__A. Zocco, G.G. Plunk and P. Xanthopoulos, Physics of Plasmas 27, 022507 (2020)__

A non-perturbative analysis for the study non-axisymmetric (3D) effects on the linear ion-temperature-gradient driven mode is introduced. Perturbations and equilibria are considered to be global on the flux surface, yet radially local. The analysis is valid for systems arbitrarily far from axisymmetry. It is found that finite Larmor radius effects can suppress the global (on the surface) instability, in analogy with the local analysis but shift its poloidal location from the position of the greatest local instability. Fourier spectra of the instability whose width grow for increasingly non-axisymmetric systems are predicted. Results are in qualitative agreement with numerical global (on the surface) gyrokinetic simulations.

__J.A. Alcuson, P. Xanthopoulos, G.G. Plunk, P. Helander et al., Plasma Physics and Controlled Fusion 62, 035005 (2020)__

We demonstrate favorable stability properties of maximum-J stellarators, exemplified by the Wendelstein 7-X (W7-X) device, in scenarios with low plasma beta. A large number of electrostatic linear gyrokinetic simulations are conducted to scan the relevant parameter space for different configurations, resulting in stability maps that account for the key micro-instabilities thought to drive turbulent transport. These maps exhibit a ``stability valley'' in the region where the normalized ion temperature gradient is roughly equal to the normalized density gradient. In this valley, the electrostatic instabilities are partly suppressed thanks to the maximum-J property of the W7-X field [...]

__I.J. McKinney, M.J. Pueschel, B.J. Faber et al. -- Journal of Plasma Physics 85(5) (2019)__

Ion-temperature-gradient-driven (ITG) turbulence is compared for two quasi-symmetric (QS) stellarator configurations to determine the relationship between linear growth rates and nonlinear heat fluxes. We focus on the quasi-helically symmetric (QHS) stellarator HSX and the quasi-axisymmetric (QAS) stellarator NCSX. In normalized units, HSX exhibits higher growth rates than NCSX, while heat fluxes in gyro-Bohm units are lower in HSX. These results hold for simulations made with both adiabatic and kinetic electrons. The results show that HSX has a larger number of subdominant modes than NCSX and that eigenmodes are more spatially extended in HSX. We conclude that the consideration of nonlinear physics is necessary to accurately assess the heat flux due to ITG turbulence when comparing QS stellarator equilibria.

__I. Predebon, P. Xanthopoulos and M. Gobbin -- Plasma Physics and Controlled Fusion 61(5), 055011 (2019)__

We describe the occurrence of electron temperature gradient driven micro-instabilities in the helical states of RFX-mod reversed-field pinch plasmas. These plasmas are usually characterized by the presence of large, radially localized electron temperature gradients. Using realistic geometry and profiles, micro-tearing modes and electron temperature gradient modes turn out to regularly co-exist in the region of the temperature barrier. In the paper, we discuss the main features of such instabilities, especially focusing on the distinctive aspects of the helical geometry with respect to the toroidal symmetry.

__S.A. Lazerson, P. Xanthopoulos, H. Mynick and D. Gates -- Physics of plasmas 26, 022509 (2019)__

The large flexibility of the proposed QUASAR facility [Gates et al., Nucl. Fusion 57, 126064 (2017)] is leveraged in order to explore the effect of magnetic shear on adiabatic Ion Temperature Gradient (ITG) turbulence. The QUASAR facility is a reimagining of the National Compact Stellarator Experiment utilizing and expanding upon the already constructed coil set. Recent work using fixed boundary optimization of the LI383 equilibrium (upon which QUASAR is based) has suggested possible improvements to ITG turbulence [Mynick et al., Plasma Phys. Controlled Fusion 56, 094001 (2014)]. In this work, a different approach is taken, wherein a series of self-consistent free boundary VMEC equilibria are developed for QUASAR. These equilibria assume temperature and density profiles consistent with 2% beta and ohmic current drive. In each configuration, the toroidal field coils are energized to different values and the STELLOPT code is used to vary the modular coil current and net toroidal current.

__G. G. Plunk, P. Xanthopoulos, G. M. Weir, S. A. Bozhenkov, A. Dinklage, G. Fuchert, J. Geiger, M. Hirsch, U. Hoefel, M. Jakubowski, A. Langenberg, N. Pablant, E. Pasch, T. Stange, D. Zhang, and the W7-X Team -- Physical Review Letters 122, 035002 (2019)__

Electron temperature gradient (ETG)-driven turbulence, despite its ultrafine scale, is thought to drive significant thermal losses in magnetic fusion devices—but what role does it play in stellarators? The first numerical simulations of ETG turbulence for the Wendelstein 7-X stellarator, together with power balance analysis from its initial experimental operation phase, suggest that the associated transport should be negligible compared to other channels. The effect, we argue, originates essentially from the geometric constraint of multiple field periods, a generic feature of stellarators.

