TN-04_Task211DFluidModelTests
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   :description: technical note

   :keywords: Excalibur-Neptune,report,2047356-TN-04-2,Task,2.1,1D,ﬂuid,model,tests,Ben,Dudson,,Peter,Hill,,Ed,Higgins,,David,Dickinson,,and,Steven,Wright,University,of,York,David,Moxey,University,of,Exeter,June,16,,2021,Contents,1,Executive,summary,2,Single,species,2.1,Nonlinear,heat,conduction,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,2.2,Uniform,source,,outﬂow,boundary,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,2.3,Half,source,,outﬂow,boundary,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,2.4,Half,source,,sheath,boundary,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,3,Recycling,3.1,Recycling,sources,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,3.2,Fluid,neutral,gas,species,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,1,2,2,4,5,5,7,8,9,3.3,Kinetic,neutral,gas,species,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,11,4,Multiple,species,12,4.1,Separate,electron,and,ion,temperatures,.,.,.,.,.,.,.,.,.,.,.,.,.,.,12,4.2,Neutral,gas,model,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,13,4.3,Impurities,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,13,5,References,14,1,Executive,summary,The,deliverable,for,this,task,is,to,deﬁne,”Test,cases,for,system,2-3”,,a,1D,ﬂuid,solver,with,UQ,and,realistic,boundary,conditions.,The,aims,of,this,is,to,provide,a,set,of,problems,relevant,to,the,Neptune,use,case,,which,run,quickly,enough,to,be,used,as,part,of,a,fast,development,cycle.,These,will,be,used,to,compare,model,implementations,,and,test,approaches,to,Uncertainty,Quantiﬁcation,(UQ),and,preconditioning,of,complex,nonlinear,systems,involving,two-way,coupling,be-,1,tween,plasma,dynamics,,atomic,reactions,,and,plasma-wall,interactions.,2,Single,species,These,tests,exercise,individual,parts,of,the,ﬂuid,solver,,excluding,interactions,between,ﬂuids,(e.g.,plasma,,neutrals),which,complicate,the,full,system,of,equa-,tions.,2.1,Nonlinear,heat,conduction,Heat,diﬀusion,is,a,standard,problem,in,numerical,methods,,but,heat,conduction,in,plasmas,has,some,features,which,can,introduce,challenges:,•,The,diﬀusion,is,strongly,anisotropic,,many,(>,6),orders,of,magnitude,diﬀerence,between,direction,parallel,and,perpendicular,to,the,magnetic,ﬁeld.,•,The,diﬀusion,is,nonlinear:,In,the,collisional,(Spitzer-Harm,/,Braginskii,limit),,the,thermal,diﬀusion,coeﬃcient,depends,on,the,temperature,T,5/2.,Some,useful,scenarios,which,are,relevant,to,the,simulation,of,the,tokamak,edge,and,scrape-oﬀ,layer,are:,1.,A,domain,with,Dirichlet,boundary,conditions,on,both,sides,,ﬁxing,the,temperature,at,both,sides.,The,equation,for,the,heat,ﬂux,is,q,=,−κ0T,5/2b,·,∇T,=,const,where,κ0,is,a,constant,,calculated,by,integrating,over,a,perturbed,Maxwellian,distribution,function,(see,e.g,Braginskii).,This,heat,ﬂux,is,along,the,mag-,netic,ﬁeld,,which,has,a,unit,vector,b,=,B/,|B|.,With,a,heat,ﬂux,q,constant,along,the,magnetic,ﬁeld,,the,temperature,T,(l),has,an,analytic,solution,T