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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.

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 electron cyclotron heating-driven counter torque are discussed in some detail. Symmetry breaking by lower single null versus upper single null asymmetry is related to the origin of intrinsic torque at the separatrix. © 2013 IAEA, Vienna.

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* = ρi/a values, ratio of the gyro-radius over the minor radius. Still, the radial correlation length scales with ρi, leading to the gyro-Bohm 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 wavelength 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.

Treatment of binary Coulomb collisions when the full gyrokinetic distribution function is evolved is discussed here. A spectrum of different collision operators is presented, differing through both the physics that can be addressed and the numerics they are based on. Eulerian-like (semi-Lagrangian) and particle in cell (PIC) (Monte-Carlo) schemes are successfully cross-compared, and a detailed confrontation to neoclassical theory is shown.

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.

This paper analyzes the properties of a two-field critical gradient model that couples a heat equation to an evolution equation for the turbulence intensity. It is shown that the dynamics of a perturbation is ballistic or diffusive depending on the shape of the pulse and also on the distance of the temperature gradient to the instability threshold. This dual character appears in the linear response of this model for a wave packet. It is recovered when investigating the nonlinear solutions of this system. Both self-similar diffusive fronts and ballistic fronts are shown to exist. When the propagation is ballistic, it is found that the front velocity is the geometric mean between the turbulent diffusion coefficient and a microinstability growth rate. © 2007 American Institute of Physics.

This study is motivated by experiments on Tore Supra and WEST tokamaks where a deepening of the E × B velocity—governed by the radial electric field Er —near the edge is observed when the safety factor decreases in L-mode plasmas. Flux-driven global simulations of ion temperature gradient turbulence recover qualitatively the trend observed in the experiments, i.e. the E × B velocity increases when decreasing the safety factor. From these simulations, multiple clues point out the role of turbulence in the establishment of the radial electric field even though the turbulent intensity increases with the safety factor. The proposed mechanism to elucidate this phenomenon, backed up by a reduced model, is that the damping of the poloidal flow, governed by the neoclassical friction, increases more strongly with the safety factor than the turbulent drive for Er, due to the (r , θ) component of the Reynolds stress. [ABSTRACT FROM AUTHOR]

The effect of toroidal rotation on both turbulent and neoclassical transport of tungsten (W) in tokamaks is investigated using the flux-driven, global, nonlinear 5D gyrokinetic code GYSELA. Nonlinear simulations are carried out with different levels of momentum injection that drive W into the supersonic regime, while the toroidal velocity of the main ions remains in the subsonic regime. The numerical simulations demonstrate that toroidal rotation induces centrifugal forces that cause W to accumulate in the outboard region, generating an in–out poloidal asymmetry. This asymmetry enhances neoclassical inward convection, which can lead to central accumulation of W in cases of strong plasma rotation. The core accumulation of W is mainly driven by inward neoclassical convection. However, as momentum injection continues, roto-diffusion, proportional to the radial gradient of the toroidal velocity, becomes significant and generates outward turbulent flux in the case of ion temperature gradient turbulence. Overall, the numerical results from nonlinear GYSELA simulations are in qualitative agreement with the theoretical predictions for impurity transport. [ABSTRACT FROM AUTHOR]

Trapped ion modes (TIMs) belong to the family of ion temperature gradient (ITG) modes, which are one of the important ingredients in heat turbulent transport at the ion scale in tokamak plasmas. A global linear analysis of a reduced gyro-bounce kinetic model for trapped particle modes is performed, and a spectral method is proposed to solve the dispersion relation. Importantly, the radial profile of the particle drift velocity is taken into account in the linear analysis by considering the magnetic flux ψ dependency of the equilibrium Hamiltonian H e q (ψ) in both the quasi-neutrality equation and equilibrium gyro-bounce averaged distribution function F e q . Using this spectral method, linear growth rates of TIM instability in the presence of different temperature profiles and precession frequencies of trapped ions, with an approximated constant Hamiltonian and the exact ψ dependent equilibrium Hamiltonian, are investigated. The growth rate depends on the logarithmic gradient of temperature κ T , density κ n and equilibrium Hamiltonian κ Λ . With the exact ψ dependent Hamiltonian, the growth rates and potential profiles are modified significantly, compared to the cases with an approximated constant Hamiltonian. All the results from the global linear analysis agree with a semi-Lagrangian based linear Vlasov solver with good accuracy. This spectral method is very fast and requires much less computation resources compared to a linear version of the Vlasov-solver based on a semi-Lagrangian scheme. [ABSTRACT FROM AUTHOR]

