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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.
