Neoclassical transport coefficients for general axisymmetric equilibria in the banana regime

Transcription

1 PHYSICS OF PLASMAS VOLUME 7, NUMBER 4 APRIL 2000 Nocassca transport coffcnts for gnra axsymmtrc qubra n th banana rgm C. Angon a) and O. Sautr Cntr d Rchrchs n Physqu ds Pasmas, Assocaton EURATOM-Confédératon Suss, Eco Poytchnqu Fédéra d Lausann, 1015 Lausann, Swtzrand Rcvd 6 Octobr 1999; accptd 3 January 2000 Usng th standard approach of nocassca thory, a st of ratvy smp kntc quatons has bn obtand, sutd for an mpmntaton n a numrca cod to comput a ratd st of dstrbuton functons. Th transport coffcnts ar thn xprssd by smp ntgras of ths functons and thy can b asy computd numrcay. Th cod CQL3D R. W. Harvy and M. G. McCoy, n Procdngs of IAEA Tchnca Commtt Mtng on Advancs n Smuaton and Modng of Thrmonucar Pasmas, Montra, 1992 Intrnatona Atomc Enrgy Agncy, Vnna, 1993, pp , whch uss th fu coson oprator and consdrs th rastc axsymmtrc confguraton of th magntc surfacs, has bn modfd to sov th bounc-avragd vrson of ths quatons. Th coffcnts hav thn bn computd for a wd varty of qubrum paramtrs, hgh-ghtng ntrstng faturs of th nfunc of gomtry at sma aspct rato. Dffrncs wth th most rcnt formuas for th on nocassca hat conductvty ar pontd out. A st of formuas, whch ft th cod rsuts, s obtand to asy vauat a th nocassca transport coffcnts n th banana rgm, at a aspct ratos, n gnra axsymmtrc qubra. Ths work xtnds to a th othr transport coffcnts, at ast n th banana rgm, th work of Sautr t a. O. Sautr, C. Angon, and Y. R. Ln-Lu, Phys. Pasmas 6, whch vauats th nocassca conductvty and a th bootstrap currnt coffcnts. Formuas for arbtrary cosonaty rgm ar proposd, obtand combnng our rsuts for th banana rgm wth th rsuts of Hnton and Haztn F. L. Hnton and R. D. Haztn, Rv. Mod. Phys. 48, , adaptd for sma aspct rato Amrcan Insttut of Physcs. S X I. INTRODUCTION Rcnt mprovmnts n nocassca transport thory ar amost compty ddcatd to para transport. In partcuar, n Rf. 2, w hav computd th nocassca conductvty and a th bootstrap currnt coffcnts, takng nto account th fu coson oprator and ncudng th advcton para to th magntc fd, consdrng th rastc axsymmtrc magntc confguraton of th fux surfac. W hav gvn ratvy smp formuas vad for gnra axsymmtrc qubra and arbtrary cosonaty rgms. For th othr transport coffcnts, mprovmnts hav bn don ony on th on thrma conductvty n th banana rgm 4 6 and for varous coson frquncs. 7 Important rcnt nw rsuts hav bn prsntd n Rf. 8, sovng a st of mutspcs fud quatons: ths nabs on to comput th nocassca conductvty, th bootstrap currnt and th partc and hat nocassca fuxs, for arbtrary cosonaty and aspct rato. Howvr, a st of quatons must b sovd n a spcfc cod, whch s not practcay sutd for numrca mpmntaton n on-dmnsona tokamak transport modng cods. Morovr, a ths attr rsuts 4 8 us an approxmat coson oprator, usuay foowng th xpanson mthod of Hrshman and Sgmar. 9 Ths a Author to whom corrspondnc shoud b addrssd. Ectronc ma: Cmnt.Angon@pf.ch mthod was aso usd to comput th k partc cosons contrbuton on th vscosty matrx, 4 and ths rsuts wr appd n Rf. 10 to comput th bootstrap currnt coffcnts at ow aspct rato. Whn compard wth Rf. 2, ths rsuts ar n vry good agrmnt n gnra, but prsnt a non-nggb rror on th bootstrap currnt coffcnts n whch th contrbuton gvn by th k partc coson oprator s partcuary mportant. 2 For a th ctron prpndcuar transport coffcnts th ony formuas avaab at sma aspct rato ar thos n Rf. 11, vad n th banana rgm, whch us th anaytca vaus of th transport coffcnts at 1 and th vaus at arg aspct rato of Rf. 12 to obtan a st of formuas wth a nar ntrpoaton btwn ths two mts, whch shoud b vad aso at sma aspct rato. In th mor rcnt nvstgatons on th on thrma conductvty, 5,6 th ntrmdat aspct rato corrctons show a dffrnc wth th rsuts of Rf. 11 of amost a factor of 2. In ths sns a compt nvstgaton of th sma aspct rato corrctons for a th nocassca transport coffcnts, takng nto account th fu coson oprator, s ncssary. It s w known that th nocassca thory can not compty xpan th prpndcuar transport n tokamaks, howvr a prcs computaton s usfu n ordr to aow a corrct vauaton of th anomaous contrbuton by mans of th comparson wth th xprmnta data. Ths s bcomng vn mor mportant wth th rcnt mprovd X/2000/7(4)/1224/11/$ Amrcan Insttut of Physcs Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