__ D. Kennedy, A. Mischchenko, P. Xanthopoulos and P. Helander -- Journal of Plasma Physics 84(6), 905840606 (2018)__

Gyrokinetic stability of plasmas in different magnetic geometries is studied numerically using the GENE code. We examine the stability of plasmas, varying the mass ratio between the positive and negative charge carriers, from conventional hydrogen plasmas through to electron–positron plasmas. Stability is studied for prescribed temperature and density gradients in different magnetic geometries: (i) An axisymmetric, circular flux surface, low 𝛽 (tokamak) configuration. (ii) A non-axisymmetric quasi-isodynamic (optimised stellarator) configuration using the geometry of the stellarator Wendelstein 7-X. We also present the analytic theory of trapped particle modes in electron–positron plasmas. We found similar behaviour of the growth rate and real frequency compared to previous studies on the tokamak case.

__A. Zocco, P. Xanthopoulos, H. Doerk, J.W. Connor and P. Helander -- Journal of Plasma Physics 84(1), 715840101 (2018)__

The threshold for the resonant destabilisation of ion-temperature-gradient (ITG) driven instabilities that render the modes ubiquitous in both tokamaks and stellarators is investigated. We discover remarkably similar results for both confinement concepts if care is taken in the analysis of the effect of the global shear . We revisit, analytically and by means of gyrokinetic simulations, accepted tokamak results and discover inadequacies of some aspects of their theoretical interpretation. In particular, for standard tokamak configurations, we find that global shear effects on the critical gradient cannot be attributed to the wave–particle resonance destabilising mechanism of Hahm & Tang (Phys. Plasmas, vol. 1, 1989, pp. 1185–1192), but are consistent with a stabilising contribution predicted by Biglari et al. (Phys. Plasmas, vol. 1, 1989, pp. 109–118).

__A. Zocco, K. Aleynikova and P. Xanthopoulos -- Journal of Plasma Physics 84(3), 745840303 (2018)__

Kinetic ballooning modes in magnetically confined toroidal plasmas are investigated putting emphasis on specific stellarator features. In particular, we propose a Mercier criterion which is purposely designed to allow for direct comparison with local flux-tube gyrokinetics simulations. We investigate the influence on the marginal frequency of the mode of a magnetic curvature which is inhomogeneous on the magnetic flux surface due to the fieldline-label dependence. This is a typical (surface) global effect present in non-axisymmetry. Finally, we propose an artificial equilibrium model that explicitly retains the fieldline-label dependence in the magnetic drift, and analyse the stability of the system by introducing a representation of the perturbations similar to the flux-bundle model of Sugama et al. (Plasma Fusion Res., vol. 7, 2012, 2403094). The coupling of flux bundles is shown to have a stabilising effect on the most unstable local flux-tube mode.

__K. Aleynikova, A. Zocco, P. Xanthopoulos, P. Helander and C. Nührenberg -- Journal of Plasma Physics vol. 84, 745840602 (2018)__

Kinetic ballooning modes (KBMs) are investigated by means of linear electromagnetic gyrokinetic (GK) simulations in the stellarator Wendelstein 7-X (W7-X), for high-β plasmas, where β is the ratio of thermal to magnetic plasma pressure. The analysis shows suppression of ion-temperature-gradient (ITG) and trapped particle modes (TEM) by finite-β effects and destabilization of KBMs at high β. The results are compared with a generic tokamak case.

__M.F. Martin, M. Landreman, P. Xanthopoulos, N.R. Mandell and W. Dorland -- PPCF 60, 095008 (2018)__

Flux tube simulations of plasma turbulence in stellarators and tokamaks typically employ coordinates which are aligned with the magnetic field lines. Anisotropic turbulent fluctuations can be represented in such field-aligned coordinates very efficiently, but the resulting non-trivial boundary conditions involve all three spatial directions, and must be handled with care. The standard 'twist-and-shift' formulation of the boundary conditions (Beer et al 1995 Phys. Plasmas 2 2687) was derived assuming axisymmetry and is widely used because it is efficient, as long as the global magnetic shear is not too small. A generalization of this formulation is presented, appropriate for studies of non-axisymmetric, stellarator symmetric configurations, as well as for axisymmetric configurations with small global shear. The key idea is to replace the 'twist' of the standard approach (which accounts only for global shear) with the integrated local shear.

