Analysis of beam ion driven Alfvén eigenmode stability induced by Tungsten contamination in EAST

Alfvén eigenmodes (AE) activity is observed in the EAST high β N and low BT discharge 93910, operation scenario dedicated to explore the ITER baseline scenario. AEs are triggered after the plasma is contaminated by Tungsten that causes an abrupt variation of the thermal plasma and energetic particles (EPs) profiles. The aim of the present study is to analyze the AE stability in the 93910 discharge using the gyro-fluid code FAR3d, identifying the AE stability trends by comparing the plasma before and after the Tungsten contamination. Tungsten contamination causes the destabilization of Toroidal AEs (TAE) and Energetic particle modes (EPMs) in the same frequency range and radial location with respect to the experimental observation and M3D-K/GTAW code results. Next, a set of parametric studies are performed to analyze the effect of the thermal plasma and EP parameters on the AE stability. The analysis indicates a lower EP β threshold for the AEs destabilization if the EP energy increases, an improved AE stability of on-axis NBI configurations due to the stronger continuum damping in the inner plasma region as well as a large enhancement of the EP drive as the thermal ion density increases due to a higher ratio of the EP and Alfven velocities. Consequently, the simulations indicate the increment of the thermal ion density after the Tungsten contamination could be the main cause of the AE/EPM destabilization.


Introduction
Energetic particle (EP) driven instability is a critical issue in future burning plasma experiments.EPs as fusion-born alpha particles or energetic ions generated by Neutral Beam Injectors (NBI) and Ion Cyclotron Heating (ICH) can reach energies up to the MeV range [1,2].Such EPs can resonate with Alfvén waves (AW) and destabilize Alfvén eigenmodes (AEs), leading to a decrease of the fast ion confinement and a lower plasma heating efficiency [3][4][5][6][7][8].
Resonance occurs when the characteristic velocity of EPs' orbits is similar to the phase velocity of Alfven waves, thus the EP energy is transferred to the AW and the AEs can be destabilized [1,9,10].Nevertheless, the AW dispersion is continuous in a cylindrical system thus the perturbations are damped and the AEs are stable.This effect is called continuum damping [11,12].On the other hand, due to the magnetic field's asymmetry within the same magnetic surface of a toroidal system, the continuum is disrupted, leading to the formation of gaps.Inside the gaps the continuum damping is weakened and the AEs inside the gap would be destabilized easier compared to the modes inside the continuum.Consequently, AEs are triggered in discrete frequency ranges and radial locations [13,14].Different AE families exists with respect to the frequency gaps such as the toroidicity induced AE (TAE) [3,[15][16][17][18] coupling two adjacent poloidal modes; reverse shear AE (RSAE) [19][20][21] triggered at the minimum of non-monotonic q profiles; beta induced AE (BAE) [22][23][24] induced by finite compressibility effects.Besides AEs, there is another instability triggered by EPs called energetic particle mode (EPM) [1,2,14] destabilized inside the continuum if the EP driving is large enough to overcome the continuum damping effect.
The Experimental Advanced Superconducting Tokamak (EAST) is a superconducting tokamak with major radius R ≈ 1.85 m and minor radius a ≈ 0.45 m.It has a ITER like magnetic configuration and it is designed to study the physics and engineering issues of long pulses and fully non-inductive operations.EAST plasmas are heated by Lower Hybrid Waves (LHW), Electron Cyclotron Resonance Heating (ECRH) and Ion Cyclotron Resonance Heating (ICRH) systems as well as two tangential NBI beam lines with four ion sources.Each NBI ion source can provide up to 4 MW power with a voltage about 80 kV [25].Several numerical studies are dedicated to analyze the AE activity in EAST operation scenarios, particularly for the identification of the unstable modes and their kinetic features [26,27].AEs triggered by EPs are hardly observed in EAST discharges, mainly destabilized during accidental events of impurity contamination that induce large fluctuation of the plasma parameters.Consequently, a further exploration of the AE activity in EAST plasma requires numerical methods to identify those configurations particularity unstable to EP driven instabilities.
The aim of the present study is to analyze the AE/EPM activity in the EAST discharge 93910.In the frequency range between 50 and 100 kHz, the plasma is contaminated by Tungsten.The study is performed using the gyro-fluid code FAR3d [28][29][30], reproducing the AE/EPM activity observed in the experiment.In addition, AE/EPM stability dependencies with respect to EP β, energy and density profile as well as the thermal plasma density profile and safety factor profile is studied by performing a set of parametric studies, identifying the AE/EPM stability trends with respect to the NBI operational regime, thermal plasma and magnetic configuration properties.Such parametric studies provide useful information to EAST operation scenarios showing weak AE activity and improved heating performance as well as configurations with robust AE activity dedicated to the study of EP driven modes triggered by the NBI and ICRF system.Moreover, the parametric studies provide a new tool to calibrate EP profiles in EAST configurations, improving the accuracy of EP density and energy profiles calculated by ONETWO/NUBEAM to reproduce the AE/EPM activity observed in the experiment.
FAR3d is a gyro-fluid code that solves a set of linear and nonlinear reduced resistive MHD equations coupled with moments of the gyro-kinetic equation, particularly the energetic ion density and parallel velocity, adding the effect of the Landau growth/damping [28][29][30].Continuum damping effect is implicitly included in the model.Because of the fixed EP distribution function we have chosen, the destabilizing effect of the EP resonances is analyzed with respect to the ratio between EP and Alfven velocity.For a fixed thermal plasma density, magnetic field intensity and EP density profile, the velocity ratio provides an approximation of the drive caused by EP populations with different energies.The FAR3d code is routinely applied to study the AE/EPM stability in several devices such as DIII-D [31][32][33][34][35][36], LHD [30,[37][38][39]59], TJ-II [40][41][42], and Heliotron J [38,60].
This paper is organized as follows.Section 2 provides the numerical scheme and the experiment setup in this study.Section 3 is dedicated to analyze the AE stability before and after the Tungsten contamination.Section 4 studies the AE stability with respect to the EP configuration and calibration of the EP model profiles.Section 5 shows the conclusions and discussion of the analysis results.

