TN-01_1DDriftKineticPeriodicBoundaryConditions ============================================== .. meta:: :description: technical note :keywords: REPORT,2047357-TN-01-02,M1.1,1,1D,drift,kinetic,models,with,periodic,boundary,conditions,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,22,February,2021),1.,Introduction,In,this,report,,we,propose,1D,drift,kinetic,equations,to,test,the,possibility,of,extracting,low,order,moments,from,the,distribution,functions,for,implicit,methods.,The,model,that,we,present,here,has,periodic,boundary,conditions,,adequate,for,the,closed,field,line,region,of,the,edge.,We,will,address,wall,boundary,conditions,for,open,field,lines,in,the,reports,for,milestones,M1.3,,M2.4,and,M2.5.,2.,1D,electrostatic,drift,kinetics,We,consider,a,plasma,with,one,ion,species,with,charge,e,and,mass,mi,,electrons,with,charge,e,and,mass,me,,and,one,species,of,neutrals,with,mass,−,mn,=,mi.,(2.1),The,plasma,is,magnetized,by,a,constant,magnetic,field,B,=,Bˆz,,and,we,assume,that,the,plasma,only,varies,along,magnetic,field,lines.,In,this,case,,the,electric,field,produced,by,the,plasma,is,electrostatic,,E,=,(∂φ/∂z)ˆz.,The,potential,φ(z,,t),depends,on,the,position,along,magnetic,field,lines,z,and,on,time,t.,−,If,we,assume,that,the,gyroradii,are,small,compared,to,the,length,scales,of,interest,,and,that,the,gyrofrequencies,are,much,larger,than,the,frequencies,that,we,want,to,model,,,t),of,the,different,species,s,=,(Hazeltine,1973),,the,distribution,functions,fs(z,,v,(cid:107),i,,e,,n,only,depend,on,the,component,of,the,velocity,parallel,to,the,magnetic,field,v,and,the,magnitude,of,the,velocity,perpendicular,to,the,magnetic,field,v,,,and,are,independent,of,the,direction,of,the,velocity,perpendicular,to,the,magnetic,field.,Thus,,the,distribution,functions,that,in,general,can,depend,on,three,spatial,variables,r,,three,components,of,the,velocity,v,and,the,time,t,depend,only,on,z,,v,and,t,,,,v,,,v,⊥,⊥,(cid:107),fs(r,,v,,t),=,fs(z,,v,(cid:107),(cid:107),,,v,⊥,,,t).,⊥,(2.2),(2.3),The,equations,for,the,distribution,functions,of,the,different,species,are,∂fi,∂t,+,v,∂fi,∂z,−,e,mi,∂φ,∂z,(cid:107),∂fi,∂v,(cid:107),=,Cii[fi],+,Cin[fi,,fn],,∂fe,∂t,+,v,∂fe,∂z,(cid:107),+,e,me,∂φ,∂z,∂fe,∂v,(cid:107),and,=,Cee[fe],+,Cei[fe,,fi],+,Cen[fe,,fn],(2.4),∂fn,∂t,+,v,∂fn,∂z,(cid:107),=,Cni[fn,,fi].,(2.5),Here,we,have,included,ion-ion,and,electron-electron,collisions,,modeled,by,the,Fokker-,Planck,collision