Simulations of beta-induced Alfvén eigenmode mitigation by off-axis energetic particle distribution

The effect of different off-axis energetic particle (EP) slowing down distribution on beta-induced Alfvén eigenmode (BAE), driven by the on-axis EP distribution, is systematically studied using kinetic-magnetohydrodynamic code M3D-K. The aim is to analyze the optimal parameter region for controlling AEs via varying EP distribution parameters. The simulation results reveal that by modifying the gradients of the EP distribution, the off-axis EP can further destabilize or mitigate the on-axis EP driven BAE, depending on the off-axis EP distribution’s parameters: deposition profile, EP beta, pitch angle, injection velocity and direction. When the off-axis EP is deposited outside the mode center, and its injection velocity is sufficiently large to satisfy the resonance with BAE, the stabilization of BAE is achieved. This stabilizing effect is directly proportional to the off-axis EP beta, while excessive off-axis EP beta can trigger a new EP-driven instability located outside the BAE. Furthermore, to achieve a stronger stabilizing effect, the pitch angle distribution and velocity direction of the off-axis EP should be close to those of the on-axis EP. For instance, compared to the off-axis counter-passing EP, the off-axis co-passing EP can lead to a more effective mitigation of the BAE driven by the on-axis co-passing EP.


Introduction
Alfvén eigenmodes (AEs), existing inside the spectral gaps of the shear Alfvén continua and extensively observed in Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.fusion experiments, are crucial to the performance of burning plasma [1,2].For example, the beta-induced Alfvén eigenmode (BAE), located in the continuum gaps caused by the combination of finite plasma beta and geodesic curvature [3,4], and the toroidal Alfvén eigenmode (TAE), located in the toroidicity-induced continuum gaps [5,6].These AEs can be driven unstable by the energetic particle (EP) that are produced by fusion reactions and auxiliary heating, and in turn induce large transport and losses of EP.The EP transport and losses could deteriorate the plasma confinement and reduce the heating efficiency, even relatively small EP losses could damage the first wall and must be avoided [7,8].In the early experimental studies of TAE, over 50% of the injected beam power was lost in the presence of violent TAE activity, which leads to a significant ablation of plasma facing components [9].In future burning plasma, the AEs are prone to being excited by the alpha particles [10,11], potentially leading to unacceptable consequences if not mitigated.Hence, it is important to mitigate or even fully suppress AEs in present and future fusion devices.
In recent years, the fusion community has made significant efforts to develop the AE control techniques [12], which are based on the main AE drive mechanisms, i.e.EP distribution [13], and damping mechanisms, i.e.Alfvén continuum or thermal kinetic effects [10,14,15].The majority of AE control techniques rely on modifying the EP distribution or equilibrium profiles to change the wave drive or damping rates [12].For example, localized electron cyclotron current drive (ECCD) [16,17], which can change the AE damping mechanisms by modifying the equilibrium magnetic helicity; localized electron resonant heating (ECRH) [18,19], externally applied resonant magnetic perturbation (RMP) [20,21], and using different neutral beam injection (NBI) launching configurations [22,23], can be useful to modify the EP distribution and thus the AE drive.Indeed, the modifications of AE drive and damping mechanisms by these techniques are closely linked to each other and are briefly reviewed in [12].
The NBI is a flexible external EP source in tokamak plasmas, producing a highly anisotropic EP distribution in contrast to the fusion-born alpha particle distribution.Varying the injection parameters of NBI may lead to changes in the EP slowing down distribution, and so to minimize the wave drive from spatial gradient and/or to maximize the wave (Landau) damping from energy gradient, thereby modifying the AE activity.As a promising technique, varying beam injection parameter has been successfully used to mitigate AEs in many experiments [22][23][24][25][26][27].In DIII-D tokamak [22], the effect of the EP's spatial gradient on AE activity is experimentally investigated by injecting different combinations of on-axis and offaxis NBI.It is found that varying beam injection parameter can alter the EP gradient, and then reduces the amplitude of RSAE activity and stabilizes the core TAEs; the injection angle also exhibits a weak effect on the stability of TAEs.Recent experiments on NSTX show that the global Alfvén eigenmode (GAE), driven by the in-board NBI heating, is effectively suppressed by the out-board NBI heating.This suppression is attributed to the increase in the population of low pitch, deeply passing particles with small Larmor radius [23].Further, a technique called 'Variable Beam Perveance' (VBP) is applied in DIII-D to achieve plasmas with different levels of AEs [26,27].VPB modifies existing neutral beams to enable timevariable current and voltage control, which allows for active control of EP velocity space distribution and thus changes the drive of AEs.Varying beam injection parameter can not only alter both the spatial and velocity distribution of EP, thus affecting the wave-particle resonances, but also modify the equilibrium profiles.As a result, the effect of varying beam injection parameter on AE activity is quite complicated, and a systematic numerical study is essential to help identify the key physics mechanisms underlying the AE mitigation and suppression, and thus to develop effective AE control techniques for future burning plasmas.
Motivated by the aforementioned, in this work, we carry out a systematic linear simulation to investigate the mitigation of the on-axis EP driven BAE by the off-axis EP.This numerical study, consisting of the fixed on-axis EP distribution and the different off-axis EP distribution, is performed using the kinetic-magnetohydrodynamic (MHD) code M3D-K [28,29].The stabilizing effect of varying the parameters of the off-axis EP slowing down distribution function is considered.As theoretically expected [10,13], reducing the spatial gradient of EP is stabilizing for BAE, thus, the strength of the stabilizing effect depends on the deposition profile and EP beta of the off-axis EP.Moreover, the energy/velocity and launching direction of the beam are crucial factors influencing the mitigation of the on-axis EP driven BAE.In this paper, we aim to perform a fundamental and general investigation of the on-axis EP driven BAE mitigated by the off-axis EP.The simulation results, concerning how to choose the parameters of off-axis EP for the effective AE mitigation, could provide a useful reference for future experiments.
This paper begins with a brief description of the physic model of M3D-K and the simulation parameters used in this work (section 2).The simulation results are presented in section 3. Finally, the discussion and conclusion are given in section 4.

