Excitation of toroidally localized harmonics of global Alfvén eigenmodes

The spherical tokamak NSTX and the upgraded, higher toroidal field, NSTX-U, routinely create plasmas with large un-thermalized populations of super-Alfvénic fast ions. This population of non-thermal energetic ions excites a broad spectrum of Alfvénic waves, from the lower frequency (≈30 kHz–≈200 kHz) toroidal Alfvén eigenmodes (Cheng and Chance 1986 Phys. Fluids 29 3695), to the higher frequency (400 kHz–3 MHz) global Alfvén eigenmodes (GAEs) and compressional Alfvén eigenmodes (Goedbloed 1975 Phys. Fluids 18 1258). In this paper we present evidence that the GAE non-linearly excite modes, presumably GAE, at frequencies consistent with non-linear or 3-wave coupling. The observation of the excitation of 2nd harmonic GAE through the intrinsic non-linearity of Global Alfvén modes demonstrates that the non-linear terms can act as an exciter-antenna inside the plasma, broadcasting at harmonics of the mode frequency and with concomitantly shorter wavelengths. As with experiments using external antenna to excite otherwise weakly stable Alfvénic modes, this data can provide information about the stability of modes at harmonics of the GAE. It may also provide information on the nature of the non-linearities in the wave dispersion equation or a direct measure of the mode amplitude (Smith et al 2006 Phys. Plasmas 13 042504). We also report that the short wavelength 2nd harmonic GAE can be strongly toroidally localized. The observation that the shorter wavelength (n ≈ 20) 2nd harmonic modes can be toroidally localized potentially has implications for the impact of fast-ion driven instabilities on fast-ion confinement in ITER and future fusion reactors (Gorelenkov et al 2014 Nucl. Fusion 54 125001).


Introduction
Alfvénic instabilities with frequencies above the toroidal Alfvén eigenmode (TAE) frequency [1] and up to the ion cyclotron frequency range are commonly observed in NSTX and NSTX-U plasmas. NSTX and NSTX-U plasmas are Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. heated with neutral beams at energies up to 100 keV, and the resulting non-thermal fast-ion population is capable of resonantly exciting a broad spectrum of Alfvénic waves. Specifically, global Alfvén eigenmodes (GAEs) are seen in most beam heated plasmas in NSTX [2]. In NSTX-U neutral beam sources were added which suppressed the GAE [1], but GAE may still be seen in the absence of those new sources. The stability and scaling of the mode numbers and frequencies of unstable GAE has been extensively studied and modeled [3][4][5][6][7][8]. In this paper we study the much weaker fluctuations seen at twice the GAE frequencies (cf figure 1). While we will refer to these higher frequency fluctuations as '2 nd harmonic GAE', it is not necessarily the case that they are actually modes in that they may just be the non-linear components of the fundamental GAE.
The simplest explanation for the fluctuations at the 2 nd harmonic are that they are a non-linear component of the Global Alfvén waves. A single, non-linear GAE would generate, through non-linearities in the wave equations, a second order perturbation at twice the frequency and half the wavelength [9]. Many observations of harmonics of strong Alfvénic waves have been reported [10][11][12][13][14][15][16]. Without damping or drive terms, or other resonances, the amplitude of the 2 nd harmonic perturbations should scale roughly as the square of the fundamental mode amplitude for a simple quadratic non-linearity, as would appear to be in keeping with the next order terms to the linearized two-fluid equations for Alfvén waves. While the 2 nd harmonic fluctuations are weak, in some cases the signalto-noise ratio is high enough to examine some of these predictions and many examples show behavior that is not completely consistent with this model. Particularly in the lower field, NSTX, plasmas, analysis of the bursts suggest that the 2 nd harmonic GAE are indeed weakly damped GAE excited through non-linear or three-wave coupling with the fundamental GAE. This is likely due in part because the GAE in low field plasmas exhibit strong frequency chirping, allowing the non-linear drive to sweep through resonances in the 2 nd harmonic frequency range. In this paper we present the experimental evidence which demonstrates that not all of the 2 nd harmonic fluctuation behavior can be explained as simple non-linear components of the GAE. A probable explanation for this could be that the effects described below are the result of the non-linear terms exciting weakly stable modes in the 2 nd harmonic frequency band in the manner of experiments with actively driven external antenna [10,17].
