1. Intermittent Oscillations Generated by ITG-driven Turbulence US-Japan JIFT Workshop December 15 th -17 th, 2003 Kyoto University Kazuo Takeda, Sadruddin Benkadda*, Satoshi Hamaguchi and Masahiro Wakatani Graduate School of Energy Science, Kyoto University *CNRS URA 773, Universit é de Provence, France

2. Table of Contents 1.Motivation Background and earlier studies 2.Model equations Extension of the earlier model 3.Highlight data Our low degree-of-freedom model 4.Summary

3. Motivation 1.Background i. Ion-temperature-gradient (ITG)-driven turbulence and self- generated sheared plasma flows ii. Low degree-of-freedom models are useful to understand nonlinear physics. 2.Earlier studies i. 11 ODE model given by Hu and Horton G. Hu and W. Horton, phys. Plasmas 4, 3262 (1997) 3.Extension i. Increasing the degree of freedom from 11 to 18 in order to include 3 rd harmonics

4. 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

5. Normalization The standard drift wave units used for normalization. And are assumed for numerical calculation.

6. Linear Stability 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.

7. Linear Growth Rate

8. Low-degree-of-freedom Model If are neglected, these agree with the 11 ODE model. G. Hu and W. Horton, Phys. Plasmas 4, 3262 (1997) Low-order modes to describe nonlinear behaviour of a toroidal ITG mode in a simple slab model are as follows,

9. Numerical Procedures For studying the anomalous transport induced by the nonlinear ITG mode,Nusselt number have been calculated. The vorticity equation contains the Reynolds stress which generates sheared flows.

10. Bifurcation Process I-I Time evolution of (a) kinetic energy K and (b) Nusselt number N u in the case of K i =0.3, where The system converges to a steady state.

11. Bifurcation Process I-II Phase space ‘K 0 (m=0 mode)-N u ’ and power spectrum of K 1 (m=1 mode) for K i =0.3, where

12. Bifurcation Process II-I Time evolution of (a) kinetic energy K and (b) Nusselt number N u in the case of K i =0.4, where The system converges to a periodic oscillation.

13. Bifurcation Process II-II Phase space ‘K 0 (m=0 mode)-N u ’ and power spectrum of K 1 (m=1 mode) for K i =0.4, where

14. Bifurcation Process III-I Time evolution of (a) kinetic energy K and (b) Nusselt number N u in the case of K i =0.5, where More modes are excited.

15. Bifurcation Process III-II Phase space ‘K 0 (m=0 mode)-N u ’ and power spectrum of K 1 (m=1 mode) for K i =0.5, where

16. Bifurcation Process IV-I Time evolution of (a) kinetic energy K and (b) Nusselt number N u in the case of K i =0.6, where Chaotic oscillations appear.

17. Bifurcation Process IV-II Phase space ‘K 0 (m=0 mode)-N u ’ and power spectrum of K 1 (m=1 mode) for K i =0.6, where

18. Intermittent Behaviour I Time evolution of kinetic energy K for K i =4, where Intermittent bursts (so called avalanche) are observed.

19. Intermittent Behaviour II Time evolution of (a) Nusselt number N u and (b) Reynolds stress S R in the case of K i =4. N u and S R burst at the time when the ITG mode grows rapidly.

20. Intermittent Behaviour III Real space contours of at (a) bursting, (b) laminar and (c) tilting phase for K i =4, where

21. Intermittent Behaviour IV This intermittency is caused by the competition of the following 3 factors; 1. Generation of sheared flows due to nonlinear coupling between higher harmonics, and suppression of the ITG turbulence by the sheared flows. 2. Gradual reduction of sheared flows due to viscosity. 3. Rapid re-growth of the ITG modes due to the reduction of the stabilizing effect by the sheared flows.

22. Scaling Law The scaling law log Nu ∝ 3 log Ki is obtained. It is suggested that the improved confinement is related to the intermittency of the system.

23. Summary 1.Sheared flows are generated by the nonlinear mode coupling, and a bifurcation corresponding to an L-H transition has been obtained. 2.In the strongly turbulent regime, an intermittent behaviour appears. This intermittency is caused by the competition of the 3 factors. 3.A scaling law, log Nu ∝ 3 log Ki, has been obtained between Nu and Ki. Improved confinements are related to the intermittency of the system. 4.Essential nonlinear behaviour of the system can be at least qualitatively accounted for by nonlinear interaction of several low order harmonics. New physical phenomena can be obtained by using 18 ODE model.

24. Intermittent Behaviour V Phase space ‘K 0 -N u ’ at (a) bursting, (b) laminar and (c) tilting phase in the case of K i =4.

25. 18 ODEs I

26. 18 ODEs II where