Nonlinear neoclassical transport in toroidal edge plasmas

Transcription

1 Nonlnear neoclasscal transport n torodal edge plasmas. Fülöp and P. Helander Ctaton: Phys. Plasmas 8, 3305 (00); do: 0.063/.3779 Vew onlne: Vew able o Contents: Publshed by the Amercan Insttute o Physcs. Related Artcles Propertes o convectve cells generated n magnetzed torodal plasmas Phys. Plasmas 9, (0) wo-dmensonal plasma expanson n a magnetc nozzle: Separaton due to electron nerta Phys. Plasmas 9, (0) Generaton o dust projectles passng over an obstacle n the plasma sheath Phys. Plasmas 9, (0) hree-dmensonal numercal nvestgaton o electron transport wth rotatng spoke n a cylndrcal anode layer Hall plasma accelerator Phys. Plasmas 9, (0) Revsted global drt lud model or lnear devces Phys. Plasmas 9, 0737 (0) Addtonal normaton on Phys. Plasmas Journal Homepage: Journal Inormaton: op downloads: Inormaton or Authors: Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

2 PHYSICS OF PLASMAS VOLUME 8, NUMBER 7 JULY 00 Nonlnear neoclasscal transport n torodal edge plasmas. Fülöp a) Department o Electromagnetcs, Chalmers Unversty o echnology, Gothenburg 4 96, Sweden P. Helander UKAEA/Euratom Fuson Assocaton, Culham Scence Centre, Abngdon, Oxon, OX4 3DB, Unted Kngdom Receved 8 December 000; accepted 6 March 00 In conventonal neoclasscal theory, the densty and temperature gradents are not allowed to be as steep as requently observed n the tokamak edge. In ths paper the theory o neoclasscal transport n a collsonal, mpure plasma s extended to allow or steeper proles than normally assumed. he dynamcs o hghly charged mpurty ons then becomes nonlnear, whch aects the transport o all speces. As earler ound n the banana regme, when the bulk plasma gradents are large the mpurty ons undergo a polodal redstrbuton, whch reduces ther parallel rcton wth the bulk ons and suppresses the neoclasscal on partcle lux. he neoclasscal connement s thus mproved n regons wth large radal gradents. When the plasma s collsonal and the gradents are large, the mpurtes accumulate on the nboard sde o the torus. 00 Amercan Insttute o Physcs. DOI: 0.063/.3779 I. INRODUCION From years o experments on the world s premer tokamaks t s well known that the plasma edge has a decsve nluence on tokamak perormance. Numercal smulatons o core plasma turbulence also suggest that the edge plasma plays an mportant role n regulatng the overall connement. he present theoretcal understandng o the edge plasma s, however, arly ncomplete. Not even ts neoclasscal transport propertes can be predcted wth any condence, snce the plasma proles are requently too steep or conventonal neoclasscal theory to be vald, especally n the H-mode pedestal regon. In ths regon, very strong polodal mpurty asymmetres have been observed n recent experments n the Alcator C-Mod,,3 but the mechansm gvng rse to these asymmetres and ts mplcatons or the transport are not known. One o the possble causes or the polodal accumulaton o the mpurtes s the strong on-mpurty rcton caused by large radal densty and temperature gradents n ths regon. he parallel on low s proportonal to these gradents. I the parallel on-mpurty rcton s large enough to compete wth the parallel mpurty pressure gradent, then the mpurtes are not able to move reely over the lux surace, and thereore accumulaton o mpurtes wll arse, as was shown recently n Res. 4 and 5. However, these papers consdered the case where the bulk plasma ons and electrons are nearly collsonless, whle the edge collsonalty n the Alcator C-Mod s much hgher. hs has motvated the study presented n ths paper, where the nonlnear neoclasscal transport theory o Res. 4 and 5 s extended to the Prsch Schlüter hghcollsonalty regme. Nonlnear transport n ths regme has been examned