CD-EXCALIBUR-FMS0074-M6c.3_ExcaliburFusionModellingSystemComplementaryActionsCodeIntegrationAcceptanceOpera
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ExCALIBUR,Complementary,actions.,Code,integration,,ac-,ceptance,and,operation,3.,M6c.3,Version,1.01,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M6c.3.,It,provides,material,concerning,the,implementation,of,the,HERMES-3,equations,in,NEKTAR++,in,support,of,recent,work,by,the,holder,of,the,grant,T/AW085/22,,covering,the,initial,version,written,by,the,grant,holder,and,further,developments,in,order,to,improve,the,efficiency,of,the,solver;,also,some,attempts,to,verify,the,outputs,of,the,aforementioned,codes,are,outlined.,In,view,of,the,need,to,couple,other,physics,codes,to,NEKTAR++,,the,interface,for,adding,source,terms,into,the,NEKTAR++,incompressible,Navier-Stokes,solver,is,demonstrated.,Additional,work,out-,lines,an,implementation,of,convection,problems,within,the,FIREDRAKE,PDE,solver,framework,using,a,simple,continuation,method,to,access,flows,at,higher,values,of,the,Rayleigh,number;,this,code,is,verified,against,some,recent,results,from,the,literature.,An,overview,of,relevant,grantee,work,is,provided,in,a,digest,of,reports,received,over,the,past,year.,Finally,,aspects,of,the,kinetic,two-stream,instability,are,investigated,,notably,a,method,for,generating,analytic,solutions,to,the,equation,system.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/ExCALIBUR,-FMS/0074,Issue:,Date:,1.01,23,December,2023,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Ed,Threlfall,Owen,Parry,Wayne,Arter,Signature,N/A,N/A,N/A,Date,23,March,2023,23,March,2023,23,December,2023,CD,Reviewed,By:,Wayne,Arter,23,December,2023,CD,1,1,Introduction,The,development,of,high,order,(spectrally),accurate,finite,element,models,is,seen,as,critical,for,the,production,of,efficient,Exascale,software.,Deliverable,6c,is,designed,to,enable,an,uninterrupted,programme,of,development,,providing,continuity,for,external,work,on,spectral,elements,and,an,advanced,referent,model.,Section,2,of,this,report,is,concerned,with,a,spectral,/,hp,element,implementation,of,a,set,of,plasma,fluid,equations.,Specifically,,System,2-6,of,the,revised,NEPTUNE,proxy-app,equations,in,[1],is,a,spatially,two-dimensional,plasma,model,incorporating,velocity-space,effects.,This,is,intended,to,be,simulated,using,two-dimensional,finite,elements,with,kinetic,effects,incorporated,eg.,by,coupling,to,a,particle,representation,that,acts,to,introduce,source,terms,into,the,fluid,model,(intended,to,represent,eg.,ionization,of,kinetic,neutrals).,Physically,this,picture,represents,a,col-,lisional,plasma,interacting,with,a,kinetic,population,of,particles,(eg.,fusion-born,ions,or,neutrals,sourced,from,the,reactor,walls,having,been,created,by,bombardment,of,fusion-born,ions).,Kinetic,species’,models,are,also,needed,for,the,interaction,of,the,plasma,with,the,first,wall,of,the,reactor,(the,so-called,plasma,‘sheath’,terms),,though,note,that,,in,the,equations,described,in,this,report,,a,much,simpler,representation,of,the,sheath,terms,is,used.,The,plasma,fluid,model,described,in,the,current,version,of,[1],mirrors,that,implemented,in,the,existing,HERMES-3,multispecies,drift-reduced,simulation,code.,HERMES-3,is,based,on,the,BOUT++,framework,and,it,uses,a,finite-difference,spatial,discretization;,one,target,of,the,current,work,is,an,exploration,of,the,benefits,of,an,imple-,mentation,using,spectral,/,hp,methods.,A,discussion,of,initial,attempts,to,implement,a,subset,of,the,HERMES-3,capability,within,NEKTAR++,is,followed,by,a,description,of,some,possible,validation,tests,applicable,to,the,implementation.,There,is,also,a,small,amount,of,material,explaining,how,source,terms,might,be,implemented,in,the,NEKTAR++,framework.,The,FIREDRAKE,PDE,solver,framework,is,an,alternative,to,NEKTAR++,and,this,can,provide,ex-,pedient,implementations,of,relevant,physics,problems,with,the,flexibility,of,its,user,interface,a,major,strength.,Section,3,contains,an,explanation,of,how,to,obtain,the,Nusselt,number,for,laminar,solutions,to,a,convecting,system,,including,a,comparison,with,recent,results,from,the,literature.,A,simple,continuation,method,for,obtaining,solutions,for,large,values,of,the,Rayleigh,number,is,presented.,Section,4,summarizes,the,various,reports,from,grantees,that,are,relevant,to,this,work,stream,,with,subsections,covering,reports,from,King’s,College,London,and,the,University,of,Oxford.,An,appendix,contains,a,small,amount,of,mathematical,analysis,for,a,kinetic,model,(the,two-stream,instability),and,a,small,study,of,how,this,kinetic,model,behaves,differently,to,a,classical,fluid.,This,is,of,course,in,keeping,with,a,key,theme,of,NEPTUNE,which,is,the,relationship,between,the,two,types,of,model,(kinetic,and,fluid),and,how,to,couple,them,,hence,it,is,expected,that,this,sort,of,study,will,aid,in,the,future,coupling,of,these,systems.,2,HERMES-3,equations,implementation,in,NEKTAR++,It,has,been,decided,to,exchange,the,plasma,fluid,equations,to,the,system,used,in,the,HERMES-3,multifluid,drift,reduced,model,(a,particular,limit,of,the,physics,governing,the,plasma,that,is,relevant,2,Figure,1:,Plots,showing,the,electron,density,after,15,,30,,45,time,steps,in,the,HERMES-3,blob2d,example.,in,realistic,tokamak,scenarios).,HERMES-3,is,an,established,plasma,simulation,code,with,an,existing,user,base,,which,should,mean,that,a,degree,of,verification,is,possible.,The,ultimate,goal,is,a,full,three-dimensional,tur-,bulence,simulation,code;,here,,an,incremental,approach,is,adopted,with,development,proceeding,via,the,implementation,and,testing,of,reduced,versions,of,the,equations.,As,a,starting,point,,attention,will,be,focussed,on,the,blob2d,example,which,can,be,found,at,[2].,This,is,intended,to,be,a,two-dimensional,model,of,extended,filamentary,structures,commonly,observed,in,tokamak,experiments,,referred,to,as,plasma,filaments,or,‘blobs’,(one,reference,is,[3]).,An,important,more,recent,reference,on,modelling,the,motion,of,blobs,is,[4].,The,equations,,taken,from,the,HERMES-3,documentation,,are,∂ω,∂t,∂ne,∂t,(cid:19),B2,∇⊥ϕ,(cid:18),1,∇,·,=,−∇,·,(ωvE×B),+,∇,pe∇,×,(cid:18),(cid:19),b,B,+,∇,·,jsh;,=,−∇,·,(nevE×B),+,1,e,∇,·,jsh;,=,ω;,pe,=,eneTe.,Here,ne,is,the,electron,density,,ω,is,the,plasma,vorticity,,ϕ,is,the,electrostatic,potential.,The,electron,temperature,Te,is,taken,to,be,a,constant,and,pe,is,the,electron,pressure.,The,strength,of,the,magnetic,field,perpendicular,to,the,plane,is,B.,The,so-called,‘E,cross,B,drift’,velocity,is,given,by,(1),vE×B,=,E,×,B,B2,(2),where,the,electric,field,is,purely,electrostatic,,E,=,−∇ϕ.,3,The,x,axis,in,this,model,corresponds,to,the,radial,coordinate,in,the,tokamak,(the,left,is,the,inboard,side),and,the,(x,,y),plane,is,a,slice,at,constant,toroidal,angle.,A,simple,(non-kinetic),sheath,closure,is,used,∇,·,jsh,=,neϕ,L∥,(3),in,which,L∥,is,the,connection,length;,this,is,supposed,to,model,the,effect,of,interaction,with,the,material,surfaces,on,which,the,magnetic,field,lines,end.,Running,the,basic,blob2d,example,and,processing,the,output,in,the,suggested,manner,generates,Fig.1.,It,can,be,seen,that,an,initial,region,of,electron,density,undergoes,non-linear,evolution,to,a,turbulent,state.,Note,that,this,example,runs,in,c.5,minutes,using,one,core,on,a,modern,laptop.,2.1,Implementation,of,blob2d,example,in,NEKTAR++,2.1.1,Preliminary,implementation,An,initial,implementation,followed,the,same,pattern,as,the,Nektar-Driftwave,example,available,at,[5],in,using,an,explicit,time-evolution,scheme,with,no,physical,dissipative,terms.,Indeed,,once,written,out,explicitly,,the,equations,are,somewhat,similar,to,the,Hasegawa-Wakatani,system,used,in,the,aforementioned,proxyapp,,being,∂n,∂t,−,1,B,∂ω,∂t,−,(cid:18),∂ϕ,∂x,(cid:18),∂ϕ,∂x,∂ne,∂y,∂ω,∂y,1,B,−,−,(cid:19),∂ϕ,∂y,∂ϕ,∂y,;,(cid:19),=,neϕ,∂ne,L,∂x,∂ne,∂ω,∂x,∂y,∇2ϕ,=,B2ω.,=,ϵ,+,neϕ,L,;,(4),The,term,∂ne,∂y,represents,the,so-called,diamagnetic,drift,and,is,caused,by,the,curvature,of,the,magnetic,field,perpendicular,to,the,drift,plane,(it,is,absent,if,the,magnetic,field,is,assumed,to,be,uniform).,The,terms,neϕ,L,called,connection,terms,are,representative,of,the,divergence,of,the,sheath,current,as,indicated,above,(Equation,(3)),and,are,intended,to,model,what,happens,at,the,termini,of,the,field,lines,perpendicular,to,the,drift-plane.,This,implementation,was,developed,by,the,holder,of,Grant,T/AW085/22,in,collaboration,with,UKAEA,,and,is,available,at,[6].,A,simple,test,involves,an,initial,blob,of,electron,density,and,this,evolves,in,time,under,the,influence,of,a,force,coming,from,the,diamagnetic,drift,(basically,a,cen-,trifugal,force),and,the,output,is,shown,in,Fig.2.,Note,that,the,implementation,contains,no,explicit,viscosity,or,number,density,diffusivity,,and,the,spectral,/,hp,method,means,that,there,is,relatively,little,numerical,viscosity,or,diffusivity.,The,physics,therefore,closely,approximates,that,of,an,ideal,(Euler),fluid,and,this,manifests,in,the,generation,of,structure,at,all,lengthscales,,leading,the,flow,to,develop,a,fractal-like,appearance.,This,is,not,observed,in,the,equivalent,output,of,the,HERMES-3,version,(Fig.1).,4,Figure,2:,Plots,showing,the,time,evolution,of,the,initial,NEKTAR++,blob2d,example.,It,would,seem,natural,to,add,an,explicit,viscosity,term,and,an,explicit,diffusivity,term,into,these,equations,in,order,to,bring,the,outputs,closer,to,those,observed,in,HERMES-3,(note,that,at,this,point,it,is,not,particularly,clear,which,of,the,two,types,of,dissipation,-,temperature,or,momentum,-,is,more,important).,One,notable,problem,with,this,,however,,is,that,the,explicit,time-evolution,is,expected,to,suffer,from,a,very,restricted,maximum,stable,timestep,once,viscosity,or,diffusivity,is,added,(this,stability,problem,is,present,for,hyperbolic,systems,,typically,in,the,form,|v|∆t,∆x,≤,1,for,maximum,velocity,|v|,,but,is,worse,for,parabolic,systems,as,there,the,stability,criterion,is,of,the,form,D∆t,(∆x)2,≤,1,for,diffusivity,D).,For,this,reason,an,alternative,approach,was,considered,and,this,is,explained,in,the,sequel.,2.1.2,Implementation,as,a,fluid,convection,problem,-,semi-implicit,Given,the,restriction,on,the,timestep,associated,to,the,diffusion,terms,,it,is,natural,to,seek,at,least,a,semi-implicit,formulation.,It,is,possible,to,reformulate,the,system,4,as,a,convection,problem,,in,which,the,diamagnetic,drift,term,represents,the,buoyancy,force,in,the,Boussinesq,approximation,(in,this,picture,,the,gravitational,field,would,be,horizontal,and,pulling,to,the,left).,This,will,mean,that,the,existing,incompressible,Navier-Stokes,solver,of,NEKTAR++,can,be,used,as,a,starting,point.,The,Navier-Stokes,equations,,coupled,to,thermally-driven,buoyancy,(assuming,also,the,Boussi-,nesq,approximation),acting,in,the,positive,x-direction,,are,∂Ω,∂t,∂T,∂t,+,v,·,∇Ω,=,∇2Ω,−,Gr,∂T,∂y,;,+,v,·,∇T,=,1,Pr,∇2ψ,=,−Ω.,∇2T,;,(5),where,Ω,=,∇,×,v,is,the,vorticity,,T,is,the,temperature,,and,ψ,is,the,streamfunction,(which,implies,the,incompressibility,constraint,since,v,=,∇,×,ψ).,The,non-dimensionalization,corresponds,to,Section,B.3;,the,timescale,is,set,by,the,kinematic,viscosity,,the,quantity,Gr,is,the,Grashof,number,5,which,measures,the,strength,of,the,buoyancy,force,,and,the,Prandtl,number,Pr,term,is,the,ratio,of,kinematic,viscosity,to,temperature,diffusivity.,Note,that,the,vorticity,formulation,is,used,here,,whereas,the,Navier-Stokes,implementation,in,NEKTAR++,uses,the,velocity,formulation.,The,fluid,analogy,further,proceeds,by,the,identifications,Ω,=,−ω;,T,=,n;,ϕ,=,ψ;,(6),where,the,quantities,on,the,right,hand,side,are,refer,to,the,plasma,and,those,on,the,left,to,an,ordinary,fluid.,The,connection,terms,are,not,included,in,this,initial,work,and,so,the,first,step,is,to,examine,the,behaviour,of,the,HERMES-3,equations,in,which,the,connection,terms,are,suppressed.,This,is,done,by,setting,the,connection,length,to,a,large,value.,It,is,found,that,this,changes,the,evolution,of,the,blob,-,Fig.3.,The,execution,time,also,increased,from,c.5,minutes,to,c.22,minutes,with,L∥,=,104,replacing,the,value,L∥,=,10,found,in,the,example.,This,gives,a,picture,of,blob,evolution,in,the,absence,of,the,connection,terms,(ie.,a,pure,Navier-Stokes,system),and,so,outputs,of,NEKTAR++,can,be,compared,with,this.,An,initial,experiment,gave,outputs,bearing,superficial,resemblance,to,the,outputs,of,HERMES-3,blob2d,with,the,connection,terms,suppressed,by,setting,L∥,=,104,-,see,fig.4.,Note,that,this,run,took,c.20,minutes,using,one,core,of,a,modern,laptop.,Two,values,of,Pr,were,tried.,The,value,of,Gr,was,determined,empirically,ie.,to,generate,something,giving,a,rough,resemblance,to,the,HERMES-3,output,(the,specific,value,used,was,Gr,=,5,×,109).,The,equations,can,be,seen,to,reproduce,some,features,of,the,HERMES-3,outputs,though,the,Grashof,number,appropriate,to,the,numerical,viscosity,in,HERMES-3,must,be,guessed.,This,is,because,there,is,a,grid-dependent,numerical,viscosity,in,HERMES-3,(see,[7],,A.10.2).,Also,the,effective,diffusivity,for,the,temperature,is,presumably,also,a,grid-dependent,numerical,effect.,In,fact,,experiments,with,HERMES-3,show,that,,at,least,for,time-stationary,fields,(zero,ve-,locity),there,is,zero,temperature,diffusivity;,this,is,shown,by,the,fact,that,the,blob,initial,state,is,time-invariant,if,the,force,term,bxcvz,is,set,to,zero,(it,does,not,diffuse),(Fig.5),and,also,a,pre-,liminary,HERMES-3,implementation,of,the,‘hot,layer’,example,in,the,following,section,does,not,show,any,diffusion,(in,fact,it,is,stationary,unless,some,perturbation,is,seeded,into,the,initial,data).,Note,that,zero,temperature,diffusivity,is,not,consistent,with,the,appearance,of,HERMES-3,outputs,(Fig.4),because,as,Pr,is,made,smaller,,the,system,develops,structure,at,small,scales,and,so,one,is,forced,to,conclude,that,there,is,some,sort,of,velocity-dependent,diffusivity,acting,to,smear,out,the,time,evolution,of,the,temperature.,Note,also,that,the,NEKTAR++,incompressible,Navier-,Stokes,solver,cannot,be,run,with,zero,temperature,diffusivity,,due,to,the,presence,of,the,code,line,1.0/aii,Dt/m,diffCoeff[i],in,the,source,code,(VelocityCorrectionScheme.cpp),-,so,zero,diffusivity,here,leads,to,a,divide,by,zero.,6,Figure,3:,Plots,showing,the,electron,density,after,15,,30,,45,time,steps,in,the,HERMES-3,blob2d,example.,The,top,sequence,has,L∥,=,10,(default),and,lower,one,has,L∥,=,104,,the,latter,sup-,pressing,the,connection,terms.,Figure,4:,Initial,experiments,with,the,NEKTAR++,blob2d,example,(NEKTAR++,temperature,field,shown).,These,used,Gr,=,5,×,109,and,Pr,=,1.0,(L),and,Pr,=,0.1,(R).,7,Figure,5:,Plots,showing,the,electron,density,after,15,,30,,45,time,steps,in,the,HERMES-3,blob2d,example,with,the,force,term,bxcvz,set,to,zero,(otherwise,parameters,are,as,in,the,original,blob2d,example).,It,is,seen,that,no,diffusion,occurs,in,the,initial,temperature,profile.,The,same,example,was,also,run,for,100,times,the,default,duration,(by,changing,the,number,of,timesteps,nout,and,amending,the,output,script,to,grab,fields,at,different,times),and,again,no,discernible,change,was,seen,in,the,output.,2.2,Implementation,of,connection,terms,The,connection,terms,are,the,simplest,implementation,of,the,effect,of,the,plasma,sheath,on,which,the,magnetic,field,lines,terminate.,These,are,the,terms,Equation,(3),in,the,system,Equation,(1);,the,vorticity,equation,and,the,density,equation,are,modified,by,the,addition,of,a,forcing,term,nϕ,L∥,.,Techniques,are,required,to,convert,between,the,velocity,formulation,of,the,NEKTAR++,incompress-,ible,Navier-Stokes,solver,and,the,vorticity,formulation,of,the,HERMES-3,equations.,The,potential,can,be,found,by,solving,the,scalar,Poisson,problem,with,the,vorticity,as,the,source,term.,The,forcing,terms,needed,for,the,velocity,components,ux,and,uy,can,be,derived,by,solving,so,that,then,if,where,∂ux,∂t,∂uy,∂t,=,fx;,=,fy,,∂ω,∂t,=,∂fy,∂x,−,∂fx,∂y,(fx,,fy),=,∇,×,g,∇2g,=,nϕ,L∥,;,8,(7),(8),(9),(10),(11),the,function,g,can,be,found,by,a,further,Poisson,inversion,and,its,curl,taken,to,give,a,manifestly,solenoidal,forcing,term,(fx,,fy),in,the,velocity,evolution,equation,,reproducing,the,connection,term,in,the,vorticity,update.,The,code,in,nektar\solvers\IncNavierStokesSolver\EquationSystems,\VelocityCorrectionScheme.cpp,is,modified,to,incorporate,the,above,as,an,additional,forcing,term.,The,above,modifications,were,inspected,briefly,by,one,of,the,senior,developers,of,NEKTAR++,(David,Moxey),and,were,judged,to,be,correct.,2.3,Simple,verification,of,linear,evolution,The,initial,time-evolution,of,a,blob,in,the,limit,of,small,diamagnetic,drift,can,be,calculated,by,a,pertubative,expansion,,with,the,assumption,of,zero,temperature,diffusivity.,This,analysis,also,applies,on,timescales,shorter,than,the,timescales,for,diffusion.,Expanding,about,a,circularly-symmetric,initial,solution,to,4,n0,using,n,=,n0,+,ϵn1,,ω,=,ϵω1,,ϕ,=,ϵϕ1,one,has,,where,all,quantities,are,the,first,order,ones,(subscript,dropped),except,n0,,and,the,prime,is,radial,derivative,,∂n,∂t,+,n′,0,Br,∂ϕ,∂φ,∂ω,∂t,=,n0ϕ,L,;,=,n′,0,sin,φ,+,n0ϕ,L,;,∇2ϕ,=,B2ω.,(12),Now,inspecting,the,departure,from,the,initial,condition,,take,all,quantities,to,be,Taylor,series,in,time,t,n,=,n(1)t,+,n(2)t2;,ω,=,ω(1)t,+,ω(2)t2;,ϕ,=,ϕ(1)t,+,ϕ(2)t2.,(13),One,sees,immediately,n(1),=,0,and,the,initial,time,evolution,of,n(2),can,be,determined,by,solving,0,sin,φ;,ω(1),=,n′,∇2ϕ(1),=,B2n′,(cid:32),0,sin,φ;,n0ϕ(1),L,1,2,n(2),=,−,n′,0,Br,∂ϕ(1),∂φ,(cid:33),.,(14),It,is,easy,to,work,this,out,for,plausible,circularly-symmetric,blob,initial,states,(now,dropping,the,superscripts,,and,assuming,ϕ,∝,sin,φ).,The,main,step,is,evaluation,of,the,potential,from,the,Poisson,equation,9,Figure,6:,Forcing,terms,added,into,VelocityCorrectionScheme.cpp,NEKTAR++,source,file,in,order,to,give,the,connection,terms.,10,ϕ′′,+,ϕ′,−,1,r,1,r2,ϕ,=,B2n′,0.,(15),The,initial,evolution,of,the,hot,fluid,blob,can,be,inspected,by,running,the,Hermes-3,example,with,a,small,value,for,the,diamagnetic,drift,term,,here,setting,bxcvz,0.001,,and,changing,timestep,to,0.01,(this,is,the,length,of,each,timestep,in,program,units),and,nout,to,20,(this,is,the,number,of,timesteps,to,simulate),(nothing,else,is,changed,except,the,initial,data).,The,vorticity,is,shown,(called,‘Vort’,in,the,output,script),for,the,Gaussian,blob,as,specified,above.,The,initial,state,has,the,expected,dipolar,form.,The,vorticity,amplitude,grows,linearly,with,time,at,the,beginning,of,the,calculation.,In,this,regime,,the,amplitude,can,be,calculated,as,ω,=,n′,0(r),sin,φ,Gr,t,(16),where,n0(r),is,the,(circularly-symmetric),initial,data,and,φ,is,the,usual,polar,angle.,This,relation,will,be,used,to,predict,the,initial,linear,evolution,of,blob2d.,A,list,of,relevant,parameters,in,blob2d,is:,number,of,timesteps,nout=,50,,timestep=,50,,Lrad=,0.05,,Lpol=,0.05,,poloidal,magnetic,field,Bpxy=,0.35,,major,radius,Rxy=,1.5,,bxcvz=,1,5.,Then,the,program,time,unit,in,seconds,is,calculated,(and,reported),by,the,program,as,t0,=,mp,2.98277,×,10−8s.,e,Bpxy,=,1.52,,,temperature=,In,order,to,access,the,linear,regime,the,following,modifications,were,made:,nout=,20,,timestep=,0.01,,and,bxcvz=,0.001.,The,amplitude,after,10,steps,will,be,inspected.,Now,the,initial,data,is,specified,as,n0,=,1.0,+,0.5,exp(−,(x,−,0.25)2,+,y2,0.052,),(17),2e−0.5,=,8.578.,The,ex-,and,the,gradient,of,this,function,has,a,maximum,absolute,value,of,10,pected,vorticity,amplitude,in,the,linear,regime,is,then,,running,for,say,10,timesteps,(and,note,the,0.001,is,the,value,of,bxcvz);,also,the,final,factor,of,2,is,due,to,an,approximate,formula,used,for,∇,×,b,B,times,the,curvature,(thanks,to,Ben,Dudson,for,this,clarification),B,≈,2,√,ω,=,10,×,0.01,×,t0,×,5,×,1.5,0.05,×,0.35,×,8.578,×,0.001,×,2,=,2.19,×,10−8.,(18),This,amplitude,is,consistent,with,the,output,shown,in,Fig.7.,In,most,cases,it,is,obvious,what,the,various,factors,are,doing,eg.,the,multiplier,of,5,is,because,the,diamagnetic,(force),term,scales,with,the,electron,temperature;,the,divide,by,0.05,is,to,scale,the,maximum,gradient,(from,8.578),to,the,actual,domain,size,0.05,×,0.05,square,metres.,Of,course,,all,that,has,really,been,achieved,is,to,fix,a,value,for,the,product,Gr,t,,where,t,is,the,calculation,duration,,that,applies,in,the,linear,regime.,For,the,above,computation,,one,has,11,Figure,7:,Plots,showing,the,electron,density,after,0,,10,,20,(L-R),time,steps,modified,as,discussed,in,the,main,text.,The,figure,for,10,steps,shows,amplitude,agrees,reasonably,well,with,the,value,calculated,in,Eq.18.,The,20-step,plot,confirms,that,the,solution,is,within,the,linear,growth,regime.,Gr,t,=,2.19,×,10−8,×,0.05,8.578,.,(19),A,computation,matching,the,above,initial,vorticity,can,be,performed,in,NEKTAR++.,The,calcula-,tion,duration,is,set,the,same,as,the,HERMES-3,duration,in,seconds,ie.,2.98277,×,10−9,(100,time,steps,were,used),and,the,corresponding,Grashof,number,needs,to,be,2.19×10−8,2.98277×10−9,divided,by,the,maximum,gradient,8.578,,giving,Gr,=,0.85714.,In,fig.8,this,gives,the,expected,amplitude.,Note,the,vorticity,is,computed,in,PARAVIEW,from,the,velocity,components,u,and,v,output,by,NEKTAR++:,add,a,Calculator,for,a,vector,quantity,u,×,iHat,+,v,×,jHat,and,then,use,the,Compute,Derivatives,filter,,selecting,Vorticity,for,the,Output,Vector,Type.,Now,one,must,try,to,establish,the,value,of,Gr,and,this,can,be,done,only,by,matching,the,behaviour,at,longer,times,beyond,the,linear,evolution,regime,(for,now,assuming,that,Pr