Symmetry breaking effects of toroidicity on toroidal momentum transport. Abstract

Transcription

1 Symmy bakg ffcs of oodcy o oodal momum aso. TH/8.9 J. Wlad, R. Sg, H. Nodma,. Kaw, A.G. s ad D. Sz. Dam of Rado ad Sac Scc, Calms. vsy of Tcology ad Euaom-VR Assocao, S96 Gobug, Swd. Isu fo lasma Rsac, a Gadaga 8 8, Ida. C fo Fuso, Sac ad Asoyscs, yscs dam, vsy of Wawck, CV 7AL, Covy K. Dam of Elccal ad Comu Egg, Naoal Tccal vsy of As, GR-57 7 As, Gc, Assocao Euaom Hllc Rulc Absac A dvao of symmy bakg oodcy ffcs o oodal momum aso as b mad fom sss so. Ts ffc s usually sog a symmy bakg causd by flows mslvs o gfuco. T modl obad galzs a c dvao of dagoal aso lms fom sss so o covcv lms of ubul quao o molcc ys. Ts gvs ossbly fo ao of sam y of ffcs vously obad fom a as sac cosvg ola gyokc quao,.

2 . Ioduco T s s momum aso -8 as cly focusd o symmy bakg ffcs,,7 o oodal momum aso. T fs ffc of s y was dfd, as ffc of a asymmc g-fuco o avag of aalll mod umb s dd of aalll momum aso. Fo accal uoss w may aoxma oodal momum by aalll momum. Rcly also symmy bakg ffcs of oodcy w foud 7. Fo a movd udsadg of suc ffcs ad fo a coss dvao of flud quaos fo momum aso s gly dsabl o also mak flud dvao of suc ffcs. Sc magc df s o a flud df suc ffcs ogas fom sss so flud quaos. W a us facd w a comlcad ask of cludg magc cuvau ffcs fom sss so. O o ad gyokc dvaos av o sa fom a as sac cosvg fomulao 9 of gyokc quao. Tus qu ambos aoacs a dd bo gyokc 7, 8 ad flud dvaos. T flud dvao s, owv, mo sag fowad, sowg sg of flud oy dvg advacd dyamc quaos. T sss so s moa maly fo low fqucy oma magzd lasmas. I was alady usd fo dvg dagoal magc df ms. I collsolss cas sss so cobuos a assocad w f gyoadus ffcs fo moo dcula o magc, fld. H fs suls cludd gyo vscous cacllaos bw covcv damagc ffcs. A s m also kc calculaos fo df ad MHD y mods ad jus sad ad full Vlasov dscos w smlfd gomy w usd. T agm fo lows od FLR ffc bw flud ad kc dvaos, could us b gadd as a sgfca acvm a s m. Wl FLR ffc, fo dcula moo s obad oug a xaso ao of caacsc mod fqucy ad cycloo fqucy, ad us usually ad as small, oodal ffcs fom sss so as od ε L / R aalll momum quao. Aloug ε ca b ad small dg ad s ycally of od co. Tus s ffc s moa fo aalll moo. A sml way o oba dagoal cuvau ffcs s o dv quaos of moo fo gudg cs gyo flud quao. Suc quaos av magc dfs cludd as gudg c dfs ad quao of moo ca b obad by akg moms of a gyo kc quao. T dagoal cuvau ms aa as du o a covcv magc df sum of gad ad cuvau ms w a faco as a covco of aalll momum flow. I s wok w wll sa fom agsk flud quao ad dv symmy bakg ffcs cosodg o Rf. 7, usg ly flud quaos fo oodal o mau gad mod ITG. II. Toodal momum flow T gal momum quao s obad fom o ad lco momum quaos ad couy quao:

3 v mn N E N E a b N N S c H s o sss so gv by agsk 7 s adx ad S s dsy souc m. W us galzd co-oda sysm ˆ, bˆ, ˆ wc s d o magc fld. H ˆ / s u vco alog magc fld ls, ˆ s oogoal o magc sufac, ad bˆ ˆ ˆ. T co-odas basc ˆ, bˆ, ˆ a lad o flux co-odas,,, w s olodal magc flux, galzd olodal agl ad s oodal agl. o co-odas sysm a lad as ˆ ˆ, bˆ ˆ ˆ, ˆ ˆ ˆ Fo lag asc ao ε / R << ad oodal symmy.., /, dffal oaos ca b w as: ˆ ˆ ~ ~ L ~ qr, θ ˆ b ~ qr ~ θ H /, J, J, Jacoba of asfomao,,, ν / ~ / R, c of fld ls, q ν d, safy faco ad fo lag asc ao ε / R <, R ε cos ad ε cos, w ad R mo ad majo ad, scvly. T sum of oodal comos of Eqs a ad b ca b w as: ~ mn ˆ ˆ J c W las m cosods o adal agula momum flux dv by backgoud ubulc ad ˆ.

