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Validation of theory based transport models in tokamak plasmas [Elektronische Ressource] / Giovanni Tardini

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Max-Planck-Institut fur PlasmaphysikValidation of theory based transportmodels in tokamak plasmasGiovanni TardiniTechnische Universit at Munc hen2003Technische Universit at Munc henValidation of theory based transportmodels in tokamak plasmasGiovanni TardiniVollst andiger Abdruck der von der Fakult at fur Physik der TechnischenUniversit at Munc hen zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof. Ph. D. F. KochPrufer der Dissertation: 1. Hon.-Prof. Dr. R. Wilhelm2. Univ.-Prof. Dr. H. FriedrichDie Dissertation wurde am 19. 02. 2003 bei der Technischen Universit at Munc heneingereicht und durch die Fakult at fur Physik am 27. 05. 03 angenommen.AbstractHeat transport in tokamaks is several orders of magnitude higher than predicted by thecollisional theory. A physical understanding of the phenomena limiting energy con ne-ment is a basic requirement in order to make reliable predictions about the fusion gain ofthe future tokamak reactor ITER and to investigate new scenarios and plasma regimes.Since decades anomalous transport has been addressed by means of empirical and semi-empirical models.

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Published 01 January 2003
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k-InstitutMax-Planc

ysikPlasmaphur¨f

Validationmodelsofintoktheoryamakbasedplasmastransport

Gioanniv

hehniscecT

ardiniT

atersit¨Univ

2003

henunc¨M

TechnischeUniversit¨atM¨unchen

Validationmodelsofintoktheoryamakbasedplasmastransport

vGioardiniTanni

Vollst¨andigerAbdruckdervonderFakult¨atf¨urPhysikderTechnischen
Universit¨atM¨unchenzurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften(Dr.rer.nat.)
Dissertation.genehmigten

Vorsitzender:Univ.-Prof.Ph.D.F.Koch
Pr¨uferderDissertation:1.Hon.-Prof.Dr.R.Wilhelm
2.Univ.-Prof.Dr.H.Friedrich

DieDissertationwurdeam19.02.2003beiderTechnischenUniversit¨atM¨unchen
eingereichtunddurchdieFakult¨atf¨urPhysikam27.05.03angenommen.

Abstract

Heattransportintokamaksisseveralordersofmagnitudehigherthanpredictedbythe
collisionaltheory.Aphysicalunderstandingofthephenomenalimitingenergyconfine-
mentisabasicrequirementinordertomakereliablepredictionsaboutthefusiongainof
thefuturetokamakreactorITERandtoinvestigatenewscenariosandplasmaregimes.

Sincedecadesanomaloustransporthasbeenaddressedbymeansofempiricalandsemi-
empiricalmodels.Thetheoreticalresearchassumesthatmicro-instabilitiesdrivenby
plasmaturbulenceareresponsiblefortheconfinementdegradation,butthefullsimula-
tionswiththecomprehensivetheoreticalcodesrequirefartoolongcomputingtimeto
affordavalidationagainstanextensiveexperimentaldatabase.
Onlyintheearlyninetiesonedimensional,theorybasedmodelshavebeendeveloped,
relyingonthefluidapproach.Thesemodelsallowcomparisonswiththeexperimental
resultswithoutanyadhocadjustmentofempiricalfittingparametersand,ontheother
hand,withouttoolongcomputingtime.

Inthepresentworkthemostcommonlyacceptedmodelsarevalidatedagainstalarge
databaseofselectedASDEXUpgradedischarges.ImportantresultsfromJET,thelargest
tokamakbuiltsofar,areincludedaswell,totestthemodelsonadifferentsizeddevice
andtoincreaseconfidenceintransportpredictionsandinextrapolationstoITER.

asAsaquanresulttitativofetheevaluationssystematicofthecomparisonpredictingwithcapabilittheayvofailablethemodata,delsarequalitativpreseneasted.wellA
physicsinterpretationoftheheattransportphenomenologyintheconventionalscenario
discussed.andosedpropis

1

2

Danksagung

Ichm¨ochtemichzuerstbeiProf.Wilhelmf¨urseineakademischeBetreuungbedanken.
MeinDankgiltvorallemanDr.ArthurPeeters,dessenZusammenarbeitstetswertvoll
undermutigendgewesenist.UnsereTransportanalysisGruppehatnichtnureinebekan-
ntlicheinfacheStruktur,sondernaucheinsch¨onumg¨anglichesArbeitsklima.Physikzu
lernen,zurFusionsforscungbeizutragenunddabeinochSpasszuhabenistbeimirGrund
eit.DankbarkderIchdankeDr.GrigoryPereverzev,derdengrossenVerdiensthat,denASTRACode
geschriebenzuhabenundimmerwiederzuaktualisieren.SeineHilfsbereitschaftbei
denvielenmeinenFragenundASTRA-W¨unschenistbeeindruckend.Dankef¨urdas
“physikalischste”IPP-Kolloquium,dasichbishergeh¨orthabe.
DerAustauschmitDr.ClementeAngioniwarbesondershilfsreich.Vonihmhabeich
gelernt,wieweitmaneine“blackbox”aufmachenunduntersuchenkann.Esistviel
interessanter,eineForschungzuzweitzuverfolgen.
DerganzenTokamakabteilungdankeichherzlichf¨urdieGesellschaftunddieinteressan-
tenphysikalischenDiskussionen.Prof.LacknerhatwesentlichzumeinerEntscheidung
f¨urdieseThemenstellungbeigetragen.
Besondersverpflichtetf¨uhleichmichHerrnDr.FrancoisRyter:ohneihnw¨urdendie
bedeutsamstenexperimentellenBefundef¨urmeineDoktorarbeitnichtexistieren.Diese
ZusammenarbeitmitaltenWurzelnundimmerfruchtbaremMeinungsaustauschistim
RahmenderunofiziellenTask-Force“GME”(GemeinsamMittagEssen)zustandegekom-
men.AndieserStellemussichmichnochbeiden¨ubrigen“GME”-Mitgliedernbedanken,f¨ur
diegem¨utlicheUnterhaltungunddiephysikalischeBereicherung:Dr.EmanuelePoli,Dr.
FritzLeuterer,StefanoRiondato,Dr.DietmarWagner,Dr.AdrianoManiniundDr.
v.KiroKrassimirDasganzeASDEXUpgradeTEAMverdientmehralseinDankwortf¨urdieDurchf¨uhrung
derExperimentesowief¨urdiesorgf¨altigeDiagnostikverwaltung,durchdiemirumfangre-
icheDatenzurVerf¨ugunggestelltwurden.DieTheorieistzwarnichtgrau,wiemansie
bezeichnet,aberersteinstetigerVergleichmitdemExperimentverleihtihrBestandund
eit.assigkerl¨ZuvZumSchlussm¨ochteichmichbeiallenFreundenbedanken,dennohneFreundschaft
¨uberlebeichkeinedreiWochen,umsowenigerdreiJahre.
Diegr¨ossteDankbarkeitgiltLucia,meinerFrau,diedieLauneunddieWitzeeinesPhysik-
ersschonlangeertr¨agt.DurchihreGegenwartwerdeichimmerwiederinderWirklichkeit
festgehalten.

3

4

Zusammenfassung

DieKernfusionisteinevielversprechendeL¨osungf¨urdenweltweitsteigendenEnergiebe-
darf,diedieUmweltwenigerbelastetundimVergleichzuKernspaltungeindeutlichgerin-
geresradiologischesProblemdarstellt.DieReaktionerfolgtsichdurchdieVerschmelzung
zweileichterNuklidezueinemschwererenAtomkern.DerMassendefektsorgtumEn-
ergiegewinn,aberdieCoulomb-BarrierezwischendenpositivgeladenenNuklidenmuss
erst¨uberwundenwerden.Dief¨ureinenReaktornotwendigeFusionsrateistdurchhohe
TemperaturundTeilchendichtebedingt;beideninFragekommendenTemperaturenist
dieMaterievollionisiertundbefindetsichsomitimPlasmazustand.
DieerfolgreichsteAnlagef¨urdenEinschlusshochenergetischergeladenerTeilchenistder
Tokamak,derdasPlasmaineinerg¨unstigenMagnetfeldkonfigurationeinschliesst.Trotz-
demsindEnergieverlustemehrereGr¨osseordnungenh¨oheralsdiekollisionaleTheorie
vorhersagt.InHinblickaufeinenFusionsreaktoristdasVerst¨andnisderdenEnergiein-
schlussbegrenzendenPh¨anomeneeineunabdingbareVoraussetzung,dieseitJahrzehn-
tendieFusionsforschungmith¨ochstemVorrangbesch¨aftigt.Einebesserethermische
Isolierungw¨urdezueinemkleinerenTokamakreaktormitdemgleichenFusionsgewinn
f¨uhren,wasbedeutendetechnischeund¨okonomischeVorteilemitsichbringenw¨urde.
JahrzehntelanghatmandenW¨armetransportinTokamaksmittelsempirischerbzw.halb-
empirischerModelleinterpretiert.DashatimmerwiederneueExperimentestimuliert.
DennochkanneinephysikalischeErkl¨arungderbeobachtetenTransporteigenschaftenauf
dieserWeisenurnahegelegt,nichtbewiesenwerden,weilvieleunterschiedlicheAns¨atze
zumgleichenErgebnisf¨uhren.InsbesonderesinddieVorhersagenf¨ureinenk¨unftigen
Tokamakoderf¨ureinneuesSzenarionurdannzuverl¨assig,wenneinModellkeinadhoc
Parameterenth¨alt,dassichzugunstenderneuenexperimentellenSituationeichenl¨asst.
DieExtrapolationenf¨urden“InternationalTokamakExperimentalReactor”(ITER)
gewinnenwedereineBest¨atigungnocheinevertauensw¨urdigeKorrekturvonsolchenem-
dellen.MohenpiriscSeitetwa30JahrengehtdietheoretischeForschungdavonaus,dassMikroinst¨abilit¨aten
imPlasmaf¨urdenanomalenW¨armetransportverantwortlichsind,indemsieTurbulenz
erzeugen,dieeineWellenl¨angeimBereichdesGyrationsradiusderIonenumdasMagnet-
feldbesitzt.AbererstdieEntwicklungvonvereinfachten,abertheoretischbegr¨undeten
1D-ModellenhatesAnfangder90erJahreerm¨oglicht,einenquantitativenVergleichzwis-
chenExperimentenundtheoretischenVorhersagenohnejeglicheAnpassungandieexperi-
mentellenBefundeundohnezulangeRechenzeitdurchzuf¨uhren.DieseM¨oglichkeitwurde
jedochbishernurbegrenztgenutzt.
ZieldieserDissertationist,mitHilfederumfangreichenASDEXUpgradeDatenbasisund
ducrhEinbeziehungwichtigerErgebnissevonderweltweitgr¨osstenTokamakanlage“Joint
EuropeanTorus”(JET)zueinersystematischenValidierungderexistierendenthereotis-

5

chenTransportmodellezugelangen.AlsErgebnisderArbeitwurdenwichtigeErkentnisse
zumanomalenTransporterzielt,diezurFusionsforschunginsbesondereinHinblickauf
einenk¨unftigenTokamakreaktorwesentlichvoranbringenk¨onnten.
DazuwurdeeineumfassendeundgezielteDatenbankzusammengestellt,dieeserlaubt,die
EigenschaftenderModelleimDetailzu¨uberpr¨ufen.DurchHinzunahmevonEntladungen
mit“verbessertemRandeinschluss”(H-mode)imsogenannten“konventionellenSzenario”,
sowie¨uberExperimentemitvorwiegenderElektronenheizung(¨uberHochfrequenzver-
fahren)andererseitswerdengetrennteAussagen¨uberIonen-undElektronenw¨armetransport
h.oglicm¨F¨urdiegenauereAnalysewurdendieModelleineinenflexiblenTransportCodeeingebaut
undaufdieausgew¨ahltenEntladungenangewandt.Analysismittelnzurqualitativensowie
quantitativenAuswertungderVorhersagef¨ahigkeitderjeweiligenModellesindeingef¨uhrt.
Somiterg¨anzensichverschiedeneBeitr¨agezumVerst¨andnisvomanomalenTransport.
DieBegrenzungenderAnwendbarkeitderModellewerdenvorgestelltunddieHaupt-
abh¨angigkeitenhervorgehoben.DiephysikalischenEffekte,welchedurchdieVariation
vonwichtigenPlasmaparameternzustandekommen,sindsorgf¨altigdurchgezielteParam-
eterscansuntersucht.
ImErgebnisf¨uhrtdiesersystematischeVergleichzwischenTheorieundExperimentzu
wichtigenAussageninBezugaufIonen-sowieElektronenw¨armetransport.Diephysikalis-
chenAnnahmenderModellewerdenanhandderexperimentellenErgebnissebeurteilt.
DieresultierendenSchlussfolgerungenaufdiedenTransporterh¨ohendendenTurbulen-
zph¨anomenewerdenaufgef¨uhrtundimDetaildiskutiert.Insbesonderestelltsichdiedurch
denIonentemperaturgradientengetriebeneModealsdiedominierendeInstabilit¨atf¨urden
vergleichsweisestarkenIonentransportinH-modeEntladungenheraus.DieKombination
dieserModemitderf¨urdenElektronentransportmassgebenden“TrappedElectronMode”
(durchgefangeneElektronenverursachteInstabilit¨at)erkl¨artohneweitereAnnahmen
dieganzeVielfaltderexperimentellenErgebnisse.BeideTurbulenzph¨anomenenzusam-
mengenommengebeneineZuverl¨assigeInterpretationf¨urdenanomalenW¨armetransport
amaks.okTin

6

Ioveggioben,chegiammainonsisazia
nostrointelletto,se’lvernonloillustra,
difuordalqualnessunverosispazia.
Posasiinessocomeferainlustra
tostochegiuntol’ha,egiugnerpollo,
senon,ognidisiosarebbefrustra;
nasceperquello,aguisadirampollo,
apie´delveroildubbio,ede`natura
ch’alsommopingenoidicolloincollo.

(Dante,Commedia,ParadisoIV,vv.124-132)

WellIperceivethatneversatedis
OurintellectunlesstheTruthillumeit,
Beyondwhichnothingtrueexpandsitself.
Itreststherein,aswildbeastinhislair,
Whenitattainsit;anditcanattainit;
Ifnot,theneachdesirewouldfrustratebe.
Thereforespringsup,infashionofashoot,
Doubtatthefootoftruth;andthisisnature,
Whichtothetopfromheighttoheightimpelsus

7

8

Contsten

1ductiontroIn11.1Controlledthermonuclearfusion........................1
1.2Transportintokamaks.............................3
1.2.1Confinementandignition.......................4
1.2.2Particleorbits..............................5
1.2.3Neoclassictransport..........................7
1.2.4Anomaloustransport..........................8
1.3Steadystate:powerbalanceanalysis.....................9
1.4Transientstate:heatpulseanalysis......................10
1.5Contentofthisthesis..............................13

2Transportmodels15
2.1Theiontemperaturegradientdriveninstability...............15
2.2SimplepictureofthetoroidalITGinstability.................15
2.3ITGinstabilityandprofilestiffness......................18
2.4Theorybasedtransportmodels........................20
2.4.1TheIFS/PPPLmodel.........................21
2.4.2TheWeilandmodel...........................21
2.4.3TheGLF23model...........................22
2.4.4TheCDBMmodel...........................22
2.4.5Maindependenceswithinthemodels.................23

3ThetheoryoftheWeilandmodel31
3.1Introduction...................................31
3.2Assumptions...................................32
3.3TheBraginskiiequations............................33
3.4Derivationofthedispersionrelation......................34
3.5Quasi-lineartransport.............................37
3.6Transportcoefficients..............................40

i

4

5

6

7

IonandelectronheattransportinstandardH-modedischarges43
4.1Diagnosticsystems...............................43
4.2NBIheatedH-modedischarges........................44
4.2.1DischargeswithNBIandECH.....................45
4.3Experimentalresults..............................46
4.3.1Ionprofilestiffness...........................46
4.3.2Electronheattransport.........................48
4.3.3Theexperimentalplasmaenergy....................49
4.4Simulationresults................................51
4.4.1TheASTRAcode............................51
4.4.2Modelscomparison...........................52
4.5Summary....................................59

HeattransportinECHdominateddischarges63
5.1TheElectronCyclotronHeating........................64
5.1.1PrincipleoftheECH..........................64
5.1.2TheECHsystemonASDEXUpgrade................65
5.2Diagnosticsystems...............................65
5.3Simulationset-up................................66
5.4ECHpowerscan................................66
5.5Severalharmonicstransportanalysis.....................66
5.6EffectsofECHontransport..........................68
5.7ExperimentswithconstantECHpower....................71
5.8Discussion:resultsandprofilestiffness....................72
5.9Summary....................................74

ModellingofJETdata77
6.1Diagnosticsemployed..............................78
6.2Experimentalresults..............................78
6.3Modellingresults................................81
6.3.1Modellingsetup.............................81
6.3.2Comparisonwiththeexperiment...................81
6.4Summary....................................83

85okoutloandConclusions7.1Summary....................................85
7.2Thisthesis’contribution............................86
7.2.1Database................................86
7.2.2Developmentofanalysistools.....................87
7.2.3Overviewofmodellingresults.....................88

ii

7.3Aglancebeyond................................89

yBibliograph

Appendix

AListoffrequentlyusedabbreviations

93

97

97

BComplementstothederivationoftheWeilandmodel99
B.1Usefulrelations.................................99
B.2Estimateoftheparallelionmotion......................99
B.3Thedispersionrelation.............................100
B.3.1Curvaturerelations...........................100
B.3.2Derivationofthediamagneticheatflux................103
B.3.3Thetemperatureperturbation.....................104
B.3.4Thecontributionfromthestresstensordrift.............105
B.3.5Thepolarisationdrift..........................107
B.3.6Thedensityperturbations.......................108
B.4Quasi-lineardiffusion..............................109
B.4.1Transportmatrix............................110

iii

iv

1Chapter

ductiontroIn

Theparadoxisonlyaconflictbetween
realityandyourfeelingofwhat
realityoughttobe(R.Feynmann)

1.1Controlledthermonuclearfusion

Theincreasingworldwideenergydemandstimulatesthedevelopmentofalternativeen-
ergysources.Mostpowerplantsnowadaysproduceelectricitybyburninggasolineand
carbonbut,whereastheamountoffossilfuelisexpectedtocovertheenergyneedsforthe
intermediatetime,seriousquestionshavebeenraisedabouttheimpactontheecosystem
climate.theandNuclearpoweraswellasrenewableenergysourcesareseenasvalidalternativesforalong
termstrategy.Energycanbegainedfromthemasslossafteranuclearreaction,suchas
thesplittingofaheavynucleusintotwolighterones,orthefusionofhydrogenisotopesto
givehelium.Nuclearfissionisdevelopedenoughtodeliveralreadyasignificantamount
ofenergyintheindustrialisedcountries.However,radioactivewastewithanextremely
longlifetimecannotbeavoided,anditssafestoringisamajorconcerntoday.Besides,
fissionreactorsworknearcriticality,whichcanbepreventedonlybymeansofanartificial
regulation.Nuclearfusionwouldhavealsowaste,sincetheinnerwallsofareactorwould
beactivatedbyneutrons.However,asubstantialimprovementisachievedbythechoice
ofappropriatewallandstructurematerialsoflowactivation.Inaddition,anaccident
wouldnottriggeranuncontrolledenergyemission,becausethefusionreactorcontains
onlyalittleamountoffuel.
Toproduceenergythroughthermonuclearcontrolledfusionisstillachallenginggoal.
Themainhurdleonthewaytowardsaneconomicallyattractivefusionpowerplantisthe
verylowcrosssectionofthereaction,duetotheCoulombianbarrierbetweencharged
nuclei.Toovercomethisinhibitionitisnecessarytoincreasetheaveragekineticenergy

1

ofthereactingnucleiuptovaluesofseveralkeV(seeFig.1.1).Akineticenergyof1keV

T+T

-2110D+T3D+ He-2210/s)D+D3-23v> (m10T+Tσ<-2410-25101000100101T (keV)Figure1.1:Reactionparameter<σv>asafunctionofTifordifferentfusionreactions

correspondsroughlytoatemperatureof10millionsK.Atsuchtemperaturesmatteris
fullyionisedandthemixtureofelectronsandionsiscalledplasma.Onearth,themost
feasiblefusionreactionsarelistedinTable1.1.
Fortemperaturesbelow500keVtheD-Treactionexhibitsthehighestcrosssection

D+D→n(2.45MeV)+3He(0.82MeV)[50%]
D+D→p(3.02MeV)+T(1.01MeV)[50%]
D+T→n(14.04MeV)+4He(3.54MeV)
D+3He→p(14.64MeV)+4He(3.71MeV)
Table1.1:Mostrelevantfusionreactionswithdeuterium.Theparticleenergyafterthe
reactionisreportedintheroundbrackets.

andaverygoodenergyproductionrate(17MeVperreaction).Therefore,itisthebest
candidateforafusionreactor,inspiteoftheunavailabilityoftritiuminnature.Thelater
problemcanbesolvedusingtheneutronsproducedintheD-Treaction.Surroundingthe
plasmawithLithium,asecondreactioncanproduceTritiumwhichcanbegatheredand
usedforplasmafuelling.TheD-Treactionratestartstoberelevantatparticleenergies

2

of10keVormore.Theaimistohaveaself-sustainedreaction,wherethefusionenergy
isenoughtoinducethenextreactions,compensatingtheheatlossesduetoradiationand
transportphenomena:thisconditioniscalledignitionintheliterature.
Fusionistheenergysourceofallstars.Inthiscase,theconfinementoftheplasmais
ensuredbythegravitationalforce.Toobtainignitiononearth,twoapproacheshave
beendeveloped:“inertialfusion”createsverydenseplasmasforashorttime,“magnetic
fusion”confinesarelativelyrarefied,hotplasmaforalongtimethroughtheuseofa
field.magneticInthepresentfusionexperimentsignitionhasnotyetbeenreached:itisnecessaryto
supplypowertoobtainthedesiredhightemperature.Oneofthemainproblemsisthe
relativelyfasttransportoftheheatoutoftheplasma.Themostsuccessfulconfigura-
tionformagneticplasmaconfinementistheTokamak,realisedforthefirsttimebyL.
Artsimovichin1952.ThenamecomesfromtheRussianacronymTOroidalnayaKAmera
MAgnitnymiKatushkami,toroidalchamberwithmagneticcoils.Thetokamakconfigura-
tionisillustratedinFig.1.2.Thedominantmagneticfieldcomponentisthetoroidalone,
producedbyexternalcoils.Apoloidalfieldiscreatedmainlybyatransformerinduced
toroidalcurrentintheplasma,withvaryingcontributionfromnoninductivecurrentdrive.
Thisensurestheplasmaequilibriumandimprovesconfinement.Thecurrentinducedis
atthesametimeatooltoionisethegasgeneratingtheplasmathroughohmicheating.
Theresultingfieldlinesintheidealtokamakconfigurationarehelicalandlieonnested
toroidalmagneticsurfaces(seeFig.1.2);thesafetyfactorq(r)isameasureofthenumber
oftoroidalwindingsofafieldlinerequiredtocompleteonepoloidallooponagivencross
section.

1.2Transportintokamaks
Oneofthemainproblemsinfusionresearchistheunderstandingofthemechanisms
governingheattransportinthedirectionperpendiculartothemagneticsurfaces.Abetter
energyconfinementwouldallowforthesamefusiongaininasmallersizedreactor,with
alltheeconomicalandtechnicalbenefitswhichthisimplies.Inparticular,itisexpensive
tosupplyastrongenoughmagneticfieldoveralargevolume;besides,themechanical
stressesduetoasuddenlossoftheplasmacausedbyinstabilitiesincreasewithgrowing
machinesize.Alargedeviceisalsolessflexible,leadingforinstancetolongshutdowns
tosubstituteacomponentofthereactor.

3

Coil for plasma shaping

Toroidal field coils

Magnetic surface

Plasma current

Transformer coil

Field line

Figure1.2:Tokamakconfiguration:theinnermostcylinderisthetransformercoilto
inducethetoroidalcurrentandthusthepoloidalmagneticfield.TheringofD-shaped
coilscreatesthetoroidalmagneticfield.Theresultingfieldlinesrunhelicoidallyonthe
nestedtoroidalsurfaces.Thetwohorizontalcoilsgenerateaverticalfieldfortheradial
plasmaequilibriumaswellasforplasmashaping.

ignitionandtConfinemen1.2.1Thethermalenergystoredinaplasma(see[1])is
W=23VkB(neTe+niTi)dV,
wherethelabelsrefertoelectronsandions,njandTjrepresenttheparticledensityand
temperature,respectively.Inthefollowing,theBoltzmannconstantkBwillbeomitted
forsimplicity.Inthesteadystate,theenergylossesPoutareequaltotheinputpowerPin,
suppliedbydifferentheatingmechanisms.Ameasureofthethermalisolationcapability
ofaplasmaisitsconfinementtimeτE,whichcorrespondstotheheatrecyclingrate:
WτE=Pout.
Thereactionrateisproportionaltotheplasmadensity,andthesocalledtripleproduct
neTiτEisafigureofmeritforthefusionefficiency;ithasbeenshown[1]thatignitionis
reachedwhenthefollowingrelationisfulfilled:

4

neTiτE>3×1021m−3keVs.
Withoutlossofgeneralitywecanintroducethesocalledpowerbalancetransportcoeffi-
cientofthespeciesj:PBqj
χj=−njTi,(1.1)
whereqjistheabsolutevalueoftheheatfluxqj,i.e.theamountofenergyflowing
throughamagneticsurfaceintheunittimeperunitsurface.Therelationbetweenthe
confinementtimeandtheheattransportcoefficientissimply:
njTjVnjTjaVa2
τEj∝qjS∝njχjPBTjS,∝χjPB
beingaistheminorradiusofthetokamak.Therefore,χjPBprovidesameasureofthe
effectiveperpendicularheattransport,regardlessofitsphysicalmeaningandofthemech-
anismgoverningtransport.
orbitsarticleP1.2.2Beforewecandiscussdifferenttransportmechanismswemustfirstlookattheparticle
orbitsinatokamakreactor.Atokamakisadeviceformagneticconfinementofplasma.
Theprincipleissimple:chargedparticlesinauniformmagneticfieldgyratearoundthe
magneticfielddirectionwithafixedradius,calledLarmorradiusandgivenby
Tjρj=2mjΩ2cj,(1.2)
whereΩcj=ejB/mjisthegyrationfrequencyoftheparticlespeciesjinpresenceofa
magneticfieldB;ejistheparticleelectricchargeandmjitsrelativisticmass.Ifρja
particleareconfined.TheycanthenonlyleavetheplasmathroughCoulombiancollisions
(classicandneoclassictransport)orthroughfluctuatingelectromagneticfields.
driftsarticlePActually,inatokamakthemagneticfieldstrengthisnothomogeneousandinaddition
thefieldlinesarenotstraight.Boththecurvatureradiusaswellasthegradientlength
LBareoftheorderofthemajorradiusRandthereforemuchlargerthanρj.However,
everyforceintheplasmaisassociatedwithaparticledrift.Wetrytogiveasimple
physicalpictureofthedriftassociatedwithB.ForsimplicityweassumethatB=Bzˆ
andthatthegradientisperpendiculartothefieldlines,B=B(x)(seeFig.1.3).The
particleinitsgyrationorbitexperiencesdifferentmagneticfieldsandcorrespondinglyits
Larmorradiuswillbesmaller(closerorbit)wherethefieldishigher,broaderwhereB
5

getssmaller.Attheendofagyrationtheparticleisdisplacedperpendicularlytothefield
andtoitsgradient,sothattheorbitisnotclosed.AsFig.1.3shows,the“center”ofthe
gyration(theguidecenter)driftsfromtheidealfield-linefollowingtrajectory.Notethat
electronsandionsdriftinoppositedirectionsbecausethegyrationmotionaroundBis
anticlockwiseforionsandclockwiseforelectrons.

