LA-8700-C N O Proceedings of the Third Symposium on the Physics

Home , Compact toroid, Compact Toroidal Hybrid

LA-8700-C n Conference

Proceedings of the Third Symposium on the Physics and Technology of Compact Toroids in the Magnetic Fusion Energy Program

Held at the Los Alamos National Laboratory

Los Alamos, New Mexico

December 2—4, 1980

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LOS ALAMOS SCIENTIFIC LABORATORY Post Office Box 1663 Los Alamos. New Mexico 87545 An Affirmative Aution/f-qual Opportunity Fmployei

This report was not edited by the Technical Information staff.

This work was supported by the US Depart- ment of Energy, Office of Fusion Energy.

DISCLAIM) R This report WJJ prepared as jn JLUOUIH of work sponsored by jn agency of ihc Untied Slates (.ovcrn- rneni Neither the United Suit's (iovci.iment nor anv a^cmy thereof, nor any HI theu employees, makes Jn> warranty, express or in,Hied, o( assumes any legal liability 01 responsibility for the jn-ur- aty. completeness, or usefulness of any information, apparatus, product, 01 process disiiosed, or represents thai its use would not infringe privately owned rights. Reference herein to any specifu- com- mercial product process, or service by tradr name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommcniidlK>n, or favoring by the United Stales Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state 01 reflect those nf llic United Stales Government or any agency thereof.

UfJITED STATES DEPARTMENT OF ENERGY CONTRACT W-7405-ENG. 36 LA-8700-C Conference UC-20 issued: March 1981

Proceedings of the Third Symposium on the Physics and Technology of Compact Toroids in the Magnetic Fusion Energy Program

Heki at the Los Alamos National Laboratory

Los Alamos, New Mexico

Decamber 2—4, 1980

Compiled by

Richard E. Siemon

- DISCLAIMtP Preface

These Proceedings document the third in a series of Compact Toroid (CT) symposia. Although all three symposia focused on the same topic, they differed substantially in format- This change in format reflects the change in the type of interaction desired by the symposium participants. Initially the interest was in getting acquainted with each others' work, but as communication and a sense of community increased, the Interest has shifted to the discussion of issues of common interest. The first symposium, held at Princeton in June 1978, was quite informal and consisted of oral presentations of new untested ideas as well as existing theoretical and experimental results. The only record of the conference consisted of copies of Vu-Graphs assembled following the meeting. The result was an increased awareness of the work and planning in the various facets of what was to become the CT community.

The second CT symposium, held at Princeton in December 1979, was more formal with 60 four-page papers being published in the Procee ".ings as a documentation of the oral presentations. These papers and the attendance of over 100 participants reflected the substantially increased activity in the CT program. The meeting was also a joint symposium with the Japanese, and in addition contained presentations on topics related to Compact Torolds such as the Reversed Field Pinch.

The third CT symposium is more narrowly focused on the issues of CT physics and reactor concepts. The contributed papers have been assembled in these Proceedings prior to the meeting to allow their use as resource material. In contrast with last year's symposium, the contributed papers are not being presented orally. As outlined in the enclosed Agenda, review talks are beirg given on physics and related reactor topics followed by substantial discussion periods. This change In format relies on the more than 60 papers (including 4 invited talks) presented at the APS meeting and the 51 contributed papers in these Proceedings to report the individual research progress. Furthermore, the sense of community established in part by the prior symposia makes it possible to shift the emphasis in this year's symposium to in-depth discussions of the major research issues.

For organizational purposes, both the Agenda and the contributed papers have been divided into categories. This division emphasizes the breadth and diversity of the CT community, but it Is not Intended to artificially divide the community which shares an important commonality of physics. These Proceedings should provide a valuable campliation of recent advances in CT research, as last year's Proceedings have done. However, since this document will have a limited distribution, the authors should be encouraged to publish their results in the refereed literature in order for these important advances to have the widest possible exposure in the fusion community.

Rulon K. LInford November 18, 1980 Compact Torold Symposium Agenda

December 2-4, 1980

Tuesday Morning: Reactors

General Session Chairman: W. C. Condit

8:30 Welcome H. Dreicer Introductory Comments W. F. Dove

Invited 8:55 Stationary Systems Speakers: J. G. Gilligan 9:25 Coffee 9:40 Moving Ring Systems R. L. Hagenson 10:10 Utility Requirements for Fusion R. J. Vondrasek 10:30 General Discussion 10:45 10 Minute Break

Group Discussion Session Chairmen:

10:55 Stationary Systems R. A. Krakowski 10:55 Moving Ring Systems A. C. Smith

12:00 Lunch

Tuesday Afternoon: Particle Rings

General Session Chairman: R. F. Post

Invited 1:00 Ion Rings Speakers: C. A. Kapetanakos 1:30 Electron Rings H. H. Fleischmann 2:00 Coffee 2:15 Hybrid Systems R. N. Sudan 2:45 General Discussion 3:00 15 Minute Break

Group Discussion Session Chairmen:

3:15 Ion and Electron Rings J. D. Sethian 3:15 Hybrid Systems N. Rostoker

Sumtnarv Session Chairman: G. H. Miley

4:30 Five minute summaries from the Group Discussion Chairmen(Krakowski, Smith, Sethian, Rostoker)

5:30 Cocktail Hour Wednesday Morning: Spheromak Theci

General Session Chairman: J. L. Johnson

Invited 3:30 Slow Formation & Equilibrium Speakers: C. K. Chu 9:00 Fast Formation & Equilibrium B. McNamara 9:30 Coffee 9:45 Stability J. H. Hammer 10:15 General Discussion 10:30 15 Minute Break

Group Discussion Session Chairmen:

10:45 Formation & Equilibrium Co W- Hartman ] 0: ur, Stability M. Okabavashi

12:00 Lunch

Wednesday Afternoon:_ Spheromak Experiment

CJpnera] Session Chairman: J. Marshall

Invited I : 00 Slow Formation Speakers: M. Yamada 1:30 Fast Formation W. C. Turner 2:00 Coffee 2:15 Equilibrium & Stability T. R. Jarbce 2:45 Spheromak Experiments in Jap^n Y. Nogi 3:00 General Discussion 3:10 10 Minute Break

Group Discussion Session Chairmen:

3:20 Formation A. R, Sherwood 3:20 Equilibrium & Stability G. C. Goldenbaum

Summary Session Chairman: A. E. Robson

4:30 Five minute summaries from the Group Discussion Chairmen (Hartman, Okabayashi, Sherwood, Goldenbaum) -vii-

Thursday Morning: FRC Theory

General Session Chairman: H. Berk

Invited 30 Formation and Scaling Speakers: L. C. Steinhauer :00 Stability D. A. Barnes 30 Coffee 45 Transport W. Grossmann 10:15 General Discussion 10:30 15 Minute Break

Group Discussion Session Chairmen:

10:45 Formation and Scaling R. E. Siemon 10:45 E. St'.'ity C. E. Seyler 10:45 Transpi ••'• N. A. Krall

12:00 Lunch

Thursday Afternoon: FRC Experiment

General Session Chairman: G. C. Vlases

Invited 1:00 Formation Speakers: W. T. Armstrong 1:30 Confinement J. Lipson 2:00 Coffee 2:15 FRC Experiments in Japan H. Ito 2:35 General Discussion 2:50 10 Minute Break

Group Discussion Session Chairmen:

3:00 Formation A- L. Hoffman 3:00 Confinement F. L. Ribe

Summarv Session Chairman: A. W. DeSilva

4:15 Five minute summarier from the Group Discussion Chairmen (Siemon, Seyler, Krall, Hoffman, Ribe) CONTENTS

Page

A- Reactors

Al Random Gapped Wall Stabilization for Travelling Mirror Compressors 1 P. M. Bellan California Institute of Technology

A2 Nuclear Elastic Scattering Effects on Fusion Burns in Compact Tori A J. Galambos, J. DeVeaux, E. Greenspan, and G. H. Miley University of Illinois

A3 A Compact-Toroid Fusion Reactor Based on the Field- Reversed Theta Pinch 8 R. L. Hagenson, SA1, and R. A- Krakowski, LASL

A4 The Moving-Ring Field-Reversed Mirror Prototype Reactor 12 A. C Smith, Jr., PG and Es G. A. Carlson, LLNL, H. H. Fleischmann, Cornell University, W. Grossmann, New York University, T. Kammash, University of Michigan, K. R. Schultz, General Atomic Company, and D. M. Woodall, University of New Mexico

A5 A Linus CT Fusion Reactor Based on Axisymmetric Implosion of Tangentially-Injected Liquid Metal 18 P. J. Turchi, Research Development Associates, A. L. Cooper, D. J. Jenkins, NRL, E. P. Scannell, JAYCOR, Inc.

A6 Utility Requirements for Fusion Power 24 R. J. Vondrasek and R. N. Cherdack Burns and Roe, Inc.

A7 TRACT Fusion Reactor Studies 2 7 H. J. Willenberg, L. C. Steinhauer, and A- L. Hoffman Mathematical Sciences Northwest, Inc.

B. Particle Rings

Bl On the Use of Intense Electron and Ion Beams and Rings in Mixed-CT Configurations 31 H. H. Fleischmann Cornell University

B2 Cornell Ion Ring Experimental Program 36 J. B. Greenly, P. L. Dreike, D. A. Hammer, P. M. Lyster, Y. Nakagawa, and R. N. Sudan Cornell University B3 Kink Motion of Long Field-Reversing Ion Layers 40 D. S. Harned University of California, Berkeley

B4 Propagation of an Intense Ion Beam Transverse

to a Magnetic Field # 4 3 H. Ishizuka and S. Robertson University of California, Irvine

B5 Theory of Plasma Injection into a Magnetic Field 4 7 W. Peter and N- Rostocker University of California, Irvine

C. Spheromak Theory

Cl Classical Transport in Field Reversed Mirrors: Reactor Implications 52 S. P. Auerbach and W. C. Condit Lawrence Livermore National Laboratory

C2 Calculation of Tilting Modes in a Spheromak 56 M. S. Chance, R. L. Dewar, R. C. Grimm, S. C. Jardin, J. L. Johnson, and D. A. Monticello Princeton University

C3 Critical Bias Fields for Tilting Stability in the Beta-II Experiment 60 H. E. Dalhed Lawrence Livermore National Laboratory

C4 External Tilting of Spheromaks with Line Tying 64 J. M. Finn, NRL, and A. Reiman, University of Maryland

C5 Turbulent Relaxation of Compact Torolds 68 E. Hameiri, New York University, and J. H. Hammer Lawrence Livermore National Laboratory

C6 Ideal MHD Tilting Modes for Arbitrary Plasma Pressure and Configuration 7 2 J. H. Hammer Lawrence Livermore National Laboratory

C7 Simulation of the Formation of the Princeton Spheromak 7 6 H. C. Lui, C. K. Chu, and A. Aydemir Columbia University

C8 Confinement Requirements for Ohmic-Compressive Ignition of a Spheromak Plasma 80 R. Olson, J. Gilligan, and G. Miley University of Illinois C9 A Computational Compact Torus Experiment 82 B. McNamara, J. L. Eddleman, J. K. Nash, J. W. Shearer, and W. C- Turner Lawrence Livermore National Laboratory

CIO Formation and Evolution of the PS-1 Spheromak 85 A. G. Sgro, LASL, H. C. Lui and C. K- Chu, Columbia University, and D. Winske, University of Maryland

Cll Reversej-Field Mirror Transport Code 89 D. E. Shomaker, J. K. Boyd, S. P. Auerbach, and B. McNamara Lawrence Livermore National Laboratory

C12 Tilting-Mode-Stable Spherotnak Configurations 93 K. Yama7,aki Princeton University

0. Spheromak Experiment

01 PS-1 Spheromak Experiment 9 7 H. Bruhns, Y. P. Chong, G. C. Goldenbaum, G. W. Hart, and R. A. Hess University of Maryland

D2 Physical Properties of Compact Toroids Generated by a Coaxial Source 101 I. Henins, H. W. Hoida, T. R. Jarboe, R. K. Linford, J. Marshall, K. F. McKenna, D. A. Platts, and A. R. Sherwood Los Alamos Scientific Laboratory

D3 A Cojipact Torus Configuration Generated by a Rotating Magnetic Field: the Rotamak 105 W. N. Hugrass, I. R. Jones, and K. F- McKenna The Flinders University of South Australia

D4 Experiments of Spheromak and Reversed Field Configuration in 2M Theta Pinch 109 Y. Nogl, S. Shimamura, H. Ogura, Y. Osanai, K. Saito, S. Shiiaa, and H. Yoshimura Nihon University, Japan

D5 Formation of Compact Toroidal Plasmas by Magnetized Coaxial Plasma Gun Injection into an Oblate Flux Conserver 113 W. C- Turner, G. C. Goldenbaum, E. H. A. Granneman, C. W. Hartman, D. S. Prono, J. Taska, LLNL, and A. C. Smith, PG and E -xi-

D6 Hydrodynamic Confinement of Thermonuclear Plasmas TRISOPS IIX 119 D. R. Wells, P. Ziajka, J. L. Tunstall University of Miami

D7 Formation of the Spheromak Plasma by a Slow Magnetic Induction Scheme 124 M. Yamada, H. P. Furth, W. Heidbrink, A. Janos, S. Jardin, M. Okabayashi, E. Salberta, J. Sinnis, and F. Wysocki Princeton University

E. FRC Theory

El Ballooning Mode Growth Rate Dependence on Separatrix Shape for Idealized Equilibria of a Field Reversed Theta Pinch .... 130 D. V. Anderson, LLNL, H. L. Berk, University of Texas, J. H. Hammer, LLNL

E2 Linear and Non-linear Computations of the Ideal MHD Tilting Mode in the FRX-B Configuration 134 D. C. Barnes and A. Y. Aydemir, University of Texas, D. V. Anderson and A. I. Shestakov, LLNL, D. D. Schnack, LASL

E3 Two-D Transport Model for FRC Plasmas 138 R. N. Byrne, SAI, and W. Grossmann, NYU

E4 Particle Confinement in FR8P with Loss-Cone-Like Scattering i 44 Q. T. Fang and G. H. Miley University of Illinois

E5 Reconnection During the Implosion Phasa of Field-Reversed Configurations 148 D. W. Hewett and C. E. Seyler Los Alamos Scientific Laboratory

E6 One-Dimensional Transport Modeling of Field Reversed Experiment 152 S. Hamasaki and N. A. Krall JAYCOR

E7 Transport, Stability, and Reactor Aspects of & Hill's Vortex Field-Reversed Geometry 156 T. Kammash University of Michigan

E8 Two-Dimensional MHD Simulation of Field- Reversed Plasma Formation 161 R. D. Milroy and J. U. Brackbill Mathematical Sciences Northwest, Inc. E9 Field Line Reconnection at thp End of a Field Reversed Theta Pinch 165 Z. A. Pietrzyk University of Washington

E10 Flux Loss During the Formation of FRC 169 A. G. Sgro Los Alamos Scientific Laboratory

Ell Scaling Laws for FRC Formation and Prediction of FRX-C Parameters 172 R. E. Siemon and R. R. Bartsch Los Alamos Scientific Laboratory

ElL Axial Shock Heating of Field-Reversed Plasmas 176 L. C. Steinhauer and A. L. Hofiinan Mathematical Sciences Northwest, Inc.

F. FRC Experiment

Fl FRC Studies on FRX-B 180 W. T. Armstrong, J. C. Cochrane, J. Lipson, R. K. Linford, K. F. McKenna, A. G. Sgro, E. G. Sherwood, R. E. Siemon, and M. Tuszewski Los Alamos Scientific Laboratory

F2 The Initial Ionization Stage of FRC Formation 184 R. J. Connaisso, W. T. Armstrong, J. C Cochrane, C. A. Ekdahl, R. K. Linford, J. Lipson, E. G. Sherwood, R. E. Siemon, and M. Tuszewski Los Alamos Scientific Laboratory

F3 Trapping of Reversed Bias Flux in a Fast-Rising Theta Pinch • 188 S. 0. Knox, H. Meuth, F. L. Ribe, and E. Sevillano University of Washington

F4 A keV Compact Toroidal Plasma 192 S. Chi, S. Okada, M. Tanjo, Y- Ito, T. Ishimura, and H. Ito Osaka University, Japan

G. General

Gl Experimental Studies of Low-Level Oscillations in a Globally-Stable EXTRAP Field Reversed Configuration 196 J. R. Drake Royal Institute of Technology, Sweden

G2 The Bumpy Z-Pinch 200 T. H. Jensen and M. S. Chu General Atomic Co. -Xlll-

G3 The Plasmak: Its Unique Structure, the Mantle 204 P. M. Koloc and J. Ogden Prometheus II

G4 Induced Surface Current Effects on Plasma Stability in Presence of an Inhomogeneous Magnetic Vacuum Field 208 B. Lehnert Royal Institute of Technology, Sweden

Author List 2 I1. PROCEEDINGS OF THE THT/RD SYMPOSIUM ON THE PHYSICS AND TECHNOLOGY v' OF COMPACT TOROTD'3 IN THE MAGNETIC FUSION ENERGY PROGRAM

Held at tr.e Los Alamos National Laboratory Los Alamos, New Mexico December 2 — 4, 1980

Compiled by

Richard E. Simmon

ABSTRACT

This document coiaj,ns papers contr ilu!te..i bv the p-trt ic > pan t y o the Third Symposium on Physics and Technnlo^-- of Dunprict Turoids in the Magnetic Fusion Eneigy Program.

Subjects include rc.n tor aspects of I'ompait c urn. J.s, energetic particle rings, spheioraak conii^urations (a mixture <;f toroidal and poloidal fit" Ids'1, and field-reversed conf igutvit i OPS (FRC'S that contain purely poloidal field). -1-

RANDOM GAPPED WALL STABILIZATION FOR TRAVELLING MIRROR COMPRESSORS

Paul M. Bellan, Caltech, Pasadena CA 91125

A technique for producing travelling magnetic nirrori in a particularly simple way has recently been denomtrated-t Tbe scheme also provides the possi xlity of three-dimensional, constant-fieId-energy adiabatic compression of the magnetic mirror, and herce compression and heating of airror-confined reversed field configurations such as a compact torus, a reversed field theta pinch, or a reversed field ion ring. The travelling mirror is produced by a double-hump current impulse [ left,Fig.1

In order for the travelling mirror to confine reversed field configurations, unstable motion away from the symmetry axis must be prevented; i.e. the mirror must be stabilized. Conven- tionally, external conducting walls or quadrupol e' fields are used to stabilize mirrors; however for reasons given below neither of these methods is suitable for the travelling mirror compressor.

Conventional wall stabilization will not work because an azimuthally continuous wall prevents field lines of the time- dependant travelling mirror from penetrating the wall; i.e. the wall constitutes a short-circuited secondary for the solenoid- delay-lme coils. Quadrupole stabilization might work for fieId -revecsed plasmas, but would cause excessive particle losses for reversed-field ion rings-s A propagating quadrupole could, in principle, be arranged by putting loffe-1ike^wind ings electrically in series with the solenoid coils, so as to generate a propagating quadrupole field synchronous with the travelling mirror field, but this would be very cumbersome. A new stabilization 1 1 l : ; n n u r Z " * - ' " ? - i " " ' 1 i . .. , • .. .,• i , • ., :j n :, i. :i .: : will .-. .- a--

1 i- /. •' t :, f L l .1 V " •-. L . ;• r V i I j • '. • ' L i .i i u •: i n ^ a u y c> J the a Do v e r; I.- i PIE I tli 6 •? c Ii eme C .".• u 1 d i ! r o b f? '.X i o provide stabil iza-

!-_ !• •,.. t; f ; • T • ;• o: C '' -i> i' rut n n t nnptic f l •.* 1 d n , sucli as

: « ; •' , • ' ' • i • • • M • • ..; i : ' - I I- f[ 1 [ j C I d ;> P II t C I .1 " -2-

The stabilizing image current of an azimuthally con- tinuoue [c.f. Fig.3(a)], perfectly conducting vail consists of two componentr, called "static" and "dynamic" respectively. The static component ic a z imu'j hal ly symmetric (m»0), whereas the dynamic component is proportional j the displacement of the configuration off .axis and so has m-1 dependence. If the wall is slit parallel to the symmetry axis to form a gap (c.f. Fig. 3(b)] so as to allow the travelling mirror field to penetrate inside the wall, the dyviamic term is unaffected vhile the static term disappears. This is because an n-0 current cannot flow across a gap, whereas an m«l current can reconnect at the gap by having return currents flow parallel to the gap. However, because this gap is at a specific azimuthal angle, the fields associated with this simple gapped wall are not axi&ymmetric this can enhance particle losses3. If the wall is split into short axial sections, each with a gap oriented at a random azimuthal eagle [c.f. Fig.3(c)], then the fields become axisymmetric. This configuration again provides the dynamic stabilization term and again allows the travelling mirror to penetrate inside. Thus, this randomly-gapped wall provides exactly what is needed to stabilize the travelling mirror field (or any otu?r time dependant confinement field for which axisymmetry is desired). Further details and a description of a table top demonstration experiment of random gap wall stabilization are given in Ref.5.

References

1. Paul H. Bellan, Phys. Rev. Lett. 43,858(1979).

2. M. S. Ioffe and R. I. Sobolev, At. Energ. 1_7 , 366(1964) [Sov. At. Energy 1_7 ,111 2 ( 1 964) ] .

3. S. C. Luclthardt and H. H. Fleiechmann, Phys. Rev. Lett. 3_9,747( 1977) ; Ronald H. Cohen, David V. Anderson, and Carolyn B. Sharp, Phys. Rev. Lett. 4^,1305(1978).

4. D. M. Woodall, B. H. Fleischmann, and H. L. Berk, Phys. Rev. Lett. Jj4, 260(1975); J. W. Beal, M. Bret t Schneider, K. C. Chris- tofilos, R. E. Hester, W. A. Lamb, W. A. Sherwood, R. L.Spoerlein, P. S. Weiss, and R. E. Wright, 3rd Int. Conf. Plasma Physics and Controlled Fusion Research (Proc. Conf. Novosibirsk, 1969) J., IAEA, Vienna(1969), p.967.

5. Paul M. Bellan. to be published. -3-

•'t'f f f input pulse delay line circuit wove form

(orb units)

fjmoll radius (large turns no Ismail A

owl posrton (meiers)

Fit.2

(0/

fit.l Fig.l(a) Left:icput to colenoid delay line, right: circuit of •olenoid delay line; (b) coil geometry of solenoid delay line coapressor. Fig.2 Typical observed propagating, compressing mirror field on a 2.5 m long table-top compressor. Initial coil bore radius-1.25 en, final (compressed) coil bore radius»0.2 cm.

Fig.3(a) Dngapped vail; (b) simple gapped wall; (c) randomly gapped vail. -4- < >••••). NUCLEAR LLASTIC SCATTERING tFFtCrs. ON FUSION BURNS IN COMPACT TORI . ..by J. Galambqs, J. PeVeaux, E. Greenspan, G.. !l. Miley Fusion Studies Laboratory HucVear Engineering Prp,gra.Ti University of 111inois Urbana, minois 61801 'he high-8 aspect of compact-tori plasmas is a favorable characteristic for ddvanced-fuel due to reduced cyclotron losses and external magnetic-field requirements. As shown by previous work (See, for example, Rets. 1 and 2), the Nuclear Elastic Scattering (NES) probability of fusion products becomes especially significant at the high temperatures required for advanced-fuel r>:jrns. Nuclear elastic scattering reactions are expected to have both positive and negative effects on the performance of compact tori. On the positive side, these reactions increase the fraction of fusion products (fp) energy that goes to the ions and thus increase the plasma ions temperature relative \'j that of the electrons. Moreover, being associated with la^ge energy transfer, NES reactions create a population of high energy knock-on ions which 'ncreise the average fusion reactivity. On the other hand, the large energy transfer reactions, which are associated with large directional changes, are ejected to increase the rate of loss of high energy ions from the plasma '''roij-jh ;.itch-angle scattering of both the fp ^d the knock-ons) thus impair- • !,-, t>ie energy balance of the compact-tori plasmas. Another consequent of the ",: V rcic* i ons is the acceleration of t"he fusion-products' slowing down^). This work reports on preliminary results from two independent, though related studies of NES effects on characteristics of advanced-fuel compact '. "f : P) The effect of NES on the startup of a Field Reversed •• pinch FROP) Moving Plasmoid Reactor (MPR), which emphasizes the time-dependent effects of NES, and (?) Spatial effects of NES in Field Reversed Mirrors (FRM). In particular the effect of NES on fusion-product orbits and the fraction of their energy being deposited in the plasma. Time-dependent NES effects were modeled bv incorporating NES into the code PHEPUS used in previous FROP-MPR studies'3). PIOUS is a 0-D, time- dependent, niijlti specie (p, D, T, ^He, ^He, e) code using a mu Hi group structure to model the slowing down of fast ions due to Coulomb and NES collisions accounting for fusion during slowing down. The slowing down model used for this study assumes an average energy transfer in a NES event and that the background ions have a Maxwellian distribution. The enhancement of reactivity due to hardening of the background-ion spectrum and large-angle Coulomb collisions are neglected. Consequently, the results obtained using this modi'l are expected to underestimate the effects of NES reactions (and large energy transfer reactions, in general). Figure 1 illustrates thp effect of NES on the startup of a Cat-D FR9P-

•!" 'he ,:• .-. \ • , -?L'.. . • • .' .' . ' . ' • . .. nil.; i.i \ Ni- S ' ak en into account (! i <;. } > , i;;.: i iq. [ !•, • ••• • ..• • .••: [..•;. !ne startup scenario assumes heating the cold (1 t.eV) '•.••• •••:'•. v/i*:h neutral beams. The results of Fig. 1 pertain to a ISO a^'ps uf •'•: : '••••; '!fjijfei-0MS operating for 59 msec. Following the beam shut-down, the ' .'•-.• •!•",•••: ''.ms' t'?npo'"(it ure keeps incrfMSi ng for a while due, primarily, to -5-

finite slowing down tine of the beam ions. The ions temperature then drops, as the fusion density can not support it. Whereas without Nl_S effects the plasma temperature keeps decaying (Fng. lh), with TiCS we observe an initiation of a thermal runaway (Fig. la), further analysis is required in order to identify the contribution of the different NFS effects on the improved energy balance of the FR/;P-MPR considered. The particular FRf:P-MPR concept examined so far does not appear to have a promise to provide high Q values. This is due, primarily, to the large energy investment required for bringing the plasma to operating temperatures, in the neutral beam heating scheme considered, along with the relatively fast loss of plasma due to leakage (no refueling is assumed). Future work will examine compact-tori schemes promising more favorable energy balance, and the effect of MES on their performance. One of the crucial questions regarding the viability of compact-tori rusion devices deals with the amount of fusion-product (fp) energy that can be successfully regained in a burning plasma. Since fp orUfs dr^ generally large compared to the scale lengths of compact -*". ori devices such as the f'RV, it was previously believed that little fp heating could be retained, '-'ecen*. work has shown, however, that the fp dre expected, when slowed down by Coulomb collisions, to deposit most of their energy in the Yk'f< plasma. The present study is aimed at investigating the effec's nc ','•_? on fp transport and c:\pr-jy deposition. A space-dependent code i1^ being developed '.<:< vim1.;! a'»._• the transport of the 1p and their knock-ons in Fp'-i configuration. "'us code :s based on the recently developed !-'onto Carlo *P Tins:"'"'. •;od'_i '•''.!"!'•'. stochastic "'ode1 which accounts for '.;'" d • sere's..1 ri.it.uri1 • •( fht> V -; i'e :•; t :;•'•• was developed for th's Purpose. Preliminary illustrative results on the effect of ";_S on fp orhits n- j D-JHe FRM are shown in Fig. 2. Figure ?a shows a totally confined orbit r' a 14.7 MeV proton which is slowed down by Coulomb drag only. An orhit the same fusion product may have when subjected to "i[S is shown in rig. i'b. "he 'iF3 event forced this fp into an unconfined orbit, it ought !.;;• he realize-'! that due to the stochastic nature of the NF.S reactions, the orbit of rin. 2P is only one out of a distribution of orbits the same fp can have, '-'ore-'"-* e>~, "JF'S might turn unconfined orbits into confined ones, leave confined o^bit confined etc. Hgure ?z shows a totally confined orbit nf a 14.7 MoV proton which underwent two 'IE S collisions. It appears, as expected, that trie lenntn of the trajectory of this proton is shorter than the trajectory of the proton of Fig. 2a. It is expected that the slowing down time of Fin. ?c proton is also shorter than that of Fig. 2a proton. Future work will investigate the composite effect of ')! S (as well as of large-angle Coulomb scattering) on the orbits and average energy deposition of the fp and of the knock-ons in FRMs. The results to be obtained will be used as an input for a global study of the energy balance of compact t in which will account for the other NFS effects as well.

References 1. F. Greenspan, Princeton Plasma Physics Lab. Report MAT I - 123b (H7t>).

'?.. Ci. Shuy, University of Wisconsin Report, UWF!)M-357 (1O:J.O).

3. J. G. Galambos, et al., IE t [I, Paper 2A5, Madison, WI , May 19-21, 1980. -6-

0.2

TIME (sec) (a)

TIME (sec) (b)

Figure 1. Evolution of ion (T-) and electron (T ) temperatures in a FR'iP MPR with, (Fig. la), and without, (Fig. lb), ,'lf.S effects. -7-

_ 1 2 o

1—I o 0.8 a. _j I*

DI A 0.4

0 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 AXIAL POSITION Figure Za. Confined orbit of 14.7-MeV proton in FRM which is slowed down by Coulomb drag only.

-1.2 -0.8 -0.4 0 0.4 0.8 1.2 AXIAL POSITION Figure 2b. An orbit of a 14.7-MeV proton knocked out of confinement after a NES event.

-1.2 -0.8 -0.4 0 0.4 0.8 1.2 AXIAL POSITION Figure 2c. Confined orbit of 14.7-MeV proton which underwent two NES events. -8-

A COMPACT-TOROID FUSION REACTOR BASED ON THE FIELD-REVERSED THETA PINCH* R.L. Hagenson** and R.A. Krakowski, Los Alamos Scientific Laboratory ABSTRACT Early scoping studies based on approximate, analytic models have been extended! 1) on the basis of a dynamic plasma model and an overall systems approach to examine a Compact Toroid (CTOR) reactor embodiment that uses a Field-Reversed Theta Pinch as a plasma source. The field-reversed plasmoid would be formed and compressionally heated to ignition prior to injection into and translation through a linear burn chamber, thereby removing the high-techrology plasmoid source from the hostile reactor environment. Stabilization of the field-reversed plasmoid would be provided by a passive conducting shell located outside the high-temperature blanket but within the low-field superconducting magnets and associated radiation shielding. On the basis of this batch-burn but thermally steady-state approach a reactor concept emerges with a length below ~ 40 m that generates 300-400 MWe of net electrical power with a recirculating power fraction less than 0.15.

K _P LASMA ENGINEERING AND CTOR DESIGN P01 NT The CTOR would use a Field-Reversed Theta Pinch (FReP) "to produce external to the reactor an FRC plasmoid that is subsequently heated and translated through a linear burn chamber. The high-voltage plasmoid source and compres- siondl heater are removed from the burn chamber to a less hostile environment. The stabilizing conducting shell would be positioned between the blanket and shield. Translation of the ignited plasmoid, shown schematically on Fig. 1, allows portions of the conducting shell that have not experienced flux diffusion f.n be continually "exposed". A nearly (thermal) steady-state operation of the first wall and blanket is possible for appropriate piasmoid speeds and injection rates. Locating the stabilizing conduction shell outside the blanket permits room temperature operation and minimizes the translational power, which appears as joule losses in the exo-blanket shell; these losses can be supplied directly by alpha-particle heating through modest radial expansion of the plasmoid inside a slightly flared conducting shell, blanket and first wall. Transiational runaway is prevented by the presence of a thin (~1 mm) first wall "shell" that is highly permeable to magnetic flux penetration but which nevertheless stabilizes the linear motion. Superconducting coils are located outside the blanket, conducting shell and shield to provide a continuous bias field that is compressed between the conducting shell and the plasmoid; MHD stability would thereby be provided throughout the burn without invoking active feedback stabilization. .The plasma simulation code used to model the CTOR is based on a three-particle, time-dependent "point-plasma" model that incorporates an analytical equilibrium expression(2), allowing three-dimensional spatial variations to be followed in time. Starting with the post-implosion (FR6P) phase, the plasma trajectory is followed through the tapered compression chamber into the burn section where conducting shell losses (translational drag) are supplied by radial plasma expansion that in turn is driven by alpha-particle heating.

•Work performed under the auspices of the U.S. Department of Energy. **Science Applications, Inc., Aiies, Iowa. -9-

w, • 6.180 MJ (-SUPERCONDUCTING COILS"

SHIELD REGION ^CONDUCTING SHELL

£ 1 1 ^ \ . PLASMA t \oPFN ON SEPARATRIX flELD CLOSED LINE FIELD LINES

Fig.1.Compact toroid geometry show- Fig.2.Energy balance for the com- ing radius of conduction shell, rc- pact toroid reactor (CTOR), includes first wall, rw, separatrix, rs and the FR6P capacitor bank, WBANK* homo- plasmoid length, £. This FRC plas- polar motor/generator, VJrjQMP> aux~ moid would be stabilized by a pass- iliary, I.'AUX' total thermal, WJH, ively conducting shell of current gross electric, Wjr-r, net electric, and radius rc, and thickness & that is VJp circulating energy, WQ. located outside a breeding blanket of thickness Ab=0.5m.

Referring to Fig.l the required radius of the conducting shell, rc, which is positioned outside a Ab=0.5 m thick blanket, and the plasmoid length, .?, are defined by experimental results (xs = rs/rc > 0.5 and £/rc >3.5). In addition to the plasma burn dynamics, an overall energy balance is performed (see Fig. 2; along with a spatial calculation of thermal and structural response of the first wal1. Parameter studies using the plasma simulation code were performed for a range of plasmoid radii, reactor lengths, plasma densities and confinement time scalings. A plasmoid is produced by the FRGP at 1.6 keV and is subsequently compressed to 8 keV in 0.1 s, requiring a radial compression factor of ~2.9 and an axial reduction of ~1.9. The ignited plasmoid enters the burn chamber with an initial plasmoid velocity equal to 2-5 times £/TS, where the electrical skin time of the stabilizing shell, TS, describes the decay of magnetic flux within the annular area between the first wall and the plasma separatrix. The velocity of the plasmoid is reduced during the translation by tailoring the flare of the burn chamber in order to maintain a constant first-wall neutron current along the burn chamber. The plasmoid velocity varies approximately as v = Pa/i"w, where Pa(W) is the instantaneous alpha-particle power. Motion proceeds until V/(£/TS) < 1 at which time the translation is terminated and plasma expansion/quench occurs. Results from a typical burn trajectory are shown in Fig.3 for an energy confinement equal to 200 Bohm times. This energy loss rate is extrapolated from tokamak experiments(l). The plasmoid is assumed to lose no particles during its 1-2 s trajectory down the linear burn chamber. In effect, if the -10-

particle confinement time is on the order of TC, complete particle recycle with a cold ga3 blanket is assumed. A thermally-stable burn results at a nearly optimal temperature (see Fig. 3A) of T^ - 10-14 keV, achieving a fuel burnup of fo - 0.17 in Tg= 1.96 s for this sample case. The burn is terminated as fuel depletion, alpha-particle buildup and plasmoid expansion result in losses that ultimately overcome alpha-particle heating. The taper required of the conducting shell to overcome transiational drag (Joule) losses in the stabilizing shell is shown in Fig. 3C for both an actual scale model and an exaggerated scale, the latter better illustrating radial variations. The first wall radius increases from 1.2 m to 1.64 m over a total reactor (burn section) length of 40m for a conducting shell thickness of 6 = 0.05 m. Spec- ifying the first-wall neutron loading to be uniform over the reactor length requires the plasmoid velocity to decrease from 38 m/s to 10 m/s at the outlet (Fig.33) where the ratio rv, of actual velocity to minimum allowed velocity (2/TS) is also plotted. The reactor length traversed, as defined by the trail- ing edge of the FRC, is also given in Fig. 3B.

200 BOHM CASE

FHSP COMPRESSOR REACTOR BURN SOURCE CHAMBER

Fig. 3. Sample CTOR plasma burn. -11-

The energy yields per FRC burn are shown in Fig. 2 along with the system energy flows required to evaluate the engineering Q-value, Qr. The system power is specified by choosing the injection time.Tj. Taking ij = 5.8s to give a 14.1 MeV neutron wall loading of 2 MW/m^, a therraa] output of 1050 MWt results with a net electric power of 310 Mwe for Q^ = 6.8 and njH = 0.35. II. CONCLUSIONS Generally, the CTOR is represented as a high-Q system of modest size. The pulsed energy storage requirements are only -60 MJ of capacitive energy for the FR9P and a 175 MJ homopolar generator Energy recovery is achieved in the quench region without the use of opening switches, with the plasma motion providing the necessary switching characteristics. The high-voltage and active source elements have been completely removed from the totally passive burn section. The linear system configuration simplifies maintenance and construction procedures. A natural divertor is also presented by the open- field line geometry outside the separatrix. The realization of this attractive system is contingent upon the transport properties assumed for the plasma. Systems with high losses (T£ ~ 0.1 s for the design plant) will either require higher operating densities (leading to higher first-wall thermal cycle) or systems of larger radial dimensions. Since the size o^ the source requirements increase as ~ nr^, larger pulsed power requirements are imposed. Particle transport may also have adverse effects on the burn cycle. The batch burn system used here assumes little change in the particle inventory, during the ~ 2 s burn. Particle loss is likely to occur along with iniestion of gas streaming to the plasmoid from the quench region. The competition of these two processes will determine the time-dependent particle inventory, a process that requires more detailed model ing. REFERENCES 1. R.L. Hagenson and R.A. Krakowski, "A Compact-Toroid Fusion Reactor Based on the Field-Reversed Theta Pinch: Reactor Scaling and Optimization for CTOR," ANS Topical on the Technology of Controlled Nuclear Fusion, King of Prussia, PA (1980). 2. W.T. Armstrong, R.K. Linford, J.Lipson, D.A. Platts and E.G. Sherwood, "Field Reversed Experiments (FRX) on Compact Toroids," submitted to Physics of Fluids (1980). -12-

THE MOVING RING FIELD-REVERSED MIRROR PROTOTYPE REACTOR*

A. C. Smith, Jr., Pacific Gas and Electric Company, San Francisco, CA 94106 G. A. Carlson, Lawrence Livermore National Laboratory, Livermore. CA 94550 H. H. Fleischmann, Cornell University, Ithaca, NY 14853 W. Grossman, Jr., New York University, New York, NY 10012 T. Kammash, University of Michigan, Ann Arbor, MI 48109 K. 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 Reactor1 produces electric power by burning fusion fuel in magnetically field-reversed plasma rings. These rings move continuously through a straight, cylindrical reactor burner chamber. The concept has the potential to be relatively simple and might be amenable to small unit size (50-100 MW(e)). The objective of our project is to design a prototype fusion reactor based on magnetic field-reversed ("FRM") plasma confinement. "Prototype" means ar- intermediate step between between an experimental pilot plant and a lead commercial plant. We are ;eeking a design that shows promise for upgrade for commercial use. Therefore, a result of this work will be a set of physics, technology, and mechanical design criteria needed to make this concept attractive. These criteria can then be used as targets for experiments and technology programs. Six major criteria guide the commercial prototype design. The prototype > 7TO of p must: (1) Produce net electricity decisively (Pnet qros,sj' ^ Scale to an economical commercial plant and have small physical size; (3) Have all features required of a commercial upgrade plant (^H breeding, etc.); (4) Minimize exotic technology and maintenance complexity; (5) Promise significantly lower safety hazards than fission plants (environmentally and socially acceptable); and (6) Be modular in design to permit repetitive production of components. We plan to perform the prototype design in three iterative steps. This paper summarizes tha results of the first iteration, shown in Figure 1. Ring Generator and Compressor: A hollow, coaxial plasma gun generates the field-reversed plasma ring in the relatively low magnetic field (= 0.26 T) just beyond the end of the burner. This is similar to the start-up for the beam-sustained FRM2. The hole through the inner gun electrode permits uninhibited passage of the escaping plasma stream to direct convertors located behind the gun. A local "moving mirror" provided by sequentially energizing "push coils" located near the conical-shaped wall of the compressor rapidly forces the plasma ring into the high (6.5 T) solenoidal magnetic field of the burner section. In this way, the ring is efficiently compressed and heated to initial burn temperatures (50-75 keV). Rings of the type proposed in this reactor have been observed experimentally to "tilt" under some conditions. One of the schemes currently being considered to stabilize the tilt is to replace some of the ring's plasma current w'i'n axis-encircling particles, since this instability is not observed in purely astron-like experimental configurations. Therefore, the current reactor desigr includes an option to "hybridize" the plasma ring with axis-encircling particles prior to the compression heating3. Whether such a scheme will work--and ••••»'-" it is actually required in the reactor--awaits exrrnmental proof. -13-

PROTOTYPE MOVING-RING FRM REACTOR

-DIRECT CONVERTOR AND PULSED PARTICLE CRYOGENIC VACUUM PUMPS 'INJECTOR FLUX CIRCULARIZING ,,-' COIL (2) , \ ^FLUXCONSERVFH PLASMA RING (4 SHOWN) PLASMA- LOBEC ALUMINUM FIRST WALL GUN LOW-ACTIVATION TRITIUM BREEDING BLANKET

SOLENOID — M^LTIPOLE MAGNETS (12)

COLD FUEL INJECTORS us)

RADIATION SHIELD

F1CXD TRIM COIL (21 "I MAGNETIC COMPRESSOR COILS (20) -MAGNETIC EXPANDEh PULSED PARTICLE COILS INJECTOR

••• 1

REACTOR MODULE (TYPICAL OF 6) Length — 53m Width — 10 0 m DIRECT CONVERTOR AND CRYOGENIC VACUUM PUMPS- Figure 1.

The r.ardware to provide the high energy particles is shown schematically in Figure 1. If protons are used, ~ 200-500 kJ bean's of ~ 12-15 MeV particles would be needed if the axis-encircling particles must carry ~ 1/3 - 1/2 of the total field-reversing current. The gun plasma combines with the particle ring in the space between the gun muzzle and the first compression coil. The details of how these rings might combine have not yet been explored. Compression scaling laws from M.HD field-reversed equilibrium codes4 were used to design the compressor. The main scaling laws, assuming no toroidal a 1 field in the plasma ring, were T.,- <* B and rn-nq B" ^. The current design calls for - 20 coils in the compressor. The coil radii range from ~ 50 cm near the throat of the burner section to - 250 cm near the plasma gun. If we assume the initial plasma temperature to be = 3 keV, a field compression ratio of - 25 is required to bring the plasma to initial ion temperatures of 75 keV. Slotted conducting walls are located just outside the compressor coils. These walls stabilize the plasma ring dynamically during compression aginst the precessional/radisl instability^. Current estimates are that the ring compression requires = 2-5 tns. Pulsed currents in the compressor coils range from 0.5 - 2.8 MA. The overall compression efficiency (plasma energy)/(total compressor energy) is calculated to be = 851. An efficiency of 751 is used to compute reactor power flows. Burner Section: Plasma Considerations. A 0-D, non-thermal, multi-species Fokker-Plancktransient plasma burn analysis6 has been used to model the -14-

fusion plasma. Results from ongoing plasma equilibrium and stability calculations will improve the current burn model assumptions on density profiles, diffusion processes, heat transfer, and permissible plasma siz°. We have investigated criteria for plasma ignition using a variety of ad hoc particle and energy confinement assumptions scaled from classical ion energy transport T^ ~ [a/]2x.. where "a" is the plasma minor radius, is the species-averaged ion gyroradius, and T^ is the ion-ion collision time. We have usually assumed that the particle confinement time is equal to the ion energy confinement time and that the electrons have an energy confinement tine that is one-tenth of the ion energy confinement time. This design uses ignited plasmas with T^t^) = 2Tg(t=0) = 75 keV. The initial plasma minor radius is 16 ion gyroradii across. (At these temperatures, the ring minor radius must be at least 12 ion gyroradii across for ignition under our current energy and particle confinement models.) The plasma does not have a toroidal magnetic field and is assumed to have an average beta of 1.3. (The assumption that plasma rings of this size are stable without an internal toroidal magnetic field may be optimistic.) The plasma attains a total fusion plasma Q of 40 in 1.6 s, growing from an initial major radius of 31 cm to a final major radius of 53 cm (Figure 2). The plasma ring is assumed to be flux-conserving. The initial and ond-of-burn ring field-reversal strengths are 5.2 and 2.0, respectively. The final ring current is 12.4 MA. The ignited ring is refueled by laser-driven DT pellets during its transit of the burner. The code simulates the periodic pellet refueling process with eighteen pulses of "cold" plasma (3 keV deuterons, 5 keV tritons). The refueling pulses are spaced 83 ms apart and have a pulsewidth of 5 ms. The refueling rate during each pulse is regulated to maintain a nearly constant total ring radiated power of = 320 MW (Figure 3).

H ; < I .! f i i i0 - £ I

PLASMA RADIUS | TOTAL RADIATED POWER PER RING , | TOTAL DEUTERONS (OR TRITONS) PER RING j

Figure 2. Figure 3.

The rings are pushed through the burner at a constant speed of 15 m-sec"1 by an axial gradient in the burner magnetic field. The gradient was tailored so that the yVB driving force balances the retarding force from the eddy currents induced in the conducting aluminum wall by the moving plasma rings7. -15-

The first wall is segmented into electrically-insulated slices, or "lobes". This decreases its magnetic field penetration time to = £5 ms. We found that a field gradient averaging 0.021 T-m"1 matched the axial drag force within 15% et four selected positions in the burner. The relative error arises from the mechanics of the force calculation; in practice the two forces will balance exactly. The axial drag fcrce calculated for our geometry range' from 0.09 MNt at the start of the burn to 0.25 MNt at the end of the burn. The total energy dissipated by ohmic heating in the wall is approximately 3.9 MJ. This energy comes from the work done by the plasma ring as it expands against the decreasing field during the burn. The plasma rings attract one another electromagnetically. Without active mirror coils to keep the rings separated, we avoid an axial "bunching" instability through the combination of (i) a flux-conserving wall with an average radius of 2.8 m and a radial thickness of 5 cm located just behind the breeding zone, and (ii) a ring-to-ring spacing of 8 m. The flux conserver diminishes the magnetic field from one ring as viewed from the position of its nearest neighbors. It also decreases the axial wall drag. (All the drag calculations took the flux-conserving wall into account.) A stability analysis showed that a ring-to-ring spacing of 8 m permitted a E>% variation in strength from one ring to the next. This inter-ring distance prevented the two rings from bunching together before the leading ring's burn was completed. Stabilization against the bunching mode also means that the rings must be well on the "stable" side of the axial wall drag force cu. v^ i.e. where axial drag increases linearly with ring speed. This requirement places an upper limit on the product of ring speed, first wall thickness, and first wall resistivity7. Burner Section: First Wall, Blanket, and Shield. High-purity aluminum alloy T6063) is used for the first wall and blanket structural material to minimize the induced radioactivity in these regions. Dose rates two weeks after shutdown are 3-4 orders of magnitude lower than for comparable steel structures. Further advantages expected ot aluminum are high thermal conductivity, relatively low tritium permeability (below 400 C), lower cost, ease of fabr' tion, and ready availability. These advantages outweigh the design complications (such as insulation and water coolant channels) needed to keep the aluminum below 120 C during reactor operation. The lobe walls are 1.1 cm thick. The average inner radius forming the first wall is 2 m. The average wall load is 2.75 MW-nf2. The LioO breeding material is housed in non-load bearing tubes of SiC. SiC is stable under neutron irradiation, has high thermal conductivity, and has low tritium permeability at temperatures below 1000 C. A high pressure (28 at.m) helium stream removes heat from the blanket; an independent helium loop flows through the SiC tubes to purge the bred tritium from the LipO. Transient heat load calculations show the maximum temperature variation of the aluminum first wall (caused by the moving plasma rings) is < 5 C. This induces a negligible thermal stress of < 4.5 MPa in the material. The maximum temperature variation in the SiC tubes is < 0.5 C, also an acceptable level. The Li"20/SiC breeding zone is 60 cm thick with a 10 cm graphite reflec- tor. The total blanket energy multiplication is 1.05, lower than conventional metallic blankets because of the reduced exothermic (n.y) reaction in the low- Z ceramic material. With the shield removed, the dose rates at the back of the blanket two weeks after shutdown are about 0.2 and 8 mrem/hr for one day's and two years' operation, respectively. Dose rates this low allow "hands-on" maintenance at the outer region of the blanket and throughout the shield and magnets. -16-

Contact maintenance ir.side the vacuum chamber is not possible. Magnetic Field Design. The graded axial guide field is produced by a set of 14 NbTi superconducting coils. Both a stabilizing radial field and an axial guide field is produced by adding a set of eight alternating bends to a simple circular coil to form a shape dubbed the !'cleric collar" coil. The square superconductor current bundles are 0.68 m on a side and have an average radius of 4.3 m; the radial well depth from the 1.7 m longitudinal segments in the coils is 0.03% at a radius of 0.5 m. The elliptical magnetic flux bundles leaving the reactor are spread and circularized by auxilliary superconducting coils at either end. The total electric power required to refrigerate the liquid nitrogen and helium for the magnets is about 2.8 MW. Vacuum System. Arrays of liquid-helium-cooled cryopanels vacuum-pump the reactor\ Th~ese~panels, located in the end tanks, pump both helium as well as deuterium and tritium. The hydrogen isotopes are cryocondensed in the usual way. The helium is cryotrapped in argon sprayed directly onto the cryopanels. Such a pump, built for the TSTA, has successfully been tested8. The total cryopanel areas required are 30 and 300 m2 at the injector and exhaust ends, respectively. At any one time, two-thirds of the cryopanels are pumping while the remaining third are defrosting. The total electric power to refrigerate tne liquid helium and liquid nitrogen for the vacuum pumps is approximately 1.6 MW. Refueling. Refueling 75 keV plasma rings with densities = 10lb cm"3 requires a pel let velocity of = 10 cm-sec"1. The pellets are ablatively accelerated with CO? lasers and have radii of = 0.09 cm. Five pellets are injected in each refueling "burst"; the total refueling power (18 injectors) is estimated to be = 2.5 MW. Maintenance. The burner consists of six modules, each of which is 5.3 m long. A module is removed from the burner by deflating annular "omega" pressure seals located on each end of the module. The modules are made "portable" by pressurizing an air-cushion located in the base of the module block. Dowl-pins in the floor position and help secure the module blocks. The first wall and blanket may be slid out from the shield's interior when the module has b?en winched to the containment building service area for repai rs. Power Conversion System. Electricity is produced from the reactor's thermal output By a conventional steam cycle power conversion system very similar to that used in the HTGR. Hot, high pressure helium (686 C, 28 atm) leaves the reactor blanket and is ducted to a steam generator inside the containment. A 350 MW(e) steam turbogenerator, located in a turbine building outside the containment, converts the plant's thermal power with a cycle efficiency of 35%. Figure 4 shows a summary power flow diagram for this reactor design. High-energy beams, if required, would raise the recirculating power from = 13% to = 16%.