Equilibrium description and simulations parameters
The AE stability modeling is performed for the shot 93910.The plasma heating in this discharge is provided by tangential NBIs and 4.6 GHz LHW.The main parameters of the shot are set as: magnetic field intensity (B T ) of 1.6 T, toroidal plasma current (I p ) of 400 kA, thermal electron line averaged density n e ≈ 2.5 × 10 19 m −3 and q 95 ≈ 4.0.Two ion sources of the NBI system are used in the discharge for a total power injection of 5 MW with a voltage of 65 kV (below the maximum injection power and voltage capability of the system).LHW power is 1 MW.There is a Tungsten contamination caused by a small piece of tile at t = 6.0 s of the discharge time once several AEs are excited [16].The analysis is performed at the discharge times t = 4.5 and 6.5 s, before the Tungsten contamination and after the Tungsten contamination once the maximum AE activity is observed.
The FAR3d code solves the linear and nonlinear reduced resistive MHD equations for the thermal plasma coupled with equations of the EP density and parallel moments [28][29][30], including the effect of the linear wave-particle resonance required for Landau damping/growth as well as the response of the thermal plasma parallel moments for the coupling to the geodesic acoustic waves.The model calibration requires the derivation of Landau closure coefficients obtained from gyrokinetic simulations, matching the analytic TAE growth rates of the two-pole approximation of the plasma dispersion function, consistent with a Lorentzian energy distribution function for the EPs.The lowest order Lorentzian can be matched either to a Maxwellian or to a slowing-down distribution by choosing an equivalent average energy [43] .Please see the [44,45] for further details of the model equations and numerical scheme.The eigensolver version of the code is used, providing the growth rate and frequency of dominant (largest growth rate) and sub-dominant modes.It should be mentioned that the analysis is dedicated to calculate the linear stability of AE/EPM, thus there is no feedback of the AE/EPM in the equilibria and the model profiles are fixed.In addition, the equilibria is not recalculated consistently with the EP β of the simulation, that is to say, the effect of the EP pressure is not considered in the equilibria reconstruction.
The plasma equilibrium of the discharge 93910, Deuterium plasma heated by Deuterium beams, is calculated using the kinetic-EFIT code [46,47] and transferred to VMEC code format [48].The kinetic profiles used in the kinetic-EFIT are measured by the EAST diagnostics and the pressure balance is constrained by ONETWO/NUBEAM code [49,50].The electron temperature (T e ) is measured using Thomson scattering and electron cyclotron emission (ECE) [51,52] with high temporal and spatial resolution.The electron density is measured by polarimeter interferometer (POINT) system [53] and T i is measured by Charge Exchange Recombination Spectroscopy (CXRS) [54].The simulations equilibria is not recalculated in the parametric studies modifying the EP β and energy for simplicity.
Figure 1 indicates the main profiles used in the FAR3d model before (black line) and after (red line) the Tungsten contamination.Panel (a) indicates the q profile, panel (b) the thermal plasma density, panel (c) the thermal plasma temperature (solid line electrons and dashed lines ions) and panel (d) the EP β profile.Panels (e) and (f ) show the continuum gaps calculated using Stellgap [55] before and after the Tungsten contamination, indicating similar structures with wide continuum bands between the frequency ranges of 40 to 100 kHz in the inner-middle plasma region.The reference case is the model for t = 6.5 s.
The EP profiles are calculated using the ONETWO/ NUBEAM code and calibrated by parametric studies with respect to the EP β, EP energy and EP density profile.EP density profile modeling is performed using the following analytic expression: The location of the EP density gradient is controlled by the parameter (r peak ) and the stiffness by the parameter (δ r ). Figure 2 show the normalized EP profile used in the reference case.
It should be noted that this model would produce a flat EP density profile in the core region but not a hollow one.This helps to remove from the analysis marginal unstable AEs but the contribution of the hollow profile is neglected in this simulation.
The EP energy in the reference model is 30 keV (V f0 /V A0 = 0.28).The Alfven velocity in the magnetic axis is v a = 5.29 × 10 6 m s −1 .The analysis of the EP energy effect on the AE stability is further discussed in section 4. Previous analysis indicate the resonance V f /V A = 1/3 leads to the destabilization of TAEs.The resonance linked to the AE/EPM destabilization can be expressed as ω = nω ζ − sω θ .
Eigenfunction(f) in FAR3d code are presented in terms of sine and cosine component, using real variables: In this paper, the cosine component of the eigenfunction is indicated by positive mode number and the sine component by negative component.FAR3d simulations are performed using a polytropic index of 1.66 except the reference case of 1.0 (1.0 for Stellgap simulations), 1000 radial grid point and a magnetic Lundquist number of 10 7 .
The discussion of the main differences between the simulations performed using FAR3d and M3D-K/GTAW codes is performed in the appendix.

