P/362 USA

Experiments on the Ohmic Heating and Confinement ofPlasma in a Stellarator

By T. Coor,* S. P. Cunningham,! R. A. Ellis,* M. A. Heald* and A. Z. Kranz*

In 1951 Spitzer proposed the stellarator as a devicefor confining plasma and heating it to thermonucleartemperatures.1 Since that time, a series of devices hasbeen constructed and operated at Princeton Univer-sity for experimental research in such systems. Thispaper summarizes experimental results in the funda-mental areas of heating and confinement. Accom-panying papers present evidence for the importanceof cooperative processes in the plasmas of thesedevices.2- 3

STELLARATOR PRINCIPLES

ConfinementConfinement of plasma in a stellarator is provided

solely by externally produced magnetic fields. Asdescribed in detail elsewhere, the magnetic configura-tion is that of a torus with rotational transform.4- б

This geometry gives plasma confinement of a higherorder than that given by a simple toroidal field. Itdoes so by virtue of providing a path, along magneticlines of force, by which currents may easily flow toneutralize charge accumulations produced by particledrifts in the general toroidal field. Two methods havebeen used to give rotational transform to the Prince-ton devices. The first, illustrated in Fig. 1, distorts thetorus into the form of a "figure eight", while thesecond employs currents in auxiliary helical conductorson the surface of a regular toroidal tube. In both casesthe general "axial" field is provided by currentflowing in copper coils surrounding the tubes. In thisand in the accompanying papers,2- 3 the results pre-sented were obtained on equipment employing thefigure-eight geometry only.

Even with the rotational transform, plasma con-finement will not be perfect. Collisions between unlikeparticles in the plasma will always give rise to adiffusion across Unes of force. In a quiescent plasmain which cooperative oscillations are negligible, thisclassical collision diffusion will be the dominantprocess. If instabilities and other cooperative pheno-mena are present, particles may reach the wall evenmore rapidly than in a quiescent plasma.

Following Spitzer6 it can be shown that, assuming

mean confinement

* Project Matterhorn, Princeton University.f General Dynamics Corporation.

classical collision diffusion, thetime, TC, for helium is given by

r c ~ (20B27W)/# sec, (1)

where В is the confining field, T the temperature, Rthe plasma radius, and n the particle density. Insert-ing typical values for our devices, В = 3 x 104 gauss,T = 105 °K, R = 1 cm, n = 1013 cm-3, we findтс ~ 0.6 sec. Therefore, if virtually quiescent condi-tions can be found, confinement times of the order of ahalf second might be expected.

Bohm,7 on the other hand, has suggested that somesort of turbulence with randomly varying electrostaticfields exists even in "quiescent" plasmas and givesfor the average macroscopic diffusion velocity

v± X —4x10"»

nBVp cm/sec, (2)

where p is the plasma pressure. This gives for a meanconfinement time

r c ~ (0.002 BR?)I T sec. (3)

201

For helium under the conditions assumed above, thiswould give a confinement time of 600 /xsec. Eventhough Bohm and his collaborators adduced experi-mental evidence in support of this rate of diffusion,Simon and Neidigh8 at Oak Ridge have refuted theirresult and claim that there is good evidence that Bohmdiffusion does not exist, at least in a "quiescent"plasma. Nevertheless it is reasonable to assume thatfluctuating electric fields associated with instabilitiesand other cooperative phenomena can, under someconditions, cause enhanced diffusion across magneticlines of force.

Heating

Ionization and preliminary heating of plasma con-fined in stellarator geometry is accomplished by makingthé closed tube of plasma the secondary of a trans-former, as shown in Fig. 2. Flux changes in the trans-former iron produce an axial electric field in theplasma which accelerates the electrons and to a muchlesser extent the ions. The energy thus given to theparticles is randomized and used in completing theionization of the gas and "heating" the resultingplasma. It is to be emphasized that this ohmic heating

202 SESSION A-9 P/362 T. COOR et al.

Figure 1. Distortion of torus into "figure-eight" geometryhaving a rotational transform, and simplified confining field

circuitry

PLASMA CONFINED IN FIGURE EIGHT GEOMETRY

OPTIONALCONSTANT CURRENT

RESISTANCE

Figure 2. Ohmlc heating apparatus and simplified circuitry

operates at very low fields (~ 0.1 v/cm) and makes nouse of dynamical heating processes that are exploitedin pinch-effect devices, nor is any use made of pinchfields produced by plasma currents for confinement.The existence of the hydromagnetic instability,predicted by Kruskal,9 which limits the maximumplasma current density to a definite value, and thefact that 4he conductivity of the plasma rises rapidlywith temperature, place an upper bound on the tem-perature that can be achieved by ohmic heating.

