5=0.4, losses, and f may be smaller. Note that the product CII =2.8. f CII appears in both Eqs. (4.32) and (4.33). For example, if CH =4.2 rather than 2.8, we may drop from 0.8 The higher values of CH and q&, are consistent with the f to 0. i.e. halve the power leaving the plasma con- very high mode of confinement seen in the DIII-0 4, , by duction, and alleviate the divertor problem. tokamak (Jackson et al. , 1991). These values yield The aspect ratio dependence of Eq. (4.32) is very weak, P~ -1900 M%', B=9.2 T, a = 1.51 m, R =6.78 m, I=9.8 . (1—a /R ) '/[(R/a)0. . (1.17—0.65a/R)'"]=O. S9 MA, (n„T„)=2.3, (P) =2.2%. for R/a =2.5 —5.0. For x=1.5~2.0, the ~-dependent An advantage of this mode of operation is the high terms are given by 0.3+12%%uo. Assuming a /a= 1.1, bootstrap current fraction. Assuming q(0) = 1.S, „=2.6 (very weak dependence), n20=1.5 (weak depen- IBs/I =0 67 from Eq. .(4.8) assuming the fast alpha pres- p dence), 5, =0.8, =0.73 we find sure does not contribute to the bootstrap current. A 2;=2. f~ nDT/n, problem for this mode is the requirement of a large field 1030 1.11 1.79 on the toroidal coil. Assuming the inner blanket, shield, (4.35) and Dewar have a thickness 6=1.5, the field ratio, Eq. is (2.40) f~ =0.535 and the field on the coil is B,„=17.2 The minor radius is T for this model. 7 (3) Higher-beta reactor. Second stable operation is 4. 1C~1. a- m . (4.36) projected for higher q& operation. For example, let 056g090(P)05(R /a)05 g=5.0 for q&, =6.0, q&(0)=1.5, x.=2 with the same parameters as case 2. The reactor parameters for 6=1.5 By contrast the field (B) does depend on (R/a), and are g=5.0, Pz-3440 MW, B =7.2 T, fbi =O.S9, the factor (R/a) (1 —a /R ) /(1. 17—0.65a/R) B,„=12.3 T. The higher beta factor allows the use of a varies from 1.53 to 2.77 as R/a varies from 2.5 to 5.0. lower field coil, but at the price of a bigger unit power. However, the field ratio increases with increasing R/a, The bootstrap current is about 100%%uo of the total current. offsetting this e6ect. For the ARIES-1-like case, the

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1069 minimum field on the toroidal coil occurs at R/a=4. 5, (b) Interesting tokamak reactors appear to be achiev- though it varies little in the range R /a =3.5 —5.0. Other able provided present levels of con6nement and beta can factors, notably the positioning and size of the inner po- be sustained without impurity accumulation, with low loidal coils, then set the aspect ratio. divertor replacement rates, and without disruptions. (5) Low-aspect-ratio reactor. The approximate param- However, the present physics base leads to a —1-GW(e) eters of the low-aspect-ratio tokamak, described by Peng reactor with a relatively high magnetic 6eld on the and Hicks (1990), may be derived by using the parame- toroidal coils —16 T or more. ters a„/a= 1.1, p~„=10 MWm, v=2.7, q&, =5, (c) With the present physics base, if higher-field coils R/a=1. 22, f =1.0, nDT/ne=0 73,. A;=2.5, n20=4.4, turn out to be impractical, according to our present un- and CH=4.0 in Eqs. (4.32) and (4.33). The multiplier on derstanding of economic factors (mass power density), the beta limit formula, where beta is normalized to the the economic react'or will be driven to a much larger unit vacuum field, must be set at g =20, assuming the benefit size than —1 GW(e). of the total field is above the vacuum level by a factor of (d) If beta levels can be improved, for example, in the about 2.9. Such reactors are very interesting from the second stable region, lower fields will be permitted at the point of view of higher power density and lower —1 GW(e) size. cost/kWe than conventional tokamaks. They pay the (e) If better confinement can be achieved, either smaller price of relatively high recirculating power and require reactors or reactors with a lower conducted heat Aux will the central column of the copper toroidal coils to be re- be possible. placed regularly. (6) Low-impurity reactor. The allowable fusion power V. STELLARATOR REACTORS is strongly dependent on the depletion of the D-T fuel, (nDT/n, )' and the radiated power, which is accounted A. Introduction for in f . Only if confinement is very good, i.e., CH large, can significant impurity levels be accommodated. A concern is that under conditions of better confinement A number of reactor studies have been made of a it is often observed that impurities accumulate [for exam- variety of stellarator configurations: Badger et al. , 1982; Uo et al. Miller et al. Volkov et ple, with counter neutral injection (Stacy et al. , 1985)], , 1983; , 1983; al. , 1984, and edge-localized modes may be necessary to avoid ac- 1987; Lacatski et al. , 1985; Harmeyer et al. , 1986; Hu- cumulation in the H mode (Post et al. 1991). The ability bener and Maurer, 1987; Kazawa et al. , 1987; Painter to remove hehum effectively is clearly very important. and Lyon, 1991;Wobig et al. , 1991, 1993. Parameters of some of the recent studies are listed in Table XIII. The K. Conclusions studies tend to be less detailed, and to include more ad hoc assumptions, than recent tokamak studies, rejecting (a) Parameters achieved independently in tokamaks are the much smaller data base available for projecting reac- close to reactor levels; see Table I. tor performance. However, there is an emerging data

TABLE XIII. Representative stellarator reactors. Reactor H-1 ATR-2 HSR Kazawa et al., 1987 Lyon, 1989 Reference Wobig et al., 1991 m =15, 1=2 m =9, l=2 Configuration Helias Heliotron Torsatron

Major radius R (m) 20.0 16.0 10.5 Aspect ratio R/a 12.5 8.0 4.66 Field 8(T) 5.0 5.0 5.0 Maximum field at 10.5 10.0 13.5 coil 8 (T) Edge (center) -0.9 -0.6 (1.5) 0.24 (0.97) transform j Confinement LGS N.C.+Alcator N.C.+const. anom. model Average beta (P) (%) 5 7.3 6.3 P „(MWm 1.6 3.1 2.7 Divertor Open semi- Helical Helical helical Fusion power 2800 5400 [MW(t)] Electric power [MW(e) net]

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1070 John Sheffield: Physics of magnetic fusion reactors base empirical similar to those 1 supporting scaling s =Boz +— bII&(lar)sinl (8—az), and Pz g developed for tokamaks (Sudo et a/. , 1990; Lackner 1 Gottardi, 1990). And major facilities are in construction — (LHD) or proposed (WVII-X), which should be powerful 8„=g lb&It'(lar)sinl (8 az), l enough to identify more clearly the best route to a viable stellarator reactor. 1 Be= b&I&(1ar)cosl (0—az), g CXP' — — B. Stellarator characteristics 8, =80 g lb&I&(lar)cosl (8 az), l

where l is the number of poles in the magnetic field, i.e., include Mi- Reviews of stellarator research those of in a poloidal plane the number of separatrices formed at yamoto (1978), Shohet (198la, 1981b), Carreras, Grieger, the plasma edge by a field component acting on its own. et al. (1988), and Lyon (1990). The parameters of some The helical pitch parameter is denoted by a, and bl gives representative stellarators are shown in Table XIV. Stel- the magnitude of the lth component of the field. Il is the larators typically achieve parameters comparable to modified Bessel function of the lth order. The magnetic those of a tokamak of similar scale and power density. In ~ ~ flux g is represented by today's devices, ( n20 ) S 3, T,o 3 keV, T,o 2 keV, (P) 52%, rE 50.03 s, 8 S2.5 T. cur A stellarator, in efFect, is produced by a loosely wound $=8o rg b&It'(—lar)cosl (8—az) . (5.2) toroidal solenoid. The open character of the helical I windings leads to an average poloidal field and a helical ripple superimposed upon the background toroidal field Representative magnetic surfaces are shown in Fig. 46, (80) of the solenoid (Miyamoto, 1978; Shohet, 1981a, for l=1, 2, and 3 configurations. Note that, while the 1981b). A simple picture of the e6'ect of the helical com- last closed Aux surface is often referred to as a separatrix, ponents can be seen in the case of a linear stellarator, in it does not have the same degree of definition as in an ax- which the scalar potential P~ and the magnetic field isymmetric system because toroidal efFects tend to create 8=V/~ are given in cylindrical coordinates by a region of ergodic field lines.

TABLE XIV. Representative stellarator experiments. Operating CHS Heliotron-E H-1 &VII-AS Uragan 2M experiment l=2, l=2, l=2, l =1,2, 3, l=2, Configuration m=8 m=19 m=3 m=5 m=4 heliotron heliotron heliac mod. stell. torsatron

Major radius R (m) 2.1 1.0 2.2 1.0 2.0 1.7 Aspect ratio 8, /a 7.8 5.0 11.0 4.5 10.0 7.8 Field B(T) 2.0 1.5 2.0 1.0 2.5 2.4 Center (edge) transform g ' 0.35 (0.97) 0.33 (1.2) 0.5 (2.5) 0.57 (0.75)

Present best Future LHD TJ-II 'W VII-X parameters (not experiment l=2 l=2 l= 1,2,3 simultaneous) Con6guration m=10 m=4 m=5 (n„)-3 heliotron heliac helias T;0-2 keV (P)-2 ~E-0.03 s Major radius A (m) 3.9 R (rn) Aspect ratio A/a 6.0 R/a Field B(T) 3.o (4.o) 1.0 3.0 Center (edge) 0.35 (0.98) 52.5 -0.84 transform g Operation 1998 1994 Proposed Reference Motojima et al., Alejadre et al., Cxrieger et al. , 1991 1990 1992

'Nominal values are shown; g'profile and level can be varied.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1071

theory and computer models and in their connection to experiment, and significant achievements in experimental plasmas have made it possible to focus this diversity of opportunity on two basis types of stellarator. In each approach, a tradeofF is made in the simultaneous optimization of beta, neoclassical transport (anomalous transport), and divertor and coil design. (1) Stellarators with a spatially varying (helical) axis and a dominant l=l component of the magnetic field FIG. 46. Representative magnetic Aux surfaces for 1=1,2, and form one class. They are characterized by a helical axis, 3 stellarators. low shear, and a magnetic well that is moderately deep but broad in minor radius. The "helias" device, which exemplifies this configuration (Nuhrenberg and Zille, The poloidally averaged toroidal transform 4=i/2m = 1/q is given in this approximation by 2 ZI' d 4—0.5 (5.3) : Bo 3 dx x In general, 4 is in the opposite direction to that of a tokamak, varying from being lowest on axis (highest q) and rising towards the plasma boundary (lowest q), to being nearly constant. Stellarators cover a wider range of configuration parameter space than do tokamaks, as illustrated in Fig. 47, which plots global shear (~/4) versus central transform l(0) (4=1/q) for a variety of present-day experiments. Within this space, widely varying combinations of magnetic shear, magnetic well, and nonplanar axes are em- ployed in the search for the optimum configuration. The principal factors that guide stellarator design are the following: (i) Neoclassical effects in the I/v regime of transport, electric-field effects, energetic-particle confinement (the loss cone), and impurity transport, as discussed in Sec. III.B. (ii) Equilibrium and stability limits, including second stable operation; Pfirsch-Schluter and bootstrap currents; and the robustness of Aux surfaces. (iii) Practical aspects such as anomalous transport, impurity and particle control, helium ash accumulation, and coil design, e.g., the use of modular coils. Good progress has been made in the development of

H-E

W VII IJJ 0 ~—0

CA W VII-AS q(a)=1.5

CO O OKAMAKS C9 (0)= 1 S q(a )=ca / —2 0 0.2 0.4 0.6 0.8 1.0 AVERAGE TRANSFORM, (4&

FIG. 47. Shear vs central rotational transform, showing the operating region for a number of existing and proposed stellara- FIG. 48. Stellarator configurations: (a) helias, (b) tors and for tokamaks with q(0) = l. torsatron/heliotron, (c}modular torsatron.

