Accretion Theory of Spontaneous Rotation in Toroidal Plasmas

Transcription

1 B. Co Accrton Thory of Sontanous Rotaton n Torodal Plasmas Massachustts Insttut of Tchnology, Cambrdg, MA, USA -mal: COPPI@MIT.EU Abstract. Th accrton thory of sontanous torodal rotaton conncts ths hnomnon to th nrgy and artcl transort rorts of th lasma column and to th rlant collct mods. Th consqunt rdcton that an nrson of th locty drcton n th transton from th H to th L rgm should occur has bn rfd by th xrmnts. Th thory s consstnt also wth th obsraton that th locty s drssd whn a ad dnsty rofl aars as a rsult of a transort barrr. Th fact that th rotaton locty ncrass wth th total nrgy contnt s xland by th fact that th nflow of angular momntum, whos sourc s at th dg of th lasma column, rsults from th xctaton of mods drn by th lasma rssur gradnt. A quas-lnar draton of th angular momntum transort roducd by ths mods, whos nol fatur s th nflow, s gn and a modl of th rlant (transort) quaton s sold. Fluctuatons at th dg of th lasma column ar consdrd rsonsbl for th scattrng, out of confnmnt, of artcls that transfr, to th surroundng matral wall, angular momntum n th sam drcton as that of th has locty of th ralnt mods. Thus th fact that th lasma rotats n th drcton of th on damagntc locty n th H-rgm, whn th ralnt mods ar xctd to ha has locty n th drcton of th lctron damagntc locty, can b xland and th rat of rotaton dcras, as th lasma currnt s ncrasd, can b justfd. 1. Ln Btwn Sontanous Rotaton and Transort Th accrton thory [1] of th sontanous rotaton hnomnon [] obsrd n axsymmtrc torodal lasma xrmnts has td ths hnomnon to th transort rorts of th lasma column and to th mods rsonsbl for thm. In artcular, th consqunt rdcton that th rotaton locty should nrt whn a drastc chang n th confnmnt charactrstcs s nducd as n th transton from th so-calld L-rgm to th H-rgm, has bn confrmd by th xrmnts [3,4]. Th mor rcnt obsraton [5] of a sgnfcant chang n th locty of rotaton corrsondng to th onst of a transort barrr n th man body of th lasma column s anothr confrmaton of th sam fatur. A y xrmntal fndng s that sontanous rotaton s not assocatd wth th njcton of RF was [3] as ths s sn both n th H-rgm and n th L- rgm whn only Ohmc hatng s rsnt (Fg.1). Thus th nflunc of Ion Cyclotron hatng on rotaton s ndrct n that th lasma thrmal nrgy and consquntly th on rssur gradnt s ncrasd by ths addtonal hatng. In fact th accrton thory whch assocats th rotaton wth ths gradnt s consstnt wth th obsraton that th rat of rotaton φ W th /I whr W th s th total lasma thrmal nrgy and I s th lasma currnt.. Flow of Angular Momntum In th contxt of th accrton thory, th sourc of th angular momntum acqurd by th cntral art of th lasma column s at th outr dg of t [6]. Thrfor a rocss for angular momntum nflow ( nch ) has to b ntroducd and assocatd wth collct mods that ar drn by th lasma rssur gradnt. Th transfr of oost angular momntum to th matral wall surroundng th lasma s consdrd to balanc that acqurd by th lasma column. In artcular, artcls ntractng wth mods localzd at th dg of ths can b scattrd to th wall transfrrng, to t, angular momntum n th sam drcton as th mod has locty. Whn th lasma s n th H rgm a strong artcl dnsty gradnt s formd at th dg of th lasma column and t s rasonabl to assum that on tmratur gradnt drn mods, whch ha a has locty n th drcton of th on damagntc locty, ar not xctd. Th strong dnsty gradnt on th othr hand faors th onst of radally localzd mods whos frquncy

