Tailoring resonant magnetic perturbation to optimize fast-ion confinement during ELM control in KSTAR

3D resonant magnetic perturbation (RMP) is one promising way to control edge localized modes that can cause excessive material erosion of tokamak first walls. However, RMP can lead to undesired degradation of plasma confinement, including fast-particle losses, which can impact the performance and safety of the reactor. This work investigates the optimization of the poloidal spectrum of the 3D field to optimize fast ion confinement during edge localized mode (ELM) suppression. In the initial step, the validity of the modeling framework is tested against experimental data. Simulations successfully replicate an increase in poloidal limiter temperature with different poloidal spectra. Then, the simulation shows improvement of fast ion confinement with a reduction of core resonant response, while edge resonant magnetic fields are maintained above the threshold to sustain the ELM suppression. Reduction of the core resonant fields keeps the Kolmogorov–Arnold–Moser surface and reduces the fast particle losses due to the stochastic magnetic field lines. The results highlight the potential of edge localization of the resonant fields to enhance the performance of fusion reactors, but further investigation is needed to improve the validation of this approach.


Introduction
A small non-axisymmetric field δB is one promising way to suppress edge localized modes (ELMs) [1][2][3][4][5][6][7][8][9][10] that should be mitigated to prevent excessive material erosion [11] for the high confinement mode (H-mode) in the fusion reactors.Therefore, ITER plans to have a flexible set of external 3D coils to prevent ELMs in high-confinement regimes [12].Meanwhile, 3D fields can lead to unnecessary degradation of confinement of fast particles, which can carry a significant amount of energy in fusion reactors.This loss can affect the heating, torque, and thermal confinement of the tokamak plasmas and can also lead to damage to the plasma-facing component (PFC).
The loss of fast particles presents challenges in maintaining the temperature of PFCs within a safe operating range, particularly for superconducting tokamak ITER, where the plasma durations will be longer than 100 s.For example, this issue was observed during the t = 80 s operations of the superconducting tokamak KSTAR.When the poloidal limiter's temperature exceeds six hundred degrees, discharge should be terminated to maintain the machine's safety of KSTAR. Figure 1(a) shows that the discharge is terminated around t = 80 s by the excessive temperature increases of the poloidal limiter, as depicted in figure 1.In the resonant magnetic perturbation (RMP) ELM suppression experiments, the heating of the poloidal limiter becomes an even more significant concern, as the presence of 3D magnetic fields can further increase the fast ion losses to the poloidal limiter.
Considering the role of fast ions, the optimal 3D field would aim to control ELMs while minimizing fast particle losses, thus maximizing plasma performance.However, separating these two effects can be challenging due to the strong coupling between ELM suppression and undesired confinement degradations [7].In particular, both ELM suppression and fast particle losses caused by the 3D field are known to be primarily influenced by the resonant layers at the plasma edge [6,9,10,13,14], which adds complexity to their separation.Note that the resonant layers in [13,14] (so-called 'ERTLs' there) are locations in energetic particle phase-space where particular resonant conditions are met.In practice, the optimal 3D field can take advantage of the degree of freedom offered by the external 3D coils, such as amplitude, and relative phase difference of coil currents, which are equipped with multiple coil arrays.The modification of the poloidal spectrum has been found to impact the response of ELM significantly, as observed in DIII-D [7], KSTAR [9] ASDEX-U [3].Similarly, the investigation in DIII-D [15], ASDEX-U [13], and ITER [14] also studied the influence of the poloidal spectrum on fast particle losses.However, this post-evaluating approach may become less effective when additional degrees of freedom are introduced to the 3D coils.This is because the number of possible scenarios grows exponentially with each dimension.For example, devices like ITER and KSTAR, which feature three rows of external coil arrays, provide six-dimensional variables for a single toroidal harmonic.Although this dimensionality can be reduced through constraints on coil current ratio [14] or phasing, there is a desire for greater flexibility to fully utilize the available degrees of freedom.
A more systematic optimization of the 3D field can be achieved by specifying particular parameters as targets for the optimization process.For example, decoupling the core and edge resonant fields can be used as targets and simplify the 3D field optimization by using null space projection [16,17] without the constraint of coil current ratio or phasing.In addition, this approach can provide the most efficient field distribution to improve the design of 3D coils [16,18] without being constrained by the a priori choice of the coil geometries.This edge localization of the resonant field has been shown to reduce unnecessary core resonant fields that can drive disruptive locked modes in KSTAR while maintaining ELM suppression [9,17].
This paper is structured as follows.Section 2 introduces the modeling framework to predict 3D field-induced fast ion losses.Then, section 3 shows the comparison of KSTAR experiments and simulation results with different 3D spectra.Section 4 shows the localization of the resonant magnetic field by changing the poloidal spectrum of the 3D magnetic field.Then, section 5 shows simulation results on the change of fast ion losses due to the localization of the resonant field.Finally, a summary is given in section 6.