Reactor Design Summary. This first design meets most of the six major criteria for the project. Tfie low-activation aluminum first wall and SiC/LioO breeding blanket strongly influenced the design. This choice permits hands-on reactor maintenance outside the breeding zone. It also results in relatively little long-lived radioactive waste to be disposed of. The net-to-gross power ratio of - 87% is very good. The reactor uses common materials. It is modular for ease of maintenance. The tritium breeding ratio, 1.05, is adequate. The axis-encircling beams, if needed, complicate—but do not funda- mentally alter--the design. -17-

Many of the plasma physics assumptions required for this prototype design to be attractive await experimental confirmation. For example, we have assumed that plasma rings with L a.^p^ = 16-23 are MHD stable without • v. an imbedded toroidal magnetic field. The tilt, for another, must also be resolved. These issues will be considered in our designs as new information becomes available.

Choices made in the course or this design iteration lea to a much larger size than we want for a prototype. One of the objectives for the remaining two design iterations will be to try to bring the size of the prototype down to the 50 - 100 MW(e) [net] range with significant reduction in physical size.

4 . MRFRMR Power Mews. (3 rings; powers in MW)

}A. C. Smith, Jr., et al., Preliminary Conceptual Design of the Moving Ring Fi_e 1 d-Reversed Mirror Reactor, PGandE Company Report 78FUS-1~(1978) ~ ^."A. Carlson, et al., Field-Reversed Mirror Pilot Reactor, LLNL Report UCID 18550, February, 1980. JH. H. Fleischmann, et al., Joint U.S./Japan Symposium on Compact Toroids, Princeton, NJ (19791 UW. Grossman, Jr., et al., (to be published). 5P. M. Bellan, California Institute of Technology, Pasadena, CA, private communication (1980). bT. Chu and A. C. Smith, Jr., "RFOT: A Fokker-Planck Code for Pulsed Fusion Plasma Analysis", LLNL Report (in press). 7D. J. Rej, et al., "Resistive Wall Interaction of Axially Moving Field- Reversed E-Layers or Plasma Rings" (to be published in J. App. Phys., Oc- tober, 1980). 8T. H. Batzer, et at•, "A TSTA Compound Cryopump", LLNL Report UCRL 84456, August, 1980.

*This work is being performed for the Electric Power Research Institute under contract No. RP922. -18-

A I.[XL'S CT FUSION REACTOR BASKD OX AX ISYMMLTRIC IMPLOSION OF TANCFXTTAI.I.Y-IXJF.CTFD LIQUID MF.TAL I1. !. TURCIU* A. L. COOF'F.R** D. .J . JL'XKIXS,** AM) h. \>. SL'ANNLLL

In the NRL LINUS concept for fusion reactors, a liquid metal hollow cylinder, or "liner", is formed and stabilized against the growth of Rayleigh-Taylor modes, by tangentially injecting the liner or directly rotating the cylindrical pressure vessel. The rotating liner is then imploded by high pressure gas acting .on pistois in contact with the liner outer surface. During the imoiosion, a closed-field D-T plasma is created either in situ, or formed externally, and then transported into the hollow liner. The plasmoid is compressed to a final state where a thermonuclear burn occurs during the short dwell time at peak compression. A schematic design is shown of a LINUS fusion power reactor system in fig. 1, based on axisymmetric (annular) piston implosion of tangentially injected liners. The reactor consists of two oppositely-directed annular pistons driven by high pressure helium ana displacing liquid metal both radially and axially. The pistons are arranged to act in a pilot-valve fashion, sealing the drive-gas reservoir or releasing drive-gas to act on the full piston in response to evacuation or pressurization, respectively, of the small volume initially (and finally) just behind the piston (as indicated by the double-headed arrow). The liquid metal is formed continually into a cylindrical liner by tangential injection at the periphery of the liner volume and by axial extraction near the inner surface. For illustrative purposes, a compact toroid plasma is shown injected through an endwall port by a theta-pinch (gun/guidefield) arrangement. A port in the opposite endwall is provided for evacuation of the implosion chamber. The angle of the duct channelling the liner flow and the angle of the piston faces are arranged to provide sufficient axial speed both to follow the axial contraction of the compact toroid and to allow the liner material to return radially beyond the radius of the port before reaching the endwall of the implosion chamber. -19-

The radial and axial compression of the compact toroid increases the plasma temperature and density resulting in a rapid increase in neutron production rate near the time of minimum liner radius. At this time (turn-around), the plasma is surrounded almost completely by a thick layer of liquid metal. Neutrons from the plasma deposit essentially all of their energy in the liquid liner, so the permanent structure of the reactor is shielded from high energy neutron irradiation. By using lithium-bearing liner material, tritium can be produced in the liner itself, without the requirement for an additional blanket (and the consequent need for a structural interface exposed to high energy neutrons). Tritium is then recovered by chemical processing of the circulating liquid liner flow. Energy is provided as heat by neutron deposition and nuclear reactions in the liner, resistive dissipation daring magnetic flux compression, plasma radiation, and viscous dissipation associated with liner motion. This heat is recovered by circulation of the liner material through heat exchangers and is converted to work by an appropriate thermodynamic cycle. A portion of the work obtained in this way is used by the pumps required to circulate the liner material, by the power system for plasmoid generation and transport, and by systems for vacuum, tritium handling, etc. Some power might also be needed to re-establish the helium driver-gas energy. In principle, however, sufficient additional energy should be obtained from the pressure of fusion alpha-particles on the re-expanding inner surface of the liner, to restore the pressure and energy of the helium drive-gas reservoirs directly by the return motion of the drive pistons. In this way, a portion of the total nuclear energy produced is directly converted to work, allowing operation of a LINUS reactor at reduced Q-values. Such a reduction in Q- values results in smaller reactor dimensions, lower drive-pressure requirements, and more attractive (i.e., lower) net output powers. -20-

rhe special features of the present design include: 1. Use of only two major moving parts and simple sections which improves mechanical reliability. 2. Tangential injection which eliminates need for rotary seals and bearings. The liner in a sense acts as the bearing fluid. 3. Axial motion of liner material which allows liner energy to follow the contracting plasmoid (i.e., a quasi-spherical imploding blanket), permits use of simple ports for plasma injection, and reduces the neutron flux to the end sections. 4. Axial convergence which reduces water hammer effects and pulsed pressure loadings. 5- Simple larges ports which provide high conductance for vacuum pumping. 6. Use of puffed gas and plasma injection which reduces problems with background neutrals and provides convection of magnetic flux into the implosion chamber.

Utilizing scaling relationships obtained from previous experimental and numerical results, a self-consistent set of operating parameters for a LINUS reactor as shown in Fig. 1 is given in Table I. It should, of course, be noted that the values given in Table I are representative and cannot be considered definitive until certain aspects of the system, both in plasma physics and liner technology are investigated more thoroughly. These items are discussed in Ref. 1, while the NRL research effort has been reviewed in Refs. 2 and 3, with the more recent results in Refs. 4-8. -21-

References

1. Turchi, P.J., A.L. Cooper, D.J. Jenkins, and E.P. Scannell, "A LINUS Fusion Reactor Design Based on Axisymmetric Implosion of Tangentially-Injected Liquid Metal," NRL Memo Report 4388, Sept 1980. 2. Turchi, P.J., A. L. Cooper, R.D. Ford, D.J. Jenkins, and R.L. Burton, "Review of the NRL Liner Implosion Program," in Megagauss Physics and Technology, P.J. Turchi, ed., Plenum Press, N.Y., p. 375 C980). 3. Turchi, P.J., R.L. Burton, A.L. Cooper, R.D. Ford, D.J. Jenkins, J. Cameron, and R. Lanham, "Development of Imploding Liner Systems for the NRL LINUS Program," NRL Memo Report 4092, Sept. 1979. 4. Turchi, P.J., et. al., Bull. Am. Phys. Soc., 24, 1038 (1979). 5. Cooper, A.L., et. al., Bull. Am. Phys. Soc., 2_4, 1038 (1979). 6. Turchi, P.J., et. al., Proc. IEEE 1980 Int'l Conf. on Plasma Science, Paper 4A3, Madison, Wise, May 1980. 7. Scannell, E.P., et. al., Bull. Am. Phys. Soc., 2J5, 832 (1980). 8. Cooper, A.L., et. al., Bull. Am. Phys. Soc., 2_5, 833 (1980).

Research Development Associates, Arlington, VA. **NTaval Research Laboratory "'JAYCOR, Inc., Alexandria, VA. T

ENERGY OUTPUT

HEAT EXCHANGERS HEAT EXCHANGERS THERMOELECTRIC GENERATORS THERMOELECTRIC GENERATORS LINER PUMPING CHEMICAL PROCESSING ^J MKMCIRATION

ME MN HtOtCAL cc«w« M>RH» CD UTHKM IP1MY

PULSED POWER TOR INITIAL PLASMA V' MAGNET ]" MAGNET POWER r j POWER , I 1

Tig. 1 — Artist's drawing of LINUS fusion reactor luscii on two opposing nnnuUir free-pistons driving a liner of Pb-Li alloy which is circulated continuously through the nuctor. A compact lorui.j pUsma created by theia-pinch techniques i.s shown schematically. TABLE I SAMPLE LINUS REACTOR DESIGN

DESICN CHOICES

Liner Material: Pb-Li , CALCULATED PERFORMANCE: Compression Ratio: a a 10 T Output: Thermal Power: ?H " 1790 Xls (H) @ V = 1 rlz Compressed Plasma Temperature: T ra 15 keV Linor Rotation Power: ?R = 19.1 MW(e) Reaction Profile Parameter; F(p) = 0.3 Liner Transport Power: P ° 2.8 NW(e) Driva Pressure: pD = 3000 psi Plasmoid Source-Power: P » 33 KW(e), (39) i

DERIVED VALUES Total Electric. Power: PT = 597 KW(e) @ 6., = 0.33

Compressed Field: B » 0.54 KG Minimum Circ. Fraction: C =9.2%, (10.2) m Operating Q-Value: Q => 1.55 Allowed Circ. Fraction: C <= 15% Initial Plasma Temperature: T. a 377 eV (446) Np.t Electric Power:: ?N = 507 MW(c) Initial Plasna Radius', r = 1,9 m o Total Reactor Radius: r => 5.1 m

Initial Plasma Length: I <= 7.8 m Cost Per kW (Thermal): Z = 148$/kV(K)

Compressed Plasma Length: X *> 3.1 m Total Cost: $ = 265 M$

Initial Plasmoid Energy: E( « 13 MJ (15.4)

Plasnoid Supply Energy: E ° 66 HJ (78) ^u^.bers in Pcirenthcscs ( ) are Corrections for Lover Compression of Plnsr.r. if Full DifCunicn is Assumed. -24-

UTILITY REQUIREMENTS FOR FUSION POWER

R. J. Vondrasek, R. N. Cherdack; Burns and Roe Inc.; Oradell, New Jersey 07649

The results described herein were obtained as part of an on-going EPRI fusion project. The project objectives are to develop a set of utility requirements for fusion power plants and to formulate a method for evaluating the relative merit of fusion concepts which incorporates consideration of these requirements. This paper addresses the utility requirements for commercial fusion identified during this project. The work was performed with the assistance of Public Service Electric & Gas, Northeast Utilities, and Dr. Robert Gross of Columbia University.

To begin, a list of utility requirements was formulated. The project team produced a preliminary list of requirements and associated factors based upon their experience. This was followed by a literature search. It was determined that utility requirements would be placed in two categories: those associated with cost-of-energy, and all other considerations which cannot be described by direct economic formulation. The second category of requirements was labeled "non-quantifiables" because of the difficulty in assigning a dollar value to their impacts. The project team had an understanding of the significance of cost-of-energy considerations, the techniques to calculate cost-of-energy and all the components that comprise it. However, it was determined that a procedure was required to identify the non-quantifiable utility requirements, their definitions, and their relative importance. Many of the non- quantif iables are factors in the cost-of-energy calculation, but also have impacts exclusive of the cost-of-energy. It is only the latter impacts which were considered in the non- quantifiable utility requirements.

Twenty-seven non-quantifiable utility requirements were identified. They were grouped into four categories: (1) Utility Planning and Finance; (2) Safety, Siting and Licensing; (3) Utility Operations; and (4) Manufacturing and Resources. In defining the utility requirements, the project team was influenced by industry experience and practice, projected future utility industry needs, an awareness of "Three-Mile Island" implications, and projected expectations of the future social and industrial environment. Fusion options are anticipated to be licensed by the NRC or its future equivalent, and hence definitions were heavily influenced by current experience with light water reactors. This was particularly true in the areas of safety and licensing. Since fusion plants are generally per- ceived to be base loaded, current utility practice concerning the operation of central station units also affected definitions of the requirements. -25-

The method used to obtain the opinion of the utility industry on the requirements and their weightings was to conduct a quesrion- naire survey. To accomplish this a questionnaire form was prepared. Each category of requirement questions was a self-contained package with a detailed set of instructions to facilitate responses. Each requirements was defined and, provision made for lespondents to assign "importance" weights and to write comments. The survey questionnaire participants were asked to redefine or modify requirements, and identify additional requirements as they saw fit. The survey was mailed to over 100 utilities, manufacturers and architect-engineering firms. Forty-three responses were re- ceived.

The results from the survey were considered by participants at an EPRI workshop to further develop the utility requirements. The results of the workshop were subject to final review by a com- mittee of utility industry executives.

Table 1 summarizes the results of the utility renuirements survey, the results of the EPRI workshop and finalized importance weightings of the utility requirements. The details of the utility requirement selection process, the survey statistical results, the workshop process and finalization of the detailed reauirements and their component factors will be documented in a forthcoming EPRI report. -26-

TABLE 1 UTILITY REQUIREMENTS SUMMARY

OVERALL WFIGHTING* SURVEY WORKSHOP FINAL

A. UTILITY PLANNING AND FINANCE

1. Plant Capital Cost 4.1 4** 5 2. Plant O&M and Fuel Costs 1.3 0 3 4 3. Forced Outage Rate 3.7 4 4. Planned Outage Rate 3.3 4 5. Plant Life 3.4 3 3 6. Dlant Construction Time 3.7 4 4 7. Financial Liability 3.9 4 5 8. Unit Rating 2.8 2 2

B. SAFETY, SITING AND LICENSING

1. Plant Efficiency 0.5 0.5 0 2. Plant Safety 4.3 4.3 5 3. Dependence on Other Systems 0.1 0.0 0 4. Flexibility of Siting 3.7 3.7 4 5. Waste Handling and Disposal 3.8 3.8 4 6. Decommissioning 2.7 2.7 3 7. Licensability 4.4 4.4 5 8. Weapons Proliferation 3.0 3.0 3

C. UTILITY OPERATIONS

1. Plant Operating Requirements 3.7 4 4 2. Plant Maintenance Requirements 2.6 4 4 3. Electrical Performance 3.9 4 4 4. Capability for Load Change 2.9 2 2 5. Part Load Efficiency 2.4 2 2 6, Minimum Load 2.7 2 2 7. Startup Power Requirements 2.8 3 3

D. MANUFACTURING AND RESOURCES

1. Hardware Materials Availability 3.3 4 4 2. Industrial Base 3.4 4 4 3. Natural Resource Requirements 0.4 0 0 4. Fuel and Fertile Material Available 4.1 0 4

Notes * Meaning Code Unimportant 0 Slightly Important 1 Moderately Important 2 Important 3 Very Important 4 Vital 5

**0verall weight 1 for fission/fusion -2 7-

TRACT FUSION REACTOR STUDIES H.J. Willenberg, L.C. Steinhauer, and A.L. Hoffman Mathematical Sciences Northwest, Inc. Bellevue, Washinqton 98004

A plasma concept has been icertified which appears to offer the promise of small fusion reactors based on technology which is achievable in the current decade. This concept, known as TRACT for Triqqered Reconnection, Adiabat^cally Compressed Torus, extrapolates the method of raDid axial compression of a compact torus which has been experimentally observed to produce plasmas which are very stable to maqnetohydrodynamic decay.^ TRACT plasma formation, hean'na, and stability issues are currently under active experimental and theoreti al investigation at Mathematical Sciences Northwest, Inc. (MSNW). A preJiminary conceptual design study of a TRACT pilot plant reactor has been carried out, and key technoloqy arid utility-oriented issues have been investigated.^ If the plasma stability and confinement properties scale as predicted, a pilot plant reactor which produces net electric power would be only nine meters high and six meters in diameter. A complete test reactor facility, includinq all buildings and area required for operation and maintenance would have an area of 2000 nr , of which only 13 percent needs to be heavily shielded.

The TRACT differs from conventional reversed-field theta Dinches primarily in the manner- in which the plasma is formed. Rathor than heatinq primarily by radial shock compression induced by a high voltage discharge which is the standard theta pinch approach, '^e TRACT plasma is heated largely by a strong axial compression resulting from jnetic field line reconnection. This significantly reduces the hiqn voltage requirements. Larger diameter plasmas can also be created which enhances both plasma stability and the overal", power balance. Plas- mas formed by axial compression have shown remarkable stability. Es'kov et alJ report plasmas of several hundred electron volts which are stable for time^ in excess of 100 us - the decay time of the confining magnetic field. An experiment to reproduce these results and extend them to much larqe>~ confinement times is under construction at MSNW.3 A fusion reactor which utilizes a TRACT plasma would have a high power density because of the relatively dense plasma after compression and because of the lack of inboard structures. This high power density, along with a low recirm- lating power fraction, offers the potential for small power reactors. A small TRACT test reactor operating in a utility network could deliver a nominal amount of power, yet still be small enough so that shutdowns due to the newness of the technology would not be a constraint. The successful development and demonstration of such a net power producing reactor would accelerate the acceptance of this technology and could lead to the development of more efficient and more advanced fusion reactors.^ Such a concept would enable the fusion program to proceed with a nur.ber of developmental facilities, each representing a modest extrapolation in physics and engineering over preceding facilities. A program of this kind could reduce the financial risk associated with a pilot plant power station from the billion dollar ranqe to the $100-200 million dollar range.

The attractiveness of TRACT as a viable reactor concept depends in part on the ability to heat the compact torus plasma to ignition without the need for high electric fields in the burn chamber. Heating of a TRACT plasma is accomplished primarily by axial compression which mitigates the need for a stronq -28- radial implosion driven by high £g. The reactor size is somewhat dictated by the rate of flux loss in the conducting wall.6 These issues have been investigated in detail and the results are summarized here. A global plasma formation and heatinq model, described elsewhere, has been developed.7 In a typical reactor-grade plasma example,8 only about five percent of the ignition temperature is achieved in the initial, radial shock implosion, with an additional four percent resulting from further radial compression. Axial contraction increases the plasma temperature by a factor of four, to about 3 keV. Adiabau'c compression resulting from raising the magnetic field to five Tesla raises the plasma temperature to its 8 keV ignition temperature. The conditions required to achieve ignition in a small fusion reactor have been investigated with the global plasma formation and heating model. These are summarized in Figure 1 for a plasma chamber with a five meter length and a one meter diameter, enclosed by a conducting coil with a 1.06 m diameter. The parameter 3b/B* measures the degree to which poloidal flux is retained during the reconnection process. B* is the maximum field that can be retained if plasma-wall contact occurs during field reversal,9 Ja B*(T) = 0.19 E^ (kV/cm)[AiPo(mTorr)] Es'kov et al. report almost complete flux retention can be achieved for Bb/B* = 2 when pulsed barrier fields are utilized to prevent such wall contact. The parameter Eg in Figure 1 is the peak electric field required to drive the radial implosion. Figure 1 indicates the required fill pressure to achieve cin 8 keV temperature showing the effect of different EQ and Bfc,/B*. Evidently, reduced Eg can be made up either by reducing the fill pressure (while holding Bb/B* fixed) or, if allowed by physics considerations, increasing Bb/B* (while holding p0 fixed). The penalty for the former is reduced plasma size (with reduced confinement time and fusion power output) and for the latter the penalty is increased capacitor bank size (with increased capital cost). The TRACT reactor utilizes a compound magnet, consisting of a normal conducting coil nested coaxially within a superconduct iig solenoid, to minimize power requirements. For most of the duty cycle the superconducting solenoid provides a 5.25 Tesla forward flux in the reactor chamber and the normal coil is passive, i.e., the switch connecting the normal coil to its external power supply is open. The normal coil is activated to reverse the field in the plasma chamber during preionization and to radially shock heat the ionized plasma. It is only activated for a few milliseconds during each power cycle, so external current is only supplied one percent of the time. Aside from its use in a compound magnet to null out the superconducting field during toroid formation, the close fitting inner conducting shell is needed to provide equilibrium and stability. Even during the relatively quiescent burn phase, strong currents are induced in the conducting shell to preserve a constant value of external flux. These currents are shown on Figure 2 where a simple sharp boundary plasma model is assumed. The confining magnetic field in the region of tlie toroid is Bn, = B0/(l -x|) and the induced eddy current line density in the inner coil is -29-

NORMALIZED 3EPARATRIX RADIUS r Ir

Figure 1 (Left). Plasma Radius and Length and Required Fill Pressure to achieve 8 keV. Figure 2 (Above). Sharp Boundary Equilibrium Flux Diffusion Model.

C.I •

NORMALIZED BIAS FIELD B It. where Xs~rslrc "*s the normalized separatrix radius. The open field line flux loss is

where R£ = 2fmrc/6 for a thin conducting shell. A characteristic flux loss time can be expressed as ( =)

For a 50 cm radius coil and 6 uil-cm resistivity copper shell (4 times room temperature resistivity) i\_ is 0.5 seconds for xs = 0.6. This is sufficient for the desired burn time. The toroid will expand radially due to the loss of the open field line flux and will encounter the wall in a time TL if the open field strength is maintained at Bm. This open field strength can be maintained either through toroid axial compression or alpha particle heating. For reactor grade plasmas, the alpha particle production rate is more than sufficient to hoth maintain plasma temperature and to supply the energy (pdV work) necessary to expand the plasma against the field Bm. For a simple one-dimensional radial expansion at cor- stant Bm, this pdV work is twice the open field line flux loss energy. The pdV work is partially deposited into the conducting shell as Joule losses and partially converted into additional magnetic field energy residing between the conducting shell and the outer superconductor. For xs< 0.707, more than half of the pdV work is converted into additional field energy, and less than half is lost to Joule heating. -30-

The portion of pdV work converted to field energy in the superconductor can be used for direct conversion and can be applied to making up for switching or energy transfer losses in the normal inner magnet. This can significantly reduce the recirculating power fraction of the reactor, especially if the absorbed alpha particle energy is sufficient to produce some axial expansion in addition to the radial expansion. Thus, on a reactor scale, Joule losses in the inner conductor neither reduce the burn time to unacceptably low values nor are they a significant energy sink on the system.

REFER:NCES 1. A.G. Es'kov et al ., "Features of Plasma Heating and Confinement In A Compact Toroidal Configuration," 7th IAEA Conference on Plasma Physics and Controlled Thermonuclear Research, Innsbruck, 1978. 2. H.J. Will en berg et al., Definition and Conceptual Design of a Small Fusion Reactor: Phase I Summary Report, Submitted for publication to Electric Power Research Institute, Palo Alto, California, April 1980. 3. A .L. Hoffman, CT-TRX1 : A Triggered-Reconnection Compact Toroid Experiment, MSNSNW 80-1144-3", Mathematical Sciences Northwest, Bel'ovue, Washington, Mav 1980. 4. C.P. Ashworth, Small Pilot Plants - Let's Bring Fusion Commercialization Down To Earth, 80 FUS-1, Pacific Gas and Electric Company, San Francisco, California, 8 January 1980. 5. H.J. Willenberg et al. , "TRACT: A Small Fusion Reactor Based On A Compact Torus Plasma," U.S.-Japan Joint Symposium on Compact Toruses and Energetic Particle Injection, Princeton, New Jersey, December 1979. 6. R.A. Krakowski and R.L. Hageman, "Preliminary Reactor Implications of Com- pact Tori: How Small is Compact?," U.S.-Japan Joint Symposium on Compact Toruses and Energetic Particle Injection, Princeton, New Jersey, December 1979. 7. L.C. Steinhauer and A.L. Hoffman, "Axial Shock Heating of Field Reversed Plasmas," Third Symposium on the Physics and Technology of Compact Toroids In The Magnetic Fusion Energy Program, Los Alamos, New Mexico, December 1980. 8. L.C. Steinhauer and H.J. Willenberg, "Technology Issues of TRACT Plasma Engineering," Fourth ANS Topical Meeting on the Technology of Controlled Nuclear Fusion, King of Prussia, Pennsylvania, October 1980. 9. T.S. Green and A.A. Newton, Phys. Fluids 9_: 1386 (1966).

ACKNOWLEDGEMENTS This research is supported by the Electric Power Research Institute under Contract RP922-4. -31--

ON THE USE OF INTENSE ELECTRON AND ION BEAMS AND RINGS

IN MIXED-CT CONFIGURATIONS

Hans H. Fleischmann School of Applied and Engineering Physics Cornell University Ithaca, New York 14853

Generally, three distinct.lv separate ring configurations are being considered for Compact Toroid (CT) applications, including the "smal1-orbit" field-reversed mirror (FRM)1 rings without toroidal field, fhe near-force- free Spheromak*' configuration and the "large-orbit" field-reversed Ion King Compressor^ or Astron1* concept. All of these share the basic potential advantages of requiring only a simple external magnetic field configuration and ot not requiring material coils linking through the plasma ring. Unfor- tunately, however, some characteristics have surfaced in each of these schemes which tend to reduce the reactor value of these schemes. In recog- nition of this situation, various mixed-CT configurations involving the use of intense high-energy electron and/or ion beams have been suggested last year' with the aim to avoid some of the more disadvantageous characteristics that have been found and are presently ascribed to the idealized basic concepts. In particulf .", these include (i) the use of large-orbit particles for the stabilization of the tilt mode of FR>1 or Spheromak configurations' (a stabilization of flat modes in FRM's falls in the same category) and (ii) the use of relativistic electron beams to maintain the potentially quite resistive poloidal surface currents required in Spheromak configurations. In addition, (iii) high-energy electrons or ions also have been considered' >° as an aid for rapid plasma heating in Spheromaks or FRM's. The present paper contains brief systems-type discussions of the conditions >.:id parameters pertaining to such mixed--CT configurations. It will follow the sequence of configurations given above.

(i) Stabilization of Spheromak and FRM Rings by Large-Orbit Particles:

(a) Tilt Mode: Theoretical analyses so far ascribe to Spheromak excellent stability characteristics with the notable exception of a (m=l)- tilt mode which appears quite difficult to overcome except by conducting shell closely fitting around the ring and an oblate shaping of the rings. Present experiments^»10 with such configurations agree with this prediction in showing a pronounced tendency of the rings to tilt, and in preventing this tendency when closely fitting copper shells around the rings are used. Similarly, such tilt instability has been predicted also for FRM rings but was not yet observed.

While it is not yet clear if this tilt mode will really be unstable for all useful equilibria without the presence of n conducting shell, a final requirement for such a shell would seriously reduce the reactor attractiveness of these configurations: In addition to making necessary the use of quite elaborate feedback stabilization systems, it would probably preclude the use of the quite advantageous moving-ring arrangement, and it is not clear if the very positive features of that scheme could be recovered economically in other arrangements. -32-

Correspondingly, it appears important to investigate the tilt- stabilization using large-orbit particles. As indicated in the adjacent figure, such large-orbit particles, in principle, could be included in

the ring either orT~Tfhe outer part of the ring (where the fast current would be parallel to the ring current) or on the inside (with anti-parallel currents); however, general orbital and stability considerations clearly seem to exclude the latter possibility.

Such a stabilization scheme is suggested by the non-observance of any tilt motion in our field-reversed electron ring RECi.-experiments. » 12 According to our present physics understanding, supported by some first theoretical indications,1' these rings are tilt-stabilized by a large-orbit ..•(Hipling of a tiJt-motion to a widening of the minor *"ing cross section, the latter resulting from the enhanced axial momentum spread (parallel to B,) "I" the fast particles in the tilted ring. Similar effects can be expected ilsu when a sizable fraction a of the toroidal ring current in a Spheromak ;>r a FKM is carried by large-orbit particles. An additional stabilizing effect could be generated by the angular momentum of the fast particles, an effect which probably is important mainly for small a where the strong plasma- turrent-generated ring fields may suppress the above coupling effect. (Corresponding stabilizing terms are reported from the small-a treatment of Ref. 6.) In contrast to the described coupling between internal and external parameters, it appears that this latter mechanism still can lead to a tilting of the rings if the "wobbling" energy of the tilted, and thus precessing, ring is expended viscously. First estimates including resistive wall losses indicate that the respective time scales may not be ajdWlj^ient for reactor applications. Clearly, it is important to obtain a'aJBHMiiStigate equilibria with large a-values and to determine stability limitJKHpfespective equilibria are presently being computed. ,

Taking ring parameters expected for reactor applications, the use of high-energy electrons will be precluded—as in the original Astron scheme— b> their synchrotron radiation losses, and protons with energies of a few 100 MeV will have to be employed. In this case, the energy losses of the protons^ due to drag clearly have to be minimized, i.e., the density overlap between the fast-ion beam and the confined plasma has to be reduced as far as possible; the fast beam has to be centered mainly on the outer fringe of the confined plasma. First estimates indicate that an average plasma density of about 20-30% of peak density may be permissible in the beam region for ax 0.3, while further reduced densities will be required if larger ax-values are needed for stabilization. Our present equilibrium calculations emphasize this point. -33-

Similar conditions apply to a large-orbit tilt stabilization of FRM rings; due to the higher 8-values of these rings, however, somewhat higher plasma densities are permissible at the fast-beam position.

In addition to its basic equilibria and stability characteristics, the generation and maintenance of such a mixed-CT cor.figuration, either based on a Spheromak or an FRM ring, obviously is a major problem. In experimental situations, the ring generation probably can be accomplished best by injection of fast particles (relativistic electrons or high-energy ions) into plasma rings either during their formation phase or while they pass by an intense-beam injector or by the generation of strong plasma currents in existing large-orbit rings. Alternately, a plasma ring could be combined, "stacked," with an already existing fast-particle ring as it has been accomplished with two electron rings in the RECE-experiments.iL | The applicabilr.ty of all these schemes to reactor conditions will have to be investigated. Theoretical predictions on this problem clearly are made difficult by the need to include high-6 effects with large-amplitude Alfven waves which appeared seriously important in our earlier beam injection experiments into pre--formed plasmas.1^

Quite importantly, the slow-down time of the fast protons may be longer than thi; anticipated life of the overall ring, and additional high- energy protons may have to be added to the hot ring (probably best by stacking). Cl^ariy, the anticipated position of the fast protons on the outer fringes of the plasma rings appears an advantage in this regard.

(ii) Relativistic Surface Currents in Spheromaks:

In the anticipated Spheromak configurations, strong poloidal currents are expected to flow on the ring surface and are expected to increase the stability char icteristics of the rings. If not unexpectedly good heat insulation fro.n the surrounding walls and the cold plasma, or unexpectedly good plasma confinement on the purely poloidal field lines outside the normal plasma ;ore can be achieved, these surface layers will be quite cold in comparison with the plasma core. In this case, the plasma resistivity in this region will lead to very sizable energy losses and to a corresponding shortening of ::he ring lifetime.16

In these cases, it can be advantageous if these surface currents are carried by relativistic electrons. Assuming a required current density 2 j (in Alfven ) a plasma temperature Te (in keV) and a plasma density n (in electrons/cm3) in this region, the resistive energy loss density would be approximately

dP . -^ = 2 x 10-6 z^^- j2[w/cm3]

In comparison, the normal electron drag fieldJ F on the relativistic electrons''' would lead to a loss -34-

dP ~^- = rj = l(r17nj [Watt/cm3]

dP / dP

An order-of-magnitude improvement in these losses would occur for 13 3 2 rcM.-tur conditions like n = 3xlO cm- , Te = 500 eV, and j = 500 A/cm = t'vt-n for ^e f f 1 .

Major questions to be answered obviously include the injection of these currents and their stability. Injection probably again can be ui-cumpl isiied during the formation phase of the rings. Later injection ,'Iearlv is difficult as recent calculations18 indicate; however, these calculations did not yet contain the influence of finite amplitude Alfven waves which again could help quite significantly in the trapping of intense electron beams. The questions of equilibrium and stability of such rings v.ill need investigation.

Acknow i ed_gemen t s

This work was supported by the Department of Energy under contract UK-ASO2-76KT53O17 and by EPRI Contract RP922-6.

References

1. E.g., W. 0. Condit, Jr., et al. UCRL 52008, Lawrence Livtrmore Laboratory, February 19 76.

2. E.g., M. N. Bussac et al., IAEA Conference, Innsbruck, 1978, Vol. II, p. 249.

3. H. H. Fleischmann, Conf. on Electrost. and Electromag. Confinement of Plasmas and Phenomenology of Relativ. Electron Beams (New York, March 1975), Ann. N. Y. Acad. Sci. 251, 472 (1975).

•4. N. C. Christofilos, 2nd United Nat. Intern. Conf. Peaceful Up.t. of Atomic Energy, Geneva 1958, Vol. 32, p. 279.

5. H. H. Fleischmann, Proc. of the US-Japan Joint Symposium on Compact Toruses and Energetic Particle Injection, Princeton, December 1979, p. 41.

6. Recently a first theoretical small-current treatment was reported by R. N. Sudan and P. Kaw (Annual Plasma Physics Division Meeting, San Diego, November 1980, paper 7D3) indicating some stabilizing terms.

7. E.g., V. L. Teofilo, J. Benford and V. Biley, Annual Plasma Physics Division Meeting, San Diego, November 1980, papers 5S16/17 and earlier papers. -35-

8. M. Yamada, private communication.

9. T. R. Jarboe et al., US-Japan Joint Symposium on Compact Toruses and Energetic Particle Injection, Princeton, December 1979, p. 53.

10. W. C. Turner et al., Annual Plasma Physics Division Meeting, San Diego, November 1980, papers 2Q5-8.

11. E.g., D. A. Phelps et al., Phys. Fluids 17_, 2226 (1974).

12. E.g., M. Tuszewski et al., Phys. Rev. Lett. U3_, 449 (1979).

13. R. V. Lovelace, private communication.

14. H. A. Davis et al., Phys. Rev. Lett. 39, 744 (1977).

15. A. C. Smith, Jr., et al., Appl. Phys. Lett. 32., 133 (1978).

16. Respective calculations are in progress.

17. Electrons clearly are preferable for this purpose as compared with ions.

18. i. S. T. Young et al., Annual Plasma Physics Division Meeting, San Diego, November 1980, paper 7D4. -36-

Cornell Ion Ring Experimental Program * J. B. Greenly, P. L. Dreike, D. A. Hammer, P. M. Lyster, Y. Nakagawa, and R. N. Sudan Laboratory of Plasma Studies, Cornell University, Ithaca, New York 14853 The Cornell ion ring program consists of experiments,!.2 numerical simulation, ^ and theory designed to show the way toward, and eventually achieve, the goal of providing ion rings for creating and/or maintaining, stabilizing, and heating compact toroidal confinement devices. We will summarize the experimental results to date and briefly mention simulation results directly relevant to the experiments.

There are two ongoing experiments, IREX and LONGSHOT. Both use intense pulsed magnetically insulated ion sources to produce rotating diamagnetic proton layers by axial (cusp) injection of an annular beam into a magnetic mirror. Intense pulsed ion sources have the advantage of high output (>1 x 1(.)1? protons are needed for field reversal) in a time (0.1-1 ysec) much shorter than energy confinement times in C-T configurations. Axial injection through a magnetic cusp has potential advantages over tangential (Astron-type) injection. First, a large area is available for beam input, and second, asym- metric injection which can disturb ring formation and stability is avoided. In IREX (Fig. 1) £5 x 10 protons at >430 keV are injected through a full cusp in a 90 nsec pulse to give nearly axisymmetric ion orbits in a short, thin annular layer. By contrast, in LONGSHOT (Fig. 2) >2 x 10i6 protons at <100 keV are injected through a half-cusp in a 700 ns pulse giving ion orbits that closely approach the magnetic axis, in a radially thick layer. We now briefly describe the present status of these two ion ring formation .schemes, starting with IREX.

The IREX system is shown schematically in Fig. 1. The proton source is an annular magnetically insulated diode powered by a 90 nsec, 450-600 kV peak voltage, 50-90 kA peak current pulse from the Neptune 800 Generator. The beam passes through a 2 ym mylar foil (with -130 keV energy loss) into a 2-1/2 m long, 40 cm diameter drift chamber containing <400 mTorr of hydrogen, helium, nitrogen, or air. The beam is caused to rotate by a cusp-like transition between the field produced by coils within the anode, which provide the insulating magnetic field, and the main solenoidal coil. The rotating proton beam thus formed is to be trapped in the mirror by inductive coupling of excess axial energy to the resistive wall located within the mirror.

When the proton beam is injected into vacuum (<10 ' torr), the beam is radially and axially dispersed over a 1/2 m propagation. However, when the beam is injected into >15 mT air or ;>60 mT hydrogen, up to 831 of the typically 390 J total kinetic energy of the beam is converted into rotational

Permanent address: Sandia Laboratories, Albuquerque, New Mexico 87185. ^Permanent address: Osaka City University, Osaka 558, Japan. -37- energy, the axial velocity dispersion is small, and the beam is sharply defined radially with inner and outer radii of 7 cm and 13 cm, respectively. The beam is 90-100% axially current neutralized by currents induced in the beam generated plasma. Azimuthal plasma currents, on the other hand, are observed only in the heavier gases - air, nitrogen, and helium - but not in hydrogen or deuterium. In hydrogen, the rotating proton beam forms a ring having a length about 30 cm with sufficiently small axial velocity dispersion that it is held together axially by its own 31 diamagnetic well as it propagates 2 m in an 8 kG solenoidal magnetic field. Such a ring contains about 5 x 1()15 protons, and it dissipates about \% of its axial energy in the resistive wall. Figure 3 shows a set of diamagnetic signals from an unusual shot which shows, in addition to the primary peak, a second, smaller peak that separates from the main ring. The weaker ring propagates to about the 172 cm point with an axial velocity of 2.5 x 10^ cm/sec, reflects coherently from the downstream mirror, and propagates back out of the system (there was no upstream mirror on this shot.), "living" for nearly 1.5 psec. The main ring is not reflected by the 1.23 doi.nstream mirror because its axial energy lo:;s is inadequate. The last two traces in Fig. 2 comprre a Faraday cup and a diamagnetic signal at the same location, demonstrating that the diamagnetic signal shape.1 in hydrogen is an indication of the current of high energy protons. In air, diamagnetic signals are broader than the Faraday cup traces, and peaks tend to be smaller and flat-topped, indicating the presence of azimuthal plasma currents. Up to 15° of the beam's axial energy is induc- tively coupled to these plasma currents, resulting in more efficient reflection of the proton beams by the downstream mirror in air, >50li, as compared to in hydrogen. A peak ring diamagnetism of 875 G was observed 50 cm from the diode using II? fill at a point where the applied field was 11 kG (near the peak of an upstream mirror - see Fig. 1 - of 1.33 mirror ratioj. This ring contained an estimated 1 x 101f^ protons Since inductive energy coupling to a resistive wall is proportional to N-, it is anticipated that a ring with 2 x 10^ protons can be trapped in IREX by optimizing the wall resistor and the magnetic mirror shape. The LONGSHOT experiment'' is shown schematically in Fig. 2. The annular ion injector is driven directly by a 1 (Ml-200 kY, <14 kJ, I i"j Marx generator. The pulse length is determined by the duration of magnetic insulation of the accelerating gap. Presently, ;2 x lO^ protons at 75-K'O keV, with \X/V|, - 1 are typically injected into the half-cusp region of the downstream ramped solenoidal field.

Experiments to date have shown that the beam traverses the half-cusp in vacuum without significant disruption and that bunching and partial reflection of the beam occur in the ramped field. Evidence for good cusp propagation in vacuum comes from non-dispersive propagation beyond the cusp, with radial extent consistent with single-particle orbits in tae magnetic field, and from good agreement of the magnitude of beam diamagnetism and radial profile with simulation. Beam diamagnetism is measured by B loops at 1 the wall. Simulation of injection of 2 x LO^ protons indicates peak 5B/B0 of 3.7°s on axis, and a factor of 10 less at the wall. Experimentally measured 6B/B0 at the wall, multiplied by 10, gives 3.5°6 on axis bO cm beyond the cusp. Further evidence comes from experiments in which^the cusp is filled with plasma from a titanium washer gun (ne ~ ltll-VcnP versus injected -38-

beam nj ~ 7 x iO ) that shovi no change in beam diamagnetism. If space charge were a problem in vacuum, the plasma fill should give improved diamagnetism. In Fig. 4(a), we see maximum beam 6B occur nearer to the entrance to the ramp as the ramp field strength is increased, indicating that the beam bunches axially in the ramp. A second example with a different injector voltage pulse shape (in Fig. 4(b)), shows bunching and partial reflection. The 6R signals narrow and steepen as the beam proceeds further along the 1 m ramp. The double peak at the first position shows the reflected component. In a simu- 5 lation with this ramp field strength, <20 o of the beam ions are reflected.

Let us now compare the results of LONGSHOT and IREX and their implications for trapping an ion ring, including a discussion of relevant computer simulations.

First, the LONGSHOT beam propagates well through the cusp in vacuum (2 x 10"5 torr), while in IREX, neutral gas (10-400 mtorr) must be provided in the cusp to avoid severe space-charge disruption. Despite the higher total ion output of LONGSHOT, the slower rise in ion density allows effective charge neutralization by electrons from external surfaces. Second, the LONGSHOT injector achieves higher total output at less than one-tenth the peak power necessary for the IREX injector, thus offering the promise of easier scale-up to the output necessary for field reversal and high repetition rate. However, IREX has the advantage, shown in simulations,^ that the spatially short beam with rapidly increasing diamagnetism interacts more strongly with a resistive wall to give axial energy dissipation adequate to trap more than 'JOo of the beam in a mirror well. By contrast preliminary results of simulations of the LONGSHOT beam show thai: the close approach of the ion orbits to the axis leads to lccally time varying magnetic field caused by spatial bunching which can trap some fraction of the injected population without explicit resistive damping. In these simulations an upstream mirror is ided at the entrance to the ramped solencidal magnetic field shown in Fig. 2. The incoming ions pass through this mirror during the first phase-coherent close approach to the magnetic axis because of beam self-field enhancement. After reflection in the ramp, the ions return to this mirror at essentially random orbit phases (radial positions), and a large fraction are trapped in the system.

In summary, the IREX and LONGSHOT experiments have both shown successful formation of diamagnetic proton layers which propagate without disruption by s* "'ce charge. The layers have been axially compressed and reflected in magnetic mirrors; in IREX clear evidence of propagation as a self-confined ion ring has been seen. Both experiments agree well with simulation, giving confidence in the implications of simulations of stronger beams. The con- tinuing program for both these experiments is to increase the output of their respective ion injectors to allow investigation of the mechanisms shown in simulations for trapping strong ion rings.

This research supported by U.S. DOE and ONR.

\v. L. Dreike, et al., Cornell University LPS Report 283 (1980). 2J. B. Greenly, Cornell University LPS Report 286 (1980). 3See, for example, A. Mankofsky, et al., Cornell Univ. LPS Report 245 (1980). -39-

Fig. 2. L0NG5H0T SYSTEM

E1E] El [S3 I3KEX3

EDE3 ESI E)

2 on axis (kG)

Fig. 1. IREX: A, to the Neptune generator; B, anode; AXIAL MAGNETIC FIELD C, external coils; D, cathode, including 2 \m mylar foil; E, typical

proton orbit; F, upstream nirror coil; C, wall resistor wiTes; H, axis

diagnostic assembly with pidoip locps; I, vacuum chamber wall.

200 ns/div 200 ns/div JO..

32cm ;152cm

0 , 172 cm 3.57J

A") 72 cm 192cm

100 «!«• > •H 92cm 212cm XI x u o Al LTl 112.'cm SB \. 260 44 FC B) :' '^132 cm Fig. 3. Diamagnetic signals for IREX Fig. A. LONGSHOT diamagnef.ic signals, shot in 390 mTorr H,. Insert compares (a) left: B =4.8 kG, right: B =6.9 kG. faraday cup signal with CB at 52 cm. (b) B =5.5 RG. 0 BQ=8 kG. -40-

Kink Motion of Long Field-Reversing Ion Layers

Douglas S. Harned Klectronics Research Laboratory University of California Berkeley. California 94720

Kink instabilities have been studied in long field-reversing ion layers. The configuration is shown in tig. I. An ion beam, at radius R and of thickness a. crosses an external magnetic Held B,=B'.:. The beam current, ./,;„, produces a self-magnetic field, B\ which tends to reverse the tola! magnetic field on axis. The layer is immersed in a uniform background plasma such that n,,«n,., where /;,, and /;,, are the densities of the beam and plasma, respectively. There is assumed to be no variation in the axial direction (i.e. d/6r=0). Kink modes in these la>ers correspond to perturbations of the form e=t,exp(//iW-/a»/)/;. with n^2. Long layer kink modes have been studied by Lovelace1. Analytic results were obtained lor uniform layers with sharp boundaries within the approximations (a/ R)2< <\, iiu/n iH:<"< I. and r inconstant, wriere II is the layer rotation frequency and r, the plasma Alfven velocity. Stability thresholds were obtained for two cases: 1.) ar« (r Ja 1 and 2.) in << i Jn)- with ai;«ar, where w, and w are the real and imaginary parts of the frequency. In hoih cases the condition T),

self-magnetic field index. If £ is defined to be the loading factor ((,~\B'{r=0)/B'\), then T?S is of order (2R/a)(.. In order tci have a useful comparison for our simulation results it was necessary to gen- eralize ihe preceding theoretical analysis by eliminating the assumptions w<(i',/f/)',

The instabilities have been observed to grov. and saturate without destining the field-reversed layer. Kink instabilities instead result in a thicker field-reversed slate with manj non-axis- encircling particles. The final state appears to be stable t.ul with higher electric fields and substantial electron currents.

1. R.V. Lovelace. Phys. Fluids 22, 708 (1979).

Instability Thresholds

H = 2 ;;=3 n=4 «=5 .00250 0.4 2.2 5.0 8.5 .00125 1.0 4.2 8.0 12.0 .000625 1.8 5.0 j 9.5 15.0 .0003125 2.0 6.0 11.0 18.0

TBL. 1. Instability threshold values of the self-magnetic field index.

FIG. 1. Schematic of an infinitely long field-reversing ion layer. -42-

FJG. 2. Initial panicle positions in the r- FIG. 3. Particle positions after ten ion- plane for an ion layer with £=1.32. cyclotron periods. An n=3 instability has grown lo a large amplitude and nonlinear effects have become important.

