CD-EXCALIBUR-FMS0058-M2.8.2_TechnicalPhysicsModelSelection
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   :description: technical note

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ExCALIBUR,Technical,report,on,Physics,model,selection,M2.8.2,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Milestone,2.8.2.,This,report,provides,a,digest,of,the,reports,produced,by,the,Oxford,group,regarding,the,develop-,ment,of,a,drift,kinetic,model,for,plasma,in,the,tokamak,edge.,There,are,eight,reports,to,date,,ﬁve,of,which,concern,the,development,of,the,model:,“Physics,in,the,edge,of,fusion,devices,(2047357-,TN-03-02)”,[1];,“1D,drift,kinetic,models,with,periodic,boundary,conditions,(2047357-TN-01-02),[2];,“1D,drift,kinetic,models,with,wall,boundary,conditions,(2047357-TN-05),[3]”;,“2D,drift,kinetic,model,with,wall,boundary,conditions,(2047357-TN-07-01)”,[4];,and,“2D,drift,kinetic,model,with,periodic,boundary,conditions,(2047357-TN-09-01)”,[5].,The,remaining,three,reports,concern,the,numerical,implementation,of,the,model.,The,ﬁve,theory,reports,are,appended,to,this,report,in,their,logical,reading,order:,TN-03,,TN-01,,TN-05,,TN-07,,TN-09.,This,report,summarizes,the,ﬁve,reports,on,theory,development,,discussing,the,state,of,the,ﬁeld,before,this,work,in,Section,1,,and,the,development,of,the,model,in,Section,2.,The,work,is,sum-,marized,and,future,work,is,discussed,in,Section,3.,1,Background,The,ﬁrst,report,[1],describes,the,important,physical,regimes,in,the,tokamak,edge,plasma,and,reviews,the,state-of-the-art,of,edge,plasma,codes.,The,report,focusses,on,known,problems,of,ﬂuid,models,,but,emphasises,the,importance,of,ﬂuid,models,and,states,that,research,with,ﬂuid,model,should,proceed,in,parallel,to,research,on,kinetic,models.,In,particular,,one,challenge,it,notes,for,kinetic,modelling,is,to,match,to,ﬂuid,models,efﬁciently,,and,to,determine,which,spatial,regions,may,be,treated,using,ﬂuid,models.,The,report,contains,four,topic,sections:,a,discussion,of,characteristic,time,and,length,scales,in,the,edge;,an,overview,of,the,existing,ﬂuid,models,for,the,edge;,a,list,physical,phenomena,that,cannot,be,captured,by,ﬂuid,models,(and,an,explanation,of,how,these,phenomena,are,currently,treated,by,ﬂuid,models);,and,a,discussion,of,the,proposed,work,to,develop,a,complete,set,of,kinetic,equations,for,the,edge.,There,are,two,key,limitations,of,using,ﬂuid,codes,to,model,edge,plasma.,The,ﬁrst,is,that,the,calculated,diffusion,heat,ﬂux,is,accurate,only,in,a,very,collisional,plasma.,This,is,because,the,heat,ﬂux,is,predominantly,due,to,energetic,particles,which,are,much,less,collisional,than,thermal,particles.,Fluid,models,overestimate,the,number,of,energetic,particles,present,in,hot,plasmas,,while,underestimating,the,number,of,energetic,particles,present,in,cold,plasmas.,This,results,in,inaccurate,heat,ﬂuxes.,The,second,limitation,concerns,boundary,conditions.,At,the,wall,,ions,and,electron,recombine,,meaning,that,near,the,wall,,the,distribution,function,vanishes,for,velocities,corresponding,to,motion,into,the,wall.,Thus,the,distribution,function,of,charged,particles,is,highly,non-Maxwellian,near,the,wall,and,must,be,treated,kinetically.,Together,,these,two,effects,result,in,an,incorrect,description,of,energetic,electrons,,which,in,turn,yields,incorrect,heat,transport,and,thus,incorrect,density,and,temperature,proﬁles.,Such,effects,have,been,seen,in,1D,kinetic,simulations,[6,,7].,Kinetic,models,for,neutrals,have,long,been,used,by,the,community,in,codes,such,as,EIRENE,[8],or,DEGAS,[9].,However,,energetic,electrons,also,mediate,important,collisional,processes,in,neu-,tral,models,,like,radiation,,ionization,and,recombination,,meaning,that,ﬂuid-plasma,simulations,of,models,containing,neutrals,can,be,inaccurate,,even,when,kinetic,models,are,used,for,the,neutrals.,The,community,is,also,starting,to,develop,edge,code,that,treat,charged,particles,kinetically,,such,as,XGC,[10],,GKEYLL,[11],and,COGENT,[12].,However,,all,these,models,are,based,on,δf,gy-,rokinetics,[13,,14],,originally,developed,for,the,tokamak,core,,and,which,assume,that,turbulence,structures,are,larger,than,the,ion,gyroradius,,and,that,density,and,temperature,are,slowly,varying.,These,assumption,no,longer,hold,at,the,edge,,and,it,is,necessary,instead,to,use,“full,f,”,gyroki-,netics,[15,,16],where,the,distribution,function,is,no,longer,taken,to,be,a,small,perturbation,about,a,Maxwellian.,Full,f,gyrokinetics,is,more,general,formulation,,and,by,imposing,appropriate,addi-,tional,orderings,,one,may,recover,either,δf,gyrokinetics,for,the,core,,or,a,regime,appropriate,for,describing,the,edge,plasma.,The,latter,is,essentially,drift,kinetics,but,with,corrections,due,to,ﬁnite,gyroradius,effects.,These,ﬁnite,gyroradius,effects,are,important,in,two,places.,Firstly,,damping,terms,appear,in,the,kinetic,equation,which,stabilize,the,small,wavelength,instabilities,that,appear,in,drift,kinetics.,Secondly,,a,“polarization,density”,term,appears,in,the,ﬁeld,equations,(quasineu-,trality,and,Amp,´ere’s,law).,This,term,is,often,small,,but,can,become,important,to,determine,the,electromagnetic,ﬁelds,in,certain,scenarios,,such,as,in,shear,Alfv,´en,waves.,While,implementing,the,damping,terms,is,relatively,straightforward,in,PIC,codes,,doing,so,has,not,been,attempted,in,full,f,continuum,codes.,Moreover,,the,correction,terms,in,the,ﬁeld,equations,can,lead,to,signiﬁcant,complications.,For,example,,to,include,these,corrections,while,also,ensuring,the,scheme,conserves,energy,and,momentum,would,lead,to,the,need,to,solve,a,nonlinear,equation,at,every,point,in,velocity,space.,In,addition,to,these,problems,,there,are,other,issues,a,model,must,address.,There,is,as,yet,no,gyrokinetic,formulation,for,the,wall,boundary,conditions,(only,a,much,less,general,formulation,for,ions,on,magnetic,ﬁeld,lines,that,intersect,the,wall,at,a,shallow,angle,[17]).,It,also,remains,to,develop,a,model,for,collisions,between,charged,particles,and,neutrals,,which,is,challenging,in,a,gyrokinetic,framework,as,gyrophase,can,be,important,in,the,process.,Finally,,gyrokinetics,assumes,ﬂuctuations,have,a,characteristic,scale,of,the,ion,gyroradius.,In,order,to,describe,Edge,Localized,Modes,–,signiﬁcant,disruptions,in,the,plasma,at,the,tokamak,edge,–,it,is,necessary,to,extend,gyrokinetics,to,described,ﬂuctuations,on,larger,MHD,scales.,2,1.1,Workplan,It,is,therefore,established,that,accurate,modelling,of,the,tokamak,edge,requires,the,inclusion,of,kinetic,effects.,A,tractable,model,will,also,use,gyroaveraging,to,remove,the,fastest,timescales,from,the,problem.,There,is,however,,no,existing,systematic,derivation,containing,both,of,these,for,plasma,in,the,tokamak,edge.,Making,such,a,derivation,is,the,main,thrust,of,this,work,package.,The,main,difﬁculty,is,performing,the,derivation,while,ensuring,that,the,equations,contain,all,the,information,required,to,completely,determine,the,system.,One,speciﬁc,problem,is,that,the,elec-,trostatic,potential,must,be,determined,from,the,quasineutrality,condition.,However,,even,small,deviations,from,charge,neutrality,cause,large,variations,in,the,electrostatic,potential,,leading,to,numerical,instabilities.,Thus,the,derived,system,must,be,amenable,to,implicit,time,stepping,,as,solving,quasineutrality,for,the,potential,explicitly,strongly,limits,possible,timesteps,[18].,The,derivation,introduces,a,novel,“moment,kinetic”,approach,,which,evolves,the,ﬂuid,moments,of,the,distribution,function,separately,from,the,rest,of,the,distribution,function.,As,well,as,allowing,efﬁcient,matching,between,ﬂuid,and,kinetic,models,,this,particular,decomposition,proves,vital,for,the,derivation,of,a,tractable,system,in,2D.,The,work,plan,is,to,derive,this,model,,ﬁrst,in,1D,with,periodic,and,wall,boundary,conditions,ap-,propriate,for,closed,and,open,ﬁeld,line,regions,of,the,edge,respectively.,The,model,will,then,be,extended,to,3D,,but,using,an,axisymmetric,helical,ﬁeld.,This,will,provide,a,model,that,is,improves,upon,models,implemented,in,existing,continuum,edge,codes.,The,model,is,limited,by,being,es-,sentially,2D,,and,for,using,a,simpliﬁed,helical,ﬁeld,,rather,than,a,realistic,tokamak,ﬁeld.,Both,of,these,are,candidates,for,future,work,,as,discussed,in,Section,3.1.,2,Theory,development,2.1,1D,drift,kinetics,with,periodic,boundary,conditions,The,ﬁrst,report,“1D,drift,kinetic,models,with,periodic,boundary,conditions,(2047357-TN-01-02),[2],develops,a,proof-of-concept,for,the,moment,kinetic,approach.,It,uses,a,simple,1D,drift,kinetic,model,with,periodic,boundary,conditions,,appropriate,for,the,edge,region,inside,the,separatrix,that,has,closed,ﬁeld,lines.,In,the,standard,kinetic,approaches,,one,solves,for,perturbations,about,a,Maxwellian,that,has,a,single,global,reference,density,,bulk,velocity,and,thermal,velocity.,In,the,moment-kinetic,approach,,one,instead,perturbs,about,a,Maxwellian,where,the,density,,bulk,velocity,and,thermal,velocity,are,themselves,variables,to,be,evolved,using,self-consistent,ﬂuid,equations.,Thus,one,replaces,a,single,kinetic,equation,for,the,distribution,function,with,a,set,of,a,ﬂuid,equations,and,a,kinetic,equation,for,a,modiﬁed,distribution,function.,The,main,motivation,for,this,approach,is,the,hope,that,it,would,allow,the,electrostatic,potential,to,be,derived,through,a,vorticity,equation,(using,ﬂuid,variables),rather,than,through,a,ﬁeld,solve,(requiring,kinetic,variables).,This,simpliﬁes,the,solution,procedure,,but,requires,further,theory,de-,velopment.,A,secondary,beneﬁt,of,splitting,out,the,moments,is,that,it,helps,the,numerics,when,3,dealing,with,dramatic,change,in,thermal,velocity,from,pedestal,to,divertor.,This,approach,normal-,izes,the,velocity,variable,in,the,modiﬁed,distribution,function,to,the,local,thermal,velocity,,meaning,no,regridding,is,necessary,for,different,regimes,in,the,plasma.,Finally,,this,approach,also,allows,one,to,develop,an,adaptive,model,which,would,automatically,switch,from,solving,the,ﬂuid+modiﬁed,kinetic,equations,to,only,ﬂuid,equations,when,the,distribution,function,was,sufﬁciently,close,to,Maxwellian.,This,may,prove,to,be,a,very,natural,and,computationally,tight,way,to,couple,a,ﬂuid,model,to,a,kinetic,model.,2.2,1D,drift,kinetics,with,wall,boundary,conditions,In,the,second,report,[3],,the,1+1D,drift,kinetic,model,is,extended,so,that,it,can,treat,the,wall,bound-,ary,conditions,of,the,open,ﬁeld,line,region,of,the,edge.,The,wall,is,assumed,to,be,perpendicular,to,the,magnetic,ﬁeld,lines,so,that,simpliﬁed,boundary,conditions,,the,logical,sheath,boundary,conditions,[19],can,be,used.,The,logical,boundary,conditions,account,for,a,thin,sheath,of,non-neutral,plasma,of,width,order,Debye,length,in,front,of,the,wall.,There,is,a,potential,drop,across,this,sheath,which,repels,electrons,–,otherwise,electrons,(with,their,lower,mass,and,higher,thermal,velocity),would,ﬂow,into,the,wall,at,a,greater,rate,than,ions,,violating,quasineutrality.,Requiring,current,into,the,sheath,to,vanish,gives,a,condition,for,the,electrostatic,potential,on,the,boundaries.,This,,in,conjunction,with,the,electron,parallel,momentum,equation,,can,be,used,to,ﬁnd,the,electrostatic,potential,across,the,whole,domain.,As,before,,the,low,moments,of,the,distribution,function,are,treated,separately,so,that,for,ions,and,neutrals,three,ﬂuid,equations,and,one,kinetic,equation,for,the,modiﬁed,distribution,function,must,be,evolved.,For,electrons,,two,ﬂuid,equations,and,a,kinetic,equation,must,be,evolved,,with,the,third,ﬂuid,equation,for,parallel,momentum,being,used,to,determine,the,electrostatic,potential.,2.3,2D,drift,kinetic,model,with,wall,boundary,conditions,The,third,report,[4],extends,the,previous,model,with,wall,boundary,conditions,to,a,2D,model,with,an,axisymmetric,helical,magnetic,ﬁeld.,While,being,simpler,,the,helical,ﬁeld,has,similarities,to,the,magnetic,ﬁeld,in,the,edge,of,a,tokamak.,As,the,magnetic,ﬁeld,is,now,not,perpendicular,to,the,walls,,it,is,in,principle,necessary,to,consider,the,magnetic,presheath,[20].,However,,the,boundary,conditions,that,this,would,entail,in,drift,kinet-,ics,are,both,complicated,and,an,area,of,open,research,[17,,21].,To,avoid,this,,it,is,assumed,that,the,electron,gyroradius,is,much,smaller,than,the,Debye,length,,in,which,case,boundary,conditions,similar,to,the,logical,boundary,conditions,[19],can,again,be,applied.,In,this,conﬁguration,,the,ion,and,electron,bulk,properties,only,vary,along,ﬁeld,lines,,so,as,before,,the,system,comprises,three,ﬂuid,equations,and,a,kinetic,equations,for,ions,,and,two,ﬂuid,equa-,tions,and,a,kinetic,equation,for,electrons.,As,before,,the,parallel,electron,momentum,equation,is,used,along,with,boundary,conditions,to,determine,the,electrostatic,potential.,Now,however,,the,distribution,functions,depend,on,two,spatial,and,two,velocity,space,coordinates.,The,velocity,coor-,4,dinates,are,parallel,and,perpendicular,to,the,magnetic,ﬁeld,,with,the,third,direction,,the,gyrophase,,removed,by,gyroaveraging.,As,neutrals,are,unaffected,by,the,magnetic,ﬁeld,,the,problem,for,neutrals,is,now,essentially,in,2+3D.,There,are,thus,ﬁve,ﬂuid,equations,to,solve,(for,density,and,thermal,velocity,,plus,three,bulk,velocity,components).,The,kinetic,equation,for,neutrals,now,also,depends,on,two,spatial,directions,and,three,velocity,space,directions.,2.4,2D,drift,kinetic,model,with,periodic,boundary,conditions,The,fourth,report,[5],derives,a,2D,model,with,periodic,boundary,conditions,relevant,for,closed,ﬂux,surfaces,,to,complement,the,wall,boundary,conditions,derived,in,a,previous,report.,That,report,described,how,to,obtain,the,electrostatic,potential,and,the,parallel,electron,bulk,velocity,with,wall,boundary,conditions.,The,same,prescription,with,periodic,boundary,conditions,only,allows,one,to,ﬁnd,the,electron,velocity;,to,obtain,the,electrostatic,potential,,higher-order,terms,must,be,included,in,the,current,conservation,equations.,The,main,work,of,this,report,is,deriving,these,higher-order,corrections,,and,consequently,a,ﬁnal,,additional,equation,(for,current,conservation,at,higher,order),to,solve,the,electrostatic,potential.,The,derivation,of,this,additional,equation,is,only,possible,in,the,moment-kinetic,framework,,which,evolves,the,ion,and,electron,densities,independently,of,their,normalized,distribution,functions.,With,unnormalized,distribution,functions,,the,corresponding,higher-order,current,conservation,equation,is,not,consistent,with,the,density,found,in,the,lower-order,kinetic,equations.,3,Summary,This,body,of,work,has,rigorously,and,systematically,derived,2D,drift,kinetic,equations,,appropriate,for,the,kinetic,treatment,of,ions,,electrons,and,neutral,in,the,tokamak,edge.,There,is,one,more,report,on,theory,development,to,be,submitted,under,this,grant,,“2D,drift,kinetics,in,a,helical,ﬁeld,with,a,‘separatrix”’,(which,is,currently,under,review),and,which,combines,the,2D,drift,kinetic,models,for,the,open,and,closed,ﬁeld,line,regions,into,a,single,model.,This,will,complete,the,theoretical,work,under,the,current,grant.,The,theoretical,developments,have,been,accompanied,by,the,numerical,implementation,of,these,equations,,using,a,Chebyshev,spectral,ﬁnite,element,implementation,in,Julia.,The,code,develop-,ments,are,described,in,separate,reports,[22,,23,,24],,and,the,software,is,publicly,available,[25].,At,present,,this,code,only,implements,a,subset,of,the,models,presented,here;,future,reports,will,discuss,the,software,developments,when,the,code,is,more,mature.,3.1,Future,directions,There,are,a,number,of,directions,for,future,work.,The,ﬁrst,is,to,extend,the,model,into,3D,with,realistic,tokamak,geometry.,The,main,difﬁculty,here,is,the,inclusion,of,drifts,in,the,direction,per-,pendicular,to,the,ﬂux,surfaces.,These,lead,to,ﬁnite-width,particle,orbits,which,can,signiﬁcantly,5,alter,the,physics,in,quiescent,2D,[26],and,turbulent,3D,[27],plasmas.,The,rigorous,derivation,of,these,drifts,requires,further,work.,The,resulting,3D,model,is,also,expected,to,have,instabilities,on,the,grid,scale.,While,the,model,could,be,stabilized,using,numerical,dissipation,on,the,grid,scale,,further,work,is,required,to,prop-,erly,derive,the,ﬁnite,ion,gyroradius,effects,that,determine,behaviour,of,ﬂuctuations,at,this,scale.,This,is,particularly,important,for,studying,high,conﬁnement,mode,(H-mode),where,the,stabiliza-,tion,of,large-scale,turbulence,is,not,well,understood.,In,H-mode,,turbulence,at,the,ion,gyroscale,and,smaller,is,important,[27],,and,so,the,accurate,capture,of,ﬁnite,ion,gyroradius,effects,is,vital.,Including,these,ﬁnite,gyroradius,effects,has,the,potential,to,be,both,theoretically,and,numerically,challenging.,The,wall,boundary,conditions,in,the,model,are,also,simpliﬁed,conditions,which,assume,the,mag-,netic,ﬁeld,is,perpendicular,to,the,wall.,In,the,edge,,this,is,often,not,the,case,,with,magnetic,ﬁeld,lines,meeting,the,wall,at,very,shallow,angles.,The,boundary,conditions,are,formally,valid,in,the,limit,of,electron,gyroradius,being,much,smaller,than,the,Debye,length,,but,relaxing,this,condition,requires,further,study.,This,is,currently,being,pursued,by,Parra,and,coworkers,outside,ExCAL-,IBUR.,Finally,,it,would,be,beneﬁcial,to,extend,the,model,to,treat,ﬂuctuations,in,the,magnetic,ﬁeld,,as,well,as,electric,ﬁeld.,While,electrostatic,models,are,often,a,good,approximation,for,edge,plasmas,,including,electromagnetic,effects,would,allow,this,model,to,connect,with,the,MHD,regime,,and,so,to,study,large-scale,plasma,disturbances.,6,Acknowledgement,The,support,of,the,UK,Meteorological,Ofﬁce,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],F.,I.,Parra,,M.,Barnes,,and,M.,Hardman.,Physics,in,the,edge,of,fusion,devices.,Tech-,nical,Report,2047357-TN-03-02,,UKAEA,Project,Neptune,,2021.,https://github.com/,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047357/TN-03.pdf.,[2],F.I,Parra,,M.,Barnes,,and,M.R.,Hardman.,1D,drift,kinetic,models,with,periodic,boundary,conditions.,Technical,Report,2047357-TN-01-02,,UKAEA,Project,Neptune,,2021.,https:,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047357/TN-01.pdf.,[3],F.,I.,Parra,,M.,Barnes,,and,M.,Hardman.,1D,drift,kinetic,models,with,wall,boundary,conditions.,Technical,Report,2047357-TN-05-01,,UKAEA,Project,Neptune,,2021.,https://github.com/,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047357/TN-05.pdf.,[4],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.,[5],F.,I.,Parra,,M.,Barnes,,and,M.,Hardman.,2D,drift,kinetic,models,with,periodic,boundary,conditions.,Technical,Report,2047357-TN-09-01,,UKAEA,Project,Neptune,,2021.,https:,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047357/TN-09.pdf.,[6],D.,Tskhakaya,,S.,Jachmich,,T.,Eich,,W.,Fundamenski,,and,JET,EFDA,contributors.,Interpre-,tation,of,divertor,Langmuir,probe,measurements,during,the,ELMs,at,JET.,Journal,of,Nuclear,Materials,,415(1):S860–S864,,2011.,[7],A.,V.,Chankin,and,D.,P.,Coster.,On,the,locality,of,parallel,transport,of,heat,carrying,electrons,in,the,SOL.,Journal,of,Nuclear,Materials,,463:498–501,,2015.,[8],D.,Reiter,,M.,Baelmans,,and,P.,Boerner.,The,EIRENE,and,B2-EIRENE,codes.,Fusion,Science,and,Technology,,47(2):172–186,,2005.,[9],D.,Stotler,and,C.,Karney.,Neutral,gas,transport,modeling,with,DEGAS,2.,Contributions,to,Plasma,Physics,,34(2-3):392–397,,1994.,[10],S.,Ku,,R.,Hager,,C.-S.,Chang,,J.,M.,Kwon,,and,S.,E.,Parker.,A,new,hybrid-Lagrangian,nu-,merical,scheme,for,gyrokinetic,simulation,of,tokamak,edge,plasma.,Journal,of,Computational,Physics,,315:467–475,,2016.,[11],A.,H.,Hakim,,N.,R.,Mandell,,T.,N.,Bernard,,M.,Francisquez,,G.,W.,Hammett,,and,E.,L.,Shi.,Continuum,electromagnetic,gyrokinetic,simulations,of,turbulence,in,the,tokamak,scrape-off,layer,and,laboratory,devices.,Physics,of,Plasmas,,27(4):042304,,2020.,7,[12],M.,A.,Dorf,,M.,R.,Dorr,,J.,A.,Hittinger,,R.,H.,Cohen,,and,T.,D.,Rognlien.,Continuum,kinetic,modeling,of,the,tokamak,plasma,edge.,Physics,of,Plasmas,,23(5):056102,,2016.,[13],P.,J.,Catto.,Linearized,gyro-kinetics.,Plasma,Physics,,20(7):719,,1978.,[14],E.,A.,Frieman,and,L.,Chen.,Nonlinear,gyrokinetic,equations,for,low-frequency,electromag-,netic,waves,in,general,plasma,equilibria.,The,Physics,of,Fluids,,25(3):502–508,,1982.,[15],F.,I.,Parra,and,P.,J.,Catto.,Limitations,of,gyrokinetics,on,transport,time,scales.,Plasma,Physics,and,Controlled,Fusion,,50(6):065014,,2008.,[16],A.,J.,Brizard,and,T.,S.,Hahm.,Foundations,of,nonlinear,gyrokinetic,theory.,Reviews,of,modern,physics,,79(2):421–468,,2007.,[17],A.,Geraldini,,F.I.,Parra,,and,F.,Militello.,Dependence,on,ion,temperature,of,shallow-angle,magnetic,presheaths,with,adiabatic,electrons.,Journal,of,Plasma,Physics,,85(6),,2019.,[18],W.W.,Lee.,Gyrokinetic,particle,simulation,model.,Journal,of,Computational,Physics,,72(1):243–269,,1987.,[19],S.,E.,Parker,,R.,J.,Procassini,,C.,K.,Birdsall,,and,B.,I.,Cohen.,A,suitable,boundary,condition,for,bounded,plasma,simulation,without,sheath,resolution.,Journal,of,Computational,Physics,,104(1):41–49,,1993.,[20],R.,Chodura.,Plasma-wall,transition,in,an,oblique,magnetic,ﬁeld.,The,Physics,of,Fluids,,25(9):1628–1633,,1982.,[21],A.,Geraldini.,Large,gyro-orbit,model,of,ion,velocity,distribution,in,plasma,near,a,wall,in,a,grazing-angle,magnetic,ﬁeld.,Journal,of,Plasma,Physics,,87(1),,2021.,[22],M.,Barnes,,F.,I.,Parra,,and,M.,R.,Hardman.,Numerical,study,of,1D,drift,kinetic,models,with,periodic,boundary,conditions.,Technical,Report,2047357-TN-02-02,,UKAEA,Project,Nep-,tune,,2021.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047357/TN-02.pdf.,[23],M.,Barnes,,F.,I.,Parra,,M.,R.,Hardman,,and,J.,Omotani.,Numerical,study,of,1+1D,,moment-,based,drift,kinetic,models,with,periodic,boundary,conditions.,Technical,Report,2047357-TN-,04,,UKAEA,Project,Neptune,,2021.,https://github.com/ExCALIBUR-NEPTUNE/Documents/,blob/main/reports/2047357/TN-04.pdf.,[24],M.,Barnes,,F.,I.,Parra,,M.,R.,Hardman,,and,J.,Omotani.,Numerical,study,of,1+1D,drift,kinetic,model,with,wall,boundary,conditions.,Technical,Report,2047357-TN-08,,UKAEA,Project,Neptune,,2021.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047357/TN-08.pdf.,[25],M.,A.,Barnes,and,J.,T.,Omotani.,moment,kinetics,GitHub,repository.,https://github.com/,mabarnes/moment_kinetics,,2021.,Accessed:,December,2021.,[26],G.,Kagan,and,P.,J.,Catto.,Arbitrary,poloidal,gyroradius,effects,in,tokamak,pedestals,and,transport,barriers.,Plasma,Physics,and,Controlled,Fusion,,50(8):085010,,2008.,8,[27],J.,F.,Parisi,,F.,I.,Parra,,C.,M.,Roach,,C.,Giroud,,W.,Dorland,,D.,R.,Hatch,,M.,Barnes,,J.,C.,Hillesheim,,N.,Aiba,,J.,Ball,,et,al.,Toroidal,and,slab,ETG,instability,dominance,in,the,linear,spectrum,of,JET-ILW,pedestals.,Nuclear,Fusion,,60(12):126045,,2020.