RMP ffcts on pdstal plasma transport J.D. Calln, Univrsity of Wisconsin, Madison, WI 5376-169 in collaboration with A.J. Col, C.C. Hgna, S. Mordijck, R.A. Moyr Issu To B Addrssd: CEMM Mting, Univrsity of Wisconsin, Madison, WI, Jun 12-13, 212 Rol of RMP-inducd fluttr transport 1 at top of H-mod pdstals. 2 Thss: At pdstal top 2 RMPs rduc T mor than n, with changs Icoil 2 ; thy also incras plasma toroidal rotation Ω t. Flow-scrning avrts stochasticity-inducd 3 but not fluttr 1 xport. RMP-fluttr-inducd transport 1 might caus obsrvd transport. 2 Modl implications ar diffrnt at low 4 and high 5 collisionality. M3D-C1 and NIMROD should contribut to ths studis. 1 J.D. Calln, Drift-Wav Turbulnc Effcts on Magntic Structur and Plasma Transport in Tokamaks, Phys. Rv. Ltt. 39, 154 (1977). 2 J.D. Calln, A.J. Col, C.C. Hgna, S. Mordijck, R.A. Moyr, RMP ffcts on pdstal structur and ELMs, UW-CPTC 11-13R (to b pub. in NF). 3 A.B. Rchstr and M.N. Rosnbluth, Elctron hat transport in a tokamak with dstroyd magntic surfacs, Phys. Rv. Ltt. 4, 38 (1978). 4 T.E. Evans t al., RMP ELM supprssion in DIII-D plasmas with ITER similar shaps and collisionalitis, Nucl. Fusion 48, 242 (28). 5 W. Suttrop t al., Studis of dg localizd mod mitigation with nw activ in-vssl saddl coils in ASDEX Upgrad, PPCF 53, 12414 (211). JDC/CEMM talk, UW-Madison Jun 13, 212, p 1
RMPs Rduc Prssur Gradint At Pdstal Top RMP-inducd rductions in P ar: small in cor, 2.5 x 14 2 no RMP largst at th pdstal top, and small (incras!?) at th dg. total prssur (Pa) 1.5 1 ELM supprssd Ky transport issu for ELM supprssion is: How do RMPs rduc P at th pdstal top?.5 cor.82.84.86.88.9.92.94.96.98 1 normalizd flux N pdstal top dg Figur 1: Edg prssur profil without/with RMP ELM supprssion. Courtsy of O. Schmitz, R. Nazikian, 211. JDC/CEMM talk, UW-Madison Jun 13, 212, p 2
RMPs Incras T, n Gradint Lngths At Pdstal Top T, n gradint lngth ratios with RMPs to wo (sym) χ, D: L T ] RMP χrmp L T ] sym + χ sym χ sym ; plus a similar formula for L n, D RMP. Apparntly RMPs incras: χ by < 6, D by < 2. Changs pak at pdstal top:.93 < Ψ N <.97, nar 11/3 surfac. L RMP L sym 8 7 6 5 4 3 2 1 5.2 ka 12644 / ka 126443 9/3 1/3 11/3 12/3 13/3 small ffcts.9.92.94.96.98 1 Ψ N L L T n dnsity pumpout Figur 2: Ratio of T and n gradint scal lngths with RMPs to without (sym) vrsus radius. 2 JDC/CEMM talk, UW-Madison Jun 13, 212, p 3
Pak Of RMP-inducd Extra Transport I 2 coil Pak T, n gradint lngth ratios scal approximatly with I 2 coil. Pak L T ratio incrass 3 mor than pak L n ratio dos, which indicats D RMP /χ RMP 1/3. 15 pak pak Ln ratio L T ratio 1 5 1 1 2 2 3 4 4 4.8 2 2 5.22 5.82 6.22 I coil Figur 3: Pak ratios of L T and L n with to wo RMPs vs. squar of I-coil currnt. 2 JDC/CEMM talk, UW-Madison Jun 13, 212, p 4
