[en] Linearized operators describing inter-species and like-species collisions have been discretized and implemented in the gyrokinetic Particle-In-Cell (PIC) code ORB5 [S. Jolliet, Comp. Phys. Comm. 177, 409 (2007)] based on the delta-f approach. Simulation results for neoclassical transport are compared with both analytical predictions as well as results from other codes. This new version of ORB5 including collisional dynamics thus makes it possible to carry out simulations of microturbulence starting from a global neoclassical equilibrium including self-consistent electric fields. First results of ITG microturbulence simulations carried out in this way are presented. The issue of numerical noise, inherent to the PIC approach and further accentuated by the implementation of collisions in the delta-f scheme, is addressed. It is shown how a noise reduction scheme based on a coarse graining procedure [Y. Chen and S. E. Parker, Phys. Plasmas 14, 082301 (2007)] ensures the physical relevance of such simulations. Furthermore, a novel delta-f algorithm is presented, which switches between a canonical and a local Maxwellian background for carrying out the collisonless and collisional dynamics respectively, and its advantages are discussed.

[en] Collisionless delta-f gyrokinetic particle-in-cell simulations suffer from the entropy paradox, in which the entropy grows linearly in time while low-order moments are saturated. As a consequence, these simulations do not reach a steady state and are unsuited to make quantitative predictions. A solution to this issue is the introduction of artificial dissipation. The notion of steady state in gyrokinetic simulations is studied by deriving an evolution equation for the fluctuation entropy and applying it to the global collisionless particle-in-cell code ORB5 [S. Jolliet et al., Comput. Phys. Commun. 177, 409 (2007)]. It is shown that a recently implemented noise-control algorithm [B. F. McMillan et al., Phys. Plasmas 15, 052308 (2008)] based on a W-stat provides the necessary dissipation to reach a steady state. The two interesting situations of decaying and driven turbulence are considered. In addition, it is shown that a separate heating algorithm, not based on a W-stat, does not lead to a statistical steady state.

[en] We derive explicit local transport relations for the global gyrokinetic formalism at arbitrary wavelength. This is an extension of the analysis in Scott et. al. 2010 where this was examined in the long-wavelength limit. Deriving a local expression for the fluxes requires that the gyroaveraging operator is symmetric, so that if point B is on the gyroring around A, point A is on the gyroring around B, for the same value of the magnetic moment. An algorithm for constructing a symmetric gyroring in a global code is described. Finally, using a simple 2D gyrokinetic code, we demonstrate the application of the momentum transport relation in a model problem with the full gyroaveraging operator without any long-wavelength approximation.

[en] The implementation of linearized operators describing inter- and like-species collisions in the global gyrokinetic particle-in-cell code ORB5[S. Jolliet, Comput. Phys. Commun. 177, 409 (2007)] is presented. A neoclassical axisymmetric equilibrium with self-consistent electric field can be obtained with no assumption made on the radial width of the particle trajectories. The formulation thus makes it possible to study collisional transport in regions where the neoclassical approximation breaks down such as near the magnetic axis. The numerical model is validated against both analytical results as well as other simulation codes. The effects of the poloidally asymmetric Fourier modes of the electric field are discussed, and the contribution of collisional kinetic electrons is studied. In view of subsequent gyrokinetic simulations of turbulence started from a neoclassical equilibrium, the problem of numerical noise inherent to the particle-in-cell approach is addressed. A novel algorithm for collisional gyrokinetic simulation switching between a local and a canonical Maxwellian background for, respectively, carrying out the collisional and collisionless dynamics is proposed, and its beneficial effects together with a coarse graining procedure [Y. Chen and S. E. Parker, Phys. Plasmas 14, 082301 (2007)] on noise and weight spreading reduction are discussed.

[en] Shear flows have a profound influence on turbulence-driven transport in tokamaks. The introduction of arbitrary initial flow profiles into the code ORB5 [Jolliet et al., Comput. Phys. Commun. 177, 409 (2007)] allows the convenient study of how flows on all length scales both influence transport levels and self-consistently evolve. A formulation is presented which preserves the canonical structure of the background particle distribution when either toroidal or poloidal flows are introduced. Turbulence suppression is possible above a certain shearing rate magnitude for homogeneous shear flows, and little evolution of the shearing rate is seen. However, when a flow with a zone boundary, where the shearing rate reverses at mid-radius, is introduced, the shear flow evolves substantially during the simulation. E x B shear flows with a zone boundary of a positive sign decay to a saturation amplitude, consistent with the well known saturation of turbulently generated zonal flows. Unlike the E x B flow, the parallel flows relax diffusively.

