We report the detection of nonstationary quadratic coupling between toroidicity-induced Alfvén eigenmodes (TAEs) on sub-millisecond time scales. Identification of phase coherency between multiple TAEs and nonlinearly generated modes is facilitated by wavelet-based bicoherence analysis of time-series from inductive coils, taken from a DIII-D discharge heated by neutral beam injection (NBI). Characterization of nonlinear three-wave interaction is inferred by stationary local bispectrum phase (biphase) and confirmed via bandpass filtering. Biphase dynamics associated with prominent bispectral features are well-resolved in time and consistent with transient quadratic coupling. Onset and duration of nonlinearity are correlated with enhanced amplitude of participating TAEs; coincident changes in amplitude are observed for modes at difference frequency $ | f TAE , 1 \u2212 f TAE , 2 |$.

## I. INTRODUCTION

Decades of research have been applied to understanding the toroidicity-induced Alfvén eigenmode (TAE). Theoretical enlightenment was first provided by refs.^{1–5} which explained the instability as a weakly damped “gap mode,” not unlike localized eigenmodes facilitated by defects in a crystal.^{6} Early experiments^{7–11} in the DIII-D tokamak^{12} noted a significant correlation between TAE amplitude and energetic particle (EP) transport, and similar observations were successively made in the Tokamak Fusion Test Reactor (TFTR),^{13} JT-60U,^{14} JET,^{15} AUG,^{16} and NSTX.^{17} Additional work^{18} confirmed that TAEs can severely degrade the confinement of EPs, posing an existential threat to future reactors such as ITER.^{19}

The important and complementary role of wave–wave coupling has been noted in many global simulations^{20–25} and laboratory experiments (see, e.g., Refs. 26 and 27). Furthermore, it has been shown^{18,28} that the mode amplitude of TAE is sensitive to nonlinear wave–particle and wave–wave interactions.

In this study, we leverage wavelet-based bispectral analysis of magnetic fluctuation data to identify likely candidates for nonlinear three-wave coupling, which are subsequently investigated with bandpass filtering. The participating waves, two TAEs and a low-frequency magnetohydrodynamic (MHD) mode, are seen to satisfy coupling conditions in both frequency and toroidal wavenumber, consistent with nonlinear generation. We conclude that the detection of quadratic nonlinearities on sub-millisecond time scales is possible with this technique.

The paper is organized as follows: Sec. II provides an overview of the relevant theory of TAE, while Sec. III contextualizes the experiment and describes its data. Section IV provides a primer on bicoherence analysis, in addition to simple examples of its implementation. Section V applies time-resolved bispectral analysis to the experimental data, focusing on the detection of quadratic nonlinearities. Section VI summarizes our work and considers the next steps of the analysis.

## II. THEORY

*r*, via

^{1}

^{,}

*B*

_{0}as the equilibrium magnetic field and

*ρ*as the mass density. Note that a radial perturbation of finite extent will tend to be sheared away by the radially dependent phase velocity $ \omega A / k \u2225$. This is known as continuum damping.

_{m}^{29}In toroidal geometry, the parallel wavevector is given approximately by

^{1,30,31}

*R*for the major radius, and

*n*and

*m*for the toroidal and poloidal mode number, respectively (the symbol

*r*should now be understood as the minor radius). The safety factor $ q = d \varphi / d \theta $ characterizes the field line pitch and thus relates the toroidal angle $\varphi $ and poloidal angle

*θ*.

^{32}The frequency in the gap (i.e., the TAE frequency) is

^{2,5,33–35}

^{5}with the same

*n*and adjacent

*m*. Setting $ k \u2225 m ( r 0 ) = \u2212 k \u2225 ( m + 1 ) ( r 0 )$, Eq. (2) implies $ q TAE ( r 0 ) = ( m + 1 / 2 ) / n$. In the laboratory frame, the TAE frequency is Doppler shifted by toroidal rotation of the plasma, and thus,

^{36,37}

^{38}A necessary condition for resonant energy exchange is consequently

^{39}

^{,}

*l*is an arbitrary integer. In practice, $ l = \xb1 1$ provides the most appreciable resonances, though higher harmonics are more important in strongly shaped discharges.

^{38}

*W*, magnetic moment

*μ*, and canonical toroidal momentum $ p \varphi $, we see that the growth rate is

^{38,40}

For more comprehensive information on the subject, especially with regard to its implications for EP transport, the reader is directed to the excellent introductions in refs,^{38,40} and references therein.