Rev. Mod. Phys. , Vol. 66, Np. 3, July 1994 t072 John Sheffield: Physics of magnetic fusion reactors

1986 and 1988), is designed to minimize Pfirsch-Schliiter bination of electron cyclotron heating, ion cyclotron currents so that the magnetic geometry varies little as the heating, and neutral beams. In a reactor, alpha power pressure is raised. The fields are configured to emphasize will sustain the plasma. See Sec. V.H for an example. helical symmetry and suppress effects due to toroidicity. Stellarators with five, and six were four, periods studied. C. Stellarator transport A five-period configuration was found to have the optimum combination reduced low-collisionality trans- of Neoclassical transport in toroidal devices is introduced shown in 51 confinement al- port, Fig. below, good of in Sec. III.C. In a stellarator, the magnitude of the field phas, maximum beta ( =5%), and optimized reactor (P } along a field line is modulated by both the toroidal varia- size. A modular coil set, generated a by computer code, tion and the helical variation (see Fig. 49). This magnetic produces the configuration shown in 48(a). An alter- Fig. geometry leads to three classes of particles (Gibson and native the "heliac, " Furth et approach, originated by al. Taylor, 1967; Miyamoto, 1978; Kovrizhnykh, 1984). As has modular coils, a beta but (1966), higher capability, in a tokamak, there are toroidally circulating particles less transport optimization. and particles trapped in the magnetic mirrors of the The l=2 heliotron/torsatron Gourdon (2) (Uo, 1961; high-field region on the inside of the torus. The third et al. 1969) is the second class of device studied exten- , class are particles trapped in the helical field and sively as a reactor. It is characterized by moderate re Aected from its local mirror regions. A helically and/or high shear and moderate well, and has the virtue trapped particle that is also trapped toroidally has an or- that the stable beta is inversely proportional to the aspect bit described as a "super banana*' (see Fig. 50). An excel- ratio. In addition, such devices at moderate ratio aspect lent review, comparing tokamak and stellarator trans- may enter the second stable region (Carreras et al. , port, has been given by Wagner and Stroh (1993). 1983). Maintaining fiux surfaces at low aspect ratio, Helically trapped particles have drift surfaces that are which are robust under the action increasing of pressure, not closely related to the magnetic surfaces, and they is accomplished careful tailoring of the helical coils, by may escape. The characteristics of the transport depend coupled with detailed control of currents in poloidal field upon the relative magnitude of the toroidal windings (Carreras et al. , 1984). Relatively low-aspect- ratio torsatrons have been designed with R/a —3. The +max +min (5.4} main limit on lowering the aspect ratio is set by neoclas- 8,„+S sical transport. The peak in the transport coefBcient in the 1/v regime is reduced by electric-field efFects (see Fig. and helical e& ripple. Super banana orbits exist when 51 below). However, the loss cone for energetic particles ez ) e&, which generally occurs in large-aspect-ratio stel- (alphas) is not afFected, and a tradeofF must be made be- larators. The major new feature of the transport, for tween this factor and the other benefits of the system. ez ~ e&, compared to an axisymmetric system, is a sub- An example configuration is shown in Fig. 48(b). The de- stantially increasing particle transport as collisionality vice may also be modularized, though this configuration decreases, starting near the plateau-banana transition. requires relatively sharp bends in the modular coils; see For this region Galeev and Sagdeev (1968) derived the Fig. 48(c). diffusion coefBcient While stellarator reactor designs with continuous coils 0 have been studied, the preferred option is to use modular 3/2~1/2 (5.5) coils. In principle, any stellarator magnetic configuration &ei e may be constructed from a set of nonplanar toroidal coils, as in the case of the helias. The goal of the program is to find configurations that satisfy the need for a —T divertor and have good access for maintenance while q80r minimizing the complexity of the coil set. On the experimental front, modular stellarators have been operated A schematic diagram showing the form of the diffusion successfully, showing a performance as good as any continuous-coil stellarator, notably, IMS (Doerner et aI 1986}and WVII-AS (Jaenicke et a/. 1993). The field ratio for most stellarators is in the range f~-0.5 for R/a) 7. Sharp bends in modularized coils may reduce ~B. Since stellarator fields are produced by external coils, the device is inherently steady state, and a plasma will be maintained as long as auxiliary heating (or alpha heating) is applied. In modern stellarators, startup is accom- FICx. 49. Variation of the magnetic of the field along a field line plished using electron cyclotron heating at the first or in a toroidal stellarator, showing toroidal (e&) and helical rip- second harmonics. The plasma is sustained with a com- ple.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1073

Some progress has been made in characterizing edge transport in a comparative study with the TEXT tokamak (Wootton, 1991). However, for the plasma core, transport mechanisms are not known, and empirical formulas must be used in combination with neoclassical transport. The radial electric field, produced by the (a) otherwise nonambipolar transport, must be included to ensure an accurate assessment of neoclassical losses FIG. 50. Trajectories of (a) untrapped bananas and (b) trapped (Mynick and Hitchon, 1983). banana (superbanana). In general, there are two modes of operation of the reactor, depending on whether ion or electron transport coeKcient as a function of collision frequency is shown in dominates. They vary in the sign of electric field re- Fig. 51. The large level of transport at low collisionali- quired to restrain the flow of the dominant species. The ties would make it very dificult to design an economic electric field may be determined self-consistently using reactor were it not for the ability to tailor the magnetic the transport equations and a procedure for selecting the geometry {Mynick, Chu, and Boozer, 1982; Nuhrenberg correct root {Hastings et al. , 1985). The two roots are and Zille, 1986, 1988;). These analyses of Mynick et al. shown in Fig. 52 for a reference heliotron/torsatron reac- and of Nuhrenberg and Zille led to the development of tor. the "helias" concept, in which the tailoring of the mag- In the 1/v regime, ion losses predominate and a nega- netic field not only improves neoclassical confinement but also optimizes the equilibrium, minimizing the plasma- driven currents and maximizing beta, in a way that is of great interest to reactor design (Nuhrenberg and Zille, 1986, 1988; Grieger et al. , 1992; Beidler et al. , 1992). Alternatively, the electric fields may reduce transport, for, in contrast to a tokamak, stellarator transport does not lead inherently to the same cruxes for ions and electrons. Consequently large radial electric fields may be set up to equalize fIows. A simpli6ed formula for the ion thermal difFusivity derived by Shaing (1984) is given in Eq. (3.23). As in the case of the tokamak, the bulk of the transport in anomalous, though in higher-T„ lower-n cases neoclassical transport can dominate, as seen in stellara- ) tors such as ATF, Heliotron-E, and WVII-AS. CO

LL I I II I I 10 100 - (b) IX UJ AX I SYMMETRI C I- HINTON-HAZELTINE UJ )02 I- D UJ

L 10 O

s CJ

/ /

/ I I' 0-$ \ I I I I I I I I I I I —6 -4 —2 0 2 4 NORMALIZED ELECTRIC POTENTIAL, (o I I 10 10 10 )01 10 FIG. 52. Effective thermal diffusivity vs normalized electric potential for an R /a = 5 heliotron-torsatron reactor: (a) at FIG. 51. Typical behavior of the diffusion coefBcient D for a moderate collisionality ( ( T) = 10 keV, ( n ) =2 X 10 m ); stellarator as the collisionality v* is varied. The effects of radial and (b) at low collisionality (( T) =30 keV, (n ) =7.5X10'9 potential (P) and magnetic field tailoring (helias) are illustrated. m ) (Painter and Lyon, 1991).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors tive electric Geld is formed —the ion root. In a reactor negative electric Aeld of the ion root, while impurities this root tends to occur at high density () 10 m 5) and may be expelled with the electron root (Shaing, 1983). low temperature ( —10 keV). There is some experimental evidence for this behavior in In the collisionless detrapping regime, electron losses Heliotron-E (Qbiki et al. , 1989}. More work is required dominate, producing a positive electric field —the elec- to find. the optimum transport conditions that avoid im- tron root. In a reactor, this root tends to occur at low purity accumulation. density ( 5 10 m ) and high temperature ( ~ 20 keV). In the case of the torsatron-heliotron it is possible to Because diferent parts of the plasma may be in difFerent modulate the poloidal 6eld so as to form a central separa- regimes, the electric 6eld may vary in sign across a ra- trix periodically, and temporarily enhance transport to dius. eject helium ash and impurities (Lyon et al. , 1986). The experimental data from modern stellarators fits empirical scalings similar to those found in tokamaks D. MHD and density limits [Eq. (4.21)]. The so-called large helical device (LHD) scaling (Sudo The general MHD behavior of toroidal systems is in- et al. , 1990) is troduced in Sec. III.C. The dominant mode for a stel- 0. 0.58 +HD & 69g0.84&2~0.75' larator is an interchange instability. E 0 17'E+ 20 p (8) (5.6) Ballooning instabilities, which occur locally in the outer regions of poor where n20 is the line-averaged density (10 m ), 8 (T) is magnetic curvature, are the primary limit on stability in the axial field, a (m) is the average minor radius of the the helias, with resistive effects coming into play at lower plasma, R (m} is the major radius, and P (MW) is the heat beta in the colder plasma edge. In the torsatron- deposited in the plasma. The data show better-than- heliotron, ballooning modes also play an important role average performance; a factor CE is included to allow for and the interchange modes are unstable in the edge re- enhanced modes of con6nement being established on a gion owing to the lack of a magnetic well. The plasma more regular basis, as in the tokamak H, VH, and su- equilibrium beta is limited by the shift of the plasma axis pershot modes. There are encouraging data from WIII- caused by P6rsch-Schluter currents, which increase with AS (Jaenicke et al. , 1993) showing signs of improved increasing plasma pressure. Optimized designs, which performance and H-mode-like characteristics. minimize P6rsch-Schluter currents, maximize the equi- The formula has some interesting similarities to that librium limit, and minimize MHD activity, have predict- expected from trapped-particle drift-wave turbulence and ed beta limits in the range of -4—5% (Boozer, 1983; ripple-induced neoclassical transport (Painter and Lyon, Carreras et al. 1983; Nuhrenberg and Zille, 1986, 1988). 1991). In terms of dimensional scaling [see Eq. (3.47)j, The Pfirsch-Schluter efFect does not yield a net current: FzHD ~8 ' rather than the correct 8 '0. Therefore . 2 cos0 dP modifications may be expected as more data are pro- '&=gps (5.9) duced. a The Lackner-Gottardi (1990) plateau scaling, devel- On the other hand, the bootstrap current does. Shaing oped for tokamaks, is a good fit to WVII-A and WVII- et al. (1989) give the formula A.S data. It has the virtue of being dimensionally correct: f~ qp '=0 JBS 3 Gb (5.10} + 17C (n )'Y' Bg P 05 XB aR P0 (s} (5.7} where f, and f, are the fractions of trapped and recirculating particles and Gb is a magnetic geometry factor (for The evidence for a positive A; dependence is very limit- a tokamak Gb=1). Measurements of these currents, ed, and in calculations below the term (A;/1. 5) is set showing agreement with theory, were made by Treffert equal to unity. et al. (1984). The two scalings above are gyroreduced Bohrn in Care must be taken to ensure that the bootstrap form, as discussed at the end of Sec. III.C. A formula current does not become large enough to degrade the that its ATF data is magnetic con6guration. By careful adjustment of the magnetic geometry the current can be made zero or re- r" - = 35(n )"Z"a'Z )-"yP" [0 p "(~ I verse sign (Murakami et al. , 1990). —0. 174 0.314 The resistive interchange modes can lead to increased (p ) l (5.8) transport, as has been observed in ATF (Harris et al. , The improvements in con6nement with decreasing col- 1993). lisionality and increasing (P) are seen in experiments in The density limit in stellarators is related to having which one is modulated and the other held constant (Mu- enough heating power to overcome radiation and con- rakarni et a/. , 1993). duction losses. At quite modest fields, 5 2 T, densities of As in the case of the tokamak, impurity accumulation greater than 10 m are obtained. If inadequate power may be a problem. Theoretically this may occur with the is applied, the energy content decays when the density is