Two time slices in linear and saturated Ohmic confinement (LOC and SOC) regimes in a Tore Supra Ohmic L-mode discharge are analyzed using nonlinear global full-f gyrokinetic simulations, and qualitative features of the LOC–SOC transition are reproduced. The exhaust of carbon impurity ions is caused by ion mixing, which is driven by the toroidal field stress. The intrinsic rotation develops in the opposite direction between the LOC and SOC phases, which is characterized by different features of the mode asymmetry between trapped electron modes (TEMs) in the LOC phase and ion temperature gradient-driven modes in the SOC phase, leading to the change of the profile shear stress. Here, the mode asymmetry or the ballooning angle depends not only on the profile shear and the Er shear but also on the radial electric field Er itself. The energy fluxes of electrons and deuterium ions are dominant in the LOC and SOC phases, respectively, and the ratio of the energy confinement time between the two phases agree with the experimental value. Turbulent frequency spectra are characterized by quasi-coherent modes (QCMs) and broad-band spectra in the LOC and SOC phases, respectively. The QCMs are produced by a split of the toroidal mode number spectra of TEMs induced by the Doppler shift of poloidal E × B rotation due to Er, which is sustained in the electron diamagnetic direction by the ripple induced counter-current rotation, and enhanced by higher temperature in the LOC phase. [ABSTRACT FROM AUTHOR]

Reynolds stress is a key facet of turbulence self-organization. In the magnetized plasmas of controlled fusion devices, the zonal flows that are driven by the averaged Reynolds stress modify the confinement performance. We address this problem with full-f gyrokinetic simulations of ion temperature gradient-driven turbulence. From the detailed analysis of the three-dimensional electric potential and transverse pressure fields, we show that the diamagnetic contribution to the Reynolds stress—stemming from finite Larmor radius effects—exceeds the electrostatic contribution by a factor of about two. Both contributions are in phase, indicating that pressure does not behave as a passive scalar. In addition, the Reynolds stress induced by the electric drift velocity is found to be mainly governed by the gradient of the phase of the electric potential modes rather than by their magnitude. By decoupling Reynolds stress drive and turbulence intensity, this property indicates that a careful analysis of phase dynamics is crucial in the interpretation of experiments and simulations. [ABSTRACT FROM AUTHOR]

A model of E × B staircases is proposed based on a wave kinetic equation coupled to a poloidal momentum equation. A staircase pattern is idealized as a periodic radial structure of zonal shear layers that bound regions of propagating wave packets, viewed as avalanches. Wave packets are trapped in shear flow layers due to refraction. In this model, an E × B staircase motif emerges due to the interaction between propagating wave packets (avalanches) and trapped waves in the presence of an instability drive. Amplitude, shape, and spatial period of the staircase E × B flow are predicted as functions of the background fluctuation spectrum and the growth rate of drift waves. The zonal flow velocity radial profile is found to peak near its maxima and to flatten near its minima. The optimum configuration for staircase formation is a growth rate, that is, maximum at zero radial wave number. A mean shear flow is responsible for a preferential propagation speed of avalanches. It is not a mandatory condition for the existence of staircase solutions, but has an impact on their spatial period. [ABSTRACT FROM AUTHOR]

Impurity transport is numerically investigated for different types of impurity, such as helium (He), argon (Ar), and tungsten (W). Both turbulent and neoclassical transports are treated self-consistently using the full-f gyrokinetic software GYSELA. For a light impurity (He), the transport is mainly controlled by turbulence, while neoclassical transport is found to be dominant in the case of a heavy impurity (W). The impact of a poloidal asymmetry of the impurity density is also studied in detail and it is found to be strong in case of a high charge impurity, due to its Boltzmann-type response. Such strong asymmetry might lead to a core accumulation of heavy impurities by reducing the thermal screening factor of neoclassical transport. The two main contributions to neoclassical transport—Pfirsch–Schlüter (PS) flux and banana–plateau (BP) flux—are also studied. Depending on their mass (A) and charge (Z), the magnitudes of each flux are determined accordingly. Tungsten shows a strong PS flux compared to the other impurities, while BP flux is dominant in the case of argon. An analytical model including the effect of poloidal asymmetry is compared with the numerical simulation and a good agreement is found between them. [ABSTRACT FROM AUTHOR]

The problem of the fluid closure for the collisionless linear Vlasov system is investigated using a perspective from control theory and model order reduction. The balanced truncation method is applied to the 1D–1V Vlasov system. The first few reduction singular values are well-separated, indicating potentially low-dimensional dynamics. To avoid large-dimensional numerical work, a reduced model is formulated using rational interpolation, generalizing the seminal work from Hammett and Perkins. The resulting models are found to outperform the state-of-the-art models for thermal phase velocities. Thanks to the versatility of this formulation, an application to toroidal gyrokinetic dynamics is discussed. [ABSTRACT FROM AUTHOR]

