CD-EXCALIBUR-FMS0021-1.24-M1.2.1_YearOneSummaryReport
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   :description: technical note

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ExCALIBUR,Equations,for,ExCALIBUR/NEPTUNE,Proxyapps,Version,1.24,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:,1.24,23,January,2023,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Name,and,Department,Signature,Date,Prepared,By:,Wayne,Arter,23,January,2023,BD,Reviewed,By:,Rob,Akers,23,January,2023,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,23,January,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).,Supporting,information,regarding,Braginskii’s,transport,coefficients,for,plasma,in,a,strong,mag-,netic,field,appears,in,Section,2.,A,description,of,sources,of,atomic,and,molecular,radiation,is,given,as,an,annex,in,Section,A.,A,second,annex,Section,B,tabulates,the,mathematical,symbols,(with,their,units),used,in,the,current,document.,1.1,Overall,Plan,In,the,Science,Plan,[2],the,description,of,work,extending,beyond,Y3,(early,2022),is,deliberately,vague,on,the,subject,of,“gyrokinetics”,,as,no,widely,accepted,model,for,the,tokamak,edge,appears,to,be,available,as,of,early,Y2,,and,even,should,one,appear,,it,might,not,be,suitable,for,use,in,Y3,in,NEPTUNE.,As,listed,,the,proxyapps,correspond,to,Plan,A,,which,assumes,that,no,suitable,gyroaveraged,model,will,emerge,in,time,,hence,kinetic,implies,Particle-in-Cell,(PIC).,PIC,approaches,,where,charge,conservation,is,vital,to,control,errors,,can,anyway,usefully,be,pursued,for,modelling,low,collisionality,plasma,species,,and,should,simplify,nicely,to,treat,neutral,species,with,long,mean-,free-paths,in,tokamak,edge,problems,where,mass,conservation,has,been,discovered,to,be,critical.,Moreover,in,the,context,of,classical,fluid,dynamics,,the,transition,from,fluid,to,particles,,or,short,to,long,mean-free-path,,has,been,well-studied,because,of,the,application,to,the,space,vehicle,2,re-entry,problem,and,related,hypersonic,situations.,Thus,the,hybrid,fluid/PIC,approach,might,be,regarded,as,a,relatively,low-risk,route,to,achieving,a,robust,numerical,algorithm.,Evidently,full-orbit,PIC,has,the,potential,to,be,extremely,inefficient,relative,to,gyroaveraged,kinetic,theory,because,of,the,need,to,follow,gyro-orbits,in,detail.,Hence,if,this,can,be,avoided,,either,through,gyroaveraging,or,clever,numerics,or,indeed,a,combination,of,both,,then,Plan,B,will,see,kinetic,imply,gyroaveraged,kinetic,theory,for,modelling,plasma,species,in,the,proxyapps.,Regarding,implementation,at,the,Exascale,there,is,also,a,conservative,Plan,A,approach,which,sees,the,use,of,relatively,simple,data,structures,such,as,scalar,and,vector,arrays,to,transfer,data,,and,consequent,use,of,existing,code-coupling,technologies.,Plan,B,is,an,aggressive,approach,to,implementation,which,sees,custom,data,structures,allowing,for,all,physical,data,(particle,arrays,and,fluid,field,vectors),colocated,near,a,point,to,be,held,close,in,memory,,permitting,very,tight,custom,code-coupling.,As,with,the,kinetic,options,,this,Plan,B,promises,significantly,faster,solu-,tions,than,the,corresponding,Plan,A,,but,its,adoption,depends,on,the,outcome,of,research,work,to,de-risk.,1.2,Proxyapps,Summary,The,overall,thinking,behind,the,proxyapps,is,to,explore,potential,‘roadblocks’,to,the,Exascale,as,early,and,in,as,simple,a,context,as,is,possible,,beginning,with,algorithmic,roadblocks.,NEPTUNE,is,directed,towards,producing,‘actionable’,code,as,the,basis,for,large,procurements,,whereas,more,physics-focussed,software,projects,conducted,by,the,worldwide,nuclear,fusion,have,already,advanced,to,greater,complexity,,minimising,the,risk,that,unexpected,problems,will,appear,in,the,full,model.,The,numbering,of,proxyapps,below,corresponds,to,the,Science,Plan.,2-1,2-D,model,of,anisotropic,heat,transport.,It,is,important,to,determine,early,the,degree,of,anisotropy,that,high-order,elements,can,treat,without,special,coding.,If,this,is,unsatisfactorily,small,,then,there,are,implications,for,geometry,input,as,well,as,algorithmic,developments,that,are,best,addressed,as,early,as,possible.,2-2,2-D,elliptic,solver,in,complex,geometry.,One,of,the,indicated,elliptic,solvers,is,Grad-Shafranov,to,produce,high,order,(‘spectrally’),accurate,magnetic,fields,for,use,in,many,other,proxyapps.,Since,Sovinec,[3],has,already,produced,a,spectral,element,Grad-Shafranov,code,,the,corre-,sponding,NEPTUNE,development,should,mainly,serve,to,identify,practical,issues,concern-,ing,implementing,high,order,fe,models.,The,second,solver,additionally,presents,a,chance,to,explore,comparatively,novel,meshing,techniques,developed,under,Activity,A2.1,,and,later,the,preconditioning,techniques,of,A2.7.,2-3,1-D,fluid,solver,with,simplified,physics,but,with,UQ,and,realistic,boundary,conditions.,This,will,determine,the,capability,of,spectral/hp,element,to,handle,sonic,outflow,boundary,condi-,tions,needed,to,represent,sheaths,,together,with,large,source,terms,,as,well,as,identifying,practical,issues,concerning,intrusive,UQ.,This,software,is,already,potentially,useful,in,its,own,right,for,example,in,modelling,MAST-U,divertor,,and,other,workers,might,be,drawn,in,to,add,additional,physical,effects,to,this,end.,3,2-4,Spatially,1-D,plasma,model,incorporating,velocity,space,effects.,From,the,numerical,analytic,point-of-view,,this,is,a,key,demonstration,of,spectral/hp,element,capability,to,handle,particle,interactions.,However,,again,this,could,be,a,basis,for,divertor,modelling,,to,explore,sheath,effects,depending,on,fieldline,incidence,on,surface,,and,with,minor,modification,the,spread,of,particle,energy,around,tile,edges,and,corners,as,performed,by,Gunn,et,al,[4,,5],for,ITER,application.,2-5,Spatially,1-D,multispecies,plasma,model.,Multispecies,throws,up,a,surprising,number,of,issues,concerning,data,definitions,(eg.,changes,to,the,Coulomb,logarithm),,structures,to,deal,with,different,number,of,species,,and,perhaps,most,significantly,,complicated,inter-species,interaction,terms,both,within,and,at,the,domain,boundaries.,This,is,also,an,opportunity,to,mix,fluid,and,kinetic,representations,of,different,species,within,the,volume.,2-6,Spatially,2-D,plasma,model,incorporating,velocity,space,effects.,With,the,1-D,multispecies,fluid,work’s,having,made,the,generalisation,to,2-D,straightforward,,the,challenge,here,is,to,start,writing,a,complex,proxyapp,in,production,mode,,incorporating,the,research,put,into,design,,documentation,,code,generation,and,benchmarking.,There,is,an,opportunity,to,study,species,with,both,fluid,and,kinetic,representations,depending,on,location,relative,to,the,wall.,Again,this,is,potentially,a,useful,tool,in,its,own,right,,capable,of,revealing,deficiencies,in,previous,2-D,modelling,work.,2-7,Interaction,between,models,of,different,dimensionality.,This,should,verify,that,the,design,has,the,right,data,structures,to,handle,additional,further,complexity,beyond,intrusive,and,ensemble-based,UQ,and,model,order,reduction.,The,hopefully,burgeoning,NEPTUNE,com-,munity,could,develop,this,into,a,design,tool,with,a,capability,both,to,explore,a,large,area,of,tokamak,edge,parameter,space,quickly,in,0-D,or,1-D,and,also,to,focus,on,relatively,small,but,critical,2-D,features,,such,as,tile,edges.,2-8,Spatially,3-D,plasma,kinetic,models.,These,will,represent,the,full,fluid,model,produced,by,the,5-year,NEPTUNE,project,,incorporating,features,of,2-D,fluid,and,kinetic,work,in,a,3-D,code.,3-1,2-D,particle-based,model,of,neutral,gas,&,impurities,with,critical,physics.,This,will,be,a,2d3v,code,(ie.,spatially,2-D,distribution,of,particles,with,3,velocity,components),designed,from,the,outset,to,interact,with,a,high-order,finite-element,fluid,model,of,plasma.,It,gives,an,opportunity,to,check,out,ideas,on,optimal,usage,of,particles.,3-2,2-D,moment-based,model,of,neutral,gas,&,impurities.,Constructing,a,2-D,fluid,code,of,neutral,gas,from,the,Nektar++,software,should,be,a,valuable,educational,exercise,,whilst,providing,scope,for,cross-validating,the,2-D,particle,model.,3-3,Interaction,with,2-D,plasma,model,when,available.,Building,on,the,1-D,multispecies,fluid,work,,the,challenge,here,is,to,incorporate,in,the,fluid,code,of,Proxyapp,PA2-6,,particle,effects,from,PA3-1,,which,will,in,the,higher,dimensional,space,be,more,subject,to,lack,of,numerical,resolution,or,‘noise’.,Should,PA3-3,be,accelerated,,it,could,usefully,treat,both,plasma,and,neutrals,via,particle,models.,3-4,3-D,model,of,neutral,gas,&,impurities.,This,is,now,at,full,dimensional,complexity,,incorporat-,ing,selected,ideas,on,