Active control of Alfvén eigenmodes by external magnetic perturbations with different spatial spectra

Alfvén eigenmodes have been suppressed and excited in tokamak plasmas by (just) modifying the poloidal spectra of externally applied static magnetic perturbations. This effect is observed experimentally when toroidal spectra of n = 2, n = 4 as well as a mixed spectrum of n = 2 and n = 4 is applied. Under the n = 2 magnetic perturbations, the modes are excited or suppressed by modifying the coil phasing between the upper and the lower set of coils. Regardless of the absolute rotation, an even parity for the n = 4 perturbation is observed to reduce the amplitude of the Alfvénic instabilities, while an odd parity amplifies it. To combine the stabilizing (and destabilizing) effect of n = 2 and n = 4, a mixed spectrum is applied, finding similar reduction (and amplification) trends. However, the impact on the mode amplitude is more subtle, due to the reduced coil current required for a mixed spectrum. The signal level on the fast-ion loss detector is sensitive to the applied poloidal spectrum, which is consistent with Hamiltonian full-orbit modelling of an edge resonant transport layer activated by the 3D perturbative fields. An internal redistribution of the fast-ion population is induced, modifying the phase-space gradients driving the Alfvénic instabilities, and ultimately determining their existence. The calculated edge resonant layers for both n = 2 and n = 4 toroidal spectra are consistent with the observed suppressed and excited phases. Moreover, hybrid kinetic-magnetohydrodynamic (MHD) simulations reveal that this edge resonant transport layer overlaps in phase-space with the population responsible for the fast-ion drive. The results presented here may help to control fast-ion driven Alfvénic instabilities in future burning plasmas with a significant fusion born alpha particle population.


Introduction
Toroidal symmetry is the basis of magnetically confined tokamak fusion devices.Nested flux surfaces and particle constants of motion ensure plasma confinement in tokamaks and quasisymmetric stellarators.Symmetry breaking 3D fields can, however, modify the overall plasma confinement and stability [1].The population of supra-thermal particles are especially sensitive to these 3D fields due to their relatively long mean free path and slowing down times [2,3].On the other hand, the gradients in phase-space associated with the fast particle population pose a reservoir of free energy capable to drive a wide variety of Alfvén Eigenmodes (AEs) via wave-particle resonant interactions.In tokamaks, static magnetic field perturbations are proven to work as an external actuator to mitigate AEs by manipulating the supra-thermal particle population and its phase-space gradients [4,5]; such 3D fields may also be mandatory to preserve the device integrity against other intolerable plasma fluctuations such as locked modes [6], Resistive Wall Modes (RWMs) [7,8] and Edge Localized Modes (ELMs) [9][10][11].
In this contribution we report on the enhancing or suppressing effect that relatively small variations of the spatial spectrum of the static, magnetic 3D perturbations have on toroidicity-induced AEs (TAEs).Maintaining the ELM suppression level, the applied poloidal spectrum could trigger detrimental AEs or it could suppress the alpha particle transport at the mid-radius.This could create an internal transport barrier for NBI fast-ions (FI) and alphas, and improve the overall confinement and performance of the fusion device.The rest of this manuscript is organized as follows.Section 2 discusses the experimental observations when applying n = 2 perturbations (section 2.1), n = 4 (section 2.2) and the mixed spectrum of n = 2 and n = 4 (section 2.3).Section 2.4 analyzes signals from FI diagnostics, which are found to be consistent with an internal fast-ion redistribution.Section 3 describes modeling work carried out with the ASCOT code (section 3.1) and how the MEGA code is applied to simulate the modes (section 3.2) as well as particle-wave energy exchange (section 3.3).Section 4 provides a summary and discusses future applications of this work.