earler n Re. 6, but only or an sothermal a Electronc mal: tunde@elmagn.chalmers.se plasma n a smpled geometry. We nd that an up down asymmetry s ndeed expected or parameters typcal o the plasma edge, and ths asymmetry has the same qualtatve eatures as those observed n the experments. However, the measured eect s substantally larger than the theoretcal predcton. As n the banana regme, the calculated polodal mpurty redstrbuton turns out to have nterestng mplcatons or the cross-eld neoclasscal transport. he neoclasscal on partcle lux becomes nonlnear and nonmonotonc as a uncton o the local gradents, and the heat lux s also aected. Conventonal neoclasscal theory 7 9 s based on the requrement that the expanson parameter /L should be nntesmally small, where s the polodal on gyroradus and L the radal scale length assocated wth the densty and temperature proles. As the radal scale length decreases, polodal asymmetres arse n the plasma. 0 In an mpure plasma, typcally the rst plasma parameter to develop a polodal varaton s the mpurty densty, n z, 6 whose polodal modulaton s o the order ñ z /n z z ˆ z, where ˆ L / s the collsonalty, wth the mean-ree path or the bulk ons and L the connecton length. In conventonal neoclasscal theory z s assumed to be small. In ths paper, we adopt the orderng, z O(), whch s more realstc or the tokamak edge. Let us pause brely to dscuss the meanng o ths orderng. As n all neoclasscal theory, our basc expanson parameter s, whle a subsdary expanson s perormed n the largeness o the collsonalty parameter ˆ. Allowng z ˆ z to be o order unty thus mples that z O( / ) s ormally regarded as large, so that we are lmted to consderng heavy mpurtes. It should be noted that z s only assumed to be o order unty on the scale set by, and ths does not preclude the possblty o z beng X/00/8(7)/3305/9/$ Amercan Insttute o Physcs Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

3 3306 Phys. Plasmas, Vol. 8, No. 7, July 00. Fülöp and P. Helander numercally larger than unty, as long as t does not become as large as z, or example. Even stronger gradents were recently treated n Re. 3, whch consders Prsch Schlüter transport n a collsonal plasma wthout mpurtes and eectvely adopts the orderng ˆ O(), whch would correspond to z O(z ). hus, whle Re. 3 consders gradents that are so steep that the parallel dynamcs o the bulk ons becomes nonlnear, we are content wth consderng the case o weaker gradents where only heavy mpurtes are nonlnear. Whle even the parameter ˆ s lkely to approach unty n the pedestal regme, t s clear that the parameter z becomes large long beore ˆ does, heavy mpurtes are present. hus the modcatons to neoclasscal theory presented n ths paper are lkely to be sgncant n the plasma edge where mpurtes are abundant. he reason why we cannot allow z to be as large as z s that we choose to keep the aspect rato arbtrary. I z O(z ) and O(), the bulk on densty begns to vary sgncantly over each surace, and the transport becomes mathematcally ntractable. he authors o Re. 3 overcome ths problem by nstead expandng n. Another dculty wth treatng very steep gradents s that the usual separaton n tme scales between parallel and perpendcular transport breaks down when z becomes large enough. When ths occurs, the precse condton or whch was derved n Re. 4, the transport cannot be analyzed on one lux surace at a tme, and thus loses ts local nature. We also restrct our attenton to the case o a hydrogen plasma () wth a sngle speces o hghly charged (z) mpurty ons. Both these speces are taken to be collsonal, as s typcally the case n the edge o hgh-densty dscharges. We evaluate the partcle and heat luxes n two opposte lmts: that o trace mpurtes, Z e n z z /n, and the Lorentz lmt n z z /n. In the rst case the