Mitigation of Alfvénic activity by 3D magnetic perturbations on NSTX

Observations on the National Spherical Torus Experiment (NSTX) indicate that externally applied non-axisymmetric magnetic perturbations (MP) can reduce the amplitude of toroidal Alfvén eigenmodes (TAE) and global Alfvén eigenmodes (GAE) in response to pulsed n  =  3 non-resonant fields. From full-orbit following Monte Carlo simulations with the one- and two-fluid resistive MHD plasma response to the magnetic perturbation included, it was found that in response to MP pulses the fast-ion losses increased and the fast-ion drive for the GAEs was reduced. The MP did not affect the fast-ion drive for the TAEs significantly but the Alfvén continuum at the plasma edge was found to be altered due to the toroidal symmetry breaking which leads to coupling of different toroidal harmonics. The TAE gap was reduced at the edge creating enhanced continuum damping of the global TAEs, which is consistent with the observations. The results suggest that optimized non-axisymmetric MP might be exploited to control and mitigate Alfvén instabilities by tailoring the fast-ion distribution function and/or continuum structure.


Introduction
As the fusion program develops in support and preparation of burning plasma operation, the role of fast ions is becoming central for scenario development. Not only is proper confine ment of energetic ions from neutral beam injection (NBI) key for heating but it also enables the use of beam ions for cur rent drive, and hence better control of magnetic safety factor or q profile. Concerns exist for the rapid loss of energetic ions induced by Alfvénic instabilities, that may jeopardize the plasma performance or, if localized, it can damage vessel components [1,2].
In magnetic confinement devices Alfvénic instabilities are commonly observed which are often driven by the fast ion population itself and manifest themselves with a strongly nonlinear dynamic, characterized by bursting and frequency chirping [3,4]. Those instabilities can span multiple fre quency bands up to the ioncyclotron frequency and have been observed in tokamaks, spherical tokamaks, and stellarators in which energeticparticle populations are created with, among others NBI, ioncyclotron and electroncyclotron heating. Coupled 'avalanches' of multiple bursting modes were also observed [5] and in many cases accompanied by substantial fast ion losses.
To maximize the performance of a fusion device, viable methods to suppress or mitigate the effect of such modes are highly desirable. Since instabilities grow to a finite size when the growth rate exceed the damping rate, the mitigation or sup pression of AEs can be achieved by reducing the drive for the instability or by enhancing the damping mechanism. As the chirping dynamic is in some cases associated with coherent structures that propagate in phase space, mitigation could also be achieved by interfering with the formation and/or propaga tion of such structures.
In several experiments the mitigation of energeticparticle driven instabilities was obtained with different forms of aux iliary wave heating. In an experiment performed in a small magnetic dipole device [6,7], the application of electron cyclotron waves to the energetic electrons that were driving violent, bursting, rapidlychirping interchange modes con verted the modes into a smaller amplitude mode of nearly constant frequency. Electron cyclotron heating was tested in DIIID as possible tool to control reversed shear Alfvén eigen modes, finding reduced activity when EC power was deposited near the location of reversal of safety factor shear [8]. More recently, in the stellarator TJII observations of strong impact of EC heating on nonlinear behaviour of Alfvén modes were reported, up to complete suppression [9]. During periods of fastwave heating, reduction or suppression of Alfvénic activity has been reported in NSTX [10][11][12]. While some of these observations could be interpreted as scatter of coherent phase space structures [6,7,10], it is extremely challenging to draw conclusions about the mechanisms responsible for the observed behaviour. This is partly due to the difficulty of developing actuators that affect a single aspect of the mode dynamic, i.e. a specific mechanism for driving or damping modes.
Mitigation can be achieved indirectly, for example when Alfvénic instabilities are coupled with other MHD activity as was observed among other on JT60U [13] and TFTR [14]. An example is the occurrence of corelocalized Toroidicity induced Alvén eigenmodes (TAE) before and Ellipticity induced Alvén eigenmodes after giant sawtooth crashes [15]. By using electron cyclotron waves to reduce the amplitude and increase the frequency of sawtooth crashes the TAEs can be suppressed [16].
