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 f TAE , 2 |.

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 experiments7–11 in the DIII-D tokamak12 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 work18 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 simulations20–25 and laboratory experiments (see, e.g., Refs. 26 and 27). Furthermore, it has been shown18,28 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.

In a cylindrical device, the angular frequency of a shear Alfvén wave is dependent on radius r, via1,
ω A ( r ) = k ( r ) v A ( r ) ,
(1)
where v A = B 0 / 4 π ρ m is the Alfvén velocity, using B0 as the equilibrium magnetic field and ρm as the mass density. Note that a radial perturbation of finite extent will tend to be sheared away by the radially dependent phase velocity ω A / k . This is known as continuum damping.29 In toroidal geometry, the parallel wavevector is given approximately by1,30,31
k m ( r ) = 1 R ( n m q ( r ) ) ,
(2)
using 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 ϕ / d θ characterizes the field line pitch and thus relates the toroidal angle ϕ and poloidal angle θ.
Following a magnetic field line as it travels helically from the high-field side to the low-field side of a tokamak ( B 1 / R), there exists a periodic modulation of the Alfvén velocity. Equation (1) implies that the phase velocity, and hence the index of refraction, is likewise modulated, leading to a Bragg gap in the otherwise continuous spectrum.32 The frequency in the gap (i.e., the TAE frequency) is2,5,33–35
f TAE = v A 4 π q TAE R ,
(3)
where the critical value of safety factor q TAE is deduced by considering counter-propagating modes5 with the same n and adjacent m. Setting k m ( r 0 ) = k ( 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
f TAE , lab = f TAE + n f rot ,
(4)
where f rot is the plasma rotation frequency.
TAEs are thus weakly damped eigenmodes intrinsic to a toroidal confinement device. In order to transfer energy to these instabilities, EPs must satisfy
v d · E d t 0 ,
(5)
when evaluated over many gyroperiods (here, v d is the fast-ion perpendicular drift velocity, and E is the perpendicular electric field).38 A necessary condition for resonant energy exchange is consequently39,
ω + ( m + l ) ω θ n ω ϕ 0 ,
(6)
where ω θ and ω ϕ are the angular frequencies of poloidal and toroidal circulation, respectively, and l is an arbitrary integer. In practice, l = ± 1 provides the most appreciable resonances, though higher harmonics are more important in strongly shaped discharges.38 
The free energy for wave growth is supplied by a gradient in configuration space. Writing the distribution function in terms of energy W, magnetic moment μ, and canonical toroidal momentum p ϕ, we see that the growth rate is
γ ω f W + n f p ϕ ,
(7)
where f ( W , p ϕ , μ ) is the distribution function, and neglecting a contribution f / μ. For the slowing-down distribution observed for fast ions, the first term f / W < 0 is stabilizing; however, because the density is peaked on the magnetic axis, the second term f / p ϕ f / r acts to destabilize the wave.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.

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 2000 cm2 and are distributed toroidally42 around the device, just inside the inner wall on the outer midplane (θ = 0). Coils used in this study are located at toroidal angle ϕ = { 20 ° , 67 ° , 97 ° , 127 ° , 132 ° , 137 ° , 157 ° , 200 ° }; unless otherwise noted, our analysis uses data from ϕ = 67 °. Through Faraday's law, the electromotive force induced in the coil is E d B / d t B ̇. Sampling is done at a cadence of f samp = 500 kHz, enabling assessment of fluctuations with frequencies 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 980 ms), the injected power was modulated around an average P 0 3.8 MW, alternating between approximately 3.5 and 4.1 MW with a period of 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.

FIG. 1.

Programmed NBI power (blue) and observed neutron flux (red), illustrating correspondence between beam program and fusion byproducts. P inj data have been smoothed by 250-sample window (semi-transparent trace shows raw data), while neutron data use a 15-sample window. Fluctuations in signal are due to beam power modulation.

FIG. 1.

Programmed NBI power (blue) and observed neutron flux (red), illustrating correspondence between beam program and fusion byproducts. P inj data have been smoothed by 250-sample window (semi-transparent trace shows raw data), while neutron data use a 15-sample window. Fluctuations in signal are due to beam power modulation.

Close modal
FIG. 2.