Effect of the Tungsten contamination on the AE stability
This section is dedicated to reproduce the instabilities observed in the experiment and the characterization of the AE/EPM activity before (t = 4.5 s) and after (t = 6.5 s) the Tungsten contamination.
Figure 3 shows the AE activity observed in the discharge from t = 6 s to 7 s.Panel (a) indicates the Mirnov coil data and panel (b) indicates the toroidal number of the modes.Panel (c) indicates the ECE data at ρ = 0.46.The toroidal mode number of the AEs are calculated using the toroidal Mirnov coils data [56].The AE activity is divided in two branches.The first branch has almost constant frequency around 90 kHz and there is second branch showing a frequency up-shift from 60 kHz to 90 kHz.In addition, there is another instability around 60 kHz.The stability of the toroidal families n = 1 to 4 are analyzed, although only n = 2 and 3 AE/EPM are unstable.The study is limited to the mode with the largest growth rate (dominant mode) and one sub-dominant mode (panel (a)) because the free energy available to trigger instabilities is channeled towards the modes with the largest growth rates.It should be noted that the FLR effects are not added in the simulations for simplicity, reason why the n = 3 EPM/TAE shows a growth rate larger compared to the n = 2 EPM and TAE.The growth rate of the n = 3 EPM/TAE is largely reduced by the FLR damping effect due to its rather slender eigenfunction (see the appendix for further information).The eigenfunction in panel (e) shows a transitional mode between 6/3 EPM and 6/3-7/3 TAE that may cause an overestimation of the mode's growth rate.The n = 2 dominant mode has a frequency around 90 kHz and the sub-dominant mode around 60 kHz, similar to the experimental observations.The eigenfunction of the dominant mode (panel (c)) indicates an transitional mode of 3/2-4/2 TAE and 4/2 EPM, reason why the mode has a partially intersection with the continuum.It should be noted that the radial location of the 2/3 − 2/4 TAE at ρ ≈ 0.45 is very close to the perturbation measured by the ECE diagnostic at ρ ∼ [0.45, 0.5].The eigenfunction of the sub-dominant mode (panel (d)) also indicates an instability located at ρ ≈ 0.5 although triggered by the single 4/2 mode.Again, the radial location of the instability is consistent with the ECE diagnostic measuring a perturbation at ρ ≈ 0.47.The radial location and frequency range of the modes with respect of the continuum (panel (b)) indicates the dominant mode is triggered inside the TAE gap in the middle plasma region, although the sub-dominant mode is destabilized inside the continuum, thus the sub-dominant mode is an EPM.
The simulations before the Tungsten contamination indicate n = 1 to 4 AE and EPM are stable, consistent result with the experiment observations.Consequently, the AE/EPM destabilization after the Tungsten contamination is caused by  an enhancement of the EP drive as the plasma thermal ion density increases, explained by a larger ratio of the EP and Alfven velocities (EP drive is stronger as the EP velocity is closer to the Alfven velocity).Despite low thermal plasma temperature, higher thermal ion density does lead to increased e-i Landau damping.However, this increment is insignificant compared to the enhancement of EP drive.Nevertheless, in a plasma with a higher thermal temperature, these effects should be taken into consideration.
The next step of the analysis is to identify the AE stability trends with respect to the EP and thermal plasma main profiles.

