Mechanisms of intrinsic toroidal rotation tested against ASDEX Upgrade observations

Transcription

1 Mechanisms of intrinsic toroidal rotation tested against ASDEX Upgrade observations William A. Hornsby C. Angioni, E. Fable, P. Manas, R. McDermott, Z.X. Lu, S. Grosshauser 2, A. G. Peeters 2 and the ASDEX Upgrade Team Max-Planck-Institut für Plasmaphysik, EURATOM Association, Boltzmannstrasse 2, Garching, Germany 2 University of Bayreuth, Bayreuth, Germany

2 Introduction Flows and flow shear regulate turbulence and can stabilise large scale MHD modes. Intrinsic rotation - No external torque, profiles determined purely by transport processes -> Still many open questions. Flow reversals - as seen on TCV, C-Mod, ASDEX Upgrade, MAST and KSTAR Can observed flow gradients be achieved? Which mechanisms are dominant? Database of ~9 ASDEX Upgrade profiles provides a testbed for the generation mechanisms of intrinsic flow McDermott et al, Nucl. Fusion (24), Hornsby et al, Nucl Fusion (27). Gyro-kinetic study using GKW (gkw.bitbucket.com). Many profiles encourages a quasi-linear approach, supported by nonlinear turbulence simulations. 2

3 Momentum transport Total toroidal momentum flux: R v th n i = u + RV u + neo + res Neoclassical background flow - Barnes, PRL 3, J Lee NF 4 Parallel velocity shear Peeters PoP 5 Coriolis pinch Peeters PRL 7 Hahm PoP 7 ExB shearing Dominguez, PFB 93 Garbet PoP 2 Gurcan PoP 7 Casson PoP 9 Waltz PoP 9 Up-down equilibrium asymmetry - Camenen PRL 9 Higher order parallel derivatives - Sung PoP 3, Stoltzfus-Dueck PoP 7 Profile shearing - Camenen, NF, Buchholz PoP 4, Waltz PoP Intensity profile effects Gurcan et al, PoP 2 Magnetic shear profile effects ZX Lu NF 5 3

4 Local 4

5 Introduction Intrinsic Momentum linked to breaking of symmetry along the field line in a Tokamak. Systematically investigate symmetry breaking mechanisms: Local Neoclassical background flows - Gyro kinetic code GKW interfaced with neoclassical code, NEO (Belli et al, PPCF 28, Belli et al, PPCF 22) Up-down equilibrium asymmetry Coriolis pinch Higher order parallel derivatives Perpendicular ExB flow shear Radially Global Profile shearing - Radial variations of Temperature, density and their gradients. Intensity profile effects - Radial variation of turbulence. Magnetic shear profile effects - Radial variation of the q-profile 5

6 GKW-NEO - Predicted u and experimental profiles R Total momentum flux: v th n i = u + RV u + neo + res = This term Balances with these terms Heat flux: Q i = it Prandtl number: Can maintain a predicted rotation gradient: u = i Q i Pr R L Ti Pr = / i Prandtl number varies weakly with plasma parameters:

7 AUG Ohmic L-mode database and modes V [km/s T n [ 9 m 3 φ e & T i [kev e u at = T =.35 Peaked (u >) Co-current (u>) Hollow (u < ) Counter-current (u < ) ν i* Wide range of plasma parameters - densities, temperatures, collisionality Flux-tube simulations with: Kinetic electrons Boron impurity species Realistic geometry Collisions 5 simulations per database point for two radial positions. Calc. Prandtl number Calc. Coriolis pinch Calc. up-down asymmetry Calc. Neoclassical effects Calc. Frequency, ω [v th i /R =.35 =.5 ITG TEM ν * [ν ii q/ε 3/2 φ 2 k i =.42 2 Pred. ITG Pred. TEM 3 k ρ 7

8 Toroidal flows» Radial force balance - n i Z i e( ~ E + ~ V i ~ B)=rP i =r n i Z i e( E ~ + V ~ B) ~ rp» with some manipulation, the toroidal flow is: t = ~ V p + n i q i dp i d p q ~ V r ExB flow Diamagnetic flow Poloidal flow Determined by neoclassical physics Collisions + Geometry 8

9 Toroidal flows» Radial force balance - n i Z i e( ~ E + ~ V i ~ B)=rP i =r n i Z i e( E ~ + V ~ B) ~ rp» with some manipulation, the toroidal flow is: t = ~ V r + dp Usually neglected i q V ~ because: p n i q i d = p i /R << ExB flow Diamagnetic flow Poloidal flow Determined by neoclassical physics Collisions + Geometry 9

10 Momentum transport - with background flows - symmetry breaking In very simple form, the total momentum flux can be written like: s = E,s u s + V E,s u + V d,s! neo + res,s ExB terms have equivalent terms with respect to the neoclassical flows u = R v thi E ExB flow u neo = R v thi! neo u = E v Parallel velocity gradient u neo = neo v Since equilibrium (time static) quantities, they are grouped with residual stress.

