Hall-Effect Thruster Simulations with 2-D Electron Transport and Hydrodynamic Ions

Transcription

1 Hall-Effct Thust Smulatons wth -D Elcton Tanspot and Hydodynamc Ions IEPC Psntd at th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan Ann Abo, Mchgan USA Sptmb 4, 9 Ioanns G. Mkllds, * Ia Kat, Rchad R. Hof and Dan M. Gobl Jt Populson Laboatoy, Calfona Insttut of Tchnology, Pasadna, CA, 9119 Abstact: A computatonal appoach that has bn usd xtnsvly n th last two dcads fo Hall thust smulatons s to solv a dffuson quaton and ngy consvaton law fo th lctons n a dcton that s ppndcula to th magntc fld, and us dsct-patcl mthods fo th havy spcs. Ths hybd appoach has allowd fo th captu of bulk plasma phnomna nsd ths thusts wthn asonabl computatonal tms. Rgons of th thust wth complx magntc fld aangmnts (such as thos na odd walls and magnt pol pcs) and/o ducd Hall paamt (such as thos na th anod and th cathod plum) challng th valdty of th quas-ondmnsonal assumpton fo th lctons. Ths pap pots on th dvlopmnt of a comput cod that solvs numcally th -D axsymmtc vcto fom of Ohm s law, wth no assumptons gadng th at of lcton tanspot n th paalll and ppndcula dctons. Th numcal challngs latd to th lag dspaty of th tanspot coffcnts n th two dctons a mt by solvng th quatons n a computatonal msh that s algnd wth th magntc fld. Th fully-d appoach allows fo a lag physcal doman that xtnds mo than fv tms th thust channl lngth n th axal dcton, and ncompasss th cathod bounday. Ions a tatd as an sothmal, cold (latv to th lctons) flud, accountng fo chag-xchang and multpl-onaton collsons n th momntum quatons. A fst ss of smulatons of two Hall thusts, th BPT-4 and a 6 kw laboatoy thust, quantfs th sgnfcanc of on dffuson n th anod gon and th mpotanc of th xtndd physcal doman on studs latd to th mpact of th tanspot coffcnts on th lcton flow fld. B magntc nducton fld c patcl thmal (o andom) vlocty D man atomc damt fo xnon E lctc fld lcton chag F total spcfc foc on ons f on vlocty dstbuton functon ( f & ) c at of chang of f du to collsons wth oth Nomnclatu βˆ magntc nducton fld unt vcto β () () componnt of magntc nducton fld unt vcto A sufac aa of a fnt-lmnt dg t tm ncmnt ε contbutons to Ohm s law fom th lcton pssu and on cunt dnsty * Mmb of th Tchncal Staff, Elctc Populson Goup, Ioanns.G.Mkllds@jpl.nasa.gov. Goup Supvso, Elctc Populson Goup, Ia Kat@jpl.nasa.gov. Mmb of th Tchncal Staff, Elctc Populson Goup, Rchad.R.Hof@jpl.nasa.gov. Scton Staff, Populson and Matals Engnng Scton, Dan.M.Gobl@jpl.nasa.gov. 1 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9 Copyght (c) 9 by th Calfona Insttut of Tchnology. Publshd by th Elctc Rockt Populson Socty wth pmsson.

2 spcs j () on (lcton) cunt dnsty k B Bolmann s constant L lngth of th acclaton channl ln(λ) coulomb logathm m () mass of on (lcton) n () numb dnsty of on (lctons) n n numb dnsty of atoms (nutals) nˆ nomal unt vcto n& lcton-mpact onaton at n& total on gnaton at fo collsons that poduc on I fom anoth havy patcl p () on (lcton) pssu q on chag (Z) Q T thmal hatng R () on (lcton) dag foc dnsty, adal and axal coodnats ˆ, ˆ unt vctos n adal and axal dctons T () on (lcton) tmpatu t tm u () man vlocty of ons (lctons) u n man vlocty of atoms u T, on thmal spd (k B T /m ) ½ v patcl vlocty Z on chag stat Gk Symbols α facto that contols th magntud of th Bohm collson fquncy ε pmttvty n vacuum ε s onaton potntal of spcs s η total o ffctv lctcal sstvty η lcton-on (-) lctcal sstvty η classcal lctcal sstvty κ lcton thmal conductvty λ on-on collson man f path λ n on-nutal collson man f path assocatd wth chag xchang λ nn nutal-nutal collson man f path µ classcal lcton moblty ν B Bohm collson fquncy ν lcton-on (-) collson fquncy ν total lcton-on (-) collson fquncy ν n lcton-nutal (-n) collson fquncy I ν lcton-nutal (mpact) onaton at n ν w lcton-wall (-w) collson at ν s collson fquncy of ons wth spcs s σ n on-nutal chag-xchang collson coss scton τ coulomb collson laxaton tm fo lctons τ thmal qulbaton tm btwn lctons T τ coulomb collson laxaton tm fo ons φ plasma potntal χ magntc-fld potntal functon ψ magntc-fld stam functon ω c lcton cycloton fquncy I. Intoducton HE numcal smulaton of Hall thusts spans mo than two dcads. In fact, th fst thotcal modls of T th patally-ond gas n Statonay Plasma Thusts (SPT) w potd n th 197s by Mooov and collagus. 1,,3 Hakawa 4,5,6 dvlopd on of th fst numcal modls of an SPT n th-dmnsons. Elctons and sngly-chagd ons w smulatd usng a Patcl-In-Cll (PIC) schm that was combnd wth a Mont- Calo Collson modl (MCC). Th lctc fld was dtmnd by solvng Posson s quaton. A computatonal appoach that has bn usd xtnsvly n th last two dcads o so to smulat th patally-ond gas n Hall thusts s to solv th flud (nta-lss) momntum and ngy consvaton laws fo th lctons but us dsct-patcl mthods to tack th voluton of th havy spcs. Ths hybd appoach allowd fo th captu of bulk plasma phnomna and on kntcs n th thust wthn asonabl computatonal tms and, as a sult, gand consdabl populaty. On of th fst modls to follow ths appoach was dvlopd by Ff and Matín- Sánch. 7 Th modl, dubbd HPHall (Hybd-PIC Hall), uss a PIC-MCC mthod fo ons n -D axsymmtc gomty and t appas that t was th fst to poduc th so-calld bathng mod oscllatons n Hall thusts, n two dmnsons. 8 Intptatons of ths oscllatons w povdd (aound th sam tm) by Ff and Matín- Sánch usng an dald -D modl, 7,8 and by Bouf and Gagus 9 usng a 1-D tm-dpndnt modl wth a hybd tatmnt of lctons and ons. In Ff s wok, a modl fo anomalous lcton moblty was mployd n th ognal (SPT-7) smulatons that was basd on Bohm s scalng 1 fo th anomalous (o noclasscal) collson fquncy, ν B ~B/16. Th pcs numcal valu usd n th smulatons was gudd by xpmnts. Snc th lat 9s HPHall has bn usd to smulat sval oth thusts and, natually, ts numcal and physcal modls hav undgon sval mpovmnts and xtnsons. Rcntly th modl was upgadd to HPHall- ** by Paa and ** All smulaton sults psntd n ths pap that a tmd as HPHall sults w gnatd wth th HPHall- vson of th cod as modfd at th Jt Populson Laboatoy [Rfs 13, 14]. Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

