EP) Driven Geodesic Acoustic Modes Research Articles

Nonlinear excitation of energetic particle driven geodesic acoustic mode by Alfvén instability in ASDEX-Upgrade Tokamak

Recently, the coexistence of multiple energetic particle driven instabilities was observed in experiments on the ASDEX-Upgrade tokamak (Lauber et al 2018 27th IAEA Fusion Energy Conf.). A hybrid simulation using the MEGA code was performed to investigate the properties of those instabilities. The basic mode properties obtained in the simulations, such as mode frequencies, mode numbers, and inward energetic particle (EP) redistribution, are in good agreement with the experiments. It is found that the energetic particle driven geodesic acoustic mode (EGAM) is initially stable, then zonal flow gradually occurs with the growth of the Alfvén instability, and finally, the EGAM is nonlinearly excited and the amplitude exceeds the Alfvén instability. The dependence of EGAM properties on EP pressure and pitch angle distribution is analyzed. The EGAM amplitude increases with EP pressure. The nonlinearly excited EGAM is a high-frequency branch that appears even under the condition of a slowing-down EP distribution. The resonant particles are also analyzed, but the dominant resonant particles of the EGAM in the linear growth phase are not found because the EGAM does not grow in the linear regime. In the phase space of pitch angle variable Λ and energy E, it is found that initially the Alfvén instability is excited by EPs with poloidal frequency 70 kHz, then, after the saturation of the Alfvén instability, the resonance region moves towards lower energy and touches the EGAM resonance line, and finally, EGAM is excited by the particles with poloidal frequency 50 kHz . This process is a kind of resonance overlap.

Destabilization of geodesic acoustic-like mode in the presence of poloidally inhomogeneous heat sources in tokamak plasmas

The effects of poloidally inhomogeneous heat sources are investigated through a gyrokinetic formula in collisionless toroidal plasmas. A gyrokinetic dispersion relation is newly derived under the assumption that equilibrium parallel heat flows are generated to remove the injected poloidally nonuniform heat source. The dispersion relation is numerically solved, considering both inboard and outboard heat source injections. In the case of the inboard source injection, both Stringer spin-up and geodesic acoustic mode (GAM) are excited. Conversely, outboard injection leads to the emergence of a heat source-driven GAM (referred to as Q-GAM), featuring a frequency around half that of the standard GAM. Various physical quantities of the Q-GAM, such as mode frequency and source threshold, are analyzed through parametric scans. The Q-GAM exhibits similarities with the energetic-particle-driven GAM (EGAM), particularly in its frequency range, and both belong to one of the strong Landau damped poles. Despite having distinct driving mechanisms and structural differences in parallel velocity and poloidal coordinates, the response function of the perturbed parallel pressure to the potential, mainly contributing to the destabilization of each mode around half of the GAM frequency, is derived to have a similar form for both the Q-GAM and EGAM cases.

Nonlinear excitation of energetic-particle-driven geodesic acoustic mode by ions drift waves

Nonlinear excitation of geodesic acoustic mode (GAM) by ions drift waves in the presence of energetic particles (EPs) is investigated through the hybrid model of fluid description and gyro-kinetic description, while the bulk ions temperature anisotropy and toroidal rotation are taken into account simultaneously. The coupling relation of EPs and the nonlinear effects on GAM is analytically derived and discussed. It is found that the nonlinear excitation can split inherent energetic-particle-driven GAM (EGAM) branches into different growth rates. For the intrinsic stable high frequency EGAM branch, the nonlinear effect generates a growth/damping rate. For the low frequency branches with inherent growth/damping rates due to the EPs, the nonlinearity cooperates with EPs in the excitation process.

Gyrokinetic modelling of the Alfvén mode and EGAM activity in ASDEX Upgrade

Energetic particles present in tokamak machines can drive through resonant wave-particle interaction different plasma instabilities, e.g Alfvén modes and energetic particle-driven geodesic acoustic modes (EGAMs). While the former are potentially detrimental as they can enhance the energetic particle transport and damage the machine wall, the latter are axisymmetric, possibly benign modes that can act to regulate turbulence. A unique scenario, the so-called NLED-AUG case, has been developed in ASDEX Upgrade by tuning the plasma parameters so that the energetic particle kinetic energy is 100 times higher than that of the background plasma, like in ITER. An intense energetic particle-driven activity is observed, most prominently various Alfvén mode bursts triggering chirping EGAMs. The present work reports studies on the Alfvén mode and EGAM dynamics showing, for the first time, many toroidal mode gyrokinetic simulations with ORB5 where the NLED-AUG case scenario is considered. We study the mode dynamics modelling the energetic particles with different equilibrium distribution functions, such as: isotropic slowing-down, double-bump-on-tail and equivalent Maxwellian. We retain, at the beginning, the nonlinearities only in the energetic particle dynamics. Later, also the background plasma species nonlinearities are taken into account.

