Nuclear power—fusion

T. Kenneth Fowler

University of California, Berkeley, 4167 Etcheverry Hall, Berkeley, California 94720-1730

In the 1990s, experiments in tokamak magnetic fusion devices have finally approached ‘‘breakeven’’—power out equal to power in—at fusion power levels exceeding 10 MW, and great progress has alsobeen made with inertial-confinement fusion laser experiments. Based on these results, therequirements to achieve ignition and high-energy gain are now fairly clear for both approaches. Thisarticle focuses on developments in modern plasma physics that led to these achievements and outlinesthe historical development of the field. Topics include the stability of magnetic fields, fieldreconnection and the magnetic dynamo, turbulent heat transport, and plasma absorption of intensebeams of light. The article concludes with a brief discussion of future research directions.[S0034-6861(99)00902-2]

I. INTRODUCTION

The earliest speculations about nuclear power—first,about nuclear fusion—followed soon after the publica-tion of Albert Einstein’s special theory of relativity. Astory related by Edward Teller tells of the young GeorgeGamow’s being offered, in 1929, the nightly use of thefull electric power grid of Leningrad if he would under-take to create in the laboratory the fusion energy thatAtkinson and Houtermans were claiming to be sufficientto explain stars (Teller, 1981). Then, when fission wasdiscovered in 1939, fusion took a back seat as the morereadily exploitable fission process forged ahead, culmi-nating in the first fission power reactors in the 1950s.

While the early success of fission reactors came at daz-zling speed, the story of fusion power—still in the re-search stage—is one of persistent determination drivenon the one hand by the alluring goal of virtually unlim-ited and environmentally attractive nuclear power, andon the other by the intellectual appeal of unprecedentedtechnical and scientific challenges that have created thefield of modern plasma physics.

Both the allure and the challenges of fusion arise fromthe nature of the fusion process. Fusion fuel is abundantand cheap, the most easily exploitable fuels being deu-terium, occurring naturally in all water, and tritium,which can easily be manufactured inside the fusion reac-tor by the neutron bombardment of lithium, also abun-dant in nature. And fusion does not produce nuclearwaste directly, though tritium is mildly radioactive andneutron activation of the reactor chamber dictates whichstructural materials are most useful to minimize wastedisposal of components discarded in maintenance or theentire reactor assembly at the end of its life. However,whereas fission occurs at normal temperatures, fusionoccurs only at the extreme temperatures characteristicof stars (aside from muon catalysis, which does not re-quire high temperatures but thus far poses other un-solved problems). The fuel with the lowest kindlingpoint is a mixture of deuterium and tritium that ignitesat temperatures around 50 keV or 50 million degreesKelvin. At such high temperatures, the fuel becomes afully ionized gas, or plasma, hence the prominence ofplasma physics in fusion research.

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Two approaches to obtaining high-temperature plas-mas have dominated the field. One is the confinement ofthe fuel at moderate pressure by means of magneticfields, and the other—called inertial-confinement fusion(ICF)—utilizes solid DT targets heated by intense laserbeams or ion beams. Impressive progress has beenmade. While self-sustaining ignition has not yet beenachieved, experiments in tokamak magnetic fusion de-vices have approached ‘‘breakeven’’—power out equalto the power in—at fusion power levels exceeding 10MW (Strachan et al., 1994; JET Team, 1997), and greatprogress has also been made with ICF laser experiments(Lindl, 1995). Based on these results, the physics re-quirements for achieving ignition are now fairly clear,for both approaches (Fowler, 1997).

In this article, we shall focus on the scientific develop-ments that led to these achievements. Largely throughthe impetus of fusion research, plasma physics hasreached a level of sophistication comparable to olderfields of applied science, such as solid-state physics andfluid mechanics. Mastery of plasma physics at a leveladequate for understanding fusion plasmas requires acomplete synthesis of classical physics. A resurgence ofinterest in this fundamental discipline has benefited as-trophysics, space physics, and applied mathematics andhas trained many scientists and engineers who havemade outstanding contributions in industry and aca-demia.

II. CREATING MAGNETIC FUSION SCIENCE

Magnetic fusion research began in the 1950s, initiallyin secret but soon declassified, in 1958, in recognition ofthe fact that the research would benefit greatly from aconcerted world effort and had little connection withnuclear weapons technology or weapons proliferation.Research on the ICF approach began about a decadelater and remained classified for a longer time, especiallyin the U.S., but it too is now largely declassified. Thevalue of international cooperation in fusion researchcannot be overstated, in terms both of science and of itscontributions to East-West communication during theCold War. A famous event in fusion history, which her-alded the dominant role of the Russian tokamak in mag-netic fusion research, was the ‘‘airlift’’ to Moscow, in

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1969, of a British research team using their own equip-ment to verify Russian claims that they had achievednew records of plasma confinement and the then-unprecedented temperature of 10 million degreesKelvin. This event, soon followed by confirming experi-ments in the U.S. and Europe, paved the way for thelarge tokamak facilities constructed in aftermath of theoil crises of the 1970s—the facilities that have nowachieved near-breakeven in the 1990s.

