Intrinsic rotation due to non- Maxwellian equilibria in tokamak plasmas. Jungpyo (J.P.) Lee (Part 1) Michael Barnes (Part 2) Felix I.

Transcription

1 Intrinsic rotation due to non- Maxwellian equilibria in tokamak plasmas Jungpyo (J.P.) Lee (Part 1) Michael Barnes (Part 2) Felix I. Parra MIT Plasma Science & Fusion Center. 1

2 Outlines Introduction to intrinsic rotation Part 1: Turbulent momentum pinch of diamagnetic flows à Peaked rotation profile due to strong pedestal in H- mode Part 2: Sign reversal of the momentum transport by non- Maxwellian equilibria à From peaked to hollow profile due to the change in collisionality 2

3 Introduction 3

4 Measurement of intrinsic toroidal rotation (1) H- mode: peaked profile in the co- current direction L- mode: both peaked and hollow profiles A sign change at mid- radius due to an internal momentum redistribution Low flow regime (subsonic)

5 Measurement of intrinsic toroidal rotation (2) Ohmic : Rotation reversal from co- current direction to counter- current direction (peaked à hollow) by changing plasma parameters (increased density or decreased plasma current) Rice et. al.(2011) NF

6 Intrinsic momentum transport determines the radial profile of rotation Without external source, the radial transport of the toroidal angular momentum is = int {z} 6=0 P n i m i RV {z } pinch n i m {z } di usion =0

7 Intrinsic momentum transport determines the radial profile of rotation Without external source, the radial transport of the toroidal angular momentum is à = int {z} 6=0 V (0) = V (a)exp P n i m i RV {z } pinch P r n i m {z } di usion Z r=a r=0 =0 int n i m i R exp P r Fixed velocity boundary condition at r=a : Momentum source and sink at the wall

8 Intrinsic momentum transport determines the radial profile of rotation Without external source, the radial transport of the toroidal angular momentum is à = int {z} 6=0 V (0) = V (a)exp P n i m i RV {z } pinch P r n i m {z } di usion Z r=a r=0 =0 int n i m i R exp P r Fixed velocity boundary condition at r=a : Momentum source and sink at the wall P > 0 > 0 Inward pinch and diffusion out The sign of intrinsic flux at outer radius determines the velocity direction in the core (V = 0) > 0 V (r = 0) < 0 (If around the edge, then )

9 No intrinsic momentum flux in the lowest order No intrinsic toroidal angular momentum flux in the lowest order for an up- down symmetric tokamak Example: ballooning ITG 1 (, v k,k x )= 1 (, v k, k x ) 1 / Re[ik y 1 h 1 v k ] where Parra et. al. PoP (2011)

10 Symmetry of the momentum transport Rv B Rv

11 Types of intrinsic momentum transport Phenomena that break the symmetry Up- down asymmetric tokamak Global profile effect on turbulence Neoclassical flow effect on turbulence u s / r d ln p dr v th B B L p v th Rv Rv

12 Higher- order effects in Gyrokinetic equation dg s dt + v k g s Z s e F 0s + v? E E F0s + v Cs + v Ms + v E?? g s Z s e F 0s E = v Ms g s Z s e F D E 0s v k E? g s E v? g D E s E v k E F 0s v? E F1s gs + Z s e v k + v Ms? E + F 1s E v nc E? g s + Z s ev k nc g s E + -profile variation

13 Neoclassical parallel heat flow can also break the symmetry of turbulence Neoclassical distribution function in the higher order distribution function : F 1 = F u k 1 + F q k 1 + F other 1 B B F 0 13

14 Neoclassical parallel heat flow can also break the symmetry of turbulence Neoclassical distribution function in the higher order distribution function : F 1 = F u k 1 + F q k 1 + F1 other B F 0 B Neoclassical parallel particle flow piece: where F u k 1 = mv ku k F 0 T = 1 n Z dv 3 v k F u k 1 14

15 Neoclassical parallel heat flow can also break the symmetry of turbulence Neoclassical distribution function in the higher order distribution function : F 1 = F u k 1 + F q k 1 + F1 other B F 0 B Neoclassical parallel particle flow piece: where F u k 1 = mv ku k F 0 T = 1 n Z dv 3 v k F u k 1 Neoclassical parallel heat flow piece: F q k 1 = 2 5 mv k q k PT mv 2 2T 5 2 F 0 @r, = 1 n Z dv 3 mv 2 v k 2 F q k 1 15

16 Neoclassical parallel heat flow can also break the symmetry of turbulence Neoclassical distribution function in the higher order distribution function : F 1 = F u k 1 + F q k 1 + F1 other B F 0 B Neoclassical parallel particle flow piece: where F u k 1 = mv ku k F 0 T = 1 n Z dv 3 v k F u k 1 Neoclassical parallel heat flow piece: F q k 1 = 2 5 mv k q k PT mv 2 2T 5 2 F 0 @r, = 1 n Z dv 3 mv 2 v k 2 F q k 1 F @2 2, 16

