Y. Kishimoto Naka Fusion Research Establishment, Japan Atomic Energy Research Institute US-Japan JIFT workshop, December 15-17, Kyoto University, Kyoto, Japan Local and non-local gyro-fluid simulation of ITG and ETG turbulence and statistical properties In collaboration with J. Q. Li, N. Miyato, T. Matsumoto, and Y. Idomura Li et al., the 13 th Toki conf. Miyato et al., 13 th Toki conf. Matsumoto et al., 13 th Toki conf. Idomura et al., 13 th Toki conf.

Contents Background and motivation Fluctuation dynamics of micro-scale ETG turbulence Summary Fluctuation = turbulent part + laminar-like flow part Control the fluctuation by changing the partition Hierarchical interaction among different scale fluctuation Enhanced ETG-driven zonal flow dynamics based gyro-fluid model (cf. Hamaguchi-Horton equation + electromagnetic effect) Statistical properties of fluctuation such as fractal dimension and PDF Fluctuation dynamics of meso-scale ITG turbulence Nonlinear Global gyro-Landau fluid simulation Toroidal and electromagnetic effect on zonal flow

[Idomura et al., ’00] [Matsumoto, Naitoh, PoP, ’03] Nonlinear fluctuation dynamics Local inverse/normal cascade Mixed turbulent/zonal fluctuation system Internal kink event MHD-driven Er-field Zonal-  [Idomura, PoP, ’00] ETG streamers found near threshold are essentially linear structures whose nonlinear interaction is weak. [Dorland, et al., IAEA, ’02] MHD ion electron skin size [Jenko-Kendel,PoP, ’02] Wendelsteien 7AS simulation [Kendel, PoP, ’03] Nonlinearly generated “convective cell mode”

Nonlinear turbulent-convective cell system with complex “activator” and “suppressor” roles Nonlinear free energy source Maternal fluctuation Transport Low m/n drive Flow driven tertiary nonlinear instability GAM : Stringer-Winsor : Kelvin-Helmholtz mode GKH mode collisonal damping p-profile q-profile streamer Neo-classical mean shear flow [Kim-Diamond, PoP, ’03]

Trigger of barrier formation Global 2-fluid nonlinear EM simulation [Thyagaraja, PPCF,’00] “profile-turbulence interaction” Long wavelength EM modes induce “corrugations”, modifying the evolution of electric field and bootstrap current Reduced MHD equation [Ichiguchi, et al., IAEA,’02] Resistive interchange modes induce a staircase structure, leading to a linearly unstable high-  profike

Turbulent de-correlation by flow and transport dynamics [Hahm-Burrel, PoP, ’02, Hahm, et al., PoP, ’99] Time varying Random shearing Scattering to high-k [Hahm, et al.,PoP, ’99 ] Heat flux PDF : almost Gaussian process

Nonlinear free energy source MHD ion electron skin size Various “Zonal modes” are exited through modulational instability Flow :Field : Pressure : [Holland-Diamond, PoP, ’02, Jenco et al., IAEA, ’02, Miyato, et al., PPCF, ’02] “Reynolds stress” “Maxwell stress” “Collisional damping” “Pressure anisotropy (Stringer-Winsor term)” [Lin, et al., PRL, ’99, Kim, et al., PRL, ’03] [Hallatshek-Biskamp PRL, ’01] Small scale pressure corrugations are hardly controllable  SOC dynamics Large scale component may change the q-profile

ITG transport modulation due to small scale flow [Li-Kishimoto, PRL, ’02, PoP, ’03] GF-ITG simulation with micro-scale ETG driven flows Upper state Lower state high-k low-k Non-local mode coupling and associated energy transfer channel to high kx damped region No flow Micro-scale flow intermittently quenches ITG turbulence [Li-Kishimoto, PRL, ’02, Idomura, et al., NF, ’02 ]

[ Smolyakov, et al., PoP, ’00, Malkov, et al., PoP, ’01, Li-Kishimoto, PoP,’02] Modulational instability and zonal flow ITG case (adiabatic electron except k || =0) ETG case (adiabatic ion) (b) Large grow rate for Streamer-like anisotropic pump wave : Parameter to change the ratio of “turbulence” part and “zonal” part (a)

ETG-driven zonal flow spectrum Modulational instability analysis : 3 and 5 fields H-M model Slow or marginal process Instability increases in small kx regime cf. saturation at low level by spectrum change : Slab ETG-mode :   x k  weak s ˆ broader narrower x   Zonal flow instability in weak magnetic shear regime pump wave :

(A) S=0.2 (B) S=0.1 Self-organization to flow dominated fluctuations disappearance of anomaly in high pressure state Weak magnetic shear increases linear instability sources, but nonlinearly transfers energy to zonal components [Kishimoto,Li, et al., IAEA ’02] [Kendel, Scott, et al., PoP, ’03] Zonal flow energy Drift-Alfven turbulence in edge plasma total energy turbulent energy

[Koshyk-Hamilton, JAS, 01] [courtesy of Earth simulator center] turbulent energy zonal flow energy Energy partition change due to zonal flow excitation sun Earth environment sun Earth environment ??? Change of fluctuation characteristics in high pressure state

