Numerical studies of current profile control in the reversed-field pinch

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reversed-eld pin h

JON-ERIK DAHLIN

Do toral Thesis

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ISSN 1653-5146

ISBN 91-7178-500-0

ISBN 978-91-7178-500-8

KTHS hoolofEle tri alEngineering

SE-10044Sto kholm

SWEDEN

AkademiskavhandlingsommedtillståndavKunglTekniskahögskolanframlägges

till oentlig granskning för avläggande av teknologie doktorsexamen i fysikalisk

elektroteknik torsdagenden30november2006klo kan10.00iKollegiesalen(F3) .

©Jon-ErikDahlin,30November2006

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TheReversed-FieldPin h(RFP)isoneofthemajoralternativesforrealizingenergy

produ tion from thermonu lear fusion. Compared to alternative ongurations

(su hasthetokamakandthestellarator)ithassomeadvantagesthatsuggestthat

anRFPrea tormaybemoree onomi . However,the onventionalRFPisawed

withanomalouslylargeenergyandparti letransport(whi hleadstouna eptably

lowenergy onnement)duetoaphenomenon alledtheRFPdynamo.

Thedynamoisdrivenbythegradientintheplasma urrentintheplasma ore,

andit has beenshownthat attening of theplasma urrentprole quen hes the

dynamoandin reases onnement. Variousformsof urrentprole ontrols hemes

havebeendevelopedandtestedinbothnumeri alsimulationsand experiments.

Inthisthesisanautomati urrentprole ontrolroutinehasbeendevelopedfor

the three-dimensional, non-linear resistive magnetohydrodynami omputer ode

DEBSP.Theroutineutilizesa tivefeedba kofthedynamoasso iatedu tuating

ele tri eld,andisoptimizedforrepla ingitwithanexternallysuppliedeldwhile

maintainingeldreversal. Byintrodu ingasemi-automati feedba ks heme, the

numberoffreeparametersisredu ed,makingaparameters anfeasible. As aling

studywasperformedands alinglawsforthe onnementoftheadvan edRFP(an

RFPwithenhan ed onnementdueto urrentprole ontrol)havebeenobtained.

The on lusions from this resear h proje t arethat energy onnement is en-

han edsubstantiallyintheadvan edRFPandthatpoloidalbetavaluesarepossible

beyondtheprevioustheoreti allimit

β

θ

<

¹₂^. ^S
alings^toward^the^rea
tor^regime^in-

di atestronglyenhan ed onnementas omparedto onventionalRFPs enarios,

butthequestionofrea torviabilityremainsopen.

Des riptors

Reversed-FieldPin h, RFP, CurrentProleControl,CPC,DEBS, DEBSP,a tive

ontrol,feedba k,MHD.

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Den Reverserade Fält-Pin hen (RFP) är ett av de främsta alternativen för att

förverkligatermonukleärfusionförenergiproduktion.Jämförtmedalternativakon-

gurationer(som tokamaken o h stellaratorn)har den ett antal fördelarsom an-

tyder att en RFP-reaktor borde kunna konstrueras till lägre kostnad.Emellertid

ärRFP:nbehäftadmedanomalthögenergi-o hpartikeltransport(vilketledertill

oa eptabelt låg energiinneslutning) på grund av ett fenomen som kallas RFP-

dynamon.

Drivkraftentilldynamonkommerfråndenbrantaplasmaströmsgradientenidet

inreavplasmat.Detharvisatsattomströmprolenatasutsåredu erasdynamon

o hinneslutningenförbättras.Olikametoderförströmprolkontrollharutve klats

o hprovatsbådeinumeriskasimuleringaro hi experiment.

