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


Introduction
Energetic particle (EP) driven instabilities are important for magnetic confinement fusion research because of their effects on plasma confinement.During the activity of EP driven instabilities, the radial transport of EPs is enhanced, resulting in the loss of EPs and a deterioration of the heating performance.As a result, the confinement level decreases.The Alfvén instability is a typical dangerous EP driven instability.For example, in DIII-D, during Alfvén eigenmode (AE) activities more than half of the EPs were lost [1,2].When the EP pressure is comparable to the pressure of the thermal plasma, a stronger Alfvén instability, the energetic-particlemode (EPM), appears on the Alfvén continuum [3,4].The energetic particle driven geodesic acoustic mode (EGAM) also enhances the radial transport of EPs, although its toroidal mode number n is 0. Also in DIII-D, strong neutron emission drops have been observed during EGAM activities [5].In addition to the negative effects on confinement, EP driven instabilities can also have some positive effects, such as anomalous heating.In LHD, anomalous bulk ion heating was observed during EGAM activities, which is caused by the energy channel created by EGAM [6,7].Then, the investigation of EP driven instabilities, such as EPM or EGAM, attracted great interest from the community.
The coexistence of multi-modes (e.g.EPM and EGAM) complicates the situation.In the case of multi-modes, modemode interaction is possible.Linear interaction between multimodes usually means that one mode competes with other modes, and the strengthening of one mode is often accompanied by the weakening of other modes.But nonlinear interaction can cause subcritical instability which may not be captured by linear analysis but has a significant impact on the nonlinear behavior of the system, and it becomes dangerous because of the sudden increase in growth rate.The nonlinear interaction enhances mode activity and particle transport, and also makes the plasma behavior more unpredictable and makes the plasma more difficult to control.Recently, a series of experiments demonstrated the coexistence of Alfvén instability and EGAM in the ASDEX-Upgrade (AUG) tokamak.In the AUG scenarios of Non-Linear Energetic-particle Dynamics (NLED-AUG) and some similar AUG scenarios, the appearance of EGAM closely follows that of the Alfvén instability.This prompt observation has led experimentalists to hypothesise that the Alfvén instability triggers the EGAM.Several simulations of the NLED-AUG case using different codes show that the EGAM is not present during the linear growth phase, suggesting that the Alfvén instability triggers the EGAM in a nonlinear phase [8].Indeed, many simulations have been performed to investigate the NLED-AUG case.For example, Vannini et al demonstrated the effect of EP concentrations on the AE and EGAM under a condition of bump-on-tail distribution, and discussed the excitation of the AE by the EGAM.Additionally, based on a slowingdown distribution, they successfully reproduced frequency chirping phenomena and the results closely matched those observed in the experiments [9][10][11].Rettino et al studied how the mode properties vary with the distribution function and other parameters [12].Wang et al reproduced the excitation of EGAM by Alfvén instability under a condition of slowingdown distribution, and the physical mechanism of nonlinear excitation is explained from the perspective of resonance overlap [13].
Even if a single mode, the EGAM, is still an interesting topic to study.For example, most of the observed EGAMs are low-frequency branches with frequencies lower than those of the conventional geodesic acoustic modes (GAMs), but sometimes high-frequency branches with frequencies higher than those of the conventional GAMs are also observed, as in LHD [14].Fu gave the frequencies of EGAMs analytically and predicted the existence of high-frequency EGAMs [15].However, under the condition of slowing-down EP distribution, the highfrequency EGAM does not grow.Qiu et al also gave the highfrequency beam branch and the low-frequency GAM branch analytically [16].Wang et al reproduced the high-frequency EGAM on LHD by simulation, and pointed out that the bumpon-tail EP distribution is a necessary condition to excite the high-frequency EGAM [17].The above studies are conducted in the linear growth phase and do not include the nonlinear excitation of EGAM.
In fact, a lot of gaps remain in [13].For example, the details of the evolution of EGAM with time are not shown, the effects of EP pressure β EP and pitch angle Λ distribution on the EGAM amplitude are not analyzed, and the energy transfer in the pitch angle and energy two-dimensional phase space is not investigated.The present paper will fill in the above gaps and it is organized as follows.In section 2 the simulation model and parameters are described.In section 3 the linear and nonlinear properties of the simulated modes are investigated, and the resonant particles are analyzed in detail.In section 4 the main conclusions are summarized.

