Study of chirping toroidicity-induced Alfvén eigenmodes in the National Spherical Torus Experiment

Chirping toroidicity-induced Alfvén eigenmodes (TAEs) are destabilized during neutral beam injection on the National Spherical Torus Experiment (NSTX (Ono M. et al 2000 Nucl. Fusion 40 557)) by super-Alfvénic ions with velocities up to five times larger than the Alfvén velocity. TAEs exhibit repeated bursts in amplitude and down-chirps in frequency. Larger bursts, so-called TAE avalanches, are eventually observed and correlate with a loss of fast ions up to 30% over ∼1 ms. Frequency, amplitude and radial structure of TAEs are characterized via magnetic pickup coils and a multi-channel reflectometer system. The modes have a broad radial structure, which appears to be unaffected by the large frequency and amplitude variations. However, the large mode amplitude does impact the modes' dynamics by favouring the coupling among different modes. In addition, the coupling involves kink-like modes and can therefore degrade the thermal plasma confinement. In spite of the non-linear regime characterizing the TAE dynamics, the measured properties are found to be in reasonable agreement with solutions from the ideal MHD code NOVA.


Introduction
The enhanced fast ion transport caused by multiple toroidicityinduced Alfvén eigenmodes (TAEs [1]) is believed to be one of the main loss mechanisms for fast ions in ITER [2]. Enhanced losses reduce the fusion efficiency and may cause harm to invessel structures. Understanding this phenomenon in order to limit, or possibly avoid, its deleterious effects in future reactors is therefore paramount for fast ion research [3]. One key aspect for predicting TAE behaviour and associated losses is the type of wave-particles interaction in velocity space. Single TAEs have already been observed to interact strongly with the fast ion population, resulting in the splitting of the original mode into multiple branches [4], or in periodic variations of its frequency (chirps) [5,6]. Of more concern is the overlap of multiple resonances in phase space [7], because broad regions of the fast ion distribution can provide enough free energy to sustain an explosive growth of the instabilities [8]. These rapid events are characterized by short time scales, 1 ms, for which no possibility of external control (e.g. through tailored injection of neutral beams or rf waves) presently exists. Therefore, they will be a particularly severe issue for future devices. In the following, these two cases are referred to as single-mode and multi-mode regimes.
Typical scenarios on the National Spherical Torus Experiment (NSTX) [9] are considerably different from those predicted for ITER and future reactors. Nonetheless, they provide a good test case to improve the present understanding of the basic physics of bursting TAEs. TAE bursts with duration ∼1 ms are commonly observed in NSTX neutral beam-heated plasmas and can cause up to 30% loss of the confined fast ion population in a single event [10,11]. NSTX plasmas can therefore be used to challenge present theories and numerical codes, with the overall goal of improving our predictive capabilities.
Comparisons between experiments and codes are usually done in terms of the modes' properties and dynamics for varying plasma conditions. A detailed study of TAE properties on NSTX, such as frequency, mode number and radial structure, and of their eventual modification during the bursting/chirping phase, is the main goal of this paper. In addition, previous results indicate that mode-mode coupling is at play during large TAE bursts [12], thus determining the possible transition from the single-to multi-mode regime. When coupling occurs, low-frequency ( 30 kHz) modes with toroidal mode number n = 1, 2 are often observed and can lead to a further enhancement of fast ion and thermal transport. The nature of these low-frequency fluctuations is briefly discussed in this paper, showing the potential for TAEs to interact with other classes of MHD modes.
The paper is organized as follows. The experimental scenario used for TAE studies on NSTX is described in section 2. General properties of individual TAE modes are then discussed in section 3. A more global picture of the TAE dynamics is then presented in section 4, where evidence of the coupling among TAE modes, and between TAE and low-frequency MHD modes, is also discussed. In section 5, a numerical analysis of the measured TAE properties (i.e. mode number, frequency and radial structure) is performed through the ideal MHD code NOVA [13]. Qualitative observations on the drive and damping mechanisms at play for TAE in NSTX are then discussed in section 6. Section 7 summarizes the main finding of this work and concludes the paper.

