Local gyrokinetic turbulence simulations with realistic tokamak geometries towards ITER and JT-60SA

Transcription

1 Local gyrokinetic turbulence simulations with realistic tokamak geometries towards ITER and JT-6SA M. Nakata, A. Matsuyama, N. Aiba, S. Maeyama in collaboration with T. -H. Watanabe, H. Sugama, and M. Nunami Japan Atomic Energy Agency National Institute for Fusion Science NEXT workshop at Kyoto univ. August 9-3th, 3

2 Outline Introduction Current status of local gyrokinetic Vlasov code GKV Interface between MEUDAS(D equilibrium solver) and GKV Application to realistic plasma shapes on JT-6SA Linear ITG mode stability Residual zonal flow levels Nonlinear zonal flow generations Summary

3 Introduction - Turbulent transport properties can be strongly affected by the plasma shaping effects, through the change of the linear frequencies, mode-structures, and trapped/passing boundary. Impact of the negative triangularity on stabilization of TEMs: global-gk analyses for TCV shape. Camenen et al., NF7 Enhanced residual-zf levels by plasma elongation: Analytic analyses for Solove v like equilibrium. Xiao and Catto, PoP6 - Implementation of realistic shaped MHD-equilibrium to gyrokinetic code is useful for analyzing plasma shape effects, and for validation/prediction study against experiments. ---> In this study, tokamak equilibria calculated by a free-boundary D Grad-Shafranov solver MEUDAS are incorporated into a local fluxtube gyrokinetic code GKV. ---> Micro-stability and zonal flow generation in several equilibria on JT-6SA are investigated.

4 Current status of GKV code - Local fluxtube 5D gyrokinetic solver, originally developed by [T. -H. Watanabe, NF6] ---> Solving the evolution of delta-f in 5D phase space ---> Eulerian (or Vlasov) solver: spectral in D k-space, Finite-Difference in 3D (z, v, μ)-space fluxtube ---> Electro-static, Electro-magnetic, Arbitrary numbers of species ---> Helical geometries from VMEC, Tokamak geometries from MEUDAS(this talk) ---> Entropy balance/transfer diagnostics [T. -H. Watanabe PRL8, M. Nunami PFR, M. Nakata PoP, S. Maeyama CPC3] - GKV code is now ready for turbulence simulations of burning plasmas composed of D, T, e, He-ash, C, W, etc. Tokamak ETG <--- Tokamak ITG Helical ITG --->

5 Advanced parallelization towards PETA/EXA computing 3-5D domain-decompositions with MPI and Thread parallelization with OPENMP Decomposition in (z, v, μ) Decomposition in k-space (kx or ky) + Species µ + k y k y z v + D#FFT k x Data transpose often degrades scalability. D#FFT - Parallel D FFT (including data-transpose) for the nonlinear ExB advection term is the most time-consuming. ---> Segmented 3D rank-allocation technique (for MESH/TORUS network) ---> Communication/Computation Overlap technique (for FFT and/or FD) k x For each species [ S. Maeyama, JSST Y. Idomura, SC ]

6 Fantastic scaling over.6m cores on K computer 4 Problem size for Multi-scale turbulence simulations (e.g. ITG-ETG) N x =4, N y =4, N z =8, N v =64, N µ =3 74 billion grids P k =8-64, P z =6, P v =8, P µ =4, 8 threads > 6 thousand cores Effects of Rank-Mapping and Overlap Performance [TFlops] 5 5 Performance [TFlops] Strong scaling on K computer Number of cores [ 3 ] Number of cores [x 3 ] [ S. Maeyama, SC3] Fantastic scalability over.6mcores has been achieved with % efficiency. ---> Multi-scale/Multi-species turbulence simulation is now accessible with GKV.

