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Global electromagnetic turbulence is simulated in stellarator geometry using the gyrokinetic particle-in-cell code EUTERPE. The evolution of the turbulent electromagnetic field and the plasma profiles is considered at different values of the plasma beta and for different magnetic configurations. It is found that turbulence is linearly driven at relatively high toroidal mode numbers. In the nonlinear regime, lower toroidal mode numbers, including zonal flows, are excited resulting in a quench of the linear instability drive. The turbulent heat flux is outward and leads to the nonlinear relaxation of the plasma temperature profile. The particle flux is inward for the parameters considered. The effect of the parallel perturbation of the magnetic field on the stellarator turbulence is addressed.

A numerical approach is devised to calculate the shear Alfvén continuum inside and outside magnetic islands in cylindrical and stellarator plasmas. Equations for an appropriate set of coordinates and the arising equations for the continuum are derived and implemented in the CONTI code. An experiment-oriented representation of the results is chosen to allow a radial localization of the modes and a comparison of different magnetic configurations. Comparison is made with results of earlier analytic work for validation. Agreement is good but more details of the spectrum, such as the generation of island induced gaps inside and outside the separatrix, are found. While the code is easily usable and can be applied to any magnetic equilibrium accessible with VMEC, the calculations are plagued with convergence issues close to the separatrix. A calculation for a realistic W7-X equilibrium with islands is done where the island width is estimated with the HINT code.

In this work, we compare gyrokinetic simulations in stellarators using different computational domains, namely, flux tube, full-flux-surface, and radially global domains. Two problems are studied: the linear relaxation of zonal flows and the linear stability of ion temperature gradient (ITG) modes. Simulations are carried out with the codes EUTERPE, GENE, GENE-3D, and stella in magnetic configurations of LHD and W7-X using adiabatic electrons. The zonal flow relaxation properties obtained in different flux tubes are found to differ with each other and with the radially global result, except for sufficiently long flux tubes, in general. The flux tube length required for convergence is configuration-dependent. Similarly, for ITG instabilities, different flux tubes provide different results, but the discrepancy between them diminishes with increasing flux tube length. Full-flux-surface and flux tube simulations show good agreement in the calculation of the growth rate and frequency of the most unstable modes in LHD, while for W7-X differences in the growth rates are found between the flux tube and the full-flux-surface domains. Radially global simulations provide results close to the full-flux-surface ones. The radial scale of unstable ITG modes is studied in global and flux tube simulations finding that in W7-X, the radial scale of the most unstable modes depends on the binormal wavenumber, while in LHD no clear dependency is found., submitted to Nuclear Fusion

Magnetic reconnection in W7-X is studied by means of global numerical simulations in a series of models of increasing complexity. The magnetic geometry ranges from that of a cylinder to the full three-dimensional field of W7-X, and the equations solved range from ideal magnetohydrodynamics (MHD) to gyrokinetics. We simulate plasmas from the first operation phase with electron cyclotron current drive (ECCD). These are characterized by an equilibrium magnetic field featuring an ECCD-distorted ‘humped’ profile of the rotational transformι, withι= 1 in two radial locations. Such plasmas generally show sawtooth activity, hence motivating the present study. We pay particular attention to the role of equilibrium current density gradients in the destabilization of reconnecting modes. When the equilibrium temperature and density gradients are artificially suppressed (to eliminate the pressure gradient drive), the perturbed electrostatic potential is radially localized between the locations at whichι= 1. This is shown with a purely collisionless gyrokinetic model, in cylindrical geometry. In the real toroidal geometry of W7-X, for a non-ideal MHD model including a uniform resistivity, electron inertia and (numerical) viscosity, the same qualitative behaviour is observed. In particular, even if a resonant (m,n) = (1, −1) perturbation is initialized, the most unstable mode is the (m,n) = (−4, 4), wheremandnare the poloidal and toroidal mode numbers, respectively. Other modes are destabilized due to geometric coupling. The growth rate of this instability scales asη1/3, whereηis the plasma resistivity, thus suggesting that ECCD drives ideal MHD stable W7-X plasmas towards non-ideal marginality. An ideal magnetohydrodynamic analysis confirms the result. A fluid-kinetic hybrid version of theEUTERPEcode shows that gyrokinetic ions have a stabilizing effect on these modes. For W7-X relevant collisionalities, the growth rate scales linearly with the electron skin depth,de. Implications of our results for sawtoothing W7-X operation are discussed.