Impurity turbulent transport in tokamak plasmas is investigated in the case of trapped ion mode (TIM) and trapped electron mode (TEM) using a non-linear bounce-averaged gyrokinetic model. In particular, we focus on the thermo-diffusive pinch and the curvature-driven pinch observed in non-linear simulations. Thermodiffusion is analyzed independently of curvature pinch by disabling the latter artificially. A parameter scan shows that the direction of thermodiffusion depends mainly on two factors. First, it depends on the sign of the turbulence phase velocity, and the resulting transport is inward in the case of TEM whereas it is outward in the case of TIM. Second, it depends on the sign of the impurity temperature gradient. Moreover, the magnitude of the thermodiffusion tends to decrease with respect to the impurity charge Z, while it is proportional to the intensity of background turbulence. Lastly, the direction of the curvature pinch is investigated as a function of the magnetic shear. The results of non-linear simulations are compared with quasi-linear theory. The Kubo number is calculated for our non-linear simulations to justify this comparison. Quasi-linear theory is found to be in qualitative agreement with our gyrokinetic numerical observations. [ABSTRACT FROM AUTHOR]

Barely trapped and passing electrons have been recently predicted to strongly enhance the damping rate of Geodesic Acoustic Modes (GAMs) in tokamak plasmas, while keeping their real frequency almost unchanged as compared to the case with adiabatic electrons. In this paper, dedicated gyrokinetic simulations are successfully compared with these analytical predictions. Specifically, the scaling of the GAM damping rate with respect to the ion to electron mass ratio, to the electron to ion temperature ratio, to the safety factor, and to the aspect ratio is recovered in most regions of the relevant parameter space. [ABSTRACT FROM AUTHOR]

The diffusive impurity transport as a function of charge and mass numbers is investigated in ion driven or electron driven turbulence, in the limit of zero impurity temperature gradient. It is found that the impurity transport decreases slightly with increasing mass number and depends much more strongly on the charge number. Moreover, this transport depends on the nature of the instability that drives turbulence. The impurity flux due to Trapped Electron Mode (TEM) turbulence increases with the charge number Z. In contrast, it is found to decrease with Z when the Trapped Ion Mode (TIM) dominated. In order to explain these observations, the quasilinear flux is derived and is compared with the results obtained from the nonlinear simulations. Quasilinear theory qualitatively reproduces the gyrokinetic numerical observations. [ABSTRACT FROM AUTHOR]

Hotspot generation by lower‐hybrid (LH) launchers is found to be governed by a resonance in the plasma electric field response to the external drive. The kinetic analysis in 1D‐1V in the parallel direction allows one to compute the amplification effect for small amplitude of the external drive. The resonant Lorentzian response distorts the distribution function. An island structure is formed in the suprathermal part at the phase velocity of the external electrostatic drive. The non‐linear features enhance the plasma response, driving overlap effects between multiple waves at rather low amplitude. The onset of a plateau in the distribution function, with extent reaching one thermal velocity, is already obtained when the standard overlap condition is achieved. The sensitivity of the resonance to plasma parameters and large variation of the amplification magnitude can compensate the fast radial decay of the small‐scale features generated by the LH launchers, which are responsible for the interaction with the cold electrons. This mechanism can trigger hotspot generation further in the scrape‐off layer than otherwise expected. [ABSTRACT FROM AUTHOR]

The effect of the presence of an impurity species on the trapped particle turbulence is studied using the gyro-bounce kinetic code TERESA, which allows the study of Trapped Electron Modes and Trapped Ion Modes. The impurity species is treated self-consistently and its influence on the nature of the turbulence, ion driven or electron driven, is investigated. It is found that the presence of heavy impurities with a flat density profile tends to stabilize the both electron and ion modes, whereas a peaked or hollow impurity density profile can change the turbulence from an electron driven turbulence to an ion driven turbulence. The effect of the turbulence regime on impurity transport is studied. [ABSTRACT FROM AUTHOR]

The Projection on Proper elements (PoPe) is a novel method of code control dedicated to (1) checking the correct implementation of models, (2) determining the convergence of numerical methods, and (3) characterizing the residual errors of any given solution at very low cost. The basic idea is to establish a bijection between a simulation and a set of equations that generate it. Recovering equations is direct and relies on a statistical measure of the weight of the various operators. This method can be used in any number of dimensions and any regime, including chaotic ones. This method also provides a procedure to design reduced models and quantify its ratio of cost to benefit. PoPe is applied to a kinetic and a fluid code of plasma turbulence. [ABSTRACT FROM AUTHOR]