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1075 raised —a "soft disruption" (Isler et al. , 1991). An E. Alpha-particle effects empirical formula for the limit has been developed by Sudo et al. (1990), The general effects of energetic alphas are discussed in Sec. II.F. The primary differences from the tokamak are (5.11) the lack of the kink instability and the addition of a po- Beta self-stabilization of Mercier modes is possible in tentially severe loss cone owing to the nonaxisymmetric stellarators with shear (Carreras et al. , 1983; Shafranov, field. The potentially severe losses of alphas are a strong 1983). The condition for stability against interchange driver in stellarator reactor design, and careful optimiza- modes may be written as tion of the magnetic configuration must be used to reduce both thermal losses and alpha-particles losses (Nuhren- shear vacuum field stabilization curvature berg and Zille, 1988; Grieger et al., 1989; Painter and Lyon, 1989, 1991). In the helias reactor, it is possible to d 3 reduce alpha losses to a very low level (Lotz et al. 1992). 2+ P c gg2y d(P4 ) & &p , 4 g2g2 dp dp gp3 The reduction in losses occurs because, as beta is raised, the increase in poloidal magnetic drift improves the pressure gradient axis confinement of trapped alpha particles; see Fig. 54. For driving effect on V" the losses are more severe. How- (5.12) the torsatron-heliotron ever, the presence of a loss cone may be an advantage if Self-stabilization occurs when the axis shift is suf6cient to the majority of the contained energetic alphas are lost overcome the effects of vacuum field curvature and when they start to scatter into the loss regions, thereby suppress the pressure gradient driving term. This effect reducing helium accumulation problems. On the down- may allow access to the so-called second stable region, as side, protection must be provided for the wall in regions illustrated in calculations for fixed profiles for ATF, bombarded by the energetic alphas to prevent unaccept- shown in Fig. 53. The suppression of interchange modes able damage An example of the effect of alpha losses, as beta is raised has been observed in ATF, for a case both direct and scattered, is shown for a reference with field errors and extremely peaked pressure profiles torsatron-heliotron reactor in Fig. 55. The alpha-driven (Harris et al. , 1989, 1990). However, stability to bal- instabilities discussed for the tokamak in Sec. IV.G may looning modes can limit access (Cooper et al. , 1989), and also be important for stellarators —toroidal Alfven further optimization is required, including the effects on modes for v ~ vz, and modified ballooning mode stabili- stability of plasma How. ty.

"BROAD" PROFILE "NARROW" PROFILE Pc/(P) = 2.6 Pci(1)) = 5.0

I I I

III I

15— FIG. 53. Second stability region in the standard ATF configuration (a) at high beta for the broad pressure profile assumed in the original design study and (b) at the lower beta for the peaked pressure profiles obtained in the 1988 —10— experiments. Beta self-stabilization (stability increasing with increasing beta for Mercier modes) occurs for P(0)-4% in (a) and for P(0) —1% in (b) (Lyon, 1990). Ballooning mode predictions for optimized pressure EQU ILI BR I UM profiles in the standard configuration limit the BOUNDARY attainable beta (Cooper et al., 1989). The maximum experimental beta achieved for both cases, with moderate heating power, is indicat- MAXIMUM ed. exp Po

-15 -10 -5 0 5 -15 -10 -5 0 5 VACUUM AXIS SHIFT (cm) VACUUM AXIS SHIFT (cm)

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1076 John Sheffield: Physics of magnetic fusion reactors

F. Power and particle control

Generic issues for power and particle handling are discussed in Sec. II.G. As in the case of tokamaks, the divertor is the preferred solution for the stellarator. For a continuous-coil system, such as the heliotron and/or torsatron, the plasma Aux is naturally diverted between the continuous helical coils (Uo et al. , 1983; Iiyoshi et al. , 1990; see Fig. 56}. The separatrix is not as clearly de6ned as in a tokamak, owing to Geld ergodization at the boundary. Nevertheless, diverted Aux is observed clearly in experiments (Anderson and Bartlitt, 1989; Obiki et al. , 1990; Matsuura et a/. , 1992). The situation for a modular stellarator is more complex, but here, too, techniques for forming a helical divertor and for diverting Qux bundles have been formulated (Strumberger, 2 4 VOLUME-AVERAGED DENSITY, (102e m s} 1992; Grieger et al. , 1992). (n)

FIG. 55. ES'ect of alpha-particle losses on the ignition contour 100 for a reference torsatron reactor operating in the electron or ion root mode: dashed curves, without alpha-particle losses; solid Lost [%] curves, all trapped alpha particles lost; chain-dashed curves, only scattered alpha-particle losses (Painter and Lyon, 1991).

50— Stellarator control is in a much earher stage of development than that for tokamaks, and tests in existing and proposed facilities are crucial to establishing viable con- ~ ~ ~ ~ ~ ~ trol ~~t ~ systems. Fueling requirements are sin.ilar to those in tokamaks, and pellet injectio~ has been used on ATF, Heliotron-E, 1 I I I and %'VII-A. 10 1O ' 10 ' 10 10 10 t [s] G. Compoter modeling (b) ~-CONTouRS WITH RISING P Codes are available similar to those used for tokamaks e.g., a revised version of the wHIsT code (Houlberg, 1982; 0.025 0.05 Lacatski et al. , 1985), which incorporates a self- consistent calculation of the electric 6eld, and HTRANS (Yamazuki and Amano, 1990). An example set of plasma operating contours (POPCONs) is shown in Fig. 57. H. Stellarator reactor options The parameters of some representative reactor designs are given in Table XIII. The options may be understood by using reference values for with the Lackner- FIG. 54. Characteristics of a-particle losses and magnetic yE Gottardi geometry in a Helias configuration, as a function of beta. (a) scaling, Eqs. (5.7), (2.27), and (2.42), and various Collisionless a-particle losses in Helias 508 one of the values of beta (2.5 to 10%) in Eqs. (2.45) and (2.46), with configurations obtained while optimizing helias configurations a fixed value for B,„and f~ =0.5, for R/a ~5: for W7-X and characterized by a special structure of 8 with (P) =0 ( o ), 0.024 ( X ), and 0.049 (A ); a particles started at aspect ratio 40 and the loss is shown as a function of the collisionless time of Qight. Each symbol indicates the loss of one particle. The number of re6ected particles is 100 in all cases. Alto- gether there are 265 particles, i.e., 165 passing particles, (Lotz et al., 1992). (b) Constant-J contours in Helias 50B with P=O, 0.024, and 0.049. Shown is a &s, 0 plane with s the Aux label and 8 the poloidal magnetic coordinate. Dashed lines indicate (c) region close to maxima, dotted lines those close to minima. The reQection value of B is a constant and defines the rejected parti- FIG. 56. Examples of baNe plate designs for the Large Helical cles considered as moderately deeply trapped (Lotz et aI., 1992). Device divertor (Iiyoshi et al., 1990).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1077

TABLE XV. Values of B(T)for typical stellarators. TABLE XVI. Values of P+ (GW) for typical stellarators, (P) =5%. (p)% X Ci.o 1 1.5 2.0 2.5 8.5 9.3 9.9 R /ap %Cga 1 1.5 2.0 5.0 5.6 6.2 6.6 5.0 6.6 3.2 1.9 7.5 4.4 4.9 5.2 7.5 7.4 3.5 2.1 10.0 8.0 3.8 2.3 12.5 8.5 4.1 2.4

' ' 734(a /a) '"(R/a ) '(p ) ""(P) " PF= 1.27 (MW) (5.13) "DT l. 82( — )1.' 09' 0.' 727f 0.727 CI.'6 +20 a )le and )0.91 )0.263( — )0.263( — )0.136(0.091f0.09 1 Co.228 7 72(R /aap (a+ / ~ Pgppg +20 ~ I6 ' 0.341 (5.14) (p)0.591 +e

For more peaked pressure profiles, the performance Since fJ3 —0.5, if (P) =S%%uo, B~~„=13T. These parame- would be better; e.g., if there were a 20% higher PF for a ters and those in Table XVI encompass the parameters of given beta, the multipliers in Eqs. (5.13) and (5.14) would the torsatron-heliotrons shown in Table XIII. be, respectively, 661 and 7.46. (2) The fusion power Pz depends strongly on the Since n20 and (p) may be specified independently, it is enhancement factor CLG. For (P) =5% and the param- necessary to confirm for a particular set of values that the eters above, enhancement ( —1.5 to 2.0 times) is needed average temperature T ~ 7.5 keV to justify the use of the to bring the power down to a respectable level ( S4 GW) formula for D-T fusion power. as shown in Table XVI. The formula in Eqs. (5.13) and (1) Note first that for the Lackner-Gottardi scaling the (5.14) may be used to calculate helias reactor parameters. field (B) depends weakly on most parameters except (P). For R/a =12.5, 4=0.9, p„„=1.6 MW/m, nDT/n, For example, if a /a=1. l, 4=0.7, p „=2.5 MWm =0.80, f =0.9, a /a=1. l, (P) =5%, and including R/ap=75 "2o 75 "2o 2.5, f =0.8, nDT/n, =0.73, the atomic mass term ( A; /1. 5 ) in the Lackner- and (P) and CzG are varied, B has the values shown in Gottardi scaling with A; =2.5 leads to a reactor of major Table XV. A,./1. 5 has been set at unity in Eq. (5.7). radius R = 19 m and fusion power PF = 1830 MW. (3) Care must be taken to ensure that g, &g;&„Eq. (3.23). For the example above with R/a =7.5, Cz =2, (P ) =5%, the stellarator parameters are R = 11.0 m, ' dip 1 46 m Tk 1 1 6 keV TE 1 63 s, and y E —0.50 E 3.2 m s '. To ensuregE &g,.z, requires fz=P/T&2. (4) The density and temperature dependences of Large Helical Device scaling factor the ion i.e. high den- 100 root, , 2.4- sity and moderate temperature. To obtain reasonable CQ 80 values of typically requires —1.5 to 2. Note that UJ PF CLG Cl C) the ion root should be more satisfactory for alleviating UJ divertor but limited accu- (3 1.6 problems, may be by impurity CC 40 mulation UJ problems. O (5) The auxiliary power required to raise the plasma UJ temperature to ignition may be calculated using Eq. 0.8- 20 0 (2.75). [The simple formula (2.77) may not be used because the temperature dependences of Lackner-Gottardi losses and alpha heating are similar. ] To obtain a refer- I I l ence value of power, the temperature =3.5 is used 8 12 16 20 ( Tk ) for the example above, R/a =7. =2, =S%%uo, DENSITY-AVERAGED TEMPERATURE, (T) {keV) 5, Czo (P) Z,8=1.8 and nDT/n, =0.8, FICx. 57. Plasma operating contour {POPCON) plot for a torsatron obtained with the global LHD confinement model and — CE = 1.5 (Painter and Lyon, 1991). P =27+54 41=40 MW .

