PPPL Compact Toruses 1979

PWl-1755

PROCEEDINGS o r T H E ^ S ^ PM JOINT SYMPOSIUM ON COflPACT TORU3ES MB ENERGETIC PARTICLE

I N J E C T I O N , - ^” PRINCETON, NEW JERSEY, 12^12-1^1/79

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PLASMA JWSICS LABORATORY

ONCOMPACT TORUSES AND ENERGETIC PARTICLE INJECTION

HOSTED BV

-*PROCEEDINGS OF THE US-JAPAN JOINT SYMPOSIUM

PLASMA PHYSICS LABORATORY PRINCETON UNIVERSITY PRINCETON, NEW JERSEY 035¥l

12-14 DECEMBER 1979

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The six;y papers contained in these Proceedings serve to convey some

idea of the highly original and diversified discussions that cook place at the US-Japan Joint Symposium on Compact Toruses and Energetic Particle Injection.

Perhaps the most striking feature of the Symposium, however, was not

its diversity, but its essential unity. Researchers from such seemingly disparate fields as tokamaks, mirrors, theta pinches, zed-pinches, and relativistic-beam injection found themselves confronting identical problems of physics and converging towards a similar reactor goal.

The participants were also pleased to experience a second form of con

vergence: the joining of the Japanese and United States fusion programs in a -ollaborative effort. This collaboration promises to be particularly fruitful in the area of compact torusas and energetic particle injection — where so much depends on the emergence of new ideas and new experimental techniques.

Harold P. Furth 21 January- 1980

PREFACE

Title Page Preface Table of Contents

Introductory Remarks by J. F. Clarke The Compact Torus Concept and the Spheromak by H. P. Furth Injection of Relativistic Electron Beam into Toroidal Systems by

A. Mohri, K. Narihara, Y. Tomita

The LASL Compact Torus Program by R. EC. Linford and CT Staff Initial Results of Field Reversed Plasma Gun Experiment by

W. C. Turner. C. W. Hartman, J. Taska

Experiment on Plasma Confinement by Intense Relativistic Electron

3eam Ring by Y. Tomita, K. Narihara, T. Tsuzuki, M. Hasegawa, K. Ikuta, A. Mohri

Intense Relativistic Electron Beams in Toroidal Magnetic Geometries by V. Bailey, J. Benford, R. Cooper, 3. Ecker, H. Helava

Compact Torus Research at U.C.I, by A. Fisher, S. Robertson,

M. Rostoker

The Longshot Injector: A 3/4-fcJ, 120-keV Pu]sed Source of 3 * 1 016 Ions for Ion Ring Formation by J. B. Greenly, D. A. Hammer, R. S. Sudan

Reversed-Field Configuration with Rotating Relativistic Electron

Beams by J. D. Sethian, K. A. Gerber, D. S. Spector, A. E. Robson

Results and Present Status of the Relativistic Electron Ring Experiments and Their Application to Spheromak Problems by H. H. Fleischmann

Reversed Field Configurations Generated by Proton Pulses by

J. A. Pasour, J. Golden, J. Harsh, C. A. Kapetanakos Thermal Background Effects on the Kink Instability of a Field-

Reversing Ion Layer by S. J. Yakura, T. Kammash

TABLE OF CONTENTS

iii

Hot Electrons by C. W. Hartman, M A. Levine

Particle-Fluid Hybrid Simulation of Field Reversal in a tfirror

Plasma by B. I. Cohen, T. A. 3rengle

Two-Uiaiensional Time-Dependent Transport in Field Reversed Equilibria by S. P. Auerback, H. L. 3erk, J. K. 3oyd, 3. McMamara, D. Shumaker

A Steady-State Beam Driven Field-Reversed Mirror by J. H. Hammer,

H.‘L. 3erk

Page

12’

i ii iii

Magnetized Gun Experiments by T. R. Jarboe, I. Renins, H. ’-;. Hoida,

Formation of a Compact Torus using a Toroidal Plasma Gun by

J. Marshall, A. R. Sherwood

M. A. Levine, P. A. Pincosy

Reconnection Conditions for Flowing Field-Reversed Plasma from a

Plasma Gun by J. w. Shearer, J. L, Eddleman, J. R. Ferguson . .. 61

Physics of the OHTE by T. Ohkawa and the OHTE Group Formation of Toroidal Plasma Confinement Configurations by using

, * 82 . .. S6

Calculation of Ideal MHD Growth Rates and Eigenfunctions in

Field Reversed >tirrors in the Large Toroidal Mode Number Limit by D. V. Anderson, W. A. Newcorab

Toroidal Reversed Field-Pinch Experiments by D. A. Baker Some Properties of the Heating and Confinement in the RF?

Configuration by S. Ortolan!

Relaxation of Toroidal Discharges by L. Turner Effects of Impurity Radiation on Reversed-Field Pinch Evolution

by E. J. Caramana, 7. W. Perkins

Field Reversal Experiments, FRX-A and FS5-B Results by

W. T. Armstrong, R. K. Linford, J. Lipson, D. A. Platts, E. G. Sherwood

FRX-C and Multiple-Cell Eicperiments by E. E. Siemon and LASL

Compact Torus Staff

Compact Torus Theory — XKD Equilibrium and Stability by

D. C. Barnes, C, E. Seyler

Two-Dimensional Simulation of Compact Torus Formation by

D. W, Hewett

Two-Dimensional Compression

E. Hameiri, W. Grossmann

Tearing-Mode S t a b i l i ty A n a l y s is

in General Compact T o ri by

to a C y l i n d r i c al Plasma by

About the Cause for Current Step (Down) by S. Yoshikawa

Dynamically Formed Spheromak Plasma (PS-1) by G. C. Goldenbaum,

Y. P. Chong, G. Hart, J. H. Irby … *.

Spheromak Equilibrium and Stability and Numerical Studies of a

Spheromak Formation Scheme by M. Okabayashi, S. Jardin, H. Okuda, T. Sato, G. Sheffield, A. Todd

Design and Fabrication of the S-l Spheromak Device by H. Yamada,

J. Sinnis, H. P. Furtb, M. Okabayashi, G. Sheffield, T. H. Stix, A. M, M. Todd

Two-Dimensional Simulation of the Formation of the»?PPL Spheromak

by A. Aydemir, C. K. Chu, H. C. Lui

Bifurcation of Tornidal Plasma in a Poloidal Ouadrupole Field by

H- Ikezi, K. F. Schwarzenegger

H. L, 3 e r k, J. Saver, D. D. Schnack

The T i l t i ng Mode in t he R e v e r s e d - F i e ld Theta Pinch by A. I. Shestakov, D. D. Schnack, J. K i l l e en

P e r i o d ic Field-Reversed E q u i l i b r ia for a M u l t i p l e - C e ll Linear

Theta Pinch by H. Meuth, F. L. Ribe

Zero-Dimensional Modeling of Field-RaVersed Theta-Pinch Machines

by E. H. Klevans

Spheromak Formation by Theca Pinch by Y. S o g i, H. Ogura,

Y. Osanal, K. S a i c o, S. S h i i n a, fl. Yoshimura

Field-Reversed Plasma’ Gun Based on t he I n v e r s e - P i n ch Discharge

by U. D. Getty

. 1 26

A Trizgered-Reconnection Compacr Tiroid Experiment by

A. L. Hoffman, G. C. Vlases

Plasma Rotation in Field-Reversed Theta Pir.ches by I. C. Steinhauer . . 151 Stellaraiak a Hybrid Stellarator — Spheromak by C. W, Hartman … .. 155 utilization of Electron Coils for an Advanced Tokamak and Conjecture

IIS

106

110

16:

159

176

135

171

Startup Scenario of Compact Tori Based on R£3-Injection Developed

in SPAC Group by £. Ikuta

The S?S Compact Torus Experiment by A. DeSilva Minimum Energy Equilibria by A. Reiman, R. S. Sudan Compact Toroidal Plasma Equilibrium and Implications on Stability

by G. K. Morikawa

Radio-Frequency Flux Control of Toroidal Plasmas by S. Inoue, K. Itoh . . 197 Field-Reversed Configurations: Theoretical Considerations and

Reactor Applications by G. H. Miley

The Holomak — A Toroidal Spheronak by T. H. Stix, A. M. M. Todd The LISOS Reactor: Compression of a Compact Torus by a Liquid Metal

Liner by A. E. Robson

Preliminary Studies of Spheronak Reactors by M. Katsurai, M. Yamada … 212 The All Plasma Spheromak: The Plasmak by P. Koloc, J. Ogden

Neutral Beam Sustained, Field-Rever3ed Mirror Reactors by

G. A. Carlson, K. R. Schultz, A. C. Smith, Jr

The Moving-Ring Field-Reversed Mirror Reactor Concept by A. C. Smith,

Jr,, G. A. Carlson, H. Et. Fleischmann, T. Kammash, EC. R. Schultz, D. M. Woodall

Preliminary Reactor Implications of Compact Tori: How Small is

Compact? by R. A. Krakowskl, R. L. Hagenson

TRACT: A Small Fusion Reactor Based on a Compact Torus Plasma by

H. J. tfillenberg, A. L. Hoffman, L. C. Steinhauer, P. H. Rose… * 233

List of Attendees

224

220

204

186

I7JTR0DUCT0RY REMARKS

John F. Clarke, Deputy Director, Office of Fusion Energy, Department of Energy, Washington, D.C. 20545

I wish to add my welcome and that of the (Department of Energy to Mel’s.

Exchange visits of the past have proven to be valuable to the participants from both countries, and it is gratifying that this is the first to be held urder the newly inaugurated series with the Government of Japan.

It may not be coincidental that the topic for this conference is in the area of alternative concepts. The diversity of our national program is probably matched only by the Japanese program among the several national fusion pro’jrams worldwide. Further, the focus of the meeting, what we call compact toroids, is indicative of the readiness of the worldwide fusion community to deal with worthy, innovative ideas.

In the U.S. program we continuously evaluate alternate confinement

approaches for development as fusion power systems. How do we arrive at a decision to launch development of an approach such as compact toroids and neglect others?

This is a difficult technical management issue that we are frequently

asfced to addresc. The answer is not simple and ultimately must rely on professional judgments. The process for selecting concepts has evolved as technical successes were obtained in the development of the mainline confine ment concepts. That introduces a key element in the selection process; we examine and select alternate confinement approaches by taking f’ull account of ;he status of the principle confinement approaches, and we do not undertake selection of alternate fusion confinement as an abstract exercise.

Historically, the mainline confinement approaches, the tokamak and irirror, have reached a prominent role in the development program because of demonstrated experimental success in confining hot, fusion-quality plasma earlier than other approaches. However, the flexibility implicit in the physical principles that unite all confinement concepts has oermitted a great variety of feasible fusion approaches to be conceived and prODosed for develop ment. These are collectively identified as alternate concepts. Any number of them might be chosen for development and ultimately lead to successful fusion power systems. It is taken as an imperative by the Office of Fusion Energy that any attempt to develop all of them with equal emphasis would be detrimental to the eventual technical success of the most promising and an irresponsible utilization of valuable resources.

-1-

Several courses of action are conceivable in order to optimize the develop

ment of the highest potential of fusion for commercial aaplication. The one approach that is clearly unsound is to ignore potential advantages that can come to the development program from ideas outside the mainline effort. It ‘s precisely this factor that is employed in the review and selection process for alternate concepts.

The selection of alternate confinement approaches for development involves

three factors; potential reactor advantages with respect to the mainline

approaches, technical feasibility, and the readiness (or timeliness) of undertaking the development with respect to the status of worldwide fusion development. The analysis of these factors is a continuing process; the level of performance by which the concepts are evaluated becomes increasingly -ore demanding as the concepts mature (and as the successes of the mainline affort elevate the general standard for all fusion development). Novel alternate concepts such as compact torolds must be perceived by the fusion community as having the potential of significant reactor advantages with *-espect to the mainline concepts. As the conceot matures and becomes ready -or proof-of-orinciple level tests, the perception must be replaced by quantitative studies supporting the reactor advantages. The compact toroid concept is clearly in an embryonic state. As the research we will near about during the next several days evolves, this concept will be submitted to this scrutiny also.

We wish the research success for it is eoually clear that this approach

has significant potential benefits for users. Despite the enthusiasm for such a fusion system it still takes the dedicated work of the fusion community to make the promise of it a reality.

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THE COMPACT T03US CONCEPT AND THE SPHEROMAK

H. P. Furch, Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 0854<i

Photographs of solar activity, using polarized filters [1J, have pro vided indirect evidence of the emission of long-lived toroidal plasma con figurations confined by linked poloidal and toroidal magnetic fluxes. Plasma configurations of this sort were first produced in the laboratory by H. Alfven [2) and his coworkers (cf. Fig. 1) and reported at the Second International Conference on Peaceful Uses of Atomic Energy in 1958.

The injection of a current layer of highly energetic electrons (an

“E-iayer”) to produce a ste?dy-state toroidal plasma confined by pololdal field was proposed at the same conference by S. Christofiios [3]. A moderr. variant of this idea, using a neutral-beam-injected, medium-energy ion cur- [], is shovn in Fig. 2. Reversed-poloidal-field configurations of che r em same form have long been produced successfully by means of ordinary plasma currents in theta pinched — both with and without toroidal magnetic field component [5,6] (Tig. 3) *

The common faacvre of all these plasma confinement schemes is chat che

magnetic field lines are closed, yet the field generating coil system is not required to iiak the plasma toiroid. The principal variants of this ”compact torus” concept are tabulated in Fig. 4. The present paper is con cerned mainly with a brief review of the case 3pol < ^plasma- where che current carriers have poloidal-field gyroradii that are smaller than che scale height of che plasma.

Experimental tbeta-pinch plasmas with essentially null toroidal-field

component ( 3p oi » B t0 T) have exhibited remarkable longevity in earns oi che characteristic time scale for MHD instability [7,8]. VJhile the ideal .’!HD theory for this configuration does not actually allow all modes to be stabilized, the elimination of th«S strongest instabilities by axial elonga tion, combined with the finiteness of the plasma ion gyroradius (cpgi.

a-piasma) n a? be providing effective stability in the experiments. Far Che reaccor applicaeion, the null-Bt variant has the obvious advantage of very high beta-value Ui) £ 15, but the possible drawback, of insufficiently stable confinement, since there may be a contradiction between the require ments for stability against gross modes and microinstabilities. The fast time scales and high voltages of the fieId-reversed cheta-pinch formation method (Fig. 2) could be avoided in the beam-injected fie Id-reversed mirror approach (Fig, 3 ), provided that the problem of ion-current cancelation by the electron-drag current can be resolved satisfactorily [4].

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The compact torus with comparable toroidal and poloidal field components

(Fig. 1 ). which has lately coot :o be called che “jpheronak,” has long re ceived theoretical attention 19,10,11]. the toroidal field is supported by poloidal currents flowing inside che plasma and oust, of course, vanish out side the plasma boundary. Ideal MHB stability is achievable when chp. plasma

is the field strength OQ the mag

encity is somewhat oblate, as in Fig. 5, and is surrounded by a moderately close-fitting conducting shell (12]. Under these conditions, the s t a b i l i ty limit for 3g 3 8TT ( { D2^ )1’2/ B| (where 30 netic axis) is typically of order 2-5%, but it can be several times greater in highly optimized configurations [13]. The basic advantage of the finita- 3t or regime is that s t a b i l i ty against a ll ideal MHD modes can be ensured =ven for the case Bpol <e a p ] _a s m a, which is congenial co good microstability properties. The main drawbacks are that special provision must be made for generating the toroidal flux and that beta must remain well below unity. It should be noted, however, that a limiting value of aj for the spheromak gives as much if the maximum field strength at Che magnet coils is the same in the two cases. This is because the spheromak field is maximal at the plasma center (Fig. 5) while the tokamak field is maximal at the coils.

as a tokamak 8£-value of 50-1002,

in the 5-102 range

The spheromak is closely related to the conventional reversed-field Z-

it corresponds to the particular case of null field-

pinch (RFP) [14): reversal and fairly low aspect r a t i o. The RFP has more shear, and thus is abla to tolerate somewhat higher limiting beta values, but has the drawback of requiring plasma linkage by external toroidal-field coils. Experimentally [15]. Che null-field case of the KFP is found to l ie precisely at the t r a n s i tion point between the “quiescent” state that is achieved with external Bt or t 0 and the turbulent state that prevails for external

3t or > 0.

The f i n i t e - r e s i s t i v i ty MHD kink modes of the spheromak have been studied

for large aspect ratio : s t a b i l i ty is found to be realizable with optimal current profiles and very close-fitting shells [16] — quite comparable Co the case of the conventional RFP. The low-aspect-ratio limit appears to hav; similar resistive i n s t a b i l i ty characteristics [12], but has not yet been treated for optimized profiles. The stabilization of che resistive change mode in the spheromak depends mainly on collisionlessness and modera tion of the beta value [17]. torus, ;he special case of axisymraetric modes [18].)

the resistive MHD analysis has been carried out thus far only for

(For the Bt or * 0 version of the compact

( ( p ^ ) )1’*

inter

It The experimental study of spheromak plasmas began with Aliven [2]- has been resumed recently, by means of the same coaxial-gun technique [19, 20] (51g. 1), as well as by means of a theta-pinch formation process £?ig- 3) that includes toroidal-field generation [21]. References 22-24 describe a new type of “quasi-static” spharomak-fomation technique chat is ainied at avoiding the high pulsed powers associated with a dynamic forming process in plasmas of reactor size-

The degree of gross s t a b i l i ty observed during the limited pulse time of the theta-pinch spheromak experiment [21] has been c :cellent — or even too good, since the ideal MHD theory clearly predicts a t i l t i ng mode [12] for prolate plasmas of the type ‘..f Fig. 3, whether a toroidal field component is present or absent. The antitheoretical s t a b i l i ty against t i l t i ng has also been noted experimentally in the B_ol >> 3c or case [7,3]. Very recently, however, the injection of a gun-produced spheromak plasma into a prolata conducting shell has exhibited the predicted t i l t i ng [20].

This work supported by US Department of Energy Contract So. S Y - 7 6 -O

ACKNOWLEDGMENT

02-3073.

REFERENCES

The suppression of the t i l t i ng node, as well as of higher surface modes,

may be a consequence of the presence of hot plasma at the separatrix (cl. Fig. 3) and jus:, outside i t. An effect might be expected vhen the ion gyro- radii are large or when the field outside che sec^ratrix recaius finite shear, due to the presence of external axial plasma currents or helical windings [12]. External-plasma stabilization could turn out to be very Important for the compact-torus reactor application, since s t a b i l i s a t i on by close-fitting con ducting shells would be inconveniently r e s t r i c t i v e, AS is shown in several reactor studies [23-27], one of the attractions of the compact toi’us is i ts potential a b i l i ty to undergo compression, expansion, or displacement, frae of mechanical constraints.

[I] [21

[4]

[3j

RIDDLE, A. C, Solar Physics 1^ (1970) 448-457. ALFVEN, H., Proc, 2nd Int. Conf. on Peaceful Us.es of Atomic Energy _3_i (1958) 3.

[3] CHRISTOFTLOS, N.. ?roc. 2nd Int. Conf. on Peaceful Uses of Atomic Energy

.32 (1958) 279. COHEN, 3. I., 3RENGL;, T. A., Particle-Fluid Hybrid Simulation of Field Reversal in A Mirror Plasma, this conference.

[5] KOLB, A. C, DOBBIE, C. B., GSIEM, H. R., Phys. Rev. Lett. 3. (1959; 5. [6] KOLB, A., et al., in Plasma Physics and Controlled Nuclear Fusion Re

search (Proc. 3rd Int. Conf., Novosibirsk., 1968) II_ (IAEA, Vienna, 1963) 567.

[7] ES’KOV, A,. 0., et al., in Controlled F u s im and Plasma Physics (?rcc.

7th European Conf., Lausanne, 1975) I_, 55. LINFORD, R. K., and CT Staff, Tho LASL Compact Torus Program, this conference.

[9] LUST, R., SCHLUTER, A., Z. astrophys. 3* (1954) 263. [10] CHAHDRASEKHAS., S., in Proc. of the National Academy of Science? A2 (.1956’.

[II] MORIKAWA, G. K. , et al., Phys. Fluids _l£ (1969) 1643. [12] 3USSAC, M. N., e_t al. , in plasma Physics and Controlled Nuclear Fusion

Research (Proc. 7th Int. Conf., Innsbruck, 1973) III. (IAEA, Vienna, 1979) 249.

[13] GAUTIER, P., et al., in Controlled Fusion and Plasma Physics (Proc. 9th

European Conf., Oxford, 1979) paper EP 30, to be published.

[14] BAKER, D. A., Toroidal Reversed Field-Pinch Experiments, this conference. [15] 0RTOLAK1, S., Some Properties o* the Heating and Confinement in the RF?

Configuration, thi.3 conference.

[161 GLASSER, A., SELBE?>G, H., to be published. {171 3USSAC, M. S., Bull. ;jn. Phys. Soc. .23 (1979) 872. [IS] BERK, H. L., &- <l_. , Tearing-Mode Stability Analysis ar a Cylindrical

Plasma, this contjrence.

[19] TURNER, W. C, et. al., Initial Results of Field Reversed Plasma Gun

[20] JAR30E, T. R., et al., Magnetized Gun Experiments, this conference. [21] GOL0ENBAUM, G. C, et al., Dynamically Formed Spheromak Plasma (?S-1),

[22] OKABAYASHI, M., &t_ al., Spheromak Equilibrium and Stability and Numeri

cal Studies of i Spheromak Formation Scheme, this conference.

[23] YAMADA, Ji., et. al., Design and Fabrication of the S-l Spheromak Device,

this conference. AYDEMIR, A., !t al., Two-Dimensional Simulation of the Formation of the PPPL Spheromak, this conference.

[25] MILEY, G. a., Field-Reversed Configurations: Theoretical Considerations

[26] KATSURAI, M., YAMADA, M., Preliminary Studies of Spheromak Reactors,

[27] SMITH, A. C, J r ., ejt^ a l ., The Moving-Ring Field-Reversed I l i r r or Reactor

[Zi]

t h is conference.

Concept, this conference.

and Reactor Applications, this conference.

experiment, this conference.

this conference.

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Neutral beams

FIGURES

£NS smMB

IBIPir li’.‘j’^ r\ ~\ *---JV^sr c

m§?9*

Fig. 1. Spheromak genera tion, using a coaxial plasma gun with poloidal field at the muzzle.

Neutral beams

rig. 2. Neutral-beam-driven field- reversed mirror machine.

f « E l 0 N I 2 A T I 0N

HtlD HUE ccr.rjECTioa

AXIAL CONTRACTION

S I AS FIELD

duo PLASMA

*^—dUADVZ TUBE

1IIE7A PlllCa c o il

Fig. 4. Tabulation of compact torases .

Fig. 3- Field-reversed tiieta pinch.

Fig. 5. Oblace spheromak. Plasma currenc is localized within the solid-fiald-line region.

(PPL 786431)

Akihiro Mohri, Kazunari Narihara, Yukihiro Tomita Institute of Plasma Physics, Nagoya University Nagoya 4S4, J A P AN

In recent years, the injection of high current

relativistic electron beams into toroidal systetus has been a subject of absorbing interest in connection with compact torus and OH assist startup for large tokamak. Injection methods tried up to the present are surveyed here and a method using Plasma Anode, which is adopted for REB injection into SPAC.is reviewed.

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INJ”£CTION OF RELATIVTSTIC ELECTRON BEAM INTO TOROIDAL SYSTEMS

use a cusp field*

use a guiding field and the drift motion of REB.

The first question we have is whether an electron ring is

suited for confinement of fusion plasma in the practical sense. The electron energy of S-layer of Astron should be so high as the relativist:.c factorc-> lOO in order to produce a strong magnetic field and a sufficient volume for confinement. The* resultant strong synchrotron radiation brings a serious problem concerning the energy loss. However, if we use an appropriate mechanism to keep the ring radius large enough at lower &* in the strong external field for equilibrium, such a problem will not be fatal. When a high current REB ring closely in a force-free state is formed, its major radius can’be controlled with an apf-ied vertical field like Astron-Spherator. We do not need the use of ion-ring in this case. Besides, generator of 1—10 MeV REB is now at the commercial base. When the main toroidal current is generated by REB injec tion from outside, we can neglect any other current-driving equip ment and,thereby, toroidal configurations cf very small aspect ratio are realizable.

The concept mentioned above much depends on the success of

SEB injection into toroidal systems. Methods examined up to the present are summarized as follows;

ASTRON ( L1L }, RECE-BERTA and RECS-CHRISTA ( Cornell Univ. )

inject REB obliquely to the mirror axis and stop the axial motion by dissipation.

Cornell uniy.( Gilad, Kusse, Lockner )

divert toroidal field lines to the cathode of a diode by using self-field of the diode current.

Physics International (Benford, Ecker, Bailey )
Cornell Univ. and Maryland Univ.

all of above methods are for injection into neutral qas. However, adjustment of the initial density of plasma at the time of the injection becomes necessary so as to control thea value of thus confined plasma for stability. In the case of the injection into neutral <?as, the range of the gas pressure should be chosen from the condition of space charge neutrality. Usually, the oressure is 0.1 to 1 torr. For the reason, REB injection into a plasma of appropriate density is required, where ions of plasma easily cancel the space charge of the injected beam elect rons.

the

In order to catch a ring immediately after its formation in

an equilibrium position, we have to suppress the induced return currents which mask the poloidsl field of the ring. Besides, we can not change the vertical field within a time comparable with the pulse duration of injection. These difficult points can be solved if we use Plasma Anode instead of usual foil anode. Figure 1 shows a schematic explanation of the function of plasma anode.

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ct-jatoer vaii

csthsde iilasaa

Fig, 1

A cathode is set inside the toroidal chamber so that its face is directed parallel to the toroidal field. The toroidal and the vertical fields are applied before FEB injection and then the chamber is filled with a plasma. The plasma contacts both the cathode and the chamber wall. When a negative-pulse voltage is applied on the cathode, ions of the plasma are accelerated towards the cathode and bombard the cathode surface where multi-layers of gas molecules are oresent. Thus, a high density plasma is producsc on the surface. There appeals a plasma sheath of double layer type and the electrons of the cathode plasma a~e accelerated in the sheath. The olasma-anode is very superior to the conventional

foil anode in that there is no need to exchange foils and thus a repetitive operation becomes possible. Owing to the thin sheath, fairly high current density is obtainable for REB without causing a pinch, and the beam electrons are ejected in the same direction. The induced returr. current can not flow towards the cathode, and the current is forced to take its path towards the chamber wall along the magnetic field lines. inside the beam channel there exists a strong self-maguetic field due to the beam and the resultant magnetic field becomes helix because of the presense of the toroidal magnetic field. The channel winds round the toroidal major axis. Another possible path of the return current is on the just outside of the beam channel as shown in Fig.l. This return current wraps the beam channel like a thin sleeve and masks the self-field of the beam. Since the skin time of the sleeve is very short, neighboring paths of the beam channel merge themselves as the self-field appears out of the sleeve. Finally, an axially symmetric REB ring with a single magnetic axis is formed. The rise time of the poloidal field is therefore very short. The time is about 150 ns in the case of SPAC-V, as shown in Fig.2. Strong plasma heating could be expected

Fig-

Rises of poloidal fields on the upper and the lower sides of a REB ring in SPAC-V.

pining

M M B U HH WMMMMMWmm

-10-

5OOG uM^MMW.

50 NSEC/D

500G

during this merging phase.

While the plasma-anode is working, the cathode is subject to

ion bombardment. During the time, the material of the cathode is evaporated or sputtered ind may enter the confined plasma region as impurities. Figure 3 pr-i-sants photographs of the cathode surface, taken with an electron-scanning-microscope. the surface like chunk3 are observed. Materials stronger against the ion bombardment should be used such as Mo. In this injection of REB, the cathode plasma is easily produced by ionizing attached molecules on the cathode surface. The density of the molecules is more than 1 014 cm-2 which is sufficient for the purpose. It does not need to use the plasma of the cathode material itse.f*

Damage of

Irradiation power of protons: xQ7 W/cm2, : ‘85 shots, Number of irradiation : 30 J/cm2. Total irradiated energy : SUS-304 Cathode

-11-

50 jj

Fig, 3 Damage of the cathode surface.

t “1

a S D n e r o n a k.

F> °. 0 G ft At’, OUTLINE

The LASL CT P r o g r am

S p n e r o m a ks a nd F R C ‘s f or ’ a c i l i ty

TH E LASL COMPACT TORUS PROGRAM**

ft. X. Linford and CT Staff’”’-, Los Alamos Scientific Laboratory, Los Aiamos. New Mexico 875A5

INTRODUCTION

t h ..

T h u s,

(CT) c o n c e pt

The C o m o a ct T o r us c o n f i g u r a t i o n, w h i ch d o es n ot ” l o le t o r u s- c y l i n d r i c al or s p h e r i c al g e o m e t ry f or Tokar.iaks a nd R F P ’ s. e n g i n e e r i ng a d v a n t a g es f i n e m e nt p r o p e r t i es a f f o r d ed by an a x i s y n s n e tr ic The c r o ss s e c t i on 5 - f i e l ds a nd p l a s ma d i m e n s i o n s. bi

The w e l l - k n o wn A s c r on c o n f i g u r a t i on

r a d i us p. a nd

i on g y ro

r i s i ng

t he

f i n ed S h e r c n s Ki a nd F i e ld R e v e r s ed C o n f i g u r a t i o ns e v e r, 3

v a c u um v e s s el , e t c ., h a ve a s i m c - le

t h r o u gh

p l a s ma t he

r e q u i r ed

s u b s t a n t i al

t he g o od c o n

t h r ee m a j or t o r o i d al

is d e

f i e l d)

t o r o i d al p l a s m a - F i e ld g e o m e t r y. t he

f e a t u r es of

t wo o t h er c l a s s es of C T ’ s.

I ).

t he

The

f i g.

= B

( s ee

to B

/a >

=0 .

-12-

in CT

; . /a <

r e s u l ts

in o r d er

( F i g. 2}

t o r o i d al

., or as

r e t a i n i ng

i n s t e ad of

t h is s t u dy

f o c u s ed on

f or an FRC

in c o n t r a st

t he s t u dy of

is a CT w i th

t h is p a p er 0

t he e s s e n t i al

to d e v e l op v i a b le

(maximum o o l o i d al

t o r o i d al g e o m e t ry

( F R C ’ s ), b o th h a ve

is u n d er c o n s t r u c t i o n.

l i n k i ng of a ny m a t e r i al

T h is s i m p l i f i ed g e o m e t ry

r e a c t or e m b o d i m e n ts w h i le

i n c l u d es a ny a x i s y m m e t r ic

in F i g. 1 of a p r o l a te CT s h o ws

r e q u i re t he m a g n et < - o i l s,

i n to t he m a x i m um l a r ge

C T ‘s c an be c l a s s i f i ed t h^ m a g n i t u de Of

r e a c t or e m b o d i m e n t s. The

t he J. S. D e p a r t n e nt of E n e r g v.

J. S h e r . v o o d, ?. E. G : « - on

t he a u s p i c i es of

: t c K e r . n a,

t wo

f or

How

f i e ld

t y p es

of ” r a in

j e r ^ o r - i ed u n d er

V o rk ’ * * U. T. A r m s t r o n g, R. R. B a r t S C h, ?.. J, C y . H s s o. Z. A. £ k H a t - . ’. S. J.

A. ? i a t t s, A. R. S h e r w o o d, £.

J. H oi d a, T. P.. J a r b o e, J. L i p s o n. J,

” a r s n a l l. J r ..

.<.

< e i \ n s.

t he D h y s i c al p r o p e r t i es

The t y p es of C T ‘s

w i ll be aroduced bv the Magnetized j un (SoheromakJ and c.ie “ftX-C (FRC) . E v e n t u a l l y, b o th sources w i ll be a b le to i n j e ct CT’s i n to t he c e n t r al CTX tank where dc - n i r r or S t a b i l i t y, s t u d i es w i ll be c a r r i ed o ut on these s i n g le c e il c o n f i g u r a t i o n s. S t a b i l i ty and t r a n s p o rt s t u d i es of m u l t i p le c e ll c o n f i g u r a t i o ns w i ll a l so be c a r r i ed o ut in a m o d i f i ed v e r s i on of t he 5 ”> long S c y l la I V -P nheta p i n c h.

CTX

« . W lH * — ™ i » H >? OH

- 7’

-**o»

UUL* * ’ -£

1 3-

3ss

_ _ < —j

IHMMcf’ ’ MM

m -C GKMTIM Mx-c cowntuctiw

they w i ll be trapped in a

f i e ld of up to 10 kG. t r a n s p o r t, and h e a t i ng

r e p e a t a b le o p e r a t i on of t he gun at nuch

f l ux c o n s e r v er w i th no guide

f i ll d e n s i t i es of 3 x 10

r e s u l ts and f u t u re plans

f i e ld of the Spheromak.

in rnese proceedings

forms t he p o l o i d al

in the gun b a r r el

to 2 x 10

f i e ld .

I n i t i al

’ ’ ’

lower

f i e ld

than have

niarr

In a r e o a r a t i on f or these e x p e r i

ments, Spneronaks a re b e i ng produced in f a c i ’ i ty and FRC’s a re a Gun P-otocype being s t u d i ed in two I m long cheta ai-ich s v s t e n s, FSX-A and FRX-8. F i g u re 3 shows che time s c a l es f or these e x p e r i m e n t s. As i n d i c a t e d, t he Gun w i ll be t he f i r st p i- sma source the CTX t a n k. The FR.X-C w i ll be o p e r a t ed as a s e p a r a te At the t r a n s l a t i on of t he FRC plasma planned experiments on FRX-B systems aaper is devoted to a b r i ef s t a t us Companion papers mental and t h e o r e t i c al

l a t e r. The remainder of Che r e p o r t . on t he Gun, CTx , and FRX e x a e r i m e n t s. d e s c r i be in more d e t a il t he e x p e r i f or these systems.

f i s c al y e a r, the t r a n s i t i on s e c t i on w i ll be added to a l l ow i n to t ht CTX t a n k. The s i g n i f i c a n ce of the

f i g u re 3. CT Program

t h e . a nd of t h at

f a c i l i ty d u r i ng FY 3 7,

is d i s c u s s ed

t e s t ed in

Plan

SJN PROTOTYPE

A 70 cm long Marshal 1-type c o a x i al plasma gun has been c o n s t r u c t ed w i th

f i e ld a c r o ss Che muzzle of t he gun, which is s t r e t c h ed by the emerging

r a d ii of 15 cm and 10 cm. Slow r i s e t i ne magnet c o i ls have been p l a c ed These c o i ls nraduce

e l e c t r o de i n s i de t he i n n er e l e c t r o de and o u t s i de the o u t er e l e c t r o d e. a r a d i al plasna and e v e n t u a Mv these c o i ls produce a b i as which a l l o ws o t h e r w i se p o s s i b l e. been used s u c c e s s f u l ly w i th bank e n e r g i es of about 30 k J. Magnetic i n d i c a te diameter s t a i n l e ss s t e el

t h at Spheromaks nave been

formed by t he gun and stopped in a 45 cm

cm f i e ld orooes

(between the e l e c t r o d e s)

f i ll pressures

ir. a d d i t i o n,

The -»ain CTX tank is i n s t a l l ed and has been pumped down to I x 10

/»pth one of the t h r ee c r yo punos . The dc magnets have a l so been i n s t a l l ed b ut ars n ot y et connected to the power s u o p l y. The 10 kv ban* f or the gun -nagnets and the 60 kV main gun bank a re in olace a w a i t i ng power SUDDIV and c o n t r ol s ‘S tern

ins ta I I at i o n.

FRX SYSTEMS

The

research c a n n ot be summarized h e r e. However, d i s c u s s e d, which

n o t i v a t i on

i m p o r t a nt FRC issues a re

from t he FRC

r e s u l ts and a l so p r o v i de some of

’ !) Gross S t a b i l i ty

The oaserved g r o ss s t a b i l i ty

the absence of _oro>dal t h e re

t r a n s it of a * 0, and i n c l u de R/a) < i e ld r e s u l ts

than 100 A l f v en

t h at as a

t he s a f e ty

i d e al MHD c a l c u l a t i o n s, low aspect

the observed gross s t a b i l i t y.

the FRC is u s u a l ly “node number) mode. The e x c e o t i o ns are

t e r m i n a t ed bv a

few n o n - r e p e a t a b Is shots on FRX-B. A l t h o u gh

in c o n t r a st

tc r o t a t i o n a l ly d ri ven^n = 2 ( t o r o i d al K u r t m u l t a ev o b s e r v a t i o ns ars not c o m p l e t e ly u n d e r s t o o d, F o l l o w i ng quency 7:. magnetic is dominated by p a r t i c le geometry, i m D o r t a nt Tent of t h at aDOut h a lf

l a r g e ly c o n s i s t e nt w i th

ion r o t a t i o n al

f r e ion d i a-

the

t h is

t h e re are . o t h er e f f e c ts a s s o c i a t ed w i th The most the eqer’gy c o n f i n e-

, p r e d i c ts

the mode g o ei u n s t a b l e.

f2)

The t r a n s p o r t, which t u r b u l a n ce does n ot appear

1H0 decreases a F i n i t e - L a r m o r - R a d i us l i f e t i m e. However, ! - ! / <* 0 t r a n s p o rt code w ’ £h . t r a n s p o rt c o e f f i c i e nt the

to be anomalous.

f i x ed R and

increases observed r e s u l ts of a

i n s t a p i l i ty d o m i n a t i ng

There a re s e v e r al

i m p o r t a nt oecause A / P. s t r o n g ly a f f e c ts

r e s u lt v e ry is c o n t r o l l ed by d u c t i ng w a l l. and most n ot o n ly because of w a ll s t a b i l i z a t i o n, but because p o r t. the r o t a t i o n al mode

As * consequence,

t r a n s p o rt s c a l i ng p a r a m e t e r s- V a r i a t i o ns l t u r n. 2

l e n g th In

the r a d ’ us of is

the

the c o n l a r g e st f a v o r a b le

I) are in s l o w er

is decreased and the o n s et of

A m a j or purpose of

p r o d u c i ng p r o d u c t i on of p i n ch d u r i ng

t e c h n i q u es

l i f e t i m e.

The the cheta f l ux w i ll be

f or

1 4-

the

t h is

f i e ld

t h r ee

and a

r e s u l ts

they a re

f or more

is s t a b le

in F i g. 3-

f r e q u e n c y.

. The mode

t h r e s h o ld of

l o ss a l t h o u gh

The s o u r ce of

the e x p e r i m e n ts

i n c l u de some of

r e s u l ts the

i d e ai MHD s t a b i l i t y,

the FRC, t he S is h i g h,

t h at are c o n s i s t e nt w i th

( h i gh e l o n g a t i on b / a, and

the p r o p er geometry e f f e c ts

t h e o r e t i c al is k e pt below

is no s h e a r. However, d e t a i l ed

r o t a t i o n, w h i ch d r i v es ?.. beyond

these the e x p e r i m e n ts o u t l i n ed

if , ; . / ?* » 1 . 4, where ?.* is

l a r ge body of e x p e r i m e n t al and t h e o r e t i c al d a ta a v a i l a b le

in t i m e s ’ a p p e a rs at odds w i th s i m p le MHO e s t i m a t e s. Note

f l ux a n n i h i l a t i o n, and e l e c t r o n - i on energy e q u i l i b r a t i o n. r e s u l t, however, ’ is

l i f e t i me appears i m p o r t a nt because d e c r e a s i ng (FLrl) s t a b i l i z a t i on but

i m p o r t a nt the usual s u r f a c e - t o - v o l u me e f f e c t s.

the LHD * ’ ’ ” i n s n o r t. is .— a / : ., which

the planned FRX e x p e r i m e n ts thus e x t e n d i ng /r

is s u p p o r t ed bv e x p e r i m e n ts t r a n s p o rt b e f o re

the o b s e r v ed s c a l i ng the

;

f at plasmas

(r f at plasmas f i e ld

the d e n s i ty n, R, and r It

f a v o r a b le v a l ue p o s s i b l e.

the a n g u l ar a c c e l e r a t i on

is the - l a s na

t r a p p i ng of more b i as

l o w e r - h y D ri d - dr i f t

is c o n s i s t e nt w i th

T r a n s p o rt S c a l i ng

r e q u i r es

The d e n s i ty s c a le

the mode does n ot

‘n p a r t i c u l a r, as

res’ul :’

Two techniques

the FRC. This

t r a o p i ng more

r e s u l t, which

r /r — 1 , 1 /

is a e l a y e d.

f at plasmas

the plasma

to examine

the e.iergy

r e v e r s a l.

where r

l o st by

I) and

l i m i ts

I’LHD)

l i m it

Thus

( r ’ /r

t h at

f l ux

the

f or

the

r a t io

these

w h i ch

l i m i t,

r e s u lt

f a c t or

in R is

t r a n s

f or

/ ;.

t e s t e d. .vi c.1 an octacc’e o a r r i er f i r st on n i ll be t r i ed to r e st o e i ^o m o d i f i ed If < u r : m u l ’ a e v. r”iea a c c o r d i n g l y.

the Bias

t e c h n i q ue is

The FRX-8

t e c h n i q u es of

r e v e r s a l, FRX-C w i ll oe Tiodi-

The M u l t i p le C e ll e x p e r i m e nt

in e f f e c ts of open f i e ld

the m u l t i p le c e ll geometry- i n c r e a s i ng f i e ld

i l i n e s.

t e st In p a r t i c u l a r, a study w i ll be made of

t r a n s p o rt and S t a b i l i ty

t he

/ p. by m u l t i p l e - m i r r or c o n f i n e m e nt of plasms on

(3)

T r a n s l a t i on and

t ie dc

in f i e ld of seen demonstrated I n s t e ad of a s t a t ic This t e c h n i q ue scnene on FRX-A (see F i g. 3 ), t i on and of CTX

trie necessary data

in FY 32.

REFERENCES

to tie s t u d i ed

is t r a n s l a t i on have a l r e a dy to use a p u f f ed gas i n to a vacuum.

it the FRC can be in c o n j u n c t i on w i th a q u a d r a p o le p r e i o n i z a t i on t r a n s l a

the

t r a p p i ng e x p e r i m e nt on FRX-3. These e x p e r i m e n ts s h o u ld produce a ll

r r a n s l a c i on of FRX-C plasmas

inco Che

T. R. Jarooe e_t j_L. “Magnetized Gun Experiments,” these proceedings.
A. T. Armstrong <jt a_., “Field Reversal Experiments, and FRX-A and
R. 5. Sietnon and LASL Compact Torus Staff, “FRX-C and Multiple Cell

’-. 3. C. Barnes ei a I *. ‘Compact Torus Theory—,4HD Equilibrium and

FRX-3 Results,” these proceedings

Experiments,” these proceedings.

Stability,” these proceedings.

A. u. Es’kov si a_i., “Principles of Pla-na Heating and Confinement in a ;omoact Toroidal Configuration,” Plasma Phys. and Contr, M u d. Fusion Research, Innsoruck, Vol. II t1373) 187.
S. Hamasaki ana R. K. Linford, Bull. Am. Phys. Soc. IU (1575J 1081.

f or

the

f a st

-1.5 -

r e v e r s al

The f a st

t r a p p i ng

rlowever,

i m p o r t a nt

t r a n s l a t ed

is designed

i n c o r p o r a t ed

’ ’ I ’* r e v e r s i ng

the s u c c e s s f ul

the FRC plasma

The r e s u l ts w i ll be

f i e ld d u r i ng r e v e r s a l . ”

t r a p p i ng a re necessary

f i e ld and r e c o n n e ct i on

i ts 250 ‘<V loop v o l t a g e.

T r a n s l a t i on and T r a p p i ng

f i e ld more r a p i d ly ,-arn; (2)

The f o r m a t i on and is

the FRX-C u s i ng the b a r r i er these prove s u p e r i or

the CTX t a n k. in FRX-A. t h at f i ll so is b e i ng s t u d i ed

the

f i ll

f l ux

INITIAL RSSOLTS OF FTtLD REVISS2D FLASMA 3’JN E.T7E5IMENT*

V. C. Turner, C. W. Hartaan, and J, Taska Lawrence Liveraore Laboratory, Livermore, California 94550

A. C. Smith, Jr. Pacific Gas and Slactric Company, San Francisco, California 94106

‘*e have begun experiments using a magnetized co-axial plasma gun to

produce field reversed piasaa>^‘2) Beta II facility at Livermore, The experimental program consists of three stages: (1) formation, (2) translation and magnetic mirror trapping and (3) neutral beats heating of field reversed plasma. Experiments to date pertain to the first stage and have demonstrated production of field reversed plasma.

jj,e ^ j, iajeccs axially into the

The experimental apparatus is shown in Figure 1. The co-axial gun

electrodes are i.5 a long. The diameter of Che inner (outer) electrode is .15 ra (.30 m ). The electrodes are fitted with solenoid coils that, together with the guide coils of Che 3eta II device, form a magnetic cusp at ths gun muzzle. The inner electrode solenoid provides magnetic flux that is opposite the guide field (field reversed flux). The outer electrode solenoid is used to control the bias field between electrodes.

Diagnostics include an array of magnetic loop probes and a Cu

calorimeter. The external magnetic loops consist of four single turn diamagnetic flux loops, two Sogowski belts and a loop that measures the azimuthally averaged component of radial magnetic fieic at one axial location. An axial cagnecic probe can be scanned vertically through the interior of the plasma and contains small loop probes to measure local components of magnetic field. The magnetic probes shown in Figure i ara center tapped for.differential readout, stainless steel jacketed, and compensated with passive XC networks. The probes are calibrated with pulsed solenoid and Helmhoitz coils.

The data in this paper were obtained by choosing a guide field

strength 4.8 kG, a net inner solenoid flux 0.90 kG - cm-, a plenum gas fill of 50 ata - cm3 Di’3”*1 c h en scanning the gun bank charge voltage (VG) and bias field between electrodes (3i,ias) for production of fieid reversal. These values turned out to be VQ » 35 kV and 3;,ias * 2.25 kG in the same direction as the guiie field, opposite the field inside the inner electrode.

to find optiauo vaiuas

Gun discharge voltage and current are shown in Figure 2. The g-jn

current peaks at 940 kA, producing a magnetic field of 25 kG at the surf e of the -.nner electrode that is accelerating plasma out of the gun. The .ime integral of voltage and current shown at Che bottom of Figure 2 gives 130 kj energy input to the gun terminals.

Time of flight data from the four diamagnetic flux loops show an

initial plasma front velocity in the range 82 to 130 x ICcm/sec, filling the region between ths gun and calorimeter with plasma in approximately 2 microseconds. Thereafter signals on “he diamagnetic loops persist for about 25 usee. For the conditions described above the Sogowski belts meas… - isro net axial current dra-.-n from the gun to the calorimeter and returning ir. the vacuum chamber walls. Figure

i shows an axial magnetic probe signal giving

L6-

*Work performed by ILL for USDOE under contract V-7405-tng-43.

a peak change in axial magnetic field strength £jiz * 18.9 kG and field reversal f acxor _3,.‘3Q - 4.2 in the 4.3 kC guide field. The hoc lata half of Figure 3 shows separately signals from each half of the center zapped probe. The two components are equal in magnitude, opposite in sign, verifying their magnetic origin. This verification is equivalent to flipping probe orientation or changing sign of all magnetic fields and observing opposite polarity probe signals. Electrostatic signals would have the same sign for each half of the probe and obviously have been reduced to negligible proportions by shielding and isolation. Typically a time integrated energy deposition of 15 kJ is recorded on the calorimeter.

A radial .«can of the axial component of magnetic field strength is

shown in figure 4. Data are plotted on successive shots, moving the probe between shots. Each data point plotted is a five microsecond average over the peak of the recorded waveform. A multi-channel radial probe has besn constructed ;o verify the profile in Figure 4 on a single shoe but no data have been taken yet with chis probe. Several points can be made from Figure 4. First, a field reversed region with Bz * -10 leG, corresponding to J3z/3() * 3.1, extend out to a radius 0,10 m. Second the radius of the axial field null is 0.15 m. Third, the magnetic flux inside Che null is 4.9 x lfl3 ^c - electrode. A flux amplification effect has been noted earlier by Alfven and his eo-uor’ers. <- ><3) Fourth, the radius enclosing :ero flax is 0.23 o. Fifth, the radial profile in Figure 4 conserves the vacuum flux out co the wall at 0.75 m. Sixth, the axial magnetic field energy inside the separatrix is ;8 fcJ/a. Equilibrium of che profile in Figure 4 requires existence or a ceroids 1 field component since the axial field pressure inside exceeds that outside the plasma by a factor of four. Toroidal field components have been observed but a radial profile verifying the eauilibrium has not been obtained yet.

c mit , factor 5.4 times the net flux inside the inner

A scan, of .13Z on axis versus net flux in the inner gun electrode is

shown in Figure 5. For negative values of net fiux che inner electrode solenoid is not pulsed strongly enough co reverse the guide field flux, and the plasma produced does not reverse che guide field. Oace :he net flux is positive large field reversals are readily produced, but if che net flux is too large, field reversed plasma does noc emerge from che gun.

-17-

This reporl *34 prepared is in iccounl ol’ work sponsored by the United Sulci Government. Sinner :he United Sulci not ihe Untied Suies Department ot Energy. not iny ol :neir employees, nor iny *-’ :.leu :onr:ac:ors. uioiortractorv 31 :!ieii employees, milees w> *arrjnTv express jt implied, jr assumes inv -essi !upiJir. or responsibility ‘^t :r.t icr-jiacy. in!jrmation. apparatus prouuc: n froucn jiwiuwa. 31 .eprdiirirt :!ul :s me *ouid not inirincc pnvaie!y«su.nea rieim.

Refcience to 1 ;on-,p.un sr produc rat Joes not ‘.mpis jpntovjl it recommend-ation si :he ^rjduL’ ?i IK ’.‘m.er«:> 01 Cmtornu ot :he i. J Department 01 Enercy 10 :<ie J\wiu«ull M .“inert ir.ji may ac suiuoie.

.omplerertcs. Jr ,*»ertiinest v

MOTICI

jny

(2) 3. Alfven, L. Lindberg, ?. Mitlid, J. Nuel. Energy, l, 116 C1960). (3) L. Lindberg, C. Jacobsen, Astro. J., 133, 1043 <196l7.

(1) H. Alfven, ?roc. Second Int’l. Conf. on Peaceful Uses of Atomic Energy,

21, 145^(1958).

TYPICAL MAGNETIZED PLASMA GUM VOLTAGE AND CURRENT

1—

-M

LC=L:ZI|

Plitwia IMMJIIH lo luavu

crm

.j,—^-jrfe-^

fitp i—te^i

o OT m H C *o

> c z

> a z o

Tunc, miciosecnmJs

TYPICAL FK’LD REVERSAL StGNAL FROM BALANCED MAGNETIC PROBE

RADIAL SCAN OF A X I AL MAGNETIC FIELD

iai 0 < r < C 2 5 em

Vacuum field

/ / / / t J m / *

R, cm

(bl 20 < r < 50 cm j

Flux conserving wall —

Radius of diamagnelic loops

R, CD!

Figure 4

A-B

^ - V ”^ A only it». B only i — —.

*»

-5

I .,

5.5

-10

-15

i **!----

i 9-

—

„ - » » M Mm

_ Of-

Figure 3

10/9/79 Shot 25

11/1/79, ^^i!2.^79

p—e—sr

,. l: ii ^ Sensitivity « 6.5 KG/voit Vacuum B0 * 4.5 kG

A X I AL MAGNETIC FIELD PROSE SIGNAL VERSUS PLASMA GUN INNEFI ELECTRODE FLUX

Guide field = 4.8 k<3 130 kj input to gun terminals

^^ —

s- S,K - 4.8 kG

, Bias field * 1.5 — 2.3 kG between gun electrodes

Inner electrode flux, kC — c m2

1 5 2 0 2 5 30

AS L

Figure 5

11/6/79 data

-i x 103

2X 103

; x I O3

/ *

5 -

- - -j

”

at R * 75 cm

4 5 50

EXPERIMENT ON PLASMA CONFINEMENT BY INTENSE RELATIVISTIC ELECTION BEAM RING

Y. Tomita, K. Narihara, T. Tsuzuki, M. Hasegawa, K. Ikuta, and A. Mohri, Institute of Plasma Physics, Nagoya University, Nagcya 464, Japan

INTRODUCTION

Toroidal magnetic configurations formed by injection of intense relativistic electron beans (REB) have many benefits for plasma confinement. The formation time is so short that the state can pass over to a final stable one, quickly going through dangerous, unstable regions. When a beam is injected into a preformed plasma, the plasma is effectively heated in a short time. Since no equipment to generate toroidal currents is necessary around the major axis, compact toroidal configurations are realizable.

An experimental program named SPAC has been conducted to inves tigate the possibilities mentioned above [1-3]. In SPAC-V, REB rings with a current of 30 kA and milliseconds life were success fully formed using a new injection method in which plasma acts as anode and no anode foil is needed. Encouraged by the results from the SPAC-V experiments, we have constructed a scale-up toroidal device SPAC-VI which was designed aiming at longer life and higher current of REB rings. In the new toroidal device SPAC-VI, the strong adiabatic compression is easily applicable, when the beam electrons and the plasma are both energized.

In the next section (Section 2) a summary of the results of the SPAC-V experiment is given. In Section 3 the design and experi mental results from SPAC-VI are presented.

SPAC-V EXPERIMENT

Experimental setup is schematic*_ly shown in Fig. 1. The end of the magnetically insulated transmission line (MITL), which is connected to pulsed high voltage generators PHOEBUS II or III, is inserted inside the vacuum vessel where the cathode is set with the surface towards the toroidal direction. After applying the magnetic fields, the vacuum vessel is filled with a prepared plasma. The plasma contacts with the surfaces of the cathode and the vessel. On the application of the voltage, a plasma sheath forms in front of the cathode. Across this sheath, elec trons are accelerated to the applied voltage and enter into the plasma. This method of beam injection, which we call “Plasna- Anode Method,” is superior to the conventional foil anode in the following points: (I) There is no need for exchanging foils and highly repetitive operation is possible. (2) Electrons are

-20-

scarcely momentum s c a t t e r ed by the anode m a t e r i a l, changing the plasma density and the area of cathode, impedance is e a s i ly changed- enables f a i r ly high current density beam emission without causing a pinch. the cathode was 4-6 cm in diameter and the plasp’. density around the cathode was l O ^ - l O11 cm” !.

(4) The small thickness of sheath

In the experiment,

(3) By the diode

Features of the REB rings thus formed depend l a r g e ly on the ex t e r n al magnetic f i e ld s t r e n g t hs and on the plasma density before REB i n j e c t i o n. When REB are injected a r i ng current as high as 150 kA was obtained. The however, was as short as 20 microseconds.

i n to the denser plasma, l i f e t i m e,

IR,

The major radius decreased in the r i s i ng phase of

and unen the radius was nearly constant for about 1 msec. the f i n al

The lifetime of the REB ring has been prolonged up to 1.6 msec. Figure 2(a) presents time dependences of poloidal magnetic f i e l ds at the center Bp^ and the major radius of the ring Rp. The ring current reached 42 kA 300 nsec a f t er the i n j e c t i o n, and it decayed f a i r ly quickly for 200 microseconds. Then, the current decayed slowly u n t il a f i n al occurred Bv A t r a in of stepwise damps of current were observed during f a st decay of the ring current in the e a r ly s t a g e. At the large d i s r u p t i o n, region and a p o s i t i ve spike appeared in the poloidal f i e ld at the center. These facts suggest t h at ‘the collapse of the ring vas accompanied with the f a st shrinkage of the major r a d i u s. Figure 2(b) presents was adjusted at a f a s t er t h at the l i fe of the ring is determined by the d-scay of t o r o i d al magnetic

f i e ld In t h is case, d i s r u p t i on occurred

time than in the previous c a s e. This fact suggests

intense x-ray flash emitted from the c e n t r al

the case where the t o r o i d al magnetic

large d i s r u p t i on

The v e r t i c al magnetic f i e ld necessary for the equilibrium

2 1-

the ring c u r r e nt

is the e f f e c t i ve poloidal beta which

is the minor radius of the r i n g, 1^

a, we can estimate 3p eff”.

i* ’ IA = Alfven c u r r e n t.

to decay f a s t e r.

the i n t e r n al

fY2- D1 /2

f i e l d s.

the

where ductance, and 30 e?f cludes the centrifugal force e f f e ct of the c i r c u l a t i ng e l e c t r o n s. From the experimentally measured values of Bv, iR, Rp, and the In the roughly estimated value of

i n

3p a ff

case of Fig. 2(a), 3o e££ increased from 0.9 at t = 0.4 msec to l.S at t = 1.2 mSecT This increment could be caused by the heating of plasma due to beam plasma interaction and/or the ac celeration of the beam electrons embedded in the plasma.

From a micro-wave interfero.-r.etry , the averaged density of the confined plasma was estimated to range around several times ,n13 - 3 .

From the Doppler broadening of impurity lines ion temperature was estimated to be about 100 ev at the early phase of the ring current evolution.

3 a

-22-

10 cm

tts paloidal r-jq-ietic r i e iJ. ‘3 1. and the T.ajor

net current fl J.

radius (flJ

» <» vi

Fig—

rig. i. sefit«*eic dijr:*3 th« t s r « i d *l d a v l ca SPAC-V.

a£

. Tina variations of the external eiaenetic f i e l 4 - ( Dv, 3c) «id th«

in eh« u ae of <a> slower decay and (b» faster decay of Bf c.

3. Design and Experimental Results of SPAC-VI

A new ioroiial davice S?AC-VI was ccnsCruc^ad contiguously

from previous experiments * SPAC-V. This new toroidal device was designed with aiming ac plasma confinement by REB rings of longer life and higher current and stronger major radial adia- batic compression. For this purpose,the vacuum chamber was made larger and vertical magnetic field coils were distributed radially.

Figure 3 shows the schematic structure of SPAC-VI. The

toroidal field Bt is generated by the current(1.2MA maximum) in a single center conductor sat along the major axis of the torus and its flat top time within ±7% change is 20msec. The verti cal magnatic field coils are installed to keep the decay index less than 1.1 for stabilizing the positional instability of RGB rings extending the whole space of plasma confiment and a maxi mum vertical magnetic field” is generated 12kG. For the purpose

-23-

Schematic structure of SPAC-VI

Fig- 3

of major radial compression of plasma, the wave form a£ the vertical magnetic field Bv is able to be adjusted. The vacuum ’.“assal fsr plasma car.fir.er.ent is mada of thin stainless steel with a shell effect. This vessel 1.28m in outer diameter does not have a flat lid aiming at adiabatic compression of confined plasma. In this’experiment, for RES injection “The Plasma Anode Method” was adopted to form KE3 rings. Pulse of negatively high voltage delivered from a Maxx generator PHOEBUS-HI (Pulserad-445Wr 1.8£lV,37QkA,5QkJ) is sent to a cathode through a magnetically insulated transmission line(MITL). This cathode is inserted inside the vacuura vessel and is set with the sur face towards the toroidal direction. The MITL can be operated in ultra high vacuum and the transmission efficiency is more than 3G% at the pulse strength of 1.6MV and 190S«A of SOnsec du ration. A partialis ionized cold hydrogen plasma, which is prepared by a coaxial plasma gun, serves for the plasiaa-anode.

The experiment of plasma confinement by REB rings using

SPAC-VT and PH0E3US-:n (Pulserad-445W) was firstly carried out to have a long lifetime. The external magnetic fields of SPAC- VI can be maintained ten times as long as in the case of SPAC- V where the lifetime of H£B rings had been determined by that of the external magnetic fields. In a high q mode operation (q>2), the ring current IR continued for 10msec which was ten The peak ring tines the previous record”of SPAC-V{Fig_4) .

