TN-08_Numerical11DDriftKineticModelWallBoundaryConditions
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   :description: technical note

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Report,2047357-TN-02,,M2.4,Numerical,study,of,1+1D,drift,kinetic,model,with,wall,boundary,conditions,M.,Barnes1,,F.,I.,Parra1,,M.,R.,Hardman1,and,J.,Omotani2,1,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Clarendon,Laboratory,,Parks,Road,,Oxford,OX1,3PU,,United,Kingdom,2,Culham,Centre,for,Fusion,Energy,,Culham,Science,Centre,,Abingdon,,Oxon,,OX14,3DB,,United,Kingdom,E-mail:,michael.barnes@physics.ox.ac.uk,1.,Introduction,Magnetic,ﬁeld,lines,beyond,the,last-closed-ﬂux-surface,begin,and,end,on,the,vessel,wall.,For,these,ﬁeld,lines,,the,periodic,boundary,condition,along,the,ﬁeld,line,employed,in,our,previous,numerical,treatment,of,parallel,dynamics,[1,,2,,3],must,be,modiﬁed.,To,that,end,we,summarise,in,this,document,the,numerical,algorithm,we,employ,to,solve,parallel-,to-the-ﬁeld,dynamics,for,a,standard,,i.e.,,not,moment-based,,drift,kinetic,model,with,wall,boundary,conditions.,A,brief,summary,of,the,1+1D,model,and,the,wall,boundary,conditions,is,given,before,describing,code,normalisation,and,implementation,details.,An,analytical,solution,is,provided,to,serve,as,a,code,benchmark,,and,results,are,presented,to,demonstrate,that,the,code,behaves,as,expected.,2.,Model,equations,A,detailed,derivation,of,the,drift,kinetic,model,we,consider,is,provided,in,our,May,2021,report,[4].,For,the,Reader’s,convenience,we,produce,an,overview,of,this,drift,kinetic,model,,noting,any,additional,simplifying,assumptions.,The,system,we,consider,consists,of,a,single,ion,species,of,charge,e,and,mass,mi,,a,single,neutral,species,of,mass,mn,=,mi,,and,an,electron,species,modelled,as,having,a,Boltzmann,response,,all,immersed,in,a,straight,,uniform,magnetic,ﬁeld,in,the,z,direction.,We,allow,for,charge,exchange,collisions,between,ions,and,neutrals,and,ionization,collisions,involving,ions,,electrons,and,neutrals,,but,do,not,account,for,intra-species,collisions.,Finally,,we,assume,that,the,plasma,is,homogeneous,in,the,plane,perpendicular,to,the,magnetic,ﬁeld.,With,these,assumptions,,our,model,system,of,equations,is,∂fi,∂t,+,v(cid:107),∂fi,∂z,−,e,mi,∂φ,∂z,∂fi,∂v(cid:107),=,−Rin,(nnfi,−,nifn),+,Rionnefn,,∂fn,∂t,+,v(cid:107),∂fn,∂z,=,−Rin,(nifn,−,nnfi),−,Rionnefn,,1,(1),(2),Numerical,study,of,drift,kinetic,model,with,wall,BCs,and,ns(z,,t),=,(cid:90),∞,−∞,dv(cid:107)fs(z,,v(cid:107),,t),,ni,=,ne,=,Ne,exp,(cid:19),,,(cid:18),eφ,Te,2,(3),(4),.,=,(cid:82),dϑdv⊥v⊥Fs,the,marginalized,particle,distribution,