Sction 2 Plasma Physics - COnnecting REpositories · Chapter 1. Plasma Dynamics Project Staff Professor George Bekefi, Professor Jonathan S. Wurtele, Ivan Mastovsky, Dr. Chiping Chen,

Sction 2 Plasma Physics

Chapter 1 Plasma Dynamics

178 RLE Progress Report Number 134

Chapter 1. Plasma Dynamics


Academic and Research Staff

Professor George Bekefi, Professor Abraham Bers, Professor Bruno Coppi, Professor Miklos Porkolab, Pro-fessor Jonathan S. Wurtele, Dr. Chiping Chen, Dr. Shien-Chi Chen, Dr. Ronald C. Englade, Dr. StefanoMigliuolo, Dr. Abhay K. Ram, Dr. Barrett Rogers, Dr. Linda E. Sugiyama, Ivan Mastovsky

Visiting Scientists and Research Affiliates

Dr. Jean-Loup Delcroix,' Dr. Cesar Meirelles Filho,2 Dr. Lazar Friedland, 3 Dr. Vladimir Fuchs,4 Dr. EliJerby,5 Dr. Pallavi Jha,6 Dr. Christopher Lashmore-Davies,7 Dr. Marco Nassi, 8 Dr. Leonid E. Zakharov,9

Toru Hara, 10 Takahide Mizuno"

Graduate Students

Neer Asherie, Riccardo Betti, Palmyra E. Catravas, Carson C. Chow, Jeffrey A. Colborn, Manoel E. Conde,Darin Ernst, Wenian Hu, Mark K. Jablonski, Kenneth C. Kupfer, Michael C. Moldoveanu, Greg Penn,Steven D. Schultz, Gennady Shvets, Jared P. Squire, Richard E. Stoner, Luigi Vacca, Jesus NoelVillasenor, Pavel P. Volfbeyn

Undergraduate Students

Mustafa K. Ahmed, Daniel J.H. Chung, Jonathan C. Doan, Colin J. Taylor, Sasha K. Wood

Technical and Support Staff

Felicia G. Brady, Laura B. Doughty, Edward W. Fitzgerald, Kerry L. Gafney, Catherine Lorusso

1.1 Relativistic Electron Beams

Sponsors U.S. Army - Harry Diamond LaboratoriesContract DAAL02-89-K-0084

National Science Foundation U.S. Department of EnergyGrant ECS 89-02990 U.S. Department of EnergyGrant ECS 89-02990 Contract DE-AC02-90ER40591

U.S. Air Force - Office of Scientific ResearchGrant AFOSR 89-0082-B U.S. Navy - Office of Naval Research

Grant N00014-90-J-4130

1 Professor, University of Paris, Orsay, and Ecole Sup6rieure d'Electricit6, France.

2 Universidade de Sbo Paulo, Brazil.

3 Professor, Hebrew University of Jerusalem, Israel.

4 Centre Canadien de Fusion Magn6tique (CCFM), Quebec, Canada.

5 Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel.

6 Physics Department, Lucknow University, Lucknow, India.

7 Visiting Scientist, AEA Fusion, Culham Laboratory, United Kingdom.

8 Politecnico di Milano, Milan, Italy.

9 Kurchatov Institute of Atomic Energy, Moscow, U.S.S.R.

10 Department of Nuclear Engineering, The University of Tokyo, Japan.

11 The Institute of Space and Astronautical Science, Sagamihara, Kanagawa, Japan.

179


Project Staff

Professor George Bekefi, Professor Jonathan S.Wurtele, Ivan Mastovsky, Dr. Chiping Chen, Dr.Shien-Chi Chen, Toru Hara, Dr. Eli Jerby, Dr.Pallavi Jha, Takahide Mizuno, Palmyra E. Catravas,Manoel E. Conde, Wenian Hu, Gennady Shvets,Richard E. Stoner, Pavel P. Volfbeyn, Jonathan C.Doan, Daniel J.H. Chung, Colin J. Taylor, Sasha K.Wood, Felicia G. Brady

1.1.1 Experimental Study of a 33.3GHz Free Electron Laser Amplifierwith a Reversed Axial GuideMagnetic Field

The free electron laser (FEL) operating in a com-bined axial guide magnetic field and a helicalwiggler field has been studied experimentally 12 andtheoretically13 over a period of many years, both inlinear and nonlinear regimes. In all these studies,the axial magnetic field Bz is oriented so that thecyclotron rotation of the beam electrons is in thesame direction as the rotation imposed by thehelical wiggler field B, This leads to an increasel4of the transverse electron velocity vi compared towhat it would be in the absence of Bz, with poten-tial benefits such as an enhanced radiation growthrate and efficiency. Indeed, when the cyclotronwavelength in the axial field c = 2nvz/Qz,approaches the wiggler periodicity Iw the trans-verse electron excursions can become too large,the electrons strike the drift tube wall and are lost(uz = eBz/moy is the cyclotron frequency in theguide field and y = [1 - (vz/c) 2 - (vl/c)2]- 1/ 2 isthe relativistic energy factor). Thus, the "reso-nance" A, = Iw becomes a dividing line for conven-tional FEL operation: at relatively weak axial fields,Ac > Iw, we have the so called Group I regime, andfor stronger fields such that Ac < Iw, the Group IIregime.

We report measurements using a new, hithertounexplored configuration with a reversed axialmagnetic field. The rotation of the electrons in thehelical wiggler field Bw is now opposed by thepresence of the guide field Bz and there is nolonger the resonance at A, = Iw. The transverseelectron velocity v, is diminished compared to

what it would be in the absence of Bz (however,the latter reduction is partially compensated in ourexperiments by increasing Bw). We will show thata reversal of Bz yields higher radiation intensityand efficiency compared to what we were able toachieve with the conventional orientation of theaxial magnetic field.

The three regimes of FEL operation (Group I,Group II, and Reversed Field) are displayed infigure 1 based on a particle trajectory calculationof v,/c which neglects space charge and radiationand assumes that the electrons are undergoingideal helical orbits in the combined Bz and Bwfields. The solid points illustrate the three param-eter regimes of Bz where maximum radiation hasbeen observed in our experiments.

A schematic of the FEL amplifier is shown in figure2. A mildly relativistic electron beam (750+ 50keV) is generated by a Marx capacitor bank(Physics International Pulserad 110 A). The elec-trons are emitted from a hemispherical graphitecathode by an explosive field emission process.The graphite anode acts as an emittance selector,allowing only a small fraction of the current to

-12 -6 0 6 12

axial magnetic field Bz (kG)

Figure 1. Ideal equilibrium electron orbits calculatedfor two different values of wiggler magnetic field. Thesolid points show the values of Bz where maximumpower is observed for Group I, Group II and ReversedField regimes.

12 S.H. Gold, D.L. Hardesty, A.K. Kinkead, L.R. Barnett, and V.L. Granatstein, Phys. Rev. Lett. 52: 1218 (1984); J.Fajans, J.S. Wurtele, G. Bekefi, D.S. Knowles, and K. Xu, Phys. Rev. Lett. 57: 579 (1986); also, J. Fajans, G.Bekefi, Y.Z. Yin, and B. Lax, Phys. Fluids 28: 1995 (1985).

13 A.K. Ganguly and H.P. Freund, IEEE Trans. Plasma Sci. 16: 167 (1988).

14 L. Friedland, Phys. Fluids 23: 2376 (1980); P. Diament, Phys. Rev. A 23: 2537 (1981); H.P. Freund and A.K.Ganguly, IEEE J. Quantum Electron. QE-21: 1073 (1985).



Figure 2. Free electron laser experimental setup.

propagate through its 2.54 mm radius and 62 mmlong aperture. The electron beam current down-stream from the emittance selector, and in theabsence of the wiggler field, is illustrated in figure3a, showing saturation at high B, where all avail-able electrons from the gun have made it throughthe anode hole. Using the technique of Prosnitzand Scharlemann, 15 we estimate the normalizedRMS beam emittance to be n < 4.4 x 10- 2 cm-radand the corresponding RMS axial energy spread tobe Ay/y, < 1.5 x 10-2 . We observe from figure3b that when the wiggler is turned on, anexpected, very pronounced current loss occursnear resonance a, = Iw with the conventional ori-entation of Bz. This happens because the trans-verse velocity of the electrons become too largeand eventually they are lost to the waveguidewalls. No significant current loss (also asexpected) is seen with the magnetic field reversed(figure 3c).

The 50 period bifilar helical wiggler produced bycurrent carrying helical wires has a period of 3.18cm and provides a magnetic field whose magni-tude on axis is adjustable up to 1.8 kG. Thewiggler field intensity is slowly increased over theinitial six periods, providing an adiabatic input forthe electron beam. The system, including the gun,is immersed in a uniform axial magnetic field gen-erated by a solenoid. The intensity of this fieldcan be varied up to a maximum of 11.6 kG.

The two-meter long stainless steel drift tube hasan internal radius of 0.51 cm and acts as a cylin-drical waveguide whose fundamental TE11 modehas a cutoff frequency of 17.2 G Hz. The system isdesigned to operate in this lowest waveguidemode.

To Spectrum Anolyzer

15 D. Prosnitz and E.T. Scharlemann, LLNL ATA Note No. 229, February 1984.

181

, 300 B =0 ** I 4

200

loo **•

100ooOm

.. (a)

S300 B = 630 G 0 II

S Bz positive group II200

o O o°

100 group I }Q *" (b)

0

3 300 - B = 1.47 kG 0 ( * -Bz reversedz

Q) 200 0

100 -

*i (c)

00 3 6 9 12


Figure 3. Electron beam current in the FEL as a func-tion of the axial guide magnetic field Bz; (a) no wigglermagnetic field; (b) wiggler field Bw = 630 G and Bz inthe conventional direction; (c) wiggler field Bw = 1.47kG and Bz in the reversed direction.

A high power magnetron operating at 33.39 GHzis the input power source for the FEL amplifier.The wave launcher consists of a short section ofcircular waveguide of radius 0.31 cm into which17 kW are coupled from a standard Ka-band rec-tangular waveguide. This section of circular wave-guide supports only the fundamental TE11 modefor the operating frequency. Its radius is thenadiabatically increased to the radius of the drifttube. A linearly polarized wave is thereby injectedinto the interaction region. Half of the incidentpower, with the correct rotation of the electric fieldvector, participates in the FEL interaction.

A parameter scan of the output power has beencarried out in order to discover the optimum oper-ating conditions for our three regimes, Group I,Group II and Reversed Field. Figure 4a illustrates

MagnetronAnode


the output power as a function of Bw at constantBz, and figures 4b and 4c show how the powervaries with Bz at constant Bw. It is seen that themaximum output power obtained in Group I andGroup II regimes is approximately the same, 5MW; however, the efficiency is much higher forthe Group I regime since here the beam current issmaller. The output power for the reversed fieldcase is higher by an order of magnitude andreaches a level of 61 MW.

The spatial growth of the electromagnetic waveintensity is determined from the measurement ofthe output power as a function of the length of theinteraction region. This length is varied bychanging the distance that the electron beam isallowed to propagate in the drift tube. Applicationof a strong magnetic field is sufficient to deflectthe electrons into the waveguide wall and therebyterminate the interaction at that point. Figure 5shows the result of this measurement for the threedifferent regimes. In the Group I regime the powerlevel reaches saturation at 5.8 MW correspondingto an efficiency of 9 percent. Operation in GroupII shows the lowest efficiency (2 percent). TheReversed Field operation has by far the highestefficiency (27 percent), and exhibits no power sat-uration (Table 1).

Table 1. Summary of Experimental Results.

Group Group ReversedParameter I II Field

Frequency 33.39 33.39 33.39(GHz)

Beam energy 750 750 750(keV)

Beam current 90 300 300(A)

Guide field 4.06 10.9 -10.9(kG)

Wiggler field 0.63 0.63 1.47(kG)

Output 5.8 4.2 61power (MW)

Efficiency 9 2 27(%)

In conclusion, we have found that our free elec-tron laser shows highest efficiency and highestpower output when operated with reversed axialguide magnetic field. Table I summarizes ourfindings. We believe that the 27 percent efficiencyof our system exceeds the efficiency of previousFELs with untapered wigglers. If the ~ 1 dB/mattenuation due to the stainless steel waveguidewere subtracted from the measurements, thepower and efficiency could be substantially higher.Moreover, figure 5c shows that our wiggler is tooshort to reach saturation, and much higher effi-ciency may well be possible with longer and/ortapered magnetic fields.

We note that our FEL falls into the so calledRaman parameter regime where RF space chargeeffects must be allowed for, both in order toaccount for the radiation frequency and the power

100 .

S10 B = 10.9 kG = -10.9 kG1

o

0.1S(a)0.01

0.0 0.5 1.0 1.5 2.0

wiggler magnetic field Bw (kG)

100 -B = 630 Gw

B10 positive

, 1 *1

o •

0.1

0.01 (b)

100 B = 1.47 kG oeaoooB reversed

a 10 Z

o .