2 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 Nocassca transport coffcnts for gnra confnmnt mods of opraton, wth ntrna transport barrrs and ratvy sma anomaous transport. In Sc. II w dscrb th approach to obtan th nar drft-kntc quatons sutab for mpmntaton n a Fokkr Panck cod and th xprssons to comput th transport coffcnt as smp ntgras of th dstrbuton functons. Th ratd bounc-avragd quatons n th banana rgm ar thn obtand, and th Lorntz mod s nvstgatd anaytcay. In Sc. III w show th numrca rsuts for th banana rgm, computd wth th Fokkr Panck cod CQL3D, 1 whch sovs th narzd drft kntc bounc-avragd quaton wth th fu coson oprator and consdrng th rastc axsymmtrc confguraton of th magntc surfacs. Som bnchmarks ar consdrd to vadat th rsuts, and th comparson wth som prvous numrca and anaytca rsuts s shown. In Sc. IV w gv a st of formuas whch ft our numrca rsuts and aow to asy vauat a th nocassca transport coffcnts n gnra axsymmtrc qubra for arbtrary aspct rato and on charg n th banana rgm. Combnd formuas for arbtrary cosonaty rgm ar thn proposd n th ast subscton. A n and A n and th ft-hand sd vctors ar thr conjugatd nocassca fuxs, rfrrd as B n and B n. and Q ar th partc and hat fuxs of spcs, j s th tota para ctrc currnt, j S s th Sptzr currnt, j R s th on contrbuton to th so-cad rturn currnt, E E F (q n ) 1 s th ffctv ctrc fd and F s th frcton forc btwn ons and ctrons. W hav cad a gnrc rada coordnat and dnots th fux surfac avrag. Th functon K (), of th magntc pooda fux, s ratd to th fux surfac avrag j R B by th quaton: j R Bq K ()B 2. Not that on and ctron forcs and fuxs ar mutuay dpndnt T B 1 In A T 4, B 4 In A 1. 2 Th kntc xprssons of th conjugatd nocassca thrmodynamc fuxs can b wrttn n th foowng form: B n dvv bˆ n n1,2,3,4, G n n A n f 0, 3 II. KINETIC THEORY A. Transport coffcnts Our approach foows th standard nocassca thory, n partcuar th on of Rf. 3. Th fux surfac avragd thrmodynamc forcs and thr conjugatd nocassca fuxs ar chosn as foows: d d Q d L L11 12 L 13 L 14 T d L 21 L 22 L 23 L 24 j B T j SB L T 31 L 32 L 33 L 34 L 41 L 42 L 43 L 44 IE Bn B 2 1 p p 1 p p 1 T T 1a E B B 2 j RB T Q d T d L11 L 21 L 12 L 22 q n T K E B B 2 1 T T I B2,. 1b Th rght-hand sd vctors of Eqs. 1a and 1b ar, rspctvy, th ctron and on forcs, whch w b rfrrd as B n dv n G 2 A 2 f 0 n whch w hav ntroducd th functons 1 Iv, 2 1 v , v T 3 v f s f 0 B, 4 1 B 2 B 2, 1 q v T B, 2 v bˆ 2, 2 Iv v v T 2, n1,2, 4 Th dstrbuton functons G and G ar ratd to th frstordr prturbatons, f 1 and f 1, by mans of two sutab transformatons. Th frst on foows Eqs and 5.43 of Rf. 3, adng to th functons H and H. Thn th scond on, smpy G H n n A n f 0 and G H 2 A 2 f 0, aows to wrt th Lnarzd Drft Kntc Fokkr Panck quatons, Rf. 3, Eqs , nth foowng smp form: v bˆ G C 0 G n C 0 n f 0 A n, v bˆ G C G 1 f 0 A 1 C 2 f 0 A 2, n whch th coson oprators C 0 and C ar, rspctvy, dfnd n Eqs and 1.19 of Rf. 3. Not that ths quatons ar partcuary usfu, as a th coffcnts of th thrmodynamc forcs n th sourc trms can b vauatd anaytcay 13 C 0 1 f 0 Z 0 v 1 f 0, 5 6 Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

3 1226 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 C. Angon and O. Sautr C 0 2 f 0 Z 0 v 2 f 0 0 vh v v T 1 f 0, C 0 3 f 0 q v B f T 0, 7 C 0 4 f 0 Z 0 v 4 f 0, C 2 f 0 0 vh v v T Iv f 0, wth 0 v v Z 3v 3 T 4Z v 3, 0 v 3v 3 T 2& v, 3 hx104x 2 rfx10x rfx, and, accordng to dfntons gvn n Rf n 4 n 1 m 1/2 3/2, 4 T 3 n Z 4 4 n m 1/2 3/2. 8 T Introducng th st of functons g n, n such a way that G n g n A n, Eqs. 5 and 6 can b nary dcoupd v bˆ g n C 0 g n C 0 n f 0, v bˆ g 1 C g 1 1 f 0, v bˆ g 2 C g 2 C 2 f 0. n1,2,3, a 10b Th kntc dfntons of th thrmodynamc fuxs, Eqs. 3 and 4, wrttn n trms of th functons g n, aow to rady dntfy th transport coffcnts, ntroducd n Eq. 1 L mn dv m C 0 n f 0 dv g m f 0 C 0 L 1n dvg n 1, n f 0 n1,2,, L 21 dv g 1 C f 2 f 0 0 L 22 dv 2 C 2 f 0, n,m1,2,3,4, dv g 2 C f 2 f In ths way w hav obtand a smp st of xprssons to comput a th nocassca transport coffcnts, onc w hav sovd th drft-kntc quatons, Eqs. 9 and 10, to obtan th dstrbuton functons g n and g n. W s that vry xprsson s composd of th sum of two trms: Th frst on s an ntgra that can b computd anaytcay and that, for som coffcnts, s dntcay zro; th scond on has an ntgrand n whch th ony trm to b computd numrcay s th dstrbuton functon g n or g n. Not that n th banana rgm th frst trm, computd anaytcay, gvs drcty th vau of th transport coffcnt at 1, whn a th partcs ar trappd; th scond trm gvs th rducton of transport du to th prsnc of passng partcs. In Appndx A w show that Eqs. 11 and 12 satsfy th Onsagr ratons of symmtry