Experimental observations
The experiments discussed here correspond to ELMy H-mode plasmas in the ASDEX Upgrade (AUG) tokamak.These plasmas have a current of I p = 0.6 MA, a magnetic field of B t = 2.5 T, a pressure on axis of P 0 ∼ = 50 kPa, resulting on a normalized ratio of plasma pressure to magnetic pressure of ⟨β N ⟩ = 1.2 and collisionallity (ν * e ≈ 0.2).In these pulses, the early application of two sources (2.5 MW each) of Neutral Beam Injection (NBI) together with 1.5 MW of Electron Cyclotron Current Drive (ECCD) injected on axis and counter current help to keep an elevated q-profile and low collisionallity through the discharge.
The externally applied static 3D perturbations, also called Resonant Magnetic Perturbations (RMPs), are induced by two rows of eight saddle coils, one above the midplane and one below [12] (see figure 1(a)).These coils are typically used for ELM control.Each coil is powered independently, so each row can produce a sinusoidal perturbation with a toroidal spectra of n = 2 (figure 1(b)) and n = 4 (see figures 1(c) and (d)).The poloidal spectrum of the n = 2 RMP can be modified by turning the phase of the upper set (ϕ U ) with respect to the lower (ϕ L ), inducing a phase shift between the upper and the lower coil sets (∆ϕ UL = ϕ U − ϕ L ).The observed effect of the coil phasing of the n = 2 RMPs will be discussed on section 2.1.On the other hand, as illustrated on figures 1(c) and (d)), the n = 4 pattern of the applied RMPs can be produced by reversing the current flowing through each saddle coil (of the total of 8) along the toroidal direction.For each row, there are therefore only two possible configurations, that come from reversing the currents on every coil.Considering both rows of 8 saddle coils, there are four possible configurations of the n = 4 MPs, given the two possible absolute phasing of the lower row (ϕ L = 0 • and ϕ L = 180 • ) and the two differential phasings between the upper and the lower row (∆ϕ UL = 0 and ∆ϕ UL = 180 • ).Note that, as it will be further discussed on section 2.2, the applied poloidal spectrum is only affected by the differential phasing ∆ϕ UL .Since each coil is powered independently, the n = 2 and n = 4 RMPs can be combined by reducing the coil current of each toroidal spectra.The effect of this mixed spectrum on AEs will be discussed in section 2.3.

Experiments with n = 2 applied 3D fields
With the goal of probing different fast-ion phase-space volumes, a tangential beam is applied together with a radial one.With a toroidicity of n = 2, the differential phase ∆ϕ UL is scanned by fixing the lower row and rotating the upper one at 1 Hz.This scan is slow enough to minimize the screening of the perturbation by the Passive Stabilizing Loop (PSL) [13].Figure 2 depicts the applied coil phasing as a grey solid line, and their values are depicted on the right axis.In that plot, on a solid black line, the induced modulation on the signal fast-ion loss detector (FILD) [14] is overlayed.This experiment was used to identify the coil phasing that minimizes and maximizes the induced losses.The minimum on the FILD signal is indicated by a blue arrow, as well at the coil phasing applied at that time point,∆ϕ UL = −50 • .The coil phasing producing the first maximum on the FILD signal at 1.85 s is identified as ∆ϕ UL = 100 • , and indicated with red arrows.
Figure 3 depicts how a train of RMP pulses of 200 ms are induced in two comparable, consecutive plasmas whose main difference is the applied poloidal spectrum, corresponding to the two configurations identified in figure 2. Figure 3(c) shows  that both plasmas have the same line integrated electron densities on axis and at the edge.It can be observed that both applied poloidal spectra produce the same level of density pump out, ensuring that the birth profile of the NBI is unaffected by the applied coil phasing.In both discharges, when the RMPs are applied, the amplitude of the ELMs, as monitored by the D α diode diagnostic, is reduced to half of its value.Therefore, the poloidal spectrum does not seem to alter the ELM mitigation in these cases.Figure 3(d) illustrates that similar relative variations on the temperature profiles are induced by both applied spatial spectra.The electron temperature, measured by the Electron Cyclotron Emission (ECE) diagnostic, is, however, sightly higher on pulse #34571, even prior to the application of RMPs.This might be explained by a small difference in wall conditioning/recycling.The same behavior is captured on the measured neutron yield (figure 3(e)).The suprathermal deuterium population generated by the NBI drives a variety of Alfvén eigenmodes during the flattop phase of these discharges.These include toroidicityinduced AEs (TAEs) with frequencies 80−110 kHz, which are depicted in the spectrograms of the Mirnov coil figures 4(a) and (b) for applied coil phasing ∆ϕ UL = 100 • and ∆ϕ UL = −50 • respectively.Analysis of Mirnov coils placed at different toroidal locations reveal that the toroidicity of these instabilities range from n = 2 to n = 5.One can observe in figure 4(a) that when the ∆ϕ UL = 100 • RMP pulses are induced, the amplitude of the TAEs are reduced and even some of the weaker modes (e.g. the mode at 85 kHz) disappear.Later in time, at 3 seconds, another radial source replaces the tangential beam and the safety factor relaxes (due to current diffusion).This brings the TAEs closer to their marginal stability, so they become fully suppressed by the application of the RMPs. Figure 4(c) shows that each time a coil current is applied, the fast signal recorded by a photo-multiplier tube of the FILD shows an abrupt increase.The TAEs on discharge #34571 are marginally stable, which is attributed to the sightly higher radiative damping (which is reported to be the dominant damping mechanism for AEs in similar plasmas [16]) produced by the larger background ion temperature [17].These marginally stable TAEs are practically unobserved until the RMPs are switched on.The perturbation ∆ϕ UL = −50 • produce a more modest increase in the FILD signal when compared to ∆ϕ UL = 100 • , and the TAEs amplitude is increased and visible on the spectrograms.This amplifying effect induced by the ∆ϕ UL = −50 • RMP suggests a rise in the fast-ion drive with respect to the case with ∆ϕ UL = 100 • .On discharge #34570, the full suppression of the amplitude of the TAEs occurs much faster than the observed variation of the global plasma parameters such as the density pump-out.On the contrary, when the RMPs are turned off in the same discharge, the TAEs persist for several tens of milliseconds, at smaller amplitude.This evolution indicates that the fast-ion distribution requires collisional time scales to recover and drive the modes.Indeed, on virtually identical discharges, NBI pulses at different duty cycles revealed that overcoming TAE marginal stability requires a relaxed FI slowed-down distribution.This indicates that the fast-ion drive associated with these low frequency AEs is induced mainly from the radial gradient, which dominates over the energy gradient.
As it will be further discussed on section 4, the RMPs produce a partial reduction (or increase) on the fast-ion drive to the TAE.Whether this drive reduction produces a mitigation (as visualized on the early phase of figure 4) or a full suppression (late phase of figure 4) will depend on the balance of drive and damping associated with a specific mode, which includes the drive produced by phase-space gradients not affected by the application of the RMPs.