Simulation model and parameters
A hybrid simulation code, MEGA [18][19][20], for EPs interacting with a magnetohydrodynamic (MHD) fluid is conducted for the simulations of the coexistence of the Alfvén instability and EGAM in AUG.The hybrid model is widely applied in the fusion community [21].The physical models used by MEGA code can be roughly classified into two categories.In the version of conventional model, only the EPs are described by the kinetic equations, while in the version of extended model, both the EPs and the thermal ions are described kinetically.In the present work, the version of the conventional model is applied.The MHD equations with the energetic-ion effects are given by where µ 0 is the vacuum magnetic permeability, γ is the adiabatic constant, ν and ν n are artificial viscosity and diffusion coefficients, chosen to maintain numerical stability, and all the other physical quantities are conventional.The subscript h denotes hot particles, and in the present work, the hot particles represent EPs.The subscript EP is not used in the present paper in order to keep consistent with previous literature.The subscript 'eq' represents equilibrium variables.The above MHD equations are solved using a fourth order finite difference scheme.The EPs are coupled into the MHD through current coupling schemes.The energetic particles are described by the drift-kinetic equations [22].The guiding-center velocity u is given by ) where v ∥ is the velocity parallel to the magnetic field, µ is the magnetic moment, m h is energetic particle mass and Z h e is energetic particle charge.The energetic particle current density j ′ h in equation ( 2) is given by with energetic particle distribution function f.The energetic particle current density includes the contributions from parallel velocity, magnetic curvature and gradient drifts, and magnetization current.The energetic particles are described by the drift-kinetic equations.The modules of the gyrokinetic approach are included in MEGA, but usually the linear properties of the mode such as frequency and mode structure are only weakly affected by the finite Larmor radius effect [23,24], and thus, in order to save computational resources, the gyrokinetic module is switched off in the present simulations.The δf particle-in-cell (PIC) method is applied, and the equations of motion for each marker particle are solved using a fourth order Runge-Kutta method.More details of the equations of MEGA are given in [7,[18][19][20].The phase space volume each computational particle occupies is in proportion to the Jacobian R at the initial location.The phase space volume for the ith particle V i is given by V i = αR i,t=0 with a normalization factor α. The phase space volume V i is constant in time (dV i /dt = 0) for collisionless energetic particles.The normalization factor α is determined by for t = 0, where P h∥0 and P h⊥0 are the initial energetic particle parallel and perpendicular pressures and N is the total number of computational particles used.The time evolution of the weight of the ith particle is described by and the initial condition is w i,t=0 = 0.A realistic equilibrium calculated with the EFIT code is used for the simulation.This equilibrium data is based on AUG shot #34924 at time t = 1.90 s.From t = 1.90 s to t = 2.01 s, the plasma parameters and profiles in the experiment remain constant within the error bars.
In the AUG shot #34924, according to NUBEAM data, the EP distribution function is large in the low energy region and small in the high energy region, and thus, it is roughly a slowing-down type distribution in velocity phase space.Also, in v ∥ /v phase space, it is roughly distributed symmetrically based on v ∥ /v = 0.8, and thus, it can be considered as a Gaussian type distribution in pitch angle phase space.Similar types of distribution are then assumed in the present simulation.The EP velocity distribution is , where v is the velocity and v c = 1.637 × 10 6 m s −1 is the critical velocity.The EP pitch angle distribution is g where Λ is defined by µB 0 /E, µ is the magnetic moment, B 0 is the magnetic strength on the axis, E is the energy, Λ peak = 0.4 is the pitch angle for the distribution peak, and ∆Λ = 0.1 is a parameter to control the width of the distribution.In order to investigate the effects of the Λ peak on mode properties, other values of Λ peak are also used in the present work for comparison.
Additionally, in shot #34924, the EP radial profile peaks around r/a = 0.5, and consequently, the radial distribution in the present simulation is where ψ is the poloidal magnetic flux, ψ nrm is ψ normalized by the maximum value, ψ peak = 0.73 is a parameter to control the radial peak location, and ∆ψ = 0.274 is another parameter to control the radial width.In order to investigate the effects of the radial EP profile on mode excitation, other types of EP profiles are also used in the present simulation for comparison.
The following parameters for the simulation are based on AUG shot #34924: And finally, (7) the safety factor q profile obtained from the EFIT code has a weak shear in the core region, with a value of 2.3 at the magnetic axis and 6.43 at the plasma edge.
The number of computational particles is 4.19 million.Cylindrical coordinates (R, ϕ, z) are employed, and the number of grid points in (R, ϕ, z) directions is (128, 32, 256).

