Global gyrokinetic particle simulation of turbulence and transport in realistic tokamak geometry

Transcription

1 Insttute of Physcs Publshng Jounal of Physcs: Confeence Sees 16 ( do: / /16/1/008 ScDAC 2005 Global gyoknetc patcle smulaton of tubulence and tanspot n ealstc tokamak geomety WXWang 1, Z Ln 2,WMTang 1, W W Lee 1, S Ethe 1, J L V Lewandowsk 1, G Rewoldt 1, T S Hahm 1 and J Manckam 1 1 Pnceton Plasma Physcs Laboatoy, Pnceton Unvesty, P.O. Box 451, Pnceton, New Jesey 08543, USA 2 Depatment of Physcs and Astonomy, Unvesty of Calfona, Ivne, Calfona 92697, USA E-mal: wwang@pppl.gov Abstact. A geneal geomety model has been developed fo the gyoknetc toodal code [1] wth a numbe of hghly desable featues ncludng a systematc teatment of plasma otaton and equlbum E B flow, ealstc plasma pofles and coespondng magnetohydodanamc (MHD equlba. A symmety coodnate system s used to constuct a elatvely egula mesh n eal space fo stongly shaped toodal plasmas, whch also facltates staghtfowad vsualzaton. By escalng the adal coodnate, gd sze s coelated wth the local gyoadus whch may vay substantally fom the coe to the edge. Gyoknetc tansfomaton of potental and chage densty between patcle and gudng cente postons n geneal geomety s caefully teated, takng nto account the fnte ato of the polodal to the total feld (B θ /B. The appled equlbum E B flow, whch s beleved to play an mpotant ole n detemnng the tubulence level, s calculated fom ou global neoclasscal patcle code -Neo[2]. In the lage aspect ato ccula geomety lmt, coss benchmaks wth the ognal code show good ageement n eal fequency, gowth ate, steady-state heat flux and zonal flow ampltude fo the on tempeatue gadent dven mconstabltes (ITG modes. 1. Intoducton Undestandng tubulence and assocated tanspot n toodal plasmas [3-5] s one of the key ssues n magnetc fuson eseach. In the past decade as compute esouces apdly ncease and advanced numecal algothms ae developed, sgnfcant pogess has been made wth the most pomnent beng the fst-pncples based gyoknetc patcle appoach [6]. The smulaton study caed out by the gyoknetc toodal code [1] epesents one of the moe poductve examples. was ognally developed to focus on fundamental nonlnea tubulence physcs. It s a full-tous global code usng a feld-lne-followng mesh and a eal space feld solve. Global tubulence smulatons fo toodal plasmas ae hghly demandng fo the followng easons: ( the tubulence-geneated zonal flow has a typcal adal scale as lage as the system sze, even though tubulence tself s on the much smalle scale of the gyoadus; ( the equlbum E B shea flow whch also plays an mpotant ole n detemnng tubulence levels typcally has the lage scale sze of the plasma mno adus; ( tubulence speadng to the lnealy stable zone esults n nonlocal tanspot whch s a tuly global phenomenon. To pose the smplest poblem whle keepng the mpotant global physcs popetes, a smplfed model was utlzed n the 2005 IOP Publshng Ltd 59