function,for,species,s,,with,fs,v(cid:107),and,v⊥,the,components,of,the,particle,velocity,parallel,and,perpendicular,to,the,magnetic,ﬁeld,,respectively,,ϑ,the,gyro-angle,,t,the,time,,φ,the,electrostatic,potential,,and,Rin,and,Rion,charge,exchange,and,ionization,collision,frequency,factors.,2.1.,Wall,BCs,Complicated,dynamics,can,occur,very,close,to,the,wall,,leading,to,a,hierarchy,of,narrow,layers,with,diﬀerent,dynamics.,Derivation,of,the,equations,that,describe,these,‘sheaths’,is,an,ongoing,research,activity,in,the,magnetic,conﬁnement,fusion,community.,We,avoid,such,complications,here,by,considering,a,plasma,domain,whose,boundaries,in,z,are,the,entrances,to,the,sheaths,furthest,from,the,wall.,Ions,that,exit,the,simulation,domain,are,assumed,to,continue,on,to,the,wall,,where,they,recombine.,As,a,result,,no,ions,enter,the,domain,from,the,walls,,giving,a,zero,incoming,BC,for,the,ions:,Fi(z,=,0,,v(cid:107),>,0,,v⊥,,t),=,0,=,Fi(z,=,Lz,,v(cid:107),<,0,,v⊥,,t).,(5),Neutrals,that,leave,the,domain,are,assumed,to,hit,the,wall,and,thermalise,at,the,temperature,of,the,wall,,Tw.,Ions,that,recombine,at,the,wall,also,re-enter,as,neutrals.,The,resulting,boundary,condition,on,the,neutrals,is,Fn(z,=,0,,v(cid:107),>,0,,v⊥,,t),=,Γ0FKw(v(cid:107),,v⊥),,Fn(z,=,Lz,,v(cid:107),<,0,,v⊥,,t),=,ΓLz,FKw(v(cid:107),,v⊥),,(6),where,FKw(v(cid:107),,v⊥),(cid:19)2,.,=,3,π,(cid:18),mi,2Tw,(cid:12),(cid:12),(cid:12)v(cid:107),(cid:12),(cid:107),+,v2,v2,⊥,(cid:113),,exp,−,(cid:16),mi,(cid:17),,,(cid:107),+,v2,v2,⊥,2Tw,is,the,Knudsen,cosine,distribution,,and,.,=,Γ0,(cid:88),2π,(cid:90),0,(cid:90),∞,dv(cid:107),s=i,n,−∞,0,dv⊥v⊥,(cid:12),(cid:12)v(cid:107),(cid:12),(cid:12),Fs(z,=,0,,v(cid:107),,v⊥,,t),and,(cid:88),.,=,2π,ΓLz,(cid:90),∞,(cid:90),∞,dv(cid:107),s=i,n,0,0,dv⊥v⊥,(cid:12),(cid:12)v(cid:107),(cid:12),(cid:12),Fs(z,=,Lz,,v(cid:107),,v⊥,,t),(7),(8),(9),are,the,combined,ﬂuxes,of,neutrals,and,ions,towards,the,walls,at,z,=,0,and,z,=,Lz,,respectively.,Numerical,study,of,drift,kinetic,model,with,wall,BCs,3,We,next,marginalise,the,above,distribution,functions,by,integrating,over,gyro-angle,and,v⊥,to,get,fi(z,=,0,,v(cid:107),>,0,,v⊥,,t),=,0,=,fi(z,=,Lz,,v(cid:107),<,0,,v⊥,,t),(10),and,with,fn(z,=,0,,v(cid:107),>,0,,t),=,Γ0fKw(v(cid:107)),,fn(z,=,Lz,,v(cid:107),<,0,,t),=,ΓLz,fKw(v(cid:107)),,(11),fKw(v(cid:107)),.,=,2π,(cid:90),∞,0,dv⊥v⊥FKw(v(cid:107),,v⊥),=,3,(cid:19)3/2,√,π,(cid:18),mi,2Tw,(cid:12),(cid:12)v(cid:107),(cid:12),(cid:12),erfc,(cid:18)(cid:114),mi,2Tw,(cid:19),(cid:12),(cid:12)v(cid:107),(cid:12),(cid:12),(12),and,erfc,the,complementary,error,function.,The,ﬂuxes,toward,the,wall,at,the