Simulation of 3D field induced fast ion losses
The simulation is performed under the experimentally relevant condition using the equilibrium and 3D coils of KSTAR.The 3D field in KSTAR is applied by in-vessel current coils with three rows of 3D coils, which are at, above, and below the midplane.Each row consists of four coils, as shown in figure 2. These coils can generate an n = 1 field with an arbitrary phase and different poloidal-harmonic distributions.The resonant magnetic field applied by the 3D coils is determined through IPEC [19] simulation, which considers the ideal magnetohydrodynamic (MHD) plasma response to the applied 3D fields.This resonant field, computed by IPEC, originates from the shielding currents at the rational surfaces, and its purpose is to counteract and suppress the formation of magnetic islands caused by the ideal MHD plasma response.The resonant field obtained by IPEC has proven to be a reliable criterion for explaining the formation of magnetic islands [9].
Then, the simulations of fast particle losses are done with the NuBDeC code [20].The NuBDeC code calculates the deposition of fast ions by neutral beam injection (NBI) and follows the guiding center orbits of the fast ions.NuBDeC has the capability to calculate fast particle loss with a perturbed magnetic field calculated by IPEC code, as shown in [21].Therefore, the outputs from IPEC are directly plugged into the NuBDeC to estimate fast ion losses due to the different poloidal spectra of 3D fields.The NuBDeC simulation considers fast ion deposition due to NBI-A and NBI-B in KSTAR, with a beam energy at 100 keV and 80 keV and the beam power at 1.5 MW.NuBDeC calculates the deposition of 200 000 Monte Carlo NBI ions for KSTAR and assumes a full energy beam source.

Validation of plasma response modeling
This section compares different poloidal spectra of the 3D field before systematically designing edge-localized RMP (ERMP).One of the main goals is to investigate the model's validity using two different 3D spectra.Unlike amplitude scans with the same poloidal spectrum, these different spectra can yield contrasting results depending on the model being used.For doing this, we calculated the resonant fields with and without plasma response using the single equilibrium input.The target discharge is a KSTAR ELM suppressed H-mode discharge (#26027, t = 5.0 s), which is operated with plasma current (I P ) of 0.51 MA, toroidal magnetic field (B T ) of 1.8 T and edge safety factor (q 95 ) of 5.1.The resonant field profiles are shown in figure 3.With vacuum superposition, discharges #26027 and #26026 can be considered as good comparative cases for edge localization as indicated by its vacuum resonant fields shown in figure 3(b).The two cases have similar edge resonant fields, but the core resonant fields are much larger in #26026 than #26027.By including plasma response, on the other hand, the resonant fields are generally larger in #26027 than #26026.Note that this contrasting result is due to the differences in the poloidal spectra shown in figure 4(a), which exhibit a higher sensitivity to specific components or distributions of the applied 3D field depending on the model.
Then, figure 5 presents the time traces of experiments with the two different 3D spectra.These 3D field spectra are applied from t = 2.4 s to t = 11 s, and the plasma current flat top of both discharges ends at t = 11 s.These spectra lead to different degradations of plasma performance.The beta degradation due to 3D fields is much more substantial in discharge #26027, which agrees better with the resonant field profiles calculated using an ideal MHD plasma response shown in figure 3(a).While vacuum resonant fields are well localized in these two cases at similar edge resonant fields, this result proves the necessity to optimize resonant field profiles using an ideal MHD plasma response model as investigated in previous studies [9,22].Based on this result, we use a resonant field that includes the ideal MHD plasma response for the remainder of this work.