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors

Additional powcI' 1s icqu1Icd to raise thc tcIIlpc1RtuI'c 1Il a reasonable length of tiIDC, say +20 MW, giving a total I', =60 MW. PLENUM In loss varies (6) stellarators, alpha power from negligi- PLASMA ble (helias) to -40% in some torsatron-heliotron designs. VAN)V Configurations with a high loss remove essentially all of Vr/ÃW the alphas before they thermalize. Therefore there should be less dilution of the I3-T fuel fraction owing to SHIELD WAX/i I accumulation of helium (assumed to lead to a 10—20% WZÃA, dilution in ITER). Assuming the high-loss cases have negligible dilution, the reactor size might have to be increased by 2% to 10% for 40% losses, when compared to a 5% loss case, depending on design. FIRST WALL/BLANKET

I l I I. Conclusions 3 4 5 R (m) (1) Stellarator reactors at the I-GWe level, are viable from an ignition point of view provided. FIG. 58. Isometric sketch of CRFPR fusion power core. The (a) An enhancement factor of CLo -—(1.5 —2) can be ob- reactor torus, de6ned as the plasma chamber, first wall, limiter, tained for I.ackner-Gottardi scaling and 3,./1. 5 = 1. blanket, shield/structure, and toroid field coils, is installed and (b) There is a radial potential factor fz ~ 2 for torsatron- replaced as a single 400-tonne unit, drained of Li-Pb coolant heliotron reactors. (Hagenson and Krakowski, 1981). (c) (P) )5% for B,„S13 T. (2) For the Lackner-Cxottardi scaling the ion root is favored (higher density and moderate temperature), growth of interest in the reversed-field pinch (RFP). Ex- which is good for divertor operation. Further research is cellent reviews of the subject have been written by Baker required to determine the optimum conditions which will and Quinn (1981), Bodin et al. (1986), Taylor (1986), be free of impurity accumulation. Baker (1988), Miyamoto (1988), Ortolani (1988, 1989), (3) The development of a stellarator reactor is on a and Bodin (1990). An isometric sketch of the CRFPR longer time scale than that for tokamaks because TFTR- reactor is shown in Fig. 58. scale devices have not yet been operated. Successful operation of large devices such as the large Helical De- B. RFP characteristics vice and WVII-X would give confidence in the underlying physics and permit consideration of a 0-T-burning The RFP is an axisymmetric magnetic-IIield con- stellarator. The combined output of such a stellarator figuration in which B&&8& outside the plasma and and ITER would be required to proceed to a demonstra- Bs-8& inside the—plasma. The poloidal field (Bs) is gen- tion stellarator reactor. erated by a toroidal plasma current. The toroidal field (8&) is produced mainly by currents ffowing in the plas- VI. REVERSEI3-FIELD-PINCH REACTORS ma and partly by external coils. The reversed toroidal field shown in Fig. 59 is generated by a relaxation process, and it is sustained the continued Cohorts of the A. Introduction by configuration to remain in the minimum. energy state- the dynamo efFect. The 6rst type of magnetic fusion reactor to be patented was a pinch device (Thomson and Blackman, 1952). It was R simple toI'01dal p1nch witli R tl Rvcllng"wave Rp" I. Plasma startup proach proposed to drive the toroidal current. Such devices were found to su6'er from severe instabilities. The The RFP plasma is created in a thin-walled metal stability wa. improved by the addition of R weak toIoidal torus using an inductively driven plasma current. Nor- magnetic field. However, the plasma remained turbulent mally, a thick-walled metal liner surrounds the torus and RIld con6ncmcnt was pooI'. Development of Rn 1ntcrcst- provides short-terIn stability. In early experiments, the ing concept came from the discovery of a quiescent mode pulse length was relatively short ( (ms), and the plasma in the ZETA pinch experiment (Robinson and King, characteristics depended strongly on the initial hydrogen 1969). Subsequently, Taylor (1974, 1976), showed that filling pressure. In modern experiments the pulse length the mode was a result of the Inagnetic field and plasma has been extended to tens of ms and the plasma density configuration's relaxing to a maximum energy state in has been controlled using wall-conditioning techniques, which the toroidal magnetic field reversed sign as a func- gas puffing, and pellet injection (Wurden et al. , 1987). In tion of the minor radius. This work accompanied a some experiments the thin metal wall has been protected

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1079

TABLE XVII. Parameters of representative RFPs. Experiment ZT-40 HBTX PBE-IR(M 15) RFXb

Major radius R (m) 1.14 1.50 0.8 0.72 2.0 Major radius a (m) 0.20 0.52 0.26 0.135 0.50 Peak current I (MA) 0.45 0.6(1.0) 0.50 0.2 2.0 Pulse length 7p ] (ms) &40 &80 514 ~ 0.8 250 Average density n (X10 m ) 0.1—1.0 0.1 0.1 —1.0 0.1 —1.0 Axial electron temperature 500 350 450 600 T,o (keV) Axial ion temperature T;p (keV) 500 300 450 600 Poloidal beta Pe 0.1 —0.2 & 0.17 0.1 —0.2 -0.1 Energy confinement time ~E (ms) ~ 0.8 1.5 ~ 0.4 -0.1 'Initial operation. Beginning operation. by graphite tiles. The parameters of some modern exper- poloidal Aux converted to toroidal field by the dynamo iments are shown in Table XVII. action and maximize the poloidal flux inductively con- A weak 8& field is applied prior to initiation of the verted to toroidal plasma current. Because the reversal is plasma current. It is common practice, but not essential, sustained naturally by the dynamo eff'ect (see below), and to assist the field reversal by, for example, control of the the plasma current provides the heating, the pulse length toroidal field. This action can minimize the use of the is determined by volt-second consumption, unless a noninductive current drive is used. It is observed that the volt-seconds used in startup increase with decreasing rate of rise of plasma current (Bodin et al. , 1986; Phillips et al. , 1988). CC te exceeds (4. QJ ~ The resistivity (g) the Spitzer value, Eq. 5), ~ 2 times) the correction due impuri- I— by more ( than to 0.1 D ties. There is evidence (Bodin, 1990) that the excess resis- UJ tance may depend on edge e6ects connected to field er- LL O I~ rors and the equilibrium position of the plasma. For ex-

LIJ ample, the plasma resistance increased when a carbon tile was inserted into the plasma, in proportion to the U BpJI ( amount of Aux intercepted. Some theoretical models in- / voke helicity dissipation in the plasma edge (Jarboe and Alper, 1987; Ho et al. , 1989; Ho and Prager, 1991) and 0 I momentum transport by energetic electrons (Moses I(&) et al. , 1988), to explain the phenomenon. Situations in 0 10 20 30 which there is strong ion heating, and even T; /T, ) 1, RAD I US (cm) & Bp"p (VI) appear to be a consequence of the anomalous resistivity (see Fig. 60). The power density may be written as

I I I I I I I I I I I I BESSELFUNCITON MODEL 500

400 cd 2 (Z LU 300 C3 I— 200 0 I I I I I I I I 1 I l I 0 10 20

RAD IUS (cm) 100 I I I I 0.15 0.20 0.25 0.30 0.35 0.40 FIG. 59. Comparison of theoretical and experimental PF magnetic-Geld profiles for an RFP. (a) Experimental RFP magnetic-field profiles and profiles from the Bessel-function FIG. 60. Ion temperature T vs fluctuating power absorption P& model; (b) corresponding p profiles. in ZT-40 (Schoenberg et al., 1989).

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&s(~) Vo al (6.1) (6.6) Resistive Dynamo, &a~& 2 y where the first term heats the electrons and the second Here P is the toroidal fiux. Field reversal occurs when heats the ions. The latter effect should ease the attain- 9&1.2. It is customary to describe RFP operation in ment of ignition. Alternative explanations of the terms of 8 and I', the field-reversal parameter, phenomenon center on viscous damping of tearing Auc- &p(a) tuations and nonviscous damping of small-scale waves ' (6.7) (&q& (Mattor et al. , 1992). A consequence of the theory is that if the current is sus- 2. Relaxed states tained at a high enough level there will be continuous reversed-field generation (dynamo eFect). This is ob- The presence of turbulence and a small resistivity a1- served experiinentally; see Fig. 62 (Caramana and Baker, low magnetic-field lines in a plasma to reconnect and 1984). The nonlinear MHD calculations of Caramana self-organize into a state of minimum magnetic energy, et al. (1983) and Caramana and Schnack (1986) have when subject to certain constraints (Taylor, 1986). The been very successful in explaining the detailed dynamics force-free fields, current everywhere parallel to the mag- of the dynamo and fiuctuations (Watt and Nebel, 1983). netic field, with p=0, are described by The experimental field profiles do not have p=constant near the plasma edge [@=@0[1 (r/a—) ]], owing to V'X S=pS, (6.2) finite beta and resistivity effects. Nevertheless, the theory predicts the experimental equilibria well. where a uniform p represents the lowest magnetic energy in a closed system with constant total magnetic helicity (Woltjer, 1958). In particular, in a toroidal pinch sur- 3. Equilibrium rounded by an ideal conducting she11, the toroidal IfIux is conserved and a minimum energy state can be found In a large-aspect-ratio RFP the radial pressure balance where the only other constraint for a resistive plasma is 1s that of total magnetic helicity invariance. An example of Bg+8~ Bg helicity-conserving reconnection is shown in Fig. 61. p+ + ——=0. (6.8) The total magnetic helicity is given by dt 2po por Ko= A.BdV, (6.3) With zero pressure at the plasma boundary (a) this fVo reduces to where the vector potential is given by 8=7'X A. The solution of the equations for uniform p in a large- aspect-ratio torus of circular cross section is referred to as the Bessel-function mode1. For 1ow beta, '2 1 =1+ I' (6 9) &s =BsDJi (pr), 8~ =B~Jo(pr), 8„=0 . g2 For finite Ps, More accurate calculations yield ps-—2p (Bodin et al. , 2 1986), where p is the beta, normalized to the total field. e. The poloidal beta is typically in the range 0.1 —0.2, which is lower than the limits predicted by resistive MHD The parameter (p) and the radius of the fiux-conserving theory. The tendency of the plasma to expand in major boundary are related by 10 (6.5) X 8 where the pinch parameter is given by X) ~a ~~4a ~) - THEORY D (NO FIFLD GENERATION)

0 I I 0 10 15 20 TIME (ms) FIG-. 62. Time variation of toroidal flux: measurement and calculation with no reversed-field generation (no dynamo) for ZT- FIG. 61. Example of helicity-conserving reconnection between 40M, illustrating the dynamo effect (Caramana and Baker, two Aux tubes (Qrtolani, 1988). 1984).