mpurtes do not aect the knetcs o the bulk plasma, whle n the second case the requency o on-mpurty collsons exceeds that o on on collsons. he nonlnear behavor o the mpurtes turns out to be qute smlar n these two lmts. he rest o the paper s organzed as ollows. Secton II s devoted to the analyss o the parallel mpurty dynamcs and the dervaton o the equaton governng the polodal mpurty dstrbuton n the lmts o trace mpurty and large mpurty concentraton, respectvely. Furthermore, the eect o the on electron heat-exchange s dscussed n the trace mpurty lmt. In Sec. III the neoclasscal partcle and heat luxes are calculated, and the conclusons are summarzed n Sec. IV. II. PARALLEL IMPURIY DYNAMICS he polodal mpurty dstrbuton s governed by the parallel momentum equaton, m z n z b V z V z zn z e p z b" I z R z, where bäb/b, s the electrostatc potental, R z s the mpurty-on parallel rcton orce and I z s the vscosty. In Eq. nerta and parallel vscosty can be neglected, beng smaller than the parallel pressure gradent by the actor /(zˆ ). As a consequence o strong on-mpurty temperature equlbraton, the mpurty pressure s equal to p z n z, where s the bulk on temperature, whch s nearly constant on each lux surace; see Re. 4. hese smplcatons reduce Eq. to zn z e n z R z. Snce the mpurtes are hghly charged, ther perpendcular velocty s domnated by the EÃB drt, V z bã /B. As the electrostatc potental s nearly constant on lux suraces, 0 (), whch can be vered a posteror, t ollows rom the mpurty contnuty equaton, (n z V z )0, that there s a parallel mpurty return low equal to V z I d 0 B d K zb, 3 n z where K z () s an ntegraton constant proportonal to the polodal low velocty and BI() s the magnetc eld, so that s the polodal lux. he on-mpurty parallel rcton orce R z can be calculated rom R z m v C z d 3 v, where the subscrpt on the on dstrbuton uncton reers to the orderng n. In lowest order, the on dstrbuton uncton s Maxwellan and constant on lux suraces, 0 0 (), so the next-order correcton,, s needed to evaluate the rcton. Snce the mass rato s large, m z /m, on-mpurty collsons are descrbed by the operator C l z z L (m v / ) V z 0, where z s the onmpurty collson requency, and the Lorentz scatterng operator s dened as L v v B v, wth v /(Bv ). he expresson or the rcton orce s thus smpled to R z m v z m v V z 0 d 3 v. 5 Next, we need to determne the rst-order on dstrbuton uncton, whch then, rom the parallel momentum equaton, Eq., enables us to derve an equaton governng the polodal densty varaton o mpurtes n z (). he on dstrbuton uncton s governed by the drt knetc equaton, v v D 0 ev C C z, 6 whch, n the Prsch Schlüter regme, s solved by an expanson n the mean-ree path parameter ˆ, ollowng Hazeltne. 4 Here, v D v bã (v / ) s the drt velocty and eb/m s the gyrorequency o the bulk ons. 4 Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

4 Phys. Plasmas, Vol. 8, No. 7, July 00 Nonlnear neoclasscal transport n torodal edge plasmas 3307 Beore solvng ths equaton, whch s done separately n the next three subsectons n the lmts o small and large mpurty concentraton, t s useul to nspect ts moments. As n Re. 4, the number and energy moments mply that the parallel partcle and heat luxes must be o the orm n V Ip eb d ln p d e d d K B, 7 and q 5Ip d eb d L B, 8 where the ntegraton constants K () and L () are lux unctons. In the expresson or the heat lux we have assumed that the energy exchange between ons and electrons can be neglected. In Sec. II B we analyze the consequences o nte on electron energy exchange. Energy exchange between the bulk and mpurty ons can be neglected n the orderng we have adopted. A. race mpurtes n a collsonal plasma We rst consder the lmt n z z /n, n whch the mpurtes do not aect the parallel knetcs o the