The effect of the plasma response on peeling–ballooning modes during edge localized modes mitigated by resonant magnetic perturbations

The effects of resonant magnetic perturbation (RMP) fields on peeling–ballooning (P–B) modes are studied with the experimental equilibria of EAST based on the four-field model in BOUT++ code. As the two basic types of plasma responses, the magnetic and the transport response to RMP are considered in our simulation to reveal the roles of the plasma response during edge localized mode mitigation. On the one hand, the results show that RMP can reduce the linear growth rates of the P–B modes and the pedestal energy loss in the nonlinear process by directly coupling with the P–B modes. The magnetic response can weaken the impacts of RMPs on the P–B modes by partially screening the applied RMP fields more precisely the resonant components. On the other hand, RMP can further reduce the linear growth rates of the P–B modes and the pedestal energy loss by changing the equilibrium pressure profiles through the transport response. More detailed analysis suggests that, compared with other resonant components of RMPs, the components whose corresponding rational surfaces are located at the top of the pedestal can lead to stronger reductions in the linear growth rates of the P–B modes, and can reduce pedestal energy loss more significantly by enhancing multi-mode coupling in the nonlinear process. Finally, the multi-mode coupling increases with the strength of the resonant components, so one can change the RMP poloidal spectrum by adjusting the phase difference Δϕ between the upper and low RMP coils from 0 to 360∘ , and hence obtain the optimal coil phase difference that leads to the strongest reductions in the linear growth rates of the P–B modes and the pedestal energy loss through maximizing the strength of resonant components, especially the resonant components whose corresponding rational surfaces are located at the top of the pedestal.


Introduction
High confinement mode (H mode) is normally accompanied by the occurrence of a steep edge gradient of pressure and a strong bootstrap current, which can drive the ballooning mode and peeling mode (P-B), resulting in the coupled P-B mode in the pedestal region [1]. The P-B mode instability may cause pedestal crash, during which enormous thermal flux and particles are periodically expelled out from the plasma edge, forming the well-known type-I edge localized mode (ELM) [1,2]. Uncontrolled type-I ELM is expected to be a serious issue in rapidly eroding divertor plates in ELMy H mode plasmas in future devices like ITER [3]. Frequent and small amplitude ELMs may bring the benefit of impurity flushing and meanwhile avoid divertor damage. Therefore, as one of the robust ELM control techniques, external applied resonant magnetic perturbation (RMP) coils have been widely installed in DIII-D [4,5], JET [6,7], MAST [8,9], ASDEX Upgrade [10,11], EAST [12][13][14], KSTAR [15] and HL-2A [16] tokamaks to reduce the amplitude and increase the frequency of type-I ELM, i.e. ELM mitigation. There are plans to install RMP coils in ITER [3] since success has been achieved in the present experiments on ELM control.
In order to help provide predictions for the experiments on ELM mitigation and guidance for optimization of the coil configurations in experiments, extensive efforts have been devoted to understanding the mechanism of ELM mitigation. Experimental efforts [8,9,[17][18][19][20] have proposed a potential interpretation that the combined effect of RMP-induced plasma braking and 3D corrugation of the plasma boundary and lobes near the X-point is to significantly degrade ballooning stability. The pedestal recovers to this lower stability boundary more rapidly after the previous ELM, and so the ELM frequency increases. Another proposal, from JOREK modeling efforts [21,22], is that the ELM mitigation is related to the nonlinear coupling process, during which moderate ELMs, which consist of the modes nonlinearly driven by RMPs, replace the large ELM crash due to the domination of the most unstable mode. In addition, extensive efforts in modeling the plasma response for tokamak devices, EAST [23], MAST [24], ASDEX-Upgrade [25,26], JET [27] and HL-2A [28], suggest that ELM mitigation is closely correlated to the edge-peeling response which often manifests as a significant plasma displacement near the X-point region [28,29].
However, the roles of the plasma response in ELM mitigation by RMPs are incompletely understood. It is of interest to bring further supplementary and perfection for providing more accurate predictions in relevant experiments and guidance for optimization of the coil configurations to achieve the best ELM mitigation. The purpose of this work is to further investigate the physical mechanism of ELM mitigation with RMPs through studying the roles of the plasma response, including the magnetic and transport responses [30,31], during pedestal crash. It is known that the RMP field is composed of various resonant components. The resonant component distribution of the actual RMP field in plasma, namely the RMP poloidal mode spectrum, is closely related to the magnetic response, which mainly depends on RMP coil configuration, such as coil phase ∆ϕ . Adressing the question of how the magnetic response affects ELM mitigation by generating different RMP poloidal mode spectra has been less exploited in the previous work and is therefore one of the focus points in this study. Additionally, the effects of the transport response on the evolution process of P-B mode are also studied.
In our work, the effects of some representative resonant components, whose resonant surfaces locate in pedestal region, in vacuum and with the magnetic response on P-B mode which is successfully used to explained type-I ELM, are separately studied based on BOUT++ four-field model. We first try to point out which resonant components have the main or stronger effect on the pedestal energy loss during the ELM crash process. Subsequently, in order to systematically explore the role of the magnetic response in the mitigation of pedestal energy loss, one of the most important parameters of the RMP coil configuration parameterscoil phase (∆ϕ = 0 ∼ 360 • )-is scanned and the actual RMP fields related to it are separately coupled to P-B mode. Finally, to take more physical factors into ELM mitigation, the effects of the transport response on the evolution process of P-B mode are studied.
The rest of this paper is organized as follows. The BOUT++ four-field model that couples the RMP field to P-B mode and the adopted equilibria, which are from shot 52 340 in the EAST device, are presented in section 2. Section 3 presents the linear growth rates of P-B modes and loss characteristics of the pedestal energy during ELM crash with RMPs. The additional effects of the changes in equilibrium parameter profiles due to the transport response are presented in section 4, followed by the summary and conclusion in section 5.

Physical model and equilibria
There are two main steps in our modeling work to simulate the evolution of P-B modes with RMP in EAST. Firstly, MARS-F code [32] is used to calculate the RMP field in vacuum and with the magnetic response in EAST. Secondly, the RMP fields calculated from MARS-F are coupled to P-B modes with a BOUT++ four-field model [33] to analyze the linear and nonlinear evolution of P-B modes. In this section, we will briefly introduce the physics model the of BOUT++ four-field model and the adopted equilibria.

Physical model
As a well-known and robust linear response code, MARS-F has been widely used to calculate the RMP fields in tokamaks, such as EAST [23], MAST [24], ASDEX-Upgrade [25,26], JET [27] and HL-2A [28]. The modeling results show that MARS-F may relatively accurately describe the linear plasma response in tokamaks. Hence, it was adopted to calculate the RMP fields in vacuum and with the plasma response in this work.
The RMP fields obtained from MARS-F are coupled to P-B modes based on the BOUT++ four-field model [33] which is presented in detail in the appendix.
RMP field ⃗ B rmp is added to the original equilibrium magnetic field ⃗ B ′ 0 in the BOUT++ four-field model, that is, the new equilibrium magnetic field ⃗ B 0 after coupling the RMP field is as follows, As , some quantities involved in the equilibrium magnetic field ⃗ B 0 in the BOUT+ four-field model, shown in equations (A1)-(A8) in Appendix A, can be rewritten as, B 0 is the amplitude of the equilibrium magnetic field and ⃗ b 0 is the unit equilibrium magnetic vector. Thus, the unit mag- The field-aligned flux coordinate system (x, y, z) [34] adopted in this simulation is derived from an orthogonal toroidal coordinate system (ψ , θ, ζ). ψ is the poloidal flux, θ is the poloidal angle and ζ is the toroidal angle. The relation between coordinate systems (x, y, z) and (ψ , θ, ζ) is in equation (4): ν is the local field-line pitch given by ν (ψ , θ) = B · ∇ζ/B · ∇θ = B ζ h θ /B θ R, where h θ = 1/ |∇θ| is the scale factor for θ. As the simulation region is symmetric in toroidal, only one 'n'th of the torus is calculated where the 'n' depends on the RMP toroidal number to economize the computing resources. The grid-point space size of the simulation region ψ = 0.6 ∼ 1.0 is set to be (N x , N y , N z ) = (132, 128, 128) in linear simulation and (N x , N y , N z ) = (260, 128, 128) in nonlinear simulation (4)

Equilibria
The electron density, equilibrium toroidal rotation and the Spitzer resistivity determined by the electron temperature play important roles in the plasma response and may significantly affect the actual RMP poloidal mode spectrum; hence, the equilibrium profiles in this simulation work are from the diagnostic data of shot 52 340 in EAST. Figure 1 briefly presents the discharge trajectory of shot 52 340. It is shown in figure 1(a) that the ELM frequency increases, and the amplitude of ELM crash significantly drops after the application of RMP. This means that strong ELM mitigation is achieved with RMP. The phase with strong ELM mitigation is marked by the green dash square in figure 1(a) and more details about shot 52340 in EAST have been reported [14]. As shown in figure 1(a), 3.15 s and 3.45 s are without and with strong ELM mitigation and correspond to before and after the application of RMP, respectively. In this work, the equilibrium profiles at 3.15 s and 3.45 s of shot 52 340 are adopted to investigate the effect of the plasma response during ELM mitigation by RMPs. Firstly, electron density, Spitzer resistivity and equilibrium toroidal rotation are adopted to calculate the RMP fields in vacuum and with the magnetic response based on MARS-F, as shown in figures 2(a)-(c). Secondly, in order to simulate the effect of RMPs on P-B modes, some important parameter profiles, adopted in BOUT++, are used from the experimental diagnostic data at the 3.15 s and 3.45 s of shot 52 340 in EAST to reconstruct the equilibria by Corsica [35]. The pressure, safety factor profiles and parallel current density are shown in figures 2(d)-(f ), respectively. The density is set as a constant n 0 = 3 × 10 19 m −3 , which is close to the electron density in shot 52340 [14], for improving the computational efficiency. It is worth noting that the same equilibrium safety factor, rotation and resistivity profiles are used both in MARS-F and BOUT++ for self-consistency.

