bij het proefschrift: THE SCREW-PINCH APPROACH TO CONTROLLED FUSION J.A. Hoekzema, 6 december 1976

S T E L L I N G E N bij het proefschrift: THE SCREW-PINCH APPROACH TO CONTROLLED FUSION J.A. Hoekzema, 6 december 1976

Lang voordat Clark e.a. met beschouwingen kwamen over "Flux Conserving Tokamaks" en nog langer voordat de additionele verhitting in tokatnaks tot flux-behoudende evenwichten kan leiden, zijn deze evenwichten reeds beschreven en experimenteel aangetoond in toroïdale schroefpinches. J.D. Callen et al., Proo. 6th Int. Conf. on Plasma Phya. and Contv. Nuol. Fusion Res., Berohteagaden (1976) IAEA CN-3S/B10. C. Bobeldijk et al., Proo. 3rd Int. Conf. on Plamna Phys. and Contr. Nuol. Fusion flee., Novosibirsk (1968) 1^ p. 28?. II De zorgvuldige constructie van SPICA maakt deze opstelling zeer geschikt voor het bestuderen van spontane omkeer van het toroïdale magnetische veld aan de wand zoals optreedt in de Reversed Field Pinch (RFP), Omdat bovendien de buisdoorsnede van SPICA groter is dan die van bestaande RFF-experimenten, zou een op veldomkeer gericht onderzoek in SPICA een waardevolle voorbereiding betekenen voor een in Engeland voorgesteld groot RFP-experiment. III Doordat in een reactorplasma van betrekkelijk lage temperatuur voornamelijk de hoogenergetische ionen verantwoordelijk zijn voor de energieproduktie, zijn de berekende parameters van een lineaire fusiereactor waarin de opsluiting van het plasma wordt bereikt met behulp van een groot aantal zwakke magnetische spiegels te optimistisch ais voor de ionen een snelheidsverdeling volgens Maxwell wordt aangenomen. IV Het criterium van Richardson is niet voldoende ou> te voorspellen of de stroming in een wervelkamer waarin de rotatie van het gas wordt verkregen door gastoevoer en afvoer aan de omtrek al of niet turbulent is. R.W. Polman, Rijnhuizen Report RR 76-97 (1976).

Hoewel buiten het .oppervlak van het Ringboog-plasma geen lijnstraling in de Lymanreeks gedetecteerd kan worden, is deze straling grotendeels verantwoordelijk voor het energietransport uit het centrum van de ontlading. Ringboog-team, Proa. Int. Symp. on Plasma Wall Interaction, Jülioh (.1976).

VI De anomale stroompenetratie ten gevolge van verzadigde stroomgedreven ion-akoestische turbulentie is weinig afhankelijk van het verzadigingsmechanisme. VII Het door Eberhagen beschreven voorionisatiesysteem is minder effectief naarmate de ontladingsruimte nauwer wordt omsloten door een metalen schild. Goede voorionisatie blijft mogelijk door in plaats van enkelvoudige draden banden of een aantal naast elkaar liggende draden te gebruiken. Als de afstand tussen ontladingsruimte en schild erg klein is, verdient het meervoudige draadsysteem de voorkeur. A. Eberhagen, Max-Planak-Inatitut fur Plaamophysik, Garahing, Report IPP 1/114 (1970). VIII Hamasaki en Krall brengen ten onrechte het vrije-deeltjesmodel en het sneeuwploegmodel in discrediet door de resultaten die deze modellen geven bij verwaarloosbare stijgtijd van het magnetisch veld te vergelijken met numerieke berekeningen, gebaseerd op een veel ingewikkelder model, voor een stijgtijd van het magnetisch veld die langer is dan de implosietijd van het plasma. De gegeven resultaten voor het sneeuwploegmodel zijn bovendien onjuist. S. Hamasaki, iï.A. Krall, üuol. Fusion 16 4 (1976) 599. IX Van de twee mogelijke weerstandsnetwerken waarmee axiaal-symmetrische magnetische velden kunnen worden nagebootst, beschrijft Knoepfel het minst praktische. H. Knoepfel, Pulsed High Magnetic Fields, North Holland Publishing Company, Amsterdam-London, 1970.

Het opzetten van databank-systemen bij thermonucleair onderzoek, waardoor technische en fysische informatie voor de partner-laboratoria toegankelijk wordt gemaakt via een datanetwerk, l;an, mits het vergezeld gaat van een zorgvuldig opgezette bibliotheek voor dataregistra<:i<>., een grote stimulans betekenen voor de deelnemende laboratoria en leiden tot een versnelling in de thermonucleaire ontwikkeling. XI In een laboratorium net parallel werkende experimenten is een rekenfaciliteit met de mogelijkheid de experimentele gegevens parallel in "timesharing" te verwerken, onontbeerlijk. XII Voor Euro-fusie is Euro-visie nodig.

THE SCREW PINCH APPROACH TO CONTROLLED FUSION

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DF WISKUNDE EN NATUURWETENSCHAPPEN AAN DE RIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. A. VERHOEFF, VOLGENS BESLUIT VAN HET COLLEGE VAN DECANEN IN HET OPENBAAR TE VERDEDIGEN OP MAANDAG 6 DECEMBER 1976 DES NAMIDDAGS TE 2.45 UUR

DOOR

JACOBUS ALFRED HOEKZEMA

GEBOREN OP 22 SEPTEMBER 1947 TE ARNHEM

DRUKKERU ELINKWUK B.V. - UTRECHT

PROMOTOR: PROF. DR. CM. BRAAMS

Dit proefschrift werd bewerkt in het FOM-Instituut voor Plasmafysica te Jutphaas in de werkgroep die aanvankelijk onder leiding stond van Dr. Ir. P.C.T. van der Laan en later van Dr. C. Bobeldijk.

CONTENTS

I.

Inleiding,

samenvatting

II.

Summary

III.

Local electron density measurements in a screw pinch by means of a Michelson interferometer,by J.A. Hoekzema, P.J. Busch and W.J. Mastop, Rijnhuize.n Report RR 76-96 (1976) • AI*^OA.

IV.

Toroidal equilibrium of non-circular sharp boundary plasmas surrounded by force-free fields, by J.A. Hoekzema, Proc. 3rd Topical Conference on Pulsed High-Beta Plasmas, Culham (1975), to appear in Plasma Physics.

V.

Decay and profile of the toroidal plasma current in a screw pinch, by J.A. Hoekzema, Proc. 3rd Topical Conference on Pulsed High-Beta Plasmas, Culham (1975), to appear in Plasma Physics.

VI.

Heating of a pinch at intermediate g-values, by J.A. Hoekzema, C. Bobeldijk, P.C.T. van der Laan and W. Schuurman, Rijnhuizen Report RR 76-98. (1976). Ot^S/t^*'

VII.

Current decay and stability in SPICA, by C. Bobeldijk, J.A. Hoekzema, M. Mimura, D. Oepts and A.A.M. Oomens, Proc. 6th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res., Berchtesgaden (1976), IAEA-CN-35/E6.

VIII.

Parameter study of a screw-pinch reactor, by C. Bobeldijk, M. Bustraan, G.C. Damstra, W.M.P. Franken. J.A. Hoekzema, H.J. Klein Nibbelink, H.Th. Klippel, P.C.T. van der Laan, M. Muysken, W. Schuurman and K.A. Verschuur, Proc. 9th Symp. on Fusion Technology, Garmisch Partenkirchen (1976).

V O O R W O O R D

In de afgelopen jaren hebben velen bijgedragen aan het uiteindelijk totstandkomen van dit proefschrift. Een aantal wil ik graag met name noemen: In de eerste plaats dank ik mijn promotor Prof. Dr. C.M. Braams voor zijn interesse in het werk en voor de stimulerende discussies. Veel dank ben ik verschuldigd aan Dr. C. Bobeldijk, Dr.Ir. P.C.T. van der Laan en Dr. W. Schuurman, die steeds bereid waren mij met deskundige raad en daad bij te staan. Alle medewerkers van de pinch-groep ben ik zeer dankbaar voor de prettige contacten en de goede samanwer king. In het bijzonder dank ik Dr. R.J.J. van Heijningen, Dr. A.A.M. Oomens en Ir. J.W.A. Zwart voor de samenwerking op fysisch gebied, P.J. Busch voor het samen experimenteren en het vele goede technische werk en W. Kooijman voor de samenwerking op numeriek gebied. De bijeenkomsten van de reactorgroep, onder bekwame leiding van Drs. M. Bustraan, waren steeds zeer stimulerend. Het typen en correctiewerk was in goede handen bij Mevr. H.J.C. Thoden van Velzen, Mevr. H. Toft-Betke en Mevr. J.M. Hamers-Smit. Het tekenwerk en fotowerk werd voortreffelijk verzorgd door P.A. van Kuyk, A. Steen, W. Tukker en W. van Zanten. Dit werk werd verricht in het kader van het associatiecontract van Euratom en de "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) met financiële steun van de "Nederlandse Organisatie voor ZuiverWetenschappelijk Onderzoek" (ZWO) en Euratom.