Autopower spectrum of magnetic fluctuations B ̇ derived from inductive coil time-series; note logarithmic scale. Differences in spectral density due to beam program are conspicuous. See Sec. IV for information on the production of such figures.

FIG. 2.

Autopower spectrum of magnetic fluctuations B ̇ derived from inductive coil time-series; note logarithmic scale. Differences in spectral density due to beam program are conspicuous. See Sec. IV for information on the production of such figures.

Close modal

The experiment used co-injected beams with ion energies 62 and 80 keV, and pitch v / v 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 RABBIT47 is presented in Fig. 3 (these calculations assume no EP transport by the instabilities).

FIG. 3.

Spatio-temporally resolved fast-ion pressure as estimated by RABBIT (ρ is the square root of the normalized toroidal flux). Note correspondence between beam program and enhancement/depression of EP pressure.

FIG. 3.

Spatio-temporally resolved fast-ion pressure as estimated by RABBIT (ρ is the square root of the normalized toroidal flux). Note correspondence between beam program and enhancement/depression of EP pressure.

Close modal

MHD equilibrium reconstructions from multiple diagnostics are gleaned via EFIT.48, Figure 4 displays a representative safety factor profile as constrained by motional Stark effect49 data. Figure 5 presents an estimate of the Alfvén continuum29 from equilibrium reconstructions of density and toroidal rotation50,51 profiles. Radial localization of modes is informed by beam emission spectroscopy (BES);52,53 see Fig. 6.

FIG. 4.

Experimental safety factor profile as reconstructed by EFIT for DIII-D shot #184936, for t = 980 ms. Note region of low magnetic shear from ρ 0.3 to ρ 0.5.

FIG. 4.

Experimental safety factor profile as reconstructed by EFIT for DIII-D shot #184936, for t = 980 ms. Note region of low magnetic shear from ρ 0.3 to ρ 0.5.

Close modal
FIG. 5.

Estimated Alfvén continuum for toroidal mode numbers n = 1 , 2 , 3 , 4, at t = 980 ms. Emphasis on n = 3, 4 continua anticipates analysis of Sec. V. Note that Doppler shift has been accounted for using (4), i.e., this represents the laboratory frame.

FIG. 5.

Estimated Alfvén continuum for toroidal mode numbers n = 1 , 2 , 3 , 4, at t = 980 ms. Emphasis on n = 3, 4 continua anticipates analysis of Sec. V. Note that Doppler shift has been accounted for using (4), i.e., this represents the laboratory frame.

Close modal
FIG. 6.

Spatially resolved fluctuations in BES signal, averaged over t [ 975 , 985 ] ms. Green boxes highlight an ensemble of modes participating in nonlinear coupling (see Sec. V). Safety factor profile (see Fig. 4) has been overplotted in gray for context.

FIG. 6.

Spatially resolved fluctuations in BES signal, averaged over t [ 975 , 985 ] ms. Green boxes highlight an ensemble of modes participating in nonlinear coupling (see Sec. V). Safety factor profile (see Fig. 4) has been overplotted in gray for context.

Close modal

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

FIG. 7.

Plasma current (blue) and line-integrated electron number density (red) of experimental shot.

FIG. 7.

Plasma current (blue) and line-integrated electron number density (red) of experimental shot.

Close modal
Despite the utility of a standard Fourier analysis, the classical power spectrum | x ̂ ( f ) | 2 occludes underlying phase relationships between Fourier coefficients x ̂ ( f ). In the case of two processes x 1 ( t ) , x 2 ( t ), the cross-spectrum x ̂ 1 ( f ) ¯ x 2 ( f ) is a popular tool41 for the assessment of linear phase coupling. However, for more general couplings, the theory of higher-order spectra54–56 is required. To investigate second-order phase coupling, we consider the triple correlation57,
C x 1 x 2 x 3 ( t 1 , t 2 ) = d τ x 1 ( t 1 + τ ) x 2 ( t 2 + τ ) x 3 ( τ ) .
(8)
It may be shown that the Fourier transform of the triple correlation is x ̂ 1 ( f 1 ) x ̂ 2 ( f 2 ) x ̂ 3 ( f 1 + f 2 ) ¯; the so-called cross-bispectrum is the ensemble average of this quantity:58,59
B x 1 x 2 x 3 ( f 1 , f 2 ) = x ̂ 1 ( f 1 ) x ̂ 2 ( f 2 ) x ̂ 3 ( f 1 + f 2 ) ¯ .
(9)
(Note that for discrete data, A i N A i / N, where 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
b x 1 x 2 x 3 2 ( f 1 , f 2 ) = | B x 1 x 2 x 3 ( f 1 , f 2 ) | 2 | x ̂ 1 ( f 1 ) x ̂ 2 ( f 2 ) | 2 | x ̂ 3 ( f 1 + f 2 ) | 2 .
(10)
Note that while the cross-bispectrum is complex-valued, the cross-bicoherence spectrum is purely real and constrained to the interval [0,1].