Validation of fast ion loss modeling
Next, the fast ion orbit loss modeling is compared with experimental data to validate its predictive capabilities.Before examining the impact of the 3D field, the simulation first explores the prompt loss of neutral beam injected fast ions in the absence of 3D fields.This simulation aims to investigate fast ion losses at two different equilibria, which have different performance degradation by the different 3D fields.The simulation targets t = 5.0 s for the discharge #26027, and t = 5.2 s for the discharge #26026 at the stationary phase.Figures 6(a), (b) and (e) show simulated total #26026 to PFCs and loss to the poloidal limiters (PL1, PL2, PL3) at the different toroidal locations at ϕ = 145 • , 170 • , and 190 • .A poloidal location of poloidal limiter is shown in figure 1(c), and the more precise location of PL1, PL2, and PL3 are described in [20].The simulations show that #26026 without the 3D field effect are small and almost similar for the two cases.Note that the simulation result is sensitive to the physical location of the limiter [20].
Then, experimental 3D coil currents are used to simulate the fast ion losses for these two different equilibria.The resonant field profiles and distribution of normal magnetic fields of two discharges are shown in figure 4. Figure 4 shows that the resonant field and normal magnetic field in discharge #26026 are larger than discharge #26027.As a result, the simulated fast ion losses for discharge #26026 are much larger than #26027 as shown in figures 6(c), (d) and (f ).Also, the simulated maximum loss to the poloidal limiter, loss at PL3, is much larger in discharge #26026 than in #26027.Note that this result highly relies on the amplification by plasma response at different equilibria.For example, more resonant field amplification is expected for discharge #26026 than the other because it has a higher plasma beta.This suggests that a systematic comparison of edge localization should consider this difference in resonant field amplification.Nevertheless, these simulation results are valuable for validating fast ion losses at different 3D spectra.
To compare the experiment and simulation results, we use the temperature of the poloidal limiter, shown in figure 1(c).The temperature of the poloidal limiter is measured with a thermocouple sensor located behind the tile.This temperature increase is not a direct measurement of fast ions but is connected to fast ion losses, especially when plasma has a diverted shaping.For the diverted shaping, interaction with thermal plasma mainly affects the diverters rather than limiters, and the fast ion losses can be considered to be the primary source of poloidal limiter heating [20].Therefore, this temperature can be an efficient way to estimate the fast ion losses in relatively longer pulse (t > 20 s) KSTAR discharges, which is still shorter than the typical pulse length of ITER (t > 100 s).
Figure 5(d) demonstrates the discrepancy in the slopes of the poloidal limiter temperature increase between the two discharges with two different 3D spectra.One of the discharges exhibits a poloidal limiter temperature increase over five times faster than the other.This discrepancy highlights the significant contrast in the poloidal limiter temperature increase rate between the two 3D spectra.Here, the experimental poloidal limiter temperature measurement shows its maximum value among all thermocouple measurements at different toroidal locations.Note that both discharges transitioned to the diverted shaping around t = 1.6 s, even before the 3D field and the full injection of NBI power.This time trace suggests that the temperature of the poloidal limiter is influenced primarily by fast ion losses.
The five times faster increase of the poloidal limiter temperature in discharge #26026 agrees with the modeling result, which shows around five times more fast-ion losses due to the different applied 3D fields.This qualitative agreement for the behavior of the poloidal limiter temperature increase proves the validity of this modeling framework.In addition, this validation result shows that the proper plasma response model and the resonant field amplification must be considered for validating edge localization.Therefore, to improve this validation approach, a more systematic edge localization will be shown in the following sections.

