JP Sta,onary Density Profiles in Alcator C Mod

Transcription

1 JP Sta,onary Density Profiles in Alcator C Mod 1

2 In the absence of an internal particle source, plasma turbulence will impose an intrinsic relationship between an inwards pinch and an outwards diffusion resulting in a stationary density profile determined by the turbulent equipartition (TEP) theory, i.e. curvature pinch. The Alcator C-Mod tokamak utilizes RF heating and current drive so that fueling only occurs in the vicinity of the separatrix. Density is determined from Thomson scattering. Discharges that transition from L-mode to I- mode are seen to maintain a stationary profile. For reversed shear discharges maintained by non-inductive current drive (V loop "0) a drop of density is observed in the vicinity of the axis consistent with an observed rise in q, although error in the measurement precludes making this observation definitive. 2

3 Is tokamak equilibria over constrained? Turbulence will intrinsically establish a stationary density pro4ile as a balance of turbulent pinch & diffusion. Fuelling is con4ined to plasma edge. Stationary pro4ile depends on details of turbulent spectrum. Turbulent transport in combination with self heating will establish the pressure and bootstrap current pro4ile. Power source requires bootstrap fraction > 80%. For power source, resulting equilibrium must fall in a window between ignition and stability to disruptions Design and Control knobs High 4ield raises disruption limit so as to enlarge window. Feedback can enlarge window. Current drive adjusts current pro4ile; but ef4iciency is low Fueling: Gas puf4ing at edge. Can density pro4ile be adjusted? (Pellet heating?)

4 Turbulent equipartition: Assume ω <ω b < Ω c F-P eq. with conservation of µ and j : " "t f = "# " "f µ, jd($,#) "# µ, j "(µ, j)=#d $ %f! D=g(")h(#) Set. λ is the pitch angle variable (λ=µ/ε). Simple case: DIPOLE Interchange modes, no λ dependence: D D 0 Integrate noting "=#D 0$ %(nv') %$ and! "" dµdj= " dl/b " d 3 v V'= dvolume dψ = d /B Turbulence will tend to flatten gradients in N=n e V leading to a peaked density profile (equal # per unit flux). %$ µ, j! "#0, n e $1/V' Ref: Baker and Rosenbluth, PoP 5 (1998), Baker et al., NF 40 (2000)

5 LDX: strong pinch observed during levitation Turbulent D drives pinch "=#D d(nv') d$ V'= dl/b % (Eφ, D = R2 E 2 " # corr $0.047 V 2 /s τ corr from edge probes) Probe measurements match pinch time of ~20 ms. A.C. Boxer, et al., Nature Phys. 6, 207 (2010). 5

6 Tokamak: pitch angle dependent D(ψ,λ). Assume "(µ, j)=#d $ %f %$ µ, j g p is ratio of passing to trapped particle diffusion For steady state η=η(g Γ 0 dlnn e (dln(qh)η ) p ) (below) For Γ 0 n e 1/V' η(g p ) dρ D=h(ψ)g(λ,g p ) V' dl/b=( Vol/ ψ)=( φ/ ψ)( Vol/ φ) R 0 q(ψ)/b 0 n e 1/q(ψ) η dρ Ref: Baker et al., NF 40 (2000). 6

7 Ware pinch: No pinch when E φ ~0 Curvature pinch with Tokamak: Compare to Ware Pinch V W =!ce! /B "!"#D $n V' e =#D(V' $n $! $! +n $V' e $! )%V " =#Dn e $V' $! D QL =R 2 E! 2! " cor V'=!(Vol)/!(pol flux)=2"r 0 q/b 0 Turbulent curvature pinch is in the q direction V κ q ψ Turbulent pinch will dominate when E!!2"a "lnq (E φ 0 with LHCD) "# DQL 7

8 Quasilinear deriva,on Results from GS2 run in ES, limit**, agree with quasilinear theory) Write the 4lux as Terms: 1 st diffusive; 2 nd thermo diffusion; 3 rd curvature pinch Coef4icients obtained from nonlinear gyrokinetic codes. From QL theory: Γ QL P = n ed QL ( 1 n e n e ρ + C T D QL (ψ, E, λ) ΣA n φ n 2 C T Im C P 1 Im k ρ ( d 3 vf M E 3/2 iγ+ω k v ω d Ê Im( d3 vf M 1 iγ+ω k v ω d Ê ( d 3 vf M ω+iγ iγ+ω k v ω d Ê Im( d3 vf M 1 iγ+ω k v ω d Ê 1 T e T e ρ C P R Ware pinch is relatively small and proportional to E φ : Γ Ware n e (ce φ /B θ ) ) ** Ref E. Fable et al., PPCF 50, (2008). 8 ) ) ) )