The development of plasma physics for magnetic fu-sion research has been strongly influenced by the re-quirements for achieving ignition. It is useful to think ofignition requirements in two steps—first, the creation ofa stable magnetic configuration to confine the fuelplasma, and, second, doing so at a critical size largeenough so that the fusion reactions heat the fuel fasterthan the heat can leak away. Examining the history ofthe tokamak in light of these two requirements willserve to illustrate how and why fusion science developedas it did. A more thorough discussion of tokamaks andother magnetic configurations can be found in Teller(1981) and Sheffield (1994) and a discussion of fusionnuclear engineering in Holdren et al. (1988), which com-pares safety and nuclear waste characteristics of fusionand fission reactors.

The starting point is a magnetic configuration to con-fine the plasma in a state of equilibrium between mag-netic forces and pressure forces. Whereas gravitationalforces are symmetrical, so that stars are spheres, themagnetic force is two-dimensional, acting only perpen-dicular to a current, so that a magnetically confinedplasma is a cylinder. The tokamak, invented by IgorTamm and Andrei Sakharov in the Soviet Union, is de-scended from the linear ‘‘pinch,’’ a plasma column car-rying currents along its length whose mutual attractionconstricts the plasma away from the walls of the tubethat contains it, as discovered by Willard Bennett in1934. Early linear pinch experiments at Los Alamos andelsewhere proved to be unstable, and heat leaked outthe ends, defects remedied in the tokamak by bendingthe cylinder into a closed ring or torus, stabilized by astrong field generated by a solenoid wrapped around thetoroidally shaped vacuum vessel. Bending the currentchannel into a circle requires an additional ‘‘vertical’’field perpendicular to the plane of the torus. Thus thetokamak solves the requirement of stable confinementusing three sources of magnetic field—the vertical fieldto confine the current, the current to confine the plasma,and the solenoid to stabilize the current channel.

The stability of the tokamak follows from its magneticgeometry, in which the field lines produced by the tor-oidal solenoid are given a helical twist by the current.Ideally, these twisting field lines trace out symmetric,closed toroidal surfaces—called flux surfaces—nestedone inside the other. A similar concept, not requiringcurrents in the plasma, is the stellarator, invented byLyman Spitzer in the early 1950s. A major theoreticalachievement, published by Bernstein, Frieman, Kruskal,and Kulsrud (1958), is the energy principle, whereby thestability of tokamaks or any other magnetic configura-

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tion can be determined exactly within the constraints ofmagnetohydrodynamic (MHD) theory borrowed fromastrophysics, in which the plasma is treated as a fluidrepresented by averaging the equations of motion overall particles in the plasma.

By the late 1960s, fusion scientists had repaid theirdebt to astrophysics by their own extensions of thetheory including the effects of resistivity due to Cou-lomb collisions between electrons and ions. In fusion de-vices carrying current, the magnetic structure can be dis-rupted by breaking or ‘‘tearing’’ of the field lines due toresistivity—an example of magnetic reconnection preva-lent in many astrophysical phenomena and in planetarymagnetic fields, and an early example of ‘‘chaos’’ inwhich the current channel breaks up into filaments thatcreate islands in the field structure or field lines wander-ing out of the machine. For tokamaks, the study of tear-ing was motivated by occasional violent disruptions ofthe current channel, which must be understood and con-trolled to avoid severe damage to the machine. In othermagnetic confinement geometries, discussed below, tear-ing can actually serve the useful purpose of self-organization of plasma currents into a stable configura-tion. Thus we see how, in concentrating on the creationof stable magnetic configurations to meet a basic igni-tion requirement, fusion science has evolved a funda-mental understanding of magnetized plasmas that hassimultaneously contributed to solving a practical prob-lem in tokamak design, shed light on phenomena ubiq-uitous in nature, contributed to the development ofchaos theory in applied mathematics and many fields ofphysics, and stimulated new inventions in fusion re-search.