The effects of a radial electric field, which is ubiquitous in stellarators, are considered when a system of reduced-MHD equations is derived from a gyro-kinetic Vlasov–Maxwell system. The resulting equations for the MHD continuum have been implemented into the continuum code CONTI. For a tokamak case, the MHD continuum is calculated and compared with a gyro-kinetic continuum calculated using the EUTERPE code, where good agreement of the obtained continua could be shown. Finally, MHD continua including the radial electric field are calculated for Wendelstein 7-X. Interestingly, for these examples, most of the gaps in the continuous spectrum of Alfvén waves below 200 kHz disappear.

The effects of 3D geometry are explored in TJ-II from two relevant points of view: neoclassical transport and modification of stability and dispersion relation of waves. Particle fuelling and impurity transport are studied considering the 3D transport properties, paying attention to both neoclassical transport and other possible mechanisms. The effects of the 3D magnetic topology on stability, confinement and Alfvén Eigenmodes properties are also explored, showing the possibility of controlling Alfvén modes by modifying the configuration; the onset of modes similar to geodesic acoustic modes are driven by fast electrons or fast ions; and the weak effect of magnetic well on confinement. Finally, we show innovative power exhaust scenarios using liquid metals.

Due to their capability to reduce turbulent transport in magnetized plasmas, understanding the dynamics of zonal flows is an important problem in the fusion programme. Since the pioneering work by Rosenbluth and Hinton in axisymmetric tokamaks, it is known that studying the linear and collisionless relaxation of zonal flow perturbations gives valuable information and physical insight. Recently, the problem has been investigated in stellarators and it has been found that in these devices the relaxation process exhibits a characteristic feature: a damped oscillation. The frequency of this oscillation might be a relevant parameter in the regulation of turbulent transport, and therefore its efficient and accurate calculation is important. Although an analytical expression can be derived for the frequency, its numerical evaluation is not simple and has not been exploited systematically so far. Here, a numerical method for its evaluation is considered, and the results are compared with those obtained by calculating the frequency from gyrokinetic simulations. This "semianalytical" approach for the determination of the zonal-flow frequency reveals accurate and faster than the one based on gyrokinetic simulations., 30 pages, 14 figures

Measurements of the relaxation of a zonal electrostatic potential perturbation in a non-axisymmetric magnetically confined plasma are presented. A sudden perturbation of the plasma equilibrium is induced by the injection of a cryogenic hydrogen pellet in the TJ-II stellarator, which is observed to be followed by a damped oscillation in the electrostatic potential. The waveform of the relaxation is consistent with theoretical calculations of zonal potential relaxation in a non-axisymmetric magnetic geometry. The turbulent transport properties of a magnetic confinement configuration are expected to depend on the features of the collisionless damping of zonal flows, of which the present letter is the first direct observation.

Drift-wave turbulence in the linear VINETA [1] device was studied using the adapted version [2] of the threedimensional gyrofluid code GEM3 [3]. The density and parallel velocity gyrofluid equations of electrons and ions for the cylindrical annulus are solved. It is shown that the code is able to reproduce the main experimental signatures of intermittency. The influence of the interaction of the plasma with neutrals – the so-called ‘neutral wind’ – on the intermittent turbulence in VINETA is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The resonant interaction of shear Alfven waves with energetic particles is investigated numerically in tokamak and stellarator geometry using a non-perturbative MHD-kinetic hybrid approach. The focus lies on toroidicity-induced Alfven eigenmodes (TAEs), which are most easily destabilized by a fast-particle population in fusion plasmas. While the background plasma is treated within the framework of an ideal-MHD theory, the drive of the fast particles, as well as Landau damping of the background plasma, is modelled using the drift-kinetic Vlasov equation without collisions. Building on analytical theory, a fast numerical tool, STAE-K, has been developed to solve the resulting eigenvalue problem using a Riccati shooting method. The code, which can be used for parameter scans, is applied to tokamaks and the stellarator Wendelstein 7-X. High energetic-ion pressure leads to large growth rates of the TAEs and to their conversion into kinetically modified TAEs and kinetic Alfven waves via continuum interaction. To better understand the physics of this conversion mechanism, the connections between TAEs and the shear Alfven wave continuum are examined. It is shown that, when energetic particles are present, the continuum deforms substantially and the TAE frequency can leave the continuum gap. The interaction of the TAE with the continuum leads to singularities in the eigenfunctions. To further advance the physical model and also to eliminate the MHD continuum together with the singularities in the eigenfunctions, a fourth-order term connected to radiative damping has been included. The radiative damping term is connected to non-ideal effects of the bulk plasma and introduces higher-order derivatives to the model. Thus, it has the potential to substantially change the nature of the solution. For the first time, the fast-particle drive, Landau damping, continuum damping, and radiative damping have been modelled together in tokamak- as well as in stellarator geometry.