I N L E I D I N G

EN

S A M E N V A T T I N G

Dit proefschrift bestaat uit een aantal publikaties, die alle betrekking hebben op de toroïdale schroefpinch. In schroefpinchexperimenten wordt een toroïdaal plasma samengeknepen en opgesloten door een schroefvormig magnetisch veld. De schroefvorm van de veldlijnen komt tot stand doordat een extern aangelegd toroïdaal veld wordt gecombineerd met een poloïdaal veld, afkomstig van een in het plasma geinduceerde stroom. In Fig. 1 zijn enkele kenmerken van een schroefpinchopstelling geschetst. Beide componenten van het magnetisch veld worden snel aangelegd: een karakteristieke stijgtijd is 10 ysec. Het plasma, dat na voor-ionisatie in de hele torus aanwezig is, implodeert onder invloed van de snel stijgende magnetische druk. Deze implosie kan, mits zij snel genoeg verloopt, een efficiënte manier van plasmaverhitting zijn. De korte stijgtijd van het veld wordt bereikt door ontlading van hoogspanningscondensatorbatterijen. Deze batterijen worden na een kwart periode, als de stromen hun maximale waarde hebben bereikt, kortgesloten, waardoor de experimenteertijd wordt verlengd. Vanwege de snelle veldaanleg moet het torusvormige ontladingsvat van een elektrisch isolerend materiaal gemaakt zijn. Hiervoor wordt veelal kwarts of aluminiumoxyde gebruikt, omdat deze materialen een hoog smeltpunt hebben, zodat het plasma zo weinig mogelijk verontreinigd wordt met materiaal van de wand. Om de isolerende torus is meestal een goed geleidende metalen mantel (schild) aangebracht, die zowel in toroïdale als in poloïdale richting is onderbroken. Dit schild speelt een belangrijke rol bij evenwicht en stabiliteit van het plasma. Als gevolg van de toroïdale stroom, de inhomogeniteit van het toroïdale magnetische veld en de kinetische druk van het plasma, werken naar buiten drijvende krachten op de plasmakolom . Bij toenemende verplaatsing ontstaat echter een tegenkracht doordat het poloïdale veld wordt samengeperst tussen de plasmakolom en het geleidende schild. Als resultaat neemt de kolom een excentrische evenwichtspositie aan. De gevaarlijkste instabiliteiten voor de schroefpinch zijn de kinkinstabiliteiten, waarbij de plasmakolom schroefvormig vervormd wordt. Slechts onder zeer bepaalde voorwaarden is het plasma stabiel èn de evenwichtspositie niet te ver excentrisch. Deze voorwaarden

Fig. 1. De schroefpinch - schematische voorstelling. 1. 2. 3. 4. 5. 6.

plasmakolom, kwarts torus, koperen schild, aanvoerflens primaire toroïdale stroom, poloïdale stroomwinding, diagnostiektuit: er zijn tuiten voor diagnostiekdoeleinden, voor vacuüm pompen en voor gasinlaat, 7. onderbreking in schild,

6. peperbus: zowel in horizontale a l s in verticale richting i s een r i j openingen in het koperen schild aangebracht, waardoor het plasma gefotografeerd kan worden. laten zich het best beschrijven met behulp van de parameters q en $. De veiligheidsfactor q is een maat voor de verhouding van de beide magnetische veldcompon-nten: 2ïïp

'

waarbij p de afstand is tot de hoofdas van de torus en geïntegreerd wordt langs een magnetische veldlijn tot deze een rondgang in poloïdale richting heeft voltooid. De magnetische veldlijn gaat daarbij dus q ma-"; in toroïdale richting rond. De parameter B wordt gedefinieerd als de verhouding tussen de kinetische druk van het plasma en de magnetische druk van het opsluitende veld: 0 2 B2

Experimenteel kan de grootte van 6 beïnvloed worden door het aanleggen van een magnetisch toroïdaal 'bias'-veld in de torus, voordat het plasma gevormd en geïmplodeerd wordt. In stationaire toestand is er

evenwicht bereikt tussen enerzijds de kinetische druk van het plasma en de magnetische druk van het 'ingevroren' bias-veld en anderzijds de externe magnetische velddruk. Hoewel in theorie stabiliteit mogelijk is voor waarden van q lager dan 1, ligt het meest gunstige gebied bij g > 1 (Ref. 2 ) . In dat geval is echter, zoals uit de definitie van q blijkt, het pololdale veld relatief zwak. Daardoor ligt de evenwichtspositie, die tot stand komt via compressie van dit veld tegen het geleidende schild, al voor lage waarden van 0 erg excentrisch. Aangezien dit vooral geldt voor een slanke torus, wordt in schroefpinchexperimenten de aspectverhouding A = R/b zo laag mogelijk gekozen. Van veel belang is het verloop van q buiten de centrale plasmakolom. Voor lokale stabiliteit is het nodig dat q > 1 op elk magnetisch oppervlak. De gunstigste situatie wordt bereikt als q in het buitengebied constant is. Dan is het mogelijk een macroscopisch stabiele plasmakolom in evenwicht te houden voor waarden van 0 tot maximaal 20% (tiij een aspectvtsrhouding A = 3) . Experimenteel is het mogelijk het profiel van q te beïnvloeden, doordat buiten de centrale plasmakolom een goedgeleidend plasma var. lage dichtheid achterblijft . In het buitengebied kunnen daardoor stromen geïnduceerd worden, die vanwege de lage dichtheid van het plasma krachtvrij aijn (de stroom loopt evenwijdig met het magnetisch veld). De q-waarde van een naar binnen bewegende magnetische veldlijn verandert, op een tijdschaal die kort is vergeleken met de diffusietijd, niet meer nadat de veldlijn het ijle plasma is binnengetreden. Hierdoor is het q-profiel in het buitengebied een afspiegeling van de aan de wand aangeboden q als functie van de tijd: Als de verhouding tussen de beide veldcomponenten aan de wand constant is in de tijd, zal het q-profiel vlak zijn. Dank zij de aanwezigheid van het ijle plasma kan B bijna een orde van grootte hoger z\ijn dan bij een vacuüm buitengebied mogelijk zou zijn. Een belangrijk onderdeel van het schroefpinchonderzoek is gewijd aan de schroefpinch met niet-circulaire doorsnede. Doorgaans wordt hierbij een toroïds gebruikt met rechthoekige of renbaanvormige kleine doorsnede, die in axiale (verticale) richting langer is dan in radiale (horizontale) richting. Vanwege de vorm die dan ontstaat wordt een dergelijk experiment ook wel beltpinch genoemd. Bij gelijke waarde van q is in de beltpinchgeometrie het poloïdale magnetische veld relatief veel sterker dan in de schroefpinch met ronde doorsnede. Daardoor zijn de evenwichtseigenschappen van de beltpinch gunstig. De stabiliteitstheorie van de beltpinch is veel minder ver ontwikkeld.dan van de schroefpinch met ronde doorsnede. Zowel experimenteel als theoretisch zijn er echter aanwijzingen dat weliswaar q in een beltpinch iets

hoger gekozen moet worden, maar dat toch een aanzienlijke verbetering mogelijk is ten opzichte van de ronde schroefpinch. Schroefpinchonderzoek maakt sinds 1963 deel uit van het programma van het FOM-Instituut voor Plasmafysica. Nadat in een aantal kleine experimenten goede resultaten waren behaald, heeft schaalvergroting plaatsgevonden met de bedoeling de tijdsduur van de ontlading te verlengen. Op het ogenblik worden in SPICA (zie Fig. 2) schroefpinchplasma's gemaakt met centrale e~-dichtheid * 10 2 z m~3 en temperatuur == 50 eV (g = 0,2) , die gedurende 100 ysec macroscopisch stabiel opgesloten blijven. Daarnaast wordt onderzoek gedaan aan een kleine beltpinch (Fig. 3 ) . In het thermonucleaire programma neemt de schroefpinch een tweeledige plaats in. Enerzijds worden de reactormogelijkheden onderzocht en anderzijds kan het schroefpinchonderzoek als tokamakondersteunend worden gezien. De 'tokamak' wordt op het ogenblik beschouwd als de belangrijkste reactorkandidaat en heeft veel overeenkomsten met de schroefpinch. Belangrijke verschillen zijn: In een tokamak is het toroïdale magnetische veld (verreweg de grootste component) quasi-stationair. Er vindt geen implosie plaats; de configuratie wordt langzaam opgebouwd. Het plasma wordt vrijgehouden van de wand doordat in de torus een diafragma (een materiële 'limiter' of een magnetische 'divertor') is aangebracht. Omdat de configuratie langzaam wordt opgebouwd, kan een metalen (dun roestvrij staal) ontladingsvat gebruikt worden. De bereikte dichtheden zijn in een tokamak veel lager (^ 10 1 9 m~3) en de temperatuur die tot stand komt door ohmse verhitting veel hoger (^ 1 keV). Terwijl in een schroefpinch, vanwege het optreden van krachtvrije stromen in het buitengebied, het stroomprofiel veel breder is dan het drukprofiel, hebben in een tokamak beide profielen ongeveer hetzelfde verloop. De evenwichtseigenschappen van een tokamak zijn daardoor veel minder gunstig. Meestal wordt daarom een extra verticaal magnetisch veld aangelegd, dat het poloïdale veld tussen plasma en buitenwand van de torus versterkt. In schroefpinches wordt deze methode zelden toegepast, omdat het gevolg is dat veel magnetische oppervlakken de wand snijden, zodat daar geen krachtvrije stromen meer kunnen lopen. Tokamaks hebben tot nu toe bij zeer lage B (£ 1%) gewerkt. Voor het bereiken van hogere fi-waarden, die zeer gewenst zijn in een uiteindelijke reactor, zullen andere verhittingsmethoden moeten worden gebruikt. Mogelijk kan intussen onderzoek aan de schroefpinch (ook wel hoog-6 tokamak genoemd) de hoog-B limiet (de maximale waarde van 3, waarbij het plasma nog stabiel blijft) van de tokamak bepalen.

Fig. 2. Overzicht van het SPICA-experiment. Aan de linkerkant is een deel van de condensatorbatterij zichtbaar. De totale energie-i.nhoud van de 50 kv batterijen is 1 MJ. De torus heeft een grote straal R = 0,6 m en een kleine straal b = 0,2 m.

Fig. 3. De kwarts torus van het beltpinchexperiment SP IVb. De hoogte van de torus is 0,45 m.