__F. Warmer, P. Xanthopoulos, J.H.E. Proll, C.D. Beidler, Y. Turkin and R.C. Wolf -- Nuclear Fusion 58, 016017 (2018)__

Due to foreseen improvement of neoclassical confinement in optimised stellarators—like the newly commissioned Wendelstein 7-X (W7-X) experiment in Greifswald, Germany—it is expected that turbulence will significantly contribute to the heat and particle transport, thus posing a limit to the performance of such devices. In order to develop discharge scenarios, it is thus necessary to develop a model which could reliably capture the basic characteristics of turbulence and try to predict the levels thereof. The outcome will not only be affordable, using only a fraction of the computational cost which is normally required for repetitive direct turbulence simulations, but would also highlight important physics. In this model, we seek to describe the ion heat flux caused by ion temperature gradient (ITG) micro-turbulence, which, in certain heating scenarios, can be a strong source of free energy.

__D.A. Gates et al. -- Nuclear Fusion 57, 126064 (2017)__

Computational optimization has revolutionized the field of stellarator design. To date, optimizations have focused primarily on optimization of neoclassical confinement and ideal MHD stability, although limited optimization of other parameters has also been performed. The purpose of this paper is to outline a select set of new concepts for stellarator optimization that, when taken as a group, present a significant step forward in the stellarator concept. One of the criticisms that has been leveled at existing methods of design is the complexity of the resultant field coils. Recently, a new coil optimization code—COILOPT++, which uses a spline insteadof a Fourier representation of the coils,—was written and included in the STELLOPT suite of codes.

__M. Zuin, S. Dal Bello, L. Marrelli, M.E. Puiatti, P. Agostinetti et al. -- Nuclear Fusion 57, 102012 (2017)__

This paper reports the main recent results of the RFX-mod fusion science activity. The RFX-mod device is characterized by a unique flexibility in terms of accessible magnetic configurations. Axisymmetric and helically shaped reversed-field pinch equilibria have been studied, along with tokamak plasmas in a wide range of q(a) regimes (spanning from 4 down to 1.2 values). The full range of magnetic configurations in between the two, the so-called ultra-low q ones, has been explored, with the aim of studying specific physical issues common to all equilibria, such as, for example, the density limit phenomenon. [...]

Transport studies of pellet fuelling experiments on LHD are reported. Spatio-temporal evolutions after pellet injection into LHD discharges show cases where central density increases on the time scale of particle transport processes. Both the temperature gradient and the density gradient change during the density relaxation, the latter even in sign. The resulting thermodynamic forces influence radial electric fields—both as a driving term but also by, e.g. affecting the Er dependence of ion transport. Magnetic fluctuations have been found to be induced by pellet injection but die out with the relaxation of the pressure profile.

In the complex 3D magnetic fields of stellarators, ion-temperature-gradient turbulence is shown to have two distinct saturation regimes, as revealed by petascale numerical simulations and explained by a simple turbulence theory. The first regime is marked by strong zonal flows and matches previous observations in tokamaks. The newly observed second regime, in contrast, exhibits small-scale quasi-two-dimensional turbulence, negligible zonal flows, and, surprisingly, a weaker heat flux scaling. Our findings suggest that key details of the magnetic geometry control turbulence in stellarators.

Quasi-symmetric configurations have a better neoclassical confinement compared to that of standard stellarators. The reduction of the neoclassical viscosity along the direction of quasi-symmetry should facilitate the self-generation of zonal flows and, consequently, the mitigation of turbulent fluctuations and the ensuing radial transport. Therefore, it is expected that quasi-symmetries should also result in better confinement properties regarding radial turbulent transport. In this paper we show that, at least for quasi-poloidal configurations, the influence of quasi-symmetry on radial transport exceeds the expected reduction of fluctuation levels and associated effective transport coefficients, and that the intimate nature of transport itself is affected. [...]

The effects of a non-axisymmetric (3D) equilibrium magnetic field on the linear ion-temperature-gradient (ITG) driven mode are investigated. We consider the strongly driven, toroidal branch of the instability in a global (on the magnetic surface) setting. Previous studies have focused on particular features of non-axisymmetric systems, such as strong local shear or magnetic ripple, that introduce inhomogeneity in the coordinate along the magnetic field. In contrast, here we include non-axisymmetry explicitly via the dependence of the magnetic drift on the field line label a, i.e., across the magnetic field, but within the magnetic flux surface. [...]

