Gyrokinetic Turbulence Research Articles

High-Frequency Nongyrokinetic Turbulence at Tokamak Edge Parameters

First of a kind 6D-Vlasov computer simulations of high frequency ion Bernstein wave turbulence for parameters relevant to the tokamak edge show transport comparable to sub-Larmor-frequency gyrokinetic turbulence. The customary restriction of magnetized plasma turbulence studies to the gyrokinetic approximation may not be based on physics but only on a practical constraint due to computational cost. Deciphering turbulent transport is crucial since edge turbulence significantly influences the confinement properties of magnetically confined plasmas. Despite the high computational costs, performing 6D kinetic simulations is essential for understanding the limitations of our current models. Published by the American Physical Society 2024

Impurity convection reversal caused by edge localized turbulence for generating a stationary edge radiative layer in the HL-2A tokamak

Abstract Effect of impurity on edge turbulence was investigated on the HL-2A tokamak by using Neon supersonic molecular beam injection (SMBI). For the first time, a stationary radiative layer accompanied by edge localized turbulence lasting more than 100ms is observed at the edge region ρ=0.7-0.9. This duration is much longer than the SMBI pulse length (1.2ms), indicating that the impurity diffusion is on the whole cancelled out by the impurity convection without spreading of the localized turbulence during this period. This implies that the impurity convection is reversed from positive to negative impurity density gradient region. This reversal could be explained by the parallel impurity compression mechanism. The observed turbulence is a broadband electrostatic turbulence (25-70 kHz) propagating in the electron diamagnetic drift direction with normalized poloidal wavenumber of 0.1≤ kθρs≤0.4. It has been found that the excitation of the edge localized turbulence strongly depends on the local collisionality. This feature is qualitatively predicted by the spectral multiscale gyrokinetic turbulence simulation, which suggests that the turbulence could be a dissipative trapped electron mode (DTEM). This experiment demonstrates that mutual interaction between externally injected impurity ions and edge turbulence can generate a stationary edge radiative layer and avoid impurity core accumulation. These results could provide a potential approach for the formation of a stable impurity radiative layer by impurity injection in tokamaks.

GX: a GPU-native gyrokinetic turbulence code for tokamak and stellarator design

GX is a code designed to solve the nonlinear gyrokinetic system for low-frequency turbulence in magnetized plasmas, particularly tokamaks and stellarators. In GX, our primary motivation and target is a fast gyrokinetic solver that can be used for fusion reactor design and optimization along with wide-ranging physics exploration. This has led to several code and algorithm design decisions, specifically chosen to prioritize time to solution. First, we have used a discretization algorithm that is pseudospectral in the entire phase space, including a Laguerre–Hermite pseudospectral formulation of velocity space, which allows for smooth interpolation between coarse gyrofluid-like resolutions and finer conventional gyrokinetic resolutions and efficient evaluation of a model collision operator. Additionally, we have built GX to natively target graphics processors (GPUs), which are among the fastest computational platforms available today. Finally, we have taken advantage of the reactor-relevant limit of small $\rho _*$ by using the radially local flux-tube approach. In this paper we present details about the gyrokinetic system and the numerical algorithms used in GX to solve the system. We then present several numerical benchmarks against established gyrokinetic codes in both tokamak and stellarator magnetic geometries to verify that GX correctly simulates gyrokinetic turbulence in the small $\rho _*$ limit. Moreover, we show that the convergence properties of the Laguerre–Hermite spectral velocity formulation are quite favourable for nonlinear problems of interest. Coupled with GPU acceleration, which we also investigate with scaling studies, this enables GX to be able to produce useful turbulence simulations in minutes on one (or a few) GPUs and higher fidelity results in a few hours using several GPUs. GX is open-source software that is ready for fusion reactor design studies.

