Amrcan Journal of Modrn Physcs 4; 3(5: -6 Pulshd onln Smr 3, 4 (h://www.scnculshnggrou.com/j/ajm do:.648/j.ajm.435. ISSN: 36-8867 (Prn; ISSN: 36-889 (Onln Dffusvy scalng on shar flow hong-tan Wang,, h-xong H, Ja-Q Dong, han-hu Wang, Shao-Yong Chn, Chang-Jan Tang Souhwsrn Insu of Physcs, Chngdu, Schuan, 64, Chna Collg of Physcs Scnc and Tchnology, Schuan Unvrsy, Chngdu, Schuan, 665, Chna mal addrss: wangz@sw.ac.cn (hong-tan Wang To c hs arcl: hong-tan Wang, h-xong H, Ja-Q Dong, han-hu Wang, Shao-Yong Chn, Chang-Jan Tang. Dffusvy Scalng on Shar Flow. Amrcan Journal of Modrn Physcs. Vol. 3, No. 5, 4,. -6. do:.648/j.ajm.435. Asrac: Dffusvy scalng on shar flow s nvsgad. adal lcrcal fld s h drv of h flow. Th urnng ons of h rad arcl ar no on h drf surfac, u modfd y h radal lcrcal fld. For h frs m, an analycal xrsson of h anana wdh n rsnc of shar flow s accuraly drvd. Th arcl dffusvy gvn y osnluh s rroducd u wh h shar flow modfcaon. Kywords: Tokamak Plasma, Dffusvy Scalng, Shar Flow, Trad Parcl, Gudng-Cnr. Inroducon I s gnrally clamd ha h shar flow lays an moran rol n h ons of h ranson from h L mod o H-mod. xrmnal vdnc also showd ha h lasma roas radly n h mrovd confnmn rgm, mlyng ha h radal lcrc fld s gnrad. Noclasscal on ransor n roang axsymmrc lasma has n sysmacally nvsgad y Hnon. Howvr, h nrgy loss somms n h xrmn s smallr han h sandard noclasscal rdcon. Th mrovmn s arud o h squzng facor 3 rad as h shar flow n hs ar. Thr ar rnwd nrss n h noclasscal fluxs 4-6 nsd h hrmal arrrs. Th dffusvy scalng on h shar flow rsnd y Cao 6 and Shang 5 ar dffrn. Thr ar nns argumns 6 on hs oc. I s moran caus s rlad o h rol of h shar flow n h hrmal arrr. Physcss should gv a dfn answr. Th dscrancy wn Cao 6 and Shang 5 coms from h rad arcl dynamcs whch s mhaszd n hs ar. Accura anana wdh, ounc frquncy, and h urnng on oson n h rsnc of h shar flow ar rsnd. Drf knc quaon s solvd wh a arcl-consrvd Krook collson oraor. osnluh s rsul 7 s rroducd u wh h shar flow modfcaon. Th fracon of h rad arcls smad hr s h sam as Shang s 5, howvr, dffusvy scalng on shar flow s dffrn. In scon, a s of canoncal gudng-cnr varals s drvd y ara-consrvd ransformaon. Dynamcs for h rad arcls s gvn n scon 3. Th drf knc quaon s solvd and h dffusvy scalng s drvd n scon 4. Summary s rsnd n h las scon.. Canoncal Gudng-Cnr Varals For a okamak confguraon, h Hamlonan of a chargd arcl can xrssd as: H + ( P [( P M A A + ( P / ] + Φ A, ( whr A, A, and A ar h vcor onal comonns of h magnc fld, Φ h lcrcal onal assumd o a funcon of h olodal magnc flux Ψ, M h mass of h chargd arcl s qual o uny for smlcy, and h charg. P, P, and P ar h canoncal momnum n h cylndrcal coordnas,, and, rscvly, whch ar as followng: P + A, ( P + A, (3 P + A. (4