The experimental data from NSTX-U is from higher field plasmas and with an extended toroidal array of fast magnetic sensors (NSTX-U), but has limited equilibrium diagnostic coverage. In the higher field plasmas (≈6 kG) the fundamental GAE bursts typically do not chirp and in these plasmas the 2 nd harmonic GAE are found to be mostly consistent with their interpretation as the simple non-linear component of the fundamental GAE. However, the extended toroidal array on NSTX-U has found that the 2 nd harmonic GAE in the higher field cases are strongly localized toroidally with a toroidal amplitude variation of at least a factor of five. This strong toroidal localization in turn suggests that weakly damped, toroidally localized, modes are being excited in the 2 nd harmonic GAE frequency band. The equilibrium parameter responsible for localizing the modes has not been identified, whether it is due to error fields, beam injection asymmetries, toroidal asymmetries in the plasma-facing components or other factors. The strong toroidal localization would likely affect stability and the transport of fast ions from these modes. The only diagnostics which have detected the 2 nd harmonic GAE in NSTX(-U) are the high bandwidth Mirnov coils on the outer vacuum vessel wall, and there is no information about the internal localization of the modes. Localized trapping of TAE (another predominantly shear-Alfvén mode) in low frequency kink modes (3-wave coupling) has previously been reported [4,11,13,18]. The expected shorter wavelength TAE in ITER [19] may be more susceptible to toroidal localization.

Experimental conditions
The data presented below come from plasmas made in the NSTX and NSTX-U devices. The typical plasma major radius is ≈1 m with a minor radius of ≈0.45 m, and typically with an elongation of κ ≈ 2.2. The principle difference between NSTX and NSTX-U [20] in terms of physical dimensions is that NSTX-U has a larger diameter 'center stack' or core (the NSTX-U center stack diameter is ≈63 cm vs. ≈37 cm for NSTX). NSTX was operated at nominal toroidal fields (vacuum field at R = 1.0 m) from 2.2 kG up to 4.65 kG and plasma currents up to 1.4 MA. The only NSTX-U campaign, which was dedicated to the validation of the engineering design, ran almost exclusively with a nominal toroidal field of 5.9 kG and had limited plasma diagnostics. The maximum neutral beam power used in NSTX was ≈7 MW at energies up to 100 keV from three beam sources. An additional three sources were added to NSTX-U providing higher maximum power, however in the single campaign the maximum beam power reached was also ≈7 MW with the highest beam voltage of ≈90 kV. The data studied for this paper comes from plasmas at varied densities, plasma currents and beam heating scenarios.
The principal diagnostic used in this study are two toroidal arrays of magnetic sensors. The high-n (HN) array of magnetic sensors consists of a toroidal array of 12 pick-up coils oriented to measure the vertical field mounted on the outboard vacuum vessel wall below the midplane. The number of sensors in the high-frequency array (HF) increased from a single coil in the very early operation of NSTX up to 15 in NSTX-U, arranged to make a toroidal array of 11 coils and an abbreviated poloidal array of 5 coils (which shares a coil with the toroidal array). The frequency response of the HF sensors was carefully documented and the system was designed for a roughly flat response up to ≈2 MHz in NSTX and increased to ≈4 MHz in NSTX-U. For the lower field NSTX shots, both the HN and HF arrays had sufficient bandwidth and a fast enough acquisition rate for the study of the fundamental and 2 nd harmonic GAE. Unfortunately, the installation of a dynamic error field correction system powered by unshielded switching-poweramplifiers (SPAs) introduces broadband noise into the test cell, corrupting the data from both arrays after 2005. Thus only data from before the addition of the SPAs is fully useful for studies of the 2 nd harmonic GAE in NSTX. The SPA noise on the HF data from NSTX acquired after the SPA installation can be mitigated by combining data from one coil with that of a nearby inverted coil to cancel the noise. This provides only a single clean measurement, not the multiple measurements needed for calculation of mode numbers.
In NSTX-U the HN array acquisition rate was reduced to 1 MHz to extend the acquisition time range for the longer shots and is primarily used only for the study of lower frequency modes (e.g. kinks, tearing modes and TAE). The HF sensors were replaced with coils with fewer turns in NSTX-U providing higher bandwidth, and additional sensors were added to the HF array to provide better toroidal coverage. Pairs of coils on NSTX-U can be combined to mitigate the noise from the SPAs and provide mode number measurements relatively free of SPA noise. While the HF array on NSTX-U has a nominal bandwidth of ≈4 MHz, the array has been used to measure the toroidal mode numbers of ion-cyclotron emission (ICE) at frequencies greater than 15 MHz [16]. However, these high frequency mode number measurements require data with a good signal-to-noise ratio, which is not always the case for the weak 2 nd harmonic GAE. Further, the 2 nd harmonic GAE appear to have very high n-numbers (short wavelengths) which make the toroidal mode number measurements more challenging.