B∇By

+-iexFigure1.3:ParticledriftduetoB.

trappingarticlePAnotherimportanteffectofthemagneticfieldinhomogeneityisparticletrapping.The
magneticfieldstrengthincreasestowardsthetokamakaxis.Sincethefieldlineistwisted
aroundatoroidalsurface,theparticleexperiencesdifferentmagneticfieldstrengthduring
itsmotionalongthefieldline(seeFig.1.4).Thisimpliesthatparticleswithhighv⊥and
lowvarereflectedwhenreachingacriticalvalueBrefofthemagneticfield;this2isknown
as“mirroreffect”.Makinguseoftheconservationofthekineticenergy(E=mv/2)and
ofthemagneticmoment(µ=mv⊥2/2B),whichisanadiabaticinvariant,thetrapping
readscondition12mv20<µBmax−µBmin,
whereBmaxisthemaximumfieldseenbyapassingparticleandv0istheparallelvelocity
attheplacewhereB=Bmin.Foranisotropicdistributionfunction,inatokamakwe
e:vhav20v20Bmax−Bmin2r2
v02−v20=v⊥20<Bmin≈Ro−r=1−,

6

where=r/Roisthelocalinverseaspectratioofthetokamakattheradialpositionr.
Thetrappedparticlesfractionis
ntv0v02
ft=n≈arcsinv0limit≈v0|limit≈1+.
TheprojectionofatrappedparticletrajectoryonapoloidalcrosssectionisplottedinFig.
1.4.Actually,ifparticledriftswerenotpresent,theorbitwouldbealignedtothefieldline

BrefB∇Fluxsurface Fieldline

xIpl

Poloidal projectionof a banana orbit Figure1.4:Trappedparticlesrunalong

andwouldhavezerowidth.ButsinceBincreasestowardsthetorusaxis,bothspecies
driftintheverticaldirection,asmentionedinthepreviousparagraph.AsshowninFig.
1.3ionsdriftupwards.Abovetheequatorialplane(theplaneinwhichthemagneticaxis
lies)theydriftinthepositiveradialdirection,belowtheequatorialplanetheydriftinthe
oppositedirection.Thismakesthatthepoloidalprojectionoftheorbitisclosedbuthas
afinitewidth,takingthecharacteristicbananashape,asshowninFig.1.4.

orttranspclassicNeo1.2.3Thefirstattemptstoestimateheattransportinplasmasperpendiculartothemagnetic
surfaceswerebasedonCoulombcollisions,whichscatterparticlesfromtheirtrajectory
closetothefieldlines.Energyistransferredfromonemagneticsurfacestoanearone
withanaveragedisplacementΔxintheperpendiculardirectioneveryτjcollseconds.Here,
τjcollistheinverseofthe90oscatteringfrequenciesνjj,resultingfrommanysmallangle
scattering.Thetypicalradialstepforions’andelectrons’randomwalkareassumedof
theorderoftherespectiveLarmorradii.Heatdiffusivitiesarederivedfroma“random
walk”Ansatz:
χicl≈ρi2νii≈miρe2νee≈miχecl
mmee

7

andarepredictedtobeoftheorder10−3m2/sand10−4-10−5m2/sforionsandelectrons,
.elyectivrespHowever,theassumptionofuniformmagneticfielddoesnotholdfortokamaks.Amore
comprehensivetheoryhasbeendeveloped[2],whichtakesintoaccounttheeffectsof
tokamakgeometry.Thisisthesocalledcalledneoclassictheory.Themainnewingredient
isthetreatmentofthebananaorbits.Definingthethermalvelocityasvjth=Tj/mj,
thebananafrequencyresultstobeνb≈vjth/qR.Ifνbislargerthantheeffectivecollision
frequencyνeff,j≈νej/2,trappedparticlesrunseveraltimesontheirbananaorbitsbefore
theyarescattered,sothatthebananawidthcanbetakenasrandomwalkstepandthe
collisionfrequencyisstilltheinversetimeconstant.Theradialstepismuchlargerthan
theLarmorradiusandhencetransportisincreased:
Δx≈rb≈q√ρj
q2ρ2νq2
χeneo≈ftrb2νeff∼3e/2ee≈χecl3/2
Forinstance,=1/4andq=3yieldχeneo≈70χecl.Theheatconductivitiesarepredicted
e:btoχeneo≈0.005m2/s
χineo≈χeneomi/me≈0.2m2/s
orttranspAnomalous1.2.4Althoughneoclassicaltransportcoefficientsareconsiderablyhigherthanthosepredicted
bytheclassicaltheoryalone,theexperimentalvaluesarestillmuchhigher:
χeexp≈χiexp≈1−5m2/s
Theneoclassicalpredictionfailsquantitatively,underestimatingtransportbyonetosev-
eralordersofmagnitude.Inotherwords,confinementisinrealplasmasmuchworsethan
accordingtothistheory.Alsoqualitativelytheneoclassicalpredictionisnotinagreement
withtheexperimentalmeasurements,whichexhibitaratioχi/χeclosetooneinsteadof
thepredictedsquarerootofthemassratio.Thismeansthatparticularlyforelectrons
othertransportphenomenadeterioratedramaticallytheenergyconfinementtime,while
iontransportislessaffected.
Scalinglawshavebeenextractedfromtheexperimentaldatabasesofdifferenttokamaks
[3]buttheirunderstandingintermsofphysicsisstillbeinginvestigated.Inparticularthe
powerdegradationofconfinement(thedecreaseofconfinementwithincreasingheating
power)cannotbeexplainedbymeansofneoclassicaltheory.Theadditionaltransport
hasbeencalledanomalousandhasalwaysbeeninvestigatedbythefusioncommunity

8

terest.inparticularwithRecentstudieshaveshownthattheionthermaldiffusivitycanbereducedtoneoclassical
valuesatleastinanarrowplasmaregion,thesocalledInternalTransportBarrier(ITB)
[4].However,thissuppressionofanomaloustransportcansofarbeproducedforalimited
durationonly.ElectronsITBhavealsobeendemonstrated[5],withtemperatureprofiles
assteepasintheionbarriers,buttheneoclassicalvalueismuchlowerthanforionsand
hasnotyetbeenreached.
Nowadaysitiscommonlybelievedthatturbulenceduetomicro-instabilitiescausesanoma-
loustransport:fluctuationsintemperatureanddensitycombinedwithplasmadrifts
wouldcarryanetheatfluxwhichisconsiderablylargerthanneoclassical.Inthelast
decadetheorybasedtransportmodelshavebeendevelopedbasedonplasmaturbulence,
leadingtoconfinementpredictions.Thereisagreementthatthemaincandidatetoex-
plainiontransportistheIonTemperatureGradient(ITG)driventurbulence,possibly
coupledtotheTrappedElectronMode(TEM).Thestabilisingcontributionofthesheared
plasmarotationωE×B[6]isalsocommonlyaccepted.Aphysicalpictureofthetoroidal
ITGinstabilitymechanismisgiveninsection2.2.However,differentmodelsbasedon
thesamebasicmodesyielddifferentpredictions,dependingontheclosureofthesetof
equations,onthestabilisinganddestabilisingtermsincludedandontheapproximations
made.Forelectrontransportuntilafewyearsagomostofthemodellingattemptsreliedon
dels.mosemi-empiricalandempirical

1.3Steadystate:powerbalanceanalysis
Thegoalofthesteadystatetransportanalysisistoinvestigatetheamountofheatflux
crossingthedifferentmagneticsurfaceswhentheplasmahasreachedthesteadystate.
ThiscanbedonebymeasuringtheeffectiveheatdiffusivityχPB,whichprovidesadirect
measureoftheenergyconfinementperformanceofatokamak,asmentionedinsection
1.2.Theenergyequationinitssimplestformreads
nj∂Tj=−∙qj+Sj=njχjPBTj+Sj.(1.3)
t∂Therefore,χjPBisalsocalled“effectiveheatdiffusivity”,where“effective”referstothe
factthatχjPBmightstilldependonplasmaparameters,thusmakingequation1.3not
e.diffusivpurelyInthesteadystate,energyconservation1.3yields:
∙qj=SQjin−SQjout,(1.4)

9

(1.4)

whereSQjinsummarisesthepositivepowersourcesandSQjoutthenegative.Toreconstruct
theexperimentalheatflux,thepowerdensityprofileoftheauxiliaryheatinghastobe
wn.knoToinputrealisticpowersourcesinthetransportequation,furthermeasuredorrecon-
structedprofilesarerequired,suchastheplasmacurrentdensity,themagneticfield
strength,plasmadensityandbothtemperatureprofiles.TheOhmicheatingisapower
sourcedependingontheelectrontemperatureprofile.Radiation(Bremsstrahlung)isa
typicalheatsink;butthewarmerspecieshasanotherheatlossduetocollisionswiththe
colderspecies.Forhighdensitydischargesthisisanimportantuncertaintysourceinthe
powerbalanceanalysis.
Integratingthesteadystatetransportequation1.4overamagneticsurfacelabelledby
theradiusr,weobtainthelocalheatfluxas:
qj(r)=|q|(r)=0rdrdrdVSQjin(r)−SQjout(r),
j4π2Ror
wherewehaveconsideredacircularplasmacrosssectionforthesakeofsimplicity.
IntheliteratureχPBisoftenreferredtoaspowerbalancediffusivity.However,itisin
principleafunctionoflocalplasmaparametersasnj,Tj,Tj,thesafetyfactorq,Zeff,
themagneticshearsˆ,theplasmacollisionality,theratioTe/Ti.Therefore,thesimple
proportionalitybetweenheatfluxandtheproductneTemaynothold.If,forinstance,
significantcontributionstotheheatfluxaredrivenbythelocaltemperature(andnot
byitsgradient),χPBdoesnotrepresentanymorethethermaldiffusivityandexhibits
acomplicateddependenceonthetemperaturegradient.Thesamedifficultyoccursif
transportistriggeredbytemperaturePBgradientsbutonlyaboveacertainthreshold(see
Fig.1.5).Whileitiseasytomeasureχ,inreality
•therearelargeuncertaintiesundercircumstances
•itisquitecomplicatedtointerpreteitsbehaviourandimproveourunderstanding
.phenomenologyorttranspof

1.4Transientstate:heatpulseanalysis
Sinceseveralyearsadifferentkindofexperimentalapproachhasbeendevelopedtogain
moreinformationaboutheattransport.Theideaistoperturbtheplasmatemperature
profileandtomeasuretheheatfluxvariation.Themainquantityforthisanalysisisthe
perturbativetransportcoefficient,definedbytherelation(see[7])
q˜j=−njχjpertT˜j,(1.5)

10

High qe

54High qe3q [a.u.]e2∇Τcrβ
Low q1αe00246810
T [a.u.]∇n ee

Figure1.5:Exampleoftransportmodelwithcriticaltemperaturegradient;χePB=tan(α),
χepert=tan(β).BelowthecriticalthresholdTcr,i.e.forlowqe,χePB≈χepert.Above
Tcr(highqe)χPeB<χepert
wherethetildeindicatestheperturbedquantitiesandjtheplasmaspecies.Itisalways
possibletoclosetheenergyequation,withoutlossofgenerality,with
q=−nχPB(T,α)T
ifoneallowsχPBtodependonTandonanarbitraryplasmaparameterα.Thespecies
labelhasbeendroppedforsimplicity.Linearisingthisclosureyieldstherelationbetween
χPBandχpert:
BPBPχpert=χPB+∂∂(χT)T+∂∂χαα˜˜T.
TThelasttermisoftenassumedtobenegligiblebecausethetemperaturegradientisthe
quantitywiththestrongestvariationswhenatemperatureperturbationpropagatesin
plasma.theSofar,theperturbativetransportcoefficienthasbeenmeasuredonlyforelectrons.A
possibleperturbationarisesfromthepropagationofasawtoothcrash,orcanbeinduced
toartificiallymeasuretheheatingsmalltheplasmaincrementswithbaecausetheradio-frequencyexperimensourcetal[7].errorHobarswever,areitisquitedifficultlarge,
gettingdramaticifonewantstodiagnosevariationsofthetemperaturegradient.In
ordertoreducetheexperimentaluncertaintiesandhaveareliablemeasurementofχjpert,
itisusefultodealwithperiodicperturbationsandperformaFourieranalysisofthe
propagatingheatpulse[8].Actually,theincrementaltransportcoefficientisrelatedto
thepropagationvelocityoftheheatwave,asweshowinthefollowing.
Wecanlinearisethetransportequation1.3:
˜23no∂∂Tt=−∙q˜+S˜.(1.6)

11

(1.6)

Splittingthepowertermsintosourcesandsinks,thelatterturnouttobeproportional
toT˜ifweassumecollisionalenergytransferasmainsink.Forhomogeneousχpertthe
ecomes:b1.6equation3∂T˜=χpert2T˜−T˜+S˜.(1.7)
2∂tτeff
AwavesolutionoftheformT˜=Tωeiωtmustthereforefulfil
3iω+1Tω−χpertTω=Sω.(1.8)
τ2Iftheperiodicsourcesarespatiallylocalised,onecanrestricttheanalysistoaregionof
sourcefreeplasma.ThisiswellsatisfiedinthecaseofElectronCyclotronHeating(ECH)
(seesection5.1).TheFouriercoefficientTωhasaspatialdependence,duetothepulse
propagationanddamping.Inaninfiniteslabgeometrythewaveisassumedtopropagate
injustonedirectionxwithnoboundaryeffects,sothatwecanwritetheperturbed
temperatureinpolarnotationasTω(x)=A(x)eiϕ(x)withA∈+andϕ∈.The
exponentialformTω(x)=ezxz∈Creduces1.8toanalgebraicequationwithsolution
z=±3ω√1+Δ2+Δ1/2+i√1+Δ2−Δ1/2,
tperχ4whereΔ:=2/(3ωτ)isthedampingstrength.Solvingforχperttherealandimaginary
ofpartz=Tω=A+iϕ
ATω

relations:wingfollotheyields3ω√3ω√3ω
χpert=4(A/A)21+Δ2+Δ=4ϕ21+Δ2−Δ=4ϕA/A.(1.9)
Thedetailsofthesolutionofequation1.8andthecorrectionsduetothetoroidalgeometry
conandvbentionoundarytoploteffectsasarephasetreatedprofileinthe[8].quanInthetityφrest=of−ϕthissothatthesispweositivadoptephasethemostdifferencesusual
correspondtoadelay.
Sinceboththeamplitudeandphaseprofilesofthetemperatureperturbationcanbe
measured,conductivitythe.Althoughrelation1.9thisproresultvidesisanapproexpximateerimentalandmakestimateesuseforofthesevperalerturbativassumptions,eheat
italloincremenwstotalrelatediffusivitayfast.wAsaveshowninpropagationFig.(flat1.5,χperphasetmighandtdeviateamplitudefromprofiles)χPB,withaindicatinghigh
thatheattransportisnotsimplydiffusive.Andactuallyintheexperimenttheyare
abobservoutedthetobtranspeortdifferenmectandhanism.theirInratiotheisframepowoferthisdepwendenork,tthe[14],estimateproviding1.9hasnotinformationbeen

12

appliedtoevaluatethecoefficientsquantitatively,buttheheatwavepropagationhasbeen
predictedbymeansoftheorybasedmodels,withalltheirintrinsicdependencesofχon
plasmaparameters.Nevertheless,thegraphicalrepresentationsofAandϕallowadirect
terpretation.in

1.5Contentofthisthesis
Tounderstandanomaloustransportintokamaksremainsaprimarychallengeinfusion
orientedplasmaresearch.Recentprogressesinthedevelopmentofturbulencecodesand
increasingknowledgeoftheexperimentalbehaviourofheattransportencourageaquanti-
tativecomparisonbetweentheoryandmeasurements.Inthisthesisseveraltheorybased
modelstestedfortheITER(InternationalThermonuclearExperimentalReactor)con-
finementdatabaseareappliedtodifferentexperimentalsituations.
IngetherChapterwith2,anaovsimpleerviewphofysicsthemainpictureofingredienthetsoftoroidaltheITGtheoryinstabilitcontainedyisinpresentheted,modelsto-
consideredintheframeofthisthesis.
AfollowsdetailedinderivChapteration3.ofThistheisWmeaneilandtasmoandel’sexampledispofersionhowarelationfluidandmodeltranspisortbuilt,cofromefficienthets
basicBraginskiiequations[9]uptotransportestimates,goingthroughthedetailsand
theChapterappro4conximationstainsoftheonesimofulationthemostresultsused1Dobtainedtranspforortmostandarddels.H-modedischarges.Ion
theandmoelectrondelsareheatcomparedtransportwithisthediscussedexperimenwithtparticularqualitativelyattenandtionquantoprofiletitatively.stiffness,and
Theorybasedmodelshavenowbeenimplementedintocodeswhichareabletodeliver
predictionsonTeprofiles[10][11][12][13].Recentprogressesontheexperimentalside[14]
aswellasthedevelopmentofthesetheoreticalmodelshavebroughtsomeclarifications
andproposedphysicalexplanationsofelectrontransport.Inseveralmachinesathreshold
behaviourofelectrontransportisobserved[14][15][16],andtotheactualknowledgethe
underlyingplasmainstabilitiescouldbeeithertheElectronTemperatureGradientdriven
mode(ETG)ortheTEM.InChapter5theTEMbasedWeilandandGLF23modelsare
appliedtoelectroncyclotronheateddischarges,bothforthesteadystatetemperature
profileaswellastotheheatpulsepropagation.
Preliminarymodellingofdischargesperformedonthe”JointEuropeanTorus”(JET)is
presentedinChapter6withparticularfocusingonthevariationofthecoreheatfluxand
oftheUpgraderatioandTe/TJETi.Thiswithisresptheectbtoeginningprofilesofanstiffnessinter-macandvhinealidationcomparisonofbtheoreticaletweenmoASDEXdels.
FFinallyurther,devtheelopmenresultstsareofthesummarisedpresentwandorkaretheirpropimpactosedatonthetransportconclusionresearcofhthisismandiscussed.uscript.

13

14

2Chapter

moortranspTdels

2.1Theiontemperaturegradientdriveninstability

Theheatlossesintokamakslargelyexceedthepredictionsfromtheneoclassicaltheory.
Plasmafluctuationshavebeenobserved[17]andmaderesponsibleforenhancedtrans-
port.Sinceafewdecadesitiscommonlybelievedthatthisturbulentbehaviourarises
frommicro-instabilitiesdrivenbythebackgrounddensityandtemperaturegradients.For
atokamakreactor,tounderstandlowfrequencymicro-instabilitiesastheITGmodeis
important,becausetheylimitconsiderablytheconfinementperformanceofthedevice
andthusitsfusionefficiency.
Inthequasi-lineartheoryoneassumesthattransportisdeterminedbythefastestgrow-
ingmode,andneglectsthemutualinteractionsofthemodes.AsimpleAnsatzforthe
transportcoefficientisthenprovidedbythemixinglengthestimate,whichleadstothe
scaling2γxΔχ∼Δt∼k⊥2,
whereγisthelineargrowthrateoftheinstability.Withrespecttoconfinement,the
worstinstabilitiesarethosewithlongwavelengthandhighgrowthrate.Thisisofcoursea
simplifiedmodel.Thetheoreticallimitsandinconsistenciesofthisapproacharediscussed
indetailintheintroductionof[18].

2.2SimplepictureofthetoroidalITGinstability

TheITGdriveninstabilityiscommonlybelievedtobethemostsignificantmechanism
limitingionenergyconfinementinpresenttokamaksaswellasinproposedlargerfusion
devicesITER.ashsucTypically,thefastestgrowingITGdrivenmodesexhibitaperpendicularscaleoftheorder

15

oftheionLarmorradius,satisfyingk⊥2ρi2≈0.1.ThismeansthattheITGspatialscale
issmallcomparedtothetokamakminorradius,butmuchlargerthantheDebyelength.
Frequenciesareintherangeofthediamagneticdriftfrequencyω,thereforemuchlower
.frequencyplasmathethanThetoroidalversionofthismodeisbasicallyduetothecombinationofthecurvature
andBiondriftsontheonehandandtheE×Bdriftontheother.Driftvelocities
andfrequenciesaredefinedinTable2.1.Asimplifiedpictureoftheinstabilityintoroidal
geometrycanbegivenassumingalocalapproximation,whereeachFouriercomponentof
theperturbationisconsideredindependent,thusneglectingthepossiblecouplingeffectof
plasmainhomogeneities[18].Theinstabilitygrowsintheunfavourablecurvatureregion,
whereTiandBareparallel;thisisthecaseinthelowfieldsideoftokamaks.A
magneticsurfacehasconstantTi,unlessaperturbationoccursforsomereason(seeFig.
2.1).Thepoloidaliondrift,duetobothBaswellasmagneticfieldcurvature,is

T.iBT∇iB∇

Equilibrium TiPerturbed T , low field sideiFigure2.1:Tiperturbationinthelowfieldsideofatokamak:BTi

proportionaltoTi.Therefore,ifTiisperturbed,thisdriftleadstoacompressionof
iondensityinthepoloidaldirection,withqueueswherethepoloidalvelocitygradient
isnegativeandrarefactioninthezoneswithpositivevelocitygradient(seeFig.2.2).
Quasi-neutralityforcesacorrespondingelectrondensityperturbationwhichisequaland
hencealsoinphasewiththeiondensityperturbation.Neglectingelectrontrappingfor
yieldingsimplicityan,theelectrostaticelectronspcanotentialassumedproptoortionalbetoadiabaticthebdensityecausepoferturbationtheirfastandwithdynamics,the
samephase:eφeφ
n˜e=ne−neo≈neoexpTe−1≈neoTe.(2.2)
Thedensityperturbationis,therefore,associatedwithapoloidalvariationofφ,which
givesrisetoaradialelectrostaticdrift(seeFig.2.3).Thisperpendicularionfluidmotion
carriesplasmafromthesidewithlowerBtothecoldspotandfromthehigherBregion
tothewarmspot.IfthebackgroundTiisparalleltoB,coldplasmaiscarriedtothe

16

efDQuantityinition

vEvpjvjπvjvvcurvBvjD

ωjωjD

Drifttype

e×φ,E×2BE×Borelectrostatic
BBE∂1B0Ωcj∂t+(v∙)EPolarisation
e×∙πjStresstensororanisotropy
ejnjB0
pBj×ejnjB2Diamagnetic
mj<v2>B×BCurvature
2BBejmj<v⊥2>B×BB
2ejBB2
2TjB×Bvcurv+vB
2BBej

neticiamagDeaturvurCBvcurv+vB

k⊥TjnjDiamagneticfrequency
Bnejj2k⊥TjBCurvatureandBfrequency
BeB

(2.1)

Table2.1:Definitionofthedriftvelocitiesandfrequencies.Thesubscriptjlabelsthe
ecies.spplasma

17

vD ∝ T iB x ∇B
ne

v∇Max Dv∇Min D(queue)

φ ∝∼ n i≈∼e n > 0
Figure2.2:nicompressionduetomagneticcurvatureandBdrifts

2.3.coldAregion,netthamounustofamplifyingheatistheptransperturbationortedastoandflattendrivingthethebacinstabilitkgroundy,asionshotempwnineratureFig.

> 0φ < 0φ

E

vE =E x 2B
B

Figure2.3:AmplificationoftheTiperturbationthroughtheelectrostaticdrift,laststep
oftheni−Tiinstabilityloop

gradient:inFig.2.3,theheatflowstowardstherighthandside.Wenoticethat,since
vEandn˜earephaseshiftedbyπ/2,thereisnonetparticletransportaccordingtothis
del.mosimplified

2.3ITGinstabilityandprofilestiffness
AccordingtotheITGmodels,turbulenttransportisexcitedwhentheinversegradient
length1/LTi=−lnTiexceedsacertaincriticalvalue1/LTcr.Asimilarthreshold
mechanismcanbeinvokedtoexplainelectrontransport,forinstancethroughtheso
calledTrappedElectronMode(TEM).Chapter3providesanexampleofhowacritical

18

thresholdeventuallyarisesaccordingtothetheory,withapredictionforITGandTEM
bined.comIftheincreaseoftransportabovetheturbulenceonsetisstrongenough,aclearlyover-
criticaltemperaturegradientcanbeinducedonlybyalargeheatflux.Sincetheheating
powerislimited,thelogarithmictemperaturegradienttendsnottooexceedsignificantly
thecriticalvalue.Ontheotherhand,ifthegradientisunder-critical,noanomalous
transportoccurs,andtheremainingneoclassiccontributionissmallandallowsTto
steepen,until1/LTcrisreached.Thismeansthattemperatureprofilestendtobe“stiff”,
keepingmoreorlessboundtothecriticalT/Tprofile,almostindependentoftheamount
ofheatdepositedandoftheshapeofthepowerdensityprofile.TheITGandTEM
physicsdonotapplyinthepedestalregion,whereothermodesdeterminethedensity
andtemperatureprofiles.Therefore,eveninascenariowithstrongprofilestiffnessthe
boundarytemperaturehasfreedomtovary.However,ifthecriticalgradientlengthdoes
notdependstronglyonplasmaparameters,inthecoreplasmathetemperatureprofileis
fixed:r
T(r)=T(ped)exp−1(r)dr=α(r)T(ped).(2.3)
LpedTcrGradientscanbesteepenedjustbyenhancingthepedestaltemperature,regardlesswhether
theinputpowerisincreasedornot.Ontheotherhand,aheatfluxvariationdoesnot
affectthetemperaturegradient;intermsofχ,thisleadstoastrongpowerdegradationof
theeffectiveheatdiffusivity:χ∝PH−1eat.Thisbehaviourisquitedifferentfromthepredic-
tionrelyingonasimplediffusivemodel,whereχisconstantandtransportisdetermined
bytheFourier’slaw
q=−nχT(2.4)
Inthiscase,thetemperaturegradientisproportionaltothelocalheatfluxanddoesnot
dependatallonthepedestaltemperaturevalue.Thedifferentbehavioursareillustrated
inFig.2.4.Asequation2.3shows,ifthecriticalthresholddoesnotchangetoomuch
fromdischargetodischarge,thentheratioofcoretopedestaltemperatureremainsfairly
constantthroughoutthedatabase.Thisisanexperimentalevidenceofprofilestiffness,
althoughitisnotaproof:intheexperimentitisnotalwayseasytodecouplethesit-
uationsa)andb)ofFig.2.4,inparticularwhenperformingapowerscan,becausethe
pedestaltemperatureincreaseswiththeheatingpower.Itiseasiertodistinguishbetween
a)andb)bymeansofadensityscan,wheretheboundarytemperatureischangedby
factorsof3-4withoutaffectingthetotalamountofheatdeposited.
AmeasureofthetendencyofT/Ttostayclosetothecriticalinversegradientlength
ishowsteeplyχincreasesafterR/LTexceedsitscriticalvalue,i.e.thequantity
∂χ/∂(R/LT),whichiscloselyrelatedtothedefinitionofincrementaltransportinequa-
tion1.5.Thisquantitycanbemeasuredforelectronsbymeansofheatwaves,asdiscussed
insection1.4,providingdirectinformationabouteventualprofilestiffness.

