1. Effects of Flow on Radial Electric Fields Shaojie Wang Department of Physics, Fudan University Institute of Plasma Physics, Chinese Academy of Sciences

2. Outline Introduction Basic equations Zonal flows in rotating systems Summary

3. 1. Introduction The dynamics of zonal flows (ZFs) is very important in tokamak fusion plasma physics researches, because the flow shear can suppress the drift-type turbulence that degrades the confinement performance. ZFs are electrostatic perturbations with the spatial structure of toroidal symmetry and poloidal symmetry. Two branches of ZFs: the low-frequency branch (ω~0), and the high-frequency branch (ω~c_s/R). that is also known as the Geodesic Acoustic Mode (GAM).

4. In a non-rotating system, ZFs are linearly stable and the GAMs are standing waves. There exists an Equilibrium Toroidal Rotation Flow in a tokamak plasma. Clearly, it is of great interest to investigate the effects of ETRF on ZFs and GAMs.

5. 2. Basic Equations governing equations

6. The equilibrium solution with an ETRF

7. Two components of the momentum equation

8. Linearized Equations

9. 3. Zonal Flows in Rotating Systems Perturbation form Large-aspect-ratio tokamak Ordering ansatz

10. Eigenmode Equation

12. Dispersion relation

13. Solution to the dispersion relation

15. When the speed of ETRF approaches the sound speed, the GAM frequency is significantly reduced and becomes sensitive to the safety factor. The low-frequency branch of zonal flows is linearly unstable in a rotating system, while it is linearly stable in a non-rotating system. When the speed of ETRF approaches the sound speed, the linear growth-rate of the ZFs in a rotating system can exceed the SW frequency, which is comparable to the collisional damping rate of ZFs.

16. Eigenfunction

17. SWs, GAMs and ZFs are poloidal standing waves in a non-rotating system. In a rotating system SWs and GAMs can propagate in the poloidal direction.

18. 4. Summary The low-frequency branch of zonal flows, which is linearly stable in the non-rotating system [2], becomes linearly unstable in a rotating system due to the centrifugal force and the induced poloidal asymmetry of the equilibrium plasma pressure distribution. This new result may be applied to analyze the physics of transport barrier control by tangential neutral beam injection.

19. GAM frequency in a rotating system is lower than in a non-rotating system; in the regime of sonic toroidal rotation, the GAM frequency is significantly reduced and becomes sensitive to the safety factor. This new result may be applied to resolve the GAM frequency scaling issue raised by recent experimental observations.

20. SWs and GAMs, which are poloidal standing waves in a non-rotating system, can propagate in the poloidal direction in a rotating system.