-26-

Wave form of a REB ring current

Fig. 4

current was 4 3kA and the major radius of REB rings, which was decided from measurement of poloidal field near the wall using magnetic probes, was 23cm, where the toroidal magnetic field B*- was 5.4kG. Irthis case, the slowest decay time [dij/(lR-dtl ]”’” was 50msec. The injected RE3 had the particle energy of -t.lMeV and the current of T-SOJCA. However, highly repetitive short bursts of boch x-rays and impurity lines appeared during the decay of the ring current. Origin of the phenomena is now in full pursuit.

The major subject of this experiment using the toroidal device SPAC-VI is to form long-lived REB rings with very low q-value and to apply a strong adiabatic compression on them, because this would lead to a compact torus reactor- Compression ratio of 2.5 with respect to the major radius is possible in SPAC-VI. Experimental results in these operations will be reported. Diagnostic instruments which can work in strong x-ray environ ments are under commission, so that detailed parameters of con fined plasma will be found out.

REFERENCES

[1! MOHRI, A., MASUZAKI, M., HARIHARA, K.r HAMANAKA, K.,

IKCTA, K., in Plasma Physics and Controlled Nuclear Fusion Research (Proceedings 6th International Conference, Berchtesgaden, 1976) Vol. ITI, IAEA, Vienna (1977) 395.

[2] MOHRI, A., SARIIIARA, K., TSUZUXI, T-, KKBOTA, Y., TOMITA, Y., IKUTA, K., MASUZAKI, M., in Plasma Physics and Con trolled Nuclear Fusion Research (Proceedings 7th Inter national Conference, Innsbruck, 1978) Vol. Ill, IAEA, Vienna (19 79) 311.

[3] NARIHARA, K., TOMITA, Y., TSUZUKI, T.-, MOHSI, A., in Pro

ceedings 3rd International Topical Conference on High- Power Elect, and Ion Beam Research and Technology (Novosibirsk, 1979) to be published.

-25-

INTENSE RELATIVISTIC ELECTRON BEAMS IN TOROIDAL MAGNETIC GEOMETRIES

V. 3ai1ey, J. Benford, R. Cooper, B. Ecker, and H. Helava, Physics International Company, San Leandro, CA. 34577

in toroidal maintaining the current in a steady-state t o r o i d al r e a c t o r , ’ ”’ ing the confining magnetic f i e l ds for the p l a s m a ,3”6 (3) rapidly heating the plasma to i g n i t i on conditions,7 (4) providing s t a r t - up plasma heating and current maintenance f or a steady-state, compact, high $ t o r u s . 8 ,9

(2) produc

functions (1) s t a r t i ng up and/or

to the t o r o i d al magnetic f i e ld and loss

trapped in the toroidal plasma column by d r i f t - i n j e c t i o n, energy trapping J0” This technique, which has been demonstrated e x p e r i m e n t a l l y ,^ allows m u l t i - t u rn i n j e c t i o n, whereby the i n j e c t i on time many times the single t r a n s it been shown experimentally to be e f f i c i e n t. This method uses t o r o i d a l- plus-betatron-type f i e ld geometry with trapping r e s u l t i ng from p a r t i al beam energy loss by self-magnetic f i e ld generation and/or by heating of the plasma.

time of electrons around the t o r u s, and has

The increase in the net toroidal current causad by the REB is

to the inductance per u n it length of the beam-

an increasing function of the k i n e t ic energy of the injected REB and is inversely proportional plasma system. The predicted scaling of the increase in net t o r o i d al current compares well with recent experimental data. The time scale t h is increase is s i g n i f i c a n t ly shorter than the normal L/R decay time of the plasma. The f r a c t i on of the injected REB energy that is converted to poloidal magnetic f i e ld energy or plasma thermal energy can be adjusted by charging the parameters of the injected beam. This allows a great deal of f l e x i b i l i ty in using the RES f or plasma heating and c u r r e n t - d r i v e.

i n i t i a l ly

The REB is injected p a r a l l el

-26-

fusion devices. Among these are:

Intense r e l a t i v i s t ic beams (REBs} can perform important

in a s t e c j y - s t a te

for

toroidal reactor is only s l i g h t ly

The average ” 3 power level required to maintain the t o r o i d al larger less than

current than the ohmica1ly dissipated power and is s i g n i f i c a n t ly that required by rf current d r i v e. For a t y p i c al r a t io of the t o r o i d al current to the average REB power required to maintain t h at current is 10 A/W. The generally accepted value of r a t io f or rf current drive is 0.1 A/W.T4 having an average power of 1.5 MM could maintain a t o r o i d al current of 15.6 MA in a tokamak reactor, while 150 WW would be reauired if rf current drive were used. beam energy is converted to increased poloidal magnetic f i e ld energy. The higher e f f i c i e n cy of the RE3 current drive permits the larger reactor

In t h is example approximately SO percent of the

i n j e c t i on of a 6.6 MV REB

tokamak reactor the

injected

t h is

L’sing the high-beta equilibrium of Peng and Oory, me have selected

a r-ppact steady-state toroidal reactor driven by fleas.8 The reactor has a low aspect ratio [2.0), high beta (0.18), a 1.6 m major radius, ana 250 MWe net outnut. It is started with 225 oulses of a 1 MV, 0.4 MA, 2 -JS beam at 100 Hz, and current is sustained b\ the same beam operatinq at 0.33 Hz.

Q-value (Qmax = fusion power/current drive power) for a steady-state reactor.

StabTe beta of tofcamaks can be increased by either Towering the aspect ratio A or by intensive heating during the formative stages of the discharge to control the growth of the ballooning modes.15>>6 Jassbyl? used the small-aspect-ratio technique in his design for “SMARTOR” and PITR, and Peng9 achieved sim’lar results by compressing a large-aspect-ratio plasma. The intense heating technique is used at Columbia in TORUS II and at Nagoya in SPAC-V to obtain a high-beta (> 10 percent) stable equilibrium. These high-beta configurations may best be realized by repetitive injection and trapping of a REB in a torus. Starting with a cold plasma and no toroidal current, a pulse sequence can be defined to bring the reactor to ignition at constant poloidal beta, without ohmic heating coils. In principle, this beta can be chosen large enough to be in the stable regine for the balloon ing modes,

-27-

REFERENCES

K. Ikuta, Jap. J. Appl. Phys., M_, 1684 (1972).

‘1. Bailey, Steady-State Current Drive in Tokamaks, woriunop Summary, DOE/ET-0077, National Technical Information Service (NTIS), Springfield, Virginia (February 1979).

1. voshikawa, Phys. Rev. Letters, 26, 295 (1971).
S. Yoshikawa and H. Christofilos, Proceedings of the Fourth

International Conference on Plasma Physics and Controlled Nuclear Fusion Research, vlol. 1, p. 357 held in Madison, Wisconsin (1971); published by International Atomic Energy Aoency, Vienna, Austria (1972).

C. h. Hartman, Phys. Rev. Letters, 267_, 14, 826 (1971).
S. Yoshikawa, 3ull. Am. Phys. Socl, p. 996 (November 1971).
1. rammer ar.c K. Paoadoooulos, Nuclear Fusion, 1_S_, 977 (19”5).
V. Bailey, R. Cooper, and T.S.T. Young, Third International Topical Conference on High Power Electron and Ion 3eam Research and Tech nology, Novosibirsk, USSR (July 3-6, 1979).

9. Y-K. M. Peng and R. A. Dory, ORNL/TM-6535, Oak Ridge National

J. Benford. 3. Ecker, and V. Bailey, Phys. Rev. L e t t e r s, 3X 574 (1974).

M. Masuzaki el a l . ., Jap. J. Appl. Phys., H_, 1413 (1975).
V. Bailey, Proceedings of the International Conference on Synchrotron

Radiation and Runaway Electrons in Tokamaks, held in College Park, Maryland (1977).

A. Mohri si a i -, Proceedings of the F i f th

International Conference

on Plasma Physics and Controlled Fusion Research, IAEA-CN-37-X-5, held in Innsbruck, Austria (1978).

A. Bers, Steady-State Current Drive in Tokamaks, Workshop Summary,

Information Service (NTIS), S p r i n g f i e l d,

-28-

DOE/ET-0077, National Technical V i r g i n ia (February 1979)

Laboratory (October 1978).

REFERENCES (Cont.)

Princeton Plasma Physics Laboratory.

B. Coppi H a l . ., Nucl. Fusion J i, 715 (1979).
H. Strauss Laboratory.

j_t a j . ., PPPL-1535 (1979), Princeton Plasma Physics

D. L. Jassby et .a]_., PPPL-1371 (1977) and PPPL-1453 (1978),

C O M P A CT T O R US R E S E A R CH AT U . C . I.

A. F i s h e r, S, R o b e r t s o n, and S. aostoker D e p a r t m e nt of P h y s i c s, U n i v e r s i ty of C a l i f o r n i a, I r v i n e, C a l i f o r n ia 9 2 7 17

Compact t o r us r e s e a r ch at C . C . I, c o n s i s ts at s e v e r al e x p e r i m e n t al a nd

t h e o r e t i c al p r o g r a ms i n v o l v i ng intense e l e c t r on a nd i on b e a m s, f i e l d - r e v e r s ed m i r r o r s, and t o r o i d al c o n f i n e m e nt s y s t e m s. T h e se p r o g r a ms a re b a s ed u p on an e a r l i er p r o g r am of r e s e a r ch into field r e v e r s al a nd h e a t i ng w i th i n c e n se electron beams.1

Spheromak Generation by Beam Induced Currents.2

In this experiment a closed, field-reversed magnetic configuration Is

created by return currents induced in initially neutral gas by a rotating relativistie electron beam. The electron beam (1 MV, 100 kA, 50 nsec) is xade to rotate by locating the cathode at the end of a helically wound supporting shank which generates a cusp-like field only during t..e beam pulse. Initially, a. cylindrical metal vacuum chamber (20 cm dia x 40 cm) is filled with 200 aTorr neutral hydrogen gas. One end of the chamber is made of thin, lev resistivity aluminum foil. All of the walls conserve flux, i.e.. the magnetic diffusion time through the walls is longer than the tj&e scale of the experiment. Therefore no field lines may pass through the container during the course of the ssperiment. This elementary property of conductors assures that any magnetic field liaes created within the container -mist close uoon. themselves. The electron beam is injected through the metal foil into the gas causing it to become Ionized and conducting. The magnetic fields due to the beam current are then “frozen in” to the plasma which then carries a return current induced by the decay of the beam current (Fig. 1 ). The present experimental parameters are: peak toroidal field — 1.5 kG, peak poloidal field as l icG, decay constant » 5 usee, and lifetime * 15 usee. In an earlier version of the experiment the plasma parameters were found to be n — 10”cl”!, T a 3 - 17 eV.

If J is parallel to B at the boundaries then no currents should flow to

the walls, i.e., the plasma current (like the 8 field) should close upon it self. Measurement of the current profile indicates tbit 32 kA flows “up” the major axis with 16 kA returning in the walls and 16 kA returning along closed current loops within the plasma. obtained and are being compared to theoretically derived force-free equilibria with cylindrical boundary conditions. In future experiments the cylindrical container will be replaced by oblate and prolate spherical containers in order co investigate the equilibria and stability of spheromaks of various elonga tions.

Detailed nroriles of Ba and Bz have been

*29-

Zlectrons.

II. Sohercmak Generation by Relativistlc

In this experiment the risetime of the rotating electron beam (630 kV,

19 kA, 1 usee) is longer than the Aifven time of the taigec plasma (ie * 1 0!2 - lO^cm”-’, B2 - 1 kG) so that, unlike the previous experiment, no return currents are generated. A reversal of the field by d factor of

two is observed; however, the lifetime is limited to the 1 -sec beam dura tion if the electrons are not confined. At. the downstream end of the 1 m long plasma the electrons have been successfully contained by a 3:1 mag netic mirror. Reflection of the electron beam is observed. gate coil is being tested to reduce the loss of electrons at the upstream end.

A fast pulsed

IIL Intense Ion 3earn Injection.

Neutral atomic beams have proved effective as a means of heating tokamaka and minimum B mirrors; however, their efficiency decreases as they are scaled Co the higher voltages required by reactor-sized devices. Re cently experiments at three laboratories (university at California, Irvine,” Naval Research Laboratory,’ University of Tsulcuba6) hav& shown that a suffi ciently intense beam of ions that is space-charge neutralized by electrons will penetrate into a magnetic field. Our experiment in progress is to determine whether or not such beams can be efficiently trapped in a target plasma. The experiment is performed with a magnetically insulated diode ion source (100 kV, 15 A/cm2, 1 ysec) and the U.C.I. Tokamak (R - 60 cm, a * 15 cm, B » 6 k G ).

Intense ion beams are space-charge neutralized by electrons and thus may

be treated as plasma describable by the usual parameters of Debye length, plasma frequency, drift velocity, and thermal velocity. The beam plasma enters the magnetic field by self-polarization and E * B drift.’ The transi tion to collective behavior occurs where there is more than sufficient beam energy density to set up the required field:

-30-

i nm.V2 > > E2/8TT . 2 J.

4iTnin,c2/B2 - £ - 1 - a ,2n .2 > > 1

(2)

(1)

where -. is the plasma dielectric constant. At a field of 5 T (a typical value for conceptual reactor designs) the critical beam density at 100 kV corre sponds to a current density of 9.2 A/cm: which is well within the capabilities of incense ion diodes.

In our experiment the io-i. beam is injected radially through a 15 cm *

30 cm injection port into the tokamak with only the toroidal field operated. Beam propagation is measured by Faraday cups, calorimeters, and floating potential probes. At the peak field of 6 feG, ’-^pi2/^1 =» 10 and 10 - 40% of the beam is detected calorimetrically at a point four lannor radii into the field. Propagation is reduced when a shorting conductor is placed across the field lines near the injection port. Floating potential probes indicate potentials of + 60 - 100 kV within the field. One dimensional3 and two dimen sional3 theoretical models have been developed which describe cross-field propagation. A parallel program of ion source development has demonstrated repetitive operation13 and impurity control.11

This expression may also be written

IV. Electron Injection Into Closed Confinement Systems.

If an electron bean is used to sustain currents or to heat a fusion reactor with closed magnetic surfaces, the electron injector must be located outside the region of hot plasma and the beam must move into the plasma across magnetic surfaces by neans of some collective or nonlinear effect, such as by means of image currents in the vails of the reactor. If the beam is entirely charge and current neutralized Mithin the beam channel (as in 5ectioa I above), there will be no currents induced in the walls. We are therefore considering injection of a long pulse beam (as is Section II above) nr a beam of variable rise and fall time into a tokamak or field reversed pio’-h. If the beam electrons carry angular momentum the stability proper ties of the field reversed pinch may be improved.12

The D.C.I, research staff includes G. Endo, W. Peter, G. Saenz,

J- Schneider, and F. Vessel.

This research is supported by the United States Eepansnent of Energy.

-31-

References

S. Robertson et al., Bull. Am. Phys. Soc. 24_, 1098 (1979).
J. A. Fasour et al., Bull. Am. Phys. Soc. 23, 816 (1978).
K. Kamada et al., J. Phys. Soc., Japan £6, 1693 (1979).

z. Ott and W. Xanheimer, Nucl. Fusion 12., 1057 (1977).

W. Peter et al., Phys. Fluids 22, 1471 (1979).
W. Peter and S. Rostoker, Bull. Am. Phys. Soc. 24, 1J9S (1979).

10- F. tfessel and S. Robertson, Appl. Phys. Lett. 34, 7 (1979).

G. Endo and S, Robertson, Appl. Phys. Lett. 35, 8&9 (1979).
S. Rostoker and A. C. Kolb, Phys. Rev. 124, 965 (1951).
K. Molvig and S, Rostoker, Phys. Fluids .20, 494 (1977); C. W. Roberson et al., Appl. Phys. Lett. 32, 214 (1978); C. W. Roberson, Nucl. Fusion 18, 1693 (1973); S. Robertson et al., Phys. Fluids, to appear.
G. Endo and S. Robertson, Bull. Am. Phys. Soc. .24, 1021 (1979).
G. Saenz and C. W. Roberson, Bull. Am. Phys. Soc. .23, 764 (1978); G. Saenz and A. Fisher, Bull. Am. Phys. Soc. 24, 1070 (1979).

lb) aiM’

^ t , Jt V

(e>

OBLATE S P H £ 3 0 M AK I STABLE)

PROLATE S P H S S O M AK tUNSTABLE]

SPIRAL, j CATHODE SHANK

3 2-

u w f ei

— SATO

AU1MINUM FOIL

VAfllAflLfi ASPECT

CONOOCTI V W UJ

(a) Before beam is fired -lo f i e ld

iptie.tQraa.ls. start-up

conducting w a l l s.

in p lastaa.

technique

Figure 1-

Right:

Lefc:

(b) Rotating beam propagates through flux-conserving

l i n es are present.

I l l u s t r a t i on of chrae experiments to examine splieromak e q u i l i b r ia and s t a b i l i t y.

sufwws

L1MITE3 ELIMINATES *WALL STABILIZATION

Laboratory o£ Plasma Studies, Cornell University, Ithaca, N’ew York

Formation of a contact toroidal magnetic configuration using the

diamagnetic current of large Larmor radius ions to produce a field- reversed ion ring was first suggested many years ago [1]. Experimental programs in ion ring foiination have been undertaken at Cornell [2] and the Naval Research Laboratory [3], and diamagnetic proton layers have been formed, their propagation in magnetic mirror systems studied [4], and transient field reversal by a rotating proton beam observed r Nevertheless, the goal of a trapped, field-reversed ion ring is far from being attorned in these experiments because the/ are limited by the ion sources which have been available to them.

The short-pulse [100 nsec) magnetically insulated source used in the IREX experiments at Cornell produces only -1.5 mcoul of protons per pulse, limiting the diamagnetism of the rotating beam (<SB) to a few percent of the applied field CSQ) [4], Also, the short-pulse beam does not propagate well across the strong cusp magnetic field which induces diamagnetic rotation, unless neutral gas (>30 mtorr) is introduced into the cusp region. The gas is needed to produce plasma in the beam channel which can respond to and . neutralize the space potential Lhat would disrupt an unneutralited proton

beam.

-33-

L3].

J. B. Greenly, D. A. Hammer, and R. M. Sudan

THE LONGSHCT INJECTOR: A 5/4 kj, 120 fceV PULSED SOURCE Cf 3 x 1 016 IONS FOR ION RIMG FORMATION

The NRL experiment, using a reflex-type ion source and a much larger

pulse power generator (Gamble II), is able’to produce, and transport through a veak cusp, enough protons (-10 mcoul) to produce transient reversal CiB/Bq - 1.2) of the weak guide field downstream of the cusp [5]. length of this beam is shorter than in the IREX experiment, and it seems likely that in using a strong cusp to slow the axial motion of the beam, as in IREX, beam propagation problems will appear. If neutral gas is required, it must either be isolated in the cusp or will, as Ln I.1E.X, limit the ring lifetime to a few microseconds by ion-neutral collisions.

The pulse

To overcome these limitations, we are developing at Cornell a new long

pulse intense ion beam source, L0NG5H0T, based on the “Pulselac” injector built at Sandia Laboratories [6]. The present LONGSHQT injector produces 5 mcoul at 100-150 teV (0.3-0.75 kJ) of ions in a 600 nsec pulse, with a pulse power generator considerably smaller than either the IREX or NRL experiment, and this long-pulie beam propagates well across a strong cusp in vacuum, without injection o£ neutral gas or provision of any active elec tron sources by charge neutralization.

The LONGSHOT injector is shown schematically in Fig. 1. It is an annular, magnetically insulated diode, with racial insulating field produced by coils in the grounded cathode structure. The insulating field is shaped by flux-excluding surfaces behind the anode, which is of the surface-flashover

type. A virtual cathode is formed by electrons emitted along the insulating field lines connecting the shaped cathode-defining surfaces. The annular ion beam passes between concentric drift tubes and exits the injector 13 cm from the anode-cathode gap. The injector is driven directly by a low induc tance [0.6 uH) Marx Generator.

Typical diode voltage and current waveforms, and current of the extracted

ion beam, are shown in Fig. 2. The current rises throughout the pulse, reaching >13 kA at the end of the voltage pulse. The ion current density 15 cm from the gap reaches >100 A/cm*, a factor of 30 above Child-Langmuir space charge limited flow. This ion current density over the 140 cm^ anode area accounts for essentially all the diode current, indicating very low electron current losses. The long pulse is made possible by good confinement of electrons in the gap, which also allows the large enhancement of ion current over Child-Langmuir flow.

Preliminary results of injection of the LONGSH0T beam across a cusp (vj/v - .5^ into a solenoid to form a diamagnetic layer are shown in Fig. 3. The observed beam diamagnetism, as well as the beam shape seen with damage targets, indicate that nearly all of the beam propagates across the cusp in vacuum (2 x 10”5 torr) without space-charge disruption. The mechanism proposed [6] for the observed charge neutralisation is that electrons emitted from surfaces provided outside the beam channel are able to equilibrate in the beam by some dissipation mechanism, and to depress the space potential to near zero on a time scale of -100 nsec. Thus, a small amount of the beginning of the long-pulse beam is lost to these surfaces, secondary elec trons are emitted, and the rest of the beam propagates undisturbed in the resulting charge-neutral channel. The time required to reach neutralization is apparently too long to allow complete neutralization of short pulse (<100 nsec] beams.

The loss of the head of the long-pulse beam can be seen in the current

density trace of Fig. I, which (offset to eliminate time-of-flight) lags behind the rising diode current for the first -100 nsec. Further beam loss in traversing the cusp appears to be small since the observed diamagnetic signal requires >7S$ if the total diode current in the rotating beam. Also the damage targets show the expected radial convergence of the beam indicating insignificant residual space charge. Once through the cusp, electrons can travel along field lines with the beam, which thus can remain charge-neutral as in the IREX experiments, without further losses [4].

-34-

The present LONGSHOT beam consists of >50% protons and <50% C ions from the “flashboard”’ anode, and is of 1.5 m axial length in~the downstream solenoid, giving <SB/B of ~3%. Experiments aTe underlay to axially compress the diamagnetic layer in a downstream mirror, and 6B/B > 20% would be expected if the present 5 mcoul pulse can be shortened axially to a length comparable to its 10 cm radius.

It appears practical, by sxraightforward scaling up of the present

injector, to produce the 20-30 mcoul necessary for a fully field-reversed ring, (6B/3Q > 1) with a relatively small and inexpensive device. Of equal importance are the superior propagation characteristics of the long pulse

bea~. Furthermore, the LCNGSHOT injector configuration is compatible with a h ig life, single ion species plasma gun anode source [6], and thus has the basic features necessary for development to a high efficiency, high repetition rate source of a pure proton or detiteron beam. Since the physics of the injection and ring formation process is dependent on the source characteristics (e.g., pulse length, as we have described), it is important to study ion ring formation with a source type that is compatible with the lifetime and repetition rate requirements of future applications.

This work supported by the U.S. Department of Energy.

References

N. C. Christofilos, Progress in Nuclear Energy; Series XI Physics and Controlled Nuclear Fusion 1, 576 (1959).

Plasma

P. L. Dreike, D. A. Hammer, R. N. Sudan, and I. G. Wilev, Phys. Rev. Lett. 41, 1323 (1973).

J. Golden, Bull. Am, Phys. Soc. 24, 966 [1979).
P. L. Dreike, J. 3. Greenly, D. A. Hammer, and R. N. Sudan, 3ull.

.Am. Phys. Soc. 24, 1152 [1979).

C. Kapetanakos, Proceedings this conference.
S. Humphries, Jr:, J. R. Freeman, J. Greenly, G. W. Kuswa, C. W. Mendel, J. W. Poukev, and D. W. Woodall, Sandia Laboratories Report 79-1673 (.August 1979).

-35-

7\ ^ ///////////////////

*WiWl

Figure 1. The LONGSHCT injector: . ^ i n s u l a t i ng field coils, 3) virtual cathode-defining electrodes, 4) “flasbboartf anode, 3) field-shaping flux excluders, 6) stainless steel drift tubes, ”) snode support and connection to pulse power generator, 5) barrier blocking external breakdown, path. dots,

End flange is IS cm from virtual cathode.

outer flux excluding vacuun /essel,

Insulating field shewn by

Figure 5(a). Diamagnetic signal at flux-excluding wall, showing time delay after diode voltage pulse due to time-of-flight, and peaks due to protons and carbon.

t.SfcA ^

So A^«v

Figure 2. Typical diode voltage and current, and current density measured by biased charge collector IS cm from gap. Current density trace has been offset to eliminate 50 nsec time-of- flight between gap and charge collector.

-36-

0-5 unt

T a r a t ti

Oa»i»«t

SakroiA

Cdlf

Figure 5(b). Qualitative damage patterns showing propagation of rotating beam. Canonical angular momentum is nearly ;ero, so ion orbits move inward, barely encircling the axis. Protons should be closest to axis at -50 cm, C+ moves inward slightly over this length. Velocity spread in entering beam mixes orbit phases by 75 an.

REVEkiiD-FIELD CONFIGURATION WITH ROTATING REUTI VIS ITIC ELECTRON SEAMS

J.D. Sethian, K.A. Gerber, D.N. Spector and A.E. Robson Naval Research Laboratory, Washington, D.C. 20375

The use of rotating reiativistic electron beams to produce a

field reversed configuration was first proposed by ChristofiIcs. In his Astron concept, the fie Id-reversing currents were provided by the reiativistic electrons. This paper describes a different method of achieving a reversed field geometry, in which the beam has been used to induce plasma currents, in originally neutral hydrogen gas, whfch-then maintain the fields long after the beam has left the system. axial components, both axial (poloidal) and azimuths! (toroidal) fields are induced, and thus the configuration can be topologically similar to a Compact Toroid.

Because the beam has both azimuthal and

-37-

In this experiment, the rotating beam is produced by first forming an annular beam (V » 900 kV, I =» 80 kA, T - 100 nsec.) in a field-eraission diode immersed in a uniform axial magnetic field, and then passing this beam through a “half cusp” in which the field is brought to zero in a distance short compared to the electron Larmor radius. When injected into neutral hydrogen at a pressure of about 150 mTorr, the plasma produced by the beam is sufficiently dense to neutra1ize “the beam charge, but not sufficiently dense to neutralize the beam current. As there is no applied magnetic field, the radial equilibrium of the beam is determined solely by the axial and azimuthal self-magnetic fields and by currents induced in the cube wall. Since the total axial flux inside the tube muse oe zero ac all times, the axial fields ins ice and outside the beam ars anti-parallel, automatically forming the reversed field con figuration. The passage of the beam increases the plasma electron temperature. Thus, the induced magnetic fields must be frozen in the plasma, and, when the beam leaves Che system, they are maintained by the induced plasma currents.

After the beam transits the system, the initial hydrogen gas is fully ionized and the electron temperature is typically 5 eV. as measured by Thomson scattering. The efficiency of conversion of beam energy to magnetic energy has been measured to be as high as 53%- That the configuration is maintained by plasma currents (rather than trapped beam electrons) has been shown by x-ray studies. The system decays over a period of 20usec. During the first Susec the system re-arranges itself, converting toroidal magnetic field along its length into polofdal magnetic field at one

end. This is similar to the manner of a coil tending to maximize its inductance by axially contracting. After the re-arrangement process is stooped (presumably due to axial pressure gradients), the entire configu^ation decays uniformly with a time constant of about Busec, which is consistent with classical resistive dissipation of the plasma currents, at the electron temperature of 7 eV measured by Thomson scattering. (The dissipation of the currents tends to heat the electrons, and in similar experiments an increase in electron temperature was observed several microseconds after the beam had left the system.)

In order to understand how the configuration is formed, a model

Particles emitted from the diode are reflected

which uses conservation of flux, particles, and canonical angular momentum to describe the beam propagation has been developed and experi mentally verified. at the beam head, giving a small fraction of their energy to the self-magnetic field. The axial motion of the beam front is retarded by the “inductive load” of the reversed field configuration, and, tnerefore, the beam front velocity, v -~ .O’lc, which is much less than the individual beam particle velocity (—c). As the particles return to the diode they are reflected by the electrostatic mirror in the anode-cathode gap, and are sent back into the beam. Because of the multiple reflection process, the beam density, and hence ‘jht, is increased to a value of approximately c^Vr”25. Con servation of canonical angular momentum through the cusp shows that the axial flux enclosed by the beam is conserved to the order of (v/y)* Furthermore, the beam current pitch angle and radius equal those of the plasma to the same approximation.

-38-

The configuration, in its present form, is not a genuine Cormiact

Toroid, even though it contains both a poloidal and a toroidal field. This, is because a net poloidal current flows from one end wall, through the plasma, to the other end wall, and returns along the side wall of ?he system. (See Fig. 1.) Because the System is maintained by plasma currents, this flow of current to the wall represents a loss mechanism. In contrast, a Compact Toroid would have the poloidal current completely enclosing the plasma, as shown in Fig. lb. (Note that what differentiates the present system from the desired configuration is that the poloida! current lines, not the poloidal field lines, are not closed in the plasmaj The reason for the existence of this net poloidal current can be seen by considering how the configuration is formed: As the electron beam propagates from one end of the system to the other, it carries a net axial current. Because the current must eventually return back to the electron beam generator, preferably by the lowest inductance path available, it chooses to flow radially Outward from the beam head (carried by secondary electrons) and then back along the drift tube wall. This produces a net toroidal field between the beam, and after it passes, the plasir-a, and the tube wall.

To eliminate the net current, zhsn, would require the elimination of this large net toroidal field. This cannot be achieved through simple interruption of the poloidal current path or reflection of the beam electrons, as large inductive voltages will arise to drive plasma currents that maintain

It is possible, however, that the current lines can be closed

with two caunterstreamlng beams. (£qui I ibrtum for the two beaflis has been predicted by extrapolating the single beam model.) This process is envisaged schematically in Fig. 2. Initially, a reversed field configuration identical to the present one is formed with one beam. After the beam has passed out of the system, a second rotating beam, whose net poloidal current is equal to that flowing in the plasma, is injected at a larger radius and from the opposite end of the tube. As this beam is rotating in the same sense of the first, the poloidal field is amplified. The opposing net polcida! current, however, nulls the toroidal field external to the plasma. As the poloidal current now flows along one beam path and returns on the other, the configuration is fres to contract axially without interrupting any current paths. Such a contraction is expected to take place as it is energetically favorable. Moreover, axial contractions have been cbserved in previous reversed fJeid configurations such as reversed field uheta pinches. In addition to closing the field lines, if the current of the second beam exceeds that of the first, it could also be used to heat the plasma. Sy adjusting the plasma density so that the second beam current is completely neutralized, save that necessary to cancel the initial poloidal current, the un-neutt-al ized portion of the current will complete the configuration, and the return currents neutralizing the remainder will heat the plasma as they are dissipated. Experiments to test these concepts are currently underway.

-39-

the field.

M.C. Christofilos, Proc. 2nd Int. Conf. on Peaceful Use of Atomic

Energy (United nations, Geneva 1958) Vol. 32, p. 279.

J.Q. Sethian, K.A. Gerber, O.N. Spector, and A.E. Robson, Phys.

Rev. Lett. jM,, 798 (1978).

J.D. Sethian, K.A. Gerber, O.N. Spector, and A.E. Robson, submitted

to Phys. Fluids.

References

a.,,/0

CONDUCTING WALL

BEAM I: GENERATES CONFIGURATION

o o o o o o 41

BEAM II: GENERATES A SECOND CONFIGURATION AT LARGER RADIUS, ELIMINATING B, OUTSIDE PLASMA

) d. Lb *%

id J:

Closure of P c l o l d al Current

CURRENT AND FItLO LINES

two CountersLrcamlng

CLOSE INSIDE PLASMA

E l e c t r on Beams

Lines w i th

FIGURE 2

I * * J’

(LOVltft)

1 -^

(UPPER)

e >

’

PRESENT CONFIGURATION;

PLASMA

O CK O O Q O O

CUHRENT ROTH-

D E S I R ED CONFIGURATION:

f S o o o o

C *

P r c i e ni C w i F i r j u r d t i on Cwmuict Ttxroiil Con f U j uv a v i on

RESULTS AND PRESENT STATUS OF THE RELATIVISTIC ELECTRON RING EXPERIMENTS AND THEIR APPLICATION TO SPHEROMAK PROBLEMS*

H. R. Fleischmann School of Applied and Engineering Physics, Cornell University, Ithaca, ST 14853

This paper is to summarise briefly the results and present status of the

Cornell RECE program on field-reversing electron rings in particular . tth regard to potential applications of these results to other Compact Toroid cancapts. Since its initiation in the early 70’s, this program is aimed mainly at testing various physics questions related to the use of field- reversing fast-particle rings for the confinement and/or heating of a fusion plasma as envisioned by various authors (e.g., Refs. 1-3). In the related experiments, fast-electron rings are generated by injection of intense rela- tivistic electron beams; their radius R normally ii comparable to Larmor radius R^ of an electron having the maximum accelerator energy4 in the applied magnetic field, as it had been envisioned in the original Astron^ and the Ion Ring Compressor^ concepts. Correspondingly, they may be considered as a large-orbit limit of the normal Field-Reversed Mirror (T8M)5 scheme; they differ, however, in this regard from the similar SPAC-experiments^ at Hagoya for which R/Rj^a few). TJith field-reversal times of up to 1 msec, these rings, together with the 3PAC rings,” constitute the longest-living Compact Toroid rings at this time. He envision, also, a number of related future applications of fast-electron and/or ions in the Compact Toroid area including (i) the injection of relativistic electrons into SpheromakT rings either for providing and/or maintaining the poloidal surface currents that otherwise may lead to a rapid resistive decay of these rings or (and) for help in the stabilization of the predicted? tilting modes, which may result through a change in coupling between the tilting motion and the internal deformation of the rings, as well as from the angular momentum of the fast particles. (ii) In addition, our estimates indicate that such electron or ion rings also may become useful for Che generation of other CT rings in present experiments as well as for ceactor applications.

-41-

SECE 2speriments have been performed and field-reversing rings have

been obtained so far in three separate devices: (i) RECE-Christa [see Fig. 1, tangential Injection, nominal beam energy presently 3 Metf-5 Me? available, Bz=500(1000) Gauss, toroidal field Bg=(l-2)B2, k-7-15 cm, ring current I2< 20 kA, field-reversal time < 1.1 msec 1; (ii) XECE-Berta9 (tangent, inj., 0.4 MeV. 200 Gauss, no Be, 8-10 cm, <5 kA, few usee, respec tively); and (lii) C H I P10 (cusp inj., 0.4 MeV, 600 Gauss, 3-6 cm, no 39, < 3 kA, few usee, respectively).

In all devices and under a wide variety of conditions, the observed

electron ring behavior in most cases is now well understood and consistent with relatively simple cheoretical analyses. Generally interesting are,in particular, the following results:

(1) Electron rings with field-reversals of up to 190% (RECi-Berta), 1702 (RECE-Christa), and 1203 (CHIP) have been abesined, and these rings generally show a smooch, 3table decay well consistent with a purely colli- jional decay due Co multiple scattering of Che fast electrons in the back ground gas. In particular, we have not observed so far any tilting or wobbling Instability. We feel that chis is due to a specific coupling of the tilting motion with a widening of che minor ring cross section (due

(iii) The application of weak loffe fields in RECE-Berta

generated

anomalous fast-particle losses which are explained in full detail by a break age of the normal fast-particle confinement due to conservation of the canonical angular momentum Pg. This breakage becomes particularly serious for some resonant particle orbit. Quite importantly, the related theory predicts similar effects to occur quite strongly around the field-reversal point for ring geometries of the 2XIIB typd. Such particle effects clearly will need careful examination in any non-symmetric system.

to the large orbitals of the fast electrons) so that the energy balance becomes unfavorable for this mode. It appears that a similar effect may scill be exploited in mixed systems with fast-electron and plasma currents.

(ii) The precessional m o d e11 of the rings could be stabilized in RECE- B e r t a12 either by conducting walls, toroidal fields or multlpole/Ioffe fields with Che stability limits in all cases agreeing well (10-20%) with theoretical analyses (including also significant increases due to partial flux penetration). In addition, studies with vertical field gradients13 showed stable ring equil ibria with the rings leaning against the conducting wall which open the possi bility to provide radial feedback stabilization in reactor-type CT rings.

-42-

Civ) In RECE-Christa, field-reversing rings were moved successfully in good agreement with theory axially over distances of up to 1.5 m in a con stant gas density as well as out of a high-density gas puff into low-density gas. Axial ring movement as it is planned in some reactor projections clearly wilj not constitute a serious obstacle.

(v) Field-reversing rings -have also been compressed

in RECE-Christa ’-*ithout the occurrence of instabilities and in good agreement with theoreti cal calculations. Also, it was observed 17 that rings of roughly equal strength would readily combine, “stack,” to produce a field single field- reversing ring.

<vi) Scrong electron rings in RECE-Christa also have been observed in

low-density gas (^1-2 mTorr) without indications of microinstabilities. In this case, the collisional rate of plasma electrons with gas molecules is con siderable smaller than the collision rate with plasma particles,-although gas collisions still will maintain a low plasma temperature.

Overall, these results clearly indicate good prospects for the usage of large-orbit particles either alone or in combination with plasma currents as anvisioned in various CT schemes. On a reactor scale where the ring life- tines will be larger than the flux conservation times of conducting walls, Che radial ring stability will have to be provided either by feedback (recent calculations1-8 indicate that this could be done quite easily on field penetra tion time scales) or by a rapid axial movement of the rings.

Regarding the trapping of fast particles in plasma rings like Spheromaks it will be important to provide some means for a change of Che magnetic field configuration. In the RECE experiments, normally this is done by providing gas densities of 0.2-1 Torr in the trapping region. In this case, measure ments1^ indicate that beam trapping occurs via fast field changes both before ionization of the gas and subsequently resulting from the high resistivity of the weakly ionized plasma. Alternate experiments were performed in fi£CE-3erta wlch beam injection into a gun-injected plasma.-“in this case, detailed measurements and analyses showed the generation of large-amplitude Alfvrn i.aves with the electrons being trapped and separated from the injector by

and

these waves. In agreement with this analysis, best results were obtained for the lowest useful plasma densities, n=10licm~^, which were comparable to the beam density at injection. Unfortunately, Alfven transit times across the ring still were longer than beam injection tines, and only rings with 5<30? could be obtained. Estimates, however, indicate that a similar trapping mechanism may be possible for beam injection into Spheromaics.

The mala open question presently concerns the plasma confinement charac teristics of the field-reversing rings, and the effort of the RECE program now mostly is directed at this poine. For this purpose, various changes were and are being incorporated into the R£CE-Christa device to obtain for at least a few msec, the low gas density (3xl0~^ Torr) around the ring and the high energy input into the ring plasma required to overcome the radiation barrier. These changes include (see Pig. 2 ): (i) a change to puffing of ethane which both is slower and has a sufficiently low vapor pressure at liquid nitrogen temperatures, (ii) the addition of an LX cooled cryopump along the tank wall, ( H i) a lengthening of the tank by 1.5 m, and (iv) the addition of a small axial heating coil which is to induce low-frequency plasma currents I <0.1 Ig in the ring. Assuming full success and good plasma confinement, the plasma and ring parameters shown in Table I are expected.

Preliminary experiments were performed at the prasent bears energy to

test various subquestions. In these quite successful tests, very strong layers with s-2 have been trapped in ethane clouds, and rings with 6-1 have been moved into the downstream well. In further preliminary tests, weak plasma currents (few percent of Z$) have been induced in the rings without serious.deterioration of the ring stability aad lifetimes; also, some heat ing of the plasma electrons is indicated by optical line observations. Further experiments will be performed after completion of the present recon struction phase in the next months.

-43-

Anticipated Plasma Confinement

*i Te I„ ”- 1/10

n T. 3x10 l3rm~3

Experiments

*v 50-80 kA

”- 2-5 msec

”< 1.5-2.0

**< 5-8 kG

Table I

-, 3-4 kG

TE i” ”»

ohmlc ”^

100 kW

ring ”-

j Sec

50-100 eV

Plasma Parameters

Electron Ring Parameters (Full accelerator energy and compression)

s ”- 5-10 MeV R ’- 6-10 cm

3t or

Tri£ig T

*tfork performed under DOE Contract EY-76-S-02-2.119.A0O1.

10 kA

N. C. Chriatofilos, 2nd United Hat. Intern. Conf. Peaceful Uses of Atomic Energy (Geneva 1958), Vol. 32, p. 279. H. H. Fleischmann, Conf. on Electrost. and Electromag. Confinement of Plasmas and Phenomenology of Relativ. Electron Beams (New York, March 1975), Ann. N.7. Acad. Sci. 251. 472 (1975). S. The The typical energy of the actually trapped electrons is poorly known so so far and is expected considerably smaller. y. E.R

X, Bussac et al., IAEA Conference (Innsbruck 1978), Vol. II, p.249.

E.R 9) E.R

E.R E-R R. < E.R 0.

H. A. Davis et el., Phys, Rev. lete. 37,, 542 (1976). D. A. Phelps et al., Phys. Fluids 17, 2226 (1974). 10} R. Kribel et al., Plasma Phys. 16., 113 (1974). H. P. Furth, Phys. Fluids 3., 2020 (1965). 11) D. M. Hoodall, Ph.D. thesis, Cornell University, 1975. 12) Meger et al., Phys. Fluids 17., 2x00 (1974). 13) S. C. Luckhardt, Ph.D. thesis, Cornell University, 1979. 14) Rej et al., Appl. Phys. Lett. 3_3, 910 (1978). 15) Tuszewski et al., Phys. Rev. Lett. 43^, 449 (1979). 16) M. ” A. Davis et al., Phys. Rev. Lett. 39, 744 (1977). 17) . H. Z. Kribel et al., Bull. Am. Phys. Soc. .24, 928 (1979). 13) i R. C. Smith, Jr., Fh.D. thesis, Cornell University, 1977. 19) A. < C. Smith, Jr., et al., Appl. Phys. Lett. 32, 133 (1978). 20) A. (

-44-

A. Mohri, tb±s conference.

Yoshikawa, Phys. Rev. Lett. 26, 705 (1971).

C. Condit, Jr., ee al., UCRL 52008 (February 1976).

-n

^ f^

PLASMA CONFINEMENT H RECE-CHRISTA

JOULI UEITIWI SPLCWOIO COIL

EQUIPMENT CHANGE5 FOR

, puf, V*L»CI n n s i i T na

M I. Ttrnt ^ E X T E M S KW

Fig. i.

9]m i W An

/sou J sotcHtm cats

COMP«Ct$IOM CO^.

Sa B a)

WICUWM TANK

5TEEL LIMCR

COHDUCTOffS

( E T H A H C)

-mjecran

COILS

COIL

flltW

.^H

/ si eaa

HAfflltCN

tflAPPf o C L e C T MM RIMn

MOVTP ELCCTflON

C O M M C f &O E1.ECTWON HMO

Fig. 2. Recent Changes in

RECE-Christa for Plasma Confine ment Experiments

3asic Schematic of RECS-Christa Device

Introduction

REVERSED FIELD CONFIGURATIONS GENERATED BY PROTON PULSES*

J. A. Pasour, J. Golden, J. Marsh* and C. A. Kapetanakos, Naval Research Laboratory, Washington, D. C. 20375

II.

-45-

Magnetic field configurations with closed field lines are very

promising for confining plasmas for thermonuclear applications because: (1) tfie plasma confinement time is considerably enhanced since the trapped plasma would have to cross the closed magnetic field lines in order to escape, (ii) the closed minimum -B magnetic field configuration provides hydrodynamic stability to the plasma,and (iii) the beta of the plasma can exceed unity resulting in high power density reactors.

Magnetic field configurations with closed field lines generated by energetic ion rings or layers have the additional advantage that the gyrating particles of the ring serve as an internal energy source for heating the confined plasma. The objective of the Ion Ring Program at NRL is to test the feasibility of forming strong ion rings that reverse the applied magnetic field by trapping about 1 017 protons of 1-2 MeV energy inside a magnetic mirror.

Description of the Experiment and Results

A schematic.of the experiment is shown in fig. 1. The 5 meter long

system consists of a low inductance inverse reflex tetrode(IRT), a magnetic cusp, a compressing magnetic field, a gate field and a magnetic mirror. In recent experiments the applied magnetic field has temporarily been modified Dy replacing the compressing field with a uniform field that is term?nated in a single mirror. The proton source has been described previously.1 In our best shots the total number of protons measured with the resonant reaction3 l 3c{p.-f)i aN(a+)i ac using carbon targets that are located 15 cm from the anode is in excess’of 8 x 1 0lS per pulse. This number must be considered as a lower bound since protons with energy less than 470 keV do not activate the target and the number of counts is not corrected for target blowoff. The peak proton current is ~ 500 kA when the ramp shaped (corrected) voltage that is applied to the anode of the IRT increases from 0.6 MV to 1.7 MV within _~ 50 nsec. As a result of the ramp shaped voltage and the somersault effect5, the ion pulse bunches axially. Magnetic probes that are located in the 1.5 kG, 75 cm long field plateau along the axis of the system show that the rotating ion pulse reverses the external field. The field reversal factor exceeds 120%. In addition, the signals of magnetic probes show that the ion pulse propagates with an axial speed v which is ~ 1.1 - 1.2 cm/nsec. The velocity of rotation v« is only 1/5 of v . “The ratio v./v is sensitive to the ratio of the magnetic field at the cuip S to the crftiCa; field of the cusp B . The parameter a a a v,/v , in order to avoid excessive losses at the exit of the C U S D. It appears that these losses are related to the long transition width j of the cusp. In the present series of experiments 5 is greater than the proton gyroradius because the cusp sharpening ferromagnetic disc has been removed.

teP* small, and so the ratio

c/B

w as

III.

IV.

In Eq. (1) d is anode-cathode gap, T is the t r a n s it time of the i o n s, V tne applied voltage on the anode and the i n t e g r al is along the p a r t i c le o r o i t. For d = 2 cm, 8„ = iO* Gauss = const., V„ = 1 MV, Eq. (1) gives ;

.—= 163.

In very recent experiments the ” f i l l i n g - i n” of the beam has been s u b s t a n t i a l ly reduced by t i l t i ng both the cathode and anode with respect to a v e r t i c al plane in such a way that the protons obtain an outward directed radial v e l o c i t y, which compensates the inward radial v e l o c i ty of the pinching force. The s o l id l i ne in F i g. 2 shows the radial p r o f i le of the cm from the anode f or a t i l t i ng angle of 10°. beam at 15

TVn’s was necessary because of the substantial reduction in the inner and outer beam radii between the anode and tne cusp. The radial profile of the beam and thus the change of its inner and outer radii is determined by individually counting small segments of a carbon target after it is activated by the beam. The dashed line in figure 2 shows the radial profile of the aroton beam 15 cm from the anode. It is apparent that the thickness of the oeam is appreciably greater than that of the anode foil. The observed ‘*fflling-fn” of the proton beam is due mainly to the qv B, pfnchfng force that acts on the ions inside the anode-cathode gap. Thl azimuthal magnetic field B, is generated by the ion current and the net electron current in the gap. it is estimated from the equations of motion, neglecting the radial electric field v = v sine, where v is the velocity corresponding to the applied voltage ahd

that at the cathode the protons have a radial velocity

-46-

sin e * (d/TV0) / B£ dz.

Conclusions

References

(1)

—

,r Work fs supported by ONR and DOE. +

C. A. Kapetanakos, J. Golden, J. A. Pasour, S. J. Marsh and R. A.

Sachs-Freeman Associates, Inc. Bladensburg, MD 20710.

Mahaffey, NRL Memorandum Report i 4135, 1979. F. C. Young, J. Golde^and C. A. Kapetanakos, Rev. S c i. Instrum. 48, 432(1977).

S. J. Marsh, A. T. Drobot, J. Golden,and C. A. Kapetanakos, Phys. Rev.

L e t t. 39, 705(1977).

The elimination of the ” f i l l i n g - i n” w i ll make possible the operation

of the system in higher magnetic f i e l d s, thus achieving higher v e l o c i ty r a t io v „ /v without appreciable losses at the cusp. Trapping studies of the proton oolss using the magnetic f i e ld configuration of F i g. 1 w i ll s t a rt in the near f u t u r e.

CO/E3 * S3 * E3

fa * * u2 a a JL

PUMP

EC~*1I

licj=> g

MAGNETS

AXIAL POSITION (METERS)

experiment (too) and applied g e n e t ic

f i e ld conr’Tguraticn Jbot’.oir).

rig. 1. Schematic of the

X 2.0

X .0

3.0

4.0

5.0

o 15

y 10 Ll_ O H 5 LLI

O < n

-5

ANODE

~ CVJ

4 8-

15 T-

(cm)

/ -J

Fig. 2. Number of protons per unit area striking a carbon target 15 cm from the anode as a function of radius. The dashed line shows the radial distribution when the anode and cathode planes are perpendicular the symmetry axis of the system. The solid line shews the radial distribution v.nen the cathode and anode planes intersect the sytRratry axis at an angle of 10G°.

T 30 p

Kink Instability of a Field-Reversing ron Layer*

(UD) examined the stability properties of a field-

Several papers have recently been written on the stability of electromagnetic modes in field-reversing ion layers immersed in cold background plasmas. Using a rigid rotor model Uhm and Davidson’ reversed xon layer by considering an infinitely long, thin layer situated in a cold magnetized background plasma which was treated by a fluid model. With a kinetic treatment of the layer ions these authors found that flute-like kink modes -with frequencies near multiples of the mean ior.-layer rotational frequency, t.Og are un.s.iable over certain ranges of positive and negative values of the compression ratio, and that the system with a dense back ground plasma can be stabilized by a sufriciently large transverse temperature of the layer ions.

In this paper we extend the work of (UD) by including the

thermal effects of the background plasma and examining its impact on the stability of the kink modes. Because of the length and complexity of the analysis, the reader is advised to consult Reference 1 for the details and the notation; only the major modifications associated with the inclusion of the thermal effects will be presented hers.

A.S in the above-mentioned reference we consider a space- charge neutralized ion layer with infinite axial extent and a mean radius and radial thickness of “R„ (= -^ C &i +* &i) ) and 2a respectively. We assuae that the equilibrium properties are azimuthally symmetric and that the layer thickness is much smaller than its radius. The layer is taken to be immersed in a warm background plasma and the whole system is placed ir. an equilibrium

-49-

Thermal Background Effects Cn The

S. j. yakura and T. Xammash University of Michigan Ann Arbor, Mien. 48109

*Work supported by EOE

which can be utilized to define the compression ratio

which in turn is related to Budker’s parameter >/ by Vi O-‘l) /(it*)) A vlasov treatment is used for the layer ions, while the remaining species including the background plasma are treated by a fluid model for which the momentum equation is given by

magnetic field given by

where

-50-

7 = 3^ O = £ , ) /&

clu-i.-l .\t+r0-%.)}

ci n ci_ — L_ _r Y’ - A’*-)) t^)

. ft

<2>

(5)

(4)

with o^y being the thermal velocity of the background plasma. If we take the equilibrium background plasma density to be uniform across the layer, and assume that the radial variations in its perturbed velocities to be small, we car. linearize the fluid equations and calculate the perturbed radial current density which, along with the appropriate Maxwell equations and a Poiason equation that contains the layer’s contribution to the perturbed surface charge density, give rise to an eigenvalue problem for the perturbed azimuthal electric fJeid. The equation in question has the form

and A(?L) is a lengthy term which contains the thermal effects to lowest order but which we will not list here due to space limita tions. In the absence of A ( A) Eq. (4) reduces to that derived in CUD). When Eq. (4) is solved subject to the boundary condi tions that the potential = <- ‘l^p/Z layer ( ,. ^ £„ ) while its derivative is discontinuous, the wave

is continuous across the

admittance of the form

can be calculated from which a dispersion equation , i

is obtained with t-v being the layer ion thermal velocity. The thermal effects are hidden in the term iM *^‘L’ej and if we specialize to the case of a dense background plasma that results in a strong electromagnetic coupling ’ we find that ths eigen functions of Eq. (4) are the ordinary Bessel functions ‘J^ CV^)with -P and ‘f given by Eq. (5).

The effects of the background plasma temperature on stability

can be assessed by the sign of those terms in j) (^kf-) which are associated with the thermal effects. A positive contribution to the right hand side of Eq. (6) is stabilizing and such contributions are denoted by the shaded areas in Figs. 1 and 2. The first figure shows the thermally stabilizing regions for the X=2- kink mode as a function of the compression ratio and the density parameter when compared to Pig. (8) of reference 1 it is seen that at “j = -<>’- the stabilizing effects occur at those values of correspond to the maximum growth rates. In Fig. 2 we observe the thermally stabilized regions for negative compression ratios. Although ’/*=-’ i.e., complete field reversal is excluded from this analysis we note nonetheless the dramatic stabilising affects of the background plasma temperature on layers with large field reversal. This figure should be viewed with Fig. 12 of reference 1 for comparison with the cold background case.

which

K. S. Ohm and R. C. Davidson, Phys. Fluids 2_2, 1329 (1979)-

„

. ~L

-51-

, - **

{^-fj^o/cj

- ‘0

40f J™

fOr

1.0

5 2-

C — i ,,

regions

F\miF i

EHJ7PE 1

itcompfessron ratio}

Shaded areas are r.’-.err.ai^v stamli?*,} r e l a t i ve to ;fce jold plas:ss ?aso.

1 t’cofTiprps.ii’on

f^%)

£*2 *‘P,

p*vor

\ LINc

-0.5

tttic}

vtn -7-^.05-5

BEYOND THIS

regions

Shaded areas are tfcenaaHy stabilised relative to ’.Se cold pXa=3.^a ess*.

T. R. Jarboe, I. Renins, E. W. Solda, J. Marshall, A. R. Sherwood Los Alamos Scientific Laboratory, Los Alamos, Sew Mexico 67545

-53-

MAGNETI1ED GUN EXPERIMENTS

In the Los Alamos Magnetized Gun Experiment we are attempting to produce a compact torus in a manner similar to an earlier experiment of Alfven.^ In our experiment a soleaoidal coil is placed inside the Inner electrode of a coaxial plasma gun. This coil produces an axial magnetic field inside the inner electrode which diverges and becomes a largely radial field in front of the gun muzzle. The idea is that when tho gun is fired, the plasma escaping from the gun stretches these radial Slelds along the axial direction away from the gun, and these field lines can reconnect behind the plasma forming the pololdal field of the compact corus. The magnetic fieli generated by the gun current becomes the toroidal field and the major axis of the corapacc torus will be the same as the axis of the coaxial gun. Recent interest in this possible method of compact corus generation was stimulated by C. Eartman, and the appruacb is also being pursued in the field-reversed plasma gun experiment at LLL.

The objective of our initial experiments is to study the generation of compact tori hopefully formed in Che above-described manner when the magnetized plasma gun is fired into J flux-conserving cylindrical shall. Preliminary results are reported for injection into a 0.46-m diameter, 1.2-m long, l.e-mm thick stainless steel, flux-conserving shell placed 0.13 m from the gun muzzle. In these experiments we have demonscrateu that the addition of a magnetic “bias” field between the gun electrodes parallel to the axis of the gun allows it to be operated at much lower fill p.-essure and makes the gun discharges much more repeatable* In the results discussed here, the tocal gas puffed into the gun is 0.75 atin an? of , The gun is energized with a 37-vF capacitor bank charged to 45 kV. The length of the gun is 0.70 m and tiva electrodes ha-j-e 0.10—a ana 0.15-m radii. Tiie current peaks at a value of about 0.8 S& and reverses at about 4.5 us. The gun current is about one-third of its peak vaiue at 3.5 us when the current sheath reaches the muzzle. The /Vide shows that the gun absorbs almost all of the bank’s So far, trie only energy during the first 2.5 us of the discharge. diagnostics used in the flux conserver are magnetic field probes. When tne plasma emerges tne gun muzzle, these probes see a disturbance wnicn propagates at a velocity of about 10° cm/s into the resistive flux conserver. The strength of the initial radial magnetic field is found to strongly affect the downstream magnetic field pattern. For low rad.‘al fields the gun plasaa pushes the magnetic field configuration completely tirough the shell, whereas for high £ield3 the configuration barely penetrates. I-.-r intermediate values the configuration stops within Che flux conserver. This effect is best shown by observing the axial component of the magnetic field on the axis of the flux conserver for various radial field values. (See Fig. 1.) fur the rest of che data reported here the initial magnetic field inside tne inner electrode of the gun was set at 8 kG. This is che average value over the 20-ca diameter inner electrode. With this field strength tne field disturbance propagates into che flux conserver, scops wich reconneccion apparently occurring, and reaains wich little or no axial motion. This is possible, since che plasma can be almost stopped by the work required to stretch trie

Fig. I. Bz in the flux coaserver at 10 us after the voltage is applied to cne gun for initial magnetic fields of a) 4.5 kG, b) 10 kG and c) 19 kG inside the inner electrode of the gun.

field lines and the remaining motion can be arrested like the current ring in Astron because Che flux conserver is resistive.

He have a sec of 20 probes in one jacket mounted perpendicular Co the

axis of the flux conserver. By rotating the probes ve can measure either B or 3g. These field directions are with respect to the natural cylindrical coordinate system for the magnetised Marshall gun and flux conserver. Figures 2 and 3 snow 3z vs r and Bg vs r at various times as measured by these probes. These measurements snow an overall change in the magnetic field profile occurring over a period of roughly 10 us. A possible interpretation of these data is shown in Tig. 4. In this interpretation the compact torus is generated with its major axis parallel r.o the axis of the gun- 3ut then the axis of che corus rotates 90° to .aake a more “oblate” compact Corus. This tipping of a prolate compact torus has been predicted. The axis at apparent rotation is orthogonal to the probes’ jacket in che daca snown here and thus if Che probes are measuring poloidal field before the rotation as in Fig. 2, chey are measuring poloidal field after cne rocacion also. The same is true for toroidal field in Fig. 3.

-54-

3 vs r at v a r i o us flux c o n s e r v e r. The probes a re measuring p o l o i d al

tlmei measured 0.49 m from

f i e l d.

RADIUS

( c m)

the

Tig. 2.

the gun eati of

’ 1

^ 44

4 9 —**

Fig- 3. 09 vs r at various times measured 0.49 m from the gun end of ct>«

FLUX CONSERVER

— 0 a m -|

/-y-i-i4„.« j

-20

R A D I US (cnl

T0R01DAC

BT0R0I0AL

Q ^G UN B.

flux conserver. The probes are measuring toroidal field.

( A)

( B)

Fig. 4. One interpretation of the daca shown la Fi^s. 2 arid 3- ;’. Initially L.ie compact torus axis is parallel to the axis cf :r.e flux eanserver and the compact torus Is prolate, b) Later die axis of the torus Is perpendicular to the axis of the flux canserver and che compact torus Is oblate.

Finally, ve have another group of probes that have three coils at eacn position so chat all components of B” can be measured at each of seven locations. This group of probes Is all la the axial plane which is orthogonal to the jacket of the 20-probe set. Figure 5 shows data from these seven positions for the same shot as shown in Fig. 2. These probes gave very similar data on the shot used for Fig* 3 except that the apparent rotation occurred earlier in this case. The shot from which data for Figs. 2 and 5 came was not typical la that the rotation occurred very late. On many shots the field profile has the signature of a rotated torus a few microseconds after the 20-position probe receives a signal and a prolate compact corus signature is never observed. Although che time history may vary from snot to shot, tne apparent direction of the axis of che final configuration tends to be the same on most shots. However, on same shots the axis of rotation

PLASM* j

SUN

(C)

(A)

. r . r . r

-SUGGESTED INITIAL

*^FLUX CONSERVEB

FIELD CONFIGURATION

FIEL3 CONFIGURATION

-.SUGGESTED ROTATED

£ _r £ £

l_,

Fig. 5. Magnetic field data from seven multidirectional probes which are in the same plane* The + is the probe location and the line is the *‘ector representation of t pointing away from tne +T A) The initial poloidal fields are consistent uicti a prolate compact torus- B) This is during the transition period frcm prolate to “oblate;1 C) These final fields are the toroidal fields of the 90° rotated “ablate” compact torus. The * represents the position of the Jacket of the set of 20 probes.

appears to be In a different direction but the data still seem to be consistent with a rotated compact torus. The data shown in Fig. 5 are consistent with the interpretation of the stopping of a prolate compact torus and its subsequent rotation of 90°. For these probes the axis of rotation is io tne plane they define. Therefore, since this plane contains poioldal fields initially, it will contain toroidal fields finally. Once the final configuration Is reached it appears to be MHD stable and cne toroidal field decays away with a time constant of 40 us. The L/R time of the flux conserver is about 300 US-

Alt hough these data are preliminary, it appears that ttiere is considerable evidence for rotation of the configuration. Additional probes are being constructed so that a more detailed understanding of the history and spatial pattern of the magnetic field can be determined-

References

H. Alfven et al-,J- tticl. Energy, Part C: ?!* ?hys.

lib (I960).

