Influence of the far non-resonant components of high-n resonant magnetic perturbations on energetic passing ions loss

The influence on the loss of energetic passing ions by the n = 4 resonant magnetic perturbation (RMP) is investigated through numerical simulations (here, n is the toroidal mode number of the RMP field). Dedicated efforts have been made to uncover how the plasma response modifies the loss fractions and underlying mechanisms. The stochastization of the drift surfaces and the particle loss fractions increase significantly under the response field, in which the resonant component is significantly shielded. In order to better understand how the response field contributes a considerable drift island width, the respective effect of each poloidal component mb on the outermost drift island mp/n=9/4 are compared (here, mb and mp are poloidal mode numbers of the RMP components and drift island, respectively). Contrary to the intuition that the components with mb=mp±1 should dominate the sideband resonance, some of the components which fulfill |mp−mb|≫1 have the dominant contributions under the response field. There are mainly two reasons accounting for this phenomenon, i.e. the drift motion of particles and the enhancement of amplitudes of non-resonant high-m components by the plasma response near the edge due to the resonant field amplification (RFA) effect. The former made it possible for particles to resonate with the far non-resonant components, and the latter significantly enhanced the perturbation field experienced by particles. The above results imply that the RFA effect is more critical than the stochastization of magnetic topology in the fast ion losses under RMP.


Introduction
Edge localized mode (ELM), which is destabilized by the steep edge pressure gradients inherent to the high-confinement mode (H-mode) [1], must be kept under control.This is due to the fact that ELMs can induce intermittent bursts of heat and particle flux that may be intolerable in future tokamak devices like ITER [2].One of the leading efforts aiming at ELM mitigation is to apply resonant magnetic perturbation (RMP) generated by in-vessel coils, which has been proven to be an effective method to suppress ELM [3][4][5][6][7][8].However, RMP can break the toroidal symmetry of the tokamak magnetic field and thus enhance the energetic particle (EP) transport, which may threaten the safety of tokamak devices [9].
Since EPs play essential roles in heating, momentum input, and current drive, investigations on how RMP affects their transport have concerned many tokamaks.Plenty of experiments conducted in ASDEX Upgrade (AUG) [10][11][12], KSTAR [11,13], DIII-D [14,15], and MAST [16] reveal that RMP can induce extra losses of EPs.Furthermore, a series of numerical simulations based on orbit-following codes have been performed to reproduce experimental results and investigate how RMP influences EP confinement [10,[12][13][14][15][16][17][18][19][20][21][22].To date, two optimization strategies aim to minimize the EP losses induced by RMP.One is to modify magnetic spectra by changing the phase difference between RMP coils [12,15,[22][23][24], and the other one is to search for an optimized toroidal phase with respect to the target fast-ion population [25].Meanwhile, certain studies have proposed some application prospects of RMP in addition to ELM control, like phase-space engineering and energy-selective transport [26,27], which suggests RMP can be a method to actively control instabilities driven by fast ions [28][29][30].Therefore, it is crucial to understand the influence mechanisms of RMP on EPs.
Regarding the mechanisms leading to the RMP-induced EP transport, resonances between the particle motions and the RMP fields have always been one of the focus points.Both linear [19] and non-linear [12] resonances have been considered in previous studies.In AUG, the resonance distribution in a thin layer (edge resonant transport layer, ERTL) near the boundary is well studied [12].The different EP loss fractions under RMPs with different δφ UL are explained by the time-averaged variation of the toroidal canonical momentum of EP through resonant interactions (here, δφ UL is the phase difference between the upper and lower RMP coils) [12].For ITER plasmas, the resonant behaviors are also of key importance, and drift orbit islands of EPs have been particularly investigated using the REORBIT module in MARS-F [31].In EAST, resonant interactions between fast ions and RMP have also been investigated, and it is found that resonances are still influential even when the broken magnetic topology is healed by the plasma response [22,23].By analyzing the drift islands contributed by the non-resonant components of RMP, the enhancement of EP losses induced by the plasma response has been preliminarily interpreted [23].However, some essential details involving EP resonances under RMP have not been well investigated in depth.
To a first approximation, the guiding centers of passing ions in tokamaks simply move along the magnetic field lines.This way, particles can only resonate with perturbation which has the same helicity as the particle orbits.However, due to their considerable drift motions, energetic passing ions can resonate with the non-resonant components, namely the sideband resonance [26,27].Because of this mechanism, the topology of EP trajectories, namely the drift island structures, can significantly differ from that of the magnetic field lines [23].Although the sideband resonance has been successfully used to interpret the increment of EP losses induced by the plasma response, its enhancement under the response field has yet to be explained.Besides, the radial drift motion of the particle is thought to be why the sideband resonance exists, while the toroidal drift motion is previously neglected [23].Moreover, the ITER Project intends to apply high-n RMP, which offers several benefits over low-n configuration, such as an ELMy suppression window with less energy confinement deterioration and the ability to exhaust impurities [8].Compared to previous simulation studies [22,23] for low-n RMPs, it is rather necessary to provide a more in-depth simulation study of the mechanism related to high-n RMPs, which is the reason why we provide this work.
The primary purpose of this work is to reveal the significant contribution to the drift island structure from the magnetic perturbation components with their poloidal number m b significantly differ from that of the island chain m p , which provides an explanation of why the response field significantly enhances the loss of passing ions and thus provides a better understanding of resonant interactions between RMP and energetic ions.We apply a helpful method based on tailoring the Fourier spectrum of the perturbation fields, which was proposed in a previous study about sideband resonance [23].The m p = 9 drift islands under RMP components with different m b are compared.Because of the inward and outward radial drift of EPs during one poloidal transit, which contributes a cos θ coupling effect, it is generally believed that sideband resonant contributions are dominated by m b = m p ± 1 components [32][33][34].Interestingly, the sideband resonance peaks at those components with |m b − m p | ≫ 1 in our simulations, and the resonance intensity is higher than that of the primary resonance.To explain the dominant role played by the far non-resonant components, both the radial and toroidal drift motions are considered and also shown to be necessary.Besides, the role played by the resonant field amplification (RFA) effect in the response field is also analyzed, and it is shown that the radial distribution of the RMP field is the determining factor of the drift island width.
The remainder of this paper is organized as follows.The simulation models are described in section 2. The losses of EPs and the comparison of the drift islands are presented in section 3. Section 4 discusses the relation among the sideband resonance, particle drift motions, and the RFA effect.Finally, a summary is given in section 5.