optimal,usage,of,particles.,4,3-5,Interaction,with,3-D,plasma,model.,This,will,represent,the,full,model,produced,by,the,5-year,NEPTUNE,project,,a,coupling,of,fluid,and,kinetic,software,developed,under,the,FM-WP2,work-package,as,PA2-8,,incorporating,features,of,Proxyapps,3-1,to,3-4,,and,allowing,for,additional,input,from,PA3-6.,3-6,Staged,introduction,of,additional,neutral,gas/impurity,physics.,It,is,expected,that,the,NEP-,TUNE,community,will,join,in,to,supplement,the,software,with,a,wide-ranging,capability,to,treat,a,wide,range,of,additional,nuclear,,atomic,and,molecular,effects.,2,Braginskii,coefficients,Braginskii’s,transport,coefficients,are,widely,used,in,tokamak,edge,modelling.,Object-oriented,Fortran,code,to,compute,the,Braginskii,coefficients,is,available,at,the,web-site,[6].,Note,that,the,constant,k,is,introduced,such,that,k,=,kB,or,k,=,|e|,(1),where,kB,is,Boltzmann’s,constant,and,|e|,is,the,unit,of,charge,,depending,whether,T,is,measured,in,Kelvin,or,eV,,,so,that,kT,is,in,energy,units.,2.1,General,Except,in,Boltzmann’s,constant,kB,,suffix,B,denotes,a,quantity,from,the,Plasma,Formulary,[7].,See,also,Braginskii’s,paper,[8].,The,subsequent,corrections,by,Epperlein,and,Haines,,and,by,Mikhailovski,and,Tsypin,are,not,relevant,to,this,work.,In,a,magnetic,field,,the,direction,of,which,is,given,by,unit,vector,b,,Goedbloed,and,Poedts,[9],define,three,auxilliary,vectors,for,a,vector,v,,viz.,v∥,=,b(b,·,v),,v∧,=,b,×,v,and,v⊥,=,(b,×,v),×,b,If,v,=,(v1,,v2,,v∥),and,b,is,aligned,with,the,3-axis,in,a,Cartesian,coordinate,system,,then,v∥,=,(0,,0,,v∥),,v∧,=,(−v2,,v1,,0),and,v⊥,=,(v1,,v2,,0),(2),(3),It,may,be,shown,that,a,tensor,T,which,is,symmetric,under,rotation,about,b,has,the,form,(in,Cartesians),T,=,so,that,,,T⊥,−T∧,T⊥,T∧,0,0,,,0,0,T∥,T,·,v,=,T∥v∥,+,T∧v∧,+,T⊥v⊥,5,(4),(5),2.2,Conduction,,Viscous,and,Resistive,Coefficients,The,electron,parallel,thermal,conductivity,in,the,Braginskii,theory,is,given,as,[7],Ke∥,=,3.2,N,kTe,me,τe,(6),a,formula,valid,in,either,cgs,or,SI,units,,where,τe,is,the,electron,relaxation,time,(measured,in,seconds),,defined,below.,The,notation,is,standard,,with,N,the,number,density,of,electrons,,ap-,proximately,the,same,as,the,number,density,of,ions,,me,the,electron,mass,,and,Tα,,s,=,i,,e,the,temperature,of,species,α.,The,perpendicular,electron,thermal,conductivity,satisfies,similarly,Ke⊥,=,4.7,N,kTe,me,τe,·,1,(ωceτe)2,where,the,electron,cyclotron,frequency,Equivalent,expressions,for,ions,are,where,the,ion,cyclotron,frequency,ωce,=,e,me,·,B,Ki∥,=,3.9,N,kTi,mi,τi,Ki⊥,=,2,N,kTi,mi,τi,·,1,(ωciτi)2,ωci,=,ZeB,mi,=,e,mp,·,ZB,A,(7),(8),(9),(10),(11),where,Z,is,the,charge,state,of,the,ion,and,A,its,atomic,mass.,The,definitions,above,have,to,be,interpreted,in,the,context,of,the,equations,given,in,[7],,so,that,thermal,diffusivities,are,obtained,by,dividing,by,3nα/2,where,α,=,i,,e,is,the,species,index.,It,is,also,convenient,to,introduce,the,dimensionless,factors,xe,=,ωceτe,xi,=,ωciτi,Kinematic,viscosities,in,the,Braginskii,theory,may,be,taken,as,νe∥,=,0.73N,kTeτe/(N,me),=,0.73,kTe,me,τe,νe⊥,=,0.51N,kTeτe/(N,me),1,x2,e,=,0.51,τe,1,x2,e,νi∥,=,0.96N,kTiτi/(N,mi),=,0.96,τi,kTe,me,kTi,mi,kTi,mi,τi,1,x2,i,νi⊥,=,0.3N,kTiτi/(N,mi),1,x2,i,=,0.3,6,(12),(13),(14),(15),(16),(17),Key,quantities,in,the,calculation,of,all,these,terms,are,τα,,α,=,i,,e.,The,first,step,in,their,calculation,is,to,convert,their,formulas,,usually,given,in,cgs,,to,SI,units,,giving,√,τe,=,6,2π3,ϵ2,0,√,me,e4,√,τi,=,12,√,ϵ2,0,mp,e4,π3,(kTe)3/2,Z2N,Λ,√,(kTi)3/2,Z4N,Λ,=,3.44,×,10−7,(Te)3/2,Z2(N/1018)Λ,A,=,2.09,×,10−5,(Ti)3/2,√,A,Z4(N/1018)Λ,(18),(19),where,the,notation,is,standard,with,Te,and,Ti,measured,in,eV,,,except,possibly,the,use,of,Λ,for,the,Coulomb,logarithm.,The,above,check,with,expressions,in,Wesson,[10,,§,14].,Note,that,Z2τi,differs,from,τe,in,being,larger,by,a,factor,of,(cid:112)2mi/me,≈,60,A,(also,substituting,Ti,for,Te,is,necessary).,The,factors,in,Z,are,taken,from,the,original,Braginskii,paper,[8].,√,It,follows,that,the,xα,factors,may,be,conveniently,written,xe,=,6.05,×,104,(Te)3/2B,Z2(N/1018)Λ,xi,=,1997,(Ti)3/2B,√,Z3(N/1018),AΛ,(20),(21),The,large,coefficients,in,Equations(20),and,(21),explain,why,classical,transport,is,so,anisotropic.,Substituting,the,explicit,expression,for,τe,in,Equations(6),and,(9),gives,respectively,,the,thermal,parallel,diffusivities,are,κe∥,=,13,√,2π3,1,√,me,√,κi∥,=,16,π3,1,√,mp,ϵ2,0,e4,·,ϵ2,0,e4,·,(kTe)5/2,Z2N,Λ,(kTi)5/2,√,Z4N,Λ,A,and,the,ratios,are,xe,=,√,2π3ϵ2,6,0,mee3,·,√,√,π3ϵ2,0,mpe3,·,An,expression,for,the,perpendicular,ion,conductivity,,maintaining,the,fixed,physical,factors,is,of,interest,(kTi)3/2B,Z3N,Λ,A,(kTe)3/2B,Z2N,Λ,xi,=,12,√,(25),(24),√,κi⊥,=,e2√,√,9,mp,π3ϵ2,0,√,·,Z2N,Λ,A,(kTi)1/2B2,Assuming,Ti,is,measured,in,eV,,,and,N,in,units,of,1018,m−3,,then,and,κi⊥,=,6.67,×,10−4,·,Z2(N/1018)Λ,(Ti)1/2B2,√,A,m2s−1,κe⊥,=,5.26,×,10−5,·,Z2(N/1018)Λ,(Te)1/2B2,m2s−1,7,(22),(23),(26),(27),(28),The,plasma,resistivity,is,taken,as,η,=,ηB/µ0,=,Assuming,Te,is,measured,in,eV,,,then,0.51,√,6,√,mee2,2π3µ0ϵ2,0,·,ZΛ,(kTe)3/2,η,=,ηB/µ0,=,41.9,·,ZΛ,(Te)3/2,m2s−1,(29),(30),2.3,Prandtl,Numbers,The,above,expressions,(except,for,the,resistivity),apply,strictly,only,when,there,are,separate,equa-,tions,for,ion,and,electron,transport,,so,decisions,have,to,be,taken,about,how,to,combine,the,trans-,port,coefficients,to,treat,the,plasma,as,a,single,fluid.,For,the,thermal,transport,,since,pressures,pe,≈,pi,,it,is,sufficient,to,add,the,κα.,However,,the,values,for,ions,and,electrons,are,so,disparate,because,mp,≫,me,that,one,or,other,might,be,neglected,,assuming,B,is,of,order,unity,(in,Tesla),and,Te,≈,Ti,,thus,κe∥,≫,κi∥,and,hence,κ∥,≈,κe∥,,since,It,also,follows,that,(cid:19)2,(cid:18),xi,xe,=,2me,Z2Amp,(cid:18),Ti,Te,(cid:19)3,κe⊥,κi⊥,=,0.078,(cid:18),Ti,Te,(cid:19)1/2,1,√,A,(31),(32),thus,κ⊥,≈,κi⊥.,There,is,the,caveat,that,if,Ti,is,approximately,spatially,constant,radially,,then,κe⊥,might,become,relevant.,As,for,viscosity,,since,the,ion,momentum,is,so,much,greater,than,the,electron,momentum,,then,ν,≈,νi.,For,interchange,motions,where,flows,are,perpendicular,to,the,field,,take,κ,=,κi⊥,,then,on,the,Cambridge,definition,,the,magnetic,Prandtl,number,is,ζ,=,η,κi⊥,=,0.765,√,2,1,µ0,(cid:114),me,mp,·,B2,√,ZN,(kTi)1/2,(kTe)3/2,A,which,evaluates,as,(Tα,in,eV,,,N,in,units,of,1018,m−3),ζ,=,η,κi⊥,=,62,700,·,B2,Z(N/1018),(Ti)1/2,(Te)3/2,√,A,(33),(34),It,may,be,argued,that,it,is,more,appropriate,to,use,the,‘anomalous’,value,of,1,m2s−1,,in,which,case,Equation,(30),without,units,gives,the,‘Cambridge’,magnetic,Prandtl,number.,The,usual,Prandtl,number,is,νi⊥,κi⊥,Note,that,P.H.Roberts,[11],defines,the,magnetic,Prandtl,number,as,P,rM,=,ν/η,=,P,r/ζ,,and,his,definition,is,more,widely,used.,=,0.23,P,r,=,(35),8,3,System,2-1:,2-D,model,of,anisotropic,heat,transport,The,model,for,time,evolution,of,the,temperature,field,T,is,thermal,diffusion,,which,in,a,plasma,gives,where,the,thermal,conductivity,tensor,is,K.,(Compare,the,model,for,a,solid,3,2,N,∂T,∂t,=,∇,·,K∇T,ρmcp,∂T,∂t,=,∇,·,kc∇T,(36),(37),where,the,thermal,conductivity,tensor,is,kc,,ρm,is,the,mass,density,of,the,medium,and,cp,is,its,specific,heat,at,constant,pressure,,implying,that,the,thermal,diffusivity,tensor,is,κ,=,kc/ρmcp.),Introducing,vector,components,as,in,Section,1,,thermal,diffusion,in,a,plasma,after,Braginskii,is,thus,=,∇,·,(cid:0)K∥b[b.∇T,],+,K⊥(∇T,−,b[b.