Experimental scenario, diagnostics and analysis techniques
NSTX operates with a toroidal field ∼0.5 T, with typical density (3-10) × 10 19 m −3 , temperature T e ≈ T i 1.5 keV and central plasma rotation f rot 40 kHz. Neutral beam (NB) injection is the main heating system. The maximum total power is P NB = 7 MW from three sources with adjustable injection energy E inj = 60-95 keV. The resulting fast ion population is super-Alfvénic with velocities 1 < v fast /v Alfven < 5. Fast ions provide the drive for a variety of Alfvénic instabilities, including TAEs [14]. The latter have toroidal mode number up to n = 8 and frequency 60 < f < 250 kHz in the laboratory frame. An example of the L-mode plasma scenario used for TAE studies is shown in figure 1. Deuterium plasmas with 1-2.5 MW of injected NB power are mainly used. The level of injected NB power, along with the low density of these L-mode plasmas (n e ∼ (3-4)×10 19 m −3 ), leads to a high ratio between fast ion and thermal plasma pressures, β fi /β th (β is the ratio between kinetic and magnetic energies), as inferred from the TRANSP code 5 ( figure 1(d)). The reversed-shear safety factor profile, q(R), evolves in time. Its minimum, q min , decreases from 4 to ≈1 during the discharge ( figure 1(c)). The safety factor is reconstructed through the Grad-Shafranov equilibrium code LRDFIT [15], constrained by the motional Stark effect (MSE [16]) measurements of the Time-resolved profiles of electron/ion density and temperature are obtained through Thomson scattering [17] and charge-exchange recombination spectroscopy [18] diagnostics that cover the full plasma minor radius. Time resolution is 16 ms and 10 ms, respectively. Magnetic fluctuations are measured by an array of 11 toroidally distributed Mirnov coils, located at the vessel wall on the low-field side. Sampling frequency is 4 MHz. When a time resolution 1 ms is sufficient, Mirnov coils data are analysed with standard techniques based on the fast Fourier transform (FFT). This provides time-dependent spectra of the modes' frequency and toroidal mode number, n, inferred from the phase variation as a function of the toroidal angle.
Higher-order Fourier analysis techniques, such as bicoherence analysis [19], are used to investigate mode-mode interactions. The bicoherence, b(f 1 , f 2 |f 3 ), quantifies the degree of correlation between triplets of modes at frequency f 1 , f 2 and f 3 via a phase-weighted average of the complex Fourier bispectrum. Frequencies of each triplet are such that f 3 = f 1 ± f 2 . Because of the normalization to the spectral amplitude, the analysis is independent of the actual mode amplitude and can reveal coupling among very weak modes, not directly observed in the Fourier spectrum. By summing b(f 1 , f 2 |f 3 ) over all pairs (f 1 , f 2 ) that contribute to a same f 3 , the total bispectrum is obtained. This quantity is an estimate of how much spectral power at a given frequency is associated with bi-linear interactions between modes.
Because of its limitation in temporal resolution due to the lack of statistics for events of short duration, FFT analysis has limited use in the study of transient events lasting hundreds of microseconds or less. Instead, based on the naturally occurring frequency separation between modes (see section 3.1), an analysis based on band-pass filtering of the time series from Mirnov coils is used to resolve fast events. Signals are band-pass filtered around the frequency of each mode to obtain the corresponding fluctuations,ṡ n (t), and the B-field fluctuation signals, s n (t), through software integration ofṡ n (t). The evolution of frequency and amplitude, f n and A n , is then calculated from the peak-to-peak amplitude and peak separation of the resulting (sinusoidal) s n (t).
This analysis can be further developed to study modemode coupling over short time scales. In essence, the signal s n (t) is converted into its analytical (or complex) equivalent h n (t) = s n (t) + iH {s n (t)}, where i is the imaginary unit and H {·} denotes Hilbert transform. Then, sub-samples of h n (t) containing only a few fluctuation cycles for all potential frequencies (or n's) of interest are used to construct a pseudospectrum with amplitude |ĥ n | and phase tan −1 ( {ĥ n }/ {ĥ n }) ( {·} and {·} are the imaginary and real parts). The pseudospectrum is finally used to compute the bicoherence, with the advantage that short time series provide sufficient statistics to obtain reliable results over 100-200 µs intervals.