Hamiltonian guiding center drift equations
In this work, particle motion is calculated by the ORBIT-RF code [36], which is based on the Hamiltonian guiding center equations [37].For a charged particle moving in an electromagnetic field, the guiding center Lagrangian [38] is expressed as with ⃗ A the vector potential, ρ ∥ = v ∥ /B the 'parallel radius', ⃗ v the guiding center velocity, µ the magnetic moment, ξ the gyro-phase, and H = ρ 2 ∥ b 2 /2 + µB + Φ the Hamiltonian with Φ the electric potential.The equilibrium magnetic field is written in terms of the PEST coordinates as B = δ∇ψ p + I∇θ + g∇ζ.Magnetic perturbation is described through δ ⃗ A = α ⃗ B where α is a scalar function of ψ p , θ, and ζ.Then the equations of motion read [36] with the determinant factor D = gq Since RMP is set to be static in this work, ∂ t α is indeed zero here.
In the equilibrium field of tokamaks, particle trajectory can be determined by three constants of motion: the energy E, the magnetic moment µ, and the canonical momentum P ζ = ρ ∥ g − ψ p .From Noether's theorem, the conservation of P ζ can be simply inferred by the axisymmetric nature of tokamak.This conservation will be destroyed when a static toroidal asymmetric perturbation is applied.Hence P ζ is no longer an invariant under the RMP field, while E and µ will remain constant.
In our simulation, an ensemble of 128 000 particles is used, and both the slowing-down and pitch-angle scattering effects caused by the background plasma are included.The test ions are followed until they drift outside the boundary or until they slow down to the thermal energy.The initial distribution in phase space (P ζ , Λ = µB 0 /E) of test ions, which is calculated by the beam ionization module in the SOFT code [24], is shown in figure 1.Most particles are born with high Λ, which affects the distribution of lost particles to some extent.Some particles are generated in the first-orbit loss region and excluded in the following discussions.

