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

Adopting dedicated phase-space diagnostics, we analyze the nonlinear dynamic evolution of an energetic particle mode (EPM) due to nonlinear wave–particle interactions, including nonlinear frequency chirping and secular particle motion due to trapping and de-trapping. By tracking representative particle orbits, we show that the nonlinear evolution is dominated by continuous turnover of the distinct resonant populations via self-consistent trapping and de-trapping, rather than by adiabatic frequency sweeping of a fixed cohort. The resulting clump motion in phase space follows the instantaneous low-frequency resonance contour, leading to pronounced downward frequency chirping. The measured EPM frequency chirping rate scales linearly with the mode amplitude, as predicted by general theory.

We have derived analytically, via variational-WKB analyses, the radiative damping rate of reversed shear Alfv & eacute;n eigenmode (RSAE) in tokamak plasmas. Here, the radiative damping occurs via "tunnelling" coupling to radially propagating convectively damped kinetic Alfv & eacute;n waves. Our analytical result is found to be consistent with one expression conjectured based on simulations and, thereby, resolves ambiguities among the various conjectured expressions. Furthermore, it is found that the radiative damping rate of toroidal Alfv & eacute;n eigenmode (TAE) could, typically, be significantly larger than that of RSAEs, implying, thus, energetic particles should excite RSAEs more readily than TAEs in tokamak fusion plasmas.

Toroidal Alfv & eacute;n Eigenmodes (TAE) are global MHD instabilities that can be excited in fusion devices by energetic particles, such as those produced by neutral beam injection or fusion reactions. Their presence can significantly enhance the radial transport of fast ions, leading to increased particle losses. Understanding and controlling TAE-driven transport is crucial for optimizing confinement in future experiments like ITER and the Divertor Tokamak Test (DTT), where minimizing EP losses is essential for sustaining efficient plasma heating and stability. In this paper, we investigate the interaction between 510 keV fast ions injected by the neutral beam injector system in DTT and TAE modes through numerical simulations using the Hamiltonian guiding-center code ORBIT. Additionally, we estimate the heat flux at the separatrix by mapping the deposited power by lost energetic ions.

Energetic particle (EP)-driven Alfv & eacute;nic fluctuations pose a significant challenge to plasma confinement and stability in future reactor relevant tokamak plasmas like Divertor Tokamak Test (DTT) facility. While previous studies often relied on isotropic EP distributions, this work presents a novel investigation into the stability of toroidal Alfv & eacute;n eigenmodes (TAEs) in DTT, focusing on anisotropic EP distribution functions characteristic of negative neutral beam injection (NNBI) and ion cyclotron resonance heating (ICRH) schemes. Utilizing the local gyrokinetic code DAS (Drift Alfv & eacute;n Stability), which employs a ballooning-mode representation and incorporates realistic magnetic equilibria, we model NNBI-driven EPs with an anisotropic slowing-down distribution and ICRH-driven EPs with an anisotropic gamma distribution. Our analysis reveals the distinct TAE stability characteristics for NNBI and ICRH scenarios, leading to different optimal toroidal mode numbers and growth rate dependencies. Parameter scans for ICRH demonstrate that TAE stability is profoundly influenced by the minority species, minority concentration, effective temperature, and critically, pitch-angle anisotropy. The numerical results are well consistent with our physical understanding based on the analyses of wave-particle resonance condition and coupling strength developed in our previous work Wei et al (2025 Nucl. Fusion 65 106035). More specifically, TAE instabilities driven by NNBI are generally weaker than those driven by ICRH, and for ICRH heating, using 3He minority with relatively high concentration and prevalence of barely trapped particle population may yield to the suppression of TAE instabilities. This study provides crucial insights for optimizing heating strategies and EP management in DTT and other next-generation fusion devices, offering a more predictive understanding of EP-driven instabilities for robust plasma performance.

