CD-EXCALIBUR-FMS0021-1.20-M1.2.1_YearOneSummaryReport
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   :description: technical note

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ExCALIBUR,Equations,for,ExCALIBUR/NEPTUNE,Proxyapps,Version,1.20,Abstract,The,report,describes,equations,for,ExCALIBUR,project,NEPTUNE,Proxyapps,,together,with,supporting,information,regarding,Braginskii’s,transport,coefﬁcients,and,sources,of,atomic,and,molecular,radiation.,The,numbering,of,the,systems,follows,that,of,the,NEPTUNE,Science,Plan,,so,that,those,listed,under,FM-WP2,are,denoted,2-1,,2-2,,etc.,,and,under,FM-WP3,as,3-1,,3-2,,etc.,It,is,a,living,document,to,which,further,equation,systems,will,be,added,throughout,the,course,of,the,project.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/021,Issue:,Date:,1.20,23,December,2021,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Name,and,Department,Signature,Date,Prepared,By:,Wayne,Arter,23,December,2021,BD,Reviewed,By:,Rob,Akers,23,December,2021,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,23,December,2021,Advanced,Dept.,Manager,Computing,1,1,Introduction,Rather,than,attempting,to,develop,a,fully,3-D,Exascale,targeted,plasma,edge,(or,boundary),code,from,day,one,,project,NEPTUNE,will,ﬁrst,focus,upon,the,development,of,“proxyapps”,[1],,devel-,oped,by,partners,across,the,project,through,a,series,of,Grant,calls.,These,must,be,designed,and,encoded,to,pave,the,way,to,the,fully,3-D,,actionable,and,performant,NEPTUNE,code,(or,codes),outlined,in,the,Science,Plan.,As,such,,all,NEPTUNE,proxyapps,must,capture,the,functionality,and,performance/scalability,characteristics,of,the,eventual,infrastructure,as,much,as,possible.,In,addi-,tion,,all,of,the,solutions,across,the,NEPTUNE,programme,must,eventually,be,synergistic,,leading,to,an,integrated,solution,for,the,eventual,code(s),–,this,will,require,close,cooperation,through,co-,design,across,all,partner,organisations.,The,baseline,proxyapps,for,the,initial,years,of,the,project,are,described,brieﬂy,in,the,Science,Plan,[2],,and,expanded,upon,below,to,give,the,“baseline,plan”,model,equations,,geometry,and,boundary,conditions.,Note,,the,baseline,plan,does,not,preclude,additional,functionality,that,any,bidder,may,deem,useful,(or,even,essential),to,the,project.,Bidders,are,encouraged,in,their,response,to,calls,to,be,creative,and,ambitious,and,to,describe,their,own,ideas,and,plans,for,delivering,above,and,beyond,core,scope,,provided,the,aim,is,to,increase,im-,pact,,quality,,reduce,risk,and/or,accelerate,delivery,(and,that,deliverables,are,fully,aligned,with,the,goals,of,the,NEPTUNE,Science,Plan,[2]).,At,baseline,,proxyapps,target,x86,(and,ideally,IBM,POWER,and,ARM,CPU),architectures,(multi-,core,and,multiple,node),for,scalability,to,ﬁrst,generation,Exascale,hardware.,Proxyapps,might,also,target,other,Exascale,candidate,architectures,(eg.,GPGPU),and/or,demonstrate,a,capability,to,explore,the,use,of,novel,hardware,as,it,becomes,available,to,the,ExCALIBUR,project,as,part,of,the,novel,test-bed,programme.,In,order,to,execute,efﬁciently,on,parallel,architectures,,proxyapps,are,expected,to,examine,use,of,MPI,,OpenMP,or,some,other,software,technology,(ideally,with,a,focus,upon,performance,portability).,Supporting,information,regarding,Braginskii’s,transport,coefﬁcients,for,plasma,in,a,strong,mag-,netic,ﬁeld,appears,in,Section,2.,A,description,of,sources,of,atomic,and,molecular,radiation,is,given,as,an,annex,in,Section,A.,A,second,annex,Section,B,tabulates,the,mathematical,symbols,(with,their,units),used,in,the,current,document.,1.1,Overall,Plan,In,the,Science,Plan,[2],the,description,of,work,extending,beyond,Y3,(early,2022),is,deliberately,vague,on,the,subject,of,“gyrokinetics”,,as,no,widely,accepted,model,for,the,tokamak,edge,appears,to,be,available,as,of,early,Y2,,and,even,should,one,appear,,it,might,not,be,suitable,for,use,in,Y3,in,NEPTUNE.,As,listed,,the,proxyapps,correspond,to,Plan,A,,which,assumes,that,no,suitable,gyroaveraged,model,will,emerge,in,time,,hence,kinetic,implies,Particle-in-Cell,(PIC).,PIC,approaches,,where,charge,conservation,is,vital,to,control,errors,,can,anyway,usefully,be,pursued,for,modelling,low,collisionality,plasma,species,,and,should,simplify,nicely,to,treat,neutral,species,with,long,mean-,free-paths,in,tokamak,edge,problems,where,mass,conservation,has,been,discovered,to,be,critical.,Moreover,in,the,context,of,classical,ﬂuid,dynamics,,the,transition,from,ﬂuid,to,particles,,or,short,to,long,mean-free-path,,has,been,well-studied,because,of,the,application,to,the,space,vehicle,2,re-entry,problem,and,related,hypersonic,situations.,Thus,the,hybrid,ﬂuid/PIC,approach,might,be,regarded,as,a,relatively,low-risk,route,to,achieving,a,robust,numerical,algorithm.,Evidently,full-orbit,PIC,has,the,potential,to,be,extremely,inefﬁcient,relative,to,gyroaveraged,kinetic,theory,because,of,the,need,to,follow,gyro-orbits,in,detail.,Hence,if,this,can,be,avoided,,either,through,gyroaveraging,or,clever,numerics,or,indeed,a,combination,of,both,,then,Plan,B,will,see,kinetic,imply,gyroaveraged,kinetic,theory,for,modelling,plasma,species,in,the,proxyapps.,Regarding,implementation,at,the,Exascale,there,is,also,a,conservative,Plan,A,approach,which,sees,the,use,of,relatively,simple,data,structures,such,as,scalar,and,vector,arrays,to,transfer,data,,and,consequent,use,of,existing,code-coupling,technologies.,Plan,B,is,an,aggressive,approach,to,implementation,which,sees,custom,data,structures,allowing,for,all,physical,data,(particle,arrays,and,ﬂuid,ﬁeld,vectors),colocated,near,a,point,to,be,held,close,in,memory,,permitting,very,tight,custom,code-coupling.,As,with,the,kinetic,options,,this,Plan,B,promises,signiﬁcantly,faster,solu-,tions,than,the,corresponding,Plan,A,,but,its,adoption,depends,on,the,outcome,of,research,work,to,de-risk.,1.2,Proxyapps,Summary,The,overall,thinking,behind,the,proxyapps,is,to,explore,potential,‘roadblocks’,to,the,Exascale,as,early,and,in,as,simple,a,context,as,is,possible,,beginning,with,algorithmic,roadblocks.,NEPTUNE,is,directed,towards,producing,‘actionable’,code,as,the,basis,for,large,procurements,,whereas,more,physics-focussed,software,projects,conducted,by,the,worldwide,nuclear,fusion,have,already,advanced,to,greater,complexity,,minimising,the,risk,that,unexpected,problems,will,appear,in,the,full,model.,The,numbering,of,proxyapps,below,corresponds,to,the,Science,Plan.,2-1,2-D,model,of,anisotropic,heat,transport.,It,is,important,to,determine,early,the,degree,of,anisotropy,that,high-order,elements,can,treat,without,special,coding.,If,this,is,unsatisfactorily,small,,then,there,are,implications,for,geometry,input,as,well,as,algorithmic,developments,that,are,best,addressed,as,early,as,possible.,2-2,2-D,elliptic,solver,in,complex,geometry.,One,of,the,indicated,elliptic,solvers,is,Grad-Shafranov,to,produce,high,order,(‘spectrally’),accurate,magnetic,ﬁelds,for,use,in,many,other,proxyapps.,Since,Sovinec,[3],has,already,produced,a,spectral,element,Grad-Shafranov,code,,the,corre-,sponding,NEPTUNE,development,should,mainly,serve,to,identify,practical,issues,concern-,ing,implementing,high,order,fe,models.,The,second,solver,additionally,presents,a,chance,to,explore,comparatively,novel,meshing,techniques,developed,under,Activity,A2.1,,and,later,the,preconditioning,techniques,of,A2.7.,2-3,1-D,ﬂuid,solver,with,simpliﬁed,physics,but,with,UQ,and,realistic,boundary,conditions.,This,will,determine,the,capability,of,spectral/hp,element,to,handle,sonic,outﬂow,boundary,condi-,tions,needed,to,represent,sheaths,,together,with,large,source,terms,,as,well,as,identifying,practical,issues,concerning,intrusive,UQ.,This,software,is,already,potentially,useful,in,its,own,right,for,example,in,modelling,MAST-U,divertor,,and,other,workers,might,be,drawn,in,to,add,additional,physical,effects,to,this,end.,3,2-4,Spatially,1-D,plasma,model,incorporating,velocity,space,effects.,From,the,numerical,analytic,point-of-view,,this,is,a,key,demonstration,of,spectral/hp,element,capability,to,handle,particle,interactions.,However,,again,this,could,be,a,basis,for,divertor,modelling,,to,explore,sheath,effects,depending,on,ﬁeldline,incidence,on,surface,,and,with,minor,modiﬁcation,the,spread,of,particle,energy,around,tile,edges,and,corners,as,performed,by,Gunn,et,al,[4,,5],for,ITER,application.,2-5,Spatially,1-D,multispecies,plasma,model.,Multispecies,throws,up,a,surprising,number,of,issues,concerning,data,deﬁnitions,(eg.,changes,to,the,Coulomb,logarithm),,structures,to,deal,with,different,number,of,species,,and,perhaps,most,signiﬁcantly,,complicated,inter-species,interaction,terms,both,within,and,at,the,domain,boundaries.,This,is,also,an,opportunity,to,mix,ﬂuid,and,kinetic,representations,of,different,species,