The physic model of M3D-K and parameters setup
The kinetic-MHD hybrid code M3D-K [28,29] is employed in this work.The physical model of M3D-K code consists of the bulk plasma and energetic ions (EPs hereafter), the former is treated as a single fluid, while the latter is treated as driftkinetic particles or gyro-kinetic particles.The M3D-K code solves the full set of resistive MHD equations: The EP's kinetic effects enter through the particle pressure tensor P h in the momentum equation (2), and P h is written in the Chew-Goldberger-Low (CGL) form [30]: where b is the unit vector of the magnetic field, P h,∥ and P h,⊥ are the parallel and perpendicular EP pressure, respectively: The EP pressures are calculated from the particle distribution function F in the gyro-center coordinates (X, v ∥ , µ): The EP's motion follow from the gyro-kinetic equations: here, the variables B ⋆ and B ⋆⋆ are given by The detailed descriptions for M3D-K code can be found in [29].The M3D-K code has been extensively applied to investigate the interaction between EP and MHD instabilities, e.g.tearing mode [31][32][33], fishbone [34,35], BAE [36,37], TAE [38,39], reversed shear Alfvén eigenmode [40,41].
A circular tokamak equilibrium and parameters are chosen based on HL-2A-like conditions, where the aspect ratio is R 0 /a = 1/ϵ = 4.125.The parameters setup are based on the specific discharge in HL-2A tokamak [42] and are similar to our previous simulation [36].The spatial profile of the safety factor q is plotted in figure 1.The Alfvén speed v A = B 0 /(µ 0 ρ 0 ) 1/2 = 4.63 × 10 6 m s −1 , and the Alfvén time τ A = R 0 /v A = 3.56 × 10 −7 s.The total beta β = β b + β h , the ratio of plasma pressure to magnetic pressure, including the fixed thermal plasma component β b = 1.4 × 10 −2 exp (−ψ/0.3)and the EP component β h .Here, ψ is the normalized poloidal magnetic flux varying from 0 at the axis to 1 at the plasma edge.In the present work, the EP model in M3D-K includes the fixed on-axis EP distribution and the off-axis EP distribution: . The spatial profiles of fixed β h,on (β on = 1.4%) and β h,off with β off = 0.28%  The total EP beta β h = β h,on + β h,off .Where the red and blue dotted lines represent the locations of q = 1.5 and q = 2.0 rational surfaces.are depicted in figure 2. In addition to the spatial distribution, the EP follow an anisotropic slowing-down distribution in energy.The form is given by where the subscripts 'on' and 'off' denote the variables of on-axis EP and off-axis EP, respectively.c is a normalization factor, H is the step function, v 0 is the injection velocity or cutoff velocity (v 0,on = √ 2E 0,on /m D = 0.45v A and ρ h,on = v 0,on /Ω c = 0.08a for the on-axis deuterium NBI).v c = 0.58v 0,on is the critical velocity.Λ ≡ µB 0 /E is the pitch angle, where µ is the magnetic moment, B 0 is strength of the magnetic field on the magnetic axis and E is the EP's energy.Λ 0 is the central pitch angle, and ∆Λ = 0.3 is the width of pitch angle distribution.∆ψ is the spatial width of distribution, and ψ c represents the deposition location of off-axis EP distribution.Based on the previous studies [36,42] and for simplicity, only the toroidal mode number of n = 2 is retained in simulations.The variables in simulation are normalized as follows: v to ϵv A , r to a, t to τ A , and ω to ω A = v A /R 0 .