as xpctd. B. Banana rgm: bounc-avragd quatons Whn th coson frquncy s much smar than th bounc frquncy b, th dstrbuton functons g n can b xpandd as foows: g n g n b g n O 2 b, and anaogousy for th on dstrbuton functons. A somwhat standard drvaton 3,12 0 shows that th functons g n ar ndpndnt of th pooda ang p, and that thy ar zro n th trappd partc rgon of vocty spac. In th passng partc rgon, th functons g n 0 satsfy th foowng bounc-avragd quatons: B d p v C 0g n S n, n1,2,3,4, 13a B d p v C g n S n, n1,2, 13b wth S n 2BC n v,b f 0, n1,2,3,4, 13c S 1 2 q B 2 f T 0, S 2 2BC 2 v,b f 0, 13d whr v /v and whr w hav ntroducd th st of functons C n (v,b), dfnd as foows: C n v,bc n f 0 /v f 0. Th anaytca xprssons of ths functons can asy b obtand from Eqs. 7. Not that BC 4 (v,b) BC 0 1 (v,b), so that th functons g 1 0 and g 4 sov th sam quaton n th banana rgm; n partcuar t foows that L 44 L 14 : Ths s a consqunc of our choc of thrmodynamc forcs and fuxs. W s that at 1, whn a 0 th partcs ar trappd, th dstrbuton functons g n ar zro vrywhr, and th frst trms n th xprssons for th transport coffcnts, Eqs. 11 and 12, gv drcty th ntr coffcnt. Th cod CQL3D has bn modfd to sov Eqs. 13 n gnra axsymmtrc qubra and wth th fu coson oprator. C. Lorntz mod For th Lorntz gas mod, Z 1, th st of Eqs. 13a s sovd anaytcay. 3,14 In fact, as cosons btwn ctrons can b ngctd, th coson oprator can b approxmatd by th ptch-ang scattrng oprator: Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

4 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 Nocassca transport coffcnts for gnra C 0 vl v and v v. Th soutons of Eq. 13a n ths approxmaton can b wrttn n th foowng form: 0 v 2 v BC n v,b f 0 g n c d 1B 1/2 H c, 14 whr (1 2 )/B, c 1/B max and H(x) s th Havsd functon. Introducng Eq. 14 n th xprssons for th coffcnts Eqs. 11, a th ctron transport coffcnts can b wrttn as ntgras n th absout vau of vocty v L mn 4 v dv m v C n f 0 0 v 4 dv v BC m BC n f 0, n,m1,2,3,4. Th ntgras of v can asy b computd anaytcay, and th rsuts hav bn compard wth Rf. 3, Eqs , , fndng a compt agrmnt. As th Lorntz mod coffcnts w b usd not ony as a bnchmark for th rsuts of CQL3D, but aso to anayz th rsuts of th cod wth th fu coson oprator and n dffrnt axsymmtrc confguratons, w rport a th ctron transport coffcnts, whch can b wrttn n th foowng smp form: L L d B 2 0 B 2 f d t, 15a L L d B 2 0 B 2 f d t, L L d B 2 0 B 2 f d t, L 31 L 34 L b f t, L 32 0, L L B 2 0 B 2 f t, L 41 L L d B 2 0 B 2 1 f t, L L d B 2 0 B 2 1 f t. wth L d n 2 p 15b 15c 15d d d 2, L b In, L n q 2 B 2 m T 0, 16 and whr w hav ntroducd two dfntons for th trappd fracton f d t 1 4B I, f t 1 4B 3 2 I, 17 I c d 0 1B 1/2. Th scond on, f t, s th usua dfnton for th trappd partc fracton. 4 Not that th ntgra I can asy b vauatd usng th formuas n Rf. 15. Th pooda gyroradus p of spcs s gvn by FIG. 1. Transport coffcnts L 1n and L 2n for an amost cyndrca qubrum, computd by CQL3D n th approxmaton of th Lorntz mod crcs, and compard wth th anaytca rsuts, Eq. 15. p v T p 2m T q B p0, 18 k n Rf. 3, Eq , whr th pooda magntc fd B po () s dfnd by B po (d/d)b 0 ()/I(), and B 0 () s an arbtrary chosn functon ntroducd to normaz th magntc fd on a gvn fux surfac. Not that th fux surfac avragd ntgras I 11, I 13, and I 33 whch appar n th rsuts of Rf. 3 can b rducd to ony th two trappd fractons, Eq. 17, wth th foowng ratons: I 11 3B B 2 f d t, I f t, I 33 3B B 2 f t. 19 W s, thrfor, that a th coffcnts n th Lorntz mod dpnd ssntay on f d t and f t. W sha show n th nxt Sctons that n th gnra cas ths proprty rmans tru, namy that a th qubrum ffcts on th nocassca transport coffcnts ar functons of ony ths two trappd fractons. III. NUMERICAL RESULTS A. Bnchmarks As th Lorntz mod gvs an anaytca souton, t can b usd as a frst bnchmark for th numrca rsuts. In Fg. 1 w show th transport coffcnts L 1n and L 2n ratv to th sourcs S 1 and S 2, Eqs. 13, computd by CQL3D n th approxmaton of th Lorntz mod: vry good agrmnt s obtand for a. Th coffcnts n th Fgur, ndcatd by L mn, ar pottd normazd by th ratv factors L d or L b gvn n Eq. 16. Ths normazaton for th ctron transport coffcnts s aso kpt n Fgs. 2 and 4, Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

