Chapter 1 Plasma Dynamics. - COnnecting REpositories · understanding of coherent generation by...

Plasma Physics

Chapter 1 Plasma Dynamics.

153

Section 2

154 RLE Progress Report Number 131

Chapter 1. Plasma Dynamics


Academic and Research StaffProfessor George Bekefi, Professor Abraham Bers, Professor Bruno Coppi, Professor MiklosPorkolab, Professor Jonathan S. Wurtele, Dr. Kuo-in Chen, Dr. Shien-Chi Chen, Dr. ThomasDupree, Dr. Ronald C. Englade, Dr. Stanley C. Luckhardt, Dr. Stefano Migliuolo, Dr. Abhay K.Ram, Dr. Linda Sugiyama, Ivan Mastovsky

Visiting ScientistsPaolo Detragiache,' Vladimir Fuchs, 2 Lazar Friedland,3 Dr. Chaim Leibovitch, 4 Dr. Kongyi Xu5

Graduate StudentsAla Alryyes, Ricardo Betti, Carson Chow, Stefano Coda, Jeffrey A. Colborn, Manoel E. Conde,Christian E. de Graff, Anthony C. DiRienzo, Robert J. Kirkwood, Kenneth C. Kupfer, Yi-KangPu, Jared P. Squire, Richard E. Stoner, Jesus N.S. Villasenor

Undergraduate StudentsDaniel P. Aalberts, George Chen, SalvatoreNerses, Kurt A. Schroder

1.1 Relativistic ElectronBeams

SponsorsLawrence Livermore National Laboratory

(Subcontract 6264005)National Science Foundation

(Grants ECS 84-13173 and ECS85-14517)

U.S. Air Force - Office of Scientifc Research(Contract AFOSR 84-0026)

U.S. Army - Harry Diamond Laboratories(Contract DAAL02-86-C-0050)

U.S. Navy - Office of Naval Research(Contract N00014-87-K-2001)

DiCecca, Marc Kaufman, Weng-Yew Ko, Nora

Project StaffProfessor George Bekefi, Professor JonathanS. Wurtele, Manoel E. Conde, Christian E. deGraff, Richard E. Stoner, Anthony C.DiRienzo, Daniel P. Aalberts, SalvatoreDiCecca, Dr. Kongyi Xu, Dr. ChaimLeibovitch, Ivan Mastovsky, Dr. Shien-ChiChen

1.1.1 Coherent, Free-ElectronRadiation Sources

The primary objective of the group is todevelop a basic experimental and theoreticalunderstanding of coherent generation by freeelectrons for wavelengths in the 1 um to 10cm range. Particular emphasis is placed onfree electron lasers, Cerenkov sources, rel-

1 University of Turin, Torino, Italy.

2 IREQ, Quebec, Canada.

3 Hebrew University of Jerusalem, Israel.

4 Rafael Laboratory, Haifa, Israel.

5 China University of Electronic Science and Technology, Chengdu, People's Republic of China.

155


ativistic magnetrons, and other novel radi-ation sources.

The experiments are carried out on four high-voltage pulsed accelerators available in ourlaboratory. Their characteristics are summa-rized in table 1.

Table 1. High-Voltage Pulsed Accelerators.Pulse RepititionAccelerator Voltage Current Length RepititionLength Range

PhysicsInterna-tional 1.5 MV 20 kA 30 nsec n/aPulserad110A

NereusNereus 600 kV 150 kA 30 nsec n/aAccelerator

PhysicsInterna-tional 500 kV 4 kA 1i sec n/a615MRPulserad

HighVoltage 700 kV 600 A 1p sec 1 ppsModulator

During the past year, workout in the following areas.

has been carried

Observations of Field ProfileModifications in a Raman FreeElectron Laser Amplifier

Free electron laser (FEL) amplification is theconsequence of a resonant interactionbetween an incident electromagnetic waveand a co-propagating electron beam that hasbeen injected into a periodic "wiggler" mag-netic field. This can lead to high outputgain, and high efficiency of convertingelectron beam kinetic energy into radiation.Remarkably, it also leads to large phase shiftsin the amplified electromagnetic wave.Under proper circumstances this phase shiftcan have a sign such that the electromag-netic wave is refracted towards the axis ofthe electron beam in a manner somewhatakin to the guiding properties of an optical

fiber. Such "optical guiding" would mitigatethe effects of diffraction, and thereby allowthe length of FEL wigglers to exceed theRayleigh range. Long wigglers are needed iffree electron lasers are to operate either inthe vacuum-ultraviolet or at high efficienciesin the infrared wavelength regime.

It comes as no surprise that strong FELactivity as described above should be accom-panied by significant modification of thespatial distribution of the RF fields withinand in the immediate vicinity of the electronbeam. To be sure, one way of confirmingoptical guiding is by observation of thechanges in the transverse spatial profile ofthe co-propagating amplified wave. Suchfield probing can be quite difficult at shortwavelengths (infrared and visible). However,the microwave regime, in which our exper-iments are conducted, offers a relativelysimple and direct way. We allow small,movable electric dipole antennas to traversethe waveguide in which the interaction takesplace. Using antennas sensitive to differentpolarizations, we are then able to distinguishbetween wave types.6

Tunable Micro-Wigglers for FreeElectron Lasers

Short period (1-10 mm) wigglers for freeelectron laser applications have been asubject of considerable interest. Besidestheir compactness, such systems have theadvantage of producing higher frequencyradiation with a given electron energy, orconversely, they reduce the electron energyrequired to access a given wavelength.

The wigglers proposed in previous workslack a simple but precise field tuning mech-anism so important in numerous applications.We have constructed7 a novel micro-wigglerwith this unique flexibility. The 30-periodmicro-wiggler, with a periodicity of 2.4 mm,is shown in figure 1 a. The wiggler consists

6 F. Hartemann, K. Xu, G. Bekefi, J.S. Wurtele, and J. Fajans, "Wave Profile (Optical Guiding) Induced by the FreeElectron Laser Interaction," Phys. Rev. Lett. 59(11):1177 (1987); K. Xu, G. Bekefi, and C. Leibovitch, "Observa-tions of Field Profile Modifications in a Raman Free Electron Laser Amplifier," submitted to Phys. Fluids (1989).

7 S-C. Chen, G. Bekefi, S. DiCecca, and R. Temkin, "Tunable Micro-Wigglers for Free Electron Lasers," submitted toAppl. Phys. Lett. 54:1299 (1989).



Figure 1. Microwigglers for free electron laser applica-tions.

of a periodic assembly of tiny electromagnetseach of which is energized by its own currentwindings (see figure 1 b). The magnetic fieldin each half-period is independently control-lable and the experimentally demonstratedtunability provides a versatile means forachieving the wiggler field required foroptimal FEL operation, such as reduction ofrandom field errors, field tapering, opticalklystron configurations, and electron beammatching at the wiggler entrance. The meas-ured operating characteristics of themicrowiggler in the linear regime (that is,well below saturation of the magnet material)agree very well with both analytical pred-

(b) 3mm x

(c) Iomm

/,

8 C. Leibovitch, K. Xu, and G. Bekefi, "Effects of Electron Prebunching on the Radiation Growth Rate of a Collective(Raman) Free Electron Laser Amplifier," IEEEJ. Quantum Elect. 24:1825 (1988); J.S. Wurtele, G. Bekefi, R. Chu,and K. Xu, "Prebunching in a Collective Raman Free Electron Laser Amplifier," submitted to Phys. Fluids (1989).

157

ictions and numerical models. The saturationbehavior in the magnet structure is alsostudied experimentally and by nonlinear finiteelement simulations.

An alternate and better design under consid-eration is shown in figure 1 c.

Effects of Electron Prebunching onthe Radiation Growth Rate of aCollective (RAMAN) Free-ElectronLaser

The concept of prebunching electron beamshas a long and venerable history; the klystronis a famous example. Prebunching is alsoused to improve the performance of othersources of coherent radiation. It is used intraveling wave tubes (TWT) to suppressparasitic feedback oscillations caused bywave reflections at or near the output of thedevice, and it has been used in travelingwave tubes and klystrons to achieve radiationat high frequencies by arranging for thebunched beam to emit on high harmonics.In more recent applications, prebunching thebeams in gyrotrons (the so-called gyrokly-stron configuration) can lead to improvedefficiency. In the case of free electron lasers(FEL's), systems with prebunching arereferred to as optical klystrons. The concepthas been employed successfully to increasethe overall optical gain both at the funda-mental and at high harmonics. Such gainenhancement is of particular importance atshort wavelengths (visible and VUV) whereFEL gains are of necessity small.

