9 th IAEA TCM on Hmode physics and

9 th IAEA TCM on H-mode physics and transport Catamaran Resort Hotel, Sep. 24 -26, 2003 Transport within transport barriers : theorist’s view of the feature Theoretical understandings of transport barrier as a complex system Y. Kishimoto Naka Fusion Research Establishment Japan Atomic Energy Research Institute (JAERI)

Contents 1. Introduction 2. Background and motivation 2. Fluctuation dynamics in wide frequency and wave number space Ø Key issues of nonlinear fluctuation dynamic essential for transport barrier physics Ø Possibility of “control” of fluctuation and related transport 3. Summary Discussion • • • Prof. P. Diamond : Furture direction in transport barrier Physics Prof. K. Itoh : Prospect of transport physics in science Dr. X. Q. Xu : Prospect for Edge physics Acknowledgement : T. S. Hahm, K. Itoh, S-I. Itoh, M. Yagi, Z. Lin, P. Diamond, E-J. Kim, C. Holland, A. K. Wong, R. White, D. R. Ernst J. Q. Li, Y. Idomura, N. Miyato, T. Matsumoto

High performance is realized by having “structure” Current Pressure [Itoh S. I. , et al. , J. Nucl. Materials, ’ 95, Zohm, PPCF, ’ 96, Burrell, Po. P, ’ 97, Ida, PPCF, ’ 98 0. 5 0 L-mode 1. 0 Inductive current dominant H-mode High bp BS current dominant High bp-H RS-mode 0 0. 5 Steady state 1. 0 High confinement • • • • • • H-mode : ASTEX (1982) CNTA-NBI mode Core H-mode Da. H-mode Reversed (Negative) Shear mode Enhanced Reversed Shear mode High Density H-mode Helical Electron ITB High bp mode High bp-H mode High li-mode High Ti mode I-mode Improved Ohmic Confinement mode Lower-Hybrid Heating mode Pellet Ehhanced Performance H-mode Radiation Improved Mode Super shot VH-mode etc ………. Understanding the “selection rule” of the distinct states and the control

Complex nonlinear loop system of structural plasma 1. 2. 3. 4. Neoclassical dynamics Fluctuation and self-organization dynamics Global linkage as nonlinear loop system, and “control” Identification of the degree of complexity of the state

11 Contribution papers 1. Neoclassical dynamics (E 8: Yagi, E 10: Ernst) 2. Fluctuation and self-organization dynamics E 1: Diamond E 2: Hahm E 3: Kim E 4: Holland E 5: Wang E 6: K-Itoh E 7: White 3. Global linkage as a nonlinear loop, and control E 8: Yagi E 9: S. Itoh E 10: Ernst E 11: Xu (E 1: Diamond) 4. Identification of the degree of complexity of the state (E 9 : S. Itoh) (E 3: Kim)

Fluctuation dynamics in wide wave number space 1. Linear free energy source in complicated magnetic field • Mode structure in reversed/weak magnetic shear cf. Failure of ballooning picture electron 2. Nonlinear free energy source • Normal/inverse spectrum cascade • Fluctuation due to nonlinear/turbulent dissipation. cf. CDBM • Secondary and higher order nonlinear instabilities with different time and spatial scales ion skin electron size skin size MHD cf. Generalized zonal/streamer mode, Zonal mode driven KH mode, etc 3. Interaction and interference among activities with different time and spatial scales, and through spatial dimension • 　Non-local, non-diffusive “new energy/information transfer channel” using not only spectral-space and “spatial dimension” [Diamond, Hahm, et al. H-WS, ’ 03]

Linear free energy source electron reversed shear ETG skin size normal shear ETG negative shear ITG ion (※) normal shear ITG • Global linear gyro-kinetic dispersion in reversed shear plasma [Idomura, et al. , NF, ’ 02] Gap structure Slab mode-like structure • Short wavelength ITG mode (shear-less slab) [Smolyakov, Yagi, et al. , PRL, ’ 02] [Idomura, et al. , Po. P, ’ 02, Kishimoto, et al. , PPCF, ’ 99] [Voitsekhovitch, Garbet, et al. , Po. P, ’ 02] • Short wavelength ITG mode in current carrying plasma [Wang, H-WS, ’ 03]