On a number of tokamaks it has been shown that external applied threedimensional magnetic perturbations can enhance and/or redistribute the fast ions [17]. A novel method to manipulate and control the Alfvén Eigenmode AE activity is by using external applied static threedimensional magnetic perturbation (MP) fields [18]. Extensive modeling suggested that a reduction of the fastion drive for the instabilities could be responsible for the mitigation observed in these scenarios. The study of the role of fastion transport in the perturbed equilibria requires accounting for the plasma response to the applied 3D fields. The plasma response as obtained from the ideal MHD code IPEC [19] was used to address the role of fastion transport due to MP fields in NSTX. In this paper we further extend the investigation of the fastion drive in these scenarios with the inclusion of the resistive MHD plasma response to the applied 3D magnetic perturbation. This allows for a more accurate modeling of the fastion transport and a more quantitive comparison can be made with the exper imental results.
The role of enhanced continuum damping as a possible mechanism for the observed reduction of toroidal Alfvén eigenmodes in presence of n = 3 static fields on NSTX is also explored in this paper by determining the 3D Alfvén con tinuum for the 3D perturbed equilibrium using 3D codes that were developed for stellarators.
The remainder of the paper is organized as follows: the experimental evidence for mitigation of TAE and GAE in dis charges with pulsed MP is presented in section 2. In section 3 the simulations that were performed to determine the plasma response to applied MP and time dependent transport of fast ions induced by the total perturbation fields is presented for the relevant experimental cases. Possible effects of changes of the distribution function and toroidal mode coupling associ ated with the MP on the AE stability are studied in section 4 while a summary of the results and a perspective for the use of this technique in fusion devices is presented in the concluding section.

Experimental methods and results
The observations discussed in this work were obtained in experiments conducted on the National Spherical Tokamak experiment (NSTX) [20]. NSTX is a medium size device (major radius R = 0.85 m, minor radius ≈ a 0.6 m, elonga tion κ = 1.8-2.2), that operates at toroidal magnetic fields of = B 0.3 t -0.5 T and plasma currents up to = I 1 p MA. Neutral beam injection (NBI) is the main auxiliary heating system, composed of three independent sources with adjustable acceleration energy, E b , between 60 and 90 keV, delivering a maximum total injected power of = P 7 NBI MW. The three beam lines inject deuterium neutrals in the direction of the plasma current with tangency radii of = R 0.7 tan , 0.6 and 0.5 m, providing the main source of the nonthermal ion popula tion. NSTX is equipped with a system of six twoturn coils that produce static magneticfield perturbations. The coils are located at the equatorial plane (see figure 1). Coil pairs can be energized separately, to obtain perturbing fields with domi nant toroidal periodicity n = 1, 2 or 3.
Since the velocity of beam ions are up to five times the Alfvén velocity, fast ions provide a strong drive for different types of Alfvénic instabilities that develop throughout the discharge. In this study the AE instabilities were excited by fast ions from the 90 kV deuterium NBI neutrals that were injected into an ELMfree Hmode plasma with lithium coated carbon walls [21]. Traces from two representative dis charges ( =

Mitigation of low frequency TAE
Mitigation of TAE activity was observed for nonchirping TAEs in Hmode plasmas where pulsed n = 3 fields were uti lized to pace ELMs. The modes were visible (NSTX pulse 133838) in a spectrogram of a magnetic probe (figure 2(d)) showing a cluster of coherent fluctuations at frequencies between 60 and 90 kHz which are Dopplershifted TAEs with toroidal numbers n = 2 (60 kHz), n = 3 (70 kHz), and n = 4 (80 kHz). From reflectometer density fluctuation measure ments, a radial displacement of 0.14 mm was found in the plasma edge for the n = 2 mode. The density fluctuations could not be measured with reflectometry beyond the pedestal because of the hollow density profile which is characteristic of NSTX Hmodes. Ideal MHD computations with the NOVA code [22] indicate that the TAE gap is open up to the last closed flux surface when no MP fields are present. These simulations also showed that these are global TAEs [23] stretching from well inside the plasma to the edge whereby a large number of poloidal harmonics couple to create a finite TAE amplitude at the edge. In response to the applied n = 3 MP fields the TAE ampl itude drops rapidly as can be seen on the Mirnov coil signal in figure 2(e) which is bandpass filtered around the n = 2 mode frequency. The n = 2 signal decreases rapidly after the MP coils are energized but before the ELM crash which occurs approximately 2 ms after the coil current is stepped up. Notice that there is no increase in the neutron signal when the TAE amplitude is reduced due to the MP fields except for pulse at 0.50 s where the TAE amplitude is small and the MP fields are successful in suppressing the mode fully.