2 s roortonal to th lctron drft mod frquncy that s roortonal to th dnsty gradnt and dcrass wth th olodal fld B I a, a bng th lasma mnor radus. Rfrrng to th smlst, larg asct rato torodal confguraton, th rlant lctrc fld fluctuatons can b rrsntd as Eˆ E( r,ϑ) cos ( ω m ϑ + n φ) whr ϑ and φ ar th olodal and torodal angl, rsctly. Th torodal has locty s ω R n, R bng th torus major radus. In artcular, ω R n ct /( B ) d ln n / dr. Thus, whn th lasma s n th H-rgm, th / man body of t should acqur angular momntum n th drcton oost to that of, that s n th drcton of th on damagntc locty, a fatur that s clarly sn n th xrmnts. W not that, th xstnc of an act rgon wth rlatly strong fluctuatons [7] nar th outr dg of th lasma column, and th rsnc of a st dnsty gradnt, has bn rfd n xrmnts by th Alcator C-Mod machn, n th H-mod rgm. In artcular a mod whos has locty has a drcton and alu consstnt wth * has bn dntfd [8]. W also not that th obsraton that φ dcrass as B ncrass s consstnt wth th fact that th radal xcurson of th sngl artcl orbts dcrass as B ncrass as wll as wth th fact that th mod amltud can b xctd to b corrlatd wth th alu of *. Followng th sam ln of rasonng w argu that whn th lasma column s n th L-rgm th hgh alus that th thrmal dffusty acqurs at th dg of th lasma column ar th rsult of th xctaton of on tmratur gradnt (ITG) drn mods. Th has locty of ths mods of whch thr ar two nds, on that can b found n a on dmnsonal gomtry [9,1] and on that dnds on th rsnc of a torodal curatur [11], s n th drcton of th on damagntc locty = c /( B n) d dr. W consdr that th ffct of ths mods s / ralnt at th dg of th lasma column and that n ths rgon artcls ntractng wth ths mods, ar thn scattrd to th wall. Thn th man body of th lasma column wll rotat n th oost drcton of, that of th lctron damagntc locty. Thus a rrsal of th torodal locty of th cntral art of th lasma column should b obsrd n th transton from th L-rgm to th H-rgm. Ths has bn clarly sn n xrmnts wth ohmc hatng only carrd out by th Alcator C-Mod machn [3]. Th sam rrsal has bn obsrd clarly n xrmnts whr ICRH hatng s ralnt by th Tor Sura machn [4] n th transton from th L-mod rgm to an nhancd confnmnt rgm wth ad dnsty rofls. Th alus of th locty n th ohmc H-rgm roducd by Alcator C-Mod ha bn n th rang 3-8 m/sc. 3. Modl Transort Equaton Th xrmnts ndcat clarly that th transort of angular momntum s fastr than rdctd by th collsonal transort thory, whn th sourc of hatng that mantans th lasma thrmal nrgy s sharly dcrasd []. Anothr xrmntal ndcaton s that th torodal locty xtnds to th cntr of th lasma column wth rofls that ar ad [] or that ar flat [5] locally. Consdrng that, n th accrton thory, th sourc of angular momntum s nar th dg of th lasma column, a smlfd transort modl can b adotd, to dscrb som of th obsrd faturs, that s smlar to th wll nown on ntroducd arlr [1] for th artcl transort. Th lattr rroducs th obsrd cntrally ad dnsty rofls whn a sourc s rsnt at th dg of th lasma column (.g., du to gas njcton). Th smlfd modl balancs an n-