The magnetic surfaces of modern stellarators are characterized by complex, carefully optimized shaping and exhibit locally compressed regions of strong turbulence drive. Massively parallel computer simulations of plasma turbulence reveal, however, that stellarators also possess two intrinsic mechanisms to mitigate the effect of this drive. In the regime where the length scale of the turbulence is very small compared to the equilibrium scale set by the variation of the magnetic field, the strongest fluctuations form narrow bandlike structures on the magnetic surfaces. Thanks to this localization, the average transport through the surface is significantly smaller than that predicted at locations of peak turbulence. [...]

With the advent of neoclassically optimised stellarators, optimising stellarators for turbulent transport is an important next step. The reduction of ion-temperature-gradient-driven turbulence has been achieved via shaping of the magnetic field, and the reduction of trapped-electron mode (TEM) turbulence is addressed in the present paper. Recent analytical and numerical findings suggest TEMs are stabilised when a large fraction of trapped particles experiences favourable bounce-averaged curvature. This is the case for example in Wendelstein 7-X (Beidler et al. 1990 Fusion Technol. 17-148) and other Helias-type stellarators. [...]

Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. [...]

Turbulence induced by the ion temperature gradient (ITG) is investigated in the helical and axisymmetric plasma states of a reversed field pinch device by means of gyrokinetic calculations. The two magnetic configurations are systematically compared, both linearly and nonlinearly, in order to evaluate the impact of the geometry on the instability and its ensuing transport, as well as on the production of zonal flows. Despite its enhanced confinement, the high-current helical state demonstrates a lower ITG stability threshold compared to the axisymmetric state, and ITG turbulence is expected to become an important contributor to the total heat transport.

Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in stellarators than in tokamaks. [...]

The RFX-mod device is operated both as a reversed field pinch (RFP), where advanced regimes featuring helical shape develop, and as a tokamak. Due to its flexibility, RFX-mod is contributing to the solution of key issues in the roadmap to ITER and DEMO, including MHD instability control, internal transport barriers, edge transport and turbulence, isotopic effect, high density limit and three-dimensional (3D) non-linear MHD modelling. This paper reports recent advancements in the understanding of the self-organized helical states, featuring a strong electron transport barrier, in the RFP configuration; the physical mechanism driving the residual transport at the barrier has been investigated. [...]

In order to study and design next-step fusion devices such as DEMO, comprehensive systems codes are commonly employed. In this work HELIAS-specific models are proposed which are designed to be compatible with systems codes. The subsequently developed models include: a geometry model based on Fourier coefficients which can represent the complex 3-D plasma shape, a basic island divertor model which assumes diffusive cross-field transport and high radiation at the X-point, and a coil model which combines scaling aspects based on the HELIAS 5-B reactor design in combination with analytic inductance and field calculations. In addition, stellarator-specific plasma transport is discussed. [...]

In order to study design points of next-step fusion devices such as DEMO, comprehensive systems codes are commonly employed. The code package PROCESS is such a tool, widely used for tokamak systems studies. In this work, the implementation and verification of a HELIAS module into PROCESS is addressed. These HELIAS models include: a plasma geometry model based on Fourier coefficients, a basic island divertor model, as well as a coil model which combines scaling aspects based on the Helias 5-B reactor design in combination with analytic inductance and field calculations. The models are verified firstly with respect to W7-X. [...]

The turbulence observed in the scrape-off-layer of a tokamak is often characterized by intermittent events of bursty nature, a feature which raises concerns about the prediction of heat loads on the physical boundaries of the device. It appears thus necessary to delve into the statistical properties of turbulent physical fields such as density, electrostatic potential, and temperature, focusing on the mathematical expression of tails of the probability distribution functions. The method followed here is to generate statistical information from time-traces of the plasma density stemming from Braginskii-type fluid simulations and check this against a first-principles theoretical model. [...]

Turbulence is widely expected to limit the confinement and, thus, the overall performance of modern neoclassically optimized stellarators. We employ novel petaflop-scale gyrokinetic simulations to predict the distribution of turbulence fluctuations and the related transport scaling on entire stellarator magnetic surfaces and reveal striking differences to tokamaks. Using a stochastic global-search optimization method, we derive the first turbulence-optimized stellarator configuration stemming from an existing quasiomnigenous design.

