Simulation of the interaction between plasma turbulence and neutrals in linear devices

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Simulation of the interaction between plasma turbulence and

neutrals in linear devices

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Leddy, Jarrod, Willett, Hannah Victoria and Dudson, Benjamin Daniel

orcid.org/0000-0002-0094-4867 (2016) Simulation of the interaction between plasma

turbulence and neutrals in linear devices. Nuclear Materials and Energy. ISSN 2352-1791

https://doi.org/10.1016/j.nme.2016.09.020

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ContentslistsavailableatScienceDirect

Nuclear

Materials

and

Energy

journalhomepage:www.elsevier.com/locate/nme

Simulation

of

the

interaction

between

plasma

turbulence

and

neutrals

in

linear

devices

J.

Leddy

∗

_,

_B.

_Dudson

_,

_H.

_Willett

YorkPlasmaInstitute,UniversityofYork,HeslingtonUK,YO105DD,UnitedKingdom

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received29June2016 Revised26August2016 Accepted2September2016 Availableonlinexxx

a

b

s

t

r

a

c

t

Theinteraction betweenplasmaand neutrals withinatokamakdominates the behaviourofthe edge plasma,especiallyinthedivertorregion.Thisareaisnotquiescent,buthassignificantperturbationsin thedensityandtemperatureduetoturbulentfluctuations.Investigatingtheinteractionbetweenthe neu-tralsandplasmaisimportantforaccuratelysimulatingandunderstandingprocessessuchasdetachment intokamaks.Forsimplicity,yetmotivatedbytokamakedgeplasma,wesimulatealinearplasmadevice andcomparethesourcesandsinksduetoionisation,recombination,andchargeexchangeforcaseswith andwithoutturbulence.Interestingly,theturbulencesystematicallystrengthenstheinteraction,creating strongersourcesand sinksfortheplasmaandneutrals.Notonlydoesthestrengthoftheinteractions increase,butthelocationoftheseprocessesalsochanges.Therecombinationandchargeexchangehave relativelyshortmeanfreepaths,sotheseprocessesoccuronthescaleoftheeddyfluctuations,whilethe ionisationismostlyunaffectedbytheturbulence.

© 2016TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

In tokamaks heat is exhausted from the core into the edge plasmawhereitisthenconductedalongthemagneticfieldontoa verythinregiononthedivertorplates.Futurefusiondevicessuch asITERwill depositso muchpowerontothedivertorplatesthat, without intervention, severe melting could occur [1] . In addition toheatflux,particlefluxthroughelasticcollisionsofmoleculeson the wallsanddivertor plays an important role indivertor condi-tions[2] ;however,onlyparticlefluxisincludedhere,and molecu-larfluxisconsideredaspotentialfuturework.Todecreasetheheat flux,theplasmacanbeforcedintoadetachedregimewheremuch ofthethermalandkineticenergyisradiatedfromtheedgeplasma and spread volumetrically. This occurs when the relatively low temperature plasma in the edge of a tokamak interacts strongly with neutrals nearthe divertor through charge exchange, ionisa-tion, and recombination [3] . These atomic processes have cross-sectionsthatarefunctionsoftheplasmatemperature,density,and neutral densitymeaningthat turbulence, whichprovides fluctua-tions inthese quantities, will havea local effect on the plasma-neutralinteractions.Toexploretheseinteractionsqualitativelyand quantitatively,simulationsareperformedofalinearplasmadevice.

∗ _{Correspondingauthor.}

E-mailaddress:[email protected](J.Leddy).

Thisprovidesasimplegeometryinwhichtoexploretheeffectof plasmaturbulenceon theplasma-neutralinteraction.Previous ef-forthasbeenmadeinthisareawithstatisticallygenerated turbu-lence[4] andwithtwo-dimensional SOLturbulenceinTOKAM2D

[5] ; however, we seek to use self-consistent, 3D fluid turbulent simulations.

2. Physicsmodels

Inthetokamakdivertorofexistingdevices,theplasmais usu-allycollisionalandcan thereforebe treatedwithafluid approxi-mation.ThemodelusedisbasedononedescribedbySimakovand Catto[6] ,buthasbeen modifiedandputintofinite volumeform toconserveparticlenumberandenergy.Thismodel,describedin thenextsection,wasthenimplementedinBOUT++[7] ,whichisa numericalframeworkforsolvingsystems ofdifferentialequations inarangeofgeometries.