within,the,volume.,2-6,Spatially,2-D,plasma,model,incorporating,velocity,space,effects.,With,the,1-D,multispecies,ﬂuid,work’s,having,made,the,generalisation,to,2-D,straightforward,,the,challenge,here,is,to,start,writing,a,complex,proxyapp,in,production,mode,,incorporating,the,research,put,into,design,,documentation,,code,generation,and,benchmarking.,There,is,an,opportunity,to,study,species,with,both,ﬂuid,and,kinetic,representations,depending,on,location,relative,to,the,wall.,Again,this,is,potentially,a,useful,tool,in,its,own,right,,capable,of,revealing,deﬁciencies,in,previous,2-D,modelling,work.,2-7,Interaction,between,models,of,different,dimensionality.,This,should,verify,that,the,design,has,the,right,data,structures,to,handle,additional,further,complexity,beyond,intrusive,and,ensemble-based,UQ,and,model,order,reduction.,The,hopefully,burgeoning,NEPTUNE,com-,munity,could,develop,this,into,a,design,tool,with,a,capability,both,to,explore,a,large,area,of,tokamak,edge,parameter,space,quickly,in,0-D,or,1-D,and,also,to,focus,on,relatively,small,but,critical,2-D,features,,such,as,tile,edges.,2-8,Spatially,3-D,plasma,kinetic,models.,These,will,represent,the,full,ﬂuid,model,produced,by,the,5-year,NEPTUNE,project,,incorporating,features,of,2-D,ﬂuid,and,kinetic,work,in,a,3-D,code.,3-1,2-D,particle-based,model,of,neutral,gas,&,impurities,with,critical,physics.,This,will,be,a,2d3v,code,(ie.,spatially,2-D,distribution,of,particles,with,3,velocity,components),designed,from,the,outset,to,interact,with,a,high-order,ﬁnite-element,ﬂuid,model,of,plasma.,It,gives,an,opportunity,to,check,out,ideas,on,optimal,usage,of,particles.,3-2,2-D,moment-based,model,of,neutral,gas,&,impurities.,Constructing,a,2-D,ﬂuid,code,of,neutral,gas,from,the,Nektar++,software,should,be,a,valuable,educational,exercise,,whilst,providing,scope,for,cross-validating,the,2-D,particle,model.,3-3,Interaction,with,2-D,plasma,model,when,available.,Building,on,the,1-D,multispecies,ﬂuid,work,,the,challenge,here,is,to,incorporate,in,the,ﬂuid,code,of,Proxyapp,PA2-6,,particle,effects,from,PA3-1,,which,will,in,the,higher,dimensional,space,be,more,subject,to,lack,of,numerical,resolution,or,‘noise’.,Should,PA3-3,be,accelerated,,it,could,usefully,treat,both,plasma,and,neutrals,via,particle,models.,3-4,3-D,model,of,neutral,gas,&,impurities.,This,is,now,at,full,dimensional,complexity,,incorporat-,ing,selected,ideas,on,optimal,usage,of,particles.,4,3-5,Interaction,with,3-D,plasma,model.,This,will,represent,the,full,model,produced,by,the,5-year,NEPTUNE,project,,a,coupling,of,ﬂuid,and,kinetic,software,developed,under,the,FM-WP2,work-package,as,PA2-8,,incorporating,features,of,Proxyapps,3-1,to,3-4,,and,allowing,for,additional,input,from,PA3-6.,3-6,Staged,introduction,of,additional,neutral,gas/impurity,physics.,It,is,expected,that,the,NEP-,TUNE,community,will,join,in,to,supplement,the,software,with,a,wide-ranging,capability,to,treat,a,wide,range,of,additional,nuclear,,atomic,and,molecular,effects.,2,Braginskii,coefﬁcients,Braginskii’s,transport,coefﬁcients,are,widely,used,in,tokamak,edge,modelling.,Object-oriented,Fortran,code,to,compute,the,Braginskii,coefﬁcients,is,available,at,the,web-site,[6].,Note,that,the,constant,k,is,introduced,such,that,k,=,kB,or,k,=,|e|,(1),where,kB,is,Boltzmann’s,constant,and,|e|,is,the,unit,of,charge,,depending,whether,T,is,measured,in,Kelvin,or,eV,,,so,that,kT,is,in,energy,units.,2.1,General,Except,in,Boltzmann’s,constant,kB,,sufﬁx,B,denotes,a,quantity,from,the,Plasma,Formulary,[7].,See,also,Braginskii’s,paper,[8].,The,subsequent,corrections,by,Epperlein,and,Haines,,and,by,Mikhailovski,and,Tsypin,are,not,relevant,to,this,work.,In,a,magnetic,ﬁeld,,the,direction,of,which,is,given,by,unit,vector,b,,Goedbloed,and,Poedts,[9],deﬁne,three,auxilliary,vectors,for,a,vector,v,,viz.,v(cid:107),=,b(b,·,v),,v∧,=,b,×,v,and,v⊥,=,(b,×,v),×,b,If,v,=,(v1,,v2,,v(cid:107)),and,b,is,aligned,with,the,3-axis,in,a,Cartesian,coordinate,system,,then,v(cid:107),=,(0,,0,,v(cid:107)),,v∧,=,(−v2,,v1,,0),and,v⊥,=,(v1,,v2,,0),(2),(3),It,may,be,shown,that,a,tensor,T,which,is,symmetric,under,rotation,about,b,has,the,form,(in,Cartesians),T,=,so,that,,,T⊥,−T∧,T⊥,T∧,0,0,,,0,0,T(cid:107),T,·,v,=,T(cid:107)v(cid:107),+,T∧v∧,+,T⊥v⊥,5,(4),(5),2.2,Conduction,,Viscous,and,Resistive,Coefﬁcients,The,electron,parallel,thermal,conductivity,in,the,Braginskii,theory,is,given,as,[7],Ke(cid:107),=,3.2,N,kTe,me,τe,(6),a,formula,valid,in,either,cgs,or,SI,units,,where,τe,is,the,electron,relaxation,time,(measured,in,seconds),,deﬁned,below.,The,notation,is,standard,,with,N,the,number,density,of,electrons,,ap-,proximately,the,same,as,the,number,density,of,ions,,me,the,electron,mass,,and,Tα,,s,=,i,,e,the,temperature,of,species,α.,The,perpendicular,electron,thermal,conductivity,satisﬁes,similarly,Ke⊥,=,4.7,N,kTe,me,τe,·,1,(ωceτe)2,where,the,electron,cyclotron,frequency,Equivalent,expressions,for,ions,are,where,the,ion,cyclotron,frequency,ωce,=,e,me,·,B,Ki(cid:107),=,3.9,N,kTi,mi,τi,Ki⊥,=,2,N,kTi,mi,τi,·,1,(ωciτi)2,ωci,=,ZeB,mi,=,e,mp,·,ZB,A,(7),(8),(9),(10),(11),where,Z,is,the,charge,state,of,the,ion,and,A,its,atomic,mass.,The,deﬁnitions,above,have,to,be,interpreted,in,the,context,of,the,equations,given,in,[7],,so,that,thermal,diffusivities,are,obtained,by,dividing,by,3nα/2,where,α,=,i,,e,is,the,species,index.,It,is,also,convenient,to,introduce,the,dimensionless,factors,xe,=,ωceτe,xi,=,ωciτi,Kinematic,viscosities,in,the,Braginskii,theory,may,be,taken,as,νe(cid:107),=,0.73N,kTeτe/(N,me),=,0.73,kTe,me,τe,νe⊥,=,0.51N,kTeτe/(N,me),1,x2,e,=,0.51,τe,1,x2,e,νi(cid:107),=,0.96N,kTiτi/(N,mi),=,0.96,τi,kTe,me,kTi,mi,kTi,mi,τi,1,x2,i,νi⊥,=,0.3N,kTiτi/(N,mi),1,x2,i,=,0.3,6,(12),(13),(14),(15),(16),(17),Key,quantities,in,the,calculation,of,all,these,terms,are,τα,,α,=,i,,e.,The,ﬁrst,step,in,their,calculation,is,to,convert,their,formulas,,usually,given,in,cgs,,to,SI,units,,giving,√,τe,=,6,2π3,(cid:15)2,0,√,me,e4,√,τi,=,12,√,(cid:15)2,0,mp,e4,π3,(kTe)3/2,Z2N,Λ,√,(kTi)3/2,Z4N,Λ,=,3.44,×,10−7,(Te)3/2,Z2(N/1018)Λ,A,=,2.09,×,10−5,(Ti)3/2,√,A,Z4(N/1018)Λ,(18),(19),where,the,notation,is,standard,with,Te,and,Ti,measured,in,eV,,,except,possibly,the,use,of,Λ,for,the,Coulomb,logarithm.,The,above,check,with,expressions,in,Wesson,[10,,§,14].,Note,that,Z2τi,differs,from,τe,in,being,larger,by,a,factor,of,(cid:112)2mi/me,≈,60,A,(also,substituting,Ti,for,Te,is,necessary).,The,factors,in,Z,are,taken,from,the,original,Braginskii,paper,[8].,√,It,follows,that,the,xα,factors,may,be,conveniently,written,xe,=,6.05,×,104,(Te)3/2B,Z2(N/1018)Λ,xi,=,1997,(Ti)3/2B,√,Z3(N/1018),AΛ,(20),(21),The,large,coefﬁcients,in,Equations(20),and,(21),explain,why,classical,transport,is,so,anisotropic.,Substituting,the,explicit,expression,for,τe,in,Equations(6),and,(9),gives,respectively,,the,thermal,parallel,diffusivities,are,κe(cid:107),=,13,√,2π3,1,√,me,√,κi(cid:107),=,16,π3,1,√,mp,(cid:15)2,0,e4,·,(cid:15)2,0,e4,·,(kTe)5/2,Z2N,Λ,(kTi)5/2,√,Z4N,Λ,A,and,the,ratios,are,xe,=,√,2π3(cid:15)2,6,0,mee3,·,√,√,π3(cid:15)2,0,mpe3,·,An,expression,for,the,perpendicular,ion,conductivity,,maintaining,the,ﬁxed,physical,factors,is,of,interest,(kTi)3/2B,Z3N,Λ,A,(kTe)3/2B,Z2N,Λ,xi,=,12,√,(25),(24),√,κi⊥,=,e2√,√,9,mp,π3(cid:15)2,0,√,·,Z2N,Λ,A,(kTi)1/2B2,Assuming,Ti,is,measured,in,eV,,,and,N,in,units,of,1018,m−3,,then,and,κi⊥,=,6.67,×,10−4,·,Z2(N/1018)Λ,(Ti)1/2B2,√,A,m2s−1,κe⊥,=,5.26,×,10−5,·,Z2(N/1018)Λ,(Te)1/2B2,m2s−1,7,(22),(23),(26),(27),(28),The,plasma,resistivity,is,taken,as,η,=,ηB/µ0,=,Assuming,Te,is,measured,in,eV,,,then,0.51,√,6,√,mee2,2π3µ0(cid:15)2,0,·,ZΛ,(kTe)3/2,η,=,ηB/µ0,=,41.9,·,ZΛ,(Te)3/2,m2s−1,(29),(30),2.3,Prandtl,Numbers,The,above,expressions,(except,for,the,resistivity),apply,strictly,only,when,there,are,separate,equa-,tions,for,ion,and,electron,transport,,so,decisions,have,to,be,taken,about,how,to,combine,the,trans-,port,coefﬁcients,to,treat,the,plasma,as,a,single,ﬂuid.,For,the,thermal,transport,,since,pressures,pe,≈,pi,,it,is,sufﬁcient,to,add,the,κα.,However,,the,values,for,ions,and,electrons,are,so,disparate,because,mp,(cid:29),me,that,one,or,other,might,be,neglected,,assuming,B,is,of,order,unity,(in,Tesla),and,Te,≈,Ti,,thus,κe(cid:107),(cid:29),κi(cid:107),and,hence,κ(cid:107),≈,κe(cid:107),,since,It,also,follows,that,(cid:19)2,(cid:18),xi,xe,=,2me,Z2Amp,(cid:18),Ti,Te,(cid:19)3,κe⊥,κi⊥,=,0.078,(cid:18),Ti,Te,(cid:19)1/2,1,√,A,(31),(32),thus,κ⊥,≈,κi⊥.,There,is,the,caveat,that,if,Ti,is,approximately,spatially,constant,radially,,then,κe⊥,might,become,relevant.,As,for,viscosity,,since,the,ion,momentum,is,so,much,greater,than,the,electron,momentum,,then,ν,≈,νi.,For,interchange,motions,where,ﬂows,are,perpendicular,to,the,ﬁeld,,take,κ,=,κi⊥,,then,on,the,Cambridge,deﬁnition,,the,magnetic,Prandtl,number,is,ζ,=,η,κi⊥,=,0.765,√,2,1,µ0,(cid:114),me,mp,·,B2,√,ZN,(kTi)1/2,(kTe)3/2,A,which,evaluates,as,(Tα,in,eV,,,N,in,units,of,1018,m−3),ζ,=,η,κi⊥,=,62,700,·,B2,Z(N/1018),(Ti)1/2,(Te)3/2,√,A,(33),(34),It,may,be,argued,that,it,is,more,appropriate,to,use,the,‘anomalous’,value,of,1,m2s−1,,in,which,case,Equation,(30),without,units,gives,the,‘Cambridge’,magnetic,Prandtl,number.,The,usual,Prandtl,number,is,νi⊥,κi⊥,Note,that,P.H.Roberts,[11],deﬁnes,the,magnetic,Prandtl,number,as,P,rM,=,ν/η,=,P,r/ζ,,and,his,deﬁnition,is,more,widely,used.,=,0.23,P,r,=,(35),8,3,System,2-1:,2-D,model,of,anisotropic,heat,transport,The,model,for,time,evolution,of,the,temperature,ﬁeld,T,is,thermal,diffusion,,which,in,a,plasma,gives,where,the,thermal,conductivity,tensor,is,K.,(Compare,the,model,for,a,solid,3,2,N,∂T,∂t,=,∇,·,K∇T,ρmcp,∂T,∂t,=,∇,·,kc∇T,(36),(37),where,the,thermal,conductivity,tensor,is,kc,,ρm,is,the,mass,density,of,the,medium,and,cp,is,its,speciﬁc,heat,at,constant,pressure,,implying,that,the,thermal,diffusivity,tensor,is,κ,=,kc/ρmcp.),Introducing,vector,components,as,in,Section,1,,thermal,diffusion,in,a,plasma,after,Braginskii,is,thus,=,∇,·,(cid:0)K(cid:107)b[b.