TN-06_BenchmarksBasicTurbulenceCasesUpdated
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   :description: technical note

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2,BLOB2D,Title:,Authors:,Date:,Report,code:,Benchmarks,for,basic,turbulence,cases,-,updated,(Report,on,Test,Cases,and,Proxy-App),Christopher,Ridgers,,Mike,Kryjak,,University,of,York,31st,January,2024,2067270-TN-06-01,1,BOUT++,,Hermes-3,and,the,benchmark,cases,In,this,section,we,will,summarise,the,equations,to,be,solved.,Full,details,of,the,first,system,,denoted,‘Blob2D’,,.,The,second,is,the,‘Hasegawa-Wakatani’,system,described,in,Ref.,[3,,4].,We,will,also,summarise,the,methodology,by,which,we,will,produce,the,benchmark,solutions,to,these,equations.,In,particular,we,will,outline,the,code,used,,i.e.,codes,built,on,the,BOUT++,framework,,,most,notably,the,Hermes-3,fluid,edge,modelling,code,,,as,well,as,the,setup,of,the,benchmark,problems.,Hermes-,3[12],is,a,multi-dimensional,and,multi-species,code,built,on,top,of,the,BOUT++,framework,[11].,In,1D,,it,allows,fast,simulations,of,SOL,dynamics,and,detachment,physics,,extending,the,capabilities,of,the,1D,BOUT++,code,SD1D,[13].,In,2D,,its,capabilities,are,similar,to,the,code,UEDGE,[14],but,with,a,more,advanced,fluid,neutral,model,,while,in,3D,it,is,similar,to,the,BOUT++,code,STORM,[15],but,with,several,improvements:,a,finite,volume,instead,of,a,finite,difference,scheme,to,improve,conservation,,the,ability,to,solve,the,ion,energy,equation,and,an,advanced,fluid,neutral,model.,In,this,report,,Hermes-3,is,used,to,simulate,the,first,case,,the,toy,2D,turbulence,system,‘Blob2D‘,as,found,in,Ref.,[1],and,described,extensively,in,the,literature,[2].,The,second,case,is,a,3D,implementation,of,the,Hasegawa-Wakatani,system.,Due,to,the,simplicity,of,the,equations,and,boundary,conditions,,an,existing,BOUT++,implementation,is,used,and,solved,using,the,Hermes-3,default,solver,to,ensure,that,the,performance,is,equivalent.,In,the,final,section,,the,more,complex,LAPD,system,is,given,a,brief,description.,LAPD,simulations,are,subject,to,future,work.,2,Blob2D,2.1,Equations,This,system,consists,of,the,following,equations.,(cid:19),jsh,(cid:18),1,e,∂ne,∂t,∂ω,∂t,=,−∇,·,(nevE×B),+,∇,·,(cid:18),=,−∇,·,(ωvE×B),+,∇,(cid:18),1,pe,=,eneTe,ω,=,∇,·,pe∇,×,(cid:19),B2,∇⊥ϕ,(cid:19),b,B,+,∇,·,jSH,∇,·,jSH,=,neϕ,L∥,(1),(2),(3),These,equations,describe,the,dynamics,of,a,localised,‘blob’,in,the,2D,drift,plane.,We,simplify,to,a,local,slab,geometry,with,effective,radial,coordinate,x,,effective,poloidal,coordinate,y,and,effective,coordinate,z,following,the,magnetic,field.,For,a,full,description,of,the,derivation,of,these,equations,see,the,review,in,reference,[2].,ne,,Te,and,pe,are,the,electron,number,density,,temperature,and,pressure,respectively.,ω,is,the,vorticity,and,vExB,is,the,E,×,B,drift,velocity.,The,electrostatic,1,2.2,Solving,in,Hermes-3,2,BLOB2D,potential,ϕ,is,determined,by,the,sheath,current,jSH,,,i.e.,the,system,is,sheath-limited.,The,magnetic,field,is,B,=,Bb,and,the,L∥,is,the,connection,length.,2.2,Solving,in,Hermes-3,The,Blob2D,problem,is,provided,as,an,example,case,in,the,BOUT++,code,Hermes-3,[1].,Hermes-3,simulation,results,can,be,read,using,the,Python,package,xHermes,[21],which,automatically,handles,the,normalisation,of,state,variables,and,dimensions,,providing,the,user,with,data,in,SI,units.,Figure,1:,Hermes-3,simulation,of,the,blob,problem,,i.e.,equations,1-3.,Density,profiles,at,the,first,,middle,and,final,timestep,with,and,without,additional,dissipation.,Hermes-3,simulations,of,the,blob,problem,were,run,with,the,following,parameters.,The,electron,density,was,normalised,to,n0,=,2,×,1018,m−3,and,the,electron,temperature,was,set,to,5,eV.The,domain,was,set,to,be,x0,×,y0,=,0.05,m×,0.05,m,and,discretised,using,260,×,256,grid,cells.,The,magnetic,field,was,set,to,0.35,T,,the,major,radius,to,R,=,1.5,m,(gives,the,curvature,as,1/R2),and,the,connection,length,to,10,m.,A,gaussian,blob,was,then,initialised,with,height,0.5n0,and,width,0.05x0.,Figure,1,shows,a,hermes-3,simulation,results,for,the,evolution,of,the,blob,(equations,1-3).,The,simulation,was,run,for,where,2500/ωci,in,steps,of,50/ωci.,2.3,Artificial,dissipation,The,fine,scale,features,present,in,the,simulation,present,a,problem,for,a,spectral,code.,Equations,1-3,do,not,include,explicit,dissipation,and,so,finer,and,finer,features,will,continue,to,evolve.,We,have,therefore,added,an,explicit,diffusion,term,to,equation,1.,∂ne,∂t,=,−∇,·,(nevE×B),+,∇,·,(cid:19),jsh,(cid:18),1,e,+,D∇2ne,(4),2,3,HASEGAWA-WAKATANI,3D,Figure,2:,Hermes-3,simulation,of,the,blob,problem.,Vorticity,profiles,at,the,first,,middle,and,final,timestep,with,and,without,additional,dissipation.,The,constant,diffusion,coefficient,was,set,to,3,×,10−4,m2s−1.,The,bottom,rows,of,figures,1,and,2,show,the,result,of,including,this,diffusion,term.,As,expected,,the,development,of,fine,scale,features,is,prevented.,3,Hasegawa-Wakatani,3D,Due,to,the,simplicity,of,the,equations,and,boundary,conditions,,the,Hasegawa-Wakatani,problem,wouldn’t,see,significant,benefit,from,being,implemented,into,Hermes-3,apart,from,the,finite,volume,operators.,Instead,,an,existing,finite,difference,BOUT++,implementation,[22],is,used,and,solved,using,the,Hermes-3,default,solver,CVODE,to,ensure,that,the,performance,is,as,close,as,possible.,3.1,Equations,The,3D,Hasegawa-Wakatani,equation,system,solves,for,density,n,and,vorticity,∇2ϕ:,∂n,∂t,=,−κ,∂ϕ,∂y,−,∇,·,{bα(∂∥ϕ,−,∂∥n)},+,[n,,ϕ],+,D∇2n,∇2ϕ,=,−∇,·,{bα(∂∥ϕ,−,∂∥n)},+,[∇2ϕ,,ϕ],+,µ∇2(∇2ϕ),∂,∂t,(5),(6),Both,the,number,density,n,and,potential,ϕ,are,fluctuations,about,a,background.,A,slab,geometry,is,used,so,that,∂∥,is,parallel,to,the,direction,of,the,magnetic,field.,The,Poisson,brackets,[A,,B],are,defined,in,consistency,with,the,BOUT++,coordinate,convention,,i.e.,[A,,B],=,(∂A/∂x)(∂B/∂z),−,3,3.2,Normalisation,3,HASEGAWA-WAKATANI,3D,(∂A/∂z)(∂B/∂x),(see,another,BOUT++,implementation,in,[16]),but,note,that,the,code,implements,this,in,the,equivalent,form,−[B,,A].,Both,equations,feature,diffusion,terms,to,aid,numerical,stability,with,user-set,particle,diffusivity,D,and,viscosity,µ.,The,dimensionless,adiabaticity,parameter,α,determines,the,strength,of,coupling,between,n,and,ϕ,as,well,as,the,degree,to,which,electrons,can,move,rapidly,and,establish,a,Boltzmann,response,[4].,Increasing,α,makes,the,solution,tend,more,towards,zonal,flows.,These,are,coherent,spatial,structures,which,can,arise,in,a,turbulent,system,where,the,flow,organises,itself,in,a,way,to,make,it,more,efficient,to,absorb,available,system,free,energy,,with,many,examples,in,nature,such,as,the,"great,red,spot",in,the,Jovian,atmosphere.,In,tokamaks,,these,structures,facilitate,the,L,to,H,mode,transition,and,suppress,cross-field,transport,[17].,α,=,T0,n0eηωci,(7),Where,T0,is,the,background,electron,temperature,in,eV,,n0,is,the,background,electron,density,in,m−3,,e,the,electron,charge,,η,the,electron,resistivity,and,ωci,the,ion,cyclotron,frequency,ωci,=,eB,.,mi,The,electron,resistivity,can,be,estimated,using,the,Spitzer,formula,[18],η,=,5.2,×,10−5ZlnΛ/T,3/2,where,the,Coulomb,logarithm,lnΛ,can,be,calculated,using,the,NRL,formulary,([19],,p.34).,0,The,formulation,for,α,equation,is,distinct,from,the,one,used,in,the,2D,HW,system,which,depends,on,an,assumption,of,a,parallel,wavenumber,k,due,to,the,domain,not,extending,in,the,parallel,direction:,α,=,k2T0,n0eηωci,(8),In,order,to,compare,the,3D,solution,to,2D,solutions,in,literature,,a,"2D-equivalent,α",or,α2D,can,be,computed,by,substituting,the,parallel,wavenumber,for,the,parallel,domain,length,in,metres:,α,=,L2,parT0,n0eηωci,(9),The,normalised,parameter,κ,controls,the,radial,density,gradient,scale,length,,which,can,be,estimated,from,the,scrape-off,layer,density,width,λn:,κ,=,ρs0,n0,∂N,∂x,=,ρs0,λn,(10),Note,that,κ,is,a,requirement,of,the,Hasegawa-Wakatani,system,and,is,present,in,the,BOUT++,HW3D,example,code,as,well,as,other,3D,implementations,[5].,The,physical,quantities,are,normalised,as,follows:,ˆϕ,=,ϕ/Te,,ˆn,=,n/n0,,ˆt,=,ωcit,and,ˆl,=,l/ρs0,where,l,is,either,x,,y,or,z.,ωci,is,the,ion,gyrofrequency,and,ρs0,=,miTe/eB,is,the,hybrid,Larmor,radius.,Normalised,vorticity,is,obtained,by,multiplying,it,by,en0,since,it,represents,charge,density.,For,more,details,on,this,system,see,references,[3],and,[4],as,well,as,reference,[16],for,a,HW2D,implementation,in,BOUT++.,√,3.2,Normalisation,All,of,the,state,variables,and,dimensions,will,be,in,their,normalised,form,and,must,be,converted,into,SI,units.,The,normalisation,parameters,are,provided,in,the,results,file,with,the,following,variables:,4,3,HASEGAWA-WAKATANI,3D,3.3,Initial,conditions,T0,=,T,norm,,n0,=,N,norm,,ρs0,=,rho_s0,,ωci,=,Omega_ci.,The,simulations,can,be,read,in,using,the,package,xBOUT,[23].,There,are,notebook,examples,of,xBOUT,post-processing,workflows,available,including,a,Hasegawa-Wakatani,case,[24].,3.3,Initial,conditions,The,density,equation,is,initialised,to,0,,while,the,vorticity,equation,is,initialised,with,a,fluctuation,field,given,by:,vort0,=,0.1(mixmode(2πx),∗,mixmode(z,−,y)),(11),Where,x,is,the,radial,position,between,0,and,1,while,y,and,z,are,the,parallel,and,binormal,positions,between,0,and,2π,respectively.,The,function,is,a,standard,BOUT++,feature,,see,documentation,in,[25].,Figure,3:,cells).,Initial,phi,and,vorticity,fields,in,normalised,units,for,the,baseline,test,case,(excl.,guard,3.4,Boundary,conditions,The,y,and,z,directions,feature,periodic,boundary,conditions,,while,the,in,the,x,direction,the,bound-,ary,conditions,for,density,,potential,and,vorticity,are,set,to,a,dirichlet,(constant,value),of,zero.,It,is,possible,to,set,the,x,direction,boundary,periodic,as,well,,but,it,requires,a,special,treatment,for,the,Laplacian,inversion,[26].