9 v Approximate F M e 2 /2T +v 2 /2T Take to account for weak passing particle response. T <T θ is the ballooning angle. θ 0 de4ines the Gaussian trial function. ω d = k ρ( cos(θ) +ŝ θsin(θ) α sin(θ 2 ) ) " = " G d# / $ G d# G =exp[%# 2 /4# 2 $ 0] k = 1 q d dθ ˆk(a/Rq)/(2θ 0 ) Ê v 2 + v2 /2 9

10 Results from gyrokine,c code agree with quasilinear theory** GS2 gyrokinetic code run in electrostatic, linear limit with gyrokinetic electrons and ions. Γ QL P ** Ref: E. Fable et al., PPCF 50, (2008). = n ed QL ( 1 n e n e ρ + C T ε=r/r=0.11 q=3, s=rq /q= 0.7 determined α= q 2 R dr dβ by calculation T e /T i =2.8 1 T e T e ρ C ) P R Thermo diffusive pinch 10

11 Example: A=3, R=0.7, L t =0.1, L n =0.1, q=2, k=1, kρ=0.1, s=1, α=0.5, γ=0.1 ITG TEM C Outwards T ω Inwards C P T /T " =0.5 Curvature pinch(c P ) in direction. "n e Thermal pinch (C T ) changes sign in TEM (ω>0) region. 11

12 C Mod I mode C mod density pro4iles from Thomson scattering for a discharge ( ). Plasma shown in a) L mode, b)i mode and c)h mode respectively. Solid line is with η=0.7. n e "1/V' #

13 V Loop P LHCD LHCD power (solid) and loop voltage (dashed) vs. time for discharge

14 Region of reversed shear is observed near axis MSE constrained q profile at t=1.34 s and experimental values derived from Thomson scattering density measurements ( ) vs R (magnetic axis at R=0.67 m). q"1/n1/# e 14

15 Density pro4iles from Thomson scattering, including error bars ( ) 15

16 Density x m 3 (x) and temperature (kev) (*) averaged for 5 discharges ( ) to reduce scatter Density inverts during LHCD. Temperature remains peaked. 16

17 TS Electron density ( ) 8 t=1 sec 8 t=1.2 sec Electron Density (10 19 m -3 ) Electron Density (10 19 m -3 ) Radius (m) 8 t=1.3 sec Radius (m) 8 t=1.4 sec Electron Density (10 19 m -3 ) Electron Density (10 19 m -3 ) Radius (m) Radius (m) Dashed lines indicate the magnetic axis (R~0.68 m) and the separatrix (R~0.88$ m). Error bars represent Monte Carlo error analysis of the 3itted pro3iles, i.e. the standard deviation of the 17 3it based on 100 trials in which the data points are varied according to their experimental error bars.

18 Electron Density (10 19 m -3 ) TS Electron density ( ) t=1s Electron Density (10 20 m -3 ) t=1.2s. Electron Density (10 19 m -3 ) Radius (m) t=1.3s Radius (m) Electron Density (10 20 m -3 ) Radius (m) t=1.4s Radius (m) 18

19 Turbulence rises during LHCD time(s)" time(s)" 19

20 Turn off of Ware pinch during LHCD (E φ ~0) means turbulence will determine density pro4ile o Outer s>0 region consistent with turbulent pinch Inner region s<0. Two interpretations Quiescent region produces a 4lat n e pro4ile. (can argue that density inversion falls within error bars) s<0 turns off TEM (but TEM branch does not provide pinch) Inverted density pro4ile in RS region requires an inverse pinch such as curvature pinch. 20

21 General result: Low frequency electrosta,c turbulence (with trapped par,cles): Turbulence tends to create density pro4iles with equal number of particles per unit 4lux, i.e. n e ~1/V o Creates inwards n e pinch when fueling is at the edge o Observed in C Mod in L mode and I mode Tends to form constant entropy density pro4iles pv' " =constant. V'= " d!/b o Augments outwards energy transport in tokamaks since core heating drives ( )<0. pv' " 21

22 Conclusions Turbulent pinch can explain peaked density pro4iles observed in L and I mode. o Peaked density pro4iles are necessary for an energy source L mode and I mode pro4iles are predicted by turbulent equipartition o DIII-D observed similar L-mode profiles to C-Mod [ref: Baker and Rosenbluth, PoP 5 (1998), Baker et al., NF 40 (2000)] C Mod exhibits density 4lattening/dip in reversed shear region during LHCD when V loop ~0, consistent with turbulent equipartition o JET does not observe reversed shear density dip [Ref: Weisen et al., PPCF 46, 751 (2004)] 22