Of even greater impact on plasma physics, and on ourunderstanding of turbulence in fluids, has been the ex-tensive body of experimental and theoretical workaimed at the second ignition requirement, to determinethe critical size at which fusion power production ex-ceeds heat transport out of the plasma. Given a stablemagnetic structure, heat can still be transported by en-tropy generation associated with processes not includedin the energy principle, which assumes perfect conduc-tivity along field lines. One such process, already men-tioned, is resistivity, for which the entropy generationrate can be calculated accurately. ‘‘Classical’’ resistivetransport, due to Coulomb collisions, is relatively weakand diminishes greatly at high temperatures. More im-portant but more difficult to calculate is transport due tomicroscopic turbulence associated with the buildup ofweak electric fields parallel to the magnetic field B.Though usually too weak to affect the resistivity in fu-sion plasmas, these weak parallel electric fields also im-ply components perpendicular to B that cause plasmaparticles to execute cycloidal orbits drifting between fluxsurfaces. Small perturbations grow into turbulence if thedrifting motion is amplified, as can be true in tokamaksfor perturbation wavelengths a few times the orbital ra-dius of ions spinning in the magnetic field. At this time,the main information about turbulent transport is ob-tained empirically, by fitting formulas to the results of

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numerous tokamak experiments, guided in part by di-mensional analysis to suggest scaling laws appropriatefor particular physical processes. Computer codes, calledparticle-in-cell (PIC) codes, have been used to simulatedrift motion turbulence (drift waves) by following thedetailed motion of thousands of particles representingcharge clouds generated by the turbulence. Other codes,focusing on magnetic turbulence, follow the nonlinearevolution of ‘‘tearing’’ modes. Calibration of code re-sults with experimental data shows promise, though ma-chine designers still must rely heavily on the empiricalapproach.

Theoretically, the potential for microturbulence isstudied by examining the stability properties of the Vla-sov equation, in which ions and electrons are repre-sented by distribution functions f(x,v,t) in the phasespace of position x and velocity v. The Vlasov equationis just the continuity equation in this phase space—aLiouville equation with Hamiltonian forces, coupled toMaxwell’s equations in which the charge and currentdensities are obtained by velocity averages of the Vlasovdistribution function. The MHD fluid equations can bederived as velocity moments of the Vlasov equation.Stability is studied by searching for growing eigenmodesof the Vlasov equation linearized around an equilibriumdistribution. It can also be shown that, as for the MHDequations, there must exist a corresponding ‘‘energyprinciple’’ for the linearized Vlasov equation; corre-spondingly, the nonlinear theory should possess a gener-alized entropy and associated ‘‘free energy’’ from whichtransport could be derived. Though useful conceptually,this approach has not yet yielded many calculational re-sults (Fowler, 1968).

At the time this article appears, a promising newdirection—already being exploited experimentally—isthe reduction of transport by the deliberate introductionof sheared flows and ‘‘reversed’’ magnetic shear thatbreak up the collective motions produced by turbulence.Initially discovered experimentally in the 1980s as the‘‘H mode’’ of operation with reduced transport at theplasma edge, with theoretical guidance this techniquehas now been extended throughout the plasma volume,resulting in heat transport associated with the ions at theminimum rates allowed by Coulomb collisions, thoughelectron-related transport still appears to be governedby turbulence.

As a final example of fusion-inspired plasma physics,we return to the tokamak and its requirement for astrong current circulating around the torus. In existingtokamaks, the current is induced by the changing flux ofa transformer, the plasma ring itself acting as the sec-ondary winding, but already methods have been demon-strated that can drive a steady current (energetic atomicbeams, microwaves, etc.). All such methods would betoo inefficient, producing unwanted heating, were it notfor the fact that, miraculously, the tokamak can generatemost of its own current, called the ‘‘bootstrap’’ current.Again the reason lies in the equation of motion, nowhaving to do with the nonuniformity of the magneticfield in a tokamak, which causes additional oscillatory

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drift motion due to changes in the orbital radius as par-ticles spin around field lines. The enhanced transport ofparticles due to collisions among these magneticallydrifting orbits, called ‘‘neoclassical’’ transport, drives adynamo-like response as the conducting plasma flowsacross magnetic field lines, and this dynamo drives thebootstrap current. It is neoclassical heat transport byions that gives the minimum possible rate of entropygeneration and the minimum possible heat transport in atokamak.

A different mechanism of current generation is themagnetic dynamo, now arising from the statistical aver-age of the v3B force in a magnetic field undergoingturbulent fluctuations due to tearing and reconnection.Whereas the collisional bootstrap current creates mag-netic flux, this magnetic dynamo mainly reorganizes thefield due to the approximate conservation of a quantitycalled ‘‘helicity,’’ given by an integral of the scalar prod-uct of B and the vector potential A. Though of limitedimportance in tokamaks, in which tearing is largely sup-pressed by the strong toroidal field, in other concepts,such as the reversed-field pinch (RFP), the plasma gen-erates its own toroidal field as it relaxes toward a state ofminimum magnetic energy at fixed helicity—known as‘‘Taylor relaxation’’ (Taylor, 1986). An open question atthis time is the extent to which turbulent relaxation cre-ates unacceptable heat transport as the field continuallyreadjusts to compensate for resistive decay near theplasma boundary where the temperature is lowest andthe resistivity is highest.

