Bounce-averaged gyrokinetic simulations of trapped electron turbulence in elongated tokamak plasmas Lei Qi a, Jaemin Kwon a, T. S. Hahm a,b and Sumin Yi a a National Fusion Research Institute (NFRI), Daejeon, Korea b Seoul National University (SNU), Seoul, Korea Email address: qileister@nfri.re.kr 1
2 Outlines I. Motivation II. Nonlinear verification of bounce-averaged gyrokinetic simulation III. Elongation effects on CTEM driven turbulence and transport Transport Zonal flow Turbulence IV. Conclusions and future work
3 I. Motivation ITG Adiabatic electrons low computational cost (large time step, simple Poisson eqn., etc.) TEM kinetic trapped electrons large computational cost (small time step, complicate Poisson eqn., etc.) TEM Bounce-averaged kinetic trapped electron reduce computation cost (time step issue is resolved) Previous trial of bounce-averaged gyrokinetic simulation scheme is limited on linear simulations [Y. Idomura et al., J. Plasma Fusion, Res. (2004); G. Rewoldt et al., CPC (2007)]. No nonlinear reports. Recently, both linear [L. Qi, J.-M. Kwon, T. S. Hahm et al., PoP (2016)] and nonlinear [J.-M. Kwon, L. Qi, T. S. Hahm et al., CPC (2017)] bounced-averaged gyrokinetic simulations are achieved. efficient ITG-TEM simulations Empirical ITER IPB98(y,2) H-mode confinement scaling τ E κ 0.78 (κ is elongation) This motivated previous GK simulation on ITG [Angelino et al., PRL (2009); Villard et al., PPCF (2013)]. No work yet is devoted to investigating elongation effects on TEM.
II-1 Toward Nonlinear Simulation In nonlinear state, particles are scattered by fluctuation in phase space i.e. trap passing Coulomb collision is another important mechanism in the mixing We need to model passing electrons and their mixing with trapped population subcycling drift kinetic electrons, which are passively following the evolution of fluctuation v Trapped particles v 2 v 2 = 1 B B max Passing particles Passing particles v 4
5 II-2 Coulomb Collision Electron pitch angle (λ = v /v) scattering by Lorentz collision 1 C L f e = ν e 2 λ 1 λ2 λ f e passing and trapped particle distributions are smoothly connected Analytic Maxwellian distribution Numerical electron distribution with collision
6 II-3 Nonlinear verification Particle number convergence test (a) Electron heat diffusivity (b) Zonal flow shearing rate
7 II-3 Nonlinear verification (Cont.) Linear TEM growth rates [J.-M. Kwon, L. Qi, T. S. Hahm et al., CPC (2017)] Nonlinear results: (a) electron heat diffusivity; (b) zonal flow shearing rate
8 II-4 Efficiency Overall distribution of computational costs 200 ~ 300% increase of simulation time compared to adiabatic ITG case Component Computational Cost Ions 17% Trapped electrons 27% Passing electrons 17% Deposition and Poisson solver 34% Collision 5%
9 II-5 Code summary Code: GyroKinetic Plasma Simulation Program (gkpsp), linear and nonlinear version δf, Global gyrokinetic code (GK ions) Bounce-averaged kinetic trapped electrons Particle-In-Cell (PIC) code Zonal flow conserving Krook operator to control noise Heating source to control density and temperature profiles 3D tokamak geometry including circular, analytical shaping, and experimental shaping equilibria
III-1 Setup We select a Collisionless Trapped Electron Mode (CTEM). Main parameters are the following at reference plane r = 0.5a: (initial setup) R 0 = 1.86m, a = 0.666m, B 0 = 1.91T, T e = 2.5keV, n e = 1.46 10 20 m 3, m i m e = 1836 R 0 L Te = 6.9, R 0 L Ti = 2.2 R 0 L n = 2.2 a ρ i = 250 T i T e = 1 q = 0.58 + 1.09r a + 1.09(r/a) 2 s = 0.78 10 Gradients locate at [0.25a, 0.75a]
11 III-2 Setup (Cont.) Linear gkpsp simulations show that for elongations κ = 1.0 and 2.0 growth rates of this CTEM case are nearly the same in terms of effective poloidal wave numbers. To separate linear and nonlinear effects. In the viewpoint of linear growth rates, the same linear drive, differences in the nonlinear saturation mechanism could be investigated. Nonlinear setup: Particle markers: N i = 200 per mode, N trapped electron = 40 per mode Simulation domain: r 0.1a, 1.0a, dr~ρ i Run with κ = 1.0, maximum toroidal mode number n max = 124, 1/4 of torus 32 modes Run with κ = 2.0, maximum toroidal mode number n max = 248, 1/8 of torus 32 modes (maximum effective poloidal wave number k θ ρ i /κ~1.4)
III-3 Linear background ITG-TEM mode actually feels the effective poloidal wave number k θ ρ i /κ. 12 Frequency of ITG-TEM mode as a function of the poloidal wave number k θ ρ i (Left figure) and the effective poloidal wave number k θ ρ i /κ (Right figure) for elongations of κ = 1.0, 1.25 and 1.5. Ref: Lei Qi, J.M. Kwon et al, PoP2016 k θ eff θ = 0 = nq r 1 κ 1+ 1 2 ε2 nq r 1 κ = k θ κ.