The role of the magnetic response in the influence of RMPs on P-B modes
In this section, the magnetic response is first calculated in MARS-F to provide RMP fields for subsequent coupling with P-B modes, and the role of the magnetic response in RMP-ELM mitigation is presented in this section. Note that the changes of equilibria resulting from the transport response are not considered for the purpose of singly investigating the effect of the magnetic response by directly coupling RMP fields to P-B modes. The role of the transport response will be considered through contrastively studying the evolution characteristics of the P-B modes that are driven by the distinct equilibrium profiles at 3.15 s and 3.45 s in EAST shot 52 340. The specific details and results about the transport response are presented in section 4. The actual RMP field is composed of various resonant components in experiments, and extensive results [17,20,25,28,32] have reported that the amplitude strength of resonant components plays an important and crucial role on ELM mitigation. In this section, we first separately study the effects of individual resonant components on P-B modes and explore which components have the main or stronger effect on the linear growth rates of P-B modes and ELM crash. The components in vacuum and with the magnetic response are separately used to affect the P-B modes to investigate the role of the magnetic response. Finally, as a primarily important coil configuration parameter in determining the poloidal spectrum of the actual RMP fields through the magnetic response, coil phase varies from ∆ϕ = 0 to 360 • and its role is investigated by coupling the actual RMPs with all resonant components to P-B modes.

The magnetic response to RMPs in MARS-F
To solely explore the role of the magnetic response, RMPs fields in vacuum and with the plasma response are calculated in MARS-F, and they are directly coupled to P-B modes in a BOUT++ four-field model without any consideration of changes, due to the transport response, of the equilibrium profiles. The equilibrium parameters, such as the resistivity and toroidal rotation used in MARS-F, are from the experimental diagnosis at 3.15 s in EAST shot 52 340, as shown in figure 2. The coil phase of n = 2 or 3 RMPs with current I coil = 10kAt are scanned in MARS-F and the vacuum and response fields are shown in figure 3.
The computed amplitudes of the last resonant radial field components for the vacuum RMP field (blue lines) and the total field including the plasma response (red lines) of the n = 2 and 3 RMPs with scanning coil phase 0 ∼ 360 • are presented in figures 3(a) and (c). The amplitude of the last resonant radial response field component of the n = 2 and 3 RMPs is very strong when the coil phase reaches ∆ϕ = 270 • and 120 • , respectively, which are the optimal coil phases to achieve best ELM control according to the well-known edge-peeling response criterion [28,29]. Therefore, n = 2, ∆ϕ = 270 o and n = 3, ∆ϕ = 120 o RMPs are used to study the effect of individual resonant components on P-B modes. Figures 3(b) and (d) compare the various computed resonant radial field components in vacuum and with the magnetic response for the n = 2, ∆ϕ = 270 • and n = 3, ∆ϕ = 120 • RMPs, respectively. Compared with the resonant radial field components in vacuum |b n,m vac |, the amplitude with a response |b n,m res | is obviously weakened by the shielding effect of the plasma. And |b n,m res | significantly decreases with the enhanced shielding effect due to the toroidal rotation, leading to partial screening of resonant components, and resistivity, allowing partial penetration of fields.
We mainly focus on the effect of the resonant components whose corresponding rational surfaces are located in the pedestal region, as shown in figure 4, because P-B modes localize in the pedestal and the amplitudes of other components, like m/n = 13/3, 12/3 . . . 3/3, m/n = 7/2, 6/2 . . . 2/2 presented in figures 3(b) and (d), with the magnetic response are too low when the corresponding rational surfaces are inside far from the plasma edge region. Poloidal mode decomposition is performed on n = 2, ∆ϕ = 270 • and n = 3, ∆ϕ = 120 • RMPs to get individual resonant components, and the distribution of some resonant components (m/n = 15/3, 18/3, 21/3) with the magnetic response in poloidal cross section are presented in figure 5.

The characteristics of linear P-B modes and nonlinear ELM crash with the various single resonant components of RMPs
Various resonant components, such as m/n = 14/3, 15/3, . . . 21/3 of n = 3, ∆ϕ = 120 • RMPs obtained from poloidal mode decomposition as described in section 3.1 are separately coupled with the P-B modes based on the BOUT++ four-field model to investigate the effect of the individual resonant components on the linear instability of P-B modes and ELM crash. In this modeling work, equilibrium parameter profiles at 3.15 s, including pressure, current, safety factor, toroidal rotation and resistivity profiles, are adopted, as shown in figure 2. The value of the hyper-Lundquist parameter is α H ≃ 10 −4 − 10 −6 Figure 3. The amplitudes of the perturbed resonant radial field components at the last rational surface close to the plasma boundary vs the phase difference between the upper and lower coils, ∆ϕ , for n = 2 (a) and n = 3 (c) RMPs. The amplitudes of the resonant radial field of various components for ∆ϕ = 270 • , n = 2 RMP (b) and ∆ϕ = 120 • , n = 3 (d). The resonant field strengths in vacuum and with the plasma response are indicated by blue squares and red circles, respectively.   in typical tokamaks [36], and it is set to α H = 1.0*10 −4 in our linear simulation. The amplitude of single resonant components, m/n = 14/3, 15/3, . . . 21/3 of n = 3, ∆ϕ = 120 • RMP, with coil current I coil = 10kAt as shown in figures 3(d) and 5 are around 10 −6 − 10 −5 T, which are too low to have obvious effects on the P-B modes. Therefore, coil currents I coil in MARS-F are first adjusted to make the resonant radial components in vacuum |B n | = 1*10 −3 T at the corresponding rational surfaces. Second, the coil currents I coil are used to calculate the resonant components with the magnetic response. Finally, the various individual components in vacuum and with response are separately coupled to P-B modes in BOUT++. The results on the linear instability of P-B modes with/without individual resonant components of RMPs are presented in figure 6.
It is found that the resonant components whose corresponding rational surfaces are located in the pedestal region have significantly different effects on P-B modes-the resonant components such as m/n = 14/3 ∼ 15/3 lead to obviously stronger reductions on the growth rate of P-B modes' linear instability when the corresponding rational surface is at the top of the pedestal, as shown in figure 6(a). However, other resonant components have negligible effects on the linear growth of P-B modes when the corresponding rational surfaces are located in other positions, such as the middle (m/n = 17/3 ∼ 18/3) and bottom (m/n = 20/3∼21/3) of the pedestal as shown in figures 6(b) and (c), respectively. Compared with the resonant components of the vacuum RMPs, those with the magnetic response lead to obviously weaker reductions in the linear growth rates of P-B modes, as shown in figure 6(a). This suggests that the magnetic response may weaken the influence of resonant components on the linear growth rates of P-B modes through the shielding effect by partially screening the applied vacuum RMP fields more precisely the resonant components. In addition, various resonant components of n = 2, ∆ϕ = 270 • RMP (such as m/n = 14/2 ∼ 8/2) were also coupled to P-B modes to analyze their effects on the linear instability of P-B modes, and the results are presented in the figure B1 of Appendix B. The results are consistent with the conclusion of n = 3 RMP. Now we start to investigate the nonlinear process of ELM crash with various single resonant components. If not otherwise specified, the parameters adopted here are the same as the linear simulation above. The non-ideal effects, including ion diamagnetic drift, E × B drift, finite resistivity and hyper-resistivity are considered in all nonlinear simulation. The hyper-Lundquist parameter is set to α_H = 2.0 * 10 −4 , which is slightly larger than α_H = 1.0 * 10 −4 in the linear simulation, to improve numerical stability. First, a small initial vorticity perturbation with toroidal number n = 21, which is the most unstable mode in the linear growth rate spectrum, is set to U 1 = 1.0 * 10 −4 in our simulation. Then, the P-B modes gradually start growing under the drive of the pressure gradient and parallel current in the pedestal region, leading to a pedestal crash and energy release. The 'ELM_size' is defined as the ratio of the energy loss to the pedestal stored energy [35]. It is represented as: Here, ψ in = 0.6 is the inner boundary in the simulation, ψ out is the radial position of the peak pressure gradient and the symbol ⟨⟩ ζ means the average over the bi-normal periodic coordinate.
Some resonant components, m/n = 15/3, 18/3, 21/3, in vacuum and with the magnetic response are coupled with P-B modes separately. The 'ELM_size' evolutions in the nonlinear process of P-B modes are presented in figure 7 and they can usually be divided into three phases. Firstly, the perturbation grows gradually and the pedestal does not crash as the initial most unstable mode is not large enough to break the equilibrium in the linear growth phase (about t < 50τ A in figure 7(a)). Secondly, tremendous energy is released out from the pedestal in the rapid initial crash phase, which manifests as the sharp increase of 'ELM_size' (t ≈ 50 ∼ 70τ A in figure 7(a)). The third one is the nonlinear turbulence transport phase of P-B modes, in which pedestal energy is further transported out slowly and 'ELM_size' reaches saturation gradually (about t > 70τ A in figure 7(a)). We will next discuss in detail the feature of 'ELM_size' when the resonant components are separately coupled to P-B modes in the nonlinear evolution process.
Stage I-In figure 7(a), 'ELM_size' evolutions with/without RMP are presented. It is found that the 'ELM_size' is significantly reduced by the components of  RMPs. As is known, the initial crash (t ≈ 50 ∼ 70τ A in figure 7(a)) is the link from the linear growth phase to the nonlinear turbulence transport phase, and thus its features are closely related to linear growth. The intensity of the initial crash depends mostly on the amplitude peak value of the dominant mode, which is usually the most unstable mode n = 21 in the linear growth rate spectrum, and its amplitude peak value is related in part to the linear growth rate. As shown in figure 6, compared with other components m/n = 18/3 and 21/3 in the pedestal region, the m/n = 15/3 resonant component, whose corresponding rational surface is located at the top of the pedestal, leads to an obviously stronger reduction in the linear growth rates of P-B modes, which is why this resonant component can significantly mitigate the initial pedestal crash (t ≈ 50 ∼ 70τ A in figure 7(a)). Almost all of these resonant components m/n = 15/3, 18/3, 21/3 can lead to obvious mitigation of the following pedestal crash (t ≈ 70 ∼ 180τ A in figure 7(a)), in particular the m/n = 15/3 component sustains a clearly stronger reduction on 'ELM_size' when P-B modes come into the early turbulence transport phase, called 'stage I' in this report.
Stage II-To further investigate the effect of the components of RMPs on P-B modes, we go on to simulate the nonlinear evolution of P-B modes and calculate the 'ELM_size' without/with the components of RMPs in the late turbulence transport phase (t > 200τ A ), called 'stage II' in this work. It is a relatively long time from the pedestal crash to this stage, which usually means the pedestal has been rebuilt by heating source in experiments. Although the influence of RMPs on P-B modes in this stage does not necessarily appear in realistic experiments of RMP-ELM mitigation, studying it can still help us probe into and understand the effect of the components of RMPs. The results are shown in figure 8(c). It is found that the mitigating effects of certain components m/n = 18/3 and 21/3 are negligible when P-B modes enter 'stage II', as shown in figure 8(c). However, it is interesting that the 'ELM_size' with m/n = 15/3 component obviously increases, which means more stored energy is transported out in 'stage II'. All evolutions of 'ELM_size' without/with the resonant components of RMPs in the nonlinear process are given in figure 8(a), which is the composite of the results from figures 7(a) and 8(c). Additionally, the role of the magnetic response is partially revealed by comparatively studying the effects of some resonant components, m/n = 15/3 and 21/3, in vacuum and with the magnetic response on 'ELM_size', as shown in figures 7(a) and 8(c). Compared to the components with the magnetic response, the components of vacuum RMP fields are obviously stronger, whether from the 'ELM_size' mitigation in 'stage I' or the 'ELM_size' increase in 'stage II'. This phenomenon may result from the shielding effect of the plasma. The magnetic response can reduce the strength of the resonant components through partially screening the applied vacuum RMP fields, and hence weaken the influence of the resonant components on 'ELM_size'.
Finally, the resonant components of n = 2 RMP fieldsm/n = 10/2 and 14/2-with magnetic response are separately coupled to P-B modes to further investigate the effects of different resonant components on pedestal energy loss. As presented in figures 7(b) and 8(b), the results of n = 2 RMP can further help us draw a clear conclusion that the resonant components whose corresponding rational surfaces are located at the top of the pedestal have the main or stronger influences, including the 'ELM_size' mitigation in 'stage I' and the 'ELM_size' increase in 'stage II', on the pedestal energy loss. The shielding effect of the plasma can weaken the influence of the resonant components on 'ELM_size'. This is consistent with the result of n = 3 RMP cases.