The reproducing of the experimental phenomenon
The coexistence of both Alfvén instability and EGAM is reproduced using the MEGA code, as shown in figure 1. Figure 1(a) shows the radial velocity v r frequency spectrum of the m/n = 3/ − 1 mode simulated with the MEGA code, and figure 1(b) shows the poloidal velocity v θ frequency spectrum of the m/n = 0/0 mode in the simulation.Here, m is the poloidal mode number and n is the toroidal mode number.In figure 1(a), the mode appears at about t = 0.2ms with a linear frequency of 100kHz, then it becomes saturated and the frequency starts to chirp up and down.The frequency is the same as the Alfvén instability observed in the experiment [25].In addition, as shown in figure 2(a), the m/n = 2/ − 1  component is also strong, in other words, the simulated mode has two dominant poloidal harmonics, m = 3 and 2. With time evolution, the phase of the m/n = 3/ − 1 component and the m/n = 2/ − 1 component are exactly the same, indicating that the 3/ − 1 and 2/ − 1 components constitute the same eigenmode.The mode number is also the same as the Alfvén instability in experiment [25].Then, the mode in figure 1(a) is identified as an Alfvén instability which is the same as that in the experiment.In figure 1(b), the EGAM appears at about Figure 3.The Alfvén continuum (blue) and safety factor q profile (magenta).A TAE gap is located at r/a = 0.45.The simulated mode, as marked by the red bar, is very close to the Alfvén continuum and almost intersects it.t = 0.4ms with a frequency of 50kHz, then it becomes saturated and the frequency starts to chirp up and down, although the chirping rate is very low.The mode structure is shown in figure 2(b) at t = 0.622ms.The frequency is the same as the EGAM observed in the experiment.It is also found that the mode number of pressure perturbation is m/n = 1/0 and that of magnetic perturbation is m/n = 2/0.The mode number is also the same as the EGAM in the experiment.Then, the mode in figure 1(b) is identified as EGAM which is the same as that in the experiment.
In order to identify the mode, the Alfvén continuum is shown in figure 3. A toroidal Alfvén eigenmode (TAE) gap is located around r/a = 0.45 with a safety factor of q = 2.5.Although the TAE gap is almost the same location as the simulated m/n = 2/ − 1 component, it is a little far from the simulated m/n = 3/ − 1 component.Also, the gap frequency is close to 200 kHz, much higher than the Alfvén mode frequency in the present work.The simulated mode, as marked by the red bar, is very close to the Alfvén continuum and almost intersects it, and the mode may be an EPM.
In figure 1(b), EGAM is initially stable, and then, grows in the nonlinear phase.However, the detailed process of the transition from the stable state to the nonlinear unstable state is not clear.Figure 4 shows the details.In figure 4(a), although the amplitude is still very weak, it is clear that the EPM starts to grow exponentially.In contrast, instead of growing, the EGAM is damped and the amplitude becomes weaker.In figure 4(b), the EPM amplitude is gradually large, and the periodic perturbations and exponential growth are very clear.At the same time, the m/n = 0/0 mode also starts to grow.However, according to the frequency spectrum in figure 1(b), at t = 0.35 ms, the frequency of the m/n = 0/0 component is almost zero, suggesting that the excited mode is a zero frequency zonal flow.The EPM excited zonal flow has been found in other simulations [19,26], and theoretically investigated in [27][28][29].Figure 4(c) shows the whole process of the mode evolution.The zonal flow starts to grow when the EPM amplitude becomes large, and then, in the nonlinear saturation phase of the EPM, the EGAM grows exponentially, and the amplitude of the EGAM is much larger than that of the EPM and the zonal flow.
In [25], the fast ion D − α signal is used to detect the radial redistribution of EPs.By comparison with the TRANSP code calculation, it is clear that the EPs redistribute inwards.A similar result is obtained in the simulation, as shown in figure 5.This agreement between simulation and experiment confirms again the reliability of the present simulation.The redistribution during the EPM and zonal flow activity is relatively weak, but during the EGAM activity, the redistribution is strong.The difference in EP redistribution between the EPM and EGAM is mainly due to the very different amplitudes of these two modes.