,7/2,0,−,T,7/2,=,7,2,ql,κ0,2,where,l,is,the,distance,along,the,domain,,and,T0,is,a,constant.,2.,A,ﬁxed,(low),temperature,at,the,”target”,end,of,the,domain,,and,a,ﬁxed,input,power,q,in,the,”upstream”,boundary.,This,power,input,can,be,implemented,either,as,a,Robin,boundary,condition,,setting,the,gradient,of,temperature,T,depending,on,the,temperature,at,the,boundary,,or,as,a,zero-gradient,(Neumann),boundary,condition,with,a,source,of,power,over,an,extended,region.,3.,A,model,which,captures,some,of,the,eﬀects,of,impurity,and,atomic,inter-,actions,,and,allows,the,heat,ﬂux,q,(l),to,be,a,function,of,distance,along,the,domain,l,,is,a,variation,on,the,Lengyel,model,[1],,which,is,widely,used,in,divertor,modelling.,The,main,feature,of,this,approach,is,that,by,assum-,ing,that,the,radiated,power,is,a,function,only,of,temperature,,an,analytic,solution,can,be,obtained.,The,divergence,of,the,heat,ﬂux,is,related,to,the,radiation,loss:,∇,·,bq,(cid:39),dq,dl,=,−n2Q,(T,),where,n,is,the,plasma,density,,and,the,equality,is,exact,in,a,constant,magnetic,ﬁeld.,By,assuming,that,the,pressure,nT,=,p,is,constant,,density,is,also,a,function,of,temperature,only.,q,=,−κ0T,5/2,dT,dl,(cid:124)(cid:123)(cid:122)(cid:125),dq,dT,dq,dl,=,κ0T,5/2n2Q,(T,),dT,dq,and,so,(cid:90),upstream,qdq,=,κ0p2,(cid:90),upstream,target,target,T,1/2Q,(T,),dT,where,the,target,and,upstream,are,usually,taken,to,be,the,wall,and,the,tokamak,outboard,midplane,or,divertor,entrance,respectively,,but,can,be,any,two,points,along,the,ﬁeld,line.,Labelling,”target”,with,subscript,”t”,,and,”upstream”,with,subscript,”u”:,u,−,q2,q2,t,=,2κ0p2,(cid:90),Tu,Tt,T,1/2Q,(T,),dT,By,choosing,an,suitable,cooling,curve,function,R,(T,),,this,equation,can,be,compared,to,numerical,solutions.,One,possible,choice,is,the,approximation,3,for,nitrogen,radiation,used,in,Lipschultz,2016[2]:,Q,=,5.9,×,10−34,(T,−,1eV)1/2,(80eV,−,T,),1,+,3.1,×,10−3,(T,−,1eV)2,Wm3,for,1eV,<,T,<,80eV,,and,Q,=,0,outside,this,range.,Note,that,the,above,model,assumes,a,constant,magnetic,ﬁeld,,so,that,∇,·,b,=,0.,For,conventional,large,aspect-ratio,tokamaks,this,can,be,a,reasonable,approximation,,but,for,spherical,tokamaks,(such,as,MAST-U,,STEP),the,variation,in,B,has,a,signiﬁcant,impact.,The,Lengyel,model,is,only,a,very,rough,guide,to,real,experiments,,but,is,useful,here,as,an,analytic,solution,to,test,against.,Moving,beyond,1D,into,2D,and,3D,tokamak,geometry,test,cases:,Analytic,tokamak,equilibria,with,X-points,can,be,created,based,on,work,by,Cerfon,and,Freidberg,(a,useful,tool,was,created,by,John,Omotani[3].,These,equilibria,were,used,in,[4],to,develop,benchmark,cases,,and,test,a,range,of,numerical,schemes.,2.2,Uniform,source,,outﬂow,boundary,Now,adding,ﬂuid,equations,,for,a,single,species,particle,density,,energy,(or,pressure,or,temperature),,and,momentum,along,the,magnetic,ﬁeld.,A,1D,domain,,with,two,boundaries:,•,No-ﬂow,upstream.,This,can,be,implemented,as,a,symmetry,boundary:,Zero-gradient,density,,pressure,and,temperature,,and,zero-value,ﬂow,ve-,locity,and,heat,ﬂux,boundary,conditions.,•,Free,outﬂow.,This,can,