The two-moment version of the three-dimensional gyrofluid code GEM3 has been adapted for the simulation of drift-wave turbulence in the linear device VINETA [1]. In the modified GEM3 code gyrofluid equations for the density and parallel velocity of electrons and ions are solved for the cylindrical annulus. Computations are done for the electrostatic case using an exponential background density profile observed experimentally.

The whistler wave dispersion relation in the transition region between unbounded and bounded plasma geometry is investigated experimentally and numerically. Measurements are done in a linear magnetized helicon plasma covering the large frequency range from 100–800 MHz, corresponding to 0.06–0.5fce. Small wavelength wave propagation (λ≪d: plasma diameter) is well explained by unbounded plasma whistler wave dispersion. In contrast to previously reported measurements [Franck et al., Phys. Plasmas 9, 3254 (2002)], the experimental findings are compared to numerical results obtained from the differential equations of a plasma-filled waveguide. Long wavelength wave measurements show that there is only qualitative agreement even with dispersion theory of whistler wave propagation in bounded plasmas. This is attributed to the perpendicular wave mode structure that influences the parallel wavelengths. Measurements of the perpendicular wave mode structure shows that it is basically given by the diameter of the plas...

The linear stability of the ion-temperature-gradient modes (ITG) in the electrostatic limit is examined in the short wavelength region by using a two fluid reactive model in fully three-dimensional Wendelstein 7-X (W7-X) stellarator [G. Grieger et al., Plasma Physics and Controlled Nuclear Fusion Research, 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 3, p. 525] geometry. The spectrum of stable and unstable modes and their real frequencies and eigenfunctions are calculated. The effects of density gradients, temperature gradients, temperature ratios, wavevector, ballooning angle, curvature and local magnetic shear on the ITG mode are also investigated. The frequency and growth rate of the most unstable ITG mode is calculated and visualized for a specific magnetic flux surface. For the equilibrium under investigation both localized and extended eigenmodes are found. The effect of small and large temperature ratios, small and large density gradients as well as large local magnetic shear are a...

A global nonlinear simulation code for the time evolution of ion-temperature-gradient-driven modes in theta-pinch geometry as a first approximation to the stellarator Wendelstein 7-X (W7-X) [Grieger , 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 deltaf 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 deltaf 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. (C) 2002 American Institute of Physics.

It is shown that a considerable improvement in the global gyrokinetic electromagnetic simulations can be achieved by a slight modification of the simulation scheme. The new scheme is verified, simulating a Toroidal Alfv\'en Eigenmode in tokamak geometry at low perpendicular mode numbers, the so-called "MHD limit". Also, an electromagnetic drift mode has been successfully simulated in a stellarator., Comment: Paper submitted for publication

The first detailed comparison between gyrokinetic and gyrofluid simulations of collisionless magnetic reconnection has been carried out. Both the linear and nonlinear evolution of the collisionless tearing mode have been analyzed. In the linear regime, we have found a good agreement between the two approaches over the whole spectrum of linearly unstable wave numbers, both in the drift kinetic limit and for finite ion temperature. Nonlinearly, focusing on the small-$\Delta '$ regime, with $\Delta '$ indicating the standard tearing stability parameter, we have compared relevant observables such as the evolution and saturation of the island width, as well as the island oscillation frequency in the saturated phase.The results are basically the same, with small discrepancies only in the value of the saturated island width for moderately high values of $\Delta '$. Therefore, in the regimes investigated here, the gyrofluid approach can describe the collisionless reconnection process as well as the more complete gyrokinetic model., Comment: Accepted for publication on Physics of Plasmas

A new approach to electromagnetic gyrokinetic simulations based on modified gyrokinetic theory is described. The method is validated using a particle-in-cell code. The Toroidal Alfven Eigenmode at low perpendicular mode numbers, the so-called “magnetohydrodynamical limit,” has been successfully simulated using this method.

The internal kink mode is an important plasma instability responsible for a broad class of undesirable phenomena in tokamaks, including the sawtooth cycle and fishbones. To predict and discover ways to mitigate this behaviour in current and future devices, numerical simulations are necessary. The internal kink mode can be modelled by reduced magnetohydrodynamics (MHD). Fishbone modes are an inherently kinetic and non-linear phenomenon based on the n = 1 Energetic Particle Mode (EPM), and have been studied using hybrid codes that combine a reduced MHD bulk plasma model with a kinetic treatment of fast ions. In this work, linear simulations are presented using a hybrid model which couples a fluid treatment of electrons with a gyrokinetic treatment of both bulk and fast ions. Studies of the internal kink mode in geometry relevant to large tokamak experiments are presented and the effect of gyrokinetic ions is considered. Interaction of the kink with gyrokinetic fast ions is also considered, including the destabilisation of the linear n = 1 EPM underlying the fishbone.