∇T,]),+,K∧b,×,∇T,(cid:1),(38),3,2,N,∂T,∂t,Henceforth,,the,‘wedge’,transport,ie.,in,due,to,the,term,in,K∧,is,neglected,for,the,reason,that,it,may,be,rearranged,to,give,a,convection-like,term,,via,the,identity,where,∇,·,K∧b,×,∇T,=,∇,·,(u∧T,),u∧,=,∇,×,(K∧b),(39),(40),(In,any,event,,if,K∧,is,a,function,purely,of,T,,,and,∇,·,b,=,0,,then,the,terms,in,Equation,(39),vanish.),Expressions,for,the,thermal,diffusivities,κ⊥,and,κ∥,for,the,different,species,are,given,in,Section,2.2,,2,N,,,ie.,κ(⊥,∥)e,i,=,K(⊥,∥)/(,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∥.,However,an,order,of,magnitude,estimate,for,tile,dimensions,is,one,metre,,discharge,timescale,is,one,second,upwards.,For,plasma,properties,assume,N,=,1018,m−3,,Ti,=,Te,=,10,eV,,Z,=,A,=,1,,B,=,3,T,and,solid,temperatures,say,500o,C.,9,Figure,1:,Sketch,of,the,test,configuration,,showing,fieldlines,in,direction,b,and,the,boundary,between,anisotropic,conductor,and,perfect,insulator.,In,Figure,1,,T,=,T0,>,0,over,the,interval,on,the,left,hand,boundary,that,is,connected,by,field,in,direction,b,with,the,thick,red,line,,elsewhere,on,the,blue,fieldlines,,T,=,0.,The,red,region,lies,on,a,black,line,which,denotes,the,boundary,between,anisotropic,conductor,and,perfect,insulator.,The,exact,steady-state,solution,has,T,=,constant,along,fieldlines,,but,numerical,diffusion,will,result,in,non-zero,T,in,the,region,of,blue,fieldlines.,The,relative,size,of,this,numerical,diffusion,must,be,estimated,as,a,function,of,incidence,angle,θ,,where,interest,attaches,to,small,θ,≤,2o.,3.1.2,Intermediate,Case,To,test,curvature,effects,the,whole,2-D,domain,of,Figure,1,could,be,distorted,by,conformal,map-,ping,(which,preserves,angles).,3.1.3,Realistic,Case,This,needs,to,be,3-D,and,involve,JET,divertor,tile,descriptions,derived,from,the,output,of,the,CAD,design,tool,,together,with,information,describing,the,magnetic,field,as,a,function,of,position,,which,will,be,supplied.,The,magnetic,equilibrium,may,be,supplied,analytically,after,Solovev,,but,the,usual,input,is,as,an,.eqdsk,file.,The,EQDSK,G,format,is,a,“non-standard”,standard,for,solutions,ψ(R,,Z),,p(ψ),,I(ψ),of,the,Grad-Shafranov,equation,,where,ψ,is,the,magnetic,flux,and,(R,,Z),are,cylindrical,coordinates,in,planes,normal,to,the,toroidal,direction.,The,functions,p,and,I,give,the,variation,of,the,pressure,and,toroidal,field,respectively.,The,basic,standard,for,EQDSK,G,may,be,found,at:,https://fusion.gat.com/theory/Efitgeqdsk,(which,may,be,password-protected),or,else,at,https://w3.pppl.gov/ntcc/TORAY/G_EQDSK.pdf,The,flux,ψ(R,,Z),is,sampled,at,uni-,formly,spaced,points,on,a,direct,product,grid,,for,which,the,.eqdsk,header,defines,the,mesh-size,,10,as,well,as,other,useful,information,,such,as,the,flux,on,axis,and,at,boundary.,Unfortunately,the,strict,EQDSK,G,standard,uses,a,Fortran,format,that,does,not,require,spaces,between,samples,,hence,there,are,many,variants,for,languages,that,cannot,handle,this,situation,,that,have,intro-,duced,other,features,such,as,mistakes,in,field,helicity,,factors,of,2π,in,the,flux,,etc.,Routines,that,calculate,magnetic,field,B,using,cubic,spline,interpolation,could,be,made,available.,It,would,be,desirable,for,the,output,of,System,2-2,to,be,used.,3.1.4,Extended,Case,An,extended,test,would,allow,for,heat,transfer,in,the,solid,surface,sketched,at,bottom,of,Figure,1,,taking,say,thermal,diffusivity,for,Tungsten,κ,≈,3,×,10−5,m2,s−1.,11,Figure,2:,Sketch,of,the,test,configuration,,showing,tokamak,cross-section,and,the,boundary,of,the,Last,Closed,Flux,Surface,(LCFS).,4,System,2-2:,2-D,elliptic,solver,in,complex,geometry,The,geometry,will,be,representative,of,a,tokamak,cross-section,,possibly,omitting,the,region,con-,taining,the,central,hot,plasma,,so,that,topologically,it,will,be,at,most,as,complex,as,an,annulus,(one-hole).,Figure,2,provides,an,example.,The,Last,Closed,Flux,Surface,(LCFS),may,be,param-,eterised,by,arc-length,in,the,cross-sectional,plane,of,projection.,The,elliptic,equations,to,be,considered,now,follow.,4.1,Simplified,Grad-Shafranov,equation,This,elliptic,equation,is,a,simplified,version,of,the,Grad-Shafranov,equation,,see,[12],R2∇,·,1,R2,∇ψ,=,−2µ0Rjϕ,(41),where,ψ,is,the,poloidal,magnetic,flux,and,(R,,Z),are,cylindrical,coordinates,in,planes,normal,to,the,toroidal,direction,ϕ,,with,the,toroidal,current,jϕ,=,R,dp(ψ),dψ,+,I,µ0R,dI(ψ),dψ,+,jext(R,,Z),(42),The,functions,p,and,I,give,the,variation,as,functions,of,ψ,of,the,pressure,and,toroidal,field,respec-,tively,,and,jext(R,,Z),may,be,produced,in,several,ways,,of,which,the,commonest,is,by,poloidal,field,circuits,,ie.,localised,current,sources,in,cross-section.,Note,that,the,operator,in,Equation,(41),simplifies,to,∂,∂R,which,implies,that,mathematically,,ψ,satisfies,a,steady-state,2-D,advection-diffusion,equation,cor-,responding,to,unit,diffusivity,in,the,flow,uR,=,1/R.,1,R,(43),∂2,∂R2,−,∂2,∂Z2,+,12,To,provide,the,simple,test,case,,take,jϕ,=,jext,only,with,localised,current,sources.,The,boundary,conditions,are,ψ,=,0,on,the,LCFS,and,ψ,∝,1/(cid:112)(R2,+,Z2),as,R,,Z,→,∞.,Note,that,the,Grad-Shafranov,equation,has,been,solved,using,spectral,elements,by,others,,eg.,Sovinec,[3].,4.2,Simplified,non-Boussinesq,vorticity,equation,A,simplified,version,of,the,non-Boussinesq,vorticity,equation,to,be,solved,for,the,scalar,field,Φ(R,,Z),in,cylindrical,polar,coordinates,(R,,Z),is,∇⊥,·,(cid:18),1,B2,∇⊥Φ,(cid:19),=,n,(44),where,B,=,|B|,is,the,amplitude,of,the,imposed,magnetic,field,,density,n,acts,as,a,source,term,,and,the,elliptic,is,to,be,solved,for,Φ,,subject,to,boundary,conditions,Φ,=,0.,The,operator,∇⊥,,ignor-,ing,the,components,of,magnetic,field,directed,within,the,(R,,Z),plane,,reduces,to,the,usual,gradi-,ent,∇,in,cylindrical,polars,of,axisymmetric,fields,,hence,mathematically,Equation,(44),is,equivalent,to,Equation,(41).,n,will,be,set,so,that,n,=,n0(si),on,the,boundaries,,where,arc-length,si,parameterises,the,inner,boundary,if,i,=,1,and,the,outer,if,i,=,2.,n,and,|B|,will,be,specified,functions,of,(R,,Z),that,capture,features,of,the,number,density,n,distribution,and,magnetic,field,intensity,distribution,expected,in,a,tokamak,,Ideally,|B|,would,represent,a,solution,of,the,Grad-Shafranov,equation,from,Section,4.1.,13,5,System,2-3:,1-D,fluid,solver,with,simplified,physics,but,with,UQ,and,realistic,boundary,conditions,5.1,Plasma,Equations,It,is,assumed,that,the,single,spatial,dimension,of,the,problem,corresponds,to,the,arc,length,dis-,tance,along,a,fieldline.,Starting,from,the,two-fluid,model,of,Braginskii,[8],,a,set,of,equations,resembling,those,of,classical,(compressible),hydrodynamics,may,be,derived,by,summing,Bragin-,skii’s,equations,for,number,density,,momentum,and,energy.,Using,the,notation,of,ref,[13],,intro-,ducing,Td,=,Ti,+,Te,,neglecting,the,stress,tensor,terms,(implicitly,setting,δpi,=,0),,and,assuming,B,is,independent,of,time,,the,resulting,system,is,Ud,∂,∂t,(N/B),+,(N,U/B),=,LsSn,B,Ud,∂,∂t,(N,U/B),+,(N,KM,Td/B),+,Ud,(cid:18),3,2,∂,∂t,(cid:18),5,2,∂,∂s,(N,U,KM,Td/B),+,(N,U,3/B),∂,∂s,∂,∂s,1,2,1,2,(N,U,2/B),(cid:19),(cid:19),(N,U,2/B),=,−,1,miB,∂,∂s,(Pi,+,Pe),+,LsSu,miB,+,=,−,1,miB,∂,∂s,(qi,+,qe),+,Ls(SE,i,+,SE,e,),miB,where,Ls,=,∂s∥,∂s,(45),(46),(47),(48),(49),Ud,=,Ls/t0,is,a,speed,measuring,the,importance,of,the,transient,term,,s,parameterises,distance,along,the,fieldline,,and,some,variables,from,[13],have,been,promoted,to,capitals,to,indicate,that,they,retain,their,physical,dimensions.,For,the,case,of,a,fieldline,connecting,two,walls,at,s∥,=,±L,,and,so,Ls,=,2L.,The,constant,KM,is,such,that,s∥,=,L(2s,−,1),KM,=,kB,mi,or,KM,=,|e|,mi,(50),(51),where,kB,is,Boltzmann’s,constant,and,|e|,is,the,unit,of,charge,,depending,whether,T,is,measured,in,Kelvin,or,eV,.,Note,that,in,adding,Eqs.(3),and,(4),of,[13],,equipartition,and,collision,terms,cancel,to,give,Equation,(48).,The,perfect,gas,equation,of,state,will,be,assumed,,so,that,pi,+,pe,mi,=,N,KM,T,The,boundary,conditions,are,that,|U,|,=,|Ms|CS,at,s,=,0,,1,where,the,sound,speed,CS,=,(cid:112),KM,Td,14,(52),(53),and,|Ms|,is,the,Mach,number,,since,Ms,will,be,allowed,to,take,either,sign.,Normally,|Ms|,=,1,so,that,M0,=,−1,and,M1=1,where,the,subscript,corresponds,to,value,of,s.,The,combined,energy,flux,at,each,boundary,has,|Q∥|,=,miCSN,(δeKM,Te,+,δiKM,Ti),≈,miCSN,δKM,Td,(54),if,δ,≈,δe,≈,δi.,For,definiteness,,δ,=,1,2,(δe,+,δi),will,be,assumed.,5.2,Fluid,Equations,with,Sources,It,is,convenient,to,replace,the,source,terms,in,Equations(45)–,(48),by,equivalent,fluxes,F,n(s),=,F,u(s),=,(cid:90),s,0,(cid:90),s,ds∥,Sn,B,Su,miB,i,+,SE,SE,e,miB,ds∥,0,(cid:90),s,0,F,E,(s),=,2,ds∥,and,write,F,Q(s),=,−,1,miB,(qi,+,qe),(55),(56),(57),(58),The,convention,with,respect,to,limits,of,integration,is,that,they,are,specified,in,terms,of,parame-,terised,length,and,to,use,the,appropriate,relation,for,s∥(s),,thus,in,the,case,of,Equation,(50),,the,lower,limit,of,0,corresponds,to,s∥(s),=,−L.,The,forms,the,sources,take,are,discussed,below,in,Section,5.3.,Observing,the,identity,1,B,∂,∂s,BnBKM,Td,=,∂,∂s,(nBKM,Td),+,nBKM,Td,∂,∂s,(ln,B),(59),and,the,frequent,appearance,of,nB,=,N/B,,the,governing,equations,in,dimensional,form,become,Ud,∂,∂t,nB,+,Ud,∂,∂t,(nBU,),+,(nBKM,Td),+,(nBU,KM,Td),+,F,n,(nBU,),=,∂,∂s,∂,(nBU,2),=,−,∂s,(cid:19),∂,∂s,∂,∂s,1,(nBU,2),2,1,2,(nBU,3),(cid:19),+,=,−,∂,∂s,F,Q,+,1,2,∂,∂s,F,E,(nBKM,Td),+,∂,∂s,F,u,Ud,(cid:18),3,2,∂,∂t,(cid:18),5,2,∂,∂s,(60),(61),(62),(63),where,the,derivative,of,ln,B,has,been,neglected.,The,boundary,conditions,on,U,are,unchanged,and,|Q∥|,=,miCSnBδKM,Td,The,Equations(60)–,(63),together,with,boundary,condition,Equation,(64),are,in,units,such,that,equivalence,may,easily,be,established,with,those,of,ref,[13],(by,setting,Ud,=,1,and,identifying,s,with,s∥).,To,proceed,,it,may,be,helpful,to,make,the,preceding,system,of,equations,dimensionless,,by,scaling,nB,with,respect,to,N0/B0,,Td,with,respect,to,T0,,U,with,respect,to,C0,and,B,with,respect,to,B0.,If,the,subscript,0,corresponds,to,the,value,of,a,variable,at,s,=,0,,then,it,is,convenient,to,KM,T0.,The,resulting,system,may,be,deduced,from,the,coupled,model,in,Section,5.4.,take,C0,=,(64),√,15,5.3,Explicit,Sources,The,above,work,concentrates,on,the,case,where,the,source,terms,are,regarded,as,given,,however,it,is,worth,describing,the,form,of,the,additional,sources,that,may,be,at,least,locally,important.,From,ref,[13],,the,plasma,sources,are,given,by,(with,the,convention,that,suffix,‘n’,denotes,neutral,species),Sn,=,NnN,⟨σv⟩ION,−,N,2⟨σv⟩REC,+,Sn,⊥,Su,=,NnN,⟨σv⟩ION,Un,−,N,2⟨σv⟩RECU,+,NnN,(Un,−,U,)⟨σv⟩CX,SE,=,SE,i,+,SE,e,=,NnN,⟨σv⟩ION,(,3,2,kTn,+,1,2,−,N,2⟨σv⟩REC(,3,kTi,+,2,(cid:18),3,2,⊥i,+,SE,−,NnN,kQH,+,SE,⊥e,+,NnN,⟨σv⟩CX,n,−,kIH,),mnU,2,1,2,miU,2),k(Tn,−,Ti),+,(cid:19),mn(U,2,n,−,U,2),1,2,(65),(66),(67),(68),Here,suffix,⊥,denotes,the,effectively,given,source,terms,arising,from,cross-field,contributions,,suffices,ION,,,REC,and,CX,denote,respectively,reaction,rates,⟨σv⟩,for,ionisation,,recombination,and,charge-exchange,,IH,is,the,Hydrogen,reionisation,potential,,and,QH,is,the,cooling,rate,due,to,excitation.,Since,the,sources,appear,in,the,analysis,primarily,as,integrals,starting,at,s,=,0,,study,of,Equa-,tions(65)–,(67),concentrates,on,this,region,,where,plasma,velocity,U,<,0,and,neutral,velocity,Un,>,0,with,the,two,having,approximately,the,same,magnitude.,There,,Equation,(65),has,only,one,negative,term,,due,to,recombination,,but,from,the,cross-section,data,in,ref,[13],,this,could,dominate,only,below,2,eV.,All,terms,in,Equation,(66),are,positive,near,s,=,0,as,the,two,velocities,reinforce.,Equation,(67),contains,two,terms,which,are,always,negative,and,an,ionisation,term,which,is,also,negative,below,IH,/2,≈,7,eV,,thus,for,example,,the,cross-field,source,terms,S⊥i,e,must,be,positive,for,SE,>,0.,The,sources,of,neutrals,may,be,deduced,from,the,ionisation,and,charge-exchange,terms,in,Equa-,tions(65)–,(67),,viz.,n,=,−NnN,⟨σv⟩ION,+,Sn,Sn,⊥n,Su,n,=,−NnN,⟨σv⟩ION,Un,−,NnN,(Un,−,U,)⟨σv⟩CX,3,2,n,=,−NnN,⟨σv⟩ION,(,n,−,kIH,),mnU,2,kTn,+,SE,1,2,(cid:18),3,2,k(Tn,−,Ti),+,1,2,(cid:19),mn(U,2,n,−,U,2),−,Nn⟨σv⟩CX,+,SE,⊥n,(69),(70),(71),(72),(73),The,S⊥n,terms,are,hard,to,quantify,,but,if,these,are,neglected,,it,is,clear,that,Sn,the,obverse,of,the,positive,plasma,sources.,Similarly,it,is,likely,that,SE,n,<,0,if,SE,>,0,n,<,0,and,Su,n,<,0,is,16,5.4,Coupled,System,Working,within,the,flux-tube,geometry,,the,equations,for,neutral,transport,take,the,same,form,as,those,used,for,plasma,above,,however,the,boundary,conditions,are,different.,They,become,at,s,=,0,,supposing,that,T(0),=,τ,2T0,and,masses,mn,=,mi,,in,dimensionless,units,,n,=,R2,=,|M0|,Rp,τ,u,=,−τ,T,=,τ,2,(74),(75),(76),where,n,,u,and,T,are,neutral,density,,velocity,and,temperature,made,dimensionless,with,respect,to,the,same,N0,and,T0,as,the,corresponding,plasma,quantities,,and,Rp,is,the,recycling,coefficient.,Note,the,usage,of,a,sans-serif,font,to,denote,dimensionless,neutral,species,quantities,,and,that,n,does,not,however,include,a,factor,of,˜b,=,B/B0,,as,the,neutral,flux,is,not,constrained,by,the,flux,tube.,A,system,explicitly,modelling,the,coupling,between,plasma,and,neutrals,may,be,derived,by,making,non-dimensionless,the,sources,set,out,in,Section,5.3,,and,assuming,mn,=,mi,,T,=,Ti,=,τ,2T,,,giving,5,equations,for,the,evolution,of,plasma,density,,velocity,,total,temperature,,neutral,density,and,neutral,velocity:,ϵr,ϵr,∂,∂t,(nu),+,∂,∂t,∂,∂s,n,+,∂,∂s,(nu),=,σI,nn,+,∂,∂s,f,n,(nu2,+,nT,),=,σI,nnu,+,σCnn(u,−,u),+,∂,∂s,f,u,ϵr,∂,∂t,(cid:0)(g,−,2)nT,+,nu2)(cid:1),+,∂,∂s,(cid:0)gnuT,+,nu3(cid:1),=,σI,nn(3[τ,2T,−,TH,],+,u2),+,σCnn(u2,−,u2),∂,∂s,−,σEnn,+,f,E,ϵr,n,+,∂,∂s,(nu),=,−σI,nn,∂,∂t,(nu2,+,nτ,2T,),=,−σI,nnu,−,σCnn(u,−,u),ϵr,∂,∂t,(nu),+,∂,∂s,(77),(78),(79),(80),(81),where,ϵr,=,Ls/(t0C0),with,t0,a,characteristic,timescale.,The,reaction,reaction,rates,are,made,dimensionless,by,division,by,C0/(LsN0)(≃,3.1×10−16,m3s−1,for,representative,parameter,values),,so,that,σI,=,σC,=,σE,=,TH,=,⟨σv⟩ION,C0/LSN0,⟨σv⟩CX,C0/LSN0,2QH,C0T0/LSN0,2IH,3T0,≃,9/T0(in,eV),17,(82),(83),(84),(85),where,IH,and,QH,are,as,defined,in,ref,[13].,It,is,of,interest,to,allow,a,stochastic,(‘turbulent’),contribution,to,the,terms,f,n,u,E.,5.5,Neutrals,as,2-D,or,3-D,fluid,The,above,Equations(80),and,(81),are,a,reduction,to,1-D,of,the,system,ϵr,∂,∂t,n,+,∇,·,(nu),=,−σI,nn,ϵr,∂,∂t,(nu),+,∇,·,(nuu),+,∇nT,=,−σI,nnu,−,σCnn(u,−,u),(86),(87),5.6,Uncertainty,Quantification,Polynomial,chaos,(PC),refers,to,a,situation,whereby,probability,functions,are,expanded,as,Hermite,polynomials,,and,generalised,Polynomial,chaos,(gPC),to,expansions,using,Hermite,and,other,polynomial,sets,{Ψj(ξ)},[14].,Thus,for,example,,suppose,θ,to,denote,a,random,event,,and,the,number,density,field,to,have,the,following,representation,in,terms,of,a,finite,number,P,of,such,modes.,PC(cid:88),n(x,,t,,θ),=,nj(x,,t)Ψj,(ξ(θ)),(88),j=0,where,{nj(x,,t)},is,the,set,of,deterministic,coefficients,of,the,“random,trial,basis”,,ie.,the,set,{Ψj,(ξ(θ))},where,ξ(θ),is,a,multi-dimensional,random,variable,with,a,specific,probability,distribution,as,a,func-,tion,of,the,random,parameter,0,≤,θ,≤,1.,Note,that,the,Ψj,are,the,set,of,multi-dimensional,Hermite,polynomials,if,ξ,is,a,vector.,Typically,but,not,necessarily,the,ξ(θ),will,be,Gaussians.,Expressions,like,Equation,(89),may,be,substituted,into,a,governing,equation,of,say,advection,type,for,n,,and,the,result,simplifies,because,spatial,operators,do,not,interact,with,the,random,variables,,then,taking,the,inner,product,with,Ψk,(ξ(θ)),yields,∂nk,∂t,+,PC(cid:88),PC(cid:88),i=0,j=0,eijk,∂uinj,∂s,=,0,(89),where,eijk,is,a,weighted,integral,of,triple,products,of,Ψi.,Hence,there,are,now,PC,equations,instead,of,one,for,n.,There,is,an,alternative,treatment,by,Polynomial,Chaos,Expansion,(PCE),which,is,non-intrusive,and,therefore,preferred.,It,relies,on,projecting,a,set,of,randomly,selected,solutions,onto,a,Hermite,or,similar,basis,,see,further,description,in,ref,[15,,end,§,2.1.1].,18,6,System,2-4:,Spatially,1-D,plasma,model,incorporating,velocity,space,effects,The,following,simple,model,is,after,Taitano,et,al,[16,,17],∂fe,∂t,∂fi,∂t,+,vex,+,vix,∂fe,∂x,∂fi,∂x,+,+,ϵ0,∂E,∂t,(cid:88),+,α,qαnuα,−,qe,me,qi,mi,(cid:88),α,E,·,E,·,∂fe,∂v,∂fi,∂v,=,0,=,0,(90),qαnuα,=,0,Equations,90,are,the,electron,and,ion,Vlasov,equations,and,Ampere’s,equation,respectively.,The,quantities,me,,mi,,fe,,fi,,ve,,vi,,qe,,qi,,E,,ϵ0,,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,(90)represents,a,generalisation,of,the,system,in,ref,[16],,where,for,vector,quantities,,the,x-component,is,always,implied,,in,the,usual,notation,the,original,system,is,1d1v,rather,than,1d3v,as,above,,where,particles,move,according,to,dx,dt,dvα,dt,=,vαx,=,qα,mα,E,(91),(Motion,in,(y,,z),is,neglected,,the,3-D,electromagnetic,version,of,Equation,(91),appears,in,Sec-,tion,10).,The,(cid:80),·,term,denotes,for,example,a,spatially,averaged,summed,quantity,and,is,included,to,en-,force,Galilean,invariance.,The,solutions,fe,and,fi,of,the,Vlasov,equations,are,functions,of,space,variable,x,,velocity,v,,and,time.,Ampere’s,equation,is,solved,for,the,self-consistent,electric,field,E,,which,is,a,function,of,space,variable,x,and,time.,The,boundary,conditions,used,are,periodicity,in,x,,and,zero,at,infinity,in,|v|.,Initial,conditions,which,might,be,used,for,the,distribution,functions,are,from,ref,[16],,f0(x,,v,,u0,,T0),=,n0(x),(cid:112)2πkT0/m,(cid:18),exp,−,(cid:19),m(v,−,u0)2,2kT0,n0(x),=,n(t,=,0,,x),=,1,+,αn,cos(kwx),(92),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,(93),where,fα,is,the,distribution,function,for,species,α,at,time,t.,replaced,by,a,sum,over,particles.),(In,practice,the,integral,would,be,19,Note,that,periodic,boundary,conditions,are,of,limited,value,in,practice,,and,attention,should,be,given,to,minimal,modifications,of,the,above,problem,where,there,is,1.,a,flux,of,momentum,across,the,domain,(inflow,and,outflow,boundary,conditions),2.,reflection,of,particles,at,the,boundaries,3.,a,source,of,plasma,within,the,domain,,and,outflow,boundaries,4.,and,where,the,spatial,dimension,corresponds,to,arc,length,s,along,a,fieldline,(implies,n,replaced,by,n/|B|,,cf.,Section,5.1).,20,7,System,2-5:,Spatially,1-D,multispecies,plasma,model,7.1,Fluid,model,For,a,multispecies,plasma,,there,is,a,system,of,Boltzmann,equations,to,be,solved,,one,for,each,species,,each,of,form,in,3,spatial,dimensions,L7fα,=,(cid:88),β,Q(fα,,fβ),+,Sα,(94),where,L7,is,the,7-D,Lie,derivative,(space,,velocity-space,and,time,make,up,the,3,+,3,+,1,=,7,dimensions,,α,,β,are,