Experiments with n = 4 applied 3D fields
n = 4 MPs produce a weaker perturbation, minimizing the impact on the kinetic profiles, but this perturbation can still affect the TAEs and fast-ion losses.Since the number of possible configurations of the n = 4 MPs is limited to 4, it takes a relative small number of discharges to assess its impact on Alfvénic activity and fast-ion losses.Figure 5 shows the time traces of two identical discharges which are used to study the effect of n = 4 RMPs on AEs.The main difference between those two discharges is the absolute phase of the lower set of coils (ϕ L ).
Both discharges have 5 MW of NBI and about 2.7 MW produced by 4 ECRH gyrotrons.These two discharges have similar densities both on axis and at the location of the TAEs.Intermittent RMP phases of 200 ms are applied in which the MPs differential phase (∆ϕ UL ) is reversed.This helps studying the effect of coil phasing on TAEs and FI losses while keeping the density as constant as possible.This is reflected on the steady frequency of the destabilized TAEs depicted in figure 6.These experiments do not aim at controlling the TAEs in the early phase (t < 1 s) of the discharge, where elevated q-profile and fast-ion pressure produce a drive (γ EP ) too high to control the modes.To study the effect of the RMPs on TAEs driven by different fast-ion phase-spaces, the beam configuration was preprogrammed to change from two tangential sources (Q6+Q7) injecting ions with very similar geometry, to the combination of a radial and a tangential source (Q7+Q8).The latter produced a broader yet less steep fastion profile, that reduced the amount of fast-ion drive, bringing the TAEs closer to their marginal stability.The spectrogram shows how the TAEs become weaker and their amplitude is strongly modulated by the applied coil phasing ∆ϕ UL .Green arrows indicate that the TAEs are being amplified by the applied ∆ϕ UL = 180 • on both discharges.As figure 6 depicts the entire discharges to illustrate the mode evolution during the different NBI and RMP phases, it might be too busy to properly visualize the details of how the TAEs respond to the applied RMPs.To illustrate this more clearly, figure 7 shows the spectrograms of both discharges for the regions where the TAE is modulated by the applied ∆ϕ UL (green rectangles on figure 6).In the marginally stable phase, the only existing TAEs are identified as n = 2 and n = 3.A simple frequency tracking algorithm is used to calculate the evolution of the amplitude of the n = 2 TAEs, which is depicted in figures 7(c) and (d).One can observe here more clearly that the mode amplitude is increased with ∆ϕ UL = 180 • .At the end of figure 7(d) the MPs are turned off and the TAE amplitude decreases.In discharge #37698, the FILD signal is also modulated by the applied ∆ϕ UL .This is not reproduced in case of failure of one of the coils.For instance, discharge #36371 in 2019 was almost identical to #37698, but one of the coils was miswired, reversing its polarity.This produced a leakage of the MPs to n = 1 component, creating additional geometrical resonances that in turn prevent the observation of a modulated FILD signal.From these experiments one can conclude that, while n = 4 MPs are too small to produce a significant variations on the kinetic profiles, they are sufficiently large to affect the fast-ion population and marginally stable TAEs, being a good case of study to further investigate the mechanisms responsible for TAE control.