3 Ahdo. 11 Addtonal algothm advancmnts ncludng a nw oson sub-modl w compltd at th Jt Populson Laboatoy. 1,13,14 A smla hybd appoach has bn followd by Fnand and Cappll that ld to th dvlopmnt of a smla modl of Hall thusts and s potd n Rfs 15,16. Haglaa, t al. 17,18 also followd a hybd appoach but nstad of Bohm dffuson usd mpcal paamts to account fo addtonal anomalous lcton tanspot 19 and ngy loss phnomna. It s ntstng to not that dspt th appant populaty of th hybd appoach n cnt yas, th alst attmpts to modl th havy spcs followd puly hydodynamc fomalsms (also potd n Rf. 1). A hydodynamc appoach fo all spcs n th thust was also appld lat n -D gomts by Kda and Boyd. Bcaus th fundamntal pncpl bhnd th acclaton of ons n Hall thusts s basd on opatng th acclato at hgh lcton Hall paamt (Ω >1), th dffuson of hat and mass fo th lcton flow n th dcton paalll to th magntc fld s much gat (by ~Ω ) than that n th ppndcula dcton fo most of th channl gon. Ths lads to a smplfcaton, th so-calld quas-on-dmnsonal appoxmaton: stamlns of th magntc vcto fld a also lns of constant lcton tmpatu and constant thmald potntal φ * φ- T ln(n ). Numcally, th assumpton allows fo th soluton of th plasma potntal and lcton tmpatu n a (quadlatal) computatonal cll that s boundd by two adjacnt lns of foc ath than on wth abtay dmnsons. Ths s th appoach followd n HPHall. Modlng gons of th thust wth complx magntc fld aangmnts (such as thos na odd walls and magnts) and/o ducd Hall paamt (such as thos na th anod and th cathod plum) challng th valdty of th quas-on-dmnsonal assumpton fo th lctons. In ths pap w psnt a -D computatonal modl of th patally-ond gas n a Hall thust that mploys th full vcto fom of Ohm s law, wth no assumptons gadng th at of lcton tanspot n th paalll and ppndcula dctons of th magntc fld. Th modl s a dscndant of OCaD, a -D computatonal modl of lctc populson hollow cathods that mploys a mx of mplct and xplct algothms to solv numcally th consvaton laws fo th patally-ond gas n ths dvss. 3,4 Numcal dffuson du to th lag dspaty of th tanspot coffcnts n th two dctons s vadd by solvng th quatons n a computatonal msh that s algnd wth th magntc fld. Th mploymnt of fld-algnd mshs s a long-standng computatonal appoach (dats back mo than 15 ys) fo smulatng hghly ansotopc plasmas, and s wdly usd nowadays spcally by th sustand fuson ngy communty 5,6,7,8 (.g. tokamak dvto tchnologs), and fo a vaty of spac plasma poblms dalng wth th popagaton of sha Alfvn wavs. 9 Also, mo cntly, fld-algnd mshng was mployd n aly vsons of a -D modl of th dschag chamb n an on ngn. 3 It was found that th complxty of th magntc fld na th ng cusps mad th fld-algnd-msh gnaton tchnqu pohbtvly snstv to small changs n th magntc fld stngth o gomty. Ths mad th applcaton of th modl fo th dsgn and study of a wd ang of thust aangmnts cumbsom, so th fld-algnd msh was vntually placd wth a smpl othogonal msh. 31 Th fully-d appoach followd h allows fo a lag physcal doman that xtnds mo than fv tms th thust channl lngth n th axal dcton, and ncompasss th cathod bounday and th axs of symmty. Th man motvaton s to xtnd th soluton to gons of th Hall thust that othws could not b modld accuatly du to th quas-1d assumpton. Th modl also ncopoats a nw algothm fo th soluton of th (collsonlss) nutal gas dnsty, basd on ln-of-sght fomulatons, that lmnats th nhnt statstcal fluctuatons of convntonal patcl mthods. Th appoach fo th nutal gas s psntd n a companon pap 3 and wll not b dscbd h. Th ons a modld usng a fully hydodynamc appoach that, n addton to th nlastc collson tms assocatd wth th onaton, tan both th on pssu and th on-nutal chagxchang dag n th momntum quaton. Th pap s ogand as follows. Scton II povds a dscpton of th physcal modls and numcal mthodologs fo th ons (II.A) and lctons (II.B). Scton III psnts sults fom numcal smulatons of two thusts, th BPT-4 and a 6 kw laboatoy Hall thust. Th numcal smulatons of th BPT-4 (III.A) hav bn pfomd manly fo bnchmakng puposs and th sults a compad wth thos fom cnt HPHall smulatons of th sam thust. 13 Plmnay studs to btt undstand th mpact of th mposd Bohm collson fquncy n th BPT-4 bnchmak smulatons motvatd an xtndd nvstgaton n th 6 kw laboatoy thust; th sults fom ths studs a psntd n III.B. Fo th sak of bvty whn compang sults wth thos fom HPHall w f to th nwly-dvlopd modl as Hall D (wth D fng to lcton flow n two dmnsons). 3 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

4 II. Th Computatonal Modl Th computatonal gon n Hall D xtnds sval channl lngths (L) downstam of th thust xt. A schmatc of th physcal doman wth namng convntons fo th vaous boundas a povdd n Fgu 1. Th typcal xtnt of th computatonal gon n an HPHall smulaton s also shown fo compason. Ions a tatd as an sothmal, cold (latv to th lctons) flud, accountng fo chag-xchang and multpl-onaton collsons n th momntum quatons. Although lcton-mpact collsons that lad to th onaton of an atom can b fqunt by compason to ts tanst tm nsd th channl, fo most Hall thusts, collsons btwn th atomc spcs ( nutals ) a a. A popula numcal mthod fo smulatng th flow of nutals n Hall thusts s PIC combnd wth DSMC to account fo onaton collsons. An nhnt dsadvantag of patcl mthods lk PIC s th nos that s gnatd du to th patcl statstcs, whch can b ducd by ncludng mo patcls but at th xpns of ncasd computaton tm. Th mthod followd n Hall D s basd on wdly-usd mthods to modl poblms such as photon tanspot n adaton hat tansf poblms 6 and s advancmnt ov a pvous algothm that has bn usd to modl gons of collsonlss flows n lctc populson hollow cathods. 3,4 Th mthod assums that patcls stkng a sufac a fully accommodatd and that th facton of thos patcls that s -mttd follows a cosn dstbuton. Th patcl flux on any gvn sufac dpnds thn on th vw facto btwn that sufac and all oth sufacs that mt patcls. Bcaus th bass fo computng patcl dstbutons n a gon boundd by msson sufacs a th vw factos th poblm thn bcoms ssntally a poblm n gomty. Th vw factos can b computd at th p-pocssng phas of th smulaton thby contbutng an nsgnfcant amount to th total computaton tm. Th appoach fo th nutal gas s psntd n a companon pap 3 and wll not b dscbd h. A -D fom of Ohm s law and th lcton ngy quaton a solvd fo th lctons and th quatons a dsctd on a fld-algnd computatonal msh. Ohm s law s combnd wth th cunt consvaton quaton to yld th plasma potntal. Th bounday condtons latd to th shath along th dlctc walls, and condtons fo th manng boundas a povdd n nsung sctons. cathod to plat plum bounday cathod bounday hoontal fa plum bounday anod bounday dlctc wall bounday xtnt of th HPHall computatonal gon vtcal fa plum bounday dlctc wall bounday cathod bounday axs of symmty Fgu 1. Schmatc of th computatonal gon and namng convntons fo th bounday condtons. A. Ions 1) Physcs modl Bcaus th tatmnt of ons, spcfcally th computatonal mthods mployd to dtmn th voluton nsd th acclaton channl, has bn wd-angng du lagly to th assumptons mad on th chaactstc collson scals, w outln ou stmats of th lvant chaactstc ss fo th ons n som dtal. Th two chaactstc tms fo laxaton to a flud, τ fo lctons and τ fo ons, a: τ τ ν ν π ( π) 3 / ε m ( k T ) n Z 3 / ε m 4 4 nz 4 ln Λ ( k T ) ln Λ 3 / B 3 / B m m 1/ T T 3 / τ Z (II-1) 4 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