Simulation results and analysis
In this section, the simulation results of the effect of different off-axis EP distribution on the fixed on-axis EP driven BAE are presented, with the aim of identifying the optimal region to minimize AE growth rate by varying EP distribution parameters.Firstly, for the above-mentioned equilibrium parameters and the fixed on-axis EP's parameters: β on = 1.4%, as depicted by the black dashed curve in figure 2, Λ 0,on = 0.0 and ∆ψ on = 0.3, we established a BAE baseline case with the on-axis EP alone.Then, the simulations with the on-axis + off-axis EP are performed.The parameters of the off-axis EP distribution, including deposition location ψ c , EP beta β off , spatial distribution width ∆ψ off , central pitch angle Λ 0,off and injection velocity v 0,off , are systematically scanned to investigate the effect of different off-axis EP distribution on on-axis EP driven BAE.Here, it is assumed that the parallel velocity (v ∥ ) directions of the on-axis and off-axis EPs are same as that of the plasma toroidal current, i.e. co-passing EP (v ∥ < 0).Finally, the effect of the velocity direction (v ∥ ) of off-axis EP is discussed.For each β h,off profile, the equilibrium (β = β b + β h,on + β h,off ) used as the initial conditions for M3D-K code is synchronously and self-consistently updated via the equilibrium code VMEC [43] (for each case, the equilibrium is consistently modified according to the different EP beta).The parameter scanning is performed keeping the others fixed.

On-axis EP alone
Firstly, the BAE excited by the on-axis EP alone is presented.Figure 3(a) shows the mode structure of the stream function U, which is the incompressible part of the velocity.One can see that this on-axis EP driven instability exhibits a clear mode structure with the poloidal/toroidal mode number m/n = 3/2, mainly located inside the q = 1.5 rational surface.In addition, the Alfvén continuum calculated by the NOVA code [44], as shown in figure 3(b), reveals that this mode frequency is located inside the BAE-gap and thus can be identified as BAE.The destabilization mechanism of the BAE is then studied.The EP drive is proportional to the EP beta, as well as to the gradients in spatial and energy distribution [2,13]: where γ and ω are the mode's growth rate and frequency, and is the toroidal angular momentum.The EP's slowing down distribution function has negative gradient in energy (∂f/∂E < 0).Thus, the energy distribution damps the wave, and wave is usually drive by the free energy in the spatial gradient.In this work, the BAE driven by the on-axis peaked energetic ion's spatial gradient propagates to −ϕ direction (the plasma current's direction) and −θ directions (the ion diamagnetic drift direction) [45], where ϕ , χ is the related to the compressible part and v ϕ is the toroidal velocity.The dashed curve shows the location of q = 1.5 rational surface.(b) The radial structure and mode frequency with the n = 2 Aflvén continuous spectrum, the spatial coordinate at maximum amplitude (mode center) ψm = 0.1, here, the lowest gap that below the first curve is the BAE-gap, which is formed by the combination of finite plasma beta and geodesic curvature.The gap above the BAE-gap (between the two curves) is the TAE-gap.and θ represent the toroidal angle and poloidal angle, respectively.The linear resonance condition can be written as [46,47]: nω ϕ − pω θ = ω, where p = m + l is an integer and l is the Fourier component of the poloidal motion of EPs; ω ϕ and ω θ represent the toroidal and poloidal orbit frequencies, respectively.The EP energy perturbation δE, as well as the locations of the EP satisfying the resonance condition are plotted in figure 4. One can see that the energy perturbation exhibits hole (blue color) and clump (red color) structures in the P ϕ direction, corresponding to the resonant EPs that lose energy and gain energy respectively, which indicates the release of free energy in the spatial gradient of EP distribution function.The location of these large δE structures is consistent with that of the particles satisfying the resonance condition 2ω ϕ + 2ω θ = ω, i.e. the BAE is driven by co-passing EP via the p = 2 resonance condition.