In addition to the information on fluctuations at the plasma edge from Mirnov coils, internal measurements of the radial structure of Alfvénic modes and of its temporal evolution are available from a multi-channel reflectometer system [20]. It measures path length variations of a probe microwave signal resulting from plasma density fluctuations. The system has 16 channels, tuned at different frequencies.
Depending on the density profile, hence on the location of the cut-off radius for each specific frequency (channel), density fluctuations at up to 16 radial points can be measured. For the L-mode scenario investigated here, there are typically 9-11 radial points available. The analysis of reflectometer data is performed assuming small density fluctuations around an unperturbed density profile and linear propagation of the ordinary-mode polarized probe waves based on a onedimensional model. Reflectometer's data are inverted using Gaussian basis functions centred around each measurement location to model density fluctuations. Each basis function is scaled until the simulated and measured path length variations match at all locations. Because density profile and its radial gradient are available every 16 ms, their possible variation on faster time scales is not accounted for in the analysis. This is the main source of uncertainties for the calculated density fluctuations, which can be as high as 20% of the relative fluctuations for events such as avalanches.

Frequency and amplitude evolution
So far, most of the TAE studies on NSTX have been conducted in NB-heated, L-mode plasmas. These discharges are characterized by a high ratio β fi /β th (figure 1(d)), which provides enough drive to destabilize TAEs. A spectrum of magnetic fluctuations, measured by Mirnov coils for the discharge in figure 1, is shown in figure 2(a). Bursts of TAEs correlate with frequency variations <10 kHz for t < 480 ms. Bursts repeat with a period of 0.5-2 ms, whose duration is rather independent of thermal plasma parameters or injected NB power. Eventually, larger bursts with frequency downchirp >10 kHz lead to a so-called TAE avalanche [11]. Avalanches cause significant fast ion losses [10], as inferred from drops in the neutron rate, cf figure 2 The spectrum of toroidal mode numbers, n, is shown in figure 3. The dominant modes for this discharge have n = 3, 4. Higher n's are observed at increasing frequencies, with very weak n = 6, 7 modes detected up to f ∼ 200 kHz.
Taking the frequency variation as indicative of the severity of a burst, the relationship between measured frequency chirp and β fi /β th is shown in figure 4 for a set of discharges with different density and injected NB power. Only cases without large, low-frequency MHD activity associated with kink-like modes, causing transient and non-reproducible drops in β fi and β th , are included. The fractional frequency chirp is calculated with respect to the mode frequency in the plasma frame. Large frequency chirps and amplitude excursions are mostly observed for larger values of β fi /β th . However, there is still a considerable fraction of bursts with relatively mild amplitude and frequency excursions at high β fi /β th . This suggests that the fast ion drive, which is roughly proportional to β fi , is only one of the parameters regulating the TAE dynamics, as expected for a semi-chaotic regime in which multiple drive and damping terms compete [8,21]. It should be noted that, because NB injection provides central fuelling and heating in addition to being the fast ion source, it is in practice difficult to separate unambiguously the effects of β fi and of β th in these discharges. This can be appreciated from figure 4(d), showing a roughly linear relationship between β fi and β th .
It should also be noted that, in addition to TAEs, modes with n = 1 and f ∼ 25 kHz (that is, well below the TAE gap) are often detected during strong TAE bursts. These fluctuations have a duration comparable to, or larger than, that of the avalanches, indicating that they may also play a role in the fast ion loss process [12]. More details on the nature and role of these low-frequency fluctuations are given in section 4.