When a single process is considered, x 1 = x 2 = x 3 x, we refer to B x ( f 1 , f 2 ) B xxx ( f 1 , f 2 ) as the auto-bispectrum of x(t), and b x 2 ( f 1 , f 2 ) 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 ) 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 , f 2 ) ¯. The principal domain of auto-bicoherence is the region satisfying f 1 f 2 and f 1 + f 2 f Nyq, where f Nyq f samp / 2 is the Nyquist frequency.

Evidently, the bispectrum assesses the phase coherency of Fourier components at f1, f2, and f 1 + f 2. To see this, notice that the phase of the Fourier-transformed triple correlation (i.e., the biphase62) is
β ( f 1 , f 2 ) = φ ( f 1 ) + φ ( f 2 ) φ ( f 1 + f 2 ) ,
(11)
where the phase φ ( f ) satisfies x ̂ ( f ) = | x ̂ ( f ) | e i φ ( f ). If the amplitude of Fourier coefficients | x ̂ ( f ) | is constant, it is clear that
B x ( f 1 , f 2 ) e i β ( f 1 , f 2 ) .
(12)
Thus, for stationary processes with relatively sparse spectral content, the interpretation of bispectral analysis is straightforward.63 Namely, the bispectrum will remain finite wherever the biphase is stationary, and tend to vanish otherwise. Random distributions of the phasor e i β ( f 1 , f 2 ) therefore imply B x ( f 1 , f 2 ) 0 in the stationary case. Figure 8 demonstrates this in detail: As the (time-dependent) phasor associated with a single point (f1, f2) 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 β ( t ) t is tantamount to a uniform distribution β [ 0 , 2 π ).)
FIG. 8.

Visualization of phasor e i β ( f 1 , f 2 ) (left) and accumulating bispectrum (right) for linear biphase β ( t ) = 2 π t. Color is used to show changes in time.

FIG. 8.

Visualization of phasor e i β ( f 1 , f 2 ) (left) and accumulating bispectrum (right) for linear biphase β ( t ) = 2 π t. Color is used to show changes in time.

Close modal

Unfortunately, phase or amplitude modulation requires more careful interpretation. If the biphase is modulated, then the phasor e i β ( 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.

FIG. 9.

Phasor e i β ( f 1 , f 2 ) (left) and accumulating bispectrum (right) for biphase modulation β ( t ) = π [ 2 t + sin ( 2 π t ) / 2 ]. This implies that phase modulation is not generally associated with a vanishing value of bispectrum.

FIG. 9.

Phasor e i β ( f 1 , f 2 ) (left) and accumulating bispectrum (right) for biphase modulation β ( t ) = π [ 2 t + sin ( 2 π t ) / 2 ]. This implies that phase modulation is not generally associated with a vanishing value of bispectrum.

Close modal
In the case of amplitude modulation, Eq. (12) is no longer valid, and the effect of varying bispectral amplitude must be considered. In fact, nonstationary processes may produce spurious bispectral features65 (see Fig. 10). Writing the auto-bispectrum as
B x ( f 1 , f 2 ) = B ̃ x ( f 1 , f 2 ) ,
(13)
where B ̃ x is the “local,”66 or instantaneous64 auto-bispectrum, we note that B ̃ x = | B ̃ x | e i β. Amplitude modulation of a participating fluctuation will invariably lead to a modulation in B ̃ 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.
FIG. 10.

Example local bispectrum B ̃ x ( f 1 , f 2 ) (left) and accumulating bispectrum (right) for linear biphase β ( t ) = 2 π t with bispectral amplitude modulation d | B ̃ x | / d t 0. Notice that despite the absence of phase coherency by construction, a naïve bispectral analysis will report its existence.