Amrcan Journal of Modrn Physcs 4; 3(5: -6 3 Th magnc fld n okamaks can xrssd as: Ψ + I, (5 whr I s rlad o h olodal currn. Th vcor onal comonns ar A Ψ, A Iln, A. (6 As sady Lljohn,hr has n a gradual voluon ovr h yars away from h avragng aroach and owards h ransformaon aroach 8. W nroduc a gnrang funcon 9 o chang h cylndrcal varals o gudng-cnr varals: Ω X X F x( (ln gα X (7 Ω Ω whr C X Ωln, (8 Ω s h orodal gyrofrquncy a h magnc axs, ρ h Larmor radus, α h gyrohas, and h suscrs o and c rfr o h valus a h magnc axs and h gudng cnr, rscvly. X and α ar h nw coordnas conjuga o h nw momna: ρ P X + ρsnα + snα, (9 4 P α C ΩC ρ, ( whr P X s h gudng cnr of h coordna, c. Th momnum s ofn ransformd no h coordna durng ara-consrvd canoncal ransformaon 9. Th Hamlonan s rwrn as: H Ω [( C CPα sn α + cos α ] + [ P + Ψ ] + Φ. ( ϕ Th canoncal ransformaon maks h Hamlonan xac n h nw coordnas. Th canoncal gudng-cnr varals, Pα, P, PX, α,, X, ar drvd and sasfy h Hamlonan quaons: H Pɺ, ( q H qɺ, (3 P whr h P and q ar known as h gnralzd momna and coordnas. Th Jacoan s uny for h ara-consrvd ransformaon 9, ha s, J dpα dpx dpd αdxd. (4 For h okamaks, h ordrng s δ ~ ρ / r~ r/ /, (5 whr r and ar h mnor and major rad rscvly and + rcos. To h frs ordr, h gyro-avragd Hamlonan n q. ( s aroxmaly xrssd as: H P Ω + ( + Ψ α c P + Φ ξ c c. (6 W hav a s of quaons of moon n h gudng-cnr sysm: d Ψ, (7 ( + z d, (8 ϕ whr ϕ ( P + Ψ c /, Φ, and Ψ, d ( Ω C µ + Ψ s n h radal drcon whras s Ω n h olodal drcon. quaons (7 and (8 ar h gnralzd vrsons of h quaons of moon oand y alscu. 3. Dynamcs for h Trad Parcls For h rad arcls, h orodal vlocy s smallr han h rndcular vlocy, / /( Ω P c α δ. (9 In h roaon fram, w can consruc a Hamlonan n h dvlod canoncal varals, Pα, P, PX, α,, X, H Ω P + P + Ψ u + u X X X Ω Ω Ω α ϕ os, ( " P + Ψ ρφ ( r whr u, u, S +, T S S ρ s h olodal Larmor radus, / s h ordr of δ, T/ M s h arcl hrmal vlocy, S s h squzng facor, h shar flow. Th dffrn forms of h Hamlonan dscr h sam moon. qs.(7 and (8 can rroducd from q.(. Wh h small nvrs asc rao aroxmaon, w hav
4 hong-tan Wang al.: Dffusvy Scalng on Shar Flow u u sn k, ( [ α ( ], ( u H Ω P r εu u k (. (3 Th r should anana cnr oson. Th urnng ons of h anana or ar dcdd y h followng formula, k sn. (4 Onc w hav k and umax ( ε u for h rad arcls. W s Ψ as h anana cnr surfac whr urnng ons ar valuad and hn w xand Ψ and nar Ψ. Th anana wdh and h oson of anana cnr surfac ar oand afr h xanson, u whr + Ψ + Ω P S P + Ψ S s h anana wdh, u u S S, (5 k sn, (6 Ω S Ω Ψ Ψ + /, (7 whr Ψ P / s drf surfac and s h valu of on h anana cnr surfac. W can s ha h urnng ons of h rad arcl ar no on h drf surfac u shfd o anana cnr surfac du o h radal lcrcal fld. W form an nvaran 9 varal, Ω Ω P dx d r d, (8 Π X c c c whch s acually h flux nclosd y anana or. Now w nroduc a nw angl whch sasfs h quaon, hn, β, (9 sn k sn 4 ( sn k sn d k sn S Ω S Ω ru Π q u k β dβ β,(3 Ω α 8 q ( P [ ( k ( k K ( k ] S whr k k, K and ar coml llc funcons. Th ounc frquncy of h rad arcl s H Π ( εs (. (3 q K( k qs. (-3 gv, for h frs m, a so clar cur of h rad arcl moon dndn on