”>. M. N. Rosenbluth and M. N. 3uasac, Sucl. Fusion 19, 489 (1979).

Myers, Levine and Plncosy report results using,a toroidal plasma gun.

The device differs from the usual coaxial plasma gun” in the use of a strong toroidal bias current for enhanced efficiency, a pair of disk-like accelera ting electrodes for reduced viscosity and a fast pulsed toroidal gas valve for more -jffective use of the injected gas sample. In addition, a technique is used for generating a toroidal current in the plasma ring. The combination offers an opportunity co deliver a plasma with a large amount of energy and to vary the density and relative toroidal and pololdal magnetic field inten sities over a range of values. It is the purpose of this paper to report further experimental results, to project the gun’s applications to the forma tion of a compact torus,and to propose a simple modification of the present apparatus as a test.

The gun consists of a pulsed toroidal gas valve (opening time - 50 micro

seconds) that injects a syjnaecric ring of gas between two annular plates. After a delay of about 100 microseconds, to allow time for the gas to distribute it self between the plates, a 4.4 microfarad capacitor charged to a maximum vol tage of 50 kV is connected across the plates resulting in a.half cycle time of 0.75 microseconds. Gas breakdown occurs in the breech of the gun in the region where the gas is injected and J. x J3 forces accelerate a plasma ring outward. This acceleration is enhanced by the application of a bias toroidal magnetic field and its direction is determined by she relative direction of the current in the plasma and the bias current. Thus, normally during the first half cycle of the capacitor discharge the plasma is accelerated radially outward toward Che muzzle of the gun,and during the second half cycle the gas in the gun is accelerated inward toward the breach which is left open in the gun.

-57-

Morton A. Levine and Philip A. Pincosy

FORMATION OF A COMPACT TORUS as DIG A TOROIDAL PLASMA GOT

Lawrence Berkeley Laboratory University of California Berkeley, California 94720

Tor,id»l Fi«lJ

I f»t «*» ViUc

L3t-10238

*This work was supported by the U. S. Department of Energy, Office of Fusion

Energy, under contract No. V-7405~E3G-68.

The total number of particles in. the plL-ma. ring is regulated by the fill

ing pressure °t the plenum in the gas valve and in practice can be varied from 1 0la co 1 019 particles (hydrogen).

It has been shown ia a computer program which solves a sec of coupled

equations for the driving circuit chit the dynamics of the gun are such that Che total plasma kinetic energy after leaving the gun is roughly constant (currently ber.

503 capacitor stored energy) over a wide range of particle num

(2>

-58-

if 1+1

v + I R + V

it j

JpUm r t ) T,

(4, £ -§

(3) f-v.

at M(R>|WI ” V 3R

R(c) is che radius of the plasma sheet, 7

is the pocential due to the

pt’loidal magnecic field, X0 is the initial radius at t * o, v(c) is che velo city of che sncvplow. M(R) is che total mass svepc up by the saowplov when it reaches radius R, m(R) is the initial mass distribution per unit length be tween the plates, and I(t) and I0 are the plasma and bias current respectively. Ks and L$ are stray resistance and inductance and d is the separation of the plates. C is the capacitance of the driver with a voltage V (t).

Early data was taken with accelerating plates 38 ci O.O., 19 cm 1.3. and

with a 3 a separation. The capacitor driver had a SC’flLAC type spark gap with a 2.1 \if capacitor, 50 kV and 150 nH. The bias current was a 0.3 aeg amperes.

In the earlier low energy test, plasma velocities of 15 cm/microsec to

20 cm/microgec were obtained. The accelerated plasma had 10*° to 6 * 10^ particles and had a Mach number of the order of 2. In this configuration the plasma could be stopped and crapped using a poloidal magnetic field in the form of a toroidal bicusp.

Measurements were made using magnetic probes, radiation detectors, Fara

day cups and a roveable laser interferometer. Particle densities and velo cities were inferred by assuming cylindrical symmetry from che interferometer data.

Velocities were measured from a series of shots in which che interfero- aeter was moved. Particle count before the plasma was stopped was inferred from the measured line density of electrons and a velocity inferred from the shoe series. These measurements indicated chat over 60S of the particles zams out of the gun during tha first burst. The magnetic probi neasurements showed Che first burst to contain a paramagnetic plasma with 2 kiloamperes to & ‘iviloaaiperes of toroidal current.

5topping and trapping the plasma in a poloidal magnetic field depends on

coroidal symmetry to close the currents generated in moving Into such a field. In fact, when che plasma is known to be assjrmetric, crapping does noc occur. Interferometer data measuring the trapped plasma indicates that essentially all the particles were trapped. This result coupled with the magnetic probe results seemed to justify the assumptions of toroidal symmetry.

More recently, in order to increase the plasma energy, a larger, low

inductance driver was installed in the system. This system has a cocal induc tance of 13.5 nanohenries for capacitors, switch, leads and gun; so that che new system shows efficiencies of greater than. 501 of the energy in the capa citor delivered to the first burst of plasma. This system under limited tests has given a plasma velocity of 45 cm/microsecond for * burst of 8 * l O1^ parti cles corresponding to a Maeh 6 plasma.

The use of a biad magnetic field parallel to che driving magnetic field

is a plasma gun has the advantage that it is inherently mora efficient and less dirty- This can be seen from Equation (1) where the driver term is pro- protional to (1^ + 2 1 1 ). Since the plasma current is introduced through electrodes, this interaction is reduced when I is reduced. Also apparent from Equation (5), small values of I lead to higher electrical efficiency for the same drive.

is the bias current and I the plasma current.

The disadvantage to using a bias current is Chat che current is closed

outside the plasma and the plasma is created within a coil structure. To form a compact torus it is necessary to remove che plasma from the coil by passing through spokes.

Removing the plasma from che coil may not be as difficult as it first

seems. Predicting Che behavior of the plasma is simplified if the plasma is moving at supersonic velocities. In the experiments so far performed with thj toroidal plasma gun there has been a range of Hach numbers of tioa 2 to 6. Higher energy plasmas can be expected to give even higher Mach numbers.

-39-

Consider a toroidal plasma with a toroidal field moving with a Mach

number, H. The toroidal field is produced by a current I flowing through the center of the system and returned through n_ bars each carrying I /n abamperes. We make the approximation that the magnetic field near the bars is given by B ~ 21 /rn Gauss where r is the distance from che center of each bar. The toroidal magnetic field in the plasma is approximately 3 - 21 /S Gauss. (The actual field is slightly higher due to a paramagnetic aifect in the plasma as is characteristic of the Tokamak configuration.)

AS plasma approaches the bars che plasma is compressed. However, che

disturbance cannot be communicated to the rest of the plasma because of che supersonic flow. A characteristic bow shock then forms wich a stand off distance such chat che magnetic field pressure near the bar balances^ the directed momentum of the plasma, i.e., 1/2 ;v* » 3 / 8^ or v - 3//4ic~ where v is the velocity of che plasma. Thus going through the bars disturbs a plasma of width 2r where r » 2I0/Bn. tn the limit of high Mach number and a chin plasma the fraction of plasma disturbed by the bars is f * 2nr/2TR . Using the fact chat S/B0 - M we find f » 1/Mff. Thus for large Mach number, the bulk; of che plasma passes through the bars unaffected. However, it is interesting

to speculate on the integrity of the magnetic configuration after passing thiough the bars.

Alfven’s theorem states that in a frame of reference, moving with and

within a plasma, the magnetic field remains constant. Thus changes in mag netic field geomecry are confined to the perturbances created at the bars where the plasma is cut. Since the cut is very thin, we can assume at some point down stream that the toroidal magnetic field will be symmetric and reconnection will have taken place. The question is thus one of asking in what distance or in what time this reconnection takes place. One notes that as the plasma flows by the bars, the field lines are beat, the volume available for the plasma to fill by flow along magnetic field lines becomes larger and larger. One also notes that there is a magnetic x-point or line in this region behind the bars. It has been shown that in the region of an x-point reconnection can take place at the Alfven velocity if it is accompanied by flow along magnetic field lines which drains plasma from the region of recon nection. Thus, in the region behind the bars reconnection can be expected to proceed at the Alfven velocity. Since the plasma in this region can be ex pected to be subsonic and reconnection fast, this implies that the magnetic configuration will be effectively unchanged as the plasma passes through the bars.

As mentioned above, the current apparatus includes a large toroidal field

coil surrounding a smaller pcloidal field coil which is used in the plasma trapping experiments. The toroidal field coil has 16 turns. These turns can be shorted using thin rods near the mu2zle of the gun. This would then leave the bias current and the poloidal field coil structure intact. Tl—<s, a plasma produced by the gun would have to pass through the coil structure before interacting with the poloidal field. However, after passing through the coil, if the plasma maintains its configurational integrity it will not be able to penetrate the poloidal field and a sharp boundary of penetration of the plasma will be observed. However, unlike the current experiment where the plasma is trapped in place, in the proposed experiment the plasma can be expected to bounce back toward the gun. If this experiment is successful a very simple extension will lead to the formation of a compact torus.

-60-

Lawrence Berkeley Laboratory, Report LBL-9507

~Z. Marshall ia ”Plasma Acceleration (S. Kash, ed.) Stanford University Press,

icanford, California (1959).

v. lunas, Review of Geophysics and Space Physics, L3, 303 (1975).

RECONNECTION CONDITIONS FOR FLOWING FIELD-REVERSED PLASMA FROM A PLASMA GUN

J. W. Shearer, J. L. Eddleman, and J. P.. Ferguson, Lawrence Liver-more Laboratory*

Although various theoretical models are known for magnetic

reconnection, U)»(2) none of them are applicable to the BETA-II experiment because they do not model the plasma flow conditions. The coaxial gun produces a plasma stream with an axial v e l o c i ty of —lO^ cm/sec. to move in the opposite direction at this sane v e l o c i t y.

In the plasma rest frame an aperture or snipper c o il would appear

A more suitable model for the 8ETA-II reconnection problem was

formulated based on the equations of f l u id flow through a nozzle. Assuming a sharp-boundary rnc^l adiabatic f l u id

leads to an equation for the axial plasma velocity u:

for the plasma { f i g. 1), B e r n o u i l l i ‘s

law for an

u - um [l — (P/Ps) U - l / - ) J l /2

where um is the maximum v e l o c i ty (at ? = 0 ), and Ps is the stagnation pressure. The plasma density p arid stagnation density os are given by

where f

flux

:*.

-51-

-s = fj

( P / P ) l / 2. and

and f i e ld B is B = (oQ + 2 S ^ / T I^

is the open f i e ld

2 ’ r 3 D /a

B)1 /2

’

Using the long thin approximation for the magnetic f i e l d, the plasma pressure P for the sharp boundary model equals the magnetic pressure

P = B ^ / 8T

where R is the wall radius, o0 is the field-reversed f l u x. The mean radius r of the plasma i s:

line f l u x, and ~j

r = (R/y’l) (l - VT

!4;

The plasma thickness 2a is assumed to be small compared to the radius r, and is obtained from the equation of continuity:

a = F/(4-rr u a)

is the mass flow rate (gm/secj.

The diffusion of the magnetic f i e ld causes -a loss of

f i e l d - r e v e r s al

c^. From Faraday’s law one obtains

= i =

“.iorie” oerforned under the auspices of the U.S. Qeoartment of Energy oy the Lawrence Livermore Laboratory under contract number ’!-7A05-ENG-<i8.”

(5)

(6)

(3)

(emu)

where the diffusivity D = n /4 - , i being the electric resistivity. A simple approximation for n is

where the term in brackets is the classical collisional resistivity/4) Te is the electron temperature in eV, VQ is the electron drift velocity, and v-j is the ion thermal velocity. The exponential term approximates the effect of unstable anomalous conductivity effects near their threshold. As these turbulent effects try to grow, the increased diffusivity 0 causes the current sheath to broaden, thus lowering the electron drift velocity vn, and stabilizing the current sheath.(5) The electron temperature Te is fixed at 50eV, in agreement with plasma gun experiments. It is assumed to be constant throughout the flov due to its high parallel thermal conductivity. The ion temperature Ti is much higher (T-j >> Te), so that the adiabatic equations (1) - (3) apply to the ions only.

These equations were combined into a differential equation for the

rate of change of pressure with axial distance z.

-62-

8 r D P

/’ P \1 /Z

term approximates the clipper coil effect, and the

In this example,

d °o

—r

) m

(i _ |-

,ft, ^

(9)

Figure 2 presents r e s u l ts for one example, where Y = 5 / 3, um = 100cm/u sec, zm = 2C0cm, R = 20cm, F = 6gm/sec, Ps = 2 atm, P0 = .25 atm, and Te = 50eV.

(MG -cm2) = 4* -L- n?

The largest flux losses are found to occur in the low field regions where the plasma density is lowest (Figure 2 ). Additional calculations showed that the flux losses increase strongly with decreasing mass flow rate F. These trends were traced to the sheath-broadening effect of the anomalous resistivity factor in eq. (7^.

Figure 3 shows a two-dimensional MHD calculation^) 0f a similar

problem, but here the clipper coil flux *a is time-dependent. At t = 11.2 ^ sec one of the plotted flux surfaces has reconnecved near z = 152 cm due to magnetic diffusion downstream from the clipper coil. The velocity plot shows strong axial velocity divergence in that region because the upstream flow ;; being stagnated by the clipper coil. This lowers the mass flow rate F near z = 152, thus enhancing the anomalous diffusion effects, with consequent enhanced reconnection.

—z - T^r ( —)

where the f i r st second term is the diffusive e f7e c t.

NOZZLE ANALOGY

In sunmary, low density and low flow rate regions experience

enhanced d i f f u s i on which broadens the current sheaths. When the width of these current sheaths approaches the scale size of the plasma, reconnection upstream flow, but the reconnection region is downstream where the flow rate

is rapid. A clipper c o il compresses and stagnates

is minimal.

References

Vytenis M. Vasyliunas, Reviews of Geophysics and Space Physics 13,

No 1, pp 303-336 (1975).

V. D. Shafranov, Nuclear Fusion 19, No 2, pp 187-193 (1979).
E.R.C. Miles, pp 1-30 of “Supersonic Aerodynamics,” Oover

Publications New York (1961).

Lyman Spitzer, Or.,p 143 of “Physics of Fully

Ionized Gases,” 2nd

ed. Interscience, New York (1962).

S. Hamasaki, R. C. Davidson, N. A. KraTT, P. C. L i e f e r, Nuclear

Fusion 14, No 1, pp 27-33 (1974). J. L. Eddleman, C. W. Hartman, J. W. Shearer, B u l l e t in of the American Physical Society 23, No 7, p 343 (1978~T

the

6 3-

F1GURE 3

2D MHD CALCULATIONS OF REVERSED-FIELD PLASMA FLOW

Aperture-, Mpercure-.

Plasma gun muzzle

. ..

*S-M

-64-

EI £ i

Axis

2= 152

8 » i i s s ; | j ° s 33

—’-’ : » ; Y--f-‘4L-J

i i i i i i i i i i i i i i i i i i i i ii

0j = 1.0 MG-cm2

0n = 14.1 Sin £(t-10.4)

z* 142 ’ _ » j u« ’ « * ’ .’ 17 »_; f « « * 1

Clipper coil /—supper con j ; ; n i i y v i < i r M t 8 i i m m H j i i i,

3 i i ! ! 3 U ! l i ! i l i i 3 ! HH ; ) i i i t : c i t i i i a i ; : * : t i ii

i i ! ! ! ! i : i i i H ! H H ! I iI ! i m i ! i i i i i i 9 i : ! ! : ( in

U i n n H i i i H H U n i i Mi

!33333!«3J3333HmiUHH

!333333m33333333!3333333

3 i J 3 3 I 3 1 3 3 i ! i 3 3 H I 3 i 3 I ! lI

. - , . . a - .J. . 9 - . t . . ^ - , , … * * > . -.

2= 142

z= 152

~ , . r - i » , . . ?i

. « . - . >. « <-

« Wl

’« HI I

rAg

: t = io.8

3 t= 11.2

It is commonly known that stable toroidal magnetic plasma confinement

requires both poloidal and toroidal magnetic fields. In axisymraetric con finement, the poloidal field is generated by toroidally directed plasma currents, while the toroidal field is supplied by external coils (tokaraak), plasma paramagnetism (spheromak), or a combination of the two (reversed- field pinch). In tokamaks the dangerous m~l current-driven kink is stabilized by periodicity (qil) while average magnetic well stabilizes the interchange modes. This q-limic drives tokamak designs toward low aspect ratios in order to obtain 3 values sufficient for fusion. In reversed-field pinches (RFPs) a conducting shell and magnetic shear f q< il) provide stability. Since toroidal periodicity is irrelevant in RFPs, ”cylindrical” tori of high aspect rati: are preferred. Relatively compact fusion reactors have thus been predicted, taking advantage of the high engineering 3 and the likelihood of ohmically heating the plasma to ignition.

The poloidal field may also be produced independently of plasma currents

in helically symmetric torcidal systems by the combination of the magnetic field from a helical winding (helical field) with a strong externally generated toroidal field to provide rotational transform. This is the basis of stella- rator and torsatron devices. Like tokamaks, they are limited to q^l and to low

It is also possible to provide transfora through the combination o poloidal field ( q « l ), such as that the basis of OHTE confinement. The these toroidal systems are illustra

a net toroidal field by translational f a helical field with a predominantly of a paramagnetic pinch, and this is similarities and differences among ted in Figure 1. Ideal MHD stability

q «l

PHYSICS OF IKE OHTE

T. Ohkawa and the OHTE Croup General Atomic Company, San Diego, California 92138

AXISYHMETHIC

TOKAIUH

TOROIDAL MAGNETIC CONFINEMENT

Fig. 1: A comparison of tokamak, stellarator. RFP, and O H TE configurations.

in low-q systems at high -: requires high shear throughout the plasma. This is achieved only if q(r) decreases memo- tonically and reverses, and in fact, RFP experiments yield stable plasmas at higher J than stabilized pinches with out reversal. The reversal of q in RFPs is achieved either by fast programming of the toroidal field during plasma setup or by self- reversal when the pinch ratio 3 = BB/<8 > at the plasma bounBary exceeds a profile-dependent value typically between 1 and 2.

However, in KFP experiments self-reversal is driven either by turbulence (hence poor confinement) or by a reversed toroidal E-£ield (Poynting vector out of the plasma, hence a transient situation). In OHTE a steady-state d-reversal is independently imposed on the outer portion of the plasma by the evanescent helical field; q in the interior is determined by the plasma current profile and paramagnetism as in a conventional £FP. Since the OHTE aspect ratio may be large, it is technically feasible to drive large plasma currents inductively for long times. Calculations show that ohmic heating to ignition will be possible if energy loss is not much worse than Alcator n a2 empirical confinement. Because of the large shear, S is expected Co be high. Therefore, an OHTE reactor should be smaller and simpler than a takamak.

Ideal MHD numerical calculations in the straight helical plasma approxi

mation with varying degrees of paramagnetism and a range of profile parameters have been performed for 1=2 and i»3 configurations to map the parameter space of OHTE equilibria. Figures 2 and 3 illustrate typical 1=2 OHTE flux surfaces and pitch (F=qR) versus helical flux profile, respectively. Numerical equilibria are being tastsd for local, pressure-driven ideal interchange stability using the criteria given by Mercier (1) and Greene and Johnson (2). The minimum energy force-frte helical pinch analytical equilibrium given by Taylor (3) is a special case of che 2=1 OHTE.

A+

) ’-

F:g. 2: Typical 5=2 calculated flux surfaces.

OHTE

Fig. 3: P!ot of a typical pitch versus flux profile. Fc and F0 are centra! and plasma boundary fluxes, respectively.

Field line tracing was used to study the loss of q-reversed volume owing

to error fields. both a uniform vertical field and a local stray field up to 7”, of the poloidal field at the separatrix.

In this calculation, the transform reversal persisted for

Free boundary current-driven i n s t a b i l i t i es in helical plasmas have been in Che

investigated by two techniques. The first circular mode equation =iven by Robinson (J). enhanced pitch reversal tends to draw resonant surfaces transforming some kinks into tearing modes. A stabilizing contribution for kink modes is a l-o obtained for a given current profile. The second technique is to analyze

It is found that helically inward,

is to use averaged fields

the helical free-boundary force-free equilibrium TxB»uS in a circular con ducting cylinder, adopting the approach of Rosenbluth and 3 .ssac (5). The analysis can be formulated analytically when the perturbation has the same periodicicy as the plasma. A shift of the mode resonant surfaces is again found, and the stability condition is the sane as for a mode of the same helicity in a cylindrical equilibrium.

The OHTE experiment is under construction and is expecced to begin

operation late in 1980. It consists of a. single, thin stainless steel bellows vacuum chamber bakeable to 125 C and pumped by turbomolecular pumps. The major radius is 1.24 m and the maximum internal minor radius is 0.19 m. The vacuum chamber is Surrounded by an aluminum alloy conducting shell 15 inn chick, with internal ninor radius 0.20 a, which provides stabili zation against plasma kinking. The flux penetration time through the shell is approximately -iO ms. The shall also supports and aligns the helical winding, which is assembled by bolting together machined copper half-turn sejiuents- An c»3, m«16 helical winding driven by a motor-generator-reccifier system was selected for the initial experiments. A toroidal field up to O.i tasla in either direction can be produced by supplying unequal currents to the positive and negative helical windings. A separate vertical field system is used to null the vartical field resulting from this mode of toroidal field generation, maintain plasma radial equilibrium for long discharges., and cancel the local stray fields at the conducting shell gap when radial equilibrium is maintained by fcnage currents in the shell. Plasma current is driven inductively by an independent coil, which is rated for a 2.25 weber flux swing {unid’rac- tional current) to allow for ar least 200 ms discharges at 300 kA. The coil is designed for either inductive storage or externally driven taodes of operatirr.

It is expected that Che filling gas will break down inside a saail dia

meter i-3 stellarator confinement volume produced by the strong helical fi^ld and a weak toroidal field. Current rise time will be 2 to 10 is, and gas puffing will be used in attempting to program plasma density. If the elec tron energy loss obeys the empirical Alcator law, it is expected that car.rral plasma temperatures will be vj. keV.

J. M. Greene and J. 1. Johnson, Phys. Fluids _5, 510 (1962).

J- 3. Taylor, Third Topical Conference on Pulsed High Beta Plasmas (Culham, 9-12 September 1975), Pergamon Press, 59.

S. C. Robinson, Nucl. Fusion 1%_, 939 (1978)”-

‘A- ::. Rosenbluth and y.. M. 3ussac, Sucl. Fusion _19, 439 0197?^.

References

f-. Mercier, Mucl. Fusion _1, 47 C1960) .

FOHMAIIO’t OF TOROIDAL PLASMA CONFINEMENT CONFIGURATIONS S^ USING KOT £L£C?30:iS *

C. Vi. Hartman, Lawrence Livermore Laboratory, and W. A. L e v i n s, Lawrence Berkeley Laboratory

hot e l e c t r o ns t i on c u r r e nt ean be employed u r a t i o n s, as ia E 3 T ,’ or to i i w r e a se t he a l l o w a b le S of a c l o s ed s y s t em as in T o r i a e .3 A l t e r n a t i v e l y, hot e l e c t r on c u r r e nt a l o ng B, n e a r ly d i s- sir.ation2.e33 if c l a s s i c a l, can be used fi__ds as in SPIBE.4 to be iariune from u s u al MHD i n s t a b i l i ty can p l ay an i n p o r t a nt

to shape hydromaanetieally u n s t a b le f i e ld c o n f i g

the tendency of h ot e l e c t r on p l a s aa

t he h ot e l e c t r on d i a a a g n a t ic

to “construct™

r o l e.

t o r o i d a l, p i a e e - l i ke

f or e c o l o y i ng

p i n c h - l i Ke c o n f i g u r a t i o ns with hot e l e c t r o ns s t a r t i ng from a c o l l i s i o n- l e s s, mirror-confined-, hot e l e c t r on plasma. The f i n al c o n f i g u r a t i o ns a re d i r e c t ly r e l a t ed

to Spheromak, Tormac, and t he s t a b i l i z ed Z - p i a e i i.

t o r o i d al

I I.

6 8-

iaTaODICTIOH

In e i t h er case

Thers e x i st a v a r i e ty of r e a s o ns (and c o n f i g u r a t i o n s)

This r.ote d i s c u s s es p o s s i b le means of producing c l o s e d,

I n i t i al Plasma and F i e ld Shape

to _lrj s i tu h e a t i ng of dense p l a s m a ,’

in t o r o i d al or m i r r o r - t o r o i d al h y b r id c o n f i n e m e n t.

the t o r o i d al hexapole mirror ( F i g. 1 a ),

I I I. T o r o i d al Pinch Formation

T y p i c a l l y, at

S t a r t i ng with

In a d dJ-

t he f o r a a c i on Of

to c u r r e nt o t h er

than by hot e l e c t r o n s . ” A d r i v i ng f l ux

the p o i o i d al f i e ld c o n f i g u r a t i on of F i g. 1a to

a hot e l e c t r on pinch has been d i s c u s s ed e a r l i er n e g l e c t i ng a ll c o n t r i b u i n t r o t i o ns duced i n s i de t he x - p o i nt r a d i us which induces to modify which i3 q u a l i t a t i v e ly subsequent buildup of c u r r e nt and c o l l i s i o n l e ss r e c o n n e c t i on of l i n es in an a n i s o t r o p ic plasma can be seen from a s i n g le p a r t i c le v i e w p o i n t. Guiding c e n t e r _ t r a j e c t o r i es shown in F i g. 2 d r i ft p e r p e n d i c u l ar 3? with v e l o c i ty VQ = Et X 3p/ 3| as ia Ware pinching in

the same as t h at obtained in t he T o k a p o l e .D The

t o r o i d al p i n ch c u r r e nt

tokamaks

f i s ld

t h at shown in F i £. 2

“rfc.-k performed under t he a u s p i c es of

the Lawrence L i v e r s o re Laboratory under c o n t r a ct number W-7A05-H2IC—^iS.

trie U.S. Department of Energy by

The ” p r e h e a t” phase c o n s i s ts of e s t a b l i s h i ng a m i r r or c o n f i n ed h ot e l e c t r on plasma e i t h er by well-known ECR h e a t i ng or by i n j e c t i ng e n e r g e t ic e l e c t r o ns from a pulse s o u r c e. f i g u r a t i o ns a re shown in F i g. 1.

P o s s i b le a x i s y a r a e t r ic i n i t i al m i r r or c o n

t he end of ECS h e a t i ng or i n j e c t i o n, a h ot e l e c t r on

plasma could be produced with p a r a m e t e r s, neh = 1 011 - Teh = 0 . ’ - 0 .5 MeV, and & a 0 . 2 - 0 . 7. to c a r ry c u r r e nt at VQ = c, c u r r e nt d e n s i t i es of 500-5COO A/cm^ would be a c h i e v e d.

If a ll of nen c o u ld be d i r e c t ed

l O1^ cm-Sj

it the X-ooitit the bannana undergoes

further drift c a r r i es the bannana into

when the driving flux ‘1/ is changed, a topological change (Fig. 2c). tokaaak-iiKe orbits which drift orbits as show: in Fig. 2b. Passing occurs when e i t h er the points merge or when the bannana flattens to eneonpa33 the O-pointt, depending en the particle gyroradius and the field d i s t r i b u t i o n. Zlee- trar.s on passing orbits are then accelerated by St contributing current to pinch Suildup. Outside the closed region alaost a ll electrons o s c i l late in v and j fore that field reconnection would be uninhibited by the hot e l e c t r o n s.

towards the 0-point to e a t er passing

is a purely diamagnetic current.

It would appear t h e r e

r e f l e c t i on

-69-

IV. Spheromak/Tormac Fornatlon

the pinch would be a t t r a c t ed

the external toroidal f i e l d,

In ic the gun would be fired with the emerging plaaaa and ee’oeddsd 3-r

The formation of a Spheronak/Toraac configuration begins wit’n the Levitron-like, Min 3 mirror shown in Fig. 1b or the gun configuration of Fig. 1 c. To fom an isolated, closed field region in F i g. in the two small radius coils would be increased as showi in r i g. 3. Fig. stretching the field guiding center orbits (similar for both Fig. lb and is) undergo topolog ical transitions as i l l u s t r a t ed in Fig. 3. Additional e l e c t r o ns trapped against the toroidal field gradient, have bannana o r b i ts wcich do -ot intersect the aidplane. These orbits undergo the t r a n s i t i o ns described: e a r l i er into trapped, tofcacalc-lilce bannanas and bannaaas trapped on the small wajor radius side of the midplane ring.

to cause reconnecticn and an i s o l a t ed r i n g,

lb the current

The orbits shown in Fig. 3 a ll make a transition to passing electrons

in the pinch region and are accelerated by the flux change of the s a a ll radius rings to increase the pinch current. current if Fig. lb is turned off, outer rir-s and the outer s-point would flatten foraing a Toraac bi’tusp configuration iabedded in a hot electron, mirror confined plasma.

If the s a a ll radius ring to the

If the outer ring is levitated, plasaa foraation without a vacuus toroidal field becomes possible. For this case the e l e c t r on guiding center orbits collapse to zero poloidal width and nearly coincide with poioidsi flux surfaces. The transition in orbit topology at the x—point takes place by nonadiabatic scattering at the x-point field lines reconnect in the presence of both hot electrons and cold plasma. The resulting pinch (or field-reversed-atirror) current vould be carried largely by diamagnetic current of the hot e l e c t r o n s. The ring without 3^, would be expected to be hydromagnetieaslly u n s t a b l e. However s t a b i l i ty may be achieved by the tendency of hot electrons to be isaune from MH) and other i n s t a b i l i t i es in the presence of dense cold plasaa when the precessionai drift frequency exceeds the ion gyrofrequsney.

’ 30 - Q; and the

electron

the

Alternatively, if a toroidal field is necessary for s t a b i l i t y,

potential of very long l i f s t i ae of the hot electron ring3 (‘-10 seconds) makes it possible to turn off remove t-.e winding, and reaove the ring on a technically nondeaanding tiicescais-

0.A. Anderson et a l ., Fourth IAEA Conf. Piasaa Phys. Contr. Nuc. Pus. Hea., 1. p. 103 097T).

H.A. Dandl et a l ., 7th IAEA Conf. on Plasma P’ays. Contr. Mue. Fus. Hea. (1978).

M-A. uevine et a l ., Seventh IAEA Conf. on Plasna I’nys. Contr. Hue.

Fus. Res., I I, p. 81 (1978).

C.W. Hartman, Phys. Rev. Letta. 26. 326 C197D.

A. P. Biddle et a l ., OT report C00-2387-103, C1979).

IS|

7 0-

Haferencss

iip:% mirror ‘:r-M V i suaactrt eui-r»nt laao* ina 3*. ‘tj ‘Mi 3” ai^rfl** it !fl {*) and JcJ nirr-cr

itLi To«Vd«J Ht»*pflk H”..w

fO ^mn«ix»iC»,«i«l Gu«.

F*g. 1 la)

fa.} Gu.id»«-C««i”«r “7t-Vtc.tof-y

71-

’ 1

1^>W

I u i i ’ i i f n^ Canter d r i f ts

<ji*-r»;fc

f ar F i n.

-j

l a.

=’

*’

‘I

’*%.

„ f >> r & *

‘{g. 3 SuiJlng-C3ni«r Orbits ’ ir F i j. !b.

PARTICLE-FLUID HYBRID SIMULATION OF FIELD REVERSAL IN A MIRROR DLASMA

I. Cohen and T. A. Brangle, Lawrence Livermore Laboratory

Livermore, C a l i f o r n ia 94550

Faraday’s

A one-dimensional, e y l i n d r i c a l ly synroetric hybrid simulation code has

been developed to investigate f i e ld reve-sal in a mirror plasma. The f l u id

momentum equation is used to determine the electron current density with

n m0dv0/dt = 0 and n

integrating p a r t i c le equations of motion. Momentum-conserving electron-ion

drag is included.

Ja + j|

is used in Ampere’s law to determine the magnetic flux V = r Aa, and from

law £„ = -3Aa/c3t. On open f i e ld Tines

l^eD’< * or 5 I . and

e l e c t r o s t a t ic effects on ion orbits are negligible for T « T..

However, alectron temperature clamping and shorting of e l e c t r o s t a t ic

need not occur on closed f i e ld

lines. Good parallel electron conductivity

f o r c e s; * ’ - { y ); the code incorporates this by flux-tube averaging:?” dJE.3” ’.

‘4-J + 3E/?t) = 0 -* = **(’*). Of particular

in the simulation are

-elaxation of elect-on return currents apposing ion-beam i n j e c t i o n, the role

of Ohkawa currents in sustaining f i e ld reversal, and large ion orbit effects

on the diffusion and relaxation of an i n i t i a l ly f i e ld reversed equilibrium.

-72-

Ion current

n ,. Large-orbit

ions are followed by

is accumulated on a radial g r i d.

interest

f i e ^s

*Open f i e ld

e°

*‘r2 - TM5

P^ysics ingredients

‘Axisymmetry (?/3e * 0) and 35 = 0

P a r t i c le ions obeying Newton-Lorentz equations -** J

“Masslass” f l u id electrons with drag on ions - d * ^ , *)

*Limited use of q u a s i - n e u t r a l i ty

7 3-

c Je

lines

lines —

in code:

loss hole

l i n e - t y i ng

< <7,i o ne v1 o n>

ta ~ =>t

in electron momentum eq-

l i m i t er and c o n t i n u i ty of A

conducting radial

A., i> - E, 3*

ions

and

Electron temperature evolution

ion velocity-^pace

*Closed f i e ld

Simultaneous solution of

*Injection of hot

*Boundary conditions:

and f at seoaratr ix.

J kJ 7; [ 4^ J1 0” + Je(Ae.=> - ^ 7r 2,] = o.

Code eauations:

particle ions:

“massless” f l u id electrons:

where Pfi = n je.

j nA

-74-

.**

4 / 2 T “nea

.- = ^—==

( xs, vx) - Js> ns

f + n ^ T ^- T^) bb

ue ic; V s( <^- V]

Eulerian mesh s = species index

Q = -en E -en v x S/c - 7- p„ e e

.» _ .» „ !!| * v(E * vs / c) + ™sV vV 3 at

T = e CT ( f l u x, time) due to convection and

J = n.eV, = nonlinear function of E and 3

Coulomb c o l l i s i o ns with ions on closed

( c l a s s i c a l] or anomalous

or Pe = Pe (J-,t)

( l i n e - t y i n g)

b = 8/B

ne = c

lines

*pold

f 1 e ld

and

] in es

i /g

Q p an

f i e ld eauations:

For axisynwietry (3/33 = 0) and B; = 0, S = --- and

f Boltzmann-like on open f i e ld

/ determined on closed lines by

-» ** B = 7 x A9

and K-. electron momentum equation-*

minuscule for e $ / T ^o ld = d ( l)

-**

^ Es

*we.,

1 3*

lines

= * c “3t A9

on open f i e ld

lines - negligible for Tr’°

secondary dependence

= ”‘“a ^

e ff”

So far only a Up_ radial version of the code exists.

mesh is used.

m v^on

Solve equations i t e r a t i v e l y. We guarantee continuity of Aa and 3 at

separatrix. There is a conducting wall at the radial

l i m i t e r. An Eulenart

Work performed by Lawrence Livermore Laboratory for U.S. Department of Energy under contract W-7405-Eng-48

TWO DIMENSIONAL TIME-DEPENDENT TRANSPORT IN FIELD REVERSED EQUILIBRIA

S. P. Ajerbach, H. L. 3er’<, J. :<. 3oyd, B. Mo’lamara, D. Shumaker Lawrence Livermore Laboratory, Livermore, CA 94550

Calculate transport coefficients and flutes relative to (moving) magnetic surfaces using transport model and equilibrium fields

fentrosy and safety factor? and v a ^e

Goal: Study buildup, steady state and decay of field-reversed mirrors, with emphasis on studying effects of neutral beams.

Method: Our method is very similar to H. Grad’s “V-i-Q” method, with some techniques from N. 3yrne and H. Klein’s G2M code. Each time step is carried out in two stages, a transport stage d.id-an equilibrium stage, as outlined next.

-76-

TRANSPORT STAGE

EQUILIBRIUM STAGE

Update adiabat-ic variables of j !; at 0—Point (*; = 0 on plasma boundary and z-axis)

i , using Grad’s Ps 0 method

Fig. 1 Computational domain

is r.o trarsDort.

tnere

The computation domain is shown in Fig. 1.

0-point —

The transport stsaq solves equations for

the adiabatic variables

p a r t i c le nurse”, q i t r : - ;/ conserved if

jna safety -actor. The adnbatic variables ?.r;

In c e t a i l, »& solve the f o l l o w i n g:

Calculate new equilibrium with updated adiabatic variables and range of

< ;Qr

“r

’*

P a r t i c le Number:

EntroDy:

Safety Factor:

In these equations

D/Dt

<cr>

<*>%>

<qr-?V>

<ur’ f i>

5/3

-77-

ld i f’

. ar

t < »r

d V2 /3(l

_± 4«

4^, JZl

drr d^

dV, dip’

0-point 0-point

*Rr + as

m V y :\

3 W

= 3/Stl^, .

. re = l i. D

S = P r

2 d I d?

Flux at 0-aoint

fur d y V

°2. - __L-_dV <r.g> i ll <F-«>

3 r S

denotes averaging over a constant -{J surface

f l ux across (moving) constant-* surface.

= p a r t i c le source due to beams

= entropy source due to beams

= work against drag force R

= tempe.-ature e q u i l i b r a t i on

= p a r t i c le

= species

= heat

label

term

f l ux

a a

is used.

The last & terms above are computed from whatever transport model At present, we have in place the f u ll surface averaged a r a g i n s k ii equations

s poloidal f l u x, V = volume ins’de a constant-J’ surface.

The e q u i l i b r i um stage alternates between solving the 2-0 equilibrium

equation (3; = 0 for simplicity):

and the surface averaaed eauilibrium equation

with

7 8-

” dV

P = P(*)

S = S(U0

f = <KV)

= Hr,z)

elliptic boundary conditions

Boundary conditions at plasma edge and at 0-point

h ««>$- “f (m [|»5’3)

We iterate between these until consistency is achieved.

Some special features of the code: a)

2-0 equation solved with 5-point ICCG algorithm on non-uniform r-z grid. We allow boundary conditions at infinity, or specified coil currents. We can deal with moving plasma boundary, and separatrix touching the r-axis. Surface averages near the 0-point are computed by fitting a polynomial to li near the 0-point, and using analytic formulae for averages.

Present status of the code:

A) Equilibrium: the equilibrium solver and the lh D algorithm have been

extensively checked, using values of S( ^) computed analytically for Hill’s vortex; i.e., we put in S{<!/) and checked that the self-consistent solution reproduce Hill’s vortex.

B) Transport: the transport stage is working, though more testing is

required.

C) Interface: the interface between the transport and eoui1’brium stages

is still being tested and debugged.

Acknowledgement: We would like to thank Dr. H. Grad for many useful conversations.

“Work performed under the au^m’ces of the U.S. Department of Energy by the Lawrence Livennors Laboratory under contract number W-7405-ENG-43.”

-79-

A STEAOY STATE BEAM DRIVEN FIELD-REVERSED MIRROR

J. H. Hammer and H. L. Berk, Lawrence Livermore Laboratory, University of California, Livermore, California, USA

< J- en

We present here a method for sustaining a current in a beam driven field

reversed mirror. It has been shownl that attempts to drive steady state field reversed configurations (or Tokamaks, Spheromaks) with neutral beams may be foiled by electron currents. The plasma electrons are accelerated by collisional interaction with the beam-produced ion current, causing the net current to vanish asymptotically in time.

We show that the current cancellation can be avoided by weakly breaking

the toroidal symmetry of the device, for instance by the addition of cjuadrupole magnetic fields. The synmetry breaking allows angular momentum transfer between the electrons and the external coil structures, without having plasma in contact with material surfaces. The momentum transfer is manifested as an additional drag force on the electrons, preventing then from accelerating to the ion velocity and cancelling the current.

The detailed mechanism for transferring momentum is magnetic pumping (synonymous with parallel viscosity, non-adiabatic pressure effects). The injected beam causes a toroidal flow which convects tubes of poloidal flux Choroidal = ° f °r this analysis) in the i> direction. The flux tubes undergo a periodic compression-decompression as they move past the tnroidally varying structure, causing electron pressure modulation. If the electron collision frequency, double-adiabatically, i.e., the magnetic moment and parallel velocity are conserved, at least in the long field line limit. If the electron collision frequency were large, ve/ ^ .o t, i on > > 1, the pressure variation would again be adiabatic, Pe - nes/3. In the intermediate regime, 0 < ve/.>0tation < =>, the pressure may vary non-adiabatically, causing a net transfer of energy from the flow to electron thermal energy. Corresponding to the energy transfer is a force which damps the flow. This force is the extra drag force on electrons that prevents current cancellation.

>Qi were zero, the pressure would vary

To see how the extra force arises, examine electron force balance:

-80-

i * 7P. > + E, + u„B = ^Jo $

J = flow normal to flux surfaces n

‘This work was performed under the auspices of the U.S./ Departrtent of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG—18.

If we take the toroidal average of V e ne x (toroidal component) of this equation ne obtain:

Pg + ene(E_ + ug x 3) = menevei(uj - u_e)

This is always of the correct sign to prevent

is the usual Ohm’s law except f or the pressure term, whlch^might be

This described as a thermo-electric e f f e c t. For symmetric systems, $ * 7 Pie = 0, so t h is ne would be 90° out of phase giving zero average. When c o l l i s i o ns are included, a phase s h i ft arises that gives a non-zero average. We f i n d:

term vanishes. For adiabatic non-symmetric systems, % * 7 Ple and

8 1-

3e n u.

f or v . r / u. < < 1

current cancellation.

Using a fluid ion model and the long, thin approximation self-consistent

steady state equilibria have been obtained by including the extra drag term in Ohm’s law. The size of the synroetry breaking perturbation, e, must typically be of order 10*1 to obtain an equilibrium. The directed velocity of the beam must be of the order of the ion thermal velocity and the required particle input rate of the beam corresponds to a particle loss rate close to the classical diffusion rate. Explicit substitution of experimental parameters into the theory for devices at LLL (Beta II, MF17) indicates that steady state equilibria may be maintained using the present beam current and voltage capabilities. The necessary symmetry breaking can be provided by the existing yin-yang coils. A limitation on available facilities, however, is the need to run the experiment for times longer than the resistive decay time in order to test the existence cf a steady state.

Two other considerations ara necessary to complete the theory. First,

in the region within an electron orbit size of the field null, the electron fluid equations break down. This is an important region s4nce any ecin’librium must have finite current at the null. Near the field null it -in be shown that she3>- viscosity affects become important. Estimates

idicate that the shear viscous force alone is sufficient to prevent current

c-nee nation in this region.. Second, the quadrupole (or other symmetry :reaking) field must be matched to the solution. Implicit in the a.iaiysis nas been the assumption that the quadrupole field in the bulk of the plasma is negligible Penetration of the quadrupole field is deleterious to any field-reversed scheme since the field lines are opened up and the advantages of closec flux surfaces for confinement are lost. In this regard the equilibrium flow plays an important role- The toroidal flow “sweeps” the ouadrupole field out of the main body of the plasma, confining it to a boundary layer of order r^l yfT^ in thickness. cp is the plasma radius and Rn is the magnetic Reynolds number Rra - u0rpA i, typically R,n > >1. A boundary layer analysis shows that the vacuum ouadrupole fields can be smoothly matched to the interior solution where thu

D. r Baldwin and T. K. Fowler, Lawrence Livermore Laboratory Seoort ‘JC1- 17691 (19771.

idrupole field vanishes.

Ref -snces

and

CALCJLATION OF IDEAL MHO GROWTH RATES AND E I G E N R J N C T I O NS IN FIELD REVERSED ERRORS IS THE LARGE TOROIDAL MODE NUMBER LIMIT

V. Anderson and W. A. Newcomb, Lawrence Livermore Laboratory, University ;-’ California, Liver-more, California, USA,- and D. C. Barnes, Los Alamos Sc-enti-’-ic -iboratory, Los Alamos, New Mexico, USA

The observation of MHD stable configurations in field reversed theta pinches has led us to review the ideal MHD stability theory. We produce 2D r,z equilibria from assorted models, including Hill’s vortex, which are intended to match various profiles measure! in experiments such as the FRX series at Los Aijmos. These equilibria are used as the starting point for several stability calculations:

MALICE Several 3D ideal MHO simulations of stationary and

rotatinc plasmas have yielded good qualitative agreement with experiment.1

RIPPLE VI 3D linear ideal modes with m = 1 ‘.ave been evolved
ALIMO 2D m = 0 linear axisymmetric modes are seen to be stable or
MHC-2D 2D non-linear modes seem stable.3’**
STA3GR0W Growth rates Are calculated from the generalized energy ’ principle. This report is limited to the subject of STABGROW.

-82-

have very small growth rates.3

showing an unstable tilting mode.2’3

4irr , r Je

. 3 *!/ 1 30 _

-j, = rA2

i’Z‘“ij

(1)

Eou*1ibrium Model

J- = or 4^

Pressure balance and Ampere’s Law are expressed by

‘This «or< as pei-‘ormec’ nder the ausaices of the U.S. Dnpar-me”. z* by tre Lawrence -ivermore Laboratory jnder contrjct number it-7S”,5-^!G—?.

a2-u r? + 77 * 7 ??

Speci-Meation of the equilibrium is completed by giving the pressure

function taken to be

P(-J) = 0 - H ) P , b ~ -} + HP, {l + cos [* (a n tanh(a*+ b}}]} . (3)

A special case of this H = 0, *s = 0, m = 1 is H i l l ‘s vortex:

P() = P2.

Conventional Energy Principle

From Bernstein^ a generalized Sturm-Liouville equation

|_ { i Hi + { A.

( 5)

r b

-83-

P,0 )x =

y DX ds

-^ 1/B* + 1/YP

^) * ABZ »- a- (

§ ds ji * ‘2

1 -* I - -x

( B Z>2j

zL{k)2

”

the value of A cannot be related to growth rates.

Energy Principle with Growth Rates

To get growth rates we need to use the kinetic energy norm

1 * K - JT / f

(4)

(6)

( 7)

is used to determine stability (instability) if the smallest eigenvalue A is positive (negative). Tnis equation is obtained by taking the toroidal mode number n - *> which is the least stable” mode in ideal theory. However, since the above equation is obtained in the norm

N«JW the parallel displacement Z cannot be analytically eliminated from the energy p r i n c i p l e. After minimizing with respect to X and Z we obtain

( —^

’- “‘s 1/B2 + l /fP

Here the growth rates are obtained from J-z, = A. AS before -,fn

gives stability.

With the substitutions U s f r- and M ! g ?r + cX we arrive at the

“irst order system

.Reduction to First Order Equations

The form of the above equations is

|_ (f ||) + hx - c -|

[s !f> + kZ - - Is w

-84-

3Z 3S

W -

, h « h(.M

k - “<A>

I = U/f

— = — B 3S

(9)

> °

(10)

fill

l UJ

are solved using a shooting method which adjusts A until all boundary conditions are satisfied.

Properties of the numerical Solution Obtained in STABGROw

The equilibria are accurately known either from the equilibrium code

CYLEQ (which employs a bi-cubic spline representation with the ICCG algorithm) or from the analytic Hill’s vortex formula. A bi-cubic spline representation is also used in the stability code such that

*i>, B. 7B, k. * - b, « 7 b. 7 * B i 0

are calculated analytically. From these the field lines and the coefficients in the above stability equations are known to arbitrary precision within the spline representation. These Sturm-Liouville like equations are solved by shooting first with a finite difference integrator and then when near convergence with the GEAR method. The GEAR method allows finite difference errors to be reduced arbitrarily to ‘cundoff levels.

It follows from the foregoing that, within the spline representation,

one can calculate the exact eigenfunctions and eigenvalues to roundoff orecision.

;u 3s ~

£- = —

These e""ations together with the equations defining a field line

References

D. V. Anderson, 0. C. Barnes, w. A. Newcomb, and C. E. Seyler, Paper 2A4, Proc. of Annual Controlled Fusion Theory Conference, Mt. Pocono, PA., USA, April 18-20, 1979-

Z, A. I, Shestakov, 0. 0- SthnaeW, and 0. Killeen, Symposium on Compact

Toruses and Energetic Particle Injection. December 12-14, 1979, Plasma Physics Laboratory, Princeton, New Jersey.

J. Killeen, 0. 0. Schnack, and A. I. Shestakov, Proc. of 4th IRIA Intl.

Symp. on Computing Methods in Applied Sciences and Engineering, Versailles, France, Dec. 10-14, 1979. Also as Lawrence Livermore Laboratory report UCRL-83332.

D. 0. Schnack and J. Killeen, J. Coup. Physics, accepted for publication.
1. B. 3ernstein, E. A. Frieman, R. M. Kulsrud, and M. D. Kruskal, Proc.

Royal Society A244, 17 (1953).

-85-

This report vai orcpared u an iccoutu ot “/oik tponsoted by :tie United Suit Gavtmmtni. Neither trie United States nor Jie United States Depattmetit 01 Erwtfy. not any of then employees, nor any of their eotutictors. subcwuftctots. or ‘iietr employees, make* any warranty. itpress Qr implied, or assumes any legal liaoUity of responsibility ‘or the accuracy, sontpletencs* or usefulness of any ullormation. apparatus, oroouc’. or process discloscr:. or represents that its use *ould not mfrinee pnvaicly-owned rtEhis.

Reference ‘o J company or product name aoes nor imply approval or recommendation or :he product by the University of Oiitonua it :he C.S. Department ji Energy 0 the cidusion at others :hat may be sulUDIe.

VOTICE

IOROIDAL REVERSED FIELD-? II! CH EJ^SflLlEHTS*

Compiled by D. A. BAKER Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

Small Experiments

The small Reversed Field Pinch (H.FP) experiments arc listed in Table I. These experiments demonstrated the favorable stability of the reversed-fiald profiles, which were predicted by MHD theory. Self reversal of tie toroidal magnetic field has been demonstrated in many machines but Che important feature of the reduced fluctuation period (quiescent period) collswing the turbulent start up vaa aat observed- The best results oa ZT-3 *-ere obtained by programming, i. e-, by forcing Che field reversal by reversing the current is che tciioidal field windings. Hell-behaved pinches were observed for

40 us. The pinch terminated in an instability when the fields profiles diffused and beta became coo high.

Recently a pitch programming mode has ‘.bees 3tudled on ZT~S in which the toroidal and pcloldal fields rise together. The method has been successful in that decreasing values of pitch (? * r3 J’ZQ) programmed ac the wall resulted ia favorable »onotonii_ally decreasing values of pitch vs_ radius in che plasna. This method can give rise to reduced diffusion and a smaller energy loss during the set up of the RF?. When done slowly trie method requires gross stability during the programming; methods for accomplishing this are being scudied theoretically.

ETA SETA II

The ETA BETA II results are discussed in detail in the following paper aad In Ret. (II. It is worth emphasizing here that .this experiment Is significant in chat it has shown that the transport and radiation losses tan be overcome in an intermediate-sized experiment (bore - 25 ca’j using a 0.’. ns rise time. The experiment has reproduced many of tile results of the much larger Zaca experiment (I m bore)-and has reached electron temperatures above 130 eV by ohmxc neating following a turbulent startup.

2T-4Q

5 6-

the ITS Department of Energy.

s t a t e s, Che e l e c t r on

the 2.28 a and

ceaperacure,

time was

snowing

the ion

to in

Shown

the

t.iac

The ZT-40 experiment

15F?; che plasma ceaperacurss and confinement previous S.FP experiments, generation, d e v i c es R.SX and LCX.

is una of a nev g e n e r a t i on or r e v e r s e d - f i e ld pincnes Includes ETA 3ETA II and H3TK-LV. These experiments w i ll extend aoove and w i ll e s t a b l i sh Che physics 3ase for c-ie next

times by an order of magnicuae

The 2T-40 device has a major diameter for

a 0 — a minor is designed co cne

for

the [1]

o p e r a t i on

t he vacuum wall (99.5? aiumina). The experiment

dlamecer o p e r a te over a range of plasma current n s e c l m e s, 0.01 to 0.7 a s. During i n i t i al t h at of ETA 3ETA II was s v e r c o o e. The

r i s e t i me of plasma current in which the l i m i t a t i on of oxygen

about 200 d i s c h a r g es with c u r r e n ts up to 400 *.k, decay increased c o n d u c t i v i ty and a reduction aisuriarge cleaning of *”.-3charges wi;n c u r r e n ts up to tne c u ll design c u r r e nt waver oris c o n s i s t i ng of a coupled s et of equations governing cne deuterium ana i o n i z a t i on

device has operated in Che f i ll pressure range of 3-10 a t o r r. After the i n i t i al - 0.15 ns c u r r e nt an i n c r e a se In the plasma as t e st value A sample 400 *A. A Z e r o - c i a e n s i j n ai plasma aocei impurity the

i m p u r i t i es the vacuum cnamber occurred- There nave been a few

is ™- 0.1 ms, comparaoie i a p u r i cy

:e=peracura, and

in F i g. 1 ( a ).

0.25 ms

r a d i a t i on

energy

l o s s es

due

*Uork performed under the a u s p i c es of

clrcuit behavior is used to evaluate the ZT-40 performance. A primary impurity, as determined spectroscoplcally, is oxygen.- Line radiation can explain the observed current decay tlme9 if oxygen is present ia amounts of

7 » 1 0lz c a-^ (*» U at 10 mtorr deuterium fill pressure). The aodel also indicates that with ttie expected further reduction la the impuricy level and/or a lower deuterium gas density a “burn through” at the oxygen line radiation will occur and the plasma temperature will rise aoove 100 eV with the present passive crowbars to extend the current.

The initial values of the pinch parameter (9 - B9H ai i /3

W J) have ranged

from 1.5 to 3-5. The toroidal field winding Is crowbarrea at its maximum current. Becau: e of trie small amount of toroidal flux outside the discharge tube in this device, self-reversal is readily obtained without any programmed reversal of the current in the rorolial field windings [see Fig. 1(b) J.

A multi-channel

simultaneously measures the line— integrated electron density along seven parallel chords across the discharge A sample display of the data is shown in Fig. 1(d). After a period of tube. large density excursions, a reduction la Che total density and in the fractional fluctuation level is often odserved starting shortly after current peak [e.g. at *» 0. U ms in Fig. 1(d)]. This period is sustained duriag the current decay until tne reversed field disappears I - 0.21 as in Fig. 1(b)]. The necessary conditions for this plasma behavior are coroidal currents above 500 ‘*A and initial fining pressures below 6 mtorr. This interval of reduced activity is similar to chat observed in ZETA and ETA SETA II.

Further discharge cleaning techniques will be used o reduce the iapurity level and the expected burn through of OV and OVI ioniiation states will be observed spectroscopicaily and the increased electron emperacure will be measured by Thomson scattering, ^n situ baking of the alumina torus can oe used to further reduce che impurity level.

A power crowbar, to be installed in early 1980, will ext&ad cue current duration to tlaea > 2 ms. The plaima behavior for longer current risetimes will be investigated when che impurity radiation barrier is overcome.

-87-

interferometer

SHIM wams .-”**! ;.:s

CX’iTW ^ cs

SHALL IF? ExrEJi.resTs

«0 ::: u.

REFERENCES

Table I

5: -:;

WrfVVn

.”AJM ‘JUHUS

t. i s ii i s ii

*-» - r;

-:-:3

.F-.i

„;;.!

i0.3

~ itT

c«)

5] 5]

7.7

L - 1.

J .:

‘J’<

ill 3LTSA, A., COSTA, S., DE AHGELIS, R.. Dl MA3.C0, J. 3., G0IDICOTTI, L., MALZSASI, G. , VALESSO, G. ?., ORTOLA.1I, S., SCARU, P., “Firsc Results from the ETA BETA II RF? Experiment,” Siacn European Conference on Controlled Fusion and Plasma Physics, Oxford, Englana, Septemocr 1979.

(b)

J {d}

”*

*”

’

;

1—

- - —>

’

^ j < —H~

—:—1

R-3a-

_ < > _

7 .9 oi

jr”**—****’^

—

— I— 1

’ ’

-88-

” > - - *—

S-S«» 1».7 sn

0.2 —

RWte’ 16.1 ea

R-»o» 12.6 o

6 - * o. S.t CM

— _ _ 0

_’ ’ ’

-»-<& 0.

— n

* ” o t

»-#»* - 5 .5 s» R-*o- 8.5 ca

—

£S5*3?

TIME AS

:i|. 1. Plots showing trie tine depeadeace of (a) toroidal current, (o) toroidal magnetic field at the wall, (c) average toroidal nagnecic field, (d) Integral of the elecrron density along seven chords normal to the torus oidplane. the radial distance or each chord fron tne toroidal 21.-.or axis Is sriourt ia cne insert’

-89-

SOME PROPERTIES OF THE HEATING AND CONFINEMENT IN THE RFP CONFIGURATION

S. Ortolani, Centro di Studio Sui Gas Iooiszati, CNR-Universita’dl Padova, Association CNR-EURATOM, Padova, Italy

In this paper some properties of RFP plasmas as observed in the ETA-BETA IX experiment1 are discussed; and some comments on scaling laws In Fig. 1 the main experimental appropriate Co Che R F P2 are presented. parameters of ETA-BETA II are summarized and the time evolution tne toroidal plasma current, I,, and of the toroidal field at the vacuum wall, B ., is sketched. The results obtained on ETA-BETA II can be summarized as At high filling pressure (> 10 mtorr) radiation losses by light follows. elements (oxygen in particular) dominate and prevent the plasma heating beyond a few tens of eV. The detailed study of the variation of the current decay time and of the electron temperature as a function of the filling pressure is discussed in Rex* 1 and here only a comparison between a RFP discharge and a nonreversed (MR) discharge in the nonradiation-aomlnated regime is discussed ’” some details. In Fig. 2 some results for the NR plasma (Fig. 2a) and for RF? plasma (Fig. 2b) are compared. In particular in the figure are shown a function of tlmer from top to bottom, the toroidal current, OIV and OIV jible line intensities, the dl./dt signal, and, only for c.ie 35? pi.isna., a str:a^ picture and tne local dB/dt signal as measured by a magnetic probe measuring tiie poloiaal flux at ~ 3 cm inside the inner conjoint ion of tae stainless steel liner. The results shown in Fig. 2 can oe discussed as follows. At low filling pressure (< 6 mtorr) burning through of oxygen impurities is observed but in MR discharges complete overcome of the radiation barrier is not quite achieved because the temperature is probably held down by turbulent transport associated with the high level of fluctuations as measured on the toroidal current trace. When a RFP configuration is formed, ni^her temperatures are obtained with current e-folding times of up to 1.5 ass lasting for about 0.7 ms. This “is associated with a distinct reduction in cne fluctuations of the dl^/dt signal observed in the RFP as compared tc tne MS. discharges. Internal magnetic probes (dB/dt) snow a significant reduetic . in the fluctuation level during the slow current decay phase of a RFP discharge. Temperature measurements have also been made’ and abou- 50 eV are measured on During the slow current decay tne axla at the time of current peak. temperature increases up to ** 100 eV before current termination. -J; jsing the filling density value a 3g of -0.1-0.2 is estimated a: the time of the current peak. If the density remains about constant in time, 6g will increase by more tnan a factor of 2 due to the increase in temperature and to tne decay of the plasma current. An instability due to too hign beta is tnen likely tne cause of the current termination. The behaviour, in time, of the SFP plasmas on the F, 8 diagram (where F - B JW/ and 6 - Bgw/<3(.> are computed just inside the metal vacuum liner) is drawn in Fig. 3. Several discharges are superimposed is the figure and they all indicate that tne RFP state is characterized by ( F |< 0-5 and 1.5 < 3 < 2.5. The importance of the external vacuum region separating the plasma from tne flux conserving wall and the comparison with tneoretlcal calculations is described by Turner in anotner paper at this meeting.3

The results, shortly discussed here, obtained on STA-3ETA II are rather encouraging and indicate good confinement properties of tne RFP configuration, but tne following questions are usually asked about the overall interest of tne RFP as a magnetic confinement fusion experiment.

where B . is che toroidal magnetic field and R is the major radius of the torus. By using the q parameter this limit can be vricten in terms of Che average current density as

why che cemperacura In tha RFP is so low as compared to a cokamak or
how can the coo high beta (which eventually terminates the discharge) be

To possibly clarify these points a short discussion of some scaling laws

appropriate to the RFP and a comparison with tokamaks is given below.

