TN-07_2DDriftKineticWallBoundaryConditions ========================================== .. meta:: :description: technical note :keywords: 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.,AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGhiZnd3MzBrJhi/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXEAuujet+O7mV1bX1jfxmYWt7Z3evuH/Q0FGiGNZZJCLVCqhGwSXWDTcCW7FCGgYCm8Hoduo3H1FpHsl7M47RD+lA8j5n1Fip9tQtltyyOwNZJl5GSpCh2i1+dXoRS0KUhgmqddtzY+OnVBnOBE4KnURjTNmIDrBtqaQhaj+dHTohJ1bpkX6kbElDZurviZSGWo/DwHaG1Az1ojcV//Paielf+ymXcWJQsvmifiKIicj0a9LjCpkRY0soU9zeStiQKsqMzaZgQ/AWX14mjbOyd1k+r12UKjdZHHk4gmM4BQ+uoAJ3UIU6MEB4hld4cx6cF+fd+Zi35pxs5hD+wPn8AejVjQQ=xAAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomKeix68diC/YA2lM120q7dbMLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVx71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJRvyxXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MH6lmNBQ==yAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGhiZnd3MzJrghi/w4kFjvPpJ3vwbB9iDgpV0UqnqTndXEAuujet+O7mV1bX1jfxmYWt7Z3evuH/Q0FGiGNZZJCLVCqhGwSXWDTcCW7FCGgYCm8Hoduo3H1FpHsl7M47RD+lA8j5n1Fip9tQtltyyOwNZJl5GSpCh2i1+dXoRS0KUhgmqddtzY+OnVBnOBE4KnURjTNmIDrBtqaQhaj+dHTohJ1bpkX6kbElDZurviZSGWo/DwHaG1Az1ojcV//Paielf+ymXcWJQsvmifiKIicj0a9LjCpkRY0soU9zeStiQKsqMzaZgQ/AWX14mjbOyd1k+r12UKjdZHHk4gmM4BQ+uoAJ3UIU6MEB4hld4cx6cF+fd+Zi35pxs5hD+wPn8AevdjQY=zAAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomKeix68diC/YA2lM120q7dbMLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVVa9UdivuDGSZeDkpQ45ar/TV7ccsjVAaJqjWHc9NjJ9RZTgTOCl2U40JZSM6wI6lkkao/Wx26IScWqVPwljZkobM1N8TGY20HkeB7YyoGepFbyr+53VSE974GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7QheIsvL5PmecW7qlzUL8vV2zyOAhzDCZyBB9dQhXuoQQMYIDzDK7w5j86L8+58zFtXnHzmCP7A+fwB372M/g==rAAAB7HicbVBNS8NAEN3Ur1q/qh69LBbBU0lU1GPRi8cKpi20oWy203bpZhN2J0IN/Q1ePCji1R/kzX/jts1BWx8MPN6bYWZemEhh0HW/ncLK6tr6RnGztLW9s7tX3j9omDjVHHwey1i3QmZACgU+CpTQSjSwKJTQDEe3U7/5CNqIWD3gOIEgYgMl+oIztJLfeQJk3XLFrboz0GXi5aRCctS75a9OL+ZpBAq5ZMa0PTfBIGMaBZcwKXVSAwnjIzaAtqWKRWCCbHbshJ5YpUf7sbalkM7U3xMZi4wZR6HtjBgOzaI3Ff/z2in2r4NMqCRFUHy+qJ9KijGdfk57QgNHObaEcS3srZQPmWYcbT4lG4K3+PIyaZxVvcvq+f1FpXaTx1EkR+SYnBKPXJEauSN14hNOBHkmr+TNUc6L8+58zFsLTj5zSP7A+fwB6xqOxA==⇣,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,∂r,∂φ,∂z,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. :pdfembed:`src:_static/TN-07_2DDriftKineticWallBoundaryConditions.pdf, height:1600, width:1100, align:middle`