Further information on the properties of chirping TAEs can be gathered from a detailed analysis of the amplitude and frequency behaviour during the bursts for single modes. An example is illustrated in figure 5(a) for a n = 3 mode. A first conclusion is that frequency and amplitude evolutions are indeed correlated, with the peak amplitude of each burst measured at about half the frequency sweep duration. Then, amplitude decays toward the noise level while the frequency continues to decrease. Eventually, a new burst begins with a rapid amplitude growth. At this point, the frequency jumps up by several kHz's and the cycle repeats. Different models have been proposed to explain the repeated bursts and chirps of the modes. Arguably, the most successful in reproducing the features observed experimentally are those based on generalizations of the bump-on-tail problem (cf [8,21,22] and references therein), which consider the formation of holeclump pairs in the fast ion velocity space. The response of the modes to this perturbation results in a characteristic evolution of the frequency, namely f ∝ ± √ t for both up-and down-chirping modes. Other models, originating from the so-called tip model [23], extend the hole-clump model to the case of modes that interact strongly with the TAE continuum during their frequency down-chirp [24]. In this case, the mode can undergo an asymptotic explosive growth with a time dependence A comparison of these two classes of models with the experiments shows a qualitative agreement, see the example in figures 5(b)-(c). For instance, the mode's growth is compatible with a t −1 dependence, at least in the time window following the initial, slow amplitude increase and before saturation inhibits a further growth (consistently with the hypotheses in [24]). However, other time dependencies, such as a simple exponential, provide an equally good (if not better) fit to the data. Similarly, the √ t dependence resulting from the hole-clump models is consistent with the observed frequency evolution, but other functional forms are, in fact, equally close to the data. A second conclusion on the amplitude and frequency evolution is therefore that the models mentioned above are indeed compatible with the experiments, but they may not be uniquely identifiable, in the sense that other time dependencies could actually provide an equally good description of the data. For instance, the inclusion of additional physics can modify the temporal behaviour of TAEs resulting from the models. Experimentally, very limited or no information is presently available to investigate the fast ion profile variations on short ( 1 ms) time scales and with sufficiently good spatial resolution. Different methods can, however, be utilized to study mode-mode coupling and the resulting multi-mode dynamics, see section 4.

Mode structure
A third important property of TAEs is their radial structure. For example, details on the mode structure can be used to distinguish between different classes of modes, such as reversedshear Alfvén eigenmodes (RSAEs) and TAEs [25], and their transition from ideal MHD modes into energetic particle modes (EPMs). In addition, the radial extent of the modes partly determines the potential of the modes to cause fast ion losses from the confinement region, in contrast to redistribution within the main plasma in either real or velocity space.
An example of density fluctuations, δn/n, associated with a n = 3 mode at f ≈ 89 kHz (cf figure 2(a)) is shown in figure 6. It demonstrates the potential of the reflectometer system for δn/n results with high temporal resolution, as required for transient events such as TAE bursts and avalanches. The general features observed in figure 6 are common to low-n modes observed on NSTX. The modes have a broad radial structure, extending over a good fraction of the minor radius. The peak δn/n is measured outside mid-radius, R 120 cm (the magnetic axis being at R ≈ 100 cm). Noticeably, a non-vanishing δn/n is found even for the outermost channels, which are just a few centimetres inside the last closed flux surface, indicating a finite mode amplitude in the proximity of the plasma boundary. The observed TAEs are therefore global modes, which are capable of interacting with the fast ion population over most of the plasma volume.
As a further step, the possible distortion of the mode structure during an avalanche is investigated (figure 7). Data refer to the times corresponding to the maximum mode activity of the three bursts that actually form the avalanche, as seen from the high time resolution spectrogram in figure 7(a). The    structure does not vary significantly within the experimental uncertainties. Consistently with the model discussed in [26], the broad radial structure of the modes relative to the scalelengths of the fast ion density profile is believed to inhibit here a macroscopic radial propagation of the mode.

Multi-mode TAE dynamics and interaction with low-frequency MHD modes
The TAE properties illustrated in the previous section refer to single, individual chirping modes. In this section, the interaction between modes and the overall multi-mode dynamics are considered. For example, a common feature of TAEs in NSTX L-mode plasmas, which is apparent in figure 3 for n = 2, . . . , 5, is a roughly constant frequency separation between modes with consecutive n's [27]. This suggests that the modes share a common frequency in the plasma frame, f TAE 0 , such that f TAE lab,n = f TAE 0 + n f TAE Doppler [28]. Here f TAE lab,n is the frequency for the toroidal mode number n in the laboratory frame and f TAE Doppler the Doppler shift caused by plasma rotation. Knowing f TAE Doppler and the plasma rotation profile f rot (R, t), measured through chargeexchange recombination spectroscopy (CHERS) of carbon rotation [18], the radius R TAE such that f TAE Doppler = f rot (R TAE ) can be calculated. In the following, R TAE is referred to as the characteristic radius of the cluster of TAEs. For modes whose radial extension is much smaller than the minor radius, R TAE represents the actual radial position of the cluster of modes and is expected to coincide with the radius of maximum mode amplitude [28]. However, this intuitive interpretation of R TAE becomes less clear for more global modes which extend over a considerable fraction of the minor radius, such as the TAEs commonly observed on NSTX. In this case, R TAE appears to be more related to the location of strongest drive of the modes, rather than to the exact radius of peak amplitude of each TAE (cf figures 6 and 9 in [27]). In general, one can interpret R TAE as the characteristic radius where the local parameters (e.g. of the thermal plasma, fast ion population and/or of other instabilities) have the strongest influence on the evolution of the cluster of TAEs. The importance of R TAE to understand the coupled dynamics of multiple TAEs is further discussed in section 6.