19

2

1

[a.u.]02 T

1

00

ρtor

(a)

(b)

0.8

Figure2.4:Exampleoftemperatureprofile(thicksolidline)anditsmodificationsaccord-
ingFixedtoathebconstanoundarytχmoT,delincreasing(thinsolidtheline)heatingandptoowaer.(b)stiffness-moFixeddeltheinput(dashedpowline)er,cwhen:hanging(a)
.Tedestalpthe

2.4Theorybasedtransportmodels

IntheITERconfinementdatabaseseveral1Dtransportmodelsareemployedtogive
thempredictionsareabclassifiedouttheaspsemi-empirical,erformancesoftheothersnextarederivgenerationedfromfusionfirstdeviceprinciples.[19].WeSomehavofe
studiedthelatter,inordertoimprovethephysicsunderstandingandtovalidatethe
underlyingphysicsagainstASDEXUpgradeandJETdata.Inthefollowingparagraphs
weprovideageneraloverviewonthefourmodelsemployedforourtransportstudies:
threeofthemarebasedonthephysicsoftheITGdriventurbulencecoupledtotheTEM,
onnamelyadifferenthetphIFS/PPPLysics,i.[20e.],Wtheeiland[12self-sustained]andGLF23turbulence[13]modrivdel,enbywhereasthetheCurrentfourthDiffusivreliese
BallooningMode(CDBM)[21].
OnlytheMulti-ModeModel(MMM)[22]hasnotbeenappliedsofar,sinceithasbeen
implementedonlyrecentlyintotheASTRAtransportcode.However,wefocusedour
studiesinthecoreplasmaandinthatregionthemulti-modemodelincludesaversionof
theWeilandmodelfortheITGandTEMinstabilities.
Noneofthemodelsconsideredhere,intheversionsusedforourstudies,hasadjustable

20

parametersoradhocassumptionsforaparticulartokamak.
ThestabilisingcontributionoftheωE×Bshearisincludedinallmodelsfollowingdifferent
4.4.1).Section(seetationsimplemen

delmoIFS/PPPLThe2.4.1Themodelhasbeendevelopedintheearlynineties.Itsnameisrelatedtotheinstitutes
whereitwasdeveloped:InstituteforFusionStudies(Austin,Texas)andPrincetonPlasma
PhysicsLaboratory(Princeton,NewJersey)[20].
Themodelreliesonfirstprinciplesnonlinearfluidsimulations,correctedandcompleted
byacomprehensivelineargyrokineticcode[23].Analyticexpressionsforthetransport
coefficientsarederivedbyfittingthetheoreticalsimulations:
•theformulaforχiisgivenbygyrofluidsimulationswithcorrectionsfromthequasi-
lineargyrokineticestimate,inordertoreducethelargeamountofnon-lineargy-
rofluidruns,whichrequirealongcomputingtime.
•“Theelectronχeisobtainedfromtheratioofthequasi-linearelectronandionheat
fluxesfoundwiththecomprehensivelinearcode”([20],pag.2383);theratioisfound
tobeaweakfunctionofplasmaparameters,sothationandelectrontransportscale
.similarly•AnomaloustransportduetoITGistriggeredonlyiftheiontemperaturegradients
exceedacertainthreshold.Thethresholdisobtainedfromthelineargyrokinetic
de.co•Besidetheiontemperaturegradient,thethresholddependsonthedensitygradient,
themagneticshear,thesafetyfactor,collisionality,theratioTe/Tiandtheeffective
ionchargeZeff.
•Profilestiffnesscanbemeasuredbythehighincreaseofχiabovethecriticalgradi-
ent;thecoefficientbeforetheHeavisidefunctionhasanimportantTi3/2dependence,
factor.Gyro-Bohmcalledsothe

2.4.2TheWeilandmodel
ThisisafluidmodelbasedonITGandTEMcoupling[12].Thederivationofthemodel’s
equationsandofthequasi-lineartransportcoefficientsispresentedinChapter3.Now
weshortlysummarisethemostrelevantapproximations:

•Therearethresholdsinbothionandelectrontemperaturegradientlengths;ifthe
gradientsarebelowthecriticalvalue,transportisneoclassic.

21

•Theclosureisobtainedtakingtheheatfluxasthediamagneticheatfluxwith
erature.tempisotropic•Inalthoughtheversionmorerecenconsideredtversionshereof(7themoequations),deldoconsiderelectromagneticthemtoeffectso.Theareionneglected,parallel
motionandsheareffectsareincluded,buttheirdescriptionissimplified.
•Thebackgroundelectrostaticfieldisassumedtobezero,exceptwhenconsidering
thestabilisingtermωE×B.

delmoGLF23The2.4.3TheGLF23(Gyro-LandauFluid)modelisalsobasedonthefluidequations[13].However:
•TheclosureisdifferentfromthatoftheWeilandmodel.“Thecomplexcoefficients
oftheselinearcombinations[ofthelowermoments]arechosentobestfitthegeneral
kineticplasmaresponsefunctionoverthefullrangefrom2smallandlargevaluesin
allthekineticparameters:thegyroradiusparameter(k⊥ρ)/2;theparallelmotion
parameterkvth/ω;andthecurvaturedriftparameterωD/Ω.”([24],pag.3138).
•“Thetoroidaliontemperaturegradient(ITG)mode,thecollisionlesstodissipative
trappedelectrondriftmodes,andtheidealmagnetohydrodynamic(MHD)balloon-
ingmodes,aswellastheedgeresistivewallmodes,areincluded”([13],pag.2482).
•Themodelassumesamagneticshear(sˆ)-Shafranovshift(α)stabilisation.Landau
dampingistakenintoaccount.

delmoCDBMThe2.4.4TheCurrentDiffusiveBallooningModemodelisbasedondifferentphysicsthanthe
previousthreemodels[21].
•Thenonlinearinstabilitiesgenerateanomaloustransport.“Inthisnewtheoretical
approach,instabilitiesarecausedbyanomaloustransportitself”([21],pag.1743),
i.e.turbulenceisself-sustained.“Inthisnon-lineardestabilisationmechanism,the
roleofthecurrentdiffusivityisessential”(ibidem).
•Themodelhasinprinciplenothresholdintheiontemperaturegradient.
•Electronandionheatdiffusivitiesareassumedtobeequal.

22

2.4.5Maindependenceswithinthemodels
Aquantitativecomparisonbetweenthetransportmodelsandtheexperimentsmightnot
beconclusivetojudgewhichphysicsunderliestheexperimentalphenomenology;itis
muchmorereliabletoobserveacorrectdependenceontherelevantphysicalparameters.
Inaddition,somemodelsareconstructedortestedonlyforcertainparametersranges
anditisimportanttochecktowhichextenteachmodelisapplicabletosimulatethe
experiments.Forthisreasonweaddressnowtotheinvestigationofthemodels’reaction
tothephysicalquantitieswithoutcaringaboutthereproductionoftheexperimentalpro-
files,inthatwerunthemodelsstand-alone.Theonlylinktotheexperimentisthatwe
chooseasbackgroundparametersthevaluesathalfradiusofatypicalcaseofNBIheated
H-modedischargeaswellasoneon-axisECHdischarge.Eachparameteristhenvaried
separately,althoughinconcreteexperimentalsituationsitisnotalwayspossibletoscan
asinglequantity,sinceotherparametersmightbecoupledandchangeaswell.Thetwo

DischargeR/LTiR/LTeR/LneqsˆZeffTiTeTe/Tine
130427.45.33.21.80.91.13.42.80.83.7
1355829.10.41.70.91.10.72.43.22.0
Table2.2:Backgroundparametersforthestand-alonerunsofthemodels.Tiisnot
measuredfor#13558.ThearbitraryvaluesTi=0.7,R/LTi=2aretaken.

standardsetsofbackgroundparametersarepresentedinTable2.2.Giventherespective
valuesofneandTe,thecollisionalityinthetwodischargesisroughlythesameandfairly
w.loWesplittheexercisefortheITG/TEMmodelsononehandandfortheCDBMmodel
ontheotherhand,sincetheycontaindifferentdependences.
AlthoughtheIFS/PPPL,GLF23andWeilandmodelsarebasedonthesameinstability
mechanisms,theyyielddifferenttransportpredictions,alsoqualitatively,duetothedif-
ferentapproachesandapproximations.IntheWeilandmodel,thecollisionsontrapped
electronsareswitchedoff,inagreementwiththestandardsetupassumedinthisthesis(see
Section4.4.1)andmostcommonintheliterature.ThestabilisingeffectofωE×Bhasbeen
neglectedforallmodels,inordertodistinguishmoreclearlythephysicseffectswithin
themodels.TheITG/TEMmodelsareprimarilysensitivetoR/LTi,R/LTe,R/Lne,the
safetyfactorq,sˆ,theeffectiveionchargeZeff,theratioTe/Tiandthetemperaturevalues.
Thescannedquantitycanvarythetransportlevelfortwocontributionswhicharenot
decouple:toeasy•increasingthatparameterenhancestheturbulentheatflux

23

•thecriticalgradientlengthisshiftedbecauseitdependsonthescannedparameter
Soasafirststepweanalysethethresholdbehaviourofthemodelswithrespecttothe
temperaturegradients.Thisinvestigationisalsoadirecttesttodeterminehow“stiff”are
thetemperatureprofilesaccordingtotheITG/TEMmodels,inthespiritoftheexpla-
nationinSection2.3.Profilestiffnessisactuallyacrucialquestionintransportresearch
nowadays.Figure2.5(a)illustratesthedifferentbehavioursoftheITGturbulence:the

# 13558# 13042100500(b)(a)8040060300q /(n T ) [m/s]q /(n T ) [m/s]ii200ee40
ei2010000051015051015
R/LTiR/LTe

Figure2.5:Dependenceoftheheatfluxesontheinversetemperaturegradientlengths
accordingtotheGLF23(solidline),Weiland(thindashedline)andIFS/PPPL(thick
dashedline)models.(a)Ionheatfluxversusiontemperaturegradient,background
parametersfromTable2.2,firstrow.(b)Electronheatfluxversuselectrontemperature
gradient,backgroundparametersfromTable2.2,secondrow.Theverticallinerepresents
theexperimentalvalueofthescannedquantityasreportedinTable2.2.TheIFS/PPPL
modelisnotplottedin(b)becauseitdoesnotdependonR/LTe.

WeilandandGLF23modelshaveasimilarITGstabilitythresholdaroundR/LTi=4,
IFS/PPPLsomewhatlower.However,theGLF23modelhassignificantresidualtransport
evenforflatTiprofiles,duetothehighdensitygradient(seealsoFig.2.6).TheGLF23
andIFS/PPPLmodelsexhibitasteepincreaseoftransportabovethethreshold,whereas
theWeilandmodelreturnsasmoothergrowth.Inotherwords,Tiprofilesarepredictedto
be“stiffer”forIFS/PPPL,lessforWeiland;GLF23hasnotaclearthresholdbehaviour
undercircumstances(e.g.highni),beingtheresidualtransportalreadyquitehigh.
TheabsolutevaluesoftheionheatfluxaremuchhigherforGLF23andIFS/PPPLthan
forWeiland,atleastforthebackgroundparametersasinTable2.2.However,thisisnot
sosignificantbecauseaslightlylowerinputgradientreducesthetransportlevelconsider-
ably:runningthemodelsinpredictivemode,thecalculatedprofilescanstillbewithinthe
experimentaluncertainty.Forthisreason,intheframeofanITGdominatedtransport
itisnotusefultoperformalocalanalysisintermsofχ’s,beingR/LTcrthedetermining

24

.ytitquanInFig.2.5(b)theIFS/PPPLmodelisnotshownsinceithasnoexplicitdependenceon
R/LTe.TheGLF23andWeilandmodelexhibitsimilarbehaviouranddonotpredicta
strongTeprofilestiffness,astheheatfluxincreasessmoothlyabovethecriticalthreshold.
However,therearesignificantdifferencesbetweenthemodelswhichwillbediscussedin
5.8.SectionindetailInthetheorythedensitygradientsarecloselyrelatedtoR/LTiandR/LTe,whichenter
theequationsalmostalwaysinthecombinationηi=Lni/LTiandηe=Lne/LTe.Also
theTEMstabilitythresholdisinfluencedbythedensitygradient,asitwillbeclarified
inEquation3.19andFig.3.1.Inaddition,sincetheexperimentaluncertaintiesonthe
experimentalprofilescanleadtolargeerrorsinthedensitygradient,itisnecessaryto
evaluatethesensitivityofeachmodeltothisparameter.Figure2.6illustratesitsimpact
ontransport.Boththeionaswellastheelectrondensitygradientlengtharevaried,
keepingLni=Lneinordertopreservequasi-neutrality.WhiletheIFS/PPPLandWei-
# 13558# 13042100100500(a)(b)80804006060300q /(n T ) [m/s]i10020q /(n T ) [m/s]e20
ii20040ee40
005R/Ln10150005R/L1015
nFigure2.6:Ion(a)andelectron(b)heatfluxversusthedensitygradientlength.Lines
andbackgroundparametersasinFig.2.5.NoticethedifferentscalefortheWeiland
output.delmolandmodelsreactweaklytodensitygradientchanges,GLF23isaffecteddramaticallyin
bothcases.Thisparticularsensitivityiswellknownandwillbediscussedindetailwhen
evaluatingtheactualsimulationresults,inSection4.4.2.Weonlyremarkthatthelocal
analysismayyieldlargeerrorsbecauseweusetheexperimentaldensityprofiles.There-
fore,themodellingmusttakeplaceovertheentirecoreplasmaregion,inordertoreduce
onaveragetheerrorduetothedensitygradient.
FortheWeilandmodel,theTEM(dominatingtransportinFig.2.6(b))shouldbesta-
bilisedbyahighelectrondensitygradient;thisisobservedifonevariesonlyR/Lne.
However,sinceinFig.2.6alsoR/Lniisvariedforconsistency,othermodesdrivenbythe
iondensitygradientcoupletothepureTEMandincreasetheturbulentheatflux.
25

TheIFS/PPPLformulasforχ’sarenotconstructedforR/Lne>6,thereforeforsteeper
densitygradientsnoinfluenceonheattransportisassumedandthemodelisnotexpected
toreproducethetemperatureprofiles.
AnotherimportantparameterfortheITGturbulenceistheratioτ=Te/Ti;itsdirect

# 13558# 13042150500(a)(b)400100300eq /(n T ) [m/s]iq /(n T ) [m/s]ei20050ei1000024681000246810
T [keV]T [keV]ei

TiFigure(b)2.7:ElectronDepheatendencefluxofversustheTITGe.moLinesdelsandonbacthetempkgrounderature.parameters(a)IonasinheatFig.flux2.5.versus

impactontransportcanbeinvestigatedvaryingitartificially,keepingconstantbothTi
andTe.Instead,wereportinFig.2.7theexperiment-relevantscans,whereeitherTi(a)
orTe(b)arevaried.Actuallythetemperaturevaluesstronglyaffecttransport,deter-
miningtowhichextentthetemperatureprofilesareexpectedtobestiff,asthemodels
containaT1.5dependencebeforethetransportcoefficients-thesocalledGyro-Bohm
factor.Noticethatthetemperaturegradientlengthsarekeptfixedtothevaluereported
inTable2.2.Fortheioncase,boththeWeilandandGLF23modelsreachamaximum
destabilisationofthemodewhenTiequalsTeandthenbendtoalowertransportlevel;
theIFS/PPPLmodelpredictstheturbulentfluxtoincreasemonotonicallywithTi.Hot
electrontemperaturesenhancetheTEMturbulenceaccordingtoallmodels.
ImpuritiesaretreateddifferentlybythemodelsasillustratedinFig.2.8.In(a)itappears
thattheIFS/PPPLandWeilandmodeldescribealsoimpuritytransport,whereasGLF23
intheversionwehaveusedtakessimplyadilutionapproximationwhichcausesZeff
toalwaysmoderatetheITG.IntheIFS/PPPLmodeltheITGcarbonbranchbecomes
overwhelmingforZeff>4.FortheTEMmode(seeFig.2.8(b))Zeffhasnearlyno
effect-weremindthatwithintheIFS/PPPLmodelχeistakenasroughlyproportional
.χtoiFinallywereportthedependencesonthesafetyfactorq(Fig.2.9)andthemagnetic
shear(Fig.2.10).TheWeilandmodeldependsveryweaklyonq,whichinthecore
plasmarangesbetween1and4;mostprobablytheassumptionfortheq-dependenceis
toosimplified.Ingeneral,qisobservedtoraisetheturbulencelevel.

26

# 13042(a)

# 13558(b)

# 13558# 1304225400(b)(a)2030015ei200q /(n T ) [m/s]i100q /(n T ) [m/s]e5
ei1000123456123456
ZZeffeff

Figure2.8:Ion(a)andelectron(b)heatfluxversustheeffectiveioncharge.Linesand
backgroundparametersasinFig.2.5.

# 13558# 1304250050(b)(a)4004030030q /(n T ) [m/s]q /(n T ) [m/s]ii200ee20
ie1001000012345012345
q (safety factor)q (safety factor)

Figure2.9:Ion(a)andelectron(b)heatfluxversusthesafetyfactorq.Linesand
backgroundparametersasinFig.2.5.

TheIFS/PPPLformulasforχ’sarenottestedforlowandnegativeshear,below0.5,
asreportedbytheauthors[20].TheWeilandmodelassumesadependenceonlyon|sˆ|,
astrongapproximationwhichmakesthemodelinadequatetodescribenegativeshear
scenarios.Inaddition,thedependenceonsˆisobservedtobelinear.TheGLF23model
hasamoreaccuratedescription,withgoodoverlappingwiththeIFS/PPPLmodelinthe
validityrangeofthelatter.Themagneticshearisfoundtostabilisetheturbulencefor
highaswellasnegativevalues.
TheCDBMmodeldependsmainlyonq,sˆandcollisionality.Intheversionavailable,
ionandelectronheattransportareassumedtobeequal.Themaximumdestabilisation
forionheateddischarges(seeFig.2.11(a))isobtainedfor1.5<q<2,buttransport
remainshighoverthewholeplasmacore.Forelectronheatingdominateddischarges,the

27

# 13042

# 13558

(b)

# 1355810050050(b)(a)80400406030030eq /(n T ) [m/s]i10020q /(n T ) [m/s]e10
ii20040e20
000- 2- 10∧123- 2- 10∧123
ss

Figure2.10:Ion(a)andelectron(b)heatfluxversusthemagneticshearsˆ.Linesand
backgroundparametersasinFig.2.5.

# 13558

# 13558# 130422580206015q /(n T ) [m/s]q /(n T ) [m/s]ii40ee10
ie20500012345012345
q (safety factor)q (safety factor)

Figure2.11:DependenceoftheCDBMmodelonthesafetyfactorq.

turbulentfluxdrivenbytheCDBMislowerandreachesitsmaximumaroundq=4,
whichbelongstothepedestalregionfortypicalASDEXUpgradedischarges.
ThedependenceonthemagneticshearisshowninFig.2.12andhasaquitedifferent
shape.Themodeisstabilisedforlowandnegativesˆ,thentransportexperiencesastep-
likeincrease;thisis,however,muchsmallerinthecaseofthebackgroundparameters
takenfromtheelectronheateddischarge.
TheoveralllowertransportlevelfortheECHdischargecanbeprobablyexplained
throughtheloweriontemperature.Figure2.13illustratesthebehaviourofheattrans-
portforincreasingtemperature.
Consideringthedependencesonqandsˆ,itisinterestingtolookattheeffectoftheplasma
currentontheentiretemperatureprofiles.Thedischarge#13042withPNBI=5MW,
Ipl=1MAandmediumdensityissimulatedwiththeCDBMmodel,settingthebound-

28

1008060q /(n T ) [m/s]ii40i200

# 13042

# 13558

100543eq /(n T ) [m/s]e2e10- 2- 10∧123- 2- 10∧123
ss

Figure2.12:DependenceoftheCDBMmodelonthemagneticshearsˆ.

# 13558# 1304240803060q /(n T ) [m/s]i20q /(n T ) [m/s]e10
ii40ee20
0024681000246810
T [keV]T [keV]ie

Figure2.13:DependenceoftheCDBMmodelonthetemperaturevalues.

aryconditionforthetemperaturesatρtor=0.8,whereρtoristhenormalisedtoroidalflux
coordinate.Theinputcurrentisvariedartificiallyfrom0.8to1.2MA,correspondingtoa
40%variation.Thischangessignificantlytheedgeqvaluesfromq(ρtor=0.95)=4.8(for
0.8MA)toq(0.95)=3.0(with1.2MA).Whentheqprofileisstationary,themodelled
temperaturesappearnottobeaffectedbythechangeintheplasmacurrent:Ti(0.4)=2.49
keVandTe(0.4)=2.06keVinthecasewith0.8MA,Ti(0.4)=2.48keVandTe(0.4)=2.07
keVinthecasewith1.2MA.Onereasonisthatqvaluesarelessmodifiedinthecore
region,wherethetemperatureprofilesaremodelled.Besides,Fig.2.13showsaresilience
ofthetemperaturetodeviatefromagivenvalue:assoonasthisisexceeded,highheat
transportisgeneratedandrecoversthepreviousvalue.PlottingtheanalogousofFig.
2.13(a)inthecasesof0.8MAand1.2MAyieldsindeedsimilarresults.

29

30

3Chapter

ThetheoryoftheWeilandmodel

ductiontroIn3.1

InthefollowingwederivetheequationsoftheWeilandmodelandthesubsequentquasi-
linearestimateforheatandparticlefluxes.Althoughnonewphysicsisaddedtothe
originalWeilandmodel,thisreviewunifiesresultsfromdifferentworks,suchas[25]and
[12].Mostfinalresultsarepublished,buthaveneverbeenderivedexplicitelyinthe
publications.Thissectionalsoprovidesanattempttolisttogetheralltheassumptions
containedatdifferentstagesinthemodel,givinganoverviewonthephysicsapproach
andonthemodel’sconsistency.
ThemodelwederivehereisthesimplestonecontainingITGandTEMmodescoupled,
whichcorrespondstothefourequationsversion.
Accordingtotheequationsandtransportcoefficientsderivedinthefollowing,asimplified
modelhasbeenimplementedasaFortrancodeinordertobenchmarkthemorecomplete
modelandallowamoredirectcontrolonthebehaviourofthedifferentphysicsingre-
dients.ModellingresultsofJETdischargeswiththissimplifiedversionoftheWeiland
modelcanbefoundinChapter6togetherwiththesimulationsperformedwiththemodels
2.4.sectionintedpresenAmongtheseveraltransportmodelsconsidered,theWeilandmodelhasafairlytrans-
parentphysics,basedonthefluidequationsandquasi-lineartransportcoefficients.It
containsonlyalgebraicequationsandtheformulasareanalytic.Nevertheless,itpredicts
temperatureprofilesingoodagreementwiththemeasurementsinavarietyofexperimen-
conditions.talComplementstothefollowingderivationcanbefoundinAppendixB.

31

Assumptions3.2TheWeilandmodelisafluidmodelbasedonthephysicsofiontemperaturegradients
andtrappedelectronmode.Transportisinducedwhenacriticalthresholdintheinverse
gradientlength1/LTjisexceeded.
Besidesthedriftvelocitiesandfrequencies(seeTable2.1),itisimportanttodefinethe
tities:quanwingfolloe=BB0
0ηj=Lnj
LTjn=2Ln
LBTe=csmiβj=4πpj
2BTeρs=miΩ2ci
Te=τTiNj=ω2−5ωωDj+10ω2Dj.(3.1)
33Derivingtheequations,itisassumedforsimplicitythatthereisonlyoneionspeciesand
thatitisanisotopeofhydrogen.Therefore,eitherj=iorj=eandZ=1.Impurities
arenottreatedandtheparallelionmotionisneglected.
Themodelreliesbasicallyonthefollowingphysicsassumptions:
•k/k⊥1,andintheperpendiculardirectionforthelinearcasekxky,where
xreferstotheradialandytothepoloidaldirection.Thiscorrespondstothefact
thatthelinearmodeswhichtransportmostoftheheatexhibitelongatedstructures
intheradialdirection.Asaconsequence,≈ikyandk∙vD≈ωD.However,
whentheinstabilitysaturatesduetononlinearterms,itisassumedtobeisotropic:
kx≈ky.
•Finiteβeffectsareneglected,inparticular
δB≈−µ0δ2p1.
BB00

32

•Passingelectronsareassumedtobeadiabatic:
δnep=eφ.(3.2)
Tneep•Thetrappedelectronsdynamicsissymmetrictotheiondynamics.
•Theplasmaisquasi-neutral:ne≈niissatisfiedalsoatthefirstorder,δne≈δni.
•Theheatfluxissetequaltothediamagneticheatflux(seeAppendixB.3.2).This
meansthattheBraginskiiequationsareretaineduptothesecondmoment,with
closurecomingfromthethirdmomentbyassumingthatthedistributionfunction
ellian.Maxwis•ThereisnobackgroundEexceptwhenthestabilisingeffectofωE×Bisimplemented
(notself-consistently).E⊥isassumedtobeelectrostatic.Electromagneticeffects
arenotconsideredintheversionofthemodelemployedforthisthesis.
•Thedriftordering(see[26],paragraph3.V)applies:
ρscsδneφ
δ=L⊥=L⊥Ωci∼n≈T1,
whereρsandcsaredefinedin3.1andrepresentrespectivelytheionLarmorradius
timesTe/Tiandthesoundwavevelocity.L⊥isthesmallestgradientlengthof
abackgroundquantity,suchastemperatureordensity.Forfrequenciesthesame
reads:orderingc∂∂t∼ω∼vE∙∼Ls⊥Ωci.
Practically,itisassumedk2ρs2≈0.1whichyieldsρs/L⊥kρs/2π≈0.05.Finite
Larmorradiuseffectsareretaineduptok2ρs2;electronLarmorradiuseffectsare
neglected.

equationsBraginskiiThe3.3Themostconvenientandcommonwaytostudyheattransportintokamaksisthefluid
approach.Thismethodallowsinparticulartodefinemeasurablequantitiesfortheheat
transportanalysisanddiagnosetheenergyconfinementcapabilityofaplasma.
ThefluidequationsasderivedbyBraginskii(see[9])arethemomentsoftheVlasov
equation∂∂ftj+w∙fj+mej(E+w×B)∙∂∂fwj=Cj+Sj(3.3)
jwhereCjistheCoulombcollisionoperatorandSjanexternalparticlesource.Taking
thezeroth,firstandsecondmomentweobtain:

33

QuantityDefinitionPhysicalmeaning
njd3wfjParticledensity
vjd3wfjw/njFluidvelocity
Snj3d3wSjParticlesource
Fj3dwmmjj|ww−Cvjj(|2fj)Collisionalmomentum
pjdw3fjPlasmapressure
πjd3wmjfj(w−vj)(w−vj)−pjIStresstensor
qjd3w(w−vj)×m2j|w−vj|2fjHeatflux
SEjd3wmj2wSjEnergyfromparticlesource
Qjd3w21mj|w−v2j|2Cj(fj)Collisionalheatexchangerate
Table3.1:Definitionofthefluidsquantities.Thelabelreferstotheplasmaspecies.

(3.4)

•Continuityequation:
n∂j∂t+∙(njvj)=Snj(3.4)
equation:balancetumMomen•mjnj∂vj+mjnjvj∙vj=njqj(E+vj×B)+Fj−pj−∙πj(3.5)
t∂equation:Energy•∂1315
∂t2mjnjvj2+2pj+∙2mjnjvj2+2pjvj+πj∙vj+qj=
(qjnjE+Fj)∙vj+Qj+SEj(3.6)
wherethementionedquantitiesaredefinedinTable3.1.Subtractingfromequation3.6
theequation3.4timesmvj2/2and3.5timesvj∙oneobtains:
5∂32∂t+vj∙pj+2pj∙vj+π:vj+∙qj=SQj(3.7)
wherethesourcetermSQjisthesumofdifferentcontributions:SQj=Qj+SEj−
mjvj2Snj/2.