Localization of resonant magnetic field
In this section, the localization of the resonant field is investigated by using the equilibrium and 3D coils of KSTAR used in the previous sections.The KSTAR is particularly suitable for these poloidal spectrum optimizations since it has three rows of 3D coils (as will ITER).In addition to the experimental validation shown in the previous section, the IPEC edge resonant field has demonstrated its effectiveness as a reliable indicator for explaining the thresholds of ELM suppression in KSTAR using the n = 1 3D field as validated by examining various poloidal spectra [9,22].Therefore, based on the validations of the n = 1 ELM suppression threshold in KSTAR, our investigation focuses on analyzing the n = 1 3D field of KSTAR discharges to explore the localization of the resonant field at the plasma edge.The target equilibrium is the identical equilibrium (discharge #26027, t = 5.0 s) used in section 3.1.
The n = 1 3D coil currents are adjusted to maintain ELM suppression for this KSTAR target discharge, to provide a sufficiently large n = 1 edge resonant field at the edge pedestal region, 0.9 < ψ N < 1, above the experimental ELM suppression threshold.The poloidal spectrum of the reference case is adjusted using two 3D coil rows at the top and the midplane, without applying currents at the bottom 3D coils as shown in figure 2(a).Figure 7 shows the poloidal harmonic spectrum of the externally applied normal field on the plasma boundary by these 3D coil currents which can show the variation of the poloidal spectrum using the flexible KSTAR 3D coils.The calculated normal field distribution and resonant field profile by this 3D field are shown in figure 2 and 8(a).
For the second case, the 3D coil variables are adjusted to the 90 • phasing, which has the same coil currents, I T = I M = I B (coil currents at the top, midplane, and bottom row), and the same toroidal phase difference (phasing), ϕ TB = ϕ MB , for all three coil rows.This configuration has been used as standard 3D configuration for ELM suppression in KSTAR [5,9,22].For this case, coil currents (I T = I M = I B ) are set to 2 kA.This current drives the same n = 1 edge resonant field as in the reference case and thus is expected to suppress ELMs for the target discharge.On the other hand, this change of the poloidal spectrum leads to the 16% reduction of the core resonant field, as shown in figure 8(a).
The third case uses an edge localization scheme [16,17] to have an optimized n = 1 poloidal spectrum for safe ELM suppression.This case maintains the same edge resonant field as in other cases for ELM suppression but has a reduced level of the core resonant field than the other two cases, as shown in figure 8(a).As a result, this case has a 32% reduction of the core resonant fields from the reference case.Note that these changes of coil variables are within the engineering limit of existing KSTAR coils, maximum n = 1 amplitude of 5 kA for each coil row to allow flexible phasing.While the core resonant field still exists, as it represents the highest level of edge localization achievable with the given flexibility of 3D coils, it will be referred to as the ERMP.A further change of the poloidal spectrum is possible by relaxing these constraints, but this is beyond the scope of this work.