0.1

0.01

0 3 6 9 12


Figure 4. FEL output power as a function of Bz andBw; (a) Bw scan for fixed Bz = 10.9 kG in each direc-tion; (b) Bz scan in the conventional direction for fixedBw = 630 G; (c) Bz scan in the reversed direction forfixed Bw = 1.47 kG.



output. A detailed theoretical understanding ofthe Reversed Field FEL is as yet unavailable due inpart to some lack of understanding of the electrontrajectories themselves. This is clearly shown infigure 4c, where a large, unexpected dip in outputpower occurs near Iw - Ac (Bz - - 7.6kG), butno such dip occurs either in the electron current(figure 3c) or in calculations based on ideal orbittheory. Thus, the ideal orbit model on whichfigure 1 is based needs substantial modificationwhen B, is reversed from its usual orientation.Such studies are now in progress.16

100 group I

10

a 0.1 44 dB/m

S(a)0.01

100 - group II

10 -

10

0.1 = 38 dB/m

** (b)0.01 (b)

100reversed field '--

o* -

I 0.1 P=41.dB/m

0.01 (c)

0 60 120 180

interaction length (cm)

Figure 5. FEL output power as a function of inter-action length; (a) Group I regime; (b) Group II regime;(c) Reversed Field regime. The Fs represent growthrate estimates over interaction regions indicated by thestraight lines.

1.2 Plasma WaveInteractions--RF Heating andCurrent Generation

Project Staff

Professor Abraham Bers, Dr. Abhay K. Ram, Dr.Jean-Loup Delcroix, Dr. Vladimir Fuchs, Dr. LazarFriedland, Dr. Christopher Lashmore- Davies,Mustafa K. Ahmed, Carson C. Chow, Mark K.Jablonski, Kenneth C. Kupfer, Michael C.Moldoveanu, Steven D. Schultz, Luigi Vacca

1.2.1 Introduction

The research work of this group is concerned withstudies on the electrodynamics of plasmas andtheir applications. Attention is directed towardunderstanding the nonlinear dynamics of plasmasdriven by high-frequency electromagnetic fields(as in RF heating and current drive of magneticallyconfined plasmas or in laser-plasma interactions)and the generation and propagation of unstableradiations from laser-plasma interactions andanisotropic electron distributions in space andastrophysical plasmas.

1.2.2 Spatio-Temporal Chaos in theSaturation of an Unstable Wave

Sponsors

Lawrence Livermore National LaboratorySubcontract B-160456

National Science FoundationGrant ECS 88-22475

U.S. Department of EnergyContract DE-FG02-91 -ER-54109

We have completed a major part of our study onspatio-temporal chaos (STC) in the nonconserva-tive, nonlinear three-wave interaction (3WI).17 Inparticular, we have found that STC arises in thesaturation of an unstable wave coupled to twodamped waves at lower frequencies. The ensuingsaturated state exhibits the chaotic behavior ofcoherent structures and is characterized by a corre-lation length which is small compared to thesystem size L; in turn, the system size is larger thanthe excitation length E (at which energy is

16 K.R. Chu and A.T. Lin, UCLA Report PPG-1373, September 1991, forthcoming; G. Shvets and J.S. Wurtele(private communication).

17 C.C. Chow, Spatiotemporal Chaos in the Nonlinear Three Wave Interaction, Ph.D. diss., Dept. of Physics, MIT,1991.

183


injected into the system by the unstable wave) andalso larger than the dissipation length eD (at whichenergy leaves the system through the dampedwaves). Thus STC characterized by the conditions

< L and L > IE eD is a new nonlinear statefor a saturated instability in an extended medium,which is very different from either low-dimensionalchaos or fully developed turbulence.18 In theformer case, > L so that the system is completelyspatially correlated, and, in the latter case,L > eE > eD so that energy is injected at somelarge scale and dissipated at a much smaller lengthscale (the inertial range lies in between theselength scales) with coherent structures existing ata large number of scale lengths.

The main features of the STC that ensues in thesaturation of an unstable wavepacket by modecoupling was described in RLE Progress Report133. During the past year, we have attempted todevelop an analytical basis for understanding andpredicting this STC state. The usual perturbationanalyses that start from linear instability (e.g.,weak turbulence theory) cannot be used to accessa description of STC. In STC, the nonlinearity isnot weak; it is essential in establishing thecoherent structures.

The approach we found useful was to start fromthe nonlinear state of the conservative 3WI andconsider the nonconservative aspects as a pertur-bation. The conservative nonlinear 3WI is exactlyintegrable by inverse scattering (IST). 19 In partic-ular, when the high-frequency (hf) wave has themiddle group velocity of the three nonlinearlyinteracting waves, the solutions involve theexchange of envelope solitons between the threewaves. In the non-conservative case, assumingthe high-frequency wave is unstable for longwavelengths and stable (by diffusion-typedamping) at short wavelengths, the dynamicsproceed as follows. When the unstable waveenvelope reaches an area threshold, it transferssolitons to the low-frequency (If) waves whenthese are of non-zero amplitude, and, by thistransfer, its envelope wavelengths become shorter;these shorter wavelengths, if not damped, begin togrow until they reach their soliton transferthreshold, and the process of depletion andgrowth repeats. Thus, from any random initialconditions, small scales are smoothed out by thediffusion-type damping and long scales grow untila soliton depletion threshold is reached which inturn generates smaller scales that may be damped.The steady state is arrived at when one attains a

balance between the nonlinear conversion of longlength scales to small length scales and the lineardamping of the small scales. Thus, if the unstablewave has a growth rate yi and its diffusive-typedamping is characterized by a diffusion constantD, its steady-state correlation length will be of theorder

S 2,t D27 71

(1)

The low-frequency damped envelopes are in astate in which they are continuously fed withsolitons (by the unstable high-frequency wave)which propagate with their group velocities andare damped; we identify these as quasi-solitons.From IST we can show that the size of thesesolitons are related to the IST bound stateeigenvalue q/ of the high-frequency unstableenvelope:

2 ^ 2/q/ (2)

In the simplest case of an envelope with oneextremum, the bound state eigenvalue r/ is givenby

2 2d(11 - )dx = 7(n + 2

where Q1, is the Zakharov-Shabat (ZS) potential(proportional to the unstable wave envelope) andthe integral is over the classical turning points ofthis potential.19 In the STC state, the depleted,unstable high-frequency envelope will have someremaining area (IST "radiation") from which itgrows before depleting again, and the low-frequency quasi-soliton envelope colliding withthe high-frequency envelopes causes theirdepletion. Thus, two questions arise: Given thatthe hf, unstable envelope depletes from someinitial envelope area (i.e., ZS eigenvalue), what isthe remaining envelope area after depletion? (i.e.,the leftover radiation); and what threshold area(i.e., threshold eigenvalue) is required fordepletion? To address the first question, amultiple-time scale analysis around the IST sol-ution for soliton decay was carried out assumingthat growth and dissipation occur on a slower timescale than soliton depletion. The answer to thesecond question required a perturbation expansionin the Zakharov-Manakov (ZM) scattering spaceof the 3WI. 19 The mathematical details and results

18 P.C. Hohenberg and B.I. Shraiman, Physica D 37: 109 (1989).

19 D.J. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51: 915 (1979).



from these IST perturbation analyses20 gave theperturbed state eigenvalue of the unstableenvelope as

threshold, eigenvaluefL< I an I >dx, where anamplitude of wave n =equations show that

/. With <Un>the complex envelope2, 3, the conservation

S,, 4 11y1 + 2y2

where 72 is the damping rate of the If wave, andthe time required for the hf unstable envelope togrow after soliton depletion to the threshold fordepletion as

1 2qtg 71 In (2 + y1 )1n3

With these results one can obtain the quasi-solitonwidth given by (2) and the characteristic cyclingtime which is given by (5) plus the (generally veryshort) soliton depletion time.

The long-time behavior in the STC of this non-linear 3WI was found to be characterized approxi-mately by the diffusive-type damping associatedwith the unstable wave. Thus, the long-time scalefor the hf unstable wave, T1, can be estimated as adiffusion time across the correlation length (1),

D (27)2I 2 -- -- (6)

S 71

and for the If damped wave, characterized by thequasi-soliton width, (2) with (4),

4T2 -2 (7)q2D

The results (1)-(7) describe all the characteristicscales in the STC ensuing from the saturation ofan hf unstable wave by nonlinear coupling to Ifdamped waves in a soliton transfer interaction (i.e.,the hf unstable wave having the middle groupvelocity). The validity of these results were cor-roborated by extensive numerical simulations. 20

Finally, there remains only to estimate the averageenergy densities in the saturated state of the STC.These are obtained from the conservation equa-tions for the nonlinearly interacting waves and the

2 71<U2> i <U 1 > (8)

3 72

<U 3 > =2 <U 2 > (9)73

with 7y the growth rate of the hf wave, and 72 andy3 the damping rates of the If waves. In the satu-rated state, the hf unstable envelope has an ampli-tude related to its threshold eigenvalue r, whichtogether with the eigenvalue quantization rule for asoliton, (3) with n = 0, allows us to estimate theaverage energy density of the saturated unstablewave as

<U1 > 1 2 )2]L 2 q 21

These estimates of the average energy densities inSTC have been found to correspond closely to theamplitudes of the spectral energy densities deter-mined from the correlation functions in the numer-ical simulations. 20

The results of these studies have been presented atseveral conferences21 and are being prepared forpublication.

1.2.3 Current Drive by Lower Hybridand Fast Alfven Waves

Sponsor

U.S. Department of EnergyContract DE-FG02-91 -ER-54109

The most effective way of driving current usingradio-frequency waves in present day tokamakshas been with lower hybrid waves (LHW).However, as is well known, in a high density andhigh temperature tokamak (ITER-type) these slow

20 C.C. Chow, Spatiotemporal Chaos in the Nonlinear Three Wave Interaction, Ph.D. diss., Dept. of Physics, MIT,1991.

21 C.C. Chow, A. Bers, and A.K. Ram, "Spatio-Temporal Chaos in the Saturation of an Unstable Wavepacket," Pro-ceedings of the International Sherwood Fusion Conference, April 22-24, 1991, Seattle, Washington, Paper 2D08;C.C. Chow, A. Bers, and A.K. Ram, "Spatiotemporal Chaos in the Three Wave Interaction," Bull. Am. Phys. Soc.36: 2407 (1991); C.C. Chow, A. Bers, and A.K. Ram, Spatiotemporal Chaos in the Nonlinear Three Wave Inter-action, Proceedings of the III Potsdam-V Kiev International Workshop on Nonlinear Processes in Physics,Clarkson University, Potsdam, New York, August 1 -11, 1991, forthcoming.

185


LHWs will not be able to penetrate into the centerof the plasma. They would damp away theirenergy and momentum onto electrons which aretowards the edge of the plasma. Consequently, itis important to determine the effectiveness of otherrf means of current drive by waves which can pen-etrate into the center of the plasma. Fast Alfv6nwaves (FAW) with frequencies below the lowerhybrid frequency are able to penetrate into thecore of a high temperature plasma and have beenproposed as one of the means for driving currents.

Recent experiments on rf current drive in JET havebrought forth some very interesting results. Thecurrent drive efficiency was found to be signif-icantly enhanced at higher plasma temperatureswhen LHWs were used in conjunction with theFAWs. The FAW spectrum was such that these rfwaves could not drive any current without LHWs.However, the current drive efficiency wasenhanced only when the phase velocities of theFAW spectrum overlapped with the phase veloci-ties of the LH spectrum.

We have begun a detailed analysis of current driveby LHWs in combination with FAWs in order tobetter understand the current drive efficiency in thepresence of both types of waves. Some of theresults were presented at the 1991 APS-DPPmeeting 22 and in an invited presentation at theIAEA meeting in Aries, France.23

The LHW generated electron tail has an effectiveperpendicular energy (or perpendicular temper-ature) which can be an order of magnitude (ormore) bigger than the bulk electron temperature. 24

This is due to the pitch angle scattering of theenergetic electrons to higher perpendicularmomenta. It has been shown that, when theseeffective increases in perpedicular temperature aretaken into account, the theoretically calculatedcurrent drive efficiency is more than that evaluatedin models where the tail electrons are assumed to

have the same perpendicular temperature as thebulk electrons. In our initial calculations, weassumed that the effect of the FAW on these tailelectrons was to increase their perpendicularenergy. This was then shown to lead to anincrease in the current drive efficiency. Two-dimensional Fokker-Planck simulations with amodel FAW diffusion coefficient and the LHW dif-fusion coefficient showed that the FAW didenhance the perpendicular tail temperature.25 Thisenhancement depended on the strength of theFAW diffusion coefficient, increasing as the dif-fusion coefficient strength was increased. Therewas a corresponding increase in the current driveefficiency. Below, we present some of theseresults.