In de eerste publikatie wordt een methode beschreven voor meting van het iijdafhankelijke elektronendichtheidsprofiel in eer» schroefpinch. De metingen zijn gedaan in de kleine schroefpinchopstelling SP II, die begin 1974 werd ontmanteld. De invloed, die de krachtvrije stromen hebben op de evenwichtseigenschappen van de pinch, wordt aangetoond in de tweede publikatie. Dank zij de aanwezigheid van de krachtvrije stromen kan (5 bijna een orde van grootte hoger zijn dan zonder deze stromen het geval zou zijn. Voor hogere waarden van (5 (^ 20%) blijkt het plasma een D-vormige doorsnede aan te nemen. Ook in de beltpinchgeometrie blijken de krachtvrije stromen een zeer gunstige invloed te hebben op de evenwichtseigenschappen. Weliswaar heeft een sterk elliptisch plasma, omgeven door vacuüm, ook slechts een geringe evenwichtsverplaatsing ', maar vooral bij hoge compressie van het centrale plasma heeft dit plasma, wanneer het wordt omringd door een vacuüm buitengebied, de neiging zich in axiale richting samen te trekken, waardoor de gunstige evenwichtseigenschappen verloren gaan. Zoals in de tweede publikatie wordt aangetoond, treedt axiale contractie veel minder op als krachtvrije stromen in het buitengebied geïnduceerd kunnen worden. Theoretisch is zowel onder de Kruskal-Shafranov-limiet (q > 1) als erboven (q < 1) stabiliteit mogelijk . i n het laatste geval moet het q-profiel vrijwel vlak verlopen tot aan de wand. In de derde publikatie wordt aangetoond dat onder invloed van de eindige resistiviteit van het ijle plasma de stromen dicht bij de wand het eerst verdwijnen. Daarmee verdwijnen tevens de stabiliteitsgebieden boven de Kruskal-Shafranov-limiet. De vierde publikatie behandelt de verhitting van het plasma door de irreversibele implosie. Deze studie is verricht om betrouwbare voorspellingen te kunnen doen over de te verwachten temperatuur in grotere schroefpinchexperimenten of in de schroefpinchreactor. De vijfde publikatie is de meest recente conferentiebijdrage over experimentele resultaten in SPICA. Een beschrijving van SPICA is te vinden in Ref. 6. Vroegere verslaggeving van de resultaten in SPICA is gedaan in de referenties 7, 8 en 9. Omdat SPICA een grote en ingewikkelde opstelling is, wordt er in teamverband aan gewerkt. De bijdrage van de auteur is in bedoelde publikatie moeilijk aan te geven, omdat veel dingen in c .derling overleg en in samenwerking zijn gedaan. De speciale interesse voor en bemoeienis met de dataverwerking komen tot uiting in de uitwerking en presentatie van de meetresultaten. In de laatste publikatie worden de vooruitzichten van een schroefpincbreactor aan de orde gesteld. Een aantal parameters van deze reactor wordt beperkt door technische mogelijkheden. De overige 10

parameters worden geoptimaliseerd. Hierbij is uitgegaan van plasmaverhitting door een snelle implosie, gevolgd door adiabatische compressie. De publikatie geeft een voorlopige rapportering van de werkzaamheden van een reactorgroep waaraan, behalve door het Instituut voor Plasmafysica, wordt deelgenomen door het ECN (voorheen RCN) en de KEMA. Eerdere berekeningen aan een schroefpinchreactor zijn gerapporteerd in Ref. 10. De auteur heeft veel bijgedragen aan de fysica, die gebruikt is bij deze berekeningen en hij heeft een groot deel van het numerieke werk gedaan. Uit de resultaten blijkt dat het rendement van de schroefpinchreactor zeer nadelig wordt beïnvloed door ohmse verliezen in de spoelen en in het systeem voor energie-opslag en -overdracht. Verbetering is mogelijk door een hogere maximale waarde van g te kiezen {hetgeen misschien mogelijk is in de beltpinchgeometrie), door het kiezen van andere verhittingsmethoden en door het toepassen van de 'sustained field'-methode, waarbij het met behulp van supergeleidende spoelen opgewekte magnetische veld gedurende korte tijd wordt opgeheven door het aanleggen van een tegengesteld gericht veld met behulp van normale spoelen. De 'sustained field'-methode stuit echter op zeer grote technische problemen. Referenties 1. P.C.T. van der Laan, Proefschrift Rijksuniversiteit Utrecht (1964); ook Rijnhuizen Report RR 64-16. 2. D.A. D'Ippolito, J.P. Freidberg, J.P. Goedbloed en J. Rem, Proc. 6th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res., Berchtesgaden (1976). 3. C. Bobeldijk, Proefschrift Rijksuniversiteit Utrecht (1968); ook Rijnhuizen Report RR 68-45. 4. J.A. Hoekzema en W. Schuurman, MHD-equilibrium of a belt-shaped plasma in a Trkal field, Rijnhuizen Report RR 73-78 (1973). 5. R.P. de Vries, Proefschrift Rijksuniversiteit Utrecht (1969); ook Rijnhuizen Report RR 69-52. 6. R.J.J. van Heijningen, P.C.T. van der Laan, D.J. Maris, B.J.H. Meddens, A.A.M. Oomens en A.B. Sterk, Proc. 8th Symp. on Fusion Technology, Noordwijkerhout (1974) 341. 7. C. Bobeldijk, R.J.J. van Heijningen, J.A. Hoekzema, W. Kooijman, P.C.T. van der Laan, D.J. Maris, D. Oepts en A.A.M. Oomens, Proc. 5th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res., Tokyo (1974) Vol. Ill, 409. 8. R.J.J. van Heijningen, C. Bobeldijk, J.A. Hoekzema, P.C.T. van der Laan, D. Oepts, J.W.A. Zwart, W. Kooijman en D.J. Maris, Proc. 7th Eur. Conf. on Contr. Fusion and Plasma Phys., Lausanne (1975) Vol. I, 38. 11

C. Bobeldijk, R.J.J. van Heijningen, J.A. Hoekzema, P.C.T. van der Laan, D. Oepts, A.A.M. Oomens, J.W.A. Zwart, P.J. Busch, D.J. Maris en W. Kooijman, Proc. 3rd Topical Conf. on Pulsed High-Beta Plasmas, Culham (1975) , wordt gepubliceerd in Plasma Physics. 10. W. Schuurman, J.A. Hoekzema, P.C.T. van der Laan en C. Bobeldijk, Proc. 3rd Topical Conf. on Pulsed High-Beta Plasmas, Culham (1975), wordt gepubliceerd in Plasma Physics.

12

S U M M A R Y

This thesis is a collection of publications on the toroidal screw pinch. In a screw-pinch experiment the plasma is pinched and confined by a helical magnetic field, whose toroidal and poloidal component are applied simultaneously. In the region outside the main plasma column a plasma of low density remains behind which can sustain large currents. Because the pressure of this plasma is negligible, these currents are force-free. The force-free currents have a very favouraole influence on both the stability and the equilibrium behaviour of the pinch. A tightly fitting metal shell often surrounds the discharge tube to minimize the flux which intersects the wall, thereby maximizing the region where force-free currents can be induced. An increasingly important part of screw-pinch research is dedicated to the screw pinch with elongated cross-section (belt pinch), because it exhibits better equilibrium characteristics than the normal screw pinch (circular cross-section). The presence of a metal shell around the torus has as a consequence that most screw-pinch experiments are not very accessible to diagnostics. In Part III a method is described to measure the electron density profile on a line through the plasma, using two small diagnostic ports. A highly compressed belt pinch, surrounded by vacuum, contracts in axial direction, thereby losing its favourable equilibrium characteristics. In Part IV it is shown that the axial contraction is much reduced i.f force-free currents are induced in the outside region. In circular geometry the plasma becomes P-shaped for high p-values. Stable equilibrium of screw pinches is theoretically possible, both below and above the Kruskal-Shafranov limit. It is pointed out in Part V that if the dilute plasma in the outside region has a finite resistivity, the currents in the region near the wall first disappear. The stability regions above the Kruskal-Shafranov limit then shrink in size. A pinch is mainly heated by the irreversible implosion. A simple model to calculate the heating of a pinch if a bias field is initially present in the plasma, is presented in Part VI. The latest experimental results of the screw-pinch experiment 13

SPICA are given in Part VII. Macroscopic stability is observed for about 100 ps at (3-values up to 0.2. Finally, the reactor prospects of the screw pinch are discussed in Part VIII. The plasma is heated by fast implosion followed by much slower adiabatic compression. The total efficiency of the reactor turns out to be rather low, due to ohmic losses in the coils and in the METS (magnetic energy transfer and storage) system. Improvement is possible by choosing a higher maximum 8-value (which may be possible in the belt geometry), by using other heating methods and by application of the sustained field scheme.

14

C U R R I C U L U M

V I T A E

De auteur van dit proefschrift werd geboren op 22 september 1947 te Arnhem. Hij behaalde in 1964 het diploma HBS-B aan het Nijmeegs Lyceum. In datzelfde jaar begon hij de studie in de wiskunde en natuurkunde aan de Rijksuniversiteit te Utrecht. Het kandidaatsexamen, met als bijvak sterrenkunde, werd in 1967 afgelegd. In 1971 volgde het doctoraal examen experimentele natuurkunde met als bijvakken wiskunde en theoretische natuurkunde. Sinds december 1970 is hij in dienst van de Stichting FOM en werkzaam op het FOM-Instituut voor Plasmafysica te Jutphaas. Aanvankelijk nam hij als wetenschappelijk assistent deel aan het onderzoek in de werkgroep die de wisselwerking tussen plasma's en hoogfrequente velden bestudeerde. Na het behalen van het doctoraal examen werd hij als wetenschappelijk medewerker geplaatst bij de werkgroep pinch-onderzoek, waar het in dit proefschrift samengevatte werk tot stand kwam.