,domain,boundaries,can,be,recast,in,terms,of,the,marginalised,distribution,function:,Γ0,=,(cid:90),0,(cid:88),s=i,n,−∞,dv(cid:107),(cid:12),(cid:12)v(cid:107),(cid:12),(cid:12),fs(z,=,0,,v(cid:107),,t),ΓLz,=,(cid:90),∞,(cid:88),s=i,n,0,dv(cid:107),(cid:12),(cid:12)v(cid:107),(cid:12),(cid:12),fs(z,=,Lz,,v(cid:107),,t).,(13),(14),and,2.2.,Density,evolution,If,we,consider,the,evolution,of,the,line-averaged,species,densities,,ns,,we,obtain,∂ns,∂t,=,1,Lz,(Γs(z,=,0),−,Γs(z,=,Lz)),±,Rionninn,,(15),with,the,+,and,-,signs,corresponding,to,ions,and,to,neutrals,,respectively.,For,the,ions,and,Γi(0),=,Γi(Lz),=,(cid:90),0,−∞,(cid:90),∞,0,dv(cid:107)v(cid:107)fi(z,=,0,,v(cid:107)),<,0,,dv(cid:107)v(cid:107)fi(z,=,Lz,,v(cid:107)),>,0,,where,we,have,used,the,boundary,conditions,on,fi,given,by,expression,(10).,For,the,neutrals,and,Γn(0),=,=,(cid:90),0,−∞,(cid:90),0,−∞,dv(cid:107)v(cid:107)fn(z,=,0,,v(cid:107)),+,Γ0,(cid:90),∞,0,dv(cid:107)v(cid:107)fKw(v(cid:107)),dv(cid:107)v(cid:107)fn(z,=,0,,v(cid:107)),+,Γ0,,Γn(Lz),=,=,(cid:90),∞,0,(cid:90),∞,0,dv(cid:107)v(cid:107)fn(z,=,Lz,,v(cid:107)),+,ΓLz,(cid:90),0,−∞,dv(cid:107)v(cid:107)fKw(v(cid:107)),dv(cid:107)v(cid:107)fn(z,=,Lz,,v(cid:107)),−,ΓLz,,,(16),(17),(18),(19),Numerical,study,of,drift,kinetic,model,with,wall,BCs,4,where,we,have,used,the,boundary,conditions,on,fn,given,by,expression,(11),and,the,fact,that,(cid:82),∞,0,dv(cid:107)v(cid:107)fKw,=,−,(cid:82),0,−∞,dv(cid:107)v(cid:107)fKw,=,1.,Combining,the,results,for,the,ion,and,neutral,densities,and,noting,that,the,ionization,contributions,to,the,ion,and,neutral,densities,cancel,,we,obtain,a,constraint,on,the,evolution,of,the,total,(species-summed),,line-averaged,density:,∂,∂t,(cid:88),s=i,n,ns,=,=,1,Lz,1,Lz,=,1,Lz,=,0.,(cid:88),(Γs(z,=,0),−,Γs(z,=,Lz)),s=i,n,(cid:32),(cid:88),(cid:90),0,s=i,n,−∞,v(cid:107)fs(z,=,0,,v(cid:107)),+,Γ0,−,v(cid:107)fs(z,=,Lz,,v(cid:107)),+,ΓLz,(cid:33),(cid:90),∞,(cid:88),s=i,n,0,(−Γ0,+,Γ0,−,ΓLz,+,ΓLz,),(20),This,is,a,consequence,of,the,assumed,wall,boundary,condition,,which,dictates,that,any,particles,leaving,the,domain,re-enter,as,neutrals,(and,thus,total,density,is,conserved).,It,can,(and,will),be,used,to,test,the,numerical,implementation,of,the,wall,boundary,condition.,2.3.,Normalisation,We,normalize,Eqs.,(1)-(4),by,deﬁning,˜fs,.,=,fs,√,cs,π,Ne,,,.,=,t,˜t,.,=,˜z,.,=,˜v(cid:107),.,=,˜ns,.,=,˜φ,,,,,,,cs,Lz,z,Lz,v(cid:107),cs,ns,Ne,eφ,Te,NeLz,cs,,,,,˜Rin,.,=,Rin,,,and,˜Rion,.,=,Rion,NeLz,cs,,,(21),(22),(23),(24),(25),(26),(27),(28),Numerical,study,of,drift,kinetic,model,with,wall,BCs,with,cs,.,=,(cid:112)2Te/ms.,In,terms,of,these,normalised