Experiments with a mixed toroidal spectrum of applied 3D fields
As TAEs and FI are modulated by the applied spectrum for n = 2 and n = 4, one could apply both simultaneously to combine the effect of both spectrums.Due to the limitations on the coil current, the toroidicities are mixed by combining 0.5 kA for n = 2 and 0.5 kA for the n = 4. Intermittent perturbations are applied in which the ∆ϕ UL of each n is modified.The ∆ϕ UL that are observed to suppress the TAEs (∆ϕ UL = 100 • for n = 2 and ∆ϕ UL = 0 • for n = 4) are applied together, while the ∆ϕ UL observed to amplify the TAEs (∆ϕ UL = −50 • for n = 2 and ∆ϕ UL = 180 • for n = 4) are combined on the other perturbations.The result of this modulation on the TAEs is depicted in figure 8. Similar TAEs as in discharges #37698 and #37699 are found.As expected, the amplitude of the TAEs is amplified when the destabilizing ∆ϕ UL is applied.This amplification effect is however not observed as many times as when a single toroidicity is applied.From these results one can conclude that each n has an impact on the TAE, but this effect is weaker due to the reduced coil current, being the resulting impact of a mixed spectrum weaker than single n perturbative fields.As it has been reported that fast-ion transport induced by MPs has upper and lower coil current thresholds [18,19], one  could speculate that such thresholds also exist on AE control.Dedicated experiments are, however, still required to provide further insight on these current thresholds and whether they are responsible for the weaker effect of the mixed spectrum perturbative fields on AE stability.

Indications of internal fast-ion redistribution
The velocity-space of FI losses induced by RMPs in discharges with n = 2 is obtained by the FILD diagnostic and depicted in figure 9(a).The tangential source dominates the signal, as the full and half energy can be identified as gyroradii of ∼3.25 cm and ∼2.25 cm respectively.Above 4 cm, one can observe the ELM-induced high energy tail [20].There is an additional spot at larger pitch angle and full energy that corresponds to the radial source.With this information, the orbits for the tangential and radial sources at full energy are traced backwards in time from the position of the FILD probe head, and their poloidal projections are depicted in figure 9(b).One can see that the spot from the tangential beam corresponds to a trapped orbit (depicted in green) while the spot from the radial beam corresponds to deeply trapped orbits (depicted in black).Both of these orbits explore the TAE on their inner banana leg and the externally applied magnetic perturbations on their outer leg.The applied coil phasing does not affect the velocityspace, indicating that the transport is induced by the same geometrical resonances.This is in line with experimental [21] and numerical [22] studies that attributed the transport to an edge resonant transport layer (ERTL).This ERTL was found to transport the injected NBI ions inwards or outwards depending on the applied coil phasing.Indeed, figure 4(c) shows that the amplitude of the measured fast-ion flux is strongly affected by the applied coil phasing.That dependence of the FI loss flux with the applied coil phasing is identified as an indication of an internal FI phase-space redistribution.According to [21,22], the ERTL highly depends on specific ion orbit (i.e.energy and local pitch angle), and thus, the induced redistribution is not measurable by diagnostics with a weighting function that is broad in velocity-space, for instance fast-ion D α (FIDA) spectroscopy [23] or the neutron detectors.Indeed, figure 3(e) shows that the neutron signals suffer the same level drop regardless of the applied coil spectrum.Figure 9(c) shows that the FIDA diagnostic behaves in a similar manner, as the measured level of MP-induced fast-ion drop lies within its standard deviation under both spatial spectra, across the minor radius.This drop is larger near the core due to higher FI density.

Hamiltonian full-orbit ASCOT modelling
The Hamiltonian full-orbit ASCOT code [24] is applied on [22] to investigate the ERTL produced by the RMPs on the injected fast-ions.This code realistically includes the NBI deposition profile, a collisional operator and a numerical first wall in 3D geometry that intersects the simulated orbits.It is extensively validated using FI measurements (e.g.[25,26]).The relaxed, slowed down FI population is calculated under both ∆ϕ UL , and the resulting radial profiles are plotted in figure 10(a).One can observe that both coil phasings, produce a small, similar fast-ion drop, the one induced by ∆ϕ UL = 100 • being sightly higher, which agrees with the decrease of the FI profiles measured by FIDA.
However, when these radial profiles include only the small pitch range around the beam injection and are measured by FILD (figure 10(b)) (v ∥ /v ∈ [0.7, 0.8]), a larger difference is found closer to the edge.This explains the fact that the measured FILD signals are dependent on the applied poloidal spectrum (figure 4(c)).The fact that the pitch-filtered profiles show a larger reduction than the profiles that integrate over the entire FI population, is a clear indication that the redistribution induced by the MPs is localized in velocity-space, which is also observed by FILD, but not visible on the FIDA and neutron flux signals, which do not have such velocity-space resolution.
To visualize the ERTL generated during the n = 2 experiments, a scan of energy (E) and major radius at the midplane (R) is launched at the pitch angle injected by beam.The particles are followed for 300 µs, and the time averaged deviation of their toroidal canonical momentum (⟨δP ϕ ⟩) is plotted in figure 11, being P ϕ = Zeψ + mRv ϕ , with Ze the ion charge, ψ the poloidal flux, m the ion mass, R the major radius and v ϕ the toroial projection of the ion velocity.⟨δP ϕ ⟩ can be interpreted as a proxy for radial transport, as when ⟨δP ϕ ⟩ > 0 (⟨δP ϕ ⟩ < 0) the particle moves inwards (outwards).In view of figure 11, one can conclude that the ERTL induces an outwards transport in #34570, maximizing the FILD signal and relaxing the FI radial gradients, while a confinement improvement takes place in #34571.On the other hand, the ERTL calculated for the n = 4 RMPs (see figure 12), also show that ⟨δP ϕ ⟩ > 0 for the phases where the TAE amplitude is being amplified (∆ϕ UL = 180 • ) and ⟨δP ϕ ⟩ < 0 for the phases where the TAE is being reduced (∆ϕ UL = 0 • ).