3.4Derivationofthedispersionrelation
ThestartingpointfortheWeilandmodelistheBraginskiisetofequations3.4,3.5and3.6,
neglectingsources(SnjandSEj)andCoulombiancollisionterms(QjandFj).Wemulti-
plythe3.5timesmj1nje×andmakeuseofthevectoridentity(2)from[27].Furthermore,

34

weassumetheelectrostaticapproximationB=B0e.
dtde×vj=mqje×E+B0vje∙e−ee∙vj−m1ne×(pj+∙πj).

jjj(3.8)Sincevje∙e−ee∙vjisv⊥j,dividingbyΩciwefind(see[25],(2.2)):
11d1
v⊥j=B0E×e+Ωcjdte×vj+qjnjB0e×(pj+∙πj)=vE+vpj+vj+vπj.
(3.9)wherevE,vpj,vjandvπjarethedriftvelocitiesdefinedinTable2.1.Thesecondterm
actuallyreducestovπjundertheassumptionthattheelectrostaticdriftvelocityvEis
themaincontributiontov⊥j.Theioncontinuityequationthenreads:
∂ni=−∙(nivi)−∙(nivE)−∙(nivpi)−∙(nivπi)−∙niv.(3.10)
t∂Inthefollowingderivationweneglectforsimplicitytheparallelionmotionterm∙niv,
althoughtheWeilandmodeltreatsitinthestrongballooningapproximation[28].An
estimateofthisquantitycanbefoundinappendixB.2.Weintroduceinthecontinuity
equationanharmonicperturbationn˜j=nj−nj(0)bydefining
n˜j=δnjei(k∙r−ωt)(3.11)
Thefrequencyisingeneralcomplex,ω=ωr+iγ.Inthisconvention,aninstabilityoccurs
ifγ>0.Thelinearisedfirstorderequationhastheform:
∂n˜i=−ni(0)∙v˜drift−ni(0)∙v˜drift−n˜i∙vdr(0)ift−n˜i∙vdr(0)ift.(3.12)
t∂•vDiisnofluiddrift,soitdoesnotappearinthefluidequationsanddoesnot
contributedirectlyto∙(nvi).
•n˜i/nicanbeassumedtobeorder1atthesaturation,butinthelinearcaseit
small.arbitrarilyis•Theproductsn˜i∙vE(0)andn˜i∙vE(0)shouldbeconsidered,butweassumethatthere
isnobackgroundE.Inthefollowing,wewritevEinsteadofδvEforsimplicity.
toreduces3.12equationSo−iωδni=−ni(o)∙vE−ni(o)∙vE+Firstorder{−∙[ni(vpi+vπi)]−∙(nivi)}.
(3.13)InappendixB.3allfirstordercontributionsareevaluatedusingtheassumptionsmen-
tionedinSection3.2.Inthisway,equationsforthedensityperturbationsofions,trapped

35

electronsandpassingelectronsareobtained.AstheequationsB.68andB.69show,the
frequenciesareoftheorderofthecurvaturedriftfrequencyωDe,thereforeitmakessense
tonormalisethemtothisphysicalquantity:ωˆ=ω/ωDe,Nˆj=Nj/ω2De.Thedispersion
relationresultsfromquasi-neutrality:
NˆiNˆenTeδni−ftδnet−(1−ft)δnep=0.(3.14)
eφninetnep
SubstitutingtheresultsB.68,B.69and3.2thedependenceofthedispersionrelationon
thephysicalplasmaparameterscanbemadeexplicit:
Nˆe−ωˆ2k2ρs2n+ωˆ1−n−k2ρs25n−k2ρs21+ηi+
17553ττ
−τηi−3+3n−k2ρs23τ2(1+ηi)=
=(1−ft)NˆiNˆen+ftNˆiωˆ(1−n)+ηe−7+5n.(3.15)
33Theequationisafourthdegreepolynomialinω,whichallowsuptotworootswith
positiveimaginarypart.Intermsofphysics,thismeansthatthereareuptotwounstable
des.moIfnisaround1(Ln≈R/2),themodesareratherindependentandpropagateinopposite
directions.Thedispersionrelationisthenwellapproximatedneglectingthepartwiththe
largerNj(see[25],par.5.11.2).IfNiNe,theTEMbecomesdecoupledanddominates
thedispersionrelation.Thisisthecaseforfrequenciesclosetotheresonantvalues:
√ω=ωDe5±10.(3.16)
3Sinceusually|ω|>|ωDj|,thecondition3.16canbesatisfiedif|ωDe|>|ωDi|,i.e.for
Te>Ti.TheformulasfortheTEMstabilitythresholdandthegrowthrateofthemode
form:simpletheassume10757
ωˆ2(1−ft)n+ωˆ−3n+ft+3ftn+3n+ftηe−3=0.(3.17)
Ifthereisanunstableroot,theimaginarypartofthesolutionis
f√γˆ=n(1t−ft)ηe−ηth
101−ftft122
ηth=9nft+1−ft4n(1−n)+3.(3.18)
Remindingtharηe/n=R/2LTe,thestabilityconditioncanbeexpressedintermsofthe
temperaturegradientlength:
ftRR
γˆ=2(1−ft)LTe−LTcr

36

R201−ffR22R
LTcr=9ftt+2(1−tft)2Ln−1+3Ln.(3.19)
ThecriticalthresholdoftheTEMasafunctionofR/LnisillustratedinFig.3.1for
differentradiiandhencefordifferenttrappedelectronsfraction.Thisapproximated
formulawillbeusedforcomparisonwiththeexperimentalinversegradientlengthin
5.8.Section

20

15

r/a = 0.110crR/LT0.2e50.40.60.800246810
R/LneFigure3.1:CriticalthresholdforR/LTeasafunctionofR/Lne,forr/a=0.1,0.2,0.4,
0.6and0.8.TheshadedregionisthecommonrangeofmeasuredR/Lnevalues.

orttranspQuasi-linear3.5Thequasilinearparticleandheatfluxesarisingfromtheturbulencearetheaverageover
harmonictimeandspacevariationsofthefluctuation.Thispointsouttheimportance
ofthephaseshiftbetweentheradialelectrostaticdriftandthedensityortemperature
perturbation.Thecontributionofallinstabilitiesshouldbesummed,butthequasilinear
approachusesthesimplifyingassumptionthattransportisdeterminedbythefastest
growingmode.Inaddition,themodelislocalandthusneglectsthecouplingbetween
differentharmonicsduetoplasmainhomogeneities.
Γ=<˜nvE>
ΓTj=<T˜jvE>
ThecorrespondingdiffusioncoefficientscomefromtheFick’sandFourier’slaws:
D=−Γn

37

χi=ΓTTi
iχe=netΓTe(3.20)
TneeActually,theadiabaticelectronsdonotcontributetoparticletransport,becausevEis
alwaysphaseshiftedbyπ/2withrespecttoδnepandthereforeintegrationoveraperiod
returnszero(theyarenormaltoeachother).Thequantitiesδnniiandδnneearegivenby
theB.68,B.69and3.2.Inaninhomogeneousplasma,theamplitudeofthemodeshasan
additionalslowspacevariation:
T˜=1δT(x)e−i(ωt−k∙r)+C.C..(3.21)
2Theheatfluxresultstobe
ΓTj=1δTjv¯E+CC(3.22)
2kwhereCCmeans“complexconjugate”.TheproductsδTvEe−2i(ωt−k∙r)andδ¯Tv¯Ee2i(ωt−k∙r)
vanishinthetimeaverage.
Wefocusonthecontributionofasinglemodetotransport.Saturationisreachedwhen
thedominantnonlinearity(theconvectiveE×Bone)balancesthelineargrowth:from
thecontinuityequationforinstance
n∂∂t∼vE∙n(3.23)
γδneik∙r=vE∙δneik∙r(3.24)
Thegradientcanbeexpressedasthecharacteristicinverselengthintheradialdirection,
i.e.kx,whichisassumedtobeapproximatelyequaltokywhentheinstabilitysaturates
e.i.turbulence),(isotropicγvE=−ikx.
yieldsB.2relationtheSubstitutingγveφEωeTe=iLn=kxLn.
Theionheatfluxcanbenowevaluatedas
ΓTi≈21ikγxδTi+CC=ReikγxδTi.(3.25)
Duetoquasineutrality,accordingtoequationsB.49,B.69and3.2thetemperatureper-
is:turbationTiγ2ω2
δTi=ω−ωDi5/3kxLn3ωe(1−ft+ftAe)+ηi−3,
38

where175
Ae=Nˆωˆ(1−n)+ηe−3+3n.(3.26)
enSubstitutingthisresultin3.25,aftersomealgebra(seeappendixB.4)thequasilinear
heatdiffusivityresultstobe:
1210n2γˆ3ωDe/kx2
χi=ηiηi−3−(1−ft)9τ−3ftΔi(ωˆr+5/3τ)2+γˆ2,(3.27)
Δi=ˆ1|ωˆ|2|ωˆ|2(n−1)+ωˆr314−2ηe−310n+35−311+2ηe+37n+

5N5502575
−3τ1+ηe−3n+9τ(1−n)ωˆr+9τηe−3+3n,(3.28)
ˆwhereNis105252
Nˆ=|Nˆe|2=ωˆr2−γˆ2−3ωˆr+3+4ωˆrγˆ−3γˆ.(3.29)
Theexpressionforχeisslightlysimpler,sinceonlythetrappedelectronscontributeto
transport:similarlytotheions’case
2ω2γTδTe=ω−5ωeDe/3kxLn3ωeAe+ηe−3.(3.30)
Thequasilinearelectronthermaldiffusivityhastheform:
122γˆ3ωDe/kx2
χe=ηeftηe−3−3Δe(ωˆr−5/3)2+γˆ2.(3.31)
ThequantityΔeisalmostidenticaltoΔi,exceptthatωDereplacesωDiinthe3.28and
thereforefactors−1/τvanish,giving:
11410582
Δe=Nˆ|ωˆ|2|ωˆ|2(n−1)+ωˆr3−2ηe−3n+3−3+3ηe+3n+
502575
−9(1−n)ωˆr−9ηe−3+3n.(3.32)
Forparticlesweneedtodeterminethedensityperturbation;takingforinstancethe
dynamics:electronedtrappeφδnet=ftneTeAe,(3.33)
1γγeφ
Γ=2(v¯Eδnet+CC)=Reikxδnet=−ImkxftneTeAe.(3.34)
Asusualweconsideronlythefastestmode:
Γγneeφγ2
D=−n=ImkxftneTeAe=Im−kx2ωeftAe=
32=Im−γˆkω2DeftnAe=−γˆkω2DeftΔn,(3.35)
xx39

whereΔncanbeimmediatelyderivedfromequationB.76:
1214105117
Δn=Nˆ|ωˆ|(n−1)+ωˆr3−2ηe−3n+3−3+2ηe+3n.(3.36)
3.6Transportcoefficients
Withtheexpressions3.27,3.31and3.35thefluxesarefullydetermined,throughthe
effectivediffusivitiesχi,χeandD.However,itisnumericallyunconvenienttodivideby
gradients,whichcanbecomeclosetozeroduringtheprofileevolution,asforinstance
afterasawtoothcrash.ThetransportmatrixinthetransportcodeASTRAallowsby
defaulttosplitthecontributiontothefluxesfromthedifferentdrivinggradients.
Thetransportequationcanbewritteninthematrixform
qi/niTia11a12a13Ti/TiUi

qe/neTe=−a21a22a23Te/Te+Ue,(3.37)
Γe/nea31a32a33ne/neUn
whereUi,UeandUnhavethemeaningofanomalouspinchvelocities.Ina13weneglect
Zeffandassumeni/ni≈ne/ne.Definingtheusefulcoefficients
3c1=γˆωDe
2kxfct1c2=(ωr−5/3)2+γˆ2
thetransportcoefficientsderivedinB.4.1read:
a=c1
11(ωr+5/3τ)2+γˆ2
ˆa12=−a112ft|ωˆ|2−2ωˆr+10−5+25
32Nft223143τ955τ5ft150175
a13=−a1131+Nˆ|ωˆ|−|ωˆ|+3ωˆr−9−3τ+Nˆτ9ωˆr−27
45ft22103525ft150125
Ui=−a113R3τ(1−ft)+Nˆ|ωˆ||ωˆ|−3ωˆr+9+9τ+Nˆτ−9ωˆr+27
(3.38)252a21=0
a22=c21−3Nˆ|ωˆ|2(−2ωˆr+5)−9
2|ωˆ|221440150175(3.39)
a23=−c231−Nˆ−|ωˆ|+3ωˆr−9+Nˆ−9ωˆr+27
Ue=−c241|ωˆ|2|ωˆ|2−10ωˆr+10+50ωˆr−125
3RNˆ39927

40

a31=0
ft10
Na32=−c1ˆ−2ωˆr+3
ft1455
93a33=−c1ˆ−|ωˆ|2+ωˆr−
fN21035
ˆ93RNUn=−c1t|ωˆ|2−ωˆr+
Itisinterestingtonotethatalltransportcoefficientscontainthefactor

ωDek⊥ρscsρs2csTe1.5mi−1.5Te1.5mi0.5
≈2=2=2≈2.
k2xk⊥2LBρsk⊥LBΩ2ciρsk⊥LBe2B2ρsk⊥R

(3.40)

Assumingthatρsk⊥isconstant,thisfactorcontainsasignificanttemperaturedependence.
Thephysicsconsequenceisasteeperincreaseoftransportbeyondthecriticalthreshold
forhigherTeandconsequentlystrongerprofilestiffness.

41

42

4Chapter

Ionandelectronheattransportin
hargesdiscdeH-mostandard

PerasperaadASTRA

Inthischapterweapplythemodelspresentedinchapter2toASDEXUpgradeH-mode
dischargesheatedwithNeutralBeamInjection(NBI).Fortheseexperimentsthetemper-
atureprofileshavebeenobservedtobestiff[29][30][31],theirgradientlengthbeingclose
toacriticalthreshold,particularlyforTi.Theprofileshapetendstoremainthesame
fromdischargetodischarge(profileresilience)andtheratiobetweencoreandpedestal
temperatureisconstant,regardlessoftheheatabsorptionprofile.
OurisolateinthisterestisfeaturetotestfromITGotherphphysics,ysicswitheffects.Fparticularorthisfocuspurponoseweprofilehavestiffness,selectedsoawedatabasetryto
inwhichonlyoneplasmaparameterisvariedfromagivenstandardsetofparameters.
AdditionaldischargesfeaturingalsoElectronCyclotronHeating(ECH)aremodelled,to
checkthepredictionswhentheelectronheatfluxbecomesrelevantandtolookatthe
influenceoftheratioTe/Tionbothionandelectrontransport.
Thecomparisonbetweentheorybasedmodelsandexperimentaldataallowstoorderand
interpretethedata.Besides,itmakespossibletoevaluatethepredictivecapabilityofthe
models,findingtheirrangesofapplicabilityandjudgingsomeoftheirassumptions.

Diagnostic4.1systems

ThetokamakASDEXUpgradeisequippedwithacomprehensivesetofdiagnosticsys-
tems,plasmacoveringequilibriumalltheaswellasmeasuremenseveraltsexprequirederimenoftalamoprofilesdernhatokvetoamak.beFordiagnosed,ourpurppartlyoses,
forcomparisonwiththetheoreticalpredictions,partlyasinputforthesimulationswhich

43

arenotfullyself-consistent.Inparticular,ionandelectrontemperatureanddensitypro-
filesmeasurementsarenecessary,aswellastoroidalvelocity(vtor),effectivecharge(Zeff)
andradiatedpower.Alsoscalarparametersarerequired,suchasplasmacurrent,toroidal
magneticfield,NBIpowerforeachsourceandgeometricparameterslikeelongationand
.ytriangularitTheiontemperatureprofile,theplasmatoroidalrotationandtheeffectivecharge(Zeff)
aremeasuredwiththeChargeeXchangeRecombinationSpectroscopy(CXRS)[32].The
toroidalCXRSdiagnostichassight-lineslyingontheequatorialplaneofASDEXUpgrade,
measuringiontemperatureat16radialpositionsintheplasma.Thespatialresolutionis
1-2cm,dependingonthesight-line:thebestresolutionisobtainedathalfradius,whereas
intheplasmacenterandattheedgetheuncertaintyislarger.Theerrorsaremoderate
inthecaseofiontemperature,butarerelevantforZeff,whichinfluencesthesimulations
sincethemodelsdependonZeff,asdiscussedinsection2.4.5.Thebasicmechanismof
theCXRSdiagnosticisthemeasurementoftheDopplershiftedandbroadenedcarbon
recombination,whichisdetectableonlyinpresenceofNBI.
ElectrontemperatureprofilesmeasurementsareperformedwiththeElectronCyclotron
Emission(ECE)system[33].TheECEinthemillimetrewavelengthrangeismeasured
byaheterodyneradiometersystem,whichdetectsthesecondharmonicoftheX-mode
electroncyclotronradiation.Thesystemhas60outputchannels,allowingafinespatial
coverageoftheTeprofilemeasurement;thespatialresolutionisabout5-10mm,thesam-
kHz.31.25rateplingElectrondensityisdiagnosedbythecombinationofline-averagedinterferometryand
Lithium-beamdiagnostics[34].TheMach-ZehnderinterferometerusesaDCNlaser
(wavelength195µm)aslightsource.Thesystemhas8outputchannels,correspond-
ingto5horizontaland3verticalsightlines.ThesignalisthenAbelinverted,delivering
profile.ydensitaAuxiliaryheatingintheselecteddischargesisalwayssuppliedbyNeutralBeamInjection
(NBI),usuallywith60kVbeams,butinsomecasesapartofthebeamisaccelerated
kV.93with

4.2NBIheatedH-modedischarges
Adatabaseof70ASDEXUpgradedeuteriumdischargeswithlowtriangularity(δ<0.2)
isestablished.TheyareH-modedischarges,withlowtomoderateplasmadensityand
noInternalTransportBarriers(ITBs).InTable4.1theparametersofthedischargesare
eaclisted;htheconsideredexperimenscanttalwodataquanaretitiesgroupareedkinepttoafixedpowtoer,theadensitstandardyandvaalues,currentwhichscan.areIn5
MWfortheNBIheatingpower,1MAforthetotalplasmacurrentand6to8×1019

44

m−3fortheplasmadensity.Thethirdquantityisvariedoverthewidestrangepossible.
Inthisway,differentphysicseffectscanbedistinguishedandeventuallyberelatedto
profilestiffness.ThecurrentscaninfluencesthepedestalpressurethroughaMHDlimit,
theheatingpoweraffectstheheatfluxintheplasmacoreaswellasthepedestalpressure,
adensityvariationchangestheboundarytemperatureleavingpressureunchanged.

ScanPNBIIpl19ne−3
)m(10(MA)(MW)PNBI1.80.86.18
7.851.02.57.131.05.06.611.07.57.311.010.07.041.012.5Ipl5.00.43.15≤ne≤4.34
4.600.65.04.980.85.07.131.05.07.136.55,1.25.0ne5.01.03.75≤ne≤8.07
Table4.1:ParametersoftheselectedASDEXUpgradedischarges.

4.2.1DischargeswithNBIandECH
Anadditionalsetof7dischargeswithbothNBIandECHaremodelledtostudythecase
wheretheelectronheatfluxbecomeslargerandTe/Tiissomewhatdifferent.Again,the
NBIheatingpowerisroughly5MWandtheplasmacurrent1MA.Inaddition,ECH
powerisappliedvaryingbetween0.8and1.6MW.Foradescriptionoftheheatingscheme
andoftheASDEXUpgradeECHsystem,seeSection5.1.Theplasmadensityofthese
dischargesislowandratherconstant,closeto3.51019m−3;inmostcasestheheatis
absorbedclosetothemagneticaxis.Thedepositionradiiarecalculatedwithabeam
tracingtechniqueusingtheTORBEAMcode[35].

45

PECRHρdep19ne−3
(MW))m(103.660.010.83.530.011.23.730.331.23.770.100.83.500.011.63.300.201.23.290.071.6Table4.2:ParametersofthedischargeswithbothECHandNBI.

4.3resultstalerimenExp

stiffnessprofileIon4.3.1Thebehaviouroftheexperimentalcoretemperatureasafunctionoftheedgevalueis
reportedinFig.4.1.Asedgepositionρtor=0.8ischosen.Inthiswaywemakesure
thatweexcludefromtheanalysisthepedestalregion,whereMHDinstabilitiesandother
modes,differentfromITG,playasignificantrole.Thecoreexperimentaltemperatureis
takenatρtor=0.4,becausefurtherinsidethemeasurementisinfluencedbysawteeth.
TherelationbetweencoreandedgeTiisclearlylinear.Theproportionalityisobservedfor
8n scan ( #11197)eCurrent scanP scan6NBINBI+ECHT (0.4) [keV]4expi2

001234
T (0.8) [keV]i

Figure4.1:Experimentalioncoretemperatureasafunctionoftheedgetemperature.
Diamondsrefertothedensityscan,trianglestothepowerscanandstarstotheplasma
currentscandischarges.Thelineisaguidefortheeye.

allscansperformedandwiththesamefactor.Thismeansthattheiontemperatureprofile

46

isdeterminedbythepedestalvalue,withthelogarithmictemperaturegradientTi/Ti
keepingalmostconstantfromshottoshot.Thisbehaviourisknownasprofilestiffness(see
[36])andindicatessomeinstabilitymechanismwithacriticalthresholdinTi/Ti,which
keepstheexperimentalinversegradientlengthclosetothecriticalvalueasdiscussedin
Section2.3.Suchstrictproportionalityis,therefore,aclearargumentinfavourofthe
ITGinstabilityasthemechanismcontrollingionheattransport.Indeed,varyingheating
power,currentorplasmadensityaffecttheplasmaindifferentways,butapparently
onlythepedestalTivalueinfluencesthecoreTiprofile.Forinstance,currentvariations
changethepedestalpressuresignificantly(factor4inthepresentdatabase),whereas
densitychangesdonot.Thescanoverheatingpowerisanevidenceforprofilestiffness,
becauseheatfluxesareenhancedbyfactorsupto5,andwithadifferentdistribution
betweenelectronsandions(seeFig.4.2)buttheaveragegradientlengthsremainthe
same.Anotherstrongargumentisprovidedbythedensityscan,whichcoversaverywide

2.01.5

1.0Qi(0.5)Qe0.5

0.0-0.502468101214
P [MW]NBI

Figure4.2:ExperimentalratiooftheionandelectronheatfluxesversusNBIheating
powerforthepowerscandischarges.
rangeofTi(0.8),thusmakingtheevidenceforalinearrelationbetweencoreandedge
Tireliable.Furthermore,thepartitionofNBIpowerbetweenionsandelectronsandthe
penetrationofthebeamsdependstronglyonneandTe(whichisalmostproportionalto
1/ne,sincecurrentandheatingpowerdonotchange),leadingtoradicalchangesinthe
localionheatflux(seeFig.4.4).Inspiteofthat,thegradientlengthsremainthesame,
asthediamondsinFig.4.1show.
Figure4.1showsalsothatinallECHdischargestheaverageiontemperaturegradients
aresmallerthaninthepureNBIshots,althoughtheionheatfluxislarger.Namely,for
highTetheheatexchangewiththeelectronsisreducedandalargerfractionoftheNBI
powergoestotheions.ThetrendofECHexperimentsisanotherevidenceinfavourofa
thresholdbehaviour.ThefactthattheratioTi(0.4)/Ti(0.8)isslightlybutsystematically

47

lowerthanforpureNBIdischargescanbeexplainedthroughthedependenceofthe
aboutinstabilit1.5yinthethresholdcaseonoftheNBIratioalone.Te/TSuci,hawhicdephisendencecloseisto1intreatedbyNBI+ECHalldiscITG/TEMhargesbasedand
transportmodelsandhasbeendiscussedindetailinSection2.4.5.

Electron4.3.2orttranspheat

5n scan ( #11197)eCurrent scan4P scanNBINBI+ECH3T (0.4) [keV]2expe1

000.511.522.5
T (0.8) [keV]e

Figure4.3:Experimentalelectroncoretemperatureasafunctionoftheedgetemperature.
SymbolslikeinFig.4.1,inadditionfulldiamondsmarkthepointsrelatedtothedischarge
#11197.Thelinesareguidesfortheeye.
CoreelectrontemperatureisproportionaltothepedestalvalueonlyatlowTe(seeFig.
4.3).Alsointhisrange,however,theproportionalityholdsstrictlyonlyforthepower
scan(opentriangles),sothatforthesedischargesthegradientlengthremainsthesame,
inagreementwiththeresultsobtainedwithanECHpowerscan[14]onASDEXUpgrade.
Interestingly,thepowerscandischargesexhibitahigherratioTe(0.4)/Te(0.8),showing
thatitispossibletoinducehighergradientlengthsbyincreasingtheheatingpower,at
leastforlowTe.Thisindicatesthatifelectrontransportisgovernedbyathreshold
mechanism,thisisnotasstrictasithasbeenobservedfortheions.Inaddition,already
forTe>0.7keVtheplotofcoreversuspedestalTetendstospreadandtobendtowards
flatterprofiles,withexceptionofthepointsrelatedtothedischarge#11197,wherethe
shortgradientlengthislikelytobelinkedtotheunusuallystrongdensitypeaking.
IntheselectedNBIexperimentsionandelectronheatfluxesarecomparable,witha
tendencytobelargerfortheionsatlowdensitybecausetheelectronheatingthrough
neutralbeamsbecomeslesseffectiveathighTe(seeFig.4.4),andbecauseofthereduced
heatexchange.ThesituationchangeswhenECHisappliedtoo:theelectronheatfluxgets
largerthaninthepureNBIcase,sincetheECHcontributionisonlypartlycompensated
bythereductionoftheheatgainedfromtheions.Figure4.3showsthatalsoECH

48

dischargesbelongtothelowerbranchintheplotofcoreversusedgeTe.Itappearsthat
thehigherratioTe(0.4)/Te(0.8)canbeenhancedbyincreasingtheheatingpoweronlyfor
lowtemperatures.Thiscanbeunderstoodintermsofweakstiffness,withamodelsimilar
toFig.1.5.Actually,forhighertemperatures,togetagivenTe/Teonehastosustaina
highertemperaturegradient,sothatalargeamountofpowerisrequired.Therefore,for
hightemperaturestheprofilestendtostayclosertothecriticalthreshold.
Alsotheexperimentalobservationofthedischarge#11197isconsistentwithamodel
likeinFig.1.5.Asalreadydiscussed,forhightemperaturesthecriticalvalueofTe/Te
canhardlybeexceededbyraisingtheheatingpower.However,theTeprofilecanbecome
steeperifthecriticalgradientisraisedtoo,whichappearstobethecaseforstrongdensity
peaking.Indeed,theTEMtheorypredictsadependenceofthestabilitythresholdonthe
densitygradient;intheWeilandmodel,peakeddensityprofilesactstabilising,asdiscussed
inSection3.4(seeFig.3.1).
In[37]thecouplingwiththeTiprofilesisinvokedtoexplaintheproportionalityofcore

3

2Qi(0.5)Qe1

0

-1

0

21T (0.5) [keV]e

3

Figure4.4:Ratioofionandelectronheatfluxesfromthepowerbalanceanalysisforthe
scan.ydensitandedgeTeprofilesinthehighdensityrange.However,inthepresentdatabasethe
densityisneversohigh,thatelectronandiontemperatureprofilescoincide;moreover,
theproportionalityfactorbetweenTe(0.4)andTe(0.8)inthemoderatedensityrangeis
higherthanforTi.