Gyrokinetic investigation of the nonlinear interaction of Alfvén instabilities and energetic particle-driven geodesic acoustic modes

This paper presents a study of the interaction between Alfvén modes and zonal structures, considering a realistic ASDEX Upgrade equilibrium. The results of gyrokinetic simulations with the global, electromagnetic, particle-in-cell code ORB5 are presented, where the modes are driven unstable by energetic particles with a bump-on-tail equilibrium distribution function, with radial density gradient. Two regimes have been observed. At low energetic particle concentration, the zonal structure (identified as an energetic particle-driven geodesic acoustic mode) is more unstable than the Alfvén mode. In the regime at high energetic particle concentration, the Alfvén mode is more unstable than the zonal structure. The interplay between the modes leads to a modification of their growth rates as well as to a modification of their saturation levels. The theoretical explanation of the mode interaction is given in terms of three-wave coupling of the energetic particle-driven geodesic acoustic mode and Alfvén mode, mediated by the curvature–pressure coupling term of the energetic particles.

The systematic investigation of energetic-particle-driven geodesic acoustic mode channeling using MEGA code

Energetic-particle-driven geodesic acoustic modes (EGAMs) channeling in the Large Helical Device (LHD) plasmas are systematically investigated for the first time using MEGA code. MEGA is a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic (MHD) fluid. In the present work, both the energetic particles and the bulk ions are described kinetically. The EGAM profiles in the three-dimensional form is illustrated. Then, EGAM channeling behaviors are analyzed under different conditions. During the EGAM activities without frequency chirping, EGAM channeling occurs in the linear growth stage but terminates in the decay stage after the saturation. During the EGAM activities with frequency chirping, EGAM channeling occurs continuously. Also, low-frequency EGAM makes the energy transfer efficiency () higher, and this is confirmed by changing the energetic particle pressure, energetic particle beam velocity, and energetic particle pitch angle. Moreover, higher bulk ion temperature makes the energy transfer efficiency higher. In addition, under a certain condition, the energy transfer efficiency in the deuterium plasma is lower than that in the hydrogen plasma.

Nonlinear dynamics of energetic-particle driven geodesic acoustic modes in ASDEX Upgrade

Turbulence in tokamaks generates radially sheared zonal flows. Their oscillatory counterparts, geodesic acoustic modes (GAMs), appear due to the action of the magnetic field curvature. The GAMs can be driven unstable by an anisotropic energetic particle (EP) population leading to the formation of global radial structures, called energetic-particle-driven geodesic acoustic modes (EGAMs). The EGAMs can redistribute EP energy to the bulk plasma through collisionless wave-particle interaction. In such a way, the EGAMs might contribute to the plasma heating. Thus, investigation of EGAM properties, especially in the velocity space, is necessary for precise understanding of the transport phenomena in tokamak plasmas. In this work, the nonlinear dynamics of EGAMs without considering the mode interaction with the turbulence is investigated with the help of a Mode-Particle-Resonance (MPR) diagnostic implemented in the global gyrokinetic particle-in-cell code ORB5. An ASDEX Upgrade discharge is chosen as a reference case for this investigation due to its rich EP nonlinear dynamics. An experimentally relevant magnetic field configuration, thermal species profiles, and an EP density profile are taken for EGAM chirping modeling and its comparison with available empirical data. The same magnetic configuration is used to explore energy transfer by the mode from the energetic particles to the thermal plasma including kinetic electron effects. For a given EGAM level, the plasma heating by the mode can be significantly enhanced by varying the EP parameters. Electron dynamics decreases the EGAM saturation amplitude and consequently reduces the plasma heating, even though the mode transfers its energy to thermal ions much more than to electrons.

