Intermittent Transport and Relaxation Oscillations of Nonlinear Reduced Models for Fusion Plasmas S. Hamaguchi, 1 K. Takeda, 2 A. Bierwage, 2 S. Tsurimaki, 2 H. Sato, 2 T. Unemura, 2 S. Benkadda, 3 and M. Wakatani 2 1 Osaka University, Japan 2 Kyoto University, Japan 3 CNRS/Université de Provence, France Dedicated to Prof. Masahiro Wakatani

Outline Intermittent Thermal Transport due to ITG turbulence –a low degree-of-freedom model: how low can it be? –full PDE Intermittent Particle Transport due to resistive drift turbulence

Toroidal ITG Equations where Toroidal ITG mode is described by the following equations, W. Horton, D-I. Choi and W. Tang, Phys. Fluids 24, 1077 (1981) : electrostatic potential (fluctuation) : ion temperature gradient : viscosity : ion pressure (fluctuation) : effective gravity : thermal conductivity

Low Degree-of-Freedom Model with 18 ODEs Low-order modes in a slab geometry If are neglected, the above model agrees with the 11 ODE model by G. Hu and W. Horton [Phys. Plasmas 4, 3262 (1997)]

Characteristic quantities The mean velocity is driven by the Reynolds stress S R. Nusselt number Nu :

Periodic Oscillation (a)Phase space ‘ K 0 -Nu ’ and (b)Power spectrum of K 1. Time evolutions of (a)kinetic energy K and (b)Nusselt number Nu for Ki=0.4. A periodic oscillation appears.

Chaotic Oscillations (a)Phase space ‘ K 0 -Nu ’ and (b)Power spectrum of K 1. Time evolutions of (a)kinetic energy K and (b)Nusselt number Nu for Ki=0.6. Chaotic oscillations appear.

Intermittent Bursts Time evolution of kinetic energy for Ki=4. Intermittent bursts occur. Time evolution of (a)Nusselt number N u (b)Reynolds stress S R N u and S R burst when the ITG modes grow rapidly.

Intermittent Transport: 3 Steps 1.Turbulence generates a mean flow through Reynolds stress. 2.The mean flow suppresses instability. 3.The mean flow becomes weak due to viscous damping.

Scaling Law -18 ODEs Transition of transport scaling due to intermittency. Dependence of the time averaged Nusselt number on the ion temperature gradient is expressed as

Intermittent oscillations in full PDE model Time evolutions of kinetic energy K (left) and Nusselt number Nu (right) for Ki=5. Intermittent oscillations appear.

Scaling law — full PDE simulation

 Mode structures in the radial direction that are essential for the formation of intermittency 1.Low degree ODE models (18 ODE) 2.ITG (PDE): Transport scaling: 3.Resistive drift (PDE) : Transport scaling: Summary

Velocity-Space Structures of Distribution Function in Toroidal Ion Temperature Gradient Turbulence T.-H. Watanabe and H. Sugama National Institute for Fusion Science/ The Graduate University for Advanced Studies

Motivation How are the velocity-space structures of f associated with turbulent transport and zonal flows ? How are the velocity-space structures of f associated with turbulent transport and zonal flows ? Kinetics of the zonal flow and GAM Kinetics of the zonal flow and GAM Structures of f providing the steady ITG turbulent transport Structures of f providing the steady ITG turbulent transport focusing on entropy balance and generation of fine-scale structures of f through the phase mixing. => Gyrokinetic-Vlasov simulation of f in multi- dimensional phase-space with high resolution (Flux tube configuration is employed).

Entropy Balance in the Toroidal ITG System (Entropy Variable) (Potential Energy) (Heat Transport Flux) (Collisional Dissipation)

Steady State v.s. Quasi-steady State In the quasi-steady state for the collisionless case, In the quasi-steady state for the collisionless case, time The two limiting states of turbulence have been extensively investigated by the slab ITG simulations. (Watanabe & Sugama, PoP 9, 2002 & PoP 11, 2004) In the steady state for the weakly collisional case, In the steady state for the weakly collisional case,

Collisionless Damping of Zonal Flow and GAM Initial value problem for n=0 mode with  f| t=0 =F M Initial value problem for n=0 mode with  f| t=0 =F M The residual zonal flow level agrees with Rosenbluth & Hinton theory. The residual zonal flow level agrees with Rosenbluth & Hinton theory. What is the velocity- space structure of f ? What is the velocity- space structure of f ? How is the entropy balance satisfied ? How is the entropy balance satisfied ? Cyclone DIII-D base case

Conservation Law for the Zonal Flow (n=0) Components From the gyrokinetic equation for k y =0, From the gyrokinetic equation for k y =0, … Subset of the entropy balance equation Entropy variable  S increases during the zonal flow damping.