15

Reprinted from PULSED HIGH BETA PLASMAS Edited by D. E. EVANS PERGAMON PRESS—OXFORD AND NEW YORK • 1976

D2.3

DECAY AND PROFILE OF THE TOROIDAL PLASMA CURRENT IN A SCREW PINCH J. A. HOEKZEMA Association Euratom-FOM, FOM-Instituut voor Plasmafysica, Rijnhuizen, Jutphaas, The Netherlands

Abstract In a screw pinch the magnetic field, applied at the wall, usually has a pitch that is approximately constant in time. If the low-density plasma outside the main plasma column has a high conductivity, the induced toroidal plasma current density in this region is almost homogeneous for practical q-values. Experimental results show, however, that in the region just in front of the wall this current, if induced at all, decays rapidly. The role of resistivity is demonstrated by comparison with numerical model calculations. If the current density decreases towards the outer wall, the stability and equilibrium properties of the pinch are affected. The stability regions above the Kruskal-Shafranov limit shrink in size. Below this limit the equilibrium shift is increased, but stable equilibrium remains possible for high 3 values. Introduction Outside the main plasma column, formed by implosion, a low-density plasma is observed. If this low-density plasma has a high conductivity, the rotational transform of a magnetic field line in a toroidal device is conserved. The magnetic field configuration is then determined by the time history at the wall. In a screw pinch the magnetic field at the wall is usually applied with approximately constant pitch. Since in the low-density plasma inertia and pressure gradients are negligible, the induced currents are force-free (jxB • 0 ) . Neglecting toroidal effects, the longitudinal current density for a constant pitch configuration is then given by: j, « (1 + r 2 /q 2 R2)" 2 , where q - ^ . z RB e The current-carrying low-density plasma has a very favourable influence on both stability and equilibrium of the pinch. Fairly large values of (3 are theoretically allowed at q-values of 0.7 and 1.4 [I]. For these q-values the current density is almost homogeneous (except for very low values of the aspect ratio, A=R/b). If the primary circuits are ideally crowbarred, only the resistivity, n, of the plasma causes the currents to decay. For uniform n and a current density that remains homogeneous, the decay time of the toroidal plasma current, I , is zp given by: T » u b2/4n .

541

542

J. A , H o e k z e m a

In the experiments the external circuits after crowbarring cannot be neglected [2],[3]. Numerical calculations that take the external circuits into account, but still assume that j_ remains homogeneous and that i| is uniform, give a reasonable agreement with the experimentally observed plasma current, I z , for an n-value of 5xlO~s Sim. If this effective n-value is a measure for the electron temperature in the outside region, this temperature is approximately 5 eV (Spitzer resistivity). For this high value of o the assumption that the plasma has a high conductivity is not justified on the time-scale of the experiments (10 - 100 ps). Therefore, the induced current distribution may even depend on the radial r;profile and will change as a function of time in the direction of a mir.jmuir. energy configuration. In order to get a better understanding of the processes that take place, a simple numerical code was developed. Numerical current profiles The basic equations governing the behaviour of the currents in the outside region are (inertia and pressure gradients are neglected): SB E+vxB-nj» POJ-VXB, £_ - - V x E, j x B = 0 . As an example of a numerical solution of these equations in cylindrical geometry the time-dependence of the toroidal current density profile is given in Fig. I. In this case it was assumed that the initially induced current is homogeneous, that n is uniform and constant in time, and that the external circuits are ideally crowbarred. The force-free fields extend from the axis to the wall.

r (m)

200

Fig. 1. Numerical study of the resistive decay of the initially homogeneous toroidal current density. The primary circuits are ideally crowbarred. The resistivity is assumed to be uniform and constant in time: n = 3*10~5 (3m. Inertia and pressure gradients are neglected throughout the tube. The total toroidal plasma current, I , and the poloidal magnetic flux, <|), as a function of time, are also given. The initial value of q at the wall is 1.6.

Since clearly the decay time of the current density depends on the radius, there is no simple relation between the decay of I z p and the resistivity. This is also apparent from the different time behaviour of the poloidal flux, <|>, vhich would decay in the same vay as I z p for a homogeneous current density. The fast decay of the current density next to the wall becomes even more enhanced

Decay and Profile of the Toroidal Plasma Current

543

if the resistivity increases towards tbe wall. In that case, in addition to current loss, a transfer of current takes place to the less resistive regions and a temporary increase of the current density there, is possible. Non-ideal crowbarring of the external circuits has complicated effects on the current density [2], The Poyoting flux nay be directed outwards. In that case a thin layer next to the wall is assumed to have infinite resistivity, so that the plasma velocity in this layer reduces to zero, preventing the plasma froa crossing the wall. In the experiment losses in the external z-circuit often have a dominating influence. If that is the case, the current at the wall reverses and a region of reversed current slowly penetrates the tube until I_p becomes negative, while j z on the axis may still be quite large. It should be noted that the time-scale for the diffusion process scales quadratically with the minor radius of the tube. For n-values in the IO~5 range this diffusion is therefore, in the small screwpinch experiments, already important during the formation phase of the pinch. This phase is only described correctly if the B-value of the plasma is very low. More elaborate calculations with the ATHENE codes [4] also give ultimately a peaked current distribution, but the tune-scale is much longer, because the resistivity in the low-density plasma quickly decreases because of ohoic heating. The possible influence of neutrals in the outside region is, however, as yet not included in this code. Moreover, the effective 7) may also be determined by turbulent processes or instabilities in the outside region or by toroidal effects (the horizontal motion of tbe main plasma column, as it oscillates around its equilibrium position, may push field lines through the quartz wall). Experimental results If toroidal effects are not very important, experimental current-density profiles can be determined from magnetic probe measurements. An example of experimental results is given in Fig. 2 for a lcw-g discharge in the small SP II device (R/b » 0.16/0.04). It is seen that in the region near the wall the currents are induced, but decay on a rather short timescale. In this experiment both external circuits are provided with power crowbar systems that keep I and the e-.-ternal 6-current constant for aibut 25 Vs. from the probe measurements it is seen that not only I , but also the j -profile remains constant after powercrowbarring. Comparing these j -profiles with numerical results, reasonable agreement was found for rj « 7 + 60(r/b) 3 uftm. In normal 0.02 0.04 high-g discharges it is observed that the cur-r<m) rents near the wall decay on an even faster time-scale and sometimes are hardly induced at all. This is also true for the large SPICA-ex- Fig. 2. Experimental profiles periment. Possibly, the remaining density in of the toroidal current density the outside region, after the snow-plough has at different times. In this case passed, is very low and anomalous resistivity 8 is very low and toroidal efoccurs [5]. Future experiments in SPICA will fects are negligible. The mean establish whether a uniform current density q-value at the wall is 1.4. At can be induced and how it decays as a function t « 0 a bias field, I0Z of the of radius. Meanwhile, it is of interest to in- maximum field, is present.After vestigate the influence on stability and equi- t « 10 ys the current profile librium of a peaked current distribution. hardly changes for about 20 us.

544

J. A . H c e k i e m a

Stability and equilibrium In order to get an iapression of bow important the currents Bear the wall are for stability, a noraal mode, marginal stability analysis for Che dangerous «•I aode was done for two different profiles of the toroidal current density. The configuration is assiaed to be cylindrical, but toroidal effects are introduced by a periodicity condition. The result is given in Fig. 3. Fro» these results it seeat likely that operation below the Krufkal-Shafranov liait is necessary to obtain stable high-B screw pinches. This region is not affected by the current profile. It should be noted, however, that the value of q at the wall is higher in the configuration with parabolic current profile. In the experinents a sure stable behaviour was indeed found when operating below the KruskalShafranov limit. For a sore definite judgement, the growth rates of the instability have to be calculated. In the q > 1 region the • » " ' • " value of B to be allowed, is restricted by the equilibrium properties of the plasaa. Calculations in first order, that assume a plasaa with homogeneous pressure, surrounded by a constant-pitch region separated from the wall by a vacuum region, show little harmful influence on the equilibrium by this vacuum region [6]. If the current density gradually decreases towards the wall, the influence is sonewhat larger, as is seen from Fig. 4 (see [7] for a description of these calculations). High 6 values (20Z) remain, however, possible. I.U y<

unstable

0.5 \

.

O 1.0

1

V"

\.

0.5

/ \.y

\

unstabf*

f \n.2 >

^ m «1

\ \n.1

n

OS

1

1.5 1
Fig. 3. Results of a marginal stability analysis for the m-I mode for two different p r o files of the toroidal plasma current. The pressure profile is gaussian. Toroidal effects are introduced by a periodicity condition. 8 is defined by

B-

p(o) B|(o) p(o)+

Decay and Profile of the Toroidal Plasma Current

545

1.0

£ ID

O5

1

r~

Fig. 4. Equilibrium configurations in the SPICA geometry calculated as in Ref. [7]. a. The magnetic field in the outside region has approximately constant pitch: y j/B « 1.8 m~ l . i|i = 8.053 Voltsec. w b. The current density decreases towards the wall: v j/B > l,8x

(i-M|i V Voltsec.