Simulation of energetic particle driven geodesic acoustic modes and the energy channeling in the Large Helical Device plasmas

Energetic particle driven geodesic acoustic modes (EGAMs) in Large Helical Device plasmas are investigated using MEGA code. MEGA is a hybrid simulation code for energetic particles interacting with a magneto-hydrodynamic (MHD) fluid and in the present work, both the energetic particles and bulk ions are described by the kinetic equations. The low frequency EGAMs are reproduced. Also, the energy transfer is analyzed and the bulk ion heating during the EGAM activity is observed. The ions obtain energy when the energetic particles lose energy, and this indicates that an energy channel is established by the EGAM. EGAM channeling is reproduced by simulation with realistic parameters for the first time. The heating power to bulk ions is 3.4 . It is found that sideband resonance is dominant during the energy transfer from EGAM to the bulk ions, and the transit frequencies of resonant bulk ions are one-half of the EGAM frequency.

Nonlinear velocity redistribution caused by energetic-particle-driven geodesic acoustic modes, mapped with the beam-plasma system

The nonlinear dynamics of energetic-particle (EP) driven geodesic acoustic modes (EGAM) in tokamaks is investigated, and compared with the beam-plasma system (BPS). The EGAM is studied with the global gyrokinetic (GK) particle-in-cell code ORB5, treating the thermal ions and EP (in this case, fast ions) as GK and neglecting the kinetic effects of the electrons. The wave–particle nonlinearity is only considered in the EGAM nonlinear dynamics. The BPS is studied with a one-dimensional code where the thermal plasma is treated as a linear dielectric, and the EP (in this case, fast electrons) with an N-body Hamiltonian formulation. A one-to-one mapping between the EGAM and the BPS is described. The focus is on understanding and predicting the EP redistribution in phase space. We identify here two distinct regimes for the mapping: in the low-drive regime, the BPS mapping with the EGAM is found to be complete, and in the high-drive regime, the EGAM dynamics and the BPS dynamics are found to differ. The transition is described with the presence of a non-negligible frequency chirping, which affects the EGAM but not the BPS, above the identified drive threshold. The difference can be resolved by adding an ad hoc frequency modification to the BPS model. As a main result, the formula for the prediction of the nonlinear width of the velocity redistribution around the resonance velocity is provided. This article is written as the second of a series of articles (the first being Biancalani et al. (J. Plasma Phys., vol. 83 (6), 2017, 725830602)) on the saturation of EGAMs due to wave–particle nonlinearity.

Chirping and Sudden Excitation of Energetic-Particle-Driven Geodesic Acoustic Modes in a Large Helical Device Experiment.

Energetic-particle-driven geodesic acoustic modes (EGAMs) observed in a Large Helical Device experiment are investigated using a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic (MHD) fluid. The frequency chirping of the primary mode and the sudden excitation of the half-frequency secondary mode are reproduced for the first time with the hybrid simulation using the realistic physical condition and the three-dimensional equilibrium. Both EGAMs have global spatial profiles which are consistent with the experimental measurements. For the secondary mode, the bulk pressure perturbation and the energetic particle pressure perturbation cancel each other out, and thus the frequency is lower than the primary mode. It is found that the excitation of the secondary mode does not depend on the nonlinear MHD coupling. The secondary mode is excited by energetic particles that satisfy the linear and nonlinear resonance conditions, respectively, for the primary and secondary modes.

Geodesic acoustic mode driven by energetic particles with bump-on-tail distribution

Energetic-particle-driven geodesic acoustic mode (EGAM) is analytically investigated by adopting the bump-on-tail distribution for energetic particles (EPs), which is created by the fact that the charge exchange time () is sufficiently shorter than the slowing down time (). The dispersion relation is derived in the use of gyro-kinetic equations. Due to the finite ratio of the critical energy and the initial energy of EPs, defined as , the dispersion relation is numerically evaluated and the effect of finite is examined. Following relative simulation and experimental work, we specifically considered two cases: and . The pitch angle is shown to significantly enhance the growth rate and meanwhile, the real frequency is dramatically decreased with increasing pitch angle. The excitation of high-frequency EGAM is found, and this is consistent with both the experiment and the simulation. The number density effect of energetic particles, represented by , is checked by fixing the pitch angle to 0.1. It is found that for , the growth rate of scan is not monotonic but the real frequency is monotonically decreased. For , the scan shows the growth rate keeps almost unchange under the condition of low value. As increases, the growth rate is rapidly enlarged.