4.3.3Theexperimentalplasmaenergy
InASDEXUpgradedischargesdensityprofilesaregenerallynotstiff[36].However,in
theconsidereddischargesthegradientsaremoderateintheregionofρtorbetween0.4and
0.8,withvaluesforne(0.4)/ne(0.8)between1and1.5.Therefore,alsotheionpressure

49

(pi)profilesresemblestiffness(seeFig.4.5).Thestoredplasmaenergy(seeFig.4.6)
isproportionaltothetotalpedestalpressure(p(0.8))ingoodapproximationforNBI
discharges.Theenergycontentreportedhereisnotmeasureddirectly,itrepresentsthe
thermalenergyresultingfromtheintegrationofthemeasuredpressureprofile.The

25n scan ( #11197)e- 3Current scan20P scanNBI19NBI+ECH15p (0.4) [keV 10 m ]105expi0

0246810
- 319expp (0.8) [keV 10 m ]i

Figure4.5:Measuredcoreversuspedestalionpressure.SymbolslikeinFig.4.1.

0.8n scan ( #11197)eCurrent scanP scan0.6NBINBI+ECHW [MJ]0.4exp0.2

0.0

0exp51019- 315
p (0.8) [keV 10 m ]Figure4.6:Measuredtotalstoredenergyversustotalpedestalpressure.Symbolslikein
Fig.4.1.Thelineisaguidefortheeye.

proportionalityholdsalthoughTeprofilesexhibitnostrictlinearrelationbetweencore
andedge.FlatterTeprofilesoccurforhighTedischarges,butinthatcasetheions
contributemoretoplasmaenergy,becauseTiinthisrangeisalmosttwiceTe.Thisisnot

50

thecaseforECH+NBIH-modedischarges,whichpresentcorrespondinglyalowerenergy
contentforagivenpedestalpressure(seecrossesinFig.4.6).Furthermore,themoderate
densitypeakingatlowdensitiesleadstoimprovedionconfinement.Thiscompensates
theenergylackcomingfromtheelectronchannel.Theproportionalitybetweenthermal
energyandpedestalpressureallowstoorderthedataintermsofglobalconfinementwith
asimplelinearrelation.Thishoweverdoesnotprovideanysimplescalinglawinterms
ofengineeringparametersbecausethepedestalpressureitselfdependsonthem.

resultsulationSim4.4decoASTRAThe4.4.1AllsimulationsareperformedwiththeASTRAtransportcode[38],keepingthedensities
equaltothemeasuredprofiles,thusswitchingoffparticletransportinthemodelswhich
containit,suchastheWeilandandtheGLF23models.Theexperimentaltemperatures
atρtor=0.8aretakenasboundaryconditionsforthecalculation,becauseneithertheITG
northeCDBMphysicsareexpectedtodescribethepedestalregioncorrectly.Current
diffusioniscalculatedateverytimestepaccordingtotheneoclassicaltheory[39],the
bootstrapcurrentbeingcalculatedasinRef.[40].Sawteetharetakenintoaccountwith
aKadomtsevfullreconnectionmodel[41],theperiodbeingexperimentallydetermined.
Measuredaswellasmodelledtemperatureprofilesareaveragedoverasawtoothperiod.
Heatdiffusivitiesarethesumoftheneoclassical[42][43][44][45]andturbulentcontri-
butions,thelatterdeterminedfromtheconsideredtransportmodel.Thesubroutinesfor
eachmodelareprovidedbytheauthors,withnofreeparameterstobeadjustedforthe
modelling.IntheWeilandmodel,thecollisionsontrappedelectronshavebeenswitched
offbysettingthecollisionfrequencyequaltozero.
InallfourmodelsthestabilisingtermarisingfromtheE×Bshear(ωE×B)isincluded.
Thedifferentimplementationsofthisshearingratearealsoprovidedbytheauthors;the
correspondingdescriptionscanbefoundin[46],[6]and[47].Theshearingrateisob-
tainedfromthemeasuredtoroidalvelocity,theneoclassicalpoloidalrotation[48]andthe
pressuregradientcontribution.Thelatteriscalculatedfromthemodelledprofiles.
TheNBIheatingpowerdistributionisimplementedasasubroutineintheASTRAcode.
Thedepositionprofilesdependonplasmaparameters:higherplasmadensityleadsto
lesspenetrationofthebeams,highTevaluesbringmostofthepowertotheions(see
Fig.4.4).WetakethecomputedtemperatureprofilesasinputfortheNBIroutinefor
.consistency

51

comparisondelsMo4.4.2WiththissetupthedischargesdescribedinSection4.2aremodelledthroughtimeinte-
grationuntilthesteadystateisreached.AtypicalresultisshowninFig.4.7forelectron
andiontemperatureprofiles,aspredictedbythefourmodels.InFig.4.7(a),theinverse

CDBM

# 13154 2.1 - 2.5 s88CDBMGLF23WeilandIFS/PPPLT[keV] e[keV] Te
001010[keV] i[keV] Ti
T1000.3805
(a)(b)-1-3(c)(d)
iTωE x B619q
[10 s ]n [10 m ]R/LγeωR/L0Tcr000
0ρtor10ρtor10ρtor10ρtor1

Figure4.7:TemperatureprofilesofatypicalASDEXUpgradeH-modedischargefrom
experiment(points)andfrommodelling(solidlines).Nomodellingoutsideρtor=0.8.In
thefromthirdtheWrow:eiland(a)Rmo/Ldel;Tiand(c)RExp/LTcrerimenprofilestaldensitfromytheprofileIFS/PPPL(d)qwithmodel;Te(b)profileωE×Bfromandtheγ
del.moCDBMiontemperaturegradientlengthisplottedcomparedtothecriticalprofileaccordingto
thetheIFS/PPPLmodel.InFig.(b)thegrowthrateisdisplayedtogetherwiththero-
tationalshearingrateω.Figure(c)showsthemeasureddensityprofileandin(d)the
safetyfactorqisreportedE×Bfromthesimulatedcurrentdiffusion,theTeprofilebeingtaken
fromtheCDBMmodelling.Outsideρtor=0.8theprofilesarenotmodelledbutsetequal
totheexperimentaldata.ThegradientlengthfortheIFS/PPPLmodeldeviatesfrom
thecriticalvalueinspiteofthestrongstiffnessbecausetherotationalshearingrateωE×B
stabilisestheITGaroundthecriticalvalue.Closetotheaxistheneoclassicaltransport
TheandtheplotsacorrespwteethalloondingwtotheFig.gradien4.1tforlengthmotodelledstaycorebeloandwtheedgeTicriticalisvrepalue.ortedinFig.4.8.

52

TheITGmodelsreproducetheexperimentaldatasatisfactorily:profilestiffnessandweak
88(b)(a)6644simT (0.4) [keV]i2simT (0.4) [keV]i2
WeilandIFS/PPPL001234001234
8iT (0.8) [keV]8iT (0.8) [keV]
(d)(c)6644simT (0.4) [keV]i2simT (0.4) [keV]i2
CDBMGLF23001234001234
T (0.8) [keV]T (0.8) [keV]iiFigure4.8:SimulatedcoretoedgeTiaccordingtothedifferentmodels:a)IFS/PPPL
b)Weilandc)GLF23d)CDBM.Thelinereproducestheexperimentalpoints,beingthe
sameasinFig.4.1.SymbolsrefertothescansasinFig.4.1.
dependenceofLTcronplasmaparametersundertheexperimentalconditionsoccurinthe
ITGmodels.AlsotheWeilandmodel,whichexhibitsalowervalueof∂χi/∂(R/LTi)atthe
turbulenceonset,keepstheprofilesstiffenoughtoreproducetheproportionalitybetween
coreandpedestalTishownbytheexperiments.TheITGmodelspresentnevertheless
deviationsfromaperfectlylinearrelationbetweencoreandedgeTi,duetochangesin
LTcr(magneticshear,ratioTi/Teoreffectiveioncharge)ortovariableωE×Bstabilisation.
AllofthempredictsteeperTiprofileforthedischarge#11197withhighlypeakedden-
sity,butintheexperimentthisisnotobserved(seeFig.4.1).Ontheotherhand,ECH
dischargesarecharacterisedbyalowerratioTi(0.4)/Ti(0.8),andthisfeatureisreturned
correctlybyallITGmodels.
TheCDBMmodelfailsreproducingTiprofilestiffness.Onlyinthehighdensityrange,
53

correspondingtoTi(0.8)<0.8keV,therelationbetweencoreandpedestalTiispredicted
toberoughlylinear(seeFig.4.8).Theexperimentaldatarelatedtothepowerand
currentscansreproducethisproportionality,althoughtheratioislowerthanintheex-
periment.Inthepowerscan,TiandTiareraisedsimultaneously.Infact,Tiincreases
withtheheatfluxandintheexperimentitisnotpossibletodecoupletheheatingpower
fromanincreaseinpedestalTi,unlessdensityorcurrentarechangedtoo.Indeed,forthe
densityscantheCDBMreturnsaclearflatteningintherelationbetweencoreandedge
Ti,incontradictionwiththeclearlinearityobservedintheexperiment.
Thebehaviourofelectrontemperatureinthepresentdatabaseisalsowellreproducedby
theITG/TEMbasedmodelsasshowninFig.4.9.Inthisfigure,thelinesareguidesfor
55(b)(a)4433simT (0.4) [keV]esimT (0.4) [keV]e
2211WeilandIFS/PPPL000.511.522.5000.511.522.5
5eT (0.8) [keV]5eT (0.8) [keV]
(d)(c)4433simT (0.4) [keV]e1simT (0.4) [keV]e1
22CDBMGLF23000.511.522.5000.511.522.5
T (0.8) [keV]T (0.8) [keV]eeFigure4.9:SimulatedcoretoedgeTeaccordingtothedifferentmodels:a)IFS/PPPL
b)Weilandc)GLF23d)CDBM.Thelinesreproducetheexperimentalpoints,beingthe
sameasinFig.4.3.SymbolsrefertothescansasinFig.4.1.
theeyeandrepresentthefitoftheexperimentaldatainthelowandhighedgeTeranges,
theyarethesameasinFig.4.3.Allmodelsreproducethebendingtowardsflatterprofile
inthelowcollisionalitycornerofthedatabase.
54

ThechangeintheratioofcoretoedgeTehasbeenexplainedintroducingothertransport
mechanismswhichwouldcausetheobserved“switch”inelectrontransport[37];however,
inthepresentdatabasenoadditionaltransportfeatureisrequiredtoreproducethisbe-
viour.haFortheCDBMmodel,however,thesimulationresultsarefarfromtheexperimentaldata,
andthechangeoftheslopeinFig.4.9isrelatedtothesameflatteningintheiontemper-
aturesplot(Fig.4.8),sinceionandheatdiffusivitiesaretakentobeequal[21].Forthe
GLF23modelthebendingisstrongerthanfortheexperiment,leadingtooverestimated
coreTeforTe(0.8)below1keVandunderpredictedTeabovethisvalue.Thedischarge
#11197isnotreproducedcorrectly:thehighdensitygradientinthismodelgeneratesa
largeheatflux,incontradictiontotheenhancedstabilityobservedintheexperiment.It
isimportanttoremindthatthedensityprofileisnotmodelledself-consistentlybuttaken
fromthemeasurement.Asmallexperimentalerrorcanhavealargeinfluenceonthe
modelledtemperatureindischargeswherethedensityispeaked,asreportedin[13]pag.
2493:“Themodelis,infact,verysensitivetodensitygradientsasthetrappedelectron
modesonsetwithincreasinglypeakeddensity”.Thismodellingset-upmightbeadis-
advantagefortheGLF23modelundercertainexperimentalconditions,becausesolving
particletransportself-consistently“leadstoadensityprofilethatisalmostimperceptibly
differentfromthegivenexperimentalprofiles,yettheself-consistentdensityprofilegives
amuchbetterfittothetemperaturedatathanusingtheexperimentaldensityprofile”
([13]pag.2493).TheWeilandmodeloverpredictscoreTeinlowcurrentdischarges.
ThedecreaseintheratioTe(0.4)/Te(0.8)iswellreproducedbecausetheinversegradi-
entlengthisclearlybeyondthecriticalthreshold,sothatforhighTethepowerisnot
enoughtoenhancefurtherthetemperaturegradientandkeepaconstantTe/Te.The
IFS/PPPLreproducestheTeprofileflatteningatlowdensitiesbecauseχeisassumedto
beroughlyproportionaltoχi[20].IfoneconsidersthatTi/Tiisfixed(whichisthe
caseinthesimulationswiththeIFS/PPPLmodel),itfollowsthattheratioTe/Teis
proportionaltoqe/qi(Ti/Te)0.6.Athighdensitiestheratioqe/qi(Ti/Te)0.6remainsalmost
constant,andthecoreelectrontemperaturesisroughlyproportionaltotheedgevalue.
Atlowerdensities,theionandelectronheatfluxesarelesscoupledandqi/qeislarger
than1duetothelowerelectronNBIheating.Te/Teisreduced,yieldingasmaller
ratioTe(0.4)/Te(0.8)inthemodelledprofiles.Thismight,however,onlypartlyexplain
theexperimentalobservations.Inthepowerscantheratioqi/qeincreaseswithheating
power(seeFig.4.2)buttheratiobetweenexperimentalcoreandpedestalTedoesnot
flatten(seetrianglesinFig.4.3).Asignificanttestofthehypothesisχe∝χiisprovided
bytheECHdischarges,withhotelectrons.TheIFS/PPPLmodelpredictsfartoohigh
coretemperaturesandturnsouttobeinadequateformodellingECHexperiments.Also
forthedischarge#11197energyconfinementisoverpredicted.Actually,thisdischargeis
characterisedbylowcollisionalityandpeakeddensitygradient,atthelimitofthevalidity

55

rangeforwhichthemodelwastested:theinterpolationwasperformedfor0<R/Lne<6.
Thecaseoflowcollisionalityandstrongdensitygradientisdiscussedin[20],warningthat
afurthertrappedelectroninstabilityoccursinthereferencecode,whichisnotincludedin
theformulasforsimplicity.SotheIFS/PPPLmodelisexpectedtoreturnhigherstability
thanthesourcecodeundertheseexperimentalconditions.
FiguresofmerithavebeenintroducedaccordingtothedefinitionbyConnorandcowork-
ers[19],inordertoevaluatequantitativelythereproductionoftheexperimentaldataby
themodels.Inparticularthestoredenergyfromthemodellingiscomparedwiththemea-
suredvalue.Hereonlythethermalenergyisconsidered,i.e.theintegralofthepressure
profile,andthevolumeistakenbetweenρtor=0.2andρtor=0.8.Theenergyamount
p(0.8)×volumeisaboutonehalfofthetotalthermalenergyanddoesnotdependonthe
modelsincetheboundaryconditionsforthetemperatureprofilesarefixedatρtor=0.8.
Thisenergyfractionhasbeensubtractedfrombothexperimentalandcalculatedenergy,
leadingtothequantitiesWjsimandWjexp(jbeingthespecieslabel),whichareplotted
inFigures4.10and4.11.TheWeilandmodelfailsreproducingthecurrentscan:the
simulationyieldstoohighelectronconfinementforlowcurrentdischarges.TheGLF23
modelshowsatrendtooverestimateplasmaenergyatlowedgetemperaturesandunder-
estimateittowardscollisionlessplasmas;thedeviationsfromtheexperimentalvalueare
small,butthereisacleartrendagainstthepedestaltemperature.Lookingatthetrans-
portcoefficient,adramaticenhancement(evenstrongerintheelectronchannel)occursin
dischargeswithsignificantdensitypeaking.Thereisaclearspatialcorrelationbetween
theregionswithhighn/nandflatTeandTi,inagreementwith[13].Afewdischarges
athighcollisionalitybehavedifferentlyfromtheoveralltrend,withthemodelledenergy
belowtheexperimentalvalue:theyarelowcurrentshots,possiblyduetothesensitivity
ofthemodeltoqandsˆprofiles.TheCDBMmodelsystematicallyunderpredictsthe
experimentalenergyof20to40%,withatrendtoworsentowardshighpedestaltemper-
ature.ThedeviationsaresummarisedinTable4.3,defining
2σj=1NWjexpsim−1
WNj1wherethesumismeantoverthedischargesinthedatabaseandNistheamountof
shotsanalysed.Relyingonlyonthisfigureofmerititisdifficulttodetectpossible
systematictrendsorbadpredictionsinthecentralregion:thevolumethereissmall
andcontributeslesstothestoredenergy.Moreover,evenifthemodelledtemperature
gradientwerejustflatfromρtor=0.5inwards,butthetemperatureprofilewererealistic
outsidethisregion,thelackwouldhardlybeobservedalsobecausethecontribution
njTj(0.5)×V(0.2≤ρtor≤0.5)dominatesthecentralregion.Assuggestedin[19],
weextendtheanalysisreportingalsothestandarddeviationofthetemperatureprofile,

56

1.61.6(b)(a)1.41.4expi1.2expi1.2
1.01.0simW / Wi0.8simW / Wi0.8
0.60.6WeilandIFS/PPPL0.4012340.401234
1.6iT (0.8) [keV]1.6iT (0.8) [keV]
(c)(d)CDBM1.41.4expi1.2expi1.2
1.01.0simW / Wi0.8simW / Wi0.8
0.60.6GLF230.4012340.401234
T (0.8) [keV]T (0.8) [keV]iiFigure4.10:Simulatedtoexperimentalionenergyratiovs.edgeTiaccordingtothe
models:a)IFS/PPPLb)Weilandc)GLF23d)CDBM.Thepedestalpressurecontribution
issubtractedfrombothexperimentalandcalculatedenergy.Symbolsrefertothescans
4.1.Fig.inasasdefinednT−Texp2
Tnstdj(c)=1njexpj2
j1wherejisthespecieslabelandthesumisperformedovernequidistantradiallocations;
thesuperscript(c)referstothecentralregion,fromρtor=0.2toρtor=0.5.Notethat
thisisthestandarddeviationforasingledischarge;inTable4.3thisquantityisreported
incapitalletterstoremarkthatithasbeenaveragedoverthedatabase.Tohaveageneral
insightwhetherdataaresystematicallyunder-oroverpredictedbyamodel,wereport
alsotheoffsetofthetemperatureprofiles,againfortheregionbetweenρtor=0.2and
57

1.61.6(b)(a)1.41.4expe1.2expe1.2
1.01.0simW / We0.8simW / We0.8
0.60.6WeilandIFS/PPPL0.400.511.522.50.400.511.522.5
1.6eT (0.8) [keV]1.6eT (0.8) [keV]
(d)(c)CDBM1.41.4expe1.2expe1.2
1.01.0simW / We0.8simW / We0.8
0.60.6GLF230.400.511.522.50.400.511.522.5
T (0.8) [keV]T (0.8) [keV]eeFigure4.11:Simulatedtoexperimentalelectronenergyratiovs.edgeTeaccordingtothe
models:a)IFS/PPPLb)Weilandc)GLF23d)CDBM.Thepedestalpressurecontribution
issubtractedfrombothexperimentalandcalculatedenergy.Symbolsrefertothescans
4.1.Fig.inasρtor=0.5:exp
offj(c)=1nTj−Tj
n1nTjexp2
Toaveragethisquantityoverthedatabaseisofcoursealossofinformation,because
positiveandnegativecontributionmigh(tc)cancel.However,theamountofthedeviationis
alreadydescribedbythequantitySTDi.
TheglobalenergycontentiswellpredictedbytheITGmodels,alsosubtractingthe
pedestalcontribution;ifoneconsidersthefullthermalenergybothfromexperimentand
modelling,thedeviationdropbyafactorofroughly1.5.Ontheotherhand,thepre-
dictionisimprovedbyfixingthedensitytotheexperimentalprofileinsteadofmodelling
itself-consistently.TheITG-TEMbasedmodelsareabletoreproducetheexperimental
58

CDBMGLF23eilandWIFS/PPPLσtot0.1770.1330.2100.367
σi0.2110.1480.2440.372
σe0.1620.1780.2160.366
STDi(c)0.1980.1530.1900.369
STD(ec)0.1220.1070.1750.339
OFFi(c)-0.154-0.1130.013-0.350
OFF(ec)-0.0290.054-0.016-0.325
Table4.3:DeviationofcalculatedfrommeasuredstoredenergyforNBIheateddischarges,
withoutpedestalcontribution.StandarddeviationandoffsetofTeandTiprofiles.All
quantitiesaveragedoverthedatabaseofNBIheateddischarges.

energywithin20%,theWeilandmodelyieldingthebestpredictions.TheCDBMmodel’s
deviationsexceed35%andreturnsystematicallyplasmaenergybelowtheexperimental
alues.vThesameanalysisisappliedtoECH+NBIheateddischargesandtheresultsaresum-
marisedinTable4.4.TheWeilandandGLF23modelperformevenbetterthaninthe
pureNBIcase,withionandelectronenergypredictedwithin10%.
TheIFS/PPPLandCDBMmodelsprovetobeinadequatetodescribeNBI+ECHexper-
iments.FortheCDBMmodelthereproductionofthesedatasetisinlinewiththetrend
shownforNBIheateddischarges.TheIFS/PPPLmodelyieldsstillverygoodpredictions
foriontransportbutfailstreatingelectrontemperature,indicatingthattheassumption
ofχe∝χiholdsonlyunderparticularconditionsandisnotageneralpropertyoftoka-
makcoreplasmas.Whentheionheatfluxissmall,theITGmodeisnotdrivenandχi
islow.Asaconsequence,χeissmallaswell;inpresenceofelectronheating,thisleads
toextremelyoverpredictedelectrontemperaturegradients.

Summary4.5Summarising,theiontemperatureprofilesonASDEXUpgradeinconventionalscenarios
arestiffoveralargerangeofplasmadensity,current,NBIheatingpowerandpedestal
temperature.However,NBI+ECHdischargesexhibitslightlylowervaluesofTi/Ti,
indicatingthateitherthestabilitythresholdorthedrivenfluxdependsontheratio
Te/Ti.Actually,alsoforthedensityscantheratioTe/Tiexhibitasignificantvariation,
gettingsmallertowardslowerdensitiestillvaluesaround0.5.Itseemsthatthestronger
transportdrivenbyhightemperatures(duetotheGyro-Bohmpre-factorofthetransport

59

CDBMGLF23eilandWIFS/PPPLσ(totr)0.1490.0540.0710.369
σ(r)0.1360.0810.1160.349
σe(ir)0.3910.0780.1050.387
STDi(c)0.1630.1180.0740.359
STDe(c)0.3450.0820.0930.368
OFFi(c)-0.138-0.0900.014-0.334
OFFe(c)0.2840.063-0.060-0.350
Table4.4:DeviationofcalculatedfrommeasuredstoredenergyforNBI+ECHheated
discharges,withoutpedestalcontribution.StandarddeviationandoffsetofTeandTi
profiles.AllquantitiesaveragedoverthedatabaseofECH+NBIdischarges.

coefficients)iscompensatedbythe(stabilising)deviationofTe/Tifromunity.Thisisnot
thecaseforNBI+ECHdischarges,whereTe/Tiiskeptroughlyequaltoone.Theelectron
temperaturebehavessimilarlytotheiontemperatureformoderatedensitybutnotfor
lowerdensities,wherethedataofcoreTeagainstpedestalTearemorespreadandtend
toflatten.Forlowcollisionality,ahigherratioTe(0.4)/Te(0.8)cannotbeobtainedby
additionalpower(asECH),butcanbeobtainedthroughthestabilisingeffectofpeaked
profiles.ydensitTheITG-TEMbasedmodelsreproducetheexperimentalionandelectrontemperature
profileswell.Therefore,thepredictionoftheglobalconfinementresultstobeverygood,
within20%,aftersubtractingthepedestalenergycontribution.TheWeilandmodelyields
inaveragethebestpredictions,althoughelectrontemperatureispoorlyreproducedfor
lowcurrentdischarges.TheIFS/PPPLpredictstoomuchstoredenergyforthecasewith
verylowcollisionalityandstronglypeakeddensitygradientandwrongelectrontransport
inNBI+ECHdischarges.TheGLF23modelunderpredictsthetemperatureandenergy
dataoflowdensityshots,withalocalstrongenhancementintransportintheregions
withsignificantdensitypeaking.Theerrorontheglobalconfinementismoderatebut
thereisatrendwithdecreasingdensity.TheCDBMalwayspredictstemperatureand
energyvaluesfarbelowtheexperimentalones,andfailsqualitativelysinceitdoesnot
reproducethelinearrelationbetweencoreandedgeiontemperature.
Intermsofthephysicscontainedinthemodels,theITG/TEMparadigmresultstobe
satisfactory.Themodeappearstobedominantlyiontemperaturegradientdriveninpure
NBIdischarges(hencethegoodagreementoftheIFS/PPPLmodel).However,foran
accuratedescriptionoftheelectronchannelunderallcircumstancesamodelhastoretain
bothdensityandelectrontemperaturegradientlengthswhendeterminingtheonsetof

60

anomalous

GLF23

mo

electron

dels

ort.transp

yields

go

o

d

The

coupling

tagreemen

with

eenwetb

the

61

exp

ITG

and

talerimen

TEM

data.

of

the

eilandW

and

62

5Chapter

HeattransportinECHdominated
discharges

Intheory,thereisnodifferencebetweentheoryandpractice.
But,inpractice,thereis.(R.M.Nixon)

InChapter4heattransporthasbeeninvestigatedforH-modedischargesheatedthrough
NBIandafewcasesfeaturingalsoECH.ItisappealingtoswitchtopureECHdischarges,
tocheckthemodels’predictionsforthecasewhereelectronheatingbecomesdominant.
Actually,inthiscasetheITGdoesnotplayamajorroleanditispossibletotestthe
models’behaviourforpureTEM.ECHpowercanbemodulatedintimegivingriseto
heatpulses.Observingtheirpropagationintheplasmaitispossibletoinvestigateenergy
transportalsotransiently,providingaconstrainingtestforthemodelsandameasureof
stiffness.profileEvidenceforathresholdbehaviourofelectrontransportisobservedonASDEXUpgrade.
Indeed,anECHpowerscanexperimentexhibitsTeprofileswithalmostconstantTe/Te
intheconfinementregion[14].However,apowerscancouldmimicthiseffectbyen-
hancingsimultaneouslyTeandTe.ExperimentsonASDEXUpgradewithECHpower
modulation(MECH)confirmanon-linearrelationbetweentheelectronheatfluxandthe
temperaturegradient:χepertisclosetoχePBinsidethedepositionlayer,wheretheheat
fluxissmall,andlargeroutside[49].
ThemodellingwithempiricaltransportmodelsbasedonathresholdinTe/Teyields
promisingresults;thebestagreementisfoundassumingthatthegradientlengthsare
notkeptclosetothecriticalthreshold[16].Toimprovethephysicalunderstandingof
electrontransport,acomparisonwiththeorybasedmodelsisrequired,inordertorelate
transportpropertiestothephysicsmechanisms.ModellingMECHdischargesprovides
averyconstrainingtestforthemodelitself,whichhastoreproducesimultaneouslythe

63

steadystateprofilesandtheheatpulsepropagation[50].TheWeilandandGLF23mod-
els[12][13]areappliedheretoverydifferentplasmaconditionscomparedtoChapter4,
namelywithdominantelectronheatingprovidedbyECH.TheIFS/PPPLmodelisnot
consideredsincetheTEMispoorlytreated.TheCDBMmodelisnotappliedsinceit
clearlyfailstodescribetheNBIheateddischarges.