The dependence of the mode properties on EP
Both the EPM and the EGAM are EP driven instabilities, and thus, their properties must be affected by EP. Figure 6 shows the dependence of the mode properties on EP pressure β EP .Figure 6(a) is the case of EPM.Although the EPM frequency slightly decreases as EP pressure increases, it can still be concluded that the EPM frequency is not sensitive to changes in EP pressure.The result of figure 6(a) is as expected.The conventional GAM frequency is about 40 kHz.Theoretically, the EGAM frequency is close to the conventional GAM frequency, and adjusted by the contribution of EP.The contribution of EP can be either positive or negative, but under the slowing-down distribution, the high-frequency branch of EGAM remains stable [15].The EGAM is excited by EPs, and the mode frequency is determined by the EP transit frequency.Normally, for the slowingdown EP distribution, the dominant resonant particle energy is relatively low, and the EP transit frequency is lower than the GAM frequency.Also, in figure 1 of [30], the energy and transit frequency of the resonant particles are low in the unstable region.As a result, the EP contributes negatively to the EGAM frequency, and thus, the EGAM frequency in the linear growth phase is lower than the conventional GAM frequency [14][15][16][17].While for the bump-on-tail EP distribution, the dominant resonant particle energy is relatively high.Also, in figure 1 of [30], the stable region does not exist and almost all the particles are in the unstable region.Thus, the EP transit frequency and EGAM frequency in the linear growth phase are higher than the GAM frequency [14,17].However, in the nonlinear saturated phase, the relationship between EP distribution and EGAM frequency is not clear.In figure 6(b), the EGAM frequency is higher than the conventional GAM frequency, suggesting that the nonlinear excited EGAM is a high-frequency EGAM.In general, the EGAM frequency in figure 6(b) increases with the EP pressure, and this confirms that the EP contributes positively to the EGAM frequency.The EGAM properties are very different from those in [15] and [17] because the EGAM in the present work is nonlinearly excited.According to figure 3(b) of [13], a very clear bump-on-tail EP distribution appears in a very localized region, which is the EGAM resonance region of the phase space.Although the EP distribution function is still slowingdown in general, the localized bump-on-tail EP distribution may be the reason why the high-frequency EGAM is excited.It is difficult to calculate the growth rate of EGAM, and thus, the maximum amplitude of EGAM is analyzed by instead.The maximum amplitude of EGAM increases with EP pressure, and this result is as expected.
Under the condition of slowing-down distribution, an anisotropic distribution in the pitch angle variable Λ phase space is required to linearly excite the EGAM.Although the EGAM in the present work is nonlinearly excited, an analysis of the dependence of mode properties on the EP pitch angle distribution is made, and the results are shown in figure 7. Figure 7(a) is the case of EPM.Although the EPM frequency and linear growth rate slightly decrease as the pitch angle Λ peak increases, it can still be concluded that the EPM frequency and linear growth rate are not sensitive to changes in EP pitch angle.Figure 7(b) is the case of EGAM.In general, the EGAM frequency in figure 7(b) decreases with Λ peak , because the EP transit frequency decreases.This result is similar to that in [17], and it is as expected.It seems that the maximum EGAM amplitude does not depend on the EP pitch angle.This may be due to the following two reasons.First, as mentioned in [13], the EGAM in the present case is a subcritical instability which is triggered by EPM but driven by EP free energy.The EP pitch angle does not affect the EP free energy, and thus, it does not affect the maximum EGAM amplitude.Second, the EGAM in the present case is nonlinearly excited through the resonance overlap [13].The resonance overlap has already been used to explain the nonlinear excitation of AEs [31].In the present work, in phase space, particles with EPM, and as the EPM amplitude grows, the size of the resonance region expands in phase space.Eventually, the EPM resonance region becomes very large, and reaches the EGAM resonance region.The EPM resonance region overlaps with the EGAM resonance region, and this resonance overlap is directly illustrated in figure 4 of [13].In figure 7(b), when the EGAM starts to grow, the distribution in phase space is already highly modified by EPM due to the resonance overlap.Then, the initial EP distribution in phase space does not directly affect the EGAM amplitude.
The dependence of the EPM properties on the EP radial distribution width is also analyzed, and shown in figure 8.Although the EPM frequency slightly decreases as the radial distribution width ∆Ψ 2 increases, it can still be concluded that the EPM frequency is not sensitive to changes in the EP radial distribution width.The growth rate appears to decrease as ∆Ψ 2 increases, because a narrower distribution represents a larger gradient.The results in figure 8 are as expected.The properties of the nonlinearly excited EGAM may not be directly determined by the initial EP distribution, and thus, it is not analyzed in order to save the computational resources.