Ion-temperature- and density-gradient-driven instabilities and turbulence in Wendelstein 7-X close to the stability threshold

Electrostatic gyrokinetic instabilities and turbulence in the Wendelstein 7-X stellarator are studied. Particular attention is paid to the ion-temperature-gradient (ITG) instability and its character close to marginal stability [Floquet-type turbulence (Zocco et al., Phys. Rev. E, vol. 106, 2022, p. L013202) with no electron temperature gradient]. The flux tube version of the $\delta f$ code stella (Barnes et al., J. Comput. Phys., vol. 391, 2019, pp. 365–380) is used to run linear and nonlinear gyrokinetic simulations with kinetic electrons. The nature of the dominant instability depends on the wavelength perpendicular to the magnetic field, and the results are conveniently displayed in stability diagrams that take this dependence into account. This approach highlights the presence of universal instabilities, which are less unstable but have longer wavelengths than other modes. A quasi-linear estimate of the heat flux suggests they are relevant for transport. Close to the stability threshold, the linear eigenmodes and turbulence form highly extended structures along the computational domain if the magnetic shear is small. Numerical experiments and diagnostics are undertaken to assess the resulting radial localisation of the turbulence, which affects the interaction of the latter with zonal flows. Increasing the amplitude of the magnetic shear (e.g.through current drive) has a stabilising effect on the turbulence and, thus, reduces the nonlinear energy transport.

Numerical dissipation induced by the low-pass filtering in nonlinear gyrokinetic simulations

De-aliasing is an essential procedure for eliminating the aliasing error in nonlinear simulations, such as nonlinear gyrokinetic turbulence simulations. An ideal approach to de-aliasing in the periodic dimension is the Fourier truncation. Finite difference low-pass filtering applied in the non-periodic direction strongly dampens aliasing modes. At the same time, it induces numerical dissipation in the region of the physically realistic solution. It is shown analytically that the long-wave dissipation coefficient is proportional to the (Np−3) power of the wavenumber under desirable constraints satisfying the highest order of accuracy, where Np is the number of filter points. Numerical results after applying the optimized low-pass filtering to the nonlinear gyrokinetic turbulence simulation suggest that the nine-point format preserves intact mesoscopic zonal structures in tokamak plasma, and is therefore suitable for long-time nonlinear turbulence simulations.

Core-edge modeling of gyrokinetic turbulence by coupling the delta-f and total-f models in the XGC code

Whole device gyrokinetic turbulence is modeled using a delta-f model in the core coupled to a total-f model in the edge. This new model is implemented in the gyrokinetic code X-Point Included Gyrokinetic Code (XGC) and works within a single simulation. It employs a canonical Maxwellian background in the core and the usual local Maxwellian in the edge. These different core and edge backgrounds are coupled over an overlap region. A thermal-bath operator is used to keep δf small in the core.

Self-Organized Evolution of the Internal Transport Barrier in Ion-Temperature-Gradient Driven Gyrokinetic Turbulence.

Understanding the self-organization of the most promising internal transport barrier in fusion plasmas needs a long-time nonlinear gyrokinetic global simulation. The neighboring equilibrium update method is proposed, which solves the secularity problem in a perturbative simulation and speeds up the numerical computation by more than 10 times. It is found that the internal transport barrier emerges at the magnetic axis due to inward propagated turbulence avalanche, and its outward expansion is the catastrophe of self-organized structure induced by outward propagated avalanche.

Gyrokinetic simulation of magnetic-island-induced electric potential vortex mode

Ion temperature gradient (ITG)-driven turbulence with embedded static magnetic islands is simulated by utilizing a gyrokinetic theory-based global turbulence transport code (GKNET) in this work. Different from the traditional equilibrium circular magnetic-surface average (EMSA) method, an advanced algorithm that calculates the perturbed magnetic-surface average (PMSA) of the electric potential has been developed to precisely deal with the zonal flow component in a non-circular magnetic surface perturbed by magnetic islands. Simulations show that the electric potential vortex structure inside islands induced by the magnetic islands is usually of odd parity when using the EMSA method. It is found that the odd symmetry vortex can transfer into an even one after a steep zonal flow gradient, i.e. the flow shear has been built in the vicinity of the magnetic islands by adopting the PMSA algorithm. The phase of the potential vortex in the poloidal cross section is coupled with the zonal flow shear. Such an electric potential vortex mode may be of essential importance in wide topics, such as the turbulence spreading across magnetic islands, neoclassical tearing mode physics, and also the interaction dynamics between the micro-turbulence and MHD activities.