HeatingCyclotronElectronThe5.1ECHtheofPrinciple5.1.1TheECHmechanismisbasedontheenergyexchangeattheresonancebetweenthe
launchedwaveandthegyrationmotionofelectronsaroundthemagneticfieldwiththe
characteristiccyclotronfrequencyΩce.Thewavesources,calledgyrotrons,operateata
givenfrequency.Theresonancetakesplaceatthelayerwherethemagneticfieldstrength
issuchthatthegyrationfrequencyΩceorahigherharmonicequalsthewavefrequency.
SincethefieldstrengthisdescribedingoodapproximationbytherelationB=BoRo/R,
itisconstantonacylindricalsurfacearoundthetorusaxis,whichcorrespondstoavertical
lineinthepoloidalprojectionshowninFig.5.1.Thefigureillustratesthelocationof
B = 2.5 T layer if B = 2.4 ToB = 2.5 T for central depositionoMovable mirror:poloidally (on / off axis)and toroidally

R = 1.65 ma = 0.5 m = 1.6κ

31 kHz, 60 chan. ECE rad. Δr < 1 cm⇒ Te
ECRH Power

Figure5.1:TheECHsystemonASDEXUpgrade(poloidalsection).Thewaveenters
theplasmaafterreflectiononasteadyandamovablemirror.Thepoloidalangletogether
withthecentralmagneticfielddeterminethedepositionradius.Fastparallelenergy
transportheatsrapidlythewholemagneticsurface(dashedellipsis).
theheatpowerdeposition,atthesectionbetweenthebeamtrajectoryandtheresonant
layer.Intokamaksheattransportparalleltothemagneticfieldisveryfast,sothatthe

64

wholemagneticsurface(thedashedellipsisinFig.5.1)israpidlyheatedandwecan
definethedepositionradiusρdepastheradiallabelofthissurface,innormalisedtoroidal
fluxcoordinate.Evenifthefocushappenstobesetexactlyattheresonantlayer,the
depositionwidthremainsfinitebecausethebeamfocusingislimitedbydiffractiveeffects
[35].Besides,relativisticelectronsslightlybroadentheabsorptiontowardsthehighfield
side,sincethegyrationfrequencydecreaseswithincreasingrelativisticmass.

5.1.2TheECHsystemonASDEXUpgrade
OnASDEXUpgradethereare4gyrotronsoperatingatafrequencyof140GHz,fora
secondharmonicX-modeabsorptionattheB=2.5Tlayer.Theelectronsabsorb100
%ofthepowerinasinglepass,whichisimportantinordertoavoidspuriousenergy
sourcesatotherplasmalocations.Eachgyrotrondeliversupto0.4MWtotheplasma,
givingamaximumECHpowerof1.6MW.Thebeamsarefocusedbymirrors,yieldinga
narrowpowerdepositionwidth,roughly3cm,correspondingtolessthan10%ofASDEX
Upgrade’sminorradius.IfBo=2.5Tandthebeamtravelsalongtheequatorialplane,
thentheplasmacenterisheated.Off-axisheatingcanbeobtainedeitherbychanging
thecentralmagneticfieldorbyvaryingthepoloidalangleofthemovablemirror(seethe
dashedbeaminFig.5.1).Thelocalisedpowerdepositionisamajoradvantageofthe
ECheatingconcerningtransportstudies,becauseitallowsanaccuratereconstruction
oftheheatflux,whichisalmoststep-like.Anotherpowerfulfeatureisthepossibilityto
modulatetheheatpowersourceswithdifferentfrequencies,providingtheidealconditions
foratransientstateanalysis,sincethelocalisedpowerdepositionensuresawideregionof
source-freeplasma.Inaddition,highflexibilityisprovidedbytheECHsystemavailable
onASDEXUpgrade:the4gyrotronscanbedecoupledinordertodepositenergyinto
differentlocations.Inthisway,theeffectsofenhancedheatfluxandeventuallyenhanced
temperaturegradientscanbeinvestigated.Itisalsopossibletoapplypowermodulation
tosomeofthegyrotronsandcontinuouspowertotheothers,sothattheeffectsofheating
ontransportcanbedetectedalsobymeansofthetransientstateanalysis.

systemsDiagnostic5.2TheelectrontemperatureprofilesaremeasuredbytheECEdiagnostics,describedinSec-
tion4.1.Thetimeresolutionandthesamplingratearemuchhigherthanthemodulation
frequency,thusallowingtoapplytheFourieranalysistotheECEtimetrace.Inthisway,
theamplitudeandphaseprofilesofthepropagatingheatwavecanbemeasured.
TheelectrondensityismeasuredthroughthecombinationofinterferometryandLithium
beamdiagnostics,alsopresentedinSection4.1.TheECHdepositionradiiaremeasured
incaseofpowermodulation,otherwisetheyarecalculatedwiththeTORBEAMcode

65

[35].

Sim5.3set-upulation

TheversionoftheWeilandmodelusedhereisofJune1998with7equations,including
impurities(dilutionapproximation)andparallelionmotion.Again,collisionsontrapped
electronsareswitchedoffbysettingthecollisionfrequencyequaltozero.Alsoforthe
GLF23modelthesameversionisusedasfortheiontransportanalysis(seeSection4.4.1).
TheboundaryconditionsforTeandTiaretakenatthelastclosedfluxsurface.Inthis
way,theheatwavepropagationisaffectedonlyattheveryedge.Thevalueisadjusted
byhandinordertomatchtheoutermostexperimentalchanneloftheECEmeasurement.
SinceTiprofilesarenotmeasured,Ti(1)issetequaltoTe(1).Plasmadensityistaken
equaltotheexperimentalprofile,i.e.particletransportisnotconsidered.Themodelsare
implementedassubroutinesintheASTRAcode[38],returningtheanomaloustransport
coefficientsateverytimestep.

5.4ECHpowerscan

Asetofdischargeshasbeenselectedwithon-axisECHandfairlylowdensity,around
21019m−3.InFig.5.2thecomparisonbetweenexperimentandmodelsisshownfor
thedischarges#13557and#13558,with0.8and1.6MWECpower,respectively.The
experimentalaswellasthemodelledelectrontemperatureprofilesareaveragedoversev-
eralmodulationperiods.AlthoughTiprofilesarenotdiagnosedforthesedischarges,the
centralTiismeasuredtobeabout1keV.ThemeasurementistakenwiththeNeutral
ParticleAnalysers.Consideringtheexperimentaluncertaintyandsomedegreeofarbi-
trarinesscomingfromtheboundaryconditionTi(1)=Te(1),bothmodelsareconsistent
withthemeasurement.AsFig.5.2shows,theheatgoesalmostentirelytotheelectrons.
Theanomalouselectrontransportreturnedbythemodelsincreasesacrossthedeposition
layer(seeFig.5.2b)),sincetheelectrontemperaturegradientisenhancedanddrives
theTEMturbulenceunstable.Bothmodelsreproducetheexperimentaldatawithgood
.accuracy

5.5Severalharmonicstransportanalysis

Thegoodresultsofthesteadystatemodellingencourageafurther,morestringentcom-
parisonbetweentheoryandexperiment,involvingalsothetransientbehaviour.The
comparisonbetweenexperimentandmodelisshownforthedischarge13722,withaline

66

5

# 13557

# 13558

(b)

85(b)(a)T [keV]T [keV]eeECHECH00521.5Qe2 GLFχT [keV]iQ [MW]χe WN [m /s]χe
eQi0000ρtor10ρtor1
Figure5.2:Steadystateprofiles:(a)Teprofileofthedischarge#13557:experimental
(crosses),fromtheWeilandmodel(solidline),fromtheGLF23model(dashedline).
TheECHpowerdensityprofileisplottedtoo(a.u.).Below:Tiprofilesaccordingtothe
models,comparedwiththemeasuredvalueofcentralTi=1.05keV.(b)Teprofileofthe
discharge#13558(symbolsasin(a)).Below:integratedelectronandionpowerprofiles
andelectronheatdiffusivitiesfromWeiland(WN)andGLF23models.

averageddensityaround41019m−3andanaverageECHpowerof1.5MWdeposited
off-axis,atρdep=0.32.Theplasmacurrentis0.6MA.Thereproductionofthesteady
stateisshowninFig.5.3.TheexperimentalTeiswellpredictedbytheWeilandmodel
overthewholeprofile.TheGLF23modelhasanoveralllowerTe,possiblyduetothe
lowplasmacurrent,asthemodelexhibitshighertransportatlowcurrent(seeFigures
4.11).and4.10Thegyrotronsaremodulatedwithaperiodof34ms,dutycycle=0.85and100%(on-off)
powermodulation.Inthisway,heatwavesareinducedandtravelacrosstheplasma.
Thepulsepropagationissource-freeoverawideradialextension,becausetheabsorption
layerisquitethin(w/a<0.1)andtheotherheatsourcesarenotperiodic.Severalhar-
monicsofthetemperatureperturbationexhibitagoodsignaltonoiseratio,allowingto
studytheheatwavepropagationatdifferentfrequencieswithidenticalplasmaconditions
(seeFig.5.4).TheamplitudeandphaseprofilesofT˜ecanbecomparedtothemodel

67

# 13722

3T [keV]eMECH06-319n [10 m ]e010ρtor

Figure5.3:Steadystateprofilesofthedischarge13722:experimentalTeprofile(points)
comparedtothepredictionsbytheWeilandmodel(solidline)andtheGLF23model
(dashedline).Below:experimentaldensityprofile.

predictionsforfrequenciesof29.4,58.8,88.2and107.6Hz.AsdiscussedinSection1.4,
flatamplitudeandphaseprofilesareassociatedwithhighincrementaltransportandfast
heatpulsepropagation.TheWeilandmodelmatchesthedatawellatallfourharmonics,
andforbothamplitudeandphase(seeFig.5.4).Inparticulartheprofiles’asymmetryis
wellreproducedqualitatively,withsteeperprofiles(lowerheatpulsediffusivity)insidethe
depositionthanoutsideasintheexperiment,aswellasquantitatively,sincetheslopes
coincidewiththemeasuredonesatallharmonics.Moreover,thecorrectabsolutevalues
oftheamplitudeandthephaseatρdepareobtained.Forthephase,itmeansthatthe
timedelayoftheTeprofilereactiontotheheatingiswellpredicted.
TheGLF23modelpredictstheheatpulsepropagationwithlessaccuracybutstillthe
.satisfactoryistagreemen

5.6EffectsofECHontransport
Inthedischarge12935theECHpowerismodulatedatρMECH≈0.75,withatime-
averagedpowerofroughly0.2MWandaplasmacurrentof0.8MA.Twofurthergy-
rotronsareswitchedonlaterinthedischarge,delivering0.8MWathalfradiuswithout
powermodulation.AtthistimemostoftheECHpowerisappliedatadifferentposition
withrespecttothesourceoftheheatwave[49].Inthisway,theimpactofheatflux

68

# 13722300Phase

0100IIIIIAmp.10

IV

300180PhasePhase [deg]ϕ00100III100IIIIV
Amp.A [eV]1010Amp.10ρtor10ρtor110ρtor10ρtor1
Figure5.4:AmplitudeandphaseprofilesofT˜eforthedischarge13722atthefrequencies
ν=29.4,58.8,88.2and117.6Hz:fromexperiment(points),Weilandmodel(solidlines)
andGLF23model(dashedlines).Theshadedregionsmarktheregionwithlowsignalto
ratio.noise

variationsontransportcanbeobserveddecoupledfromtheanalysistool.Inparticular
thisexperimentallowstotestthemodelsintwodifferentrangesofheatflux.Insidethe
innermostdepositionradius,theelectronheatfluxisnearlyzerobecausetheonlysourceis
ohmicandmostoftheheatistransferredtotheions,whicharecolderthantheelectrons.
CrossingtheECHlayer,theheatfluxhasastepandbecomeslargeoutside.Reminding
Fig.1.5,itisclearthatfornonlinearqe−Teschemetheincrementaltransporthas
stronglydifferentbehavioursforhighandlowheatfluxes,sincetheslopeoftherelation
betweenqeandTe/Techangesradically.
Theexperimentalsteadystatetemperatureprofilesarereproducedwellintheohmic
phase,asFig.5.5(a)shows.IntheECHphase,outsideρdepthepredictionisstill
satisfactory(Fig.5.5(c)),theWeilandmodelyieldingthebetteragreement.However,
insidethedepositionradiusthemodelledTeprofilesaretooflat.Inthisregionanomalous
transportisnotdrivenbyTe,sothatsmallerrorsinthereconstructedheatfluxprofile
introducelargeuncertaintiesonthecomputedtemperaturegradients.
Theexperimentalamplitudeandphaseprofiles(seethepointsinFig.5.5(d))change
theirslopewhencrossingtheECHdepositionradius.Theprofilesareobservedtobecome
flatteroutsideρdep,wheretheheatfluxislarger[49].SincetheECHsourceathalfradius
isnotmodulated,theslopechangebetweeninsideandoutsideρdepisnotanartifactof
powermodulation,butindicatesindeedachangeintheheatwavepropagation.The
Weilandmodelreproducestheamplitudeandphaseprofilesverywellbothintheohmic
aswellasintheECHphase.Themostimportantfeature,i.e.theslopeflattening
outsideρdep,isreproducedwithaccuracy(seeFig.5.5(d)).Noteintheohmiccasethe

69

21018032OHMIC# 12935OHMIC# 12935ECH# 12935ECH# 12935ECH
[deg]MECHMECHT [keV]eϕT [keV]eϕ [deg]
00005(a)100(b)5100ECH
(d)(c)-3-319191010n [10 m ]eA [eV]MECHn [10 m ]eA [eV]MECH
11000ρtor10ρtor10ρtor10ρtor1

Figure5.5:Discharge12935:experimental(points),solidline(Weilandmodel)anddashed
line(GLF23model).Overplotted,theECHandMECHpowerdensityprofiles.Ohmic
case:(a)AverageTeprofile.Below:neprofile.(b)AmplitudeandphaseprofilesofT˜e
atν=29.4Hz.AfterswitchingonECHathalfradius:(c)AverageTeprofile.Below:ne
profile.(d)Amplitudeandphaseprofiles.Theverticallinemarkstheslopechange.The
shadedregionhasalowsignaltonoiseratio.

steepeningoftheexperimentalamplitudeandphaseprofilesaroundρtor=0.3observed
inFig.5.5(b).ThisisnicelyreproducedbytheWeilandmodel.Duetothesawteeth
instability,thetemperatureprofileisfairlyflatinsideρtor=0.25,sothatthetemperature
gradientiskeptbelowtheTEMcriticalthreshold.Thisislikelytobetheexplanationof
theexperimentalobservation:theheatpulsepropagationisslowbecausethegradients
aretoolow(duetosawtoothoscillations)todrivetheturbulenceunstable.Switchingon
ECHathalfradiusreducestheelectronheatfluxinsideρdepduetotheexchangewiththe
thermalions;hence,thetemperatureprofileinthatregiongetsflattertoo,withrespect
totheohmiccase.Inthisway,theTEMthresholdisreachedfurtheroutside,namely
wherethestrongECHisapplied.BelowtheTEMonset,however,transportisnotpurely
neoclassic,sincethereissomeresidualanomaloustransportaswewilldiscussindetailin
5.8.SectionTheGLF23modelpredictstheheatwavepropagationratherpoorlyalreadyintheohmic
case:theamplitudefallstoosteeplytowardstheplasmacenter,thephaseistooflat
outsideρMECH.WhenECHisappliedathalfradius,themodellingreturnsfartooflat
phaseprofile,withawrongabsolutephasevalueatρMECH.Theamplitudedropstovery
smallvaluesinsideρdep,makingtheanalysisofthephaseprofilesunreliableinthatregion.
However,ajumpinthephaseandachangeinthelogarithmicslopeoftheamplitudeshow
thatthereisatleastasmallqualitativeeffectcausedbytheadditionalheating.

70

5.7ExperimentswithconstantECHpower
AnECHpowerscanmightnotbeanadequatetooltoestimatetowhichextenttheTe
profilearestiff,becauseenhancingtheglobalheatingpowerdeterminesalsoanincrease
ofthepedestalTe.Therefore,auniformvalueforTe/Teisreturnedalsoassuming
weakprofilestiffnessornothresholdbehaviouratall,atleastinalimitedrangeof
temperatures.However,thelocalisedECHabsorptionandthepossibilitytodecouplethe
depositionradiiforthe4gyrotronsensurethehighestflexibilitytoachievelargevariations
inthelocalheatfluxwithaconstantECHpower.Takingadvantageofthisfeature,a
setofdischargesisperformedwiththesametemperatureprofileoutsideρtor=0.7and
differentheatfluxesathalfradius[52].Thisisobtainedheatingtheplasmaattwo
differentlocationsρ1≈0.35andρ2≈0.70.Inthedischarge#14793allgyrotronsheat
atρ1,the#14794hastwosourcesinρ1andtwoinρ2;finally,forthe#14796ECHis
depositedonlyattheouterlocationρ2,thusyieldingaverylowheatfluxbetweenρ1
andρ2,whichistheregionofinterest.Ineachdischargetwosources(heatingatthe
sameradiallocation)aretimemodulated,addingusefulinformationaboutperturbative
transportandprofilestiffness.Themodulationperiodis33ms,withdutycycleequalto
0.5andabout65%ofthefullpowerduringtheoff-phase,foranaverageheatingrate
≈1.4MW.Plasmadensityisverylowandalmostconstant,around2.21019m−3,the
plasmacurrentbeing0.8MA.AsFig.5.6shows,theedgeTevaluekeepsconstantin
4# 14793# 14794ECH13# 147962T [keV]ECH2expe1

00.00.20.4ρ0.60.81.0
torFigure5.6:ExperimentalTeprofilesofthedischarges#14793,#14794and#14796.The
heatingisput:allintotheinnerlocation(#14793),halfatρ1andhalfinρ2(#14794),
allatρ2(#14796).ThetotalECHpoweriskeptconstant.
thethreedischarges.Thisisbecausetheglobalpowerinputisthesame.Acheckwith
theThomsonscatteringdiagnosticconfirmsthattheTeprofileoutsideρ2isthesamefor
thethreedischarges[52].Ifoneassumesstrictprofilestiffness,giventhesameboundary
conditiontheprofilesshouldalmostcoincideoverthewholeradius,regardlessofthe
shapeoftheheatdepositionprofile.Ontheotherhand,inthepictureofpurelydiffusive

71

transportwithχeinsensitivetotheplasmaparameters,gradientsshouldsteepenbythe
samefactorastheheatflux.Themeasuredtemperaturegradientsareinbetweenthese
twoextremecases.Thisconfirmstheexistenceofanon-linearityforanomaloustransport
butshowsthatprofilesareonlyweaklystiff,inagreementwiththetheexperimental
observationsandthemodellingresultsalreadypresentedinthepresentChapterandin
Chapter4.TheTEMphysicsappearstoreproducethesteadystatetemperatureprofiles,
4# 147934# 147943# 14796
MECHT [keV]eT [keV]eECHT [keV]e
MECHMECH030303
-3-3-3191919n [10 m ]en [10 m ]en [10 m ]e
0000ρtor10ρtor10ρtor1
Figure5.7:Teprofilesofthedischarges#14793,#14794and#14796fromexperiment
(points),Weilandmodel(solidlines)andGLF23model(dashedlines).Overplotted,the
ECHandMECHpowerdensityprofiles.Below:neprofiles.
astheWeilandandGLF23modelsareabletoreproduceall3caseswithgoodagreement
5.7).Fig.(see5.8Discussion:resultsandprofilestiffness
ThegoodagreementbetweentheWeilandmodelandtheexperimentallowstoextractuse-
fulinformationaboutthepropertiesrequiredbyatransportmodeltopredictsuccessfully
thebehaviourofASDEXUpgradeECHdominateddischarges,andabouttheunderlying
physics.Consideringtheexperimentalresults[14][49],itisimportanttocheckhowfar
theexperimentalandmodelledinversegradientlengthsareabovethecriticalthreshold.
Thedischarges#13558and#12935withstrongECHpowerandlowdensityareselected
forthisstudy.InFig.5.8thecriticalthresholdprofilesareshownforexperimentalback-
groundprofiles.ThestarsgivethecriticalvalueofR/LTefromtheapproximatedformula
3.19.TheexperimentalR/LTeprofile(solidline)exceedsclearlythepredictedstability
72

# 1355830(a)Te20R/L Tcr≈ 3
ECHR/L10

# 1293530(b)20eECHTR/L10

ECHMECH

000.20.40.60.81000.20.40.60.81
ρρtortor

Figure5.8:Normalisedinversegradientlength:TEMcriticalthresholdfromtheapprox-
imatedformulawithexperimentalbackgroundplasmaparameters(stars);experimental
profile(solidline).(a)Discharge#13558.(b)Discharge#12935intheECHphase.

thresholdintheconfinementregion,byafactor2to3.Thismeansthatprofilesarenot
closetomarginalstability,atleastwithstrongcentralelectronheating.Indeed,within
themodeltheelectronheatdiffusivitydoesnotincreaseverysteeplybeyondthecritical
gradient(Fig.5.9).Thisfact,combinedwiththelowvalueofthestabilitythreshold,
allowstoreachgradientsbeyondthecriticalthresholdthroughanincreaseoftheheating
power.Thedischarge#12935showsaninterestingbehaviour:forR/LTebetween2and
5theTEMisnotyetthemodewiththehighestgrowthrate(infactitisanITGmode)
andtheslopeoftheheatfluxwithrespecttotheinversegradientlengthisflatter.This
explainswhythemodelledheatwavepropagationisslowerinsidethedepositionradius
(seethesolidlineinFig.5.5):theexperimentalR/LTeatρdepisaround5andincreases
outsidethedepositionlayer.Weremindthatthesteepertheheatflux,thefasterthe
heatwavepropagation,theflattertheamplitudeandphaseprofiles.Consideringthe
quantitativeagreementwiththeexperimentalamplitudeandphaseprofiles(Fig.5.5),
thisobservationmightprovideaphysicalexplanationofthemeasuredslopechange.In
thedischarge#13558theeffectisnotobservedbecauseoftheflatdensityprofile,which
ort.transphigheresgivTheGLF23modeldoesnotreproducesimultaneouslythesteadystateandtheincremen-
taltransport.Inmostcasesitreproducestheaveragetemperatureprofilescorrectlybut
underestimatesthespeedoftheheatpulsepropagation,predictingsteepamplitudepro-
files.Onthecontrary,thedischarge#13722,withlowplasmacurrent,isreturnedwith
satisfactoryagreementconcerningphaseandamplitude,butthesteadystatetemperature
profileispredictedtobetooflat.Asageneraltrend,theratioχpert/χPBislowerthan
intheexperiment.Thisisconfirmedifonelooksatthebehaviouroftheheatfluxwith

73

ρ#13558 = 0.5tor

ρ#12935 = 0.5tor

(b)

tortor1050(b)8(a)40630q /(n T ) [m/s]e10q /(n T ) [m/s]e2
ee20ee4
0005R/LTe101505R/LT1015
eFigure5.9:qedependenceonR/LTefromtheWeilandmodel,otherparametersfrom
theexperimentatρ=0.5.ThedashedthinlineistheonsetoftheTEM,thethick
dashedmarkstheTevalueatwhichthemodebecomesdominant,thesolidlineisthe
experimentalR/LTe.(a)Discharge#13558.(b)Discharge#12935intheECHphase.

increasingtemperaturegradients(Fig.5.10).Theslopeismoderatelysteep,comparable
χtoperttheisWcloseeilandχPmoBsodel,thatbutintherespiteisnotofahighsignificananomaloustoffsetoftransptransporttheort.Asaphenomenologyconsequence,is
differentfromthatofprofilestiffness.
ThefigureshowsalsothatthestabilitythresholdoftheTEMandtheanomaloustrans-
ptheseortdepdiscendharges,significantheretlyisaonLTcertaini.SincedegreetheofiontemparbitrarinesseraturedueprofilestothearecnothoiceofmeasuredtheLTin
ialue.v

Summary5.9TheWeilandmodelsucceedsinthesimultaneousreproductionofthesteadystateaswell
asthetransienttransportinavarietyofexperimentalconditions.Forthesteadystate,
thegoodpredictionofχePBholdsfordifferentdensities,depositionradiiandECHpower.
Thetransientbehaviourisalsowellreproduced,sincetheslopesoftheamplitudeand
phaseprofilesmatchtheexperimentalresults,showingthatχepertisclosetothemea-
suredone.Amplitudeandphasearereproducedatallharmonicssimultaneously,sothe
frequencydependenceoftheheatwavepropagationfollowstheexperimentalone.The
profilesasymmetryoccurringintheexperimentisobservedalsointhemodelling,with
quantitativeagreementbothinsideandoutsideρdep.Theabsolutevaluesofamplitudeand
phaseareobtainedaswell.Theeffectsofheatingontransienttransportarereproduced,
astheslopesflattenoutsideρdepafterswitchingonthefurthergyrotrons,decoupledfrom

74

= 0.5ρ#13558 tor50

40R/L = 5Ti30eq /(n T ) [m/ s]e20e10R/L = 2Ti0150105R/LTe

R/L = 2Ti

Figure5.10:qedependenceonR/LTefromtheGLF23model,otherparametersfromthe
experimentatρ=0.5,discharge#13558.DifferentvaluesofR/LTiarechosen.The
verticallinerepresentstheexperimentalvalueofR/LTe.

theheatwavesource.
Electrontemperatureprofilearenotstronglystiff:atagivenpedestaltemperature,an
increaseoftheheatfluxatmid-radiuscorrespondstoasteepertemperatureprofile.A
comparisonwiththestabilitythresholdoftheTEMintheWeilandmodelshowsthat
theexperimentalgradientlengthsare2-3timeshigherthanthecriticalthresholdifthe
heatingisstrongenough.ThisisbecauseaccordingtotheWeilandmodelqeincreases
withTestartingfromR/LTcr,butnotverysteeply,ifwecompareitforinstancewith
qidrivenbytheITGmodeforionheateddischarges.
TheTEMphysicsappearstobeabletopredictthefeaturesofelectronheattransport
observedonASDEXUpgrade,includingthetransientbehaviour.Therefore,theTEM
islikelytobethemechanismgoverningelectrontransportinthecaseofstrongelectron
heating.TheGLF23modelreproducesthetimeaveragedtemperaturebutunderpredictsthespeed
ofpropagationoftheheatpulse.Inadischargewithlowplasmacurrent,theheatwave
iswellpredictedbuttheaveragetemperatureistooflat.Inallcases,theratioofthe
transienttothesteadystatetransportcoefficientiscloseto1andhencelowerthaninthe
experiment.Thisisbecausethemodelfeaturesanomaloustransportwithoutasignificant
offsetintermsofTe,sothattherelationbetweentheheatfluxandthetemperature
linear.roughlyistgradien

75

76

6Chapter

ModataJETofdelling

OnASDEXUpgradetheiontemperatureprofileshavebeenobservedtobestiff.The
ITG/TEMphysicscontainedintheIFS/PPPL,Weiland-NordmanandGLF23modelsis
abletopredictthisbehaviourandtoyieldaquantitativeagreementwiththeexperiment
4).Chapter(seeElectrontemperatureprofilesexhibitprofileconsistencyundersomecircumstances,butit
seemsthatthisisnotageneralproperty.Indeed,athresholdfortheonsetofanomalous
transportisobservedandMECHexperimentsconfirmit(seeChapter5).However,the
inversegradientlengthiswellabovethecriticalthresholdanddoesnotkeepconstantover
adensityscan.TheTEMphysicsascontainedintheWeilandreproducesthisfeature,
andisinexcellentagreementwithASDEXUpgradedataalsoquantitatively.TheGLF23
modelalsoreproduceselectrontransport,butfeaturestoosmalloffsetintheqe−Te
scheme,sothattheagreementwiththeexperimentdoesnotholdforbothsteadystate
andtransientanalysisinECHdischarges.
Aninter-machinecomparisonbetweendifferentsizedtokamaksisveryhelpfulinorder
tovalidatethetheorybasedtransportmodelsconsideredsofar(seesection2.4),which
arebelievedtobethecandidatestopredicttheconfinementperformanceofITER.Itis
challengingtoapplythemodelstoJETdischarges,becausetherearenofreeparameters
tobeadjustedandtheirphysicsisderivedintermsofdimensionlessparameters.JET
isthelargesttokamakbuiltsofar,withamajorradiusof2.96mandaminorradius
of1.25m,yieldingasomewhatloweraspectratiocomparedtoASDEXUpgrade.The
largeminorradiusmakesJEToptimalforcoretransportanalysisandrepresentsastep
intokamaksizetowardsthelargerdimensionsofITER.Theflexibleheatingmethods
allowtostudytransportpropertieswithlocallyvariableheatflux,whichisakeypoint
toinvestigateeventualprofilestiffness;inJETitispossibletohaveboththehotionand
thehotelectronregime,allowingtoexplorethedependenceoftransportandstiffnesson
./TTie

