TN-03_PhysicsEdgeFusion
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REPORT,2047357-TN-03-02,M1.2,1,Physics,in,the,edge,of,fusion,devices,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Rudolf,Peierls,Centre,for,Theoretical,Physics,,University,of,Oxford,,Oxford,OX1,3PU,,UK,(This,version,is,of,23,April,2021),1.,Introduction,We,present,a,brief,overview,of,the,current,state,of,fusion,device,edge,modeling.,This,re-,port,is,written,for,ExCALIBUR,NEPTUNE,contract,T/NA085/20.,This,contract,stated,mission,is,the,development,of,kinetic,models,for,the,edge.,Thus,,this,report,will,focus,on,the,known,problems,of,fluid,models,without,making,much,emphasis,on,their,many,suc-,cesses.,This,lack,of,praise,for,fluid,models,is,not,the,objective,of,this,report,and,hence,we,would,like,to,start,by,reassuring,the,reader,that,we,believe,current,fluid,codes,have,much,to,offer,and,need,to,be,pursued,in,parallel,to,kinetic,modeling.,One,of,the,challenges,for,kinetic,modeling,is,to,devise,methods,to,match,with,fluid,simulations,efficiently.,With,this,capability,,the,edge,can,be,split,into,non-overlapping,spatial,regions,with,different,levels,of,kinetic,sophistication.,The,remainder,of,this,reports,is,organized,as,follows.,In,section,2,we,explain,what,the,characteristic,time,and,length,scales,are,in,the,edge.,In,section,3,,we,give,an,overview,of,the,existing,fluid,models,for,the,edge.,In,section,4,we,list,physical,phenomena,that,cannot,be,captured,by,fluid,models,,and,we,explain,how,these,phenomena,have,been,addressed,so,far.,Finally,,we,discuss,our,proposal,to,develop,a,complete,set,of,kinetic,equations,for,the,edge,in,section,5.,2.,The,edge,In,this,report,,we,call,the,edge,the,plasma,that,surrounds,the,separatrix.,The,separatrix,(represented,as,a,red,line,in,figure,1),is,the,flux,surface,that,separates,the,region,where,magnetic,field,lines,are,in,contact,with,the,walls,of,the,vessel,from,the,region,in,which,magnetic,field,lines,form,nested,toroidal,flux,surfaces.,This,means,that,the,edge,includes,both,open,field,lines,(those,who,are,in,contact,with,walls),and,closed,field,lines,(those,that,form,nested,toroidal,flux,surfaces).,We,need,to,distinguish,the,core,,where,the,fusion,reactions,are,supposed,to,take,place,,from,the,edge.,In,this,report,,we,will,use,a,theoretical,criterion,to,distinguish,one,from,the,other:,whether,the,characteristic,transit,times,in,the,directions,parallel,and,perpendicular,to,the,magnetic,field,are,comparable,or,not.,In,the,core,,both,ions,and,electron,closely,follow,magnetic,field,lines,and,can,travel,around,the,device,many,times,before,collisions,,turbulent,fluctuations,or,other,effects,drive,them,away,from,the,magnetic,field,line,on,which,they,started.,As,a,result,,density,and,temperature,are,almost,constant,along,magnetic,field,lines,,and,one,only,worries,about,small,fluctuations,around,the,mostly,quiescent,profiles,of,density,and,temperature.,This,is,,of,course,,a,highly,idealized,situation,that,ignores,violent,events,taking,place,in,the,core,,such,as,sawteeth,(Hastie,1997),,but,it,is,a,useful,one.,In,contrast,,in,the,edge,,charged,particles,that,were,well-confined,in,the,core,region,cross,the,separatrix,and,eventually,reach,the,wall,by,following,magnetic,field,lines.,The,displacements,of,ions,and,electrons,away,from,the,magnetic,field,line,in,which,they,2,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Figure,1.,Sketch,of,a,tokamak,edge,on,a,plane,that,contains,the,axis,of,symmetry,(represented,here,by,a,dash-dot,vertical,line).,The,edge,region,is,in,blue,,and,the,separatrix,is,the,red,line.,started,are,still,small,compared,to,the,size,of,the,device,,but,they,are,not,small,compared,with,the,characteristic,lengths,of,the,density,and,temperature,in,the,edge,region.,If,the,characteristic,lengths,of,density,and,temperature,along,and,across,magnetic,field,lines,are,L(cid:107),and,L⊥,,the,edge,is,characterized,by,|v(cid:107)|,L(cid:107),∼,|vd|,L⊥,,,(2.1),where,v(cid:107),is,the,characteristic,parallel,velocity,of,the,particles,,and,vd,their,small,drift,perpendicular,to,the,magnetic,field.,Since,|v(cid:107)|,(cid:29),|vd|,,L⊥,is,much,smaller,than,L(cid:107).,In,the,core,,for,comparison,,we,find,that,L(cid:107),and,L⊥,are,of,similar,order,and,comparable,to,the,machine,size,,giving,|v(cid:107)|,L(cid:107),(cid:29),|vd|,L⊥,.,(2.2),There,is,another,aspect,that,makes,the,edge,very,different,from,the,core.,The,tem-,perature,of,both,electrons,and,ions,is,kept,low,near,the,wall,because,the,wall,is,a,very,effective,sink,of,energy,(wall,materials,that,prevent,slow,Hydrogen,particles,from,going,back,into,the,plasma,are,notable,exceptions,where,the,temperature,of,the,plasma,can,be,large,near,the,wall;,see,,for,example,,Schmitt,et,al.,(2015),for,liquid,Lithium,divertors,,or,Jackson,et,al.,(1991),for,boronized,walls).,At,low,plasma,temperatures,,the,plasma,is,partially,ionized,and,collisions,between,the,charged,particles,in,the,plasma,and,neutrals,become,important.,In,extreme,limits,,detachment,occurs,,that,is,,the,plasma,tempera-,ture,decreases,sufficiently,due,to,radiation,that,the,plasma,recombines,and,a,cushion,of,neutrals,appears,in,front,of,the,walls,protecting,them,(Krasheninnikov,&,Kukushkin,2017).,In,addition,to,limiting,the,temperature,,the,presence,of,the,wall,controls,the,size,of,the,electric,field,and,the,flows,in,the,open,field,line,region.,We,will,discuss,these,effects,in,more,detail,in,section,3.,We,finish,this,section,by,calculating,a,few,characteristic,time,and,length,scales.,In,Militello,&,Fundamenski,(2011),,one,can,find,a,concise,summary,of,typical,values,of,plasma,characteristics,in,the,edge,of,tokamaks.,In,current,tokamaks,,the,magnitude,of,Closedfieldline,regionOpen,field,line,regionWalls,Physics,in,the,edge,of,fusion,devices,3,the,magnetic,field,B,is,typically,2,T,,and,the,characteristic,size,of,the,device,is,a,few,meters.,The,plasma,temperature,T,ranges,from,1,keV,in,the,closed,magnetic,field,line,part,of,the,edge,to,100,eV,at,the,separatrix,down,to,10,eV,near,the,wall.,In,detached,plasmas,,the,temperature,drops,to,1,eV,near,the,wall.,The,electron,density,ne,ranges,from,1019,m−3,in,the,open,field,line,region,to,1020,m−3,in,the,closed,field,line,region.,Neutral,density,nn,is,usually,smaller,than,ne,and,it,ranges,from,1015,m−3,in,the,closed,field,line,region,to,1019,m−3,in,the,open,field,line,region,(Colchin,et,al.,2000;,Scotti,et,al.,2021).,With,these,quantities,,we,calculate,several,characteristic,frequencies,of,interest:,•,The,gyrofrequencies,of,both,Deuterium,and,electrons,,Ωi,:=,eB/mD,and,Ωe,:=,eB/me,,are,the,characteristic,frequencies,of,the,nearly,circular,motion,of,the,charged,particles,around,magnetic,field,lines.,Here,e,is,the,proton,charge,,and,mD,and,me,are,the,Deuterium,and,electron,masses.,•,The,transit,frequencies,for,Deuterium,ions,and,electrons,,vtD/L(cid:107),and,vte/L(cid:107),,are,the,inverse,of,the,time,that,it,takes,charged,particles,to,move,along,a,magnetic,field,line,from,one,wall,to,another,in,the,open,field,line,region,,and,the,inverse,of,the,time,that,it,takes,charged,particles,to,sample,a,flux,surface,in,the,closed,field,line,region.,Here,vtD,:=,(cid:112)2T,/mD,and,vte,:=,(cid:112)2T,/me,are,the,Deuterium,and,electron,thermal,speeds,,and,L(cid:107),is,the,characteristic,length,of,magnetic,field,lines,in,the,edge,,which,we,take,to,be,10,m.,•,The,transit,frequency,for,Deuterium,neutrals,,vtD/L⊥,,is,the,inverse,of,the,time,that,it,takes,neutral,atoms,to,cross,the,edge,region.,The,characteristic,scale,of,variation,of,density,and,temperature,across,magnetic,field,lines,,L⊥,,is,of,the,order,of,5,cm,or,larger,in,the,closed,field,line,region,(Sugihara,et,al.,2000),,and,of,the,order,of,1,cm,or,larger,in,the,open,field,line,region,(Goldston,2012).,•,The,collision,frequencies,and,νii,:=,√,4,π,3,e4ne,ln,Λ,(4π(cid:15)0)2m1/2,D,T,3/2,νep,:=,√,4,2π,3,e4ne,ln,Λ,(4π(cid:15)0)2m1/2,e,T,3/2,(2.3),(2.4),describe,how,often,Deuterium,ions,collide,with,each,other,or,electrons,collide,with,other,electrons,or,Deuterium,ions,,respectively.,Here,ln,Λ,≈,15,is,the,Coulomb,logarithm,and,(cid:15)0,2,difference,between,νii,and,νep,is,a,convention,is,the,vacuum,permittivity.,The,factor,of,introduced,by,Braginskii,(Braginskii,1958).,√,•,We,also,need,the,collision,frequencies,that,describe,how,often,Deuterium,ions,collide,with,Deuterium,neutral,atoms,,νin,:=,nnvtDσin,,and,how,often,Deuterium,neutral,atoms,collide,with,Deuterium,ions,,νni,:=,nivtDσin.,Here,,the,ion-neutral,cross,section,σin,is,of,order,10−18,m2,(Lindsay,&,Stebbings,2005).,Similarly,,we,need,the,collision,frequencies,that,describes,how,often,electrons,collide,with,Deuterium,neutral,atoms,,νen,:=,nnvteσen,,and,how,often,neutrals,are,ionized,,νion,:=,nevteσion.,Here,,the,electron-neutral,collision,cross,section,σen,is,of,order,10−19,m2,(Brackmann,et,al.,1958),and,the,ionization,collision,cross,section,σion,is,of,order,10−20,m2,(Zel’dovich,&,Raizer,2013).,All,these,frequencies,are,shown,in,table,1.,It,is,clear,that,the,collision,frequencies,are,the,ones,that,change,the,most,across,the,edge.,For,ions,and,electrons,,collisions,become,very,important,in,the,cooler,plasma,of,the,open,field,line,region,,but,are,infrequent,in,the,closed,field,line,region.,Conversely,,for,neutrals,,collision,are,more,frequent,in,the,closed,field,line,region,than,in,the,open,field,line,region.,We,also,calculate,a,few,length,scales,of,interest:,4,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,Ion,freq.,[104,s−1],Ωi,Region,Closed,lines,9400,9400,Open,lines,vtD/L(cid:107),3.1,0.31,νin,νii,0.16,0.031,16,31,Neutral,freq.,[104,s−1],vtD/L⊥,νni,3100,31,νion,1800,18,610,310,Electron,freq.,[106,s−1],Ωe,νen,vte/L(cid:107),νep,320000,320000,0.13,0.0018,13,1.8,0.18,1.8,Table,1.,Characteristic,frequencies,for,the,closed,field,line,region,(B,=,2,T,,ne,=,1020,m−3,,nn,=,1015,m−3,,T,=,1,keV),and,the,open,field,line,region,(B,=,2,T,,ne,=,1019,m−3,,nn,=,1019,m−3,,T,=,10,eV).,Parallel,lengths,[1,m],L(cid:107),λii,Region,Closed,lines,10,190,1000,140,10000,Open,lines,λen,L⊥,5,1,λin,λep,10,0.19,0.14,0.1,1,Perpendicular,lengths,[1,cm],ρe,ρi,0.33,0.0056,0.033,0.00056,λni,1,10,λion,1.7,17,Table,2.,Characteristic,lengths,for,the,closed,field,line,region,(B,=,2,T,,ne,=,1020,m−3,,nn,=,1015,m−3,,T,=,1,keV),and,the,open,field,line,region,(B,=,2,T,,ne,=,1019,m−3,,nn,=,1019,m−3,,T,=,10,eV).,•,The,mean,free,paths,λii,:=,vtD/νii,and,λin,:=,vtD/νin,are,the,distances,that,a,Deu-,terium,ion,can,travel,before,colliding,with,another,Deuterium,ion,or,with,a,Deuterium,atom,,respectively.,Similarly,,the,mean,free,paths,λni,:=,vtD/νni,and,λion,:=,vtD/νion,are,the,distances,that,a,Deuterium,neutral,atom,can,move,before,colliding,with,an,ion,or,getting,ionized,,respectively,,and,λep,:=,vte/νep,and,λen,:=,vte/νen,are,the,distances,that,an,electron,can,move,before,colliding,with,another,charged,particle,or,with,a,neutral,,respectively.,•,The,Deuterium,and,electron,gyroradii,,ρi,:=,vtD/Ωi,and,ρe,:=,vte/Ωe,,are,the,characteristic,size,of,the,gyration,of,charged,particles,around,magnetic,field,lines.,All,these,lengths,are,given,in,table,2.,Unsurprisingly,,we,see,that,the,mean,free,paths,,inversely,proportional,to,the,collision,frequencies,,are,the,characteristic,lengths,that,change,the,most,across,the,edge.,In,the,direction,perpendicular,to,the,magnetic,field,,we,see,that,the,gyroradii,are,small,compared,to,the,characteristic,lengths.,3.,Drift-ordered,fluid,models,Due,to,the,large,collision,frequencies,in,the,open,field,line,region,,the,bulk,of,the,ion,and,electron,distribution,functions,is,Maxwellian.,For,this,reason,,much,of,the,edge,modeling,has,been,based,on,plasma,fluid,equations,derived,in,the,limit,of,large,Coulomb,collisions,(Braginskii,1958).,Since,the,most,readily,available,fluid,equations,for,magnetized,plasmas,assume,that,there,are,no,neutrals,,most,fluid,models,ignore,neutrals,,with,some,notable,exceptions.,The,main,difference,between,the,fluid,equations,used,in,edge,models,and,the,usual,fluid,equations,is,that,diffusivities,are,anisotropic.,As,demonstrated,by,table,1,,charged,parti-,cles,gyrate,around,magnetic,field,lines,many,times,before,having,a,collision.,Thus,,parti-,cles,barely,move,across,magnetic,field,lines,and,the,diffusivity,across,magnetic,field,lines,is,much,smaller,than,the,diffusivity,along,them.,To,capture,the,effect,of,this,anisotropy,correctly,and,efficiently,,one,has,to,either,use,flux,coordinates,that,follow,magnetic,field,lines,(Beer,et,al.,1995),or,employ,appropriate,discretizations,(Hariri,&,Ottaviani,2013).,Physics,in,the,edge,of,fusion,devices,5,The,problem,with,following,magnetic,field,lines,is,that,it,is,difficult,to,find,a,grid,that,both,aligns,with,the,magnetic,field,lines,and,extends,to,the,walls,of,the,vessel.,Unfor-,tunately,,the,wall,geometry,is,important,because,it,determines,how,far,neutrals,leaving,the,wall,travel,into,the,plasma,(Wiesen,et,al.,2018).,Finding,grids,that,adjust,to,the,wall,geometry,has,become,a,problem,of,great,interest,in,recent,years,,and,the,community,has,been,trying,exotic,methods,to,address,it,(Isoardi,et,al.,2010;,Bufferand,et,al.,2019).,An,important,aspect,of,the,edge,fluid,models,is,the,size,of,the,flow.,The,potential,difference,between,the,wall,and,the,plasma,is,controlled,by,the,non-neutral,Debye,sheath,that,forms,around,the,walls,(Riemann,1991).,Typically,,the,potential,drop,between,the,wall,and,the,plasma,has,to,be,several,times,the,electron,temperature,because,otherwise,a,large,electron,current,flows,into,the,wall,,breaking,the,neutrality,of,the,plasma.,If,one,assumes,that,the,wall,in,contact,with,the,plasma,is,a,conductor,and,hence,the,potential,is,constant,across,its,volume,,the,potential,differences,within,the,plasma,are,restricted,to,be,of,the,order,of,the,electron,temperature,,φ,∼,T,/e.,This,gives,an,electric,field,E,=,−∇φ,of,the,order,of,T,/eL⊥.,The,perpendicular,velocity,of,the,fluid,is,determined,by,the,balance,between,the,electric,and,magnetic,forces,,This,equation,gives,a,perpendicular,velocity,of,order,u⊥,×,B,∼,E.,u⊥,∼,|E|,B,∼,ρi,L⊥,vtD.,(3.1),(3.2),Thus,,according,to,table,2,,u⊥,is,significantly,smaller,than,vtD.,The,fluid,equations,obtained,with,the,ordering,(3.2),are,known,as,drift-ordered,equations,because,the,flow,is,of,the,same,order,as,the,slow,particle,drifts,–,the,other,possible,ordering,is,the,high,flow,ordering,that,assumes,that,the,perpendicular,velocity,is,sonic.,Importantly,,for,fluid,velocities,of,the,size,given,in,equation,(3.2),,one,needs,to,keep,terms,that,depend,on,the,gradient,of,the,temperature,and,the,pressure,in,the,stress,tensor,to,be,completely,consistent,(Mikhailovskii,&,Tsypin,1971;,Simakov,&,Catto,2003;,Catto,&,Simakov,2004).,The,system,of,drift-ordered,fluid,equations,is,usually,comprised,of,•,one,continuity,equation,per,ion,species,(the,electron,density,need,not,be,calculated,because,it,is,determined,by,quasineutrality),,•,one,conservation,equation,for,the,component,of,the,plasma,momentum,parallel,to,the,magnetic,field,,•,a,vorticity,equation,that,determines,the,electrostatic,potential,,•,in,electromagnetic,models,,Amp´ere’s,law,,and,•,one