2 60 smulatons, such as smple magnetc geomety wth a lage aspect ato ccula concentc coss secton and the assumpton of unfom pessue. Ths poved to be an effectve means of ganng key nsghts nto the complexty of toodal tubulence system. As a esult, smulaton has led to a numbe of mpotant fndngs wth egad to zonal flow effects, tanspot scalng wth devce sze, tubulence speadng, etc [1,7-9]. Whle such a smplfed model s a necessay and useful tool to sepaate and clafy fundamental physcs ssues, moe ealstc featues ae needed as the eseach moves fowad. Patculaly fo smulatng tubulence phenomena n tokamak expements, a moe compehensve model s needed whch consstently ncopoates the nfluence of geneal geomety, ealstc plasma pofles, plasma flow, neoclasscal equlbum, Coulomb collsons and othe featues. In ths pape we pesent such a model wth emphass on the geneal geomety capablty, whch has now been mplemented nto. The geneal geomety s ntefaced wth TRANSP, a wdely used expemental data analyss softwae tool fo specfyng expemental plasma pofles of tempeatue, densty and toodal angula velocty, and also wth vaous numecal MHD equlbum codes, ncludng the Jsolve, qsolve and ESC codes. Ths s a majo poject wthn the ScDAC Cente fo Gyoknetc Patcle Smulaton. 2. Coodnate system and mesh constucton Magnetc flux coodnates n whch the adal coodnate labels magnetc sufaces ae geneally used fo a toodal system and assocated wth MHD equlba. Ou gyoknetc smulaton n pncple can use abtay flux coodnates wth staght feld lnes. In the flux coodnates, the feld-lne-followng mesh, whch possesses the hghest effcency by captung the flute-type chaacte of the dft wave tubulence n toodal plasma, can be easly constucted. A pefeable flux coodnate can be chosen n tems of a dffeent equement. A symmetc coodnate system n whch the toodal angle ϕ s chosen to be the azmuthal angle of cylndcal coodnates s pefeable n many cases. These coodnates ae elatvely unfom compaed to othe whch have been pevously used, and advantageous fo constuctng a elatvely egula mesh n eal space fo stongly shaped plasmas. It also facltates staghtfowad vsualzaton wth the polodal plane defned wth the physcal angle ϕ. The adal coodnate s defned as = ψ/ψ e whee ψ and ψ e ae the toodal flux and ts value on the last magnetc suface, espectvely. Ths same adal coodnate s wdely used n the expemental communty. Because of the flute-type chaacte of dft wave tubulence n toodal plasmas, wth k k, whee k and k ae the paallel and pependcula wave numbes, espectvely, uses the feld-lne-followng mesh whch shows hgh effcency fo calculatng the tubulent feld. Fo dft wave tubulence, the spatal scale length n the pependcula decton s geneally n coelaton wth the local gyoadus ρ j T j, whch may vay substantally fom the coe to the edge of the plasma. Fo nstance, t s common n Natonal Sphecal Tous (NSTX plasmas that the on tempeatue changes fom KeV n the coe to 10eV nea the sepaatx egon at the plasma edge. Theefoe, fo a global smulaton, whch ncludes the ente adal doman, t s mpotant to use a nonunfom gd wth gd sze n the pependcula decton coelated wth the local gyoadus fo mpoved spatal esoluton and effcency. To ths end, we e-scale the adal coodnate by defnng ρ as follows: dρ d = T c /T (, (1 whee T c s the tempeatue at a efeence adal locaton. Wokng wth the new coodnate ρ, we use an even spaced adal gd, whch offes geat convenence fo fequent opeatons such as patcle sotng, chage deposton, gatheng, etc. Ths allows the gd sze n eal space to be coelated wth the local gyoadus: T (/T c. In the polodal decton, the gd sze θ( s unfom on a flux suface, whle vayng ove dffeent flux sufaces. The gd sze θ(

3 61 s detemned so as to make the polodal ac length l θ nea the md-plane coelated wth ρ. An example of such a gd on the toodal plane ϕ = 0 s shown n Fg.1. Geneally, a twodmensonal mesh on the ϕ = 0 plane s set up fst. A thee-dmensonal mesh s constucted by followng each (appoxmate feld lne whch stats at a gd pont on the ϕ = 0 plane and has q(θ ϕ =constant, wth q slghtly changed fom the usual safety facto q( so that the appoxmate feld lnes wll lead back to one of the gd ponts on the ϕ = 0 plane. The gyoknetc patcles ae followed n geneal flux coodnates usng gudng cente Lagangan equatons nstead of Hamltonan equatons, whch eque constucton of canoncal vaables[10] that ae complcated foms n geneal geomety and nconvenent to use. The gudng cente Lagangan obtaned by Lttlejohn has the followng nomalzed fom[10,11] L(x, ẋ; t =(A + ρ B v H, (2 wth the gudng cente Hamltonan H = ρ 2 B2 /2+µB+Φ. Hee, the magnetc feld B = A, ρ = v /B s the paallel gyoadus, µ s the magnetc moment, and Φ s the electc potental. The ndependent vaables ae x =(, θ, ϕ, ρ. The patcle gudng dft moton s govened by the Lagangan equaton ( d L dt x x L =0. (3 The obtaned equatons fo dx/dt ae sutable fo any genealzed flux coodnates. 0.8 gd sze n coelaton wth local ρ Fgue 1. An example of nonunfom gd on a polodal plane showng gd sze n coelaton wth local on gyoadus. 3. Smulaton esults In ths secton we pesent new scentfc esults fom ou pesent studes. Detals on the fom of the basc equatons and the assocated gyoknetc tansfomaton n geneal geomety wll be pesented n a sepaate publcaton Benchmaks The geneal geomety veson of has been caefully benchmaked, n the lage aspect ato ccula geomety lmt, aganst the ognal code, whch uses a smple analytcal MHD equlbum. Fo ths benchmak, a coespondng numecal equlbum s poduced fo the geneal geomety. The numecal equlbum ncludes a small Shafanov shft due to nonzeo plasma beta and hghe ode (n small nvese aspect ato coectons, whch ae neglected n the analytcal equlbum. The benchmaks ae caed out fo on tempeatue gadent