Mitigation of high frequency GAE
In discharge 138146 a MP configuration with toroidal peri odicity n = 3 was employed with coils pulses of 3 ms long and a repetition rate of 40 Hz. The coil current amplitude was chosen to be just below the threshold for repetitive triggering of ELMs ('ELM pacing' [24]). In discharges such as the one shown in figure 3, ELM pacing does not occur, and the plasma remains virtually ELMfree. The coil pulses, however, cause rapid 8% reductions in the volumeintegrated neutron emis sion whose secular decrease is associated with deuterium dilution from carbon accumulation. For these conditions, the neutron rate is dominated by beamplasma reactions so the neutron reduction indicates a degradation of the fastion confinement.
Persistent bursting modes at frequencies between 400-700 kHz were detected on magnetic probes in these dis charges. The modes have toroidal mode numbers of n = 7-9 and based on their mode numbers and frequency evolution The Mirnov signal at the TAE frequency reduces strongly after the coil current is ramped up and before the ELM crash. [25] are thought to be global Alfvén eigenmodes (GAE) [26]. Both GAEs and compressional Alfvén eigenmodes (CAE) [27] can be present in NSTX in the 0.5-2.0 MHz frequency range with very similar experimental characteristics but CAEs need a potential well [28] which is not present at the exper imental parameters of the observed mode and therefore, we identify those modes as GAEs. CAEs were observed in this discharge before 0.3 s around 2 MHz but ceased to exist after 0.3 s. In the absence of MP fields the GAEs exhibit a bursting and chirping behavior with a bursting rate of ∼4 kHz and a frequency sweep of ∼100 kHz (figure 4).
When the MP fields are applied the frequency sweep, rep etition period, and mode amplitude all drop by a factor of 2-3. Those effects occur in ∼0.1 ms which is the typical fastion poloidal transit timescale and is much shorter than the time scale for the formation of a fastion slowingdown popula tion which occurs on the collisional time scale of the energy slowing down time of about 20 ms. In contrast, when the MP current is turned off the bursting stays small for several mil liseconds which is typical for the NBI refueling timescale. In this case the mitigation effects are decoupled from the ELM behavior and are most readily observed during ELMfree phases of the discharge. Edge soft xray signals are barely perturbed by the MP pulses suggesting that the MP have a minor effect on the plasma edge density and temperature pro files leaving the beam deposition unaltered.
The phenomena shown in figures 3 and 4 are observed consistently in a set of 20 discharges of this type, with vari able MP duration (2-6 ms), repetition rate (20-40 Hz) and coil current (0.8-2.2 kA per turn). Pulses with MP coil currents below 1 kA per turn have little or no effect. Concurrent, large amplitude, lowfrequency MHD masks or weakens the miti gation effect.
In contrast to the TAE case, a clear reduction in the neutron rate is observed in response to applied 3D fields, in discharges with only highfrequency GAE and CAE activity. This sug gests that the fastion distribution in the discharges with TAE activity is already affected by the modes in such a way that any additional changes to the fastion population due to the MP fields are not detectable.

Simulations
In order to understand the effects of MP fields on the fast ions as presented in the previous section we have used stateofthe art simulations with experimental equilibrium profiles as input for NSTX discharge 138146 (figures 3 and 4). The results of these simulations can be compared directly with observations. We have chosen to study the changes in the fastion population in this NSTX discharge at 0.5 s because it is free of lowfre quency TAE activity which is known to be able to redistribute fast ions efficiently [29] and hence, mask the effects of the MP fields on the fast ions. The high frequency CAE and GAE activity that is visible on the Mirnov coil signals (figure 3) is not harmful for the fastion population because they are weak despite their visibility. Mirnov coils are sensitive to the time derivative of the magnetic fluctuations and therefore, the sig nals are linearly weighted to highfrequency activity.