species,labels,and,Q,is,the,Boltzmann,collision,operator.,The,multispecies,equations,are,derived,following,Grad,[18,,§,6],by,substituting,in,Equation,(94),fα,=,exp(−λHα)Fα(x,,v,,t),(95),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,(96),The,resulting,system,is,linearised,and,solved,by,iteration,to,give,the,multispecies,plasma,fluid,equations,in,Zhdanov,[18,,§,6].,There,are,believed,to,be,typographical,errors,in,Zhdanov,,so,cross-checking,is,needed.,To,see,Grad’s,approach,applied,to,classical,fluids,see,for,example,[19,,§,8].,7.2,Coupling,to,particles,Other,,less,collisional,species,are,to,be,treated,as,particles,as,in,Section,6,and,coupled,via,Sα.,Mathematical,forms,for,Sα,will,be,guided,by,the,emerging,results,from,particles’,method,research.,21,8,System,2-6:,Spatially,2-D,plasma,model,incorporating,velocity,space,effects,With,reference,to,the,Hermes,web-site,[20],,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,=,Σα̸=eZαnα,=,Zini,∂ne,∂t,∂pe,∂t,∂pi,∂t,∂ω,∂t,=,−∇,·,(nevE×B),+,∇,·,jsh,−,necs,L∥,+,Sn,e,=,−∇,·,(pevE×B),−,=,−∇,·,(pivE×B),−,+,Sp,e,+,Df,pe∇,·,(κe⊥ne∇⊥kTe),+,Sp,i,+,Df,pi∇,·,(κi⊥ni∇⊥kTi),1,|qe|,δepecs,L∥,δipics,L∥,=,−∇,·,(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|necs,|qe|Φ,kTref,L∥,∇,·,jsh,=,−,vE×B,=,B,×,∇Φ,B2,(97),(98),(99),(100),(101),(102),(103),(104),(105),(106),(107),(108),where,it,has,been,assumed,that,there,is,a,single,ion,species,i.,(Cases,with,Zi,̸=,1,need,checking,,so,the,ion,species,has,to,be,assumed,Hydrogenic,Zi,=,1,for,now.),The,sheath,heat,transmission,coefficients,are,from,Stangeby,[21,,§,2.8],(who,uses,the,notation,‘γα’),δe,=,6.5,,δi,=,2.0,(109),Notes,1.,The,system,is,defined,in,two,space,dimensions,,so,both,number,ne,and,charge,(ω),density,are,defined,per,square,metre,,and,pressure,as,force,per,unit,length.,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),have,been,added,so,that,Hermes-3,simulations,may,be,compared,minimising,complications,due,to,sub-grid-scale,effects.,22,3.,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.,Successive,simplifications,for,the,pressure,‘dissipation’,replace,Df,pe∇,·,(κe⊥ne∇⊥kTe),→,Df,pe∇,·,(κe⊥∇⊥pe),Df,pi∇,·,(κi⊥ni∇⊥kTi),→,Df,pi∇,·,(κi⊥∇⊥pi),and,then,with,adjusted,numerical,multiplication,factors,Dκ,Df,pe∇,·,(κe⊥∇⊥pe),→,Dκ,Df,pi∇,·,(κi⊥∇⊥pi),→,Dκ,f,pi:,f,pe,,Dκ,f,pe∇,·,∇⊥pe,f,pi∇,·,∇⊥pi,(110),(111),(112),(113),(114),which,are,increasingly,inaccurate,but,increasingly,easier,to,implement.,8.1,Dimensionless,units,NEPTUNE,will,not,use,dimensionless,units,because,of,the,potential,for,confusion.,However,,the,Hermes-3,models,are,expressed,in,dimensionless,variables,,and,it,is,necessary,to,understand,how,this,has,been,done.,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.,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.,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.,(It,is,suggested,that,in,keeping,with,the,application,to,the,Exascale,,the,pure,number,Nref,=,1018.),In,the,Hermes-3,equations,,variable,‘cs’,becomes,the,Mach,number,,variable,‘e’=-1,,L∥,is,in,units,of,ωci/cs,ref,.,(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.,23,8.2,Boundary,conditions,Boundary,conditions,employed,are,typically,Neumann,or,Dirichlet.,8.3,Initial,conditions,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,(115),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′),=,1,−,t′/tH,,,t′,=,t,−,tR,(116),(117),where,mathematically,8.4,Kinetic,effects,A,particle,(PIC),model,,see,Section,9,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,11,for,treatment,of,particle,inter-,actions.,(Necessary,Monte-Carlo,techniques,for,particle,production,and,motion,are,described,in,Section,10.),e,,,Sp,24,9,Systems,3:,Kinetic,models,The,following,generic,transport,equation,[22,,§,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),(118),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,(118),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),(119),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,[23],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.,9.1,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,[23,,§,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),(120),(121),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),25,(122),The,fundamental,import,of,(2),is,that,E(xm),=,Σ′,mV,eG(xm,,x′,m)ρc(x′,m),(123),where,G(xm,,x′,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′,F(xp),=,q2,αΣmΣ′,mW,(xp,−,xm)V,eG(xm,,x′,m)W,(xp,−,x′,m),(124),Only,the,electrostatic,field,E,is,shown,in,Equation,(120),but,the,above,analysis,holds,more,gener-,ally,for,the,Lorentz,force.,10,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.,10.1,Prerequisites,10.1.1,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.,10.1.2,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,[24],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.,26,10.1.3,Sampling,from,a,specified,distribution,Generally,Textbooks,such,as,Kalos,and,Whitlock,[25],(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′)dx′,(125),Equation,(125),may,be,solved,explicitly,for,x,in,many,important,cases,Gaussian,distribution,This,may,be,sampled,using,the,Box-Muller,method,[25,,§,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,(126),(127),Knudsen,cosine,Sample,by,accept-reject,from,the,Knudsen,cosine,distribution,below,Equa-,tion,(128),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).,10.2,Sources,and,sinks,of,neutrals,10.2.1,Knudsen,distribution,As,suggested,in,the,TN-07,Neptune,report,by,Parra,,Barnes,and,Hardman,[26],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,(128),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,[26],,so,despite,the,”f,”,notation,,it,has,different,units,to,the,other,distributions,f,.,27,10.2.2,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.,10.2.3,Recycling,This,cannot,be,properly,treated,without,coupling,to,a,plasma,model,,hence,is,treated,in,Sec-,tion,11.4.1.,10.2.4,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.,10.3,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.,10.3.1,Perfect,Absorption,Any,particles,that,reach,a,domain,boundary,are,deleted.,10.3.2,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,(129),10.3.3,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.,28,Figure,3:,Extract,from,publication,indicated,in,the,text.,10.3.4,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,negelcted,in,an,initial,investigation.,11,System,3-3:,Interaction,with,2-D,plasma,model,11.1,Prerequisites,11.1.1,Cross-section,data,input,Cross-section,data,will,be,obtained,from,the,ADAS,library,[27].,Cross-section,data,⟨σv⟩,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.,[13].,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,(130),(131),29,Figure,4:,Configuration,for,ionisation.,Neutrals,,indicated,by,arrows,,moving,against,a,background,of,thermal,electrons,shown,as,dots.,11.2,Physical,Models,11.2.1,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,(132),(133),where,RION,(⟨σv⟩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),(134),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,[28],,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),(135),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|},(136),x,E,30,then,for,every,particle,,provided,all,particles,have,the,same,weight,(ie.,identical,superparticles),,approximately,pp,=,pT,=,1,−,exp,(−νσmaxt),(137),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),(138),11.2.2,Detailed,model,For,sources,in,the,fluid,equations,due,to,ionisation,of,neutrals,by,electrons,,the,formulae,,cf.,Equations(69)–,(71),([13,,Eqs.