To date, experimental and theoretical studiesof the optical klystron have been carried outin the low gain, single particle (Compton)regime applicable to very short radiationwavelengths (visible and ultraviolet) whereelectron beam energies in excess of severalhundred MeV are used. In contrast, ourexperiments are made at microwave frequen-cies using mildly relativistic electrons (-200keV). In this collective (Raman) regime, thegains are high and the effects of spacecharge cannot be neglected. We find8 that


prebunching does indeed increase thegrowth rate of the radiation quite dramat-ically as compared with the case where pre-bunching has not been incorporated.

A 35 GHz Cyclotrom AutoresonanceMaser Amplifier

The cyclotron autoresonance maser (CARM)has been subjected to extensive theoreticalstudies and numerical simulations. However,unlike the gyrotron and the free electronlaser, its capabilities as a source of coherentmillimeter wavelength radiation remain virtu-ally untested in the laboratory. To the bestof our knowledge, only CARM oscillatorexperiments have been reported in the litera-ture.

Figure 2 illustrates what we believe to be thefirst CARM amplifier experiment. 9 We havemeasured a small signal gain of 90 dB/m anda saturated RF power of 10 MW in the fre-quency band of 35+1 GHz. The electronicefficiency of converting electron beam energyto radiation is 3 percent.

1.2 Plasma Wave Interactions- RF Heating and CurrentGeneration

Sponsors

U.S. Department of Energy(Contract DE-AC02-78-ET-51013)

National Science Foundation(Grant ECS 85-1 5032)

Project Staff

Professor Abraham Bers, Dr. Abhay K. Ram,Ala Alryyes, George Chen, Carson Chow,Lazar Friedland, Vladimir Fuchs, Weng-YewKo, and Kenneth C. Kupfer

Figure 2. a) Schematic of a cyclotron autoresonancemaser amplifier; b) Detail of the RF launcher.

1.2.1 Introduction

An overall description of the work of thisgroup was given in Progress Report Number130.

Our work of the past year has focused ontwo aspects:

1. Diffusion in transport induced bycoherent waves in plasmas; this workextends our nonlinear studies of inducedstochasticity and chaos to couplingvelocity and configuration spacedynamics. It is reported in sections1.2.2, "Spatial Stochasticity Induced byCoherent Wavepackets" on page 159and 1.2.3, "Transport in RF Heating andCurrent Drive" on page 160.

2. Kinetic wave propagation and mode con-version in magnetized plasmas; this workentails extending ray tracing to kineticdescriptions of plasmas in complex mag-netic configurations, and developingreduced-order analyses at singular layers

9 G. Bekefi, A. DiRienzo, C. Leibovitch, and B.G. Danly, "A 35 GHz Cyclotron Autoresonance Maser Amplifier,"submitted to Appl. Phys. Lett. 54:1302 (1989); A.C. DiRienzo, G. Bekefi, C. Leibovitch, and B.G. Danly, "Radi-ation Measurements from a 35 GHz Cyclotron Autoresonance (CARM) Amplifier," SPIE (1988).


WAVE LAUNCHER BIFILAR HELICAL CYLINDRICALWIGGLER DRIFT TUBE

TOARX-i

ACCELERATOR

\ANODEFIELD EMISSION AND SLENOIDCATHODE EMITTANCE

SELECTOR

(a)

RF INPUTPORT I

GRAPHITE ANODEAND .- RF POWER

EMITTANCE IGRAPHITE SELECTORCATODE DRIFT TUE

,A ET

RFTUNER


in such plasmas. This work is describedin sections 1.2.4, "Kinetic Ray TracingCode - Phase and Amplitude" onpage 163 and 1.2.5, "Reduced-OrderAnalyses for Ion Cyclotron Heating*" onpage 164.

1.2.2 Spatial Stochasticity Inducedby Coherent Wavepackets

As a first step towards the understanding ofparticle transport induced by RF waves, it isnecessary to know the phase-space behaviorof those particles that interact with thecoherent field spectrum. In particular, we areinterested in the phase-space behavior ofcharged particles moving in one-dimensionalconfiguration space (along the magnetic fieldlines) and interacting periodically with anelectrostatic wave. Such a situation wouldrepresent the interaction of tail electrons witha lower hybrid wave. Towards this end, wehave developed and analyzed a model for theperiodic interaction of charged particles witha spatially localized (delta-function), time-dependent electrostatic force.10 The (normal-ized) equations of motion are:

dv = e cos(2t) 6(z-n)n=-oo

(1)

where s is the field strength of theelectrostatic wave, v is the velocity and dz/dt= v. This system lends itself to an extensiveanalytical and numerical analysis primarilybecause the mapping equations (representingthe change in the energy of the particles aftereach interaction with the impulsive force)can be explicitly derived. The mappingequations are:

Un+ 1 = Un+2E cos2t n

1tn+ 1 -- t n +

- n+1

mod 1

where un+, is the energy of the particle after ithas received its n-th impulse and tn+ is thetime (or the phase of the wave) at which itarrives at the position where it receives its(n+l)-th impulse; a is either 1, or 0, or -1and is related to the direction of the velocityof the particle. These mapping equationshave been extensively analyzed to determinethe conditions under which the motion of thecharged particles becomes chaotic and thedomain of phase-space that becomeschaotic."

One of the features of this map is that thechaotic phase-space is always finite for anyfinite amplitude, e. This will be the genericbehavior of all charged particles interactingperiodically with any spatially imposedcoherent field. In other words, boundedness

0.3-

- - - - - - - - - - - - - - - - - - -- -

0.2

0.1

00 0.5 1.0

Figure 3. Phase-space plot for 8 = 10- 3.

10 A.K. Ram, A. Bers, and K. Kupfer, Periodic Interactions of Charged Particles with Spatially Localized Fields. PlasmaFusion Center Report, PFC/JA-88-41. MIT, 1988 (accepted for publication in Phys. Lett. A); A.K. Ram, A. Bers,and K. Kupfer, "The 'Spatial' Standard Map," Bull. Am. Phys. Soc. 33:1877 (1988).

11 Ibid.

159


Figure 4a. Dn versus n for e = 10 -5 < Ec(8 l10-3).

of the chaotic phase-space will occur in anyphysical situation of the type describedabove. Under these circumstances a distribu-tion of particles in the chaotic phase-spacewill not diffuse forever. If we define:

Dn = <(un-uo) 2>2n

where <...> denotes ensemble average over aset of initial conditions with energy uo dis-tributed randomly in t, then in the limitn - oo, for a bounded region of stochasticityDn 0. (In an unbounded or periodicstochastic phase-space the limit n - oowould give the conventional diffusion coeffi-cient.) It is important to know the diffusioncoefficient so that the evolution of the dis-tribution function of particles can bedescribed by the Fokker-Planck description.By a detailed study of the mappingequations, we find that the concept of dif-fusion can be used to describe the dynamicsin the stochastic region only for amplitudesbelow a certain critical amplitude. The crit-ical amplitude is determined by the width of

12 A.N. Kaufman, Phys. Fluids 15:1063 (1972); H.E. MynickNo. PPPL-2534. Princeton, New Jersey: 1988.

Figure 4b.the value oftc zero.

Dn versus n for E = 10 - 2 > Ec. nc indicatesn beyond which correlations have decayed

the stochastic region and the correlationtime. At the critical amplitude within a corre-lation time a majority of the particles in abeam will have reached the boundaries of thestochastic region. For amplitudes above thecritical amplitude, the dynamics do not seemto be described by the usual concept of dif-fusion. In this case, even though the distribu-tion function may have evolved to a steadystate, its evolution to that state may not bedescribed by the Fokker-Planck equation.