FIG. 10.

Example local bispectrum B ̃ x ( f 1 , f 2 ) (left) and accumulating bispectrum (right) for linear biphase β ( t ) = 2 π t with bispectral amplitude modulation d | B ̃ x | / d t 0. Notice that despite the absence of phase coherency by construction, a naïve bispectral analysis will report its existence.

Close modal
Notice that a quadratic nonlinearity contributes coherent sum and difference frequencies, via
2 cos ( ω a t + φ a ) cos ( ω b t + φ b ) = cos [ ( ω a + ω b ) t + φ a + φ b ] + cos [ ( ω a ω b ) t + φ a φ b ] .
(14)
Hence, if x ( t ) = cos ( 2 π f a t + φ a ) + cos ( 2 π f b t + φ b ) + A cos ( 2 π f a t + φ a ) cos ( 2 π f b t + φ b ), the bispectrum will have peaks in the principal domain at ( f 1 , f 2 ) = ( f a , f b ) and ( f 1 , f 2 ) = ( f b , f a f b ), assuming fa > fb and f b > f a f b. At these points in bi-frequency space, the biphase satisfies
β ( f a , f b ) = φ a + φ b ( φ a + φ b ) = 0 , β ( f b , f a f b ) = ( φ a φ b ) + φ b φ a = 0 ,
(15)
for arbitrary phases φ a , φ b, and coupling coefficient A = 1. More generally, for complex coupling coefficient A = | A | e i δ, we find that β ( f a , f b ) = β ( f b , f a f b ) = δ. Clearly, if A is real-valued, the parity of δ / π is wholly dependent on the sign of A; this condition is satisfied even if φ a , φ b and | A | are functions of time.
To efficiently estimate the bispectrum or bicoherence spectrum, we note that Eqs. (9) and (10) are trivially parallelizable. Indeed, to estimate a single value of the local auto-bispectrum B ̃ x ( f 1 , f 2 ), we require only three values from a complex time-frequency representation (TFR). Undoubtedly, the most common TFR is the short-time Fourier transform (STFT), given by67 
X ( f , t ) d τ x ( τ ) w ( τ t ) e 2 π if τ ,
(16)
where w ( · ) is an integrable window function. (Note we have presented the square modulus of STFT, i.e., the spectrogram, in Fig. 2). An illuminating example is given by analysis of a contrived signal. Consider periodic quadratic coupling
x ( t ) = cos ( 2 π f a t ) + cos ( 2 π f b t ) A ( t ) cos ( 2 π f a t ) cos ( 2 π f b t ) ,
(17)
where A ( t ) = sin 4 ( 2 π f AM t ), and f AM f a , f b. We choose sampling rate f samp = 500 Hz, time interval t [ 0 , 100 ] s, and ( f a , f b , f AM ) = ( 97 , 84 , 0.05 ) Hz; Fig. 11 depicts the bispectral analysis of x ( t ) + x noise, where the latter term represents uniform noise x noise [ 1 , 1 ].
FIG. 11.

Bicoherence analysis of contrived time-series. Gray triangle delineates the principal domain f 1 f 2 f Nyq f 1. Spectrogram | X ( t , f ) | 2 produced via STFT with a subinterval of 512 samples, stepped by 128 samples; bicoherence calculation involves all 387 resulting subintervals. Window given by Hann function.68 Light green region outlines a single interval of coupling. Lower right panel depicts power spectrum | X ( f ) | 2 , where red regions highlight sum and difference (i.e., nonlinearly generated) modes.

FIG. 11.

Bicoherence analysis of contrived time-series. Gray triangle delineates the principal domain f 1 f 2 f Nyq f 1. Spectrogram | X ( t , f ) | 2 produced via STFT with a subinterval of 512 samples, stepped by 128 samples; bicoherence calculation involves all 387 resulting subintervals. Window given by Hann function.68 Light green region outlines a single interval of coupling. Lower right panel depicts power spectrum | X ( f ) | 2 , where red regions highlight sum and difference (i.e., nonlinearly generated) modes.

Close modal

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 ̃ x ( f 1 , f 2 , t ) = X ( f 1 , t ) X ( f 2 , t ) X ( f 1 + f 2 , t ) ¯ are plotted in Fig. 12. Of salience are the periods of stationary biphase coincident with enhanced bispectral amplitude.