When the heating field is applied to the dischargethe initial conditions are in general, a neutral hydro-gen or helium gas pressure of the order of 10~3 mmHg, pre-ionized to the extent of a few per cent. Thetemperature of the gas and plasma is essentially thatof the walls of the apparatus. The applied electricfield rapidly accelerates the initially existing electronswhich in turn make collisions with other electrons,ions, and neutral gas atoms or molecules. As theelectrons become energetic enough they will inducemolecular break-up, radiative excitation and ioniza-tion in the gas. If the initial ionization is high and ifthe applied electric field is low, the electron energydistribution will remain essentially Maxwellian. Theinelastic processes will then be executed by thoseelectrons at the high energy end of the distribution.

Detailed calculations have been made by Bergeret al.,10 on the ohmic heating of hydrogen and heliumin stellarators. These calculations, in which thefractional ionization and electron and ion tempera-tures are followed, allow for the important atomic andsingle-particle processes that might take place in thedischarge but assume that the energy distributionsstay Maxwellian. Some doubt is cast on the applica-bility of these results by the fact that, in practice, theapplied electric heating fields are large enough todistort the electron energy distribution away fromMaxwellian. In particular, the group of runawayelectrons (electrons which are gaining energy from thefield faster than they lose energy by collisions) whicharise early in the heating pulse are capable of morerapid ionization of the neutral gas than the mainbody of thermal electrons. At the same time, therunaway may be capable of exciting some cooperativemechanism that will transfer energy to the ions at amore rapid rate than the simple electron-ion Coulombcollisions assumed in the Berger model.

To make an exact calculation of the progress of adischarge, in which the electron energy distribution isallowed to be non-Maxwellian and in which coopera-tive processes are included, is of course impossible atthe present time, since the nature of many of theprocesses is unknown. The best we can hope for in acomparison of the experimental results with heatingtheory is an indication of how important a part theunknown processes are playing in the heating.

EXPERIMENTAL APPARATUS

This and the accompanying papers2» 3 are concernedprincipally with results obtained in the operation ofthe B-l stellarator. This machine uses "figure-eight"geometry and ohmic heating. A schematic diagram ofthe B-l device with its associated diagnostic instru-mentation4s shown in Fig. 3. The discharge tube isfabricated from stainless steel, is 5 cm in diameter,and 450 cm in axial length. A short ceramic (alu-mina) section is welded into the otherwise conductingdischarge tube at one point. This ceramic break pre-

STAINLESS STEELDISCHARGE TUBE

AUDIO FREQUENCYRESISTANCEMEASUREMENT

ТЕМ

WOLFRAMAPERTURELIMITER

MICROWAVE PHASE-SHIFTMEASUREMENT SYSTEMAND NOISE RECEIVER

OPTICAL ANDSPECTROSCOPIC

OBSERVATIONWINDOW

Figure 3. Schematic diagram of instrumentation on the Model B-1stellarator. Not shown are the coils which produce the axialmagnetic confining field, and the two iron transformer cores for

ohmlc heating (Fig. 2)

OHMIC HEATING AND CONFINEMENT 203

vents the discharge tube from short-circuiting theheating transformer and provides a convenient pointfor applying a 250-kilocycle radio-frequency pulse forinitial breakdown.

Pulsed magnetic confining fields up to 30 kilogaussare used in the B-l device. Energy storage for this fieldis obtained from a capacitor bank of 10e joules. Switch-ing is performed by ignitrons of standard type.

Subsequent to the initial breakdown pulse, ioniza-tion is completed and the plasma is heated by meansof a unidirectional axial electric field produced by twolaminated iron transformer cores. The duration of thisfield is limited by saturation of the iron (~ 0.1 v-sec).Under normal operation the ohmic heating voltagewaveform approximates a rectangular pulse; this istermed "constant-voltage" operation. Because of theKruskal hydromagnetic instability which occurs at adefinite critical current, it is sometimes useful toinsert resistance in the primary circuit of the ohmicheating supply, thereby giving approximately "con-stant-current" operation at a point below the criticalvalue. The ohmic heating field can be removed quicklyat a pre-set time by means of a "crowbar" circuit,shown in Fig. 2, consisting of an ignitrón switch toshort-circuit the primary of the heating-fieldtransformer.