∇T,],+,K⊥(∇T,−,b[b.∇T,]),+,K∧b,×,∇T,(cid:1),(38),3,2,N,∂T,∂t,Henceforth,,the,‘wedge’,transport,ie.,in,due,to,the,term,in,K∧,is,neglected,for,the,reason,that,it,may,be,rearranged,to,give,a,convection-like,term,,via,the,identity,where,∇,·,K∧b,×,∇T,=,∇,·,(u∧T,),u∧,=,∇,×,(K∧b),(39),(40),(In,any,event,,if,K∧,is,a,function,purely,of,T,,,and,∇,·,b,=,0,,then,the,terms,in,Equation,(39),vanish.),Expressions,for,the,thermal,diffusivities,κ⊥,and,κ(cid:107),for,the,different,species,are,given,in,Section,2.2,,2,N,,,ie.,κ(⊥,(cid:107))e,i,=,K(⊥,(cid:107))/(,3,where,they,incorporate,the,factor,3,2,N,).,3.1,Test,Cases,The,aim,of,the,work,is,to,calculate,in,a,series,of,calculations,that,increasingly,approach,the,re-,alistic,model,,the,magnitude,of,the,spurious,numerical,diffusion,perpendicular,to,the,magnetic,ﬁeld,direction,b.,The,main,interest,concerns,how,much,diffuses,in,the,plasma,,not,the,solid,sur-,face,,even,though,the,deposition,of,power,is,on,the,surface,,reason:,all,sorts,of,complicated,extra,physics,come,into,play,in,the,plasma,especially,near,surfaces.,3.1.1,Starting,Case,For,the,2-D,test,case,illustrated,in,Figure,1,,it,is,suggested,that,κ⊥,=,0,so,that,any,perpendicular,diffusion,is,numerical,in,origin.,Given,this,,the,problem,can,be,analysed,using,any,spatial,scale,and,any,convenient,κ(cid:107).,However,an,order,of,magnitude,estimate,for,tile,dimensions,is,one,metre,,discharge,timescale,is,one,second,upwards.,For,plasma,properties,assume,N,=,1018,m−3,,Ti,=,Te,=,10,eV,,Z,=,A,=,1,,B,=,3,T,and,solid,temperatures,say,500o,C.,9,Figure,1:,Sketch,of,the,test,conﬁguration,,showing,ﬁeldlines,in,direction,b,and,the,boundary,between,anisotropic,conductor,and,perfect,insulator.,In,Figure,1,,T,=,T0,>,0,over,the,interval,on,the,left,hand,boundary,that,is,connected,by,ﬁeld,in,direction,b,with,the,thick,red,line,,elsewhere,on,the,blue,ﬁeldlines,,T,=,0.,The,red,region,lies,on,a,black,line,which,denotes,the,boundary,between,anisotropic,conductor,and,perfect,insulator.,The,exact,steady-state,solution,has,T,=,constant,along,ﬁeldlines,,but,numerical,diffusion,will,result,in,non-zero,T,in,the,region,of,blue,ﬁeldlines.,The,relative,size,of,this,numerical,diffusion,must,be,estimated,as,a,function,of,incidence,angle,θ,,where,interest,attaches,to,small,θ,≤,2o.,3.1.2,Intermediate,Case,To,test,curvature,effects,the,whole,2-D,domain,of,Figure,1,could,be,distorted,by,conformal,map-,ping,(which,preserves,angles).,3.1.3,Realistic,Case,This,needs,to,be,3-D,and,involve,JET,divertor,tile,descriptions,derived,from,the,output,of,the,CAD,design,tool,,together,with,information,describing,the,magnetic,ﬁeld,as,a,function,of,position,,which,will,be,supplied.,The,magnetic,equilibrium,may,be,supplied,analytically,after,Solovev,,but,the,usual,input,is,as,an,.eqdsk,ﬁle.,The,EQDSK,G,format,is,a,“non-standard”,standard,for,solutions,ψ(R,,Z),,p(ψ),,I(ψ),of,the,Grad-Shafranov,equation,,where,ψ,is,the,magnetic,ﬂux,and,(R,,Z),are,cylindrical,coordinates,in,planes,normal,to,the,toroidal,direction.,The,functions,p,and,I,give,the,variation,of,the,pressure,and,toroidal,ﬁeld,respectively.,The,basic,standard,for,EQDSK,G,may,be,found,at:,https://fusion.gat.com/theory/Efitgeqdsk,(which,may,be,password-protected),or,else,at,https://w3.pppl.gov/ntcc/TORAY/G_EQDSK.pdf,The,ﬂux,ψ(R,,Z),is,sampled,at,uni-,formly,spaced,points,on,a,direct,product,grid,,for,which,the,.eqdsk,header,deﬁnes,the,mesh-size,,10,as,well,as,other,useful,information,,such,as,the,ﬂux,on,axis,and,at,boundary.,Unfortunately,the,strict,EQDSK,G,standard,uses,a,Fortran,format,that,does,not,require,spaces,between,samples,,hence,there,are,many,variants,for,languages,that,cannot,handle,this,situation,,that,have,intro-,duced,other,features,such,as,mistakes,in,ﬁeld,helicity,,factors,of,2π,in,the,ﬂux,,etc.,Routines,that,calculate,magnetic,ﬁeld,B,using,cubic,spline,interpolation,could,be,made,available.,It,would,be,desirable,for,the,output,of,System,2-2,to,be,used.,3.1.4,Extended,Case,An,extended,test,would,allow,for,heat,transfer,in,the,solid,surface,sketched,at,bottom,of,Figure,1,,taking,say,thermal,diffusivity,for,Tungsten,κ,≈,3,×,10−5,m2,s−1.,11,Figure,2:,Sketch,of,the,test,conﬁguration,,showing,tokamak,cross-section,and,the,boundary,of,the,Last,Closed,Flux,Surface,(LCFS).,4,System,2-2:,2-D,elliptic,solver,in,complex,geometry,The,geometry,will,be,representative,of,a,tokamak,cross-section,,possibly,omitting,the,region,con-,taining,the,central,hot,plasma,,so,that,topologically,it,will,be,at,most,as,complex,as,an,annulus,(one-hole).,Figure,2,provides,an,example.,The,Last,Closed,Flux,Surface,(LCFS),may,be,param-,eterised,by,arc-length,in,the,cross-sectional,plane,of,projection.,The,elliptic,equations,to,be,considered,now,follow.,4.1,Simpliﬁed,Grad-Shafranov,equation,This,elliptic,equation,is,a,simpliﬁed,version,of,the,Grad-Shafranov,equation,,see,[12],R2∇,·,1,R2,∇ψ,=,−2µ0Rjφ,(41),where,ψ,is,the,poloidal,magnetic,ﬂux,and,(R,,Z),are,cylindrical,coordinates,in,planes,normal,to,the,toroidal,direction,φ,,with,the,toroidal,current,jφ,=,R,dp(ψ),dψ,+,I,µ0R,dI(ψ),dψ,+,jext(R,,Z),(42),The,functions,p,and,I,give,the,variation,as,functions,of,ψ,of,the,pressure,and,toroidal,ﬁeld,respec-,tively,,and,jext(R,,Z),may,be,produced,in,several,ways,,of,which,the,commonest,is,by,poloidal,ﬁeld,circuits,,ie.,localised,current,sources,in,cross-section.,Note,that,the,operator,in,Equation,(41),simpliﬁes,to,∂,∂R,which,implies,that,mathematically,,ψ,satisﬁes,a,steady-state,2-D,advection-diffusion,equation,cor-,responding,to,unit,diffusivity,in,the,ﬂow,uR,=,1/R.,1,R,(43),∂2,∂R2,−,∂2,∂Z2,+,12,To,provide,the,simple,test,case,,take,jφ,=,jext,only,with,localised,current,sources.,The,boundary,conditions,are,ψ,=,0,on,the,LCFS,and,ψ,∝,1/(cid:112)(R2,+,Z2),as,R,,Z,→,∞.,Note,that,the,Grad-Shafranov,equation,has,been,solved,using,spectral,elements,by,others,,eg.,Sovinec,[3].,4.2,Simpliﬁed,non-Boussinesq,vorticity,equation,A,simpliﬁed,version,of,the,non-Boussinesq,vorticity,equation,to,be,solved,for,the,scalar,ﬁeld,Φ(R,,Z),in,cylindrical,polar,coordinates,(R,,Z),is,∇⊥,·,(cid:18),1,B2,∇⊥Φ,(cid:19),=,n,(44),where,B,=,|B|,is,the,amplitude,of,the,imposed,magnetic,ﬁeld,,density,n,acts,as,a,source,term,,and,the,elliptic,is,to,be,solved,for,Φ,,subject,to,boundary,conditions,Φ,=,0.,The,operator,∇⊥,,ignor-,ing,the,components,of,magnetic,ﬁeld,directed,within,the,(R,,Z),plane,,reduces,to,the,usual,gradi-,ent,∇,in,cylindrical,polars,of,axisymmetric,ﬁelds,,hence,mathematically,Equation,(44),is,equivalent,to,Equation,(41).,n,will,be,set,so,that,n,=,n0(si),on,the,boundaries,,where,arc-length,si,parameterises,the,inner,boundary,if,i,=,1,and,the,outer,if,i,=,2.,n,and,|B|,will,be,speciﬁed,functions,of,(R,,Z),that,capture,features,of,the,number,density,n,distribution,and,magnetic,ﬁeld,intensity,distribution,expected,in,a,tokamak,,Ideally,|B|,would,represent,a,solution,of,the,Grad-Shafranov,equation,from,Section,4.1.,13,5,System,2-3:,1-D,ﬂuid,solver,with,simpliﬁed,physics,but,with,UQ,and,realistic,boundary,conditions,5.1,Plasma,Equations,It,is,assumed,that,the,single,spatial,dimension,of,the,problem,corresponds,to,the,arc,length,dis-,tance,along,a,ﬁeldline.,Starting,from,the,two-ﬂuid,model,of,Braginskii,[8],,a,set,of,equations,resembling,those,of,classical,(compressible),hydrodynamics,may,be,derived,by,summing,Bragin-,skii’s,equations,for,number,density,,momentum,and,energy.,Using,the,notation,of,ref,[13],,intro-,ducing,Td,=,Ti,+,Te,,neglecting,the,stress,tensor,terms,(implicitly,setting,δpi,=,0),,and,assuming,B,is,independent,of,time,,the,resulting,system,is,Ud,∂,∂t,(N/B),+,(N,U/B),=,LsSn,B,Ud,∂,∂t,(N,U/B),+,(N,KM,Td/B),+,Ud,(cid:18),3,2,∂,∂t,(cid:18),5,2,∂,∂s,(N,U,KM,Td/B),+,(N,U,3/B),∂,∂s,∂,∂s,1,2,1,2,(N,U,2/B),(cid:19),(cid:19),(N,U,2/B),=,−,1,miB,∂,∂s,(Pi,+,Pe),+,LsSu,miB,+,=,−,1,miB,∂,∂s,(qi,+,qe),+,Ls(SE,i,+,SE,e,),miB,where,Ls,=,∂s(cid:107),∂s,(45),(46),(47),(48),(49),Ud,=,Ls/t0,is,a,speed,measuring,the,importance,of,the,transient,term,,s,parameterises,distance,along,the,ﬁeldline,,and,some,variables,from,[13],have,been,promoted,to,capitals,to,indicate,that,they,retain,their,physical,dimensions.,For,the,case,of,a,ﬁeldline,connecting,two,walls,at,s(cid:107),=,±L,,and,so,Ls,=,2L.,The,constant,KM,is,such,that,s(cid:107),=,L(2s,−,1),KM,=,kB,mi,or,KM,=,|e|,mi,(50),(51),where,kB,is,Boltzmann’s,constant,and,|e|,is,the,unit,of,charge,,depending,whether,T,is,measured,in,Kelvin,or,eV,.,Note,that,in,adding,Eqs.