Equilibrium and RMP spectrum
The equilibrium used here is generated by the code EFIT [39] with the basic parameters and experimental configurations from the EAST discharge # 85 920.The magnetic field at the magnetic axis is B 0 = 1.5 T, the safety factor at 95% of the normalized poloidal flux is q 95 = 3.65, and the ratio of plasma pressure to magnetic pressure is β N = 1.39.
RMP is included through two different approaches in this work.One is called the vacuum field, which is calculated directly by Biot-Savart's law using the MAPS code [40] with currents in the actual RMP coils in EAST.The other one is called the response field, which considers not only perturbation fields caused by RMP coils but also the spontaneous response of plasma.The latter is achieved by solving linearized single fluid magneto-hydrodynamic equations using the MARS-F code [41].RMP is added in the simulations by the vector potential δ ⃗ A. This approach has been benchmarked by comparing the kinetic Poincare plot of low-energy passing ions and the magnetic topology at the same toroidal position [42].
In the perturbation model used here, the external perturbation field can be equivalently expanded into a two-dimensional Fourier series over θ and ζ, which can be written as where m b and n are the poloidal and toroidal mode numbers, respectively.Because of the features of the spatial distribution of RMP coils, n is chosen to be a single positive value, while m b has a large range of −64 ⩽ m b ⩽ 64.

Loss of EPs and variation of P ζ under RMP
In order to evaluate the impact of RMP on the confinement of energetic passing ions, a series of slowing-down simulations are carried out.An n = 4 RMP with I RMP = 4 kA is used, δϕ UL is scanned to study the spectral effect more systematically, and both vacuum and response RMPs are considered.
The particle loss fractions under the vacuum and response fields are shown in figure 2, where the 'passing' and 'trapped' are determined when the particles drift out of the plasma boundary.Both loss fractions show a sine-like dependence on δϕ UL .The losses under the vacuum field peak at around δϕ UL = 90 • , while the losses under the response field peak at around δϕ UL = 180 • .It is interesting to note that the losses of ions are even worse and enhanced by 1%-9%.In this way, even the minimum loss under the response field stays at the same level as the worst situation under the vacuum field.More importantly, it should be noted that these increments mostly come from passing particles, as shown by figures 2(b) and (c).
The distributions of lost particles in phase space under δϕ UL = 90 • RMP are shown in figure 3(a) (without plasma response) and figure 3(b) (with plasma response included).Particles near the trapped-passing boundary account for most of the lost particles under both kinds of perturbations, which is closely related to the birth profile of beam ions in figure 1.The difference induced by the plasma response is given in figure 3(c), where the color dashed lines denote the resonances of the particles.As the primary source of lost particles, the barely passing particle also dominates the increase of the loss fraction induced by plasma response.Another essential difference comes from the deep passing particles, which lose barely under the vacuum field but significantly under the response field.Therefore, the following discussion will use the passing ions as an example to analyze the variation induced by the plasma response.
The increment of lost passing ions induced by the plasma response is inconsistent with the variation of the magnetic surface breakage.At the corresponding rational surface, the intensity of the RMP component is usually reduced to a low level when considering the plasma response, which heals the broken magnetic topology.Figure 4 shows the comparison between the vacuum and response fields under the n = 4 RMP with phase difference δϕ UL = 90 • .Under the response field, the magnetic island widths are observably smaller than that under the vacuum field.Besides, the outermost unbroken rational surface under the response field locates around ρ p = 0.89, while that under the vacuum field locates around ρ p = 0.83.Therefore, the stochastic area is also narrower under the response field.However, the ratio of particle loss is higher under the response field, which is contrary to the intuitive expectation that better magnetic topology means better confinement of EPs.
For energetic ions, the cross-field drift can be significant, and hence the corresponding orbital topology differs from the magnetic topology.Therefore, it is necessary to investigate the drift island structure of energetic ions influenced by RMP. Figure 5 illustrates the drift island structures of 60 keV copassing ions, where (a) is under the vacuum field and (b) is under the response field.The phase differences between the upper and lower coils are both 90 • .As shown in the figure, the outermost drift island chain has its poloidal mode number m p = 9.The disappearance of rational drift surfaces with higher m p is attributed to the magnetic drift, which causes distortion of the Kolmogorov-Arnold-Moser (KAM) surfaces and ultimately results in first-orbit losses of particles in outer orbits.The drift islands with m p = 9 maintain its shape under the vacuum field but become stochastic under the response field.The stochastization of the outermost drift island chain is obviously more severe under the response field than under the vacuum field, which is consistent with the particle loss ratio shown in figure 2 and confirms its significant impact on particle loss.
For further understanding the difference in resonances with EPs between the vacuum and response fields, it is beneficial to isolate the Fourier components of RMP and compare their contributions to the same drift orbit.Here, we use m p /n = 9/4 drift island as an example to study the respective effects of the Fourier components of vacuum and response fields.To achieve this, we kept the corresponding Fourier component of magnetic perturbation unchanged while setting the other ones to zero.Figures 6(a  of the unperturbed particle orbit is m/n = 9/4, the m b = 9 perturbation component contributes to the drift island width through the primary resonance.In contrast, other components contribute to the island width through the sideband resonance [26,27].
The difference between the drift island widths under the vacuum and response fields interprets the enhancement of the overall resonance when the plasma response is included.As shown in figure 6, the drift island widths start to decrease under the vacuum field as the m b exceeds 13, while the contributions from m b = 14-16 components increase significantly Therefore, the increment of the contribution by those far nonresonant components is the main reason for the considerable overall resonance under the response field.Figure 6 shows the drift islands under the δϕ UL = 90 • because this phase difference corresponds to the maximum loss under the vacuum field while the loss fraction under the response field is still higher, as shown in figure 2. The contribution by the far nonresonant components also plays the dominant role under the response field with other phase differences.
Due to the characteristic that particles drift inward and outward for once in the radial direction during one poloidal cycle, it seems more natural that particles resonate principally with the m b = m p ± 1 components.However, the dominance of the far non-resonant components under the response field implies that a more comprehensive mechanism may lie under the resonances between particle motions and applied fields or electromagnetic instabilities.

