Advances in the study of intrinsic rotation with flux tube gyrokinetics

Transcription

1 Adans n th study o ntrns rotaton wth lux tub gyroknts F.I. Parra and M. arns Unrsty o Oxord Wolgang Paul Insttut, Vnna, Aprl 0

2 Introduton In th absn o obous momntum nput (apart rom th dg), tokamak plasmas rotat Last yar, prsntd low low trms that an xplan ntrns rotaton Ths talk: Show that low low ordrng s ndd: symmtry o transport o momntum n tokamaks Improd ordrng that gs nw ontrbutons: nw salng o turbuln wth / Wolgang Paul Insttut, Vnna, Aprl 0

3 Exprmntal dn ω φ (krad/s) ICF, I p =.6MA ICF, I p =.7MA Ohm, I p =.7MA ms 90ms 90ms 40ms 70ms 30ms (m) JG Cor Ohm: hollow ountr-rotatng Cor ICF Hgh I p : pakd, orotatng Low I p : hollow, ountr-rotatng Edg: o-rotatng ndpndnt o snaro Courtsy o M. F. Na Wolgang Paul Insttut, Vnna, Aprl 0 3

4 Gomtry Hr, s n th o-urrnt drton a Flux sura arag = (V ) dd ( )/ Wolgang Paul Insttut, Vnna, Aprl 0 4

5 Intrns rotaton Assum ltrostats to smply: E = φ Consraton o total torodal angular momntum J or anshs du to axsymmtry and n = n t n V M = V ( V Π) ( J ) Hr Π = ( )( ) 3 d M /t = 0, no momntum nput Π = 0 Wolgang Paul Insttut, Vnna, Aprl 0 5

6 Hgh low ordrng Wolgang Paul Insttut, Vnna, Aprl 0 6

7 Hgh low ordrng To mak = 0 V = Φ b V V = V = Ω ~ t Φ Wolgang Paul Insttut, Vnna, Aprl 0 7

8 δ smulatons Sn lutuaton <<, us = F, φ = φ 0 φ Varabls D Dt = E φ Φ Ω 0 ε =, μ = M M Smlar quaton or ltrons l M E 0 o E ( ) ( ) { b C } M M φ t I Potntal ound rom d 3 d 3 = 0 Ω E Ω M ε Wolgang Paul Insttut, Vnna, Aprl 0 8

9 Wolgang Paul Insttut, Vnna, Aprl 0 9 Momntum transport or hgh lows Turbuln wth k ρ ~, qulbrum wth k ρ << M, M /ε = onstant n omputatonal doman Fourr analyz, and φ n Momntum transport ( ) ( ) Π Ω Ω Ω Ω = Π = Π χ φ ud T n T P T n T M d,,,...,, ;, 3 b

10 Symmtry o momntum transport In up-down symmtr tokamak Ω Ω Ω,,, k, Ω,,, φ,,, k, φ Ω Ω Thus, Ω, Ω, Π Π Ω In addton, Ω = 0 = Π = 0 Ω Hn, or subson Π P Ω χ Wolgang Paul Insttut, Vnna, Aprl 0 0,

11 Physal ntrprtaton Paralll moton o partls ( ),, Π = ( ) 0 Smlar ptur or prpndular damagnt drt Changs sgn du to k k Wolgang Paul Insttut, Vnna, Aprl 0

12 Numral dn (I) Ω = 0 Ω = 0. Wolgang Paul Insttut, Vnna, Aprl 0

13 Numral dn (II) Ω = 0 Ω = 0. Wolgang Paul Insttut, Vnna, Aprl 0 3

14 Intrns rotaton wth hgh low In th hgh low ordrng, only up-down asymmtry an produ ntrns rotaton wth nough strutur P Ω P Ω = Ω χ 0 xp d χ Up-down asymmtry only strong nar dg Up-down asymmtry sms unabl to xplan sgn hangs n rotaton and dpndns wth hatng Stll at ara o rsarh Nd to ontnu to nxt ordr Wolgang Paul Insttut, Vnna, Aprl 0 4