Analysis of the AE stability trends and EP model calibration
This section is dedicated to studying the AE stability trends by performing a set of parametric scans.In addition, the EP profiles are calibrated comparing the FAR3d simulation results and experiment observations, reason why the analysis begins with the profiles calculated by ONETWO/NUBEAM code.The AE stability is analyzed with respect to the EP energy, EP β, EP radial density profile, thermal ion density and magnetic configuration (q profile).

Effect of the EP β on the AE stability
First, the AE stability is analyzed with respect to the EP β.A higher EP population leads to a stronger plasma perturbation.Such enhancement of the EP drive is reproduced in the simulation as an increment of the EP β, parameter linked to the injection power of the NBI.An increase of the EP β corresponds to an upscale of the EP density because the EP energy of the simulations is fixed.The high EP energy in the plasma periphery is caused by setting Zeff constant, leading to an increase of fast ion pressure in the plasma edge [46].Nevertheless, we could see in figure 1(d) that the fast ion pressure and EP density gradient are rather small, thus the AE stability of the system is not affected (AEs are not triggered in the plasma periphery).
Low frequency modes, below 20 kHz, are observed if the EP β is smaller than 0.065 (panels (b) and (c)), pointing out the destabilization of BAE showing a lower threshold than the TAE.On the other hand, once the TAE is triggered, the TAE's growth rate is larger and the low frequency modes, still unstable, are sub-dominant.The EP β required for the destabilization of a mode in the frequency range of the experiment is 0.065, triggering a 2/3 − 2/4 TAE with a frequency of 71 kHz in the middle plasma region.An EP β of 0.065 is 3 times larger compared to the value calculated by the ONETWO/NUBEAM code (≈ 0.02).In the following sections such inconsistency is analyzed performing parametric studies with respect to the EP energy and radial density profile.

Effects of EP energy on the AE stability
Now the AE stability is analyzed with the respect to the EP energy.When we change the β f , the T f remains unchanged.This implies increasing the EP beta leads to an upscaling of the EP density profile.On the other hand, in the simulations with different EP energies, for a given EP beta, the EP density decreases/increases proportionally to the EP energy variation.Thus, simulations with different EP energy have the same EP density if cases with a proportional increase of the EP beta are compared.The EP drive/damping is analyzed with respect to the resonance 'intensity' or 'efficiency', linked to the ratio between the EP velocity (V f ) to the Alfvén velocity (V a ).For a fixed EP density profile, simulations are conducted by modifying the EP energy (T f ), which provides information of the EP drive caused by different energy levels.As the velocity ratio increases, the EP drive is stronger, maximum once the ratio is the unit.It should be noted that the EP drive/damping is determined by the gradient of the phase space distribution and the gradient depends on the phase space shape of the distribution function of the EP.Thus, the velocity ratio should be understood as simplified index for the resonance description.Nevertheless, this information is useful for future optimization studies.The ratio of V f /V a for each T f is showed in table 1.The EP β threshold for the destabilization of n = 2 and n = 3 AE/EPM with frequencies similar to the experiment decreases from 0.065 to 0.045 as the EP energy increases.In addition, the AE/EPM growth rate increases with the EP energy for the same EP β.Consequently, the EP drive is stronger as the EP energy increases leading to a lower EP β threshold for the AE/EPM destabilization.