0.0 .003 00

FIG. 4. Growth rales, u>,, for ion layers FIG. 5. Growth rales of /i=2 modes for ion with (R/a)~5 and (v,,/c)-.OO125 as a layers with (R/a)~5 and £=.65 as a func- function of the loading factor. tion of background plasma Alfven speed. -43-

PROPAGATION OF AN INTENSE ION BEAM TRANSVERSE TO A MAGNETIC FIELD

Hiroshi Ishizuka and Scott Robertson Department of Physics, University of California, Irvine, California 92717

Energetic particle beams have been used for decades to heat and drive current in plasmas confined in magnetic mirrors. Closed confinement systems provide better plasma containment but are thought to prevent the direct injection of energetic particles. We present the results of an experiment which shows that an energetic ion beam of sufficient intensity will cross a magnetic field by the polarization drift. It has been suggested that an ion beam injected in this way could be used to heat-"- or drive currents^ in plasmas confined in toroidal magnetic fields. Collective motion of intense ion beams incident upon magnetic fields has recently been investigated in several laboratories. Ours is the first with sufficient diagnostics to identify transverse polarization as the mechanism of penetration.

The experimental apparatus is shown schematically in Fig. 1. The 15 cm id annular magnetically-insulated ion source^-9 is contained within a 25 cm dia. stainless steel tube mounted on the side of a Marx generator. At a distance of 22.5 cm downstream from the diode the beam enters a 45 cm dia. X 50 cm cylindrical glass chamber. A pair of rectangular magnetic field coils (20 cm inside X 30 cm outside X 23 cm separation) is mounted within the glass cylinder to provide a magnetic field of 0 - 3 kG oriented perpendicular to the beam direction. The primary diagnostics are a diode voltage monitor, a Rogowski coil to measure the net current, of the diode, movable biased Faraday cups to measure the beam current density, a Langmuir probe, and plastic film to determine the bean, location by means of damage patterns.

A plot of the current density on axis as a function of distance from the source (Fig. 2) shows that the initially hollow channel becomes filled in between 15 and 20 cm from the diode and has peak current density of 43 A/cm^. A radial scan at z = 40 cm showed that the current density was uniform to within 10% out to a radius of 8 cm. The deflection of the ion beam by the magnetic field was determined by damage patterns on cellulose acetate plastic sheets which were placed perpendicular and parallel to the beam path as shown in Fig. 3. The outlines of the damage patterns are shown in Fig. 4. The diameter of the beam channel at z = 54 cm is 20 cm. The diameter of the pattern is reduced to 16 cm when a metal mesh Is placed at z = 18 cm. This indicates that the additional charge and current neutralization provided by the mesh reduces the edge divergence to below 1 . With an applied magnetic field of 2 kG and with the mesh removed, fish-like patterns (Fig. 4) are obtained. The tail of the fish is at the location calculated for protons deflected by the magnetic field. The non-uniform field acts like a cylindrical lens and focusses the deflected beam to a smaller cross section (Fig. 3). A mirror image of the pattern is obtained when the direction of the field is reversed. The central region of the -44-

damage patterns indicate that part of the beam is undeflected by the field. There are three possible explanations: (1) the beam contains neutrals, (2) the beam contains carbon ions which are only weakly deflected by the fields or (3) part of the beam crosses the field undeflected due to the polarization drift. The pressure in the apparatus is too low for there to be a significant fraction of neutrals. To distinguish between the second and third possibilities, a metal screen was placed in the beam channel perpendicular to the magnetic field so as to short out a transverse electric field. This caused the central pattern to become faint and reduced in size and caused the tail to become more pronounced and wider. The faint central pattern is then due to carbon impurity since the cross-field propagation of protons was prevented. The central pattern without the shorting screen is therefore composed of some carbon, but primarily is protons which have crossed the field. With the shorting screen these protons are transferred to the tail section.

In order to reduce the fraction of the beam which is deflected, a neutralizing metal mesh was placed at z = 18 cm. This resulted in the complete disappearance of the tail section of the pattern (Fig. 4). When a metal plate was placed in the beam channel transverse to the field, the transmitted beam disappeared and was replaced by a weak pattern in the tail location.

In order to obtain a more quantitative measure of the beam transmission a Faraday cup was placed on the beam axis at z = 60 cm. The signals with the metal mesh are shown in Fig 5 at fields of 0 - 3 kG. The peak signal with the mesh and with no applied field is 6A/cm2 which is lower than that obtained without the mesh (Fig. 2) due to the mesh transparency. At applied fields up to 2 kG the peak current density is independent of field. After the peak current the transmitted beam intensity falls slightly more quickly with time when the field is applied. At 2.5 kG the peak intensity is reduced to 4 A/cm2 and at 3 kG it is reduced to 1.5 A/cm2. Without the neutralizing metal mesh the beam was attenuated at lower values of applied field . Also, a shorting plate placed perpendicular to the field caused greater attenuation. Thus the data from the Faraday cup on axis agrees with that t'rom the plastic damage patterns.

In order to measure the electrostatic potentials created by the beam, an electrostatic probe**>? was inserted into the beam channel from an off- axis location. The probe tip was located_at z = 35 cm near the peak in the magnetic field and was scanned along the V X B direction. The transverse electric field determined by the difference between the probe signals at two locations is shown in Fig. 6. At 1 kG, the electric field is initially above the value given by - V X B, then falls below this value later in the pulse. At 2 kG, the measured field does not quite reach the calculated value before decaying away. The decay of the electric field is consistent with the reduction in transmission late in the beam pulse indicated by the Faraday cup data.

The appearance of the transverse electric field indicates that the mechanism by which the beam crosses the magnetic field is the polarization drift derived by Schmidt. •*• A condition proved necessary (but not -45-

sufficient) for the drift is that € > > 1 where € Is the beam plasma dielectric constant. At the peak of the applied field the current density is 10 A/cm^. At a field of 2 kG this current density corresponds to € — 700 At 3 kG where the beam is poorly transmitted 6 s; 300 . This indicates that € > > 10 is a more accurate minimum condition. This is in agreement with a recent theoretical prediction-'-- that *i > > JW/m is required, where M is the ion mass and m is the electron mass. Our data indicate that the more strict Rosenbluth sheath condition**^, 13 £ >> M/m is not required.

This research was sponsored by the United States Department of Energy.

References

E. Ott and W. Manheimer, Nucl. Fusion 17_, 1057 (1977). W. Manheimer and N. Winsor, to be published. K. Kamada, C. Okada, T. Ikehata, H. Ishizuka, and S. Miyoshi, J. Phys. Soc. Japan 46, 1963 (1979). J. Pasour, R. Mahaffey, J. Golden, and C. Kapetanakos, Bull. Am. Phys. Soc. 23^ 816 (1978). 5. M. Wickham and S. Robertson, Bull. Am. Phys. Soc. _25, 1007 (1980). 6. Wessel and S. Robertson, Bull. Am. Phys. Soc. 25.> 1007 (1980). 7 _ Robertson and F. Wessel, Appl. Phys. Lett. 3]_, 151 (1980). 8. Humphries, Jr. and G. Kuswa, Appl. Phys. Lett. 15, 13 (1979). 9. Humphries, Jr.et al., J. Appl. Phys. 51, 1876 (1980). 10. Schmidt, Phys. Fluids 3_» 961 (1960). 11. Peter, Ph.D. Thesis, to be published. 12. L. Longmlre, Elementary Plasma Physics, Wiley Interscience, New York 1963, Chapter V. 13. W. Peter, A. Ron, and N. Rostoker, Phys. Fluids 2_2, 1471 (1979).

3LASS TUBE

3N SOURCE

/-COIL

TO ;, MARX 3,

• t : -ARA0AY CUP '0 PUMP

0 cm

Fig. 1 Schematic diagram of the apparatus. -46-

,.- COIL BOUNDARY 50

40 j r 1 30 \ I N 1 j

20 1 j j 10 i 1

i i i i 1 1 0 20 "0 60 PLASTIC SHEET cm Figure 3. Figure 2. Diagram showing the placeoent of the plastic uea* current density as a function of film relative to the field coils. The arrows iis'ance from the source. indicate the trajectories of 100 keV protons in a 2 kG peak, field. Contours of constant magnetic field are also shown.

90°BEND

Figure 3. /scil1jgrams jf the Faraday cup signal ob- :ained at 2 =- 60 cm for five values of appiisd aia^necic field. The vertical scale :s 5 A/cm-/div and the horizontal scale is 0. J; .sec .' div .

3 r 8 = I kG

E o 0

> a B = 2kG

Figure 4. Damage patterns- obtained with the cellulose acetate film. (a) No applied field with 02 04 06 08 metal mesh. (b) Applied field of 2 kG without metal mesh. (c) Same as (b) but with a shorting plate (horizontal line) transverse to the field. (d) Applied field of 2 kG with Figure 6. a metal mesh. (e) Same as (d) but with a shorting plate. 'L-its if the transverse electric field as a -'.-.-:• i.3n ot time for applied magnetic fields f . and 2 kG. The dotted line is the value • • -metric field given by - V X B. -47-

THEORY OF PLASMA INJECTION INTO A MAGNETIC FIELD

William Peter and Norm.m Rostoke.r Department of Physics, University of California, Irvine, California 92717

We summarize recent theoretical results describing cross-field plasma motion for small- and large-gyroradius plasma beams. For a small-gyroradius beam, preliminary calculations show that the cross-field motion is driven by a dynamic Raylaigh-Taylor instability. For a large-gyroradius beam, the plasma propagar.es across; the field by means of an EXB plasma drift. ~ ~

The objective of the present investigation is to assess the possibility of injecting pulsed, neutralized ion beams into toroidal fields for the purpose of driving currents cr of supplementary tokamak heating. ' Previous cross-field injection experiments have been divided into two categories: (1) those in which the bean radius R is much greater than an ion gyroradius a-L andj (2) those in which R < a^. Condition (1) is usually applicable to beams injected from plasma »uns while condition (2) is usually applicable to high-energy neutralized ion beams injected from ion diodes.

In the case that R > > a^, the beam can be considered to resemble a one-dimensional slab of plasn,a which is incident upon the -magnetic field. We have previously studied an equilibrium based on a generalization of the Rosenbluth-Ferraro sheath. Additional work on this one-dimensional model has been conducted during the past year to determine the stability of the equilibrium. This was consideied important because many investigators have reported flute-type instabilities in cross-field injection experiments."1

Q Q In the case that R ^ a^, we have extended previous calculations ' of the two-dimensional case to determine the behavior of the propagating plasma. We have generalized the vork of previous workers both in the U.S.A. and the U.S.S.R. to arrive at analytic solutions to our model equations. In this case, provided there is no shorting of the electric field, the beam can cross the field by means of an jE XB^drift. A new lower limit on the plasma parameter € > >(M/m)!'^ has been derived, and is found to be in better agreement with experiments than the limit £ > > 1 found in previous studies. We discuss these two cases in more detail in the following sections.

A. Small-Gyroradius Beam In the case that R >> a^, we have undertaken to extend our one-dimensional equilibrium model to real experimental situations. One of the primary efforts was to analyze the stability of this equilibrium, and if possible to give theoretical justification for the appearance of flutes in many previous experiments. When the lew-beta equilibrium was analyzed, it was found that this configuration was stable. The stabilizing effect was due to the presence of an equilibrium electric field produced by an excess surface charge of ions that is characteristic of the equilibrium. In equilibrium there are electron and ion diifts Vrj" = - UQ yM/m and -48-

This is quite similar to the classic case of the Rayleigh- Taylor instability. When the surface is perturbed, since |Vp~ 1 >> Vp+ charge accumulates on the perturbed surface. The resulting electric field causes both electrons and ions to drift in such a direction as to make the perturbation grow. However, in this case there is also an electric field normal to the unperturbed boundary caused by an excess of ions. This causes ions and electrons to drift in the same direction. However, the excess of ions dominates. The electrons drifting with velocity VD and the excess ions drifting in the same direction because of their self-electric field, are respectively de-stabilizing and stabilizing. The net result is stability for wavelengths longer than the boundary layer thickness which is of the order of the hybrid gyro-radius a, . Short wavelengths are stabilized by several other effects in addition, including finite gyro-radius.•'-

The equilibrium model is stable and the fundamental reason is that there is no acceleration. To account for the observed flute instabilities, we consider perturbations about a non-equilibrium state which is a much better description of the actual experiments.

We solved the one-dimensional equations following an individual fluid particle (the Langrangian formulation). We then investigated the possibility that as the plasma moved in to the magnetic field region, perturbations on the moving plasma-field boundary would cause an instability which would allow cross-field plasma motion. This ia not a conventional instability since the plasma never reaches an equilibrium state. Preliminary results show that perturbations on the moving boundary grow like a Rayleigh-Taylor instability with w2 = - k|g| = -kfijxg (1) The quantity Xg is the instantaneous coordinate of the moving boundary and the domain of validity of the present calculation is Xg ^ a^. The small gyro-radius experiments^ >°'I-* often show cross-field propagation of plasma from plasma guns. Evidence for flutes such as spatial variations in the electric polarization has been observed.6 We believe that in these experiments an equilibrium similar to the Rosenbluth-Ferraro sheath is never established. A dynamic Rayleigh-Taylor instability prevents this and accounts for cross-field propagation. In some experiments the charge polarization accompanying the Rayleigh-Taylor instability may have been shorted through conducting walls and no cross-field propagation would be observed.

B. Large-Gyroradius Plasma In this case, the finite width of the beam becomes important as polarization effects now influence the propagation of the plasma. We have extended our previous calculations by considering the full one-dimensional equations in the Langrangian formulation. This has allowed us to obtain exact analytic solutions to the cross-field injection problem which reduce to approximate results we discussed previously.9 in addition, we have examined the conditions necessary for the assumption of plasma quasineutrality to be valid in our model. From this we have determined a more stringent condition on the plasma dielectric constant for the beam to propagate across the field. If £ > >«/M/m the plasma will remain quasineutral -49-

and the beam will propagate through the field. If this condition is not satisfied, we expect that a region of positive space charge will form that will reflect the beam. The details of this calculation can be found in the Ph.D. dissertation of W. Peter.-^ In a recently completed UCI experiment with € °"U0, such a space charge region was indeed observed.6 In Figure 1 we show a schematic drawing of this charge separation region formed near the toroidal field boundary of the UCI tokamak. In contrast to this we show in Figure 2 the charge structure that would form when £ > > (/M/m and the beam propagates across the field.

In Table 1 we cite the known large-gyroradius experiments. It is apparent that, with one exception, those satisfying the condition £ > > «/M/m were successful in propagating through the field. In the Marcovic and Scott experiment where this was not the case, there is a possibility that the polarization electric field was shorted out by the special experimental geometry. 4 The most recent UCI experiment-^ lends support to our more stringent condition, since beam propagation was seen for values of € 2 300, whereas previous experimental results showed no propagation for € ^ 40.

C. Discussion We have summarized some of the theoretical results we have obtained within the last year on the study of cross-field propagation of intense plasma beams. We have found it convenient to consider two specific cases: the case when the beam radius is much greater than an ion gyroradius, and the case when it is not. Plasma motion across the field is allowable in both cases, although the physical mechanisms are quite different. For a small-gyroradius beam, the plasma appears to cross the field by means of a dynamic Rayleigh-Taylor instability; for a large-gyroradius beam, the plasma motion is driven by an EX B drift.

There are many important applications of intense ion beam technology to magnetic fusion; for example, (1) steady state beam driven field-reversed mirror^'', (2) steady state tokamak-'-S, (3) sustained compact torus and (4) plasma heating. These possibilities are still quite speculative because several basic physical questions must first be resolved. The first question concerns cross field transport which is unique to neutralized ion beams and has now been resolved. The second question concerns current produced in a plasma by an ion beam and concerns either neutralized ion beams or neutral atom beams. In spite of optimistic proposals of reactor applications for neutralized ion beams,19 they may be frustrated without basic understanding of the second question . For example, serious experimental efforts to produce a beam driven field-reversed mirror were frustrated by plasma electron currents. The experimental effort at LLL was not successful. To date the only successful application is plasma heating which has been accomplished with neutral atom beams and could probably also be done with neutralized ion beams.

All of the other applications depend on our understanding of the currents produced by the beam in a plasma. The theoretical models are of two varieties. -50-

(i) The beam is treated neglecting particle discreteness. Macroscopic electric fields result from turning on the beam^O and the reaction of a resistive plasma is determined. (ii) The beam is considered in a steady state and coupling with the plasma is entirely by discrete particle effects. "'I In the first case a return current neutralizes the plasma unless the beam rise time is long compared to the Alfven wave propagation time. The plasma magnetic field is considered. In the second case the main effect is beam ions "pushing" plasma electrons. The net current can be zero or in the opposite direction to the ion current. Plasma magnetic fields are neglected.

We believe that a model that would be useful for interpreting experiments must contain both the macroscopic fields, and the discrete particle effects. We plan to carry out a coordinated experimental and theoretical effort on this problem.

References

1. E. Ott and W. M. Manheimer, Nucl. Fusion 17, 1057 (1977). Z. W. Peter, A. Ron, S. Robertson, and F. Vessel, Proc. IEEE Trans, on Plasma Sci. June, 1978, Monterey, California. 3. W. Peter, A. Ron, and N. Rostoker, Phys. Fluids _22, 1471 (1979). A. P. D. Marcovic and F. R. Scott, Phys. Fluids 1^, 1742 (1971). 5. I. I. Demidenko, N. S. Lomino, V. G. Padalka, Sov. Phys. Tec. Phys. JL6, 1096 (1972). 6. F. Wessel, Ph.D. Dissertation, Univ. of California, Irvine, 1980. 7. C. E. Speck, J. J. Lee, and 0. K. Mawardi, IEEE Trans. Plasma Sci. PS-3, 3 (1977). 8. K. D. Sinelnikov and B. IN. Rutkevich, Sov. Phys. Tech. Phys. \2_, 37 (1967). 9. W. Peter and N. Rostoker, UCI Physics Dept. Tech. Report #79-69. 10. G. Schmidt, Phys. Fluids 2» 961 (1960). 11. M. N. Rosenbluth, N. A. Krall, and N. Rostoker, Conference on Plasma Physics and Controlled Nuclear Fusion Research, Salzburg, September 1961 (unpublished), Paper CN-10/170. 12. H. Ishizuka and S. Robertson, Bull. Am. Phys. Soc. Z5, 1007 (1980). 13. D. A. Baker and J. E. Hammel, Phys. Rev. Lett. 8^, 157 (1962); Phys. Fluids 8_, 713 (1965). 14. K. Kamada, C. Okada, T. Ikehata, H. Ishizuka, and S. Miyoshi, J. Phys. Soc, Japan ^6, 1963 (1979). 15. S. Robertson and M. Wickham, to be published. 16. W. Peter, Ph.D. dissertation, Univ. of California, Irvine, 1980. 17. D. E. Baldwin and T. K. Fowler, Lawrence Livermore Laboratory Report UCID 17691 (1977). 18. T. Ohkawa, Nucl. Fusion 1£, 185 (1970), 19. W. Manheimer and N. Winsor, to be published. 20. H. L. Berk and D. Pearlstein, Phys. Fluids ^9_, 1831 (1976). 21. T. G. Cordey, E. M. Jones, D. F. H. Start, A. R. Curtis, and I. P. Jones, Nucl. Fusion 19, 249 (1979). -51-

TABLE 1

Results and parameters of some large-gyroradius experiments. Unless otherwise noted, the data is taken at 2 kG. R/a, Experiment Type of Beam j£_ Result

Wessel and Robertsonc ion beam 1.5 26 no propagation; boundary layer formed. 14 Kamada et al. ion beam 0.1 208 E x S drift

Ishizuka and ion beam 1.5 300 E X B drift Robertsonl^

Wickham and potassium 0.7 1.3X106 E x B drift Robertson-'--' plasma 750 G)

Marcovic and Scott helium 0.4 3 • io7 no propagation plasma 50 G) boundary layer formed.

oe,

Figure 1. Figure 2. Schematic drawing of the positive Polarization of a bounded plasma space charge region formed near the in a magnetic field for the case toroidal field boundary. Such longi- R'-iaj. The resulting ExB drift tudinal space charge separation is allows the beam to cross the expected when the plasma dielectric field relatively unimpeded. constant does not satisfy the charge neutrality condition discussed in the text. -52-

CLASSICAL TRANSPORT IN FIELD REVERSED MIRRORS: REACTOR IMPLICATIONS* Steven P. Auerbach and William C. Condit Assuming that the field-reversed mirror (or the closely related spheromak) turns out to be stable, the next crucial issue is transport of particles and heat. Of particular concern is the field null on axis (the X-point), which at first glance seems to allow particles to flow out unhindered. We have evaluated the classical diffusion coefficients for particles and heat in field-reversed mirrors, with particular reference to a class of hill's vortex models. Two fairly surprising results emerge from this study. First, the diffusion-driven flow of particles and heat is finite at the X-points. This may be traced to the geometrical constraint that the current (and hence the ion-electron drag force, which causes cross-field transport) must vanish on axis. This conclusion holds for any transport model. Second, the classical diffusion coefficient D(^), which governs both particle and heat flux, is finite on the separatrix. Indeed, in a wide class of Hill's vortex equilibria (spherical, oblate, or prolate) D( = DW dP/d*~here ~

This result for the classical diffusion coefficient has two immediate consequences. First, D('l') is finite on the separatrix, since the effect of the field null on axis is cancelled by the factor of r?. Second, D(tp) is finite at the O-point; in this case both di, and B are proportional to distance from the O-point, and their ratio is finite. It is instructive to examine the fluid velocity near the X-point in some detail, to understand why the X-point does not cause very rapid losses. For a general transport model, we examine the moment equation relating cross-field flow of one species (take the ions, for example) to the frictional force due = r to collisions with other species: Xi '^ ( c/ne)R9, where e is the electron charge and RQ is the 0-component of the friction due to collisions between ions and the other species. (Ro = ennJo for classical ion-electron collisions.) Clearly, geometrical constraints (Rg(-r) = -Re(r)) force Ro to vanish on axis, and v, must be finite at the X-point. It is clear from this discussion that the X^-point will not be of concern in any transport model. We have evaluated D(

We use this result for D(^) to provide an exact solution to the particle transport equation for Hill's vortex. Since n scales as T-3/2, we observe that if n ~ ^3/5, j ~ ^2/5 then P ~ * which is consistent with the Hill's r vortex equilibrium and p will be constant. This gives a solution to the transport equation for a vortex fueled at the O-point, with n = T = 0 on the boundary. The particle confinement time Tp which follows from this flux is

= 4.41 (2) where Ro is the O-point radius, cr0 is the Sp'tzer conductivity at the O-point, and ^ is the elongation. There &re several features of this result that deserve comment. For a prolate plasma (A > l)xp is essentially independent of elongation A, and scales as the square of the radial size. For a plasma which is quite oblate (say,A <, l/3)xp scales as the square of the axial size. However, for fixed O-point radius and central temperature, there is little penalty for making the vortex mildly oblate. For example, a vortex with A = .8 would have a lifetime only 10% shorter than a spherical vortex. This is of interest because there is evidence that oblateness may stabilize the tilting instability. In practical units Eq. 2 reads

7 A2 T3^2 x = 4.26xlO'X e ° P with T in eV, and £n/V = 24 - An(ne 1/2/Te). For vypical FRX-B data (17 mtorr fill pressure) Te = 110 eV, Ro = 3.8, A » 1, this yields ip = 164 usec/Z compared to experimental Tp - 40 usec. Although Z was not measured, it appears that there is not a great discrepancy between Eq. 2 and the data. Equation 2 can be written

where pe is the electron gyro-radius based on the temperature at the O-point, and the field at r = Z = 0, and x^e is the ion-electron collision time. This result is independent of elongation. The factor A2/(l + 4A?) in Eq. 2 disappeared, due to the equilibrium relation B2 = (32TTA?P )/ (1 + 4A2). The energy confinement time x^ can be calculated in a similar way, with the result 2 x = .04 Ei ^i ' (5) where P^ is the ion gyro-radius, defined using the ion temperature at the O-point and the magnetic field B, and (xii)o ^s tne ion-ion collision time -54- at the O-point. This calculation was performed with a profile n(4') ~ l T() ^2/3, which gives a classical heat flux which is independent of i|». Our estimates for classical confinement times are substantially more pessimistic than previous heuristic estimates. In order to understand the implication of these results we discuss the power balance in a reactor fueled with equal amounts of D and T, with the density and temperature profiles just given. For these profiles we can analytically calculate the fusion power output and energy losses due to ion heat conduction, Bremsstrahlung and synchrotron radiation. Using the approximation nj = (-229T - .001T2) x 10-16 cm3/sec for the D-T fusion cross section (T in keV) the fusion power is

30 3 PF = 6.23xl(T n^0 T0(l - 3,34xlO" To)Vs watts {6) where neo,To are the electron density and temperature at the 0-point. The power lost to Bremsstrahlung (Pg) and to synchrotron radiation (Ps) is -31 2 1/2 P = 2.19x10 n T V watts B eo o s (7)

33 2 Ps = 3.66xl0- ne T2Vs watts (8)

We find Pg/Pp = .035/To 1/2, PS/PF = TQ/1702; thus both Bremmsstrahlung and synchrotron radiation are negligible. We define Q* = pF/(rQO + r0j) where Tq$ is the heat flux for species S; we find R 2 Q* = 2.04xl0"4 T 3/2{l - 3.34xlO~3T )[•£-) 0 ° P Do (9) where P0O is the deuteron gyro-radius as defined previously. Assuming good beam absorption, and neglecting ct-partide heating, the input beam power must balance the ion heat loss. Thus Q* * Q, the true energy multiplication factor. If we no* specify a neutron wall loading W megawatts/square meter, and a wall at radius a x 2 Ro, we can solve for the reactor parameters R d B :

15 a 3 non - 1.126xl0 i/J Vn I/-, cm' eo T 1/6 rl/3 nl/3 To C Q (10) 2/3 RQ = 33.59

3 .952 (ow)!/ T c B = TTT tesia Q1/6 (12) -55-

3 where C = (1 - 3.34 x 10" To)* 1 Assuming that finite gyro-radius stabilization requires a plasma length L 1 we find a given by L/Ro >_ Ro/PDo minimum length Lmin and power

7/6 L - 2,342 Q min (;w)l/3 c5/6

. 124 (aW)1/3 Q11/6 " _7/6 ,25/12 m C To (14)

For example, for To = 50 keV, Q = 5 we find Ro = 5.84 cm, Pn.o = .59 cm, B = 5.49 tesla, neo = 7.84xiO^, Lm-jn = 57.8 cm, P^n = 1.70 MW. This is a very attractive reactor, obviously. To investigate the effects of an anomalous resistivity, we calculate the ohmic heating rate Po for an anomalous resistivity na, which is taken as constant throughout the plasma, and whose magnitude is na = Ai"ici, where nci is the classical resistivity at the O-point, and A is an anomalous enhancement factor. Ohmic heating transfers energy from the magnetic field to the electrons. We assume that the electrons immediately lose this energy, either by an anomalous loss mechanism or by radiation, so that the Ohmic heating is an additional energy loss. The ohmic heating power is Po = na / d-3 x fi. This may be calculated analytically, with the result P_Q = .136Ax(ion heat flux). The energy multiplication factor Q is therefore reduced by the factor (1 + .136A), if the reactor parameters are fixed. Alternatively, to produce a given Q, if the parameter «W is fixed (fixed wall loading x wall radius T- plasma radius) Ro/PDo must be increased by a 2 factor (1 + .136A)1/ as follows from Eq. 9. Similarly Pm-jn increases by (1 + .136 A)H/6. For example, for A = 50, T = 50 keV, we find the following parameters: Ro = 24.80 cm, PQ0 = .95 cm, B = 3.40 tesla, 4 3 ne0 = 2.87 x 10* cm" , Lmin = 646 cm, Pmin = 50.4 MW. Obviously, this is a much larger device than the previous example based on classical ion heat-conduction losses alone; however, it still seems to be a reasonable reactor. Further increases in A would produce devices with extremely large power outputs, which would negate the main advantage of a field reversed mirror compared to a tokamak. For example, A = 200 gives a reactor with Pmin ~ 500 MW. REFERENCES 1. D. V. Anderson et al., in Proc. 8th Internat'l. Conf. on Plasma Phys. and Contr. Nuc. Fusion Research, Brussels 1980, to be published by IAEA. -56-

Cs.icuiation of Tilting Modes in a Spheromak

M. S. Chance, R. L. Dewar, R. C. Grimm, S. C. Jardin, J. L. Johnson"1", and D. A. Manticello

Plasma Physics Laboratory, Princeton University Princeton, New Jersey

One of the most troublesome and fundamental instabilities of the spheromak contigaration arisej from the tendency of the plasma to tilt so as r.o align its magnetic moment with the externally imposed equilibrium field. The free energy driving tnis instability ->.s large, arising from the interaction of the toroidal plasma current with the currents in the external field coils. The global, almost rigid nature of the instability makes stabilization due to magnetic shear or finite gyration radius effects seem unlikely.

Anadyric studies of this mode have been confined to treatment of 1 2 snail departures from a spherical configuration or to cylindrical models. Since the mode could have serious consequences, it is important to understand its parametric dependence on the plasma shape, size of the central flux hole, the presence or strength of an external toroidal field, the q-profile, and the extent to which conductors in the vacuum region can provide stabilization. The application of linear ideal MHD stability programs ' to numerically generated equilibrium ' solutions should provide this understanding. Several features of the PEST-1 stability code keep this from being a straightforward exercise. This code was designed with special consideration to the numerical problems associated with MHD modes in low - g tokamaks with modest aspect ratio. Thus both the choice of the coordinate system and the representation of the displacement vector are far from optimal for the spheromak configuration. In particular, the representation problem leads to "numerical pollution" of 1 • •, when the toroidal field is small, so that there is a spurious stabilizing effect from compressibility unless many Fourier components of tne displacement vector £ are retained. Tilting instabilities have been found with PEST-1 by keeping a small but finite toroidal vacuum field, keeping a modest aspect ratio, employing a high resolution equilibrium grid and including sufficient terms in the Fourier expansion for £.

The difficulties associated with PEST-1 are avoided in the PEST-2 ideal MHD stability code which has been developed specifically to solve the ideal MHD equations at marginal stability, and thus avoids spurious cornpressive stabilization by imposing V«£ = 0 analytically for nonaxisymmetric k'n#0) modes. Representation problems are avoided by working only with the component £>V

to represent the perturbation 6. For instance, choosing 6 to be proportional to the arc length on the pololdal cross section of a magnetic surface reduces the number of Fourier modes needed by at least a factor of 3 over that needed when the PEST-1 coordinate system is used. The results presented in this work were all obtained from PEST-2, using a coordinate system with equal arc lengths in 6.

Our choice of parameterizing the shape of the equilibrium we study is motivated by the exact spherical solution. The outer boundary, in spherical coordinates, is given by A = - r sin9 -j ^ (r) P (cosB), where 0

A typical spheromak result is given in Fig. 1 where the "growth rates" for n = 1 modes are given for a low - B device with q varying from 1.0 at the axis to 0.01 at the plasma surface as functions of the shape of the plasma cross section. For a classical spheromak plasma with a small flux hole, A = 0.1, the system is unstable with respect to a tilting mode which is primarily a superposition of m = ± 1, n = 1 modes of equal amplitude but of opposite phase. As the plasma shape becomes oblate the growth rate of this mode decreases, but at some point a second, shifting, instability sets in. This shifting instability is essentially a rigid horizontal translation or a superposition of equal amplitude m = ± 1, n = 1 modes with the same phase. As the plasma is made more oblate the growth rate of the shifting mode continues to increase while the ti.lting mode growth rate decreases and finally becomes stable.

It is interesting to note the "mode mixing" that occurs in Fig. (1) for 1 <6 <3. Examination of the two eigenfunctions for 6=1 shows a pure tilt mode and a pure shift mode. As 6 is increased and the two growth rates become comparable these modes mix to form pure rotating m = 1, n = 1 and m = - 1, n = 1 modes at 6 =2. Increasing 6 further causes the modes to change again into a pure tilt and a pure shift, but with interchanged identity.

These results were obtained without a conducting shell. It is clear that placing a perfectly conducting wall just outside the plasma should provide stabilization. The geometry of the spheromak necessitates that such a shell give rise to topologicaliy spherical boundary conditions rather than the usual toroidal ones in the vacuum region. The shell must be fairly close, on the average, po that the eddy currents generated in the shell can affect the bulk of the plasma.

For a given size flux hole, or aspect ratio, the mean wall radius needed for stabilization is roughly proportional to the inverse of the maximum growth rate without a wall, althouqh there is a tendency for the mean wall radius needed to stabilize tilting modes to be less than that required to stabilize shifting modes with the same infinite wall growth rates. Also, the larger the flux hole, the farther away the wall can be to provide stabilization. For the configuration denoted by 6 = 2 in Fig. 1, a spherical wall of average distance 0.7 minor -58-

radii from the plasma will provide stabilization. Increasing the flux hole to A = 0.5, or R/a~2, allows the stabilizing wail to be 1.2 minor radii.

We have also investigated the possibility of providing srabi-lization by a purely vertical or a purely horizontal wail. The tilting instability is stabilized with either of these with almost equal effectiveness, while the shifting instability is more easily stabilized by a horizontal wail.

We have studied the effect of an axial current down the symmetry axis on tilting and shifting modes. The presence of this current, 1, • _ , causes the toroidal field in the vacuum to be non-zero. As I ^ is raised from zero, keeping q at the magnetic axis fixed at 1, the eigenfunctions of the unstable modes change continuously into an aiiriost pure m = 1, n = 1 free boundary kink mode. When I, • _ > /? •troroidai' q on the edge exceeds unity and a stable configuration, the small aspect ratio tokamak, results.

It seems possible that the presence of a conducting fluid in the region between the plasma surface and the wail could provide significant stabilization due to line tying. We are at present investigating techniques for modifying the PEST-2 code to incorporate such effects. We are also considering the possibility of modifying the PEST-1 code to use the more general coordinate system of PEST-2 and a better projection of £.

* Supported by the U.S. Department of Energy Contract No. DE-ACO2-76- CH03073. +• On loan from Westinghouse Research and Development Center

1. M. Bussac and M. N. Rosenbluth, Nuclear Fusion, VZ_ (1979), 489 2. J. Finn, W. Manheimer, and E. Ott (submitted to Phys. Fluids) 3. R. C. Grimm, J. M. Greene, and J. L. Johnson in Methods in Computational Physics, 16 ed. J. Killeen (Academic Press, New York 1973) p. 293 4. R. L. Dewar, R. C. Grimm, J. Jfenickam, and M. 3. Chance, Bull. Am. Phys. Soc, _25_ (1980), 864 5. S. C. Jardin and W. Park, PPPL-1706 (submitted to Phys. Fluids) 6. J. Dalucia, S. C. Jardin, and A. Todd, J. Comput. Phys. 3J7_, (1980) 183 -59-

0.12

2 _

0.0 8 -

0.0 4 -

0 I 4 5 PROLATE OBLATE

Fig. 1. Free boundary, n = 1, mode is a force free spheromak with q varying from 1.0 at the magnetic axis to 0.01 at the plasma surface. The shape of the surface is denoted by <5 as shown. -60- CRITICAL BIAS FIELDS FOR TILTING STABILITY IN THE BETA-II EXPERIMENT* Hoi 1 is E. Dalhed Lawrence Livermore National Laboratory University of California Livermore, CA 94550

The PESTt1) equilibrium code and the GATC)(2) ideal MHD stability code have been modified to study stability properties of Spheromak configurations. Of particular interest is the effect on tilting modes'3' of perfectly conducting walls which do not link the plasma. This paper makes use of equilibria and conducting walls specifically designed to model the BETA-II experimental 3t LLNL. Onset of the tilting mode is determined as a function of the bias magnetic field. Comparison with available experimental data shows promising agreement with the numerical results. The equilibria used for this study are generated numerically in the usual manner'l) by specifying the toroidal field function as (h - * X Ah - v "RB,

where To is adjusted to maintain a desired total toroidal discharge current Ip; subscripts I and n refer to the plasma limiting surface and magnetic axis respectively. All results in this paper were obtained for zero pressure. The exponent a is fixed at 1.1, inferred from experimental data from BETA-II.(5; Fixed boundary conditions for the equilibrium are imposed along the edge of the computational mesh (which corresponds with the physical flux conserver in BETA-II beyond 17 cm radius) as i'$ = ^B^R^, where Bg is a uniform bias magnetic field in the Z-direction. The separatrix surface, dividing open and closed field lines, is defined by i> - 0. The plasma limiting surface, at which 7{i>) goes to zero, is defined by ^ = 6i|jn (0 < 6 < 1) where 5 measures the "flux hole" or extent of vacuum region linking the plasma. Figure la shows the poloidal flux contours for an equilibrium with Ip = 330 kA, bias field BQ = 500 G, and <$ = 0.01. Figures 1b and lc show the corresponding magnetic field and current density profiles. Figure Id shows the plasma limiting surface for this case, and the location of the conducting wall used for the stability calculations. This conducting wall coincides with the flux conserver in BETA-II (except for closure of the ends), and remains fixed for all the stability calculations. Critical bias fields for tilting stability are determined by converging growth rates for N,/ = 25, 30, and 40 (N^ is the number of flux surfaces for the stability calculations) to "zero" mesh size. Three finite values of converged eigenvalues are then extrapolated to zero growth rate as a function of bias field. Figure 2 shows the critical bias fields as a function of toroidal discharge current Ip for various values of the flux

*Work supported by U.S. DOE contract #W-7405-ENG-48. -61-

hole parameter 6. The dashed line in Figure 2 is, an extrapolation of the critical fields for <5 = 0.01, 0.05, 0.10, and 0.20 to 6 = 0. Figure 3 shows a typical "tilting" eigenmode for the case & = 0.01, Ip = 500 kA, and BQ = 1000 G. Figure 4 shows poloidal flux contours for equilibria with differing values of <5 and with BQ adjusted to the predictea critical values for tilting stability in each case. All of the equilibria have Ip = 500 kA. Of particular interest from this figure is the similarity in location of the magnetic axis and the location and shape of the separatrix surface, irrespective of the flux hole size.

To compare to the BETA-II experiment, Figure 5a shows profiles of the poloidal (Bz) and toroidal (BR) magnetic fields taken along the Z = 0 inidplane of the BETA-II flux conserver at approximately 50 Msec. Figure 5b shows the magnitude of the poloidal magnetic field on the axis of symmetry at Z = 0, with a tilt of the ring apparently taking place at approximately 95 Msec. The equilibrium shown in Figure 1 was used to model this set of data at 50 usec; profiles from this numerical equilibrium are superimposed as dashed lines in Figure 5a. The magnitude of the discharge current is inferred from the magnitude of the poloidal magnetic field. Experimentally the fields are seen to decay until, at 95 Msec, the ring tilts. In terms of Figure 2, this decay is interpreted as a reduction of discharge current, which is linearly related to the magnitude of the poloidal magnetic field, to approximately 150 kA; the points corresponding to 330 kA and 150 kA are shown in Figure 2 as points A and B respectively. It is seen that point A falls well within the predicted stable region while point B has fallen into the unstable region. It must be pointed out that in inferring Ip for point B, it was implicitly assumed that the magnetic field profiles did not change shape during the decay. That tnis is true is far from certain, and has direct bearing on the inferred discharge current. The effect of different profile shapes, the inclusion of finite pressure, and the effect of new shapes of flux conservers being installed in BETA-II are currently being investigated.

REFERENCES

1. Johnson, J. L., et al., J. Comp. Phys. 32 (1979) 212. 2. Bernard, L. C, et al., in Proc. Ninth Conf. Num. Sim. Plas., Northwestern University, 1980, Paper OD-7. 3. Rosenbluth, M. N. and Bussac, M. N., Nuc. Fus. 19 (1979) 489. 4. Papers 2Q5-2Q8 presented at APS-DPP 22nd Annual Meeting, November 1980, San Diego. 5. Hartman, C. W., private communication.

FIGURES

1. Poloidal flux contours (a), magnetic field and current density profiles (b,c), and location of conducting wall (d) for typical BETA-II equilibrium. 2. Critical bias field magnitude versus toroidal current Ip. 3. Typical tilting eigenmode. 4. Poloidal flux contours for equilibria of differing flux hole sizes near critical tilting point. 5. Experimental and numerical magnetic profiles (a) and magnitude of poloidal magnetic field versus time (b) for a particular BETA-II shot. -62-

TWOIML IMC POLftltW. nfSWTIC morute cwwon Twin martin ;

Fig. la F1g. lb Fig. lc

1.8 CONDUCTING WALL UNSTABLE 1.6 6 • 1.4 PLASMA 'il/// SURFACE 6 = {U)5 77//Y JSSC BG 6 » O.Ol/ (kG) c OF SYMMETRY 0.6 Fig. Id 0.4

0.2 STABLE

0.2 0.4 0.6

Ip (MA)

F1g. 2 -63-

0.01 0.05 <5 - 0.10 0.20

F1g. 4

TCTonw. •« mjicn. narn: raa

F1g. 5a Fig. 5b -64-

EXTERNAL TILTING OF SPHEROMAKS WITH LINE TYING

John M. Finn*, NRL; Allan Reiman, Univ. of Maryland

In their study of spheromak stability, Rosenbluth and Bussac[l]

found that a spheromak must be oblate in order to stabilize the

tilting mode with a conducting wall at the separatrix, i.e. the

plasma boundary. Moreover, an oblate spheromak is unstable to

•') free boundary tiltinq mode in the absense of a conducting wall.

Rosenbluth and Bussac came to the rather pessimistic conclusion

•nat. --i shell of radius R(l-.2e) is required to stabilize the

external mode, where R is the plasma radius and e is the ellip-

' i •' u y of the plasma surface. This result is for nearly spherical

T.-qui 1 ibr ia, i.e. small e. In addition, their analysis is

limited to equilibrium profiles having j = UB inside the plasma,

where UL is a constant, independent of position.

We have numerically studied the wall stabilization of the free

boundary tilting mode for general shapes and current, profiles.

We find that stability to external tilting improves dramatically

w:-h tailoring of the current profile. We also see a dramatic

improvement in stability when e is not small.

We take a force free equilibrium satisfying j = U,B, where

Lc is a function of the poloidal flux yt.(By taking LL=O on the

open field lines, we include the effects of line tying.) We depart

only slightly from the state which has minimum internal energy,

and thus is optimal to internal modes. For this purpose we find

y it convenient to choose U.( y^) = }i

is nearly constant except near the separatrix \b = 0. The

conducting wall is cylindrical, i.e. at r=a, z=0,L. (Fig. 1)

This is the geometry of the moving ring reactor, with L >> a. -65- • :: • 1 i v : •'•:•- .:••! tT. iri.'. - : .'.• > ."-"i '.'•< - '.''.:'. . ". ' TTX-'':':'• + o stucv the stability of these equilibria. In this code we integrate the perturbed vector potential Al (= ^xBO) in time rather than the displacement £ to avoid numerical difficulties near the points where B = 0. That is, we integrate ) /ji.TAl = f BO /o)x

( /^(^)Bl-jl) , where ^ represents components perpendicular to the equilibrium field BO. The growth rate is computed as f— £w/Ti "*~ where £w is the usual potential energy and T = [ dxfiAl /DO .• 2 •

To speed up cor.verqence to the eiqenmode, we taXe BO = Q ;-\n<\ add y Al to the equation of motion.

The displacement ¥ for an unstable free boundary til* is shown in Fiq. 2 for the equilibrium of F'iq. 1, with e=-.4.

The physics responsible for the instabi1ityr1], namely that a rnarqinally stable internal tilt yields a magnetic presssure imbalance near the pel curbed X-points, is evident in this fiqure. In Fiq. 3 we see the axial wall position required for stability as a function off. The radial wall is near the plasma. For

4=0, the wall must be closer axially than indicated in Ref. 1.

This is not surprising, since the wall here is not the same shape as the plasma. For Q > 0 the axial wall position increases dramatic

-illy, so that for ft =.12, the wall can be twice the plasma length.

This result seems reasonable, since the drivinq force for the instability is localized near the separatrix.

Figure 4 shows how the required axial wall position varies with oblateness. Again the axial wall is near the plasma. Note again that for e small Zw/Zp is less than l-.2e. However, at e=-.2 stability begins to improve rapidly. The scaling of Zw/Zp with e becomes approximately quadratic. At e=-.4 the axial wall stabilizes at about 3 times the plasma length. *Present address: Science Applications. This work supported by D.O.E. 1 M. N. Rosenbluth and M. N. Bussac, Nucl. Fusion 19,489(1979). -66-

Flg. 1. Flux surfaces of the equllib- riun.

r • — i— •• • • i • •!• • • •> i i

•.« 4it4*t***''

•*-v-^^ •/////*•••• •.1

< < *•<

•.I ..T . . . ..t.ttTr.TTTt1 M 3

Fig. 2. Poloidal displacenent ^. The X-polnta are at z-.13,.87. -67-

WAUU

/.©•

0.T

.az. >oi « .

Fig. 3. Axial wall position required for stability vs. h

Fig. A. A*Ial wall position vs. ellioticity. -68- TURBULE:NT RELAXATION OF COMPACT TOROIDS

Eliezer Hameiri Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 and James H. Hammer Lawrence Livermore National Laboratory,University of California Livermore, CA 94550

We propose a maximum entropy theory of the turbulent relaxation of compressible plasmas. The entropy is maximized subject to a small set of constraints determined to be the only ones consistent with a general turbulent state. Taylor's minimum energy theory is recovered as a special case and relations to ideal MHD and tearing mode stability are discussed. We find good agreement between field reversed 6-pinch experiments and the theory. In analogy to the incompressible case,l we assume that MHD turbulence provides an irreversible drive toward a relaxed state consistent with thermodynamics and the constancy of invariants of the motion. We assume an isolated system, so that this means maximizing the total entropy, S =y"p-i dV (,j = £n P/pY for an ideal gas, p = mass density, P = particle pressure, y = ratio of specific heats), rather than postulating the Gibbs' distribution as is done for the incompressible case. The maximization must be performed subject to some constraints in order to yield nontrivial results. For an isolated system the total energy and mass are held fixed. A component of the magnetic flux is also held fixed if conducting boundaries are assumed. To determine what other constraints exist, consider an ideal plasma undergoing a turbulent phase. In general the time evolution involves the formation of shocks,^ including possible magnetic topology changes. Only those quantities which are conserved in the presence of shocks, i.e. conserved in the zero diffusion limit of a diffusive system, can be acceptable invariants. This and other requirements (e.g., gauge invariance) reduces the infinity of invariants associated with shock-free ideal plasma motion to a small discrete set. An acceptable invariant, fq dV, must be related to a conservation law H + V - G = 0 (1)

where £ is a corresponding flux. The jump condition across shock fronts that follows from Eq. (1) must be consistent with the conservation laws for mass, momentum, etc. In addition, the flux through the system boundary should be zero, f G_ • n ds = 0 where n is a unit normal. The jump conditions that follow from Eq. (1) in general are given by - U[g] + n • [G] = 0 (2) -69- with U the normal speed of the shock and n its unit normal, and the brackets denote changes across the shock front. We demonstrate the procedure by applying it to the magnetic heiicity A_ • B^, where A is the vector potential of the magnetic field JB. The ideal Ohm's law is

|f A + V(J) + B x u = 0 , (3) with arbitrary potential . Combined with Faraday's law this yields

^ f n _ n\ i n ^ f n. .. f .. .. n \ 1 nj.\ _ rt (4) which is.of the form of Eq. (1) and yields a jump condition of, the form of Ea. (2). Thjs^jump condition can be shown by straightforward manipulation to be consistent with the condition - U[A] = -n [<(>] corresponding to Eq. (3) as well as [B • n] = 0 and [if x A] = 0 which follow from V • B = 0 and B = V x A. Thus f /\ • B dV meets the shock condition. For quantities satisfying the shock requirement, the conditions of square integrability of g and invariance under arbitrary, uncorrelated ideal magnetic topology changes (which implies gauge invariance) further restricts the class of acceptable invariants. The number of invariants is larger if the turbulent process is symmetric (e.g., axially or helically symmetric). If a vector £(xj can be found L-uch that p£ • _§ is the conserved component of mechanical momentum (e.g., £ = r6 in axisymmetry and £= z in plane symmetry) then the set of acceptable invariants (subject to which the entropy,fo 4dV, is maximized) consists of volume integrals of the following quantities: pu_2/2 + pe(p,4) + $-/2 (the energy, where e = specific internal energy); p; A * J3; pA • K\ PU_ • £; pu_ • jj_ k . _£ ; as well as total magnetic flux inside a perfectly conducting container and total toroidal angular momentum if the container is axially symmetric. This set is complete with regard to invariants not involving the flow, and is possibly complete with regard to the flow invariants (/u_ • ]$ dV used as an invariant for incompressible flow^ is not constant in the compressible case since the entropy is not constant along field lines). The pA_ • £ invariant we name the frigidity since it expresses the extent to which the magnetic field is tied to the fluid. If no symmetry exists then the quantities involving C. are not invariant. It is also possible that the turbulence may be localized (by shear, for instance) so that different frigidity invariants exist in bands where the turbulence has a given helical symmetry, e.g. tearing about rational surfaces in a Tokamak.