4 62 (ITG modes wth a smplfed adabatc esponse fo the electon dynamcs. The epesentatve paametes fo the famla cyclone case[12] ae used hee: nvese aspect ato a/r 0 =0.358, on tempeatue pofle R 0 /L T =6.92 exp{ [( 0.5/0.28] 6 }, T e /T =1,q = , and L T /L n = The lnea benchmak smulatons ae caed out n a adal doman fom 0.2a to 0.8a, and the ITG nstablty s measued at = 0.5 whee the tempeatue gadent s peaked. As llustated n Fg.2, good ageement s obtaned fo the eal fequency ω, whle the gowth ate γ of the hghe-n modes fom the new geneal geomety veson of ae slghtly lowe. The oveall dffeence n fequency magntude ω s less than 5%. The contou plots of the electostatc potental petubaton on a polodal plane show qute smla egenmode stuctues fom the two smulatons. ω (V /L T γ (V /L T geneal geomety model geomety 0.01 Fgue 2. Real fequency ω (left and gowth ate γ (ght of ITG nstablty vesus polodal wave numbe k θ, compaed wth the ognal calculaton. k θ ρ k θ ρ In the nonlnea ITG benchmak, the velocty space nonlneaty s ncluded, whch may have consdeable effect on tubulence dynamcs[13]. Flat tempeatue and densty pofles ae used n the geneal geomety fo benchmaks n ode to be consstent wth the ognal. The adal smulaton doman hee s fom =0.1a to =0.9a. As shown n Fg.3, the nonlnea benchmak esults ae n good ageement fo both steady state heat flux and zonal flow ove the ente adal doman. Some dffeence s found n the lnea phase and n the appoach to steady state. It s also found that the self-geneated zonal flow n ITG tubulence has a global scale of ode the mno adus, wth a oughly odd paty. In the Fg.3 the enegy flux, whch s the sum of the heat flux and the convectve flux due to the patcle flux, s also plotted. The esult that the enegy and heat flux ae ght on the top of each othe ndcates that no patcle flux s poduced a well known fact about electostatc tubulence wth adabatc electons. It epesents a goous test fo such a complex smulaton. heat (enegy flux geneal geomety /a=0.54 (model geomety heat flux geneal geomety (model geomety zonal flow dφ 00 /d steady state zonal flow (t=8000 geneal geomety (model geomety tme Fgue 3. Tme hstoy of on heat flux at =0.54 (left, steady state heat flux pofle (mddle and zonal flow pofle (ght of nonlnea ITG smulaton wth smple magnetc geomety, compaed wth the ognal esult Applcatons Most lnea analyses of mconstablty ae caed out locally, neglectng the adal vaaton of equlbum quanttes such as pessue gadent and pessue tself. Whle s a global

