Masanori NUNAMI¹.²)  Tomo-Hiko WATANABE¹.²) and  Hideo SUGAMA¹.²) 1)National Institute for Fusion Science, Toki 509-5292, Japan ²)Graduate University for Advanced Studies, Toki 509-5292, Japan (Received 26 January 2013 / Accepted 6 February 2013) Relation among the ion temperature gradient (ITG) turbulence, zonal flows, and the transport in helical plasmas is investigated by nonlinear gyrokinetic simulations.  Local gyrokinetic simulations for helical field configurations are carried out employing various parameters such as the density and temperature gradients and local shears. From the simulation results, we construct a simple model function to represent ion heat diffusivity in terms of the turbulent fluctuations and zonal flow amplitude in helical plasmas. ✆ 2013 The Japan Society of Plasma Science and Nuclear Fusion Research Keywords:  ITG turbulence, gyrokinetic simulation, Large Helical Device DOI: 10.1585/pfr.8.1203019 Turbulent transport has been considered to be one of the most critical issues in the magnetically confined fu- sion plasma research.  For the study of plasma confine- ment properties, it is very important to evaluate the trans- port level associated with turbulence caused by micro- instabilities such as the ion temperature gradient (ITG) mode.  To date, a large number of gyrokinetic simulation studies have been made for clarifying the turbulent trans- port physics [1].  Owing to recent great progress in the development of computational resources, direct compar- isons of numerical simulation with experimental data are possible.  However, only a few validations of the gyroki- netic simulations in helical systems such as the Large He- lical Device (LHD) [2] have been performed, in contrast to the case for tokamaks in which studies against the experi- ments have been extensively promoted [3]. This is because gyrokinetic simulations for non-axisymmetric systems re- quire a higher spatial resolution than those for tokamaks. In addition to the physical clarification and validation, it is also necessary to construct a reduced transport model for integrated simulations [4].  Therefore, we should clar- ify the intermediate relation bridging the turbulence simu- lation and transport modeling. In our previous paper [5], nonlinear local flux-tube gyrokinetic Vlasov simulations of ITG turbulent transport were carried out by using the GKV-X code [6] for the high ion temperature (high-T i ) LHD plasma in shot number 88343 [7]. The simulation results reproduced the turbulent ion heat transport level and the wavenumber spectra of the density fluctuations observed in the experiment.  Figure 1 shows that the simulation results are in reasonable agree- ment with the experimental values of turbulent ion heat Fig. l Radial profiles of ion heat flux P i in the high-T i LHD plasma #88343 obtained from the experiment (dotted curve) and the simulations (red symbols).  Solid curve shows the turbulent contribution of the experimental re- sult obtained by subtracting neoclassical contribution from the observation. flux obtained by subtracting the neoclassical contribution from the observation.  The turbulent transport level is de- termined through the interactions of turbulent fluctuations and zonal flows [8]. In the present study, in order to quan- titatively clarify the effects of turbulence and zonal flows on the transport level in helical plasmas, we define two nonlinear quantities, the squared turbulent potential fluc- tuation T  ≡  (1/2) ∑ k , k , k , ≠ 0 (l Ø k , k , k , l ² ), and the squared am- plitude of zonal flow potential乙≡ (1/2) ∑ . 〈| Õ k. k.=0| 〉 Here φ is the electrostatic potential fluctuation normalized by T i ρ i / eR 0 with the ion thermal gyro radius ρ ii , and the major radius of the field R 0 .  The flux-surface average is denoted by 〈· · 〉, and ( k x , k y ) represent wavenumbers in radial and poloidal directions, respectively. Figure 2 shows the time evolutions of T , Z and the ion heat transport coefficient in the gyro-Bohm unit χ i /χ GB for two cases, where χ GB  = ρ π ν ti/R 0 and τ ii  =  √ t i/m i. © 2013 The Japan Society of Plasma One of the simulation results indicated by red curves in Fig. 2 corresponds to the point for ρ = 0.65 in Fig. 1 using the magnetic configuration obtained from the MHD equi- librium calculation for the high-T i LHD plasma #88343, in which the magnetic axis is at R 0 = 3.75 m.  