Resonant particles
Both the EPM and the EGAM are driven by EP, and thus, it is important to investigate the resonance between these modes and the EPs.The resonance condition is given by where ω mode is the frequency of EPM or EGAM, n is the toroidal mode number, ω ϕ and ω θ are the EP toroidal and poloidal frequencies, and L is an arbitrary integer.Since the toroidal mode numbers of EPM and EGAM are n = −1 and n = 0 respectively, the above equation for EPM and EGAM can be re-written as and The L values of EPM and EGAM in the linear growth phase are analyzed, and shown in figure 9. L is calculated by (ω EPM + ω ϕ )/ω θ and ω EGAM /ω θ , respectively.However, for better comparison with the experiment, the angular frequency ω is re-written as 2π f, and in figure 9, f is shown, but not ω.Note that the EPM frequency is actually −103.5 kHz and not 103.5 kHz, although the differences between the negative and positive signs cannot be distinguished from the frequency spectrum in figure 1(a).Then, in figure 9(a), 128 particles with largest δf values are analyzed and plotted as red circles, and 512 particles with largest δf values are analyzed and plotted as green circles.For the 128 particle case, many red circles are located around the position of L = 0 and L = −1, in other words, the dominant L value in equation ( 19) can be both 0 and −1 for EPM.If the particle number is increased to 512, many green circles are also located around the position of L = 1, in other words, the dominant L value in equation (19) can also be 1 for EPM.In fact, if the number of particles is increased  to 4096, it is found that the number of particle in the intervals [0.9, 1.1], [−0.1, 1], and [−1.1, −0.9] is 1689, 798, and 110.In other words, L = 1 and L = 0 are more important for the EPM resonance than L = −1, and this is consistent with the analysis in [13].However, 4096 particles are too many to plot, and thus, only 512 particles are shown in figure 9(a).It can also be seen that as the number of particles increases, the gaps between the integer L values are gradually filled in.This is similar to that shown in [32] and [17], and it implies the existence of the fractional resonances [4,33,34].Moreover, the toroidal frequency of the particles with L = 1 and L = −1 is approximately 170 kHz and 70 kHz, and thus, it is easy to conclude that the poloidal transit frequency of these particles is approximately 70 kHz and 30 kHz.In figure 9(b), 512 particles with the largest δf values are analyzed and plotted as blue circles.These particles obviously do not accumulate near the region where L is an integer.This is as expected because EGAM does not grow in linear phase, and thus, the resonance between EGAM and EPs is not strong.
The analysis of resonant particles in 2-dimensional (Λ,E) phase space is important and informative for EGAM [30].In figure 10, the energy transfer from EPs to the modes in (Λ,E) phase space is shown at t = 0.30 ms, t = 0.40 ms, t = 0.56 ms, and t = 0.66 ms.The blue color represents δf × dE/dt < 0, in other words, the EPs lose energy, and the energy is transfered from EPs to the mode and the mode is destabilized.The red color represents δf × dE/dt > 0, and the modes are damped.The solid curves in figure 10 represent the constant poloidal transit frequency defined by f tr = √ 1 − Λv/(2π qR 0 ), where v is the particle velocity.The transit frequency calculated in this way is not very accurate, but can be used for qualitative analysis.In figure 10(a), it looks noisy because in the linear growth phase the energy transfer from the EPs to the mode is very strong.But even in this noisy figure, it can still be seen that the energy transfer occurs at f tr = 70 kHz.This implies that the energy transfer is mainly occurs between EPM and EP, because the resonant particle poloidal frequency of EPM is 70 kHz for L EPM = 1, as mentioned in figure 9(a).In figure 10(b), EPM is saturated and the energy transfer in phase space looks smooth.Same as that in figure 10(a), the transit frequency of the resonant particles are mainly 70 kHz.In figure 10(c), the resonance region (purple region) extends downwards.The lower part of the purple region reaches the line of f tr = 50 kHz, which means that a 50 kHz mode can be excited.Indeed, in figure 4(c), it can be seen that at t = 0.56 ms the m/n = 0/0 mode has transitioned from zonal flow to a periodically perturbed EGAM with frequency 50 kHz.In figure 10(d), the purple region at f tr = 50 kHz is very clear.This means that the 50 kHz EGAM has been saturated.This can be confirmed in figures 1(b) and 4(c).It can also be seen that the pitch angle of the dominant resonance region is around 0.45.This is due to the following two reasons.First, it is mentioned in section 2 that the peak value of the pitch angle is Λ peak = 0.4, and thus, the pitch angle of the dominant resonant particles is around this region.Second, as explained in [30], in (Λ,E) phase space, the destabilization region is on the right hand side of Λ peak in order to satisfy the condition of positive ∂f/∂E.