Recent progress in ‘turbulent optimization’ of toroidal configurations is described, using a method recently developed for evolving such configurations to ones having reduced turbulent transport. The method uses the GENE gyrokinetic code to compute the radial heat flux Qgk, and the STELLOPT optimization code with a theory-based ‘proxy’ figure of merit Qpr to stand in for Qgk for computational speed. Improved expressions for Qpr have been developed, involving further geometric quantities beyond those in the original proxy, which can also be used as ‘control knobs’ to reduce Qgk. [...]

We investigate the linear theory of the ion-temperature-gradient (ITG) mode, with the goal of developing a general understanding that may be applied to stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and kinetic models that follow closely from existing literature are reviewed and two new first-principle models are presented and compared with results from direct numerical simulation. One model investigates the effect of regions of strong localized shear, which are generic to stellarator equilibria. These “shear spikes” are found to have a potentially significant stabilizing affect on the mode; however, the effect is strongest at short wavelengths perpendicular to the magnetic field, and it is found to be significant only for the fastest growing modes in W7-X. [...]

__J.H.E. Proll -- Ernst-Moritz-Arndt-Universität Greifswald (2014) __

In this thesis, collisionless electrostatic microinstabilities are studied both analytically and numerically. Magnetic configurations where the action integral of trapped-particle bounce motion, J, only depends on the radial position in the plasma and where its maximum is in the plasma centre, so-called maximum-J configurations, are of special interest. This condition can be achieved approximately in quasi-isodynamic stellarators, for example Wendelstein 7-X. In such configurations the precessional drift of the trapped particles is in the opposite direction from the direction of propagation of drift waves. Instabilities that are driven by the trapped particles usually rely on a resonance between these two frequencies. Here it is shown analytically by analysing the electrostatic energy transfer between the particles and the instability that, thanks to the absence of the resonance, a particle species draws energy from the mode if the frequency of the mode is well below the charateristic bounce frequency. Due to the low electron mass and the fast bounce motion, electrons are almost always found to be stabilising.

Microinstabilities exhibit a rich variety of behavior in stellarators due to the many degrees of freedom in the magnetic geometry. It has recently been found that certain stellarators (quasi-isodynamic ones with maximum-J geometry) are partly resilient to trapped-particle instabilities, because fast-bouncing particles tend to extract energy from these modes near marginal stability. In reality, stellarators are never perfectly quasi-isodynamic, and the question thus arises whether they still benefit from enhanced stability. Here, the stability properties of Wendelstein 7-X and a more quasi-isodynamic configuration, QIPC, are investigated numerically and compared with the National Compact Stellarator Experiment and the DIII-D tokamak.

The next step in the Wendelstein stellarator line is the large superconducting device Wendelstein 7-X, currently under construction in Greifswald, Germany. Steady-state operation is an intrinsic feature of stellarators, and one key element of the Wendelstein 7-X mission is to demonstrate steady-state operation under plasma conditions relevant for a fusion power plant. Steady-state operation of a fusion device, on the one hand, requires the implementation of special technologies, giving rise to technical challenges during the design, fabrication and assembly of such a device. On the other hand, also the physics development of steady-state operation at high plasma performance poses a challenge and careful preparation. [...]

The gyrokinetic turbulence code GS2 was used to investigate the effects of plasma beta on linear, collisionless ion temperature gradient (ITG) modes and trapped electron modes (TEM) Stellarator Experiment (NCSX) geometry. Plasma beta affects stability in two ways: through the equilibrium and through magnetic fluctuations. The first was studied here by comparing ITG and TEM stability in two NCSX equilibria of differing beta values, revealing that the high beta equilibrium was marginally more stable than the low beta equilibrium in the adiabatic-electron ITG mode case. However, the high beta case had a lower kinetic-electron ITG mode critical gradient. [...]

Recent theoretical findings suggest that in perfectly quasi-isodynamic stellarators, the trapped-particle instability as well as the ordinary electron-density-gradient-driven trapped-electron mode are stable in the electrostatic and collisionless approximation. In these configurations contours of constant magnetic field strength B are poloidally closed, the second adiabatic invariant J is constant on flux surfaces and peaks in the centre. It follows that the diamagnetic drift frequency ω∗a and the bounce-averaged magnetic drift frequency ωda are in opposite directions, ω∗a·ωda<0, everywhere on the flux surface. [...]