Idennaavhandlingharenautomatiskströmprolkontrollrutintilldentredimen-

sionella,i ke-linjäraresistivamagnetohydrodynamiskadatorkodenDEBSPutve k-

lats. Rutinen använder sig av aktiv återkoppling av det uktuerande elektriska

fältsom hörsamman med dynamon, o h är optimeradför attersätta detta sam-

tidigtsomdetupprätthållerfältreverseringen.Genomattinföraenhalvautomatisk

återkoppling kan antalet fria parametrar redu eras, vilket möjliggör en parame-

teröversyn.En skalningsstudiehargenomförts somresulteradei skalningslagarför

inneslutningen i den avan erade RFP:n (det vill säga en RFP vars inneslutning

förbättratsmedhjälpavströmprolkontroll).

Slutsatsernafråndettaforskningsprojektärattenergiinneslutningenäravsevärt

högrei denavan eradeRFP:n än iden konventionella RFP:n o h attpoloidalbe-

tavärdensomnärmarsigettärteoretisktmöjligaattuppnå(faktumäratt

β

θ

>

¹₂

visasvaraidealttillåtet).Dåparametrarskalas motdetreaktorrelevantaområdet

erhålles my ket högre inneslutning än tidigare studier har visat, men frågan om

huruvidaenRFP-reaktorär realiserbarförbliröppen.

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Thereare manypeople I would liketo express my gratitudeto, who havehelped

andinspiredmein variouswaysduringmyPhDstudies.

I wantto expressmygreatestgratitudetoDr. JanS heelfor beingaperfe t

supervisorandmentor. Iappre iatehispatien eandhisabilitytoguideandinspire.

IamgratefultoDr. DaltonS hna kforenlighteningdis ussionsonRFPphysi s

andtheDEBSP ode,whi hhavegivenmeaninvaluablese ondpointofview.

Thanks goes to Dr. Jay Anderson for broadening my view in dis ussions on

experimentalimplementationof urrentprole ontrol.

IalsowanttothankthePhDstudentsduringmytimeattheAlfvénLaboratory

for being en ouraging dis ussion partners and good friends. Thanks to Tommy

Bergkvistforsolvingmy omputerproblems,toKerstinHolmströmfortaking are

of our undergraduate students and to Tomas, Thomas, Martin and Dmitriy for

leadingthewayandshowinghowtoa quirethedo toraldegree.

IwouldliketothankallthepeopleontheAlfvénLaboratorywhohavehelped

meinpra ti almattersandprovideda onstru tiveatmospheretoworkin.

I want to thank my family and friends for being supportive and for showing

interestin mywork.

Finally,Iwouldliketoexpressmydeepestappre iationandadmirationtoJenny

foralwaysbeingsupportivenomatterhowhardtimeshavebeen.

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Theworkinthisthesisisbasedontheworkpresentedinthefollowingpapers.

I. J.-E.DahlinandJ.S heel

Feedba k urrentprole ontrolintheadvan edRFP

Pro eedings of the31st EPS Plasma Physi s Conferen e, June28 - July 2,

(2004),London,UnitedKingdom

II. J.-E.DahlinandJ.S heel

Anovelfeedba kalgorithmforsimulating ontrolleddynami sand onne-

mentintheadvan edreversed-eldpin h

Physi s of Plasmas,12(2005)p62502-1-5

III. J.S heel andJ.-E.Dahlin

Connements alingintheadvan edreversed-eldpin h

Plasma Physi s andControlledFusion,48(2006)L97-L104

IV. J.-E.DahlinandJ.S heel

Numeri alstudiesof onnements alingsforthedynamo-freereversed-eld

pin h

A eptedforpubli ationin Nu learFusion,(2006)

V. J.-E.DahlinandJ.S heel

Ultrahighbetainnumeri alsimulationsofatearing-moderedu edreversed-

eldpin h

SubmittedtoPhysi al ReviewLetters,(2006)

VI. J.-E.Dahlin,J.S heelandJ.K.Anderson

Numeri alstudiesofa tive urrentprole ontrolinthereversed-eldpin h

SubmittedtoPlasma Physi s andControlled Fusion,(2006)

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Papersthatarenotin ludedin thisthesis.