The nonlinear evolution of P-B modes with various single resonant components
In section 3.3, the features of 'ELM_size' evolution due to the resonant components of RMP fields have been discussed in detail. It is worth noting that the resonant components especially m/n = 15/3 and 10/2, with corresponding rational surfaces located at the top of the pedestal, have a significant and interesting impact on 'ELM_size'. As is well known, the energy will transfer from the initial most unstable mode to other modes by mode-coupling in the nonlinear turbulence transport phase, which leads to competitive development among multi-modes, and the pedestal energy loss 'ELM_size' mainly depends on this nonlinear process. Thus, we will next analyze the nonlinear evolution of P-B modes in the turbulence transport phase to investigate the physical mechanisms behind the interesting phenomena above involving the impact of resonant components on P-B modes.
Firstly, figures 9 and 10 present the various toroidal modes of pressure perturbations corresponding to the m/n = 15/3 and 21/3 RMP cases of figure 8(a). As shown in figures 9(a) and 10(a), the most unstable mode n = 21, namely the toroidal mode with the maximum growth rate in figure 6, is set as the initial perturbation and grows rapidly under the drive of the pressure gradient and the parallel current density. Compared with the m/n = 21/3 component, m/n = 15/3 can lead to stronger reductions in the linear growth rates of P-B modes, thus the initial most unstable mode coupled with this component reaches a lower amplitude peak value at about t ≈ 60τ A in figure 9(a), which leads to a weaker initial pedestal crash at t ≈ 60 ∼ 70τ A in figure 8(a).
Then P-B modes come into the aforementioned 'stage I'. At the beginning of 'stage I', n = 21 is the dominant mode that causes the pedestal crash. Figures 9 and 10 show that the initial most unstable mode decreases and multi-modes grow more rapidly in the transport turbulence phase when the m/n = 21/3 and 15/3 resonant components are coupled to P-B modes. This reveals that the resonant components of RMPs can enhance multi-mode coupling by directly coupling to P-B modes in the turbulence transport phase, leading to energy transfer from the dominant mode n = 21 to other modes such as n = 3, 6, 9 . . . very rapidly. The enhanced muti-mode coupling process may cause the initial most unstable mode n = 21 to lose its dominant role during the pedestal crash and multimodes grow rapidly. In other words, a moderate pedestal crash resulting from multi-mode coupling significantly enhanced by RMPs replaces the dramatic energy loss due to the domination of a single mode in the nonlinear process [21,22,37,38], and is responsible for the mitigation of 'ELM_size' in 'stage I'. Interestingly, the enhancement of multi-mode coupling by the resonant components of RMPs is significantly affected by the radial position of the corresponding rational surfaces. In comparison to the m/n = 21/3 resonant component whose corresponding rational surface is located at the bottom of the pedestal shown in figures 10(a)-(c), m/n = 15/3 whose corresponding rational surface is located at the top of the pedestal may lead to stronger multi-mode coupling, as shown in figures 9(a)-(c). Therefore, the 'ELM_size' mitigation by m/n = 15/3 resonant component is stronger than other resonant components such as m/n = 21/3 . In addition, as presented in figures 9(a) and 10(a), the initial most unstable mode n = 21 with the resonant components m/n = 15/3 and m/n = 21/3 of vacuum RMPs (blue square lines) decreases and hence loses its dominant role more rapidly than that with the response field components (blue solid lines) in the multi-mode coupling process. This suggests that the enhancement of multi-mode coupling induced by RMPs is significantly weakened by the shielding of the plasma. Thus, compared with response RMP fields, the vacuum resonant components can lead to stronger reductions on ELM_size in the nonlinear phase, as shown in figures 7 and 8.
We go on evolving the nonlinear process of P-B modes with the resonant components of RMPs in 'stage II' to further probe and help understand the roles of enhanced multi-mode coupling by the resonant components, although the impacts of RMPs in this stage does not necessarily appear in realistic experiments because it has been a relatively long time since the pedestal crash. As shown in figures 9(a) and 10(a), the initial most unstable mode n = 21 has lost its dominant role on the pedestal crash during the multi-mode coupling process in 'stage II'. Low-n modes such as n = 3, 6, 9 in figure 9(b) grow further and n = 0 mode in figure 9(d) decreases rapidly due to the enhanced multi-mode coupling, leading to more energy being transported out of the pedestal when coupled with the m/n = 15/3 component, and this results in the 'ELM_size' increase in 'stage II'. The resonant component m/n = 21/3 has a negligible effect on 'ELM_size' in 'stage II' due to its weaker enhancement on multi-mode coupling. The results confirmed that the multi-mode coupling enhanced by RMPs appears throughout the entire nonlinear process of P-B modes. They also further verify that the component m/n = 15/3 whose corresponding rational surface is located at the pedestal top can obviously lead to stronger multi-mode coupling than other components.
To visually present the multi-mode coupling without/with RMP components, the toroidal mode spectrum evolutions of pressure fluctuation at the peak gradient location at the outer midplane are shown in figure 11, which correspond to the cases in figure 8(a). The mode spectrum strength is normalized by the maximum value at each time step. The n = 0 part is not included here. As shown in figures 11(b) and (c), the resonant components, especially m/n = 15/3 , can obviously enhance multi-mode coupling. It makes multi-modes grow more rapidly and broadens the mode spectrum earlier in 'stage I'. Therefore, 'ELM_size' is significantly reduced by m/n = 15/3 in this stage. However, in 'stage II', more energy of all these modes continuously transfers to low-n modes such as n = 3, 6, 9 when the resonant components, especially m/n = 15/3 , are added. It causes narrowing of the mode spectrum and hence increasing energy is further transported outside the pedestal.
To sum up, some physics may be obtained from the above analysis on the evolution of various modes and the normalized mode spectrum. Multi-mode coupling enhanced by RMPs plays a crucial role in 'ELM_size' mitigation and it makes the mode spectrum wider and the pedestal crash moderate. In other words, a moderate pedestal crash that results from the enhanced multi-mode coupling caused by RMPs replaces the dramatic energy loss due to the domination of a single mode in the nonlinear phase, which is responsible for mitigation of 'ELM_size'. The enhancement of multi-mode coupling may be weakened by the shielding effects of the plasma, which means the resonant components of response RMPs lead to an obviously smaller reduction on ELM_size than those of vacuum RMPs. The enhancement of multi-mode coupling caused by components, such as m/n = 15/3 , whose corresponding rational surfaces are located locate at the top of the pedestal is obviously stronger than that of other resonant components such as m/n = 21/3. Hence, the resonant components of RMPs whose corresponding rational surfaces are located at the pedestal top may lead to obviously stronger reductions on ELM_size during ELM mitigation with RMPs, which suggests that these resonant components will play the main or crucial roles in RMP-ELM mitigation.