1

; I|J = 0.04

Acknowledgement All members of the Pinch Group collaborated in the acquisition of the experimental material underlying this study. Especially C. Bobeldijk, F.C.T. van der Laan, and J.H.A. Zwart contributed by many discussions. This work was performed under the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek" (ZWO) and Euratom. References [1] [2] [3] [4] [5] [6] [7]

Schuurman W, Bobeldijk C, and Vries R.F. de, Plasma Physics l± 495 (1969). Laan P.C.T. van der et al., IAEA Proc. 4th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Research Madison (1971) J_, 217. Oomens A A M and Meddens B J H, Proc. Sec. Topical Conf. Pulsed High Beta Plasmas, Garching (1972) IPP 1/127,201. Lister G G, this conference, paper B.2.9. Bobeldijk C et al., this conference, paper D.I.2. Bobeldijk C et al., IAEA Proc. 5th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Research, Tokyo (1974) Ë.9.1. Hoekzema J A, this conference, paper D.2.3a.

Reprinted from PULSED HIGH BETA PLASMAS Edited by D. E. EVANS PERGAMON PRESS—OXFORD AND NEW YORK • 1976

D2.3

TOROIDAL EQUILIBRIUM OF NON-CIRCULAR SHARP BOUNDARY PLASMAS SURROUNDED BY FORCE-FREE FIELDS J. A. HOEKZEMA Association Euralom-FOM, FOMInstituul voor Plasmafysica, Rijnhuizen, Jutphaas, The Netherlands

Abstract

The HHD equilibrium of an axisymmetric free-boundary plasma with homogeneous pressure and an .iternal bias field is studied numerically. The plasma is surrounded jy force-free fields such as commonly produced during the formation of a pinïh. Results are presented for configurations enclosed by a metal shell with either circular or elongated cross-section. In the circular geometry a pronounced elongation of the plasma is found even for moderate equilibrium displacements. In the belt geometry, equilibria with highly elongated plasma cross-section are possible, also at high compression ratios. In experimental studies of pinch discharges it has been observed that in the region outside the main plasma column a low-density plasma is present which may support strong currents [1]. Because the pressure of this plasma is very low, the pressure gradient may be neglected and the magnetic field is approximately force-free: j*B • 0. If the conductivity of the plasma is high, the magnetic field configuration only depends on the time history at the wall. In screw-pinch discharges the magnetic field is often programmed to have approximately constant pitch outside the main plasma column. The presence of force-free currents there may have a favourable effect on both stability and equilibrium [2],[3]. Equilibrium calculations,that take force-free fields into account,have been done earlier by assuming that the magnetic surfaces have circular cross-sections and by expanding in powers of the inverse aspect ratio [2], [4]. The 'development of a numerical equilibrium code was motivated by the inadequacy of these low-order calculations for high-0 plasmas (in SFICA [5] the actual equilibrium displacement was surprisingly small), but also by a growing interest in pinch experiments with non-circular cross-section. In this numerical study it is assumed that the magnetic field in the outside region is characterized by constant a (V*B « ciB). Constant pitch configurations can often be approximated if a is chosen properly. For a • 0 the plasma is surrounded by a vacuum region. In the force-free field region the equation for the poloidal flux (in cylindrical coordinates r,<j>,z) reads:

0

+ f^f - i | | • «<«* + C) - 0 , where » - ƒ B.fcir'dr'

and C is a constant of integration that is determined by the boundary conditions. This equation can be solved numerically (in discrete points) by iteration on a rectangular grid. For a ^-dependent a [6] the poloidal flux equation includes terms of higher order in t|i and iteration takes more computing time. In535

536

J. A . H o e k z e m a

side the plasma a homogeneous kinetic pressure, p, is assumed; the bias field in the toroidal direction has a vacuum configuration: B*£ « b/r. At the interface between the dense plasma column and the force-free field region pressure balance must held: 2

Bf_

b

2u * p 2u r2 • o o The numerical method to find an equilibrium configuration consists of successive iteration of the poloidal flux equation for the outside region and of the position of part of the plasma surface until pressure balance holds. The calculation is ended when there is pressure balance simultaneously in 90 evenly spaced points on the plasma surface. The force-free field region extends to a surrounding metal wall. Results are presented for two different cross-sections of this metal wall. The reason for choosing these particular cross-sections (see Figs. 1, 2, and 3) stems from the experimental programme of the institute, where a screw pinch with circular cross-section (SFICA) is in operation and a beltpinch (SP IVb [7]) is under construction. Some equilibrium configurations in the SPICA geometry are presented in Fig. I. For stability reasons the poloidal flux, IJI , was chosen such that the value of the safety factor is nowhere below unity. Except for the first example, where the plasma is surrounded by vacuum, the value of a was adapted to approximate a constant pitch configuration. Comparison of Che first and second example illustrates the importance of force-free currents for the equilibrium. Especially at higher values of ($ the plasma shows a pronounced flattening against the poloidal magnetic field buffer (D-shape). In the elongated configuration, the equilibrium displacement is small, even if the plasma is surrounded by vacuum. This is illustrated in Fig. 2. The ellipticity of the plasma, which is defined as the ratio between its largest dimensions in z- and r-direction, is only large if the compression ratio is not very high. A 6 = 0.2 plasma, surrounded by vacuum, exhibits an ellipticity higher than 4 as long as the compression ratio is below 10. If the plasma is surrounded by force-free fields with approximately constant pitch, the ellipticity is much higher, especially for large compression ratios. An example is given in Fig. 3. For very large compression ratios the ellipticity goes to infinity, while in a vacuum configuration the plasma cross-section becomes circular. This property may be especially important for a reactor, where the plasma after a fast implosion is adiabatically compressed to large compression ratios [8]. Conclusions 1. Both in the circular and in the elongated geometry constant pitch can be approximated with an accuracy of about \0X. 2. The plasma cross-section in the circular geometry becomes D-shaped for high values of 6. 3. Because of this effect the equilibrium displacement at higher values of 8 is much less than would follow from the first-order calculations. 4. In a beltpinch, equilibrium configurations with a plasma cross-section that has a higher ellipticity than the cross-section of the surrounding metal wall exist, even if the plasma is surrounded by vacuum. 5. Especially for large compression ratios, the ellipticity of the plasma in a beltpinch surrounded by constant pitch magnetic fields is much higher than that of a plasma surrounded by vacuum. 6. The cross-section of a highly elongated plasma resembles more a racetrack than an ellipse. This is a favourable property for stability. In both geometries the force-free currents therefore have a favourable influence on the equilibrium characteristics.

537

Toroidal Equilibrium 6-5°/.

1O

oe 0.6 04 02 0 -0.2 t 0

'•«•0 055

08

06 04 02 0 -02

©

1O OB

06 04 02 O -02

o.18

E"* • „.006

P = 40V.

®-

10 08 06 04

r

02 0

04

05

06

O?

OB

r (m) Fig, I. Examples of equilibrium configurations in the SPICA «eomeLry. In the figures on the left i|)-constant-lines are drawn. The outer flux surface (ijj - i|i > represents the surrounding metal wall. The plasma region (ifi « 0, homogeneous pressure) is shaded. In the figures on the right the toroidal and poloidal magnetic field components in the horizontal plane (z = 0 ) , Bi and B , are plotted as a function of r. Also plotted is the safety factor: q = f ?Z n » where the integration path is the cross-section of a magnetic surface. The figures illustrate the influence of the forcefree currents (Fig. a-b) and of (3 (Fig. b - d ) . 3 is defined here as the ratio between the plasma kinetic pressure and the external magnetic field pressure at the innermost point of the plasma surface. The mean value of g is therefore somewhat higher.

M F":

f:1

538

J. A. Hoekzema

Acknowledgements

'H

U

s

The author thanks C. Bobeldijk and P.C.T. van der Laan for useful discussions. This work was perforu.?d as part of the research programme of the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek" (ZWO) and Euratom.

0.25

0.20 -

0.15 -

0.10 -

rei

0.05 -

CD 9-

Fig. 2. Example of an equilibrium configuration in a beltpinch. The plasma is surrounded by vacuum.

Fig. 3. Example of an equilibrium configuration where the plasma is surrounded by force-free fields with approximately constant pitch.

References [1]

Bobeldijk C et al., Froc. 3rd Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Research, Novosibirsk, j_ 287 (1968).

[2]

Bobeldijk C, Rijnhuizen Report 68-45 (1968).

[3]

Goedbloed J P, Rijnhuizen Report 70-64 (1970).

;,!;

;

Toroidal Equilibrium

i"j

'"

[4]

(••;

f

[5]

Schuurman W, Rijnhuizen Report 72-72 (1972). Bobeldijk C e t a l . , t h i s conference, paper D1.2.

],.j

j

[6]

Hoekzema J A, t h i s conference, paper D2.3-b.

'f~:

>

[7]

Rijnhuizen Annual Report 1974.

p:-'_

I

[8]

Schuurman W e t a l . , t h i s conference, paper D 3 . 3 .

wIL:

539

FOM-Instituut voor Plasmafysica

I.R. 76/023 June 1976

Rijnhuizen

PREPRINT

Jutphaas

PARAMETER STUDY OF A SCREW-PINCH REACTOR

by C. Bobeldijk, M. Bustraan, G.C. Damstra, W.M.P. Franken, J.A. Hoekzema, H.J. Klein Nibbelink, H.Th. Klippel, P.C.T. van der Laan, M. Muysken, W. Schuurman; and K.A. Verschuur

This paper was presented at the 9th Symposium on Fusion Technology in Garmisch Partenkirchen, June 14-18, 1976.

FOM-Instituut voor Plasmafysica Rijnhuizen Jutphaas

I.R. 76/023 June 1976

PARAMETER STUDY OF A SCREW-PINCH REACTOR by C. Bobeldijka, M. Bustraanb, G.C. Damstrac, W.M.P. Frankenb, J.A. Hoekzemaa, H.J. Klein Nibbelinkc, H.Th. Klippelb, P.C.T. van der Laand, M. Muyskenb, W. Schuurmana, and K.A. Verschuur13 a. Association Euratom-FOM, FOM-Institute for Plasma Physics, Rijnhuizen, Jutphaas . b. Reactor Centre Netherlands, Petten c. KEMA, Arnhem d. FOM, now Los Alamos Scientific Laboratory, Los Alamos, USA.