## III. EXPERIMENTAL DATA

The time-series data analyzed in this work are gathered from an array of inductive coils in the DIII-D tokamak. Each coil is a mineral-shielded (MgO) coaxial cable wrapped around alumina rods in a “racetrack” cross section.^{41} The coils provide an effective area of $ \u223c 2000$ cm^{2} and are distributed toroidally^{42} around the device, just inside the inner wall on the outer midplane (*θ* = 0). Coils used in this study are located at toroidal angle $ \varphi = { 20 \xb0 , 67 \xb0 , 97 \xb0 , 127 \xb0 , 132 \xb0 , 137 \xb0 , 157 \xb0 , 200 \xb0}$; unless otherwise noted, our analysis uses data from $ \varphi = 67 \xb0$. Through Faraday's law, the electromotive force induced in the coil is $ E \u221d d B / d t \u2261 B \u0307$. Sampling is done at a cadence of $ f samp = 500$ kHz, enabling assessment of fluctuations with frequencies $\u2264$250 kHz.

We focus on DIII-D discharge #184936, which employed an upper single-null, diverted H-mode configuration; heating was provided by neutral beam injection (NBI).^{43,44} During the time of interest ( $ t \u223c 980$ ms), the injected power was modulated around an average $ P 0 \u2248 3.8$ MW, alternating between approximately 3.5 and 4.1 MW with a period of $ \u223c 20$ ms. Figure 1 displays both the programmed NBI power and the neutron flux as measured by ZnS scintillators.^{45} Figure 2 depicts the spectral density of magnetic fluctuations as a function of time.

The experiment used co-injected beams with ion energies 62 and 80 keV, and pitch $ v \u2225 / v \u223c 0.65$.^{46} Both the plasma and NBI employed deuterium as the working gas. A time-resolved estimation of fast-ion pressure as given by RABBIT^{47} is presented in Fig. 3 (these calculations assume no EP transport by the instabilities).

MHD equilibrium reconstructions from multiple diagnostics are gleaned via EFIT.^{48}^{,} Figure 4 displays a representative safety factor profile as constrained by motional Stark effect^{49} data. Figure 5 presents an estimate of the Alfvén continuum^{29} from equilibrium reconstructions of density and toroidal rotation^{50,51} profiles. Radial localization of modes is informed by beam emission spectroscopy (BES);^{52,53} see Fig. 6.

Typical plasma parameters are electron density $ n e \u2248 2.5 \xd7 10 13$ cm^{−3}, temperature $ T \u223c 2 \u2212 3$ keV, equilibrium field $ B 0 \u2248 1.25$ T, and plasma current $ I p \u2248 0.6$ MA. In this regime, the velocity of an 80 keV deuteron *v _{fi}* is a significant fraction of the Alfvén speed, $ v f i / v A \u2248 0.6$. See Fig. 7 for traces of the plasma current and line-integrated electron density.

## IV. BICOHERENCE PRIMER

^{41}for the assessment of

*linear*phase coupling. However, for more general couplings, the theory of higher-order spectra

^{54–56}is required. To investigate second-order phase coupling, we consider the triple correlation

^{57}

^{,}

*cross-bispectrum*is the ensemble average of this quantity:

^{58,59}

*N*is the number of realizations; i.e., we assume each realization is equally likely.) Furthermore, the cross-bispectrum may be normalized using the Cauchy–Schwarz inequality, yielding the (squared)

*cross-bicoherence spectrum,*

^{60,61}

When a single process is considered, $ x 1 = x 2 = x 3 \u2192 x$, we refer to $ B x ( f 1 , f 2 ) \u2261 B xxx ( f 1 , f 2 )$ as the *auto-bispectrum* of *x*(*t*), and $ b x 2 ( f 1 , f 2 ) \u2261 b xxx 2 ( f 1 , f 2 )$ as the *auto-bicoherence spectrum*. By inspection of (9), we find that for real-valued signals $ x ( t ) \u2208 \mathbb{R}$ both the auto-bispectrum and auto-bicoherence spectrum exhibit many symmetries, e.g., $ B x ( f 1 , f 2 ) = B x ( f 2 , f 1 ) = B x ( f 1 + f 2 , \u2212 f 2 ) \xaf$. The *principal domain* of auto-bicoherence is the region satisfying $ f 1 \u2265 f 2$ and $ f 1 + f 2 \u2264 f Nyq$, where $ f Nyq \u2261 f samp / 2$ is the Nyquist frequency.