,operators,Cii[fi],and,Cee[fe],(Rosenbluth,et,al.,1957),,elastic,electron-ion,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,and,electron-neutral,collisions,,modeled,by,the,simplified,Fokker-Planck,collision,operator,Cei[fe,,fi],(Braginskii,1958),and,the,Boltzmann,collision,operator,Cen[fe,,fn],,and,charge-,exchange,collisions,,represented,by,the,simplified,Boltzmann,collision,operators,and,Cin[fi,,fn],=,Cni[fn,,fi],=,(cid:90),(cid:90),−,−,v,Rin(,|,v(cid:48),),[fi(v)fn(v(cid:48)),|,−,−,fi(v(cid:48))fn(v)],d3v(cid:48),(2.6),Rin(,v,|,v(cid:48),|,−,),[fn(v)fi(v(cid:48)),−,fn(v(cid:48))fi(v)],d3v(cid:48),,(2.7),To,simplify,our,equations,,we,assume,that,the,function,Rin,is,constant,(Connor,1977;,Hazeltine,et,al.,1992;,Catto,1994),,finding,and,where,the,densities,are,Cin[fi,,fn],=,Rin,(nnfi,−,−,nifn),Cni[fn,,fi],=,Rin,(nifn,−,−,nnfi),,,ns(z,,t),:=,2π,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,v,fs(z,,v,(cid:107),⊥,⊥,,,v,⊥,,,t).,(2.8),(2.9),(2.10),Note,that,we,can,neglect,the,effect,of,electron,collisions,on,ions,and,on,neutrals,due,to,the,smallness,of,the,electron,mass,(Braginskii,1958).,We,have,also,neglected,neutral-neutral,collisions,because,,in,current,fusion,devices,,the,neutral,density,is,sufficiently,small,that,the,neutral-neutral,collisions,are,rare.,The,kinetic,equations,will,be,solved,in,the,interval,z,periodic,boundary,conditions,at,z,=,0,and,z,=,L,,∈,[0,,L],,and,we,will,impose,fs(z,=,0,,v,(cid:107),,,v,⊥,,,t),=,fs(z,=,L,,v,(cid:107),,,v,⊥,,,t).,Finally,,the,potential,φ(z,,t),is,determined,by,the,quasineutrality,equation,ni,=,ne.,(2.11),(2.12),To,solve,this,equation,,we,need,to,treat,the,equations,implicitly,as,the,potential,enters,only,via,its,effect,on,∂fi/∂t,and,∂fe/∂t.,The,need,to,use,implicit,methods,is,one,of,the,reasons,why,we,are,trying,to,extract,some,of,the,low,order,moments,from,the,distribution,function,,notably,the,density.,Before,we,treat,the,complete,problem,,we,will,simplify,the,treatment,of,electrons,to,obtain,a,system,of,equations,that,can,be,solved,with,an,explicit,time,advance,so,that,we,can,compare,our,implicit,schemes,with,an,explicit,numerical,method.,Instead,of,solving,for,fe,,we,will,use,a,Maxwell-Boltzmann,response,,ne(z,,t),=,Ne,exp,(cid:18),eφ(z,,t),Te,(cid:19),,,(2.13),where,Ne,and,Te,are,constants,(see,Appendix,A,for,a,derivation,of,the,Maxwell-Boltzmann,response).,Moreover,,the,full,Fokker-Planck,ion-ion,collision,operator