2.1. Plasma model

Themodeliselectromagnetic,andevolveselectrondensity n e, electron pressure p e, parallel ionmomentum nv ||i, plasma vortic-ity

ω

, andpoloidalflux

ψ

.There isno separationbetween back-ground and fluctuations in this model, since we wish to study regions such as the tokamak edge where these are of similar magnitude.Inthelineardevicetobesimulatedthemagneticfield

http://dx.doi.org/10.1016/j.nme.2016.09.020

2352-1791/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

2 J.Leddyetal./NuclearMaterialsandEnergy000(2016)1–5

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isconstantinspaceandtime,somagneticdriftsandcurvature ef-fectsareomittedfromthismodel.

∂n

e

∂t

=−

∇

·

neV E×B+neb

v

_e

₊

∇

_·

₍

D_⊥

∇

_⊥n_e

₎

₊S_n−S (1)

3 2

∂

pe

∂t

=−

∇

·

₃

2peV E×B+ 5 2peb

v

||e

−pe

∇

·V E×B+

v

_||e

∂

_||p_e

+

∇

||

κ

e||

∂

||Te

+0.71

∇

||

Tej||

−0.71j||

∂

||Te+

ν

nj 2 || +

∇

·

₃

2D⊥Te

∇

⊥ne

+

∇

·

(

χ⊥

ne

∇

⊥Te

)

+Sp− 2 3Q (2)

∂ω

∂t

=−

∇

·

(

ω

V E×B

)

+

∇

||j||+

∇

·

(

µ

⊥

∇

⊥

ω

)

(3)

∂

∂t

ne

v

_||i

=−

∇

·

ne

v

_||i

V _E×B+b

v

_||i

−

∂

||pe+

∇

·

D⊥

v

||i

∇

⊥n

−F (4)

∂

∂t

1 2

βe

ψ

−

me mi

j||

ne

=

ν

j|| ne

+

∂

||

φ

−

1 ne

∂

||pe−0.71

∂

||Te

+me

mi

V E×B+b

v

_||i

·

∇

j|| ne

(5)

with E ×B driftgivenby:

V E×B=

b ×

∇

φ

B (6)

Thevorticityisgivenby

ω

=

∇

·

_n0

B2

∇

⊥

φ

(7)

wheretheBoussinesq approximation isused,replacing n e witha constant n 0inthevorticityequation.Thisexpressionisinvertedto

findtheplasmapotential

φ

.Acold-ionapproximationisusedsuch that p i=0.Theplasmadensitysourceisgivenby S nandpressure (energy)sourceby S p.Theneutralinteractionsdetailedinthenext sectionprovidesourcesfordensity S ,pressure Q ,andflux(friction)

F .Thedensityandthermalperpendiculardiffusioncoefficientsare takentobe classical[8] .Theparallel heat conductivityisdefined astheSpitzervalue,

κ

e= n

v

th2

τ

e [9] ,where

τ

e istheelectron col-lisiontime.The resistivity,

ν

,istakentobethe Spitzerresistivity

[10] .Hereweusethenotation

∂

||f =b·

∇

fand

∇

||f =

∇

·

(

bf

)

.

2.2. Neutral model

Akineticapproachisoftenpreferredforthedescriptionof neu-tral behaviour in the edge of tokamaks through codes such as EIRENE[11] becausetheneutralmean-freepathcanbelargerthan themachinesize.This,however,iscomputationallyexpensive,and goodqualitativeagreementhasbeenseenbetweensimulationand experimentwhenfluidneutralmodelshavebeenusedwithcodes such asUEDGE [12] . For our simulations, a fluid-diffusive model isused such that theneutrals are treatedasfluid parallel to the fieldlinesso thatparallel momentumisconserved intheplasma neutralinteractions,withperpendicular motiondictatedby diffu-sion[13,14] .This systemevolves neutralgas density n n,pressure

p n andparallelvelocity v ||n:

∂n

n

∂t

=−

∇

·

nnb

v

_||n+n_nv _⊥n

₊S

∂

∂t

nn

v

_||n

₌₋

∇

_·

n_n

v

_||nb

v

_||n+n_n

v

_||nv _⊥n

₋

∂

_||p_n

+

∇

||

Dnnnn

∂

||

v

||n

+F

Fig.1. Thecross-sectionratesforionisation,recombination,andchargeexchange arepre-calculatedforhydrogenspeciesasafunctionofthe plasmatemperature

[15,16].