The two-dimensional, relativistic Fokker-Planckcode 26 was used to determine the effect of FAWon an electron distribution function which has anasymmetric tail in pll due to LH waves.22 The FAWdiffusion coefficient was taken to be of the formgiven by Giruzzi and Fidone,27 with the FAW spec-trum assumed to be a symmetric Gaussian cen-tered at n1l = 0 with a half width of An,, = 2.3. Forthe LH waves, we assumed DbH/Dc = 50, where Dcis the collisional diffusion coefficient with the LHspectrum extending from v1I/vTe = v1 to v2 wherevl = 4 and V2 = C/(VTena) (na is the lowest acces-sible nil) . For na = 1.6 and Te = 2 keV, v2 = 10while for Te = 6 keV, v2 = 5.77. When consid-ering the effect of the FAW, the FAW diffusioncoefficient was assumed to be such thatD6W/Dc = 0.1 when I v1 /Vre > 4.0 and zero, other-wise. [This allows us to consider only the inter-action of the FAW with the LH spectrum. The lowphase velocity FAW are strongly affected by theexistence of magnetically trapped electrons; theycan also directly heat bulk electrons which wouldhappen if we allow the FAW diffusion coefficientto extend into the bulk of the electron distributionfunction. However, our current code does notaccount for trapped electrons nor does it evolve

22 A.K. Ram, A. Bers, V. Fuchs, and M.M. Shoucri, Bull. Am. Phys. Soc. 36: 2339 (1991).

23 A. Bers and A.K. Ram, in Proceedings of the IAEA Technical Meeting on Fast Wave Current Drive in Reactor ScaleTokamaks (Synergy and Complementarity with LHCD and ECRH), Aries, France, September 23-25, 1991.

24 V. Fuchs, R.A. Cairns, M.M. Shoucri, K. Hizanidis, and A. Bers, Phys. Fluids 28: 3619 (1985).

25 A.K. Ram, A. Bers, V. Fuchs, and M.M. Shoucri, Bull. Am. Phys. Soc. 36 (1991) 2339; A. Bers and A.K. Ram, inProceedings of the IAEA Technical Meeting on Fast Wave Current Drive in Reactor Scale Tokamaks (Synergy andComplementarity with LHCD and ECRH), Aries, France, September 23-25, 1991.

26 M.M. Shoucri, V. Fuchs, and A. Bers, Comput. Phys. Comm. 46: 337 (1987).

27 G. Giruzzi and I. Fidone, in "Controlled Fusion and Plasma Heating," Proceedings of the 17th European Confer-ence, Amsterdam, 1990, ed. G. Briffod, A. Nijsen-Vis, and F.C. Schuller (European Physical Society, 1990), Vol.14B, Part III, pp. 1279-1282.



the bulk temperature, and the conditions for itsvalidity would breakdown if the diffusion coeffi-cient were allowed to extend into the bulk of thedistribution function.]

From the numerical code, we evaluated the currentdrive efficiency, i, and the effective perpendiculartemperature in the LH induced tail of the distribu-tion function:

T _

Tp= T,where Ti =

3

ip I fd 2 fo

dpLpLfo

The other parameters used in the numericalprogram were: Z, = 1.6, electron density = 2.4 x1019 m - 3 , B0 = 3.3 Tesla, LH frequency = 3.7GHz, and FAW frequency = 48 MHz (parameterssimilar to those in the JET experiment).

When the Fokker-Planck equation was solved withjust the LH diffusion coefficient, we obtained thefollowing results:

for Te = 2 keV:

for Te = 6 keV :

T 34, r 0.434(2)

Tp a 10, q/ = 0.544(3)

When the FAW diffusion coefficient was includedalong with the LH diffusion coefficient, weobtained:

for Te = 2 keV : Tp , 55, r 0.608 (4)

for Te= 6 keV : T, a 16, r q 0.683 (5)

Comparing (2) with (4), and (3) with (5), weclearly see that the effect of the FAW is toenhance the TI in the LH induced tail of the dis-tribution function. This also gives an enhancementin the current drive efficiency. These increases inTI and r can be understood in terms of a simpleanalytical model developed primarily for the LHcurrent drive. 28 It must, however, be remarked that

this explanation of the observations on currentdrive by LHWs and FAWs is to be taken as onlypreliminary. For one thing, in the JET experimentsthe FAW not only interacts with the LHCD gener-ated electron tail, but it also raises the electrontemperature of the plasma. Our numerical simu-lations do not allow for the bulk electron temper-ature to evolve, and thus cannot account for thechange in the current drive efficiency due electronbulk heating.

1.2.4 Space-Time Propagation ofElectromagnetic Instabilites Acrossthe Magnetic Field in AuroralRegions

Sponsor

National Aeronautics and Space AdministrationGrant NAGW-2048

We have shown that the space-time analysis ofinstabilities leads to distinctly observable featuresthat can identify whether an instability is absoluteor convective.29 An absolute instability has anarrow frequency bandwidth while a convectiveinstability typically has a broad band emission.We have considered instabilities that propagateacross an ambient, steady-state magnetic field, andare generated by relativistic, highly anisotropic,electron distribution functions. Such distributionfunctions arise in the auroral regions and are con-sidered to be sources of the observed auroral kilo-metric radiation. 30 We have studied thepropagation of the ordinary and the extraordinary(x) modes. The instabilities are generated at theelectron cyclotron frequency (Oce) and its har-monics. For low electron densities, the x-mode isan absolute instability at (ce and a convectiveinstability at the harmonics. As the density isincreased the instability at 60ce becomes convectivewhile that at 2Oce becomes absolute, thereby,leading to harmonic generation of radiation by alinear mechanism. Detailed results for these insta-bilities and the effect of a cold background plasmaon the propagation of these instabilities are cur-rently being studied.

28 V. Fuchs, R.A. Cairns, M.M. Shoucri, K. Hizanidis, and A. Bers, Phys. Fluids 28: 3619 (1985).

29 A.K. Ram and A. Bers, in Physics of Space Plasmas, Proceedings of the 1990 Cambridge Workshop in GeoplasmaPhysics, eds. T. Chang, G.B. Crew, and J.R. Jasperse (Cambridge, Massachusetts: Scientific Publishers, 1990).

30 C.S. Wu and L.C. Lee, Astrophys. J. 230: 621 (1979).

187


1.2.5 Linear Instability Analysis ofthe Double Stream Cyclotron Maser

Sponsors

National Aeronautics and Space AdministrationGrant NAGW-2048


The double stream cyclotron maser in which twocopropagating beams, with different beam veloci-ties, gyrate in a uniform axial magnetic field hasbeen proposed as a source of millimeter wave-length radiation.31 The interaction which leads tothe radiation is between the slow cyclotron spacecharge of one beam and the fast cyclotron spacecharge of the other beam. This interaction leads tohigh frequency bunching. We have carried out alinear stability analysis of an electrostaticdispersion function derived for two "cold" beamsfrom the fully relativistic Vlasov equation. 32 Thislinear analysis of the interaction shows that insta-bilities are generated near harmonics of theDoppler shifted electron cyclotron frequency (Wce)and also near the Doppler shifted half harmonics((n + 1/2)coce for n any integer). The region ofinstability depends on the values of k1l (the axialcomponent of the wave vector). Including axialtemperatures in the two beams eliminates the largekll instabilities. Regions of interest to either spaceplasma physics or coherent em amplifying devicesare currently being explored by further numericalstudies.

1.2.6 Coupling of Positive andNegative Energy Waves in aNonuniform Plasma

Sponsors


U.S.-Israel Binational Science FoundationGrant 87-0057

Recently developed techniques for findingembedded pairwise mode couplings in high order

systems and calculating the transmission coeffi-cient in linear mode conversion for passive, stableplasmas have been extended to the treatment ofcoupling of positive and negative energy waves innonuniform plasmas. In particular, the coupling ofpositive and negative energy waves in co-streaming, nonuniform electron beams has beentreated in detail and the ensuing spatial amplifi-cation determined. A journal article on this workhas been recently accepted for publication. 33

1.2.7 Publications

Bers, A., V. Fuchs, and C.C. Chow. MaximizingAbsorption in Ion-Cyclotron Heating ofTokamak Plasmas. Plasma Fusion CenterReport PFC/JA-91-10, MIT, 1991.

Bers, A., and A.K. Ram. Lower Hybrid and FastWave Current Drive-Status of Theory. PlasmaFusion Center Report PFC/JA-92-3, MIT,1992.

Bers, A., and A.K. Ram. "Signature of Absoluteand Convective Instabilities with Application toSpace Plasmas." Bull. Am. Phys. Soc. 36: 2462(1991).

Bers, A., V. Fuchs, and C.C. Chow. "MaximizingAbsorption in Ion-Cyclotron Heating ofTokamak Plasmas." Proceedings of the EPS18th Conference on Controlled Fusion andPlasma Physics, Berlin, Germany, June 3-7,1991. Forthcoming.

Bers, A., and A.K. Ram. "Lower Hybrid and FastWave Current Drive-Status of Theory." Pro-ceedings of the IAEA Technical Meeting onFast Wave Current in Reactor Scale Tokamaks(Synergy and Complementarity with LHCD andECRH), Aries, France, September 23-25, 1991.Forthcoming.

Chow, C., A. Bers, and A.K. Ram. "SpatiotemporalChaos in the Nonlinear Three WaveInteraction." Proceedings of the /// Potsdam-VKiev International Workshop on Nonlinear Pro-cesses in Physics, August 1-11, 1991.

31 G. Bekefi, J. Appl. Phys., forthcoming 1992.

32 A. Bers and C.E. Speck,1965; C.E. Speck and A.January 1966; G. Bekefi,California, January 1992.

Quarterly Progress Report 78: 110-114, Research Laboratory of Electronics, MIT, JulyBers, Quarterly Progress Report 80: 159-161, Research Laboratory of Electronics, MIT,A.K. Ram, A. Bers, and C. Chen, SPIE Proc. 1992 OE LASE Conference, Los Angeles,

33 L. Friedland and A. Bers, "Hermitian Description of Interacting Inhomogeneous Electron Beams," Phys. Fluids B,forthcoming 1992.



Potsdam, New York: Clarkson University.Forthcoming.

Chow, C.C., A. Bers, and A.K. Ram. "The ThreeWave Interaction and Spatiotemporal Chaos."Paper presented at the 1991 Cambridge Work-shop on Theoretical Geoplasma Physics.Physics of Space Plasmas (1991), SPI Confer-ence Proceedings and Reprint Series. Forth-coming.

Chow, C.C., A. Bers, and A.K. Ram. Spatio-temporal Chaos in the Nonlinear Three WaveInteraction. Plasma Fusion Center ReportPFC/JA-92-2, MIT, 1992.

Chow, C.C., A. Bers, and A.K. Ram. "Spatio-Temporal Chaos in the Saturation of anUnstable Wavepacket." Proceedings of theInternational Sherwood Fusion Conference,Seattle, Washington, April 22-24, 1991, Paper2D08.

Chow, C.C., A.K. Ram, and A. Bers. "Spatio-temporal Chaos in the Nonlinear Three WaveInteraction." Research Trends in Physics:Chaotic Dynamics and Transport in Fluids andPlasmas. Eds. I. Prigogine et al. New York:American Institute of Physics, 1992.

Chow, C.C., A. Bers, and A.K. Ram. "Spatio-temporal Chaos in the Three Wave Interaction."Bull. Am. Phys. Soc. 36: 2407 (1991).

Chow, C.C., A. Bers, and A.K. Ram. The ThreeWave Interaction and Spatiotemporal Chaos.Plasma Fusion Center Report PFC/JA-92-1,MIT, January 1992.

Friedland, L., and A. Bers. "Hermitian Descriptionof Interacting Inhomogeneous ElectronBeams." Phys. Fluids B. Forthcoming.

Kupfer, K., and A. Bers. "Fast Electron TransportDuring Lower-Hybrid Current Drive." Pro-ceedings of the EPS 18th Conference on Con-trolled Fusion and Plasma Physics, Berlin,Germany, June 3-7, 1991. Forthcoming.

Kupfer, K., and A. Bers. Fast Electron Transport inLower-Hybrid Current Drive. Plasma FusionCenter Report PFC/JA-91-13, MIT, 1991.

Kupfer, K., A. Bers, and A.K. Ram. "Fast ElectronTransport in Current Drive." Proceedings of theInternational Sherwood Fusion Conference,

Seattle, Washington, April 22-24, 1991, Paper2B05.

Kupfer, K., A. Bers, and A.K. Ram. "Fast ElectronTransport During Lower-Hybrid Current Drive."Phys. Fluids B 3: 2783 (1991).

Kupfer, K., A. Bers, and A.K. Ram. Fast ElectronTransport During Lower-Hybrid Current Drive.Plasma Fusion Center Report PFC/JA-91-9,MIT, 1991.

Ram, A.K., A. Bers, V. Fuchs, R.W. Harvey, andM.G. McCoy. "Current Drive by Lower HybridWaves in Combination with Fast Alfv6nWaves," Proceedings of the InternationalSherwood Fusion Conference, April 6-8, 1992.Forthcoming.

Ram, A.K., and A. Bers. "Comments on Absoluteand Convective Instabilities." Geophys. Res.Lett. 19: 143 (1992).

Ram, A.K., and A. Bers. "Space-Time Propagationof Electromagnetic Instabilities Across theMagnetic Field in Auroral Regions," EOS Trans.Amer. Geophys. Union 72: (1991).

