A systematic, constructive and self-consistent procedure to quantify nonlocal, nondiffusive action at a distance in plasma turbulence is exposed and applied to turbulent heat fluxes computed from the state-of-the-art full- f, flux-driven gyrokinetic GYSELA and XGC1 codes. A striking commonality is found: heat transport below a dynamically selected mesoscale has the structure of a Lévy distribution, is strongly nonlocal, nondiffusive, scale-free, and avalanche mediated; at larger scales, we report the observation of a self-organized flow structure which we call the " E × B staircase" after its planetary analog.

Turbulence in hot magnetized plasmas is shown to generate permeable localized transport barriers that globally organize into the so-called "ExB staircase" [G. Dif-Pradalier et al., Phys. Rev. E, 82, 025401(R) (2010)]. Its domain of existence and dependence with key plasma parameters is discussed theoretically. Based on these predictions, staircases are observed experimentally in the Tore Supra tokamak by means of high-resolution fast-sweeping X-mode reflectometry. This observation strongly emphasizes the critical role of mesoscale self-organization in plasma turbulence and may have far-reaching consequences for turbulent transport models and their validation. A puzzling result in recent years in plasma turbulence has arguably been the discovery of the quasiregular pattern of E × B flows and interacting avalanches that we have come to call the "E × B staircase," or the "plasma staircase" in short [1]. This structure may be defined as a spontaneously formed, self-organizing pattern of quasiregular, long-lived, localized shear flow and stress layers coinciding with similarly long-lived pressure corrugations and interspersed between regions of turbulent avalanching. The plasma staircase exemplifies how a systematic organization of turbulent fluctuations may lead to the onset of strongly correlated flows on magnetic flux surfaces.Flow patterning is a prominent topic in many fluidrelated systems and hot magnetized plasmas are no exception to that. In fact the "staircase" name is borrowed from the vast literature in planetary flows motivated by the desire to explain the banded structure of observed atmospheres in our Solar System-including Earth [2] or Jupiter [3]-and of terrestrial oceans [4]. Just as in the geophysical or astrophysical systems where the planetary staircase strongly influences the general circulation, the plasma staircase plays an important role in organizing the heat transport [1]: avalanches and the staircase interplay, statistically interrupting at mesoscales the long-range radial avalanching that could otherwise expand over the whole system. The nonlocal heat transport thus remains contained at the mesoscale staircase step spacing, resulting in a beneficial scaling of confinement with machine size. This flow patterning is primarily a spontaneous mean zonal shear patterning. "Zonal" denotes the axisymmetric n ¼ m ¼ 0 component of the E × B flows [5], n and m respectively being the toroidal and poloidal mode numbers while "mean" refers to the ensemble-averaged part of the zonal flows. Remarkably, the plasma spontaneously generates robust shear patterns that endure despite the strong background turbulence and retain their coherence over long (several milliseconds) to very long (hundreds of milliseconds) periods of time. The results presented throughout this Letter are based on state-of-the-art flux-driven gyrokinetic [6] computations using the GYSELA code [7] with realistic tokamak plasma parameters. Systematic features of the plasma staircase can be inferred from extensive computational scans, see ...

An overview of the physics of intrinsic torque is presented, with special emphasis on the phenomenology of intrinsic toroidal rotation in tokamaks, its theoretical understanding, and the variety of momentum transport bifurcation dynamics. Ohmic reversals and ECH-driven counter torque are discussed in some detail. Symmetry breaking by LSN vs. USN asymmetry is related to the origin of intrinsic torque at the separatrix.

This paper addresses non-linear gyrokinetic simulations of ion temperature gradient (ITG) turbulence in tokamak plasmas. The electrostatic Gysela code is one of the few international 5D gyrokinetic codes able to perform global, full-f and flux-driven simulations. Its has also the numerical originality of being based on a semi-Lagrangian (SL) method. This reference paper for the Gysela code presents a complete description of its multi-ion species version including: (i) numerical scheme, (ii) high level of parallelism up to 500k cores and (iii) conservation law properties.

The impact on turbulent transport of geodesic acoustic modes excited by energetic particles is evidenced for the first time in flux-driven 5D gyrokinetic simulations using the Gysela code. Energetic geodesic acoustic modes (EGAMs) are excited in a regime with a transport barrier in the outer radial region. The interaction between EGAMs and turbulence is such that turbulent transport can be enhanced in the presence of EGAMs, with the subsequent destruction of the transport barrier. This scenario could be particularly critical in those plasmas, such as burning plasmas, exhibiting a rich population of suprathermal particles capable of exciting energetic modes.Understanding turbulent transport is crucial in numerous plasma physics frameworks, ranging from plasma laboratories such as nuclear fusion devices 1 to astrophysical systems such as the solar tachocline 2 or the atmospheres 3 . In this letter, we focus on the turbulent transport in toroidal nuclear fusion devices (tokamaks), where accurate predictions are essential on the route towards the steady-state production of energy. Together with turbulence, energetic particles (EPs) constitute a ubiquitous component of current and future tokamaks, due to both nuclear reactions and heating systems. EPs are characterized by energies larger than the thermal energy. Whereas the impact of turbulence on EP transport has been analyzed and found to be weak 4 , the effect of EPs on turbulence has not been much studied so far (see e.g. Ref. 5) and represents the aim of our study. This analysis is done via the excitation by EPs of a class of modes naturally existing in tokamaks: the geodesic acoustic modes (GAMs) 6 , which are the oscillatory component of large scale E × B zonal flows. The EP-driven GAMs are called EGAMs. These modes have been predicted theoretically 7,8 , observed experimentally 9,10 and reported very recently numerically in the absence of turbulence 11 in gyrokinetic simulations with the 5D Gysela code 12 . The motivation of the present work relies upon fluid simulations where the turbulence level was controlled by GAMs in the core/edge transitional regime 13 . In addition, experimental evidence of the role of GAMs in the edgeturbulence suppression has been reported for the first time during the analysis of the L-H transition in the AS-DEX Upgrade tokamak 14 . However, in the context of core-turbulence suppression, the role of GAMs is less evident for several reasons. First, these modes are Landau damped in the core plasma. Second, since they are nonlinearly generated by turbulence, their external control a) Electronic mail:david.zarzoso-fernandez@polytechnique.org; Current address: Max-Planck-Institut für Plasmaphysik, EU-RATOM Association, Boltzmannstr. 2, Garching D-85748, Germany has proven difficult. Last, their frequency ω GAM is close to the characteristic turbulence frequency ω turb , which means that the shearing rate provided by GAMs might be large compared to the autocorrelation time. In that respect, theoretical predictions of the shearing effect ...

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.

Conservation equations are derived for the gyrocenter toroidal momentum density and the polarization field. These equations are derived from the gyrokinetic model as it is implemented in simulation codes. In view of predicting the toroidal rotation in future fusion devices such as ITER, where external momentum input will be small, accurate simulations of momentum transport are crucial. The evolution equation for gyrocenter toroidal momentum density involves the divergence of the off-diagonal components of the Reynolds and generalized Maxwell stress, while the source term is the radial current of gyrocenters. The time evolution of the polarization field is the opposite of the gyrocenter current. Hence, an evolution equation for the total momentum density, i.e., the sum of gyrocenter and polarization field toroidal momentum density can be written. The force balance equation and the toroidal momentum conservation equations have been numerically tested with the gysela code. They are satisfied with a high level of accuracy.

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.