,3.5,Domain,and,simulation,time,The,domain,was,discretised,with,64,domain,cells,plus,4,guard,cells,in,the,x,direction,,making,nx,×,ny,×,nz,=,68,×,64,×,64.,Note,that,if,the,z,and,y,directions,were,not,periodic,,they,would,have,the,same,ny,and,nz,value:,Following,BOUT++,convention,,the,definition,of,nx,includes,guard,(ghost),cells,while,ny,and,nz,do,not,[20],due,to,historical,reasons.,Note,that,any,non-periodic,direction,will,always,include,guard,cells,,and,that,while,there,are,two,guard,cells,on,each,end,of,5,3.6,Input,parameters,3,HASEGAWA-WAKATANI,3D,the,domain,,the,outer,cell,is,used,only,for,higher,order,boundary,conditions,not,present,in,these,models.,The,grid,widths,set,to,dx,×,dy,×,dz,=,0.2,×,120.9,×,0.2,,which,makes,the,domain,size,18.2,x,1000,x,18.2mm.,The,simulation,was,run,for,500,output,timesteps,where,each,timestep,represented,1,×,10−6Ωci,normalised,time,units,,or,1µs,of,simulation,time.,3.6,Input,parameters,Case,T0,[eV],1,2,3,40,40,40,n0,[m−3],1.59,×1019,1.45,×1018,1.33,×1017,B,[T],Z,[-],0.5,0.5,0.5,1,1,1,λn,[m],0.00646,0.00646,0.00646,κ,[-],0.2,0.2,0.2,α2D,[-],1,×10−1,1,×10+0,1,×10+1,α,[-],2.54,×10−1,2.54,×100,2.54,×101,Table,1:,Input,parameters,for,the,three,HW3D,cases.,The,model,inputs,are,shown,in,1.,All,cases,were,ran,with,density,and,vorticity,diffusivities,set,to,1,×10−4.,The,parameters,in,Case,1,were,chosen,to,approximate,typical,conditions,in,the,MAST-U,scrape-off,layer,,with,λn,reduced,to,reach,a,κ,value,of,0.2,for,comparison,with,other,studies,in,literature,[16].,Cases,2,and,3,had,their,α,parameter,increased,to,move,the,solution,into,more,zonal,driven,regimes,(see,[6],for,an,in-depth,study,of,the,influence,of,α,and,κ,parameters,on,the,regimes).,3.7,Results,3.7.1,Performance,Figure,4:,Last,solver,timestep.,The,simulations,were,ran,on,the,Viking2,cluster,at,the,University,of,York,using,32,cores,on,one,node,with,a,runtime,of,57s,,4m,25s,and,10m,41s,for,cases,1,,2,and,3,respectively,using,the,CVODE,solver,with,default,settings.,Figure,4,shows,the,final,solver,timesteps,for,each,of,the,output,timesteps,for,all,cases.,6,3,HASEGAWA-WAKATANI,3D,3.7,Results,Figure,5:,Simulation,wall,time,for,case,1,on,different,core,counts,on,CSD3.,An,additional,performance,benchmark,was,undertaken,on,the,CSD3,cluster,where,case,2,was,ran,on,2,,4,,8,,16,and,32,cores,on,a,single,node,of,the,cclake,partition.,The,results,are,shown,in,fig.,5.,3.7.2,System,evolution,Contours,of,density,fluctuations,are,shown,in,fig.,6.,As,the,alpha,parameter,increases,,the,turbulence,becomes,more,zonal,,leading,to,finer,scale,structures.,The,3D,implementation,of,Hasegawa-Wakatani,will,always,tend,towards,a,zonal,solution.,Since,the,damping,depends,on,the,parallel,potential,gradient,,extending,the,model,to,explicitly,simulate,the,parallel/poloidal,direction,allows,a,scenario,where,there,a,flux,surface,has,a,high,potential,but,one,without,a,parallel,gradient,,which,prevents,damping.,This,undamped,mode,leads,to,the,growth,of,zonal,flows,in,contrast,to,the,classical,Hasegawa-Wakatani,where,an,assumption,is,made,on,the,parallel,dynamics,through,a,constant,wavenumber,,but,in,a,similar,way,to,that,in,modified,Hasegawa-Wakatani,[16],where,the,zonal,component,is,taken,out,of,the,damping,term,explicitly