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0058,Issue:,Date:,1.0,9,December,2021,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Joseph,Parker,Wayne,Arter,Signature,N/A,N/A,Date,9,December,2021,9,December,2021,BD,Reviewed,By:,Rob,Akers,9,December,2021,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,9,December,2021,Advanced,Dept.,Manager,Computing,9,REPORT,2047357-TN-03-02,M1.2,1,Physics,in,the,edge,of,fusion,devices,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,23,April,2021),1.,Introduction,We,present,a,brief,overview,of,the,current,state,of,fusion,device,edge,modeling.,This,re-,port,is,written,for,ExCALIBUR,NEPTUNE,contract,T/NA085/20.,This,contract,stated,mission,is,the,development,of,kinetic,models,for,the,edge.,Thus,,this,report,will,focus,on,the,known,problems,of,ﬂuid,models,without,making,much,emphasis,on,their,many,suc-,cesses.,This,lack,of,praise,for,ﬂuid,models,is,not,the,objective,of,this,report,and,hence,we,would,like,to,start,by,reassuring,the,reader,that,we,believe,current,ﬂuid,codes,have,much,to,oﬀer,and,need,to,be,pursued,in,parallel,to,kinetic,modeling.,One,of,the,challenges,for,kinetic,modeling,is,to,devise,methods,to,match,with,ﬂuid,simulations,eﬃciently.,With,this,capability,,the,edge,can,be,split,into,non-overlapping,spatial,regions,with,diﬀerent,levels,of,kinetic,sophistication.,The,remainder,of,this,reports,is,organized,as,follows.,In,section,2,we,explain,what,the,characteristic,time,and,length,scales,are,in,the,edge.,In,section,3,,we,give,an,overview,of,the,existing,ﬂuid,models,for,the,edge.,In,section,4,we,list,physical,phenomena,that,cannot,be,captured,by,ﬂuid,models,,and,we,explain,how,these,phenomena,have,been,addressed,so,far.,Finally,,we,discuss,our,proposal,to,develop,a,complete,set,of,kinetic,equations,for,the,edge,in,section,5.,2.,The,edge,In,this,report,,we,call,the,edge,the,plasma,that,surrounds,the,separatrix.,The,separatrix,(represented,as,a,red,line,in,ﬁgure,1),is,the,ﬂux,surface,that,separates,the,region,where,magnetic,ﬁeld,lines,are,in,contact,with,the,walls,of,the,vessel,from,the,region,in,which,magnetic,ﬁeld,lines,form,nested,toroidal,ﬂux,surfaces.,This,means,that,the,edge,includes,both,open,ﬁeld,lines,(those,who,are,in,contact,with,walls),and,closed,ﬁeld,lines,(those,that,form,nested,toroidal,ﬂux,surfaces).,We,need,to,distinguish,the,core,,where,the,fusion,reactions,are,supposed,to,take,place,,from,the,edge.,In,this,report,,we,will,use,a,theoretical,criterion,to,distinguish,one,from,the,other:,whether,the,characteristic,transit,times,in,the,directions,parallel,and,perpendicular,to,the,magnetic,ﬁeld,are,comparable,or,not.,In,the,core,,both,ions,and,electron,closely,follow,magnetic,ﬁeld,lines,and,can,travel,around,the,device,many,times,before,collisions,,turbulent,ﬂuctuations,or,other,eﬀects,drive,them,away,from,the,magnetic,ﬁeld,line,on,which,they,started.,As,a,result,,density,and,temperature,are,almost,constant,along,magnetic,ﬁeld,lines,,and,one,only,worries,about,small,ﬂuctuations,around,the,mostly,quiescent,proﬁles,of,density,and,temperature.,This,is,,of,course,,a,highly,idealized,situation,that,ignores,violent,events,taking,place,in,the,core,,such,as,sawteeth,(Hastie,1997),,but,it,is,a,useful,one.,In,contrast,,in,the,edge,,charged,particles,that,were,well-conﬁned,in,the,core,region,cross,the,separatrix,and,eventually,reach,the,wall,by,following,magnetic,ﬁeld,lines.,The,displacements,of,ions,and,electrons,away,from,the,magnetic,ﬁeld,line,in,which,they,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Figure,1.,Sketch,of,a,tokamak,edge,on,a,plane,that,contains,the,axis,of,symmetry,(represented,here,by,a,dash-dot,vertical,line).,The,edge,region,is,in,blue,,and,the,separatrix,is,the,red,line.,started,are,still,small,compared,to,the,size,of,the,device,,but,they,are,not,small,compared,with,the,characteristic,lengths,of,the,density,and,temperature,in,the,edge,region.,If,the,characteristic,lengths,of,density,and,temperature,along,and,across,magnetic,ﬁeld,lines,are,L(cid:107),and,L⊥,,the,edge,is,characterized,by,|v(cid:107)|,L(cid:107),∼,|vd|,L⊥,,,(2.1),where,v(cid:107),is,the,characteristic,parallel,velocity,of,the,particles,,and,vd,their,small,drift,perpendicular,to,the,magnetic,ﬁeld.,Since,|v(cid:107)|,(cid:29),|vd|,,L⊥,is,much,smaller,than,L(cid:107).,In,the,core,,for,comparison,,we,ﬁnd,that,L(cid:107),and,L⊥,are,of,similar,order,and,comparable,to,the,machine,size,,giving,|v(cid:107)|,L(cid:107),(cid:29),|vd|,L⊥,.,(2.2),There,is,another,aspect,that,makes,the,edge,very,diﬀerent,from,the,core.,The,tem-,perature,of,both,electrons,and,ions,is,kept,low,near,the,wall,because,the,wall,is,a,very,eﬀective,sink,of,energy,(wall,materials,that,prevent,slow,Hydrogen,particles,from,going,back,into,the,plasma,are,notable,exceptions,where,the,temperature,of,the,plasma,can,be,large,near,the,wall;,see,,for,example,,Schmitt,et,al.,(2015),for,liquid,Lithium,divertors,,or,Jackson,et,al.,(1991),for,boronized,walls).,At,low,plasma,temperatures,,the,plasma,is,partially,ionized,and,collisions,between,the,charged,particles,in,the,plasma,and,neutrals,become,important.,In,extreme,limits,,detachment,occurs,,that,is,,the,plasma,tempera-,ture,decreases,suﬃciently,due,to,radiation,that,the,plasma,recombines,and,a,cushion,of,neutrals,appears,in,front,of,the,walls,protecting,them,(Krasheninnikov,&,Kukushkin,2017).,In,addition,to,limiting,the,temperature,,the,presence,of,the,wall,controls,the,size,of,the,electric,ﬁeld,and,the,ﬂows,in,the,open,ﬁeld,line,region.,We,will,discuss,these,eﬀects,in,more,detail,in,section,3.,We,ﬁnish,this,section,by,calculating,a,few,characteristic,time,and,length,scales.,In,Militello,&,Fundamenski,(2011),,one,can,ﬁnd,a,concise,summary,of,typical,values,of,plasma,characteristics,in,the,edge,of,tokamaks.,In,current,tokamaks,,the,magnitude,of,Closedfieldline,regionOpen,field,line,regionWalls,Physics,in,the,edge,of,fusion,devices,3,the,magnetic,ﬁeld,B,is,typically,2,T,,and,the,characteristic,size,of,the,device,is,a,few,meters.,The,plasma,temperature,T,ranges,from,1,keV,in,the,closed,magnetic,ﬁeld,line,part,of,the,edge,to,100,eV,at,the,separatrix,down,to,10,eV,near,the,wall.,In,detached,plasmas,,the,temperature,drops,to,1,eV,near,the,wall.,The,electron,density,ne,ranges,from,1019,m−3,in,the,open,ﬁeld,line,region,to,1020,m−3,in,the,closed,ﬁeld,line,region.,Neutral,density,nn,is,usually,smaller,than,ne,and,it,ranges,from,1015,m−3,in,the,closed,ﬁeld,line,region,to,1019,m−3,in,the,open,ﬁeld,line,region,(Colchin,et,al.,2000;,Scotti,et,al.,2021).,With,these,quantities,,we,calculate,several,characteristic,frequencies,of,interest:,•,The,gyrofrequencies,of,both,Deuterium,and,electrons,,Ωi,:=,eB/mD,and,Ωe,:=,eB/me,,are,the,characteristic,frequencies,of,the,nearly,circular,motion,of,the,charged,particles,around,magnetic,ﬁeld,lines.,Here,e,is,the,proton,charge,,and,mD,and,me,are,the,Deuterium,and,electron,masses.,•,The,transit,frequencies,for,Deuterium,ions,and,electrons,,vtD/L(cid:107),and,vte/L(cid:107),,are,the,inverse,of,the,time,that,it,takes,charged,particles,to,move,along,a,magnetic,ﬁeld,line,from,one,wall,to,another,in,the,open,ﬁeld,line,region,,and,the,inverse,of,the,time,that,it,takes,charged,particles,to,sample,a,ﬂux,surface,in,the,closed,ﬁeld,line,region.,Here,vtD,:=,(cid:112)2T,/mD,and,vte,:=,(cid:112)2T,/me,are,the,Deuterium,and,electron,thermal,speeds,,and,L(cid:107),is,the,characteristic,length,of,magnetic,ﬁeld,lines,in,the,edge,,which,we,take,to,be,10,m.,•,The,transit,frequency,for,Deuterium,neutrals,,vtD/L⊥,,is,the,inverse,of,the,time,that,it,takes,neutral,atoms,to,cross,the,edge,region.,The,characteristic,scale,of,variation,of,density,and,temperature,across,magnetic,ﬁeld,lines,,L⊥,,is,of,the,order,of,5,cm,or,larger,in,the,closed,ﬁeld,line,region,(Sugihara,et,al.,2000),,and,of,the,order,of,1,cm,or,larger,in,the,open,ﬁeld,line,region,(Goldston,2012).,•,The,collision,frequencies,and,νii,:=,√,4,π,3,e4ne,ln,Λ,(4π(cid:15)0)2m1/2,D,T,3/2,νep,:=,√,4,2π,3,e4ne,ln,Λ,(4π(cid:15)0)2m1/2,e,T,3/2,(2.3),(2.4),describe,how,often,Deuterium,ions,collide,with,each,other,or,electrons,collide,with,other,electrons,or,Deuterium,ions,,respectively.,Here,ln,Λ,≈,15,is,the,Coulomb,logarithm,and,(cid:15)0,2,diﬀerence,between,νii,and,νep,is,a,convention,is,the,vacuum,permittivity.,The,factor,of,introduced,by,Braginskii,(Braginskii,1958).,√,•,We,also,need,the,collision,frequencies,that,describe,how,often,Deuterium,ions,collide,with,Deuterium,neutral,atoms,,νin,:=,nnvtDσin,,and,how,often,Deuterium,neutral,atoms,collide,with,Deuterium,ions,,νni,:=,nivtDσin.,Here,,the,ion-neutral,cross,section,σin,is,of,order,10−18,m2,(Lindsay,&,Stebbings,2005).,Similarly,,we,need,the,collision,frequencies,that,describes,how,often,electrons,collide,with,Deuterium,neutral,atoms,,νen,:=,nnvteσen,,and,how,often,neutrals,are,ionized,,νion,:=,nevteσion.,Here,,the,electron-neutral,collision,cross,section,σen,is,of,order,10−19,m2,(Brackmann,et,al.,1958),and,the,ionization,collision,cross,section,σion,is,of,order,10−20,m2,(Zel’dovich,&,Raizer,2013).,All,these,frequencies,are,shown,in,table,1.,It,is,clear,that,the,collision,frequencies,are,the,ones,that,change,the,most,across,the,edge.,For,ions,and,electrons,,collisions,become,very,important,in,the,cooler,plasma,of,the,open,ﬁeld,line,region,,but,are,infrequent,in,the,closed,ﬁeld,line,region.,Conversely,,for,neutrals,,collision,are,more,frequent,in,the,closed,ﬁeld,line,region,than,in,the,open,ﬁeld,line,region.,We,also,calculate,a,few,length,scales,of,interest:,4,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Ion,freq.,[104,s−1],Ωi,Region,Closed,lines,9400,9400,Open,lines,vtD/L(cid:107),3.1,0.31,νin,νii,0.16,0.031,16,31,Neutral,freq.,[104,s−1],vtD/L⊥,νni,3100,31,νion,1800,18,610,310,Electron,freq.,[106,s−1],Ωe,νen,vte/L(cid:107),νep,320000,320000,0.13,0.0018,13,1.8,0.18,1.8,Table,1.,Characteristic,frequencies,for,the,closed,ﬁeld,line,region,(B,=,2,T,,ne,=,1020,m−3,,nn,=,1015,m−3,,T,=,1,keV),and,the,open,ﬁeld,line,region,(B,=,2,T,,ne,=,1019,m−3,,nn,=,1019,m−3,,T,=,10,eV).,Parallel,lengths,[1,m],L(cid:107),λii,Region,Closed,lines,10,190,1000,140,10000,Open,lines,λen,L⊥,5,1,λin,λep,10,0.19,0.14,0.1,1,Perpendicular,lengths,[1,cm],ρe,ρi,0.33,0.0056,0.033,0.00056,λni,1,10,λion,1.7,17,Table,2.,Characteristic,lengths,for,the,closed,ﬁeld,line,region,(B,=,2,T,,ne,=,1020,m−3,,nn,=,1015,m−3,,T,=,1,keV),and,the,open,ﬁeld,line,region,(B,=,2,T,,ne,=,1019,m−3,,nn,=,1019,m−3,,T,=,10,eV).,•,The,mean,free,paths,λii,:=,vtD/νii,and,λin,:=,vtD/νin,are,the,distances,that,a,Deu-,terium,ion,can,travel,before,colliding,with,another,Deuterium,ion,or,with,a,Deuterium,atom,,respectively.,Similarly,,the,mean,free,paths,λni,:=,vtD/νni,and,λion,:=,vtD/νion,are,the,distances,that,a,Deuterium,neutral,atom,can,move,before,colliding,with,an,ion,or,getting,ionized,,respectively,,and,λep,:=,vte/νep,and,λen,:=,vte/νen,are,the,distances,that,an,electron,can,move,before,colliding,with,another,charged,particle,or,with,a,neutral,,respectively.,•,The,Deuterium,and,electron,gyroradii,,ρi,:=,vtD/Ωi,and,ρe,:=,vte/Ωe,,are,the,characteristic,size,of,the,gyration,of,charged,particles,around,magnetic,ﬁeld,lines.,All,these,lengths,are,given,in,table,2.,Unsurprisingly,,we,see,that,the,mean,free,paths,,inversely,proportional,to,the,collision,frequencies,,are,the,characteristic,lengths,that,change,the,most,across,the,edge.,In,the,direction,perpendicular,to,the,magnetic,ﬁeld,,we,see,that,the,gyroradii,are,small,compared,to,the,characteristic,lengths.,3.,Drift-ordered,ﬂuid,models,Due,to,the,large,collision,frequencies,in,the,open,ﬁeld,line,region,,the,bulk,of,the,ion,and,electron,distribution,functions,is,Maxwellian.,For,this,reason,,much,of,the,edge,modeling,has,been,based,on,plasma,ﬂuid,equations,derived,in,the,limit,of,large,Coulomb,collisions,(Braginskii,1958).,Since,the,most,readily,available,ﬂuid,equations,for,magnetized,plasmas,assume,that,there,are,no,neutrals,,most,ﬂuid,models,ignore,neutrals,,with,some,notable,exceptions.,The,main,diﬀerence,between,the,ﬂuid,equations,used,in,edge,models,and,the,usual,ﬂuid,equations,is,that,diﬀusivities,are,anisotropic.,As,demonstrated,by,table,1,,charged,parti-,cles,gyrate,around,magnetic,ﬁeld,lines,many,times,before,having,a,collision.,Thus,,parti-,cles,barely,move,across,magnetic,ﬁeld,lines,and,the,diﬀusivity,across,magnetic,ﬁeld,lines,is,much,smaller,than,the,diﬀusivity,along,them.,To,capture,the,eﬀect,of,this,anisotropy,correctly,and,eﬃciently,,one,has,to,either,use,ﬂux,coordinates,that,follow,magnetic,ﬁeld,lines,(Beer,et,al.,1995),or,employ,appropriate,discretizations,(Hariri,&,Ottaviani,2013).,Physics,in,the,edge,of,fusion,devices,5,The,problem,with,following,magnetic,ﬁeld,lines,is,that,it,is,diﬃcult,to,ﬁnd,a,grid,that,both,aligns,with,the,magnetic,ﬁeld,lines,and,extends,to,the,walls,of,the,vessel.,Unfor-,tunately,,the,wall,geometry,is,important,because,it,determines,how,far,neutrals,leaving,the,wall,travel,into,the,plasma,(Wiesen,et,al.,2018).,Finding,grids,that,adjust,to,the,wall,geometry,has,become,a,problem,of,great,interest,in,recent,years,,and,the,community,has,been,trying,exotic,methods,to,address,it,(Isoardi,et,al.,2010;,Buﬀerand,et,al.,2019).,An,important,aspect,of,the,edge,ﬂuid,models,is,the,size,of,the,ﬂow.,The,potential,diﬀerence,between,the,wall,and,the,plasma,is,controlled,by,the,non-neutral,Debye,sheath,that,forms,around,the,walls,(Riemann,1991).,Typically,,the,potential,drop,between,the,wall,and,the,plasma,has,to,be,several,times,the,electron,temperature,because,otherwise,a,large,electron,current,ﬂows,into,the,wall,,breaking,the,neutrality,of,the,plasma.,If,one,assumes,that,the,wall,in,contact,with,the,plasma,is,a,conductor,and,hence,the,potential,is,constant,across,its,volume,,the,potential,diﬀerences,within,the,plasma,are,restricted,to,be,of,the,order,of,the,electron,temperature,,φ,∼,T,/e.,This,gives,an,electric,ﬁeld,E,=,−∇φ,of,the,order,of,T,/eL⊥.,The,perpendicular,velocity,of,the,ﬂuid,is,determined,by,the,balance,between,the,electric,and,magnetic,forces,,This,equation,gives,a,perpendicular,velocity,of,order,u⊥,×,B,∼,E.,u⊥,∼,|E|,B,∼,ρi,L⊥,vtD.,(3.1),(3.2),Thus,,according,to,table,2,,u⊥,is,signiﬁcantly,smaller,than,vtD.,The,ﬂuid,equations,obtained,with,the,ordering,(3.2),are,known,as,drift-ordered,equations,because,the,ﬂow,is,of,the,same,order,as,the,slow,particle,drifts,–,the,other,possible,ordering,is,the,high,ﬂow,ordering,that,assumes,that,the,perpendicular,velocity,is,sonic.,Importantly,,for,ﬂuid,velocities,of,the,size,given,in,equation,(3.2),,one,needs,to,keep,terms,that,depend,on,the,gradient,of,the,temperature,and,the,pressure,in,the,stress,tensor,to,be,completely,consistent,(Mikhailovskii,&,Tsypin,1971;,Simakov,&,Catto,2003;,Catto,&,Simakov,2004).,The,system,of,drift-ordered,ﬂuid,equations,is,usually,comprised,of,•,one,continuity,equation,per,ion,species,(the,electron,density,need,not,be,calculated,because,it,is,determined,by,quasineutrality),,•,one,conservation,equation,for,the,component,of,the,plasma,momentum,parallel,to,the,magnetic,ﬁeld,,•,a,vorticity,equation,that,determines,the,electrostatic,potential,,•,in,electromagnetic,models,,Amp´ere’s,law,,and,•,one,conservation,equation,for,the,energy,of,all,ion,species,and,another,one,for,the,energy,of,the,electrons.,Note,that,there,is,one,single,conservation,equation,for,the,whole,plasma,parallel,momen-,tum,and,one,single,conservation,equation,for,the,energy,of,all,the,ion,species,,and,not,several,conservation,equations,,one,per,ion,species.,The,reason,why,all,ion,species,must,be,considered,as,one,in,these,ﬂuid,equations,is,that,,within,the,large,collision,frequency,approximation,,all,ion,species,have,the,same,temperature,and,all,charged,species,have,the,same,average,ﬂow.,The,electron,temperature,can,be,diﬀerent,from,the,ion,temperature,due,to,the,mass,diﬀerence,between,the,two,species.,The,temperature,diﬀerences,between,the,diﬀerent,ion,species,are,of,the,same,order,as,kinetic,eﬀects,that,are,neglected.,The,diﬀerences,between,the,parallel,ﬂows,are,calculated,and,used,in,the,vorticity,equation,,where,they,are,needed,because,the,electric,current,enters,in,the,Lorenz,force.,As,we,mentioned,at,the,start,,the,drift-ordered,ﬂuid,equations,usually,implemented,6,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,in,edge,codes,(Mikhailovskii,&,Tsypin,1971;,Simakov,&,Catto,2003;,Catto,&,Simakov,2004),ignore,neutrals.,There,have,been,attempts,to,include,neutrals,in,the,limit,where,both,ions,and,neutrals,are,suﬃciently,collisional,that,both,species,can,be,treated,as,ﬂuids,(Hazeltine,et,al.,1992;,Catto,1994;,Helander,et,al.,1994).,The,advantage,of,these,models,is,that,they,account,for,the,eﬀect,that,ion-neutral,collisions,have,on,the,diﬀusivities,,but,it,is,generally,believed,that,these,models,are,insuﬃcient,because,neutrals,have,long,mean,free,paths,in,large,regions,of,the,edge,(see,tables,1,and,2).,There,have,been,some,attempts,at,simpliﬁed,kinetic,treatments,of,the,neutrals,(Wersal,&,Ricci,2015),and,there,exist,sophisticated,Monte-Carlo,approaches,with,a,large,collection,of,collisions,such,as,EIRENE,(Reiter,et,al.,2005),or,DEGAS,(Stotler,&,Karney,1994).,Note,that,these,Monte-,Carlo,treatments,are,not,completely,self-consistent,because,they,assume,that,the,ion,and,electron,distribution,functions,are,Maxwellian,and,,depending,on,the,version,of,the,code,,they,ignore,elastic,collisions,and,they,average,over,the,dependence,of,the,diﬀerential,cross,section,on,the,scattering,angle.,Moreover,,these,neutral,kinetic,codes,are,sensitive,to,the,parameters,that,determine,how,particles,interact,with,the,wall,(Chankin,et,al.,2021),,and,these,parameters,are,not,well,known.,There,are,two,types,of,problems,in,which,the,drift-ordered,ﬂuid,equations,are,used:,2D,proﬁles,and,turbulence.,The,objective,of,2D,ﬂuid,solvers,such,as,SOLPS,(Wiesen,et,al.,2015),,SolEdge2D,(Buﬀerand,et,al.,2015),,UEDGE,(Rognlien,et,al.,2007),or,EDGE2D,(Simonini,et,al.,1994),is,to,determine,the,toroidally,averaged,density,and,temperature,proﬁles,in,the,edge.,These,codes,cannot,model,turbulent,ﬂuctuation,because,they,are,missing,the,third,dimension.,For,this,reason,,instead,of,the,collisional,perpendicular,diﬀusion,coeﬃcients,,these,codes,use,enhanced,perpendicular,diﬀusion,coeﬃcients,that,are,chosen,to,match,the,experimental,observations.,These,2D,codes,are,meant,to,provide,detailed,understanding,of,transport,along,magnetic,ﬁeld,lines,and,of,neutrals,,as,some,of,them,are,coupled,to,Monte-Carlo,neutral,codes,such,as,EIRENE,(Reiter,et,al.,2005),or,DEGAS,(Stotler,&,Karney,1994).,Turbulence,codes,such,as,GBS,(Halpern,et,al.,2016),,TOKAM3X,(Tamain,et,al.,2016),,Hermes,(Dudson,&,Leddy,2017),or,GRILLIX,(Stegmeir,et,al.,2018),are,3D,ﬂuid,codes.,Originally,,ﬂuid,turbulence,codes,assumed,that,ions,were,much,colder,than,electrons,and,that,the,turbulent,ﬂuctuations,were,small,compared,to,an,almost,constant,background,(Zeiler,et,al.,1996).,However,,it,was,soon,realized,that,this,treatment,is,not,appropriate,for,the,edge.,Ions,are,not,cold,and,the,density,and,temperature,proﬁles,cannot,be,easily,split,into,a,slowly,varying,piece,plus,small,ﬂuctuations,due,to,the,presence,of,the,wall.,The,same,wall,boundary,conditions,that,constrain,the,ﬂuid,velocity,perpendicular,to,the,magnetic,ﬁeld,to,be,subsonic,,as,shown,in,equation,(3.2),,require,that,the,ﬂuid,velocity,parallel,to,the,magnetic,ﬁeld,be,sonic,near,the,wall,(Chodura,1982).,The,pressure,drops,along,magnetic,ﬁeld,lines,have,to,be,signiﬁcant,to,ensure,that,the,parallel,ﬂow,is,accelerated,suﬃciently.,This,is,incompatible,with,the,assumption,that,the,ﬂuctuations,are,small.,Thus,,most,current,edge,turbulence,codes,have,tried,to,lift,the,assumption,of,small,ﬂuctuations,,and,they,have,done,so,by,diﬀering,degrees,depending,on,the,code,or,the,version,of,the,code,in,use.,Overall,,ﬂuid,codes,are,maturing,,and,although,there,is,still,work,to,be,done,,there,is,starting,to,be,a,consensus,on,the,physics,that,they,must,include.,The,same,cannot,be,said,about,kinetic,eﬀects,in,the,edge.,4.,Kinetic,eﬀects,Tables,1,and,2,are,evidence,that,kinetic,eﬀects,must,be,taken,into,account:,in,the,closed,magnetic,ﬁeld,line,region,and,in,part,of,the,open,ﬁeld,line,region,,ions,and,electrons,can,Physics,in,the,edge,of,fusion,devices,7,travel,a,distance,of,the,size,of,the,device,without,suﬀering,a,single,collision.,Moreover,,kinetic,eﬀects,are,surprisingly,important,even,in,the,regions,where,the,collision,frequency,is,high.,There,are,two,reasons,for,this.,•,The,diﬀusive,heat,ﬂux,calculated,in,the,ﬂuid,limit,is,only,accurate,if,νep,(cid:38),100,vte/L(cid:107),because,the,heat,ﬂux,is,dominated,by,energetic,particles,that,collide,much,less,than,the,thermal,particles,(Gurevich,&,Istomin,1979;,Gray,&,Kilkenny,1980).,The,ﬂuid,solution,assumes,that,there,are,too,many,energetic,particles,in,the,hot,plasma,regions,,and,predicts,too,few,energetic,particles,in,the,cooler,regions.,This,is,a,well,known,issue,,and,in,2D,ﬂuid,simulations,,it,is,resolved,by,imposing,an,upper,bound,for,the,electron,heat,ﬂux,that,is,of,order,neT,vte.,•,The,other,region,where,kinetic,eﬀects,are,important,is,near,the,wall.,The,ions,and,electrons,that,reach,the,wall,recombine,and,do,not,come,back,as,charged,particles.,As,a,result,of,this,recombination,,the,charged,particle,distribution,functions,vanish,for,signiﬁcant,parts,of,the,velocity,space,and,cannot,be,approximated,by,Maxwellians,,as,one,would,need,to,be,the,case,to,be,able,to,use,ﬂuid,equations.,Thus,,the,treatment,of,charged,particles,near,the,wall,has,to,be,kinetic,(Loizu,et,al.,2011;,Geraldini,et,al.,2018).,The,combination,of,these,two,issues,can,manifest,in,a,population,of,energetic,electrons,that,cannot,be,predicted,by,ﬂuid,models.,In,turn,,these,energetic,electrons,aﬀect,the,density,and,temperature,by,suppressing,or,enhancing,heat,transport,and,by,changing,the,potential,diﬀerence,across,the,Debye,sheaths.,Evidence,of,these,eﬀects,has,been,found,in,direct,kinetic,simulations,of,1D,problems,(Tskhakaya,et,al.,2011;,Chankin,&,Coster,2015).,The,fact,that,kinetic,eﬀects,are,important,has,been,recognized,for,neutrals,,leading,to,kinetic,codes,for,them,(Reiter,et,al.,2005;,Stotler,&,Karney,1994).,However,,as,we,have,pointed,out,before,,these,codes,ignore,any,possible,non-Maxwellian,features,in,the,ion,and,electron,distribution,functions,,are,sensitive,to,the,parameters,that,determine,how,particles,interact,with,the,wall,and,,depending,on,the,version,,do,not,include,many,elastic,collisions,or,detailed,collision,physics.,The,non-Maxwellian,features,of,the,electron,dis-,tribution,function,are,particularly,important,as,electrons,mediate,many,of,the,processes,considered,important,in,these,neutral,models:,radiation,,ionization,,recombination,,etc.,For,ions,and,electrons,,the,community,is,starting,to,construct,edge,kinetic,codes,such,as,XGC,(Ku,et,al.,2016),,GKEYLL,(Hakim,et,al.,2020),or,COGENT,(Dorf,et,al.,2016).,These,codes,are,broadly,based,on,the,same,gyroaveraged,kinetic,models,used,in,δf,gyrokinetic,codes,such,as,GS2,(Kotschenreuther,et,al.,1995;,Dorland,et,al.,2000),,GENE,(Dannert,&,Jenko,2005),or,stella,(Barnes,et,al.,2019),,but,are,very,diﬀerent,from,them,because,of,the,edge,particularities,,as,we,proceed,to,explain.,The,idea,behind,gyroaveraged,kinetic,models,is,to,average,over,the,fast,gyrofrequency,time,scale,to,avoid,a,cripplingly,small,time,step.,All,gyroaveraged,kinetic,models,are,based,on,an,asymptotic,expansion,in,ρi/L⊥,(cid:28),1,(see,table,2,for,values,of,ρi,and,L⊥).,The,simplest,possible,approach,is,drift,kinetics,(Hazeltine,1973),that,assumes,that,the,size,of,all,turbulent,structures,is,much,larger,than,ρi.,Unfortunately,,in,the,presence,of,temperature,and,density,gradients,,drift,kinetics,develops,instabilities,at,the,grid,scale.,These,instabilities,can,be,stabilized,by,numerical,dissipation,,and,in,the,real,word,,they,are,stabilized,by,ﬁnite,gyroradius,eﬀects,that,drift,kinetics,neglects.,Gyrokinetics,(Catto,1978;,Frieman,&,Chen,1982),was,developed,to,solve,this,problem,in,the,core,of,the,tokamak.,Initially,,δf,gyrokinetics,assumed,that,turbulent,ﬂuctuations,had,a,characteristic,size,of,the,order,of,ρi,and,their,amplitude,was,small,by,a,factor,of,ρi/L⊥,(cid:28),1.,This,ensures,that,the,gradients,of,the,ﬂuctuations,are,comparable,to,the,background,gradient,and,not,larger.,However,,as,we,have,explained,in,section,3,,in,the,edge,it,is,not,possible,to,assume,that,density,and,temperature,are,slowly,varying,quantities,plus,small,ﬂuctuations.,This,fact,8,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,was,part,of,the,justiﬁcation,to,develop,what,is,know,as,full,f,gyrokinetics,,in,which,the,distribution,function,is,not,assumed,to,be,composed,of,a,slowly,varying,piece,plus,small,ﬂuctuations.,See,Parra,&,Catto,(2008),and,Brizard,&,Hahm,(2007),for,two,diﬀerent,types,of,derivations,of,full,f,gyrokinetics.,Overall,,full,f,gyrokinetics,can,be,seen,as,a,mixture,of,drift,kinetics,and,the,original,δf,gyrokinetics.,The,distribution,function,is,allowed,to,have,wavelengths,that,range,from,L⊥,to,ρi,,but,the,size,of,these,diﬀerent,Fourier,components,has,to,be,suﬃciently,small,that,the,gradient,of,the,distribution,function,is,never,larger,than,1/L⊥.,Assuming,a,core,ordering,(2.2),,one,naturally,recovers,δf,gyrokinetics,,showing,that,the,distribution,function,has,to,be,a,slowly,varying,piece,plus,small,ﬂuctuations,(Parra,&,Catto,2010;,Calvo,&,Parra,2012).,In,the,edge,ordering,(2.1),,full,f,gyrokinetics,fundamentally,becomes,drift,kinetics,with,some,small,corrections.,These,corrections,are,of,two,types.,•,The,electromagnetic,ﬁelds,appear,in,the,kinetic,equation,averaged,over,circular,gyro-,orbits.,Thus,,electromagnetic,ﬂuctuations,that,have,characteristic,lengths,much,smaller,than,the,gyroradius,are,averaged,over,and,do,not,drive,ﬂuctuations,in,the,distribution,function.,Since,these,short,wavelength,electromagnetic,ﬂuctuations,can,only,survive,if,there,are,charge,and,current,ﬂuctuations,of,similar,wavelength,,which,they,are,not,able,to,drive,,they,eventually,damp,and,disappear.,•,The,ﬁeld,equations,,quasineutrality,and,Amp´ere’s,law,,contain,densities,and,currents,that,one,calculates,from,the,ion,and,electron,distribution,functions.,As,a,result,of,the,gyrokinetic,expansion,,these,distribution,functions,have,ﬁnite,gyroradius,corrections,that,give,terms,that,are,small,in,ρi/L⊥,(cid:28),1.,One,of,these,small,terms,in,particular,,the,polarization,density,in,the,quasineutrality,equation,,is,important,because,,despite,its,small,size,,it,can,determine,the,electric,ﬁeld,in,diﬀerent,situations.,For,example,,for,shear,Alfven,waves,,the,polarization,density,is,small,,but,so,are,the,rest,of,the,contributions,to,the,density,,so,in,the,end,a,balance,between,the,polarization,density,and,another,term,determines,the,ﬂuctuating,electric,ﬁeld.,Another,example,is,the,component,of,the,electric,ﬁeld,perpendicular,to,the,ﬂux,surfaces,,which,is,also,determined,by,a,balance,between,the,polarization,density,and,other,terms,(Parra,&,Catto,2009).,The,small,ﬁnite,gyroradius,terms,in,the,full,f,gyrokinetic,equations,are,important,to,stabilize,the,short,wavelength,instabilities,,and,some,of,them,(e.g.,the,polarization,density),can,also,be,important,for,certain,aspects,of,the,physics.,Keeping,the,ﬁnite,gyro-,radius,eﬀects,in,the,kinetic,equation,is,relatively,straightforward,in,PIC,codes,,although,one,has,to,be,careful,with,the,accuracy,of,the,average,(Guadagni,&,Cerfon,2017).,We,are,not,aware,of,any,full,f,edge,gyrokinetic,code,that,retains,ﬁnite,gyroradius,eﬀects,in,the,kinetic,equation.,The,corrections,to,the,ﬁeld,equations,are,much,more,diﬃcult,to,retain,–,formulations,that,explicitly,try,to,conserve,energy,and,momentum,exactly,require,solving,nonlinear,equations,for,every,element,in,velocity,space,,for,instance.,This,has,driven,the,community,towards,simplifying,these,terms.,As,a,result,of,these,simpliﬁcations,,several,edge,codes,solve,drift,kinetics,with,some,ad,hoc,additions,to,the,ﬁeld,equations,,such,as,a,simpliﬁed,polarization,density.,In,addition,to,these,fundamental,issues,,edge,drift,kinetics,and,gyrokinetics,need,to,address,other,problems.,As,gyrokinetics,was,devised,to,model,turbulent,ﬂuctuations,with,a,spatial,size,of,the,order,of,ρi,in,tokamak,cores,,most,available,models,do,not,include,features,that,are,important,for,edge,physics.,•,There,is,no,gyrokinetic,formulation,for,the,wall,boundary,conditions.,Only,recently,one,such,formulation,was,developed,for,drift,kinetic,ions,in,magnetic,ﬁelds,that,reach,the,wall,with,a,grazing,angle,(Geraldini,et,al.,2018).,This,work,has,to,be,generalized,to,Physics,in,the,edge,of,fusion,devices,9,electrons,,more,general,angles,between,the,wall,and,the,magnetic,ﬁeld,,and,eventually,to,gyrokinetics.,•,Gyrokinetics,does,not,usually,include,collisions,with,neutrals,,and,such,collisions,are,important,because,neutrals,break,the,symmetry,introduced,by,the,fast,charged,par-,ticle,gyration,around,magnetic,ﬁeld,lines.,In,other,words,,the,dependence,on,gyrophase,,neglected,in,gyrokinetics,,can,become,important.,•,Gyrokinetics,was,developed,for,ﬂuctuations,with,characteristic,scales,of,the,order,of,the,ion,gyroradius.,At,these,scales,,the,magnetic,ﬁeld,ﬂuctuations,have,a,very,constrained,form.,For,this,reason,,the,extension,of,gyrokinetics,to,include,MagnetoHydroDynamic,(MHD),modes,is,non-trivial,and,an,active,area,of,research.,See,Zheng,et,al.,(2007),for,a,theoretical,treatment,,and,Collar,et,al.,(2020),for,recent,numerical,work,in,this,area.,A,gyrokinetic,model,of,the,edge,should,be,able,to,reproduce,MHD,results,,as,MHD,modes,are,believed,to,be,the,main,drive,of,the,eruptions,known,as,Edge,Localized,Modes,(ELMs),(Ham,et,al.,2020).,5.,Discussion,A,complete,edge,description,requires,kinetic,eﬀects.,Current,attempts,to,model,kinetic,eﬀects,in,the,edge,rely,heavily,on,gyroaveraged,kinetic,models,that,average,over,the,very,fast,gyrofrequency,timescale.,In,our,opinion,,there,does,not,exist,a,systematic,procedure,to,choose,the,relevant,ﬁnite,gyroradius,eﬀects,to,be,kept,in,the,kinetic,and,ﬁeld,equations.,The,ﬁrst,objective,of,a,kinetic,modeling,eﬀort,for,the,edge,must,be,to,establish,the,ﬁnite,gyroradius,eﬀects,to,be,kept,in,the,equations.,To,do,so,,for,contract,T/NA085/20,,we,have,proposed,as,a,ﬁrst,attempt,to,construct,a,drift,kinetic,model.,This,drift,kinetic,model,will,be,unstable,at,grid,scales,,but,hyperviscosity,might,be,enough,to,stabilize,such,scales,if,they,do,not,contribute,much,to,transport,(as,one,would,expect,due,to,their,small,size).,For,the,contract,work,,we,will,use,the,model,only,in,1D,and,2D,conﬁgurations,that,cannot,develop,these,grid,scale,instabilities.,We,will,then,determine,analytically,which,ﬁnite,gyroradius,eﬀects,must,be,kept,in,the,equations,to,determine,every,part,of,the,problem,,and,in,particular,the,component,of,the,electric,ﬁeld,perpendicular,to,the,ﬂux,surfaces.,A,well,known,issue,arises,when,keeping,ﬁnite,gyroradius,eﬀects,in,the,ﬁeld,equations.,In,quasineutrality,,the,only,term,that,contains,the,electric,potential,explicitly,is,the,small,ﬁnite,gyroradius,correction.,Thus,,unless,an,implicit,time,stepping,algorithm,is,employed,,one,needs,to,solve,for,the,potential,by,inverting,a,small,term,in,the,equation.,This,procedure,limits,the,time,step,size,severely,(Lee,1987;,Barnes,et,al.,2019).,Thus,,in,addition,to,keeping,ﬁnite,gyroradius,eﬀects,,we,need,to,make,sure,that,the,kinetic,equations,that,we,obtain,are,amenable,to,implicit,time,stepping,methods.,In,addition,to,studying,ﬁnite,gyroradius,eﬀects,,we,will,determine,the,eﬀect,that,collisions,with,neutrals,have,on,the,gyrokinetic,formalism,by,introducing,charge,exchange,collisions,and,ionization,collisions.,The,ﬁnal,deliverable,of,contract,T/NA085/20,will,be,a,drift,kinetic,model,with,neutrals,and,the,ﬁnite,gyroradius,terms,that,are,needed,to,calculate,the,component,of,the,electric,ﬁeld,that,is,perpendicular,to,the,ﬂux,surfaces.,Both,of,these,features,will,be,improvements,on,the,models,implemented,in,existing,continuum,edge,codes.,By,the,end,of,the,contract,,the,model,will,have,been,tested,in,1D,and,2D,problems,,and,even,though,it,will,have,a,3D,version,,this,3D,version,will,not,have,been,tested,in,the,turbulent,regime.,Moreover,,all,the,work,will,have,been,performed,in,a,helical,ﬁeld,and,not,in,a,diverted,tokamak,ﬁeld,because,the,helical,ﬁeld,is,the,state-of-the-art,for,continuum,edge,kinetic,codes,(only,recently,,in,the,2020,Annual,APS,DPP,meeting,,the,10,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,two,leading,continuum,codes,,GKEYLL,and,COGENT,,have,reported,the,ﬁrst,runs,with,more,complex,tokamak,geometry).,The,limitations,of,the,model,delivered,at,the,end,of,contract,T/NA085/20,leave,two,obvious,avenues,for,future,work.,•,The,main,theoretical,diﬃculty,added,by,tokamak,magnetic,ﬁelds,is,the,presence,of,a,component,of,the,drifts,in,the,direction,perpendicular,to,the,ﬂux,surface,that,does,not,exist,in,the,case,of,a,helical,ﬁeld.,This,component,of,the,drifts,leads,to,ﬁnite,orbit,widths,and,can,signiﬁcantly,modify,some,of,the,physics,of,both,quiescent,2D,plasmas,(Kagan,&,Catto,2008;,Landreman,et,al.,2014),and,3D,turbulence,(Parisi,et,al.,2020).,Adding,this,component,of,the,particle,drift,to,the,equations,will,require,extra,work.,•,As,we,have,explained,above,,the,3D,model,is,expected,to,have,grid,scale,instabilities,that,can,be,stabilized,by,hyperviscosity,or,other,numerical,methods,of,dissipation.,To,properly,capture,the,ﬂuctuations,at,the,ion,gyroradius,scale,,one,would,have,to,include,more,precise,ﬁnite,gyroradius,eﬀects.,This,is,an,obvious,extension,of,the,work,in,contract,T/NA085/20,,and,it,is,very,important,for,High,conﬁnement,mode,(H-mode),where,the,turbulence,at,scales,larger,than,the,ion,gyroradius,is,stabilized,for,still,unclear,reasons.,In,this,regime,,turbulence,at,scales,of,the,order,of,or,smaller,than,the,ion,gyroradius,is,important,(Hillesheim,et,al.,2016;,Hatch,et,al.,2017;,Parisi,et,al.,2020),,and,the,ﬁnite,gyroradius,eﬀects,become,crucial,for,ions.,We,foresee,that,adding,more,detailed,ﬁnite,gyroradius,eﬀects,will,be,theoretically,and,numerically,challenging,and,will,require,dedicated,work.,Two,other,aspects,of,an,edge,kinetic,model,are,beyond,the,scope,of,contract,T/NA085/20.,•,To,test,the,eﬀect,of,wall,boundary,conditions,on,drift,kinetics,,we,will,impose,a,simpliﬁed,version,of,the,boundary,conditions,that,are,valid,in,the,limit,in,which,the,electron,gyroradius,is,much,smaller,than,the,Debye,length.,However,,more,detailed,boundary,conditions,must,be,found,because,usually,the,electron,gyroradius,is,larger,than,or,comparable,to,the,Debye,length.,This,is,work,that,is,being,pursued,by,one,of,the,authors,of,this,report,(F.I.P.),with,other,sources,of,funding.,•,We,pointed,out,above,that,it,would,be,desirable,to,be,able,to,recover,MHD,modes,with,the,edge,kinetic,model.,In,the,work,for,contract,T/NA085/20,,the,ﬂuctuations,in,the,magnetic,ﬁeld,will,be,neglected,(this,is,a,good,approximation,for,many,edge,plasmas),,and,hence,it,will,not,be,possible,to,explore,connections,with,MHD.,This,is,an,area,of,research,where,ExCALIBUR,NEPTUNE,could,beneﬁt,from,collaboration,with,the,EPSRC,Programme,Grant,‘Turbulent,Dynamics,of,Tokamak,Plasmas,(TDoTP)’.,Several,PIs,in,the,ExCALIBUR,NEPTUNE,project,also,belong,to,TDoTP.,REFERENCES,Barnes,,M.,,Parra,,F.I.,&,Landreman,,M.,2019,stella:,An,operator-split,,implicit-explicit,δf,-gyrokinetic,code,for,general,magnetic,ﬁeld,conﬁgurations.,J.,Comput.,Phys.,391,,365.,Beer,,M.A.,,Cowley,,S.C.,&,Hammett,,G.W.,1995,Field-aligned,coordinates,for,nonlinear,simulations,of,tokamak,turbulence.,Phys.,Plasmas,2,,2687.,Brackmann,,R.T.,,Fite,,W.L.,&,Neynaber,,R.H.,1958,Collisions,of,Electrons,with,Hydrogen,Atoms.,III.,Elastic,Scattering.,Phys.,Rev.,112,,1157.,Braginskii,,S.I.,1958,Transport,phenomena,in,a,completely,ionized,two-temperature,plasma.,Sov.,Phys.,JETP,6,,358.,Brizard,,A.J.,&,Hahm,,T.S.,2007,Foundations,of,nonlinear,gyrokinetic,theory.,Rev.,Mod.,Phys.,79,,421.,Bufferand,,H.,,Ciraolo,,G.,,Marandet,,Y.,,Bucalossi,,J.,,Ghendrih,,Ph.,,Gunn,,J.,,Mellet,,N.,,Tamain,,P.,,Leybros,,R.,,Fedorczak,,N.,,Schwander,,F.,&,Serre,,E.,2015,Numerical,modelling,for,divertor,design,of,the,WEST,device,with,a,focus,on,plasmawall,interactions.,Nucl.,Fusion,55,,053025.,Bufferand,,H.,,Tamain,,P.,,Baschetti,,S.,,Bucalossi,,J.,,Ciraolo,,G.,,Fedorczak,,N.,,Physics,in,the,edge,of,fusion,devices,11,Ghendrih,,Ph.,,Nespoli,,F.,,Schwander,,F.,&,E.,Serre,and,,Y.,Marandet,2019,Three-dimensional,modelling,of,edge,multi-component,plasma,taking,into,account,realistic,wall,geometry.,Nucl.,Mat.,Energy,18,,82.,Calvo,,I.,&,Parra,,F.I.,2012,Long-wavelength,limit,of,gyrokinetics,in,a,turbulent,tokamak,and,its,intrinsic,ambipolarity.,Plasma,Phys.,Control.,Fusion,54,,115007.,Catto,,P.J.,1978,Linearized,gyro-kinetics.,Plasma,Phys.,20,,719.,Catto,,P.J.,1994,A,short,mean-free,path,,coupled,neutral-ion,transport,description,of,a,toka-,mak,edge,plasma.,Phys.,Plasmas,1,,1936.,Catto,,P.J.,&,Simakov,,A.N.,2004,A,drift,ordered,short,mean,free,path,description,for,magnetized,plasma,allowing,strong,spatial,anisotropy.,Phys.,Plasmas,11,,90.,Chankin,,A.V.,,Corrigan,,G.,&,JET,Contributors,2021,EDGE2D-EIRENE,modeling,of,the,impact,of,wall,materials,on,core,edge,,scrape-oﬀ,layer,and,divertor,parameters.,Plasma,Phys.,Control.,Fusion,63,,035010.,Chankin,,A.V.,&,Coster,,D.P.,2015,On,the,locality,of,parallel,transport,of,heat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agnetohydrodynamics,and,missing,ﬁnite,Larmor,radius,eﬀects.,Phys.,Plasmas,14,,072505.,REPORT,2047357-TN-01-02,M1.1,1,1D,drift,kinetic,models,with,periodic,boundary,conditions,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,22,February,2021),1.,Introduction,In,this,report,,we,propose,1D,drift,kinetic,equations,to,test,the,possibility,of,extracting,low,order,moments,from,the,distribution,functions,for,implicit,methods.,The,model,that,we,present,here,has,periodic,boundary,conditions,,adequate,for,the,closed,ﬁeld,line,region,of,the,edge.,We,will,address,wall,boundary,conditions,for,open,ﬁeld,lines,in,the,reports,for,milestones,M1.3,,M2.4,and,M2.5.,2.,1D,electrostatic,drift,kinetics,We,consider,a,plasma,with,one,ion,species,with,charge,e,and,mass,mi,,electrons,with,charge,e,and,mass,me,,and,one,species,of,neutrals,with,mass,−,mn,=,mi.,(2.1),The,plasma,is,magnetized,by,a,constant,magnetic,ﬁeld,B,=,Bˆz,,and,we,assume,that,the,plasma,only,varies,along,magnetic,ﬁeld,lines.,In,this,case,,the,electric,ﬁeld,produced,by,the,plasma,is,electrostatic,,E,=,(∂φ/∂z)ˆz.,The,potential,φ(z,,t),depends,on,the,position,along,magnetic,ﬁeld,lines,z,and,on,time,t.,−,If,we,assume,that,the,gyroradii,are,small,compared,to,the,length,scales,of,interest,,and,that,the,gyrofrequencies,are,much,larger,than,the,frequencies,that,we,want,to,model,,,t),of,the,diﬀerent,species,s,=,(Hazeltine,1973),,the,distribution,functions,fs(z,,v,(cid:107),i,,e,,n,only,depend,on,the,component,of,the,velocity,parallel,to,the,magnetic,ﬁeld,v,and,the,magnitude,of,the,velocity,perpendicular,to,the,magnetic,ﬁeld,v,,,and,are,independent,of,the,direction,of,the,velocity,perpendicular,to,the,magnetic,ﬁeld.,Thus,,the,distribution,functions,that,in,general,can,depend,on,three,spatial,variables,r,,three,components,of,the,velocity,v,and,the,time,t,depend,only,on,z,,v,and,t,,,,v,,,v,⊥,⊥,(cid:107),fs(r,,v,,t),=,fs(z,,v,(cid:107),(cid:107),,,v,⊥,,,t).,⊥,(2.2),(2.3),The,equations,for,the,distribution,functions,of,the,diﬀerent,species,are,∂fi,∂t,+,v,∂fi,∂z,−,e,mi,∂φ,∂z,(cid:107),∂fi,∂v,(cid:107),=,Cii[fi],+,Cin[fi,,fn],,∂fe,∂t,+,v,∂fe,∂z,(cid:107),+,e,me,∂φ,∂z,∂fe,∂v,(cid:107),and,=,Cee[fe],+,Cei[fe,,fi],+,Cen[fe,,fn],(2.4),∂fn,∂t,+,v,∂fn,∂z,(cid:107),=,Cni[fn,,fi].,(2.5),Here,we,have,included,ion-ion,and,electron-electron,collisions,,modeled,by,the,Fokker-,Planck,collision,operators,Cii[fi],and,Cee[fe],(Rosenbluth,et,al.,1957),,elastic,electron-ion,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,and,electron-neutral,collisions,,modeled,by,the,simpliﬁed,Fokker-Planck,collision,operator,Cei[fe,,fi],(Braginskii,1958),and,the,Boltzmann,collision,operator,Cen[fe,,fn],,and,charge-,exchange,collisions,,represented,by,the,simpliﬁed,Boltzmann,collision,operators,and,Cin[fi,,fn],=,Cni[fn,,fi],=,(cid:90),(cid:90),−,−,v,Rin(,|,v(cid:48),),[fi(v)fn(v(cid:48)),|,−,−,fi(v(cid:48))fn(v)],d3v(cid:48),(2.6),Rin(,v,|,v(cid:48),|,−,),[fn(v)fi(v(cid:48)),−,fn(v(cid:48))fi(v)],d3v(cid:48),,(2.7),To,simplify,our,equations,,we,assume,that,the,function,Rin,is,constant,(Connor,1977;,Hazeltine,et,al.,1992;,Catto,1994),,ﬁnding,and,where,the,densities,are,Cin[fi,,fn],=,Rin,(nnfi,−,−,nifn),Cni[fn,,fi],=,Rin,(nifn,−,−,nnfi),,,ns(z,,t),:=,2π,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,v,fs(z,,v,(cid:107),⊥,⊥,,,v,⊥,,,t).,(2.8),(2.9),(2.10),Note,that,we,can,neglect,the,eﬀect,of,electron,collisions,on,ions,and,on,neutrals,due,to,the,smallness,of,the,electron,mass,(Braginskii,1958).,We,have,also,neglected,neutral-neutral,collisions,because,,in,current,fusion,devices,,the,neutral,density,is,suﬃciently,small,that,the,neutral-neutral,collisions,are,rare.,The,kinetic,equations,will,be,solved,in,the,interval,z,periodic,boundary,conditions,at,z,=,0,and,z,=,L,,∈,[0,,L],,and,we,will,impose,fs(z,=,0,,v,(cid:107),,,v,⊥,,,t),=,fs(z,=,L,,v,(cid:107),,,v,⊥,,,t).,Finally,,the,potential,φ(z,,t),is,determined,by,the,quasineutrality,equation,ni,=,ne.,(2.11),(2.12),To,solve,this,equation,,we,need,to,treat,the,equations,implicitly,as,the,potential,enters,only,via,its,eﬀect,on,∂fi/∂t,and,∂fe/∂t.,The,need,to,use,implicit,methods,is,one,of,the,reasons,why,we,are,trying,to,extract,some,of,the,low,order,moments,from,the,distribution,function,,notably,the,density.,Before,we,treat,the,complete,problem,,we,will,simplify,the,treatment,of,electrons,to,obtain,a,system,of,equations,that,can,be,solved,with,an,explicit,time,advance,so,that,we,can,compare,our,implicit,schemes,with,an,explicit,numerical,method.,Instead,of,solving,for,fe,,we,will,use,a,Maxwell-Boltzmann,response,,ne(z,,t),=,Ne,exp,(cid:18),eφ(z,,t),Te,(cid:19),,,(2.13),where,Ne,and,Te,are,constants,(see,Appendix,A,for,a,derivation,of,the,Maxwell-Boltzmann,response).,Moreover,,the,full,Fokker-Planck,ion-ion,collision,operator,Cii[fi],is,a,com-,plicated,integro-diﬀerential,operator,that,we,will,not,implement,in,the,ﬁrst,versions,of,our,drift,kinetic,code,,so,we,do,not,include,it,in,the,equations,for,now.,Thus,,the,ﬁnal,,,v,simpliﬁed,model,for,fi(z,,v,(cid:107),,,t),and,φ(z,,t),is,given,by,the,equations,,,v,⊥,∂fi,∂t,+,v,(cid:107),⊥,∂fi,∂z,−,,,t),,fn(z,,v,(cid:107),∂fi,∂v,(cid:107),e,mi,∂φ,∂z,=,Rin(nnfi,−,−,nifn),,(2.14),1D,drift,kinetic,models,with,periodic,boundary,conditions,∂fn,∂t,+,v,∂fn,∂z,(cid:107),=,Rin(nifn,−,−,nnfi),and,ni,=,Ne,exp,(cid:19),,,(cid:18),eφ,Te,3,(2.15),(2.16),with,periodic,boundary,conditions,(2.11).,This,system,of,equations,can,be,solved,explic-,itly,because,the,simple,electron,model,allows,one,to,obtain,φ,as,a,function,of,ni.,3.,1D,moment,drift,kinetics,Instead,of,solving,for,fs(z,,v,(cid:107),v3,ts(z,,t),ns(z,,t),Fs(z,,w,(cid:107),,,t),:=,,,w,,,v,⊥,⊥,,,t),,we,solve,for,(cid:16),fs,z,,us,(cid:107),(z,,t),+,vts(z,,t)w,(cid:107),,,vts(z,,t)w,(cid:17),,,t,,,⊥,where,we,have,deﬁned,the,normalized,velocities,w,(z,,v,(cid:107),(cid:107),,,t),:=,v,(z,,t),(cid:107),−,us,(cid:107),vts(z,,t),and,the,average,parallel,velocity,w,⊥,(z,,v,⊥,,,t),:=,v,⊥,vts(z,,t),,,(z,,t),:=,us,(cid:107),2π,ns,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,v,v,fs(z,,v,(cid:107),(cid:107),⊥,⊥,,,v,⊥,,,t),and,the,thermal,speed,vts(z,,t),:=,(cid:115),4π,3ns,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,[(v,v,⊥,⊥,(cid:107),−,us,(cid:107),(z,,t))2,+,v2,⊥,]fs(z,,v,(cid:107),,,v,⊥,,,t).,(3.5),According,to,its,deﬁnition,,Fs(z,,w,(cid:107),(cid:90),∞,(cid:90),∞,,,w,⊥,,,t),must,satisfy,the,conditions,2π,dw,(cid:107),0,dw,⊥,w,⊥,Fs(z,,w,(cid:107),,,w,⊥,,,t),=,1,,−∞,(cid:90),∞,dw,2π,−∞,(cid:90),∞,(cid:107),0,dw,w,⊥,w,Fs(z,,w,(cid:107),(cid:107),⊥,,,w,⊥,,,t),=,0,(cid:90),∞,2π,(cid:90),∞,0,dw,(cid:107),dw,⊥,w,⊥,(w2,(cid:107),+,w2,⊥,)Fs(z,,w,(cid:107),,,w,⊥,,,t),=,3,2,and,at,every,point,z,and,time,t.,−∞,The,equations,for,ions,become,∂ni,∂t,+,∂,∂z,(cid:0)niui,(cid:107),(cid:1),=,0,,nimi,(cid:18),∂ui,∂t,(cid:107),+,ui,(cid:19),∂ui,(cid:107),∂z,(cid:107),=,∂pi,(cid:107),∂z,−,−,eni,∂φ,∂z,+,ninnmiRin(un,),,ui,(cid:107),(cid:107),−,(3.10),(3.1),(3.2),(3.3),(3.4),(3.6),(3.7),(3.8),(3.9),4,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,3,2,nimivti,(cid:18),∂vti,∂t,+,ui,(cid:107),∂vti,∂z,(cid:19),=,∂qi,(cid:107),∂z,−,∂ui,(cid:107),∂z,pi,(cid:107),−,+,3,4,ninnmiRin(v2,tn,−,v2,ti),+,1,2,ninnmiRin(un,)2,ui,(cid:107),(cid:107),−,and,∂Fi,∂t,+,˙zi,∂Fi,∂z,+,˙w,i,(cid:107),Here,,we,have,deﬁned,the,coeﬃcients,∂Fi,∂w,(cid:107),+,˙w,⊥,i,∂Fi,∂w,⊥,=,˙Fi,+,Cin.,˙zs[Fs](z,,w,(cid:107),,,t),:=,us,(cid:107),+,vtsw,,,(cid:107),(3.11),(3.12),(3.13),˙w,s[Fs](z,,w,(cid:107),(cid:107),,,t),:=,1,nsmsvts,∂ps,(cid:107),∂z,+,2w,(cid:107),3nsmsv2,ts,(cid:20),∂qs,(cid:107),∂z,(cid:18),+,ps,(cid:107),−,3,2,nsmsv2,ts,(cid:21),(cid:19),∂us,∂z,(cid:107),∂vts,∂z,,,w2,(cid:107),−,˙w,⊥,s[Fs](z,,w,(cid:107),,,w,⊥,,,t),:=,2w,3nsmsv2,ts,⊥,(cid:18),∂qs,(cid:107),∂z,+,ps,(cid:107),∂us,(cid:107),∂z,(cid:19),w,w,(cid:107),⊥,−,∂vts,∂z,(3.14),(3.15),and,˙Fs[Fs](z,,w,(cid:107),,,w,⊥,,,t),:=,w,(cid:107),(cid:34),(cid:18),3,2,nsmsv2,ts,−,the,parallel,pressure,∂vts,∂z,−,vts,ns,(cid:18),∂qs,(cid:107),∂z,(cid:19),∂ns,∂z,(cid:18),+,ps,(cid:107),−,1,2,nsmsv2,ts,(cid:19),(cid:35),(cid:19),∂us,∂z,(cid:107),Fs,,(3.16),[Fs](z,,t),:=,2πnsmsv2,ts,ps,(cid:107),(cid:90),∞,−∞,(cid:90),∞,0,dw,(cid:107),dw,⊥,w,⊥,w2,(cid:107),Fs(z,,w,(cid:107),,,w,⊥,,,t),(3.17),the,parallel,heat,ﬂux,[Fs](z,,t),:=,πnsmsv3,ts,qs,(cid:107),(cid:90),∞,−∞,dw,(cid:90),∞,(cid:107),0,dw,⊥,w,w,(cid:107),(w2,(cid:107),+,w2,⊥,⊥,)Fs(z,,w,(cid:107),,,w,⊥,,,t),,(3.18),and,the,modiﬁed,charge,exchange,collision,operator,Cin[Fi,,Fn,,nn,,ui,:=,nnRin,(cid:107),−,(cid:107),,,un,(cid:20),Fi,−,∂,∂w,(cid:107),∂,∂w,⊥,,,vti,,vtn](z,,w,(cid:107),(cid:18),Fn,z,,v3,ti,v3,tn,(cid:20)(cid:18),un,ui,(cid:107),(cid:107),−,vti,(cid:18),v2,tn,v2,ti,−,(cid:20),w2,⊥,2,,,t),,,w,ui,⊥,(cid:107),−,vtn,w,(cid:107),2,+,1,+,,,vti,vtn,2(un,un,(cid:107),+,vti,vtn,w,(cid:107),(cid:18),v2,tn,v2,ti,−,2(un,(cid:107),−,3v2,ti,1,+,(cid:19),)2,ui,(cid:107),+,nnRin,+,nnRin,w,⊥,(cid:19)(cid:21),,,t,⊥,w,(cid:19)(cid:19),)2,(cid:107),(cid:21),Fi,ui,(cid:107),−,3v2,ti,(cid:21),Fi,.,(3.19),Note,that,the,diﬀerential,terms,in,this,modiﬁed,collision,operator,could,have,been,in-,i,and,˙Fi,,but,we,have,decided,to,make,cluded,in,the,deﬁnitions,of,the,coeﬃcients,˙w,(cid:107),them,part,of,a,modiﬁed,collision,operator,instead,to,separate,the,eﬀect,of,collisions,clearly.,This,split,should,not,be,taken,as,a,suggestion,on,how,to,implement,these,terms,in,a,code.,˙w,i,,⊥,1D,drift,kinetic,models,with,periodic,boundary,conditions,5,The,equations,for,the,neutrals,are,∂nn,∂t,+,∂,∂z,(cid:0)nnun,(cid:107),(cid:1),=,0,,nnmi,(cid:18),∂un,∂t,(cid:107),+,un,(cid:19),∂un,∂z,(cid:107),(cid:107),=,∂pn,∂z,(cid:107),−,+,ninnmiRin(ui,(cid:107),−,un,),,(cid:107),(3.20),(3.21),3,2,nnmivtn,(cid:18),∂vtn,∂t,+,un,∂vtn,∂z,(cid:107),(cid:19),=,∂qn,(cid:107),∂z,−,−,∂un,∂z,pn,(cid:107),3,4,nnnimiRin(v2,ti,−,v2,tn),nnnimiRin(un,(cid:107),−,)2,ui,(cid:107),(3.22),(cid:107),+,+,1,2,and,Cni.,Here,,we,have,deﬁned,the,modiﬁed,charge,exchange,collision,operator,+,˙w,(cid:107),+,˙zn,+,˙w,⊥,⊥,n,n,=,˙Fn,+,∂Fn,∂t,∂Fn,∂z,∂Fn,∂w,(cid:107),∂Fn,∂w,(3.23),Cni[Fn,,Fi,,ni,,un,niRin,:=,(cid:107),−,(cid:107),,,ui,(cid:20),Fn,−,∂,∂w,(cid:107),∂,∂w,⊥,,,t),,,w,un,⊥,,,vtn,,vti](z,,w,(cid:107),(cid:18),v3,tn,v3,ti,(cid:20)(cid:18),ui,un,Fi,z,,+,(cid:107),(cid:107),−,vtn,(cid:18),v2,ti,v2,tn,−,(cid:20),w2,⊥,2,(cid:107),−,vti,w,(cid:107),2,1,+,,,vtn,vti,2(un,ui,(cid:107),+,vtn,vti,w,(cid:107),(cid:18),v2,ti,v2,tn,−,2(un,(cid:107),−,3v2,tn,1,+,(cid:19),)2,ui,(cid:107),+,niRin,+,niRin,w,⊥,(cid:19)(cid:21),,,t,⊥,w,(cid:19)(cid:19),)2,(cid:21),Fn,ui,(cid:107),(cid:107),−,3v2,tn,(cid:21),Fn,.,(3.24),Equations,(3.12),and,(3.23),for,Fi,and,Fn,are,constructed,such,that,conditions,(3.6),,(3.7),and,(3.8),are,satisﬁed,at,all,times,if,they,are,satisﬁed,at,t,=,0.,4.,Linear,test,One,possible,test,for,the,sets,of,1D,equations,described,above,is,the,evolution,of,small,perturbations,to,a,uniform,Maxwellian,equilibrium.,We,assume,the,following,form,for,the,ion,and,neutral,distribution,functions,,fs(z,,v,(cid:107),,,v,⊥,,,t),=,fM,s(v,,,v,⊥,(cid:107),),+,fs1(v,,,v,⊥,(cid:107),)[exp(ik,z,(cid:107),−,iωt),+,complex,conjugate],,(4.1),where,fM,s(v,,,v,⊥,(cid:107),),=,ns,(cid:19)3/2,(cid:18),mi,2πTh,(cid:32),−,+,v2,⊥,mi(v2,(cid:107),2Th,(cid:33),),.,exp,(4.2),Note,that,both,species,share,the,same,constant,temperature,Th.,To,ensure,that,the,potential,is,small,,we,assume,ni,=,Ne.,Since,the,perturbations,fs1(v,can,be,linearized,to,give,,,v,⊥,(cid:107),),and,φ,are,small,,equations,(2.14),,(2.15),and,(2.16),v,(k,(cid:107),(cid:107),−,ω,−,innRin)fi1,+,iniRinfn1,=,eφ,Th,−,k,v,(cid:107),(cid:107),fM,i,+,iRin(nn1fM,i,−,ni1fM,n),,(4.3),innRinfi1,+,(k,(cid:107),v,ω,iniRin)fn1,=,iRin(ni1fM,n,−,−,(cid:107),−,nn1fM,i),and,ni1,ni,=,eφ,Te,.,(4.4),(4.5),6,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Figure,1.,Solutions,to,the,dispersion,relation,(4.7):,acoustic,waves,(solid,lines),and,non-prop-,agating,modes,(dashed,lines).,(a),Real,frequency,ωr,:=,Re(ω),and,(b),growth,rate,γ,:=,Im(ω),as,functions,of,the,charge,exchange,collision,frequency,(ni,+,nn)Rin,for,Te/Th,=,1,and,several,values,of,the,parameter,ni/(ni,+,nn).,(c),Real,frequency,ωr,and,(d),growth,rate,γ,as,functions,of,the,the,charge,exchange,collision,frequency,(ni,+,nn)Rin,for,ni/(ni,+,nn),=,1/2,and,several,values,of,the,parameter,Te/Th.,Here,,we,have,deﬁned,the,perturbations,to,the,density,as,ns1,=,2π,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,fs1.,v,⊥,⊥,(4.6),Solving,for,the,functions,fi1,and,fn1,as,functions,of,ni1,and,nn1,and,then,integrating,fi1,and,fn1,over,velocity,space,,we,ﬁnd,the,equations,(cid:18),Aii,Ain,Ani,Ann,(cid:19),(cid:18),ni1,nn1,(cid:19),(cid:18),0,0,=,(cid:19),,,(4.7),where,the,elements,of,the,matrix,are,Te,Th,ni,ni,+,nn,Aii,=,1,+,Te,Th,+,ζZ(ζ),+,nn,ni,+,nn,(cid:20)(cid:18),1,+,(cid:19),Te,Th,ζin,−,ζ,(cid:21),Z(ζin),(4.8),Ain,=,−,(cid:26)(cid:20)(cid:18),ni,(ζin,−,ni,+,nn,(cid:19),ζ)Z(ζin),(cid:21),Ani,=,nn,ni,+,nn,−,1,+,Te,Th,ζin,−,ζ,Z(ζin),(cid:27),ζZ(ζ),Te,Th,−,and,Here,,we,have,deﬁned,Ann,=,1,+,ni,ni,+,nn,(ζin,−,ζ)Z(ζin).,(4.9),(4.10),(4.11),with,vth,:=,(cid:112),k,|,2Th/mi,,and,we,have,used,the,plasma,dispersion,function,(Fried,&,Conte,vth,(cid:107)|,(cid:107)|,k,|,ζ,:=,ω,vth,,,ζin,:=,ω,+,i(ni,+,nn)Rin,,,(4.12),00.511.5200.511.513/41/21/4100.511.52-2.5-2-1.5-1-0.5000.511.5211.21.41.61.821/41/212400.511.52-2.5-2-1.5-1-0.5000.511.5200.511.513/41/21/4100.511.52-2.5-2-1.5-1-0.5000.511.5211.21.41.61.821/41/212400.511.52-2.5-2-1.5-1-0.50<latexit,sha1_base64="OvM8PmVf0P9y3U2YSoOEWA7eWfY=">AAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmEhm/gpJD0tMtGkvb0pwZyPPk9pZyb0oOXfnhfJ1VkcO7INDUAQuuARlcAsqoAoIeATP4BW8WU/Wi/VufUxHF6xsZw/8gfX5A2NImXk=</latexit>(ni+nn)Rin/|kk|vth<latexit,sha1_base64="OvM8PmVf0P9y3U2YSoOEWA7eWfY=">AAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmEhm/gpJD0tMtGkvb0pwZyPPk9pZyb0oOXfnhfJ1VkcO7INDUAQuuARlcAsqoAoIeATP4BW8WU/Wi/VufUxHF6xsZw/8gfX5A2NImXk=</latexit>(ni+nn)Rin/|kk|vth<latexit,sha1_base64="OvM8PmVf0P9y3U2YSoOEWA7eWfY=">AAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmE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periodic,boundary,conditions,7,Z(ζ),:=,exp(,(cid:32),ζ,2),i√π,−,(cid:90),ζ,2,−,0,(cid:33),exp(y2),dy,.,(4.13),By,setting,the,determinant,of,the,matrix,in,equation,(4.7),to,zero,,we,can,calculate,the,frequency,ω,of,the,modes,for,an,initial,k,(cid:107),We,show,two,diﬀerent,types,of,solutions,to,the,dispersion,relation,in,ﬁgure,1:,acoustic,waves,that,have,both,real,frequency,ωr,:=,Re(ω),and,damping,rate,γ,:=,Im(ω),,and,non-,propagating,modes,with,ωr,=,0.,In,the,ﬁgure,,we,plot,the,real,frequency,and,damping,rate,for,the,acoustic,waves,as,solid,lines,,whereas,for,the,non-propagating,modes,,we,only,plot,the,damping,rates,as,dashed,lines.,We,can,use,these,solutions,to,benchmark,the,implementation,of,the,equations,in,our,code.,.,5.,Conclusions,We,have,identiﬁed,the,ﬁrst,set,of,equations,that,we,will,use,to,test,a,new,approach,to,drift,kinetics,that,extracts,the,low,order,moments,from,the,distribution,function.,The,chosen,model,can,be,integrated,without,employing,implicit,time-stepping,methods.,This,is,a,choice,that,we,have,made,to,ensure,that,we,can,compare,the,new,model,with,the,well-established,drift,kinetic,model.,We,have,also,developed,an,analytical,benchmark,for,the,equations.,The,calculation,ignores,ion-ion,collisions,and,it,is,hence,not,relevant,to,all,edge,operational,regimes,,but,it,allows,us,to,test,the,implementation,of,the,equations,with,and,without,collisions.,Similar,calculations,can,be,performed,including,the,full,ion-ion,collision,operator,and,model,collision,operators.,REFERENCES,Braginskii,,S.I.,1958,Transport,phenomena,in,a,completely,ionized,two-temperature,plasma.,Sov.,Phys.,JETP.,6,,358.,Catto,,P.J.,1994,A,short,mean-free,path,,coupled,neutral-ion,transport,description,of,a,toka-,mak,edge,plasma.,Phys.,Plasmas,1,,1936.,Connor,,J.W.,1977,An,analytic,solution,for,the,distribution,function,of,neutral,particles,in,a,Maxwellian,plasma,using,the,method,of,singular,eignfunctions.,Plasma,Phys.,19,,853.,Fried,,B.D.,&,Conte,,S.D.,1961,The,Plasma,Dispersion,Function:,The,Hilbert,transform,of,the,Gaussian.,Academic,Press.,Hazeltine,,R.D.,1973,Recursive,derivation,of,drift-kinetic,equation.,Plasma,Phys.,15,,77–80.,Hazeltine,,R.D.,,Calvin,,M.D.,,Valanju,,P.M.,&,Solano,,E.R.,1992,Analytical,calculation,of,neutral,transport,and,its,eﬀect,on,ions.,Nucl.,Fusion,32,,3.,Rosenbluth,,M.N.,,MacDonald,,W.M.,&,Judd,,D.L.,1957,Fokker-Planck,Equation,for,an,Inverse-Square,Force.,Phys.,Rev.,107,,1.,Appendix,A.,The,Maxwell-Boltzmann,response,The,Maxwell-Boltzmann,response,in,equation,(2.13),is,the,solution,to,electron,drift,1.,The,expansion,in,the,mass,ratio,is,me/mi,(cid:28),kinetic,equation,(2.4),in,the,limit,based,on,the,fact,that,the,species,within,the,plasma,tend,to,thermalize,due,to,collisions,,and,hence,the,diﬀerent,species,have,in,general,similar,average,kinetic,energies.,Thus,,1/2,the,characteristic,thermal,speeds,of,the,ions,and,neutrals,,vti,and,vtn,,scale,as,m−,,,i,whereas,the,electron,thermal,speed,scales,as,m−,e,(cid:112),1/2,vtn.,We,assume,that,the,massive,ions,and,neutrals,control,the,dynamics,of,interest,,giving,,,giving,vte,(cid:29),vti,∼,8,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,the,estimate,∂,∂t,∼,vti,L,.,Thus,,the,time,derivative,in,equation,(2.4),is,negligible,compared,to,terms,like,∂fe,∂z,∼,v,(cid:107),fe,vte,L,.,(A,1),(A,2),Hence,,we,can,neglect,the,time,derivative,to,ﬁnd,∂fe,∂z,v,(cid:107),+,e,me,∂φ,∂z,∂fe,∂v,(cid:107),=,Cee[fe],+,Cei[fe,,fi],+,Cen[fe,,fn].,(A,3),To,solve,this,equation,,we,need,to,use,the,properties,of,the,collision,operators.,The,electron-electron,collision,operator,satisﬁes,an,H-theorem:,the,entropy,production,(cid:90),−,ln,fe,Cee[fe],d3v,(cid:62),0,(A,4),is,always,positive,and,it,only,vanishes,if,fe,is,a,Maxwellian.,The,elastic,collision,operators,Cei[fe,,fi],and,Cen[fe,,fn],also,satisfy,H-theorems,,but,they,are,much,more,complicated,as,in,general,these,theorems,involve,the,ions,and,the,neutrals.,Luckily,,if,we,perform,the,expansion,1,,Cei[fe,,fi],and,Cen[fe,,fn],satisfy,simpliﬁed,versions,of,their,H-theorems,,namely,,the,entropy,productions,me/mi,(cid:28),(cid:112),(cid:90),−,ln,fe,Cei[fe,,fi],d3v,(cid:62),0,and,(cid:90),−,ln,fe,Cen[fe,,fn],d3v,(cid:62),0,(A,5),are,always,positive,,and,they,only,vanish,if,fe,is,isotropic.,Note,that,,in,the,limit,(cid:112),me/mi,(cid:28),Armed,with,these,properties,,we,multiply,equation,(A,3),by,1,,these,operators,do,not,impose,conditions,on,fi,or,fn.,over,velocity,space,to,obtain,∂,∂z,(cid:20),(cid:90),−,(fe,ln,fe,−,fe),v,(cid:107),(cid:21),d3v,=,(cid:90),(cid:90),−,−,ln,fe,Cee[fe],d3v,(cid:90),−,ln,fe,Cen[fe,,fn],d3v.,ln,fe,and,we,integrate,−,ln,fe,Cei[fe,,fi],d3v,(A,6),Integrating,this,equation,over,z,and,using,the,periodic,boundary,conditions,,we,ﬁnally,obtain,0,=,−,(cid:90),L,0,(cid:90),L,−,0,(cid:90),(cid:90),dz,dz,ln,fe,Cee[fe],d3v,(cid:90),L,(cid:90),dz,−,0,ln,fe,Cei[fe,,fi],d3v,ln,fe,Cen[fe,,fn],d3v.,(A,7),Since,the,entropy,production,of,each,collision,operator,is,always,positive,,this,equation,can,only,be,satisﬁed,if,each,of,the,entropy,productions,vanish,at,every,z.,This,implies,that,,at,every,z,,fe,is,Maxwellian,and,isotropic,,fe(z,,v,(cid:107),,,v,⊥,),=,fM,e(z,,v,(cid:107),,,v,⊥,,,t),:=,ne(z,,t),(cid:18),me,(cid:19)3/2,2πTe(z,,t),(cid:32),−,+,v2,me(v2,⊥,(cid:107),2Te(z,,t),(cid:33),),.,exp,(A,8),We,need,to,determine,the,dependence,of,ne(z,,t),and,Te(z,,t),on,z.,Substituting,the,1D,drift,kinetic,models,with,periodic,boundary,conditions,9,,,v,⊥,solution,fM,e(z,,v,(cid:107),(cid:34),v,(cid:18),∂,∂z,(cid:107),ln,ne,−,,,t),on,equation,(A,3),,we,ﬁnd,(cid:19),e,Te,∂φ,∂z,+,v,(cid:107),(cid:32),me(v2,+,v2,⊥,(cid:107),2Te,(cid:33),),3,2,−,∂,∂z,(cid:35),ln,Te,fM,e,=,0.,(A,9),,,Te(t),cannot,depend,Since,this,equation,has,to,be,satisﬁed,for,every,value,of,v,on,z,,and,ne(z,,t),=,Ne(t),exp(eφ(z,,t)/Te(t)).,Thus,,we,ﬁnd,equation,(2.13),with,Ne,and,Te,being,in,general,functions,of,t.,For,simplicity,,we,choose,them,to,be,constants.,and,v,⊥,(cid:107),REPORT,2047357-TN-05-02,M1.3,1,1D,drift,kinetic,models,with,wall,boundary,conditions,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,27,August,2021),1.,Introduction,In,previous,reports,,we,proposed,1D,drift,kinetic,equations,with,periodic,boundary,conditions,,adequate,for,the,closed,ﬁeld,line,region,of,the,edge.,In,this,report,,we,discuss,a,minimal,1D,drift,kinetic,model,with,wall,boundary,conditions,that,represents,open,ﬁeld,lines.,The,basic,drift,kinetic,model,is,presented,in,section,2,,and,the,wall,boundary,conditions,are,discussed,in,section,3.,We,then,proceed,to,determine,the,novel,moment,drift,kinetic,equations,for,ions,and,neutrals,in,section,4,,and,for,electrons,in,section,5.,The,electrons,have,to,be,treated,diﬀerently,so,that,we,can,exploit,the,expansion,in,the,electron-ion,mass,ratio.,We,ﬁnish,with,a,discussion,in,section,6.,Some,of,the,details,of,the,collision,operators,in,the,moment,drift,kinetic,formulation,are,relegated,to,appendices,to,make,the,main,text,easier,to,read.,2.,1D,electrostatic,drift,kinetics,We,consider,a,plasma,with,one,ion,species,with,charge,e,and,mass,mi,,electrons,with,charge,−e,and,mass,me,,and,one,species,of,neutrals,with,mass,mn,=,mi.,(2.1),The,plasma,is,magnetized,by,a,constant,magnetic,ﬁeld,B,=,Bˆz,,and,we,assume,that,the,plasma,only,varies,along,magnetic,ﬁeld,lines.,In,this,case,,the,electric,ﬁeld,produced,by,the,plasma,is,electrostatic,,E,=,−(∂φ/∂z)ˆz.,The,potential,φ(z,,t),depends,on,the,position,along,magnetic,ﬁeld,lines,z,and,on,time,t.,If,we,assume,that,the,gyroradii,are,small,compared,to,the,length,scales,of,interest,,and,that,the,gyrofrequencies,are,much,larger,than,the,frequencies,that,we,want,to,model,(Hazeltine,1973),,the,distribution,functions,fs(z,,v(cid:107),,v⊥,,t),of,the,diﬀerent,species,s,=,i,,e,,n,only,depend,on,the,component,of,the,velocity,parallel,to,the,magnetic,ﬁeld,v(cid:107),and,the,magnitude,of,the,velocity,perpendicular,to,the,magnetic,ﬁeld,v⊥,,and,are,independent,of,the,direction,of,the,velocity,perpendicular,to,the,magnetic,ﬁeld.,Thus,,the,distribution,functions,that,in,general,can,depend,on,three,spatial,variables,r,,three,components,of,the,velocity,v,and,the,time,t,depend,only,on,z,,v(cid:107),,v⊥,and,t,,fs(r,,v,,t),=,fs(z,,v(cid:107),,v⊥,,t).,(2.2),The,equations,for,the,distribution,functions,of,the,diﬀerent,species,are,∂fi,∂t,+,v(cid:107),∂fi,∂z,−,e,mi,∂φ,∂z,∂fi,∂v(cid:107),=,Cii[fi],+,Cin[fi,,fn],+,Ci,ion[fe,,fn],+,Cie[fi,,fe],+,Si,,(2.3),2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,∂fe,∂t,+,v(cid:107),∂fe,∂z,+,e,me,∂φ,∂z,∂fe,∂v(cid:107),=,Cee[fe],+,Cei[fe,,fi],1,+,O,(cid:20),(cid:19)(cid:21),(cid:18),me,mi,(cid:20),+Cen[fe,,fn],1,+,O,(cid:19)(cid:21),(cid:18),me,mi,+,Ce,ion[fe,,fn],+,Se,(2.4),and,∂fn,∂t,+,v(cid:107),∂fn,∂z,=,Cni[fn,,fi],+,Cne[fn,,fe],+,Cn,ion[fn,,fe],+,Sn.,(2.5),The,sources,Ss(z,,w(cid:107),,w⊥,,t),with,s,=,i,,e,,n,represent,heating,,fueling,and,the,eﬀect,of,transport,perpendicular,to,the,magnetic,ﬁeld,line.,We,have,included,the,following,collisions.,•,Ion-ion,and,electron-electron,collisions,are,modeled,by,the,Fokker-Planck,collision,operators,Cii[fi],and,Cee[fe],(Rosenbluth,et,al.,1957),,Css[fs],:=,2πe4,ln,Λ,(4π(cid:15)0)2m2,s,∇v,·,(D[fs],·,∇vfs,+,P[fs]fs),,,(2.6),where,the,matrix,D,is,D[fs],:=,(cid:90),|v,−,v(cid:48)|2I,−,(v,−,v(cid:48))(v,−,v(cid:48)),|v,−,v(cid:48)|3,fs(v(cid:48)),d3v(cid:48),and,the,vector,P,is,P[fs],:=,−2,(cid:90),v,−,v(cid:48),|v,−,v(cid:48)|3,fs(v(cid:48)),d3v(cid:48).,(2.7),(2.8),Here,,I,is,the,3D,unit,matrix,,(cid:15)0,the,vacuum,permittivity,and,ln,Λ,≈,15,the,Coulomb,logarithm.,•,The,eﬀect,of,electron-ion,and,elastic,electron-neutral,collisions,on,the,electron,distri-,bution,function,can,be,simpliﬁed,in,the,limit,of,small,electron-ion,mass,ratio,,me/mi,(cid:28),1.,With,this,expansion,,we,ﬁnd,the,simpliﬁed,Fokker-Planck,collision,operator,Cei[fe,,fi],:=,2πe4ni,ln,Λ,(4π(cid:15)0)2m2,e,∇v,·,(cid:20),|v,−,ui|2I,−,(v,−,ui)(v,−,ui),|v,−,ui|3,(cid:21),·,∇vfe,(2.9),for,electron-ion,collisions,(Braginskii,1958),,and,the,simpliﬁed,Boltzmann,collision,oper-,ator,Cen[fe,,fn],:=,nn,4π,(cid:90),π,dχ,(cid:90),2π,0,0,dϕ,sin,χRen(|v,−,un|,,χ),[fe(v(v,,χ,,ϕ,,un)),−,fe(v)],(2.10),(2.11),for,electron-neutral,collisions.,Here,ns(z,,t),:=,2π,(cid:90),∞,(cid:90),∞,dv(cid:107),−∞,0,dv⊥,v⊥fs(z,,v(cid:107),,v⊥,,t).,is,the,density,of,species,s,,us,:=,n−1,s,(cid:82),vfs,d3v,is,the,average,velocity,of,species,s,,v(v,,χ,,ϕ,,un),:=,un,+,cos,χ(v,−,un),+,|v,−,un|,sin,χ(cos,ϕ,ˆe1,+,sin,ϕ,ˆe2),(2.12),is,a,rotation,of,the,vector,v,centered,around,un,,Ren(|v−un|,,χ),is,a,function,determined,by,the,physics,of,the,electron-neutral,collisions,,and,the,unit,vectors,ˆe1,and,ˆe2,are,chosen,to,form,an,orthonormal,basis,with,the,vector,(v,−,un)/|v,−,un|.,In,equation,(2.4),,we,have,indicated,that,both,Cei,and,Cen,are,missing,pieces,small,in,me/mi.,These,pieces,can,become,important,because,they,represent,collisional,energy,exchange,and,collisional,heating,,but,they,are,cumbersome.,The,moment,method,that,we,propose,in,1D,drift,kinetic,models,with,wall,boundary,conditions,3,this,document,will,allow,us,to,keep,these,important,eﬀects,even,with,the,simpliﬁed,collision,operators,(2.9),and,(2.10).,•,The,expansion,in,electron-ion,mass,ratio,also,implies,electron-ion,collisions,and,electron-neutral,collisions,have,a,very,small,eﬀect,on,fi,and,fn,–,the,terms,Cie,and,Cne,in,equations,(2.3),and,(2.5),are,small,compared,with,Cii,and,Cni,by,a,factor,of,(cid:112)me/mi,(cid:28),1,,Cie,∼,(cid:114),me,mi,Cii,,Cne,∼,(cid:114),me,mi,Cni.,(2.13),Like,the,mass,ratio,corrections,to,Cei,and,Cen,,these,terms,can,become,important,because,they,contain,the,collisional,energy,exchange,between,electrons,and,the,heavier,species.,We,will,keep,these,eﬀects,in,a,simpliﬁed,form,in,our,moment,formulation.,•,Charge-exchange,collisions,are,represented,by,the,simpliﬁed,Boltzmann,collision,operators,and,Cin[fi,,fn],:=,−,Cni[fn,,fi],:=,−,(cid:90),(cid:90),Rin(|v,−,v(cid:48)|),[fi(v)fn(v(cid:48)),−,fi(v(cid:48))fn(v)],d3v(cid:48),(2.14),Rin(|v,−,v(cid:48)|),[fn(v)fi(v(cid:48)),−,fn(v(cid:48))fi(v)],d3v(cid:48).,(2.15),•,To,model,ionization,,we,use,the,collision,operators,Ci,ion[fe,,fn],:=,fn,(cid:90),Rion(v(cid:48))fe(v(cid:48)),d3v(cid:48),and,Cn,ion[fe,,fn],:=,−fn,(cid:90),Rion(v(cid:48))fe(v(cid:48)),d3v(cid:48).,(2.16),(2.17),We,also,need,to,include,a,collision,operator,Ce,ion,in,the,electron,equation,to,model,the,increase,in,the,number,of,electrons,and,the,energy,loss,due,to,ionization.,This,operator,is,complicated,because,it,involves,three,particles,(the,resulting,ion,and,two,electrons),,but,we,will,be,able,to,avoid,giving,it,a,deﬁnite,form.,Instead,,we,will,use,the,expansion,in,me/mi,(cid:28),1,and,the,fact,that,Ce,ion[fe,,fn],∼,nnRionfe.,(2.18),•,We,have,neglected,neutral-neutral,collisions,because,,in,current,fusion,devices,,the,neutral,density,is,suﬃciently,small,that,the,neutral-neutral,collisions,are,rare.,It,is,pos-,sible,that,the,higher,densities,expected,in,fusion,reactors,will,make,neutral-neutral,colli-,sions,more,relevant.,To,include,neutral-neutral,collisions,,a,Boltzmann,collision,operator,is,in,principle,required,,but,using,a,simpliﬁed,collision,may,be,possible,if,the,exact,shape,of,the,neutral,distribution,function,is,not,important,for,the,physics,of,interest.,To,simplify,our,equations,,we,assume,that,the,functions,Ren,,Rin,and,Rion,are,constant,(Connor,1977;,Hazeltine,et,al.,1992;,Catto,1994),,ﬁnding,Cen[fe,,fn],=,nnRen,(cid:34),1,2,(cid:90),π,0,fe(z,,un(cid:107),+,|v,−,un|,cos,χ,,|v,−,un|,sin,χ,,t),sin,χ,dχ,(cid:35),−fe(z,,v(cid:107),,v⊥,,t),,,(2.19),4,with,|v,−,un|,=,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,(cid:113),(v(cid:107),−,un(cid:107))2,+,v2,⊥,,Cin[fi,,fn],=,−Rin,(nnfi,−,nifn),,,Cni[fn,,fi],=,−Rin,(nifn,−,nnfi),,,Ci,ion[fe,,fn],=,fnneRion,Cn,ion[fe,,fn],=,−fnneRion.,and,The,potential,φ(z,,t),is,determined,by,the,quasineutrality,equation,ni,=,ne.,(2.20),(2.21),(2.22),(2.23),(2.24),To,solve,this,equation,,we,need,to,treat,the,equations,implicitly,as,the,potential,enters,only,via,its,eﬀect,on,∂fi/∂t,and,∂fe/∂t.,The,need,to,use,implicit,methods,is,one,of,the,reasons,why,we,are,trying,to,extract,some,of,the,low,order,moments,from,the,distribution,function.,3.,Wall,boundary,conditions,The,kinetic,equations,will,be,solved,in,the,interval,z,∈,[0,,L],,and,we,will,impose,wall,boundary,conditions,at,z,=,0,and,z,=,L.,We,assume,the,wall,to,be,exactly,perpendicular,to,the,magnetic,ﬁeld,lines,to,be,able,to,impose,a,set,of,simpliﬁed,boundary,conditions:,the,logical,sheath,boundary,conditions,of,Parker,et,al.,(1993).,When,the,magnetic,ﬁeld,is,at,an,angle,to,the,wall,,one,needs,to,consider,a,thin,boundary,layer,with,a,width,of,the,order,of,the,ion,gyroradius,that,forms,on,the,wall,and,is,known,as,the,magnetic,presheath,(Chodura,1982).,The,complicated,boundary,conditions,that,this,layer,imposes,on,drift,kinetic,models,are,an,active,area,of,research,(Geraldini,et,al.,2017,,2018,,2019;,Geraldini,2021).,These,works,indicate,that,the,magnetic,presheath,and,the,Debye,sheath,must,be,solved,in,conjunction,with,the,quasineutral,plasma,,but,this,is,not,necessarily,computationally,expensive,as,the,presheath,and,sheath,models,are,1D,or,at,most,2D,,and,hence,cheaper,than,the,5D,drift,kinetic,models,that,one,needs,for,edge,turbulence.,The,technique,proposed,by,Geraldini,et,al.,(2018),,for,example,,solves,the,magnetic,presheath,in,a,single,processor,in,seconds,,and,this,technique,can,be,parallelized.,Logical,sheath,boundary,conditions,make,use,of,the,fact,that,a,thin,sheath,of,non-,neutral,plasma,with,a,width,of,the,order,of,the,Debye,length,forms,on,walls,to,ensure,quasineutrality.,The,potential,drop,across,this,sheath,repels,electrons,away,from,the,wall,because,otherwise,electrons,would,ﬂow,to,the,wall,at,much,greater,rate,than,ions,due,to,their,lower,mass,and,higher,thermal,speed.,In,our,model,,φ(0,,t),and,φ(L,,t),are,not,the,potential,of,the,wall,,but,the,potential,at,the,entrance,of,the,sheath.,In,this,report,,we,choose,the,potential,of,the,wall,at,z,=,0,to,be,0,without,loss,of,generality.,We,denote,the,potential,of,the,wall,at,z,=,L,as,φw.,Then,,for,the,sheaths,to,repel,electrons,,φ(0,,t),must,be,larger,than,0,and,φ(L,,t),must,be,larger,than,φw.,The,value,of,the,potential,at,z,=,0,and,z,=,L,is,determined,by,requiring,that,the,current,towards,the,wall,at,both,z,=,0,and,z,=,L,vanishes.,We,consider,the,sheath,at,z,=,L,ﬁrst,,and,we,will,then,apply,the,results,that,we,obtain,to,the,sheath,at,z,=,0.,Since,the,thin,sheath,at,z,=,L,imposes,a,large,electric,ﬁeld,perpendicular,to,the,wall,,which,in,this,case,is,along,the,magnetic,ﬁeld,B,=,Bˆz,,the,sheath,only,modiﬁes,the,parallel,velocity,of,electrons.,Within,the,sheath,,the,parallel,energy,E(cid:107),:=,mev2,(cid:107)/2,−,eφ,is,conserved,,and,as,a,result,an,electron,that,has,velocity,v(cid:107),at,the,entrance,of,the,sheath,1D,drift,kinetic,models,with,wall,boundary,conditions,5,(cid:113),v2,(cid:107),−,2e(φ(L,,t),−,φw)/me,when,it,reaches,the,is,slowed,down,to,a,parallel,velocity,wall.,Thus,,electrons,with,parallel,velocity,larger,than,(cid:112)2e(φ(L,,t),−,φw)/me,reach,the,wall,,where,they,recombine,with,ions,,whereas,electrons,with,parallel,velocity,smaller,than,(cid:112)2e(φ(L,,t),−,φw)/me,are,repelled,back,into,the,quasineutral,plasma.,Thus,,the,boundary,condition,on,the,electron,distribution,function,at,z,=,L,is,(cid:26),fe(L,,−v(cid:107),,v⊥,,t),for,v(cid:107),(cid:62),−(cid:112)2e(φ(L,,t),−,φw)/me,,for,v(cid:107),<,−(cid:112)2e(φ(L,,t),−,φw)/me,,fe(L,,v(cid:107),<,0,,v⊥,,t),=,(3.1),0,i.e.,the,electron,distribution,is,mirrored,with,respect,to,v(cid:107),=,0,and,a,cut-oﬀ,is,imposed,for,suﬃciently,negative,parallel,velocities.,Note,that,no,boundary,condition,is,needed,for,v(cid:107),>,0,at,z,=,L,because,of,the,direction,of,the,characteristics,of,the,kinetic,equation.,Expression,(3.1),also,gives,the,electron,current,density,towards,the,wall,at,the,entrance,of,the,sheath,at,z,=,L,,Je(cid:107)(L,,t),=,−2πe,(cid:90),∞,√,2e(φ(L,t)−φw)/me,dv(cid:107),(cid:90),∞,0,dv⊥,v⊥v(cid:107)fe(L,,v(cid:107),,v⊥,,t).,(3.2),This,is,the,electron,current,density,at,the,wall,because,electron,ﬂow,is,conserved,across,the,sheath.,Imposing,that,the,electron,current,cancels,the,ion,current,gives,us,a,nonlinear,equation,for,the,potential,diﬀerence,φ(L,,t),−,φw,,(cid:90),∞,√,2π,2e(φ(L,t)−φw)/me,(cid:90),∞,=,2π,(cid:90),∞,0,(cid:90),∞,dv(cid:107),dv(cid:107),dv⊥,v⊥v(cid:107)fe(L,,v(cid:107),,v⊥,,t),dv⊥,v⊥v(cid:107)fi(L,,v(cid:107),,v⊥,,t).,(3.3),0,0,To,obtain,the,ion,current,,we,have,used,the,fact,that,the,sheath,attracts,ions,and,hence,no,ions,can,have,negative,parallel,velocity,at,the,entrance,of,the,sheath,at,z,=,L.,The,conditions,at,z,=,0,for,the,electron,distribution,and,the,potential,are,similar,to,those,for,z,=,L.,For,the,electron,distribution,function,,we,ﬁnd,fe(0,,v(cid:107),>,0,,v⊥,,t),=,(cid:26),fe(0,,−v(cid:107),,v⊥,,t),0,for,v(cid:107),(cid:54),(cid:112)2eφ(0,,t)/me,,for,v(cid:107),>,(cid:112)2eφ(0,,t)/me,,(3.4),and,for,the,potential,we,obtain,√,(cid:90),−,−∞,2π,2eφ(0,t)/me,(cid:90),0,=,2π,(cid:90),∞,0,(cid:90),∞,dv(cid:107),dv(cid:107),−∞,0,dv⊥,v⊥v(cid:107)fe(0,,v(cid:107),,v⊥,,t),dv⊥,v⊥v(cid:107)fi(0,,v(cid:107),,v⊥,,t),,(3.5),Note,that,conditions,(3.3),and,(3.5),imply,that,no,net,electrical,current,is,leaving,the,system.,Thus,,the,total,source,of,charge,in,the,magnetic,ﬁeld,line,of,interest,must,be,zero,,(cid:90),L,(cid:90),dz,0,Si,d3v,=,(cid:90),L,(cid:90),dz,0,Se,d3v.,(3.6),We,still,need,boundary,conditions,for,the,ion,and,neutral,distribution,functions.,Ions,recombine,when,they,hit,the,wall,,so,no,ions,come,back,,giving,fi(0,,v(cid:107),>,0,,v⊥,,t),=,0,,fi(L,,v(cid:107),<,0,,v⊥,,t),=,0.,(3.7),The,neutrals,hit,the,wall,and,thermalize,at,the,temperature,of,the,wall,Tw,,while,also,6,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,receiving,back,the,ions,that,have,recombined,at,the,wall,,fn(0,,v(cid:107),>,0,,v⊥,,t),=,Γ0fKw(v(cid:107),,v⊥),,fn(L,,v(cid:107),<,0,,v⊥,,t),=,ΓLfKw(v(cid:107),,v⊥),,(3.8),where,and,Γ0,:=,(cid:88),2π,(cid:90),0,(cid:90),∞,dv(cid:107),s=i,n,−∞,0,ΓL,:=,(cid:88),2π,(cid:90),∞,(cid:90),∞,dv(cid:107),s=i,n,0,0,dv⊥,v⊥|v(cid:107)|fs(0,,v(cid:107),,v⊥,,t),(3.9),dv⊥,v⊥v(cid:107)fs(L,,v(cid:107),,v⊥,,t),(3.10),are,the,ﬂuxes,of,neutrals,and,ions,towards,the,walls,at,z,=,0,and,z,=,L.,Here,,fKw(v(cid:107),,v⊥),:=,(cid:19)2,3,4π,(cid:18),mi,Tw,(cid:113),|v(cid:107)|,(cid:107),+,v2,v2,⊥,(cid:32),exp,−,(cid:33),mi(v2,(cid:107),+,v2,⊥),2Tw,(3.11),is,the,Knudsen,cosine,distribution,(Knudsen,1916),that,assumes,that,the,particles,have,entered,the,wall,lattice,,have,reached,thermodynamic,equilibrium,with,it,,and,have,then,left,the,wall.,Knudsen,showed,that,this,distribution,function,ﬁts,experimental,measure-,ments,well.,4.,1D,moment,drift,kinetics,for,ions,and,neutrals,Instead,of,solving,for,fs(z,,v(cid:107),,v⊥,,t),with,s,=,i,,n,,we,solve,for,Fs(z,,w(cid:107),,w⊥,,t),:=,v3,ts(z,,t),ns(z,,t),(cid:16),fs,z,,us(cid:107)(z,,t),+,vts(z,,t)w(cid:107),,vts(z,,t)w⊥,,t,(cid:17),,,(4.1),where,we,have,deﬁned,the,normalized,velocities,w(cid:107)(z,,v(cid:107),,t),:=,v(cid:107),−,us(cid:107)(z,,t),vts(z,,t),w⊥(z,,v⊥,,t),:=,v⊥,vts(z,,t),,,and,the,average,parallel,velocity,(4.2),(4.3),us(cid:107)(z,,t),:=,2π,ns,(cid:90),∞,(cid:90),∞,dv(cid:107),−∞,0,dv⊥,v⊥v(cid:107)fs(z,,v(cid:107),,v⊥,,t),(4.4),and,the,thermal,speed,with,vts(z,,t),:=,(cid:115),2Ts(z,,t),ms,,,(4.5),Ts(z,,t),:=,2π,ns,(cid:90),∞,(cid:90),∞,dv(cid:107),−∞,0,dv⊥,v⊥,ms[(v(cid:107),−,us(cid:107)(z,,t))2,+,v2,⊥],3,fs(z,,v(cid:107),,v⊥,,t),(4.6),the,temperature,of,species,s.,According,to,its,deﬁnition,,Fs(z,,w(cid:107),,w⊥,,t),must,satisfy,the,conditions,2π,dw(cid:107),dw⊥,w⊥Fs(z,,w(cid:107),,w⊥,,t),=,1,,(4.7),(cid:90),∞,(cid:90),∞,−∞,0,1D,drift,kinetic,models,with,wall,boundary,conditions,(cid:90),∞,2π,dw(cid:107),(cid:90),∞,−∞,0,dw⊥,w⊥w(cid:107)Fs(z,,w(cid:107),,w⊥,,t),=,0,and,(cid:90),∞,2π,dw(cid:107),(cid:90),∞,−∞,0,at,every,point,z,and,time,t.,dw⊥,w⊥(w2,(cid:107),+,w2,⊥)Fs(z,,w(cid:107),,w⊥,,t),=,3,2,The,equations,for,ni,,ui(cid:107),and,Ti,are,4.1.,Ion,equations,∂ni,∂t,+,∂,∂z,(cid:0)niui(cid:107),(cid:1),=,nnneRion,+,(cid:90),Si,d3v,,7,(4.8),(4.9),(4.10),nimi,(cid:18),∂ui(cid:107),∂t,+,ui(cid:107),(cid:19),∂ui(cid:107),∂z,=,−,∂pi(cid:107),∂z,−,eni,∂φ,∂z,+,nimi(nnRin,+,neRion)(un(cid:107),−,ui(cid:107)),(cid:90),+,mi(v(cid:107),−,ui(cid:107))Si,d3v,(4.11),and,3,2,ni,(cid:18),∂Ti,∂t,+,ui(cid:107),(cid:19),∂Ti,∂z,=,−,∂qi(cid:107),∂z,−,pi(cid:107),∂ui(cid:107),∂z,+,3,2,ni(nnRin,+,neRion)(Tn,−,Ti),+,1,2,nimi(nnRin,+,neRion)(un(cid:107),−,ui(cid:107))2,+,(cid:90),1,2,mi|v,−,ui(cid:107)ˆz|2Cie,d3v,(cid:90),(cid:18),1,2,+,mi|v,−,ui(cid:107)ˆz|2,−,(cid:19),3,2,Ti,Si,d3v.,(4.12),Here,,we,have,deﬁned,the,parallel,pressure,(cid:90),∞,ps(cid:107)[Fs,,ns,,vts](z,,t),:=,2πnsmsv2,ts,(cid:90),∞,dw(cid:107),−∞,0,dw⊥,w⊥w2,(cid:107)Fs(z,,w(cid:107),,w⊥,,t),(4.13),and,the,parallel,heat,ﬂux,qs(cid:107)[Fs,,ns,,vts](z,,t),:=,πnsmsv3,ts,(cid:90),∞,(cid:90),∞,dw(cid:107),−∞,0,dw⊥,w⊥w(cid:107)(w2,(cid:107),+,w2,⊥)Fs(z,,w(cid:107),,w⊥,,t).,(4.14),We,have,also,included,the,term,(1/2),(cid:82),mi|v,−,ui(cid:107)ˆz|2Cie,d3v,due,to,collisions,with,elec-,trons.,Collisions,with,electrons,are,negligible,to,lowest,order,in,(cid:112)me/mi,in,the,ion,kinetic,equation,and,thus,cannot,determine,the,lowest,order,distribution,function,Fi,,but,when,collisions,are,suﬃciently,frequent,that,νiiL/vti,(cid:38),(cid:112)mi/me,(cid:29),1,,the,term,(1/2),(