,conservation,equation,for,the,energy,of,all,ion,species,and,another,one,for,the,energy,of,the,electrons.,Note,that,there,is,one,single,conservation,equation,for,the,whole,plasma,parallel,momen-,tum,and,one,single,conservation,equation,for,the,energy,of,all,the,ion,species,,and,not,several,conservation,equations,,one,per,ion,species.,The,reason,why,all,ion,species,must,be,considered,as,one,in,these,fluid,equations,is,that,,within,the,large,collision,frequency,approximation,,all,ion,species,have,the,same,temperature,and,all,charged,species,have,the,same,average,flow.,The,electron,temperature,can,be,different,from,the,ion,temperature,due,to,the,mass,difference,between,the,two,species.,The,temperature,differences,between,the,different,ion,species,are,of,the,same,order,as,kinetic,effects,that,are,neglected.,The,differences,between,the,parallel,flows,are,calculated,and,used,in,the,vorticity,equation,,where,they,are,needed,because,the,electric,current,enters,in,the,Lorenz,force.,As,we,mentioned,at,the,start,,the,drift-ordered,fluid,equations,usually,implemented,6,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,in,edge,codes,(Mikhailovskii,&,Tsypin,1971;,Simakov,&,Catto,2003;,Catto,&,Simakov,2004),ignore,neutrals.,There,have,been,attempts,to,include,neutrals,in,the,limit,where,both,ions,and,neutrals,are,sufficiently,collisional,that,both,species,can,be,treated,as,fluids,(Hazeltine,et,al.,1992;,Catto,1994;,Helander,et,al.,1994).,The,advantage,of,these,models,is,that,they,account,for,the,effect,that,ion-neutral,collisions,have,on,the,diffusivities,,but,it,is,generally,believed,that,these,models,are,insufficient,because,neutrals,have,long,mean,free,paths,in,large,regions,of,the,edge,(see,tables,1,and,2).,There,have,been,some,attempts,at,simplified,kinetic,treatments,of,the,neutrals,(Wersal,&,Ricci,2015),and,there,exist,sophisticated,Monte-Carlo,approaches,with,a,large,collection,of,collisions,such,as,EIRENE,(Reiter,et,al.,2005),or,DEGAS,(Stotler,&,Karney,1994).,Note,that,these,Monte-,Carlo,treatments,are,not,completely,self-consistent,because,they,assume,that,the,ion,and,electron,distribution,functions,are,Maxwellian,and,,depending,on,the,version,of,the,code,,they,ignore,elastic,collisions,and,they,average,over,the,dependence,of,the,differential,cross,section,on,the,scattering,angle.,Moreover,,these,neutral,kinetic,codes,are,sensitive,to,the,parameters,that,determine,how,particles,interact,with,the,wall,(Chankin,et,al.,2021),,and,these,parameters,are,not,well,known.,There,are,two,types,of,problems,in,which,the,drift-ordered,fluid,equations,are,used:,2D,profiles,and,turbulence.,The,objective,of,2D,fluid,solvers,such,as,SOLPS,(Wiesen,et,al.,2015),,SolEdge2D,(Bufferand,et,al.,2015),,UEDGE,(Rognlien,et,al.,2007),or,EDGE2D,(Simonini,et,al.,1994),is,to,determine,the,toroidally,averaged,density,and,temperature,profiles,in,the,edge.,These,codes,cannot,model,turbulent,fluctuation,because,they,are,missing,the,third,dimension.,For,this,reason,,instead,of,the,collisional,perpendicular,diffusion,coefficients,,these,codes,use,enhanced,perpendicular,diffusion,coefficients,that,are,chosen,to,match,the,experimental,observations.,These,2D,codes,are,meant,to,provide,detailed,understanding,of,transport,along,magnetic,field,lines,and,of,neutrals,,as,some,of,them,are,coupled,to,Monte-Carlo,neutral,codes,such,as,EIRENE,(Reiter,et,al.,2005),or,DEGAS,(Stotler,&,Karney,1994).,Turbulence,codes,such,as,GBS,(Halpern,et,al.,2016),,TOKAM3X,(Tamain,et,al.,2016),,Hermes,(Dudson,&,Leddy,2017),or,GRILLIX,(Stegmeir,et,al.,2018),are,3D,fluid,codes.,Originally,,fluid,turbulence,codes,assumed,that,ions,were,much,colder,than,electrons,and,that,the,turbulent,fluctuations,were,small,compared,to,an,almost,constant,background,(Zeiler,et,al.,1996).,However,,it,was,soon,realized,that,this,treatment,is,not,appropriate,for,the,edge.,Ions,are,not,cold,and,the,density,and,temperature,profiles,cannot,be,easily,split,into,a,slowly,varying,piece,plus,small,fluctuations,due,to,the,presence,of,the,wall.,The,same,wall,boundary,conditions,that,constrain,the,fluid,velocity,perpendicular,to,the,magnetic,field,to,be,subsonic,,as,shown,in,equation,(3.2),,require,that,the,fluid,velocity,parallel,to,the,magnetic,field,be,sonic,near,the,wall,(Chodura,1982).,The,pressure,drops,along,magnetic,field,lines,have,to,be,significant,to,ensure,that,the,parallel,flow,is,accelerated,sufficiently.,This,is,incompatible,with,the,assumption,that,the,fluctuations,are,small.,Thus,,most,current,edge,turbulence,codes,have,tried,to,lift,the,assumption,of,small,fluctuations,,and,they,have,done,so,by,differing,degrees,depending,on,the,code,or,the,version,of,the,code,in,use.,Overall,,fluid,codes,are,maturing,,and,although,there,is,still,work,to,be,done,,there,is,starting,to,be,a,consensus,on,the,physics,that,they,must,include.,The,same,cannot,be,said,about,kinetic,effects,in,the,edge.,4.,Kinetic,effects,Tables,1,and,2,are,evidence,that,kinetic,effects,must,be,taken,into,account:,in,the,closed,magnetic,field,line,