Effects of the EP radial density profile on the AE stability
Next, the effect of the EP radial density profile on the AE stability is analyzed.The EP density profile depends on the radial location where the NBI is deposited as well as the penetration of the beam inside the plasma.The AEs are triggered in the plasma region with large EP density gradients.The parametric analysis is performed modifying the radial location of the EP density gradient (r peak ) linked to the NBI deposition region as well as the stiffness of the profile (δ r ) related with the beam penetration.If the EP density gradient is located nearby the axis the model reproduces an on-axis NBI operation.If the EP density profile is located outwards, the NBI operation is offaxis.On the other hand, if the stiffness of the profile increases and the gradient is larger, the model represent an NBI operation with an efficient beam penetration leading to a radially localized EP population.To avoid the destabilization of secondary AE/EPM with lower growth rate compared with the dominant instability, the EP density profile is artificially flattened away from the radial plasma location where the analysis is performed.
Figure 7 shows the parametric analysis with respect to the radial location of the EP density gradient for the n = 2 AE/EPM.Panel (a) shows EP profiles with the same gradient stiffness (δ r = 3) although different radial locations (r peak = 0.05, 0.15 and 0.25).Panels (b) and (c) show the frequency and growth rate of the dominant modes in simulations with different r peak and EP β values.
Panel (c) indicates an increment of the dominant mode growth rate as the EP density profile gradient moves from the magnetic axis to 0.25, showing that an off-axis NBI injection lead to a stronger destabilization of the AEs.The improved AE stability in the on-axis NBI injection (r peak = 0.05) is explained by a stronger continuum damping nearby the magnetic axis due to the presence of wide continuum band, decreasing as the NBI is deposited off-axis (see figure 7 panel (b)).Likewise, the frequency of the dominant modes in the off-axis simulations (r peak = 0.15 and 0.25) is similar to the experiment for a lower EP β compared to the simulation with on-axis NBI injection.
Figure 7 panels (d)-(f ) show the eigenfunction of three EP density profiles with β f = 0.065.The perturbation moves outwards as the NBI is deposited further off-axis.The simulation for r peak = 0.05 indicates a dominant 3/2-4/2 TAE in the middle plasma as well as a lower amplitude 2/2-3/2 TAE near the magnetic axis.On the other hand, in the r peak = 0.25 case the lower amplitude mode is a 4/2-5/2 TAE at the plasma periphery.
In the following analysis the effect of the EP density stiffness on the AE stability is analyzed.The growth rate of the dominant mode (panel (c)) increases as the EP density gradient enhances, consistent with the increment of the free energy available to trigger AEs.In addition, dominant modes with a frequency similar to the experiment are observed at lower EP β values, closer to ONETWO/NUBEAM profiles prediction.
Summarizing, the parametric studies performed with respect to the EP β, energy and density profile provide a method to calibrate the EP profiles obtained from ONETWO/NUBEAM simulations, leading to an improved configuration of the EP model and more accurate simulations outcome with respect to the experimental observations.

Effects of thermal ion density on the AE stability
This section is dedicated to analyze the effect of thermal ion density on the AE/EPM stability linked to the Tungsten contamination.The simulations are performed using an EP β = 0.06, larger that the experimental value, to show more clearly the effect of the thermal ion density on the AE stability.Thermal ion density influences the AE/EPM stability by changing the ratio of fast ion and Alfven velocities.
Figure 9, panel (a), shows the thermal ion density profiles used in the simulations.Panel (b) indicates the growth rate and frequency of the dominant n = 2 and 3 AE/EPM for different thermal ion density profiles.The AE growth rate is 400% larger if the thermal ion density increases by 50%, thus the EP β threshold required for the AE destabilization is lower in the simulations with larger thermal ion density.This is explained by a enhancement of the EP drive as the ratio of fast ion and Alfven velocities increases.Panels (c) and (d) shows the continuum gaps of the n = 2 and 3 toroidal mode families for different thermal ion density profiles, respectively.There is a decrease of the continuum gaps frequency range as the thermal ion density increases that explains the decrease of the AE/EPM frequency observed in the simulations.