The effect of actual RMP fields with different toroidal flow profiles on linear P-B modes and nonlinear ELM crash
As is well known, extensive experiments [12,14,[39][40][41] in tokamaks show that the field penetration, which is closely related to the E × B flow or perpendicular electron flow near the pedestal top, plays a crucial role in ELM suppression or its strong mitigation with RMPs. In particular, recent experiments in DIII-D find that ELM suppression is closely related to the zero-crossing point of the perpendicular electron flow, and changing the flow can lead to a close/far-top motion of the zero-crossing point to achieve/lose ELM suppression. As shown in sections 3.2 and 3.3, it is found that the resonant components of RMPs whose corresponding surface is located at the pedestal top may lead to stronger reductions in the linear growth rates of P-B modes and pedestal energy loss in the nonlinear phase. This suggests that the resonant components of RMPs, whose corresponding rational surfaces are located at the pedestal top, may play the main or crucial roles in RMP-ELM mitigation. Therefore, the field penetration due to the flow effect near the pedestal top may be crucial in RMP-ELM control.
In this section, we will adopt different flow profiles to investigate and discuss the roles of the resonant radial components, whose corresponding rational surfaces are located at the pedestal top, and the influence of the flow effect. Because the present model that calculates the response RMP fields does not contain the perpendicular electron flow model, we still use toroidal flow to study the field penetration and flow effect. The simulation process contains two steps.
Step 1-Firstly, we adjusted three kinds of toroidal flow profiles, V 1 , V 2 , V 3 , as shown in figure 12(a). The amplitudes of various radial resonant components of ∆ϕ = 45 • , n = 3 response RMP with V 1 , V 2 , V 3 toroidal flow profiles are shown in figure 12(b). Figure 12(c) exhibits the amplitudes of some resonant radial components of figure 12(b), and the corresponding rational surfaces are m/n = 14 ∼ 15/3, 17 ∼ 18/3, 20 ∼ 21/3, respectively. The RMP coil current is given by I coil = 10kAt. In addition, we increased the resistivity at 3.15 s by a factor of ten to enhance the penetration of resonant components. The 3.15 s equilibrium profiles are adopted in the calculation if not otherwise specified.
(1) V 1 toroidal flow profiles are finite and relatively strong in the whole pedestal region, which means that all the resonant components in the pedestal region will be very weak, presented as the cyan line in figure 12(b). (2) V 2 toroidal flow is very low at the pedestal bottom, and it allows the resonant components whose corresponding rational surfaces are near the pedestal bottom to penetrate, presented as the orange line in figure 12(b).  To sum up, as shown in figure 12(c), almost all the resonant components are very weak in the whole pedestal region with the V 1 toroidal flow profile. The resonant components whose corresponding rational surfaces are located at the pedestal bottom are very strong with the V 2 toroidal flow profile. With the V 3 toroidal flow, the resonant component m/n = 14/3 can penetrate near the pedestal top and m/n = 15/3 also gets significantly stronger than in the V 1 and V 2 flow profiles.
Step 2-Now we start to couple the RMP fields, which are calculated under V 1 , V 2 , V 3 flow profiles, to P-B modes based on the BOUT++ four-field model. First, the linear growth rates of P-B modes with a single resonant harmonic whose corresponding rational surfaces are located at the top, middle and bottom of the pedestal are separately calculated. Note here, because the amplitudes of the single resonant components (m/n = 14 ∼ 15/3, 17 ∼ 18/3, 20 ∼ 21/3 RMP) with coil current I coil = 10kAt as shown in figure 12(b), are around 10 −6 ∼ 10 −5 T, they are too low to have obvious effects on the linear growth rates of P-B modes. Therefore, the coil currents adopted in MARS-F are first enhanced to I coil = 80kAt, which, although far larger than the coil current in experiments, can still help us separately explore the roles of different resonant components. Additionally, as we know, toroidal rotation itself may directly influence the instability of P-B modes in RMP-ELM mitigation. On the other hand, rotation may partly screen the resonant components of RMP and hence influence the instability of P-B modes through the shielding effect of the flow. In this paper, we try to focus on the influence of the shielding effect of the flow on P-B modes and exclude the interference of toroidal flow itself on P-B modes. Therefore, the toroidal rotation profiles in all the linear simulations without/with RMP are V 2 profiles.
The results on the linear instability of P-B modes with/without a single resonant component of RMPs are presented in figure 13.   Then, the actual response RMP fields with all components are coupled to P-B modes in the BOUT++ four-field model. The linear growth rates of P-B modes and pedestal energy loss are calculated. Here, the toroidal rotation profile at 3.15 s is adopted in all the following simulations of P-B modes and ELM crash. The linear growth rates and 'ELM_size' without/with n = 3 response RMPs are shown in figures 14(a) and (b) respectively. The flow effect is as follows.
wo-RMP-V 1 -res cases: The flow of the V 1 profile is very strong in the whole pedestal region and hence almost all the resonant radial components are very weak. Therefore, the reduction of growth rates of P-B modes is very small, presented as the cyan line in figure 14(a). Similarly, the corresponding response RMP leads to a little reduction on 'ELM_size', presented as the cyan line in figure 14(b). V 1 -res-V 2 -res cases: Compared with the V 1 profile, the flow of V 2 profile is very low at the pedestal bottom and hence the resonant radial components whose corresponding rational surfaces are located at the pedestal bottom get very strong. The reduction of growth rates of P-B modes increased slightly, presented as the orange line in figure 14(a). The corresponding response RMP leads to a moderate reduction on 'ELM_size', presented as the orange line in figure 14(b).
V 2 -res-V 3 -res cases: Compared with the V 2 profile, the flow of theV 3 profile is very close to zero at the pedestal top and hence the corresponding resonant radial components m/n = 14 ∼ 15/3 obviously get stronger. The corresponding response RMP leads to a significant reduction in linear growth rates and 'ELM_size' in 'stage I', presented as the red lines in figures 14(a) and (b). The response RMP would lead to a significant increase in 'ELM_size' in 'stage II', which is consistent with the conclusion in section 3.2.
In conclusion, the results further indicate that the resonant radial components whose corresponding rational surfaces are located at the pedestal top have the main or stronger influences on pedestal energy loss. The toroidal flow plays an important role in RMP-ELM mitigation by affecting the field penetration, especially near the pedestal top, of the resonant components of RMP.
Additionally, as the results show above, the toroidal flow would partly screen the resonant components of RMP and hence weaken the influence of RMP on the instability of P-B modes through the shielding effect of the flow. Besides this, toroidal rotation itself may directly influence the instability of P-B modes in RMP-ELM mitigation. In this paper, we have mainly focused on the influence of the shielding effect of the flow on P-B modes and excluded the interference of the toroidal flow itself on P-B modes. Therefore, all the toroidal rotation profiles in the BOUT++ simulations without/with RMP are kept the same. The role of toroidal rotation itself in the linear process of P-B modes and nonlinear ELM crash has been widely studied in previous studies [42]. The influence of the toroidal rotation itself on the instability of P-B modes during RMP-ELM mitigation will be discussed in future work.