5 63 toodal code, n ode to focus on the smplest poblems nvolvng shea effects due to the adal vaaton of the pessue gadent, plasma tempeatue and densty wee assumed constant n the ognal smulatons. Whle such a teatment appeas necessay and useful to sepaate and clafy fundamental physcs ssues, t does not ealstcally captue the compehensve global physcs. In Fg.4 we pesent an example of global ITG nstablty fom the geneal geomety smulaton, takng nto account the adal vaaton of tempeatue and densty consstent wth the gadent pofles. Compaed to the smulaton wth constant T and n, the ITG gowth ate s consdeably educed, wth no sgnfcant change n the eal fequency. Meanwhle, the contou plot of the electc potental on a polodal plane shows that the egenmode stuctue s twsted n the polodal decton. T pofle 4 ω (V /L T usng nonunfom T, n usng unfom T, n k θ ρ R / L pofle 0 T γ (V /L T usng nonunfom T, n 0.01 usng unfom T, n 0 k θ ρ Fgue 4. Smulaton esults of a nonlocal ITG nstablty usng adally vayng tempeatue and densty, compaed wth the smulaton wth unfom tempeatue and densty: eal fequency ω (uppe-left and gowth ate γ (lowe-left vs k θ ρ ; contou plot of electc potental fom smulaton wth nonunfom (uppe-mddle T and n and unfom T and n (lowe-mddle; on tempeatue pofle (uppe-ght and coespondng gadent pofle(lowe-ght used n the smulatons. We have also appled ou new nonlnea ITG smulaton capablty to a shaped toodal plasma, the DIII-D expement at Geneal Atomcs, San Dego, wth the same model pofles as n Fg.4. The snapshots of the tubulence ntensty pofle ae plotted n Fg.5, whch shows that the tubulence, whch gows ntally n the lnealy unstable egon, speads n both the nwad and outwad adal dectons nto the stable egons, leadng to adally global tubulence and tanspot nonlocalty. Also pesented ae thee snapshots of the electc potental contou plot on a polodal plane, whch llustate the dynamc pocess of tubulence development: n the ealy phase the elongated adal steame stuctue of the egenmode s lnealy dven n the localzed unstable zone, then s boken up as self-geneated zonal flow s establshed, and eventually envolves nto wdely spead global tubulence. 4. Summay We have developed a moe ealstc model fo smulatng plasma tubulence n actual tokamak expements. It ncopoates the compehensve nfluence of geneal geomety, ealstc plasma pofles, plasma otaton, neoclasscal (equlbum electc feld, and Coulomb collsons. The

6 2 tubulence ntensty <δφ > t = t= Fgue 5. Snapshots of tubulence ntensty pofile (left, and contou plot of electc potental on a polodal plane (bottom fom a smulaton of a shaped plasma wth typcal DIII-D paametes. t=1500 t=2100 t=12600 caeful benchmaks of the lnea and nonlnea chaactes of the well known on tempeatue gadent nstablty have been caed out to valdate the new capablty. Applcaton of ths code to DIII-D szed shaped plasma has poduced nteestng esults wth egad to nonlnea dynamcs of ITG tubulence speadng and assocated enegy cascadng to the longe wavelength (low-n modes. Ths hghly desable capablty makes a poweful tool to nvestgate the popetes of plasma tanspot and confinement of actual tokamak devces. Snce finte beta (ato of plasma pessue to magnetc pessue toodal plasma electomagnetc effects become moe mpotant n detemnng the tanspot level n bunng plasma expements such as ITER, futhe extensons of the poject to smulate electomagnetc tubulence ae cuently beng actvely pusued. Acknowledgments Ths wok was suppoted by the US DoE Scentfic Dscovey though Advanced Computng (ScDAC Cente fo Gyoknetc Patcle Smulaton and by the DoE Contact No DE-AC02CHO Refeences [1] Ln Z, Hahm T, Lee W, Tang W and Whte R 1998 Scence [2] Wang W X, Tang W M, Hnton F L, Zakhaov L E, Whte R B and Manckam J 2004 Comput. Phys. Commun [3] Tang W M 1978 Nucl. Fuson [4] Conno J W and Wlson H R 1994 Plasma Phys. Contol. Fuson [5] Hoton W 1999 Rev. Mod. Phys [6] Lee W W 1983 Phys. Fluds [7] Ln Z Ethe S Hahm T S and Tang W M 2002 Phys. Rev. Lett [8] Ln Z and Hahm T S 2004 Phys. Plasmas [9] Hahm T S, Damond P H, Ln Z, Rewoldt G, Gucan O and Ethe S submtted to Phys. Plasmas [10] Whte R and Zakhaov L E 2003 Phys. Plasmas [11] Lttlejohn R J 1981 Phys. Fluds [12] Dmts A M et al Phys. Plasmas [13] Lee W W th APS-DPP FI1B 3 Savannah Geoga