Toroidal Rotation Of Carbon Incrass With I coil Whn ELMs ar supprssd with I coil > 4. ka in DIII-D ISS dischargs, 4 carbon toroidal rotation at pdstal top jumps up to plasma toroidal flow V tor > 2 1 4 m/s (Ω t V tor /R 1 4 s 1 ). Incrass in V tor inducd by RMPs do not systmatically incras with I coil, but ar always larg whn ELMs ar supprssd (I coil 4.7). Changs ar largst at th pdstal top: nar Ψ N.96. v tor a.u.] 5 4 3 2 1 1. 4. 5.2 6.2 4.7 5.8 2.9.91.92.93.94.95.96.97.98.99 1 Ψ N Figur 4: Carbon toroidal rotation in dg as function of I coil. Fig. 4.19b in S. Mordijck thsis, UCSD 211. JDC/CEMM talk, UW-Madison Jun 13, 212, p 5
Expt. Summary: 2 Thr Ar Som Ky RMP-inducd Plasma Transport Effcts That Nd To B Explaind RMPs incras gradint scal lngths and lctron diffusivitis: L T, L n I 2 coil implis χrmp, D RMP Icoil 2 (χrmp L T /L n 1/3 implis D RMP χ RMP /3, both ar largr than valus without RMPs (χ sym 4 m 2 /s at I coil = 5.2 ka),.6 m 2 /s), and thir incrass ar localizd to pdstal top rgion (.93 < Ψ N <.97). In DIII-D low collisionality ISS dischargs 4 plasma toroidal rotation Ω t incrass at pdstal top whn thr is ELM supprssion. RMP ffcts apparntly dpnd on collisionality: low 4 (DIII-D) ELMs ar supprssd in narrow q 95 rsonanc windows, high 5 (ASDEX-U) ELMs mitigatd at high dnsity, no rsonanc ffcts. JDC/CEMM talk, UW-Madison Jun 13, 212, p 6
Transport Effcts Of RMPs: Fluttr Or Stochasticity? RMP-inducd radial (ρ) magntic prturbations δb ρ : mostly just non-rsonantly spatially fluttr th fild lins, flux surfacs, but can induc stochasticity if islands ar cratd and ovrlap (Chirikov). Transport can b inducd by magntic fluttr and stochasticity: fluttr causs 1 χ δb v T λ (δb ρ /B ) 2 via finit collisional ( Braginskii) paralll lctron hat conduction lctron collision-inducd irrvrsibility; stochasticity causs (RR 3 ) χ δb st v T L c (δb st /B ) 2 via motion along B st. But flow scrning of RMP filds inhibits rconnction, island formation & ovrlap, and hnc stochasticity nxt 2 viwgraphs. Nonthlss, δb ρ off rational surfacs inducs fluttr xport. 1 JDC/CEMM talk, UW-Madison Jun 13, 212, p 7
Flow Scrning Rducs δb ρ At Rational Surfacs, But RMPs Induc Many δb ρ m/n At Pdstal Top Rsonant δb ρ 1/3 scrnd ( 3) at 1/3 surfac = no island. But, othr δb ρ m/n /B 3.3 1 4 MPs ar nonzro thr. 8 7 m = 1 solid lins m = 12 dashd lins 6 5 δb ρ 4 Rsonant 3 (m/n = 1/3, 12/3) 2 1 (a)..2.4.6.8 1. N 1/3 m = -1 solid lins m = -12 dashd lins Non-rsonant (~ vacuum) vac -6-7 -8 idal q (b)..2.4.6.8 1. Figur 5: MARS-F idal/rs. RMP-inducd radial δb ρ m/n profils: flow-scrnd (lft), vacuum (right). Fig. 3 in M.S. Chu t al., Nucl. Fus. 51, 7336 (211). N JDC/CEMM talk, UW-Madison Jun 13, 212, p 8
Two-fluid Modling 6 Yilds Lss Flow-Scrning M3D-C1 visco-rsistiv 2-fluid modling 6 is mor ralistic. In Fig. 6 solid lins ar with flow and th dashd lins ar in vacuum. 6 N.M. Frraro, Calculations of two-fluid linar rspons to non-axisymmtric filds in tokamaks, DPP invitd talk, Phys. Plasmas 19, 5615 (212). Faturs of say 11/3 (purpl) RMP fild in rotating 12644 plasma: it is scrnd by 5 nar 11/3 surfac, its amplitud grows linarly away from minimum, it xtnds ovr many m/n rational surfacs. 8/3, 9/3 RMP filds ar not flow scrnd sinc V at Ψ.86. 8/3 9/3 1/3 11/3 12/3 Figur 6: Spatial variation of δb ρ m/3 (G/kA) for 12644. Courtsy of N. Frraro, March 212. JDC/CEMM talk, UW-Madison Jun 13, 212, p 9