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00 radius may be counteracted by image currents in a con- I I I I I IIII I I I I I IIIl I I I I I IIII I I I I I IIII I ducting shell or by an applied vertical field (B„). The radial displacement in steady state, typically a few percent EXPERIMENTAL POINTS TITAN O- of the minor radius, is O DESIGN POINTS — a C (/; 1) Q a, pe+ 1 — +ln (6.10) 2R Q C a 102 = CONSTA where a, is the radius of the conducting shell (Shafranov, Ps 1966). Field errors, owing to the use of an imperfect conducting shell —poloidal gaps, ports —and coil-induced MS ripple are observed to degrade the equilibrium 103 = ZT-40M o configuration and plasma performance. This is an impor- ' ~ HBTX-1 B tant consideration in reactor design (Bodin et a/. , 1986). HTE ~ + TPE-1RM &0-4 E-1 RM C. Transpart BETA II Neoclassical transport in an RFP is even lower than in a tokamak (see Sec. III.C), owing to improved orbits &0-' I I I I I I II I I I I I I I II I I I I I I I II I I I I I I I I (Boozer, 1983). As in the case of the tokamak, however, 103 &0-' io-' &0' &0" &0' the observed transport exceeds neoclassical levels, a2[3/2([/N)3/2 [/n2 MA3/2 (1P-14 A m}3/2] presumably due to the resistive-mode turbulence which produces the field reversal. FICi. 63. Confinement time for RFP experiments Plotted vs the Connor-Taylor 1988) (and up- A principal issue for the RFP is at what scale (current, constant p/b scaling (DiMarco, dates). size} the resistive-mode activity, necessary to sustain the configuration, will be weak enough to allow reactor-grade temperatures to be reached. Two theoretical models higher density —versus the scaling parameter I /(an' ) have been used to characterize the anomalous losses, is shown in Fig. 63. Support for the thesis that beta both based on the efTects of resistive Quid turbulence. confinement adjusts to keep beta constant comes from ex- Connor and Taylor (1984) use the invariance proper- periments in which krypton was added to the plasma ties of the reduced resistive MHD equations to develop a (Pickrell et a/ , 1984). A. s the fraction of radiated power scaling for transport, which indicates that beta should be in ZT-40M increased, beta remained roughly constant approximately constant, as observed in some experiments and the Ohmic power increased only slightly, showing an (Phillips et a/. , 1984). They derive a quantitative esti- increase in ~E (decrease in conduction). mate for resistive g-mode losses, However, there are experimental data showing Pe de- 1/2 creasing with increasing current (Wurden et a/. , 1985; Alper et a/. , 1987). This efFect may be the result of 6r/ m, p' changes in plasma profiles, T, /T;, or radiation levels. This result is also consistent with a resistive interchange I for Ohmic heating . (6.11) turbulence model (Carreras and Diamond, 1989), in which the restriction to equal scaling for electrostatic Because Ohmic power is much less than alpha power in and magnetic Quctuations, used by Connor and Taylor, is an ignited reactor, it is necessary to rewrite this scaling removed. In the Carreras-Diamond model, formula in terms of total power delivered to the plasma. I2 CD ~ in 1. ~ for Ohmic heating . (6.12) For heating general the form E I / A a n ( A; na P) ma'y be obtained. An analysis of experimental data by DiMarco (1988) and an analysis by For heating in general this has the form ~E Qrtolani and Rostagni (1983) for operation near the den- ~ I' R '/( A ' n P'). When benchmarked sity limit support this result. A plot of the best energy against present experiments (DiMarco, 1988), the confinement times —typically higher beta is achieved at Connor-Taylor version becomes

—3 cT [I(MA)] /[n(X10 m )]' [a(m)] (s) . (6.13)

The Carreras-Diamond model benchmarked against the same data becomes

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1082 John Sheffield: Physics of magnetic fusion reactors

2.' 8 X 10 rP= [I(MA)] [n(X10 m )][a(m} ] (s} . (6.14)

For example, the TITAN reactor has I= 17.8 MA, It is important to realize that resistive MHD activity in n =9X10 m, a=0.6 m, A;=2.6, Z, =2, and represent-day experiments may be masking other sources quires ~z =0.22 s. In the Connor- Taylor model, of turbulence that might emerge as important in the reac- ~E —0.61 s, while in the Carreras-Diamond model, tor regime. The ion temperature-gradient mode has been zz —0.072 s. For confinement formulas written in terms invoked as a candidate mechanism in the edge region of total power, i.e., not Ohmic power alone [see Eqs. (Schnack et al. , 1985), thought it was not observed in (6.19) and (6.20)], the equivalent confinement times, using ZT-40M (Schoenberg et al. , 1989). the full conduction power, are 0.31 and 0.06 s, respective- There is a density limit for RFP s similar to that ob- ly. If a large amount of power were radiated, and a lower served in tokamaks (Ortolani and Rostagni, 1983), conduction power were appropriate for use in the formulas, the confinement times would improve. The conse- na — —1 1014 A 1m (6.15) quences of using the two formulas are discussed in Sec. IV.H. Electrostatic fluctuations can account for particle However, in contrast to tokamaks, no major or density- transport in the edge but not heat transport (Ji et al. , limiting disruptions have been seen, even with high radia- 1991; Rempel et al. 1991; Tsui et al. , 1991). Energetic tion fractions (0.95) and large pellet fueling (Wurden electrons from the plasma core traveling along the field et al. , 1987). lines remove 35—40% of the input power in ZT-40 The Connor-Taylor confinement model may be used to (Weber et al. , 1991). These losses are consistent with show that Ohmic heating to ignition may be possible in the Rech ester-Rosenbluth 1978 model of stochastic an RFP reactor. Substituting from Eq. (6.13), with magnetic-field transport (Schoenberg and Moses, 1991). yE-a /6&E in Eq. (2.75), yields

1.21X10 a RTk n20 0. 11Ra Zegn 2pTk dt n20 m conduction losses bremsstrahlung 0. 2 08y~ Z,gRn 20 I io-'Ra' n' ' T" 2 1.5 +1.9X (MW) a TI n2o ne L Ohmic heating Alpha heating (6.16)

For the TITAN reactor, R=3.9 m, a=0.6 m, and I Fig. 64. n2p, and Tk are raised together during startup to the final At the TITAN operating point the alpha heating values I =17.8 MA, n20=9X10 m, Tk =10 keV. exceeds the losses, for both confinement scalings. This Setting yii =1.5, Z,&=2, and nDT/n, =0.83, assuming would permit the reactor parameters to be relaxed or im- no helium ash buildup during startup, and assuming a purities to be added, as in the TITAN case, to radiate and relieve linear rise of current and density, I /n zo = 1.98 power the divertor problems. The test done MA/(10 m ), we find that the power balance becomes in ZT-40M in which krypton was added supports this hy- pothesis (Pickrell et al., 1984; also see Werley, 1991}. Tk d 8' 2 0.5 Such behavior implies an underlying confinement that 19.4 —0.31n 2oTk n P. 5 may be better than the scalings indicate.

10.2n 2p +, +0.018n 20Tk (MW) . (6.17) Tk D. MHD limits Balancing conduction losses against Ohmic heating, we find that Po &P„„d, provided n20/Tk &1.3. At about Owing to finite resistivity, the magnetic-field diffusion Tk-4. 5 keV, alpha heating dominates over the losses, alters the configuration. Resistive instabilities act to re- and the plasma ignites. A plot of the required current as store the configuration to the Taylor minimum energy a function of T and N for the TITAN model is shown in state. The increased plasma resistivity at the plasma

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1083

and resistive current-driven modes (m=O and 1; Ho et al., 1989, Ito and Prager, 1991). Tearing modes may be stabilized by plasma rotation (Bodin, 1990). In experiments on HBTX and OHTE, the current-driven modes were observed to grow when rz„», &r,» (Alper et al., 1989). The growth rate was proportional to r„,» (Alper et al., 1987}. More information is required to determine whether a reactor could use a resistive wall and whether external feedback stabilization of the plasma would be possible. A comprehensive study of feedback control of such instabilities with a resistive wall has been made by Zita et al. (1992). In the TITAN study it was assumed that the first wall, liquid lithium coolant, and blanket materials would act as a conducting wall.

E. Current drive

For the condition proposed for an RFP reactor, e.g., TITAN, n20=9, T-10 keV, and Pe-0.2, the bootstrap current is expected to be small; see Eq. (4.8). The noninductive current-drive techniques which use neutral beams and RF, discussed in Sec. IV.C, will be too inefficient at FICx. 64. The required power for power balance as a function of such high densities. A possible solution, to extend pulse TITAN and plasma density temperature. The steady-state lengths beyond a few 100 s, is to use magnetic helicity in- TITAN plasma is denoted a filled square and the saddle by jection, either directly, as has been used in Spheromaks point by a filled circle. The TITAN startup path to ignition and (Barnes, 1986) or the oscillating-field current-drive burn is located above this point and passes close to the saddle by (Bevir and Schoenberg point (Najmabadi et al., 1990). (OFCD) approach Grey, 1982; et al., 1984}. In the latter approach, oscillating, out-of- phase fields are applied through the poloidal and toroidal edge leads to increased activity in that region to coun- circuits. The rate of helicity injection is given by 2/V&, teract the field diffusion. where dP/dt =—Ve. If V& = V& coscot and Ve = Veo sin( wt +0), and it is as- re- 1. infinitely conducting shell sumed that the plasma will continually adjust to its laxed state, the system acts as if it has a steady loop volt- Configurations with tailored shear and pressure age V„ profiles have been found which are completely stable to V&sinO ideal modes for the reversed state (Robinson, 1971; V, ==1 (6.18) 2 f (/ (r)]— Goedblod and Sakanaka, 1974; Freidberg, 1982},provid- Q V~ ed Pe%0.5 and (a, /a) (3~B&(0)~B&(a„)~ and the total where f 5 1 is a constant dependent on field distribution; toroidal fiux P &0. dglg and sin8 are the fractional oscillations in P and V&. Stable configurations have also been found for resistive The relaxation occurs on an MHD time scale, which is tearing modes for P =0.2 (Robinson, 1978; Schnack much shorter than the oscillating-field period. Calcula- et a/. , 1985). These modes have large toroidal mode tions for the TITAN reactor have been made using a 1D numbers, n &20, and m=0 and 1; such modes are ob- MHD model for oscillating-field current drive (Najma- served. In experiments, RPF configurations with 1 Pe-0. badi et a/. , 1990). are calculated to be stable or close to stability for resis- The requirement that (5K/dt ) =0 results in (5$/$0) tive tearing modes (Bodin, 1990). However, the (5V&/V&)=2. Toroidal fiux oscillations, 5$/$~0. 05, configurations are unstable the local resistive inter- to are required to avoid the loss of toroidal field reversal; (or resistive mode) (Hender and Robinson, change g therefore the ac toroidal voltage needed to sustain the 1983). current may be -40 times greater than that required inductively. Similarly, there is a lower bound at which 5I& 2. Resistive shell becomes too large (It, becomes too small, so that reversal is lost), leaving a narrow window for 5$/P. Modeling of The use of a resistive shell is an attractive option be- the currents, including the conducting wall, showed that cause it allows the use of plasma control with an applied the wall must contain insulating breaks to avoid large Bz and also minimizes field errors froxn gaps and ports. power dissipation, but not so great as to impair wall sta- Unfortunately, it is predicted to destabilize both ideal bilization. The operating window shrinks with decreas-