cid:82),mi|v,−,ui(cid:107)ˆz|2Cie,d3v,becomes,comparable,to,the,other,terms,in,the,energy,equa-,tion.,Here,,√,8,νii,:=,2π,3,e4ni,ln,Λ,(4π(cid:15)0)2m2,i,v3,ti,(4.15),is,the,ion-ion,collision,frequency,as,deﬁned,by,Braginskii,(Braginskii,1958).,At,the,large,collision,frequencies,required,for,the,term,(1/2),(cid:82),mi|v,−,ui(cid:107)ˆz|2Cie,d3v,to,be,relevant,,the,ion,and,electron,distribution,functions,become,close,to,a,Maxwellian,,fs,(cid:39),fM,s,:=,ns,π3/2v3,ts,(cid:18),exp,−,(v(cid:107),−,us(cid:107))2,+,v2,⊥,v2,ts,(cid:19),.,(4.16),8,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Thus,,we,can,use,the,approximation,(cid:90),1,2,mi|v,−,ui(cid:107)ˆz|2Cie[fi,,fe],d3v,(cid:39),(cid:90),1,2,mi|v,−,ui(cid:107)ˆz|2Cie[fM,i,,fM,e],d3v,(cid:39),3nemeνei,mi,(Te,−,Ti),,(4.17),where,νei,:=,π,√,16,3,e4ni,ln,Λ,(4π(cid:15)0)2m2,ev3,te,(4.18),is,the,electron-ion,collision,frequency,as,deﬁned,by,Braginskii,(Braginskii,1958).,Note,that,Braginskii’s,deﬁnitions,of,νii,and,νei,diﬀer,by,a,factor,of,√,2.,The,ion,kinetic,equation,is,∂Fi,∂t,+,˙zi,∂Fi,∂z,+,˙w(cid:107)i,∂Fi,∂w(cid:107),+,˙w⊥i,∂Fi,∂w⊥,=,˙Fi,+,Cii,+,Cin,+,Ci,ion,+,Si.,(4.19),Here,,we,have,deﬁned,the,coeﬃcients,˙zs[Fs,,us(cid:107),,vts](z,,w(cid:107),,t),:=,us(cid:107),+,vtsw(cid:107),,(4.20),˙w(cid:107)s[Fs,,ns,,us(cid:107),,vts](z,,w(cid:107),,t),:=,+,2w(cid:107),3nsmsv2,ts,∂ps(cid:107),∂z,(cid:18),1,nsmsvts,(cid:20),∂qs(cid:107),∂z,+,ps(cid:107),−,3,2,nsmsv2,ts,(cid:21),(cid:19),∂us(cid:107),∂z,−,w2,(cid:107),∂vts,∂z,,,˙w⊥s[Fs,,ns,,us(cid:107),,vts](z,,w(cid:107),,w⊥,,t),:=,2w⊥,3nsmsv2,ts,(cid:18),∂qs(cid:107),∂z,+,ps(cid:107),(cid:19),∂us(cid:107),∂z,−,w(cid:107)w⊥,∂vts,∂z,and,˙Fs[Fs,,ns,,us(cid:107),,vts](z,,w(cid:107),,w⊥,,t),:=,w(cid:107),(cid:34),(cid:18),3,∂vts,∂z,−,vts,ns,∂ns,∂z,(cid:19),(4.21),(4.22),−,2,nsmsv2,ts,(cid:18),∂qs(cid:107),∂z,(cid:18),+,ps(cid:107),−,1,2,nsmsv2,ts,(cid:19),(cid:35),(cid:19),∂us(cid:107),∂z,Fs.,(4.23),We,have,also,deﬁned,a,modiﬁed,source,Si,and,several,modiﬁed,collision,operators.,The,modiﬁed,source,is,given,by,Ss[Ss,,Fs,,ns,,us(cid:107),,vts](z,,w(cid:107),,w⊥,,t),(cid:90),Ss,d3v,−,(cid:21),Ss(z,,us(cid:107),+,vtsw(cid:107),,vtsw⊥,,t),v3,ts,ns,(cid:90),:=,−,+,+,(cid:20),Fs,ns,∂,∂w(cid:107),1,w⊥,(cid:18),1,(cid:20),Fs,nsvts,(cid:20),w2,⊥Fs,3nsv2,ts,∂,∂w⊥,(v(cid:107),−,us(cid:107))Ss,d3v,+,(cid:90),(cid:18),|v,−,us(cid:107)ˆz|2,−,3,2,v2,ts,Ss,d3v,w(cid:107),3nsv2,ts,(cid:19),(cid:90),(cid:18),|v,−,us(cid:107)ˆz|2,−,(cid:19),3,2,v2,ts,(cid:19)(cid:21),Ss,d3v,(cid:21),.,(4.24),Note,that,the,diﬀerential,terms,in,this,modiﬁed,source,could,have,been,included,in,the,˙w⊥i,and,˙Fi,,but,we,have,decided,to,make,them,part,deﬁnitions,of,the,coeﬃcients,˙w(cid:107)i,,of,a,modiﬁed,source,instead,to,separate,the,eﬀect,of,the,source,clearly.,We,will,do,the,same,for,collisions.,This,split,should,not,be,taken,as,a,suggestion,on,how,to,implement,these,terms,in,a,code.,The,modiﬁed,collisions,operators,are,described,in,Appendix,A.,1D,drift,kinetic,models,with,wall,boundary,conditions,9,The,ﬂuid,equations,for,the,neutrals,are,4.2.,Neutral,equations,∂nn,∂t,+,∂,∂z,(cid:0)nnun(cid:107),(cid:1),=,−nnneRion,+,(cid:90),Sn,d3v,,(4.25),nnmi,(cid:18),∂un(cid:107),∂t,+,un(cid:107),(cid:19),∂un(cid:107),∂z,=,−,∂pn(cid:107),∂z,+,nnminiRin(ui(cid:107),−,un(cid:107)),(cid:90),+,mi(v(cid:107),−,un(cid:107))Sn,d3v,(4.26),and,3,2,nn,(cid:18),∂Tn,∂t,+,un(cid:107),(cid:19),∂Tn,∂z,=,−,∂qn(cid:107),∂z,−,pn(cid:107),+,1,2,nnminiRin(un(cid:107),−,ui(cid:107))2,+,∂un(cid:107),∂z,(cid:90),1,2,+,3,2,nnniRin(Ti,−,Tn),mi|v,−,un(cid:107)ˆz|2Cne,d3v,(cid:90),(cid:18),1,2,As,with,electron-ion,collisions,,the,term,(1/2),(cid:82),mi|v,−,un(cid:107)ˆz|2Cne,d3v,only,becomes,im-,portant,when,the,collisions,are,suﬃciently,frequent,that,the,distribution,functions,are,close,to,Maxwellians,,giving,mi|v,−,un(cid:107)ˆz|2,−,Sn,d3v.,(4.27),3,2,Tn,+,(cid:19),(cid:90),1,2,mi|v,−,un(cid:107)ˆz|2Cne[fn,,fe],d3v,(cid:39),(cid:90),1,2,mi|v,−,un(cid:107)ˆz|2Cne[fM,n,,fM,e],d3v,(cid:39),3nemennRen,mi,(Te,−,Tn),,(4.28),The,neutral,kinetic,equation,is,∂Fn,∂t,+,˙zn,∂Fn,∂z,+,˙w(cid:107)n,∂Fn,∂w(cid:107),+,˙w⊥n,∂Fn,∂w⊥,=,˙Fn,+,Cni,+,Sn.,(4.29),The,modiﬁed,charge,exchange,collision,operator,Cni,is,described,in,Appendix,B.,Equations,(4.19),and,(4.29),for,Fi,and,Fn,are,constructed,such,that,conditions,(4.7),,(4.8),and,(4.9),are,satisﬁed,at,all,times,if,they,are,satisﬁed,at,t,=,0.,In,practice,,this,property,has,to,be,enforced,in,the,numerical,method.,We,have,found,an,algorithm,that,works,well,and,we,have,discussed,it,in,report,2047357-TN-04-02,M2.2.,4.3.,Boundary,conditions,These,equations,for,ions,and,neutrals,have,to,be,solved,with,the,boundary,conditions,in,equations,(3.7),and,(3.8).,For,ns,,us(cid:107),,vts,and,Fs,known,at,time,t,,we,can,construct,fs,at,z,=,0,and,z,=,L,,and,we,can,apply,boundary,conditions,(3.7),and,(3.8).,We,can,then,use,the,resulting,fs,to,obtain,ns,,us(cid:107),,vts,and,Fs,,and,to,calculate,ps(cid:107),and,qs(cid:107),,closing,the,system,of,equations.,We,note,that,this,is,the,only,place,where,the,full,distribution,function,fs,is,needed.,Depending,on,the,numerical,method,chosen,to,solve,these,equations,,reconstructing,fs,could,be,expensive.,This,is,one,of,the,problems,that,needs,to,be,addressed,in,the,proxy,apps.,10,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,5.,1D,moment,drift,kinetics,for,electrons,The,equation,for,Fe,can,be,signiﬁcantly,simpliﬁed,using,the,expansion,in,(cid:112)me/mi.,The,electron,source,Se,is,usually,of,order,Se,∼,vtife,L,(5.1),because,the,ions,control,the,dynamics,due,to,their,large,mass,,and,the,electrons,adapt,to,the,ions,because,of,collisions,and,quasineutrality.,Thus,,Se,is,a,small,term,in,equa-,tion,(2.4),by,a,factor,of,vti/vte,∼,(cid:112)me/mi,(cid:28),1.,Moreover,,the,electron,parallel,ﬂow,ue(cid:107),is,of,the,order,of,the,ion,parallel,ﬂow,because,boundary,conditions,(3.3),and,(3.5),impose,that,these,two,velocities,be,equal,at,z,=,0,and,z,=,L,,and,quasineutrality,keeps,the,diﬀerence,of,the,order,of,vti.,As,a,result,,ue(cid:107),∼,ui(cid:107),∼,vti,(cid:28),vte,and,we,can,neglect,ue(cid:107),to,lowest,order,in,most,terms,in,equation,(2.4),–,the,two,exceptions,in,which,ue(cid:107),cannot,be,neglected,are,given,in,Appendix,C.,Finally,,the,eﬀect,of,ionization,,modeled,by,Ce,ion,∼,nnRionfe,is,,according,to,equation,(4.10),,of,the,order,of,fevti/L,and,thus,also,small.,5.1.,Electron,kinetic,equation,Employing,the,expansion,in,(cid:112)me/mi,(cid:28),1,,the,kinetic,equation,for,electrons,becomes,∂Fe,∂w⊥,=,˙Fe,+,Cee,+,Cei,+,Cen,,∂Fe,∂z,∂Fe,∂w(cid:107),+,˙w⊥e,+,˙w(cid:107)e,˙ze,(5.2),where,and,˙ze[Fe,,vte](z,,w(cid:107),,t),:=,vtew(cid:107),,˙w(cid:107)e[Fe,,ue(cid:107),,vte](z,,w(cid:107),,t),:=,1,nemevte,∂pe(cid:107),∂z,+,2w(cid:107),3nemev2,te,∂qe(cid:107),∂z,−,w2,(cid:107),∂vte,∂z,,,˙w⊥e[Fe,,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,2w⊥,3nemev2,te,∂qe(cid:107),∂z,−,w(cid:107)w⊥,∂vte,∂z,(5.3),(5.4),(5.5),˙Fe[Fe,,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,w(cid:107),(cid:34),(cid:18),3,∂vte,∂z,−,vte,ne,∂ne,∂z,(cid:19),−,2,nemev2,te,∂qe(cid:107),∂z,(cid:35),Fe.,(5.6),The,modiﬁed,collision,operators,Cee,,Cei,and,Cen,are,described,in,Appendix,C.,Note,that,,in,equations,(C,1),and,(C,6),,we,have,kept,small,terms,that,scale,with,(us(cid:107),−,ue(cid:107))/vte,∼,(cid:112)me/mi,(cid:28),1,,where,s,=,i,,n.,These,terms,are,kept,to,ensure,that,we,recover,the,Braginskii,equations,in,the,appropriate,limit,(see,subsection,5.4).,We,have,ensured,that,equation,(5.2),is,compatible,with,conditions,(4.7),,(4.8),and,(4.9),by,keeping,terms,that,are,second,order,in,ue(cid:107),in,the,collision,operators,in,Appendix,C.,Indeed,,multiplying,equation,(5.2),by,1,,w(cid:107),and,w2,⊥,and,integrating,over,velocities,gives,0,=,0.,Despite,the,fact,that,we,do,not,keep,all,possible,terms,that,are,second,order,in,(cid:112)me/mi,in,the,collision,operator,,we,will,see,in,subsection,5.4,that,proposed,model,recovers,the,regimes,of,interest.,(cid:107),+,w2,Conditions,(4.7),,(4.8),and,(4.9),have,to,be,imposed,on,Fe,when,solving,the,kinetic,equation,(5.2).,One,possible,way,to,impose,these,conditions,is,to,include,the,term,∂Fe/∂t,in,equation,(5.2),so,that,we,can,evolve,Fe,to,a,steady,state,solution.,With,this,approach,,if,Fe,satisﬁes,conditions,(4.7),,(4.8),and,(4.9),at,t,=,0,,it,will,satisfy,them,at,all,times.,1D,drift,kinetic,models,with,wall,boundary,conditions,11,Once,we,know,Fe,,we,can,calculate,the,ﬂuid,equations,for,electrons.,•,The,electron,continuity,equation,is,5.2.,Electron,ﬂuid,equations,∂ne,∂t,+,∂,∂z,(cid:0)neue(cid:107),(cid:1),=,−nennRion,+,(cid:90),Se,d3v.,(5.7),Subtracting,this,equation,from,equation,(4.10),and,using,quasineutrality,,we,obtain,the,current,conservation,equation,(cid:2)ne,(cid:0)ui(cid:107),−,ue(cid:107),(cid:1)(cid:3),=,(cid:90),(cid:90),Si,d3v,−,Se,d3v.,∂,∂z,(5.8),This,equation,can,be,used,to,calculate,ue(cid:107).,•,The,electron,parallel,momentum,equation,simpliﬁes,to,0,=,−,∂pe(cid:107),∂z,+,ene,∂φ,∂z,+,Fei(cid:107),+,nemennRen(un(cid:107),−,ue(cid:107)),,(5.9),where,Fei(cid:107)[Fe,,ne,,ni,,ue(cid:107),,ue(cid:107),,vte,,vti](z,,t),8π2e4neni,ln,Λ,(4π(cid:15)0)2mev2,te,:=,−,dw(cid:107),(cid:90),∞,−∞,(cid:90),∞,0,dw⊥,w⊥(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)Fe,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,(5.10),is,the,friction,force,between,electrons,and,ions.,Equation,(5.9),can,be,used,to,calculate,the,potential,φ.,Since,the,potential,at,φ(0,,t),is,determined,by,equation,(3.5),,we,can,start,integrating,φ,at,z,=,0.,With,the,potential,φ(L,,t),that,we,obtain,from,this,integration,and,condition,(3.3),,we,can,calculate,the,potential,diﬀerence,between,the,two,walls,,φw.,•,The,electron,energy,equation,is,3,2,ne,(cid:18),∂Te,∂t,+,ue(cid:107),(cid:19),∂Te,∂z,=,−,∂qe(cid:107),∂z,−,pe(cid:107),∂ue(cid:107),∂z,+,(cid:90),1,2,me|v,−,ue(cid:107)ˆz|2Ce,ion,d3v,+,+,(cid:90),1,2,(cid:90),1,2,me|v,−,ue(cid:107)ˆz|2Cei,me|v,−,ue(cid:107)ˆz|2Cen,(cid:20),1,+,O,(cid:20),1,+,O,(cid:18),me,mi,(cid:18),me,mi,(cid:19),Te,(cid:19)(cid:21),(cid:19)(cid:21),d3v,d3v,Se,d3v.,(5.11),(cid:90),(cid:18),1,2,+,me|v,−,ue(cid:107)ˆz|2,−,3,2,For,the,integral,over,the,ionization,collision,operator,,we,use,the,model,(cid:90),1,2,me|v,−,ue(cid:107)ˆz|2Ce,ion,d3v,=,−nennRionEion,,(5.12),where,Eion,is,the,ionization,energy,cost,that,includes,in,it,radiation,from,excited,states.,The,integrals,over,Cei,and,Cen,are,only,suﬃciently,large,when,collisions,are,large.,In,this,limit,,all,the,species,are,Maxwellian,and,we,can,easily,calculate,the,integrals,over,Cei,and,Cen,to,higher,order,in,the,mass,ratio,expansion,,ﬁnding,(cid:90),1,2,me|v,−,ue(cid:107)ˆz|2Cei[fe,,fi],(cid:20),1,+,O,(cid:19)(cid:21),(cid:18),me,mi,d3v,(cid:39),3nemeνei,mi,(Ti,−,Te),+Fei(cid:107)(ui(cid:107),−,ue(cid:107)),(5.13),12,and,(cid:90),1,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,me|v,−,ue(cid:107)ˆz|2Cen[fe,,fn],(cid:20),1,+,O,(cid:19)(cid:21),(cid:18),me,mi,d3v,(cid:39),3nemennRen,mi,(Tn,−,Te),+nemennRen(un(cid:107),−,ue(cid:107))2.,(5.14),5.3.,Boundary,conditions,These,equations,for,electrons,have,to,be,solved,with,the,boundary,conditions,in,equa-,tions,(3.1),and,(3.4).,As,for,ions,and,neutrals,,the,best,way,to,impose,these,boundary,conditions,is,to,transform,back,to,fe,,apply,the,boundary,conditions,and,then,calculate,Fe,,ne,,me,,vte,,pe(cid:107),and,qe(cid:107),from,fe.,We,ﬁnish,by,showing,that,these,equations,recover,the,desired,result,in,the,two,limits,of,interest:,a,modiﬁed,Braginskii,limit,with,neutrals,,and,a,collisionless,limit.,5.4.,Some,limits,of,interest,5.4.1.,Braginskii-like,equations,For,the,Braginskii-like,limit,,we,use,the,orderings,suggested,in,(Catto,1994;,Helander,et,al.,1994):,we,assume,that,the,ion-ion,,ion-neutral,,electron-electron,,electron-ion,and,electron-neutral,collisions,are,so,frequent,that,their,collision,frequencies,νss(cid:48),satisfy,νss(cid:48)L,vts,∼,(cid:114),mi,me,(cid:29),1,,(5.15),whereas,the,ionization,frequencies,neRion,and,nnRion,are,of,order,vti/L.,In,this,limit,,the,charge,exchange,collisional,terms,dominate,in,the,ion,and,neutral,momentum,and,energy,equations,,forcing,ui(cid:107),=,un(cid:107),and,Ti,=,Tn.,We,use,uh(cid:107),and,Th,to,denote,the,average,ﬂow,and,temperature,of,the,heavy,species.,By,summing,the,ion,and,neutral,momentum,equations,(4.11),and,(4.26),,we,ﬁnd,the,equation,for,uh(cid:107),,(ni,+,nn)mi,(cid:18),∂uh(cid:107),∂t,+,uh(cid:107),(cid:19),(cid:90),∂uh(cid:107),∂z,+,=,−,∂,∂z,(pi(cid:107),+,pn(cid:107)),−,eni,∂φ,∂z,mi(v(cid:107),−,uh(cid:107))(Si,+,Sn),d3v.,(5.16),By,summing,the,ion,and,neutral,energy,equations,(4.12),and,(4.27),,we,ﬁnd,the,equation,for,Th,,3,2,(ni,+,nn),+,uh(cid:107),(cid:18),∂Th,∂t,3neme(νei,+,nnRen),mi,+,∂Th,∂z,(cid:19),=,−,∂,∂z,(Te,−,Th),+,(qi(cid:107),+,qn(cid:107)),−,(pi(cid:107),+,pn(cid:107)),(cid:90),(cid:18),1,2,mi|v,−,uh(cid:107)ˆz|2,−,3,2,∂uh(cid:107),∂z,(cid:19),Th,(Si,+,Sn),d3v.,(5.17),Note,that,in,this,energy,equation,,the,terms,due,to,electron-ion,and,electron-neutral,collisions,are,of,the,same,order,as,the,other,terms.,In,the,electron,ﬂuid,equations,,the,collisional,friction,terms,in,the,electron,momentum,equation,(5.9),are,comparable,to,the,pressure,and,electric,ﬁeld,terms.,The,electron,heat,ﬂux,term,and,the,terms,related,to,the,electron-ion,and,electron-neutral,collisions,in,the,electron,energy,equation,(5.11),are,also,comparable,to,the,rest,of,the,terms.,In,the,kinetic,equations,,the,collisions,dominate,and,lead,to,distribution,functions,that,are,Maxwellian,to,lowest,order.,We,can,use,the,kinetic,equations,to,ﬁnd,the,corrections,1D,drift,kinetic,models,with,wall,boundary,conditions,13,to,the,Maxwellian.,For,ions,and,neutrals,,the,corrections,to,the,Maxwellian,do,not,give,large,contributions,to,either,the,parallel,pressure,or,the,heat,ﬂux.,For,the,electrons,,how-,ever,,the,correction,gives,important,contributions,to,the,friction,force,in,the,momentum,equation,(5.9),and,to,the,electron,heat,ﬂux,in,the,energy,equation,(5.11).,If,we,write,Fe,=,FM,+,Fe1,+,.,.,.,,with,FM,:=,π−3/2,exp(−w2,⊥),,we,can,ﬁnd,the,equation,for,Fe1,(Braginskii,1958),,(cid:107),−,w2,ee,[Fe1],+,C((cid:96)),C((cid:96)),ei,(cid:20),Fe1,−,2(uh(cid:107),−,ue(cid:107))w(cid:107),vte,(cid:21),FM,+,C((cid:96)),en,(cid:20),Fe1,−,16π2e4ni,ln,Λ,ev3,(4π(cid:15)0)2m2,te,(cid:90),∞,w(cid:107)FM,+,(cid:34),=,vtew(cid:107),(cid:18),w2,(cid:107),+,w2,⊥,−,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,√,32,ln,Te,+,−∞,(cid:19),∂,∂z,5,2,(cid:21),w(cid:48),FM,⊥w(cid:48),2(uh(cid:107),−,ue(cid:107))w(cid:107),vte,(cid:107)Fe1(z,,w(cid:48),(cid:107),+,w(cid:48)2,(w(cid:48)2,πe4ni,ln,Λ(uh(cid:107),−,ue(cid:107))w(cid:107),(cid:107),,w(cid:48),⊥)3/2,⊥,,t),3(4π(cid:15)0)2m2,ev4,te,+,2nnRen(uh(cid:107),−,ue(cid:107))w(cid:107),vte,(cid:35),FM,.,(5.18),Here,we,have,neglected,the,electron,heat,ﬂux,qe(cid:107),because,it,is,proportional,to,Fe1.,The,collision,operators,C((cid:96)),en,are,the,linearized,collision,operators,,given,in,Appendix,D.,Note,that,the,terms,proportional,to,uh(cid:107),−,ue(cid:107),,needed,to,recover,the,friction,force,and,electron,heat,ﬂux,in,Braginskii,(1958),(see,subsection,5.4),,come,from,the,electron-ion,and,electron-electron,collision,operators,in,equations,(C,1),and,(C,6).,ei,and,C((cid:96)),ee,,,C((cid:96)),We,ﬁnish,by,pointing,out,that,Fe,satisﬁes,conditions,(4.7),,(4.8),and,(4.9),to,the,order,that,we,have,calculated,it.,Since,FM,satisﬁes,these,conditions,,the,conditions,for,Fe1,become,(cid:90),∞,(cid:90),∞,2π,dw(cid:107),dw⊥,w⊥Fe1(z,,w(cid:107),,w⊥,,t),=,0,,−∞,(cid:90),∞,2π,dw(cid:107),0,(cid:90),∞,−∞,0,dw⊥,w⊥w(cid:107)Fe1(z,,w(cid:107),,w⊥,,t),=,0,and,(cid:90),∞,2π,dw(cid:107),(cid:90),∞,−∞,0,dw⊥,w⊥(w2,(cid:107),+,w2,⊥)Fe1(z,,w(cid:107),,w⊥,,t),=,0.,(5.19),(5.20),(5.21),These,conditions,determine,the,pieces,of,Fe1,that,are,in,the,kernel,of,the,operators,in,the,left,side,of,equation,(5.18).,5.4.2.,Collisionless,electron,equations,We,call,this,limit,collisionless,in,contraposition,to,the,collisional,Braginsikii-like,limit,discussed,above,,but,we,still,keep,collisions.,We,assume,that,ion-ion,,ion-neutral,,electron-,electron,,electron-ion,,electron-neutral,and,ion-neutral,collisions,satisfy,νss(cid:48)L,vts,The,ionization,frequencies,neRion,and,nnRion,are,still,assumed,to,be,of,order,vti/L.,∼,1.,(5.22),In,this,limit,,we,can,neglect,the,collisional,coupling,between,the,heavy,species,(ions,and,neutrals),and,electrons,in,the,ﬂuid,equations.,In,the,electron,energy,equation,(5.11),,the,dominant,term,is,∂qe(cid:107)/∂z,,giving,∂qe(cid:107),∂z,(cid:39),0,(5.23),14,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,In,the,electron,momentum,equation,(5.9),,the,friction,forces,are,negligible,,giving,∂pe(cid:107),∂z,(cid:39),ene,∂φ,∂z,.,(5.24),Using,these,results,in,the,electron,kinetic,equation,(5.2),and,neglecting,ui(cid:107),−,ue(cid:107),and,un(cid:107),−,ue(cid:107),in,the,collision,operators,in,equation,(C,1),and,(C,6),,we,ﬁnd,(cid:20),vtew(cid:107),∂,∂z,+,(cid:18),e,mevte,∂φ,∂z,−,w2,(cid:107),∂vte,∂z,(cid:19),∂,∂w(cid:107),−,w(cid:107)w⊥,=,ne,v3,te,(cid:19),∂,∂vte,∂w⊥,∂z,(cid:16),Cee,+,C((cid:96)),(cid:21),(cid:18),neFe,v3,te,ei,+,C((cid:96)),en,(cid:17),.,(5.25),The,collision,operators,C((cid:96)),ei,and,C((cid:96)),en,are,deﬁned,in,Appendix,D.,We,can,use,entropy,production,of,the,electron-electron,,electron-ion,and,electron-,neutral,collision,operators,to,solve,equation,(5.25).,We,multiply,equation,(5.25),by,−,ln(neFe/v3,(cid:20),vtew(cid:107),te),to,ﬁnd,(cid:18),e,−,w(cid:107)w⊥,1,−,ln,(cid:19)(cid:19)(cid:21),+,(cid:18),−,w2,(cid:107),∂,∂z,∂φ,∂z,mevte,∂vte,∂z,(cid:19),∂,∂w(cid:107),(cid:18),neFe,v3,te,∂vte,∂z,∂,∂w⊥,=,−,ne,v3,te,ln,(cid:21),(cid:20),neFe,v3,te,(cid:18),neFe,(cid:19),(cid:16),v3,te,Cee,+,C((cid:96)),ei,+,C((cid:96)),en,(cid:17),.,(5.26),Multiplying,by,v3,(cid:34),te,and,integrating,over,velocity,space,,we,obtain,(cid:19)(cid:19),(cid:90),(cid:18),(cid:35),∂,∂z,nevte,1,−,ln,(cid:90),=,−ne,(cid:18),neFe,v3,te,(cid:18),neFe,v3,te,ln,Few(cid:107)w⊥,dw(cid:107),dw⊥,(cid:19),(cid:16),Cee,+,C((cid:96)),ei,+,C((cid:96)),en,(cid:17),w⊥,dw(cid:107),dw⊥.,(5.27),Integrating,equation,(5.27),over,z,gives,(cid:34),nevte,(cid:90),(cid:18),1,−,ln,(cid:19)(cid:19),Few(cid:107)w⊥,dw(cid:107),dw⊥,(cid:35)z=L,z=0,(cid:18),neFe,v3,te,(cid:90),=,−ne,dz,ln,(cid:90),L,0,(cid:19),(cid:16),(cid:18),neFe,v3,te,Cee,+,C((cid:96)),ei,+,C((cid:96)),en,(cid:17),w⊥,dw(cid:107),dw⊥.,(5.28),Conditions,(3.3),and,(3.5),impose,that,only,a,few,electrons,leave,the,system,towards,the,wall.,The,number,of,electrons,that,leave,is,small,by,a,factor,of,(cid:112)me/mi,(cid:28),1,and,thus,,the,left,side,of,equation,(5.28),can,be,neglected,,ﬁnally,giving,(cid:18),neFe,v3,te,w⊥,dw(cid:107),dw⊥,(cid:39),0.,Cee,+,C((cid:96)),ei,+,C((cid:96)),en,(5.29),(cid:19),(cid:16),(cid:90),L,dz,ln,−,(cid:17),(cid:90),0,The,integrand,under,the,integral,over,z,is,the,entropy,production,,and,it,is,positive,unless,the,distribution,function,Fe,is,the,Maxwellian,FM,.,Hence,,Fe,is,a,Maxwellian,to,lowest,order,in,the,expansion,in,(cid:112)me/mi.,Substituting,the,Maxwellian,into,equation,(5.25),,we,ﬁnd,that,the,right,side,of,the,equation,vanishes.,For,the,left,side,of,the,equation,to,be,zero,for,all,w(cid:107),and,w⊥,,we,need,∂vte/∂z,=,0,and,ne(z,,t),=,Ne(t),exp,(cid:18),eφ(z,,t),Te(t),(cid:19),.,(5.30),1D,drift,kinetic,models,with,wall,boundary,conditions,15,The,value,of,Ne(t),is,calculated,from,quasineutrality,,Ne(t),(cid:90),L,0,exp,(cid:19),(cid:18),eφ(z,,t),Te(t),dz,=,(cid:90),L,0,ni(z,,t),dz.,(5.31),The,value,of,Te(t),is,obtained,from,the,electron,energy,equation.,Integrating,the,energy,equation,(5.11),over,z,,we,ﬁnd,3,2,d,dt,+,(cid:34),NeTe,(cid:90),L,0,exp,(cid:19),(cid:18),eφ,Te,(cid:35),dz,=,−,(cid:90),L,(cid:18),0,ene,∂φ,∂z,−,nennRionEion,+,(cid:20),5,2,(cid:90),1,2,(cid:21)z=L,neue(cid:107)Te,+,qe(cid:107),(cid:19),z=0,dz.,mev2Se,d3v,(5.32),It,might,seem,that,the,heat,ﬂux,should,vanish,here,because,the,distribution,function,is,a,Maxwellian,to,lowest,order.,In,reality,,the,real,Fe,is,suﬃciently,far,from,a,Maxwellian,at,large,velocities,to,induce,a,signiﬁcant,heat,ﬂux.,Indeed,,to,satisfy,boundary,condi-,tions,(3.1),and,(3.4),,the,distribution,function,must,vanish,exactly,in,certain,regions,of,phase,space.,Since,these,regions,are,at,large,velocities,,the,techniques,used,to,calculate,the,collisional,losses,into,the,loss,cones,of,mirror,machines,can,be,used,(Pastukhov,1974).,6.,Discussion,The,model,that,we,propose,is,comprised,of:,•,the,three,ﬂuid,equations,(4.10),,(4.11),and,(4.12),for,ions,that,have,to,be,solved,in,conjunction,with,the,ion,kinetic,equation,(4.19);,•,the,three,ﬂuid,equations,(4.25),,(4.26),and,(4.27),for,neutrals,that,have,to,be,solved,in,conjunction,with,the,neutral,kinetic,equation,(4.29);,•,the,two,ﬂuid,equations,(5.8),and,(5.11),for,electrons,that,have,to,be,solved,in,conjunction,with,the,electron,kinetic,equation,(5.2),with,the,modiﬁed,coeﬃcients,(5.3),,(5.4),,(5.5),and,(5.6);,and,•,the,electron,parallel,momentum,equation,(5.9),for,the,potential.,The,boundary,conditions,for,this,system,of,equations,are,described,in,section,3.,To,test,the,model,proposed,in,this,report,,we,will,ﬁrst,extend,the,existing,code,based,on,adiabatic,electrons,,which,we,prove,to,a,be,a,good,approximation,for,collisionless,plasmas,in,section,5.4,,to,wall,boundary,conditions.,We,will,then,explore,the,eﬀect,of,adding,electrons.,For,most,physics,of,interest,,it,is,suﬃcient,to,use,simpliﬁed,ion-ion,and,electron-electron,collision,operators,,and,for,this,reason,we,do,not,expect,to,implement,a,full,Fokker-Planck,collision,operator.,Appendix,A.,Modiﬁed,collision,operators,for,the,ion,kinetic,equation,The,modiﬁed,Fokker-Planck,like-particle,collision,operator,is,(cid:40),:=,Css[Fs,ns,,vts](z,,w(cid:107),,w⊥,,t),2πe4ns,ln,Λ,sv3,(4π(cid:15)0)2m2,ts,(cid:20),w⊥,∂,∂w(cid:107),+,(cid:18),1,w⊥,∂,∂w⊥,(cid:18),D(cid:107)(cid:107)[Fs],∂Fs,∂w(cid:107),+,D(cid:107)⊥[Fs],(cid:19),+,P(cid:107)[Fs]Fs,∂Fs,∂w⊥,D(cid:107)⊥[Fs],∂Fs,∂w(cid:107),+,D⊥⊥[Fs],∂Fs,∂w⊥,(cid:19)(cid:21),(cid:41),+,P⊥[Fs]Fs,.,(A,1),16,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,The,coeﬃcients,needed,for,this,collision,operator,are,D(cid:107)(cid:107)[Fs](z,,w(cid:107),,w⊥,,t),:=,4,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),w(cid:48),⊥,(cid:32),×,K(κ),−,(w(cid:107),−,w(cid:48),(cid:107))2E(κ),(cid:107))2,+,(w⊥,−,w(cid:48),⊥)2,(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,+,w(cid:48),(w(cid:107),−,w(cid:48),(cid:33),⊥)2,Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),,D(cid:107)⊥[Fs](z,,w(cid:107),,w⊥,,t),:=,2,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),w⊥,(cid:32),[(w(cid:107),−,w(cid:48),(w(cid:107),−,w(cid:48),×,⊥,+,w(cid:48)2,(cid:107))2,−,w2,(cid:107))2,+,(w⊥,−,w(cid:48),⊥]E(κ),⊥)2,−,K(κ),w(cid:48),⊥(w(cid:107),−,w(cid:48),(cid:107)),(cid:107))2,+,(w⊥,+,w(cid:48),(w(cid:107),−,w(cid:48),(cid:33),⊥)2,Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),,D⊥⊥[Fs](z,,w(cid:107),,w⊥,,t),:=,2,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,dw(cid:48),⊥,(cid:113),w2,⊥,w(cid:48),⊥,(cid:107))2,+,(w⊥,+,w(cid:48),⊥)2,(w(cid:107),−,w(cid:48),0,(cid:34),(cid:40),×,2w⊥,w⊥(w(cid:107),−,w(cid:48),(cid:107))2,(cid:107))2,+,(w⊥,−,w⊥)2,−,w(cid:48),⊥,(w(cid:107),−,w(cid:48),(cid:35),E(κ),+[(w(cid:107),−,w(cid:48),(cid:107))2,+,w2,⊥,+,w(cid:48)2,⊥][K(κ),−,E(κ)],Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),,(cid:41),P(cid:107)[Fs](z,,w(cid:107),,w⊥,,t),:=,8,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),(w(cid:107),−,w(cid:48),w(cid:48),⊥(w(cid:107),−,w(cid:48),(cid:107)),(cid:107))2,+,(w⊥,+,w(cid:48),(cid:33),E(κ),⊥)2,K(κ),−,E(κ),(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,+,w(cid:48),(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,−,w(cid:48),⊥)2,⊥)2,−,Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),(cid:32),×,and,P⊥[Fs](z,,w(cid:107),,w⊥,,t),:=,4,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),w⊥,(cid:32),[(w(cid:107),−,w(cid:48),(w(cid:107),−,w(cid:48),×,(cid:107))2,−,w2,⊥,+,w(cid:48)2,(cid:107))2,+,(w⊥,−,w(cid:48),⊥]E(κ),⊥)2,−,K(κ),w(cid:48),⊥,(cid:107))2,+,(w⊥,+,w(cid:48),(w(cid:107),−,w(cid:48),(cid:33),⊥)2,Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t).,Here,,K(κ),:=,(cid:82),π/2,elliptic,integrals,,and,the,function,κ,is,0,(1,−,κ2,sin2,α)−1/2,dα,and,E(κ),:=,(cid:82),π/2,0,(1,−,κ2,sin2,α)1/2,dα,are,the,κ(w(cid:107),,w⊥,,w(cid