Ram, A.K., A. Bers, V. Fuchs, L. Vacca, and M.Shoucri. "Current Drive by Fast Alfv6n Wavesand in Combination with Lower HybridWaves." Bull. Am. Phys. Soc. 36: 2339 (1991).

Ram, A.K., and A. Bers. "Propagation andDamping of Mode Converted Ion-BernsteinWaves in Toroidal Plasmas." Phys. Fluids B 3:1059 (1991).

Ram, A.K., and A. Bers. "Absolute Versus Convec-tive Analysis of Instabilities in Space Plasmas."Invited paper presented at the 1990 CambridgeWorkshop on Theoretical Geoplasma Physics.In Physics of Space Plasmas (1990), SPI Con-ference Proceedings and Reprint Series,Number 10. Eds. T. Chang, G.B. Crew, and J.R.Jasperse. Cambridge, Massachusetts: Scien-tific Publishers, 1991.

Ram, A.K., and A. Bers. Comments on Absoluteand Convective Instabilities. Plasma FusionCenter Report PFC/JA-91-8, MIT, 1991.

Ram, A.K., and A. Bers. Absolute Versus Convec-tive Analysis of Instabilities in Space Plasmas.Plasma Fusion Center Report PFC/JA-91-4,MIT, 1991.

189


1.3 Physics of ThermonuclearPlasmas

Sponsor

U.S. Department of EnergyContract DE-FG02-91 ER-54109

Project Staff

Professor Bruno Coppi, Dr. Ronald C. Englade, Dr.Stefano Migliuolo, Dr. Marco Nassi, Dr. BarrettRogers, Dr. Linda E. Sugiyama, Dr. Leonid E.Zakharov, Riccardo Betti, Darin Ernst

The main theme of this program is the theoreticalstudy of magnetically confined plasmas in regimesof thermonuclear interest. A variety of physicalregimes that fall in this category characterize bothpresent-day experiments on toroidal plasmas (e.g.,Alcator, TFTR, JET) as well as future experimentsthat will contain ignited plasmas. These will eitherinvolve first generation fuels, namely a deuterium-tritium mixture (Ignitor), or more advanced fuelssuch as deuterium-deuterium or deuterium-heliummixtures (Candor).

We are participating in a coordinated effort of col-laboration between the design group of a U.S.compact ignition experiment and that of theEuropean experiment (Ignitor). At MIT, theAlcator C-MOD experiment that combines thefavorable features of an elongated plasma crosssection with a high magnetic field is under con-struction. These features, which are also beingplanned for Ignitor, were originally proposed for amachine called Megator, which we designed in theearly 1 970s.

Presently, our research program follows two majoravenues. First, the basic physical processes ofthermonuclear plasmas (equilibrium, stability,transport, etc.) are being studied as they apply toexisting or near-term future systems. In this effort,

we closely collaborate with our experimental col-!eagues, as well as theorists from other researchgroups (e.g., Joint European Undertaking (JET),Princeton, Columbia). This work also involvestime-dependent simulations of plasma dischargesin the planned D-T burning Ignitor experiment,with particular attention being focused on the evo-lution of spatial profiles of plasma current andtemperature. Collaboration with our colleagues atthe Italian laboratories of Energia Nucleare eEnergie Alternative (E.N.E.A.), as well as in-housecode development by visiting scientists from Italy,plays a major role in this endeavor. Second, weexplore advanced regimes of thermonuclearburning, including those employing low neutronyield fuels (3D-He, and "catalyzed" D-D). We con-sider both the design of machines that will containthese very high temperature plasmas as well as thephysics that govern their behavior.

We present below some of the salient results onwork completed or presently being worked on bymembers of our research group.

1.3.1 Physics of Compact,Field Ignition Experiments

High

Magnetically confined, toroidal experiments toinvestigate deuterium-tritium ignition conditions ina plasma can be designed on the basis of the pres-ently known experimental results and theoreticalunderstanding of plasma behavior. The mostadvantageous and least expensive designs incor-porate an interlocking set of characteristics: 34 tightaspect ratio, relatively small size with significantvertical elongation, high toroidal and poloidalmagnetic fields, large plasma currents, high plasmadensities, good plasma purity, strong ohmicheating, good plasma and oc-particle confinement,and robustness against ideal MHD and resistiveplasma instabilities. We have proposed and devel-oped the physics 35 and engineering ideas36 behind

34 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on c-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992).

35 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on cc-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992); L. Sugiyama and M. Nassi, Free Boundary Current Ramp and Current Profile in a D-T IgnitionExperiment, RLE Report PTP 90/8, Res. Lab. of Electron., MIT, 1991, Nucl. Fusion, forthcoming (1992); B. Coppi,R. Englade, M. Nassi, F. Pegoraro, and L. Sugiyama, "Current Density Transport, Confinement and Fusion BurnConditions," Proceedings of the 13th International Conference on Plasma Physics and Controlled Nuclear FusionResearch, Washington D.C., 1990 (Vienna, Austria: I.A.E.A., 1991), paper CN-53/D4-14.

36 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on a-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992); B. Coppi, R. Englade, M. Nassi, F. Pegoraro, and L. Sugiyama, "Current Density Transport, Con-finement and Fusion Burn Conditions," Proceedings of the 13th International Conference on Plasma Physics and



such a design (the Ignitor Ult,37 shown in figure6), as part of an ongoing program of research intoignition experiments. In addition, we have studiedthe feasibility of using similar characteristics todesign an experiment to study the next level offusion burning, the D-3He reaction.

Basic Physics Considerations for D-TIgnition

The ignition of a 50:50 deuterium-tritium plasmarequires a minimum value of the parameternoE 4 x 1020 sec /m3 in order to achieveignition with To < 15 keV, where no is the peakplasma (electron) density, To the peak temperature,and TE the energy replacement time. Here ignitionis defined to be the point when the plasma heatingdue to fusion a-particles, P., equals the plasmathermal losses PL. Relatively high values of theplasma density, no - 1021 m- 3 , then allows ignitionto be considered at moderate values of TE, whose

Figure 6. Cross section of the Ignitor Ult experiment, showing the major components of the machine.

Controlled Nuclear Fusion Research, Washington D.C., 1990 (Vienna, Austria: I.A.E.A., 1991), paperCN-53/D4-14; B. Coppi, "High Current Density Tritium Burner," RLE Report PTP 75/18, Res. Lab. of Electron.,MIT, 1975; B. Coppi, Comm. Plasma Phys. Cont. Fusion, 3: 2 (1977); B. Coppi and The Ignitor Group, "Charac-teristics and Expected Performance of the Ignitor-U Experiment," Proceedings of the Twelfth International Confer-ence on Plasma Physics and Controlled Nuclear Fusion Research, Nice, France, 1988 (Vienna, Austria: I.A.E.A.,1989), Vol. 3, p. 357.

37 B. Coppi, R. Englade, M. Nassi, F. Pegoraro, and L. Sugiyama, "Current Density Transport, Confinement andFusion Burn Conditions," Proceedings of the 13th International Conference on Plasma Physics and ControlledNuclear Fusion Research, Washington D.C., 1990 (Vienna, Austria: I.A.E.A., 1991), paper CN-53/D4-14.

191

S640oo

) PLASMA CHAMBER

() TOROIDAL MAGNET

() SHAPING COILS

0 EOUIUBRIUM COILS

(OUTER TRANSFORMER COIL) EQUATORIAL AND VERTICAL PORTS

( CENTRAL SOLENOID

@ SHAPING TRANSFORMER COIL

(J) AXIAL PRESS

SCENTRAL POST

SC-CLAMPSHRINK RING

(* TENSIONING WEDGES

0 SUPPORTING LEGS

SCRYOSTATRoo 1300 mma @ 470 mmb a 870 mm


magnitude is less easy to predict with certainty.Both of these values should be achievable, basedon the favorable confinement properties of highdensity plasmas that have been demonstrated by aseries of high field experiments, the Alcator A andAlcator C at MIT and the FT/FTU devices atFrascati, Italy. The maximum plasma density nothat can be supported correlates experimentallywith the ratio BT/Ro, where BT is the toroidal mag-netic field at the center of the plasma column, atmajor radius R = Ro. On the basis of the Alcator Cmachine, where no- 2 x 1021 m- 3 was achievedwith BT -- 12.5 T and Ro = 0.64 m, and the TFTRmachine at Princeton, where even larger ratios ofnoRo/BT were achieved, a configuration withRo- 1.3m and B- 13T should be able tosustain reliably densities of 1021m- 3. Given BT, thesmallest Ro consistent with the structural andelectromechanical requirements is advantageous.

A strong toroidal field also supports a highpoloidal magnetic field and correspondingly largeplasma current. A vertical elongation, e.g.,K_ 1.8, can be used to substantially increase theplasma current that can be carried for a given BTand R. If the density actually correlates with the(volume) averaged toroidal current density,< J0 >, then a value somewhat less than 1kA/cm2

should offer a considerable margin to obtain thedesired peak density no - 1021m - 3 . High values ofthe poloidal field Bp produce a strong rate ofohmic heating, while large toroidal plasma currentsI, tightly confine the fast ca-particles produced bythe fusion reactions, so that they deposit theirenergy inside the central region of the plasma.The degradation of the plasma energy confinementthat is commonly observed when externallyinjected (nonohmic) heating is applied is reducedat higher plasma current. In addition, the poloidalplasma beta fi, can be kept small at ignition, toimprove the plasma stability and in particular tostabilize ideal MHD modes with mode numbersm = 1, n = 1 associated with sawtooth oscil-lations.

Large plasma density combined with good ohmicheating allows ignition at low plasma temper-atures. This increases the overall margin of plasmastability due to low beta. It reduces the fusion

power required for ignition, and, therefore, thethermal wall loads.

Plasma purity has been shown to improve withincreasing plasma density, the effective chargeZeff = XniZi/ne decreasing monotonically with ne, inan extensive series of experiments starting with theAlcator A.38 The major effect of impurities is todilute the concentration of fusing nuclei, while asecondary effect is the increase the radiative powerlost due to bremsstrahlung. If auxiliary heating isnot used, Zeff cannot exceed about 1.6 for D-Tignition, as shown in the next section.

Relatively high plasma edge densities also help toconfine impurities to the scrape off layer, wherethe induced radiation helps to distribute thethermal wall loading more uniformly over theplasma chamber surface. The low ignition temper-atures further help to keep the plasma clean, byreducing the thermal wall loading.

Peaked plasma density profiles should be main-tained by external means such as a pellet injector ifnecessary. Peaked profiles maintain stability to qimodes that enhance the ion thermal transport.3 9

Since the neoclassical (Ware) particle pinch is rel-atively strong in a tight aspect ratio, high fieldconfiguration and considering that an anomalousinflow process is also present, pellets that pene-trate partway into the plasma can be successfullyused to produce peaked profiles near ignition.40

Plasma configurations, such as x-points, that con-centrate the thermal (particle) heat flux on local-ized areas of the vessel wall, limit the amount offusion power that can be handled. In addition,they limit the maximum possible plasma currentthat can be sustained. However, it may be desir-able to operate with a detached plasma or anx-point, in order to limit the degradation of theplasma confinement caused by nonohmic heatingby creating the conditions known to produce"H-mode" operation. These configurations can beobtained by sacrificing relatively little in the plasmaand magnet parameters.

Divertors represent a more severe compromise,since they alter the design of the plasma chamberand the toroidal magnet. The major radius must be

38 G.J. Boxman, B. Coppi, L.C.J.M. de Koch, et al., "Low and High Density Operation of Alcator," Proceedings of

the Seventh European Conference on Plasma Physics, Lausanne, Switzerland, 1975 (Ecole Polytechnique F6d6ralede Lausanne: Lausanne, Switzerland, 1976), Vol. 2, p. 14.

39 B. Coppi, S. Cowley, P. Detragiache, R. Kulsrud, F. Pegoraro, and W.M. Tang, Proceedings of the Tenth Interna-

tional Conference on Plasma Physics and Controlled Nuclear Fusion Research, London, 1984 (Vienna, Austria:I.A.E.A., 1985), Vol. 2, p. 93; B. Coppi and W.M. Tang, Phys. Fluids 31: 2683 (1988).

40 W. Houlberg, private communication (1991).



increased to accommodate a reliable divertor,reducing the ratio BT/R and therefore the maximumplasma density. The magnetic fields are alsoreduced, lowering the plasma current and theohmic heating, and increasing the plasma beta. Alarge injected heating system becomes necessaryto replace the ohmic heating, and the resultingdegradation of the plasma confinement makes lowtemperature ignition difficult. The divertor platesthen would have to handle large thermal heatfluxes. There is no demonstrated advantage tousing divertors in high density plasmas and thecumulative disadvantages make ignition difficult toattain and remove much of the rationale for usinga compact, high field machine.

Shown in Table 2 is a detailed reference design forthe Ignitor that incorporates the above features.41

Time Dependent D-T Ignition

Transient Processes and the CurrentDensity Evolution

Our research 42 has underlined the importance ofthe transient nature of the ignition process. Wehave considered the time evolution of a freeboundary plasma from the initial current rampphase, when the plasma current, density, size, andtoroidal magnetic field are raised to their maximumvalues, through ignition, using numerical simu-lation with the TSC code,43 for the specificexample of the Ignitor Ult device of Table 2.