*f*

_{1},

*f*

_{2}, and $ f 1 + f 2$. To see this, notice that the phase of the Fourier-transformed triple correlation (i.e., the

*biphase*

^{62}) is

^{63}Namely, the bispectrum will remain finite wherever the biphase is stationary, and tend to vanish otherwise. Random distributions of the phasor $ e i \beta ( f 1 , f 2 )$ therefore imply $ B x ( f 1 , f 2 ) \u2192 0$ in the stationary case. Figure 8 demonstrates this in detail: As the (time-dependent) phasor associated with a single point (

*f*

_{1},

*f*

_{2}) in bi-frequency space randomly samples the unit circle over a given time interval, the corresponding phasor average [i.e., $ B x ( f 1 , f 2 )$] is nullified, due to the absence of any preferential direction. (Note that the effect of a linear biphase $ \beta ( t ) \u221d t$ is tantamount to a uniform distribution $ \beta \u2208 [ 0 , 2 \pi )$.)

Unfortunately, phase or amplitude modulation requires more careful interpretation. If the biphase is modulated, then the phasor $ e i \beta ( f 1 , f 2 )$ *may* be preferentially distributed along some portion of the unit circle, and the ensemble average (i.e., the bispectrum) is nonvanishing. See Fig. 9 for a visualization. While analysis of this type of system is interesting in its own right (*see, e.g.,* Ref. 64, p. 63), biphase modulation is not consistent with quadratic nonlinearities, and thus, we shall neglect it, moving forward.

^{65}(see Fig. 10). Writing the auto-bispectrum as

^{66}or

*instantaneous*

^{64}auto-bispectrum, we note that $ B \u0303 x = | B \u0303 x | e i \beta $. Amplitude modulation of a participating fluctuation will invariably lead to a modulation in $ B \u0303 x$, and thus, a randomly distributed biphase is therefore not generally sufficient to guarantee a vanishing value of $ B x ( f 1 , f 2 )$. This motivates our consideration of a three-wave coupling process, which implicitly provides a stationary biphase.

*f*>

_{a}*f*and $ f b > f a \u2212 f b$. At these points in bi-frequency space, the biphase satisfies

_{b}*A*= 1. More generally, for complex coupling coefficient $ A = | A | e i \delta $, we find that $ \beta ( f a , f b ) = \beta ( f b , f a \u2212 f b ) = \delta $. Clearly, if

*A*is real-valued, the parity of $ \delta / \pi $ is wholly dependent on the sign of

*A*; this condition is satisfied even if $ \phi a , \phi b$ and $ | A |$ are functions of time.

^{67}

*i.e.,*the spectrogram, in Fig. 2). An illuminating example is given by analysis of a contrived signal. Consider periodic quadratic coupling

As expected, peaks are observed in the bicoherence spectrum at $ ( f 1 , f 2 ) = ( 97 , 84 )$ Hz and (84, 13) Hz, corresponding to phase coherency involving the sum and difference frequency, respectively. Anticipating the approach of Sec. V, line-outs of the local auto-bispectrum $ B \u0303 x ( f 1 , f 2 , t ) = X ( f 1 , t ) X ( f 2 , t ) X ( f 1 + f 2 , t ) \xaf$ are plotted in Fig. 12. Of salience are the periods of stationary biphase coincident with enhanced bispectral amplitude.

^{66,69–71}Using a Morlet wavelet,

^{72}we define the continuous wavelet transform (CWT) as a complex TFR given by

*σ*governs the resolution in both time and frequency. (We find that choosing $ \sigma \u223c \pi \Delta $, where $ \Delta \u2261 t end \u2212 t start$ is the duration of process

*x*(

*t*), furnishes the optimal trade-off.) The unitless quantity

*α*corresponds to a frequency $ f = \alpha f 0$, where

*f*

_{0}is typically the smallest resolvable frequency, i.e., $ f 0 \u2261 f samp / N$ for a time-series with

*N*samples. The wavelet auto-bispectrum is then

^{73}(FFT).

