R.Fitzpatrick,Phys.Plasmas,092502F.Hofmann,M.J.Dutch,D.J.Ward,M.Anton, - PDF document

R.Fitzpatrick,Phys.Plasmas,092502F.Hofmann,M.J.Dutch,D.J.Ward,M.Anton,I.Furno,J.B.Lister,andJ.-M.Moret,Nucl.Fusion,681D.A.HumphreysandA.G.Kellman,Phys.Plasmas,2742R.Fitzpatrick,Phys.Plasmas,062505J.P.Freidberg,IdealMagnetohydrodynamicsSpringer,Berlin,1987L.E.Zakharov,Phys.Plasmas,062507V.Riccardo,T.C.Hender,P.J.Lomas,B.Alper,T.Bolzonella,P.deVries,G.P.Maddison,andtheJETEFDAContributers,PlasmaPhys.ControlledFusion,925012506-10R.FitzpatrickPhys.Plasmas,012506 parametersarethesameasthethoseusedintheaxisymmet-ricVDEsimulationshowninFig..ItcanbeseenthatthenonaxisymmetricVDEsimulationisverysimilartothecor-respondingaxisymmetricone,exceptinthetimeperioddur-ingwhichthelattersimulationindicatesanunstablekinkmode.AccordingtoFig.,theunstablekinkmodetrig-gersanintenseburstofbothaxisymmetricandnonaxisym-metrichalocurrent,withanassociatedspikeinthemaxi-mumhalocurrentforceexertedonthelimiter.Notethatthetoroidalpeakingfactorforthehalocurrentforcequicklyattainsitsmaximumallowedvalueof2duringtheperiodof=1instability.ThesimulationsuggeststhattheonsetofkinkinstabilityduringaVDEcangiverisetoadangerousincreaseinthehalocurrentforce.X.SUMMARYANDDISCUSSIONWehavedevelopedasimplemodelofaxisymmetricver-ticaldisruptioneventsintokamaksinwhichthehalocurrentforceexertedonthevacuumvesselorwhateverrigidcon-ductorlimitstheplasmaiscalculateddirectlyfromlinear,marginallystable,ideal-MHDstabilityanalysis.ThebasicpremiseofourmodelisthatthehalocurrentforcemodiÞespressurebalanceattheedgeoftheplasma,andthereforealsomodiÞesideal-MHDplasmastability.Inordertopreventtheidealverticalinstability,responsiblefortheVDE,fromgrowingontheveryshortAlfvŽntimescale,thehalocurrentforcemustadjustitselfsuchthattheinstabilityisrenderedmarginallystable.Thisallowstheverticalinstabilitytode-veloponarelativelylongtimescaledeterminedbyresistivediffusionofmagneticßux-surfacesthroughthescape-offlayerandvacuumvessel,seeSec.VIII.Consequently,theplasmaremainsinanapproximateaxisymmetricequilibriumthroughoutthedurationoftheVDE.Thisexplainswhyitisvalidtoemployideal-MHDtocharacterizetheverticalstabilityoftheplasmaateachstageoftheVDE.WehaveusedourmodeltoperformanumberofcrudesimulationsofaxisymmetricVDEsintokamaks,seeSec.VII.ThesesimulationspredicthalocurrentsandhalocurrentforceswhicharesimilarinmagnitudetothoseobservedinInaddition,thesimulationsindicatethatthehalocurrentincreasesinmagnitudeastheedge-andthatan=1kinkmodeistriggerediftheedge-comestooclosetounity.Theseconclusionsarealsoinbroadagreementwithexperimentaldata.Wehavegeneralizedourmodeltodealwithahalocurrentinducedbythe=1kinkmode,seeSec.IX.Thisextendedtheoryallowsustopredictthepeakhalocurrentforceexertedonthevacuumvessel.However,ournonaxisymmetricmodelrequiressomeimprovementinordertomakeittrulyrigorous.Oneimportanteffectwhichisabsentfromourmodelisthemoderatinginßuenceofeddycurrentsßowinginexternalconductors,suchas,thevacuumvessel,ontheverticalinsta-bility.SomejustiÞcationfortheomissionofthiseffectinthelatterstagesofaVDEisgiveninSec.VIII.Forthesakeofsimplicity,wehavealsoneglectedanychangeintheplasmashapeassociatedwiththegrowthoftheverticalmode.Fi-nally,ithasbeenassumedthattheinductiveelectricÞeldsgeneratedbyaVDEaresufÞcientlylargetocausethebreak-downofanysheathsattheplasma/limiterboundary,and,consequently,thatthehalocurrentmagnitudeisnotlimitedbytheionpolarizationcurrent.TheauthorwouldliketothankL.Zahkarovprovidingthecrucialinsightwhichmadethispaperpossible.ThisresearchwasfundedbytheU.S.DepartmentofEnergyunderContractNo.DE-FG05-96ER-54346.J.A.Wesson,R.D.Gill,M.Hugon,F.C.Schuller,J.A.Snipes,D.J.Ward,D.V.Bartlett,D.J.Campbell,P.A.Duperrex,A.W.Edwards,R.S.Granetz,N.A.O.Gottardi,T.C.Hender,E.Lazzaro,P.J.Lomas,N.L.Cardozo,K.F.Mast,M.F.F.Nave,N.A.Salmon,P.Smeulders,P.R.Thomas,B.J.D.Tubbing,M.F.Turner,andA.Weller,Nucl.FusionE.J.Strait,L.L.Lao,J.L.Luxon,andE.E.Reis,Nucl.Fusion,527K.Ioki,G.Johnson,K.Shimizu,andD.Williamson,FusionEng.Des.,39S.Mirnov,J.Wesley,N.Fujisawa,Yu.Gribov,O.Gruber,T.Hender,N.Ivanov,S.Jardin,J.Lister,F.Perkins,M.Rosenbluth,N.Sauthoff,T.Taylor,S.Tokuda,K.Yamazaki,R.Yoshino,A.Bondeson,J.Conner,E.Fredrickson,D.Gates,R.Granetz,R.LaHaye,J.Neuhauser,F.Porcelli,D.E.Post,N.A.Uckan,M.Azumi,D.J.Campbell,M.Wakatani,W.M.Nevins,M.Shimada,andJ.VanDam,Nucl.Fusion,2251T.C.Hender,J.C.Wesley,J.Bialek,A.Bondeson,A.H.Boozer,R.J.Buttery,A.Garofalo,T.P.Goodman,R.S.Granetz,Y.Gribov,O.Gruber,M.Gryaznevich,G.Giruzzi,S.GŸnter,N.Hayashi,P.Helander,C.C.Hegna,D.F.Howell,D.A.Humphreys,G.T.A.Huysmans,A.W.Hyatt,A.Isayama,S.C.Jardin,Y.Kawano,A.Kellman,C.Kessel,H.R.Koslowski,R.J.LaHaye,E.Lazzaro,Y.Q.Liu,V.Lukash,J.Manickam,S.Medvedev,V.Mertens,S.V.Mirnov,Y.Nakamura,G.Navratil,M.Okabayashi,T.Ozeki,R.Paccagnella,G.Pautasso,F.Porcelli,V.D.Pustovitov,V.Riccardo,M.Sato,O.Sauter,M.J.Schaffer,M.Shimada,P.Sonato,E.J.Strait,M.Sugihara,M.Takechi,A.D.Turnbull,E.Westerhof,D.G.Whyte,R.Yoshino,H.Zohm,andtheITPAMHD,Disruption,andMagneticControlTopicalGroup,Nucl.Fusion,S128T.H.JensenandD.G.Skinner,Phys.FluidsB,2358J.P.FreidbergandF.Haas,Phys.Fluids,440 FIG.5.NonaxisymmetricVDEsimulationperformedwith=2.0,=0.15,=5.04,=0.3,=0.30,=0.31,and=0.60.TheÞrstpanelsolidcurveand10dashedcurveasfunctionsof.Thesecondpanelshowssolidcurvedottedcurve,anddashedcurve.Thethirdpanelshowssolidcurvedashedcurve.Finally,thefourthpanelshowstheeigenvalueofthemostunstable=1orhybridmode,excludingthemarginallystablemode.012506-9AsimpleidealmagnetohydrodynamicalmodelPhys.Plasmas,012506 Now,the=1componentofthelimiterforcegivesrisebetweenmodeswhosetoroidalmodenumbersdifferbyunity.Inthepresenceofsuchaforce,astraightfor-wardgeneralizationoftheanalysisinSec.Vyieldsthefol-lowingeigenvalueequationwhichgovernsthestabilityofthecoupled=0and=1modes:isthe=0forcematrix,=1forcematrix,=0eigenfunction,=1eigenfunction,and.Moreover,thecouplingmatrix,,hasthe 2h  haexp*
m
md Intheabove,wehaveneglectedthecouplingofthemodetothe=2mode,forthesakeofsimplicity.,where=1.Here,arethemeanamplitudesofthe=0and=1modes,respectively.Wecanwriterepresentthenormalizedeigenfunctionsof=0and=1forcematrices,respectively.Thecorre-spondingeigenvalueswhichareassumedtobethemostnegativeones.Ofcourse,boththeseeigenval-uesarefunctionsoftheaxisymmetrichalocurrentforcepa-rameter,.Here,forthesakeofsimplicity,wehavene-glectedanychangeinshapeofthe=0andeigenfunctionsduetothecouplingproducedbythenonaxi-symmetrichalocurrent.reducestoThisexpressiondescribeshowthenonaxisymmetrichalocurrentcouplesthe=0and=1modestoproducetwo=1modes.TheconditionforoneofthesemodestobemarginallystableisTheeigenvalueoftheothermodeisthenNow,forthemarginallystablemode 0=0 However,itisreasonabletosupposethatthe=1andlimiterforcesareinapproximatelythesameratioasthecor-respondingdisplacements,i.e., Pˆ01 whichleadsto Pˆ0 20 Finally,weexpectthelimiterforcetobeapositivedenitequantity.Inotherwords,theplasmaisabletopushoutwardonthelimiter,butnottopullinward,since,inthelattercase,theplasmawouldinsteadlosecontactwiththelimiter.Inoursimplemodel,inwhichthelimiterforceonlyhasanandan=1component,thisrestrictionimpliesthati.e.,thetoroidalpeakingfactorforthehalocurrentforcecannotexceed2,seeEq..Wecanincorporatetherestric-tionintoEq.bywriting