region,and,in,part,of,the,open,field,line,region,,ions,and,electrons,can,Physics,in,the,edge,of,fusion,devices,7,travel,a,distance,of,the,size,of,the,device,without,suffering,a,single,collision.,Moreover,,kinetic,effects,are,surprisingly,important,even,in,the,regions,where,the,collision,frequency,is,high.,There,are,two,reasons,for,this.,•,The,diffusive,heat,flux,calculated,in,the,fluid,limit,is,only,accurate,if,νep,(cid:38),100,vte/L(cid:107),because,the,heat,flux,is,dominated,by,energetic,particles,that,collide,much,less,than,the,thermal,particles,(Gurevich,&,Istomin,1979;,Gray,&,Kilkenny,1980).,The,fluid,solution,assumes,that,there,are,too,many,energetic,particles,in,the,hot,plasma,regions,,and,predicts,too,few,energetic,particles,in,the,cooler,regions.,This,is,a,well,known,issue,,and,in,2D,fluid,simulations,,it,is,resolved,by,imposing,an,upper,bound,for,the,electron,heat,flux,that,is,of,order,neT,vte.,•,The,other,region,where,kinetic,effects,are,important,is,near,the,wall.,The,ions,and,electrons,that,reach,the,wall,recombine,and,do,not,come,back,as,charged,particles.,As,a,result,of,this,recombination,,the,charged,particle,distribution,functions,vanish,for,significant,parts,of,the,velocity,space,and,cannot,be,approximated,by,Maxwellians,,as,one,would,need,to,be,the,case,to,be,able,to,use,fluid,equations.,Thus,,the,treatment,of,charged,particles,near,the,wall,has,to,be,kinetic,(Loizu,et,al.,2011;,Geraldini,et,al.,2018).,The,combination,of,these,two,issues,can,manifest,in,a,population,of,energetic,electrons,that,cannot,be,predicted,by,fluid,models.,In,turn,,these,energetic,electrons,affect,the,density,and,temperature,by,suppressing,or,enhancing,heat,transport,and,by,changing,the,potential,difference,across,the,Debye,sheaths.,Evidence,of,these,effects,has,been,found,in,direct,kinetic,simulations,of,1D,problems,(Tskhakaya,et,al.,2011;,Chankin,&,Coster,2015).,The,fact,that,kinetic,effects,are,important,has,been,recognized,for,neutrals,,leading,to,kinetic,codes,for,them,(Reiter,et,al.,2005;,Stotler,&,Karney,1994).,However,,as,we,have,pointed,out,before,,these,codes,ignore,any,possible,non-Maxwellian,features,in,the,ion,and,electron,distribution,functions,,are,sensitive,to,the,parameters,that,determine,how,particles,interact,with,the,wall,and,,depending,on,the,version,,do,not,include,many,elastic,collisions,or,detailed,collision,physics.,The,non-Maxwellian,features,of,the,electron,dis-,tribution,function,are,particularly,important,as,electrons,mediate,many,of,the,processes,considered,important,in,these,neutral,models:,radiation,,ionization,,recombination,,etc.,For,ions,and,electrons,,the,community,is,starting,to,construct,edge,kinetic,codes,such,as,XGC,(Ku,et,al.,2016),,GKEYLL,(Hakim,et,al.,2020),or,COGENT,(Dorf,et,al.,2016).,These,codes,are,broadly,based,on,the,same,gyroaveraged,kinetic,models,used,in,δf,gyrokinetic,codes,such,as,GS2,(Kotschenreuther,et,al.,1995;,Dorland,et,al.,2000),,GENE,(Dannert,&,Jenko,2005),or,stella,(Barnes,et,al.,2019),,but,are,very,different,from,them,because,of,the,edge,particularities,,as,we,proceed,to,explain.,The,idea,behind,gyroaveraged,kinetic,models,is,to,average,over,the,fast,gyrofrequency,time,scale,to,avoid,a,cripplingly,small,time,step.,All,gyroaveraged,kinetic,models,are,based,on,an,asymptotic,expansion,in,ρi/L⊥,(cid:28),1,(see,table,2,for,values,of,ρi,and,L⊥).,The,simplest,possible,approach,is,drift,kinetics,(Hazeltine,1973),that,assumes,that,the,size,of,all,turbulent,structures,is,much,larger,than,ρi.,Unfortunately,,in,the,presence,of,temperature,and,density,gradients,,drift,kinetics,develops,instabilities,at,the,grid,scale.,These,instabilities,can,be,stabilized,by,numerical,dissipation,,and,in,the,real,word,,they,are,stabilized,by,finite,gyroradius,effects,that,drift,kinetics,neglects.,Gyrokinetics,(Catto,1978;,Frieman,&,Chen,1982),was,developed,to,solve,this,problem,in,the,core,of,the,tokamak.,Initially,,δf,gyrokinetics,assumed,that,turbulent,fluctuations,had,a,characteristic,size,of,the,order,of,ρi,and,their,amplitude,was,small,by,a,factor,of,ρi/L⊥,(cid:28),1.,This,ensures,that,the,gradients,of,the,fluctuations,are,comparable,to,the,background,gradient,and,not,larger.,However,,as,we,have,explained,in,section,3,,in,the,edge,it,is,not,possible,to,assume,that,density,and,temperature,are,slowly,varying,quantities,plus,small,fluctuations.,This,fact,8,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,was,part,of,the,justification,to,develop,what,is,know,as,full,f,gyrokinetics,,in,which,the,distribution,function,is,not,assumed,to,be,composed,of,a,slowly,varying,piece,plus,small,fluctuations.,See,Parra,&,Catto,(2008),and,Brizard,&,Hahm,(2007),for,two,different,types,of,derivations,of,full,f,gyrokinetics.,Overall,,full,f,gyrokinetics,can,be,seen,as,a,mixture,of,drift,kinetics,and,the,original,δf,gyrokinetics.,The,distribution,function,is,allowed,to,have,wavelengths,that,range,from,L⊥,to,ρi,,but,the,size,of,these,different,Fourier,components,has,to,be,sufficiently,small,that,the,gradient,of,the