The nonlinear saturation of toroidal Alfven eigenmode (TAE) due to thermal plasma nonlinearities is investigated using gyrokinetic particle-in-cell simulations and theoretical analysis. In the single toroidal mode number simulations with zonal fields filtered out, we find that the saturation level of TAE is governed by thermal plasma nonlinearities for gamma_L/omega_n > 0.47

Pressure gradient-driven instabilities are investigated in tokamak plasmas using the global gyrokinetic code EUTERPE emphasizing the role of moderate high mode numbers. As the normalized plasma pressure β is increased, there is a well-known, sudden transition from ion-temperature-gradient (ITG) instabilities to kinetic ballooning modes (KBM), if the magnetohydrodynamic (MHD) geometry is held fixed. However, if the equilibrium field is recomputed for each value of β, so that the equilibrium is consistent with the stability calculation, the transition can disappear. In a number of cases, we are only able to find an ITG-KBM transition if inconsistent equilibria are used. In the MHD unstable regime, gyrokinetic simulations and MHD stability calculations show good agreement for moderate ratios of ion temperature gradient to density gradient and small values of the ion gyro-radius. Otherwise, non-MHD contributions are important, such as the diamagnetic stabilization and ion wave–particle resonant effects.

ORB5X is a C++17/Kokkos AI-assisted translation of ORB5, a global electromagnetic gyrokinetic particle-in-cell code for toroidal confined plasmas. The translation preserves the mathematical model and the main numerical algorithms of the Fortran implementation, i.e. a Lagrangian gyrokinetic formulation, marker-particle representation of the distribution functions, finite-element B-spline representation of the fields, Fourier filtering in the angular directions, MPI domain decomposition and cloning, and HDF5 diagnostic output. The main software change is the replacement of Fortran modules and allocatable arrays by typed C++ classes, structs, namespaces, and Kokkos Views and kernels for performance portability across multicore CPU and GPU backends. This paper summarizes the physical models, the numerical scheme, the distributed and shared-memory parallel design, the AI-assisted code translation workflow, and the validation path used to compare ORB5X against the original Fortran reference codebase. The initial tests for ITG, ITPA, and chirping show an agreement of almost machine accuracy between ORB5X and ORB5 for the electrostatic and electromagnetic field components. Furthermore, we demonstrate that with the modern CMake and Kokkos, ORB5X can be easily compiled and efficiently run on personal Linux and Mac laptops, as well as on high-performance computing clusters such as LUMI-G, Daint-ALPS, CINECA, and Discoverer.

A new field solver has been implemented in the global electromagnetic total-f gyrokinetic particle-in-cell code XGC to extend the code's capability to large-scale reduced MHD-type instabilities in tokamak plasma. While XGC's regular field solver is accurate at typical microturbulence scales of the order of the ion Larmor radius in tokamaks with arbitrary aspect ratio, a more accurate field solver is required for large-scale (i.e., low toroidal mode number) MHD-type modes such as internal kink, tearing and peeling modes. The higher accuracy of the new field solver is achieved by dropping the (large aspect ratio) assumption that the poloidal magnetic field is much smaller than the toroidal magnetic field, while its numerical complexity is controlled by using a spectral discretization in the toroidal direction. To cover the entire spectrum from large-scale MHD-type modes to small-scale microturbulence, the regular and the new field solver can be run alongside each other. This work details the derivation of the new field solver, analyzes the differences between the XGC's regular and new field solvers, and verifies the new field solver against analytic predictions and the gyrokinetic code ORB5 and the MHD code NIMROD.