,Cii[fi],is,a,com-,plicated,integro-differential,operator,that,we,will,not,implement,in,the,first,versions,of,our,drift,kinetic,code,,so,we,do,not,include,it,in,the,equations,for,now.,Thus,,the,final,,,v,simplified,model,for,fi(z,,v,(cid:107),,,t),and,φ(z,,t),is,given,by,the,equations,,,v,⊥,∂fi,∂t,+,v,(cid:107),⊥,∂fi,∂z,−,,,t),,fn(z,,v,(cid:107),∂fi,∂v,(cid:107),e,mi,∂φ,∂z,=,Rin(nnfi,−,−,nifn),,(2.14),1D,drift,kinetic,models,with,periodic,boundary,conditions,∂fn,∂t,+,v,∂fn,∂z,(cid:107),=,Rin(nifn,−,−,nnfi),and,ni,=,Ne,exp,(cid:19),,,(cid:18),eφ,Te,3,(2.15),(2.16),with,periodic,boundary,conditions,(2.11).,This,system,of,equations,can,be,solved,explic-,itly,because,the,simple,electron,model,allows,one,to,obtain,φ,as,a,function,of,ni.,3.,1D,moment,drift,kinetics,Instead,of,solving,for,fs(z,,v,(cid:107),v3,ts(z,,t),ns(z,,t),Fs(z,,w,(cid:107),,,t),:=,,,w,,,v,⊥,⊥,,,t),,we,solve,for,(cid:16),fs,z,,us,(cid:107),(z,,t),+,vts(z,,t)w,(cid:107),,,vts(z,,t)w,(cid:17),,,t,,,⊥,where,we,have,defined,the,normalized,velocities,w,(z,,v,(cid:107),(cid:107),,,t),:=,v,(z,,t),(cid:107),−,us,(cid:107),vts(z,,t),and,the,average,parallel,velocity,w,⊥,(z,,v,⊥,,,t),:=,v,⊥,vts(z,,t),,,(z,,t),:=,us,(cid:107),2π,ns,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,v,v,fs(z,,v,(cid:107),(cid:107),⊥,⊥,,,v,⊥,,,t),and,the,thermal,speed,vts(z,,t),:=,(cid:115),4π,3ns,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,[(v,v,⊥,⊥,(cid:107),−,us,(cid:107),(z,,t))2,+,v2,⊥,]fs(z,,v,(cid:107),,,v,⊥,,,t).,(3.5),According,to,its,definition,,Fs(z,,w,(cid:107),(cid:90),∞,(cid:90),∞,,,w,⊥,,,t),must,satisfy,the,conditions,2π,dw,(cid:107),0,dw,⊥,w,⊥,Fs(z,,w,(cid:107),,,w,⊥,,,t),=,1,,−∞,(cid:90),∞,dw,2π,−∞,(cid:90),∞,(cid:107),0,dw,w,⊥,w,Fs(z,,w,(cid:107),(cid:107),⊥,,,w,⊥,,,t),=,0,(cid:90),∞,2π,(cid:90),∞,0,dw,(cid:107),dw,⊥,w,⊥,(w2,(cid:107),+,w2,⊥,)Fs(z,,w,(cid:107),,,w,⊥,,,t),=,3,2,and,at,every,point,z,and,time,t.,−∞,The,equations,for,ions,become,∂ni,∂t,+,∂,∂z,(cid:0)niui,(cid:107),(cid:1),=,0,,nimi,(cid:18),∂ui,∂t,(cid:107),+,ui,(cid:19),∂ui,(cid:107),∂z,(cid:107),=,∂pi,(cid:107),∂z,−,−,eni,∂φ,∂z,+,ninnmiRin(un,),,ui,(cid:107),(cid:107),−,(3.10),(3.1),(3.2),(3.3),(3.4),(3.6),(3.7),(3.8),(3.9),4,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,3,2,nimivti,(cid:18),∂vti,∂t,+,ui,(cid:107),∂vti,∂z,(cid:19),=,∂qi,(cid:107),∂z,−,∂ui,(cid:107),∂z,pi,(cid:107),−,+,3,4,ninnmiRin(v2,tn,−,v2,ti),+,1,2,ninnmiRin(un,)2,