Neoclassical transport processes in weakly collisional plasmas with fractured velocity distribution functions

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1 Neolaial tanpot poee in weakly olliional plama with fatued veloity ditibution funtion AA KAANTSEV, F DRISOLL, DHE DUIN and YA TSIDULKO Univeity of alifonia at San Diego, La Jolla, A 993, USA udke Intitute of Nulea Phyi, Novoibik 639, Ruian Fedeation Abtat Ripple in magneti o eletotati onfinement field give ie to tapping epaatie, and onventional neolaial tanpot theoy deibe the olliional tapping/detapping of patile with fatued ditibution funtion Ou expeiment and novel theoy have now haateized a new kind of neolaial tanpot poee aiing fom haoti (nominally olliionle) epaatix oing, whih ou due to E plama otation along uffled o wave-petubed epaatie Thi haoti neolaial tanpot beome dominant at low olliionality when the olliional peading of patile enegy duing the dynamial peiod i le than the epaatix enegy uffle PAS: 55Fi, 57Aj Intodution All laboatoy devie fo plama onfinement have ome level of tapping epaatie, aiing fom vaiation in magneti field tength and extenal potential The loally-tapped and paing population of patile may then have ignifiantly diffeent dift obit, (o alled neolaial effet), giving ie to lage diipative dynamial tep when epaatix oing ou Neolaial tanpot (NT) theoy analyze the patile tanpot and wave damping effet aiing fom olliional atteing (at ate ) ao a igidly unifom epaatix in a vaiety of geometie [-4], and expeimental ooboation ha been obtained in ome egime of a modeate olliionality [5, 6] Hee, we deibe the enhaned tanpot whih ou when the ipple epaatix i itelf uffled in the ( ) dietion of plama dift, o when the epaatix flutuate due to peene of wave/tubulene in the plama In uh ae the difting patile ee a time-vaying epaatix baie, and they an haotially tanit between loally-tapped and paing obit nea the epaatix A a eult, the NT ate doe not depend dietly on the olliion fequeny Howeve, olliion ae till impotant hee in the ene that thei tohati input ha to be tong enough fo a patile pefoming it next epaatix oing to foget about the peviou one (making oneutive oing tatitially independent) In ou expeiment with uffled epaatie, thee haoti oing lead to onideably enhaned neolaial ipple tanpot diven by a global magneti tilt aymmety The expeiment utilize extenally ontolled eletotati uffle o flutuation on the epaatix, and an thu identify the novel haoti neolaial ipple tanpot aling a / /, and thu ditint fom olliional neolaial ipple tanpot aling a Thi haoti tanpot mehanim an ubtantially modify patile tanpot in low olliionality egime aoiated with fuion plama, though it ha peviouly been onideed to be ineffetive due to a peumed ymmety of uh tanition [7] Indeed, one hould ealitially expet tong tanpot even with nea-zeo olliionality Expeimental etup The expeiment utilize a ylindial Penning-Malmbeg tap to onfine quieent, low-olliionality pue eleton plama [8-] Eleton ae onfined adially by a nominally unifom axial magneti field z ˆ, with The 4 T ; and ae onfined axially by voltage V V on end ylinde of adiu R 35m 7 3 eleton olumn have length L 49m, and adial denity pofile n () with ental denity n 6 m 7 and line denity NL Rpn 6 m The unneutalized hage eult in an equilibium potential enegy e() with e 8eV at (hee, all ' ae in enegy unit) Thi give an E otation fequeny fe() whih deeae monotonially fom fe 3kHz ( /kg) The bulk eleton have a nea-maxwellian veloity ditibution FM ( E) with themal enegy T ev, giving axial boune fequeny fb / L 43kHz and igidity paamete R f / f ( /kg) Expeiment ae done in ultahigh vauum having a bae peue of b E to, o the eleton-neutal olliion ate / e eleton-eleton olliion ate / e en i negligible in ompaion with W