17 Part 1. Turbulent momentum pinch of diamagnetic flows 17

18 Momentum transport for ExB flow and diagmagnetic flow are different There are two types of toroidal rotation (Assume small poloidal rotation) = 0 Zen {z } {z } E B flow diamagnetic flow Difference between two types of rotation Toroidal rota*on source Origin Time scale Effects on par*cle mo*on Radial electric fields Force balance Momentum transport Pressure gradient Energy and par*cle transport Energy transport Change in orbit and energy No change in orbit and energy. Only par*cle flux difference 18

19 Intrinsic momentum transport occurs when two types of rotations cancel each other Different pinch and diffusion coefficients for two types of rotation generate intrinsic momentum transport = 0 int m i R 2 [P,E,E + P,p,p ] m i R @,p 19

20 Intrinsic momentum transport occurs when two types of rotations cancel each other Different pinch and diffusion coefficients for two types of rotation generate intrinsic momentum transport = = 0 int m i R 2 [P,E,E + P,p,p ] m i R @,p apple 0 int m i R 2 P,E P,p (,E ) m i {z } apple m i R 2 P,E + P,p (,E +,p ) 2 int applem i R 2,E @r 20

21 Intrinsic momentum transport occurs when two types of rotations cancel each other Different pinch and diffusion coefficients for two types of rotation generate intrinsic momentum transport = = 0 int m i R 2 [P,E,E + P,p,p ] m i R @,p apple 0 int m i R 2 P,E P,p (,E ) m i {z } apple m i R 2 P,E + P,p (,E +,p ) 2 int =,p int applem i @2 p 2,E @r 21

22 Momentum pinch for diamagnetic flow is bigger than that for a ExB flow ( /Q)/( GB/Q GB) The ratio of momentum pinch ( ) to heat flux = P m i R 2 Ω ζ,e = Ω ζx, Ω ζ,p =0 Ω ζ,p = Ω ζx, Ω ζ,e =0 Ω ζ,e =-Ω ζx, Ω ζ,p = Ω ζx (Ω ζx R 0 )/v ti Linearity on the rotation holds Different factors of rotation peaking due to the momentum = P R/L T =9.0, R/L n =9.0,q =2.5, r/a =0.8, R/a =3.0, ŝ =0.8 Pr i =

23 Momentum pinch for diamagnetic flow is bigger than that for a ExB flow ( /Q)/( GB/Q GB) The ratio of momentum pinch ( ) to heat flux = P m i R 2 Ω ζ,e = Ω ζx, Ω ζ,p =0 Ω ζ,p = Ω ζx, Ω ζ,e =0 Ω ζ,e =-Ω ζx, Ω ζ,p = Ω ζx (Ω ζx R 0 )/v ti Linearity on the rotation holds Different factors of rotation peaking due to the momentum pinches Radial electric field driven P,E / ' 2.9/R 0 Pressure gradient driven R/L T =9.0, R/L n =9.0,q =2.5, r/a =0.8, R/a =3.0, ŝ =0.8 P,p / ' 3.5/R 0 Pr i =0.517 P,p 23

24 Gyrokinetic equations are different for different type of rotations In lab frame, there is an additional acceleration term due to the radial electric field à Origin of the different pinches: the energy of the particle changes not by pressure gradient but by radial electric field

25 Gyrokinetic equations are different for different type of rotations In lab frame, there is an additional acceleration term due to the radial electric field à Origin of the different pinches: the energy of the particle changes not by pressure gradient but by radial electric field =,E +,p In the frame rotating with total toroidal flow, the Coriolis terms due to the total flow, but energy correction + Rˆ r f (R) tb + v kˆb,p B ˆb r + v M + v C,p,p c B rh tb i ˆb rf (R) tb Ze m i c,pv M = c B rh tb i ˆb rf (R) 0 + Ze [v kˆb + vm + v C ] rh tb 0 m + C(f i ) Where v C = 2v k i ˆb [(rr ) ˆb] is the drift due to the Coriolis force. 25

26 Rotation peaking due to positive diamagnetic flow and negative ExB flow varies by parameters Rotation peaking ( ) when,p =,E +,p =0 (,E R = 0.3 and,p R =0.3 ) (d /dr)(a/, p) R/L n R/L T q

27 Strong pressure gradient in the pedestal results in a large inward intrinsic momentum transport(h- mode) In the pedestal, a negative radial electric field is generated to balance the strong radial pressure drop,e,p < 0 Rotation peaking can be significant due to the strong pressure drop (e.g. for a pedestal with T i 1keV,, and B 0.5T à,p R 0 200km/s at the pedestal à R 0 at the R 0 0.4,p R 0 80km/s 27 r 1cm

28 The acceleration due to radial electric field results in intrinsic momentum transport The difference between two momentum pinches are caused by acceleration term Ze m i c v M Ω ζ, p R 0 =0.3v ti, Ω ζ, E R 0 =-0.3v ti 1.0 Ω ζ, E R 0 =0.3v ti /max ( ) v [v ti ] θ[radian] 28

29 Part 2. Sign reversal of the momentum transport by non- Maxwellian equilibria 29