An immediate consequence of the first variation of the entropy is that the temperature, T(~ 3e/3i) is constant in the relaxed state. If we do not assume symmetry, then the only invariants are the total mass, energy, helicity and toroidal flux. It follows that ~ ' <. *re constant throughout the plasma, p ib detennm. .., . ,c mass ^on. , _• J IS determined by e(J,o) = [E - B_2/2 dV]/M, where E and M are me total energy and mass respectively. For fixed P, e is monotone increasi m; i" _-, so maximizing i (and hence the total entropy S) is equivalent to inimmir ing / _B^/2 dV, subject to the helicity and flux constraints. W>- th;;s recover Taylor's minimum energy formulation^ which provided an appeal ir"j e'.ol nation of self-reversal phenomena in z-pinches. -70-

A state which is of maximum entropy can be shown to be stable to ideal MHD modes. If the entropy is maximized subject to the full ideal constraints, the second variation yields: 6S = - 1/T {fiW + (y - 1) [pgy (J"PV • £ dV)2 - /P(V • K)2 dv]} (5) where £ is a displacement vector characterizing the ideal perturbations^ and <5W is the familiar stability integral from ideal MHD. Use of a Schwartz inequality shows that the expression in the square brackets cannot be positive hence 6S £ 0 implies ) = ug/r is also linear in ty). For Be = 0, this pressure profile is known as the "rigid rotor" profile,6 corresponding to rigid rotation of the current-carrying species (we would prefer the term "frigid rotor"). The conservation of our invariants allows us to uniquely relate initial and final states of a system that undergoes a turbulent transition. In principle, once the fill pressure, bias field, capacitor-bank energy, etc., for an experiment are known, the resulting plasma parameters can be predicted, although lossy processes (e.g., impurity radiation, heat flow to walls) outside of the theory may intrude and degrade the invariants. The axisymmetric theory outlined above can be applied to field-reversed 6-pinch experiments. The toroidal field is zero so that helical disturbances are not expected (A • J3 = 0 in this case, but we retain the axisymmetric frigidity). The existence of open field lines in these devices prompts us to model the experiments by a two stage process. After a uniform plasma with imbedded bias field is radially imploded by the 0-pinch coil, we assume a radial rigid rotor profile with a separatrix is established preserving our invariants. This occurs on a radial Alfven time scale (~1, J ys typically). On a longer axial Alfven time scale, the matter on open field lines exhausts and the plasma adjusts to axial force balance. To model this second phase, *we replace energy conservation by axial force balance and preserve frigidity and mass inside the separatrix only. The resulting "truncated rigid rotor" equilibrium (a rigid rotor inside the separatrix and zero density outside) is uniquely related to the initial state; in particular we find the vortex (magnetic null) radius as a function of bias flux as shown in Fig. 1 in comparison with experimental results (the curve is quite insensitive to the initial energy)6. The discrepancy between some of the experiments and theory may be due to pre-ionization techniques that do a poor job of imbedding the bias flux in the initial state. -71- The authors wish to thank professors H. Berk, H. Grad, H. Weitzner, and Dr. M. Mond for extensive discussions. Part of this work was done while one of the authors (E.H.) was visiting the Lawrence Livermore National Laboratory and he is grateful for the hospitality extended to him. This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48 and by the Courant Institute of Mathematical Sciences under Contract No. DE-ACO2-76ERO3O77; and under the auspices of the U.S. Air Force Office of Scientific Research by the Courant Institute under Contract No. 76-2881. REFERENCES

1. D. Montgomery, Proc. Indian Acad. Sci. A86, 87 (1977). 2. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, Philadelphia, 1973). 3. L. Woltjer, Astroph. J. 130, 405 (1959). 4. J. B. Taylor, Phys. Rev. Lett. !33, 1139 (1974). 5. W, A. Newcomb, Nucl. fusion, Suppl. Pt. 2, 451 (1962). 6. W. T. Armstrong, et al., Los Alamos National Laboratory Report, No. LA-UR-80-1585 (submitted to Phys. Fluids).

FIGURE CAPTIONS

The ratio of vortex radius to wall radius vs the ratio of bias flux in the initial state to the final value of flux at the wall for field-reversed 6 pinches. The curve is the theoretical result and the circles are experimental results from Naval Research Lab (NRL), Garching (G), Los Alamos (FRXA, FRXB), and Kurchatov (K).

0 0.1 0.2 0.3 C4 0.5 0.6 0.7 0.8 0.9

Figure 1 -72- IDEAL MHD TILTING MODES FOR ARBITRARY PLASMA PRESSURE AND CONFIGURATION* James H. Hammer University of California, Lawrence Livermore National Laboratory Livermore, CA 94550

The MHD tilting instability has been predicted to be a severe limitation on compact torus equilibria, requiring a close-fitting conducting shell for its suppression [1]. Here I describe the extension of the earlier work to plasmas of arbitrary shape, and pressure with the limitation that the separatrix and conducting wall shapes are close to spherical. The basic analytical technique, first used in Ref.[l], is an expansion about a spherical system. An equilibrium with a spherical separatrix (the plasma need not extend to the separatrix) and a spherical conducting shell at the same radius as the separatrix has a zero frequency eigenmode: a rigid rotation about any axis, £ = ^ = co x •£.,wher e £ is the displacement, u is a constant vector, £ is the radius vector. In the vacuum region (if any), a vector potential, Aw = ^wx S_, would accompany the displacement £_ . If there is a deviation from a spherical separatrix and wall by an amount 0[z), e « 1, then it is expected that i = j^ + k\ and A = Aw+ Aj, with |Cl|/5aj ~ |Ai|/| A^l =<^(e) and £i, Aj obeying appropriate boundary conditions. The usual MHD energy principle (Ref. [2]) is expanded to first order in £, allowing reduction ofSW to a surface integral over the conducting wall

^ is a unit normal pointing into the wall. Note that ^\ and Aj have dropped out completely. This is a consequence of the self-adjointness of the energy principle. Since £,0 and A^ are known functions, the sign of 6W in Eq. 1 is determined completely by equilibrium properties; in fact it turns out that the separatrix, wall shape, and the zeroth order |JB| at the wall are the only quantities necessary. The wall and separatrix shapes are determined by:

Ps ^ s (2) and

P = pLl - e{e) - 6(6)J; n = Vpw where p, 6, and $ are spherical coordinates, e(e) is the separatrix shape factor and 6(9) is the fractional separation of wall and separatrix. Both e(8) and 6(e) are small and of the same order. Equations (1) and (2) and the property v • By = 0 can be combined to yield

TTD^ /"IT ' /

: JQ 'j-T^p^ /

+ 6(e)[(,99 ^/ +\A\sin2e . (3) *Work supported by U.S. DOL contract #W-7405-ENG-48. -73-

through first order in e. BQ is the zeroth order (spherical) equilibrium field evaluated at pg, the zeroth-order radius of the separatrix. As an example of an application of Eq. (3), set e = r cos^e, Bg = Bg sin 0, and 6(6) = 6, a constant, to recover the Rosenbluth-Bussac result: 1 < -0.2 I" (4) for stability [2]. This result indicates that_the stabilizing wall must be quite close to an ellipticajjy oblate plasma (e < 0). Equation (4) can predict stability even for e > 0 (prolate) if the plasma is slightly removed from the separatrix, i.e. a small flux hole exists. It shows that an elliptically prolate plasma can be stable if the wall is inside the separatrix. Equation (4) has meaning for arbitrary plasma pressure since a family of arbitrary 3 spherical equilibria exists with the property BQ <* sin 6 (Ref. [3]). 8 A general result apparent from Eq. (3) is that the extension of the wall inside the separatrix is stabilizing (the term multiplying 6(9) is always negative). Another feature of Eq. (3) is that the e(6) and 6(9) terms are not coupled, so that the separatrix shape and relative separatrix-wall spacing can be optimized separately; i.e. 5W = <5We + 6W5. Consider the term containing e(6). If equilibria with BQ = Bg sin 6 are used we find that

6W = £- Bn / sined eE(e)(3 cos <3 - 1) (5) c u jQ This indicates that c[e) should be positive for |cos 9| < l/\/3 and negative for I cos 6| >l/vT to ensure 5W_ > 0. The optimal separatrix shape given the constraint |e(9)| < F/2, is shown in Fig. 1. The optimal shape is oblate but has "indentations"_near S = cos~l +1/V1. The stability condition for this shape with 6(6) = "6 is given by

^ =O.29F , (6) so a 50% improvement is possible for fixed oblateness (e fixed) over the elliptical shape. Next consider the term proportional to 5(6). For BQ = Bg sine, 6W\. becomess

o 2 J r sin 6 6(6) (cos2 8 + 1) d9 6W6 = - -y-^ BQ J sin 6 6(6) (cos 8 + 1) d9 . (7) The optimal wall spacing will depend on what other constraints exist. As an example we can maximize 6W5 with a fixed wall surface area and the constraint 0 < 6 (8) < 8. Through first order in e, this means that f sin 66(e) de is held fixed. The optimal wall becomes cose < a (8) •i cose > a where a is determined by the value off sin 66(e) de. This indicates that walls that fit closely at the poles (cos e = ±1) are best. Because of the -74- siowly varying integrand in Eq. (7), however, the difference between the best and worst wall shapes for fixed wall area is at most a factor of 2 in the magnitude of 6Wx. Finally, the effect of changing Bg due to different equilibrium profiles and varying flux hole size can be investigated. The angular variation Be1* sin 8 came from a broad equilibrium current profile that extends up to tne separatrix (dP/di/j = constant, fdf/dij; = \ty, where P is the pressure,

f°r these equilibria [3]). For equilibria with the current peaked more sharply on the separatrix, Bg is expected to be a flatter function of 9, since in the limit of a sharp-boundary flux-excluded equilibrium, the separatrix approaches a constant |Bj surface except at 9 = 0, TT , In the other limit of current peaked near the vortex point because of steep pressure and field profiles, a more peaked behavior of Bg(6) near 6 = TT/2 is expected. All of these effects can be modeled roughly by taking

n Be-sin 6 , (9) where n < 1 corresponds to current peaked near the separatrix and n > 1 corresponds to current peaked near the vortex. The critical wall position for an elliptical separatrix is now given by 7 < =£" e (10) (n + 1) (2n + 3) for stability. The coefficient of e in Eq. (10) is maximized at n = (VT3 + l)/2 giving

s" < -0.256 7 , (11) which is only a 25% improvement on the n = 1 result. The optimized separatrix shape for arbitrary n is similar to that shown in Fig. 1 except that the indentations occur at |cos &\- 1/V2n + 1 in _ general. The critical wall position for the optimized shape and 6(9) = 6 is a monotonically increasing functon of n which asymptotes at large n to the result: < -0.97 7 , (12) for stability. A current profile peaked sharply near the vortex point and an appropriately shaped separatrix can therefore yield wall-stabilization conditions that are several times more optimistic than those for broad current profiles and elliptical separatrices. Assorted other results of the nearly-spherical tilting analysis include: placing a conducting post downjthe symmetry axis is stabilizing, yielding the stability condition r^ > 16/19 efor an elliptical separatrix, where r is the radius of the post; if the equilibrium includes a slow rigid rotation about the symmetry axis at frequency ft, then a sufficient condition for tilting tne stat c stability is ft > Ystatic i growth rate. In conclusion, I have extended the ideal MHD stability analysis of tilting modes to arbitrary plasma pressure and shape with the limitation that the separatrix and wall shapes must be nearly spherical. I find that a -75- conducting wall must in general be quite close to the separatrix (or even extend inside it) for stability. The stability conditions found show *hat improvements over the previous result, Eq. (4), can be made by appropriate shaping of the separatrix and wall and by providing equilibrium current profiles peaked near the vortex point. Even under the best conditions, however, the theory indicates that conducting walls must be^(e) distant from the separatrix to suppress tilting. REFERENCES 1. Rosenblutn, M. N., and Bussac, M. N., Nucl,. Fusion 19 (1979) 489. 2. Bernstein, I., Frienan, E. A,, Kruskal, M. D., Kulsrud, R. M., Proc. Roy. Soc. Lond. A-244 (1958) 17.. 3. Weitzner, H., Berk, H. L., Hammer, J. H., Lawrence Livermore National Laboratory, private communication (Sept. 15, 1979). 4. Hammer, J. H., "The Problimak; a Non-tilting Field Reversed Equilibrium, (Prpc. of the Sherwood Meeting, Tucson, Arizona, April 1980). FIGURE CAPTIONS 1. Optimized separatrix shape for Bg « sin 6 at the separatrix.

Figure 1 ilMIJLATION OF THK l'r'RMATKJ:

PRINCETON SP.MHKi MAK

ii. ;''. Lui , ('. K. >':.u, :

Columbia Univ:;i , ;. •

We simulate several schemes with dj ;' r. r. -i.t ! d structur.. and current • . i ..lining for the Princeton S-l Spherorriak , ;•.•; r h '.... i:n of helping in its i :.-. and future operation. The method, i .if.cd ...n ;:• rically solving the i -•"! f —fluid compressible hydromagnetic . gu.-i i .i oi •-, •.-. idissipation, has b.een . .r.fully used in the last few yeats to .•i.-.-.u.i lie i-w : Lnchi-5, belt i.'. !:-.:s, reversed-f ield pinches, etc., \ i ••> d;:.-, ^. ... i n . IOOQ agreement 1 measurements (see e.g. [1]). The | icsnn 'ii-i:' ton repre.yflits the t '.-ompli cati-d geometry attempted to date. .• . l..i :: I.he flux fujiction boun- . nii'lit. ion:; to the externally cunt r- i 1 • . ";••;!.. i.'iu: rent .-•, for example, •-.••<• difticult and non-obvious. A -:• -t .J : J ••; u> .. j i Lou .if I'-.i:, formulation . iv. n ,it ,( tin 'Vir,u:; S'/Hlposium ['.:.] i liid ..'• .: .ii . ; r< -i --at 11 !ien-; i n- !, '.<•• i-ti.il! cotiQ'titratf: on the r< suiis .,:,;: •'• .; el .-T.i t !.•

i-i-is. t(b) and 2(b) show schema i. i < • .-.:! arijiW ••nii.rit:', in tlif i da I piano of the S-l geometry, thi f i : ;•: puir: Ci'i 1.-. in tfiL j;las- ':....• latter without pinching coils. T;: .t.:., in the ext-.-rnal coils 1 ii 1 li.rium coils) are considered as i>C, ady turned on before the in.i i :-. formed. A core, approximated 1. ,• -on i :: our calculations, i: ih~ a .solenoidal coil with current ! . toroidal ouil with current I ! rmer induces toroidal fields and pol ir! tits in the ; lasma. while c ..i't<-r inducts poloidal fields and tor .rj. i r,ts. A key feature in our ..bit ions is that the toroidal coil cur i.; initially large, and then !\se:i in direction (figs. 1 (a) and .!(ai i ; .-; done for two reasons: t, a strong initial poloidal field avo occurrence of the x-f>oint he midplane; our previous calculations that this occurrence results IK: plasma collapsing and re-expanding, t. h yielding a poor spheromak ,'. -on'otry. Second, the reversal induces J h.i,.i urrent in the plasma, which

rir..~hes che plasma and compresses the- tei;o i da 'ield :nore effectively.

Typical current histories are shown in f ui,.i (a) and 2 (a) . In the former the pinching coil current I,., i:; also included, and the solenoidal current has not been shown in either case. ince our method requires pres- -„ uii i_nc; V..^J.C; tinj ij.i on the pinching coil) and the toroidal field B, on the core as input boundary conditions, the currents are all calculated and not prescribed,- the wiggles in the resulting current histories correspond to the bouncing of the plasma.

* Work supported by USDOE

# Present address: Los Alamos Scientific Laboratory -77-

In the case with pinching coils, fig. l(c) shows the development of the flux surfaces, the spheromak plasma being the closed surfaces on the left-. We used an initial filling density of 2.4 x 101-' cm~^ for our calculations. At 60 sec, the plasma density inside the closed flux surfaces reached 6 time. the initial, the temperature reached some 30 eV, and the toroidal field was compressed to 12 kG. This represents a satisfactorily formed spheromak plasma. Another case was run with the toroidal field on the core crowbarred at the peak value, but the resulting flux contours and the subsequent compression uiJ not differ significantly.

In the case without pinching coils, an equally successful spheromak plasma is formed as shown in fig. 2(c)< The removal of the pinching coil does require an additional equilibrium field coil (compare figs. 2(b) and !(:.. to improve operation. The resulting density inside is a bit lower than bt-i.^ri , being about 5 times the initial, and the temperature is about the same; \:n:-:- ever, the compressed toroidal field is significantly lower, being only about- 600 kG or half of the previous case.

We also made another calculation without pinching coils, in which the fluxes were prescribed to rise at the same rate as in fig. 2(a) , but the toroidal field on the core was 50% greater. This spoils the operatic:-: the stronger fields near the midplane now splits the spheromak plasma into two pieces, resulting in a doublet like configuration.

In both cases (with or without pinching coils), we have also seen r.:ny other modes of failure to form a spheromak. Either the plasma comprer.sea rapidly and then reexpands, and does not form a spheromak plasma at all, or the density inside the closed flux lines becomes smaller than the ambient, giving a partial vacuum rather than a compressed plasma. The two cases pr< - sented in figs. 1 and 2 were achieved after considerable trial and error in programming currents and fields. This is one illustration of the use of computer simulation in the design and prediction of an experimental device.

References:

1. H. C. Lui, C. K. Chu, Phys. Fluids 1£, 1277 (1975); 19_, 1947 (1976).

2. H. C. Lui, C. K. Chu, Proc. US-Japan Symp. Compact Torus (Princeton 1979), p. 176. /xsec

t = 0 400

107 G-cm2

t = 20

20 30 40 50 60 (/xsec)

ao I

(b) Schematic geometry t = 40

(a) Fluxes and t = 60 currents 40cm

E 'c G (c) Development of the )00cm spheromak flux surfaces

Fig. 1. Spheromak formation with pinching coil. 60{jj.sec)

(b) Schematic geometry

W -400h (a) Fluxes and 40cm currents -500F (c) Development of the flux surfaces 'if 100 cm

Fiq. 2. Sfheromak formation without ] mclunti coil. -80-

i ')<]') m! (,. I'll i i.-y fusion Mudios I.at,oratory inclfcir ! ngirieori n

"•'•,.' '•''.,i'!'] Plaor'oid HL'.'IC1'or f''i'K) is an attractive •! 11 ernat i ve ' •: -1 • ] r • i •,:•;,'', v.:>uv;e in which S:.-hen' s/iV plasroids are envisioned tu be for:."?''!, . ressed, burned, and expanded as the plasnoids translate through a spr liriHir n-aLti-ir no,1;i 1 es • - . Altho'jqh auxiliary heatinq of r.h'.' - - "-li ds pi,:y h-? j'Ossiblf', ! rio '.\'-'V scenario wO'jl'i be especially \ nt en's: i 'Av;-r iiT.ay -nri coi'ipresr, i r:.n dione were sufficient to heat the ;:1 d-^'K:i •: ."i i ';nif i un t.e^'i'erat ire. !n the present work, we will study r.ho :''sr •";r* con.Ii •: ions jnder which n Sjilieronak plasnoid could be expected ' !r:h ignition ,'ia a conbi nation of ohnic and compression heatiirj.

1 '.!.'. le es i';;at''S of the ef t ec'i/en< ss nf r.hrir. hivifim; v; ,< L;r:Lr"j- : '••,-,.1 ." jr, ('<•"• obtai iod by wri'."in;j • r,e power l.ilance in t he for:11

WM ' r>l'_M " M ' t

.vh;c.h .i1,, i', trip tot..-;l piasn'jiii ridijnet. IC rnerny, <.-"> is ,> i^easjrL1 c? ! '. rage luc.il plasna beta, ;.i is the classical ::•;! rj not i c deciy tine, . i n • 1 is the ;•• 1 rss"*'a energy c^nfinenerif time. Since the MHii-stahle beta is i ted to thi1 4-'i% ranqe'"', Equaticn (1) inplios t h d t for q;.iod i sma hcdt. i r>• i:

rhis criterion is S'newhat. i^orn i nf urn,it i ve when ex|iressei in ter>'is of a

'quality of confinement" paraipeter, oc, necessary for heating of the plasma:

f- T WZ 3 i: Qc > 3 x 10- .--^fL- . (3) TB c B(T) z lnA

where 'g is the Bohm time and Qc is the minimum nuuher of Bohm times required for power balance.

A parameter study of Qc vs. Te indicates that a typical reactor scale Spheromak plasma would be expected to heat rapidly (i.e., Qc < 1) into the 100-500 oV range due to the classical ohmic decay of W^. At some point just past this level, however, Qc rises quickly into the 10-100 Bohm time range and it is conceivable that an ultimate temperature limit might be -81-

n. posed by transport limitations. One possible method of increasing the ; 1 =?srna temperature beyond this point would be adiabatic plasma compression.

It can be shown that, as the externally applied magnetic field Bo is

raised, Wf.i increases proportional to Bo ' , Te increases in p.ro- r .•irtion to lie, and n:^ increases as Be'. mr power balance during expression then becomes

u ff ' - 1) <>'->>W <°,>vl f M [ B } M M B o + o o,.^

where the zero subscripts indicate values before compression, :c is the compression tine, and the compression factor, fy •" Be/13e . The juality of confinement parameter including compression becomes

•'•<••<'ii we see, for example, that with an adiabatic compression of fj; " 4 and

c .001 M. it v/ould be possible to ignite the plasma under much poorer confinement conditions (i.e., Qc 1/^0 L)r ) than would be possible with ohnic heating alone. <-ie are in the'process of studying more •!f• *"ai led ()- requirements for various compress i on-to-i gnition scenarios by i •n.'irporat ing several possible plasma transport novels into the framework of i i 1/2-P Spheronak computational model.

•>. •'. • J1 s o'' ". "!. '\ Todd, J. Ci. '""• i 11 i T a: >, an:; :;. 'I. Mi ley, Bull. r'hys. Soc. , /A, 1H?2 (1979).

f M. i|,abayashi and A. M. M. !odd, '.'uclear fusion, 20, 571 (19^0). -82-

A COMPUTATIONAL COMPACT TORUS EXPERIMENT* B. McNamara, J. L. Eddleman, J. K. Nash, J. W. Shearer, W. C. Turner Lawrence Livermore National Laboratory, University of California Livermore, California 94550

INTRODUCTION The 2D (r, z, t) rnagnetohydrodynamic code, HAM(l), has been used to model + he performance of the magnetized, coaxial plasma gun injecting a plasma ring in a copper flux conserving chamber, as used in the Beta-II experiment'^) (Fig. 1). The qualitative results of the calculation appear correct but a quantitative calibration of the numerica1 against the laboratory experiment remains to be done, Two principal defects are immediately evident in the laboratory: the code has no model of the sputtering of impurities from the gun, and these appear to be important in providing energy losses from the plasma and enhancing the resistivity. The second defect of the simulation is that, being axisymmetric, it is unable to model MHD instabilities. Stability calculations by Dalhed(3) and the laboratory results show that the guide field of 500 gauss used here could lead to rapid tilting instability after the toroid is formed. Some of the more important observations are as follows: (i) The gun produces two plasmas, a very dense one pressed radially outwards on the anode, and a lower density plasma around the cathode which accelerates ahead of the dense plasma to become the compact toroid (Fig. 2). The dense plasma never really separates from the gun and falls back into it at the end of the discharge. Laboratory diagnostics are not yet sufficient to determine if this is correct. (ii) The field line reconnection seems to be affected by plasma flow through the gap between the gun and the flux conserver (of Fig. 2) and even more by plasma flow around the curvature of the expansion chamber (Fig. 3). (iii) The plasma resistivity is anomalous in the reconnection regions -- using a model due to Sgro'4) -- and, although theory would indicate that the reconnection rate should be independent of resistivity, this point needs to be checked in more elementary simulations. (iv) The efficiency of converting the capacitor bank energy into plasma and magnetic energy trapped in the toroid is only a few percent. This could apparently be improved by varying the power imput to the gun and better control over the reconnection process. (v) The current and voltage variations have considerable effects on th° final toroid produced, which needs to be understood in ter.ns of some simpler model.

*Work supported by U.S. DOE contract #W-74O5-ENG-4 -83-

I. Initial arid Final Conditions: The computational grid was a 223 x 46 mesh of one-centimeter squares. The external guide field has a value of 500 Gauss and the poloidal flux in the annular gun barrel is -1.94 x 10^ Gauss -cm?. The gas fill was 3.125 Atm - cc. of deuterium, initially distributed according to a Gaussian profile centered about the mid-point of the gun p(z) = 5.1272 x 1015 exp {-.00281*5 (z - 50)2} „

In the guide space, a uniform background density of 1.87 x 10^2 was assumed. The plasma temperature in the grid was initialized at .1 eV for both the ions and electrons, except in the region 22 cm £ z £ 28 cm where the temperature was increased to 5 eV to initiate the discharge problem. A Saha ionization model is employed and the plasma resistivity is the sum of Spitzer and electron-neutral resistivity and Sgro's anomolous resistivity. The compact toroid is shown just forming in Fig. (3) and by 15.6 usecs is isolated from the main body of the plasma which returns to the gun. The parameters of the final toroid -- at 19.2 usecs -- are summarized in Table I. The overall efficiency of the gun is about 3% in energy delivered to the toroid and about 30% in the number of particles. TABLE I. CHARACTERISTICS OF THE COMPACT TOROID Time = 19.2 ysecs Toroid Thermal Energy 0.602 kJ Toroid Kinetic Energy 0.0252 kJ Toroid Poloidal Field Energy 0.369 kJ Toroid Toroidal Field Energy 0.0067 kJ Total Energy in Toroid 1.00 kJ Bank Energy Supplied 35.00 kJ I - Ring Current -75.6 kA Density Average 3.89 x 1014 cm-3 Total Particles 3.12 x 10*9 Mean Electron Temperature 39.0 ev Mean Ion Temperature 41.7 ev Axial Velocity at 0-pcint -1.25 cm/usec Field on Axis -1.05 kg Reversed Flux at Axis -17.2 x 102 kg cm? Ring Volume 80.1 L Toroidal Field at 0-point -706.0 gauss.

REFERENCES 1. M. Stephen Maxon, J. L. ERddleman, Phys. Fluids 21., 1856 (1978). 2. D. V. Anderson, et al., in Proc. 8th Internat'l. Conf. on Plasma Phys. and Contr. Nuc. Fusin Research, Brussels 1980. (UCRL 83518) 3. Hollis E. Dalhed, 1980 3rd Symposium on the Physics and Technology of Compact Toroids in the Magnetic Fusion Energy Program, LASL. 4. A. G. Sgro, Phys. Fluids 2±, 1410 (1978). -84-

FIGURES

„ 1

«3 O O" > o — ru to r> o

Fig. 1. Gun and Flux conserver in the Beta-II Experiment. Solid Lines are conductors, lighter lines enclose insulating zones.

S f S S

Fig. 2. Density contours at the end of the gun rundown at 3.9 ysecs. Low density, high speed plasma near the inner electrode becomes the compact toroid.

Fig. 3. Plasma density at 13.8 ysecs as compact toroid disconnects. Resistivity in the reconnecting region is anomalous. -85- FORMATION AND EVOLUTION OF THE PS-1 SPHEROMAK

A. G. Sgro Los Alamos Scientific Laboratory H. C. Lui and C. K. Chu Columbia University D. Winske University of Maryland

A numerical analysis of spheromak formation and evolution gives insight into the physical processes involved and the equilibrium configuration achieved. By varying the initial and boundary conditions, methods for achieving suable equilibria may be predicted. In this report, preliminary results of a numerical analysis of the PS-1 spheromak are presented. The numerical model used is a 2-D MHD code developed by C. K. Chu and collaborators at Columbia University.1'2 The resistivity is chosen to be

2 u /2 n = mev/ne , v = Pi[* ~ exp(vd/3vg)l where vd • |JI/ne and vg =• (ykTg/n^)* . The thermal conductivity < = 0. The initial conditions are p = 2.3 > 10~s g

3 2 cm" , To » 4 x \Qh K, (both spatially uniform), BzQ - 650 [l-? ]G, BeQ = 850 ' 2 1/2 (l-2/3£ ) G (both uniform in z), and Br = 0. The implosion is driven by

specifying B2(t,z) and BQ(t,z) at the outer boundary. B2 rises sinusoidally to its final value in 2 us and is fixed in time after that. The final value is 2.5

kG for 0 < z < zmax/2 and 3.2 kG for zmax/2 < z < zmax- BQ is independent of z. It remains fixed for the first 4 ps and then falls to zero in 2 us. The usual boundary conditio 3 are applied at the axis. The configuration is assumed to be

z B a 9 3z = z z = symmetric about z = 0. At z = max» r */ 0» 9x/<* = 0> 3p/9 °> and 3v/3z « 0 where I|J and \ are tne toroidal and poloidal flux.

By 2 us a noticeable density enhancement has been formed. In Fig. la contours of equal poloidal flux and density in the r-z plane are plotted. A is the lowest density ani I is the highest. In this plot I * 3.46 pQ, and the contours are uniformly spaced in p. It may be seen that the peak density occurs

in the region .5 < r/S^jj < .6 and .4 < z/zmax < 1.3. In Fig. lb, the experimental equiflux contours are plotted. At this time both experiment and theory show that reconnection has occurred =nd an island th it extends to z = 0 has been formed. The simulation does not show the island

z near max. most probably due to the simplified boundary conditions chosen to represent this boundary. -86- As time evolves the density peak moves toward the midplane. By 4 ys p is

peaked in the region 0 < z/z__v < .7. Inertial effects have not yet become small, and between 5 and 8 us the plasma bounces out and then recontracts. By 10 ys the spheromak has contracted axially and occupies a region •2 < r/R < .5 0 < z/z___ < .4, as may be seen in Fig. 2a. The experimental plot in Fig. 2b shows the island to be larger and at larger radius because the experimental boundary field has fallen off by this time, whereas the field in the computation did not. In conclusion, the initial results of the simulations approximately matched the experimental data. Reconnection near the ends and subsequent axial contraction of the regions of peak density have been demonstrated. At late times, the density peaks on the midplane (z = 0).

References

1. H. C. Lui and C. K. Chu, Phys. Fluids _18, 1277 (1975).

2. H. C. Lui and C. K. Chu, Phys. Fluids _1^, 1942 (1976). -87-

1-.45

1.09 —

M 0.72U-

0.35 —

0.0C 0.00 0.22 0.45 0.67 0.90 0.00 0.22 0.45 0.G7 0.90 R R Figure la

Figure lb FLUX Z« 1.65E-03 A=. 4.44E-01 DENS !• 3.95E*CO 1.45,

:.oe —

0.72 —

0.36 —

o.od 0.00 0.22 0.45 0.67 0.90 0.00 0.22 0.45 0.67 0.9

Figure 2a

Figure 2b Reversed-Field Mirror Transport Code*

D. E. Shumaker

National Magnetic Fusion Energy Computer Center Lawrence Livermore National Laboratory Livermore, California

J. K. Boyd, S. P. Auerbach and B. McNamara

Lawrence Livermore National Laboratory Livermore, California

A code (FRT) has bt -a written to describe the evolution of a axisymmetric field-reversed mirror. '^-.-i transport equation, which are 1-D, describe the plasma inside the separatrix. The code is being used to model the Beta II magnetized coaxial plasma gun at Lawrence Livermore Laboratory. The plasma is assumed to consist of hydrogen ion and electrons. A small percentage of impurity ion with fixed density profile is also included in the calculation.

The four quanities which are advanced by the transport calculation are,

= Nisi

5/3 0 - pP bS Q2 - e l

Q3 =

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Hvermore National Laboratory under contract number W-7405-ENG-48. -90-

where the flux surface integrates are,

J |V|

y IVPI r2

These Qj^'s are adiabatic quantities, that is, in the absence of diffusion they are constants. The independent variable used in the transport equations is,

Where YQ is the o-point (magnetic axis) value of ip, and ij^ is the boundary (separatrix) value of y.

The transport equation for the Qj^'s are obtained by doing a flux surface average of the transport pquations of Braginskii. The transport equation are of the form

d f F + Gj I m 1,2,3,4 dt dp

The transport coefficients of Braginskii are contained in the F^ and Go.

The physical effects which are included in the code are: -91-

(1) Classical transport (Braginskii)

(2) Collieional transfer of energy between ions and electrons

(3) Joule heating of electrons

(4) Radiation cooling of electrons due to impurities.

(5) Neutral beam heating of ions

(6) Charge exchange loss of ion energy.

(7) Anomalous electron thermal conductivity

The radiation cooling of electrons is modeled by assuming a given profile of impurity ions. The radiation energy loss rates are given in Ref. (1) which are obtained from a coronal equilibrium.

The calculation of the neutral beam heating of the ions uses a code developed by J. K. Boyd^ '.

The approximate calculation of the charge exchange ion energy loss is done by using a given profile of cold neutral gas.

Anomalous electron thermal conductivity is modeled by multiplying the classical electron thermal conductivity by some given factor.

The evolution of the plasma is computed by two alternating steps, the 1-D transport calculation which advances Q,, CM, QO and Q-, and the 2-D equilibrium calculation which computes the poloidal magnetic flux function ij>. The 2-D equilibrium calculation is done by a code written by J. K. Boyd.^ ^. -92-

References

1. D. E. Post, R. V. Jensen, C. B. Tarter. W. H. Gasberger, and W. A. Lokke, "Steady State Radiation Cooling Rates for Low-Density High Temperature Plasmas", PPPL-1352 (1977).

2. J. K. Boyd, S. P. Auerbach, J. L. Berk, B. McNamara, . A. Willmann, "Adiabatic Compression and Neutral Beam Buildup in a I- >':ld Reversed Mirror Configuration", Sherwood Meeting, Tucson (1980).

3. J. K. Boyd, S. P. Auerbach, P. A. Willmann, H. L. Berk, and B. McNamara, "Computational Methods for Reversed-Field Equilibrium", UCRL-84044 (1980). -93-

Tilting-Mode-Stable Spheromak Configuration

K. Yamazaki

Plasma Physics Laboratory, Princeton Urirersity Princeton, New Jersey 08544 (* On leave from Institute of Plasma Physics, Nagoya University, Nagoya 464, Japan)

1. Introduction The Spheromak has no external toroidal coil topologically linking the plasma and has many advantages from the view-point of simplifying reactor engineering. The MHD stability properties of this configuration have been recently discussed in Ref. [1-4]. Among global MHD modes expected in Spheromak tilting instabilities are peculiar to Spheromak and compact torus configurations. Rosenbluth and Bussac first pointed out the tilting-stability criterion for elongated spheromak [2] , and the author has recently extended their approach to the internal-tilting-mode stability of the arbitrary shaped spheromak [4]. The work described here is to check external mode criterion for arbitrary shaped spheromak and to find the optimum spheromak configuration stablro against both internal and external tilting instabilities.

2. Internal Tilting Mode Rosenbluth and Bussac [2] showed that a horizontally elongated spheromak (oblimak) is stable against internal tilting modes for force-free configurations; 3 - k i\

The author has extend this approach to the arbitrary shaped spheromak by using the following plasma shape:

00 OO r = r (1 + E e sinjz.8 + I e cos£e] % = 1 l l = 1 * where spherical co-ordinates (r,9,) are used and all sinusoidal coefficients e. and ££„ are assumed smaller than one. The eigenvalue of the shaped spheromak has a small deviation from the exact spherical value kQ = 4.493;

k = k + 6k . o Taking account of the boundary condition,

B . Vr = 0 , o -94-

we can obtain the eigenvalue fraction of the n = 0 configuration, _6k 9TT s s £ + — k 'n = 0 32 1 32 £3

3 C 3 c 1 c 5 £2 35 ! 105 e6 The eigenvalue of the n = 1 configuration also can be obtained and we can check the internal tilting mode stability by means of the constraint of

6k|n = 0 < 6k|n = 1 ' which leads to the following criterion,

c 45TT S 15TT S ^ c _1_ c 16 c £2 + 128 £3 128 G1 7 £4 7 £6 231 £8 ' ' Detailed discussions on this results were presented in Ref. [4]. It should be noted here that triangular and trapesoidal shapings of the half-cross-section assist the internal tilting stability as well as does an oblate shaping (Fig. 1).

3. External Tilting Mode Next, we should take external mode into consideration. According to Rosenbluth and Bussac's physical picture, the most dangerous tilting mode for oblimak is caused by displacement and perturbation field with the following normal components;

?n = ? P2 (cos6) e^ and B = —^ j'fk r ) P (cose] e ' , n 5r l o o' 1 l ' o respectively. The radial perturbation of magnetic field should be zero on the conductive wall (rw = rQ (1 + 6);. Finally, the energy intergal for this surface mode is expressed by

25 ^ o kr dr i ^k n = 0 > -I o The stability condition 6w > o yields

<-? i.-„>>« o For oblate spheromak, stability condition is given by

6 < 0.6 ( - e°) .

The coefficient 0.6 is different from Ref. [2] because of the difference of the definition of plasma boundary. Effects of rectangular and hexagonal shapings on the tilting stability of elliptically elonga*-^d spheromak are shown in Fig. 2(a) and (b), respectively. Region (i) means wall-assisted stability domain and within the ordering of 6 ~ o (e) we can obtain wall-free stability region which is shown as area (ii). The triangular and trapesoidal -95-

shapings of the half—cross-section again have stabilizing effects on external tilting mode.

4. Conclusions We studied both internal and external -_ilting-mode stability in slightly non-spherical spheromak as a extension of Rosenbluth and Bussac's stability approach and came to the following conclusions:

(1) The oblate spheromak with near conductive wall is stabilized against both internal and external tilting modes by adding the triangular or trapesoidal shaping of the plasma half cross-section. (2) The prolate spheromak is also stabilized by the same shaping even if there is no conductive wall near the plasma boundary.

References

[1] BUSSAC, M. N., FURTH, H. P. OKABAYASHI, M. , ROSEHBLUTH, M. N., TODD, A. M. , in Plasma Physics and Controlled Nuclear Fusion Research (Proc. 7th Int. Conf. Innsbrtuck, 1978, Vol. 3, IAEA, Vienna (1979) 249.

[2] ROSENBLUTH, M. N. , BUSSAC, M. N., Nucl. Fusion J19_, (1979) 489.

f3] OKABAYASHI, M. , TODD, A. M. , Nucl. Fusion _20_ (1980) 571.

[4] YAMAZAKI, K., Princeton Plasma Physics Laboratory Report, PPPL-1665 (1980) (to be appeared in Nucl. Fusion 20 No. 11).

Fig. 1. Plasma boundaries of Spheromak stable against internal tilting modes. -96-

(o) I \(;

Fig. 2. Shaping effects on tilting mode stability, (a) rectangular effects of elliptical spheromak, (b) hexctgonal pff"-*3 of elliptical spheromak. Stability regions are shaded. (i) denotes wall-assisted domain and (ii) wall-free domain. -97-

PS-1 SPHEROMAK EXPERIMENT

H. Bruhns*, Y. P. Chong, G. C. Goldenbaum, G. W. Hart, and R. A. Hess Department of Physics and otronoay University of Maryland College Park, Maryland 20742

T. Introduction^

Our method of producir^ a spheronak configuration plasma Involves producing currents by capacitor bank discharges in the region where the plasma is to be confined. A second method of production being pursued in other laboratories has a configuration being injected from outside the contained region. Both resulting plasmas \ave been shown to be susceptible tr, a tilting instability if the length-to-radius ratio (L/R) in a cylindrical configuration • ' is greater than approximately 1.7. This is consistent with earliest theoretical predictions' which indicated that oblate spheroidal plasmas should be stable to tilting. Two different approaches exist toward naking the configuration oblate: either by fitting the plasma into a conducting shell with the desired cross-sectional shape or by shaping the applied field with an array of exterr.d coils. We are trying the latter approach using a mirror coil system which causes both flux surface closure .ind axial compression to make the configuration obi ate.

Historically, most of the experiments have focused on maximizing plasma heating by rapid radial compression. This tends to make a prolate- shaped plasma. If there is toroidal field causing the gyroradil to be small, this tends to produce tilting. Less radial compression and some axial compression can make the flux surface shape more oblate and reduce the tendency to tilt. The net result of this is to reduce strong nonadiabatic corapressional heating ind thus require other heating means to be invoked immediately. A large source of thermal heating is the ohmic dissipation in the toroidal and poloidal currents. The major difficulty with ohmic heating is to overcome the radiation barrier associated with impurity Ion line radiation.

11. _RadjL_at ion_ and J>hjllc_ flea_t_i.n&

We have Investigated the impurity problem both experimentally and theoretically with one result being the replacement of our original tungsLen-copper alloy electrodes with carbon electrodes. While impurity transport In spheromak configurations is ill understood at this time, it seems to be a reasonable approach ,_o reduce the number density of high Z Ions as much as possible. Our present apprjach Is to eliminate as much high Z material as possible utilizing carbon for electrodes, end plates, and if possible, wall liners. To understand the impurity requirements, we have developed a zero-dimensional model of the ohmic heating process from which carbon ionization rate processes are calculated and their radiation -98-

is taken into account in a thermal energy equation. In Fig. 1 we show two cases for typical conditions: n = 1 x 10 cm ", 5% carbon impurity, and current densities of 1 kA.'cm and 2 kA/cm . We see that the one case is just below the radiation barrier; the other is above. We conclude that 5% levels are not unrealistic and will allow reasonable spheromak conditions to be ohmically heated to just below keV temperatures. Our present experiment is below the radiation barrier.

111 • HI} ing and Shifting

Axial compression has been seen in our experiment to impede It" not LO eliminate the tilting instability. In Fig. 2, we display an experimentally determined tlux surface plot showing an axlally compressed configuration. At this stage it is difficult to control the plasma shape over long periods of time for several reasons. Because of the low field strength and relatively high density at this time, the currents diffuse Into the plasma rapidly, allowing it expansion and shape change. Also, as we are experimenting to find an optimum combination of coil location, risetime, and decay time, we have not yet utilized our power crowbar banks and are presently limited to rapid applied fields of decay time of 30-50 ps. Presently, we have Increased the axial compression to obtain a lower L/R ratio corresponding to a more oblate shape (Fig. 3). However, these magnetic probe measurements Indicate at later times the presence of an n=l sideways shift of the torus as well as a possible tilt. This shift is probably related to previously observed shifts In mirror-confined plasmas. Should the effect of axial compression to the mirror coil be to reduce the minimum B well, a possible cure is to install multipole field conductors on the outside of the vacuum chamber.

This work is supported by ^he Department of Energy.

References

* Institut fuer Angewandte Physlk, Univ. Heidelberg; supported by the Humbcldt Foundation, Germany. 1 G. C. Goldenbauii, J. H. Irby, Y. P. Chong, and G. W. Hart, Phys. Rev. Letts. kk_, 393, (1980). Z- G. An et al. , Proc. Eighth International Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Brussels, IAEA-CN-38/R-3-2 (1980). 3 Z. G. An et al. , Univ. of Maryland Report PP #81-029 (1980). T. R. Jarboe et al. Phys. Rev. Lett._45^ 1264 (1980). 5 M. N. Rosenbluth and M. !,'. Bussac, Nucl. Fusion J^, 489 (1979). -99-

Figure 1. Time dependence of calculated electron temperature (eV) tor 5X carboq impurity and current densities of (a) 1 kA/cm and (b) 2 kA/cm2 Case (a) is below the radiation barrier.

Figure 2. Contours of constant poloidal flux determined from magnetic probe data at 7 ys. This case shows lowei axial compression. Contour Intervals are 30 kG-cm . -100-

POLOID°_ run

Figure 3. Constant poloidal flux contours determined from magnetic probe data at 5 ps. Higher axial compression results in a more oblate configuration. Contour intervals are 60 kG-cm . -101-

PHYSICAL PROPERTIES OF COMPACT TOROIDS GENERATED BY A COAXIAL SOURCE

I. Henins, H. W. Hoida, T. R. Jarboe, R. K. Linford, J. Marshall, K. F. McKenna, D. A. Platts, and A. R. Sherwood

In the CTX facility we nave been studying CTs generated with a magnetized coaxial plasma gun. CTs have been generated and trapped in prolate and oblate cylindrically symmetric metallic flux conservers. The plasma and magnetic field properties are studied through the use of magnetic probes, Thomson scattering, interferometry, and spertroscopy.

In the prolate case a simple circular cylinder is used as the flux conserver. The initial poloidal field strength of the coaxial source can be adjusted so that the CT configuration comes to rest within the flux conserver and separates from the gun. In some cases the CT stops with its axis parallel to the common axis of t'.e source and the flux conserver, and then it rotates (as predicted by Rosenbluth and Bussac) until its axis is orthogonal to the axis of the flux conserver. In most cases the stopped CT when first observed has already partially rotated. The rotated CT appears to be MHD stable and its magnetic fields (= 2 kG for the prelate case) decay with about a 100 us time constant. Interferometric measurements show an Initial density of about 1014 cm and a density lifetime similar to that of the magnetic field. Details of the verification of this tilting as well as details of the gun operation and geometry are reported elsewhere. ' We have produced compact toroids when an initial axial magnetic "guide" field was established within the cylindrical stainless steel flux conserver. In the presence of this field the compact toroid is observed to rotate 180°. The characteristic decay time of the magnetic field configuration was only 10-15 us, i.e., much shorter than comparable conditions without the guide field. We speculate that the rapid destruction in the guide field case is due to reconnection of magnetic field lines in the high shear regions which occur after the toroid rotates, opening previously closed field lines.

In an attempt to stabilize the tilting of compact toroids produced by the magnetized gun, we have generated compact toroids in other flux conservers having oblate regions incorporated into their geometry. Cross sections of such flux conservers are shown In Fig. 1. The plasma from the magnetized gun is injected from the left through the 0.34-m diameter entrance cylinder into the confining region. With the geometry of Fig. lb the tilting no longer occurs and the configuration is stable throughout its lifetime. In order to achieve this stability we increased the amount of initial gas puffed into the gun from 0.5 to 3 torr liters. We also tried a 0.46-m diameter entrance cylinder and found that the compact toroid then tilted again. With the elimination of the complication of tilting, three distinct time scales emerge. The first (~ 1 us) is the time required to fill the flux conserver with magnetic field and plasma. The second (~ 12 us) ie the time for the decay of the fields in the entrance cylinder. Figure 2a shows this decay. We interpret this decay as being due to field line reconnection which is completed in about 30 us. The third time (~ 150 us) Is the characteristic time for the decay of the fields in the flux conserver measured after reconnection has occurred. Figure 2b shows this decay. It is interesting to observe that the three time scales 1 us, 12 us, and 150 us have the proper relative values -102-

GUIDE FIE1D GUIDE HELP _WITH0UT WITH WITHOUT WITH

TILTED TILTED STABLE TILTEt (o)

STABLE SUBLE

Fig. 1. Various flux conserver and magnetic field geometries tried in order to stabilize the tilting.

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0 Z (m) Z (m) Fig. 2. Plots of the axial component of the magnetic field on axis at various times. Figure 2a shows plots at various times during the decay of the field in the entrance cylinder. The time elapsed oetween plots is 5 iis. Figure 2b shows plots at various times during the decay of the compact toroid and the time between plots here is 50 us. The gun discharge is initiated at t=0 and the plot labeled t=0 shows the value of the axial component of the magnetic field in this region due to the coils which supply the initial axial flux for the plasma gun.

to be an Alfyen time, a resistive tearing time, and a resistive decay time respectively.

If the CT does not tilt one expects no transverse components of B on axis. For the case in Fig. 1 the peak transverse components are measured to be less than 15% of the peak B and are not shown. The measurement of all components of the magnetic fields on the axis of the flux conserver is a powerful means of determining the extent of tilting. We have spent considerable time searching for a field and flux conserver configuration which gives a stable CT with guide field. Fig. 1 shows a summary of the -103-

geometries tried with comments as to the stability of the CTs created in thdse geometries- Some properties of the CT plasma have been measured for the geometry shown in Fig. lb without external guide field. The density as a function of time as measured by a 3.39 um wavelength HeNe interferometer is shown in Fig. 3. Observe that the time behavior of the magnetic field and density are similar. Spectroscopic data are taken with a 5-channel polychromatcr., a monochromator, and a spectrometer. The polychromator has five channels spaced one angstrom apart. Photomultipliers are used for time resolution. The monochromator has a two-angstrom spectral width. Using both the polychromator and the monochromator two regions of wavelength can be monitored on each shot. The spectrometer which covers a region of approximately 100 A, is time integrated, and is used to identify spectral lines. Nickel and iron impurities have been identified. The region of the CV triplet (2270.91 A, 2277.25 A, 2277.92 A) has been studied extensively with all three instruments. A line suitably close (within the resolution of the polychromator and spectrometer ~ 1/2 A) to the CV line (2270.91 A) was identified at approximately 2270 A and a similar time history was observed for data taken in the 2277 A region. Unfortunately, these lines are all very close (within 1 A) to Nill lines - 2270.213 A, 2277.28 A, and 2278.77 A. The radiation appeared 50-100 |JS after the initiation of the discharge, and it often disappeared in a short time. One interpretation is that the temperature rises due to ohmic heating causing the CV line to appear. However, this time history also correlated well to two other nickel II lines outside of the region of interest (2264.457 A and 2287.084 A). Thus the ~ 2270 and the ~ 2277 lines are more likely Nill than CV lines. Sc an alternate interpretation is that the plasma was cooling causing the Nill line to appear. OIV was also detected at 2781.05 A indicating a temperature on the order of or higher than 2C oV. This line appears immediately and typically lasts for about 20 us. CIII radiation at 2298 A appears immediately and lasts for the duration of the plasma. Typically there is an initial spike (~ 20 ps) after which the radiation settles down to roughly half the peak and slowly decays. Thomson scattering has been used to measure the electron temperature at various times. We have taken some preliminary data at a position 5 cm from the midplane of the confinement region and at a radius equal to two-thirds of that of the confinement region. There is a large shot-to-shot variation of the temperature and density at this position especially early in time. However, the plasma does appear to be fairly cool (Te < 10 eV) for the times shown in Fig. 2b (t > 50 us). Early in time (t ~ 5 ys) temperatures as high as 60 eV have been measured. Quartz pressure probe^ data show a rapid drop early in time and are consistent with the Thomson scattering temperature measurements. The rapid cooling during reconnection is probably due to transport along open field lines. It may be possible to lower the energy lost on open field lines by shortening the time for reconnection. One method of doing this is to apply a fast rising external field in the entrance region.