In order to minimize direct wall bombardment bythe plasma, and control the size of the dischargecolumn, an aperture limiter was introduced in manyexperiments. The limiter is a wolfram mask with3.2 cm diameter hole centred on the magnetic axis ofthe stellarator. The limiter has incidental uses inconnection with diagnostic instrumentation, providinga reference potential for Langmuir probe measure-ments, and a localized target with high atomic numberfor the production of X-rays by fast electrons losingconfinement from the discharge.

The instantaneous heating field is inferred from thevoltage measured across the ceramic break in thestainless steel discharge tube. The plasma current ismeasured by means of a toroidally-wound pickup loopencircling the tube, the output of which is electronicallyintegrated. These and other diagnostic signals areobserved on multiple-channel synchronized oscillo-scopes.

It is not possible within the scope of this paper todiscuss in detail or justify the many types of instru-mentation which have been used to analyze the pro-gress of discharges under various conditions. Electrontemperature is inferred from the plasma conductivityderived from observed heating voltage, plasmacurrent, and aperture; from relative strengths ofrelated singlet and triplet lines in the neutral heliumspectrum; and from the amplitude of an inducedsignal coupled by the plasma between toroidally-wound transmitter and receiver coils, operating at afrequency low enough that skin effect can be neg-lected. Ion 'temperature is inferred from Dopplerbroadening of spectral Hnes of the working gas and ofhighly ionized impurities. Confinement effects areinferred from the maximum energy of X-rays pro-

RF.BREAKDOWN OHMIC HEATING

О 10 20 30 4 0TIME (MILLISECONDS)

Figure 4. Relative time scales of axial magnetic confining field andvoltage externally applied to stellarator loop

duced by fast electrons; from time variations inelectron density, derived from phase-shift of a trans-mitted millimeter microwave beam11; and fromLangmuir probes located outside the active dischargeaperture. Instability processes of various types andwall effects are inferred from spectral lines of impurityatoms; from streak photography; and from emissionof X-rays and high-level non-thermal microwavenoise. Langmuir (electric) and magnetic probes havenot been found useful in the active region of thedischarge. Because of the rotational transform, anobstacle such as a probe intercepts in principle allplasma outside the magnetic surface adjacent to theinner tip of the obstacle, thus becoming an aperturelimiter. While probe studies were nevertheless possiblein early devices in which high impurity levels existedbecause of wall outgassing, it has been found that thehigh plasma temperatures and abundant runawayelectrons of our baked devices consistently destroythe probe in one or two pulses.

The results of early experimentation led to the con-clusion that the behavior of discharges in devices ofthis type was completely dominated by outgassing ofthe walls if conventional vacuum systems were used.Therefore, since 1956 all devices have been constructedso as to be fully bakable to 450 °C. With baking, abase pressure of the order of 2 x lO"10 mm Hg hasbeen achieved, and the influx of impurities has beenreduced by about two orders of magnitude. In spiteof these efforts, material from the wall is still a signi-ficant factor in the behavior of these devices undermost operating conditions. During operation, spectro-scopically pure gas is introduced continuously througha controlled leak to maintain an operating pressurebetween 10~4 and 10~3 mm Hg. While the behaviorof hydrogen and deuterium discharges has beenstudied in some detail, more extensive work has beendone with helium, because of better characteristics foroptical spectroscopic analysis and greater freedomfrom complex molecular and chemical phenomena.

The time scale and sequence of operation is shownin Fig. 4. The magnetic confining field is essentiallyconstant during the interesting portions of the dis-

204 SESSION A-9 P/362 T. COOR et al.

ST E LLAR ATO R.LOOP

VOLTAGE

( I . , l 2 v / d i v , r ,24 v/div )

Figure 5. Representative constant-voltage ohmic-heating dis-charges in helium

Confining field, 27 k-gauss; pressure 5 x 1 0 ~ 4 mm Hg

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Confining field, 27 k-gauss; pressure, 5 x 1 0 ~ 4 mm Hg

charge. The timing of all operations is controlledby pre-set counters driven by a 100-kc/sec masteroscillator. Operation is limited to about one pulse perminute by capacitor bank charging and heat dissipa-tion of the field coils.

EXPERIMENTAL RESULTS

In this section the major experimental observa-tions at various stages of the discharge are sum-marized. The principal variables are the value ofconfining field (10 to 30 k-gauss), the ohmic-heatingfield (0.04 to 0.5 v/cm), and the initial gas pressure(lO"4 to 10~3 mm Hg). Other parameters are the

amplitude, duration, and timing of the radio-frequencybreakdown pulse and the shape of the heating fieldwaveform (constant-current or constant-voltageoperation, and termination by crowbar). The follow-ing discussion refers to helium discharges unlessotherwise stated.

The Current Rise

Upon application of the ohmic-heating field, theplasma current and electron density rise rapidly.Representative cases of current and voltage behaviorfor constant-voltage operation are shown in Fig. 5.Figure 6 shows the current rise for several values ofheating fields. At high heating fields, the currentreaches the Kruskal instability limiting current in100 /xsec or less. The limiting current for the B-l

0.05 0.10

HEATING FIELD (VOLTS /CM)

Figure 7. Maximum of first current plateau as a function ofheating field and pressure

Confining field, 27 k-gauss; helium gas filling

TIME (MILLISECONDS)Figure 8. Plasma current, average electron density, and spectro-scopic lines during a low-pressure, constant-voltage discharge inhelium. The heating field (.067 v/cm) lasts about 4 msec.

Confining field, 27 k-gauss; pressure, 3 x 1 0 ~ 4 mm Hg

OHMIC HEATING AND CONFINEMENT 205

¡\ He II (th)

TIME (micro,ecoi.,1.)

Figure 9. Comparison of experimental plasma current, ionized helium light, and electron temperaturedata with theoretical values

Heating field, 0.11 v/cm; confining field, 27 k-gauss; helium 8x10 - 4mm Hg. For this case, the plasmaaperture was estimated to be 8 cm2

device with 4.1 cm diameter geometrical aperture isabout 2200 amp at a confining field of 27 k-gauss.At moderate fields the current typically rises to avalue (< 500 amp) well below the Kruskal limit,remains at this plateau value for as much as severalmilliseconds, and then rises rapidly to the Kruskallimit. In general, the second rise always proceeds tothe Kruskal limit although at low heating fields{Fig. 6, case A) the field ends before a second rise isevident. The plateau currents are essentially indepen-dent of confining field, and are shown as a function ofheating field and pressure in Fig. 7. The plateau isaccompanied by X-ray production and intense non-,thermal microwave noise generation.

Typical spectroscopic data obtained by Photo-multiplier techniques are shown in Fig. 8. The neutralhelium line peaks early in the current rise and thenfalls slowly. The slowness of fall may result from theinflux of cold gas from the external portions of thevacuum system, in particular the observation1 port,and perhaps also from wall outgassing. Ionizedhelium light peaks slightly later during the currentrise and then falls to low intensity. Spectroscopicevidence indicates first ionization in excess of 95%,and probably a high degree of second ionization.Spectral lines of impurity atoms, such as C i n 4647 Âand O114415 Â, emerge well before the Kruskal insta-bility current is reached. Preliminary observations ofDoppler broadening at the time of the current plateausof Fig. 6 (this feature is barely visible at about 70%of maximum current in Fig. 8) indicate ion tempera-tures far in excess of what would be expected from

ohmic heating alone. These measurements, of greatpotential significance, must be carried much farthefbefore dependable conclusions can be drawn.

In Fig. 8 the average electron density, inferredfrom 8.6 mm micro-wave phase-shift data, begins tofall before the Kruskal critical current is reached,indicating imperfect confinement of the plasma.

A comparison of theoretical calculations12 with theobserved time dependence of plasma current, electrontemperature, and ionic (4686 A) He11 light is shown inFig. 9 for a typical case where two current plateausare observed. The temperature was inferred fromplasma conductivity data. The agreement betweentheory and experiment is good in the early phase ofthe discharge. The theory predicts the existence of thefirst current plateau, and approximately predicts theobserved value of current and electron temperature.However, the duration of the first current plateau isconsiderably longer than theoretically predicted; thiseffect is presumably caused by the influx of cold gasinto the discharge region. Furthermore, some He11

is observed much earlier than theory predicts, at atime when first ionization of helium is not complete.

Operation at the Kruskal Limiting Current2

When the current reaches the Kruskal kink-instability critical value, both plasma current andvoltage show violent fluctuations, as indicated inFigs. 5 and 8; the average current then decreasessomewhat with time. No X-rays are produced at thistime, indicating that single-particle confinement is tooshort for runaway electrons to reach high energies.