ui,(cid:107),(cid:107),−,and,∂Fi,∂t,+,˙zi,∂Fi,∂z,+,˙w,i,(cid:107),Here,,we,have,defined,the,coefficients,∂Fi,∂w,(cid:107),+,˙w,⊥,i,∂Fi,∂w,⊥,=,˙Fi,+,Cin.,˙zs[Fs](z,,w,(cid:107),,,t),:=,us,(cid:107),+,vtsw,,,(cid:107),(3.11),(3.12),(3.13),˙w,s[Fs](z,,w,(cid:107),(cid:107),,,t),:=,1,nsmsvts,∂ps,(cid:107),∂z,+,2w,(cid:107),3nsmsv2,ts,(cid:20),∂qs,(cid:107),∂z,(cid:18),+,ps,(cid:107),−,3,2,nsmsv2,ts,(cid:21),(cid:19),∂us,∂z,(cid:107),∂vts,∂z,,,w2,(cid:107),−,˙w,⊥,s[Fs](z,,w,(cid:107),,,w,⊥,,,t),:=,2w,3nsmsv2,ts,⊥,(cid:18),∂qs,(cid:107),∂z,+,ps,(cid:107),∂us,(cid:107),∂z,(cid:19),w,w,(cid:107),⊥,−,∂vts,∂z,(3.14),(3.15),and,˙Fs[Fs](z,,w,(cid:107),,,w,⊥,,,t),:=,w,(cid:107),(cid:34),(cid:18),3,2,nsmsv2,ts,−,the,parallel,pressure,∂vts,∂z,−,vts,ns,(cid:18),∂qs,(cid:107),∂z,(cid:19),∂ns,∂z,(cid:18),+,ps,(cid:107),−,1,2,nsmsv2,ts,(cid:19),(cid:35),(cid:19),∂us,∂z,(cid:107),Fs,,(3.16),[Fs](z,,t),:=,2πnsmsv2,ts,ps,(cid:107),(cid:90),∞,−∞,(cid:90),∞,0,dw,(cid:107),dw,⊥,w,⊥,w2,(cid:107),Fs(z,,w,(cid:107),,,w,⊥,,,t),(3.17),the,parallel,heat,flux,[Fs](z,,t),:=,πnsmsv3,ts,qs,(cid:107),(cid:90),∞,−∞,dw,(cid:90),∞,(cid:107),0,dw,⊥,w,w,(cid:107),(w2,(cid:107),+,w2,⊥,⊥,)Fs(z,,w,(cid:107),,,w,⊥,,,t),,(3.18),and,the,modified,charge,exchange,collision,operator,Cin[Fi,,Fn,,nn,,ui,:=,nnRin,(cid:107),−,(cid:107),,,un,(cid:20),Fi,−,∂,∂w,(cid:107),∂,∂w,⊥,,,vti,,vtn](z,,w,(cid:107),(cid:18),Fn,z,,v3,ti,v3,tn,(cid:20)(cid:18),un,ui,(cid:107),(cid:107),−,vti,(cid:18),v2,tn,v2,ti,−,(cid:20),w2,⊥,2,,,t),,,w,ui,⊥,(cid:107),−,vtn,w,(cid:107),2,+,1,+,,,vti,vtn,2(un,un,(cid:107),+,vti,vtn,w,(cid:107),(cid:18),v2,tn,v2,ti,−,2(un,(cid:107),−,3v2,ti,1,+,(cid:19),)2,ui,(cid:107),+,nnRin,+,nnRin,w,⊥,(cid:19)(cid:21),,,t,⊥,w,(cid:19)(cid:19),)2,(cid:107),(cid:21),Fi,ui,(cid:107),−,3v2,ti,(cid:21),Fi,.,(3.19),Note,that,the,differential,terms,in,this,modified,collision,operator,could,have,been,in-,i,and,˙Fi,,but,we,have,decided,to,make,cluded,in,the,definitions,of,the,coefficients,˙w,(cid:107),them,part,of,a,modified,collision,operator,instead,to,separate,the,effect,of,collisions,clearly.,This,split,should,not,be,taken,as,a,suggestion,on,how,to,implement,these,terms,in,a,code.,˙w,i,,⊥,1D,drift,kinetic,models,with,periodic,boundary,conditions,5,The,equations,for,the,neutrals,are,∂nn,∂t,+,∂,∂z,(cid:0)nnun,(cid:107),(cid:1),=,0,,nnmi,(cid:18),∂un,∂t,(cid:107),+,un,(cid:19),∂un,∂z,(cid:107),(cid:107),=,∂pn,∂z,(cid:107),−,+,ninnmiRin(ui,(cid:107),−,un,),,(cid:107),(3.20),(3.21),3,2,nnmivtn,(cid:18),∂vtn,∂t,+,un,∂vtn,∂z,(cid:107),(cid:19),=,∂qn,(cid:107),∂z,−,−,∂un,∂z,pn,(cid:107),3,4,nnnimiRin(v2,ti,−,v2,tn),nnnimiRin(un,(cid:107),−,)2,ui,(cid:107),(3.22),(cid:107),+,+,1,2,and,Cni.,Here,,we,have,defined,the,modified,charge,exchange,collision,operator,+,˙w,(cid:107),+,˙zn,+,˙w,⊥,⊥,n,n,=,˙Fn,+,∂Fn,∂t,∂Fn,∂z,∂Fn,∂w,(cid:107),∂Fn,∂w,(3.23),Cni[Fn,,Fi,,ni,,un,niRin,:=,(cid:107),−,(cid:107),,,ui,(cid:20),Fn,−,∂,∂w,(cid:107),∂,∂w,⊥,,,t),,,w,un,⊥,,,vtn,,vti](z,,w,(cid:107),(cid:18),v3,tn,v3,ti,(cid:20)(cid:18),ui,un,Fi,z,,+,(cid:107),(cid:107),−,vtn,(cid:18),v2,ti,v2,tn,−,(cid:20),w2,⊥,2,(cid:107),−,vti,w,(cid:107),2,1,+,,,vtn,vti,2(un,ui,(cid:107),+,vtn,vti,w,(cid:107),(cid:18),v2,ti,v2,tn,−,2(un,(cid:107),−,3v2,tn,1,+,(cid:19),)2,ui,(cid:107),+,niRin,+,niRin,w,⊥,(cid:19)(cid:21),,,t,⊥,w,(cid:19)(cid:19),)2,(cid:21),Fn,ui,(cid:107),(cid:107),−,3v2,tn,(cid:21),Fn,.,(3.24),Equations,(3.12),and,(3.23),for,Fi,and,Fn,are,constructed,such,that,conditions,(3.6),,(3.7),and,(3.8),are,satisfied,at,all,times,if,they,are,satisfied,at,t,=,0.,4.,Linear,test,One,possible,test,for,the,sets,of,1D,equations,described,above,is,the,evolution,of,small,perturbations,to,a,uniform,Maxwellian,equilibrium.,We,assume,the,following,form,for,the,ion,and,neutral,distribution,functions,,fs(z,,v,(cid:107),,,v,⊥,,,t),=,fM,s(v,,,v,⊥,(cid:107),),+,fs1(v,,,v,⊥,(cid:107),)[exp(ik,z,(cid:107),−,iωt),+,complex,conjugate],,(4.1),where,fM,s(v,,,v,⊥,(cid:107),),=,ns,(cid:19)3/2,(cid:18),mi,2πTh,(cid:32),−,+,v2,⊥,mi(v2,(cid:107),2Th,(cid:33),),.,exp,(4.2),Note,that,both,species,share,the,same,constant,temperature,Th.,To,ensure,that,the,potential,is,small,,we,assume,ni,=,Ne.,Since,the,perturbations,fs1(v,can,be,linearized,to,give,,,v,⊥,(cid:107),),and,φ,are,small,,equations,(2.14),,(2.15),and,(2.16),v,(k,(cid:107),(cid:107),−,ω,−,innRin)fi1,+,iniRinfn1,=,eφ,Th,−,k,v,(cid:107),(cid:107),fM,i,+,iRin(nn1fM,i,−,ni1fM,n),,(4.3),innRinfi1,+,(k,(cid:107),v,ω,iniRin)fn1,=,iRin(ni1fM,n,−,−,(cid:107),−,nn1fM,i),and,ni1,ni,=,eφ,Te,.,(4.4),(4.5),6,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Figure,1.,Solutions,to,the,dispersion,relation,(4.7):,acoustic,waves,(solid,lines),and,non-prop-,agating,modes,(dashed,lines).,(a),Real,frequency,ωr,:=,Re(ω),and,(b),growth,rate,γ,:=,Im(ω),as,functions,of,the,charge,exchange,collision,frequency,(ni,+,nn)Rin,for,Te/