A linearized multi-species collision operator has been developed for an efficient implementation in gyrokinetic codes. This operator satisfies the main expected properties: particle, momentum, and energy conservation, and existence of an H-theorem. A gyrokinetic version is then calculated, which involves derivatives with respect to the gyrocenter position, parallel velocity, and magnetic momentum. An isotropic version in the velocity space can be constructed for the specific problem of trace impurities colliding with a main species. A simpler version that involves derivatives with parallel velocity only has been developed. This reduced version has been implemented in the GYSELA gyrokinetic code, and is shown to comply with particle, momentum, and energy conservation laws. Moreover, the interspecies relaxation rates for momentum and energy agree very well with the theoretical values. [ABSTRACT FROM AUTHOR]

This paper presents a reduced kinetic model, which describes simultaneously trapped-ion (TIM) and trapped-electron (TEM) driven modes. Interestingly, the model enables the study of a full f problem for ion and electron trapped particles at very low numerical cost. The linear growth rate obtained with the full f nonlinear code Trapped Element REduction in Semi Lagrangian Approach is successfully compared with analytical predictions. Moreover, nonlinear results show some basic properties of collisionless TEM and TIM turbulence in tokamaks. A competition between streamer-like structures and zonal flows is observed for TEM and TIM turbulence. Zonal flows are shown to play an important role in suppressing the nonlinear transport and strongly depend on the temperature ratio Te/Ti. [ABSTRACT FROM AUTHOR]

In a dedicated collisionality scan in Tore Supra, the geodesic acoustic mode (GAM) is detected and identified with the Doppler backscattering technique. Observations are compared to the results of a simulation with the gyrokinetic code GYSELA. We found that the GAM frequency in experiments is lower than predicted by simulation and theory. Moreover, the disagreement is higher in the low collisionality scenario. Bursts of non harmonic GAM oscillations have been characterized with filtering techniques, such as the Hilbert-Huang transform. When comparing this dynamical behaviour between experiments and simulation, the probability density function of GAM amplitude and the burst autocorrelation time are found to be remarkably similar. In the simulation, where the radial profile of GAM frequency is continuous, we observed a phenomenon of radial phase mixing of the GAM oscillations, which could influence the burst autocorrelation time. [ABSTRACT FROM AUTHOR]

One important issue in turbulence self-organization is the interplay between the Kelvin-Helmholtz (KH) instability and streamers and/or zonal flows. This question has been debated for a long time. The effects of the KH instability and its position in the sequence of events between streamers, turbulence, and zonal flows have been investigated with a reduced gyro-bounce averaged kinetic code devoted to study the primary ion temperature gradient (ITG) instability linked to trapped ion modes (TIM). In toroidal geometry, the specific dynamics of TIM linked to trapped particles becomes important when the frequency of ITG modes falls below the ion bounce frequency, allowing one to average on both the cyclotron and bounce motion fast time scales. This reduction of the number of degrees of freedom leads to a strong reduction of computer resources (memory and computation time). Bounce-averaged gyrokinetic code can be considered as a toy model able to describe basic structures of turbulent transport in tokamak devices. In particular, by means of this code, we have observed that the energy injected in the system by the TIM instability is exchanged between streamers and zonal flows by means of KH vortices that grow along these structures in the nonlinear phase. The energy transfer occurs throughout the relaxation phase of the streamer growth leading to a modification of the KH modes and to the generation of the zonal flows. [ABSTRACT FROM AUTHOR]

A Flux-Coordinate Independent (FCI) approach for anisotropic systems, not based on magnetic flux coordinates, has been introduced in Hariri and Ottaviani [Comput. Phys. Commun. 184, 2419 (2013)]. In this paper, we show that the approach can tackle magnetic configurations including X-points. Using the code FENICIA, an equilibrium with a magnetic island has been used to show the robustness of the FCI approach to cases in which a magnetic separatrix is present in the system, either by design or as a consequence of instabilities. Numerical results are in good agreement with the analytic solutions of the sound-wave propagation problem. Conservation properties are verified. Finally, the critical gain of the FCI approach in situations including the magnetic separatrix with an X-point is demonstrated by a fast convergence of the code with the numerical resolution in the direction of symmetry. The results highlighted in this paper show that the FCI approach can efficiently deal with X-point geometries. [ABSTRACT FROM AUTHOR]