From tokamak research we know that there is an upper density limit associated with the problems of radiation losses and of plasma disruptions* This limit Is usually written In terms of the average particle density as:

which for high field tokamaks (q - 2-3) can simply be written as:

mi »“w”

i uUiSiL b j j H ^ — *—

similar current?

avoided?

n < 5 - 1 01 9^!

J/n - I/N > —

I/M > 1 0 ~U A«m

9 0-

1 0- 14 A-a

Indeed, che high field tokamak experiments (as Ale?,cor and F T ), aiming at large otimlc heating and long energy confinement times, do operate in this range of parameters. Operation at lower densities is easily achieved In tokamak experiments but shorter energy confinement times are obtained in this case.

There is also evidence that, to obtain a good performance of an RFF

plasma, experiments must operate above a minimum value of I/N.

In fact, in ETA-BETA II, to overcome the line radiation barrier due to light Impurity elements, we had to operate at I - 200 kA and pQ < 6 mtorr. Only under these conditions was Cha good RFP performance (slow current decay and temperature increase v|_ time) observed in ETA-BETA II. ’ These parameters correspond to I/N as given in Eq. (3). Optimum conditions for quiescence in ZETA were obtained with I > 0.42 MA and n = S ’ L O^ s^‘V* corresponding again to I/N as given in Eq. (3).

If’ igaiciou Is eventually co be reached by ohmic heating alone, a minimum critical value for J/n will be required to overcome bremsstrahlung radiation; with classical resistivity this value is easdly found to be described by Eq. (3).

The value of J/n can then define a class of experiments capable of scaling CO ignition conditions by ohmtc heating alone, provided the overall losses remain acceptable.5

As in cae cokamak, tnere are arguments in the R?P posing Limitations on

cne upper value for I/N at which an experiment can operate.

(1)

(2)

(3)

Oa ZETA, quiescence was not obtained at filling pressures lover than » L.5 mtorr and streaming induced microiustabllity could have enhanced the transport and limited the achievable temperature. Values of I/N ~ 2-lO”1 A>m were not reached with quiescent performance. **

Streaming induced mieroinstabilities should arise at high I/N values and

a limit on I/N can be set as

1/3 < 3 . 5 « 1 0 ~U VT X T, ,

with T in keV.

All these considerations suggest then that a value of I/N substantially larger than 2,10~’-* A*m can be incompatible with achieving good confinement of the plasma. The following optimum range of values:

1 0 ~U A’m < I/H 5 2 - 1 0 “14 Ain ,

is then indicated by the previous discussion.

Pressure balance simply requires for the average temperature:

T = 1.5-LQ8 &eH(l/N)2 fceV ,

T = l . Ws 8gl(I/tf) keV ,

with T in kaV”; with ttie above discussed limitations on the values of I/a, tnis gives:

ar.d

*91-

tokamak with s i m i l ar c u r r e n t, lower by PgRFp/Sffroit ” O - t - 0 . 2. As an example, a

0 . 6 4 - 1 06 a

to a l e a st

(j)

(5)

(4)

(7)

(1.0)

(9j

{$)

These figures are of course inversely proportional to Sg, but once trie radiation losses are overcome and transport losses dominate, the a-r?ntual $, will be limited, and one way co be conservative about the transport losses is to conceive operation at Sg ~ 10%.

There are reasons to believe that an RFP will have a larger and more persistent level of fluctuations than a tokamak, because of the large number cf resonant surfaces which characterizes the RFP as compared to the Eew

T * (1-2)«1.5’10”6 S8I .

Compared t h ea be at of

I 3S

1 keV with I/N - 1 0- 14 A in- r e q u i r e s:

which, with Se - 0 . 1, p r e s c r i b e s:

I - 6.5 MA

N - 6 . 5 - I 020 m_1

ttie temperature la ttve aff w i ll temperature

resonant surfaces that characterize the tolcamak configuration. Moreover, the Ion heat conduction should be more relevant in the pinch than la the tokamak because of the lower magnetic field>

All these co aside ratio as support the ilea that a very high 3Q will not be

reached la large-current experiments.

The study at high temperatures of the resistive stability properties of the RFP will be a major objective of the future experiments, but the temperature regime in which these studies can be done must first be achieved. If we base our estimates on 69 ~ 302, ve assume to already know that high temperature stabilizing effect will lead to low fluctuations and to transport properties at lease as good as In cokamaka. Compared to tokamaks, RFP’s do not require high toroidal fields, but It can be expected that large toroidal currents (even taking into account the poloidal currents) will be necessary to overcome the losses.

The discussed scaling is plotted in Fig. 4 where graphs of I vs_ N and of

I v sS as given by Eqs. (5) and (5), are drawn for Sg - 0.1-0.2.

There is clear evidence that high current (> 2 MA) experiments are required for a significant step coward high temperature (0.5-1 keV) RFP plasmas.

References

Buffa, A.., eje al., 9th Europ. Conf., Oxford, Sept. 1979.
Ortolani, S., Rostagni G., internal note G.I.P. 79-20.
Turner, L., contribution tD this meeting.
Newton, A. A., Proc. RFP Workshop, UPee 78-08, Padova, Sept. 1978, Section

II.

Ortolani, S., internal note G.I.P. 78-2G.

-92-

-WbjOftMQIUS

t Alt > v W« CU**£MT af »w

MLUWQ Pft«*U« 6, I t f t M l t t UM

’ *

-93-

ETA BETA II £xPEai\1!WtA|.*ftflAMETt«S

T!«.U

RELAXATION OF TOROIDAL DISCHARGES*

Leaf Turner, Los Alamos S c i e n t i f ic Laboratory, Los Alamos, New Mexico 87545

^ ^ ( r t - f f lV

The v a r i a t i o n al p r i n c i p le of T a y l o r1 provides the guiding s p i r it

c a l c u l a t i o ns concerning

The p r i n c i p le

turbulent plasma discharges.

higb-magnetic-Reynolds-number

plasma confined by a p e r f e c t ly conducting s h e ll w i ll achieve the s t a te of

minlmun magnetic energy consistent with s p e c i f i ed magnetic h e l i c l ty and

magnetic

il where,

reversad-fleld pinch geometries, the eigenvalue A is a continuous variable

that s p e c i f i es the r a t io of net t o r o i d al current to net t o r o i d al magnetic

f l u x.

j u s t i f i c a t i o n,

experimental features of the B r i t i sh r e v e r s e d - f i e ld pinch experiment ZETA.

The quiescent ZETA plasma was observed to EoH.au c l o s e ly

t h e o r e t i c al

F-8

Fig. 1. [r(r) - Bz( r ) / B av8 ( r ’ < R) ,

quiescent ZETA plasma with f i e ld reversal, 1.2 < 8 < 1.6. On the basis

of the Taylor p r i n c i p l e,

is explained as that j o t at beyond

which the symmetric (m ” 0, kz - 0) ica.ee is no longer the minimum energy

s t a t e; the lower l i m it

is that point at which f i e ld reversal f i r st occurs.

the

p r i n c i p l e .2

For

yet

the

9 4-

t h is

able

ouch

f l u x.

s t a t es

that a

by use

received

Although

has not

p r i n c i p le

predicted

reproduce

has been

It y i e l ds

turbulent,

trajectory

the Euler

the Taylor

equation ? x f«

the relaxation of

8(r) - B8( r > / B f B ( r’ < B)]}

the corresponding aagcietle h e l i c l ty

the ease of three-dimensional

that was q iescent

It was observed

the upper limit

spectrum, and

ensemble was

c o e f f i c i e n t s.

established

s p e c i f i ed

toward

that

chat

the

( I)

for

{See

gross

rigorous

s t a t e;

s t a te

corresponding p r e c i s e ly to the minimum energy s t a te of Taylor was the only

s t a te

in the sense: <v2>-« 0. The study suggested a

physical mechanism causing evolution

the minimum energy

namely, r e s i s t i ve d i s s i p a t i o n, which, acting predominantly ac nijn wave

nunbera, more readily d i s s i p a t es magnetic energy

than magnetic h e l i c i cy

because

t.ie magnetic energy spectrum peaks at higher wa”e numbers

than

(2) Che natural

The Taylor p r i n c i p le received some t h e o r e t i c al support in the work, of

Montgomery, Turner, and Vahala. 3 By representing the v e l o c i ty and magnetic

f i e l ds

in terms of a truncated expansion of Chandrasekhar-Kendall vector

eigenfunctlons1’ whose time-dependent c o e f f i c i e n ts obey a L i o u v i l le theorem

ideal magneco.lydrodynamics, an absolute

for

equilibrium

d i s t r i b u t i on

tne probability

the

chat of magnetic hellclcy flow 19 toward lower «aveuumb*r»,

dlrection of magnetic energy flaw Is toward higher wave m-obers whereas

Recently thei-e have bean concernss*6 regarding the possibility that

che continuous nature of the eigenvalue spectrum of the ml nlnniip energy

state la a pathological result of ttie assumption of a circular plasma

cross section. To allay these fears, Rasband and Turner7 demons traced

that the continuous spectrum far X is neither an artifact of the straight

cylindrical geometry nor o? the circular nature of the cross section*

They proved chat for any value of k, the solution of Che above Euler

equation la either a straight cylindrical or a toroidal geometry with

arbitrary croas section can be reduced to the the solution of either

lahomogeneous Haloholci equation or an inhomogeoeoue Grad-Sbafranov

equation with simple boundary conditions. Using standard Green’s function

theory tney obtained analytical solutions for the case of rectangular

cross sections, with relevance to force-free regions within cusped

geometries* Furtner support for the presence of che continuous spectrum

cooes from tbe current numerical work of Bak.ec and Mann wno are solving

the innomogeneous Grad-Shafranov equation for a constant pressure plasma

utilizing che LASL antisymmetric toroidal magnetohydrodynamic equilibrium

code.9 It is crucial to remark here that there is only a discrete

eigenvalue spectrum for plasmas contained in slaply-connected rations;

-95-

The most recent reversed-field piacft experiment to achieve the

qulescent-lllce results of ZEIA Is the Padua experiment ETA-BETA II. 9 Ttiis

experiment has a plasma column confined to a radius of 12.5 cm by a

stainless steel liner. A vacuum occupies the region between cne liner and

cne flux-conserving shell at radius 17.8 cm. In an attempt to explain the

experimental data, we have made a generalization’11 of the variational

principle of laylor to the case where an insulating or resistive liner ac

radius r0 maintains a vacuum region betveen the plasma column and Che

foe arbitrary variations o38(r), £Bz(r) in the plasma and for arbitrary

perfectly conducting shell at radius R.

e.g., spheromak plasaas.

We demand that

==JBVf- ”

variacions of the independent components of the vacuum magnetic field.

The parameters X, y, and n are undetermined Lagrange multipliers* Si ace

the total magnetic flux * within the flux-conserving she]1 ts time-

independent and prescribed, we define K re be the total magnetic belicicy

inalde che flux-conserving shell in order that the field equations remain

gauge-invariant* This invariance is physically required by the dynamical

isolation of the plasma from magnetic fields external to the flux-

conserving shell* The jump in the normal component of ff at the liner

[Br(r£) - B

E(ro )] ls """trained to be zero. Vte then solve Eq. (1) for

the state of minimum magnetic energy within the flux-conserving shell, e,

with prescribed values of K and 4.

We find that la the minimum energy state the confined plasma is

force-free and enveloped by a skin current. (See Fig- 2, for example.)

The F - 8 trajectory evaluated at che flux-conserving shell with ETA-3EXA

II parameters is depicted la Fig. (1). The 6 interval discussed above has

become in this case 1-4 < 8 < 2.2.

Mote chat skin currents must also occur at the surfaces of relaxed

(force-free) spheromak plasmas where a liner maintains a vacuum region

between the plasma and the conducting shell*

-96-

(1979).

(L97S).

G. tfahala, “Stability and Force-Free’ Fields in an Ellipcical

Cylinder,” (Sherwood Meeting on Theoretical Aspects of Controlled

Thermonuclear Research, Mount Pocono, Pennsylvania) paper 2B4a

E. P. Butt, A* A* Newton, and A. J. 1* Verhage, Pulsed High 3eta

Plasmas (Proceedings of the Third Topical Conference, Culham, 1975)

ad. by D. E. Evans (Pergamoa Press, Oxford, 1976) paper C2.1, 419

D. Montgomery, L* Turner, G. Vahala, Phys. Fluids U., 757 (1973).
S. Chandraaekiiar aod P. C. Kendall, Astrophys. J. 126. 457 (1957).
J. B. Taylor, Pnys. Rev. Lett. .33. 1139 (1974).

10. Leaf Turner, in preparation.

*ttork performed under cue auspices of the USPQE.

C. Chu and M. Chu. “Force-Free Hagnaeic F i e l ds

la Na act c o l l ar

Cylinders,”

(Annual Meetlag at D i v i s i on of Plasma Physics, Boscoo)

paper 6E6 (1979).

S. S e ll Raaband and Leaf Turner, “Solutions of the Helmholcs Equation

with

LA-OR-79-a3l6, to be published-

F i e l d s ,”

*. A. Baker and L. W. Mann, Pulsed High Beca Plasmas (.Proceedings of

the Second Topical Conference, Garchlng) paper B7 (1972).

?. A. Buffa et a l. “First Results from Che ETA-BETA II UTS Experiment,”

(Ninth European Conference on Controlled Fusion and Plasna Physics,

Oxford, U.K..) post-deadline paper (1979).

for

9 7-

Boundary Conditions

Force-Free Magnetic

-98-

Plasma Physics Laboratory,

E. J. Caramana* and F. W. Perkins

E F F E C TS OF I M P U R I TY R A D I A T I ON

P. 0. Sox 451, Princeton, N. J, 08544

ON R E V E R S E D - F I E LD P I N CH E V O L U T I ON

coaling by oxygen impurity acorns which peaks

Spheromak and reversed-field pinch plasmas share

che property that Ohmic heating must overcome strong radiative at a temperacure T^ ; 25eV. We presenc results of trans port code studies of this phenomenon, for several reversed- field pinch devices, and summarize these results in terms of two nondimertsional parameters «hich parametrize the impurity content and the device size.

8ir n e

(const.)

LA J-^—

v2)

2 2

a -

3otb sphercmak and reversed field pinch plasmas depend on Ohmic heating

to raise the plasma temperature past the peak of the radiative cooling caused by the presence of oxygen impurity atoms. The peak of this cooling cur/e’ lies near T0 occur: First the Ohmic heacing pcaer l(T0)j- must exceed the aajcimum

: 25eV. Two conditions 3ust be satisfied for this co

cooling rate C ( T0) nQne [C(T0) - 9 x 1 0- 19 erg cm3/sec]. In other words

Che inequality

must be satisfied over a good portion of the plasma. Secondly, the energy expended in heating the plasma to a temperacure T. : iOeV, muse be a small

fraction of the magnetic energy stored in che poloidal magnetic field or else a serious erosion of the poloidal field will result. The first criterion is evidently a criterion involving impurity concentration, plasma density, and current density. The second criterion, which can be written as

involves device size and plasma density.

*Present Address: Los Alamos Scientific Laboratory

A transport coda for reverged-field pinches, which embodies classical

transport and impurity radiation loses, has been developed2 and applied to the three devices listed in Table 1. The results are summarized in Figure 1. The evolutionary parameter T IS given in terms of the loss of poloidal field energy Wp

The definition of 6T is

vhere B is the internal kinetic energy content of the plasma. The para

\ is defined by r

where the c > brackets indicate a mass-weighted average. is units, the formula for r

In practical

meter

-99-

(0.005)

( - ’- \ < J

1 ClJ < j >2

3Mp(t-0)

C (TQ) < nQ > < nfi >

x. 11*3^21

vacuus w»U jiadiua

i . ^ ’ a -3

wall riadiua

? t s p » r * t u B*

Conducting

I Ci»trn

la «v

T a c o i c Ul

40QC C

On . \ * t»

On A x il

5 e n » i ty

l.~, «B

I n i t i al

i n i t i al

13.SOD

505 j»

s.:«

10 «v

20 cat

10 **

:: c»

: T - 4O

r i « ld

i J1’^

. 1 1»

’ J.

1.9

’.

[

;

(6)

(5)

C3)

? -*

60 cm

S l ta

. :» « j

5153 s.

l ln

*-« *

In terms of these nondiaensional evolutionary parameters, each of the three devices is quite simi lar. Figure Id provides an empirical fit to many transport code runs, giving the 3

which vill be achieved for a given expenditure of magnetic field energy (parameterized by t) as a function of impurity concen tration (parameterized by V ). The location of various devices (with 3 being the 8_ value

at 40 eV) is also shown.

Figure Id is intended to provide a design guide for reversed-field pinch and spheromak. devices. For a given design, one computes E from Eq. (5) or

(6) and the 3_ value corresponding to T » 40eV from Eq. (4) and plots the

resulting point on Figure Id. A design falling in region I ( 20> 0.45)

Ij&fi**‘9?”’

will not break through the radiation barrier. Designs in region XI (0.25 < \r < 0.45) will experience substantial difficulty in breaking

through che radiation barrier and should be regarded as risky. Region III

l <* 0.25, B_ > &_ (t » 0.05) J

corresponds to devices that have too

small a size for their density. In the shaded region IV, the plasma should safely break through the radiation barrier. Within region IV, the value t corresponding to a design point gives, via Eq. ( 3 ), fraction of the initial poloidal field energy which must be expended to achieve tempera_ura of 40eV.

According to Figure Id, ZT-S was much too high a density (or too snail

a device) to break through the radiation barrier. The ZT-40 device is pre dicted to work well for oxygen concentrations below 0.431, but to e::nerienee grave difficulties at l.&S (given the density of Table 1 ). The RFX para meters produce a successful device at 2% oxygen impurity concentration due to its low density.

Clearly, oxygen impurity radiation cooling remains a key issue for

reversed field pinch and spheramak devices.

This work was supported by the Air Force Office of Scientific Research, Contract F49620-76-C-O005 with University of Colorado and by The Department of Energy, U. S. DoE Contract EY-7&-C-Q2-3073 with Princeton University.

-100-

FPPL - 1352 (July 1977).

REFERENCES

D. E, Post, R. V. Jensen, C. B, Tarter, Vf. H. Grasburger, and W. A. Lokke, “S.eady State Cooling Rates for Low Density, High * Temperature Plasmas”

E. J. Caramana and F. W. Perkins, “Effects of Impurity Radiation

on Reversed-Field Pinch Evolution”, PPPL - 1626 (Feb. 1979).

lo)

°0

0.20,

0.05

(So

5!i0

0.01

0.03

0.02

0.4%M

-101-

Zy-QAt

lo’OM

-Device Sice-

/U’0.11

1—Substantial —** Radiation LOSS

-I S

0.04

0.08

0.06

0.10

0.3

0.4

0.1

“0

0.02

/ y U T . 01 -‘2„=a45

0.04

00=0-3

I Radiotion- Limited

0,05

f 792511)

0.6

0.5

Fig. 1. (a-ft) Sondimansional Evolution of the reversed-fiald pinch

plasmas listed in Table 1, for indicated concentrations of oxygen im purities; (a) ZT-S, (b) ZT-40, (c) RFX.

Fig. Id. Design guide for reversed-iield pinch plasnas. Full, half and quarter-shaded syabols correspond to the saKiaum, half-oiaxiaiua, and quartar-aaximra oxygen concentrations of Figs. 1(a), K b ), and 1(c). The i represent RFX, 0 represent ZT-40, and * represent ZT-S. (Sote that ZT-S is off-scale.)

I. Introduction

W. T. Armstrong, R. K. Linford, J. Lipson, D. A. Platts. and E. 0. Sherwood

The equilibrium, stability, and confinement properties of the Field Reversed Configuration (FRC) are being studied In two theta pinch facilities referred to as FRX-A, and FRX-B. The configuration is a toroidal plasma confined *n a purely pololdal field configuration containing both closed and open field lines (see Fig. I ). The HUT system produces highly elongated tori with major radius R-3-5 cm, minor radius a ~ 2 cm, and a full length * ~ 35-50 on- ?lasma conditions riave ranged from Tg ~ iso ev, Tj. ~ 800 eV, and nm ax 100 eV, Tt - 150 nV, and a^g^ ~ 4 x l oI 5/ c m3. The plasma remains in a stable equilibrium for up to 50 ua followed by an n«2 rotational Instability which results In termination of the FRC- The plasma behavior with respect to equilibrium, stability, and rotation is consistent with recent theoretical worle la these areas,

1 01 5/ « n3 to Te ~

-102-

Fleld Reversal Experiments. FRX-A and FRX-B Results*

length-

Independently driven, crowbarred mirror colls were

recently included on FRX-A. With optimum mirror timing, it was found that the particle inventory increased as compared with plasmas created with passive mirrors or without any mirrors, the increase being largely due cc greater plasma It Is thought that Che mirrors produce a more localized tearing and reconnection of field lines during the formation phase by providing more favorable field curvature at the ends- This should contribute to increased Inventory. After the first few microseconds, the ndrrors act more like an extension of the implosion coll rfhich would also increase plasma length and Inventory. An Increase la plasma lifetime from 20 to about 35 us is also observed.

An experiment in plasma translation was performed by disconnecting one of the driven mirrors. Where the remaining mirror was activated 0-5 ys before the implosion, the plasma was observed Co translate tcwards the opposite end with an initial drift velocity of 3-3 cai/ys, exti’txig the coll in 15 us. This time is* intermediate between the a nd an Alfven transit time (t^) so that She plasma stable time (t.) plasma can undergo a collective translation prior to instability. A &32S & fractional fringe interferometer,.external field probes, and a side on streak camera all indicated a well defined plasma column forming in the center of Che implosion coil and Chen drifting uniformly out of the system. End-on framing photography Indicated retention of an annular equilibrium as the plasma moves axially-

*Work performed under the auspices of the 0. S. Department of Energy.

II. FRX-A Recent Results

IIT. FBX-B Spatial Scans

Radial scans with Thomson scattering have recently been made on FRX-B. The scattering system has all Its components rigidly mounted to a single table surface such Chat ail relative alignments remain fixed- To shift the instrument for either radial, or axial scans of the plasma. the table is translated by means of air bearings. This versatility Is permitted because a three grating polychromator1 has been employed’ This instrument has sufficient rejection at 6943 A to obviate the necessity of a bean or viewing dump. Observations were made on the axial midplane at a constant filling pressure of 17 mtorr. Over the stable lifetime of the plasma, the electron temperature was essentially ad space at about 100 eV for radii Interior to coastant in both time the seperatrlx- Relative density was also fairly coostaat as a function of time at these radii.

Line-integrated density across a dlaoerer and excluded flux radius

fit axial

have been determined aa a function Measurements were taken along a half-length of ”he coil at 17 mtorr fill pressure. plasma half-length of about 22 cm- The excluded flux radius near the midplaae Is found to be close to the separatrtx radius, ae inferred from Thomson This correlation Is scattering and image convetler measurements. expected from several equilibria models. Plasma shape modeling is required to unfold the data in the region z > 20 cm due to field curv ture effects in this region.

coordinate

indicate

(z)-

-103-

measurements

Interferometry

Scattering

Thomson

and

Electron temperature measurements were made at the axial midplane and r * A cm with Thomson scattering- loo temperatures were deduced from C7(227lA) Doppler broadening aeasareaents. The carbon impurity radiation was collected along a diameter In the midplane of the plasma configuration. At all fill pressures, the electron temperature appears Ion rather constant in the range of 100-150 eV. temperature approaches the electrons on an approximately classical time scale- Tj (measured at 10 us) varies from ~500 eV with Po 9 ntorr, to ~t80 eV with Po - 21 mtorr.

Whereas, the

Interferometer,

Doppler broadening measurements were used to examine pressure balance with the measured external field as a function of time aad fill pressure. The errors associated with the determination of the temperatures and peak density

Scaling of plasma parameters with fill pressure have been made for 0 - 9, 13, 17, 21 mtorr- Density measurements were made using Thomson scattering and two fractional fringe interferometers operated at different wavelengths- The Thomson scattering measurement was made near the mldplane with r * 4 cm, the appriximata value of the tuaj or radius. The peak density (corresponding to r - 4 cm) was unfolded from the interferometers integral measurement through profile information from excluded flux and luminosity measurements. The peak density (determined at 10 us) varies from 1-8 x 10*5 en -3 with Fo - 9 tntorr, to 4.2 x 1 013 cm”3 with Po - 21 mtorr.

IV. FXX-B Scaling Studies

result In a cumulative uncertainty In the plasma pressure of 40S- Preaaure balance is satisfied within the3e uncertainties.

The plasma lifetime is limited by the onset of an n-2 rotational instability which has been treated theoretically by Seyler2 using a Vlasov fluid code, and by Freidberg and Fearlstein3 using an FIR expansion. In Seyler’s model -«>r/n* . 1.64 where w,. is the real part of the perturbation frequency, and SI* is the dlaoagnetle drift frequency- For FRX, this ratio Is about 2.0 where u>r ls determined by end on framing photography. The Vlasov fluid model also indicates a critical value of a = - SS/SJ* - 1-40 for stability wheze SJ is the ion rotational frequency. of at experimentally using the Doppier shifted profile of the 2271 A line of CV. The measured value of 0.4 is highly suspect for the following reason* The ion fluid equilibrium equation requires that when CV and D+ ions are both rotationally and thermally equilibrated the CV density profile must be highly peaked at the major radius. The experimentally distribution is ia total disagreement with this requirement- The failure of the data to satisfy theoretical criterion w < nf. is further evidence that the the assumption of rotational equilibration la faulty The critical Si however has been observed to scale with f2* as predicted.

Attempts have been made to establish the value

determined

radial

The most plausible explanation for the spin-up of the plasma has also been advanced by Seyler^ where the preferential loss of particles with negative angular momentum ia considered. These particles are found to be located either outside the seperatrix or encircling the axis, precisely the locations of anticipated large losses. This mechanism requires ~502 plasma loss for a » l.5. Inventory estimates of particles were made from combined measurements of plasma parameters at 17 mtorr- Particle loss at the time of the rotational Instability was ~40?. The i…ierent . uncertainties in the determination of the particle inventory prevent a definitive correlation with Seyler’s hypothesis, however, the results are not inconsistent with this model.

v. Rotation

*104-

VI. Transport/Scaling

The energy confinement time during the stable period ls of the order of tens of microseconds which ls significantly shorter Chan a classical diffusion time. One candidate for enhanced transport is anomalous diffusion due to the lower hybrid drift (LHD) instability which is driven by strong pressure gradients. A 113 hybrid code developed by Hamasaki5 to consider both classical and anomalous transport has yielded preliminary results indicating significant LHD transport exists near the separatrlx. From this cede, times for 5C? particle loss are calculated. The scaling of this loss time with 1/a and 8-Vo compares favorably with stable time scaling from FRX-A and FRX-B (see Fig. 2 ).

Experfaentally, th# study of transport and scaling is limited by the relatively restricted parameter range of existing machines- A larger experiment called FRX-C has been proposed to alleviate Che difficulty.

VII* Conclusions

Considerable progress has been made In defining til* equilibrium, stability, and rotational properties of the FRC. These efforts have been aided by the addition of Thomson scattering and excluded flux measurements co F&&-B, and by substantial nev results from tne theoretical community. Many key issues however remain unresolved both experimentally and theoretically. The range of equilibria for wnich the FRC will remain MfiD stable is stl!l poorly defined. The stability of the plasma to resistive codes has H OC yae been cttaxactetiied either by experiment or theory* are regsrd to aaoataloua Inadequately processes* the next generation of experiments combined with more iopbistlcateil 10 computer codes vill hopefully elucidate these key paints.

scallos

and

References

R. Siemon, Appl. Opelcs 1J., 697 (1974) i. C. E. Seyler, LASL Bpt. LA-UR-79-1S5 (1979) 3- J. P. Freidberg, L. D. Faarlstein, Phys. Fluids ^1_, 1207
C. E. Seyler to R. K. Linford, private eomnunlcation
S. Hamasalti, Conf. Record of 1979 IEEE, Intl. Conf. on Plasma

(1978)

Science, p. 142

-105-

with

understood

particularly

Finally, transport

Figure 2

Figure

Field Reversed Configuration

Scaling of the Stable Period - Open circle is FRX-A data, dots and crosses are old, and recent i-KX-B data respectively. R = i cm.

Two nev compact torus experiments have recently been proposed

and one (FRX-C) Is now under construction- The proposed experiments are based on recent theoretical advances and experimental results obtained Jlch the LASL TRX-A and FRX-B field-reversed theta-plach devices.3 Thesis recent experiments demonstrate a formation method and establish some of the confinement properties for one type of plasma compact torus—an elongated prolate plasma torus confined by purely pololdal magnetic field. The plasma displays a quiescent phase, free of any gross MHD instabilities, that persists tnucn longer than either characteristic MHD times or open-field line loss cimes. The quiescent phase is terminated by an n-2 mode tb?c is correlated experimentally with a steadily increasing plasma rotation that eventually exceeds the theoretically predicted threshold for Instability (approximately 1.5 3 where tJ Is the angular diamagoetic drift frequency}. The roLation is interpreted as resulting from a preferential loss of ions tnat carry angular momentum in a particular direction leaving the plasma to spin In tne opposite direction. According to this point of view a significant fraction (about “ene-half) of the plasma must be lost to the closed-field configuration before the plasma reaches the threshold for instability, and therefore the n-2 disruption can be viewed as a secondary consequence of the more important cross-field transport processes. Both the FRX-C and Multiple-Cell experiments are intended to Investigate ways of reducing cross-field transport.

Background

FRX-C AND MULTIPLE-CELL EXPERIMENTS

R.E. Slemon, and LASL Compact Torus Staff Los Alamos Scientific Laboratory, Los Alamos, New Mexico 37545

FRX-C

-106-

The FRX-C experiment is a scaling study wnich will extrapolate the FRX-3 results coward tne higher temperature and longer lifetime needed for fusion. As shown in Fig. 1, the FRX-C coil dimension^ are approximately twice tnosa of FRX-B. Two nign-voltage banks in a dual-feed arrangement are used to produce strong implosion heating. Larger variation of plasma density and temperature should be possible as a result of the higher voltage, because the nigher voltage allows a larger variation in initial filling pressure as snown in Table 1. For the same temperature in FRX-C as in FRX-B, the larger diamecjr coll results in an increase of a/p1, where a is the plasma minor radius and Pt is the ion gyro radius. plasma size will be tested.

Thus, the important scaling of confinement with

Another important design objective of FRX-C is to use a small amount of a „?batic compression, and thereby to produce a “fat” plasma cnaracterized by a large ratio of the separacrlx radius to the metal wall radius, r3/ ru. As a general rule, better confinement in this type of field-reversed configuration has been observed experimentally when r3/ rw is maae larger (see discussion of literature in Ref. 1 ). The reason for improved confinement with large values of r_/rw (typically > 0.4) is, theoretically, a result of reduced pressure on

The

th* separatrix and a corresponding reduction of crosa-field transport ar. the boundary of che plasma- It is a accessary consequence of pressure ‘“fiance tnac tne plasm* pressure at the separatrix is reduced as rs/ rv Increases.

The minimum amount of adlabatic compression (maximum rs/ rw) results wnen

the maximum reversed flux Is trapped in the plasma- Trapping the maximum flux in the ordinary theta pinch requires reversing the magnetic field quickly, because during the reversal phase the preiouized plasma tends to expand, interact with the wall, and allot- flux Co be lost. The high voltage of FRX-C produces a rapid magnatic field reversal, and the resulting ratio ra/ rw should be larger than In FRX-A or FBX-B.

The FRX-C experiment is located adjacent Co, and on a centerline wich, the CTX vacuum tank and d.c. magnetic mirror colls. Thus, the device is 7>Lao&ed to seir^e as a plasms source for confinement studies of CI’a containing no toroidal field. As much flexibility as possible 13 being designed in the FRX-C system to allow variation of the basic configuration. Important options are che addition of electrodes for lz to include toroidal field,* and the incroauccion of barrier fields and trigger coils for study of the Kurtmullaev mode of operation-

Multiple-Cell Experiment

la addition co possible reactor implications, a linear array of compact’

In a single-cell experiment such as FSX-B, T.0.& plasma on open field

corus plasma cells makes possible interesting studies of the confinement system. lines is unconflhed, and loos leave in approximately one thermal transit time* The radial pressure L .ofile presumably drops abruptly at the separacrlx producing enhanced cross-field transport. In the proposed Multiple-Cell Experiment, the confinement of plasma on open field lines is expected to be markedly improved by the influence of multiple magnetic mirrors created by che linear array of plasma cells as shown in Fig. 2. The resulting pressure profile should h« smoother and the nlaams. confinement in the cloead-flaid configuration should improve*

improved confinement on tne open magnetic field lines is a consequence of multiple magnetic mirror effects that become important when the classical ion mean free path becomes comparable to the cell length. For cne Is plasma parameters expected In the experiment, the mean free path approximately 30 cm, and the required condition is satisfied. Consequently, cne plasma decay time on the open fields is increased from approximately 3 us in FRX-B (T ~ L/vt tl where i la the half-length and vch is tne ion thermal speed) to approximately 60 us in the proposed Multiple-Cell Experiment (T -3 L AT-ji where M is the magnetic mirror ratio, and A is the ae-.n free pajn).

An experiment to test the effects of multiple plasma cells can be done by a straightforward modification of Che existing 5-m Scylla IV-P theta pinch. The coll, dlscnarge cube, and capacitor bank would be modified to macen the FRX-B experiment la all respects except length. Providing H e ld line rearing and reconaectioo can be conciciied to generate a linear array of cells in tne

-107-

-108-

Coll diameter (cm) C o il lengtn (cm) Source voltage <kV) Magnetic f i e ld (kcG) Field quarter peiiod (us)

TABLE I. FRX-C Experimental Parameters

2 .4 1.2 * L 0;o

1100

Anticipated Plasma Parameters

Tnata-Plach Parameters

shoving hi&h-volcage

150

9 L.6

and discnarge

t r a n s l a t i o n,

caeca-pined

torus

c o i l,

IQ13

Fig. L. FR2-C experiment c o l l e c t or p l a c e s, arrangement makes optimum use of the a v a i l a b le space adjacent CO cne CXX, a vacuum Lank and mirror c o ll and confinement s t u d i o.

capacitor banks, dual-faed wing”

trapping,

compact

“Gull

cube.

for

I n i t i al f i l l i ng pressure (atorr) Plasma density (em”3) Electron temperature (eV) Ion cemparacure (eV)

6.8 x

200

-109-

, MfTAL COIL

MAGNETIC HELD LINE

ec a l ., Proc. of

the Seventh

rig- 2. Scnemari.c drawing of magnetic fields surrounding linear array of contact torus plasos cells.

5-n coll, the influence of multiple cells can be studied for a plasma witn cne known slagle-cell c’ araccerlscies of FKX-B. It has been pointed out cnac weak aagnecic mirrors in ctie external coll structure nay noc produce tearing;& prellialnary cests of aethods for controlling tearing are being planned for FRX-A or FRX-B.

Ref arences

H. Oreicer ec a i ., Proposal for FSX-C and M u l t i p l e - C e ll Compact Torus Experiments, Los Alaooa S c i e n t i f ic Laboratory Heporc LA-8045-? <L979).

2- D.C. Barnes, chess Proceedings.

‘J. T. Armstrong, ec a l ., Procedlags this conference.

G. C, ucldanbaum, Bull. Am. Ptiys. Sac. 2±, 1072 (1979).
A. G. Ss’itov

IAEA Conference on Flaaisa

Physics and Controlled Thermonuclear Research, 3E 15 (Innsbruck, 1973).

A. E. Sobson, privace conmunicacioa.

COWPACT TORUS THEORS—MHD EQUILIBRIUM AHD STAMLITT

D. C. Barnes and C. E. Sayler, Los Alamos Scientific Laboratory) Los Alamos. H. M. 87545 and D. V. Anderson, Lawrr.nce Live m o ra Laboratory, Livermore, CA ¥4550

Field reversed ttieta pinches at Los Alamos and elsewhere l ,z have

the production and confinement

toroidal demonstrated configurations these observations, the plasma Is eltnex lost by diffusion or by tne loss of ttu. applied field or la disrupted by en n - 2 (where n is cne toroidal mode number) rotating instability only after 30-100 MED d o e s, wnea the configuration begins to rotate rigidly above a critical speed*

surprisingly good MHD stability.

of compeer

Tries e experiments have led us to investigate the equilibrium, stability, and rotation of i. very elongated, toroidally axlsymoetrlc configuration with no toroidal field. Many of the above observations are explained by recent results of these investigations which are summarized here.

the MHD equilibrium of a very elongated,.field reversed configuration has been considered by a number of authors»^“J Suppose an elongated plasma is reflection symmetric about a mldplane and Is terminated by a separatrlx wrich is of radius r at the mldplane and which extends to the axis at the plasma en-’ Assume further that this equilibrium Is contained in a uniform c> .ndrical conductor of radius r which extends axially veil beyond the plasma end.

Integrating the equilibrium equations over the volume bounded by th<i

oidplaae, the cylindrical vail, and an eadplioe leads to

-110-

with

Ei -illlbrlum

Introduction

fv r dr ( P - 1 B2 )

t dr ( ? - la* >

* 1 _ JL.

midplaue

(Z)

where 3 is the {predominantly axial) magnetic field and F is the plasma pressure. At the eodplane, P « 0, JJ » Be which 13 approximately constant In r. At the mldplane, P + — B2 * -TBQ, where BQ is the external magnetic field. finally, since the same vacuum flux passes through the aidplane and the endplane, BgC^-r*-) - Ber*. Using cnese, Eq. (1) becomes

aadplane

where - 2/r* / a r dr (P / - B $ ), is the average 6 and vnert e - rs/r„.

Tor apeciEic pressure profiles, the relative pressure at the separaerix, & * PC^/Pm-,. - Bfr,), may be expressed as a function of the radial -onpression, <- Two examples are shown In Fig* 1. As K * 0 (very compressed plasma), Eq. (2) Indicates that <3> * I, the central 3. Thus, In tnls limit, the profile becomes very flat, & approaches I, and the details of the profile are less important. This is Indicated by the convergence of tn« two curves as K * 0. Experimentally, < * -5, a * .7, which is consistent with either profile considered.

The above analysis gives no details of the axial structure of the equilibrium. Elongated solutions have been found numerically using a These solutions Dave two-dimensional, r-z, equilibrium code, CYLEQ. gradual surfaces. associated Experimentally, che z variation occurs over only a fraction of the plasma length and the overall picture is of s more racetrack, like equilibrium. This feature Is not veil understood, but might be explained by considering configurations vith internal islands.

and

The simplest modes to consider are gross displacements of the plasma. For very elongated configurations, the applied field has a maximum in the midplane. Thus, sideways displacements of the equilibrium (normal to tne axis) are stable. Axial displacements are unstable withouc a conducting wall but are neutral for a saooch cylindrical boundary and become stable if the wall is moved closer.

To examine

incompressible modes <SW may be written as

<5W6.

-111-

flux

Stability

elliptical

z variation

stability’ in more detail, consider

B - 1; d J dx { ^ $2

L y - \i* a

B 2IT + ‘x 3

** -f -

n* J 3x J 3x

»2^2

-p” *2

4¥-

+”

( 3)

For

and where the notation of Ref. 6 has been used.

Since all eigenvalues of LQ are negative, the last tern in Eq. (3) is negative definite. An examination of the remaining terms indicates that, for elongated configurations, SVJ is large and positive, unless X * xr3 oc X - xrBz, where x is a slowly varying function, i.e., does not have rapid variation near the turning points of the field lines. The choices 3 and

where Y and f are functicuals of X defined by

Sz the a x i al mode

lead to a x i al or radial displacements r e s p e c t i v e l y.

** - \ ,’ dV J d* i «2t| [ p ^ p C g )* + 1 « ( ^ )1 ] + « }

The second tern oa cue rigor above Is small for elliptical, elongated equilibria. The Cirsc t en vanishes for x« x(*>. Thus, we find Chat axial, Internal displacements are unstable for all n for sufficiently elongated configurations vita elliptical flux surfaces. For racetrack, equilibria, this scaling does not apply, and the possibility of stabilising small n modes exists. Similarly, for radial modes, the stability of small n modes is determined by tne external boundary conditions.

Tnese conclusions are reproduced by numerical simulations. 13 sin a

equilibria generated by CYLEQ as Initial conditions, tne MALICE7 code has been used to study stability of large scale modes. Equilibria completely stable to a 1 3 have been constructed. When < 13 made too small, however, the a * 2 radial mode is found to be unstable. These results are shown in Fig. 2- In independent writ, Sbestalcov et a. nave observed an n - 1 internal, axial instability. This la related to the spherooak tilting node but becomes nearly an azisuthally varying axial translation of the plasma for the very elongated case considered.

Local modes with n + «* are of two types. For the interchange mode, the normal displacement Is approximately constant along a field line and the plasraa compressibility must be taken into account. Interchange stable solutions have been constructed by analytic” and numerical means. For tne remaining co—interchange modes, compressibility is less Important. In an Independent paper, ve report recent work on these local modes.

The rotational stability of infinitely long field reversed theta pinches has bees studied by PearLateta1*, Treiaberj , and SeyLei * Freidberg used the finite Larmor radius equations in wnich he treated the field null in an ad hoc manner. The predicted value of the critical rotation velocity vas about Q * 1.6 a^, where u* 1? -he ion diaoagnetic drift frequency. Using the exact Vlasov-Fluld equac is in which no field null singularities occur, Scyler found the critical rotation velocity to be somewhat lower, Q * 1.3-1.5 u^, depending upon the profiles. Bo en results, however, are in qualitative agreement, the threshold for n * 2 rotational instability is Q > u». Direct experimental measurement of the critical rotation velocity is extremely difficult, however, the real part of tne elgenf requency has been measured and is in agreement vitn the theoretical prediction, thus lending indirect experimental support for the theory.

-112’

the above

Rotation

«)

Foe

The mechanism »hich causes the plasma to spin up to this instability threshold has not been positively identified. However, ctirea types of explanations have been proposed. First, the plasma may Initially rotate and because of prof 11a changes rclatad to transport later reach the critical rotation. Trie magnitude of Initial rotation for this explanation to hold requires the Ions to carry a large fraction of the currenc initially. Sine* this Is inconsistent with observations in other chata pincnes, this explanation seams unlikely.

A second possible explanation of plasma rotation night be end shorting on the open field lines and viscous transfer of rotation to the remainder of cae plasma. This la unlikely to be the full explanation because the energy avallabl« in the open field region is much less than the final rotational energy of the plasma.

Finally, particle lass from Che closed field region might result in the loss of angular momentum from the system and corresponding plasma rocatlon. This appears to be the most likely cause of rotation. There are five types of particle orbits In a field reversed configuration and those encircling the axis or outside. th.e sepitatrtx have, negative aechanical angular monencjn. These particles are lose is a efceraal transit time, resulting in a nat gain of angular momentum by the confinad plasma. In fact, for a reutral plasma, the total mechanical angular momentum of all particles is a constant of the motion. Thus, it a fraction F of particles are lose from a plasma ulch no initial rotation, this conservation law is approximately

*U3-

(l-F)miNR2 + Frn^r2!!^* - 0

’

where m1 is the ion mass, S 13 the total number of particles, a is the plasma major radius, u>d is the delft frequency of an ion at radius r wnere it is lost, and <: > indicates an average over the particle distribution. For a rigid rotor distribution with cold electrons, tne rotational velocity reaching the instability threshold requires the loss of about one half of che plasma. Experimental measurement of the particle confinement tine gives a value of about 352 ±15Z for tne fraction of particles lost at the d oe the rotaflonal instability Is observed. This result is somewhat inconclusive in determining the rocacion mechanism, nouever, other factors could account for rotation in addition to the particle transport mecnanlsm.

B.. R. Llnford et al.. Proc. Seventh IAEA Conf. on Plasma Physics and

Controlled rtuclear Fusion Research, Innsbruck, Austria (1973).

A. 6. Ea’ltov et. a ^, ibid.

jC. Wright, a. D. Medfocd, and 3- chambers. Plasma Physics .12.,

J. (1961).

(6)

2 42

6. I. B. Bernstein , ec al., P T O C. Roy. Soc. A 244, 17 (19585.

A. I. Sheatakov, D. D. Schnack, J. KJLlleen, these proceedings.
D- C. Barnes and C. E. Sevier, LASL Beport, LA-UH.-79-13, January
0* ?* Anderson, D. G. Barnes, W« A. Newcomb, these proceedings.
L. D. Pearlstein et_ al^, Paper Ho- CN-37-5-1-2, Proc. Seventh IAEA Conf. on Plasma Phyaica and Controlled Nuclear Fusion Research, Innsbruck, Austria (1978).
C. E. Seyler, submitted to Phys. Fluids.
J. U. Brac&blll, Heeh. in comp. Phys. 16, I (1976).
A. Kadistt, Phys. Fluids .22,, 2248 (1979).
B- Linford, private communication.

-114-

J- P- Freidberg, private communication.

RJG10 A O r oA

Figure 1 - Horoalized separatrlx pressure, D, as a function of normalized plasma radius X for two profiles. Hill’s vortex curve for P’(j/) » - i (conscant). Rigid rotor profiles ::r ?’(v) - a ex? (- <?/*a) .

Figure 2 - Results of three-dimensional simulations for l”.rfo different elongated plasmas. Curves Indicate P in mldplaue as a function of r for two cases. Conducting wall posirlon Is Indicated by shaded line. Density contours at late times indicate n - 2 ins ^ability for compressed case, stability for less compressed case.

TWO-D SESSIONAL SIMULATION OF COMPACT TORCS FORMATION

D. W. Hevett, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

A recently-developed two-dimensional algorithm1 Is now being used In formation studies of the FRX configuration at LASL. The nev procedure Is part of a quasi-neutral R—Z hybrid code which, by representing the Ion component by the particle-in-cell technique, correctly represents the complicated orbits exhibited by the ions field-reversed configurations’ The eleceron and Maxwell’s equations are combined in the new part of the algorithm in the zero-electron-lnertia limit to eliminate electron time scales. The resu’tlr.g combination is solved so that electron source terms and the electric and magnetic fields are provided self-consistently even in strongly Inhomogeneous plasmas vlth arbitrary plasma-vacuum lnreimlxlng- As will be seen, this last feature makes simulation of ruch highly dynamic configurations as the TRX much more tractable.

Thi FRX formation studies are accomplished by simulating the else evolutijn of a homogeneous plasma vieh a negative bias B -field (-500 gauss) which is subjected at t * 0 to an external implosion B^-fiejd (5 The impinsion field drives the plasma toward the axis where it KG). exhibits rapid radial oscillations for a period of about 1.5 us- After this Initial chaotic behavior, the plasma settles into a configuration which, while not yet in thermal equilibrium, attains the macroscopic appearance of an infinitely long field reversed 3-pinch configuration. Figure I shows the Ion particle positions in R-Z space at this time. The surrounding magnetic flux contour is also shown.

After the gross radial motion has subsided, the plasma develops inhonageneities along the outer surface of the configuration tn the Given the restrictions of R-Z geometry, these perturbations Z-direction. are nec&jsarily m - 0 modes. these perturbations continue to grow, and after approximately 3 vs from the t * 0 Implosion break the axially homogeneous configuration into elongated current Tings or compact toroids with dimensions of roughly 2.5 cm in minor radius In the H direction and 10 cm in minor radius in the The initial corfiguration had a 10 cm radius and has a Z-direction. periodicity length in Z of SO cm. Also shown in Fig. 2 are the contourJ of constant toroidal flux, which now indicate reconnection.

As is apparent in Fig.

The t - 0 temperature of both electrons and ions was 1 eV and at t ” 3 us the ions were roughly 4 IceV. A self-consistent electron temperature has not yet bac-n Included in the model. For this run T was artifically raised from 1 eff to SO eV linearly during the first 500 is of the simulation and then held constant. The electrical resistivity is the Spitzer value consistent with this electron temperature.

such

-115-

D. W. Fewett, “A Global Method of Solving the Electron-Field Equations in a Zera-Iaertia-Electron-Hybrid Plasma Simulation Code”, accepted for publication in J. CourD. Physics, (1979).

50.0

25.0

0.0

1 -

FieuitE 1

T = 1.5 us

ION POSITIONS AND I|‘CONTOURS

5.0 R CH

10.0

> 2r>.0

0.0

T a 3 (IS

FIGURE 2

|0M POMTIOHS AHD H’ConTCllIRS

JO.O

lourant Institute, N’ew York University, Mew Yo-** , S. V. 10012

Introduction

Field-reversed B-pinches are observed to rotate, presumably as a result of

:f the loss of particles. If the rotating plasma is compressed, conservation of angular momentum dictates that the rotation frequency will increase by approximately the square of the radial compression ratio. Thus, even a small initial rotation may affect the equilibrium state and stability properties of the compressed plasma.

In this work we generalize the adiabatic compression code of H. Srad [1]

to include the effects of rotation. The situation envisioned is a compression process much slower than acoustical transit times across the major dimensions of the plasma. The plasma waves then cause fast equilibration and on the :om?ressicr. time scale the plasma is seen as creeping from one equilibrium state to the next. The different states are related to each other by maintain ing constants of the motion of the exact dynamical system, namely: the magnetic flux and the T.ass and angular momentum within each flux tube. The compression itself is created either by increasing the magnetic flux in the vacuum region, :r oy moving the external wall to a new radial position (to simulate the liner experiment).

Tor the work described here, transport processes have bean neglected and

the plasma is assumed to obey the ideal MHD equations. We also assume that the plasma is :ur;tained completely within the closed separatrijc. This represents a slight departure from reality since experimental measurements have shown a small but finite pressure on the separatrix, it offers, however, a substantial numerical advantage in eliminating surface currents which may appear in a compressed plasma which spills outside the separatrix. In addition, the assumption makes it possible to guarantee that the computed equilibria will be interchange stable. (The interchange stability criterion was generalized to -over rotating systems.)

-118-

E, Hameiri and W. Grossmanr.

Tno-Simer.sior.ai Compression in General Compact Tori

Ir. a rotating axisymmetric equilibrium state, if the magnetic field h=s

a torcidai component 3a , compression of the plasma will generate both toroidal and coloidal flews. To avoid this complication and consider purely toroidal flows we assume Bg = 0 if the plasma rotates. In the static case we allow 3j i 0.

The rotating equilibrium state can be described by a pclcidal flux func

tion -j such that 3 = 8* 7iJ>/r, and three functions of ‘Ji: -he entropy s(>), the toroidal rotation frequency Td’n), and the Bernoulli function (-‘otice that each field line Tiust rotate like a rigid body.) Ths equilibrium, stuatior.s are then

‘d(-o).

Ice Ecuilibrium State

- div (7m/r2) = r oflfl+ pH - prs/(>-i)

YspV”-/CY-l) = H + r V /2

with the dot derivative denoting d/dili. Given the value of * on the plasma boundary, and using equation (2) to solve for the density c » o(r,t|i)j Eq. (1) becomes a standard elliptic problem. Notice that <2) can be easily solved for Y = n/{n-i), w-‘th n some integer, since then y-l = 1/Cn-l). The correct y value (5/3) we use y s 2.

between a = 2 (v = 2) and ts = 3 \y~ 3/2). Here, for simplicity,

During compression, s(i)i) is unchanged but H and G vary. To find them it

is convenient to present a variational formulation for the rotating equilibrium (along the lines of a known static formulation [2]).

(I) <Li)

V(#)

-119- *

r a S3 dv = L(^)

l e v el curves in t he plasma region are assumed.

«r >)b. This e x p r e s s i on

implies u = 1 -r (r

t he bcuncary

i n s e r t ed

Minimize E = [ B2/2 + p r2!J2/2 + spY/(.Y-l)

subject to | o dv - M(ip) , ‘Aw;

ar.d the range of <!> in the plasma is also given. 7(I(I) is the domain within a particular j> surface. The two constraints represent conservation of mass and angular momentum, and the functions s( Jr) ,M(tli), L(ip) are unchanged during compression.

“allowing 3rad [1], we introduce a surface average, <f> - ) - dS/l’v|,

with the function V(x) describing the volume within any u-curve.- “rfe also define the inductance function X(V) = <|wl^/y*>. After differentiating the constraints with respect to V and defining u(i|/J = S Hr , () = f.-’/?l and s s M Vu ($’ = di|>/d7), tna variational problem takes the form

Minimize * » i KCV)*‘2 + i A2ji’/<r2u> + U’i, Y<uY>/(Y-l)

subject to = 1 .

Assuming for :he time being that the geometry is Vnown, i.e., 7(g) is given b’-it -ct nOg), X(V) is then known and averages can be taken. The ;uler ecuaticr. for u yields (for Y = 2)

u * a(v) T r2b(v) .

= i y i e l d i ng an unknown function b ( v) r e s p e ct an a l g e b r a ic cubic equation for b ).

to **!) and b by s o l v i ng t he corresponding Euler e q u a t i o ns (an ODE f or ’!>,

i n s t e ad of u ( x ). E now is minimized with

ia.-> (<*),

iiutierical Algorithm

I n i t i al ’*>(’!) is obtained by s o l v i ng a mixed QEE-algebraic system as d e s c r i b ee b s f o r e. The shooting method is used to s a t i s fy of given * values it V = 0 and at t he t o t al plasma volume.

zcr.cL’Lons

( 3)

(1)

(2)

<5)

(v)

% ’ *-

(iii) A comparison of (2) aud (5) enables us to determine H, a at each point in

(iv) The right-hand side of (1) is evaluated as a function of x. In particu

lar it is set to zero in the assumed vacuum region, So.. (1) is solved as a linear Poisson problem. The boundary conditions used are a constant value of i|r at the outer wall and t|> = const, T at the two ends of the device. This last condition equals the behavior of y in the absence of any plasma. The geometry of the new solution is determined. Steps (ii)-(v) are repeated until convergence is achieved.

Interchange Stability

When B9 = 0, all field lines within the plasma region are closed, regard-

lass of rotation. Thus, the relative positions of fluid elements carrying scecific mass u and angular momentum A, cannot be predetermined but have to be found as part of the solution. Specifically, it means that instead of specifying the functions u(v)jA(id> tions but allows “permutations”. Presumably, the plasma will pick a permutation corresponding to the lowest energy state. From a numerical point of view, however, this state is rather difficult to find. We treat the problem as a stability problem (the usual misconception; where we specify u(ty),A(i|i) and then check if the energy decreases after local permutations. After some experi mentation it is possible to find states which remain stable throughout the compression.

o ne gives the set of values of these func

The criterion for stability can be obtained in the following way: s-ecify

functions u0C*),AQ(), and let u() = u0(f(«)), MJ = A0(f{tf)>. ft) expresses the original flux coordinate of the particle now residing in >!>* 3ecause of the flux conservation, f is required to be measure preserving, with the measure expressing total flux. Thus raeas{f’1(E)} = meas{E} for every set Z in the J» interval. For a given magnetic field and density, the energy has the form Z - | F(f(if),iJ/) d* and has to be minimized with respect to all meas ure preserving f. It can be shown that a necessary condition for the function f(ji) s -\i to be a local minimum (local in the maximum norm sense}, is to have 32r73f3t|) <_ 0 at f = * , for all iji in the plasma. In terms of the variables in F.c. ’.->), stability requires

’ -120-

space.

V { u ( < uY> *, y - 1) ’ / C Y - l) * A A (l/<r’u>)’} <_ 0 .

(5)

It is difficult to decide analytically when the criterion will be satisfied except for the static case A s 3. Then u = 1 u i” <_ 7. This is recognised as the well known interchange stability criterion, wr.er 1- is pointed out that 2wl>’ = (J d H / H )-1 arid u VY is the pressure. Ir. .; is. and region of a reverse field configuraticn it appears that always

sr-< we have the condition

0, he-..ce we need u to be

ttcndecreasing t.-.e -ressure can still vanish at the plasma boundary if V -+* 0 there. r:rt’jr.ately, this is the case since K ** ” at the boundary due to the singularity associated with the separatrix while X V, which is the total tcrciial current, remains finite. The current density however depends on the derivative of singular quantities and will tend to infinity towards the se:ar=trix. This “skin current” appears regardless of the rotation only if we insist on interchange stable equilibria. The tcinputaticn sisc suggests that

towards the boundary. Notice that

Results

it is favorable for stability co have l2 decrease towards the separatrix.

Typical compressions by increase of the magnetic flux in the vacuum region, tend to make flux surfaces circular while preserving the aspect ratio [defined as the ratio of the plasma radius to tha radius of the magnetic axis).
Typical liner compressions tend to elongate the flux surfaces.
In all cases B increases by a few percent during compression.

i8 is defined in terms of volume integrals over the plasma region.)

i*. The Hach number of the rotation (with respect to the relevant fast

magnetoaeoustic wave) increases by only a few percent during compression, thus removing the possibility of shocked solutions as the result of the increase in rotation frequency.

The evolution of the ratio a = ft/$* , with fift the diamagnetic frequency

(fift = 7px|/(2raeB2)), was monitored during compression, a < 1 is known M to be a sufficient condition for stability within tV a 7LR model. ‘We find in the few cases computed, Chat a decreases during compression thus suggesting that the compression itself will not give rise to rotational instabilities.

Acknowledgment

?.eferences

3”S9 (19”:).

The authors wish to thank J. Saltzman and 3. Stevens for assistance in the computations. Work performed under Contract Mo. EY-76-2C-3077 with the Department of Energy.

H. Grad, ?. S. Ku and 3. Stevens, Proc. .Wat. Acad. Sci. USA, Vol. 1Z, 10,

*/.. :. Kruskal and S. H. Kuisrud, ?hys. “luiis 1, >£5 (1953).