2 + queeze eto e tapped paing tapped e V Vq Vm o( m ) o V tilt z Figue Shemati of eleton plama with tilt and a tapping baie in a ylindial Penning-Malmbeg tap An eletotati tapping baie (epaatix) (, ) i eated by a queeze wall voltage V q (ee Fig) Hee, we inlude the effet of adjutable -eto (uffle) voltage (, ) ( ) o[ m ( )] Hee we fou motly on z 4m nea the V, giving ontollable inteio epaatix enegy m m m uffle, eated by voltage applied to fou 6 eto, extending ove z ente At evey plama adiu, low enegy patile ae tapped in eithe the left o ight end, wheea highe enegy paing patile tanit the entie length of the olumn Ruffle pead the haateiti epaatix enegy by ( ) ( / ) m m m W m V R, omewhat edued by plama hielding Patile hange fom ipple tapped to paing (and vie vea) due to binay olliion at ate, o due to tempoal flutuation (), due to dift- t in the epaatix enegy The eleton- otation ao -uffle vaiation m eleton olliionality in the peent expeiment i elatively low ( otation peiod pead the epaatix by an enegy width / / / E / e), and olliion ating fo a dift- W T( / f ) ( / T) ev ( /kg) Thu, the haoti (de)tapping poee will be impotant when m() W, o when () t W We diagnoe the bulk adial expanion of the plama, haateized by the mean-quae adiu d n( ) and the oveall expanion ate Fotunately, amplitude defined a d dt an be auately and eadily obtained fom the ontinuouly meaued fequeny f () t of a mall (/ ) / m diooton mode, a f df dt The bulk expanion ate i an integal meaue of the full adial flux that inlude both mobility and diffuive ontibution, both being popotional to the adial diffuion oeffiient D () 3 Neolaial tanpot fom olliional and haoti epaatix tep Radial patile tanpot i onveniently diven by a global magneti tilt aymmety with ontolled magnitude / and hoen tilt dietion z tan ( y / z) (thu, otated by m V () elative to the uffle) Thi tilt i equivalent to applying z-antiymmeti wall voltage Va ( RW,, z) z( enl / R W )o( ), whih aue inteio Debye-hielded z-aymmeti potential ( z a, ) Fo lage field, giving igidity R, imple z-boune-aveaged theoy uffie to deibe the epaatixindued tanpot and wave-damping The tilt-indued z-aymmeti eo field ( z, ) ha boune aveage value a

3 L and R fo left- and ight-end tapped patile nea the epaatix enegy, with paing patile expeiening zeo boune-aveage eo field The dift obit fo left- and ight-end tapped tajetoie ae then offet adially by () L R e Random tanition between tapped and paing population ae aued by olliion (); by dift otation along the o( m ) epaatix uffle (m); and by tempoal flutuation in the epaatix enegy (t) If the fation of patile tanitioned in a otation peiod i, then the adial diffuion oeffiient i expeted to be D f (3) ~ E Fo olliion alone, onventional neolaial ipple tanpot give the epaatix indued fatuing of the veloity ditibution to an enegy width the uffle m and tempoal t poee will be een to be independent of both and / / W FM ( ) olliion limit / W T( / f E) In ontat, With plama otation ao a epaatix uffled by, a detailed analyi [, ] of tanition between the left and ight tapping egion give neolaial diffuion oeffiient D ( ) F ( ) f 4 M E W D A DA in oth the olliional boune-aveaged tanpot oeffiient D and the A (4) m uffle oeffiient D A alulated in [] a funtion of the nomalized uffle tength / W ae hown in Fig The uffle-indued tanpot oeffiient D A i nealy independent of / W, giving D In ontat, the olliional oeffiient DA how a fat deline a haoti epaatix oing beome dominant and mooth out the olliion-indued fatue of F M In the ae of pefetly aligned aymmetie ( in ), uffle may enable ome uppeion of olliional neolaial ipple tanpot, until the boune-eonant tanpot poee beome dominant on thei own D D A A D A D A / Figue alulated olliional D and uffle indued D oeffiient veu the nomalized uffle tength A W A

4 a) b) tapped tapped z > z > z < z < paing paing Figue 3 Sketh of plit E dift obit nea the m uffled epaatix a) b) Fo the magneti tilt aymmety the tapped potion of the obit ae the patial ile hifted along the tilt dietion Indeed, pio theoy [7] onideed only o, in whih ae the phae-dependent pat of the diffuion oeffiient i exatly zeo The eaon fo thi anellation an be qualitatively undetood fom Fig3a, whih how a keth of plit E dift obit nea the m uffled epaatix Fom the magneti tilt aymmety the tapped potion of the dift obit ae patial ile hifted along the tilt dietion If thi dietion oinide with the zeo phae of epaatix uffle, the left-ight ymmety implie patile tanit fom tapped to paing and bak at the ame adiu, o the dift obit i loed and thee i no net adial tep Howeve, when o, the ymmety i boken and patile obit ae tapped and detapped at diffeent adii, leading to adial tep (Fig3b) Of oue, fo o the diffuion doe not ompletely vanih; olliional effet not kept in the above analyi yield finite diffuion onitent with the one obtained in [7] Nevethele, thi pefet tellaato ymmety with epet to all global aymmetie i ommonly aumed in tellaato optimization theoy Figue 4 how the pedited tanition fom pedominantly olliional neolaial diffuion to the haoti egime (uffle dominated, A athe onveniently appoximated a D D ) Fo ompaion with the expeiment, the effetive tanpot oeffiient D an be D W D D A A in 88 W 4( in 88 We ) (5) D W in Figue 4 ombined neolaial ipple tanpot oeffiient D veu the nomalized uffle tength The dahed line i the appoximate fom (5) The full adial flux ha both mobility and diffuive ontibution, a D N (6) e, with D N T and N dzn Thi flux definition implie D, and then onnet it to the bulk expanion ate a 88 / W W D 4( 88 We ) onvenient appoximation