11 Flux-Tube - Predicted and experimental u.2 v =,u = t = ~ V p + n i q i dp i d p q V ~ r.2 GKW QL turbulent flux used to calculate the ExB flow gradient. u [R 2 /v th i T =.35 ν * [ν ii q/ε 3/2 T =.35 GKW NEO, u ExB EXP, u tot Can maintain a R predicted rotation gradient: Total momentum flux: = u Residual from updown + RV u + neo + res v th n i asymmetry u = i Q i Pr s = E,s u s + V E,s u d,s R L Ti Heat Prandtl number: + V d,s! neo + res,s Q i = it Pr = / i

12 Flux-Tube - Predicted and experimental u.2 v =,u = u ExB +u dia t = ~ V p + n i q i dp i d p q V ~ r u tot.2 u Exper data22 GKW QL turbulent flux used to calculate the ExB flow gradient. u [R 2 /v thi Corrected by Boron diamagnetic flow gradient We also have to add this (as calculated by NEO).2 T =.35 Total momentum flux: s = E,s u s + V E,s u + V d,s! neo + res,s.4 ν * [ν ii q/ε 3/2 Can maintain a R predicted rotation gradient: = u Residual from updown + RV u + neo + res v th n i asymmetry u = i Q i Pr R L Ti Heat flux: Prandtl number: Q i = it Pr = / i 2

13 Flux-Tube - Predicted and experimental u - Coriolis, finite R s v th n i = E,s u s + RV E,s u + neo + res,s... res and from up-down asymmetry r parallel derivatives..3 = i /R.8.5 T =.35 Predicted flow gradient, u [R 2 /v th i Neoclassical Coriolis Up down equil Neoclassical (total flow) Finite rho parallel ν = ν q/ε 3/2 * ii Coriolis pinch rhostar corrections Terms usually ignored in fluxtube geometry for ordering Up-down EQ asymmetry z [m r [m 3

14 Flux-Tube - Predicted and experimental u - Coriolis, finite R s v th n i = E,s u s + RV E,s u + neo + res,s... res and from up-down asymmetry r parallel derivatives. = i /R T =.35.2 Predicted flow gradient, u [R 2 /v th i Neoclassical Coriolis Up down equil Neoclassical (total flow) Finite rho parallel ν * = ν ii q/ε 3/2 u [R 2 /v th i T =.35 ν * [ν ii q/ε 3/2 GKW NEO, u ExB EXP, u tot GKW NEO, u tot Combined total flow gradient, u under-predicted by.5-. depending on collisionality. Terms are all same magnitude. Hornsby et al, NF 27 =.35 4

15 Flux-Tube - Density profile variations Neoclassical flow gradient is the dominant neo T 2.5. ρ t =.35 ρ t =.5 Using a fit over the database - Predict the 2nd der. required = 2P rr/l ne AR/L Ti u E B u ExB [v th /R Where = 2 n e n Fit gradient, A.5.5 ω d,ζ / r [v th i /R 2 5

16 Flux-Tube - Density profile variations n e [ Neoclassical flow gradient is the dominant neo T 2 (R 2 /n e ) 2 n e / r ρ =.5 2nd der. meas T =.5 need =.35 need ρ =.35 meas. T (R 2 /ne ) 2 n e / r s s s s s s s s ρ p High resolution reflectometry measurements ρ#3236 p Many thanks: A Medvedeva et al x larger 2nd derivatives required from nominal params. Corrugations in ne exist in the region of interest. Need to 5- times larger. Amplitudes too small to generate flows large enough to explain observations. n e /n e ( =.35) n e n e +err n e err 2nd der. =.35 =.35 needed 2nd der. =.5 =.5 needed u exp

17 Background ExB shearing u = i Q i Pr R L Ti u can also be increased by reduction on Pr For a selection of points, nonlinear flux tube simulations, with and without perpendicular ExB shearing. ExB shearing rate given by experimental u ; E =( /q)u ExB Effective Prandtl number reduction by perpendicular flow shear also small - Approx -2% from Nonlinear simulations. Momentum diffusion term can be written, i = m i n i u + M E Prandtl number, P eff,r = χ φ /χ i ,eff = +( /q)m Pr eff =,eff / i Dashed lines are QL calc value used in database evaluation γ E = (ε/q) u ExB 7