A Modl Fild Will B Usd For Flow-Scrnd δb ρ Ky charactristics of flow-scrnd RMP m/n filds ar: on rational surfacs thir magnituds ar rducd from thir vacuum valus, thy incras linarly with distanc x = ρ ρ m/n off rational surfacs, idal MHD rsponss outsid dissipativ layr (δ µη.5 cm) grow linarly until thy rach vacuum valus at cylindrical analytic stimat of x 1/k θ. Dfin a flow scrning factor as th ratio of vacuum δb ρ m/n to its flow-scrnd valu on th rational flux surfac q(ρ m/n ) m/n: f scr ] δb vac ρ m/n δb plasma ρ m/n ρ m/n flow scrning factor 2 in th plasma ( 1); (idal MHD), 3 (rs. MHD), 4 (2-fluid 6 ). Cyl. modl of flow-scrnd m/n RMP in plasma is 2 ( x < 1/k θ ) δb plasma ρ m/n (x) δbρ vac m/n (ρ m/n) = 1/f 2 scr + k2 θ x2 { 1/fscr, x 1/k θ f scr, k θ x, 1/k θ f scr x 1/k θ. JDC/CEMM talk, UW-Madison Jun 13, 212, p 1
What Physics Causs Magntic Fluttr Transport? 1,2 Considr paralll hat conduction in a prturbd B fild: Rprsnt magntic fild by axisymmtric B plus RMP fild: B = B +δ B. Radial prturbation δ ˆB ρ m/n (x) cos(mθ nζ) inducs finit hat flow along B nar m/n surfac. For T = T (ρ) and nglcting magntic shar k (x)v T < ν] δ q m/n n χ ( B B B 2 T ) n χ B δ ˆB ρ m/n B 2 cos(mθ nζ) dt dρ B + δ B B δq m/n. Avrag radial (ˆ ρ ) hat flow inducd by collisional hat flow along B is ˆ ρ q ˆ ρ B + δ B B δq m/n = δ ˆB ρ m/n cos(mθ nζ) δq m/n. This rsults in a radial lctron thrmal diffusivity of 1,2 χ m/n = 1 2 ( δ ˆB plasma ρ m/n B ) 2 χ, which is to b summd ovr all mn magntic prturbations. Vry small RMP-inducd filds can induc significant χ m/n For DIII-D pdstal 2 1 m2 χ 1 s, so δ ˆB plasma ρ m/n /B 2 > 1 5 yilds χ m/n valus: But magntic shar rducs ffctiv χ for x > δ.2 cm. > 1 m2 s. JDC/CEMM talk, UW-Madison Jun 13, 212, p 11
Various Spatial Scal Lngths Can B Important Radial distancs from 11/3 rational surfac in trms of th radial coordinat ρ ψ t /πb t for DIII-D pdstal top paramtrs 2 (distancs on outboard mid-plan ar about half ths numbrs): ion sound gyroradius width for small magntic shar ffcts on χ rsistiv MHD layr width visco-rsistiv (2-fluid 6 ) MHD layr width χ causs T to not follow island topology 7 magntic island half-width (for f scr = 4) distanc btwn rational surfacs Wc T ρ S.2 cm δ.2 cm δ η.2 cm δ µη.5 cm /2.5 cm W/2.7 cm 1/nq 2.8 cm radial xtnt of δ ˆB plasma ρ m/n (x) incrasing with x 1/k θ 6.7 cm 7 R. Fitzpatrick, Hlical tmpratur prturbations associatd with taring mods in tokamak plasmas, Phys. Plasmas 2, 825 (1995). JDC/CEMM talk, UW-Madison Jun 13, 212, p 12