Here, T j  =  2.16 keV and ρ j  =  1.8 × 10 − 3 m at ρ  =  0.65.  For the other results indicated by blue curves in Fig. 2, the tem- perature and density gradients are the same as in the case mentioned above, although we use the vacuum magnetic configuration without finite β effects such as the Shafranov shift and the magnetic axis is located at R 0 = 3.6m.  In the latter case, the radial drift velocities of helical ripple trapped particles decrease, consequently, the neoclassical transport is reduced [9] while the zonal flow response is enhanced [10].  The squared turbulent fluctuations T  for the two cases are comparable, which corresponds to the fact that the maximum linear growth rates of ITG modes are almost the same, Ymax = 0.246 v ii/R 0 for the configu- ration with R 0 = 3.75m and 0.252 v ii/R 0 for R 0 = 3.6m. On the other hand, the squared zonal flow amplitude Z for R 0  =  3.6 m is greater than that for R 0 = 3.75 m.  Con- sequently, the transport level is reduced by the enhanced zonal flow generation for R 0 = 3.6 m. To investigate the dependence of χ i on ( T , Z ), we have performed 17 nonlinear ITG turbulent transport sim- ulations making use of GKV-X for helical plasmas with changing values of various parameters such as the tem- perature gradient (7.5 < R 0 / L T i < 23.9), radial position (0.46 < ρ < 0.83), and safety factor (1.1 < q < 2.2). Then, it is found that the simulation data at a wide range of the parameters are well represented by the following the model function
where α  =  0.55, C 1  =  0.055, and C 2  =  0.063. In the definition of F(T,Z) given by Eq.(1), the numera- tor C 1 T α indicates the enhancement of χ i with increas- ing the turbulent fluctuations while Z 1 ⁄ 2 in the denomi- nator represents the regulation of χ i due to zonal flows as shown in Fig. 2.  Comparisons between χ i /χ ; B  obtained from the simulations and the functions are done. In Fig. 3, F ( T , Z ) agrees with the simulation results better than an- other function G(T ) defined without the zonal flow effects as G(T ) ≡ C o T δ with C 0 = 0.26 and δ = 0.69. The relation in Eq. (1) will contribute to constructing a reduced transport model for the ITG turbulence in helical plasmas that can be applied to an integrated transport code such as TASK3D [4]. Indeed, the squared turbulent fluctu- ation T  is strongly correlated with the linear growth rates of the ITG mode, and the squared zonal flow amplitude Z can be related to the linear zonal flow responses.  To con- struct the model, we should clarify the relations between the results of the nonlinear simulations and linear analyses, which will be reported elsewhere.  Because the number of samples used here may still not be sufficient to evaluate the validity of the formula in Eq. (1), we should perform more nonlinear simulations and investigate the dependencies on other parameters, such as collisionality and T i / T e which are not considered in this paper. Extension of the physical model to include trapped electrons and electromagnetic ef- fects should also be pursued in future study. This study was supported in part by the Japanese Min- istry of Education, Culture, Sports, Science and Technol- ogy (Grant Nos. 22760660, 21560861, and 24561030), in part by National Institute for Fusion Science (NIFS) Col- laborative Research Program (KNST039, KNXN229, and KNTT015), and by the use of the Helios system at the In- ternational Fusion Energy Research Center (project code: VLDGK and GTNAXIS). [1] X. Garbet et al. , Nucl. Fusion 50 , 043002 (2010). [2]  A. Komori et al. , Fusion Sci. Technol. 58 , 1 (2010). [3] T.L. Rhodes et al. , Nucl. Fusion 51 , 063022 (2011). [4] M. Yokoyama et al. , Plasma Fusion Res. 7, 2403011 (2012). (2005).
[9] S. Murakami et al. , Nucl. Fusion 42, L19 (2002). [10]  T.-H. Watanabe et al. , Phys. Rev. Lett. 100 , 195002 (2008).