5 d (7) 4 Expeiment Figue 5 how the meaued expanion ate the maximal tanpot alignment in a a funtion of the m uffle voltage V at the wall fo It ha eentially the ame fitting funtion a the alulated one in Fig4, / W (4 / 3) V /V Thi i loe to it giving the nomalized adially aveaged uffle tength a etimated value Thu, at 6 kg and V 3V, the effetive uffle width i appoximately 4 the olliional ( 4 W ) width, and the tanpot ate ha hanged by a fato of 4 aodingly 3 V 66V ( V 66 e ) [/] in V [V] V q 45mad 6kG 6V Figue 5 Meaued expanion ate a a funtion of the uffle voltage at the wall V 5 V V V 6V q 6kG mad [/] 5 ( bkg ) 5 5 Figue 6 Meaued tanpot ate a [ ] in at fixed V V, howing haoti pat of neolaial tanpot vaying, and -independent olliional ipple tanpot

6 6 V q mad 4kG 6V V V 8 [/] ( ) bkg 5 5 [ ] Figue 7 Meaued tanpot ate V in Figue 6 i a plot of meaued expanion ate at fixed, howing haoti pat of neolaial tanpot vaying a, and -independent olliional ipple tanpot veu the magneti tilt angle at a fixed wall uffle value V V The uffled-indued pat how an unambiguou on elative angle m, (hee, /4) with the magnitude popotional to in tep of /veifie the dependene on elative angle only Similaly, Figue 7 i a plot of meaued expanion ate applied wall uffle tength tanpot ate how it unambiguou V, now at the fixed tilt value in in ignatue, togethe with epaate ontol of veu magneti tilt oientation angle, fo vaiou tilt tength in dependene V Sepaate data vaying, fo vaiou One again, the uffled-indued pat of the V ignatue, but now with magnitude popotional to The ditintive V and, enable lea expeimental identifiation of z-boune-aveaged neolaial tanpot poee epaately fom z-kineti poee We model the full tanpot ate a ( V ) V in A A V ( bkg ) K K whee A and A epeent the adial integal of Eqn (4); K and K epeent olliional Kineti (bouneeonant) tanpot diven by and ( bkg ) V a z-dependent eo" field [3-5]; and mall aie fom unontolled bakgound tilt, epaatie, and omnipeent uffle Hee, fo dimenional impliity, /mrad and V V /Volt, (8)

7 9 6 V V V 6V q 6kG V A [/ ] 3 A ( V ) A () Figue 8 Meaued aling fo the A and ( ) A V neolaial ipple tanpot tem at 6kG Evey A make hee (not hown fo A() and ) i the eult of ( ) A V ae eadily obtained fom the ( in ) a b fit like thoe hown in Fig 6 vaying give the expeted aling, a hown in Fig 8 Data taken with ompaing () A to (V) 9 / e A and uing the DA in dependene uh a hown in Fig 6 and 7; and ( / ) V define () 33/ e A W data fom Fig, one an get anothe / W (4 / 3) V /V ; and thi i onitent with etimate on the adially aveaged uffle tength, a the peviou onluion baed on the eult in Fig 4 and 5 ( bkg ) Data taken with how a offet and a paaboli dependene on the applied V, giving K Vaying then elet A and K ; thee tem ae ditinguihed by thei -aling (diued next), and by the fat that the z-antiymmeti boune-aveage in A equie the epaatix, wheea the kineti K depend only weakly on the applied queeze voltage In Fig 7, K (4kG) 3/ e, giving elevated in minima fo lage V ; the depeed minima fo V 33V ae fom uffle-uppeion of DA( / W ) (ee Fig ); and ( bkg ) / e [mrad] Figue 9 how the meaued tanpot ate oeffiient, A A and K veu magneti field, with expeimental aling (dahed), ompaed to theoy (olid line) At high, the meaued haoti and olliional epaatix tanpot / poee agee loely with theoy, aling a and epetively Hee the auay of ompaion i limited by expeimental tempeatue unetainty, by enitivity of alulation to the plama edge denity gadient, and by F ( ) tanpot indued modifiation of M We note that pio tanpot aling expeiment have been onfued by the peene of unontolled epaatie and uffle, a well a by ovelapping tanpot egime [8]