be,implemented,in,ﬁnite,diﬀerence/ﬁnite,volume,methods,by,extrapolating,all,quantities,with,e.g.,constant,gradient.,To,preserve,positive,deﬁnite,density,and,temperatures,,it,can,be,useful,to,extrapolate,the,logarithms,of,these,thermodynamic,quantities.,Inside,the,domain,a,uniform,source,of,particles,and,(internal),energy,,which,then,ﬂow,towards,the,free,outﬂow,boundary.,4,This,system,should,evolve,to,a,steady,state,in,which,the,particle,ﬂux,increases,linearly,with,distance,from,the,no-ﬂow,boundary.,2.3,Half,source,,outﬂow,boundary,This,is,a,variation,on,the,above,test,,which,tests,the,ability,of,the,numerical,scheme,to,resolve,abrupt,changes,in,sources,,and,the,resulting,abrupt,changes,in,gradients.,The,boundary,conditions,are,the,same,as,for,the,previous,test,,but,the,sources,only,ﬁll,half,of,the,domain,closest,to,the,no-ﬂow,boundary.,In,this,case,the,particle,source,is,uniform,in,the,ﬁrst,half,of,the,domain,,so,the,particle,ﬂux,starts,from,zero,at,the,no-ﬂow,boundary,,and,increases,linearly,with,distance,until,reaching,the,end,of,the,source,region,,and,is,then,constant,between,the,end,of,the,source,region,and,the,free,outﬂow,boundary.,Figure,1:,Sources,and,particle,ﬂux,in,”Half,source,,outﬂow,boundary”,case,2.4,Half,source,,sheath,boundary,This,has,the,same,sources,as,the,previous,case,,but,now,tests,the,implementation,of,a,sheath,boundary,condition.,Two,important,components,of,this,boundary,condition,in,a,ﬂuid,model,are:,5,DistancealongdomainlParticlesourceParticleﬂuxNo-ﬂow(symmetry)boundaryFreeoutﬂowboundary,1.,Imposed,out-ﬂow,velocity,at,the,sound,speed,,generally,called,the,Bohm,condition,,or,Bohm-Chodura,in,a,magnetised,plasma.,The,ﬂow,speed,into,the,sheath,depends,in,general,factors,such,as,the,currents,and,voltages,,whether,species,should,be,considered,isothermal,or,adiabatic,,and,can,be,quite,complex,in,situations,with,multiple,species,(see,e.g.,[5]).,A,simple,and,widely,used,approximation,is,the,isothermal,sound,speed,(see,e.g.,Stangeby):,(cid:115),cs,=,e,(Te,+,Ti),mi,where,the,electron,and,ion,temperatures,Te,and,Ti,are,in,units,of,eV,,mi,is,the,ion,mass,(in,kg),,and,the,sound,speed,cs,is,in,units,of,m/s.,2.,Heat,conduction,to,the,target.,In,addition,to,the,energy,loss,expected,because,the,ﬂuid,is,ﬂowing,to,the,target,,there,is,also,typically,an,addi-,tional,loss,of,energy.,Physically,what,is,happening,at,the,target,is,that,slow,electrons,are,reﬂected,,while,energetic,electrons,reach,the,target,,so,that,the,electrons,are,rapidly,cooled.,Some,of,this,energy,is,transferred,to,accelerating,ions,into,the,target,,via,the,sheath,voltage.,The,energy,ﬂow,through,the,sheath,is,often,characterised,by,the,energy,ﬂux:,qsheath,=,γshenT,cs,with,a,diﬀerent,γsh,for,each,species.,The,actual,power,ﬂow,to,the,target,is,complex,,depending,like,the,ﬂow,speed,on,the,electric,ﬁelds,and,currents,,and,the,sheath,heat,loss,coeﬃcientγsh,can,vary,considerably,(by,an,order,of,magnitude),,particularly,during,transients.,If,solving,a,single,ﬂuid,,i.e,a,single,density,,energy,and,momentum,equation,,a,common,assumption,