The equilibrium reconstruction used as input for the resis tive one and twofluid M3DC 1 calculations [30][31][32] was computed with the EFIT code [33] and included the effects of kinetic profiles. The vacuum n = 3 MP fields were obtained from a Biot-Savart calculation for the currents in the MP coils in NSTX. The vacuum fields are rather small and localized at the lowfield side midplane as is shown in figure 5(a). The one and twofluid plasma responses, however, are strong as is shown in figures 5(b) and (c) and are due to the high value of β = 4.2 N . The MP field was imposed via the boundary condi tions whereby the normal component of the perturbed field at the computational domain boundary was set to be equal to the contribution from the nonaxisymmetric coils while the contribution from the plasma was ignored because the com putational domain boundary was put at such a location that the plasma response was small there. In the onefluid case the × E B rotation changes sign at two places near the plasma edge (figure 6(a)) where the internal kink response is strong. Similarly, in the twofluid simulations the electron diamagn etic frequency changes sign near ψ = 0.6 resulting in a strong kink response there as can be seen from the flux surface aver aged magnetic perturbation shown in figure 6(b).
The fastion distribution was created in the simulation by continuously injecting the three energy components of two neutral beams with full energies of 90.0 and 89.5 keV. These ions then slowdown and scatter on the bulk plasma when they thermalize with a typical energy slowingdown time of 25 ms forming a slowingdown distribution. In the simulations the slowingdown distribution was calculated with the SPIRAL code [34] from the fastion deposition profiles that included the full, half, and third energy components as calculated with the NUBEAM package in the TRANSP code [35,36] resulting in a typical ( ) distribution as shown in figure 7 whereby the parallel and perpendicular velocity components are taken at the particle location. In this representation of the fastion velocity space particles with constant energy form semicircles while particles with constant pitch are on straight lines going through the origin. The advantage of using the ( ) phase space representation over the more commonly used (energy,pitch) representation is that TAE and counter propagating GAE resonances for passing particles are straight lines at constant ∥ v . The slowingdown distribution was generated in the simulations by drawing particles from the birth profiles and injecting them at a constant rate of 8000 particles per ms and in a 25 ms time interval. Those particles were then followed until 25 ms or until they got lost to the wall or reached thermal velocities. The resulting ensemble of confined particles is then representative of the slowingdown distribution before the MP turns on. The effects of the MP fields on the fast ions was then studied by introducing the 3D perturbed magnetic fields from different models and continuing the simulations with the same constant fueling rate as before whereby the fastion dis tribution was recorded at regular time intervals (0.2 ms) for further analysis. In all SPIRAL calculations, the full particle orbit was calculated (i.e. including gyromotion) because the typical size of the perturbed fields is on the order or smaller than the fastion Larmor radius. Moreover, because of the low equilibrium magnetic fields in spherical tori, the criteria for applicability of guidingcenter models are not well satisfied, as was shown in [37].
Linear calculations were performed with the M3DC 1 code, and therefore, the vacuum and plasma response fields were known up to an arbitrary scaling factor. Moreover, the MP coils are located outside the vacuum vessel and it takes on the  order of 5 ms for the perturbing fields to penetrate fully into the vacuum vessel. This wall response time is on the same order of the duration of the applied pulses in the experiment (3-5 ms). In order to get a realistic time behavior of the per turbed fields inside the vessel and in the plasma we have nor malized the fields as obtained from M3DC1 to the signal of a magnetic probe that was located inside the vacuum vessel and far from the plasma so that it only registered the vacuum field response. The distortions of the invessel perturbed magn etic fields due to the vacuum vessel response time can be seen clearly in figure 8(a) where the probe response is com pared with the applied MP coil current. The plasma response is scaled with the same factor because the plasma response occurs on the Alfvénic time scale of about 10 μs. Simulations were performed for four cases: (1) with no MP fields, (2) with the vacuum response only, (3) with the onefluid plasma response and (4) twofluid plasma response.
When the MP fields are applied the fastion losses increase from the promptloss level of 8% of the injected power to transiently up to 21% at the end of the current pulse ( figure  8(c)). This transient occurs because the MP fields increase the loss cone in phase space thereby ejecting marginally confined particles. In this case, the coils were switched off before a new phasespace equilibrium with the MP fields included was established. After the coils were switched off the enhanced losses disappear on a time scale comparable to the decay rate of the perturbed fields.