Fluttr Transport Modl Employs Som Assumptions 1) RMP-inducd prturbations ar gyroradius small: δb ρ /B ρ S /R < 1 3. Effcts ar linarly indpndnt, quasilinar-typ. 2) Us flow-scrnd RMPs from linar xtndd MHD cods xplor slf-consistncy with nonlinar RMP-inducd torqu latr. 3) If magntic islands ar prsnt, thy ar thin, isolatd ons that do not ovrlap. (Modl okay outsid islands that do occur.) 4) Intrplay btwn ffctiv lctron collision frquncy ν ff and paralll straming k (x) v T causs T ρ dt /dρ, i.., T to b roughly constant on axisymmtric (not hlical) flux surfacs. 5) Significant flow-scrnd RMP-fluttr rsponss xtnd radially ovr a numbr of rational surfacs: m 2nq /k θ ( 5). JDC/CEMM talk, UW-Madison Jun 13, 212, p 13
RMP-fluttr Inducs Radial Elctron Hat Transport 2 A cylindrical scrw pinch modl of th radial plasma transport inducd by RMP-inducd fluttr has bn dvlopd; 2 it yilds χ RMP mn χ m/n, D RMP < χrmp /2.5, χ m/n χ rf m/n F m/n(x), rfrnc χ rf m/n tims spatial factor F m/n, χ rf m/n 1 2 δ ˆB vac ρ m/n B ] 2 χ rf c kθ δ c ] 2 ρ m/n.1 m2 /s at DIII-D pdstal top, 2 χ rf c 5 4 nut δ c nut n F m/n (x) n k θ δ c ] 2 ] 3 ν λ 2 1.45 19 m 2 /s, Braginskii with ut particls, ] 2 2 LS.22 cm, λ v T, lctron collision lngth, k θ λ ν 1 δ ˆB plasma ρ m/n (x) ] 2 χ ff (x) δ ˆB vac ρ m/n (ρ m/n) χ rf c } {{ } G c 1/(1+x 2 /δ 2 c ) 1/k2 θ f 2 scr + x2 δ 2 c + x2, in which x ρ ρ m/n is radial distanc off th rational surfac. JDC/CEMM talk, UW-Madison Jun 13, 212, p 14
Spatial Factor F m/n (x) Varis A Lot And Is Important F m/n (x) has diffrnt scalings in various radial rgions ( x < 1/k θ ): F m/n (x) (δ ˆB plasma ρ m/n /δ ˆB ρ vac m/n )2 magntic shar ffct = 1/f scr 2 k2 θ + x2 δ c 2 + x2 larg, x δ c, x 2, transition, 1, x 1/f scr k θ. This causs diffrnt bhavior for various scrning modls: Rsistiv MHD with f scr 3, δ c 1/f scr k θ = F m/n 1 = constant. Visco-rs. 2-fluid 6 MHD with f scr 4 = highly pakd, F m/n () 6. Collisional pdstal 5 with δ c 1/f scr k θ 2 cm = also F m/n 1. This spatial factor has a significant ffct on th T gradint: 2 dt dρ P /V ρ 2 n χ RMP = P /V ρ 2 n mn χrf m/n F m/n(x) 1 mn F m/n(x). JDC/CEMM talk, UW-Madison Jun 13, 212, p 15
Spatial Profil Of F m/n Modifis Prdictd T Profil 2 Obtain T profil by intgrating dt /dρ inward from 11/3 surfac: ρ T (ρ) = T (ρ 11/3 ) dρ P /(n χrf m/n V ρ 2 ) ρ 11/3 mn F, avrag χ m/n(x) T / ρ, which yilds 2 an avrag χ RMP of.5 m 2 /s for f scr = 3, 1.2 m 2 /s for f scr = 4. 1 1.15 1 f scr = 4 T 1.1 1.5.1 f scr = 3 1/3 11/3 X 5 1 15 2 25 1. 5 1 15 2 25 X 1/3 11/3 Figur 7: χ RMP radial variation btwn 1/3, 11/3 surfacs for cyl. (dashd) & toroidal (solid) modls (X x/δ t ). Figur 8: T profils btwn 1/3 and 11/3 surfacs for cylindrical (dashd) & toroidal (solid) modls for f scr = 4. JDC/CEMM talk, UW-Madison Jun 13, 212, p 16