background ons. he background-on dstrbuton uncton can thereore be obtaned by solvng the drt-knetc equaton, v v D ev C, 9 where we have neglected on-mpurty collsons, beng less requent than on on collsons by the actor n z z /n. o order ( ) we have C ( )0, whch mples p m v Ṽ p x 5 0, 0 where the subscrpt and superscrpt on reer to the orderngs n and, respectvely, xv/v wth v ( /m ) /, and the notaton p p,,ṽ V s used or smplcty. o order ( 0 ) we obtan C 0 v ev 0, whch, by multplyng by m v and ntegratng over velocty space, gves p n e 0. It ollows rom Eq. 7 that Ṽ K ()B/n, where K () s an ntegraton constant determnng the polodal rotaton o the plasma. Usng Eqs. and 7 then smples Eq. to C 0 v x 5 0 m v P K ln B 0, 3 where P ()(3 )/ and v /v. Furthermore, we note that the lux-surace average o the m v B-moment o Eq. 9 gves p p Bn eb 0, 4 where denotes lux-surace averagng. Snce ñ /n and e / are both o order, the second term s O( ) and p p B0 5 holds to O(), leadng to the concluson that the P () component o 0 has to vansh. Snce ths component s proportonal to K (), see Eq. 3, t then ollows that K ()0 and we thus have C 0 v x Solvng ths Sptzer problem as n Re. 4 gves 0 hom part, 7 where the homogeneous soluton represents a Maxwellan perturbaton, hom p 8 m v V p whle the partcular soluton, x 5 0, part 5 x 4 x 0v 48 0, 9 where the on collson tme s dened as 3/ 0 m / 3/ /n e 4 ln, carres the heat lux, q 5 3 n m. 0 Combnng ths equaton wth Eq. 8 gves 6 I d ln 5 d b, where the magnetc eld strength has been normalzed so that bb/b /. Insertng the results 7, 8 and 9 n Eq. 5 gves R z m n z V V z 5 6 m, where z (n /n z z ), so that z 3 / /4 z x 3. he parallel velocty V o the bulk ons can be determned by calculatng the parallel vscosty assocated wth, whch s obtaned by gong to the next order n. hs calculaton s presented n Re. 4 and leads to the result V V z I m d ln p d d ln d.8b 0.05 b ln b K zb, 3 b n z whch s not aected by the presence o trace mpurtes. Also, Eq. agrees wth the correspondng expresson n Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

5 3308 Phys. Plasmas, Vol. 8, No. 7, July 00. Fülöp and P. Helander Re. 9. he only derences between the present case and the conventonal theory o Prsch Schlüter transport n an mpure plasma s that the mpurty densty n z vares over the lux surace and the polodal mpurty rotaton K z s derent, as we shall see shortly. Returnng to the parallel momentum equaton that governs the polodal mpurty dstrbuton, we note that Eq. mples that the on temperature ndeed vares less over the lux surace than the magnetc eld strength, as assumed n Eq.. he polodal electrc eld,, n Eq. s determned by quasneutralty, zn z e zn z e e 6 5z e e Ip z d ln d b, 4 and s O(z ) smaller than the rcton orce. Usng these results n Eq. then gves n m n I e z n z B" n d ln p d.8nb d ln eb K d In z z, d ln d 5 where the mpurty densty and the polodal angle have been normalzed, nn z /n z and d/db" /B", and the term nvolvng the small coecent 0.05 n Eq. 3 has been neglected. Integratng Eq. 5 over allows us to determne the polodal mpurty low, K z In z d ln n B e d.8nb d ln d, 6 and the equaton or the polodal dstrbuton o the normalzed mpurty densty thus nally becomes n gnb nnb b. 7 Here.8(ln )/(ln n ) s a constant, the prme denotes a dervatve wth respect to and g m n I e z n z B" d ln p d d ln d O z measures the steepness o the bulk on densty and temperature proles. Equaton 7 has exactly the same orm as was ound n the mxed-collsonalty regme, 4 where the bulk ons and electrons are collsonless and the mpurtes collsonal. Only the constants g and are derent. As n that regme, the mpurtes accumulate on the nboard sde o the tokamak when the gradents are steep, g, thereby reducng the mpurty-on parallel rcton. Indeed, n the lmt when the pressure or temperature gradents are so large that g we can expand the soluton o Eq. 7 n g : nn 0 n O(g ), gvng the ollowng expresson to lowest order: n 0 b b b, 8 whch ndcates that the mpurtes are pushed toward the nboard sde o the torus where b s large. Conventonal neoclasscal theory assumes g and thereore neglects these eects. B. Ion electron heat exchange I the collsonalty s hgh, on electron energy transer may nluence the on dynamcs, as ponted out by Engelmann and Nocentn. 