The interaction between energetic particles (EPs) and ion temperature gradient (ITG) modes is studied using the global particle in cell ORB5 code. In this work, we extend previous studies to a broader range of EP temperatures, including the burning plasma regime and to wider variety of EP distribution functions. Two main stabilization mechanisms are found to be effective in ITG stabilization confirming previous studies: direct dispersion relation modification (DDRM) effective only at intermediate EP temperatures and dilution effect (DE) which is independent of EP temperature and becomes dominant in burning plasma regime (T_f > 50T_i). The study is further extended to slowing-down EP distributions which in contrast exhibit no DDRM-related stabilization. The findings are further validated in an ITER pre-fusion operation scenario and additionally compared with electromagnetic effects. In this scenario EP stabilization is found to be weaker than β-stabilization. Overall, these results provide better understanding of EP-ITG interactions over a wider range of EP parameters relevant to burning plasma regime which is important for predicting turbulence and confinement in future devices such as ITER.

This paper presents a novel approach for simulating plasma instabilities in tokamak plasmas using the piecewise field-aligned finite element method in combination with the particle-in-cell method. Our method traditionally aligns the computational grid, but defines the basis functions in piecewise field-aligned coordinates to avoid grid deformation while naturally representing the field-aligned mode structures. This scheme is formulated and implemented numerically. It also applies to the unstructured triangular meshes in principle. We have conducted linear benchmark tests, which agree well with previous results and traditional schemes. Furthermore, multiple- $n$ simulations are also carried out as a proof of principle, demonstrating the efficiency of this scheme in nonlinear turbulence simulations within the framework of the finite element method.

In this work, the non-ideal Ohm's law is integrated in the mixed variable-pullback scheme for the gyrokinetic particle simulations. This scheme captures the evolution of the symplectic solution of the gyrokinetic model accurately not only in the MHD limit but also in the electrostatic limit. This scheme also provides a pure symplectic (v_) scheme for electromagnetic gyrokinetic particle simulations without causing the traditional cancellation problem in the pure Hamiltonian scheme. Various mixed variable schemes have been comprehensively analyzed for the 1D shear Alfvèn wave problem with kinetic electrons, with the connection to the traditional pure Hamiltonian scheme and the symplectic scheme. It is demonstrated that the pure v_ form with the non-ideal Ohm's law has comparable performance to the widely used mixed variable-pullback scheme with the ideal Ohm's law. The mixed variable-pullback scheme without Ohm's law is also proposed as a feasible improvement of the traditional pure Hamiltonian scheme with minimum modification and considerable performance improvement in terms of a marker number reduction.

We report a universal mechanism for turbulence-driven magnetic islands in fusion plasmas. Using gyrokinetic simulations of a linearly stable tearing mode in a large-aspect-ratio toroidal geometry under collisionless conditions, we demonstrate that micro-instabilities generate an E×B flow that drives magnetic field line reconnection. This process forms multiple small-scale islands along the resonant surface, which interact nonlinearly and eventually coalesce into large-scale magnetic islands. These islands are capable of significantly flattening the equilibrium profile across the island O-points and thus acting as a seed for their further neoclassical growth. Notably, the mechanism operates independently of the parity of the destabilizing micro-instability.

The triangular mesh-based gyrokinetic scheme enables comprehensive axis-to-edge studies across the entire plasma volume. Our approach employs triangular finite elements with first-derivative continuity (C1), building on previous work to facilitate gyrokinetic simulations. Additionally, we have adopted the mixed variable/pullback scheme for gyrokinetic electromagnetic particle simulations. The filter-free treatment in the poloidal cross-section with triangular meshes introduces unique features and challenges compared to previous treatments using structured meshes. Our implementation has been validated through benchmarks using ITPA-toroidicity-induced Alfv & eacute;n eigenmode parameters, showing its capability in moderate to small electron skin depth regimes. Additional examinations using experimental parameters confirm its applicability to realistic plasma conditions.

In this Letter, we demonstrate by comparison of computer simulations and experimental results that excitation of turbulence-driven electromagnetic secondary instabilities can serve as a paradigm to explain the observed activity of otherwise stable modes in fusion devices. In particular, we discuss the excitation of Alfvén eigenmodes in electron cyclotron heated plasmas in the Wendelstein 7-X (W7-X) stellarator. In more than 700 experimental programs of W7-X Operational Phase 1.2b (from which 25 were selected and discussed in this paper), poloidal magnetic fluctuation measurements with a clear Alfvénic nature are seen to correlate with turbulence density fluctuations. In response, we perform nonlinear, global, gyrokinetic, electromagnetic simulations using profiles consistent with the experimental conditions. The results agree remarkably well with the measurements and reveal excitation of zonal flow activity and long wavelength Alfvénic modes by ion temperature gradient turbulence. Integer ratios between the primary and secondary growth rates suggest an excitation mechanism similar to the forced-driven excitation of zonal flow by Alfvén eigenmodes.