RMP Fluttr Inducs Toroidal Torqus On Edg Plasma Plasma toroidal momntum balanc 8 for L t m i n i R 2 Ω t is: L t t = T ζ, torqus T ζ q s Γ s ψ p causd by non-ambipolar fluxs Γ s. Ion & lctron 3D dnsity fluxs caus oppositly dirctd torqus: Γ i (NTV, rippl) crat 8 countr-currnt torqus bcaus q i = + (J ρ < ), whras lctron dnsity fluxs crat co-currnt torqus bcaus q =. RMP-fluttr inducs a non-ambipolar radial lctron dnsity flux and hnc a co-currnt torqu (for offst frquncy Ω fl > Ω t ): T RMP ζ Γ RMP ψ p n m i R 2 L 2 B S p 2 ν 6ρ 2 S B 2 fluttr offst frq. Ω fl 1 n ( δ ˆB vac 2 mn ρ m/n B 2 F m/n )(Ω fl Ω t ), dp 1 4 s 1 (> = V t dψ c T t dt p B dρ ). Low ν RMP-inducd torqu is localizd nar rational surfacs. 8 Eqs. (3) (7) in J.D. Calln, Effcts of 3D magntic prturbations on toroidal plasmas, Nucl. Fusion 51, 9426 (211). JDC/CEMM talk, UW-Madison Jun 13, 212, p 17
Cylindrical Modl Has Bn Compard 2 To DIII-D Rsults Fluttr prdictions in qualitativ agrmnt with xprimnt ar: scaling of χ RMP and D RMP with δ ˆB 2 ρ I2 coil, ratio of D RMP /χ RMP < 1/3, and RMP-inducd incras in toroidal rotation (E ρ ) to Ω t 1 4 s 1 >. But cylindrical χ RMP is smallr than xprimnt at pdstal top: in DIII-D χ RMP xp 4 m2 /s (#12644, 5.2 ka) whil χ sym xp.6 m2 /s (#126443), vrsus χ RMP (5 12)(1 7 ) (1 6 ) <.5 1.2 m2 /s; but should us and sum flow-scrnd δ ˆB plasma ρ m/n (x) + all possibl mn filds. Additional routs to improvd agrmnt with xprimnt: toroidal kintics modl and/or δ ˆB plasma ρ m/n could incras magnitud of χrmp, with q 95 3.5 rsonanc if 1/3, 11/3, 12/3 RMP rsponss ar modifid. JDC/CEMM talk, UW-Madison Jun 13, 212, p 18
Toroidal 9 Modl Yilds Flows & Radial Transport Fluxs RMP-inducd paralll lctron flow and conductiv hat flux ar 9 (y v 2 /v 2 T ) ] n δv = δq u d 3 v v L (3/2) L (3/2) 1 T ] δf = 2π B B max v 3 1 dv T (y 5 2 ) ] 1 dλ R{δh u }. Ths RMP-inducd flows caus radial lctron dnsity and hat fluxs: δγ m/n ρ δυ m/n ρ ] = ] δ Γ ρ = d 3 1 v δf δ q ρ u T (y 5 2 ) ] n δv δ B ρ = n δq B t D m/n χ m/n n ] v gc ρ D m/n T χ m/n ] = d 3 1 δ B ρ v δh u v u T (y 5 2 ) B ] ] d ln p /dρ (/T )dφ /dρ. dt /dρ Transport cofficints hr ar dfind by kintic cofficints matrix K ij : D m/n χ m/n n K K 1 K 1 K 11 D m/n T χ m/n ] ] = B t B max ( ) 2 ] = v2 T 1 δ ˆB ρ m/n K K 1, ν 2 K 1 K 11 B t dy y 3 y 2 π 1 dλ R{Λ} 1 y 5 2 y 5 2 (y 5 2 )2 ], in which y v 2 /v 2 T. 9 J.D. Calln, A.J. Col and C.C. Hgna, Rsonant-magntic-prturbation-inducd plasma transport in H-mod pdstals, UW-CPTC 11-15, May 1, 212 (submittd to Phys. Plasmas, availabl at http://www.cptc.wisc.du). JDC/CEMM talk, UW-Madison Jun 13, 212, p 19
Kintic Matrix Cofficints K ij Rsult From Λ Solutions Rsultant kintic cofficints and thir rgims of applicability ar k : k : K K 1 K 1 K 11 K K 1 K 1 K 11 ] ] = 2 3 π f c k = = k for y > y min max B max B t 6 9 9 75/2 1 B t /B max X t 3/2 8 π v λ=1 /v } { 1 X t 1/2, 1 X 1/2 crit ], for y < 1 X t 1/2 in which X t x δ t, 1 3/2 3/2 13/4 ],, X crit B t 2B max λ Rq 22 (so νk λ ff < ω ut). Normalizing th kintic cofficints to (13/4)K k λ 11 and using nrgy smoothing 1 to dvlop a Padé approximation yilds for th kintic cofficint matrix K K 1 ] = 13 32 π K 1 K 11 tot ] G G 1 4 = G 1 G 11 13 X t 3/2 B t /B max v λ=1 /v ( Xt 3/2 c t G G 1 G 1 G 11 1/ Xt 1/2 dy y 3 y + ], in which cofficint.38 and matrix is y min dy y ) 1 y 5 2 y 5 2 (y 5 2 )2 ], c t.94. Proprtis of th gomtric cofficint matrix that dtrmin diffusivitis ar lim G 11 = 15 = χ rf t 3.3 1 1, for X t, X 13 c t G G 11 4 13 Dm/n = χ m/n 1 3.25. 1 K.T. Tsang and J.D. Calln, Smooth transition of noclassical diffusion from th banana to Pfirsch-Schlütr rgim, Phys. Fluids 19, 667 (1976). JDC/CEMM talk, UW-Madison Jun 13, 212, p 2