3 flow locty of th scfc angular momntum dnsty m n, whr = Rφ, and an outward dffuson, n th ntror of th lasma column, as xmlfd by a flux Γ of th form Γ mn r + r (1) to b usd wthn th consraton quaton mn + Γ = S () t whr S s a sourc localzd at th dg. Th rato s tan to b an ncrasng functon of th radus n th man body of th lasma column, as n th cas of th artcl transort modl quaton, and n artcular r α (3) a whr α const. nar th cntr of th lasma column. Clarly, whn α 1 th statonary rofl rsultng from Γ n ths rgon, ( 1 α r / a ), s ad, whl f α s small th rofl s flat. In ordr to smulat th nds of rofls that ha bn obsrd xrmntally, Eq. () has bn sold numrcally (by I. mo and S. Kurbayash) usng an ad hoc sourc of angular momntum localzd at r = a for dffrnt alus of α and modls for (Fg.) consdrng that transort analyss of wll confnd lasmas ndcat that th rlant thrmal coffcnts ar strongly ncrasng functons of th lasma radus. 4. Quas-lnar Thory Hr w dmonstrat that lctrostatc mods drn by th on rssur gradnt can roduc an nflow locty of angular momntum and fnd an xrsson for Γ from quas-lnar thory that can lnd suort to Eq. (). For th sa of smlcty w rfr to a on dmnsonal lan gomtry whr th magntc fld s along th drcton. A flow locty x, along th magntc fld s rsnt and s smallr that th on thrmal locty th. W not that, n th cas of dutrons, th 31 m/sc ( T / 1 V) 1/. Snc th maxmum obsrd flow locty n th Alcator C-Mod xrmnts dos not xcd 1 m/sc n th cntr of th lasma column whr T s hghr than 1 V, ths s wll blow th. W consdr lctrostatc mods, rrsntd by ˆ φ φ ( x ) x( ωt + y y + z), that ar localzd around a surfac x = x and dfn th olr shftd frquncy ϖ ω ( x ). For ϖ > th th rturbd longtudnal momntum consraton quaton s m ( ϖ dx) n + d = nφ, whr = E yc / B = yc φ B and ϖ d / dx. In artcular, w consdr ϖ d dx d dx mn ( ) ( ) and d dx c ( Bn) > ϖ y. Thn y c d d 1 φ + ϖ B dx ϖ mn dx W not that th locty gradnt trm dx s a dstablzng factor for mods n th frquncy rang d th < ϖ < th whn [13,14] y d z. (4) ( ) z dx > and y < s th sgn that w

4 consdr rfrrng to th cas whr d / dx < for > as ths als to th narly statonary locty rofls stablshd wthn th lasma column. In artcular, f w rfr to mods for whch sgn ϖ = sgn y such as ϖ y* whr * ct ( B) d ln n dx, w s that th trm roortonal to d dx n Eq. (4) s of oost sgn to that of dx. To llustrat ths ont furthr w not that th rturbd on gudng cntr consraton quaton s n t + dn dx + nˆ + nˆ and w nclud [13,14] a ry small dssat trm ( ) ˆ = n Eq. (4) rrsntng, for nstanc, th ffcts of fnt longtudnal scosty. Thn f w ta nˆ nˆ and, snc ϖ < th, n n ˆ(1 φ ε ) T whr ε << 1 tas nto account, gnrcally, dffrnt factors ncludng nonlnar ntractons that can dr th consdrd mod unstabl. Th rsultng dsrson rlaton s ct d d ϖ 1 ϖ (1 ε ) y* y + (5) B ϖ dx ϖ mn dx and w argu that condton y d d dx > s th on for whch th mod s asr to xct. ( ) ( ) ( ) ( ) ˆ Th rlant quaslnar flux Γ = m n ˆ ˆ + ˆ whn usng, for smlcty, Eq. (4) s Γ ϖ d mn + dx ϖ γ () ˆ d dx. (6) Hr γ = Imϖ s th mod growth rat and not that ( γ ϖ ) s th famlar quaslnar dffuson coffcnt. W s that th trm d