In this work, we present a systematic analysis of the mixed variable-pullback scheme and the pure symplectic v parallel to scheme used in electromagnetic gyrokinetic particle simulations. The non-ideal Ohm's law is integrated in the mixed variable-pullback scheme for the gyrokinetic particle simulations. This scheme captures the evolution of the symplectic solution of the gyrokinetic model accurately not only in the MHD limit but also in the electrostatic limit. This scheme also provides a pure symplectic ( v parallel to) scheme for electromagnetic gyrokinetic particle simulations without causing the traditional cancelation problem in the pure Hamiltonian scheme. Various mixed variable schemes have been comprehensively analyzed for the 1D shear Alfv & egrave;n wave problem with kinetic electrons, with the connection to the traditional pure Hamiltonian scheme and the symplectic scheme. It is demonstrated that the pure v parallel to form with the non-ideal Ohm's law has comparable performance to the widely used mixed variable-pullback scheme with the ideal Ohm's law. The mixed variable-pullback scheme without Ohm's law is also proposed as a feasible improvement of the traditional pure Hamiltonian scheme with minimum modification and considerable performance improvement in terms of a marker number reduction.

After a long device enhancement phase, scientific operation resumed in 2022. The main new device components are the water cooling of all plasma facing components and the new water-cooled high heat flux divertor units. Water cooling allowed for the first long-pulse operation campaign. A maximum discharge length of 8 min was achieved with a total heating energy of 1.3 GJ. Safe divertor operation was demonstrated in attached and detached mode. Stable detachment is readily achieved in some magnetic configurations but requires impurity seeding in configurations with small magnetic pitch angle within the edge islands. Progress was made in the characterization of transport mechanisms across edge magnetic islands: Measurement of the potential distribution and flow pattern reveals that the islands are associated with a strong poloidal drift, which leads to rapid convection of energy and particles from the last closed flux surface into the scrape-off layer. Using the upgraded plasma heating systems, advanced heating scenarios were developed, which provide improved energy confinement comparable to the scenario, in which the record triple product for stellarators was achieved in the previous operation campaign. However, a magnetic configuration-dependent critical heating power limit of the electron cyclotron resonance heating was observed. Exceeding the respective power limit leads to a degradation of the confinement.

In this work, we use the global electromagnetic and electrostatic gyro kinetic approaches to investigate the effects of zonal flows forced-driven by Alfvén modes due to their excitation by energetic particles (EPs), on the dynamics of ITG (Ion temperature gradient) instabilities. The equilibrium of the 92416 JET tokamak shot is considered. The linear and nonlinear Alfvén modes dynamics, as well as the zonal flow dynamics, are investigated and their respective radial structures and saturation levels are reported. ITG dynamics in the presence of the zonal flows excited by these Alfvén modes are also investigated. We find that, the zonal flows forced-driven by Alfvén modes can significantly impact the ITG dynamics. A zonal flow amplitude scan reveals the existence of an inverse relation between the zonal flow amplitude and the ITG growth rate. These results show that, forced-driven zonal flows can be an important indirect part of turbulence mitigation due to the injection of energetic particles.

The current state of the EUTERPE code is described with emphasis on the implemented models and their numerical implementation. The code solves the multi-species electromagnetic gyrokinetic equations in the full volume of a three-dimensional domain. Noise reduction of the particle-in-cell method is achieved by using a δf-method and Fourier filters. The field equations are discretized with B-splines and the resulting system of equations is solved iteratively. For linear simulations a phase-factor transformation is applied in order to strongly reduce the necessary grid resolution. Apart from the full gyrokinetic model, other numerically less expensive hybrid models are also implemented. They are mainly tailored for comparison with fluid theory and for studying the interaction of the bulk plasma with fast particles. The code is parallelized for CPUs by particle and domain decomposition. Good scalability up to several thousand nodes is demonstrated.