77

edyemploDiagnostics6.1Themodellingrequiresexperimentalmeasurementseitherasinputforthesimulationsor
as4.1inreferenceordertoforthereconstructmodels’thepredictions.equilibriumTheandsamethedataheatarefluxesneededforbasothionsdiscussedandinelectrons.section
Inonantaddition,Heatingthepo(ICRH)weranddeliveredNBItocannotthebeplasmacomputedwithacomwiththebinationexistingofIonASTRACyclotronroutinesRes-
coanddehas[53]toandbegivreliesenonastheinputexpprofile.erimentalThetemperaturereconstructionandisdensitdoneybyprofiles.meansofthePION
TheelectrontemperatureismeasuredwiththeECEsystem,basedontheMichelsonin-
terferometer.TheChargeeXchangeSpectroscopyisemployedfortheiontemperatureprofileaswell
asfortheiontoroidalvelocity.Theeffectivechargemeasurementisnotavailableforthe
presentdischarges,sothatforthesimulationsitischosentobe2.5,anarbitraryyet
alue.vrealisticThedensityprofileisobtainedwiththeLIDARdiagnostics,whichisbasedontheThom-
scattering.son

resultstalerimenExp6.2Inordertostudyprofilestiffness,adatabaseofdischargeswithconstantedgeheatflux
isselected.Inthisway,thepedestaltemperatureiskeptratherconstantandprofile
stiffnesscanbecheckedbychangingtheheatdepositionprofile.ThecombinationofNBI
andICRHheatsonaxisinsomecasesandaroundhalfradiusinothers,althoughthede-
positionisnotwelllocalised.Thetotalinputpowerisabout12MW,theplasmacurrent
2.8MAandthetoroidalmagneticfield2.75T.Thelineaverageddensityrangesbetween
4.5and61019m−3.
NBIpre-heatingensuresthatstrongICRHcreatesasignificantsupra-thermalions’popu-
lation;theslowingdownoftheseparticlestransfersenergymainlytothethermalelectrons,
whereaslessenergeticparticlesaresloweddownbythethermalions.Soitispossibleto
raisetheratioTe/TibyapplyingNBIpre-heatingandhigherICRHpower.Actually,in
thedischarge52097theNBIpowerisreducedto4MWandICRHisenhancedupto9
MW,sothatthetotalpowerisstillthesameasfortheotherselecteddischarges.
Inaddition,two“similarityshots”areanalysed:thesearedischargesbuilttomatchmost
oftherelevantdimensionlessparametersofcorrespondingdischargesperformedonan-
othertokamak,inthiscaseASDEXUpgrade.Theparametersinourcaseareρ=ρs/a,
ν=νieqR/1.5ve,thandβ.Itwouldbeastringenttestforthemodels,whichcontain
onlydimensionlessparameters,representingadirectcheckofthemodels’physicsandof
thechoiceoftheexperimentalsimilarityparameters.Unfortunatelythecorresponding

78

dischargesofASDEXUpgradewerenotyetavailable,sothemodellingresultsofthese
JETdischargesarenotshowninthepresentthesis.Thesimilarityshotshavemuchlower
plasmacurrent(1MA),toroidalfield(1.1T),density(about2.41019m−3)andNBI
heatingpower(3.5MW).NoICRHisapplied.
InFig.6.1themeasuredcoreiontemperatureisplottedagainstthepedestaltempera-
4Similarity shotsQ scan3T (0.4) [keV]2expi1

000.5exp1T (0.8) [keV]1.52
iFigure6.1:Experimentalioncoretemperatureasafunctionoftheedgetemperature.
Thelineisaguidefortheeye.
ture.Theexperimentaldatabaseiscertainlytoosmalltomakeanyconclusivestatement.
However,theresultsresembletheclearlinearrelationobservedforASDEXUpgradestan-
dardH-modedischarges(seeFig.4.1).Therangeofedgetemperaturesisfairlywideand
theratioTi(0.4)/Ti(0.8)isclosetothevalue≈2foundonASDEXUpgrade.Thelinear
trendofFig.6.1needstobeconfirmedbyincreasingthedatabaseandbroadeningthe
range.eraturetempThebehaviourofelectrontemperatureisalsoconsistentwiththeobservationofASDEX
4Similarity shotsQ scan3# 52097T (0.4) [keV]2expe1

000.5exp11.52
T (0.8) [keV]eFigure6.2:Experimentalelectroncoretemperatureasafunctionoftheedgetemperature.
Upgrade(seeFig.4.3),howeverthebendingtoalowerratioofTe(0.4)/Te(0.8)needsto

79

beestablishedonthebasisofseveraladditionaldischarges.
WeshowinFig.6.3theplasmadensityprofilesaswellasthepowerdepositionprofiles
reconstructedwiththePIONcodeforthreerepresentativedischarges:onewithcentral
heating,anoff-axisheatedcaseandadischargewithprevalentelectronheating.The
simulationresultswillbediscussedinSection6.3.2.Thedensityprofilesarequitesimi-

# 50624# 52097# 506210.380.5828
-3iP-3-3ne-3-3-3
ne1919ne19
P [MW m ]n [10 m ]eP [MW m ]iPn [10 m ]eP [MW m ]PeiPn [10 m ]e
PPee000.1000.1000.1
Qe-2-2-2QiQiQi
QeQ [MW m ]QeQ [MW m ]Q [MW m ]
0000ρtor10ρtor10ρtor1

Figure6.3:Experimentalprofilesofdischarge50621,50624and52097.Above:powerden-
sityprofilesaccordingtoPIONreconstruction.Electrondensity(points)isoverplotted.
Below:ionandelectronheatfluxprofiles.

lar,whereastheionandelectronheatfluxesvarysignificantlywithinthedatabase.The
discharge52097inparticularhasstrongelectronheatinginthewholeconfinementregion,
thusresemblingtheexperimentalsituationoftheASDEXUpgradepowerscan,analysed
inChapter4.ThecaseisdifferentfromASDEXUpgradedischargeswithbothECH
andNBIheating,inthattheboundarytemperatureislower.Interestingly,thedischarge
52097exhibitsthehighestratioTe(0.4)/Te(0.8)forgivenboundarytemperature,confirm-
ingtheexperimentaltrendobservedinASDEXUpgradeforthepowerscan.Thisalso
pointsinthedirectionofweakprofilestiffnessoftheelectrons,whereadditionalelectron
heatingleadstosteepertemperaturegradients.Thelargemachinesizelimitstherange
ofheatfluxreachablewithrespecttosmallersizedtokamakssuchasASDEXUpgrade.
However,theobservationinthedischarge52097indicatesthatthe“strength”ofTeprofile
stiffnesscanbeinvestigatedonJETaswell.

80

resultsdellingMo6.3

setupdellingMo6.3.1WemodelthedatabymeansofthesamemodelsemployedforASDEXUpgrade.The
implementationintheASTRAcodeisthesame,i.e.thesameroutinesareused.Ac-
tually,forJETdatawedonotshowthesimulationresultsobtainedwiththeCDBM
model,becauseagainthemodelhasratherpoorpredictivecapability.Instead,were-
portthemodellingwiththesimplifiedversionoftheWeilandmodelderivedinChapter
3.ItcorrespondsroughlytotheversionoftheWeilandmodelwith4equations,with
neitherimpuritieseffectsnorparallelionmotion.Forthismodel,theroutinehasbeen
programmedandimplementedbytheuserandnotbytheauthors,withthesamestabil-
isationmechanismarisingfromωE×BasfortheWeilandmodel.Theadvantageis,that
theformulasareexplicite,sothatalltermscanbeeasilycontrolledandthephysicseffects
canberecognisedstraightforwardly.
Theboundaryconditionfortheheattransportequationsareagainatρtor=0.8forboth
TiandTe.Theparticletransportisnotmodelled;instead,theexperimentaldensitypro-
filesaretakenasinputforthesimulation.Forneoclassictransportandcurrentdiffusion
thesamemodelsareassumedasin4.4.1.
ThedifferencewithrespecttothemodellingofASDEXUpgradedischargesisthatwe
cannotcomputethecombinedheatingdeliveredbyICRHandNBIwiththeexistingAS-
TRAroutines.Therefore,wetaketheoutputprofilesofthePIONcodeasexperimental
reconstructionofthepowerdeposition,whichisthecommonprocedureadoptedbythe
modellersofJETdata.Inthisway,thedepositionprofileskeepconstantintime,since
theydonotfollowtheevolutionofthecalculatedtemperatureprofiles.

6.3.2Comparisonwiththeexperiment
Withtheseassumptionsandsetupweapplythemodelstothethreecasesmentionedin
Section6.2.Thedischarge50621featuresoff-axisheating,deliveredmainlytotheion
channel;the50624hasmorecentrallypeakedpowerdeposition,the52097ischaracterised
bydominantelectronheating.ThepredictionsofthemodelsareillustratedinFigures6.4,
6.5and6.6.ThebestagreementforiontransportisobtainedwiththeGLF23model,
butactuallyallmodelspredictthedatawithgoodaccuracy.TheWeilandand“reduced
Weiland”modelsdonotreproducetheprofilesteepeningoccurringaroundρtor=0.4.In
thecasewithdominantelectronheatingtheWeilandandGLF23modelsyieldthebest
predictionsforTiwhereastheIFS/PPPLmodelunderpredictsthecentralTi.
Asforelectrontemperature,thereducedWeilandmodelyieldsthebestpredictions.Also
theGLF23andWeilandmodels,bothcontainingTEMphysics,reproducetheexperimen-

81

# 50621WeilandIFS/PPPL

Weilandreduced

8WeilandWeilandIFS/PPPLGLF23reducedT [keV]e08T [keV]i00ρtor10ρtor10ρtor10ρtor1
Figure6.4:Discharge50621:calculated(solidlines)andmeasured(points)temperature
profiles.Nomodellingoutsideρtor=0.8.

Weilandreduced

# 506248WeilandIFS/PPPLGLF23WeilandreducedT [keV]e08T [keV]i00ρtor10ρtor10ρtor10ρtor1
Figure6.5:Discharge50624:calculated(solidlines)andmeasured(points)temperature
profiles.Nomodellingoutsideρtor=0.8.

82

# 52097WeilandIFS/PPPL

Weilandreduced

10WeilandWeilandIFS/PPPLGLF23reducedT [keV]e05T [keV]i00ρtor10ρtor10ρtor10ρtor1
Figure6.6:Discharge52097:calculated(solidlines)andmeasured(points)temperature
profiles.Nomodellingoutsideρtor=0.8.
talprofiles.WeobservethatGLF23slightlyunderpredictsthetemperaturegradients,
WeilandtendstokeepabovethemeasuredTeprofile.ItseemsthattheGLF23model
worksbetterifthecoreelectronheatfluxissignificant.Consideringthepossibleerrorsof
thepowerdepositionprofiles,bothmodelshavetobeconsideredtopredictthedatacor-
rectly.TheIFS/PPPLisabletopredicttheTeprofilewhentheelectronheatfluxislow
(discharge50621),butitclearlyoverpredictsthecoretemperatureassoonaselectronsare
heatedtoo.Thisindicatesthattheassumptionχe∝χidoesnotholdingeneral,because
forlowionheatfluxes(andthereforelowχi,duetoprofilestiffness)electrontransportis
clearlyunderestimatedbytheIFS/PPPLmodel.

Summary6.4AninitialdatabasetostudytransportandprofilestiffnessissuesonJEThasbeenes-
tablished.Theheatdepositionlocationisshiftedkeepingtheedgeheatfluxconstant.
ToextendthecomparisonwithASDEXUpgraderesults,twosimilarityshotshavebeen
analysedalthoughnotyetmodelled.Thedatabaseneedstobeincreasedandthecorre-
spondingASDEXUpgradedischargesofthesimilarityshotshavestilltobeinvestigated.
Strongervariationsofthemostrelevantplasmaparametersarerequiredbeforeanyconclu-
sivestatementispossible.However,preliminaryobservationsconfirmsofarthebehaviour
ofionandelectrontransportasobservedonASDEXUpgrade:
•Coreiontemperatureisproportionaltothepedestalvalue,withtheratioTi(0.4)/Ti(0.8)
2.toclose

83

Forelectrontemperaturethisratioisusuallyhigherandtendstobendtolower
valuesathighertemperatures.Again,strongerheatingleadstosteepergradients
bygivenedgeTe,indicatingthatstiffnessshouldbeweak.Experimentsintransient
statewithTemodulationareexpectedtoprovideadditionalinformation.

IontransportisingeneralwelldescribedbyITGphysics:theIFS/PPPL,GLF23
andtensionWofeilandthemodelsdatabasereprowithduceresptheectiontobtempoundaryeratureTivprofilesaluesishighlysatisfactorily.desirableAnex-to
checktowhichextentprofilesarestiff.

ThecombinationofITGandTEMphysicsascontainedintheWeilandandGLF23
modelsreproducesJETdatawithintheexperimentaluncertainties.Furtherranges
ofdensitygradient,plasmacurrentandaveragedensitycouldbeexploredtoprovide
informationontransportandonthestabilisingeffectofdensitypeakingobserved
onASDEXUpgrade.Theassumptionχe∝χiprovestobeinadequatetomodel
ort.transpelectron

84

7Chapter

okoutloandConclusions

Summary7.1

Itiseasytoobtainconfirmations,
orverifications,fornearlyeverytheory-
ifwelookforconfirmations(K.Popper)

ThefusionpowerobtainedinpresenttokamakdevicesandinITERisstronglyrelatedto
theirenergyconfinementcapability.Understandingandcontrollingheatlossesisamajor
challengeforfusionresearchsincedecades.Inter-machinecomparisonsprovidescaling
lawsforextrapolationstothelargerdimensionsofITER,thefirsttokamakreactor,but
aphysicalunderstandingisnecessarytohavemoreconfidenceintheextrapolation,to
optimisethedesignofreactorscenariosandtoreliablypredictthetransportphenomenol-
ogylimitingtheenergyconfinement.Forthispurpose,modelsrelyingonadjustmentsby
meansofadhocparameterscannotbeapplied,althoughtheymayprovetobeusefulto
grouptheexperimentaldataandgivefeedbacktoexperimentandtheory.Theory-based
modelsarerequiredwhichareintrinsicallydimensionlessandareconstructedwithoutany
fittingtoexistingexperimentaldata.
Althoughthereiscommonagreementthatanomaloustransportisdrivenbyplasmatur-
bulencecausedbymicro-instabilities,predictionsarefarfrombeingunivoque,starting
fromtheindividuationofthemodewhichisbelievedtodrivemostoftheheatlosses.
Evenmodelsbasedonthesameinstabilitymechanismsreturnquitedifferentresponses
[31][54][51].Itisthusnecessarytovalidatethetransportmodelsagainsttheexperiments
ossible.passystematicallyas

85

tributionconthesis’This7.2Inthisthesissuchasystematiccomparisonispresented.Theworkhasimprovedour
areas.tdiffereninunderstanding

TheoryAlthoughnonewtheorymodelsaredevelopedwithintheframeworkofthisthesis,the
applicationofthemodelshasincreasedourunderstandingoftheirmainproperties.This
isnotrivialoutcome:themodelsareverycomplexintheirbehavioursanddependon
manydifferentparameters.Ithasbeenpossibletoidentifythekeydependencesandto
understandfailuresincertainparameterregionsduetothelackofessentialphysics.

terimenExpThemodellingprovidesa“newlanguage”todiscusstheexperimentalresults.Itallowsto
orderthedataandtounderstandseeminglycontradictoryexperimentalevidences.This
pointshouldnotbeunderestimated.Fordecadesthedescriptionoftransportwasbased
onempiricaldescriptions.Althoughthiskindofmodellinghelpedtoidentifydifferent
phenomenology,thisapproachislimitedbythefactthatacompletelydifferentempirical
Ansatzcandescribetheexperimentsequallywell.Besides,mostoftheempiricalmodels
provedtobeverysuccessfulonagiventokamakforcertainkindofdischarges,buthadto
beadaptedoradjustedforothermachinesorexperimentalregimes,withoutanypossibility
todetermineaprioriavalidityrange.Thisallowsnoconfidenceinthepredictionofa
newexperiment.Therefore,reliableresultscanintheendonlybeobtainedthrougha
descriptionthatfindsitsrootsinthetheory.

Comparisontheoryversusexperiment
Finally,onegetsanoverviewhowwellthecurrentmodelsdescribetheexperiment.
Throughthecomparisonofthemodelsonehassomeinsightwhichphysicsisessential
tokeep,whatrequiresmoreaccuratedescriptionandtowhichfeaturetheexperimentis
sensitive.Inthisway,feedbackisprovidedforthetheorytoimprovethephysicsdescrip-
tionwhereitisnecessary;newexperimentalactivityisstimulatedtotestthetheoryin
thecrucialparameterranges.Wediscussinthefollowingtheachievementsofthepresent
detail.inorkwthesis

Database7.2.1Adedicatedextensivedatabaseof91ASDEXUpgradedischargeshasbeenconstructed
andtranslatedintoASTRAformat;thisallowstochallenge,distinguishandpossibly

86

falsifytheavailableandfuturetransportmodels.

•Therearesingleparameterscansoverrangesaswideaspossible,totestthemodels
dependencesandnotonlythegoodagreementofthetransportcoefficients.

•Somedischargesfeaturedominantionheating,otherselectronheating.
•Inmostelectronheateddischarges,steadystatetransportanalysisaswellasthe
analysisofthetransientstateisperformed.

•Theeffectsoflocalisedelectronheatingisstudied.
•DischargewithconstantpedestalTeandtotalECHpowerprovideadirectmeasure
stiffness.profileTofe•Aninitialdatabaseof7JETdischargesisbuilt,including2similarityshotslinked
toASDEXUpgradedischarges.Theinter-machinecomparisonchallengesthemod-
elstopredicttheperformancesofdifferentsizeddivertortokamaksindischarges
withidenticaldimensionlessparameters.Nofreeparametersareincludedinthe
theoreticalmodels.However,thedatabaseneedstobeincreasedandthesimilarity
shotshavetobecompared.

7.2.2Developmentofanalysistools
•Thetheorybasedmodelshavebeenprovidedbytheauthorsasroutines.Interfaces
betweentheroutinesandtheASTRAcodehavebeenadjustedandtested,enabling
themtorunontheavailableplatforms.

•Routinesforthequantitativeevaluationoftheprofiles’predictionhavebeenwritten
accordingtothecriteriabyConnorandcoworkers[19].Theprofileanalysisreduces
onaveragethelargeuncertaintiestypicalforthelocalanalysis,inparticularlythose
arisingfromthelargeerrorsinneandfromprofilestiffness.Fortranprograms
havebeenwrittentohandlethestatistics.

•ToolsfortheheatpulseanalysisinASTRAhavebeenimplemented.
•Stand-aloneversionsofthemodelshavebeenbuilt,allowingtostudytheparametric
dependencesofthemodelsandeventuallyfindoutanexperimentumcruciswhere
thepredictionsdeviate.Asystematicvalidationofthemodelswiththeoretical
transportcodesisstilltobedone.

•AreducedversionoftheWeilandmodel,correspondingtothe4equationsversion,
hasbeenre-derivedandimplemented.

87

7.2.3Overviewofmodellingresults
IFS/PPPLTheIFS/PPPLmodelprovestopredicttheiontransportwithgoodaccuracy.Profile
stiffnessispredictedandtheenergycontentisinquantitativeagreementwiththeex-
perimentalvalue.Electrontransportiswellpredictedonlyinpresenceofdominantion
heating.Themodelfailsdescribingelectrontransportassoonastheionheatfluxgetssmaller
andtheelectronheatfluxincreases.Duetoprofilestiffness,lowionheatfluxmeanslow
χi.Sincethemodelassumesχe∝χi,electrontransportisreducedtooandalmostno
anomaloustransportoccurs,unlikeintheexperiment.
TheITGphysicsislikelytobethemainmechanismgoverningcoreiontransport.The
generalassumptionχe∝χiiscontradictedbytheexperimentalobservations.Theonset
foranomalouselectrontransportinelectronheateddischargesappearstohaveathreshold
inTe/TeratherthaninTi/Ti.
GLF23TheGLF23modelsucceedswhenpredictingiontransportandTiprofilestiffness.Electron
steadystatetransportisalsowellpredictedformostASDEXUpgradedischarges,both
NBIandECHheated.ForJETdischargesthemodelgivesthebestpredictions.
Thedependenceonneisindisagreementwiththeexperiment:strongdensitypeaking
destabilisesthetrappedelectronmode,enhancingtransportdramatically.ForASDEX
UpgradeECHdischarges,theratioχepert/χePBisbelowthemeasuredvalue.Thesteady
andthetransientstateareneversimultaneouslywellpredicted:iftheaveragedTeprofile
isaccurate,theheatpulseisslowerthanintheexperiment;iftheheatwaveiswell
described,thesteadystateTeprofileistooflat.
ThephysicsofITGandTEmodesappearstobesuitedtodescribeheattransportin
thetokamaksASDEXUpgradeand,accordingtoapreliminaryanalysis,onJETaswell.
Themodel’ssensitivitytoneisincontradictionwiththeexperimentalresult,where
densitypeakinghasnotsuchadramaticimpactontransportandinanycaseitacts
stabilising.Thispartofthemodelshouldbeimproved,althoughforITERthedensity
profileisexpectedtobequiteflat.Theratioχepert/χePBindicatesthatinrealitythere
issomemoresignificantoffsetforanomalouselectronheattransportthanintheGLF23
model.Thisisanotherevidencethattheresidualtransportdrivenbydensitygradients
appearstobemuchlargerthanintheexperiment.Thisproblemisknowntotheauthors
andhasbeenexplainedastheoccurrenceofaninstability,butthereasonofthestrong
anomaloustransportdrivenbythismodeisnotyetclarified.

88

eilandWForH-modedischargestheWeilandmodelyieldsquantitativeagreementwiththeex-
periment.Thepowerandthedensityscansarereproducedcorrectly.Themodelalso
predictselectrontransportwithhighaccuracy,boththetimeaveragedtemperaturepro-
filesaswellastheamplitudeandphaseofthepropagatingheatwave.Severalharmonics
ofthefundamentalwavefrequencyarewelldescribedandtheeffectofadditionalheating
.correctlypredictedisHowever,dischargeswithlowcurrent(400kA)arepredictedtohavemuchhighercon-
finementthantheexperiments.Thisindicatesthatthedependenceonqandsˆisnot
accurate.Indeed,theimplementationoftheqandsˆdependencesintheWeilandmodel
appearstobetoosimplified.Negativeshearisnottreated.Therefore,thisversionofthe
modelshouldnotbeappliedtoscenarioswithflatqornegativeshear.
ThissimplemodelconfirmsthatthecombinationofITGandTEmodesislikelytogovern
heattransportonASDEXUpgradeandpossiblyonJET.Thesuccessfulreproductionof
ionheateddischargesinspiteofmoderatestiffnessclarifiesthatASDEXUpgradeexper-
imentalresultsarecompatiblewithawiderangeofTiprofilestiffness:fromIFS/PPPL
(strong)toWeiland(medium),theincreaseofχibeyondR/LTiisdifferentbyanorderof
magnitude.TheqeversusTeoffsetschemeofthemodel,fittingdifferentconstraining
experimentalsituations,providesarealisticestimatefortheTEMstabilitythresholdand
forprofilestiffness.Electrontemperatureprofilesareweaklystiffandtheexperimental
gradientsexceedthecriticalthresholdbyfactorsupto3inpresenceofstrongcentralEC
heating.

CDBMThequalitativedisagreementwiththeexperimentshowsthatthismodeldoesnotcontain
thephysicsnecessarytodescribeionheattransport.Inparticular,ionprofilestiffnessis
notpredicted.Thelowertransportlevelobservedinstand-alonerunswiththebackground
parametersfromanelectronheateddischargesuggeststheCDBMmodelasapossible
alternativedescriptionofECHdischarges.Asystematiccomparisonbetweenexperiment
andtheory,involvingalsotheheatpulseanalysis,isstilltobedone.However,the
differenceobservedintheexperimentbetweenionandelectronheattransportproves
thattheassumptionχe=χiisinadequate.