1.2.3 Transport in RF Heating andCurrent Drive

We are investigating the effects of coherentRF wave-fields on the transport of particles.This is in contrast to usual transport analyseswhich assume that the fields are of randomphase. 12 If one assumes that the fields arecoherent, then the random phase assumptioninherent in the previous analyses is of ques-tionable validity. For example, it has beenshown that correlations in two-dimensional

Princeton University Plasma Physics Laboratory Report


10 x 10-s

°°]

50 100-n

I


hamiltonian chaos drive the local diffusioncoefficient into oscillations about the randomphase value;13 such models have been pro-posed for studying the velocity space dif-fusion of tail particles in an RF heatedplasmas.14 We are extending these studies bycoupling the chaos in velocity space to con-figuration space, thereby obtaining thespatial transport of particles across flux sur-faces.

In developing this problem, we have consid-ered the following simplified model of super-thermal electrons. The field structure isconsidered localized on a flux surface whereelectrons cycle through it periodically. Anelectron's interaction with the field on eachtransit is taken to be

dz- =v (1)

dt -

dv= e z2 /d2cos(z-t), (2)dt

where a = (eElk/mwo2) is the normalized fieldstrength, d specifies the width of the inter-action region, and v is the parallel velocity ofthe particle; distance is normalized to thewave-number k and time to the frequency Cw.The electron's radial guiding center drift issimply

dr = (v2+v2 /2) 2 sin( 0),

dt L L(3)

where the radial position, r, is normalized toPo0,

_ _ Lkwoce 27R o

L is the normalized path length along onepoloidal circuit of the field line, and 80 is the

poloidal angle about which the wave-packetis localized. In this approximation, theelectron's perpendicular velocity, v1, is con-served. Note that equation (3) ignores theperpendicular RF fields which also contributeto radial motion.

If after an electron's nth transit through thewave-packet, the time is tn and the parallelvelocity is vn, then integrating equations (1)and (2) in z from -L/2 to L/2 give the valuesof ton, and vn, . This mapping can be iteratedfor a given initial condition (to,vo) and plottedin the appropriate surface of section. Forexample, the map is area preserving when v2

is plotted against tn(modulus 27). WhenLad>4, the map contains a connected regionof stochasticity, bounded in parallel velocityby two KAM surfaces on either side of theresonance at v = 1. For oad l and L>d,these KAM surfaces are accurately located bythe two values of v for which K(v) = 1,

Lad 7 e _ 1 )2d2/4K(v) - ( 3v

V(4)

where K(v) is the local standard map param-eter.15 Large primary islands are embedded inthe upper portion of the stochastic phasespace where K dips below 4 so that the firstorder fixed points are elliptic.

The radial motion is obtained by integratingequation (3) as the electron transits thewave-packet and its parallel velocity fluctu-ates. In general, as an electron diffuses inthe stochastic region of parallel phase spaceit also diffuses radially. The radial diffusionis found to be extremely sensitive to thelocalization of the fields; field structuresextending over a broad range of poloidalangles diffuse the particles across flux sur-faces at a much faster rate. This effect canbe visualized by allowing the field envelopeon the right-hand side of equation (2) to

13 A.J. Lichtenberg and M.A. Lieberman. University of California at Berkeley Report No. UCB/ERL M88/5.Berkeley, California: 1988; A.K. Ram, A. Bers, and K. Kupfer. MIT Plasma Fusion Center Report No.PFC/JA-88-41. MIT, 1988.

14 V. Fuchs, V. Krapchev, A. Ram, and A. Bers, Physica 14D:141 (1985); S.J. Tanaka, Plasma Phys. Contr. Fusion29-9:067 (1987).

15 V. Fuchs, V. Krapchev, A. Ram, and A. Bers, Physica 14D:141 (1985).

161


become a delta-function. In this case, theradial equation is easy to integrate and onefinds that

N

ArN = Avn cos n

n=1

ArN is the change in average radial positionresulting from N instantaneous jumps in par-allel velocity, where the nth jump, Avn, occursat the poloidal angle On. Note that weignored v2/2 compared to v2 when integratingequation (3) to obtain equation (5) in thisparticularly simple form; retaining this termdoes not affect the following argument. If allthe impulsive kicks occur at the samepoloidal angle, then cos On can be pulled outof the sum in equation (5). The remainingsum is simply vN-vo, which is bounded asN -- oo because the stochastic orbits cannotdiffuse past the last KAM surface in parallelphase space. Thus an electron being scat-tered in an infinitely narrow range of poloidalangle remains within a scale length po of itsinitial radial position so that there is no radialdiffusion. On the other hand, if the impulsivekicks occur at two or more values of On, thenthe sum in equation (5) is not bounded andone has the following equation for radial dif-fusion

lim 2N <Ar>/N = qD,N -, oo (6)

where < > denote an ensemble averageover an even distribution of initial conditionsin the stochastic phase space, D is theensemble averaged diffusion coefficient inparallel velocity, and q is the average value ofcos 28 weighted by the poloidal distribution ofthe field intensity. The numerical value of Dand details concerning the structure ofvelocity space diffusion (when the fieldenvelope is a delta-function) are given in ourrelated work,1 6 described in section 1.2.2,"Spatial Stochasticity Induced by CoherentWavepackets" on page 159.

VLOC ITY VS. CYCLES

POSI0511 ION VS. CYCL S

Figure 5. Parallel velocity (top figure) and the radialposition (bottom) as a function of the number oftransits through the wave packet. The radial positionhas a positive slope when the particle is in the vicinityof an island and a negative slope when it is diffusingbetween islands.

Finally, we have found and characterizedsome distinctive behavior associated with theintegration of the radial drift in equation (3)

16 A.K. Ram, A. Bers, and K. Kupfer, Periodic Interactions of Charged Particles with Spatially Localized Fields. PlasmaFusion Center Report, PFC/JA-88-41. MIT, 1988 (accepted for publication in Phys. Lett. A).



through a finite width field structure as givenby our model equations (1) and (2). In par-ticular, as stochastic orbits diffuse throughthe portion of phase space, where4 > K(v) > 1 and v > 1,they are convectedradially. The effect is entirely due to thelarge primary islands embedded in this regionwhich prevent a dynamical average of theradial jumps from vanishing. The convectionscales like (d/L)2p0 per transit and can beinward or outward depending on the poloidalangle about which the RF field is localized.

1.2.4 Kinetic Ray Tracing Code -Phase and Amplitude

This study was motivated by experimentalobservations of significant electron heatingwhen the fast Alfv6n wave (FAW) in the ion-cyclotron range of frequencies is launchedinto tokamak plasmas. Ordinarily oneassumes that the slowing down of theminority ion tail heats the electrons.However, in some recent experiments onJET, this could not entirely account for theintense electron heating observed. In simpletheoretical analyses the electrons could getheated by transit-time damping of the FAW,but this was not sufficient to explain theobservations. We decided to study the fate ofthe ion-Bernstein wave (IBW) as it propa-gated in a toroidal plasma to see if that couldbe an explanation for the observed electronheating. The IBW is a natural consequenceof the mode-conversion process that occursnear the ion-ion hybrid resonance (intwo-ion species plasmas). The conversionprocess is efficient for low kl's (magnitude ofthe wave-vector parallel to the magneticfield) which carry significant power fromsingle loop antennas into the plasma. Tostudy the propagation of IBW, we developed

a fully kinetic (Vlasov) ray trajectory codewhich followed IBWs in toroidal geometry.1 7

A detailed numerical analysis revealed thatthe k,'s along the IBW upshifted rapidly as itpropagated away from the mode-conversionlayer. The upshift could lead to electronLandau damping. The code was for a hotMaxwellian plasma containing profiles indensity, temperature, poloidal and toroidalmagnetic fields. The code results were wellsubstantiated by simple analytical models forthe propagation of an IBW.1 8

Since that work, we have found that thecode is general enough to be used to studythe propagation of waves of any frequency intokamaks. Furthermore, we have recentlysupplemented the code to calculate theamplitude of the electric field along the raytrajectory. To this end, we have to includethe solution of the following differentialequations:19

dU + IUV * + * * I2 = 0 (1)dt k

where

U= 1 (co) 28 8o0

is the energy density associated with themode of amplitude c, and aH is the hermitianpart of the conductivity tensor. The secondterm in (1) describes the effect of the nearbyrays on the particular ray trajectory being fol-lowed (i.e., it includes the effects offocusing), and the third term in (1) describesthe damping along the ray. In order to eval-uate the second term in (1) along the rays,we need to solve the following six coupledequations for dyadic, Vk:

17 A.K. Ram and A. Bers, In Applications of Radio-Frequency Power to Plasmas. AlP Conference Proceedings 159,eds. S. Bernabei and R.W. Motley, 402-405. New York: American Institute of Physics: 1987.