Abstract

In the framework of system studies on pulsed high-g fusion reactors a parameter study of a reactor based on a screw-pinch configuration has been performed. The plasma is heated in two stages. In order to guarantee pitch conservation of the inward moving magnetic field lines, the cold plasma is heated by fast implosion, the theory of which has been generalized to a g < 1 plasma. After implosion an adiabatic compression heats the plasma to the ignition temperature, where a-particle heating takes over. For reasons of stability (3 is kept below 0.25. Further constraints imposed on the calculations are a wall loading of 2 MW/m2, a maximum poloidal electric field at the first wall of 3x105 V/m, a blanket thickness of 0.8 m, and a down time of 8 s between two burn phases to allow for pumping and refuelling. A computer programme searched by iteration on the filling density and the bias field, for optimal reactors, i.e. for which at a given thermal output the net efficiency is a maximum. Numerical examples are given. Some conclusions are: 1. The net efficiency, although increasing with output energy, is low because of ohmic losses in the compression coil. 2. The application of sustained fields generated by superconducting coils to reduce these ohmic losses, is problematical.

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I.R. 76/023

3. A belt-shaped screw pinch, in which higher values of f? may be reached, improves the net efficiency and alleviates the technological requirements. 4. Heating by adiabatic compression of a 3 < 1 plasma is inefficient. Therefore, other means of heating the plasma to ignition may be attractive. 1. Introduction The work presented here is the first result of a system study on pulsed hi'gh-3 reactors that began early in 1975. Stimulating examples of reactor designs like the RTPR-design at Los Alamos, KRAKOWSKI [ lj, and the Reversed-Field Pinch Reactor design at' Culham, BODIN [ 2] , were already at hand. At Jutphaas, a third pulsed high-g configuration, the screw pinch SPICA, showed good experimental results and theoretical expectations, BOBELDIJK [ 3] . As the extrapolation to the reactor regime was lagging behind the other devices, an investigation in that direction seemed appropriate. As a first step to a Conceptual Screw-Pinch Reactor, a parameter study was performed. As heating method fast implosion followed by adiabatic compression was chosen like in the RTPR. This choice was made because it can guarantee the pitch conservation of the magnetic field lines at low plasma temperature. The considerations were mainly devoted to a torus with circular cross-section. Some attention was given to a belt-shaped vessel (allowing larger maximum (3), and to the sustained-field concept using superconducting coils. Section 2 describes the physical and technological model, while Section 3 shows the way in which the reactor parameters were calculated. Sections 4 and 5 give the results and their discussion respectively. A more detailed version of this parameter study will be published. 2. The model The aspect ratio of the torus was given the small value of 3 (this and following numbers refer to the reference reactor), because for a well-centered screw-pinch plasma f5A should not exceed 0.75. .The time history of some physical quantities during

-3-

I.R. 76/023

one operation cycle is shown in Fig. 1. In a preionized 50% D-T plasma a bias field B is generated by the adiabatic compression field plus a counterfield from the implosion circuit. A fast rising magnetic field (^ 105 Ts"1) is applied which makes the plasma implode. The implosion process has been described by a free-particle model modified to take account of the bias field. The maximum electric field strength at the first wall, E , is an important constraint (3 x 105 V/m). After the implosion thermalization of ions and electrons takes place. An adiabatic compression field (* 50 Ts"1) then heats the plasma up to the ignition temperature. The a-particles further heat the plasma raising 3 at the same time. At maximum 3 burn with control of 3 and T is achieved by injection of cold gas-and programmed B-field. After the burn a down time of 8 s allows for ash removal, inlet of new fuel, etc. The implosion coil, fed by a capacitor bank, is situated between the blanket, thickness 0.8 m, and the compression coil at sufficient distance from the latter to avoid a largeinductive coupling. For the periodic supply of the adiabatic compression energy an inertial storage and transfer system (METS) seems appropriate, with an assumed efficiency of 98% (RTPR-value). The compression coil, thickness 2.3 m, is partially tapered as a compromise between reduction of ohmic losses and accessibility. The copper coils have a filling factor of 0.7 and a resistivity of 2 x 10~ e üm (65 C ) . The thermal conversion efficiency n., is assumed to be 40%. Appropriate changes in the model have been made for a belt-shaped screw pinch and for the case of a sustained field generated by superconducting coils. In the latter case shaping of the field for implosion and adiabatic compression is done with normally conducting coils. In all computations various loss processes have been considered. 3. Calculation of reactor parameters a. Implosion. In order to describe the implosion phase one has to solve the equations for the plasma circuit, the implosion circuit, and the compression circuit. The total magnetic flux enclosed by the compression coil is assumed constant. Furthermore, the equations of motion of the plasma sheath should be solved to determine the maximum electric field strength at the first wall. We used a modified free-particle model after having

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I.R. 76/023

verified that more sophisticated models give approximately the same results. After the implosion electrons and ions equilibrate, and the final position and temperature of the plasma are determined by the energy balance and the pressure balance. The energy of the implosion bank is considered as being lost. Eddy-current losses in the blanket have been neglected. b. Adiabatic compression. The parameter set corresponding to the reference reactor (to be given in Section 4) has been calculated with time-dependent equations. They comprise the particle balance for fuel ions, a-particles, impurity ions, and electrons, the pressure balance, and the energy balance. In the latter various losses, such as bremsstrahlung, cyclotron radiation, line and recombinative radiation, have been taken into account. c. Burn phase. In the burn phase we distinguish two parts. In the first part the temperature is increasing because the energy production by a-particles exceeds the radiation losses. The plasma (3, that stayed below 0.25 during the preceding stages, now reaches this value and the second part of the burn phase begins. By injection of impurities further increase of f3 is prevented. Quenching of the plasma is avoided by field control,as is shown in Fig. 1. d. Computer programme and constraints. The mathematical relations describing the phases a to c have been translated into a computer programme. The physical and technological constraints entering as input parameters are listed in the left-hand column of the table. In the numerical calculations some checking, iteration (on n and B ) , and optimization procedures have been built in. The computer programme has been extended in order to treat also belt pinches in rectangular reactor vessels. Moreover, for both geometries the parameters of a sustained-field reactor can be computed. In this case major changes occur in the circuit and energy equations. Losses in the superconducting coil have been neglected.

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4. Results The computer programme has been run with a set of input parameters including the constraints mentioned in Section 2. Together with the output parameters they build up what can be called a Reference Screw Pinch Reactor (RSPR). Input and output parameters of the RSPR are listed in the table. The influence of varying input parameters has been examined in such a way that in all computer runs only one parameter deviated from its reference value. The effects of 3 m a x , A, n M , P w , B ^ ^ , and Pfch on W , W / B , and n . are shown in Fig. 2. Some other effects s a a net are just touched upon in the discussion. 5. Discussion The RSPR shows a relatively low net efficiency.This is predominantly caused by the ohmic losses in the compression coil during the long burn period (even then a thick, partly tapered coil is imperative). To reach ignition, some overcompression is required and this badly influences the ohmic losses. The parameter to which these losses are most sensitive is the maximum 3 of the plasma. Figure 2a clearly shows the profits if 3 =50%, possibly attainable in a belt screw pinch (it should be noted that a calculation for a rectangular cross-section slightly changes the parameters in an unfavourable sense)• Some calculations were done for the sustained-field method. Here also, large ohmic losses are avoided and the net efficiency becomes 34%. However, this is achieved at the cost of a large increase of adiabatic compression energy and a more complex structure of the reactor. The maximum in the curve of n vs A together with the physics limitation of f3A < 0.7 5 and the technical limitations demonstrate the correctness of the choice A = 3. Figure 2c clearly shows the necessity of a very high n„. From Fig. 2d it can be seen that, apart from the influence on the life-time of the first wall, variation of P leads either to a large B-field or a high implosion energy. A further increase of Ê , above 3xlO5 V/m (Fig. 2e), although improving n t» does not seem attractive because of the steep increase of W . The net efficiency increases with the thermal power of the reactor (Fig. 2f). However, above 6 GW the relative gain becomes small, while the size of the reactor with its

-6-

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large volume of electrical equipment becomes unpractical. If ntj1 could be enhanced (topping cycle) the total efficiency of the reactor would be enhanced by the same absolute amount. A larger thickness of the compression coil would help in reducing the ohmic losses but the value 2.3 m is already formidable (45,000 tons). The coil should be tapered 50% along the minor radius as a compromise between accessibility and low ohmic losses. General conclusion. In view of the ohmic losses the net efficiency of a screw-pinch reactor with circular cross-section is too low to be of economical interest. A belt-shaped reactor vessel will give considerable improvement (provided 3 can be raised to 50%). A sustained-field reactor, also acting against ohmic losses will probably meet with serious difficulties in its construction. Fast implosion and adiabatic compression at 3 < 1 is inefficient and other heating methods may well be worth-while considering. References [1] KRAKOWSKI, R.A. et al., "A Technological Assessment of the Reference Theta-Pinch Reactor (RTPR)", Proc. 8th Symposium . on Fusion Technology, Noordwijkerhout, The Netherlands (1974) p. 623. [2] BODIN, H.A.B, et al., "Further Considerations of the Toroidal Diffuse Pich Reactor", Workshop on Fusion Reactor Design Problems, Culham, U.K., 1974. [3] BOBELDIJK, C. et al., "Experimental Results of the ScrewPinch Experiment SPICA", Proc. 5th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res., Tokyo (1974) Vol. Ill, p. 409.