7.3Aglancebeyond
Onecandotheexercisetocheckthemodels’predictionsaboutITER’senergyconfinement
andgainedfusionpower[54].“Stiffer”modelsgetmoreoptimisticwithhigherpedestal
temperatures,buthavestrongerpowerdegradation(theeffortofadditionalheatingis

89

almostuseless).Ourstudyhasprovideddetailedevidenceforathresholdbehaviourof
heattransport,findingweakstiffnessforTeprofilesandmoderatetostrongstiffnessforTi
profiles.ThisresultremarksthattheconfinementperformanceofITERdependsstrongly
onthepedestalvaluesofTiandTe,withsomemorefreedomforelectronheating.As
reportedin[54],itisprematuretomakerealisticstatementsforITERsofar,asthereis
stilllargeuncertaintyaboutthepedestalvaluesofTiandTe.Theunderstandingofedge
transportisthereforeofprimaryinterest.
TheinfluenceofZeffistreatedinthemodelsbuthasnotbeeninvestigatedexperimen-
tallysofar.DedicateddischargesscheduledforthenextASDEXUpgradecampaignwill
providethedatatobecomparedwiththetheoreticalpredictions.Theinter-machine
comparisonshouldbeextended.TheJETdatabaseistoosmalltodrawconclusions.Itis
necessarytoincludemoredischargesandtoextendtheintervalsofthescanparameters
likeTe/Ti,Ipl,totalpowerandplasmadensity.Anoptimaltestforthedimensionless
characterofthemodelsisthesimulationofthesimilarityshots,dischargesfromdifferent
tokamakswheretherelevantdimensionlessparameterarechosentobeequal.Thiscom-
parisonaswellasageneralextensionofthedatabaseareinprogress.
Atheorymodelisnotsuccessfulifitjustprovidesagoodguessofthetransportlevelincer-
tainsituations:itisexpectedtoapproximatelyreproducethephysicsoftheexperiment.
Forinstance,theIFS/PPPLmodelmatcheselectrontransportforNBIheateddischarges,
buttheanomaloustransportisdrivenbyTiandnotbyTe,leadingtowrongpredic-
tionsforECHdischarges(actuallythemodelisnotdevelopedfortheseexperiments).
Forthisreasonwehavecheckedthemodels’dependencesandperformedsingleparameter
scans.However,acomplementaryapproachconsistsinvalidatingthemodelsagainstmore
completetheoreticalcodes.Suchcodesdoexistintheformoffullkineticcalculationsor
realturbulencedescriptions.Themorecompletemodelsarecomputer-timedemanding
andcannotbeusedforastudyasextensiveaspresentedinthisthesis.However,they
canandshouldbeusedtobenchmarkthesimplifiedmodels,assessingthelimitationsof
thelatter.SuchavalidationishighlydesirablefortheITG/TEMmodels.
ItiscommonlybelievedthatωE×BhasabeneficialeffectinthatitstabilisestheITG
modes.SincevtorisamajorcontributiontoωE×B,thequestionariseshowtheangu-
larvelocitydoespropagateradiallyacrosstheplasma.Thedevelopmentofmomentum
transportmodelsandthemodellingofexperimentalprofilesareexpectedtobringclarity
future.neartheinTheIFS/PPPLandWeilandmodelsarenotconstructedforlowandnegativemagnetic
shearplasmas.Ingeneral,themodelsrelyingontheballooningapproximation(asGLF23,
WeilandandIFS/PPPL)arenotsuitedtodescribearegionwithzeroornearlyzeromag-
neticshear[55].ThisisthecasefordischargeswithInternalTransportBarriers(ITBs).
Thedevelopmentofmodelsbasedonotherapproachescanprovideausefulbenchmark
andalsocomplementaryinformation.InparticulartheWentzel-Kramers-Brillouin

90

(WKB)method[55]hasbeendevelopedforITGmodesandcouldleadsoontoanexten-
sive1D-modellingofITBsandreversesheardischarges.
Itseemsfeasibletoperformsoonexperimentswithionheatpulsesbyheatingtangentially
withmodulatedNBI(lowpower).Thispowerfulanalysistoolwouldprovideafurther
testforthemodelsandadirectmeasureofTiprofilestiffness,whichhasbeenobserved
butnotdefinitivelyevaluated.BoththeWeilandandtheIFS/PPPLmodelsshowgood
agreementwiththedataalthoughtheydifferbyanorderofmagnitudeinthequantity
∂χi/∂(R/LTi).
Anextensionofourapproachistomodelalsothedensityprofileself-consistently.This
wouldreducethelargeerrorsontheheatfluxesduetotheexperimentaluncertaintiesof
ne,particularlyfortheGLF23model.Studyingthebehaviourofthedensityprofilesis
atopicofmajorinterest,becauseparticletransportiscrucialforenergyconfinementand
fortheignitioncondition.Thesestudieshavealreadystartedshowingpromisingresults
[57].[56]

91

92

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95

96

AppAendix

Listoffrequentlyusedabbreviations

EC(R)HMECHNBIICRHECECXRSITERASDEXJETMHDITBITGETGTEMχPB,χpert
ρtorASTRAIFS/PPPLGLF23CDBM

systemsdiagnosticsandatingHeMoElectrondulatedCyclotronElectronCyclotron(Resonance)HeatingHeating
InjectionBeamNeutralHeatingResonanceCyclotronIonEmissionCyclotronElectronChargeeXchangeRecombinationSpectroscopy
esdevicerimentalExpInternationalThermonuclearExperimentalReactor
Axis-SymmetricDivertorEXperiment
JointEuropeanTorus
eviationsabbrPhysicsMagnetoHydroDynamicsInternalTransportBarrier
IonTemperatureGradient
ElectronTemperatureGradient
TrappedElectronMode
HeattransportcoefficientfromPowerBalance,perturbativeanalysis
ordinatecofluxoroidalTTransportcodeandmodels
AutomatedSystemforTRansportAnalysis
InstituteforFusionStudiesandPrincetonPlasmaPhysicsLaboratory
GyroLandauFluid(from2dimensionsand3dimensionssimulations)
CurrentDiffusiveBallooningMode

97

98

BendixAppComplementstothederivationof
delmoeilandWtherelationsUsefulB.1Inthederivationofthemodel,somerelationsareveryusefultorewritetheequationsin
asimplerform.Wereportthemostrelevant:
2ρs2=Ωc2s=2ρi2τ(B.1)
civEr≈EpolB≈−ikyφ≈−ieφkyTe=−ieφωeLn(B.2)
B2BTeeBTe
ωe=−kyTemine=−kycs2ne=k⊥ρscs(B.3)
mieBneΩcineLn
ωDe=2k⊥ρscs(B.4)
LBωe=−τωi(B.5)
ωωDe=−τωDi(B.6)
Dj=n(B.7)
ωjB.2Estimateoftheparallelionmotion
Theequationfortheparallelionmotionisobtainedmultiplyingequation(3.5)bye∙:
∂vie1cs2cs2
∂t+vi∙vi=mie∙E−minie∙pi=−Tee∙(eφ)−pee∙pi(B.8)
SincetheequilibriumTeandpeareperpendiculartoe,Teandpearefreetomove
insidee∙inthefirstorderequation.ThelinearisedandFouriertransformedequation
99

(B.9)

is:vi≈−ics2e∙eφ+1δpi(B.9)
ωTeτpi
Weareinterestedinthequantity∙nv:
∙nivi=vi∙n˜i+ni∙vi∙2BB≈niB∙vi∙2B+nivi∙2B∙B≈

BBB
≈nikvi≈nik2cs2eφ+1δpi=−iωDeni|sˆ|eφ+1δpi(B.10)
ωTeτpi2qTeτpi
Thelastpassageisduetothestrongballooningapproximation;thederivationisnot
immediateandcanbefoundin[28].Thecontributionfromtheparallelionmotionis
notneglectedintheWeilandmodelasitisafirstorderperturbativeterm.However,a
commonsimplificationtreatingdriftwavesistoneglecttheparalleliondynamics,sothat
“thediamagneticmusttakeplacemuchfasterthansoundwavetimescale”([26],pag.
5.4),whichmeanskcsω∼ω.
relationersiondispTheB.3relationsatureCurvB.3.1Toderivetheusefulrelations

1∙(njvj)=TvDj∙δpj(B.11)
jq2∙vE=eB0TvD∙φ(B.12)
westartrewritingtheidealMHDequation:
2µ0p=µ0j×B=(×B)×B=−(B)∙B+(B∙)B=−B+(B∙)B(B.13)
2whichleadstothepressurebalance:
112p+2µ0B=µ0(B∙)B(B.14)
where(B∙)Bisthefieldcurvature.SinceB0≈B0/Rcandk∼1/R(estimatevalid
forquasi-flutemodes),theperturbationofthecurvaturetermyields:
δ(B∙)B≈B0kδB+(δB∙)B0∼kB0δB(B.15)

100

whichismuchsmallerthanδB2∼k⊥B0δB.Therefore,
δp+2µ1δB2=0
0ButδB2=δ(B0+δB)2≈2B0∙δB=2B0δB,andhence
δB=−µ0Bδ0p

Besides,thefollowingidentityholds:


(B.16)

(B.17)

e×(v×B)=e∙Bv−e∙vB≈Bv−ve=Bv⊥=B01+δBv⊥

B0(B.18)whereatermvδB⊥hasbeenneglected.δB⊥=0isduetothecommonapproximation
thatE⊥iselectrostatic.Thedivergenceofjis:
δB1δB
∙(nivi−neve)1−B0≈∙eB01−B0e×p=

=11−δB∙e×p+11−δB∙e×p(B.19)
eB0B0eB0B0
wherewehaveusedquasi-neutrality.ThelinearisedrighthandsideofequationB.19is:
1∙e×δp−δB2∙e×p0+

eBeB00−1δB∙e×p0+1∙e×δp(B.20)
eB0B0eB0
Inthesecondterm,thederivativeofthedenominatorisnegligiblecomparedtothatof
thenumeratorifweassumewithWeiland([25]pag.82)
B0δB(B.21)
BδB0

negligible:istermthirdThe∙e×p0=p0∙×e−e∙×(p0)=
=p0∙1(×B0)+p0∙1×B0=µ0p0∙j0−p0∙B20×B0=
B0B0B0B0
=−B0∙e×p0(B.22)
B0

101

andaccordingtoassumptionB.21
δB3B0∙e×p0δB2∙e×p0(B.23)
eBeB00ThentheB.20isapproximatelyequalto:
1e×B0∙δp+1×e∙δp+µ0e×p0∙δp(B.24)
eB0B0eB0eB03
wherewehavesubstitutedfromB.17andusedthevectoridentities(1),(9)and(15)from
[27].Thequantity×ecanbesplittedintoperpendicularandparallelcomponent:
×e⊥=−e×e××e=e×e∙e(B.25)
e∙×e=B1e∙(×B)−e∙BB20×B0=B1e∙(×B)(B.26)
0Theparallelcomponentof×eisthereforeassociatedwithabackgroundcurrent,
andresultstobenegligiblebecausekisassumedtobesmall.Thecurvaturevectoris
therefore:κ=e∙e=−RRc2(B.27)
cFromtheB.14appliedtothebackgroundfields
B0=−µ0p0+e∙B0=−µ0p0+B0e∙e+e∙B0e≈
B0B0
≈−µ0p0+B0e∙e(B.28)
B0onecansubstituteintheB.24toobtain
∙(nivi−neve)1−δBB≈eB2e×κ∙δp(B.29)
00that:noteeW1.Finiteβ(p0terms)termscancel,justifyingalowβtreatment.
2.Thedivergenceofthediamagneticdriftflux(diamagneticcurrent)isacurvature
effect.IfweputvEinsteadofnivi−neve,wehaveφ/eninsteadofpandwecanmakeuse
ofφ0=0:
∙vE1−δB=2qvκ∙φ(B.30)
B0eB0T
wherewehaveintroducedaneffectivecurvaturevelocityvκj=2qjTBj0e×κ.Thelowβ
approximationB.16isactuallyassumedintheWeilandmodel,sothatvκjreducestovDj
andB.29holdsforeachspecies.Sinceδβ/Botermsareneglected,B.29andB.30reduce
tothesimplerformB.11andB.12.
102

B.3.2Derivationofthediamagneticheatflux
TheclosureoftheWeilandmodelisobtainedfromthethirdmomentumofthekinetic
equation

0=∂d3wfmiw2w+∙d3ωfmiw2ww+
∂t22
+qid3wmiw2wE∙∂f+qid3wmiw2w(w×B)∙∂f(B.31)
mi2∂wmi2∂w
neglectingcollisions(firstterm).Besides,wesetE=0toisolatethediamagneticpart.
Thefourthtermcanbeintegratedbyparts,andsince(w×B)xcontainsonlywyandwz,
itisnotaffectedby∂/∂wx.Sowemustcalculate
d3wmif∂w2w∙(w×B)=
2∂w
d3wmifw2I+2w∙(w×B)=d3wmifw2I∙(w×B)(B.32)
22Thefourthmomentum(secondterminB.31)isevaluatedsettingfMaxwellian(see[26],
page2.35).Inparticularfisisotropic,sothati=jyieldsw2wiwjf=0andwi2canbe
replacedby1/3w2:
mm1πm+∞
∙d3wiw2wwf=i∙Id3ωw4f=idww6f(B.33)
33220ThisisaknownGaussianintegral:
+∞dww6αe−aw2=α15π(B.34)
3aa160αcanbedeterminedfromthedefinitionoftheiondensity:
ni=d3wf=2πdww2αe−aw2=2πα−∂1π=πα√π1a−3/2
+∞
0∂a2a2
a3/2
α=2niπ(B.35)
Inourcase,a=mi/2T,sothewholeB.33is
∙d3wmiw2wwf=5min=5pT(B.36)
28a22mi
EquationB.31reducestotheform:
5piTi=qid3wmifw2w×B(B.37)
2mimi2

103

(B.36)

(B.37)

(B.40)

Theterminroundbracketsisknowntobe(see[26],pag.2.161)
2d3ωmifw2w=qi+3niTi+pi+nimivivi(B.38)
222ThefluidvelocityreducestothediamagneticbecausewesetE=0.Isotropyensures
pi=niTiandinadditionweassumemiv2iTi,soitremains:
5piTi=qi5pivi+qi×B(B.39)
2mimi2
whichmultipliedtimesqimBi2B×yields:
5152pivi+qi=qiB2B×2piTi
p5iqi=2qiB2B×Ti(B.40)
B.3.3Thetemperatureperturbation
WewanttocalculateδTi/Ti.Wecanrewrite3.7as:
3∂51
2∂t+vj∙pj+2pj∙vj+π:vj+∙qj=Qj+SEj−2mjvj2Snj(B.41)
Ofcoursetherearenoexternalsourceswhichcontributeatthefirstperturbativeorder,
becauseeithertheydonotvaryintimeortheydo,butmuchslowlier.Neglectingthe
Coulombiantermsaswellasthecontributionfromthestresstensor,itremains:
∂32ni∂t+vi∙Ti+Pi∙vi=−∙qi(B.42)
TheRighi-LeducordiamagneticflowqihastheformB.40.
InanalogywiththeequationB.22,assumingTi⊥j,itis
Pi∙B×Ti=PiTi∙1×B=−2niTiTi∙B×B=niTi∙vDi

eB2eB2eBB2
(B.43)Therefore,thedivergenceofthediamagneticheatfluxcanberewrittenthroughthevector
identities(9),(15)and(1)ofRef.[27]intheform:
5Pi55PiB
∙qi=2∙eBe×Ti=2eBPi∙e×Ti+2e∙B2×Ti=
5555
2eBTi∙Pi×e+2niTi∙vDi=−2nvi∙Ti+2nvDi∙Ti(B.44)

104

Combiningittothe3.6wehave
3∂55
2ni∂t+vi∙Ti+Pi∙vi=2nivi∙Ti−2nivDi∙Ti(B.45)
Weneglectthetermscontainingk2ρs2(thatis,vi≈vE+vi),anduse∙(nivi)=−∂∂nti
fromthecontinuityequation;furthermore,23nivi∙Ti−Tivi∙nicancelswith25nivi∙Ti
sincebydefinitionvi∙(niTi)=0.TheB.45becomes:
53∂Ti3
−2nivDi∙Ti=2ni∂t+2nivE∙Ti+Ti[−∙(nivi)−vE∙ni]=
3∂Ti32∂ni
2ni∂t+2vE∙niTi−3Tini−Ti∂t(B.46)
Thelinearisedequation,multipliedby2/(3niTi)andFouriertransformed,reads:
iωδTi−iωeeφ∙ηi−2−2iωδni=5iωDiδTi(B.47)
TiTe33ni3Ti
aluatedevewwhere−ni∙vE=−niikyφ=−Teniikyeφ≈−iωenieφ(B.48)
BeBTeTe
Thisdeliverstherelativetemperatureperturbation
δTiω2δniωe2eφ
Ti=ω−5ωDi/33ni+ωηi−3Te(B.49)
B.3.4Thecontributionfromthestresstensordrift
Anestimateofthedivergenceofthepressureanisotropydriftisrequired.Largemode
numbersareassumed,i.e.kκp=|lnp0|.Besides,κp=0andthemagneticfield
istakentobehomogeneous,suchthatzˆ=B/Bisuniform.Thestresstensorisdefined
[9]):of(2.21)equation(seeπxy=πyx=p∂vx−∂vy+1∂qx−∂qy
2Ωc∂x∂y4Ωc∂x∂y
πyy=−πxx=p∂vy+∂vx+1∂qy+∂qx(B.50)
2Ωc∂x∂y4Ωc∂x∂y
Theqtermsareformallyidenticaltothevones,sotheywillbedroppedforsimplicityand
recoveredfinally.Thebackgrounddensityandtemperaturehavenopoloidalvariation,so
inournotationdno/dy=dTo/dy=0.
(∙π)x=∂πxx+∂πxy=−p∂2vy+∂2vx−1∂vy+∂vxdp+
∂x∂y2Ωc∂x2∂x∂y2Ωc∂x∂ydx

105

p∂2vx∂2vypp∂vy∂vx
+2Ωc∂x∂y−∂y2=−2Ωcvy+2Ωcκp∂x+∂y
(∙π)=∂πyx+∂πyy=p∂2vx−∂2vy+1∂vx−∂vydp+
y∂x∂y2Ωc∂x2∂x∂y2Ωc∂x∂ydx
p∂2vy∂2vxpp∂vx∂vy
+2Ωc∂x∂y+∂y2=2Ωcvx−2Ωcκp∂x−∂y
Writinginamorecompactform,andreintroducingq:
1p∙π=2Ωc[zˆ×⊥v+κp(vy−zˆ×vx)]+4Ωczˆ×⊥q⊥(B.51)
whereq⊥isthepartofqcorrespondingtoafluxofperpendicularenergy(reference[2.12]
eiland’s):Winq⊥=2p⊥(zˆ×T⊥)(B.52)
ΩmcOmittingtermscontainingρi/LB,theanisotropydriftfluxcanbewrittenas
111nvπ=eBzˆ×∙π=2mΩ2−p⊥v+pκp(zˆ×vy+vx)−2⊥q⊥(B.53)
cWenotethatsincewehavenotconsideredtheB-curvature
∙(zˆ×vy)=vy∙(×zˆ)−zˆ∙(×vy)=0(B.54)
Weneglecttermswhicharenotlinearinκp,becauseκpk.Bcanbetakenoutofall
’sforthelowβapproximationB.21.
11∙(nvπ)=2mΩ2−p∙⊥v−p∙⊥v+pκpvx−2∙⊥q⊥=
c11⊥=2mΩc2−p∙⊥v−p⊥∙v+pκpvx−2⊥∙q(B.55)
InabsenceofB-curvature,thedivergenceoftheelectrostaticdriftvelocityzero(seeB.12);
thedivergenceofthediamagneticdriftvelocityanddiamagneticheatfluxareofhigher
orderinδ:
∙v=1∙zˆ×p=11∙(zˆ×p)=−n∙v∝δnv≈0(B.56)
nneBneB1∙q=eBp∙(zˆ×T)=−T∙v∝δTv≈0
Atorderzeropisintheradialdirectionandthereforep/p∙=−κp∂/∂x.Equation
B.55canbesimplified:1
∙(nvπ)=mΩ2κpvx
c

106

driftolarisationpTheB.3.5Thepolarisationdriftinthesimplifiedgeometryisdefinedas
∂1vp=Ωc∂t+v∙(zˆ×v)(B.57)
Duetothelargemodenumberapproximation,onlyperturbedvelocitiesenterthelast
v.Inthelinearapproximation,theonlyrelevantconvectivederivativev∙istheone
containingthebackgroundv.AsusualwesetE=0andweisolatethediamagneticdrift,
whichresultstobetheonlybackgroundvelocity.Thecontributionsto∙(nvp)are:
Ω1nvj∂j∂i(zˆ×v)i=Ωn(v∙)∙(zˆ×v)(B.58)
ccΩ1∂invj∂j(zˆ×v)i=Ω1(nv):(zˆ×v)(B.59)
cn∂c
∙Ωc∂t(zˆ×v)(B.60)
WeevaluatetheB.58andtheB.59:
nn∂vx∂vy1∂∂vx∂vy
Ωc(v∙)∙(zˆ×v)=Ωcv∙∂y−∂x=−mΩc2pκp∂y∂y−∂x(B.61)
Ω1∂invj∂j(zˆ×v)i=−Ω1eB(pκp)∙∂∂y(zˆ×v)≈0(B.62)
ccNowweputtheseresultstogetherwiththedivergenceofthefluxrelatedtotheanisotropy
drift.

n1∂∂vx∂vy
∙(nvπ)+Ωc(v∙)∙(zˆ×v)=mΩ2κpvx−κp∂y∂y−∂x=
c1∂2vx∂2vy1∂
=mΩ2κpvx−κpvx−∂x2−∂x∂y=mΩ2κp∂x∙v≈0(B.63)
ccTotheleadingorderthelasttermiszeroduetoequationB.56andbecause∙vE≈0.
“Theconvectivediamagneticcontributionto∙(nvp)areexactlycancelledbythestress
tensorcontribution∙(nvπ).”([25],pag.21).Thefollowingrelationholds:
∂n∙[n(vp+vπ)]=∙(zˆ×v)(B.64)
t∂ΩcSubstitutingtheleadingordersdriftsaccordingtoequationsB.11andB.12gives:
ni∂ni∂∂niTi∂eφ2∂eφ
Ωci∂t∙(zˆ×vE)=−Ωci∂t∂xvE,x=−mΩ2∂tTi=−2niρi∂tTi
ci∂∂n2Ω∂t∙(zˆ×vi)=−2niρi∂tδpi
ci

107

Wechooseaparticulardensityresponse,thesimplestleadingorderofflutemodes,that
istheconvectivecontribution,sothatthepressureperturbationresultstobe:
δpi=−ωiTeφ(B.65)
TωpiiRememberingthatρs2=Te/miΩ2ci=2ρi2τ,weobtaintheusefulrelation
eφeφeφ∙[n(vpi+vπi)]≈−2nρi2ik2T−ik2ωiTT=−ink2ρs2T(ω−ωiT)(B.66)
eiiB.3.6Thedensityperturbations
Weneedtoevaluatethefourremainingcontributionstoequation3.12:
(I)−ni∙vE≈−iωeniTeφ
e(II)−ni∙vE=−nievDi∙φ=−iωDinieφ≈−iωDeneeφ
TiTiTe
eφ(III)−∙[ni(vpi+vπi)]=inik2ρs2(ω−ωiT)T
e11(IV)−∙(nivi)=−TivDi∙δPi=−TivDi∙(Tiδni+niδTi)=
ω2δniωe2eφ
=−iωDiδni+niω−5ωDi/33ni+ωηi−3Te
WehavemadeuseoftheequationB.48for(I),B.12for(II),B.66for(III)andfinally
equationsB.11andB.49fortherelation(IV).Wenotethatthetermk2ρs2isstabilising
andquadraticink,sothattooshortwavelengthsarenotaccessible.Theiondensity
perturbation,givenby−iωδni=I+II+III+IV,timesi/niistherefore
δni2ωωDi
niω−ωDi−3ω−5ωDi/3=
=eφ−k2ρs2(ω−ωiT)+ωDiωeηi−2+ωe−ωDe(B.67)
Teω−5ωDi/33
WemultiplytheB.67timesω−5ωDi/3andthendividebyNi=ω2−ωDi5/3+ωD2i10/3:
δni=eφ11ωD2e−ωˆ2k2ρs2n+ωˆ1−n−5k2ρs2n−k2ρs21+ηi+
175225
niTeNin3ττ
−τηi−3+3n−kρs3τ2(1+ηi)(B.68)

108

Asimilarequationholdsfortrappedelectrons:
δneteφ1ωe75
net=TeNeωDeωωDe(1−n)+ωD2eηe−3+3n=
eφ1175eφ
=TeNˆenωˆ(1−n)+ηe−3+3n=TeAe(B.69)
wherethedefinitionofAe3.26isused.

diffusionQuasi-linearB.4WealreadyknowthetemperatureperturbationB.49,nowwesubstitutethereδni/ni
fromB.69and3.2viaquasi-neutrality.Thisallowstoclosethesystemofequationsand
toestimatetheionheatflux3.25andthereforethequasilinearionheatdiffusivity:
ΓTiγ1Tiγ2ω2
χi=−Ti=−Reikxω−35ωDiTikxLn3ωe(1−ft+ftAe)+ηi−3=
γ21ω¯−5ωDi/32ω2
=Reik2ηi(ω−5ω/3)2+γ23ωe(1−ft+ftAe)+ηi−3
iDrxWenormalisefrequenciesasusualtoωDeanddefineω¯=ω¯/ωDe.
χi=γˆ2ωDe1
222225ωˆ52
kxηi(ωˆr+5/3τ)+γˆ
−Im3n|ωˆ|+3τ(1−ft+ftAe)+ω¯+3τηi−3(B.70)
Let’sfocuson−Im[]:
−Im[]=−2nft|ωˆ|2Im(Ae)−10γˆn(1−ft)−10nftIm(ω¯Ae)+ˆγηi−2=
210239τ9τ3
n=γˆηi−3−(1−ft)9τ−3ftΔi(B.71)
where1255
Δi=γˆ|ˆω|+3τωˆrIm(nAe)+3τRe(nAe)(B.72)
Sowehaveforχiexactlytheexpression3.27,providedwederivethecorrespondent
expression3.28forΔi.Weremindthat
175
nAe=Nˆωˆ(1−n)+ηe−3+3n(B.73)
eLet’srationalise1/Nˆe:
1=ω¯2−ω¯10/3+5/3=ωˆr2−γˆ2−ωˆr10/3+5/3+iγˆ(−2ωˆr+10/3)(B.74)
NˆeNˆNˆ

109

WewanttoevaluateReandIm:
NˆRe(nAe)=ωˆr2−γˆ2−10ωˆr+5ωˆr(1−n)+ηe−7+5n+
3333
−γˆ2(1−n)−2ωˆr+10=(1−n)ωˆr3+ωˆrγˆ2−10ωˆr2−10γˆ2+5ωˆr+
3333
+ωˆr2−γˆ2−10ωˆr+5ηe−7+5n(B.75)
3333NˆIm(nAe)=ωˆr2−γˆ2−10ωˆr+5ˆγ(1−n)+
103375
+γˆ−2ωˆr+3ωˆr(1−n)+ηe−3+3n=
51075
=γˆ(1−n)−ωˆr2−γˆ2+3+γˆ−2ωˆr+3ηe−3+3n(B.76)
SosubstitutinginB.72aftersomealgebraweobtaintheexpressionforΔicontainedin
3.28:equationΔi=1|ωˆ|2−|ωˆ|2(1−n)+5(1−n)+2−ωˆr+5ηe−7+5n+

ˆ33335N5751075
+3τ−|ωˆ|2ωˆr(1−n)+3ωˆr(1−n)−2ωˆr2ηe−3+3n+3ˆωrηe−3+3n+
+(1−n)|ωˆ|2ωˆr−10|ωˆ|2+5ωˆr+ωˆr2−γˆ2−10ωˆr+5ηe−7+5n=

333333
122755117
=Nˆ|ωˆ||ωˆ|(n−1)−2ωˆrηe−3+3n+32ηe−3+3n+
55502575
+3τ−ηe+1+3n+9τ(1−n)+9τηe−3+3n(B.77)
matrixortranspTB.4.1Wecansplitthepre-factorofχiasgivenin3.28inordertoisolatethecontributionsof
thedifferentdrivinggradients:
1−2−(1−ft)10n−2ft1|ωˆ|2|ωˆ|2(n−1)+ωˆr14−2ηe−10n+
511755502575
3ηi9τηi3ηiNˆ33
+3−3+2ηe+3n−3τ1+ηe−3n+9τ(1−n)ωˆr−9τ3−ηe−3n=
2ft2214555ft50175
=1−3ηi1+Nˆ|ωˆ|−|ωˆ|+3ωˆr−9−3τ+Nˆ9τωˆr−27τ+
−2n5(1−ft)+ft|ωˆ|2|ωˆ|2−10ωˆr+35+25+ft−50ωˆr+125+

2ftηe10525
3ηi3τNˆ399τNˆ9τ27τ
−3Nˆηi|ωˆ|2−2ωˆr+3−3τ+9τ=

110

+1=−Tini21+ft|ωˆ|2−|ωˆ|2+14ωˆr−55−5+ft150ωˆr−175+
Tini3Nˆ393τNˆτ927
+TTii34R35τ(1−ft)+fNˆt|ωˆ|2|ωˆ|2−310ωˆr+935+925τ+fNˆtτ1−950ωˆr+27125+
−TTiTTe32fNˆt|ωˆ|2−2ωˆr+310−35τ+925τ(B.78)
eiWerecognisethediffusivecontribution,theconvectivepartofthefluxdrivenbyni,the
Wpincehdotermtheandsametheforelectronoff-diagonalheattermtranspproport,ortionalremindingtoTe.equation3.32:
1−32Nˆ|ωˆ|2(−2ωˆr+5)−925+
2
−TTeennee321−|ωˆNˆ|−|ωˆ|2+314ωˆr−940+Nˆ1−950ωˆr+27175+
+TTe34Rˆ1|ωˆ|2|ωˆ|2−310ωˆr+910+509ωˆr−27125(B.79)
NeThereisadiffusivecontribution,aconvectivefluxdrivenbyneandapinchterm;no
contributioncomesfromTi.
Theparticletransportisalsobuiltupofseveralcontributions.Fromequation3.36:
|ωˆ|2(n−1)+ωˆr314−2ηe−310n+35−311+2ηe+37n=
5514=−|ωˆ|2+3ωˆr−9+
−nneR2|ωˆ|2−310ωˆr+935+
e+TTenne−2ωˆr+310(B.80)
eeIfweequationswriteB.78,theB.79transportandB.80equationsforimmediatelyparticlesleadandtotheheattranspfluxesortincotheefficienmatrixtsform3.38,3.37,3.39
3.40.and

111