Understanding nonlinear saturation of reversed-shear Alfvén eigenmodes (RSAEs) in tokamaks is crucial for high-performance burning plasmas. Employing both nonlinear gyrokinetic simulations and theoretical analyses, we have discovered the novel result that, with energetic particle dynamics kept linear, the nonlinear suppression and eventual saturation of RSAE occur via the downward frequency chirping induced by the beat-driven zonal current. More specifically, as the mode frequency chirps downward, there is enhanced mode conversion to radially propagating electron Landau-damped kinetic Alfvén waves; resulting in enhanced convective (radiative) damping and, thereby, its suppression and saturation. Theoretical results are in good agreement with simulations both qualitatively and quantitatively. This Letter, thus, establishes a fundamental framework for zonal-current mediated saturation of Alfvén eigenmodes in burning plasmas.

The indirect nonlinear interactions between toroidal Alfvén eigenmode (TAE) and ion temperature gradient mode (ITG) are investigated using nonlinear gyrokinetic theory and ballooning mode formalism. More specifically, the local nonlinear ITG mode equation is derived adopting the fluid-ion approximation, with the contributions of zonal field structure and phase space zonal structure beat-driven by finite amplitude TAE accounted for on the same footing. The obtained nonlinear ITG mode equation is solved both analytically and numerically, and it is found that, the zonal structure beat-driven by TAE has only weakly destabilizing effects on ITG, contrary to usual speculations and existing numerical results.

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.

By employing both nonlinear gyrokinetic simulation and analytical theory, we have investigated the effects of zonal (electromagnetic) fields on the energetic particle's (EPs) drive of reversed-shear Alfven eigenmodes (AEs) in tokamak plasmas. Contrary to the conventional expectation, simulations with zonal fields that are turned on and off in the EP dynamics while keeping the full nonlinear dynamics of the thermal plasma indicate that zonal fields further enhance the instability drive and thus lead to a higher saturation level. These puzzling simulation results can be understood analytically in terms of the general fishbone-like dispersion relation with the correspondingly different EP phase-space structures induced by the zonal fields. Analytical expressions for the zonal fields that are beat driven by the reversed-shear AEs are also derived, and shown to be in good agreement with the simulation results.

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.

Employing both nonlinear gyrokinetic simulations and theoretical analyses, we have discovered the novel result that, with energetic particle dynamics kept linear, the nonlinear suppression and eventual saturation of reversed-shear Alfvén eigenmode occur via the downward frequency chirping induced by the beat-driven zonal current. More specifically, as the mode frequency chirps downward, there is enhanced mode conversion to radially propagating electron Landau-damped kinetic Alfvén waves; resulting in enhanced convective (radiative) damping and, thereby, its suppression and saturation. Theoretical results are in good agreement with simulations both qualitatively and quantitatively.

Energetic particle(EP)related physics,including fusion alpha particles,are expected to play crucial roles in burning plasma of future reactors.In particular,EPs,and the collective oscillations driven by EPs,act as mediators of cross-scale couplings among the rich spatiotemporal scales characterizing burning plasmas.In this work,taking the well-known toroidal Alfvén eigenmode(TAE)as example,it is shown that meso-scale TAE can interact with micro-scale drift wave turbulence(DW)via both direct and in-direct channels.Three examples of cross-scale couplings among TAEs and DWs are given,including,(1)TAE instabilities can be effectively reduced or even suppressed by ambient stationary DW due to stimulated absorption via direct-scatterings into short-wavelength electron Landau damped kinetic Alfvén waves(KAWs);(2)in the"reverse"process of DW scattering by ambient stationary TAEs with typical amplitude expected in tokamak experiment,direct nonlinear scatterings to short-wavelength KAWs have a negligible damping effect on DW due to the cancellation of stimulated absorption via the upper-sideband KAW and spontaneous emission via the lower-sideband KAW;and(3)TAE can indirectly regulate the DW intensity via the nonlinearly excited zonal structures,which,in the local limit,yield an unexpected weak destabilizing effect,contrary to usual speculations.These findings illuminate the richness and complexity of nonlinear plasma physics that underlying these processes,required for extrapolating to future reactor burning plasmas.Future works along this line are also addressed.

A linear gyrokinetic eigenvalue code is developed to study the stability of toroidal Alfvén eigenmode (TAE) in general axisymmetric toroidal geometry, with a self-consistent treatment of energetic particle drive and core plasma Landau damping in a non-perturbative way. The general particle responses of both circulating and trapped particles are incorporated in the calculation by means of the action-angle approach, and, in particular, the finite Larmor radius and orbit width effects of energetic particles are fully taken into account. The ballooning-mode representation is adopted to solve the eigenmode equations in order to reduce the computational resource while obtaining a high resolution of the fine radial structure. Furthermore, the code is able to study the physics of wave-particle interaction in great detail, thanks to the development of systematic theory-based numerical diagnostics, including effective mode structure and phase space resonance structure. As an application of the code, we perform an in-depth study of the triangularity effect on TAE stability based on the reference equilibrium of the Divertor Tokamak Test facility. It is demonstrated that TAE growth rate can be affected by the triangularity through the modifications of geometric couplings, resonance condition, as well as mode frequency and mode structure. As a result, negative triangularity can either stabilize or destabilize the energetic particle driven TAE depending on the dominant mechanism. The relative importance of these mechanisms under different circumstances is systematically analyzed, providing clear physical insights. The overall effect of negative triangularity for a specific tokamak scenario can be assessed based on these studies.

After a brief introduction on present day simulation and modeling of burning plasmas in magnetically confined devices, the results of a non-linear benchmark are presented, undertaken among several state-of-the-art codes available to study the self-consistent interaction of an Energetic Particle (EP) population with shear Alfvén waves: HYMAGYC, MEGA, ORB5 and XTOR-K. The first two codes, HYMAGYC and MEGA, are hybrid codes: HYMAGYC is a MHD-Gyrokinetic code (the bulk plasma is represented by MHD equations, while the EP species is treated using the gyrokinetic formalism), while MEGA is an MHD-Drift-Kinetic code (the bulk plasma is represented by MHD equations, while the EP species is treated using the drift-kinetic formalism, with the possibility of an ad-hoc gyroaveraging); ORB5 is a global electromagnetic gyrokinetic code (both bulk and EP species are treated using the gyrokinetic formalism); XTOR-K is a non-linear kinetic-MHD code (the bulk plasma is described by a set of non-linear resistive two-fluid MHD equations, extended to include kinetic effects of multiple ion species with a fully kinetic PIC module). The equilibrium of the so-called NLED-AUG reference case, in its version with peaked off-axis EP density profile, has been used, while considering a |n|=1 perturbation. This non-linear benchmark is the natural continuation of the linear benchmark already considered in the recent past, and represent a first-ever code comparison in the deep non-linear stage. In the present study fluid non-linearities are omitted, and the focus will be on comparing wave-particle interactions effects across codes. Characteristics of the non-linear saturation of the mode, self-consistent modification to the EP density profile and other features are compared among the considered codes. This brief review presents the state of the art of gyrokinetic and hybrid MHD-kinetic codes emerging as tools for studying reactor-relevant burning plasmas in realistic conditions.

The Van Allen radiation belts contain relativistic electrons trapped by Earth's magnetic field, posing serious risks to spacecraft. Chorus waves are known to accelerate these electrons via resonant interactions, but these interactions are inherently nonlinear and coherent. How such processes shape large‐scale electron dynamics remains unresolved. Two competing paradigms, nonlinear advection and diffusive transport, have been debated for decades. Here, we address this controversy using large‐scale first‐principles simulations that self‐consistently generate realistic chorus wave fields, coupled with test particle modeling. We find that electron motion is coherent on short timescales—comparable to or less than a bounce period—but becomes stochastic over longer timescales due to decorrelation. The resulting transport coefficients support the use of quasilinear diffusion theory for long‐term evolution. This work bridges microscopic nonlinear physics with macroscopic modeling frameworks, offering a unified explanation of radiation belt dynamics and advancing the foundation for space weather forecasting.

In nonuniform plasmas, toroidal Alfvén eigenmode (TAE) is shown to decay into high toroidal mode number modes due to ion induced scattering (Cheng et al 2024 Nucl. Fusion 64 066031). In this work, the three wave parametric decay model is generalized into reactor-relevant multiple-mode case, and the wave-kinetic equation describing the TAE spectrum evolution in k _θ space due to ion induced scattering is derived. Here, k _θ is the poloidal wavenumber. By numerically solving the wave-kinetic equation with reactor-relevant parameters, the nonlinear evolution and saturation process of TAE spectral intensity is derived, which yields a saturation spectrum consistent with the fixed point solution. It is found that the magnetic perturbation amplitude induced by the TAE saturation spectrum is lower than the uniform plasma result due to the enhanced nonlinear coupling by plasma nonuniformity. Potential impact on intrinsic plasma rotation is also discussed.