Effects of q profile up-shift on the AE stability
This section shows the effect of the q profile up-shift on the AE/EPM stability.It should be noted that this part of the study does not intend to reproduce the variation of the q profile predicted before after the Tungsten contamination, because such effect was already included in the analysis performed in section 3 (see figure 1, panel (a)).The aim of this analysis is the identification of optimization trends with respect to the q profile and the AE/EPM stability.
Figure 10, panel (a), shows the q profile in the simulations.Panels (b) and (c) show the frequency and growth rate of the dominant n = 2 and 3 modes.The up-shifted q profile  decreased the EP β threshold for AE destabilization.The continuum in panels (d) and (e) show the up-shifted q profiles lead to a TAE gap at a lower frequency range and TAEs with larger growth rates.In addition, up shifted q profiles avoids the 1/1 mode resonance and the destabilization of sawtooth and fishbones [57,58].Consequently, EAST discharges with a higher q minima may lead to an stronger destabilization of the AE instability.

Summary of the AE stability trends
The parametric studies indicate the variation of the thermal ion density after the Tungsten contamination may cause the AE/EPM destabilization.On the other hand, the evolution of the thermal electron temperature leads to a minor effect on the AE/EPM stability towards the variation of the plasma resistivity, because the e-i landau damping effect is negligible in a plasma with low thermal temperature.
The up-shifted q profile is found to decrease the AE destabilization threshold by significantly modifying the continuum while in the real experiment the q profile varies less than the parametric study.The enhancement of the EP drive as the thermal ion density increases is the dominant effects of the destabilization.
The analysis indicates an improvement of EAST heating efficiency if the NBI injection is on axis and the NBI voltage is reduced, leading to a increment of the EP β threshold required for the AE/EPM destabilization, 0.02 after the Tungsten contamination.

Conclusions
FAR3d simulations reproduce and characterize the instabilities observed in the EAST shot 93910.The analysis also explains the destabilization of AEs after the Tungsten contamination and identifies optimization trends with respect to the AE stability required to improve the heating efficiency of the EAST plasma heated by tangential NBIs.FAR3d simulations results show a reasonable agreement with the calculations performed using M3D-K/GTAW code [16].It should be noted that the optimization is based on the weakening or stabilization of the AE activity.The configuration with improved AE stability may show an improved performance with respect to the plasma heating.Moreover, the nonlinear simulations are required to explore the impact on the EP transport in the heating performance and this will be the topic of future studies.
The simulations indicate the instabilities observed after the Tungsten contamination, at the discharge time t = 6.5 s, correspond to a 2/3 − 2/4 TAE with a frequency around 90 kHz and a 2/4 EPM with a frequency around 60 kHz.The 2/3 − 2/4 TAE radial location (ρ ≈ 0.45) is similar to the perturbation measured by the ECE diagnostic (ρ ∼ [0.45, 0.5]).The 2/4 EPM is also destabilized in the middle plasma, at ρ ≈ 0.5, radial location consistent with the ECE data that observes a perturbation at ρ ≈ 0.47.The simulations performed before the Tungsten contamination indicate stable n = 1 to 4 AE/EPMs.
The increase of the thermal ion density after the Tungsten contamination may explain the destabilization of the AE/EPM along the 93910 discharge.The increment of the thermal ion plasma leads to a decrease of the plasma Alfvén velocity that enhances the EP drive and reduces the EP β threshold for the AE destabilization.The AE/EPM growth rate increases 400% if the thermal ion density is 50% larger.On the hand, the decrease of the thermal plasma temperature shows a minor effect on the AE/EPM stability towards an increment of the plasma resistivity.Likewise, the up-shifted q profile leads to a further destabilization of the AEs.
The AE stability optimization trends indicate that, an increment of the NBI voltage (EP energy) leads to a further destabilization of the AE/EPM.If the tangential NBI power injection is strong enough to increase the EP β ⩾ 0.02, n = 2 AE/EPM are destabilized.In addition, the simulations also predict the destabilization of n = 3 TAE/EPM.
The simulations indicate an on-axis NBI injection may improved the AE/EPM stability compared to the off-axis NBI injection, explained by a stronger continuum damping at the inner plasma region due to a wide continuum band.However, the simulation did not take into account the influence of a hollow EP density profile, which is commonly observed in the real off-axis NBI heating regime.This will be addressed in the future work.The simulation of q profile also shows that the up-lifted q profile would make the AE destabilized much easier.Experimentally, in the low B t and high β N discharge on EAST, a q profile with lower q min would help to suppress the AE activity by avoid the resonance happened in the wide continuum gap.
The parametric studies provides a method to calibrate the EP profiles calculated by ONETWO/ NUBEAM code.The improved EP profiles provide a better fit of the simulations results compared to the experiment observations.
Previously, various codes such as MEGA and M3D-K have been used to analyze the AE instability on EAST [26,27].Present study is dedicated to investigate the influence of the plasma contamination by Tungsten on AE activity in order to address current issues in EAST.Additionally, the AE stability is analyzed with respect to the NBI operational regime, thermal plasma and magnetic field configuration properties of EAST device, identifying scenarios with low AE activity and improved plasma heating efficiency.Future analysis will be dedicated to characterize the AE stability in EAST discharges heated by a combination of tangential NBIs and ICRF [61].Such plasma will host multiple EP populations that may lead to a destabilization of AEs/EPMs, thus further studies are required to optimize the EAST heating efficiency.