The MP induced losses are small (up to 2%) for the vacuum fields only and increase significantly (up to 12 and 8%) when the one or twofluid plasma response is included. The MPinduced fastion losses (both prompt and resonant) occur mainly at the lowfield side midplane in a n = 3 pat tern which is quite different from the MPinduced losses on conventional tokamaks where the losses occur in the lower divertor region [38]. The low equilibrium fields in NSTX give rise to significantly outward shifted cogoing orbits and in combination with large Larmor radii, particles on those orbits are easily lost at the LFS midplane and don't reach the strongly perturbed fields near the Xpoints. In conventional   v v , phase space where particles with constant energy form semicircles and the angle is the particle pitch. Energy slowingsdown follows radii of constant pitch (red arrow) and pitch angle scattering occurs at semicircles with constant energy. The full, half, and third energy components are indicated. The thermal plasma is located at the pink semicircle near the origin. tokamaks the outward orbit shift and Larmor radii are much smaller and the fast ions can reach the perturbed fields near the Xpoint where they become ripple trapped and lost to the divertor.
Without the MP fields present the losses in phase space occur mainly at the full and half energy components as prompt losses as can be seen in figure 9(a). Once confined, the pitch angle scattering is small enough not to induce additional significant losses. When the MP fields are present, however, resonant losses occur for highenergy particles between the full and half energies while diffusive losses become dominant below the half energy component as can be seen in figure 9(b). Note also that without MP fields the thirdenergy component which is deposited close to the edge is well confined but when the MP fields are present a significant amount of thirdenergy particles are lost.
The resonant condition for fast ions with the applied static MP fields can easily be deduced graphically from figure 10(a) as: with ω b the particle bounce and ω p the particle preces sional frequency and k, a (rational) bounce harmonic. When inspecting the orbits of the resonant particles we find three resonant trapped particle orbits with k = 1, 1/2, and 2/3 and one passing particle orbit with k = 1/3 as shown in figure 10 and indicated in figure 9(b).
The changes in the neutron rate can be calculated from the simulated confined fastion distribution and the thermal plasma profiles as measured with the NSTX Thomson scat tring system [41,42] by using some elementary equations for the energydependent deuterondeuteron cross sections as given in [43]. In contrast to the fastion losses which peak at the maximum of the perturbed field amplitude, the neutron losses peak 5 ms after the perturbation was applied (figure 8(c)) while the neutron rate recovers on the slowingdown time scale which is in good agreement with the measure ments ( figure 11). The amplitude of the predicted neutron rate drop depends significantly on the perturbed fields that were used. When only the vacuum fields are included the peak neutron loss is 1% which is five times lower than the observed one while the one and twofluid plasma response fields give losses that are comparable to the observed ones as can be seen in figures 8(c) and 11. Ideal MHD computa tions of the plasma response using the IPEC code [19] that were reported in [18] found a larger MP amplitude in the core, which resulted in an overestimate of the simulated neu tron rate drop. Hence, the inclusion of resistivity effects in the M3DC 1 calculations via the Spitzer resistivity as was described in [39], provides a more accurate representation of the plasma response in this scenario. Therefore, we con clude that for simulating meaningful MPinduced loss rates the resistive plasma response has to be included for lown MP fields.

Effects of MP fields on AEs
It was found experimentally that applied MP pulses can affect the dynamic of fastion driven TAEs and GAEs (figures 2 and 4). The resonant drive for TAEs from passing particles is given by / with v A the Alfvén velocity [40]. For trapped particles the resonant drive emerges when the TAE frequency, ω TAE , equals ω ω + p j p b with ω p the precession frequency, p an integer precession harmonic, ω b the bounce frequency, and j an integer bounce harmonic. The drive for the highfrequency counterpropagating GAEs as were observed in the experiments comes from particles that satisfy the GAE esonant condition: that, together with the dispersion relation for Alfvén eigen modes, with ω c the ion cyclotron frequency, ω GAE the GAE frequency and l an integer bounce harmonic. The wave passingparticle  v v , phase space. In the following we will use semiqualitative arguments to show how the MP fields can influence the TAE and GAE activity via changes in the fastion phase space distribution to reduce the AE drive. For GAEs this change in drive is the major factor for the observed change in activity but for TAEs changes in the continuum damping rate at the plasma edge due to the MP fields can also contribute to the observed TAE suppression.
The bursting and chirping behavior of modes such as the observed GAE (figure 3) is often ascribed to the eruptive dynamics which is characteristic of marginally stable modes: modes for which the total growth rate only slightly exceeds the total damping rate [45]. Under such conditions one can expect that relatively small changes of the drive or damping, possibly caused by the interaction of the mode with the fastion distri bution, might alter significantly the nonlinear dynamics of the mode. Although those nonlinear physics mechanisms are studied extensively (see [3,46] and references therein), direct comparisons between theory and experiments are still sparse because accurate measurements of the fast ions in phase space are lacking. In the following we investigate whether the fastion transport can be changed sufficiently with externally applied 3D fields to alter the GAE drive significantly.