5 1228 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 C. Angon and O. Sautr As w hav sad n Sc. I, th most rcnt nvstgatons on prpndcuar nocassca transport wr ddcatd ony to th on thrma conductvty. 5,6 In Fg. 3 w compar our rsuts for th coffcnt L 22, obtand wth an amost cyndrca qubrum, wth th rsuts of Rfs. 5, 6, and 11. As mntond at th nd of th prvous paragraph, a corrct gomtrca paramtr must b chosn to pot a gvn transport coffcnt. As t can b nfrrd from th Lorntz mod rsuts for th ctron coffcnt L 22, Eq. 15b, and as t w b prsntd atr, th on hat conductvty s a functon of th trappd fracton f d t, Eq. 17. W fnd good agrmnt wth th most rcnt formuas of Rfs. 5 and 6. Ths rsuts nab to fnay rsov th dscrpancy btwn th formuas gvn n Rfs. 5 and 6, obtand wth approxmatd coson oprators. It turns out that th rsuts wth th fu coson oprator, CQL3D, ar n btwn th prvous r- FIG. 2. Onsagr symmtry s corrcty rspctd by th numrca rsuts. a Th transport coffcnt L 13 symbos, pottd vs f t, s w agnd wth th formua 14a of Rf. 1 for th bootstrap currnt coffcnt L 31 sod n. Th dffrnt symbos rfr to dffrnt qubra n a th fgurs. Not that th rsuts gvn by th dffrnt qubra ar prfcty ovrappd, as thy ar pottd vs f t. b Transport coffcnt L 12 symbos and L 12 sod ns computd wth four dffrnt qubra, pottd vs 1/2. Not that whn th compt coffcnts ar pottd vs th nvrs aspct rato, a strong dpndnc on qubra appars at sma aspct rato. TABLE I. Equbra spcfcatons and ratd symbos usd n th fgurs. Symbo R mag m R go m a m k FIG. 3. Th on hat conductvty, transport coffcnt L 22, computd by CQL3D wth an amost cyndrca qubrum sod crcs, dvdd by th fux surfac avrag B 2 0 B 2, pottd vs th trappd partc fracton f d t and compard wth formuas of Rf. 3, CH 82 dashd dottd n, Rf. 4, T 88 dashd n, and Rf. 9 modfd, HHR 73 sod n. and anaogousy n Fg. 3 for th on transport coffcnts. Th compt dfnton of a st of dmnsonss coffcnts w b gvn n th nxt scton. Whn th fu coson oprator s usd, th symmtry of th transport matrx gvs a scond bnchmark of th numrca rsuts: ndd, ach off-dagona coffcnt can b computd n two dffrnt ways, L nm and L mn. Not that w hav arady computd 2 th nocassca rsstvty and th bootstrap currnt coffcnts L 3n, n1,2,4. In Fg. 2a w pot th rsuts for th coffcnt L 13, computd sovng th kntc quaton wth th sourc S 1, Eqs. 13a and 13c, n1: thy ar prfcty agnd wth th sod n gvn by Eq. 14a of Rf. 2, whch fts th cod rsuts for th bootstrap currnt coffcnt L 31, hnc computd wth th sourc S 3, Eqs. 13a and 13c, n3. Th xact ratons btwn th bootstrap currnt coffcnts dfnd n Rf. 2 and th transport coffcnts dfnd n ths papr w b prsntd n th nxt scton. In Fg. 2b w pot th two coffcnts L 12 and L 21, computd consdrng four dffrnt qubra, as shown n Fg. 1 of Rf. 2, and whos man spcfcatons and ratd symbos, fu or opn, usd n a th fgurs, ar gvn n Tab I. Th coffcnt L 12, obtand sovng Eqs. 13a and 13c, n1, s pottd wth symbos, th coffcnt L 21, obtand sovng Eqs. 13a and 13c, n2, s pottd wth sod ns: w fnd a vry good agrmnt btwn th two coffcnts, wthn 1% for 0.1. W s aso that th bhavor of th transport coffcnt strongy dpnds on th qubrum at sma aspct rato. Prvous formuas, whch gv th transport coffcnts wth an xpanson n powrs of 1/2, ar corrct ony for amost cyndrca qubra and ar of practca ntrst n gnra qubra ony for 0.1: ths must b takn nto account whn comparng wth our rsuts. It ndcats that for ach transport coffcnt an approprat gomtrca paramtr, k f t for L 31 and L 13 n Fg. 2a, nds to b usd nstad of, as t w b shown n th nxt subscton. B. Comparson wth prvous rsuts and bhavor at sma aspct rato Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

6 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 Nocassca transport coffcnts for gnra FIG. 4. Th transport coffcnt L 22, man contrbuton to th ctron hat conductvty, computd by CQL3D wth four dffrnt qubra: a Th compt coffcnt s pottd vs 1/2, a strong dpndnc on th dffrnt qubra appars at sma aspct rato; b th coffcnt s dvdd by an approprat fux surfac avrag B 2 0 B 2, and pottd vs th corrct gomtrca paramtr, f d t, whch aows to prfcty agn a th ponts of th dffrnt qubra. suts. Not that th pottd formua of Rf. 11 has bn modfd, kpng n th xprsson for th transport coffcnt th fux surfac avrag of th magntc fd, whch wr corrcty computd n th rfrnc to obtan th mt at 1, but thn not takn nto account n th fna formuas. Whn dffrnt axsymmtrc qubra ar consdrd n th numrca cacuatons, th transport coffcnts prsnt partcuar faturs of th nfunc of gomtry at sma aspct rato, as arady hghghtd n Fg. 2b. In Fg. 4 w aso show th ctron coffcnt L 22, computd wth th four dffrnt qubra of Tab I. In Fg. 4a, L 22 s pottd vrsus 1/2 : w s dffrncs up to 30% arady at In Fg. 4b w show th sam coffcnt L 22 dvdd by th fux surfac avrag (B 2 0 B 2 ) and pottd vrsus th trappd partc fracton f d t, dfnd n Eq. 17, as suggstd by th rsuts of th Lorntz mod: th ponts ar w-agnd at a f d t,.., at a. Th sam bhavor s obtand for a th othr transport coffcnts: th coffcnts must b normazd by a sutab fux surfac avrag and a corrct gomtrca paramtr must b usd to ncapsuat th ffcts of th dffrnt qubra. Not that th dfnton of a nw trappd partc fracton, f d t, Eq. 17, s ffctvy ncssary, as suggstd by th Lorntz mod, to corrcty dscrb th gomtrca bhavor of crtan coffcnts, n partcuar a th partc and hat conductvts, for whch th usua on, f t, turns out to b nadquat. From Fg. 4 t s aso vdnt that, whn pottd vrsus th corrct paramtr, th transport coffcnts turn out to b vry smp functons, amost proportona to th approprat trappd fracton. Ths xpans th ratvy smp formuas of Rf. 2 for th ctrca conductvty and th bootstrap currnt coffcnts, and w b compty prsntd n th nxt scton and shown n Fg. 5, for a th prpndcuar FIG. 5. Computd vaus of th dmnsonss transport coffcnts K mn symbos, compard wth th fttng formuas, Eqs. 23 and 24 sod ns. a Coffcnts K 11 pottd vs f d t sod symbos, and K 14 pottd vs f t opn symbos. b Coffcnts K 12 and K 21, pottd vs f d t sod symbos, and K 24 pottd vs f t opn symbos. c Coffcnt K 22 pottd vs f t d sod symbos. d Coffcnt K 22 pottd vs f d t sod symbos. transport coffcnts. Hnc th study of th ffcts of pasma shap on th nocassca transport can b smpy obtand consdrng th dpndnc of th trappd fracton on pasma ongaton and tranguarty at a gvn aspct rato. Both th two gvn xprssons for th trappd fracton, f t and f t d, Eq. 17, turn out to b amost ndpndnt of ongaton, and ncrasng whn dcrasng tranguarty. In ths sns, n th nocassca transport, at a gvn vau of th aspct rato, a hghy tranguar pasma shap s favorab for confnmnt, as shown n Fgs. 2b and 4a symbo, but not favorab for drvng bootstrap currnt. IV. TRANSPORT COEFFICIENTS FORMULAS A. Anaytca fts to th numrca rsuts for th banana rgm Consdrng th rsuts of th prvous scton, w can ntroduc a st of dmnsonss ctron and on transport coffcnts K mn L nm L d B 2 0 B 2 K nm f d t, n,m1,2, 20a L n3 L b K n3 f t, n1,2,4, 20b L n4 L d B 2 0 B 2 1 K n4 f t, n1,2, 20c L 33 L B 2 0 B 2 K 33 f t, L 11 L 12 L B 2 0 B 2 K 11 f t, L b K 12 f t, 20d 20 20f Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

7 1230 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 C. Angon and O. Sautr L 22 L d B 2 0 B 2 K 22 f d t, 20g and anaogousy a thr symmtrcs, whr L d, L b, and L ar dfnd by Eq. 16; th on normazaton factors, L d, L b, and L, ar dfnd as foows: L d n 2 p d, L b In, L n q 2 B 2 m T 0. 2 d 21 Th dmnsonss coffcnts ar functons of ony on sutab gomtrca paramtr,.., a trappd fracton, whch compty ncapsuats th ffcts of th varous qubra: n ths way th cod rsuts for th coffcnts K mn can b fttd n trms of th approprat trappd fracton, f t or f d t,as thy prfcty ovrap, rgardss th qubrum consdrd n th cacuaton, vn hghy noncrcuar and at sma aspct rato. Not that ony n ths way ratvy smp formuas vad n gnra axsymmtrc qubra and at a aspct ratos can b gvn. W hav arady computd th nocassca conductvty and th bootstrap currnt coffcnts, n Rf. 2, sovng th sam kntc quaton of Eq. 13a, n3, and computng th transport coffcnts wth th sam ntgras gvn by Eq. 11, n3, and Eq. 12, n 2, m1, whch wr frst obtand usng an adjont formasm 13 adaptd to cacuat ony th bootstrap currnt: th gnra kntc rsuts of Sc. II show, howvr, that th adjont formuaton s not ncssary. Th ratons btwn th transport coffcnt L 33 and th nocassca conductvty no, Eq. 13a n Rf. 2, and btwn th dmnsonss coffcnts K 3m, m1,2,4, and th bootstrap currnt coffcnts L bs 3m, Eqs. 14 and 15 n Rf. 2, rad L 33 no Sptz T B 2, K 3m L bs 3m. 22 Th coffcnt K 12 s ratd to th coffcnt, Eqs. 17a and 17b n Rf. 2, by Eq. 25. Not: shoud b rpacd by n Eq. 17b. W hav run th cod CQL3D wth dffrnt qubra and for th ctron coffcnts w hav aso vard th on charg, to obtan th dpndnc on th ffctv charg Z. Our da s that, at ast for th ctron transport coffcnts, an ffctv charg approxmaton for mutspcs cass shoud st b vad: cosons btwn ctrons and man ons, or btwn ctrons and mpurty ons ar amost of th sam knd, nvovng bascay th ptch-ang scattrng. In any cas, th comparson wth th rsuts of mutspcs cods 8 shoud nab on to dtrmn th corrct form of Z, nstad of th usua dfnton of Z ff, to b usd n our formuas, as arady mntond n Rf. 2. For th ons, th prsnc of on havy mpurty spcs ads to cosons btwn man ons and mpurty ons whch nvov bascay th ptch-ang scattrng, and whch ar compty dffrnt from k partc cosons. In ths cas, as shown n Rf. 