Hybrid kinetic-magnetohydrodynamic (MHD) MEGA simulations
Hybrid kinetic-MHD simulations using the resistive, 3D, nonlinear MEGA code [27][28][29][30] of discharges #34570 and #34571 are performed to study the underlying mechanisms connecting the RMP-induced FI redistribution to the dependence of the TAE amplitude.
The inputs of the simulations are plotted in figure 13(a), including the electron density and temperature profiles as well as the q-profile of the equilibrium reconstruction performed with the CLISTE code [31].This q-profile has a reversed shear   consistent with motional stark effect (MSE) measurements [32] and MHD markers.These discharges have a significant magnetic shear in the region ρ pol > 0.6.The parameters of the built-in MEGA distribution are tuned to capture the main features of the population calculated by the TRANSP code using the NUBEAM module [33].The MEGA distribution used in these simulations is an anisotropic slowing-down FI population, equivalent to the one employed in [34][35][36].Figures 13(c) and (d) compare the poloidal distribution from NUBEAM and the reproduced by MEGA.
The same toroidal modes as observed in the experiment are destabilized (n = 2−5).These modes propagate in the same direction as the ion diamagnetic drift, and saturate within ≈0.1 ms.The spatial location of the modes is inferred from the experiment by the ECE diagnostic [37].In figure 14(a   The magnetic fields from the RMPs are added to the same fields from CLISTE using the vacuum approximation, before the MHD force balance is evaluated by MEGA as the initial simulation integration step.This produces a static 3D background in which the TAEs are destabilized once kinetic effects are included.The simulated TAE mode frequency, location and poloidal structure are not affected by the inclusion of these RMP fields.Nevertheless, as shown in figure 15 and table 1, the applied poloidal spectrum modifies the growth rates, as ∆ϕ UL = −50 • (#34571) produces sightly higher growth rate than ∆ϕ UL = 100 • (#34570).The resulting differences in growth rate are smaller than what it is observed in the experiment.This can be explained by the fact that these simulations use a δf version of the MEGA code, which does not include a constant particle source, required to reproduce the amplitudes during the non-linear phase.Using the full f method, including such source, could reproduce the RMP-induced redistribution of the injected fast-ions constantly.Unfortunately, those multi-phase simulations including the entire toroidal geometry would require an amount of computational resources that are not available for this project at the moment, and it is proposed for future work.Still, the fact that the growth rates when the RMPs of #34571 are applied are larger than when using #34570 is consistent with the trend observed in the experiment.Moreover, these differences scale up with β EP , indicating that are modifications on the drive resulting from the redistribution of the initial fast-ion population, and not by the damping mechanisms from the interaction of the mode with the bulk plasma, that would produce the same growth rate difference when scanning β EP .Additionally, MEGA is used to calculate the plasma response to the externally applied 3D fields.The methodology, convergence tests and benchmark against other codes are described in detailed in appendix.The resulting plasma response fields are used as the equilibrium of hybrid simulations.It is observed that including the response of the plasma to the RMPs on the initial equilibrium does not affect the instability growth rate, when compared to the simulations that directly assume the vacuum approximation as the equilibrium (figure 15).