Resonant interactions between particles and perturbation fields
In order to better understand the dominant contribution of far non-resonant components, the magnetic perturbation in the particle's perspective that really matters in drift orbit width is studied in detail.Both roles played by the magnetic drift motions and RFA effect are investigated to clarify the reasons that facilitate resonances between EPs and non-resonant components.

The perturbation fields in the perspective of EPs
Using the canonical equations of Hamilton, the time evolution of Here the Hamiltonian can be written as [43].Due to the toroidal symmetry of tokamaks, H 0 keeps constant in the ζ direction.Therefore, when particles move on a certain orbit, the parallel component of the magnetic potential of the perturbation field will determine the variations of their canonical toroidal momenta.Evaluating the parallel magnetic potential A ∥ hence becomes crucial in analyzing the underlying mechanisms of the primary and sideband resonances.
In the PEST coordinate, the magnetic potential of the RMP field can be described by a Fourier decomposition.As mentioned previously, the toroidal mode number is chosen to be a single value, namely n = 4. Therefore, the Fourier decomposition is solely expanded in the poloidal direction as For low-energy particles moving basically along the magnetic field lines, equation ( 5) can express the magnetic field they explore.However, when the particle has enormous energy so that the magnetic drift should not be neglected, its orbit is no longer straight but curved in the straight-field-line coordinate.Therefore, equation ( 5) is no longer suitable for evaluating the resonance between the EPs and the magnetic field.
A new straight-line coordinate for the particle trajectory (r p , θ p , ζ p ) is required for investigating the magnetic field experienced by EPs [44].The bounce resonance and toroidal magnetic drift effects should be included to make the particle's trajectories straight and uniform, hence θ and ζ should be replaced by the phase angle of the transit motion θ p = θ p (θ, κ 2 ) and the shifted field line label ζ p = ζ + ∆, respectively.Besides, to take into account the radial drift motion, the radial position of the particle becomes r p = r 0 + δr [27].In this new coordinate, the magnetic potential reads In this way, the m p th term of the Fourier decomposition of A m b (r p ) e im b θ e −in∆ dθ p (7) can ultimately represent the resonance amplitude of the perturbation component in the EP's perspective.In the coordinate transformation, θ p and ∆ are defined via , respectively [44].