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

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

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

Zonal pressure and b-increase Reduced MHD equation [Ichiguchi, et al. , IAEA, ’ 02] [Carreras, et al. , Po. P, ’ 01] • 　Resistive interchange modes induce a staircase structure, leading to a linearly unstable high-b profile

Impact of zonal flow on transport (1) Gyro-kinetic PIC simulation using global profile effect and canonical Maxwellan particle distribution [Idomura, et al. , NF, 2002] linear saturation Quasi-steady state Zonal fluctuation Turbulent fluctuation : m/n=0/0 GAM fluctuation : m/n=1/0 • Macro-scale “mean flow”, same level as that of the equilibrium, regulated by equilibrium profile [Lin, et al. , Science, ’ 98]

Modulational instability and zonal flow Parameter to change the ratio of “turbulence” part and “zonal” part [Smolyakov, et al. , Po. P, ’ 00, Malkov, et al. , Po. P, ’ 01, Li-Kishimoto, Po. P, ’ 02] [White, et al. , H-WS, ’ 03] • ITG case (adiabatic electron except k||=0) • ETG case (adiabatic ion) (b) Large grow rate for Streamer-like anisotropic pump wave :

Self-organization to flow dominated fluctuations • Weak magnetic shear increases linear instability sources, but nonlinearly transfers energy to zonal components disappearance of anomaly in high pressure state KH-mode like instability Zonal flow energy (A) S=0. 2 (B) S=0. 1 [Kishimoto, Li, et al. , IAEA ’ 02] • Drift-Alfven turbulence in edge plasma [Kendel, Scott, et al. , Po. P, ’ 03]

Characteristics of flow dominated fluctuations Time averaged spectrum w/o zonal flow wavelet analysis 1. 0 0. 8 0. 6 0. 4 ETG 0. 2 with zonal flow GKH ? DW 1. 0 0. 8 0. 6 0. 4 0. 2 Condensation to KH-mode KH ZF Near marginal and quasi-linear process [Kim-Diamond, Po. P ’ 02]

Statistical nature of turbulence-zonal fluctuation system “Fractal dimension” and “PFD” : Probability Distribution Function rate No flow case rate strong flow case • Shrinking dimensionality due to coherent structure • Coherency increases with near Gaussian PDF of flux [Matsumoto, et al. , Toki-conf, ’ 03]

Statistical nature of turbulence 1. Fractal dimension • TEXTOR: Signal from Langmuir probes [Budaeev, et al. , PPCF, ’ 93] d= 12 -16 (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) 2. Probability Distribution • Size distribution of heat pulse from GK simulation [Nevince, ’ 00, Holland, et al. , IAEA, ’ 02] PDF of density fluctuation of PISCES-A linear device and So. L of the Tore Supra “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 [S-Itoh, et al. , 　Kim-Diamond, H-WS, ’ 03] [Antar, et al. , PRL, ’ 01]

Interference among different scale fluctuations l Interaction through quasi-coherent zonal modes electron skin size [Hahm-Burrel, Po. P, ’ 02, Hahm, et al. , Po. P, ’ 99] [Li-Kishimoto, PRL, ’ 02, ion Idomura, et al. , NF, ’ 02] • Time varying Random shearing • Scattering to high-k MHD l Interaction through micro-scale structure, eddy viscosity, noise, etc. [Itoh, et al. , PPCF, ’ 01, Yagi, et al. , IAEA, ’ 02] Open new nonlinear energy transfer channel Trigger problem

ITG transport modulation due to small scale flow GF-ITG simulation with micro-scale ETG driven flows (b) Upper state Probabilistic damping trigger No flow (a) Lower state • Non-local mode coupling and associated energy transfer to high kx damped region • Micro-scale flow intermittently quenches ITG turbulence low-k high-k L-state H-state [Li-Kishimoto, PRL, ’ 02, Po. P, ’ 03] [courtesy of Miura and JFT-2 M group]

Multiple-scale turbulence and bifurcation Langevan approach for 2 -scale plasma turbulence system Semi-Micro (cf. ITG) Micro. Mode (cf. skin/ETG) Semi-micro mode amplitude • For micro-mode dominated solution, semi-micro mode is quenched, and vise-versa. • Mechanism of ITB formation with different ion and electron dynamics [Yagi, et al. , IAEA, ’ 02, Itoh-Itoh, PPCF, ’ 01] cf. Distinct dynamics between ion and electron [Koide, et al. , PPCF, ’ 98]