Particle-wave energy exchange
A synthetic diagnostic is implemented in MEGA to calculate the particle-wave energy exchange as reported in [35,38].
The energy exchange is found to be localized around magnetic moment µ = 2 × 10 −15 kgm 2 s −2 T −1 while the instability grows linearly.Similar ASCOT simulations as reported on figure 11 are performed to depict the ERTL in phase-space.This time, the magnetic moment is fixed at µ = 2 × 10 −15 kgm 2 s −2 T −1 while each pixel represent a different value of the other motion invariants, energy (E) and toroidal canonical momentum (P ϕ ).The time-averaged variation of the toroidal canonical momentum ⟨δP ϕ ⟩ is depicted for both poloidal spectra in figures 16(a) and (b).The trend observed in these figures, is similar to what is shown in figure 11, meaning that ∆ϕ UL = 100 • locally reduces the confinement while ∆ϕ UL = −50 • locally improves it.The externally applied 3D fields induce a radial transport of ⟨δP ϕ ⟩ ≈ 1%.This deflection of the orbits is too small to reach the tokamak wall or to cross the topological boundary to become a lost orbit, but big enough to induce an internal redistribution, affecting the P ϕ gradient.One can conclude that particles captured by FILD are initially closer to the plasma edge, not exploring the TAE, explaining the fact that FILD does not measure coherent losses at the frequency of those TAEs.By using the phase-space gradients of the fast-ion distribution (f FI ), it is possible to estimate locally the driving and damping contributions (γ TAE ) before ruining the hybrid simulation, as follows, Figure 16(c) depicts the analytical drive for the initial FI distribution.Figure 16(d) illustrates the calculated energy transfer during the linear phase of the hybrid kinetic-MHD simulations using the synthetic diagnostic described on [35,38].On both figures 16(c) and (d), the solid black contour lines represent the FI population around the selected µ.On all the plots in figure 16, the green solid lines correspond to the particles that overlap the TAE location when crossing the outer midplane, providing a spatial reference.Comparing figures 16(c) and (d), it can be observed that the analytical prediction for the drive and the synthetic diagnostic implemented on the simulations are located in the same phase-space region (around 60 keV), providing a verification of the result.This FI phasespace region driving the TAE is located at the same location as the ERTL depicted in figures 16(a) and (b).Moreover, as also depicted on figure 11, the value of the ⟨δP ϕ ⟩ associated with the ERTL depends on the applied spectrum.In the phase-space location where the FI drive the TAEs, the RMPs with ∆ϕ UL = 100 • produce a ⟨δP ϕ ⟩ negative, which locally flattens the FI gradients.Contrarily, the poloidal spectrum generated with ∆ϕ UL = −50 • produce a ⟨δP ϕ ⟩ positive, increasing the gradient and thus the FI drive.That is a plausible explanation for the fact that while MPs cause a signal drop on FIDA and neutrons, the poloidal spectra of the applied RMP can modify the local FI redistribution at the same phase-space region where the TAE is located, affecting the mode drive and thus its stability.

Summary and conclusions
This manuscript introduces experimental evidence as well as supporting modelling of supression and excitation of the amplitude of TAEs by modifying the poloidal spectrum of static, magnetic 3D perturbations.These experiments cover externally applied 3D fields with toroidal spectra of n = 2, n = 4 and a mixed spectrum of n = 2 and n = 4.The global FI measurements provided by the FIDA and neutron diagnostic are equally reduced by the applied poloidal spectrum and TAE amplitude.On the other hand, the coil phasing produce different values of FI loss as measured by the FILD diagnostic on specific velocity-space regions.ASCOT simulations explain the losses and identify an ERTL producing an internal FI redistribution, and locally modifying the radial (P ϕ ) gradients of the FI distribution.Hybrid kinetic-MHD simulations reproduce the radial profile and frequency of the measured instabilities.A synthetic diagnostic implemented in MEGA show that energetic particle drive overlaps in phase-space with the induced ERTL, producing slight variations on the resulting instability growth rate.The response of the plasma to these static magnetic perturbations is computed by MEGA and benchmarked against VMEC and MARS-F, but in this case, it does not alter the obtained TAEs.Hybrid simulations show an overlap of the ERTL and the TAE drive in phase space.
This study shows that in future fusion devices (e.g.ITER) designed to use RMPs to suppress ELMs and other detrimental perturbations, RMPs could amplify AEs, and their subsequent loss of confinement.On the other hand, small modifications on the RMP spectrum could reverse this effect, locally relaxing the FI gradients and thus suppressing the AEs.The spatial spectrum of applied 3D fields can be further optimized to produce this local FI gradient relaxation while increasing the overall FI confinement [39].This could improve the plasma performance while suppressing AEs, as achieved in KSTAR for ELMs and disruptions [40].In future burning plasmas, this method to locally modify the phase-space gradients paves the way to suppress the transport associated with AEs, which could be applied to improve the confinement of NBI fast-ions as well as fusion born alphas, while maintaining the same level of pedestal degradation required by ELM suppression.