206 SESSION A-9 P/362 T. COOR et al.

PLASMACURRENT

(500amp/div)

VOLTAGE(12 volt/div)

Hel492l

He IL 4686

INTEGRATEDX-RAY

INTENSITY

3 6 2 Z m s / d i v i s i o n

Figure 10. Representative constant-current discharge in helium.Confining field, 27 k-gauss; pressure, 5 x 1 0 ~ 4 mm Hg

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Figure 11. Variation of average electron density during constant-current heating

Current, approx. 1000 amp; helium, 5x10~ 4 mm Hg

PLASMACURRENT

(5OOomp/div)

Figure 12. Constant-current discharge showing abrupt currentend. Confining field, 20 k-gauss; helium, 5x10~ 4 mm Hg

Depending on the outcome of the competition betweenthe driving of ions into the walls and the resultingintense outgassing of the walls, the charged particledensity either falls, as in Fig. 8, or remains high. Thetime constant for decay of electron density in Fig. 8is about 300 /¿sec. Spectral lines of impurities (princi-pally carbon, oxygen, and hydrogen) are very intense,particularly in the case where outgassing dominatesand the electron density remains high. Although noquantitative measure of the abundance of impurityions in the discharge exists, it is clear that they are amajor factor even under ultra-high vacuum conditions.

Operation at "Constant Current"By placing a relatively high resistance in series with

the heating-field power supply, it is possible to limitthe plasma current to a value well below the kinklimit. Typical current and voltage behavior is shownin Fig. 10. Under these conditions the charged particlesare again rapidly lost from the discharge. Figure 11shows the variation in electron density during theseconstant current discharges.

A brief X-ray burst appears at the time of thecurrent rise. As the discharge progresses and thecharged particle density falls, there is at first little andthen increasingly intense X-ray production. It isprobable that a significant part of the current in thefinal stages of the discharge is carried by runawayelectrons. In the usual case, as the ohmic-heatingtransformer cores saturate, the current falls slowly,often in a series of small steps accompanied by X-raybursts. Occasionally, the current and electron densityboth drop abruptly, with a consequent sudden rise inthe heating voltage. This effect, illustrated in Fig. 12,is presumed to be some type.of plasma instability,perhaps associated with runaway electrons.

At sufficiently high initial gas pressures the wallemission rate is high enough to maintain the charged-particle density in the discharge. To study wallemission effects, data similar to Fig. 11 were taken asa function of pressure. Electron density loss rates,tangents to observed decay curves at 5 x 1012 electrons/cm3, are shown in Fig. 13 for hydrogen. An extra-polation to zero pressure suggests a confinement timeof the order of 100 fisec in the absence of wall feedbackeffects.13

The current and voltage data of Fig. 10 can be usedto compute plasma resistivity if an effective plasmadiameter is assumed. This diameter can be estimatedfrom the Kruskal instability current or from geometri-cal considerations, these being in satisfactory agree-ment.2 For the Model B-l stellarator the effectivediameter without wolfram limiter is 3.6 cm with anestimated accuracy of 15%. One infers the electrontemperature, using the theoretical resistivity re-lationship which assumes Maxwellian electrondistributions,14

rj = К (1пЛ)Ге-* . (4)

(the electron kinetic temperature Te is in degreesKelvin; In Л is a slowly varying function of tempera-

OHMIC HEATING AND CONFINEMENT 207

ture and density of the order of ten in our experi-ments; the constant К is 6.5 x 103 ohm cm for hydro-gen, 11.1 x 103 for doubly-ionized helium). For thedata of Fig. 10 the resistance is relatively constant atabout 5 milliohms during the latter two-thirds of the•discharge, corresponding to a temperature of about:90 electron-volts. This computation of temperature issubject to question because a significant fraction ofthe total current may be carried by runaway electrons,and indeed the "thermal" electrons may not have aMaxwellian velocity distribution. Furthermore, theactual current channel may be broken up or distortedby cooperative processes. Low-frequency ac con-ductivity measurements cannot be made during heat-ing because fluctuations in the heating current producetoo high a noise level.

Electron temperature measurements based on therelative intensities of corresponding singlet andtriplet lines in the helium spectrum are inaccuratebecause of poor knowledge of the excitation cross-sections, the non-Maxwellian distribution, and low-experimental light intensities. However, such measure-ments give better than order-of-magnitude agreementwith resistivity data.

