CD-EXCALIBUR-FMS0021-1.30-M1.2.1_YearOneSummaryReport
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ExCALIBUR,Equations,for,ExCALIBUR/NEPTUNE,Proxyapps,Version,1.30,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:,1.30,1,June,2023,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Name,and,Department,Signature,Date,Prepared,By:,Wayne,Arter,1,June,2023,BD,Reviewed,By:,Rob,Akers,1,June,2023,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,1,June,2023,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).,The,question,of,what,physical,units,to,be,employed,is,discussed,later,in,this,section,,see,Sec-,tion,1.3.,Supporting,information,regarding,Braginskii’s,transport,coefﬁcients,for,plasma,in,a,strong,magnetic,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,2,to,long,mean-free-path,,has,been,well-studied,because,of,the,application,to,the,space,vehicle,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,3,right,for,example,in,modelling,MAST-U,divertor,,and,other,workers,might,be,drawn,in,to,add,additional,physical,effects,to,this,end.,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.,4,3-4,3-D,model,of,neutral,gas,&,impurities.,This,is,now,at,full,dimensional,complexity,,incorporat-,ing,selected,ideas,on,optimal,usage,of,particles.,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.,1.3,Physical,Units,and,Internal,Scalings,As,far,as,is,compatible,with,numerical,stability,and,accuracy,,NEPTUNE,will,employ,SI,units,,with,the,main,exception,that,temperature,may,be,speciﬁed,in,electron-Volts,eV,.,Algorithmically,the,ex-,ception,is,easily,handled,by,always,writing,kT,for,temperature,T,where,the,dimensional,quantity,k,ensures,that,the,product,kT,has,units,of,energy.,(So,k,=,|e|,if,T,is,given,in,eV,or,Boltzmann’s,constant,if,T,is,measured,in,Kelvin.),The,units,system,will,be,referred,to,as,‘SIx2’,units,,since,a,further,practical,exemption,arises,from,the,need,to,interface,with,CAD,systems,,which,typically,work,in,millimetres,mm,,where,the,recommendation,is,to,scale,geometry,to,metres,as,soon,as,is,practicable.,NEPTUNE,will,not,accept,models,expressed,in,dimensionless,units,because,of,the,potential,for,confusion,when,a,range,of,different,physical,time,and,length-scales,are,involved.,It,is,of,course,,entirely,reasonable,for,theorists,to,pick,scalings,appropriate,to,their,particular,problem,,and,indeed,to,incorporate,purely,numerical,factors,in,the,scalings,to,simplify,the,coefﬁcients,of,the,equations,,and,hence,their,analytic,work.,Why,this,causes,problems,for,NEPTUNE,can,be,understood,by,considering,the,situation,when,a,user,or,developer,wants,to,introduce,an,additional,physical,effect,not,considered,by,the,original,theorist,,for,then,the,newcomer,needs,to,know,precisely,what,was,done,before,,and,then,has,to,scale,terms,representing,the,new,effect,before,their,introduction,into,the,software.,It,is,worthwhile,emphasising,the,word,‘precisely’,-,the,inclusion,of,purely,numerical,factors,in,the,scaling,may,well,introduce,ambiguity.,Moreover,,there,are,only,a,small,number,of,fundamental,dimensional,quantities,,namely,time,,length,,charge,and,mass,,so,a,limited,number,of,ways,in,which,terms,can,be,non-dimensionalised,unambiguously.,(Temperature,in,Kelvin,and,angular,quantities,are,dimensionless,fundamental,quantities.),An,example,of,the,problem,of,ambiguity,in,the,scaling,,occurs,when,augmenting,Magnetohydro-,dynamics,(MHD),with,electrostatic,phenomena.,In,MHD,,scaling,magnetic,ﬁeld,by,a,reference,value,B0,causes,no,difﬁculty,,but,as,soon,as,electric,charge,is,introduced,and,scaled,,typically,by,the,charge,on,the,electron,|e|,,since,in,SI,units,magnetic,induction,has,unit,1,T=,kgs−1C−1,,there,may,arise,the,ambiguity,as,to,whether,B0,or,say,mu/(|e|ts),is,appropriate,for,non-dimensionalising,a,new,magnetic,ﬁeld,term.,Internally,,however,,NEPTUNE,needs,to,scale,ﬁelds,so,that,their,values,are,expressible,efﬁciently,in,ﬂoating,point,,meaning,that,large,exponents,of,either,sign,are,to,be,avoided.,The,suggested,5,limit,is,to,exponents,smaller,than,38,in,absolute,value,,a,criterion,set,by,the,standard,single,pre-,cision,representation,of,ﬂoating-point,numbers.,This,is,expected,to,be,best,achieved,by,non-,dimensionalising,,but,‘under,the,hood’,,ie.,with,minimal,user,interaction,,as,explained,in,Section,2,below.,There,will,be,allowed,one,permissible,exception,to,the,rule,that,no,purely,numerical,factors,should,appear,in,scalings,,namely,to,introduce,a,factor,Nref,to,reduce,number,densities,to,more,reasonable,magnitudes,,and,a,factor,wref,to,allow,for,the,representation,of,many,plasma,parti-,cles,by,a,single,computational,super-particle,,in,effect,to,allow,for,the,smallness,of,an,individual,particle’s,charge,and,mass.,With,that,,the,key,scalings,are,1.,time,in,units,of,ts,,2.,length,in,units,of,Ls,,3.,mass,in,units,of,mu,,the,atomic,mass,unit,4.,electric,charge,in,units,of,|e|,=,|qe|,,the,absolute,value,of,the,charge,on,the,electron,5.,number,densities,such,as,ne,made,dimensionless,with,respect,to,an,additional,factor,of,Nref,as,well,as,L−D,s,where,D,is,the,spatial,dimensionality,of,the,problem,6.,superparticles,carry,weight,wref,,,eg.,a,charge,wref,qe,is,carried,by,each,‘super-’electron,The,above,scalings,combine,to,give,an,energy,scaling,of,mu(Ls/ts)2,,from,which,follows,a,tem-,perature,scale,given,by,kTs,=,mu(Ls/ts)2.,Scalings,also,follow,for,the,electric,and,magnetic,ﬁelds.,In,practice,,the,ratio,of,absolute,electron,charge,to,atomic,mass,unit,((cid:39),proton,mass),ap-,pears,frequently,enough,to,warrant,its,own,symbol,KM,=,|e|,≈,108,,eg.,the,electric,ﬁeld,unit,mu,becomes,U,2,s,LsKM,and,the,magnetic,ﬁeld,unit,becomes,Us,LsKM,.,The,NEPTUNE,website,has,a,section,on,“Physical,properties,of,the,edge,plasma”,which,gives,likely,values,for,ts,and,Ls.,Equivalently,Ls,and,the,velocity,scale,Us,may,be,speciﬁed,instead,,thus,depending,on,the,relevant,physics,Us,=,103,−,107,ms−1,,Ls,=,10−5,−,10,m.,These,two,scalings,could,be,set,by,the,user,(together,with,Nref,and,wref,),,once,for,a,given,simulation.,In,keeping,with,the,application,to,the,Exascale,,the,pure,number,Nref,=,1018,,whilst,taking,wref,=,1010,seems,most,convenient.,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.,6,2.1,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,=,,,T⊥,−T∧,T⊥,T∧,0,0,,,0,0,T(cid:107),so,that,T,·,v,=,T(cid:107)v(cid:107),+,T∧v∧,+,T⊥v⊥,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,(4),(5),(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,ωce,=,e,me,·,B,Ki(cid:107),=,3.9,N,kTi,mi,τi,Ki⊥,=,2,N,kTi,mi,τi,·,1,(ωciτi)2,7,(7),(8),(9),(10),where,the,ion,cyclotron,frequency,ωci,=,ZeB,mi,=,e,mp,·,ZB,A,(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,kTe,me,τe,1,x2,e,νi(cid:107),=,0.96N,kTiτi/(N,mi),=,0.96,kTi,mi,τi,νi⊥,=,0.3N,kTiτi/(N,mi),1,x2,i,=,0.3,kTi,mi,τi,1,x2,i,(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,(kTe)3/2,Z2N,Λ,=,3.44,×,10−7,(Te)3/2,Z2(N/1018)Λ,√,τi,=,12,√,(cid:15)2,0,mp,e4,π3,√,(kTi)3/2,Z4N,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.,8,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,√,(24),(25),√,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,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,(22),(23),(26),(27),(28),(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,(cid:19)2,(cid:18),xi,xe,=,2me,Z2Amp,(cid:18),Ti,Te,(cid:19)3,9,(31),It,also,follows,that,κe⊥,κi⊥,=,0.078,(cid:18),Ti,Te,(cid:19)1/2,1,√,A,(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,(viscous),Prandtl,number,is,P,r,=,νi⊥,κi⊥,=,0.23,(35),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.,2.4,Using,Braginskii,and,Other,Coefﬁcients,2.4.1,Introduction,to,Scaling,The,Branginskii,coefﬁcients,might,be,used,in,a,multispecies,context,over,a,range,of,different,timescales.,Ultimately,ﬁnite,element,codes,,indeed,most,numerical,schemes,,depend,on,solution,of,a,linear,algebra,problem,Ax,=,b,(36),where,x,are,ﬁeld,values,to,be,determined,,and,A,and,b,follow,from,the,model.,Take,the,simplest,case,of,a,2,×,2,model,A,=,(cid:19),(cid:18)a11,a12,a21,a22,,,b,=,(b1,b2)T,,,x,=,(x1,x2)T,(37),where,the,demands,of,multispecies,and,multiscale,may,lead,to,widely,different,values,of,all,coef-,ﬁcients,and,unknowns.,To,make,these,values,comparable,,which,is,desirable,from,the,numerical,standpoint,,exploit,the,fact,that,it,is,possible,to,multiply,each,each,equation,separately,without,10,changing,the,value,of,the,solution,x.,Further,,provided,the,coefﬁcients,aij,are,appropriately,ad-,justed,,each,component,xj,of,x,may,be,scaled,separately.,Thus,and,where,(cid:18)a11,a12,a21,a22,(cid:19),(cid:18)x1,x2,(cid:19),=,(cid:19),(cid:18)b1,b2,(cid:18)s1a11t1,s2a21t1,s1a12t2,s2a22t2,(cid:19),(cid:18)˜x1,˜x2,(cid:19),=,(cid:19),(cid:18)˜b1,˜b2,˜x,=,(x1/t1,x2/t2)T,,,˜b,=,(s1b1,s2b2)T,(38),(39),(40),give,the,same,x,,although,likely,by,solving,equations,with,very,different,sized,coefﬁcients,for,very,different,sized,unknowns,˜x.,This,approach,is,written,more,elegantly,in,matrix,notation,as,where,S,and,T,are,diagonal,matrices,SAT,−1T,˜x,=,S,˜b,S,=,(cid:18)s1,0,(cid:19),0,s2,,,T,=,(cid:18)1/t1,0,(cid:19),0,1/t2,(41),(42),(so,that,T,may,be,viewed,as,an,elementary,preconditioner).,Scaling,each,row,of,the,matrix,and,each,unknown,becomes,very,expensive,for,a,nonlinear,and/or,timestepping,problem,where,A,and,b,are,large,and,continually,changing.,Thus,it,is,helpful,on,this,account,only,to,scale,by,equation,subsystem,and,by,ﬁeld,,where,by,subsystem,is,meant,eg.,momentum,balance,for,species,X,,and,ﬁeld,is,meant,say,number,density,of,the,coupled,species,Y.,Hence,it,is,convenient,to,regard,x1,as,corresponding,to,an,entire,vector,of,discrete,values,of,the,number,density,of,species,X,and,x2,similarly,to,represent,the,density,of,species,Y,,so,that,the,coefﬁcients,aij,for,each,i,and,each,j,separately,represent,a,possibly,very,large,matrix.,Such,a,grouping,is,convenient,for,making,the,equations,dimensionless,,in,that,s1,may,compensate,for,the,dimensions,of,the,equation,and,s2,for,the,dimensions,of,a11x1,+,a12x2,=,b1,a21x1,+,a22x2,=,b2,(43),(44),whereas,tj,may,serve,make,dimensionless,xj,for,each,j,=,1,,2.,There,should,be,a,reduced,,hopefully,avoidable,,need,to,rescale,equations,as,ﬁelds,evolve,,and,properly,dimensioned,values,should,be,easily,restored,as,needed.,2.4.2,Fluid,Models,In,the,above,context,,there,is,no,need,,having,evaluated,the,Braginskii,coefﬁcients,to,convert,them,to,