Pˆ0 2=0 ThesolutiontotheproblemisobtainedbysearchingforasolutionofEqs.forwhichtheeigenvalueispositive.Thisimpliesthatthehalocurrentrendersoneofthehybridmodesmarginallystable,andtheotherstable.Ofcourse,themarginallystablemodeactuallygrowsonare-sistivetimescale.NotethatwehavemadenoattempttoÞndarigoroussolutionofthe=1eigenvalueequation.Therea-sonforthisisthatouranalysisdoesnottakeintoaccountthefactthataslowlygrowinghybrid=1modeeventuallymakestheplasmaequilibriumthree-dimensional.Previously,thiswasnotaproblem,becauseaslowlygrowingpure=0modeconvertsanaxisymmetricplasmaequilibriumintoanotheraxisymmetricequilibrium.Itfollowsthattheaboveanalysisissomewhatheuristic,sinceanaccuratetreat-mentofanonaxisymmetrichalocurrentwouldrequireforce-matrixcalculationsmadeusingkink-distortedplasmashowsanexamplenonaxisymmetricVDEsimulation.Thesimulationparametersare=2.0,=0.15,=5.04correspondingtoaninitialedge-of4.0=0.3,=0.30,=0.31,and=0.60.Ofcourse,these012506-8R.FitzpatrickPhys.Plasmas,012506 componentoftheinductiveelectricÞeldavailabletodrivethecurrentis BoEBo
a2 qaR.
Here,qaBo istheedgesafetyfactor.Itfollowsthatthehalocurrentdensityisistheaverageelectricalconductivityofthehalo.Thus,thetoroidalandpoloidalhalocurrentsare qaR,
IhIh istheradialthicknessofthehalo.Notethati.e.,thehaloisrelativelythin.Afternormalization,thecurrentstaketheform .AccordingtoEq.,thelimiterforcepa-rameterbecomes 2 ha2h
3/2 Now,thestabilitytheoryofSec.VimpliesthattheverticalinstabilityisrenderedmarginallystablewhensincealltheotherparametersinthetheoryareorderunityHence,thehalocurrentmoderatedverticalinstabilitygrowsonthetimescale ThistimescaleiscertainlymuchlongerthananAlfvŽntime,aswasassumedearlier.Thus,amodegrowingonsuchatimescaleiseffectivelymarginallystableasfarasideal-MHDstabilitytheoryisconcerned.Thegrowthtimeisalsolonger,byafactor,thanthetimeofthehalo.Thissuggeststhatanyinductivetimedelayinset-tingupthehalocurrentisnegligible.Finally,theVDEgrowthtimeisprobablylongerthanthetimeofthevacuumvesselwhichisoforder,wherearethevesselconductivity,minorradius,andradialthickness,respectivelyunlessthevesselisboththickandhighlyconducting.ThislastobservationofferssomejustiÞ-cationfortheneglectofvacuumvesseleddycurrentsinouranalysisÑatleastinthelatterstagesofaVDEwhentheplasmahasmadecontractwiththelimiterÑsinceitimpliesthathalocurrentsaremoreeffectiveatmoderatingthegrowthoftheidealverticalinstabilitythanvesseleddycur-IX.NONAXISYMMETRICHALOCURRENTSAshasalreadybeenmentioned,weexpectthehalocur-renttodevelopacomponentassoonasthemostnegativeeigenvalueofthe=1force-matrixlatedinthepresenceofanaxisymmetrichalocurrentwhichissuchastorenderthe=0modemarginallystablecomesnegative.Now,anonaxisymmetrichalocurrentisas-sociatedwithtoroidalpeakingofthelimiterforce.Wecanmeasureofthedegreeoftoroidalpeakingviaaso-calledtoroidalpeakingfactor,whichisdeÞnedastheratioofthemaximumtothemeanlimiterforce.Another,moreconven-tional,deÞnitionofthetoroidalpeakingfactoristheratioofthemaximumtothemeanpoloidallimiterhalocurrent.designoftheITERvacuumvesselisextremelysensitivetotheassumedmagnitudeofthetoroidalpeakingfactor.Thus,itisofparamountimportancetopredictthevalueofthisparameter.Inthefollowing,weoutlinehowthisgoalcanbeThecurrentsheetdensityassociatedwithan=1halocurrenthasthefollowingcomponents: Bobcos
,
ih=P1 intheSOL,and Bobcos*
,
ih=P1 inthelimiter.Here,isaconstant,and 2