Condensation of turbulent energy in flow dominated plasma Isotropic spectra at short wavelength, but energy condensation to narrow k y region : with zonal flow w/o zonal flow Isotropic spectra in short and long wavelength region

KH mode weakly unstable in an enhanced zonal flow Weakly unstable KH Marginally unstable KH Linear analysis intoroducing ETG-driven zonal flow pattern Zonal flow instability and KH mode instability DW ZF KH Near marginal and quasi-linear process [Kim-Diamond, PoP ’02]

Turbulent structure in an enhanced zonal flow Spatial correlation function: With zonal flowW/O zonal flow   Coherent in y-direction   Incoherent in x-direction

Size distribution of heat pulse from GK simulation [Nevince,’00, Holland, et al., IAEA,’02] TEXTOR: Signal from Langmuir probes [Budaeev, et al., PPCF, ’93] d= (attached) d=6-7 (detached) d=30 (from 15) (induced H-mode) CHS : Electron density fluctuation [Komori, et al., PRL, ’94] d~ 6.1 (RF heating) d~6.2 (NBI heating) d~8.4 (RF+NBI) 1. 1.Fractal dimension 2. Probability Distribution PDF of density fluctuation of PISCES-A linear device and SoL of the Tore Supra [Antar,et al.,PRL,’01] Statistical nature of turbulence “Noise forcing by coherent structure” Non-Gaussian PDF for the Reynolds stress and hest flux [Kim, et al., IAEA,’02] Probabilistic view of L-H transition

Statistical nature of turbulence-zonal fluctuation system “Fractal dimension” and “PFD” rate strong flow case Shrinking dimensionality due to coherent structure [Matsumoto, et al., Toki-conf, ’03] Heat flux No flow case rate

Electromagnetic effect on turbulent transport Finite b-stabilization consistent to with Okawa-scaling [Labit-Ottaviani, PoP, ’03, Okawa, Phys. Lett., ’78] Reduction of zonal flows due to the cancellation of the Reynolds stress by the Maxwell stress [Li-Kishimoto, PoP, submitted] Cancellation between Reynolds stress and Maxwell stress  =0  =1.5%  =3.0%  =7.5%

Zonal flow in toroidal geometry B. D. Scott, 2003, Edge turbulence simulation by DALF3 Geodesic curvature effect, i.e. coupling between pressure an-isotropy and vorticity, plays an important role for the zonal flow generation

Electromagnetic Landau fluid global simulation [N. Miyato, et al, Toki-conf., ’03] Density equation Vorticity equation Ion parallel velocity equation Ohm’s law [c.f. Electrostatic toroidal simulation by Garcia, Leboeuf, et al, IAEA, ’00 ]

Electromagnetic Landau fluid global simulation Ion temperature equation With R/a=4, ρ i /a=0.0125, Te=Ti, D ～ m 4, η=4×10 - ５ With definition : [cf. Snyder, et al., PoP, ’02] r/a N 0 / N c T 0 / T c q

Nonlinear EM toroidal simulation [Rewoldt, ’87, Zonca, ’01, Falchetto, ’02] Onset of KBM above a critical beta: [cf. Nonlinear GK simulation, Snyder, et al., PoP, ’02, Candy, et al. PoP, ‘02] Growth rate time

Zonal flow: Heat flux: Zonal flow: Heat flux: Nonlinear EM toroidal simulation Nearly stationary zonal flow in inner low-q region Oscillatory zonal flow dominated in outer high-q region : [Hallatschek-Biskamp ’01, Schoch ’02, Ramisch et. al. ’03, McKee et. al. ’03] time

Reynolds 17.1 Maxwell GAM Reynolds 6.84 Maxwell GAM Nonlinear EM toroidal simulation time

Energy loop of DW-ZF system GAM term is a sink [B Scott 2003 (4-field drift-Alfvén)] or a source [K Hallatschek and D Biskamp 2001(Electrostatic Braginskii)] for zonal flow ? Our results shows that GAM term is a sink. Drift wave turbulence Zonal flow   Pressure asymmetry p 10 Reynolds stress [φ, ∇ ⊥ 2 φ] [φ,p] Toroidal coupling

Unified MHD-ITG turbulence simulation MH D io n electro n ski n siz e Potential q KBM(n=20,β=1.5%) Positive magnetic shear Potential Toroidal ITG(n=20) Positive magnetic shear q Vector potential q Double tearing mode negative magnetic shear Zonal field and associated flattening of q-profile emerge at an lower rational surface, c.f. q=1.5

Unified MHD-ITG turbulence simulation Rich activities in macro-scale MHD and micro– mesoscale fluctuation Physical challenges Anomalous transport — MHD/Microturbulence solved Slow evolution —Transport equation solved Plasma shaping — Realistic geometry High temperature — Large “Reynolds’ numbers” Collisionality — Resistive MHD Strong magnetic field — Highly anisotropic transport Resistive wall — Non-ideal boundary conditions Computational challenges   Wide spatial-temporal scales — non-adiabatic ion/electron fluid response; internal boundary layers; small dissipation; mesoscale …   Extreme anisotropy — Special direction determined by strong magnetic field