:48),(cid:107),,w(cid:48),⊥),:=,(cid:115),4w⊥w(cid:48),⊥,(cid:107))2,+,(w⊥,+,w(cid:48),⊥)2,.,(w(cid:107),−,w(cid:48),(A,7),(A,2),(A,3),(A,4),(A,5),(A,6),1D,drift,kinetic,models,with,wall,boundary,conditions,17,The,modiﬁed,charge,exchange,collision,operator,for,the,ion,kinetic,equation,is,Cin[Fi,,Fn,,nn,,ui(cid:107),,un(cid:107),,vti,,vtn](z,,w(cid:107),,w⊥,,t),:=,−,nnRin,(cid:20),Fi,−,(cid:18),Fn,z,,v3,ti,v3,tn,+,nnRin,+,nnRin,w⊥,∂,∂w(cid:107),∂,∂w⊥,(cid:20)(cid:18),un(cid:107),−,ui(cid:107),vti,(cid:18),v2,tn,v2,ti,(cid:20),w2,⊥,2,ui(cid:107),−,un(cid:107),vtn,w(cid:107),2,+,(cid:18),v2,tn,v2,ti,+,vti,vtn,w(cid:107),,−,1,+,−,1,+,2(un(cid:107),−,ui(cid:107))2,3v2,ti,(cid:19)(cid:21),w⊥,,t,vti,vtn,2(un(cid:107),−,ui(cid:107))2,3v2,ti,(cid:21),(cid:19),(cid:19)(cid:19),(cid:21),Fi,Fi,.,(A,8),Finally,,the,modiﬁed,ionization,collision,operator,for,the,ion,kinetic,equation,is,Ci,ion[Fn,,ne,,ui(cid:107),,un(cid:107),,vti,,vtn](z,,w(cid:107),,w⊥,,t),:=,−,neRion,(cid:20),Fi,−,(cid:18),Fn,z,,v3,ti,v3,tn,+,neRion,+,neRion,w⊥,∂,∂w(cid:107),∂,∂w⊥,(cid:20)(cid:18),un(cid:107),−,ui(cid:107),vti,(cid:18),v2,tn,v2,ti,(cid:20),w2,⊥,2,ui(cid:107),−,un(cid:107),vtn,w(cid:107),2,+,(cid:18),v2,tn,v2,ti,+,vti,vtn,w(cid:107),,−,1,+,−,1,+,2(un(cid:107),−,ui(cid:107))2,3v2,ti,(cid:19)(cid:21),w⊥,,t,vti,vtn,2(un(cid:107),−,ui(cid:107))2,3v2,ti,(cid:21),(cid:19),(cid:19)(cid:19),(cid:21),Fi,Fi,.,(A,9),Appendix,B.,Modiﬁed,collision,operators,for,the,neutral,kinetic,equation,The,modiﬁed,charge,exchange,collision,operator,for,the,neutral,kinetic,equation,is,Cni[Fn,,Fi,,ni,,un(cid:107),,ui(cid:107),,vtn,,vti](z,,w(cid:107),,w⊥,,t),:=,−,niRin,(cid:20),Fn,−,(cid:18),Fi,z,,v3,tn,v3,ti,+,niRin,+,niRin,w⊥,∂,∂w(cid:107),∂,∂w⊥,(cid:20)(cid:18),ui(cid:107),−,un(cid:107),vtn,(cid:18),v2,ti,v2,tn,(cid:20),w2,⊥,2,un(cid:107),−,ui(cid:107),vti,w(cid:107),2,+,(cid:18),v2,ti,v2,tn,+,vtn,vti,w(cid:107),,−,1,+,−,1,+,2(un(cid:107),−,ui(cid:107))2,3v2,tn,(cid:19)(cid:21),w⊥,,t,vtn,vti,2(un(cid:107),−,ui(cid:107))2,3v2,tn,(cid:21),(cid:19),(cid:19)(cid:19),(cid:21),Fn,Fn,.,(B,1),Appendix,C.,Modiﬁed,collision,operators,for,the,electron,kinetic,equation,The,electron-electron,collision,operator,is,described,in,equation,(A,1).,The,modiﬁed,ion-electron,collision,operator,is,Cei[Fe,,ni,,ui(cid:107),,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,(cid:40),2πe4ni,ln,Λ,ev3,(4π(cid:15)0)2m2,te,(cid:20),w⊥,+,∂,∂w⊥,1,w⊥,∂,∂w(cid:107),(cid:20),M(cid:107)(cid:107),∂Fe,∂w(cid:107),+,M(cid:107)⊥,(cid:18),+,1,+,∂Fe,∂w⊥,2(ui(cid:107),−,ue(cid:107))w(cid:107),3vte,(cid:19),(cid:21),F(cid:107)Fe,(cid:18),M(cid:107)⊥,∂Fe,∂w(cid:107),+,M⊥⊥,∂Fe,∂w⊥,+,2(ui(cid:107),−,ue(cid:107))w⊥,3vte,(cid:19)(cid:21),(cid:41),,,F(cid:107)Fe,(C,1),18,where,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,M(cid:107)(cid:107)[ue(cid:107),,vte,,ui(cid:107)](z,,w(cid:107),,w⊥,,t),:=,w2,⊥,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,,,(C,2),M(cid:107)⊥[ue(cid:107),,vte,,ui(cid:107)](z,,w(cid:107),,w⊥,,t),:=,−,(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2w⊥,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,,,(C,3),M⊥⊥[ue(cid:107),,vte,,ui(cid:107)](z,,w(cid:107),,w⊥,,t),:=,(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,(C,4),and,F(cid:107)[Fe,,ue(cid:107),,vte,,ui(cid:107)](z,,t),:=,−4π,(cid:90),∞,(cid:90),∞,dw(cid:107),dw⊥,−∞,0,w⊥[w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte]Fe(z,,w(cid:107),,w⊥,,t),[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,.,(C,5),The,modiﬁed,electron-neutral,collision,operator,is,Cen[Fe,,nn,,un(cid:107),,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,−nnRen,Fe,−,(cid:20),1,2,+,nnRen(un(cid:107),−,ue(cid:107)),vte,∂,∂w(cid:107),(cid:90),π,0,(cid:20)(cid:18),sin,χFe(z,,w(cid:107),,w⊥,,t),dχ,1,+,2(un(cid:107),−,ue(cid:107))w(cid:107),3vte,(cid:19),Fe,(cid:21),(cid:21),+,2nnRen(un(cid:107),−,ue(cid:107))2,tew⊥,3v2,∂,∂w⊥,(cid:0)w2,⊥Fe,(cid:1),,,(C,6),where,w(cid:107)[ue(cid:107),,vte,,un(cid:107)](χ,,z,,w(cid:107),,w⊥,,t),:=,un(cid:107),−,ue(cid:107),vte,(cid:115)(cid:18),+,cos,χ,w(cid:107),−,(cid:19)2,un(cid:107),−,ue(cid:107),vte,+,w2,⊥,(C,7),and,w⊥[ue(cid:107),,vte,,un(cid:107)](χ,,z,,w(cid:107),,w⊥,,t),:=,sin,χ,w(cid:107),−,(cid:115)(cid:18),(cid:19)2,un(cid:107),−,ue(cid:107),vte,+,w2,⊥.,(C,8),Appendix,D.,Linearized,collision,operators,for,electrons,The,linearized,electron-electron,collision,operator,is,given,by,C((cid:96)),ee,[Fe1,,ne,,vte](z,,w(cid:107),,w⊥,,t),2πe4ne,ln,Λ,ev3,(4π(cid:15)0)2m2,te,∂,∂w(cid:107),:=,(cid:40),(cid:32),D(cid:107)(cid:107)[FM,],∂Fe1,∂w(cid:107),+,D(cid:107)⊥[FM,],+,P(cid:107)[FM,]Fe1,∂Fe1,∂w⊥,(cid:33),−,2w(cid:107)D(cid:107)(cid:107)[Fe1]FM,−,2w⊥D(cid:107)⊥[Fe1]FM,+,P(cid:107)[Fe1]FM,+,1,w⊥,∂,∂w⊥,(cid:34),(cid:32),w⊥,D(cid:107)⊥[FM,],∂Fe1,∂w(cid:107),+,D⊥⊥[FM,],∂Fe1,∂w⊥,+,P⊥[FM,]Fe1,−,2w(cid:107)D(cid:107)⊥[Fe1]FM,−,2w⊥D⊥⊥[Fe1]FM,+,P⊥[Fe1]FM,(cid:33)(cid:35)(cid:41),.,(D,1),1D,drift,kinetic,models,with,wall,boundary,conditions,19,The,coeﬃcients,are,deﬁned,in,Appendix,A.,The,linearized,electron-ion,collision,operator,is,C((cid:96)),ei,[Fe1,,ni,,vte](z,,w(cid:107),,w⊥,,t),(cid:40),2πe4ni,ln,Λ,ev3,(4π(cid:15)0)2m2,te,(cid:34),(cid:32),:=,+,∂,∂w(cid:107),(cid:34),w2,⊥,(cid:107),+,w2,⊥)3/2,(w2,∂Fe1,∂w(cid:107),−,w(cid:107)w⊥,(cid:107),+,w2,(w2,⊥)3/2,1,w⊥,∂,∂w⊥,w⊥,−,w(cid:107)w⊥,(cid:107),+,w2,(w2,⊥)3/2,∂Fe1,∂w(cid:107),+,w2,(cid:107),(cid:107),+,w2,(w2,⊥)3/2,∂Fe1,∂w⊥,(cid:35),∂Fe1,∂w⊥,(cid:33)(cid:35),(cid:41),.,(D,2),Finally,,the,linearized,electron-neutral,collision,operator,is,C((cid:96)),en,[Fe1,,nn,,vte](z,,w(cid:107),,w⊥,,t),:=,−nnRen,Fe1,(cid:34),−,1,2,(cid:90),π,0,sin,χFe1,(cid:16),z,,cos,χ,(cid:113),w2,(cid:107),+,w2,⊥,,sin,χ,(cid:113),w2,(cid:107),+,w2,(cid:17),⊥,,t,(cid:35),dχ,.,(D,3),REFERENCES,Braginskii,,S.I.,1958,Transport,phenomena,in,a,completely,ionized,two-temperature,plasma.,Sov.,Phys.,JETP,6,,358.,Catto,,P.J.,1994,A,short,mean-free,path,,coupled,neutral-ion,transport,description,of,a,toka-,mak,edge,plasma.,Phys.,Plasmas,1,,1936.,Chodura,,R.,1982,Plasma-wall,transition,in,an,oblique,magnetic,ﬁeld.,Phys.,Fluids,25,,1628.,Connor,,J.W.,1977,An,analytic,solution,for,the,distribution,function,of,neutral,particles,in,a,Maxwellian,plasma,using,the,method,of,singular,eigenfunctions.,Plasma,Phys.,19,,853.,Geraldini,,A.,2021,Large,gyro-orbit,model,of,ion,velocity,distribution,in,plasma,near,a,wall,in,a,grazing-angle,magnetic,ﬁeld.,J.,Plasma,Phys.,87,,905870113.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2017,Gyrokinetic,treatment,of,a,grazing,angle,magnetic,presheath.,Plasma,Phys.,Control.,Fusion,59,,025015.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2018,Solution,to,a,collisionless,shallow-angle,magnetic,presheath,with,kinetic,ions.,Plasma,Phys.,Control.,Fusion,60,,125002.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2019,Dependence,on,ion,temperature,of,shallow-,angle,magnetic,presheaths,with,adiabatic,electrons.,J.,Plasma,Phys.,85,,795850601.,Hazeltine,,R.D.,1973,Recursive,derivation,of,drift-kinetic,equation.,Plasma,Phys.,15,,77–80.,Hazeltine,,R.D.,,Calvin,,M.D.,,Valanju,,P.M.,&,Solano,,E.R.,1992,Analytical,calculation,of,neutral,transport,and,its,eﬀect,on,ions.,Nucl.,Fusion,32,,3.,Helander,,P.,,Krasheninnikov,,S.I.,&,Catto,,P.J.,1994,Fluid,equations,for,a,partially,ionized,plasma.,Phys.,Plasmas,1,,3174.,Knudsen,,M.,1916,Das,Cosinusgesetz,in,der,kinetischen,Gastheorie.,Annal.,Phys.,353,,1113.,Parker,,S.E.,,Procassini,,R.J.,&,Birdsall,,C.K.,1993,A,Suitable,Boundary,Condition,for,Bounded,Plasma,Simulation,without,Sheath,Resolution.,J.,Comput.,Phys.,104,,41.,Pastukhov,,V.P.,1974,Collisional,losses,of,electrons,from,an,adiabatic,trap,in,a,plasma,with,a,positive,potential.,Nucl.,Fusion,14,,3.,Rosenbluth,,M.N.,,MacDonald,,W.M.,&,Judd,,D.L.,1957,Fokker-Planck,Equation,for,an,Inverse-Square,Force.,Phys.,Rev.,107,,1.,REPORT,2047357-TN-07-01,M1.4,1,2D,drift,kinetic,model,with,wall,boundary,conditions,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,31,July,2021),1.,Introduction,In,previous,reports,,we,proposed,1D,drift,kinetic,equations,with,periodic,boundary,conditions,,adequate,for,the,closed,ﬁeld,line,region,of,the,edge,,and,wall,boundary,con-,ditions.,In,this,report,,we,build,a,2D,drift,kinetic,model,for,a,helical,magnetic,ﬁeld.,The,helical,magnetic,ﬁeld,has,similarities,with,the,magnetic,ﬁeld,in,the,tokamak,edge.,2.,Helical,magnetic,ﬁeld,To,describe,the,geometry,of,the,magnetic,ﬁeld,,we,use,the,cylindrical,coordinates,{r,,z,,ζ},(see,ﬁgure,1).,In,this,coordinates,,the,helical,ﬁeld,is,B(r,,ζ),:=,Bz(r)ˆz,+,Bζ(r)ˆζ(ζ),,where,ˆz,and,ˆζ,are,the,unit,vectors,in,the,direction,of,∇z,and,∇ζ.,Note,that,the,compo-,nents,Bz,and,Bζ,only,depend,on,the,radial,position,r.,(2.1),In,principle,,one,can,use,any,Bz(r),and,Bζ(r).,There,is,a,particular,choice,that,is,more,physical.,In,the,edge,,the,magnetic,ﬁeld,is,determined,by,currents,running,through,the,core,plasma,or,through,external,magnets.,Thus,,according,to,Amp´ere’s,law,,the,magnetic,ﬁeld,in,the,edge,should,satisfy,∇,×,B,(cid:39),0.,This,condition,imposes,that,Bz,be,a,constant,and,that,Bζ,decay,as,1/r,,where,I,is,a,constant,determined,by,the,vertical,current,through,the,core,plasma.,Bζ(r),=,I,r,,,(2.2),3.,Geometry,and,orderings,We,consider,a,magnetized,plasma,with,one,ion,species,with,charge,e,and,mass,mi,,electrons,with,charge,−e,and,mass,me,,and,one,species,of,neutrals,with,mass,mn,=,mi.,(3.1),The,plasma,is,magnetized,by,a,helical,magnetic,ﬁeld,like,the,one,described,in,the,previous,section,,and,we,assume,that,the,plasma,only,varies,in,r,and,z.,We,assume,that,the,electric,ﬁeld,produced,by,the,plasma,is,electrostatic,,E,=,−(∂φ/∂r)ˆr,−,(∂φ/∂z)ˆz,,where,ˆr,is,the,unit,vector,in,the,direction,∇r.,The,potential,φ(r,,z,,t),depends,on,the,coordinates,r,and,z,and,on,time,t.,There,are,two,conducting,walls,at,z,=,0,and,z,=,Lz.,In,the,radial,direction,,we,consider,the,interval,between,r,=,r0,and,r,=,r0,+,Lr.,The,length,Lr,is,determined,by,a,balance,between,the,fast,parallel,velocity,of,the,particles,along,magnetic,ﬁeld,lines,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Figure,1.,Two,magnetic,ﬁeld,lines,(in,blue,and,red),of,the,helical,magnetic,ﬁeld.,Note,that,the,direction,of,the,angle,ζ,is,opposite,to,the,direction,usually,chosen,in,cylindrical,coordinates.,and,their,slow,drift,across,them.,The,characteristic,length,between,the,two,walls,along,a,magnetic,ﬁeld,line,is,of,order,B,Bz,Thus,,the,typical,time,that,it,takes,an,ion,to,move,from,wall,to,wall,is,L(cid:107)/vti,∼,(B/Bz)(Lz/vti),,where,vti,:=,(cid:112)2Ti/mi,is,the,ion,thermal,speed,and,Ti,is,the,ion,tem-,perature.,For,a,potential,φ,of,the,order,of,Ti/e,,where,e,is,the,proton,charge,,the,radial,E,×,B,drift,is,L(cid:107),∼,(3.2),Lz.,vEr,:=,−,Bζ,B2,∂φ,∂z,∼,ρi,Lz,vti,,(3.3),:=,vti/Ωi,is,the,characteristic,ion,gyroradius,and,Ωi,where,ρi,:=,eB/mi,is,the,ion,gyrofrequency.,Thus,,the,time,it,takes,for,an,ion,to,cross,the,domain,in,the,radial,direction,is,Lr/vEr,∼,LrLz/ρivti.,By,making,this,time,of,the,same,order,as,L(cid:107)/vti,,we,solve,for,Lr,to,ﬁnd,To,simplify,the,problem,to,a,tractable,drift,kinetic,form,,we,assume,that,ρi,is,much,smaller,than,Lr.,This,implies,that,Lr,∼,B,Bz,ρi,(3.4),ρi,Lr,∼,Bz,B,∼,Bz,Bζ,(cid:28),1,,(3.5),that,is,,we,will,limit,our,model,to,magnetic,ﬁelds,that,are,mostly,azimuthal,and,have,a,very,small,vertical,component.,This,is,an,approximation,that,is,consistent,with,magnetic,ﬁeld,geometry,in,conventional,tokamaks,and,also,in,the,edge,of,many,shots,in,spherical,tokamaks.,We,also,assume,that,r0,∼,Lz,(cid:29),Lr.,Since,r0,is,the,characteristic,length,of,variation,of,the,magnetic,ﬁeld,B,,the,magnetic,ﬁeld,barely,changes,across,the,domain,[r0,,r0,+,Lr].,Thus,,within,our,ordering,,we,assume,B,to,be,uniform,in,the,domain,of,interest.,Our,orderings,above,rest,on,the,assumption,φ,∼,Ti/e.,This,ordering,is,a,result,of,the,wall,boundary,conditions,that,impose,φ,∼,Te/e,(see,section,5),and,the,fact,that,Ti,∼,Te,due,to,collisional,temperature,equilibration.,<latexit,sha1_base64="xYZka1EHgeBx11XDNEAJIBtNWIk=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGhiZnd3MzBrJhi/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXEAuujet+O7mV1bX1jfxmYWt7Z3evuH/Q0FGiGNZZJCLVCqhGwSXWDTcCW7FCGgYCm8Hoduo3H1FpHsl7M47RD+lA8j5n1Fip9tQtltyyOwNZJl5GSpCh2i1+dXoRS0KUhgmqddtzY+OnVBnOBE4KnURjTNmIDrBtqaQhaj+dHTohJ1bpkX6kbElDZurviZSGWo/DwHaG1Az1ojcV//Paielf+ymXcWJQsvmifiKIicj0a9LjCpkRY0soU9zeStiQKsqMzaZgQ/AWX14mjbOyd1k+r12UKjdZHHk4gmM4BQ+uoAJ3UIU6MEB4hld4cx6cF+fd+Zi35pxs5hD+wPn8AejVjQQ=</latexit>x<latexit,sha1_base64="fSFHYI9Zr8JNHToPEkr14C8WnIo=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomKeix68diC/YA2lM120q7dbMLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVx71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJRvyxXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MH6lmNBQ==</latexit>y<latexit,sha1_base64="xF2+1Dnriudo0aLfowFKMmKS2HQ=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGhiZnd3MzJrghi/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXEAuujet+O7mV1bX1jfxmYWt7Z3evuH/Q0FGiGNZZJCLVCqhGwSXWDTcCW7FCGgYCm8Hoduo3H1FpHsl7M47RD+lA8j5n1Fip9tQtltyyOwNZJl5GSpCh2i1+dXoRS0KUhgmqddtzY+OnVBnOBE4KnURjTNmIDrBtqaQhaj+dHTohJ1bpkX6kbElDZurviZSGWo/DwHaG1Az1ojcV//Paielf+ymXcWJQsvmifiKIicj0a9LjCpkRY0soU9zeStiQKsqMzaZgQ/AWX14mjbOyd1k+r12UKjdZHHk4gmM4BQ+uoAJ3UIU6MEB4hld4cx6cF+fd+Zi35pxs5hD+wPn8AevdjQY=</latexit>z<latexit,sha1_base64="wOxQZJSIQt2N9AsBQ1FB2xh7I4s=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomKeix68diC/YA2lM120q7dbMLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVVa9UdivuDGSZeDkpQ45ar/TV7ccsjVAaJqjWHc9NjJ9RZTgTOCl2U40JZSM6wI6lkkao/Wx26IScWqVPwljZkobM1N8TGY20HkeB7YyoGepFbyr+53VSE974GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7QheIsvL5PmecW7qlzUL8vV2zyOAhzDCZyBB9dQhXuoQQMYIDzDK7w5j86L8+58zFtXnHzmCP7A+fwB372M/g==</latexit>r<latexit,sha1_base64="DEY+8YYy4sCq49kJ630ZcrF0gEM=">AAAB7HicbVBNS8NAEN3Ur1q/qh69LBbBU0lU1GPRi8cKpi20oWy203bpZhN2J0IN/Q1ePCji1R/kzX/jts1BWx8MPN6bYWZemEhh0HW/ncLK6tr6RnGztLW9s7tX3j9omDjVHHwey1i3QmZACgU+CpTQSjSwKJTQDEe3U7/5CNqIWD3gOIEgYgMl+oIztJLfeQJk3XLFrboz0GXi5aRCctS75a9OL+ZpBAq5ZMa0PTfBIGMaBZcwKXVSAwnjIzaAtqWKRWCCbHbshJ5YpUf7sbalkM7U3xMZi4wZR6HtjBgOzaI3Ff/z2in2r4NMqCRFUHy+qJ9KijGdfk57QgNHObaEcS3srZQPmWYcbT4lG4K3+PIyaZxVvcvq+f1FpXaTx1EkR+SYnBKPXJEauSN14hNOBHkmr+TNUc6L8+58zFsLTj5zSP7A+fwB6xqOxA==</latexit>⇣,2D,drift,kinetic,model,with,wall,boundary,conditions,3,4.,2D,electrostatic,drift,kinetics,We,assume,that,the,gyroradii,are,small,compared,to,the,length,scales,of,interest,,and,that,the,gyrofrequencies,are,much,larger,than,the,frequencies,that,we,want,to,model.,Thus,,the,distribution,functions,fs(r,,z,,v(cid:107),,v⊥,,t),of,the,charged,species,s,=,i,,e,only,depend,on,the,component,of,the,velocity,parallel,to,the,magnetic,ﬁeld,v(cid:107),and,the,mag-,nitude,of,the,velocity,perpendicular,to,the,magnetic,ﬁeld,v⊥,,and,are,independent,of,the,direction,of,the,velocity,perpendicular,to,the,magnetic,ﬁeld,(Hazeltine,1973).,The,distribution,functions,of,ions,and,electrons,(s,=,i,,e),that,in,general,can,depend,on,three,spatial,variables,r,,three,components,of,the,velocity,v,and,the,time,t,depend,only,on,r,,z,,v(cid:107),,v⊥,and,t,,fs(r,,v,,t),=,fs(r,,z,,v(cid:107),,v⊥,,t).,The,neutral,distribution,function,depends,in,general,on,the,three,velocity,components,,(4.1),fn(r,,v,,t),=,fn(r,,z,,vr,,vz,,vζ,,t).,(4.2),We,remind,the,reader,that,the,model,is,2D,because,we,have,assumed,that,the,plasma,parameters,do,not,depend,on,the,angle,ζ.,The,equations,for,the,distribution,functions,of,the,diﬀerent,species,are,∂fi,∂t,−,(cid:18),v(cid:107)Bz,B,+,∂φ,∂z,∂fi,∂r,1,B,=,Cii[fi],+,(cid:104)Cin[fi,,fn](cid:105),+,(cid:104)Ci,ion[fe,,fn](cid:105),+,Cie[fi,,fe],+,Si,,eBz,miB,∂fi,∂v(cid:107),∂φ,∂z,∂φ,∂r,1,B,+,−,(cid:19),∂fi,∂z,(4.3),∂fe,∂t,−,1,B,∂φ,∂z,+,Cei[fe,,fi],+,∂fe,∂r,(cid:20),1,+,O,(cid:18),v(cid:107)Bz,B,(cid:18),me,mi,+,1,B,(cid:19)(cid:21),(cid:19),∂fe,∂z,∂φ,∂r,(cid:28),+,+,Cen[fe,,fn],∂φ,∂z,eBz,meB,(cid:20),1,+,O,∂fe,∂v(cid:107),(cid:18),me,mi,=,Cee[fe],(cid:19)(cid:21)(cid:29),+,(cid:104)Ce,ion[fe,,fn](cid:105),+,Se,(4.4),and,∂fn,∂t,+,vr,∂fn,∂r,+,vz,∂fn,∂z,=,Cni[fn,,fi],+,Cne[fn,,fe],+,Cn,ion[fn,,fe],+,Sn.,(4.5),The,triangular,brackets,on,a,function,G(r,,z,,vr,,vz,,vζ,,t),indicate,gyoraverage,,(cid:104)G(cid:105)(r,,z,,v(cid:107),,v⊥,,t),:=,1,2π,(cid:90),2π,0,G,(cid:0)r,,z,,v⊥,cos,ϕ,,v⊥,sin,ϕ,,v(cid:107),,t(cid:1),dϕ.,(4.6),The,sources,Ss(r,,z,,v,,t),with,s,=,i,,e,,n,represent,heating,,fueling,and,the,eﬀect,of,turbulence.,The,ion,and,electron,particle,sources,satisfy,(cid:90),(cid:90),Si,d3v,=,Se,d3v.,(4.7),Note,that,we,have,neglected,the,curvature,and,∇B,drifts,in,equations,(4.3),and,(4.4).,These,drifts,point,in,the,z-,and,ζ-direction.,The,ζ-direction,is,unimportant,because,it,is,a,direction,of,symmetry,,whereas,in,the,z-direction,,the,magnetic,drifts,can,be,neglected,compared,to,the,much,larger,terms,due,to,the,parallel,velocity,,v(cid:107)Bz/B,,and,the,z-component,of,the,E,×,B,drift,,vEz,(cid:39),B−1(∂φ/∂r).,We,have,included,the,following,collisions.,•,Ion-ion,and,electron-electron,collisions,are,modeled,by,the,Fokker-Planck,collision,4,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,operators,Cii[fi],and,Cee[fe],(Rosenbluth,et,al.,1957),,Css[fs],:=,2πe4,ln,Λ,(4π(cid:15)0)2m2,s,∇v,·,(D[fs],·,∇vfs,+,P[fs]fs),,,(4.8),where,the,matrix,D,is,D[fs],:=,(cid:90),|v,−,v(cid:48)|2I,−,(v,−,v(cid:48))(v,−,v(cid:48)),|v,−,v(cid:48)|3,fs(v(cid:48)),d3v(cid:48),and,the,vector,P,is,P[fs],:=,−2,(cid:90),v,−,v(cid:48),|v,−,v(cid:48)|3,fs(v(cid:48)),d3v(cid:48).,(4.9),(4.10),Here,,I,is,the,3D,unit,matrix,,(cid:15)0,the,vacuum,permittivity,and,ln,Λ,≈,15,the,Coulomb,logarithm.,•,The,eﬀect,of,electron-ion,and,elastic,electron-neutral,collisions,on,the,electron,distri-,bution,function,can,be,simpliﬁed,in,the,limit,of,small,electron-ion,mass,ratio,,me/mi,(cid:28),1.,With,this,expansion,,we,ﬁnd,the,simpliﬁed,Fokker-Planck,collision,operator,(cid:21),Cei[fe,,fi],:=,2πe4ni,ln,Λ,(4π(cid:15)0)2m2,e,∇v,·,(cid:20),|v,−,ui|2I,−,(v,−,ui)(v,−,ui),|v,−,ui|3,·,∇vfe,(4.11),for,electron-ion,collisions,(Braginskii,1958),,and,the,simpliﬁed,Boltzmann,collision,oper-,ator,Cen[fe,,fn],:=,nn,4π,(cid:90),π,dχ,(cid:90),2π,0,0,dϕ,sin,χRen(|v,−,un|,,χ),[fe(v(v,,χ,,ϕ,,un)),−,fe(v)],for,electron-neutral,collisions.,Here,(cid:90),ns(r,,z,,t),:=,fs(r,,z,,v,,t),d3v.,is,the,density,of,species,s,,us(r,,z,,t),:=,(cid:90),1,ns,vfs(r,,z,,v,,t),d3v,is,the,average,velocity,of,species,s,,(4.12),(4.13),(4.14),v(v,,χ,,ϕ,,un),:=,un,+,cos,χ(v,−,un),+,|v,−,un|,sin,χ(cos,ϕ,ˆe1,+,sin,ϕ,ˆe2),(4.15),is,a,rotation,of,the,vector,v,centered,around,un,,Ren(|v−un|,,χ),is,a,function,determined,by,the,physics,of,the,electron-neutral,collisions,,and,the,unit,vectors,ˆe1,and,ˆe2,are,chosen,to,form,an,orthonormal,basis,with,the,vector,(v,−,un)/|v,−,un|.,In,equation,(4.4),,we,have,indicated,that,both,Cei,and,Cen,are,missing,pieces,small,in,me/mi.,These,pieces,can,become,important,because,they,represent,collisional,energy,exchange,and,collisional,heating,,but,they,are,cumbersome.,We,showed,in,report,2047357-TN-05-01,M1.3,that,the,moment,method,that,we,use,allows,us,to,keep,these,important,eﬀects,in,the,moment,equations,even,with,the,simpliﬁed,collision,operators,(4.11),and,(4.12).,•,The,expansion,in,electron-ion,mass,ratio,also,implies,electron-ion,collisions,and,electron-neutral,collisions,have,a,very,small,eﬀect,on,fi,and,fn,–,the,terms,Cie,and,Cne,in,equations,(4.3),and,(4.5),are,small,compared,with,Cii,and,Cni,by,a,factor,of,(cid:112),me/mi,(cid:28),1,,Cie,∼,(cid:114),me,mi,Cii,,Cne,∼,(cid:114),me,mi,Cni.,(4.16),2D,drift,kinetic,model,with,wall,boundary,conditions,5,Like,the,mass,ratio,corrections,to,Cei,and,Cen,,these,terms,can,become,important,because,they,contain,the,collisional,energy,exchange,between,electrons,and,the,heavier,species.,We,will,keep,these,eﬀects,in,a,simpliﬁed,form,in,our,moment,formulation.,•,Charge-exchange,collisions,are,represented,by,the,simpliﬁed,Boltzmann,collision,operators,and,Cin[fi,,fn],:=,−,Cni[fn,,fi],:=,−,(cid:90),(cid:90),Rin(|v,−,v(cid:48)|),[fi(v)fn(v(cid:48)),−,fi(v(cid:48))fn(v)],d3v(cid:48),(4.17),Rin(|v,−,v(cid:48)|),[fn(v)fi(v(cid:48)),−,fn(v(cid:48))fi(v)],d3v(cid:48).,(4.18),•,To,model,ionization,,we,use,the,collision,operators,Ci,ion[fe,,fn],:=,fn,(cid:90),Rion(v(cid:48))fe(v(cid:48)),d3v(cid:48),and,Cn,ion[fe,,fn],:=,−fn,(cid:90),Rion(v(cid:48))fe(v(cid:48)),d3v(cid:48).,(4.19),(4.20),We,also,need,to,include,a,collision,operator,Ce,ion,in,the,electron,equation,to,model,the,increase,in,the,number,of,electrons,and,the,energy,loss,due,to,ionization.,This,operator,is,complicated,because,it,involves,three,particles,(the,resulting,ion,and,two,electrons),,but,we,will,be,able,to,avoid,giving,it,a,deﬁnite,form.,Instead,,we,will,use,the,expansion,in,me/mi,(cid:28),1,and,the,fact,that,Ce,ion[fe,,fn],∼,nnRionfe.,(4.21),See,report,2047357-TN-05-01,M1.3,for,more,details.,•,We,have,neglected,neutral-neutral,collisions,because,,in,current,fusion,devices,,the,neutral,density,is,suﬃciently,small,that,the,neutral-neutral,collisions,are,rare.,To,simplify,our,equations,,we,assume,that,the,functions,Ren,,Rin,and,Rion,are,constant,(Connor,1977;,Hazeltine,et,al.,1992;,Catto,1994),,ﬁnding,(cid:104)Cen[fe,,fn](cid:105),=,−,nnRen,fe(r,,z,,v(cid:107),,v⊥,,t),(cid:34),−,1,8π2,(cid:90),π,dχ,(cid:90),2π,dϕ,(cid:90),2π,0,0,0,dϕ(cid:48),sin,χfe,(cid:0)r,,z,,v(cid:107),,v⊥,,t(cid:1),(cid:35),,,v(cid:107)(v(cid:107),,v⊥,,χ,,ϕ(cid:48),,un),:=,un(cid:107),+,ven,cos,χ,,v⊥(v(cid:107),,v⊥,,χ,,ϕ,,ϕ(cid:48),,un),:=,(cid:113),n⊥,+,v2,u2,en,sin2,χ,−,2un⊥ven,sin,χ,cos,ϕ,ven(v(cid:107),,v⊥,,ϕ(cid:48),,un),:=,(cid:113),(v(cid:107),−,un(cid:107))2,+,v2,⊥,+,u2,n⊥,−,2v⊥un⊥,cos,ϕ(cid:48),,(cid:104)Cin[fi,,fn](cid:105),=,−Rin,(nnfi,−,ni(cid:104)fn(cid:105)),,,Cni[fn,,fi],=,−Rin,(nifn,−,nnfi),,,(cid:104)Ci,ion[fe,,fn](cid:105),=,(cid:104)fn(cid:105)neRion,Cn,ion[fe,,fn],=,−fnneRion.,with,and,and,(4.22),(4.23),(4.24),(4.25),(4.26),(4.27),(4.28),(4.29),6,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,The,potential,φ(r,,z,,t),is,determined,by,the,quasineutrality,equation,ni,=,ne.,(4.30),To,solve,this,equation,,we,need,to,treat,the,equations,implicitly,as,the,potential,enters,only,via,its,eﬀect,on,∂fi/∂t,and,∂fe/∂t.,The,need,to,use,implicit,methods,is,one,of,the,reasons,why,we,are,trying,to,extract,some,of,the,low,order,moments,from,the,distribution,function.,5.,Wall,boundary,conditions,We,impose,wall,boundary,conditions,at,z,=,0,and,z,=,Lz.,In,principle,,we,need,to,consider,the,eﬀect,of,the,magnetic,presheath,(Chodura,1982),because,the,magnetic,ﬁeld,is,not,perpendicular,to,the,wall.,However,,the,complicated,boundary,conditions,that,the,magnetic,presheath,imposes,on,drift,kinetic,models,are,an,active,area,of,research,(Geraldini,et,al.,2017,,2018,,2019;,Geraldini,2021).,To,avoid,this,complication,,we,assume,that,the,electron,gyroradius,is,much,smaller,than,the,Debye,length,,thus,ensuring,that,electrons,are,magnetized,even,within,the,thin,sheath,of,non-neutral,plasma,with,a,width,of,the,order,of,the,Debye,length,that,forms,on,walls,to,ensure,quasineutrality.,With,this,assumption,,we,can,impose,boundary,conditions,similar,to,those,proposed,by,Parker,et,al.,(1993).,The,boundary,conditions,that,we,propose,make,use,of,the,fact,that,the,potential,drop,across,the,magnetic,presheath,and,the,Debye,sheath,repels,electrons,away,from,the,wall,because,otherwise,electrons,would,ﬂow,to,the,wall,at,much,greater,rate,than,ions,due,to,their,lower,mass,and,higher,thermal,speed.,In,our,model,,φ(r,,0,,t),and,φ(r,,Lz,,t),are,not,the,potential,of,the,wall,,but,the,potential,at,the,entrance,of,the,magnetic,presheath.,In,this,report,,we,choose,the,potential,of,the,wall,at,z,=,0,to,be,0,without,loss,of,generality.,We,denote,the,potential,of,the,wall,at,z,=,Lz,as,φw.,Then,,for,the,magnetic,presheaths,and,Debye,sheaths,to,repel,electrons,,φ(r,,0,,t),must,be,larger,than,0,and,φ(r,,Lz,,t),must,be,larger,than,φw.,The,value,of,the,potential,at,z,=,0,and,z,=,Lz,is,determined,by,the,relationship,between,the,current,crossing,the,magnetic,presheath,and,the,Debye,sheath,and,the,total,potential,drop,across,these,layers.,We,consider,the,magnetic,presheath,and,the,Debye,sheath,at,z,=,Lz,ﬁrst,,and,we,will,then,apply,the,results,that,we,obtain,to,the,magnetic,presheath,and,the,Debye,sheath,at,z,=,0.,Ions,recombine,when,they,hit,the,wall,,so,no,ions,come