qa,

=0ha

 Theaboveexpressionsareobtainedbyrequiringthatthe=1halocurrentßowalongmagneticÞeld-linesintheSOL,ßowalongtheshortestpathinthelimiter,andconservecharge.Itfollowsthat,inthepresenceof=0and=1halocurrents,theradiallyintegratedlimiterpressuretakestheOfcourse,iszerothroughouttheSOL.Thetoroidalpeak-ingfactorforthehalocurrentforceisthus Thetoroidalpeakingfactorforthepoloidallimiterhalocur-rentis012506-7AsimpleidealmagnetohydrodynamicalmodelPhys.Plasmas,012506 thepresence,ofthelimiterforce.TheÞgureconÞrmsthatthe=0modeisindeedaverticalinstability.Moreover,itcanbeseenthat,inthemarginallystablecase,thedisplacementislocallyreducedinthelimiterregion,presumablyasaconse-quenceofthereactionforceexertedontheplasma.Notethat,aslongastheinstabilitygrowsonaresistivetimescale,thedisplacementdoesnothavetobezerointhelimiterregion.Indeed,anonzerodisplacementindicatesthediffusionofplasmamagneticßux-surfacesintothelimiter.showsanexampleaxisymmetricVDEsimula-tionperformedusingthefollowingparameters:=2.0,=0.15,=5.04correspondingtoaninitialedge-=0.3,=0.30,=0.31,and=0.60.Itcanbeseenthat,astimeprogresses,andthehigh-plasmagraduallyshrinksinsizeduetoitsinteractionwiththelimiter,theedgesafety-factor,whichactuallyrepresentsthesafety-factoratthelastclosedßux-surfacedecreases.Thisisanaturalconsequenceoftheshrinkageofthehigh-plasmaatconstanttoroidalplasmacurrent.Thefallintheedge-continuesuntilthestartofthecurrentquench.Duringthecurrentquench,thereductionintheplasmacurrentissufÞ-cientlyrapidtooffsettheshrinkageofthehigh-andtheedge-increases,eventuallybecomingverylargeastheplasmacurrentdecaystozero.Thetoroidalandpoloidalhalocurrentscanbeseentoriseastheedge-decreases,andfallasitincreases,andthusattaintheirpeakvaluesatthestartofthecurrentquench.Notethatthepeaktoroidalandpoloidalhalocurrentsareabout20%oftheinitialtoroidalplasmacurrent.Thelimiterforcecanbeseentoriseandfallinproportiontothepoloidalhalocurrent,andthusattainsitspeakvalueatthestartofthecurrentquench.Thisimpliesthatthelimiterforceneededtomakethemodemarginallystablegenerallyincreasesastheedge-creases.Finally,itcanbeseenthatthemostnegativeeigen-valueofthe=1force-matrixisinitiallypositive,indicatingthattheplasmaisinitiallystabletotheideal=1kinkmode.However,theeigenvaluedecreasesastheedge-falls,andeventuallybecomesnegative,indicatingkinkmodeinstabil-ity.Now,inthepresenceofanunstablekinkmode,weex-pectthehalocurrenttodevelopan=1componentseeSec.,whichinvalidatesoneofthecentralassumptionsofouri.e.,thatthehalocurrentisaxisymmetric.Fortu-nately,theperiodofinstabilityiscomparativelybrief,andisterminatedbytheonsetofthethermalandcurrentquenches.showsanexampleaxisymmetricVDEsimula-tionperformedusingthefollowingparameters:=2.0,=0.15,=5.04correspondingtoaninitialedge-=0.3,=0.20,=0.21,and=0.50.Thissimula-tionisidenticaltothatshowninFig.,exceptthatthether-malandcurrentquenchescommencesomewhatearlier.Itcanbeseenthattheearlierquenchonsetpreventstheedge-fromfallingtoaslowavalueasintheÞrstsimulation.Consequently,themaximumhalocurrentandlim-iterforcearesigniÞcantlyreduced.Moreover,the=1moderemainsrobustlystableduringthesimulation.VIII.GROWTH-RATEOFTHEVERTICALINSTABILITYInourmodel,itisassumedthatthehalocurrentforceconsiderablyslowsdownthegrowthoftheverticalinstabil-ity.Letusnowestimatethemoderatedgrowth-rateofthisWeshallassumethatthehalocurrentispredominantlyaconsequenceoftheinductiveelectricÞeldgeneratedbythereductioninthetoroidalmagneticßuxlinkedbythehaloloop,asitshrinksinsize.ApplyingFaradayÕslawtoapo-loidalcircuitofthehalo,weobtain