Spontaneous transport barrier generation at the edge of a magnetically confined plasma is reproduced in flux-driven three-dimensional fluid simulations of electrostatic turbulence. Here, the role on the radial electric field of collisional friction between trapped and passing particles is shown to be the key ingredient. Especially, accounting for the self-consistent and precise dependence of the friction term on the actual plasma temperature allows for the triggering of a transport barrier, provided that the input power exceeds some threshold. In addition, the barrier is found to experience quasi-periodic relaxation events, reminiscent of edge localised modes. These results put forward a possible key player, namely, neoclassical physics via radial force balance, for the low- to high-confinement regime transition observed in most of controlled fusion devices. [ABSTRACT FROM AUTHOR]

In this paper, the nature of the primary instability present in the pedestal forming region prior to the transition into H mode is analysed using a gyrokinetic code on JET-ILW profiles. The linear analysis shows that the primary instability is of resistive nature, and can therefore be stabilized by increased temperature, hence power. The unstable modes are identified as being resistive ballooning modes. Their growth rates decrease for temperatures increasing towards the experimentally measured temperature at the L–H transition. The growth rates are larger for lower effective charge Zeff. This dependence is shown to be in qualitative agreement with recent and past experimental observations of reduced Zeff associated with lower L–H power thresholds. [ABSTRACT FROM AUTHOR]

Evidence for the impact of energetic particles (EPs) on turbulence is given in this paper. Firstly, the excitation of electrostatic instabilities in linear gyrokinetic simulations performed with the global GYSELA code by introducing distribution functions typical of fast ions in tokamak plasmas is presented. The obtained mode is unambiguously characterized as an EGAM, i.e. a geodesic acoustic mode (GAM) excited by EPs. The influence of EGAMs on turbulence and associated transport is then analyzed by implementing a source adapted to the inclusion of fast particle populations in non-linear simulations. This source successfully excites EGAMs in the presence of turbulence, which leads to a drastic reduction of the turbulent transport. However, this reduction is only transient; it is followed by an increase of the turbulent activity, characterized by a complex interaction between the EGAMs and the turbulence. In the subsequent steady-state regime, turbulent transport appears to be modulated at the EGAM frequency. [ABSTRACT FROM AUTHOR]

The generation and dynamics of transport barriers governed by sheared poloidal flows are analyzed in flux-driven 5D gyrokinetic simulations of ion temperature gradient driven turbulence in tokamak plasmas. The transport barrier is triggered by a vorticity source that polarizes the system. The chosen source captures characteristic features of some experimental scenarios, namely, the generation of a sheared electric field coupled to anisotropic heating. For sufficiently large shearing rates, turbulent transport is suppressed and a transport barrier builds up, in agreement with the common understanding of transport barriers. The vorticity source also governs a secondary instability- driven by the temperature anisotropy (T∣∣ ≠ T⊥). Turbulence and its associated zonal flows are generated in the vicinity of the barrier, destroying the latter due to the screening of the polarization source by the zonal flows. These barrier relaxations occur quasiperiodically, and generically result from the decoupling between the dynamics of the barner generation, triggered by the source driven sheared flow, and that of the crash, triggered by the secondary instability. This result underlines that barriers triggered by sheared flows are prone to relaxations whenever secondary instabilities come into play. [ABSTRACT FROM AUTHOR]

The interplay between turbulent bursts and transport barriers is analyzed with a simplified model of interchange turbulence in magnetically confined plasmas. The turbulent bursts spread into the transport barriers and, depending on the competing magnitude of the burst and stopping capability of the barrier, can burn through. Simulations of two models of transport barriers are presented: a hard barrier where interchange turbulence modes are stable in a prescribed region and a soft barrier with external plasma biasing. The response of the transport barriers to the non-linear perturbations of the turbulent bursts, addressed in a predator-prey approach, indicates that the barriers monitor an amplification factor of the turbulent bursts, with amplification smaller than one for most bursts and, in some cases, amplification factors that can significantly exceed unity. The weak barriers in corrugated profiles and magnetic structures, as well as the standard barriers, are characterized by these transmission properties, which then regulate the turbulent burst transport properties. The interplays of barriers and turbulent bursts are modeled as competing stochastic processes. For different classes of the probability density function (PDF) of these processes, one can predict the heavy tail properties of the bursts downstream from the barrier, either exponential for a leaky barrier, or with power laws for a tight barrier. The intrinsic probing of the transport barriers by the turbulent bursts thus gives access to the properties of the barriers. The main stochastic variables are the barrier width and the spreading distance of the turbulent bursts within the barrier, together with their level of correlation. One finds that in the case of a barrier with volumetric losses, such as radiation or particle losses as addressed in our present simulations, the stochastic model predicts a leaky behavior with an exponential PDF of escaping turbulent bursts in agreement with the simulation data. [ABSTRACT FROM AUTHOR]