(34)-(36)]),are,Sn,e,=,NnN,⟨σv⟩ION,Sv,n,=,NnN,⟨σv⟩ION,vn,Sp,NnN,⟨σv⟩ION,kIH,2,3,e,=,−,2,3,Sp,i,=,NnN,⟨σv⟩ION,(,3,2,kTn,+,1,2,mnv2,n),(139),(140),(141),(142),Note,that,other,effects,due,to,charge-exchange,and,recombination,may,be,deduced,from,the,equations,in,Section,5.3.,Monte,Carlo,calculation,may,then,proceed,using,a,total,cross-section,for,all,three,interactions.,11.2.3,Simplified,model,For,an,exploratory,calculation,,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,Equa-,tions(139)–,(142),above,become,Sn,e,=,NnN,⟨σv⟩ION,Sv,n,=,0,Sp,e,=,0,2,Sp,i,=,3,NnN,⟨σv⟩ION,(,3,2,kTn,+,1,2,mnv2,n),(143),(144),(145),(146),Compared,to,the,introductory,model,in,Section,11.2.1,,spatial,variations,in,background,density,and,cross-section,are,handled,by,the,null,collision,method,,see,Section,11.5.2.,31,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,(147),(148),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,(149),(150),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.,11.3,Initial,conditions,These,are,defined,separately,for,the,fluid,(‘continuum’),species,and,the,particle,species.,11.4,Boundary,conditions,Only,conditions,coupling,both,particle,and,fluid,species,are,to,specified,here,,otherwise,see,sep-,arate,treatments.,11.4.1,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.),32,11.5,Calculating,particle,interactions,11.5.1,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,(151),(152),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),(153),(154),(155),11.5.2,Simplified,models,The,PIC-MCC,software,[28],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.,11.5.3,Preferred,approach,The,“Direct,Sampling”,approach,of,Brown,and,Martin,[29],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,[30,,31].,33,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′)ds′,We,assume,only,that,σ,is,finite,and,σ(s),≥,0.,Note,that,dτ,ds,=,σ(s),(156),(157),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,(∞)),(158),Then,the,probability,density,function,(pdf),for,a,collision,occurring,after,a,particle,has,traveled,a,distance,s,through,the,medium,is,given,by,[32,,§,7],p(s),=,PN,Cδ(s,−,s∞),+,dτ,ds,exp,(−τ,(s)),(159),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,(159),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′)ds′,(160),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,0,34,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,(161),,note,that,g(s′),=,1,G,dτ,ds′,exp,(−τ,(s′)),G,=,(1,−,PN,C),(cid:90),s,0,g(s′)ds′,=,1,G,(cid:90),τ,(s),0,exp,(−τ,)dτ,Using,Equations(160),and,(161),,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,(156),,viz.,τs,−,ln(1,−,Gξ),τs,=,(cid:90),s,0,σ(s′)ds′,(161),(162),(163),(164),(165),(166),When,σ(s′),has,a,simple,functional,form,,Equation,(166),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,(166),,however,,can,be,readily,solved,numerically,for,s,using,Newton,iteration,with,f,=,(cid:82),sσ(s′)ds′,−,s,,starting,with,an,initial,estimate,0,s0,=,τs/σ(0),[29].,Because,df,/ds,≤,0,,f,is,monotone,and,there,can,be,at,most,one,root.,For,cases,where,σ(s′),≥,0,,the,Newton,iteration,is,guaranteed,to,converge.,However,,if,σ(s′),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′).,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|,35,(167),(168),12,Summary,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,Valu-,able,input,from,Ben,Dudson,is,also,acknowledged.,Stuart,Henderson,provided,helpful,advice,concerning,plasma,radiation.,References,[1],Proxy,Applications.,https://proxyapps.exascaleproject.org,,2020.,Online;,accessed,May,2020.,[2],W.,Arter,,L.,Anton,,D.,Samaddar,,and,R.,Akers.,ExCALIBUR,Fusion,Modelling,System,Science,Plan.,Technical,Report,CD/EXCALIBUR-FMS/0001,,UKAEA,,11,2019.,https://www.metoffice.gov.uk/binaries/content/assets/metofficegovuk/pdf/,research/spf/ukaea-excalibur-fms-scienceplan.pdf.,[3],E.C.,Howell,and,C.R.,Sovinec.,Solving,the,Grad–Shafranov,equation,with,spectral,ele-,ments.,Computer,Physics,Communications,,185(5):1415–1421,,2014.,http://dx.doi.org/,10.1016/j.cpc.2014.02.008.,[4],J.P.,Gunn,,T.,Hirai,,Y.,Corre,,F.,Escourbiac,,A.,Grosjean,,and,R.A.,Pitts.,A,study,of,planar,toroidal–poloidal,beveling,of,monoblocks,on,the,ITER,divertor,outer,vertical,target.,Nuclear,Fusion,,59(12):126043,,2019.,[5],R.,Dejarnac,,J.P.,Gunn,,P.,Vondracek,,M.,Komm,,R.,Panek,,and,R.A.,Pitts.,Physics,of,toroidal,gap,heat,loading,on,castellated,plasma-facing,components.,Nuclear,Materials,and,Energy,,19:19–27,,2019.,[6],Code,generation,QPROG,style,,example,of,Braginskii,plasma,transport,coefficients.,https:,//github.com/wayne-arter/smardda-misc,,2017.,Accessed:,December,2020.,[7],J.D.,Huba.,NRL,Plasma,Formulary.,Technical,Report,NRL/PU/6790–07-500,,Naval,Research,Laboratory,,Washington,,2007.,Online,version,dated,2009,at,https://apps.dtic.mil/dtic/,tr/fulltext/u2/a499299.pdf.,[8],S.I.,Braginskii.,Transport,Processes,in,a,Plasma.,In,M.A.Leontovich,,editor,,Reviews,of,Plasma,Physics,Vol.1,,pages,205–311.,Consultants,Bureau,,New,York,,1965.,[9],J.P.,Goedbloed,and,S.,Poedts.,Principles,of,magnetohydrodynamics:,with,applications,to,laboratory,and,astrophysical,plasmas.,Cambridge,University,Press,,2004.,[10],J.A.,Wesson.,Tokamaks,,3rd,Edition.,Clarendon,Press,,Oxford,,2003.,[11],P.H.,Roberts.,An,Introduction,to,Magnetohydrodynamics.,Longmans,,London,,1967.,36,[12],L.C.,Appel,,I.,Lupelli,,and,JET,Contributors.,Equilibrium,reconstruction,in,an,iron,core,toka-,mak,using,a,deterministic,magnetisation,model.,Computer,Physics,Communications,,223:1–,17,,2018.,[13],E.,Havl´ıˇckov,´a,,W.,Fundamenski,,F.,Subba,,D.,Coster,,M.,Wischmeier,,and,G.,Fishpool.,Benchmarking,of,a,1D,scrape-off,layer,code,SOLF1D,with,SOLPS,and,its,use,in,modelling,long-legged,divertors.,Plasma,Physics,and,Controlled,Fusion,,55(6):065004,,2013.,[14],D.,Xiu,and,G.,E.,Karniadakis.,Modeling,uncertainty,in,flow,simulations,via,generalized,poly-,nomial,chaos.,Journal,of,computational,physics,,187(1):137–167,,2003.,[15],W.,Arter,,E.,Threlfall,,and,J.,Parker.,layer,design,for,Uncer-,tainty,Quantification.,Technical,Report,CD/EXCALIBUR-FMS/0024-M3.1.3,,UKAEA,,10,2020.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_,reports/CD-EXCALIBUR-FMS0024-M3.1.3.pdf.,Report,on,user,[16],W.T.,Taitano,,D.A.,Knoll,,L.,Chac,´on,,and,G.,Chen.,Development,of,a,Consistent,and,Stable,Fully,Implicit,Moment,Method,for,Vlasov–Amp,`ere,Particle,in,Cell,(PIC),System.,SIAM,Journal,on,Scientific,Computing,,35(5):S126–S149,,2013.,[17],G.,Chen,,L.,Chac,´on,,and,D.C.,Barnes.,An,energy-and,charge-conserving,,implicit,,electro-,static,particle-in-cell,algorithm.,Journal,of,Computational,Physics,,230(18):7018–7036,,2011.,[18],V.M.,Zhdanov.,Transport,processes,in,multicomponent,plasma.,CRC,Press,,2002.,[19],W.B.,Thompson.,An,Introduction,to,Plasma,Physics.,Pergamon,,1962.,[20],Hermes,plasma,edge,simulation,model:,Hermes-3,,a,hot,ion,multifluid,drift-reduced,model.,https://github.com/bendudson/hermes-3,,2021.,Accessed:,June,2021.,[21],P.C.,Stangeby.,The,plasma,boundary,of,magnetic,fusion,devices.,Taylor,&,Francis,,New,York,,2000.,[22],J.J.,Duderstadt,and,W.R.,Martin.,Transport,Theory.,Wiley,,1979.,[23],R.W.,Hockney,and,J.W.,Eastwood.,Computer,Simulation,Using,Particles.,IOP,Publishing,,1988.,[24],H.,Niederreiter.,Random,Number,Generation,and,Quasi-Monte,Carlo,Methods.,Society,for,Industrial,Mathematics,,1992.,[25],M.H.,Kalos,and,P.A.,Whitlock.,Monte,Carlo,Methods.,Vol.,1:,basics.,Wiley-Interscience,New,York,,NY,,USA,,1986.,[26],F.,I.,Parra,,M.,Barnes,,and,M.,Hardman.,2D,drift,kinetic,models,with,wall,boundary,conditions.,Technical,Report,2047357-TN-07-01,,UKAEA,Project,Neptune,,2021.,https://github.com/,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047357/TN-07.pdf.,[27],The,ADAS,Project.,Atomic,Data,and,Analysis,Structure,ADAS,website.,http://www.adas.,ac.uk/,,2020.,Accessed:,July,2020.,37,[28],J.P.,Verboncoeur.,Particle,simulation,of,plasmas:,review,and,advances.,Plasma,Physics,and,Controlled,Fusion,,47(5A):A231–A260,,2005.,[29],F.B.,Brown,and,W.R.,Martin.,Direct,Sampling,of,Monte,Carlo,Flight,Paths,in,Media,with,Con-,In,ANS,Mathematics,and,Computation,Topical,Meeting,,tinuously,Varying,Cross-sections.,Gatlinburg,,TN,,April,,pages,6–11,,2003.,[30],Z.,Li,,K.,Wang,,and,X.,Zhang.,Research,on,applying,neutron,transport,Monte,Carlo,method,in,materials,with,continuously,varying,cross,sections.,2011.,https://inis.iaea.org/,collection/NCLCollectionStore/_Public/48/022/48022323.pdf.,[31],H.,Belanger,,D.,Mancusi,,and,A.,Zoia.,Review,of,Monte,Carlo,methods,for,particle,transport,in,continuously-varying,media.,The,European,Physical,Journal,Plus,,135(11):1–22,,2020.,[32],E.E.,Lewis,and,W.F.,Miller,Jr.,Computational,Methods,of,Neutron,Transport.,American,Nu-,clear,Society,,La,Grange,Park,,Il,,1993.,[33],L.,Golub,and,J.M.,Pasachoff.,The,solar,corona,,2nd,Edition.,Cambridge,University,Press,,2010.,[34],V.A.,Ambartsumyan,,editor.,Theoretical,Astrophysics,,translated,by,J.B.Sykes.,Pergamon,Press,,1958.,[35],M.,O’Mullane.,Atomic,Processes,in,Plasmas.,Slides,from,45th,Culham,Plasma,Physics,Summer,School,,2008.,CCFE,intranet,,not,publicly,available.,[36],R.J.,Bray,and,R.E.,Loughhead.,Sunspots.,Dover,Publications,,1964.,[37],The,ADAS,Project.,Dissemination,of,atomic,data,and,analysis,structure,adas,website.,https:,//open.adas.ac.uk/,,2020.,Accessed:,July,2020.,[38],Gwyn,Williams.,Electron,Binding,Energies.,https://userweb.jlab.org/~gwyn/ebindene.,html,,2000.,Accessed:,July,2020.,[39],H.,Summers,,M.,O’Mullane,,and,A.,Whiteford.,ADAS:,Atomic,data,,modelling,and,analysis,for,fusion.,https://icamdata2006.obspm.fr/documents/icam_pdf/Summers.pdf,,2006.,Slides,from,Meudon.,[40],N.R.,Badnell,,M.G.,O’Mullane,,H.P.,Summers,,Z.,Altun,,M.A.,Bautista,,J.,Colgan,,T.W.,Gorczyca,,D.M.,Mitnik,,M.S.,Pindzola,,and,O.,Zatsarinny.,Dielectronic,recombination,data,for,dynamic,finite-density,plasmas-I.,Goals,and,methodology.,Astronomy,&,Astrophysics,,406(3):1151–1165,,2003.,[41],J.