Toroidal Rsults 9 Ar Similar To Cylindrical Modl Paralll lctron thrmal diffusivity on a rational surfac: toroidal, Lorntz collision valu is largr χ rf t 3.3 1 1 m 2 /s, whil χ rf c 1.5 19 m 2 /s. Layr width byond which magntic shar ffcts dominat: toroidal width is smallr δ t.11 cm, whil δ c.22. Spatial dcay of ffctiv paralll diffusivitis away from rational surfac: toroidal dcays slowr with x δ toroidal G ij x 3/2, whil cylindrical G c x 2. Magnitud of a singl χ m/n using δ ˆB plasma ρ m/n midway btwn rational surfacs: (x) on p 1, thy ar about qual χm/n t.23 m 2 /s, whil χ m/n c.21 m 2 /s; and using δ ˆB plasma ρ 11/3 (x) profil from Fig. 6, both stimats giv χ11/3 2 m 2 /s at Ψ N.955. Ratio of RMP-inducd lctron dnsity to lctron thrmal diffusivity: rsults ar similar D m/n t /χ m/n t 1/3.25, whil D m/n c /χm/n c 1/2.5. Radial RMP-fluttr-inducd lctron transport fluxs: toroidal fluxs hav off-diagonal K ij matrix lmnts, whil cylindrical modl fluxs don t. JDC/CEMM talk, UW-Madison Jun 13, 212, p 21
Som Prdictions Ar Diffrnt At High Collisionality ASDEX-U 5 lctron collision frquncy ν is > 1 gratr which 1) incrass shar-ffcts width paramtr by a factor 1 to δ 2 cm, 2) incrass rconnction layr width by a factor 2 to δ µν 1 cm, 3) causs most smoothing procsss to xcd half of th distanc btwn rational surfacs and hnc ovrlaps th ffcts around various m/n surfacs this causs q 95 rsonanc ffcts and magntic islands to b lss likly. Modl prdictions for approximat ASDEX-U conditions ar: 1) χ RMP ν L 2 S 2 mn ] δ ˆB ρ vac 2 m/n > B 1 m2 /s, L S R q s magntic shar lngth, 2) which rducs gradints throughout th pdstal if it xcds a typical lvl of D η ν δ 2 transport thr and yilds an ELM mitigation critrion: δ 2 c2 ω 2 p 3 119 1 6 < L 2 S n 2 mn δ ˆB vac ρ m/n B ] 2 = n > 5 119 m 3? JDC/CEMM talk, UW-Madison Jun 13, 212, p 22
Summary Exprimntal ffcts of ELM-supprssing RMPs on pdstals ar: 2 rducd P at top via incrasing L T ( < 6) & L n ( < 2) I2 coil, Ω t chang. Flow scrning inhibits RMPs in forming islands, stochasticity. But RMP-fluttr inducs 1,2 radial transport at pdstal top: both cylindrical 2 scrw pinch and toroidal 9 modls hav bn dvlopd rsults ar qualitativly similar and diffr quantitativly mostly by < 2; in high collisionality pdstals q 95 rsonancs and islands ar lss likly. Comparison btwn fluttr modl prdictions and DIII-D data: 2 χ RMP I 2 coil, DRMP /χ RMP < 1/3 and Ω t changs agr qualitativly, but nd incras of cyl. χ RMP by > 3 and q 95 3.5 rsonanc via modifid 1/3, 11/3, 12/3 RMP rsponss (δ ˆB plasma ρ m/n profils in Fig. 6 may do both). JDC/CEMM talk, UW-Madison Jun 13, 212, p 23