Ion temperatures, from Doppler line widths, are.difficult to measure during heating because of thelow light intensity. Under typical conditions, He 1 1

5 I 0 X I 0 - 4

PRESSURE (MM-HG)

¿Figure 13. Electron density loss rates during heating as a functionof operating pressure

Heating current, 1500 amp; confining field, 27 k-gauss

TIME (MILLISECONDS)

Figure 14. Average electron density following crowbar termina-tion of heating field

Heating field (0.2 v/cm) applied 0.75 msec prior to crowbar;helium, 5x 10~4 mm Hg

PLASMACURRENT,

(IOO amp/div)

Figure 15. Stepwise decay of plasma current following removal ofheating fieíd by crowbar 1.8 msec after start of oscilloscope traceConcave curvature of baseline is instrumental; current reacheszero 12.6 msec after start of trace. Heating field, 0.3 v/cm;

confining field, 27 k-gauss; helium, 4x10~4 mm Hg

4686 Â appears cold ( < 5 ev) in the later stages of thedischarge when one might expect the temperature tobe a maximum. The possibility of mass motion, due to

- cooperative processes, and the possibility that coldgas continually entering the discharge may contributedisproportionately to the radiating ions limit thevalidity of these observations. Light observations arenormally made transverse to the stellarator axis, anunfavorable geometry for observing volume ratherthan surface radiation.

Discharge Behavior after HeatingMany discharges terminate when the charged-

particle density falls so low that the electron energiescan no longer be moderated and all particles run away.If wall emission has been high enough that substantialdensity is left when the heating field ends, the densityfalls rapidly at first and then more slowly, with a timeconstant of the order of a millisecond. To retain asignificant density at the heating pulse end, the fieldcan be removed by crowbar before much gas has beenlost and before the walls are heavily bombarded.Typical data are shown in Fig. 14. Again the densitydrops quickly at first and then more slowly. Decaytime constants of the order of 6 msec have beenobserved under these conditions. However, low-frequency conductivity measurements made at thistime indicate a completely cold plasma. A very large

208 SESSION A-9 P/362 T. COOR et al.

prolonged burst of X-rays is observed some 10 msecafter the discharge, when the confining field is betweenone-half and one-third of its peak value.

A very striking phenomenon, characteristic of thecrowbarred heating-field, is that the current does notfall abruptly, but rather decays slowly in a series ofabrupt "steps" and plateaus. This effect is shownin Fig. 15. The steps are accompanied by X-rays andmicrowave noise. For some conditions the electrondensity is observed to increase suddenly, by as muchas a factor of two, at the time of a current step. Sincethe after-current would appear to consist principallyof runaway electrons, this period of the discharge, likethe rise of current, may be controlled largely byassociated plasma instabilities. These effects areconsidered in greater detail in the accompanyingpaper.3

Comparison of Hydrogen and Helium13

The behavior of the B-l device operated with thetwo atomic species is qualitatively similar. The rate ofdecay of the plasma density during ohmic heating isfrom two to five times greater in hydrogen than inhelium, presumably because of the greater binding ofhydrogen on the walls. The plateaus on the rise of thecurrent are not observed in hydrogen. A furtherdifference occurs during the radio-frequency break-down discharge. Hydrogen is removed very rapidlyduring the discharge, to the extent that the ohmicheating discharge is inhibited for several millisecondsafterwards. Helium, on the other hand, shows negli-gible net loss of gas during the breakdown discharge.This difference may result from dissociative recom-bination and charge exchange processes in the feebly-ionized radio-frequency discharge. No significantdifferences between hydrogen and deuterium havebeen detected.

CONCLUSIONS

Although many details of the operation of stellara-tors are yet to be explained in full, most generalfeatures of the observations described above appear tobe understood. One of the most clear-cut resultsobtained in the operation of these devices is theeffectiveness of the figure-eight geometry in providingsingle-particle confinement. Runaway electrons per-sist for many milliseconds after the heating pulseterminates, as evidenced by the late burst of X-rayswhich occurs when the confining field has diminishedmarkedly. This persistence is clear evidence that therotational transform of the stellarator produces con-finement. These electrons make about 5 x 106 circuitsof the loop during a time when there is negligible axialcurrent along the discharge tube to produce anyrotational transform, a phenomenon which would notbe possible in a simple torus, where the drifts are of theorder of one gyration radius per revolution.