Appendix. FAR3d and M3D-K/GTAW codes comparison
The analysis performed by the codes FAR3d and M3D-K/GTAW are compared, showing consistent results.Nonetheless, different code implementations and inputs lead to some deviations between the simulations outcome.FAR3d and Stellgap code uses a VMEC equilibria transformed from an EFIT, thus q profile (figure 11) and continuum gaps show small differences.Thermal plasma profiles are identical in both codes' simulations.
Figure 12 shows the EP density profile in FAR3d simulations obtained after the parametric study and EP profiles calibration.The EP density profile is flattened nearby the magnetic axis and the plasma periphery to remove from the analysis marginal unstable AEs and focus the study in the modes triggered in the inner-middle plasma region.
The main discrepancy between the codes results is the identification of an unstable 2/3 EPM by M3D-K/GTAW simulation although FAR3d simulation finds the 6/3 EPM or 6/3 − 7/3 TAE is the fastest growing mode.Nevertheless, n = 3 is a transitional mode caused by the code limitation to distinguish between individual EPM and TAE, leading to an overestimation of the mode growth rate.2/4 − 2/3 TAE is the second fastest growing mode.On the other hand, the strong n = 3 dominant mode destabilization is partially consequence of not including the effect of the EP and thermal ion FLR damping effects in the simulation, that causes a large decrease of the n = 3 EPM/TAE growth rate due to the slender width of the eigenfunction.Figure 13 shows the growth rate of n = 2 and 3 dominant mode (purple and orange box in figure 4 panel (a)) including the effect of the FLR thermal ion and EP dampings.In addition, the n = 3 mode is damped to be stable in figure 13

Figure 1 .
Figure 1.Profiles before and after Tungsten contamination.t = 4.5 s of red line and t = 6.5 s of black line (a) q profiles in VMEC (b) Electron density profiles.(c) Thermal temperature for ion(dashed line) and electron(solid line).(d) EP β used in FAR3d simulation.(e) Alfvén continuum gaps calculated by Stellgap for t = 4.5 s and (f ) for t = 6.5 s, Black dot for n = 1, red for n = 2, blue for n = 3 and cyan for n = 4.

Figure 4 ,
panel (a), shows the growth rate and frequency of the modes identified in FAR3d simulations after the Tungsten contamination.Panel (b) indicates the continuum gaps including the radial location and frequency range of the modes calculated in the simulations.Panels (c) and (d) show the eigenfunction of the n = 2 modes highlighted in the panel (a): purple box in panel (c) and pink box in panel (d).

Figure 4 .
Figure 4. (a) Growth rate and frequency of the n = 2 dominant mode (purple box) and sub-dominant mode with the largest growth rate (pink box) Purple dot of n = 2 and orange dot of n = 3.(b) Continuum gaps including the radial location and frequency range of the modes highlighted in panel (a).Black dot for n = 1, red for n = 2, blue for n = 3 and cyan for n = 4. (c) Eigenfunction of the n = 2 dominant mode.(d) Eigenfunction of the sub-dominant mode with the largest growth rate.(e) Eigenfunction of the n = 3 dominant mode.