For the observed GAE the resonances include the l = 0 resonance which intersects the passing particle population marginally in the core and not in the edge as is shown in fig  ures 12(a) and (b) while the l = −1 resonance intersects both the core and edge passingparticle population. When the MP fields are applied the passing particle distribution changes in the core and at the edge as can be seen in figures 12(c) and (d) when the onefluid plasma response is taken into account and in figures 12(e) and (f) for the twofluid response. For both the one and two fluid response a depletion of the distribution is found in the core for ∥ < v 10 6 m s −1 while at the edge the passing particle distribution is enhanced.
The GAE instability depends on the perpendicularvelocity space gradient of the distribution function along the resonance as was shown in [26]. The phase space distributions along the l = −1 resonance (at = ⊥ v 2.08 10 6 m s −1 ) are shown in figures 13(a) and (b) for the core (r/a < 0.6) and the edge (r/a > 0.6) respectively where it can be seen that both the one and two fluid response modify the distribution. For the excita tion of the GAEs the perpendicular velocity gradient, / ∂ ∂ ⊥ F v , is the determining factor. Compared to the unperturbed gra dient both the gradients for the one and two fluid are reduced in the core and at the edge as is shown in figures 13(c) and (d), indicating that the MP fields can reduce the GAE fastion drive. Trapped particles can resonate with the GAE [26] when For a quantitative analysis of the GAE drive the spatial dis tribution of the eigenmode in the plasma is needed and can be obtained from reflectometer measurements as was shown in [44]. Unfortunately, these measurements were not available in this experiment because of the steep edge density gradients and the flat or hollow density profiles in the core as is shown in figure 14 and therefore, an accurate quantitative estimate of the change in GAE fastion drive is not possible.
The applied MP fields can not only influence the drive due to changes in the fastion distribution but they can also affect the damping of the AEs in two ways. The kinetic profiles such as density, electron, and ion temperature profiles can be modified by the applied magnetic perturbations, especially in cases where the vacuum MP is strongly amplified by the plasma response. This can lead to changes in the Landau damping on thermal electrons and ions. However, in the present experiments the profile changes due to the MP fields are small as can be seen in figure 14 so it is unlikely that MPinduced profile changes modify the balance between drive and damping terms in a significant way.
A more direct damping mechanism emerges from the toroidal symmetry breaking of the MP fields. This symmetry breaking introduces a source for toroidal coupling. In a non axisymmetric equilibrium the MHD equations cannot be solved for each toroidal Fourier component separately and the eigenmodes will emerge as a combination of multiple finite toroidal and poloidal components. These aspects have been extensively studied for stellarators where due to the highly nonaxisymmetric configurations perturbative methods are not viable [47].
In particular, the results in [47] imply that, when per turbing an axisymmetric equilibrium with MP fields of a low toroidal field period ( = n 2, 3 MP ), an eigenmode with toroidal number n 0 will generate a family of strongly coupled toroidal modes with mode numbers =± ± n n kn 0 M P with k = 0, 1, 2, .... Since this holds for the continuum solutions of the MHD equations as well, the application of external 3D fields is  expected to alter the frequency, amplitude and radial extent of the Alfvén continuum gap structure.
To study this effect, the Alfvén continuum has been com puted using the STELLGAP code [47] for a = n 3 MP per turbed equilibrium corresponding to the experimental one of NSTX pulse 133838 at t = 0.5 s just before the MP is applied when a strong n = 2 TAE is excited as can be seen in figure 2. The 3dimensional equilibrium was calculated with the nonlinear equilibrium code VMEC [48]. Two up down symmetric VMEC equilibria were calculated by only including cosinelike terms in the representation of B and R and only sinelike terms were included for Z. For the first equilibrium, only the n = 0 component of the fields was included in the computation, so as to provide an axisym metric reference configuration. For the second equilibrium the n = 3 MP fields were included.