7 Outline Introduction Current status of local gyrokinetic Vlasov code GKV Interface between MEUDAS(D equilibrium solver) and GKV Application to realistic plasma shapes on JT-6SA Linear ITG mode stability Residual zonal flow levels Nonlinear zonal flow generations Summary

8 Gyrokinetic equation in flux coordinates 5 - Gyrokinetic and Poisson equations: electro-static limit, multi-species + v µ kb r + ik? v ds b f sk? + N ( f sk?, sk? ) m k >< >: k? = F Ms ik? v Ts ik? v ds v k b r e s sk? + C s (h sk? ) T s k? =4 X applez e s k? e s dvj s f sk? n s ( s) T s s - Fluxtube coordinates (x, y, z) defined by general SFL flux coordinates (,, ) : coordinates: magnetic field: x = c x ( ), y = c y [q( ) ], z = B = c b rx ry, c b = /c x c y = ( ) : poloidal flux = ( ) : toroidal flux Jacobian: Derivatives: safety factor and magnetic shear: p gxyz =(rx ry rz) =(c x c y ) = + q( ) q( ) = c ŝ( ) = q @ +

9 Gyrokinetic equation in flux coordinates - Metric tensor g xx = rx rx = c xg g xy = rx ry = c x c y [ q g + qg g ] g xz = rx rz = c x g g yy = ry ry = c y[( q ) g + q g + g +q q g qg q g ] g yz = ry rz = c y [ q g + qg g ] g zz = rz rz = g 6 - Advection operators in general forms for arbitrary flux coordinate systems: c b r k? = kxg xx +k x k y g xy + kyg yy, B p g xyz k? v ds = m sv k + µb e s c b [K x k x + K y k y ], k? v s = T s e s c b K x = gxz g xy g xx g ln K y = gxz g yy g xy g yz B /c b " = d ln n s L ns dx, = d ln T s L Ts dx!# + L ns L ln m s vk + µb 3 T s T s ln B /c b ---> Only B-intensity, and its derivatives, and the contra-variant metric components are necessary to calculate these operators. k y

10 Construction of SFL flux coordinates from MEUDAS Data flow 7 MEUDAS (developed in JAEA) D Free-boundary Grad-Shafranov solver = (R, Z) IGS [Matsuyama] Flux coordinate Interface Interpolation, tracing flux surfaces Constructing Axisymmetric, Hamada, and Boozer coordinates P = P ( ), I = I( ) The radial q label : q = / edge, = / edge The minor q radius : q a = V edge / R ax, a = edge /B ax, q a = edge/b ax GKV Local fluxtube code Calculating metric components and advection operators Turbulence simulation R = R[, ], Z = Z[, ], = q( )G[, ] B = B[, ], db/d, db/d Co-variant metric components are calculated by g @Z @u j u i = {,, }, then converted to contra-variant components.

11 Verification of flux coordinates with solovev equilibrium 8 - Analytic expression of solovev equilibrium with finite elongation, triangularity, and up-down asymmetry: ( R, Z) = apple ( d) R Z + d Z edge a E + 4 ( R ) + asym ( R ) Z, R = R/Rax, Z = Z/R ax a (R, Z) (R[, ],Z[, ]) (R[, ],Z[, ]) Original D equilibrium data with E =.6, d =.5, a =.5, asym =.3 Boozer coord. with =( / edge) Hamada coord. Less deformation of constant -surface near the edge for Hamada coord. (especially, out-board side)

12 Verification of flux coordinates with solovev equilibrium 9 - Consistency is verified by the following analytical expressions: p ] (R, = rr rz Relative error to analytical expressions Axisymmetric coord. Boozer coord. Hamada coord. All the constructed coordinate systems show sufficient accuracy(up to ~ -8 ) for turbulent transport analyses.

13 Outline Introduction Current status of local gyrokinetic Vlasov code GKV Interface between MEUDAS(D equilibrium solver) and GKV Application to realistic plasma shapes on JT-6SA Linear ITG mode stability Residual zonal flow levels Nonlinear zonal flow generations Summary

14 Application to realistic plasmas on JT-6SA - Two L-mode plasmas are considered (The MHD stability has been well investigated: N. Aiba PFR7 ): ITER-like plasma with single null: IT Highly-Shaped plasma with quasi double null: HS (R, Z) (R, Z) Boozer Boozer R ax =3.m, a =.9m, V = 9.7m 3, B ax =.64T, q ax =.8, q 95 =3.88, I p =.59MA, S = q 95 I p /B ax a =.95 R ax =3.5m, a =.53m, V = 9.5m 3, B ax =.88T, q ax =.4, q 95 =3.59, I p =5.MA, S = q 95 I p /B ax a =4.7