The accumulation of modes around the same radial location is one of the conditions potentially leading to coupling between modes. The total bispectrum for the shot in figure 2 is shown in figure 8, with details of the bicoherence calculated before and during a TAE avalanche (figures 8(a) and (b)). The bicoherence is small for small amplitude oscillations, indicating a weak or null coupling between modes which, nevertheless, still exhibit a bursting/chirping character. Conversely, values 40% are obtained during the burst for frequencies corresponding to the TAEs and to other modes at f < 60 kHz and f > 160 kHz ( figure 8(b)), indicating strong coupling at this time. The increase in the overall modemode coupling with time is observed in the total bispectrum (figure 8(c)), which shows enhanced values at all frequencies after t ≈ 460 ms that culminate in the TAE avalanche.
A detail of the bicoherence evolution during short events is shown in figure 9. The analysis shows a finite bicoherence during some of the bursts and vanishing values for other times, in spite of the presence of multiple modes. This suggests that the mode-mode coupling strength varies on submillisecond time scales, thus confirming previous observations from NSTX [12]. Another result that was anticipated in [12] is the clear evidence of the coupling between TAE modes and low-frequency n = 1, 2 fluctuations. From the comparison with NOVA results, these modes are identified as kink-like modes (see the next section and [29] for a more exhaustive analysis).
For an efficient coupling, three-wave matching conditions, including phase coherence among the three waves, must be maintained for a sufficiently long time, of the order of tens of wave cycles of the primary (pump) modes, as observed in the experiments during the larger bursts. The particular relationship between frequencies (and, therefore, their clustering around a characteristic radius) of TAEs with subsequent n's, along with the presence of a seed n = 1 mode and a sufficiently strong drive, are thought to be other important ingredients for the development of bursting/chirping TAEs into avalanches. This hypothesis, supported by the results shown in figure 8, is tested in figures 10 and 11 for discharges where no TAE avalanches are observed in spite of the large number of chirping modes. In the first discharge (figure 10), strong low-frequency MHD activity is present but differs from what was discussed before. The dominant low-frequency mode has n = 2. Based on the flattening of thermal plasma profile around the q = 2 rational surface, it is tentatively identified as a large m/n = 3/2 island rotating at ∼20 kHz. Bicoherence and total bispectrum indicate that TAEs are interacting with these low-frequency modes, but no significant coupling among TAEs is measured, in contrast with the results shown in figure 8. This suggests that the criteria for an efficient mode-mode coupling, eventually leading to an avalanche, are not satisfied in this case. For instance, TAE modes may be localized around different radii and have mode structures that overlap only in part. The second discharge ( figure 11) presents all the signatures that concur to originate an avalanche. Although relatively large frequency sweeps are observed, however, no large amplitude bursts and sudden drops in the neutron rate are measured. In this case, the missing ingredient is believed to be a strong enough drive of the modes, which keeps the TAEs close to marginal stability (cf section 6 and figure 17).
As a concluding remark for this section, we note that a recent theory on coupling between Alfvénic modes due    to perturbations of the axial symmetry [30] is qualitatively consistent with the coupling processes described above. A more quantitative study is left as future work.

