1. 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 1/17 Gyrokinetic Theory and Simulation of Zonal Flows and Turbulence in Helical Systems T.-H. Watanabe and H. Sugama National Institute for Fusion Science / The Graduate University for Advanced Studies (Sokendai)

2. 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 2/17 Introduction In the LHD experiments, better confinement is observed in the inward-shifted magnetic configurations, where the pressure- gradient drives instability stronger while better neoclassical ripple transport. In the LHD experiments, better confinement is observed in the inward-shifted magnetic configurations, where the pressure- gradient drives instability stronger while better neoclassical ripple transport. Anomalous transport is also improved in the inward shifted configuration. Anomalous transport is also improved in the inward shifted configuration. H. Yamada et al. Gyrokinetic Simulations of the ITG Turbulence Gyrokinetic Simulations of the ITG Turbulence Zonal flow is a key ingredient to regulate turbulent transport in magnetically confined plasmas. Zonal flow is a key ingredient to regulate turbulent transport in magnetically confined plasmas.

3. 3/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Outline This work deals with gyrokinetic theory and simulations of turbulent transport and related zonal flow dynamics in helical systems. This work deals with gyrokinetic theory and simulations of turbulent transport and related zonal flow dynamics in helical systems. Theoretical analysis and numerical simulations of the linear response of zonal flows in helical systems. Theoretical analysis and numerical simulations of the linear response of zonal flows in helical systems. Gyrokinetic-Vlasov (GKV) simulations of the ITG turbulence in helical systems. Gyrokinetic-Vlasov (GKV) simulations of the ITG turbulence in helical systems.

4. 4/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Collisionless Damping of Zonal Flow and GAM in Tokamaks 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 is considered to be important in regulating turbulent transport. The residual zonal flow is considered to be important in regulating turbulent transport. Cyclone DIII-D base case Distribution Function Positive Negative Residual Zonal Flow (response kernel)

5. 5/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 The result is useful to optimize configurations for enhancing zonal-flow generation and accordingly reducing turbulent transport. The result is useful to optimize configurations for enhancing zonal-flow generation and accordingly reducing turbulent transport. It is suggested that reduction of ripple-trapped particles’ drift not only improves the neoclassical transport but also enhances the zonal flows. It is suggested that reduction of ripple-trapped particles’ drift not only improves the neoclassical transport but also enhances the zonal flows. Zonal-Flow Potential Response Kernel Long-time Response Function Collisionless Response of Zonal Flows in Helical Systems Collisionless Response of Zonal Flows in Helical Systems

6. 6/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 ( q = 1.5,  t = 0.1, k r a i =0.131 ) ( L = 2, M = 10 ) Simulation of Zonal Flow Damping in Helical Systems Simulation Long-Time  Response Kernel K L (t) Validity of the theoretical analysis is verified by GKV code. Validity of the theoretical analysis is verified by GKV code. Radial drift of helical-ripple- trapped particles is identified. Radial drift of helical-ripple- trapped particles is identified. Velocity distribution function for  =8  /13 at t = 6.23 R 0 /v ti.

7. 7/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Limiting Form of the Long-Time Response Kernel K L (t) The long-time limit of K L (t) depends on the depth of helical ripples  H as well as on the radial wave number k r. The long-time limit of K L (t) depends on the depth of helical ripples  H as well as on the radial wave number k r. Low-k r High-k r Lower residual flow (smaller K > ) is obtained for lower k r. (longer radial wavelength). Lower residual flow (smaller K > ) is obtained for lower k r. (longer radial wavelength).

8. 8/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Gyrokinetic Simulations of ITG Turbulence in Helical Systems GK ordering + Flux tube model + Periodic (x,y) GK ordering + Flux tube model + Periodic (x,y) Co-centric Flux Surface with Constant Shear and Gradients Co-centric Flux Surface with Constant Shear and Gradients Quasi-Neutrality + Adiabatic Electron Quasi-Neutrality + Adiabatic Electron

9. 9/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Model of the ITG Turbulence Simulation in Helical Systems Effects of the helical field are introduced through |B|. Effects of the helical field are introduced through |B|. The mirror force term also involves the helical components of |B|. The mirror force term also involves the helical components of |B|. The simulation is done on a torus with an effective minor radius r 0, where  T =  B 0 r 0 2. The simulation is done on a torus with an effective minor radius r 0, where  T =  B 0 r 0 2.

10. 10/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 ITG Instability in Helical Systems ITG mode is more unstable in the inward-shifted configuration while slower radial drift of helical-ripple-trapped particles. ITG mode is more unstable in the inward-shifted configuration while slower radial drift of helical-ripple-trapped particles. Standard Configuration Inward-Shifted Configuration  t =  h =0.1,   =-0.2  t,  3,10 =0  t =  h =0.1,  1,10 =-0.8  t,  3,10 =-0.2  t  =0.145v ti /L n Re(  ) Im(  )  =0.0883v ti /L n Re(  ) Im(  )

11. 11/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 ITG Turbulence Simulation in Helical Systems Standard Configuration Inward-Shifted Configuration

12. 12/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Transport Coefficient, Growth Rates, and Turbulent Spectrum Observed transport coefficients are comparable between the two cases, while ~60% differences in their linear growth rates and their different saturation levels in the turbulence energy. Observed transport coefficients are comparable between the two cases, while ~60% differences in their linear growth rates and their different saturation levels in the turbulence energy.  i ~2.5  2 v t /L n  i ~2.9  2 v t /L n Growth Rates Turbulence Transport

13. 13/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Zonal Flows in Helical ITG Turbulence The stronger zonal flows generated in the inward-shifted model configuration regulate the turbulent transport with  i comparable to the standard model case. The stronger zonal flows generated in the inward-shifted model configuration regulate the turbulent transport with  i comparable to the standard model case. Amplitudes of Zonal Flows