The structure of the n = 2 continuum solutions without MP is shown in figure 15(a) where wide TAE (10-90 kHz) and EAE (100-150 kHz) gaps are found. When the = n 3 MP MP are included, coupling between the = ± ± n 2, 2 3, 2 6, toroidal harmonics is expected. In figure 15(b) the continuum structure is shown, when the n = 2 and n = 5 components are included. In this figure the blue and red dots indicate continuum solutions with domi nant toroidal component n = 2 or n = 5 respectively. The gap structure is strongly modified in a region close to the plasma edge in the Hmode pedestal region. It can be seen that the TAE gap is reduced and the EAE gap at the edge is closed due to 3D effects. This is due to the presence of a new class of solutions (figure 15(c)) that are characterized by finite n = 2 and n = 5 components, indicating a strong toroidal coupling in the pedestal region.
At present, the STELLGAP code does not include the effect of rotation which has been shown to be important for NSTX strongly rotating plasmas [49]. Hence, the results cannot be used for a direct quantitative comparison with the NSTX experiments. Nevertheless, it is interesting to notice that the upper bound of the TAE gap is reduced down to a frequency of 50 kHz, which suggest a possible interaction between the observed n = 2 global TAE at 60 kHz with a finite amplitude near the plasma boundary and the MPperturbed continuum. This suggests that the mode amplitude reduction of this global TAE might be attributed to a strongly increased continuum damping near the edge due to the MP fields. In [50] the con tinuum damping for TAEs is calculated in 3D magnetic field configurations at negligible β which might be extended to high β plasmas which are routinely created in NSTX.

Discussion and conclusion
Suppression of Alfvén eigenmode activity was observed when threedimensional magnetic perturbation fields were applied on NSTX. Complete suppression of TAEs due to short MP pulses was observed whereas GAE activity was significantly reduced. Two mechanisms for the reduction of AE activity were identified, involving respectively changes in the fastion drive and continuum damping. Extensive stateoftheart fullorbit fastion modeling in which the plasma response to the MP fields was included has shown that the fastion distribution is affected in such a way that the drive for the observed GAEs is reduced. Simulations of the experimental scenario found that the inclusion of the plasma response as calculated with the resistive M3DC 1 code was crucial to reproduce the measured drop in neu tron production rate associated with the applied magnetic perturbations. When only the vacuum fields of the MP were included, the simulated neutron drop was five times smaller than the observed one whereas using ideal MHD to calculate the plasma response resulted in a significant overestimate of the neutron rate drop [18]. From fullorbit modeling with the SPIRAL code it was found that the MP fields are able to reduce the fastion velocity gradients, which drive the GAEs, along the waveparticle resonances in phase space indicating that the GAE drive can be modified with the MP fields.
The reduction of the TAE amplitude was observed with only minor changes in the neutron rate, indicating a lesser role of the fastion drive. In this case the TAE suppression was attributed to changes in the magnetic topology at the plasma edge when the coils are energized. When the MP fields are present the toroidal symmetry is broken and hence, toroidal modes can couple. The STELLGAP code was used to com pute the TAE gap in a perturbed magnetic equilibrium and the results indicated that the toroidal mode coupling reduces the TAE gap near the edge, suggesting an increased continuum damping of the observed global TAEs. A more accurate com parison with the experiments, however, requires the inclu sion of toroidal rotation effects which were neglected in the STELLGAP simulations presented here.
The observations and the result of the modeling suggest that applied 3D perturbations can be used to effectively control the insurgence of Alfvén instabilities. An effective mitigation scheme has to reduce the mode amplitudes while preserving good fastion confinement. In the experiments presented here it was shown that mode amplitudes could be reduced but the fastion confinement was compromised.
Although fastion transport is very sensitive to 3D fields [51], fine tuning the MP spectrum and amplitude might be possible to obtain a favorable minimum in the fastion losses as the MP fields reduce the AE activity and their effects on the fastion losses while inducing fastion losses themselves. In the current experiments this minimum was not found as indicated by the drop in the neutron rate when the MP fields were applied. The natural amplification of MP fields provided by the plasma can be exploited to obtain radially localized internal MPs by applying small external fields with a single row of MP coils as was done in the current experiment or with multiple rows of MP coils [52,53]. In this respect, the pos sibility of adjusting the radial location of the plasma response by optimizing the MP spectrum and the plasma profiles, in particular the electron diamagnetic frequency, holds inter esting possibilities.
While significant theoretical and experimental advances have been seen recently, in the physics of the plasma response to resonant and nonresonant applied MP, dedicated experiments are needed for an accurate study the effect of the MP on AE stability in controlled and well diagnosed scenarios.