Refereed Journals

VII. J.-E.DahlinandJ.S heel

Self- onsistentzero-dimensionalnumeri alsimulationofamagnetizedtarget

fusion onguration

Physi aS ripta, 70(2004)p310-316

Conferen e Contributions

Chronologi al order

VIII. J.-E.Dahlin

Rea torpotentialformagnetizedtargetfusion

AnnualMeetingoftheFusionResear hUnitofSweden,RUSA-2002,Novem-

ber4-5,(2002),Göteborg,Sweden

IX. J.S heel,J.-E.Dahlin,D.D.S hna kandJ.R.Drake

Energy onnementintheadvan edRFP

45th Ameri an Physi al So iety Annual Meeting of the Division of Plasma

Physi s, 2003, O tober 27-31, (2003), Albuquerque, New Mexi o, United

States

X. J.-E.Dahlin,J.S heelandD.D.S hna k

Feedba k urrentprole ontrolintheadvan edreversed eldpin h

Annual Meeting of the Fusion Resear h Unit of Sweden, RUSA-2004, April

20-21,(2004),Studsvik,Nyköping,Sweden

XI. J.-E.DahlinandJ.S heel

Advan edreversed-eldpin h onnements alinglaws

Pro eedingsof the32nd EPSPlasma Physi s Conferen e, June27 -July 1,

(2005),Tarragona,Spain

XII. J.-E.DahlinandJ.S heel

Advan edreversed-eldpin h onnements alinglaws

APS: 47th Ameri an Physi al So iety Annual Meeting of the Division of

Plasma Physi s, 2005, O tober 24-28, (2005), Denver, Colorado, United

States

XIII. J.-E.DahlinandJ.S heel

Improved omputer simulations of energy onnement in the advan ed

reversed-eldpin h

Pro eedingsofthe33rdEPSPlasmaPhysi s Conferen e,June19-23,(2006),

Rome,Italy

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1 Introdu tion 1

1.1 Energy risis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Thermonu learfusion . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Magneti onnement . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Basi plasma physi s 7 2.1 Plasmamodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Magnetohydrodynami s . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The reversed-eld pin h 11 3.1 RFPequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 RFPstability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 TheRFPdynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Currentprole ontrol . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Numeri alsimulations 17 4.1 TheDEBSP- ode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Currentprole ontrolin DEBSP . . . . . . . . . . . . . . . . . . . . 19

5 Results 23 5.1 In reasedenergy onnementduetoa tiveCPC . . . . . . . . . . . 23

5.2 StationaryCPCstudies . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Operationduring quasi-steadystate . . . . . . . . . . . . . . . . . . 27

5.4 Modeanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5 Analyti investigationoftheenergy onnementtime . . . . . . . . 30

5.6 Ultrahighbetaoperation . . . . . . . . . . . . . . . . . . . . . . . . 32

5.7 S alinglawsfortheadvan edRFP . . . . . . . . . . . . . . . . . . . 36

5.8 Experimentalviability . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Dis ussion 43

7 Con lusions 45

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Bibliography 49

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1.1 Bindingenergypernu leon . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Fusion ross-se tions . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Fusionrea tivitiesforaplasmainthermodynami equilibrium . . 5

3.1 Magneti elds fortheBFM-equilibrium . . . . . . . . . . . . . . 12

3.2 Safetyfa tor

q

^for^theBFM-equilibrium . . . . . . . . . . . . . . . 12

5.1 Evolutionof

τ

E^,

β

θ^,

< B

_r²

>

^and

F

^for^CPC¹ ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²⁵

5.2 Radialprolesfor

q

^,

µ

^,

E

f ^and

p

^for^CPC¹ ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²⁶

5.3

E

f ^for^theun ontrolled aseandwithCPC . . . . . . . . . . . . . 27

5.4

τ

E ^for^a^series^ofsimulationswith

E

a ^frozen^at^various^times ^. ^. ^. ²⁸

5.5

τ

E ^for^a^series^ofsimulationswithdierent

E

_f^w ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²⁹

5.6 Modespe tra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.7 ExternalinputpoweroverSpitzerheatingpowerratio . . . . . . . 33