Comparison with linear analysis from NOVA
In this section, the agreement between the measured mode properties and linear MHD theory is explored via the NOVA Figure 13. NOVA calculation of (a) TAE continuum and (b) radial displacement, ξ , for the two dominant poloidal harmonics (characterized by their poloidal mode number, m) for a n = 3 mode at two different times. The calculated mode shows a RSAE character early in the discharge and a transition into a TAE as β fi increases. Only the harmonic structure inside the TAE gap is shown. Vertical dashed lines in (a) indicate where the modes intersect the lower TAE continuum. code ( [13] and references therein). First, mode frequency and radial structure for the observed toroidal mode numbers are examined. Then, the analysis is extended in section 6 to a qualitative study of the damping terms from linear theory and fast ion drive from experimental observations. NOVA analysis is based on the experimental plasma profiles at a fixed time. Toroidal rotation is included as a Doppler shift contribution to the mode frequency. The calculated n = 3 TAE continuum for the discharge in figure 2 at t = 470 ms, i.e. just before the TAE avalanche occurs, is illustrated in figure 12. The eigenmodes found by NOVA are shown in the same figure for the frequency range 60-140 kHz, with the measured mode frequency for n = 3 being 90-100 kHz. The code finds a large number of modes, which can be classified as TAE/RSAE and continuum modes. Unlike TAE/RSAE modes, the latter modes are characterized by a strong interaction with the TAE continuum (resulting in a discontinuity in the mode structure in ideal MHD theory). In practice, they are expected to be stable in the experiments and are not further discussed in the following.
From the many (ideal) eigenmodes found by NOVA, the best fit to the experimental mode is selected based on the agreement with the measured mode number, frequency and δn/n profile, as detailed in [11]. The procedure can be repeated at different times to model the evolution of a single mode as the plasma parameters evolve. An example is shown in figure 13, which compares the dominant poloidal harmonics of a n = 3 mode at two different times. In both cases the mode intersects  the continuum around mid-radius. The total mode structure is similar for the two times. By looking at the poloidal harmonic composition, one can see that the mode has a dominant RSAE character early in time, with a single dominant m, then evolves into a TAE, characterized by two harmonics with comparable amplitude. This evolution is quite common on NSTX, and is attributed to a suppression of the RSAE modes as the plasma β th increases [25], thus favouring the appearance of TAE modes instead.
By knowing the experimental profiles and the mode structure from NOVA, the corresponding (normalized) density perturbations can be calculated and rescaled to the δn/n measured by the reflectometer. The comparison between measured and modelled density fluctuations is reported in figure 14. Considering that the ideal model does not include the self-consistent kinetic effects from fast ions, the modelled δn/n is in fair agreement with the measurements for modes with n = 2, . . . , 5. In turn, poor agreement is found for the low-frequency n = 1 mode.
Based on the upgraded reflectometer system, the density fluctuation resulting from the n = 1 mode has been measured and compared with NOVA calculations for ideal kink modes. No solutions are found for the standard, fixed-boundary version of the code, which assumes vanishing mode amplitude at the plasma edge (separatrix). When the fixed-boundary constraint is relaxed, a kink mode is found, see figure 15. The mode causes large edge perturbations, which is qualitatively consistent with the experimental observations ( figure 16). This supports the interpretation of the n = 1 fluctuation as a kinklike mode. Additional analysis discussed in [29] strengthen this interpretation. Further studies will include fast ion effects in the simulation to investigate whether the mode is an ideal kink or a energetic-particle driven fishbone mode.

Role of drive and damping
As discussed in the previous sections, the observed TAEs are in a saturated, possibly non-linear regime characterized by a large variety of mutual interactions among TAEs themselves and between TAEs and other types of modes. The main factor determining this scenario is the drive provided by fast ions from NB injection. A first example of dependence of mode dynamics on fast ion population is given in figure 17, where the fast ion profile is perturbed using different heating schemes, including NB only and NB + rf injection. The reference discharge has NB injection only ( figure 17(a)). The companion discharges have both NB+rf injection, with a notch of 20 ms in the NB power ( figure 17(b)) and in the rf power ( figure 17(c)). As the figures show, NB injection alone is insufficient to drive TAEs unstable in this case. When rf is added, the fast ion population slightly increases, as observed from the raise in neutron rate ( figure 17(d)), and the modes are destabilized. It is plausible to assume that TAEs remain close to marginal stability, as suggested by the fact that their amplitude drops to noise level as soon as either NB or rf injection is interrupted. A noticeable result is the prompt onset of the bursting/chirping regime, indicating how easily the modes enter in this phase, at least for NSTX plasmas, once they are driven unstable.