2 nd harmonic GAE in low field plasmas
In NSTX plasmas with GAE, weaker fluctuations are typically seen in a frequency band covering roughly twice the frequency band of the GAE. An example of this is shown in figure 1 where the 1 st and 2 nd harmonic GAE activity, and at lower frequency the TAEs, are seen in a spectrogram of the timederivative of magnetic fluctuations (i.e. the signal from a magnetic sensor coil). While TAE activity is commonly present in these shots and can affect the GAE, the TAEs in this example do not appear to interact significantly with the GAE. For the purpose of illustration, figure 1 is a composite spectrogram where the sensitivity of the 1 st harmonic GAE frequency band is increased by a factor of 5 and the sensitivity in the 2 nd harmonic GAE frequency band is increased by a factor of 50 relative to that for the TAE frequency band. As is generally the case, the GAE are weaker than the TAE, and the 2 nd harmonic GAE are weaker still.
The fundamental GAE bursts typically consist of two or more modes split by 50 kHz-100 kHz, roughly consistent with the splitting expected for poloidal harmonics as estimated using a simple dispersion relation for GAE. Chirping GAE bursts are seen with frequencies in the range from 0.45 MHz up to 0.7 MHz. The 2 nd harmonic GAE are seen in the frequency band from 0.9 MHz to 1.4 MHz. In the 2 nd harmonic band, fluctuations are seen corresponding to frequency doubling of the fundamental modes, but also fluctuations are also seen at the sum of the two 1 st harmonic mode frequencies. The nominal toroidal field for this NSTX shot is 2.57 kG. The plasma was heated with 2.2 MW of deuterium neutral beams with a full energy of up to 97 keV. The GAE bursting shown here is during the current ramp phase. Over the time range covered by the spectrogram in figure 1 the plasma current increases from 0.51 MA to 0.59 MA and the central electron density increases from ≈1.7 × 10 13 cm −3 to ≈2.05 × 10 13 cm −3 .
The frequencies and mode numbers of these 2 nd harmonic GAE are consistent with an assumption that they result from a non-linearity in the wave equation which generates higher harmonic perturbations (occasionally even the third harmonics can be seen). These include non-linear interactions where two GAE non-linearly couple to drive a third wave at the sum of their frequencies. In this paper we show that, while the perturbations are qualitatively consistent with this non-linear model, details indicate that the 2 nd harmonic perturbations are significantly affected by the background plasma, possibly through resonances with weakly damped GAE that might be present in this frequency band.
We begin by examining in detail the GAE burst seen in figure 1 between 0.115 s and 0.116 s. Figures 2(b) and (c) show the expanded spectrograms of the 2 nd and 1 st harmonic bursts, respectively, and the overlaid curves show the center frequency and approximate frequency band over which the root-mean-square mode amplitudes are calculated. Figure 2(a) shows the simulated 2 nd harmonic fluctuations calculated by digitally filtering the raw data to include only the fundamental GAEs and then squaring that data to simulate the non-linear 2 nd harmonic drive term. The actual drive term should be normalized by equilibrium parameters which would depend on which non-linearity in the wave equations was dominant [9,21]. The nonlinear drive term would be radially dependent, as well as the 2 nd harmonic response to the drive term. The limited equilibrium diagnostic data for the early NSTX shots and for the NSTX-U shots analyzed below, mean that the approximations needed to estimate the equilibrium contributions to the drive term are so large as to render the effort largely meaningless. Here we assume that the equilibrium parameters do not change significantly over the short period of the burst and compare just the amplitude of the 2 nd harmonic response to the square of the fundamental amplitude (or the product of the two fundamental mode amplitudes).