∂

pn

∂t

=−

∇

·

pnb

v

_||n+pnv ⊥n

− 2 3pn

∇

·

b

v

_||n

+

∇

·

(

Dnnnn

∇

⊥Tn

)

+ 2 3Q with

v

_⊥n=−D_nn 1

pn

∇

⊥pn

Theneutraldensity,momentum,andenergysources(S, F ,and

Q respectively) are calculated with expressions for the plasma-neutralinteractionthroughionisation,radiativeand3-body recom-bination, radiation, andcharge exchange. Although molecular re-combinationcanplayanimportantroleinrealdevices,only radia-tiveand3-bodyrecombinationareconsideredinthismodel,which isessentially the sameasthat used inUEDGE [17] .The ratesfor theseprocessesaregivenby

Rrc=n2

σ

v

_rc R_iz=nn_n

σ

v

_iz R_cx=nn_n

σ

v

_cx (8)

where n isthe electrondensity, n n isthe neutraldensity,

σ

v

is the rate coefficient for the given reaction (in m3_s−1_{) which is a}

functionofthelocalelectrontemperature(anddensityfor recom-bination).Theseratecoefficientsarecalculatedthroughafit func-tion of a database [15,16] , which are shown in Fig. 1 .Given the ratesin Eq. (8) , thedensitysource fortheneutralscan be calcu-lated:

S=Rrc−Riz (9)

Themomentumtransfertotheneutralsisgivenby

F=mi

v

iRrc−mi

v

nRiz+mi

v

_i−

v

_n

_{Rcx (10)}

Finally,theenergytransfertotheneutralsis

Q= 3 2TeRrc−

3 2TnRiz+

3

2

(

Te−Tn

)

Rcx (11)

Fig.2. Asimpleschematic(left)ofalineardevice(Pilot-PSIinthiscase[18])showingthebasicgeometryofthesystem.Themagneticfieldgeneratedbythecoilsisroughly constant,pointingtotheright.Theplasmasourceontheleftproducesaplasmathatstreamsalongthefield-linesuntilimpactingthetargetontheright.Thedensity contoursalongtheplasmadeviceareshown(right)foraturbulentcasewiththelineardevicedimensionsindicatedinmeters.

Fig.3. Theelectrondensity(left),temperature(middle),andneutraldensity(right)contoursinthedriftplaneatasnapshotintime.

3. Lineardeviceandturbulence

A linear device consists of a series of magnetic coils in a straight line configuration providing a constant magnetic field pointingalong thedevice. Aplasmasourceis thenplaced atone end,theplasmastreams downthedevice,andhitsatargetatthe farend(seeFig. 2 ).Thelineardevicesimulatedherehasalength of 1.2 m and magnetic field of 0.15 T. The plasma density and powersourcesaresetasspatialdistributionsthatgeneratea Gaus-sian plasmaprofile witha standard deviation of 3.3 cm, a peak temperatureof3eV,andapeakdensityof1020 _m3_atthesource.

Theboundaryconditionsatthesourceendarezerogradient allow-ingoutflowofplasmaandneutrals,whilesheathboundary condi-tions[19] areusedatthetarget.Fig. 2 showsabasicschematicof thesimulatedlineardevicealongwithatypicalturbulent density contour.Theplasmadensityisatamaximumnearthesourceand decreasesalongthedevicetowardsthetargetplate.Inthe simula-tions presentedhere, asinglespeciespureDeuteriumplasmaand neutralgasareassumed.

3.1. Turbulence in the linear device

The turbulenceinthe simulationsofthislineardeviceis elec-tromagnetic in nature, driven by drift-waves that are a resultof theradialpressuregradientsfromthelocalisedplasmasource. Af-terevolvingfor60,000

ω

−1

ci (roughly1.25ms),theturbulence sat-urates. Fig. 3 shows theneutral andplasmadensity inthe drift-plane (1mfromthesourcefortheplasmaandatthetarget plate

fortheneutrals) ata singlepointin time.The neutraldensityat the target is highly correlated with the electron density due to the recycling, which is the dominant source for the neutrals. In realmachines,gas leakagefromtheplasmasource contributesto the ambientgas density andpressure, butin this simulation re-cyclingand recombinationare the only neutralsources. Interest-ingly,thepeak plasmadensitycanlieoff-axisdespitethedensity sourcehavingamaximumonaxis,whichisindicativeoflarge per-turbationsoverthebackground.Theneutraldensitydoesnotshow fine scale features seen in the electron densityfor two reasons: firstly,the fluid modelis diffusive,smoothingout features in the drift plane. In addition, the average lifetimes of the plasma and neutralsbeforerecombinationorionisation(1/n

σ

v

rangingfrom 100 µs to 0.1 s), are larger than the turbulence correlation time (625

ω

ci≈13

µ

s),meaning the neutrals essentially ‘see’a

time-averaged plasma. Using the instantaneous and local temperature to calculate the rate coefficients is justified because the longest ionisationandrecombination processescomplete inroughly 1

µ

s [20] ,whichisanorderofmagnitudebelowthe turbulence corre-lationtime.Inthistime,anaverageneutralparticlemovesroughly 0.1cm, which is an order of magnitudesmaller than the turbu-lencecorrelationlength.Neglectingthetransportofexcited states isthereforeareasonableassumption.