Appendix. Plasma response in MEGA
Prior to the time integration of the system, MEGA calculates the MHD mismatch ∂ρ/∂t t=0 = 0, ∂v/∂t t=0 = 0, ∂p/∂t t=0 = 0 and E t=0 = 0 as the right hand side terms of equations ( 1)-( 3) and ( 5) from [29].These terms are subtracted to the calculated time-derivatives during time integration, ensuring that the system remains in equilibrium and the MHD variables do not evolve unless an initial random seed perturbation and fast-ion gradients are applied.
Since both pressure and magnetic field are given by an axisymetric equilibrium, the MHD mismatch of the initial system is generally small.However, the externally applied 3D magnetic fields perturb this equilibrium.Therefore different results will be obtained depending whether these 3D fields are included before or after the initial MHD balance.If they are applied before, the MHD mismatch accounts for these discrepancies, and the system remains in equilibrium unless fastions destabilize AEs (that will propagate in a 3D equilibrium, figure 15).On the other hand, if these 3D fields are applied after the initial MHD balance, the effect of the fields is not included on the MHD mismatch and MEGA calculates the response of the plasma to these externally applied 3D fields.
In the plasma response simulations reported here, kinetic effects are neglected, and thus the evolution of the markers is not calculated in MEGA to save computational time.A binary mask is applied near the edge (ρ pol = 0.98) ensuring that the magnetic field at the plasma boundary remains constant, acting as an internal simulation boundary.However, when toroidal rotation is applied, this boundary requires the modification of the input plasma rotation to ensure that ω ϕ = 0 and ∂ω ϕ ∂/ρ pol = 0 at the position of the binary mask, to avoid artificial electric fields at the boundary that may terminate the simulation.
To validate the plasma response simulations, the applied coil configuration from discharge #34570 (n = 2, ∆ϕ UL = −50 • ) is employed.All the components different than n = 2 are filtered out to reduce numerical noise.Different simulations were run applying this filter each 10, 100, 200 and 500 time steps finding the same results.Convergence on the simulation time-step is found as long as Courant-Friedrichs-Lewy condition is satisfied.Figure A1 shows the temporal evolution of the kinetic (a) and magnetic energies (b) when the poloidal resolution is modified while maintaining N ϕ × 32.In these n = 2 simulations, only half of the toroidal geometry is simulated.One can observe that a resolution larger than N R = N Z = 256 is required to find stabilization of the plasma response and that convergence is achieved with resolutions above N R = N Z = 512.The time derivatives of these energies are not strictly zero at the end of the simulation.However, their value is very small, and the poloidal structure of the associated perturbations (shown in figure A2) is observed to remain constant.Therefore, this final state is assumed to be a stabilized plasma response.
The values employed on the diffusive parameters also have an impact on the simulation.Figure A1(c) shows the evolution of the kinetic energy for different values of normalized resistivity η/v A R 0 µ 0 .These simulations are set with N R × N ϕ × N Z = 512 × 32 × 512 and the applied n = 4 poloidal spectrum from discharge #37698.The other diffusive parameters are set to ν = 1 • 10 −7 v A R 0 and χ = 5 • 10 −7 v A R 0 .One can observe that the results η/(v A R 0 µ 0 ) = 1 • 10 −10 and η/(v A R 0 µ 0 ) = 1 • 10 −9 are identical.This is produced by the intrinsic numerical damping of MEGA, that limits the minimum achievable resistivity for a given spatial resolution of the MHD grid.A simulation using Spitzer resistivity (η S ) would require unfeasible spatial resolution, therefore, a resistivity η/(v A R 0 µ 0 ) = 1 • 10 −8 is employed in these simulations.
The toroidal flow of the bulk plasma (ω ϕ ) is proven to screen the B r perturbation.This phenomenon is typically reproduced by numerical studies [41].To observe this screening effect in MEGA, a scan of 6 simulations was performed.Each run uses a plasma rotation (ω ϕ ) that is obtained by multiplying the experimental rotation profile ω exp by a constant.The final value of the radial magnetic field is shown in figure A1(d), where one can observe how it is mitigated when the rotation profile is artificially increased.
To further validate these plasma response simulations, the resulting MEGA fields are compared with calculations performed with the MARS-F [7,42] and VMEC [43] codes, using the same equilibrium and kinetic profiles.VMEC is an ideal MHD code and thus provides solutions with nested flux surfaces only.On the other hand, MARS-F is a linear resistive code that solves MHD equations in toroidal geometry.The MARS-F code is run with and without parallel slow wave damping.The poloidal spectrum calculated by these codes is shown in figure A3 as the radial magnetic field B r along the minor radius (ρ pol ) for all the different poloidal harmonics (m).The overplotted dashed lines correspond to the product of the safety factor (q) and the toroidicity of the applied 3D fields (n), indicating the regions of the spectrum that are pitchaligned with the bulk plasma.The applied binary mask is represented by a horizontal magenta dashed line on the MEGA spectrum of figure A3(a).One can observe that all results are similar, showing an internal kink at (ρ pol = 0.7) and an edgepeeling response near the plasma boundary.However, the ideal MHD spectrum calculated by VMEC agrees with MARS-F without parallel slow wave damping.On the other hand, when this damping mechanism is included in MARS-F, it resembles more the MEGA spectrum.This can be explained by the fact that MEGA uses a larger resistivity and has a finite damping associated with its integration scheme.

Figure 1 .
Figure 1.Illustration of the two rows of 8 coils each inducing static magnetic perturbative fields on the plasma (a).Radial field induced on the last closed flux surface by the n = 2 RMP configuration used on discharge #34570, with a differential coil phasing of ∆ϕ UL = 100 • (b).Radial field induced the last closed flux surface by the n = 4 RMP configuration used on discharges #37698 with a differential coil phasing of ∆ϕ UL = 0 • (c) and ∆ϕ UL = 0 • (d).