Enhancement and suppression of turbulence by energetic-particle-driven geodesic acoustic modes

We propose a novel mechanism of enhancement of turbulence by energetic-particle-driven geodesic acoustic modes (EGAMs). The dynamics of drift-wave-type turbulence in the phase space is investigated by wave-kinetic equation. Spatially inhomogeneous turbulence in the presence of a transport barrier is considered. We discovered that trapping of turbulence clumps by the EGAMs is the key parameter that determines either suppress or enhance turbulence. In regions where turbulence is unstable, EGAM suppresses the turbulence. In contrast, in the stable region, EGAM traps clumps of turbulence and carries them across the transport barrier, so that the turbulence can be enhanced. The turbulence trapped by EGAMs can propagate independent of the gradients of density and temperature, which leads to non-Fickian transport. Hence, there appear a new global characteristic velocity, the phase velocity of GAMs, for turbulence dynamics, in addition to the local group velocity and that of the turbulence spreading. With these effect, EGAMs can deteriorate transport barriers and affect turbulence substantially. This manuscript provides a basis to consider whether a coherent wave breaks or strengthen transport barriers.

Saturation of energetic-particle-driven geodesic acoustic modes due to wave–particle nonlinearity

The nonlinear dynamics of energetic-particle (EP) driven geodesic acoustic modes (EGAM) is investigated here. A numerical analysis with the global gyrokinetic particle-in-cell code ORB5 is performed, and the results are interpreted with the analytical theory, in close comparison with the theory of the beam-plasma instability. Only axisymmetric modes are considered, with a nonlinear dynamics determined by wave–particle interaction. Quadratic scalings of the saturated electric field with respect to the linear growth rate are found for the case of interest. As a main result, the formula for the saturation level is provided. Near the saturation, we observe a transition from adiabatic to non-adiabatic dynamics, i.e. the frequency chirping rate becomes comparable to the resonant EP bounce frequency. The numerical analysis is performed here with electrostatic simulations with circular flux surfaces, and kinetic effects of the electrons are neglected.

Nonlinear interaction between energetic particles and turbulence in gyro-kinetic simulations and impact on turbulence properties

The modification of radial structure, frequency and intensity of turbulent transport in the presence of energetic-particle-driven geodesic acoustic modes (EGAMs) is analysed by means of full-f global gyro-kinetic simulations using Gysela code. It is observed that turbulence leads to a smoother evolution of the distribution function, less pronounced flattening of the distribution function in velocity space during the nonlinear saturation of EGAMs and reduced saturation level of electrostatic potential with respect to the case where turbulence is artificially suppressed. It is shown that EGAMs are excited and impact turbulent transport in the region where the EP is localised, fading away the staircase structure observed in the absence of energetic particles. For the first time, evidences of a three-wave coupling between turbulent modes and EGAMs in gyro-kinetic simulations are provided by means of bispectral analysis using wavelet transform in time. The coupling evolves from the standard self-regulation of turbulence by the zero-frequency zonal component to a steady-state regime where turbulence dynamics is dominated by the zonal component oscillating at the EGAM frequency.

Observation of subcritical geodesic acoustic mode excitation in the large helical device

The abrupt and strong excitation of the geodesic acoustic mode (GAM) has been found in the large helical device (LHD), when the frequency of a chirping energetic particle-driven GAM (EGAM) approaches twice that of the GAM frequency. The temporal evolution of the phase relation between the abrupt GAM and the chirping EGAM is common in all events. The result indicates a coupling between the GAM and the EGAM. In addition, the nonlinear evolution of the growth rate of the GAM is observed, and there is a threshold in the amplitude of the GAM for the appearance of nonlinear behavior. A threshold in the amplitude of the EGAM for the abrupt excitation of the GAM is also observed. According to one theory (Lesur et al 2016 Phys. Rev. Lett. 116 015003, Itoh et al 2016 Plasma Phys. Rep. 42 418) the observed abrupt phenomenon can be interpreted as the excitation of the subcritical instability of the GAM. The excitation of a subcritical instability requires a trigger and a seed with sufficient amplitude. The observed threshold in the amplitude of the GAM seems to correspond with the threshold in the seed, and the threshold in the amplitude of the EGAM seems to correspond with the threshold in the magnitude of the trigger. Thus, the observed threshold supports the interpretation that the abrupt phenomenon is the excitation of a subcritical instability of the GAM.