non-dimensional,form.,Selected,terms,from,the,energy,evolution,equation,demonstrate,how,the,above,scalings,are,to,implemented,in,practice.,For,demonstration,purposes,,the,factor,of,11,B,can,be,omitted,from,System,2-3,[12].,The,following,terms,in,the,energy,evolution,equation,are,representative,∂,∂t,(cid:18),3,2,(cid:19),(N,kTd),+,.,.,.,=,N,∂,∂s(cid:107),κe(cid:107),∂kTe,∂s(cid:107),+,.,.,.,(45),·,L3,The,overall,scaling,of,this,equation,to,render,its,terms,dimensionless,is,evidently,ts,s,Nref,kTs,unknowns,to,be,solved,for,are,N,and,Td,,which,have,respectively,dimensions/scalings,Nref,and,Ts.,L3,s,The,time,dependent,term,in,Equation,(45),involves,both,unknowns,,hence,needs,re-expressing,as,and,the,∂,∂t,(cid:18),3,2,(cid:19),(N,kTd),=,3,2,N,k,∂Td,∂t,+,3,2,kTd,∂N,∂t,(46),2,N,k,and,3,so,that,3,2,kTd,are,equivalents,of,coefﬁcients,aT,and,aN,of,matrix,A.,The,partial,deriva-,tive,∂/∂t,leads,to,discrete,operators,˜∆t/∆t,where,˜∆t,is,a,(dimensionless),difference,operator,so,the,scaling,factorisations,have,and,3,2,N,k,∆t,3,2,kTd,∆t,˜∆t(T,),→,L3,sts,Nref,kTs,·,3,2,N,k,∆t,·,Ts,˜∆t,(cid:19),(cid:18),Td,Ts,˜∆t(N,),→,L3,sts,Nref,kTs,·,3,2,kTd,∆t,·,Nref,L3,s,˜∆t(n),(47),(48),where,˜∆t,operates,on,unknowns,of,scaled,temperature,Td/Ts,and,density,n,=,L3,spectively.,The,partial,derivative,∂2/∂s2,(dimensionless),difference,operator,so,the,scaling,factorisation,for,aκ,is,(cid:107),leads,to,a,discrete,operator,˜∆2,sN/Nref,re-,s/(∆s(cid:107))2,where,˜∆s,is,a,L3,sts,Nref,kTs,·,N,kκe(cid:107),(∆s(cid:107))2,·,Ts,˜∆2,s,(cid:19),(cid:18),T,Ts,(49),where,˜∆2,s,operates,on,the,unknown,of,scaled,electron,temperature,Te/Ts.,The,algorithm,above,producing,the,scaled,matrix,coefﬁcients,involves,multiplying,3,factors,to-,gether.,If,T,is,measured,in,eV,,,then,k,=,|e|.,Suppose,Ls,=,10,m,and,ts,=,Ls/105,ms−1,=,10−4,s,,(Ts,≈,100,eV,),N,=,1016,m−3,,Td,=,Te,=,10,eV,,and,approximating,|e|,→,10−19,,3/2,→,1,,taking,∆t,=,0.1ts,and,∆s(cid:107),=,0.1Ls,,then,numerically,L3,sts,Nref,kTs,kTd,∆t,·,L3,sts,Nref,kTs,L3,sts,Nref,kTs,·,·,·,Ts,→,103−4−18+19−2,·,10−19+16+5,·,102,N,k,∆t,Nref,L3,s,N,kκe(cid:107),(∆s(cid:107))2,·,Ts,→,103−4−18+19−2,·,1016−19κe(cid:107),·,102,→,103−4−18+19−2,·,10−19+1+5,·,1018−3,·,(50),(51),(52),so,that,each,factor,is,not,unmanageably,different,from,unity,when,using,single,precision,arithmetic.,(The,largest,numerical,value,is,the,number,density,scaling,of,1015,if,κe(cid:107),is,estimated,as,109,m2s−1.),12,Ultimately,,the,scaled,values,of,the,coefﬁcients,aT,,,aN,and,aκ,are,easily,shown,to,be,equal,to,respectively,˜aT,=,tsn,∆t,,,˜aN,=,ts,∆t,Td,Ts,,,˜aκ,=,which,are,all,evidently,of,order,unity.,tsκe(cid:107),(∆s(cid:107))2,n,(53),If,it,is,necessary,to,iterate,to,a,solution,for,Te,or,Ti,,then,the,coefﬁcient,should,be,evaluated,for,T,=,Ts.,For,example,κe(cid:107),,which,varies,proportional,to,T,5/2,,should,be,evaluated,as,κe(cid:107)(T,),=,κe(cid:107)(Ts),(cid:19),5,2,(cid:18),T,Ts,(54),and,then,any,further,work,,for,example,linearisation,,may,proceed,using,the,dimensionless,group,T,/Ts.,2.4.3,Scalings,for,Particle,Models,The,above,considerations,also,apply,in,the,case,of,particle,evolution.,Given,the,choice,of,units,,if,the,equations,of,motion,are,posed,in,terms,of,superparticles,of,weight,wp,,wpmα,dv,dt,=,wpqα(E,+,v,×,B),(55),It,is,convenient,to,introduce,a,reference,particle,weight,wref,,,cf.,Nref,,,to,make,scalings,more,manageable,numerically.,The,overall,equation,scaling,to,render,the,terms,dimensionless,is,then,and,the,unknown,to,be,solved,for,,namely,v,,has,dimensions,Ls,.,evidently,ts,Consider,the,acceleration,in,Equation,(55),,which,leads,to,discrete,coefﬁcient,terms,of,the,form,mα,∆t,,,then,the,scaling,factorisation,is,t2,s,wref,Lsmu,ts,wref,mu,·,ts,Ls,=,wpmα,∆t,v,→,t2,s,Lswref,mu,·,wpmα,∆t,·,Ls,ts,to,be,applied,to,dimensionless,velocity,v,→,ts,Ls,v,(56),(57),As,explained,above,,the,numerical,values,of,the,3,factors,should,be,estimated.,For,deﬁniteness,,suppose,|v|,=,105,ms−1,,B,=,1,T,,then,the,gyro-radius,is,approx.,10−3,m,,so,take,Ls,=,10−3,m,as,a,typical,lengthscale,,then,ts,=,10−3−5,=,10−8,s,,and,a,timestep,size,∆t,of,order,10−9,is,indicated.,Approximate,the,amu→,10−27,kg,,and,if,A,=,Z,=,1,and,suppose,wp,=,wref,=,1010,,the,resulting,factors,follow,as,t2,s,Lswref,mu,·,wpmα,∆t,·,Ls,ts,→,10−16+3−10+27,·,1010−27+9,·,10−3+8,(58),so,that,each,factor,is,not,unmanageably,different,from,unity,when,using,single,precision,arithmetic.,(The,smallest,numerical,value,is,10−8.),The,scaled,coefﬁcient,is,easily,shown,to,simplify,as,˜a,=,t2,s,wref,muLs,·,wpmα,∆t,·,Ls,ts,=,tsA,∆t,(59),13,if,wref,=,wp,,which,is,clearly,of,order,unity.,Evidently,˜a,has,a,numerical,value,of,order,the,number,of,timesteps,chosen,to,discretise,a,gyro-period,,typically,a,number,of,order,unity.,The,magnetic,ﬁeld,term,may,similarly,be,treated,(60),(61),(62),wp|qαv,×,B|,→,t2,s,Lswref,mu,·,wp|qα|B,·,Ls,ts,whence,Ls,ts,ie.,the,smallest,numerical,factor,is,10−9,,and,t2,s,Lswref,mu,·,wp|qα|B,·,→,10−16+3−10+27,·,1010−19,·,105,˜a,=,tsZKM,|B|,≈,10−8+8,=,100,=,1,14,3,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,(63),(64),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),(65),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),(66),(67),(In,any,event,,if,K∧,is,a,function,purely,of,T,,,and,∇,·,b,=,0,,then,the,terms,in,Equation,(66),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.,15,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,,16,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.,17,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,[13],R2∇,·,1,R2,∇ψ,=,−2µ0Rjφ,(68),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),(69),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,(68),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,(70),∂2,∂R2,−,∂2,∂Z2,+,18,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,(71),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,(71),is,equivalent,to,Equation,(68).,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.,19,5,System,2-3:,1-D,ﬂuid,solver,with,simpliﬁed,physics,but,with,UQ,and,realistic,boundary,conditions,5.1,Plasma,Equations,Starting,from,the,two-ﬂuid,model,of,Braginskii,[8],,a,set,of,equations,resembling,those,of,clas-,sical,(compressible),hydrodynamics,may,be,derived,by,summing,Braginskii’s,equations,for,num-,ber,density,,momentum,and,energy,[14].,Using,the,standard,notation,of,Section,B,,introducing,Td,=,Ti,+,Te,,neglecting,the,stress,tensor,terms,(implicitly,setting,δpi,=,0),,and,assuming,B,is,independent,of,time,,the,resulting,system,is,∂,∂t,(,N,B,),+,(,(,∂,N,U,∂s(cid:107),B,miN,U,2,B,miN,U,2,B,miN,U,3,B,(cid:19),),(cid:19),),∂,∂s(cid:107),1,2,1,2,(,(,(,miN,U,B,∂,∂t,(cid:18),3,N,kTd,2,B,N,U,kTd,B,∂,∂t,(cid:18),5,2,(,(,),+,),+,),+,∂,∂s(cid:107),),=,Sn,B,),=,−,1,B,∂,∂s(cid:107),(pi,+,pe),+,Su,B,+,=,−,1,B,∂,∂s(cid:107),(qi(cid:107),+,qe(cid:107)),+,(SE,i,+,SE,e,),miB,(72),(73),(74),where,s(cid:107),is,distance,along,the,ﬁeldline,,and,all,variables,retain,their,physical,dimension.,(Some,variables,from,[14],have,been,promoted,to,capitals,to,indicate,that,they,retain,their,physical,di-,mensions.),Ion,mass,is,deﬁned,as,mi,=,Amu,where,A,is,atomic,mass,of,the,ion,and,mu,is,the,atomic,mass,unit.,Note,that,in,adding,Eqs.(3),and,(4),of,[14],,equipartition,and,collision,terms,cancel,to,give,Equa-,tion,(74).,The,perfect,gas,equation,of,state,will,be,assumed,,so,that,The,thermal,conduction,ﬂuxes,are,pi,+,pe,=,N,kT,qα(cid:107),=,−κα,(cid:107),∂kTα,∂s(cid:107),,,α,=,e,,i,(75),(76),where,the,κα,conduction,dominates,(cid:107),take,Braginskii,values,,see,Section,2,,thus,for,usual,situation,in,which,the,electron,qi(cid:107),+,qe(cid:107),=,κe,(cid:107),∂kTe,∂s(cid:107),(77),and,Te,has,be,expressed,in,terms,of,Td,,eg.,Te,=,Td/2,or,Te,=,(1,−,τ,2)Td,in,terms,of,an,arbitrary,0,<,τ,<,1,that,determines,the,ion,temperature.,To,give,an,easier,test,problem,,the,conduction,term,may,be,accounted,for,by,augmenting,the,advective,energy,ﬂux,,5/2,→,g.,The,boundary,conditions,are,that,|U,|,=,|Ms|CS,at,s,=,0,,1,where,the,sound,speed,CS,=,(cid:112)kTd/mi,20,(78),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)|,=,1,2,CSN,(δekTe,+,δikTi),≈,1,2,miCSN,δkTd,(79),if,δ,≈,δe,≈,δi.,For,deﬁniteness,,δ,=,1,2,(δe,+,δi),will,be,assumed.,Values,of,δα,from,the,literature,imply,δ,=,4.25.,The,energy,ﬂux,factor,g,is,chosen,such,that,the,easier,model,has,the,same,energy,ﬂux.,For,mathematical,analysis,,it,is,convenient,to,replace,the,source,terms,in,Equations(72)–,(74),with,equivalent,ﬂuxes,,however,this,is,unnecessary,for,computational,purposes.,The,forms,the,sources,take,are,discussed,below,in,Section,5.2.,5.2,Explicit,Sources,The,above,work,considers,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,[14],,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,mi,SE,=,SE,i,+,SE,e,=,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,=,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,(80),(81),(82),(83),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(80)–,(82),concentrates,on,this,region,,where,plasma,velocity,U,<,0,and,neutral,velocity,Un,>,0,with,the,two,having,approximately,the,same,magnitude.,There,,Equation,(80),has,only,one,negative,term,,due,to,recombination,,but,from,the,cross-section,data,in,ref,[14],,this,could,dominate,only,below,2,eV.,All,terms,in,Equation,(81),are,positive,near,s,=,0,as,the,two,velocities,reinforce.,Equation,(82),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,in,steady,state.,21,The,sources,of,neutrals,may,be,deduced,from,the,ionisation,and,charge-exchange,terms,in,Equa-,tions(80)–,(82),,viz.,Sn,n,=,−NnN,(cid:104)σv(cid:105)ION,+,Sn,⊥n,Su,n,=,−NnN,(cid:104)σv(cid:105)ION,Un,−,NnN,(Un,−,U,)(cid:104)σv(cid:105)CX,+,Su,⊥n,mn,SE,n,=,−NnN,(cid:104)σv(cid:105)ION,(,n,−,kIH,),mnU,2,kTn,+,3,2,1,2,(cid:18),3,2,−,Nn(cid:104)σv(cid:105)CX,+,SE,⊥n,k(Tn,−,Ti),+,1,2,(cid:19),mn(U,2,n,−,U,2),(84),(85),(86),(87),(88),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,The,boundary,conditions,on,the,neutrals,[14,,Table,4],are,(1),that,the,ﬂux,of,neutrals,is,set,by,recycling,of,the,plasma,,so,where,Rp,is,the,recycling,coefﬁcient.,and,(2),of,close,to,sonic,outﬂow,NnUn,=,−RpN,U,Un,=,M0,(cid:114),kTn,mn,(89),(90),where,M0,is,the,signed,Mach,number,for,the,neutrals,,and,Tn,is,the,neutral,temperature,in,the,volume,,which,assumed,constant,throughout,a,calculation.,5.2.1,Symmetries,and,Constraints,Solutions,that,could,be,used,for,testing,purposes,are,described,in,the,separate,dedicated,doc-,ument.,Here,it,is,brieﬂy,noted,that,symmetry,could,be,used,to,test,code,validity.,The,model,supports,a,solution,symmetric,about,the,domain,mid-point,in,density,and,temperature,(antisym-,metric,in,ﬂow,velocity),,provided,any,applied,sources,have,the,corresponding,symmetries.,There,are,also,point,relations,which,must,be,satisﬁed,at,the,midpoint.,In,steady,state,,there,is,the,physical,,integrated,constraint,that,sources,must,balance,total,ﬂuxes,of,plasma,and,neutral,mass,across,the,boundaries.,Further,,in,absence,of,diffusive,terms,,if,the,sources,vanish,at,the,boundaries,then,steady-state,has,the,additional,constraint,of,boundary,con-,ditions,of,zero,gradient,(Neumann,conditions).,Care,is,required,though,,as,this,will,not,generally,be,true,since,the,more,realistic,representation,of,the,sources,in,the,coupled,system,typically,leads,to,non-zero,values,at,the,boundaries.,22,6,System,2-4:,Spatially,1-D,plasma,model,incorporating,velocity,space,effects,The,following,simple,model,is,after,Taitano,et,al,[15,,16],∂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,(91),qαnuα,=,0,Equations,91,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,(91)represents,a,generalisation,of,the,system,in,ref,[15],,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,(92),(Motion,in,(y,,z),is,neglected,,the,3-D,electromagnetic,version,of,Equation,(92),appears,in,Sec-,tion,11).,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,[15],,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),(93),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,(94),where,fα,is,the,distribution,function,for,species,α,at,time,t.,replaced,by,a,sum,over,particles.),(In,practice,the,integral,would,be,23,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).,24,7,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,(95),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,[17,,§,6],by,substituting,in,Equation,(95),fα,=,exp(−λHα)Fα(x,,v,,t),(96),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,(97),The,resulting,system,is,linearised,and,solved,by,iteration,to,give,the,multispecies,plasma,ﬂuid,equations,in,Zhdanov,[17,,§,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,[18,,§,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.,25,8,System,2-6:,Spatially,2-D,plasma,model,incorporating,velocity,space,effects,With,reference,to,the,Hermes,web-site,[19],,the,following,2-D,time,dependent,model,of,plasma,evolution,may,be,derived,,expressed,using,the,agreed,notation,for,plasma,quantities,,see,Sec-,tion,B,,note,in,particular,that,plasma,‘vorticity’,has,dimensions,of,charge,density.,p,=,ρm,=,(cid:88),α,(cid:88),nαkTα,Aαmunα,cs,=,α,(cid:114),p,ρm,ne,=,Σα(cid:54)=eZαnα,=,Zini,∂ne,∂t,∂pe,∂t,∂pi,∂t,∂ω,∂t,=,−∇,·,(nevE×B),+,∇,·,=,−∇,·,(pevE×B),−,=,−∇,·,(pivE×B),−,1,|qe|,δepecs,L(cid:107),δipics,L(cid:107),jsh,−,necs,L(cid:107),+,Sn,e,+,∇,·,De∇⊥ne,+,Sp,e,+,Df,pe∇,·,(κe⊥∇⊥kTe),+,Sp,i,+,Df,pi∇,·,(κi⊥∇⊥kTi),=,−∇,·,(ωvE×B),+,∇,·,(pe,+,pi),∇,×,(cid:20),(cid:21),b,B,+,∇,·,jsh,+,Df,vs∇,·,ν∇⊥ω,∇,·,nref,|qe|Φ,+,(cid:19)(cid:21),1,Zi,pi,=,ω,(cid:18),(cid:20),mi,Zi|qe|B2,∇⊥,|qe|Φ,|qe|necs,kTref,L(cid:107),∇,·,jsh,=,−,vE×B,=,B,×,∇Φ,B2,(98),(99),(100),(101),(102),(103),(104),(105),(106),(107),(108),(109),where,it,has,been,assumed,that,there,is,a,single,ion,species,i.,(Cases,with,Zi,(cid:54)=,1,need,checking,,so,the,ion,species,has,to,be,assumed,Hydrogenic,Zi,=,1,for,now.),The,sheath,heat,transmission,coefﬁcients,are,from,Stangeby,[20,,§,2.8],(who,uses,the,notation,‘γα’),δe,=,6.5,,δi,=,2.0,(110),Stangeby,[20,,§,2.1],deﬁnes,cs,=,(cid:113),k(Te+Ti),mi,Notes,which,agrees,with,Equation,(100),to,within,order,me/(Aimu).,1.,Although,the,system,is,described,as,2-D,,it,is,conventional,to,treat,densities,as,measured,per,cubic,metre,rather,than,per,square,metre,,and,consistently,pressure,as,force,per,unit,area.,2.,The,dissipative,terms,with,coefﬁcients,ν,,κe⊥,and,κi⊥,(Braginskii,functional,forms,to,be,used,from,Section,2,,enhanced,if,necessary,by,numerical,multiplication,factors,Df,pe,,Df,pi,,Df,vs),26,have,been,added,so,that,Hermes-3,simulations,may,be,compared,minimising,complications,due,to,sub-grid-scale,effects.,3.,Thanks,to,the,use,of,equations,for,pressure,instead,of,energy,density,(Eα,=,3,2,pα,,α,=,e,,i),,Braginskii,values,correspond,to,Df,pe,=,Df,pi,=,2,3,4.,Similarly,the,pressure,sources,are,to,be,reduced,Sp,α,=,2,3,Qα,,α,=,e,,i.,5.,The,same,expressions,may,be,taken,for,the,source,terms,as,those,listed,in,System,2-3,in,Section,5,6.,A,potentially,useful,simpliﬁcation,for,the,pressure,‘dissipation’,replaces,Df,pe∇,·,(κe⊥∇⊥kTe),→,Dκ,Df,pi∇,·,(κi⊥∇⊥kTi),→,Dκ,f,pe∇,·,∇⊥pe,f,pi∇,·,∇⊥pi),(111),(112),where,Dκ,but,easier,to,implement.,f,pe,and,Dκ,f,pi,have,dimensions,of,thermal,diffusion.,This,replacement,is,inaccurate,8.1,Boundary,conditions,Except,for,Φ,(and,consequently,ω),,all,boundary,conditions,are,Neuman,with,values,speciﬁed,by,the,initial,conditions.,For,Φ,,zero-gradient,is,speciﬁed,,except,that,Φ,=,0,is,set,at,the,core,boundary,(assumed,to,lie,at,small,x),and,also,on,the,extreme,radial,boundary.,8.2,Dimensionless,units,As,explained,in,Section,1.3,,NEPTUNE,will,not,use,dimensionless,units,because,of,the,potential,for,confusion.,A,prominent,source,of,confusion,is,the,introduction,of,purely,numerical,factors,when,transforming,,for,which,there,is,one,permissible,exception,namely,,as,in,the,Hermes-3,model,,where,the,opportunity,has,been,taken,to,introduce,a,factor,Nref,to,reduce,number,densities,to,more,reasonable,magnitudes.,(It,is,suggested,that,in,keeping,with,the,application,to,the,Exascale,,the,pure,number,Nref,=,1018.),Unfortunately,for,the,project,,the,Hermes-3,models,are,expressed,in,dimensionless,variables,,and,it,is,necessary,to,understand,how,this,has,been,done.,The,key,scalings,are,1.,time,in,units,of,1/ωci,,2.,length,in,units,of,cs,ref,/ωci,,where,the,speed,satisﬁes,kTref,=,mic2,s,ref,3.,electric,potential,in,units,of,kTref,/|qe|,4.,magnetic,ﬁeld,in,units,of,B0,used,to,deﬁne,ωci,=,Zi|qe|B0/mi,(remembering,Zi,=,1,in,the,model).,5.,ne,,pi,,pe,,ω,made,dimensionless,with,respect,to,an,additional,factor,of,Nref,×,the,expected,scalings,of,(ωci/cs,ref,)2,,miω2,ci,,miω2,ci,and,|qe|,·,(ωci/cs,ref,)2,respectively.,27,6.,consequently,dimensional,source,terms,have,to,be,scaled,(divided),by,Nref,ωciScale,where,Scale,is,the,appropriate,factor,for,the,equation,from,the,preceding,list.,In,the,Hermes-3,equations,,variable,‘cs’,becomes,the,Mach,number,after,scaling,by,Us,,vari-,able,‘e’=-1,,L(cid:107),is,in,units,of,cs,ref,/ωci.,(Quoted,variables,indicate,that,they,are,not,NEPTUNE,approved,symbols.),The,NEPTUNE,website,has,a,section,on,“Physical,properties,of,the,edge,plasma”,which,gives,representative,values,for,the,units,employed,above.,8.3,Practical,implementation,For,practical,purposes,,it,is,easier,to,work,with,dimensional,quantities,,where,a,good,source,for,parameter,values,is,the,experimental,data,itself.,It,is,necessary,to,begin,with,a,checklist,to,ensure,that,all,necessary,physical,parameter,values,are,speciﬁed,,all,ﬁelds,initialised,and,their,boundary,conditions,set,,see,Section,8.3.1.,Thereafter,,based,on,knowledge,of,the,likely,physics,behaviour,,numerical,parameters,are,set,,see,Section,8.3.2,and,Section,8.3.3.,8.3.1,Physics,problem,parameters,1.,Take,Zi,=,1,,Ai,=,1,→,mi,=,Aimu.,2.,Further,given,nref,or,equivalently,Nref,/L3,s,,Tref,or,equivalently,Ts,,blob,function,FBlob(x,,y)†,(a),Initialise,ne,,pe,,pi:,(b),Te,=,Ti,=,Tref,(c),ne,=,nref,FBlob(x,,y),(d),ni,=,ne,(e),pe,and,pi,follow,as,pα,=,nαkTα,(f),cs,=,(cid:112)p/ρm,3.,Further,given,speciﬁcation,for,Φ,=,0,and,B**,(a),Initialise,ω,using,Equation,(107),4.,Further,given,L(cid:107),and,setting,sources,Sn,p,α,=,0,completes,speciﬁcation,of,physics,problem,†Blob,function,is,Gaussian,FBlob(x,,y),=,1.0,+,ncon,exp(−,(x,−,x0)2,+,(y,−,y0)2,r2,0,),(113),where,ncon,is,the,blob,contrast,,(x0,,y0),is,the,initial,position,of,the,blob,and,r0,measures,its,radius,(strictly,a,Gaussain,spreading).,**Specify,only,vacuum,ﬁeld,,in,z-direction,,B,=,B0R0/(R0,+,a,+,x),,noting,that,R0,is,major,radius,of,tokamak,28,Case,Hermes-3,[19],C-MOD,[21],NSTX,[21],Ai,,Zi,·,,·,1,,1,1,,1,1,,1,nref,1018,m−3,1,100,5.8,Tref,eV,5,47,19,BT,,,R0,,a,T,,,m,,m,?,,0.56,,0.23,5.2,,0.67,,0.22,0.44,,0.854,,0.646,ncon,,r0,·,,mm,0.5,,50?,0.6,1.3,1.6,7,L(cid:107),m,10,,50,Table,1:,Parameters,that,require,setting,to,specify,problem.,Note,plasma,species,has,to,be,assumed,to,be,Hydrogen,,ie.,not,D,or,T.,Myra,et,al,[21],give,Zef,f,=,2.,8.3.2,Boundary,conditions,parameters,This,problem,,see,Section,8.1,,requires,no,extra,parameters,to,specify,boundary,conditions.,8.3.3,Numerical,problem,parameters,Numerical,solution,requires,speciﬁcation,of,computational,domain,and,its,mesh,,together,with,a,means,of,estimating,timestep,∆t.,Assume,a,uniform,rectangular,mesh,of,elements,in,the,region,of,Cartesian,space,[0,,L0],×,[0,,L0],coordinates,(x,,y),with,nodes,uniformly,spaced,at,separation,h,in,each,coordinate,direction.,The,blob,should,be,initially,positioned,at,the,centre,of,the,domain,in,y,,so,y0,=,L0/2,and,x0,≥,3r0,in,order,that,the,density,perturbation,is,negligible,at,x,=,0.,Take,L0,≥,6r0,,suppose,Nsamp,≥,10,nodes,per,e-folding,length,r0,,then,there,needs,be,at,least,60,nodes,in,each,direction.,Timestep,∆t,<,h/Us,,where,Us,is,an,estimated,value,of,maximum,ﬂow,speed,,if,an,explicit,scheme,is,used.,The,Hermes-3,website,gives,nref,=,1018,m−3,,L(cid:107),=,50,m,(or,10,m),,Tref,=,5,eV.,A,COMPASS-like,equilibrium,is,mentioned,,but,B0,is,not,given,,nor,is,the,blob,size,in,physical,units.,There,are,several,works,which,quote,relevant,experimental,parameters,,see,Table,1.,Plots,in,Myra,et,al,[21],show,blob,motion,of,about,2.5,cm,radially,in,10,frames,equating,to,25,µs,elapsed,,speed,Us,=,1000,ms−1.,Values,of,anomalous,diffusion,are,typically,required,in,practical,calculations,[22],,usually,given,by,taking,ﬁxed,,uniform,values,for,Dκ,f,pα,,Df,usν,and,De.,To,ensure,smoothness,of,numerical,solution,,diffusion,should,dominate,over,advection,on,the,mesh,scale,h,,implying,mesh,Peclet,number,P,eh,=,Ush/D,≤,1,where,D,is,a,diffusion,coefﬁcient,(see,ref,[23,,§,2.2],for,a,more,detailed,discussion,,note,this,is,a,conservative,estimate,in,a,solenoidal,ﬂow),,thus,D,≥,Ush.,Values,of,anomalous,diffusion,quoted,by,Dudson,et,al,[22],are,Dκ,f,pα,=,0.2,m2s−1,,with,a,particle,diffusivity,of,De,=,0.