Table 2. Design Parameters of the IgnitorU It.

Ro - 1.30 m Major radius

a x b-0.47 x 0.87m2 Minor radii

6G 0.4 Triangularity

1, 12 MA Toroidal plasma current

BT < 13 T Vacuum toroidal fieldat Ro

< Jo > <0.93 kA/cm 2 Volume-averagetoroidal current density

B < 3.75 T Mean poloidal field

q, 3.6 Edge magnetic safetyfactor

tR 3-4 sec Ramp up time for I,

tFT 4 sec Flat top time atmaximum I, and BT

PINJ < 16 MW Injected heating power(ICRH at f - 130MHz)

The results show that the initial current ramp,when the toroidal current, plasma density, toroidalfield, and plasma cross section are increasedsimultaneously, has important effects on theplasma energy balance and stability at ignition.42

These effects arise from the relatively slow inwarddiffusion of the plasma current compared to thegrowth of the central temperature. The currentramp generates a inhomogeneous electric field inthe plasma that is peaked off axis and allows largevalues of ohmic heating at high central temper-ature. The magnetic safety factor q (the inverse ofthe winding number of the magnetic field linesdivided by 2r) can be easily maintained aboveunity or held to a very small q < 1 region duringthe current ramp. This is important to minimize theeffects of plasma instabilities with poloidal andtoroidal mode numbers m/n = q = 1/1 that can

41 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on a-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992); L. Sugiyama and M. Nassi, "Free Boundary Current Ramp and Current Profile in a D-T IgnitionExperiment," Report PTP 90/8, Res. Lab. of Electron., MIT, 1991, Nucl. Fusion, forthcoming (1992); B. Coppi, R.Englade, M. Nassi, F. Pegoraro, and L. Sugiyama, "Current Density Transport, Confinement and Fusion Burn Con-ditions," Proceedings of the 13th International Conference on Plasma Physics and Controlled Nuclear FusionResearch, Washington D.C., 1990 (Vienna, Austria: I.A.E.A., 1991), paper CN-53/D4-14.

42 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on a-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992); L. Sugiyama and M. Nassi, "Free Boundary Current Ramp and Current Profile in a D-T IgnitionExperiment," Report PTP 90/8, Res. Lab. of Electron., MIT, 1991, Nucl. Fusion, forthcoming (1992).

43 S.C. Jardin, N. Pomphrey, and J. DeLucia, "Dynamic Modelling of Transport and Positional Control of Tokamaks,"J. Comp. Phys. 66: 481 (1986).

193


destroy the good confinement in the center of theplasma. A more careful study42 shows that theq < 1 region can also be kept small after the ramp,at least until the central temperature reaches highvalues and fusion alpha particles begin to appear,both of which have stabilizing effects for m = 1modes. Furthermore, small amounts of externallyinjected heating (e.g., 5 MW) during the currentramp can maintain the very small size of the q < 1region until well past ignition if central temper-atures can be raised to < 10 keV by the end of thecurrent ramp through the "freezing-in" of thecentral current density as the rate of inward,resistive diffusion decreases with temperature.Injected heating also reduces the magnetic fluxconsumption required to reach ignition, particularlyif ignition occurs during the current ramp.

We have also been able to show that it is possibleto simultaneously maintain q profiles that increasemonotonically from the plasma center, withoutlarge regions of low magnetic shear (flat or evenhollow q profiles) and with values of q at the edgeof the plasma maintained within 3 < qa < 4during the current ramp, to beyond ignition. Thisavoids instabilities associated with internal plasmamodes (e.g., "locked" or quasistationary modes)that are triggered during the current ramp and canlead to disruptions. Since hollow q profiles areusually associated with the excitation of internalmacroscopic modes and enhanced or "anomalous"current penetration (cf. Sugiyama and Nassi44),while ignition is aided by a slow current pene-tration that keeps q0 > 1 for as long as possible,these precautions are not superfluous.

Energy Confinement at Ignition

A major question that faces ignition experiments isthe degree of degradation expected in the plasmaenergy confinement near ignition. D-T ignition iseasily achieved if the confinement remains at theoptimal, ohmic heating levels. The strategy for ahigh field experiment is to maintain a high level ofohmic heating up to ignition, P, < 2 POH, to reducethe degree of degradation. Since fusion alpha-particle heating possesses two important charac-teristics of ohmic heating that are not shared byany presently available form of injectedheating-axisymmetric deposition and generationin the center of the plasma column-we expect thatthe degradation should not be as severe.

The requirement of relatively low edge q (highplasma current) means that special care must be

devoted to maintaining q > 1 up to ignition. Ifonly ohmic heating is contemplated, the steadilyincreasing size of the q < 1 region after the end ofthe current ramp imposes a more severe limit onthe time in which the plasma can ignite and on therequired energy confinement level, than the energybalance alone, assuming that the sawtooth oscil-lations that destroy the central peaking of the tem-perature cannot be avoided. Thus, ignition in thereference case, where To - 11 keV, requiresTE ~ 0.66 sec and ignition within approximately1.5 sec of the end of the current ramp. Our the-oretical analysis, on the other hand, indicates thatIgnitor remains, in all regimes, within the stabilitylimits of the ideal MHD and resistive m = 1, n = 1modes that are presently known. On the otherhand, moderate amounts of auxiliary heating,PICRH < 15 MW, allow ignition down to theexpected limits required for noTE, i.e., TE < 0.4 sec,while maintaining very small q=l 1 regions wellbeyond ignition. Similarly, if the requirement ofsmall q < 1 region is dropped in the ohmic case,either on the basis of existing theoretical analysesor by envisioning an external means of stabilizingsawtooth oscillations, ignition can also occur atthese values of TE and times of t, - 5-5.5 sec.

The importance of ohmic heating during theignition sequence at high field and density meansthat a model for the electron thermal transport, likethe one that has been adopted here, shouldinclude a diffusion coefficient that simulates ohmicregimes. It should reproduce at least typicaltoroidal loop voltages in steady state ohmic exper-iments, that are observed to be approximately a"universal" constant. In addition, the total dif-fusion coefficient should increase with injectedheating and reproduce the degraded confinementobserved in present experiments that are domi-nated by injected heating.

Plasma Density

Another major question for ignition is the effect ofvariation in the plasma density and its profile,since pellets injected to raise the density areunlikely to fully penetrate a high density plasma.

For a given level of thermal transport, there is anoptimum density for fastest ignition. Higherdensity is more favorable under degraded condi-tions. Higher density, however, also increases thetoroidal current penetration at a given time by low-ering the electron temperature, producing largerq < 1 regions earlier than at lower density. A

44 L. Sugiyama and M. Nassi, "Free Boundary CurrentReport PTP 90/8, Res. Lab. of Electron., MIT, 1991.

Ramp and Current Profile in a D-T Ignition Experiment,"



similar effect operates in the outer part of theplasma radius, when density profiles are broad-ened. Thus, for neo = 1.1 x 1021 m - 3 , profilepeaking factors an ne,,/ < ne > = 1.9-2.9, where< ne > is the volume average, with a fixed ratio ofedge to central density nea/neo = 0.1, give similarresults for ignition, requiring TE > 0.67 sec, att, 4.5 sec, with q = 1 radius ri/a 1/3 on thehorizontal midplane. Lower peaking factorsrapidly lead to degraded ignition, for examplean = 1.5 yields TE = 0.70 sec, t, = 4.7 sec, andrl/a - 0.45. A lower central density, 6.5 x 10 20m - 3

at the end of the current ramp, increasing to8-9 x 1020 by ignition, allows much broader pro-files (an < 1).

For related reasons, increasing the plasma densityafter the current ramp, is more advantageous thanincreasing the density during the ramp.

At high density, broad regions of low magneticshear (flat q at q < 1) develop in the mid-region ofthe minor radius. This region may be unstablewhen its value of q approaches unity, since idealMHD instabilities with m = 1 can occur. This isthe major limit on the broad density profile cases,since at the reference thermal transport level evena very broad profile, an - 0.1, will ignite at thelower density.

Burning Conditions

Fusion ignition is thermally unstable in the lowtemperature interval we consider, due to the tem-perature dependence of the fusion reaction crosssection. At ignition, by definition, P, = PL, so thatthe plasma thermal energy is increasing asdW/dt = POH + PINJ. Provision should be made tolimit possible temperature excursions. However,there are intrinsic processes such as the rise of thecentral pressure that can overcome the stabilitythreshold for m = 1 , n = 1 modes and limit thetemperature excursion naturally. Marginal ignitionand sustained subignited states, partly supportedby ohmic heating, are also of interest for studyingfusion burning. An example has been given.45

Plasma Stability

A major concern for ignition is the stability of theplasma to central, sawtooth oscillations. The sta-bility of modes with dominant poloidal modenumber m = 1 has been discussed previously. 46

Additional considerations for instabilities driven bythe current density profile have been discussedabove. Other instabilities, such as shear Alfv6nmodes destabilized by fusion a particles, have beendiscussed,45 based on the analysis reported in Bettiand Friedberg. 47

Engineering Design of a D-T IgnitionExperiment

The main engineering problems encountered indesigning a high field D-T ignition experimentarise from the need to:

* create and control different plasma configura-tions;

* induce the toroidal plasma current and main-tain the plasma discharge for a time >2 10 E atignition, where TE is the energy replacementtime;

* operate with an acceptable thermal loading onthe first wall;

* withstand the static, dynamic, electromag-netic, thermal, and disruptive loads with suffi-cient margin to provide an adequate reliability;

* provide access for the plasma diagnostics,pellet injector, radio frequency antennae,vacuum system, remote maintainance, etc.

Poloidal Magnetic Field System

The Ignitor Ult uses a highly optimized set of 14up-down symmetric poloidal field coils, placed inproximity to the plasma column (figure 6), toinduce the plasma current, create the desiredplasma configurations, and maintain them againstradial and vertical motions of the plasma.

Copper OHFC has been selected as the material forthe central solenoid in order to maximize the

45 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on a-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992).

46 B. Coppi, "High Current Density Tritium Burner," RLE Report PTP 75/18, Res. Lab. of Electron., MIT, 1975.

47 R. Betti and J.P. Friedberg, "Stability of Alfv6n Gap Modes in Burning Plasmas," Report PFC/JA-91-25(Cambridge: MIT Plasma Fusion Center, 1991), submitted to Phys. Fluids.

195


current that it can carry, while a material withenhanced mechanical properties (GLIDCOP) isused for the other poloidal field coils in order toproduce self-supporting structures. These othercoils are located in a region of the machine wheremore space is available and carry high current foronly a limited time during each discharge, so thatelectrical conductivity is not the major constraint.

The central solenoid consists of a double array ofcopper coils, each wrapped 32 times around thecentral steel pole of the machine. Each coil is pro-vided with a cooling channel at its center, cooledby He gas. The initial temperature before a plasmadischarge is about 30 K. The final temperature canreach a maximum of about 230 K, after a longpulse at the maximum plasma current I,.

An assessment 48 of the magnetic flux variationlinked with the plasma column has been carriedout, and the results have been checked against thetime dependent numerical analysis performedusing the TSC code. The volt-second requirementat ignition is found to be less than 32 V s, under arange of assumptions on the values of the ion andelectron thermal conductivity, effective charge,etc., for plasmas that reach ignition during theconstant current ("flat top") phase of the dis-charge, at the maximum plasma current. The volt-second consumption during the flat top is about1.5 V s per second, due to resistive losses and anincrease in the plasma internal inductance. Wenote that use of injected heating, when the plasmareaches ignition during the current ramp, canreduce the volt-second requirement to as low as_25 V s, due to the lower plasma current atignition and the higher plasma temperature thatreduces the resistive losses. The poloidal fieldsystem has been designed to produce a flux vari-ation of about 32 V s.

Several plasma configurations have beenanalyzed:49

limiter configurations that fill the entire plasmachamber are useful to distribute the thermalloading on the chamber walls uniformly;

* transient double x-point configurations, usedto reproduce the characteristics of the so-called H-regime of plasma confinement, whereTE is only slightly degraded in the presence ofinjected heating, relative to ohmic heatingregimes. They require I, to be kept well belowits design value. In addition, when the localthermal wall loading at the x-points exceedsdesirable limits, the equilibrium can bechanged to a limiter configuration.

* detached limiter configurations, which sepa-rate the plasma boundary from the first wall bya distance larger than a/10, to obtain theH-regime, a procedure suggested and con-firmed by a significant set of experiments.50

The radial feedback system for the plasma size andposition is mainly used to:

* control the plasma current density distributionand the magnetic safety factor during theplasma current ramp by changing the size andshape of the plasma cross section;

* maintain the plasma position with respect tothe first wall and, possibly, control the transi-tion between x-point and limiter configura-tions;

* control the radial position during vertical dis-ruptions or other rapid change of the plasmainternal inductance (li) or the poloidal beta(Op).