Nonlinear transfer channel of fluctuation In spectral space Energy transfer among different scale fluctuations through local/non-local cascade or inverse cascade process In real position space Radial energy transfer through propagation and/or spread skin size ion electro n • Inverse cascade of “radial” shorter wavelength modes • Radial “diffusion” and/or “convection” from linearly unstable region to stable zone [Diamond-Hahm-Lin, et al. H-WS, ’ 03] [White, et al. , H-WS, ’ 03] Modulational approach with spatial dimension • Successive excitation of secondary and tertiary instabilities using spatial dimension

Nonlinear transfer channel of fluctuation • Toroidal linear coupling • Nonlinear mode coupling “convection” “conduction” [Rewoldt-Shirai, et al. , NF, ’ 02] JT-60 U E 29728 t=6. 0 3 g(105 sec-1) Garbet, et al. , NF, ’ 94 A turbulent zone spreading radially in such a way that its level is no longer directly related to local plasma parameter • Coupling through equilibrium profile Newmann, Diamond, et al. ITB dynamics based on turbulent-transport equation system x(m 2/s) Mattor-Diamond, Rep. UCRL, ’ 93 g P 2 Full code (w/rotation) kq. Pi=0. 53 1 0 10 1 xi xe 0. 1 0 0. 5 r/a 1 cf. Transport phenomena hardly explained from linear analysis

Spatial convection of instability q(r) q-min surface • Increase of linear instability source for reversed shear plasma -2 cm 0 rdius +2 cm ① t=2 ms • Turbulent energy is “nonlinearly” converted to flow component through “spatial dimension”. KH ③ t=6 ms ETG KH KH-ZF ② t=2 ms ETG-ZF ④ t=8 ms Origin of “structure” is anomalous transport!! ① ② ③ [Idomura, et al. , Po. P, ’ 00] ④ Transport suppression Strong turbulence

Turbulent spreading and diffusion Some evidence from numerical simulation Garbet, et al. , NF, ’ 94, Sydora, et al. , PPCF, ’ 96, Parker et al. , Po. P, ’ 96, Lin, et al. , IAEA, ’ 02, also PRL, ’ 02 ITG GK simulation ETG GF simulation linear phase steady state phase saturation phase [Sydora, et al. , PPCF, ’ 96] [courtesy of J-Li] no rational surface (no damping)

Turbulent spreading and size scaling Discussion about transport size scaling (B or GB) • No device size dependence of radial eddy length : Dx~7 ri : Scale size is “microscopic” • Radial spreading of fluctuation into stable zone Nonlinear model of turbulent propagation Front like solution I Linear damping region r 0+D r PDF of particle diffusion : close to “Gaussian” with no significant tail, suggesting “diffusive transport” [Hahm, et al. , H-mode WS, ’ 03] [Lin, et al. , IAEA, ’ 02]

Size scaling of transport • GB scaling well above the instability threshold • Break of the GB scaling to Bohm scaling (worse) near threshold Stabilization effect due to shearing in the ballooning phase velocities due to global profile variation • Non-local transport where local diffusivity depend on finite radial length : [Waltz-Candy, et al. , Po. P, ’ 02, also IAEA, ’ 02]

Summary : prospect for future direction • The physics of key elements dominating the transport barrier, specifically nonlinear process, is extensively studied, and the understandings have been developed. • Interaction and/or interference among different time and scale fluctuations, not only in wide frequency/wave number space, but also real space dynamics, becoming crucially important. • Statistical approaches to identify the degree of complexity of the state and transition dynamics are becoming necessary. • Numerical approach to handle wider dynamical range is becoming tough, for example, micro-scale electron dynamics, but continuous efforts are crucial. • Methodology to control the nonlinear loop system is becoming necessary. cf. integration of key element. • Close interplay and interference among theory, simulation and experiment is desirable.

Role of H-mode and ITB physics in science ? [courtesy of Earth simulator center] [Koshyk-Hamilton, JAS, 01]