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Nonlinear state of turbulence and zonal flow
K. Itoh, K. Hallatschek1), S. Toda, H. Sanuki and S.-I. Itoh2)
National Institute for Fusion Science, Toki 509-5292, Japan
It is well known that the radial electric field structure plays a crucial role in the establishment of the improved confinement [1]. There arises a question whether the turbulence-driven microscopic radial electric field turns to be a global radial electric field. A large-scale kink-soliton solution has been discussed in [2]. However, evolution to stable state has not been studied yet. We here study the nonlinear state of turbulence and zonal flow. Equation of the zonal flow velocity has been derived by calculating the change of the drift-wave action. The second order term with respect to the zonal flow velocity has been discussed in [2]. Calculating up to the third order response and keeping the damping of the secondary flow [3], the nonlinear equation for the zonal flow is obtained [4]. In this derivation, the cut-off wave number of zonal flow instability is taken into account by noting the dispersion of the waves [5]. From this equation, we have the stable and stationary solution for the fluctuations and zonal flow. Characteristic saturation velocity and radial scale length are obtained. This result shows an important role of the secondary toroidal flow. Comparison with direct nonlinear simulation is discussed. Influence of collisional damping is also studied.
References
[1] K. Itoh, S.-I. Itoh, A. Fukuyama: Transport and Structure Formation in Plasmas (IOP, England, 1999)
[2] A. I. Smolyakov, et al.: Phys. Rev. Lett. 84 (2000) 491 [3] K. Hallatschek: invited paper at APS Conference (2002); also submitted to Phys. Rev. Lett. [4] K. Itoh, et al.: submitted to Phys. Rev. Lett. [5] Liu Chen et al.: Phys. Plasmas 8 (2000) 3129
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