The most significant measurement of plasmaconfinement appears to be the confinement time duringohmic heating in hydrogen, obtained from Fig. 13.The result, approximately 120 jtisec at 27 k-gauss, is

much shorter than the classical collision diffusiontime and is nearer the value predicted by Bohm'sequation, a result which may be fortuitous. • Theconfining field dependence is not yet known. We haveno knowledge of the process that is responsible forthis rate of plasma disappearance from our devices.The runaway electrons or non-Maxwellian particledistributions that exist during ohmic heating mayproduce plasma oscillations which give rise to en-hanced diffusion across Unes of force. Also there existsthe possibility of interchange instability or highermodes of the kink instability of Kruskal. Evaluationsof the ratio of material pressure to magnetic energydensity, a parameter which appears in the theory ofthe interchange type of hydromagnetic instability,16

yield values of the order of 10~6 for discharges undervarious conditions. Theoretical estimates of thecritical value of this parameter for the B-l device arenot available because of the complex geometry. Whilethe data are not inconsistent with the occurrence of aninterchange type of instability, its presence cannot beassumed without some more direct evidence.

The observations on the plasma decay after theohmic heating is terminated unfortunately give nogood evidence as to the confinement of a "quiescent"plasma. The rapid fall in electron temperature after theend of the ohmic pulse is clear evidence that enoughimpurities have entered the plasma to catalyze recom-bination. However, the runaway electrons whichcontinue to exist in the plasma during this period haveenough energy content to keep the ionization processgoing and maintain the plasma in spite of recombina-tion. There is some difficulty in explaining the rate ofionization, but since the transverse velocities of run-away electrons may also be large, the runaway electronflux effective in ionization could be considerablygreater than that detected by axial current measure-ments.

The existence of runaway electrons and non-Maxwellian particle distributions makes detailedcomparison with the simple theory of ohmic heatingunrewarding, but the general behavior of the ioniza-tion and heating process is about as predicted. It hasbeen shown to be possible with relatively low electricfields to ionize a gas almost completely and to heat itto maximum temperatures of the order of 100 electron-volts in spite of cooperative processes that limit theconfinement time during heating and in spite of anappreciable influx of impurities from the walls.

ACKNOWLEDGEMENTS

We wish to acknowledge the assistance of R. G.Tuckfield and H. J. Winthrop in this work.

REFERENCES

1. L. Spitzer, A Proposed Stellarator, AEC Report No. NYO-993 (PM-S-1, 1951).

2. M. D. Kruskal, J. L. Johnson, M. B. Gottlieb and L. M.Goldman, Hydromagnetic Instability in a Stellarator,P/364, Vol. 32, these Proceedings.

3. W. Bernstein, F. F. Chen, M. A. Heald and A. Z. Kranz.

OHMIC HEATING AND CONFINEMENT 209

Runaway Electrons and Cooperative Phenomena in B-lStellarator Discharges, P/358, Vol. 32, these Proceedings.

4. L. Spitzer, The Stellarator Concept, P/2170, Vol. 32, theseProceedings.

5. M. D. Kruskal and R. M. Kulsrud, Equilibrium of aMagnetically Confined Plasma in a Toroid, P/1876, Vol. 31,these Proceedings.

€. L. Spitzer, Physics of Fully Ionized Gases, p. 38, Inter-science Publishers, New York (1956).

7. A. Guthrie and R. K. Wakerling, éd., The Characteristicsof Electrical Discharges in Magnetic Fields, Chap. 2,Sec. 5, McGraw-Hill Book Co., New York (1949).

S. A. Simon and R. V. Neidigh, Diffusion of Ions in a PlasmaAcross a Magnetic Field, AEC Report No. ORNL-1890(1955). ,.

9. M. D. Kruskal, Large-scale Plasma Instability in theStellarator, AEC Report No. NYO-6045 (PM-S-12, 1954).

10. J. M. Berger, I. B. Bernstein, E. A. Frieman, J. Dawsonand R. M. Kulsrud, On the Ionization and Ohmic Heatingof a Helium Plasma, P/363, Vol. 32, these Proceedings.

11. С. В. Wharton, Bull. Am. Phys. Soc, 3, 86 (1958); UCRLReport 4836, in press; R. F. Whitmer, Phys. Rev., 104,572 (1956); M. A. Heald, Record of IRE National Con-vention (1958).

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