Th,=,1,and,several,values,of,the,parameter,ni/(ni,+,nn).,(c),Real,frequency,ωr,and,(d),growth,rate,γ,as,functions,of,the,the,charge,exchange,collision,frequency,(ni,+,nn)Rin,for,ni/(ni,+,nn),=,1/2,and,several,values,of,the,parameter,Te/Th.,Here,,we,have,defined,the,perturbations,to,the,density,as,ns1,=,2π,(cid:90),∞,−∞,dv,(cid:90),∞,(cid:107),0,dv,fs1.,v,⊥,⊥,(4.6),Solving,for,the,functions,fi1,and,fn1,as,functions,of,ni1,and,nn1,and,then,integrating,fi1,and,fn1,over,velocity,space,,we,find,the,equations,(cid:18),Aii,Ain,Ani,Ann,(cid:19),(cid:18),ni1,nn1,(cid:19),(cid:18),0,0,=,(cid:19),,,(4.7),where,the,elements,of,the,matrix,are,Te,Th,ni,ni,+,nn,Aii,=,1,+,Te,Th,+,ζZ(ζ),+,nn,ni,+,nn,(cid:20)(cid:18),1,+,(cid:19),Te,Th,ζin,−,ζ,(cid:21),Z(ζin),(4.8),Ain,=,−,(cid:26)(cid:20)(cid:18),ni,(ζin,−,ni,+,nn,(cid:19),ζ)Z(ζin),(cid:21),Ani,=,nn,ni,+,nn,−,1,+,Te,Th,ζin,−,ζ,Z(ζin),(cid:27),ζZ(ζ),Te,Th,−,and,Here,,we,have,defined,Ann,=,1,+,ni,ni,+,nn,(ζin,−,ζ)Z(ζin).,(4.9),(4.10),(4.11),with,vth,:=,(cid:112),k,|,2Th/mi,,and,we,have,used,the,plasma,dispersion,function,(Fried,&,Conte,vth,(cid:107)|,(cid:107)|,k,|,ζ,:=,ω,vth,,,ζin,:=,ω,+,i(ni,+,nn)Rin,,,(4.12),00.511.5200.511.513/41/21/4100.511.52-2.5-2-1.5-1-0.5000.511.5211.21.41.61.821/41/212400.511.52-2.5-2-1.5-1-0.5000.511.5200.511.513/41/21/4100.511.52-2.5-2-1.5-1-0.5000.511.5211.21.41.61.821/41/212400.511.52-2.5-2-1.5-1-0.50AAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmEhm/gpJD0tMtGkvb0pwZyPPk9pZyb0oOXfnhfJ1VkcO7INDUAQuuARlcAsqoAoIeATP4BW8WU/Wi/VufUxHF6xsZw/8gfX5A2NImXk=(ni+nn)Rin/|kk|vthAAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmEhm/gpJD0tMtGkvb0pwZyPPk9pZyb0oOXfnhfJ1VkcO7INDUAQuuARlcAsqoAoIeATP4BW8WU/Wi/VufUxHF6xsZw/8gfX5A2NImXk=(ni+nn)Rin/|kk|vthAAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmEhm/gpJD0tMtGkvb0pwZyPPk9pZyb0oOXfnhfJ1VkcO7INDUAQuuARlcAsqoAoIeATP4BW8WU/Wi/VufUxHF6xsZw/8gfX5A2NImXk=(ni+nn)Rin/|kk|vthAAACCnicbVDLSgMxFM34rPU16tJNtAgVoc6IqMuiG5dV7APaIWTStA3NZIYkUyjTWbvxV9y4UMStX+DOvzFtZ6GtBwKHc+7l5hw/4kxpx/m2FhaXlldWc2v59Y3NrW17Z7emwlgSWiUhD2XDx4pyJmhVM81pI5IUBz6ndb9/M/brAyoVC8WDHkbUC3BXsA4jWBsJ2QdFgRg8gQKJY3iPEibS01EftUZwBAco0b0U2QWn5EwA54mbkQLIUEH2V6sdkjigQhOOlWq6TqS9BEvNCKdpvhUrGmHSx13aNFTggCovmURJ4ZFR2rATSvOEhhP190aCA6WGgW8mA6x7atYbi/95zVh3rjwTL4o1FWR6qBNzqEM47gW2maRE86EhmEhm/gpJD0tMtGkvb0pwZyPPk9pZyb0oOXfnhfJ1VkcO7INDUAQuuARlcAsqoAoIeATP4BW8WU/Wi/VufUxHF6xsZw/8gfX5A2NImXk=(ni+nn)Rin/|kk|vth