A link between mode dynamics and fast ion population can be established experimentally by measuring the fast ion profile and its temporal evolution, e.g. through fast ion D-alpha (FIDA) spectroscopy [31,32]. The correlation between TAEs and their drive is discussed in figure 18(a), where the calculated R TAE (cf section 3.1) is compared with the position of steepest fast ion gradient. The two locations are similar, indicating that the shape of the fast ion profile plays a primary role in determining the localization (and, possibly, the dynamics) of TAEs. These observations may be used to design tools to affect the fast ion, hence the TAE, dynamics, for instance by varying the NB deposition profile and/or by means of localized rf deposition. (The maximum toroidal rotation shear is also located at the same radius, although it has been shown that rotation shear does not affect the macroscopic mode dynamics, cf [27]). The dependence upon the location of steepest fast ion gradient is believed to be responsible for the clustering of multiple modes around the same radius. The strong NB source term can efficiently sustain a steep, localized fast ion gradient, thus paving the way for mutual interactions between strongly unstable modes [12].
A departure of the characteristic radius of the TAE cluster, R TAE , from the steepest fast ion gradient region is observed during the avalanche phase around t = 480 ms, (figures 18(a) and 19). At this time, the calculated R TAE shifts towards the magnetic axis and f TAE 0 (R TAE ) → 0, with f TAE lab,n ≈ n f TAE Doppler (R TAE ), see figure 19(a). The comparison of measured f TAE lab,n against the values inferred from the fit f TAE 0 + n f TAE Doppler shows that the simple, linear relationship for the mode frequencies can reproduce satisfactorily the measured f TAE lab,n . Note that f TAE 0 → 0 is satisfied at R = R TAE because of a change in R TAE , which does not necessarily imply a large change of the actual TAE frequency or radial mode structure, consistently with the results shown in figure 7.
Because toroidal rotation can undergo rapid, transient changes during large mode chirps due to the non-ambipolar loss of fast ions [33,34], the sensitivity of these results to the fast evolution of f rot is studied. By comparing figures 19(a) and (b), frequency changes do not appear to be directly synchronized with the neutron rate drop, hence the sudden reduction in rotation [33], as it would be expected if the observed change in f TAE lab,n is caused by a decrease in f rot (or, equivalently, in f TAE Doppler ) only. The inferred R TAE is also quite robust against uncertainties in the exact evolution of f rot (R, t) (figures 19(c) and (d)), which is not directly measurable because of the long integration time (10 ms) of the CHERS system.
Based on the available data, the proposed interpretation for the evolution of f TAE 0 and f TAE Doppler is that the dependence of R TAE on the fast ion profile is partly lost during the avalanche. The cluster of TAEs locks instead on a perturbation which starts at the magnetic axis and then move outward (i.e. R TAE shifts from the axis towards the edge, i.e. to regions with lower rotation) in the ∼10 ms following the TAE burst. This scenario, suggestive of strong coupling between TAEs and lowfrequency MHD, is indeed supported by the analysis of modemode coupling discussed in section 4. Further improvements (or corrections) to this interpretation would require data that are not presently available for NSTX plasmas. For instance, a fast CHERS system for measurements of f rot on sub-millisecond time scales has recently been developed [35] and will be used to complement future investigations of the TAE dynamics during avalanches.
Along with the radial gradient, details of the fast ion distribution in velocity space, F (E, p) (here E is the energy and p the pitch), also contribute to the TAE drive. However, a full experimental characterization of F (E, p) is still missing and modelling is required. For MHD-quiescent plasmas, where fast ions are expected to behave classically, modelling can be accomplished with numerical tools such as the TRANSP/NUBEAM module [36]. An example is reported in figure 20, from which the complex nature of F (E, p) for a realistic case with NB injection can be appreciated. Problems arise when fast ions and waves are expected to mutually influence each other and the modelling is required to be selfconsistent, as in the case of strongly resonant TAEs. In this case, F (E, p) will experience some distortion in either real or velocity space (or both). Neither the NUBEAM module mentioned above nor NOVA can model this interaction selfconsistently. An anomalous fast ion diffusion coefficient can be introduced in TRANSP, but it rather represents a tool to mimic ad hoc losses in the simulation than a truly physicsdriven model. For instance, rapid loss events may be collective rather than diffusive. Along with improvements in the fast ion  diagnostics utilized in today's experiments, more sophisticated theories and codes are required. The comparison between experiments, quasi-linear modelling of the wave-particles interaction [37,38] and/or non-linear and self-consistent codes (e.g. M3D-K [39,40]) is beyond the scope of this paper, thus leaving the quantification of the F (E, p) contribution to the TAE drive as an open issue.