0 Startng rom the on drt knetc equaton and expandng n the usual way n the smallness o the mean-ree path, we can nclude the on electron energy transer by treatng C e /C as O(). hen, to lowest order we have C 0, 9 and to rst order we have v ev C 0 C e, 30 nstead o Eq.. Snce C e ( ) s even n v, the odd part o 0 s gven by 0 odd v I d ln 5 d b x 4 x 0 0 m v V 0, 3 as obtaned n conventonal theory, see Eq. 9. he (m v /5 /) moment o the drt knetc equaton becomes B 5Ip d eb d 3m e m e B e, 3 where 3.9 /m s the heat conductvty correspondng to Eq. 0. Snce the correspondng conductvty or electrons s hgher by a square root o the mass rato m /m e, the electron temperature e on the rght o Eq. 3 s practcally constant over the lux surace. For consstency, e must be equal to the lux-surace average o the on temperature, as ollows rom the average o Eq. 3 unless an on heat source or snk s also ncluded n ths equaton. o solve the equaton explctly, t s necessary to know the magnetc equlbrum. I the rght-hand sde s neglected one obtans the usual Prsch Schlüter soluton, whle the coecent on the rght hand sde s large, then s small and the energy transport s suppressed. For a standard, crcular, large-aspect-rato equlbrum Eq. 3 becomes qrb wth the result 5Ip eb cos 3m e qr m e B, 33 Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

6 Phys. Plasmas, Vol. 8, No. 7, July 00 Nonlnear neoclasscal transport n torodal edge plasmas Ip /eb 0 d /d sn, /qrb 3m e qr/m e B 34 and t s clear that s smply reduced by a constant actor, 3m e m q R e R m e m / e 3/, 35 where v. hus the on electron heat exchange has a neglgble eect on the plasma transport unless R/ (m /m e ) /4 ( e / ) 3/4. Unless, the polodal dstrbuton o mpurtes wll be aected, snce s reduced by the actor, whch n turn aects the controllng parameters g and o Eq. 7 n the ollowng way: and g m n I e z n z B" d ln p d d ln d 36.8/ln ln p ln /. 37 In the tokamak edge, the densty prole s typcally about twce as steep as the temperature prole, and t appears that the eect o on electron energy exchange does not aect the mpurty dynamcs very much. Even, so that the parameter g grows by a hal and alls to about a hal compared to the case when, ths has no dramatc eect on the mpurtes. he numercal soluton o 7 shows that, as s reduced, the magntude o the n out asymmetry grows. As we shall see n the next secton, ths n out asymmetry s assocated wth reduced neoclasscal transport, and ths eect s thus somewhat enhanced n the very hgh collsonalty regme where on-electron energy equlbraton occurs. C. Lorentz lmt: Z e š In the Lorentz lmt n z z /n, mpurty on collsons are more requent than on on collsons, C C z, and thereore sgncantly aect the bulk on low. he soluton o the drt-knetc equaton, v v D C C z C z, 38 s then the same as Eq. 0 n the order, but n the order 0 we obtan v ev 0 z L 0 m v V z he soluton o Eq. 39 can be wrtten as 0 m v V z 0 4 z 3 v 3 x p e p x Usng parallel on momentum conservaton, the on-mpurty rcton orce can be wrtten as R z p n e, 4 where we need to express the parallel gradents as unctons o radal gradents. he partcle and heat luxes assocated wth 0 are gvng n V q / v x 5/ 0 d 3 v, 4 n V V z q / p z m l l l l a a, 43 wth l l l l 3/3 6, /3 and where a p /p e / and a / are the usual thermodynamc orces. Combnng Eqs. 7 and 3 gves Bnn V V z 0 Ip e whch determnes K (): so that d ln p d K K zn n z nb, 45 K nb K zn n z Ip d ln p e d, 46 n V V z Ip d ln p eb d b nb K zbn n z nb n. 47 From Eq. 43 t ollows that Bnq 0, and usng Eq. 8, we can determne L (): L nb 5I d e d, so that the parallel heat lux becomes q 5Ip d eb d b. nb Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