Simulated fast ion loss with ERMP
This section details the simulation of fast particle loss due to the different resonant magnetic fields shown in figures 7 and 8(a).Figure 8(b) illustrates the reduction in toroidally averaged total fast ion loss resulting from the decreased core resonant field.This is an average loss at 8 toroidal angles, shown in figure 8(c).The simulation results show that gradually reducing the core resonant fields decreases fast ion losses, indicating an improvement in fast ion confinement.This result suggests that reducing the core resonant field can positively impact the overall confinement of fast ions in the plasma.Figure 8(c) also compares the toroidal angle dependence of fast ion losses, as fast ion losses due to RMPs is known to have a strong toroidal angle dependence [13,14,21].Here, the toroidal phase of midplane coils, the closest 3D coils to the plasma boundary, is identical for all three cases, and the poloidal spectra are modified using the toroidal phase of the upper and lower coil rows.Although these simulation results show different toroidal variations with the different poloidal field spectra, the total fast ion losses are minimal with reduced core resonant field case for any toroidal angle, as shown in figure 8(c).This finding indicates that improved fast ion confinement can be maintained at any toroidal angle while maintaining ELM suppression through localization of RMPs.
The phase mapping of lost particles in figures 9 and 10 supports the reduction of fast ion losses due to the ERMP.Here, µ represents magnetic moment, B 0 represents magnetic field at the magnetic axis, E represents particle energy, P ϕ represents toroidal canonical angular momentum, and ψ W represents poloidal flux at the last closed flux surface.Also, q eff = ω ϕ /ω θ represents an effective helicity, where ω ϕ and ω θ represent toroidal and poloidal transit frequencies of particles, respectively.Figure 9 shows that particles with q eff > 2.5 are lost for 100 keV and 80 keV NBIs for the reference case.Most particles have low-pitch angles, covering 0 < µB 0 /E < 0.6.On the other hand, particles with q eff > 3 are lost by the ERMP, as shown in figure 10.These different loss properties in 2.5 < q eff < 3 show that fast particle injection at this region will lead to the fast ion loss to the reference case, unlike the ERMP case.These are consistent with the edge localization of RMPs at ψ ∼ 0.425, (q = 2) ψ ∼ 0.75, (q = 3), shown in figure 8(a).In addition, figures 9 and 10 also show that µB 0 /E ranges of two different spectra are different.While µB 0 /E for both cases indicates that most lost particles are passing particles, the reference case covers relatively higher µB 0 /E, around 0 < µB 0 /E < 0.6.This difference implies the potential loss of the higher pitch-angle particles with the increased core RMP.
The reduced fast ion losses at the inner q eff regions are also consistent with the Poincaré map of the energetic ions shown in figure 11, plotted with 100 keV fast particles.The distinct differences are the presence of the Kolmogorov-Arnold-Moser (KAM) surface around q eff = 2.5 and P ϕ = 0.1 with the ERMP as shown in figure 11(c).On the other hand, the region is more stochastic for the reference case, as shown in figure 11(a).This is consistent with the birth-loss phase space reconstruction in figures 9 and 10, and also with edge localization of RMPs shown in figure 8(d).These indicate that edge localization can prevent the disappearance of the KAM surface, while still keeping the edge resonant field at the narrower pedestal region to suppress ELMs.
In addition, the amplitude scans of applied 3D fields (Amps * B pert ) show the different properties of fast particle losses and the disappearance of KAM surfaces for different spectra and amplitude of applied 3D fields.Here, B pert is the applied 3D field shown in figure 8.For example, at lower B pert at 0.1B pert and 0.3B pert , there is a relatively minor fast ion loss than higher B pert cases as shown in figure 12. Additionally,  all three cases have good KAM surface at q eff = 2.5 surface, as shown in figure 13.Interestingly, all three cases have a similar fast ion losses and slope until 0.5B pert .The implication is that the edge resonant field plays a significant role as the primary source of fast ion loss at lower levels of the applied field, especially before the disappearance of the specific KAM surface.However, above the 0.6B pert , three cases have different slopes and losses.The reference case starts to lead to more fast-ion loss than other cases, as shown in figure 12. Interestingly, figure 13 shows that the KAM surface around q eff > 2.5 disappears at 0.7B pert in the reference case, showing that an increase in the core resonant field eventually breaks KAM surfaces.On the other hand, this KAM surface still remains for ERMP as shown in figures 11 and 13.This breaking of the KAM surface at inner integral q eff indicates that KAM surface breaking at these regions leads to more fast ion losses.The difference in fast ion loss in figure 12 and KAM surface in figure 13 at 0.7B pert clearly indicates a correlation between the threshold in the additional fast ion losses and the threshold for KAM surface breaking as discussed in [21].Also note that the magnetic field line looks slightly different with 0.3B pert at more edge regions, q eff > 3.5, as shown in figures 13(d)-(f ), but this leads to similar fast ion losses as shown in figure 12.This can also imply a more dominant role of the edge resonant field and KAM surface in the fast ion losses.