Compact toroids are generated which tear free from the source in ~ 30 ps. Prolate CTs are subject to the tilting instability in a straight cylindrical flux conserver. CTs formed with an oblate geometry can be stable to this tilt. Density and magnetic field measurements show that the CTs have a lifetime of about 150 us. Thomson scattering, spectroscopy, and a pressure -104-

'e ?u

Fig. 3. Density and magnetic field vs time. These data were taken with an interferometer and a magnetic probe located near the midplane of the confinement region of the flux conserver. The density is the line average on a diameter and the magnetic field is Bz on axis. The density and Bz measurements are from different shots with similar initial conditions.

probe indicate that the temperature decreases to about 10 eV on the time scale of the reconnection (~ 15 us).

REFERENCES

1. M. N. Rosenbluth and M. N. Bussac, Nucl. Fusion Xi, 489 (1979).

2. T. R. Jarboe, I. Henins, H. W. Hoida, J. Marshall, D. A. Platts, and A. R. Sherwood, "The Motion of a Compact Toroid Inside a Cylindrical Flux Conserver," Phys. Rev. Lett. 4_5, 1264 (1980).

3. T. R. Jarboe, I. Henins, H. W. Hoida, J. Marshall, and A. R. Sherwood, "Magnetized Gun Experiments," Symposium on Compact Toruses and Energetic Particle Injection, Princeton Plasma Physics Laboratory, Princeton, NJ, December 12-14, 1979.

4. H. P. Furth, J. Killeen, and M. N. Rosenbluth, Phys. Fluids j>, 459 (1963).

5. K. F. McKenna, R. R. Bjrtsch, R. J. Commisso, C. Ekdahl, W. E. Quinn, and R. E. Siemon, "Collisionless Flow and End Loss from a High-Energy Theta-Pinch Plasma," Phys. Fluids 23, 1443 (1980). -105-

A Compact Torus Configuration Generated by a Rotating Magnetic Field: The Rotamak

W. N. Hugrass, I. R. Jones, and K. F. McKenna** The Flinders University of South Australia 5042 South Australia

In the Rotamak. concept, a rotating magnetic field (rotating in planes normal to the z-axis; see Fig. 1) is used to drive the steady toroidal current in a compact torus device. Analytical, computational, and experimental details concerning this novel method of driving plasma currents can be found in Refs. 2-6. The magnetic field configuration of the Rotamak consists of the combination of a poloidal field, due to the toroidal plasma currents and an X-POINT externally generated 'vertical' field required for equilibrium, with the applied rotating field which, in a sense, represents a time varying MAGNETIC AXIS toroidal field. To clarify this Fig. point, Fig. 2 shows a computed stereoscopic pair of two total field lines, at one instant of time, due to a steady ring current of 6 kA and a rotating field of 70 gauss. These field lines are being viewed along the z-axis. It is seen that the toroidal magnetic configuration is cut by two, diametrically opposed cusps. The magnetic helix has a different sense on each semicircular segment. As time progresses, the field pattern shown in Fig. 2 spins about the major toroidal axis (the z-axis in the present context) with the frequency of the rotating field. Open field lines enter one cusp region and leave the other. It is not clear, however, that the difficulties usually encountered with static open lines arise in a time-varying situation where associated electric fields exist. A first experimental investigation"6 of the rotamak configuration has been undertaken in the apparatus shown in Fig. 3. The Pyrex discharge vessel is spherical in shape (inner radius of 64 mm) and is equipped on the outside with a pair of orthogonally oriented Helmholtz coils. rf Fig. 2. currents of the same frequency (0.67 MHz) and amplitude (~2 kA), but dephased by 90°, are passed through these coils to produce a magnetic field (~600 gauss on axis) which rotates about the z-axis. The vertical field needed for equilibrium is produced by a pair of coils located on the z-axis as shown in Fig. 3. In a modified version of the rotamak device, an applied toroidal

*0n leave from Cairo University, Cairo, Egypt **0n leave from Los Alamos Scientific Laboratory, Los Alamos, NM -106-

tr.ugnetic field is generated by allowing a steady current to pass through a conducting rod aligned along the z-axis. Argon was used in these experiments borli to ensure that the necessary inequality (required for no net ion motion), e B.. was well satisfied and take advantage of its relative ease of ionization at the low filling jiR's.surf which was used (]./(' mTorrj. The quantities :::i-;i.si:rfd were the total toroidal current, I,, driven by the rotating field; the .'. i •-; L ri but ion of the magnetic r it..-Ids; and the power !. ransturred between the rf line >.;t-ner.U ors and the load, Hgures 4(a,b, and c) show t.ht- various In (t ) oscillograms obt. lined with different i ur.ibina Lions of the steady :.;a>;nt-t i c fields. For reference, Fi g. 4(d) shows the characteristic of the rf pulses ishapc, frequency, and duration) used in these • ' x:ie riment s . In the absence of vertical Hg. 3. i ield, the plasma ring is not magnetically confined and a comparison of ! if\s. 4(a) and 4(b) shows that tlie total toroidal current is substantially reduiod under this condition.

with no toroidal magnetic field (kA) For the case of applied vertical field, but no toroidal field, Fig. 4(b) shows that I initially increases to a peak value. Thereafter, during the interval. 10-16 us, I, remains at the substantially constant value of 6.7 kA. The eventual final decrease of I, 9 coincides with the termination of the rl pulse. We refer to the initial (b) period u-10 us as tlie 'formation' phase and to the interval 10-16 ps as the 'steady' phase. Experience showed that steady phase only if the [j, exhibited a (c) amplitude of the initially imposed vertical field was chosen to lie within a narrow range of values for a given filling pressure. For the 1.7 mTorr experiments reported here, the steady phase was obtained for vertical fields (at r = 0, z = 0) lying in the range (d) 340-380 gauss. The magnetic configuration was 5/x«/DIV investigated by measuring the z-component of the total poloidal field Fig. 4. (i.e., the sum of the vertical field and the field due to I.) along both the r -107-

and z axes. The distribution of this component along these axes is shown for three instants of time (o 10 us, + 14 us, • 16 ps) in Figs. 5(a) and 5(b). The interpretation of this data is facilitated by reference to Fig. I. Appropriate integration of the profiles shown in Fig. 5(a) enables one to determine the poloidal flux and hence the position of ihe separatrix. Figure 6 shows the variation with time of the positions of the separatrix (rg), the magnetic axis (r^A, and the neutral point (zx). For the first 8 us of the discharge the separatrix remains outside i n,NEe the wall of the discharge vessel. However, for the interval 10-16 us corresponding to the observed steady phase of the discharge, the separatrix lies ~10 mm inside the vessel wall. It is also seen that the neutral point on the o^ z-axis lies well within the vessel during the i discharge. These measurements indicate that the steady phase of the discharge is associated with t he generation of a well-defined, oblate

(zx/ra ~ 0.75), compact torus configuration lying entirely within the discharge vessel. The rf energy transferred from one of the two generators to the load was obtained by measuring both the current, i(t), in one of the Helmholtz coils and voltage, V(t), generated across its terminals. The quantity t

W(t) = j ic(t') V(t') dt' o Fig. 5. was calculated and is shown as a function of time in Figs 7(a) and 7(b) for a vacuum shot and a plasma shot respectively. Since the generator 8 was used in a constant current mode, ? the energy dissipated in the coil resistance remains the same fcr both "£ situations and the amount of energy -H absorbed by the plasma W (t)a is obtained directly by subtracting the two quantities shown in Figs. 7(a) ,-nd 7(b). The slope of the graph 10 12 shown in Fig. 7(c) yields the rf power delivered to the plasma by one t (jis) generator. It is seen that the two Fig. 6. rf generators feed energy to the plasma at a total constant rate of 1.1 MW. No direct measurements of the plasma properties have been made in these experiments. However, calculations made with a zero-dimensional plasma model which has been adapted for argon and which calculated the temporal temperature behavior of an ohmically heated, uniform plasma suggest that, for the experimental conditions considered here, the electron temperature reaches a steady-state value of about 7 eV. In this steady state, it is predicted that energy is lost from the argon plasma by line radiation at a rate of 1.4 MW. Given the approximate nature of the calculations, we conclude that during the steady phase of the discharge, an oblate compact torus configuration lying entirely within the discharge vessel is generated for which the ohmic power input is approximately balanced bv line radiation. -108-

Experiments with a toroidal magnetic field Experiments were undertaken in a rotamak device which had been modified to allow the generation of an additional steady toroidal field by means of a current flowing along the z-axis. For this case, the total magnetic field lines have a more complicated structure than that shown in Fig. 2, An oscillogram of the toroidal current, I, , obtained with both an initially imposed vertical field of 360 gauss (at r = 0, z = 0) and a toroidal field generated by 7.5 kA flowing along the z-axis is shown in Fig. 4(c). It is seen that I , remains at a substantially constant value for the duration of the rf pulse. Figure 8 shows the radial distribution of the poloidal field and the toroidal field at t = 14 us. Comparison with Fig. 5(a) shows that the application of the steady toroidal magnetic field broadens the poloidal field profile. Moreover, the toroidal field is increased above its vacuum value by paramagnetic poloidal currents flowing in the plasma. It is estimated that about 5 kA of poloidal current is generated in this experiment.

We are grateful to S. Ortolani, M. G. R. Phillips, R. G. Storer, and H. Tuczek for their contributions to this work. The work was supported by grants from the National Energy Research, Development, and Demonstration Council, the Fig. 7. Australian Institute of Nuclear Science and Engineering, and the Australian Research Grants Commi ttee.

References (G) 1. I. R. Jones, Flinders University Report I2OO No. FUPH-R-151, 1979. Bz(r)

2. I. R. Jones and W. N. Hugrass, Flinders B+(r). VACUUM IO00 University Report No. FUPH-R-166, 1980. B^(r). PLASMA] 3. W. N. Hugrass and R. C. Grimm, Flinders 800 University Report No. FUPH-R-167, 1980. 4. W. N. Hugrass, I. R. Jones, and 600 M. G. R. Phillips, Flinders University Report No. FUPH-R-168, 1980. 400 5. W. N. Hugrass, I. R. Jones, and M. G. R. Phillips, Nucl. Fusion _li, 1546 200 (1979). 6. W. N. Hugrass, I. R. Jones, K. F. McKenna, • (mm) 10 20 30 7*" M. G. R. Phillips, R. G. Storer, and 50 60 200- MAGNETIC7 H. Tuczek, Phys. Rev. Lett. 44, 1676 AXIS (1980). 4001- 7. S. Ortolani, Los Alamos Scientific Laboratory report LA-8261-MS, 1980. Fig. 8. -109-

EXPERIMENTS OF SPHERGMAK AND REVERSED FIELD CONFIGURATION IN 2M THETA PINCH

Y.Nogi, S. Shimamura, H.Ogura, Y.Osanai, K.Sa-'tu, S.Shiina and H. Yoshir.iura

Department of Physics, College of Science and Technology, Uiho.i Uni .1 <" v Krnda-Sur..^, _ Chlyoda-ku, Tokyo 101, Japan

5 1 Introduction

Compact tori are divided into 3 spheromak and a versed field configuration (RFC), which by confinement field with and without toroidal field. Each configuration shows different stability behaviors. Tne spheromak i:; unstable for " tilting node " on a prolate plasma, 1,2) whereas the prolate plasma in ^he RFC is stable for the mode, 3 JJ O This opposite tendency is not system- atically understood. ( ManManyy RFRFC experiments rhow " n=2 rotational instability'.y''' ^ ^ , 55))) j\j\ f few explanation of the gen- erating mechanism has bee l proposed.-''1 However, experimental identification of the mechanism is net sufficient. Since the toroidal field has stabilization effect for the kink instability, it is expected that the n=2 elongation of the plasma is suppressed by addition of tne toroidal field to the RFC. However, the configuration having toroidal field will become unstable for the tilting mode. It is necessary to study how the toroidal field stabilizes the rotational instability or destabilizes the prolate plasma. As a theta pinch combined with a zrdischarge system is able to produce the toroidal field,7»8) above problems will be easily studied. As the firs'- 3tep, the behaviors of the plasma produced by each configuration are stuctied.

§ 2 Spheromak Configuration

The spheromak formation technique has been improved compared with that reported In Ref.8. The first improvement is almost perfect separation of the z-current flow to the plasma from the power supply as soon as the spheromak Is formed. For this purpose, negative z-current Is supplied from the power supply soon after the initial current is compressed by the theta pinch. Return current to the power supply Is shown in Fig.2. In the early experiment, 70% of the z-current is trapped in the plasma to produce the toroidal field and the remaining 30? returns to the power supply through electrodes (see Fig.2-2). Full trap of the current is realized due to counterbalance of the remaining current and the additional negative current from the power supply (see Fig.2-3). The second improvement is that cusp coils are equipped at the ends of -110-

:e the*,';, pinch coil as shown in Fig.l. Since the z-current L:,v.'L' r a: ••"•magnet ically in the bias field, it strengthens the •it'll field not only in the theta pinch coil region, but . ;:o near' the electrodes. Especially the axial field near ;•.- eleo'-r.-'de is unfavorable fci the field reconnection to T.'iuco '.he- RFC. Magnetic probe measurement reveals that ie :-<-o:<:v.iec •; ion happens earlier in 'he central region than ;•::;• .id';- M.H theta rich coil. Magnetic probe signals and .0 •.:•:'-•'.;i:\.-it.od field evolutions are shown in Fig. 3 and Fig. 4, •• :r c •.-_•.••., i v.; ly . Experimental parameters are descrived in §3- . the "."her hand, the "usp configuration has magnetic field ,:u. s cr-::sirv: a discharge tube at line "usps even if the -:• irrc-nr. flows paranagnetlcally. These magnetic field .;•";(.-.-:• -.-.re useful ror the reconnection of the negative bias ••I : -:M'j the theta pinch field n t the coil ends. .• :'<•:"•..'.•.•, the spheromak configuration will be easily fo^mea •j-.r-;::./; *,-• the procedure illustrated in Fig. 5. The final .:.f L.-y.u-'jt 1 ~j\: of Ft~.5 has teen confirmed by magnetic probe ••LS'ir-.-rr.ent. Diamagnetic signal also shows the good effect :.:'.-.• .r.isp Lias field on the formation of the spheromak :.''.' I r; iv! ~, '..or.. The Fiys.6-a and 6-b are obtained at the ;•.• • •-;•':. vi-Imer'tal conditions except the CUSD bias field.

:: ;' :-.'.-•.•• 'j'sed Field Configuration

T::L' J;;FC is formed v/ith the :ame machine as the sphero- :.:'F: •: onf Lrurat ion i;pc ,b;;t chu "-'lisciiMrge Is used only as ;• ."•••-iono^'ition of the plasma. The cusp bias coils are re- :r:veu. It is also possible that experiments are carried • •it with various mirror ratios (R!T1=1.1 - 1.7). After •.:;-.:• discharge tube is exhausted till 3x10-7 mmHg by a turbo molecular vacuum pump, the tube is filled with 10 mtorr hydrogen gas. The theta pinch coil produces 800 G negati bias ;"ield, and 10 kG main field having 2 ps rise time and -0 ps decay time. Schematic of the machine and arrangement {':-,r diagonostics are depicted in Fig.l. Typical diamagnetic signals of the plasma are shown in Fig.?. The signals damp slowly with the external field for 30 )J 3. After then they damn exponentially. These data are obtained at Rm^l-l- The plasma radius analyzed from Uie signal at position A is 2 cm and nearly constant during 30 (js (see Fig. 8). End on photographs show clear growth of a n=2 rotational instability. Start of tne exponential decay of the diamagnetic signal coincides with the time when the plasma contacts with the discharge tube. Electron density measurement using He-He laser interferometer is made for more precise information concerning the instability. The density integrated along the axis shows behavior having periodic change corresponding to the rotational instability (see Fig.9). These data also show that the generation time of tne instability is very early ; 20 ys after the RFC is formed. The rotation period is about constant over one shot, but slightly different every shot. -111-

§ 4 kj ummary

Since the z-current produces ;;K paramagnetic i i t i a near the electrodes, the spnerorr.ak formation is more- cult in th'r' straight bias field. In der to help the re- 1 connection at the coil ends, the cus; blas coils -i; e ': ' I i::' O to the both ends of the straight coil Th.en L 11 e r:-].-:-V-.:•'/•s configuration is formed and the rlasm; -. >-. r: } r iS CO fined f -_'i J — 1 ••-•• us. On the other hand, the RrC continues for about in case of the straight b;as fiel;. The confinement is limited by the rotational instability. A It ho... start time of the instability is not clear, the eion .s detected in 15-.^-

every snot. Detaile instability ours u C' d .

Fig.l

Schematic of Machine (Upper figure i1lus- trates probe position^

1. Initial Current

2. Compressed Current Position

3. Compressed plus Added Current

Theta Pinch Field Fig.3

Magnetic Probe Signals for hig.2 Z-current Flow Spheromak Operation -112-

Fiq.5 Field Reconnection witn '-):.•- Field Reconnection Cusp Bias Field

;i(jf...-i .-inch

Fig.o Plasma Radius

!jianidvjnetic Current of Spheromak

— i 2, 3.

:ig.7 Diamagnetic Current of RFC Fig.9 Electron Density

(1) M.H.Rosenbluth and M.N.Bussac ; Nuclear Fusion ]9_ (1979) 489 (2) T.R.Jarooe et a 1. ; Phys. Rev. Lett. 45^ (1930) 1264 (3) C.Bartori end T .S.Green ; Nuclear Fusion 3 (1963) 84 (4) R.K.Linford et al . ; Proc. 7th International Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Innsbruck (1978) CN-37/X-1 (5) A.Eberhagen et al. ; Z. Physik 248 (1971) 130 (6) D.C.Barnes and C.E.Seyler ; Proc. the US-Japan Joint Symposium on Compact Toruses and Energetic Particle Injection,Princeton (1979)130 (7) G.C.Goidenbaum et al. ; Phys. Rev. Lett. 44 (1980) 393 (8) Y.Nogi et al. J. Phys. Soc. Japan 49 (1980) 710 -113-

FORMATION OF COMPACT TOROIDAL PLASMAS 5Y MAGNETIZED COAXIAL PLASMA GUN INJECTION INTO AN OBLATE FLUX CONSERVER

W. C. Turner, G. C. Goldenbaumi, E.H.A. Grannemani, C. W. Hartman, D. S. Prono, J. Taska : Lawrence Livemore National Laboratory Livermore, CA 94550

A. C. Smith, Jr. Pacific Gas and Electric Company San Francisco, CA 94106

In this paper we report our initial results on the formation of compact toroidal plasmas in an oblate shaped metallic flux conserver similar to that used previously by Jarboe, PI. al'^' at Los Alanos Scientific Laboratory. A schematic of the experimental apparatus is shown in Fig. 1. The plasma injector is a coaxial plasma gur with solenoid coils wound on the inner and outer electrodes. The electrode length is 100 cm, the diameter of the inner (outer) electrode is 19.3 cm (32.4 era). Deuterium gas is puffed into the region between electrodes by eight pulsed valves located on the outer electrode 50 en from the end of the gun. The gur. injects into a cylindrica 1 ly symmetrical copper shell (wall thickness = 1.6 m) which acts as a flux conserver for the time scale of experiments reported here. The copper shell consists of a transition cylinder 30 cm long, 34 cm in diameter, a cylindrical oblate pill box 40 cm long, 75 cm in diameter and a downstream cylinder 30 cm long, 30 cm in diameter. The gap between the gun and transition, cylinder is 6 cm. An axial array of coils outside the vacuum chamber car. be used to establish an initial uniform bias field.

Axial and radial arrays of probes shown in Fig. 1 are used for measurements ot internal magnetic field profiles. A He-Me (6328X) interferometer has been used to measure line density. Photographs of the gun discharge have been taken with a framing camera. Recently we have added a number of additional diagnostics; bolometer, VJV spectrometer, collimated secondary electron emitting diode, radial Langmuir probe, radial piezoelectric pressure probe, PIN diodes, 2 mm microwave interferometer and magnetic loops on the outside of the flux conserver.

We will first summarize qualitatively the main results obtained with the apparatus in Fig. 1 and then follow with detailed examples from the experiment. Initially a uniform bias field (which may be zero) is established in the flux conserver. The inner electrode solenoid is pulsed on to produce a relatively strong field inside the inner electrode, a radial field at the end of the gun and a return field in the gap between gun electrodes. The vacuum field strength of the inner electrode solenoid falls off rapidly with increasing distance from the end of the gun — decreasing from several kilo Gauss at the end of the gur to less than 100 Gauss at the center of the transition region of the flux conserver shown in Fig. 1. The gun discharge is initiated and the j x B force uxially elongates the inner t on leave from University of Maryland + on leave from FOM-Institute for Atomic and Molecular Physics, Amsterdam, The Netherlands -114-

solenoid field lines until the volume of the transition region and flux conserver are filled with plasma currents and several kilo Gauss magnetic fields. The field strength in the transition region then decays rapidly to zero while persisting for a much longer time in the oblate region of the flux conserver. From this we infer that some of the elongated field lines have undergone reconnection and resulted in formation of an isolated closed flux surface toroidal plasma in the oblate region of the flux conserver. With zero initial bias flux, the major axis of the toroidal plasma is along the symmetry axis of the experiment and the magnetic axis encircles the symmetry axis. Toroidal and poloidal magnetic field components are present in a nearly force free configuration analogous to "Taylor states" for the reversed field pinch. '2) As the bias flux (with orientation opposing the inner electrode solenoid field) is increased from zero, the radius of the magnetic axis shrinks until a critical ratio of bias flux to closed field line poloidal flux is reached. Above this ratio, the oblate shape of the flux conserver is no longer effective for stabilizing the n = 1 tilting mode(3) and the major axis of the toroidal plasma flips.

We now turn to the detailed discussion of experimental results. Figures 2, 3 and 4 show the plasma gun discharge characteristic and magnetic field profiles for a particular shot. For this shot, the initial bias field in the flux conserver was zero, the inner electrode flux was 1465 kG-cm~2 ancj the outer electrode solenoid was shorted, producing a return field 1.4 kG in the gap between electrodes. The gun discharge capacitor bank (232 uF) was charged to 30 kV. The total gas fill in the eight gas valve plenum chambers was 29 Torr- iD2. The discharge was initiated 225 ps after opening the gas valves. The gun discharge current reaches a peak value 830 kA and falls to 650 kA when plasma begins to leave the gun muzzle (inferred from downstream magnetic signals). The initial magnetic pulse propagates axially with a velocity 5 x 10' cm/sec. Profiles of the B2 component of magnetic field along the symmetry axis are shown in Figure 3 at t = 6 usec and 36 ysec after plssma leaves the gun. An outline of the axial locations of the gun and flux conserver is also shown in Figure 3. At 6 usec the peak amplitude of magnetic signal extends back into the center of the transition region indicating elongation of field lines from the inner electrode solenoid. At 36 us the field in the transition region has decayed to zero while fields of 6.5 kG persist in the oblate region of the flux conserver. For reference the vacuum field on the symmetry axis due to the inner electrode solenoid of the gun is 5.0 kG at the gas valves, 2.5 kG at the gun muzzle and .05 kG at the position of the first probe measurement in Figure 3. The profile at t = 36 us in Figure 3 is asymmetrical with respect to z = 0 because of insertion of a truncated 45° cone in the exit cylinder of the flux conserver. As will be discussed below this cone has some influence on stabilizing the tilting instability when bias flux is added to the flux conserver. Radial profiles of Bz and Bx at z = 0 cm are shown in Figure 4. These profiles have the symmetry expected for a toroidal plasma with major axis along the symmetry axis of the flux conserver and magnetic axis at a radius Ro = 22 - 24 cm (Bz = poloidal component, Bx = toroidal component). The Bz component is 5 kG maximum at y = 0 cm, is zero at y = ^ 23 cm, and reverses sign near the flux conserver wall. Within experimental error the net Bz flux inside the copper shell at r = 37.5 cm is zero, as expected for a flux conserving boundary. The Bx component is zero at y = 0, reaches a peak value 4 kG and has opposite polarity above and below the symmetry axis. Taken together -115-

Figures 3 and 4 are evidence that a toroidal closed field line plasma has been formed in the oblate region of the flux conserver and is detached from the plasma gun electrodes.

From the magnetic profiles in Figure 4 we estimate the following; toroidal

current, Itor = 175 kA, poloidal current Ipoi = 260 kA, poloidal flux f' = 3800 kG-cm2, toroidal flux 1000 kG-cm2, poloidal field energy = 3.7 kj, toroidal field energy 1.7 kj.

Figure 5 illustrates the time behavior of magnetic signals at two positions on the symmetry axis for the experimental conditions of Figures 2 to 4. The probe in Figure 5 (a) is at the center of the transition region near where reconnection occurs and the plasma becomes detached from the gun discharge. The probe in Figure 5 (b) is at the center of the oblate region of the flux conserver. Ten microseconds after the first appearance of signals the field in the transition section has dropped to zero while the field in the oblate section is at its maximum value 8.6 kG and then decays nearly linearly to zero. The time fB fOr the signal in Figure 5 (b) to decay to 1. of its peak value is about 110 psec.

He-Ne interferometer measurements of plasma line density have been made along horizontal lines of sight just above the probe positions shown in Figure 5 — i.e. at the center of the transition section and center of the oblate section of the flux conserver. For the conditions of Figures 2 to 5 the peak line density at the center of the transition section is 6 x 10^ cm"2 and at the center of the oblate section is 1.7 x 10^ cm"2. At the center of the oblate section this corresponds to a chord average electron density ne = 2.2 x 10*5 cm"3. The line density decays much more slowly and with different time behavior than the magnetic signals shown in Figure 5. Roughly plasma density decays to zero in the transition region in ~ 200 ysec and in the oblate region in ~ 500 ysec. There is no apparent abrupt change in line density in the transition region when reconnection occurs.

Measurements of plasma thermal pressure with a piezo electric crystal coupled to a quartz acoustic delay line inserted in the plasma give an upper 4 3 15 3 limit P = n^ kT£ + ne kTe < 10 Joules/m . For ne = 2. x 10 cm" and ti£_j= ne this gives an upper limit on plasma temperature T{ = Te < 15 eV. For .B = 5 kG the corresponding local plasma beta is less than 10% and we conclude that the plasma configuration is essentially force free. The profiles in Figure 4 are in agreement with 2-D equilibrium calculations of pressureless force free equilibria.^4'

We have taken some data with an axial bias field opposing the field in the inner electrode of the plasma gun. Figures 6 and 7 indicate, first schematically and then with experimental data, a particular case where the plasma is initially formed with its major axis along the symmetry axis and then rotates 90° about the vertical axis. In this particular case the bias field was 200 G and the ratio of bias flux trapped in the flux conserver to closed poloidal flux in Figure 7 (a) is 0.23. The plasma begins to rotate almost immediately after the snapshot of Figure 7 (a) until the profiles in Figure 7 (b) are obtained 30 sec later. Oscilloscope photographs of B2 at the center of the flux conserver indicate that after an additional 40 use~ the plasma has rotated 180° and then decays with T§ = 70 ps. The data in -116-

Figure 7 were obtained without the 45° cone placed in the exit cylinder of the flux conserver, as sketched in Figure 2. If the cone is in place the plasma decays without tilting for a bias field of 100 G. When the bias field is raised to 200 G the plasma tilts after a delay of 130 us instead of beginning to tilt almost immediately after formation as in Figure 7.

At the present time our measurements do not identify the factors responsible for the field decay rate TJ = 110 psec. For resistive decay of poloidal flux we can calculate the decay time T of the lowest energy force free normal mode in a cylindrical flux conservere''^'

P R2 - _2 T n

where n = plasma resistivity, R = radius of flux conserver = .375 m and L = length of flux conserver = .40 m. For Z = 1 the classical resistivity is given by

.51 x 10 J-nA n rrr— ohm-m T (eVy'2 e where

n 'cm ; *.nA = 24 - log • T (eV) e

Inserting Te = 15 eV, which is an upper limit from the pressure measurements, and ne = 2 x 10 ^ cm"-' gives t = 950 p s or a decay anomaly of approximately 9. For Te = 4 eV we get T^ - 130 us, approximately in agreement with the experiment.

We would like to thank the Los Alamos plasma gun group for many helpful discussions related to the work reported in this paper.

REFERENCES 1. T R. Jarboe, I. Henins, H. W. Hoida, J. Marshall, D. Platz, A. Sherwood, "Gun Generated Compact Tori" in Proc. of Reversed Pinch Theory Workshop, Los Alamos, Apr. 28 - May 2, 1980.

2. J. B. Taylor, Phys. Rev. Lett., 22 > 1139 1974.

3. M. N. Rosenbluth, M. N. Bussac, Nuclear Fusion. j_9, 489 1979.

4. H. E. Dalhed, Bull. Am. Phys. Soc, Z5, 1025 1980 and H. E. Dalhed, "Critical Bias Fields for Tilting Stability in the Beta-II Experiment," contributed to this workshop.

5. Z. G. An, A. Bondeson H. Bruhns, H. H. Chen, Y. P. Chong, G. C. Goldenbaum, et. al. (to be published). -117-

EXPEFIIMENTAL C •JFIGURATION .15

•- -- - A Flui constnter ;- Pl«m« gun / i r - 0 o !, - o 75 cm a o o o a o o t o 0 o I •r- IT" \ -100 cm- • t \ I) i '• \ -40cm— J A. z i • -• • »/ I / a Bt. 8, probt \ O B^protK m

Figure 1 •

•

¥ s—:

Figure 2

VERTICAL PROFILES OF B, AND 8, IN THE FLUX CONSERVER MIDPLANE " [jj MAGNETIC FIELD PROFILES.ON THE FLUX CONSERVER AXIS 9/18/80 S-18

t • Oflt ptnmi IMV« gun ' [a) B (pot(Wd*l component) A t • Otis pljsma leavei gun I K 1 OI-3SM>K I 1 • 4 / \ O - oT

(It) B. (tocc>kW component)

2 a £ 0 - «• V

i Figure 3 -40-20 0 +20 *40 y-cin

Figure -118-

ILLUSTRATION OF OECAV OF MAGNETIC FIELD AT TWrO AXIAL LOCATIONS £3 (A) CMMS: at I/iruflion ngion (Z " 22 cm from tht plauna gunl

B. «5s 110 KG _£

(6) Ctnur of flux eonurvtr (Z - 57 cm From thi plasma jun)

1.72 nG T. » 111*1

Figure 5

SCHEMATIC OF PROBE SIGNATURE FOR 90' ROTATION ABOUT PROBE AXIS

Figure 6

gXAMPLt Of BIAS FLUX INDUCED TILTING

V1S/NS-12 BiMfWd-2000 (A!f«,a.»>0* •is IHu • • ~ ' r *" • i • - \ 4 i: I —/ -. o ~. i.

i • _

I I * .' 0 \ '- m' 0 \/ , . i . -40-20 0 20 40 -40-20 02040

Figure 7 -119-

Hydrodynamic confinement of thermonuclear plasmas TRISOPS IIX

Daniel R. Wells, Paul Ziajka, Jack L. Tunstall Department of Physics University of Miami Goral Gables, Florida 33124

Abstract

Experiments on fusion reactions produced by adiabatic compression of plasma vortex structures are discussed. The TRISOPS machine at the Uni- versity of Miami has been modified by improving the preionization of the plasma and increasing the ring frequency of the conical theta-pinch coils. It has been possible, with this modified machine, TRISOPS IIX, to obtain ion temperatures of 1 kev before secondary magnetic compression and without any magnetic guide field. Ion temperatures of over 6 kev are obtained with secondary magnetic compression fields of 30,000 G. The plasma pressure in both instances, must be balanced by hydrodynamic forces. Ion temperatures and densities were measured by three different methods. All methods yeild essentially the same results. The plasma was held in stable equilibrium for 100 ysecs and neutrons were produced for 70 ysecs.

1. Co-rotational and contra-rotational plasma structures

The first important work done on rotational MHD fluids was accomplished by Busemann in the late 1940 's. (1) Consider a vortex filament in a surrounding conducting fluid which is permeated by a constant magnetic field. Simple considerations of the equilibrium indicate that there must be two separate and distinct types of structures (see Fig. 1). One, called a co-rotational^ vortex filament, moves anti-parallel to the magnetic field § with velocity v. A conduction current j flows along the vortex filament and j is parallel to C, where i is the vorticity which is given by r = curl v. The other type, called a contra-rotational vortex filament, moves parallel to the magnetic guide field B and has its conduction current density 3 anti-parallel to its vorticity 5. Thus there are two basic, separate and distinct types of MHD vortex, one moving parallel to the background magnetic field and the other anti-parallel to the background magnetic field. The corresponding balance of forces for each vortex shown in Fig. 1 accounts for the equilibrium of each simple vortex structure.

2. Vortex rings, force-free fields and stability

The step from the simple plasma filaments described in Fig. 1 to a set of co-rotational and contra-rotational vortex rings is simply one of going from a two-dimensional description of these structures to a three dimensional description in spheroidal or toroidal geometry. The details of the theory of this transition are given elsewhere. (2)(3)(4) It turns out that a contra- rotational ring does indeed propagate paralle to a magnetic guide field and a co-rotational ring propagates anti-parallel to the field. -120-

The equilibrium of these rings is self-evident, but the question of stability has to be considered separately. It is at this point in a discussion of these configurations that the work of Chan.drasekhar and Woltjer plays a major role. (5)(6)(7) Chandrasekhar asked what configuration the velocity and magnetic induction fluids in a rotational plasma structure must take if the structure is to have minimum total energy or, in the case of a compressible fluid, free energy. One answer to this question is trie following set of Fuler-Lagrange equations which, govern the flow and magnetic fields in such a structure: curl B = ,(D (1 ) v = t :-.D . _(2) Equation 1 specifies a Lorentz force-free equation. The jxB forces are everywhere zero both inside the structure and in the surrounding fluid. Equation 2 states t.iat the mass flow field must be everywhere parallel or anti-parallel to the local :r.a •:.• r. e 11 c induction field. But Eq. 2 is jus-- the equation which describes an;! co- and contra-rotational plasma ring. Adopting r'.q. 1 tc the set of rings is slightly more difficult, but the rosv.it is surprisingly simple. It is a set of force-free col- I : f:f:,; r rl:; : ~. These are shown in Fig. 2. The core of the rings has a purely toroidal flow and corresponding current density ar.: voriticity, i.e., they are directed around the center of the ri;uT. At the surface of the ring the flow has no toroidal component at all, but the velocity, current density, magnetic and vortex fields all circle the ring cross section as a poloidai field which, of course, is necessary to satisfy the boundary con- iitior.s, since the rings are moving parallel and anti-parallel to the r ?. -r.ctie guide field in the laboratory frame of reference.

'.' reject ion of collincj.r force-free structures in the

A simple- conical theta pinch immersed in a magnetic field parallel to the center line of the pinch coil will generate the ::r:ctures considered here, this is shown in Fig. 2. It shows :: double-ended conical theta pinch coil with a magnetic field

If *J ceil is placed with a magnetic guide field parallel to its axis, and if a current is pul. d through such a coil, a pair of plasma rings will be generate.i, one at each end of the coil. It is easily shown that if the guide field is directed toward the left the right hand ring will be contra-rotational, and the left hand ring will be co-rotational. The toroidal components of the magnetic field trapped in the ring are directly generated by induction. The rings simply act as secondary turns of the transformer whose primary is the conical coil itself. The toroidal components of the trapped field in the rings are generated by Hall currents. The gradients of the magnetic field produced by the conical theta pinch coil effectively produce a charge separation which drives poloidal currents. These, in turn, generate the toroidal magnetic field. (5) -121-

If the double ended conical theta pinch is now cut in half and the two ends reversed in position in the magnetic guide field as shown, the result will be c. pair of co- and contra-rotating ring structures approaching each other. Theory predicts that two co-rotational or two contra-rotational rings will superpose to form one stronger ring while a co-rotational and contra-rotational ring will interact.. This interaction is the result of the fund a- mental nonlinear character of the partial differential equat ion s describing the flow. In 1962, Wells (2) invest.igated these structures, Firs t bv propagating them along a magnetic solenoid and then by olaci na the halves of the two ended theta pinch inside a magnetic mirror which acted as the quide field replacing the solenoid.

4. Heating plasma rings by adiabatic compression

Ion temperatures of 1000 ev have been measured in the rings after they meet at the center of the mirror guide field. In order to achieve themonuclear temperatures some auxiliary heatinq method is required. In 1973, a set of secondary mirror coils was added to the machine. The configuration is shown in Fig. 3. Initial experiments indicated that the structure of the rings was not des- troyed by a large secondary magnetic compression. Ion temperatures as high as 170 ev were observed. In order to increase the magnetic field intensity, the compression coils were reduced in diameter from C" to 3 1/2". This allowed the production of much higher magnetic compression fields without increasing the size of the 230,000 joule capacitor bank that powers these coils. The vacuum chamber was decreased in diameter to fit the smaller coils. With this new configuration, 150,000 G fields have been produced at the conjugate of the mirror. The size of the conical theta pinches was also reduced to fit the now geometry. The machine was run in the static mode by filling the vacuum chamber with 300 microns of deuterium and then sealing off the system. Pulsed gas valves were not used. The pre-ioner consisted of a single turn coil energized by a single GE clamshell capacitor. This system rang at 500 kc. The capacitors were charged to 20 kv. The conical theta pincher. were powered by GE capacitors. These coils rang at a frequency of 500 kc at 20 kv. The compression mirror coils had a quarter cycle rise time of 10 i:sec. These compression coils were crowbarred at peak current to give an effective "field on" time of 8 jjsec. The vacuum chamber was pyrex. It was attached to the ceramic cones which acted as liners for the conical theta pinch coils. The large mirror coils which act as a guide field for the plasma rings before compression were powered by a rectifier and maintained a steady magnetic field of 500 G during the experiment. Results given are in reference (10). Ion temperatures as high as 4.5 kev were measured optically using the 4686 A He-II line and C-III and IV lines at 3170 A and 3938 A, respectively (11) .

5. Variation of machine parameters

After completion of the experiments described above, the capacitor bank that powers the secondary compression coils was -122-

replaced by a new bank with a much higher capacitance and lower operating voltage. The old bank consisted of 36 modules of two capacitors each, all in parallel. Each capacitor was rate at 16 pf and 20 kv. The new bank consists of 18 modules of two capacitors each. Each capacitor is rated 176 uf at 10 kv. The old machine is designated as TRISOPS VII, the new machine is TRISOPS IIX. The quarter cycle rise time of VII was 10 ysec. The rise time of IIX is 42 psec. Figure 4 is a plot of ion temperature and density vs. time for TRISOPS VII. Figure 5 is a similar plot for TRISOPS IIX. Figure 5 also shows neutron production and plasma density vs. time. As an independent check on plasma density and ion temperature, simultaneous measurements of the time history of neutron flux and 4686 A Helium line broadening were made. It should be noted that for both machines the ion temperature curve has a characteristic shape. For both machines, the measured electron temperature is less than 300 ev at all times in the cycle. Hydromagnetic confinement The minimum free energy plasmoid is a force-free structure. Unfortunately, as is well known, a force-free structure cannot support a high density, high temperature plasma. Both the Lorentz force and Magnus force are identically zero in such a structure. In order to explain the high temperature, high density plasmas actually attained in TRISOPS, there must either be a force bearing current structure or H torce bearing flow structure surrounding the force-free core or the structure is quasi force-free. The choice between a force bearing current structure in which Lorentz forces p]ay the major role in containing the 1000 atmosphere pressures measured in TRIOPS and a force bearing flow structure in which Magnus forces play the major role had not really been made up ho this time. There was simply not enough experimental evidence on wnich a well informed decision might be made. Data taken from TRISOPS IIX definitely established the validity of previously obtained measurements of temperature and density. It also yields a definite time history of the ion temperature and density, allowing a final decision between the two containment models. Perhaps the most interesting result is that with greatly improved preionization of the plasma the precompression ion temperature reaches 1.0 kev. This temperature is obtained with essentially zero magnetic guide field intensity. The vortex structures are formed by conical theta pinch guns. The mechanisms of formation ensures entrapment of magnetic fields in the structure. Recent experiments utilizing much more efficient preionization indicate that the precompression ion temperature is almost an order of magnitude higher with zero external magnetic guide fields than similar temperatures measured with guide field strengths from zero to 1500 G. This suggests a model in which a colinear force-free core is surrounded by a vortex flow structure that supports the high pressure hot dense plasmas. This surrounding flow structure, in turn, must be supported by the vacuum chamber walls. Since the plasma pressure, nkT., at these levels is of the order of 200 atmospheres, a more detailed examination of the mechanism is necessary. -123-

References 1. Busemann, A , 1962 NASA SP-25, 1. 2. Wells, D.R., 1962 Phys. Fluids, 5_, 1016. 3. Wells, D.R., 1964 Phys. Fluids, ]_, 826. 4. Wells, D.R., Norwood, J., 1969 Jour, of Plasma Physics, Vol. 3, pare 1. 5. Bostick, W.H. and Wells, D.R., 1963 Phys. Fluids 6_, 1325.

Figures

COROTATSNG CONTRAROTATING

B ' V Br

MOVES WITH-Bo MOVES >..',}* *B0 1. fling structure in frame ol LH rinj FIG. 1 2. Ring structure in frame of RH r,ng, «**

TRISOPS IIX - DC MIRROR COILS TEMPERATURE i, D£WSJTY VS.TJME , 2 CONICAL THETA PINCHES — 8 comp., 36 kG peak ^COMPRESSION COILS-- I \ PROBE PORT /

niziJ Lean

TO VACUUM PUMP / OPTICAL PORT / Z 1x10" PREONIZER ' 7 10 20 30 40 GAS VALVES TIME-MICROSECONDS

FIG. 3 -124-

Formation of the Spheromak Plasma by a Slow Magnetic Induction Scheme*

M. Yamada, K. P. Furth, W. Heidbrink, A. Janos, S. Jardin, M. Okabayashi, E. Salberta, J. Sinnis, and F. Wysocki

Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544

I. Introduction

The S-l experiment will develop an electrodeless "slow" spheromak formation technique to produce a 500 kA toroid of a = 25 cm, R = 40 cm. The formation scheme is based on a transformation of poloidal and toroidal magnetic flux into a plasma from a flux core. Physics background and engineering aspects of the S-l apparatus have been presented earlier.U»2]

The experimental realization of the spheromak configuration to date has followed two main lines of approach: the coaxial plasma-gun scheme-^ and the field-reversed theta pinch with center-column discharge. These are both "dynamic" schemes, occurring on a time scale comparable to the Alfven-wave transit time (T^JO , and involving the passage of large plasma currents through electrodes.

The attractiveness of the spheromak as a fusion reactor concept would be enhanced by a slow formation scheme in which the required forming power could be kept moderate even for reactor plasma parameters. The elimination of electrodes would also be a favorable development. The S-l program-'- is designed to demonstrate an appropriately slow (quasistatic) and electrodless formation scheme. The present paper describes initial results from the Proto-S-1, the first experimental demonstration of this type of approach.

The S-l scheme of Fig. 1, was developed in the course of extensive two-dimensional resistive MHD simulations.5>">' An initial poloidal field is generated by toroidal current inside a ring-shaped (toroidal) flux core, and is weakened on the small-major-radius side of the core by the superposition of an externally generated vertical field. The core also contains a toroidal solenoid, which generates a toroidal field on its interior, and therefore generates an equal and opposite toroidal flux change on its exterior. When the toroidal solenoid is energized, it induces a poloidal current in a sleeve-shaped plasma surrounding the ring. The associated toroidal field distends the poloidal-field sleeve, stretching it towards the magnetic axis where the poloidal field is weakest. When the toroidal core current is reduced through zero and 'crowbarred' at a negative value, an increasingly large toroidal current is induced in the pTasma. Magnetic reconnection of the poloidal field then occurs, on a time scale that is slow compared with the dynamic time , but rapid compared with the resistive diffusion time T, .-.-, and a -125-

separated plasma toroid is created on the small-major-radius side of the flux core. This toroid, the desired spheroinak configuration, is held in subsequent equilibrium by the externally generated steady-state field.

II. Experimental Arrangement

The machine configuration and the main components of the Proto S-l device are scaled down (1/6) versions of the main S-l device, which will be completed at the end of 1982. The flux core of 15 cm major radius and 3 cm minor radius contains the Poloidal Flux Coil (PF, 3-turn toroidal winding) and the Toroidal Flux Coil (TF, 40-turn poloidal winding). The core is covered by 3-mil-thick metallic liner (stainless steel), which provides a vacuum enclosure around the core and also tends to symmetrize the induced fields during the initial breakdown stage. The core is supported against induced forces and is powered by 2 sets of electrical leads. The external vacuum vessel is made of 3/8" stainless steel with no insulating breaks. The externally generated vertical field is of order 1 kG or less, and is steady-state. The PF; and TF coils are driven by fast capacitor banks.

III. Experimental Results

When the two coil currents are drawn into the flux core with th>: sequence shown in Fig. 2, a plasma discharge is created. Framing- camera observations show that a few microseconds after the initiation of the; TF current a plasma sleeve is created around the core, which chen expands on its small-major-radius side and finally transforms into a localized plasma in the intended spheromak equilibrium position. Many plasma discharges have been made in H2, He, and Ar gases with best results obtained by filling the vacuum vessel with 20-50 m Torr Helium gas.

In order to provide a conclusive demonstration of the formation of the spheromak field configuration, the time evolution of the magnetic fields has been measured directly by movable magnetic probes (2 mm diam. 20-turn loops immersed into the plasma). The success of this method is due to the excellent reproducibility of the field configuration (within 5% deviation). The evolution of the toroidal and poloidal magnetic field has been measured and described in the earlier communication. The spheroinak equilibrium configuration with the poloidal field reversing as a function of major radius R and the toroidal field vanishing at the plasma edge, is established 12-14 psec after the start of the plasma discharge (TF-current start). This fully formed configuration remains intact for about 12 ysec. Figure 3 shows the measured contours of the toroidal field in a polcidal plane at t = 16 ysec. A time evolution of the toroidal field contours is shown in Fig. 4.

Following the establishment of the desired configuration, the spheroinak plasma shrinks gradually to S = 3-4 cm, while the poloidaj. and toroidal fluxes trapped in the plasma are decaying by a resistive diffusion. At about t = 25 psec, the sudden appearance of a nonuniformity in the framing camera pictures suggests the onset of a nonaxisymmetric instability. -126-

Th e plasma density and temperature have been monitored by double Langmuir probes (voltage-swept) and CO2 laser interferometry. For helium discharges, the plasma density, measured at R = 5 cm, t = 16 ysec

is 1.2 + 0.6 * 1()15 cm~3, an(j the central temperature reaches its highest value 25 ± 5 eV at t = 15-18 usec.

We have performed a computer simulation for the present spheromak formation experiment by a resistive two-dimensional MHD code." The qualitative agreement between the experimental data and the simulation results is excellent.2

In conclusion, the feasibility and effectiveness of the quasistatic S-l spheromak formation scheme has been verified experimentally. The resultant spheromak configuration lasts about 15-20 ysec (< 100 f^if), which is significantly long, since the classical magnetic diffusion time of the plasma (Te = 20 eV) is expected to be of the same order (50 ysec). Utilizing about 1/4-1/3 of the flux change from the core, the maximum toroidal and poloidal plasma currents are found to be roughly 20 kA and 50 kA respectively. "Larger experiments in the S-l program are expected to provide more detailed information on the MHD-stability and transport characteristics of the spheromak configuration.

We thank G. Wurden for his help in CO2 laser interferometry and P. Heitzenroeder, D. Herron, T. Holoman, R. Labaw, P. Larue, and other PPPL personnel for their excellent technical assistance.

^Supported by US Department of Energy Contract No. DE-AC02-76-CHO3073.

Figure Captions

Fig. 1 (a,b,c). S-l Spheromak Formation Scheme. Fig. 2. Time evolution of Coil Currents. PF (top) and TF (bottom). Abscissa: 10 psec. Ordinate: PF 30-kA turn/div; TF: 400- kA turn/div. Fig. 3. Measured toroidal field contours at t = 16 ysec. Fig. 4. Time evolution of Toroidal Field Contours at t = 12 ys, 16 ys, and 20 ys. TF current is crowbarred at time t = 10 ysec.