FIG. 12.

Line-outs of local bispectrum B ̃ ( f 1 , f 2 , t ) for four points in bi-frequency space near (f1, f2) = (84,13) Hz. Upper panel is phase angle of B ̃ (biphase) divided by π; lower panel is modulus | B ̃ |. Note that biphase is stationary at odd multiple of π (coupling coefficient is negative) when bispectral amplitude becomes significant, and slips randomly otherwise. Cyan region highlights a single pulse of coupling, dotted green line traces β = 5 π.

FIG. 12.

Line-outs of local bispectrum B ̃ ( f 1 , f 2 , t ) for four points in bi-frequency space near (f1, f2) = (84,13) Hz. Upper panel is phase angle of B ̃ (biphase) divided by π; lower panel is modulus | B ̃ |. Note that biphase is stationary at odd multiple of π (coupling coefficient is negative) when bispectral amplitude becomes significant, and slips randomly otherwise. Cyan region highlights a single pulse of coupling, dotted green line traces β = 5 π.

Close modal
To enhance our control of simultaneous time and frequency resolution, we utilize a wavelet-based bispectrum in the spirit of refs.66,69–71 Using a Morlet wavelet,72 we define the continuous wavelet transform (CWT) as a complex TFR given by
W ( α , t ) = π 1 / 4 2 σ F 1 { x ̂ ( f ) exp [ 2 π 2 σ 2 ( f α f 0 ) 2 ] } ( t ) ,
(18)
where F 1 { · } ( t ) represents an inverse Fourier transform, and the time-like parameter σ governs the resolution in both time and frequency. (We find that choosing σ π Δ, where Δ t end t start is the duration of process x(t), furnishes the optimal trade-off.) The unitless quantity α corresponds to a frequency f = α f 0, where f0 is typically the smallest resolvable frequency, i.e., f 0 f samp / N for a time-series with N samples. The wavelet auto-bispectrum is then
B x ( α 1 , α 2 ) = W ( α 1 , t ) W ( α 2 , t ) W ( α 1 + α 2 , t ) ¯ .
(19)
Consequently, the wavelet auto-bicoherence spectrum is
b x 2 ( α 1 , α 2 ) = | B x ( α 1 , α 2 ) | 2 | W ( α 1 , t ) W ( α 2 , t ) | 2 | W ( α 1 + α 2 , t ) | 2 .
(20)
Note that, in practice, the STFT or CWT may be efficiently calculated using the fast Fourier transform transform73 (FFT).

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 b2 for each point in the principal domain entails an average over N = 480 complex values. Maximum frequency resolution is f 0 500 Hz. The entire range t [ 300 , 2000 ] ms is considered, stepped in 1 ms increments. Figure 13 exhibits a typical bicoherence analysis of a single interval.

FIG. 13.

Wavelet bicoherence analysis of magnetics data for DIII-D shot #184936, for t [ 980 , 982 ] ms. Squared auto-bicoherence spectrum (left) reports peak at ( f 1 , f 2 ) ( 86 , 21 ) kHz, with b 2 0.5 (denoted by cyan ×). Logarithmic plot of CWT square modulus | W ( f , t ) | 2 (upper right) and its ensemble average | W ( f ) | 2 (lower right) are also displayed. Dotted red lines correspond to participating fluctuations as implied by peak in b2.

FIG. 13.

Wavelet bicoherence analysis of magnetics data for DIII-D shot #184936, for t [ 980 , 982 ] ms. Squared auto-bicoherence spectrum (left) reports peak at ( f 1 , f 2 ) ( 86 , 21 ) kHz, with b 2 0.5 (denoted by cyan ×). Logarithmic plot of CWT square modulus | W ( f , t ) | 2 (upper right) and its ensemble average | W ( f ) | 2 (lower right) are also displayed. Dotted red lines correspond to participating fluctuations as implied by peak in b2.

Close modal

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

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

FIG. 14.

Line-outs of local bispectrum B ̃ ( f 1 , f 2 , t ) for four points in bi-frequency space near (f1, f2) = (87,21) kHz. Upper panel is phase angle of B ̃ (biphase) divided by π; lower panel is modulus | B ̃ |. Cyan region guides the eye toward time interval where modulus is relatively significant, and biphase has coincident period of stationarity near multiple of π. Dotted green line in upper panel traces β = 2 π.