Summary and conclusions
In summary, the coexistence of Alfvén instability and EGAM in the AUG is successfully reproduced in the MEGA simulation.The basic mode properties such as mode frequencies and mode numbers are in agreement with the experiment.The radially inward redistribution of EPs during the mode activity is also similar to that observed in the experiment.Through careful analysis of the time evolution of the m/n = 0/0 harmonic, it is found that the EGAM is initially stable, then zonal flow gradually occurs with the growth of the EPM, and finally the EGAM is nonlinearly excited and the amplitude exceeds the EPM.A parameter scan is performed to better understand the mode properties.The frequencies of EPM and EGAM are neither sensitive to EP pressure β EP nor EP pitch angle Λ peak .Although EGAM is triggered nonlinearly by EPM rather than linearly driven by EP, the amplitude of EGAM still increases with EP pressure.It is interesting to note that the high-frequency branch of EGAM appears even under the condition of slowing-down EP distribution.This means that there are still many issues that need to be understood about highfrequency EGAM.Analytically, the earlier literature of G. Morales and T. O'Neil used the condition of simultaneous conservation of momentum and energy to explain the frequency decrease of electron plasma waves [35].Although it did not directly study EGAM, it provided a very valuable idea.Furthr computational and experimental studies of nonlinearly excited EGAM are still needed.The resonant particles are also analyzed.It is found that for EPM the dominant resonance condition is ω EPM = −ω ϕ + ω θ , and another resonance condition ω EPM = −ω ϕ is also important.These resonance conditions are consistent with those shown in the (P ϕ ,E) phase space in [13].By contrast, the dominant resonance particles of EGAM in the linear growth phase are not found because EGAM does not grow.In the pitch angle variable Λ and energy E phase space, it is found that initially EPM is excited by EPs with poloidal frequency 70 kHz, in other words, with resonance condition L = 1.Then, after the saturation of EPM, the resonant particles move downwards in the energy space and touch the EGAM resonance line, in other words, the line of f tr = 50 kHz.Then, the EGAM is excited by the particles with poloidal frequency 50 kHz.The above process is not as straightforward and accurate as that shown in the (P ϕ ,E) phase space of [13], but the story of the resonance overlap is similar.In addition, it is found that the pitch angle of the dominant resonant particles is around Λ = 0.45, because this is the destabilization region in the pitch angle phase space, and also, the value of the particle distribution f is large.Some further investigations are still needed.In the present work, the parameter scan was based on the same equilibrium file.In the future, it is worth trying to reconstruct the existing equilibrium, and then, more equilibrium parameters can be scanned, such as the background equilibrium beta, safety factor, etc.Also, the bulk ion kinetic effect is not considered, but this should be addressed in the future.
(1) The plasma major radius R 0 = 1.6857 m. (2) The magnetic field strength on the magnetic axis B 0 = 2.4944 T. (3) The electron density in the plasma center n e = 1.78 × 10 19 m −3 .(4) The electron temperature in the plasma center T e = 1.5 keV.(5) The injected neutral beam energy, or the birth energy of the neutral beam, is E NBI = 93 keV.(6) Both the bulk plasma and the EPs are deuterium.