III. INERTIAL-CONFINEMENT FUSION

Turning briefly to the ICF approach, we find entirelydifferent physics issues, reflecting an entirely differentsolution to the basic requirements of a stable assemblyof fuel and the critical size to achieve ignition. Thoughheating the solid target also forms a plasma, its density isso high that plasma turbulence is irrelevant in calculat-ing the critical size. However, achieving a useful criticalsize requires that the fuel first be compressed to densi-ties many hundreds of times that of ordinary materials.This is accomplished by heating the spherical target uni-formly from all sides, whereby the intense heating of thesurface creates an inward implosion of the fuel as thesurface layer, called the ablator, explodes outward. Atthe intensity of giant lasers now available, implosionpressures of millions of atmospheres are created, com-pressing the fuel to 100 times liquid density, and a 1000-fold or more compression should be possible. The phys-ics issues concern mainly the uniformity of illuminationand target design requirements to suppress hydrody-namic instability that amplifies imperfections in the sur-face finish.

Plasma physics enters mainly in ensuring efficient ab-sorption of the laser energy before the beams are re-flected at the ‘‘cutoff’’ density at which the laser fre-quency matches the ‘‘plasma frequency’’ (the samecondition as that for the reflection of light from an ordi-nary mirror). Two methods are employed, the direct-

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drive approach, in which laser beams shine directly onthe target, and indirect drive, in which the target ismounted inside a tiny metal cylinder, called a hohlraum,which converts laser light to x rays that in turn irradiatethe target. For both approaches, efficient absorption—either in a plasma cloud surrounding the ablator for di-rect drive, or in the metallic plasma formed where laserbeams strike the hohlraum wall for indirect drive—requires the use of ultraviolet light to penetrate to den-sities where collisional absorption dominates over col-lective ‘‘laser-plasma interactions’’ (Lindl, 1995). Theinvention at the University of Rochester in the late1970s of efficient methods to convert the infrared lightproduced by glass lasers into ultraviolet light was an im-portant milestone in ICF research.

IV. FUTURE DIRECTIONS

Magnetic fusion research is now focused on an inter-national effort to achieve ignition in a tokamak and onimprovements in the concept, including other means forcreating the nested toroidal flux surfaces so successfullyutilized to confine plasmas in tokamaks. A central issueis to what extent one should rely on internal currents, asthe tokamak does. As noted earlier, the stellaratoravoids internal currents altogether by creating closed to-roidal flux surfaces. Its external helical coils impart atwist to the field lines as current does. At the oppositeextreme are the reversed-field pinch devices, with only aweak external toroidal field, and a very compact devicecalled the spheromak, which has no external toroidalfield and relies totally on Taylor relaxation to create thedesired field configuration (Taylor, 1986). This is morethan an intellectual exercise, since the size and cost oftoroidal confinement devices tends to increase as theyrely more heavily on externally generated toroidal fields,requiring large coils looping through the plasma torus.

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The spherical tokamak is an innovative exception de-signed to minimize this difficulty (Peng and Strickler,1986).

For ICF, the immediate goal is ignition, in the Na-tional Ignition Facility now under construction in theU.S. Application of the concept to electric power pro-duction will require new laser technology, or perhapsion beams, capable of rapid repetition—several timesper second—and greater efficiency than existing glasslasers. An innovative means for reducing the laser en-ergy required for compression is the ‘‘fast ignitor,’’ usinga small but very-high-power laser to ignite the targetafter it has been compressed Tabak et al. (1994).

REFERENCES

Bernstein, I. B., E. A. Frieman, M. D. Kruskal, and R. M.Kulsrud, Proc. Soc. London, Ser. A 244, 17 (1958).

Fowler, T. K., 1968, in Advances in Plasma Physics, edited byA. Simon and W. B. Thompson (Interscience, New York),Vol. 1, p. 201.

Fowler, T. K., 1997, The Fusion Quest (Johns Hopkins Univer-sity, Baltimore, MD).

Holdren, J. P., D. H. Berwald, R. J. Budnitz, J. G. Crocker, J.G. Delene, R. D. Endicott, M. S. Kazimi, R. A. Krakowski,B. G. Logan, and K. R. Schutty, 1988, Fusion Technol. 13, 7.

JET Team, 1997, Plasma Phys. Controlled Fusion (Suppl.) 2B,1.

Lindl, John, 1995, Phys. Plasmas 2, 3933.Peng, Y.-K., and D. J. Strickler, 1986, Nucl. Fusion 26, 769.Sheffield, John, 1994, Rev. Mod. Phys. 66, 1015.Strachen, J. D., and TFTR Team, 1994, Phys. Rev. Lett. 72,

3526.Tabak, Max, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C.

Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J.Mason, 1994, Phys. Plasmas 1, 1626.

Taylor, J. B., 1986, Rev. Mod. Phys. 58, 741.Teller, E., 1981, Ed., Fusion (Academic, New York), Vol. I,

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