Effects of the deposition profile of off-axis EP
In this subsection, Λ 0,off = 0.0 and v 0,off = 0.45v A are fixed, the effect of the off-axis EP's deposition profile parameters, including deposition location ψ c , EP beta β off and spatial width ∆ψ off , is investigated.Firstly, the simulation results with different deposition locations ψ c of β h,off profile (β off = 0.28% and ∆ψ off = 0.3, see the solid curves in figure 2) are presented in figure 5, where the blue curve represents the simulations with the synchronously modified equilibrium (β = β b + β h,on + β h,off ) for each ψ c , which means the equilibrium is consistently modified for each β h,off , and the red curve represents the simulations with the fixed equilibrium (β = β b + β h,on ).It is found that the effect of the equilibrium modification, induced by the additional offaxis EP, is weak.The growth rate of BAE, normalized by that of the on-axis EP alone driven BAE, first decreases and then increases with ψ c , reaching the minimum at ψ c ∼ 0.35.The EP spatial gradient (∇β h = ∇β h,on + ∇β h,off ) at ψ = ψ m versus ψ c is also plotted as the green dashed curve in figure 5, where ψ m = 0.1 represents the radial location of maximum amplitude (mode center) of BAE.One can see that the absolute value of ∇β h at ψ = ψ m (being negative when β off = 0.28%) has a positive correlation to the growth rate, suggesting that the stabilization of BAE is directly caused by the off-axis EP induced spatial gradient modification.The strength of the offaxis EP induced stabilizing effect is proportional to the value of ∇β h,off at the BAE mode center ψ = ψ m (∇β h,off | ψ =ψm , being positive when ψ c > ψ m ), and it reaches the maximum when the off-axis EP's spatial gradient at the mode center is largest (satisfying ∂∇β h,off | ψ =ψm (ψ c )/∂ψ c = 0).Thus, it can The effect of the off-axis EP beta β off is then studied (∆ψ off = 0.3).As shown in figure 6, the growth rate of BAE remains almost unchanged when ψ c = ψ m ; when ψ c is located outside the mode center, the BAE can be mitigated by the off-axis EP.This stabilizing strength is proportional to the off-axis EP beta β off .On the other hand, when β off exceeds the threshold of 0.42% (see green curves in figure 6), there is a potential for the excitation of a new type high-frequency instability.As shown in figure 7, this high-frequency instability's mode structure lies between the rational surfaces of q = 1.5 and q = 2.0, and its mode frequency is located inside the TAE-gap.Hence, this new high-frequency instability, located outside the BAE, can be identified as TAE.The excitation of the TAE is attributed to the increase of the spatial gradient of EP at the mode center, and thus TAE's growth rate is proportional to the off-axis EP beta β off .The wave-particle interaction is analyzed in figure 8 9, when β off grows to 0.56%, |∇β h | TAE is larger than |∇β h | BAE at ψ c = 0.17 ∼ 0.37, and thus, the excited mode alters from BAE to TAE.These results are consistent with those shown in figure 6.The critical value of the off-axis EP beta depends on the deposition profile and the modes' locations.
Figure 10 shows the linear growth rate of the BAE versus ψ c with different spatial width ∆ψ off of β h,off profile.It can be seen that as ∆ψ off decreases, the change of BAE growth rate versus ψ c becomes more apparent, and the stabilizing effect of off-axis EP can be stronger.In addition, the deposition location where the stabilizing strength of off-axis EP reaches the maximum, ψ c,m , increases with the increasing ∆ψ off , agreeing well with the above relation ψ c,m ∼ ψ m + √ 2/2∆ψ off .Here, it should be pointed out that, the values of ψ c,m calculated  To further understand the dependence of the stabilizing effect on the mode location, we adjust the location of q = 1.5 rational surface by shifting the q profile, as shown in figure 11.For both downward-shifted and upward-shifted q profiles, the simulations with the on-axis EP alone are first performed, and the results are presented in figure 12.As expected, the mode structures of BAE are all localized around the q = 1.5 rational surface (at ψ = 0.3 for q down and 0.1 for q up ).Subsequently, the results of simulation with the on-axis + off-axis EP for different q profiles are compared in figure 13.One can see that the deposition location with the maximum stabilizing strength,   ψ c,m , is relative to the location of mode center ψ m : ψ c,m increases proportionally with ψ m .These results are also consistent with the previous relation.Moreover, the black curve in figure 13 indicates that the BAE could be further destabilized when the off-axis EP is deposited inside the mode center.
Finally, it can be briefly summarized that, the off-axis EP could (i) further destabilize the on-axis EP driven BAE when ψ c is located inside the BAE mode center; (ii) stabilize the BAE when ψ c is located outside the BAE mode center.The deposition location with the maximum stabilizing strength, ψ c,m , is related to the mode center location and the spatial  The growth rate of BAE, normalized by the growth rate of the on-axis EP alone driven BAE for each q profile, versus the deposition location ψc of the off-axis EP with different q profiles.The blue, red and black regions denote the radial location of BAE for the upward-shifted, initial and downward-shifted q profiles, respectively.distribution width, and can be approximately estimated by the theoretical relation ψ c,m ∼ ψ m + √ 2/2∆ψ off .As a result, achieving optimal mitigation of BAE necessitates careful selection of the deposition profile of the off-axis EP, which can be roughly obtained by calculating ∇β h at the mode center.