Velocity-Space Structures of f in the Zonal Flow Damping Decrease of W kx with the invariant G means increase of  S kx as well as generation of fine- scale structures of f due to phase mixing by passing particles Entropy transfer in the v- space Decrease of W kx with the invariant G means increase of  S kx as well as generation of fine- scale structures of f due to phase mixing by passing particles Entropy transfer in the v- space Coherent structures for trapped particles Coherent structures for trapped particles Neoclassical Polarization Neoclassical Polarization (2   v // passing trapped Bounced Averaged Solution

ITG Turbulence Simulation k x,min = , k y,min = 0.175, = (Lenard-Berstein collision model) Cyclone DIII-D base case

Entropy Balance in Toroidal ITG Turbulence Entropy balance is satisfied (  /D i <~ 8%). Entropy balance is satisfied (  /D i <~ 8%). Statistically steady state of turbulence:  i Q i ~=  D i => Entropy production balances with dissipation. Statistically steady state of turbulence:  i Q i ~=  D i => Entropy production balances with dissipation.

Velocity-Space Structures of f in Toroidal ITG Turbulence Fine-scale structures generated by the phase mixing appear in the stable mode, and are dissipated by collisions, while the transport is driven by macro-scale vortices. Fine-scale structures generated by the phase mixing appear in the stable mode, and are dissipated by collisions, while the transport is driven by macro-scale vortices. => Entropy variable transferred in the phase space High-velocity space resolution is necessary. High-velocity space resolution is necessary. Long wavelength mode Linearly stable mode (2   v //

Production, Transfer and Dissipation of Entropy Variable Entropy variable produced on macro-scale is transferred in the phase space, and is dissipated in micro velocity scale. Entropy variable produced on macro-scale is transferred in the phase space, and is dissipated in micro velocity scale. The result confirms the statistical theory of the entropy spectrum for the slab ITG turbulence (Watanabe & Sugama, PoP 11, 2004). The result confirms the statistical theory of the entropy spectrum for the slab ITG turbulence (Watanabe & Sugama, PoP 11, 2004). Production Dissipation Transfer Phase Mixing Nonlinearity Small Velocity Scale Small Spatial Scale Transport Collisionless Slab ITG Transport (WS 2004)

Summary Gyrokinetic-Vlasov simulations of ITG turbulence and ZF with high velocity-space resolution. Gyrokinetic-Vlasov simulations of ITG turbulence and ZF with high velocity-space resolution. Collisionless damping process of ZF & GAM is represented by the entropy transfer from macro to micro velocity scales. Collisionless damping process of ZF & GAM is represented by the entropy transfer from macro to micro velocity scales. Neoclassical polarization effect is identified. Neoclassical polarization effect is identified. Statistically steady state of toroidal ITG turbulence is verified in terms of the entropy balance. Statistically steady state of toroidal ITG turbulence is verified in terms of the entropy balance. Production, transfer, and dissipation processes of the entropy variable are also confirmed for the toroidal ITG turbulent transport. Production, transfer, and dissipation processes of the entropy variable are also confirmed for the toroidal ITG turbulent transport.

Flux Tube Code Toroidal Flux Tube Code Toroidal Flux Tube Code Linear Benchmark Test AB CDFE 0 22   B z yy The gyrokinetic equation is directly solved in the 5-D phase space.

Gyrokinetic Equation GK ordering + Flute Reduction GK ordering + Flute Reduction Co-centric & Circular Flux Surface with Constant Shear and Gradients Co-centric & Circular Flux Surface with Constant Shear and Gradients Quasi-Neutrality + Adiabatic Electron Response Quasi-Neutrality + Adiabatic Electron Response

Selected Parameters for our simulations ・ box size ： Lx=Ly=10.0 Lz=100.0 ・ initial perturbations: ・ # of grid points in the x direction : 200 ・ mode numbers: -10  m  10, -2  n  2 （ total: 105modes ） -15  m  15, -3  n  3 （ total: 341modes) 、 ・ density profile: （ L n ： density gradient scale length ）

Quasi-Linear saturation Time evolutions of (a)kinetic energy K and (b)Nusselt number Nu for Ki=0.3. (a)Phase space ‘ K 0 -Nu ’ and (b)Power spectrum of K 1.

Periodic Oscillations (a)Phase space ‘ K 0 -Nu ’ and (b)Power spectrum of K 1. Time evolutions of (a)kinetic energy K and (b)Nusselt number Nu for Ki=0.5. More modes are excited.

Linear Analysis The most unstable wave number is estimated as In this study, the following wave number is assumed, From the dispersion relation, the linear growth rate is as follows.

Normalization The standard drift wave units are used for normalization.

Linear growth rate

Increase of Accuracy 1.18 ODE: Two modes (m=0 and 1) in the y direction and 3 modes (l=1,2, and 3) in the x direction. 2.1D: Two modes (m=0 and 1) in the y direction and full modes in the x direction. 3.PDE: full modes in the x and y directions.

Intermittent oscillations in 1D model Time evolutions of kinetic energy K (left) and Nusselt number Nu (right) for Ki =5.

Scaling Laws ODE, 1D, and Full Simulations 1D model Full PDE simulation 18 ODE

Intermittent Particle Transport in Resistive Drift Turbulence

Particle Flux (a)’ 0 < t < 10000(b)’ 0 < t < (a) 0 < t < 10000(b) 0 < t < 50000

Particle flux vs. Resistivity particle flux Γ resistivity η