5 (wth tmpatu xpssd n K). Hon, ou convnton wll xclud th backts fom man valus of th collson fquncy, that s ν <ν>. Fo slow-movng ons th (Spt) thmal qulbaton tm btwn snglychagd ons and lctons whn T T may b appoxmatd by m τ T τ. m (II-) Usng stmatd valus n th 6 kw laboatoy Hall thust th on tanst tm τ u L/u can ang appoxmatly fom (.3 m)/( 1 4 m/s)1.5 µs fo thos ons that a acclatd downstam of th channl to (.1 m)/(5 1 m/s)1 µs fo thos gnatd na th anod gon and lost to th walls. In compason, th thmal qulbaton tm btwn lctons and ons angs.3-.5 s. nsd th channl. Ths mpls that th ons man cold latv to th lctons. Th (thmal) man-f-path (mfp) fo on-on collsons λ, (II-3) ut,τ s plottd n Fgu -lft along th mddl of th acclaton channl of th 6 kw Hall thust fo vaous (assumd) valus of th on tmpatu. Th pofls hav usd th HPHall-computd valus 14 fo th plasma dnsty and lcton tmpatu. It wll b shown lat that th on dnsty may n fact b substantally hgh n th anod gon than th valus pdctd by HPHall, whch suggsts vn small collson mfps fo ons n ths gon than thos plottd n Fgu -lft. Also, cnt Las-Inducd Fluoscnc masumnts of X + nsd th 6 kw Hall thust hav shown that ons follow vy closly th qulbum dstbuton functon, 33 whch futh stngthns th contnuum assumpton fo th ons n ths gon. Fgu -ght dpcts th chag-xchang collson mfp fo ons wth nutals as stmatd by, ( σ ) 1 λ, (II-4) n nn n and s plottd fo two valus of th chag-xchang coss scton σ n : 5 Å and 1 Å. Basd on th masumnts of Mll t. al., 34 th two valus cov th ang of typcal on ngs attand n th channl, <1 V to 3 V, wth th hgh valu of th coss scton psntng th lowst ngy ons. Fo compason, th chaactstc mfp fo slf collsons btwn nutals ( πn D ) 1 λ, (II-5) nn n s also plottd n Fgu 1-ght usng a man atomc damt fo xnon of D.6 Å. 35 Ion-on coulomb collson mfp/channl hght 1E+ 1E+1 1E+ 1E-1 1E- 1E-3 1E-4 3 K 6 K 1 K 5 K 1 V /L Fgu. Collsons man f paths at th mddl of th acclaton channl n th 6 kw laboatoy Hall thust. Lft: Ion-on coulomb collsons fo dffnt valus of th on tmpatu. Rght: Ion-nutal collsons that lad to th xchang of chag and nutal-nutal collsons. 5 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9 Collson mfp/channl hght 1E+3 1E+ 1E+1 1E+ 1E-1 1E- on-nu (5 A^) on-nu (1 A^) nu-nu /L

6 Basd on th stmats n Fgu th appoach followd h s to tat ons as a flud, and nclud chagxchang collsons as a contbuton to th (lastc) fcton o dag foc, whas nutals wll b smulatd usng a collsonlss appoach. 3 It s notd that th addton of chag-xchang collsons can bcom ncasngly mpotant n th anod gon snc th lctc foc can b nglgbly small th as by suggstd by cnt masumnts of th plasma potntal. 36 Th fomulatons that lad to th momntum consvaton law bgn wth Boltmann s quaton fo th dstbuton functon of ons f (t,,v) ( f& ) c f + v f + F v f (II-6) t wh F s th total spcfc foc (foc/mass) on th ons contanng th lctc and Lont focs. Th tm on th ght xpsss th at of chang of th dstbuton functon as a sult of collsons btwn ons and spcs s and, n pncpl, t may b composd of both lastc and nlastc componnts. By takng th poduct of Eq. (II-6) wth th on momntum m v and ntgatng ov vlocty spac on obtans th consvaton law fo th tanspot of momntum: t ( nm v ) + ( nm vv ) n m ( F v ) v m v( f & ) dv c (II-7) Rcall th dfntons of th lvant vlocts: v s th patcl vlocty (wth spct to th laboatoy fam of fnc), u <v>n -1 vfdv s th man patcl vlocty and c v-u s th patcl thmal vlocty. Hnaft, th subscpt fom th spatal opato shall b xcludd. Accodngly, th vaous tms n Eq. (II-7) n th absnc of th Lont foc bcom t n m ( nm v ) ( nmu) t ( nm vv ) ( nm cc + nmuu) ( F v ) v n qe mv( f & ) dv R c (II-8) Th dag foc dnsty R, dfnd n tms of th collson tm on th ght n Eq. (II-7), may b bokn up nto two pats to dstngush th momntum xchang btwn spcs by lastc collsons fom that by nlastc collsons: R mv( f & ) dv mv( f& + ) dv c c lastc nlastc. (II-9) Bcaus th dynamcs of th dct and nvs lastc collsons a th sam th tm fo collsons wth spcs s may b may b appoxmatd n tms of a man collson fquncy ν btwn ons and oth spcs s ( ) dv nmνs( u us ) c m v f &. lastc s (II-1) Th momntum consvaton law fo ons may thfo b xpssd n consvatv fom as follows: t ( nmu ) + ( nmuu) qne p + R (II-11) wh w hav nglctd th vscous tms n th pssu tnso p nm<cc> and hav assumd that p p I (wth I bng th unt o dlta tnso). Equaton (II-11) may b combnd wth th on contnuty 6 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

7 ( nm) t + ( nm v ) m ( & ) f c m n& dv (II-1) to yld th momntum quaton n non-consvatv fom u nm + ( u ) u qne p + R t Du nm qne nmνs Dt s ( u us ) p + S u m n& (II-13) wh S ncluds all th nlastc contbutons to th tanspot of on momntum, ( & ) dv m u n& S m v f. (II-14) c It s notd that th a two tms n S and thy a mathmatcally dstnct. Th fst appas as a dct consqunc of takng th fst momnt of Eq. (II-6) to obtan th consvatv fom of th momntum quaton (II-11). Th scond tm appas bcaus Eqs (II-11) and (II-1) w combnd to obtan th non-consvatv fom of th momntum quaton (II-13). Fo a quas-nutal plasma wth only sngly-chagd ons and no combnaton, S taks th smpl fom S m u n& m u n& m n ν n I n ( u u ) n (II-15) and, assumng only chag-xchang collsons fo th lastc contbutons n Eq. (II-1), Eq. (II-13) s smplfd as follows: I ( ν + ν )( u u ) Du nm qne p nm n n n (II-16) Dt to yld th on vlocty fld. Multply-chagd ons may b accountd fo by followng th sam fomulatons outlnd abov whl ncludng th lvant onaton collson fquncs fo th hgh chag stats. Th last tm on th ght of Eq. (II-16) may n fact b xpssd mo gnally n tms of an on gnaton at that ncluds both chag-xchang and lcton mpact-onaton collsons as follows: m < n& ( u u ) (II-17) wth dnotng th dcton of th acton. Fo xampl, fo collsons that gnat sngly-chagd ons fom nutals, X X +, thn and 1 and so foth and so on. In th psnt modl w account fo th followng actons lstd n Eq. (II-18) wth all collson coss sctons spcfd basd on avalabl data. Th dag btwn n Z Z Z Z+ 1 ν u u has not yt bn ncludd n Eq. (II-16). multply-chagd ons ~ ( ) m X + X, + X 3 + X, + X 4 + X X + X + X ++ + X X + X X + X, X + X X ++ + X (II-18) Fo th sults psntd n ths pap a constant tmpatu of 5 o C has bn assumd fo th ons. Although no snstvty calculatons hav bn pfomd yt to assss th mpact of th assumpton on th soluton, t s not xpctd to hav a sgnfcant ffct snc th on tmpatu affcts only th on pssu and th on-nutal 7 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

8 collson fquncs. On-avag th latt has a squa-oot dpndnc on th tmpatu and th on-pssu gadnt foc s nglgbl compad to th oth focs; ths wll b futh quantfd n an nsung scton. Also, n th cas of th havy spcs wth dffnt masss, m n xpsson (II-17) would b popotonal to th ducd mass, m m /(m +m ), but snc h only xnon ons and atoms a psnt m dnots th mass of th havy spcs. Th systm of consvaton laws fo th ons s closd wth condtons spcfd at all boundas n Fgu 1. At th anod and dlctc-wall boundas th Bohm condton s pscbd fo th spd wth whch th ons xt th physcal doman (.. at nty to th shath). At th plum boundas th ons a allowd to flow out of th systm fly (gadnts of th two vlocty componnts a st to o). Rflcton bounday condtons a st at th axs of symmty. Psntly no on flux s spcfd to flow fom th cathod bounday nto th physcal doman. ) Numcal appoach Equaton (II-16) s solvd usng a fst-od upwnd schm fo th vlocty fld at th vtcs, u v. Snc th smulaton doman s compsd of quadlatal computatonal clls of abtay shap as shown n Fgu 3-lft th schm accounts fo th suoundng ght vtcs (opn ccls) to dtmn th upwnd dcton. Bcaus all th consvaton laws n Hall D a dsctd usng fnt-volum dffncng, scalas (such as th numb dnsty n cl ) a computd at th cll cnts and vctos (such as focs F dg and fluxs) a computd at th cll dgs. Th on momntum quaton s th only consvaton law solvd n non-consvatv fashon so t qus that both vctos and scalas b known at vtcs. Rfng to Fgu 3-mddl and -ght, th algothm uss blna ntpolaton at ach vtx fo dg-cntd (mddl) and cll-cntd quantts (ght) usng th quadlatal dfnd by th dashd lns. Equatons (II-1) and (II-16) a machd xplctly fom tm t to yld nw valus fo th on dnsty and vlocty fld at t+ t u v 7 6 F v F dg n cl n v n cl 1 8 n cl n cl Fgu 3. Lft: Th on momntum quaton s solvd n non-consvatv fom fo th vlocty fld at th vtcs usng a fst-od upwnd schm that taks nto account th contbutons fom a maxmum of ght suoundng vtcs. Mddl: Blna ntpolaton s usd to dfn focs at vtcs usng th pmtv valus at th cll dgs. Rght: Blna ntpolaton s usd to dfn scala quantts at vtcs usng th pmtv valus at th cll cnts. A. Elctons 1) Physcs modl Th lcton momntum quaton n th absnc of th vscous tms and assumng p p I s gvn n vcto fom by: Du nm Dt ( E + u B) p + R n (II-19) As n th cas fo ons, th fcton foc R /n fo lctons s latd to th ntgal of th collson tm n th lcton dstbuton functon and th lcton momntum. Fo th cas of a na (o quas )-Maxwllan dstbuton functon n an ansotopc, classcal plasma (.. wth no andom fluctuatons n th flds) R may b appoxmatd as follows, assumng u >>u n : 1 * 1 R nm ν( u u ) + νnu m( ν + νn ) j + ( Z ) mν Zj (II-) 8 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

9 wth th lcton and on cunt dnsts dfnd as j -n u and j q n u Zn u spctvly, and th total lcton-on (-) collson fquncy gvn by * 4 n Z lnλ ν nz. (II-1) 3 * 1 Z n ( ) ( ) π 3/ ε m k T 3/ B Unlss othws notd all fncs to - collson fquncy n th mand of ths pap shall mply th dfnton n Eq. (II-1). If th lcton nta can b nglctd thn on obtans th vcto fom of Ohm s law E j j βˆ p + η j η + ηω (II-) n wh ( ν + ν ) m n mν B 1 η η β ˆ B/ B Ω j * Zj. (II-3) n n n η Z In th fam of fnc of th magntc nducton fld wth // and dnotng paalll and ppndcula componnts spctvly, th componnts of Eq. (II-) may b wttn as p ( 1+ Ω ) j + η p E. (II-4) // // η j // + η j // E η j n n It s notd that n th absnc of th on vlocty n th lcton dag foc dnsty (Eq. (II-)), Eqs (II-4) tak a fom that s sutabl fo poblms nvolvng lcton dffuson n wakly-ond plasmas // p nµ p j µ + + // n E// j E (II-5) n 1+ Ω n Th fom of Ohm s law gvn n Eq. (II-5), usng th lcton moblty µ nstad of th sstvty (not µ 1/n η ), s th fom that has tadtonally bn mplmntd n Hall thust modls such as HPHall. 7 Equatons (II-5) mply th appoxmaton u >>u (n addton to u >>u n ) and thus R -n m ν u wth th total collson fquncy ν psntng th contbutons fom classcal collsons of lctons wth all oth spcs. It has also bn suggstd that th dffuson of lctons n Hall thusts s nhancd n a non-classcal mann,.g. by plasma tubulnc, and attmpts to captu ths nhancmnt n numcal smulatons wth HPHall hav bn mad though th us of an ffctv collson fquncy 7 basd on Bohm s 1/B scalng fo th coss-fld moblty. 1 That s, R - n m (ν +ν B )u wh, α νb ωc (II-6) 16 wth α bng a constant. Dung th amuthal dft lctons may also colld wth walls and ths has bn poposd (ognally by Mooov 1 ) to b on mo pocss that affcts th tanspot of lctons n th acclaton channl. In numcal smulatons of Hall thusts ths addtonal tanspot mchansm has bn accountd fo though th addton of anoth ffctv collson fquncy ν w. Bcaus ths fquncy s found to b mpotant only n gons wh th quas-1d assumpton s vald th appoach usd to dtmn t s th sam as that usd n HPHall. 13 Accountng fo all tanspot mchansms ffctv valus of th lctcal sstvty and th Hall paamt may b dfnd as follows: m η ( ν + ν + ν + ν ) n n w B B Ω. (II-7) n η 9 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