,back.,The,velocity,of,the,ions,perpendicular,to,the,wall,is,a,combination,of,parallel,velocity,and,E,×,B,drift,,v(cid:107)Bz/B,+,B−1(∂φ/∂r).,Thus,,the,ions,that,would,come,back,from,the,wall,must,satisfy,v(cid:107),<,−B−1,z,(∂φ/∂r),,giving,fi(r,,Lz,,v(cid:107),<,−B−1,z,(∂φ/∂r),,v⊥,,t),=,0.,(5.1),Since,the,electrons,are,magnetized,,the,potential,drop,across,the,magnetic,presheath,and,the,Debye,sheath,only,modiﬁes,the,parallel,velocity,of,electrons.,Within,these,layers,,the,parallel,energy,E(cid:107),:=,mev2,(cid:107)/2,−,eφ,is,conserved,,and,as,a,result,an,electron,that,has,velocity,v(cid:107),at,the,entrance,of,the,sheath,is,slowed,down,to,a,parallel,velocity,[v2,(cid:107),−,2e(φ(r,,Lz,,t),−,φw)/me]1/2,when,it,reaches,the,wall.,Thus,,electrons,with,a,parallel,velocity,larger,than,(cid:112)2e(φ(r,,Lz,,t),−,φw)/me,reach,the,wall,,where,they,recombine,with,ions,,whereas,electrons,with,parallel,velocity,smaller,than,(cid:112)2e(φ(r,,Lz,,t),−,φw)/me,are,repelled,back,into,the,quasineutral,plasma.,Thus,,the,boundary,condition,on,the,electron,2D,drift,kinetic,model,with,wall,boundary,conditions,7,distribution,function,at,z,=,Lz,is,fe(r,,Lz,,v(cid:107),<,0,,v⊥,,t),=,(cid:26),fe(r,,Lz,,−v(cid:107),,v⊥,,t),0,for,v(cid:107),(cid:62),−(cid:112)2e(φ(r,,Lz,,t),−,φw)/me,,for,v(cid:107),<,−(cid:112)2e(φ(r,,Lz,,t),−,φw)/me.,(5.2),Here,we,can,neglect,the,E,×,B,drift,because,it,is,small,compared,to,the,typical,electron,thermal,speed,by,a,factor,of,(cid:112),me/mi,(cid:28),1.,Expressions,(5.1),and,(5.2),give,the,parallel,ion,and,electron,current,density,towards,the,wall,at,the,entrance,of,the,sheath,at,z,=,Lz,,Ji(cid:107)(r,,Lz,,t),=,2πe,(cid:90),∞,−B−1,z,(∂φ/∂r),dv(cid:107),(cid:90),∞,0,dv⊥,v⊥v(cid:107)fi(r,,Lz,,v(cid:107),,v⊥,,t),(5.3),and,Je(cid:107)(r,,Lz,,t),=,−2πe,(cid:90),∞,√,2e(φ(r,Lz,t)−φw)/me,dv(cid:107),(cid:90),∞,0,dv⊥,v⊥v(cid:107)fe(r,,Lz,,v(cid:107),,v⊥,,t).,(5.4),Hence,,the,total,parallel,current,density,at,z,=,Lz,is,the,following,function,of,φ(r,,Lz,,t)−,φw,and,the,radial,derivative,of,φ(r,,Lz,,t),−,φw,,J(cid:107)(r,,Lz,,t),=,2πe,(cid:90),∞,−B−1,z,(∂φ/∂r),−2πe,(cid:90),∞,√,2e(φ(r,Lz,t)−φw)/me,dv(cid:107),(cid:90),∞,dv(cid:107),dv⊥,v⊥v(cid:107)fi(r,,Lz,,v(cid:107),,v⊥,,t),0,(cid:90),∞,0,dv⊥,v⊥v(cid:107)fe(r,,Lz,,v(cid:107),,v⊥,,t).,(5.5),We,assume,that,the,potential,φw,does,not,depend,on,r,and,hence,the,radial,derivative,of,φ(r,,Lz,,t),−,φw,is,simply,the,radial,derivative,of,φ(r,,Lz,,t).,The,conditions,at,z,=,0,for,the,ion,and,electron,distribution,functions,and,the,potential,are,similar,to,those,for,z,=,Lz.,For,the,ion,and,electron,distribution,functions,,we,ﬁnd,fi(r,,0,,v(cid:107),>,−B−1,z,(∂φ/∂r),,v⊥,,t),=,0,and,fe(r,,0,,v(cid:107),>,0,,v⊥,,t),=,(cid:26),fe(r,,0,,−v(cid:107),,v⊥,,t),0,for,v(cid:107),(cid:54),(cid:112)2eφ(r,,0,,t)/me,,(cid:112)2eφ(r,,0,,t)/me,,for,v(cid:107),>,(5.6),(5.7),The,relationship,between,the,parallel,current,and,the,potential,at,the,magnetic,presheath,entrance,is,J(cid:107)(r,,0,,t),=,−2πe,(cid:90),−B−1,z,(∂φ/∂r),(cid:90),∞,dv(cid:107),dv⊥,v⊥v(cid:107)fi(r,,0,,v(cid:107),,v⊥,,t),−∞,√,(cid:90),−,2eφ(r,0,t)/me,−∞,0,(cid:90),∞,0,dv(cid:107),dv⊥,v⊥v(cid:107)fe(r,,0,,v(cid:107),,v⊥,,t).,(5.8),+2πe,We,still,need,boundary,conditions,for,the,neutral,distribution,function.,The,neutrals,hit,the,wall,and,thermalize,at,the,temperature,of,the,wall,Tw,,while,also,receiving,back,the,ions,that,have,recombined,at,the,wall,,that,is,,fn(r,,0,,vr,,vz,>,0,,vζ,,t),=,Γ0fKw,(cid:113),(cid:16),vz,,r,+,v2,v2,ζ,(cid:17),and,fn(r,,Lz,,vr,,vz,<,0,,vζ,,t),=,ΓLfKw,(cid:16),vz,,(cid:113),r,+,v2,v2,ζ,(cid:17),,,(5.9),(5.10),8,where,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,fKw(vn,,vt),:=,(cid:19)2,3,4π,(cid:18),mi,Tw,|vn|,n,+,v2,v2,t,(cid:112),(cid:18),exp,−,(cid:19),mi(v2,n,+,v2,t,),2Tw,(5.11),is,the,Knudsen,cosine,distribution,(Knudsen,1916),,and,Γ0,:=,2π,(cid:90),−B−1,z,(∂φ/∂r),(cid:90),∞,dv(cid:107),dv⊥,v⊥,−∞,(cid:90),∞,+,0,dvr,(cid:90),0,dvz,(cid:12),v(cid:107)Bz,(cid:12),(cid:12),B,(cid:12),(cid:90),∞,+,1,B,∂φ,∂r,(cid:12),(cid:12),(cid:12),(cid:12),fi(r,,0,,v(cid:107),,v⊥,,t),dvζ,|vz|fn(r,,0,,vr,,vz,,vζ,,t),(5.12),and,ΓL,:=,2π,(cid:90),∞,−B−1,z,(∂φ/∂r),−∞,−∞,−∞,(cid:90),∞,dv(cid:107),dv⊥,v⊥,0,(cid:90),∞,+,(cid:90),∞,dvr,(cid:18),v(cid:107)Bz,B,(cid:90),∞,dvz,(cid:19),+,1,B,∂φ,∂r,fi(r,,Lz,,v(cid:107),,v⊥,,t),dvζ,vzfn(r,,Lz,,vr,,vz,,vζ,,t),(5.13),are,the,ﬂuxes,of,neutrals,and,ions,towards,the,walls,at,z,=,0,and,z,=,Lz.,−∞,0,−∞,6.,2D,moment,drift,kinetics,Instead,of,solving,for,fs(r,,z,,v(cid:107),,v⊥,,t),with,s,=,i,,e,,we,solve,for,Fs(r,,z,,w(cid:107),,w⊥,,t),:=,v3,ts(r,,z,,t),ns(r,,z,,t),(cid:16),fs,r,,z,,us(cid:107)(r,,z,,t),+,vts(r,,z,,t)w(cid:107),,vts(r,,z,,t)w⊥,,t,(cid:17),where,we,have,deﬁned,the,normalized,velocities,w(cid:107)(r,,z,,v(cid:107),,t),:=,v(cid:107),−,us(cid:107)(r,,z,,t),vts(r,,z,,t),w⊥(r,,z,,v⊥,,t),:=,v⊥,vts(r,,z,,t),,,and,the,average,parallel,velocity,,,(6.1),(6.2),(6.3),us(cid:107)(r,,z,,t),:=,2π,ns,(cid:90),∞,(cid:90),∞,dv(cid:107),−∞,0,dv⊥,v⊥v(cid:107)fs(r,,z,,v(cid:107),,v⊥,,t),(6.4),and,the,thermal,speed,with,vts(r,,z,,t),:=,(cid:115),2Ts(r,,z,,t),ms,,,(6.5),Ts(r,,z,,t),:=,2π,ns,(cid:90),∞,(cid:90),∞,dv(cid:107),−∞,0,dv⊥,v⊥,ms[(v(cid:107),−,us(cid:107)(r,,z,,t))2,+,v2,⊥],3,fs(r,,z,,v(cid:107),,v⊥,,t),(6.6),the,temperature,of,species,s.,According,to,its,deﬁnition,,Fs(r,,z,,w(cid:107),,w⊥,,t),must,satisfy,the,conditions,(cid:90),∞,(cid:90),∞,2π,dw(cid:107),dw⊥,w⊥Fs(r,,z,,w(cid:107),,w⊥,,t),=,1,,−∞,(cid:90),∞,2π,dw(cid:107),0,(cid:90),∞,−∞,0,dw⊥,w⊥w(cid:107)Fs(r,,z,,w(cid:107),,w⊥,,t),=,0,(6.7),(6.8),2D,drift,kinetic,model,with,wall,boundary,conditions,and,(cid:90),∞,2π,dw(cid:107),(cid:90),∞,−∞,0,dw⊥,w⊥(w2,(cid:107),+,w2,⊥)Fs(r,,z,,w(cid:107),,w⊥,,t),=,9,(6.9),3,2,at,every,point,(r,,z),and,time,t.,Similarly,,for,neutrals,,we,solve,for,,,t),:=,Fn(r,,z,,wr,,wz,,wζ,(cid:125),(cid:124),(cid:123)(cid:122),=w,v3,tn(r,,z,,t),nn(r,,z,,t),(cid:16),fn,r,,z,,un(r,,z,,t),+,vtn(r,,z,,t)w,,t,(cid:17),,,(6.10),where,we,have,deﬁned,the,neutral,temperature,Tn(r,,z,,t),:=,1,nn,(cid:90),mi|v,−,un(r,,z,,t)|2,3,fn(r,,z,,v,,t),d3v.,(6.11),According,to,its,deﬁnition,,Fn(r,,z,,w,,t),must,satisfy,the,conditions,and,(cid:90),(cid:90),(cid:90),Fn(r,,z,,w,,t),d3w,=,1,,wFn(r,,z,,w,,t),d3w,=,0,w2Fn(r,,z,,w,,t),d3w,=,3,2,(6.12),(6.13),(6.14),at,every,point,(r,,z),and,time,t.,6.1.,Ion,equations,The,equations,for,ni,,ui(cid:107),and,Ti,are,(cid:18),ni,B,∂ni,∂t,∂φ,∂z,∂,∂z,∂,∂r,ni,+,−,(cid:19),(cid:20),(cid:18),ui(cid:107)Bz,B,(cid:19)(cid:21),+,1,B,∂φ,∂r,=,nnneRion,+,(cid:90),Si,d3v,,(6.15),nimi,(cid:20),∂ui(cid:107),∂t,−,1,B,∂φ,∂z,∂ui(cid:107),∂r,+,(cid:18),ui(cid:107)Bz,B,+,1,B,∂φ,∂r,(cid:19),∂ui(cid:107),∂z,(cid:21),=,−,Bz,B,∂pi(cid:107),∂z,−,eniBz,B,∂φ,∂z,+nimi(nnRin,+,neRion)(un(cid:107),−,ui(cid:107)),+,(cid:90),mi(v(cid:107),−,ui(cid:107))Si,d3v,(6.16),and,3,2,ni,Here,,−,1,B,(cid:20),∂Ti,∂t,3,2,3nemeνei,mi,+,+,∂φ,∂z,∂Ti,∂r,+,(cid:18),ui(cid:107)Bz,B,+,1,B,(cid:19),∂Ti,∂z,(cid:21),=,−,Bz,B,∂qi(cid:107),∂z,−,pi(cid:107)Bz,B,∂ui(cid:107),∂z,ni(nnRin,+,neRion)(Tn,−,Ti),+,nimi(nnRin,+,neRion)[(un(cid:107),−,ui(cid:107))2,+,u2,n⊥],∂φ,∂r,1,2,(Te,−,Ti),+,(cid:90),(cid:18),1,2,mi|v,−,ui(cid:107)ˆz|2,−,(cid:19),3,2,Ti,Si,d3v.,(6.17),(6.18),νei,:=,√,16,3,π,e4ni,ln,Λ,(4π(cid:15)0)2m2,ev3,te,is,the,electron-ion,collision,frequency,as,deﬁned,by,Braginskii,(Braginskii,1958),,and,we,have,deﬁned,the,parallel,pressure,ps(cid:107)[Fs,,ns,,vts](r,,z,,t),:=,2πnsmsv2,ts,(cid:90),∞,(cid:90),∞,dw(cid:107),−∞,0,dw⊥,w⊥w2,(cid:107)Fs(r,,z,,w(cid:107),,w⊥,,t),(6.19),10,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,and,the,parallel,heat,ﬂux,qs(cid:107)[Fs,,ns,,vts](r,,z,,t),:=,πnsmsv3,ts,(cid:90),∞,(cid:90),∞,dw(cid:107),−∞,0,for,the,charged,species,s,=,i,,e.,The,ion,kinetic,equation,is,dw⊥,w⊥w(cid:107)(w2,(cid:107),+,w2,⊥)Fs(r,,z,,w(cid:107),,w⊥,,t),(6.20),∂Fi,∂t,+,˙ri,∂Fi,∂r,+,˙zi,∂Fi,∂z,+,˙w(cid:107)i,∂Fi,∂w(cid:107),+,˙w⊥i,∂Fi,∂w⊥,=,˙Fi,+,Cii,+,Cin,+,Ci,ion,+,Si.,(6.21),Here,,we,have,deﬁned,the,coeﬃcients,˙ri[φ](r,,z,,t),:=,−,1,B,∂φ,∂z,,,˙zi[ui(cid:107),,vti,,φ](r,,z,,w(cid:107),,t),:=,ui(cid:107)Bz,B,+,1,B,∂φ,∂r,+,vtiBz,B,w(cid:107),,(6.22),(6.23),˙w(cid:107)i[Fi,,ni,,ui(cid:107),,vti](r,,z,,w(cid:107),,t),:=,+,2w(cid:107)Bz,3nimiv2,tiB,∂pi(cid:107),∂z,Bz,nimivtiB,(cid:18),(cid:20),∂qi(cid:107),∂z,+,pi(cid:107),−,3,2,nimiv2,ti,(cid:21),(cid:19),∂ui(cid:107),∂z,˙w⊥i[Fi,,ni,,ui(cid:107),,vti](r,,z,,w(cid:107),,w⊥,,t),:=,2w⊥Bz,3nimiv2,tiB,(cid:18),∂qi(cid:107),∂z,+,pi(cid:107),∂ui(cid:107),∂z,−,(cid:19),w2,(cid:107)Bz,B,∂vti,∂z,,,(6.24),−,w(cid:107)w⊥Bz,B,∂vti,∂z,(6.25),and,˙Fi[Fi,,ni,,ui(cid:107),,vti](r,,z,,w(cid:107),,w⊥,,t),:=,(cid:34),(cid:18),w(cid:107),Bz,B,3,∂vti,∂z,−,−,2,nimiv2,ti,(cid:18),∂qi(cid:107),∂z,(cid:18),+,pi(cid:107),−,1,2,nimiv2,ti,(cid:19),(cid:35),Fi.,(6.26),(cid:19),vti,∂ni,ni,∂z,(cid:19),∂ui(cid:107),∂z,We,have,also,deﬁned,a,modiﬁed,source,Si,and,several,modiﬁed,collision,operators.,The,modiﬁed,source,for,the,charged,species,s,=,i,,e,is,given,by,Ss[Ss,,Fs,,ns,,us(cid:107),,vts](r,,z,,w(cid:107),,w⊥,,t),v3,ts,ns,(cid:90),:=,−,+,+,(cid:90),(cid:20),(cid:20),Fs,ns,∂,∂w(cid:107),1,w⊥,Ss,d3v,−,(cid:18),1,Fs,nsvts,(cid:20),w2,⊥Fs,3nsv2,ts,∂,∂w⊥,Ss(r,,z,,us(cid:107),+,vtsw(cid:107),,vtsw⊥,,t),(cid:21),(v(cid:107),−,us(cid:107))Ss,d3v,+,(cid:90),(cid:18),|v,−,us(cid:107)ˆz|2,−,3,2,w(cid:107),3nsv2,ts,(cid:19),(cid:90),(cid:18),|v,−,us(cid:107)ˆz|2,−,(cid:19),3,2,v2,ts,(cid:19)(cid:21),Ss,d3v,v2,ts,Ss,d3v,(cid:21),.,(6.27),Note,that,the,diﬀerential,terms,in,this,modiﬁed,source,could,have,been,included,in,the,˙w⊥i,and,˙Fi,,but,we,have,decided,to,make,them,part,deﬁnitions,of,the,coeﬃcients,˙w(cid:107)i,,of,a,modiﬁed,source,instead,to,separate,the,eﬀect,of,the,source,clearly.,We,will,do,the,same,for,collisions.,This,split,should,not,be,taken,as,a,suggestion,on,how,to,implement,these,terms,in,a,code.,The,modiﬁed,collisions,operators,are,described,in,Appendix,A.,2D,drift,kinetic,model,with,wall,boundary,conditions,11,6.2.,Electron,equations,For,the,electrons,,we,use,the,expansion,in,me/mi,described,in,report,2047357-TN-05-01,M1.3.,We,ﬁrst,describe,the,electron,ﬂuid,equations,and,how,to,use,them.,•,The,electron,continuity,equation,is,∂ne,∂t,−,(cid:19),∂,∂r,(cid:18),ne,B,∂φ,∂z,+,∂,∂z,(cid:20),ne,(cid:18),ue(cid:107)Bz,B,+,1,B,∂φ,∂r,(cid:19)(cid:21),=,−nennRion,+,(cid:90),Se,d3v.,(6.28),Subtracting,this,equation,from,equation,(6.15),and,using,quasineutrality,,we,obtain,the,current,conservation,equation,Bz,B,∂,∂z,(cid:2)ne,(cid:0)ui(cid:107),−,ue(cid:107),(cid:1)(cid:3),=,0,,(6.29),where,we,have,used,property,(4.7).,This,equation,can,be,used,to,calculate,ue(cid:107),once,ue(cid:107),is,known,at,z,=,0.,We,discuss,how,to,obtain,ue(cid:107),at,z,=,0,in,the,next,bullet,point.,•,The,electron,parallel,momentum,equation,simpliﬁes,to,0,=,−,Bz,B,∂pe(cid:107),∂z,+,eneBz,B,∂φ,∂z,+,Fei(cid:107),+,nemennRen(un(cid:107),−,ue(cid:107)),,(6.30),where,Fei(cid:107)[Fe,,ne,,ni,,ue(cid:107),,ue(cid:107),,vte,,vti](z,,t),8π2e4neni,ln,Λ,(4π(cid:15)0)2mev2,te,:=,−,dw(cid:107),(cid:90),∞,−∞,(cid:90),∞,0,dw⊥,w⊥(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)Fe,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,(6.31),is,the,friction,force,between,electrons,and,ions.,Equation,(6.30),can,be,used,to,calcu-,late,the,potential,diﬀerence,between,z,=,0,and,any,value,of,z,,φ(r,,z,,t),−,φ(r,,0,,t).,To,completely,determine,the,potential,,we,need,to,calculate,φ(r,,0,,t).,We,do,so,with,the,current-potential,relationships,of,the,magnetic,presheath,and,the,Debye,sheath,,given,in,equations,(5.5),and,(5.8).,We,use,two,coupled,nonlinear,equations,for,the,unknowns,φ(r,,0,,t),and,φ(r,,Lz,,t).,◦,Integrating,equation,(6.29),between,z,=,0,and,z,=,Lz,,we,obtain,the,condi-,tion,J(cid:107)(r,,Lz,,t),−,J(cid:107)(r,,0,,t),=,0.,Since,equations,(5.5),and,(5.8),give,J(cid:107)(r,,Lz,,t),and,J(cid:107)(r,,0,,t),as,functions,of,φ(r,,Lz,,t),−,φw,and,φ(r,,0,,t),,condition,J(cid:107)(r,,Lz,,t),−,J(cid:107)(r,,0,,t),=,0,is,an,equation,for,φ(r,,Lz,,t),and,φ(r,,0,,t),(we,assume,the,bias,φw,to,be,externally,determined).,◦,The,other,equation,is,the,value,of,φ(r,,Lz,,t),−,φ(r,,0,,t),obtained,by,integrating,equation,(6.30),from,z,=,0,to,z,=,Lz.,Note,that,the,value,of,φ(r,,Lz,,t)−φ(r,,0,,t),de-,pends,on,the,unknown,ue(cid:107)(r,,0,,t),(recall,that,ue(cid:107),can,be,determined,everywhere,from,equation,(6.29),for,a,given,ue(cid:107)(r,,0,,t)).,The,value,ue(cid:107)(r,,0,,t),depends,on,φ(r,,0,,t),via,equation,(5.8),and,so,,in,the,end,,equation,(6.30),gives,a,relationship,between,φ(r,,Lz,,t),−,φ(r,,0,,t),and,φ(r,,0,,t).,Once,these,two,equations,for,φ(r,,0,,t),and,φ(r,,Lz,,t),are,solved,,we,can,substitute,the,value,of,φ(r,,0,,t),in,equation,(5.8),to,calculate,ue(cid:107)(r,,0,,t),,and,then,integrate,equa-,tion,(6.29),to,ﬁnd,ue(cid:107),everywhere.,12,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,•,The,electron,energy,equation,is,(cid:18),ue(cid:107)Bz,B,(cid:20),∂Te,∂t,∂Te,∂r,∂φ,∂z,1,B,3,2,ne,+,−,+,1,B,−nennRionEion,+,∂φ,∂r,(cid:21),(cid:19),∂Te,∂z,3nemeνei,mi,=,−,Bz,B,∂qe(cid:107),∂z,−,pe(cid:107)Bz,B,∂ue(cid:107),∂z,(Ti,−,Te),+,Fei(cid:107)(ui(cid:107),−,ue(cid:107)),+,3nemennRen,mi,(Tn,−,Te),+,nemennRen[(un(cid:107),−,ue(cid:107))2,+,u2,n⊥],(cid:90),(cid:18),1,2,+,me|v,−,ue(cid:107)ˆz|2,−,(cid:19),3,2,Te,Se,d3v,,(6.32),where,Eion,is,the,ionization,energy,cost,that,includes,in,it,radiation,from,excited,states.,The,kinetic,equation,for,electrons,is,˙ze,∂Fe,∂z,+,˙w(cid:107)e,∂Fe,∂w(cid:107),+,˙w⊥e,∂Fe,∂w⊥,=,˙Fe,+,Cee,+,Cei,+,Cen,,(6.33),where,˙ze[Fe,,vte](z,,w(cid:107),,t),:=,vtew(cid:107),,˙w(cid:107)e[Fe,,ue(cid:107),,vte](z,,w(cid:107),,t),:=,Bz,nemevteB,∂pe(cid:107),∂z,+,2w(cid:107)Bz,3nemev2,teB,∂qe(cid:107),∂z,−,w2,(cid:107)Bz,B,∂vte,∂z,,,˙w⊥e[Fe,,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,2w⊥Bz,3nemev2,teB,∂qe(cid:107),∂z,−,w(cid:107)w⊥Bz,B,∂vte,∂z,(6.34),(6.35),(6.36),and,˙Fe[Fe,,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,(cid:34),(cid:18),w(cid:107),Bz,B,3,∂vte,∂z,−,vte,ne,∂ne,∂z,(cid:19),−,2,nemev2,te,∂qe(cid:107),∂z,(cid:35),Fe.,(6.37),The,modiﬁed,collision,operators,Cee,,Cei,and,Cen,are,described,in,Appendix,B.,The,ﬂuid,equations,for,the,neutrals,are,6.3.,Neutral,equations,+,∇,·,(nnun),=,−nnneRion,+,(cid:90),Sn,d3v,,∂nn,∂t,(6.38),nnmi,(cid:18),∂un,∂t,(cid:19),+,un,·,∇un,=,−∇,·,Pn,+,nnminiRin,(cid:16),ui(cid:107),ˆζ,−,un,(cid:17),(cid:90),+,mi(v,−,un)Sn,d3v,(6.39),and,3,2,nn,(cid:18),∂Tn,∂t,(cid:19),+,un,·,∇Tn,=,−∇,·,qn,−,Pn,:,∇un,+,3,2,nnniRin(Ti,−,Tn),+,1,2,nnminiRin[(un(cid:107),−,ui(cid:107))2,+,u2,(cid:90),(cid:18),1,2,+,n⊥],+,3nemennRen,mi,(Te,−,Tn),mi|v,−,un|2,−,(cid:19),Sn,d3v.,(6.40),3,2,Tn,2D,drift,kinetic,model,with,wall,boundary,conditions,13,Here,,we,have,deﬁned,the,pressure,tensor,Pn[Fn,,nn,,vtn](r,,z,,t),:=,nnmiv2,tn,(cid:90),wwFn(r,,z,,w,,t),d3w,(6.41),and,the,heat,ﬂux,qn[Fn,,nn,,vtn](z,,t),:=,(cid:90),1,2,nnmiv3,tn,w2wFn(r,,z,,w,,t),d3w.,(6.42),The,neutral,kinetic,equation,is,∂Fn,∂t,where,we,have,deﬁned,the,coeﬃcients,∂Fn,∂r,+,˙zn,+,˙rn,∂Fn,∂z,+,˙wn,·,∇wFn,=,˙Fn,+,Cni,+,Sn,,(6.43),˙rn[unr,,vtn](r,,z,,wr,,t),:=,unr,+,vtnwr,,˙zi[unz,,vtn](r,,z,,wz,,t),:=,unz,+,vtnwz,,(6.44),(6.45),˙wn[Fn,,nn,,un,,vtn](r,,z,,w,,t),:=,1,nnmivtn,∇,·,Pn,+,2w,3nnmiv2,tn,(∇,·,qn,+,Pn,:,∇un),−,w,·,∇un,−,ww,·,∇vtn,,(6.46),The,modiﬁed,charge,exchange,collision,operator,Cni,is,described,in,Appendix,C.,The,modiﬁed,source,is,Sn[Sn,,Fn,,nn,,un,,vtn](r,,z,,w,,t),:=,−,(cid:90),(cid:20),Fn,nn,(cid:20),+,∇w,·,Fn,(cid:18),1,(cid:90),nnvtn,(v,−,un)Sn,d3v,+,w,3nnv2,tn,Sn,d3v,−,v3,tn,nn,(cid:90),(cid:18),Ss(r,,z,,un,+,vtnw,,t),(cid:21),|v,−,un|2,−,(cid:19),3,2,v2,tn,Sn,d3v,(cid:19)(cid:21),.,(6.47),6.4.,Boundary,conditions,These,equations,have,to,be,solved,with,the,boundary,conditions,in,equations,(5.1),,(5.2),,(5.6),,(5.7),,(5.9),and,(5.10).,For,ns,,us(cid:107),,vts,and,Fs,known,at,time,t,,we,can,construct,fs,at,z,=,0,and,z,=,Lz,,and,we,can,apply,the,wall,boundary,conditions,to,fs.,We,can,then,use,the,resulting,fs,to,obtain,ns,,us,,vts,and,Fs,,closing,the,system,of,equations.,7.,Discussion,The,model,that,we,propose,is,comprised,of:,•,the,three,ﬂuid,equations,(6.15),,(6.16),and,(6.17),for,ions,that,have,to,be,solved,in,conjunction,with,the,ion,kinetic,equation,(6.21);,•,the,ﬁve,ﬂuid,equations,(6.38),,(6.39),and,(6.40),for,neutrals,that,have,to,be,solved,in,conjunction,with,the,neutral,kinetic,equation,(6.43);,•,the,two,ﬂuid,equations,(6.29),and,(6.32),for,electrons,that,have,to,be,solved,in,conjunction,with,the,electron,kinetic,equation,(6.33);,and,•,the,electron,parallel,momentum,equation,(6.30),for,the,potential.,The,boundary,conditions,for,this,system,of,equations,are,described,in,section,5.,To,test,the,model,proposed,in,this,report,,we,will,ﬁrst,extend,the,existing,1D,code,based,on,adiabatic,electrons,to,wall,boundary,conditions.,We,will,then,explore,the,eﬀect,14,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,of,adding,electrons.,For,most,physics,of,interest,,it,is,suﬃcient,to,use,simpliﬁed,ion-,ion,and,electron-electron,collision,operators,,and,for,this,reason,we,do,not,expect,to,implement,a,full,Fokker-Planck,collision,operator.,Appendix,A.,Modiﬁed,collision,operators,for,the,ion,kinetic,equation,The,modiﬁed,Fokker-Planck,like-particle,collision,operator,is,Css[Fs,ns,,vts](r,,z,,w(cid:107),,w⊥,,t),(cid:18),2πe4ns,ln,Λ,sv3,(4π(cid:15)0)2m2,ts,(cid:20),∂,∂w(cid:107),:=,(cid:40),(cid:18),D(cid:107)(cid:107)[Fs],∂Fs,∂w(cid:107),+,D(cid:107)⊥[Fs],(cid:19),+,P(cid:107)[Fs]Fs,∂Fs,∂w⊥,+,1,w⊥,∂,∂w⊥,w⊥,D(cid:107)⊥[Fs],∂Fs,∂w(cid:107),+,D⊥⊥[Fs],∂Fs,∂w⊥,(cid:19)(cid:21),(cid:41),+,P⊥[Fs]Fs,.,(A,1),The,coeﬃcients,needed,for,this,collision,operator,are,D(cid:107)(cid:107)[Fs](r,,z,,w(cid:107),,w⊥,,t),:=,4,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),w(cid:48),⊥,(cid:32),×,K(κ),−,(w(cid:107),−,w(cid:48),(cid:107))2E(κ),(cid:107))2,+,(w⊥,−,w(cid:48),⊥)2,(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,+,w(cid:48),(w(cid:107),−,w(cid:48),(cid:33),⊥)2,Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),,D(cid:107)⊥[Fs](r,,z,,w(cid:107),,w⊥,,t),:=,2,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:32),[(w(cid:107),−,w(cid:48),(w(cid:107),−,w(cid:48),⊥,+,w(cid:48)2,(cid:107))2,−,w2,(cid:107))2,+,(w⊥,−,w(cid:48),×,(cid:113),w(cid:48),w⊥,⊥(w(cid:107),−,w(cid:48),(cid:107)),(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,+,w(cid:48),(cid:33),⊥]E(κ),⊥)2,−,K(κ),Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥)2,⊥,,t),,D⊥⊥[Fs](r,,z,,w(cid:107),,w⊥,,t),:=,2,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,dw(cid:48),⊥,(cid:113),w2,⊥,w(cid:48),⊥,(cid:107))2,+,(w⊥,+,w(cid:48),⊥)2,(w(cid:107),−,w(cid:48),0,(cid:34),(cid:40),×,2w⊥,w⊥(w(cid:107),−,w(cid:48),(cid:107))2,(cid:107))2,+,(w⊥,−,w⊥)2,−,w(cid:48),⊥,(w(cid:107),−,w(cid:48),(cid:35),E(κ),+[(w(cid:107),−,w(cid:48),(cid:107))2,+,w2,⊥,+,w(cid:48)2,⊥][K(κ),−,E(κ)],Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),,(cid:41),P(cid:107)[Fs](r,,z,,w(cid:107),,w⊥,,t),:=,8,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),w(cid:48),⊥(w(cid:107),−,w(cid:48),(cid:107)),(cid:107))2,+,(w⊥,+,w(cid:48),⊥)2,(w(cid:107),−,w(cid:48),(A,2),(A,3),(A,4),(cid:32),×,K(κ),−,E(κ),(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,+,w(cid:48),⊥)2,−,E(κ),(cid:33),(w(cid:107),−,w(cid:48),(cid:107))2,+,(w⊥,−,w(cid:48),⊥)2,Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t),(A,5),2D,drift,kinetic,model,with,wall,boundary,conditions,15,and,P⊥[Fs](r,,z,,w(cid:107),,w⊥,,t),:=,4,(cid:90),∞,−∞,dw(cid:48),(cid:107),(cid:90),∞,0,dw(cid:48),⊥,(cid:113),w⊥,w(cid:48),⊥,(cid:107))2,+,(w⊥,+,w(cid:48),(w(cid:107),−,w(cid:48),(cid:33),⊥)2,(cid:32),[(w(cid:107),−,w(cid:48),(w(cid:107),−,w(cid:48),×,⊥,+,w(cid:48)2,(cid:107))2,−,w2,(cid:107))2,+,(w⊥,−,w(cid:48),⊥]E(κ),⊥)2,−,K(κ),Fs(z,,w(cid:48),(cid:107),,w(cid:48),⊥,,t).,(A,6),Here,,K(κ),:=,(cid:82),π/2,elliptic,integrals,,and,the,function,κ,is,(1,−,κ2,sin2,α)−1/2,dα,and,E(κ),:=,(cid:82),π/2,0,0,(1,−,κ2,sin2,α)1/2,dα,are,the,κ(w(cid:107),,w⊥,,w(cid:48),(cid:107),,w(cid:48),⊥),:=,(cid:115),4w⊥w(cid:48),⊥,(cid:107))2,+,(w⊥,+,w(cid:48),⊥)2,.,(w(cid:107),−,w(cid:48),(A,7),The,modiﬁed,charge,exchange,collision,operator,for,the,ion,kinetic,equation,is,Cin[Fi,,(cid:104)Fn(cid:105),,nn,,ui(cid:107),,un,,vti,,vtn](z,,w(cid:107),,w⊥,,t),(cid:20),Fi,−,:=,−,nnRin,(cid:104)Fn(cid:105),r,,z,,(cid:18),v3,ti,v3,tn,+,nnRin,∂,∂w(cid:107),(cid:20)(cid:18),un(cid:107),−,ui(cid:107),+,vti,vtn,w(cid:107),,vti,vtn,(cid:19)(cid:21),w⊥,,t,ui(cid:107),−,un(cid:107),vtn,(cid:18),v2,tn,v2,ti,w(cid:107),2,+,−,1,+,vti,(cid:18),v2,tn,v2,ti,−,1,+,2[(un(cid:107),−,ui(cid:107))2,+,u2,3v2,ti,n⊥],2[(un(cid:107),−,ui(cid:107))2,+,u2,3v2,ti,(cid:19),(cid:21),n⊥],(cid:19)(cid:19),(cid:21),Fi,Fi,,,(A,8),+,nnRin,w⊥,∂,∂w⊥,(cid:20),w2,⊥,2,where,(cid:104)Fn(cid:105)[Fn,,un⊥](r,,z,,w(cid:107),,w⊥,,t),:=,1,2π,(cid:90),2π,0,(cid:18),Fn,r,,z,,w⊥,cos,ϕ,−,unr,vtn,,,w⊥,sin,ϕ,−,unz,vtn,(cid:19),,,w(cid:107),,t,dϕ,(A,9),is,the,gyroaverage,of,Fn.,Finally,,the,modiﬁed,ionization,collision,operator,for,the,ion,kinetic,equation,is,Ci,ion[(cid:104)Fn(cid:105),,ne,,ui(cid:107),,un,,vti,,vtn](r,,z,,w(cid:107),,w⊥,,t),(cid:20),:=,−,neRion,Fi,−,(cid:18),(cid:104)Fn(cid:105),r,,z,,v3,ti,v3,tn,+,vti,vtn,w(cid:107),,vti,vtn,(cid:19)(cid:21),w⊥,,t,(cid:20)(cid:18),un(cid:107),−,ui(cid:107),+,neRion,∂,∂w(cid:107),+,neRion,w⊥,∂,∂w⊥,(cid:20),w2,⊥,2,vti,(cid:18),v2,tn,v2,ti,2[(un(cid:107),−,ui(cid:107))2,+,u2,3v2,ti,(cid:19),(cid:21),n⊥],(cid:19)(cid:19),(cid:21),Fi,Fi,.,(A,10),−,1,+,2[(un(cid:107),−,ui(cid:107))2,+,u2,3v2,ti,n⊥],ui(cid:107),−,un(cid:107),vtn,(cid:18),v2,tn,v2,ti,w(cid:107),2,+,−,1,+,Appendix,B.,Modiﬁed,collision,operators,for,the,electron,kinetic,equation,The,electron-electron,collision,operator,is,described,in,equation,(A,1).,16,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,The,modiﬁed,ion-electron,collision,operator,is,Cei[Fe,,ni,,ui(cid:107),,ue(cid:107),,vte](r,,z,,w(cid:107),,w⊥,,t),:=,2πe4ni,ln,Λ,ev3,(4π(cid:15)0)2m2,te,(cid:20),1,w⊥,∂,∂w⊥,(cid:40),∂,∂w(cid:107),(cid:18),w⊥,M(cid:107)⊥,(cid:20),M(cid:107)(cid:107),∂Fe,∂w(cid:107),+,M(cid:107)⊥,(cid:18),+,1,+,∂Fe,∂w⊥,2(ui(cid:107),−,ue(cid:107))w(cid:107),3vte,(cid:19),(cid:21),F(cid:107)Fe,∂Fe,∂w(cid:107),+,M⊥⊥,∂Fe,∂w⊥,+,2(ui(cid:107),−,ue(cid:107))w⊥,3vte,(cid:19)(cid:21),(cid:41),F(cid:107)Fe,,,(B,1),+,where,M(cid:107)(cid:107)[ue(cid:107),,vte,,ui(cid:107)](r,,z,,w(cid:107),,w⊥,,t),:=,w2,⊥,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,,,M(cid:107)⊥[ue(cid:107),,vte,,ui(cid:107)](r,,z,,w(cid:107),,w⊥,,t),:=,−,(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2w⊥,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,,,M⊥⊥[ue(cid:107),,vte,,ui(cid:107)](r,,z,,w(cid:107),,w⊥,,t),:=,(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,and,F(cid:107)[Fe,,ue(cid:107),,vte,,ui(cid:107)](r,,z,,t),(B,2),(B,3),(B,4),:=,−4π,(cid:90),∞,(cid:90),∞,dw(cid:107),dw⊥,−∞,0,w⊥[w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte]Fe(z,,w(cid:107),,w⊥,,t),[(w(cid:107),−,(ui(cid:107),−,ue(cid:107))/vte)2,+,w2,⊥]3/2,.,(B,5),The,modiﬁed,electron-neutral,collision,operator,is,Cen[Fe,,nn,,un,,ue(cid:107),,vte](z,,w(cid:107),,w⊥,,t),:=,−nnRen,Fe,(cid:34),−,1,8π2,(cid:90),π,0,dχ,+,nnRen,∂,∂w(cid:107),(cid:90),2π,0,(cid:20)(cid:18),1,+,dϕ,(cid:90),2π,0,dϕ(cid:48),sin,χFe(z,,w(cid:107),,w⊥,,t),(cid:35),2[(un(cid:107),−,ue(cid:107))2,+,u2,n⊥]w(cid:107),(cid:19),(cid:21),Fe,3v2,te,∂,∂w⊥,(cid:0)w2,⊥Fe,(cid:1),,,(B,6),+,where,2nnRen[(un(cid:107),−,ue(cid:107))2,+,u2,3v2,tew⊥,n⊥],w(cid:107)[ue(cid:107),,vte,,un](r,,z,,w(cid:107),,w⊥,,χ,,ϕ(cid:48),,t),:=,un(cid:107),−,ue(cid:107),vte,+,wen,cos,χ,,(B,7),w⊥[ue(cid:107),,vte,,un(cid:107)](r,,z,,w(cid:107),,w⊥,,χ,,ϕ,,ϕ(cid:48)t),:=,and,wen[ue(cid:107),,vte,,un](r,,z,,w(cid:107),,w⊥,,ϕ(cid:48),,t),(cid:115),u2,n⊥,v2,te,+,w2,en,sin2,χ,−,2un⊥wen,vte,sin,χ,cos,ϕ,(B,8),(cid:115)(cid:18),:=,w(cid:107),+,(cid:19)2,ue(cid:107),−,un(cid:107),vte,+,w2,⊥,+,u2,n⊥,v2,te,−,2un⊥w⊥,vte,cos,ϕ(cid:48).,(B,9),2D,drift,kinetic,model,with,wall,boundary,conditions,17,Appendix,C.,Modiﬁed,collision,operators,for,the,neutral,kinetic,equation,The,modiﬁed,charge,exchange,collision,operator,for,the,neutral,kinetic,equation,is,:=,−,niRin,Cni[Fn,,Fi,,ni,,un,,ui(cid:107),,vtn,,vti](r,,z,,wr,,wz,,wζ,,t),un(cid:107),−,ui(cid:107),vti,(cid:18),v2,ti,v2,tn,ˆζ,−,un,vtn,v3,tn,v3,ti,ui(cid:107),+,niRin∇w,·,(cid:20),Fn,−,w,2,r,,z,,(cid:34)(cid:32),Fi,+,(cid:18),+,vtn,vti,−,1,+,(cid:19)(cid:21),,,t,wζ,,(cid:12),(cid:12),(cid:12),(cid:12),w⊥,+,vtn,vti,un⊥,vtn,(cid:12),(cid:12),(cid:12),(cid:12),2[(un(cid:107),−,ui(cid:107))2,+,u2,3v2,tn,n⊥],where,(cid:12),(cid:12),(cid:12),(cid:12),w⊥,+,un⊥,vtn,(cid:12),(cid:12),(cid:12),(cid:12),(cid:39),(cid:115)(cid:18),wr,+,unr,vtn,(cid:19)2,(cid:18),+,wz,+,(cid:19)2,.,unz,vtn,(cid:19)(cid:33),(cid:35),Fn,,,(C,1),(C,2),REFERENCES,Braginskii,,S.I.,1958,Transport,phenomena,in,a,completely,ionized,two-temperature,plasma.,Sov.,Phys.,JETP,6,,358.,Catto,,P.J.,1994,A,short,mean-free,path,,coupled,neutral-ion,transport,description,of,a,toka-,mak,edge,plasma.,Phys.,Plasmas,1,,1936.,Chodura,,R.,1982,Plasma-wall,transition,in,an,oblique,magnetic,ﬁeld.,Phys.,Fluids,25,,1628.,Connor,,J.W.,1977,An,analytic,solution,for,the,distribution,function,of,neutral,particles,in,a,Maxwellian,plasma,using,the,method,of,singular,eigenfunctions.,Plasma,Phys.,19,,853.,Geraldini,,A.,2021,Large,gyro-orbit,model,of,ion,velocity,distribution,in,plasma,near,a,wall,in,a,grazing-angle,magnetic,ﬁeld.,J.,Plasma,Phys.,87,,905870113.