,=,1).,Assuming,Gr,takes,the,value,above,,scaled,to,match,the,force,term,in,the,original,blob2d,(ie.,use,0.85714,×,1.52×0.001,=,380.951),then,the,NEKTAR++,simulation,must,run,for,a,duration,of,50,×,50,×,2.98277,×,10−8,=,7.45692,×,10−5,in,order,to,match,the,duration,of,HERMES-3,blob2d.,1,The,problem,is,that,the,output,density,profile,shows,little,change,from,the,initial,data.,The,suspi-,cion,is,that,the,internal,units,of,HERMES-3,are,different,to,the,inputs,and,that,the,code,rescales,the,units,prior,to,performing,the,time,evolution,and,then,rescaled,them,back,afterwards,(obviously,this,would,affect,the,nonlinear,evolution,but,not,the,linear,calculation,performed,above).,Discussions,with,the,developers,of,HERMES-3,are,ongoing.,2.4,Reproduction,of,an,alternative,plasma,blob,result,from,the,literature,The,paper,[3],contains,an,analysis,of,the,same,sort,of,blob,motion,as,in,the,HERMES-3,blob2d,example.,There,,the,Rayleigh,and,Prandtl,numbers,are,made,explicit,hence,allowing,easy,repro-,duction,of,the,published,outputs.,Note,the,time,used,in,that,paper,corresponds,to,time,as,used,12,Figure,8:,Plot,from,NEKTAR++,showing,the,electron,density,after,the,same,time,duration,as,in,the,HERMES-3,case,in,fig.7.,The,figure,shows,amplitude,agreeing,reasonably,well,with,the,value,calculated,in,Eq.18.,Note,the,sign,reverse,as,indicated,in,the,fluid,mechanics-plasma,analogy,,Eqs.6.,13,in,the,NEKTAR++,set-up,multiplied,by,the,square,root,of,the,Rayleigh,number,(this,can,be,worked,out,with,reference,to,the,appendix,Section,B).,It,is,straightforward,to,replicate,the,results,of,that,paper,using,the,incompressible,Navier-Stokes,solver,of,NEKTAR++,(Fig.9).,Note,that,this,example,remains,in,a,largely,laminar,flow,regime,and,is,thus,not,subject,to,the,vagaries,of,turbulent,chaos,,which,means,the,solutions,match,nearly,exactly.,2.5,Verification,of,linear,instability,growth,rate,in,a,one-dimensional,case,This,subsection,presents,an,additional,fluid,problem,that,could,form,the,basis,for,a,meaningful,cross-validation,between,HERMES-3,and,NEKTAR++.,Instead,of,the,hot,blob,initial,condition,,it,is,possible,to,consider,for,the,temperature,a,Gaussian,density,profile,varying,only,in,the,x-coordinate.,With,the,appropriate,periodic,boundary,condition,,this,is,then,a,one-dimensional,system,but,implemented,in,two,dimensions.,In,one,dimension,,the,coupled,Navier-Stokes,/,thermal,system,is,∂u,∂t,∂T,∂t,+,u,+,u,∂u,∂x,∂T,∂x,∂u,∂x,=,∂2u,∂x2,−,∂p,∂x,+,Gr,T,;,=,0;,=,0.,(20),The,final,equation,above,is,the,incompressibility,constraint,and,it,restricts,solutions,to,spatially-,constant,u,(which,gives,a,family,of,solutions,related,by,Galileian,invariance,,with,which,in,mind,set,u,=,0).,This,means,that,an,equilibrium,of,the,form,∂p,∂x,=,Gr,T,is,a,stationary,solution,(in,one,dimension,,the,incompressibility,constraint,means,that,heat,does,not,rise,as,the,fluid,blocks,its,own,movement,,with,the,force,balance,being,maintained,by,a,pressure,gradient,countering,the,buoyancy).,Here,,the,equilibrium,will,be,taken,to,be,a,layer,of,hot,fluid,with,a,smooth,localized,temperature,profile,(eg.,a,Gaussian),added,to,a,constant,background,temperature.,Note,that,this,equilibrium,does,not,exist,if,there,is,a,non-zero,value,of,the,temperature,diffusivity,(this,would,make,the,initial,data,diffuse,,whatever,the,dimension).,In,two,dimensions,,the,equilibrium,described,above,is,still,a,stationary,solution,,but,analysis,shows,that,is,not,stable,and,eventually,the,instability,leads,to,nonlinear,evolution,as,shown,in,Fig.10.,The,associated,linear,instability,mode,and,its,exponential,growth,rate,is,easily,calculated,using,NEKTAR++,or,FIREDRAKE.,The,instability,modes,are,seen,to,be,localized,to,the,region,of,negative,temperature,gradient,,as,expected.,The,two-dimensional,system,of,equations,is,14,15,Figure,9:,Plots,from,NEKTAR++,(left),showing,time-evolution,of,a,blob,of,electron,density,in,repli-,cation,of,Fig.5,from,[3],(right),showing,very,good,agreement,in,the,outputs,(note,the,domain,sizes,and,initial,data,are,identical).,Figure,10:,A,sequence,of,frames,showing,to,nonlinear,evolution,of,the,two-dimensional,‘hot,fluid,layer’,described,in,the,main,text.,∂u,∂t,∂v,∂t,∂T,∂x,∂T,∂t,+,u,+,u,+,v,+,v,∂u,∂x,+,∂u,∂x,∂v,∂x,∂T,∂y,∂v,∂y,+,Gr,T,;,=,∇2u,−,=,∇2v,−,∂p,∂x,∂p,∂y,;,=,0;,=,0.,(21),A,linear,stability,analysis,proceeds,from,the,equilibrium,∂p0,zero),,giving,the,equations,for,the,linear,perturbation,∂x,=,Gr,T0,(all,other,quantities,being,λu,=,∇2u,−,λv,=,∇2v,−,∂p,∂x,∂p,∂y,;,+,Gr,T,;,λT,+,u,∂T0,∂x,∂v,∂y,∂u,∂x,+,=,0;,=,0.,(22),By,assuming,separability,and,assuming,wavenumber,k,in,the,periodic,y-coordinate,,one,can,ob-,tain,a,higher-order,equation,for,the,x-component,of,velocity,u,as,(cid:18),λ,+,k2,+,k2,(cid:19),Gr,λ,dT0,dx,u,=,−,d4u,dx4,+,λ,d2u,dx2,(23),dx4,+,E,d2u,(note,that,this,is,almost,reducible,to,a,Schr,¨odinger,equation:,differentiate,twice,Eu,=,−,d2u,dx2,+,V,u,to,d4u,dx2,and,observe,only,the,single,derivative,term,is,discrepant).,Note,also,that,if,k,=,0,then,the,coupling,to,the,background,temperature,gradient,vanishes,-,this,corresponds,to,the,(stable),1-D,case,where,there,can,be,no,y,dependence.,dx2,u,+,2,dV,dx,+,V,d2u,dx2,=,d2V,du,dx,16,One,can,look,for,scaling,laws,by,taking,appropriate,limits;,as,an,example,,large,λ,and,constant,α,=,Gr,λ2,gives,the,equation,(cid:18),d2u,dx2,=,k2,1,+,α,(cid:19),u,dT0,dx,(24),which,is,suggestive,of,the,scaling,law,λ,∼,Gr,to,self-adjointness),,and,small,λ,and,constant,β,=,Gr,λ,gives,1,2,(and,which,clearly,has,purely,real,eigenvalues,due,d4u,dx4,=,k2β,which,is,suggestive,of,the,scaling,law,λ,∼,Gr.,The,scaling,laws,are,observed,in,Fig.11.,Note,these,scalings,are,useful,in,determining,estimates,for,the,eigenvalues,for,inputs,as,initial,guesses,for,numerical,eigenvalue,computations,,which,can,be,dramatically,faster,for,good,initial,data.,dT0,dx,(25),−,u,2,sech2,x,The,two-dimensional,case,was,simulated,in,the,incompressible,Navier-Stokes,solver,of,NEK-,TAR++.,The,initial,temperature,condition,was,exchanged,for,a,P,¨oschl-Teller-like,(sech2),profile,so,that,the,initial,pressure,profile,can,be,provided,analytically,to,NEKTAR++,which,currently,supports,expressions,provided,by,the,cmath,library,(for,example,,erf(x),is,not,available);,the,temperature,profile,was,specifically,T,=,1,a,with,a,=,0.05,and,no,additional,perturbation,was,applied.,For,a,cross-validation,,it,was,found,that,simulations,performed,for,a,selection,of,values,exhibited,lin-,ear,growth,rates,largely,consistent,with,those,output,by,FIREDRAKE.,Note,the,NEKTAR++,growth,was,tracked,using,the,growth,of,the,kinetic,energy,(because,this,involves,a,quadratic,function,of,the,mode,,the,energy,growth,rate,is,twice,the,linear,coefficient);,it,was,necessary,to,modify,the,FilterModalEnergy,filter,of,NEKTAR++,for,this,example,as,the,supplied,version,counts,any,ad-,vected,scalars,(here,temperature),as,additional,components,of,velocity,and,so,misreports,the,fluid,kinetic,energy.,A,word,on,temperature,diffusivity,in,the,NEKTAR++,incompressible,Navier-Stokes,solver,is,also,indicated,in,that,it,is,necessary,to,specify,the,diffusivity,for,the,T,field,(which,is,formally,an,additional,advected,scalar,in,the,solver),explicitly,in,the,session,file,,or,a,(unit),default,value,will,be,applied,which,causes,diffusion,in,the,initial,condition.,The,appropriate,lines,are,<FUNCTION,NAME="DiffusionCoefficient">,<E,VAR="T",VALUE="1.0e-8",/>,</FUNCTION>,It,is,not,possible,to,specify,precisely,zero,diffusivity,here,as,this,results,in,a,numerical,error,(as,noted,earlier,,the,time-evolution,scheme,involves,division,by,the,diffusivity).,The,NEKTAR++,sim-,ulations,described,in,this,section,were,therefore,performed,by,taking,a,small,value,(10−16),for,the,thermal,diffusivity.,This,problem,is,related,to,stability,analyses,of,Rayleigh-B,´enard,convection,first,performed,by,Lord,Rayleigh.,Note,that,the,example,here,is,not,stable,for,negative,values,of,the,Grashof,number,(unlike,Rayleigh’s,example),as,it,has,a,positive,as,well,as,a,negative,temperature,gradient,region;,the,stability,characteristics,are,actually,invariant,under,a,change,in,the,sign,of,Gr.,Along,similar,lines,,a,layer,of,this,type,is,always,unstable,,whereas,the,Rayleigh,case,is,stable,up,to,a,certain,critical,value,of,Rayleigh,number,Ra,(which,is,independent,of,the,Prandtl,number),beyond,which,the,system,undergoes,a,Hopf,bifurcation.,Rayleigh’s,problem,is,also,different,in,having,a,finite,17,Figure,11:,The,dependence,of,the,linear,instability,growth,rate,for,the,hot,layer,on,the,Grashof,number,Gr.,The,orange,dashed,line,is,a,linear,fit,,which,applies,at,small,Gr;,the,grey,dashed,line,has,gradient,1,2,and,corresponds,to,the,square-root,dependence,observed,at,large,Gr.,Prandtl,number,,which,cannot,be,introduced,here,as,it,would,cause,the,initial,data,to,diffuse,(the,problem,discussed,here,exists,in,the,P,r,→,∞,limit).,It,is,straightforward,to,run,the,FIREDRAKE,eigen-analysis,repeatedly,and,thus,obtain,the,largest,growth,rate,as,a,function,of,the,Grashof,number.,This,can,also,be,done,using,the,time-domain,incompressible,Navier-Stokes,solver,of,NEKTAR++,though,this,is,a,little,less,convenient,due,to,the,need,to,adjust,the,calculation,duration,and,timestep,size,for,each,simulation,(the,linear,regime,must,be,reached,from,an,initial,quiescent,state,and,the,timestep,size,must,be,such,that,the,simula-,tion,is,stable).,A,comparison,of,numerical,eigenvalues,obtained,using,both,frameworks,is,exhibited,in,Table1.,Such,analysis,reveals,the,dependence,of,the,largest,eigenvalue,on,Gr,(Fig.11),and,that,this,scales,according,to,the,power,laws,described,earlier.,A,more,detailed,inspection,of,this,curve,reveals,discontinuities,in,the,derivatives,when,the,most,unstable,mode,changes,values,of,the,transverse,wavenumber,k,(Figs.12,,13);,the,system,exhibits,a,response,surface,of,discontinuous,gradient.,The,same,problem,can,be,studied,in,three,dimensions,and,in,that,case,there,is,a,richer,spectrum,of,transverse,harmonics,determined,by,the,container,geometry;,see,Fig.14.,The,3-D,case,is,more,18,Figure,12:,The,dependence,of,the,linear,instability,growth,rate,for,the,hot,layer,on,the,Grashof,number,Gr,near,the,transition,between,the,transverse,k,=,1,and,k,=,2,modes,(the,two,colour,maps,correspond,to,the,temperature,component,of,the,mode,at,Gr,=,7000,(leftmost,colourmap),and,Gr,=,8000,(rightmost)),.,Figure,13:,The,dependence,of,the,linear,instability,growth,rate,for,the,hot,layer,on,the,Grashof,number,Gr:,the,difference,between,subsequent,values,has,been,taken,in,order,to,show,the,deriva-,tive,discontinuities,as,jumps;,these,occur,at,a,change,in,the,transverse,wavenumber,k,of,the,most,unstable,mode.,19,log10,Gr,λ,(FIREDRAKE,