A global nonlinear simulation code for the time evolution of ion-temperature-gradient-driven modes in θ-pinch geometry as a first approximation to the stellarator Wendelstein 7-X (W7-X) [Grieger et al., Proceedings of the 13th International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Washington, DC, 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 3, p. 525] has been developed. A δf particle-in-cell (PIC) method is used to solve the coupled system of gyrokinetic equations for the ions, in the electrostatic approximation, and the quasineutrality equation, assuming adiabatically responding electrons. The focus has been on adherence to conservation laws, i.e., particle number and energy conservation. Besides other improvements it has been shown that a well-chosen initial distribution of the markers in reduced phase space makes optimal use of the δf PIC method to reduce the statistical noise for a given number of markers. In a model including all (1351) physically relevant modes, it has been possible to achieve energy conservation beyond the saturation of the instability.

The turbulent transport governed by the toroidal ion temperature gradient driven instability is analysed with the full-f global gyrokinetic code GYSELA (Grandgirard et al 2007 Plasma Phys. Control. Fusion 49 B173) when the system is driven by a prescribed heat source. Weak, yet finite, collisionality governs a neoclassical ion heat flux that can compete with the turbulent driven transport. In turn, the ratio of turbulent to neoclassical transport increases with the source magnitude, resulting in the degradation of confinement with additional power. The turbulent flux exhibits avalanche-like events, characterized by intermittent outbursts which propagate ballistically roughly at the diamagnetic velocity. Locally, the temperature gradient can drop well below the linear stability threshold. Large outbursts are found to correlate with streamer-like structures of the convection cells albeit their Fourier spectrum departs significantly from that of the most unstable linear modes. Last, the poloidal rotation of turbulent eddies is essentially governed by the radial electric field at moderate density gradient.

Flux-driven global gyrokinetic codes are now mature enough to make predictions in terms of turbulence and transport in tokamak plasmas. Some of the recent breakthroughs of three such codes, namely GYSELA, ORB5 and XGC1, are reported and compared wherever appropriate. In all three codes, turbulent transport appears to be mediated by avalanche-like events, for a broad range of ρ * = ρ i /a values, ratio of the gyro-radius over the minor radius. Still, the radial correlation length scales with ρ i , leading to the gyroBohm scaling of the effective transport coefficient below ρ * ≈ 1/300. The possible explanation could be due to the fact that avalanches remain meso-scale due to the interaction with zonal flows, whose characteristic radial wave-length appears to be almost independent of the system size. As a result of the radial corrugation of the turbulence driven zonal and mean flows, the shear of the radial electric field can be significantly underestimated if poloidal rotation is assumed to be governed by the neoclassical theory, , especially at low collisionality. Indeed, the turbulence contribution to the poloidal rotation increases when collisionality decreases. Finally, the numerical verification of toroidal momentum balance shows that both neoclassical and turbulent contributions to the Reynolds' stress tensor play the dominant role. The phase space analysis further reveals that barely passing supra-thermal particles mostly contribute to the toroidal flow generation, consistently with quasi-linear predictions.