A vertically elongated plasma configuration ispotentially unstable to vertical motion. We havecarried out extensive numerical simulations5' withthe TSC code of the plasma dynamics and the sta-bility of the plasma/vacuum vessel/poloidal fieldcoils system, taking into account, in the axisym-metric geometry of the system, the position of thediagnostic magnetic pick up coils, and the voltagelimitations and time delay in the response of thepower supplies. The vertical displacement time(z,), the inverse of the exponential growth rate ofthe plasma vertical position, has been estimated tobe about 13.5 ms, taking into account the pres-ence of the thick vacuum vessel and the poloidal

48 M. Nassi, Volt-sec Requirement and Relevant MarginElectron., MIT, 1990.

in the Ignitor Device, RLE Report PTP-90/1, Res. Lab. of

49 G. Cenacchi, B. Coppi, L. Lanzavecchia, and M. Rulli, Ideal MHD Equilibria and Poloidal Field Magnet System forCompact Ignition Experiments, RLE Report PTP-88/14, Res. Lab. of Electron., MIT, 1988.

50 N. Suzuki, A. Aikawa, K. Hoshino, et al., Proceedings of the Twelfth International Conference on Plasma Physicsand Controlled Nuclear Fusion Research, Nice, France, 1988 (Vienna, Austria: I.A.E.A., 1989), Vol. 1, p. 207.

51 M. Nassi, S.C. Jardin, C.E. Kessel, and N. Pomphrey, Vertical Stability Analysis for Ignitor, RLE Report PTP-91/10,Res. Lab. of Electron., MIT, 1991.



field coils connected in antiseries (so that currentsinduced in coils by the plasma can flow antisym-metrically).

Our results show that / , and K have only a weakeffect on the plasma vertical motion. The weakinfluence of K (for variations AK = + 10%) is dueto the presence of a close-fitting vacuum vessel.52

Toroidal Magnetic Field System

The Ignitor design has always used toroidalmagnets made of copper plates, connected inseries externally and supported by an appropriatesteel structure against the vertical (axial) and hori-zontal (radial) electrodynamic forces. The loadson the inner leg of the toroidal magnet are sup-ported by bucking between the toroidal andpoloidal field coils (a sliding surface is provided atthe interface between the toroidal magnet and theair core transformer), by wedging in the inner partof the toroidal field coils, and by an external struc-ture which consists of:

* A set of steel plates or "C-clamps" sur-rounding each of the 24 modules of thetoroidal magnet. The C-clamps are wedged onthe outside to allow the unwedged part torotate around an effective hinge under theeffect of the bracing rings. Thus, only a smallfraction of the vertical separating force isunloaded onto the central leg of the toroidalmagnet.

* Two bracing rings maintaining the plateassembly and transferring the vertical sepa-rating force produced by the toroidal magnetto the effective outer shell formed by the steelplates.

* A central post filling the bore of the air coretransformer that absorbs the centripetal forceacting on the inner leg of the toroidal magnetand acts as a component of a central press.Vertical cuts are made in the post to reducethe effect of the induced currents.

* A vertical electromagnetic press connected tothe central post that applies a compressionpreload on the inner leg of the toroidalmagnet to reduce the electromagnetic load.The press is deactivated as soon as thermal

expansion in the toroidal magnet becomessignificant, or whenever the machine is oper-ated with magnetic fields below the maximumvalue.

A hybrid cryogenic system 53 is adopted for coolingthe toroidal magnet, using liquid N2 and liquid He.After a current pulse corresponding to themaximum plasma current scenario, the temperaturereaches about 230 K in the region facing the trans-former (one-third of its volume) and about 95 K inthe remaining part. The cooling of toroidalmagnets with liquid N2 is a well established proce-dure. The step from 80 K to 30 K then requires theuse of He. Heat transfer takes place throughforced convection in conduits within the coils.The same helium stream is passed through themagnet several times and recooled between pas-sages. The estimated cooling time for the toroidalmagnet is shorter than that required by the centralsolenoid.

Plasma Chamber

The plasma chamber has been designed to with-stand both static and dynamic loads with goodreliability, while minimizing weight, constructionand assembly difficulties, and the overall cost. Theresult is a relatively thick chamber (17 mm in theinboard side and 26 mm in the outboard side),made of Inconel 625, divided in 24 sectors thatcan be assembled and joined by welding.

The chamber is mechanically supported andrestrained by C-clamps, with long horizontal portducts that allow for freedom of deformation underelectromagnetic and thermal loads. It has beendesigned to withstand both vertical and axisym-metric disruptions with plasma current decay ratesfrom 1 to 5 MA/ms. A two-dimensional numericalanalysis54 shows that the stresses are below thelimit imposed by ASME rules. The plasmachamber also acts as a support for the first wallsystem. Vertical and equatorial access ports forthe plasma diagnostics, the vacuum system, thepellet injector, and the auxiliary heating are pro-vided. Six of the twelve equatorial ports(0.8 x 0.165 m 2) connect to recessed "pockets"(0.8 x 0.5 m 2) facing the plasma, which house theantennae of the ion cyclotron radio frequency (RF)heating system.

52 C.E. Kessel and S.C. Jardin, "Vertical and Radial Plasma Position Control for BPX," Proceedings of the FourteenthSymposium on Fusion Engineering, San Diego, California, 1991.

53 A. Angelini and H. Quack, U.S. Patent No. 4,884,409, December 5, 1989.

54 Consorzio CITIF, Report CTFFIGNN5251, Turin, Italy (1991).

197


First Wall System

The first wall is made of 20 mm thick tiles, thatcover the entire inner surface of the plasmachamber, which can be replaced using the remotehandling system. In principle, the entire first wallcan function as a limiter to increase the area overwhich the heat load may be spread. Graphite rein-forced by carbon fibers has been selected as thefirst wall material, primarily on the basis of itsability to withstand the expected effects of dis-ruptions.

Numerical simulation55 has shown that Ignitor canachieve ignition at low peak temperature(To 2 11 keV) and moderate a-particle heatingpower (P, 18 MW). About 26% of this power islost by radiative processes (PR - 4.6 MW) from themain plasma, and the remaining 13.4 MW (74%) istransfered to the scrape off layer (SOL), where it isreleased to the first wall by radiative andconvective-conductive energy transfer. Simplephysical models56 and experimental data from highdensity, high magnetic field experiments predict ahigh density, low temperature plasma in the SOL.This cold, dense SOL plasma shields the coreplasma from impurity nuclei and experiences astrong radiative cooling. In particular, on the basisof an extensive series of observations made onhigh field machines, we estimate that at least 50percent of the power transfered into the SOL willbe released by radiation. Thus, - 63% (11.3 MW)of the total power is uniformly distributed to thefirst wall by radiation in the main plasma and inthe SOL. The remaining ~ 37% (6.7 MW) is non-uniformly distributed (with an estimated maximumpeaking factor of 3) due to convection and con-duction in the SOL.

We estimate the maximum local thermal wallloading to be about 0.9 MW/m 2 at ignition for alimiter plasma filling the entire plasma chamberand more if the plasma interacts only with theinner part of the first wall. If we assume that theignited plasma eventually reaches To < 15 keV,

where about 40 MW arepower and the radiativeplasma are about PR L 6wall loading is 2 MW/m 2.

supplied by the inputlosses from the main

MW, the corresponding

Auxiliary Systems

Since ignition can be attained by ohmic heatingalone, injected heating systems in compact highfield experiments have a backup role to suppress, ifnecessary, the possible onset of sawtooth oscil-lations, control the temperature evolution and thecurrent density profiles, and accelerate ignition.

An ICRH system with a frequency ~ 130 MHz andmaximum power delivered to the plasmaP 16 MW, has been adopted for Ignitor becauseof its demonstrated effectiveness in relatively highdensity plasmas. The antennae are placed in sixhousings inset into the vacuum chamber walls,each delivering 2.5 to 4.0 MW.

An injector of deuterium or deuterium-tritiumpellets (~- 4 mm diameter) is considered, in addi-tion to gas injection ("puffing"),57 to create andmaintain the desired plasma density profile. Pelletvelocities of 2 km/s or higher that are required toreach the center of the plasma have already beenachieved with existing technologies. Another usefor a pellet injector that has been demonstratedrecently is to condition the first wall by launchinglithium pellets into the plasma column prior toregular hydrogenic discharges.58

1.3.2 Momentum Transport andPlasma Collective Modes

It is widely known that the transport properties ofplasmas are not, with the exception of relativelyfew cases, adequately described by collisional pro-cesses alone. Rather, the presence of fine-scalecollective modes typically gives rise to transportcoefficients exceeding the collisional ones by two

55 B. Coppi, M. Nassi, and L. Sugiyama, "Physics Basis for Compact Ignition Experiments," Proceedings of the 1991International Atomic Energy Agency Conference on a-particle Physics, Goteborg, Sweden, Physica Scripta, forth-coming (1992); L. Sugiyama and M. Nassi, Free Boundary Current Ramp and Current Profile in a D-T IgnitionExperiment, Report PTP 90/8, Res. Lab. of Electron., MIT, 1991, Nucl. Fusion, forthcoming (1992).

56 A. Angelini and H. Quack, U.S. Patent No. 4,884,409, December 5, 1989; Consorzio CITIF, ReportCTFFIGNN5251, Turin, Italy (1991).

57 L. Sugiyama and M. Nassi, Free Boundary Current Ramp and Current Profile in a D-T Ignition Experiment, RLEReport PTP 90/8, Res. Lab. of Electron., MIT, 1991, Nucl. Fusion, forthcoming (1992).

58 C. Ferro and R. Zanino, "Edge Parameters and Scrape-off Layer (SOL) Characteristics in Ignitor," ENEA ReportRT/NUCL/90/31, ENEA CRE Frascati, Italy, 1990, Proceedings of the Ninth International Conference on PlasmaSurface Interaction, Bournemouth, United Kingdom, 1990.



or more orders of magnitude. Although the so-called "anomalous" transport of particles andthermal energy has been studied extensively bymany authors, relatively little attention has beenpaid to the anomalous transport of fluidmomentum in plasmas. Over the last fifteen years,an abundance of cases have arisen in which theknown collisional theories could not account forthe observed momentum transport in laboratoryand space as well as in astrophysical plasmas. Ourwork has focused on this problem in each of thethree regimes.

In the auroral F-region of the earth's ionospere,regions in which the plasma flow velocity isstrongly sheared tend to develop near auroralarcs.59 Dynamics Explorer 2 satellite data has beenused to study the spectral characteristics of densityand electric field fluctuations in these regions, 60

and it was found that the canonical Kelvin-Helmholz instability can only be excited withwavelengths much larger than those in the 1 kmrange observed. We have proposed an explanationof the fluctuations in a new theory of collisional,electrostatic modes that are driven unstable by thesheared, field-aligned ion flow velocity61 in par-tially ionized plasmas. The modes can be drivenunstable by a moderate amount of velocity shear(- 3 sec- 1) with wavelengths in the sub-km toseveral meter range. The resulting (anomalous)viscosity is derived from quasilinear theory, and amodel for the nonlinear evolution of the mode isproposed. For the first time, an equation for thenonlinearly evolving normal mode amplitude isderived and solved. We then argued that thesespatially localized modes give rise to a flattening ofthe profile of Vill, the ion flow velocity parallel tothe magnetic field. This flattening effect begins inlocalized regions where the growth rate is largestand spreads by creating additional maxima andminima in dVil/dx at the edges of these regions.New normal modes then grow at the extrema,propagating the effect outward from the initialpoint of growth.

This work has been extended to include not onlythe presence of a flow Vi- A 0 across the magnetic

field, but also its shear dVi_/dx # 0.62 The profile ofVi1 (x) has strong effects on the spatial topology ofthe aforementioned modes. In fact, it had beenpreviously suggested that no normal modes local-ized in the x-direction (transverse to the magneticfield) can be found when dVi-/dx # 0. However,this leaves no reasonable explanation for theobserved fluctuations near auroral arcs in whichshears are present in both Vll and V1_. After furtherinvestigation, we have found that indeed there arerealistic profiles of Vii = VExB(X) for which signif-icant x-localized, normal mode solutions exist. Wehave described these modes in detail and haveproposed a saturation model for the Kelvin-Helmholz instability, which is driven by dVExB/dx,to explain the observed profiles of VExB. Wesuggest two relevant mechanisms: quasilinear flat-tening and coupling to the short-wavelength colli-sional modes that are driven by dV1 l/dx. Thus theK-H modes may provide the "seed" for theobserved short wavelength fluctuations. Wesuggest, in addition, that "composite" modesviewed as spatial sequences of these modes arerelevant to extended regions over which the shearof VExB is depressed. We have indeed shown theexistence of localized, unstable modes driven bydVii/dx in both collisional and collisionlessregimes. All of the modes are expected to have apronounced effect on the spatial profile of Villthrough anomalous momentum transport.