What Ar Nxt Stps For Modl Tsting, Validation? 1) Us M3D-C1 flow-scrnd δ ˆB plasma ρ m/n (ρ) to prdict RMP-fluttrinducd transport and compar to ONETWO intrprtiv transport modling of lctron thrmal and dnsity diffusivitis, to s if modl magnitud, scaling ar corrct & it can captur q 95 rsonancs. 2) Explor low vrsus high collisionality RMP cass with M3D-C1 RMP modling plus ONETWO intrprtiv transport modling, to undrstand diffrncs btwn RMP ffcts at low, high collisionality. 3) Employ quasilinar fluttr modl toroidal torqu in M3D-C1 to xplor its ffcts on toroidal rotation at th pdstal top, to dtrmin if fluttr-inducd torqu plays a significant rol in RMP ffcts. 4) Ultimatly, trat toroidal rotation (radial lctric fild) on par with n, T transport by coupling M3D-C1 to prdictiv ONETWO, to dvlop a prdictiv capability for RMP ffcts in ITER plasmas. JDC/CEMM talk, UW-Madison Jun 13, 212, p 24
Linar Aras Whr M3D-C1 & NIMROD Can Hlp What ar major influncs on linar RMP rsponss δ ˆB ρ m/n (x)? what dtrmins th dgr of flow-scrning in th visco-rsistiv rgim? Col-Fitzpatrick-typ modl 11 apparntly prdicts f scr ωτ δ 1? why dos δ ˆB ρ m/n (x) incras to RMP vacuum lvl by nxt rational surfac is this causd by toroidal mod coupling and/or nar-sparatrix shaping? dos th larg positiv gradint of th paralll currnt inducd by th dg bootstrap currnt play a significant rol in th structur of δ ˆB ρ m/n (x)? would a M3D-C1, NIMROD bnchmarking xrcis b hlpful, dsirabl? Kintic ffcts produc som diffrncs in linar RMP rsponss: only untrappd particls carry paralll lctron currnt and hat flux this rducs χ rf but incrass layr width by > 4; fix by η, χ? magntic rconnction and island formation occur at V 3/2 dt B dρ > fix by changing p in Ohm s law to p (1 + 3/2 1+1/η ) in which η d ln T d ln n? ρ m inrtia incrass by q 2 /ɛ 1/2 for ω > ν i /ɛ which incrass layr width by 2. 11 A.J. Col and R. Fitzpatrick, Drift-magntohydrodynamical modl of rror-fild pntration in tokamak plasmas, Phys. Plasmas 13, 3253 (26). JDC/CEMM talk, UW-Madison Jun 13, 212, p 25
Nonlinar Aras Whr M3D-C1 & NIMROD Can Hlp Thr ar 4 fundamntal nonlinar xtndd MHD RMP issus: is th dgr of flow-scrning modifid by th nonlinar ffcts? ar th RMP-inducd δ ˆB ρ m/n (x) profils diffrnt from th linar ons? ar th radial T and n diffusivitis and toroidal torqu as prdictd? 2,9 how do RMPs caus Ω t to jump up in ELM-supprssd pdstals? inducd torqu or nar-sparatrix magntic-stochasticity lctron losss? Som ky ingrdints for xtndd MHD nonlinar studis ar idntification of main contributors to structur of linar δ ˆB ρ m/n (x) profils, implmntation of low collisionality collisional viscous forcs 12 in th cods, dtrmination of RMP-inducd p, T and Ω t profils on transport tim scal, and ultimatly computd kintic-basd closurs for paralll hat flux tc. 12 J.D. Calln, Viscous Forcs Du To Collisional Paralll Strsss For Extndd MHD Cods, UW-CPTC 9-6R, Fbruary 1, 21. Availabl via http://www.cptc.wisc.du and as supplmntary matrial in Rf. 11] in J.D. Calln, C.C. Hgna and A.J. Col, Phys. Plasmas 17, 56113 (21). JDC/CEMM talk, UW-Madison Jun 13, 212, p 26