1,m2s−1,[22],,and,no,value,is,quoted,for,ν,,These,non-zero,D,values,would,seem,appropriate,to,an,h,smaller,by,a,factor,of,order,ten,,if,r0,=,7,mm,,than,the,value,of,r0/10,chosen,above,,and,indeed,a,smaller,value,of,h,would,allow,for,better,resolution,of,the,smaller,scales,which,seemingly,emerge,as,a,blob,evolves.,(Note,that,P,eh,=,1,implies,equality,of,the,maximum,timesteps,allowed,for,explicit,treatment,of,respectively,advective,and,diffusive,terms.),29,8.4,Special,treatment,of,sources,Initial,conditions,rely,on,simple,analytic,responses,to,source,terms,that,may,be,either,of,vol-,ume,form,meaning,imposed,contributions,such,as,Sn,e,above,(representing,eg.,gas-pufﬁng),,or,expressed,as,inﬂows,,either,from,the,hot,central,plasmas,or,due,to,recycling,of,plasma,at,the,ﬁrst,wall.,By,default,the,source,terms,will,each,have,a,Maxwellian,distribution.,The,presence,of,large,,possibly,dominant,source,terms,implies,a,need,to,ramp,up,solutions.,A,two,stage,process,is,sug-,gested,,whereby,ﬁrst,an,analytic,approximation,Sana,to,the,expected,source,say,Sn,α,is,speciﬁed,,so,that,for,t,<,tR,α(t),≈,Sn,Sn,num,=,Sana,(114),t,tR,thereafter,introduce,a,hand-off,function,wH,(t,−,tR),falling,linearly,from,1,to,0,over,a,time,tH,so,that,num,=,wH,Sana,+,(1,−,wH,),Sn,Sn,α,wH,(t(cid:48)),=,1,−,t(cid:48)/tH,,,t(cid:48),=,t,−,tR,(115),(116),where,mathematically,8.5,Kinetic,effects,A,particle,(PIC),model,,see,Section,10,may,be,used,to,compute,models,to,represent,the,sheath,instead,of,the,terms,in,∇,·,jsh,above.,As,an,alternative,(or,additional),means,of,including,particle,effects,,the,source,terms,Sn,α,,α,=,e,,i,may,be,calculated,by,Monte-Carlo,techniques,,see,Section,12,for,treatment,of,particle,inter-,actions.,(Necessary,Monte-Carlo,techniques,for,particle,production,and,motion,are,described,in,Section,11.),e,,,Sp,30,9,System,2-7:,Interaction,between,models,of,different,dimensional-,ity,To,begin,with,,the,section,deﬁnes,the,3-D,equations,for,plasma,transport,supplied,by,Ben,Dud-,son,(private,communication),,using,the,standard,notation,of,Section,B.,9.1,3-D,plasma,transport,equations,The,equations,for,the,electron,species,density,ne,,parallel,velocity,ve(cid:107),≡,b,·,ve,and,pressure,pe,are:,∂ne,∂t,(cid:0)meneve(cid:107),∂,∂t,=,−∇,·,(cid:2)(cid:0)bve(cid:107),+,vE×B,+,ve∇B,(cid:1),ne,(cid:3),(cid:1),=,−∇,·,(cid:2)(cid:0)bve(cid:107),+,vE×B,+,ve∇B,(cid:1),meneve(cid:107),(cid:1),(cid:0)vi(cid:107),−,ve(cid:107),+,qeneE(cid:107),+,meneνei,pe,=,kTene,(cid:3),−,b,·,∇pe,(117),(118),(119),The,equations,for,the,ion,species,density,ni,(assumed,Hydrogenic,so,that,Zi,=,1),,parallel,velocity,vi(cid:107),≡,b,·,vi,and,pressure,pi,are:,∂,∂t,(cid:0)minivi(cid:107),ni,=,ne,(cid:1),minivi(cid:107),(cid:1),=,−∇,·,(cid:2)(cid:0)bvi(cid:107),+,vE×B,+,vi∇B,(cid:1),(cid:0)ve(cid:107),−,vi(cid:107),+,qiniE(cid:107),+,meneνei,pi,=,kTini,where,b,=,B/B,is,the,unit,vector,in,the,direction,of,the,magnetic,ﬁeld.,(cid:3),−,b,·,∇pi,(120),(121),(122),Both,Te,and,Ti,are,assumed,to,be,isothermal,at,speciﬁed,temperatures.,Electron,force,balance,is,used,to,calculate,the,parallel,electric,ﬁeld,E(cid:107),≡,b,·,E,and,so,transfer,electron,pressure,pe,forces,to,the,ions:,qeneE(cid:107),=,b,·,∇pe,+,∇,·,(cid:2)vdiﬀmeneve(cid:107),where,particle,diffusion,across,the,magnetic,ﬁeld,is,implemented,as,a,cross-ﬁeld,ion,drift,velocity,vdiﬀ,with,diffusion,coefﬁcient,Di:,(123),(cid:3),vdiﬀ,=,−Di,∇⊥ni,1,ni,where,the,gradient,in,the,plane,perpendicular,to,the,magnetic,ﬁeld,is,∇⊥,≡,∇,−,b(b,·,∇).,The,drift,contributions,to,the,velocity,ﬁelds,are,ve∇B,=,vE×B,=,vi∇B,=,∇,×,kTe,qe,b,×,∇Φ,B,kTi,qi,∇,×,b,B,b,B,31,(124),(125),(126),(127),where,Φ,is,the,electrostatic,potential.,The,deﬁnition,of,vorticity,ω,and,the,equation,for,its,time,advance,are,ω,=,∇,·,(cid:20),mi,(cid:21),|qe|B2,∇⊥(nref,|qe|Φ,+,pi),(cid:20),=,−∇,·,(ωvE×B),+,∇,·,+,|qe|b,·,∇,(cid:0)nivi(cid:107),−,neve(cid:107),(pe,+,pi)∇,×,(cid:1),+,Df,vs∇,·,ν∇⊥ω,(cid:21),b,B,∂ω,∂t,(128),(129),The,dissipative,term,with,coefﬁcient,ν,(Braginskii,functional,form,to,be,used,from,Section,2),,may,be,enhanced,if,necessary,by,a,numerical,multiplication,factor,Df,vs,,cf.,Section,8.,The,collision,frequency,of,charged,species,α,on,charged,species,β,is,given,by,[24]:,να,β,=,with,v2,α,=,2kTα/mα.,|qα||qβ|nβ,log,Λ,(1,+,mα/mβ),(cid:16),(cid:17)3/2,3π3/2(cid:15)2,0m2,α,α,+,v2,v2,β,(130),The,Coulomb,logarithm,is,different,for,electron-electron,,ion-ion,and,electron-ion,species,interac-,tions,,and,is,calculated,using,the,NRL,formulary,[7,,25,,p.,34].,Converted,to,SI,units,with,T,in,eV,,,the,Coulomb,logarithm,for,electron-ion,species,interaction,is:,log,Λe,i,=,,,,31,−,0.5,ln,ne,+,ln,Te,30,−,0.5,ln,ne,−,ln,Zi,+,1.5,ln,Te,23,−,0.5,ln,ni,+,1.5,ln,Ti,−,ln,(cid:0)Z2,i,Ai,(cid:1),if,Ti,if,Ti,me,mi,me,mi,<,10Z2,i,<,Te,<,Te,<,10Z2,i,(131),if,Te,<,Time/mi,9.2,Boundary,conditions,for,3-D,plasma,transport,equations,The,usual,computational,domain,is,an,annulus,consisting,of,a,region,of,closed,and,open,magnetic,ﬂux,surfaces,,since,the,central,,hot,“core”,of,the,plasma,generally,requires,a,different,modelling,approach.,Neumann,boundary,conditions,,ie.,of,vanishing,normal,derivative,,are,the,default,on,all,boundaries,for,all,ﬁelds.,However,the,exceptions,to,this,rule,are,important.,A,boundary,condition,must,be,imposed,at,that,innermost,closed,ﬂux,surface,where,no,boundary,physically,exists,,and,the,effects,of,the,sheaths,in,front,of,solid,wall,surfaces,must,be,accounted,for.,Typically,when,the,Dirichlet,condition,Φ,=,0,is,applied,on,radial,boundaries,,there,is,a,problem,with,local,discontinuities,,so,a,special,relaxation,procedure,is,adopted,,see,Dudson,et,al,[26].,The,following,description,is,taken,from,ref,[26]:,“The,boundary,condition,on,the,potential,Φ,is,time-evolving,,designed,to,relax,towards,a,Neumann,condition,,as,follows.,When,inverting,the,Laplacian-type,equation,for,Φ,from,the,vorticity,,the,po-,tential,is,ﬁxed,at,both,core,and,wall,boundaries.,If,a,simple,Dirichlet,(ﬁxed,zero,value),condition,is,used,then,narrow,boundary,layers,typically,form,close,to,the,boundaries,in,which,the,imposed,boundary,potential,is,matched,to,the,plasma,potential.,These,boundary,layers,can,develop,un-,physical,instabilities.,Instead,,at,every,timestep,the,value,of,the,boundary,condition,is,adjusted,towards,the,value,inside,the,domain,with,a,characteristic,timescale,that,is,set,by,default,to,1µs.,32,In,this,manner,the,electrostatic,potential,Φ,evolves,smoothly,to,solutions,that,can,have,different,potentials,on,core,and,wall,boundaries.”,At,(strictly,just,in,front,of),solid,wall,surfaces,,sheath,conditions,following,Stangeby,as,in,Sec-,tion,5.1,are,applied.,For,the,present,model,,these,amount,to,no,electron,outﬂow,ve(cid:107),=,0,,and,ion,outﬂow,at,magnitude,|vi(cid:107)|,=,cs,where,the,acoustic,speed,cs,=,(cid:112)k(Te,+,Ti)/mi,(132),33,10,Systems,3:,Kinetic,models,The,following,generic,transport,equation,[27,,§,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),(133),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,.,Ultimately,it,will,be,necessary,to,solve,a,multi-species,version,of,Equation,(133),complete,with,appropriate,source,terms,to,represent,the,physics,thought,critical,for,modelling,the,tokamak,edge.,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),(134),and,equations,for,the,evolution,of,the,electromagnetic,ﬁelds,E(x,,t),and,B(x,,t),such,as,Maxwell’s,Inevitably,choice,of,SC,is,a,function,of,lengthscale,equations,,neglecting,displacement,current.,and,timescale.,On,fast,timescales,in,a,strong,electromagnetic,ﬁeld,,the,effect,of,collisions,can,be,ignored,(collective,effects,are,felt,through,the,electromagnetic,ﬁeld),,when,the,Particle-in-Cell,or,PIC,approach,[28],is,effective.,Note,that,strictly,local,particle-particle,interactions,should,be,accounted,for,,but,these,are,expected,to,have,negligible,effect,in,a,plasma,(although,not,in,a,gravitating,system).,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.,10.1,Particle-in-Cell,(PIC),Although,PIC,codes,are,conceptually,simple,to,implement,,in,practice,there,is,often,a,problem,with,statistical,effects,,aka,noise.,Noise,is,generally,found,to,be,reduced,when,the,scheme,is,mo-,mentum,conserving,,which,is,usually,achieved,[28,,§,5-3-3],by,use,of,(1),the,same,function,in,both,charge,assigment,and,interpolation,of,force,onto,particles,,and,(2),space-centred,approximations,to,derivatives.,In,the,NEPTUNE,symbols,,assignment,of,charge,to,nodes,and,force,interpolation,to,particles,,share,a,weighting,function,W,,,such,that,∆ρc(xm),=,qα,V,e,W,(xp,−,xm),F(xp),=,=,ΣmqαW,(xp,−,xm)E(xm),(135),(136),where,xp,is,the,position,of,the,particle,,xm,is,the,location,of,a,ﬁnite,element,node.,For,consistency,with,the,Nektar++,basis,W,(x),=,φe,ξ(x),34,(137),The,fundamental,import,of,(2),is,that,E(xm),=,Σ(cid:48),mV,eG(xm,,x(cid:48),m)ρc(x(cid:48),m),(138),where,G(xm,,x(cid:48),m,,xm).,The,antisymmetry,of,G,then,ensures,vanishing,of,the,particle,self-force,and,that,the,forces,exerted,by,one,particle,on,another,are,equal,and,opposite,,since,eg.,m),=,−G(x(cid:48),F(xp),=,q2,αΣmΣ(cid:48),mW,(xp,−,xm)V,eG(xm,,x(cid:48),m)W,(xp,−,x(cid:48),m),(139),Only,the,electrostatic,ﬁeld,E,is,shown,in,Equation,(135),but,the,above,analysis,holds,more,gener-,ally,for,the,Lorentz,force.,11,System,3-1:,2-D,particle-based,model,of,neutral,gas,and,impuri-,ties,with,critical,physics,The,neutrals,are,represented,as,super-particles,that,travel,ballistically,after,introduction,and,are,In,this,context,,neutral,particle,may,include,photons.,Super-,lost,when,they,strike,the,ﬁrst,wall.,particles,have,label,p,,weight,wp,and,sample,the,point,(xp,,vp),in,6-D,position,and,velocity,space.,At,introduction,,all,particle,quantities,are,deﬁned,by,sampling,from,speciﬁed,probability,distribu-,tions.,11.1,Prerequisites,11.1.1,Parameters,It,is,necessary,to,have,parameters,describing,initial,and,boundary,conditions.,There,must,be,a,means,of,tying,boundary,conditions,to,speciﬁc,parts,of,the,surface,and,volume,geometry.,See,discussion,of,deﬁnitions,of,objects/classes,for,NEPTUNE,in,web,pages.,11.1.2,Random,number,generator,It,is,generally,important,to,test,the,properties,of,a,random,number,generator,to,ensure,there,is,an,absence,of,bias,,typically,by,producing,histograms,of,the,output,and,comparing,with,expected,curves.,If,other,routines,are,found,to,be,unsuitable,,a,technique,based,on,the,‘Mersenne,Twister’,should,give,a,satisfactory