is,that,the,ions,and,electrons,have,equal,densities,(n,=,ne,=,ni),,temperatures,(T,=,Te,=,Ti),and,velocities,(v,=,ve,=,vi).,This,means,that,the,total,pressure,p,=,enTe,+,enTi,=,2enT,.,The,sheath,velocity,is,then,cs,=,(cid:114),2eT,mi,In,a,ﬂuid,model,with,ratio,of,speciﬁc,heats,γ,=,5/3,(typical,for,plasma,simula-,6,tions),,the,energy,ﬂow,is,qf,luid,=,(cid:18),5,2,p,+,(cid:19),1,2,miv2,v,+,qcond,where,qcond,is,the,heat,conduction.,At,the,sheath,where,v,=,cs,we,have:,qf,luid,=,(cid:18),5,2,2enT,+,(cid:19),1,2,mi,2eT,mi,cs,+,qcond,=,6enT,cs,+,qcond,so,by,comparing,qf,luid,with,qsheath,,if,γsh,=,6,then,there,should,be,no,heat,conduction,through,the,sheath.,More,typical,values,are,γsh,between,6.5,and,9.,The,additional,heat,ﬂux,can,be,used,to,calculate,the,temperature,gradient,at,the,sheath,in,a,Robin-type,boundary,,or,removed,from,the,last,cell,through,the,boundary,with,the,target.,This,more,complex,system,still,has,analytic,solutions,,depending,only,on,the,given,input,power,,and,the,upstream,density.,3,Recycling,The,half,source,,sheath,boundary,test,case,is,a,reasonable,model,of,a,low,density,plasma,(what,Stangeby,describes,as,a,”simple,SOL”),,but,is,missing,a,crucial,in-,gredient,for,tokamak,edge,and,divertor,simulation:,the,interaction,with,neutral,gas.,When,plasma,meets,a,material,surface,a,sheath,is,formed,,a,small,region,with,strong,electric,ﬁelds,,which,accelerates,ions,towards,the,surface.,When,ions,hit,the,surface,they,tend,to,pick,up,electrons,and,become,neutral,atoms.,Some,of,those,atoms,will,reﬂect,from,the,surface,,others,will,stick,to,the,surface,and,perhaps,combine,into,molecules,,or,become,embedded,inside,the,surface,before,diﬀusing,out,again.,The,details,depends,on,what,the,wall,is,made,of,(e.g.,carbon,or,tungsten),,but,the,key,feature,is,that,the,majority,of,the,impinging,ions,(>,99%),will,typically,come,back,into,the,plasma,as,neutral,atoms,or,molecules.,The,atoms,and,molecules,which,come,oﬀ,the,material,surface,back,into,the,7,plasma,typically,encounter,high,electron,temperatures,,and,are,quickly,ionised,and,converted,into,ions,again.,These,ions,then,return,to,the,wall,,where,they,again,become,neutral,atoms.,The,majority,of,the,ions,ﬂowing,to,the,wall,are,typically,undergoing,this,recycling,process,,so,that,the,ﬂow,of,ions,from,upstream,(regions,away,from,the,wall,and,divertor),are,typically,small,relative,to,the,ﬂow,of,ions,to,the,surface.,This,recycling,process,of,continually,ionising,atoms,removes,energy,from,the,electrons,,both,in,overcoming,the,ionisation,potential,,and,also,in,radiation,from,the,relaxation,of,intermediate,excited,states,of,the,atoms.,This,radiation,can,also,be,critical,to,reducing,heat,ﬂows,to,the,surfaces,,and,is,enhanced,by,a,feedback,mechanism:,As,the,plasma,(electrons),cool,,the,radiated,energy,per,ionisation,rapidly,increases,at,low,temperatures,(<,5eV).,This,can,lead,to,the,plasma,”detaching”,from,the,wall,,reducing,the,heat,and,particle,ﬂuxes,,in,a,process,similar,to,the,condensation,instability,seen,in,space,plasmas.,3.1,Recycling,sources,A,straightforward