The amplitudes are calculated by summing the discrete spectral components in quadrature over the frequency band indicated by the dashed curves in figures 2(a)-(c), that is with a width of ≈±20 kHz. As there are multiple modes in the fundamental GAE burst and the modes are chirping, the frequency bands track the mode frequency in time. The amplitude of the measured 2 nd harmonic fluctuations is plotted against the fundamental mode amplitude in figure 3. As the nonlinear response should scale as the square of the fundamental mode amplitude, that would ideally produce a curve with slope of 2 in a log-log plot. In practice, the background noise level contributes significantly in the 2 nd harmonic band at low amplitude where the signal-to-noise ratio is lower than for the 1 st harmonic band from which the simulated fluctuations were calculated. The blue-dashed curve in figure 3 indicates the approximate response expected for a simple nonlinearity, including a noise floor added in quadrature. While there is significant scatter in this data, the amplitude of the 2 nd harmonic approximately tracks the expected drive amplitude during the growth period (black points) and during the first part of the amplitude decay of the fundamental mode. However, the 2 nd harmonic perturbation then shows a sudden growth partway through the decay of the fundamental GAE. During the decay period of the fundamental GAE, the relative amplitude of the 2 nd harmonic fluctuation reaches an amplitude five times higher than the ratio established during the growth period. The mode frequencies have been chirping downwards during the burst and this might indicate that the frequency of the 2 nd harmonic perturbation has become resonant, or close to resonance, with the natural frequency of a high frequency GAE and the non-linear drive term overcomes the weak damping. In this model the fundamental GAE, through the inherent non-linearities, acts as an active antenna creating perturbations at twice the fundamental frequency which then triggers the growth of an otherwise weakly damped mode when nearly resonant.
Further analysis can be done by calculating the relative phase between the simulated drive and the measured response. The spectrograms of the simulated 2 nd harmonic 'GAE drive antenna signal' and the 2 nd harmonic GAE shown in figures 2(a) and (b) appear to be qualitatively very similar, but the frequency resolution of the spectrograms is only about 5 kHz. The burst lasts for ≈800 µs at a frequency of ≈1 MHz, thus there are about 800 wave periods during this burst. A deviation in frequency by one period over 800 µs, or ≈1 kHz, would be undetectable in the spectrogram, but would be clearly visible as a net phase shift of 360 • over the duration of the burst.
The relative phase evolution is shown in figure 4(a) and shows modest variation over the duration of the burst, demonstrating that the 1 st and 2 nd harmonic perturbations are strongly coupled. (As the frequency is chirping downwards, the frequency axis has been reversed so that time proceeds from left to right in the figure.) The black points are those from times before the fundamental mode reached its peak amplitude (growth period) and the red points are during the decay in amplitude of the fundamental mode (decay period). The simple model of a driven, damped harmonic oscillator (see below) suggests that the relative phase of ≈0 • would be consistent with the drive frequency being below the natural resonance frequency.
The relative phase changes substantially during the very early growth period when the 2 nd harmonic fluctuations are significantly affected by noise. During most of the growth period, drive and response are nearly in phase. A jump of ≈60 • in the relative phase between the simulated drive and the measured response is seen roughly at the time of peak fundamental mode amplitude, or when the 2 nd harmonic frequency chirp passes through ≈1 MHz. A second phase jump is seen at the onset of the 2 nd harmonic perturbation growth when the chirp frequency passes through ≈0.96 MHz. Both of these phase jumps could possibly indicate the presence of a resonant, but weakly damped mode, although in the first case there was no corresponding deviation in the relative amplitudes. The jump in the second harmonic fluctuation amplitude as the frequency sweeps through a resonance would be consistent with the presence of a weakly damped mode, but the simple model would predict that the phase should shift from ≈180 • to ≈0 • as the drive frequency sweeps down through the resonance frequency (see discussion below).
We have been looking at the 2 nd harmonic perturbations of the dominant fundamental mode, 'f 1 ', we can also look at the fluctuations resulting from the non-linear interaction between the two dominant fundamental GAE modes (labeled 'f 1 ' and 'f 2 ' in figure 2(c)). The frequency of the non-linear drive from the two dominant fundamental GAE is labeled 'f 1 + f 2 ' in figure 2(a) and the plasma response is shown in figure 2(b). The amplitude of the drive term scales as the product of the amplitudes of the modes labeled 'f 1 ' and 'f 2 '. In figure 5 we show the relative amplitude of the 2 nd harmonic response vs.  the simulated drive amplitude. As we are here comparing the response to the drive, the expected slope in the log-log plot should have a slope of 1. As in figure 3, the ratio of the response to drive amplitudes is again higher during the decay period than during the growth period, although the deviation has similar slopes during the growth and decay periods.