Roles of radial and toroidal drift motions in resonance
In this section, the roles played by the radial and toroidal drift motions in the resonances between energetic passing ions and far non-resonant components are interpreted by comparing the magnetic field variation in the particle perspectives.
Figure 7(a) illustrates a comparison of the contributions to resonance of m b = 9-18 components with different treatments of drift motions, i.e. neglecting both radial and toroidal drift, only considering the radial drift, and considering both radial and toroidal drift.It is observed that without drift motions, the sideband resonance does not occur.When the radial drift motion is solely considered, the sideband resonance is significant for adjacent non-resonant components but decreases monotonically with increasing poloidal mode number.Only when both magnetic drifts are considered can the far nonresonant components contribute primarily to the resonance, resulting in the variation trend consistent with the drift island widths.
When both drift motions are neglected, EPs move along the magnetic field lines like low-energy particles.In the perspective of a low-energy particle, the resonant component keeps in-phase with the orbit, so its influence on the particle accumulates as the particle moves.In contrast, those non-resonant components change |m b − m p | periods in one cycle of the particle orbit.The positive and negative phases cancel each other.Hence the effect of non-resonant components cannot accumulate, resulting in a negligible influence of sideband resonance for low-energy particles.The first difference between EPs and low-energy particles comes from the radial drift motion.The amplitude of each component varies in the radial direction, which can be pretty significant due to the RFA effect.This variation is negligible for low-energy particles because they remain at the same radial position during the motion.However, the radial drift motion makes the EPs travel through different radial positions and thus experience perturbations with varying amplitude.As shown in figure 8, particles experience a high perturbation region when they pass through θ = 0.The equiphase lines of magnetic potentials are approximately straight because they are shown in the PEST coordinate.Actually, the slopes of these equiphase lines are the same between components with the same mode numbers under the vacuum and response fields, and in particular the structures of magnetic components mapped on the drift surface under the vacuum field are similar to that of the m b = 15 component under the response field shown in figure 8(b).The difference lies in the distribution of perturbation strength along each line due to the variation of magnetic components in the radial direction.
Once the amplitude of the perturbation is known, the investigation of the phase variation along the particle trajectory is required to ascertain if the perturbation can accumulate during the particle motion.Since we deal with linear resonances in this work, evaluating the perturbation fields along the unperturbed particle trajectory is enough [45].The dashed line in figure 8 corresponds to a simplified particle orbit, which neglects the toroidal drift motion and hence can be expressed as 9θ + 4ζ = 0.For adjacent non-resonant components such as m b = 10 in figure 8(a), the equiphase lines approximate this simplified orbit.The perturbation can stay positive or negative when the particle passes through the high perturbation region, although the particle does not strictly resonate with this component.When the phase changes to the other side, the perturbation amplitude has reduced, so the positive and negative phases cannot neutralize each other, resulting in the considerable sideband resonance of adjacent non-resonance components.In contrast, for far non-resonant components such as m b = 15 in figure 8(b), the perturbation always completes at least one period when the particle crosses the high perturbation region due to the evident slope difference.Therefore, the sideband resonance of far non-resonant components is negligible when only considering the radial drift.
What turns things around is another difference between EPs and low-energy particles, i.e. the phase variation induced by the toroidal drift motion.The real particle orbit is depicted by the solid line in figure 8, which is not straight but curved due to the toroidal drift, with the slope of the tangent line being steep at the low field side and flat at the high field side.Therefore, at the beginning and the end of each poloidal cycle, the far non-resonant component may have the same helicity as the particle orbit.As shown in figure 8(b), the solid line is locally parallel with the equiphase lines of the m b = 15 component near θ = 0, suggesting that the particle resonates with the component in this region.Due to the radial drift, this region is also the high perturbation region.Therefore, the synergetic effect of toroidal and radial drift motions can effectively enhance the sideband resonance of far non-resonant components.Meanwhile, enhancing the resonance with far non-resonant components means reducing the resonance with adjacent non-resonant components.As shown in figure 8(a), the distortion of the particle trajectory makes particles experience different phases of the m b = 10 component during the high perturbation region.The perturbation is hard to accumulate in this situation.Hence, the sideband resonance of adjacent non-resonant components reduces significantly while far non-resonant components dominate when the toroidal drift motion is considered.
Influences of far non-resonant components are significant for EPs in different regions of phase space, as shown in figures 7(b) and (c).Each component's contribution is compared for particle orbits with different pitch angles or energies.It can be seen that the component with the largest contribution gradually moves to higher m b as the particle's pitch angle or energy grows.This indicates that both the drift motion of EPs and the magnetic spectrum play an important role here.It is the synergistic effect of these two factors that really matters.To understand the spectrum effect, the impact of RFA induced by plasma response will be discussed in the next section.