FIG. 14.

Line-outs of local bispectrum B ̃ ( f 1 , f 2 , t ) for four points in bi-frequency space near (f1, f2) = (87,21) kHz. Upper panel is phase angle of B ̃ (biphase) divided by π; lower panel is modulus | B ̃ |. Cyan region guides the eye toward time interval where modulus is relatively significant, and biphase has coincident period of stationarity near multiple of π. Dotted green line in upper panel traces β = 2 π.

Close modal

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 ± 5) kHz ( f b ± 5) kHz, and ( f a + f b ± 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 ± 5) kHz next to the product of TAE bands ( f a ± 5) kHz × ( f a + f b ± 5) kHz, as in Fig. 16. Note that a low-pass filter is used to omit the sum-frequency component.

FIG. 15.

Absolute value of magnetic fluctuations in frequency bands 21 ± 5 kHz (blue), 87 ± 5 kHz (orange), 108 ± 5 kHz (green), in addition to the product of 87 kHz × 108 kHz fluctuations (red). 50-sample moving average is used. Coincident changes in amplitude of participating modes are apparent, with toroidal mode number as given in legend. Cyan region reproduced from Fig. 14.

FIG. 15.

Absolute value of magnetic fluctuations in frequency bands 21 ± 5 kHz (blue), 87 ± 5 kHz (orange), 108 ± 5 kHz (green), in addition to the product of 87 kHz × 108 kHz fluctuations (red). 50-sample moving average is used. Coincident changes in amplitude of participating modes are apparent, with toroidal mode number as given in legend. Cyan region reproduced from Fig. 14.

Close modal
FIG. 16.

Comparison of (low-pass filtered) 87 kHz × 108 kHz TAE fluctuations with low-frequency (21 ± 5 kHz) MHD fluctuations. Note that correspondence of phases occurs within cyan region reproduced from Figs. 14 and 15.

FIG. 16.

Comparison of (low-pass filtered) 87 kHz × 108 kHz TAE fluctuations with low-frequency (21 ± 5 kHz) MHD fluctuations. Note that correspondence of phases occurs within cyan region reproduced from Figs. 14 and 15.

Close modal

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” nonlinearity26) 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.)

FIG. 17.

Detail of phase coherence between reconstructed nonlinearity and low-frequency ( 21 kHz) MHD fluctuation. Note that data are normalized to peak value.

FIG. 17.

Detail of phase coherence between reconstructed nonlinearity and low-frequency ( 21 kHz) MHD fluctuation. Note that data are normalized to peak value.

Close modal

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.

FIG. 18.

Block diagram of bispectral analysis algorithm used in this work. Uncertainty calculation uses the approach outlined in Ref. 65.

FIG. 18.

Block diagram of bispectral analysis algorithm used in this work. Uncertainty calculation uses the approach outlined in Ref. 65.

Close modal

In addition to satisfying a matching condition in frequency, nonlinear interaction of waves with toroidal mode numbers na, nb will produce waves with n ± = | n a ± n 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 [ 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 ϕ, 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 ϕ.) While somewhat time-consuming, this process estimates n with low uncertainty; data with δ n 0.2 are plotted in Fig. 19.

FIG. 19.

Toroidal mode number spectrum gleaned from array of inductive coils. Green boxes indicate fluctuations considered by the present work.

FIG. 19.

Toroidal mode number spectrum gleaned from array of inductive coils. Green boxes indicate fluctuations considered by the present work.

Close modal

Within the interval t [ 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 [ 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 β / π 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 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 μs.

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 π. This deviation from integer values β / π 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.

FIG. 20.

Evolution of local bispectrum for ( f 1 , f 2 ) ( 87 , 21 ) kHz, in the interval t [ 980 , 982 ] ms. Values have been normalized to the maximum modulus | B ̃ x | max, as in Ref. 77. Color is used to represent evolution in time.

FIG. 20.

Evolution of local bispectrum for ( f 1 , f 2 ) ( 87 , 21 ) kHz, in the interval t [ 980 , 982 ] ms. Values have been normalized to the maximum modulus | B ̃ x | max, as in Ref. 77. Color is used to represent evolution in time.

Close modal

Analysis of the subinterval t [ 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.