In figure 6, as in figure 4, we show the relative phase and amplitude evolutions vs. frequency. While the larger relative amplitude during the decay phase also suggests the presence of a nearby weakly damped mode, the lack of frequency dependence in the phase is curious. The range of the frequency chirp in this case falls short of 0.96 MHz where the strong response was seen in figure 4 and here it may just be that the 2 nd harmonic response is much more weakly damped than the fundamental mode during the decay phase. The relative phase in this example is close to 180 • , suggestive that the drive frequency is above a natural resonance, consistent with this example being higher frequency than the example in figures 3 and 4. However the results shown in this paper are not fully consistent with the simple model discussed below and there may be other explanations.

2 nd harmonic GAE in higher field plasmas
The examples shown so far are all from an NSTX shot with relatively low toroidal field (2.57 kG). In contrast, the 2016 NSTX-U campaign was carried out entirely with a higher toroidal field of ≈5.9 kG, and an improved fast magnetic sensor array. However, since it was primarily an 'engineering' campaign, this data also has a reduced equilibrium diagnostic set. It was previously reported that the toroidal mode number and frequency of the unstable GAE increases as the toroidal field is increased [3], so it is not surprising that the GAE for the 2016 campaign have higher toroidal mode numbers and shorter wavelengths, than in the low field shots from earlier NSTX campaigns. The bursts are also shorter in duration and typically the scaling of the response amplitude to the drive amplitude is consistent with that expected for the non-linear component of the fundamental mode. This may follow from the weaker frequency chirping which makes it less likely that the 2 nd harmonic will sweep through a resonance with a high frequency weakly damped mode. A surprising observation is that the 2 nd harmonic fluctuations are typically strongly toroidally localized with up to a factor of 10 amplitude variation.
Spectrograms for the 1 st and 2 nd harmonic GAE bursts at higher field are shown in figure 7. These bursts last about 250 µs vs. ≈1000 µs for the previously shown GAE burst at lower field. While the frequency chirping over just the initial 250 µs is easily visible in the lower field GAE, there is no clear indication of frequency chirping in this example. While the burst is shorter, it is still possible to track the amplitude evolutions of the fundamental and 2 nd harmonic bursts through the growth and decay. This data is shown for the 1.42 MHz 1 st harmonic mode and its 2 nd harmonic at 2.84 MHz in figure 8. The data shows a roughly quadratic scaling of the ratio of the 1 st and 2 nd harmonic amplitudes over two orders of magnitude, consistent with that expected for the non-linear components of the GAE. As before the black points are data during the growth of the fundamental mode and the red points are those during the decay. The measured 2 nd harmonic amplitude tracks both the growth and decay of the fundamental mode, consistent with the measured response being the non-linear component of the fundamental GAE. The lack of strong frequency chirping reduces the likelihood of the 2 nd harmonic drive frequency resonating with a mode in this frequency range.
The three dominant fundamental modes in this example have toroidal mode numbers of 9, 10 and 11, from lowest to highest frequency. That would imply that the 2 nd harmonic perturbations would have toroidal mode numbers in the range 18 ⩽ n ⩽ 22. However, the mode fits for the 2 nd harmonic modes were not good. It is unlikely that this is a diagnostic bandwidth issue, as this system was used to measure the amplitudes and mode numbers of ICE at frequencies greater than  15 MHz [14]. However, the expected high toroidal mode number, short wavelength, of the 2 nd harmonic perturbations would challenge the toroidal resolution of the array. The closest toroidal spacing of the coils would correspond to roughly a one quarter wavelength separation for n = 20, and more typically the coil spacings would correspond to more than half a wavelength. A further potential contribution to the bad fits could be the strong variation in the toroidal amplitude of the 2 nd harmonic modes, that is the mode is not global, but strongly toroidally localized. This can be seen in figure 9(a) where the normalized 2 nd harmonic fluctuation amplitude is shown vs. toroidal angle. Multiple measurements have been made of the mode amplitude through the burst and each measurement was normalized to the average of all the poloidal fluctuation measurements. The error bars reflect the scatter in normalized amplitude in measurements through the burst duration. The coils located at 153 • , 158.6 • and 351.4 • show significantly higher normalized amplitudes than on the other coils (for this and many other examples). For the 1 st harmonic GAE, there is much less scatter toroidally in normalized amplitude than for the 2 nd harmonic mode. The toroidal fluctuation measurements, again normalized to the average of the poloidal amplitudes, show relatively little toroidal variation.