H. Zrii, ?hys. ?luids ]_, 1283 (196U).

J. J. Treidberg and D. Pearistein, ?hys. Fluids 21, 1207 (1973).

-121-

Typic4l quill&Pia, flux surfacsa far fi«ld r«v«n«d 9-oiasJi. tfet XBAbdad rwiial seal.

f”

—

n o t e, va u se n e i g h b o r i ng e q u i l i b r i um a r g u m e n ts o r i g i n a l ly d e v e l o p ed by

R e v e r s ed

t he q u e s t i on of s t a b i l i ty

P f i r s ch

TEARIXG-MQDE STABILITY ANALYSIS OF A CYLINDRICAL PLASMA

H. L. B e r k, J. S a y e r, D, D. S c h n a c k, U n i v e r s i ty of C a l i f o r n i a, L a w r e n ce 1 - i v e r a o re L a b o r a t o r y, L i v e n n o r e, CA, USA

a x i s.

c o n d i t i o ns a re f u l f i l l e d:

t e nd to be e x c l u d ed

t he b u lk of Che p l a s m a, and t he s e c o nd

l ie c l o se to t he

Such c o n d i t i o ns

e x p e r i m e n ts and may be r e s p o n s i b le f or

8—pinch.

t e a r i ng

i n s t a b i l i : /.

in s t e a d y — s t a te p l a s m as

w i th a s i g n i f i c a nt b o o t s t r ap e f f e c t, v h e re

l ow t h r o u g h o ut t he

l o ng

1 2 2-

t he f l ux

l i f e t i m e s,

to s h ov t h at

in r e v e r s e d - f i e ld

t e nd to be f u l f i l l ed

t h at t he p l a s n a. e d ge

is s t r o n g ly e n h a n c ed

t e a r i n g - m o de s t a b i l i ty

t he f i r st c h at t he f l ux

l a ck of o b s e r v a t i o ns a£

f i e ld 9 - p i n ch e x p e r i m e n ts e x h i b it

T h is b e n e f i c i al e f f e ct c an a l so a r i se

to t he t e a r i ng mode f r e q u e n t ly a r i s e s.

l et us c o n s i d er a c l a ss of MKD c y l i n d r i c al e q u i l i b r ia

|2 - - a2 * «i d>>

1 3’!f f i e ld 3 » — -^~.

t he G r a d - S h a f r a n ov

1 3’j ^ 2 3B

SO e

e q u a t i o n,

s a t i s fy

if t wo

t or t he

a nd

t h is

C2)

, ,,

where ‘jr is the flux at Che plasaa edge.

The solution to Eq, (2)

in the region containing plasma is

^’..‘ork performed undar the auspices of the ‘J.S. Department of Energy by the

ijvrsn:-: T.iversiore Laboratory under contract nucber ’.»-7405-iMC-43.

body of t he p l a s i a a.

F i r s t,

f l ux v t h at

where p is t he p r e s s u r e, and t he m a g n e t ic

5Te c h o o se

Here r

t he f i e ld n u l l, and the- i n n er a nd o n c er

, r e s p e c t i v e l y. We assume

b o u n d a r i es oi

the p r e s e n ce of a v a il (r

To apply

the n e i g h b o r i ng e q u i l i b r i ua method, se n e e d, to c o n s i d er

p e r t u r b a t i o ns of

t he e q u a t i on

$ » _5

e s i ah

siah. [ a C r2 - T2) / 2]

, ,,

« r

-123-

cosh

t r a ps

r0) / 2]

r2> / 2]

( a { r2 -

r J chat

sinh [ a ( r” -

[sCc2 - T Q ) / 2]

the p o s i t i on of

t he plasma are r

2 , 2 rQ - r

t he f l ux o£ etie p l a s c a ..

2 ^ 2 , 2 2. e

the f o rn £<ji * d’Jj,,(r) exp Cikz) Chat s a t i s fy

(.7 s (r - r ) * S[r - < 2 r’ - r * J1 /

<_ rw, Chen the system is s t a b le

*i-E - - a2 fl’(:J/ - * ) + a2$0

,-r. _ / , .2 _ _ 2 , 1 / 2 , J.

If we can show

2 32p r, 3 l J )Z

t e s t i n g.

1 3 . dr

- ’ ) **

i 2 i,

( 2 ),

J 3r

.2 .

JlJ

« i ch t he bcur.dary c o n d i t i on o^(r~ ” 0) » 0.

t h ac 5<JCr~) h as

p.o n u l ls be “veer. 0 <_ r

( T h is

the sana

technique used by Marx. )

We now have, frota Eq.

C3)

,,-,

. 2 . , ,, 2

The easiest case to analyze (and the nost difficult

to s t a b i l i z e)

k = 0, and ’-‘e nov limit ourselves to this case.

After salving far £a. (6), the margiaaJ. stability condition found i s,

Hence the’equation for &<>r becomes,

»r3 1 2 «; * 2 sinh CfrJ - r2>l |l - ^

a.(r-2 ~ r2) ]|

(

M th

1 d

-124-

,1 d5i,

2, „ ,, 2

2 ,. „, 2

[” ^ * %

I2 ’ ” S -° \

2, *(V - ” V T - 2 - T - ^ ^ O -^

— «^»I C

r2 7-2

fro

“e , 2

)

. C7)

<3>

which shows that Che wall can be exposentially far as a becomes Lar-je-

Physically,

increasing corresponds to increasing the flux exclusion, in

the plasaa. The flus contained in the plasaa i’s £(r } - * C r0), vhi:h froa

Eq. (3b) gives,

The ^c-st dramatic aspect of this result is the laprovansertt of the

stabilicv criteria far large B I; K < 2.

If we first consider r » 0.

we have

Then a wall outsida the plasma can stabilize only whaa — j— < 1.

The stabilising property of flux exclusion disappears i£ the pl*s:ca

edga is away froo th« axis, i.e., ar /2 £ 1. Assuming S TQ » I,

ar , Eq. C 7”.) is approximately

”*

-125-

SEFERESCES

r2 < r2. + T< w — ed

A- Eberhagen, W. Grossnan, Z. Phys. 24£, 30 C197I).

A- G. Eskov et al., Proceedings at the 7Ch IAEA Conference oa Plas.-ja

Physics and CuaCrcHed Thermonuclear Research C1978).

R. S. Linford at al., Pracee-diags at th* 7th IAEA Conference on

Plaa^a. Physics and Controlled TiieraoBUClear Research Cl978).

D. Pfirseh, 2. Nacurforshg. I_7a, 361 (1962).

K. Mars, Phys. Fluids 11, 357 (196S).

Abstract UCRL-83698

THE TILTING MODE IN THE REVERSED-FIELD THETA PINCH* A.I. Shestakov, D. D. Schnack, and J. Kllleen, National Magnatic Fusion Energy Computer Center, Lawrence Llvermore Laboratory, Llvermore, CA 94550

Introduction

There is currently active Interest in plasma confinement schemes in which

the separatrix extends to the major axis. Fusion experiments based on such configurations (e.g. the field-reversed theta pinch, the field-reversed mirror, Spheromak, etc.) are expected to avoid certain engineering difficulties associated with conventional toroidal devices. Higher values of 6 may also be attainable.

The MHD stability of such equilibria has been investigated analytically ry

Rosenbluth and Bussac [1] who used a modified energy principle to determine marginal stability to all ideal modes and most tearing modes. They found stability against magnetically driven modes for all toroidal mode numbers n>l, but found an unstable n=l mode for prolate spheroidal plasmas (prolomaks). This mode,which is characterized by a rotation of the major axis inside the separatrix, is temed the tilting mode.

In this paper we describe preliminary results of a numerical Investigation

of the resistive stability of such devices. We employ an initial value ap proach in which the linearized resistive MHD equations are used to follow the time evolution of an arbitrary perturbation in a known equilibrium configuration until an exponentially growing eigenmode appears. The fastest growing mode is thus determined. This differs from previous work [2] in that Fourier decompo sition Is performed only in the toroidal direction so that two-dimensional equilibria can be treated. The specific model is described briefly in section 2, and in more detail elsewhere [3].

He have applied our model to a numerically determined equilibrium [4] wnich contains no toroidal field and closely resembles the FRX-8 experiment at Los Alamos. We find this prolate configuration to be unstable to an n»l tilting mode, in qualitative agreement with ref. [1]. These results, and their relevance to experimental observations, are described in section 3.

-126-

c u r1 ^ V7 ) 5i+ < V* >5o. *

3 F \c u r1 Bl) **

/- IW ,

C U Pl

\ *

d lv

c u r1

[Z)

We assume that the hydromagnetic approximation is valid, that the re sistivity is isotropic; neglect pressure and inertia terms in Ohm^s law, and take the fluid to be incompressible. When a static equilibrium (v„=0) and constant density are assumed, and the effects of perturbed resistivity are neglected, the resulting linearized resistive MHD equations are:

*Work performed under the auspices of the U.S. Department of Energy by the

Lawrence Livermore Laboratory under Contract Number W-74Q5-ENG-43.

-=? ^\w-} h

“J

Computational Model

3 B*1 TT° d iv §7

«6 cbtain Eq. (3) by taking the double curl of the equation of motion.

The component equations a£e written In cylindrical coordinates and the

equilibrium 1s specified by: Ba- r Br o(r.z) + <a 8*„(r,zJ * z BZ n( r , z ). The perturbations are of the form f| (r.z.tj exp (1n»). In the following we con sider non-ax1symmetric instabilities (rtfO). The conditions (2) allow us to express v ^, and B*, in terms of their corresponding r and z components. The Reversed Field Theta Pinch (RFTP) experiment contains no toroidal field (Ba0« 0). This allows us to consider only the four variables vr-, vzi, 8ri, and Bz]. In the general case with non-zero B$„» we need to compute both the real and imaginary parts of the complex perturbations giving a total of eight unknown quantities.

It is convenient to put the equations in dimensionless form. There are

two characteristic time scales 1n the problem, the resistive diffusion time, Tn » 4ira2/c2<n7) and the hydrcroagnetic transit time, TH * a(4ir<p0>)‘/2/<R>. Tne symbol <*> denotes a characteristic value for the enclosed variable; “a” denotes a characteristic distance. He non-d1mensiona1ize time by T H, spatial coordinates by “a”, magnetic fields by <8> and velocities by v^» a/rH. The magnetic Reynolds number 1s S» TR/T^.

In the RFTP case, the above substitutions in Eqs. (1) and (3) yield four

equations of which we present only the Bri equation in detail:

-127-

/ 3v .

— I 3BT„ jF M- W1

3B,„ , vzl)

(“roW*1 * *zo W1

whtreUu)- F 3 T < #+l zV ” FT «*

Implicit Difference methods are used to solve the system of four equations. The same equations are assumed to hold Inside the plasma and in the vacuum. We presently assume constant resistivity (*1) in the plasma but set n to an arbitrarily high value (»l(ru) in the vacuum.

These assumptions allow the use of a modified leap-frog, predictor- corrector algorithm to advance the system one equation at a time. In the in-constant case under consideration, Eq. (4) does not contain Brl. Given 8zi at time nAt and (n-l)At and vj at t- nAt, Bzi is advanced by its implicit discretization at (n+1)at, and a leap-frog discretization of v, at nit. Once Bgtl 1s computed, a ?1m1lar approach 1s used 1n the analogue of Eg. (4) to compute Bp.^’. Using Bi at t»ndt and the newly computed B f y, 95|T, the vr-, vz] equations are solved by implicit methods to compute the poioidaT velocity. This describes the predictor phase of the time advancement. The fields can be corrected m a similar manner, however in practice the correction cycle is rarely used.

We are also able to compute axisymmetric (n»0) instabilities by similar

initial value methods. The equations in this case are modified and are

at— j L (B2 l, * s ¥ [^

3v .

UL.;Wfl’*’**-”

“Slffrw

described elsewhere [3]. Results for f:he RFTP indicate that those insta bilities are less dangerous than the n-1 case to be described below. 3. Preliminary Results

de apply the methods described above to a numerically obtained equilibrium with zero toroidal field [4]. The particular case has a half-length of 45 cm., and a conducting wall of rw» 12.S cm. The resulting plasma properties are n» 2 x l 016 cm-3, Bzwall* 3-”4 KG, and Ta 100 eV. V-e mldplane plasma radius is 6 cm, the vortex radius is 3.4 cm, and the separatrix radius is 4.7 cm. Ths separatrix intersects the major axis at zsep* 20.9 cm, so that the plasma resembles a prolomak with elongation zs ep / rs e p» 4.45. This configuration closely models the FRX-B experiment, and the poloidal flux surfaces are shown in figure 1. As stated above, we model the vacuum as a highly resistive region, and choose the scale length to be a* rw an * 12.5 cm. This results in a magnetic Reynolds number S s 103.

As described above, we excite an n=l perturbation and follow the solution

forward in time until an exponentially growing eigenmode apgears. „In figure 2a we display the resulting poloidal velocity field v”p= vr r + vz z, and figure 2b shows contours of constant toroidal velocity v4. We remark that the results portrayed in figures 2a,b lie on distinct 4= const, planes that differ by an angle of n/2. This is due to the description of the perturbations as Re[u(r,z,t)exp(in<j>)] and the use of £q. (2) to express v4, in terms of vr] and vz. a more detailed analysis reveals that there is net flow across the major axis corresponding to a tilting of the plasme inside the separatrix. Figure 3a „ shows the magnetic field vectors for the equilibrium configuration f0= Br& r + B2 z. This is to be contrasted with figure 3b in which we plot f = I„ + .1 Bl, thus displaying the distortion due to the presence of the eigenmode. The tilting of the major axis is evident.

While the largest displacements are clearly concentrated near the vortex,

We have found the growth rate for this mode to be UTH= 0.47, approximately independent of resistivity. This corresponds to an e-folding time of - 2ysec. However, experimental results Indicate that such configurations can persist 1n a quiescent state for times approaching 30 usee, the discharge being lost due to a rotaticially driven n«2 interchange instability. There are seve.:1 possible reasons for the lack of observation of the tilting mode [5]. K’rstly, it is thought that the experimental flux surfaces may be more “racetrack” than elliptical, as they are in our model. In that case the concentration of magnetic field in the regions of high curvature may be stabilizing. Secondly, even if the experimental flux surfaces were elliptical, nonlinear effects may cause the field in the high curvature regions to increase, leading to a reduced growth rate or complete saturation. Thirdly, the effect of finite particle

orbits on this mode 1s unknown, but is probably stabilizing.

References

M.N. Rosenbluth and M.N. Bussac, Nucl. Fusion 19, 489 (1979).
J.A. Dibiase and J. Killeen, J. Comp. Phys. 247^58 (1977).
J. Killeen, D.D. Schnack, and 4.1. Shestakov, Lawrence Livermore Laboratory

Report UCRl-83332 (1979).

D.V. Anderson, W.A. Newcomb, and D.C. Barnes, these proceedings.
D.C. Barnes, C.£. Seyler, and D.V. Anderson, these proceedings.

-128-

I/O.

r/a.

-i?9-

z/a.

I ’**..V/-

Fig. 1. Equilibrium n>0

j mini

Fig. 3b. S0 + ( . 1 )^

r/a.

z/o.

t *

rva.

r/a.

Fig, 2a. Perturbed poloidal velocity

Fig. 2b, Perturbed toroidal velocity

; , {.

…

Fig. 3a. Equilibrium B0

Introduction Analytical solutions for azimuthally symmetric reversed-field MHO equilibria were the object of several early investigations.! Recently, there has been renewed interest in these solutions, particularly in the investigation of the MHD stability of a Field-Reversed Theta Pinch.2,3 In this paper we report the possibility of producing periodic field- reversed equilibria in the High-Beta Q Machine.** We give analytical solutions within the framework of a linear flux approach:s

p’ = dp/dy = p*-(a + bv)

I2’ = dl2/a> = T2 (rfa + rhfl)

-130-

H. Meuth and F.L. Ribe University of Washington Seattle, Washington 98195

PERI00IC FIELD-REVERSED EQUILIBRIA FOR A MULTIPLE-CELL LINEAR THETA PINCH*

(1) -for b < 0:

(2) For b > 0:

? ( r , z) - £ (G ( ^ Xb; X) - 1) + J F0(r.(lc)tXJ(A|te1’t2 * A .ke -i k z)

The Problem A.

Depending on the sign of b, we can choose the scaling parameter 2 con veniently to transform (2) the Coulomb-Wave eauaticn into the form of (b>0) or into Whittaker’s standard form (b<C).s

For >?(r=0\z) = 0, and with %<b)- V ! b |” r^i^, the homogeneous part of {2} ( i . e ., ra= r ^) reads6 as follows:

the general solution of

*(r.z) - -rfr 0-r(«iXb) WK ( 0 ) j l s(ZX)) + [ M< ( k )> h( 2 X ) ^kei kz + A.ke”i^)(3a)

where <{k) = -kS/a/fbT-HXb. The M’s and W’s are the reqular and i r r e ^ l ar Whittaker functions respectively.

(1)

(3b)

where ? is the longitudinal flux, u the plasma pressure and I the longi tudinal current. All other quantities are independent of r,z. The disposable parameters a and b are constrained by certain stability criteria2 for the case where 1=0 and ’; is periodic in z.

Inside the plasma: We assume p ;0 and take the Fourier expansion of the real quantity ? * ’?+a/b. After a change of variable the d i f f e r e n t i al equation f or the f l u xi takes the f o il owing form:

to x = = r2,

If there is only one non-zero Fourier component, (4) implies for a multiple cell solution: 2Ao < Afc + A„|(.

B. Vacuum solution: For p * 0 (and also for p” * const) the k-dependent

solutions are modified Bessels functions for m * 1:

Yv a c( r , z) = AJ + B£- + r j! e ^ A JJ I , ( k r] + Bg K^kr)) + c.c. {5)

Regardinq boundary conditions, three cases are of importance: C. Boundary conditions across a current sheath ( c o i l ): For a current

j$(r,z) = *5(r-b)-expikz-j,jl >k on a cylindrical surface one has, in addition to the continuity of v^:

where n{k) » kz/4/b”~- ^XD. Here FL and G|_ are the regular and irregular Coulomb-wave function.6 For reference we give only the resulting fields on axis (r -» 0) for I « 0;

k>0

1 3 1-

(2A0

J«.k *

L lAkeikz + A_ke-*k z)

?lc(r>bw) * lint ’^(b^-e) - const /

Br = 0; Ez . 4f *-

™5 (a72 ‘*x!bc+e ” “oTI *klb=-£> ” ¥

(4)

(7)

<6>

D. Boundary condition on a metallic wall (perfect skin effect):

E. Plasma-vacuum interface: This interface is determined by the condition p »0 in the plasma and p = 0 in vacuum. This boundary condition fs non-linear.- Generally, it has to be evaluated numerically, for on the surface both flux and its derivatives have to be continuous, and (rs>zs) = ”> where ? is the first zero (w.r.t. increasing r) of p U ). In our case _(c.f. (1)), depending on the choice cf a, b, and P0 = p(i”0)/p:

Of course, we require p0 ? az/2b in the case b > 0 to ensure real ~. For preliminary demonstration, we have assumed appropriate surface values.

III. Results

The following s t a b i l i ty criteria may be applied:2 A.

Interchange s t a b i l i t y: Finite pressure en the separatr-lx, i . e ., 1<0.* With (8), this always holds true for b>0, but for b<0 -:t implies that v within the plasma never exceed a/|b|+(a2/b + 2p0/jb|):5. This is the case for multiple cell parameters.

“We choose v>0 inside the separatrix.

Acknowledgment

References

The authors wish to thank R. Sanchez f or numerous discussions and assistance w i th the computational e v a l u a t i o n. ,

B. Surface s t a b i l i z a t i o n: _Zero pressure gradient on plasma surface:

7p|s=p’T?|5 = 0, i . e ., *! = -a/b and therefore b>0 and p0= a2/ 2 b, where we used (8) and ( a ).

C. Roman candle s t a b i l i t y: A s u f f i c i e nt condition is p”<0; i . e ., b<0-

In conclusion, we see t h at not a ll conditions can be s a t i s f i ed simul taneously. For c h a r a c t e r i s t ic values of the parameters ( c g s - u n i t s ), lines {v=eonst), 3 io c a]» and pressure are depicted in Figures t, 2 and 3.

IV. Application to the High-Beta Q Machine

low density theta pinch whose plasma radius is about

This is a 3-meter, 5 cm. be excited w i th currents of a l t e r n a t i ng s i g n, before or a f t er excited in p a r a l l el the periodic vacuum bias f i e ld which might be used to form the reversed c e l ls of Figure 4a.

i n to 30 rings which can they are a ll

to form the theta pinch. Figure 4b shows schematically

f i e ld

-132’

I ts compression c o il 1s segmented a x i a l ly

= ri*{n..)/CL«(’).

A(*:)- -^(

\ = C0

V.D. Shafranov, UK

“Reviews of Plasma P h v s i c s/ V o l. 2, M.A. Leontovich ( e d . ), Consultants Bureau, Mew York (19(56.), p. 103, and r e f s. 10, 18, 27, 28 and 30 t h e r e i n; H. Grad, MJ.: Proc. of Symposia in Applied Mathematics, Vol. X V I I I, H. G r a d ’ ( e d . ), American Mathematical Society, Providence, RI (1967), p. 162; 0. Oobrott and G.K. Morikawa, Phys. Fluids ]6_, 140 (1973).

D.C. Barnes and C.E. Seyler, “Ideal MHD S t a b i l i ty of a F i e ld Reversed

Theta Pinch,” LLL Theory Seminar (1978), LASL Preprint LA-UR-79-13.

A.E. Robson, “Equilibrium of a Reversed Theta Pinch,” informal note (1979); L. Steinhauer, “Field-Reversed Plasma Rotation and Transport,” Sherwood Meeting (1979); H. Weitzner, H.L. Serk and J. Hammer, “Analytic F i e l d- i_n_: Mirror Theory Monthly, LLL Magnetic Fusion Reversed E q u i l i b r i a ,” Energy Program, Sept. 1979.
F.L. R-ibe and l. Turner, B u l l. Am. Phys. Soc. .23, 836 (1978); H. Meuth and £. S e v i l l a n o, “The Radio-Frequency Heating System for the High-Seta Q Machine,” University of Washington Report UWFPP-4, March 1979.
The essential change f or p containing t h i rd and higher oraer terms is

there then emerge more than one s o l u t i o n, w . r . t. also J.P- Freidberg and R.L. Morse, Phys. Fluids J_L, 387 (1966) and S. Fisher, Phys. Fluids 1_4, 962 (1971).

that the r.on-uniaueness, see

W. Magnus, F. Oberhettinrer, and 3.P. Soni, “Formulas and Theorems f or Special Functions of Mathematical Physics,” Sorirger, .‘lew York (1966): M. Abramowitz, i n: and I.A. S t s g u n T e d s . ), National Bureau of Standards, Washington, D.C. (1964). We use trie followino normalization, comDared with Abramowitz’ notation (A): A.R. C u r t i s, “Coulomb Wave Functions,” Royai Society Mathematical Tables, Vol. 1 1, U n i v e r s i ty Press, Cambridge (1964).

“Handbook of Mathematical Functions,” M. Abrarowit;

-,„}, CL-(*-);

FL[I,J)

f i e l d-

the

f i g. 1. FleldHnes for a»4, Ao»0.5. Ak»rt.|,«0.512S, k»*/4, and b«l ( l e f t ). b—\ ( r i g h t ). Flux scale: Inside separatrlx « - n . 2. outside 4«2.

’

»j »; * A

-133-

ij il i a *

! » » * « * *!

and i for same parameters as In Ffg. I, Ki’th S a ! and Po8. Sl o c il is not very sensitive to c.nanccs of b for -1 ; is *Z.

flux and pressure wiif. radlgs for negative and positive values of the parameter b of Eq. (1) St Z * 0.

Ffg. 3. Variation of maoretic

4 function of r

rodlus

Fig. Z. 5;0cal

B/a B gV-Q/‘p 1! a y a / a la ^ a /a a a\a/<g & a’yays a £ \B/5 a

B B Sa

\a a a/sna a s/snsi a S/SI\H SI a/sna si a/a

si\s

u u u

1 3 4-

i l ! \B/

“/Ail2

’ O^ “—M/~S

-PLASMA

~~X^)i^~~—~^(FX^

SEGMENTED ‘RESSION COIL

FIELD-REVERSED PLASMA CELLS

MULTIPLE FIELD-REVERSED MIRRORS IN THE HJGH-BETA Q MACHINE

VACUUM BIAS FIELD

\Yr sip E GsVaafa a swerJfa a Gsfe’jp a SMEWS a s-jVatip a sWaBfe a sfl/avp a ssa

5vY//’^

”Ir^v^

^‘li^‘O

*’~

Fig. 4. qualitative plots of periodic magnetic fields in the High Beta

Q machine in vacuum (lower) and after plasma compression (upper).

n « n m 3 » B

J j^S. T dt

d Pai

ZERO DIMENSIONAL MODELING OF FIELD-RE7SSSE THETA PINCH MACHINES Edward a. Klevans, The Pennsylvania State University

A zero-dimensional (0-D) time dependent computer model Is developed to describe post-implosion behavior of field-reverse theta pinch m a c h i n e s .1 -^ The model consists of six equations, representing the rate of change of plasma particle number, electron internal energy, and ion internal energy; radial pressure balance; axial tension balance; and the change of the flux inside the major radius R, where B{R) - 0. It is assumed that the length I is much larger than either R or Che minor radius a, so chat particle and heat losses are taken to be predominately radial- However, axial losses (through the X point, for ex ample) can be included by use of phenomenalogical coefficients in the particle and heat loss confinement times,

A rigid rotor m o d e l2 is chosen to describe the radial behavior of the density and 3 field:

seen” K[ (r2/RZ) - 11 (1) canh K[(r2/R2> - 11

r, r Fig. 1. n(r) vs. r.

where nm and R, shown l.n Fig-1 , an is a specified function of tine. The electron and ion temperatures are assumed to be uniform inside the separatris. Sy setting B ( rs) » -8(0), where rg is the separatrix radius, one finds that r3 * I/IR. The minor radius is defined by Linford2 through the relation

and EC are time dependent parameters, and Ba

n 2irR(2a) - f”a2vTdt

Assuming axial uniformity of n and B within the separatrix, the equations for

S » 1 2 T /rs d r m ( r) = ln(irr 2) - i:rr 2n (tanh K)/TC

‘J » i2-r(3/2)T /r3 drrn(r) - iirr 21.5T nfcanh K)/K

n_ i

-135-

(2)

which yields R/a » 4K/(1 + tanh K)

l ± l d R . I I £ i 4 . f 2. ’ 3 R dt ” 3 1 dt

, l ip drm(r)/re q(r) - [

MT ’ "" ,i ” T R dt 3 i dt

8v^7 a 1 /2 a4 Z2lnA n

, 2K Lsinh 2K

2l . 5 T ,n (canh K)/K

; ” W eq

s m

JK dt

2dR

e m

’ W

n1^

.Ti ” Te. fT ”, l

(3)

,», K0’

(4)

(5)

K ’

(6)

^’

„.

T, dt “l T,

C /% q) - I

where T, is the particle confinement time; rcjj and Tth are the electron and ion thermal conduction times; (l/r6q) is Che average of the inverse of Che electron- ion equilibration time, defined by

I _ ^ _( 1- * £ « * - *)

( 1 0)

2 _

o W, - *2ir(3/2)T, /rs drrn(r> - 2Tr 1 Q can be reduced to

1 W. n dt * ~ T ~ R dt ” T dt

i, - if*3 d r 2 » C T * j ,2

J. Assuming d i f f u s i ve

(13)

P a r t i c le confinement l o ss at the separatrix,

< ^r) rs ” “CO f f )r p

where the Spitzer expression for t (r) was used; and k„ is the Ohmic heating rate given by

e<*

with n._’, the c l a s s i c al cross f i e ld e l e c t r i c al r e s i s t i v i ty annihilation), and j

(associated with flux

” ^ C 3 Bz/ 3 r) - -(rcKBa >/2irR2)sech2 K [ ( r2/ S2) - 1)

”

r r9

-136-

- { [ C c2nt

Pn ( r3) ( Te + T1) ] / BZ( rg)} ( 3 n / 3 r )ra

is governed by T, « N/[2irr i(nu )

rvi/z < w5M

ifw li ~ T O T ^5

, the diamagnetic current d e n s i t y, given by

.2 fl ai2 1 + (T^T.) ^2 {V i)

cross

field

wnere i is the resistivity governing particle loss- Several expressions for ” ? have been tested, including classical resistivity and the lower.hybrid drift resistivity.-’ rhe former expression resulted In particle losses which were <:aa small and did not agree with experiment. Consequently, we have chosen4

r P . 2/2V2 ”* ^ ~ ~ ^T where 3± is the ion cyclotron frequency and w^ is the Ion plasma frequency. The quantity e^ is the inverse gradient length, and a^ is the ion gyro-radius. When sn is evaluated at the separatrix using a rigid rotor profile, the value En is sufficiently small that particle loss times are larger than those observed experimentally. Hamasakl^ has found from 1-D modeling that the density drops very rapidly outside the separatrix, so sn should be larger than the value ob tained from the rigid rotor profile. These uncertainties have led us to take the quantity £na£ as a parameter which can be varied. Typically we have found that tna^_ * 2 yields good agreement with experimental data.

( 1 4)

]

The treatment of thermal conduction is uncertain because the temperature is constant inside the separatrix and is unknown outside the separatrix. If it is assumed that hot ions which cross the separatrix flow out the ends without collision then there is no ion thermal conduction loss across the separatrix. Classical tsmpera- ture gradient length has also been studied. Although the differences in re sults are not great, dropping ion thermal conduction gives better agreement with experiment.

ion chermal conduction with an ion gyro-radius

For electron thermal conduction we use

Tth ” V ^ r, w h e r« ^ r, ” -K.e<3V3 r ) |‘r3 * -K.8< V” where i is the thermal sheath length and K e is the classical value for elec tron thermal conductivity. The value of 5 Is not known, but two choices were examined: S « a^ and 5 » /ajae. When 6 * a^ is used, electron thermal con duction times are so long that electron thermal conduction is unimportant. When the hybrid gyro-radius is used, electron thermal conduction is no longer negligible, and may be responsible for the observed “clamping” of the electron temperature, in spite of a continual energy input from the hotter ions. Con sequently the hybrid radius wa3 chosen for the present studies. [Radiation

C l 55

(12)

(11)

R dt

7 d?

from oxygen Impurities was also considered as a possible mechanism for electron temperature clamping, but it was found to be small.] The unknowns in this theory are n,,, R, R, I, Te and T^. Consequently three more equations are needed. The first is obtained from pressure balance: (1/n )(dn /dt) - -[1/(T + T,)](dT /de + dT /dt) + (2/B ) (dB /de) e

** The next equation represents magnetic flux change inside R:

d»/dt = (d/dt)/* drrB - /* drr(5B/3t)

Using Ampere’s Lav and Ohm’s Law to compute 3B/3tr the Integral on the right can be evaluated. The integral in the middle is evaluated using Eq. (2). After differentiation, Eq. (17) can be reduced to the expression

tj>~

where T » (“TR log cosh K)/(n *c it”) is the decay time for the flux. The final expression is obtained from axial tension balance (see Fig. 2 ).

;rw d r r[BV ,z )/S* - p(r,2 )]

/rw drr(B2(r,o)/8ir - p(r,o>]

where rw is a conducting wall, and 111 « z0 < chamber half length. We neglect particle pressure at z0, and take B(r,z0) * B(z0). Also we neglect particle

pressure beyond the separacrix at z * 0, and set B(r,o) - B^ for rs < r < c Using flux conservation, B(z0)irrw 2 - BBir(rw evaluated, resulting in the expression (R2/rH2) * l - (tanh K)/K. Mote that 2 » 2K2 - r„2 results in a maximum value for K of 1.91. Converting setting rs the last expression into a differential equation we obtain

2). Then Eq. (19) can be

2 - rg

Fig. 2. B line behavior,

(2/R)(dR/dt) - K tanh2 K/(K - canh K) - 1(dK/dt) . The set of six equations can be reduced to obtain an equation for Zi

-137-

(19)

J. Vz

l oi = °9h K ”

K <*t ” B^ dt

(T + T_ ‘TIT + <T +T,)^TI + 5 W~ + ^T “(T ” a )— 7T 1 vth

i -th

5[— +

( 2 1)

where a is a function of K and is approximately 0,5 for the LASL experiments. It is seen that t can expand or contract depending on the relative size of the heating and loss terms, and the behavior of the B field. The set of Eqs. (7), (8), (9), (16), (18) and (20) has been solved simultane ously on the computer. Comparison with a recent FBX-B set of experiments^ conducted at 17 mTorr fill pressure of D2 are shown In Figs. 3 and 4. Initial values correspond to t - 10 us in the experiment. Initial conditions are the following; 3Q - 6.8 kC, Tt - 200 eV; Ta - 120 eV; 1% - 3.6 x 1 015 cm-3; K - 1.1; S * 4 cm; I » SO cm and N - 1.3 x 101’. The magnetic field behavior Is given approximately by B » B 0(l - (t/96 us)]. Fig. 3 shows temperature behav ior. Ion temperature behavior, which is dominated by ion-electron equilibra tion, is in good agreement with experiment. Electrons gain heat from ions and and lose it slowly through electron thermal conduction. The experimental re-

’

(17)

(16)

(20)

sults suggest a greater electron energy loss, either from anamolous eleccron thermal conduction or from radiation, rig. 4 shows the decay of M and a^, both with an exponential decay time of T * 50 us. These results are within the large error bars of the experimental values. Quantities not shown include length I, which decreased from 50 cm to 46 cm in 20 us; P., wnich increased from 4.0 to 4.2 cm; and K, which increased from 1.1 to 1.2. These results are con sistent with experimental observation.

0 29,

-138-

In conclusion, a 0-D model has been constructed which predicts behavior con sistent wich behavior observed in FRX-B. Additional worfc on eleccron tempera ture modeling may be needed. The model will next be used for scaling studies and comparison wich the 1-0 model of Hamasaki.-

Acknowledgement. 1 want to thank Rulon Linford ; nd Peter Gary for helpful discussions in the development of the model, W. Tom Armstrong for supplying FRX-B data, and John McCowan for assiscance with the computational work.. Fart of the work was performed while the author was a visitor at Los Alamos Sci entific Laboratory. This work wa3 supported by the U.S. Department of Energy.

References.

R. K. Linford, tf. T. Armstrong, D. A. Platts” and t. G. Sherwood, 7th IAEA Conference on Plasma Phys. and Controlled Fusion Research, Innsbruck, 1973.
R. K. Kinford, D. A. Platts and E. G. Sherwood, LASL Controlled Thermo

nuclear Research Annual Report, Jan.-Dec. 1977. 3. W. T. Armstrong, et al. , Bull. APS, 24,. 1979 (1082). 4. S. P. Gary, “Wave-Particle Transport From Electrostatic Instabilities,”

1979, submitted to Phys. Fluids.

S. Hamasafci and R. K. Linford, Bull, APS, 24, 1979 (1081).

SPHSF.CMAX 70RMATI0;: 3Y THETA PINCH

Y.Sogi, H.Ogura, ’/.Csanai, X.Salto, S.Shlina and H.Yoshimura College of Science and Technology, MIHOM University Kanda-Surugadai, Chlyoda-fcu, TOKYO 101, JAPAN

Abstract

5pheromak configuration is formed by using a fcheta pinch

combined with a z-dlscharge. Toroidal field is generated by the z-current flowing in the plasaia and the reverse current on Its surface. The reverse mechanism Is investigated here by the analysis of z-dlscharge circuit Including the plasma. It is easy to reverse 701 of z-current on the plasma surface by using a very low external inductance circuit.

i. Introduction

Compact torus has a,number of advantages in order to sim plify the nachlne design*’ However, it Is not easy to generate the toroidal field because of the spherical design. The toroid al field can be produced by z-discharge current operated with a linear theta pinch. It is necessary for the formation of mag netic surface that most of z-current flowing in the plasma returns on its surface as soon as the reverse field configu ration is formed. Two methods are considered to produce this closed current loop. One is the return of z-current due to an inverse voltage applied from an external circuit between the electrodes?) The other is that due to an inverse voltage associated with the compression of plasma column. The latter case is investigated here from analysis of the electric circuit including the compressed plasma. The analysis shows that the large reverse ratio of z-current is obtained by the small exter nal inductance of z-dlscharge circuit and the large compression ratio of plasma. Experimental results agree well with that of the analysis.

-139-

Oeneration of Reverse Current

It was already shown that the reverse toroidal current emerged on the plassia column when the tokamak plasma was com pressed by the fast rising toroidal fieldJ’lt is expected that the reverse current can be also generated in the linear machine. The z-current is initially flowed in the plasma along the bias field through the electrodes without the self pinch. When the current is compressed by the theta pinch field, the reverse cur rent emerges on the plasma surface. The reverse current will link with the initial current at the ends where the conductivity of plasma is very lew.

Two conditions must be imposed on the generation cf reverse

current. Cne is that the field diffusion time is longer than the compression time due to the theta pinch field. The other is a condition to the 2-discharge circuit. The present experi mental circuit is shown in Fig.l-(a). Since the theta pinch field is applied to the current after the short of the 1st crew- bar circuit,it is important to take out only the right side r.et-

worlc. Its equivalent circuit 13 shown simply in ?ig.l-(b), where Le and He are the inductance and the resistance of exter nal circuit, respectively, and L_ and H^ are the inductance and the resistance of plasma. The L£ is sum of the external induct ance, l0o> and the Internal inductance, Lj_, of plasma. Then the circuit” equation is written as follows

By using the conservation of poioidal flux in plasma, reverse ratio of z-current, a , is calculated from Eq.(l) as follows 4’

(1)

[

C Le +

-140-

X„-I

(uJ/2TT)In(a/r)

wall as possible.

Le + (ji0l/2ir)ln(b/rp)

of the compressed plasma,

Lp > 1 3 + C Hs v Rp ) I - 0

where I is trapped current in plasma, I is length of plasma oluran, a is initial plasma radius, rp is compressed plasma

nl radius and b is inner radius of conducting equation the next conditions are needed for the effective gen eration of the reverse current;

shell. From this

(i; muchsmaller’external”inductance of “the circuit than that

(2) large compression ratio, (3) snail distance between the conducting shell and the tube

Ve show the reverse ratio as functions of the compression ratio, r=/a, and Le in 7ig,U, where I , a and b are values in the pre sent experiment.

Apparatus

The experimental machine consists of a theta pinch system

and a z-discharge one, as shown in Fig.2. The machine is so designed as to generate largely the reverse current. The theta pinch coil produces 300 G negative bias field and 10 kG com pression field having 2 us rise time. The discharge tube has two cylindrical electrodes at both ends, which are 230 cm apart each other. The z-current flowing in the plasma returns to the electric circuit through the conducting shell. The z-current can be obtained till 50 kA by varying the inductance L0 shown in Fig.l. The 1st crowbar circuit is connected in order to reduce the external inductance, Le. This switch is clamped at a quarter period of the current ringing. The decay behavior of the current in plas.ma is mainly determined by the inductance, Lp,*and the resistance, RD. The 2nd crowbar switch is set for decoupling the current flowing in condenser with the current in plasma. This switch is usually clamped at a half period of the current ringing. The theta pinch field is applied to compress the plasma near the time when the 2nd erowbar switch is clamped. A typical wave form of plasma current is shown in Fig.3.

experimental Results

At first the plasma is prccuced by -he z-discharge. The

magnitude of z-current must be selected to get a initial plas-a without self pinch. It is also necessary that most of the

current flows In the plasma, but does not on Its surface. The reason is that the reverse current emerges on the plasma when the plasma is compressed. The alow rise time and the crowbar of the plasma current are favorable to this condition. At the present experiment, the current is adjusted to 10 kA with the rise time of 10 us. The plasma currant shows the rapid decrease with the increase of the compression field and the slow decrease after the crowbar of compression field, as shown lr. Pig.3- At the present experiment three values of 60, 150 and 270 nH are prepared for Le. The observed values of »* 70, 62. and tig 5 coresponding to each Inductance are plotted with small circles in Fig.U. The radius predicted from the analysis is about 2.5cm which is Independent of experimental values, and Le. The similar size radius Is an anticipated result because the com pression ratio will not depend on almost the external Inductance, L9. The Internal profiles oT magnetic field are measured by a magnetic probe inserted into the plasma in case of 1260 nK. The 32 and Bg profiles at 1 us after the plasma compression are shown in Pig.5 and Pig.5,respectively, The data of three shots are plotted in these figures at every radial position. The dotted line in Pig.5 is the current density profile calculated from the observed radial profile of the average value of Be on the assumption that the z-current flows cylindrlcally along the axis. Because of bad reproducibility of Bg near the axis, the current density is not calculated there. The current density profile shows the clear generation of the reverse current. The plasma radius estimated from this profile is about 3cm. The radial position of the null current surface in the plasma agrees well with that of Bz-0. The total 3Z flux trapped in the ?la3ma equals nearly the flux produced by the compression field within 2.5cm. This radius is almost the same within experimental errors as the current flowing region calculated from 3g profile and the radius predicted from the reverse ratio of the current.

-141

Productions of the toroidal field for spheromak configura

tion are studied frcm the analysis of electric circuit c? ;-dis- sharge. The analysis shows the generation of reverse current on the plasma column with the plasma compression due to the theta pinch. Magnetic probe measurements show clearly not only the generation of reverse current, but also the establishment of spheromak configuration.

Merits of the present method are that the reverse of z-cur

rent can be done by using a simple technique like a theta pir.ch artd its magnitude can be controlled by such parameters as the external inductance of z-discharge circuit, the compression rati; and the length of plasma. The effect of the externally supply ing voltage is excluded here, it will be needed in such case that the studies of spheromak configuration are made at the un favorable condition to the generation of the reverse current.

Conclusion and Discussion

“i£.4 Current Reverse Ratio a

< < *>

ttaG)

”. **

ii ”.

(b)

-142-

1 = * J -

L^.L.sL.

discharge Circii:

“uslcn and Plasma Physics, Oxford, (1579; 129

Controlled Fusicn Research .Innsbruck, (1973)

m r [ *f. *i

Experimental Apparatus

it kA/div, 5us/div

Racial

and J-

r u ui Million

”

;

1 «

*7<?-l6

s and

ocer 3eajn : Compression Field lOKG/div, 5ns/div

aver 3eam : Plasma Currant

/ ’ *

:r.2 Oscilloscope Trace

Pig. 6 Radial Profiles

fsrences ; X.M.Bussac et al.

”, O.C ,‘jcldenbaum et al.

?th later. Ccnf. on Plasma Physic Z”-57/7.-~- ?th Suropean Ccnf. en Control

) X.‘fckoyama et al. * Plasma Physics 15 (1977’/ =“i ; “jf.Mogi et al.

Mihon University report(1979) MUP-A-7

i 1

1 f

l=1 -Dy, sfile of 3

FIELD-REVERSED PLASMA G^H BASED ON THE ISVEP.SE-PINCH DISCHARGE

W. D. Getty, Electron Physics Laboratory, Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, Michigan 48109

The inverse pinch (i.e., hard-core pinch or unpinch) dis

charge possesses several properties that make it potentially useful as the basis of a plasma gun for injecting a field-reversed plasma into a confinement device. The useful characteristics are: (1) the possibility of initial resistive heating by axial current flow followed by compression heating; (2) the controlled genera tion of toroidal and poloidal plasma currents and the associated magnetic field components; and (3) the relatively slow plasma formation time which permits the use of pulse technology on the time scaj.e of several microseconds. These characteristics are shared with the imploding-pinch concept upon which the Paramag netic Spheromak <PS-1)1 is based, but in addition the inverse pinch is theoretically stable to ideal MHD instabilities as pre dicted by the Suydam criterion2. It may therefore offer some advantages in high-power regimes of operation or for certain geo- metrical shapes.

To produce a useful haz plasma the pinch discharge must

be separated from the electrodes and translated axially to clear them before further heating by compression or other means takes place. Several recent experiments have shown that this separa tion can be done.

The experimental properties of the inverse pinch discharge were reported by Colgate and Furth3 in 1960 and by Aitken, et. al.4, in 1964. Improved MHD behavior relative to the imploding pinch was found, but non-MHD instabilities identified as the tearing mode were observed. This mode was apparently more prom inent in the Aitken experiment, a result attributable to the of the discharge tube. More recently, the longer length inverse pinch discharge was successfully used as a preionizer in a Marshall coaxial gun^ and as a plasma injector for TORMAC5. The TORMAC plasma injector have produced a detached plasma ring with many of the proper ties expected from the analysis presented in this paper6.

has recently been reported to

-143-

The basic operation of the proposed inverse-pinch plasma

gun is illustrated in Fig. 1. A discharge with an initial posi tive (or negatiye)_B bias field is accelerated radially by the inverse-pinch J x B radial force. A toroidal field (Ba) is produced in the plasma by the z-directed current flow.” As the plasma ring crosses the axial magnetic field it will develop a "" toroidal current Jg to preserve constant flux linkage by the plasma ring. Thus a toroidal and poloidal magnetic field component will be generated in the plasma. The plasma rir.g must’be separated frcm the inverse-pinch electrodes whils maintaining

this magnetic structure. From the results of the TORMAC and PS-1 experiments1’6 it appears that this separation can occur. Following separation the ring must be transported axial- ly to clear the gun electrodes. Axial transport velocities ob served in field-reversed ring experiments range from 1 to 20cm/ ;isec, limited by eddy-current drag in the adjacent metal walls?‘3 Therefore the time scale for axial transport should be a few microseconds, sufficiently tast in comparison to the pinch time and the plasma energy confinement time (expected to be 20-40 usee). Radial compression could then follow to reheat the plasma as shown in Fig. l(c,d).

Inverse pinch experiments ’ indicate that one can use

axial currents up to 300kA and produce plasmas with B and Bft values in the l-10kG range, and toroidal currents of leveral !cA. Electron temperatures are estimated to be 5 to 10 eV and densities are 1 013 to 10l6/cm3. The S for the plasmas produced in References 3 and 4 is of the order of unity. The aspect ratio R/a (see dimensions in Fig. 1) varies from =0.1 in the Aitkeri apparatus to =10 in the TORMAC gun. The range of Bq and B , and the geometry make these plasmas similar to belt-pinch of tokamak plasmas.

As shown in Figure 1, the ejected plasma may have an aspect

ratio of the order of 3 to 10, and the geometry is similar to that of a tokamak. Since stabilitv requires the safety factor

q = S 3- =1

(1)

this configuration will have Be much larger than Bz and 3^=R/a as- in tokamaks; thus g=a/Hq2 may have to be small to maintain stability. In this geometry several arimuthal coils would be pulsed to supply dB2/dz to translate the plasma ring in a manner similar to the use of radial B-field control to position a tokamakplasma vertically.

By using a smaller aspect ratio one approaches belt-pinch

-144-

<2>

with 2a^R. Thus one can obtain c=l with 3e=Bz and high 3. The axial inverse-pinch p.lasma current would control Bq and the azimuthal coil currents control Bz. The magnetic field and plasma parameters of the early experiments give q=l from Eq. (2) and 8=1.

A plasma in the belt-pinch geometry could be generated directly by using a Bz reverse-bias field in the inverse oinch. Discharges with a reverse field of 4kG inside the plasma and a positive B2 field of 3.5kG outside the plasma have been oro- duced by an inverse-pinch with 2a/R=4 at a time of 15 ys after

Geometry and

2a Ba q _ :f 5f

initiation of the discharge . By forcing reconnection of the field lines at each end, one may be able to form a reveraed- field plasma in this geometry and translate it axially. Es’kov, et. al-9, have reported doing this in a hard core device with q=l and 3=.6-.9. In their device the hard core current did not return through the plasma, as in the inverse pinch.

An inverse-pinch plasma gun utilizing belt-pinch geometry

and the field reconnection and axial transport scheme of Es’kov, et. al., ’ , is illustrated in Figure 2. A plasma cylinder with entrapped Bg and reverse-bias B2 is injected from the inverse- pinch electrodes and a positive B2 field is applied by an exter nal coil. A pulsed coil at one end forces reconnection at that end and accelerates the plasma cylinder axially. A similar pulsed coil causes reconnection at the other end before the pinch electrodes are cleared, and axially compresses the plasma. A transverse octupole “barrier” field similar to that of Es’kov, et. al. , could be used to prevent the plasma from striking the *vail. The new feature in the scheme of Figure 2 is the entrapped Ba field, which with the induced poloidal field should allow a stable h.igh-3 equilibrium to exist from the onset of the dis charge.

-145-

CONCLUSION

‘REFERENCES

G. C. Goldenbaum, Invited Paper, Bull Amer. Phys. Soc., 24:

1072 (October, 1979) (Abstract).

W. 3. Thompson, An Introduction to Plasma Physics, Peraamon

Press C1962) pp. 117-120.

S. A. Colgate and H. P. Furth, Phys. Fluids, 3:982-1000

(1960).

6:39-69 (1964).

X. L- Aitken, et. al.. Plasma Phys. (J. Nucl. Energv Part C)
P. Deschamps, et. al., Proc. IAEA Conf. on Plasma Phys. and

Controlled Nuclear Fusion, Culham, “vol. II, pp. 449-461 (1965).

1. R. Myers, M. A. Levine, and P. A. Pincosv, Bull. Amer.

Phys. Soc., 24:1037 (October, 1979). (Abstract).

D. J. Rej, et. al., Appl. Phys. Lett. 33:910-912 (1 Dec. 1978).
A. G. Es’kov, et. al., Proc. IAEA Conf. on Plasma Phys. and Controlled Nuclear Fusion Research, Innsbruck, vol. II, pp. 135-204 (1979).
A. G. Es’kov, et. al., Nucl. Fusion Supp. 1975, pp. 155-161

(IAEA Conf. on Plasma Physics and controlled KucI, Fusion Research, Tokyo).

The inverse-pinch discharge has a proven magnetic-field configuration v»ith stability and geometric properties that make it attractive for generating a reversed-field plasma. A high-S plasma cylinder or ring can be generated with frozen-in poloidal and toroidal magnetic field components. The inverse-pinch dis charge is theoretically MHD stable and has shown superior stability properties in experiments. By choice of the aspect ratio, one can produce tokamak- or belt-pinch-like geometries. In the latter case one could use azimuthal coils to force reconnection of field lines and axially compress the plasma.

(a) Inverse pinch stage with B bias field. (b) After separation from inverse pinch electrode and formation of separatrix by induced azimuthal current, (c) After axial translation and radial compression. (d) After additional compression.

-146-

G O D a G

u O G C *

?MCi3 4(Co»wikrio^ ’ M A SS

r«Awswrta*i pmic

?fvt*3t 3, 111] SOLMSIS

? < * l l 0 Cat»

2 General scheme for separation from electrodes, forced reconnection, and axial translation of an elongated plasma ring.

A TRIGGERED-RECQNNECTION COMPACT TOROID EXPERIMENT

A.L. Hoffman and G.C. Vlases Mathematical Sciences Northwest, Inc.

y* Tf

0 J «i

Field reversed theta pinches have been the subject of renewed interes4-

both in t.ie Soviet Union and the United States. The primary reas’on for this interest is the attainment of plasma configurations which are stable for many Alfven times. One particularly interesting and very promising compact torus experiment is the 8H-I device of Kurtmullaev and co-workers at the Kurchatov Institute.” This group has observed plasma lifetimes of up to 100 usee 1n a 1 m long device. The lifetime was limited by electrical circuit parameters alone, with no tendency toward rotational instability, which has been the . lifetime terminating process in all other long lived field reversed theta pinch experiments.

The principal feature of the BH-I experiment is careful control of *n» field line reconnection process sc as to maximize the trapped poloidal flux A byproduct of this technique is an increase in the non-adiabatic plasma heating through rapid axial compression, which has important implications for both current experiments and eventual reactor designs. Mathematical Sciences Northwest, Inc. (MSNW) has initiated a reversed field theta pinch experiment which is designed to incorporate and improve on the features of the Kurchatov experiment, to thoroughly explore the controlled reconnecticn process, and to investigate the full parameter space between conventional radial compression processes and the axial compression brought about by triggered reconnection. An additional principal objective of this experiment is to explore the causes and cures for plasma rotation.

A schematic of the MSNW experimental apparatus, which is closely patterned after the Soviet experiment,1 is shown in Figure 1. A bias field is created ” by an external” quasi-steady solenoid. An additional quasi-steady coil at each end is used to apply a mirror or cusp field configuration. Eiqht pairs of oppositely directed longitudinal conductors are located as shown to fon- a pulsed octopole field near the tube wall, in addition, at each end tner» H L? aesired time.

‘trigger coil” used to force reconnection a’ tne

st Pu l s ed ™r™r

-147-

ae-PlNCH” COILS ^PAST PULSE) TRIGGER COILS (FAS”

PREIONIZER PINS STEADY CUSP/MIRROR

Schematic of MSNW Experimental CT Apparatus

D D D D D

BIAS FIELD COILS {QUASI-STEADY)

GEE ffiC C8B

PULSE)

MULT1P0LE ASSEMBLY (FAST fuLSE)

Figure 1.

figure 2 shows sketches of the field line topology at various phases of

a second half cycle “Russian operating mode”. A bias field with a weak mirrcr is established, and the plasma is preionlzed (Stage 1). The main fielc is then applied, first in the direction of the bias. It rings througn the first half cycle and reaches a maximum in the opposite direction, creating a cusp in the outer field lines (Stage 2 ). During the period when the main field passes through zero, the octopole field is pulsed on in order to minimize plasma-wall contact and to thus prevent rapid cooling and loss of trapped bias field. This technique maximizes the compact toroid poloidal flux, and

”

-148’

J2J

Main Field Applied

|5?j ** ***’

BIAS

is :i.s?

figure 2 Stages of Toroid Formation Using Cusp Holdof- arc

Triggered Reconnection (second half-cycle operating mode)

nay also inhibit development of rotation. When the main field is near its value, the fast trigger coil is energized, causing rapid reconnection peak (Stage 3). The resulting toroid is not in equilibrium and strong axial compression o urs (Stage 4 ). In the Russian experiments, it was shown that this axial contraction increased the line energy density and the separatrix radius, resulting in long plasma lifetimes.

The MSNW experiment, while incorporating all of the features-of the Kurchatov experiment, has the capability of providing a higher rate of field H se so that it can explore the trade offs between radial and axial “shock” heating. It employs an axial Z discharge, so that high levels of preionization are possible and it can operate on either first or second half cycles. First and second half cycle operation 1s indicated by the 8Z plots on Figure 3a labelled “conventional operation”. Also shown 1s a mode labelled “compound magnet operation” which uses a quasi-steady outer solenoid and a fast pulsed concentric inner “theta pinch” coil set. In this second mode, the initial quasi-steady field is nulled out by the inner theta pinch coils, and operation from that point proceeds as it does in the “standard” mode. The principal experimental advantage is the ability to achieve a fast rise (10 usee) followed by a very long (- 100 msec) flat-top, rather than the 100 - 200 usee decay of conventional crowbarred 9-pinches. The principal uncertainty with the compound magnet technique lies in the lim’ted tiise to preionize, ar.d in the relatively lower rate of field rise at the zero field cross-over point. 3oth <iifficu1ties can be overcome, if need be, by a pulse shaping network driving the inner magnets.

-149-

1st HALF CYCLE OPERATION 2nd HALF CVCLE OPERATION

Figure 3. Schematic of Operating Modes

(b) Compound Magnet Operation

(a) Conventional Operation

I—PREIONIZATION

BIAS

A drawing of the magnet system is shown on Figure 4. The basic plasma tube is 1 m long and has a 20 cm bore. Fast theta pinch and trigger coils are fed through gaps between the quasi-steady bias and cut:; coils. The octopole rods can produce 0.20 tesia fields at the Inner edge of the plasma tube. Initial operation will be in the standard mode with 1 tesia theta pinch fields produced with 3 usee quarter cycle times. The compound magnet wir, be operable up to 2.0 tesia fields. Depending on the efficiency of the axial heating process, scaling analyses predicts that toroids with 1 0” cm”3 peak densities and up to 500 eV temperatures could be produced. Containment times based on classical perpendicular thermal diffusion are 10 msec, and the quasi-steady field decay time is designed to exceed this value by a factor of 10.

-150-

Figure 4. Assembly of Compound Magnet

The experiment is supported by an analytical-computational program wiiicn

incorporates a 2-D hydrodynamics code and examination of the effects of anomalous transport and impurity radiation.

REFERENCE

A.G. Es’kov, et al., in Plasma Physics and Controlled Nuclear Fusion

Research (IAEA, Vienna 1978).

PLASMA ROTATION IN FIELD-REVERSED THETA PINCHES

L C. Steinhauer Mathematical Sciences Northwest, I n c ., Bellevue, Washington 96009

Field-reversed plasmas (FRP) have f or many years been observed to spin up

and develop a destructive m-2 i n s t a b i l i t y. While the i n s t a b i l i ty threshold r o t a t i ng modes has been reasonably well characterized, the cause o f ’ r o t a t i on remains obscure. Several mechanisms have been proposed to explain the r o t a t i o n. The most promising seems to be the f o l l o w i n g: end shorting on open f i e ld causes that portion of the plasma to spin up; viscous f r i c t i on r o t a t i ng motion to plasma on closed f i e ld l i n e s. We develop t h e o r e t i c al models to describe t h is mechanism.

then t r a n s f e rs

f or

Structure and Rotational S t a b i l i ty of Field-Reversed Theta Pinches. The separatrlx in a f i e l d - r e v e r s ed pinch divides the plasma into two d i s t i n ct regions, the central closed magnetic f i e ld l i ne region and the surrounding open l i ne region. ( f i e ld vectors in r-z plane) and have negligible t o r o i d al

theta pinch, the magnetic f i e l ds are purely poloidal

In a s t r a i g ht c o il

Rotating FRP s t a b i l i ty was treated recently by Barnes, et

tl. They c a l

culate t h at the s t a b i l i ty threshold f or the commonly observed r o t a t i ng m=2 mode is .” = 1.35 to 1.55 x G* where &» * -(1/renB)Sp,/5r d r i ft frequency; i . e ., f i e ld pinches.2

the threshold is similar to t h at predicted f or non-reversed

is the ion diamagnetic

MHD Picture of Plasma Rotation. Treating the plasma as a magnetohydro-

dynamic f l u i d, the principal equations governing the r o t a t i on are the ion angular momentum law,

1 5 1-

f i e l d.

1 v ( r2n v , ^) = —7 , nr

BZ {3. <c * c: - s

V(rBe)

r B o)

(

where D/Dt is the convective d e r i v a t i v e, vj_ is the ion kinematic shear v i s c o s i t y, and S is a source term containing p r i n c i p a l ly gradient terms in the poloiaa’; f i e l d,

temperature and other plasma properties.