(3),and,(4),of,[13],,equipartition,and,collision,terms,cancel,to,give,Equation,(48).,The,perfect,gas,equation,of,state,will,be,assumed,,so,that,pi,+,pe,mi,=,N,KM,T,The,boundary,conditions,are,that,|U,|,=,|Ms|CS,at,s,=,0,,1,where,the,sound,speed,CS,=,(cid:112),KM,Td,14,(52),(53),and,|Ms|,is,the,Mach,number,,since,Ms,will,be,allowed,to,take,either,sign.,Normally,|Ms|,=,1,so,that,M0,=,−1,and,M1=1,where,the,subscript,corresponds,to,value,of,s.,The,combined,energy,ﬂux,at,each,boundary,has,|Q(cid:107)|,=,miCSN,(δeKM,Te,+,δiKM,Ti),≈,miCSN,δKM,Td,(54),if,δ,≈,δe,≈,δi.,For,deﬁniteness,,δ,=,1,2,(δe,+,δi),will,be,assumed.,5.2,Fluid,Equations,with,Sources,It,is,convenient,to,replace,the,source,terms,in,Equations(45)–,(48),by,equivalent,ﬂuxes,F,n(s),=,F,u(s),=,(cid:90),s,0,(cid:90),s,ds(cid:107),Sn,B,Su,miB,i,+,SE,SE,e,miB,ds(cid:107),0,(cid:90),s,0,F,E,(s),=,2,ds(cid:107),and,write,F,Q(s),=,−,1,miB,(qi,+,qe),(55),(56),(57),(58),The,convention,with,respect,to,limits,of,integration,is,that,they,are,speciﬁed,in,terms,of,parame-,terised,length,and,to,use,the,appropriate,relation,for,s(cid:107)(s),,thus,in,the,case,of,Equation,(50),,the,lower,limit,of,0,corresponds,to,s(cid:107)(s),=,−L.,The,forms,the,sources,take,are,discussed,below,in,Section,5.3.,Observing,the,identity,1,B,∂,∂s,BnBKM,Td,=,∂,∂s,(nBKM,Td),+,nBKM,Td,∂,∂s,(ln,B),(59),and,the,frequent,appearance,of,nB,=,N/B,,the,governing,equations,in,dimensional,form,become,Ud,∂,∂t,nB,+,Ud,∂,∂t,(nBU,),+,(nBKM,Td),+,(nBU,KM,Td),+,F,n,(nBU,),=,∂,∂s,∂,(nBU,2),=,−,∂s,(cid:19),∂,∂s,∂,∂s,1,(nBU,2),2,1,2,(nBU,3),(cid:19),+,=,−,∂,∂s,F,Q,+,1,2,∂,∂s,F,E,(nBKM,Td),+,∂,∂s,F,u,Ud,(cid:18),3,2,∂,∂t,(cid:18),5,2,∂,∂s,(60),(61),(62),(63),where,the,derivative,of,ln,B,has,been,neglected.,The,boundary,conditions,on,U,are,unchanged,and,|Q(cid:107)|,=,miCSnBδKM,Td,The,Equations(60)–,(63),together,with,boundary,condition,Equation,(64),are,in,units,such,that,equivalence,may,easily,be,established,with,those,of,ref,[13],(by,setting,Ud,=,1,and,identifying,s,with,s(cid:107)).,To,proceed,,it,may,be,helpful,to,make,the,preceding,system,of,equations,dimensionless,,by,scaling,nB,with,respect,to,N0/B0,,Td,with,respect,to,T0,,U,with,respect,to,C0,and,B,with,respect,to,B0.,If,the,subscript,0,corresponds,to,the,value,of,a,variable,at,s,=,0,,then,it,is,convenient,to,KM,T0.,The,resulting,system,may,be,deduced,from,the,coupled,model,in,Section,5.4.,take,C0,=,(64),√,15,5.3,Explicit,Sources,The,above,work,concentrates,on,the,case,where,the,source,terms,are,regarded,as,given,,however,it,is,worth,describing,the,form,of,the,additional,sources,that,may,be,at,least,locally,important.,From,ref,[13],,the,plasma,sources,are,given,by,(with,the,convention,that,sufﬁx,‘n’,denotes,neutral,species),Sn,=,NnN,(cid:104)σv(cid:105)ION,−,N,2(cid:104)σv(cid:105)REC,+,Sn,⊥,Su,=,NnN,(cid:104)σv(cid:105)ION,Un,−,N,2(cid:104)σv(cid:105)RECU,+,NnN,(Un,−,U,)(cid:104)σv(cid:105)CX,SE,=,SE,i,+,SE,e,=,NnN,(cid:104)σv(cid:105)ION,(,3,2,kTn,+,1,2,−,N,2(cid:104)σv(cid:105)REC(,3,kTi,+,2,(cid:18),3,2,⊥i,+,SE,−,NnN,kQH,+,SE,⊥e,+,NnN,(cid:104)σv(cid:105)CX,n,−,kIH,),mnU,2,1,2,miU,2),k(Tn,−,Ti),+,(cid:19),mn(U,2,n,−,U,2),1,2,(65),(66),(67),(68),Here,sufﬁx,⊥,denotes,the,effectively,given,source,terms,arising,from,cross-ﬁeld,contributions,,sufﬁces,ION,,,REC,and,CX,denote,respectively,reaction,rates,(cid:104)σv(cid:105),for,ionisation,,recombination,and,charge-exchange,,IH,is,the,Hydrogen,reionisation,potential,,and,QH,is,the,cooling,rate,due,to,excitation.,Since,the,sources,appear,in,the,analysis,primarily,as,integrals,starting,at,s,=,0,,study,of,Equa-,tions(65)–,(67),concentrates,on,this,region,,where,plasma,velocity,U,<,0,and,neutral,velocity,Un,>,0,with,the,two,having,approximately,the,same,magnitude.,There,,Equation,(65),has,only,one,negative,term,,due,to,recombination,,but,from,the,cross-section,data,in,ref,[13],,this,could,dominate,only,below,2,eV.,All,terms,in,Equation,(66),are,positive,near,s,=,0,as,the,two,velocities,reinforce.,Equation,(67),contains,two,terms,which,are,always,negative,and,an,ionisation,term,which,is,also,negative,below,IH,/2,≈,7,eV,,thus,for,example,,the,cross-ﬁeld,source,terms,S⊥i,e,must,be,positive,for,SE,>,0.,The,sources,of,neutrals,may,be,deduced,from,the,ionisation,and,charge-exchange,terms,in,Equa-,tions(65)–,(67),,viz.,n,=,−NnN,(cid:104)σv(cid:105)ION,+,Sn,Sn,⊥n,Su,n,=,−NnN,(cid:104)σv(cid:105)ION,Un,−,NnN,(Un,−,U,)(cid:104)σv(cid:105)CX,3,2,n,=,−NnN,(cid:104)σv(cid:105)ION,(,n,−,kIH,),mnU,2,kTn,+,SE,1,2,(cid:18),3,2,k(Tn,−,Ti),+,1,2,(cid:19),mn(U,2,n,−,U,2),−,Nn(cid:104)σv(cid:105)CX,+,SE,⊥n,(69),(70),(71),(72),(73),The,S⊥n,terms,are,hard,to,quantify,,but,if,these,are,neglected,,it,is,clear,that,Sn,the,obverse,of,the,positive,plasma,sources.,Similarly,it,is,likely,that,SE,n,<,0,if,SE,>,0,n,<,0,and,Su,n,<,0,is,16,5.4,Coupled,System,Working,within,the,ﬂux-tube,geometry,,the,equations,for,neutral,transport,take,the,same,form,as,those,used,for,plasma,above,,however,the,boundary,conditions,are,different.,They,become,at,s,=,0,,supposing,that,T(0),=,τ,2T0,and,masses,mn,=,mi,,in,dimensionless,units,,n,=,R2,=,|M0|,R,τ,u,=,−τ,T,=,τ,2,(74),(75),(76),where,n,,u,and,T,are,neutral,density,,velocity,and,temperature,made,dimensionless,with,respect,to,the,same,N0,and,T0,as,the,corresponding,plasma,quantities,,and,R,is,the,recycling,coefﬁcient.,Note,the,usage,of,a,sans-serif,font,to,denote,dimensionless,neutral,species,quantities,,and,that,n,does,not,however,include,a,factor,of,˜b,=,B/B0,,as,the,neutral,ﬂux,is,not,constrained,by,the,ﬂux,tube.,A,system,explicitly,modelling,the,coupling,between,plasma,and,neutrals,may,be,derived,by,making,non-dimensionless,the,sources,set,out,in,Section,5.3,,and,assuming,mn,=,mi,,T,=,Ti,=,τ,2T,,,giving,5,equations,for,the,evolution,of,plasma,density,,velocity,,total,temperature,,neutral,density,and,neutral,velocity:,(cid:15)r,(cid:15)r,∂,∂t,(nu),+,∂,∂t,∂,∂s,n,+,∂,∂s,(nu),=,σI,nn,+,∂,∂s,f,n,(nu2,+,nT,),=,σI,nnu,+,σcnn(u,−,u),+,∂,∂s,f,u,(cid:15)r,∂,∂t,(cid:0)(g,−,2)nT,+,nu2)(cid:1),+,∂,∂s,(cid:0)gnuT,+,nu3(cid:1),=,σI,nn(3[τ,2T,−,TH,],+,u2),+,σCnn(u2,−,u2),∂,∂s,−,σEnn,+,f,E,(cid:15)r,n,+,∂,∂s,(nu),=,−σI,nn,∂,∂t,(nu2,+,nτ,2T,),=,−σI,nnu,−,σcnn(u,−,u),(cid:15)r,∂,∂t,(nu),+,∂,∂s,(77),(78),(79),(80),(81),where,(cid:15)r,=,Ls/(t0C0),with,t0,a,characteristic,timescale.,The,reaction,reaction,rates,are,made,dimensionless,by,division,by,C0/(LsN0)((cid:39),3.1×10−16,m3s−1,for,representative,parameter,values),,so,that,σI,=,σC,=,σE,=,TH,=,(cid:104)σv(cid:105)ION,C0/LSN0,(cid:104)σv(cid:105)CX,C0/LSN0,2QH,C0T0/LSN0,2IH,3T0,(cid:39),9/T0(in,eV),17,(82),(83),(84),(85),where,IH,and,QH,are,as,deﬁned,in,ref,[13].,It,is,of,interest,to,allow,a,stochastic,(‘turbulent’),contribution,to,the,terms,f,n,u,E.,5.5,Uncertainty,Quantiﬁcation,Polynomial,chaos,(PC),refers,to,a,situation,whereby,probability,functions,are,expanded,as,Hermite,polynomials,,and,generalised,Polynomial,chaos,(gPC),to,expansions,using,Hermite,and,other,polynomial,sets,{Ψj(ξ)},[14].,Thus,for,example,,suppose,θ,to,denote,a,random,event,,and,the,number,density,ﬁeld,to,have,the,following,representation,in,terms,of,a,ﬁnite,number,P,of,such,modes.,PC(cid:88),n(x,,t,,θ),=,nj(x,,t)Ψj,(ξ(θ)),(86),j=0,where,{nj(x,,t)},is,the,set,of,deterministic,coefﬁcients,of,the,“random,trial,basis”,,ie.,the,set,{Ψj,(ξ(θ))},where,ξ(θ),is,a,multi-dimensional,random,variable,with,a,speciﬁc,probability,distribution,as,a,func-,tion,of,the,random,parameter,0,≤,θ,≤,1.,Note,that,the,Ψj,are,the,set,of,multi-dimensional,Hermite,polynomials,if,ξ,is,a,vector.,Typically,but,not,necessarily,the,ξ(θ),will,be,Gaussians.,Expressions,like,Equation,(87),may,be,substituted,into,a,governing,equation,of,say,advection,type,for,n,,and,the,result,simpliﬁes,because,spatial,operators,do,not,interact,with,the,random,variables,,then,taking,the,inner,product,with,Ψk,(ξ(θ)),yields,∂nk,∂t,+,PC(cid:88),PC(cid:88),i=0,j=0,eijk,∂uinj,∂s,=,0,(87),where,eijk,is,a,weighted,integral,of,triple,products,of,Ψi.,Hence,there,are,now,PC,equations,instead,of,one,for,n.,There,is,an,alternative,treatment,by,Polynomial,Chaos,Expansion,(PCE),which,is,non-intrusive,and,therefore,preferred.,It,relies,on,projecting,a,set,of,randomly,selected,solutions,onto,a,Hermite,or,similar,basis,,see,further,description,in,ref,[15,,end,§,2.1.1].,18,6,System,2-4:,Spatially,1-D,plasma,model,incorporating,velocity,space,effects,The,following,simple,model,is,after,Taitano,et,al,[16,,17],∂fe,∂t,∂fi,∂t,+,vex,+,vix,∂fe,∂x,∂fi,∂x,+,+,(cid:15)0,∂E,∂t,(cid:88),+,α,qαnuα,−,qe,me,qi,mi,(cid:88),α,E,·,E,·,∂fe,∂v,∂fi,∂v,=,0,=,0,(88),qαnuα,=,0,Equations,88,are,the,electron,and,ion,Vlasov,equations,and,Ampere’s,equation,respectively.,The,quantities,me,,mi,,fe,,fi,,ve,,vi,,qe,,qi,,E,,(cid:15)0,,and,nuα,are,the,electron,and,ion,masses,,electron,and,ion,distribution,functions,,electron,and,ion,velocities,,electron,and,ion,charges,,the,electric,ﬁeld,,permeability,constant,of,vacuum,,and,the,momentum,of,species,α,=,i,,e,,respectively.,Note,that,Equation,(88)represents,a,generalisation,of,the,system,in,ref,[16],,where,for,vector,quantities,,the,x-component,is,always,implied,,in,the,usual,notation,the,original,system,is,1d1v,rather,than,1d3v,as,above,,where,particles,move,according,to,dx,dt,dvα,dt,=,vαx,=,qα,mα,E,(89),(Motion,in,(y,,z),is,neglected,,the,3-D,electromagnetic,version,of,Equation,(89),appears,in,Sec-,tion,9).,The,(cid:80),·,term,denotes,for,example,a,spatially,averaged,summed,quantity,and,is,included,to,en-,force,Galilean,invariance.,The,solutions,fe,and,fi,of,the,Vlasov,equations,are,functions,of,space,variable,x,,velocity,v,,and,time.,Ampere’s,equation,is,solved,for,the,self-consistent,electric,ﬁeld,E,,which,is,a,function,of,space,variable,x,and,time.,The,boundary,conditions,used,are,periodicity,in,x,,and,zero,at,inﬁnity,in,|v|.,Initial,conditions,which,might,be,used,for,the,distribution,functions,are,from,ref,[16],,f0(x,,v,,u0,,T0),=,n0(x),(cid:112)2πkT0/m,(cid:18),exp,−,(cid:19),m(v,−,u0)2,2kT0,n0(x),=,n(t,=,0,,x),=,1,+,αn,cos(kwx),(90),where,n0,,u0,and,T0,are,the,initial,number,density,,initial,ﬂuid,velocity,and,initial,temperature,respectively.,The,parameter,αn,is,the,perturbation,amplitude,,kw,is,its,wave,vector,,and,m,is,the,species,mass.,It,follows,that,the,(scaled),momentum,is,given,formally,as,(cid:90),nuα(x),=,vfα(x,,v,,t)dv,(91),where,fα,is,the,distribution,function,for,species,α,at,time,t.,replaced,by,a,sum,over,particles.),(In,practice,the,integral,would,be,19,Note,that,periodic,boundary,conditions,are,of,limited,value,in,practice,,and,attention,should,be,given,to,minimal,modiﬁcations,of,the,above,problem,where,there,is,1.,a,ﬂux,of,momentum,across,the,domain,(inﬂow,and,outﬂow,boundary,conditions),2.,reﬂection,of,particles,at,the,boundaries,3.,a,source,of,plasma,within,the,domain,,and,outﬂow,boundaries,4.,and,where,the,spatial,dimension,corresponds,to,arc,length,s,along,a,ﬁeldline,(implies,n,replaced,by,n/|B|,,cf.,Section,5.1).,20,7,System,2-5:,Spatially,1-D,multispecies,plasma,model,7.1,Fluid,model,For,a,multispecies,plasma,,there,is,a,system,of,Boltzmann,equations,to,be,solved,,one,for,each,species,,each,of,form,in,3,spatial,dimensions,L7fα,=,(cid:88),β,Q(fα,,fβ),+,Sα,(92),where,L7,is,the,7-D,Lie,derivative,(space,,velocity-space,and,time,make,up,the,3,+,3,+,1,=,7,dimensions,,α,,β,are,species,labels,and,Q,is,the,Boltzmann,collision,operator.,The,multispecies,equations,are,derived,following,Grad,[18,,§,6],by,substituting,in,Equation,(92),fα,=,exp(−λHα)Fα(x,,v,,t),(93),where,the,ﬂow,of,the,Lie,derivative,is,given,by,the,Hamiltonian,Hα,and,Fα,is,a,functional,of,moments,of,fα,,to,include,(dropping,the,sufﬁx,on,f,),(cid:90),n,=,f,dv,,u0,=,(cid:90),f,vdv,,T,=,(cid:90),f,v2/2dv,(94),The,resulting,system,is,linearised,and,solved,by,iteration,to,give,the,multispecies,plasma,ﬂuid,equations,in,Zhdanov,[18,,§,6].,There,are,believed,to,be,typographical,errors,in,Zhdanov,,so,cross-checking,is,needed.,To,see,Grad’s,approach,applied,to,classical,ﬂuids,see,for,example,[19,,§,8].,7.2,Coupling,to,particles,Other,,less,collisional,species,are,to,be,treated,as,particles,as,in,Section,6,and,coupled,via,Sα.,Mathematical,forms,for,Sα,will,be,guided,by,the,emerging,results,from,particles’,method,research.,21,8,System,2-6:,Spatially,2-D,plasma,model,incorporating,velocity,space,effects,8.1,2-D,interchange,model,with,ﬂuid,neutrals,After,P.,Tamain,(private,communication),,the,equations,for,time,advance,of,respectively,electron,number,density,ne,,“vorticity”,(∇,·,E+),,electron,energy,Ee,,ion,energy,Ei,and,neutral,number,density,nn,are,respectively,∂tne,+,∇,·,(neue),=,Sn,e,−,ne,τne,∂t∇,·,E+,+,∇,·,(cid:0)∇,·,(cid:0)ui,⊗,E+(cid:1)(cid:1),=,∇,·,(cid:18),ni,(u∇Bi,+,ucx),−,−,ni,τni,+,∇,·,(cid:0)ν∇⊥,neu∇Be,1,Zi,(cid:0)∇,·,E+(cid:1)(cid:1),+,1,Zi,ne,τne,Ee,τEe,Ei,τEi,∂tEe,+,∇,·,(Eeue,+,peue),=,SE,e,−,+,Qie,+,∇,·,(κ⊥ene∇⊥kTe),∂tEi,+,∇,·,(Eiui,+,piui),=,SE,i,−,∂tnn,=,Sn,n,+,∇,·,(Dn∇⊥pn),−,Qie,+,∇,·,(κ⊥ini∇⊥kTi),(cid:19),(95),(96),(97),(98),(99),where,with,the,usual,notation,for,species,α,,pressure,pα,,temperature,in,energy,units,kTα,,charge,state,Zi,,species,mass,mα,,electric,potential,Φ,and,magnetic,ﬁeld,B,,ion,number,density,ni,=,ne,3,2,Eα,=,pα,=,modiﬁed,electric,ﬁeld,E+,=,mi,ZieB2,3,nαkTα,,α,=,i,,e,2,(cid:18),ni∇⊥Φ,+,∇⊥pi,1,Zie,Perpendicular,advection,velocities:,ue,=,uE×B,+,u∇Be,+,udiﬀ,ui,=,uE×B,+,u∇Bi,+,udiﬀ,+,ucx,with,uE×B,=,u∇Bi,=,µ2,cx,1,+,µ2,cx,µ2,cx,1,+,µ2,cx,2kTe,eB,B,×,∇Φ,B2,2kTi,ZieB,B,×,∇B,B2,(cid:18),B,×,∇B,B2,u∇Be,=,−,ucx,=,µcx,1,+,µ2,cx,1,B,udiﬀ,=,−D⊥∇ne,−∇⊥Φ,−,22,(cid:19),∇pi,1,Zieni,(cid:19),(100),(101),(102),(103),(104),(105),(106),(107),(108),(109),(110),with,the,magnetization,with,respect,to,charge,exchange,reactions,deﬁned,as,µcx,=,ωci/νcx,,ωci,=,Zi|e|B/mi,being,the,ion,cyclotron,frequency,and,νcx,=,Kcxnn,the,charge,exchange,frequency,(Kcx,(ni,,Ti),is,the,reaction,rate,of,charge,exchange,reactions).,Loss,rates,can,be,given,several,meanings,in,this,kind,of,model.,They,can,be,prescribed,as,param-,eters,,but,they,can,also,describe,sheath,losses,in,which,case,they,have,a,non-linear,dependency,with,the,plasma,conditions,,eg.,τne,=,τni,=,τEe,=,τEi,=,L(cid:107),cs,L(cid:107),cs,2,3γe,2,3γi,(cid:18),exp,Λb,−,(cid:19),eΦ,kTe,τne,τni,where,cs,=,(cid:113),kTi+ZikTe,mi,is,the,acoustic,velocity,and,Λb,=,−,(cid:20),ln,0.5,1,2,me,mi,(cid:18),1,+,Ti,Te,(cid:19)(cid:21),is,the,sheath,potential,drop,normalized,to,Te.,The,collisional,energy,equipartition,term,(from,Braginskii),is:,Qie,=,3,me,mi,ne,τe,(kTe,−,kTi),with,the,electron,collision,time,given,by:,(111),(112),(113),(114),(115),(116),3,(2π)3/2,(cid:15)2,0,niZi,and,Λ,is,the,Coulomb,logarithm,(which,has,a,weak,dependence,on,ne,and,Te,as,indicated,in,Section,2.2).,me(kTe)3/2,2e4Λ,(117),τe,=,√,Source,terms:,n,=,nennKi,(ne,,Te),−,n2,e,=,−Sn,Sn,e,=,−nennKi,(ne,,Te),Ei,(Te),−,n2,SE,eKr,(ne,,Te),eKr,(ne,,Te),3,2,kTe,i,=,−n2,SE,eKr,(ne,,Te),3,2,kTi,where,Ki,and,Kr,are,the,ionization,and,recombination,reaction,rates.,Neutral,diffusion,coefﬁcient:,Dn,=,1,mi,(niKcx(ni,,Ti),+,neKi(ne,,Te)),Other,perpendicular,diffusion,coefﬁcients,from,Braginskii,as,in,Section,2.2.,Boundary,conditions:,23,(118),(119),(120),(121),Figure,3:,Handling,a,plasma,sheath,adjacent,to,wall,at,right.,Plasma,has,ﬂuid,properties,dis-,cretised,using,FEM,indicated,using,warped,grid,,but,in,sheath,is,better,represented,as,particles,indicated,by,dots.,•,electron,density,ne:,zero,ﬂux,•,electron,energy,Ee:,zero,ﬂux,•,ion,energy,Ei:,zero,ﬂux,•,electrostatic,potential,Φ:,Dirichlet,equal,to,average,of,ΛbTe,along,the,boundary,•,neutral,density,nn:,incoming,ﬂux,ﬁxed,by,integral,of,particle,losses,in,parallel,direction,(exact,redistribution,to,be,discussed),8.2,Coupling,to,particles,The,only,appearance,of,particle,or,kinetic,effects,in,the,model,of,Section,8.1,is,via,the,sheath,boundary,condition.,A,more,detailed,treatment,of,the,sheath,uses,particle,techniques,cf.,Section,7,,where,the,different,representation,is,used,in,a,second,overlapping,region,as,indicated,in,Figure,3,taken,from,ref,[20].,24,9,System,3-1:,2-D,particle-based,model,of,neutral,gas,and,impuri-,ties,with,critical,physics,The,following,generic,transport,equation,[21,,§,1],applies,to,all,particle-based,models,for,the,time,evolution,of,the,density,distribution,function,f,(x,,v,,t),∂f,∂t,+,v,·,∂f,∂x,+,a,·,∂f,∂v,=,SC(f,),=,(cid:18),∂f,∂t,(cid:19),C,+,Sexp(x,,v,,t),(122),where,a,=,d2x/dt2,is,the,acceleration,experienced,by,a,particle,at,position,x,with,velocity,v.,This,represents,scalar,advection,in,a,6-D,space,with,an,explicit,source,Sexp(x,,v,,t),and,a,source,due,to,other,inter-particle,interactions,that,is,conventionally,written,(∂f,/∂t)C,when,it,is,localised,and,usually,depends,linearly,on,f,.,Complete,speciﬁcation,of,the