TABLE Parameters of Reference Screw Pinch Reactor Input parameters

Output parameters A = 3

aspect ratio maximum $

3max P

thermal output power

th

.small radius first wall

0.25

large radius torus

= 6 GW

neutron wall loading E = 3X10 win 0.98

maximum E-field at wall METS-efficiency thermal conversion efficiency down time

0.4 t c

8 s

blanket thickness nonconducting part of blanket energy multiplication factor blanket

%1 2m

thickness compression coil distance blanket-compression coil specific resistance compression coil filling factor compression coil

5

V/m

4.6 m

b =

R = 13.8 m

813 MJ

implosion energy

\ij

adiabatic compression energy

W = = 232

=

burn time bias field

= 0.27 T

I B after implosion B after compression

0.8 m

I T after implosion

0.1

J T during burn

1.2

; degree of ion impurities (C)

2.3 m

I minimum plasma radius

0.3 m

. filling density

p

= 2xl0~8 ftm ! burn-up

f

= 0.7

net efficiency

Q

GJ

~

Q

g

0.63 T = 7.1

T

1 keV 10 keV 0.05 0.92 m 5.3X1019 m~3 20% "net"

18

'5%

O

O

f™ l<4 • I Time history of the magnetic field strength, the plasma temperature, the fusion power and the injection of a heavy gas.

arbitrary units coiïigressiqn ohase

temperature of plasma burn

phase

B constant at 2 5 % I I

magnetic field strength

CO

fusion energy j

source impurities

implosion phase 2 microseconds

time

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(MJ) (GJ) 2000_ _500

. 400

0

300

.. 0,30

200

.. 0,20

100

.. 0,10

-2

0,10

FIG: 2a

_

0,20

0,30

0,40

0,50

\S max

Dependence of the implosion energy, the compression energy, the maximum magnetic field strength and the net efficiency on the maximum 3-value. The other input parameters are the same as in the RSPR.

-10-

w«

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B a ,max (Tes!a)

'/„et

(WIJ) (GJ) 2000 _ _500

10. .0,50

400

. 0,40

300

. 0.30

1000__

VV< 200

.. 0,20

0,10

100

o o

o. o 8

1

FIG'. 2 b

10

aspect ratio A

Dependence of the implosion energy, the compression energy, the maximum magnetic field strength and the net efficiency on the aspect ratio. The other input parameters are the same as in the RSPR.

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B a ,max (TeslaJ '/net (WIJ) (GJ)

10

500

2000

0,50

400

0,40

300

0,30

1000. net 200

0,20

100

0,10

o o

0

0,90 FIG:

2c

0,92

0,94

'METS efficiency

0,96

0-98

1,00

METS

Dependence of the implosion energy, the compression energy, the maximum magnetic field strength and the net efficiency on the METS efficiency. The other input parameters are the same as in the RSPR.

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Ws

Wa

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B a ,max

(WIJ) (GJ) 2000_

I.R.

'net

(Tesla)

500

10.

.0,50

I .. 400

.0,40

.. 300

. 0,30

200

0,20

1000.

0,10

.. 100

O

0

F!G : 2 d

0

wall

loading

0

Pw (MW/m 2 )

Dependence of the xmplosion energy, the compression energy, the maximum magnetic field strength and the net efficiency un the wall loading. The other input parameters are the same as in the RSPR.

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Ba-max (MJ) (GJ)

76/023

'/net

(Tesla)

2000_ __500

_0,50

.. 400

.. 0,40

300

.. 0,30

200

0,20

1000..

.. 0,10

.. 100

0

0

0 0,1

02

0,3

0,4

0,5

(MV/ m ) A

F I G ; 2 e maximum field strength at first wall Ew

Dependence of the implosion energy, the compression energy, the maximum magnetic field strength and the net effiency on the maximum electric field strength at the first wall. The other input parameters are the same as in the RSPR.

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ws

Wa

Ba .max

(MJ) (GJ) 2000

76/023

(Tesla)

_500

10J_0,50

.. 400

. - 0,40

. 300

- 0,30

200

.. 0,20

100

• 0,10

1000.

6

FIG: 2 f

thermal power

9 th

Dependence of the implosion energy, the compression energy, the maximum magnetic field strength and the net efficiency on the thermal power. The other input parameters are the same as in the RSPR.

FOM-Instituut voor Plasmafysica Rijnhuizen Jutphaas

X R

' ' 76/°32 Au< ust 1976 ? PREPRINT - to be pviblished in Proc. 6th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res., Berchtesgaden (1976) IAEA-CN-35/E6.

'*

CURRENT DECAY AND STABILITY IN SPICA by C. Bobeldijk, J.A. Hoekzema, M. Mimura*, .D. Oepts,and A.A.M. Ooroens Association Euratom-FOM FOM-Instituut voor Plasmafysica Rijnhuizen, Jutphaas, The Netherlands *Guest from I.P.P. Nagoya University, Japan

ABSTRACT The equilibrium and stability of the toroidal screw pinch has been studied in SPICA at q-values around 1.5. This regime is of interest because theory predicts stability there for rather high values of 3• The experimental results show that plasmas with peak S-values of 0.2 (Te = 60 eV, n e = 5<10 21 /m ; ) can be contained in stable equilibrium during 100 ys. Lew-current decay rates indicate a well-developed force-free field profile in a sufficiently hot dilute plasma region outside the main column. In unstable discharges m-2 modes appear. These discharges are characterized by faster decay and larger eccentricity of the plasma column. 1. INTRODUCTION A series of scre;;-pinch experiments in Jutphaas has revealed good equilibrium and stability properties of pinch configurations in which the central high-density column is surrounded by a constant-pitch magnetic field region [ 1 ] . The buildup of such a configuration is made possible by a dilute well-conducting plasma which is left behind during the rapid implosion of the pinch. The production of this particular configuration asks for a simultaneous rise of and a constant ratio between the toroidal plasma current and the primary poloida3 current. Furthermore, the configuration must be closely surrounded by a highly conducting shell to prevent rr.arrnetic field lines frcr' crcssir.'/ the wall. The configuratior. set up in this way is, fron' the very beginning, a minimum energy configuration in a flux-conserving- geometry [2]. In this paper experimental results of the largest screw-pinch experiment SPICA are presented. MHD high-(3 theory predicts good stability and equilibrium for constant-pitch screw pinches. Some results of this theory.

I.R. 76/032

-2applied to SPICA, are briefly described in Section 2. The experimental results shov; that equilibrium at high values of g (up to 20%) can be obtained. The outcome of an experimental study concerning the mode structure of stable and unstable discharges and correlation with the decay rate of the plasma current is given in Section 3. Although the experiment appears to confirm many of the theoretical predictions, insufficient knowledge of the limits of the theory and of the discharge properties as a function of time could lead to incorrect conclusions. This is briefly discussed in Section 4. 2. THEORY Recent theoretical work [3] on the stability of toroidal screw pinches and high-g tokamaks, based on a high-g ordering scheme, represents an improvement over the earlier MHD-stability calculations [4],[5] , which predicted the stability behaviour of a straight circular screw-pinch. Calculations similar to those of Freidberg and Haas for sharp boundary tokamaks [6] have been extended in [3] to different pressure and current profiles within circular, elliptic or race-track cross-sections* The region outside the main column, where plasma of low density is present, is a region of force-free fields for which VxB = aë". A convenient subclass of force-free configurations, including tokamak, screw pinch and reversed field pinch, is that with a being a constant. The screw pinch is characterized by a force-free field with uniform pitch, which can be approximated by a * 2/qR, where q is the safety factor and R the torus major radius. As is documented in [3] this configuration represents an optimum for MHD-stability and g-values of 0.6 e can be obtained if q* is slightly larger than 1 (sharp-boundary model). This result is presented in Fig. 1. Here e is the inverse aspect ratio of the plasma column and q* is defined by q* = 20rf)/yoRl5, where 0,^ and I^j are the toroidal flux and current in the column. In SPICA c* is a factor /1-3 smaller than q. Further results of the theory indicate that elliptic deformation of the plasma column due to the eccentricity could increase the limit of g. 3. EXPERIMENTAL RESULTS A schematic drawing of SPICA is given in Fig. 2. The actual aspect ratio of the torus^is R/b = 60 cm / 20 cm. The maximum magnetic field strength, BA, is 1.6 T, the maximum toroidal plasma current, I^n, is 400 kA. The rise time of the poloidal and toroidal primary currents is 10 us and the 1/e-decay time after crowbarring exceeds 1 msec. A more detailed description of the experiment has been given in [7] . The following set of diagnostics has been used: 1. magnetic probes to measure the magnetic field and current profile in a region which extends from the wall a few centimeters inward; 2. Rogowski coils to measure the currents; 3. a set of modified sinö - ceo.; coils to observe ;n=l motions in thu horizontal c:\d vertical direction; 4. a large nuir.ber of magnetic pick-up coils to measure B^ and B^ at the wall at different poloidal and toroidal positions; 5^. a He-Ne laser interferometer (Michelson) to measure the line density; 6. a set of two image-converter cameras which make stereoscopic streak pictures of the plasma column at