order,3,,2,,3),λ,(NEKTAR++),0,1,2,3,4,5,6,7,8,9,10,0.007810,0.07803,0.7739,7.198,51.38,293.00,1368.70,5511.75,20418.12,71489.00,241978.31,0.035,0.0827,0.7993,7.3944,50.478,292.22,1358.3,5449.2,20283,70506,238642,Table,1:,Table,of,linear,growth,rates,for,the,hot-layer,problem.,The,NEKTAR++,growth,rates,are,seen,to,be,slightly,less,than,the,FIREDRAKE,ones;,one,reason,is,the,need,to,include,a,small,amount,of,temperature,diffusivity,in,the,NEKTAR++,simulations;,also,the,way,the,boundary,con-,ditions,are,done,differs,somewhat,between,the,two,codes.,The,NEKTAR++,value,for,Gr,=,1,is,clearly,discrepant.,challenging,with,only,,say,,a,103,element,mesh,with,element,orders,up,to,three,being,feasible,on,a,32Gb,machine.,It,is,hoped,that,examples,of,this,type,can,provide,a,cross-validation,of,simulations,performed,using,HERMES-3.,The,results,obtained,lie,within,the,linear,regime,of,the,simulations,and,are,thus,not,subject,to,the,chaotic,aspects,of,nonlinear,evolution.,It,is,also,the,case,that,the,early,time-,evolution,studied,here,is,independent,of,the,details,of,viscosity,or,temperature,diffusivity,(both,of,which,operate,on,longer,timescales,than,treated,here).,Actually,performing,the,study,for,HERMES-3,has,been,delayed,due,to,the,aforementioned,prob-,lems,in,clarifying,the,units,used,in,HERMES-3.,2.6,Space-,and,time-dependent,source,terms,in,NEKTAR++,In,view,of,the,need,to,implement,source,terms,in,NEKTAR++,(eg.,for,coupling,to,a,particle-based,or,other,kinetic,code),,a,brief,test,was,made,of,the,existing,capabilities,of,the,code,to,handle,forcing,terms.,The,buoyancy,force,in,the,incompressible,Navier-Stokes,solver,is,enabled,in,the,session,file,using,the,BodyForce,capability;,this,corresponds,to,the,top,line,in,the,sequel.,It,is,also,possible,to,include,space-,and,time-dependent,source,terms,into,these,directives,,but,the,EVARS,section,must,be,left,out,as,this,forces,the,code,to,look,up,the,equation,variables,for,the,forcing,(as,done,for,the,buoyancy,force),and,to,ignore,the,space,/,time,variables,(here,x,,y,,t).,<FUNCTION,NAME="BodyForce">,<E,VAR="u",VALUE="Gr*T",EVARS="u,v,T,p",/>,<E,VAR="v",VALUE="0",EVARS="u,v,T,p",/>,20,Figure,14:,Transverse,temperature,profiles,of,instability,modes,for,Gr,=,7,×,103,(left),and,Gr,=,2,×,104,(right).,The,simulations,used,orders,3,,2,,2,for,velocity,,pressure,,and,temperature.,The,plot,artifacts,are,due,to,the,rather,low,resolution.,Note,that,these,simulations,used,Dirichlet,conditions,at,the,transverse,boundaries,,due,to,the,lack,of,a,partially-periodic,cube,mesh,in,FIREDRAKE.,(Plots,obtained,using,PARAVIEW,slice,view,at,x,=,0.55,to,coincide,with,steepest,part,of,unstable,gradient.),<E,VAR="T",VALUE="(1.0e4)*exp(-((x+0.25)^2+(y+0.25-5000.0*t)^2)/(0.05^2))",/>,<!--,space-,and,time-dependent,forcing,term,-->,</FUNCTION>,The,output,is,shown,in,Fig.15.,2.7,Concluding,remarks,The,above,sections,described,work,aiming,to,replicate,some,of,the,outputs,of,the,established,Initial,outputs,show,some,similarity,in,the,be-,plasma,fluid,code,HERMES-3,using,NEKTAR++.,Figure,15:,Time-evolution,of,a,simulation,with,a,heat,source,that,translates,upward,during,the,simulation.,Note,that,numerical,errors,are,evident:,the,third,frame,shows,a,blob,of,excessively,large,and,small,density.,21,haviour,of,plasma,blob-type,simulations,but,the,analysis,is,complicated,by,difficulty,in,clarifying,the,units,used,in,the,HERMES-3,code,,and,also,the,fact,that,the,viscosity,and,temperature,diffu-,sivity,in,HERMES-3,are,implemented,numerically,and,exhibit,grid-dependence.,This,work,is,in,progress,and,it,is,currently,intended,that,any,doubt,as,to,the,units,be,removed,by,running,the,code,within,a,debugger,in,order,to,expose,exactly,what,system,is,being,solved.,3,Analysis,of,vertical,convection,in,FIREDRAKE,It,is,straightforward,to,simulate,at,least,time-independent,convection,problems,using,the,FIRE-,DRAKE,PDE,solver,framework.,This,restricts,the,study,here,to,the,laminar,regime,,though,it,is,of,course,possible,to,implement,a,time-stepper.,Here,a,vertical,natural,convection,problem,is,studied;,a,fluid-containing,cavity,transfers,heat,from,a,hot,side,to,a,cold,one.,The,parameters,of,the,problem,are,the,dimensionless,temperature,difference,(Rayleigh,number,Ra),and,the,usual,Prandtl,number,Pr,of,the,fluid.,This,work,provides,an,additional,cross-validation,against,NEKTAR++,simulations,of,the,same,prob-,lem,as,well,as,validation,against,recent,results,in,the,literature;,the,model,used,here,follows,the,set-up,in,[8],with,a,unit,square,cavity,and,a,value,of,Pr,=,10,(the,latter,is,not,far,from,the,value,for,water,,approx.,7).,3.1,Calculation,of,Nusselt,number,The,Nusselt,number,is,easily,calculated,using,a,few,lines,of,Unified,Form,Language,(UFL,,used,in,the,FIREDRAKE,interface).,In,the,sequel,,normals,normL,and,normR,are,constant,members,of,the,vector,space,V,,,the,assemble,command,calculates,the,components,of,the,argument,(scalar,here),,and,ds(1),and,ds(2),refer,to,the,composites,labelled,by,1,and,2,,which,are,respectively,the,left-,and,right-hand,boundaries,and,are,domains,of,integration,for,computing,the,Nusselt,number,,which,is,then,0.5*(fluxL-fluxR).,normL,=,Function(V),normL,=,Constant((-1.0,0.0)),fluxL,=,assemble(inner(normL,,grad(T))*ds(1)),normR,=,Function(V),normR,=,Constant((1.0,0.0)),fluxR,=,assemble(inner(normR,,grad(T))*ds(2)),This,high-level,functionality,can,be,compared,with,adding,the,same,functionality,to,a,NEKTAR++,simulation,,in,which,a,Filter,needs,to,be,added,to,the,source,code,and,the,relevant,parts,of,the,code,recompiled;,the,filter,must,then,be,activated,by,adding,a,command,in,the,session,file.,As,a,reference,,the,AeroForces,filter,included,with,NEKTAR++,codebase,calculates,the,aero-,dynamic,forces,on,a,surface,or,body,and,can,be,easily,modified,to,compute,heat,fluxes,(the,main,step,is,to,change,the,relevant,field,to,the,temperature,and,to,take,the,derivative);,the,file,FilterAeroForces.cpp,is,of,order,1000,code,lines,in,size.,22,Figure,16:,Steady-state,temperature,profiles,from,the,case,discussed,in,the,text,for,Ra,=,7,×,105,and,element,orders,2,,1,,1,on,a,uniform,grid,(left),and,for,Ra,=,2,×,106,and,element,orders,5,,4,,4,on,the,boundary-refined,mesh.,The,meshes,are,visualized,for,illustrative,purposes.,3.2,Simple,continuation,method,An,initial,study,involves,setting,up,the,problem,with,a,direct,solver,(as,described,in,[9]),and,pro-,ceeding,to,solve,the,equations,with,zero,values,in,the,initial,state.,Using,order-two,elements,for,the,velocity,components,and,one,order,less,for,the,pressure,and,the,temperature,(this,,called,here,order,2,,1,,1,is,the,usual,Taylor-Hood,element,choice),and,a,uniform,40×40-element,grid,,it,is,found,that,the,method,works,well,up,to,Ra,=,7,×,105,but,that,it,fails,for,Ra,=,8,×,105,and,above.,It,is,found,that,increasing,element,order,gives,only,very,small,improvement,in,the,maximum,value,of,Ra,for,which,the,simulation,can,be,performed,successfully:,increasing,order,to,4,,3,,3,gives,convergence,for,Ra,=,8,×,105,but,not,Ra,=,9,×,105.,Increasing,element,order,in,conjunction,with,the,boundary-refined,mesh,it,is,possible,to,do,a,little,better,with,Ra,=,2,×,106,converging,(but,Ra,=,3,×,103,not),for,element,orders,5,,4,,4.,This,improvement,is,clearly,not,sufficient,to,reproduce,results,from,[8],which,extend,up,to,Ra,=,1014,using,a,finite-difference,code,(a,more,relevant,target,here,is,Ra,≈,5,×,1010,,beyond,which,it,is,found,[8],that,the,flow,ceases,to,be,laminar,and,stationary).,√,The,boundary-refined,mesh,was,used,with,a,simple,continuation,method,to,reach,higher,values,of,Ra.,Here,,a,sequence,of,simulations,is,performed,for,gradually-increasing,values,of,Ra,,using,the,result,of,the,previous,calculation,to,initialize,the,solver,for,the,next,one.,It,was,found,that,an,10,starting,from,Ra,=,105,gives,a,method,that,works,up,to,and,beyond,the,incremental,factor,of,limit,of,the,laminar,regime.,(It,was,,however,,noted,that,very,high,values,of,Ra,(beyond,say,1012,sometimes,resulted,in,out-of-memory,errors,on,a,32Gb,machine.),This,gives,an,efficient,method,for,obtaining,the,physically-interesting,relation,Nu(Ra),(the,formulation,as,an,elliptic,problem,re-,moves,the,need,to,simulate,until,any,transients,have,decayed).,The,relation,shows,the,expected,1,power-law,scaling,Nu,∼,Ra,4,characteristic,of,the,laminar,flow,regime,,Fig.17.,Good,agreement,with,the,values,obtained,by,[8],is,seen,up,to,where,the,laminar,regime,ceases,to,be,the,true,time-domain,solution.,23,log10,Ra,Nu,(Wang,et,al,,[8]),Nu,(orders,4,,3,,3),Nu(5,,4,,4),Nu(6,,5,,5),7,8,9,10,17.45,32.17,57.83,103.57,17.33,31.78,57.44,103.57,17.33,31.79,57.53,103.04,17.33,31.79,57.56,103.33,Table,2:,Table,of,steady-state,Nusselt,number,values.,The,values,from,[8],are,the,highest,grid,resolution,ones,quoted.,Figure,17:,Graph,of,heat,transfer,rate,represented,by,Nusselt,number,Nu,as,function,of,Rayleigh,number,Ra.,Two,datasets,from,FIREDRAKE,are,shown,,using,elements,of,order,5,,4,,4,and,4,,3,,3,respectively,-,there,is,a,small,amount,of,discrepancy,for,the,largest,Rayleigh,numbers,simulated.,These,outputs,can,be,compared,with,the,results,from,Wang,et,al,,showing,agreement,up,until,the,emergence,of,the,time-dependent,state,for,Ra,≥,5,×,1010,where,the,scaling,changes,from,the,classic,1,3,power,law,associated,to,Malkus,scaling,[10].,4,laminar,power,law,to,the,1,24,3.3,3-D,version,It,is,possible,to,perform,the,same,calculation,in,3-D,in,order,to,access,the,laminar,flow,regime,of,a,real,tank,(with,three-dimensional,no-slip,boundary,conditions).,These,3-D,flows,are,relevant,to,the,Smallab,experiments,being,performed,at,the,University,of,Leeds.,Note,that,steady,flows,in,a,3-D,geometry,with,periodicity,in,the,extra,dimension,will,be,identical,to,those,in,two,dimensions,,though,the,linear,stability,properties,would,be,expected,to,differ.,4,Task,work,by,grantees,Accepted,reports,are,held,on,the,Documents,repository,of,the,ExCALIBUR-NEPTUNE,organisa-,tion,on,GitHub,,to,which,belong,other,repos,containing,software,developed,under,project,NEP-,TUNE.,Note,that,some,of,these,repos,may,be,access-controlled,,please,email,neptune@ukaea.uk,if,difficulties,are,encountered,in,seeing,the,material.,4.1,Reports,received,under,Grant,T/NA078/21,,PO,(King’s,College,London),•,2053622-TN-03-3,-,D1.2:,Augmentation,of,the,NekMesh,generator,to,provide,quad-based,meshes,for,2D,configurations,[11],•,2053622-TN-04-2,-,D2.4:,Evaluation,of,preconditioners,of,matrix-free,operations,in,Nektar++,for,anisotropic,heat,transport,[12],4.1.1,2053622–TN–03,Report,2053622–TN–03,concerns,NEKMESH,,NEKTAR++’s,built-in,meshing,utility.,It,describes,some,of,the,current,capabilities,of,the,tool,and,an,extension,that,will,