Effects of the pitch angle of off-axis EP
The alteration of beam injection angle can change the EP population and, consequently, impact the AE activity by modifying the wave-particle resonances [22,23,48].In this subsection, ψ c = 0.3, ∆ψ off = 0.3 and v 0,off = 0.45v A are fixed, the dependence of the BAE mitigation on the central pitch angle Λ 0,off is investigated.
Firstly, the simulations with the on-axis EP alone are performed.Figure 14 shows the linear growth rate and frequency of BAE versus the on-axis EP's central pitch angle Λ 0,on .One can see that, with the increase of Λ on , the BAE's frequency remains almost unchanged, while the growth rate decreases monotonically.These results indicated that the BAE is mainly driven by the deeply passing EP (Λ ∼ 0).
Next, we systematically scan the central pitch angle Λ 0,off in the on-axis + off-axis EP simulations, with Λ 0,on = 0.0.As shown in figure 15, as Λ 0,off grows from 0 to 1, the  frequency of BAE remains almost unchanged; the BAE's growth rate initially remains nearly unchanged (Λ 0,off = 0.0 − 0.2), subsequently increases, and finally keeps almost constant (Λ 0,off = 0.7 − 1.0).These results further suggest that the BAE is mainly resonating with the deeply passing EP.Consequently, as the fraction of off-axis passing EP decreases (Λ 0,off grows), the stabilizing effect induced by the off-axis EP on BAE is diminished.Hence, to optimize the mitigation of the on-axis EP driven BAE using the off-axis EP, the pitch angle distribution of the off-axis EP should closely resemble that of the on-axis EP to facilitate more resonant EPs.