,quantities,,Eqs,(1)-(4),become,∂,˜fi,∂˜t,+,˜v(cid:107),∂,˜fi,∂,˜z,−,1,2,∂,˜φ,∂,˜z,∂,˜fi,∂˜v(cid:107),(cid:16),=,−,˜Rin,˜nn,˜fi,−,˜ni,˜fn,(cid:17),+,˜Rion˜ni,˜fn,,∂,˜fn,∂˜t,+,˜v(cid:107),∂,˜fn,∂,˜z,(cid:16),=,−,˜Rin,˜ni,˜fn,−,˜nn,˜fi,(cid:17),−,˜Rion˜ni,˜fn,,e,˜φ,=,˜ni,=,1,√,π,(cid:90),∞,−∞,d˜v(cid:107),˜fi,,˜nn,=,1,√,π,(cid:90),∞,−∞,d˜v(cid:107),˜fn.,The,normalised,forms,for,the,wall,boundary,conditions,are,˜fi,(cid:0)˜z,=,0,,˜v(cid:107),>,0,,˜t(cid:1),=,0,=,˜fi,(cid:0)˜z,=,1,,˜v(cid:107),<,0,,˜t(cid:1),and,and,5,(29),(30),(31),(32),(33),˜fn,(cid:0)˜z,=,0,,˜v(cid:107),>,0,,˜t(cid:1),=,˜Γ0(˜t),˜fKw(˜v(cid:107)),,˜fn,(cid:0)˜z,=,1,,˜v(cid:107),<,0,,˜t(cid:1),=,˜ΓLz,(˜t),˜fKw(˜v(cid:107)),,(34),where,and,˜fKw(˜v(cid:107)),=,c2,nfKw(v(cid:107)),=,√,3,π,˜T,3/2,w,(cid:12),(cid:12)˜v(cid:107),(cid:12),(cid:12),erfc,(cid:32),(cid:12),(cid:12),(cid:12)˜v(cid:107),(cid:12),(cid:112),˜Tw,(cid:33),,,˜Γ0(˜t),=,(cid:90),0,(cid:88),s=i,n,−∞,˜ΓLz,(˜t),=,(cid:90),∞,(cid:88),s=i,n,0,d˜v(cid:107),(cid:12),(cid:12)˜v(cid:107),(cid:12),(cid:12),˜fs(˜z,=,0,,˜v(cid:107),,˜t),,d˜v(cid:107),(cid:12),(cid:12)˜v(cid:107),(cid:12),(cid:12),˜fs(˜z,=,1,,˜v(cid:107),,˜t),,(35),(36),(37),with,˜Tw,.,=,Tw/Te.,3.,Analytical,solution,To,derive,an,analytical,solution,for,the,electrostatic,potential,for,our,system,,we,neglect,charge,exchange,collisions,and,replace,the,ionization,source,term,appearing,in,Eq.,(1),with,a,simpliﬁed,source.,In,particular,,we,follow,the,approach,of,Ref.,[5],and,assume,that,ionization,occurs,at,a,constant,rate,and,gives,birth,to,ions,with,zero,parallel,velocity.,The,resulting,ion,kinetic,equation,is,∂fi,∂t,+,v(cid:107),∂fi,∂z,−,e,mi,∂φ,∂z,∂fi,∂v(cid:107),=,RionN,2,e,δ(v(cid:107)),,(38),with,δ(v(cid:107)),the,Dirac,delta,distribution.,With,these,assumptions,,the,ion,kinetic,equation,decouples,from,the,kinetic,equation,for,the,neutrals,and,makes,the,problem,tractable.,Numerical,study,of,drift,kinetic,model,with,wall,BCs,6,Ions,that,are,formed,at,a,given,spatial,location,z0,are,accelerated,by,the,parallel,electric,ﬁeld,setup,by,the,wall,BC,toward,the,walls.,The,parallel,speed,of,this,ion,after,travelling,to,a,location,z,closer,to,the,wall,,v(cid:107)(z),is,obtained,via,conservation,of,energy:,miv(cid:107)(z)2,2,=,e,(φ(z0),−,φ(z)),.,(39),In,steady-state,,the,ﬂux,of,ions,with,speeds,in,the,range,v(cid:107),to,v(cid:107),+,dv(cid:107),through,the,location,z,must,equal,the,rate,at,which,these,ions,are,generated,between,z0,and,z0,+dz0:,v(cid:107)(z)fi(v(cid:107)(z))dv(cid:107),=,N,2,e,Riondz0.,Combining,Eqs.,(39),and,(40),and,imposing,quasineutrality,gives,Ne,exp,(cid:19),(cid