Evidently from ( 1 ), changes in angular momentum arise from e i t h er viscous

f i e ld torque on the plasma. The r o t a t i o n al k i n e t ic

forces or jxfi forces. The l a t t er are manifested h e u r i s t i c a l ly as magnetic l i ne tension exerting an azimuthal energy is not carried by transport processes or torsional Alfven waves3 (else the waves would decay) but instead springs out o f ” p o l a r i z a t i on energy latent in two forms; s t a t ic energy 1n the non r o t a t i ng plasma. This is p r i n c i p a l ly due to ion displacement in the presence of the e l e c t r ic to I1); and convective energy due to displacement when there is a radial pressure gradient (proportional energies appearing 1n a spinning plasma are, in the course of establishing a new plasma e q u i l i b r i u m, drawn n a t u r a l ly from the transverse k i n e t ic energy, P, and axial f i e ld energy B|/2u.

to Erd p . / d r ). The B|/2u and r o t a t i o n al k i n e t ic

(proportional ion

f i e ld

the

l i n es

( 1)

1 D(r2n) rr ~ot rs

3t \r

and a reduced form of Ohm’s law, which in s i m p l i f i ed form is

Rotation on Open Field lines. Open line plasma contact with end walls

shorts the transverse electric field. This initiates a torsional Alfven wave which imparts an angular velocity of f. = -a* to the plasma (if shorting produces £r = 0). In the dissipationless problem, the interaction of waves from opposite ends alternately sets fl to -fi., to -2a», then to zero, and so on. The dissipation (diffusivities ni/u. n,r/y* vi appearing in (1), (2), which are comparable at 3 = 1) leads to spatial spreading of approximately

where ^ is the ion larmor radius in vacuum and LA is the axial collisionality. Evidently the spreading, « P J, is small compared to the axial dimension L and would be negligible if the torsional waves had uniform velocity over the plasma cross section. high at low density and vice versa. Consequently, the rotation, Si, quickly develops an evolving closely-spaced spatially-oscillating radial structure with length scales less than o.. In short, dissipation produces a roughly uniform rotation 9. * -0, in at most a few Alfven transit times (based on the Alfven Speed at the separatrix).

is very

Even after end shorting, a net outward-pointing electric field can arise on a transient basis as observed by Ekdahl.* There is another effect which even in quasistatic conditions can cause a reversed (outward)-electric field. If the electrical potential is uniform across the plasma at the end wall, then the thermoelectric potential in the plasma interior is proportional to kT /e. If in addition, the plasma has a negative radial temperature gradient, there will be an outward electric field. Such gradients are expected because the internal closed line plasma remains hotter, not being in contact with the end walls. The resulting rotation is found by applying the appropriate electric field to Ohm’s law

a <1 ff

-152-

In fact,

5 P , ( L / xe)%

tlttty do not since their speed Vfl * Bz/Sn

2 = -a* i + r0?)0rt)’

(3)

(4)

where 5j = - T /(3T /3r) is the local electron temperature gradient length, and

s- s corresond likewise to T. and n. Clearly-ft/fL can exceed unity as a

consequence of the expected 7T . The thermoelectric potential then provides a physical basis for the outwardly oriented electric fields needed to explain observed rotational instabilities in non reversed 71eld theta pinches.2

Rotation on Closed Field Lines. In contrast to the open-line plasma,

the closed-line plasma is spun principally by viscous forces rather than jxB forces. Shorting can take place within a closed field line, but the result only produces uniform rotation along the flux line, 2 “Zi (*). This result holds so long as n, Te and T., are uniform along the flux line. Angular momentum imparted by viscous drag to plasma near the separatrix is therefore transmitted at the Alfven speed to plasma on the same flux line near the axis; this in

’ !!’

effect increases the moment of inertia of the outer layers.

The transient spinup of the closed line plasma accounting for the inertia

enhancement and ignoring cross-field particle diffusion is governed by

where R is the radius of the magnetic axis (where B = 0). This is a simple diffusion equation with spatially dependent diffusivity; since it is in fact linear, it can be solved by a normal mode analysis. The transient behavior is dominated by the lowest order (fundamental) mode, the decay time of which is

Tspin ” ^

for a deuterium plasma with T * T. - T and inA = 15; >, is v, at 2, = 1 (3i is based on the local magnetic field). The proportionality factor a depends on^the aspect ratio of the toroidal plasma; calculated values for a rigid rotor density profile are shown in Table I.

-153-

Aspect Ratio, R/a

Table I Transient Spinup Factor

.a2 , a* ,..__, _ az(cnQT3 / 2{eV) : ^ ( u s e c) ’ ’ J9 ’

W.* P (§)’»]

0.16 2.3

165

{ 6)

,,,

Several trends emerge from this result.

(1) The transient spinup time scales with plasma size (minor radius) and tempera

ture in the same way as the classical cross field particle transport time T, = L a ^ n ,; since v. s 3.2n,/_ (deuterium, T = T ) then -: „,* r. = a/1.6.

(2) ‘Ine proportionality factor a from Table I is extremely sensitive to aspect ratio. The physical basis for this is clear considering that the kinematic viscosity varies as vj_= n/3T; low aspect ratio plasmas have nign average density and low average field inside the separatrix, therefore they tend to be highly viscous; high aspect ratio plasmas have low average density and average fields almost as high as.the vacuum field, B . Of course a highly viscous plasma will rapidly spin up in response to a rotation applied at the surface, and a low viscosity plasma will spin up slowly.

(3) Evidently for R/a < 3, the transient spinup time is shorter than the plasma decay time; for R/a > 3 the plasma tends to decay faster than the spin transient.

The above analysis only t r e a ts the t r a n s i e nt aspect of the spinup.

The other side of the question is the quasisteady r o t a t i on ‘*ate t h at is approached. Trie asymptotic state has unfform Q. ( r i g id body) inside trie separatrix with r. = .”. , the r o t a t i on rate at the separatrix. greater than (.%)s according to ( 4 ) ). r a t a, :.., were r e l a t i v e ly f a s t, than the r o t a t i o n al s t a b i l i ty l i m it ( - £ / £„ 0(1)} could be reached in a time less than T , t i rs

. to reach -iV”+ * 1 is

I f, e . g ., the asymptotic r o t a t i on

In t h is case the’ stable

is equal to or s l i g h t ly

(Note, 2

. . spin

~,«

where <n*> denotes the average value Inside the separatrix. If, on the other hand, r < <dt> then the asymptotic state should itself be stable. What dis- tfngufsfies tnese two cases is the plasma profile, file of ?.t. If the radial pressure profile becomes ysry steep near the separatrix, such as may arise as a result of endloss in the open field line plasma, then r. > <r^> and T ^ nay be short indeed. If, on the other hand, the gradients near the seoaratrix become more gentle, which is the case for an anomalous radial transDort sheath, then 2 > <£*> and the asymptotic state may be rotationally stable.

the radial pro

in particular

Summary. The open field line region is spun up principally by jxB forces, and the closed line plasma by viscous forces (although jxB forces insure uniform ’> along a closed flux line). The non uniform speed of torsional Alfven waves caused by end shorting creates a highly transient, highly structured radial .“s-profile; dissipative processes smooth the profile in at most a few Alfven transient times, producing Si = -£#. The transient spinup time in the closed line plasma scales in the same way as the transverse particle diffusion- time, but has extreme sensitivity to aspect ratio; it is short for “viscous” low aspect ratio plasmas, and vice versa. Ultimately, the rotational stability depends on the ion pressure profile; endloss (which steepens the profile near the separatrix) is destabilizing, and anomalous radial transport (which flattens the profile) is stabilizing.

-154-

Vl * ^ p l n ^ ‘S

C 8)

REFERENCES

988 (1979).

ACKNOWLEDGEMENT

D.C. Barnes, C.E. Seyler and D.V. Anderson, Bull. Am. Phys. Soc. 24_,
O.P. Freidberg and L.D. Pearlstein, Phys. Fluids 21_, 1201 (1978).
A. Kadish, Phys. Fluids J£, 141 (1976).
C.A. Ekdahl, R.R. Barch, R.J. Contnisso, R.F. Gribtole, .<.F. Mc<enna,

G. Miller and R.E. Siemon, submitted to Phys. Fluids.

This work was supported by USDOE contract 0E-AcCS-76ET330Z’ (formerly EY-76-C-06-2319).

S7r-u?.:u:< A r-nrsHiD srsLLAaAros - SPHEEQMAX

Charles W. Hartaan, Lawrenoa Liveraore Laboratory, university of California, Liveraors, California 94550

Introduction

It has been known for soas tine that nonaxisyaaatric, vacuua-fieid, toroidal confinement systems can be formed by combination of axisysHetric aultipole fields acd modified S t e l l a r a t o r - l i ke f i e l d 3 .1 Thi3 “hybrid ization” can result in improved negative V!’ properties and shear. addition, simpler coil topology, as compared with a conventional S t e l l a r a t o r, is possible sines the S t e i l a r a t o r - l i ke windings need not link the torua. fields however, It is apparent that plasma current can a l so serve to provide a poloidal field which can then be transformed in the direction by helical s t e l l a r a t or windings aa has been r e c e n t ly suggested2.

I n i t i al calculations were concerned l a r g e ly with vacuum

t o r o i d al

This paper discusses hybridization of modified S t e l l a r a t o r - l i ka transform windings (T-windings) with a Spheromak3 or Fiald-asversed- Mirror configuration.^ This ccnfiguration-Stellaraak-rstains iaportant topological advantage of the Spheromak or FHM of having no plasma linking conductors or blankets. The T-windiags provide r o t a t i o n al transformation in toroidal angle of the outer poloidal f i e ld effect creating a reversed B -o r o i d 3i Spheromak or adding average Bj to the F3M producing higher shear, increased U n i t i ng 8, and possibly greater s t a b i l i ty to kinks and t i l t. The presence of f i e ld r i p p le in the toroidal direction may be sufficient to i n h i b it cancellation of directed ion current by electron drag to allow steady s t a te operation with toroidal as well as polotdal current maintained by n e u t r al beams.

l i n e s, in

the

I I. Stellaraak Configuration

The basic hybrid approach is shown in Fig. 1. Axisyamafcric

f i e l ds

the two rings, and a unifora 3Z to give the

are provided by Iz, paioldai-fiald pattern shown. -Helical windings, centered on t hs z - a x i s, provide S t e l l a r a t o r - l i ke rotational transform. The coablaed f i e l ds give numerically computed toroidal magnetic surfaces a3 shown with the rotational transform on the small-major-radius side of the surfaces provided by the axisymmetric poloidal field and the t r a r s f a ra ca the large-major-radius side provided ay the helical windings. Continuation of che rotational transform by the axisyaastric Bp eliminates the r.aed for helical windings which link the toroidal surfaces.

Application of T-windings to produce toroidal r o t a t i o n al

transform tj of the basic poloidal field of a Spheromak is shown in F i g. 2. The modular T-windings shown in Fig. 2(a) l ie on a sphere and follow loxoaroaic spiral trajectories with l a t i t u d i n al c r o s s - l i n k a. A vacuum poioidai field line near ij; = 0 passes near the axis of the Sphe-omak,

-155-

to that shown in Fig. 2(b). Numerical c a l c u l a t i o ns

saerjes at the poles, and .“stums near the spherical surface where the T-windinga induce toroidal transform modifying the basic Sphsromak configuration progress (assuming vacuum 3p) have given t ^ / t0 ” .3*! for 3 T-vind- ing modules with flux surfaces as shown in Fig. 2 ( c ). A l t e r n a t i v e ly T-wir.di.is3 for a hi&iiy prolate reveraec-field- configuration would aore nearly resemble the helices of Fig. 1 and a r b i t r a r i ly l a r ge could be obtained depending on the elongation.

t ^ / tp

ill. MHD Stability Aspects

The most striking aspect of introducing transform windings to a

Spherooafc/FRM is that shear and reversed toroidal transform can be introduced externally without hole-linking conductors. For a l a r ge aspect r a t io stabilized Z-pineh the maxiaum stable S is increased from 155 to HO? in going from a field profile having &; s 0 on the l a st <l>p surface to the usual reversed B^ configuration. A s i m i l ar factor of 3 in 3 can possibly be obtained for the Stellaraak providing ballooning limits are not encountered. Similarly, T-windi.Tgs added to the FHH caa lend shear stabilization to an otherwise MHD unstable configuration. Since a conventional s t e l l a r a t or is stable to t i lt and s l ip solid body notions, the transform windings would presumably give improved s t a b i l i ty in the Stellarmak.

IV. Equilibrium Aspects

Probably the most significant acvantage to be gained by introducing

symmetry perturbing transf ora ‘windings is the potential of s u s t a i n i ag plasma currents by injection of directed neutral beams- Electron trapping and viscous effects caused by the field ri^pxa can prevent electron cancellation of the ion directed current opening the way to a t r u ly steady state system. For a large spheromalc reactor with losses r e s i s t i v e ly decaying currents the Q based on r e s i s t i ve

i s,

-156-

On = /Pf dt/WM ~ 100 ~ 82B2

t he

tfegative-7” Toru3 ” Physics of

H.P. Furth and C.W. Sartaan, “Strong Fluids, vol. 11, No. 2 (Feb. 1963) p. 108.
T. Ohkawa, “Physics of the OHTE (Qhmically Heated Toroidal Experiment),” in these Proceedings.
M.M. Sussac et a l ., 7th I n t e r. Conf. (IA2A, Vienna, 1979). f. G. Carlson, “Meutral-Beam-Sustained, Fieid-Heversad-Xij-ror Psac-sr,” in these Proceedings.

in Plasma Phy3. 4 Cont. W>JC. Fusion, Proceedings

so that a relatively small currant-sustaining bean pcver is necessary the process is fairly efficient. This is especially t r ue if 23 is increased by a factor of 3 by enhanced s t a b i l i t y.

References

-157-

(a) Coil C o n f i g u r a t i on

Magnetic Surfaces

v v\

Kybritro-n Configuration. Inters&cf i.onS °£ magnetic surfaces with Che r-z plant [Ti.%. 2(b)] w&re computed for the coil configuration [~ig- 2(a)] with 3. = 1/r, B = 0.12, 1^ = -0.12 (at r » 1, z ** +1) and with a helical” eagnecic potential X = CAl(fcl”) cos(A(J + fcz) where A * 6 = “k and C\ = 1.

*158-

ra» w^>? rvi

NOTICE

Cb)

Fig. 2. Stellarmak Properties. Fig’. 2 ( a) shows chrae

nodular T-windings based on loxodromic s p i r a ls and l a t i t u d i n al cross l i n k s. Also shown a rs compensating loops. Fig. 2(t>> shows h e u r i s t ic fisld-line-crajeccory p r o j e c t i o ns in a_jBer±dical plana. Mso showi axe average Bp atid B~. vs. & for Spheromak and F2H c o n f i g u r a t i o n s, shows computed z i s l i - l i oa projections in equatorial plana for S nodular w i n d i n g s.

r i g- 2Cc)

tha

“Work performed under the auspicjs of the U.S. Departr am of Energy by the Laurence Liverffiore Laboratory under contract number W-740S-ENG-48.”

This report was prepared u in account of work sponsored t>y the United St3ies Government Neither the United States nor ihe United Stoics Department of Enersy, nor any of their employees, nor any of their- contractors, subcontractors, or their employees, makes any Morranty. express or implied, or assumes jny lesal liability or responsibility for the jecuracy. completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe fmatdly-oiuned rierlts.

Reference :o a company or product name does not imply approval at ’(.*commendation o; the product by the University of California or :he U.S. department at £ne:?y TO the evcluston of others chit may be suitable.

UTILIZATION OF ELECTRO»7 COILS FOR AN ADVANCED TOKAMAK AJTD CONJECTURE ABOUT THE CAr.‘SE 70R CURRENT STEP (DOWN) . Shoichi Yoshikawa, Plasma Physics Laboratory, Princeton University, Princeton, Uew Jersey 08544

Utilization of electron coils for an advanced tokamak

The use of relativistic electron beam current (REB) in

the plane of plasma current in tokamaks offers several possi ble advantages such as:

but

(ii) lowering A(= R/a) ,*

-159-

(i) reduction in q;

EOH - ab Vph nJ

w3’2 %

^rfw * np 737I

?rfw ” % V

(iii) avoidance of current driven microinstabilities.

We repeat, though, the above advantages may well be illusion- ary.

The fourth advantage is the possibility of a current-

sustained tokamak. In this scenario, the REB electron current or electron coil will be set up either by runaway discharges a l<z UCLA or by REB injection a la. IPP, Nagoya University. The resultant configuration will be kept for a while until the cold plasma trapped in this magnetic trap is heated to high temperatures leading to ignition or high Q (Fusion Power/ N.3. Power) modes. Then the REB current will be maintained by the rf wave acceleration of electrons. This should be compared with the rf maintenance of plasma current.2 The rf oower needed, Prfw* f °r maintaining the REB current i’s given by

t-\iere ni, is the beam density, W is the energy of relativistic electrons, n„ is the olasma electron density. Therefore f voh s

This should be compared with the rf power needed to couple :o superthermal electrons, ?r-^,, which is

(2)

(3)

where f may be between 1/3 to 1/10 and v is the electron thermal velocity. The rf power needed for REB is then lower than P -T by the ratio

-160-

Prfw -

PrfT ” %^Tk

7p + p = %.

e which could be easily 1/10 to 1/30.

Conjecture about the Cause for Current Step

, As Mohri har shown, the current in the SPAC experi

ment decays in step. One possible explanation of this phenomenon is the inability of the REB beam to maintain toroidal equilibrium.

REB is moving jn toroidal directions, hence there is

a centrifugal force, F = (F0, 0, 0 ). This force is not derivable from the surface function £ 0JJ) . The situation is analogous to toroidal equilibrium of rotating plasma under gravity. Thus the plasma pressure must.be finite, so that

It is conjectured that the plasma pressure decays with its own energy confinement time until when Eq. (6> could not be maintained. Then the equilibrium is lost. The resultant partial loss of RSB may contribute to the production of back ground plasma either by instability (yet unidentified) or by beam-wall interaction. The increased plasma pressure, re stores the equilibrium by satisfying ?*. (6). The process repeats itself, hence current steps. Normal resistive inter action between REB and background plasma is not expected to keep the plasma temperature high.

The experimental verifications _of this conjecture may

be done by measuring plasma pressure. ’ Also, if the plasma can be heated by rf or other means without upsetting the electron coil, the current step may be avoided. Finally Thompson scattering may enable the measurement of pressure gradient within the magnetic surface.

(6)

Referenceo

(1966).

Yoshikawa and H. Yaraato, Phys. Fluids, 9, 1314

1 6 1-

a. Mohri, this proceeding.

S. Yoshikawa (to be published).

1S. Yoshifcawa, PPPL Tech. Memo, 260.

DYSAMICALW FOFMED SPHEROMAK PLASMA (PS-1)

G. C. Goldenbaum, Y. P. Chong, G. Hart, J. H. Irby department of Physics and Astronomy, University of Maryland College Park, Maryland 20742

-162-

Formation Phase

In this paper ve discus3 our recently reported observations of the for

mation of a highly elongated spheromak configuration.(1} In the present context, we define a spheromak as a toroidal configuration with both toroidal and pololdal field components, but with no toroidal field coils and no transformer. The toroidal field vanishes on the symmetry axis and has a maximum near the magnetic axis. Both the toroidal and poloidal fields have approximately equal maximum values. Our experiment is called Paramagnetic Spheromak (PS-1). The present vacuum chamber is a circular cross section cylinder (R -11 cm) filled with deuterium gas at a pressure of 5 to 30 mlorr. A capacitor bank is discharged into a single turn solenoid surrounding the cylinder creating an axial Bz field. The field rises to 4 kG in 9 usee and is clamped to give a 100 usee L/R decay time. At about peak Bz field a second capacitor bank is discharged between annular electrodes to produce an annular shell of Iz current. This current puts the B$ toroidal flux into the plasma. The current rises to 150 kA in 4 usee. Finally a fast rising current is produced in the external solenoid creating a 9 kG Bz field oppo site in direction to the initial Bz field. This field rises in 1.5 us e c. Presently, this circuit is cl-mped so that it decays to 1/e of its maximum in 58 usec. Most of the results reported in reference 1 were obtained with out the clamp switch however, i.e. the circuit oscillated. The plasma radially implodes at approximately the snowplow speed to give an ion energy of Bz-/47rne, where Bz is fast rising piston field and ne is the density. For the conditions of this experiment, the radially directed ion energy is about 1.0 keV. Typically, in theta pinches, some of this directed energy is randomized resulting in larger ion temperature than electron temperature. The ion and electron temperatures may then equilibrate if there is no other heating or loss mechanism dominating one of the species.

Three important physics questions arise in the formation phase: 1) Mill closed poloidal magnetic surfaces form? 2) Will the toroidal field be peaked approximately in the center of the closed flux surfaces (paramagnetic toroidal field)? 3) Will heating occur as a result of the implosion? In . the prjviously referred to paper (1) all these questions are answered in the affirmative. An experimentally determined poloidal flux surface contour map is shown fn figure 2, demonstrating closed surfaces. Flux surface closure occurs within 0.5 usee of the 3tart of the reversed 3Z field. Initially, two magnetic islands form which coalesce to form the structure shown in figure 2. In reference (1). it was shown that for the conditions chosen, the maximum of toroidal field is about 6 kG and occurs near the centar of the magnetic island shown in figure 2. The poloidal field on the outer edge rises to about 9 kG. Finally. Doppler broadening spectroscopy has indicated a temperature of 190 eV for carbon impurity ions while the same cype of raea- surements on the D_ line indicatas 290 eV at 2 ysec. Thomson scattering

measurements give 3n electron, temperature of 30-50 eV at the same time. He-.Ve laser Interferons try Indicate a density of 3xl0l-5Cm~3 In the center of the island at 2 usee.

In any dynamically formed plasma, an important consideration is the

relaxation of the imploded state to the desired equilibrium configuration. An appealing feature of the spheromak is the possibility of a near force-free configuration corresponding to a minimum magnetic energy. Rosenbluth and Bussac(2) have considered a classical force-free spheroid as well as slightly prolate and oblate spheroids. They find a tilting instability for a prolate spheroid. We have no internal magnetic field data for times later than 2 usee and consequently, cannot, with certainty, say whether the magnetic sur faces exhibit any tilting after 2 usee. We do have interferometer measure ments of the integral of the density along lines parallel to the z axis so that the integrated density as a function of radius can be determined. We find chac ac 30 usee, che plasma still exhibits an annular shape vlch no indication of grossly unstable behavior. During this time there are about 40 Alfven transit times from che end to the center.

Since it Is desirable to know what the largest S value is for which the

plasma remains grossly stable, it is of interest to Iciow what che value of beta is here. We will assume an operational value of bets defined as

-163-

Equilibrium Phase

. _ 6Vp(r » 5 cm)

B2( r -U cm)

Because the ion lines used for the ion temperature measurement burn out, we Thomson scattering mea do noc know the ion temperature after about 5 \iasc. surements indicate an electron temperature of about 30 eV as late as 40 usee. Since the electron and ion equilibration time is of the order of a raw micro seconds, we have to presume the ion temperature has dropped to the electron temperature. From the interferometer measurements we get the line integral of the density at r-5 cm as a function of time. Assuming a length of 30 cm, we calculate che density and hence che pressure. In figure 3, we plot Se as a function of time. We see that as the field, density and temperature drop, the value of beta stays almost constant, in che vicinity of 0.3 to 0.4. F’om this we conclude that it is possible to confine 3 = 0.4 plasmas. The question of whether the internal magnetic field structure remains the same during this time is difficult to answer at this time. We note, however, that at t*2 usee, the radial profiles of the toroidal and poloidal fields are to a good approximation, given by Bessel functions, [Bj,«Ji(krr) and 3r ’ - O ^ r1) ]- T he Bessel fun-tion model has the property of retaining Its configuration as it diffuses. Sines the configuration decay time ( T - 4 T O / 1 C | C 2) for chese parameters is abouc 66 usee, the configuration should not drastically change, in 30 usee if only classical resistive diffusion is present.

This work is supported by che O.S. Department of Energy under contract

3E-AC05-77clT 53044.

G, C. Goldenbaum, J. H. Irby, Y. P. Chong and G. Hart, ‘Jniveraity of Maryland, Department of Physics and Astronomy, Report py 80-057, revised December 1979.

M. S. Rosenbluth and M. N. Bussac, Nuclear Fusion 19, 439 (1979).

ure 1

^ C o il

-164-

PrtiofKration

References

Bias MagntTic Fold

Eauillbrium

in PS-1.

Implosion Fiild

Rteonmction

F b st Riling

Implotion

3 —r

I , -’

x—

Schematic diagram of the formation and equilibrium phases of the spheromak c o n f i g u r a t i on

Flgure 2

Contours of constant poloidal flux at 2 psec after the start of the fast Bz field. The contours are calculated from magnetic field measurements made in the plasma and are plotted In intervals of ZOirkG-cin2.

-165-

KJJ sec)

JS.

Figure 3 Time dependence of S - At 40 usee the externally applied field

has dropped to half Its maximum.

SPHEROMAK EQUILIBRIUM AND STABILITY AND NUMERICAL STUDIES OF A 5PHEROMAK FORMATION SCHEME

M. Okabayashi, S. Jardin, H. Okuda, I. Sato*, G. Sheffield, Plasma Physics Laboratory, Princeton University, Princeton, Mew Jersey 08544, and A. Todd Grumman Aerospace Corporation, Princecon, Sew Jersey 08540

1 6 6-

Recently, intensive theoretical studies of the spheromak configuration

have been carried out to reveal possible advantages towards a commercial re actor [1-3]. These MHD analyses have shown that stable configurations exist at medium beta with respect to the externally applied equilibrium field. The focus of recent theoretical work has therefore shifted toward the study of plasma formation processes without a toroidal or inductive coil on the axisymmetric axis.

The formation scheme proposed by Princeton is quasi-static where Che

configuration formation takes place over many AlfvSn transit times (-1000), through a sequence of quasi-MHD equilibrium states. The process is controlled by three groups of C D I I S: (1) external equilibrium field coils; (2) a “flux core” consisting of toroidal coils producing poloidal flux, and a toroidal solenoid producing toroidal flux; and (3) a pair of “pinching coils.”

Initially, the equilibrium field coils are turned on and the resultant

field, which supports the final plasma equilibrium, penetrates the vacuum vessel. All subsequent field programming proceeds on a faster time scale, whereby the vacuum vessel acts as a perfect conductor. The toroidal coils in the core and the pinching coils are now turned on, to generate the vacuum field pattern of Fig. la. Breakdown is initiated around the core and the solenoid is energized. Field pressure balance causes Che plasma to expand preferentially towards the low field region on the symmetric’ axis sJ_de of the core. As the toroidal flux penetrates the plasma, a reduction of the toroical current in the core and an increase of the negative pinching coil currents induces a positive toroidal current in the plasma, leading to the equilibrium configuration of Fig’, lb. The core currents are now crowbarred while the pinching coll currents are further increased until a section of che plasma is severed from the core, as shown in Fig. lc. The core and pinching coil currents are now turned off leaving a 0.25-m minor radius spheromak plasma with an aspect ratio of two, supported by che initial ex ternal field.

The formation configurations shown here were calculated as force-free

MHD equilibrium solutions satisfying i*? - -2T2[dgC?)2/d1?], where ’« is the poloidal magnetic flux and g(V) « RBj. Such sequences are currently being advance^ in time using the circuit equations to produce a more self- consistent picture of the evolution. However, it is clear that the impor tance of resistivity with regard to flux loss and field-line reconnection requires a two-dimensional time-dependent solution.

^Permanent address: Japan.

Geophysics Research Laboratory, University of Tokyo,

Th e incompressible MHD equations a re solved on a nonorthogonal g r i d.

Grid l i n es are l i n es of couscant spect to e i t h er i or l i n d r i c al c o o r d i n a t e s,

x f { 3 x / 3 i ) ( 3 z / 3 j) - ( 3 x / 3 j H 3 z / 3 i ) J. The v e l o c i ty and magnetic f i e ld a re r e o r e s e n t ed in the form

D e r i v a t i v es with r e

or constant

B * <7 $* v$ +* 4r- g?* ,

Here, ‘Ji, g, A, and u obey the time advancement equations:

3t c 3j 31 31 3 jJ

3 i(

j K2 3j

i-<J_ -A) . X r i. ,,„ iA + & J-(adft) . £ i 3 t’ 2 ” x

n V(

’

3 jL

o ’

o .

-167-

v ’

5 3 i’

6 3 jJ

,x2 3i

T 5 3j

i 4 .x

2 * * J i.

2 3j x

3 il 2 ” * 3j

have t he o t t er held f i x e d.

( x . t . z) a re c y . i j ) -l

L ^i

J-j 5;’ x }

If 5 =» ( ji x a$ *

j then t he Jacobian

I n i t i a l l y, a c u r r e nt of 600 k amps is it time

] ). and c o n s t a nt (core) in the t o r o i d al d i r e c t i o n,

the r e s u l ts of one c a l c u l a t i on u s i ng these e q u a t i o ns

t - 3,0 usee, when the t o r o i d al core c u r r e nt

^« Is. JL. SL JA ’

Figure 2a Shows the p o l o i d aj

(3a7 and constant

l i n es now e x i s t.

the i n s i de of

flux Surfaces

t o r o i d al

3 iV

5 31

2 31

5 Sj”

S H1

s ’

’

(’? » c o n s t a nt con

t * 0, v h i le Fig. 2b shows the computational mesh ( l i n es of

t « 0 v o l t a g es a re to reduce the t o r o i d al c u r r e nt and to induce a c u r r e nt

flowing

is « 3 (x ‘g i£ _ ii iA\ +. J_f 3 jl 3j 3t

3 1v i

The core boundary c o n d i t i o ns a re chat v * fi « 0, and t h at t he v o l t a g e s, in the p o l o i d al and t o r o i d al d i r e c t i o n s, a re p r e s c r i b ed functions of

jE * dl t i n e. The equation* ara advanced’using a l e a p - f r og scheme with Dufort- f r a n k i el d i f f e r e n c i ng of

the d i f f u s i ve

terms.

Figure 3 shows contours of constant

time

(x-3/3) at i ts o r i g i n al value and c u r r e nt has been Induced in magnetic axis has formed near magnetic f i e ld

f i e ld

is almost h a lf

the plasma. Sote t h at a t he s h e ll and t h at some closed

We i l l u s t r a te

r i g s. 2 - 6. time tours) at i constant in the s h e ll applied to the s h e ll In the p o l o i d al d i r e c t i o n.

Figure 4 shows the flux surfaces at time c - 6.7 usee after two-thirds

of the current has been induced Into the plasma. More closed flux surfaces have now been foraed around the plasma magnetic axis. Figure 5 shows the flux surfaces at t » 36 usee, after all of the toroidal current has been transferred to the plasma by induction. This method of spheromak formation is very efficient in transferring current from the flux shell to the plasma. Figure 6 shows the time histories of the currents in the shell and the plasma, and the total kinetic energy in the plasma.

In addition to this incompressible code, we have developed a two-

dimensional simulation code which solves the compressible one-fluid resis tive MHD equations [4] using a two-step Lax-Wendroff algorithm. Initially a lov-beta plasma is assumed to exist everywhere between the vacuum vessel and the circular core. The toroidal and pololdal magnetic fields at the surface of the core are given as functions of time, and their penetration into the plasma is being studied. Preliminary results indicate chat the toroidal field can penetrate into a plasma, preferentially toward the axi- symietric axis, in a stable manner; hence suggesting the successful forma tion of a spheromak plasma under actual conditions.

[1]

-168-

3USSAC, M. N ., FT7RTH, H. P ., OSABAYASHI, M., ROSENBLOTH, M. N ., TODD, A. H. M., in Plasma Physics and C o n t r o l l ed Nuclear Fusion Research f ? r o c. 7th I n t. Conf., I n n s b r u c k, 1978) I I I. (IAEA, Vienna, 1979) 249.

[2] ROSENBLUTH, M. N ,, BUSSAC, M. N., Nuclear Fusion 19. (1979) 439. (31 OKABATfASHI, M., TODD, A. K., PPPL-1540 (August 1979); s u b m i t t ed

Nuclear Fusion.

[4] SATO. T ., HAYOSH, T ., Physics of F l u i ds .22 (1979) 1139.

ACKNOWLEDGMENT

02-3073.

REFERENCES

T h is w o rk s u p p o r t ed by US D e p a r t m e nt of E n e r gy C o n t r a ct N o. E Y - 7 6 - C-

06 ^

‘0

:.2

3.41-

0.4

0.«

0.6

3.*

3 2

Oil-

-169-

,e.i…

0J6 OflOIUS tn)

l ( a - c ). Fig. (PPL 7924 39)

Fig. K g - i ). (??L 79:i41)

~*>.

rixeo F i g. 1. (PPL 792640)

l ( d - f ). F i g. (PPL 7924.10)

-170-

TIMe-ifljilM

Fig. 3. (PPL 792639)

. v= H r - - - ’ *.

F i g. 5. (PPL 792641)

F i%. (PPL

,’ -

}-,.-

™:

;

-1*U«t. 6. 7926^3)

Fig. 4. (PPL 792638)

M. Yamada, J. S i n n i s, H. P. F u r t h, M. Okabayashi, r,, S h e f f i e l d,

The S-1 experiment w i ll develop an e l e c t r o d l e ss “slow” spheromak

technique

to produce a 500 kA t o r o id of a - 25 cm, R ” 40 cm.

formacion The formation scheme is based on a t r a n s f o r m a t i on of p o l o i d al and magnetic flux t i o ns of background and e n g i n e e r i ng a s p e c ts of t he S-1 experiment a r e: o b j e c t i v es of

i n to a plasma from a f l ux c o re and two-dimentional MHD c a l c u l a

t he formation p r o c e ss a re described In an accompanying p a p e r. P h y s i cs

t he S-1 a p p a r a t us a re p r e s e n t e d. The

t o r o i d al

(1) development of a slow “quasi-static” formation scheme, suitable for future scale-up to large magnetic energy contents at moderate forcing powers. (2) properties on a slow time scale of aagnecic diffusion

investigation of spheromak s t a b i l i ty and energy confinement

time.

ABSTRACT

P r i n c e t o n, Sew J e r s ey

Design and F a b r i c a t i on

1 7 1- t he S-1 Spheroraak Device

T. H. S t ix and A. y.. M. Todd* Plasma Physics L a b o r a t o r y, P r i n c e t on U n i v e r s i ty

treatment of RJTP, which finds

Introduction

1.1 The Spheromak Configuration and s t a b i l i ty

The spheromak concept is characterized by magnetic field lines that are closed — as in a tokamak — and by a coil-Wanker topology that does not link the plasma — as in a mirror machine.^>2,3 the magnetic field configuration of the spheromak (Fig. 1) includes both toroidal and poloidal components, but the toroidal component is maintained entirely by plasma currents, and therefore vanishes outside the plasma. Correspondingly, there are no t a r o i d a i - f i e l d - c o i ls — only poloidal-field coils, as in a airror machine. The outward pressure of the toroidal field and of the plasma is balanced by the inward pressure of the poloidal field.

The e q u i l i b r ia and s t a b i l i ty of the spheromak configuration has been

investigated and it was concluded that ar. oblate spheromak configuration (Oblimak) of aspect r a t io 2.0, e l l i p t i c i ty 0.5, having a peaked current profile w i ll be stable to ideal MHD modes up to 3edge * 1 5 ’. Global modes can be stabilized by a loose-fittina shell. Another theoretical approach to the problem is Taylor’s ( av/ ap i 1.5) that the idealized large-hole spheromak is stable for low beta condition. However, it should be noted that a r e s i s t i ve MRU theory demands a close- f i t t i ng shell for s t a b i l i ty against high number modes. It also predicts interchange modes growing at a ll 3 values although various s t a b i l i ty effects due to finite-gyroradius effects, non-linear self-stabiliia=ion and line-crying of the diverted edge plasma nave to be assessed ir. : he future. An encouraging feature of the experiments ‘6 of gross instability even though the walls are noc near. But che long-cine confinement characteristic of the spheromak configuration

;o dace is that there is no sizn

is yec to be proved.

‘*Grumnan Aerospace Corporation, Princeton, Sew Jersey 08540

1. 2 Spheromak Forming Scheme

In the S-l experiment, we will employ a “alt-> ’ (quasistatic) spheromak formation scheme, suitable for future scale-up to large reactor-level experi ments. The formation time scale (-100 usee) is intermediate between the dynamic time scale (T « £ =0.1 usee) and the resistive time scale (T =iii exploit a certain advantage of slow RFP (ZETA, Eta-beta) experiments.

-l msec). Ke expect that the present slow formation scheme will

The proposed scheme (Fig. 2) will create a spheromak plasma configura tion without using coils which link the compact toroidally shaped plasma. In this scheme an i n i t i al poloidal field is generated by a winding inside a coroidal ring-shaped flux core. The i n i t i al poloidal field is weakened on the small-major-radius side of the ring by superposition of an external vertical field. The ring also contains a toroidal solenoid, which is able to generate an interior toroidal flux, and is therefore able to emit an equal and opposite toroidal flux on i ts exterior. When the toroidal solenoid is energized, it induces a poloidal current in a sleeve-shaped plasma surrounding the ring. The associated toroidal field distenda the palaijiaL- field sleeve, stretching it in the direction towards the magnetic axis, where the poloidal field is weakest. When the pinch coils are energized, plasma will be pinched off from the topology linked to the flux core, producing a separated spheromak plasma configuration. The e l e c t r ic currents inside the ring can Chen be allowed to decay, while the spheromak configuration remains,

the

I I. Parameter Survey for the Spheromak Experiment.

In order to select-parameters for an experimental spheromak device, we

begin by considering the simultaneous requirements of MHD s t a b i l i ty and microscability, which impose a limitation on the line density. We then determine values of plasma current, field strength and size that should allow adequate plasma temperature.

Theoretical studies Indicate that optimal finite-fi s t a b i l i ty will be

achieved in a spheromak that is somewhat oblate and has an appreciable “flux hole” around the axis of symmetry. Conducting-wall stabilization also takes ri^ce more easily with large-hole devices. Beta-values close cc maximum are achieved for a toroidally shaped plasma of aspect ratio K/a - 2, as in Fig. 1. The significance of a central “flux hole” in practical experimental terms is that an outer flux region is maintained free of plasma current, so that the shear ax the edge of the current- carrying region becomes large. Various options for realizing this s i t u a tion are available, including appropriate i n i t i al field-programming, edge- cooling, and shaping of the “divertor flux.”

The range of appropriate plasma size density is determined basically bv the MHD g-limic and by the limit on the electron streaming velocity v”stream- The typical 2-requirement

-172-

< 0.02 .

1/2 ^

(1)

Assuming chat MHD s t a b i l i ty c*.n be achieved at this 3-Level, important, next, to avoid microinstabilities. For this purpose, the electron drift velocity associated with the strong internal currents of the magnetic field configuration has to be well below the electron thermal velocity. Ke require

it will be

= * _ 2U

thermal

£ « * «_ _ / 3^1.0 \ g n a “thermal

The conservative limit toroidal pinch experiments and noting that the onset of alasma anonalies very generally tends to occur at this level. Combining the conditions on 3 and v the large-flux-hole condition (R/a » 2), we obtain

.03 was chosen by examining data from various

cm.

-173-

l \1 |

< Q.03 .

is the toroidal current.

In the desirable temperature range Te > 100 eV,

The parameter region that is stable from the point of view of Eq. 3 is shown in Fig. 3 for the case 30 - 2%. This figure shows that in a spheroraak plassoa with R = 40 cm, a - 20 cm and a density ne » 5*101-3cm~3, one can ‘.ope to operate without generating gross MHD or current-driven microinstabilities, ., provided the beta limit is reached. Once the line-density is determined, the maximum electron temperature Co satisfy Eq. 1 and Eq. 2 muse scale a s Tea It’, where Ic a 500 kA toroidal current provides an appreciable operating margin.(Fig.i) The nature of the ohmic-heating power balance in S-l is illustrated graphically in. Fig. Above the 100 eV Level in which classical raagnetic-field decay time T5 exceeds 10 milliseconds, the ohmic heating power f a l ls below 10 MW. For ne * l d ^ c a “3, this heating oower would be sufficient one percent cf metalic Impurities, plus transport losses comparable to Bohm diffusion. The most c r i t i c al temperature limitation in the i n i t i al discharge cycle is expected co a r i se from heat losses along field lines to the ring supports. As a r e s u l t, Te will probably be “clamped” in the 50 sV ran?R. As soon as some reconnection of field lines has taken place, so as to create a “private flux” within the spheroraak section of the plasma, the local electron temperature will begin to grow again; dacached spheromak plasma will begin to rise to a new limit. cypicai evolution of flux-core currents, plasma current and plasma ^ar=ir-.eters are shown together with a sequence of field configurations during formation.

to balance the radiation coolin? due to

finally the temperature of the “hole

In ” i s. 6 3

I I I. The 5-1 Facility.

The machine configuration a ij i ts main components are shown in Figures 7. The flux ring contains the Poloidal Flux Core (PC) and the Toroidal Flux Core (TC). These two coil systems induce the toroidal and poloidal currents of the i n i t i al plasma. The pinching coils (PN) are to separate a fraction of the i n i t i al plasma and to move it towards the center of the vessel where it on the spherota&k configuration. The Equilibrium Field Coils (EF) hold the final spheromak plasma in a position near the center of the vessel.The vacuum vessel is made of 1/2” stainless steel with no insulating breaks, since it constitutes a passive element in the overall spheromak magnetic c i r c u i t.

takes

The EF coils are driven by a generator and have a relatively long pulse length; the EF is basically constant during the time of the experiment. The i a r a- ?C, TC, and PS coils are a ll driven by fast capacitor banks. Principal

(3)

with

FORMATION TIKE -IOC usee

MAXIMUM V „0 .10 kV

KUCIMCK V EQUILIBRltjrriElD - 2.2 kG

1 .6 kV

y„ S Bussact., et a l ., Cotif. Innsbruck,

in Plasma. Phvs. Cow:. VIMC. FMSIOP (Pt&c. 7 th

t a t, in (IAEA- S-37-:->l)

M. N. Rosenbluth, M. S. Bussac. Sucl. Fusion 19_ (1979) 489.
M. Okabayashi and A. M. Todd, PPPL-1530 to be p u b l i s h e d.
G. C. Goldanbaum _et, al Univ. Maryland Rep. PPS0-011.
R. K. Linford et a i ., Plasma Phys. Conf. Sue. Fusion (Proc. 7th I n t.

J. B. Taylor, Phys. Rev. L e t t. _3_2 (1974) 1139.

It can emit 0 .1 v o l t - s ec of

t o r o i d al and p o l o i d al copper windings housed in a

The flux c o re c o n t a i ns s u l a t i ng ceramic m a t e r i a l. induce a 0.5 v o l t - s ec of p o l o i d al flux change, r e q u i r i ng about a^«J at c a p a c i t or energy for each c i r c u i t. The c o re w i ll be covered by 10 mil t h i ck m a t a l ic l i n er from s p u t t e r i ng and e r o s i on but a l so w i ll smooth an induced f i e ld at t i al breakdown s t a g e. The core is s e lf supported a g a i n st is powered by s is s e ts of e l e c t r i c al

t h at w i ll not only p r o t e ct che c o r e - s u r f a ce

i n i induced forces and

( s t a i n l e s s - s t e el or i n c o n e l)

t o r o i d al flux and can

the

References 1.

Coot. Innsbruck

him ” 10° « aRlNG ’ 15 TOROIDAL RIXG CtSREST <600 IcA T POLOIDAL RIXG CURRENT <15 MA T TOTAL MAGNETIC ENERGY <1 MJ

-17i-

t h e re

l e a d s.

‘78) 2 IAEA (1979) 447.

‘78) Vol. 3. and r e f e r e n c es

i n

11 a a (I

— ^~vj» J ^ V« /

i ^ 0 ^ _- /^~~v^^~~-^. ss»l-

w *

o«

I.<

r ,i i”

F i g. 7

The S-1 Spheromak device

TWO-DIMENSIONAL SIMULATION OF THE FORMATION OF THE PPPL SPHEROMAK

A. Aydemir, C. K. Chu, H. C. Lui Columbia University, New York, NY 10027

Plasma Physics Laboratory,

-176-

This paper is a progress report on the simulation of the

formation of the PPPL spheromak with a dissipative MHD calcula tion. We present the precise formulation of the problem, empha sizing particularly the time-dependent boundary conditions. We also present some very preliminary results, which are by no means complete or conclusive, but which do indicate that the general trend is correct and that this model has the ’ .isic capa bility to make a realistic simulation.

Assuming axisymmetry, we describe the plasma in the po- loidal plane by the usual single-fluid MHD equations with dis sipation, see e.g. Ref. 1. The ohmic dissipation used is either classical Spitzer or anomalous, the thermal conductivity is usually (but not necessarily) taken to be constant, and vis cosity is neglected, except for a small numerical viscosir o remove numerical oscillations in regions of sharp grad: :;= 3r low densities. Vacuum regions are not handled explicit.y: a low density cut-off (or “density pedestal”), typically 0 1 ’. f the initial filling density, is used. Equivalently, one can allow a small flow velocity from the walls into the plasma. Simula tions of this type have been carried out successfully —. screw and belt pinches1, reversed-field pinches, and Tormac-. Agree ment with experiment has been invariably good.

Initial conditions for the plasma are a low-temp? :* \ture

(say 1 ev) conducting fluid at rest. A vacuum equiliir - “vertical” field is already present in this plasma.

The set of equations uses the variables j (dens (velocity), T (temperature), $ (the ooloidal flux fun ion;. and x = rB+(toroidal field function). Boundary condi P and v are standard for inviscid fluids, while T, are given on all boundaries (or their normal derivar ves;. Relating the boundary values of y and x t0 t ae externally con trolled fields and currents is not immediately obvious, and -.s shall describe it in some detail.

Fig. 1(a) shows the ^“ametry of the spherorcak in ..,e po-

loidal planp; detailed desc ,-iptions are given elsewhere in -‘—.s conferences, The equilibrium vertical field curr

t Ip i—

on _ind x

assumed to be d,c. and already fully established at t » 0. The toroidal current It is also already on, and then drops as shown in fig. 1(b), while the solenoidal current Is rises as shown in fig. 1(b). We have not yet simulated the pinching current I3, shown in the same figure.

On the outer boundaries, we stipulate

-p = .. … ,, and initial

X = 0, since the conducting’ shell has essentially a frozen-in field. On. the inner tube, which is also conducting, we write

- ^initial + constant on the tube but ‘Change with time, while ^initial ts fixed in time but nonuniform on both the tube and the outer walls. To relate -ji- to the current It, we must solve the prob

x * XB(t). Here both <>B and x0 are

wB( t J’ a nd

lem in the plasma, then calculate dt(// 3n on the tube boundary, which is equaL to the tangential B f.ield. It is then given ay

*t =

-177-

9 S’^i

t ne t ube surface

To relate XR to the solenoid current I , we note that x also constant everywhere inside *i:e tube, and in fact, it is w ne re N is the number of turns per unit equal to Xn - “I_, radian. Thus, the total toroidal flux $ inside the tube can be expressed in terms of the boundary value XR rent I . Now from the solution of the 5IHD proElem, we have £ = -v x B + j_/o , and thus,

a nd the cur

$(- J . x B + j y^{'} a) - d l = - d$ /dt on tiibe surface

which is the condition that lints I to x H-

Using the differential equations and boundary conditions

shown in fig. 1(c), we, obtained the results shown in fig. 2- Fig. 2(a) shows the poloidal flux, 2(b) the density, and 2(c) the toroidal field, at the peak of the toroidal field rise on the tube boundary. These results are preliminary, in that we have not yet shifted to the correct time scale (our basic code, written for pinches, deals with ^5’us while the present problem has a time scale of “vlOO us), we have not understood all the seemingly anomalous results, we have not varied the input para meters or optimized them, and we are still dealing with a coarse grid and crude geometry (e.g., the circular tube is replaced by an octagon, etc.). Nevertheless, we see that (1) the poloidal flux lines do close at the right locations, (2) the density increases in about the same area, and (3) considerable toroidal field is trapped in the closed polcidal flux region. These fea tures indicate that the model as w en as the proposed formation scheme are operating in the right direction. Itore conclusive results will be prc, Mded at a later occasion.

1=>

Refarences

H. C. Lui, C. K. Chu, Phys. Fluids 18_, 1277 (1975); 19. 1947 (1976).
W. Park. C. K. Chu Nuc. Tias. 17.’ 1 1 0° (1977); Columbia Plasma Lab. Report, No. 75 (1978) .
A. Ayd&mir, Columbia plasma Lab. Report, No. 80 (1979).
M. Okabayashi et el, paper presented this conference.

Fig. 1. (a) Schematic diagram of spheromak.

(b) Proposed time-

programmed currents.

used in this simulation.

(a)

-178-

6 us

v = 6 x 10 G-cm

V = 9 x 10 G-cn>”

on body varies from -0.1 to 3.0

(a)

-179-

*P = -0.5

Vo) Contours of p

(All units suitably norroalijad)

?ig- 2. (a) contours of -j, at 6 y S.

ooay

BI FURCATION OF TOROIDAL PLASMA I.V \ POLOIDAL QUADRUPOLE FIHLD

H. Ike;i and K. F. Schwarzenegger Bell Laboratories, MuTi-ay Hill, New Jersey 079’J

-130-

We have studied the effect of a poloidal quadrupole

field on the elongation and bifurcation of a current-carrying

toroidal plasma. The experiment was carried out by using an

argon discharge in a small tofcamak wit- the toroidal field

B =2-SKG the major radius Rn*llcm, the plasma current I =

r 2-SKA, T.^T ~10eV, and n-(5-15)xl0lj/cc.

The auadrupole field B , which increases as a qp’

function of time, is applied after the tolcamak discharge has

settled. As B increases, the plasma crossection elongates

until B reaches a critical value. The photographs in Fig. 1

were taken from the tangential direction and show the plas.ua

crossection. The crossection is circular when B =0 [Fig. I (a.]].

N’ear the critical value of B , the plasma is elongated

vertically as shown in Fig. 1(b). The elongation ratio, b/a,

reaches about 5, where b and a are the larger and the smaller

radii. The horizontal dark stripe at the middle of the cross-

section is a shadow caused by the quadrupole field winding.

When B exceeds the critical value, the plasma divides in

two within a few microseconds and each section migrates

toward the wall [Fig. 1(c)]. This splitting time is shorter

than the resistive field penetration tine which is about

20 ;i, but much longer than the Alfve’n time (0.05 -.si.

t=20

V.e have measured the horizontal component of the

poloidal field Bg on the vertical axis, y, which is crossing

the minor axis. Fig. 2 shows the profiles of BR, the

current density J. and the safety factor q. The quadrupole

field is turned on at a time t»0. As B increases and the

plasma elongates, B„ decreases, which is consistent with

the theory for elongated equilibria. B reaches the crit

ical value at t*lT us. An oscillatory profile of B~,

indicating the existence of a magnetic island, is seen at

-„s. The magnetic field reconnection occurs at this

stage.

We write The flux function of the quadrupole field

-181-

\k i2-cos 29,

where r and 6 are the minor radius and the angle around th*

ainor ax:s, respectively. Strauss has shown that the

elongated equilibria are found wher. A/J.^-A /J_~Q.0”S, rte

als.o found that b/a*2.9 when A^A . The experimentallv

observed value of A is about 5Q\ larger than the theoret-

:cal value.

The addition of the quadrupole field to the vertical

field yields a large field index (the sign of index is

negative in our case], which constitutes an unstable field

(1)

1. K. R. Strauss, Phys. Fluids r\ 1040 (1974).

M. Yain^a et al., Bull. Am. Phys. Soc. 2_4, 1022 fl9T9).

configuration with respect to the translative movement along

the vertical direction, .‘o stable discharge has been ootained,

it a DC quadrupole field is applied. We have also tried to

produce elongation in. the hori:ontal direction. The experi

ment has not been successful, however, because of the

positional instability.

The present experiment suggests some ideas for the

design of a proposed spheromak plasma source” which requires

plasma splitting.

-182-

(a) No quadrupole field, (b) Most elongated, (c) After plasma bifurcates

Fig. 1(b) -

ga’iS’i

| g > nn

ga»UTl

Fig. 1(a)

fig. 1 Photographs showing plasma crossectian.

Fig. 1 c)

-133-

critical value at t-l” i.s.

vOmclL POSITION, j Icml

.b’,[c]

fig. 2 (a) Profiles of the poloidal field. The quadri

pole field starts risng at t*0 and reaches the

tactor arid the toroidal current densitv at ts’.

The safe;-’

Start-up Scenario of Compact Tcri Based on EE3-injection

-184-

Kazunari Ikuta

Developed in SPAC-group

Institute of Plasma Physics, Nagoya

Quasi-static start-up of compact tori without toroidal field coil is reviewed thoroughly in a proposal of the S-i sphercmak. During the formation phase we should note that the rap_d heat loss from the plasma will give a bad effect for the generation of che confinement confiquration. I.i the case of fast start-up of the conf iquration plasma can safely pass over the dangerous state of the instability toward the desirable stable state with a bonus of producing hot plasma. fast start-up scenario of the compact tori based on HEB injection developed in SPAC group.

By this reason it is intended to discuss a

Veil-known coaxial plasma gun is assummed, throughout

in this story, to act as a injection chamber of RE3 when che gun is working. Consider a coaxial plasma gun with a third electrode which is a REB ejector. One of the arrangement: is shown in Fig.l-a. A low temperature plasma disc is made to drift toward the end of the inner electrode. As soon as the disc reaches the end, the strong” pulse voltage is applied to the third electrode in order to eject the 3Z3 which propagates along spiral lines of force to the disc.

As soon as the SEB is trapped LP. ^he disc plasma ’:’-;. 1-=., it heats the plasma rapidly. As the RE3 together with the hot plasma emerges downstream from the end of the ir.r.er

A. Mohri et al: in Plasma Physics and controlled Nuclear

Fusion Research (Proc.7th Int. Conf. Inusburg)

X. ikut’.: Jpn J. Appl. Physics 19(1980) in press.

electrode the RES ring is formed and settled in as shown

in Fig.l-c- Aftar adjusting and improving properly the

jaranetars of the olasma thus formed the configuration

could work as a reactor.

The impurity problem must be

solved in this device.

-185-

«>

“3 ! ~*3, 3_ :

“1

1 -<

Spiral E - t i im

C*SV:E3’;

; S C T ;;.

i .i

I — -;

I. Introduction

THE SPS COMPACT TORUS EXPERIMEHT

A. DeSilva Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

*186-

Balancing the apparent advantages of the spheromak configuration as a potential reactor are the difficulties In creating such a configuration in the first place, In particular, creating a toroidal field without material conductors threading the torus is a nontrivial exercise. We r-opose here a scheme that utilizes an intermediate energy storage within the plasma in the form of ordered kinetic energy, that may be released to drive pololdal currents and thereby create a toroidal magnetic field. The scheme is seen as a means of obtaining, in a short Cine, a spheromak configuration isolated from external current sources, for study of basic stability proper ties.

The proposed scheme rests on some unpublished observations in rotating-

plasma machines made in Berkeley in the early 1960’s, and on subsequent work on normal ionizing shock waves performed at Columbia University1>2>3 and the University of Sydney.1*‘5 In order to describe the proposal, it is necessary to first describe briefly the properties of the normal ionizing shock wave in a cylindrical geometry.

II. The Normal Ionizing Shock

Figure 1 shows the shock and driver schematically. A steady axial bias magnetic field is initially applied. Application of a high voltage pulse to the coaxial electrode structure causes the gas to break down, and a radial current flows near the end insulator. This current ionizes the gas, aid through the jr* Bz body force, sets the plasma into rotation about the tuoe axis. The current front advances at constant speed parallel to the tube axis, leaving behind a rotating, nearly fully ionized plasma. Typically the rotational kinetic energy per ion is about equal to the ionization energy. Such shocks have been studied extensively, i1*!’

III. Compact Torus Initiation

The compact torus is to be created by utilising a reverse-bias theta pinch to make the poloidal fields and to perform plasma heating, with the normal ionizi-g shock used to initially ionize the gas and to make the toroidal Bg field. Figure 2 shows schematically the sequence of events. Ionizing shocks having opposite polarities, and therefore the same sense of axial current flow, are driven from both ends. The unpublished work referred to in the introduction concerns this phase. It was observed in 3erkeley that simultaneous application of opposite polarity potentials to the end electrodes in such a geometry did indeed lead to the creation of oppositely-directed shocks. Unfortunately this observation was not pursued.

As the shocks approach one another, plasma energy is stored as rota

tional kinetic energy. Ac the time the shocks meet in the center, fuses open the driving circuits, leaving the plasma without external current linkages (Fig. 2b). At the midplane, the opposite potentials associated with the rotation short-circuit in an internal crowbar. Since Er goes to zero, vg must also go to zero. The information that Er and >Q are zero at the midplane is carried axially outward by MHD shock waves with a poloidal current structure as in Fig. 2b. The radial current flow in the shock front provides the impulse that halts rotation, and the kinetic energy of rotation is converted to Bg magnetic field energy. The process is analogous to the short-circuiting to each other of two freely spinning generators, with the resulc that a large pulse of current is driven and the generator’s kinetic energy converts to field energy associated with the short-circuit current.

As the MHD shocks propagate outward, the reversed theta pinch Is

rapidly applied, compressing and trapping the plasma through field line reco-neccion at the ends (Fig. 2c).

IV. The SPS Experiment

A schematic of the experimental apparatus is shown i_i Fig. 3. Initial

experiments are directed at determining whether the sequei.ce ‘f events pos tulated in Section II will work out in practice. If so, a theta pinch coil and capacitor hank will be added to the circuit to complete the necessary- set of drivers.

V. References *

W. 3. Kunkel and K. A. Gross, in Plasma Hvdromagnetlcs, ed by

D. Bershader (Stanford U. Press, 1962).

1. Miller, ?hys. Fluids 10_, 9 (1967).
R. T. Taussig, Fhys. Fluids ii. 1616 (1965); £, 421 (1966).

-187-

*i. M. H. Brennan, J. A. Lahane, D. D. Millar, C. N. Watson-Hunro, Aus. J.

Physics 16, 340 (1963).

R. C. Cross, R. A. Gross, X. W. James and C. S. Watson-Munro, ?hvs.

Fluids _11, i44 (1968).

VI. Figure Captions

Schematic of coaxial driver for normal ionizing shock wave.

Sequence of events to form toroidal magnetic field. (a) Normal ionizing

shocks are driven from each end. (b) Shocks meet at midplane and internal short-circuit occurs. Simultaneously, fuses open up external circuit. MH5 shocks propagate axially ouc from midplane, with radial current at shock fronts returning via axial paths along the Cube axis and near wall. (c) Application of reversed 3;, field by theta pinch coil sets up poloidal field by reconnection at ends and some compressional heating follows.

Schematic of experiment.

(Ml* rwcM can.

Figure I

J T ±^

, . : * ;* :i\ _u

’%

b m m n Sa

- *J---‘---fie * *

i. rr^RBg

g^tHsasiBgg

Wfljpwqggwjwa

B3BmsffisBK^

=»^-*-[rr7)

Figure 2

*J T

K l g n r i: ’)

T T

Laboratory of Plasma Studies, Cornell University, Ithaca, Mew York

Taylor’s theory of the relaxation of toroidal plasma discharges has accounted remarkably well for the exceTimentally observed behavior of toroidal z-pinches [1]. His analysis also leads to particularly stable magnetic con figurations which have served as the starting point for work on the speromak [2]. We present here a corrected vsrsion of Taylor’s analysis, including a term improperly left out of his expression for the energy of a force-free state. We also present force-free equilibrium solutions in cusp and higher order multipole external fields. These are to serve as the starting point for a study of modified spheromaJc configurations having improved magnetic field line curvature.

Taylor’s prescription is to minimize the magnetic energy,

-139-

MINMJM ENERGY EQUILIBRIA

A. Rsiman. and R. N- Sudan

subject to the constraints of global flux conservation and conservation of

In Eqs. (1) and (2) the integration is over the plasma volume, with plasma boundaries assumed rigid. From the method of Lagrange multipliers it follows that the s t a te of rainimura energy must satisfy

W = 4 : B-d^x,

K = j A-B dJx.

7 x B - uB,

(1)

(2)

(j)

where u is a constant independent of position. This is of course a necessary but not a sufficient condition for minimum energy.

-uuation (3) has an infinite set cf solutions. Taylor gives an egres

sion for the most general solution in a large aspect ratio torus of circular cross section [1]. We want to know which solution has the lowest energy, consistent with a given initial value of JC and of toroidal flux. Integrate by parts, using Eq. (3), to express the energy ^n terms of K plus a boundary term,

Cl/2) ’ B2 d3x - uX/2 * (1/2) f (AxB) -n d2x.

fi)

)S Here V denotes the plasma volume, S is the surface of die plasma, And n is a unit vector normal to the plasma surface.

Taylor expresses the energy difference between too solutions corre

sponding to the same K as Wa-K^ * (1/2) (ya-u-[OK. Ke then minimizes the energy by fir.ding the state of lowest u. His expression -for the energy difference is incorrect. The surface integral in Eq. (4) is in general a nonvanishing function of u.

We evaluate the surface integral in Eq. (4) explicitly, using the general

solution to Eq. (3) given in Ref. 1 for a torus of large aspect ratio and circular cross section. The torus is represented by a cylinder of radius a and length 2TTR, with its ends identified. The result is

W- (1/2) (R/a) ru((a/R)(K/«r) + JgCuaJ/J^ua)].