18 Ibid.

19 A. Bers, In Plasma Physics - Les Houches 1972, eds. C. DeWitt and J. Peyraud. London, Gordon & BreachScience Publishers: 1975; I.B. Bernstein and L. Friedland, In Handbook of Plasma Physics, eds. M.N. Rosenbluthand R.Z. Sagdeev, Vol. 1: Basic Plasma Physics I, eds. A.A. Galeev and R.N. Sudan. North-Holland Pub. Co.:1983.

163


d(Vk) 2_ a (V)a ad( = r - r - (V k) * 8dt -r r ak ak

a am - a2w(+ (Vk) .~ ) * Vk (3)ar ak akak

Results of these calculations should be forth-coming in a future publication.

1.2.5 Reduced-Order Analyses forIon Cyclotron Heating*

Mode Conversion in the Presence ofKinetic Dissipation

Reduced order analytic descriptions of thecoupling between fast Alfv6n waves (FAW)and ion-Bernstein waves (IBW) in ICH areuseful for gaining insight into the scaling ofsuch heating, and for simplifying two- andthree-dimensional kinetic codes. Even inone-dimension (in the direction x of thetoroidal field gradient), the FAW-IBW cou-pling is usually represented approximately bya fourth (or sixth) order o.d.e. derived fromthe Vlasov-Maxwell equations. This high-order o.d.e. must then be solved by numericaltechniques. In recent work, using nonreso-nant (quasimode) perturbation theory, 20 wehave shown that the FAW transmission coef-ficient (T) can be obtained in closed formfrom the solution of a first-order o.d.e.. Theseclosed form, analytic results are in excellentagreement with those obtained from numer-ical solutions of the full fourth (or sixth)-order o.d.e. description. Also recently, wehave shown that both the transmission coef-ficient (T) and the reflection coefficient (R)of the incident FAW can be obtained accu-rately from a reduced second-order o.d.e.21

We present here the completion of the

reduced analytic description by showing thatthe mode-conversion coefficient (C), in thepresence of kinetic dissipation, can bedecribed approximately by a coupled modesystem of two first-order o.d.e.'s (CM E's)which we solve in closed form.22 With theknowledge of T, R, and C, the power dissi-pated (D) per unit of incident FAW power isobtained from global energy conservation.

We start from the usual 2 x 2, zero-electron-mass, local dispersion tensor for ICRF,23 andexpand the elements to first-order in (k±ip) 2.We thus obtain

N4 K, + N2 (K0 -21K 1)-22'K0 + 2 = 0 (1)

where

2 = (1/3)- N , N = kCA/, CA

is the majority Alfv6n speed,

K0 (- 1/3)-- N2 + (q/4N1 2 )Z(a 2 ),

7-K, = (/31 /4N )Z(al ),

al, 2 = CA/1NI /1,2 Ro , i = xw cA, Ro

is the tokamak major radius, 1 refers to themajority and 2 to the minority ion species.The approximate IBW dispersion relation,exx = n,, is Ko + K1N2 = 0, and (1) can befactored into

(22-N2)(K0 + K, N )= 2.

The corresponding wave equationscoupled FAW and the IBW are

F'' + 22F = ;B

(2)

for the

(3a)

20 G. Francis et al., In Applications of Radio-Frequency Power to Plasmas. AlP Conference Proceedings 159, eds. S.Bernabei and R. Motley, 370-373. New York, American Institute of Physics: 1987; A. Bers et al., In Europ. Phys.Soc. Contributed Papers, eds. F. Englemann and J.L. Alvares Rivas, Vol. 11 D, Part iII, pp. 995A-C. 1987.

21 C.N. Lashmore-Davies, V. Fuchs, G. Francis, A.K. Ram, A. Bers, and L. Gauthier, Phys. Fluids 31:1614 (1988).

22 V. Fuchs and A. Bers, Phys. Fluids 31:12 (1988).

23 C.N. Lashmore-Davies, V. Fuchs, G. Francis, A.K. Ram, A. Bers and L. Gauthier, Phys. Fluids 31:1614 (1988).



(3b) and

where ' (d/d ). Associated with (3) is thepower conservation equation

[Ilm(F*F' - B*K1B ' )] '

= -BB* ImKo-B'B*' ImK, (4)

Global conservation laws (i.e., connectionformulae) are obtained from (4) upon inte-gration and use of asymptotic WKB solu-tions.

We now note that F and B contain variousspatial scales of variation. Away from thecoupling region they can be represented byeikonal amplitudes accounting for the inho-mogeneity in the magnetic field. In the cou-pling region they acquire an additional,slowly-varying amplitude due to couplingand dissipation. Separating these spatialevolution scales, we obtain CME's for theslowly-varying amplitudes (which we denoteby f and b) due to coupling and dissipation;further, introducing a stationary phaseexpansion around the coupling point, theCME's are

f'= -iL o exp[Dc-iK( - c) 2/2]b (5a)

b' = i4c exp[-D c + iK( - c)2/2]f (5b)

where the stationary phase point c is givenby

N2 = NB with N2 = 2(1 + N )/(1/3 + N )

and

NB = -Re(Ko/K,); 22 = 2 /4N 2K 1 c,

K = cA(1/ 3 + N )/Nc,/ 1Roo,

Dc = dC[Im(Ko/Kl)]/2N g-OO

is the damping factor. Equations (5a) and(5b) give Weber's equation, whose asymp-totics are well-known.

Thus, in high-field (HF) incidence of theFAW, we obtain the mode-conversion coeffi-cient

27 exp(- 7 ReA - 2Dc)I AI I F(- iA) 12

where A = A22/K. The transmission coefficientis obtained as T = exp(-27t ReA). In low-field (LF) incidence of the FAW, given thereflection coefficient R from the fast-waveapproximation, 24 we obtain the mode-conversion coefficient

CLF = (TR/CHF) exp( - 4Dc)

In the absence of kinetic dissipation (NI, = 0),equations (6) and (7) reduce to the well-known results: CHF = 1 -T and CLF = T(1 - T).Figures 6 and 7 give results obtained by ourreduced order analysis for heating scenariosproposed in Alcator C-Mod and CIT.

ICRF Analytic Reduction Scheme ForNondegenerate Minority Heating

The analytic theory (see "Mode Conversionin the Presence of Kinetic Dissipation" onpage 164) of dissipative coupling for ICRF,previously limited to D(H) minority heating,25

has been extended to include the importantcase of D - T - 3He heating for arbitraryconcentration ratios. It was found that astraight forward finite-Larmor radius expan-sion of the dispersion tensor for D(3He)heating does not reproduce the properties ofthe full 3 x 3 dispersion relation. A spuriousresonance in the ion-Bernstein branch

24 Ibid.

25 V. Fuchs and A. Bers, Phys. Fluids 31:12 (1988).

165

(K, B')' - KoB = AF


appears between the fast Alfv6n wave cut-offand the minority fundamental harmonic andan associated spurious propagating modeappears on the low field side. In the otherheating scenarios, the spurious singularityoccurs well outside of the coupling regionand can be ignored. A method for removing

1.00

.95

.90

65

.80

.70

.65

.60

.55

.50

.45

.40

,35

.30

.25

.20

15

.10

05

.00

the singularity and obtaining the correctfourth order dispersion relation was devel-oped and used for D(3He) heating. Excellentagreement with the results from the Imre-Weitzner boundary layer code is obtained.26

I f I LOW flELb IN'CIDoE'NCE

T -

C--

Figue c cis Td cC-

Figure 6. Power transfer coefficients T, R, C and the power dissipated, D, for low field incidence for Alcator C-Modheating. Ro0 = 64 cm, f = 80 MHz, B0o = 9T, To = 2 keV, ne= 5 x 10 20m - 3

.

26 K. Imre, Private communication



1.0

0.9 - Dg

0.8 -

0.7 R CHF

S0.6 CIT Minority Heatln(o 0.5 - D(H), = 5%

S0.4

0.3 -0.2 - DHF

0.1

0 2 4 6 8 10 12 14 16 18

K, (n1 )

Figure 7. Power transfer coefficients T, R, CHF(CLF < 0.01), and the powers dissipated, DHF and DLF,minority heating. Ro = 1.75 m, f = 95 MHz, Bo = 7T, To = 14 keV, ne = 1.3 x 1020m - 3 , D(H), = 0.05.