5.8 Evolutionof

τ

E^,

β

θ^,

< B

_r²

>

^and

F

^for^CPC¹ ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ³⁴

5.9 Evolutionof

m = 1

^modes^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ³⁵

5.10 Poin arése tion, noCPC . . . . . . . . . . . . . . . . . . . . . . . 39

5.11 Poin arése tion, withCPC . . . . . . . . . . . . . . . . . . . . . . 39

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Introdu tion

Ahundredyearsago,AlbertEinstein[1℄proposedthatmassisequivalenttoenergy,

thusit an betranslatedintootherformsofenergyin a ordan ewith

E = mc

² ^(1.1)

where

c

îs^the^speedôf^lightⁱⁿ ^va
uum¹^. Ênergy ^release^due^to^nu
leartransitions isindeedexplainedbythemassdieren ebetweenrea tantsandrea tionprodu ts.

Thisenergyis alledthebindingenergyofthenu lear onguration. Thebinding

energyofnu leiareshownin gure1.1. Astheeldofnu learphysi sunfoldedin

theearly twentieth entury, itbe ame lear that energy ould bea hievedeither

bysplitting heavynu lei or byfusinglight ones. Both pro esseso urin nature,

andthenu learfusionof hydrogenintoheliumistheverypro essfrom whi h the

sunandotherstarsdrawtheirenergy.

Forthes opeofenergyprodu tiononearth,themosta essiblefusionrea tion

would be the transmutation of one deuteron (heavy hydrogen nu leus) and one

triton(superheavyhydrogennu leus)intoonealphaparti le(heliumnu leus)and

oneneutron:

d

(

^t

,

^He

)

ⁿ ^(1.2)

Themassoftherea tantsinrea tion(1.2)ex eedsthemassoftherea tionprod-

u tsby approximately

0.4 %

^. ^This ênergy^manifests îtselfâs ^kinetiênergyⁱⁿ ^the

rea tionprodu ts,distributed as

3.5 MeV

² ^to^the^helium^nu
leus^and

14.1 MeV

^to

theneutron.

Thedeuteron andthe tritonexperien e repulsiveele tri for es andattra tive

nu lear for es. The potential barrier that results from the ompetition of those

for espeaksatadistan efromthenu leusontheorderofafemtometer

(10

⁻¹⁵

m)

^.

Thenu lei do nothaveto over ome thefull height of thebarrier but an tunnel

throughitdue to thequantum me hani alprobability distribution,but still they

1c = 299792458 m/s

21 eV = 1.602 · 10⁻¹⁹J

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2 CHAPTER1. INTRODUCTION

have to ome very lose to ea h other for the probability of the rea tion to be

reasonablyhigh.

A

E[MeV]

1 10 100

0 1 2 3 4 5 6 7 8 9

1

H

2

H

3

H

3

He

4

He

6

Li

7

Li

56

Fe

235

U

Figure1.1:Bindingenergypernu leon.

This anbedoneinanumberofways. Inthe oreofstars,gravityisso strong

thatnu leiarefor ed loseenoughtoea hothertofuse. Inproposedthermonu lear

fusionrea tors,temperatureissohigh(hundredsofmillionsofdegreesCelsius)that

nu leio asionally ollideandfusedueto theirthermalmotion.

1.1 Energy risis

Worldwide energy demand is today

14 TW

ând ^rising ^fast: ^theêstimated ênergy

demand in year

2030

^is

24 TW

³^. ^The ^main ^part ôf ^this ênergy îs ^produ
ed ^by

ombustion offossil fuels su h as oal, oiland fossil gas. The oilprodu tion will

most likely peak within a de ade and oal produ tion is expe ted to peak in a

hundredyearsorso. Thepri esofpetroleumprodu tsandfuelsarealreadyrising.