allow,it,to,generate,2-D,meshes,composed,of,purely,quadrilateral,elements.,In,previous,version,of,NEKMESH,,quadrilat-,erals,could,only,be,added,as,part,of,the,boundary,layer,meshing,functionality.,“Quad”,meshes,have,some,important,computational,advantages,over,triangular,meshes;,in,particular,,they,tend,to,use,fewer,elements,overall,and,provide,superior,performance,when,using,matrix-free,operators.,This,latter,factor,is,expected,to,be,of,critical,importance,when,applying,high-order,methods,on,ex-,ascale,platforms,,as,a,matrix-free,approach,can,reduce,memory,bandwidth,and,greatly,increase,computational,intensity.,The,same,arguments,also,apply,to,hexahedral,meshes,in,3-D,,though,the,new,developments,discussed,in,this,report,are,for,2-D,applications,only.,In,section,3,,the,authors,describe,NEKMESH’s,existing,quad/hexahedra,construction,capability,,which,is,used,in,isoparametric,boundary-layer,generation.,This,technique,extends,a,given,bound-,ary,(mesh,constraining,curve),in,the,normal,direction,,thus,creating,a,layer,of,quadrilateral,ele-,ments,either,inside,or,outside,said,boundary.,The,authors,concede,that,the,existing,implemen-,tation,does,not,consider,interactions,between,nearby,curves,when,treated,in,this,manner,(which,could,create,problems,when,adding,boundary,elements,to,nested,flux,surfaces,,or,x-point,ge-,ometries).,They,also,explain,that,a,further,difficulty,exists,with,adding,boundary,layers,to,both,25,sides,of,an,embedded,constraint,curve,(as,opposed,to,a,boundary,curve),,though,a,fix,for,this,is,currently,in-development.,Another,existing,capability,is,to,refine,the,boundary,layer,by,splitting,the,quads,along,the,direction,transverse,to,the,surface,normal;,the,splitting,can,be,graduated,so,that,the,element,density,in,the,normal,direction,increases,geometrically,toward,the,boundary,(one,current,limitation,here,is,that,this,also,can,only,be,done,on,one,side,of,a,constraint,curve).,These,methods,are,compatible,with,the,curved,meshes,,on,which,the,splitting,scheme,relieves,difficulties,inherent,in,more,general,approaches,to,anisotropic,boundary,refinement.,Subsection,3.3,gives,an,instructive,illustration,of,the,refined,boundary,layer,(on,a,3-D,mesh).,Section,4,describes,a,new,method,for,generating,fully-quadrilateral,meshes.,This,code,currently,exists,on,a,work-in-progress,branch,of,the,NEKTAR++,repository,,and,is,the,subject,of,a,recent,publication.,The,core,of,the,method,is,the,use,of,a,systematic,approach,to,detect,where,non-,valence-four,vertices,are,located.,The,authors,plan,to,validate,this,approach,for,tokamak-relevant,geometries,and,to,integrate,it,into,the,NEKMESH,workflow,,though,they,expect,it,will,only,be,applicable,to,2-D,use,cases.,Finally,,the,authors,conclude,that,structural,changes,to,the,NEKMESH,code,will,be,necessary,in,order,to,proceed,beyond,the,current,version,,which,was,designed,with,a,focus,on,aeronautical,applications,eg.,isolated,aerofoils,in,empty,space.,They,suggest,a,number,of,technical,changes,required,to,fully,integrate,the,new,method,in,NEKMESH,and,allow,others,in,project,NEPTUNE,to,make,use,of,the,functionality.,4.1.2,2053622–TN–04,Report,2053622–TN–04,provides,a,summary,of,the,preconditioner,types,currently,available,in,NEKTAR++,and,describes,work,to,develop,a,new,preconditioning,method.,It,goes,on,to,present,a,comparison,of,the,different,options,based,on,their,performance,when,applied,to,a,steady-state,,two-dimensional,,anisotropic,diffusion,problem.,A,formulation,of,the,benchmarking,problem,is,presented,in,Section,2.,The,authors,begin,by,explaining,the,form,of,the,diffusivity,tensor,in,a,magnetised,plasma,,then,derive,the,anisotropic,thermal,conductivity,tensor,that,features,in,the,problem,definition.,Section,2.1,describes,how,the,problem,is,discretised,in,variational,form,and,expressed,as,a,matrix,system.,Implementation,details,of,NEKTAR++’s,existing,preconditioners,are,discussed,in,Section,3,,along,with,a,new,method,called,‘Geometrically,Informed,Algebraic,Multigrid’,(GIAMG),,which,uses,the,Saena,library.,As,background,,the,authors,explain,that,the,algebraic,multigrid,(AMG),method,is,well,suited,to,unstructured,meshes,,but,tends,to,suffer,from,a,loss,of,sparsity,which,can,affect,performance,and,result,in,poor,scaling.,The,new,method,addresses,this,weakness,by,using,a,combination,of,‘p,coarsening’,(polynomial,order,reduction),and,‘h,coarsening’,(coarsening,of,the,mesh,itself).,Results,of,the,comparison,study,are,presented,in,Section,4.,Performance,(iteration,count,or,run,time),is,presented,as,a,function,of,both,anisotropic,ratio,and,transport,angle,for,the,full,matrix,system,and,for,the,‘statically,condensed’,version.,In,both,cases,,the,best,results,are,achieved,by,the,‘Diagonal’,preconditioner.,The,new,GIAMG,method,typically,takes,many,fewer,iterations,to,arrive,at,a,solution,,but,tends,to,have,longer,solve,times,overall.,The,authors,point,out,,however,,26,that,GIAMG,is,expected,to,scale,better,than,the,linear,space,(XX,T,),solver,for,large,problems,(>,104,cores),and,that,the,benchmark,problem,does,not,approach,this,scale.,They,also,note,that,they,expect,the,efficiency,of,the,new,method,to,improve,with,further,development.,4.2,Reports,received,under,Grant,T/NA085/21,,PO,2047357,(University,of,Oxford),•,2047357-TN-10,-,Numerical,study,of,1+1D,drift,kinetic,models,for,parallel,dynamics,in,the,plasma,edge,[13],•,2047357-TN-13-2,-,Numerical,study,of,a,reduced,model,coupling,2D+2V,drift-kinetic,ions,and,2D+3V,kinetic,neutrals,in,a,helical,magnetic,field,with,wall,boundaries,[14],•,2047357-TN-14-2,-,Overview,of,the,numerical,issues,and,findings,associated,with,the,1D,and,2D,drift,kinetic,models,[15],4.2.1,2047357–TN–10,The,report,presents,the,various,1d1v,versions,of,the,moment,kinetic,approach,,separating,out,different,numbers,of,fluid,moments,in,the,sense,that,first,only,density,is,treated,separately,from,the,rest,of,the,particle,distribution,for,the,species,,then,density,and,velocity,,and,finally,all,the,five,special,moments,of,density,,velocity,(counts,by,itself,as,3,moments),and,energy,are,treated,sepa-,rately,,effectively,giving,a,series,of,coupled,fluid,and,kinetic,models.,For,each,of,the,three,cases,,the,corresponding,boundary,conditions,are,specified.,Coordinates,are,made,dimensionless,(us-,ing,the,standard,project,notation,rather,than,that,of,the,report),based,on,a,choice,of,Ls,,Us,and,employing,Nref,to,normalise,number,density.,The,report,then,presents,Julia-derived,numerical,results,for,the,various,cases,,using,SSPRK,(Strong,Stability,Preserving,Runge-Kutta),schemes,for,the,time,advance,and,either,third,order,upwind,finite,difference,schemes,or,Chebychev,polynomials,to,represent,the,spatial,dependence.,The,latter,use,the,Gauss-Chebyshev-Lobatto,grid,that,has,nodes,at,element,edges.,Conserva-,tion,of,integral,quantities,is,enforced,by,uniform,scaling,of,the,moments,and,distribution,functions,at,each,timestep.,Realistic,boundary,conditions,at,walls,,as,well,as,the,usual,(largely,irrelevant,outside,the,plasma,separatrix),periodic,conditions,,are,implemented.,Simulations,are,presented,of,a,wall-bounded,model,and,a,periodic,model.,The,former,is,designed,to,compare,with,an,analytic,model,of,the,plasma,sheath,,and,the,latter,with,a,Landau-damped,sound,wave,,ie.,a,sound,wave,which,kinetic,effects,cause,to,be,damped.,These,simulations,exhibit,spectrally,accurate,convergence,,ie.,that,errors,diminish,exponentially,with,number,of,mesh-points,,demonstrating,the,correctness,of,the,implementation,of,the,model,and,the,spatial,and,temporal,discretizations.,There,is,discussion,as,to,the,implementation,of,plasma-neutral,collisions,,in,which,there,is,argued,the,need,for,a,separate,grid,in,velocity,space,for,the,neutrals.,27,4.2.2,2047357–TN–13–2,This,report,extends,the,work,of,ref,[16],to,two,dimensions,in,space,and,two,or,three,dimensions,in,velocity,space,,ie.,2d2v,or,2d3v.,Only,the,moment,corresponding,to,fluid,density,is,treated,In,respect,of,the,two,space,dimensions,(r,,z),,r,corresponds,to,distance,from,the,separately.,plasma,centre,and,z,to,distance,along,the,axis,of,a,circular,cylinder.,The,magnetic,field,,has,a,fixed,poloidal,component,(ie.,directed,in,the,ζ,direction,-,for,which,θ,is,the,preferred,symbol),as,well,as,having,the,expected,component,in,z,,which,is,also,taken,to,be,constant,throughout,the,model.,Poloidal,,ie.,ζ,,variation,of,quantities,is,not,allowed.,The,same,normalisation,is,used,for,the,equations,to,make,them,dimensionless.,Ion-neutral,collisions,are,however,now,included.,Computationally,,the,same,language,(Julia),and,the,same,numerical,approach,is,used,as,before,,except,that,the,spatial,distributions,are,now,represented,exclusively,as,Chebychev,polynomials,,ie.,no,finite,differences.,An,important,point,is,the,use,of,the,electron,Boltzmann,response,relation,ne,∝,exp(qeΦ/Te),(in,standard,project,notation),to,calculate,the,electric,potential,Φ,from,ne,and,hence,the,electrostatic,field,that,acts,on,the,plasma.,The,neutrals,are,represented,on,a,3-D,grid,which,,since,the,neutrals,are,unaffected,by,electromagnetic,forces,,ignores,the,presence,of,magnetic,field.,(There,is,an,interesting,discussion,of,the,issues,involved,in,having,different,‘best’,representations,for,charged,and,neutral,particles,respectively.),Implementation,details,for,the,MMS,(see,next,paragraph),are,presented.,Analytically,,a,solution,for,the,plasma,sheath,is,presented,,but,most,of,the,solutions,derived,for,testing,the,numerics,are,derived,by,use,of,the,Method,of,Manufactured,Solutions,(MMS).,MMS,may,be,illustrated,using,a,simple,ODE,,viz.,¨y,+,Ω2y,=,0,(26),MMS,entails,intelligent,selection,of,y(t),,then,forming,a,source,term,S(t),,to,appear,on,the,right-,hand,side,of,the,equation,,then,solving,¨y,+,Ω2y,=,S(t),For,example,if,is,chosen,y(t),=,cos(ωt),so,that,S(t),=,(Ω2,−,ω2),cos(ωt),(27),(28),then,there,should,be,zero,residual,if,the,modified,equation,is,solved,with,this,form,of,additional,source,term.,There,are,potential,difficulties,,in,that,the,numerical,work,may,be,of,little,value,if,y(t),is,inappropriately,chosen,,thus,a,sinusoidal,forcing,may,not,tell,much,about,solutions,of,¨y,−,Ω2y,=,0,(29),which,has,exponentially,growing,solutions.,Moreover,,resonant,effects,in,Equation,(26),could,cause,difficulties,if,ω,happened,to,have,been,chosen,close,to,the,natural,frequency,Ω.,As,in,the,1d1v,work,[16],,many,simulations,,including,all,those,with,periodic,boundary,conditions,,demonstrate,spectrally,accurate,convergence.,However,,when,an,electric,field,is,present,,other,simulations,demonstrate,not,only,poor,convergence,,but,(in,private,communications),indications,of,divergence.,There,follows,an,honest,report,of,numerical,difficulties,encountered,at,the,frontiers,of,computational,plasma,physics.,Possible,causes,of,the,difficulties,are,discussed,at,more,length,in,the,follow-up,report,[15].,28,4.2.3,2047357–TN–14–2,Work,in,the,report,[14],showed,that,problems,arose,in,2-D,kinetic,modelling,tested,using,MMS,,when,a,wall,boundary,condition,was,applied,in,the,z,coordinate.,The,report,[15],which,follows,,discusses,a,number,of,possible,causes,of,these,problems,(but,note,that,references,[7],and,[8],need,correcting,to,cite,reports,2047357-TN-10,and,2047357-TN-13,respectively).,Very,recent,work,has,served,to,identify,the,prime,cause,and,its,elimination,,so,the,report,below,is,largely,superseded.,The,manufactured,solutions,might,have,been,expected,to,be,of,wider,value,,but,subsequent,experience,has,indicated,that,they,are,the,primary,cause,of,the,numerical,difficulties,encountered.,It,is,worth,however,noting,that,sheath,conditions,give,rise,to,a,boundary,condition,on,velocity,depending,on,the,value,of,the,electric,field,,and,in,a,kinetic,model,,velocity,is,itself,a,coordinate,over,which,the,unknown,distribution,function,f,is,defined.,The,upshot,is,that,a,condition,for,f,to,vanish,is,applied,between,nodes,,so,that,f,is,not,smooth,within,the,computational,domain,and,spectral,convergence,was,impossible,without,employing,a,smoothing,algorithm.,In,addition,,the,MMS,numerical,tests,(private,communication),indicated,an,even,worse,problem,in,that,there,was,not,only,poor,convergence,,but,indications,of,divergence.,It,was,acknowledged,there,and,below,that,dealing,with,this,would,be,challenging,,because,there,was,then,no,certain,identification,of,the,cause,of,the,problem(s).,The,report,below,[15],also,notes,that,,regardless,of,the,original,research,plan,,resolving,the,numerical,difficulties,has,to,be,the,priority,objective,,a,decision,vindicated,by,subsequent,developments.,4.3,Reports,received,under,Grant,T/AW086/22,,PO,2070839,(University,of,Oxford),•,2070839-TN-01,-,Update,on,state,of,the,art,in,edge,modelling,[17],•,2070839-TN-02,-,Implementing,distributed-memory,in,the,2D-3V,drift-kinetic,edge,code,[18],4.3.1,2070389–TN–01,This,report,updates,the,earlier,document,[19],,specifically,the,review,of,the,state-of-the-art,in,edge,modelling,is,brought,up-to-date.,The,updated,report,focusses,on,developments,which,the,authors,believe,to,represent,an,advance,over,previous,edge,modelling,work,,especially,if,they,involve,kinetic,treatments.,These,leads,to,the,discussion,of,approximately,a,dozen,new,citations,,mostly,dating,from,the,last,years,2021-2022.,The,advances,turn,out,all,to,be,concerned,with,the,addition,of,further,physical,effects,,mostly,to,the,codes,directed,at,modelling,kinetic,effects.,There,is,a,description,at,the,end,of,the,still,severe,limitations,of,most,kinetic,modelling,in,treating,effects,at,the,transition,from,closed,to,open,fieldlines,at,the,plasma,edge,,ie.,across,the,plasma,boundary,defined,by,separatrix,,and,particularly,near,the,X-point,where,the,poloidal,component,of,magnetic,field,vanishes.,The,authors,expect,a,special,treatment,to,be,needed,close,to,the,latter,place.,The,hope,,in,the,context,of,the,broader,project,,is,that,unstructured,finite,element,meshes,will,not,only,enable,solution,near,the,X-point,,but,might,be,suitably,adapted,(ie.,in,conformity,with,29,the,magnetic,field,direction),so,that,co-ordinates,which,explicitly,follow,fieldlines,can,be,avoided,altogether.,4.3.2,2070839-TN–02,The,report,describes,the,further,parallelisation,of,the,2d3v,Julia,software,under,development,at,Oxford,,in,order,to,speed,turnaround,of,tests,of,the,software,and,thereby,lead,to,faster,identification,of,the,problem,issues.,The,additional,parallelisation,sees,implementation,of,distributed-memory,MPI,alongside,the,shared-memory,MPI,that,was,already,in,use;,both,MPI,features,use,the,Julia,MPI,library.,The,resulting,source,code,is,made,available,on,github.,The,distribution,of,computational,nodes,applies,only,to,2-D,position,space,,which,is,spread,across,processors,as,a,set,of,rectangular,regions,,each,point,accompanied,by,the,corresponding,3V,,ie.,3-D,velocity,space.,Spectrally,accurate,convergence,is,demonstrated,for,periodic,boundary,conditions,,using,up,to,32,×,48,cores,on,the,MARCONI,supercomputer.,5,Summary,This,report,has,laid,out,initial,work,toward,implementing,a,subset,of,the,HERMES-3,equations,in,NEKTAR++.,Some,understanding,of,the,basic,physics,of,the,blob2d,example,has,been,gained,and,some,possible,benchmark,tests,have,been,outlined.,There,are,some,difficulties,in,understanding,the,units,used,by,the,core,code,in,HERMES-3,and,other,uncertainty,comes,in,the,form,of,the,numerical,diffusivities,used,in,that,code.,A,simple,continuation,method,was,shown,with,the,result,that,higher,values,of,Rayleigh,number,can,be,accessed,by,FIREDRAKE,convection,simulations.,Grantee,work,over,the,year,was,summarized,in,Section,4.,Acknowledgement,ET,acknowledges,assistance,with,HERMES-3,from,John,Omotani,of,UKAEA,,and,Chris,Ridgers,and,Mike,Kryjak,of,the,University,of,York,,assistance,with,NEKTAR++,from,the,user,community,via,the,mailing,list;,also,assistance,with,FIREDRAKE,from,Colin,Cotter,of,Imperial,College,London,and,the,FIREDRAKE,online,user,community.,ET,also,thanks,Wayne,Arter,for,pointing,out,the,analogy,between,the,plasma,blob,examples,and,fluid,thermal,convection.,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,30,A,A,study,of,the,Vlasov-Poisson,equation,system,The,Vlasov-Poisson,system,of,equations,in,1,+,1,dimensions,(one,space,,one,velocity),,also,referred,to,as,1d1v,,is,the,simplest,example,of,a,meaningful,kinetic,problem;,technically,this,is,a,special,case,of,the,collisionless,Boltzmann,equation.,Here,a,method,of,finding,analytic,solutions,is,outlined,and,demonstrated,in,simple,cases.,The,system,is,not,a,typical,system,of,partial,differential,equations,(Cauchy,problem),because,it,contains,explicit,quadratures,over,one,of,the,dimensions.,It,will,be,seen,that,the,kinetic,problem,differs,significantly,from,the,fluid,case,and,this,provides,(albeit,limited),guidance,for,what,to,expect,when,exchanging,the,classical,fluid,picture,of,a,plasma,to,a,kinetic,one,(the,full,3,+,3-dimensional,case,involves,seven,independent,variables,and,so,cannot,easily,be,solved,,hence,the,focus,on,a,toy,example,here).,The,problems,discussed,here,represent,collisionless,systems,where,the,self-interactions,of,the,matter,are,of,the,collective,,long-range,nature,(the,matter,density,gives,rise,to,a,collective,electrostatic,field).,A.1,Example,of,an,analytic,equilibrium,solution,The,Vlasov-Poisson,system,is,,for,particles,in,a,neutralizing,background,,∂f,∂t,d2Φ(x),dx2,=,ω2,P,∂f,∂x,+,v,(cid:18),1,vN,(cid:90),+∞,−∞,+,E(x),∂f,∂v,=,0;,(cid:19),f,dv,−,1,;,E,=,−,dΦ,dx,;,(30),where,the,constant,vN,is,chosen,such,that,the,domain,contains,no,net,charge.,The,constant,ωp,is,the,plasma,frequency,,which,specifies,the,strength,of,the,coupling,and,is,the,only,parameter,in,the,problem,(excepting,initial,data).,Physically,this,translates,into,∝,e,n,for,electron,number,density,n,for,spatially-homogeneous,solutions.,The,function,f,is,the,distribution,function,(DF),for,the,charged,particles,,ρ,is,the,charge,density,,and,Φ,is,the,electrostatic,potential.,√,It,is,clear,that,any,spatially-homogeneous,solution,f,(v),and,E,=,0,is,a,time-independent,solution;,the,important,question,of,the,stability,of,such,equilibria,is,deferred,until,later.,To,find,stationary,solutions,,let,f,(x,,v),=,f,(H),where,H,=,1,2,v2,+,Φ(x).,(31),This,automatically,reduces,the,Vlasov,equation,to,∂f,/∂t,=,0,and,then,the,associated,potential,is,evaluated,by,inverting,the,nonlinear,Poisson,equation.,Note,that,H,is,just,the,Hamiltonian,for,a,unit,mass,particle,moving,in,a,potential,Φ(x),ie.,a,clas-,sical,oscillator.,The,validity,of,the,solution,is,easily,shown,by,change,of,variable;,more,generally,31,the,fact,that,a,function,of,a,conserved,quantity,solves,the,steady-state,equations,is,known,as,Jeans’,theorem.,It,is,known,from,gravitational,potential,theory,(eg.,[20]),that,this,enables,a,range,of,equilibrium,solutions,to,be,constructed,and,studied.,The,remaining,part,of,the,problem,is,the,linear,equation,for,the,charge,density,ρ,=,−ω2,P,(cid:18)(cid:90),(cid:19),−,1,f,dv,vN,(32),Let,us,seek,a,solution,where,there,is,a,maximum,value,of,permitted,energy,ie.,an,energy,scale,is,added,into,the,problem.,Thus,define,the,constant,Φ0,to,be,this,maximum,and,E,≡,Φ0,−,Φ,−,v2,1,2,(33),(note,this,automatically,satisfies,the,Vlasov,equation,by,the,reasoning,presented,earlier),so,that,E,>,0,represents,allowed,values,of,v,ie.,f,>,0,for,E,>,0,and,f,=,0,for,E,≤,0.,Further,define,Ψ,≡,Φ0,−,Φ.,Now,one,has,(cid:18),vN,1,−,ρ(x),ω2,P,(cid:19),(cid:90),=,f,>0,f,dv,=,2,(cid:90),Ψ,0,f,(E)dE,(cid:112)2(Ψ,−,E),.,This,is,clearly,of,the,form,(cid:90),g(Ψ),=,f,dv,=,(cid:90),Ψ,0,f,(E)dE,(cid:112)(Ψ,−,E),(34),(35),and,it,is,known,that,this,equation,has,a,unique,solution,for,f,given,by,the,Abel,integral,formula1:,f,(E),=,1,π,d,dE,(cid:90),E,0,g(t)dt,√,E,−,t,.,Now,ρ,can,be,expressed,in,terms,of,Ψ,using,the,Poisson,equation,(note,change,of,sign),so,the,general,Abel,inversion,is,d2Ψ,dx2,=,ρ,(cid:18),vN,1,−,1,ω2,P,d2Ψ,dx2,(cid:19),(cid:90),=,f,>0,f,dv,=,2,(cid:90),Ψ,0,f,(E)dE,(cid:112)2(Ψ,−,E),.,(36),(37),(38),1which,is,a,special,case,of,the,Radon,transform,,which,is,presumably,the,technique,of,choice,for,more,general,,higher-dimensional,cases,32,A,spatially-periodic,system,will,be,considered,from,this,point.,It,is,clear,that,the,charge,density,function,cannot,have,a,nonzero,n,=,0,Fourier,component,,because,there,must,be,zero,net,charge.,However,,the,potential,Φ(x),may,have,a,constant,component,as,this,is,compatible,with,periodicity,and,it,does,not,affect,the,dynamics.,Specialising,to,eg.,ρ,=,4π2,cos,2πx,on,[0,,1],one,has,Ψ,=,Φ0,−,cos,2πx,and,ρ,=,4π2,(Φ0,−,Ψ).,(It,is,worth,mentioning,here,that,this,is,the,main,trick,-,one,could,also,specify,ρ,implicitly,by,using,something,like,d2Φ,dx2,=,−k2,sin,Φ.),Hence,(cid:18),1,+,vN√,2,4π2,ω2,P,(cid:19),(Ψ,−,Φ0),=,(cid:90),f,>0,f,dv,=,2,(cid:90),Ψ,0,f,(E)dE,(cid:112)(Ψ,−,E),.,Applying,Abel’s,formula,gives,(cid:16),f,(E),=,vN√,2π,d,dE,(cid:90),E,0,giving,the,DF,as,,for,positive,E,(zero,otherwise),1,−,4π2Φ0,ω2,√,P,(cid:17),t,dt,+,4π2,ω2,P,.,E,−,t,f,(E),=,√,d,dE,2EvN,π,(cid:18),1,−,(cid:19),4π2Φ0,ω2,P,+,8π2E,3ω2,P,ie.,f,(x,,v),=,,,vN√,2π,(cid:113),1,−,4π2Φ0,ω2,P,Φ0,−,cos,2πx,−,1,2,v2,8π2(cid:113),Φ0,−,cos,2πx,−,1,2,v2,ω2,P,+,(39),(40),(41),(42),,,2,v2,=,Φ0,−,cos,2πx,−,1,wh.,E,≡,Φ0,−,Φ,−,1,Note,there,is,a,singularity,as,E,→,0,albeit,an,integrable,one,(cf.,(cid:82),1,0,Φ0,=,ω2,2,v2.,Here,Φ0,is,a,maximum,energy,level,(free,parameter).,dx√,1−x2,