-Ch.,Sublet,,J.W.,Eastwood,,J.G.,Morgan,,M.R.,Gilbert,,M.,Fleming,,and,W.,Arter.,FISPACT-,II:,An,Advanced,Simulation,System,for,Activation,,Transmutation,and,Material,Modelling.,Nuclear,Data,Sheets,,139:77–137,,2017.,http://dx.doi.org/10.1016/j.nds.2017.01.,002,,website:,https://fispact.ukaea.uk/.,[42],S.S.,Henderson,,M.,Bluteau,,A.,Foster,,A.,Giunta,,M.G.,O’Mullane,,T.,P,¨utterich,,and,H.P.,Summers.,Optimisation,and,assessment,of,theoretical,impurity,line,power,coefficients,rele-,vant,to,ITER,and,DEMO.,Plasma,Physics,and,Controlled,Fusion,,59(5):055010,,2017.,38,[43],S.S.,Henderson,,M.M.,Bluteau,,M.G.,O’Mullane,,and,H.P.,Summers.,Improvements,of,impu-,rity,line,ITER,and,DEMO,via,systematic,optimisation,of,atomic,structure.,Technical,Report,UKAEA-CCFE-PR(18)31,,CCFE,,2018.,[44],W.,Arter.,Equations,for,EXCALIBUR/NEPTUNE,Proxyapps.,port,CD/EXCALIBUR-FMS/0021-1.00-M1.2.1,,github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.00-M1.2.1.pdf.,UKAEA,,2020.,03,Technical,Re-,https://,[45],G.,Karniadakis,and,S.,Sherwin.,Spectral/hp,element,methods,for,computational,fluid,dy-,namics,2nd,Ed.,Oxford,University,Press,,2005.,https://doi.org/10.1093/acprof:oso/,9780198528692.001.0001.,A,Annex,A:,Atomic,and,Molecular,Effects,Often,the,radiation,emitted,and,absorbed,by,atoms,in,different,ionisation,states,must,be,accounted,for.,There,is,a,compact,introduction,in,Golub,and,Pasachoff,[33,,§,3.3.1].,This,explains,how,in,principle,,given,a,“particular,mixture,of,elements,at,a,specified,temperature,.,.,.,the,number,of,atoms,per,unit,volume,of,the,gas,which,are,in,a,particular,ionisation,state,may,be,calculated,.,.,.,then,for,that,atom,which,emission,lines,are,emitted”,and,so,on,for,all,other,constituents,of,the,mixture.,(Temperature,refers,to,a,black-body,radiation,field,in,which,the,atoms,are,assumed,to,sit.),“The,sum,total,of,all,these,bound-bound,emissions,,plus,the,bound-free,and,free-free,emissions”,is,the,spectrum,,where,it,is,explained,that,‘bound’,and,‘free’,describe,the,state,of,the,electron,involved,in,the,formation,of,the,line,with,respect,to,the,atom.,But,“in,practice,,carrying,out,this,calculation,is,.,.,.,enormously,complicated”.,The,complication,follows,from,the,range,of,competing,mechanisms,even,within,atoms,of,one,ele-,ment,,namely,the,bound-bound,mechanisms,of,decay,and,excitation,described,in,ref,[33,,§,3.2.1],,and,bound-free,of,recombination,and,photo-emission,,because,of,the,different,possible,degrees,of,ionisation,as,atomic,number,A,increases,and,because,the,proportion,of,atoms,in,each,ionisation,state,depends,on,the,proportions,in,the,others.,Ambartsumyan,[34,,§,5],explains,at,greater,length,the,calculation,in,thermal,equilibrium,of,the,proportion,of,different,ions,for,each,element,[34,,§,5.2],,then,the,bound-free,/,free-bound,coeffi-,cients,(§,5.3—§,5.5),and,free-free,(§,5.6).,In,[34,,§,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,[35],,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,[35,,slide,21].,The,Zeeman,effect,is,also,neglected,,39,although,this,might,be,expected,to,be,important,,as,from,its,use,in,sunspot,observation,[36,,§,5.2],spectral,line,splitting,by,wavelengths,of,0.1,nm,is,expected.,The,key,observation,is,from,O’Mullane,[35],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,more,involved,but,it,is,only,critical,for,diagnosticians,working,with,particular,apparatus.,Indeed,,Golub,and,Pasachoff,[33,,§,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,),(169),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,[35],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),,,(170),where,Z,is,charge,state,,the,suffix,s,on,the,density,has,been,dropped,,Zm,=,Z,−,1,,Zp,=,Z,+,1,,and,where,Z0(s),is,the,number,of,charge,states,of,species,s,included,in,the,model.,It,may,be,inferred,that,S,Zm→Z,or,S,Z→Zp,=,ionisation,coefficient,αZ→Zm,or,αZp→Z,=,partial,dielectronic,recombination,rate,coefficient,P,Z,=,radiated,power,per,atom,of,nZ,I,Z,=,power,per,atom,released,in,dielectronic,recombination,(171),(172),(173),(174),where,the,coefficients,,as,elsewhere,in,this,section,,are,expected,to,be,obtained,from,the,Atomic,Data,and,Analysis,Structure,ADAS,database,[27,,37].,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,[38].,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,40,implications,for,computation,[35],,arise,in,the,time,evolution,equations,if,each,charge,state,is,treated,separately.,This,may,be,necessary,in,a,strong,electric,field,because,each,different,ion,feels,a,different,electromagnetic,force.,An,ion,of,species,s,with,charge,state,Z,will,acquire,a,source,s,=,S,Zm→ZnenZm,−,(cid:0)αZ→Zm,+,S,Z→Zp(cid:1),nenZ,+,αZp→ZnenZp,SZ,where,again,the,suffix,s,on,the,density,has,been,dropped.,Thus,the,demands,on,atomic,data,are,not,very,different,from,those,for,the,energy,equation,,but,since,the,total,cost,of,these,additional,computations,with,Z0(s),extra,species,will,scale,at,least,as,fast,as,Zsum,=,(cid:80),s,Z0(s),(inter-species,coupling,may,add,considerably,to,the,computational,expense),,rendering,negligible,the,cost,of,inputting,a,few,thousand,coefficients,from,disc.,In,practice,a,useful,surrogate,is,produced,by,replacing,separate,ionisation,states,by,‘superstages’,(slide,17,of,ref,[39]),,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.,(175),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,[40],gives,an,idea,of,the,computational,demands.,GCR,modelling,requires,each,metastable,state,to,be,considered,separately,,since,each,has,a,separate,finite,lifetime.,It,helps,that,the,transport,of,each,atom,in,the,state,is,presumably,the,same,,but,even,so,there,is,a,need,to,solve,a,rate,equation,for,metastable,state,density,at,sample,points,throughout,the,computational,domain,The,source,terms,are,complicated,,namely,for,the,metastable,state,labelled,ρ,SZ,ρ,/ne,=,+,+,+,(cid:88),σ,(cid:88),µ,(cid:88),ν,(cid:88),σ,X,Z→Z,σ→ρ,nσ,−,(cid:88),X,Z→Z,ρ→σ,nρ,µ→ρ,nZm,S,Zm→Z,σ,µ,−,(cid:88),S,Z→Zp,ρ→ν,nρ,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,(176),(177),(178),(179),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,(180),(181),Note,that,the,expressions,in,both,ref,[40,,eq.,(9)],and,ref,[35,,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,[35,,39],that,the,number,of,metastable,states,for,a,given,ionisation,Z,is,relatively,small,(slide,9,of,ref,[39],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,(176),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,[41],experience,with,rate,equations,indicates,the,cost,of,these,additional,computations,41,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,[40,,eq.,(5)],,nZ,i,/ne,=,(cid:88),X,FiσnZ,σ,+,I,FiµnZm,µ,+,(cid:88),(cid:88),RFiνnZp,ν,(182),σ,µ,ν,where,X,I,RFiσ,are,the,coefficients,of,excitation,,ionisation,and,recombination,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,,each,is,a,function,of,ne,and,Te,with,correspond-,ing,storage,requirement,of,order,100.,Equation,(182),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,[40].,Next,,as,explained,in,the,opening,paragraph,,to,each,state,there,corresponds,a,description,of,its,spectrum,,which,may,contain,many,separate,lines,,each,described,by,its,wavelength,,relative,amplitude,and,a,profile,shape,which,may,require,several,further,parameters,to,describe.,Mitigating,the,demand,for,coefficient,data,,is,the,fact,that,the,diagnostics,need,only,be,computed,intermittently.,To,treat,atomic,physics,UQ,in,a,later,stage,of,NEPTUNE,,a,Monte-Carlo,calculation,might,be,considered,,involving,all,the,different,interactions,between,all,the,metastable,states,where,the,Maxwellian,assumption,is,relaxed,,posing,a,multiscale,multiphysics,problem.,However,,the,validity,of,this,approach,requires,further,consideration,as,Henderson,et,al,[42],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,%,[43].,B,Annex,B:,Index,of,Mathematics,42,Table,1:,TABLE,OF,MATHEMATICAL,SYMBOLS,If,no,units,are,given,,then,quantity,is,dimensionless,,or,if,the,units,are,given,as,?,,then,the,dimensions,depend,on,context.,Gener-,ally,,the,usage,of,symbols,tries,to,follow,that,from,the,Plasma,Formulary,[7],,in,SI,units,,with,temperatures,specified,as,kT,which,returns,J.,The,Formulary,also,give,the,fundamental,dimensions,of,the,SI,units,,which,should,enable,checking,of,dimensional,consistency,of,equations,,eg.,magnetic,field,in-,duction,is,in,Tesla,(T,),whence,the,fundamental,dimension,expression,gives,T,=,kgs−1C−1.,Note,that,the,symbols,are,sorted,by,font,as,well,as,alphabet,,so,that,boldface,sym-,bols,appear,immediately,after,‘b’,(backslashes,ignored).,The,main,source,for,the,symbols,is,the,Equations,document,[44],,also,included,are,those,listed,as,used,in,the,text,by,Karni-,adakis,and,Sherwin,[45],,prefaced,by,(K+S),,plus,symbols,used,in,the,report,[15].,Description,minor,radius,of,the,torus,(horizontal),atomic,mass,of,the,ion,arbitrary,finite,interval,as,suffix,is,species,label,or,index,perturbation,amplitude,partial,dielectronic,recombination,rate,coefficient,partial,dielectronic,recombination,rate,coefficient,minor,radius,of,the,torus,(vertical),used,to,make,B,dimensionless,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,acceleration,experienced,by,a,particle,magnetic,vector,potential,magnetic,field,unit,vector,giving,the,direction,of,the,magnetic,field,electric,field,modified,electric,field,force,vector,pseudo,/,thermal,velocity,component,in,flux,surface,normal,to,field,direction,generic,velocity,velocity,of,species,α,fluid,velocity,component,along,fieldline,fluid,velocity,component,normal,to,flux,surface,Units,m,m3s−1,m3s−1,m,T,T,T,m2s−1,T,m,T,V,m−1,m−2,N,ms−1,ms−1,ms−1,ms−1,ms−1,43,Symbol,a,A,[a,,b],α,αn,αZp→Z,αZ→Zm,b,B0,¯NZ,B,=,|B|,BT,β,β,a,=,d2x/dt2,A(x,,t),B(x,,t),b,E(x,,t),E+,F,u∧,v,vα,v∥,v⊥,v∧,v0,vcx,vE×B,ve,vi,v∇Be,v∇Bi,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,ue,W,ξ(θ),√,kT0,Bp,C0,=,∩,χ,χδ,χi(ξ),cp,cs,=,(cid:113),kTi+ZikTe,mi,ms−1,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,position,‘b-factors’,ref,[35,,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,(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,44,m,s−1,or,Cm−3,T,ms−1,Jkg−1K−1,ms−1,cs,=,cse,=,(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),δ(x),D,DA,De,Df,vα,Dn,Df,pα,Di,|e|,e,eijk,∅,ϵ0,ϵr,=,Ls/(t0C0),η1,,η2,,η3,ηB,ηd,=,ηB/µ0,ηen,ηen∥,ηin,ηin∥,plasma,acoustic,speed,acoustic,speed,of,electrons,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,dimensionality,of,problem,diffusion,coefficient,for,plasma,charges,in,a,background,of,neutrals,diffusion,coefficient,for,electrons,in,a,background,of,neu-,trals,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,in,a,background,of,neutrals,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,(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,from,ms−1,ms−1,ms−1,ms−1,N,m−2,m2s−1,m2s−1,m2s−1,m2s−1,C,F,m−1,Ωm,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,45,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,hj,hp(ξ),i,i,i,I,Iϕ,IH,i,,j,,k,I,Fiσ,∈,I(ψ),=,BT,/R,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),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,[13],(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,46,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,A,eV,T,m−1,I,Z,j,jext(R,,Z),jϕ,jsh,k,k,κα,κe∥,κe⊥,κi∥,κi⊥,κ,=,kc/ρmcp,kB,kc,Kcx,(ni,,Ti),Ki,KM,Kr,kT0,kTd,Kv(x),kw,λ,λ,λ,Λ,λq,Λb,λD,λαmf,p,⟨σv⟩CX,⟨σv⟩ION,⟨σv⟩REC,⟨σv⟩,L0,LNm,i,(ξ),Ls,m,M0,M1,?,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,m2s−1,JK−1,Jm−1s−1K−1,m3s−1,m3s−1,?,W,√,√,KM,T0,,KM,Td,is,an,power,per,atom,released,in,dielectronic,recombination,as,suffix,is,generic,label,electric,current,density,induced,in,plasma,by,external,coils,Am−2,Am−2,ϕ−,or,toroidal,component,of,plasma,current,density,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,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,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,length,of,fieldline,species,particle,mass,Mach,number,at,s,=,0,boundary,Mach,number,at,s,=,1,boundary,m,m3s−1,m3s−1,m3s−1,m3s−1,m,m3s−1,J,J,m−1,?,eV,m,m,kg,m,47,mα,E,Ek̸=i,l̸=j,Exk̸=i,L(u),P,Pδ,v,Eα,Ee,Ei,ER,F,Fα,I,I,δ,K∥,K,K⊥,K∧,L7,PP,(Ω),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,kg,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,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,neutral,species,particle,mass,mass,of,proton,m3s−1,m3s−1,m3s−1,m3s−1,m3s−1,?,m3s−1,m3s−1,kg,kg,kg,kg,s−1,m−1s−1,m−1s−1,m−1s−1,m−1s−1,48,mu,Ms,MS,µ,,ν,µcx,=,ωc/νcx,µm,MZ,n,nref,Nref,N,n0,∇·,∇×,∇2,Nb,nB,=,N/B,ND,Ndof,ne,Nel,Neof,ni,nj(x,,t),Nm,nn,/∈,̸⊂,np,NQ,ns,Ns,NT,ν,να,νcx,=,Kcxnn,νe0,νe∥,νe⊥,νi∥,freedom,per,dimension,,D,=,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,number,density,of,the,electrons,(K+S),Number,of,finite,elements,(K+S),Total,number,of,elemental,degrees,of,Neof,≃,NelNm,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,[27,,37],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,freedom,1.6605,×,10−27,kg,kg,m−3,m−3,1018,m−3,m−3,m−3T,−1,m−3,m−3,m−3,m−3,m−3,m−3,m2s−1,m2s−1,s−1,m2s−1,m2s−1,m2s−1,m2s−1,49,L0n0/C4,0,νi,νi0,νi⊥,ν∗,α,ν∗,c,q4,e,pϵ2,3m2,0,να,ναn,nZ,NZ,nZ,i,nZ,σ,ωce,=,|e|B/me,ωci,=,ZieB/mi,(cid:113),nq2,e,ϵ0me,(cid:113),nq2,e,ϵ0mi,ωpe,=,ωpi,=,Zi,Ω,Ωe,p(A|B),pα,∥,Q,∥E,(∂f,/∂t)C,∂Ωe,∂Ω,∂ΩD,∂ΩN,PC,pe,ϕ,Φ,ϕpq,,ϕpqr,ϕe,ξ,(FE,Basis),expansion,basis,as,a,function,of,global,position,x,p,p,=,(cid:80),p,pi,Pi,p(ψ),α,nαkTα,p,,q,,r,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,number,density,for,charge,state,Z,number,of,charge,states,for,an,ion,species,number,density,for,charge,state,Z,,excited,state,i,number,density,for,charge,state,Z,,metastable,state,σ,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,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,m2s−1,m2s−1,m2s−1,s−1,s−1,m−3,m−3,m−3,radianss−1,radianss−1,radianss−1,radianss−1,N,m−2,m−6s2,N,m−2,radians,c,V,(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,N,m−2,N,m−2,N,m−2,N,m−2,50,P,r,P,rM,ψ,ψa,p,,,ψb,Ψi,pq,,ψc,pqr,P,(T,),p(x),P,(x),P,Z,Q∥,qα,qe,Q(fα,,fβ),QH,qi,qi,qi,Qi,Qie,r,R,R0,Rp,RE,ρ,ρ,ρc,=,(cid:80),ρm,(cid:80),ρtα,α,Zα|e|nα,=,α,Aαmunα,RFiσ,s∥,s,s,Sα,SC,Sana(x,,t),Sn,ana(x,,t),Sexp(x,,v,,t),n,T,u,si,typically,multi-,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,Boltzmann,collision,operator,cooling,rate,due,to,excitation,as,defined,in,ref,[13],charge,on,an,ion,electron,energy,flux,ion,energy,flux,(FE,Basis),Quadrature,order,in,the,ith,direction,collisional,energy,equipartition,term,order,of,higher,order,term,cylindrical,coordinate,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),51,T,m2,W,m3,W,Jm−2s−1,C,C,m−6s2,Km−3s−1,C,Jm−2s−1,Jm−2s−1,kgm−1s−3,m,m,Cm−3,kgm−3,m,m,m−6s2,m−6s2,m−3s−1,?,m−3s−1,?,m−6s2,Si,σ,σ,σC,σE,σE,σI,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′,T,t0,tH,tR,T0,TKn,Tref,Tα,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,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,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),temperature,of,species,α,m2,Ω−1m−1,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,?,kgm−2s−2,kgm−2s−2,kgm−2s−2,m−3s−1,m−3s−1,s,s,eV,s,s,s,eV,eV,eV,eV,52,τ,τα,τe,τi,τen,τin,τce,=,1/fce,τci,=,1/fci,τpe,=,1/fpe,τpi,=,1/fpi,τEe,τEi,τne,τni,Td,=,Ti,+,Te,Te,TH,θ,θ,Ti,˜b,=,B/B0,˜ψa,pq,,˜ψc,p,,,˜ψb,u,U,Ud,=,Ls/t0,UA,f,e,W,e,uR,=,1/R,v,V,e,Vi,Vij,pqr,w,wjk,wp,W,x,x1,,x2,,x3,,x,xα,xe,=,ωceτe,xi,=,ωciτi,xi,ξ1,,ξ2,,ξ3,,ξ,ξi,m,s,s,s,s,s,s,s,s,s,s,s,s,s,eV,eV,radians,c,eV,ms−1,ms−1,ms−1,ms−1,ms−1,m3,ms−1,m,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,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,magnetic,field,(FE,Basis),Orthogonal,principal,functions,generic,first,velocity,component,velocity,component,(flow),along,streamline,speed,measuring,the,importance,of,the,transient,term,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,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,weighting,function,for,particle-in-cell,Cartesian,coordinate,(FE,Basis),Global,Cartesian,coordinates,collisionality,factor,of,species,α,collisionality,factor,of,electrons,collisionality,factor,of,ions,generic,parameter,or,variable,(FE,Basis),Local,Cartesian,coordinates,random,number,within,the,unit,interval,[0,,1],53,X,Fiσ,y,z,Z,Z,Z,Z0(α),Za,ζ,ζ,=,−ϕ,Zi,Zm,=,Z,−,1,Zp,=,Z,+,1,Zsum,(cid:80),α,Z0(α),coefficient,of,excitation,for,the,transition,from,metastable,state,σ,to,regular,excited,state,i,Cartesian,coordinate,Cartesian,coordinate,Cartesian,coordinate,charge,state,of,the,ion,cylindrical,coordinate,number,of,charge,states,of,species,α,included,in,the,model,Gaussian,random,process,,index,a,magnetic,Prandtl,number,as,defined,in,Cambridge,toroidal,angle,coordinate,charge,state,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,radians,c,54

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