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RUNAWAYELECTRONSANDITER(Summaryofapapersubmittedforpublication)

AllenBoozerColumbiaUniversityJuly21,2016

WorksupportedbyDoEOfficeofFusionEnergySciencegrantDe-FG02-03ER54696Theory and simulation are essential for ensuring that relativisticrunawayelectronswillnotpreventITERfromachievingitsmission.

Therunawayphenomenonisuniquebecauseof

Thepotentialfordamage,

Themagnitudeoftheextrapolation,

Theimportanceoftheatypical---onceina1000shots.Theplanedinjectionofimpuritiesisassociatedwithmagneticsurface

breakup;avoidanceissubtle;mitigationmaybeimpossible.

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RunawayElectronsMayAriseWhen

Loopvoltage!ℓ ≡$%&

$' %(≳ 2.9

-

./01/34Volts.

1. Due to a thermal quench, ~1ms, as

partofanaturaldisruption.

ITER poloidal flux 56 ≈ 70V∙s;removaltimeis50-150ms.

!ℓ ≳ 500Volts

2. As a result of a mitigation strategy to prevent the plasmadriftingintoawall.Poloidalfluxremovaltime<150ms.

!ℓ ≳ 500Volts

Ifpoloidalfluxisnotremovedthisquicklyastronghalocurrentcanarisealongwitharelativisticelectroncurrent.

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AbsenceofaMaxwellianRunaway(optimisticresult)WhenelectronsarecooledsufficientlyslowlythattheyremainclosetoMaxwellian,arunawaycannotoccurinITER.Runawayonlypossiblewhen <=|| = <@A|| >

CDEF

GHIJKL.

M3NO ≡

P

0QRG/S

3NO≈ 50,maximumkineticenergy.

NumberoftailelectronsinaMaxwellianℱ'NUV

WNO M =G H

X<YH.

ForM > M3NO,eitherlessthanoneelectronintheplasmaornotenoughpossiblee-foldstoincreasetheirnumbertobesignificant.

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ConditionforMaxwellianRunaway

RunawayrequiresS < M3NOG [

\

/.].

GX

G ^||

^E

G

≲ 4eV,whereAa ≡ <bc. A||/Aa~2×10

Y[

MaintenanceofMaxwellianrequiressufficientlyslowcooling;

Coolingtime>g'h =G40

\

HIJKL

3a0

4

0 .

iEF

≲ 25ms,jah ≡CDEF

3a.

MinimalcoolingtimeformaintainingaMaxwellianis

muchshorterthanthefastesttimerequiredforpoloidalfluxremoval,~150ms,but

muchlongerthanthermalquenchtime,~1ms.

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MagneticSurfaceBreakup

BasisofpresentITERstrategyforavoidance

If allmagnetic field lines in theplasma strike thewalls within 100’s of toroidal circuits, runawayelectrons are lost too rapidly for relativisticelectronstobeanissue(anotheroptimisticresult).Unfortunately,tubesofmagneticfluxthatdonot

interceptthewallscanremaininthecoresof

islandsandneartheplasmacenter. Izzoetal,PPCF2012

Non-interceptingfluxtubesareplaceswhereenergeticelectronscanbestoredandacceleratedtorelativisticenergies.When outer surfaces reform before the flux tubes dissipate,electronscandumpin~0.5msalonganarrowfluxtube~150cm2.

6/1/16, 9:51 AMppcf419857f03 466×430 pixels

Page 1 of 1http://iopscience.iop.org.ezproxy.cul.columbia.edu/0741-3335/54/9/095002/downloadFigure/figure/ppcf419857f03

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FastMagneticRelaxationsThevoltagespikeaccompanyingathermalquenchgives

thetimescale,~1ms,ofthemagneticrelaxationand

thecompletenessofthemagnetichelicityconservingrelaxation.Flux change during the thermal quench, k56 ≲ Ψ'/9 ≈ 13V⋅s, issufficient to accelerate electrons in non-intercepting flux tubes torelativisticenergiesandexponentiatetheirnumber.ITER operability is determined by worst event in about a year,

~1000shots.

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PlasmaCoolingCross-fieldtransport:UpperlimitisBohmtransport--tooslow.Magneticsurfacebreakup:Maxwellianmaintenanceis impossible;fluxtubesthatdonotinterceptwallsallowelectronacceleration.Radiation: In principle highly controllable and consistent with arapidITERshutdown,<<150ms,butrequiresextremecare:

1. Cooling rate can increase as Te drops, which can cause theMaxwelliantobebrokenandproducearunaway.

2. Radialprofilecontrolrequiredto(a)coolcentralTeand(b)avoidlossofmagneticsurfaces(tearinginstabilities).

3. ImpuritydeliveryconsistentwithrequiredTe(r,t)probablynotproduciblebymassivegasinjectionorshatteredpellets.

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CircumscribingQuantitiesFourquantitiescircumscribewhatispossible1. Numberofseedelectrons,

2. Kineticenergyrequiredforrunaway,

3. Poloidalfluxrequiredforane-fold,and

4. Decayrateofrelativisticelectrons.

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NumberofSeedElectronsNeedseedelectronsabovecriticalkineticenergyforrunaway,Kr,tobeginrunawayprocess. (Needexperimentstostudynumber)

Mostcrediblesourceisthepre-thermal-quenchMaxwelliantailℱ'NUV

WNO M =G H

X<YH,whereM ≡ 3o0

GL.