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1084 John SheffieId: Physics of magnetic fusion reactors ing frequency and, for TITAN, -25 Hz is required. The were used to assess the impact of field ripple on magnetic calculations for TITAN have 5$//=0. 0365, leading to a island formation. A systems code (Copenhaver et al., variation in 8 from 1.5 to 1.6 and I from —0.03 to 0.17, 1985) marrying various physics, engineering, and costing and a calculated dissipated power of 49 MW (for TITAN elements was developed for the TITAN reactor studies. II) to 57 MW (for TITAN 1) to drive 17.8 MA. H. RFP reactors Experiments on ZT-40 (Wurden, Schoenberg, et al., 1987; Schoenberg et al. , 1988), showed the expected Studies of RFP reactors have been made by Hancox features of Eq. (6.18), notably an increase or decrease or et al. (1981), Krakowski et al. (1986), and Najmabadi loss of current for the appropriate phasing, but only at et al. (1990). The recent work has emphasized the value lower input powers. At higher drive power, current in- of the RFP's moderate beta ( —10%) and high field ratio creases were masked by plasma-wall interactions, which f73 —1 —1.3, which come with this current-dominated increased the plasma resistance. A ramping of about 5% configuration and which permit a relatively higher- of the current was obtained at low power. power-density reactor core than tokamaks and stellarators, e.g., p„„up to 20 M%'m, rather than S6 MWm . To offset the F. Power and particle handling power handling problems, it is assumed in the TITAN study, based on experimental evidence (Pickrell et al. that Generic issues for power and particle handling are dis- , 1984), confinement will be suSciently good to permit added xenon ra- cussed in Sec. II.G. To date, RFP's have operated with impurities to diate -70%%uo of the core with short pulse length, in limiter-defined plasmas. Poloidal power, a substantial added fraction of the remaining power's pumped limiters and toroidal magnetic limiters have being radiated at the plasma edge. Parameters been considered for RFP reactors (Bathke and of three reactors are shown in Table XVIII. Krakowski, 1985; Najmabadi et al., 1990; see Fig. 65). Two Inodels of transport have been developed to de- In TITAN a large percentage of the core power ( -70%%uo) scribe RFP transport. have been is radiated using xenon impurities, on the assumption They benchmarked against experimental data for Ohmic that there is excess confinement. The reduced power Aux heating by DiMarco (1988). Quantitative formulas, taking into im- to the divertor, coupled with the high operating density account purity levels and atomic are in of RFP, )&10 m, leads to a low divertor plasma tem- mass, given Eqs. (6.13) and (6.14). For a reactor, Ohmic heating is small com- perature. The plasma density is sustained by pe11et fuel- pared to alpha heating, and it is necessary ing at the moderate velocities, 2 kms ', required to to rewrite the formula in terms of the total fuel past the reversal radius. power P (MW): 'C CT 6 1X10 cTI'"R""m 0.' 333 0.' 667 0.333 (6.19) G. Computer modeling A/ 0 Pl2oPm Ii. 9 5X10 CCD m g A number of coxnputer codes have been applied to CD 0.' 182 3.455 0.455 (6.20) modeling. RFP plasmas and reactors. Among the tools A &2o ~m used have been 3D MHD stability codes (Aydemir et al., I (MA) and CCT and CcD are multipliers to allow for 1985; Kusano and Sato, 1990; Schnack et al., 1985), a 2D degradation or enhancement. These formulas may be equilibrium code (Jardin et a/. , 1986), a 1D core-plasma combined with Eqs. (2.45) and (2.46) to obtain a value for transport code (Nebel, 1980), and divertor, edge-plasma the fusion power and poloidal field needed for a self- codes such as ODEssw (Prinja et a/. , 1987) and DEG&s sustained plasma. In deriving these formulas we set (Heifetz et a/. , 1982; Werley, 1987, Braams, 1987). De- ~=1, 7.z-—a /6yz, and I' =0.2f I'F, and we use Eq. tailed 3D field-line tracing codes and 3D MHD codes (2.42) for p „ to obtain

I I I I I I FW/B/S I FW/B/S FIELD LINES FIELD LINES SYMMETRIC SYMMETRIC DIVERTOR- DIVERTOR

~ ~ FIG. 65. Field-line tracing in the compact r rr &r' r', r ~ reversed-field pinch reactor equations plane for ~ y 4I r r ~or ~ r r ~ ~ r ~ ~r poloidally symmetric and bundle divertors rr ~r (Hagenson and Krakowski, 1981). r rr rr rrrr rr ~ r rrr

rr r ~ t I' i I i Ii )II I I i I i I 6 0 x (m) x (m)

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TABLE XVIII. RFP reaction design. Reactor CRFPR (5) CRFPR (20) TITAN-1 p.„(MWm-') 5 20 18 PF (MW) 2900 2740 2300 R (m) 7.6 3.8 3.9 a (m) 1.42 0.71 0.60 a (m) 1.52 0.75 0.66 n (X10 m ) 2.3 6.3 9.0 T (keV) 10.0 10.0 10.0 0.23' 0 23' 0.23' ~E (s) 0.70 0.23 0.22 Bq {T) 3.0 5.2 5.9 B,„(T) coil 2.6 4.5 5.7 I (MA) 21.6 18.4 17.8 P,„„(MW) {—) (73) 57 P„„[MW(e)] 1000 1000 970 'Includes 0.03 for alpha-particle pressure (P }=0.12.

' 5.0X10 A'i n 20 (R/a)(n /n )' P pCT (MW} (6.21) C4.0f2.66(a /a)2. 33(P )2.33 0.884n 2.21(R 42 3 6 X 106+l 20 /a)p4. pcD (MW), (6.22) C4.86f 2.65(n /n )0.442(a /a)1. 43(p )1.43 53(R /a)0 125(a /a)0 375(P )0 475 B6)= (T). pO. 125( n /n )0.5pO. 5

I (1) The Connor-Taylor scaling leads to a TITAN-like RFP is the relationship, under reactor conditions, be- reactor. For example, with A; =2.5, n20=9, R /a=6. 33, tween the maintenance of the configuration (dynamo nDT/n, =073, P&=0.21, CCT=l, f~=0.3, a /a=1. l5, effect) by turbulence and the level of transport due to this and p „=19 MW rn, we obtain PF —2370 MW, turbulence. Bs-5.8 T, I =—17.7 MA, a=0.61 m, (Tk ) =10 keV. (2) With optimistic Connor-Taylor transport scaling, Note that these parameters have sufFicient excess there is sufIIicient margin of confinement to permit a large confinement to permit a large amount of power to be ra- amount of the fusion power to be radiated by added im- diated (f =0.3), alleviating the divertor problem. purities, thereby easing the divertor problem. Under (2) By contrast, the Carreras-Diamond scaling leads to these circumstances it is beneficial to operate at high den- a large reactor with no margin to increase radiation lev- sity. However, the "conventional" radio frequency and els above the normal -20%. For example, A;=2.5, hearn current-drive schemes are then too inefFicient to be n20=3.5, R/a=6. 33, nDT/n, =0.83, P&=0.15, CcD =1, used. Because of the low poloidal beta, the bootstrap tokamak. f~=0.8, a„/a=1.15, and p„„=18 MWm leads to currents in an RFP are small compared to a injection in one form or another offers an in- PF =4790 MW, Bz —-5.9 T, I =25.5 MA, a=0.86 m, Helicity teresting opportunity to provide steady-state operation. ( Tk ) =19 keV. Decreasing from 0.8 —+0.6 doubles f Present results from experiments (F 8pumping) ar-e PF. equivocal, and further tests are essential. (3) To recover a similar TITAN-like reactor to that (3}With the more pessimistic Carreras-Diamond mod- with Connor-Taylor scaling requires Cco =4.2. el the transport margin implied by the Connor-Taylor model is absent. As a consequence it is not possible to I. Conclusions operate at high power density except with either very high fusion power () 10000 MW) or an enhancement (1) Th«FP»s the potential to make an attractive factor CCT-4. If transport adjusts to maintain beta, and reactor; however, the experimental data base is weak is effectively lower than power is radiated, then this compared to that of the tokamak. As in the case of other might provide an enhancement factor and permit sub- configurations, uncertainty remains in transport scaling, stantial radiated power. and data from RFX and subsequent large experiments (4) Experiments and theory suggest that external feed- will be necessary to confirrn the viability of the reversed- back control and a conducting shell may be required near field pinch as a reactor. Of particular importance to the the plasma to stabilize MHD activity. Providing a shell

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1086 John SheffieId: Physics of magnetic fusion reactors involves a tradeoff' between stabilization and permitting time-varying fields to be applied to the plasma. Gaps re- SHIELD quired for oscillating-field current drive should not inter- REGION CONDUCTING SHELL fere with the stabilization.

BLANKET AB REGION Vll. FIELD-REVERSED CONFIGURATIONS FIRST WALL ////////////////////////////////////////////////////////////////////////////////////'~ "c

A. Introduction f,, a The compact torus area of research includes field- R rs reversed configurations (FRC} and field-reversed mirrors PLASMA ON OPEN (B&»B& and S& 1), spheromaks (Bti-B& and S& 1), CLOSED SEPARATRIX FIELD FIELD LINES LINE and the Astron (Bs»B& and S( 1), where S is the ratio of the toroidal plasma radius to the external ion gyrora- FIG. 66. FRC reactor geometry showing the radius of a con- dius (p;, ). Recent reviews of the area have been made by ducting shell (r, ), first wall (r ), separatrix (r, ), and plasmoid Tuszewski (1988) for FRC, Jarboe (1989) for spheromaks, length (I). The conducting shell thickness (5) is located outside and Wright (1990) for both configurations. FRC reactor a breeding blanket of thickness (hb) (Hagenson and Krakowski, 1981). studies include those of Willenburg et al. (1979), Willen- berg (1980), Robson (1980}, Krakowski et al. (1981), Hagenson and Krakowski (1981), Alikhanov et al. through the blanket area using a weak gradient in the (1983), Hirano (1984), Vlases et al. (1986), Burtsev et al. solenoidal magnetic field. This approach allows the use (1987), Kernbichler et al. (1991), and Momota et al. of eKcient adiabatic compression and a separation of the (1992). The FRC represents a diff'erent class of toroidal high-technology formation phase from the burn and device than the tokamak, stellarator, and RFP, since it quench phases. has no coil down the center of the toroidal plasma. It In the steady-state system the plasma might be formed must be emphasized that, while it is an intruging candi- and then expanded into a reaction chamber, where it date for a fusion reactor, its experimental data base is far would be heated and the current sustained by neutral- smaller than that of the other three configurations. beam or rf power (Kernbichler et al., 1991; Momota et a/. , 1992}and, in some cases, by current driven by the charged fusion products (Berk et al., 1988). The magnet- B. FRC characteristics ic structure includes a natural divertor. A schematic diagram of the pulsed reactor CTOR is shown in Fig. 67. The FRC is a highly elongated torus of aspect ratio about one, formed without a toroidal field by a toroidal current sheet and an external axial field; see Fig. 66. The C. Equilibrium configuration may be formed in a linear "theta-pinch" device by creating a plasma in an insulating cylinder con- The radial pressure balance may be written as taining an initial axial magnetic field B,o. The preionization is usually done with an oscillating discharge using Bz 8e + (7.1) the theta coil. It is sometimes supplemented by an axial Pm P discharge between electrodes or by other techniques to enhance the uniformity of the initial plasma. A main bank is then the capacitor applied to coil, and the LOWER HYBRID DRIFT TRANSPORT (x = 0.75) toroidal current induced in the plasma leads to a reversal of the initial field and the formation of an elongated Ty 't c I / toroidal plasma. This plasma may be contained in the discharge chamber by mirror fields. FROP COMPRESSOR REACTOR BURN QUENCH The configuration is particularly interesting because of SOURCE CHAMBER CHAMBER I } I its very high beta, 50—90%, and small aspect ratio with 0 5 10 15 m no toroidal coil. The cylindrical geometry permits the cr) 10- PLASMA . use of a simple blanket, shield, and coil arrangement in a K EXP.ANS ION tjj reactor. The very high beta allows the use of low-field }- Uj C~ coils and may result in reactor systems of relatively small II/10 unit size compared to other configurations, provided the 0 10 20 30 40 50 60 transport scales favorably. The bulk of reactor studies LENGTH (m) have been on pulsed systems; however, steady-state sys- FIG. 67. Reactor dimensions showing plasmoid trajectory in tems are now being studied more seriously. the CTOR reactor, using lower-hybrid drift transport (Hagen- In pulsed systems the FRC plasmas may be translated son and Krakowski, 1981).

