Turbulence simulations

Not only do collisions and the inhomogeneity of the magnetic field produce transport, but also turbulent fluctuations in the plasma do. The density and temperature gradients constitute free energy sources for the development of the so-called drift-wave turbulence [Horton-99], that can manifest due to the excitation of different types of instabilities: Ion temperature gradient (ITG) turbulence is considered to be responsible of most of the turbulent transport in the core of tokamaks and stellarators. The trapped electrons can generate the so-called trapped electron mode (TEM) instability even with flat temperature profiles, when a density gradient is present. In addition, an electron-scale microinstability, electron temperature gradient (ETG) mode, can also appear in presence of large electron temperature gradient [Horton-99].

In tokamaks, the ITG and TEM instabilities have already been extensively studied from the theoretical and numerical points of view and many comparisons with experiments have already been carried out. There are fewer simulations of ETG modes because of the huge computing resources required. In stellarators these instabilities have been less studied but several differences betwen stellarators and tokamaks have already emerged in this respect [Xanthopoulos-14, Helander-12]. In the case of ITG-driven turbulence, the unstable regions are more spatially localized as a consequence of the three-dimensional equilibrium magnetic field, while the level of instability appears to be similar to that found in tokamaks [Plunk-14]. For TEM, a similar localization can occur and some results indicate that the quasi-isodynamic configuration of W7-X could be almost immune to TEM modes, due to the non-correlated localization of trapped particle regions and bad curvature regions [Proll-12].

In the last years, we have carried out a characterization of electrostatic instabilities in the stellarator TJ-II by means of gyrokinetic (GK) simulations [Sánchez14]. Signatures of ITG, TEM and ETG have been found in different TJ-II plasmas. All the instabilities are spatially localized, the local magnetic shear and magnetic field line curvature being key magnitudes controlling the instability.

Zonal flows [Diamond-05] (that is, electrostatic potential perturbations that are constant on flux surfaces) are thought to play an important role in the self-regulation of turbulence in stellarators (as in tokamaks). Linear calculations have shown that zonal flows exhibit characteristic properties in stellarators that are different from those found in tokamaks [Sugama-05, Sugama-06, Mishchenko-08, Helander-11, Monreal-15]. We have studied the linear relaxation of zonal flows in different stellarator configurations by means of GK simulations and semi-analytical calculations [Sánchez-12, Sánchez-15, Monreal-15]. There are indications that the linear properties of zonal flow relaxation (residual zonal flow level and oscillations) can have an effect on the regulation of turbulent transport [Sugama-09, Nunami-12, Nunami-13, Xanthopoulos-11].

Equilibrium radial electric fields can also influence are turbulence regulation by modifying the linear instability and also the zonal flow linear damping. Due to the three- dimensionality of the stellarator configurations, the effect of a radial electric field on the linear instability, the zonal flows [Mishchenko-12] and the non-linear saturation level is more subtle than in tokamaks and its elucidation requires a detailed study. Different oscillations have been characterized in simulations of the linear relaxation of zonal flows in TJ-II [Sánchez-12]. Their dependence on the background radial electric field has been analyzed [Velasco-13]. Finally, some recent pellet injection experiments might provide empirical evidence for the existence of low-frequency oscillations in the relaxation of zonal flows.

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