Energetic particle driven geodesic acoustic mode in a toroidally rotating tokamak plasma

Energetic particle (EP) driven geodesic acoustic modes (EGAMs) in toroidally rotating tokamak plasmas are analytically investigated using the hybrid kinetic-fluid model and gyrokinetic equations. By ignoring high-order terms and ion Landau damping, the kinetic dispersion relation is reduced to the hybrid one in the large safety factor limit. There is one high-frequency branch with a frequency larger than , the transit frequency of EPs with initial energy, which is always stable. Two low-frequency solutions with a frequency smaller than are complex conjugates in the hybrid limit. In the presence of ion Landau damping, the growth rate of the unstable branch is decreased and the damping rate of the damped branch is increased. The toroidal Mach number is shown to increase , the normalized real frequency of both branches. Although not affecting the instability critical condition, the Mach number decreases the growth rate when is larger than a critical value and enlarges the growth rate when . The ion Landau damping effect is negligible for large M. But the discrepancy between the kinetic dispersion relation and the hybrid one becomes ignorable only for .

Onset condition of the subcritical geodesic acoustic mode instability in the presence of energetic-particle-driven geodesic acoustic mode

An analytic model is developed for understanding the abrupt onset of geodesic acoustic mode (GAM) in the presence of chirping energetic-particle-driven GAM (EGAM). This abrupt excitation phenomenon has been observed on LHD plasma. Threshold conditions for the onset of abrupt growth of GAM are derived, and the period doubling phenomenon is explained. The phase relation between the mother mode (EGAM) and the daughter mode (GAM) is also discussed. This result contributes to the understanding of “trigger problems” of laboratory and nature plasmas.

Strong Destabilization of Stable Modes with a Half-Frequency Associated with Chirping Geodesic Acoustic Modes in the Large Helical Device.

Abrupt and strong excitation of a mode has been observed when the frequency of a chirping energetic-particle driven geodesic acoustic mode (EGAM) reaches twice the geodesic acoustic mode (GAM) frequency. The frequency of the secondary mode is the GAM frequency, which is a half-frequency of the primary EGAM. Based on the analysis of spatial structures, the secondary mode is identified as a GAM. The phase relation between the secondary mode and the primary EGAM is locked, and the evolution of the growth rate of the secondary mode indicates nonlinear excitation. The results suggest that the primary mode (EGAM) contributes to nonlinear destabilization of a subcritical mode.

Simulation study of high-frequency energetic particle driven geodesic acoustic mode

High-frequency energetic particle driven geodesic acoustic modes (EGAM) observed in the large helical device plasmas are investigated using a hybrid simulation code for energetic particles and magnetohydrodynamics (MHD). Energetic particle inertia is incorporated in the MHD momentum equation for the simulation where the beam ion density is comparable to the bulk plasma density. Bump-on-tail type beam ion velocity distribution created by slowing down and charge exchange is considered. It is demonstrated that EGAMs have frequencies higher than the geodesic acoustic modes and the dependence on bulk plasma temperature is weak if (1) energetic particle density is comparable to the bulk plasma density and (2) charge exchange time (τcx) is sufficiently shorter than the slowing down time (τs) to create a bump-on-tail type distribution. The frequency of high-frequency EGAM rises as the energetic particle pressure increases under the condition of high energetic particle pressure. The frequency also increases as the energetic particle pitch angle distribution shifts to higher transit frequency. It is found that there are two kinds of particles resonant with EGAM: (1) trapped particles and (2) passing particles with transit frequency close to the mode frequency. The EGAMs investigated in this work are destabilized primarily by the passing particles whose transit frequencies are close to the EGAM frequency.

Analytic dispersion relation of energetic particle driven geodesic acoustic modes and simulations with NEMORB

Recent progress regarding the excitation of energetic-particle driven geodesic acoustic modes (EGAMs) in particle-in-cell simulations is presented in this paper. The exact dispersion relation with adiabatic electrons is derived and solved. The origin of the so-called EGAM is briefly analysed and we show that its nature changes, at least, with the safety factor. A simple expression for the GAM frequency modified in the presence of a small concentration of energetic particles is given in the fluid limit. We show that gyrokinetic simulations with Nemorb in the presence of adiabatic electrons are able to reproduce the analytic predictions. Also, different energy channels are analysed by means of dedicated energy diagnostics characterizing the wave-particle interaction. Finite Larmor radius and finite orbit width effects are studied regarding the excitation of geodesic acoustic modes, showing that these effects are likely to be negligible for sufficiently high concentration of energetic particles, but significant when approaching the threshold of excitation.