References

[ 1] H. P. Furth £t al., Proc. US-Japan Symp. on Compact Toruses (Dec. 1979) p. 1, 166, 171; M. Yamada et al., Bull. Am. Phys. Soc. 24 (1979) 1022. [2] M. Yamada et al., PPPL-1723 (Submitted to Phys. Rev. Lett.). [3] H. Alfven et al_., in Proc. 2nd Int. Conf. on Peaceful Uses of Atomic Energy _3_1, (1958) 2; C. W. Hartman e_t^ al. , Lawrence Llvermore Lab. Quarterly Report #2 (1978); T. R. Jarboe et al., in Proc. US-Japan Symp. on Compact Toruses (Dec. 1979) p. 53. [4] G. Goldenbaum et^ al_., Phys. Rev. Lett. 4A (1980) 393. [5] W. Grossmann et_ aL., Proc. 8th Int. Conf. on Plasma Phys. and Cont. Nucl Fusion Res. (Burssels, 1980) Paper IAEA-CN-38. [6] S. Jardin and W. Park, PPPL-1706 (Submitted to Phys. Fluids). [7] H. C. Lui et al., submitted to Phys. Fluids. -127-

QUASISTATIC FORMATION USING INDUCTION

Flu» Core

Support Leads

8p Coil

Bp Current

Bt Generoted Reduced in Plosmo Thru Zero

(c)

Bp Current Reversed ond Crowborred

Figure 1, -123-

50 JISEC

Figure 2.

TOROIDAL FIELD PLOT

T8 0.2 kG

0.6

12 10 8 6 4 2 0 2 4 6 8 10 12 r (cm) r (cm) Figure 3. -129-

Figure 4. -no--

Ballooning Mode Growth Rate Dependence on Separatrix Shape for Idealized Equilibria of a Field Reversed Theta Pinch*

D. V. Anderson, National Magnetic Fusion Energy Computer Center, Lawrence Livermore National Laboratory,

H. L. Berk, Institute for Fusion Studies, University of Texas,

J. H. Hammer, Mirror Program, Lawrence Livermore National Laboratory.

The fastest growing ideal MHD modes in a field reversed theta pinch are in the infinite toroidal mode number limit and have low poloidal mode numbers. These are called co-interchange modes or less precisely ballooning modes. Using simple analytical models for the equilibria we can change the shape of the separatrix without changing the pressure function. For the cases examined here we find that making the separatrix more racetrack in shape reduces growth rates near the vortex point but this is partially offset by an increase of tne growth rates near the separatrix. We have chosen the pressure function associated with the elliptical Hill's vortex model as one suitable for this study. It is given by

Here a gives the axial half length of the separatrix, b gives the radius of the separatrix at the midplane, v is the poloidal flux function, and A gives the magnetic field normalization. The actual shape of the separatrix depends on the currents in the external conductors and depending on how these are arranged one can obtain the elliptical Hill's vortex configuration where

2 2 2 V TP = Ar (-1 + -z + |2). (2.)

For this case the magnetic field on the axis at the midplane is just B(0,0) = -2A. Because the separatrix radius varies as z2 near the midplane we also call this the "z2" profile. At any rate the separatrix • This work was performed under the auspices of the U. S. D. 0. E. -Ill-

is an ellipse. Profiles that are oblate, spherical, and prolate correspond to external fields characterized by mirror ratios exceeding unity, unity, and less than unity (the anti-mirror) respectively. If we make other modifications to the external conductors we can deform these ellipses to shapes that are more rectangular. One such profile, we dub as the racetrack flat vortex, is given by

•*=

b2" + b4 where this more cumbersome expression has a separatrix which varies as z4 near the midplane but ^ t i I L has the dimensions of a and b. We term this profile the "z4" type. W^ should stress the fact that the pressure function is the same for both profiles and that the configurations differ only because of the differing external conductors. For each elongation studied we keep a, b, and A fixed. To vary the cl'-'cation of the separatrix, Es = a/b, we change a only. These equilibrium profiles differ from what has been observed in field reversed theta pinch experiments such as FRX-H at Los Alamos. The experiments are characterized by having finite pressure on the separatrix which is known to enhance the MHD stability. We feel that the effects we are studying here will also occur in a qualitatively si:n;lar manner for the more realistic equilibria. We have derived equations from the energy principle for single adiabatic perturbations. In the large toroidal mode number limit the minimization can be done separately for each field line. In order to obtain growth rates from our analysis we do the minimization in the kinetic energy norm where

^ (BZ)2] (4.) c 1J rrj is used. A coupled pair of generalized Sturm-LiouvI11e eigenvalue equat ions,

U )) + ( - PP'DD - °2 ££ D ^ ds> >2B2 1/S2 + 1/>'P;B \/B2 + 1/>P 3s and (5.) d . B dZ. o , DX M/B2 + IAP ds> asM2 ; -132-

result from this analysis. Here X and '/ give the normalized d1sp1acement s perpendicular and parallel to the field I Ine, s is the 1 eng t-h a 1 ong a field line, r is the ov 1 i ndr i eaI radius, and . is the the rmodynam i c ratio of specific- heats. I) = 2k/rB is the curvature dependent factor where k is the curvature of the field 1ino at the

point s. We search the eigenvalue spectrum for mln where > 0 ( < 0 ) corresponds to stable (unstable) equilibria. The ^row t h rate , = - i . ,s ob t a l ned f roir, t he de f I n i t ; on • "" • = . Newcornb has solved I h i s s v s I em exact 1 v for flux surfaces at the vortex of an elliptical Hi I I '^ vortex configuration. lie has found a growth rat e

1/: () .(.v) = A(.v) = .ds<-) - t -1 i 1_ ' H nil ich corresponds to the inverse transit time tj f an A I f ven wave travelling around the flux line. He has also been able to eompu t e growth rates for flux .surfaces away from the vortex i)V an expansion technique. (irowth rates on other flux lines are of comparable magnitudes a i ven by let's say . ( . ) = •(.).(.,.) = • ( . ) A ( . ^ } where I y p i caI Iv 1 < . ( . ) < 1 . He have studied the Mill) s t ab l 1 I t v of each equilibrium by computing the e i gen f uncI ions and eigenvalues on ten different flux lines f rom verv near t tie vortex to I l ne s near' the separ ,i t r I x . For the elliptical Hill's vortex we obtained results in agreement with those of \ewcomb just described. For the racetrack flat vortex we find growlh rates down by about a factor of :S on lines near the vortex. On I he 70" flux surface, where (. — '. )/. — .70, the growth rates are about the same. Just inside of the separutrix we find about a 10" increase of the racet/'ack grow; ti rate compared to the elliptical Hill's vertex result. Wi performed this study for elongat ions K = 5 and [•'. = 10 and found similar results for both. The growth rate dependence on elongation is found to be , -• c/F fairly well independent of the flux surf ace. The racetrack equilibria have much more elongated flux surfaces- near the vortex than the elliptical Hill's vortex for the same separatrix elongation as can be confirmed bv comparing Fqs . '.i and :i for V • i .. One finds the elongation at the vortex, F,. , is s I mp 1 v related to that at the separatrix by -133-

Ev = 2ES (7.)

for the elliptical Hill's vortex and by

Ev = 2.31R2 (8.) for the racetrack flat vortex. We have compared the computed growth rates with the Alfven frequency for each flux 1 ine and found .'.'(v ) and >(v) to behave similarly with ..(vj being somewhat smaller. This means we can estimate growth rales from the geometrical arrangement of the equ i 1 i br i urn f1ux lines. Since the Alfven transit t ime will be proportional to the field line length -~E and inversely proportional to <-. typical B value we can understand why the racetrack flat vortex has a lower growth rate near the vortex point. In summary we have shown how increasing the rectangularlly of an equilibrium modifies the ideai MHD growth rates fo" ballooning. The results we have computed show trends and effects that can be estimated from the geometric differences of the profiles studied by very simple analyses based on the Alfven transit times.

Re f e fences :

1 . ) W. A. Newcomb, "MHD Instabi I i ty in the Neighborhood of the Vortex Point of a !•' i e I d-Re ver sed Magnetic-Mirror System," Lawrence Livermore Laboratory report ICRL-83867 ; to be published in Physics of Fluids. -IV-

Linear and Non-linear Computations of the Ideal MHD Tilting Mode in the FRX-B Configuration*

by D. C. Barnes and A. Y. Aydemir, Institute for Fusion Studies, University of Texas,

D. V. Anderson and A. 1. Shestakov, National Magnetic Fusion Energy Computer Center, Lawrence Livermore National Laboratory,

D. D. Schnack, Los Alamos Scientific Laboratory.

Experimentally, the tilting mode has not been observed in the field reversed theta pinch experiment FRX—B. Various estimates have indicated that ideal MHD theory would predict such a mode to be unstable. We have computed an equilibrium configuration very much like the typical FRX-B plasma and have used it to initialize two different evolutionary computations of the tilting displacement. A linearized MHD code, RIPPLE VI1, and the non-linear MHD code, MALICE2, both confirm the existence of the tilting mode and are in good agreement on its linear growth rate. Compressive MHD effects (as well as kinetic mechanisms ignored in this model) may non-linearly stabilize this mode. To compute an equilibrium configuration we write Ampere's law in terms of a flux function and assume scalar pressure equilibria with P = P(i/)- If we denote = d/dy then in cylindrical coordinates we get the equilibrium equation:

We have used

up (2.) to compute equilibria consistent with the observed FRX-B plasma. Here ip g and i/'mt n give V at the plasma surface and vortex respectively. The two terms allow current densities which are maximum or hollowed out at the vortex. Pt is the pressure maximum. Unlike some idealized equilibria which require a cutoff of the plasma at the separatrix, the configurations we compute include a region ei_Elasma_21liside._ilie. separatrix to better model the experiment. The • This work was performed under the auspices of the U. S. D. 0. E. -135- results of our equilibrium code, CYLEQ, are stored on disk and are available to initialize the time dependent codes MALICE and RIPPLE VI The full 3D ideal MHD equations,

dB - 4 BV.u - B.Vu = 0, dt (;i.

cd- + PV.u = 0, dt

du V. BB - dt u-u O '

F = (-.-l)ci are evolved by the MALICE code. Here c is the plasma mass density, u the flow velocity, B the magnetic induction, and 1 the specific internal energv ( temper P ture ). The artificial drag coefficient i- is used to damp the approach to a steady state configuration with velocity uQ; for the stationary equilibria we investigate here uQ = 0. Once the initial equilibrium is obtained we then set v = 0, apply perturbations to u, and then let the system evolve.

Evolution of the linearized MHD equations,

1 C — = - -V x (rV x B,) 4 V x (iij x Bo),

1 oV x (V x ~— ) = -V x : V x [(B0-V)B1 4 (B,-V)BO];,

(4.)

'[D{) = U,

V-(u,) = 0 -136-

is accomplished using the RIPPLE VI code. Here B, and u{ are the usual linearized perturbations about the known background (equilibrium) state where J = B (r,z) and u = 0. r is the resistivity which is taken very O O v ' O J small for the computation of ideal modes. In the derivation of these equations we have assumed displacements with a pure toroidal mode of the form f { r , z , t )exp( i nv) where

RIPPLE V! Refined Mesh(67x64) .038 RIPPLE VI Coarse Mesh(12x16) .024 MALICE Coarse Mesh( 12x20x12 ) .029

We attribute these discrepancies to numerical diffusion associated with the spatial discretization. We estimate, for the same grid, that MALICE has smaller errors of this sort because of its use of an almost l.agrangian algorithm in contrast to the Eulerian treatment used in RIPPLE VI . During the early part of the evolution the growth of the tilting mode is accompanied by a decrease in the magnetic field energy. This we expect physically as the dipoie is turned around into a lower magnetic energy configuration. As we follow the tilt into the non-linear regime we observe a damping of this growth and a concomitant increase of the internal energy produced : • compressive heating effects. This occurs when the tilt has proceeded about 20°. We have been unable to carry the simulations much farther than this because some shortened grid spacings of the Lagrangian mesh produced unacceptably small time steps. Attempts to remedy this problem by increasing the rezoning have led to numerical diffusion of the magnetic 'IeId which causes much of the field reversed region to be lost. One obvi JUS remedy would be the use of more refined grids in MALICE. This -ould require faster and more capacious computers than are now available. Another avenue of improvement lies in the development of more accurate numerical schemes. To this end we have been developing a -137- new version of MALICE- sometimes dubbed FALICE- which replaces the algorithm for advancing the magnetic induction with one that advances the magnetic flux in a conservative fashion. In summary, we have observed the tilting mode in an FRX-B configuration using the two codes MALICE and RIPPLE VI. Good agreement for the growth rate in the linear regime was obtained between these two calculations. Indications of non—linear stabilization by compressive heating are given in the MALICE simulation but as yet we have been unable to carry the simulations far enough to give a conclusive answer.

I ^ir^T^r^

Figure 1. Magnetic flux surfaces show the distortions characteristic of the tilt ing mode.

Ke ferenres:

1. J. Killeen, D. D. Schnack, and A. 1. 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 L'CRL-83332 .

2. .1 . U. Brackbill, Meth. in Comp. Phys. 16, 1 (1976).

'.) . A. I. Shestakov, D. D. Schnack, and J. Killeen, Symposium on Compact Toruses and Energetic Particle Injection, Princeton, New Jersey, Dec. 12-1-1, 1979. -138- TUO-D TRRNSPORT MODEL FDR FRC PLflSMRS THIRD COMPRCT TOROID SYMPOSIUM

R. N. Byrne & M. Grossmar,n«

Science Implications, Inc., La Jolla. Cfl 9201'

T) INTRODUCTION

Field reversed theta pinch (FRGP) experiments have exhibited favorable stability behavior since .1969 (Eberhagsn ^nd (irossmann , 1S73D ond mere recently this behavior has improved (Linford et al.. 3389) io the point where stable lifetimes approaching 303 us are within sight. The need for a realistic transport simulation is thus felt and the present paper describes a model for FRBP simulations.

Previous attempts usinq 0-D and 1~D models have had surprisinqly good success in representing some aspects of the FRSP evolution. In particular. the 1-D model of Hamasaki and Linford is successful in incorporating the gross features of axial and radial contraction and expansion. This J-D model also was able *o representp trivially, ths experimentally observed flatness of the profiles in the longitudinal direction.

^Permanent address: New York University, Courant Institute of Mathematical Sciences -139-

Recognizing the need for a more realistic transport model, we have developed a \- D simulation code which takes into account the complex topology of FRCP plasmas including the separatrix and with long flat profiles in the axial direction. The initial elongation can be specified arbi- trarily. The present paper describes very briefly some of the features of the model and the code* along with representative results from compression studies, transport simulations and comparisons with experiment.

ID DETRILS OF THE MODEL

Most of the 2-D plasma equilibria heretofore studied are topological ly related to a z-pinch bent round into a torus. The flux surfaces, in those that have nested onesi shrink to a line as one approaches the poloidal magnetic field null, and field lines near that null have considerable curvature as they do in the 3-D infinite z-pinch. FRX is a twc dimensional version of a theta pinch, however, and its equilibrium is different. In FRX, the field lines become straight away from the ends of the plasma, and the field null is a belt rather than a hoop, as illustrated hy the field lines plotted for the calculated equilibrium shown in Figure 3. -140-

Figure 1 Calculated Field Lines for an FRX EGuilibrium

These equilibria are characterized by profiles fi[W) which are singular at the flux minimum [field nutU as f"^1^] 2 where u is the adibalically invar iant function

P is the pressure. U the f lux, and V(W the volume enclosed uy the flux surface. i/)0 is the minimum value or iji, which is taken at the field null. It is important to realize that no directly measurable quantity such as temperature^ density, magnetic field, or current is singular or irregular for these equilibria, fllthough M is singular here it is also singular in tho simple infinite 3-D theta pinch. -141-

The adibatic equilibrium is given by solving what Grad has called a QDE r oncG V>0 and a profile ui 'W are known. The upper limit of U for which plasma is present is taken to be zero, the value on the separ?trix (and the axis). Outside the separatrix, the field lines laave the system and, whereas the sophisticated 1-Q model of Hamasaki and Linford. mentioned abovei takes account of the finite time it takes for plasma to stream to tha ends 10 derive a model for pressure evolution outside the separatrix. »3 shci!! herein simplify by assuming zero pressure outside the separatrix. ll> > 0.

Ue must still follow the evolution of u and 0. Classical transport would seem an irreducible minimum; we start there and plan to supplement it with anomalous terms in future work. For the classical terms *e use the forms given by Braginskii as modified by Hazeltine and Hinton to take account of toroidal effects.

Final ly. the one- and two-dimensional equations are reconciled by an iterative procedure, as described by G^ad . Byrne and Klein**, and others.

1) S. I. Braginskii in Reviews of Plasma Physics, Vol. i. New York (1365)

2) R. D. Hazeltine and F. L. Hinton, Physics of Fluids 16. 1882 C1972)

3) H. Grad, in Uorkshop on. 2D Transport, DOE/ET-0079, (1979)

1) R. N. Byrne and h. H. Klein, J. Computational Physics 26 (352),1978 -142-

The 2-D problem is solved, given the profiles of U, and the resulting geometric factors used to find more accurate Drofiles. The process is iter- ated (repeatedly integrating the 1-D equations forward in time) to convergence.

Ill) COMPflRTSON WITH EXPERIMENT

Results of simulation of the transport evolution of the FRX-B experiment at LflSL are described in this section, fl remarkable property of this device is that the plasma is very elongated (10:1] with flat density and temperature profiles over about 95S of its axial extent. The "slit" solutions for the equilibria described above simulate very adequately these properties of the experimental plasma. fl rapidly converging iteration scheme where the first computed 2-D ctosed flux surface just surrounds the slit has been implemented in the equilibrium solver, flny desired initial length of an FRX plasma may thus be found as a starting point for the transport simulations. The equilibria are computed using the 1 — D adibatic formulation and allow adibatic compression to be self-consisiant!y incorporated in the simulations. Constant fixed ion and electron temperatures are assumed on the separatrix! their valuos are infarred from expQri- mental observations. Pressure is finite on the separatrix. The remaining boundary conditions are- perfect conductivity on pinch tube wall, and vacuum flux distribution at the open pinch tube ends. Rs mentioned, finite particle leakage time in the real open field plasma is ignored here. The -143- evolution of the plasma profiles and geometry is simulated in time using the i — iteration algorithm assuming initial density and temperature profiles consistent with FRX measurements. Purely classical transport yields particle and energy half-lives of 200 and 50 us respectively, roughly five timos longer than what is seen in the experiment. By adjusting the transport coefficients in order to agree with experiment it is possible to obtain a measure of the degree of anomalous transport; FRX-B requires an anomaly factor of threo for olRctron thermal conductivity and fifty for particle loss.

IVD CONCLUSIONS

Initial attempts to model FRX-B plasma with the 1 — D simulator described above have been successful and have shown the plasma to exhibit somewhat anomalous behavior. He mention again that our model presently ignores the presence of a boundary layer outside the ssparatrix observed in the experiment; this layer contains significant mass flow and its contribution to energy and particle loss rates is not correctly treated by our model. Future efforts will be aimed at modelling the boundary layer and incorporating its contribution to the overall energy balance. -144-

PARTICLE CONFINEMENT IN FR6P WITH LOSS-CONE-LIKE SCATTERING

Q. T. Fang and G. II. Mi ley Fusion Studies Laboratory University of 111inoi s Urbana, IL 61801

ABSTRACT

A new model that takes into account a loss-cone-like mechanism in FR6P confinement has been investigated. Confinement times predicted by this model fall into the range of values reported from the experiments FRX-A and B.

I. Introduction

The particle confinement times observed in the experiment FRX-A and B1 are much less than that calculcated from classical diffusion or that from the lower-hybrid-drift scaling2 except one assumes a density gradient scale length to be about one ion gyroradius in the latter case.3 A new model considering the loss-cone-like scattering has been investigated.

II. Model of Calculation

A. Confinement Criteria and Loss-Cone Boundary

In an infinite circular cylindrical plasma the two adiabatic constants of motion, the Hamiltonian H and canonical angular momentum P , are given by

r2e2) and

Pe = m r 6 + q ^(r) , (2)

where is the electric potential and \\> ErAp is the flux function with A the vector potential. From Eqs. (1) and (2) we obtain: -145-

2 r = £ (E - q (fr(r))+ -^ry (Pf

• 0 The first confinement criterion is obtained by requiring r >o, giving

(4)

This corresponds to the physical boundary beyond which particles do not exist. The second confinement criteria is obtained by assuming that a particle is promptly lost due to collisions once it has E and Pg such that its orbit goes beyond the confinement radius a. Then, requiring that r^(a we have:

For convenience, Eqs. (4) and (5) are transformed to velocity space. Then the confinement boundary becomes:

2 vr _>_ 0 (6)

and

2 ,. -2-2 a ma ma — - (/) where $* - ({•(O-^a) and ty' = \p(r)-ip{a). Fig. 1 shows the confinement region. Equation (6) does not appear in this figure since it is naturally satisfied.

R. Loss-Cone Flux and Particle Confinement Time

The Boltzman equation including the particle loss in velocity space is given by

J dt 3t o ma v a i 0 i (8)

Here j^1 is the collisional flux in velocity space given by1* -146-

a/e F1 3f i f rfl/P __a (9) Ji ~ m" a " Dik 3VT Uj a K where a,3 represents particle species and i ,K represents coordinate, Fj and D-jk are the dynamic friction and diffusion tensor respectively. Integrating Eq. (8) in velocity space we obtain

• dSsJ(r)

The right hand side of Eq. (10) represents the loss-cone flux across the boundary S (Fig.l). The particle confinement time is now:

T = /ndv/ | J(r)dv

In order to simplify the calculation a cut-off Maxwellian distribution is assumed instead of solving Eq. (8) self-consistently. Also the ellipse in Fig. 1 is approximated by a circle. The electric potential and flux function are obtained from the rigid-rotor equilibrium model.5'6 The confinement radius a is chosen to be

a = m. + 6a (12) where /ZR is the separatrix radius and 6 is taken to be the gyroradius or its multiples. a III. Results From this model a typical ion particle confinement time ~ 70 ps is obtained for the case of FRX-B. This compares favorably with the experimental confinement time (~ 50-100 ys) for FRX-B. This model can in principle be applied to any field reversed confinement system although the geometry and thus the confinement boundary may become intricate. -147-

Fig. 1 Confinement boundary for ion in velocity space

References: 1. R. K. Linford, W. T. Armstrong, D. A. Pldtts and E. G. Sherwood, 7th IAEA Conference on Plasma Phys. and Controlled Fusion Research, Innsbruck, Australia, 1978. Z. S. P. Gary, Phys. of Fluids, 23, 1193, 1980. 3. E. H. Clevans, Symp. on Compact Torus and Energetic Particle Injection, Princeton, NJ, Dec. 1979. 4. B. A. Trubnikov, Review of Plasma Physics, V.IV, p. 137. Consultants Bureau, New York, 1966. 5. R. L. Morse, Los Alamos Sci. Lab. Report, LA-3844-MS, 1963. 6. Q. T. Fang and G. H. Miley, IEEE Sym. on Plasma Sci., Madison, WI, Hay 1980. -148- RECONNECTION DURING THE IMPLOSION PHASE OF FIELD REVERSED CONFIGURATIONS

D. W. Hewett and C. E. Seyler Los Alamos Scientific Laboratory Los Alamos, New Mexico 87545

Magnetic field topology changes in Field Reversed Configurations (FRC's) are essential for the formation and containment of the plasma. A significant part of the FRC research program relies upon the idea that a newly formed plasma, formed on open field lines will quickly change field topology before the rapid parallel electron thermal conduction depletes the plasma energy.

We have simulated the implosion dynamics of FRC formation using an axisTOuetric hybrid model consisting of kinetic ions and finite resistivity fluid electrons. The LASL FRC experiments are well described by our model, which assiuiit-s quasineutrality, zero electron inertia, and no electromagnetic rad iat io:i.

The simulation parameters during the initial stage of the implosion are similar to those of FRX-B. The simulation procedure assumes a homogeneous fully ionized plasma imbedded in a 1.5 kG uniform reversed bias field. At t = 0, an Vq of 45 kV is applied at the wall. Five distinct stages of dynamical evolution can be identified. They are: (1) The initial early implosion process, wherein

7 the ions are driven inward at twice the magnetic piston velocity (v. n ~ 4 x 10 era/sec), (2) The ions are reflected from the reversed bias field, during which time axial perturbations of the reversed bias field lines occur. During this stage, a significant amount of localized toroidal magnetic field (BJ is produced, however there is no net toroidal flux. (3) Nonlinear bouncing of the ions between the two regions of antiparallel fields follows, with the bias field line perturbations focusing the secondary reflected ions into the positive bias region, thereby causing clumping of ions localized near the field wall. Concurrently during this stage, small scale island structures develop. (4) Nonlinear coalescence of the small scale islands into larger islands rapidly follows. Also during this stage, the BQ field is annihilated. (5) After coalescence, a quasistationary equilibrium stage occurs, wherein no significant dynamical processes take place. -149-

Since the implosion process ij highly dynamical and very complicated, a detailed analytical description of the process is out of the question. Hov;ever there are several aspects of the process which can be explained qualitatively with quantitative estimates of the time scales involved. Perhaps the most intriguing aspect of the implosion, is the very rapid development of field line perturbations in the reversed bias region. Equilibrium-stability studies have ruled out the two most commonly used explanations, the mirror mode instability driven by highly anisotropic ion pressure and tearing mode instability driven by magnetic energy relaxation. Stability studies of these modes were pertornei4 usin<5 the Hybrid code itself and a Vlasov-fluid linear stability code. [Tie results are in agreement, these modes grow on a time scale which is an order oc magnitude too slow for explaining the observations. What we do feel is the correct explanation is a kinetic ion, fluid electron version of the Kruskal-Schwarzschild (K-S) instability driven bv ion acceleration in a region where the density gradient is favorable for instability. The region in which this explanation could apply is exactly where the bias field line perturbations are initially observed. The magnitude of ion acceleration can be estimated by considering the reflection of the ions off the reverse bias, this gives

lu cm a ~ v?/rT ~ 3.2 x 10 , . The formula for the growth rate of the K-S mode 1 L sec gives an estimate of the relevant time sci.le; this is y^ = ka - k^V?. The stabilizing term -k^V? has sufficient strength to localize the instability to the vicinity of the field null. Simulation values substituted into this formula give y"1 ~ 37r|S, a result which is more than adequate to explain the observations.

Another observation which requires explanation is the rapid small scale island formation. This process is probably a nonli ear consequence of the K-S mode for the following reason. Distortions in the reverse bias field where the K-S mode is unstable tend to focus the reflected ion bean into localized regions in the positive bias side of the magnetic null. (See Fig. 1) The ion^. then undergo a secondary reflection, the result of which is to distort the positive bias field lines such that the ions are more sharply focused into clumps near the magnetic null. (See Fig. 2) Since ion density clumping must necessarily involve magnetic islands to maintain the localization of ;he ions, this seems to be a reasonable explanation. If this picture is correct, an estimate of the time scale to form fully developed islands is obtained by calculating the ion -150-

Figure 1 Figure 2

bounce time between the regions of c imura magnetic field line perturbation. The time scale for island formation should then be about a couple of bounce times. The bounce time is calculated to be about 50ns and the island formation time is between 100 and 150r|S. Figure 3 shows the contours of poloidal flux from simulations after 150ns — clearly showing the existence of small scale magnetic islands at the field null by this time.

As these early small scale islands are followed in time, they grow in amplitude and ultimately coalesce until saturation and equilibration reveal a series of quiescent compact tori. As seen in Fig. k, the final state is a sequence of current rings with L_ =: 15 cm and cross sectional radius of = 2.5 cm at R = 5 cm. A significant feature is the existence of an internal separatrix that could conceivably enhance MHD stability over the more simply connected magnetic geometry that had previously been supposed. -151-

APEX 9/17/60 DETAILED IMPLOSION RAO .... i "Y T! j \ \" nr i I i ; ))r ! I I I I ; | { / A i i 2.6Zr- \\ | i i ' i i i i I 'j

0 .00)— i i I [T i i | \ I ! I I -2 .62 I i W \ \v i I i I ! I I j | I 0 /C.7I 6 i. -CO] 0.21 0.50 : R 10 T= : .50E-0J MICSEC

Figure Ij

AFEX 9/17/80 DETAILED IMPLOSION RAO 5.Z5

10 ; T= 7.50E-01 MICSEC

Figure 4 -152-

ONE-DIMENSIONAL TRANSPORT MODELING OF

FIELD REVERSED EXPERIMENT

S. Hamasaki and N. A. Krall

JAYCOR, 1401 Camino Del Mar, Del Mar, California, 92014

Abstract

A one-dimensional (1-D) modeling of plasma diffusion in field reversed

configurations, e.g., the LASL FRX experiment, is described, which incorporates

some interesting 2-D features including i) leakage of particles through open

field lines, ii) density and temperature equalization at the two different

radial points which are connected by the magnetic flux lines in the closed

fieid region, and iii) axial contraction or expansion. Specific numerical

calculations of anomalous diffusion in FRX are carried out including these

2-D effects. The numerical predictions of plasma lifetime due to the ano-

malous diffusion roughly agrees with the existing experimental data, if it

is assumed that the experimental plasma rotation develops because of the

plasma leakage.

1. 1-D Model ing

In FRX-type reversed field configurations, the open field lines at the outer region support the inner closed reversed field configuration. The

plasma contained in the closed field region diffuses out into the open field

region, mainly due to various microinstability-induced anomalous transports.

The plasma in this region leaks out along the open field lines, which enables the plasma edge to sustain a finite sharp gradient which, in turn, feeds the microinstabilities near the open field region, enhancing radial plasma transport. Meanwhile, the plasma contained in closed field region may expand or contract axially depending on whether magnetic diffusion or plasma leakage dominates. We have developed a one-dimensional fluid code (GlM/HYBRID) which -153- features elaborate anomalous transport coefficients. The code solves MHD like equations

, DT. 3

DT 3 £ ± n — = -nT 7 • V - ^ n (T - T.)/T . + 7 Q + H c ut e c e l ei e e

- r (1 v Dt " ' C l

E1 = u • J

1 Dn _ / T n \ --=-,. v + (2D)n

J x B

-7n(T. + Te)+ ^—- = 0 where 7 = r j/or. T . is the classical electron-ion equilibration time. Q- and Q are heat flows which contain the classical and anomalous (microinstability induced) contrihutions. H and H^ are heat source terms which again include anomalous as well as classical heating. Similarly, the resistivity, n, also contains two contributions. E' is the electric field in the moving coordinates. The terms (2D)T., (2D)T and (2D)n indicate the ion, electron temperature and density correction due to various 2-D effects, i.e., i) leakage of plasma density and temperature through open field lines, ii) density and temperature equalization at the two different radial points which are connected by the magnetic flux lines in the closed field region, and iii) axial contraction or expansion of the plasma in the closed field region. The numerical scheme to solve the above set of equations, which involves successive iterations to ensure the pressure balance at each time step is -154-

essentially the same as the method utilized by Roberts, et al.

2. Numerical Calculations Using the G1M/HYBRID code, we have simulated plasma diffusion in FRX. The 2 2 2

rigid-rotor type initial profiles are chosen; n(r) = n sech [K(r /R -1)],

B. (r) = B tanh [K(r2/R2-D] , T (r) = T and 1 (r) = T., where we fixed R =

4.5 cm, K = 1.08 and T = 100 eV. The radial pressure balance requires

B /8T = n (T. + T ). Choosing various values of T. and B , we can vary

R/p. (p- = Larnior radius), which turns out to be an important parameter to

describe anomalous diffusion.

Figures 1-a and 1-b show the profiles of B (r), n(r) at various times in

the simulation with setting T. = 400 eV and B = 14 kG. The plasma slowly

leaks out while the magnetic field diffuses. The time histories of the total

particle inventory in different simulation runs choosing various plasma para-

meters of R/-.J are shown in Figure 2. It can be clearly seen that the ano-

malous scaling of the plasma containment time is R/p., so that the plasma

lifetime becomes shorter as the ion temperature increases with fixed B ,

which is contrary to the classical picture. However, the dominant time scale

in the experiment is the time scale for buildup of plasma rotation and this

time scale roughly agrees with the anomalous diffusion time. Thus if anoma-

lous diffusion and subsequent plasma leakage is causing the rotation, the

experimental and theoretical plasma lifetimes are in agreement. Acknowledgement

The authors wish to express their appreciation for helpful discussions with

the FRX Group at LASL. Various suggestions made by Dr. R. K. Linford have

especially been a great help to this paper. The early part of this research was supported by U.S. Department of Energy Contract No. DE-AC-03-79ET53057 when one author (SH) was at SAI, and the rest was supported by LASL Contract

No. W-7405-EN6-36. -155-

I .6

I .2 -

o 0.8 -

0.4 -

-2 0.0

Figure 1. Profiles at various times for a) magnetic field and b) density, where B = 14 kG, T = 400 eV and T = 100 eV 0 1 c

I .0 i 1 1 1 1 1 0.8 R//>, = 44 0.6 - ^\ 0.4 \ 0.2 ^5.5 1 1 ! 0 0 20 40 60 80 100 120 140 TIME ( fis)

Figure 2. Time history of the total plasma inventory.

Case R/.:.= 5.5, 11, 22 and 44 correspond to

B = 3.5, 7.0, 14.0 and 23.0 (kG) with

T. =400 eV and T = 100 eV l e

References

1. R. K. Linford, et al., Los Alamos Report LA-UR-78-1783 (1978).

2. S. Hamasaki, SAI Report SAI-023-79-843LJ (1979).

3. K. V. Roberts, et al., Culham Report CLM-P442 (1975). -156-

Transport, Stability and Reactor Aspects of a Hill's Vortex Field-Reversed Geometry* T. Kammash University of Michigan Ann Arbor, Mich. 48109 1. Introduction In spite of some restrictive physical assumptions the Hill's Vortex (!) model has been used repeatedly in recent years tof2. describe plasma equilibria in field-reversed mirror systems and other compact tori(3). it has been noted for example( , that a plasma ring with an elongation of 3 to 1 can be adequately described by a spherical Hill's Vortex. In this paper we will apply this model to a steady state field-reversed geometry (with a poloidal magnetic field only) and explore with it the stability and transport of the plasma in such a system as well as its reactor aspects.

2. Classical Transport and Confinement Times When a current-carrying ion ring is placed in an externally applied magnetic field Bo > the equilibrium poloidal flux function, ^ , of the field-reversed system can be obtained from the Grad-Shafranov equation, namely

= JLA0 A ~^Q (1) where -%if/ then the so-called Hill's Vortex solution to Eq. (1) emerges" It assumes the form i# ~ /v L ° ~ ~ s J (i if the following relation is invoked (3) In the above relations "V# is the value of the flux function at the "0" point or magnetic axis whose distance from the origin is given by ft. ~= f\e , and Z = o , and oc is the ellipticity parameter. We observe that according to the Hill's Vortex model the plasma pressure is maximum at the "0" point and zero at the separatrix ( y/- o ), while the current-density is linear in the radial distance t\ . Since & -yc^l^'X 6 then it can readily be shown that, for a spherical vortex i.e.,"

*Work supported by DOE -157-

° ~ LL 8 (4)

where Ko = 1.2 ^o is the radius of the spherical vortex, ~&. is the value of the magnetic field at the separatrix ( r> = /?„ , ? - o ) , and o Using the steady state MHD equations Auerbach has shown that the diffusion coefficient across a flux surface of a Hill's Vortex can be written as ^.2 0^4-"^) <6) where \£ is the volume of the separatri^x, t. is the plasma resistivity, C is the speed of light, i3g is the magnetic field introduced earlier, and Y\ ( \f ) is the plasma density at the surface 'V . Noting that the particle flux is P C\ — J) df/d\fsw e can write for the flux at the separatrix the expression

&• • where the geometrix factor in front represents that portion of the particle flux that avoids the so-called " X -pointP" of the separatrix^'. It we denote by // the total number of the particles in the system then the particle confinement time can be written as n .-i

If we further assume that the plasma temperature is nearly constant so that the resistivity is also a constant, then Eq. (8) reduces to (6) ^ (9) v/here we recall that ^o is the radius of the spherical Vortex. Moreover, we note that Eq. (9) also gives the energy confinement time, ~Cg , since the temperature is assumed to be constant. -158-

3. Stability Considerations and Beta Value Perhaps the most serious instability which this confinement configuration may be subject to is the "interchange" mode arising from the interaction of the plasma pressure gradient and the unfavorable field curvature. Newcomb^'' has shown that in the absence of a toroidal magnetic field all interchange modes are unstable in a plasma with such an equilibrium. More- over, it has been pointed out by Barnes and Seyler(^) that these modes can be stabilized if a certain amount of pressure is allowed to exist at the separatrix. Here, we choose to demonstrate that the interchange modes can be stabilized by the addition of a small toroidal field. In some physical situations it may be easier or perhaps more convenient to generate a small toroidal field for stability purposes. Following Newcomb and writing the equilibrium solution near the "0" point as h } _ x1 -

— U ft /t . *>*-%. (12) where r0 is the pressure at the "0" point. As in reference (7), we subject the above equilibrium to the following perturbations which are the Fourier components of the displacement vector :

V - {I I (13) with" m" being the azimuthal mode number. From the varia- tional principle £ ^(T- W)ait = o , where f and W are the kinetic and potential energies respectively, we obtain the equations of motion -159-

Sn 4 - JL [ Suu ^ - ^i II which reduce to those of Ref. (7) in the absence of the toroidal field. The first two of the above equations give the axial and radial modes for which the eigenfrequency is given by while the third gives the toroidal mode which is a purely propagating mode at a frequency

It is clear from Eq. (15) that the most unstable mode i.e., Vj-o can be stabilized by the addition of a small toroidal field especially for large toroidal mode number i.e., w>>|. It is also evident from the above relations that a prolate Hill's Vortex is more stable to the interchange modes than a spherical or an oblate one.

Assuming stability can be achieved by the above approach we proceed to calculate the beta value (ratio of plasma pressure to the applied magnetic field pressure) which we need for reactor assessment. Since -f _ y>oT/- f/ V/V« where •%' is the plasma pressure at the "0" point, this quantity can be directly obtained from Eqs. (3) and (4) i.e.,

where we have assumed that Pi = 1.5 o0 . The maximum beta occurs at the "0" point and from (17) it has the value

/SnrJj^+^) (18) which for a spherical vortex i.e. , ol •= I , becomes Po a. 2. & • Since the pressure is linear in -j^ then <^^>~1.4. 4. Reactor Aspects We consider first a steady-state system. Using the confine ment times and beta values obtained above we have examined the potential of such a confinement configuration as a reactor by solving an appropriate set of global particle and energy balance equations for a D-T plasma with fast and thermal alpha particles taken into account. These equations are available elsewhere(10) and, in the interest of brevity, will not be re- produced here. Such an assessment is best carried out by obtaining the Q (ratio of fusion power to injected power) value of the system. We find that large values of Q( >> I ) are obtain- able especially if a significant fraction of the fast (fusion) alphas escape before thermalization. Similar results are obtained for a pulsed system with a characteristic time equal to or larger than the ion confinement time. In all instances, a -160-

near constant dependence of Q on beta is observed for /b^> i~0 . References 1. W. B. Thompson, An Introduction to Plasma Physics, Perga- mon Press, Oxford (1962) 2. G. A. Carlson, W. C. Condit, R. S. DeVoto, J. H. Fink, J.D. Hanson, W. S. Neef, and A. C. Smith, Jr.,UCRL-52467 (1978) 3. D. C. Barnes and C. E. Seyler, Proc. U.S.-Japan, Symp. on Compact Toruses and Energetic Particle Injection, Prince- ton, p. 110 (1979) 4. D. E. Driemeyer, Univ. of Illinois, Fusion Studies Labor- atory Report COO-2218-150/EPRI-AFR-120 (1980) 5. S. P. Auerbach, Bull. Am. Phys. Soc. 2_4, 954 (1979) 6. T. Kammash and T. Mi zoguchi, Bull. Am. Phys. Soc. 2_5, 978 (1980) 7. W. A. Newcomb, UCRL-83867 (1980) 8. D. C. Barnes and C. E. Seyler, LASL Rept. LA-UR 79-13 (1978) 9. K. Nguyen and T. Kammash, Bull. Am. Phys. Soc. 2_5, 978 (1980) 10. D. L. Galbraith and T. Kammash, Fusion Technology Pergamon Press, Vol. 1, p. 141 (1978) -161-

Two-Dimensional MHD Simulation of Field-Peversed Plasma Formation R.D. Milroy and J.U. Brackbill Mathematical Sciences Northwest, Inc. A reversed field theta-pinch has been modelled with a two dimensional (r-z) magnetohydrodynamic computer code. Resistive diffusion of the magnetic field is calculated using either classical resistivity, or a form of anomalous resistivity to be described later. Separate electron and ion temperatures are calculated and the effects of both parallel and perpendicular thermal conductivity is included. The code employs a specially rezoned Lagrangian mesh which concentrates in regions of large current to resolve sharp gradien*s in the magnetic field. Plasma in contact with the radial wall is held at 1 ev ana can flow freely parallel to the wall. Where the vacuum-plasma interface is not in contact with the wall, thermal insulation is assumed. Plasna at the open end boundary, 1/2-m from the theta-pinch end, is held at l eV, but can expand freely out of the calculation region.

Anomalous resistivity is accounted for through the specification of a plasma resistivity r. = r, + n > where n. the classical resistivity given by Spitzer, and n is an assumed anomalous resistivity. Chodura has found that the assumption of an empirically derived anomalous resistivity satisfactorily describes the Garching theta pinch. The same basic algorithm was later used by Sgro and Nielson2 to describe the ZT - 1 experiment. The anomalous resistivity is specifi?J by

van = cc ^pi[1 ! where V^ is the electron drift velocity, and Vs is the sound speed. Chodura assumed Cc = 0.5 and f = 5, while Sgro and Nielson found that C = 1 and f = 3 led to better agreement for their calculations. For the present calculations this form of anomalous resistivity can be used during the dynamic theta-pinch implosion; however during the more quiescent phase that follows, the plasma resistivity is not expected to be this large. Davidson and Krall have analytically calculated an anomalous resistivity based on the lower hybrid drift instability.

C : ^an= LH ' 7.2 x 10- | (vE/v.) (2) where B is the magnetic field in Gauss, n is the plasma density in cm"3, and V. is the ion thermal velocity. C is an adjustable constant of order one, that we use to vary the magnitude of the anomalous resistivity. This is a more appropriate form of anomalous resistivity to assume in the post implosion phase than Eq. (1).

As magnetic energy is dissipated through resistive diffusion it is added to the plasma thermal energy through the nj2 term. For classical resistivity, essentially all of this energy goes to the electrons. However, Davidson and Gladd have shown that some microinstabilities cause comparable -162-

elect'-on. and Ion heating. For all of the calculations considered in this paper, we assume that 30 percent of the magnetic energy dissipated through anomalous resistivity is deposited directly in the ions.

We have studied the formation of a reversed field theta-pinch plasna in the parameter range of current experiments (FRX-B5 at LASL and Cl-TRX-1 at MSNW.. In particular, we have studied the mechanism by which the magnetic field lines close at the theta-pinch end, and have examined the sensitivity of the confuted results on the assumed anomalous resistivity and the initial bias flux.

f n '" T_he o lowinq calculations, we have assumed a coil length of 1 m, a c-;'1 radius of 12.5 cr, and a wall radius of 10 cm. Initial reverse bias fields in the range of -0.11 T to -0.33 T have been considered. The flux, on the theta-pinch coil is raised sinusoidally to give a peak forward field of n.rJ T a* ?.' i.sec, and the" ""alls to a value of 0.7 T by 3.7 usec after which the ^lux is held constant.

figure 1 shews magnetic field lines and marker particles at several ! a 11 -\es for a typical simulation. The initial plasma density was 7 x 10 "cm" 1 anomalous resistivity given by Fq. (1) with C = 1, and f = 3 was n use c •.,r'ng the rad 'al implosion. After t = 1.25 ,jsec an anomalous res "! ' v •• t V given by Eq. (?) with C. „ = 2.5 was used. Figure 1 il 1 LJ St rat^i the mechanise by which magnetic field lines reconnect near the the t- Z\ _ ri"c- end. Reconnect ion is seen to take place very quickly beyond the end s Q * the theta pinch. However, the field strength is very low in this re^or. S'. that only a very small magnetic force is exerted on the plasma. However the sharp curvature of the inner field lines which are closed near the t.hpta pinch ends causes magnetic forces sufficient to reverse the direction of axial flow for the confined plasma. Endloss (moving outward) combined with axial contraction (moving inward) of the plasma carried on inner field lines causes a low pressure region near the theta pinch end. This allows the magnetic field lines in this region to move in radially producing steep magnetic field gradients and rapid particle-field diffusion. In this way, the axial contraction of field lines which close near the theta cinch end accelerates the reconnection of the remaining open field lines.

By 3 sec the sepa-atr;x has reached the axis of symmetry and the whole field-reversed configuration begins to contract axially. Meanv;hile the plasma beyond tne theta pinc'r, ends slowly drifts out of the region of the simulation. By 10 usec tht reversed field confiouration has contracted to a quasi -equilibrium state. At this time, the electron and ion temperatures are very uniform (within 10 percent) ovtf the entire field reversed configuration. The average electron and ion temperatures are 85 eV and 165 eV respectively.

Several calculations have been made assuming a variety of anomalous resistivity forms. Some effects of changing the anomalous resistivity are summarized in Table I. The calculation labeled "A" corresponds to the calculation discussed in the previous paragraphs. The calculation "B", is identical except the anomalous "Chodura" resistivity given by (1) is not -163-

turned off at 2 usec. Only 2 percent of the initial reverse bias flux was left by 3.5 '^sec and the entire reversed field configuration decayed away by 6.5 usec. The calculations "C", "D" and "E" assume an initial reverse bias field of -0.22 T, and lower hybrid resistivity with C^H = 2.5, 1, and 0.4, respectively [see (2)]. The results dre quite Insensitive to the assumes magnitude of the anomalous resistivity. The time of complete reconnection only varies from 2.7 psec to 2.9 usec as the magnitude of anomalous resistivity is increased by a factor of 6. This suggests that the reconnectior is "dynar-ic", i.e.., the rate is duterni ned by dynamic effects rather '. h3.-. the magnitude of the resistivity. Calculations "J", "F", and :'H" are similar excert that the initial reverse bias field is lower, -P,U T. For this cjse., the time of complete reconnect "ion increases from 3.15 L,ser "r 4.2 -sec as * he- magnitude o'; the anomalous resistivity is decreased by j ractor of 6. Thus, the reconnect ion fime is more sensitive to the magnitude of resistivity for sr..:ill reverse bias field, suggesting that "resistive" reconnection i, beginning to appear.

The sensitivity of the results to the assumed initial reverse bias flux has been investigated and is summarized i>; Table i entries "D", "F", and "G". These calculations indicate that an increased reverse bias flux leads to a significantly increased plasma temperature, and quicker reconnection of magnetic field lines. The increased temperature cai. be attributed partly to ar". incrpased amount of joule heating as the reverse bias flux decays and partly to the strong axial motion imparted to the plasma. The rate at wh: ch magnetic field lines reconnect increases with increased reverse bias flux. Tin's is because the reconnection mechanism becomes more dynamic a? inner reconnected field lines quickly contract axially evacuating the region near the the I. a pinch end. Then the outer, open fic-ld lines move radially inward intc this region and steepen the magnetic field gradients.