The second, important element in determining the TAE stability is damping. Linear stability is calculated through NOVA (figure 21). Dominant terms in the total damping rate are continuum damping and ion Landau damping. Other terms, such as electron Landau damping, are negligible. Radiative damping is not considered here, because the model used in NOVA is inaccurate for high-β, low aspect ratio NSTX plasmas [41] and results may be unreliable.
An interesting result from figure 21 is that ion Landau damping and continuum damping are comparable in magnitude, but show different trends with frequency and time. Ion Landau damping is larger at higher frequency and increases in time because of the slight increase in ion temperature and density. In contrast, continuum damping is larger at lower frequency and decreases in time as the peak mode amplitude shifts to larger radii, away from the intersection with the continuum ( figure 12). The opposite dependence of the two terms results in a minimum in frequency of the linear damping, which is in qualitative agreement with the observed mode frequency. Furthermore, the decreased damping at lower frequencies for increasing time may favour the down-chirp of the modes observed during the avalanche at t ≈ 480 ms. More analysis is required to extend this analysis to the other modes observed in the experiment and verify the generality of these observations.

Summary and conclusions
This work has investigated two main aspects of chirping TAE modes on NSTX, namely (i) the mode's properties (namely frequency, amplitude, radial structure and the qualitative dependence on drive and damping terms) and (ii) the nonlinear coupling between modes leading to the destabilization of low-frequency, kink-like activity.
Chirping parameters are rather constant among different scenarios, characterized by different schemes of additional heating (NB, rf) and confinement mode (L-versus H-mode). Bursting rates are ≈0.5-2 ms −1 . No systematic trend with thermal plasma and fast ion parameters is observed, except that larger bursts/chirps tend to be detected for increasing ratio of fast ion to thermal plasma pressure. The comparison with solutions from the NOVA code shows that, although the mode dynamics are complex, the mode structure is still reasonably well described by linear eigenmodes.
By studying the coupling between pairs of TAEs, one observes that the multi-mode TAE regime is prone to coupling with lower frequency modes, such as kinks or fishbones, which mediates the three-wave coupling between primary TAEs [12,42]. The destabilization of otherwise stable kinks causes fast ion and thermal plasma losses that add to those caused by TAEs alone. The coupling between TAEs is favoured by their clustering around a similar radius. This is plausibly caused by the steep fast ion density gradient sustained at that location by NB injection. Similar conditions could be expected in fusion devices such as ITER. As a result of such non-linear TAE dynamics, an enhancement of fast ion loss or redistribution (in both velocity and real space) is expected. The radial mode extension can be used for a rough guess of whether fast ions are actually expelled or not from the core plasma. Significant losses are likely for global, radially extended TAEs with low n's, whereas more localized TAEs with high n's are more likely to cause redistribution. The latter case is predicted for the high toroidal field, large major radius plasmas of ITER [43]. However, even if no direct loss of fast ions is expected, redistribution itself can be a deleterious process in a reactor. First, modifications of the fast ion distribution may result in unwanted variations of the non-inductive current profile [44]. Second, a relaxed fast ion pressure can impact the stability of other modes (e.g. kinks [45], sawteeth [46] and resistive wall modes [47]), thus affecting the overall plasma stability. Moreover, radial redistribution can enhance ripple-induced fast ion loss.
Future work will extend the comparison of measured mode properties and dynamics to predictions from the nonlinear, self-consistent (M3D-K [39,40]) code. For instance, preliminary simulations for NSTX-like plasmas were able to reproduce profile relaxation and significant multi-mode interactions, including the destabilization of otherwise stable, low-n modes [48].