Figure 5 ,
panel (a), indicate the radial profiles of the EP density and energy calculated by ONETWO/NUBEAM code.

Figure 5 .
Figure 5. AE stability with respect to the EP β.Circle marker for n = 2 and diamond marker for n = 3 (a) Radial profiles of the EP density and energy calculated by ONETWO/NUBEAM code.Dominant mode (b) frequency and (c) growth rate for different EP β values.(d) the eigenfunction of low frequency mode of n = 2. (e) and (f ) the eigenfunction of high frequency mode around 80 kHz of n = 2 and n = 3.

Figure 6 ,
panel (a), shows the different T f profiles used in the simulations.Panels (b) and (c) ((d) and (e)) indicate the frequency and growth rate of the n = 2 (n = 3) dominant mode in simulations with different EP energies and EP β.

Figure 6 .
Figure 6.AE stability with respect to the EP energy and β.(a) EP energy profiles, dashed line shows the location of dominant mode.Dominant n = 2 mode (b) frequency and (c) growth rate.Dominant n = 3 mode (d) frequency and (e) growth rate.

Figure 8 ,
panel (a), shows EP density profiles with the same radial location of the gradient r peak = 0.45 and different stiffness (δ r = 5, 7.5 and 10).Panels (b) and (c) indicate the frequency and growth rate of the dominant mode, respectively.

Figure 7 .
Figure 7. AE stability with respect to the radial location of the EP density gradient, n = 2 of circle marker and n = 3 for diamond marker.(a) EP density profiles if the EP density gradient is located at 0.05, 0.15 and 0.25 if δr = 3.(b) Dominant mode frequency and (c) growth rate for different EP β values and radial locations of the EP density gradient.(d)-(f ) The eigenfunction of dominant mode in three density profiles with β f = 0.065 of n = 2 and (g) their locations in continuum.

Figure 8 .
Figure 8. AE stability with respect to the stiffness of the EP density profile.(a) EP density profiles if the EP density profile stiffness is 5, 7.5 and 10 for r peak = 0.45.Dominant mode (b) frequency and (c) growth rate for different EP β values and EP density profile stiffness.

Figure 9 .
Figure 9. AE stability with respect to the thermal ion density.Black line indicates n i0 ≈ 1.7 × 10 19 m −3 , green line of n i0 ≈ 2.3 × 10 19 m −3 and orange line of n i0 ≈ 2.7 × 10 19 m −3 .(a) Thermal ion density profiles.(b) Growth rate and frequency of the dominant mode for different thermal ion density profiles, circle marker of n = 2 mode and diamond marker of n = 3. (c) Continuum gaps of n = 2 mode.(d) Continuum gaps of n = 3 mode.

Figure 10 .
Figure 10.AE stability with respect to the q profile.Circle marker for n = 2 and diamond for n = 3.(a) the up-shifted q profile.(b) growth rate and (c) frequency of up-shifted q profile.(d) and (e) the continuum of three q profile of n = 2 and n = 3.
This work is supported by the project National Key Research and Development Program of China No. 2019YFE03020004, National Natural Science Foundation of China under Grant No. 11975276, Anhui Provincial Natural Science Foundation No. 2008085J04, Anhui Provincial Key R&D Programmes No. 202104b11020003 and the Excellence Program of Hefei Science Center CAS No. 2021HSC-UE015.The work is also supported by the Comunidad de Madrid under the project 2019-T1/AMB-13648.The numerical study in this paper was performed on the ShenMa High Performance Computing Cluster in Institute of Plasma Physics, Chinese Academy of Sciences.
once the EP Larmor radius is increased larger than ρ i = 7.5 × 10 −3 m.The simulations indicate a larger decrease of the n = 3 AE/EPM growth rate compared to the n = 2 TAE.

Figure 13 .
Figure 13.Frequency (a) and growth rate (b) of FLR effects in different Larmor radius (Normalized to minor radius).Diamond marker for n = 3 and circle marker for n = 2.

Table 1 .
Ratio of V f /Va for each T f .