Effects of the injection velocity of off-axis EP
Similarly, the injection velocity v 0 is the key parameter for tailoring the EP distribution and populating or depopulating the wave-particle resonances, and thus modifying the EP drive of AEs [11].In this subsection, ψ c = 0.3, ∆ψ off = 0.3 and Λ 0,off = 0.0 are fixed, the effect of the off-axis EP's injection velocity v 0,off is analyzed.Figure 16 shows the growth rate and frequency of the BAE versus v 0,off /v 0,on , with the fixed v 0,on = 0.45v A .It can be seen that, as v 0,off /v 0,on grows from 0.2 to 1.4 (ρ h,off /a grows from 0.016 to 0.112), the BAE's growth rate first keeps almost unchanged when v 0,off /v 0,on is small (⩽0.4), then decreases, and finally increases slowly (remains nearly unchanged, v 0,off /v 0,on ⩾ 0.9).These results can be explained by the wave-particle resonance condition.As elucidated in the previous section 3.3, the BAE is mainly driven by the deeply passing EP (Λ ∼ 0, v ∥ ∼ v), and the toroidal and poloidal orbit frequencies of these EP can be approximately written as ω ϕ = v ∥ /R 0 and ω θ = v ∥ / (qR 0 ).Deriving the mode frequency from resonance condition of the BAE: (here, q = 1.5), and thus, we can establish a rough relationship between the resonant EP velocity and the BAE mode frequency: v res = qv A ω m /ω A ∼ 0.21v A ∼ 0.47v 0,on .Moreover, from figure 4, it can be obtained that the velocity of EP resonating with BAE approximately spans from 0.47v 0,on to 0.84v 0,on .This is consistent with the simulation results shown in figure 16: the off-axis EP stabilizes the BAE only when v 0,off > 0.4v 0,on , and as v 0,off rises, the stabilizing strength first increases due to the increase of the fraction of resonant EP, and then decreases slowly when v 0,off > 0.9v 0,on .The fraction of resonant EP is numerically defined κ = N r /N, where N r and N represent the resonant EP number with |δE| ⩾ 0.1%E 0,on and total EP number, respectively.Figure 17 shows κ at the linear stage with the same kinetic energy for different v 0,off cases (red curve), as well as the BAE growth rate versus v 0,off /v 0,on (blue curve).Clearly, there is a positive correlation between these two curves, indicating that the dependence of the off-axis EP induced stabilizing effect on injection velocity v 0,off is directly related to the fraction of resonant EP.Consequently, the effective mitigation of the on-axis EP driven BAE using the off-axis EP necessitates an injection velocity v 0,off that is sufficiently large (close to that of the on-axis EP), to satisfy the resonance condition with the BAE.Insufficient v 0,off results in off-axis EP failing to resonate with the BAE, thus exerting minimal influence.On the other hand, excessive v 0,off cannot lead to further BAE stabilization, it can even reduce the stabilizing effect.Finally, to investigate the effect of the off-axis EP's velocity direction (direction of v ∥ ) on BAE, we have performed the simulations with different velocity directions.As shown in figure 18(a), in comparison with the off-axis counter-passing EP, the off-axis co-passing EP exhibits a more effective stabilizing effect on the on-axis co-passing EP driven BAE, and vice versa (see figure 18(b)).That is, the better mitigation of the on-axis EP driven BAE by the off-axis EP demands the velocity direction (v ∥ ) of the off-axis EP to be the same as that of the on-axis EP.These results may be associated with the strength of wave-particle resonance, and the underlying physical mechanism will be subject to further exploration in future work.total EP number (blue curve) versus the ratio of the injection velocities of the off-axis EP and on-axis EP v 0,off /v 0,on .Here, the resonant EP are defined as those EP with perturbed energy |δE| ⩾ 0.1%E 0,on , and κ is calculated at the linear stage with the same kinetic energy for different v 0,off cases.The scales of the right y axis is decreasing.