An overview is given of physics differences between stellarators and tokamaks, including magnetohydrodynamic equilibrium, stability, fast-ion physics, plasma rotation, neoclassical and turbulent transport and edge physics. Regarding microinstabilities, it is shown that the ordinary, collisionless trapped-electron mode is stable in large parts of parameter space in stellarators that have been designed so that the parallel adiabatic invariant decreases with radius. Also, the first global, electromagnetic, gyrokinetic stability calculations performed for Wendelstein 7-X suggest that kinetic ballooning modes are more stable than in a typical tokamak.

The relation between magnetic geometry and the level of ion-temperature-gradient (ITG) driven turbulence in stellarators is explored through gyrokinetic theory and direct linear and nonlinear simulations. It is found that the ITG radial heat flux is sensitive to details of the magnetic configuration that can be understood in terms of the linear behavior of zonal flows. The results throw light on the question of how the optimization of neoclassical confinement is related to the reduction of turbulence.

The nonlinear gyrokinetic code GS2 has been extended to treat non-axisymmetric stellarator geometry. Electromagnetic perturbations and multiple trapped particle regions are allowed. Here, linear, collisionless, electrostatic simulations of the quasi-axisymmetric, three-field period national compact stellarator experiment (NCSX) design QAS3-C82 have been successfully benchmarked against the eigenvalue code FULL. Quantitatively, the linear stability calculations of GS2 and FULL agree to within 10%.

The linear response of a collisionless stellarator plasma to an applied radial electric field is calculated, both analytically and numerically. Unlike in a tokamak, the electric field and associated zonal flow develop oscillations before settling down to a stationary state, the so-called Rosenbluth–Hinton flow residual. These oscillations are caused by locally trapped particles with radially drifting bounce orbits. The particles also cause a kind of Landau damping of the oscillations that depends on the magnetic configuration. The relative importance of geodesic acoustic modes and zonal-flow oscillations therefore varies among different stellarators.

The linear response of a collisionless stellarator plasma to an applied radial electric field is calculated, both analytically and numerically. Unlike in a tokamak, the electric field and associated zonal flow develop oscillations before settling down to a stationary state, the so-called Rosenbluth–Hinton flow residual. These oscillations are caused by locally trapped particles with radially drifting bounce orbits. The particles also cause a kind of Landau damping of the oscillations that depends on the magnetic configuration. The relative importance of geodesic acoustic modes and zonal-flow oscillations therefore varies among different stellarators.

Up to now, the term ‘‘transport-optimized’’ stellarators has meant optimized to minimize neoclassical transport, while the task of also mitigating turbulent transport, usually the dominant transport channel in such designs, has not been addressed, due to the complexity of plasma turbulence in stellarators. Here, we demonstrate that stellarators can also be designed to mitigate their turbulent transport, by making use of two powerful numerical tools not available until recently, namely, gyrokinetic codes valid for 3D nonlinear simulations and stellarator optimization codes. Two initial proof-of-principle configurations are obtained, reducing the level of ion temperature gradient turbulent transport from the National Compact Stellarator Experiment baseline design by a factor of 2–2.5.

This letter provides a theoretical interpretation of numerically generated probability density functions (PDFs) of intermittent plasma transport events. Specifically, nonlinear gyrokinetic simulations of ion-temperature-gradient turbulence produce the time series of heat flux that manifestly exhibit non-Gaussian PDFs with enhanced tails. It is demonstrated that, after the removal of autocorrelations, the numerical PDFs can be matched with predictions from a fluid theoretical setup based on the instanton method. This result points to a universality in the modeling of intermittent stochastic process offering a predictive capability.

Using the nonlinear gyrokinetic code package GENE/GIST F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers, Phys. Plasmas 7, 1904 2000; P. Xanthopoulos, W. A. Cooper, F. Jenko, Yu. Turkin, A. Runov, and J. Geiger, ibid. 16, 082303 2009, we study the turbulent transport in a broad family of stellarator designs, to understand the geometry dependence of the microturbulence. By using a set of flux tubes on a given flux surface, we construct a picture of the two-dimensional structure of the microturbulence over that surface and relate this to relevant geometric quantities, such as the curvature, local shear, and effective potential in the Schrödinger-like equation governing linear drift modes.

The GENE/GIST code package is developed for the investigation of plasma microturbulence, suitable for both stellarator and tokamak configurations. The geometry module is able to process typical equilibrium files and create the interface for the gyrokinetic solver. The analytical description of the method for constructing the geometric elements is documented, together with several numerical evaluation tests. As a concrete application of this product, a cross-machine comparison of the anomalous ion heat diffusivity is presented.