In tokamak experiments heated by neutral beamco-injection,63 the plasma rotates at nearly sonic- 5 x 105 m/s speeds in the toroidal direction.The beams are injected near the magnetic axis andresult in a toroidal velocity profile that is peakednear the center of the plasma and decreasestoward the edge. The heating has a similar effecton the ion temperature profile. The poloidal flowis complicated by the possibility of shock forma-tion when the poloidal Alfv6n Mach numberexceeds unity, as the relevant Grad-Shafranovequation undergoes an elliptical-to-hyperbolictransition at this point. In the absence of suchshocks, however, the classical, strongly collisionaltheory64 predicts damping of the poloidal flow byparallel viscosity at a rate ~ (p2/a 2)(VYi/a 2vii), rapid

59 M.C. Kelley and C.W. Carlson, J. Geophys. Res. 82: 2343 (1977).

60 B. Basu and B. Coppi, Geophys. Res. Lett. 15: 417 (1988).

61 B. Basu and B. Coppi, J. Geophys. Res. 94: 5316 (1989).

62 B. Basu and B. Coppi, J. Geophys. Res. 95: 21213 (1990).

63 S.D. Scott, et al., Phys. Rev. Lett. 64: 531 (1990); S.D. Scott et al., "Plasma Physics and Controlled NuclearFusion Research 1988," Paper I.A.E.A.-CN-50/E-3-5 (Vienna, Austria: I.A.E.A., 1989).

64 J.W. Connor, S.C. Cowley, R.J. Hastie, and L.R. Pan, Plasma Phys. and Cont. Fusion 29: 919 (1987).

199


on the timescales of interest. Therefore, theremaining flow is predominantly toroidal and caneasily be shown to be constant, to lowest order,on magnetic flux surfaces64 with the magnetic fieldfrozen into the flow. Shock formation is no longerpossible as the relevant Grad-Shafranov equationis always elliptic. The collisional damping of thetoroidal flow is determined primarily by perpendic-ular viscosity with a small gyroviscous contributionat a rate ~ viipija 2, as it is obvious that parallelviscosity cannot transport toroidal momentumacross the magnetic field. The correspondingexperimental rates and diffusivities are 30-100times this65 and are therefore "anomalous." Inaddition, it is well-documented that the exper-imental cross-field diffusivities of toroidalmomentum, X0, and ion thermal energy, Xi, scalealmost identically with the minor radius of theplasma column,66 increasing toward the edge. Thishas led to suggestions by others that ion temper-ature gradient modes produce the anomaloustransport of both.

Fluctuations traveling in both the electron and theion diamagnetic directions have been observed,however, and the shape of the associated spec-trum is independent of the toroidal flow velocitywhile acquiring a Doppler shift. This suggestsanomalous transport by modes of the electron-drifttype as well, and quite plausibly, those driven bythe ion velocity gradient. We have written areport 67 describing two collective modes of theelectron-drift type68 driven unstable by the largeion flow velocity shear induced by NBI. Thetheory described is consistent with the aforemen-tioned conditions. One mode requires large flowvelocities in addition to shear, while the other,weaker instability does not require such high flowvelocities but relies on the existence of a parallelion viscosity. Recent experimental conditions inTFTR 66 are sufficient for the excitation of thisweaker instability. Both instabilities are shown to

produce momentum diffusivities X, greatlyexceeding the neoclassical result.

Following this work, we have obtained scalinglaws for Xo employing the neoclassical parallelviscosity. In addition, we have compiled scalingsfor x0 for other "anomalous viscosity" theories in aform convenient for comparison with experimentaldata obtained from interpretation codes. To thisend, transport equations describing the radial dif-fusion of toroidal momentum have been derivedfor axisymmetric systems, and a method forextracting XZ from experimental radial profile datahas been developed.69 This work is relevant to theinterpretation of data taken on the JET machine aswell as future machines in which the flux surfacesare very noncircular, and the data will be com-pared to cylindrical models.

Most recently, we have formulated arguments69

showing the ways in which spatial and other sym-metries of a given system affect the momentumtransport. It is found that, in general, spatial sym-metries simplify the form of the anomalous stresstensor as a function of the thermodynamic forces(i.e., gradients of temperature, velocity, etc.). Spe-cifically, we have shown that for an axisymmetrictorus with up-down symmetry, the only forces thatcontribute to the viscous stress are rank two,traceless, and symmetric tensors. This form isexactly that exemplified by the classical viscousstress due to velocity gradients. This result lendsplausibility to the argument that the anomalousmomentum transport in tokamak plasmas is due tovelocity gradient driven modes.

Hospitality extended to one of our group, D. Ernst,by our colleagues at the Joint European Under-taking (JET) during the summer of 1990 and bythe Institute of Fusion Studies at the University ofTexas during the summer of 1991, has been partic-ularly beneficial in this work.

65 S.D. Scott, et al., Phys. Rev. Lett. 64: 531 (1990); S.D. Scott et al., "Plasma Physics and Controlled NuclearFusion Research 1988," Paper I.A.E.A.-CN-50/E-3-5 (Vienna, Austria: I.A.E.A., 1989).

66 S.D. Scott et al., "Plasma Physics and Controlled Nuclear Fusion Research 1988," Paper I.A.E.A.-CN-50/E-3-5(Vienna, Austria: I.A.E.A., 1989).

67 B. Coppi, RLE Report PTP 89/2, Res. Lab. of Electron., MIT, 1989.

68 B. Coppi, M.N. Rosenbluth, and R.Z. Sagdeev, Phys. Fluids 10: 582 (1967).

69 D. Ernst, RLE Report PTP 91/3, Res. Lab. of Electron., MIT, 1991.



1.3.3 Collisional ElectrostaticImpurity Drift Modes and theIsotopic Effect

Experiments performed on toroidal, magneticallyconfined plasmas indicate70 that the en r gy con-finement time scales roughly with VAi whereAi = m/mp is the atomic mass of the primary ionspecies. Most theoretical models predict the so-called gyro-Bohm transport scaling where the rele-vant diffusion coefficient is:

cT PiD oc cTeB rp

where pi = ion Larmor radius oc m, and rp =scale length representative of the source of freeenergy (e.g., density or temperature scale length).Now since z oc a2/D where a = plasma cross-section, we see that these theories fail to predictthe favorable scaling with atomic mass (hereaftercalled the "isotopic effect").

There exists one class of fluctuations encounteredin collisionless and weakly collisional plasmas thatapparently gives the required transport scaling:electrostatic impurity drift waves. To be moreprecise, these are modes that occur in the presenceof impurities (i.e., additional ion species, heavierthan Hydrogen or Deuterium), propagate in theelectron diamagnetic direction (hence "driftwave") and require a gradient in the temperatureof the primary ion species (though not necessarilyin the impurity temperature).

These modes were first discovered7 1 to occur inplasmas with uniform temperatures when thedensity profiles are reversed: (dni/dx)(dni/dx) < 0,where the subscripts i and I refer to the primaryion and the impurity respectively, and later7 2 foundto occur in plasmas with regular profiles,(dni/dx)(dnl/dx) > 0, when an ion temperature gra-dient is present. Recently, it has been argued 73

that these modes occur at the edge of the plasma,and are responsible for the main part of the energytransport in that region. In that case, they ulti-mately determine the energy confinement time(energy cannot be lost faster than the speed pre-dicted by the characteristics of these modes) andprovide an explanation for the isotopic effect.

In the following, we sketch a derivation of thelinear stability properties of these modes. Webegin by considering a slab geometry, withsheared magnetic field B = Bo(ez + ex/Ls), and aset of fluid equations for the ions: 74

ni eow ni + Zi*i i (2)

2pi 2 - ke

- [)- -i( + )] T klul = 0

wu11 = klc 2 (-i-;j + ei

Ti

Ti e(wo + iwK + Zi i T,

T; T;

2(1 + i]Pi 2 e

2 3 -L Ti

2 k = 03 ll u1 = 0

where we have neglected perpendicular transportcoefficents (of order v2/Q 2 compared to parallelcoefficients) and dropped parallel collisionalviscosity, for simplicity (it contributes a small sta-bilizing term). Here wo-i = kycTi/ZieBLn is the iondiamagnetic frequency (Ln is the density gradientscale length), c? = Ti/m, pi = ci/Qi (Qi= Larmor fre-quency) and o, = 2xok&c?/3vi where XO is a con-stant coefficient of order unity. The parallelwavenumber is k1l = kyx/Ls, and the temperaturegradient parameter is j/i = (d In Ti/dx)/(d In ni/dx).

70 G. Boxman, B. Coppi, L. DeKock, B. Meddens, A. Oomens, L. Ornstein, D. Pappas, R. Parker, L. Pieroni, S. Segre,F. Schuller, and R. Taylor, 7th European Conference on Plasma Physics and Controlled Fusion, Lausanne, Vol. 4B,Part II, p. 14, (1975); A.G. Barsukov et al, in Plasma Physics and Controlled Nuclear Fusion Research (Vienna,Austria: I.A.E.A., 1983), Vol. 1, p. 83; J.G. Cordey et al, in Plasma Physics and Controlled Nuclear FusionResearch (Vienna, Austria: I.A.E.A., 1985), Vol. 1, p. 167; F. Wagner et al., in 16th European Conference onPlasma Physics and Controlled Fusion, Venice, 1989, Vol. 13B, Part I, p. 195.

71 B. Coppi, H.P. Furth, M.N. Rosenbluth, and R.Z. Sagdeev, Phys. Rev. Lett. 17: 377 (1966).

72 B. Coppi, G. Rewoldt, and T. Schep, Phys. Fluids 19: 1144 (1976) and references therein.

73 B. Coppi, in Plasma Physics and Controlled Nuclear Fusion Research (Vienna, Austria: I.A.E.A., 1991), Vol. 2, p.413.

74 A.B. Hassam, T.M. Antonsen, Jr., J.F. Drake, and P.N. Guzdar, Phys. Fluids B2: 1822 (1990).

201


Equations (2)-(4) are valid for any ion species. Inthe following, we shall consider the orderings:

klC I < < co < kllci

where X _- x/pi and

222yPi ci

2L vi). i

wv i < klcl (5 - b)

Then the impurities respond to the perturbation bysimple E x B drift, while the primary ion responseis weakly collisional (w < wo) and shorts out the

parallel electric fields (pi = -e4, to lowest order).

The electron reponse is adiabatic, ne/ne = ek/Te.Using quasi-neutrality, Zin + Z1n1 = ne, a simpleanalysis leads to the following dispersion equation:

d2 eo

dX 2 Ti

( 2 )e+7Ti

x -Z i -e Z ni - * i

6 (25 qj 3

and , -= d In ni/d In ni.

This equation admitsmodes for

qi >2 ( 1 + 5 kyPi

+ k2+ kypi

(7 - a)

(7 - b)

unstable, spatially localized

with

1 dn,=-ZkyDB ne dx x

LsPiT )2(2N + 1)-2i( i - 2/3

where N is the radial quantum number andDB = cT,/eB is the Bohm diffusion coefficient.

From (9) we notice that Re(o)/w o < 0 for a, > 0and that, for y = Im(o) < Re(co), we have

In addition, one can argue that as the mass ratiomi/m, is increased, the two thermal speeds VT andVTi get closer together and the "window of oppor-tunity"

y oc ZjYli Vi PiS

kypi

oc Z,Zimi 1 / 2 (10) VTI < w/kl < VTi

Referring to the well-known mixing len,.gLhformula, D - y/k_ we see that TE oc a2/D oc /mi ,as observed experimentally. It also predicts thattransport will not be better in Helium (Zi = 2,Ai = 3) than in Hydrogen, also an observed prop-erty.

decreases, leading to a dramatic narrowing in thespectrum of unstable modes. This has in fact beenconfirmed by a local analysis of the relevant kinetictheory 75 and further contributes to this "isotopiceffect."

75 S. Migliuolo,Nucl. Fusion,

RLE Report PTP-90/11, Res. Lab. of Electron., MIT, 1990, and JET preprint P(90)34, submitted to1991.


(5 - a) 2 4iZ°

5i

Te+ Zi

'Ti

Zln lne +

9iZ i

570

Te

Ti

ViWo*i2 2kyCi

56

(11)

2 2kyPi


1.3.4 Two-Fluid MHD Description ofthe Internal Kink Mode

Oscillations in the central electron temperature andx-ray emission (so-called "sawtooth oscillations")are an important and well-known instability intokamak experiments. It is widely believed thatthey are initiated by a mode with poloidal andtoroidal mode numbers m/n = 1/1 known as theinternal kink mode. This mode has been exten-sively studied theoretically using both fluid7 6 andkinetic descriptions;77 the results found in thesetwo types of investigations, however, differ in thehigh-temperature and finite # > 2i2,i/w0e regimes (/ is the ratio of plasma pressure to magnetic pres-sure and ,i, cOpe are the ion and electron plasmafrequencies, respectively) currently attained intoroidal machines such as Alcator, TFTR, and JET.In these regimes, the ion gyroradius exceeds thescale-length of plasma motions predicted by thefluid analysis of Ara et al., 76 which is determinedby either electron inertia (i.e., the collisionlessplasma skin depth de = c/cope ) or resistivity. As afluid description is not expected to be valid in thiscircumstance, it has therefore been argued7 8 thatonly kinetic theory can properly describe thebehavior of kink modes in such experiments.