15 B( ) B( ) Field aligned structures of B-intensity, ωd, and k IT: ITER-like, HS: Highly shaped B-intensity Magnetic drift Squared wavenumber IT: =.5 IT: =.5 IT: =.75 Circular HS: =.5.4 HS: =.5 HS: =.75 Circular d [k x =,k y =.5,v=.5,µ=] x 3 d [k x =,k y =.5,v=.5,µ=] x HS: =.5 4 HS: =.5 HS: =.75 Circular - IT: =.5 IT: =.5 IT: =.75 Circular HS: =.5.8 HS: =.5 HS: =.75 (x.5) Circular.6 - Field aligned structures in shaped plasmas deviate from those in circular plasmas (outer sides). - Stronger asymmetry in k for ITER-like plasma than that in Highly-Shaped plasma. k [k x =,k y =.5] k [k x =,k y =.5] IT: =.5 IT: =.5 IT: =.75 (x.5) Circular

16 Trapped/Passing boundaries IT: ITER-like, HS: Highly shaped 4 =.5 4 =.5 4 =.75 IT IT IT µ=.7 µ=3.3 µ=6.96 µ=.5 µ=.7 µ=3.3 µ=6.96 µ=.5 µ=.7 µ=3.3 µ=6.96 µ=.5 v /v ti v /v ti v /v ti =.5 HS µ=.7 µ=3.3 µ=6.96 µ= =.5 µ=.7 4 =.75 µ=.7 µ=3.3 µ=3.3 HS µ=6.96 µ=.5 HS µ=6.96 µ=.5 v /v ti v /v ti v /v ti More sharp boundary, but less trapped region for shaped plasmas compared with circular ones. ---> Strong impacts on ITG with kinetic elec. and TEM turbulence dynamics.

17 Linear ITG mode stability for the ITER-like plasma q( ): IT s( ): IT q( ): HS s( ): HS =.5 q- and s-profiles ITG-ae mode spectra =.5 = ( )/ ( =) ρ=.5 Real Imag IT: =.5 Circular ( )/ ( =) ITG, r /4 [v ti /R ax ] ρ=.5 Real Imag IT: =.5 IT: =.5 IT: =.75 Circular.5.5 k ti IT: =.5 Circular IT r/rax q s Broader ITG-spectra with more localized eigenmode appear especially in strongly shaped region (outer side). ITG! r /4 ( )/ ( =) ρ=.75 Real Imag IT: =.75 Circular 3 R ax /L n =3., R ax /L T =6., T e /T i =,k y(min) =.5

18 ( )/ ( =) Linear ITG mode stability for the Highly-Shaped plasma q( ): IT s( ): IT q( ): HS s( ): HS =.5 =.5 = ρ=.5 q- and s-profiles Real Imag HS: =.5 IT: = ( )/ ( =) ITG, r /4 [v ti /R ax ] ρ=.5 Real Imag ITG! r /4 ITG-ae mode spectra HS: =.5 HS: =.5 HS: = k ti HS: =.5 IT: = HS r/rax q s ITG mode growth rates in Highly-shaped case is slightly lower than those in the ITER-like cases. ( )/ ( =) ρ=.75 Real Imag HS: =.75 IT: =.75 4 R ax /L n =3., R ax /L T =6., T e /T i =,k y(min) =.5

19 (t)/ (t=) Residual ZF level Geometric dependence of residual zonal flow levels Circular(IT) IT: =.5 HS: =.5 R-H ρ= Time t [R ax /v ti ] Circular(IT) IT: =.5 HS: =.5 R-H ρ= k x ti (t)/ (t=) Residual ZF level IT: ITER-like, HS: Highly shaped - Shaping effects in IT- and HS- plasmas lead to ---> significant enhancement of residual zonal flow levels ---> stronger kx- (or kr-) dependence compared with circular plasmas... Circular(IT) IT: =.5 HS: =.5 R-H Time t [R ax /v ti ] Circular(IT) IT: =.5 HS: =.5 R-H ρ=.5 ρ= k x ti (t)/ (t=) Residual ZF level Circular(IT) IT: =.75 HS: =.75 R-H ρ=.75 HS is expected to be more favorable to ITG-stability and ZF-response! Time t [R ax /v ti ] Circular(IT) IT: =.75 HS: =.75 R-H ρ= k x ti