I.R. 76/032

-3two different toroidal positions; 7. a giant pulse ruby laser with monochromator and multichannel detector system for 90° Thomson scattering; 8. a microwave system to measure the reflection of- 8-nun waves from plasma in the outside region; 9. a simple scintillation detector for the measurement of hard X-rays; 10. a 4-rrun microwave interferometer. The regime which h*s been studied in the preceding months is given by B^ * 1.2 I, I^p « 250 kA, initial D2~pressure p o « 10 mTorr, initial bias field B o = 0.1 T. Both ringing and crowbarred currents have been applied. The discharge behaviour is sensitive to small changes in initial conditions. Especially the exact timing of the primary currents is very important. This timing affects the time dependence of q at the wall during the buildup phase. Best results are 'obtained when q W all is nearly constant in time. If this is not the case, the plasma either grows unstable or assumes an equilibrium position which is too eccentric. Also the preionization has a pronounced influence on the discharge characteristics. Figure 3a shows the time behaviour of a discharge which was programmed such that the field lines at r=b (inside copper shield) had a q of 1.7 during the rise of the currents. This value was calculated from q^all = b 2 I^ c /R 2 lAp, where Ip.c is the primary poloidal current. This is a good approximation for the usual integral expression since the local q-value varies only 1 by 30% along the minor circumference. Discharges like the one in Fig. -3a show parameters which represent a considerable improvement over those measured earlier at lower bias-field values [8]. In particular the decay time of the plasma current during the first 100 ps increased from 150-200 MS to 500-700 ps. This is a clear indication of the improved quality of the force-free current region outride the column. Information about the behaviour of the force-free currents is obtained from a comparison between the value of the "inductance" Li, measured as Li = ^V^dt/I^p [b ] , with the theoretical value Lj_f For Lit we assume that the torus volume is completely occupied by a constant pitch field apart from a vacuum region close to the copper. Skin effects in the main column have been neglected since they introduce only a minor correction in Lit especially at later times when skin currents have smoothed out. The measured value Li increases during the first 100 vis from 450 nH to 530 nH, which corresponds to a decrease of the outer region of the force-free current region from 19 to 17 cm (the inside quartz radius is 20 cm) . Using this to evaluate q in the pressureless plasma region, we find q = 1.5 constant in time. The exact correlation between current decay rate and temperature is complicated. The decay time depends not only on the resistance of the plasma column, but on the exact resistivity profile, on the decay time of the external circuit and also on the horizontal movement of the column which causes field lines to move into the quartz wall. If we take the nonideal crov.'barriiig of the external circuits into account and assume a uniform resistivity, a temperature exceeding 20 eV car. be attributed to the outside region 1.9] . This value is not very dependent on th.e resistivity in the main column. At these temperatures and a density in the order of 1% of the column density turbulent processes are not expected.

I.R. 76/032 -4A study was made of the stability behaviour of these stable discharges and of clearly unstable ones [10] . An example of an unstable discharge is shown in Fig. 3b. Although the initial conditions•and the settings of the banks were chosen identical, there is a jnarked difference in behaviour. The temperature (22 eV) and the density (3xlO21/m3) measured at 6.4 us by means of local Thomson scattering are way below those of the stable discharge (60 eV, 5xiO21/m3)» y n e line density measurements, however, indicate a density which even exceeds 5xlO 21 /m 3 . it is therefore likely that the detected scattered laser light originates from the edge of the column. This can be explained by either a large equilibrium shift or- a much more peaked pressure profile. The streak photographs, taken at ppsitions 150° apart (in the <j>direction), and the Bg pick-up coils indicate an m=2, n=l distortion. The plasma touches the waj.1 after 70 us. The current decay time is only 250 ps and both q w a n and L^ increase much faster than in the stable discharge, which points to a rapid decay of the force-free currents. A logarithmic .plot of the currents is shown in Fig. 4, together with the amplitudes of the m=l and m=2 deformations as measured by the sin-cos coils and a poloidal set of eight B Q coils. Large m=2 amplitudes correspond to fast current decay. Moreover, the variation in the fluctuation level appears to correlate well with similar variations in the decay rate (see Fig. 5 ) . 4. DISCUSSION If the effects of diffusivity, conducting walls and elliptic deformation due to an outward shift are ignored, the now prevailing MHD-theory en equilibrium and stability of a toroidal circular screw pinch predicts for 3 an upper limit Bj;,ax = 0.62 s. This limit holds for q* = 1. For the experiments described in this paper the theory predicts 6 m a x * 0.3 e, which gives 3 m a x = 0.03 for a plasma radius of 7 cm. In the experiment a peak 3 of 0.2 is measured at 6.4 us. Even under the assumption of a parabolic pressure profile we conclude that a plasma column with an average 6 equal to 0.1 can be held in equilibrium at q=1.5. This is three times as high as the theoretical limit. Since it is mainly the equilibrium restriction, determined by the circular configuration, which limits the theoretical 2 ma>: , it is likely that the observed elliptic deformation of the column is responsible for this effect. This is supported by the fact that relaxation of the theoretical constraints (a diffuse profile in the equilibrium calculations or a non-circular cross section in the stability calculations) leads to a substantial increase of 3max* In general no m=l instabilities are observed in SPICA, even in the discharges with a fast current decay. These modes, instead of the observed m=2 modes, should be the dominant components when the stability limit is exceeded. It is possible that a chancre in the initial conditions causes tiio buildup of a q-prorile fcr which m=2 is the dominant instability. Radiation cooling and heat conduction can cause a relatively fast decay of 8. Not enough is known yet about the impurity content. Introduction of impurities, however, caused the current decay rate to increase considerably.

I.R. 76/032

-55. CONCLUSION Screw-pinch discharges with q=1.5 and a peak 3 of 0.2 have been produced in 10 mTorr D2. These discharges are in equilibrium and show no MHD-instabilities during about 100 us. From the value of the plasma inductance it is evident that forcefree currents, important for equilibrium as well as stability, exist almost up to the wall. The low current decay rate indicates a high corUuctivity of the tenuous plasma in this outside region. In unstable discharges an m=2 kink instability is observed. The decay rate of the plasma current increases with the amplitude of the instability. A very accurate control of the initial conditions is necessary to avoid these unstable discharges. ACKNOWLEDGEMENT The machine has been operated by D.J. Maris with the assistance of G. van Dijk, A.C. Griffioen, and P.H.M. Smeets. The data handling was taken care of by W. Kooijman. Further assistance was given by P.J. Busch (Thomson scattering), H.W.L. Luysterbuj.g (magnetic measurements) , and W.J. Mastop (laser interferometry). A.F.G. van der Meer started spectroscopie measurements, J.W.A. Zwart took part in the measurements during 1975. This work was performed under the Euratom-FOM association agreement with financial support from ZWO and Euratom. R E F E R E N C E S [1]

BOBELDIJK, C , VAN HEIJNINGEN, R.J.J. , VAN DER LAAN, P.C.T., ORNSTEIN, L.Th.M., SCHUURMAN, W., DE VRIES, R.F., Proc. 3rd Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Research (IAEA), Novosibirsk 1_ (1968) 287. [2] TAYLOR, J.B., Phys. Rev. Lett. 3_3 (1974) 1139. [3] D'IPPOLITO, D.A., FREIDBERG, J.P., GOEDBLOED, J.P., REM, J. , this conference, E22. [4] SCHUURMAN, W., BOBELDIJK, C., DE VRIES, R.F., Plasma Phys. II (1969) 495. [5] GOEDBLOED, J.P., "Stabilization of magnetohydrodynamic instabilities by force-free magnetic fields", (Thesis Eindhoven 17-12-1970), Rijnhuizen Report 70-64. [6] FREIDBERG, J.P., HAAS, F.A., Phys. Fluids _L6 (1973) 1909. [7] VAN HEIJNINGEN, R.J.J., VAN DER LAAN, P.C.T., MARIS, D.J., MEDDENS, B.J.H., OOMENS, A.A.M., STERK, A.B., Proc. 8th Symp. on Fusion Techn., Noordwijkerhout (1974) 341. [8] BOBELDIJK, C , VAN HEIJNINGEN, R.J.J., HOEKZEMA, J.A., KOOIJMAN, W. , VAN DER LAAN, P.C.T.. , MARIS, D.J., OEPTS, D., OOMENS,'A.A.M., Proc. 5th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res., Tokyo (1974) Vol. Ill, 409. [9] HOEKZEMA, J.A., Proc.1 3rd Topical Conf. on Pulsed High-Beta Plasmas, Culham (1975). [10] MIMURA, M., KOOIJMAN, W., OOMENS, A.A.M., to be published as a Rijnhuizen Report.

I.R. 76/032

-6-

0.6 -

Fig. 1. Maximum S/e as a function of the force-free current parameter r = aqR, for a sharp-boundary model (from Ref. [3]).

Fig. 2. Schematic view of SPICA. a. Feed flange for primary I, b. Copper shield. c. Toroidal field coil. d. Plasma. e. Vacuum chamber. f. Ports for diagnostics.

I.R. 76/032 -7<

300

Je- 200 f 100 ' O

L * "

t

800 600 400 200 O

-r

top view

side view

40

-

80 t ((is)

120 0

40

8O -t

12O

Fig. 3. a. Characteristics of a stable screw-pinch discharge. The currents are not corrected for the time constant of the integrator (1.5 ms) . From top to bottoia: plasma current, primary poloidal current,- safety factor at the wall, internal inductance, line density, streak photographs. b. Characteristics of an unstable discharge, compare figure 3a.

I.R. 76/032 -80.6 0.4 m=2 0.2 0 0.2 ^~ m = 1 (s i n) O -0.2 - r^-m= 1 (cos)

:

®:

1

m= 2

1 ^.m- 1 (sin) ]/ ^ m = 1 (cos)

-04

28

< °

•

i

i

i

i

i

i

i

i

i

i

\

i

i

i_

i

i

•

f

i

i

i

i

i

i

•

<

i

i

i

i

i

t

i

r

i

i

t

t

i

i

i

t

~ 240 •e-

2001160

II

20

40

60 •

80

' i

20

6O

80

t ( us)

Fig. 4. a. Current (logarithmic) and relative magnitude of the m=l and m=2 deformations for a stable discharge. The dotted line yields an average decay time of 440 ys. After correction for the integrator this increases to 620 ps. b. Unstable discharge, compare figure 4a.

Fig. 5. Correlation of the m=2 amplitude (relative to the m=0 amplitude) with the current decay rate in an unstable discharcG.