dx s rsonsbl for th rlant nflow. Snc, n th cas w consdr, th sourc of th momntum s at th dg of th lasma column, w argu that d dx drs th locty gradnt d dx and s Γ rlatly small. W consdr th factor rsonsbl for mantanng th mod amltud to b th lasma rssur gradnt whch, n addton, drs ( ) th rlant thrmal conductty. Thus both γ and th mod amltud dnd on ths factor. Not that th nflow trm ( ϖn) d dx s about ( y ) ( B c) [ T T + ( dt dx) ( T d ln n dx) ]. Thus, n th rocss, dx should ncras as th on tmratur gradnt s ncrasd. Ths s d consstnt also wth th xrmntal obsratons that th rotaton locty ncrass whn, at constant dnsty, th lasma total thrmal nrgy s ncrasd. Th xrmntal obsraton [5] that th rotaton locty dcrass as a ad dnsty rofl s roducd n th cntral art of th lasma column, whn a transort barrr s formd that dos not chang th on tmratur nor ts rofl, s also consstnt wth th fact that th nflow trm bcoms drssd whn d ln n / dx ncrass. Consdrng th lmt whr η d ln T d ln n > ω /( th ) > 1 th dsrson rlaton (5) ylds also th on tmratur gradnt drn mod, modfd by th rsnc of d /dx. Th contrbuton of ths to th sctrum of mods that can b xctd s mortant. In artcular, n th rang of walngths whr th mod frquncy s narly ral and ϖ / ( d / dx) ( d / dx) / ( mn), (tang > so that ω / < whl y > ), th ffct of ν s suffcnt to mantan th mod unstabl. An stmat of th ffct dffuson coffcnt for th angular momntum can b dducd from xrmnts n whch th njctd ff hatng s turnd off, thus forcng th rotaton locty to dcras. Th dcay tm rortd n ( )

5 Rf. [] for Alcator-C Mod s τ 7 m/sc and for a rough stmat w may ta ff a κ ( 4τ ) whr κ s th lasma longaton. Thus, f w ta a. m and κ 1. 6, ff.3 m /sc and th dnsty fluctuaton ll that can roduc ff s rlatly low. It s a lasur to than.e. Rc for hs commnts and nsghts, L.E. Sugyama for hr gudanc n th numrcal soluton of Eq. (), I. mo and S. Kurbayash for thr analyss of t, and N. Attco for hs tmly and aluabl suggstons. Ths wor was sonsord by th U.S. artmnt of Enrgy. Rfrncs [1] B. Co, Intrnatonal Shrwood Fuson Thory Mtng, Par C (Publ. Un. Calforna, Los Angls, CA, March ) and arlr rsntatons. [].E. Rc t al., Nuclar Fuson 39 (1999) [3].E. Rc, Bull. Am. Phys. Soc. 44 (1999). [4] G. T. Hoang t al., Nuclar Fuson 4 () 9. [5].E. Rc t al., n Fuson Enrgy, Par IAEA-CN-77 EXP 5/4 (IAEA, Vnna, to b ublshd). [6] B. Co, G. Pnn and L.E. Sugyama, n Fuson Enrgy 1998 (IAEA, Vnna, ), Par IAEA-F1-CN-69-Th3/7. [7]. Sns, Prat Communcaton [8] A. Mazurno t al., Bull. Am. Phys. Soc. 44 (1999) 57. [9] L. I. Rudao and R.Z. Sagd, Sot Phys.-olady 6 (1961) 415. [1] B. Co, M.N. Rosnbluth and R.Z. Sagd, Phys. Fluds 1 (1967) 58. [11] B. Co and F. Pgoraro, Nucl. Fuson 17 (1977) 969. [1] B. Co and N. Shary, Nucl. Fuson 1 (1981) [13] B. Co, M.I.T. Rort PTP 89/, RLE, Cambrdg MA (1989), and B. Basu and B.Co,. Gohys. Rs. 34 (1989) [14] B. Co, Phys. Ltt. A, 1 (1) (1995) 66. FIGURE 1. rmntal dnc of torodal rotaton n th absnc of njctd hatng (courtsy of.e. Rc, ). FIGURE. Scfc angular momntum Ĵ rofls drd from th soluton of Eq. (1) for dffrnt alus of th nward transort locty rlat to th rlant dffuson coffcnt as rrsntd by th aramtr α ntroducd n Eq. (3). Th ad locty rofl corrsonds to a largr alu of α.