Discussion and conclusion
In this paper, we have investigated the effect of different offaxis EP distribution on the fixed on-axis EP driven BAE using the hybrid kinetic-MHD code M3D-K, in which both onaxis EP and off-axis EP described by drift kinetic equations are included.The key goal of this work is to analyze the optimal mitigation of AEs by varying EP distribution parameters.Varying the parameters of the off-axis EP distribution can lead to further destabilization of the on-axis EP driven BAE or stabilizing effects depending on the parameters: deposition location ψ c , EP beta β off , spatial distribution width ∆ψ off , central pitch angle Λ 0,off , injection velocity v 0,off and velocity direction (v ∥ ).For this reason, a systematic simulation of the off-axis EP distribution's parameter scan has been performed, wherein the equilibria used as the initial conditions are synchronously updated.
It is found that the off-axis EP induces the modifications of spatial gradient ∇β h , which can (i) further destabilize the on-axis EP driven BAE when it is deposited inside the mode center; (ii) stabilize the BAE when the deposition location is outside the mode center.The deposition location with the maximum stabilizing strength, ψ c,m , is related to the location of mode center and the spatial distribution width, and can be approximately estimated by the theoretical relation ψ c,m ∼ ψ m + √ 2/2∆ψ off .This stabilizing effect is directly proportional to the off-axis EP beta β off .On the other hand, a new EPdriven instability, located outside the BAE, could be triggered by excessive off-axis EP beta.These results agree well with the theoretical expectations [10,13] and experimental studies [22].Besides the deposition profile, the parameters of the offaxis EP in velocity-space, including the central pitch angle Λ 0,off , the injection velocity v 0,off and velocity direction, also play an important role in the mitigation of BAE.The stabilizing strength of the off-axis EP is proportional to the fraction of resonant EPs, which is linked to the wave-particle resonance strength and determined by Λ 0,off and v 0,off of the off-axis EPs.Therefore, a sufficiently large v 0,off of off-axis EP is crucial to satisfy the resonance condition and effectively mitigate the BAE.A small v 0,off has negligible effect, while an excessive v 0,off hardly further stabilizes BAE even weakens the stabilizing effect.Further simulation results indicate that the pitch angle distribution of off-axis EP should be close to that of the on-axis EPs, including more resonant EP and thus achieving optimal mitigation of BAE.Moreover, in comparison to the off-axis EP with the opposite velocity direction to the on-axis EP, the off-axis EP with the same velocity direction as the onaxis EP exhibits a stronger stabilizing effect.
In present and future fusion devices, the intense heating sources, particularly NBI, are used to reach the plasma temperature requirements of high β operation, and thus lead to the destabilization of AEs.This work has numerically confirmed that such AEs can be effectively mitigated by the offaxis EP slowing-down distribution under the following conditions: (i) deposition position located outside the mode center (∼ ψ m + √ 2/2∆ψ off ); (ii) appropriate EP beta that depends on the profile and modes locations; (iii) similar pitch angle distribution and the same direction of v ∥ as the on-axis EP; (iv) sufficiently large injection velocity (close to that of on-axis EP).One can then envisage using off-axis NBI system to generate such appropriate distribution by changing the injection energy and the pitch angle with respect to the field, as well as exploring different plasmas scenarios, and thus to mitigate AEs.Hence, in future burning plasmas such as ITER, the ICRH and/or NBI systems could be a useful tool to mitigate the alpha particle-driven AEs by modifying the EP gradients at certain phase-space locations.

Figure 1 .
Figure 1.Spatial profile of safety factor, where the dashed and dotted curves represent the locations of q = 1.5 and q = 2.0 rational surfaces respectively.

Figure 2 .
Figure 2. Spatial profiles of EP beta.(a) The peaked on-axis β h,on (βon = 1.4%, dashed curve) and peaked off-axis β h,off (β off = 0.28%, solid curves) with different deposition location.(b)The total EP beta β h = β h,on + β h,off .Where the red and blue dotted lines represent the locations of q = 1.5 and q = 2.0 rational surfaces.

Figure 3 .
Figure 3. (a) The mode structure (U) of BAE excited by the on-axis EP alone with βon = 1.4%,where U is the stream function, which represents the incompressible part of the velocity by the equation:v = R 2 0 ϵ∇ ⊥ U × ∇ϕ + ∇ ⊥ χ + v ϕ φ, χ is the related to the compressible part and v ϕ is the toroidal velocity.The dashed curve shows the location of q = 1.5 rational surface.(b) The radial structure and mode frequency with the n = 2 Aflvén continuous spectrum, the spatial coordinate at maximum amplitude (mode center) ψm = 0.1, here, the lowest gap that below the first curve is the BAE-gap, which is formed by the combination of finite plasma beta and geodesic curvature.The gap above the BAE-gap (between the two curves) is the TAE-gap.

Figure 4 .
Figure 4.For the BAE, the distribution of perturbed energy δE and the linear resonant particle locations in (E, P ϕ ) phase space around Λ = 0.0.