,modiﬁcation,to,the,1D,single,ﬂuid,model,used,in,section,2,,is,to,add,sources,to,represent,the,recycled,ﬂux,of,particles.,•,The,power,source,is,kept,away,from,the,”target”,,to,represent,the,ﬂow,of,heat,from,the,main,plasma.,•,Rather,than,the,particle,source,being,at,the,same,location,as,the,power,source,,the,particle,source,is,now,put,close,to,the,target.,A,reasonable,choice,of,particle,source,is,an,exponential,decay,,with,highest,source,at,the,target.,This,is,to,represent,the,mean,free,path,of,neutral,atoms,entering,the,plasma.,It’s,important,to,note,that,the,decay,length,for,this,source,should,be,signiﬁcantly,(10,−,100,times),longer,in,the,direction,along,the,magnetic,ﬁeld,than,the,ionisation,mean,free,path.,This,is,because,neutral,atoms,are,travelling,away,from,the,wall,,but,the,magnetic,ﬁeld,is,at,a,shallow,angle,(typically,a,few,degrees),to,the,wall.,Neutral,atoms,may,only,travel,a,short,distance,from,the,wall,,but,this,can,correspond,to,a,relatively,long,distance,along,the,magnetic,ﬁeld,in,the,1D,domain,simulated,here.,8,To,represent,the,loss,of,energy,from,ionisation,and,the,associated,excitation,radiation,,a,sink,of,energy,proportional,to,the,particle,source,can,be,added.,The,ionisation,potential,for,hydrogen,atoms,is,13.6eV,,and,dissociation,poten-,tial,for,molecules,about,4.5eV,per,molecule,,but,the,energy,lost,per,ionisation,varies,signiﬁcantly:,Neutral,species,can,enter,the,plasma,with,some,energy,(e.g,Franck-Condon,energy,,around,3.5eV,but,with,wide,variation),,lowering,the,eﬀective,energy,cost;,as,discussed,above,,excitation,radiation,increases,the,eﬀective,energy,loss,per,ionisation.,Typical,values,used,in,the,literature,are,around,30eV,,which,is,probably,a,reasonable,value,to,use,for,this,test,case.,With,this,ﬁxed,particle,source,(and,so,ﬁxed,power,sink),,some,care,should,be,taken,that,the,temperature,is,not,driven,to,zero,by,removing,more,power,than,is,put,in.,This,is,quite,straightforward,if,a,ﬁxed,energy,cost,per,ionisation,is,used,,as,then,the,total,power,sink,can,be,balanced,against,the,power,input.,If,a,temperature-dependent,ionisation,is,used,,however,,then,this,can,be,more,diﬃcult.,3.2,Fluid,neutral,gas,species,A,more,sophisticated,neutral,gas,model,represents,the,interaction,of,plasma,and,gas,as,two,ﬂuid,species,,one,for,the,plasma,and,one,for,the,neutral,atoms.,Fur-,ther,ﬂuid,species,can,also,be,added,to,represent,molecules,,short-lived,species,(eg,H−,,H+,3,),or,a,number,of,vibrationally,excited,states.,The,network,of,reac-,tions,between,these,species,can,become,quite,complicated,,even,for,something,as,”simple”,as,hydrogen.,Evolving,neutral,gas,as,a,species,modiﬁes,the,reactions:,The,ionisation,source,of,plasma,particles,is,now,evolving,with,the,system,state,,rather,than,being,imposed,as,an,input,parameter.,The,volumetric,source,S,,in,units,of,particles,per,second,per,cubic,meter,is,S,=,nena,(cid:104)σv(cid:105)iz,where,ne,is,the,number,density,of,electrons,,na,the,number,density,of,neu-,tral,atoms,,and,(cid:104)σv(cid:105)iz,is,the,reaction,rate,in,units,of,m3s−1.,This,rate