Summary
This paper demonstrates the potential benefits of localizing the resonant magnetic field for ELM suppression while minimizing fast particle loss.The paper compares the simulation and experimental results using different 3D spectra, effectively demonstrating that the ideal MHD plasma response model is required to explain the observed plasma performance degradation in the experiment.Then, it compares the modeling of fast particle loss with experimental measurements of poloidal limiter temperature in KSTAR discharges with different 3D spectra.The results validate the reliability of this modeling framework in predicting fast ion losses.
Then, the simulations show that reducing the core resonant field can be key to optimizing the poloidal spectrum of a 3D field, which leads to improved fast ion confinement while maintaining edge RMPs.More precisely, the KAM surface with ERMP leads to this confinement optimization, which prevents unnecessary fast ion losses at the core region, unlike reference 3D fields without edge localization.This result shows that edge localization of RMPs can be a promising way to optimize fast particle confinement while maintaining RMP ELM suppression in tokamaks.
Although the experimental validation is limited, the findings contribute to developing optimized 3D field configurations and 3D coil designs for future fusion reactors, improving plasma performance and reducing damage to PFCs.Meanwhile, further research is needed to explore the edge localization approach with a broader range of experimental data and to incorporate nonlinear MHD simulations for a more comprehensive understanding of the effects of different 3D spectra.

Figure 1 .
Figure 1.The time traces of (a) plasma current, (b) poloidal limiter temperature in KSTAR long pulse discharge #21706.(c) Poloidal cross-section of the poloidal limiter, and the last closed flux surface of the discharge #21706 at t = 10 s.

Figure 2 .
Figure 2. The actual geometry of KSTAR 3D coils and perturbed flux surface due to the plasma response and normalized 3D coil currents and distribution of non-axisymmetric fields in color for three different 3D fields.

Figure 3 .
Figure 3.The comparison of resonant field profiles (a) with and (b) without ideal MHD plasma response for KSTAR discharge #26026 and #26027 using the same experimental equilibrium (discharge #26027, t = 5.0 s).

Figure 4 .
Figure 4. (a) The poloidal harmonic spectrum of the externally applied normal field on the plasma boundary, and (b) resonant field profiles for discharge #26026 and #26027 using equilibrium at different plasma beta (discharge #26026 at t = 5.0 s and discharge #26027 at t = 5.2 s).The distribution of the normal magnetic field of (c) discharge #26026 and (d) #26027.

Figure 6 .
Figure 6.Simulated birth (black) and loss (red) position of the lost fast ions deposited by the KSTAR NBI system using the total number of 200 000 particles for discharge #26026 (a) without and (c) with 3D fields, and discharge #26027 (b) without and with (d) 3D field.The simulated fast ion losses of Monte-Carlo particles for discharge #26026 and #26027 at the KSTAR poloidal limiter and diverter (e) without and (f ) with 3D fields.The PL1, PL2, and PL3 represents different toroidal locations of poloidal limiters located at ϕ = 145 • , 170 • , and 190 • and more detailed descriptions are in [20].

Figure 7 .
Figure 7.The poloidal harmonic spectrum of the externally applied normal field on the plasma boundary for three different cases.

Figure 8 .
Figure 8.The comparison of (a) resonant field profiles, (b) the toroidal average of the fast ion losses of Monte-Carlo particles simulated with NuBDeC and core resonant field at the 2/1 rational surface, and (c) the fast ion losses of Monte-Carlo particles as a function of toroidal angle due to applied 3D field simulated with NuBDeC for three different 3D fields with different resonant field profiles.

Figure 9 .
Figure 9. Birth (red dots) and loss (black dots) positions of (a) 100 keV (NBI1-A) and (b) 80 keV (NBI1-B) lost particles at PFC in normalized phase space of for the reference case.The contour lines represent q eff numbers.

Figure 10 .
Figure 10.Birth (red dots) and loss (black dots) positions of (a) 100 keV (NBI1-A) and (b) 80 keV (NBI1-B) lost particles at PFC in normalized phase space for the ERMP case.The contour lines represent q eff numbers.