20

for CIT

*A portion of this work was carried out incollaboration with members of the theoryand computation group at the Tokamak deVarennes in Canada.

1.3 Physics of ThermonuclearPlasmas

SponsorU.S. Department of Energy

(Contract DE-AC02-78ET-51013)

Project StaffProfessor Bruno Coppi, Dr. Ronald C.Englade, Dr. Stefano Migliuolo, Dr. LindaSugiyama, Paolo Detragiache, Ricardo Betti,Stefano Coda, Yi-Kang Pu

The main theme of this program is the the-oretical study of magnetically confinedplasmas in regimes of thermonuclear interest.A variety of physical regimes that fall in thiscategory characterize both present-day

experiments on toroidal plasmas (Alcator,TFTR, JET, etc.) as well as future ones thatwill contain ignited plasmas employing eitherfirst generation fuels, namely a deuterium-tritium mixture (Ignitor, CIT), or moreadvanced fuels such as deuterium-deuteriumor deuterium-helium (Candor). A coordi-nated effort of collaboration between thedesign group of CIT, the U.S. compactignition experiment, and the European Ignitorhas been set up with our participation. AtMIT, the Alcator C-Mod facility is underconstruction. This experiment combines thefavorable features of elongated plasma crosssections with those of high field compactgeometries. These features have beenembodied earlier in both the Ignitor and theCIT designs, as well as in a machine (pro-posed by us in the early 1970's) calledMegator. A recently proposed MIT programcalled Versator-Upgrade, has, as its mainpurpose, to achieve the "second stability"region for ideal MHD ballooning modes. Wewere the first group to discover this stabilityregion in the winter of 1977-1978, and tooutline the experimental procedure by whichto reach it. This consists of starting with a

167


tight aspect ratio configuration, and raisingthe value of q (the inverse of the local rota-tional transform) on the magnetic axis abovea certain level, such as 1.4, while increasingthe value of the plasma pressure parameter,flp, to those typical of the second stabilityregion.

Presently, our research program follows twomajor avenues. First, the basic physicalprocesses of thermonuclear plasmas (equilib-rium, stability, transport, etc.) are beingstudied as they apply to existing or near-termfuture systems. In this effort, we closely col-laborate with our experimental colleagues.Second, we explore advanced regimes ofthermonuclear burning, including thoseemploying low neutron yield fuels (D-3 Heand "catalyzed" D-D). We consider both thedesign of machines to contain these ultra-high temperature plasmas as well as thephysics that governs their behavior.

Below, we present some salient results ontopics pursued by our group this past year,as well as from current research.

1.3.1 Anomalous Ion ThermalConductivity

Over the past few years we have undertakena comprehensive analytic and numericalstudy of the anomalous ion thermal transportassociated with a microinstability knownvariously as the ion temperature gradientmode, rq, or ion mixing mode. Many of therelevant properties of this mode were out-lined by our group 27 as part of the analysis ofthe mechanism responsible for inward par-ticle transport in toroidal devices. Subse-quently, the increased energy confinementtimes rE and record values of the confinement

parameter achieved by pellet injection28 inAlcator C have validated our previously sug-gested29 "cure," for the observed inadequateenergy confinement times in the plasmasproduced by Alcator C, that significantanomalous ion thermal conductivity could beavoided by producing peaked density pro-files.

Our recent investigations focused on thespatial profile and parameter dependence ofthe mixing mode in toroidal devices. Simu-lations30 successfully reproduced present dayregimes and have been employed in pred-ictions regarding high field experiments inignited regimes. A collaborative effort hasbeen initiated with scientists at ColumbiaUniversity where a linear containment deviceis being built for the purpose of studying thethreshold properties of the ion temperaturegradient mode. Attention will be given tothe effect of a non-uniform radial electricfield on the stability properties of this mode.

1.3.2 Profile Consistency: Globaland Nonlinear Transport

It is experimentally well established that thesteady-state electron temperature profile intokamak discharges is relatively independentof the spatial deposition of auxiliary powerand assumes a canonical shape that can berelated to that of a macroscopically stableand relaxed current density profile outsidethe region of sawtooth activity. Thisbehavior is often referred to as the "principleof profile consistency."31 In addition, local-ized heat pulse propagation is faster than thesteady state transport for Ohmic discharges,but comparable for auxiliary heated dis-charges with flat deposition profiles.

27 T. Antonsen, B. Coppi, and R. Englade, Nucl. Fusion 19:641 (1979).

28 S. Wolfe et al., Nucl. Fusion 26:329 (1986).

29 B. Coppi, In Abstracts of the 1984 Annual Controlled Fusion Theory Conference, Lawrence Livermore NationalLaboratory, Lake Tahoe, Nevada, 1984.

30 R. Englade, RLE PTP-87/12 (1987), to appear in Nucl. Fusion (1989).

31 B. Coppi, Comm. Plasma Phys. Cont. Fusion 5:261 (1980).



These considerations have led us toconclude32 that the relevant electron cross-field heat flux is the result of at least two dif-ferent transport processes described as fastand slow. The slow process can be appro-priately described by a standard diffusionterm that is linear in the temperature gra-dient, but the fast process is strongly deter-mined by the departure of the electrontemperature from the canonical profile andhas a nonlinear temperature gradient depend-ence in our formulation. In neither case isthe magnitude of transport required toexplicitly depend on the profile and strengthof the source of electron thermal energy,even though the solution of the energybalance equation adheres reasonably closelyto the "principle of profile consistency."Analytic solutions of a simplified energybalance equation verify the applicability ofthe model.

1.3.3 Sawtooth and FishboneStabilization in Auxiliary Heated andFusion Burning Plasmas

Recent experiments at JET 33 have shown thation cyclotron resonance heating (ICRH) canstabilize "sawtooth" oscillations. Shortlyafter ICRH turn-off, a sawtooth crash occurswith a delay time that is a finite fraction ofthe slowing down time of energetic ionsproduced by the RF fields.34 In collaborationwith the JET theory group, we have under-taken a study of the interaction of globalno = 1, mo - internal modes (which areresponsible for the sawtooth) with a highenergy ion population. Standard MHDtheory35 yields an instability parameter for

these internal modes, denoted AH. The ener-getic ions 1) contribute to a reduction in thisparameter iH, -,H + Re(AK) < AH, and 2)provide a viscous-like dissipation term, Im(AK)that arises from a resonance between themode and trapped energetic ions that precessaround the torus due to the curvature of theequilibrium magnetic field.

In high temperature plasmas, where finite iondiamagnetic frequency has brought the idealm'_1 kink to marginal stability, the residualgrowth is provided by electrical resistivity.This growth can be negated by hot ioneffects. Detailed calculations, using a modeldistribution function appropriate for energeticions produced by ion cyclotron resonanceheating, have shown 36 that a finite windowof stability exists in 13p -#ph space (fl, =poloidal beta of thermal plasma, ph =

poloidal beta of energetic particles) againstall m' = 1 internal modes and that, for agiven q(r) profile, there exists a maximumvalue of the thermal plasma parameter

H oc c/p above which no stabilization is pos-sible. We have also identified37 the importantrole played by AK(w = 0) on the stabilizationof the resistive internal kink. These topicshave been further considered in the contextof ignited deuterium-tritium plasmas38 wherefusion a-particles are shown to stablize idealmodes that would otherwise produce"sharkstooth oscillations" of the centralplasma column. By parametrizing the g -par-ticle functional

K = (ro/R)3/2 (VA/! -SoROD) #pot AK(/ODM)

where ro is the position of the q(r)= 1surface, So the magnetic shear at that surface,

32 B. Coppi, Phys. Lett. A 128:193 (1988).

33 D.J. Campbell et al., Phys. Rev. Lett. 60:2148 (1988).

34 F. Porcelli and F. Pegoraro, Second European Fusion Theory Meeting, Varenna, Italy, 1987.

35 M.N. Bussac, R. Pellat, D. Edery, and J.L. Soule, Phys. Rev. Lett. 35:1638 (1975).

36 B. Coppi, R.J. Hastie, S. Migliuolo, F. Pegoraro, F. Porcelli, Phys. Lett. A1 32:267 (1988).

37 F. Pegararo, F. Porcelli, B. Coppi, P. Detragiache, S. Migliuolo, in Plasma Physics and Controlled Nuclear FusionResearch 1988. International Atomic Energy Agency (1989). Forthcoming.