:18),eφ(z),Te,(cid:90),=,dv(cid:107)fi(z,,v(cid:107)),=,(cid:90),Lz/2,z,N,2,(cid:114),mi,2e,(cid:114)mi,2e,e,Rion,(cid:112)φ(z0),−,φ(z),N,2,e,Rion,(cid:112)φ(z0),−,φ(z),dz0,(cid:90),φ(Lz/2),=,φ(z),dz0,dφ(z0),dφ(z0),,(40),(41),which,is,an,implicit,equation,for,φ(z).,Setting,our,gauge,so,that,φ(Lz/2),=,0,and,.,=,−eφ(z)/Te,,we,have,deﬁning,x,.,=,−eφ(z0)/Te,and,y,where,Ne,exp,(−y),=,(cid:90),y,0,√,h(x),y,−,x,dx,,h(x),.,=,N,2,e,Rion,ci,dz0,dx,.,(42),(43),The,density,integral,appearing,in,Eq.,(42),is,of,the,form,of,an,Abel,transform,[6],,which,can,be,inverted,to,yield,h(x),=,Ne,π,d,dx,(cid:90),x,0,dy,exp,(−y),√,x,−,y,Equating,(43),and,(44),yields,an,expression,for,dz/dx:,dz,dx,=,ci,NeRionπ,d,dx,(cid:90),x,0,dy,exp,(−y),√,x,−,y,.,Integrating,with,respect,to,x,yields,an,implicit,expression,for,φ(z):,z,−,Lz,2,=,±,ci,πRionNe,=,±,2ci,πRionNe,(cid:90),x,dy,0,,(cid:115),D,,−,exp,(−y),√,x,−,y,,,,,eφ(z),Te,(44),(45),(46),with,D(a),=,exp(−a2),(cid:82),a,z,>,Lz/2,and,the,−,sign,corresponds,to,z,<,Lz/2.,0,exp(b2)db,the,Dawson,function,,the,+,sign,corresponds,to,Numerical,study,of,drift,kinetic,model,with,wall,BCs,7,4.,Numerical,implementation,A,detailed,description,of,the,time,and,space,discretisation,employed,in,the,code,is,given,in,[2],,and,the,code,itself,is,publicly,available,at,https://github.com/mabarnes/,moment_kinetics.,Here,we,focus,on,the,numerical,treatment,of,the,wall,boundary,conditions,encapsulated,in,Eqs.,(10)-(14).,Note,that,the,boundary,condition,for,the,neutrals,at,z,=,0,for,v(cid:107),>,0,depends,on,the,neutral,distribution,function,at,z,=,0,for,v(cid:107),<,0,through,Γ0.,This,distribution,function,requires,speciﬁcation,of,the,neutral,boundary,condition,for,z,=,Lz,and,v(cid:107),<,0,,which,itself,depends,on,the,distribution,function,at,z,=,Lz,for,v(cid:107),>,0.,Due,to,the,explicit,time,advance,employed,in,the,code,,this,inter-dependence,is,straightforward,to,accommodate.,The,procedure,employed,is,to,ﬁrst,solve,for,fs(z,,v(cid:107),,tm+1),at,all,(z,,v(cid:107)),locations,except,for,(z,=,0,,v(cid:107),>,0),and,(z,=,Lz,,v(cid:107),<,0).,As,the,time,advance,is,explicit,,this,can,be,achieved,given,{fs(z,,v(cid:107),,tm)}s=i,n.,The,solutions,for,fs(z,=,0,,v(cid:107),<,0,,tm+1),and,fs(z,=,Lz,,v(cid:107),>,0,,tm+1),are,then,used,to,compute,Γ0(tm+1),and,ΓLz,(tm+1).,These,ﬂuxes,can,then,be,used,in,Eq.,(11),to,compute,the,boundary,values,fn(z,=,0,,v(cid:107),>,0,,tm+1),and,fn(z,=,Lz,,v(cid:107),<,0,,tm+1).,It,is,worth,noting,that,in,order,to,ensure,the,conservation,of,the,ﬁeld-line-averaged,,species-summed,density,(as,shown,in,Sec.,2.2),,care,must,be,taken,to,ensure,that