7 330 Phys. Plasmas, Vol. 8, No. 7, July 00. Fülöp and P. Helander Usng Eqs. 43, 47 and 49 we can express the thermodynamc orces as n V V z, 50 a a m p z l l l l q / so that a / can be wrtten as a uncton o the radal gradents: 9 56 I b nb he on-mpurty rcton orce s z d ln n d 3 d ln d K zbm z n z nb n. 5 R z p a 3 3 m n V V z 6 z n, so usng Eqs. 50 and 5 we arrve at R z 75 m n 5 z I eb b nb d ln p d 3 5 d ln d K zb n z nb n he remanng unknown quantty K z governs the polodal mpurty rotaton and s determned rom the parallel vscosty constrant, B" I B mv P d v 3 0, 54 where P () denotes the Legendre polynomal P () (3 )/, wth II I z. he mpurty vscosty s unmportant, snce z n zz n z z 3/, 55 n z z n whch smples the calculaton. he P component o s obtaned rom the next order n o the knetc equaton, Employng ths result n the parallel vscosty constrant 54 then gves the ollowng expresson or the polodal mpurty low, K z In z d ln p B e d d ln d. 58 Fnally, nsertng these results n Eq. and usng quasneutralty, n e n zn z, gves the equaton or the polodal mpurty dstrbuton, n n ĝ n ˆ nb b ˆ b, 59 where Z e e /( e ), ˆ (lnp ) (ln ) / (lnp )(3/5)(ln ) and ĝ 75 5 m n I e z n z B" d ln p d 3 5 d ln d O z. When the gradents become very large so that ĝ, the square bracket on the rght-hand sde o Eq. 59 must vansh, so that the mpurty dstrbuton becomes ˆ b n ˆ /nb b, 60 whch mples that the mpurtes agan accumulate on the nboard sde o the lux surace. hs s the same basc behavor that we ound n the lmt o trace mpurtes at the end o Sec. II A, and n the banana regme nvestgated n Res. 4 and 5. D. Polodal mpurty dstrbuton he equatons governng the polodal dstrbuton o mpurtes over each lux surace are gven by Eqs. 7 and 59 n the lmts o small and large mpurty concentraton, respectvely. hese equatons are complcated ntegroderental equatons that are not amenable to analytcal soluton except n certan smple lmts. One such lmt s that o very steep radal plasma proles, where the solutons are gven by 8 and 60 and predct the accumulaton o mpurty ons on the nboard sde o the torus. Another tractable v 0 v D 0 C z L. 56 Usng L( ) P 3 P, where ( ) P denotes the P -component o ( ), Eq. 56 gves or ths component P v 9 P 0 I d ln p d x 5/ d ln d x3 ln B 4 zn z a x 5/a ln B 3 n z a x 5/a n z m K z B n z ln B B n z. 57 FIG.. Magnetc reconstructon o a lux surace close to the edge n SAR. Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

8 Phys. Plasmas, Vol. 8, No. 7, July 00 Nonlnear neoclasscal transport n torodal edge plasmas 33 FIG.. Flux surace close to the edge (r/a0.9) n the Alcator C-Mod. Fgs. and. he results are shown n Fgs. 3 4, and conrm the qualtatve behavor expressed by the analytcal solutons. An up down asymmetry orms when the radal gradents are moderate, go(), and turns nto an n-out asymmetry or larger gradents, g, rrespectve o the mpurty concentraton. In the Alcator C-Mod, an up down asymmetry s ndeed observed n the lne radaton rom heavy mpurtes near the plasma edge. Qualtatvely, these observatons agree wth our predctons: the asymmetry has the correct sgn and s reversed the torodal magnetc eld s reversed. However, the observed asymmetres are much larger than those calculated here. hs may be due to the act that the parameter g s extremely large n these experments whle we have ormally treated t as O. A more extreme orderng, e.g., go(z), may descrbe the Alcator C-Mod