,is,averaged,over,a,Maxwellian,distribution,,and,is,typically,derived,from,a,0-D,collisional-radiative,model,,as,an,eﬀective,rate,which,averages,over,a,number,of,9,processes.,These,are,tabulated,in,several,databases;,a,commonly,used,one,in,plasma,physics,is,Amjuel,[6],,used,in,the,EIRENE,monte-carlo,code1.,Reaction,2.1.5FJ,in,Amjuel,describes,ionisation,of,neutral,atoms,by,electrons:,e,+,H(1s),→,e,+,H,+,+,e,In,addition,to,the,ionisation,,other,reactions,now,need,to,be,included.,Most,important,is,charge,exchange,(CX),between,plasma,ions,and,neutrals.,In,this,reaction,,the,electron,in,a,neutral,atom,swaps,to,the,ion,,represented,as:,p,+,H(1s),→,p,+,H(1s),(Amjuel,reaction,0.1T).,The,left,and,right,side,of,this,equation,are,the,same,because,the,reaction,swaps,an,ion,and,neutral,atom,but,doesn’t,aﬀect,the,number,of,particles,of,each,species.,This,reaction,often,has,a,higher,likelihood,than,ionisation,,and,provides,a,strong,coupling,between,the,plasma,and,neutral,species.,There,is,no,net,source,or,sink,of,particles,for,the,plasma,,but,there,is,a,source,or,sink,of,energy,(equation,1),and,momentum,(equation,2),,depending,on,the,relative,temperatures,and,ﬂow,velocities,of,the,plasma,ions,(subscript,’i’),and,neutral,atoms,(subscript,’a’):,Senergy,=,e,(Ta,−,Ti),nina,(cid:104)σv(cid:105)cx,Smomentum,=,mi,(va,−,vi),nina,(cid:104)σv(cid:105)cx,3,2,(1),(2),Once,this,reaction,is,included,,the,strong,interaction,between,the,plasma,and,neutral,atom,ﬂows,will,tend,to,force,the,atoms,into,a,very,narrow,layer,close,to,the,target,,which,can,be,challenging,to,resolve.,Cross-ﬁeld,diﬀusion,of,neutral,atoms,(which,can,freely,move,across,the,magnetic,ﬁeld,,unlike,the,plasma,ions),,together,with,reﬁnement,of,the,grid,resolution,near,the,target,,are,typically,needed,to,widen,and,resolve,this,layer.,The,handling,of,charge,exchange,reactions,in,a,ﬂuid,model,is,problematic:,Charge,exchange,produces,a,population,of,fast-moving,neutral,atoms,with,a,quite,diﬀerent,energy,and,momentum,distribution,to,the,atoms,from,the,wall.,Unless,these,atoms,are,strongly,coupled,to,each,other,,which,is,rarely,the,case,1The,EIRENE,Fortran,code,reads,the,data,tables,from,the,Amjuel,LaTeX,source,ﬁle.,10,in,tokamaks,except,in,speciﬁc,high,density,locations,,then,a,single,ﬂuid,ap-,proximation,is,unlikely,to,be,good.,Occasionally,in,the,literature,another,ﬂuid,species,is,used,to,represent,the,charge,exchanged,neutrals.,In,1D,there,is,also,the,challenge,of,how,the,transport,of,neutrals,,particularly,charge-exchanged,fast,neutrals,,across,the,plasma,should,be,represented:,The,plasma,is,not,really,a,1D,tube,,but,a,relatively,thin,sheet,,from,which,neutrals,can,escape,,carrying,momentum,and,energy,with,them.,Representing,this,,and,the,resulting,interac-,tions,with,in-vessel,components,like,walls,,baﬄes,and,pumps,,requires,at,least,a,2D,representation.,3.3,Kinetic,neutral,gas,species,In,most,of,the,tokamak,the,ratio,of,the,mean,free,path,to,the,local,length-scales,of,density,,temperature,etc,(the,Knudsen,number),is,much,larger,than,1,,putting,the,transport,in,