,distribution,function,is,never,larger,than,1/L⊥.,Assuming,a,core,ordering,(2.2),,one,naturally,recovers,δf,gyrokinetics,,showing,that,the,distribution,function,has,to,be,a,slowly,varying,piece,plus,small,fluctuations,(Parra,&,Catto,2010;,Calvo,&,Parra,2012).,In,the,edge,ordering,(2.1),,full,f,gyrokinetics,fundamentally,becomes,drift,kinetics,with,some,small,corrections.,These,corrections,are,of,two,types.,•,The,electromagnetic,fields,appear,in,the,kinetic,equation,averaged,over,circular,gyro-,orbits.,Thus,,electromagnetic,fluctuations,that,have,characteristic,lengths,much,smaller,than,the,gyroradius,are,averaged,over,and,do,not,drive,fluctuations,in,the,distribution,function.,Since,these,short,wavelength,electromagnetic,fluctuations,can,only,survive,if,there,are,charge,and,current,fluctuations,of,similar,wavelength,,which,they,are,not,able,to,drive,,they,eventually,damp,and,disappear.,•,The,field,equations,,quasineutrality,and,Amp´ere’s,law,,contain,densities,and,currents,that,one,calculates,from,the,ion,and,electron,distribution,functions.,As,a,result,of,the,gyrokinetic,expansion,,these,distribution,functions,have,finite,gyroradius,corrections,that,give,terms,that,are,small,in,ρi/L⊥,(cid:28),1.,One,of,these,small,terms,in,particular,,the,polarization,density,in,the,quasineutrality,equation,,is,important,because,,despite,its,small,size,,it,can,determine,the,electric,field,in,different,situations.,For,example,,for,shear,Alfven,waves,,the,polarization,density,is,small,,but,so,are,the,rest,of,the,contributions,to,the,density,,so,in,the,end,a,balance,between,the,polarization,density,and,another,term,determines,the,fluctuating,electric,field.,Another,example,is,the,component,of,the,electric,field,perpendicular,to,the,flux,surfaces,,which,is,also,determined,by,a,balance,between,the,polarization,density,and,other,terms,(Parra,&,Catto,2009).,The,small,finite,gyroradius,terms,in,the,full,f,gyrokinetic,equations,are,important,to,stabilize,the,short,wavelength,instabilities,,and,some,of,them,(e.g.,the,polarization,density),can,also,be,important,for,certain,aspects,of,the,physics.,Keeping,the,finite,gyro-,radius,effects,in,the,kinetic,equation,is,relatively,straightforward,in,PIC,codes,,although,one,has,to,be,careful,with,the,accuracy,of,the,average,(Guadagni,&,Cerfon,2017).,We,are,not,aware,of,any,full,f,edge,gyrokinetic,code,that,retains,finite,gyroradius,effects,in,the,kinetic,equation.,The,corrections,to,the,field,equations,are,much,more,difficult,to,retain,–,formulations,that,explicitly,try,to,conserve,energy,and,momentum,exactly,require,solving,nonlinear,equations,for,every,element,in,velocity,space,,for,instance.,This,has,driven,the,community,towards,simplifying,these,terms.,As,a,result,of,these,simplifications,,several,edge,codes,solve,drift,kinetics,with,some,ad,hoc,additions,to,the,field,equations,,such,as,a,simplified,polarization,density.,In,addition,to,these,fundamental,issues,,edge,drift,kinetics,and,gyrokinetics,need,to,address,other,problems.,As,gyrokinetics,was,devised,to,model,turbulent,fluctuations,with,a,spatial,size,of,the,order,of,ρi,in,tokamak,cores,,most,available,models,do,not,include,features,that,are,important,for,edge,physics.,•,There,is,no,gyrokinetic,formulation,for,the,wall,boundary,conditions.,Only,recently,one,such,formulation,was,developed,for,drift,kinetic,ions,in,magnetic,fields,that,reach,the,wall,with,a,grazing,angle,(Geraldini,et,al.,2018).,This,work,has,to,be,generalized,to,Physics,in,the,edge,of,fusion,devices,9,electrons,,more,general,angles,between,the,wall,and,the,magnetic,field,,and,eventually,to,gyrokinetics.,•,Gyrokinetics,does,not,usually,include,collisions,with,neutrals,,and,such,collisions,are,important,because,neutrals,break,the,symmetry,introduced,by,the,fast,charged,par-,ticle,gyration,around,magnetic,field,lines.,In,other,words,,the,dependence,on,gyrophase,,neglected,in,gyrokinetics,,can,become,important.,•,Gyrokinetics,was,developed,for,fluctuations,with,characteristic,scales,of,the,order,of,the,ion,gyroradius.,At,these,scales,,the,magnetic,field,fluctuations,have,a,very,constrained,form.,For,this,reason,,the,extension,of,gyrokinetics,to,include,MagnetoHydroDynamic,(MHD),modes,is,non-trivial,and,an,active,area,of,research.,See,Zheng,et,al.,(2007),for,a,theoretical,treatment,,and,Collar,et,al.,(2020),for,recent,numerical,work,in,this,area.,A,gyrokinetic,model,of,the,edge,should,be,able,to,reproduce,MHD,results,,as,MHD,modes,are,believed,to,be,the,main,drive,of,the,eruptions,known,as,Edge,Localized,Modes,(ELMs),(Ham,et,al.,2020).,5.,Discussion,A,complete,edge,description,requires,kinetic,effects.,Current,attempts,to,model,kinetic,effects,in,the,edge,rely,heavily,on,gyroaveraged,kinetic,models,that,average,over,the,very,fast,gyrofrequency,timescale.,In,our,opinion,,there,does,not,exist,a,systematic,procedure,to,choose,the,relevant,finite,gyroradius,effects,to,be,kept,in,the,kinetic,and,field,equations.,The,first,objective,of,a,kinetic,modeling,effort,for,the,edge,must,be,to,establish