(5)

Because our expression for IV is not a monotcmic function of y, as was

Taylor’s, the determination of the minimum energy state is now more compli cated. Although the analysis using the correct expression for the energy is considerably different in some Tespects from that of Taylor, it yields the same minimum energy state for a large aspect ratio torus of circular cross section. Because of the dependence of the boundary term on v, this term cannot be removed by a gauge transformation. Whether the boundary term can alter the minimum energy state for some other geometry is at present an open question.

N’ow we show how to find the state which minimizes Eq. (S), subject to fixed K and *’ [3]. In addition to satisfying Eq. (3), B must also satisfy the boundary ccndition, 3? * 0 at r = a. The m = 0, k = 0 solution has Bp = 0 everywhere, and thus always satisfies the LOundaTy condition. (Here m is the acimuthai mode number and k the axial wave number.) If m f 0 or k f 0 the boundary condition gives an eigenvalue equation.

Only the m = 0, k = 0 mode contributes to >?. The amplitude of this mode

is therefore determined by ’?,

bQ - uo/[Z*a J^ya)].

Once we fix bn in this way, we automatically satisfy the constraint on ?. We need only minlmice

-190-

(6)

(7)

(3)

Only the m * 0, k - 0 mode contributes to the boundary term of Eq. (-1).

We therefore consider first the pure B « 0, i * 0 mode. In Fig. 1 the solid line is a plot of V(ua) vs. a{iia) for the m » 0, k = 0 solutions. For each value of 3 there is a discrete set of ua consistent with the m * 0, k = 0 mode. The lowest such value of ya corresponds to the m * 0, k * 0 ^ode with lowest energy.

Vow suppose we superpose a discrete eigenmode on the m * 0, k - 0 solu

tion. Let the coefficient of this mode be bp,. Evaluating -j. for such a we find that it is of the form a(‘ja) - ao(ua) + (b|n/bo)2am,k(-a) > with ot^^> 0. As bm goes from 0 to =°, with u fixed, a goes from aq to =. V increases lin early with increasing i. In Fig. 1, each such mixed state would correspond tc a straight line originating on the m * 0, k » 0 curve. The four dashed lines

:.i Fig. 1 correspond to four such solutions.

rode,

V(ua) = ^((a/R)C:</‘n * JgCuaj/J^ua)],

subject to a given value of

-j. = Ca/RHK/?2).

The lowest dashed line in Fig. 1 corresponds to ua ’ 3.11, which is the

lowest discrete eigenvalue. This line clearly lies below the solid line, so the pure m » 0, k * 0 mode is unstable to the ua * 3.11 mode when it can exist (that is, for s > 8.215. The lines corresponding to all other discrete eigenvalues lie above this one. We thus arrive at the same conclusion reached by Taylor,

Cur derivation of the minimum energy state has been specific to a large

aspect ratio torus of circular cross section. Are there geometries for which the minimum energy state is not the one with lowest eigenvalue? We don’t know at present. A rough estimate indicates that the ratio of the boundary term to uK is of the order S/(uV), where S is the surface area of the boundary and V is the plasma volume. This suggests that the boundary term will be most important for a geometry with large surface to volume ratio. We are led to a stucy of tori with noncircular cross section.

The eigenfunctions of Eq. (3) subject to the condition TJ-n » 0 at the

boundary have recently been determined for a large aspect ratio torus of elliptic.il cross section [4]. We can use these solutions in determining the state of mini.-T.rn energy. T1^ eigenfunctions corresponding to the continuous part of the spectrum are now sums of products of Mathieu functions. We have verified that again the discrete eigenfunctions do not contribute to ’? or to the boundary term. This allows us to carry through our analysis exactly as for a circular cross section. We have expressed W(|ja} and V(;ja) for the continuous eigenfunction as a sum of integrals over radial Mathieu functions. These expressions are being evaluated numerically to produce a graph corre sponding to that of Fig. 1. The mixed states with fixed eigenvalue will again produce straight lines on this graph.

Finally, we turn to the problem of ijaproving the field line curvature cf

the spheromak. Our motivation in doing this is to ijiiprove the stability of surface modes and to increase the 3 allowed by the .‘fercier criterion. Previous work on improving the spheromak stability has focused on the introduction of a hole to improve shear [a]. Our approach is an outgrowth of recent work on systems with mass flow.

-191’

represent a .Tiiniraum energy state for a plasma bounded by an infinitely con ducting wall, subject to the constancy of K and of .V y-B d^x [6]. Stability to surface perturbations, if the boundary is no longer assumed rigid, has recently been established for such a system in an external cusp field {”]. This suggests that in the absence of mass flow, too, we study force free equilibria in cusp-type external fields. We have constructed analytic equi librium solutions of this type, for use in studies with and without mass fiOK.

We work in sprxerical coordinates, treating the plasma boundary as a

perturbation about a spherical surface. The boundary is thus given by

It was shown by Woltjer that the foTce-fTee solutions with

v * 3/vT=r

r * rQ + <ir(6),

where Ar << ro is assumed. The general solution for the poloidal flux func tion is

p in S s5*h I w ^ v^

inside the plasma, and

W ^ iA ^-^-‘XCcose)

in the vacuum. Here P is a Legendre polynomial, j is a spherical Bessel function, aid we have assumed axial synmetry.

A vacuum cusp solution with the field coils far away corresponds to

having only a^ f 0. Requiring % jjj * 0 for 8 » ;./2, we’ find that the interior solution can have only even modes. Beginning with only m » 2, we progressively add modes until we get the kind of solution we are seeking. For only c? t 0, the boundary is spherical. The matching external solution has only a2,b2 i 0, and is a cusp at laTge T. .Adding ci f 0, C4 « c?, makes the surface oblate or prolate. With q, f 0, c,i,c$ « c?, we can choose the coefficients so that ir » z <zos~S sin2e. This gives us’the qualitative behavior we seek. It matches to external modes having m * 2, 4, 6. If we wish to sharpen up the comers further, we can do so by adding more niodes.

References

J. B. Taylor in Plasma Physics and Controlled Fusion, Tokyo 1974 (IAEA,

Vienna, 1975]. M. N. Rosenbluth and M. N. Bussac, Nucl. Fusion 1£, 489 (1979).

A. Reiman, Phys. Fluids ^3, (Jan. 1980).
R. Moses and L. Turner, private conraunication.
M. N. Bussac, H. P. Furth, M. Okabayashi, M. N. Rosenbluth, and A. M. Todd in Plasma Physics and Controlled >faclear Fusion, Innsbruck, 19~3 IAEA, Vienna, 1979).

“69 (1959).

-192-

Woltjer, Proc. Nat. Acad’. Sci. 44, 333 (195S) and 4S_, N. 5udan, Phys. Rev. Lett. 42, T?77 (1979).

Figure 1. Energy of the force free solutions. The arrows indicate the direction of increas ing ua.

CCMPACT TOROIDAL PLASMA EQUILIBRIUM AND IMPLICATIONS ON 5TA3ILITV

jecrge K, Morikawa tourant Institute of Mathematical Sciances, Mew 7ork University, -isw Yarx, New VirJc 13C12

The very enthusiastic interest and activity during the past year or

so in compact toroidal plasma containment is gratifying to those of us who have been studying various low-aspect-ratio configurations and devices fcr a number of years. From an experimental viewpoint a desir able feature af compact-torus studies is rhat relatively small-scale working devises are apparently possible flj and this fortunate stata-of- affairs nay hold, up to break-sven fusion states. This happy possibility is ir. striking contrast to the present development of the next generation :f tsjcanai-type nachines vith large external polcidal cjrrsnt-carryi.tg c=i.”.= surrounding a toroidal plasma container.

He-ever, r=y earlier [2] and present studies indicate that some

-a-’.ti;^ is necessary in the on-going process of reducing the external toroidal nagnetic field 3. to a small value in compact tight tcruses; ir. fact both present theoretical and experimental scheromak studies apparently hope to eliminate this field completely. Such configurations may be feasible for relatively fast: pinch-type low S experiments and tlasncid guns; but for future longer-lasting higher 3 devices, albeit is the federate 3 paramagnetic regime, provision for some external 3_ is necessary particularly as the diamagnetic regime is approached. “To satis?;? this modest 3- requirement we only need to provide an alecrric ;urrett -; along the ma^or axis with the initial flexibility of using aither a straight natsrial conductor or straight charged beams.

The present discussion is limited primarily to the close relation ship be—<ie«r. -he plasrea-current-derisity distribution 1 mostly polcidal J_ ’* =.?.i the need for an external axial current IQ to prevent unreal singula? behavitr -’* the plasma or plasma-vacuum interface. Consider the scnfig- uraticr. ‘shewn in “lg. 1) vhich is related to spheromaks; the toroidal tla*na <; ^_ 3) is nominally in the double-cross-hatched region, and the single-cross-hatched region represents a toroidal vacuum (or force-free pressursless plasma ? = J) region between the spherical separatrix plus ti-.s sajcr axis) and the plasma, “or axialiy symmetric magnetohydrc- iyr.anic C-E3) equilibrium < 5( )/39 = 0) in spherical polar coordinates, the xagnetic field in terras sf the poleisai flux function f(i,) is

-193-

3 sin 9 La »

M”H

Specifitally ve ;.-.=cs» ? = -v’Q)-sir.”3 f separable); zr.i alsc > = := - ; . ?: - ft, tarazietsr ^r.t ? s 5^ <c 1

: * ! ’ (* * = XV, ahere i is thu j o l c i i al current is the flux function value at the tlasna-

where «: is the pressure parameter; and

“n particular for t = 0 (or p = 0) the current density is

vacuum interface. Ve choose ? = 0 oc the spherical separate!* and along the major axis and * -^ 52 in the interior of the plasma with * attaining its T.aximum value or. the magnetic axis. In addition if we choose -ha plasma pressure 3 = <(V-62)

*/.I^.j,,#’”^*1”

h—r [<’ o2sin2i + I.f(1f)l\ j

-194-

P sin v

J = I’(¥)-B

= (l’(i)-3( p !,:M’?)-3(*),

tlm electric current density is

.For jp ? ’ , J ^ * ’ ).

I n i t i a l ly this force-free-field

-*‘1-s factor mainly In ‘the entire para

where A = A = 23.2, the lowest eigenvalue; the pressureless plasma is roieratalv paramagnetic. configura tion was studied with 2. S. Gardner as a possible model for a rnegagauss r.agnet. !fow in any viable plasma equilibrium configuration we require that the magnetic field be bounded (nonsingular). Then for^any 3 ^_ 0 the nature of the current-density distribution J given by (2) tor < > 01 depends” primarily or. the structure of I ’ ( ¥) shown in (3); this is particu larly so for the pressureless plasma. affects the pcloidal component J = (J^s magnetic regime A __ * > 0, -J„ has a singularly high value tor I. = 3 and ¥ - ? . rfhen 3 is increased” into the diamagnetic regime X < 0, 1(f) hecozses unreal unless IQ > 0; and the required Ig rapidly becomes excessive as S is increased (cf. Sef. 2 and e a r l i er work sited therein). Ir. addition current reversal) in the diamagne-ic regime as 3 increases. A:.parer-tiy there will be a severe r e s t r i c t i on on 2 for compact toroidal equilibrium ir. the diamagnetic regime; fortunately the useable range of 3 is about ; <_ 2 <_ 1/2. In the transition region between the paramagnetic ar.d diatragnetic regimes |^| < rr /u an analytic study of equilibrium configura tions with slightly anisotropic pressure ( pt J p s p) has been sade using the guiding-center-fluid (GC?) formulation. For A = 2 The lowest order plasma boundary (for 0 * 0) can be a prolate or oblate depending on whether o± > ?i or pa < tj 2. Sobrott). However for X * 1 (paramagnetic) a spherical plasaa boundary can be maintained for ?A ? p. ; and the preceding !ffiD arguments for 3 < $ « 1 appears to be valid in the full paramagr.eti: regi-e (A <_ A < *). The ir.plicaticn of these GCF calculations is -?hat a s r a li (but nonlinear) t r o i a t e - o s l a te oscillating MKS mode is stable in this paramagnetic t r a n s i tion region; and an estimate on the required size cf -^

the toroidal current density j l oj changes sign [plasna

( i n i t i a l ly studied with

spheroid

; b t a i r s.

(2)

(1)

*e conclude t h is b r i ef d i s c u s s i on v i th a s n o rt d e s c r i p t i on of a more

s l e e c r a te con-pact plasnia confinement c o n f i g u r a t i on ( i n i t i a t ed wixh ^. ?—bhan and Tyan Veh) for future c o n s i d e r a t i on { r i g. 2 ). From The viewpoint sf - t o s s i i i e. ” s t e a d y - s t a t e” ( or slowly o s c i l l a t i ng *) r.-.s i n t r o d u c t i on of a c e n t er r i n g - c o il vacuum hole of r a d i us p = a may have b e n e f i ts vhich o v e r r i de t ie design complexity. Aside from t he advantages a t a t ed in 3ef. 2 ot’r.sr b e n e f i ts of an a-vacuum hole a r e: rroro global d i s t u r b a n c es sucb as v e r t i c a l, sideways and t i l t i ng motions; for example t h e se movements can a r i se when r i ng plasmoids a re fed a x i a l iy i n to t he main body of plasma p e r i o d i c a l ly and r i ng i m p u r i t i es a re and (2) the plastna remains paramagnetic f or l a r g er B with i n c r e a s i ng a, d e c r e a s i ng t he I

tc produce a s p h e r i c al ( or s p h e r o i d) i n c r e a s ed

( l) The c o n f i g u r a t i on w i ll fce s t e a d i ed

r e q u i r e m e n t.

r e j e c t e d;

r e a c - c r,

F i n a l ly t he i m p l i c a t i on of our work is t h at a study of n o n l i n e ar

s t a b i l i ty of compact t o r o i d al c o n f i g u r a t i o ns with plasina-wacuun is needed. lagrar.gian formulation vacuum flux surface

to handle t he fr-ss boundary and tc choose The proper

i n t e r f a c es

t o r o i d al

-195-

SErEREHCiS

to Locate t he shaped e x t e r n al c u r r e n - - c a r ry

Jfunerical computations aay involve a combined a u l e r i = n-

G.C. Soldenhaum, J. H. I r b y, V, ?. Chong and Q. ». Har^,

“jniversi-ry of Maryland P r e p r i nt He. PL30-010 ( S e p t. 1579), to be p u b l i s h e d.

3- .<- Morikawa, Phys. F l u i ds 22, 13?” (Oct. 1979).

..^“Wfl”*”

Fie.!.

1 9 6-

Fn».l.TieM»Uy symmetric geomttiy of *a ftquilitmuni cort&guntion when there is no dipole K theoripn.

RAEIQ FREQUENCY FLUX CONTROL OF TOROIDAL PLASMAS

Sanae Inaue and Kimitaka Icon’ Institute for Fusion Theory, Hiroshima Cniversity, Hiroshima 730, Japan “Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-11, Japan

One of the moat important problems in tho research of thermonuclear fusion is how we confine inhomogeneous plasmas in the stable state. The series of experiments in the world has been showing that the plasma suf fers from its anomalous cross-field transport, particulcrly in particle and electron heat losses, Thara has been much work done to try to understand the anomalous losses. A spontaneous excitation of the waves, which is the consequence of instabilities, i*. believed to be a cause of anomalous loss mechanisms. This anomalous trans -ore mechanism, however, iapiies a possibility to control actively plasma confinement by RF waves. We have proposed a concept of Radio Frequency Flux Control (RFFC), a new scheme to control plasma losses [1]-

The basic concept of RFFC is tc drive inward plasma fluxes by supplying

the momentum via waves so as to compensate the outward fluxes induced by instabilities. The particle fluxes and electron heat flux due to arbitrary low frequency electromagnetic (EM) waves are given as [1,2]

-197-

gen si^” TFuTi

j Z} |E +- a LB k”c

Tm „,tu - k”u ., I- , M> ,

e </2 |lc„[v iiii+U,,u,-‘ii

’ V v^*x ~Bzl

y^hTi / i | k „ |v

) {2T + * ( £ —

e e „ ,

/ 7 ! k „ [ ve

la ZC” -

f ^- 3 k Ie

^ l k . . ! ve

Iv b

i . e .,

, k”u

k”

-”

C 1)

u )

,, .

(2)

including

.: of che p a r t i c le f l u x,

t he ion p a r t i c le flux we have a l so demonstrated

taken in t he * ’ ” - e c t i on of g r a d i e n t s, and other n o t a t i o ns a re

t he magnetic f l u c t u a t i o ns E^ t e r as a re obt t h at

where x - a x is s t a n d a r d. The g e n e r al e x p r e s s! C^n”E,.^ + l / c& * 3 )] + (c/B)n£„, e l e c t r on heat f l u x, energy and zometitum balance e q u a t i o ns lned the Concerning p a r t i c l es follow to e l e c t r o ns so as to maintain the a m b i p o i a r i ty cf p a r t i c le flux Due to a strong dependence of t u n a t e l y, ion heat the p a r t i c ie fluxes and e l a c t r on heat f l u x. The e q u a t i o ns saying t h a t, by tlie proper choice of (~4+k,,u-i,;)k < 0,

the XF waves which s a t i s fy t he wave momentum is absorbed by e l e c t r o ns v ia Landau

in t he presence of any low frequency e l e c t r o m a g n e t ic waves [ 2 ].

l o ss flux remains small [ 4 ] .” Therefore, we should c o n s i d er

t he ton heat flux on t he n o n l i n e ar e f f e c t,

..n Ref. 3. ion t he

(1) and (2) a re

t he c o n d i t i on

rs = (c/B)Re

i . e .,

f o r

iv lea

i - f ig

a re given by

t he r e s o n a nt surface In t he plasma column, t he inward

resonance near a re co be o b t a i n e d. The plasma f l u x e s, as f ar as a q u a s i l i n e ar approximation is used, a re p r o c o r t i o n al f i e ld which p a r t i c l es f e e l, \lz + (x/Ls”)tvj 2 = | ( k u / k )1 z I Ey + (w/kip) 3X - ( i vef c ;e/ c ) 3zj2, and t he wave p a r t i c le resonance r a t e, n a m e l y , ’ t he momentum and the energy exchange race between p a r t i c l es and t he waves.

to t he f l u c t u a t i ng amolltude of Che p a r a l l el e l e c t r ic

f l u x es

:l^2

The cocal plasma f l u x e s, f i r s t ly n e g l e c t i ng

i n t e r a c t i o n s,

’ ”?’!£*…

-198-

t he n o n l i n e ar

(anomalous) + T (SF)

(anomalous) + Q (RF).

l< Lt

:> rr

0.5

In RFFC sehf!!£, we induce the n e g a t i ve d i f f u s i o ns of p a r t i c le and heat f l u x e s.

In order

to u n t i l i ze RFFC, we have to examine v a r i o us k i u e c ic RF waves input

one by one to search out Che optimum c o n d i t i on power, frequency ranges and wave numbers, caking b a l a n c es c o r r e c t ly i n to c o n s i d e r a t i o n s. We examine t he EM low frequency SF waves co study t he e f f i c i e n cy and Che optimum c o n d i t i on for HJFC scheme, solving che r a d i al propagation equations and show i ts a c c e s s i b i l i ty and d e pendences of

induced f l u x e s, I”X(RF) and (JX(RF), on t he wave s t r u c t u r e s.

in parameters such chat

t he plasma energy and momencum

We have basic e q u a t i o ns for RF waves in c u r r e n t - c a r r y i ng f i n i t e -3 c y l i n

d r i c al plasma in sheared magnetic f i e l d, which a re the same as for d r i f t - C e a r i ng and EM-drift waves [ 5 ]. The r e s o n a nt surface mist be i n s i de ttie pUsma column.

Se launch a. MHD shear Alfv?n wave at

the plasma edge r » a, demanding E.,

ion Landau damping,

in-coming d r i ft waves a re s u p p r e s s e d. With

i n t e r a c t i o ns become i n- jk„v^/ai|: < 1, che wave

the plasma. The k i n e t ic In che region

i n to in che region [ k ^ / wi < 1.

0, which propagates ignorable t a r ns ouc co be. k i n e t ic d r i f t - A l f v en node e x c i t i ng outgoing d r i rt modes. Due to s t r o ng t h e se boundary c o n d i t i o ns we solve che propagation e q u a t i o n s, d i v i d i ng the plasma columa i n to MHD and k i n e t ic r e g i o ns £5 3. The r a d i al wave s t r u c t u re and the e x c i t ed fluxes for in f i g. 1. One s«es the wave a c c e s s i b i l i ty and t h at can be induced at a r b i t r a ry magnetic surfaces by R5 waves with sharp spectrum. A broad spectrum one, on che ocher hand, gives r i se to f l u x es in t he whole plasma column [ 6 ]. The required power of SF wa’e f or p a r t i c le confinement twice l a r g er e s t i m a t e d. The r a te of RF heating vliich simultaneously caused by the RF wave can a l so be c a l c u l a t ed .

time being than the one without RFFC can be

t he parameters w - 3 uf t,, m * - and n = 1 a re shown

the c c n c r o l i s b le

:.Ca)=i.3

q C J W .2

fluxes

V I

(3)

(4)

Ke* I

J « . - ’ ,/

J X 1.0 r.‘a.

In conclusion, the plasma parameters are improved through che reduction

of the losses and the incremenc of the input. We thus have additional free dom by XFFC scheme to optimize plasma confinement; one finds the possibilities to control temperature, density and current profiles, impurities, scrape-off and boundary plasmas and the global stability by employing RFFC.

The authors thank Drs. 3. Yoshikawa, T. Takizuka and X. :risnikawa for

discussions. Mark partially supported by Grant in Aids for Scientific Re search D£ MOE Japan.

(J.) K. Itoh and S. Inoue, Comments Plas. Phys. Jon. Fus. ±_ (1979) So. 6 in

(2) S. Inoue, T. Tange, K. Itoh and T. Tuda, Sucl. Fusion J£ (1979) 1232. (3) I. T3nge, 5. Inone, K. Icoh and K. Nlstilkawa, J. Phys. 3oc. Japan £6

press.

(1979) 266.

(i) .M. Shigheta, T. Tange and K. Sishikava, Phys. Rev. Letts. £ 2, 762 (1979). (5) 5. Inoue, K. Itoh and S. Voshikaua, Institute tor Fusion Theory Seport

HIFT-11 (1979. (to be published in J. Phys, Soc. Japan)

[r>) F. Inoup, K. Itoh anc” Y. Terashioia, J. Phys. Soc. Japan ^8 (1980)

Mo. 1 in press.

-199-

References

Introduction — The Motivation

A major motivation f or studying oomzaa* ^vi.

is the desire f or a small

low-capital cost fusion plant with easy access for maintenance. Access achieved by having plasma currents produce most of the confining f i e l d, external c o i l s. Small size is achieved by the combination of high-g and closed- f i e ld confinement embedded w i t h in an ooen-field to form a natural d i v e r t o r.

(cr)

-200-

that

(Size l i m i t:

Field-Reversed Configurations: Theoretical Considerations and Reactor Applications*

George H. Mi ley Fusion Studies Laboratory, Nuclear Engineering Program University of Illinois, Urbana, Illinois 61301

tneory which is thought

to require the S-factor

“he potential advantages of a small power plant require comment. For a

in S/kW-hr decreases with s i z e.

Consequently, some CT concepts run the r i sk of being

intensive p l a n t, “economy of scale” dictates that the e l e c t r i c al

caoital output cost network capacity). too small to be economical! S t i ll a low c a p i t al cost p i l ot *jnit could s i g n i f i c a n t ly speed up fusion development. This suggests an important, if not e s s e n t i a l, feature ~ addition of .-nodules [ 1 ]. This has the dual advantage that the p i l ot unit would provide a test of the basic module while the desired sizing and incroved economics would be achieved by mass production of modules [ 2 ].

increased power is possible by

(demonstration)

10-20% of

Another desirable feature of the C? concepts is that t h e ir high-3 -akes

them candidates f or burning advanced f u e l s. Smaller plants l o g i c a l ly located near the user, reducing transmission costs. Such s i t i ng would require r e l a t i v e ly clean u n i t s, suggesting the concept of a s a t e l l i te [ 3 , 4 ]. plant burning fuels such as 0-^He

.re most

In summary, - - c o n c e p ts should be judged on t h e ir a b i l i ty

to achieve:

Small size with possible modular a d d i t i o n; * Easy maintenance (minimum ob structing c o i l s ); » High-3 (favorable power density and extension to advanced f u e l s ).

Reactor Concepts — General Considerations

A ”

. eactov contains three basic components (Fig. 1): formation, heating

and burning zones. Reactor designs either into a single chamber or use a l i n e ar configuration based on the moving ring or plasmoid concept. Examples of s t a t ic burn-chamber designs are the FRM con cepts [ 4 , J] and reversed f i e ld pinch (RFP) reactors [10, 11], while configurations (MPH) [ 7 - 9] and the ion r4ng compressor (IRC) [ 1 2 ]. The MPH d i f f e rs the other concepts in several respects, includjng the use of heating rather than compression.

l i n e ar include the moving ring FRM [ 6 ], the moving Dlasmoid heater

incorporate these functions

i n j e c t i on

is reducing

from

“ypical reactor designs are summarized in Fig. 2. FRM anc RFT? con-

ceots irs quite small (< 10 MHe per module) due to f i n i te Lamor r a o i js [-’.?.) s t a b i l i z a t i on radius/ion gyro radius) -nuch more study). MHO theory, cover a range of larger sizes. Except for the FR.‘I, a ll ™ concepts to date have involved pulsed operation.

(slasna issue requiring [13,14] (a fundamental In c o n t r a s t, the Spheromak and RFP, being described by

to be < 10

‘Work Psrformed for EPRI (Contract 645-1! and JOE (EY-75-S-02-2213).

Power (r:we) *‘*At. Rad. (rn) Plasma 3 i i

; ; i i

300-1000 5-12 0.3-0.5

.Cri- rs o”i * . . _ _ i -RFP [10,11]

Burn Chamber

I Steady-State

(refuel i n g/ deashing/energy recovery)

FLR

MHD

Rotor

^ D I

I \

(Spk)

Chtfic

BASIC

Heating

Figure

C Pulsed

Inject ion

COMPONENTS

Spheromak

Formation

Compression

Pinch (RF6P)

< 10 < 0.3

Reverseci-Field Theta

f * Ion Ring (IR)

Reversed-Field Pinch (RFP)

» Field-Reversed Mirror {FRM)

Figure 2: REACTOR CHARACTERISTICS

100 - 300 1 - 5 0.3 - 1

FRM [4,5] FRM [4,5] MRFRM [5] MRFRM [5] MPH (RF9P) [7] - IRC [12] - MPH (RF9P) [7]

ion, e l e c t, energy (keV) SO, 53

Table 1: PILOT UNIT PARAMETERS

20,4

SAFFIRE

2X-II

.32

\vg.

Max. Magnetic Field (Tesla) Plasma Volume ( L i t e r s)

Fusion, Met Elect. Power (MW) 1.2,

Reactor Concepts — Examples

We b r i e f ly review four

i l l u s t r a t i ve examples of CT reactor concepts. *

SAFFIRE: a ouasi-steady state FRM

is quite small (see comparison wi.h 2X-II

SAFFIRE is a D-^He fueled FRM designed as a p i l ot unit

for s a t e l l i te operation [ 3 , 4 ]. in Table 1). Peversed-field targets generated by a conical pinch c a il are f i r st b u i l t - up in density and heated by neutral-beam i n j e c t i o n. Then, a quasi-steady state oiim is achieved using the three features of F i g. 3. 3ecause of SAFFIRE’s snail size (radius - 20 era), detailed Monte Carlo studies [15,16] of product (fps) heating were done. Despite the small in the closed f i e ld (- 103), s i g n i f i c a nt heating is obtained from fps

i r c u l i te energy (protons - 20”) aporoaching i g n i t i o n. an energy iiul t i pl i c a t i on (Q) of - 5 would result while at S = 15 i g n i t i on predicted.

that through both open and closed regions. Thus, - 50% of the alpha

in SAFFIRE goes into the c l o s e d - f i e ld plasma, Indeed the 5-factor

f r a c t i on of fps confined

is the key parameter:

f or S = 3, is

fusion

Preliminary studies of the SMPR were carried out in support of the PPPL

Spheromak project technology assumptions [ 8 ]. Physics assumptions are l i s t ed in F i g. 4.

to establish a possible operating “window” using moderate

Cold f u e l i ng maintains pressure profi le/diamagnetic currentl

t Fusion product heating supplemented by a u x i l i a ry heating

Cold plasma blanket w.-ovides divertor/energy recovery

’

-202-

I g n i t i on

Ohraic Heating to

Figure 4: S14PP. ASSUMPTIONS

Spherotnak Moving Plasmoid Reactor (SMPR)

Figure 3: SELF-SUSTAINED SAFFTFE CONCEPT

Creation of MHD-Staole, Reactor-Scale Plasma
S t a b i l i z a t i on of T i l t i ng Mode by Conducting Shell

A “Compact” Reversed-Fie’a ’*‘rich (car3)

the CRF2 concept appears to warrant

A 3 - - l u i d, 1-D coae (^F’BR.‘i; l::,.1’

Table 2: ”**?*’ “NT:::DCW”

0.5 < a < 1.0 meters

10 < Tea < 50 sec

3 < 3e < 5 Tesla

large”- ii"".:’:’:‘i”

!*= < rw a l V/ rX

jsad in p—;-r

100 < ?net <

: a s - j -;

: oa “u

/i’hile p r i or r e v e r s e d - f i e li 2”icn ‘,D.FS)

t r a t ed on larger reactors (2-3COC ’ ^ - n ), “he 5F^ c i u lc an a t t r a c t i ve small reactor. with a r b i t r a ry aspect r a t io 5ei”g stao’iized 3y snear w.” a C O ^ J C V I; «a’ I j r ^ - : on is b; ohmic heating; aoeration *s in a ‘ong z-i la ""3CS ’ 1 ” - ’ . *” s w;th either a batch or refueled

-eactor st-jd’SS Z l “j :once”- :ct9nv”2” ‘j ~aks .’-. ^cerates at n’g”. i1 > ’*’.-J-3.<;’-£na-rancv;

yjrr.

jsad here to see* a small RF° (- IOC ‘We) design tar-c ”?’*>.

- e f j ’ ts Cable 3} indicate t h a t, even wTth 3ohn-l;‘<e -’—*.’ riasonaOle energy gain factor can be maintained, lecessitate higher densities ana larger 3 - f i e ’ ds *or * g n - v . *, shorter burns. S t i l l,

’,%*“:.rrr**,”, i

i ” . es

» i . ’ - . * - *;

’„-.-»- -.:,-.j.

i | * Marginal Internal-Mode Stability During the 3urn I , ! » High-Efficiency Recovery of Magnetic Energy in Plasma 1

” modified PEST code computed equilibrium at each ti’ne step during the burn, plasma loses being assumed at a rate to satisfy stability reauirements. The £ was optimized using an ellipticity of - 0.5, a flattened current prof-le, and a center hole of - 10*. The “window” found is listed in Table 2.

References

0.75, 6.35

10, 15

528, 150

fteversefl-FieTd Tneta Pinch iRFBfr) Moving Plasmcrid Heater [WA)

Concept

F i r s t - w a ll and Major Radius: rw( m ), R(ffi)

3 u ri and Cycle Ti.-nes; -R.(S),

Total Th. and Elect. Power, PTH (MW*h), P(MWe)

Q» = (fusion energy)/(ohmic & f i e ld energy)

Preliminary studies suggest use of the MPH concept

for RF9P operation

is p o t e n t i a l ly a t t r a c t i ve m’th - 10 MWe output using a 0.5-tn d r i ft wi-n R = O.lSm, and B » 3 T [ 7 ]. A key component is neutral-beam-assisted heating which also builds up reversal currents and, qost i n s t a b i l i t i es

importantly, control?

r o t a t i o n al

tube

rc( s)

-203-

[ 2 0 ].

21, 26

3000, 750

1.5, 12.7

*ge” RFPR [10]

Table 3: COMPARISON 3F CRF? AliO PRIOR OESIKIS

to. H u d. Soc.. 32, 23 (1379).

IL report

r i] J, 3. G i l l i g a n, et a l ., “Modularization of Plasnoid Reactors, Proc. IEEE

Fusion Tech. Conf., S. F., CA (1979).

[ 2] J. 3. 511 h g a n, rusion Economics, Report EPRI-AFR-93, U. IL (1979). i”3] 3. H- Mi ley and J. G. G i l l i g a n, Energy, V o l. 4, Mo. 2, 153-170 (1979). [ 4] 3. H. Wiley, et a l ., 3ASFZ5E, EPRI 645-1 Report (1979).

Z”< 3. A. Carlson, et a l ., “Conceptual Design FSM Reactor,” JCRL-52467

(1973).

” 5] A. C. Smith, J r ., et a l ., P r e l. Concept Design MRFRM Reactor, Pacific

3as & Elec. Rpt. 78FUS-1. (1378). J. 0. ^alaratws, at a l ., Trans,

Ill [ 3] R. E. Olson, et a l ., B u l l. Am. Phys. Soc. ,“2TT 1022 (1979). [ 9] v. K. Peng, OR.NL, private cormunication, (l”579). [13] R. L. Haaenson and R. A. Krakowski, “Reactor Des gn for the RFPP.,”

LA-UK-78-2268 (1973).

[ I I] R. A. “lebel, “CRFP Oesign Parameters,” U. ir, preparation. [12] K. H. Fleischmann and T. Kamish, Nuclear Fusion, 15, 1143 (1975). [13] C. H. Miley, et a l ., FLR Effects

in a FRM, 9th European Conf. Fusion,

Oxford, UK (1979).

[ 1 4] Ed Morse. ” S t a b i l i ty of the FRM,” Ph.D. Thesis, LI. IL (1979). [15] 0. orieneyer, et a l ., “M. C. Method for Fusion Product Behavior ANS !<tg. Comp. Methods, Williamsburg, VA, 2, 7-37 (1979).

r1 5] D. Oriemeyer, “A Study of Fusion Product Effects *n Field-Reversed

in FRHs,”

M i r r o r s ,” Ph.D. Thesis, U. IL (1980]-

[17] R. A. “lebel, et a l ., IEEE I n t. Conf. Plasma S c i ., 5C1. 146 (1979). [13] 3. H. Miley, et a l ., “Reversed Field Pinch Plasma Model,” 9th European

Conference on Controlled Fusion. Oxford, UK (1979).

[19] ?.. A, ‘lebel, et a l ., “Comparison of 0- and 1-D Sum Computations f or

RF? Reactor,” LASL Report, in press.

“20] Q. T. Fang and G. ‘-iiley, Trans. Am. r i j c l. S o c. 33, 3f (1979).

CRFP

the

7HE HOLOMAK — A TOROIDAL SPHEROMAK

Thomas H. Stix and Alan M. M. Todd, Plasma Physics Laboratory Princeton Qniversity, Princeton, Sew Jersey 08544

ABSTRACT. Recent spheromak stability analysis favors a profile of plasma current concentrated in the toroidal core of the spheromak, with very low current density (and correspondingly low temperatures) in the region imme diately about the major axis. Considering a toroidal dal) spheromak therefore produces little change in the physics, but the existence of a permanently accessible central hole introduces major options in the reactor engineering. The “holomak” geometry (from holos. Greek: whole, safe) Is a low-|q| toroidal pinch which exploits these options:

{rather than spheroi

*A close-fitting toroidal stabilizing shell. *Divertor capability. *Tokam,tk-type Ohmic heating transformer. *Tokamak-type start-up with finite toroidal field, a quadrupole or

hexapole null in the poloidal field, and protection from impuri ties by a large scra?e-off region.

^Programmed toroidal field. The stabilizing shell also serves as

a transformer-coupled one-turn toroidal field coil. In a reactor, a non-superconducting shell could provide a medium-strength temporary field for start-up, followed by a weak forward or re versed field, all with low energy dissipation.

-204-

“Magnetic shear profile adjustable over a broad range. *Spheroidal blanket. *Axial disassembly.

technology.

Engineering features which would be eminently desirable in the

apparatus for a CTR reactor include axial and/or radial disassembly and a spheroidal blanket. With toroidal magnetic confinement, one also looks for high beta, closed magnetic surfaces, high shear and, possibly, divertor compatibility. Satisfying these criteria are spheromaks and low-lq[ toroi dal pinches. The latter correspond exactly to the toroidal “core” portion of tiie spheromak, but the existence of an actual accessible central hole permits a close-fitting toroidal shell, a weak external toroidal B field, and a normal-cciductor Ohmic-heating transformer. Motivating research on such “holoaak” <.‘»vices is the following: confirmation of adequate confine ment for a low-|q| pinch would provide the scientific basis for a CTR reac tor engineered with existing

The holomak design introduces a toroidal stabilizing shell which

also serves as a one-turn toroidal-field coil. (Incorporation of such a toroidal one-turn non-superconducting shell in a reactor is still consistent with axial disassembly and a spheroidal blanket!) Current to produce the external toroidal B field is induced by transformer coupling to the shell; after start-up, this field can be driven tc a low value, to zero, or can be reversed. (In a reactor after start-up, a weak toroidal field aay be main tained with modest power dissipation in the non-superconducting shell.) Quasi-steady-state current in two large outer Helmholtz coils provides the vertical field (in a reacfr, the vertical field would be produced by con stant-current superconducting coils located cuzside blanket), while inner coils form, for start-up, a temporary hexapole or

the spheroidal reactor

tjuadmpole null in this poloidal field. Afcer start-up, chese nulling cur- rer’s may also be driven co z e r% to form a conventional tsroidal z-pincb, or to reduced values, leaving the plasma with owi or two poloidal divertor lobes. la a reactor, these (normal-conductor) coils ma? also pe driven in a pulsed mode, to remove helium ash from time to time. See Figs. 3 and 4.

An ideal holomak would have, after stare—ap, zero net external

toroidal field on the plasma surface, obviating the need for steady-state current in the toroidal-field coil. But without compromising ei’e CTR engi neering technology — i.e., the axial-disassembly and spheroidal-blanket concepts — one may allow a relatively weak external toroidal field driven with modest power dissipation by current flowing poloidally in the non- superconducting toroidal shell. (A conducting shell fitting close to the plasms appears necessary for spheromaks and holomaks to suppress plasma instabilities.) The S-1 spheToitak apparatus can, without major changes, be operated in the “holomak” mode and Fig. 2 indicates the placement of an inner shell into the S-1 vacuum chamber to achieve this end. The low imped ance of the shell — considered as a single-turn inductor — suggests trans former coupling to the (higher Impedance) supply. The S-1 spheromafe core with its programmed toroidal flux provides exactly such a transformer and operation of the 5-1 apparatus in the holomak mode makes use of chis compat ibility. Pulsing the cora will induce poloidal currents in the shell with toroidal magnetic field values up to several kllogauss. The. exact value of this B field will depend on r’.ie shape of the shell and particularly on the closeness of fit of the shell to the plasma and to the core. After start up, the core can be programmed to bring the toroidal & field down to a low value, to zero, oc to a reversed field value.

Ohmlc-he’atlng current is induced by the Ohmic-heating transfer- T,

an array of coils which produces a changing magnetic flux through the h- -^ of the plasma-stabilizing shell and linking the- plasma. The shell shields the plasma itself from the OH 3-field, which is channeled around the plasma by eddy currents induced in the shell and ‘n the vacuum-chamber wall.

In equilibrium, the radial hoop force of the plasma, due to mag

netic and particle pressure, is balanced by the interaction of che toroidal plasma current and the so-called vertical field. In 5-1, the equilibrium vertical field is induced by quasi-steady-state current in two Helmholcz colls located outside the vacuum vessel and the field is chus “frozen” into both the vacuum vessel and shell. Far start-up, the vertical field is anni hilated locally by pulsing the “pinch” coils, now lowered to 12.5 cm above and below the central plane, and producing a hexapale nuil on the midplane approximately at R”57.5 cm. (Fig. 1) As plasma current start to flow, the cross-section of the closed magnetic surfaces is small and a large scrape- off region protects the interior plasma from wall impurities released ac this early stage of the discharge. (Tig. 2) As the plasma current then builds up further, radial force balance requires a stronger net vertical fl»ld and it is accessary to reduce the currsnt-in the nulling coils. At this point, several scenarios can be followed: reducing che current co zero in boch nulling coils leads co the standard toroidal z-pinch with plasma current IQ force-balanced by ehe full vertical field. Reducing the nulling- coil current to half its original value allows force balance with che toroi dal current at about &QZ of I0 and leases the cross-section still with two divertor lobes (Fig. 3 ). Or, in a reactor, occasional pulsing of che null ing colls could be used to reduce 12R loss while still removing che helium

-205-

ash as needed (Fig. 4 ).

The early work on low-lq’ and reversed-field pinches, particularly

that on the British 2eta device, shoved a promise that has been neither ade quately explored nor exploited in the laboratory. On the other hand, the theoretical calculations which argue stability for the spheromak may be ap plied to reach the saae optimistic conclusions for the lew-jq[ pinch. Holo- aak mode-operation of S-1 shares with Zeca the concept of the lov-|q[ tfroi- dai pinch, and addresses approximately the same parameter regimes of g&s density, magnetic field strength, and time scale. 3ut the differences be tween the devices are significant and include (a) an externally produced vertical field “frozen” into the shell, (b) a programmed toroidal field which, after start-up, may be driven to low positive or negative values, (c) a hexapole or quadrupole null for start-up, with a large scrape-off region which protects Che Interior plasma from wall impurities release-9 at this early stage of the discharge, (d) divertor or pulsed divertor operation, (e) an adjustable shear profile, via programming of the OH, divertor and toroidal fields, and (f) axial disassembly.

In common with other compact-torus reactor concepts, there are

major questions which are still unresolved — stability, maintenance of the q-profile, and overall reactor economics. Nevertheless, a lov-iq| pinch with adequate confinement vould form the scientific basis for a holomak CTR reactor engineered with existing technology. It will, therefore, be the experimental objective of holomak-racde research on S-1 to apply the labora tory knowledge and experience of successful cokamak research to seek experi mental confirmation of the stability indicated by recent theory, and to chart the path toward a CTR. reactor which could be built with engineering based on present technology.

This work has been supported in part by the If. S. Department of

Energy Contracc N’o. EY-76-C-02-3073

-206-

RADIUS ,‘m)

lUSiffll

0.GH

Figure 1. Vacuum field for

S-1 in holomak-mode operation. Hexapole null, for start-uo, is * placed at R-0.575, Z»0. Eaui- iibrium coils: Ir-393,000 amperes ? 3-1.5, Z-±0.8257 S«llin« coils: I[“85,370 anroeres I H-0.7, 2-;0.125. (PPPL-79243)

Figure 2. Start-up of S-1 in

holonak -ode. Flux plot with plasma current aodeled by 1-50,000 anmeres at 3-0.5 75, Z»0. Coil currents as in Tiz. 1. CPPPL-792454)

3.2 1* 0.6 a.3 10 12

“l^‘ire “i. Sketch of S-l

” i m re J,

c-l

2 0 7-

fUOtUSdnl

???I.-7930<S”

HOLOMAK-MODE OPERATION FOR S-l

H0L0MAK REACTOR-SCHEMATIC REPRESENTATION

i n t e r i or configured for holo^ak operation, show

ing addition of aluminum s h e ll and of OH transformer. vacuum chaabe- and v e r t i c al f i e ld c o i ls not shown (see Via. ”, w. Yamada gj a j. . these proceedings).

Schematic rearesentation of a holortak rsaccor. blanket encloses aluninun s t a b i l i z i ng s h e ll -which *lso s e n ss as a o n e - t un t o r o i d a l - f i e ld c o i l. Divertor shown oulsed on ‘ri?he-side) and or: ( l e f t - s i d e ). Design is coooatible with axial disassembly.

(PPPW96525)

S-^eroidai

THE LINUS REACTOR.- COMPRESSION OF A COMPACT TORUS 3? A LIQUID METAL LISER

A. E. Robson, Naval Research Laboratory, Washington, D.C. 20375

The Linus reactor concept , 3> ’ is based upon a nondestructive, reversible, compression-expansion cycle in which a rotacing liquid lithium liner is imploded mechanically, using high pressure helium as the energy source, and acts as a cylindrical piston to compress a magnetically-confined plasma adlabatically from a few hundred eV to about IjkeV. The magnetic field at peak compression is -0.4-0.5MG, corresponding to a peak density of ~10 ions, cm . In the subsequent expansion Che plasma energy and the fusion energy carried by trapped o-particlea Is directly recovered; the a-particle energy compensates for Che electrical and mechanical losses In che system, making the mechanical cycle self-sustaining (Fig. 1). The Linus reactor can thus be regarded as a fusion engine, except Chat there is no shaft output: all the nuclear energy appears in Che liner as heat. At peak compression che liner is thick enough (“-im) to absorb the fusion neutrons, and so serves as both tritium-breeding blanket and the heat-transfer r.edium, as well as pro viding a continuously-regenerated ‘first wall’ whose mean powev loading can be 3-5 Limes greater than in a reactor irith a permanent wall. The combination in one element, the liner, of functions which in olhe.r fusion concepts require separate systems, leads to a rather simple reactor design.

-208-

Fig. L Linus Heactrr Cycle

Hydrodynamically stable compression-expansion cycles are obtained by

\ number of arrange-

rotating the liner (vhieh stabilizes the inner surface against HayLeigh- Taylor instability at turnaround) and eliminating the outer free surface by aeans of a constraining mechanism (the captive liner”). aents”J,’“‘i have been devised for chis purpose, one of which is shown in Fig.2. Bere a quasi-cylindrical collapsing shell is formed by a number of long inter leaved members which can slide into one another, while the axis of the shell is maintained fixed by means of a system of connecting rods. Driven by helium at ~2000 psi such a mechanism could implode a rotating liquid lithium liner with a mean inner surface velocity of -iO’cm. sec-”’. Ihis relatively slow velocity requires that the plasma be veil confined, and an elongated compact torus configuration is required, as shown in Fig. 3. The configuration may or may not have a toroidal field for stability. The solid end valla contain the liquid lithium, and also serve to take the end pressure of the plasma, which allows an initial plasma with average 3>0.5. When compressed radially. <2> falls until it reaches a value of 0.5, after which the plasma contracts axiaily while maintaining <&> constant.

-209-

« “v ,( «P^?vi^ - vi

The principal loss mechanism during a cycle arises from the resistive

penetration of the confining magnetic field into the loner surface of the liner. As che system initial radius R is increased, the 3-particle energy increases faster than the resistive loss, so a self-sustaining cycle is always possible if the system is made large enough. Scaling laws derived elsewhere3 show that the minimum radius is related to the principal variables as follows:

„ . „, MOTOR

where ? is the driving pressure, £ the hydraulic efficiency of the system, ” che resistivity and p the density of the lir.er, and a the radial compression ratio. Although here it seems that 8, is quite insensitive to a, if a > 10 the increase in 1 due to heating of the liner, and the decrease in £ due to wave motion and compressibility in the liner, result in a rapid increase in R. Using reasonable values of che main parameters, che self-sustaining cycle requires R >. 2a.

-210-

UNEA MECHANISM

HELIUM Msenvoin /

UTHIUM MIXING CHAMIEfl AND OUTLET MANIFOLD

MAIN 7- LITHIUM \ ,V HP INNER SURFACE PUMP

I 1 T ~T

Fig. 4 Linus Reactor

ELECTRON SEAM GENERATOR

A conceptual reactor on these principles is shown in Fig. A . The

initial plasma is created inside the liner by means of a rotating relativis- tic electron beam, as described elsewhere3. The initial D-T fuel charge is injected in supersonic Jets from injectors in che end plugs directed so as co avoid the liquid surface of the liner. The electron beam is injected through an annular slic in one end, which is closed by a shutcer as che liquid liner implodes. The lithium Is circulated through the reactor in the axial direc tion by means of tvo concentric pumps: che inner surface pump replaces che hot inner surface betveen cycles, and by introducing a cool lichium scream, condenses che lithium vapor and the previous fuel charge. This pumping requirement limits the repetition rate to ~ 2Hz. The main body of the lithium is circulated more slowly, and is completely replaced after about ten reactor cycles.

The principal physical and technical characteristics of this system are

given in Table 1. A more complete description of the reactor can be found in Ref. 5.

Gross electricpower 844 MW 141 MW E-beam power 13 MW Motjr power 50 MW Power to b.o.p.- Net electric power 640 MW Mass of reactor

3500 tonnes

Liner length: Initial inner radius: 2 m Driving pressure: 2000 psi Implosion time: 24 msec

Lithium: 2.6 tonnes, m Steel shell: 25 tonnes, m~ Fluid rotation: 2.5 rpm Reaction Time: 0.85 msec

Plasma radius Central density Temperature Magnetic field < Bz > Energy (plasma + field)

E-beam energy/cycle I-itial plasma energy Compressed plasma energy Nuclear energy (22.4 MeV/DT) Repetition rats Cross thermal power

22 an l . l x l O17 cm”’ 15 keV 3o0 kG 0.375 1016 MJ

-211-

Initial

Compressed

Table 1. Linus Point Design

60 MJ 30 MJ 1016 MJ 1206 MJ 2 Hz 2532 MW

200 an 9.3 x lO^cm-*3 635 eV 6.9 kG 0.6 30 >J

Sate that the reactor operates with Q~l: this is made possible by the

efficient energy recovery in the mechanical cycle. Since the fractional burnup is small, helium accumulation in the plasma is no problem. Because of the size, and the short timescale, plasma loss during a cycle should be negligible, even allowing for anomalous transport. A3 so, because of the short timescale and the high field, contamination by lithium vapor should be con fined to the edge of the plasm?.

Probably the most significant feature of this concept is that, apart

from the electron beam generator and the pump driving meter, there are no electrical components in the reactor. The electron beam sets up both the plasma and che confining magnetic field and the latter is maintained by skin currents in the liner, making any permanent field coils unnecessary. The main shortcoming of the concept is that the size of the reactor, determined by the conditions for a self-sustaining cycle, is rather large. The size is partic ularly sensitive to 3, and to obtain the figure given above, a rather high 3 has been assumed. Note that the electron beam generator shown in Fig. 4 is based upon a conceptual inductive stornge systen: a conventional capacitor- powered generator would be substantially larger.

A.E. Robsoa and P.J. Turchi, Proc. 3rd Topical Conference on Pulsed High

3eta Plasmas, Culham UK, 9-12 Sept. 1975, p. 477.

P.J. Turchi and A.E. Robson, Proc. 6th Symposium in Engineering Problems

of Fusion Research, San Dieeo, 18-21 Sov. 1975, p. 983.

D.L. Book. e_t ^. , Plasma Physics and Controlled Nuclear Fusion Research

1976, IAEA, Vienna 1977, Vol. Ill, p. 507.

’-. I.L. Burt DO e_t a^., Proc. 7th Symposium on Engineering Problems of Fusion

Research, Knoxville, 25-28 Oct. 1977, Vol. 1, p. 225.

A.E. Robson, 2nd Int. Conf. on Megagauss Magnetic Field Generation and

Related Topics, Washington, D.C., 29 May - i June 1979.

J. D. Sethian, this conference.

PXELE1INARY STUDIES OF SPHEROMAK REACTORS

M. K a t s u r a i* and M. Yamada, Plasma Physics Labo. i t o r y, P r i n c e t on Ur.i-”ecsity, P r i n c e t o n, Sew J e r s ey 08544 (*0n l e a ve from Dept. of E l e c t r o n ic E n g i n e e r i n g, U n i v e r s i ty of Tokyo, Tokyo, Japan)

In t he spherooak r e a c t o r, a t o r o i d al fusion plasma

is ( s e l f - g e n e r a t e d) magnetic f i e ld and t he v e r t i c a l-

l i n k i ng t he plasma. Among v a r i o us

confined by i ts own i n t e r n al f i e ld Bi which is produced by c o i ls not kinds of spheromak c o n f i g u r a t i o n s, an o b l a t e - s h a p ed plasma c o n f i g u r a t i on with a c e n t er h o le c a l l ed “Oblimak”, has been shown to provide b e t t er ME2 s t a b i l i ty and higher a t t a i n a b le beca v a l ue Chan Che c l a s s i c al s p h e r i c al spheroaial, [1] In t h is paper, ve derived approximate f u n c t i o n al r e l a t i o ns between v a r i o us spheromak plasna p a r a m e t e r s, based on a s i m p l i f i ed energy p r i n c i p le of c i r c u it theory and using t h e se r e l a t i o n s h i p s, have c a r r i ed OUT -ireliminary s t u d i es of c o n c e p t u al spheromak r e a c t or with a s t a t i o n a ry plasma as well as with a moving plasma produced by Che connak type f o r a a t i on scheme employed experiment at PPPL.

in Che S-l

[3j

-* Approximate A n a l y t i c al T r e a t a e nt of

In t he framework of e l e c t r ic c i r c u it system is regarded as a kind of corresponds to a s i n g le c u r r e nt by t he e x t e r n al c o i l s, as shown in F i g. 1. The energy function of system is given by

inductance c i r c u i t, where t he spheromak plasma loop immersed

in t he magnetic f i e ld produced

t h is

t he Spheromak C o n f i g u r a t i o n. t he spheromak confinement

W - ~ Lnl ’ „2 - K I -I Qt

where L and >! are the self and mutual inductances ; and V and p are the volume and the pressure of the plasma, respectively.

The sph&rcasak configurations are characterized by the following evt

conditions (R: the major radius, a: ainor radius):

“^K

Bi - rr

T? =a~2-

Ir Che low-3 limit, where effects of plasma pressure is neglected, following relationships are derived [4J:

Adiabatic Compression. For the formation and heating-up of the

reactor gride spheromak plasmas, the adiabatic compression scheme is one of key factors. During Che compression, the ;oroidal and poloidal fluxes have to be conserved resulting in the conservation of both the average safety factor and che aspect ratio of the plasma. Considering further the ecuition of adiabaticity T V 2/3 a const, where TD is the plasma cemperacure, c’ e

following scalings for various quantities are obtained:

?w is che input power required for the uaanetic flux supply, w is t^e -lasnetic energy inside the plasma and TM is the magnetic diffusicT time.

-212-

t h e o r i e s,

I n t r o d u c t i o n.

3t o’ Bi= R”2- S ’ h “a _ 1’ VSW “R”3

T h’ V V ?

*i’-a^- - l!> , Ip - 1.9

V V V P

I H - I2 + -k * * +

( 8R ,”|

2 ^ t.

p p

2 Uu

\3 a/T

:0 h

2 pp

’ 0

’ ^)

[ 2 ],

(3)

i. Reactor S t u d i e s* Based upon Che above approximate a n a l y t i c al s o l u t i o n s,

is the t o r o i d al magnetic f i e ld at the o u t e r - e d ge of

s t u d i es a re made on the spheroraak. r e a c t o r s. When t he a s p e ct r a t io to be in t he range of about L.7 to 4, where Bta f i e ld at s e m i - s p h e r i c al f i r st wall is s p e c i f i ed by a^ and av - 1.5 a is assumed. Three types of r e a c t o rs a re i n v e s t i g a t e d: type) o p e r a t i on with a fusion gain QF of 1.7; i g n i t ed c a t a l y z e d - c y pe DD (Cat-D) r e a c t o r. and d e n s i ty a re assumed.

t he c e n t er and Ba is t he plasma as shown in r ig 2 . The l o c a t i on of ->ie

DT r e a c t or with TCT (two-component

i g n i t ed DT r e a c t or and, 3)

the r e l a t i on 3t0 - 2 .1 5e

S p a t i a l ly uniform

the magnetic

temperature

is obtained

is a s s u me

Table 1 l i s ts

the main parameters for d i f f e r e nt conceptual d e s i g ns of

is about 4T for DT i g n i t ed r e a c t o r s, which is about half

-, a spheromak r e a c t or with a s t a t i o n a ry plasma. A t o t al w a ll l o a d i ng of 4 tfK/m” and t he c e n t er beta v a l ue <80> ” 0.045 are assumed. The ourer-edge magnetic for f i e ld tokamaks. eiae. is ceduced by a f a c t or of - 4, however, t he v a il Loading is by a i n t o l e r a b le f a c t or of - 1 6. loading power d e n s i ty and d e c r e a s i ng

is found to be a major c o n s t r a i nt a g a i n st

the r e q u i r ed energy confinement

is given r i se to -8T,

t he oral! fusion

i n c r e a s ed

tha v a l ue

S t r u c t u r al problems of che spheromak r e a c t o r. Various methods have

in t he spac» between

to numerous d i f f i c u lt

the d e t e r i o r a t i on of che

t he the b l a n k e t, which a re used only to produce the spheromak plasma the sLdrt-up phase. The i n s t a l l a t i o ns of such formation components g i ve

been proposed to produce the spheromak plasma in t he r e a c t i on chamber. H 0 “e , /e r, zhey u s u a l ly r e q u i re i n s t a l l a t i o ns of v a r i o us plasma formation components such as c o i l s, c o n d u c t o r s, and sometimes e l e c t r o d es plasma a r/ at r i se r a d i a t i on s h i e l d i ng and c o o l i n g, and a l so r e s u lc neutron economy in che b l a n k et system caused by the neutron a b s o r p t i on by them. Therefore, from t he r e a c t or chamber so t h at neutrons the c o n f i g u r a t i on the syheroraak i n i t i a l ly produced by a non-reacting plasma in t he former i n j e c t ed or moved i n to che l a t t er h e a t i n g. Examples of schematic s t r u c t u re a re presented spheromak plasma is i n j e c t ed or c a r r i ed arranged pinch gun ( a ), or che coraak type e x c i t e rs <b) [ 3 ], a d d i t i o n, i ts a p p l i c a t i on with small a p p e r t u ra chamber. Successive one l a r ge spherrcmak in t he chamber by t h is method as i l l u s t r a t ed

to them a re avoided. This requirement can be f u l f i l l ed by applying s e p a r a t ed s p a t i a l l y, and

to be used cc> feed a l a r g er spheromak in the r e a c t i on i n j e c t i on of p l u r al of spheromaks w i ll be a b le co produce

is very a t t r a c t i ve because it enables r e l a t i v e ly small e x c i t er

t he merging of spheromaks a re proved p r a c t i c a l,

t he d i r e ct bombardments of r a d i a t i on and

into the r e a c t i on chamber by s p e c i a l ly

f u r t h er in F i g. 2, where

where t he production chamber

the concept of

in F i g. 3-

i s o l a t ed

*213-

t h is f i e ld

t e c h n o l o g i c al probleas a s s o c i a t ed with

i n c r e a s i ng the r e q u i r ed energy confinement

In these s t a t i o n a ry spheromaks, t he time.

to the o p e r a t i o n, whose plasma parameters for (1) [ - ].

followed by the fuel supply and

is highly recommended t h at

t h o se components a re

to the numbers (1)

t h e ir

Moving Spheromak Reactor. The spheromak plasma, fed s p e c i a l ly

the c y l i n d r i c al r e a c t i on chamber, can be loved along the fusion r e a c t i on p r o c e s s, which enables us to reduce e f f e c t i ve wall An example of t he c o n f i g u r a t i on wards according An example of l i s t ed

is shown in F i g. 3 where t he plasma soves

is axplained

in Table 2,

(4) a re

in a c o n t i n u o u s ly r e p e a t i ng c y c l e.

l o a d i n g. i n

( 4 ),

in the c e n t er a x is during

Useful discussions with Drs. H. P. Furth, D. L. Jassby, H. Maeda,

M. Okabayashi, R. E. Olson, A. M. M. Todd and S. Yoshikawa’are gracefully acknowledged. This work supported by OS Department of Energy Contract No. rY-76-C-O2-3073.

[1] Bussac, M. N. , ec_ aJL., Plasma Physics and Controlled Suclear Fusion

Research (Proc. 7th Int. Conf., Innsbruck, 1973) IAEA.