10 Unlss othws notd, all fncs to th Hall paamt n th mand of ths pap shall mply th dfnton n Eq. (II-7). In Scton III a ss of numcal smulatons wll b psntd that compa sults obtand by Hall D wth thos obtand cntly by HPHall fo two Hall thusts, th BPT-4 and th 6 kw Hall thust. 13,14 In ths smulatons th HPHall solutons ncopoat a spatally vayng α that s gudd by plasma masumnts and by th obsvd opatonal chaactstcs of th thusts (such as dschag cunt and thust). Th psnc of tubulnc, ts al ffct on lcton tanspot and th quston of whth t can b quantfd usng Bohm s fomula has bn an ongong aa of sach. As a consqunc, n modls lk HPHall th vaaton of α fom on thust smulaton to anoth s not basd on fst pncpls, whch psnts th bggst obstacl n advancng such modls to fully-pdctv dsgn tools fo Hall thusts. Th ovall systm of consvaton laws s augmntd wth an quaton fo th consvaton of cunt ( + j ) j (II-8) and th quaton fo th lcton tmpatu (xpssd n V) 3 T n t T E j + T + T T ns εs + T + Q j κ j &. (II-9) s Th last tm on th ght psnts th ngy xchang p unt tm btwn lctons and th havy spcs 37 du to dvatons fom thmal qulbum and s popotonal to n (m /m)ν (T -T ) fo ons and n (m /m)ν n (T -T n ) fo nutals. In Hall thusts t s usually a small contbuton to th total lcton ngy contnt. Th quatons fo th lctons a closd wth bounday condtons at all sufacs n Fgu 1. As t wll b shown n th nxt subscton, Eq. (II-8) s n fact combnd wth Eq. (II-4) to yld th plasma potntal, and qus bounday condtons th fo th plasma potntal, ts gadnt o fo th cunt dnsty. Fo all dlctcwall boundas a o-cunt condton s mposd, j j. At th anod a Dchlt condton spcfs dctly th voltag at ts dschag valu. Fo both smulaton cass that a psntd n ths pap ths valu s 3 V. A Dchlt condton s also mposd at th cathod wth a valu of V. Fo th lcton ngy th convctv hat loss follows th fomulatons of Hobbs and Wsson (H&W) 38 usng a ft 39 fo th H&W soluton of th shath potntal as a functon of th sconday lcton yld (SEE). Th Maxwllan-avagd SEE s also spcfd usng a ft to data fo th dlctc matal usd n ach thust. A Dchlt condton fo th lcton tmpatu s also mposd at th anod. ) Numcal appoach Th lag dspaty (> ods of magntud n gons wth hgh valus of th magntc fld) that xsts n th lcton tanspot quatons n th ppndcula vsus th paalll dctons qus spcal tatmnt. On appoach s to solv th quatons fo lctons only n th ppndcula dcton; ths quas-1d appoach s followd by HPHall and oth smla modls of Hall thusts. Ths vads th numcal dffcults assocatd wth th soluton of tanspot n both dctons and s an xcllnt appoxmaton fo most of th acclaton channl and na-plum gons. To xtnd th soluton to gons n th fa plum and/o to solv gons of th magntc fld wth complx topology qus a -D soluton of th lcton tanspot quatons. To accomplsh ths, th appoach followd h s to solv th quatons n th fam of fnc of th magntc fld, n two dmnsons ( βˆ β ˆ ˆ + β ). To dmnsh xcssv numcal dffuson all quatons a dsctd n a computatonal msh that s algnd wth th magntc fld. In ths scton th gnal appoach s outlnd usng Eq. (II-8) as th xampl quaton. Th plasma potntal n Hall D s solvd by combnng th quaton fo cunt consvaton and Ohm s law nto on quaton. Thn fo a sngl quadlatal computatonal cll wth volum V th dvgnc thom allows fo th followng dsctaton: j t+ t V 4 dg 1 4 t+ t ( j nˆ A) dg 1 dg t t+ t ( j βˆ ) βˆ βˆ ( βˆ j) + t nˆ A dg (II-3) 1 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

11 Th dot poduct n Eq. (II-3) at a dg may b xpandd as wh t t t+ t 1 t t t t ( j βˆ ) βˆ βˆ ( βˆ j) + nˆ η ( E + ε ) n + (II-31) n n β β n n + n ( β ˆ + β ˆ ) nˆ + ( β ˆ β ˆ ) 1+ Ω nˆ ( β ˆ + β ˆ ) nˆ ( β ˆ β ˆ ) nˆ β β 1+ Ω (II-3) and E- φ. Th manng tms n Eq. (II-4) nvolvng th lcton pssu and th on cunt tms a ncludd n th tm ε. Equaton (II-3) s solvd mplctly fo th plasma potntal. It s notd that a smplfcaton occus n Eqs (II-3) whn th computatonal msh s algnd wth th magntc fld, as llustatd n Fgu 4. Numcal dffuson du to th dspaty btwn th tms wth and wthout Ω s ducd by assumng that cll dgs a xactly th paalll o ppndcula to th magntc fld lns. Th accuacy of th soluton s thn dpndnt upon th xtnt of th spatal dvatons of th msh fom th tu lns of constant potntal and stam functons χ and ψ. H, χ and ψ a th commonly-usd st of conjugat hamonc functons satsfyng th Cauchy-Rmann condtons fo th adal and axal componnts of th magntc fld. dg s paalll to ψ ln dg s paalll to χ ln n β n n β n β β 1 β nˆ 1+ Ω 1 β 1+ Ω ( β ˆ + β ˆ ) nˆ + ( β ˆ β ˆ ) ( β ˆ + β ˆ ) nˆ ( β ˆ β ˆ ) nˆ ( β ˆ + β ˆ ) nˆ + ( β ˆ β ˆ ) ( β ˆ + β ˆ ) nˆ ( β ˆ β ˆ ) nˆ 1 β nˆ 1+ Ω 1 β 1+ Ω φ βˆ nˆ ( j βˆ ) βˆ βˆ ( βˆ j) j ˆ ẑ Fgu 4. Each dg of a computatonal cll n Hall D s closly algnd wth th a ln of constant potntal functon (χ) o a ln of constant stam functon (ψ). Fgu 5. Lft: A st of lns of constant stam functon (ψ) n blu (stamlns of th magntc fld) ovlad by lns of constant potntal functon (χ) n d, n th vcnty of th acclaton channl n th 6 kw Hall thust. Rght: cospondng fnt-lmnt computatonal msh. 11 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

12 Th computatonal msh s gnatd fst by supmposng lns of constant χ and ψ onto th computatonal gon usng commcally avalabl gaphcs softwa (Fgu 5-lft). Th computatonal gon boundas a spcfd by ln sgmnts that connct ponts usd to spcfy th gomty of th gon. Thn th spatal locatons of ponts along ach ln, gnatd by ntgaton n spac along ach stamln, a xtactd. Each pa of adjacnt ponts along a χ-ln (o a ψ-ln) dfns a ln sgmnt. A msh algothm thn sachs fo th ntsctons btwn all ln sgmnts ov all χ-lns, ψ-lns and bounday lns. Each ntscton dfns a vtx locaton and ths vtcs a thn usd to gnat th fnt lmnt msh shown n Fgu 5-ght. Th quaton fo th lcton tmpatu s solvd n a sm-mplct fashon. Th thmal conducton tm s mplct whas all oth tms a valuatd at th pvous tm-stp as xpssd by Eq. (II-33). 3 n t T t+ t T t t t t+ t ( T ) T κ E j + T j T j n& s εs + T + Q s (II-33) t III. Numcal Smulatons A. Bnchmak smulatons wth th BPT-4 As a fst ss of Hall D algothm tsts w pfomd compasons wth xstng numcal smulaton sults 13 obtand by HPHall fo th BPT-4 opatng at 4.5 kw. Th opatonal chaactstcs of ths thust as usd n th numcal smulatons a outlnd n Tabl 1. Th smulatons mployd th sam spatal vaatons of th Bohm collson fquncy facto α n th acclaton channl and na-plum gons as n th HPHall smulatons. It s notd that n thos smulatons th facto was sgnfcantly low nsd th acclaton channl (α.35) compad to th plum gon (α1.). Moov, byond th ffctv HPHall computatonal gon, dfnd by a na-anod magntc-fld stamln and a na-cathod stamln (as dfnd by th two stamlns n black n Fgu 6) both th Bohm collson fquncy and th Hall paamt a st to o. Fo ths fst ss of algothm tsts th spatal vaatons fo ν B and Ω n Hall D s th sam as n HPHall but wth a slghtly mo gadual ducton to o (usng a Gaussan functon) downstam of th cathod fld ln. Th on-on-on compasons along a ln that cosss th mddl of th acclaton channl a shown n Fgu 7. Th bnchmak smulatons hav also usd th sam modl fo th wall collson fquncy ν w. Th compasons n Fgu 7 and th -D contou plots n Fgu 8 show smla solutons but wth som makd dffncs. Th ovall hatng of lctons appas to b n clos agmnt btwn th two solutons, whch s xpctd snc th pak lcton tmpatu and ts spatal vaaton na ths maxmum s dvn manly by sstv hatng that s domnatd by th Bohm collson fquncy. By compason to th oth collson fquncs, ν B s at last on od of magntud hgh at th xt and na-plum gons wh th maxmum n th tmpatu s computd. Na th anod th lcton tmpatu n Hall D s dtmnd lagly by th anod Dchlt bounday condton (cuntly spcfd as 1 V) and th suoundng dlctc-wall condtons, whch pscb th sam H&W soluton 38 as n HPHall fo th convctv hat loss of lctons n th shath. Fgu 6. Contous of th Hall paamt as computd by HPHall n th 6 kw Hall thust smulatons. 14 Naanod and na-cathod stamlns dfn th ffctv HPHall computatonal doman byond whch Ω and ν B a st to o. Th sam appoach s followd n th BPT-4 smulatons Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