8 K 5 7 A, A, K [/] - A A 35 6 Figue 9 Meaued tanpot ate 5 Kineti Effet i q At low, the kineti (boune-eonant) tanpot pat, labeled ak in Fig 9, i obeved to depend tongly on 7 field ( ); but no imple powe law i theoetially expeted, ine multiple boune-otation eonane beome dominant Then, the adial flux i dominated by long-time-oelated quai-peiodi obit whih initially how adial peading popotional to time At ulta low olliionality, when W / R, patile etain infomation about many peviou olliionle oing of the uffled epaatix Uing nomalized uffle tength m A / w (hee / w WR i the enegy olliional peading duing one axial boune at the epaatix enegy) a a paamete we have done a numeial boune-mapping of the mean adial patile flux veu igidity The alulated fato fo the phae ( in ) independent pat of the adial flux i hown in Fig, inluding the tanition fom the bouneaveaged to the boune-eonant egime A numeial alulation give the independent flux oeffiient J veu igidity R, fo 4 value of the nomalized uffle tength A R (ie, a ) in the high-igidity egime R etimate of thei maxima give /3 A 5 J R, in geneal oepondene to the A The tong uffle uve ale a /, but how tong eonant enhanement at mall intege R /3 A 7 aling obeved fo K ; and an at low At lage R, the A 3 flux ale a, in oepondene with the A and the boune-aveaged theoy The / A uve ale a, in quantitative oepondene with no uffle olliional neolaial tanpot - [kg] veu atv 6V with empiial aling Solid line ae theoy pedition Oveall, thi numeial mapping how loe oepondene both with analytial eult fo boune-aveage ae (olliional and haoti) and with expeimental aling in the whole boune-aveage and boune-eonant domain

9 Figue Numeially alulated uffle tength A In the ange haoti ae, R independent dimenionle adial flux J veu igidity R fo indiated value of /3 A J A / R The ed dahed line ( A ) 6 Flutuation-indued tanpot the uve have eonant pike The dotted line ae analytial eult in the i analytial eult in the olliional ae, J ( / R ) Simila enhaned patile tanpot i obeved when thee ae tempoal vaiation in the epaatix enegy Figue illutate the immediate ineae in adial expanion ate indued when white noie ( V V, f f MHz) i applied to the -ymmeti ipple (queeze) ing, diving andom RMS E (olliionle) epaatix oing The 3 ineae in ( d / dt) ate obeved hee i onitent with a olliional epaatix laye width W 7eV flutuating by ( t ) ev / Peumably, any noie- o wave-indued flutuation whih hange patile kineti enegie elative to the epaatix enegy would be equally effetive in enhaning neolaial helial ipple tanpot Vq 6V V noie kg 6 e () t e V noie ( khz) V Time [] Figue Enhaned tanpot ate duing the two but of V (RMS) white noie () t ) applied to an eletotati ipple 7 Low-olliionality enhanement A ummay of the diffeent tanpot egime i hown in Fig, whih kethe the adial diffuion oeffiient veu olliion fequeny We emphaize that thi pepetive i uppoted by quantitative expeiment [Ref 8- and Ref theein], in ageement with detailed theoy and tageted imulation [3,4,,] Thi eeah enabled

identifiation of the diffeent egime haateized by olliional enegy peading W, uffle amplitude m, and flutuation amplitude () t ; and it laified the onnetion to kineti poee at low igidity R Thi boade

potential vaiation may ou even along field line, and that dift-kineti effet may be ignifiant Thee effet ae neeaily onideed in ylindial (open) tap, but ae often negleted in analyi of tellaato and