18 Radially Global 8

19 Introduction Intrinsic Momentum linked to breaking of symmetry along the field line in a Tokamak. Systematically investigate symmetry breaking mechanisms: Flux-tube Neoclassical background flows - Gyro kinetic code GKW interfaced with neoclassical code, NEO (Belli et al, PPCF 28, Belli et al, PPCF 22) Up-down equilibrium asymmetry Coriolis pinch Higher order parallel derivatives Perpendicular ExB flow shear Radially Global Profile shearing - Radial variations of Temperature, Density and their gradients. Intensity profile effects - Radial variation of turbulence. Magnetic shear profile effects - Radial variation of the q-profile 9

20 Global mechanisms - Setup and profiles Do global effects make up the difference seen? Concentrate on three profiles from Shot Number #27-27 Use profiles as input to Global nonlinear turbulence calculation - Significantly more expensive than flux-tube simulations. n e 6 x 9 Linearly (fluxtube) at rho=.5 - TEM -> ITG transition, at rho=.35 always ITG. T e /T i goes from 2.24 to.54, Time Radial Domain Density ramp - Increasing collisionality. = ii q/ p 3 =.5.75 Flux-tube predictions for these points at T =.35 u = -.2 u = -.27 u = -. Missing ~.4-. Do global turbulence simulations replicate the gradients observed and the flat-hollow-flat transition? Toroidal flow, v ζ u.5.5 Flow, u Flow gradient, u

21 Global mechanisms - Setup and profiles Simulation parameters: Nx = depending on Nv = 64, Nmu = 8, Ns = 6 Maximum mode number: k i =.4 Toroidal mode number spacing, -5 depending on Boundaries have damping buffers, 5-2 grid points. Circular flux surfaces. R/LTe = 5 (approximate value at centre of domain) all other profiles are experimental. Flux-tube predictions for these points at T =.35 u = -.2 u = -.27 u = -. Missing ~.4-. Toroidal flow, v ζ u.5.5 Flow, u Radial Domain Flow gradient, u Radial Domain

22 Global mechanisms - realistic normalised gyro-radius Integrated from Π i /Q i Exp. flow profile Exp. error bars GKW, u final state left buffer right buffer Adiabatic electrons. One can, early in a nonlinear simulation calculate the residual stress and estimate flow gradient u = i Q i Pr R L Ti Flow, u [v th i Prandtl number, Pr=.63 from quasi-linear simulation. Flow profile calculated from residual from this gives good experimental agreement.3 Agreement with flow profile in this case is good r/a =.5 Adiabatic electrons (Dashed lines) give peaked profiles - Flow profiles reversed. However, reversed, or flat, experimental flow profiles are not reproduced. 22

23 Global mechanisms - Flow profiles Kinetic electron simulations (Solid lines) give hollow rotation profiles. Like the majority of the experimental profiles. Adiabatic electrons (Dashed lines) give peaked profiles - Flow profiles reversed. Linear residual stress has same sign. Why reversal? u [v thi =.5 = i /R artificially large - allows a scan of parameters on expensive global simulations. With kinetic electron, scaling with is weak Weaker than adiabatic scaling with idealised profiles (Buchholz et al, POP 22) u u = -.6 to -.7 seen in the centre of the domain. Large enough to bridge the difference in observations and flux tube mechanisms Radial coordinate, ψ [r/r 23

24 Global mechanisms - realistic normalised gyro-radius Flow, u [v th i Exp. flow profile GKW, u final state Integrated from Π i /Q i left buffer right buffer One can, early in a nonlinear simulation calculate the residual stress and estimate flow gradient u = i Q i Pr Prandtl number, Pr=.7 R L Ti Flow profile calculated from residual from this gives no experimental agreement Agreement with flow profile in this case is bad r/a =.5 However, reversed, or flat, experimental flow profiles are not reproduced. 24

25 Global mechanisms - Density profile variation With adiabatic electrons - No particle flux - No density profile variation. With kinetic electrons - Particle flux - Density profile time-evolves. - Increases the value of 2 R 2 n e n 2 - Changes the residual stress, i,res = i,res (n e,n e,n e,t i,ti,ti,...) - Intrinsic flow profile slowly evolves accordingly - u increases Slice the simulation: n e /n e Heat flux, Q i [ρ * v th,i Time, t [R/v th,i Into time slices t N 4R /v th,i And radial slices Calculate mean: u r.2a R 2 n e n 2 U [v th i r/a 25

26 Global mechanisms - Density profile variation With adiabatic electrons - No particle flux - No density profile variation.. With kinetic electrons - Particle flux - Density profile time-evolves. - Increases the value of - Changes the residual stress, i,res = i,res (n e,n e,n e,t i,ti,ti,...) - Intrinsic flow profile slowly evolves accordingly - u increases Heat flux, Q i [ρ * v th,i Time, t [R/v th,i R 2 n e n 2 Slice the simulation: Into time slices t N 4R /v th,i And radial slices r.2a Calculate mean: u R 2 n e n 2 n e (r)/n i,res [m i n i v thi R 2 * u [v thi r/a 26