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1087

where B, is the external equilibrium Geld. l, /2r, —3 —10 (see Fig. 66). To match the experimental If p is a function only of the poloidal Aux variable, it is equilibria in code calculations it is necessary to invoke possible to show that r, =i/2R where R is the radius of steep pressure gradients at the separatrix (Shumaker, the field null, and r, is the separatrix major radius. From 1984). the axial equilibrium (Barnes et al. , 1979), X D. Startup (P) =1— (7.2) 2', The FRC was discovered by accident in early theta- where pinch research, when a bias field was applied in the re- verse direction to the main magnetic field, and spontane- f' ~ rdr ous formation of closed poloidal fields was observed. r, o p Subsequently, it was demonstrated that similar configurations could be formed using rotating magnetic is the volume average of beta (P=p /p ) within the fields (Hugrass et al. , 1980), with a coaxial geometry separatrix, =r, is the radius of the conduc- x, /r„and r, (Pietrzyk et al., 1987), and in the field-reversed mirror tor which conserves the Aux. Representative plasma and (FRM) approach, by using neutral-beam injection into a field profiles are shown in Fig. 68. It may also be shown magnetic mirror (Pearlstein et al., 1979). The formation that, for the range of possible poloidal profiles, the po- ~ of a plasma in a theta pinch is illustrated in Fig. 69. loidal fiux is bounded by the values (x, /i/2) P/ There are two main issues for startup of an FRC reac- (mR B,)~x, /&2 (Tuszewski et a/. , 1982). In turn, the tor. First, plasma tearing activity as a process for form- cruxes ing the plasma should be a avoided, as it can lead to 3+6 in fast for- ' asymmetries. Second, present-day systems a +s /=mr, B, (7.3) mation scheme is generally used which, in a reactor, would require large voltages on the formation coil and where e ranges from 0 to 1. The ratio of plasma scale large amounts of pulsed power. The former issue has length to ion gyroradius is defined as been handled using additional coils, which form separatrices at the end of the plasma (Hoffman et al., 1986). A & P dl' number of potential solutions have been proposed to han- (7.4) IR rs pi dle the latter issue: a coaxial slow source, which produces an annular FRC plasma between concentric coils For a radially uniform ion temperature, carrying current (Barnes et al. , 1987); a rotating magnetic-field system, which drives a toroidal current, S = (7.5) ROTOMAK (Blevin and Thonemann, 1962); the 2mr, p;,B, EXTRAP, which produces a toroidal current in an optu- p;, is the ion gyroradius in the external magnetic field. pole field (I.ehnert, 1977); and a field-reversed mirror The parameter S =R /p;, is sometimes used, and (Simonen et al., 1979). Sx, /4&2 ~ s ~ Sx, /4. Experimentally, equilibria have In the present fast formation schemes the ions and, to

CONDUCTING WALL PREIONIZATION // (o)

FIELD REVERSAL // (i) // J p'~ RADIAL COMPRESSION // 8, FIELD LINE CONNECTION (2) / MAJOR RAD IUS A XIAL CONTRACTION INSULATING (3) WALL SEPARATRIX

EQUIL I BR I V M (4)

FIT+. 69. Stages of formation of an FRC plasma in a theta FIG. 68. Representative plasma field profiles for an FRC. pinch (Ho6man et a/. , 1986}.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1088 John Sheffield: Physics of magnetic fusion reactors a lesser extent, the electrons are shock heated and resis- E. Adiabatic compression tive heating occurs during a slow radial compression. Heating also occurs due to resistive magnetic-Geld an- Adiabatic compression has been proposed for pulsed nihilation. Temperatures in present-day experiments reactors to heat to reactor conditions (T ~ 8 keV), and it range from T; —1.5 keV, T, -0.5 keV at n —8X10 might be useful as part of the formation process in a ' m, x, =0.3 to T; —T, =0.15 keV at n-2X10 m steady-state reactor. Assuming nT-B„particle conser- and x, =0.45. In reactor-relevant formation schemes, vation ( n V= constant), and the adiabatic law / shock heating will be less, though anomalous resistive (nT V-), yields T-B, and n-B, . If flux is heating may still lead to T, T,. Auxiliary heating will conserved, from Eq. (7.3), the length of the plasmoid ' + be required to raise the plasma to reactor conditions. varies as l, -r, x, ', which shows that the results The parameters of representative experiments are given of compression are affected by the profile factor e ranging in Table XIX. from 0 to 1. The adiabatic scalings may be written in Recently, studies have been made of D- He reactors, terms of the changes in the x, (where (P) =1—x, /2) RUBY (Kernbichler et al~, 1991), and ARTEMIS and the flux conserved radius r, (Spencer et al., 1983), (Momota et al. 1992). While D- He reactors are outside , 2(4+3@)/5 fi) —ii+2m)/5 2/5 the scope of this paper, it is worthwhile to indicate some S S ( c of the interesting features of this concept. A principal is- —4(3+@)/5 j g $2(1 —e)/5~ —8/5 XS X~ / C sue for steady-state operation is sustaining the plasma — — — — (7.6) 6(3+a)/5 g ~ g 2(1 e)/5 12/5 current, I=l,8/po. For the previous example, I =26 712 S X~ I C MA. Fortunately, it appears that the orbits of energetic —(3+a) —2 xs protons produced in D- He fusion lead naturally to a e C current supporting the initial plasma current, owing to For present FRC data e-0. 1 —0.3. losses associated with the large gyro-orbits (Berk et al., There are two methods of compression, Aux compres- 1988). If this efFect works, then only a modest amount of sion, in which x, =r, /r, is changed, and wall compres- noninductive current drive is required. For the alpha sion, in which r, is reduced either by moving the wall, as particles (Zb=2) in D-T fusion, the efFect would be with a linear implosion, or by moving the FRC into smaller [see Eq. (4.11)j. An issue is whether the electron smaller and smaller coils. The combination of translation viscosity is sufBcient to prevent shielding by a return and Aux compression can have corn.parable efticiency to plasm. a current. An analysis by Baldwin and Rensink wall compression, as has been demonstrated experimen- (1978) suggests that, in a closed magnetic field-line sys- tally (Rej et al. , 1992). tem, the development of large potentials can lead to signi6cant cancellation of the ion current by drifting electrons. For D- He fusion the neutron power is very low, F. Transport in the case of the RUBY reactor 6% of the fusion power of 1.4 GW. For a steady-state D-T system with x, =0.57, In FRC energy and particle transport are comparable I, /r, =6, a„/a, =1.13, nDT/n, =0.83, f =1.0, p „=6 in experiments (rE -0.5m~ ), and transport is about an or- MWm . P~=3.7X10 /CH MW. A large value of der of magnitude higher than the classical level. In the ( ~ 10), or the equivalent confinement from the real high-beta plasmas of the FRC both high-frequency insta- transport, is required to make a conventional-size reac- bilities, notably the lower-hybrid drift (LHD) mode tor. The reactors use direct conversion of the charged- (Davidson and Krall, 1977; Tuszewski and Linford, 1982) particle power leaving the plasmoid. It will be important and the low-frequency high-beta electromagnetic drift to verify the electrical ef6ciency assumed in the studies. wave (Krall, 1987), are predicted to be important contri-

TABLE XIX. Representative modern FRC experiments (Tuszewski, 1988). Configuration Coil length Coil radius Field lifetime v. Device I, (m) r, (m} B (T) I (ps) TOR 1.5 0.15 1.0 100 NUCTE 2.0 0.08 1.0 60 OCT 0.6 0.11 1.0 130 CSS 1.0 0.225 0.3 60 FRXC/LSM 2.0 0.35 0.6 450 LSX 4.5 0.45 0.8 500

Rev. Mod. Phys. , Vol. 66, No. 3, July)994 John Sheffield: Physics of magnetic fusion reactors 1089 butors to transport. The LHD mode, which is driven a L B n'/P ((P) =variable), primarily by the density gradient, is a Qutelike mode with a predicted growth rate comparable to the hybrid which are similar to tokamak and stellarator empirical gyrofrequency and a wavelength comparable to the elec- scalings. tron gyroradius. The nonlinear evolution of the mode is The amount of poloidal Aux trapped during plasma formation is because the de- not well understood. Experimental transport shows some important, FRC lifetime of the features of LHD scaling, and in present experi- pends mainly on the rate of decay of this flux (P), assum- ments loss rates are consistent with predictions. Howev- ing no additional drive mechanisms for the plasma er, the predicted negative temperature dependence T current. Optimum trapping requires that the reversal is greater than the experimental observation of T —T time r„-2BO/B be on the order of several radial Alfven Therefore, a modified LHD loss (MLHD) has been pro- times, rz -r, /U„, but not so fast that tearing instabili- posed for reactor design, which has a factor to allow for ties cause Aux loss. The trapping e%ciency is impaired more favorable scaling (HofFman and Milroy, 1983), by impurities, and techniques are used to isolate the plasma from the walls during the formation phase. Initial CHM0. 47X10 A; (p)x, S (1+C/S) Aux retention factors in the range 0.3—0.6 are obtained P (1+T, /T, )B, experimentally. (7.7) In the equilibrium phase, loss of Aux is expected to vE —0.5~p, occur owing to resistive annihilation at the field null. Ex- perimentally, the loss rate exceeds that expected from where Spitzer resistivity by a factor of 3 —10 (Slough et al., 1984; Siemon et al. , 1986). The anomalous loss has not been explained. In present-day experiments (Hoffmann and Slough, 1986).

G. MHD limits T, B,(T)l, (m) 3 T; A; T;(k Ve) 2x General features of MHD stability are discussed in Sec. III.C. The FRC might be expected to sufFer from a Various empirical transport formulas have been proposed variety of modes. However, various features promote (see Tuszewski, 1988), in the form 'n "T', r„-l;x, r, stability (Tuszewski 1988): the compressive energy of the a =0— b =2— — — 0.75, 3, c =1.25 2, (b+c =1.4 3.6), high-beta plasma is stabilizing in the absence of toroidal d =0.5 — = — 0.9, and T 0.5 to 0.5. The simplest parame- field; the region outside the separatrix has good curvature ter that scales like the measured particle confinement in the low-field regions; there is a nearby conducting times is R /p;, (Hamasaki and Krall, 1979). boundary; there is strong elongation; toroidal rotation With the aid of the power-balance equation (2.22), the and Hall e6'ects act to stabilize tearing and tilt modes: formula for rz may be rewritten as kinetic e8'ects are important at low s —1 —2. The expect- a L Bn /P ((P) =constant) ed modes are listed in Table XX. Of particular importance are the n=2, m=1 global ideal mode in the pres- or ence of rotation and the n + 1, m = 1 internal tilt mode.

TABLE XX. FRC stability: MHD theory vs experiment: Tuszewski, 1988 and Tuszewski et al. , 1991. m Experimental (toroidal) (poloidal) Mode Name observation

1. Local ideal modes 0 Interchange yes 1 —2 Axial or Ballooning maybe radial 2. Global ideal modes a. No rotation 0 Axial Roman candle no 1 Radial Sideways shift no &1 Axial Tilt yes b. Rotation 1 Radial Wobble yes' 2 Radial n=2 ye sb ~2 Radial n&2 no 3. Resistive modes 0 Radial and Tearing no' axial 'Saturates at 6nite amplitude. Stabilized by multipole fields. 'Disappeared in modern experiments.