[2] Okabayashi, M. , and Todd, A.M.M., in this proceedings.

[3] Yamada, M., 1£ al_., Soil. Am. ?hys. Soc. 24 (1979) 3; also proceed

ings of this conference.

[4] Katsurai, M., and Yamada, M., to be published.

Fig. 1. Circuit model for the

spherorcak configuration

Fig. 2. Examples of conceptual

methods co feed spherotnak plasma In the reactor chamber. Initial spherotnak is produced by the pinch gun (a) and, the cormak cype exci^ors (b).

Fig. 3. Conceptual design of the

spheronal. reactor with moving plasma and the eonaak type excitor.

-214-

FIGURE CAPTIONS

Fig. 1

]

l .t

l .)

l .t

l .J

1.)

l .S

! .*

1.1

1.2

1.1

0.5

7.4

4.1

0 .1

S»0

0.12

f&ia

s.s

0 . »3

C»«cl

0.045

1.300

t . r oo

a. Ml

»,JCO

0.011

0.0*9

O.WJ

O.if

R f i j«

St »

f i e ld IT?

=’* J”

i-s< iu« ta>

confinacent

a.J

(CTJ loading

P Teril ai.;p«e

**.2-10’ 2 . 1 . 1 0’

r’olaldal t u r r r at

<»9» C*«t*e ¥«<«

Internal u i n a t lc

A -L«ior r/idlu* (a)

I Toroidal current

BL VartLcxl e * Cn a c tc

T P l « t» G«9i.(XlV>

n U t e t rn daflilCT

10 ,„=

’ J nr

pcw«f F..TotJL ualL

l u e t ic d i j f u- ( ml IICR ela«

1 _ Tor old*i = * | n « ic flaid *[ ecfte«t(T]

cr.r=v v: H .riTi ^ * n r

1 « i

*fcroiial current ’

I.’-“A)

?o^oidal s u r e * .*

i i

irag« v « l ^l

IT 1

1 ,.=

V e s i c al

j a l a f d al

3…S

2,::o

— ; - - *;

»-

O.M

aoi

a.:

;c—*f

Z.ZBl

c:;>

(S-TJ

UJ01

fraifi

rocal

O.IS

S.M

: . 3i

’ * * * « ’>

2.0

3.2

5.3

i .a

’ - * ’>

‘J

. . - *S.

;

’

T.-3LE I

TABLE 2

THE ALL PLASMA SPHEROMAK: THE PLASMAK

P. Xoloc and J. Ogden, Prometheus II, Ltd,, Sx 222, College Park, Hd. 207&0 (301) 434-7317

*216-

There has been an evolutionary pattern established in magnetic fusion

concepts. The flow in ideas follows three directions. By extrapolating this evolutionary movement, we have anticipated the concept called Spheromak and have predicted the omega of this evolution which is called PLASMAK, or all Plasma Spheromak. The evolutionary directions are from open systems to closed systems, From zero or low dimensional compression schemes to three aimensional compression, and finally from plasma configurations without any self confining currents to a plasma configuration which is completely self confined except for tha mechanical pressure necessary to maintain the vertical field and hoop stress. The Stsllaratar represents a closed configuration requiring a very czrsiex array of external confinement coils. The Tokamak allows vast improve- Tien- in plasma parameters, lower aspect ratio, and ohinic heating from plasma “<ifi-ing toroidal currents. ATC compression ratios of - 1 may be possible. Nevertheless, the plasma is imprisoned by heavy poloidal coils, and a vacuum rfall.

T’-e Scheromak enjcya greater stability and obtains higher effective betas,

si-aiv axoressed as higher fuel amcunt per unit external confining pressure. This comes about due to the poloidal plasma currents which, in addition to the . e m c al field, exert a second confining force on the plasma torus. Sphero- -avs -sve higher interior magnetic energy than at the separ-atrix or vertical “ield current shell. The term used to describe this advantage is Pressure Leverage. Unfortunately, in Tokamaks this second confining force is wasted on Sea F i g. 1. i~e polaiJal windings and is a source for engineering problems.

Higher ATC compression ratios for Spheromaks are allowed because of the

lack of axial obstructions. Energy scales as the square of the linear com pression ratio c,

- . ~z), as it does in rokamaks; this limits the utility of lar;s ^repressions. Another consideration limiting very large compressions is the necessity of a tightly fitting shell to maintain stability.

“-is ‘.rings us to the Plasmak, or all Plasma Spheronak. The Plasmak

represents che final step in reducing the deoendence on external confining calls and for improving compressibility. In the Plasmak, the confining cur rents are all embedded in plasma with the fully ionizing energetic currents in che cutermost mantle producing the vertical field. The equilibrium force arises from the ordinary gas pressure. See F i g, 2.

The Plasmak is fully compressible. As in Spheromak, there are no axial obstructions. The currents are energetic, allowing for high conductivity con tributing to ideal MHO stability. Utilising high-voltage or EMF starting for mation schemes results in very compact Plasmaks with intensive currents. In t-e -‘isr.3 kernel (see Fig. 2 for nomenclature) the current densities result i- considerable ohmic hea-i.ng while the L/R times of the uncompressed ‘higher ir-cuctance) Plasmaks is > 1 seccix. ;r=reauisite for ideal stability.

The tightly fitting -ancle is another

Co;rp.-es3ior. is achieved by simply increasing the fluid pressurs of the

surrounding gas blanket. The first advantage is that the upper bound on -necnanical compression methods is nigher than magnetic techniques. The second advantage is that the mantle is compressed along with the internal kernel, thereby maintaining a tightly fitting shell and ideal MHO stability throughout compression. The third advantage relates the significantly lower amounts of energy neeced to compress a Plasmak as opposed to Tokamaks, or rigid shell Sphefomaks —<rere the nagnetic energy varies as the compression ratio squared iC’J. In Plasmak. the energy is proportional to the compression ratio (c). The magnetized volume varying as the compression ratio inverse cube (c~3) is responsible far this advantage.

In general, formation techniques can include a technique in which the

Plasmak is produced through MHD effects initiated by a helical current stroke in a high density gas. Such a formation technique has been proposed to the US QQE and is currently under review. This particular approach sets uo the piasma kernel and Mantle simultaneously with high density energetic currents and assumes an ideal MHO configuration from the outset. This technique relates to physical parameter regimes and phenomena not familiar to most fusion p”ysi- cists. and will not be discussed here.

Another :iass of formation techniques utilizes a step by step formation and assembly of the Plasmak. A three step scenario includes ring formation followed by capture and injection of energetic currents, “he third final step would be formation of the mantle over its external vacuum magnetic field. Gne common technique for ring formation being currently pursued is the plasma gun. Even the 3-1 Spheroraak can be considered a plasma gun. This gun differs from these described by LLL and LASL in that it utilizes an inductive startup and it has a disc shape rather than conical or cylindrical shape. Fig. J demon strates this scenario. Ir.cidently, the formation technique of G. Goidenbaum of the University of Maryland is also amenable to gun conversion since the “acpetizsd plasma kernel can be ejected through a hollow electrode at one end.

Comoression, ignition, and energy exchange can take place in a separate and reicote compartment. This is true to a limited extent for Spherorsaks. Con venience of moving Plasmaks by EM or fluid flow is assured sines lifetimes fat uncompressed Plasmaks are on the order of seconds. An exciting sngineering feature of this fusion furnace in electric power generating application is cnat the entire fusion and magnetic energy less neutrons can be dissipated in the gas blanket without afFecting the walls. This blanket heated to tens of ev can re red into inductive MHO generators achieving electrical conversion efficien cies of about 35SS. Sucn efficiencies obtain 0 values an the order of one hun dred. Reduced wall loading is also achieved by moving radiating Piasmaks and 5pheromaks. Sot less direct energy conversion schemes will not be as efficient.

-217.

High effective Betas, Pressure Leverage, favorable energy compression scal ing ratios, and a technique called 5CH or Self Compression beating along wit.t high 0 all comfcine in the Plasmak to make this approach by far ^he most *. iable vemcle to burn exotic fuels. See Fig. 4 and Table 1.

Self Compression Heating (SCH) is the beneficial effect that occurs wnen

a Plasma* has been gas compressed to ignition in a tightly fitting pressure chamber. Fusion energy driven Sremsstrahlung and alphas radiated oy ‘.he kernel plasma thermalize the surrounding hijti aensity cold compression blanket. The subsequent pressure increase cf the fusion thermaiiied blanket further drives

:» adiabatic compression of the kernel fuel increasing its temperature, den tw sity, and reaction rate. This tool is most effective for higher temperature fuel cycles, i.e., P-8. SCH is not critical for the success of Plasmak fur naces : in fact, careful application of SCH to DT fueled devices is necessary due ts the -isk of runaway reactions. SCH also improves Q, the energy effi ciency factor.

In addition to the proposed work by Spheromak Inc., a subsidiary of Pro- :;-e’.;s II, Ltd., the Westinghouse Fusion Power Systems Oivision is planning

Plssnsx fusion power generating studies in early 1980. The enthusiasm for :.-is concept is largely due to its engineering simplicity at ’ design points. 3y proceeding directly with the proposed Plasmak schenie of Spheromak, Inc., the development of a compact fusion device could be completed within five years. Soheromak Inc. is fully ready to cooperate with various interests to ts\eioo Plas-nak fusion furnaces and power generators. The implementation of :”.“ii.^ia< development requirss exceptional administrative and executive action.

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NEUTRAL BEAM SUSTAINED, FIELD-REVERSED MIRROR REACTORS*

G. A. Carlson, Lawrence Livermore Laboratory, Livermore, C a l i f o r n ia 94550 K. R. Schultz, General Atomic Company, San Diego, C a l i f o r n ia 92024 A. C. Smith, J r ., P a c i f ic Sas and E l e c t r ic Company, San Francisco,

C a l i f o r n ia 94106

We have carried out conceptual design studies f or neutral beam sus

tained, f i e l d - r e v e r s ed mirror reactors. The f i e l d - r e v e r s ed mirror has a compact t o r o i d al plasma geometry bounded by closed poloidal magnetic f i e ld in a lines generated by an azimuthal diamagnetic plasma current neariy uniform background magnetic f i e l d. The closed poloidal structure has major radius R and minor radius a. The p r i n c i p al hypotheses of the FRH plasma are as f o l l o w s:

f i e ld

t * *

f i e l d, and therefore 3 ~ 1.

Equilibrium with weak or zero t o r o i d al Magnetohydrodynamic (HHD) s t a b i l i ty achieved by f i n i te ion Larmor radius, S s a / = i — 5, combined with low aspect r a t i o, R / a - 2. Plasma positioning by weak m i r r o r s. Plasma confinement controlled by c r o s s - f i e ld Steady state plasma density, energy, and current maintained by neutral beam i n j e c t i o n;

-220-

transport.

in the p i l ot p l a n t.

to that

These hypotheses w i ll be tested in the ongoing Beta II experiment at

LLL. The objectives of Beta II are startup of a field-reversed plasma, confinement in a magnetic w e l l, and steady state sustenance by neutral beam i n j e c t i o n. For our reactor studies, we have assumed the v a l i d i ty of the FRM plasma hypotheses, and have calculated the c h a r a c t e r i s t i cs of the plasma j s i ng an analytical model described in Ref. 1.

Our reactor designs have included a m u l t i c e ll commercial f a c t or and

a single c e ll p i l ot reactor. The m u l t i c e ll reactor* has a linear arrangement of 11 c e l l s, each containing a neutral beam sustained, . l u l t i c e l! approach field-reversed plasma. Advantages of the linear include: more uniform wall magnetic

loading and better u t i l i z a t i on of background

f i e l d.

The p i l ot reactor design2 was sponsored by the E l e c t r ic Power Research I n s t i t u te and was carried out j o i n t ly by LLL, General Atomic Company, and Pacific Gas and E l e c t r ic Company. Net power production was specified to be the primary objective cf the p i l ot reactor. Another ab j e c t i ve vas the use of near-term techhnology s t r u c t i on of the p i l ot plant experimental program on field-reversed mirrors is successful. Following the operation of the p i l ot p l a n t, we would envisage the rapid development of m u l t i c e ‘l commercial reactors, each c e ll of which would contain a f i e l d - r e v e r s ed plasma i d e n t i c al

in order to permit con the present

In the l a te 1980’s,

“This work was performed for the Electric Power Research I n s t i t u te undor Contract No. RP922 and under the auspices of the U.S. Department of Enei gy Contract No. W-7405—ENG-48 with Lawrence Livermore Laboratory.

The performance of the field-reversed plasma depends most strongly on

the parameter S, the ratio of plasma minor radius a to the ion gyroadius -*j. Stability of the plasma is believed to require that S be a small nuraer, probably 10 or less. We have considered both S * 5 and 5 = 7, and have sized the reactor to accommodate either plasma. The S * 5 plasma produces 22 MW of fusion power and requires 4.4 MW of neutral beam injection power; so Q * 5. The S * 7 plasma produces 42 MW of fusion plasma and requires 3.2 MW of neutral beam injection power; so Q = 13.

The PRM pilot plant is shown in r i g. 1. The field-reversed DT plasma

is at the center of the reactor. A cylindrical blanket and shield surround the plasma. The blanket is helium-cooled and contains lithium oxida for the breeding cf tritium f u e l. Surrounding the blanket and shield is a set of niobium-titanium superconducting magnets which provide the required background magnetic f i e l d. Neutral beam injectors fuel and energetically sustain the plasma. The large tanks at the ends of the reactor house direct energy converters and vacuum pumping systems for the plasma end leakage. Also housed in one of the tanks is a coaxial plasma gun for plasma startup.

2 2 1-

Several ways of forming field-reversed plasma rings are being con sidered for FRM startup. The startup scenario chosen for this design is the use of a magnetized, coaxial plasma gun, similar to the one now being tested on the Beta II experiment at LLL. A ring-shap»d plasma is formed and axially accelerated by discharge current flowing radially across the anode-cathode gap. Beyond the end of the gun the plasma passes through a cusp-shaped magnetic f i e ld and emerges - with “captured flux” - as an annular plasma confined by an axiaUy field-reversed, closed magnetic f i e l d. We designed the pilot reactor gun based on the Seta II gun and preliminary scaling laws. A unique feature of the pilot reactor gun is the use of a hollow cathode of sufficient radius to permit uninhibited passage of the escaping plasma stream to the direct converters.

The neutral beam injector for the FRM pilot reactor must continuously

(These

deliver 11 A of 200 keV deuterium and 11 A of 200 fceV tritium. are the requirements for the S » 5 plasma; the requirements for S = 7 are somewnat less.) In this study we designed an Injector based on negative ions that are directly extracted from a source, similar in principal to that under development at the Brookhaven National Laboratory. The ior source is held at a -200 kV potential. The negative ions, extracted from the source, travel to the accel grids via a sector magnet. The sector magnet separates the negative Ions from electrons and deflects the ion beam out of the plume of neutral gas streaming out of the ion source. Subsequently the beam is accelerated and made to pass through a cesium vapor stripping c e l l, where abwt 60* of the negative ions are stripped of their extra electrons to become neutrals. A percentage of the energy carried by the un-neutraHzed fraction of the beam which leaves :de stripper is recovered via a thermal beam dump at ground potential. Based on sn analysis of all the power needs of such an injector, we estimate its efficiency to be SOS, not including the thermal recovery of the beam dump icns-

The magnet system f or the FRM p i l ot plant consists of a pair of

length Is 3.5 m. The solenoidal magnets provide a nearly uniform

superconducting solenoidal magnets and a set of four superconducting saddle c o i l s. The o v e r a ll diameter of the c o il set is 7,5 m and the overall background magnetic f i e ld p a r a l l el ground magnetic f i e ld strength is 4.8 T for f or the S = 7 plasma. The solenoidal c o i ls also provide a shallow a*ia1 magnetic well f or plasma centering. The need for a r a d i al magnetic well is not presently c l e a r, but in t h is study we assumed t h at f or plasma s t a b i l i t y. The r a d i al well

to the axis of the reactor. The back

is produced by the saddle c o i l s.

the S = S plasma and 4.6 T

it 1s necessary

The FRM plasma p a r t i c l e s, although confined in a t o r o i d a l, closed magnetic f i e ld region, d i f f u se to open f i e ld lines and escape from the ends of the reactor into the d i r e ct conversion/vacuum pumping tanks. Half of the p a r t i c l es e x it at each end. At tne end of each tank is a simple one-stage, immersed-gria d i r e ct converter. The plasma passes through a tenuous grounded g r i d, then through a negative g r id which r e f l e c ts c o l l e c t or plate held at a p o s i t i ve p o t e n t i a l. The d i r e ct recovery e f f i c i e n cy of t h is d i r e ct converter c o l l e c t or plate for the d i r e ct converter is convectiveiy coaled to remove the heat deposited by the impacted ions. For the p i l ot plant design, we have assumed that the heat w i ll be removed at a thermodynamically un i n t e r e s t i ng temperature and therefore do not route it to the thermal conversion system.

the primary electrons. The ions are then decelerated by a

is estimated to be about 505£. The

f i r st

-222-

is 1.14.

In steady s t a t e, the ions collected in the d i r e ct converter w i ll

reflux as gas and must be vacuum-pumped to maintain an operating pressure of about 5 x 10~6 t o r r. The pumping is done by an array of chevron- baffled l i q u id helium-cooled cryopanels mounted on the side walls of d i r e ct conversion/vacuum pumping tanks. The t o t al area of cryopanels double the active area so that half of the system can be defrosted and degassed while the other half is a c t i v e ly pumping. Reclaimed deuterium and t r i t i um from the cryopanels are recycled to the neutral beam i n j e c t o r s.

the is

The blanket consists of a single large module which may be removed

and replaced without c u t t i ng and welding. The module is clamped in a frame made by prestressing the b i o l o g i c al shielding vault in which the reactor cushion interposed between the module and the frame. When the cushion <s i n f l a t ed it closes a l l - m e t al knife seals to seal the central vacuum region.

Is plared. The clamping force is generated by an i n f l a t a b le

Neutron moderation and t r i t i um Breeding takes place in the c y l i n d r i c al s h e ll blanket constructed of Inconel 71S. The t r i t i um breeding material 1s l i t h i um oxide and the blanket coolant The local blanket energy m u l t i p l i c a t i on p l i c a t i o n, accounting f or beam port and end losses,

is helium. is 1.21 and the average m u l t i

We calculated the overall performance of the FRM pilot reactor for two cases: the smaller, S * 5 plasma and the larger, S * 7 plasma. The pilot reactor with the smaller plasma is just barely a net power pro ducer: 90X of the 11.6 MW gross electric power must be recirculated to operate the reactor, and the net electric power 1s only 1.2 MW. Although marginal, the pilot reactor with the smaller plasma does satisfy the net electric power criterion. The larger plasma results *In much more attractive performance: 46X of the 19.8 MK gross electric power is re circulated to operate the reactor, and the net electric power 1s 10.7 MW. Further improvements in performance would be realized 1n a multicell commercial reactor.

G. A. Carlson, W. C. Condit, ft. S. Oevoto, 0. M. Fink, J. D. Hanson,

W. F. Neef, and A. C. Smith, Jr., Conceptual Design of the Field-Reversed Mirror Reactor, LLL report UCRl-52467, Hay 1978.

2 G. A. Carlson, K. R. Schultz, and A. C. Smith, Jr., Definition and Conceptual Design of a Small Fusion Reactor, EPRI Interim Report ER-1045, April 1979, also Final Report in press.

f ”

-223-

Fig. 1. Single Cell FRM Pilot Reactor

3EM£aATOB

THE MOVIJVG-RIAIS FIELD-REVERSED MIRROR REACTOR CONCEPT*

A. C. Smith, J r ., Pacific Gas and Electric Company, San Francisco, CA 94106 G. A. Carlson, Lawrence Livermore Laboratory, Livermore, CA 94550

Ithaca, NY 14853 H. H. Fleischmann, Cornell University, T. Kammash, University of Michigan, Ann Arbor, MI 48109 ’<. R. Schultz, General Atomic Company, San Diego, CA 92024 D. M. Woodall, University of New Mexico, Albuquerque, NM 87106

The Moving-Ring Field-Reversed Mirror Reactor (MRFRMR) concept envisions production of e l e c t r ic power by burning magnetically field-reversed rings of fusion fuel which are translated continuously down tue bore of a s t r a i g h t, reactor scheme c y l i n d r i c al arises from it might be in small size (50-100 MW(e)). This reactor has been evaluated only piloted is on a preliminary basis founded on forming a more detailed and thorough design study of this concept*.

reactor burner chamber. Our i ts potential design simplicity and the hope that

f i r st evaluation*. We are currently per

following discussion

the results of

that

the

The principle

in Figure 1. The reactor i n - l i ne sections:

matically a pjasmar-ing generator, a central burner sec three of t i o n, and a spent-ring exhaust se’ctidri’. The preTiminary scoping study men tioned above concentrated c h i e f ly 0” the burner section of the resctcr.

the reactor’s burner section are shown sche is cylindrically-symrietric and consists

. 5V

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-224-

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Interest 1n this

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Figure 1

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MOVING RING FRM REACTOR

N \ I ^ ^ S -^ COLLECTOR-

General Reactor Overview: A pulsed start-up mechanism (such as an in- tense charged particle beam source or a coaxial plasma gun) could generate a field-reversed plasma ring in the relatively low magnetic field just beyond the end of the burner chamber (“Ring Generator” section of Figu.-e 1). A lo cal “moving mirror” (provided by sequentially energizing “push coils” locat ed near the wall of the reactor) into the high solenoidal magnetic field of the burner stction. Certain start-up approaches, such as the coaxial plasma nun currently being tested on the Beta II device at LLL, may provide the plasma ring with adequate axial k i netic energy to drive the ring into the high field region without the use of a u x i l i a ry c o i H. “five ririg may fnerefor* be twnpress&tl anti “neater! tt> i n i tial burn temperatures in a manner that is potentially efficient. Also, the pulsed start-up mechanism and transient nature of in Me reactor may easily permit the plasma, should plasma s t a b i l i ty or other considerations require

imbedding a toroidal magnetic field

lifetimes In

the plasma ,-ing

i t.

-225-

the rings’

then rapidly forces

the “burner section” of

lines. Although

through

Once the plasma ring has been brought to i n i t i al burn temperature, it

is the reactor, possibly by the transported local “moving mir peristaltic action of an array of sequentially-energized ror” coils positioned just inside the first wall. These pulsed coils would probably be constructed from aluminum or copper. The burner chamber con sists of a linear string of identical cylindrical modules. * Each module is itself a set of nested cylindrical components consisting of the f i r st wall, the tritium breeding blanket, the biological and magnet coil shield, and the superconducting magnet coils providing the axial solenoidal guide field.

As discussed below, the ring may be refuelled during its transit of the burner section with cold plasma (either as pellets or via low-energy plasma guns) such that the power produced by the plasma ring is held nearly con stant. Without energetic beam injection, internal plasma currents driven by the ring’s magnetic self-induct by the ance. For fusion rings of interest, classical magnetic diffusion times can be of the order of tens of seconds, considerably longer than the 0.2 - 1 s plasma burn times envisioned so far. Also, absence of energetic beams in the burner chamber means that only relatively small-bore penetrations in the first wall, blanket, and shield may be required for refueling, considerably simplifying the overall reactor design.

the confinement will be sustained

An attractive engineering design feature arising from the pulsed plasma to

in the MRFRMR is the possible elimination of multipole fields

this approach would use an electrically conducting f i r st wall to center

operation provide radial stabilization of the rings in the mirror field. ful, the rings radially rents axis of tages in terms of overall plasma confinement. Theoretical underway to explore this possible means of ring stabilization.

induced in the wall as the rings move with adequate speed down the the device. Elimination of multipole fields may also have advan

in the burner section through the action of the eddy cur

.vork is currently

If success

Depending on the fuel composition and purpose of the reactor, heat can be extracted from the tritium-breeding blanket and the radiation shield and used i;i a conventional steam cycle. When the plasma burn is nearly quenched (due to the cooling effect on the plasma by cold-ion refueling), the plasma rings may be exhausted out the expansion end of the device into magnetic and electrostatic energy recovery units. The lineir arrangement of the reactor also offers a built-in divertor-like action, since particles which have dif fused out of the burning plasma rings both ends of the device into direct converters along the open, axial sole noidal

in the reactor will probably flow out

the MRFRMR plasma burns are,

individually

field

speaking, transient events, we believe that fluctuations in the reactor wall heat load can be minimized by appropriate choices of 1nter-r1ng spacing and ring axial speed.

Pulsed Plasma Burn Calculations: To date, most of the work -v- the MRFRMR has centered on studying the characteristics of pulsed burns of DT The details of the current 0-0 F-‘okker-Planck plasma modelling plasmas. analysis have been given elsewhere’. Where passible, ongoing plasma equilibrium and stability studies will be incorporated to improve the burn model assumptions on plasma density profiles, diffusion processes, heat transfer, and permissible plasma size. Although the plasma burn model is still under development, some preliminary results illustrate what we might expect during the transient burn of a field-reversed plasma ring heated to initial temperatures T-j(t»0) = 75keV » 2Te(t=0). The confining guide field 1s 65 kG and S, the number of average ion gyroradii which fit across the ring’s minor radius, 1s chosen S(t=0) = 10. (The implication that the plasma is stable in the range of S = 10 - 12 may well be an optimistic pre sumption). Two cases are considered: (A) a plasma which proceeds to burn in a constant magnetic field with no external fuel addition to influence the burn characteristics and {B) a plasma wi+h identical initial conditions but where low energy ions (3 keV deuterons, 5 kev’ tritons) are added during the b um at a rate such that the total radiated power is held nearly constant. Burn characteristics for cases (A) and (B) are plotted in Figures 2 - 4.

Adding no cold fuel to the plasma causes the ring radiated power to drop from 49 toi to less than 10 ^a4 iii 0,5 s (Figure 2 ). Adding cold plasma fuel under the condition., of case (B) shows the ring radiated power of = 49 MW is sustained for about 0.35 s before dropping rapidly to zero.

-226-

Figure 3

Figure 2

Figure 4

. The reasons for this disparate behavior are summarized in Figures 2 - 4. As shown in Figure 3, withholding cold fuel leads to ’. brief drop in fuel temperature due to heat conduction followed by a rise as alpha particle heating takes effect. Also, as particles are lost via diffusion, the ring volume drops to maintain pressure equilibrium with the external confining i and 3, the combined effect of decreasing fuel field. As shown in Figures ion densityand plasma volume results in a net drop in ring power by about a factor of 5 at the end of 0.5 s. On the other hand, adding “cold” fuel to the burning plasma brings a dramatic drop in ion temperatures—from 75 keV at t = 0 to about 8 keV at t * 0.35 s. While such a drop in ion temperature greatly reduces <cv> for the DT reactions, the ion density has been in creased via the cold fuel addition by nearly a factor of 3 in 0.35 s to

ring power. As seen

maintain constant the plasma to which cold fuel case by about 30S. Also, eventually hovers around S = 7, while dramatic in S near tions may d i c t a te whether such a plasma parameter regime is tenable. i l l u s t r a t es some general

in Figure 4, aftar 0.3S s the Q of the no fuel addition the rings with no cold fuel added the c o l d - ’ u e l - a d d i t i on case shows a

the burn. Plasma s t a b i l i ty considera

This sample, preliminary calculation

the S-value of

the end of

features

that of

rise

l ie

in the range of 75 - 150 leeV under current assumptions.

t r a d e - o ff be influence reactor design. Because of which may Strongly tween ion temperature and density in a fixed confining f i e l d, optimal plasma 3oth Q values to t o t al power per plasma s i z e: beginning leads to I n i t i al value of S to 8 gives a Q of only 2.3 and a burn time of 0.17 s. A p r a c t i c al p i l ot plant design with commer c i al upgrade p o t e n t i al w i ll probably require plasma q values in excess of I S.

r i ng and plasma performance are c r i t i c a l ly sensitive

;he case (B) plasma burn with an S(t=0) * 12

i g n i t i o n, while decreasing the

the

‘<eV at

l i m i t ed us

the s t a rt of

Preliminary Reactor Study

temperatures were presumed

the bum and s t a b i l i ty concerns

radius of 32 cm, and blanket/shield

the MRFRM called f or a reactor burner chamber 12 m long, a

The preliminary design mentioned f i r st above f or thickness of 137 cm. The guide wall j u st outside located f i e ld of 65 kG was provided by superconducting c o i ls to be the s h i e l d. to 75/37.5 in which S(t) £ 5. The plasma r i ng major radius ranged from 9.6 cm to burns laser-acceler 6.6 cm during the burn. Plasma r e f u e l i ng was accomplished by ated OT p e l l e t s. Heat conduction from the field-reversed rings was neglect ed in t h is work under the assumption that the d i f f u s i ng plasma escaping from to achieve a plasma Q t hi c l o s e d - f i e ld confinement value nearly equal tt was nec the p a r t i c le confinement was essary in the preliminary study to assume that three times better the FRM. The plasma rings produced 10.6 MWt/ring with a burn time of 0.32 s. The rings moved down the axis of the 2 cm thick aluminum f i r st wall at a speed of to achieve r a d i al focusing with minimal d i s s i p a t i ve wall drag (5 10 kJ/m per r i n g ). Heat wa-s the 97 cm recovered thick stainless steel/BaC shield by c i r c u l a t i ng helium turbine cycle. A plant energy balance showed a net e l e c t r ic power production of 1.25 the magnetic compression and ring energy recovery processes.

l o s t. Also, the beam-driven FRM^ (Q » 5 . 5 ),

4.75 MW(e) per r i n g, depending on the e f f i c i e n c i es possible

than that needed for

the 40 cm

in a gas

that of

4u m/s

from

for

f u el

Plasma

(1978)1:

Is r a p i d ly

ion/electron

is added exceeds

thick Li7Pfa2 breeding blanket and

the Electric Power Research

i d e n t i fy (where possible)

for

U. C. Smith, J r ., et a l ., Preliminary Conceptual Design of the Moving Ring

Field-Reversed Mirror Reactor, Pacific Gas & E l e c t r ic Company Report 78FUS-1 (19”3).

2T. Chu and A. C. Smith, J r ., P.FOT: A Fokker-Planck Code for Pulsed Fusion

evaluate The goal of nology issues and conditions which w i ll to an economically commercial p l a n t.

the study

Continuing Reactor Design Work: Work is continuing on the MRFRMR to r e issues bearing most d i r e c t ly on reactor performance.

the design

Plasma Analysis, Lawrence Livermore Laboratory Report ( in press). JG. A. Carlson, et a l ., Conceptual Design of the Field-Reversed Mirror

the physics and tech lead to a p i l ot reactor which scales

f h is work under contract No. RP922.

is being performed

Reactor, LLL Report UCRL-52467, May, 1978.

I n s t i t u te

PRELIMINARY REACTOR DtPLICATIOHS OF COMPACT TORI: HOW” SMALL 15 COMPACT?

R. A. Krakouski aad R. L. Hageusaa, 0aiderslty of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

I. INTRODUCTION

The generic name “compact torus” (CT) is applied to the general class of toroidal plasma configurations in which no magnetic colls or material vails extend enrough the torus. Figure 1 schematically summarizes the “family tree”

I I.

of CT configurations chat have been subject ed either to theoretical or experimental ex amination. Two branches co cne CT family of closed-field plasmolds are evident. On going reactor studies at LA3L have focused primarily onca che left-hand branch depicted la Fig. L, with an emphasis being placed upon field-reversed etieta pinch (FR0P) as a means co form, heat and confine a CT plas- aioid in a reactor context. 3och che spheromac and che FRSP configurations are assumed to require an electrically conducting wall co provide equilibrium and stability. The purpose of this paper is to present parametrically and by means of a simple analytic model the reactor implica tions of a FROP; an electrically conducting Shell is presumed necessary for equilibrium and stability. The question of n-.nutmra power and size for this specific CT configuration is addressed.

;

««.0

i’ *s” “1

TrH1!?3

’.Cff; 3J1S3

MUlUB^yM:

7T’JFC.*SIIKE

3BU cswsi

’ “AI!0M*Y

:O»MC: TORI

-228-

; ; « i a s . ’ i » £ M ag I

\ uF: ,-’"" |„..,,i-„,

f i g u re i

”OPUS ^EiCTOIl I CTCH I

*, - ,| ’ i io ;a * « ! < » » , * ’ .’

.^rr?

3 ^ ^ y K C N O u C i tG

« . = ! . * . .-

C3MP4CT

, -« *;

’ ’ C OP

c/r

:ers

and i/r=

s increases beyond

-inent notation that torsi the basis Indicates” decreased confinement times as r

_ and che plasma becomes over- compressed. The experimental para;: along with analytic equilibrium constraints, are used to guide this study. As is observed in experiments, trie FS.3P plasaoid is assumed Co contain little or no toroidal field. The question cf plasmsid formation and heating is not addressed here, altr.oug.i these processes would occur in an ex- reactor system by neans ‘-f slow uiploslon, adiabacic neating, cue application of energetic particle beams, and/or ancic dissipation.

Schematic summary of compact corus plasma c o n f i g u r a t i o n s.

ttODEL

A. GENERAL COSSIOERATIONS Figure 2 depicts the CT model and pei

for chis analysis. Experimental evidence

:; JwUCTVlfi ‘Xi’j.

liCKTiriG SELL |

< : . .* 1

; - * ; *: riOlMi^ICi’SyU |

_JJ3£S:!

*WIIL.-TM ;

Figure 2 Schematic lavout of Coraoac: Torus Reactor

CT1V’ based on FRGP. Conducting shell is 5-ovr, located either ac first wall or out side ii blanket.

can

Given che formation, heatiag, and translation of tne FRSP plasmoid, hov best energy lie obcalned frCQ a shell-stabilized configuration, and la which way does this constraint affect che projected reactor size and power level if realistic engineering constraints are applied? In short, how small is compacc?

B. ROLE AiND LOCATION OF CONDUCTING SHELL A. central thesis of this study Is the presumpclon that a passive, elec

trically conducting shell provides both equilibrium and stability and tnat cc/ r3 must be < 2 for thi3 to occur. The heated and ignited plasmold of length X enters the linear burn chamber of length L and radius rw at a velocity v - L / T- chat is compatible with the electrical skin time, Tg, of the conducting shell positioned at radius rc > rw. Translation of the plasmoid Inside this flux-conserving shell increases the stationary magnetic field (provided by external superconducting magnets, Fig. 2 ), as magnetic flux Is constricted to a saaller volume between cae separatrix and the shell. Tha: part of the plasma pressure supported by the conducting shell results in ornate dissipation within snell which must be extracted from the translacioaal energy of L.ie plasmoid- This energy loss may be significant and generally points Co the use of a room-temperature snell located outside a blanket of thickness Ho.

As the plasmoid is translated to the burn chamber, the flux within tne snell, provided initially by che external superconducting coils, is conserved. A characteristic cime, T3, for flux penetration into the shell of resistivity 1 can be derived on the basis of this flux conservation

T3 - T n r 7( r» -rs}

”>

This expression is based upon an allowable flux loss as determined by the limit vhen trie plasmoid would contact the’ firsc wall. Placemenc of che snell at the first wall (i.e., rc « rw) will require tnat the shell thickness, a, be less than - 0.05 m fot aeutronic reasons; additionally, n will be increased oecause of tae higner operating temperature at a first-wall location. Generally, t3 is computed to be 3-4 times longer if the shell is positioned outside the blanket, inspite of che higher value of r, = ru + ib. Even wnen locatea outside the blanket, the shell thickness snould present a croas- sactlonal area that is appreciably smaller t:ian the crucial area from wnic.-i flux Is being displaced (I.e., n(rj - r*))> Furthermore, che oriole power dissipated in cne shell, PQJJJ^, wnen expressed relative to cne alpna-partlcle power, P2, is given by

fusion

useful

~~zzs~~

translating,

For

the

field within

lib * 0.5 a, Bi -” 5 T)> this ratio decreases (5 - 0.1 a)

where 3^ is the compressed magnetic reactor parameters ( rw - 1.0 m, from ~ 60 for an ex-blanket shell for a first-wall shell, again giving impetus to shell placement autsiae tne blantcet. Lastly, tne ohmic dissipation occurlng wicnin shell must be provided the kinetic or the fusion ( i . e ., directly converten aIpna-particle aitner by energy) powers associated with For a first-wall required transnational power can be v-.insiierable, and shell, the plasma expansion required to chanr.l directly the alpna-particie energy :i loss woula be prani’oitively large. On tne oasis at these supply

burning plasooid.

( 5 - 0 . 05 m)

this ohoic

typical

shell.

the

arguaents, the stabilizing conducting shell should be located at a radius rc « T.J + Ab, where fib Is expected to be ” ^.5 a. In order to provide the necessary stabilization of a. plasmoid of length t * 7 rs, the translatioual v»locicy must be v ” l/T3, again with rc/ rs simple constraints play an important role in establishing the minimum size and power of the CT reactor-

s 2. Aa will be seen, tnese

C. CONSTRAINTS IMPOSED 3Y NEUTRON HALL LOADING AND TOTAL POWER The fusion neutron wall loading, I^‘J/m’), and a Lawson-llka criterion, nig (s/m ), at this level of analysis represent important indicators of system performance. The analytic CT equilibrium relationships predict that 37% of the plasnoid volume within the separatrix radius, rs, would be filled with 3 * 1 plasma. The fraction of the burn chamber that is filled with plasma (i.e., the duty factor), is easily shown to equal tg/x-j, whsre 1/Xj is the plasaold injection rate. following expression for l^ results

Recalling that the burn time, T3 » TS(L/Z), the

( n rB)2 <ov> £„

where aks units are used, EJJ is the fusion neutron energy, rtf - r. - Ab, rr/r = 2, i/rs = 7, and xg (Sq. (I)) can similarly be expressed in terms of r^. With Che exception of Tj, the left-hand-side of Eq. (3) represents a constant chac is chosen on Che basis of confineoenc physics, nig, and desired system performance, 3^. The total system thermal power, P-jaC^l:) equals 2TTTV L IW M, where che multiplication M is typically ~ 1.42 (20 Me7/fusion) . For a given wall loading and ntg value, therefore, P-JJJ can be evaluated as a function of system dimensions (i.e., rg and L ).

structural

An additional and important constraint is imposed by the allowable thermal cycle, AT(K), experienced by the first wall. The thermal neat flux at the is expected to originate primarily from 3remsscrahlung radiacion, in that particle losses snould be directed out of che b um cnamber along open fi«ld lines in the region from radius rs to rv. If ’<(W/mK), p(kg/m3) and c (J/kg thermal conductivity, densicy, and neat capacicy of the first-wall material, the temperature rise for a “cnermally chick” first wall that is irradiated solely by Sremsstrahlung leads to che additional constraint

’£.) are,

wall

che

l .\

-230-

rw T,* 2 = , ”*

rs2 r*,2 *w ^

‘xf/2

< ^S>2 L2T a3 /2

respectively,

,* 2 r3- ib

AT VCpicp

first

(3)

(4)

3’Ti>

2.63(10)“37 ( n xB)2T1 /2

where T(keV) is the average plasmoid temperature, and the thermal irradiation time experience by any given section of first wall is taken as Tg.

III. RESULTS

Typical results are illustrated on Fig. 3, which shows che dependence of ?-.„., L, and T on the separacrlx radius tor che fixed parameters indicated. A aiaimura cocal power is shown for this case where the duty factor, f^ - T has been fixed at 0.1 and n t3 » S(lO)-1 s/m2 (i.e., a fuel burnup fraction r’3 - 0.22). The reactor power initially decreases with iacreased r3 because of tne increased Ts and correspondingly decrease la required translarional

\P„;CWII T.IOWV

1 2-

L«

the

velocity; an increased reactor length results. As r, tacraaaec beyond cne minimum at 3.83 m, tn« power power Increases because of increased plasma cross-sectional area. Only tn* magnitude buc not Che position of the on Fig. 3 minimum fixed the depends of parameters, tne £q. (3) appropriate scaling relationship. Spec ifically, combining Eq. (3) with Pra - 2mr„ L L, K gives the following explicit form for the power curve depicted o* Tig. 3.

providing

<n/«) i b3 /2

the

-231’

Figure 3

power on

X3 / 2[ ( l - x )2 - 1/4i

?M<W e’ ” 8*78 :‘J Iw/2 <0^>in

indicated choice

and the t h e m al cycle c o n s t r a i n t, AT < 30 £,

rH - 1.2 m, r, - 1. 7 a, T0 * 0*. 6 s, T„^- i .j s,

to - 3-2 a and ij correspondingly be reduced

?T H<Wt) - 5 . 5 4 U 0 ) “17 I „1 /2 A b3 /2

the minimum-power point for a

-O.U.1 mv^Li ai.m.^-cao-d>.eB^

9.4 m/s (34 lm/h;

s t a i n l e s s - s t e el

( < a v > / T1 / 2) f,

e x p l i c i t?

^c-pkn-1

copper

( n tB) /4

aim

t \12

<?>

(6)

i l l u s t c a t es

e s t a b l i s h i ng

r o le the minimized t o t al power v a r i es weakly as the square root neutron wall l o a d i n g.

whica d i r e ct coupling of plasma performance with the ex-blanket s n e l l, prominent i n t u i t i o n, f i r s t - w a ll 3, following system c h a r a c t e r i s t i cs a re p r e d i c t e d: ? _H - 330 MWt, L » 42 a, r 0.83 m, ( f, F-jg/inr2 t - 2.4 M»/m3.

thft CTOB. aiaimwa-power s c a l i n g. Secause oi we a to the tne

ub plays the minimum pqwer. Furthermore, c o n t r a ry

For the minimum power shown on F i g.

fixed at 0 4 ), v -

l.l(lO)-1 ST

rT » 6.2 s

I » 5.8 i,

3*3 I

The temperature-rise constraint

given by iq. (4) Ls expressed below in

explicit form

dependence of total power, P _, length of burn chamber, L, ana shell skin time, T , on plasmoid separatrix radius, r , for the fixed parameters indicated.

i b /r

« A b / 2 r3. Equation (5) expected minimum at x » 0.3 Foe

where x * shows ( i . e ., T3 - 0.33 for T - . 10 keV, M - 1.42 n - 2 ( 1 0 ) ”* ohm a (Cu at 300 K), the minimum power equals

ib - 0.5 m).

and

f i r st wall witn p r o p e r t i es of copper. Equation (7) nas seen p l o t t ed on F i g. 3

wnlcn p r e d i c ts 4T « 14.2 K at thermophysical for p r o p e r t i e s. ? copse ui.es. t h at T^

r st walls with both (SJ.J.J Application of

De adjusted

tnermophyslcal

r e q u i r es 05,

resulting la an optimum (i.e., minimised) reactor power of 600 MWt with L « 30 m, again for a first wall with theraophyslcal properties similar to copper. Stainless steel represents an extreme relative to tee assumed copper-like first-wall properties, represenciag an increase in 41 bj a factor of V (i t cnP’cu^i t cp’I^SS ” 4”*” *c “s e mPn a s l-z ed that the methods used to estimate the thermal-cycle constraint are highly approximate, and considerably acre analysis of tnl3 Important and often neglected problem Is warranted

IV- CONCLUSIONS

The application of simplified but realistic engineering constraints to the special class of wall-stabilized FRSP configurations leads to reactor systems that may be as small as - 30 B in length and generating a total thermal power of the order of SOO MWt.. Decreased size and power for a given at. will be accompanied by decreased performance indicators, as reflected in this study by L, and the allowable AT. tc should be noted that this analysis is based upon fixing the duty factor, fg * T tr^ac Ty rather than fj as a parameter give sorae.’*iat different optima, but the basic conclusions and results embodied la Tig. 3 are not significantly altered. The results of this simple scoping calculation will oe used to guide a more detailed modeling of important issues related to plasmoid infection, plasma transport/equilibrium/stability, burn dynamics, transient response of the Che first vail aad conducting snell, and overall system energy balance.

s^Ti’ Other approaches which

REFERENCES

LAil CTS-Oivision Staff, “Proposal for FHX-C and Multiple-Cell Compact Torus Experiments,” Los Alamos Scientific Laboratory report, LA-8045-? (1979).
i. X. Llnford, . D. A. Platts, and F. G. Shervood,

“Field-Reversed experiment (Compiled by K. 3. Thomas and 3. A- Sawyec),” Los Alamos Scientific Laboratory report LA-7471-PR (1979).

-232-

TRACT: A SMALL FUSION REACTOR BASED ON A COMPACT TORUS PLASMA

H.J. tfillenberg, A.L. Hoffman, L.C. Steinhauer and P.H. Rose Mathematical Sciences Northwest, I n c .; Bellevue, Washington 98009

A compact torus formed by triggered reconnectiart and adiabatic compres fusion

sion of a reversed-field theta-plnch plasma can provide the oasis f or power reactors which are small, r e l a t i v e ly easy to f a b r i c a te and mair.fcain, and which promise high power gain with modest r e c i’ : u l a t i ng power r e q u i r e ments. A new reactor concept, designated TRACT (Triggered Reconnection, Adiabatically Compressed Torus), has been developed based on such a confine ment scheme. The TRACT plasma is produced in a l i n e ar solenoid by a staged process i n v o l v i ng plasma formation in a reversed bias f i e l d, introduction of forward f l ux by external solenoidal shock c o i l s, triggered reconnection of the magnetic f i e ld lines r e s u l t i ng in rapid axial compression and heating of the resultant plasma torus and, f i n a l l y, some adiabatic compression of the plasma by additional experimentally observed to be stable for at least 100 Alfven

forward f l u x. Plasmas formed in t h is manner have been

-233-

t i m e s .1

is activated in the opposite p o l a r i ty

injected a x i a l ly

For plasma major r a d ii of 7 cm or more, the confinement time based on classical Darticie d i f f u s i on across the separatrix exceeds one second. Allow ing for anomalous transport across the separatrix to establish a quasi-steady state balance w i th axial transport processes leads to a confinement time which is about a factor of 30 smaller than c l a s s i c a l. Based on t h is hybrid confine ment model, a 15 keV compact torus with a major radius of 24 dm and aspect r a t io of 3.9, confined by an external f i e ld of 7 Tesla, has a plasma confine ment time of C.5 sec. Such a plasma would generate 225 HJ of neutron energy in 0.5 sec. A TRACT reactor based on t h is plasma with a r e o e t i t i on rate of one Hertz is capable of generating 90 megawatts of net e l e c t r i c al power.

A conceptual TRACT reactor design is i l l u s t r a t ed in Figure 1. The com

located near the f i r st wall during plasma formation.

plete reactor vessel, including the blanket, s h i e l d, and magnets, is 9 metars hSgh and 3.5 meters in diameter. The plasma chamber is f i ve meters long with a one meter bore. A steady 7 Tesla axial f i e ld is maintained with a suoerconducting solenoid located outside the blanket. This f i e ld is disturbed b r i e f ly by a normal c o il Prior to each pulse, deuterium-tritium gas is screen nozzle. The normal c o il superconductor to establish a reverse bias f i e ld in the plasma chamber. After s r e i o n i z a t i o n, the plasma is heated to i g n i t i on by a four stage process occur ring during the second quarter cycle of the normal c o i l. F i r s t, a rapid drop in the normal c o il current causes radial shock compression of the plasma and reversal of the magnetic f i e l d. Magnetic f i e ld l i ne reconnection is delayed with a magnetic cusp at each end while the plasma is compressed. When suf f i c i e nt forward s u l t i ng in a mirror f i e ld which rapidly triggers f i e ld l i ne reconnection. This triggered reconnection causes the plasma t o . c o n t r a ct a x i a l ly to a con f i g u r a t i on with a much less elongated torus. As the normal c o il current r e duces to zero, the plasma is then a d i a b a t i c a l ly compressed to i g n i t i on perature. During burn the plasma is confined wholly by the superconaucting f i e l d. Fusion product heating raises the temperature to 15 keV during burn.

f1ux has been introduced, the cusp current is reversed, r e

through a

to the

tem

The a x i al contraction r e s u l ts in a comoact torus w i th a high f i ll t or x, defined as the r a t io of the separatrix radius to the f i r st wall

f a c

radius.

9.0 n

BLANKET C O X*

PULSED W Sm.”

FUiL IWEC7I01.

2 3 4-

Figure 1. A 90 MWe TRACT Fusion Reactor

COOLAS”

Fill factors approaching one (rs ** rw) have been observed.1 No signs-of in stability have been observed for the circuit decay time of the confining mag- iet as long as the fill factor remains above a critical value which depends on the initial pressure, and is in the range x * 0.3-0.4.

The rapid axial contraction and magnetic field reconnection generates irreversible heating of the plasma, so that the contracted torus is hotter than if the process were adiabatic. Thus, the plas..*i has been heated by two nonadiabatic processes: radial shock heating of the conventional theta pinch type, and axial compression heating from the reconnection process. The more the plasma is heated by axial compression, the less is required from purely radial processes. This relationship is illustrated in Figure 2, where the axial heating Is described by the ratio, g, of the temperature after r e c o n n e c ts to that immediately before reconnection. The higher the value of g, the more heating 1s achieved by axial processes and the less is required of radial processes. This translates to more relaxed requirements on the azimuthal electric field Ee at the first wall for shock heating. Figure

2 shows the electric field required to achieve 6 keV at final fill factor Xf for a range of axial temperature gains. X] is the fill factor after reconnec- tion. If the critical fill factor xf« 0 . 3. then Ee<0.5 kV/cm for very modest temperature gains, g* 1.5-2.0. If the critical fill factor is Xf*0.5, then g£,10 is required for Eg * 0.5 kV/cm.

The normal coil is supplied by two separate power supplies: the slow

field reversal and adiabatlc compression is provided by an 85 MJ fast-discharg ing homopolar generator, and the radial shock field is provided by a 10 MO capacitor bank. The first wall is made of alumina to allow the fast-rising field to penetrate. The blanket concept chosen for the reference design con sists of lithium lead plates cooled by boiling water flowing axially in stain less steel tubes. The radial blanket fs divided Into two zones: the inner blanket zone must be replaced periodically because of radiation damage, and the outer zone lasts the lifetime of the reactor. A 0-60 meter thick shield of boron carbide and stainless steel protects the superconducting coil from radi ation. The entire vessel is enclosed in a vacuum chamber which can be ten meters nigh and seven meters in diameter. The plasma chamber is vacuum pumped continuously through penetrations at the ends.

-235-

The small size and simple geometry of the TRACT reactor provides a great

advantage for reactor maintenance. The reactor vessel is small enough to stanc vertically. There are no radial penetrations into the plasma chamber or inner blanket, so there are no restrictions to vertical movement during component maintenance. Since the first wall, normal coil, and inner blanket zone are free tc move axially and are small, no remote cutting must be performed to pre pare these components for removal and replacement by an overhead crane. These features will greatly decrease the time required for blanket/first wall reolace- ment comoared to other fusion reactor concepts.

The recirculating power requirements for an ignited TRACT reactor are low.

A” 1 m bore, 5 m long plasma chamber is sufficient to produce 278 MJ of thermal energy per pulse, with a blanket multiplication of 1.22. Use of the compound magnet concept limits resistive heating of the magnet to only a small fraction of the duty cycle. With a ten millisecond half-cycle time, which is achievable with fast-discharging homopolar generators, Joule heating losses are 1.9 MJ/ pulse. Assuming 95 percent switching efficiency, switching Josses in tie mag net system account for another 4.2 MJ/pulse. Plasma thermal energy at ignition is A.5 MJ, vmich is not recovered in the current design. Other losses, in cluding vacuum and coolant pumping power in the blanket as well as the super conductor, amount to less than 1 MJ/pulse. At 36 percent blanket thermal ef ficiency ^ the recirculating power fraction is 11 percent. A reactor with i 5C percent duty factor would thus produce 90 MW of net electric power.

Since the plasma volume scales as rJ, where r is the first wall radius,

and the confinement time scales as t2 , the thermal energy output per pulse scales roughly as r5. A small change in the reactor dimensions can substanti ally change the power output. J”he hybrid confinement model described above predicts that an experiment with a 0.52 meter bore would result in energy break even, and a fusion power pilot plant which produces net electrical power re- oulres only a 0.70 meter bore. The strong power dependence on size als: leads to weak reactor sensitivity to confinement scaling. Figure 3 illustrates the first wall radius required to produce 90 HWe for different confinement times. Particle confinement based on-classical radial transport would allow a ‘ir*t

wall radius of 0.39 m. Alcator scaling allows a radius of 0.56 m. Even tne p a r t i c le reouired f i r st wall radius would s t i ll be only 0.9 m.

confinement time scaled as one-tenth of the Alcator model, the

«T OKTIlMl. *Mi> . V. »-

‘T

-236-

factor Xf with

f a c t or at

i g n i t i o n.

factor

Figure 2. E l e c t r ic Field Reduction With Axial Compression. Field Required to Achieve Tf=6 keV at f i ll 7 Tesla Magnetic F i e l d. Length Compres- f i ll sion -actor = 3. X-|«radial immed”atsly a f t er reconnection.

Figure 3. Reactor Size Scaling With Confinement Time. F i r st Wall

Radius Required to Achieve 90 Mw Net E l e c t r i c al Power. Xf= radial f i ll

A.G. Es’kov et a l ., “Features of Plasma Heating and Confinement in a

Compact Toroidal Configuration,” 7th IAEA Conf. on Plasma Physics and Controlled Thermonuclear Research, Innsbruck, 1978.

REFERENCE

ACKNOWLEDGEMENT

This work was supported by the E1ectric»Power Research I n s t i t u te under Contract No. RP-922-4.

US-Japan Joint Symposium on Compact Toruses and Energetic Particle

Injection

Plasma Physics Group, raculty of Science Theoretical Research Center for Nuclear Fusion Hiroshima University Higashisenda-machi, Hiroshima, 730 JAPAN

Dr.

-237-

Attendees

Irvine

Dr. S. Inoue

I n s t i t u te of Plasma Physics Nagoya University Ch1kusa-ku, Nagoya, 46* JAPAN

Ikuta Or. K. Dr. A. Mohri Dr. K. Narihara Dr. Y. Tomita

College of Science and Technology Ninon University Chiyoda, Tokyo 101 JAPAN

i. Jtogl

Sell Telephone Laboratories, Inc. 600 Mountain Avenue Hurray Hill, NJ 07974

Dr. H. Ikezi

Department of Physics University of California at I r v i n e, CA 92717

Dr. S. Robertson Dr. M. Rostoker

University of C a l i f o r n i a, Los Angeles Los Angeles, CA 90G24

Dr. R. Taylor (Center for Plasma Physics and Fusion Engineering)

Plasma Physics Laboratory Department of Applied Physics and Nuclear Engineering Columbia University New York, NY 10027

Attendees

-238-

D r. C. K. Chu

Laboratory of Plasma Studies Cornell University, Upson Hall Ithaca, NY 14853

Dr. H. Fleischmann (Department of Applied Physics) Dr. 3. Kusse (Department of Applied Physics) Dr. J. Greenly Dr. R. Sudan Dr. A. Reiman

Office of Fusion Energy Department of Energy Washington, DC 20545

Dr. J. Clarke Dr. W. Dove (Division of Applied Plasma Physics) Dr. T. George Or. D. Thomson Dr. H. EIHs

Grumman Aerospace Corporation 101 College Road, East Princeton, .’(J 08450

Dr. A. Todd

E l e c t r ic Power Research I n s t i t u te 3412 m il view Avenue Post Office Box 10412 Palo A l t o, CA 94304

Dr. D. Paul Dr. 3. Scott

Attendees

General Atomic Company Post Office Box 31608 San Diego, CA 92138

Dr. T. Ohkawa

Fusion Studies Laboratory 214 Nuclear Engineering Laboratory University of Illinois Urbana, IL 61801

Dr. G. M1ley

Institute for Advanced Study Princeton, NJ 08540

Dr. V.-P. Ho Or. V. Rosenbluth Dr. J. VanOam

Lawrence Liveraore Laboratory Post Office Box 808 Livemore. CA 94550

-239-

Or. 3. Linford Or. S. Ortolani Or. A. Sherwood Or. R. Si’emon Dr. i Turner

Or. S. Auerback Dr. H. 3erk Or. G. Carlson Dr. T. X. Fowler Or. J. Hammer Dr. C. Hartman Dr. R. Post Or. J. Shearer Or. ‘A. Turner Or. 0. Schnack Dr. A. Shestakov Or. 0. Anderson

Or. D. Baker Dr. D. 3arnes Dr. T. Armstrong Dr. T. Jarboe Dr. R. Krakowski Dr. W. Ouinn

Los Alamos Scientific Laboratory Post Office Box 1663 Los Alamos, NM 87545

Dr. A. DeSllva Dr. G. Goldenbaum Dr. Z. An (Southwest I n s t i t u te of Physics, People’s Republic of China)

Attandees

Department of Physics and Astronomy University of Maryland College Park, MD 20742

Department of Physics Massachusetts I n s t i t u te of Technology Cambridge, HA 02139

Dr. A. Janes Dr. 0. Pappas

Inc.

-240-

fur Plasmaphysik

(Code S770)

Mathematical Sciences Northwest, Post O f f i ce Box 1837 Sellevue, WA 98009

Dr. P. Rose Dr. L. Steinhauer Dr. G. Vlases

Max-Planck I n s t i t ut D-8046 Garching Federal Republic of Germany

Dr. G. Spies

University of Michigan Department of Nuclear Engineering 145 Coolev S u i l d i ng Ann Arbor’, MI 48105

Dr. T. Kammash Dr. U. Getty

Plasma Physics D i v i s i on Naval Research Laboratory Washington, DC 20375

Dr. C. Kapetanakos Dr. J. Finn Dr. A. Robson Dr. J. Sethlan Dr. P. Turchi

Attendees

Jaycor, Inc. 205 Soutn Whiting Street Alexandria, VA 22015

Or. £. Scannell

New York University Courant I n s t i t u te of Mathematical Sciences 251 Mercer Street Sew York, NY 10012

Or. W. Grossmann Dr. G-

Oak Ridge National Laboratory Fusion Energy Division Post O f f i ce 3ox Y Oak Ridge, TJI 37330

Dr. D. Sigmar

The Pennsylvjni- * -. ’!**,-. ’ / 233 Hammonc Sui” . , University Park, * ’ - T>T”

Or. E. Klevar.i Dr. T. York

Physics International Company 2700 Merced Street San leandro, CA 94577

Dr. V. Bailey Dr. B. Ecker Dr. H. Helava

-tl

tforikawa

Department of Nuclear Engineering, Soon 2524 Pacific Gas and Eelctric Company” 77 3eal Street San Francisco, CA 9^106

Dr. A. C. Smith, Jr.

Attendees

Public Service Electric and Gas Company SO Park Place Newark, NJ 07101

Dr. 8. Jensen

Plasma Physics Laboratory Princeton University Post Office Box 451 Princeton, NJ 08544

-242-

OT. ft. Boozer Dr. H. Furth Dr. M. Gottlieb Dr. R. Jacobsen Or. J. L. Johnson Dr. f1. Katsurai Or. s. Jardi n Or. M. OkabayasM Dr. F. Perkins, Jr Dr. J. Sinnis Dr. T. Stix Dr. M. Yamada Dr. K. Yamazaki Dr. s. Yashikawa

University of Washington Seattle, WA 98195

Dr. F. Ribe Dr. H. Weuth

Department of Physics University of Miami Coral Gables, FL 33134

Dr. D. Wells

Prometheus I I, L t d. Post O f f i ce Sox 22 College Park, HO 20740

Dr. P. Koloc Dr. J. Ogden

Society to Advance Fusion Energy 10 Konrandy Lane Scarsdale, MY 10S83

Mrs. L. Slaner

Lawrence Berkeley Laboratory Plasma Physics Program Serkeley, CA 94720

Dr. 3. Fsinberg Dr. M. Levine

Stevens I n s t i t u te of Technology * Department of Physics Castle Point Station Hoboken, NO 07030

Or. H. Seidl

Attendees

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