13 Tabl 1. Opatonal chaactstcs usd n th numcal smulatons of th BPT-4 at 4.5 kw. Thust paamt Valu Dschag (o anod) cunt (A) 15 Dschag voltag (V) 3 Anod mass flow at (mg/s) 15.5 Cathod mass flow at (mg/s) 1.55 Hall paamt Hall D HPHall Ch. Ext Collson fquncy (1/s) 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 -n (Hall D) - (Hall D) - (HPHall) Bohm (Hall D) Bohm (HPHall) -wall (Hall D & HPHall) Ch. Ext 1.E /L 1.E /L Elcton tmpatu, T (V) T (Hall D) T (HPHall) ph (Hall D) ph (HPHall) Ch. Ext Plasma potntal, ph (V) Elcton (n) o nutal (nn) numb dnsty, (m -3 ) 1.E+1 1.E+ 1.E+19 1.E+18 1.E+17 n (Hall D) n ( HPHall) nn (Hall D) nn (HPHall) Ch. Ext /L 1.E /L Fgu 7. Axal slc plots fom th bnchmak numcal smulatons of th BPT-4. Th plots compa th soluton obtand by HPHall 13 wth that obtand by Hall D at th mddl of th acclaton channl. In ths bnchmak smulatons th Hall D smulatons nfoc a ducton of th Bohm collson fquncy and of th Hall paamt byond /L~1.5 to mulat th appoach followd n th HPHall smulatons. A notabl dstncton btwn th solutons fo th lcton numb dnsty and plasma potntal s vdnt n th anod gon. Ths s llustatd n Fgu 7 (bottom), Fgu 8 (top), and s mo vdnt n th axal pofls of Fgu 9 (lft). Hall D computs a hgh plasma dnsty n ths gon wth valus fo /L<. xcdng on od of magntud thos obtand by HPHall. A compason of th tms n Eq. (III-1) show that th anod gon s domnatd by on dffuson snc ths s wh th lctc fld s nglgbl. It s notd that as pat of th nhnt assumptons assocatd wth th PIC smulaton of ons th on dag tms (numbd as IV n Eq. (III-1)) a not accountd fo n HPHall. Th on pssu s also xcludd n HPHall and th Hall D smulatons confm ths to b a good appoxmaton fo th assumd on tmpatu. Th compason of all th on momntum tms n stady 13 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

14 stat s shown n Fgu 9-ght. Th ffct of msh soluton on th soluton has not yt bn quantfd but, f of any sgnfcanc, ths s xpctd to nflunc th soluton only n th vy-na anod gons wh th msh s coasst. Fgu 8. Contou plots fom th numcal smulatons of th BPT-4. Th plots compa th -D soluton 13 obtand by HPHall (lft) wth that obtand by Hall D (ght). Fo ths bnchmak smulatons Hall D and HPHall us appoxmatly th sam modl fo th spatal vaaton of th Bohm collson fquncy. Th aows pont to th maxmum adal xtnt of th HPHall computatonal gon. Top: lcton numb dnsty. Bottom: plasma potntal. u t q E + m p n m I ( u ) u + + ( ν + ν )( u u ) (I) (II) (III) (IV) n n n (III-1) A latd ffct s assocatd wth th vaaton of th lctc fld n ths gon wh masumnts 4 hav suggstd lttl to no vaaton of th plasma potntal (.. E ). Smla compasons as thos pfomd fo th on momntum may b cad out fo th domnant tms n Ohm s law, namly th sstv tm and th lcton pssu. Ths compasons suggst that th hgh plasma dnsty ducs sgnfcantly th mpotanc of ths tms n ths gon such that any dffncs btwn thm (th numato n Eq. (III-)), that would othws gnat a fnt lctc fld, a ducd. Th numato n Eq. (III-) s ducd futh by compason to th HPHAll soluton snc th total classcal collson fquncy s hgh (fst tm) and th dnsty gadnt s low 14 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

15 (scond tm). Th - collson fquncy s found to b hgh n th na-plum gons as wll, n pat du to th hgh plasma dnsty th but lagly as a sult of accountng fo th multply-chagd ons n Eq. (II-1) (though Z * ); mo notabl howv n th Hall D sults s ts contnud s downstam of th HPHall computatonal gon. Ths s s lagly du to th cold lcton tmpatu snc th fquncy s popotonal to T -3/. E p j B / m ( ) ( ν + νn ) 1+ Ω j ~ T η, (III-) n n n Plasma potntal, ph (V) ph (Hall D) ph (HPHall) n (Hall D) n ( HPHall) Ch. Ext 1.E+1 1.E+ 1.E+19 1.E+18 1.E+17 Elcton numb dnsty, n (m -3 ) II/I (nta/lctc foc) III/I (pssu gad/lctc foc) IV/I (dag foc/lctc foc) Ch. Ext 1 1.E /L /L Fgu 9. Compason of tms n th on momntum consvaton law fo sngly chagd ons (Eq. (III-1)) along th md-channl ln of th BPT-4 fo th stady-stat bnchmak smulaton cas. Th pofls on th ght dntfy th anod o on-dffuson gon (/L<~.3), th onaton gon (.3</L<.6) and th on-acclaton gon (/L>.6). Th capacty to solv ths s du to Hall D s xtndd computatonal gon pomptd a ss of plmnay smulatons to btt undstand th spons of th plasma, spcfcally that of th - collson fquncy, snc t domnats ov both th lcton-nutal (-n) and th lcton-wall (-w) fquncs byond th onaton gon. Fgu 1-lft shows th computd - collson fquncy fo th dffnt valus of th maxmum Bohm facto: α1.,. and.5. It should b clafd that w us maxmum h bcaus, as also notd al, α s vad n th physcal doman: fo th BPT-4 bnchmak smulatons n Fgu 7, α( α c n Rf. 13).35 nsd th channl, α( α p n Rf. 13)1. n th na-plum and fo /L>~1.5 α p s ducd to o. So, whn w dcas th Bohm facto n ths pap to, say, α. w mply a global ducton of th Bohm collson fquncy, that s ν B (.35.)ω c /16.7ω c /16 nsd th channl and ν B (1..)ω c /16.ω c /16 n th na-plum. Th sults n Fgu 1-lft show an ncasng - collson fquncy wth dcasng α manly bcaus of th ducd lcton tmpatu n th na-plum. Th plasma dnsty s also found to b hgh than th HPHall valus along th md-channl ln. Th computd pofls fo th lcton tmpatu a shown n Fgu 11. Th anod cunt (15 A) s obtand fo a maxmum α., an ~8% low valu than that usd n th Hall D-HPHall bnchmak smulatons. Fo ths spatal vaaton and valu of maxmum α th on (bam) cunt s found to I b (I + )+(I ++ )+(I +++ ) A. Also, th soluton fo th anod and bam cunts n Fgu 1-ght xhbts a podc chaact but on that s und-dampd; all cunt oscllatons d out wthn a fw pods. Th dampng of th oscllatons occus fast (wthn appoxmatly on pod) fo α1. and poducs ~3.5 A mo cunt to th anod than th nomnal valu. Th cas of α.5 poducs cunt oscllatons that do not d out albt at 1. A low tm-avagd anod cunt. W call that ν B dos not dpnd on any plasma paamts that could patcpat n and/o nduc plasma oscllatons, and w postulat ths to b th ason fo th dampng of th oscllatons whn ν B s th domnant collson fquncy. Low-fquncy oscllatons n th Hall D soluton at low valus of α a also obtand n th 6 kw Hall thust smulatons to b psntd n th nxt subscton. Th ncas of th - collson fquncy to valus that a compaabl to o hgh than th Bohm collson fquncy s of sgnfcanc n pat bcaus, to th bst of ou knowldg, th tanspot of lctons fom gons of th plum that xtnd byond th HPHall physcal doman has not yt bn solvd. In vw of ths lvatd valus of th - collson fquncy n th fa-plum, w attmpt n th nxt scton to lft th ducton of th Hall 15 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