isthegrowthrate,andisthemeanminorradiusofthehalo.Now,thehalocurrentisforthemostpartstrainedtoßowparalleltomagneticÞeldlines.Hence,the FIG.3.AxisymmetricVDEsimulationperformedwith=2.0,=0.15,=5.04,=0.3,=0.30,=0.31,and=0.60.TheÞrstpanelshowssolidcurveand10dashedcurveasfunctionsof.Thesecondpanelsolidcurvedottedcurve,anddashedcurve.Thethirdpanelshows.Finally,thefourthpanelshowsthemostnegativeeigen-valueofthe=1F-matrix. FIG.4.AxisymmetricVDEsimulationperformedwith=2.0,=0.15,=5.04,=0.3,=0.20,=0.21,and=0.50.TheÞrstpanelshowssolidcurveand10dashedcurveasfunctionsof.Thesecondpanelsolidcurvedottedcurve,anddashedcurve.Thethirdpanelshows.Finally,thefourthpanelshowsthemostnegativeeigen-valueofthe=1F-matrix.012506-6R.FitzpatrickPhys.Plasmas,012506 fromgrowingontheveryshortAlfvŽntimescale,thevalueoflimiterforceparameter,,mustbeadjustedsuchthatthiseigenvalueissettozeroi.e.,suchthatthemodeisrenderedmarginallystable.WealwaysÞndthattherequiredvalueof,whichcorrespondstoanoutwardforceactingonthelimiter.NotethataslongastheSOL/limiterresistivetimescaleremainsmuchlongerthantheAlfvŽntimescale,thelimiterforceparameterissolelydeterminedbylinear.Inparticular,theforceparameterisoftheSOL/limiterresistivetimescale.Thesameisalsotrueforan=0modemoderatedbyeddycurrentsßowinginexternalconductors,providedthatthesloweddowninstabil-itygrowsonatimescalewhichissigniÞcantlyshorterthantheSOL/limiterresistivetimescale.Intheoppositecase,thehalocurrentislikelytobelargelysuppressed.Now,wecancrudelysimulatethesequenceofplasmaequilibriaduringaVDEbygraduallydecreasingthescalefromunity,therebycausingaproportionaldecreaseinthepoloidaldimensionsoftheplasma.RecallthattheminorradiusoftheplasmaisThissequenceofequilib-riaisgeneratedasthehigh-plasmamovesvertically,andconsequentlyshrinksinsize,becauseanincreasingfractionofitsßux-surfacesareintersectedbythelimiter,andsobe-comepartofthelow-halo.SincethemotionoftheplasmatakesplaceonamuchlongertimescalethantheAlfvŽntimescale,wecansafelyassumethattheplasmaisalwaysclosetoanequilibriumstate.Obviously,wemustimaginethatourequilibriamoveupward,astheyshrinkinsize,sothattheirlastclosedßux-surfacesalwaysremainincontactwiththestationarylimiter.Assumingthattheplasmamovesverticallyatanap-constantvelocityonaSOL/limiterresistivetimescale,wecanwriterepresentstimenormalizedtothistimescale.Thus,=0correspondstothestartoftheVDEwhentheplasmaÞrstcontactsthelimiter,and=1totheendwhenthevol-umeofthehigh-plasmahasshrunktozeroThehalothicknessparameter,,isassumedtoinitiallygrowinproportiontothepenetrationoftheplasmaintotheinsuchamannerthatthetotalvolumeofthecentralplasmaandthelow-haloremainsÞxedthatthemaximumradialwidthofthehaloisHowever,itisalsoassumedthatthereisamaximumpossiblevaluecantakedetermined,forinstance,bythethicknessofthelimiter.Thismaximumvalueisdenoted.Thus,thehalothicknessparameterisgivenby=minFinally,itisassumedthatthehaloinitiallyformsatcon-stantplasmathermalenergydensity,i.e.,constant,andconstanttoroidalplasmacurrent,,butthatboththesequantitiesareeventuallyquenchedasaconsequence,forinstance,ofasuddeninßuxofimpurities.Experimentally,itisgenerallyfoundthatthethermalquenchtakesplacemuchmorerapidlythanthecurrentquench.Thethermalandcur-rentquenchesarecrudelysimulatedinourmodelbylinearlyrampingtheparametersfromtheirinitial,respectivelyÑtozero.Thestartofthethermalandcurrentquenchesisat.Theendofthether-malquenchisat,andtheendofthecurrentquenchat.ThemodelisnowcompletelyspeciÞed.VII.EXAMPLEAXISYMMETRICVDESIMULATIONSInthefollowing,thelimiterisassumedtolieabovetheplasma,andtoextendfrom=0.3=0.7Asabasicillustrationofourmethodfordeterminingthelimiterforce,consideraplasmaequilibriumcharacterizedby=0.20,=3.0,=2.0,and=1.0.Supposethat=0.1.Intheabsenceofalimiterforce,acalculationof=0force-matrixyieldsonenegativeeigenvalue,=1.948,indicatingthattheplasmaisideallyun-stabletothe=0mode.However,ifthelimiterforceparam-eterisadjustedsuchthat=4.075,andtheforce-matrixisrecalculated,thenthenegativeeigenvalueisincreasedtozeroandalltheothereigenvaluesremainposi-Ingeneral,thereisonlyonevalueofwhichcanachievethis.Thisindicatesthattheplasmacanberenderedmarginallystabletothe=0modebyanaxisymmetrichalocurrentwhicharrangesitselfinsuchamannerthatanormal-izedoutwardforceperunitareaof4.075isexertedonthelimiter.Figureshowsthe=0plasmadisplacementatthelastclosedßux-surfacecalculatedintheabsence,andin FIG.2.The=0plasmadisplacementatthelastclosedßux-surfaceasafunctionofpoloidalangleforaplasmaequilibriumcharacterizedby=0.20,=3.0,=2.0,=1.0,and=0.1.Thesolidcurveshowsthedisplacementintheabsenceofalimiterforce,whereasthedashedcurveshowsthedisplacementinthepresenceofalimiterforcewhichissuchastorenderthe=0modemarginallystable.Thelimiterliesbetweenthetwoverticaldashedlines.012506-5AsimpleidealmagnetohydrodynamicalmodelPhys.Plasmas,012506 Supposethatallperturbedquantitiesvarytoroidallyas,whereisthetoroidalmodenumber.ThemostgeneralsolutionofEq.whichiswellbehavedas0iswheretheareconstants.Likewise,intheabsenceofexternalconductors,themostgeneralsolutionofEq.whichiswell-behavedaswheretheareconstants.,wheretheareconstants.FollowingtheanalysisofRef.,Eqs.,andthethreematchingconditions,reducetothehomog-enousmatrixequation=0,whichmustbesolvedsubjecttotheincompressibilityconstraint=0. 2kG+2
k 2kG+k0GˆkˆH,
andG= 
i
m
md 2 ,
Gˆ= 

i
m
md 2 ,
H=s
2 ha2+ ˆcosP0 hha ei
m
md Thestabilityproblemessentiallyreducestothesolutionoftheeigenmodeequationsubjecttotheconstraint=0.Ofcourse,onlythe=0solution,whichcorrespondstothemarginalstablecase,isphysical.Thisisequivalenttosolv-ingtheunconstrainedeigenmodeequation,and .Notethatthematrixisrealandsymmetric,whichguaranteesthatallofitseigenvaluesarereal.Accordingtotheidealenergyprinciple,theplasmaisifanyoftheeigenvaluesofEq.,andstableotherwise.=0modeisaspecialcase,sincetheconstraintisautomaticallysatisÞed,duetothefactthattheareallzero.Inthissituation,theplasmaperturbationcanbemadeincompressiblebysimplysetting=0.Hence,theeigen-modeequationforthe=0modeisVI.ANIDEAL-MHDMODELOFAXISYMMETRICVDEsAverticaldisruptioneventoccurswhenanelon-gatedtokamakplasmaequilibriumissubjecttoanideal=0instabilitywhichcausesittomoveverticallytowardarigidconductorsituatedeitheraboveorbelowit.Thiscon-ductoristermedtheÒlimiter.ÓForthesakeofargument,letthelimiterlieabovetheplasma.Thus,theupwardmovementoftheplasmacausesitsoutermostßux-surfacestointersectthelimiter,andalsoinducesahalocurrentwhichßowsaroundtheintersectedsurfaces.Aswehaveseen,thehalocurrentisforce-freeinthescrape-offlayeri.e.,theplasma-Þlledsegmentoftheintersectedßux-surfaces,butßowsacrossequilibriummagneticÞeldlinesinthelimiter.Thus,thehalocurrentgivesrisetoanetelectromagneticforceactingonthelimiter.ItturnsoutthatthisforceisalwaysoutwardIngeneral,anideal=0plasmainstabilitywhichisnoteffectivelymoderatedbyeddycurrentsßowinginexternalgrowsonanAlfvéntimescalewhichistypi-cally10sinamoderntokamakHowever,assoonastheplasmaintersectsthelimiter,andahalocurrentßowsacrossit,theassociatedforcemodiÞespressurebalanceattheplasmaboundary,andis,thereby,abletomoderatethegrowthoftheinstability.Thismoderatingeffectisrepre-sentedbytheparameterinthestabilityanalysisoftheprevioussection.Ingeneral,wewouldexpecttheinstabilitytobesloweddowntosuchanextentthatitgrowsonaresistivetimescalecalculatedusingtheelectricalre-sistanceoftheSOLplasmaandthelimiter,seeSec.VIII.SuchatimescaleisinevitablyverymuchlongerthananAlfvŽntime.Infact,atypicalSOL/limiterresistivetimescaleinamoderntokamakisabout10Hence,themoderatingeffectofthelimiterforceeffectivelyrendersthemarginallystabletotheideal=0mode.Thiscrucialinsight,duetoZakharov,allowsustoconstructalinearmodelofaverticaldisruptionevent.Tobemoreexact,the=0force-matrix,,ofaplasmawhichisideallyunstabletothe=0modepossessesonenegativeeigenvalue.Inordertopreventthe=0mode012506-4R.FitzpatrickPhys.Plasmas,012506 Now,giventhattheSOLplasmaislow-sinceitisrelativelycold,duetorapidparallelheattransporttotheanycurrentsßowinginitmustbeapproximatelyi.e.,theymustßowparalleltoequilibriummag-neticÞeldlinesThus,thehalocurrentsheetdensity,hasthecomponents Bo,
ih=P0 intheSOL,whereisaconstant.Here,wehavemadethereasonableassumptionthatthepoloidalcomponentofthehalocurrentsheetdensityassociatedwithan=0verticalinstabilityis.Ofcourse,thereisnoconstraintthatthehalocurrentbeforce-freeinthelimiter,sinceanyassociatedelectromagneticforcecanbebalancedtherebymechanicalstresses.Thus,wewouldexpectthehalocurrenttoßowacrossthelimiteralongtheshortestpathi.e.,thepathofleastelectricalresistance.Forthecaseofalargeaspect-ratiotokamak,andanaxisymmetriclimiterwhichonlyextendsoverarelativelysmallrangeofpoloidalangles,theshortestpathisalmostpurelypoloidal.Itfollowsthatinthelimiterthehalocurrentßowsinthepoloidaldirection.Inotherwords, =0,inthelimiter.NotethatthepoloidalcomponentofthehalocurrentsheetdensitymustbeacrosstheSOL/limiterboundary,inordertoconservecharge,whereasthetoroidalcomponentisallowedtobediscontinuous.Thehalocurrentinducedelectromagneticpressureact-ingontheSOLiszero,bydeÞnition.Ontheotherhand,theelectromagneticpressureactingonthelimiterisNotethatthispressureisAsiseasilydemonstrated,thecomponentsofthenor-malizednethalocurrentare,andAssumingthat,itisclear,fromEqs.,thatthepoloidalhalocurrentisalwaysmuchlessthanthepoloidalplasmacurrent.V.IDEALPLASMASTABILITYConsideramarginallystable,idealplasmainstability.Assumingthattheperturbedcurrentiszeroinsidetheplasma,itfollowsthattheperturbedpressureisalsozero.Hence,wecanwritetheperturbedmagneticÞeldsbothin-sideandoutsidetheplasmaintheform,where=0.AccordingtoRef.,theperturbedmatchingconditionsattheplasmaboundary,,are=0,=0,wherethesubscriptsrefertoinsideandoutsidetheboundary,respectively,andistheedgeplasmadis-placement.TheÞrsttwomatchingconditionsensurethattheboundaryremainsamagneticßux-surface,whereastheÞnalmatchingconditionenforcespressurebalance.Ofcourse,thelimiterforceassociatedwithanaxisymmetrichalocurrentaffectspressurebalanceattheplasmaboundary.Inthepres-enceofsuchaforce,itseemsplausibletomodifyEq.isthenormalcomponentofthelimiterforcedensity.Here,=1for,and=0,otherwise.Theadditionaltermontheright-handsideoftheaboveequationrepresentsthevirtualworkdoneonthedis-placedplasmabythereactiontothelimiterforce.FollowingtheanalysisofRef.,thematchingcondi-tionsattheplasmaboundary,,reduceto =haˆ ,
iVˆo =Bˆ +ha  ˆ+Bˆ ˆ,
iBˆ +ha  VˆoaVˆi =s
2 ha2+ ˆcosPˆ0 ,and.Thepotentialsbothsatisfy 2+2Vˆi,o =0.012506-3AsimpleidealmagnetohydrodynamicalmodelPhys.Plasmas,012506 libriumtoroidalmagneticÞeld-strengthsinsideandoutsidetheplasmaarewrittenas,re-spectively,whereareconstants.Thereisnopoloi-dalmagneticÞeldwithintheplasma.However,apoloidalÞeldisgeneratedoutsidetheplasmabyacurrentsheetingontheplasmaboundary.Letthecomponentsofthisex-ternalpoloidalÞeldattheedgeoftheplasmabe=0andPressurebalanceacrosstheplasmaboundaryyields 20Bi scos 2=1 20 Bo Itisconvenienttoadoptthefollowinglargeaspect-ratio,,ratherthan,and.Here,.Makinguseoftheaboveorderings,Eq.reducestoisan,andNow,thesafety-factorattheplasmaboundaryisdeÞnedas Bˆ
d Writing,weobtain 20 ha
d betheequilibriumcurrentsheetdensityi.e.,ra-diallyintegratedcurrentdensityßowingontheplasmaboundary.Itiseasilydemonstratedthat Thus,theelectromagneticpressureexertedontheplasmaboundaryduetothecurrentsheetis Clearly,theelectromagneticpressureattheboundaryactsinward,andisexactlybalancedbytheoutwardactingplasmapressure.Throughoutthispaper,currentisnormalizedas.Hence,thenormalizedcomponentsofthenetequilibriumplasmacurrentareIV.AXISYMMETRICHALOCURRENTSSupposethattheedgeoftheabove-mentionedplasmaequilibriumisincontactwithanaxisymmetricrigidconduc-tor,whichistermedthe,andwhichextendsovertherangeofpoloidalangles.Forthesakeofsimplic-ity,letthelimiterliebetweenthetoroidalsurfaces.Thescrape-offlayerisalow-plasmawhichisalsosituatedbetweenthetoroidalsurfaces,andextendsoverallpoloidalanglesexceptthoseintherange.ThemaximumradialwidthoftheSOLis.Itisassumedthat1.Allmagneticßux-surfaceslyingwithintheSOLintersectthelimiter.ThelimiterandSOLarecollectivelyknownasthe.Thehaloissandwichedbetweenahigh-closedßux-surfacesontheinsidei.e.,theregionandavacuumontheoutsidei.e.,theregion,seeFig..Inthefollowing,anywithinthehaloisneglected,forthesakeofsimplicity.Inci-dentally,theÒlimiterÓinouranalysismightrepresentareallimiter,adivertor,orevenasectionofthevacuumvessel.Anycurrentßowingwithinthehaloistermedacurrent.Ingeneral,thereisnohalocurrentassociatedwiththeunperturbedplasmaequilibrium.Suppose,however,thattheplasmaisforcedtomovetowardthelimiterbyanideal=0,whereisthetoroidalmodenumberinstability.WewouldexpectthismovementtocausetheplasmaÕsoutermostmagneticßux-surfacestointersectthelimiter,andtoalsogenerateahalocurrentßowingaroundthesesurfaces. =µa l scrape-o
laye =1=2 µ /svacuum FIG.1.Schematicdiagramshowingthepoloidalcrosssectionofahigh-plasmasurroundedbyanaxisymmetrichaloconsistingofalow-offlayerandarigidconductinglimiter.012506-2R.FitzpatrickPhys.Plasmas,012506 +cos.Moreover,ifisthetoroidalangle,then,assumingthat 1coshistheinitialplasmaminorradius,adimension-lessscale-factor,and1.Here,isalabelforasetofnestedaxisymmetrictoroidalsurfaceswhosepoloidalcrosssectionsareverticallyelongatedellipses,andisananglelikepoloidalcoordinate.Theinnermostsurfacecorrespondsto=0,whereastheouter-inÞnitevolumesurfacecorrespondstoNotethat 1sinh AsimpleidealmagnetohydrodynamicalmodelofverticaldisruptioneventsintokamaksR.FitzpatrickInstituteforFusionStudies,DepartmentofPhysics,UniversityofTexasatAustin,Austin,Texas78712,USAReceived7August2008;accepted18December2008;publishedonline27January2009Asimplemodelofaxisymmetricverticaldisruptioneventsintokamaksispresentedinwhichthehalocurrentforceexertedonthevacuumvesseliscalculateddirectlyfromlinear,marginallystable,ideal-magnetohydrodynamicalstabilityanalysis.ThebasicpremiseofthemodelisthatthehalocurrentforcemodiÞespressurebalanceattheedgeoftheplasma,andthereforealsomodiÞesideal-MHDplasmastability.Inordertopreventtheidealverticalinstability,responsiblefortheVDE,fromgrowingontheveryshortAlfvŽntimescale,thehalocurrentforcemustadjustitselfsuchthattheinstabilityisrenderedmarginallystable.Themodelpredictshalocurrentswhicharesimilarinmagnitudetothoseobservedexperimentally.Anapproximate