,problem,even,for,SC,=,0,and,a,single,species,of,particle,requires,a,force,law,such,as,that,for,particles,of,species,α,with,charge,qα,and,mass,mα,F,=,mα,d2x,dt2,=,qα(E,+,v,×,B),(123),and,equations,for,the,evolution,of,the,electromagnetic,ﬁelds,E(x,,t),and,B(x,,t),such,as,Maxwell’s,equations,,neglecting,displacement,current.,For,neutral,particles,,when,often,a,=,0,,interest,attaches,to,SC,which,for,2-particle,interactions,is,often,the,Boltzmann,operator,for,different,species,Q(fα,,fβ),where,α,,β,are,species,labels.,It,will,be,seen,that,Equation,(122),is,a,statement,that,the,7,−,D,Lie,derivative,of,f,vanishes,,and,it,is,hoped,that,in,the,longer,term,,the,properties,of,the,Lie,derivative,might,be,exploited.,For,current,purposes,,it,is,necessary,to,state,a,multi-species,version,of,Equation,(122),complete,with,appropriate,source,terms,to,represent,the,physics,thought,critical,for,modelling,the,tokamak,Inevitably,choice,of,SC,is,a,function,of,lengthscale,and,timescale.,On,fast,timescales,in,edge.,a,strong,electromagnetic,ﬁeld,,the,effect,of,collisions,can,be,ignored,(collective,effects,are,felt,through,say,the,electrostatic,ﬁeld).,25,10,Summary,Acknowledgement,The,support,of,the,UK,Meteorological,Ofﬁce,and,Strategic,Priorities,Fund,is,acknowledged.,Valu-,able,input,from,Ben,Dudson,is,also,acknowledged.,Stuart,Henderson,provided,helpful,advice,concerning,plasma,radiation.,References,[1],Proxy,Applications.,https://proxyapps.exascaleproject.org,,2020.,Online;,accessed,May,2020.,[2],W.,Arter,,L.,Anton,,D.,Samaddar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heets,,139:77–137,,2017.,http://dx.doi.org/10.1016/j.nds.2017.01.,002,,website:,https://fispact.ukaea.uk/.,[32],S.S.,Henderson,,M.,Bluteau,,A.,Foster,,A.,Giunta,,M.G.,O’Mullane,,T.,P,¨utterich,,and,H.P.,Summers.,Optimisation,and,assessment,of,theoretical,impurity,line,power,coefﬁcients,rele-,vant,to,ITER,and,DEMO.,Plasma,Physics,and,Controlled,Fusion,,59(5):055010,,2017.,[33],S.S.,Henderson,,M.M.,Bluteau,,M.G.,O’Mullane,,and,H.P.,Summers.,Improvements,of,impu-,rity,line,ITER,and,DEMO,via,systematic,optimisation,of,atomic,structure.,Technical,Report,UKAEA-CCFE-PR(18)31,,CCFE,,2018.,A,Annex,A:,Atomic,and,Molecular,Effects,Often,the,radiation,emitted,and,absorbed,by,atoms,in,different,ionisation,states,must,be,accounted,for.,There,is,a,compact,introduction,in,Golub,and,Pasachoff,[22,,§,3.3.1].,This,explains,how,in,principle,,given,a,“particular,mixture,of,elements,at,a,speciﬁed,temperature,.,.,.,the,number,of,atoms,per,unit,volume,of,the,gas,which,are,in,a,particular,ionisation,state,may,be,calculated,.,.,.,then,for,that,atom,which,emission,lines,are,emitted”,and,so,on,for,all,other,constituents,of,the,mixture.,(Temperature,refers,to,a,black-body,radiation,ﬁeld,in,which,the,atoms,are,assumed,to,sit.),“The,sum,total,of,all,these,bound-bound,emissions,,plus,the,bound-free,and,free-free,emissions”,is,the,spectrum,,where,it,is,explained,that,‘bound’,and,‘free’,describe,the,state,of,the,electron,involved,in,the,formation,of,the,line,with,respect,to,the,atom.,But,“in,practice,,carrying,out,this,calculation,is,.,.,.,enormously,complicated”.,The,complication,follows,from,the,range,of,competing,mechanisms,even,within,atoms,of,one,ele-,ment,,namely,the,bound-bound,mechanisms,of,decay,and,excitation,described,in,ref,[22,,§,3.2.1],,and,bound-free,of,recombination,and,photo-emission,,because,of,the,different,possible,degrees,of,ionisation,as,atomic,number,A,increases,and,because,the,proportion,of,atoms,in,each,ionisation,state,depends,on,the,proportions,in,the,others.,Ambartsumyan,[23,,§,5],explains,at,greater,length,the,calculation,in,thermal,equilibrium,of,the,proportion,of,different,ions,for,each,element,[23,,§,5.2],,then,the,bound-free,/,free-bound,coefﬁ-,cients,(§,5.3—§,5.5),and,free-free,(§,5.6).,In,[23,,§,24.1–24.2],there,is,a,discussion,of,metastable,28,states,,which,in,the,astrophysical,context,are,crucial,for,the,formation,of,forbidden,lines,in,nebu-,lae,,but,may,also,be,important,in,the,context,of,fusion,because,these,metastable,atomic,states,can,survive,for,many,seconds,at,low,densities,of,matter,and,of,radiation.,By,metastable,state,is,meant,that,no,transition,to,it,from,lower,energy,levels,of,the,electrons,is,possible,except,for,the,so-called,‘forbidden’,,less,probable,electric,quadrupole,interactions,from,the,quantum-mechanical,matrix,elements.,The,above,outlines,the,main,physics,issues.,From,O’Mullane’s,slides,at,the,2008,Summer,School,[24],,the,main,difference,between,astrophysics,and,fusion,application,seems,to,be,that,in,the,plasma,context,,if,it,is,used,,Saha’s,ionisation,formula,needs,modiﬁcation,by,the,Saha-,Boltzmann,deviation,factors,bn,or,‘b-factors’,ref,[24,,slide,21].,The,Zeeman,effect,is,also,neglected,,although,this,might,be,expected,to,be,important,,as,from,its,use,in,sunspot,observation,[25,,§,5.2],spectral,line,splitting,by,wavelengths,of,0.1,nm,is,expected.,The,key,observation,is,from,O’Mullane,[24],that,in,tokamak,modelling,,there,are,two,distinct,uses,for,atomic,data,-,(1),to,calculate,source,(loss),terms,for,species,time,evolution,equations,,and,(2),to,compute,synthetic,spectra,,ie.,intensity,as,a,function,of,frequency.,The,latter,(2),is,by,far,the,Indeed,,more,involved,but,it,is,only,critical,for,diagnosticians,working,with,particular,apparatus.,Golub,and,Pasachoff,[22,,§,3.3.2],go,on,to,argue,that,for,an,optically,thin,plasma,,the,radiation,(in,W/m3),could,be,expressible,as,simply,as,ER,=,nenpP,(T,),(124),where,np,accounts,for,the,number,density,of,the,plasma,ions,and,P,(T,),is,the,emitted,power,integrated,over,all,wavelengths,for,a,plasma,with,a,speciﬁed,mix,of,elements.,The,separate,functions,used,to,compute,P,(T,),depend,mainly,on,electron,temperature,with,a,weak,dependence,on,density.,The,form,of,P,(T,),as,a,result,of,the,integration,over,spectrum,always,seems,to,be,smooth.,It,would,seem,to,be,a,prime,candidate,for,precomputation,as,a,function,of,the,fractions,of,the,major,plasma,species,,and,could,be,approximated,very,efﬁciently,because,of,the,smoothness.,O’Mullane,[24],give,a,more,detailed,result,,namely,that,there,is,a,source/sink,term,for,electron,energy,of,form,SE,=,−ER,=,ne,,(cid:88),,Z=Z0(s),(cid:88),s,Z=0,P,ZnZ,−,I,Z,(cid:0)S,Z→ZpnZ,+,αZp→ZnZp(cid:1),,,(125),where,Z,is,charge,state,,the,sufﬁx,s,on,the,density,has,been,dropped,,Zm,=,Z,−,1,,Zp,=,Z,+,1,,and,where,Z0(s),is,the,number,of,charge,states,of,species,s,included,in,the,model.,It,may,be,inferred,that,S,Zm→Z,or,S,Z→Zp,=,ionisation,coefﬁcient,αZ→Zm,or,αZp→Z,=,partial,dielectronic,recombination,rate,coefﬁcient,P,Z,=,radiated,power,per,atom,of,nZ,I,Z,=,power,per,atom,released,in,dielectronic,recombination,(126),(127),(128),(129),where,the,coefﬁcients,,as,elsewhere,in,this,section,,are,expected,to,be,obtained,from,the,Atomic,Data,and,Analysis,Structure,ADAS,database,[26,,27].,The,data,requirements,for,this,look,relatively,modest,,assuming,the,coefﬁcients,for,each,species,and,charge,state,are,smooth,functions,of,29,temperature,only.,Thus,if,say,NT,≈,20,samples,specify,these,functions,and,Zsum,=,(cid:80),s,Z0(s),,then,the,total,number,of,coefﬁcients,required,could,be,estimated,as,Zsum,×,3,×,NT,≈,20,×,3,×,10,=,600,where,if,the,number,of,different,elements,present,Ns,=,10,,and,if,the,average,number,of,charge,states,¯NZ,=,2,,then,Zsum,=,Ns,¯NZ,≈,20.,In,another,case,of,interest,,a,calculation,might,include,only,two,or,three,extra,species,if,one,were,Tungsten,(W),,so,Ns,=,4,but,then,NZ,=,22,for,W,alone,if,Te,>,40,eV,[28].,The,number,densities,nZ,for,each,charge,state,may,be,straightforwardly,calculated,by,solving,a,transport,equation,for,each,isotope,ns,and,using,the,Saha-Boltzmann,formula,modiﬁed,with,(Further,,for,heavier,elements,a,mean,b-factors,to,determine,the,distribution,of,charge,states.,atomic,mass,may,be,used,to,avoid,separate,treatment,of,isotopic,species.),Much,more,serious,implications,for,computation,[24],,arise,in,the,time,evolution,equations,if,each,charge,state,is,treated,separately.,This,may,be,necessary,in,a,strong,electric,ﬁeld,because,each,different,ion,feels,a,different,electromagnetic,force.,An,ion,of,species,s,with,charge,state,Z,will,acquire,a,source,s,=,S,Zm→ZnenZm,−,(cid:0)αZ→Zm,+,S,Z→Zp(cid:1),nenZ,+,αZp→ZnenZp,SZ,where,again,the,sufﬁx,s,on,the,density,has,been,dropped.,Thus,the,demands,on,atomic,data,are,not,very,different,from,those,for,the,energy,equation,,but,since,the,total,cost,of,these,additional,computations,with,Z0(s),extra,species,will,scale,at,least,as,fast,as,Zsum,=,(cid:80),s,Z0(s),(inter-species,coupling,may,add,considerably,to,the,computational,expense),,rendering,negligible,the,cost,of,inputting,a,few,thousand,coefﬁcients,from,disc.,In,practice,a,useful,surrogate,is,produced,by,replacing