4 Fo oodal symmy.., /, covcv m Eq. 8 ca b xssd as: Nˆ [ ] [ N N ] N l N N 5 y usg Eq. 5 ad couy quao, oal dvav of momum dsy lf ad sd of Eq. ca b xssd as: N mnˆ [ ] m N N N m N N mn l N m S m N N mn l 6 H w av mad us of a soal lao ˆ ˆ l o ˆ ˆ l, ˆ ˆ, ˆ ˆ ad S s acl souc m. No fom Adx A, dfo of s of fom αβ βα, sss foc alog oodal dco ca asly b xssd as: 7 ˆ αβ αβ αβ αβ J W cosd a lasma s collsolss gm qrν / c <, ν s o collso fqucy. W ow collc oly ad fom collsolss sss so Adx A. b b b b b b [ b κ b ]

5 [ ] l m c S m N N N m m N [ ] ˆ m c m c θ θ θ w κ H / < s assumd ad Ω /,,. Aga low β ad lag asc ao lm, m ˆ ow ca b xssd as : 8 w w usd b b b κ, κ ad aalogously fo. y usg quaos 6 ad 8 quao, oodal momum quao ca ow b xssd as 9 W ow fs dv la aalll o vlocy ubao fom Eq.9. I lm kl >> k s dcula wav vco of backgoud sably, ad by akg ubaos F F f δ [ F ad f δ a backgoud qulbum ad ubao, scvly], la fom of Eq. 9 ca b xssd as: [ ] D D E J c N T m u m N θ δ δ [ ] κ

6 H D ct T, D D ˆ / ad T. I Eq cuvau m volvg cosods o ubul quao TE T cuvau m volvg δ as wo as w mau uao a cosods o molcc covcv m. If w assum olzma lcos ad quasualy w ca w as: mn θ J c E D δu m D [ ][ δ T N ] T τ Tus ow dsy ubao a adds u o TE a ad w cov Cools c of Rf 8 af subsug δu o flux Γ < v u >. W m N < θ c < J > > m N < δu δu m N > D < δu δ E δ av oly a faco sc ou D cluds a faco makg quval o oal magc df fo low ß. Ts a ca also b covd fom Rf 9 w, owv, mo ffo was mad o saa cuvau ad gad as as wll as cludg mau asooy. T ogal TE a coms fom covcv al m,.. m backs Eq 9. W o fom way Eq. was w a oodal cuvau ffc adds u as a w symmy bakg ffc o aalll gad. Ts cass subsaally ffc of flowsa. W ca ow fomally galz gvalu soluo Rf. 8 by addg w symmy bakg m o aalll gad. Howv, also symmy o adal dco wll b sfd du o oodcy. Fom Eq. 9, quao fo ma oodal vlocy ca b obad by avagg ov magc flux sufacs. I lag asc ao lm, quao fo ma ca b w: > m N δ D δu I dvg Eq w k oly cuvau ms fom sss so. W o a w av cobuos bo fom Ryolds sss al a ad fom sss so.

7 AENDIX A: THE STRESS TENSORS I lm ν / <<, sss so 7 ca b sl o as Ω,,, A T aalll sss so o dagoal max, gyo-sss so, ad dcula sss so a gv as 8, T aalll sss so: I ˆˆ ˆ ˆ T gyo sss so: { ˆˆ bb ˆˆ[ bˆ ˆ ˆ bˆ] b ˆ ˆ b ˆ ˆ [ ˆ ˆ bˆ bˆ] }, { ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ b b b b } A,, A [ ] [ ] T dcula sss so:,, A [.. ] [.. ] w { ˆˆ bb ˆˆ[ ˆ. ˆ bˆ. bˆ] b ˆ ˆ b ˆ ˆ [ ˆ. bˆ bˆ. ˆ ]} { ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ b b b b },,. 96ν, dx - ad - f o agsk s coffcs 7, / Ω, ν / Ω, ad. Rfc. J.E. Rc, E.S. Mama, F. ombada ad L. Qu, Nucla Fuso 7, X. Gab, Y. Saaz,. Gdc, S. kadda,. y, C. Fgalla ad I Voskovc, ys. lasmas 9, 89.. J.S. dgass, K.H. ull, L.R. aylo, W. Houlbg ad J. Lo, ys. lasmas,.. A.G. s ad C. Ago, ys. lasmas, C. dvs, K.M. Raamäk, C. Goud, E. As, G. Coga, A. Eksso, M. dgf, I. Jks, H.C.M. Koos,. Maca, H. Nodma,. Sad, T.

8 TALA, J. Wlad, K-D Zasow ad JET-EFDA cobuos, lasma ys. Cool. Fuso 8, J. Wlad, A. Eksso, H. Nodma ad A. Zagoody, ys. lasmas, T.S. Ham,.H. Damod, O.D. Guca ad G. Rwold, ys. lasmas, A. G. s, C. Ago ad D. Sz, ys. Rv. L. 98, 65 7.; A.G. s, D. Sz, Y. Cam al, yscs of lasmas, o b ublsd. 9. T. S. Ham, ys. Fluds, D. Sz, A.G. s ad J. Wlad, ys. lasmas 5, R.E. Walz, R.R. Domguz ad G.W. Hamm, ys. Fluds, K.V. Robs ad J.. Taylo, ys. Rv. L. 8, M.N. Rosblu, N.A. Kall ad N. Rosok, Nucl. Fuso Sul., 96.. A. Rogs, ys. lasmas 7, J.J. Ramos, ys lasmas, J.J. Ramos, ys. lasmas, H.A. Class, H. Gaus, A. Rogs ad C. Yam, ys. lasmas 7, J. Wlad ad H. Nodma, ocdgs of d ES cofc, Rom 6, ECA Vol., -, 86.