Figure 1 .
Figure 1.The frequency spectrum of (a) the radial velocity vr of the m/n = 3/ − 1 mode, and (b) the poloidal velocity v θ of the m/n = 0/0 mode, respectively.

Figure 2 .
Figure 2. The mode structure of (a) the radial velocity vr of the m/n = 3/ − 1 and m/n = 2/ − 1 mode at t = 0.239 ms, and (b) the poloidal velocity v θ of the m/n = 0/0 mode at t = 0.622 ms, respectively.

Figure 4 .
Figure 4.The time evolution of the m/n = 3/1 harmonic of vr (green) and the m/n = 0/0 harmonic of v θ at (a) the initial stage, (b) the linear growth stage and (c) the whole stage.In figure (a), in order to better observe the change in EGAM, the m/n = 0/0 harmonic is increased by a factor of 4 when plotting.

Figure 5 .
Figure5.The EP pressure profile at t = 0 (red), t = 0.375 ms (green) when the EPM is active, and t = 0.656 ms (blue) when the EGAM is active.

Figure 6 .
Figure 6.The dependence of (a) EPM and (b) EGAM properties on EP pressure β EP .The red circles represent the mode frequencies, and the blue triangles represent the mode growth rate (EPM) or maximum amplitude (EGAM).In this figure, Λ peak = 0.4 and ∆Ψ 2 = 0.075.

Figure 6 (
Figure 6(b)  is the case of EGAM.The conventional GAM frequency is about 40 kHz.Theoretically, the EGAM frequency is close to the conventional GAM frequency, and adjusted by the contribution of EP.The contribution of EP can be either positive or negative, but under the slowing-down distribution, the high-frequency branch of EGAM remains stable[15].The

Figure 7 .
Figure 7.The dependence of (a) EPM and (b) EGAM properties on the EP distribution in the pitch angle variable Λ phase space.The red circles represent the mode frequencies, and the blue triangles represent the mode growth rate (EPM) and maximum amplitude (EGAM).In this figure, β EP = 0.4% and ∆Ψ 2 = 0.075.

Figure 8 .
Figure 8.The dependence of the EPM properties on the EP radial distribution width ∆Ψ 2 .The red circles represent the mode frequencies, and the blue triangles represent the mode growth rate.In this figure, β EP = 0.4% and Λ peak = 0.4.

Figure 9 .
Figure 9. L values of the resonance condition.In figure (a), L = (−103.5+ ω ϕ )/ω θ , the red circles represent the 128 particles with the largest δf values, and the green circles represent the 512 particles with the largest δf values.In figure (b), L = 51.5/ωθ , and the blue circles represent the 512 particles with the largest δf values.The green circles in figure (a) and the blue circles in figure (b) are the same.