experments better. lmt s that o a large aspect rato and crcular cross secton, n whch case Eqs. 7 and 59 can be solved by makng the expansons b cos O( ), nn c cos n s sn O( ). he solutons then become g n s g, 6 n c g g, 6 n the lmt o trace mpurtes,, and n the Lorentz lmt g n s ˆ g, 63 ˆ g n c ˆ g, 64 where gĝ/(). hese expressons ndcate that as the gradents represented by g and ĝ are made successvely larger, the mpurtes rst develop an up down asymmetry proportonal to g or ĝ and then an n out asymmetry proportonal to. hs agrees completely wth the behavor ound or the banana regme n Res. 4 and 5. In the general case, Eqs. 7 and 59 must be solved by numercal means. We have done so or typcal equlbra n two tokamaks: the Small ght-aspect-rato okamak SAR 5 and the Alcator C-Mod, or whch magnetc reconstructons o magnetc suraces n the edge are shown n III. NEOCLASSICAL RANSPOR We now proceed to evaluate the classcal and neoclasscal partcle luxes o the bulk ons, cl neo R R z R z eb. Usng the rcton orce R z rom Eq. 4 we obtan the average neoclasscal partcle lux across a lux surace, neo Ip zb n n eb b. 65 Evaluatng ths expresson n the two opposte lmts, we obtan neo trace Ip zb" eb n the case o trace mpurtes, and n b nb g neo Lorentz Ip zb" n eb b nb 66 ˆ 67 nb ĝ, n the Lorentz lmt. I the gradents are weak, so that g and ĝ are both small and n, these expressons reduce to the FIG. 3. Calculated mpurty densty vs polodal angle n SAR at the normalzed radus r/a0.9; the mpurty strengths are 0.0 and 5, respectvely. Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

9 33 Phys. Plasmas, Vol. 8, No. 7, July 00. Fülöp and P. Helander FIG. 4. Calculated polodal varaton o mpurty densty n the Alcator C-Mod at r/a0.9 and mpurty strength 0.0 and 5, respectvely. usual ormula or the cross-eld partcle lux n the Prsch Schlüter regme, 9 contanng the characterstc actor b. A. Large aspect rato In a plasma wth small nverse aspect rato and crcular cross secton, we may use the results dsplayed n Eqs or the mpurty dstrbuton. he partcle lux then becomes cl neo trace p z q q 3 e g / g and cl neo Lorentz 68 p z q q 3 e ĝ ˆ / ĝ. 69 In these expressons, the rst term on the rght s the classcal lux, whch s not much aected by the mpurty redstrbuton. he second term, whch nvolves the Prsch Schlüter actor q, represents the neoclasscal lux. hs lux s suppressed the pressure gradent becomes sucently steep snce the denomnator n the second term o Eqs. 68 and 69 depends quadratcally on g and ĝ, respectvely. As ound earler n the banana regme, 4,5 the classcal transport then domnates, and the total lux s a nonmonotonc uncton o the gradents. hs concluson s thus not aected by the collsonalty. In actual magnetc equlbra, the partcle lux s not necessarly nonmonotonc although the neoclasscal lux s suppressed when the radal pressure gradent s large. Fgures 5 and 6 show partcle luxes n SAR and Alcator C-Mod as unctons o the gradents, usng the numercal soluton or the mpurty dstrbuton rom the prevous secton. Note that these partcle luxes are nonlnear but monotonc. B. Heat lux he heat lux can be calculated n a smlar way by computng the heat rcton, H m v x 5 C 0 d 3 v. 70 In the case o trace mpurtes, the collson operator s domnated by on on collsons, and collsons wth mpurtes thereore hardly contrbute to the heat lux. In the Lorentz lmt, however, on-mpurty collsons are domnant, and the heat lux becomes q neo IH z eb where d ln p d 5 d ln 3 d n b K zeb I n z nb, 45 I p m n z. 5 e B z n z nb 7 7 FIG. 5. Ion partcle lux vs the normalzed gradent g n SAR dscharge No , n the Lorentz lmt, wth the mpurty strength 5. he dashed lne s the classcal lux, the dotted lne s the neoclasscal lux, and the sold lne s the sum o classcal and neoclasscal luxes. he neoclasscal lux s suppressed by large gradents, but the total lux s stll monotonc. FIG. 6. he same as Fg. 5 but or Alcator C-Mod dscharge No Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see