TaLle 1 - Summary of simulation-esults at 3.5 usec for some different in-tiai conditions ani assorted anomalous resistivity, n = initial pla;ma fill density in 10:' cm"3, c and f are used in Eq. (1). t.H = time that switch is made from Eq. (1) to Eq. (2) to evaluate •"• , C, u is used in Eq. (2), f. - fraction of initial reverse bias flux trapoed, f = traction of fill plasrv.a trapped inside

ix, and tr - time .hat complete reconnection was observed.

n C T T B c f LH f e i Vc ir 0 bias c : « P

A 0.7 -0.2? T 1 3 1.25 ' 2.5 0.136 0.69 96 216 3.C

E C.7 -C.22 T 1 3 - 0.O2 0.26 107 240 2.4 C j -0.22 T - - - 2.5 0.23 0.9 134 24B 2.7 D 0.7 -0.22 T - - 1 0.27 0.9 133 248 2.8 E 0.7 -C.22 T - - - 0.4 0.29 0.9 13b 250 2.9

F 0.7 -0.11 T - - 1 0.23 0.85 86 150 3.4 G 0.7 -0.33 T - - 1 1 0.23 0.93 186 311 2.5 H 0.7 -0.11 T - - 0.4 0.22 0.S6 86 160 4.2 0.7 -0.11 T - - - |2.5 0.20 0.71 87 160 3.15 -164-

6-pinch coil

10 t=0 Radius t = 3.0 -sec (cm) ^ 0 -50 Axial Position +50 (cm)

t = 0.5 ysec t = 4.0 usec

t =1.5 psec t = 6.5 usec

t = 2.5 ysec t = 10 psec

Figure 1. Magnetic field lines and marker particles for a typical simulation. This figure corresponds to entry "A" of Table I. REFERENCES 1. R. Chodura, Nuc. Fusion JJ5, 65 (1975). 2. A.G. Sgro and C.W. Nielson, Phys. Fluids 19., 126 (1976). 3. R.C. Davidson and N.A. Krall, Nucl. Fusion ]7_, 1313 (1977). 4. R.C. Davidson and N.T. Gladd, Phys. Fluids 18., 1327 (1975). 5. W.T. Armstrong, et al., "Field-Reversed Experiments on Compact Toruses," submitted to Phys. Fluids. -165-

FIELD LINE RECONNECTION AT THE END OF A FIELD REVERSED THETA PINCH Z.A. Pietrzyk University of Washinoton Seattle, Washington 98195 We have carried out simulations of field reversed theta pinches using ?- dimensional Lagrandian MHD code.1.2 It has been observed that initial der.s.cies lower than lO^cm3, the field lines do not reconnect for several microseconds, if only classical transport coefficients are used. This is contradictory to experimental observations, which suggests that microinstabilities are present in 9-pinch plasmas at low density. Several models for anomalous resistivities have been included in our present calculation to model these microinstabilities. The saturated lower hybrid instability is described, using the resistivity:3

n1H 2 uPe • \H We have found that this resistivity, which is very important in later phases of reversed field configuration, does not contribute to field lines reconnection significantly. A second model resistivity examined is that of Chodura3, which is related to the turbulence in the shock wave structure. It changes the calculated plasma parameters, making them closer to experimental values. This resistivity, however, does not influence the reconnection process, if used by itself in the calculation. The use of the classical formula for resistivity, when enhanced by a multi- plicative factor at low temperatures, as suggested by the experiment11, does influence the reconnection and has a very 'important influence on the start of the reconnection process. In our calculation the following formula was used: n = Snclass where s is a constant, which was varied from 2 to 10. Such a resistivity, of unknown nature, does explain the early (1-2 us) re- connectirn of field lines seen in experiments, and gives proper plasma temperature. However, it does not explain either axial separatrix contraction or axial plasma compression properly. These exercises with different resistivities suggest that the plasma ex- periences additional microinstabilities near the pinch end. We have examined various anisotropic instabilities as a possible mechanism for rapid field line reconnection. One expects that the two instabilities related to temperature anisotropy, the mirror and fire hose instabilities5, should be important near the end of a pinch. To model these instabilities we took their growth rates5, multiplied them by the classical collision frequency and assumed that this product was proportional to the anomalous collision frequency. The squares of the I| and _L velocities were used to compute Tj_ and T||. The anomalous resistivity due to the anisotropic effects used in our calculation is:

„ „ Sll(l I 1, K \> Tl/2 kT " kT " 3 ' ' -166-

2 with the threshold of (-V^ - 1 )6 - 1 > 0 IT where V is parallel and u perpendicular velocity. This model properly seems to describe the process of field lines reconnection. Field lines near the pinch end look very similar to those experimentally measured6 (see Fig. 1). The temperature calculation also corresponds closely to measured values (see Fig. 2), if in addition the Chodura or increased classical resistivity is used in the calculation. To obtain and maintain a compact toroidal plasma configuration one has to control the microinstabilities. They are needed for proper field line reconnection at the end of a pinch. They are not needed, however, in the middle of the pinch, or after the compact toroid configuration is established. As the result of these calculations, qualitative differences between different experiments can be explained. A very fast pinch, the HBQM at the University of Washington, has large 60% flux trapped in the plasma7, as compared to 25% for FRXB.8 The separatrix radius is also much different for these two experiments (see Figs. 3a and 3b for comparison). In the fast pinch, plasma is resting against the wall, during reversal phase, only for a short time. This reduces the flux lost in the reversal phase. Shock turbulence instability also lasted for shorter time, further contributing to a lower flux loss. On the other hand, for a very slow pinch (Kurchatov9 experiment) , the radial shock wave strength is smaller, thus reducing flux loss due to this effect. To prevent flux less in the reversal phase, a barrier field was used in the Kurchatov experiments. Active mirrors at the end of a pinch promote reconnection by inducing microinstabilities and add more heating near the end, which in turn produces an axial pressure gradient, which adds to the axial magnetic force to produce a strong axial shock wave, as observed in the Kurchatov experiment. This shock heating method is more effective than conventional axial contraction. It should be recognized that these calculational results are preliminary, and that more calculations Are necessary to arrive at an accurate quantitative description of the formation of FRC plasmas in pinches. This work was supported by the Magnetic Fusion Division of the U.S. Depart- ment of Energy. References: 1. A.H. Makomaski and Z.A. Pietrzyk, Phys. Fluids 23, 379 (1980). 2. Z.A. Pietrzyk, J. Appl. Phys., to be published. 3. R.C. Davidson and N.A. Krol1, Nucl. Fusion ]7_, 1313 (1977). 4. W.T. Armstrong, J. Cochran, R.J. Commisso, J. Lipson and M. Tuszewski, "Theta Pinch Ionization for FRC Formation," to be published. 5. F.F. Cap, Handbook on Plasma Instabilities, Vol. 2, pp. 504-507, Academic Press, New York (1978). 6. J.H. Irby, J.F. Drake and H.R. Griem, Phys. Rev. Lett. 1_2, 228 (1979). 7. S.O. Knox, H. Meuth, F.L. Ribe and E. Sevillano (this meeting). 8. W.T. Armstrong, R.K. Linford, J. Lipson, D.A. Platts and E.G. Sherwood, "Field Reversed Experiment on Compact Toroidal Plasmas," Bull. Am. Phys. Soc. 24, 1082, (1979). 9. A.G.Es^Kov, K. Ku. Kurtmollaev, A.P. Kreshchuck, Ya.N. Lankin, A.I. Molyutin, A.I. Markin, Ya. S. Martyushov, B.K. Minov, M.M. Orlov, A.P. Proshletsov, V.N. Semyenov and Yu. B. Sosunov, 7th IAEA Conf. on Plasma Physics and CTR, Innsbruck, (1978). -167-

FIELDLIKES

TIME = 8.701 MICROSEC

Fig. 1. Early Stage of Field Lines Configuration for HBQ (University of Washington)

1 '•';• / M ii ,\

•. • r! ' ; .* r h^i' o p.i t'on '• ••

...\

Fig. 2. Comparison of Measured FRXB (Los Alamos) and Calculated Temperature in Plasma. -168-

FIELDLINES

TIME = 2.20Q niCROSEC

FIELDLIN:G

Tire - 2.200 nicRosa:

Fig. 3. Comparison of Field Lines Configuration for A) FRXB and B) HBQ -169- FLUX LOSS DURING THE FORMATION OF FRC

A. G. Sgro Los Alamos Scientific Laboratory Los Alamos, New Mexico

One important feature of FRC formation is the loss of the reversed flux. In this note we present the results of simulations that are Intended to study this loss.

A hybrid simulation1'2 in which the ions are kinetic while the electrons are an inertialess, charge neutralizing fluid, is necessary when studying implosions in experiments such as FRX-B because of the presence of ion beams reflected off the magnetic piston. The presence of such kinetic features makes a fluid description questionable for the ions. Such a feature is illustrated in Fig. 1,

in which a vr - r cut of the ion phase space during an FRX-B implosion is presented.

The electron momentum equation implies an ohms law for the evolution of the magnetic fields. However, an electrical conductivity must be specified. By an appropriate specification, the influence of electron inertia, change separation, microturbulence, etc. may be approximately included. We use a formula 1'- which has successfully described many theta pinch and reversed field pinch experiments.

We first simulate an FRX-B implosion in which the initial density

15 3 nQ = 1.1 x 10 cm~ , TeQ = Tio = 4ev, BQ = 1.2kG and initial trapped flux = TTR^B = 4.5 x 10^G cm2. The trapped reversed flux at any subsequent time, 4i(t), is the Jlux enclosed between the geometric axis and the cylinder on which

the field null occurs. Figure 2 is a plot of $(t)/$Q. The x with error bars are experimental measurements, the lower curve is a simulation with anomalous resistivity, and the upper curve is a simulation with classical resistivity.

The curve calculated with classical resistivity is above the experimental points, 2.5 us, while the curve calculated with anomalous resistivity is consistent with the data. The implications are thai classical resistivity alone -170-

is insufficient to explain the data. The flux is lost anomalously fast. The anomalous resistivity algorithm used is adequate tt, explain the observed loss, although a slightly better fit might be obtained by slightly lowering the strength of the anomalous contribution.

After the imploding sheath has past a point near the wall, the plasma initially there is swept up and the density there is very low. The numerical algorithm does not represent this low density region well, so that additional assumptions must be made to model it. We have considered two possible models for the region. One is that vacuum fields are present, and another is that the tenuous plasma can carry a nonnegliglble current. In Fig. 2, field profiles at 1.25us are plotted. The x are the experimental -fata, the dotted line is the simulation with the vacuum field assumption, and the solid line is the simulation with the tenuous, current carrying, plasma assumption. The latter assumption is clearly a better representation of the data.

When the resistivity is anomalous, the question of the division of the joule heating among the various species must be addressed. Microinstabilities, such as the lower hybrid drift,3 which generate the anomalous resistivity, can also cause the anomalous ion heating. If the regime near the wall if assumed to contain a low density, current carrying plasma and if all the joule heating is deposited in the electrons, ~ 50 ev at 2.5 ps, compared with the measured value of 500 ev at 5ys. Evidently, the computed icn heating rate is too low. If, however, 50% of the joule heating is given directly to the ions, ~ 200 ev at 2.5ys. This heating rate is sufficient to match the data.

Perhaps the most important question to address when studying the issue of reversed flux loss during the implosion is how to minimize it. We have varied several of the initial and boundary conditions and found the loss to be insensitive to most parameters. The one parameter that did influence it was the initial bias field. In particular, (t)/<|>0 increased as I ^ias^wall' increased. When | Bblas/Bwalll = 0.04, $(t)/0 = 0.15 after the first bounce while if IBbias/Bwall( = 0.21, Q and (t)/o increase. However, if l bias/ wall ° large, the CT will not form properly. Thus, there must be an optimum value for it and we are presently searching for this value. -171-

In conclusion, the field profiles and the evolution of the trapped flux as a function of time imply that the resistivity is anomalous and that a tenuous plasma between the CT and the wall carries a nonnegligible fraction of the current. About half of the anomalous joule heating is deposited directly to the

B ions. The fractional flux retention increases with increasing l bias/BwallI, up to some optimum value.

References

1. A. G. Sgro and C. W. Nielson, Phys. Fluids _1_9, 126 (1976)

2. A. G. Sgro, Phys. Fluids 2_3, 1055 (1980)

3. R. C. Davidson and N. T. Gladd, Phys. Fluids 18, 1327 (1975)

10-

10. _r

Fiaure 1 Figure 2

2 5

Figure 3 -172-

Scaling Laws for FRC Formation and Prediction of FRX-C Parameters

R. E. Siemon and R. R. Bartsch

A semi-emperical method has been developed to extrapolate the experimental results from FRX-B, a field-reversed theta pinch which generates an FRC (Field-Reversed Configuration—a compact toroid with no toroidal field), to the larger size FRX-C. Even though there are many uncertainties about details of the dynamic processes by which an FRC is formed, the scaling exercise has proven useful in identifying limitations in the original FRX-C design and the design has been modified to have a lower voltage and larger capacitance. The goal of FRX-C remains unchanged: to test the confinement scaling of an FRC in a larger device over a wider range of temperatures. Of particular interest is the testing of possible MHD stability limits as the ratio of plasma size to gyro radius increases.

1. K'on-adiabatic implosion heating. The high-voltage capacitor bank is approximated as a capacitor (C) initially charged to voltage V connected to the coil of radius r and length L through a source inductance The total temperature (Tg + Tj ) resulting from the implosion process is

2 c/(jjnl-

where XJ_J is a dimensionless emperical constant, Lcc = ^ is the coil inductance, rtt is the discharge tube radius, c/m • is a characteristic length that can be used to estimate the sheath thickness at the initial pressure p , and e is the electron charge. This scaling is derived by ignoring the bias field and assuming that the energy is initially imparted to the ions by a moving sheath described by the snow plow equation pQv = B^/2yo where pQ is the initial mass density, v is the sheath velocity, and B is the externally applied rising magnetic field. The scaling can be justified in the presence of a small bias field or if the bias is adjusted to be proportional to V ' as discussed below with regard to the Green-Newton limit. From FRX-B data, X^ is found to have the value 0.06.

2. Initial equilibrium. Following the implosion, it is assumed that the current continues to rise until a compact toroid of length L equal to the coil length is established in an equilibrium state. The balance between plasma pressure and field line tension in an elongated equilibrium requires

= 1 - I (2)

2 Here <8> = g /B(r) 2TTT dr is the volume averaged 6 = p(r)/(B /2uQ) defined with respect to the external magnetic field, B, and x i the ratio of the separatrix radius to rw. A useful analytic form for /3(r) is a sharp-boundary profile

3(r) 0 6(r) =0 (3)

The plasma current flows at the separatrix and at the major radius R = r //27 s -17 3-

3. Trapped flux. The preionized plasma contains magnetic flux as a result of the initial bias field, Bj. Preionization, field reversal, implosion, and other formation processes result in a fairly small fraction of the initial flux being trapped in the equilibrium. The fraction Xp, about 0.13 according to the FRX-B experiment, is taken as an emperical constraint on the trapped, poloidal equilibrium flux, <$>:

It was pointed out by Green and Newton that there exists an upper bound, B on the bias field because of the finite time for field reversal,

B E /2 max = 9 ^oP0) where the electric field at the inner wall of the discharge tube is

Eg = (V/2irrw)(rt/rw) Lc/(LC+Lg). The value of bias field is taken for scaling purposes to be B^ = XjjBmax where Xg is a multiplying factor less than unity. A typical value of Xg used in the FRX-B experiment is 0.75, but Xg = 1.0 is assumed for FRX-C. Given the magnetic profile implied by Eq. (3), the poloidal flux is constrained by these arguments to be

3 2 2 i/2 r r AX h F ,1/4 172 r t B Q (5)

A. Trapped particles. Of the initial particles in the discharge tube, a fraction X (0.35 in FRX-B) is observed to be confined in the equilibrium formed. Thus the plasma volume of a compact toroid of length 9, and radius x r determines the density

n x % — X^n L (r,_/r ) . (6)

5. Adiabatic compression. For a given device and fill pressure p , the values of constants X[.[, Xp, Xg, and X with Eqs. (l)-(6) specify the value of B = B^ that is required to form a compact toroid of length I = L. To avoid end effects and to insure that the compact toroid is compressed to a length less than the coil length, it is desirable to use enough capacitance that the field increases to a final value L larger than B, . An adiabatic compression model is used to describe the compression phase despite the fact that the quarter-period rise time in typical experimental arrangements is roughly the same as the thermal transit time of particles along the length of the system. Assuming adiabatic compression pV^ = const., and the sharp boundar> profile of Eq. (3), one finds

,8/5 1 o .3/5 ZS = (7) where zg = £/L, and xgL is the value of xg when the compact toroid is first formed with length L. The magnitude of current in the coil at the end of the compression phase is approximately given by

I = Lbias (8) where is an inefficiency factor (~0.75) associated primarily with crowbar -174-

switches. Eq. (8) is equivalent to assuming that the plasma has no effect on the coil inductance, an approximation that can be justified to within a few percent despite the fact that the final B field between the separatrix and metal coil does depend on the shape of the compact toroid: B - fy -i (9)

6. Maximum values of x and p . According to both experiment and theory it is desirable to make xg as large as possible in order to minimize the pressure gradients that tend to result from the equilibrium relationship of Eq. (2).

The maximum xg is xgT (no compression) and is determined by the above equations to be 1/2 1/2 1/2 xsL =5__ ((1 + 16/c.) - 1) 2 ,rt,2 X2 xjj a = — X X * V f H It is important co note that bank parameters such as V, C, etc. do not affect the result. The small values of Xp so far observed are not fully understood, and it may be possible to increase x , by increasing Xp in future experiments. The upper limit on x corresponding to I = L is not directly observable because the non-uniform magnetic fields at the ends of the coil perturb the assumed equilibrium when I ~ L.

There also exists an upper limit on the initial fill pressure, po(max), corresponding to X. = L. For a given capacitor bank it is impossible to operate above p (max) because the plasma energy cannot be confined with the available magnetic field. For po(nax) in mTorr using MKS units,

19 2 Po(max) = (l/6.6xlO ) (4^/mp (Fc|/X^2) (CV/L) (1/(1 H-B^)) ,

7. FRX-C parameters. Plotted in Fig. 1 are predictions of FRX-C parameters based on the above equations, assuming the 3/4-bank parameters listed in Table I. The mechanical configuration has two feed slots giving an effective voltage of 110 kV from the individual 2.8-yf, 55-kV capacitors. The x value ranges .from 0.4 to 0.5 which is the same as that obtained on FRX-B. ' The scaling parameter S = R/p.[, where R is the major radius and pj is the ion gyro radius in the external field, is larger than in FRX-B as desired for the scaling

TABLE I. FRX-C Device Parameters

Coil Diameter 0.45 m Coil Length 2.0 m Equivalent Circuit Full bank 3/4-bank C 112 uf 84 pf

Ls 16.8 nH 20.0 nH V 110 kV 110 kV lev2 678 kJ 508 kJ -175-

studies. Larger values of x^ and S would he achieved if Che trapped flux exceeds 0,13 as assumed in these calculations.

References

1. W. T. Armstrong et al., ''Field-Reversed Experiments (FRX) on Compact Toroids," Phys. Fluids, to he published. 2. H. Dreicer et al., "Proposal fur FRX-' and Mil] t ipl e-Ce 11 Conpact TIT,is Experiments," Los Alamos Scii'n: ir :. Laboratory proposal LA-8D45-P, 1979. 3. T. S. Green and A. A. Newt ,••, Phys. Fluids 9, 1386 (l%fS). 4. A. G. Es'kov et ,tl., "Principles ••:' 10- Plasma lle.it ini: and Confinement i ,; > Compact Toroid Gonf ip.ur.it ion" (Proi1. Plasma Phys. and Contr. Nucl. Fusion 5~ Research 1978, Vol. II) IAEA, Innsbruck (1978), p. 187. 5. W. T. Arnstront; et al., "Compart Toroids Experiments and Theory," (Proc. Plasma Phys. and Contr. Nucl. Fusion Research 198D) 1AF.A, Brussels, to be published.

0.5

0.4

0.3

i I... 20 40

Po (mtorr)

Fig. 1. Predictions of FRX-C Parameters AXIAL SHOCK HEATING OF FI ELD-RLVERSED PLASMAS Loren C. Steinhauer and Alan L. Hoffman Mathematical Sciences Nu-thwest, Inc. A phenomenon observed in most revered field-theta pinch experiments is the axial contraction of the plasma soon after the radial implosion. Experiments in the Soviet Union^ indicate that, this contraction may cause significant shock heating if the theta pinch apparatus includes 1) a multipole field pulsed on during field reversal to Drever.t momentary plasma-wall contact, 2) cusp coils at the pinch coil ends to delay the magnetic reconnection and subsequent axial contraction, and 3) fast mirror (trigger) coils at the theta pinch ends to trigger reconnection at the peak axial field.

EXPERIMENTAL APPARATUS Figure 1 shows the configuration used to perform the above operations ano the coil firing sequence. The multipole bars set up a pulsed "barrier field", preventing plasma- V' \\)\ well contact during field reversal. r> I I i The large reversed magnetic flux ! I II - losses observed in experiments have \|f-(tf. f-• f-f ^-(^7\T\/ y been attributed to the plasma-wall •} TRIGGK ' V"""^" PLVSK^ interaction which ordinarily occurs COI. - W CHAMB-f \ , during reversed-field theta pinch -(, X. VMUMIPO.t Bi-:i P^SK- formation.2 However, experiments - •' ^-SLOTTED PIHChCC:. using pulsed multipole fields show flPINi SEQUENCE t^at such flux losses can be eliminated, a","'Oh ing large reversed magnetic *'^.> to be trapped.3 The force :r-;,ing axial contraction is the TRIGGER FIEL!\ c-rvstjre of reconnected magnetic B fielc lines; hence, the larger the trapped flux, the greater is this TIKE force, and thus the stronger is the axial shock heating. BIAS FIELUTIB

CUSP FIELD. B Figure 1. Coil structures ana firing sequence. The cusp coils (Fig. 1) delay the reconnection of reversed magnetic field lines, which is initiated later by the trigger coils. This delay enables the axial contraction (which follows reconnection) to occur at higher magnetic and plasma density, leading to stronger axial shock heating. Mathematical Sciences Northwest, Inc. has begun a DOE-supported experiment called CT-TRX1, which is designed to study the use of octopole barrier fields to maximize the trapped reverse-bias flux, cusp coils to delay reconnection, and fast trigger coils to maximize the effectivness of axial shock heating. Two important quantities are the critical bias field B* defined as the bias field at wh'ch strong wall contact and flux loss can be expected^ -177-

B*(T) = 0.19 E/' (kV/cm)[A.p (mTorr)] A

and the average temperature, T,, which would be produced in a standard theta pinch by shock heating arid adiabatic compression to a peak field B .

10 /l3 Tj = 1.61 A/ E/' (kV/cm)B^'(T)/pQ(mTorr) keV

CT-TRX1 encompasses a 1-m long theta pinch with a 2C cm plasma tube bore. The electrical parameters in its initial configuration are: bias field, B = 0 - 0.3 tesla; compression field, B =1.0 tesla - Bb; and electric field, E-. = 0.3 kV/cm. At a deuterium fill pressure of 15 mTorr the important parameters are B* = 0.24 T, and T. = 115 PV (at B = 0.75 T). The experimental emphasis will be to achieve trapped bias fluxes equal to or above B^-r'', and plasma temperatures somewhat greater than T..

THEORETICAL MODELLING OF PLASMA HEATING 4 A model, presented in detail elsewhere, has been constructed to describe the plasma formation and heating. Given such input parameters as the fill pressure, p , the initial bias field, B, , the electric field, E , and the maximum magnetic field, B , the plasma temperature, radius, length, and trapped magnetic flux can be predicted. Summary of Model. The model assumes a particular family of density profiles, a generalized version of a truncated rigid rotor profile, n = rw 1 - <• seclr [K(2r/r - 1)]- , r < r^; n - n sech'[tanh"' ( itanhK) + .'•{r/r - 1)], r > r ; where n is the ;' nsity at the magnetic axis (B=0), r. is the separatrix radius, and i, K, > are three shape parameters: spatially uniform temperature is assumed. The model determines the evolution of these parameters as well as the density, n , and temperature, T, in the course of plasma formation and heating. The process is described as a series oT steps from one discreet state to the next. Successive states (eight in all) are linked by the appropriate conservation laws. Reversed magnetic flux dissipation and the consequent resistive heating are accounted for by imposing a microstabi1ity condition, v./v. < const, where v, is the diamagnetic drift speed, v- is the ion thermal speed, and the constant (of order unity) is specified. Just enough reversed flux is dissipated at each step to satisfy this condition. The results depend primarily on the ratio of effectively trapped bias field, BL, to the critical bias field B*. The most important result will be the ratio of final temperature T to the ideal temperature T.. -178-

Prediction for a Current Experiment. Figure 2 indicates the temperature predicted for parameters achievable in the CT-TRX1 experiment. The dashed lines represent the temperature before reconnection is triggered, accounting for the radial implosion and resistive heating. At high fill pressure, the temperature falls with increasing bias field due to a weaker radial implosion. For higher Bj-,, however, it begins to increase; this is the result of the strong resistive heating that arises C.« 0.6 with large trapped flux. The resistive BIAS FIEL:, E.'B. heating is relatively weak at high fill pressure but strong at low fill pressure. The solid lines represent the temperature Figure 2. Temperatures predicted after the axial contraction. Significant for the CT-TRX1 experiment: shock heating is predicted especially for wall radius, r = 10 cm; high fill pressure. At low pressure, the coil radius, r = 11.5 cm; strong resistive heating during the radial E = 0.3 kV/cm; B = 3B*; implosion dissipated enough of the re- Q c versed magnetic flux that the subsequent deuterium. shock was much weaker. Overall, the final temperature ratio T/TT is not very sensitive to fill pressure, although the source of the heating, whether by axial shocks or resistivity. varies considerably. The upper bounds shown on the higher fill pressure curves indicate the limiting condition beyond which the plasma strikes the wall during the axial contraction. These bounds are raised for higher B /B*. Comparison with Results from FRX-B. Since the FRX-B experiment.6 did not employ delayed reconnection, the present model does not strictly apply. However, because of low bias field the axial shocks were weak and the model should give reasonable predictions. For the sake of comparison, the assumed fill pressure is reduced from 17 mTorr to p = 14.5 mTorr to o 7 account only for the preionized portion (85 percent) of the fill. Moreover, the preionized plasma region occupied only a fraction of the cross-section containing about 50 percent of the reversed flux-, thus, we take B, = 0.115 T instead of the initial 0.23 T bias field. Then the model predicts the results shown in Table I. Table I - Comparison with FRX-B Results

Temperature Fraction of Initial Fraction of Initial Exampls (eV) Flux Trapped Particles Trapped

FRX-B 155 0. 13 0.45 - 0.5 model with Vvi! £ 1/2 143 0. 41 0.82 model with Vvi !* 1/4 176 0. 19 0.78 -179-

The predicted temperature for VQ/V.,- = 1/2 is low but is in reasonable agreement considering that the model ignored additional heating from kinetic effects, and the magnetic field oscillation due to the crowbar. The flux and particle trapping predictions are somewhat high. The discrepancy in trapped flux is reduced by taking a mere stringent microstability condition, vD/v. < 1/4, and agreement with experiment could be achieved by further reducing the constant to about 0.18. However, the predicted particle trapping remains too high by a factor of 1.5 to 1.7. These discrepancies might be resolved by reducing the initial bias field and/or pressure to account for processes that are not understood.

REFERENCES 1. A.G Es'kov, et al., in Plasma Physics and Controlled Thermonuclear Research, Vol. II, (IAEA, Vienna, 1978), p. 187. 2. T.S. Green and A.A. Newton, Phys. Fluids 9_, 1386 (1966). 3. A.G. Es'kov, et al., in Proceedings of the 9th European Conference on Controlled Fusion and Plasma Physics, Culham, 1979. 4. L.C. Steinhauer, "Axial Shock Heating of Reversed-Field Theta Pinches," submitted to Phys. Fluids. 5. J.P. Freidberg, R.L. Morse, and F.L. Ribe, in Techrology of Controlled Thermonuclear Fusion Experiments and the Engineering Aspects of Fusion Reactors, USAEC Report, CONF-721111 (1972), p. 812. 6. W.T. Armstrong, et al, "Field-Reversed Experiments on Compact Toroids," submitted to Phys. Fluids. 7. W.T. Armstrong, et al., "Theta Pinch Ionization for FRC Formation," submitted to Appl. Phys. Letts. -180-

FRC Studies on FRX-B

W. T. Armstrong, J. C. Cochrane, J. Lipson, R. K. Linford, K. F. McKenna, A. G. Sgro, E. G. Sherwood, R. E. Siemon, and M. Tuszewski

Introduction Recent experimental studies of Field-Reversed Configurations (FRC) on the KRX--B* device have included 1) characterization of FRC formation with regard to loss ~,c bias flux, 2) examination of FRC equilibria through separatrix profiles, 3) formation of FRC's with different end-mirror configurations, and A) extension of FRC parameter range. Studies on loss of bias flux during the pre-ionization (PI) phase of FRC formation are presented in another paper dedicated solely to PI considerations. Loss of bias flux during the reversal phase of FRC formation is reviewed in the first section of Lhis paper. Use of barrier fields during the reversal phase to enhance trapping of bias flux is included in the third section of this pap-.r. I- addition to barrier field studies, results from different mirror configurations are also discussed in the third section. A critical diagnostic for interpretation of the results from the different machine modifications is the excluded-flux probe array. Analysis of excluded-flux measurements to obtain the FRC separatrix profile is described in the second section. Finally, preliminary results of FRX-B operation in an extended range of plasma parameters is briefly discussed in the fourth section.

Flux Annihilation during the Reversal Phase R The trapped flux of a FRC is defined by ^^ = J Bz2Tirdr where B is the magnetic field in the axial midplane, and R is the radius at which B =0. Both experiment and theory suggest that the longest lived FRC's are produced when ^ is maximized. The largest possible bias flux that can be trapped is itr^B^^as = 0> where rt is the inner radius of the discharge tube (10 cm) and Bbias *s t*le magnitude of the bias field (~2.4 kG). In past work, the final configuration has $-;/(£> o ~ 0.14. Due to the nature of the PI technique used, ' <|>J/(J> ~ 0.5 at the initiation of the main bank. During thj rise of the main field, flux may be annihilated by resistive processes. We endeavored to examine this last loss mechanism in detail. The apparatus consists of a theta-pinch coil which has a 22-cra id and is 100 cm in length. The field sequence begins with the slow rise of the negative bias field. A ringing theta-pinch PI is initiated near the bias field maxinum such that the net field passes through zero ("zero-crossing PI"^). Finally, the main bank is fired whereupon the field attains its maximum positive value ("15 kG) with a risetime of 2.6 us. Data were taken with a radial array of B probes located in the axial midplane of the device. The radial probe array also contained coils which were sensitive to toroidal field. However, no toroidal field was observed during the implosion, even when the radial array was moved off the axial midplane.

Figure 1 displays Q vs time for data averaged over five shots taken at a D2 filling pressure of 17 mtorr. The error bars correspond to shot-to-shot variations. Also plotted in Fig. 1 are the results of a computer simulation of the implosion using a 1-D hybrid code (Vlasov ions, fluid electrons). »5 The model employed either Choduia or classical resistivity. To account for the observation of current flowing between the sheath and the wall, a hot tenuous plasma created from particles being continuously emitted by the wall was included. In addition, anomalous joule heating of the ions (~50% of the ohmic -181-

heating) was enployed. The initial conditions were an electron temperature of = 4 eV, 100% ionizatio:., and <^^/^0 0.5. Though the detailed results for the simulation with Chodura resistivity diverge sonewhat from the data, the end value of

the final observed ^j/^o suggests that the resistivity must be anomalous during this time. There are preliminary indications fror; the computer simulations that <•> - /<^ evaluated at the peak of the external field is bigger for larger bias fields. Hence, a substantial improvement in <^ may be possible for operations at larger bias fields.

Separatrix Profile from Analysis of Excluded Flux Measurements At each of nine stations along the FRX-B coil, the excluded-flux radius r = r 1 1Tr B is obtained i" r/ ~'''p^ P P^ with flux loop (6 ) and Bz probe (B ) data. These diagnostics are located at a radius r . The excluded flux radius profile, r^(z), is therefore available over the entire length of the theta-pinch 'coil. In general, r/\A(z) differs from the separatrix profile

rg(z). The latter profile is important since it gives the dimensions of the FRC equilibrium. A numerical procedure has been developed to determine a separatrix profile consistent with the r^ data. Plrsma pressure on open field lines is neglected and constant flux is assumed at the inner coil surface at a given time. We give in Fig. 2 numerical analysis results for a typical shot from past experimental data. The separatrix profile (solid line) and the corresponding excluded-flux radius profile (dotted line) are shown in Fig. 2, along with the five experimental values of r^ (circles) available iron stations on half the coil. In Fig. 2. r (z) corresponds to a probable smooth profile. Several sepnratrir.es are consistent with the experimental data since, as was shown numerically, the excluded flux array can only resolve features of the separatrix that have a scale lengtti greater than about a coi] radius.

FRC Behavior for Different Barrier and Mirror Field Geometries Studies of FRC formation have been performed utilizing different barrier and mirror field geometries with a variety of main field amplitudes. Table I briefly summarizes the different configurations and the FRC behavior observed. Referring to Tabli J. , cases (1) and (2) had a solid, main coil (100 cm long with a 25-cm id) which passively produced 107 mirror fields at each end of the coil. Cases (3), (4), and (5) employed an aziraut.ia 1 ly-slotted main coil (100 cm long with a 22-cm id) which had no passive mirrors. The coil wa;; slotted to allow application of an octopole barrier field from an azirnuLhal array of lone,] tudinal conductors. In addition to 'he slotted main coil and barrier field coils, case (3) also utilized independent mirror coils at both ends of the main coil. These mirror coils produced a peak, on-axis field of ~11 kO during the rise of the main field. Furthermore, a "non-zero-crossing,"" theta-pinch PI was used in cases (2), (3), and (5) where large bias fields (--3.0 kG) complimented the high B operation. A conventional "zero-crossing"'- theta-pinch PI was u^ei" with the modest bias field levels (•~?.3 kG) in cases (1) and (4). A D-, fill ai 17 mtorr was used in all cases. Removal of passive mirrors with the installation of barrier coils resulted in FRC's which suffered a rapid decay (10 to 15 us lifetimes). End-on framing pictures indicated the configuration was often grossly turbulent and di«.J erratically. The n=2 rotational instability was absent. The short plasma lifetime appears to be associated with the production of a "poor" FRC equilibria. Excluded-flux measurements indirate that the FRC equilibrium had a length comparable to the coil length, was irregularly positioned along the axis, and displayed a tendency to move axially. Application of the independent -182-

nirror fields during FRC formation (case(3)) resulted in equilibria more consistently centered in the coil. However, the equilibrium length was still comparable to the coil length and the lifetime was still 10 to 15 us. Though the slotting of the cw'n coil and presence of barrier field coils may contribute to the rapid decay of the FRC, the lack of continuous mirror fields appears to be the most likely cause of the short FRC lifetine. The absence of mirror fields allows longer FRC equilibria. Equilibria with lengths conparable to the lengtn of the coil may have several deleterious effects on the FRC lifetime: I) field line divergence at the coil ends may induce net axial drifts of the FRC, 2) contact of the FRC with the cold end region may "result in rapid cooling leading to annihilation of the FRC, or 3) complete field li>'e reconnection inside the coil region may never occur. The octopole barrier fields were produced by energizing a set of 16 longitudinal conductors. Connecting the conductors in two different arrangements maximized either the B or the BQ component of the octopole field geometry. Both nodes were studied and gave a siiiiilar result: a modest increase in the separatrix radius was observed at early times (~10% increase ~5 us after the main field initiation was typical). The effect of increased exciuded-flux radius on late-time FRC behavior was obscured by the short plasma lifetime described above. Use of the barrier field (case (4)] is illustrated in Fig. 3.

Extended FRC Parameter Range Past scaling studies of FRC parameters employed a bias field of -2.3 kG and maximum main field of ~9 kG (case (1), Table i). These field values and associated conventional PI technique limited the formation of FRC's to a D2 fill pressure range of 5 to 23 mtorr. Recent studies with a bias field of -3.0 kG, maximum main field of -13 kG and "non-zero-crossing" PI (case (2), Table i) have extended FRC formation over a range of fill pressure from 5 to 50 mtorr. Similar FRC behavior is observed as tn the previous operation: 1) separatrix radius is ~5.5 cm, 2) stable period scales approximately linearly with fill pressure up to ~50 us, and 3) n=2 rotational instability terminates the FRC. The increase in main field generally increased T. such that only a modest increase in R/p^ is observed (p^ is the ion gyro-radius in the external field). The modest increase in stable period over the previous study (~15/o) appears consistent with a scaling of stable period with R/p-. Additional data were taken at a fill pressure of 17 mtorr over a range of main bank voltages (external magnetic fields). It was found that at similar values of R/p. the stable period was virtually independent of T- over a range 200 eV < Tt < 1180 eV.

References 1. W. T. Armstrong, R. K. Linfcrd, J. Lipson, D. A. Platts, and E. G. Sherwood, to be published in Phys. Fluids. 2. R. J. Coranisso, W. T. Armstrong, J. C. Cochrane, C. A. Ekdahl, J. Lipson, R. K. Linford, E. G. Sherwood, R. E. Siemon, and M. Tuszewski, "The Initial Ionization Stage of FRC Formation," Proceedings of this conference. 3. W, T. Armstrong, J. C. Cochrane, R. J. Commisso, J. Lipson, and M. Tuszewski, submitted to App. Phys. Lett., 1980. A. A. G. Sgro and C. W. Nielson, Phys. Fluids j_9, 126 (1976). 5. A. C. Sgro, Phys. Fluids .23, 1055 (1980). 6. R. Chodura, Nucl. Fusion _15_, 55 (1975). 7. M. Tuszewski, submitted to Phys. Fluids, 1980. -18.3-

AVERAGE OF 5 SHOTS 2557, 2558,2559, 2563, 2564

Fig. 1. 4>j^/4>0 is compared between Fig. 2. Measurements of the excluded- experiment and simulation during the flux radius vs z and the associated rise of the main field. The dashed separatrix profile are compared. and solid lines represent simulations with classical and Chodurs resistivity, respectively.

TABLE I FRX-B OPERATION IN 1980

1 9 kG PASSIVE LONG LIVED FRC. n - 2 IDATA FnOM 19791 INSTABILITY LONG LIVED FRC. n-2 INSTABILITY 3. 15 kG PULSED NOT ENtRGIZED RAPID DECAY OF FRC 4. 9 kG NONE B. OR B. «, INITIALLY INCREASED BUT F'RC DECAYED RAPIDLY

tt INITIALLY INCREASED BUT FRC DECAYED RAPIDLY

Fig. 3. End-on framing pictures are compared for barrier field operation with maximized Br (on left) and maximized Bg (on right). The oscillogram traces are the barrier field (upper) and the main Bz field (1 owe r) • -184-

Th e Initial Ionization Stage of FRC Formation

R. J. Commisso, W. T. Armstrong, J. C. Cochrane, C. A. Ekdahl, J. Lipson, R. K. Linford, E. G. Sherwood, R. E. Siemon, and M. Tuszewski

Introduction A FJeld-Reversed Configuration (FRC) is a prolate compact torus that is confine^ by poloidal fields only. Theta-pinch formation of an FRC employs an initial bias field, Bj, whose direction is opposite to that of the main theta-pinch field. Some fraction of the flux associated with this bias field eventually constitutes the closed-field-line flux of the FRC. Experimental and theoretical evidence suggest that the longest-lived FRC's are obtained when the closed flux is maximized. Because the initial ionization is done in the presence of the bias field, the actual bias flux available at the time of application of the main theta-pinch field depends strongly on the initial ionization, or "preionization," technique used. In this paper we report on experiments characterizing the previously used theta-pinch preionization technique that employed a net field (bias plus preionization) null, or "zero-crossingv" of the axial component of the magnetic field to break down the gas. We also discuss results of experiments designed to develop preionization techniques in which the gas breakdown is not accomplished by a zero-crossing.

Characterization of Zero-Crossing Preionization A study of the effect of the zero-crossing preionization scheme on the Initial bias flux was carried out on FRX-B. The FRX-B device is a 100-cm long theta pinch of 22-cm i.d. Internal magnetic field probes sensitive to the axial component of the magnetic field and compensated diama^^etic loops for measuring the plasma diamagnetism were the principal diagnostics used. The 400-kHz theta-pinch preionization was done with the zero-crossing mode (see Ref. 2 for details). The probe data indicate that at the field zero-crossing a gas breakdown occurs and a plasma ring or annulus forms at the wall and subsequently implodej. The annulus then undergoes an oscillation around an equilibrium radius determined by the amount oftrapped flux within the annulus. 0.75 A numerical code which includes 1 1 1 1 1 1 1 damping by ion-neutral collisions 0 f \> / and finite (classical) resistivity , 0.50 - \ / \ o -1 o was also developed to model the I \ / observed properties of the plasma 0 3. 0.25 - during preionization. Figure 1 is /

a comparison of the code I 1 1 1 1 1 prediction and experimental data I 8 for the ratio of flux inside the plasma annulus, <)>j, to the vacuum Fig. 1. flux associated with the bias Comparison of measured normalized trapped field, , as a function of time. o bias flux with code predictions. Circles As time proceeds, the ratio are measured values and solid line is code approaches 0.5. Thus, only half Drediction.

Permanent Address: Mission Research Corporation, 1400 San Mateo SE, Albuquerque, MM 87108 -185-

of the total possible bias flux is available at the time of the main discharge. We have assumed that the flux outside the annulus is lost during the main discharge. This assumption has been substantiated by further probe measurements. >J Evidently, the conductivity of the plasma annulus inhibits the bias field from diffusing through it.

Alternative Ionization Schemes The results outlined in the previous section suggest that a preionization scheme which does not use a net field null to break down the gas could be beneficial in optimizing the trapped bias flux. The possibility of photoionization has been evaluated with the result that very large energy-storage capabilties are required in order to provide a sufficient percentage of ionization. Because eventual translation and trapping of the FRC necessitates a puff-fill from one end, possibilities for electrode (z-pinch) type ionization are limited. Among various electrodeless discharges, we have chosen to study experimentally a quadrupole system. The induced ionizing current in this system is parallel to the bias field eliminating the need for a zero-crossing. A low amplitude (compared to the bias) theta-pinch was also studied as a part of a multistage scheme. These studies were carried out on the FRX-A device which has the same physical dimensions as FRX-B. a. Unassisted quadrupole The quadrupole experiments show a high degree of ionization (60-80%) for values of the quadrupole magn«..:i- field (BAD) comparable to the bias magnetic field, Bij however, density build-up is inhibited fof r < B The density build-up is correlated with gross compressional effects. Moreover, the plasma maintains the quadrupole symmetry. These results suggest that by increasing B^p, ionization could be achieved at higher values of Bj. However, the ionization would probably be UNASSISTED OUADRUPOLE MULTI-STAGE 1.5 kG BIAS-NO rf 3kG BIAS-36MH2 rf accompanied by excessive heating LIGHT INTENSITY and an unwanted quadrupole structure embedded in the plasma-. We therefore began experiments with QUADRUPOLE CURRENT a multistage system. 12OkA/d,v b. Multistage system The multistage system employs a sequence of discharges to provide END VIEWING adequate seed ionization for the FRAMING CAMERA low amplitude theta pinch to couple (1 MHz RATE) to the gas without the undesirable J2Z zero crossing. The discharges consist of a 36-MHz, ~10-kW RF generator; a 1-kG, 160-kHz quadrupole; and a ~2-kG, 520-kHz theta pinch. The sequencing begins with RF break down of the gas so that visible light is detected. After ~10 ps of light emission the quadrupole is fired and clamped after one half cycle. The theta pinch Is fired at the quadrupole current peak. This sequence was found to result in the most TIME AFTER l( reproducible discharges and the Fig. 2. highest level of ionization. In Comparison of unassisted quadrupole and Fig. 2 the unassisted quadrupole multistage systems. -186-

and multistage system are compared. As can be seen from the end-view framing camera photographs in the figure, the quadrupole structure in the plasma becomes circumfused when the theta pinch is added. Emission from the CIII 2296M line indicates an electron temperature < 10 eV. We found it possible to achieve sufficient preionization without the quadrupole phase of the multistage system. This result is illustrated in Fig. 3. The value of the bias field must be reduced some ~700-800 G to obtain the same ionization as is achieved when the quadrupole is included in the multistage scheme. Using this scheme without the quadrupole, ionization has been obtained for Bj < 3 kG at a fill pressure of 10 mT and for By < 3.5 kG at a 20 mT fill. The CIII emission is associated with the density buildup and again indicates an electron temperature of < 10 eV. The RF phase of this scheme is critical in obtaining ionization. This behavior is observed whether or not the quadrupole is used and is illustrated in Fig. A. When the RF does not couple to the neutral gas, as determined by visible light emission, no density build-up is observed. The optimum RF antenna configuration consisted of ~3 turns of l/4"-o.d. insulated wire positioned ~6 cm from each end of the main theta-pinch coil. In order to achieve reproducible RF coupling, an additional burst of ~70 MHz RF is sequenced to fire ~5 us before the 36 MHz unit. The 70 MHz RF is generated by a ferrite core driven to saturation by a capacitor discharge (0.25]jf at 25kV) and is coupled to the gas by a separate antenna consisting of a single turn of l/4"-o.d. insulated wire placed just inside the 36 MHz antennas.

LIGHT EMISSION (ARBITRARY) FRAMING )QOO CAMERA RF ~1 FRAME//xs) MONITOR OOQO 5/j.s/div- -L8 PINCH FIRED FRAMING MONITOR

Cm,2296A (ARBITRARY) 9 PINCH FIRED 5/i.s^div— -*- 9 PINCH FIRED

Fig. 3. Multistage operation without quadrupole. -187-

POOR COUPLING f LIGHT 1 EMISSION (ARBITRARY)

1 (ARBITRARY) T MONITOR (ARBITRARY

PINCH 9 PINCH FIRED FIRED Fig. 4. Effect of RF coupling on density build-up.

Summary Only half of the total initial bias flux is available at the time of the main discharge when the zero-crossing theta-pinch preionization scheme is used. Preionization schemes which avoid a zero-crossing to ionize the gas have been successful in producing an initial plasma in the presence of the bias field. Measurements to determine the amount of bias flux trapped in plasmas so produced are in progress.

Re ferences

1. W. T. Armstrong, R. K. Linford, J. Lipson, and E. G. Sherwood, accepted by Phys. Fluids, 1980. 2. W. T. Armstrong, J. C. Cochrane, R. J. Commisso, J. Lipson, and M. Tuszewski, submitted to App. Phys. Lett., 1980. 3. W. T. .Armstrong, J. C. Cochrane, J. Lipson, R. K. Linford, A. G. Sgro, E. G. Sherwood, R. E. Siemon, and M. Tuszewski, "FRC Studies on FRX-B," Proceedings of this conference. A. C. A. Ekdahl and R. J. Commisso, Los Alamos Scientific Laboratory report LA-UR-80-2233. 5. C. A. Ekdahl, R. J. Commisso, R. K. Linford, and E. G. Sherwood, Los Alamos Scientific Laboratory report LA-UR-80-134. 6. R. J. Commisso, C. A. Ekdahl, R. K. Linford, and E. G. Sherwood, Proceedings of 1980 IEEE Conference on Plasma Science, Madison, WI, Paper 5A10 (1980). 7. R. J. Commisso, C. A. Ekdahl, E. G. Sherwood, and R. E. Siemon, Bull. Am. Phys. Soc. 2_5, 1021 (1980). -188-

TRAPPING OF REVERSED BIAS FLUX IN A FAST-RISING THETA PINCH S.O. Knox, H. Meuth, F.L. Ribe and E. Sevillano University of Washington Seattle, Washington 98195 I. Introduction Experimental results on trapping of reversed-bias magnetic flux * i a fast theta-pinch, the high beta Q. machine (HBQM)1, are reoorted. Both pre -•nization and implosion phases have been studied with measurements of the inters 1 magnetic field, combined with excluded flux data and holographic interfer'.^etry Reverse-field configurations with large values (>0.8) of separatrix-io-coil ratio are produced. II. Experiment The HBQM is a 3-m long, 20-cm tube ID, 5-kG theta pinch. A 12O-kV implosion-heating capacitor bank is switched by low inductance multichannel cylindrical rail gaps, giving a magnetic field risetime of 350 ns in the presence of plasma (400 nsec in vacuum). Identical low-resistance rail gaps switched at current maximum passively crowbar the circuit. The subsequent L/R decay of the field has a 1/e time of approximately 10 ps. Deuterium filling pressures of p0= 11 mT and 16 mT were selected for this work. The preionization plasma is created by a ringing theta-pinch discharge (15 us prior to main discharge) with a frequency of 200 kHz and a magnetic field amplitude of 1.2 kG. The negative bias bank provides 1 kG of magnetic field with a quarter period of 24 us. Preionization and implosion phases of plasma with an embedded negative field have been studied by means of a compensated diamagnetic flux loop (), 4^-Lernal (B-j) and external (Bo) magnetic probes measuring Bz, and holographic interferometry. The internal probe is located 70 cm from one end of the coil inside a 8-mm OD quartz sheath that can be positioned at any radius. Ablation of the probe surface by plasma is negligible as confirmed by holographic data. The diamagnetic loop and its compensating external field probe are at the same axial position as B-j. A ruby laser holographic interferometer (1 fringe= 1.lxlO15cnr3) has been used for the study of the implosion dynamics and subsequent equilibrium of the FRC plasma. III. Results The effective plasma radius r determined from excluded flux data is defined in the usual way: rv

(BQ- B(r)) 27irdr , where Bp is the field at the tube outer radius, r^., B(r) is the field at rand t is the flux inside r-j-. This also gives an estimate of the separatrix radius .2 A. Preionization. A study of the preionization phase has been performed and gives results similar tothose of Ref. 3. Measurements of the internal magnetic field as a function of radius and time show the presence of a current sheath that oscillates radially in response to the applied magnetic field. In 9-pinch preionization, gas breakdown occurs at the zero crossing of the magnetic field. We have found that current sheath formation occurs at the first zero crossing for filling pressures po above 13 mT and at the -189-

second zero crossing for pressures below this. Measurements of B-j(r)for p0 > 13 mT show a region near r=Q which contains positive field. A radial scan of B-j for po = 16 mT determines that this region extends to approximately r = 3 cm. Measurements with the internal field probe positioned at r=2 cm show that this positive field disappears after the implosion phase and a negative field is then present. At p0= 11 mT there is no positive ^ield at any radius. Figure l(a) shows Bi as a function of r at three different times during the preionization and illustrate the dynamics of this phase. The addition of a pre-preionizer circuit should provide sufficient ionization to prevent the penetration of positive field to the center.3 B. Main Discharge. Oscillographic traces starting shortly before the implosion phase (sudden rise of Bo) are given in Fig. 2: Bo is the external magnetic field, $t is the total flux from a one-turn loop encircling the discharge tube, Atfi-j- is the excluded flux measured by the compensated diamagnetic loop, and B-j is the internal magnetic field. Holograms of the implosion dynamics at po=16 mT show three phases: the implosion phase takes place during the first 1.5 vs, and is followed by an oscillatory phase with a mean plasma radius of 7 cm; after -<4 us a stable configuration with a minimum density in the center and ? maximum at r=9 cm persists for several microseconds. The experiment is effectively terminated by the decay of the main field. Figure 3 shows r-x as a function of time after the implosion; this can be compared with the holographic data of Fig. 4. The probe-measured r^ and the maximum of the plasma density for times greater than 4 ys are both ^9 cm. The vertical arrows in Fig. 3 mark the times at which the holographic interferometer data of Fig. 4 were taken. From a simple consideration of trapped-flux loss during zero-crossing of the confining magnetic field1*'5 the fast n'setime of the magnetic field is expected to trap most of the initial bias flux. Radial scans of Bj (cf. Fig. 1 (b)) arc- used to calculate the internal trapped flux ^ directly. The ratio 4>i/s>o> wnere ^o 1S the negative flux inside the tube at the initiation of the main discharge, is shown in Fig. 3. Referring to the figure, 752 of the initial flux is retained at 0.5 us. During the ensuing 1.5ys this quantity decreases to 35%, probably because of the Bo decrease due to imperfect crowbar circuit clamping. For the remainder of the experiment, Ci/:i>o appears to be correlated with the decay of Bo. IV. Conclusions Fast field reversal techniques are effective in the trapping of reversed bias flux. Reverse-field configurations with large separatrix radius are obtained which are stable for the lifetime of the present experiment. The planned addition of a power crowbar will extend the useful duration of the magnetic field to about 30 us and long-term stability can then be studied. V. Acknowledgements We are grateful to l(. Bych and G. Harper for assistance in obtaining these results. This work was performed under the auspices of the Office of Fusion Energy, U.S. Department of Energy. VI. References 1. S.O. Knox, H. Meuth, F.L. Ribe, and E. Sevillano, IEEE Intl. Conf. on Plasma Science, Madison, Wisconsin. (1980) Paper 1A5. -190-

2. W.T. Armstrong, R.K. LinforJ, J. Lipson, D.A. Platts, and E.G. Sherwood, Field Reversed Experiments (FRX) on Compact Toroids. Accepted for publication in Phys. of Fluids. 3. W.T. Armstrong, J. Cochran, R.J. Commisso, J. Lipson, and QD M. Tuszewski, Theta Pinch Io- nization for FRC Formation. Submitted to Appl. Phys. Lett. 4. T.S. Green and A.A. Newton, Phys. Fluids 9, 1386 (1966). 5. S.O. Knox, H. Meuth, F.L. Ribe and E. Sevillano. To be pub- (a) 1ished.