That the mode is localized suggests that the mode has a strong standing-wave structure where the mode is localized, combined with a weaker traveling wave component. In the toroidal region of the strong standing wave, the phase would vary toroidally in a step-like manner, shifting by 180 • every half a wavelength. The colored bands in figure 9 indicate toroidal regions where it is not possible to install magnetic sensors in the toroidal array, thus the extent of the mode localization cannot be well documented given the limited toroidal coverage of the arrays.

Discussion
The data presented here strongly suggests that at least in some cases the 2 nd harmonic fluctuations which are often present during GAE activity are independent modes coupled to the 2 nd harmonic component of the fundamental GAE through the non-linear terms in the GAE dispersion relation. This conclusion is supported by the amplitude evolutions which often violate the expectation that the 2 nd harmonic perturbations should scale roughly as the square of the fundamental mode amplitudes. The variation in the measured relative phase between the simulated drive and the measured 2 nd harmonic fluctuations also supports this conclusion. If these were just the non-linear component of the fundamental mode, the relative phase should stay fixed, independent of frequency or amplitude. Finally, at higher field, the 2 nd harmonic fluctuations were strongly toroidally localized, whereas the fundamental mode was toroidally symmetric. A non-linear distortion of the GAE due to non-linear terms in the dispersion relation should generate 2 nd harmonic fluctuations locally proportional to the GAE amplitude and no toroidal localization of the 2 nd harmonic fluctuations should have occurred.
These observations suggest that the non-linearity of the GAE cause it to work in a similar fashion to the external TAE antennas used on JET [17] and C-Mod [10] to measure the damping rate of stable TAE. The advantages of the 'GAE antenna' are that the effective location of the 'antenna' is inside the plasma close to the location of the modes leading to strong coupling, that it excites modes with specific wavelengths and that it requires no additional investment in hardware. The disadvantage, of course, is that there is little direct control of the 'GAE antenna' frequency, amplitude or wave numbers. The driving perturbation is weak, but has the advantage of being generated deep in the plasma close to the mode location. Conversely, an external antenna has to deal with the rapid fall-off in perturbation amplitude between the plasma edge and the mode location. The wavelength from the GAE is fixed, but the wavelength from an external antenna is typically not well defined, further reducing the coupling to specific modes. And finally an external antenna generates a large signal at the plasma edge, complicating detection of modes excited by the 'antenna' using external coils.
Some insight into the relative phase can be gained by modeling the 2 nd harmonic response as that of an externally driven, damped oscillator, for example, a mass attached to a spring with frictional damping. This model predicts that the phase shifts by ≈180 • as the drive frequency sweeps down through the resonance frequency. The equation describing the, nontransient, behavior of this system is: Here, x is the displacement, 2γ 0 is the frictional damping term, ω 0 (=K/M) is the natural, undamped, oscillation frequency where K is the effective spring constant, M is the mass, and F/M is the driving term. The solution to this equation can be written as: In the limits ω ≪ ω 0 and ω ≫ ω 0 the solutions reduce to x (t) ≈ sin (ωt) and x (t) ≈ − sin (ωt), respectively, and the amplitude of the response peaks when ω ∼ = ω 0 . Below the resonance frequency, the drive force must be in phase with the natural restoring force to increase the oscillation frequency and above the resonance frequency the drive must oppose the natural restoring force to slow the oscillation. These solutions, however, do not capture the transient effects which may be important for these short, quickly chirping bursts, nor does a simple damped harmonic oscillator capture the complexity of the plasma medium, which might, for example, have multiple resonances nearby in frequency.
With more complete equilibrium data (e.g. measurements of the q profile) to support modeling of the GAE structure, it might be possible to make meaningful predictions about the expected ratio of the 1 st and 2 nd harmonic fluctuation amplitudes [9,21]. The ratio of the peak amplitude of the magnetic fluctuations of the 2 nd harmonic to the fundamental modes varies, but is typically in the range of 1.5%-2.5%. These measurements are of course from the edge of the plasma. The internal amplitude of the fundamental mode can be measured with the reflectometer array, but not, so far, of the 2 nd harmonic fluctuations. The amplitude of the density fluctuations measured with a multi-channel reflectometer for GAE bursts in similar shots to that used for figures 7-9 is δn ≈ (4-7) × 10 10 cm −3 and δn/n ≈ 0.2%-0.35% around the peak in mode amplitude, dropping to less than 0.1% nearer the plasma edge.