16, th thrma conductvty computd as Z ff tms th pur on conductvty s undrstmatd. Usng th rsuts of Rf. 16, whch uss th arg aspct rato mt of Rf. 17, w hav gnrazd our formua for th transport coffcnt L 22, to ncud th ffct of a sng havy mpurty spcs n th Pfrsch Schütr rgm. W hav aso adaptd th formua for th bootstrap currnt coffcnt n th banana rgm, Eq. 17a n Rf. 2, to ncud th sam ffct, usng th arg aspct rato mt of Rf. 17, and notng that at 1 not ony th pur pasma coffcnt, but aso th mpurty contrbuton must b qua to zro. Th anaytca fts to th rsuts of CQL3D for a th transport coffcnts not arady computd n Rf. 2, vad n th banana rgm, for arbtrary trappd fracton and Z, and th modfd formua for th bootstrap currnt coffcnt, rad as foows: K 11 f d t 0.5F 11 f d t, 23a K 12 f d t 0.75F 12 f d t, 23b K 14 f t K F 11 f t, 23c K 22 f d t Z & F 22 f d t, 23d K 24 f t 0.75F 12 f t, 23 K 22 F 11 X F 12 X f d t F 22 f d t, 1 Z X 1.9 Z0.5 X2 1.6 Z0.5 X3 0.6 Z0.5 X4, 1 Z X 0.95 Z0.5 X2 23f 24a 0.3 Z0.5 X Z0.5 X4, 24b F 22 X 1 Z X 0.08 Z0.5 X Z0.5 X3, 24c F 22 X I X 0.75X 2 0.7X 3 0.5X I, 24d I f t K I 1 f t f t 0.19f t 2, 25 whr I n I Z 2 I /n Z 2 s th usua mpurty strngth paramtr, and ndx I rfrs to th on mpurty spcs. Th factorzatons usd n Eqs. 23 and 24 ar such that th Lorntz mt (Z ), th ow ( f t 0) and th arg aspct rato ( f t 1) ar asy rcovrd. Morovr th functons F j hav vaus wthn 0,1. Not that K 11 and K 14, as w as K 12 and K 24 hav th sam functona dpndnc on thr rspctv trappd fracton. Ths ratons can b consdrd as th xtnson to a gnra axsymmtrc qubrum at a aspct ratos of Eqs and Eq n Rf. 3. W hav aso computd th coffcnt L 11, whch s usuay not consdrd, foowng th wak-coupng approxmaton, whch ngcts th forc A 1. For comptnss, w gv aso th ft to th cod rsuts for th transport coffcnt K 11 K 11 f t f t 1.25f 2 t 0.44f 3 t 1 1. Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

8 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 Nocassca transport coffcnts for gnra Not that Eq. 2 aows to rduc th numbr of ndpndnt thrmodynamc forcs from 6 to 4, hnc wth ony 4 conjugatd thrmodynamc fuxs. Takng th frst 3 ctron forcs and th scond on forc, whos conjugatd fuxs hav mor drct physca manng and mor drct appcaton n th fud transport quatons, th ratons whch connct fuxs wth forcs rad as foows: 3 B n m1 2 L nm 1 T L 11 L n4 F Z 2 A T In m T L 12 1 L F Z T n4 A In 2, n1,2,3, 26a L 21 B 2 1 F In m1 3 L 4m A m whr L 22 1 F T L 11 F1 Z 2 1 T In 2, T L 12 L 21 Z 2 T In 2 L 44 A 2, L 44 26b and Z s th man on charg numbr. Th condton for th vadty of th wak-coupng approxmaton s gvn by Eq n Rf. 3, and s smpy F11. Introducng th dmnsonss coffcnts K mn, ths raton rads, consstnty wth th stmat gvn n Rf. 3, Tab IV 2& m Z m 1/2 T T 3/2 K 14 K Th absout vau of th trm K 14 K 11 turns out to b smar thn 0.25, whch confrms th vadty of th wak coupng approxmaton n th banana rgm. In ths way Eq. 26 ar rducd to: 3 B n m1 B 2 L 3 21 In m1 L 22 L nm A m T L Z T n4 L 4m A m L 12 In A 2, n1,2,3, 28a T L 12 L 21 Z 2 T In 2 L 44 A 2. 28b In Fg. 5 w compar th cod rsuts for th dmnsonss transport coffcnts K nm symbos wth th agbrac formuas, Eqs. 23 and 24, whch ft th data, sod ns. B. Combnd formuas for a cosonaty rgms FIG. 6. Dpndnc on cosonaty for th transport coffcnts no, a and L 31, b, for dffrnt vaus of th trappd fracton f t, as gvn by Rf. 1 sod ns, by Rf. 12, wth th vau at 0, banana mt, corrctd wth th rsuts of Rf. 1 dashd ns, and st by Rf. 12, wth aso th cosona paramtr rscad by Eq. 29 dashd dottd ns. In ordr to comput th nocassca transport coffcnts at arbtrary cosonaty rgm, th nonbouncavragd kntc quatons, Eqs. 9 and 10, must b sovd. Ths, as arady mntond, has bn don n Rf. 2, to comput th nocassca rstvty and a th bootstrap currnt coffcnts, usng th cod CQLP, whch ncuds th advcton para to th magntc fd, wthout any assumpton on th rato btwn th coson frquncy and th bounc frquncy. In ordr to strcty compar ony th dpndnc on cosonaty, n Fg. 6 w hav pottd formuas of Rf. 2 sod ns and thos of Rf. 3, Sc. VI F dashd ns, n whch w hav rpacd th banana mt wth th corrct rsuts of th cod CQL3D. Th nocassca rsstvty s shown n Fg. 6a and th bootstrap currnt coffcnt L 31 n Fg. 6b, for thr vaus of th trappd fracton. At ow aspct rato thr s a vry good agrmnt, whch fas down at argr vaus of th trappd fracton. Ths coms from th man approxmaton adoptd to comput th banana-patau rgm, n Rf. 18, whch ngcts th nrgy scattrng n th k-partc coson oprator and whch undrstmats th nocassca transport at ow aspct rato. 