,the,properties,(cid:82),∞,−∞,dv(cid:107)v(cid:107)fKw,=,1,are,exactly,satisﬁed,by,the,numerics.,This,is,achieved,in,our,case,by,ﬁrst,obtaining,the,numerical,approximation,to,these,integrals,and,then,deﬁning,a,modiﬁed,fKw,that,is,normalised,by,this,result.,0,dv(cid:107)v(cid:107)fKw,=,−,(cid:82),0,5.,Numerical,results,We,ﬁrst,compare,our,simulation,results,to,the,analytical,solution,presented,in,Sec.,3.,The,ion,equation,being,solved,in,the,code,is,∂,˜fi,∂˜t,+,˜v(cid:107),∂,˜fi,∂,˜z,−,1,2,∂,˜φ,∂,˜z,∂,˜fi,∂˜v(cid:107),=,˜Rion,ci,vδ,exp,(cid:18),−˜v2,(cid:107),(cid:19),,,c2,i,v2,δ,(47),(cid:107)/v2,where,we,have,approximated,δ(v(cid:107)),≈,(1/,vδ,(cid:28),ci.,In,terms,of,normalised,quantities,,the,solution,(46),is,πvδ),exp(−v2,δ,),with,the,proviso,that,√,˜z,=,1,2,±,2,π,˜Rion,D,(cid:18)(cid:113),(cid:19),−,˜φ,.,(48),For,our,benchmark,simulation,,we,use,a,Chebyshev,pseudo-spectral,method,in,both,z,and,v(cid:107).,The,z,grid,consists,of,two,z,elements,,each,containing,nine,grid,points,,and,the,v(cid:107),grid,consists,of,ten,elements,,each,containing,seventeen,grid,points.,The,wall,temperature,Tw,is,taken,to,be,equal,to,Te,,the,source,width,parameter,is,vδ,=,0.1ci,,˜Rion,=,0.688,and,the,ions,are,initialised,to,a,Maxwellian,velocity,distribution,with,initial,temperature,Te,and,an,initial,density,with,Gaussian,distribution,in,z.,The,resulting,Numerical,study,of,drift,kinetic,model,with,wall,BCs,8,Figure,1.,The,steady-state,electrostatic,potential,proﬁle,obtained,by,solving,Eq.,(47),(blue,line),and,by,using,the,analytical,solution,of,Eq.,(46),(red,line).,steady-state,solution,for,˜φ(˜z),,along,with,a,comparison,to,the,analytical,solution,(48),is,given,in,Fig.,1.,Figure,2.,Time,trace,of,the,species-summed,,line-averaged,density,,normalised,by,its,initial,value.,We,next,consider,the,original,model,system,of,equations,given,in,Sec.,2.3.,We,take,the,normalised,collision,frequency,factors,to,be,˜Rion,=,˜Rin,=,2,and,otherwise,use,the,same,numerical,parameters,and,initial,conditions,as,we,did,for,the,analytical,benchmark,,except,that,we,use,eight,z,elements,rather,than,two.,A,cross-section,of,the,numerical,results,are,presented,in,Figures,2,-,4.,In,Fig.,2,we,consider,the,evolution,of,the,species-summed,,line-averaged,density.,As,shown,in,Sec.,3,,this,density,should,be,conserved,,and,this,is,indeed,the,case,in,the,simulation.,The,steady-state,electrostatic,potential,and,particle,distribution,functions,for,ions,and,neutrals,are,given,in,Figs.,3,and,4.,As,might,be,expected,,the,loss,of,electrons,to,the,wall,gives,rise,to,an,electric,Numerical,study,of,drift,kinetic,model,with,