The effect of actual RMP fields with various coil phases ∆ϕ = 0 ∼ 360 • on linear P-B modes and nonlinear ELM crash
By coupling various single resonant components of vacuum and response RMPs to P-B modes, we analyzed the linear instability of P-B modes and nonlinear ELM crash. It is found that the resonant components whose corresponding rational surfaces are located at the top of the pedestal lead to stronger reductions in the linear growth rates of P-B modes and 'ELM_size'. While the actual RMP fields in tokamaks are composed of various resonant components, and their effects on P-B modes are actually the composite results of all the above resonant components. The magnetic response usually has a significant influence on the distribution of the RMP poloidal mode spectrum, thus potentially impacting the effects of actual RMPs on P-B modes. In other words, the RMP coil configuration parameters such as coil phase that can influence the poloidal spectrum of the actual RMP field probably influence the effects of RMPs on P-B modes and pedestal energy loss. As the coil phase is the focus of RMP coil configuration parameters, we scanned the effects of actual RMP fields, containing all resonant components with various coil phases ∆ϕ = 0 ∼ 360 • , on the linear instabilities of P-B modes and nonlinear ELM crash. The modeling work still consists of two steps, as follows.
(1) Firstly, we used MARS-F to scan the n = 3 actual RMP fields with various coil phases ∆ϕ = 0 ∼ 360 • at 3.15 s. The parameter profiles adopted here remain the same as the front calculations of resonant components, as shown in figure 2. The coil current is I coil = 10kAt. The amplitudes of various radial resonant components, m/n = 3/3, 4/3, . . . 21/3, with the magnetic response are shown in figure 15(a). Figure 15(b) shows the normalized results of figure 15(a), and the strengths of the resonant components are normalized by the maximum value at each resonant component.
For n = 3 RMP fields, it is found that the resonant components, such as m/n = 14/3, 15/3, . . . 21/3 whose corresponding rational surfaces are located in the pedestal region, can partially penetrate into the plasma, and other resonant components, such as m/n = 13/3, 12/3, . . . 3/3, are obviously screened by the plasma as the corresponding rational surfaces are inside, far from the plasma edge region, as shown in figure 15(a). As exhibited in figure 15(b), the coil phases that minimize the strength of almost all the components are around ∆ϕ = 270 • . The optimal coil phase that maximizes the strength of the components, such as m/n = 16/3, . . . 21/3, is around ∆ϕ = 150 ∼ 165 • , and the optimal coil phase of other components is around ∆ϕ = 0 • . However, the optimal coil phases that maximize and minimize the strength of the resonant components whose corresponding rational surface are near the pedestal top, such as m/n = 14/3, 15/3, are obviously different. The optimal coil phases that maximize and minimize the strength of m/n = 14/3 are ∆ϕ = 285 • and 120 • , respectively, and ∆ϕ = 105 • and 300 • for m/n = 15/3.
(2) Secondly, the calculated n = 3 actual RMP fields with different coil phases are separately coupled to P-B modes, and the linear growth rates are shown in figures 15(c) and (d). The growth rates of P-B modes linear instability without RMP (blue solid line) and with ∆ϕ = 0 • (black dashed line), 270 • (red dashed line), 120 • (purple dashed line) and 180 • (green dashed line) actual RMP fields are presented in figure 15(c). It is found that the growth rates of P-B modes are significantly reduced when coil phases of n = 3 actual RMP fields are ∆ϕ = 180 • (green dashed line), which is very close to the optimal coil phase ∆ϕ = 150 ∼ 165 • that maximizes the strength of most of the resonant components such as m/n = 16/3, …21/3. The reduction of the growth rates is very small when the coil phase of actual RMP fields is ∆ϕ = 270 • (red dashed line), which is the coil phase that minimizes the strength of almost all radial resonant components m/n = 3/3, 4/3, . . . 21/3. The results suggest that the reduction of the linear growth rate of P-B modes with the actual RMPs is closely related to the strength of the radial resonant components.
To help us draw a clear general relationship between the reduction of the linear growth rates of P-B modes with the actual RMPs and the strength of the radial resonant components, the n = 3 actual RMP fields with various coil phases ∆ϕ = 0 ∼ 360 • are separately coupled to P-B modes. The linear results for the P-B modes are presented in figure 15(d). The change of the growth rate ∆γ is defined as ∆γ = γ rmp − γ wo−rmp . Here, γ rmp and γ wo−rmp represent the linear growth rates of P-B modes with/without actual RMP fields, respectively. Therefore, ∆γ < 0 means that the growth rates are reduced when RMP fields are coupled to P-B modes. As shown in figure 15(d), n = 3 actual RMP fields lead to certain reductions in the linear growth rates of P-B modes, and the strengths of the reduction change with the scanning coil phase ∆ϕ = 0 ∼ 360 o . Comparing figure 15(b) with (d), it is found that the changing trend of ∆γ versus ∆ϕ is relatively similar to the trend with the radial resonant component strength. ∆ϕ = 180 • and ∆ϕ = 270 • are relatively close to the coil phases that maximize and minimize, respectively, the strength of the resonant components such as m/n = 16/3, . . . 21/3. Correspondingly, the n = 3 RMP lead to the strongest and weakest reductions in the growth rates when coil phases are ∆ϕ = 180 • and ∆ϕ = 270 • , respectively. To sum up, the reduction of the linear growth rates of P-B modes coupled with n = 3 actual RMP fields increases with the strength of the resonant components.
In addition, the n = 2 actual RMP fields with various coil phases ∆ϕ = 0 ∼ 360 • are also separately coupled to P-B modes to further verify the conclusions above. The results are presented in figure B2 of Appendix B. Although n = 2 actual RMP fields with I coil = 10kAt lead to an obviously weaker reduction in the linear growth rates of P-B modes compared to n = 3 actual RMP fields, this can still help us draw a clear general relationship between the reduction of the linear growth rates of P-B modes and the strengths of the radial resonant components by comparing figures B2(b) with (d). The results further demonstrate that the reduction of the growth rates of P-B modes increases with the strengths of the resonant components, which is consistent with the conclusion of n = 3 RMP.
However, it is worth noting that both ∆ϕ = 90 • and 270 • RMP have minimal changes in the linear growth rates, as shown in figure 15(d), yet there is a clear difference in the |bm| profile between ∆ϕ = 90 • and 270 • in figures 15(a) and (b). After careful analysis of the |bm| of various resonant components one by one, it is surprising to find that the changes in the linear growth rate are mainly related to the sum of |bm| of m/n = 14/3 and 15/3. The relevant results are shown as follows.
As shown in figure 6, the changes in linear growth rates of P-B modes with various single resonant components m/n = 14 ∼ 15/3, 17 ∼ 18/3, 20 ∼ 21/3, whose rational surfaces are located at the top, middle and bottom of the pedestal, respectively, have been investigated in section 3.2. In section 3.2, these single resonant components (m/n = 14 ∼ 15/3, 17 ∼ 18/3, 20 ∼ 21/3) of vacuum RMPs are all kept same, and they are all |B n | = 1*10 −3 T at the corresponding rational surfaces respectively. The linear results show that the resonant components whose corresponding rational surfaces are located in  15 are the weight factors, which are defined as the relative changes in the linear growth rates of the most unstable P-B modes (n = 21), and they can be written as: ∆γ 14/3 and ∆γ 15/3 are the changes in linear growth rate of the most unstable P-B modes (n = 21) with the vacuum resonant components m/n = 14/3 and 15/3, respectively, as presented in figure 6(a). γ wo−rmp is the linear growth rate of the most unstable P-B modes (n = 21) without RMP. The weighted sum |b 14,15 | is normalized to the maximum value, as shown in figure 16(b).
Comparing figure 16(b) with figure 15(d), the weighted sum of the |bm| of m/n = 14/3∼15/3 is minimal when ∆ϕ = 90 • and 270 • . The response RMP fields have minimal changes in linear growth rate when ∆ϕ = 90 • and 300 • . The changing trend of ∆γ versus ∆ϕ is found to be similar to the trend in the weighted sum of the |bm| of m/n = 14/3 ∼ 15/3 whose corresponding rational surface is at the top of the pedestal. However, it is worth noting that the correlation is not absolutely close and perfect. For example, the changes in linear growth rate are maximal when ∆ϕ = 180 • but the phase that maximizes the weighted summation |b 14,15 | are around 210 • . There are about 30 • phase shift, and the reason may be as follows. The weighted sum method is relatively rough and mainly for qualitative discussion of the dependence between the changes in linear growth rate and the amplitudes of resonant components. Some other components, such as m/n = 17/3∼18/3 and 20/3∼21/3, have no significant changes in growth rates, leading to ∆γ that are too small to get accurate calculation values for. And some other resonant components whose rational surfaces are inside, far from the pedestal top, such as m/n = 13/3, 12/3 . . ., are not included in our qualitative calculations. Therefore, the summing method only includes m/n = 14/3∼15/3. Additionally, the weighted factors A 14 and A 15 are simply calculated from the relative changes in linear growth rates. Therefore, these reasons may cause the phase shift. However, although the sum method is relatively rough, it still shows that the changes of linear growth rate are related to the amplitudes of the resonant components of RMP. In particular, this suggests that the changing growth rate ∆γ closely depends on the amplitudes of m/n = 14/3 ∼ 15/3 whose corresponding rational surfaces are located at the top of pedestal. It further indicates that the resonant components whose corresponding rational surfaces are located at the pedestal top play the main and most important role in the influence of RMPs on P-B modes. Now we start to investigate the nonlinear ELM crash with coupling n = 3, ∆ϕ = 0 • , 30 • , 60 • . . . 330 • actual RMP fields which contain all resonant components to P-B modes. Some important parameters involved are the same as the nonlinear simulation in section 3.2. Figure 17 presents the simulation results about 'ELM_size' mitigation with the actual RMP fields. As shown in figure 17(a), the evolutions of 'ELM_size' without/with ∆ϕ = 0 • , 90 • , 120 • , 180 • RMPs with magnetic response are marked with black solid, blue dashed, black dashed, red dashed and green dashed lines, respectively. The RMP coil current is I coil = 1kAt, which is smaller than the I coil = 10kAt in experiments. Compared with ∆ϕ = 0 • , 90 • , 120 • RMPs, ∆ϕ = 180 • RMP leads to an obviously stronger mitigation on 'ELM_size' in the early turbulence transport phase. And n = 3 actual RMP field leads to a weak reduction on 'ELM_size' when ∆ϕ = 90 • .
More systematic scanning of the coil phases ∆ϕ = 0 • , 30 • , 60 • . . . 330 • with I coil = 1 and 5kAt is shown with the red and blue lines in figure 17(b). ∆ELM_size is the change of 'ELM_size', and it is defined as ∆ELM_size = ELM_size rmp − ELM_size wo−rmp . Here, ELM_size rmp and ELM_size wo−rmp represent the pedestal energy loss with and without actual RMP fields, respectively. Therefore, ∆ELM_size < 0 presents that 'ELM_size' reduced when actual RMP fields are coupled to P-B modes. As shown in figure 17 figure 16(b), it is found that the trend in ∆ELM_size versus ∆ϕ is similar to the trend in the weighted sum |b 14,15 |. Namely, the 'ELM_size' mitigation is strongest when ∆ϕ = 180 • , which is very close to the optimal coil phase ∆ϕ = 210 • that maximizes the weighted sum of the radial resonant components m/n = 14/3 ∼ 15/3. And when ∆ϕ = 270 • and 90 • , the weighted sum |b 14,15 | is weakest, and the 'ELM_size' mitigation is also weakest. In addition, the weighted sum of m/n = 14/3 ∼ 15/3 is relatively strong when ∆ϕ = 0 • , which may be responsible for n = 3, ∆ϕ = 0 • actual RMP fields also leading to a strong mitigation of 'ELM_size'. To confirm this conclusion, we increase the strength of the resonant components by enhancing the RMP coil currents from I coil = 1kAt to 5kAt. As shown in figure 17(b), the 'ELM_size' reduction increases with the RMP coil current.
In this section, coils phase ∆ϕ was found to play an important role in RMP-ELM mitigation by changing the poloidal spectrum of actual RMP fields through the magnetic response. The coil phase scanning work showed that the reductions in the linear growth rates of P-B modes and 'ELM_size' generally increase with the strength of the radial resonant components, especially with respect to the strength of the components whose corresponding rational surfaces are located in the pedestal region. The weighted sum of the |bm| of m/n = 14/3 ∼ 15/3 further indicates that the resonant components m/n = 14/3 ∼ 15/3 whose corresponding rational surface is located at the pedestal top play the main roles in RMP-ELM mitigation.