7 Howvr, for both th nocassca rsstvty and th bootstrap currnt coffcnt L 31, and aso for th coffcnt L 32 not shown hr, th Rf. 3 formuas go down to zro at smar vaus of wth rspct to th rgorous rsuts of Rf. 2, wth approxmaty th sam bhavor. Whn th cosona paramtr, dfnd n Rf. 2, Eq. 18, s rscad n trms of th trappd fracton wth th smp transformaton f 2, f t Rf. 3 formuas aow an agrmnt wthn 20% for a th bootstrap currnt coffcnts and th nocassca rsstvty dashd dottd ns, comparng wth Rf. 2. Hnc, foowng th da of Rf. 5, n whch a formua vad for a cosonaty rgms for th on hat conductvty s obtand connctng a nw banana mt, vad aso at sma aspct rato, wth th cosona dpndnc of Rf. 3, w propos to combn formuas of Rf. 3, Sc. VI F, adaptd to Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

9 1232 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 C. Angon and O. Sautr sma aspct rato, wth th rsuts of ths papr n th mt at 0. Th ctron transport coffcnts K mn, m,n1,2 can b computd at arbtrary cosonaty rgms as foows: K 11 f d t, H 11, K 12 f d t, H 12 2H 5 11, K 22 f d t, H 22 5H H 11, 30a H mn f t d, H 0 mn f d t, 0 1a mn Z 1/2 f b mn Z f d mnz f f t d 3 1 f d 6 t 1c mn Z f f t d3 d 1 f 6 t F PS, 30b whr th banana mt coffcnts H mn ( f d t, 0) can b rady vauatd usng Eqs. 23 and 24, wth H 11 K 11 f d t, H 12 K 12 f d t 2K 5 11 f d t, H 22 K 22 f d t 5K 12 f d t 25 4 K 11 f d t. 30c Th coffcnts K 4n ar gvn by K 41 f t, H 41, K 42 f t, H 42 2H 5 41, 30d H 4n f t, H 4n f t, 0 1a 1n Z f b 1n Z f d 1nZ f f t f 3 t 1c 1n Z f f t f 3 t 4 F PS f f t 30 whr anaogousy H 41 K 41 f t, H 42 K 42 f t 2K 5 41 f t, 30f and wth 1 F PS 1 B 2 B 2, F 4 PSB B g Th on thrma conductvty K 22 s gvn by K 22 f d t, K 22 1a 2 f f t d 1/2 b 2 f d 2 f f t d 3 1 f d 6 t 1c 2 f f t d3 d 1 f 6 t H PF PS, 30h wth 16 f f (11.54 I) and H P I ( I )/(11.79 I ). Th coffcnts a mn (Z), b mn (Z), c mn (Z), and d mn (Z) ar gvn n Appndx B, obtand by ntrpoaton of th data gvn n Rf. 3, Tab III for th ctron coffcnts and bow Eq for th on coffcnt. Th dpndnc on of th patau-cosona trms of th formuas of Rf. 3, frst computd n Rf. 19, has bn rscad on f t or f d t. Fnt aspct rato ffcts n ths trms hav bn takn nto account, k n Rf. 5, ntroducng th compt xprsson of th Pfrsch Schütr gomtrca factor, by mans of F PS and F (4) PS, Eq. 30g. Not that Eq. 30h for th on thrma conductvty K 22 ncuds th ffcts of a sng havy mpurty spcs n th Pfrsch Schütr rgm, accordng to Rf. 16, usng th modfd on cosonaty paramtr f and th factor H P, whch tak nto account th nhancmnt of man on thrma transport du to th prsnc of th mpurty spcs. 16 V. CONCLUSION W hav prsntd an approach for th nocassca transport thory whch aows to obtan smp quatons sutd for mpmntaton n numrca cods n ordr to comput a th nocassca transport coffcnts. Th cod CQL3D, sovng th bounc-avragd narzd drft-kntc Fokkr Panck quaton wth th fu coson oprator, has bn modfd to cacuat a ths coffcnts at a aspct ratos of varous axsymmtrc qubra n th banana rgm. W hav shown that th mts at arg and unt aspct rato ar corrcty rspctd by th numrca rsuts, as aso th Onsagr symmtry of th nondagona transport coffcnts. Invstgatng th dpndnc of th coffcnts on gomtry paramtrs, w hav shown that approprat dfntons of trappd fractons ar rqurd n ordr to ncapsuat a th gomtry ffcts n a sng varab. In ths way, a st of smp formuas can b obtand from th numrca rsuts and aow th vauaton of any transport coffcnt for vry axsymmtrc qubrum and at a aspct ratos. Our formua for th on thrma conductvty s n good agrmnt wth th most rcnt vauatons of ths coffcnt, 5,6 whch, howvr, do not us th fu coson oprator, wth rrors at fnt aspct rato of about 10% by xcss and by dfct, rspctvy. For a th othr prpndcuar transport coffcnts, n partcuar th ctron thrma conductvty, our formuas ar th ony xstng to dat and to our knowdg, computd for gnra axsymmtrc qubra takng nto account fnt aspct rato ffcts. Th transport coffcnts formuas, whch ft th numrca rsuts n th banana rgm, ar gvn by Eqs of Sc. IV A. Extnson of ths work s to comput th transport coffcnts at a cosonaty rgms: not that ths has arady bn don for th nocassca conductvty and th bootstrap currnt coffcnts n Rf. 2. Ths rsuts, compard wth th ons of Rfs. 3 and 5, hav motvatd us to propos combnd formuas for a th othr transport coffcnts, vad for arbtrary cosonaty rgm, ndd to corrcty vauat ths coffcnts ovr th who pasma mnor radus. Th thrmodynamc fuxs B 1 d d, B 2 Q T d d, B 3 j B j SB, B T T 2 Q d T d, whr s th prpndcuar ctron partc fux, Q s th ctron prpndcuar hat fux, j and j S ar th para ctrc currnt and th Sptzr currnt, and Q s th on prpndcuar hat fux, ar gvn by Eqs. 28, n th wak coupng approxmaton, whos vadty s confrmd by Eq. 27. Equatons 28 can b rordrd, and th thrmody- Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