38 B. Coppi, S. Migliuolo, F. Pegoraro, F. Porcelli, submitted to Phys. Fluids (1989).

169


VA the Alfv6n frequency, w(D,M a represen-tative39 value of the precession drift fre-quency and fl the mean poloidal betaweighted by (r/ro) 3/2, marginal stability can beexpressed in terms of the almost universalcurve AK(/D,), a curve whose principaldependency is on the q(r) profile of the rele-vant experiment.

1.3.4 Density Limit and D-T ignitionConditions

Relatively high values of the plasma densityare required to ensure a confinement param-eter nE (central electron density times energyconfinement time) sufficient to reach ignitioneven for deuterium-tritium fuel. Collisionalion thermal conductivity and bremsstrahlungradiation, both proportional to the square ofthe particle density, play a strong role in theglobal energy balance up to temperaturessomewhat above Tm, the "minimum ignitiontemperature." Near and above Tm, the pres-ence of an anomalous ion thermal conduc-tivity can have significant effects.Investigation of the microinstability known asthe ion temperature gradient mode (also ther/i or ion mixing mode) shows that it can giverise to an ion thermal conductivity that scalesas nT. /2 . In collaboration with W. Tang ofthe Princeton Plasma Physics Laboratory, wehave analyzed an approximate modelequation for the steady state energy balancein the central region of a tokamak plasmawhich includes this type of transport. 40 Multi-valued solutions for the density are obtainedas a function of temperature, ion thermalanomaly strength, and power.

If no ion anomaly is active, the density mustremain below a certain maximum value forthe plasma to attain the minimum ignition

temperature Tm, but any density is allowedbeyond Tm. In the presence of anomaloustransport, a critical heating power depositionthat increases with the magnitude of theanomaly is required to open a channelbetween regions of allowed density for tem-peratures above and below Tm. Since therate of change of the fusion cross-sectiondecreases with temperature, the density mustbe progressively increased above a minimalvalue, to heat the plasma to steady-statetemperatures considerably higher than Tm.

1.3.5 Alpha Particle InducedFishbone Oscillations in FusionBurning Plasmas

The "fishbone" is a burst of oscillations thatoccurs in some experiments in which plasmashave two components, a main plasma and anenergetic ion population. The fishbone hastoroidal n-l and poloidal m_~1 spatialdependence and a frequency close to thediamagnetic frequency of the core (mainplasma) ions. 41 Three requirements must bemet to have instability: 1) high enough tem-peratures are produced so that finitediamagnetic frequency brings the ideal modeto marginal stability; 2) the core plasma betamust exceed a threshold value; and 3) aviscous-like resonant interaction of the wavewith trapped energetic ions, that precess dueto the curvature of the magnetic field, mustexist. For parameters relevant to an Ignitordevice,42 alpha particles with energies near300 keV resonate with the fishbone oscil-lation.

Recent work43 has focused on the stabiliza-tion of this mode (as well as that of theinternal kink, see 1.3.3, "Sawtooth andFishbone Stabilization in Auxiliary Heated

39 Ibid.

40 B. Coppi and W.M. Tang, Phys. Fluids 31:2683 (1988).

41 B. COppi and F. Porcelli, Phys. Rev. Lett. 57:2272 (1986); B. Coppi, S. Migliuolo and F. Porcelli, Phys. Fluids31:1630 (1988).

42 B. Coppi, L. Lanzavecchia, and the Ignitor Design Group, Comm. Plasma Phys. Cont. Nucl. Fusion 11:47 (1987).

43 B. Coppi, S. Migliuolo, F. Pegoraro, F. Porcelli, submitted to Phys. Fluids (1989); B. Coppi and F. Porcelli, FusionTechnology 13:447 (1988); B. Coppi, S. Migliuolo and F. Porcelli, RLE PTP-88/4 (1988).



and Fusion Burning Plasmas" on page 169)as a consequence of the energetic particlecontribution, AK(W), to the effective instabilityparameter ,H + AK(). Here AH is the param-eter given by standard 44 ideal MHD theory.The real part of AK is negative for modes withCOOWdi much lower than DOM which is ameasure of the precession drift frequency ofalphas at their birth energy. Thus, the effec-tive instability parameter decreases and thefishbone restabilizes at high enougha-particle pressure. This stabilization arisesas the last step in the stabilization of theideal internal kink (see 1.3.3, "Sawtooth andFishbone Stabilization in Auxiliary Heatedand Fusion Burning Plasmas" on page 169),namely when:

_Rw.*i/V A > 'H + ReAK(w) > 0

We have also found that another mo-l oscil-lation can exist, but only at very high a pres-sure (lp,, > 1). This oscillation is at higherfrequency (OwDaM/ 3 ) and can be entirelysupported by trapped energetic ions whichprovide a destabilizing contribution:Re[2K(OO 9ODM/3)] > 0. This instability, nowknown as the "fast fishbone," was in factdiscovered previously 45 in the context offishbones observed in beam injected plasmas.

1.3.6 Transport Simulations ofThermonuclear Ignition in CompactExperiments

The attainment of ignition in a toroidaldevice depends critically on the net balancebetween bulk heating of the plasma andenergy loss due mainly to anomalouselectron and ion thermal conductivity,bremsstrahlung, synchrotron radiation, andmacroscopic processes such as sawtoothoscillations. Using a modified version of the

BALDUR one and one-half-dimensionaltransport code,4 6 we have investigated theimpact of these processes on the time evolu-tion of D-T plasmas in a compact, high fielddevice of the Ignitor type with strong ohmicheating.47

We have used a generalization of Coppi-Mazzucato-Gruber diffusion to modelelectron heat transport that includes theeffects of alpha particle heating. For ananomalous ion heat diffusion coefficientpeaked halfway out in the discharge andsawtooth repetition times down to 0.3seconds, ohmic ignition could be achievedover a fairly wide density range. A centralelectron density near 8 1014cm - 3 was optimalin the sense of allowing relatively rapidapproach to the state in which fusion heatingdominates ohmic heating. A slightly higherdensity allows faster heating to fusionburning regimes. The overall magnitude ofelectron heat transport could be increasedabout a factor of two before ignition wasprevented, while the magnitude of ion heattransport was not a sensitive parameter. Inaddition, we have studied the effects of vari-ations in the form of electrical resistivity,plasma impurity level, and flux surface equi-librium configuration on the approach toignition. Simulation of the Compact IgnitionTokamak (CIT) illustrates the rather strongauxiliary heating requirements of that design.

In collaboration with Miklos Porkolab, wehave modified our one and one-half equilib-rium and transport code to study electroncyclotron radio frequency heating prospectsin the proposed CIT by incorporating powerdeposition results from a stand-alone raytracing package. We have simulated someimportant aspects of proposed heating sce-narios. We have examined the individualeffects of electron and ion thermal diffusivitymagnitudes, sawteeth, profile consistency,

44 M.N. Bussac, R. Pellat, D. Edery, and J.L. Soule, Phys. Rev. Lett. 35:1638 (1975).

45 L. Chen, R.B. White, M.N. Rosenbluth, Phys. Rev. Lett. 52:1122 (1984).

46 G. Bateman, Spring College on Plasma Physics, Trieste, Italy, 1985.

47 R. Englade, RLE PTP-87/12 (1987), to appear in Nucl. Fusion (1989).

171


and density profile shape in achievingignition in this experiment.48

1.3.7 Experiment to InvestigateD- 3 He Burning

Recent developments in the design of highmagnetic field toroidal experiments for thestudy of D-T burning plasmas have encour-aged us to consider the possibility of anexperiment burning D-3He at higher temper-atures and pressures. We have proposed adesign, the Candor,49 that can be constructedusing present-day technology. It has aD-shaped cross section, with major radiusRo = 170 cm, a horizontal minor radius of 65cm and a vertical elongation of 1.8. Thevacuum toroidal magnetic field would reachup to 13.5 Tesla at Ro and the maximumtoroidal plasma current would be 18megamperes. The machine is intended touse D-T burning as the initial boost to reachD-3He burning conditions. We are carryingout this work with several engineering andscientific groups in the United States andEurope.