Gyrokinetic turbulence modeling of a high performance scenario in JT-60SA

Local gyrokinetic simulations are used to model turbulent transport for the first time in a representative high-performance plasma discharge projected for the new JT-60SA tokamak. The discharge features a double-null separatrix, 41 MW of combined neutral beam heating and electron cyclotron heating, and a high predicted ratio of the normalized plasma kinetic to magnetic pressure β. When considering input parameters computed from reduced transport models, gyrokinetic simulations predict a turbulent heat flux well below the injected 41 MW. Increasing the background gradients, on the other hand, can trigger a non-zonal transition (NZT), causing heat fluxes to no longer saturate. Furthermore, when considering fast ions in the simulations, a high-frequency mode is destabilized that substantially impacts the turbulence. The NZT is avoided by reducing the electron pressure by 10% below its nominal value, and the fast-ion resonance is removed by reducing the fast-ion temperature. The thus-obtained simulation features broadband frequency spectra and density and temperature fluctuation levels δn/n≈1% –2%, δT/T≈1% –6% that should be measurable with fluctuation diagnostics planned for JT-60SA. The temperature profile is fixed by the critical main-ion temperature gradient as a consequence of the high stiffness; heat fluxes increase by a factor of ten when increasing the main ion temperature gradient by 17%. Despite large gradients, it is demonstrated that, due to the large β, retaining compressional magnetic field fluctuations and in particular, the contribution of the pressure gradient in the ∇B drifts, is crucial to achieving non-zero heat fluxes.

Trapped-particle precession and modes in quasisymmetric stellarators and tokamaks: a near-axis perspective

This paper presents the calculation of the bounce-averaged drift of trapped particles in a near-axis framework for axisymmetric and quasisymmetric magnetic fields that possess up-down and stellarator symmetry, respectively. This analytic consideration provides important insight on the dependence of the bounce-averaged drift on the geometry and stability properties of the field. In particular, we show that although the maximum- $\mathcal {J}$ property is unattainable in quasisymmetric stellarators, one may approach it through increased plasma $\beta$ and triangular shaping, albeit going through a reduced precession scenario with potentially higher particle losses. The description of trapped particles allows us to calculate the available energy of trapped electrons analytically in two asymptotic regimes, providing insight into the behaviour of this measure of turbulence. It is shown that the available energy is intimately related to magnetohydrodynamics (MHD) stability, providing a potential synergy between this measure of gyrokinetic turbulence and MHD stability.

Characterizing the velocity-space signature of electron Landau damping

Plasma turbulence plays a critical role in the transport of energy from large-scale magnetic fields and plasma flows to small scales, where the dissipated turbulent energy ultimately leads to heating of the plasma species. A major goal of the broader heliophysics community is to identify the physical mechanisms responsible for the dissipation of the turbulence and to quantify the consequent rate of plasma heating. One of the mechanisms proposed to damp turbulent fluctuations in weakly collisional space and astrophysical plasmas is electron Landau damping. The velocity-space signature of electron energization by Landau damping can be identified using the recently developed field–particle correlation technique. Here, we perform a suite of gyrokinetic turbulence simulations with ion plasma beta values$\beta _i = 0.01, 0.1, 1$and$10$and use the field–particle correlation technique to characterize the features of the velocity-space signatures of electron Landau damping in turbulent plasma conditions consistent with those observed in the solar wind and planetary magnetospheres. We identify the key features of the velocity-space signatures of electron Landau damping as a function of varying plasma$\beta _i$to provide a critical framework for interpreting the results of field–particle correlation analysis ofin situspacecraft observations of plasma turbulence.

Dynamic edge transport investigations at ASDEX Upgrade using gas puff modulation

Gas puff modulation experiments are performed at ASDEX Upgrade in L-mode, EDA H-mode and quasi-continuous exhaust discharges. Plasma density and temperatures are measured and their temporal development is analyzed simultaneously, revealing that both heat and particle transport are strongly influenced by the modulation. As a consequence, the particle transport coefficients are underdetermined. In the transport modelling, the pedestal cannot be treated as a single region, but the pedestal foot must be allowed to increase its transport with gas puff modulation independently. The analysis of the temporal behaviours of the heat and particle diffusivities shows that they are strongly correlated. Considering the heat diffusivity as a proxy for the particle diffusivity, allows interpretation of the density evolution: a pinch is not required for any of the discharges. An analysis with the gyrokinetic turbulence code GENE identifies dominant instabilities and reproduces several experimentally found trends. Despite all uncertainties concerning particle transport, one can expect a future reactor featuring a weak edge density gradient even with purely diffusive transport.

Statistical Analysis of Plasma Dynamics in Gyrokinetic Simulations of Stellarator Turbulence.

A geometrical method for assessing stochastic processes in plasma turbulence is investigated in this study. The thermodynamic length methodology allows using a Riemannian metric on the phase space; thus, distances between thermodynamic states can be computed. It constitutes a geometric methodology to understand stochastic processes involved in, e.g., order-disorder transitions, where a sudden increase in distance is expected. We consider gyrokinetic simulations of ion-temperature-gradient (ITG)-mode-driven turbulence in the core region of the stellarator W7-X with realistic quasi-isodynamic topologies. In gyrokinetic plasma turbulence simulations, avalanches, e.g., of heat and particles, are often found, and in this work, a novel method for detection is investigated. This new method combines the singular spectrum analysis algorithm with a hierarchical clustering method such that the time series is decomposed into two parts: useful physical information and noise. The informative component of the time series is used for the calculation of the Hurst exponent, the information length, and the dynamic time. Based on these measures, the physical properties of the time series are revealed.

Influence of collisions on the validation of global gyrokinetic simulations in the edge and scrape-off layer of TCV

Understanding and predicting turbulent transport in the edge and scrape-off-layer (SOL) of magnetic confinement fusion devices is crucial for developing feasible fusion power plants. In this work, we present the latest improvements to the gyrokinetic turbulence code GENE-X and validate the extended model against experimental results in the TCV tokamak (“TCV-X21”). GENE-X features a full-f electromagnetic gyrokinetic model and is specifically targeted for edge and SOL simulations in diverted geometries. GENE-X can model the effect of collisions using either a basic Bhatnagar–Gross–Krook (BGK) or more sophisticated Lenard–Bernstein/Dougherty (LBD) collision operator. We present the results of a series of GENE-X simulations using the BGK or LBD collision models, contrasting them to collisionless simulations. We validate the resulting plasma profiles, power balance, and SOL heat flux against experimental measurements. The match to the experiment significantly improves with the fidelity of the collision model chosen. We analyze the characteristics of the turbulence and find that in almost all cases in the confined region the turbulence is driven by trapped electron modes (TEM). Both the simulations without collisions and those with the BGK collision operator do not accurately describe turbulence driven by TEMs. The more sophisticated LBD collision operator presents a minimum requirement for accurate gyrokinetic edge turbulence simulations.

Quantifying the role of higher order neoclassical corrections to gyrokinetics in tokamak plasmas

We implement the higher order gyrokinetic theory developed in Dudkovskaia et al (2023 Plasma Phys. Control. Fusion 65 045010), reduced to the limit of , where B 0 is the tokamak equilibrium magnetic field, and B ϑ is its poloidal component, in the local gyrokinetic turbulence code, GS2. The principal motivation for this extension is to quantify the importance of neoclassical flows in electromagnetic gyrokinetics, with a particular interest in sharp pressure gradient regions where the bootstrap current becomes dominant. To incorporate neoclassical equilibrium physics, GS2 is coupled to NEO, a multi-species drift kinetic solver. It is found that the regions where microinstabilities are most likely to be influenced by neoclassical equilibrium effects are in a pedestal plasma and a spherical tokamak core plasma.

First-principles based plasma profile predictions for optimized stellarators

In the present Letter, first-of-its-kind computer simulations predicting plasma profiles for modern optimized stellarators—while self-consistently retaining neoclassical transport, turbulent transport with 3D effects, and external physical sources—are presented. These simulations exploit a newly developed coupling framework involving the global gyrokinetic turbulence code GENE-3D, the neoclassical transport code KNOSOS, and the 1D transport solver TANGO. This framework is used to analyze the recently observed degradation of energy confinement in electron-heated plasmas in the Wendelstein 7-X stellarator, where the central ion temperature was ‘clamped’ to keV regardless of the external heating power. By performing first-principles based simulations, we provide key evidence to understand this effect, namely the inefficient thermal coupling between electrons and ions in a turbulence-dominated regime, which is exacerbated by the large ratios, and show that a more efficient ion heat source, such as direct ion heating, will increase the on-axis ion temperature. This work paves the way towards the use of high-fidelity models for the development of the next generation of stellarators, in which neoclassical and turbulent transport are optimized simultaneously.