The effects of changes in equilibrium profiles due to the transport response on P-B modes
Simulation results in section 3 reveal that RMP can directly couple to P-B modes and influence the ELM behavior in the pedestal, while changes in equilibria resulting from the transport response were not considered in section 3 for the purpose of solely investigating the effect of the magnetic response by directly coupling the RMP field to P-B modes. In section 4, the effects of changes in basic equilibrium profiles due to the transport response on P-B modes are investigated to better understand the physical mechanism of ELM mitigation.
We constructed the equilibrium at 3.15 s (before turning on the RMP current) and 3.45 s (RMP had been added for a long time and the new equilibrium was rebuilt through the slow transport response) in shot 52 340 of EAST [14]. Some important parameter profiles are shown in figure 2. By comparatively studying the ELM crash behavior at 3.15 s and 3.45 s, the role of the transport response, which leads to changes in equilibrium profiles, in ELM mitigation can be investigated. n = 3, ∆ϕ = 180 • RMP fields with the magnetic response were subsequently directly coupled with P-B modes at 3.45 s to simultaneously consider both effects of mode-coupling, related to the magnetic response as presented in section 3, and changes in equilibrium profiles, related to the transport response, on ELM crash behavior. Note that the changes in equilibrium parameters are multifarious. The role of toroidal rotation [42] and resistivity [43,44] in the pedestal crash have been widely studied in previous work. Here, resistivity and toroidal rotation were kept unchanged to comparatively study the new features brought about by the changes in some important equilibrium profiles, such as pressure profiles. If not otherwise specified, the parameters and non-ideal effects that are included in all the following simulations are the same as the linear simulation in section 3.
The linear results are shown as figure 18, in which blue and red lines are the linear growth rates of P-B modes at 3.15 s and 3.45 s, respectively. The green line is coupled with n = 3, ∆ϕ = 180 • , I coil = 10kAt response RMP fields at 3.45 s. Equilibrium profiles are rebuilt and in particular the pressure gradient becomes flatter due to the slow transport response from 3.15 s to 3.45 s. This indicates an obviously weaker drive source, leading to significant reduction in the linear growth rates of P-B modes. The linear growth rates further decrease when RMP is directly coupled to P-B modes in 3.45 s (green line).
Further simulations on the nonlinear evolutions of P-B modes were carried out. The hyper-Lundquist parameter is set to α H = 1.6*10 −4 ,and the corresponding 'ELM_size' evolutions in 'stage I' are shown in figure 19(a). The black dashed and red dashed lines are 3.15 s and 3.45s; green, blue, black and red solid lines all are 3.45s with I coil = 1, 2, 3, 5kAt response RMPs (n = 3, ∆ϕ = 180 • ), respectively. Firstly, compared with the 3.15 s case, due to the lower linear growth rate in the 3.45 s case, the initial most unstable mode n = 21 takes a longer time to grow and break the equilibrium, leading to a time delay (about ∆t = 25τ A ) of the initial pedestal crash in the 3.45 s case. The 'ELM_size' of the 3.45 s case in the entire process of P-B mode evolution is far smaller than that of the 3.15 s case, which is mainly related to a flatter equilibrium pressure gradient, resulting from the enhanced transport of RMPs in experiments [14]. Secondly, 'ELM_size' of the 3.45 s cases is further mitigated in 'stage I' when n = 3, ∆ϕ = 180 • response RMPs are coupled to P-B modes and the 'ELM_size' mitigation increases with the coil currents I coil = 1, 2, 3, 5kAt, as shown in figure 19(a).
We go on to simulate the nonlinear evolutions of P-B modes and calculating the 'ELM_size' without/with the response RMPs in 'stage II'. The hyper-Lundquist parameter is set to α H = 1.6*10 −4 , and the entire 'ELM_size' evolutions from 'stage I' to 'stage II' are shown in figure 20(a). Black and blue dashed lines are 3.15 s and 3.45 s, respectively; blue and red solid lines are all 3.45 s with I coil = 2, 5kAt response RMPs (n = 3, ∆ϕ = 180 • ), respectively. When 'stage II' is reached, the 'ELM_size' increases with the coil currents I coil = 2, 5kAt, but it is still far smaller than the 'ELM_size' in the 3.15 s case, as shown in figure 20(a).
In addition, hyper-resistivity is closely related to the numerical stability [36] and is usually adjusted through changing the hyper-Lundquist parameter α H over a range convenient for computation. We checked the robustness of the above results by calculating 3. To investigate the physical mechanism of the effects of RMPs on 'ELM_size', mode evolutions of α H = 1.6*10 −4 cases in figure 20(a) are analyzed. As we know, 'ELM_size' mainly depends on the development of the initial most unstable mode n = 21 and multi-mode coupling in the nonlinear phase. Therefore, the time evolutions of some relevant modes are presented in figure 21. The δp is the pressure perturbation with different toroidal numbers, normalized to B 2 0 /2µ 0 . Figure 21 3.15 s-3.45 s cases: By comparing the equilibrium profiles at 3.15 s and 3.45 s, the results show that new equilibrium forms through the slow transport response after adding RMP fields, in particular the pressure gradient becomes lower due to enhanced transport of particles and electron temperature, leading to significant reductions in the linear growth rate of P-B modes. Therefore, the initial most unstable mode n = 21 of the 3.45 s case (red dashed line) grows far slower than that of the 3.15 s case (blue dashed line), as shown in figure 21(a), and is responsible for the time delay (about ∆t = 25τ A ) of the initial pedestal crash in the 3.45 s case. The peak amplitude of the initial most unstable mode n = 21 of the 3.45 s case (red dashed line) is far lower than that of the 3.15 s case (blue dashed line), as shown in figure 21(a). This makes the initial pedestal crash of the 3.45 s case far weaker than that of 3.15 s. It is worth noting that there is no direct coupling between P-B modes and RMPs in the 3.45 s and 3.15s cases. Therefore, the enhanced multi-mode coupling of P-B modes is not observed and hence the initial most unstable mode n = 21 slowly decreases. The result reveals the roles of the transport response during RMP-ELM mitigation. It shows that the enhanced transport of particles and electron temperature due to the transport response may weaken the drive source of P-B modes, leading to slower growth of P-B modes. Therefore, the   time of initial pedestal crash would be delayed, and the intensity would also be far smaller. However, the enhanced multimode coupling of P-B modes is not observed because there is no direct coupling between the P-B modes and RMPs. It suggests that the transport response has no significant impact on the muti-mode coupling of P-B modes, and it mainly affects the amplitudes of P-B modes by changing the drive sources (pressure profiles, etc).
3.45 s-3.45 s with I coil = 1, 2, 3, 5kAt RMPs cases: As mentioned earlier, the energy transfer from the dominant mode n = 21 to all other modes happens more quickly in 'stage I' due to the enhanced muti-mode coupling by direct coupling between RMPs and P-B modes. It makes the dominant mode n = 21 decrease more rapidly when RMPs are directly coupled to P-B modes in the 3.4 5s case, as shown in figure 21(a). The multi-mode coupling increases with the strength of the RMPs. Therefore, when we adjusted the RMP coil currents from I coil = 1 to 5kAt, the amplitudes of n = 21 modes further decreased, as presented in figure 21(a), which leads to the result that the 'ELM_size' also further decreases with I coil in 'stage I'. When 'stage II' is reached, due to the enhanced multi-mode coupling still existing, the energy will continuously transfer to low-n modes such as n = 3, 6 and lead the low-n modes to grow further, resulting in the 'ELM_size' increase. As shown in figure 21(b), the amplitudes of n = 3, 6 modes continue to increase with the coil currents I coil = 2, 5kAt due to multi-mode coupling enhanced by RMPs, which mainly results in the 'ELM_size' also increasing with I coil in 'stage II'. The result reveals the roles of the direct coupling between RMPs and P-B modes during RMP-ELM mitigation. It shows that RMPs may enhance the multi-mode coupling of P-B modes in the nonlinear phase, leading to reductions in pedestal energy loss 'ELM_size'. The direct coupling between RMPs and P-B modes is closely related to the strength of the resonant components of RMPs, which has been shown in section 3. We can increase the strength of the resonant components of RMPs by enhancing the RMP coil currents, and hence achieve stronger multi-mode coupling, as shown in figure 21.