=,π,4π2,is,special,in,that,it,avoids,this,singularity.,2,).,The,case,where,P,It,is,easy,to,substitute,this,solution,back,into,the,charge,density,equation,to,verify,its,correctness,(as,part,of,this,,Φ0,cancels,out,once,the,integration,is,done,as,it,must,as,it,does,not,appear,in,the,charge,density,equation).,Note,that,ωP,→,∞,gives,back,the,limit,where,the,charge,density,does,not,depend,on,position,(see,this,by,evaluating,the,integral,of,f,over,velocity).,A.2,Summary,The,Vlasov-Poisson,system,is,,for,particles,in,a,neutralizing,background,,33,∂f,∂t,d2Φ(x),dx2,=,ω2,P,∂f,∂x,+,v,(cid:18),1,vN,(cid:90),+∞,−∞,+,E(x),∂f,∂v,=,0;,(cid:19),f,dv,−,1,;,E,=,−,dΦ,dx,.,A,particular,example,of,a,general,steady-state,solution,is,f,(x,,v),=,,,vN√,2π,(cid:113),1,−,4π2Φ0,ω2,P,8π2(cid:113),Φ0,−,cos,2πx,−,1,2,v2,ω2,P,+,2,v2,Φ0,−,cos,2πx,−,1,(cid:19),dv,vN,−,1,f,=,4π2,cos,2πx;,(cid:18)(cid:90),ρ(x),=,−ω2,P,d2Φ,dx2,=,−ρ.,,,(43),(44),(45),This,model,contains,only,trapped,particles,if,ωP,8π2,+,Φ0,2,v2),but,otherwise,contains,free,particles.,The,non-singular,case,is,on,the,limit,of,trapped,particles,and,this,limit,is,the,only,non-singular,case,,for,which,2,≤,1,(recall,E,=,Φ0,−,Φ,−,1,f,(x,,v),=,(cid:114),4,√,2πvN,ω2,P,ω2,P,4π2,−,cos,2πx,−,1,2,v2,(46),It,is,curious,that,this,solution,does,not,resemble,more,closely,the,states,found,by,evolving,unsta-,ble,initial,data,as,reported,in,[21];,those,solutions,seemed,to,be,characterized,by,the,formation,of,phase-space,holes,(probably,present,due,to,the,incompressibility,property,of,phase,space),,though,the,author,is,uncertain,whether,those,states,were,actually,time-independent.,This,solution,is,unlikely,to,be,a,good,candidate,for,simulation,using,spectral,/,hp,methods,due,to,the,discontinuity,in,the,derivative,where,the,density,terminates.,It,is,also,not,known,whether,the,solution,is,stable,(the,stability,analysis,is,more,involved,than,the,examples,exhibited,in,the,next,subsection,,because,the,spatial,variation,means,that,a,spatial,perturbation,is,not,a,Fourier,mode).,These,type,of,states,appear,stationary,but,it,is,important,to,keep,in,mind,that,the,model,used,here,was,collisionless;,collisional,systems,need,to,be,invariant,under,the,action,of,particle,collisions,,which,generally,leads,to,Maxwellian-like,velocity,distributions.,A.3,Stability,of,equilibrium,solutions,The,stability,of,equilibrium,solutions,is,of,interest,as,it,can,provide,tests,of,the,time-evolution,in,a,linear,regime,(in,addition,to,answering,the,obvious,question,of,whether,a,given,solution,is,likely,to,34,Figure,18:,Contour,plot,of,one,x-period,of,the,solution,for,ω2,special,choice,of,parameters,for,which,the,integrable,singularity,in,f,cancels,away.,8π2,=,1,and,Φ0,=,2.,Note,this,is,a,P,persist,under,time-evolution).,It,is,clear,that,any,spatially-homogeneous,solution,f,(v),and,E,=,0,is,a,time-independent,solution,to,Equation,(30),and,one,consideration,is,to,check,its,stability,by,computing,the,dispersion,relation,for,a,linear,perturbation,(the,following,standard,formula,applies,for,an,x-independent,background,and,a,harmonic,perturbation,with,frequency,ω,and,wavenumber,k,-,the,latter,of,which,is,quantized,if,on,a,periodic,spatial,domain),1,=,ω2,p,(cid:90),+∞,−∞,f,(v)dv,(ω,−,kv)2,.,(47),This,comes,from,linearizing,the,system,Equation,(30),with,eg.,f,=,f0,+,ϵf1,about,the,zero-electric-,field,,x-uniform,initial,distribution;,the,equation,before,integrating,by,parts,is,1,=,−,ω2,P,k,(cid:90),∞,−∞,df0(u),du,du,ω,−,u,(48),in,which,u,=,kv,-,note,the,integral,is,clearly,the,Hilbert,transform,of,df0(u),du,.,Note,that,the,integrals,over,v,mean,that,this,is,not,a,usual,linear,eigenvalue,problem.,In,the,following,,the,stability,of,several,interesting,physical,systems,will,be,discussed.,The,analysis,focusses,on,the,two-stream,instability,problems,of,the,sort,investigated,using,a,NEKTAR++,imple-,mentation,in,continuum,kinetics,in,[21],and,a,NESO-Particles,implementation,using,particle-in-cell,in,[22].,35,A.3.1,Gaussian,beams,A,stability,analysis,was,done,for,the,case,of,two,counterpropagating,Gaussian,beams,in,[21].,There,the,initial,f0,is,the,bi-Gaussian,(normed,so,that,the,whole,system,integrates,over,v,to,a,density,of,1),-,for,which,total,energy,is,v2,0,+,σ2,(think,of,beam,mean,kinetic,energy,plus,thermal,energy,and,note,the,simple,addition,of,these,is,due,to,Gaussianity),-,f0(v),=,√,(cid:18),1,8πσ2,e−,(x−v0)2,2σ2,+,e−,(x+v0)2,2σ2,(cid:19),for,which,the,dispersion,integral,can,be,done,analytically,to,1,=,−,ω2,P,2σ2k2,√,(cid:16),2,−,πAe−A2,(erfi(A),−,i),−,√,πA′e−A′2,(cid:0)erfi(A′),−,i(cid:1)(cid:17),(49),(50),with,A,≡,ω−kv0√,2σk,,,A′,≡,ω+kv0√,2σk,.,Note,using,a,single,Gaussian,,or,v0,=,0,in,the,above,,gives,,in,the,small-k,limit,,the,dispersion,relation,for,a,warm-plasma,Langmuir,wave,ω2,=,ω2,P,+,3k2σ2,using,the,series,expansion,of,the,Dawson,function.,It,is,also,possible,to,show,that,for,v0,=,0,the,system,is,stable,(this,is,a,Maxwellian,profile,in,velocity,space):,as,usual,the,boundary,of,the,stable,region,is,given,by,setting,ω,=,0,in,the,dispersion,relation.,Now,choosing,the,DF,as,f,(v),=,√,e−,v2,2σ2,1,2πσ2,(51),the,Hilbert,transform,is,trivial,in,the,case,ω,=,0,,giving,the,criterion,ω2,no,solutions,to,this,equation,for,any,positive,ωP,,,the,case,v0,=,0,is,always,stable.,P,=,−k2σ2.,Since,there,are,It,is,known,that,there,are,pure,imaginary,solutions,ω,=,iγ,to,Equation,(50),representing,purely,growing,or,decaying,modes,,though,actually,plotting,the,dispersion,ω(k),is,challenging,due,to,the,stiffness,of,the,resulting,Newton,solver,scheme.,One,interesting,feature,is,the,boundary,between,stable,initial,data,and,unstable,,which,is,found,by,inspecting,where,the,sign,of,Im(ω),changes.,With,ω,purely,imaginary,over,the,range,of,interest,,setting,ω,=,0,in,the,above,gives,the,stability,frontier,(Fig.19),ωP,kσ,=,1,(cid:112)2xD+(x),−,1,(52),(cid:113),mev2,≡,2,e−x2erfi(x).,The,initial,data,is,where,x,≡,v0√,2σ,stable,if,ωP,is,below,the,value,given,by,the,RHS.,(Note,the,cold-beam,case,can,be,obtained,from,the,σ,→,0,limit,,stable,for,ωP,<,kv0.),2kBT,and,the,Dawson,function,is,D+(x),≡,0,π,√,This,tells,that,the,system,is,always,stable,for,x,below,the,zero,at,2xD+(x),=,1,→,x,≈,0.92414,(where,2xD+(x),<,1,so,the,quantity,in,the,radical,is,negative),and,that,above,this,there,is,a,stability,boundary,where,the,system,is,stable,for,small,enough,values,of,the,normalized,coupling,36,Figure,19:,Stability,frontier,(‘phase,diagram’),for,the,Gaussian,initial,condition,two-stream,insta-,bility:,vertical,axis,is,ωP,.,Points,below,the,curve,represent,stable,initial,conditions,and,overall,scale,is,set,by,the,Gaussian,width,,physically,the,temperature.,kσ,and,horizontal,is,v0√,2σ,ωP,kσ,.,This,boundary,slopes,upwards,ie.,for,large,enough,normalized,initial,velocities,the,system,is,more,likely,to,be,stable.,The,picture,obtained,here,is,consistent,with,the,expectation,for,the,ωP,→,0,limit,,in,which,any,initial,data,must,be,stable,as,there,is,no,dynamics.,This,formula,also,predicts,evenly-spaced,instability,for,modes,of,increasing,integer,wave,number,ie.,the,n,=,2,mode,requires,twice,the,ωP,value,to,become,unstable,as,does,the,n,=,1,mode,-,hence,it,is,apparent,that,the,instability,is,a,associated,to,long,wavelengths,(the,latter,is,reminiscent,of,another,kinetic-related,linear,instability,,the,Jeans,instability).,The,above,analysis,gives,good,agreement,for,the,stability,bound,of,the,system,when,simulated,in,NEKTAR++,,predicting,instability,when,ωP,>,2.9223,in,that,case,(see,the,x-axis,intercept,in,Fig.21).,The,stability,is,here,seen,to,be,conditional,on,the,value,of,the,coupling,strength,ΩP,.,Readers,familiar,with,plasma,physics,might,ask,how,this,can,be,consistent,with,a,Penrose’s2,criterion,[23],,which,involves,only,the,initial,data,(in,the,formula,,u0,is,the,location,of,a,local,minimum,of,f,(u),,and,satisfaction,of,the,criterion,is,a,necessary,condition,for,the,system,to,be,unstable).,(cid:90),∞,−∞,f0(u),−,f,(u0),(u,−,u0)2,du,>,0.,(53),The,answer,is,that,the,Penrose,criterion,applies,to,the,non-periodic,case,in,which,there,is,no,long-wavelength,cut-off,and,in,that,case,everything,with,a,value,of,x,above,the,zero,at,2xD+(x),=,1,→,x,≈,0.92414,is,unstable,(indeed,this,condition,seems,to,come,out,of,doing,the,integral,in,the,Penrose,formula).,The,non-periodic,case,can,also,be,accessed,by,plotting,the,graph,for,the,periodic,case,and,then,rescaling,the,y,axis,by,letting,k,→,0,,the,stability,curve,will,collapse,to,the,x,axis,leaving,the,unstable,region,only,for,x,>,0.92414.,This,system,may,be,viewed,a,sort,of,‘phase,diagram’,problem,in,the,x,,y,≡,ωP,kσ,plane.,Note,that,2Oliver,Penrose,,theoretical,physicist,and,brother,of,the,more,popularly-known,Roger,37,Figure,20:,Time-evolution,of,the,two-stream,instability,as,a,sequence,of,frames,,taken,from,a,NEKTAR++,simulation.,Figure,21:,Dispersion,relation,for,the,two-stream,instability,for,various,values,of,k,obtained,from,NEKTAR++,code,(crosses),compared,to,the,analytic,solution,in,Equation,(50),(curves).,As,evident,from,Eq.52,the,ascending,spatial,harmonics,are,evenly,spaced,in,ωP,with,spacing,of,2.9223,,which,is,the,value,of,the,stability,frontier,ωP,for,σ,=,1,5,and,v0,=,1,as,used,in,the,NEKTAR++,simulation.,38,the,stability,frontier,has,two,asymptotics,,x,=,0.92414,and,y,=,√,2x.,The,stability,frontier,and,its,two,asymptotics,are,superficially,similar,to,the,phase,diagram,for,the,Lorenz,model,exhibited,in,[24],in,which,the,behaviour,of,the,Lorenz,oscillator,as,a,function,of,the,usual,Rayleigh,and,Prandtl,numbers,is,studied;,that,system,also,exhibits,an,L-shaped,stability,frontier,Figure,22.,Note,that,the,Rayleigh,number,corresponds,to,y,and,the,Prandtl,to,x,and,that,the,axes,are,interchanged,compared,to,Fig.19,ie.,the,coupling,ωP,is,analogous,to,the,Rayleigh,number,and,the,velocity,offset,v0,the,Prandtl,number,(all,scaled,to,the,system,temperature,in,the,Vlasov-Poisson,case).,A.3.2,Delta-function,(ie.,cold),beams,This,case,uses,non-differentiable,functions,so,it,is,not,straightforward,to,apply,the,Penrose,crite-,rion.,It,is,sometimes,referred,to,as,beams,of,cold,plasma,as,it,corresponds,to,zero-width,Gaussian,beams.,One,has,f0(v),=,1,2,(δ(v,−,v0),+,δ(v,+,v0)).,The,dispersion,relation,is,z2,=,(cid:16),1,2,1,+,2y2,±,(cid:112),1,+,8y2(cid:17),(54),,,y,≡,kv0,ωP,wh.,z,≡,ω,.,I,think,the,negative,sign,for,the,radical,to,be,the,correct,choice.,Considering,ωP,stability,bounds,,setting,z,=,0,one,finds,a,zero,at,y,=,1,ie.,ωP,=,kv0,and,it,seems,the,system,is,stable,for,ωP,<,kv0.,Clearly,for,k,=,0,(continuum,case),the,system,is,always,unstable.,A.3.3,Top,hat,function,beams,Normalized,to,unity,in,total,,two,top,hats,centred,on,±v0,and,both,width,=,σ.,Again,,it,is,not,straightforward,to,apply,the,Penrose,criterion.,z2,=,(cid:16),1,2,1,+,2y2,+,2s2,−,(cid:112),1,+,8y2,+,16s2y2(cid:17),(55),with,s,≡,kσ,ωP,.,One,obtains,the,fact,that,this,case,is,stable,up,to,ω2,0,−,σ2),(solve,quadratic,and,take,positive,root,to,ensure,recovery,of,s,=,0,limit),so,again,always,unstable,in,the,continuum,and,the,width,actually,makes,the,system,more,unstable,in,the,sense,that,if,the,top,hats,touch,or,overlap,the,system,is,always,unstable,even,if,periodic,(weirdly,,this,means,a,single,top,hat,centred,on,zero,is,always,unstable,and,this,is,a,function,with,no,minimum!).,The,stability,frontier,can,be,expressed,in,the,form,P,=,k2(v2,wh.,x,≡,v0,σ,.,ωP,kv0,=,√,x2,−,1,x,39,(56),Figure,22:,Phase,diagram,of,the,Lorenz,63,model,in,(Pr,,Ra),space,(in,the,figure,r,is,Rayleigh,number,and,σ,Prandtl,number).,The,black,regions,correspond,to,values,in,parameter,space,where,the,system,tends,to,a,stable,fixed,point,(ie.,there,is,no,chaotic,behaviour).,Figure,reproduced,from,[24].,40,A.3.4,Cauchy,(or,Lorentzian,,or,Breit-Wigner),beams,This,is,a,bit,questionable,as,the,second,moment,of,the,distribution,is,not,finite,(suggesting,un-,bounded,temperature);,the,distribution,is,f0(v),=,(cid:18),s,2π,1,(v,−,v0)2,+,s2,+,1,(v,+,v0)2,+,s2,(cid:19),(57),and,in,this,case,the,integration,can,be,done,thanks,to,a,formula,obtained,from,the,useful,Wolfra-,mAlpha,website,[25],(cid:90),∞,−∞,dv,(v2,+,s2)(ω,−,kv)2,=,−,π,s(ks,−,iω)2,(58),and,doing,a,linear,transformation,on,v.,The,resulting,dispersion,relation,,with,ω,set,zero,to,find,the,stability,contour,,gives,ωP,kv0,=,x2,+,1,√,x2,−,1,(59),where,as,usual,x,≡,v0,s,.,The,asymptotics,are,y,=,1,and,y,=,x,,trivial.,This,stability,contour,is,very,much,like,that,obtained,in,the,Gaussian,case,,and,indeed,in,the,k,=,0,limit,(non-periodic),the,system,is,stable,for,s,>,v0.,The,latter,concurs,with,the,non-periodic,Penrose,formula,analysis,done,at,[23].,This,case,shows,that,Gaussianity,is,not,required,for,stable,solutions,and,indeed,the,v0,limit,represents,a,single,Cauchy,distribution,centred,on,zero,-,so,the,dynamics,here,,unlike,the,collisional,case,,admit,multiple,stable,solutions,and,not,just,Maxwellians.,The,full,dispersion,relation,for,this,case,is,obtained,by,solving,the,equation,2,¯v2,=,1,(iy,+,z)2,+,1,(i¯y,+,z)2,wh.,z,≡,ω,k,,,y,≡,s,+,iv0,and,¯v,≡,ωP,k,.,The,solution,is,ω,ks,=,i,2,,(cid:115),(cid:114),−2,±,∓2x2,1,+,8y2,x2,−,4y2,−,2x2,,,(60),(61),which,can,be,plotted,(but,is,not,very,interesting,beyond,observing,the,existence,of,the,asymp-,totics).,Doubtless,if,some,QoI,associated,to,the,full,nonlinear,dynamics,were,used,,the,plot,would,be,richer,(as,is,the,Lorenz,case,in,[24]).,Note,that,this,dispersion,relation,is,easier,to,work,with,nu-,merically,than,the,one,for,the,Gaussian,case,Eq.Equation,(50),as,it,does,not,involve,multiplication,of,very,large,and,very,small,factors,(as,well,as,not,using,higher,functions).,41,Figure,23:,Dispersion,plot,for,Cauchy,case,with,here,x,≡,v0,ks,.,The,natural,logarithm,has,been,taken,to,highlight,the,stability,contour,given,by,Eq.59.,Generated,using,WolframAlpha,with,the,input,plot,log(-2+sqrt(2x,^2,sqrt(1+8y^2/x^2)-4y^2-2x^2)),from,0,to,10.,Note,similarity,to,Fig.19.,s,and,y,≡,ωP,42,A.3.5,A,simple,comparison,of,kinetic,and,fluid,two-stream,problems,One,can,ask,whether,there,is,a,fluid,analogue,of,the,two-stream,instability.,Here,,a,particular,fluid,set-up,is,considered,where,the,only,mutual,interaction,is,via,an,electrostatic,force,and,there,are,two,fluids,of,negative,charge,(with,charge,neutrality,enforced,with,a,non-dynamic,positive,charge,background),forming,one-dimensional,beams,with,bulk,velocity,±¯v,(of,course,,there,is,Maxwellian,thermal,velocity,on,scales,much,below,the,hydrodynamic,scale).,The,fluids,must,be,compressible,since,the,dynamics,of,a,one-dimensional,incompressible,fluid,are,trivial.,The,equations,of,motion,are,,noting,negative,charge,for,both,fluids,,and,the,neutralization,term,being,implicit,and,constant,in,space,and,time,∂ρ1,∂t,+,(ρ1v1)′,=,0;,∂v1,∂t,+,v1v′,1,=,−,1,ρ1,p′,1,+,q,ρ1,Φ′;,∂ρ2,∂t,+,(ρ2v2)′,=,0;,∂v2,∂t,+,v2v′,2,=,−,q,ϵ0,Φ′′,=,1,ρ2,p′,2,+,q,ρ2,Φ′;,(ρ2,+,ρ1).,(62),(63),(64),(65),(66),Now,note,the,sound,speed,is,given,by,p1,=,v2,s,ρ1,and,is,the,same,for,both,fluids.,Now,the,background,state,has,ρ1,=,ρ2,=,¯ρ,and,v1,=,−v2,=,¯v,as,constants,and,zero,electric,potential,given,the,neutralization.,Doing,a,linear,expansion,about,the,background,state,one,has,,having,time-diffentiated,the,conti-,nuity,equation,and,space-differentiated,the,momentum,equation,(cid:0)ω2,+,¯vkω,−,v2,(cid:1),ρ1,−,ω2,s,k2,−,ω2,P,P,ρ1,+,(cid:0)ω2,−,¯vkω,−,v2,s,k2,−,ω2,P,−ω2,P,ρ2,=,0,(cid:1),ρ2,=,0,(67),(68),where,ω2,P,≡,¯ρq2,ϵ0,.,The,dispersion,relation,is,obtained,from,the,usual,determinant.,The,stability,frontier,is,obtained,by,solving,in,the,ω,=,0,case,and,one,obtains,the,simple,stability,criterion,k,≥,0.,(69),This,means,that,the,charged,fluid,two-stream,case,is,always,stable,for,any,value,of,the,coupling,strength.,There,is,no,analogue,of,the,Jeans,wavenumber,that,occurs,in,the,gravitational,case,-,this,is,because,the,two,electrostatically-charged,fluids,here,experience,a,mutual,repulsion,,not,an,43,attraction.,The,electrostatic,two-stream,instability,is,seen,to,be,a,purely,kinetic,effect,(or,,at,least,,the,instability,requires,more,terms,than,are,retained,in,the,model,of,the,system,by,compressible,Navier-Stokes).,The,fluid,result,can,be,compared,with,the,delta-function,kinetic,case,in,which,the,stability,criterion,is,ωP,<,kv0.,(70),This,is,always,unstable,in,infinite,space,(but,it,may,be,stable,in,a,periodic,system,because,there,is,in,that,case,a,minimum,permitted,value,for,k).,The,full,dispersion,relation,is,easily,solved,to,(cid:18),z2,+,y2,=,(cid:19),1,2,x2,+,1,−,(cid:115)(cid:18),z2,+,(cid:19),1,4,x4,+,x2,+,1,(71),where,y,≡,ω,ωP,Equation,(50).,,,x,≡,¯vk,ωP,,,and,z,≡,vs,¯v,.,This,is,the,continuum,version,of,the,kinetic,theory,formula,Note,large,z,is,the,subsonic,limit,in,which,is,recovered,sound,waves,,ω,=,vsk.,In,the,supersonic,z,=,0,limit,there,are,no,waves,(ω,=,0).,Note,at,least,in,the,static,limit,,stability,is,expected,since,the,system,will,act,to,neutralize,a,local,perturbation,in,charge,density,and,not,to,amplify,such,a,perturbation.,B,Units,in,vertical,convection,A,sketch,of,the,derivation,of,the,equations,of,Rayleigh-B,´enard,convection,using,the,Boussinesg,approximation,is,presented,in,eg.,Tritton,[26,,§,14,Appendix],,with,result,∂u,∂t,∂T,∂t,+,u,·,∇u,=,−,1,ρ0,∇p,+,ν∇2u,+,gα(T,−,T0)ˆy;,+,u,·,∇T,=,κ∇2T,;,∇,·,u,=,0.,(72),where,ρ0,is,the,fluid,mass,density,at,a,reference,temperature,T0,,so,that,ν,is,then,the,viscous,diffusivity.,Coefficient,κ,is,thermal,diffusivity,,both,with,dimension,L2,s,,,and,note,that,p′,=,p/ρ0,has,dimension,(Ls/ts)2.,st−1,Rayleigh-B,´enard,convection,is,driven,by,a,temperature,difference,between,opposite,boundaries,,and,the,lengthscale,Ls,is,taken,to,be,the,distance,between,these,boundaries,,usually,the,layer,depth.,Even,so,,there,are,several,ways,to,non-dimensionalize,the,equations;,understanding,of,these,is,necessary,to,convert,between,the,scalings,used,in,the,literature.,The,resulting,systems,are,,of,course,,physically,equivalent.,44,B.1,Scaling,to,Rayleigh,and,Prandtl,number,using,thermal,diffusivity,timescale,Rescale,time,t,as,t,=,t0˜t,using,t0,=,L2,are,dimensionless,and,u0,=,Ls/t0.,s,κ,,,u,=,u0,˜u,,and,T,=,T0,+,∆T,˜T,where,the,tilded,quantities,The,equations,become,in,the,dimensionless,quantities,,and,dropping,the,tildes,,(cid:19),+,u,·,∇u,=,−∇p′,+,∇2u,+,Ra,T,ˆy;,1,Pr,(cid:18),∂u,∂t,∂T,∂t,+,u,·,∇T,=,∇2T,;,∇,·,u,=,0.,The,dimensionless,Rayleigh,and,Prandtl,numbers,are,respectively,P,r,≡,Ra,≡,,,ν,κ,gα∆T,L3,s,κν,.,(73),(74),(75),(76),This,formulation,is,widely,used,eg.,ref,[27],and,notably,in,the,classic,1960s,studies,by,Elder,[28],and,in,the,papers,by,Winters,[29].,B.2,Scaling,to,Rayleigh,and,Prandtl,number,using,free-fall,velocity,The,free-fall,velocity,is,u0,=,(cid:112)gLsα∆T,(77),and,the,rescaling,is,t,=,t0˜t,,u,=,u0,˜u,,and,offset,temperature,T,−,T0,=,∆T,˜T,where,the,tilded,quantities,are,dimensionless,and,t0,=,Ls/u0.,The,equations,become,(as,before,,dropping,the,tildes),∂u,∂t,∂T,∂t,+,u,·,∇u,=,−∇p′,+,+,u,·,∇T,=,√,1,Ra,Pr,∇,·,u,=,0.,(cid:114),Pr,Ra,∇2T,;,∇2u,+,T,ˆy;,(78),(79),(80),This,formulation,is,used,in,Zucatti,et,al.,[30],and,Wang,et,al.,[8],(and,is,useful,eg.,for,converting,the,length,of,calculation,needed,for,the,system,to,relax,to,the,quasi-steady,state,quoted,in,the,appendix,to,the,latter,paper).,45,B.3,Scaling,to,Grashof,number,using,viscosity,Rescale,time,t,as,t,=,t0˜t,using,t0,=,L2,are,dimensionless,and,u0,=,Ls/t0.,s,ν,,,u,=,u0,˜u,,and,T,−,T0,=,∆T,˜T,where,the,tilded,quantities,The,equations,become,,dropping,the,tildes,,and,with,Prandtl,number,defined,as,above,Equa-,tion,(73),(cid:19),+,u,·,∇u,=,−∇p′,+,∇2u,+,Gr,T,ˆy;,(cid:18),∂u,∂t,∂T,∂t,+,u,·,∇T,=,1,Pr,∇,·,u,=,0.,∇2T,;,Gr,≡,gκ∆T,L3,s,ν2,≡,Ra,Pr,.,The,Grashof,number,is,Two,relevant,cases,are,the,Pr,→,∞,limit,in,which,there,is,zero,thermal,diffusivity,,and,Pr,=,1,where,the,diffusivities,for,momentum,and,for,temperature,are,identical.,This,formulation,is,used,in,the,FIREDRAKE,Rayleigh-B,´enard,convection,example,at,[9].,B.4,Scaling,with,the,Ad,hoc,Approach,Rescale,time,t,by,ts,,velocity,by,us,=,Ls/ts,and,offset,temperature,by,Ts,,where,ts,,Ls,and,Ts,may,be,chosen,arbitrarily.,The,equations,become,,dropping,the,tildes,on,the,fields,(cid:19),+,u,·,∇u,=,−∇p′,+,˜ν∇2u,+,RaahcT,ˆy;,(cid:18),∂u,∂t,∂T,∂t,+,u,·,∇T,=,˜κ∇2T,;,∇,·,u,=,0.,The,resulting,Rayleigh,number,is,Raahc,=,gαTst2,s,Ls,×,∆T,Ts,This,formulation,is,particularly,convenient,for,engineering,problems,,where,the,diffusivities,will,normally,be,significant,functions,of,temperature,that,might,directly,introduced,,and,Ts,may,be,46,(81),(82),(83),(84),(85),(86),(87),(88),taken,in,units,of,Kelvin.,Specifically,˜κ,=,κts/L2,s,and,˜ν,=,νts/L2,s,with,equations,(cid:19),+,u,·,∇u,=,−∇p′,+,∇,·,˜ν∇u,+,RaahcT,ˆy;,(cid:18),∂u,∂t,∂T,∂t,+,u,·,∇T,=,∇,·,˜κ∇T,;,∇,·,u,=,0.,(89),(90),(91),References,[1],W.,Arter.,Equations,for,ExCALIBUR,/,NEPTUNE,proxyapps.,port,CD/EXCALIBUR-FMS/0021,,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.26-M1.2.1.pdf.,UKAEA,Project,Neptune,,2023.,Technical,Re-,https:,[2],Hermes-3.,https://hermes3.readthedocs.io/en/latest/examples.html#,d-drift-plane.,Accessed:,March,2023.,[3],O.E.,Garcia,,N.H.,Bian,,and,W.,Fundamenski.,Radial,interchange,motions,of,plasma,fila-,ments.,Physics,of,Plasmas,13,,082309,(2006),,2006.,[4],J.T.,Omotani,,F.,Militello,,L.,Easy,,and,N.R.,Walkden.,The,effects,of,shape,and,ampli-,tude,on,the,velocity,of,scrape-off,layer,filaments.,Plasma,Physics,and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