,sequence,of,pseudo-random,numbers.,For,a,range,of,applications,where,functions,or,distributions,do,not,vary,on,very,small,scales,,Quasi-,Monte,Carlo,sampling,[29],may,be,preferable,to,Monte,Carlo.,Note,that,although,the,place-name,Monte-Carlo,has,a,hyphen,,the,name,of,the,mathematical,technique,does,not,by,convention.,For,parallel,computation,,it,will,generally,be,best,to,compute,a,block,of,numbers,at,a,time.,35,11.1.3,Sampling,from,a,speciﬁed,distribution,Generally,Textbooks,such,as,Kalos,and,Whitlock,[30],(notable,for,its,treatment,of,radiation,trans-,port,in,§,6),explain,how,to,generate,samples,of,a,given,distribution,f,(x),from,random,numbers,uniformly,sampled,on,the,unit,interval.,Suppose,that,ξ,is,such,a,random,number,,then,the,corre-,sponding,value,of,x,is,given,by,solution,of,ξ,=,1,−,(cid:90),x,0,f,(x(cid:48))dx(cid:48),(140),Equation,(140),may,be,solved,explicitly,for,x,in,many,important,cases,Gaussian,distribution,This,may,be,sampled,using,the,Box-Muller,method,[30,,§,3.1].,Given,two,random,numbers,ξ1,and,ξ2,uniformly,sampled,on,the,unit,interval,,then,two,samples,of,a,Gaussian,distribution,f,(x),∝,exp(−x2/2),are,given,by,(cid:112)[−2(ln(1,−,ξ1)],cos,2πξ2,(cid:112)[−2(ln(1,−,ξ2)],cos,2πξ1,(141),(142),Knudsen,cosine,Sample,by,accept-reject,from,the,Knudsen,cosine,distribution,below,Equa-,tion,(143),to,launch,particle,trajectories.,Let,fmax,and,smax,be,the,maximum,of,the,distribution,function,in,[0,,6vth,i,Kn),and,the,associ-,ated,speed,,respectively,where,vth,i,Kn,=,(cid:112)2kTKn/mi).,Then,provide,pairs,of,random,numbers,Rf,∈,[0,,fmax],and,vR,∈,[0,,6vth,i,Kn],(and,other,components,of,velocity,similarly,sampled,for,the,tangential,components,(but,with,a,negative,velocity,ranges)).,Keep,the,particle,with,normal,speed,vR,if,Rf,<,fn,Kn(vR).,11.2,Sources,and,sinks,of,neutrals,11.2.1,Knudsen,distribution,As,suggested,in,the,TN-07,Neptune,report,by,Parra,,Barnes,and,Hardman,[31],equations,(5.9)-,(5.13),,the,source,of,neutrals,emitted,from,a,wall,can,be,described,by,the,Knudsen,cosine,distri-,bution,fn,Kn,(v;,n,·,v,>,0),=,3,4π,(cid:18),mi,kTKn,(cid:19)2,n,·,v,|v|,(cid:18),exp,−,(cid:19),miv2,2kTKn,(143),where,n,is,the,normal,to,the,wall,,the,Knudsen,distribution,is,used,for,outgoing,neutrals,for,which,n,·,v,>,0,,and,kTKn,is,a,parameter,that,controls,the,temperature,of,the,emitted,neutral,distribution.,The,Knudsen,cosine,distribution,appears,with,a,factor,of,a,particle,ﬂux,in,Equation,(5.9),of,[31],,so,despite,the,”f,”,notation,,it,has,different,units,to,the,other,distributions,f,.,36,11.2.2,Volumetric,distribution,As,a,starting,point,,the,source,of,neutrals,should,be,ﬁxed,in,time,and,given,a,prescribed,spatial,distribution,around,the,edge,of,the,simulation,domain.,A,point,source,,ie.,delta,function,in,space,,would,be,a,valid,representation,of,a,‘gas,valve’.,Volumetric,sources,consisting,of,eg.,a,Maxwellian,at,a,given,temperature),could,be,considered,for,testing,purposes.,11.2.3,Recycling,This,cannot,be,properly,treated,without,coupling,to,a,plasma,model,,hence,is,treated,in,Sec-,tion,12.4.1.,11.2.4,Particle,sink,An,explicit,volume,pumping,region,,where,neutrals,are,absorbed,when,they,reach,it.,This,might,be,speciﬁed,by,a,set,of,ﬁnite,element,identiﬁers,e.,11.3,Boundary,condition,for,neutral,particles,The,particle,simulation,domain,need,not,coincide,with,the,simulation,domain,,since,there,may,be,ﬁnite,elements,where,a,species,is,treated,as,a,ﬂuid.,11.3.1,Perfect,Absorption,Any,particles,that,reach,a,domain,boundary,are,deleted.,11.3.2,Reﬂection,These,conditions,are,very,useful,for,testing,eg.,energy,conservation.,Perfect,specular,reﬂection,might,be,needed,to,handle,symmetry.,Supposing,the,unit,surface,normal,is,n,,if,the,incident,velocity,is,vp,,then,the,reﬂected,particle,has,velocity,vp,=,vp,−,2(vp.n)n,(144),11.3.3,Periodic,boundaries,These,conditions,are,very,useful,for,testing,purposes,,and,required,for,full,or,repeat,sections,of,toroidal,geometry.,Particles,simply,leave,one,end,of,the,domain,and,re-enter,at,the,other.,In,the,case,of,a,rectilinear,grid,,periodic,boundary,conditions,are,easily,applied,by,taking,the,modulus,of,the,coordinate,value,with,respect,to,the,period,length.,37,Figure,3:,Extract,from,publication,indicated,in,the,text.,11.3.4,Imperfect,reﬂection,These,conditions,are,probably,most,appropriate,for,photons.,This,may,be,achieved,by,arranging,that,a,fraction,(1,−,Rp),of,incident,particles,are,absorbed.,If,the,particles,are,allowed,different,weights,,then,simply,reduce,the,weight,wp,of,each,reﬂected,particle,by,the,particle,recycling,factor,Rp,and,its,energy,consistent,with,recycling,factor,RE.,Sputtering,is,the,ejection,of,surface,atoms,by,impact,of,both,energetic,ions,and,neutrals.,However,,the,rates,of,sputtering,by,energetic,neutrals,are,relatively,low,below,100,eV,[10,,§,9.7],and,may,be,neglected,in,an,initial,investigation.,12,System,3-3:,Interaction,with,2-D,plasma,model,12.1,Prerequisites,12.1.1,Cross-section,data,input,Cross-section,data,will,be,obtained,from,the,ADAS,library,[32].,Cross-section,data,(cid:104)σv(cid:105),averaged,over,a,Maxwellian,velocity,distribution,suitable,for,use,by,a,ﬂuid,model,of,the,plasma,edge,is,shown,in,Figure,3,,from,Havlickova,et,al.,[14].,The,dominant,relevant,reactions,affecting,both,neutrals,and,plasma,in,the,graph,are,Ionisation,(ION),e−,+,H,→,H,+,+,e−,+,e−,Charge-exchange,(EXC),H,+,H,+,→,H,+,+,H,(145),(146),38,Figure,4:,Conﬁguration,for,ionisation.,Neutrals,,indicated,by,arrows,,moving,against,a,background,of,thermal,electrons,shown,as,dots.,12.2,Physical,Models,12.2.1,Introductory,model,Perhaps,the,simplest,model,to,consider,is,that,for,ionisation,of,neutrals,by,electron,impact.,A,simple,model,for,ionisation,can,be,written,as:,∂fn,∂t,∂ni,∂t,=,.,.,.,−,RION,nifn,=,.,.,.,+,RION,ninn,(147),(148),where,RION,((cid:104)σv(cid:105)ION,elsewhere),is,a,constant,ionisation,rate,,fn,is,the,distribution,function,for,the,neutrals,,ni,and,nn,are,the,ion,and,neutral,densities.,The,quasineutrality,assumption,implies,ne,=,ni.,So,the,‘source,term’,in,the,plasma,density,equation,is,SION,(x),≡,RION,ninn,=,RION,ni,(cid:90),d3v,fn(v),(149),the,integral,in,which,converts,,for,number,densities,,into,weighted,counts,of,the,number,of,particles,within,a,given,spatial,volume,about,point,x.,The,loss,term,in,the,neutral,density,equation,is,computed,using,Monte,Carlo,techniques,,cf.,Ver-,boncouer,[33],,conﬁguration,sketched,in,Figure,4.,The,probability,of,ionisation,of,a,particle,at,time,tn,travelling,with,velocity,vp,in,the,following,interval,of,∆t,is,pp(tn),=,1,−,exp,(−neσION,|vp|∆t),(150),where,the,cross-section,for,ionisation,is,σION,.,Provided,the,background,density,ne,is,approxi-,mately,constant,in,space,and,time,,and,σION,variation,with,energy,E,is,assumed,to,be,negligible,,taking,for,example,νσmax,=,max,{nt(x)},max,{σT,(E)|v|},(151),x,E,39,then,for,every,particle,,provided,all,particles,have,the,same,weight,(ie.,identical,superparticles),,approximately,pp,=,pT,=,1,−,exp,(−νσmaxt),(152),and,the,number,of,neutrals,that,undergo,ionisation,in,volume,∆V,is,pT,nn∆V,.,Such,particles,should,be,chosen,at,random,from,those,in,∆V,.,The,Monte,Carlo,algorithm,for,selecting,which,neutrals,turn,into,ions,,is,simple,provided,that,particles,are,distributed,at,random,throughout,∆V,,,viz.,to,obtain,a,random,number,ξ,from,the,uniform,distribution,on,the,unit,interval,(ie.,0,<,ξ,<,1),for,each,particle,in,turn,and,if,at,the,qth,ξ,<,pT,then,the,qth,particle,is,regarded,as,ionised,at,time,tn.,If,particles,are,each,allowed,a,weight,wp(t),which,varies,with,time,,then,the,weight,of,the,neutral,may,simply,be,reduced,to,account,for,the,ionisation,wp(tn,+,∆t),=,pp(tn)wp(tn),(153),12.2.2,Detailed,model,For,sources,in,the,ﬂuid,equations,due,to,ionisation,of,neutrals,by,electrons,,the,formulae,,cf.,Equations(84)–,(86),([14,,Eqs.(34)-(36)]),are,Sn,e,=,NnN,(cid:104)σv(cid:105)ION,Sv,n,mn,Sp,=,NnN,(cid:104)σv(cid:105)ION,vn,NnN,(cid:104)σv(cid:105)ION,kIH,2,3,e,=,−,2,3,Sp,i,=,NnN,(cid:104)σv(cid:105)ION,(,3,2,kTn,+,1,2,mnv2,n),(154),(155),(156),(157),Note,that,other,effects,due,to,charge-exchange,and,recombination,may,be,deduced,from,the,equations,in,Section,5.2.,Monte,Carlo,calculation,may,then,proceed,using,a,total,cross-section,for,all,three,interactions.,12.2.3,Simpliﬁed,model,For,an,exploratory,calculation,with,the,Hermes-3,equations,,the,ionisation,potential,term,is,an,unwelcome,complication,and,the,momentum,term,is,assumed,not,to,contribute,to,the,evolution,of,the,vorticity,ω.,Hence,Equations(154)–,(157),above,become,Sn,e,=,NnN,(cid:104)σv(cid:105)ION,Sv,n,=,0,Sp,e,=,0,2,Sp,i,=,3,NnN,(cid:104)σv(cid:105)ION,(,3,2,kTn,+,1,2,mnv2,n),(158),(159),(160),(161),Compared,to,the,introductory,model,in,Section,12.2.1,,spatial,variations,in,background,density,and,cross-section,are,handled,by,the,null,collision,method,,see,Section,12.5.2.,40,Introducing,the,(super-)particles,newly,ionised,in,time,interval,∆t,,occupying,positions,xp,with,label,p,and,weight,wp,and,interaction,I:,e,V,e∆t,≈,ΣpI,wpδD(x,−,xp),Sn,2,Sp,i,V,e∆t,≈,3,mnΣpI,wpδD(x,−,xp),1,2,v2,n,pI,(162),(163),where,the,contribution,to,the,energy,in,a,ﬁnite,element,e,is,a,sum,over,all,particle,interactions,I,that,have,occurred,in,e.,These,values,are,projected,on,the,ﬁnite,element,basis,as,for,charge,assignments,(δD,is,the,Dirac,delta,function),so,that,they,give,rise,to,source,terms,e,(x)V,e∆t,=,Σe,pwpφe,ξ(xp)φe,ξ(x),Sn,Sp,i,(x)V,e∆t,≈,mnΣe,pwpv2,n,pφe,ξ(xp)φe,ξ(x),1,3,(164),(165),where,φe,ξ,is,the,expansion,basis,as,a,function,of,global,position,x,,ie.,φe,ξ(x),=,φe,(ξ(x)),,and,the,mass,matrix,has,been,lumped.,12.3,Initial,conditions,These,are,deﬁned,separately,for,the,ﬂuid,(‘continuum’),species,and,the,particle,species.,12.4,Boundary,conditions,Only,conditions,coupling,both,particle,and,ﬂuid,species,are,to,speciﬁed,here,,otherwise,see,sep-,arate,treatments.,12.4.1,Recycling,Recycling,of,the,plasma,reaching,the,wall,implies,that,the,source,of,neutrals,coming,from,the,target,plates,has,a,ﬂux,(and,spatial,proﬁle),equal,to,the,ﬂux,of,ions,reaching,the,target,multiplied,by,some,recycling,coefﬁcient,Rp,(eg.,a,fraction,like,0.99).,Recycling,is,a,complicated,process,[10,,§,9.4],whereby,the,ions,penetrate,the,solid,lattice,of,the,surface,,lose,signiﬁcant,energy,before,neutralising,and,fraction,Rp,reappears,at,the,surface,with,a,relatively,low,(below,5,eV,),temperature.,The,ﬂux,of,ionised,plasma,from,the