s-3.45 s cases:
It is found that not only the initial most unstable mode n = 21, but also almost all other modes n = 3, 6, 9 . . . of the 3.45 s case are far weaker than those of the 3.15 s case, as exhibited in figures 22(a1) and (b1). This is principally due to the lower pressure gradients in the 3.45 s case, mainly resulting from the enhanced transport of particles and electron temperature with RMPs. Therefore, compared with the 'ELM_size' of the 3.15 s case, that of the 3.45 s case is significantly reduced in the entire nonlinear process including 'stage I' and 'stage II', as shown in figure 20(a). The direct coupling between RMPs and P-B modes is not involved in the two cases to singly study the impacts of the transport response 3.15 s-3.45 s with I coil = 5kAt RMP cases: When RMP fields are coupled to P-B modes, the initial most unstable mode n = 21 further decreases rapidly and other modes grow much earlier due to the enhanced multi-mode coupling. This leads to a significantly wider mode spectrum in 'stage I', as presented in figure 22(c2), which is responsible for the further reduction on the 'ELM_size'.
In conclusion, the effects of RMPs on 'ELM_size' behavior can be summed up as follows. (1) The pressure profile becomes flatter due to the enhanced transport caused by RMPs, and it means an obviously weaker drive of P-B modes yielded. By comparatively studying the 'ELM_size' behavior of the 3.15 s and 3.45 s cases, we found that the changes in equilibrium profiles, such as the pressure profile, can lead to significant mitigation of pedestal energy loss in the entire ELM crash process. (2) 'ELM_size' can be further reduced through the multi-mode coupling enhanced by direct coupling between RMPs and P-B modes in the early transport phase, called 'stage I' in this report. However, multi-mode coupling enhanced by RMPs may cause pedestal energy loss to increase in the late transport phase, here called 'stage II'. Both the 'ELM_size' reduction in 'stage I' and the 'ELM_size' increase in 'stage II' increase in a certain range of coil currents. But it is still remarkable that the increase of 'ELM_size' caused by the enhanced multi-mode coupling in 'stage II' is far lower than the reduction that is due to the changes in equilibrium profiles resulting from the transport response. Therefore, actual mitigation of pedestal energy loss is at least a composite result of the action of two factors, which are the changes in equilibrium profiles due to the transport response and the direct coupling between RMPs and P-B modes.
Finally, figure 23(a) presents the pedestal crash changes for the cases in figure 19(a). The black solid and dashed lines are the pressure profiles before and ∆t = 140τ A after the initial pedestal crash at 3.15 s without RMP. The blue solid and dashed lines are at 3.45 s without RMP. The red solid line is ∆t = 140τ A after the initial pedestal crash at 3.45 s with n = 3, I coil = 5kAt, ∆ϕ = 180 • response RMP. By comparing the gray and cyan shaded areas, it is found that both the amplitude of the pedestal crash and affected radial region are significantly reduced as a result of the changes in the equilibrium profiles due to the transport response. The direct coupling between RMPs and P-B modes may further significantly reduce the affected radial region, as the red line shows in figure 23(a). The characteristics of pedestal crash changes during the ELM mitigation phase are similar to the experimental results of shot 52 340 in EAST [14]. Additionally, to improve the computing efficiency, only one second or a third of the torus region is calculated in theBOUT++ four-field model and hence the RMP toroidal numbers are n = 2 or 3, which is distinct from the n = 1 RMP adopted in shot 52 340. But the relevant physical mechanism behind the phenomena would be the same. As exhibited in figure 23(b), the amplitudes of the resonant components of n = 1 RMP are maximized when ∆ϕ = 0 ∼ 15 • . According to the conclusion from scanning the coil phase in section 3, better ELM mitigation may be achieved when the coil phase of n = 1 RMP is around ∆ϕ = 0 ∼ 15 • , which is near to the coil phase ∆ϕ = 0 • during the strong ELM mitigation phase in EAST [14]. However, further verification and demonstration of this conclusion is required in more experiments.