Significant role of the RFA effect in the sideband resonance
Since the magnetic drift motions enable EPs to retain the disturbance of the RMP field at different radial positions, the influence of plasma response on the radial distribution of the RMP field is sometimes more important than the effect on a specific radial position.In what follows, this effect is investigated by comparing the sideband contributions to particles with different radial drift ranges.
Besides the shielding effect at the corresponding rational magnetic surfaces, the plasma response can amplify the RMP field at other positions, namely the RFA effect.This leads to significant differences in the RMP spectrum not only between vacuum and response fields but also among different poloidal components.This is shown in figure 9, where using m b = 9 For the perturbation fields used in our investigation on the 9/4 particle orbit, the RFA effect is generally more pronounced for those far non-resonant components with m b ≫ 9. Large amplitude perturbation can break down the linear theory at the edge of the plasma [46].However, this effect has a limited impact on the drift islands since the magnetic perturbation is investigated along the unperturbed particle orbit.On the other hand, the plasma response brings variations in the spectrum but no significant amplification in the intensity for the resonant component and those adjacent non-resonant components.
The role played by the RFA effect can be better interpreted by comparing the resonances between the m b = 15 and 18 components and particles with different radial drift ranges.The dark shadow region in figure 9 illustrates the radial drift range of the 9/4 orbit of deep passing particles with Λ = 0 and E = 60 keV, while the light shadow region represents the drift range of barely passing particles with Λ = 0.6, slightly larger than the deep passing case.The radial drift range extension allows particles to experience more of the amplified region of the m b = 18 component.Therefore, the contribution by the m b = 18 component increases significantly as the pitch angle of the particle grows.In contrast, the amplitude of the m b = 15 component reduces in the expanded region, so the enhancement of the contribution by the m b = 15 component is relatively inferior.Increasing the particle energy can also extend the particle's radial drift range and result in a similar phenomenon.Besides the radial drift motion, it should be noted that coupling with the toroidal drift motion is also required for the RFA effect to influence the contributions on drift islands by the RMP components.Perturbations of the resonant and adjacent non-resonant components are difficult to accumulate in the expanded radial drift region due to the toroidal drift motion.Thus, their contribution enhancement induced by particle energy and pitch angle increment is insignificant.
In general, although the shielding effect will heal the broken magnetic topology, the RFA effect can assist far non-resonant components in influencing the drift islands of EPs by coupling with the drift motions.The component with the most outstanding contribution is not necessarily the primary resonant component.However, it has to be amplified significantly by the RFA effect, and the radial drift motion of particles must also cover this amplification region.