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.

We investigate tearing modes (TM) driven by current density gradient in collisionless tokamak plasmas by using the electromagnetic gyrokinetic simulation code ORB5. We elucidate the TM width by simulations for flat profiles, as the absence of background diamagnetic flows implies a small rotation speed, while finite gradients are included to investigate the TM rotation. For flat profiles, the initial saturation width of nonlinearly driven magnetic islands is related to the TM linear growth rate; however, large islands in the initial saturation phase are prone to current density redistribution that reduces the island width in the following evolution. Island-induced ExB and diamagnetic sheared flows develop at the separatrix, able to destabilize the Kelvin-Helmholtz instability (KHI). The KHI turbulence enhances a strong quadrupole vortex flow that reinforces the island decay, resulting in a strong reduction of the island width in an eventual steady state. This process is enhanced by trapped electrons. For finite gradients profile, the TM usually rotates in the electron diamagnetic direction but can change direction when the ion temperature gradient dominates the other gradients. The reduced growth of the TM by diamagnetic effects results in a moderate island size, which remains almost unchanged after the initial saturation. At steady state, strong zonal flows are nonlinearly excited and dominate the island rotation, as expected from previous theoretical and numerical studies. When beta is increased, the TM mode is suppressed and a mode with the same helicity but with twisting parity, coupled with the neighboring poloidal harmonics, is destabilized, similar to the kinetic ballooning mode.

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.

We propose a theoretical scheme for the study of the nonlinear interaction of drift-wave-like turbulence and energetic particles in stellarators. The approach is based on gyrokinetics, and features a separation of time and scales, for electromagnetic fluctuations, inspired by linear ballooning theory. Two specific moments of the gyrokinetic equation constitute the main equations of the system, which requires a full kinetic nonlinear solution. This is found iteratively, expanding in the smallness of the bounce-average radial drift frequency, and nonlinear $\boldsymbol {E}\times \boldsymbol {B}$ drift frequency, compared with the inverse time scales of the resonantly interacting energetic particles. Our analysis is therefore valid for neoclassically optimised stellators. The resummation of all iterative and perturbative nonlinear kinetic solutions is discussed in terms of Feynman diagrams. Particular emphasis is put on the role of collisionlessly undamped large-scale structures in phase space, the kinetic equivalent of zonal flows, i.e. phase-space zonal structures, and on wave-like fluctuations generated by energetic particles.

In this paper, we summarize our recent work on gyrokinetic applications in electron–positron and non-neutral plasma. The electrostatic stability of electron–positron plasmas was investigated in dipole and slab geometries, with and without ion admixture. The gyrokinetic dispersion relation was derived and, for the slab case, extended to non-neutral plasmas. Here, we further extend the gyrokinetic formulation to the relativistic regime. Electron–positron plasmas are found to be remarkably stable as long as perfect symmetry between the two species prevails, but instabilities appear if this symmetry is broken, for instance by the introduction of impurities or magnetic curvature.

Frequency chirping of Alfvén modes, a phenomenon observed in tokamak fusion plasmas driven by energetic particles (EPs), can result in significant losses of EPs. In this study, we use the global gyrokinetic code ORB5 (Lanti et al 2020 Comput. Phys. Commun. 251 107072) to investigate the nonlinear dynamics of non-adiabatic frequency chirping EP modes (EPMs). Our results illuminate non-perturbative features of EPMs caused by the presence of EPs. Additionally, we find that, with a fixed safety factor profile and a single toroidal mode number, the frequency chirping rate is linearly proportional to the mode saturation amplitude, as predicted by the theory (Chen and Zonca 2016 Rev. Mod. Phys. 88 015008).