20 Outline Introduction Current status of local gyrokinetic Vlasov code GKV Interface between MEUDAS(D equilibrium solver) and GKV Application to realistic plasma shapes on JT-6SA Linear ITG mode stability Residual zonal flow levels Nonlinear zonal flow generations Summary

21 Nonlinear simulation and entropy balance ent. balance (non-zonal part) Highly-Shaped ρ=.5 d S trb /dt D trb - zf J trb X T error Time t [R /v ti ] ent. balance (zonal part) Highly-Shaped ρ=.5 d S zf /dt D zf zf error Time t [R /v ti ] entropy variable d S trb dt d S zf dt = J trb X T zf + D trb (flux) x (thermo. force) = zf + D zf transfer from NZ to Z 6 col. dissipation - The entropy balance relations are well satisfied both in the non-zonal and zonal parts. - Highly-Shaped cases show lower transport levels with higher critical temperature gradient. i / GB ITER-like ρ=.5 R/Ln=3 R /L Ti =6 R /L Ti =5 R /L Ti =4 R /L Ti =3 i / GB Highly-Shaped ρ=.5 R/Ln=3 R /L Ti =6 R /L Ti =5 R /L Ti =4 R /L Ti = Time t [R /v ti ] Time t [R /v ti ]

22 Nonlinear zonal flow generation IT: ITER-like, HS: Highly shaped ZF/Total IT: =.5 HS: =.5 Circular(IT) IT Snapshots of the potential fluctuation HS 7. Z/T-spectra Time t [R /v ti ] kx -.75 kx -.37 IT: nonlin. HS: nonlin. ρ=.5 - As is expected by linear ZF-damping analyses, more efficient ZF generation is observed in the nonlinear phase of the Highly- Shaped case. - Stronger kx-dependence of Z/T is also identified in the Highly- Shaped case. (Z: ZF-intensity, T: turbulence intensity) ---> Qualitative features on the amplitude and kx-dependence are well agreed with linear results shown in the previous slides.. k x y ti

23 Towards the transport modeling 8 - ITG turbulence simulations on LHD show that the ion heat diffusivity well scales with T/Z /, and GK-simulation-based transport model including zonal-flow effects has been developed. [M. Nunami PoP, 3] Scaling on LHD [M. Nunami PoP] and JT-6SA LHD IT: =.5 JT-6SA 8 HS: =.5 IT: =.5 HS: =.5 6 IT: =.75 HS: =.75 4 i / GB i/ GB CT/Z /, with C T/Z / - Turbulence spectra near the peak region are well characterized with linear spectra, ITG/ky. ---> Promising to construct the similar reduced transport model produced by the linear spectra and residual ZF levels. (currently underway...) Turbulence intensity [A.U.] - JT-6SA ρ=.5 - Good scaling between χi and Τ/Ζ / is also observed in JT-6SA case. Turbulence spectra IT: nonlin. HS: nonlin. IT: C /k HS: C /k. k y ti

24 Summary 9 - An interface code to generate flux coordinates system from realistic plasma equilibria calculated by free-boundary D Grad-Shafranov solver MEUDAS is successfully implemented to a local fluxtube code GKV. - The accuracy of the flux coordinates, i.e., Axisymmetric, Boozer, and Hamada, are verified with an analytical solovev equilibrium model. - Linear ITG-ae stability (ae: adiabatic electrons) is investigated for two types of shaped plasmas in JT-6SA, i.e., ITER-like plasma(it) and Highly shaped one(hs), then the difference from the concentric circular equilibrium, which is conventionally used in gyrokinetic code, has been clarified. ---> Highly-shaped configuration shows less ITG-driven transport than that in the standard ITER-like configuration, due to stronger generation of zonal flows. ---> The turbulent ion heat diffusivity well scales with T/Z /, which suggests the applicability of a GKsimulation based transport model. More detailed analyses including a multi-species/scales stability analysis, nonlinear simulations, and constructing the GK-based transport model are in progress.