4. Neutral-plasmainteraction

4 J.Leddyetal./NuclearMaterialsandEnergy000(2016)1–5

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Fig.4. Acomparisonbetweenaveragesourcesforturbulent(solidline)andnon-turbulent(dashedline)cases.Theresultingprofilesforionisation(left)andrecombination (middle)areplotted.Finally,thedifferencebetweentheturbulentandnon-turbulentcasesisshown(right).

Fig.5. Contoursoftheionisation(left)andrecombination(middle)densityratespercubicmetre,andchargeexchangeenergysource(right)areshown1mdownthelinear deviceforasingletimeslice.Therecombinationisnegativebecauseitactsasasinkfortheplasmadensity.Thechargeexchangedemonstratesthattheplasmaheatsthe neutralsatthecentreofthedevice,whiletheneutralsheattheplasmatowardstheedge.

of the turbulence on the sources and sinks for plasma density, momentum,andenergy?Secondly,wherearetheinteractions lo-calisedintheplasmacolumn?Thesetwoquestionsareaddressed inthefollowingsections.

4.1. Net impact of plasma turbulence

By examiningthedata intwoseparate ways,theeffectof the turbulence is isolated. First, the neutral density source is calcu-lated(Eq. (9) ) using the localdensity andtemperature andthen averagedaxially toproduceradial profiles forthe sources (turbu-lentionisation source, forexample: S ¯t= nn n

σ

v

, wherethe bar overa variableindicates anaxialaverage).Thisrepresentsthe av-eragesourcesfromturbulentprofiles.Contrastingly,radialprofiles for the sources can be obtained using axially averaged density andtemperatureprofiles - thisrepresents theaverage source for acasewherethereisnoturbulence(ionisation source,for exam-ple:S ¯=n ¯n ¯n

σ

v

₍

_{T e}

₎

_).

TheresultsfromthesetwocasesareshowninFig. 4 .These ra-dialprofiles ofthe densitysourcesare averaged over10,000

ω

−1

ci (roughly 16 fluctuation correlation times) and along the axis of thedevice.Theionisationdensitysourceisincreasedwith turbu-lenceby10%atmaximum.Therecombinationsinkisalsostronger withturbulenceby afactorof50%atmaximum.Thiscanbeseen clearlyintheplotontherightinFig. 4 .Theinteractionofthe neu-tralswiththelocalturbulentfluctuationsconsistentlygivesriseto

stronger sources andsinks than the equivalent averaged profiles would.Thisconsistent strengtheningoftheinteraction appearsto be universal dueto the nonlinear dependence of the sources on theplasmadensityandtemperature.

4.2. Localisation of interactions

5. Conclusion

Through simulationsofaplasmaturbulenceinalineardevice, the interaction betweenturbulenceand neutralshas been exam-ined. The net impact of the turbulence is to amplify the inter-actions, both sources and sinks. The local perturbations in den-sityandtemperature,combinedwithanonlinear relationship be-tween the sources andthesefields,givesrise to asystematic in-creasein theinteraction. Notonlydoesthe turbulenceaffectthe strengthoftheinteraction,butalsothelocation.The ionisationis mostlylocalisedinthecentreofthedevice,unaffectedbythe tur-bulence, but the recombinationand charge exchange conform to thefeaturesoftheturbulenceduetotheirrelativelylowmeanfree paths.Understandingthefundamentalcharacteristicsofthe turbu-lentplasma-neutral interaction is not onlyinteresting, but useful becausemanycodesfortokamakstudycurrentlyapproximatethis in2D without turbulence.Includingthe featuresof theturbulent interactioninthese2D codesisfeasible,butthiswork also moti-vatestheuseoffullturbulentcodeswhensimulatingdetachment andotherphenomenadependentonneutrals.

Acknowledgements

SomeofthesimulationswererunusingtheARCHERcomputing servicethrough thePlasmaHEC ConsortiumEPSRC grantnumber

EP/L0 0 0237/1 .Thisworkhasreceived fundingfromtheRCUK En-ergyProgramme,grantnumberEP/I501045 .Ithasbeencarriedout within theframework ofthe EUROfusionConsortiumandhas re-ceivedfundingfromtheEuratomresearchandtrainingprogramme 2014–2018undergrantagreementNo633053.Theviewsand opin-ionsexpressedherein donotnecessarilyreflectthoseofthe Euro-peanCommission.

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