the,regime,of,rareﬁed,gas,dynamics,,and,implying,that,a,kinetic,(not,ﬂuid),treatment,is,needed.,Unfortunately,in,other,regions,,typically,near,the,target,,the,Knudsen,number,can,be,much,less,than,1,,so,that,kinetic,models,become,highly,ineﬃcient,(because,they’re,simulating,a,ﬂuid,situation).,There,are,several,kinetic,plasma,neutral,models,,EIRENE,probably,being,the,most,well,known,,and,several,eﬀorts,ongoing,to,develop,hybrid,ﬂuid-kinetic,models.,Coupling,a,1D,plasma,model,to,EIRENE,would,be,a,non-trivial,task,,not,least,because,EIRENE,is,not,publicly,available,or,open,source,in,any,meaningful,way2.,It,is,likely,that,examining,the,SOLPS,code,,and,benchmarking,against,SOLPS,,would,be,the,most,direct,way,to,achieve,this.,The,main,applications,of,such,a,1D,EIRENE,coupling,would,be,to,compare,against,the,ﬂuid,model;,to,use,it,as,a,platform,for,studying,model,order,reduc-,tion,of,the,neutral,model,to,simpler,and,faster,models;,to,test,preconditioning,strategies;,uncertainty,quantiﬁcation,involving,coupled,ﬂuid-monte-carlo,cou-,pled,algorithms;,and,to,gain,experience,of,carrying,out,the,coupling,,to,apply,to,more,complex,2D,and,3D,models.,It,is,likely,however,that,there,are,quicker,,more,direct,ways,of,achieving,these,goals,,such,as,using,SOLPS,directly.,2Access,to,EIRENE,is,in,principle,available,from,Juelich,on,request,11,4,Multiple,species,Real,plasmas,of,interest,are,not,pure,hydrogen,,but,contain,a,mixture,of,diﬀer-,ent,species.,A,reactor,will,have,deuterium,and,tritium,isotopes,,helium,,wall,materials,such,as,tungsten,and,beryllium,,seeded,impurities,such,as,argon,,and,trace,impurities,of,e.g.,oxygen.,Not,all,of,these,need,to,be,simulated,in,all,cases,,but,the,capability,to,model,multiple,species,will,become,increasingly,essential.,4.1,Separate,electron,and,ion,temperatures,The,most,important,,but,also,probably,simplest,,variation,on,the,models,de-,scribed,in,previous,sections,,is,to,separate,the,ion,and,electron,temperatures.,Close,to,the,target,,the,low,temperatures,and,high,densities,often,lead,to,a,strong,coupling,between,electron,and,ion,temperatures,,but,upstream,this,cou-,pling,becomes,weaker:,Typical,measurements,on,present-day,tokamaks,ﬁnd,ion,temperatures,around,a,factor,of,two,higher,than,electron,temperatures,around,the,outboard,midplane.,Separating,out,the,ion,and,electron,temperatures,involves:,•,Evolving,separate,temperature,or,(internal),energy,equations,for,the,ions,and,electrons.,The,density,and,velocity,equations,are,still,the,same,,due,to,quasineutrality,of,the,plasma,and,the,absence,of,net,currents,in,1D,simulations.,•,Using,diﬀerent,heat,conduction,coeﬃcients,for,electrons,and,ions,•,Separate,sheath,heat,transmission,coeﬃcients,(γsh,,section,2.4),•,Carefully,tracking,the,contribution,of,reactions,to,the,energy,balance,of,each,species:,Ionisation,,for,example,,is,an,energy,loss,for,electrons,,but,an,energy,gain,for,the,ions,,as,energy,is,transferred,from,atoms,into,ions.,•,Adding,a,coupling,between,electrons,and,ions,(see,Braginskii),,due,to,collisions,between,them.,12,4.2,Neutral