10 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 Nocassca transport coffcnts for gnra namc fuxs can b xprssd drcty n trms of th ctron and on tmpratur and dnsty prpndcuar gradnts and th para ctrc fd v bˆ g n v bˆ g n C 0 g n C 0 g n C 0 n f 0, 0 A2 B n L n1 n n L n1 L n2 n T 1R p R p n T L n1 n n L E B n3 B 2, 1R p L R n1 L n4 p n1,2,3, B 2 Q d T d L n n 41 L 41 L 42 n T L 1R p n n 41 R p L E B 43 B 2 L 22 1R p 2 R p L Z 44 n T, whr R p p /p and Z s th man on charg numbr. Th prpndcuar transport coffcnts L mn, for gnra axsymmtrc qubra and arbtrary cosonaty rgm, ar gvn by Eqs. 30 of Sc. IV B n trms of th trappd fractons f t or f d t, Eq. 17, th cosonaty paramtr and th ffctv charg numbr Z. Th nocassca conductvty and th bootstrap currnt coffcnts, L 3n and, ar connctd wth formuas of Rf. 2 by Eqs. 22 and 25. ACKNOWLEDGMENTS W ar gratfu to Y.R. Ln-Lu and J. Vacavk for ntrstng dscussons. On of th authors, C.A., woud aso k to thank P. Handr for usfu dscussons. Ths work was party supportd by th Swss Natona Scnc Foundaton. APPENDIX A: ONSAGER SYMMETRY OF THE TRANSPORT COEFFICIENTS Th xprssons for th transport coffcnts gvn by Eq. 11 for th ctrons and by Eq. 12 for th ons satsfy th Onsagr ratons of symmtry, as xpctd. 3 W bgn wth th ctron cas. In Eq. 11, th frst trm, L (1) mn dv m C 0 ( n f 0 ), s symmtrc drcty from th sf-adjontnss of th coson oprator. Hnc 1 dv m C 0 dv n C 0 m f 0 L 1 nm. L mn n f 0 A1 For th scond trm, L (2) mn dvg m / f 0 C 0 ( n f 0 ), w sha rwrt t n a symmtrc form. Introducng th foowng notaton, 3 for a gnrc functon f (v), f 2 1 f ( 1)f (1) s ts vn part n v /v and f 2 1 f (1)f(1) s th odd part, so that C 0 as n1,2,3,4 w can prform th foowng drvaton: ( n f 0 ) C 0 ( n f 0 ) and C 0 ( n f 0 ) 0, L 2 mn dv g m dv g m f 0 C 0 f 0 C 0 n f 0 n f 0 dv g m v f bˆ g n 0 C 0 g n dv 1 g f m v bˆ g n g m 0 dv 1 g f m 0 dv 1 C 0 g f m C 0 g n 0, C 0 g n g n g m C 0 g n whch s a symmtrc xprsson, usng th sf-adjontnss of th coson oprator. Not that w hav usd n ths drvaton th fact that th oprator v bˆ s th adjont of th oprator v bˆ.) Hnc w can concud that L 2 mn dv 1 g f m C 0 g n 0 dv 1 g f n C 0 g m 0 L 2 nm, A3 whch shows th symmtry of th coffcnts L mn. A somwhat anaogous cacuaton can b prformd for th on coffcnts L 12 and L 21, whch shows that th two gvn x- prssons, Eq. 12, satsfy th foowng raton: L 12 L 21, A4 consstnty wth th rsut n Rf. 3, Eq APPENDIX B: COEFFICIENTS FOR THE COMBINED FORMULAS OF SEC. IV B Th coffcnts a mn (Z), b mn (Z), c mn (Z), and d mn (Z) for th ctron transport coffcnts ar dfnd as foows: a 11 Z 13Z Z, a 12 Z Z 10.5Z, a 22Z0.46, B1a b 11 Z 11.1Z 1.37Z, b 12Z 1Z 2.99Z, B1b Z b 22 Z 35.32Z, Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s

11 1234 Phys. Pasmas, Vo. 7, No. 4, Apr 2000 C. Angon and O. Sautr c 11 Z Z 1.65Z c 22 Z Z 17Z,, c 12 Z Z, 13Z B1c d 11 Z 0.23Z 13.85Z, B1d d 12 Z Z 16.1Z, d 22Z Z 10.82Z. For th on thrma conductvty, th coffcnts ar a , b , c , d B2 1 O. Sautr, C. Angon, and Y. R. Ln-Lu, Phys. Pasmas 6, S. P. Hrshman and D. J. Sgmar, Nuc. Fuson 21, C. S. Chang and F. L. Hnton, Phys. Fuds 25, M. Taguch, Pasma Phys. Controd Fuson 30, C. Boton and A. A. War, Phys. Fuds 26, W. A. Houbrg, K. C. Shang, S. P. Hrshman, and M. C. Zarnstorff, Phys. Pasmas 4, S. P. Hrshman and D. J. Sgmar, Phys. Fuds 19, S. P. Hrshman, Phys. Fuds 31, R. D. Haztn, F. L. Hnton, and M. N. Rosnbuth, Phys. Fuds 16, M. N. Rosnbuth, R. D. Haztn, and F. L. Hnton, Phys. Fuds 15, R. W. Harvy and M. G. McCoy, n Procdngs of Intrnatona Atomc Enrgy Agncy Tchnca Commtt Mtng on Advancs n Smuaton and Modng of Thrmonucar Pasmas, Montra, 1992 Intrnatona Atomc Enrgy Agncy, Vnna, 1993, pp ; J. Kn, G. D. Krb, M. G. McCoy, and A. A. Mrn, Computatona Mthods for Kntc Mods of Magntcay Confnd Pasmas Sprngr-Vrag, Nw York, F. L. Hnton and R. D. Haztn, Rv. Mod. Phys. 48, Y. R. Ln-Lu, prvat communcaton s Rf. 29 n Rf P. H. Ruthrford, Phys. Fuds 13, Y. R. Ln-Lu and R. L. Mr, Phys. Pasmas 2, C. S. Chang and F. L. Hnton, Phys. Fuds 29, S. P. Hrshman, Phys. Fuds 19, F. L. Hnton and M. N. Rosnbuth, Phys. Fuds 16, J. M. Raws, M. S. Chu, and F. L. Hnton, Phys. Fuds 18, Downoadd 31 Aug 2001 to Rdstrbuton subjct to AIP cns or copyrght, s