In recent work, we re-examined the two-fluidMHD theory of the internal kink mode. Unlike Araet al.,76 however, we did not assume that the elec-tron velocity appearing in the two-fluid form ofOhm's law is divergenceless. We found that thisassumption, which effectively eliminates the con-tribution of the Hall term in Ohm's law, is not validin the regime mentioned above. Avoiding thisassumption, furthermore, we showed that theresults obtained from the two-fluid theory are ingood agreement with those found from kinetictreatments (at least in the absence of hot parti-cles). Thus, with a careful treatment of the Hallterm, the two-fluid model appears to provide an

adequate description of the internal kink mode inthe parameter region of current tokamak exper-iments. This result may be useful in the numericalstudy of such experiments due to the relative sim-plicity of the two-fluid model compared to akinetic description.

Returning to the apparent breakdown of the fluidtheory, we note that the Hall term in our treatmentintroduces_ into the theory a new lengthPs - c/#/2 /wpi of the order of the ion gyroradius(as calculated on the basis of the sum of the elec-tron and ion temperatures), which becomes thecharacteristic scale of the plasma motion in thehigh temperature regimes referred to above. Thisscale, due to the decoupling of the ions from themagnetic field afforded by Hall's term, may greatlyexceed the scale-length of the current sheetsexcited in the resonant region during the mode'sdevelopment (e.g., the skin depth). Thus, thetwo-fluid theory does not predict that the charac-teristic scale of plasma motions falls below the iongyroradius (as previously thought), which mayaccount for the agreement between the MHD andkinetic results.

We also developed two variational approaches tocalculating the linear growth rate. One of theseformulations is in terms of a single function (thepoloidal velocity profile) and provides an upperbound estimate for the growth rate. The other is interms of two trial functions (the radial displace-ment and the helical magnetic flux perturbation)and leads to a lower bound estimate for this rate.The numerical implementation of the latterapproach is also very efficient in the regimes dis-cussed above, despite the presence of two verydifferent scale lengths. The nonlinear generaliza-tion of the second approach, which is based onlyon the largest scale length, may be especiallyuseful in numerical simulations involving theseregimes.

76 G. Ara, B. Basu, B. Coppi, G. Laval, M.N. Rosenbluth, and B.V. Waddel, Annals of Physics 112: 443 (1978).

77 J.F. Drake, Phys. Fluids 21: 1777 (1978); G.B. Crew and J.J. Ramos; Nucl. Fusion 26: 1475 (1986);F. Porcelli, and T.J. Schep, Phys. Fluids B1: 364 (1989); F. Porcelli, Phys. Rev. Lett. 66: 425 (1991S.M. Mahajan, and Y.Z. Zhang, Phys. Fluids B3: 351 (1991); B. Coppi and P. Detragiache,PTP-90/1, Res. Lab. of Electron., MIT, 1990, submitted to Phys. Rev. Lett.

78 F. Pegoraro, F. Porcelli, and T.J. Schep, Phys. Fluids B1: 364 (1989).

F. Pegoraro,); H.L. Berk,RLE Report

203


1.4 Versator II TokamakResearch Program

Sponsor

U.S Department of EnergyContract DE-AC02-78-ET-51013

Project Staff

Professor Miklos Porkolab, Jeffrey A. Colborn,Jared P. Squire, Jesus Noel Villasenor, Edward W.Fitzgerald

Versator II is a small tokamak facility (major radiusR = 40 cm, minor radius a = 13 cm) with modestplasma parameters (magnetic field Bo - 1.3 Tesla,density ne - 3 x 1013 cm - 3 , and plasma currentIP- 10-80kA) which is used for fundamentalstudies of the interaction of electromagnetic waveswith a fully ionized, nearly collisionless plasma.For this purpose, we use several high power (100 kW) microwave sources to launch waves atfrequencies near the electron-gyro frequency (f -28-35 GHz) and the lower hybrid (ion-plasma)frequency (f = 800 MHz or 2.45 GHz). In thesection below, we describe three different exper-imental projects which were carried out during thepast year.

1.4.1 High Beta-PoloidalExperiments with Advanced X-rayDiagnostics

The purpose of this project is (1) to measure thespatial and energy distributions of fast electronsduring rf current drive experiments, and (2) tostudy the equilibrium and stability of high betapoloidal plasmas in the Versator II tokamak. (Betapoloidal is the ratio of plasma pressure to poloidalmagnetic pressure and is an important parameterfor plasma equilibrium and stability.) In thesedischarges the plasma current and most of theplasma pressure are supplied by highly energeticelectrons created by launched high power lower-hybrid plasma waves.79 We have been mapping thespatial and velocity distributions of the current car-rying fast electrons. With this information we cancalculate realistic plasma current and pressure dis-tributions. Along with other basic plasma para-meters, we can then test theoretical predictionsregarding the equilibrium and stability of these

plasmas. Specifically, we would like to determineif these equilibria are in or near what is called thesecond stability regime.8 0 In addition, at high betapoloidal significant bootstrap current 8l generationis expected, and we would like to verify its exist-ence in the Versator experiment.

To accomplish this goal, we have developed twoarrays of Nal detectors which measure the hardx-ray (20-500 kev) spectrum by utilizing pulse-height analyzers. With the x-ray detectors we lookat plasma bremsstrahlung emission from the colli-sion of the energetic electrons with plasma ions.To obtain spatial distributions we first constructedan array to view the plasma vertically from thebottom of the tokamak at different major radii (seefigure 7). Next, we exploited the fact thatbremsstrahlung emission becomes asymmetric withrespect to the direction of the electron motion athigh energies. We do this by measuring the x-rayemissions at various angles to the toroidal mag-netic field. For this purpose we reconfigured thedetectors into a tangential array which viewthrough the plasma mid-plane at a full range ofangles. Due to the excellent port access, histor-ically this is the first Nal array capable of viewingall angles simultaneously (see figure 8). Duringthe tangential x-ray measurements a singlescannable detector was used to monitor the radialemission profile. The detector electronics cancollect time integrated x-ray energy spectra andenergy integrated time dependent data simultane-ously on all detectors. With this data we canmodel the distribution of energetic electrons bycalculating the expected x-ray emissions and iter-atively fitting the model to the measured data.

In the past year, we have obtained x-ray energyspectra and profiles from mostly high beta poloidalrf driven plasmas, but some data has also beentaken at lower beta poloidal for comparison. Theradial profile data show an outward shift in majorradius of the peak emission. This corresponds tothe Shafranov shift of the inner magnetic fluxsurface. The shift helps confirm that the plasma isindeed in a high poloidal beta equilibrium. Also,we have found that the x-rays coming from theouter region of the plasma are more energetic.This can be explained by noting that higher phasevelocity waves are accessible only to the lowdensity (outside) region. At lower beta poloidalthe emission profile does not have an outwardshift; rather, we observe a secondary emission

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

80 B. Coppi, A. Ferreira, J.W-K. Mark, and J.J. Ramos, Nucl. Fusion 19: 715 (1979).

81 A.A. Galeev and R.Z. Sagdeev, Sov. Phys. JETP 26: 233 (1968).



Figure 7. Perpendicular array.

peak toward the inside plasma major radius. Thismay be evidence of the tips of trapped particlebanana orbits. The peak may correspond to areflection point where the gradient in the toroidalmagnetic field has converted all the trapped elec-tron energy into perpendicular energy.

Using the tangential array, we observed a distinctasymmetry in the plasma x-ray emissions during rfcurrent drive with unidirectional launched rfwaves. There was as much as a factor of ten timesgreater high energy x-ray flux emitted in the co-current drive direction as compared to the counter-current drive direction. In the course of this study,symmetric injection of lower-hybrid waves wasalso examined. Current drive is not expected withthis configuration, but at high poloidal beta nearlythe same current was driven with symmetricinjection as with the asymmetric current drive con-figuration (see figure 9a). The x-ray flux for thesymmetric case was less by as much as a factor of5, implying a strong reduction in the current car-

2 3

TANGENTIAL HARD X-RAY ARRAY

Figure 8. A view of the tangential array from abovethe tokamak.

rying electron tail (see figure 9b). At the lowerbeta poloidal it was not possible to maintain theplasma current with the symmetric injection alone.We believe this is evidence that in high betapoloidal plasmas a major fraction of the plasmacurrent may be supplied by bootstrap current. Wehave yet to model the electron distribution basedon this data.

In 1992, we will continue analyzing the data col-lected during the past year. We will carry out the-oretical numerical analysis using the ACCOME 82

code, partially developed at the Plasma FusionCenter (PFC). Given just a few external para-meters this code calculates the self-consistentplasma equilibrium. It is the most sophisticatedcode of its kind available in the world. We willperform bremsstrahlung emission calculations witha code such as the one developed for the AlcatorC lower-hybrid experiments at the PFC.83 Informa-tion from this work should allow us to determinewhether the equilibria obtained in Versator II is insecond stability and whether bootstrap current isgenerated.

82 P.T. Bonoli et al., NucL. Fusion 30: 533 (1990).

83 S. Texter et al., Nucl Fusion 26: 1279 (1986).

205

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90 & 180 Comparison

. . .During the past year, current-drive experiments50 1 200 250 w 350 were completed on the Versator II tokamak using

E010 S 00 3electron-cyclotron (EC) and lower-hybrid (LH)waves. The experiments performed fall into threegroups: (1) studies of the electron distribution

Figure 9. (a)(top) A comparison of current driven by function during LH current drive, (2) studies of theasymmetric (90') and symmetric (1800) rf injection. current drive efficiency of combined LH and EC(b) (bottom) A comparison of the emitted x-ray spectra. waves, and (3) preionization and current drive

experiments with EC waves. The results are pres-

1.4.2 Fast Wave Current Drive ently under analysis, and attention is beingExperiments focused primarily on the following issues.Experiments

In previous progress reports, we have discussed acombined fast wave slow lower hybrid wavecurrent drive experiment where two travelingwaves are launched. Slow lower hybrid waveslaunched at 2.45 GHz by a conventional grill formsa suprathermal electron tail, which is thenexpected to absorb an 800 MHz fast wave spec-trum launched by a specially built dielectric-loadedwaveguide array. The available power at each ofthese frequencies is up to 100 kW at the sources.The fast wave antenna was successfully testedduring last year's experimental runs and low powercoupling results were consistent with theoreticalpredictions.

Recent high power experiments showed no clearsign of fast wave penetration into, and/or inter-action with the plasma interior. In particular, noevidence of rf driven current or acceleration ofsuprathermal electrons was observed from hardx-ray detectors. Our code calculations indicate

During LH current-driven discharges, a two-temperature tail in the parallel electron distributionfunction has been observed using an improvedelectron-cyclotron transmission diagnostic. Anal-ysis of this result has focused on modeling the dis-charges using a computer code, CQL3D, whichcomputes the electron distribution function underthe influence of LH waves, electric field, ECwaves, and other power sources, if desired. Theresults of the code are being compared to themeasured data to improve our understanding ofthe cause of the two-temperature feature.

Combined LH and EC current-driven dischargesare presently being analyzed to determine the effi-ciency of this type of current drive and to learn asmuch as possible about the physical mechanismsthat affect that efficiency. Our results indicatethat, in many cases, the efficiency is considerablylower than predicted theoretically on Versator II.This may be due to enhanced electron losses or


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that according to linear wave theory a significantfraction of the power should have penetrated theplasma interior and also been absorbed by fastelectrons. We conclude that the fast waves mustbe absorbed on the plasma surface, possibly bynon-linear effects such as parametric decay orscattering by low frequency edge density fluctu-ations.

To further study the physics at the edge, we haveinstalled a poloidal rf probe array on the plasmasurface to look for evidence of such non-linearphenomena. In addition, a retarding potential anal-yzer was also installed to measure the fast electrondistribution function near the plasma surface thatmay be formed by parametric decay. Early resultsshow a low energy electron tail radially concen-trated within a centimeter and located at theplasma periphery. The rf probes have also detectedasymmetric broadening at the pump frequency,indicating some level of parametric decay activity.These results are under further study.

1.4.3 Current-Drive ExperimentsUsing Electron Cyclotron Waves andLower Hybrid Waves

Ar


trapping caused by EC and LH wave combination.This analysis is ongoing.

Effective preionization and current drive at lowplasma densities has been achieved on Versator IIusing EC waves alone. The current drive efficiencyis higher than expected for the low electron tem-perature (Je; 250 eV) of the target plasmas. Pre-liminary analysis suggests that this may be due to

the presence of a hot parallel electron tail pre-formed by the tokamak's toroidal electric field.The current-drive efficiency was found to be inde-pendent of the launch angle of the EC waves, butstrongly dependent on plasma density, decreasingrapidly as the density was increased above3 x 1012 cm - 3 . A doctoral dissertation based onthese results is being prepared by Jeffrey A.Colborn.

207


Documents

Sction 2 Plasma Physics - COnnecting REpositories · Chapter 1. Plasma Dynamics Project Staff Professor George Bekefi, Professor Jonathan S. Wurtele, Ivan Mastovsky, Dr. Chiping Chen,