16 paamt and Bohm collson fquncy that was mposd as pat of th BPT-4 bnchmak smulatons. In cognton of th snstvty of th - collson fquncy on th lcton tmpatu, w nd ths scton wth a calculaton that asssss th nflunc of th fa-plum bounday condton. Fgu 1 compas th computd fquncs at th channl mdln fo two cass. Th fst cas sts T 1 V at th hoontal and vtcal fa-plum boundas (s Fgu 1) and th scond cas sts T 4 V at ths boundas. As xpctd, th na-plum and channl gons a not found to b sgnfcantly affctd whas th fa-plum gons a lagly dvn by th bounday valu. Rcnt plasma masumnts n th fa-plum gons ndcat that th tmpatu sval channl hghts downstam of th xt s 4-5 V. 41 A valu of 4 V has bn usd fo th BPT-4 bnchmak smulatons shown n Fgu 7 and Fgu 8. Th sults n Fgu 11 on th ffcts of α w obtand wth T 1 V at th faplum boundas and th qualtatv tnds assocatd wth - collson fquncy a found to b unaffctd by ths valu. Th nsung smulatons wth th 6 kw Hall thust us T 4 V at th boundas basd on cnt plasma masumnts. 14 Collson fquncy (1/s) 1.E+1 1.E+9 1.E+8 1.E+7 1.E+6 Bohm (a1.) - (a1.) Bohm (a.) - (a.) Bohm (a.5) - (a.5) Cunt (A) Anod X+ X++ X+++ 1.E /L.E+ 5.E-5 1.E-4 1.5E-4.E-4.5E-4 Tm (sc) Fgu 1. Lft: Th spons of th classcal - collson fquncy to ductons of th mposd Bohm collson fquncy (though th Bohm facto α) along a ln at th mddl of th acclaton channl. Rght: Th computd anod and bam cunts as a functon of tm fo α Hall paamt 1 1 a1. a. a.5 Elcton tmpatu, T (V) a1. a. a /L /L Fgu 11. Th spons of th Hall paamt Ω (lft) and th lcton tmpatu (ght) to ductons of th mposd Bohm collson fquncy (though th Bohm facto α) along a ln at th mddl of th acclaton channl. 16 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

17 Elcton tmpatu, T (V) T (T at fa-plum bdy 1 V) T (T at fa-plum bdy4 V) - (T at fa-plum bdy1 V) - (T at fa-plum bdy4 V) 1.E+9 1.E+8 1.E+7 1.E+6 Elcton-on collson fquncy (1/s) 1.E /L Fgu 1. Snstvty of th lcton-on collson fquncy on th fa-plum bounday condtons fo th lcton tmpatu. B. Extndd smulatons wth th 6-kW laboatoy Hall thust In ths scton w xtnd th studs of Scton III.A on th ffcts of ducd Bohm collson fquncy to a 6 kw laboatoy Hall thust. Ths thust has a smpl channl gomty compad to th BPT-4, and s bng opatd and studd fo fundamntal sach at sval nsttutons. 4,4,43,44 Also, sval masumnts hav bn obtand wth th cathod placd at th axs of symmty 4 (.g. s photo n Tabl ). Ths cathod-thust aangmnt s of gat ntst n numcal smulatons bcaus t s -D axsymmtc. Th opatonal chaactstcs fo th smulatons psntd n ths pap a outlnd n Tabl. Th magntc fld aangmnt, physcal doman and fld-algnd computatonal msh a shown n Fgu 13. Tabl. Opatonal chaactstcs usd n th numcal smulatons of th 6 kw laboatoy Hall thust. Thust paamt Valu Dschag (o anod) cunt (A) Dschag voltag (V) 3 Anod mass flow at (mg/s).98 Cathod mass flow at (mg/s) 1.47 Th sold cuvs n Fgu 14 dpct th Hall D soluton wth a ducton of th Bohm collson fquncy and Hall paamt n th fa-plum smla to that mplmntd n th BPT-4 smulatons. It s found that th maxmum valu of α, wth th pscbd spatal aangmnt shown by th sold cuv n Fgu 14, that ylds th obsvd anod cunt s.4. Ths st of smulaton sults hav bn obtand to allow fo dct compasons wth nsung smulatons that lft th constants on ν B and Ω. Th computd anod cunt and bam cunts a shown n Fgu 14-ght. Du to th small mpact on studs latd to th collson fquncs th X +++ hav bn xcludd fom ths smulatons. Th sngly- and doubly-chagd on bam cunts a I A and I A. Th computd pofl fo T appas to b n fa agmnt wth th masumnts as suggstd by Fgu 15- ght (sold cuv). Th dashd cuvs n Fgu 14 llustat th cospondng sults whn ν B, Ω and all oth plasma vaabls a dtmnd slf-consstntly wth no spatal stctons on α n th fa-plum. Th motvaton fo ths smulatons s not to mply that anomalous (Bohm) dffuson xtnds to th fa-plum; h w am to dmonstat th ffct that th ad-hoc lmnaton of th Bohm collson fquncy and Hall paamt has on th plasma soluton. Th poposd mplcaton s that n modls such as HPHall th nhnt lmtatons posd by th quas-1d assumpton on th xtnt of th computatonal doman mak th achvmnt of a numcal soluton that s unbasd by th mposd stctons byond th na-cathod ln (s Fgu 6) challngng f not mpossbl. Th 17 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

18 latvly small dffncs btwn th two smulaton cass fo th ntgatd vaabls such as anod and bam cunts (s Fgu 14-ght) falsly mply a small ffct by th mposd stctons. But th compasons fo th lcton tmpatu do not suppot such asstons. Fgu 15-ght quantfs th dffnt spatal xtnts assocatd wth th dffuson of hat n th lcton flow along th channl mdln fo th two cass, and Fgu 16 compas th -D pofls. Fgu 13. Physcal doman, magntc fld and computatonal msh fo th numcal smulatons of th 6 kw laboatoy Hall thust. Collson fquncy (1/s) 1.E+11 1.E+1 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 Bohm (Bohm ducd n fa plum) Bohm (Bohm allowd n fa plum) - (Bohm ducd n fa plum) - (Bohm allowd n fa plum) -n (Bohm ducd n fa plum) -n (Bohm allowd n fa plum) -wall (Hall D & HPHall) Cunt (A) Anod (Bohm ducd n fa plum) Anod (Bohm allowd n fa plum) X+ (Bohm ducd n fa plum) X+ (Bohm allowd n fa plum) X++ (Bohm ducd n fa plum) X++ (Bohm allowd n fa plum) 1.E E /L.E+ 1.E-4.E-4 3.E-4 4.E-4 Tm (sc) Fgu 14. Studs on th ffct of ducng th Bohm collson fquncy and th Hall paamt n th fa plum (cas shown s fo a maxmum Bohm facto of α.4). Lft: Collson fquncs along th mddl of th channl. Rght: Anod and bam cunts as a functon of numcal smulaton tm. 18 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

