TN-12-2_Tests2DDriftKineticModelPlasmaEdge
==========================================

.. meta::
   :description: technical note

   :keywords: Report,2047357-TN-12,Tests,for,a,2D,drift,kinetic,model,for,the,plasma,edge,M.,Barnes1,1,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Clarendon,Laboratory,,Parks,Road,,Oxford,OX1,3PU,,United,Kingdom,E-mail:,michael.barnes@physics.ox.ac.uk,1.,Introduction,In,this,report,we,briefly,discuss,a,strategy,for,testing,the,2D,(in,configuration,space),model,described,in,detail,in,our,previous,report,[1].,As,the,system,of,equations,we,must,solve,becomes,more,complex,,analytical,solutions,are,more,difficult,to,obtain,and,are,of,limited,utility.,To,address,this,challenge,we,propose,to,use,the,method,of,manufactured,solutions,[2,,3].,The,basic,idea,behind,this,approach,is,to,specify,the,form,for,the,solution,to,the,equations,that,we,wish,the,code,to,produce,,and,then,to,derive,an,appropriate,source,term,to,add,to,the,equations,to,make,this,solution,consistent.,In,doing,so,,the,key,is,to,choose,solutions,that,are,closed-form,,sufficiently,smooth,and,differentiable,and,that,are,able,to,test,a,variety,of,terms,appearing,in,the,equations.,2.,Model,equations,As,an,illustration,of,the,proposed,method,,we,consider,the,drift,kinetic,system,of,[1],in,the,collisionless,limit,and,with,an,assumed,Boltzmann,response,for,the,electrons:,∂fi,∂t,−,1,B,∂φ,∂z,∂fi,∂r,+,(cid:18),v(cid:107)Bz,B,+,1,B,∂φ,∂r,(cid:19),∂fi,∂z,−,eBz,miB,∂φ,∂z,∂fi,∂v(cid:107),=,Si,,and,∂fn,∂t,+,vr,∂fn,∂r,+,vz,∂fn,∂z,=,Sn,,ni,=,ne,=,Ne,exp,(cid:19),,,(cid:18),eφ,Te,(1),(2),(3),where,fs,is,the,particle,distribution,function,for,species,s,,B,is,the,magnetic,field,strength,,Bz,is,its,component,along,the,vertical,(z),direction,,φ,is,the,electrostatic,potential,,r,is,the,radial,cylindrical,coordinate,,t,is,time,,v(cid:107),is,the,component,of,the,velocity,along,the,magnetic,field,,vr,and,vz,are,the,velocity,components,along,r,and,z,,e,is,the,proton,charge,,mi,is,the,ion,mass,,ns,is,the,density,of,species,s,,Ne,and,Te,are,constants,with,the,dimensions,of,density,and,temperature,,respectively,,and,Si,and,Sn,are,source,terms,to,be,determined.,1,Tests,for,a,2D,drift,kinetic,model,for,the,plasma,edge,2,Finally,,the,particle,distribution,function,fs,is,related,to,the,particle,density,ns,via,(cid:90),ns,=,d3vfs.,3.,Manufactured,solution,We,seek,steady-state,solutions,of,the,form,with,and,ˆfs,=,ns,(cid:18),ms,2πTs,(cid:19)3/2,(cid:18),exp,−,(cid:19),,,msv2,2Ts,ns(z,,r),=,ˆns(z),exp,−,(cid:18),Ts(z,,r),=,ˆTs(z),exp,(cid:18),−,(cid:19),(cid:19),,,,,r,Lns,r,LTs,1,Lns,1,LTs,=,−,∂,ln,ns,∂r,=,−,∂,ln,Ts,∂r,=,constant,,=,constant.,(4),(5),(6),(7),(8),(9),These,choices,for,the,forms,of,the,radial,density,and,temperature,profiles,are,consistent,with,the,equilibrium,plasma,profiles,of,the,‘local’,approximation,typically,employed,in,the,closed-field-line,region,of,tokamaks.,Consequently,,they,are,expected,to,be,physically,sensible.,With,these,assumptions,,the,ion,and,neutral,drift,kinetic,equations,become,(cid:32),Si,=,ˆfi,Te,eBLTi,(cid:32),(cid:18),miv2,2Ti,(cid:18),Te,Ti,∂,ln,ˆni,∂z,Bz,B,(cid:19),(cid:32),−,3,2,∂,ln,ˆni,∂z,(cid:19),+,1,+,∂,ln,ˆTi,∂z,−,LTi,Lni,(cid:18)miv2,2Ti,(cid:33),∂,ln,ˆTi,∂z,(cid:19)(cid:33),(cid:33),−,3,2,+,v(cid:107),and,(10),(cid:32),Sn,=,ˆfn,−vr,(cid:18),1,Lnn,+,1,LTn,(cid:18),mnv2,2Tn,−,3,2,(cid:19)(cid:19),+,vz,(cid:32),∂,ln,ˆnn,∂z,+,∂,ln,ˆTn,∂z,(cid:18),mnv2,2Tn,−,3,2,(cid:19)(cid:33)(cid:33),.,(11),All,that,is,left,to,have,closed-form,expressions,for,Si,and,Sn,are,specifications,of,ˆns(z),and,ˆTs(z).,We,choose,ˆns(z),=,ns,+,δns,cos,kzz,and,ˆTs(z),=,T,s,+,δTs,cos,kzz,,with,ns,,T,s,,δns,and,δTs,independent,of,z.,Tests,for,a,2D,drift,kinetic,model,for,the,plasma,edge,3,4.,Future,work,With,manufactured,solutions,of,the,type,presented,in,this,report,,we,are,in,a,position,to,test,our,numerical,implementation,of,the,2D,model,presented,in,[1].,Furthermore,,it,should,be,straightforward,to,include,more,complicated,forms,for,ˆns,and,ˆTs,in,the,code,framework,,as,the,Julia,programming,language,in,which,the,code,is,written,supports,symbolic,manipulation,packages,that,would,allow,for,automation,of,this,process.,Tests,for,a,2D,drift,kinetic,model,for,the,plasma,edge,4,[1],F.,I.,Parra,,M.,Barnes,,and,M.,R.,Hardman.,2d,drift,kinetic,model,with,wall,boundary,conditions.,Excalibur/Neptune,Report,,7:2047357–TN–07–02,M1.4,,2021.,[2],P.,J.,Roache.,Code,verification,by,the,method,of,manufactured,solutions.,J.,Fluids,Engineering,,124:4,,2002.,[3],Benjamin,Daniel,Dudson,,Jens,Madsen,,John,Omotani,,Peter,Hill,,Luke,Easy,,and,Michael,Løiten.,Verification,of,bout++,by,the,method,of,manufactured,solutions.,Physics,of,Plasmas,,23(6):062303,,2016.

:pdfembed:`src:_static/TN-12-2_Tests2DDriftKineticModelPlasmaEdge.pdf, height:1600, width:1100, align:middle`