14. 14/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Zonal Flow Spectra in Tokamak and Helical Systems The zonal flow spectra in helical systems have relatively smaller amplitude on the low-k r side than that in tokamaks, as expected from the k r -dependence of the zonal flow response kernel,  >. The zonal flow spectra in helical systems have relatively smaller amplitude on the low-k r side than that in tokamaks, as expected from the k r -dependence of the zonal flow response kernel,  >. | Tokamak | Helical

15. 15/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Summary The response kernel of zonal flows in helical systems is analytically derived from the gyrokinetic theory by taking account of helical geometry and FOW effects. The response kernel of zonal flows in helical systems is analytically derived from the gyrokinetic theory by taking account of helical geometry and FOW effects. The GKV simulations on the ITG turbulence in helical systems show stronger instability in the inward- shifted configuration. Because of the larger zonal flows, however, the resultant transport is found in comparable magnitude to the standard configuration. The GKV simulations on the ITG turbulence in helical systems show stronger instability in the inward- shifted configuration. Because of the larger zonal flows, however, the resultant transport is found in comparable magnitude to the standard configuration. The zonal flows with longer radial wavelengths are observed with relatively smaller amplitudes than those in tokamaks as expected from the analytical theory on the zonal flow response. The zonal flows with longer radial wavelengths are observed with relatively smaller amplitudes than those in tokamaks as expected from the analytical theory on the zonal flow response.

16. 16/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Future Directions The present theoretical and numerical studies confirm that the stronger zonal flows in the inward-shifted configuration regulate the ITG turbulent transport. The present theoretical and numerical studies confirm that the stronger zonal flows in the inward-shifted configuration regulate the ITG turbulent transport. The obtained result encourages us to intensively promote the gyrokinetic simulation activities. The obtained result encourages us to intensively promote the gyrokinetic simulation activities. In order to investigate the turbulent transport physics further, the GKV simulations will be extended, step by step, so as to include the detailed equilibrium parameters, global profiles (  * -shear, variation of rotational transform etc.) as well as multi-physics and multi-scale effects. In order to investigate the turbulent transport physics further, the GKV simulations will be extended, step by step, so as to include the detailed equilibrium parameters, global profiles (  * -shear, variation of rotational transform etc.) as well as multi-physics and multi-scale effects.

17. 17/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Acknowledgments Dr. S. Ferrando i Margalet (NIFS) Dr. S. Ferrando i Margalet (NIFS) Dr. O. Yamagishi, Dr. S.Satake, Dr. M. Yokoyama, Prof. N. Nakajima and Prof. H. Yamada, and all colleagues in NIFS. Dr. O. Yamagishi, Dr. S.Satake, Dr. M. Yokoyama, Prof. N. Nakajima and Prof. H. Yamada, and all colleagues in NIFS. Prof. W. Horton (IFS@U. Texas) Prof. W. Horton (IFS@U. Texas) Prof. T. Sato (ESC@JAMSTEC) Prof. T. Sato (ESC@JAMSTEC) Gyrokinetic-Vlasov simulations are carried out by utilizing the Earth Simulator under the support by JAMSTEC and by using the Plasma Simulator at NIFS. Gyrokinetic-Vlasov simulations are carried out by utilizing the Earth Simulator under the support by JAMSTEC and by using the Plasma Simulator at NIFS. Plasma Simulator

18. 18/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Tokamak B = B 0 (1   t cos  ) Helical Systems B = B 0 [ 1   t cos   h cos (L  M  ) ] B passing trapped Classification of Particle Orbits

19. 19/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 GAM dispersion relation  =  G + i  approximate solution = 0 2 GAM response function GAM Freq. & Damping Rate (Sugama & Watanabe, 2005, 2006) Effects of FOW & Helical Ripples

20. 20/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 Simulation Model Toroidal Flux Tube Model Toroidal Flux Tube Model GKV code GKV code Directly solving GK eq. in 5-D phase space Directly solving GK eq. in 5-D phase space Zonal flow and GAM in tokamak and helical systems Zonal flow and GAM in tokamak and helical systems Entropy balance in GK turbulent transport Entropy balance in GK turbulent transport AB CDFE 0 22   B z yy

21. 21/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 GKV Turbulence Simulation on Earth Simulator Large-scale and high- speed computation Large-scale and high- speed computation 192 nodes (1536PEs) 192 nodes (1536PEs) Memory ~ 2.6TBytes Memory ~ 2.6TBytes Speed ~ 4.8-5.0TFlops Speed ~ 4.8-5.0TFlops Highly optimized code for Earth Simulator Highly optimized code for Earth Simulator 3D domain decomposition 3D domain decomposition Hybrid parallelization … Hybrid parallelization …

22. 22/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 ITG Turbulence Simulation in Helical Systems (1) Color contour of potential perturbations plotted on a flux surface and an elliptic poloidal cross-section. Model Parameters for the LHD Standard Configuration: r 0 /R 0 = 0.1, q 0 = 1.5, s = -1, R 0 /L n = 3.333,  i = 4,  e = 1, L n /v t =0.002,  t =  h =0.1,  L-1 =-0.2  t,  L+1 =0

23. 23/17 21st IAEA Fusion Energy Conf. Chengdu, China, Oct.16-21, 2006 ITG Turbulence Simulation in Helical Systems (2) Color contour of potential perturbations plotted on a flux surface and an elliptic poloidal cross-section. Model Parameters for the LHD inward-shifted Configuration: r 0 /R 0 = 0.1, q 0 = 1.5, s = -1, R 0 /L n = 3.333,  i = 4,  e = 1, L n /v t =0.002,  t =  h =0.1,  L-1 =-0.8  t,  L+1 =-0.2  t