FIG. 21.

Wavelet bicoherence analysis of magnetics data for DIII-D shot #184936, for t [ 984 , 986 ] ms. Squared auto-bicoherence spectrum (left) reports peak at ( f 1 , f 2 ) ( 93 , 15 ) kHz, with b 2 0.6 (denoted by cyan ×). Logarithmic plot of CWT square modulus | W ( f , t ) | 2 (upper right) and its ensemble average | W ( f ) | 2 (lower right) are also displayed. Dotted red lines correspond to participating fluctuations as implied by peak in b2.

FIG. 21.

Wavelet bicoherence analysis of magnetics data for DIII-D shot #184936, for t [ 984 , 986 ] ms. Squared auto-bicoherence spectrum (left) reports peak at ( f 1 , f 2 ) ( 93 , 15 ) kHz, with b 2 0.6 (denoted by cyan ×). Logarithmic plot of CWT square modulus | W ( f , t ) | 2 (upper right) and its ensemble average | W ( f ) | 2 (lower right) are also displayed. Dotted red lines correspond to participating fluctuations as implied by peak in b2.

Close modal
FIG. 22.

Line-outs of local bispectrum B ̃ ( f 1 , f 2 , t ) for four points in bi-frequency space near (f1, f2) = (93,15) kHz. Upper panel is phase angle of B ̃ (biphase) divided by π; lower panel is modulus | B ̃ |. Dotted green line in upper panel traces β = π.

FIG. 22.

Line-outs of local bispectrum B ̃ ( f 1 , f 2 , t ) for four points in bi-frequency space near (f1, f2) = (93,15) kHz. Upper panel is phase angle of B ̃ (biphase) divided by π; lower panel is modulus | B ̃ |. Dotted green line in upper panel traces β = π.

Close modal
FIG. 23.

Absolute value of magnetic fluctuations in frequency bands 15 ± 5 kHz (blue), 93 ± 5 kHz (orange), 108 ± 5 kHz (green), in addition to the product of 93 kHz × 108 kHz fluctuations (red). 50-sample moving average is used. Coincident changes in amplitude of participating modes are apparent, with n as given in legend. Cyan region reproduced from Fig. 22.

FIG. 23.

Absolute value of magnetic fluctuations in frequency bands 15 ± 5 kHz (blue), 93 ± 5 kHz (orange), 108 ± 5 kHz (green), in addition to the product of 93 kHz × 108 kHz fluctuations (red). 50-sample moving average is used. Coincident changes in amplitude of participating modes are apparent, with n as given in legend. Cyan region reproduced from Fig. 22.

Close modal
FIG. 24.

Comparison of (low-pass filtered) 93 kHz × 108 kHz TAE fluctuations with low-frequency (15 ± 5 kHz) MHD fluctuations. Note that correspondence of phases occurs within cyan region reproduced from Figs. 22 and 23.

FIG. 24.

Comparison of (low-pass filtered) 93 kHz × 108 kHz TAE fluctuations with low-frequency (15 ± 5 kHz) MHD fluctuations. Note that correspondence of phases occurs within cyan region reproduced from Figs. 22 and 23.

Close modal
FIG. 25.

Detail of phase coherence between reconstructed nonlinearity and low-frequency ( 15 kHz) MHD fluctuation. Note that data are normalized to peak value.

FIG. 25.

Detail of phase coherence between reconstructed nonlinearity and low-frequency ( 15 kHz) MHD fluctuation. Note that data are normalized to peak value.

Close modal

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 ̃ x, as in Fig. 20. We find that the biphase associated with | B ̃ 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 984.5 ms), the unwrapped biphase slips toward π, where it remains until the amplitude returns to baseline.

FIG. 26.

Evolution of local bispectrum for ( f 1 , f 2 ) ( 93 , 15 ) kHz, in the interval t [ 984 , 986 ] ms. Values have been normalized to the maximum modulus | B ̃ x | max.

FIG. 26.

Evolution of local bispectrum for ( f 1 , f 2 ) ( 93 , 15 ) kHz, in the interval t [ 984 , 986 ] ms. Values have been normalized to the maximum modulus | B ̃ x | max.

Close modal

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 [ 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.

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 ( 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 eigenmodes80,81 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 

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.

The authors have no conflicts to disclose.

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).

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

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