Existence of finite and anisotropic heavy ion parallel compressibility pinch in the gyrokinetic turbulence

Turbulence-driven heavy ion transport in hot magnetized plasma is investigated by means of the gyrokinetic theory and simulations. A finite heavy ion parallel compressibility pinch (${\mathrm{\ensuremath{\Gamma}}}_{s,\ensuremath{\parallel}}$) is found in the gyrokinetic framework, in contrast to the conventional understanding that ${\mathrm{\ensuremath{\Gamma}}}_{s,\ensuremath{\parallel}}$ is negligible. A perturbation theory clarifies the turbulence frequency dependence of ${\mathrm{\ensuremath{\Gamma}}}_{s,\ensuremath{\parallel}}$, resolving the discrepancy with experimental observations. It is also predicted by a nonlocal approach of the parallel advection term that ${\mathrm{\ensuremath{\Gamma}}}_{s,\ensuremath{\parallel}}$ is strongly anisotropic on a magnetic surface. The parameter dependence shows that decreasing the heavy ion mass ${m}_{s}$ strongly enhances ${\mathrm{\ensuremath{\Gamma}}}_{s,\ensuremath{\parallel}}$ through kinetic effects, leading to a deviation from the $1/{m}_{s}$ scaling. Moreover, the pinch direction can be reversed in nonlinear trapped electron mode turbulence through the inverse cascade.

Gyrokinetic simulations of turbulence and zonal flows driven by steep profile gradients using a delta-f approach with an evolving background Maxwellian

Long global gyrokinetic turbulence simulations are particularly challenging in situations where the system deviates strongly from its initial state and when fluctuation levels are high, for example, in strong gradient regions. For particle-in-cell simulations, statistical sampling noise accumulation from large marker weights due to large deviations from the control variate of a delta-f scheme makes such simulations often impractical. An adaptive control variate in the form of a flux-surface-averaged Maxwellian with a time-dependent temperature profile is introduced in an attempt to alleviate the former problem. Under simplified collisionless physics, this adaptive delta-f scheme is shown to reduce noise accumulation in the zonal flows and the simulated heat flux in a quasi-steady turbulent state. The method also avoids the collapse of the signal-to-noise ratio, which occurs in the standard non-adaptive scheme, and, therefore, allows one to reach numerically converged results even with lower marker numbers.

Gyrokinetic electrostatic turbulence close to marginality in the Wendelstein 7-X stellarator.

The transition from strong (fluidlike) to nearly marginal (Floquet-type) regimes of ion-temperature-gradient (ITG) driven turbulence is studied in the stellarator Wendelstein 7-X by means of numerical simulations. Close to marginality, extended (along magnetic field lines) linearly unstable modes are dominant, even in the presence of kinetic electrons, and provide a drive that results in finite turbulent transport. A total suppression of turbulence above the linear stability threshold of the ITG modes, commonly present in tokamaks and known as the "Dimits shift," is not observed. We show that this is mostly due to the peculiar radial structure of marginal turbulence, which is more localized than in the fluid case and therefore less likely to be stabilized by shearing flows.

Phase synchronization versus modulational instability for zonal flow generation and pattern formation

From global gyrokinetic simulations of toroidal ion temperature gradient-driven (ITG) turbulence, we identify two distinguished regimes where zonal flow generation and its radial pattern formation is governed by either phases or amplitudes of the turbulence spectrum. When the unstable region is wider than the correlation length of ITG modes, the zonal flow structure changes in a turbulence time scale. We newly uncover that the radial phases of ITG modes determine the evolution of the zonal flow structure. Synchronization of mode phases induced by the global zonal flow drives a fine-scale zonal flow pattern. With a narrow unstable region comparable to the mode correlation length, the phase effect almost vanishes. In this regime, we recover that the modulational instability promptly amplifies a coherent zonal flow, leading to a lower turbulence saturation level. This finding explains the reduction of turbulent transport at a narrow width of the strong gradient region, which has been attributed to the system size effects in ion-scale gyrokinetic turbulence.