1.3.8 Advanced Fuel Fusion

The strength of potentially damaging, highenergy neutrons from the D-T fusion reactionmakes other reactions, such as D-3He andD-D, more attractive for a potential reactor.We have developed a set of numerical trans-port codes that can handle the multiple ionspecies (protons, deuterons, tritons, 3He ,4He) and have used the one-dimensionalversion to study the dynamics of ignition andburn for the Candor device.50 Results showthat ignition of D-3He using a D-T start-up is

possible if the thermal transport losses do notincrease rapidly with temperature. Studies ofthe D-3He burning state have shown thatlean (i.e., 30 percent of the total ion density)3 He mixtures will burn at the lowest values ofthe plasma beta (the ratio of the plasmakinetic to magnetic energies). Effectivefusion ash removal and central fueling of theburning nuclei are essential to a sustainablereaction. Given these prerequisites, eitherburn control or operation at high beta arenecessary because the low temperatureburning state is dynamically unstable.

1.3.9 The Role of the UbiquitousMode in Anomalous Electron EnergyTransport

Electron thermal energy transport in mag-netically confined plasmas is one of the mostimportant subjects in fusion research. Exper-iments indicate that such a transport processis anomalous, namely that the observedelectron energy confinement time is an orderof magnitude smaller than the one predictedby collisional neoclassical theory. We find51

that the ubiquitous mode5 2 satisfies therequired5 3 properties for driving the anoma-lous transport:

1. It is driven by the electron temperaturegradient;

2. It is active over a significant portion ofthe plasma column and is not sharplyreduced in regions of considerablecollisionality (Ve/F1Obe );

3. Its wavelength is long enough toproduce substantial energy transport overa significant portion of the plasma;

48 R. Englade et al., International Sherwood Theory Conference, San Antonio, Texas (1989).

49 B. Coppi and L. Sugiyama, RLE PTP-88/6 (1988), submitted to Nucl. Fusion.

50 Ibid.

51 Y-K. Pu, Ph.D. diss., MIT, 1988.

52 B. Coppi and F. Pegoraro, Nuclear Fusion 17:969 (1977).

53 B. Coppi, Phys. Lett. A 128:193 (1988).



4. The energy transport is up to an order ofmagnitude larger than the correspondingparticle transport.

1.4 Tokakmak Experiments inthe High Poloidal Beta Regimeon Versator II

Sponsors

U.S. Department of Energy(Contract DE-AC02-78-ET-5101 3)

Project Staff

Professor Miklos Porkolab, Dr. Kuo-in Chen,Dr. Stanley C. Luckhardt, Stefano Coda,Jeffrey A. Colborn, Robert J. Kirkwood, NoraNerses, Kurt A. Schroder, Jared P. Squire,Jesus N.S. Villasenor

In the past year, experiments on the VersatorII tokamak have pioneered an experimentaltechnique for producing plasma equilibriawith very high values of poloidal beta.Poloidal beta (fP) is defined as the ratio ofthe volume average plasma pressure to themagnetic pressure generated by the tokamakplasma current. In toroidal equilibria at highflp instability of ideal MHD ballooning insta-bilities can limit the maximum achievableplasma pressure achievable in tokamaks.However, if these modes could be stabilized,a significant design constraint in tokamakD-T fusion reactors would be eliminated.Such high fP, operation is also an importantstep to making advanced fuel fusion reactorsfeasible.

Experiments have been carried out onVersator II in a regime of low plasma currentwith the aim of reaching high poloidal beta,fp. These experiments use the energeticelectron population created by lower hybridcurrent drive to provide the plasma currentand a dominant component of the plasmapressure. The phenomena studied includetoroidal equilibria at high flP (s/p 1.3), RFdriven current profiles with high q(0), access

to the second stability regime, and relatedmagnetic and density fluctuation phenomenaat high e ,.

Our experiments will address issues in thefollowing areas: 1) MHD equilibrium proper-ties of high /, toroidal equilibria, 2) accessto the second stability regime for ballooningmodes, and 3) related fluctuations and insta-bilities at high /,. At Versator II, we havepioneered a technique for producing suchhigh efp plasmas that the pressure is predom-inantly generated by energetic electronsmaintained by lower hybrid current drive.The energetic electron population created byRF current drive has an anisotropic pressurewith P,1 > P± where

P11 = d3v[ymefe(vi , vL)v ]

and

P = d3vl[mef(vl , v 1)v ]

are the components of the pressure tensorparallel and perpendicular to the magneticfield. The poloidal beta for such an aniso-tropic pressure plasma entering the toroidalequilibrium condition was evaluated byMondelli and Ott54 to be

<1/2PII>P 2

B2/2y0

<1/4PL>

B2/2 0o

where the brackets indicate a toroidal volumeaverage, and Ba = pol/(2a). (Note that PI asdefined above contains v2 and v terms.) Theenergetic electron velocity distribution func-tion, f(vl , v1) , generated by lower hybridcurrent drive, has been the subject of exten-sive theoretical work 55 and well founded ana-lytical models of f are available.Generally f(v) consists of a Maxwellian bulkand a flat plateau produced by quasilineardiffusion extending up to a maximum parallelvelocity v11 = v2. Using the analytic model

54 A. Mondelli and E. Ott, Phys. Fluids 17:1018 (1974).

55 N.J. Fisch, Rev. Modern Phys. 59:175 (1987); V. Fuchs et al., Phys. Fluids 28:3619 (1985).

173


discussed in Fuchs et al.,56 the above formulafor fl can be evaluated to give

TAIL Ct IA(V2) (2)P IRF

where IRF is the RF driven plasma current; IAis the Alfven current, IA = 177V2/c in kA; y isthe usual relativistic factor evaluated atv = v2, and C, is a model dependent numer-ical constant and is approximately equal tounity. Equation 2 indicates that highpoloidal beta can be produced by reducingthe RF current below the Alfven current.Experiments on Versator II have demon-strated this effect as discussed in Luckhardtet al.57 and provide a means of producinghigh poloidal beta toroidal equilibria in ourexperiment.

The following is a summary of the activitiesin the high efp experiments, theory and mod-eling of Versator II, diagnostics, and dataanalysis.

30

20

p 2

0

0

0 10

o0061r-mcuirl

;, = 2,1 SG6a

20 30 40

TIME (MSEC)

1.4.1 High Poloidal Beta Using f =2.45GHz Current Drive

During 1987, the high poloidal betaequlibrium technique was developed withcurrent drive at f=0.8GHz. Recently theseexperiments were repeated using thef=2.45GHz RF system, allowing operation athigher density. A discharge obtained withthe f=2.45GHz system is shown in figure 8.Note that fp + 1 i/2 - p by definition. Oper-ation at increased density improves the accu-racy of the density measurement andsimplifies the analysis of hard x-ray profiledata.

1.4.2 New Magnetic DiagnosticsInstalled

An array of 36 poloidal magnetic field probeswas constructed and installed and has beenused in detailed magnetic equilibrium anal-ysis (see 1.4.3, "Data Analysis with XFIT").Further magnetic diagnostics have beendesigned for measurement of fast magneticfluctuations associated with kink or possiblyballooning modes. These consist of mag-netic probes to be installed inside thevacuum chamber. Two types of probes willbe used; the first of which is a magneticpickup coil designed for Alcator C-MOD.This probe is shown in figures 9 and 10. Thefrequency response of the coil has a band-width of : 0.5MHz. The coil mount can berotated to pick up fluctuations in both thetoroidal and poloidal directions. A secondtype of pickup probe was designed whichconsists of a set of short length coils capableof detecting fluctuations with higher poloidalmode numbers (see figure 11). For qa = 27this array will detect modes with m 160and n < 6.

1.4.3 Data Analysis with XFIT

An equilibrium analysis code was developedsimilar to the EFIT code developed at GA

Figure 8.

56 V. Fuchs et al., Phys. Fluids 28:3619 (1985).

57 S.C. Luckhardt et al., Phys. Rev. Lett. 62:1508 (1989).


I


Ttubeo.o '

I

1. 3in

1TDi, iLPI _ _.i. '* -

TO 110A

i i i IIil =i I l I

"_,d ;m Tarfis; t* "+,;@r

.- Ek~,a,: :rc4,k o: iU ! : '

• II , .i ,,,..

