TN-05_1DDriftKineticWallBoundaryConditions
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   :description: technical note

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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.

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