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2017,Gyrokinetic,treatment,of,a,grazing,angle,magnetic,presheath.,Plasma,Phys.,Control.,Fusion,59,,025015.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2018,Solution,to,a,collisionless,shallow-angle,magnetic,presheath,with,kinetic,ions.,Plasma,Phys.,Control.,Fusion,60,,125002.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2019,Dependence,on,ion,temperature,of,shallow-,angle,magnetic,presheaths,with,adiabatic,electrons.,J.,Plasma,Phys.,85,,795850601.,Hazeltine,,R.D.,1973,Recursive,derivation,of,drift-kinetic,equation.,Plasma,Phys.,15,,77–80.,Hazeltine,,R.D.,,Calvin,,M.D.,,Valanju,,P.M.,&,Solano,,E.R.,1992,Analytical,calculation,of,neutral,transport,and,its,eﬀect,on,ions.,Nucl.,Fusion,32,,3.,Helander,,P.,,Krasheninnikov,,S.I.,&,Catto,,P.J.,1994,Fluid,equations,for,a,partially,ionized,plasma.,Phys.,Plasmas,1,,3174.,Knudsen,,M.,1916,Das,Cosinusgesetz,in,der,kinetischen,Gastheorie.,Annal.,Phys.,353,,1113.,Parker,,S.E.,,Procassini,,R.J.,&,Birdsall,,C.K.,1993,A,Suitable,Boundary,Condition,for,Bounded,Plasma,Simulation,without,Sheath,Resolution.,J.,Comput.,Phys.,104,,41.,Pastukhov,,V.P.,1974,Collisional,losses,of,electrons,from,an,adiabatic,trap,in,a,plasma,with,a,positive,potential.,Nucl.,Fusion,14,,3.,Rosenbluth,,M.N.,,MacDonald,,W.M.,&,Judd,,D.L.,1957,Fokker-Planck,Equation,for,an,Inverse-Square,Force.,Phys.,Rev.,107,,1.,REPORT,2047357-TN-09-01,M1.6,1,2D,drift,kinetic,model,with,periodic,boundary,conditions,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,29,September,2021),1.,Introduction,In,report,2047357-TN-07-02,,we,presented,a,2D,drift,kinetic,model,for,a,helical,mag-,netic,ﬁeld,with,wall,boundary,conditions.,The,wall,boundary,conditions,made,it,possible,to,obtain,the,potential,and,the,electron,ﬂow,from,the,conservation,of,parallel,current,and,the,electron,parallel,momentum,equation.,With,periodic,boundary,conditions,,ap-,propriate,for,closed,ﬂux,surfaces,,it,will,not,be,suﬃcient,with,conservation,of,parallel,current,,and,we,will,have,to,include,terms,small,in,the,expansion,parameter,that,we,use,,the,ion,gyroradius,over,the,characteristic,width,of,the,scrape-oﬀ,layer,(ρi/Lr).,In,this,report,,we,ﬁrst,remind,the,reader,of,the,content,in,report,2047357-TN-07-02,,and,we,then,explain,how,to,modify,the,current,conservation,equations,for,cases,with,periodic,boundary,conditions.,2.,Magnetic,ﬁeld,,geometry,and,orderings,We,use,the,cylindrical,coordinates,{r,,z,,ζ},(see,report,2047357-TN-07-02,for,the,direc-,tion,of,increase,of,ζ).,We,consider,a,magnetized,plasma,with,one,ion,species,with,charge,e,and,mass,mi,,electrons,with,charge,−e,and,mass,me,,and,one,species,of,neutrals,with,mass,The,plasma,is,magnetized,by,the,helical,magnetic,ﬁeld,mn,=,mi.,B(r,,ζ),:=,Bz(r)ˆz,+,Bζ(r)ˆζ(ζ),,(2.1),(2.2),where,ˆz,and,ˆζ,are,the,unit,vectors,in,the,direction,of,∇z,and,∇ζ.,Note,that,the,compo-,nents,Bz,and,Bζ,only,depend,on,the,radial,position,r.,We,assume,that,the,plasma,only,varies,in,r,and,z.,We,assume,that,the,electric,ﬁeld,produced,by,the,plasma,is,electrostatic,,E,=,−(∂φ/∂r)ˆr,−,(∂φ/∂z)ˆz,,where,ˆr,is,the,unit,vector,in,the,direction,∇r.,The,potential,φ(r,,z,,t),depends,on,the,coordinates,r,and,z,and,on,time,t.,We,impose,periodic,boundary,conditions,at,z,=,0,and,z,=,Lz.,In,the,radial,direction,,we,consider,the,interval,between,r,=,r0,and,r,=,r0,+,Lr.,The,length,Lr,is,determined,by,a,balance,between,the,fast,parallel,velocity,of,the,particles,along,magnetic,ﬁeld,lines,and,their,slow,drift,across,them.,The,characteristic,length,between,the,two,walls,along,a,magnetic,ﬁeld,line,is,of,order,B,Bz,Thus,,the,typical,time,that,it,takes,for,an,ion,to,move,from,wall,to,wall,is,L(cid:107)/vti,∼,L(cid:107),∼,(2.3),Lz.,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,(B/Bz)(Lz/vti),,where,vti,:=,(cid:112)2Ti/mi,is,the,ion,thermal,speed,and,Ti,is,the,ion,tem-,perature.,For,a,potential,φ,of,the,order,of,Ti/e,,where,e,is,the,proton,charge,,the,radial,E,×,B,drift,is,vEr,:=,−,Bζ,B2,∂φ,∂z,∼,ρi,Lz,vti,,(2.4),:=,vti/Ωi,is,the,characteristic,ion,gyroradius,and,Ωi,where,ρi,:=,eB/mi,is,the,ion,gyrofrequency.,Thus,,the,time,it,takes,for,an,ion,to,cross,the,domain,in,the,radial,direction,is,Lr/vEr,∼,LrLz/ρivti.,By,making,this,time,of,the,same,order,as,L(cid:107)/vti,,we,solve,for,Lr,to,ﬁnd,Lr,∼,B,Bz,ρi,(2.5),To,simplify,the,problem,to,a,tractable,drift,kinetic,form,,we,assume,that,ρi,is,much,smaller,than,Lr.,This,implies,that,ρi,Lr,∼,Bz,B,∼,Bz,Bζ,(cid:28),1,,(2.6),that,is,,we,will,limit,our,model,to,magnetic,ﬁelds,that,are,mostly,azimuthal,and,have,a,very,small,vertical,component.,This,is,an,approximation,that,is,consistent,with,magnetic,ﬁeld,geometry,in,conventional,tokamaks,and,also,in,the,edge,of,many,shots,in,spherical,tokamaks.,We,also,assume,that,r0,∼,Lz,(cid:29),Lr.,Since,r0,is,the,characteristic,length,of,variation,of,the,magnetic,ﬁeld,B,,the,magnetic,ﬁeld,barely,changes,across,the,domain,[r0,,r0,+,Lr].,Thus,,within,our,ordering,,we,assume,B,to,be,uniform,in,the,domain,of,interest.,We,assume,the,time,derivatives,to,be,of,the,same,order,as,the,parallel,and,perpendic-,ular,time,scales,that,we,have,discussed,above,,∂,∂t,∼,ρi,Lr,vti,Lz,.,(2.7),Our,orderings,above,rest,on,the,assumption,φ,∼,Ti/e.,In,the,case,with,wall,boundary,conditions,,the,wall,boundary,conditions,ensured,that,φ,remained,of,this,order.,With,periodic,boundary,conditions,,the,size,of,φ,is,controlled,by,the,momentum,input.,The,force,per,unit,volume,on,the,plasma,due,to,external,sources,,neutral-plasma,collisions,or,ionization,must,satisfy,|Fi,ext⊥|,,|Fe,ext⊥|,,|Fin⊥|,,|Fen⊥|,,|Fi,ion⊥|,,|Fe,ion⊥|,(cid:46),(cid:18),ρi,Lr,(cid:19)2,pi,Lz,,,(2.8),where,pi,is,the,ion,pressure.,This,estimate,for,the,force,per,unit,volume,comes,from,making,the,force,of,the,order,of,the,perpendicular,inertia,,∂(nimiui⊥)/∂t,,where,the,perpendicular,ion,velocity,ui⊥,is,taken,to,be,of,order,(ρi/Lr)vti,(see,equation,(4.7),below,for,a,justiﬁcation,of,this,ordering,for,ui⊥;,also,note,that,this,means,that,the,perpendicular,ﬂow,is,much,smaller,than,the,parallel,one,,which,we,assume,to,be,of,the,order,of,vti).,Equation,(2.8),might,seem,stringent,,but,the,friction,between,ions,and,neutrals,and,electron,and,neutrals,(due,to,elastic,collisions,or,ionization),is,small,in,the,closed,ﬁeld,line,region,of,the,tokamak,because,the,neutral,density,is,small,,i.e.,we,can,assume,that,nnRin,∼,nnRen,(cid:114),me,mi,∼,nnRion,(cid:46),(cid:18),ρi,Lr,(cid:19)2,vti,Lz,,,(2.9),where,nnRin,,nnRen,and,nnRion,are,the,ion-neutral,,electron-neutral,and,ionization,collision,frequencies.,2D,drift,kinetic,model,with,periodic,boundary,conditions,3,3.,Summary,of,report,2047357-TN-07-02,The,model,in,report,2047357-TN-07-02,is,comprised,of:,•,three,ﬂuid,equations,(conservation,of,particles,,parallel,momentum,and,energy),for,ions,that,have,to,be,solved,in,conjunction,with,an,ion,kinetic,equation,to,determine,the,ion,density,ni,=,ne,,the,ion,parallel,velocity,ui(cid:107),,the,ion,temperature,Ti,and,the,normalized,ion,distribution,function,Fi;,•,ﬁve,ﬂuid,equations,(conservation,of,particles,,the,three,components,of,momentum,and,energy),for,neutrals,that,have,to,be,solved,in,conjunction,with,a,neutral,kinetic,equation,to,determine,the,neutral,density,nn,,the,three,components,of,the,neutral,velocity,un,,the,neutral,temperature,Tn,and,the,normalized,neutral,distribution,function,Fn;,•,two,ﬂuid,equations,(conservation,of,parallel,current,,Bz,B,∂,∂z,(cid:2)ni,(cid:0)ui(cid:107),−,ue(cid:107),(cid:1)(cid:3),=,0,,(3.1),and,conservation,of,energy),for,electrons,that,have,to,be,solved,in,conjunction,with,an,electron,kinetic,equation,to,determine,the,electron,parallel,velocity,ue(cid:107),,the,electron,temperature,Te,and,the,electron,normalized,distribution,function,Fe;,and,•,the,electron,parallel,momentum,equation,,0,=,−,Bz,B,∂pe(cid:107),∂z,+,eneBz,B,∂φ,∂z,+,Fei(cid:107),+,nemennRen(un(cid:107),−,ue(cid:107)),,(3.2),for,the,potential,φ.,Here,,Fei(cid:107),is,the,collisional,friction,force,between,electrons,and,ions.,In,report,2047357-TN-07-02,,we,proposed,a,method,to,solve,equations,(3.1),and,(3.2),in,conjunction,with,wall,boundary,conditions.,Equation,(3.1),can,be,integrated,in,z,to,obtain,ue(cid:107)(r,,z,,t),−,ue(cid:107)(r,,0,,t),(recall,that,ni,and,ui(cid:107),are,time-advanced,using,ion,equations).,With,this,result,,equation,(3.2),can,be,integrated,in,z,to,obtain,the,diﬀerence,φ(r,,z,,t),−,φ(r,,0,,t),as,a,function,of,the,unknown,ue(cid:107)(r,,0,,t),(recall,that,pe(cid:107),is,determined,by,the,electron,energy,equation,and,the,electron,kinetic,equation,,and,that,Fei(cid:107),depends,on,ue(cid:107)).,With,wall,boundary,conditions,,we,could,solve,for,both,ue(cid:107)(r,,0,,t),and,φ(r,,0,,t).,Unfortunately,,the,same,cannot,be,said,for,periodic,boundary,conditions.,With,periodic,boundary,conditions,and,these,equations,,it,is,possible,to,ﬁnd,an,equation,for,ue(cid:107)(r,,0,,t),,but,not,for,φ(r,,0,,t).,Indeed,,dividing,equation,(3.2),by,ne,,integrating,in,z,and,using,the,periodic,boundary,conditions,for,φ,,we,ﬁnd,the,condition,(cid:90),Lz,(cid:20),−,0,=,0,Bz,neB,∂pe(cid:107),∂z,+,Fei(cid:107),ne,(cid:21),+,mennRen(un(cid:107),−,ue(cid:107)),dz.,(3.3),This,condition,is,satisﬁed,by,choosing,the,correct,value,of,ue(cid:107)(r,,0,,t).,Within,equa-,tions,(3.1),and,(3.2),,there,is,no,other,similar,condition,for,φ(r,,0,,t).,To,ﬁnd,such,a,condition,,we,need,to,modify,equation,(3.1),by,keeping,higher,order,terms,in,ρi/Lr.,4.,Current,conservation,for,periodic,boundary,conditions,Parra,&,Catto,(2008),showed,that,gyrokinetic,equations,(and,consequently,the,sub-,sidiary,limit,of,drift,kinetics),require,higher,order,terms,in,ρi/Lr,in,order,to,determine,the,component,of,the,electric,ﬁeld,perpendicular,to,the,ﬂux,surfaces,traced,by,the,mag-,netic,ﬁeld,when,the,magnetic,ﬁeld,is,axisymmetric.,In,this,case,,this,means,that,one,needs,higher,order,terms,to,determine,∂φ/∂r.,The,need,for,higher,order,terms,can,be,demonstrated,using,moments,of,the,full,kinetic,equation.,Before,we,start,taking,moments,,we,explain,what,we,mean,by,full,kinetic,distribution,function,and,full,kinetic,equation,in,this,report,,and,how,they,compare,to,4,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,the,drift,kinetic,distribution,function,and,kinetic,equation,that,we,have,used,so,far.,Importantly,,what,follows,does,not,apply,to,the,neutral,distribution,function,and,kinetic,equation,as,we,have,not,reduced,them,using,the,drift,kinetic,approximation,in,previous,reports.,We,denote,the,ion,and,electron,full,distribution,functions,by,gs(r,,z,,v,,t),,where,the,velocity,is,given,by,v,=,v(cid:107),ˆb,+,v⊥,=,v(cid:107),ˆb,+,v⊥(cos,ϕ,ˆr,+,sin,ϕ,ˆb,×,ˆr),,(4.1),Here,v(cid:107),is,the,velocity,parallel,to,the,magnetic,ﬁeld,,ˆb,:=,B/B,is,the,unit,vector,in,the,direction,to,the,magnetic,ﬁeld,,and,the,velocity,perpendicular,to,the,magnetic,ﬁeld,v⊥,is,described,by,its,magnitude,v⊥,and,its,direction,,given,by,the,gyrophase,ϕ.,The,full,kinetic,distribution,functions,for,ions,and,electrons,,gi,and,ge,,satisfy,the,full,kinetic,equations,∂gi,∂t,+,∇,·,(vgi),+,∇v,·,(cid:20),e,mi,and,∂ge,∂t,+,∇,·,(vge),+,∇v,·,−,(cid:20),e,me,(−∇φ,+,v,×,B),gi,=,Cii,+,Cin,+,Cie,+,Ci,ion,+,Qi,(4.2),(cid:21),(−∇φ,+,v,×,B),ge,=,Cee,+,Cen,+,Cei,+,Ce,ion,+,Qe.,(4.3),(cid:21),Note,that,we,have,included,like-particle,collisions,,electron-ion,collisions,,collisions,with,neutrals,and,ionization,collisions.,The,sources,Qe,represent,the,particle,,momentum,and,energy,input,into,the,plasma.,The,drift,kinetic,ion,and,electron,distribution,functions,fi,and,fe,that,we,used,in,previous,reports,are,gyroaverages,of,the,full,distribution,functions,,fs(r,,z,,v(cid:107),,v⊥,,t),:=,1,2π,(cid:90),2π,0,gs(r,,z,,v(v(cid:107),,v⊥,,ϕ),,t),dϕ.,(4.4),(Correspondingly,,the,sources,Ss,that,we,used,are,the,gyoraverage,of,the,full,sources,Qs.),For,ρi/Lr,(cid:28),1,,the,distribution,functions,gs,and,fs,are,almost,identical,,gs,(cid:39),fs,(Hazeltine,1973).,The,diﬀerence,between,these,distribution,functions,,of,order,ρi/Lr,(cid:28),1,,was,negligible,in,the,case,with,wall,boundary,conditions,,but,it,will,be,important,with,periodic,boundary,conditions.,Thus,,we,write,gs,=,fs,+,gs1,+,.,.,.,,,(4.5),where,,neglecting,the,ion-neutral,,electron-neutral,and,ionization,collisions,to,lowest,order,(recall,equation,(2.9)),,gs1,=,sin,ϕ,(cid:18),v⊥,Ωs,∂fs,∂r,−,1,B,∂φ,∂r,∂fs,∂v⊥,(cid:19),∼,ρs,Lr,fs.,(4.6),Here,Ωs,:=,ZseB/ms,is,the,gyrofrequency,of,species,s,,and,Zs,is,the,charge,number,(1,for,ions,and,−1,for,electrons).,Using,equation,(4.5),,we,can,calculate,the,perpendicular,average,velocity,of,ions,and,electrons,,us⊥,:=,(cid:90),1,ns,v⊥gs,d3v,(cid:39),(cid:90),1,ns,v⊥gs1,d3v,=,ˆb,×,ˆr,(cid:18),1,ZsensB,∂ps⊥,∂r,+,1,B,∂φ,∂r,(cid:19),∼,ρi,Lr,vti,,(4.7),where,ps⊥,:=,(cid:82),(msv2,⊥/2)fs,d3v,is,the,perpendicular,pressure,of,species,s.,Note,that,since,the,average,of,v⊥,over,the,gyrophase,vanishes,,us⊥,is,only,due,to,the,small,correction,gs1.,Thus,,us⊥,is,small.,Note,that,this,is,a,consequence,of,our,assumption,φ,∼,Ti/e.,A,larger,φ,would,have,led,to,a,larger,average,velocity.,2D,drift,kinetic,model,with,periodic,boundary,conditions,5,After,this,brief,introduction,to,the,full,distribution,functions,gs,,we,can,use,moments,of,these,distribution,functions,to,impose,current,conservation,following,the,procedure,proposed,by,Parra,&,Catto,(2009).,Current,conservation,can,be,written,as,(cid:104),(cid:105),eni(ui(cid:107),−,ue(cid:107))ˆb,∇,·,+,∇,·,J⊥,=,0,,(4.8),where,J⊥,is,the,component,of,the,current,density,perpendicular,to,the,magnetic,ﬁeld.,In,report,2047357-TN-07-02,we,could,neglect,the,term,∇,·,J⊥,because,it,is,small,in,Lr/Lz,(cid:28),1.,We,cannot,neglect,it,any,longer,,as,we,need,it,to,determine,φ(r,,0,,t).,Equation,(4.8),can,be,written,as,∂,∂z,(cid:20),eniBz,B,(ui(cid:107),−,ue(cid:107)),+,J⊥,·,ˆz,(cid:21),+,1,r,∂,∂r,(rJ⊥,·,ˆr),=,0.,(4.9),Due,to,the,periodic,boundary,conditions,,the,large,parallel,current,term,can,be,eliminated,by,integrating,in,z,,(cid:32)(cid:90),Lz,0,∂,∂r,(cid:33),J⊥,·,ˆr,dz,=,0,,(4.10),where,we,have,also,used,the,fact,that,r,(cid:39),r0,is,almost,constant.,Condition,(4.10),deter-,mines,φ(r,,0,,t).,Unfortunately,,equation,(4.7),is,not,enough,to,compute,J⊥,·,ˆr,to,the,suﬃciently,high,order,needed,to,obtain,φ(r,,0,,t).,The,perpendicular,current,density,can,,however,,be,calculated,to,very,high,order,by,taking,moments,of,the,full,kinetic,equations,for,ions,and,electrons.,Multiplying,equations,(4.2),and,(4.3),by,msv,and,integrating,over,velocity,space,,we,ﬁnd,the,ion,and,electron,total,momentum,equations,,(nimiui),+,∇,·,(cid:18)(cid:90),(cid:19),mivvgi,d3v,∂,∂t,=,−eni∇φ,+,eniui,×,B,+,Fin,+,Fie,+,Fi,ion,+,Fi,ext,(4.11),and,(nemeue),+,∇,·,(cid:18)(cid:90),(cid:19),mevvge,d3v,∂,∂t,=,ene∇φ,−,eneue,×,B,+,Fen,+,Fei,+,Fe,ion,+,Fe,ext.,(4.12),Here,,Fα,:=,(cid:82),msvCα,d3v,is,the,force,due,to,the,collision,operator,Cα,,and,Fs,ext,:=,(cid:82),msvQs,d3v,is,the,external,momentum,input.,Summing,equations,(4.11),and,(4.12),,using,quasineutrality,and,Fie+Fei,=,0,,and,neglecting,the,electron,momentum,compared,to,the,ion,momentum,,we,ﬁnd,J×B,(cid:39),∇·,(cid:18)(cid:90),mivvgi,d3v,+,(cid:90),mevvge,d3v,(cid:19),+,∂,∂t,(nimiui)−Fin,−Fi,ion,−Fext.,(4.13),where,we,have,deﬁned,the,total,external,force,Fext,:=,Fi,ext,+,Fe,ext.,Multiplying,equa-,tion,(4.13),by,(ˆr,×,ˆb)/B,,we,obtain,the,radial,component,of,J⊥,,J⊥,·,ˆr,(cid:39),−,(cid:18)(cid:90),1,B,∇,·,mivv,·,(ˆb,×,ˆr)gi,d3v,+,(cid:90),(cid:90),miv,·,∇,(cid:17),(cid:16)ˆb,×,ˆr,·,vgi,d3v,+,(cid:90),mev,·,(ˆb,×,ˆr)ge,d3v,(cid:19),mev,·,∇,(cid:17),(cid:16)ˆb,×,ˆr,·,vge,d3v,1,B,(cid:104),nimi,B,ui,·,(ˆb,×,ˆr),(cid:105),+,1,B,Fin,·,(ˆb,×,ˆr),+,1,B,Fi,ion,·,(ˆb,×,ˆr),+,1,B,Fext,·,(ˆb,×,ˆr),,(4.14),+,−,1,B,∂,∂t,6,where,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,(cid:17),(cid:16)ˆb,×,ˆr,∇,=,−,(cid:19),B2,ζ,B2,d,dr,(cid:18),Bz,Bζ,ˆrˆb,−,BzBζ,rB2,ˆbˆr,+,B2,rB2,(ˆb,×,ˆr)ˆr,∼,z,Bz,r0B,∼,ρi,Lr,1,Lz,.,(4.15),The,term,with,the,time,derivative,in,the,right,side,of,equation,(4.14),sets,the,size,of,the,equation.,For,∂/∂t,∼,ρivti/LrLz,,using,the,fact,that,|ui⊥|,∼,(ρi/Lr)vti,,we,ﬁnd,∂,∂t,(cid:104),nimi,B,(cid:105),ui,·,(ˆb,×,ˆr),∼,(cid:18),ρi,Lr,(cid:19)2,pi,BLz,.,(4.16),This,estimate,justiﬁes,the,ordering,(2.8).,Also,,estimate,(4.16),sets,the,size,of,the,terms,that,we,need,to,keep,in,the,equation.,For,example,,the,integrals,that,contain,∇(ˆb,×,ˆr),are,small,because,∇(ˆb,×,ˆr),is,itself,small,in,ρi/Lr,and,the,integrals,over,velocity,are,small,by,a,factor,of,(ρi/Lr)2,,as,can,be,checked,using,expression,(4.6),for,gi,and,ge.,Thus,,integrating,over,z,,using,equation,(4.7),for,ui⊥,and,employing,the,approximation,r,(cid:39),r0,,we,ﬁnd,(cid:90),Lz,0,J⊥,·,ˆr,dz,(cid:39),−,∂,∂t,(cid:34)(cid:90),Lz,(cid:18)(cid:90),1,B,∂,∂r,0,(cid:34)(cid:90),Lz,0,nimi,B2,(cid:18),∂φ,∂r,+,1,eni,∂pi⊥,∂r,miv,·,ˆr(v,×,ˆb),·,ˆr,gi,d3v,+,(cid:19),(cid:90),(cid:35),dz,mev,·,ˆr(v,×,ˆb),·,ˆr,ge,d3v,(cid:19),(cid:35),dz,(cid:90),Lz,0,(cid:20),1,B,Fin,·,(ˆb,×,ˆr),+,1,B,Fi,ion,·,(ˆb,×,ˆr),+,(cid:21),Fext,·,(ˆb,×,ˆr),dz.,1,B,(4.17),−,+,The,velocity,integrals,in,equation,(4.17),can,be,calculated,using,a,moment,of,equa-,i,/4eB)[(v,·,ˆr)2,−,((v,×,ˆb),·,ˆr)2],tions,(4.2),and,(4.3).,Multiplying,equation,(4.2),by,(m2,and,integrating,in,velocity,space,and,z,,we,ﬁnd,(cid:90),Lz,(cid:90),dz,0,(cid:39),+,−,−,d3v,miv,·,ˆr(v,×,ˆb),·,ˆr,gi,(cid:34)(cid:90),Lz,(cid:90),1,4Ωi,∂,∂t,1,4Ωi,(cid:90),Lz,0,1,4Ωi,0,0,(cid:34)(cid:90),Lz,(cid:90),dz,∂,∂r,0,(cid:20),∂φ,∂r,(cid:90),nimi,2B2,(cid:90),Lz,dz,d3v,mi[(v,·,ˆr)2,−,((v,×,ˆb),·,ˆr)2],gi,(cid:35),(cid:35),d3v,miv,·,ˆr[(v,·,ˆr)2,−,((v,×,ˆb),·,ˆr)2],gi,(cid:18),∂φ,∂z,+,1,eni,∂pi⊥,∂z,(cid:19),+,∂φ,∂z,(cid:18),∂φ,∂r,+,1,eni,∂pi⊥,∂r,(cid:19)(cid:21),dz,dz,d3v,mi[(v,·,ˆr)2,−,((v,×,ˆb),·,ˆr)2]Cii,,(4.18),where,we,have,neglected,most,collision,operators,due,to,our,assumption,(2.9).,Due,to,the,integrals,over,the,gyrophase,,the,terms,with,the,time,derivative,and,the,ion-ion,collision,2D,drift,kinetic,model,with,periodic,boundary,conditions,7,operator,vanish,to,the,order,of,interest,,leaving,d3v,miv,·,ˆr(v,×,ˆb),·,ˆr,gi,(cid:34)(cid:90),Lz,(cid:90),(cid:90),Lz,(cid:90),dz,0,(cid:39),−,∂,∂r,1,4Ωi,(cid:90),Lz,0,dz,d3v,miv,·,ˆr[(v,·,ˆr)2,−,((v,×,ˆb),·,ˆr)2],gi,0,(cid:20),∂φ,∂r,(cid:18),∂φ,∂z,+,1,eni,∂pi⊥,∂z,(cid:19),+,∂φ,∂z,(cid:18),∂φ,∂r,+,1,eni,∂pi⊥,∂r,(cid:19)(cid:21),dz.,(4.19),nimi,2B2,(cid:35),There,is,still,another,integral,over,velocity,to,be,calculated.,We,can,use,the,moment,(m2,i,/3eB)[((v,×,ˆb),·,ˆr)3,+,3(v,·,ˆr)2(v,×,ˆb),·,ˆr],of,equation,(4.2),to,ﬁnd,(cid:90),Lz,(cid:90),dz,0,d3v,miv,·,ˆr[(v,·,ˆr)2,−,((v,×,ˆb),·,ˆr)2],gi,(cid:39),−,(cid:90),Lz,0,2pi⊥,B,∂φ,∂z,dz.,(4.20),With,this,result,and,employing,integration,by,parts,in,z,to,write,(cid:34)(cid:90),Lz,0,∂,∂r,2pi⊥,B,∂φ,∂z,(cid:35),dz,(cid:39),=,(cid:90),Lz,0,(cid:90),Lz,0,2,B,2,B,(cid:18),∂pi⊥,∂r,(cid:18),∂pi⊥,∂r,∂φ,∂z,∂φ,∂z,+,pi⊥,(cid:19),∂2φ,∂r∂z,dz,−,∂pi⊥,∂z,∂φ,∂r,(cid:19),dz,,(4.21),equation,(4.19),simpliﬁes,to,(cid:90),Lz,(cid:90),dz,0,d3v,miv,·,ˆr(v,×,ˆb),·,ˆr,gi,=,−,(cid:90),Lz,0,nimi,B2,∂φ,∂z,(cid:18),∂φ,∂r,+,1,eni,∂pi⊥,∂r,(cid:19),dz.,(4.22),The,electron,integral,in,equation,(4.17),can,be,calculated,using,the,same,procedure,that,has,led,to,equation,(4.22),,and,it,turns,out,to,be,negligible,due,to,the,smallness,of,me/mi.,Substituting,equation,(4.22),into,equation,(4.17),and,then,equation,(4.17),into,equa-,tion,(4.10),,and,using,the,same,simpliﬁed,ion-neutral,and,ionization,collision,operators,that,we,have,used,in,previous,reports,,Cin,:=,−Rin(nngi,−,nifn),,Ci,ion,:=,neRionfn,,(4.23),we,ﬁnd,the,ﬁnal,equation,for,φ(r,,0,,t),,∂,∂t,(cid:90),Lz,0,(cid:20),nimi,B2,(cid:18),∂φ,∂r,+,1,eni,(cid:18),∂φ,∂r,(cid:19)(cid:21),dz,∂pi⊥,∂r,+,1,eni,∂pi⊥,∂r,(cid:19),(cid:35),dz,+,1,eni,∂pi⊥,∂r,(cid:19),dz,(cid:34)(cid:90),Lz,0,nimi,B2,niminnRin,B2,∂φ,∂z,(cid:18),∂φ,∂r,(cid:40),∂,∂r,−,+,−,+,1,B,∂,∂r,(cid:90),Lz,0,(cid:90),Lz,0,niminn(Rin,+,Rion)unz,B,dz,+,(cid:90),Lz,0,Fext,·,ˆz,B,(cid:41),dz,=,0.,(4.24),Note,that,in,several,places,we,have,used,the,approximation,ˆb,×,ˆr,(cid:39),ˆz.,Equation,(4.24),is,small,by,a,factor,of,(ρi/Lr)2,compared,to,the,parallel,current,term,in,the,current,conservation,equation,(4.8),,of,order,(ρi/Lr)(enevti/Lz).,Thus,,in,principle,,8,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,to,keep,these,terms,we,would,need,to,keep,terms,up,to,order,(ρi/Lr)3,in,this,equation.,Thankfully,,we,do,not,need,to,do,that,because,we,now,have,equation,(4.24),in,explicit,form.,We,can,add,equation,(4.24),to,our,set,of,equations,in,a,consistent,manner.,We,ﬁnish,by,relating,equation,(4.24),to,momentum,conservation,,a,connection,that,we,0,J⊥,·,ˆr,dz,0,J⊥,·,ˆr,dz,mentioned,at,the,end,of,section,2.,Equation,(4.24),is,a,radial,derivative,of,(cid:82),Lz,(see,equation,(4.10)).,Then,,equation,(4.24),can,be,integrated,to,show,that,(cid:82),Lz,is,a,constant,that,we,will,call,Ir,,giving,∂,∂t,(cid:90),Lz,0,nimiui⊥,·,ˆz,dz,−,(cid:32)(cid:90),Lz,0,∂,∂r,nimi,B,∂φ,∂z,(cid:33),ui⊥,·,ˆz,dz,=,−IrB,−,+,(cid:90),Lz,0,(cid:90),Lz,0,niminnRin(ui⊥,·,ˆz,−,unz),dz,niminnRionunz,dz,+,(cid:90),Lz,0,Fext,·,ˆz,dz.,(4.25),Here,we,have,used,the,ˆz,projection,of,equation,(4.7),to,rewrite,∂φ/∂r,as,a,function,of,ui⊥,·,ˆz.,Equation,(4.25),is,the,perpendicular,momentum,balance,in,the,z,direction,,and,it,includes,the,magnetic,force,due,to,the,radial,current,Ir.,This,magnetic,force,is,zero,in,tokamaks,,where,the,radial,current,vanishes.,5.,Moment,drift,kinetics,The,moment,drift,kinetic,formulation,of,the,problem,with,periodic,boundary,condi-,tions,is,thus,the,equations,in,report,2047357-TN-07-02,plus,our,new,equation,(4.24),for,φ(r,,0,,t).,The,perpendicular,pressure,appearing,in,this,equation,can,be,calculated,from,the,normalized,distribution,function,using,the,formula,ps⊥[ns,,vts,,Fs](r,,z),:=,πnsmsv2,ts,(cid:90),∞,(cid:90),∞,dw(cid:107),−∞,0,dw⊥,w3,⊥Fs(r,,z,,w(cid:107),,w⊥,,t).,(5.1),6.,Discussion,Note,that,the,addition,of,equation,(4.24),has,only,been,possible,because,we,evolve,the,densities,of,ions,and,electrons,independently,of,their,normalized,distribution,functions.,Had,we,proposed,to,evolve,the,unnormalized,ion,and,electron,distribution,functions,,it,would,not,have,been,possible,to,have,an,independent,higher-order,current,conservation,equation,because,it,would,not,be,consistent,with,the,density,that,arises,from,the,time,evolution,of,the,lowest,order,kinetic,equations.,When,implementing,the,equations,proposed,in,this,report,,it,is,important,to,ensure,that,there,is,z-variation,of,density,,temperature,and,ﬂows.,Otherwise,,the,equations,become,trivial.,When,connected,to,the,open,ﬁeld,line,region,(the,topic,of,a,future,report),,the,z-variation,will,arise,naturally,due,to,the,wall,boundary,conditions.,In,the,absence,of,open,ﬁeld,lines,,one,can,use,sources,and,sinks,with,that,are,not,uniform,in,z,(i.e.,excess,ionization,in,the,region,close,to,the,divertor,,where,most,neutrals,are).,REFERENCES,Hazeltine,,R.D.,1973,Recursive,derivation,of,drift-kinetic,equation.,Plasma,Phys.,15,,77–80.,Parra,,F.I.,&,Catto,,P.J.,2008,Limitations,of,gyrokinetics,on,transport,time,scales.,Plasma,Phys.,Control.,Fusion,50,,065014.,2D,drift,kinetic,model,with,periodic,boundary,conditions,9,Parra,,F.I.,&,Catto,,P.J.,2009,Vorticity,and,intrinsic,ambipolarity,in,turbulent,tokamaks.,Plasma,Phys.,Control.,Fusion,51,,095008.

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