,the,finite,gyroradius,effects,to,be,kept,in,the,equations.,To,do,so,,for,contract,T/NA085/20,,we,have,proposed,as,a,first,attempt,to,construct,a,drift,kinetic,model.,This,drift,kinetic,model,will,be,unstable,at,grid,scales,,but,hyperviscosity,might,be,enough,to,stabilize,such,scales,if,they,do,not,contribute,much,to,transport,(as,one,would,expect,due,to,their,small,size).,For,the,contract,work,,we,will,use,the,model,only,in,1D,and,2D,configurations,that,cannot,develop,these,grid,scale,instabilities.,We,will,then,determine,analytically,which,finite,gyroradius,effects,must,be,kept,in,the,equations,to,determine,every,part,of,the,problem,,and,in,particular,the,component,of,the,electric,field,perpendicular,to,the,flux,surfaces.,A,well,known,issue,arises,when,keeping,finite,gyroradius,effects,in,the,field,equations.,In,quasineutrality,,the,only,term,that,contains,the,electric,potential,explicitly,is,the,small,finite,gyroradius,correction.,Thus,,unless,an,implicit,time,stepping,algorithm,is,employed,,one,needs,to,solve,for,the,potential,by,inverting,a,small,term,in,the,equation.,This,procedure,limits,the,time,step,size,severely,(Lee,1987;,Barnes,et,al.,2019).,Thus,,in,addition,to,keeping,finite,gyroradius,effects,,we,need,to,make,sure,that,the,kinetic,equations,that,we,obtain,are,amenable,to,implicit,time,stepping,methods.,In,addition,to,studying,finite,gyroradius,effects,,we,will,determine,the,effect,that,collisions,with,neutrals,have,on,the,gyrokinetic,formalism,by,introducing,charge,exchange,collisions,and,ionization,collisions.,The,final,deliverable,of,contract,T/NA085/20,will,be,a,drift,kinetic,model,with,neutrals,and,the,finite,gyroradius,terms,that,are,needed,to,calculate,the,component,of,the,electric,field,that,is,perpendicular,to,the,flux,surfaces.,Both,of,these,features,will,be,improvements,on,the,models,implemented,in,existing,continuum,edge,codes.,By,the,end,of,the,contract,,the,model,will,have,been,tested,in,1D,and,2D,problems,,and,even,though,it,will,have,a,3D,version,,this,3D,version,will,not,have,been,tested,in,the,turbulent,regime.,Moreover,,all,the,work,will,have,been,performed,in,a,helical,field,and,not,in,a,diverted,tokamak,field,because,the,helical,field,is,the,state-of-the-art,for,continuum,edge,kinetic,codes,(only,recently,,in,the,2020,Annual,APS,DPP,meeting,,the,10,Felix,I.,Parra,,Michael,Barnes,and,Michael,Hardman,two,leading,continuum,codes,,GKEYLL,and,COGENT,,have,reported,the,first,runs,with,more,complex,tokamak,geometry).,The,limitations,of,the,model,delivered,at,the,end,of,contract,T/NA085/20,leave,two,obvious,avenues,for,future,work.,•,The,main,theoretical,difficulty,added,by,tokamak,magnetic,fields,is,the,presence,of,a,component,of,the,drifts,in,the,direction,perpendicular,to,the,flux,surface,that,does,not,exist,in,the,case,of,a,helical,field.,This,component,of,the,drifts,leads,to,finite,orbit,widths,and,can,significantly,modify,some,of,the,physics,of,both,quiescent,2D,plasmas,(Kagan,&,Catto,2008;,Landreman,et,al.,2014),and,3D,turbulence,(Parisi,et,al.,2020).,Adding,this,component,of,the,particle,drift,to,the,equations,will,require,extra,work.,•,As,we,have,explained,above,,the,3D,model,is,expected,to,have,grid,scale,instabilities,that,can,be,stabilized,by,hyperviscosity,or,other,numerical,methods,of,dissipation.,To,properly,capture,the,fluctuations,at,the,ion,gyroradius,scale,,one,would,have,to,include,more,precise,finite,gyroradius,effects.,This,is,an,obvious,extension,of,the,work,in,contract,T/NA085/20,,and,it,is,very,important,for,High,confinement,mode,(H-mode),where,the,turbulence,at,scales,larger,than,the,ion,gyroradius,is,stabilized,for,still,unclear,reasons.,In,this,regime,,turbulence,at,scales,of,the,order,of,or,smaller,than,the,ion,gyroradius,is,important,(Hillesheim,et,al.,2016;,Hatch,et,al.,2017;,Parisi,et,al.,2020),,and,the,finite,gyroradius,effects,become,crucial,for,ions.,We,foresee,that,adding,more,detailed,finite,gyroradius,effects,will,be,theoretically,and,numerically,challenging,and,will,require,dedicated,work.,Two,other,aspects,of,an,edge,kinetic,model,are,beyond,the,scope,of,contract,T/NA085/20.,•,To,test,the,effect,of,wall,boundary,conditions,on,drift,kinetics,,we,will,impose,a,simplified,version,of,the,boundary,conditions,that,are,valid,in,the,limit,in,which,the,electron,gyroradius,is,much,smaller,than,the,Debye,length.,However,,more,detailed,boundary,conditions,must,be,found,because,usually,the,electron,gyroradius,is,larger,than,or,comparable,to,the,Debye,length.,This,is,work,that,is,being,pursued,by,one,of,the,authors,of,this,report,(F.I.P.),with,other,sources,of,funding.,•,We,pointed,out,above,that,it,would,be,desirable,to,be,able,to,recover,MHD,modes,with,the,edge,kinetic,model.,In,the,work,for,contract,T/NA085/20,,the,fluctuations,in,the,magnetic,field,will,be,neglected,(this,is,a,good,approximation,for,many,edge,plasmas),,and,hence,it,will,not,be,possible,to,explore,connections,with,MHD.,This,is,an,area,of,research,where,ExCALIBUR,NEPTUNE,could,benefit,from,collaboration