,ﬂuid,code,is,nv,,which,translates,into,a,total,incident,number,of,particles,n|v|∆t∆S,,where,∆S,is,the,area,of,surface,impacted,,which,might,be,taken,as,the,area,of,a,ﬁnite,element,surface.,Thus,there,are,Rpn|v|∆t∆S,recycled,particles,to,be,represented,as,superparticles.,If,the,superparticles,have,ﬁxed,weight,then,it,might,be,necessary,to,use,Monte,Carlo,to,treat,‘fractional’,superparticles,,but,simple,rounding,to,the,nearest,integer,should,meet,larger,number,cases.,Otherwise,,if,there,is,a,reference,weight,wp0,then,a,set,of,superparticles,should,be,launched,each,with,a,weight,close,to,this,value.,It,will,be,assumed,that,these,‘recycled’,neutrals,are,born,with,a,Cosine-Knudsen,distribution,at,a,user-speciﬁed,temperature,TKn,of,a,few,eV,.,(Momentum,and,energy,ﬂuxes,given,by,the,ﬂuid,code,may,in,a,simple,approximation,be,disregarded.),41,12.5,Calculating,particle,interactions,12.5.1,Classical,scattering,For,interactions,in,which,there,is,signiﬁcant,momentum,or,energy,transfer,,it,is,necessary,to,do,a,classical,scattering,problem,to,account,for,the,interchange.,For,inelastic,collisions,between,two,particle,of,mass,mp,and,velocity,vp,,p,=,1,,2,,momentum,and,energy,conservation,give,m1v1,+,m2v2,=,m1v+,2,=,m1v+2,m1v2,1,+,m2v2,1,+,m2v+,2,1,+,m2v+2,2,(166),(167),where,the,velocities,v+,conservation,is,satisﬁed,if,v+,1,=,v1,−,p/m1,,v+,conservation,equation,,it,follows,that,p,=,2µm(v1,−,v2),where,reduced,mass,2,at,the,new,time,are,found,from,the,observation,that,momentum,2,=,v2,−,p/m2,for,any,p.,Substituting,in,the,energy,1,and,v+,so,that,µm,=,m1m2,m1,+,m2,v+,1,=,v1,−,2,v+,2,=,v2,+,2,m2,m1,+,m2,m1,m1,+,m2,(v1,−,v2),(v1,−,v2),(168),(169),(170),12.5.2,Simpliﬁed,models,The,PIC-MCC,software,[33],accounts,for,spatial,variations,in,background,density,and,cross-,section,by,the,null,collision,method,,also,known,as,‘delta-tracking’.,This,method,amounts,to,a,correction,to,the,introductory,model,,relying,on,the,maximum,property,of,the,rate,νσmax,,whereby,the,number,of,collisions,is,reduced,according,to,the,local,value,of,neσION,in,the,volume,∆V,(which,volume,might,well,correspond,to,that,of,ﬁnite,element,e).,The,local,value,gives,a,more,accurate,estimate,for,pq.,A,second,random,number,ξ2,is,drawn,from,the,uniform,distribution,and,the,neutral,remains,unchanged,if,ξ2,<,pT,−,pq.,12.5.3,Preferred,approach,The,“Direct,Sampling”,approach,of,Brown,and,Martin,[34],involves,the,most,arithmetic,per,particle,of,the,techniques,considered,,but,should,generally,provide,increased,accuracy,which,since,the,arithmetic,cost,is,likely,dominated,by,data,movement,,comes,essentially,“free,of,charge”.,Note,that,the,sampling,techniques,needed,to,treat,all,3,interactions,mentioned,above,are,common,to,all,,although,additional,modelling,is,needed,to,handle,momentum,and,energy,transfer,in,some,interactions.,Comparisons,of,“Direct,Sampling”,and,the,algorithm,used,in,PIC-MCC,are,made,in,refs,[35,,36].,42,Figure,5:,Conceptual,conﬁguration.,A,particle,,indicated,by,the,straight,line,,moves,against,a,background,thermal,(‘ﬂuid’),species,shown,as,dots.,In,the,reﬁnement,by,De,Esch,,interactions,take,place,at,uniformly,spaced,intervals,along,the,track,,also,indicated,by,dots,As,illustrated,in,Figure,5,,let,τ,(s),be,the,optical,depth,traversed,by,a,article,traveling,a,distance,s,through,a,medium,with,arbitrarily,speciﬁed,macroscopic,cross-section,σ(s):,τ,(s),=,(cid:90),s,0,σ(s(cid:48))ds(cid:48),We,assume,only,that,σ,is,ﬁnite,and,σ(s),≥,0.,Note,that,dτ,ds,=,σ(s),(171),(172),To,explicitly,allow,for,the,case,of,no,collision,in,a,ﬁnite,distance,of,travel,,we,deﬁne,PN,C,,the,probability,of,no,collisions,,as,PN,C,=,exp,(−τ,(∞)),(173),Then,the,probability,density,function,(pdf),for,a,collision,occurring,after,a,particle,has,traveled,a,distance,s,through,the,medium,is,given,by,[37,,§,7],p(s),=,PN,Cδ(s,−,s∞),+,dτ,ds,exp,(−τ,(s)),(174),where,dτ,ds,is,the,interaction,probability,per,unit,distance,travelled,,s∞,is,the,distance,to,the,boundary,of,the,computational,domain,and,exp,(−τ,(s)),is,the,probability,of,traversing,distance,s,without,collision.,Equation,(174),explicitly,allows,for,cases,where,τ,(∞),is,ﬁnite,,hence,there,is,a,possibility,of,traveling,an,inﬁnite,distance,without,colliding.,Unbiased,random,sampling,of,the,Monte,Carlo,path,requires,solving,the,following,for,s,,distance,along,the,path,,namely,(cid:90),s,ξ,=,p(s(cid:48))ds(cid:48),(175),where,ξ,is,sampled,from,a,uniform,random,variable,on,[0,,1).,In,the,spirit,of,De,Esch,,values,of,ξ,=,j/Nξ,,j,=,1,,.,.,.,Nξ,−,1,should,be,used,,and,the,charged,particle,weights,(effectively,the,0,43,number,of,physical,particles,each,represents),be,reduced,by,Nξ.,Values,of,Nξ,≈,10,−,100,are,suggested.,In,the,ﬁrst,step,of,the,sampling,,discrete,sampling,is,used,to,select,a,collision,with,probability,(1,−,PN,C),or,an,inﬁnite,ﬂight,with,probability,PN,C.,That,is,,if,ξ,>,PN,C,,then,there,is,a,collision.,The,second,step,is,to,sample,s,from,from,the,pdf,given,by:,where,Using,Equation,(176),,note,that,g(s(cid:48)),=,1,G,dτ,ds(cid:48),exp,(−τ,(s(cid:48))),G,=,(1,−,PN,C),(cid:90),s,0,g(s(cid:48))ds(cid:48),=,1,G,(cid:90),τ,(s),0,exp,(−τ,)dτ,Using,Equations(175),and,(176),,we,can,sample,τs,=,τ,(s),by,solving,ξ,=,1,G,(cid:90),τs,0,exp,−τ,dτ,This,is,equivalent,to,sampling,from,a,truncated,exponential,pdf,,which,has,the,solution,Pathlength,s,then,follows,from,Equation,(171),,viz.,τs,−,ln(1,−,Gξ),τs,=,(cid:90),s,0,σ(s(cid:48))ds(cid:48),(176),(177),(178),(179),(180),(181),When,σ(s(cid:48)),has,a,simple,functional,form,,Equation,(181),can,often,be,solved,analytically,for,s.,In,many,cases,which,arise,in,practice,,the,solution,may,involve,a,transcendental,equation,or,other,form,not,amenable,to,analytic,solution.,Equation,(181),,however,,can,be,readily,solved,numerically,for,s,using,Newton,iteration,with,f,=,(cid:82),sσ(s(cid:48))ds(cid:48),−,s,,starting,with,an,initial,estimate,0,s0,=,τs/σ(0),[34].,Because,df,/ds,≤,0,,f,is,monotone,and,there,can,be,at,most,one,root.,For,cases,where,σ(s(cid:48)),≥,0,,the,Newton,iteration,is,guaranteed,to,converge.,However,,if,σ(s(cid:48)),is,zero,or,very,small,over,a,portion,of,the,path,,df,/ds,may,be,0,,leading,to,numerical,difﬁculties,and,nonconvergence.,This,potential,problem,is,remedied,easily,by,combining,Newton,with,a,bisection,search,method,,such,that,bisection,is,used,if,df,/ds,is,very,small,or,zero.,Using,this,approach,,Brown,and,Martin,found,that,only,1,−,5,iterations,are,typically,needed,to,converge,s,to,within,part,in,106.,even,for,extreme,variations,in,σ(s(cid:48)).,A,ﬁnal,practical,point,concerns,the,relation,of,path,length,s,to,physical,coordinates.,If,the,particle,starts,at,x0,and,travels,in,a,direction,given,by,vp,parallel,to,unit,vector,d,then,the,particle,path,is,given,by,so,inverting,so,it,is,helpful,if,d,is,a,unit,vector.,x,=,x0,+,sd,s,=,|x,−,x0|/|d|,44,(182),(183),13,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.,Refe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,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,[39,,§,5],explains,at,greater,length,the,calculation,in,thermal,equilibrium,of,the,proportion,of,different,ions,for,each,element,[39,,§,5.2],,then,the,bound-free,/,free-bound,coefﬁ-,cients,(§,5.3—§,5.5),and,free-free,(§,5.6).,In,[39,,§,24.1–24.2],there,is,a,discussion,of,metastable,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,[40],,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,[40,,slide,21].,The,Zeeman,effect,is,also,neglected,,although,this,might,be,expected,to,be,important,,as,from,its,use,in,sunspot,observation,[41,,§,5.2],spectral,line,splitting,by,wavelengths,of,0.1,nm,is,expected.,The,key,observation,is,from,O’Mullane,[40],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,[38,,§,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,),(184),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,[40],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),,,(185),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,49,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,(186),(187),(188),(189),where,the,coefﬁcients,,as,elsewhere,in,this,section,,are,expected,to,be,obtained,from,the,Atomic,Data,and,Analysis,Structure,ADAS,database,[32,,42].,The,data,requirements,for,this,look,relatively,modest,,assuming,the,coefﬁcients,for,each,species,and,charge,state,are,smooth,functions,of,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,[43].,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,b-factors,to,determine,the,distribution,of,charge,states.,(Further,,for,heavier,elements,a,mean,atomic,mass,may,be,used,to,avoid,separate,treatment,of,isotopic,species.),Much,more,serious,implications,for,computation,[40],,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,(190),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,[44]),,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.,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,[45],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,50,are,complicated,,namely,for,the,metastable,state,labelled,ρ,(cid:88),(cid:88),SZ,ρ,/ne,=,X,Z→Z,σ→ρ,nσ,−,X,Z→Z,ρ→σ,nρ,+,+,+,σ,(cid:88),µ,(cid:88),ν,(cid:88),σ,µ→ρ,nZm,S,Zm→Z,σ,µ,−,(cid:88),S,Z→Zp,ρ→ν,nρ,ν→ρ,nZp,αZp→Z,ν,−,ν,(cid:88),αZ→Zm,ρ→µ,nρ,QZ→Z,σ→ρ,nσ,−,µ,(cid:88),σ,QZ→Z,ρ→σ,nρ,(191),(192),(193),(194),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,(195),(196),Note,that,the,expressions,in,both,ref,[45,,eq.,(9)],and,ref,[40,,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,[40,,44],that,the,number,of,metastable,states,for,a,given,ionisation,Z,is,relatively,small,(slide,9,of,ref,[44],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,(191),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,[46],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,[45,,eq.,(5)],,nZ,i,/ne,=,(cid:88),X,FiσnZ,σ,+,I,FiµnZm,µ,+,(cid:88),(cid:88),RFiνnZp,ν,(197),ν,σ,µ,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,(197),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,[45].,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,51,Maxwellian,assumption,is,relaxed,,posing,a,multiscale,multiphysics,problem.,However,,the,validity,of,this,approach,requires,further,consideration,as,Henderson,et,al,[47],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,%,[48].,B,Annex,B:,Index,of,Mathematics,Table,2:,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,sym-,bols,appear,immediately,after,‘b