Supplmntary Viwgraphs Kintic Toroidal Modl JDC/CEMM talk, UW-Madison Jun 13, 212, p 27
Toroidal: 9 RMP-Inducd Fluttr Modifis Distribution Elctron drift kintic quation for v gc v B/B + v d with B B + δ B is 2, ] f t + v B ( B + δ B) + v d f + dε dt f ε = C{f }, ε m v 2 2 Φ = m v 2 2 + µb Φ. Lowst ordr solution is a Maxwllian constant along B : f = f M (ρ, ε). For δb ρ ( x, t) mn δ ˆB ρ m/n (ρ, θ) R{ i(nα ωt+ϕ m/n ) } with hlical variabls ρ, θ, α ζ (m/n)θ, quation for δh δf (/T )δφf M inducd by δ ˆB ρ m/n is 9 v ( B θ) δĥm/n B θ + i (m nq) } {{ } (dα/dθ) ( / ζ) δĥ m/n i(ω ω d ) δĥ m/n C{δĥ m/n } = v df M δ ˆB ρ m/n B dρ. Trappd particl solution δh t vanishs bcaus bounc avrag yilds no driv sinc trappd particls don t carry any paralll flow ovr λ 2πR q. Magnituds of frquncis indicat ω and ω d can b nglctd ( ν ff ): v T /R q 3 1 6, k v T ( 8) 1 6, ω 1 4, ω d 1 4, ν ff 2 1 5. JDC/CEMM talk, UW-Madison Jun 13, 212, p 28
Bounc-avraging Yilds 9 Non-adiabatic Rspons Eq. Nglcting ω, ω d & oprating on δĥm/n qn. with π π dθb /v B θ] yilds ik v δ h u 2ν B max B t λ ( λ v v δ h u λ ) = v δ ˆB ρ m/n B t f M ρ, in which is FSA, λ µb max, ε k (x) nq(ρ) m/n] Rq k x, k k θ L S, δ h u = π π dθ 2π δh u, δ ˆB ρ m/n δ B ρ inα. Sinc δ h u is a mostly sparabl function, it is usful to writ it as: δ h u D(ρ) V (v, ρ) Λ(λ, x, v), in which D δ ˆB ρ m/n B t, V = v ν (v) d f M dρ = v4 v T d f M vt 4 ν dρ. Thn, quation abov rducs to an quation for th pitch-angl function Λ: λ ( λ v v ) ( Λ k (x) v i λ 2 ν (v) B t B max ) Λ = 1, with B.C.: 1) Λ(λ = 1) =, 2) Λ() finit. This quation will b solvd for Λ(λ) in two limits: k = on q = m/n rational surfac, which will yild Braginskii-lik rsult for χ, and k for x ρ ρ m/n δ whr magntic shar ffcts bcom dominant. JDC/CEMM talk, UW-Madison Jun 13, 212, p 29
Pitch-angl Function Λ Can B Obtaind 9 In Limits On a rational surfac k = and Λ quation can b intgratd dirctly: Λ k =(λ) 1 λ dλ v (λ, θ) /v = 1 dλ Λ k = = 1 λ dλ 1 4 Bmax 2 λ B/Bmax 3 B 2 f c. For larg k th solution will b localizd in λ nar th untrappd-trappd particl boundary whr λ < 1. Thus, dfining λ 1 λ, quation bcoms 2 Λ k λ 2 i 2 k 2 λ k k Λ 1 k = v λ=1 /v, in which k λ(x, v) k (x) v (B t /B max ) 4 ν (v) v λ=1 /v Complmntary solutions of this quation ar of form ± ±2i k λ λ = ±(1±i) k λ λ. Boundary-layr-typ particular solution that satisfis boundary conditions is Λ k = (i/2k2 λ )(k / k ) v λ=1 /v 1 k λ λ (cos k λ λ i k k sin k λ λ) Ky paramtrs of this solution can b writtn as 1/2 v2 k λ (x, v) = X t, X vt 2 t x, δ t c t δ t k λ ]. ] 1/2 = c tl S k θ λ.11 cm, c t 4 v λ=1 /v B max B t. Pitch-angl intgral of this solution for k λ λ 1 (v 2 /v 2 T 1/ X t 1/2 ) is 1 dλ R{Λ k } = 1 2k 2 λ v λ=1 /v 1 dλ k λ λ sin k λ λ 1 4 k 3 λ v λ=1 /v. JDC/CEMM talk, UW-Madison Jun 13, 212, p 3.