Gyrokinetic and kinetic-MHD simulations are performed for the fishbone instability in the DIII-D discharge #178631, chosen for validation of first-principles simulations to predict the energetic particle (EP) transport in an ITER prefusion baseline scenario. Fishbone modes are found to generate zonal flows, which dominate the fishbone saturation. The underlying mechanisms of the two-way fishbone-zonal flows nonlinear interplay are discussed in details. Numerical and analytical analyses identify the fishbone-induced EP redistribution as the dominant generation mechanism for zonal flows. The zonal flows modify the nonlinear dynamics of phase space zonal structures, which reduces the amount of EPs able to resonate with the mode, leading to an early fishbone saturation. Simulation results including zonal flows agree quantitatively with DIII-D experimental measurements of the fishbone saturation amplitude and EP transport, supporting this novel saturation mechanism by self-generated zonal flows. Moreover, the wave-particle mode-locking mechanism is shown to determine quantitatively the fishbone frequency down-chirping, as evident in GTC simulation results in agreement with predictions from analytical theory. Finally, the fishbone-induced zonal flows are possibly responsible for the formation of an ion-ITB in the DIII-D discharge. Based on the low EP transport and the large zonal flow shearing rates associated with the fishbone instability in gyrokinetic simulations of the ITER scenario, it is conjectured that high performance scenarios could be designed in ITER burning plasmas through fishbone-induced ITBs.

The self-consistent nonlinear interaction of drift wave (DW) and zonal flow (ZF) is investigated using nonlinear gyrokinetic theory, with both spontaneous excitation and beat-driven of ZF by DW treated on the same footing. DW solitons are formed in the nonlinear DW-ZF interactions and are confined between radially spaced micro-barriers. The resulting radial structures in the nonlinear DW-ZF interactions exhibit similar pattern to the ExB "staircase" observed in numerical simulations. These micro-barriers are generated by the repulsive response due to spontaneously excited ZF, which, as a general property demonstrated in this work, also generate an attractive nonlinear potential in DW equation. Meanwhile, the nonlinear potential due to beat-driven ZF is always attractive and, as such, always serve as potential well to contribute to soliton formation. For spontaneously excited ZF from initial noise, the simultaneous excitation of solitons and micro-barriers is found to be universal, due to the zero frequency nature of ZF and spatial structure of the Reynolds stress. The present analysis, thus, provides a potential first-principle-based interpretation of the ExB staircase observed in simulations, which may contribute to micro transport barriers formation and enhance plasma confinement.

For over 50 years, chorus wave frequency chirping has been regarded as a nonlinear phenomenon driven by the hot electron current density aligned with the wave magnetic field (δjB). However, recent theoretical models have challenged the crucial role played by δjB. Here, using modified first-principles particle simulations, we demonstrate that chirping can occur even in the absence of δjB. This result not only clarifies the physical mechanism governing chorus wave evolution but also has broader implications for nonlinear frequency chirping in other wave modes and plasma environments.

In this work, gyrokinetic theory of drift waves (DWs) self-regulation via the beat-driven zonal flow (ZF) is presented, and finite diamagnetic drift frequency due to plasma nonuniformity is shown to play a dominant role in the ZF beat generation. The obtained nonlinear DW equation is a nonlinear Schrödinger equation, in which the linear dispersiveness, linear growth, nonuniformity of diamagnetic drift frequency, and cubic nonlinearity induced by the feedback of beat-driven ZF to DWs are self-consistently included. The nonlinear DW equation is solved numerically in both uniform and nonuniform plasmas. It is shown that the DW envelope soliton may form due to the balance of linear dispersiveness and nonlinearity and lead to turbulence spreading to linearly stable region. It is further found that though the threshold on the DW amplitude for soliton formation is well within the relevant parameter regimes of realistic tokamak experiments, solitons cannot extend beyond the range bounded by the turning points of the wave packet when plasma nonuniformity is self-consistently accounted for.