,separate,ionisation,states,by,‘superstages’,(slide,17,of,ref,[29]),,where,one,superstage,corresponds,to,one,electron,shell,of,the,atom.,However,,even,the,smaller,number,of,7,superstages,required,for,W,might,double,or,treble,the,length,of,a,typical,computation.,(130),The,above,is,typically,as,much,detail,as,is,sensible,to,consider,under,heading,(1).,If,detailed,diagnostics,under,(2),are,required,,the,generalised,collisional-radiative,(GCR),model,[30],gives,an,idea,of,the,computational,demands.,GCR,modelling,requires,each,metastable,state,to,be,considered,separately,,since,each,has,a,separate,ﬁnite,lifetime.,It,helps,that,the,transport,of,each,atom,in,the,state,is,presumably,the,same,,but,even,so,there,is,a,need,to,solve,a,rate,equation,for,metastable,state,density,at,sample,points,throughout,the,computational,domain,The,source,terms,are,complicated,,namely,for,the,metastable,state,labelled,ρ,SZ,ρ,/ne,=,+,+,+,(cid:88),σ,(cid:88),µ,(cid:88),ν,(cid:88),σ,X,Z→Z,σ→ρ,nσ,−,(cid:88),X,Z→Z,ρ→σ,nρ,µ→ρ,nZm,S,Zm→Z,σ,µ,−,(cid:88),S,Z→Zp,ρ→ν,nρ,αZp→Z,ν→ρ,nZp,ν,−,ν,(cid:88),αZ→Zm,ρ→µ,nρ,QZ→Z,σ→ρ,nσ,−,µ,(cid:88),σ,QZ→Z,ρ→σ,nρ,(131),(132),(133),(134),where,the,superﬁx,Z,as,well,as,the,sufﬁx,s,on,the,density,has,been,dropped,and,the,new,symbols,are,X,Z→Z,QZ→Z,σ→ρ,=,generalised,collisional-radiative,(GCR),excitation,coefﬁcient,σ→ρ,=,parent-metastable,cross-coupling,coefﬁcient,(135),(136),30,Note,that,the,expressions,in,both,ref,[30,,eq.,(9)],and,ref,[24,,slide,41],appear,to,contain,typos,,and,that,the,meanings,of,X,and,Q,have,swapped.,Each,of,the,new,terms,contains,approximately,8MZ,coefﬁcients,where,MZ,is,the,number,of,metastable,states,for,species,s,(which,includes,the,ground,state).,It,may,be,inferred,from,refs,[24,,29],that,the,number,of,metastable,states,for,a,given,ionisation,Z,is,relatively,small,(slide,9,of,ref,[29],indicates,that,all,ionisation,states,for,Oxygen,have,MZ,≤,4;,slide,19,suggests,MZ,≤,6,for,W,when,Te,<,100,eV).,The,coefﬁcients,X,,,S,,Q,in,Equation,(131),are,functions,of,electron,density,as,well,as,temperature,so,may,require,at,least,100,sample,points,to,specify,,hence,the,total,data,can,be,estimated,as,Zsum×MZ,×8MZ,×100.,However,FISPACT-II,[31],experience,with,rate,equations,indicates,the,cost,of,these,additional,computations,with,MZ,metastables,far,exceeds,the,cost,of,inputting,of,order,ten,or,so,thousand,coefﬁcients,from,disc.,Where,the,demands,of,data,might,become,important,is,in,the,translation,of,the,nZ,σ,into,spectral,lines.,First,the,regular,excited,states,,because,they,equilibrate,on,the,usual,atomic,timescales,which,are,negligible,compared,to,plasma,timescales,,are,calculated,using,a,purely,algebraic,rela-,tion,[30,,eq.,(5)],,nZ,i,/ne,=,(cid:88),X,FiσnZ,σ,+,I,FiµnZm,µ,+,(cid:88),(cid:88),RFiνnZp,ν,(137),σ,µ,ν,where,X,I,RFiσ,are,the,coefﬁcients,of,excitation,,ionisation,and,recombination,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,,each,is,a,function,of,ne,and,Te,with,correspond-,ing,storage,requirement,of,order,100.,Equation,(137),requires,Zsum,×,MS,×,MZ,F,coefﬁcients,where,MS,is,the,number,of,states,,which,is,potentially,inﬁnite,,and,indeed,in,practice,could,be,as,large,as,≈,500,,necessitating,the,use,of,‘bundling’,of,the,higher,energy,states,to,reduce,the,number,to,manageable,proportions,,say,10,[30].,Next,,as,explained,in,the,opening,paragraph,,to,each,state,there,corresponds,a,description,of,its,spectrum,,which,may,contain,many,separate,lines,,each,described,by,its,wavelength,,relative,amplitude,and,a,proﬁle,shape,which,may,require,several,further,parameters,to,describe.,Mitigating,the,demand,for,coefﬁcient,data,,is,the,fact,that,the,diagnostics,need,only,be,computed,intermittently.,To,treat,atomic,physics,UQ,in,a,later,stage,of,NEPTUNE,,a,Monte-Carlo,calculation,might,be,considered,,involving,all,the,different,interactions,between,all,the,metastable,states,where,the,Maxwellian,assumption,is,relaxed,,posing,a,multiscale,multiphysics,problem.,However,,the,validity,of,this,approach,requires,further,consideration,as,Henderson,et,al,[32],also,indicates,that,even,as,recently,as,2017,,errors,of,30,%,were,present,in,important,coefﬁcients,,although,the,discrepancies,have,now,been,reduced,to,approximately,5,%,[33].,B,Annex,B:,Index,of,Mathematics,31,Table,1:,TABLE,OF,MATHEMATICAL,SYMBOLS,If,no,units,are,given,,then,quantity,is,dimensionless,,or,if,the,units,are,given,as,?,,then,the,dimensions,depend,on,context.,Gener-,ally,,the,usage,of,symbols,tries,to,follow,that,from,the,Plasma,Formulary,[7],,in,SI,units,,with,temperatures,speciﬁed,as,kT,which,returns,J.,The,Formulary,also,give,the,fundamental,dimensions,of,the,SI,units,,which,should,enable,checking,of,dimensional,consistency,of,equations,,eg.,magnetic,ﬁeld,in-,duction,is,in,Tesla,(T,),whence,the,fundamental,dimension,expression,gives,T,=,kgs−1C−1.,Note,that,the,symbols,are,sorted,by,font,as,well,as,alphabet,,so,that,boldface,symbols,appear,immediately,after,‘b’,(backslashes,ignored).,Units,m3s−1,m3s−1,T,T,T,T,Description,atomic,mass,of,the,ion,as,sufﬁx,is,species,label,or,index,perturbation,amplitude,partial,dielectronic,recombination,rate,coefﬁcient,partial,dielectronic,recombination,rate,coefﬁcient,used,to,make,B,dimensionless,average,number,of,charge,states,amplitude,of,the,imposed,magnetic,ﬁeld,amplitude,of,the,imposed,toroidal,magnetic,ﬁeld,(Glossary),Ratio,of,plasma,pressure,to,pressure,in,mag-,netic,ﬁeld,as,sufﬁx,is,species,label,acceleration,experienced,by,a,particle,magnetic,ﬁeld,unit,vector,giving,the,direction,of,the,magnetic,ﬁeld,V,m−1,electric,ﬁeld,m−2,modiﬁed,electric,ﬁeld,N,force,vector,ms−1,initial,ﬂuid,velocity,‘charge,exchange’,perpendicular,ﬂuid,velocity,component,ms−1,ms−1,‘E,cross,B’,perpendicular,ﬂuid,velocity,component,ms−1,velocity,of,the,electrons,ms−1,velocity,of,the,ion,species,‘grad,B’,perpendicular,ﬂuid,velocity,component,for,electrons,ms−1,ms−1,‘grad,B’,perpendicular,ﬂuid,velocity,component,for,ions,ms−1,‘diffusive’,perpendicular,ﬂuid,velocity,component,ms−1,velocity,of,species,α,ms−1,velocity,of,the,electrons,ms−1,ﬂuid,velocity,component,along,ﬁeldline,ms−1,generic,velocity,ms−1,velocity,of,the,ion,species,ms−1,ﬂuid,velocity,component,normal,to,ﬂux,surface,m2s−1,T,32,Symbol,A,α,αn,αZp→Z,αZ→Zm,B0,¯NZ,B,=,|B|,BT,β,β,a,=,d2x/dt2,B(x,,t),b,E(x,,t),E+,F,u0,ucx,uE×B,ue,ui,u∇Be,u∇Bi,udiﬀ,vα,ve,v(cid:107),v,vi,v⊥,√,kT0,(cid:113),kTi+ZikTe,mi,v∧,x,bn,C0,=,cp,cs,=,CS,δpi,δ,δs,2,(δe,+,δi),δe,δ,=,1,δi,Dn,|e|,eijk,(cid:15)0,(cid:15)r,=,Ls/(t0C0),ηB,η,=,ηB/µ0,f0,fα,fe,fi,f,E,F,E,f,n,F,n,f,u,F,u,f,(x,,v,,t),Hα,i,i,i,IH,I,Fiσ,I(ψ),=,BT,/R,ﬂuid,velocity,component,in,ﬂux,surface,normal,to,ﬁeld,di-,rection,position,‘b-factors’,ref,[24,,slide,21],used,to,make,velocities,dimensionless,speciﬁc,heat,at,constant,pressure,acoustic,velocity,sound,speed,coefﬁcient,in,radiation,equation,stress,tensor,Kronecker,delta,function,(Glossary),Magnetisation,parameter,,species,s,gyroradius,normalised,to,L,energy,ﬂux,factor,at,boundary,of,the,electrons,energy,ﬂux,factor,at,boundary,of,‘mean’,species,energy,ﬂux,factor,at,boundary,of,the,ion,species,neutral,diffusion,coefﬁcient,absolute,value,of,the,charge,on,the,electron,weighted,integral,of,triple,products,of,Ψi,of,the,ion,species,permittivity,of,free,space,scale,factor,for,transient,term,plasma,resistivity,after,Braginskii,plasma,resistivity,,as,diffusivity,initial,distribution,function,of,the,electrons,distribution,function,of,species,α,distribution,function,of,the,electrons,distribution,function,of,the,ion,species,ﬂux,term,(ﬁeldline,integrated,source),for,plasma,energy,ﬂux,term,(ﬁeldline,integrated,source,divided,by,ﬁeld),for,plasma,energy,ﬂux,term,(ﬁeldline,integrated,source),for,plasma,number,density,ﬂux,term,(ﬁeldline,integrated,source,divided,by,ﬁeld),for,plasma,number,density,ﬂux,term,(ﬁeldline,integrated,source),for,plasma,momentum,ﬂux,term,(ﬁeldline,integrated,source,divided,by,ﬁeld),for,plasma,momentum,generic,distribution,function,Hamiltonian,for,species,α,as,sufﬁx,denotes,ions,as,sufﬁx,denotes,regular,excited,state,as,sufﬁx,generic,label,Hydrogen,reionisation,potential,as,deﬁned,in,ref,[13],coefﬁcient,of,ionisation,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,function,giving,the,the,toroidal,ﬁeld,as,a,function,of,ψd,ms−1,m,ms−1,Jkg−1K−1,ms−1,ms−1,N,m−2,m2s−1,C,F,m−1,Ωm,m2s−1,m−6s3,m−6s3,m−6s3,m−6s3,m−1s−2C,m−3C,m−2s−1C,m−6s3,eV,T,m−1,33,I,Z,j,jext(R,,Z),jφ,k,k,κα,κe(cid:107),κe⊥,κi(cid:107),κi⊥,κ,=,kc/ρmcp,kB,kc,Kcx,(ni,,Ti),Ki,KM,Kr,kT0,kTd,kw,λsmf,p,λ,λD,Λ,Λb,(cid:104)σv(cid:105)CX,(cid:104)σv(cid:105)ION,(cid:104)σv(cid:105)REC,(cid:104)σv(cid:105),Ls,m,M0,M1,mα,Eα,Ee,Ei,ER,F,Fα,K(cid:107),K,?,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,JK−1,Jm−1s−1K−1,m3s−1,m3s−1,?,W,√,√,KM,T0,,KM,Td,is,an,power,per,atom,released