10 Phys. Plasmas, Vol. 8, No. 7, July 00 Nonlnear neoclasscal transport n torodal edge plasmas 333 FIG. 7. Neoclasscal heat lux as unctons o the gradent g n a largeaspect-rato tokamak wth a crcular cross secton. he mpurty strength s 5 and t was assumed that n /n /. Note the sgn-reversal or large gradents. Usng Eq. 58 or K z here gves the ollowng result: q neo d ln p d n b d ln d 5 3 n b 8 3nb. 73 Agan, n the absence o large gradents (g) the mpurtes are evenly dstrbuted on each lux surace (n ), so that both terms n Eq. 7 become proportonal to the Prsch Schlüter actor b, wth coecents n agreement wth Re. 9. For large gradents (g), on the other hand, the polodal redstrbuton o mpurtes aects the heat lux quanttatvely and can even reverse ts sgn under typcal condtons. Snce the densty prole n the tokamak edge s usually steeper than the temperature gradent, the neoclasscal heat lux calculated by conventonal theory s oten nward. Fgure 7 llustrates the neoclasscal heat lux as a uncton o the gradents or the case where n /n /. For small radal gradents the heat lux s nward but reverses when the radal gradents become sucently steep. hs s smlar to the behavor earler ound n the banana regme. 5 IV. CONCLUSIONS In ths paper we have nvestgated the character o collsonal transport o an mpure plasma where the pressure and temperature gradents are so large that the mpurty dynamcs s nonlnear. hs extends earler work, where the bulk ons were nstead taken to be collsonless, 4,5 to the Prsch Schlüter regme more characterstc o cool, dense plasmas. he conclusons reman broadly the same: heavy mpurty ons accumulate on the nboard sde o each lux surace, and ths reduces neoclasscal transport. In addton, they develop an up down asymmetry, whch has the same qualtatve eatures as that measured n the Alcator C-Mod. he observed asymmetry s however larger than the analytcal predcton, whch may have to do wth the extreme steepness o the gradents n the experments. Fnally, the energy exchange between the bulk ons and electrons, whch s known to sgncantly aect the on heat lux n a pure plasma, 0 does not strongly nluence the mpurty dynamcs. ACKNOWLEDGMENS hs work was supported by the U.K. Department o rade and Industry, and by Euratom under assocaton contracts wth Sweden and the U.K. M. Kotschenreuther, W. Dorland, M. A. Beer, and G. W. Hammett, Phys. Plasmas, 38995; A. M. Dmts, G. Bateman, M. A. Beer et al., bd. 7, J. E. Rce, J. L. erry, E. S. Marmar, and F. Bombarda, Nucl. Fuson 37, S. Pedersen, R. S. Granetz, A. Hubbard, E. Marmar, D. Mossessan, J. Hughes, I. H. Hutchnson, J. E. Rce, and J. erry, Proceedngs o the 7rd EPS Conerence on Plasma Physcs and Controlled Fuson, Budapest, Hungary, 6 June 000, edted by K. Szegö,. N. odd, and S. Zoletnk European Physcal Socety, Geneva, P. Helander, Phys. Plasmas 5, Fülöp and P. Helander, Phys. Plasmas 6, C.. Hsu and D. J. Sgmar, Plasma Phys. Controlled Fuson 3, F. L. Hnton and R. D. Hazeltne, Rev. Mod. Phys. 48, S. P. Hrshman and D. J. Sgmar, Nucl. Fuson, P. H. Rutherord, Phys. Fluds 7, F. Engelmann and A. Nocentn, Nucl. Fuson 7, K. H. Burrell and S. K. Wong, Nucl. Fuson 9, M. endler, n Plasma Physcs and Controlled Fuson Research, Proceedngs o the 8th Internatonal Conerence on Plasma Physcs and Controlled Fuson Research, Brussels Internatonal Atomc Energy Agency, Venna, 98, Vol., p H. A. Claassen, H. Gerhauser, A. Rogster, and C. Yarm, Phys. Plasmas 7, R. D. Hazeltne, Phys. Fluds 7, M. Gryaznevch, R. Akers, P. G. Carolan et al., Phys. Rev. Lett. 80, Downloaded 08 Aug 0 to Redstrbuton subject to AIP lcense or copyrght; see