wall,BCs,9,ﬁeld,that,pulls,ions,out,of,the,simulation,domain.,These,ions,are,replaced,by,neutrals,traveling,back,into,the,simulation,domain,,leading,to,the,regions,of,high,neutral,density,near,the,walls,(with,neutral,velocities,away,from,the,walls),visible,in,Fig.,4.,The,phase,space,structure,in,the,ion,distribution,function,is,likely,a,result,of,the,ionisation,of,neutrals,in,the,high-density,regions,near,the,walls,,followed,by,an,acceleration,of,these,ions,towards,the,nearest,wall.,This,reduces,their,initial,speed,away,from,the,walls,and,eventually,changes,their,direction,of,motion,and,accelerates,them,into,the,wall,near,which,they,were,generated.,Conversely,,ions,generated,suﬃciently,far,from,the,nearest,wall,can,cross,the,symmetry,point,in,z,and,are,accelerated,towards,the,wall,furthest,from,where,they,were,generated.,As,an,aside,,we,note,that,the,kinetic,Bohm,criterion,,which,for,the,case,of,Boltzmann,electrons,can,be,written,[5],(cid:90),dv(cid:107),c2,i,v2,(cid:107),fi(v(cid:107)),≤,2ni,,(49),is,satisﬁed,for,this,simulation.,Figure,3.,Electrostatic,potential,proﬁle,in,steady,state.,6.,Future,plans,Now,that,we,have,a,working,implementation,of,the,standard,drift,kinetic,equation,with,wall,boundary,conditions,,we,will,move,on,to,implementation,of,the,wall,boundary,conditions,for,the,moment-kinetic,system,of,equations.,Numerical,study,of,drift,kinetic,model,with,wall,BCs,10,Figure,4.,Phase,space,portraits,of,the,steady-state,,normalised,ion,(left),and,neutral,(right),particle,distribution,functions,,˜fi,and,˜fn.,Numerical,study,of,drift,kinetic,model,with,wall,BCs,11,[1],M.,Barnes,,F.,I.,Parra,,and,M.,R.,Hardman.,Numerical,study,of,1d,drift,kinetic,models,with,periodic,boundary,conditions.,Excalibur/Neptune,Report,,2:2047357–TN–01–02,M2.1,,2021.,[2],M.,Barnes,,F.,I.,Parra,,M.,R.,Hardman,,and,J.,Omotani.,Numerical,study,of,1+1d,,moment-based,drift,kinetic,models,with,periodic,boundary,conditions.,Excalibur/Neptune,Report,,4:2047357–,TN–01–02,M2.2,,2021.,[3],M.,Barnes,,F.,I.,Parra,,M.,R.,Hardman,,and,J.,Omotani.,Numerical,study,of,1+1d,,moment-based,drift,kinetic,models,with,periodic,boundary,conditions.,Excalibur/Neptune,Report,,6:2047357–,TN–01–02,M2.3,,2021.,[4],F.,I.,Parra,,M.,Barnes,,and,M.,R.,Hardman.,1d,drift,kinetic,models,with,wall,boundary,conditions.,Excalibur/Neptune,Report,,5:2047357–TN–05–01,M1.3,,2021.,[5],E.,R.,Harrison,and,W.,B.,Thompson.,The,low,pressure,plane,symmetric,discharge.,Proc.,Phys.,Soc.,,74:145,,1959.,[6],N.,H.,Abel.,Journal,f¨ur,die,reine,und,angewandte,Mathematik,,1:153,,1826.

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