13 III-4 Transport Elongation effects on transport: Nonlinear stabilizations Transport burst is observed, and persist independently of Elongation introduces more evident radial fine structures in χ i. Ion heat transport more localized than electron heat transport, elongation enhances the localization. D < χ i < χ e for both elongations Γ = D n Q i = n 0 χ i T i Q e = n tr χ e T e
14 III-5 Zonal flow (1) Zonal flow shearing Reduce turbulence transport Elongation enhances zonal flow shearing rate 1. In radially average (Fig. c) 2. Especially at outer of midradius (2) Radial fine structures Larger radial wave numbers k r 0.5, 1.0 (by TEM) [Y. Xiao et al., PoP (2010)] Elongation shows a stronger spectrum that locates at relatively larger radial wave number
15 III-6 Radial scales Scale of turbulence eddies another factor to affect transport radial correlation Two point correlation function C rζ Δr, Δζ = <δφ r+ r,ζ+δζ δφ r,ζ > <δφ 2 r+ r,ζ+δζ ><δφ 2 r,ζ >, at θ = 0 By taking maxima along the ridge of C rζ Δr, Δζ, we obtain C r Δr Two length scales in both elongations: Microscopic: L r ~4ρ i κ = 1.0 L r ~3ρ i κ = 2.0 Mesoscopic : larger L r Consistent with previous simulations [Y. Xiao et al., PRL (2009)] and experimental results [P. Hennequin et al., 42 nd EPS Conf. (2015)] Elongation reduces radial correlation in both microscopic and mesoscopic scales.
16 III-7 Turbulence spectrum For κ = 2: (around r=0.635a) A dual mode propagating in counter directions (ion diamagnetic and electron diamagnetic directions) With the occurrence of dual mode, stronger shearing rate is observed along with lower transport level. Similar observation has been made from experiments in the edge across L-H transition on DIII-D[Z. Yan, Submitted to NF (2017)]
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IV Conclusions Nonlinear bounce-averaged gyrokinetic simulation is now ready for efficient ITG- TEM investigations. Further topics can be nonlinear ITG-TEM coupling and collisional effects, etc.. Nonlinear effects of plasma elongation on turbulence, zonal flow and transport in isolation from linear physics, are explicitly explored by efficient bounceaveraged global gyrokinetic simulations for CTEM. 21 Plasma elongation can stabilize both electron and ion heat fluxes as well as particle flux in CTEM dominant turbulence transport. Higher elongation is found to enhance the shorter radial scale zonal flows, and thus to lower transport levels. Two radial characteristic scales of turbulence eddies, i.e., microscopic and mesoscopic scales are observed in semi-quantitative agreements with previous simulations [Y. Xiao et al., PRL (2009)] and experimental results [P. Hennequin et a., 42 nd EPS Conf. (2015)]. Plasma elongation reduces both radial scales. A dual mode propagating in counter directions is observed with stronger sheared zonal flow for higher elongation.
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23 III-7 Turbulence potential Turbulence potential amplitude e δφ /T, could affect transport level as well Elongation reduces turbulence amplitude
24 III-9 Global κ-scaling and role of zonal flow Global κ-scaling from gkpsp shows χ i κ 1.4, χ e κ 0.9, D κ 1.4 Due to κ itself, without shearing Empirical ITER IPB98(y,2) H-mode confinement scaling τ E κ 0.78 GYRO local simulations claimed scaling of κ 1, however mainly due to local κ shearing. It is evident that zonal flow plays significant roles in turbulence suppression with elongation