h shar flow. I s ral moon no ohr choc. For a Maxwllan dsruon funcon, h fracon of h rad arcls can asly calculad va xmax xmax y x F yd y dx, (3 whr y /, x /, and S max S u max ( ε. For h rad arcls, x s small, hus lads y F 4 dy y ( εs ( ε S, (33 whch agrs wh Shang s 5 and dsagrs wh Cao s 6. Snc h rad arcl ch angl s lmd, h cv collson frquncy ν hr should ν/( /. Thus, h dffusvy s dffrn wh Shang s 5. u ν / νρ( ε/ 4. Noclasscal Transor F u S whch s To llusra h sgnfcanc of h rad-arcl dynamcs, h arcl dffusvy s calculad. Insad of canoncal gyroknc varals, Pα, P, PX, α,, X, w us xndd has sac varals, Pα, P, r, H, α,, β,, n h drf knc quaon, f dβ f dr f + + C( f. (34 d β d W dfn an avragd angl vlocy, d β d β < > d d T, (35 d T
Amrcan Journal of Modrn Physcs 4; 3(5: -6 5 whr T s h ounc rod. W can s ha h avragd angl vlocy s h ounc frquncy n q. (3. To llusra flow shar cs on h noclasscal ransor, w ak f f + g f + g + g sn β + g cos β, (36 s c whr f s qulrum dsruon funcon n a Maxwllan form wh H a h lac of nrgy and P a h lac of oson, ( [ lnn f ] Fm r, (37 Ω r f. (38 To cach h ky ons and avod h comlxy w only consdr dnsy gradn. Tmraur gradn can chang h xrsson of dffusvy, u can no chang h scalng on h shar flow. q. (34 s rwrn as and β g sn g d C( g gm f β, (39 g m To mak q. (39 racal, w ak lnn Fm. (4 Ω r dβ dβ < >, (4 d d C( g g ν ( g g. (4 m m Furhrmor, s assumd ha k and ar small, whch mans dly rad arcls domna. In q. (36 g s and g c ar on ordr smallr han g. From q. (9 and g kν q. (39 w oan g g m, gs, g + ν. c ν Wh q. (37 and q. (38 h drf quaon of q. (39 afr h ounc avrag urns o f ν k. (43 d Fm + ν In h anana rgm, w hav ν δ. (44 Afr ngrang q. (43 ovr vlocy sac, w g h connuy quaon whr n Γ D, xmax n + Γ, (45 xmax ν k D ydy dx.75 / S d y x ν ε ρ, (46 εs whr x, y, xmax εsy. From q.(46 w can s ha osnluh s rsul 9 s rroducd u wh h shar flow modfcaon. 5. Summary Th ara-consrvd ransformaon roosd y Lchnrg and Lrman9 s mloyd. A coml s of canoncal gudng-cnr varals, Pα, P, PX, α,, X, ar drvd. Th accura rlaon wn h arcl moon and h shar flow for h rad arcls s also drvd, u u ncludng h anana wdh k sn, S Ω S Ω h oson of anana cnr surfac Ψ Ψ + / and h ounc frquncy H ( εs (. For Π q K( k a Maxwllan dsruon funcon, h fracon of h rad arcls s calculad as y 4 ( ( F dy y εs ε S whch agrs wh Shang s5 and dsagrs wh Cao s6. Snc h rad arcl ch angl s lmd, h cv collson frquncy ν hr should ν/( u /, hrfor, h / ( / dffusvy s F ν u νρ ε S whch s dffrn wh Shang s5. Drf knc quaon s solvd wh arcl-consrvd Krook collson oraor. osnluh s rsul7 s rroducd u wh shar flow modfcaon n hs ar. Acknowldgmns I s arcad ha Dr.. D. Hazln and Dr. P. Morrson chckd and vrfd h gnrang funcon whn auhor. T. Wang workd a IFS of h Unvrsy of Txas. Ths work was suord y Naural Scnc Fund No. 6437, No.553, No.535, No.546 and Th Naonal Magnc Confnmn Fuson Scnc Program (Gran No. 3G7.
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