The results of global linear gyrokinetic simulations of residual flows carried out with the code EUTERPE in the TJ-II three-dimensional geometry are reported. The linear response of the plasma to potential perturbations homogeneous in a magnetic surface shows several oscillation frequencies: a Geodesic-acoustic-mode-like frequency, in qualitative agreement with the formula given by Sugama and Watanabe (2006 Plasma Phys. 72 825), and a much lower frequency oscillation in agreement with the predictions of Mishchenko et al (2008 Phys. Plasmas 15 072309) and Helander et al (2011 Plasma Phys. Control. Fusion 53 054006) for stellarators. The dependence of both oscillations on ion and electron temperatures and the magnetic configuration is studied. The low-frequency oscillations are in the frequency range supporting the long-range correlations between potential signals experimentally observed in TJ-II.

This work deals with the results on flow dynamics in TJ-II plasmas under Li-coated wall conditions, which produces low recycling and facilitates the density control and access to improved confinement transitions. The low-density transition, characterized by the emergence of the shear flow layer, is described from first principles and within the framework of neoclassical theory. The vanishing of the neoclassical viscosity when approaching the transition from below explains the observation of a number of turbulent phenomena reported in TJ-II in recent years; a unifying picture is provided in which zonal, i.e. large scale, radially structured, perturbations are observable when the neoclassical damping is sufficiently small. Preliminary linear, collisionless gyrokinetic simulations are carried out to assess that the measured time scale of relaxation of such perturbations is reasonably understood theoretically. In higher density regimes, the physical mechanisms behind the L–H transition have been experimentally studied. The spatial, temporal and spectral structure of the interaction between turbulence and flows has been studied close to the L–H transition threshold conditions. The temporal dynamics of the turbulence-flow interaction displays a predator–prey relationship and both radial outward and inward propagation velocities of the turbulence-flow front have been measured. Finally, a non-linear relation between turbulent fluxes and gradients is observed.

Describing turbulence and microinstabilities in fusion devices is often modelled with the gyrokinetic equation. During the time evolution of the distribution function a field equation for the electrostatic potential needs to be solved. In the case of adiabatic electrons it contains a flux-surface-average term resulting in an integro-differential equation. Its numerical solution is time and memory intensive for three-dimensional configurations. Here a new algorithm is presented which only requires the numerical inversion of a simpler differential operator and a subsequent addition of a correction term. This new procedure is as fast as solving the equation without the surface average.

The linear dynamics of zonal flows is addressed in stellarator geometry in the presence of an ambient (neoclassical) radial electric field. Global gyrokinetic particle-in-cell simulations are used to study the properties of the residual flow and its dependence on the plasma parameters and magnetic geometry. Properties of the zonal flow are compared in different magnetic geometries.

While the usual way of calculating continuum damping of global Alfven modes is the introduction of a small artificial resistivity, we present a computational approach to the problem based on a suitable path of integration in the complex plane. This approach is implemented by the Riccati shooting method and it is shown that it can be transferred to the Galerkin method used in three-dimensional ideal magneto-hydrodynamics (MHD) codes. The new approach turns out to be less expensive with respect to resolution and computation time than the usual one. We present an application to large aspect ratio tokamak and stellarator equilibria retaining a few Fourier harmonics only and calculate eigenfunctions and continuum damping rates. These may serve as an input for kinetic MHD hybrid models making it possible to bypass the problem of having singularities on the path of integration on one hand and considering continuum damping on the other.

An iterative formulation of an arbitrarily short-wavelength solver for global gyrokinetic simulations is suggested. The solver is verified against solutions of the dispersion relation. It can be used to treat the nonlinear polarisation density which is important at the plasma edge. In the linear case, the solver is shown to be computationally efficient.

Electromagnetic gyrokinetic particle-in-cell simulations have been inhibited for long time by numerical problems. This paper discusses the origin of these problems. It also gives an overview and summary of the mitigation techniques.

In the linear collisionless limit, a zonal potential perturbation in a toroidal plasma relaxes, in general, to a non-zero residual value. Expressions for the residual value in tokamak and stellarator geometries, and for arbitrary wavelengths, are derived. These expressions involve averages over the lowest order particle trajectories, that typically cannot be evaluated analytically. In this work, an efficient numerical method for the evaluation of such expressions is reported. It is shown that this method is faster than direct gyrokinetic simulations performed with the GENE and EUTERPE codes. Calculations of the residual value in stellarators are provided for much shorter wavelengths than previously available in the literature. Electrons must be treated kinetically in stellarators because, unlike in tokamaks, kinetic electrons modify the residual value even at long wavelengths. This effect, that had already been predicted theoretically, is confirmed by gyrokinetic simulations., 31 pages, 15 figures