Figure 5 .
Figure 5.The growth rate of BAE, normalized by the growth rate of the on-axis EP alone driven BAE, versus the deposition location ψc of the off-axis EP (β off = 0.28%).The blue and red curves represent the simulation results with the synchronously modified equilibrium (β = β b + β h,on + β h,off ) for each ψc (named 'modified-eq') and the fixed equilibrium (β = β b + β h,on ) (named 'fixed-eq'), respectively.The green dashed curve represents the absolute value of EP's spatial gradient |∇β h | at the mode center of BAE ψm = 0.1.

Figure
Figure (a) Mode growth rate, normalized by the growth rate of the on-axis EP alone driven BAE, and (b) frequency versus the deposition location ψc of the off-axis EP with different β off .The blue and red regions in figure (a) denote the radial location of BAE and TAE, respectively.
. It can be seen that the dominant p = 2 and secondary p = 1 components of co-passing EP are responsible for the wave-particle resonances exciting the TAE.Then, we theoretically calculated the difference of spatial gradients |∇β h | BAE − |∇β h | TAE , where |∇β h | BAE and |∇β h | TAE are the absolute values of β h at the modes center of BAE (ψ = 0.1) and TAE (ψ = 0.45).As shown in figure

Figure 7 .
Figure 7.For the simulation of on-axis + off-axis EP (β off = 0.56% and ψc = 0.3).(a) The mode structure (U) of TAE, and (b) the corresponding radial structure and mode frequency with the n = 2 Alfvén continuous spectrums.The black dashed and dotted curves in figure (a) show the locations of q = 1.5 and q = 2.0 rational surfaces, respectively.

Figure 8 .
Figure 8.For the TAE, the distribution of perturbed energy δE and the linear resonant particle locations in (E, P ϕ ) phase space around Λ = 0.0.

Figure 10 .
Figure 10.β off = 0.28%.The growth rate of BAE, normalized by the growth rate of the on-axis EP alone driven BAE, versus the deposition location ψc of the off-axis EP with different ∆ψ off .

Figure 11 .
Figure 11.Spatial profiles of the downward-shifted, initial and upward-shifted safety factor, where the locations of the q = 1.5 rational surface are ψ = 0.3, 0.2 and 0.1, respectively.

Figure 12 .
Figure 12.The BAE mode structures (U) for (a) downward-shifted and (b) upward-shifted q profiles, where the black curves represent the locations of the q = 1.5 rational surfaces.

Figure 13 .
Figure13.β off = 0.28% and ∆ψ off = 0.3.The growth rate of BAE, normalized by the growth rate of the on-axis EP alone driven BAE for each q profile, versus the deposition location ψc of the off-axis EP with different q profiles.The blue, red and black regions denote the radial location of BAE for the upward-shifted, initial and downward-shifted q profiles, respectively.

Figure 14 .
Figure 14.(a) Mode growth rate, normalized by the growth rate of the on-axis driven BAE with Λ 0,on = 0.0, and (b) frequency versus Λ 0,on .

Figure 15 .
Figure 15.(a) Mode growth rate, normalized by the growth rate of the on-axis EP alone driven BAE, and (b) frequency versus Λ 0,off with different β off .

Figure 16 .
Figure 16.(a) Mode growth rate, normalized by the growth rate of the on-axis EP alone driven BAE, and (b) frequency versus the ratio of the injection velocities of the off-axis EP and on-axis EP 0,off /v 0,on with different β off , where v 0,on = 0.45v A is fixed.

Figure 17 .
Figure17.β off = 0.28%.BAE growth rate (red curve), normalized by the growth rate of the on-axis EP alone driven BAE, and the fraction of resonant EP κ = resonant EP number total EP number (blue curve) versus the ratio of the injection velocities of the off-axis EP and on-axis EP v 0,off /v 0,on .Here, the resonant EP are defined as those EP with perturbed energy |δE| ⩾ 0.1%E 0,on , and κ is calculated at the linear stage with the same kinetic energy for different v 0,off cases.The scales of the right y axis is decreasing.

Figure 18 .
Figure 18.β off = 0.28%.The BAE growth rate, normalized by the growth rate of the on-axis EP alone driven BAE, versus the deposition location ψc with different velocity directions (direction of v ∥ ) of the on-axis and off-axis EPs.(a) The on-axis EP is co-passing EP, and the off-axis EP is co-passing (red curve) or counter-passing (blue curve).(b) The on-axis EP is counter-passing EP, and the off-axis EP is counter-passing (red curve) or co-passing (blue curve).