[en] Global gyrokinetic simulations of ion temperature gradient (ITG) driven turbulence in an ideal MHD ITER equilibrium plasma are performed with the ORB5 code. The noise control and field-aligned Fourier filtering procedures implemented in ORB5 are essential in obtaining numerically healthy results with a reasonable amount of computational effort: typical simulations require 10⁹ grid points, 10⁹ particles and, despite a particle per cell ratio of unity, achieve a signal to noise ratio larger than 50. As compared with a circular concentric configuration with otherwise similar parameters (same ρ_* = 1/720), the effective heat diffusivity is considerably reduced for the ITER MHD equilibrium. A self-organized radial structure appears, with long-lived zonal flows (ZF), modulating turbulence heat transport and resulting in a corrugated temperature gradient profile. The ratio of long-lived ZF to the fluctuating ZF is markedly higher for the ITER MHD equilibrium as compared with circular configurations, thereby producing a more effective ITG turbulence suppression, in spite of a higher linear growth rate. As a result, the nonlinear critical temperature gradient, R/L_Tcrit,NL, is about twice the linear critical temperature gradient, R/L_Tcrit,lin. Moreover, the heat transport stiffness above the nonlinear threshold is considerably reduced as compared with circular cases. Plasma elongation is probably one of the essential causes of this behaviour: indeed, undamped ZF residual levels and geodesic acoustic mode damping are both increasing with elongation. Other possible causes of the difference, such as magnetic shear profile effects, are also investigated. (paper)

[en] Global gyrokinetic simulations of electrostatic temperature-gradient-driven trapped-electron-mode (TEM) turbulence using the δf particle-in-cell code ORB5 are presented. The electron response is either fully kinetic or hybrid, i.e. considering kinetic trapped and adiabatic passing electrons. A linear benchmark in the TEM regime against the Eulerian-based code GENE is presented. Two different methods for controlling the numerical noise, based, respectively, on a Krook operator and a so-called coarse-graining approach, are discussed and successfully compared. Both linear and non-linear studies are carried out for addressing the issue of finite-ρ^*-effects and finite electron collisionality on TEM turbulence. Electron collisions are found to damp TEMs through the detrapping process, while finite-ρ^*-effects turn out to be important in the non-linear regime but very small in the linear regime. Finally, the effects of zonal flows on TEM turbulence are briefly considered as well and shown to be unimportant in the temperature-gradient-driven TEM regime. Consistently, basically no difference is found between linear and non-linear critical electron temperature gradients in the TEM regime. (paper)

[en] In this paper, we present global nonlinear gyrokinetic simulations including finite β_e effects and collisions in tokamak geometry. Global electromagnetic simulations using conventional δf particle in cell methods are very demanding, with respect to numerical resources, in order to correctly describe the evolution of the non-adiabatic part of the electron distribution function. This difficulty has been overcome using an appropriate adjustable control variate method in the conventional δf scheme. Linearized inter-species and like-species collision operators have also been introduced in the model. The inclusion of the collisional dynamics makes it possible to carry out simulations of microturbulence starting from a global neoclassical equilibrium and to study the effect of collisions on the transport induced by electrostatic microinstabilities.

[en] In this paper, the influence of the parallel nonlinearity on zonal flows and heat transport in global particle-in-cell ion-temperature-gradient simulations is studied. Although this term is in theory orders of magnitude smaller than the others, several authors [L. Villard, P. Angelino, A. Bottino et al., Plasma Phys. Contr. Fusion 46, B51 (2004); L. Villard, S. J. Allfrey, A. Bottino et al., Nucl. Fusion 44, 172 (2004); J. C. Kniep, J. N. G. Leboeuf, and V. C. Decyck, Comput. Phys. Commun. 164, 98 (2004); J. Candy, R. E. Waltz, S. E. Parker et al., Phys. Plasmas 13, 074501 (2006)] found different results on its role. The study is performed using the global gyrokinetic particle-in-cell codes TORB (theta-pinch) [R. Hatzky, T. M. Tran, A. Koenies et al., Phys. Plasmas 9, 898 (2002)] and ORB5 (tokamak geometry) [S. Jolliet, A. Bottino, P. Angelino et al., Comput. Phys. Commun. 177, 409 (2007)]. In particular, it is demonstrated that the parallel nonlinearity, while important for energy conservation, affects the zonal electric field only if the simulation is noise dominated. When a proper convergence is reached, the influence of parallel nonlinearity on the zonal electric field, if any, is shown to be small for both the cases of decaying and driven turbulence.

[en] In this work, a Fourier solver [B.F. McMillan, S. Jolliet, A. Bottino, P. Angelino, T.M. Tran, L. Villard, Comp. Phys. Commun. 181 (2010) 715] is implemented in the global Eulerian gyrokinetic code GT5D [Y. Idomura, H. Urano, N. Aiba, S. Tokuda, Nucl. Fusion 49 (2009) 065029] and in the global Particle-In-Cell code ORB5 [S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T.M. Tran, B.F. McMillan, O. Sauter, K. Appert, Y. Idomura, L. Villard, Comp. Phys. Commun. 177 (2007) 409] in order to reduce the memory of the matrix associated with the field equation. This scheme is verified with linear and nonlinear simulations of turbulence. It is demonstrated that the straight-field-line angle is the coordinate that optimizes the Fourier solver, that both linear and nonlinear turbulent states are unaffected by the parallel filtering, and that the k_∥ spectrum is independent of plasma size at fixed normalized poloidal wave number.