WhenelectronswithM < Mp ≈ 9.5canrunawaywithℱ'NUV

WNO Mp ≡^||

^E,whereAa ≡ <bc,so^||

^E≈ 2×10Y[,

runawayisfastrequiringonlyatinypoloidalfluxchange,56N ≡

3a

C2qr ≈ 0.064V∙s.ApparentlyseenonTFTR.

Whenℱ'NUVWNO tu/S ≡

^||

^E<Yvw,xye-foldsrequiredforrunaway.

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KineticEnergyRequiredforRunaway

tu >3a0

G

zEF

zℓwhere!ah ≡ 2qr=ah ≈ 2.9

-

./01/34VPitch-anglescatteringandradiationcanincreaseKr.WhenKr>20kev,twosourcesofrunawaysareeliminated:(1) Tritiumdecay—max.electronenergyis18.6keV.

(2) Collisionwithan{particle—max.energyis4 3

W|t} ≈ 2keV.

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PoloidalFluxRequiredforanE-fold

5Cp~V� = ÄCp56N;simpletheorygivesÄCp = 2ÅbΛ ≈ 25.56N ≡

3a

C2qrandk Ä

o||

a=

Ñ%&

%&J

1.EnergyDistributionofRunaways

Large-anglecollisionsaddnewrunaways:

Attheminimumenergyforrunaway.Atarateproportionaltotherunawaynumberdensitybu(Ü).

Thedistributionfunctionforrelativisticrunawayelectronsisthen

à =-â(')

6160<Y6/61,whereä = ÄQcandä/ ≡ ÄCpQc.

AveragerelativisticfactorisÄ = ÄCp;effectsatÄ ≫ ÄCpirrelevant.

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2.EnergyinRunawaysEnergyinrelativisticelectronsåu = Ä − 1 éu56N ≪

P

4éΨ6,

éurunawaycurrent;P4éΨ6istheenergyinthepoloidalmagneticfield.Changeinpoloidalmagneticenergyaséu → égivenby

k P

4éΨ6 ≈ 0

4ékΨ6andkΨ6 = −xyÄCp56N.

Only1/xyfractionofthepoloidalfieldenergygoestotherelativisticelectrons;therestgoestoOhmicdissipation.

3.MaximumNumberofE-Foldsëíìî

x3NO =ï&

%ñóòôöõ=

.

úñò

ï&

%&J.

SimpletheorygivesÄCp = 2ÅbΛ ≈ 25andx3NO ≈ 40.

ForaMaxwellianseedneedM < M3NO = Mp + x3NO ≈ 53.

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4.MultipleRunawayStrikesduringOneDisruptionWhentherequiredM ≡ QRG/2Sforrunawayis≪ M3NO,onlyasmallfractionofthepoloidalfluxΨ6 ≈ 70V∙sislostinasingleacceleration.If90%oftherelativisticelectronsarelostinasinglestrike,only2.3e-foldsareneedtogiveanothersimilarstrike.

5.CalculationsofgefInstandardtheoryÄCp ∝ 1/tu,withtutherunawaykineticenergy.Sincex3NO ∝ 1/ÄCp,seriousnessofrunawayisstronglydependentonÄCp.ExistingcodescouldgiveamoreaccurateÄCp,buttheavalancheformularequiresmodificationforareliablevalue.

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DecayRateofRelativisticElectrons

Relativisticcurrentsdecaybythelossofrelativisticelectrons.

Forthecurrenttodecayneed!ℓ < !y,wheretheloopvoltagerequiredtosustainarelativisticcurrentsatisfies!y ≥ !ah.InequalitybecauseConnor-Hastievoltageomitssomedissipativeeffects.

When!y = !ah,thedecaytimeisg�CaN† ≈ 24°./01/34

-.

Importanceof!yisquestionablefortworeasons:

1.TheITERverticalfieldsystemappearsinadequate,sodissipationmustbefastcomparedto150mstoavoiddriftintothewall.

2.Theevolvingprofileoftherelativisticcurrentmustremaintearingstabletoavoidlossofmagneticsurfaces.

Mayprecludemitigationofrelativisticcurrentsbymassivegasor

shatteredpelletinjectionduetomagneticsurfacebreakup.

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Discussion

AstrongrelativisticelectroncurrentstrikingthewallsmorethanoneinathousandshotswouldprobablypreventITERfromachievingitsmission.Massivegasinjectionandshatteredpelletsrelyonmagneticsurfacebreakuptospreadeffectofimpurities,whichmayprecludeuseformitigation. Success for avoidancedepends ondissipationof non-interceptingfluxtubesbeforesurfacesre-form.Theoryandsimulationusingphysicsvalidatedinexperimentscouldadvancewhatapracticalmitigationsystemwouldlooklike.

TwopossibilitiesAfasterandmoreflexiblepelletinjector

Passivelyinducednon-axisymmetriccurrentsinchamberwalls.