10 identifiation of the diffeent egime haateized by olliional enegy peading W, uffle amplitude m, and flutuation amplitude () t ; and it laified the onnetion to kineti poee at low igidity R Thi boade pepetive may aid in intepetation of numeial imulation developed fo tellaato geomety [6] (eidle et al ), whee multiple poee ombine Moeove, eent tellaato meauement [7] (Pedoa et al 4) ugget that eleti potential vaiation may ou even along field line, and that dift-kineti effet may be ignifiant Thee effet ae neeaily onideed in ylindial (open) tap, but ae often negleted in analyi of tellaato and tokamak / In Fig, the and line (ed) epeent tandad "upebanana" olliional tanpot, with the egime pedited due to uppeed olliional tanpot fom the peumed ymmety Thi oepond to the uffleindued edution of D A in Fig, with DA being peumed ineffetive due to in in Eqn 4 Without the peumption of dynamially eveible dift, tanpot due to uffle (geen) and due to tempoal potential flutuation (oange) may dominate at low oth of thee poee ae eentially independent of At ulta low (pobably unphyially low), nonlinea effet ould limit thee tanpot mehanim Oveall, ou expeiment, theoy and ompute imulation ugget that haoti neolaial tanpot due to uffle o flutuation aling a may well be dominant in egime of low olliionality (Fig ) Thu, onideable wok may be equied to bing theoy into oepondene with low-olliionality, epaatix-uffled, flutuating fuion plama olliion ate Figue Low olliionality haoti tanpot due to uffle o due to tempoal flutuation may ineae the diffuion ate well above it olliional pedition in and / egime Mot plama onfinement devie have tapping epaatie (ipple), aiing fom vaiation in magneti field tength o extenal potential Thee epaatie ae neve pefetly ymmeti, o pefetly aligned with othe aymmetie If the epaatix itelf i aymmeti o tempoally petubed, the difting patile olliionlely hange f to enhaned aymmeti fom tapped to paing and bak, leading in the ae of low olliionality ( / E ) ipple tanpot ( ) in ompaion to the tandad neolaial ipple tanpot / / ( ) When the epaatix laye olliional width i le than it -aymmety o tempoal petubation, thi new lo mehanim beome the dominant bulk tanpot poe in ou non-neutal plama expeiment, and it ould have impotant impliation fo imila low olliionality egime in othe magneti onfement expeiment Aknowledgment Thi wok wa uppoted by National Siene Foundation Gant PHY and PHY-6374, and by Depatment of Enegy Gant DE-S45 Refeene

11 P Helande, DJ Sigma olliional Tanpot in Magnetized Plama, ambidge: ambidge Univeity Pe, MN Roenbluth, DW Ro, DP Kotomaov Stability egion of diipative tapped-ion intability// Nulea Fuion, 97, v, N, p TJ Hilabek, TMO'Neil Tapped-patile diooton mode// Phyi of Plama 3, v, N9, p DHE Dubin Theoy and imulation of eletotati field eo tanpot// Phyi of Plama 8, v 5, N7, p T Ohkawa, JR Gilleland, T Tamano Obevation of neolaial, intemediate, and Pfih-Shulte diffuion in the d otople// Phyial Review Lette 97, v 8, N7, p 7-6 M Zantoff, K MGuie, et al Paallel eleti eitivity in the TFTR tokamak// Phyi of Fluid 99, v N8, p HE Mynik Effet of olliionle detapping on non-axiymmeti tanpot in a tellaato with adial eleti field// Phyi of Fluid 983, v 6, N9, p AA Kabantev, F Dioll Tapped-patile mode and aymmety-indued tanpot in ingle-peie plama// Phyial Review Lette, v 89, N4, p AA Kabantev, TM O'Neil, YuA Tidulko, F Dioll Reonant dift-wave oupling modified by non-linea epaatix diipation// Phyial Review Lette 8, v, N6, p 65-4 AA Kabantev, F Dioll Tapped-patile mediated olliional damping of nonaxiymmeti plama wave// Phyial Review Lette 6, v 97, N9, p 95-4 DHE Dubin, F Dioll, YuA Tidulko Neolaial tanpot aued by olliionle atteing ao an aymmeti epaatix// Phyial Review Lette, v 5, N8, p DHE Dubin, YuA Tidulko Neolaial tanpot and plama mode damping aued by olliionle atteing ao an aymmeti epaatix// Phyi of Plama, v 8, N6, p DL Eggleton, TM O'Neil Theoy of aymmety-indued tanpot in a non-neutal plama// Phyi of Plama 999, V 6, N7, p DL Eggleton, JM William Magneti field dependene of aymmety-indued tanpot: a new appoah// Phyi of Plama 8, v 5, N3, p EP Gilon, J Fajan Quadupole-indued eonant-patile tanpot in pue eleton plama// Phyial Review Lette 3, v 9, N, p D eidle, K Allmaie, MYu Iaev, SV Kailov, W Kenbihle, GO Leitold, H Maaßbeg, DR Mikkelen, S Muakami, M Shmidt, DA Spong, V Tibaldo and A Wakaa enhmaking of the mono-enegeti tanpot oeffiient eult fom the Intenational ollaboation on Neolaial Tanpot in Stellaato (INTS)// Nulea Fuion, v 5, N7, p MA Pedoa, JA Alono, JM Gaía-Regaña, Hidalgo, JL Velao, I alvo, Silva, P Helande Eletotati potential vaiation along flux ufae in tellaato// axiv:4493

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