’,(backslashes,ignored).,The,main,source,for,the,symbols,is,the,Equations,document,[49],,also,included,are,those,listed,as,used,in,the,text,by,Karni-,adakis,and,Sherwin,[50],,prefaced,by,(K+S),,plus,symbols,used,in,the,report,[51].,Symbol,a,aij,A,Ai,Aα,[a,,b],α,αn,αZp→Z,αZ→Zm,b,B0,Bs,¯NZ,B,=,|B|,BT,β,β,Units,m,Description,minor,radius,of,the,torus,(horizontal),coefﬁcient,of,matrix,A,atomic,mass,of,ion,atomic,mass,of,ion,atomic,mass,of,ion,species,α,arbitrary,ﬁnite,interval,as,sufﬁx,is,species,label,or,index,perturbation,amplitude,m3s−1,partial,dielectronic,recombination,rate,coefﬁcient,m3s−1,partial,dielectronic,recombination,rate,coefﬁcient,m,minor,radius,of,the,torus,(vertical),T,used,to,make,B,dimensionless,characteristic,magnetic,ﬁeld,used,to,make,B,dimensionless,T,average,number,of,charge,states,amplitude,of,the,imposed,magnetic,ﬁeld,amplitude,of,the,imposed,toroidal,magnetic,ﬁeld,as,sufﬁx,is,species,label,(Glossary),Ratio,of,plasma,pressure,to,pressure,in,mag-,netic,ﬁeld,T,T,52,a,=,d2x/dt2,A(x,,t),B(x,,t),b,E(x,,t),Es,E+,F,u∧,v,vα,v(cid:107),v⊥,v∧,v0,vcx,vE×B,ve,vi,ve∇B,vi∇B,vdiﬀ,x,(x1,,x2,,.,.,.,,,xd),x,bn,ξ(θ),=,B,Dξ,f,e,H,H,e,L,Λ(u),Le,M,AT,M,e,n,ω,m2s−1,T,m,T,V,m−1,V,m−1,m−2,N,ms−1,acceleration,experienced,by,a,particle,magnetic,vector,potential,magnetic,ﬁeld,unit,vector,giving,the,direction,of,the,magnetic,ﬁeld,electric,ﬁeld,characteristic,electric,ﬁeld,used,to,make,E,dimensionless,modiﬁed,electric,ﬁeld,force,vector,pseudo,/,thermal,velocity,component,in,ﬂux,surface,normal,to,ﬁeld,direction,generic,velocity,velocity,of,species,α,ﬂuid,velocity,directed,along,ﬁeldline,ﬂuid,velocity,component,normal,to,ﬂux,surface,ﬂuid,velocity,component,in,ﬂux,surface,normal,to,ﬁeld,di-,rection,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,is,a,d-dimensional,vector,ms−1,ms−1,ms−1,ms−1,ms−1,position,‘b-factors’,ref,[40,,slide,21],multi-dimensional,random,variable,with,a,speciﬁc,probabil-,ity,distribution,as,a,function,of,the,random,parameter,0,≤,θ,≤,1,(K+S),Basis,matrix,(K+S),Elemental,derivative,matrix,with,respect,to,ξ,(K+S),Force,vector,of,the,eth,element,(K+S),Helmholtz,matrix,(=,AT,H,eA)),(K+S),Elemental,Helmholtz,matrix,(K+S),Laplacian,matrix,(=,AT,LeA)),(K+S),Diagonal,matrix,of,u(ξi,,ξ2),evaluated,at,quadrature,points,(K+S),Elemental,Laplacian,matrix,(K+S),Mass,matrix,(=,AT,M,eA),(K+S),Matrix,global,assembly,(K+S),Elemental,mass,matrix,(K+S),Unit,outward,normal,(K+S,and,plasma,models),Vorticity,53,m,s−1,or,Cm−3,√,KM,AT0,(cid:113),kTi+ZikTe,mi,(cid:113),p,ρm,(cid:113),kTe,me,(cid:113),kTi,mi,csi,=,CS,∪,C(xi,,xj),d,δpi,δ,δD,δe,δ,=,1,δi,δα,2,(δe,+,δi),ue,W,ξ(θ),Bp,C0,=,∩,χ,χδ,χi(ξ),cp,cs,=,cs,=,cse,=,δ(x),D,DA,De,Df,vα,Dn,Df,pα,Di,|e|,e,eijk,(K+S),Vector,containing,function,evaluated,at,quadrature,points,(K+S),Diagonal,weight,/,Jacobian,matrix,multi-dimensional,random,variable,with,a,speciﬁc,probabil-,ity,distribution,as,a,function,of,the,random,parameter,0,≤,θ,≤,1,amplitude,of,the,poloidal,magnetic,ﬁeld,used,to,make,velocities,dimensionless,(Sets),Set,intersection,(K+S),Space,of,trial,solutions,(K+S),Finite-dimensional,space,of,trial,solutions,(FE,Basis),Local,Cartesian,to,global,coordinate,mapping,speciﬁc,heat,at,constant,pressure,approx.,plasma,acoustic,speed,plasma,acoustic,speed,acoustic,speed,of,electrons,T,ms−1,Jkg−1K−1,ms−1,ms−1,ms−1,ms−1,ms−1,N,m−2,acoustic,speed,of,ions,sound,speed,coefﬁcient,in,radiation,equation,(Sets),Set,union,covariance,of,random,variables,xi,,xj,number,of,dimensions,over,which,the,integral,is,performed,stress,tensor,Kronecker,delta,Dirac,delta,function,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,(Glossary),Magnetisation,parameter,,species,α,gyroradius,normalised,to,L,Dirac,delta,function,of,continuous,real,variable,x,spatial,dimensionality,of,problem,diffusion,coefﬁcient,for,plasma,charges,in,a,background,of,neutrals,diffusion,coefﬁcient,for,electrons,,eg.,in,a,background,of,neutrals,scale,dissipation,in,equation,for,evolution,of,species,veloc-,ity,vα,neutral,diffusion,coefﬁcient,scale,dissipation,in,equation,for,evolution,of,species,pres-,sure/energy,pα,diffusion,coefﬁcient,for,ions,,eg.,in,a,background,of,neutrals,m2s−1,absolute,value,of,the,charge,on,the,electron,(K+S),Finite,element,number,1,≤,e,≤,Nel,weighted,integral,of,triple,products,of,Ψi,of,the,ion,species,m2s−1,m2s−1,m2s−1,C,54,∅,(cid:15)0,(cid:15)r,=,ts/t0,η1,,η2,,η3,ηB,ηd,=,ηB/µ0,ηen,ηen(cid:107),ηin,ηin(cid:107),f0,f0,fα,fe,fi,fij(xi,,xj),fce,=,ωce,2π,fci,=,ωci,2π,fpe,=,ωpe,2π,fpi,=,ωpi,2π,fi(xi),f,(x1,,.,.,.,,,xd),f,E,F,E,f,n,F,n,f,u,F,u,f,(x,,v,,t),fn,Kn(v),Γ(x),g(hj),G,Hα,ˆue,ˆvg,ˆvg,F,m−1,Ωm,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,m−6s3,m−6s3,m−6s3,m−6s3,s−1,s−1,s−1,s−1,m−1s−2C,m−3C,m−2s−1C,m−6s3,m−4s4,from,(Sets),Empty,set,permittivity,of,free,space,scale,factor,for,transient,term,(FE,Basis),Local,collapsed,Cartesian,coordinates,plasma,resistivity,after,Braginskii,plasma,resistivity,,as,diffusivity,contribution,to,plasma,resistivity,,as,diffusivity,,electron-neutral,interactions,contribution,to,plasma,parallel,resistivity,,as,diffusivity,,from,electron-neutral,interactions,contribution,to,plasma,resistivity,,as,diffusivity,,from,ion-,neutral,interactions,contribution,to,plasma,parallel,resistivity,,as,diffusivity,,from,ion-neutral,interactions,constant,in,the,expansion,of,f,(x1,,.,.,.,,,xd),initial,distribution,function,of,the,electrons,distribution,function,of,species,α,distribution,function,of,the,electrons,distribution,function,of,the,ion,species,coefﬁcient,in,the,expansion,of,f,(x1,,.,.,.,,,xd),electron,cyclotron,frequency,ion,cyclotron,frequency,electron,plasma,frequency,ion,plasma,frequency,coefﬁcient,in,the,expansion,of,f,(x1,,.,.,.,,,xd),joint,probability,distribution,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,Knudsen,distribution,function,gamma,function,of,continuous,variable,x,activation,function,(of,input,hj),of,a,neuron,in,a,neural,net-,work,Green’s,function,Hamiltonian,for,species,α,(K+S),Vector,of,expansion,coefﬁcients,(K+S),Global,list,of,coefﬁcients,(K+S),List,of,all,elemental,coefﬁcients,(=,ve),55,h,hj,hp(ξ),i,i,i,I,Iφ,IH,i,,j,,k,I,Fiσ,∈,I(ψ),=,BT,/R,I,Z,j,jext(R,,Z),jφ,j(cid:107),jsh,k,k,κα,κe(cid:107),κe⊥,κi(cid:107),κi⊥,κ,=,kc/ρmcp,kB,kc,Kcx,(ni,,Ti),Ki,KM,A,KM,Kr,kT0,kTd,Kv(x),kw,λ,λ,λ,Λ,m,A,eV,T,m−1,W,mesh,or,inter-node,spacing,real-number,input,to,a,neuron,in,a,neural,network,(FE,Basis),One-dimensional,Lagrange,polynomial,of,order,p,as,sufﬁx,denotes,ions,as,sufﬁx,denotes,regular,excited,state,as,sufﬁx,generic,label,as,sufﬁx,labels,Monte-Carlo,interactions,φ−,or,toroidal,component,of,plasma,current,Hydrogen,reionisation,potential,as,deﬁned,in,ref,[14],(K+S),General,summation,indices,coefﬁcient,of,ionisation,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,(Sets),Is,a,member,of;,belongs,to,function,giving,the,toroidal,ﬁeld,as,a,function,of,ψ,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,Am−2,component,of,plasma,current,density,parallel,to,ﬁeldline,Am−2,sheath,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,chosen,as,kB/mu,or,|e|/mu,so,that,(cid:112)KM,Td/A,is,an,ion,speed,recombination,reaction,rate,T0,in,energy,units,Td,in,energy,units,modiﬁed,Bessel,function,of,the,second,kind,,order,v,wavenumber,vector,arbitrary,quantity,Coulomb,logarithm,(K+S),Helmholtz,equation,constant,Coulomb,logarithm,KM,Td,is,an,ion,speed,?,?,m−1,?,m3s−1,J,J,√,?,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,JK−1,Jm−1s−1K−1,m3s−1,m3s−1,56,λq,Λb,λD,λmf,p,α,(cid:104)σv(cid:105)CX,(cid:104)σv(cid:105)ION,(cid:104)σv(cid:105)REC,(cid:104)σv(cid:105),L0,LNm,i,(ξ),Ls,L(cid:107),m,M0,M1,mα,E,Ek(cid:54)=i,l(cid:54)=j,Exk(cid:54)=i,L(u),P,Pδ,v,Eα,Ee,Ei,ER,F,Fα,I,I,δ,K(cid:107),K,K⊥,K∧,L7,PP,(Ω),m,eV,m,m,m,kg,m,m3s−1,m3s−1,m3s−1,m3s−1,m,e-folding,length,of,midplane,proﬁle,of,power,loss,when,an,exponential,is,ﬁtted,sheath,potential,drop,normalized,to,Te,(Glossary),Debye,lengthscale,above,which,local,electro-,static,ﬂuctuations,due,to,presence,of,discrete,charged,par-,ticles,are,negligible,(Glossary),Mean,free,path,of,particle,species,α,reaction,rate,for,charge,exchange,reaction,rate,for,ionisation,reaction,rate,for,recombination,generic,reaction,rate,typical,lengthscale,(FE,Basis),Two-dimensional,Lagrange,polynomial,through,Nm,nodes,ξi,typical,lengthscale,along,ﬁeldline,connection,length,of,typical,ﬁeldline,species,particle,mass,Mach,number,at,s,=,0,boundary,Mach,number,at,s,=,1,boundary,mass,of,species,α,expectation,expectation,computed,by,integrating,over,all,the,xk,except,for,xi,and,xj,expectation,computed,by,integrating,over,all,the,xk,except,for,xi,(K+S),Linear,operator,in,u,(K+S),Projection,operator,(K+S),Discrete,projection,operator,(K+S),Velocity,[u,,v,,w]T,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,(K+S),Interpolation,operator,(K+S),Discrete,interpolation,operator,parallel,thermal,conductivity,of,plasma,thermal,conductivity,of,plasma,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),(K+S),Polynomial,space,of,order,P,over,Ω,s−1,kg,m−1s−1,m−1s−1,m−1s−1,m−1s−1,57,Q,QZ→Z,σ→ρ,S,S,Zm→Z,S,Z→Zp,T,V,V,δ,X,X,Z→Z,σ→ρ,R,Var(f,),Var[Q],me,mi,mn,mp,mu,Ms,MS,µ,,ν,µcx,=,ωc/νcx,µm,MZ,n,nref,Nref,N,n0,∇·,∇×,∇2,Nb,nB,=,N/B,ND,Ndof,ncon,ne,Nel,Neof,coefﬁcient,in,radiation,equation,parent-metastable,cross-coupling,coefﬁcient,coefﬁcient,in,radiation,equation,ionisation,coefﬁcient,ionisation,coefﬁcient,generic,tensor,(K+S),Space,of,test,functions,(K+S),Finite-dimensional,space,of,test,functions,coefﬁcient,in,radiation,equation,generalised,collisional-radiative,(GCR),excitation,coefﬁ-,cient,Real,numbers,variance,of,the,distribution,of,f,computed,by,integrating,over,all,variables,xi,variance,in,random,variable,Q,mass,of,electron,mass,of,ion,species,particle,mi,=,Amu,neutral,species,particle,mass,mass,of,proton,atomic,mass,unit,Mach,number,,allowed,to,take,either,sign,number,of,energy,states,of,an,atom,(K+S),Dynamic,,kinematic,viscosities,measures,strength,of,magnetization,with,respect,to,charge,exchange,reaction,reduced,mass,of,two,particles,number,of,metastable,states,for,species,α,(which,includes,the,ground,state),number,density,reference,number,density,of,the,plasma,ions,normalising,or,reference,number,density,number,density,,may,be,scaled,by,Nref,=,1018,initial,number,density,(K+S),Divergence,(K+S),Curl,(K+S),Laplacian,(K+S),Number,of,global,boundary,degrees,of,freedom,number,density,divided,by,ﬁeld,strength,Number,of,degrees,of,1,,2,,.,.,.,6,(K+S),Number,of,global,degrees,of,freedom,blob,contrast,factor,number,density,of,the,electrons,(K+S),Number,of,ﬁnite,elements,(K+S),Total,number,of,elemental,degrees,of,Neof,(cid:39),NelNm,freedom,per,dimension,,D,=,freedom,58,m3s−1,m3s−1,m3s−1,m3s−1,m3s−1,?,m3s−1,m3s−1,kg,kg,kg,kg,1.6605,×,10−27,kg,kg,m−3,m−3,1018,m−3,m−3,m−3T,−1,m−3,ni,nj(x,,t),Nm,nn,/∈,(cid:54)⊂,np,NQ,ns,Ns,NT,ν,να,νcx,=,Kcxnn,νe0,νe(cid:107),νe⊥,νi(cid:107),νi,νi0,νi⊥,ν∗,α,ν∗,c,q4,e,p(cid:15)2,3m2,0,να,ναn,ναβ,nZ,NP,NP,α,NZ,nZ,i,nZ,σ,ωce,=,|e|B/me,ωci,=,ZieB/mi,L0n0/C4,0,(cid:113),nq2,e,(cid:15)0me,(cid:113),nq2,e,(cid:15)0mi,ωpe,=,ωpi,=,Zi,Ω,Ωe,p(A|B),pα,number,density,of,the,plasma,ions,member,of,the,set,of,deterministic,coefﬁcients,of,the,“ran-,dom,trial,basis”,(K+S),Number,of,elemental,degrees,of,freedom,neutral,density,(Sets),Is,not,a,member,of;,does,not,belong,to,(Sets),Is,not,a,subset,of,number,density,of,the,plasma,ions,(K+S),Total,number,of,quadrature,points,NQ,=,Q1Q2Q3,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,[32,,42],plasma,kinematic,viscosity,kinematic,viscosity,of,species,α,charge,exchange,‘frequency’,electron,kinematic,viscosity,caused,by,neutrals,parallel,kinematic,viscosity,of,electrons,perpendicular,kinematic,viscosity,of,electrons,parallel,kinematic,viscosity,of,ions,ion,kinematic,viscosity,ion,kinematic,viscosity,caused,by,neutrals,perpendicular,kinematic,viscosity,of,ions,(Glossary),Normalised,collision,frequency,for,species,α,Collisionality,parameter,(Glossary),Collision,frequency,for,species,α,Collision,frequency,for,species,α,with,neutrals,Collision,frequency,for,species,α,with,species,β,number,density,for,charge,state,Z,number,of,particles,in,a,calculation,number,of,particles,of,species,α,in,a,calculation,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,electron,cyclotron,angular,frequency,ion,cyclotron,angular,frequency,plasma,angular,frequency,for,electrons,plasma,angular,frequency,for,ions,(K+S),Solution,domain,(K+S),Elemental,region,conditional,probability,of,event,A,given,event,B,is,known,or,assumed,to,have,occurred,pressure,of,species,α,59,m−3,m−3,m−3,m−3,m−3,m2s−1,m2s−1,s−1,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,s−1,s−1,s−1,m−3,m−3,m−3,radianss−1,radianss−1,radianss−1,radianss−1,N,m−2,=,(cid:107),Q,(cid:107)E,(∂f,/∂t)C,∂Ωe,∂Ω,∂ΩD,∂ΩN,PC,pe,φ,Φ,φpq,,φpqr,φe,ξ,α,nαkTα,p,p,=,(cid:80),p,pi,Pi,p(ψ),p,,q,,r,P,r,P,rM,ψ,p,,,ψb,ψa,Ψi,pq,,ψc,pqr,P,(T,),p(x),P,(x),P,Z,Q(cid:107),qα,qe,Q(fα,,fβ),QH,qi,q(cid:107)e,q(cid:107)i,qe,qi,Qi,Qie,r,r0,R,the,‘energy’,norm,source,in,Boltzmann,due,to,inter-particle,interactions,(K+S),Boundary,of,Ωe,(K+S),Boundary,of,Ω,(K+S),Domain,boundary,with,Dirichlet,conditions,(K+S),Domain,boundary,with,Neumann,conditions,number,of,modes,in,basis,for,polynomial,chaos,pressure,of,the,electrons,angle,in,toroidal,direction,electr(ostat)ic,potential,(FE,Basis),Expansion,basis,(FE,Basis),expansion,basis,as,a,function,of,global,posi-,tion,x,(K+S),pressure,plasma,pressure,as,sufﬁx,labels,(super-)particles,pressure,of,the,ion,species,(FE,Basis),Polynomial,order,in,the,ith,direction,function,giving,the,pressure,as,a,function,of,ψ,of,the,mag-,netic,ﬂux,(K+S),General,summation,indices,Prandtl,number,magnetic,Prandtl,number,poloidal,magnetic,ﬂux,(FE,Basis),Modiﬁed,principal,functions,ith,member,of,a,set,of,basis,functions,,dimensional,Hermite,polynomials,emitted,power,integrated,over,all,wavelengths,probability,distributions,Cumulant,probability,distribution,radiated,power,per,atom,of,nZ,combined,energy,ﬂux,at,a,boundary,charge,on,a,particle,of,species,α,charge,on,an,electron,,negative,by,convention,Boltzmann,collision,operator,cooling,rate,due,to,excitation,as,deﬁned,in,ref,[14],charge,on,an,ion,electron,energy,ﬂux,along,ﬁeldline,ion,energy,ﬂux,along,ﬁeldline,electron,energy,ﬂux,ion,energy,ﬂux,(FE,Basis),Quadrature,order,in,the,ith,direction,collisional,energy,equipartition,term,order,of,higher,order,term,radius,used,in,initial,condition,,such,as,blob,size,cylindrical,coordinate,typically,multi-,m−6s2,N,m−2,radians,c,V,N,m−2,N,m−2,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,Jm−2s−1,Jm−2s−1,kgm−1s−3,m,m,60,R0,Rp,RE,ρ,ρ,ρc,=,(cid:80),ρm,(cid:80),ρtα,α,Zα|e|nα,=,α,Aαmunα,RFiσ,s(cid:107),s,s,Sα,SC,Sana(x,,t),Sn,ana(x,,t),Sexp(x,,v,,t),n,T,u,si,sE,sE,e,sE,i,sE,n,sE,⊥e,sE,⊥i,sE,⊥n,sn,sn,n,sn,e,sn,i,su,su,n,su,⊥n,Si,σ,σ,σC,σE,σE,σI,major,radius,of,torus,recycling,coefﬁcient,for,particles,recycling,coefﬁcient,for,particle,energy,as,sufﬁx,is,label,of,metastable,state,(K+S),Density,charge,density,of,the,medium,mass,density,of,the,medium,the,transition,from,(Glossary),Gyroradius,or,Larmor,radius,of,orbit,of,charged,particle,of,species,α,about,magnetic,ﬁeld,direction,coefﬁcient,of,recombination,for,metastable,state,σ,to,regular,excited,state,i,arclength,along,ﬁeldline,as,sufﬁx,,isotope,label,(α,preferred,for,species),parameterises,distance,along,the,ﬁeldline,0,≤,s,≤,1,source,term,in,Boltzmann,equation,for,species,α,total,source,term,in,Boltzmann,equation,explicit/analytic,source,term,in,ﬂuid,equation(s),numerically,convenient,source,term,in,ﬂuid,equation(s),explicit,source,term,in,Boltzmann,equation,neutral,density,neutral,temperature,neutral,velocity,arclength,parameter,for,boundary,(i,=,1,inner,,i,=,2,outer),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,source,term,in,neutral,density,equation,number,density,source,term,for,electrons,number,density,source,term,for,ions,source,term,in,plasma,momentum,equation,source,term,in,neutral,momentum,equation,momentum,cross-ﬁeld,source,term,for,neutrals,Sobol,sensitivity,index,,gives,a,normalised,measure,of,the,sensitivity,of,the,distribution,of,f,to,the,parameter,xi,as,sufﬁx,labels,metastable,state,reaction,cross-section,reaction,rate,for,charge,exchange,cooling,rate,due,to,excitation,electrical,conductivity,reaction,rate,for,ionisation,61,m,Cm−3,kgm−3,m,m,m−6s2,m−6s2,m−3s−1,?,m−3s−1,?,m−6s2,m2,Ω−1m−1,σi|0,s,σe|0,s,Sij,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,α,t,t(cid:48),T,t0,ts,tH,tR,T0,TKn,Tref,Ts,Tα,τ,τα,τe,τi,τen,τin,collision,cross-section,for,ions,with,neutrals,collision,cross-section,for,electrons,with,neutrals,Sobol,sensitivity,index,,gives,a,normalised,measure,of,the,sensitivity,of,the,distribution,of,f,to,the,parameters,xi,and,xj,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,(Sets),Is,a,subset,of,source,term,in,neutral,momentum,equation,momentum,cross-ﬁeld,source,term,for,neutrals,particle,source,for,ion,of,metastable,state,σ,(species,α),with,charge,state,Z,particle,source,for,ion,of,species,α,with,charge,state,Z,time,usually,in,seconds,offset,time,usually,in,seconds,plasma,temperature,characteristic,evolutionary,timescale,usually,in,seconds,characteristic,timescale,usually,in,seconds,Numerical,hand-off,time,interval,usually,in,seconds,Numerical,ramp-up,time,interval,usually,in,seconds,initial,temperature,(preﬁxed,by,k,implies,energy,in,SI),reference,temperature,of,Knudsen,distribution,(preﬁxed,by,k,implies,energy,in,SI),reference,temperature,(preﬁxed,by,k,implies,energy,in,SI),characteristic,temperature,(Ts,=,(Ls/ts)2/KM,),temperature,of,species,α,optical,depth,collision,or,relaxation,time,of,species,α,electron,collision,or,relaxation,time,ion,species,collision,or,relaxation,time,electron-neutral,collision,time,ion,species-neutral,collision,time,m2,m2,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,m−3s−1,kgm−2s−2,kgm−2s−2,kgm−2s−2,m−3s−1,m−3s−1,s,s,eV,s,s,s,s,eV,eV,eV,eV,eV,m,s,s,s,s,s,62,τce,=,1/fce,τci,=,1/fci,τpe,=,1/fpe,τpi,=,1/fpi,τEe,τEi,τne,τni,Td,=,Ti,+,Te,Te,TH,θ,θ,Ti,˜a,˜b,=,B/B0,pq,,˜ψc,p,,,˜ψb,˜ψa,u,U,Uα,Ud,=,Ls/t0,Us,=,Ls/ts,UA,f,e,W,e,uR,=,1/R,v,v(cid:107),V,e,Vi,Vij,pqr,w,wjk,wp,wα,ref,wref,W,x,x0,x1,,x2,,x3,,x,xα,xe,=,ωceτe,xi,=,ωciτi,xi,s,s,s,s,s,s,s,s,eV,eV,radians,c,eV,ms−1,ms−1,ms−1,ms−1,ms−1,ms−1,ms−1,ms−1,m3,ms−1,1010,m,m,electron,cyclotron,timescale,ion,cyclotron,timescale,plasma,timescale,for,electrons,plasma,timescale,for,ions,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,electron,temperature,(preﬁxed,by,k,implies,energy,in,SI),the,Hydrogen,reionisation,potential,angular,coordinate,random,parameter,0,≤,θ,≤,1,ion,temperature,scaled,matrix,coefﬁcient,dimensionless,magnetic,ﬁeld,(FE,Basis),Orthogonal,principal,functions,generic,ﬁrst,velocity,component,velocity,component,(ﬂow),along,ﬁeldline,velocity,component,(ﬂow),along,ﬁeldline,of,species,α,speed,measuring,the,importance,of,the,transient,term,characteristic,speed,Alfv,´en,speed,(K+S),Concatenation,of,elemental,vector,f,e,(K+S),Block-diagonal,extension,of,matrix,W,e,Radial,component,of,Grad-Shafranov,‘ﬂow’,generic,second,velocity,component,ﬂuid,velocity,component,along,ﬁeldline,spatial,volume,occupied,by,ﬁnite,element,e,variance,of,the,distribution,of,f,as,the,parameter,xi,varies,variance,of,the,distribution,of,f,as,the,parameters,xi,and,xj,vary,generic,third,velocity,component,weight,in,neural,network,indexed,by,neuron,j,and,input,k,weight,of,particle,p,normalising,or,reference,weight,of,particle,of,species,α,normalising,or,reference,number,for,superparticles,weighting,function,for,particle-in-cell,Cartesian,coordinate,coordinate,value,used,in,specifying,initial,condition,,eg.,blob,position,(FE,Basis),Global,Cartesian,coordinates,collisionality,factor,of,species,α,collisionality,factor,of,electrons,collisionality,factor,of,ions,generic,parameter,or,variable,63,ξ1,,ξ2,,ξ3,,ξ,ξi,X,Fiσ,y,y0,z,z0,Z,Z,Z,Z0(α),Za,ζ,ζ,=,−φ,Zef,f,Zi,Zα,Zm,=,Z,−,1,Zp,=,Z,+,1,Zsum,(cid:80),α,Z0(α),(FE,Basis),Local,Cartesian,coordinates,random,number,within,the,unit,interval,[0,,1],coefﬁcient,of,excitation,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,Cartesian,coordinate,coordinate,value,used,in,specifying,initial,condition,,eg.,blob,position,Cartesian,coordinate,coordinate,value,used,in,specifying,initial,condition,Cartesian,coordinate,charge,state,of,the,ion,cylindrical,coordinate,number,of,charge,states,of,species,α,included,in,the,model,Gaussian,random,process,,index,a,magnetic,Prandtl,number,as,deﬁned,in,Cambridge,toroidal,angle,coordinate,effective,charge,state,of,plasma,ions,charge,state,of,ion,charge,state,of,ion,species,α,where,Z,is,ion,charge,state,where,Z,is,ion,charge,state,where,Z0,is,number,of,charge,states,of,species,α,=,m,m,m,m,m,m,radians,c,64

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