Conclusion and discussion
The roles of the plasma response, including the magnetic response and the transport response, in ELM mitigation by RMPs have been studied by following two aspects: (1) The resonant components, with corresponding rational surfaces located in the pedestal region, have been separately coupled with P-B modes to investigate their effects on the evolution of P-B modes. It is found that the components whose corresponding rational surfaces are located at the top of the pedestal can lead to stronger reductions in the linear growth rates of P-B modes, and RMPs may enhance multi-mode coupling through directly coupling with P-B modes in the nonlinear process, and therefore it can reduce the 'ELM_size' by leading more energy transferring from dominant mode to other modes. Compared to the resonant components with the magnetic response, the components in vacuum lead to obviously stronger reductions in both the linear growth rates of P-B modes and 'ELM_size' in the early turbulence transport phase. Our results reveal that the shielding effect of the plasma may partially screen the resonant components near their corresponding rational surfaces and weaken the direct coupling between RMPs and P-B modes, leading to weaker reductions in the linear growth rates of P-B modes and 'ELM_size'. In addition, the results on the flow effect further indicate that the resonant radial components, with corresponding rational surfaces located at the pedestal top, have the main or stronger influence on the pedestal energy loss. Flow plays an important role in RMP-ELM mitigation by affecting the field penetration, especially near the pedestal top, of the resonant components of RMPs.
The actual response RMP field is composed of various components, and the effects of RMPs on ELM are the composite results of the action of all the components together. Different poloidal mode spectra of RMPs are generated under various RMP coil configuration parameters, such as coil current phase ∆ϕ , leading to distinct effects on P-B modes. As coil phase ∆ϕ is usually the focus of RMP coil configuration parameters, its role in 'ELM_size' reduction is studied by coupling n = 3 ∆ϕ = 0 • , 30 • , 60 • . . . 330 • actual RMP fields containing all resonant components to P-B modes. The results suggest that the reductions in the linear growth rates of P-B modes and 'ELM_size' both usually increase with the amplitudes of the resonant components. In particular, the weighted sum analysis on the amplitudes of the radial resonant components m/n = 14/3 ∼ 15/3 indicates that the resonant components whose corresponding rational surface is located at the top of the pedestal play the main role in reductions in linear growth rates of P-B modes and nonlinear ELM crash.
(2) Compared to the magnetic response, the transport response is a significantly slower process [30,31]. Extensive studies have found that RMP can generally enhance particle and heat flux transport, changing confined density and electron temperature [5,14,45]. The role of the transport response in the evolution of P-B modes is preliminarily investigated by adopting different equilibria at 3.15 s and 3.45 s of shot 52 340 in EAST, respectively. The results show that the lower pressure gradient in the 3.45 s case, resulting from the transport response, may reduce the linear growth rates of P-B modes. The strengths of the P-B modes with various toroidal mode numbers are obviously weaker in the 3.45 s case in the entire nonlinear process, leading to significant reduction on 'ELM_size'.
In the end, changes in equilibria due to the transport response and direct coupling between RMPs and P-B modes affected by the magnetic response are simultaneously considered in a simulation. The composite results of the action of the two factors reveal that changes in equilibria can lead to a significant mitigation on 'ELM_size' through reducing the strength of the P-B modes with various toroidal mode numbers in the entire turbulence transport process. Multimode coupling enhanced by RMPs brings further reduction in 'ELM_size' in the early turbulence transport phase through leading more energy transferring from the initial most unstable mode to other modes. The latter would increase 'ELM_size' due to the energy continuously transfering to low-n modes under the influence of the enhanced multi-mode coupling in the late turbulence transport phase. However, the increase of 'ELM_size' appears to be negligible in comparison with the 'ELM_size' reduction due to the former, and it leads to 'ELM_size' that is still obviously reduced in the late turbulence transport phase.
The multi-mode coupling enhanced by RMPs plays an important role in ELM amplitude reduction, and similar results have been reported in JOREK code [21,22]. It is significant that similar effects of enhanced multi-mode coupling are further observed using a different code, BOUT++. In this work, new effects involving multi-mode coupling are observed. The multi-mode coupling enhanced by RMPs is found to basically increase with the strength of the resonant components of RMPs. The resonant components of RMPs, whose corresponding rational surfaces are near the pedestal top, would play a main or important role in ELM mitigation by inducing stronger multi-mode coupling. The magnetic response may weaken the direct coupling between RMPs and P-B modes through screening the resonant components and hence weakening both the reductions in the linear growth rates of P-B modes and 'ELM_size'. Different RMP poloidal spectra can be generated at various coil phases ∆ϕ = 0 • , 30 • , 60 • . . . 330 • through the magnetic response, which means the optimal coil phase for reducing 'ELM_size' may be obtained by maximizing the amplitudes of the resonant components of RMPs, especially the resonant components whose corresponding rational surfaces are located at the pedestal top. In addition, the investigation on the roles of the ransport response during RMP-ELM mitigation shows that the transport response mainly leads to ELM_size reduction by reducing the amplitudes of P-B modes, while significant impact on the muti-mode coupling of P-B modes was not observed in this work.
The results above are preliminary discussions on the role of the plasma response in the mitigation of pedestal energy loss by RMPs. There are some aspects to discuss and improve upon in future work.
Three distinct kinds of timescales, involved in the corresponding physics, should be noted when studying the plasma response in the process of RMP-ELM mitigation in this effort. One is the single ELM crash timescale t ELM ≃ 100 − 200τ A in BOUT++ and it is about 10 −5 s. The other two are about the plasma response. One is the magnetic response with a relatively short response timescale t mag of about a few to tens of microseconds [30]. This response primarily includes the resonant field amplification outside the resonant surface and the partial shielding effect near the resonant surface, which can change the topology of the magnetic field and hence generate a magnetic island or a random region [30], etc. The other one is the transport response with a long response timescale t tran . This process occurs on a much slower timescale than the magnetic response, but can still be of the order of a millisecond or less [30,31]. Electron and ion temperature, density and other related parameter profiles, such as Spitzer resistivity determined by electron temperature, are changed by the significant effect of RMPs on heat and particle transport during the slow response process [5,14,45]. The timescales of the three physical processes, t ELM ≪ t mag ≪ t tran , span an evidently large range. Considering that the plasma response and one single pedestal crash are not on the same timescale, this work adopted another method-response RMP fields are first calculated based on MARS-F and subsequently coupled to P-B modes in a BOUT++ four-field model-instead of implementing the RMP as a spatial boundary condition and letting it evolve over time. It is well known that MARS-F is a linear response model. RMP fields have significant influences on toroidal rotation by baking [46,47] and can generally change Spitzer resistivity by enhancing particle and electron temperature transport [5,14]. In turn, resistivity and toroidal rotation would affect the resonant component penetration by a shielding effect. This is a complex nonlinear process, and it is a challenge for us in the calculation of plasma responses in the future.
ELM suppression with RMP is usually thought to be related to enhanced transport due to the field penetration near the pedestal top and the changes of edge magnetic topology [12,39,[48][49][50]. Some efforts [46,[51][52][53] show that the field penetration during ELM suppression usually involves a nonlinear plasma response, which cannot be described by the linear response code MARS-F. MARS-F has the known challenge that it cannot currently differentiate between ELM suppression and ELM mitigation. In our work, MARS-F is just used to calculate the RMP fields, which are then adopted in BOUT++ to simulate the effects of RMPs on P-B modes during ELM mitigation. In the modeling work, ELM crash was simulated based on BOUT++. The current model cannot describe the physics processes of RMP-ELM suppression because the nonlinear plasma response and the transport model are not included in this work. More efforts need be made in the future to describe RMP-ELM suppression as follows: (a) RMP fields with a nonlinear plasma response should be considered in the modeling work and (b) the modeling work of ELM crash in BOUT++ may include the transport model.
In addition, extensive experiments in tokamaks show that the field penetration, which is closely related to the E × B flow or perpendicular electron flow near the pedestal top, plays an important role in ELM suppression or strong mitigation with RMPs [12,14,[39][40][41]. In particular, recent experiments in DIII-D find that ELM suppression with RMP is closely related to the zero-crossing point of the perpendicular electron flow, and changing the flow can lead to a close/far-top motion of the zero-crossing point to achieve/lose ELM suppression. In our work, the results about toroidal flow also show that the resonant components of RMPs, whose corresponding surfaces are located at the pedestal top, can lead to stronger reductions in the linear growth rates of the P-B modes and pedestal energy loss in the nonlinear phase. This suggests that the penetration of the resonant components, whose corresponding rational surfaces are located at the pedestal top, may play the main or a significant role in ELM mitigation. The present model that calculated the response RMP fields does not contain the perpendicular electron flow model and further investigation of the effects of E × B flow or perpendicular electron flow on the field penetration in ELM suppression/mitigation will be reported in our next work.
It is important to notice that the level of the reduction in linear growth rates and 'ELM_size' is associated with the strength of resonant components, and especially closely related to the weighted sum of the amplitudes of the resonant components whose corresponding rational surface are near the pedestal top. In other words, it suggests that not only the RMP coil configuration, such as coil phase ∆ϕ , but also plasma parameters, such as Q95 [24][25][26][27], βn [25,27], plasma shape [26,28], toroidal rotation [23,28,53,54], resistivity [32,54,55] etc, can influence the poloidal spectrum of actual RMP fields and potentially affect ELM mitigation, which may be included in our future work. Note that our prediction on n = 1 RMP coil phase is also close to the coil phase which is adopted in EAST experiments to achieve strong ELM mitigation [14]. However, further verifications and demonstrations on this conclusion are required in more experiments.
Finally, the results of this report are systematic discussions on the roles of the magnetic response and the transport response in the influence of RMPs on P-B modes. We hope these relatively comprehensive and new considerations can be taken forward into the exploration of the physical mechanism in ELM mitigation by RMP, and to provide guidance for optimization of the RMP coil configuration and plasma parameters to achieve the best ELM mitigation in experiments.
to describe the P-B modes' evolution process with non-ideal effects, including ion diamagnetic drift, E × B drift, finite resistivity and hyper-resistivity (known as anomalous electron viscosity). ⃗ B0 and 2 ⃗ b 0 × ⃗ κ · (∇P) are the drive sources of the P-B modes, respectively. ρ 0 = n 0 m i is the plasma mass density, m i = 2.0 × 1.6726 × 10 −27 kg is the ion mass and n 0 = 3 × 10 19 m −3 is the plasma number density, which is constant in space and time for computing efficiency. J ∥ = ⃗ b · ⃗ J is the parallel current density, ⃗ b is the unit magnetic vector, ⃗ J is the current density that contains the bootstrap current and Φ is the electrostatic potential.
In the equations, all variables F can be written as F = F 0 + F 1 , where F 0 represents the equilibrium part and F 1 represents the perturbation. All length is normalized to the major radius R 0 = 1.8 m. Time is normalized to Alfvén time τ A = R 0 /v A = R 0 / B mag /(µ 0 n 0 m i ) 1 2 , about 2.779 × 10 −7 s. The magnetic field at the magnetic axis is B mag = 2.3T. Resistivity η is determined by the Lundquist number S = (µ 0 v A R 0 ) /η, which is the dimensionless ratio of an Alfvén wave crossing timescale to a resistive diffusion timescale of magnetic field. η H , known as the anomalous electron viscosity, is hyper-resistivity, which is determined by S H = (µ 0 v A R 0 ) /η H . α H = S H /S, η H = α H · η, α H is a dimensionless hyper-Lundquist parameter.

Appendix B. The linear results of n = 2 RMP
Various resonant components, such as m/n = 8/2, 9/2, . . . 14/2, of n = 2, ∆ϕ = 270 • RMPs obtained from poloidal mode decomposition as those in section 3.1 are separately coupled with the P-B modes to further investigate the effect of the individual resonant components on the linear instability of P-B modes. The linear results are shown in figure B1.
It is found that the resonant components whose corresponding rational surfaces are located in the pedestal region have significantly distinguished effects on P-B modes-the resonant components such as m/n = 8/2 ∼ 9/2, lead to obviously stronger reductions in the growth rate of P-B modes linear instability when the corresponding rational surface is at the top of pedestal, as shown in figure B1(a). However, other resonant components have negligible effects on the linear growth of P-B modes when corresponding rational surfaces are located in other positions, such as in the middle (m/n = 10/2 ∼ 12/2) and bottom (m/n = 13/2 ∼ 14/2) of the pedestal, as shown in figures B1(b) and (c) respectively. Compared with the resonant components of the vacuum RMPs, those with the magnetic response lead to obviously weaker reductions in the linear growth rates of P-B modes, as shown in figure B1(a). It suggests that the magnetic response may weaken the influence of resonant components on linear growth rates of P-B modes through a shielding effect by partially screening the applied vacuum RMP fields, more precisely the resonant components. The results are consistent with the conclusion of n = 3 RMP in section 3.2.
In addition, the n = 2 actual RMP fields with various coil phases ∆ϕ = 0 ∼ 360 • are also separately coupled to P-B modes for further verifying the conclusion above. The results are presented in figure B2. Although n = 2 actual RMP fields with I coil = 10kAt lead to obviously weaker reductions in the linear growth rates of P-B modes compared to n = 3 actual RMP fields, it still can help us draw a clear general relationship between the reductions of the linear growth rates of P-B modes and the strength of the radial resonant components by comparing figures B2(b) with (d). For n = 2 RMP, the changing trend of ∆γ versus ∆ϕ is also consistent with the varying trend of the radial resonant component strength. As shown in figure B2(b) with (d), RMP lead to the strongest reductions in the linear growth rates of P-B modes when ∆ϕ = 210 • , which is very near to the optimal phase ∆ϕ = 240 • that maximizes the strength of the radial resonant components m/n = 14/2 ∼ 8/2. Similarly, all the components are relatively weak when ∆ϕ = 120 • and hence the reductions of the linear growth rates are minimum at this coil phase, which is also close to the optimal phase ∆ϕ = 90 • that minimizes the strength of all the radial resonant components m/n = 14/2 ∼ 8/2. The reductions in the growth rates of P-B modes that are coupled with n = 2 actual RMP fields also increase with the strength of the resonant components, which is consistent with the conclusion of n = 3 RMP in section 3.2. Figure B1. The linear growth rates of P-B modes vs toroidal mode numbers (n) with single resonant harmonic whose corresponding rational surfaces are located at the (a) top, (b) middle and (c) bottom of the pedestal, and the corresponding resonant components are m/n = 8/2 ∼ 9/2, m/n = 10/2 ∼ 12/2, m/n = 13/2 ∼ 14/2, respectively. The dashed lines are the growth rates with vacuum RMP fields and the solid lines are with the magnetic response to RMP fields. Blue cross solid lines denote without RMPs.