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1090 John Sheffield: Physics of magnetic fusion reactors

1. Rotational instability H. Alpha effects

These modes have been observed, notably the n=2 The general effects of energetic alphas are discussed in mode. The origin of the rotation is not fully understood, Sec. II.F. There has been no study of such effects for though it is expected that the Hall effect and particle FRC to the level of that for tokamaks. The symmetric losses may be important. Fortunately, the mode may be FRC configuration and high currents (multi-mega-amps) stabilized using multipole fields (Ohi et al. , 1983). Exper- should have, inherently, good containment. However, iments work and are in fair agreement with theory multipole fields used to stabilize rotational modes might (Harned, 1984; Ishimura, 1986). Helical multipoles have adversely affect confinement. In the pulsed FRC reactors also proved effective. For a reactor, analysis is required considered, only fractional burn is achieved ~0.2, so to understand the possible impact of the additional field helium ash buildup would be limited even if all the ash ripple on transport and alpha-particle confinement. were confined. In steady-state systems the same issue exists as for other toroidal devices, demonstrating that helium ash diffuses out at a rate not too different. from the 2. Tilt instability fuel.

This mode was predicted to occur in CT's by Rosen- bluth and Bussac (1979). A computation of the time vari- I. Power and particle control ation of the fields during the instability is shown in Fig. 70. It has not been observed in present-day experiments Generic issues for power and particle handling are dis- on LSX with s S8 (Hoffman, 1992). However, evidence cussed in Sec. II.G. For the pulsed FRC, with its natural for the mode and its correlation with degraded divertor, clean startup, and isolation from both walls and confinement has been found in one experiment, FRX- chamber ends, wall-generated impurities should not be a C/LSM (Tuszewski et al., 1991;Rej et al., 1991). Candi- problem. In a steady-state concept continuous fueling date mechanisms to suppress the mode include rotation, will be important. It will be necessary to develop a more Hall effects, and kinetic effects. Experiments at larger s detailed divertor system, notably to isolate the plasma will help to resolve the questions. source, and experimental demonstrations will be needed. Clearly, current-drive systems will be required.

J. FRC reactor options

Various types of compact torus reactor have been studied. They are reviewed by Hagenson and Krakowski (1981). More recently, pulsed FRC reactors have been studied by Vlases et al., (1986), and steady-state reactors by Kernbichler et al. (1991) and Momota et al. (1992). Parameters of some designs are given in Table XXI. 10 For simplicity, in indicating the parameters required for a plasmoid that is self sustained by alpha heating, the FRC plasma may be treated as a cylinder of length l, and radius r„with Hat plasma profiles. In reality, with strong 20 central alpha heating the temperature profiles may be more peaked than in present-day experiments, leading to higher fusion power for a given beta. The energy confinement time of the plasma for T, = T, =T is 30 (7.8)

The alpha power is 40 7l DT =O. SX IO-"' (n,2T,')r,'I, (W) . (7.9) r. l1e

Allowing for impurities and helium ash, we set 50 (n; T; ) =0.42+(nT). For fiat profiles (n, T; =(n;T; ) . The neutron fiux to the wall (a ) is ). FKx. 70. Time sequence of magnetic-field lines from 30 MHD 0.8P~ simulation, illustrating the internal tilt mode (Milroy et al., (7.10) 1987). 2n.( l, /r, )(a„/r, )r,

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 John Sheffield: Physics of magnetic fusion reactors 1091

TABLE XXI. Representative FRC reactor options. Name CTOR FIREBIRD B RUBY ARTEMIS (Hagenson Reference and Krakowski 1981) (Vlases et al., 1986) (Kernbichler et al., 1991) (Momota et al., 1992) r, (m)' 2.25 0.80 1.25 1.12 a„(m)' 2.55 0.90 2.0 r, (m)' 3.0 1.40 2.0 I, 8..5 4.80 17.0 17.0 Burn chamber 53 67 length (m) Plasma burn 0.07 time (s) Density (nz, ) 2.1 4.26 5.2 0.66 (X102' m 3) Temperature 10 10 87.5 (Tk) (keU) Transport model LHD MLHD & 10 classical -200 classical P+ (MW average) 2600 1917 1400 1610 p „(MWm 2.0 4.0 NBI+ fusion product average) 1 MeV drive Net P, MW(e) 752 —1000 1052 Applied field (T) 2.0 4.3 6.7 Trapped field (T) 4.4 6.4 4.9 3.7 Repetition rate (s ') -0.033 0.2 Translation 25 95 velocity (m/s) Reactor type Pulsed D-T Pulsed D-T Steady-state Steady-state D- He D- He 'Initi. al parameters during burn for pulsed systems. For pulsed system. Current-drive scheme for steady-state systems. (p „-0.3 MW m ).

Combining these equations to eliminate the explicit Here I'z (MW) and P „are the instantaneous values for a dependence on r, yields pulsed reactor and f is the fraction of alpha power available for plasma heating. In a steady-state reactor with current drive, allows for the additional power. 1.21(P~) ' (n, /nDT) f Combining Eqs. (2.37) and (7.7)—(7.11) leads to formulas 25 (7.11) f (P )0 75(a /r )0 75(( /r )0 for ZF and a, .

)0.091(a )0.091( )0.364( 0.364 128(pwn w /r s Is / s )( n e /n DT p ) (MW), (7.12) 0.727C0.727(1+C/S)2. 18 4.36 fa HM s 313(p„„)075(a. )075(I, )025(n, 2 /r, /r, /nDT) 2 (p)~, (7.13)

I where (p) is in % and, for the D-T reactor, the tempera- of the Aux conserver, with an angle 50.5 degrees. The ture has been set at 10 keV (see Table XIX). flaring minimizes the drag force (Hagenson and Krakowski, 1981). The plasmoid velocity V is set so 1. Pulsed reactor that in the time I, /V~ the flux leakage is less than the external Aux between the separatrix and the wall. The design of the conducting shell must carefully balance be- The geometry of a pulsed reactor is illustrated in Fig. tween minimizing dissipation of the changing magnetic 65. The plasmoid, created by a slow process to minimize field and confining the Aux. A thin graphite shell inside coil voltages (Farengo and Brooks, 1992), may be heated the first wall has been proposed to provide stabilization to ignition by adiabatic compression. The plasmoid may (Vlases et al. , 1986). then be translated by active coils or by a modest Baring In order to obtain a sufticient energy gain, it is cus-

Rev. Mod. Phys. , Vol. 66, No. 3, July 1994 1092 John Sheffield: Physics of magnetic fusion reactors tomary to let the plasma burn for ~~ —IQ~E. Since For a pulsed reactor, key issues are the design of the ~ -2~E, it will be necessary to fuel the plasmoid as it conducting shell, the high instantaneous neutron Aux re- translates. It is tacitly assumed that under reactor condi- quired for a useful average neutron flux (-4 MWm ), tions the Aux dissipation time will be much longer, so and cyclic fatigue of the wall, shield, and blanket. that current drives will not be required. For a steady-state system current-driven power is a The total energy produced will be EF-—PF~z. If the major concern. The problem may be alleviated by a nat- time between plasmoids is ~„,the average power is ural current drive due to the orbits of the charged fusion products. The efficient conversion of charged-particle P~=P~ (MW) . (7.14) energy to electricity needs to be demonstrated. prep

The average wall loading, where L, is the burn chamber ACKNOWLEDGMENTS length, is Now that the paper is completed, I am reminded of the PF I, story of the man who was observed banging his head pwn= P&Pl„(MWrn ) . (7.15) ~ against a wall. When asked why he did it he replied, "Because it feels so nice when I stop. " Now that I am Typical values are ~z -—1 s, ~, —10 s, I, =5 m, and ~- feeling well again I can express my appreciation to L =50 m, so we see that -10PF and 100p PF p „— „. Richard Hazeltine (University of Texas) for encouraging In Sec. it was pointed out that a value re- II, typical me to undertake this review. I am grateful to Darcus quired for neutron wall loading for an attractive reactor Johnson for her patience and skill in typing and retyping was about 4 MW m on average. This indicates that an the manuscript, as various changes were made. I am also attractive pulsed FRC reactor have handle may to peak very appreciative of the advice and contributions of a loads as high as a few hundred MW m number of colleagues, including: Jurgen Niihrenberg and The substitution of representative values for =0.75 x, Horst Wobig (Institut fiir Plasma Physik, Garching); 57 = and 0. (P) 72% and 84%, respectively, and Charles Bathke, Robert Krakowski, Ron Miller, Kurt T=10 a — l, /r, =6, keV, /r, =1.13 in Eqs. (7.12) (7.15) Schoenberg, Richard Siemon, and Kenneth Werley (Los yields parameters similar to those of early studies, name- Alarnos National Laboratory), Dieter Sigmar (MIT); ly, those of Hagenson and Krakowski (1981) and Vlases Kimitaka Itoh (National Institute for Fusion Science, et a/. 1986, respectively, in which forms of LHD , — scaling Nagoya); Don Batchelor, Benjamin Carreras, James were used. An enhancement factor of CHM 10 is need- Lyon, Donald Spong, and Nermin Uckan (Oak Ridge ed (Vlases et al. , 1986) to produce a reasonable scale and National Laboratory); Douglass Post (Princeton Plasma field reactor. For the parameters above, with x, =0.57, Physics Laboratory); Kenro Miyamoto (Seikei Universi- CHM=10, nDT/n, =0 9, f~.=1.0r~=10rE, r„=5 s, ty); Alan Hoffman (University of Washington); and and L.=70 m, we obtain PF = 10000 MW PF =2000 MW, Stewart Prager (University of Wisconsin). I also thank p „=300 MWm, p „=4.0 MWm, r, =0.80 m, the many people who provided valuable figures and ' B; =6.8 T, and n =4.3 X 10 m tables. Research on this paper was sponsored by the The number of alpha particles produced during the Ofhce of Fusion Energy, U.S. Department of Energy, un- ' burn is 1V =3.7X 10 m, which is about 9% of the to- der Contract No. DE-AC05-84OR21400 with Martin tal number of electrons in the plasmoid. Consequently, Marietta Energy Systems, Inc. even if all helium ash were retained, there would still be a high percentage of D-T fuel. REFERENCES 2. Conclusions Alejadre, C., J. J. A. Gozalo, J. B. Perez, F. Maga5a, J. R. C. Diaz, J. G. Perez, A. Lopez-Fraguas, L. Garcia, V. I. Kriven- Principle issues for an FRC reactor include the re- ski, R. Martin, et al. , 1990, Fusion Technol. 17, 131. quirement for slow formation of the initial plasma and Alikaev, V. V., and V. V. Parail, 1991, Plasma Phys. Controll. the still unsettled questions of confinement scaling, Aux Fusion 33, 1639. confinement, and the tilt instability. Substantially more Alikhanov, S. G., V. P. Bakhtin, A. G. Es'kov, R. Kh. Kurt- experimental work in LSX and large-scale experiments mullaev, V. N. Semenov, E. F. Strizhov, N. P. Kozlov, V. I. will be needed to verify the reactor potential of this in- Khvesyuk, and A. V. Yaminskij, 1983, in Plasma Physics and teresting configuration. Controlled Nuclear Fusion Research: Proceedings of the 9th In ternational Conference, Baltimore, 1982 (IAEA, Vienna), Vol. The ARTEMIS reactor is neutral-beam driven. Neu- 3, p. 319. tral beams are also for Aux from proposed buildup a Alper, B., V. Antoni, M. K. Bevir, H. A. B. Bodin, C. A. Bunt- present-sized FRC. The beam ions may also provide sta- ing, P. G. Carolan, J. Cunnane, D. E. Evans, A. R. Field, S. J. bility against tilt and/or shift modes. Beams of the re- Gee, et al., 1987, in Plasma Physics and Controlled Nuclear quired energy will need a substantial development pro- Fusion Research: Proceedings of the 11th International Confer gram. ence, Baltimore, 1986 (IAEA, Vienna), Vol, 2. p. 399.

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