,with,the,EPSRC,Programme,Grant,‘Turbulent,Dynamics,of,Tokamak,Plasmas,(TDoTP)’.,Several,PIs,in,the,ExCALIBUR,NEPTUNE,project,also,belong,to,TDoTP.,REFERENCES,Barnes,,M.,,Parra,,F.I.,&,Landreman,,M.,2019,stella:,An,operator-split,,implicit-explicit,δf,-gyrokinetic,code,for,general,magnetic,field,configurations.,J.,Comput.,Phys.,391,,365.,Beer,,M.A.,,Cowley,,S.C.,&,Hammett,,G.W.,1995,Field-aligned,coordinates,for,nonlinear,simulations,of,tokamak,turbulence.,Phys.,Plasmas,2,,2687.,Brackmann,,R.T.,,Fite,,W.L.,&,Neynaber,,R.H.,1958,Collisions,of,Electrons,with,Hydrogen,Atoms.,III.,Elastic,Scattering.,Phys.,Rev.,112,,1157.,Braginskii,,S.I.,1958,Transport,phenomena,in,a,completely,ionized,two-temperature,plasma.,Sov.,Phys.,JETP,6,,358.,Brizard,,A.J.,&,Hahm,,T.S.,2007,Foundations,of,nonlinear,gyrokinetic,theory.,Rev.,Mod.,Phys.,79,,421.,Bufferand,,H.,,Ciraolo,,G.,,Marandet,,Y.,,Bucalossi,,J.,,Ghendrih,,Ph.,,Gunn,,J.,,Mellet,,N.,,Tamain,,P.,,Leybros,,R.,,Fedorczak,,N.,,Schwander,,F.,&,Serre,,E.,2015,Numerical,modelling,for,divertor,design,of,the,WEST,device,with,a,focus,on,plasmawall,interactions.,Nucl.,Fusion,55,,053025.,Bufferand,,H.,,Tamain,,P.,,Baschetti,,S.,,Bucalossi,,J.,,Ciraolo,,G.,,Fedorczak,,N.,,Physics,in,the,edge,of,fusion,devices,11,Ghendrih,,Ph.,,Nespoli,,F.,,Schwander,,F.,&,E.,Serre,and,,Y.,Marandet,2019,Three-dimensional,modelling,of,edge,multi-component,plasma,taking,into,account,realistic,wall,geometry.,Nucl.,Mat.,Energy,18,,82.,Calvo,,I.,&,Parra,,F.I.,2012,Long-wavelength,limit,of,gyrokinetics,in,a,turbulent,tokamak,and,its,intrinsic,ambipolarity.,Plasma,Phys.,Control.,Fusion,54,,115007.,Catto,,P.J.,1978,Linearized,gyro-kinetics.,Plasma,Phys.,20,,719.,Catto,,P.J.,1994,A,short,mean-free,path,,coupled,neutral-ion,transport,description,of,a,toka-,mak,edge,plasma.,Phys.,Plasmas,1,,1936.,Catto,,P.J.,&,Simakov,,A.N.,2004,A,drift,ordered,short,mean,free,path,description,for,magnetized,plasma,allowing,strong,spatial,anisotropy.,Phys.,Plasmas,11,,90.,Chankin,,A.V.,,Corrigan,,G.,&,JET,Contributors,2021,EDGE2D-EIRENE,modeling,of,the,impact,of,wall,materials,on,core,edge,,scrape-off,layer,and,divertor,parameters.,Plasma,Phys.,Control.,Fusion,63,,035010.,Chankin,,A.V.,&,Coster,,D.P.,2015,On,the,locality,of,parallel,transport,of,heat,carrying,electrons,in,the,SOL.,J.,Nucl.,Mater.,463,,498.,Chodura,,R.,1982,Plasma-wall,transition,in,an,oblique,magnetic,field.,Phys.,Fluids,25,,1628.,Colchin,,R.J.,,Maingi,,R.,,Fenstermacher,,M.E.,,Carlstrom,,T.N.,,Isler,,R.C.,,Owen,,L.W.,&,Groebner,,R.J.,2000,Measurement,of,neutral,density,near,the,X,point,in,the,DIII-D,tokamak.,Nucl.,Fusion,40,,175.,Collar,,J.P.M.,,McMillan,,B.F.,,Saarelma,,S.,&,Bottino,,A.,2020,Comparing,electro-,magnetic,instabilities,in,global,gyrokinetic,simulations,with,local,and,MHD,models.,Plasma,Phys.,Control.,Fusion,62,,095005.,Dannert,,T.,&,Jenko,,F.,2005,Gyrokinetic,simulation,of,collisionless,trapped-electron,mode,turbulence.,Phys.,Plasmas,12,,072309.,Dorf,,M.A.,,Dorr,,M.R.,,Hittinger,,J.A.,,Cohen,,R.H.,&,Rognlien,,T.D.,2016,Contin-,uum,kinetic,modeling,of,the,tokamak,plasma,edge.,Phys.,Plasmas,23,,056102.,Dorland,,W.,,Jenko,,F.,,Kotschenreuther,,M.,&,Rogers,,B.N.,2000,Electron,Temper-,ature,Gradient,Turbulence.,Phys.,Rev.,Lett.,85,,5579.,Dudson,,B.D.,&,Leddy,,J.,2017,Hermes:,global,plasma,edge,fluid,turbulence,simulations.,Plasma,Phys.,Control.,Fusion,59,,054010.,Frieman,,E.A.,&,Chen,,L.,1982,Nonlinear,gyrokinetic,equations,for,low-frequency,electro-,magnetic,waves,in,general,plasma,equilibria.,Phys.,Fluids,25,,502.,Geraldini,,A.,,Parra,,F.I.,&,Militello,,F.,2018,Solution,to,a,collisionless,shallow-angle,magnetic,presheath,with,kinetic,ions.,Plasma,Phys.,Control.,Fusion,60,,125002.,Goldston,,R.J.,2012,Heuristic,drift-based,model,of,the,power,scrape-off,width,in,low-gas-puff,H-mode,tokamaks.,Nucl.,Fusion,52,,013009.,Gray,,D.R.,&,Kilkenny,,J.D.,1980,The,measurement,of,ion,acoustic,turbulence,and,reduced,thermal,conductivity,caused,by,a,large,temperature,gradient,in,a,laser,heated,plasma.,Plasma,Phys.,22,,81.,Guadagni,,J.,&,Cerfon,,A.J.,2017,Fast,and,spectrally,accurate,evaluation,of,gyroaverages,in,non-periodic,gyrokinetic,Poisson,simulations.,J.,Plasma,Phys.,83,,905830407.,Gurevich,,A.,&,Istomin,,Ya.N.,1979,Thermal,runaway,and,convective,heat,transport,by,fast,electrons,in,a,plasma.,Sov.,Phys.,JETP,50,,470.,Hakim,,A.H.,,Mandell,,N.R.,,T.N.,Bernard,and,,M.,Francisquez,,Hammett,,G.W.,&,Shi,,E.,L.,2020,Continuum,electromagnetic,gyrokinetic,simulations,of,turbulence,in,the,tokamak,scrape-off,layer,and,laboratory,devices.,Phys.,Plasmas,27,,042304.,Halpern,,F.D.,,Ricci,,P.,,Jolliet,,S.,,Loizu,,J.,,Morales,,J.,,Mosetto,,A.,,Musil,,F.,,Riva,,F.,,Tran,,T.M.,&,Wersal,,C.,2016,The,GBS,code,for,tokamak,scrape-off,layer,simulations.,J.,Comput.,Phys.,315,,388.,Ham,,C.,,Kirk,,A.,,Pamela,,S.,&,Wilson,,H.,2020,Filamentary,plasma,eruptions,and,their,control,on,the,route,to,fusion,energy.,Nature,Rev.,Phys.,2,,159.,Hariri,,F.,&,Ottavia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