Figure 5 .
Figure 5.For discharges #37698 (solid lines) and #37699 (dashed lines), temporal evolution of: (a) auxiliary heating, (b) plasma and ELM control current, (c) core and edge electron densities, (d) core and edge electron temperature.The main difference among both discharges is the phase of the lower coil row (ϕ L ).The applied poloidal spectrum is varied between even (∆ϕ UL = 0 • ) and odd parity (∆ϕ UL = 180 • ).

Figure 6 .
Figure 6.Spectrograms of discharges #37698 and #37699 showing TAEs becoming marginally unstable and being affected by the applied coil phasing ∆ϕ UL .

Figure 7 .
Figure 7. Spectrograms of discharges #37698 and #37699 showing TAEs becoming marginally unstable and being affected by the applied coil phasing ∆φ UL .

Figure 8 .
Figure 8. Spectrogram showing TAEs when a mixed spectrum n = 2 and n = 4 is applied.

Figure 9 .
Figure 9. (a) Velocity-space pattern of the measured losses when MP are active.(b) Poloidal projection of the trajectory of the measured lost ions overlapping the imposed MP fields.(c) MP induced drop on the fast-ion profile measured by FIDA.Adapted figure with permission from [15], Copyright (2023) by the American Physical Society.

Figure 10 .
Figure 10.Integrated (a) and pitch-filtered (b) FI radial profile simulated with the ASCOT code for both coil phasing.Adapted figure with permission from [15], Copyright (2023) by the American Physical Society.
) the TAEs are observed to be near ρ pol ≈ 0.7.The frequency and radial location of these instabilities is well reproduced by these numerical simulations.In figures 14(b)-(d), modes n = 3-5 are observed together with the shear Alfvén wave continuum, depicting the instability laying in the TAE gap at the frequency

Figure 13 .
Figure 13.(a) Kinetic profiles and safety factor inferred from the experiment and employed on the MEGA simulation.(b) Space-integrated velocity-space of the employed anisotropic slowing down FI distribution employed in MEGA.Velocity-space integrated poloidal plane from NUBEAM simulation (c) and reproduced by the MEGA initial distribution (d).

Figure 14 .
Figure 14.(a) ECE-inferred TAE radial location and frequency.Location in the SAW continuum for n = 3 (b), n = 4 (c) and n = 5 (d).Location of the simulated TAE on the shear Alvén wave continuum.(e) Poloidal structure of the simulated n = 4 TAE.

Figure 15 .
Figure 15.δvr evolution of TAEs including externally applied 3D fields for different values of initial βep.

Figure 16 .
Figure 16.For particles with magnetic moment µ = 2 × 10 −15 kg m 2 s −2 T −1 , ERTL induced by MPs for coil phasing ∆ϕ UL = 100 • (a) and ∆ϕ UL = −50 • (b) producing a modification of the total gradient.Predicted particle-wave energy exchange based on gradients of the FI distribution (c) and simulated by MEGA (d) both overlaying the ERTL location.Reprinted figure with permission from [15], Copyright (2023) by the American Physical Society.
via the Euratom Research and Training Programme (Grant Agreement No. 101052200-EUROfusion).Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission.Neither the European Union nor the European Commission can be held responsible for them.The support from the Spanish Ministry of Science (Grant Nos.PID2020-116822RB-I00 and FPU15/06074) is gratefully acknowledged.J. Galdon-Quiroga acknowledges funding from the European Union under the Marie Sklodowska-Curie Grant Agreement No. 101069021.The MEGA and ASCOT simulations reported herein were performed on the MARCONI cluster under the MEGAFILD project.

Figure A1 .
Figure A1.Convergence test performed changing the poloidal resolution while maintaining N R = N Z .The kinetic (a) and magnetic (b) energies converge with a resolution of N R = N Z = 512.(c) Scan on normalized plasma resistivity (η/v A R 0 µ 0 ), for an n = 4 run with N R × N ϕ × N Z = 512 × 32 × 512, ν = 1 • 10 −7 v A R 0 and χ = 5 • 10 −7 v A R 0 .(d) Radial magnetic field obtained for different simulated toroidal flows.The total magnetic field is reduced when increasing the simulated toroidal flow (ω sim ).

Figure A2 .
Figure A2.Poloidal structure of the plasma response for magnetic field along the major radius δB R and toroidal flow velocity δv ϕ .

Figure A3 .
Figure A3.Poloidal spectrum of the radial magnetic field Br calculated with MEGA (a) VMEC (b) MARS-F (c) and MARS-F including parallel slow wave damping (d).

Table 1 .
Normalized TAEs for different values of fast-ion pressure.