## V. ANALYSIS OF MAGNETIC FLUCTUATIONS ON DIII-D

Identification of quadratic coupling of TAEs is informed by highly time-resolved auto-bicoherence analysis. Guided by previous work,^{74} which identified phase coherency between *n* = 2 and *n* = 3 TAEs during the current ramp phase, we apply the techniques outlined in Sec. IV. For DIII-D discharge #184936, we estimate the wavelet auto-bicoherence spectrum for ∼2-ms intervals. After processing, the CWT is a matrix with size 480 × 480; hence, the calculation of *b*^{2} for each point in the principal domain entails an average over *N* = 480 complex values. Maximum frequency resolution is $ f 0 \u2248 500$ Hz. The entire range $ t \u2208 [ 300 , 2000 ]$ ms is considered, stepped in 1 ms increments. Figure 13 exhibits a typical bicoherence analysis of a single interval.

Within each of these intervals, we search for local maxima in the range $ f 1 \u2208 [ 70 , 110 ]$ kHz, $ f 2 \u2208 [ 10 , 40 ]$ kHz. (This is consistent with the location of difference frequency interaction found in previous work.) Once located, the coordinates (*f _{a}*,

*f*) associated with this peak identify the frequencies most likely to participate in three-wave coupling within the given time interval. For brevity, the present analysis will be limited to a single bispectral feature per interval. Interpretation of other statistically significant peaks in these bicoherence spectra is left for future work.

_{b}After confirmation that the value of bicoherence well exceeds the noise floor,^{65} line-outs of the local wavelet auto-bispectrum $ B \u0303 ( \alpha a , \alpha b , t ) = W ( \alpha a , t ) W ( \alpha b , t ) W ( \alpha a + \alpha b , t ) \xaf$ are calculated for $ ( \alpha a , \alpha b ) = ( f a , f b ) / f 0$; plots of the biphase and $ | B \u0303 ( \alpha a , \alpha b , t ) |$ vs time are subsequently produced (see Fig. 14). Of particular interest are intervals where the local bispectral modulus is peaked and $ \beta / \pi $ is close to an integer, as the interpretation is easily couched within our simplified framework of quadratic coupling (see Sec. IV).

To document the relative changes in amplitude of fluctuations at $ f a , f b , f a + f b$, bandpass filters are applied to the original signal,^{75} yielding three new time-series with spectral content limited to ( $ f a \xb1 5$) kHz ( $ f b \xb1 5$) kHz, and ( $ f a + f b \xb1 5$) kHz. This typically represents TAE, a low-frequency MHD mode, and TAE, respectively. (We note that the low-frequency fluctuation is consistent with a kink mode.^{76}) The fluctuation amplitude for each frequency band is visualized after performing a 50-sample moving average on the absolute value of the band-passed data (see Fig. 15). Finally, we plot the low-frequency band ( $ f b \xb1 5$) kHz next to the product of TAE bands ( $ f a \xb1 5$) kHz × ( $ f a + f b \xb1 5$) kHz, as in Fig. 16. Note that a low-pass filter is used to omit the sum-frequency component.

In principle, this plot should clearly articulate consistent phase relationships between observed low-frequency fluctuations and a quadratic nonlinearity formed between TAEs. In the absence of such a quadratic term, it is unlikely that extended correspondence would be observed between the product of band-passed data at the TAE frequencies (i.e., a “reconstructed” nonlinearity^{26}) and the fluctuations existing at the difference frequency. A detail of this correspondence is given in Fig. 17. Essentially, this picture lends credence to our identification of nonlinearity formed *between* TAEs, as coupling between the low-frequency mode and a single TAE would not provide the dynamics observed in Figs. 16 and 17. (We do not, however, exclude the possibility of additional such nonlinearities existing contemporaneously.)

As many thousands of each type of plot were produced for a given discharge, an exhaustive search was required to identify quadratic coupling (we consider automation of this process in Sec. VI). A diagram of our algorithm is shown in Fig. 18.

In addition to satisfying a matching condition in frequency, nonlinear interaction of waves with toroidal mode numbers *n _{a}*,

*n*will produce waves with $ n \xb1 = | n a \xb1 n b |$.

_{b}^{36}Thus, once time intervals containing candidates for quadratic coupling are characterized, the identification of toroidal mode number is crucial for the confirmation of nonlinearity. As an example, Fig. 19 displays the mode number analysis of $ t \u2208 [ 970 , 995 ]$ ms. This mode number spectrum is estimated by calculating the STFT (subinterval = 1024, step = 16) for data from inductive coils at eight toroidal positions along the tokamak midplane, then applying a linear regression to the unwrapped phases as a function of toroidal angle $\varphi $, for each value of frequency and time.