*~ I

.11' St~e Las

1----. --I I

I :1f\. .~I*Ii~

* N~ . II

ao1 . ii lrln ' i i liH ,120"; ;A V 0i%

W •

rrs,, I it'I 1 i; 1 , , !'

,.~ ~ , ; I..:/A,. I "I

LJ . i,, mbI... . O~. , ....I.III I :.. I, :. 1 bJ;;:! " iil[ .l . .':

i i

i I iII ,i Oi," . i ,i

I I

i i. , Iii.t ' " " ." '

" i i ! , : fr01 Wif , ,.W.:!i i !Iii; :

II

-I

Figure 9.

Technologies.5 8 In Versator experiments mea-surements of the x-ray emissivity profileusing collimated PHA detectors providedcrucial information about the width of the RFcurrent profile. Current profile widths aremuch more difficult to obtain in ohmicallydriven plasmas and this width measurementin these experiments allows approximatevalues for 1i and q(0) to be found.

A data analysis code, XFIT, was developedusing PC-MATLAB software on an IBMPS/2 model 80 computer. XFIT includesequilibrium modeling of magnetic probe arraydata, density profile and hard x-ray profiledata. The x-ray measurements in conjunc-

tion with the density profile give informationabout the current width allowing determi-nation of the plasma internal inductance andapproximate values for q(0). The code flowchart is shown in figure 12, and figure 13shows an example of a best fit model equi-librium RF current profile, density profile, andx-ray emissivity profile. The results of thisfitting procedure are values of 1j, q(0), fl,,magnetic axis location, position and shape ofthe outer flux surface, and a best fit modelMHD equilibrium with JO(r,z) and i/f(r,z).Figure 14 shows the time evolution of theflux surfaces for the discharge shown infigure 8. Note that the current profile widthis determined from x-ray and density profiles

58 L.L. Lao et al., Nucl Fusion 25:1611 (1985).

175

aO6Z

nai -~--~- I --~ ~___ __

.I~-------

~1

iI: I


,COILSSTAINLESS STEEL TUBE

-- TEFLON DISK

SEAL

YMACOH PIJG-,-

ROTA1 ED

I

Figure 10. The test coils in the vacuum chamber.

during the steady state RF-driven portion ofthe discharge. Current profile widths cannotbe determined accurately during the initialohmic inductive phase. This time sequenceshows that the separatrix does not enter thelimiter volume at the highest fl, and, hence,these plasmas have not reached an equilib-rium limit. Further, the outward motion ofthe plasma during the current ramp-downcauses the current to continue to decay afterthe initiation of the RF pulse, but later in thesequence the equilibrium is seen to re-centerand fill the volume to the limiter radius.

1.4.4 Comparison with HighPoloidal Beta MHD EquilibriumTheory

The outward shift of the magnetic axis as afunction of fl for high poloidal beta toroidalequilibria has been calculated analytically byJ. Friedberg5 9 using model high fl equilibria.In this series of experiments, the magneticaxis was compared to the observed outwardshift of the density profile, and the magneticprobe array fit of the magnetic axis position.

Figure 11. Design of the next test coil.

The observed shifts (see figure 15) werefound in agreement with predictions of highsfp equilibrium theory. This result providesconfirmation of equilibrium theory at high efpwhere little or no data was available.

1.4.5 MHD Modeling and Criteriafor Transition to the SecondStability Regime

Theoretical calculations were carried out bythe PFC theory group including Drs. JayKesner, Barton Lane and Stefano Migliuolo.Regarding the ballooning mode stability con-dition for anisotropic pressure plasmas, theyshowed that when P, > P1 the stability con-dition is the same as for a scalar pressureplasma (PII > PI). This result suggests thatthe present experiments with anisotropicpressure are relevant indicators of possiblefuture machines with high efp and scalarpressure.

With regard to transition to the second sta-bility region: in the initial stage of theseexperiments we had begun using simple ana-

59 J.P. Friedberg, Rev. Mod. Phys. 54:801 (1982).


PLASMA

m


Figure 12.

lytical criteria available in the literature, e.g.,the Troyon criterion, the Wesson Sykes con-dition, etc., to assess the stability of theseequilibria. This analysis has been consider-ably improved in the past year. Now thepredictions of ideal MHD equilibrium andstability theory modeling are obtained usingthe Princeton-Grumman PEST code. ModelMHD equilibria were constructed which bestfit the experimental equilibria as determinedfrom XFIT. The model MHD equilibria werethen tested for stability to high n ballooningmodes using the PEST code. It was foundthat equilibria which best modeled the exper-iments had approximately circular plasmaboundaries and values of q(0) ranging up toq(0)= 6. The numerical modeling found astable transition from the first to the secondregime when q(0) > 5, q, = 20, and E8lp > 0.4.Here we define the transition to second sta-bility as occuring at the least stable or mostunstable value of fl. For lower q(0), higherq,, or broader pressure profiles, an unstablerange of efp exists for high n modes, but athigher Efp these modes restabilize. For q(0)

6 (as indicated for these experiments byXFIT) and qa = 28 these modes restabilizedabove Eflp = 0.7. Thus for s/p > 0.7 these

Figure 13.

plasmas are predicted to enter the secondstable region. The experimental model equi-libria were plotted on a stability diagram forhigh n ballooning modes and the result isshown in figure 16. The data points for thethree equilibria at highest f , appear to be inthe second stability region. Thus, the codemodeling of our experiment suggests thatsome of the equilibria obtained were near orbeyond the transition to the second stabilityregion.

These experiments have been reported inPhysical Review Letters 62:3 (1989) andpresented in an invited talk at the 1988 APSDivision of Plasma Physics Meeting in Holly-wood, Florida. The magnetic flux surfacereconstruction was first presented at theIAEA TCM on Research Using SmallTokamaks on October 10, 1988, in Nice,France, and a paper on these results will be

177

XFIT EQUILIBRIUM FITTING PROGRAM

MODEL RF CURRENT DENSITY J(R,Z)IMODEL PARAMETERS A1,A2,...,A71T

- - 1

SCALCULATE POLOIDAL FLUX, TAIL DENSITY,BULK DENISTY SELF CONSISTENT WITH J(R,Z)

CALCULATE MEASURED QUANTITIESIMAGENTIC FIELD AT PROBES, DENSITY PROFILE,

X-RAY EMISSIVITY PROFILEI

COMPARE DATA WITH CALCULATION

LOOP BACK AND ADJUST PARAMETERSOPTIMIZE FIT TO PROBE AND X-RAY PROFILE DATA

FINAL BEST FIT EOUILIBRIUM PARAMETERS

OUTPUT BEST FIT MODEL EQUILIBRIUMIPOLOIDAL FLUX, CENTRAL SAFETY FACTOR.

INTERNAL INDUCTANCE]

RF CuRReN T peNsrTY

uLk DEN-ri T

, FIT

~, w/vryrr~RD x-R~Y


Figure 14.

published in the proceedings of that meeting.Another paper containing the experimental

0.4

0.2

0.00 1

Figure 15.

2 3

fp

4 5 6

HIGH N BALLOONING STABIUTY (PEST)

2 3

Pp

Figure 16.


a = 3.1

20

p 0 D

P 4-0 10 20 30 40 so

TIME (MSEC)

* DENSITY PEAKTHEORY J. FREIDBERC. Rev. Mod. Phys. (1982)

A MHD CODEV MAGNETIC PROBE DATA

AA *4, - ~ ;i*~C

results and theory contributions was pre-sented at the 12th International Conferenceon Plasma Physics and Controlled Fusion inNice, France (1988). The results of exper-iments using the 2.45GHz current drivesystem to produce high flP plasmas were pre-sented at the U.S./Japan Workshop onAdvanced Current Drive Methods at KyotoUniversity, Kyoto, Japan (1988), and a paperwill appear in the proceedings of that work-shop.

The equilibrium characteristics of these highflp plasmas has been the subject of the bulkof the experimental work to date. The bal-looning mode stability of these equilibria hasbeen assessed by MHD code modeling.Taking into account information on thecurrent profile width coming from the x-rayprofile measurement, the important parame-ters 1i and q(0) are known to some degree.Although the theoretical modeling of theseplasmas suggests that the second stabilityregime was entered, it is desirable to findmore experimental evidence that this isindeed the case. In the coming year, we planto improve our x-ray and density profilediagnostics, and install diagnostics to lookfor ballooning mode activity in theseplasmas.

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