-e- oh

CD'O\~

(b)

Fig. 1. Radial profiles of the inter- Fig. 2. Oscillographic traces (Pq=H nal magnetic field during the (a) pre- mT) of (a) external magnetic field, inization and (b) main discharge for (b) excluded flu*, (c) total flux, p0 = 11 mT. (d) internal magnetic field at r=5cm -19.1-

Figg. 3. Effective plasma radius r, from excluded flux measurements, and percentage of trapped flux (-p-j/4>0) from internal magnetic field probe data for filling pressures of (a) 16 mT and (b) 11 mT.

(cm)

Fig. 4. Radial profiles of electron density from holographic interferometry data for po=16 mT. -192-

A keV Compact Toroidal Plasma

S. Ohi, S. Okada, M. Tan j o, Y. Ito, T. Ishimura and H, Ito

Plasma Physics Laboratory, Faculty of Engineering Osaka University Yamada-Kami, Suita, O.-ak': 565, Japan Introduction

A compact toroidil plasma with field reversal configuration (FRC) is realized by the similar production method as that of a linear ti.cta pinch with a negative bias field. The ion temperature of a linear theta pinch plasma produced in apparatuses of usual scale reaches to keV although its decay time is only the order of microseconds. On the other hand, a typica1 confinement time of the FRC plasmas reported already is severs! tens microseconds but the ion temperature is somewhat lower than those of linear theta pinch plasmas.02^ It seems to be very important for us to understand differences" between FRC and linear theta pinch plasmas experimentally ana theoretically. The confinement of a KeV plasma with FRC has been studied on the machine PIACE.3)

Exper iment

A schematic picture and the dimension of the apparatus is shown in Fig.l. Diameters of the main coil and the discharge tube were extended recently from 10cm and 8cm to 15cm and 12cm respectively. The decay time of main current of 45ys was also improved to 70us. Almost all experimental results presented here were obtained on the unmodified apparatus except those in measurements of the diamagnetic flux and the reversed field inside the plasma. The production method10 :)is the theta compression of the initial plasma made by the encounter of two counter streaming plasma flows each of which is ejected from the conical theta pinch gun mounted near the end of the main compression coil. Two plasma flows are adjusted to have the same travelling velocity for the encounter to occur at the midpiane of the main coil. At an appropriate time after the encounter, a rapidly increasing magnetic field with the onposite polarity to the guiding field is imposed on the encounter plasma and an FRC plasma is produced and sustained. The ion temperature T± of Deuterium or Hydrogen is inferred from the Doppler broadening of CV line (2271A). The neutron yield was also measured as a proof for the realization of high temperature. The electron temperature Te or density ne were measured by Thomson scattering of ruby laser temporally and spatially. The detection of the reversed field on the axis was done by magnetic probes. -193-

Experimental Results According tc the signals of compensated diamagnetic loons arranged along the discharge tube, tearing instability is not observed in cur experiment.6^ The longest duration time of the diamagnet ic flux signal at the midpl? J is about 30 ^ 40)JS. Any kind of trials to detect the reversed field on the axis by inserting magnetic probes across the field lines disrupt the plasma immediately. However, an experimental proof of the establishment of FRC was given by the detection of the reversed field on the axis by magnetic probes concealed inside of a thin stainless tube with 0.3cm diareter stretched along the axis of the discharge tube. The temporal behavior of the reversed field at the midplane is given in Fie.'1. The reversed field exists about 15-;s until the diamagnetic fLux signal disappears. The intensity of the reversed field is about 70s of that of the external one. Without th.- magnetic pi obe, the diamagnetic signal continues for about 4 0IJS although the cross section of the plasma becomes to deform into an ellipse from a circle and rotates.around the axis. According to an analysis, our FRC seems to be nearly conserved even after the deformation of the plasma section as long as the diamagnetic signal continues. The ion temperature Tj reaches 1.SkeV at tne maximum after the field reconnection as shown in Fig. 3 and the neutron signal from the plasma continues up to lOvis. Spatial distributions of ne and Te at 5ps after the implosion are shown in Fig.4. Ac- cording to such measurements, radial profiles of ne and Te are nearly flat and any remarkable decrease is not found near the 15 axis. Maximum values of Te and ne are about ?>0 0eV and 6 * 10 electron/cm3 respectively.

Discussion According to streak pictures or the measurement of the reversed field on the axis, the field reconnection time is about 2\is. The initial plasma is made by th:? encounter of two counter streaming plasmas. Through our previous studies7), it is already known that the high $ parts of plasma flows nass through each other without decrease of their initial velocities. Since the initial velocity of each flow is 1 * 107cm/sec and the half length of the FRC plasma is 20cm, the observed reconnect ion time is just the passing through time of plasma flows. As this time is short, the FRC plasma seems to attain its equilibrium state very early so that the diameter of the plasma is nearly constant (2 ^ 3cm) and the axial contraction is relatively mild. Most important behavior of the FRC plasma is the n = 2 instability near the end of the diamagnetic signal. The rotation of the plasma column is clearly observed after the deformation of the section of the plasma. According to streak pictures of the plasma, temporal variations of the section (ellipse) and the rotational frequency are easily obtained as shown in Fig.5. -194-

Th e ratio of the major radius b to minor one a of the ellipse often reaches 4. The rotational frequency oo is not zero at the beginning time of the observation already. Then it decreases slowlv as shown in the same figure. According to our observation, the area of the ellipse seems to be conserved within the experimental error. These facts suggest to us that the plasma is rotating already even when the cross section of the plasma is a circle, although we could not know the onset time of the rotation. As stated before, the plar .a section deforms into an ellipse in order to keep its field reversal configuration as the plasma current decreases. However, as the plasma rotates still, tin- opposite torque against the rotation might be produced. Then, the rotational frequency w is considered to decrease eradua11v.

References 1. H. K. Li 11 ford et al. Pro. 7th Int. Ccnf. on Plasma Phys. and '.'.ov-.tv . Nucl. Fusion, Insbruck , (1978) VOL.II p.447 2. IV. I. Arastrcnn et al. Pro. 8th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion, Brussels, (1980) R-3-1

7>. .'".. Ohi et ai. Pro. Int. Symp on Phys. in Open Ended Fusion Systems, Tsukuba, (1980) p.273 4. H. Kishnnoto et al. Phys. Rev. Letters 31 (1973) p.1120

S H. Ito et al. Pro. Sth «.onf. on Plasma Phys. and Contr. Nucl. Fusion, Tokyo, (1974) VOL. El p.389

6. A. Kaleck et al. Pro. 4th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion, Novosibirsk, (1968) VOL. E p". 581 7. S. Goto et al. J. Phys. Soc. Japan. (1977) VOL.43 p.2042 n

Gun Coil Compression Coil

Brriax ?CKG (Rlsetirne nPs) ^ig.l- Schematic picture of the apparatus

B Decay t,me 70Ps after a modification. Cor I. D • t* 6 cm Tube I.D. i? 3cm -195-

Deuterium P'.asra

( 2J..0) S as F:r'i E, = ? r':<-0 I

0 0 O o

9 10 20 30 0 12 3 5 6 7 8

p Flg 3 ig.2. Diamagnetic flux(upper) and " - *°" temperature T£ inferred from reversed field on the axis of ?°Pp1^ prOflle °f CT Xl"? the plasma column with vacuum (upper) and neutron yield(lower) field(lower-,. ,j/, )\ versus time, . . versus time. (5us/d)

ne (x 10 /cm ) 8

20 « IP5) 1 r(cm) ' 3 Fig.5. Time evolutions of the ellipticity of plasma cross section Fig.U. Radial profiles of ne and Te and rotational frequency ob- at 5ps after implosion. served by streak photograph. a:minor radius, b:major radius, c:frequency -196-

Experimental studies oi leu.—level oscillations

in a '.-I oba i ly-s t ab ! e KXTRAi' Field Reversed Configuration'

J.R. Drake

Royal Institute of Technology, S-10044 Stockholm, Sweden

Experiments have been carried out on the linear EXTRAP apparatus. In

this device, the plasma is basically a straight /-pinch, but the axial plasma

current channel is along the axial null field line of an octupole vacuum magnetic field. The vacuum magnetic field is generated by currents in four rods.

T're rod current, J , is ant lparal Is; 1 to the plasma current, J . All magnetic

fields are transverse (poloidal) relative to the longitudinal axis of the discharge and the device becomes a field-reversed configuration if the plasma current is of sufficient magnitude and sufficiently localised.

The experimental device is shown in Fig 1. The parameters of the experiment are shown in Table I.

At the start of the discharge, the breakdown between the electrodes occurred along the zero field region. The plasma discharge current waveform had a 4-i,sec risetime at the start followed by a 15-.:sec, nearly-constant flat top period where quasi-steady state conditions existed. The currents in the rods producing the vacuum field were nearly constant during the plasma discharge.

Theoretical discussions of the equilibrium and stability of the Extrap system are discussed in Ref. 1-3 and a presentation of experimental results can be found in Ref. 4. In this paper we present additional experimental observations of the system. The presence of low-level oscillations in globally-stable discharges, diagnosed using magnetic probes, has been reported on in Ref.4.

During the course of a discharge, three phases were observed:

1) At the start of the discharge, the discharge column excecuted one or

two long-wave-length kink oscillations (A greater than electrode

separation), as the plasma current and density built up.

2) After this start phase, the discharge was quiet for a period,

the duration depending on the parameters of that discharge. -197-

3) This quiet period was sometimes followed by the onset of

kink-like oscillations with signal amplitudes more than an

order of magnitude higher than those sesn during the quiet

period. However, the plasma column was conatined if

.1 was not too large, and these oscillations P gave fluctuating probe signals. Furthermore, if the plasma current

magnitude was under a certain threshold, the quiet period extended

to the end of the discharge v.'ith the limitation being set by the

power supplies.

In Ref. 4, curves o[ the signal amplitudes versus J indicated P that the vacuum "lagnetir field totally suppressed the onset of these global oscillations if J /J was sufficiently large. After a threshold

at J /J -8 , the amplitude of the probe signals increased rapidly with

increasing J . These experiments were done with a filling pressure of

]3Q mTorr and J =36 kA. v In the experiments reported on now, the duration of the quiet period preceding the onset of the contained kink-like oscillations was examined.

The duration of this period was dependent on the plasma current and the filling pressure. In Fig 2, the duration of the quiet phase is shown versus

J fo- three filling pressures, 80, 130, and 200 mTorr. Again the rod-current was J =36 kA. The duration increased with decreasing J and with increasing filling pressure. The transition from the quiet condition to

the oscillating state occurred very rapidly in that the first fluctuation, indicating the onset of the low level oscillating state7had full amplitude.The parametric dependence of this transition is the subject of additional investigations . In addition to the observations presented above, local measurements of the magnetic field strength induced by the plasma current were made. Scans are made using a search coil with a diameter of 4mm housed in an 8-mm diameter glass tube. The path of the scan is shown in Fig 1. A profile for a stable-discharge case with J =4 kA and J =36 kA and a filling pressu-re of 130 mTorr is presented in Fig. 3. Shown is the meajured field strength due to the plasma discharge current and, for comparison, the -198- vacuum field due to the current in the rods, and a calculated curve representing the field from the plasma current assuming it was a line current down the center null poiqt of the vacuum flux plot. The amplitude and r dependence of the measured field indicate that the current channel was located inside the separatrix in the reversed-field region created by the plasma current. For this case, the separatrix was located at r=ll mm. The actual shape of the plasma current density profile and plasma density profile have not been measured. However it has been observed that the presence of probes inside the separatrix perturb the discharge resulting in an increased fluctuation level in the magnetic probe signals. These measurements demonstrate that a stable field reversed configuration has been generated in the relatively simple linear EXTRAP configuration described here and that the stabilizing mechanisms of such configurations can be studied.

References j_lj LEHNERT, B., Physica Scripta 10/1974)139 and J_6 (1977) 147; Roy.Inst.of Tech., Stockholm, TRITA-PFU-7Q-08(1979) and TRITA-PFU-80-04(1980).

J2j HELLSTEN, T., Roy.Inst.of Tech., Stockholm, TRITA-EPr "'4-23(1974).

[3j LEHNERT, B., Physica Scripta j_3(1979)250;Roy.Inst. of Tech., Stockholm,TRITA-PFU-79-10(1979) and TRITA-PFU-80-03(1980).

J4J DRAKE, J.R., HELLSTEN, T., LANDBERG, R., LEHNERT, B., WILNER, B., 8th International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Brussels, 1-10 July 1980, paper IAEA-CN-38/AA-3.

This work has been supported by the European Community under an association contract between Euratom and Sweden. -199-

fKh TABLE I. ;.1TRAP Parameter?

radial, distance Co rods 2.S cm.

plasma discharge radial discance Co vacuun vail 7.5 cm.

electrode separation 2 2. cm

plasma current, J 2-10 kA P rod current, J 12-3= kA magnetic probe' v

filling pressure 20-200 cTo.-r

p 1 JSDd den.s i Ly

election renpet j:ure

siamless steel vacuum vessel

mcgne'.ic probe

0.50

Fig.l. Extrap device.

(power supply limit ,15jjsecjl

10 15 20 25 30 r(mm) •Plasma Current (kA)

Fig.2. Duration of quiet period of Fig.3. Magnetic field versus radial discharge versus plasma current distance from axis, fcr various filling pressures. -200- r.:-; :':::rv V.-?I::L;\ by Toik.il ii. Junsen and Mi;iji Sheng Chu General Atomic Co., San Diego, CA.

Abstract

Tiie 'Bunpy Z-rir.ch" is a contact torus-like ran:' ivtur.it ion which nay be easy to establish .u^d us i nt;i!. n indefinitely. A si-:; ..? version i f the configuration is ax i s\".; necric . It re nfuns regioni; of. c]'-:e/! a:v.! o;;en field lines. In r. region of closed field linos tiie topol T;,-v is much like that of n r.oka-iak; such r.'»ions of closed field lines link a :.;.uon of open field lines around the axis of r vr.inet jrv. Assuming that the v-l.isna spontaneously .-.laintains a Taylor equi libr iun, *" it is possible to esta-ilish an.! maintain indef i;;i t<;l y re«;ons of closed field lines by driving an axial current through the pl.as:'.a in the region of open field lines. The ratio het-.ven the total ax;ai driven current and the total a.tinuthal current in a tokar.ak-like region of closed field lines .nay in principle be nade arbifarily suall.

! . En1: roduc t ion

It was found experimentally by Robinson and King that the axial magnetic field sDrntaneouslv reverses at the plasna surface whenever the parameter IR, v /i = k R, !."^e'~-:s ;: certai.:i value; here, I is the total axial current, R. the radios of a conduct Lne, -ube surrounding the p:as-ia, i the total axial magnetic f! ux throui;!'. the tube and \s is the vacm.n perneabi- iity. This result was explained theoretically by Taylor" as c-iused by resistive instabilities.

Consider a pinch arrangement for which the conducting tube surrounding the plasna is bunpy and has a radius Rj(Z) which, depends on the axial distance Z along the tube. Assume furthermore that provisions are made that an axial magnetic field WLCVI field lines not intersecting the tube wall can be established before tr.e pinch, discharge. For such P. pinch arrangement the axial current, I, and the axial flux, $, do not depone on 7., but since the governing parameter, k R,(Z), does, one expects that circumstances may exist under which the field is reversed in regions of Z where R^(Z) is large but not in regions of small R,(Z). Under such circumstances, regions of closed field lines exist. One can imagine that the closed field lines may be populated by hot plasna while the open field lines, which nay terminate on electrodes may be populated by cold plasma. Alternatively one can imagine the whole configuration bent into - "skinny" torus; in this case the "open" field lines arc really closed and linked by the regions of "closed" field lines.

?.. Model

In order to examine closer the idea presented in the previous section we construct the following model. let the shape of the bvmpy conducting tube surrounding the plasma be determined by its radius

Rj(Z) = RQ [l - b (1 - cos < Z) ] (1) - 2 01 -

v):::r-.: R. Is ti:e max ;::•:;•- r:dvi3, the c:;ast..nt b i ^ the ''bumpyncss" of tlie tube and -.•.•'ncr-s •: is the "a:-:ial wave rur.bcr". As mentioned above, the constant k is proportional to the rario bvtverjn the axial current, I, and the axial flex, J, through

V. = Iun/* (2)

The Taylor equilibrium, specified by the vector potential, A, is determined by

vilh the bouidarv condition at the wall that the nzin;;thal component of the vector potential is given by A,-. - 1>/2"TP. and that the other components_ vanish; this ilcternires the gauge of the vector potential such that 7«A = 0.

If the bouniary condition had beed that A_vanishes ,ir the conducting wall, En .' '>) W:JI_:M only have had the solution A - 0, except for special values of k, the cieenvalues. for which solutions A 4= 0 exist. For the case in question \/i':h the ab ive given boundary condition of specified ani'iuthal cc:-,pon;--it of fp,e v.-ctor potential, the solution of Eq . ( 3) is '..'ell defined for values of k such that. |k| < I kr, ! where k_ is the numerically smallest eigenvalue. In this case the solution is also nxisymmetric and one finds that Eq.(3) can be simplified into

f Ll+ k2 y = 0 (4) here, the flux function ^ ^ RA,^ and the boundary conditions are that for R = 0 and •{• = ?/2n at the wall.

|k| is approaching Ikr| from below, the solution A blows up; the magnetic energy increases toward infinity and therefore the voltage between' the electrodes through which the axial current is driven must increase toward infinity. It the e igenf unc t i on associated with kr is axi svr.m^t ric , the configuration when |k| - |kr! may he an attractive one: one finds readily that the ratio between the axial current driven through the electrodes and the azimuthal current of the closed field regions approaches zero. If on the other hand the eigenfunetion associated with kr is nonaxi.symmetric, the configuration will switch to this nonaxisyr.irne t ric solution as !k| * |kF!; this situation seems undesirable and akin to the onset of a resistive kink instability.

It is easy to solve r.q.('») in the special cases

the kR - b-plane. For kR < k.R no closed field lines exist; the line kjRo is shown in Fl$>. 2 for KRQ =°0, 2 TT/3 and *. Similarly, lines corresponding to the lowest eigenvalue of F.q.(4), k R , are also shown in Fig. 2. The line k^R indicates for ~R - 0 and thhe cicunstances under which the separatrix lifts off the wall. r

0 ' 0

FIGURE 1. Contour plots of the flux function in the R/R - Z/RQ - plane for KR = 2TT/3 and various values of kR and b. The three possible field line topologies are illustrated by I: no closed field lines; II: closed field lines, separatrix at the wall; III: closed field lines, separatrix separated from the wall. The step-like shape of the wall is due to the finite grid used in the calculations.

Not shown in Fig. 2 are lines indicating the lowest eigenvalue of Eq.(3), kERQ, because these eigenvalues have not been computed yet. We know however from Taylor's work that for b = 0, k£R = 3.11; since it is shown in Fig. 2 that in this case k R = 3.83 (independent of the configuration is similar to the Spheroraak configuration . From the Spheromak theory we know that the axisymmetric state is stable when the configuration is axially FIGURE Location of the lines compressed relative to a sphere and k,R and k.R 1 o 2 o -203-

un.1; table when elongated. This is interpreted to mean that for the axially compressed case, '< R < krR and visa versa for the elongated c?sc. For the Bunpy Z-Pinch. the analogy to axially compressed is a large value of

3. Discussion

The most attractive feature of the Bunpy Z-Pinch nay be that it suggests a practical way to establish and maintain indefinitely a compact torus configuration. This feature is dependent on the assumption that the plasraa spontaneously approach the Taylor equilibrium. Maintenance of the configuration must, when transport is considered, take place through resistive instabilities. It is not known whether the associated turbulence will reduce the confinement drastically or mildly. The usefulness of the Bumpy Z-Pinch for fusion depends on that this reduction is tolerable. Also, the Taylor equilibrium, which is stable to all ideal and resistive instabilities can of course not be realized; one must expect the current density to approach zero at the plasma surface; again one may hope that such deviations from the Taylor equilibrium do not give rise to detrimental instabilities. Finally the Taylor equilibrium is pressure free which is useless for fusion applications; one may hope that nearby equilibria with finite pressures are sufficiently stable.

Acknowledgement

The authors wish to express their appreciation to F. W. McClain for the nunerical calculations he expertly performed. Also, the authors wish to acknowledge the significance for this paper of several discussions with T. Ohkawa.

References

1. The Bunpy Z-Pinch by T.H. Jensen and M.S. Chu, GA-A15811, March 1980. To

appear in J. Plasraa Phys.

2. J.B. Taylor, Phys. Rev. Lett, 21 P'1139 (1974).

3. D.C. Robinson, R.E. King, Proc. of Meeting on Plasma Physics and Controlled Nuclear Fusion Research, Novosibirsk, 1963, Vol.1 p.263. 4. M.N. Bussac, H.P. Furth, M. Okabayashi, M.N. Rosenbluth, A.M. Tobb, Proc. of Meeting on Plasma Physics and Controlled Nuclear Fusion Research, Tnsbruck, 1973, Vol.3 p.249. -204-

TH£ PLASMAK: ITS UNIQUE SiRUCTURE, THE MANTLE

P. M. Koloc and J. Ogden, Prometheus II, Ltd., Bx 222, College Park, Md. 20740 (301-434-7317)

Introduc t. ion

The SpheroTnak plasma ring (Kernel plasma here) is confined hv it:.; own i 'ter- aal self - generated magnetic field, and the vertical field is nr.-Juceii by rigid coils and sometin.es also image current surfaces vhich do not link the plasma. This embodiment of the Spheromak, including the necessary so] id vacuuir. v?l]> i., referred to here as a Rigid Shell Spherotnak. Although such plasma configurations appear to be highly desirable as fusion power generators'", they are not without problems. In fact, these problems constitute several of the fundamental issues thai confront the national Compact Toroid Development program. These issues are scattered throughout both theoretical and engineering areas. They include tipping and sliding instabilities, increasing lifetime (L/R times), excessive reci.rculation power, impurities, energy recovery (wall loading), and complexity of hurn chamber or drift tube topology. On the other hand, we have proper: _j a Spheromak embodiment which, if successful, totally obviates or significantly ameliorates these concerns within the CT program. This invention is a compound plasma, or Plasmak, confined entirely by self-generated currents. The vertical field arises from energetic image currents flowing in a plasma Mantle on or near the separatrix. Pressure balance is achieved by ordinary gas pressure. A question which immediately comes to mind relates to the raaintericiiice of both the highly conducting currents and the plasma of thit. structure .

I. Vt^'J'j£t±c±> ±1 the Plasmak A. General Description The simple Plasmak (Fig. lc) includes a toroidal plasma or Kernel plasma similar to tht* classical force free Spheromak. The region within the Mantle plasma including vacuum field and plasma ring is designated Kernel. The Kernel plasma is confined by the vertical field generated in a closely fitting, highly conducting ellipsoidal shell or Mantle within a gas blanket. This contrasts with other Sphero- mak and CT configurations (FRP, FRM. S-l) in which external vertical field coils are used to confine the plasma.

6. The Nature of the Mantle The Mantle plasma is formed at the ellipsoidal interface between the vacuum magnetic field of the inner current ring and the surrounding cold neutral gas. The Mantle conductivity, as explained below, is due to relativistic electron currents moving in a thin layer of fully ionized cold plasma. These currents are essentially image currents (toroidal) of those in the Kernel ring. Currents in the Mantle are maintained by the E.M.F. of the decaying Kernel magnetic field. Ionization of the Mantle is maintained even in proximity to surrounding cold neutral gas because of the extremely good ionizing properties of the Bremsstrahlung from the Kernel plasma ring.

The Mantle currents are focused into a thin sheet because of interactions with the electric dipole layer at the interface of the vacuum magnetic field and the surrounding cold plasma gas. The dipole layer of the Mantle arises naturally from the differential penetration of electrons and ions into the vacuum wgne.ic field -205-

region. Assuming that the cold plasma electrons and ions have the same temperature, then because of their greater momentum the ions move radially inward and penetrate further Into the increasing magnetic field than the electron0;. The electrons have higher mobility. As -> result, the iens spend more time deeper in the field than the electrons, and \ nt-t positive charge layer develops at the average 'on penetration distance. Si-îlar2v, a:i outer electron layer develops, forrî.:g the dipole layer shown in Fig. Ih. '/:ie extent and intensity of the electric field at the dipole layei" depends upon Lhe temperature and density of the ions and electrons (pressure) and the pressure gradient of the equilibrium, magnetic field at the Mantle field interface. rcr example, we can estimate the dipole separation by taking the differenc in the ion and electron lareor radii in the boundary magnetic field. Assuning a cold plasir.a with a temperature of 1 eV and a boundary fie±d c: 5 KG which balances an t.-vternal gas pressure of 1 ATM, the dipole layers are separated by abcu;. 0.08 cm. The vacuur, magnetic field drops quicK.lv across the dipole layer (Fig. 2). The g£S pressure opposes the magnetic pressure. Ignoring the curvature eff.cts and the kinetic inîtia of the particles (uv'/r term) in tne -adial force balance field equations fcr electrons and ions, we tind that E-/8- ^ B*/8- + P =• constant. Assu- ming an electric field due to a dipole ]?yer, we have fields with representative profiles showi. in Figs. 2 & 3, where I. is the dipole separation. The electron currents flow within the dipole layer. The radial electric (Fig. .3) of the dipole layer Lends to force the current inward. This force is balanced by the outward v x B magnetic force on the currents. So momentun of the energetic electrons also contributes. In addition to these two radial forces, the relativistic currents experience a radially differential resistive drag proportional to the density of the ions (Figs. 2, 4). Balancing this effect, tne currents also undergo acceleration fror> tiie radially differential E.V.F. (Fig. 4) of the decay:ng Kernel fêld (Fig. 2). This results in the energetic current distribution in Fig. 5.

II. Advantages of the Mantle A. Theoretical 1. Ideal MHD Stability - The Mantle is a highly conducting closely fitting shell. It has been shown3 that the presence of such a shell is desirable for Sphero- mak Ideal MHD stable topology. 2. Lowest Energy Configuration - The Plasmak Mantle is fluid and forms a free boundary. The presence of the surrounding neutral gas requires the Mantle to be an isobaric surface, and j x B = t"P = k everywhere. On any other contour j x Is = VP(r6<£) everywheie, ,->nd this implies that if the surface were non rigid it would move to form an isobaric one. When the j x B forces on the Mantle are uniform, the energy of the Spheromak or Plasmak configuration is the lowest attainable for a given volume and flux, implying greater stability.

3. Tipping Instability, Sliding Instability - The tipDing and sliding instabilities occur in Spheromaks with externally driven coils and without a closely fitting conducting shell. The presence of the Mantle would stabilize against both tipping and sliding instabilities. In this embodiment the Mantle remains tightly fitting even during adiabatic compression.

B. Engineering Advantages

Concepts of a high density gas blanket h.we been advocated by a number of authors4. Such a blanket is especially functional with advanced fuels since it not -206-

only protects walls and acts as an ash sink, but it can also efficiently transport the bulk of fusion released energy into inductive MHD electric power producticr. Furthermore, the fusion heated blanket allows the use of advanced fuels and also promotes excellent fuel consumption and high Q through the mechanism of Self Com- pression Heating. The Plasmak is confined entirely through gas pressure and self gener^ced internally trapped fields. Ordinary fluid flow is all that is needed for transport and manipulation. Thus, need for complicated raultipole or solenoidal cells is eliminated. Even specially shaped conducting drift tubes are not needed. Be- cause of ni=ai-ly complftp Tqp'-iif of the poJojd^l fuv -f the Kern' 1 pTasma bv the highly conducting Mantle, the magnetic energy losses due to wall eddy current indjction are also eliminated. Recirculating power can be kept to a very low value. Inductance resistance ratios are adequate to accomplish formation, transport, and burn without the need Tor auxiliary heating except for natural ohrr.ic heating and mechanically driven adiabatic compression.

III. Mantle Properties A. Conductivity Many of the desirable aspects of the Mantle derived from the fact that it is a highly conducting tightly fitting fluid shell surrounding the Kernel plasma and Kernel vacuum field region. The resistivity of the Mantle can be calculated for a typical Plasmak. For energetic electrons, TI is given by:

n » 5.2 * 10~3 * Z * In A ohm * cm erf

where /. = 740 * {T (eV)}^ * 3.55 * 106 * T

In the Kernel plasma the energetic electrons drag on botf thermal protons electrons so the conductivity is reduced by the ratio of the energetic current ej.^c trons to the thermal particles. In the Mantle, however, there is no such thermal electron drag predicted by cur hypothetical model since the energetic currents ere flowing in the channel formed by the differential maximum penetration of the electrons and protons. Mantle resistivities for a 100 KJ Plasmak with 2 MeV currencs- in an atmospheric pressure blanket are on the order of 5 * J.0"1] ohm cm, yielding a Mantle resistance of 10"9 ohm. These values seem appropriate considering values estimated for runaway electrons in Tokamak discharges.6 B. lonization Assume a cold plasma of temperature 1 eV. The iieat loss to the surrounding neutral gas is determined by the heat conduction and convection at the outer edge of the Mantle. A typical convection cell site in air is .6 to 1.0 cm. We assume that the temperature drops from 1 eV to ambient 300°K in this distance. The power loss per cm2 via heat conduction, using diffusivity of plasma at 1 ATM is:

P/A - (1.65 * 10-1* watts / °K * cm) * 1.16 * 104 °K / 0.6 cm

• 3.2 watts / cm2 -207-

2 For a Plasmak of radius 30 em, the total power loss is Pt^ 4 * TT * P/A * R = 36 KW. The above power loss rate should be regarded as a very conservative upper bound since the radial electron thermal conduction is greatly reduced by the presence of a weak magnetic field. The ratio of the heat conduction with the field to the heat conduction without the magnetic fi.eid 1c|| / K_L is on the order of 10 for our parameters. Assuming 1c±*is confined to the poiar regions (about 10°< of surface area), the expected conduction losses would be about 4000 watts. Bremsstrahlung from the uncompressed Kernel plasma is about 20 KW. Compression increases available power per Mantle particle aDnroximately by the compression ratio to the fourth pswer. * unit vector perpendicular to Mantle surface References

1. H. W. Hoida, I. Henins, T. R. Jarboe, et al., Bull.Amer.Phys.Soc. _25 (1980) 1022. 2. H.P. Furth, "The Compact Torus", Vacuum Society Meeting, October 1980. 3. M.K. Rosenbluth and M.N. Bussac, Nucl Fusion _19 (1979) 489. 4. B. Lehnert, IAEA CN-37 , Innsbruck, 1978. G. Miley, Proc. US-Japan Joint S.mposium on Compact Toiuses and Energetic Parti- cle Injection (Dec. 1979)200. 5. P. Koloc and J. Ogden, Proc. L'S-Japan Joint Symposium or Compact Toruses and Energetic Particle Injection (Dec. 1979) 216. 6. J. Strachan, Princeton PPL, Private Communication.

Vac-cium Field

Manfle plasma ion Layer ke.-nel plasmd -ernel vacuum Electron Layer

Plasma

Cas Blanket J. '• , • ,r •" •• Tig. la PLASMAK CUTAWAY Fig. lb MAMLE BLOWUP

Fig.

Fig.U Profiles Determine the Forces in the Current Direction: / Accelerating CMF is proportional r to dB/dt of Kernel field, n o r +L o resistive drag on current is proportional to number of Fig. 5 collisions, which is proportional to cold ion density.

ro -208-

INIHX.;-.:) HCRF.-WK CURKK.VI trrhcrs ON PLASMA STABIF.ITY IN !^i':sJ2ilJL OF AN jNHOMOGF-NKOrS MA^'KTIC VACUUM FIKI.L

B. i.ehnert

Ro'.-.i [ Institute ..if IV L hno ! ogv , S- 10044 Stockholm 70, Sweden

Ah ;= t r :ic t . liu- expression for the change in potential energy of a conlin.ed p:,]s::i.i has been re i ornml at ed to Lake induce-' surface current effects explicitly i:;to account . An externally imposed i nhoinogeneous magnetic vacuum field lias a s! •':;] i iy.)\\i[ •. iiect .in frut'-surl jfc- modes, ben:.!; especial ly pronounced m the '• .•i,.--l.;;ive 1 in; I .

1 . I; 11 i o i u • • L i i •::

!i.e Mhl) j !!.••;an i 1 l 1 i es in ,i magnet lzed plasn.i con:- i:'t nl free-surface modes

.i: : i:'iu i"!:..i ::i>-..;'<.• . in i.i.-mv t onf i iH-men t sclieires, such as compact toroiJs, tne

i1 ! :.-::.'! • ::::'.'', be ;-;"iniied i)v ;• condiu; t i nfi wall but has a free surface. Stabiii-

-••.t:.-.i . ! ; r>. t. •• -;ui'! .ice :r:''Ces t in n heconies a crui ial pro!) 1 ein.

i- : i|" . I!I •. a..-:e •• t •; t - s I .'ii!: 1 :/..it ion ol . ,-ee-su;" t ace elei 1 I'IMJ'JK' t i c Miii:

;.. :• v- :- n. ::..;•• •-,'••.: I'::,.TII ;:ei,eou T;iaf.'.ne L i c "vacuum iiclc" due t(j external

• ••;::..•!:•! i '.:::•:;••., a • prv ]io -ec i •. I be i x t : ap" co::ccp t 1,2 .

•v.i , i - : i; i:.i t 11'.;.. -iiiC A.'.s; .:;p ; ] ;:.-:

A ,: ; :.s i p .il i . >n- ! r ee p ! .i ,i: . i .-••: w 1 u::,i '. .;•:.: : • . i J : i . i; :• ;^ surf ace S with the

u:.:t no:"i:;al \: is assumed to lie .•>•:.! iue.i in equ ! ! i b r i Lin by 3 nia>.;iie t i c lie].:

.4 -'"B +B . The pi a;:;;,] current :<••. :t. i • :li ; ,. ,c: .' K is due to currents in <•• -p -v - J. p ••'

s,!i.! fixtd external conductors, 'urtbei '.'\t - .'; • !•: v/hi-re p is t'ne plasr.'.a

prii--:irr. We assume' j , '. p and n to vanish .ft S, wliich is a relevant - t) " O il

;:upi. xi:::.il :•':: i i. r.'.iu'-' expe r i ;;;••:! I s .

In a perliU'bec! state, arbitrary fluid displacements \ Linluce a surface

Liirier.t .it:i-:;Ly ..' at S. With A-:,-IS the change • U' in potential energy -3

i s i *:vv"r i L ter. as *

;l ! • \<--• ,;c+ Wv i 1/2: ).".'.'(v, + w ,)dV (])

- •• -• , ^ ~ :•;. :W..= .. r 'A-JdS- •-'n- ' i :' • V)B : • AdS-///di vi | ( v •T .J) li i 'AJdV ."//w dV (j) J o S ob -- - s - - - - -V - v - - -V - v S il<-re 6W is the contribution from t he induced surface currents at S and w=A-curl A-.- :,-j .-.curlA+iJ ~-V(j -A)+,. ,p (div^)~ (3)

\ - — O'" —O — (J— A) — O O - represents contributions from the volume V where > is the specific heat r a t i o . -209-

3. The Kffects of the Magnetic Field Gradients

To separate the effects of the magnetic field derivatives from those of the fluid t i isp utccntnt, we reformulate the expression tor w +w , thereby intro- S v due ing a frame of rectangular coordinates. Kith A = :"VE and A =:,-B we haw -p ••• -p -v - -v

+w -CciirlA )~+(curlA )-(ouri.\ >+;. •' • ( B •-v —v -p' o-- •

; w B = •'. - '• ." i H -vj * -(•'•'.• ) —!iv • '-( •' • . i —B n v = (?> • "•') ' -V> d : v' "f "< B •'.") ; -B di v; : • (3 •'.' / .• -ii di v7 - <: • 7)B . v - _ ...v _ . —v v ^_ ._.p • p .- - - - ._,.

• 'i ;'. •'.•)"-!•; di v "-(•"••. )1J • ( :-•, ••.') \-ii di v^" *. ' • [ '• •'." i ;" • i .. -i. - - •(-• - • '--^r- - -v- •-'- —v -. o - --i.i - -- -

p i d i v ' ) ~ •<• { •' • B )• (i; • "ii"jr i •' +B ;•• (clivu)-j (div;)-" -V(b • ' ' • •' - c o - - -.. L - -.• - —o -' - ' o - ''^ • -ii

. ( •' • ^ Mi ' + '!' d i v',- i :•. • . i ' • cm ]('•!> i * i : • ii > • . \ : •'.' ) • :i • 7 - (>• — v -•'..• -- -i.i -v' L. ' •-' --V-

"((>•'. i .-..( -i.i. /-x.;' ; •.--., ) fcr r .)••: vecUL' i" i •.• I d Q. Thus w, contains -: - ) , ' . i K • -- B K 1 !er : v; it :i 1 iv m

: i i".-, •'•<•• at 1 \' I' •f ! i . .1 i .j: p> • l :i I s s <.,•:: i d :u- observed:

,i> i:.c. u e t •'.'. . !•face nir i t> 'f ' W v;;:. i ^ Ii when \_ vdiiisi.i's at

t , • : V.'. 1 u v! \- • • • i h • rt h . i i i . Jli.' i ::di:. i d ;" i u i d )',-ru v i > ' • i; i v.-ip. i sin's for e ! ec i ros t a t i c pe ]• t;: r:-at 1 > :; s

si:. ':• ,is t i i:t i •- l vpc in, -.i.l (.• •, . '...'t- restrict t!:o d i sous s * on on sucii p-- rt i:i r>a t : oi: s

11 : it- t •.•> 1 i ov. i ii.i.1, p. ii ii t ( vi ) . liii) l-'.-r ::ititi('.is .-icioss |-i , inluccd cirrent effects arise botr. from tile i nhoiiii>s.-,r"i' i I v of B and of •". . Tile first term of cxpres'ion (.5) represents one part oi tlie former effect, being due to 'W i r, L'q . (i) and contributing to slabiliiv. I':;r i i]hiniRi;:ei;e i t v effects of f. ht-l n;.; included in expression ((•>) can scMiietines contribute to stability. On the other hand, Lh «• two induced current effects may cancel for certain types of displacements, as described later under (vi). 2~ (vi) 1 he contribution trum A-cur!*~_A_ to cW has earlier been cast into a form

of volume ami surface currents [_4j from which the stabilizing effect of a -210-

strong longitudinal field B SB in a linear pinch can readily be seen, (v) The terms of eqs. (5) and (6) can be estimated by introducing the characteristic moduli B , B of B , B and lengths L = B 1/ 3B /3x. I, pc vc ~ZP v P ~P —P k1 L =JB /j 3B /3x, |, L =|c|/ 3£/3x, where 3/3x. are derivatives in relevant v '—v ' —v k' £ '—' — k k directions, Eqs. (4)-(7) then contain the parameters B L /B L , L /L and ^ pc v vc p v C B /B . With restrictions (i) and (ii), the long wave-length limit L >>L , L yields jw |/|w |=0(L /L )<<1. We particularly consider a linear pinch where a strongly inhomogenpous transverse field IJ makes B L /B L = = !C'7)B |/j(i.*V_)B I smaller than unity, such as in Extrap (_1,2 j . Stability for lonj-wavc modes (including kinks) is then secured by the term \(^'^)^ ~] in eq. (5), regardless of the zero field at the axis which does not affect the surface integral (2). The stabilizing contribution from a purely trans- 2 verse inhomogeneous field B then becomes about (B L /B L ) >>1 times —v vc £ z v greater than that from a purely longitudinal field B of the same magnitude (compare Ref:-. |l,2|). A similar result applies to axisymmetric systems with a weakly inhomogeneoud toroidal vacuum field.

(vi) There sometimes exist displacements €, for which the curlA terms of — —v eq. (4) become small. For a linear pinch immersed in a vacuum field _B -B +B with a transverse strongly inhomogeneous part B and a longitudi- v —va —z —vx nal part B

B :curlA =F(B -V)C-B divtl -(S-7)B (8)

In the long-wave limit L >>L , and when B =0, the second term of the 5 Vj. —z right-hand member remains finite and dominates over the first (square bracket) for all displacements £ being directed across B . However, when introducing a strong longitudinal field, |]} |>>|B j, the components of _B permit perturbations for which the first term of eq. (8) can be made to nearly cancel the second, thus making B small also in the long-wave limit. The longitudinal field J5 then weakens the stabilizing effect of the transverse field j} . In other words B +J5 +J3 then yields a helical field structure allowing for flute-like helical perturbations £ with small electromagnetic induction effects (compare also Ref. \i\) • -211-

A. Conclusions The present results are summarized as follows:

(i) The electric currents which are induced by fluid displacements in a magnetized plasma have two possible sources, namely the inhomogeneities of the magnetic field and those of the displacements. In the case of high electrical conductivity, surface currents will arise from free-surface motions across an externally imposed inhomogeneous magnetic "vacuum field". These currents can produce a restoring force and an increase in magnetic energy. Consequently, an imposed strongly inhomogeneous vacuum field provides means for stabilizing a magnetically confined plasma against a large class of free-surface modes.

(ii) In the particular case of a straight Z-pinch, an imposed strongly inhomogeneous transverse vacuum field becomes more effective in stabilizing long-wave surface modes than a homogeneous longitudinal field of the same order of magnitude. In addition, a superimposed longitudinal field weakens the stabilizing effect of the transverse field.

5. References

[l] Lehnert, B., Physica Scripta K)(1974)139; K3( 1976) 250; 16/1977)147.

!2J Drake, J.R., Hellsten, T., Landberg, R. , Lehnert, B., and Wilner, B., Eighth International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Brussels, 1-10 July (1980), paper IAEA-CN-38/AA-3.

[3] Bernstein, I.B., Frieman, E.A., Kruskal, M.D. and Kulsrud, R.M., Proc.Roy.Soc.A., 244(1958)17.

] Lehnert, B. , Royal Institute of Technology, Stockholm, TRITA-PFU-80-04(1980).

Stockholm, October 6, 198C -212-

AL'TIIOR LIST

Anderson, D. V. 130,134 Gilligan, J. G. 80 Armstrong, W. T. 180,184 Goldenbaira, G. C. 97,113 Auerbach, S. P. 52,89 Granneman, E. H- A. 113 Ay.lerair, A. Y. 76,134 Greenly, J. B. 36 Barnes, D. C. 134 Greenspan, E. 4 Bartsch, R. R. 172 G r Li! in, P. • C . 5 6 Bel Km, P. M. 1 Grossmann, W. 12,138 Berk, H. L. 130 Hagenson, R. L. 8 Boyd, J. K. 8 9 Hamasaki, S. 152 Brackbill, J. U. 161 Hameiri, E. 68 Bruhns, H. 97 Hammer, D. A. 36 Byrne, X. X. 138 Hammer, J. H. 68,72,130 Carlson, G. A. 12 Harned, D. S. 40 Chance, M. S. 5 6 Hare, G. W. 97 Cherdack, H. N. 24 Hartraan, C. W. 113 Chon^, Y. P. 97 He idbrink, W. 124 Chu, C. K. 7 6,85 Henins, I. 101 Chu, M. S. 2 00 Hess, R. A. 97 Cochrjne, J. C. 180,184 Hewett, D. W. 148 Coumisso, F<. J. 184 Hoffman, A. L. 27,176 Condit, W. C., Jr. 52 Hoida, H. W. 101 Cooper, A. L. 18 Hugrass, W. N. 105 Dalhed, H. E. , Jr. 60 Ishimura, T. 192 DeVeaux, J. 4 Ishizuka, H. 43 Dewar, R. L. 56 Ito, H. 192 Drake, J. R. 196 Ito, Y. 192 Dreike, P. L. 36 Janos, A. 124 Eddleman, J. L. 82 Jarboe, T. K. 101 Ekdahl, C. A. 184 Jardin, S. C. 56,124 Fang, Q. T. 144 Jenkins, D. J. 18 Finn, J. M. 64 Jensen, T. H. 200 Fleischmann, H. H. 12,31 Johnson, J. L. 56 Furth, H. P. 124 Jones, I. R. 105 Galambos, J. * Kammash, T- 12,156 -213-

Knox, S. 0. 188 Scannell, E. P. 18 Koloc, P. M. 2 04 Schnack, D. D., Jr. 134 Krakowski, R. A. 8 Schultz, K. R. 12 Krall, N. A. 152 Sevillano, E. 188 Lehnert, B. P. 208 Seyler, C. E. 148 Linford, R. K. 101,180,184 Sgro, A. C. 85,169,180 Li pson, J. 180,184 Shearer, J. W. 82 Lui, H. C. 76,85 Sherwood, A. R. 101 Lyster, P. M. 36 Sherwood, E. G. 180,184 Marshall, J. 101 Shestakov, A. I. 134 McKenna, K. F. 101,105,130 Shiina, S. 109 McNamara, B. 82,89 Shiraamura, S. 109 Meuth, H. 188 Shuraaker, D. E. 89 Mi ley, G. H. 4,80,144 Si onion, i<. E, 17 2,180,184 Milroy, R. D. 161 Sinnis, J. 124 Moncicello, D. A. 56 Snith, A. C., Jr. 12,113 Nakagawa, Y. 36 Steinhauer, L. C. 27,176 Na s h, J . K. 82 Sudan, R. ". 36 Nogi, Y. 109 Tanjo, M. 192 O^den, J. 2 04 Tnska, J. 113 Ogura, H. 109 Tunsrall, J. L. 119 Ohi, S. 192 Turchi, P. J. 18 Okabayashi, M. 124 Turner, W. C. 8 2,113 Okada, S. 192 Tuszewski, M. 180,184 Olson, R. 80 Vondrasek, R. J. 24 Osanai, Y. 109 Wells, D. R. 119 Peter, W. 47 Willenberg, H. J. 27 Pietrzyk, Z. A. 165 Winske, D. 85 Platts, D. A. 101 Woodall, D. M. 12 Prono, D. S. 113 Wysocki, F. 124 Reiman, A. H. 64 Yamada, M. 124 Ribe, F. L. 188 Yamazaki, K. 93 Robertson, S. 43 Yoshimura, H. 109 Ro stoker, N. 47 Ziajka, P. 119 Saito, K. 109 Salberta, E. 124

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