^{41}(For the considered mode numbers (

*n*< 10), aliasing is not a problem due to nonuniform distribution of coils in $\varphi $.) While somewhat time-consuming, this process estimates

*n*with low uncertainty; data with $ \delta n \u2264 0.2$ are plotted in Fig. 19.

Within the interval $ t \u2208 [ 970 , 995 ]$ ms, we highlight two 2-ms subintervals containing characteristics of quadratic coupling. (These subintervals are not the only data exhibiting signatures of three-wave coupling, but have been chosen due to their clarity in interpretation.) The first such subinterval, $ t \u2208 [ 980 , 982 ]$, has been thoroughly analyzed in Figs. 13–17. As changes in TAE amplitude are observed to be contemporaneous with enhanced local bispectral modulus and stationary biphase, we conclude that the experimental data are consistent with sub-millisecond quadratic coupling between *n* = 3 and *n* = 4 TAEs. (Note that Fig. 19 suggests the participating low-frequency mode has the required *n* = 1.) We gather from Fig. 15 that near *t* = 981.35 ms, all three modes begin increasing in amplitude. At *t* = 981.45 ms, both the *n* = 3 and *n* = 4 TAE continue to grow, seemingly at the expense of the low-frequency *n* = 1 MHD mode.

Crucially, we observe a correspondence between the parity of $ \beta / \pi $ and the sign of the coupling coefficient; i.e., the in-phase behavior of the reconstructed nonlinearity and low-frequency fluctuations observed in Fig. 16 is fundamentally related to the stationary biphase at an even multiple of *π* (see Fig. 14, upper panel). In fact, we see in Fig. 16 that near-perfect phase coherency is maintained until roughly *t* = 981.6 ms, where the amplitude of 21 kHz fluctuations is reduced to $ \u223c 20 %$ its peak value. After this point, the low-frequency *n* = 1 fluctuations resume, but perfectly out of phase with the reconstructed TAE nonlinearity. The duration of phase coherency is approximately 300 *μ*s.

The finer details of this interaction may be succinctly understood with a plot such as Fig. 20. We see that the local bispectrum travels along the real axis until just past *t* = 981.4 ms, when the biphase begins to slip away from $ 2 \pi $. This deviation from integer values $ \beta / \pi \u2208 \mathbb{N}$ could be a consequence of a complex-valued coupling coefficient, a coexisting quadratic interaction (directly between the low-frequency wave and *n* = 3 TAE, say), or simply an additional mode that happens to be phase-coherent over the subinterval.

Analysis of the subinterval $ t \u2208 [ 984 , 986 ]$ ms yields a similar, but distinct, conclusion. Figure 21 displays the bicoherence analysis of this subinterval, wherein a peak at $ ( f 1 , f 2 ) = ( 93 , 15 )$ kHz is detected. Following the espoused methodology, Figs. 22–25 sequentially detail the dynamics of local bispectrum, the evolution of mode amplitudes, and the phase coherency of low-frequency *n* = 1 MHD fluctuations with respect to the reconstructed nonlinearity.

Once again, an MHD mode (now at 15 kHz) initially grows with the TAEs, before losing energy to coupling (see Figs. 23 and 24). Moreover, as the biphase is seen to correspond to an odd multiple of *π* (Fig. 22), the reconstructed nonlinearity formed between *n* = 3 and *n* = 4 TAEs is now out of phase with the MHD mode (Figs. 24 and 25).

Alternatively, we may directly interpret the evolution of $ B \u0303 x$, as in Fig. 20. We find that the biphase associated with $ | B \u0303 x | max$ is now nearly purely real (see Fig. 26), in contrast with the previous subinterval. As the modulus begins to increase, the local bispectrum moves outward from the origin, parallel to the positive imaginary axis. Soon after ( $ t \u2248 984.5$ ms), the unwrapped biphase slips *toward* $ \u2212 \pi $, where it remains until the amplitude returns to baseline.