27 Global mechanisms - Density profile variation Residual stress is a function of many variables Here we vary only i,res = i,res (n e,n e,n e,t i,t i,t i,...) n e,n e,n e with time. (The Krook operator fixes the background temperature profile) Steepening of profile is seen periodically in measurements. Profile curvature notoriously difficult to measure..7.2 u.8 u [v th /R u.8 2 n e n u R 2 n 2 n t [s (R 2 /n)d 2 n/dr 2 T =.5 Time traces of u at from Shot number #27 27

28 Global mechanisms - Density profile variation Residual stress is a function of many variables Here we vary only background temperature profile).4.2 i,res = i,res (n e,n e,n e,t i,t i,t i,...) n e,n e,n e with time (The Krook operator fixes the..9 R 2 n 2 n 2 5 (R 2 /n e ) 2 n e / r =.5 2nd der. meas =.5 need =.35 need =.35 meas..8 2 u [v th /R Consistent with measurements, gives: u u exp.6.8 u (R 2 /n)d 2 n/dr n e n Mechanism much stronger than neoclassical flow effects. u. n e /n e ( =.35) n e n e +err n e err 2nd der. =.35 =.35 needed 2nd der. =.5 =.5 needed

29 In conclusion A systematic study of intrinsic momentum generation mechanisms was performed on a large database of AUG Ohmic L-mode shots Linear and nonlinear gyro kinetic simulations in both flux-tube and global geometry were compared with experimental observations. Combined mechanisms described by flux tube simulations falls short of experimental u by a factor 3-5. Global simulations with kinetic electrons to quantify profile shearing effects show significant flow gradients - u =.6-.2 Density profile curvature is seen in the global and flux-tube cases, to be the significant parameter. However, profile shearing is seen to be much more sensitive to curvature. Parameterisation gives good qualitative agreement with experiment. Going forward - Global-local analysis (c.f Zhixin Lu tomorrow morning) -> Towards a quasi-linear model. Many thanks 29

30 GKW-NEO - Implementation Implementation of Barnes-Parra model M. Barnes, F. I. Parra, et. al, Phys. Rev. Lett.,555 Interface of Neoclassical transport code NEO with GKW Neoclassical distribution function is a correction to the equilibrium Maxwellian f = F + f.35.3 GKW NEO GS2 NEO a( k a ) GKW NL Region of database interest F = F M + f NEO = NEO + f NEO Neoclassical part calculated by Eulerian code, NEO Turbulent part solved for by non-linear Gyro-kinetic code, GKW 2 with corrections due to the perturbed background distribution function. f Flux ratio, Π i /Q i Symmetry breaking mechanism that produces momentum transport. E A Belli and J Candy 22 Plasma Phys. Control. Fusion A.G Peeters et al. Computer Physics Communications, 8, 265 (29) 2 Normalised collision frequency, ν = ν q/ε 3/2 * ii 3

31 Global mechanisms - Flow profiles and exp. comparison k measures the level of mode asymmetry along the field line n Momentum flux here is given by: Residual stress can be calculated by damping the nonlinearly generated n= flow or early in the nonlinear phase. u / <k > <k R >= s = u (Nonlinear global) 5x<k > linear flux tube, k θ ρ i =.42 <k > (nonlinear global) R/L n <k > (nonlinear global) ds E,s u s + res,s u [v th i u gkw = -.6 u gkw = -.6 u gkw = u exp = -.4 u exp = -.2 u exp = ψ = r/r ψ = r/r ψ = r/r.2.25 Boundary conditions (no slip u = at edges) restrict shape of flow. Restrict comparison to middle of the domain. u large enough to reproduce experiment in some cases u (r) <k > R L n ψ [r/r k profile well reproduced by series of radial flux tube simulations input tilt angle and nonlinear u Profile shearing important component Density gradient variation -> Large change in i /Q i 3

32 Backup 2 - Integrated profiles from u u = i Q i Pr R L Ti.2 u II GKW u exp u from Π/Q.4.2 u from Π/Q u from GKW Where we assume Pr=.7.2 u is well reproduced u [v th i.4.6 u U GKW U exp U integrated from Π/Q.6.4 u from Π/Q u from GKW ψ = r/r.6..2 ψ = r/r.9.8 u integrated from Π/Q u GKW final u exp.5 u from Π/Q u GKW final U.5.4 u.2 u [v thi u ψ = r/r ψ = r/r ψ = r/r ψ = r/r APS DPP 26 San Jose, California 32