,gas,model,As,mentioned,in,section,3.2,,the,plasma,chemistry,of,even,pure,hydrogen,plas-,mas,can,be,quite,complicated:,There,are,a,number,of,diﬀerent,pathways,by,which,a,ionisation,(for,example),can,occur,,some,involving,catalytic,interactions,between,molecules,,atoms,and,plasma,ions.,Atomic,reaction,rates,are,sensitive,to,the,starting,state,of,the,molecule,(e.g.,high,vibrational,states,are,more,likely,to,dissociate),or,atom,(highly,excited,states,are,more,likely,to,ionise).,In,some,cases,it,is,likely,that,these,states,can,persist,for,long,enough,(metastable,states),that,they,can,be,transported,around,the,domain,and,should,be,tracked,as,sep-,arate,species.,It,is,currently,not,well,understood,in,the,plasma,community,how,complex,the,model,needs,to,be,,and,what,errors,are,made,when,simpliﬁed,models,are,used.,4.3,Impurities,Finally,,more,complex,models,can,be,built,to,study,the,transport,of,multiple,species,,and,their,multiple,charge,states.,For,low-Z,species,each,charge,state,is,typically,evolved,as,a,separate,species,,with,the,reactions,between,them,(typi-,cally,ionisation,,recombination,,and,charge,exchange,with,hydrogen);,higher,Z,materials,like,tungsten,often,need,a,charge-state,”bundling”,treatment,,where,a,range,of,charge,states,is,evolved,as,a,single,ﬂuid,,with,eﬀective,reaction,rates,between,bundled,states.,The,model,reduction,needed,to,do,this,accurately,and,eﬃciently,is,an,active,area,of,research.,Once,multiple,species,are,included,,particularly,where,their,masses,are,com-,parable,to,each,other,,and,where,their,concentrations,are,not,”trace”,level,,the,calculation,of,collisions,between,species,becomes,complicated.,Hirshman,and,Sigmar,published,models;,Zhdanov,is,a,well,known,model;,and,there,has,been,some,work,recently,on,improving,these,models,and,implementing,them,in,simulation,codes.,Unfortunately,once,models,become,this,complex,,analytic,test,cases,can,no,longer,be,found.,Instead,the,best,option,is,probably,a,Method,of,Manufactured,Solutions,(MMS),test.,13,5,References,[1],L,L,Lengyel.,Analysis,of,Radiating,Plasma,Boundary,Layers,(IPP,1/191).,http://hdl.handle.net/11858/00-001M-0000-0027-6A04-1.,[2],Bruce,Lipschultz,,Felix,I.,Parra,,and,Ian,H.,Hutchinson.,Sensitivity,of,de-,tachment,extent,to,magnetic,conﬁguration,and,external,parameters.,Nucl.,Fusion,,56:056007,,2016,,doi:10.1088/0029-5515/56/5/056007.,[3],John,Omotani.,Cerfon-freidberg,geometry,generator,in,python.,https:,//github.com/johnomotani/CerfonFreidbergGeometry.,[4],M.Held,,M.Wiesenberger,,and,A.Stegmeir.,Three,discontinuous,Galerkin,schemes,for,the,anisotropic,heat,conduction,equation,on,non-aligned,grids.,Comp.,Phys.,Comm.,,199:29–39,,2016,,doi:10.1016/j.cpc.2015.10.009.,[5],D.Tskhakay,and,S.Kuhn.,Boundary,conditions,for,the,multi-ion,mag-,netized,plasma-wall,transition.,J.,Nucl.,Mat.,,337–339:405–409,,2005,,doi:10.1016/j.jnucmat.2004.10.073.,[6],D.,Reiter.,The,data,ﬁle,AMJUEL:Additional,Atomic,and,Molecular,Data,for,EIRENE.,http://www.eirene.de/html/amjuel.html,,2020.,14

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