19 Hall paamt Bohm allowd n fa plum Bohm ducd n fa plum Elcton tmpatu, T (V) T (Bohm ducd n fa plum) T (Bohm allowd n fa plum) T (Expmnt) /L /L Fgu 15. Studs on th ffct of ducng th Bohm collson fquncy and th Hall paamt n th fa plum (cas shown s fo a maxmum Bohm facto of α.4). Th plots compa solutons along th mddl of th channl. Fgu 16. Studs on th ffct of ducng th Bohm collson fquncy and th Hall paamt n th fa plum (cas shown s fo a maxmum Bohm facto of α.4). Th plots compa -D contous of th lcton tmpatu. Lft: ν B and Ω ducd byond /L~1.5. Rght: No ductons of ν B and Ω n th fa plum. A latd sult s n Fgu 17, whch shows -D contous of th Hall paamt ovlad by unt vctos of th lcton flux fo th cas of no ductons of α n th fa-plum gon. Slctd stamlns of th magntc fld a also shown. Th plot suggsts th fomaton of an ffctv funnlng gon fo th lctons that xtnds fa byond th nd of th HPHall computatonal gon. Th gons of lowst Hall paamt a closly algnd wth th on bam as shown n Fgu 17-ght; ths s wh th dag foc dnsty R on th lctons has th lagst contbutons by th classcal - collson fquncy. Howv, bcaus th Bohm collson fo ths cas mans th domnant fquncy thoughout th plum gon, th pcs lcton paths spcally n th aas na th magnts (.. th lobs of hgh Hall paamt shown n Fgu 17) cannot yt b dntfd. Th poxmty of th valus of th two collson fquncs, - and Bohm, n th fa-plum gons and th appant nsgnfcanc of th latt n most of th acclaton channl motvatd th nxt ss of numcal smulatons. In ths smulatons w am to lmnat th Bohm collson fquncy nsd th channl whl lowng th valu of α n th plum vn futh. Two cass a compad n Fgu 18-lft and Fgu 19-lft cospondng to two valus of th Bohm facto: α.15 and α.75. Th Bohm collson fquncy nsd th channl fo both cass has bn lmnatd as shown n Fgu 19-lft. Th compason wth plasma masumnts fo th cas of α.15 s shown Fgu 19-ght. As n th BPT-4 cass, t s found that by ducng ν B n th plum gon th dampng of th low-fquncy oscllatons s also dmnshd. Fgu 18-ght shows oscllatons n 19 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

20 th computd cunts wth a fquncy of ~8.5 kh, whch s about.5 tms low than th obsvd valu of ~ kh. 14,36 Th avag anod cunt n both cass s also low than th obsvd valu, by ~3.5 A. Ths smulatons combnd wth th pvous sults on th BPT-4 llustat claly th dpndnc of th spatal and tmpoal pats of th soluton on th magntud and spatal dstbuton chosn fo th Bohm collson fquncy. Psntly, ths makd dpndnc nds xtndd compasons wth tm-avagd plasma masumnts and thust pfomanc usful only n tms of boad gudanc on futh studs latd to lcton tanspot. Thus fa, th tnds fom all th plmnay smulatons n ths pap suggst that lowng α vn futh may ncas th ampltud of th low-fquncy oscllatons but th ffct on th tm-avagd cunts s not yt known. Also, Fgu 17-lft suggsts that lowng α futh could hav a sgnfcant ffct on th funnlng of lctons fom th plum gons, spcally n laton to th xpctd ncas of th Hall paamt na th magnts and th dffuson of lctons n ths gons. Futh studs on α a plannd as pat of th dvlopmnt and valdaton of Hall D. Fgu 17. Lft: contous of th Hall paamt ovlad by unt vctos of th lcton patcl flux and by slctd stamlns of th magntc fld (shown n blu). Rght-top: contous of th lcton numb dnsty ovlad by th sngly-chagd on vlocty fld. Rght-bottom: contous of th - collson fquncy Anod X+ X++ Anod cunt (A) 15 1 Cunt (A) a.15 5 a.75.e+ 5.E-5 1.E-4 1.5E-4.E-4.5E-4 3.E-4 Tm (sc).e+.e-4 4.E-4 6.E-4 8.E-4 1.E-3 Tm (sc) Fgu 18. Numcal smulatons of th 6 kw Hall thust wth no Bohm dffuson nsd th acclaton channl. Lft: anod cunt fo two valus of α:.15 and.75. Rght: anod and bam cunts fo α.75 (showng long smulaton tm). Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9

21 Collson fquncy (1/s) 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 Bohm (a.15) Bohm (a.75, t.85 ms) - (a.15) - (a.75, t.85 ms) -n (a.15) -wall (Hall D & HPHall) Elcton tmpatu, T (V) T (a.15) T (Expmnt) ph (a.15) ph (Expmnt) Plasma potntal, ph (V) 1.E /L /L Fgu 19. Numcal smulatons of th 6 kw Hall thust wth no Bohm dffuson nsd th acclaton channl. Lft: - and Bohm collson fquncs a compad fo two valus of α:.15 and.75. Rght: compasons wth plasma masumnts along th mddl of th acclaton channl fo th cas of α.15. IV. Concluson Ths pap has potd on th dvlopmnt of a -D axsymmtc computatonal modl fo Hall thusts that has th followng man fatus: (1) soluton of lcton tanspot paalll and ppndcula to th magntc fld, mad possbl by th dsctaton of th lcton flud quatons on a fld-algnd msh, () mplct soluton fo th conducton of lcton cunt and hat, (3) lag computatonal gon (by compason to stat-of-th-at hybd smulaton modls) that xtnds sval tms th channl s downstam of th thust xt, (4) soluton of th hydodynamc quatons of contnuty and momntum fo sothmal ons, accountng fo multpl onaton and chag-xchang collsons wth nutals, (5) an algothm fo th (collsonlss) nutal gas that dos not dpnd on dsct patcls. Fatus (4) and (5) allow fo qut smulatons of Hall thusts by lmnatng th statstcal nos that s nhnt n dsct-patcl mthods, Rsults fom a fst ss of smulatons of th BPT-4 and of a 6 kw laboatoy Hall thust suggst that th anod gon s domnatd by on dffuson wth th lctc fld pdctd to b nglgbly small th. Th sults show a sgnfcantly hgh plasma dnsty n ths gon by compason to th hybd smulaton sults of HPHall. Studs to assss th mpotanc of th Bohm collson fquncy (that s usually mposd n th HPHall smulatons) on th spatal and tmpoal bhavo of th plasma suggst that th soluton s not nsnstv to th condtons mposd byond th na-cathod magntc fld ln. Although t has bn known (at last fo th two thusts studd h) that th mposd spatal dstbuton of th Bohm collson fquncy n HPHall has a sgnfcant ffct on th plasma soluton n th vcnty of th channl xt, th sgnfcanc of th condtons mposd downstam of th na-cathod stamln could not b assssd du, n pat, to th lmtd xtnt of th HPHall doman. By solvng fo th lcton flow wth th Hall D xtndd gon t s found that th flow fld xhbts chaactstcs that a puly two-dmnsonal and cannot b solvd wth a quas-on-dmnsonal lcton modl. In gads to th tmpoal bhavo of th soluton t s found that th Bohm collson fquncy also nfluncs sgnfcantly th dampng of th cunt oscllatons. If lowd suffcntly, to valus that na th - collson fquncy n gons of th plum, only thn dos th plasma bgn to xhbt th tmpoal bhavo that s known to psst n Hall thusts (th wll-known bathng mod ). It has bn hypothsd n ths pap that th dampng of ths oscllatons computd at th hgh-bohm valus s du to ts domnanc ov all oth collson fquncs; th fst s ndpndnt of th plasma condtons whas th scond s not and can thfo patcpat n tansnt plasma bhavo. Although th qustons on tubulnc and ts sgnfcanc on th tanspot of lctons man unanswd by th smulaton sults psntd hn, t appas that th path to th answ cannot xclud th two-dmnsonal chaact of th lcton flow fld n th thust plum. 1 Th 31st Intnatonal Elctc Populson Confnc, Unvsty of Mchgan, USA Sptmb 4, 9