Summary and conclusion
In this work, the effects of n = 4 RMP on the energetic passing ions are studied through the drift island structure.When RMP is set in a high-n configuration, the influence of the plasma response can increase the stochastization of the drift islands but heals the magnetic topology.The difference is mainly caused by the contribution of sideband resonance [23,26,27].In order to provide a better understanding of the overall strength of the sideband resonance, we take the Fourier components of the RMP field apart and compare their contributions on the same drift island chain and find out that the plasma response significantly increases the sideband resonance of those far non-resonant components.
We make coordinate transformations to analyze the perturbation field in the particle's perspective for interpreting why the far non-resonant components can dominate the resonance between the RMP field and energetic passing ions.It is found that the drift motions of the particles and the RFA effect induced by the plasma response are the two main reasons.The toroidal drift motion enables the particles to resonate with far non-resonant components.Meanwhile, the radial drift motion coupling with the radial variation of the RMP field makes the particles experience a relatively large perturbation during this resonant portion.Their synergy effect induces the significant influence of the RFA effect on the confinement of EPs.
The magnetic topology was once believed to decide passing particle losses because particles move approximately along the magnetic field lines.This idea is correct to a certain extent for thermal particles whose drift motion is negligible.However, the analyses above reveal that the drift motions enable the internal EPs to be disturbed by the boundary perturbation when the RMP is set in a high-n configuration.Consequently, large drift islands are formed by the resonances between EPs and multiple non-resonant components close to the boundary.Therefore, the enhanced shielding effect due to the lower resistivity in future tokamak devices does not necessarily improve the EP confinement.In contrast, the significant RFA effect in the high beta scenario may lead to more considerable EP losses.Finding a solution to prevent this issue is essential for applying RMP in future devices like ITER.
In addition, attention should not only be paid to the minimization of the detrimental impact of RMP on the EP confinement.Resonances between the EPs and the RMP field sensitively depend on various parameters of the plasma and the RMP, implying the feasibility of utilizing the externally applied 3D fields to exert targeted control over the EP profile [15].In particular, the active control of Alfvén eigenmodes by the RMP field has been experimentally obtained, and simulation has revealed that the EP redistribution induced by the RMP is responsible for this achievement [30].Since the dominant sideband resonant components are selective for EPs with different energy and pitch angles, modifying the corresponding component may be a potential method to alter the phase space distribution of EPs, which may be of value in developing the active control of EP-driven instabilities.

Figure 1 .
Figure 1.The phase-space distribution of test particles.Here, the toroidal momentum P ζ is normalized to the poloidal flux at the last closed flux surface.The phase-space topology at E = 60 keV is shown as solid lines [35], of which the red one denotes the trapped-passing boundary.The shaded region denotes the prompt loss region.The dashed lines depict the resonances for passing particles.

Figure 2 .
Figure 2. The loss fractions of energetic particles under vacuum and response fields.Circles denote the total lost particles, while squares and triangles denote the lost passing particles and trapped particles, respectively.
)-(h) show the m p = 9 drift islands of 60 keV co-passing particles solely under the m b = 9-16 components of the vacuum field, respectively, while figures 6(i)-(p) are under the response field.Since the helicity

Figure 3 .
Figure 3.The distributions of lost particles under (a) vacuum and (b) response fields with δϕ UL = 90 • .(c) Illustrates the variation due to the plasma response.The phase-space topology at E = 60 keV is shown as black lines [35], of which the dashed one denotes the trapped-passing boundary.

Figure 4 .
Figure 4.The edge magnetic topology under n = 4 RMP (a) without and (b) with plasma response.Both phase differences are 90 • .

Figure 6 .
Figure 6.Comparison of the contributions of m b = 9-16 perturbation components on the mp = 9 drift islands under n = 4 RMP (a)-(h) without and (i)-(p) with plasma response.Co-passing ions are set as E = 60 keV and Λ = 0.The most significant contribution under the vacuum field comes from the resonant component.In contrast, the two most considerable contributions under the response field come from the m b = 15 and 16 components, whose poloidal mode number fulfills |mp − m b | ≫ 1.

Figure 7 .
Figure 7. Contributions to the Âmp by each poloidal RMP component in the perspective of particles with different (a) drift motions, (b) pitch angles, and (c) energies.The purple circles in subplot (a) denote the normalized drift island widths of particles with E = 60 keV and Λ = 0. Here, the drift orbit width ∆P ζ is normalized to the poloidal flux of last closed flux surface Ψw.

Figure 8 .
Figure 8.The magnetic potential, Am b , of (a) m b = 10 and (b) m b = 15 components mapped on the drift surface for particles with E = 60 keV and Λ = 0.The dashed line is a simplified particle orbit on this drift surface, which neglects the toroidal drift motion and fulfills 9θ + 4ζ = 0.The solid line is the real unperturbed particle trajectory.

Figure 9 .
Figure 9.The radial distribution of RMP components with poloidal mode number (a) m = 9, (b) m = 10, (c) m = 15, and (d) m = 18.The phase difference between upper and lower coils is 90 • .The solid blue and red lines correspond to the vacuum and response fields.The dashed lines mark the radial position of the rational surface corresponding to each poloidal mode number.The shadow region shows the radial range of particle drift.