,,allowing,zonal,flows,to,grow.,Higher,values,of,α,increase,total,system,damping,,which,results,in,a,greater,prominence,of,the,undamped,zonal,modes.,This,progression,is,clearly,visible,in,fig.,6,where,Case,1,features,large,blobs,and,less,defined,zonal,structures,while,Case,3,features,stronger,zonal,structures,which,break,up,the,eddies,into,smaller,ones.,To,capture,the,evolution,of,the,turbulence,in,a,more,quantitative,way,,the,total,system,energy,and,enstrophy,were,calculated,according,to,the,methodology,in,[7],but,in,SI,units,,resulting,in,equations,12,,13,and,14.,Note,that,n,is,the,density,fluctuation,about,a,background,,n0,the,constant,background,,and,T0,is,in,eV.,EK,=,(cid:90),1,2,(cid:104),E,×,B,B2,(cid:105)2,n0mi,dv,EP,=,(cid:90),3,2,n2,n0,T0e,dv,7,(12),(13),3.7,Results,3,HASEGAWA-WAKATANI,3D,W,=,(cid:90),1,2,(ne,−,∇2ϕ),dv,(14),As,shown,in,fig.,9,,each,case,has,been,run,for,long,enough,to,get,close,to,steady,state.,Increased,values,of,α2D,result,in,lower,energy,and,enstrophy,as,the,damping,increases.,The,change,in,the,kinetic,energy,profile,in,Case,2,at,around,200µs,corresponds,to,a,transition,from,a,more,turbulent,to,a,more,zonal,regime.,Figure,10,shows,the,power,spectrum,of,the,density,field,fluctuations,in,the,three,simulations.,There,is,a,clear,difference,between,the,more,turbulent,case,1,which,features,larger,blobs,and,cases,2,and,3,where,the,zonal,flows,lead,to,many,more,smaller,scale,features,as,the,larger,eddies,are,broken,up.,The,RHS,cutoff,of,the,plot,represents,the,smallest,length,scale,fluctuation,resolved,by,the,grid.,Figure,6:,Density,fluctuation,contours,for,the,three,cases,across,three,time,slices,halfway,along,the,Y,direction,in,the,XZ,plane.,8,3,HASEGAWA-WAKATANI,3D,3.7,Results,Figure,7:,Potential,fluctuation,contours,for,the,three,cases,across,three,time,slices,halfway,along,the,Y,direction,in,the,XZ,plane.,9,3.7,Results,3,HASEGAWA-WAKATANI,3D,Figure,8:,Vorticity,fluctuation,contours,for,the,three,cases,across,three,time,slices,halfway,along,the,Y,direction,in,the,XZ,plane.,10,3,HASEGAWA-WAKATANI,3D,3.7,Results,Figure,9:,Energy,and,enstrophy,system,integrals,over,time,for,each,simulation.,Figure,10:,Power,density,spectrum,of,density,fluctuations.,11,REFERENCES,4,LAPD,The,LAPD,system,is,not,a,major,focus,of,the,present,work,and,no,benchmark,simulations,have,been,done,so,far.,Instead,,a,brief,description,of,the,system,is,provided,herein.,The,Large,Plasma,Device,(LAPD),is,a,linear,plasma,device,at,UCLA,[8].,With,a,plasma,length,of,17m,,a,vacuum,vessel,radius,of,50cm,and,typical,plasma,radius,of,30cm,,LAPD,primarily,features,helium,plasmas,[9].,The,plasma,itself,is,created,by,electrodes,at,each,end,of,the,device,creating,a,stream,of,electrons,in,the,centre,of,the,plasma,which,also,acts,as,an,energy,source,of,the,system.,This,leads,to,high,electron,temperatures,and,lower,temperatures,of,the,ions,which,must,obtain,their,energy,through,electron-,ion,collisions.,Due,to,the,system,featuring,relatively,low,densities,,collisionalities,are,low,and,lead,to,a,large,ion-electron,temperature,disparity,as,well,as,weak,interactions,with,the,neutrals.,A,simple,Hermes-3,implementation,features,evolved,electron,energy,,constant,temperature,or,constant,profile,ion,temperatures,along,with,a,optional,addition,of,an,isotropic,neutral,background,along,with,a,charge,exchange,reaction,rate.,The,domain,is,annular,with,zero-gradient,radial,boundary,conditions,and,sheath,boundaries,at,each,longitudinal,end,[10].,5,Conclusions,1.,BOUT++,codes,can,be,used,to,provide,an,effective,benchmark,for,blob,transport,and,simple,edge,turbulence,cases,2.,In,the,blob,case,artificial,diffusion,must,be,added,to,facilitate,comparison,with,spectral,codes.,3.,The,Hasegawa-Wakatani,3D,turbulence,system,runs,well,for,MAST-U,type,parameters,in,BOUT++,and,can,be,used,to,investigate,different,turbulence,regimes.,Runtimes,are,on,the,order,of,minutes,on,32,cores.,6,Supplementary,material,Please,see,the,GitHub,repo,in,[27],for,additional,content.,Simulation,files,are,not,provided,due,to,their,size,but,are,available,to,send,upon,request.,The,repo,contains:,1.,All,of,the,plots,in,the,report,along,with,all,of,the,code,used,to,generate,them,in,a,Jupyter,Notebook,(note:,requires,xBOUT,and,xHermes),2.,Code,to,unnormalise,the,simulations,and,calculate,all,of,the,post-processing,and,input,quan-,tities,3.,Code,to,generate,animations,from,the,simulations,as,well,as,gif,animations,of,density,,potential,and,vorticity,for,all,three,cases,4.,Input,files,for,all,three,cases,References,[1],Example,‘Blob2d’,taken,from,hermes-3,manual,(https://hermes3.readthedocs.io/en/latest/examples.html),[2],S.I.,Krasheninnikov,,D.A.,D’Ippolito,&,J.R.,Myra,,J.,Plasma,Physics,,74.,679,(2008).,[3],A.,Hasegawa,&,M.,Wakatani,,Phys.,Rev.,Lett.,50,,682,(1983),12,REFERENCES,REFERENCES,[4],J.,Anderson,&,B.,Hnat,,Phys.,Plasmas,,24,,062301,(2017).,[5],B.,Friedman,,T.,A.,Carter,,Phys.,Plasmas,,22,,012307,(2015),[6],R.,Numata,,R.,Ball,,R.,Dewar,Phys.,Plasmas,14,,102312,(2007),[7],S.,B.,Korsholm,,P.,K.,Michelsen,,V.,Naulin,Phys.,Plasmas,6,,2401–2408,(1999),[8],W.,Gekelman,et,al.,Rev.,Sci.,Instrum.,62,,2875–2883,(1991),[9],P.,Popovich,et,al.,Phys.,Plasmas,17,,122312,(2010),[10],B.,Dudson,,personal,communication.,[11],B.D.,Dudson,et,al.,,Computer,Physics,Communications,,180,,1467,(2009),[12],B.D.,Dudson,et,al.,,arXiv:2303.12131,[13],B,D,Dudson,et,al,Plasma,Phys.,Control.,Fusion,61,065008,(2009),[14],T.D,Rognlien,et,al,J.,Nucl.,Mater.,196–198,347–51,(1992),[15],L.,Easy,et,al,Phys.,Plasmas,21,,122515,(2014),[16],T.,Gheorghiu,,F.,Militello,,J.,Juul,Rasmussen,,Phys.,Plasmas,31,,013901,(2024),[17],Ö,D,Gürcan,and,P,H,Diamond,,J.,Phys.,A:,Math.,Theor.,48,293001,(2015),[18],F.,Chen,,Introduction,to,Plasma,Physics,and,Controlled,Fusion,,ISBN,978-3-319-22308-7,(2016),[19],https://library.psfc.mit.edu/catalog/online_pubs/NRL_FORMULARY_19.pdf,[20],https://bout-dev.readthedocs.io/en/latest/user_docs/bout_options.html,[21],https://github.com/boutproject/xhermes,[22],https://github.com/boutproject/BOUT-dev/tree/master/examples/hasegawa-wakatani-3d,[23],https://github.com/boutproject/xBOUT,[24],https://github.com/boutproject/xBOUT-examples/tree/master,[25],https://bout-dev.readthedocs.io/en/stable/user_docs/variable_init.html#expressions,[26],https://bout-dev.readthedocs.io/en/stable/userdocs/inputgrids.htmlperiodic,−,x,−,domainshttps,:,//github.com/mikekryjak/neptune,−,docs,13

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