Thus, we hypothesize that quadratic coupling between TAEs provides a conduit for energy exchange with existing low-frequency MHD modes. (Though it is beyond the scope of the present work to model this exchange, we assume bilinear interactions between mode pairs.^{26,78,79}) Consistent with this interpretation, changes in TAE amplitude appear to be coincident with periods of enhanced bispectral modulus and stationary biphase, and hence with the duration of quadratic nonlinearity. In both intervals, the radial overlap of interacting modes is supported by analysis of BES data; Fig. 6 highlights the modes participating during the interval $ t \u2208 [ 980 , 982 ]$ ms. Note also that the inferred radial localization of TAEs approximately coincides with gaps in the Alfvén continuum, as predicted by Fig. 5.

## VI. DISCUSSION AND FUTURE WORK

In this work, we present evidence of nonstationary nonlinear coupling between TAEs in the DIII-D tokamak. The experimental data comprise an NBI-heated discharge exhibiting low magnetic shear. Characteristic local bispectral presentation of transient quadratic coupling is demonstrated for simple system of sinusoids.

Highly time-resolved, wavelet-based auto-bicoherence analysis is used to detect phase coherency in magnetic field fluctuations, as measured by an array of inductive coils. Partially automated analysis yields portfolio of candidates for quadratic coupling, which are interrogated using bandpass filtering. Toroidal mode number is inferred via linear regression of unwrapped Fourier transform phase vs coil toroidal angle. Radial localization of participating modes is gleaned from BES data.

Nonlinear three-wave coupling involving TAEs is repeatedly documented in the experimental data. Onset of nonlinearity is precipitated by growth of TAEs, and subsequent changes in TAE amplitudes are coincident with local bispectral signatures of quadratic nonlinearity (i.e., enhanced local bispectral modulus and stationary biphase). We conclude that the experimental data are consistent with sub-millisecond, quadratic coupling between *n* = 3 and *n* = 4 TAEs. Energy exchange with low-frequency ( $ \u223c 20$ kHz), *n* = 1 MHD modes is hypothesized to be facilitated by nonlinearities.

Most importantly, the present work allows a more detailed and time-resolved analysis of three-wave nonlinear interactions from experimental observations, investigating the processes that are theoretically and numerically predicted to play crucial roles. Indeed, signatures consistent with quadratic coupling appear to be ubiquitous in the experimental data, and presumably exist in any discharge where multiple, overlapping TAEs are driven unstable.

The methodology described herein is amenable to further automation, and thus may be relevant to ongoing applications of machine learning to tokamak research, such as real-time characterization of Alfvén eigenmodes^{80,81} or active feedback control.^{82} Additionally, we aim to categorize more types of coupling in DIII-D than the (conceptually simple) quadratic nonlinearity. The compilation of a “catalog” of bispectral signatures would be illuminating to these ends. The general-purpose bicoherence analysis routines used herein are derived from the open-source PyBic module;^{83} a forthcoming publication describes this software in more detail.^{84}

## ACKNOWLEDGMENTS

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award(s) DE-NA0003874, DE-SC0021404, and DE-FC02-04ER54698. G.R. and M.K. express deep gratitude to the DIII-D team for necessary support and impactful collaboration during this project.

Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**G. Riggs:** Conceptualization (supporting); Data curation (supporting); Formal analysis (lead); Investigation (equal); Methodology (lead); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **M. Koepke:** Conceptualization (lead); Data curation (supporting); Funding acquisition (lead); Investigation (equal); Methodology (supporting); Project administration (lead); Resources (equal); Supervision (lead); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). **W. Heidbrink:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (supporting); Project administration (supporting); Resources (equal); Software (supporting); Validation (equal); Visualization (supporting); Writing – review & editing (equal). **M. A. Van Zeeland:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (supporting); Project administration (supporting); Resources (equal); Software (supporting); Validation (equal); Visualization (supporting); Writing – review & editing (equal). **D. Spong:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (supporting); Resources (equal); Software (supporting); Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*n*ideal and resistive shear Alfvén waves in tokamaks

*n*shear Alfvén spectra in axisymmetric toroidal plasmas

*Applicability of Bispectral Analysis to Unstable Plasma Waves*

*Interpretations of Bicoherence in Space & Lab Plasma Dynamics*

*Characterizing Intermittent Turbulent Wave Kinetics and Energy Transfer via Three-Wave Coupling in Dipole-Confined Plasma*

*Issues in Acoustic Signal-Image Processing and Recognition*

_{2}interferometer data on the DIII-D national fusion facility