,in,dielectronic,recombination,as,sufﬁx,is,generic,label,electric,current,density,induced,in,plasma,by,external,coils,Am−2,Am−2,φ−,or,toroidal,component,of,plasma,current,density,as,sufﬁx,is,generic,label,chosen,to,scale,so,that,kT0,,kTd,is,an,energy,thermal,diffusivity,of,species,α,parallel,thermal,diffusivity,of,electrons,perpendicular,thermal,diffusivity,of,electrons,parallel,thermal,diffusivity,of,ions,perpendicular,thermal,diffusivity,of,ions,thermal,diffusivity,tensor,of,solid,Boltzmann’s,constant,thermal,conductivity,tensor,reaction,rate,of,charge,exchange,reactions,ionization,reaction,rate,chosen,as,kB/mi,or,|e|/mi,so,that,ion,speed,recombination,reaction,rate,T0,in,energy,units,Td,in,energy,units,wavenumber,vector,(Glossary),Mean,free,path,of,particle,species,s,arbitrary,quantity,(Glossary),Debye,lengthscale,above,which,local,electro-,static,ﬂuctuations,due,to,presence,of,discrete,charged,par-,ticles,are,negligible,Coulomb,logarithm,sheath,potential,drop,normalized,to,Te,reaction,rate,for,charge-exchange,reaction,rate,for,ionisation,reaction,rate,for,recombination,generic,reaction,rate,length,of,ﬁeldline,species,particle,mass,Mach,number,at,s,=,0,boundary,Mach,number,at,s,=,1,boundary,kg,mass,of,species,α,Jm−3,energy,of,species,α,Jm−3,energy,of,the,electrons,Jm−3,energy,of,the,ion,species,W,m−3,total,plasma,radiation,generic,coefﬁcient,of,excitation,,ionisation,or,recombination,m3s−1,m−6s3,functional,of,moments,of,species,α,m−1s−1,parallel,thermal,conductivity,of,plasma,m−1s−1,thermal,conductivity,of,plasma,m3s−1,J,J,m−1,m,?,m,eV,m3s−1,m3s−1,m3s−1,m3s−1,m,kg,34,K⊥,K∧,L7,Q,QZ→Z,σ→ρ,S,S,Zm→Z,S,Z→Zp,T,X,X,Z→Z,σ→ρ,me,mi,mn,mp,Ms,MS,µcx,=,ωc/νcx,MZ,n,N,n0,nB,=,N/B,ne,ni,nj(x,,t),nn,np,ns,Ns,NT,ν,να,νcx,=,Kcxnn,νe(cid:107),νe⊥,νi(cid:107),νi,thermal,conductivity,of,plasma,perpendicular,to,ﬁeld,and,ﬂux,surface,thermal,conductivity,of,plasma,perpendicular,to,ﬁeld,in,ﬂux,surface,7-D,Lie,derivative,(space,,velocity-space,and,time,make,up,the,3,+,3,+,1,=,7,dimensions),coefﬁcient,in,radiation,equation,parent-metastable,cross-coupling,coefﬁcient,coefﬁcient,in,radiation,equation,ionisation,coefﬁcient,ionisation,coefﬁcient,generic,tensor,coefﬁcient,in,radiation,equation,generalised,collisional-radiative,(GCR),excitation,coefﬁ-,cient,mass,of,electron,mass,of,ion,species,particle,neutral,species,particle,mass,mass,of,proton,Mach,Number,,allowed,to,take,either,sign,number,of,energy,states,of,an,atom,measures,strength,of,magnetization,with,respect,to,charge,exchange,reaction,number,of,metastable,states,for,species,s,(which,includes,the,ground,state),number,density,number,density,,may,be,in,units,of,1018,m−3,initial,number,density,number,density,divided,by,ﬁeld,strength,number,density,of,the,electrons,number,density,of,the,plasma,ions,member,of,the,set,of,deterministic,coefﬁcients,of,the,“ran-,dom,trial,basis”,neutral,density,number,density,of,the,plasma,ions,number,density,of,isotope,s,number,density,of,isotope,s,number,of,samples,in,temperature,used,to,deﬁne,typically,a,crossection,in,the,ADAS,database,[26,,27],plasma,kinematic,viscosity,kinematic,viscosity,of,species,α,charge,exchange,‘frequency’,parallel,kinematic,viscosity,of,electrons,perpendicular,kinematic,viscosity,of,electrons,parallel,kinematic,viscosity,of,ions,ion,kinematic,viscosity,m−1s−1,m−1s−1,s−1,m3s−1,m3s−1,m3s−1,m3s−1,m3s−1,?,m3s−1,m3s−1,kg,kg,kg,kg,m−3,m−3,m−3,m−3T,−1,m−3,m−3,m−3,m−3,m−3,m−3,m2s−1,m2s−1,s−1,m2s−1,m2s−1,m2s−1,m2s−1,35,νi⊥,νs,ν∗,s,nZ,NZ,nZ,i,nZ,σ,ωci,=,ZieB/mi,ωce,=,|e|B/me,p,pα,(∂f,/∂t)C,PC,pe,φ,Φ,pi,p(ψ),P,r,P,rM,ψ,Ψi,P,(T,),P,Z,Q(cid:107),qα,qe,Q(fα,,fβ),QH,qi,qi,qi,Qie,R,R,ρ,ρts,ρm,RFiσ,s(cid:107),s,s,perpendicular,kinematic,viscosity,of,ions,(Glossary),Collision,frequency,for,species,s,(Glossary),Normalised,collision,frequency,for,species,s,number,density,for,charge,state,Z,number,of,charge,states,for,an,ion,species,number,density,for,charge,state,Z,,excited,state,i,number,density,for,charge,state,Z,,metastable,state,σ,ion,cyclotron,frequency,electron,cyclotron,frequency,pressure,pressure,of,species,α,source,in,Boltzmann,due,to,inter-particle,interactions,number,of,modes,in,basis,for,polynomial,chaos,pressure,of,the,electrons,angle,in,toroidal,direction,electric,potential,pressure,of,the,ion,species,function,giving,the,pressure,as,a,function,of,ψ,of,the,mag-,netic,ﬂux,Prandtl,number,magnetic,Prandtl,number,poloidal,magnetic,ﬂux,ith,member,of,a,set,of,basis,functions,,dimensional,Hermite,polynomials,emitted,power,integrated,over,all,wavelengths,radiated,power,per,atom,of,nZ,combined,energy,ﬂux,at,a,boundary,charge,on,a,particle,of,species,α,charge,on,an,electron,Boltzmann,collision,operator,cooling,rate,due,to,excitation,as,deﬁned,in,ref,[13],charge,on,an,ion,electron,energy,ﬂux,ion,energy,ﬂux,collisional,energy,equipartition,term,cylindrical,coordinate,recycling,coefﬁcient,as,sufﬁx,is,label,of,metastable,state,(Glossary),Gyroradius,or,Larmor,radius,of,orbit,of,charged,particle,of,species,s,about,magnetic,ﬁeld,direction,mass,density,of,the,mediuma,coefﬁcient,of,metastable,state,σ,to,regular,excited,state,i,arclength,along,ﬁeldline,as,sufﬁx,,species,label,parameterises,distance,along,the,ﬁeldline,0,≤,s,≤,1,the,transition,from,recombination,for,typically,multi-,36,m2s−1,s−1,m−3,m−3,m−3,s−1,s−1,N,m−2,N,m−2,m−6s2,N,m−2,radians,c,V,N,m−2,N,m−2,T,m2,W,m3,W,Jm−2s−1,C,C,m−6s2,Km−3s−1,C,Jm−2s−1,Jm−2s−1,kgm−1s−3,m,m,kgm−3,m,Sα,SC,Sexp(x,,v,,t),n,T,u,si,σ,σ,σC,σE,σE,σI,SE,SE,e,SE,i,SE,n,SE,⊥e,SE,⊥i,SE,⊥n,Sn,Sn,e,Sn,i,Sn,n,Sn,⊥n,Sn,⊥,S⊥n,Su,Su,n,Su,⊥n,SZ,ρ,SZ,s,t,T,t0,T0,Tα,τα,τe,τi,τEe,τEi,τne,τni,Td,=,Ti,+,Te,source,term,in,Boltzmann,equation,for,species,α,total,source,term,in,Boltzmann,equation,explicit,source,term,in,Boltzmann,equation,neutral,density,neutral,temperature,neutral,velocity,arclength,parameter,for,boundary,(i,=,1,inner,,i,=,2,outer),as,sufﬁx,labels,metastable,state,reaction,cross-section,reaction,rate,for,charge-exchange,cooling,rate,due,to,excitation,electrical,conductivity,reaction,rate,for,ionisation,source,term,in,plasma,energy,equation,energy,density,source,term,for,electrons,energy,density,source,term,for,ions,source,term,in,neutral,energy,equation,energy,cross-ﬁeld,source,term,for,electrons,energy,cross-ﬁeld,source,term,for,ions,energy,cross-ﬁeld,source,term,for,neutrals,source,term,in,plasma,density,equation,number,density,source,term,for,electrons,number,density,source,term,for,ions,source,term,in,neutral,density,equation,number,density,cross-ﬁeld,source,term,for,neutrals,number,density,cross-ﬁeld,source,term,for,plasma,generic,cross-ﬁeld,source,term,for,neutrals,source,term,in,plasma,momentum,equation,source,term,in,neutral,momentum,equation,momentum,cross-ﬁeld,source,term,for,neutrals,particle,source,for,ion,of,metastable,state,σ,(species,s),with,charge,state,Z,particle,source,for,ion,of,species,s,with,charge,state,Z,time,usually,in,seconds,plasma,temperature,characteristic,timescale,usually,in,seconds,initial,temperature,temperature,of,species,α,collision,or,relaxation,time,of,species,α,electron,collision,or,relaxation,time,ion,species,collision,or,relaxation,time,loss,time,of,energy,density,for,electrons,loss,time,of,energy,density,for,ions,loss,time,of,number,density,for,electrons,loss,time,of,number,density,for,ions,combined,temperature,of,the,electrons,and,ions,m−6s2,m−6s2,m−6s2,m2,Ω−1m−1,kgm−1s−3,kgm−1s−3,kgm−1s−3,kgm−1s−3,kgm−1s−3,kgm−1s−3,kgm−1s−3,m−3s−1,m−3s−1,m−3s−1,m−3s−1,m−3s−1,m−3s−1,?,kgm−2s−2,kgm−2s−2,kgm−2s−2,m−3s−1,m−3s−1,s,eV,s,eV,eV,s,s,s,s,s,s,s,eV,37,Te,TH,θ,θ,Ti,˜b,=,B/B0,U,Ud,=,Ls/t0,uR,=,1/R,x,xα,xe,=,ωceτe,xi,=,ωciτi,ξ(θ),X,Fiσ,y,z,Z,Z,Z,Z0(s),ζ,ζ,=,−φ,Zi,Zm,=,Z,−,1,Zp,=,Z,+,1,Zsum,=,(cid:80),electron,temperature,the,Hydrogen,reionisation,potential,angular,coordinate,random,parameter,0,≤,θ,≤,1,ion,temperature,magnetic,ﬁeld,velocity,component,(ﬂow),along,streamline,speed,measuring,the,importance,of,the,transient,term,Radial,component,of,Grad-Shafranov,‘ﬂow’,Cartesian,coordinate,collisionality,factor,of,species,α,collisionality,factor,of,electrons,collisionality,factor,of,ions,multi-dimensional,random,variable,with,a,speciﬁc,probabil-,ity,distribution,as,a,function,of,the,random,parameter,0,≤,θ,≤,1,coefﬁcient,of,excitation,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,Cartesian,coordinate,Cartesian,coordinate,Cartesian,coordinate,charge,state,of,the,ion,cylindrical,coordinate,number,of,charge,states,of,species,s,included,in,the,model,magnetic,Prandtl,number,as,deﬁned,in,Cambridge,toroidal,angle,coordinate,charge,state,where,Z,is,ion,charge,state,where,Z,is,ion,charge,state,eV,radians,c,eV,ms−1,ms−1,m,m,m,m,m,radians,c,s,Z0(s),where,Z0,is,number,of,charge,states,of,species,s,38

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