15 Low low ordrng Wolgang Paul Insttut, Vnna, Aprl 0 5

16 Low low ordrng V = n b φ 0 b p V, = V V = Ω K ~ V, ρ a ( ) t T = K ( ) Wolgang Paul Insttut, Vnna, Aprl 0 6

17 Changs n momntum lux Nd to alulat to an ordr hghr n δ = ρ /a Π ~ D g r ( n M V { δ t ) << Π at hgh low Two drnt ontrbutons Nw trms n th xprsson or Nw trms n th gyroknt quaton Wolgang Paul Insttut, Vnna, Aprl 0 7

18 Wolgang Paul Insttut, Vnna, Aprl 0 8 Turbulnt momntum transport Usng momnts o th ull Fokkr-Plank quaton and gnorng nolassal transport o momntum ( ) ( ) ( ) ( ) ( ) { } ( ) φ φ φ Π = ) ( b b b F C d M d V V M d I M t p M d M l

19 Nw trms M V V ( ) ( ) 3 φ b d M p t... ~ d dt ( ) Causd by hangs n wdth o drt orbts M = = onst. gs drt orbt ( ) wdth ( ) nras n wdns drt orbts mods n = n nw Ω Wolgang Paul Insttut, Vnna, Aprl 0 9

20 Wdth o drt orbts For nrasng ( ) E E n = n ( ) ( ) n n r r GK polarzaton E no polarzaton ΔΩ Wolgang Paul Insttut, Vnna, Aprl 0 0

21 Wolgang Paul Insttut, Vnna, Aprl 0 Gyroknt quaton To lowst ordr, du to symmtry To nxt ordr Problmat baus w nd,, φ! Nd hghr ordr trms n gyroknt quaton ( ) ( ) 0 3 = φ b d M ( ) ( ) ( ) ( ) φ φ b b 3 3 d d M

22 Wolgang Paul Insttut, Vnna, Aprl 0 Hghr ordr gyroknt quaton Flux tub quaton to sond ordr ( ) { } ( ) ( ) ( ) { } 0 () () () () () ) ( = E E C E E E F t M F E t M T C t M M n n E M M M M E E M & & & & l b b b b araton prol radal φ φ φ φ φ φ

23 Hghr ordr gyroknt quaton Flux tub quaton to sond ordr Nolassal qulbrum t on turbuln Flow and hat low wthn lux sura Slowly aryng nlop o th turbuln n th polodal drton Paralll nonlnarty adal prol araton Hghr ordr orrtons to gyroknt quaton Wolgang Paul Insttut, Vnna, Aprl 0 3

24 Frst proposd smplaton For / << & turbuln NOT salng wth / ( ) { } ( l) n b M E C = E F... t Nolassal paralll lows and hat lows largr than turbulnt ts by ρ p /ρ ~ / >> Usng Π t n dτ E F Ω,,, P Ω χ Ω K T L Ω T T T Wolgang Paul Insttut, Vnna, Aprl 0 4

25 Numral rsults Wolgang Paul Insttut, Vnna, Aprl 0 5

26 Salng o turbuln wth / Important trms n gyroknt quaton... b E E M = 0 qδ ~ a l ρ φ T ~ ~ M M ~ l a Maxmum Δ ~ φ T ~ ~ M M ~ ρ, a l ~ ρ Wolgang Paul Insttut, Vnna, Aprl 0 6

27 Numral salngs Wolgang Paul Insttut, Vnna, Aprl 0 7

28 Whn ths / salng works Nd to onsdr Slowly aryng nlop o th turbuln n th polodal drton Paralll nonlnarty adal prol araton Hghr ordr orrtons to gyroknt quaton an b ngltd Stll studyng undr what rumstans ths salng works Wolgang Paul Insttut, Vnna, Aprl 0 8

29 Wolgang Paul Insttut, Vnna, Aprl 0 9 Intrns rotaton Addng all th ontrbutons In ordr o magntud omparabl to obsratons 0 sours hatng = Ω Ω Π χ T Z T n N n M T K T L P ~ r T r V

30 Conlusons Nw sl-onsstnt modl or ntrns rotaton that an xplan hang o sgn n rotaton Intrns rotaton dpnds on gradnts o dnsty and tmpratur o both ltron and ons, and on th hatng sour Possbl sours o ntrns rotaton not studd hr ar up-down symmtry and rppl Wolgang Paul Insttut, Vnna, Aprl 0 30