Ifnoalternativeenergysour eisdeveloped, apableofrepla ingfossilfuelsas the

foundationofenergyprodu tioninso iety,theworldwillundoubtedlybeexposed

to massivee onomi rises.

Todayitis a eptedinmostparts ofthes ienti so ietythatthe ombustion

of fossil fuels will ause global limate hanges, and that this will ertainly hap-

pen within atime s aleshorter than the lifetime of these fuels based upon their

availability. Conne tedtothese hanges,whi hmaybeseverealreadyearlyinthis

3

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entury, are hangesin thee osystems. Thiswillput demands onagri ultureand

mayde rease rops. Againwefa eane onomi problemwithhuge dimensions.

The problem with energy produ tion based on onsumption of fossil fuels is

hen etwofold: thefuelavailabilitywill eventuallyde reaseandthe ombustion of

oalleadstosevere limate hanges. Bothproblemswillleadtoe onomi stresson

so iety andit is thus enormouslyimportant toalter thefo usfrom fossil fuelsto

alternativeenergysour es.

Todaytherearenoavailablealternativesthatwouldhavethe apa ityofrepla -

ingfossilfuels. Hydropowerisrelativelyenvironmentallyfriendly;itisarenewable

energysour eandhasthe apabilityoflarge-s aleenergyprodu tionbutisunfor-

tunatelyimpossibleto expandto thes aleof fossilfuelenergyprodu tion. Other

renewable energy sour es su h as wind power, solar power, wave power and en-

ergy rops fa e the problems of largeareal demands, largeinvestment osts and

possibly high impa t on e osystems. Nu lear (ssion) power is also in prin iple

environmentally friendly and may indeed be possible to expand. Nu lear power

ishoweverproblemati for several reasons: Conventional nu lear power produ es

largeamounts of waste that is noteasily managed; mining of uranium (whi h is

usedasfuel)isperformedin awaythat resultsintremendouslo alenvironmental

problems;ssion rea torsofthetypethat ismainly used todayareusing thefuel

veryune onomi ally,whi hwillleadtothesamedepletionproblemsaswithfossil

fuels.

Fusionpower ould ertainlybethesolution. Itisnotarenewableenergysour e,

buttheavailabilityofthefuelsisalmostunlimited: tritium anbeprodu edwithin

therea torfromlithiumanddeuterium anbee onomi allyextra tedfromseawa-

ter. Lithium isabundantontheplanetin largeamountsand an bee onomi ally

mined. Thefusionfuelsavailableon theplanet willnotbedepletedwith the ur-

rentrateofenergy onsumptioninthousandsorevenmillionsofyears,whynu lear

fusion anberegardedasustainableenergysour e. Itshouldhoweverbenotedthat

fusionpowerisnotfreefrom drawba ks. Itsdisadvantagesin ludetheradioa tive

loadingofwallmaterial(whi hhastobeentaken areof, eventhoughtheamount

ofradioa tivewasteisfarlessthanfromnu learssionpowerplants,andthede ay

timeisaround100yearsas omparedtotensofmillenniaforssionwaste),therisks

onne tedtohandlingradioa tivefuel(iftritiumisused)andthequitelarge osts

fordevelopingthete hnologyand onstru tionoffa ilities. Asitappears,anideal

energysour e isnoteasily on eivedand inthe ompetition betweenalternatives,

fusion ertainlytakesaleadingposition.

1.2 Thermonu lear fusion

Twomainpathshavebeendevelopedtondaworkingfusion on ept. InMagneti

Fusion Energy (MFE), a plasma ontaining fusion fuels (most often deuterium

and tritium) is onned by a strong magneti eld and heated to thermonu lear