This paper presents the experimental identification of the dynamic interaction between three fluctuations excited by different thermodynamical forces in a linear plasma. The observed fluctuations are characterized as an axial symmetric mode, a drift wave, and an azimuthally symmetric potential fluctuation from the spatiotemporal structures measured with Langmuir probe arrays. The intermittent burst of the axial symmetric mode is observed by the instantaneous wave number spectrum. The energy transfer analysis revealed that the axially symmetric mode gains energy from the other mode during the burst while the drift wave loses energy. The intermittent burst synchronizes with the azimuthal symmetric potential fluctuation.

## I. INTRODUCTION

In magnetically confined plasmas, several types of fluctuations can be linearly excited by different free energy sources and will saturate via transport and nonlinear interactions between them. It has been universally recognized that ion and electron pressure gradients destabilize drift wave (DW) instabilities.^{1} Strong flow shear can destabilize Kelvin–Helmholtz type instability.^{2–4} Not only the free energy due to plasma inhomogeneities but also large fluctuations can excite other fluctuations, such as Reynolds stress driven zonal flow as a secondary instability and generalized Kelvin–Helmholtz instability excited via zonal flow as a tertiary instability.^{5} These different types of fluctuations could inherently coexist and interfere with each other in plasmas. Thus, energy partition and energy transfer between fluctuations get more complex. The competition and coexistence of fluctuations with different origins are observed in basic plasma experiments and simulations. The coexistence of drift waves and the other fluctuations were reported in linear plasmas, such as CSDX,^{6} Mirabelle,^{7} and LMD-U.^{8} In the QT-Upgrade machine, nonlinear coupling between drift wave and electron temperature gradient mode was observed.^{9} Cross-ferroic transport is driven by coexistence of drift wave and parallel velocity gradient mode in PANTA,^{10,11} and the related theoretical and simulation studies were carried out.^{12,13} Identification of mutual interaction and energy transfer between fluctuations with multiple origins is, thus, important to understand where sources, sinks, and dissipative structures are conflicting and cooperating, and dynamic behaviors of the open system.

This paper reports the identification of interactions between a drift wave, an axial symmetric mode, and an azimuthal symmetric potential fluctuation in a linear plasma. These interactions between three fluctuations are shown to be time varying. Intermittent energy transfer between drift wave and axial symmetric mode synchronizing with the azimuthal symmetric potential fluctuation is discovered.

## II. EXPERIMENTAL SETUP

The experiment was carried out in a linear magnetized plasma device, PANTA,^{10} which has a vacuum vessel with a length of 4 m and a diameter of 450 mm. Argon plasma is generated by an RF helicon source (6 kW, 7 MHz) consisting of a double loop antenna at a quartz tube, which has a diameter of 100 mm. This experiment is characterized by larger input power compared to other similar helicon plasma experiments.^{14} Neutral gas pressure is controlled at 0.5 Pa by a mass flow controller. A homogeneous axial magnetic field of $B=$ 0.13 T is created by 17 Helmholtz magnetic coils surrounding the vacuum vessel. Central electron temperature and density are ∼3 eV and $\u223c1\xd71019\u2009m\u22123$, respectively.

An azimuthal 64ch probe array and a radially movable probe are used for diagnosing plasma fluctuations. The 64ch probe is installed at 1875 mm from the helicon source, and tungsten tips are aligned azimuthally at radial position of 40 mm from the center. Fluctuations of ion saturation current ($Iis$) and floating potential ($Vf$) are measured by the probe tips at each azimuthal position alternately,^{8} which are considered to be indexes of fluctuations of plasma density and potential, respectively. The radially movable probe, which has three tungsten tips aligned azimuthally with 5 mm intervals, is located at 1375 mm from the source, and it allows us to observe radial structures of plasma profiles and fluctuations.

Figures 1(a) and 1(b) show radial profiles of electron temperature and electron density measured by the radial movable probe. Strong density gradient is formed around r = 30–60 mm. While the density gradient-scale length becomes minimum at r = 40 mm, the ratio of the gradient-scale-lengths of density and temperature is about 0.5. In this case, drift wave is unstable, but the electron temperature gradient mode is stable. Profiles of floating potential and space potential are shown in Figs. 1(c) and 1(d). Space potential is estimated from electron temperature and floating potential as $Vs=Vf+5.2Te/e$, which is a model equation for Argon. The profile of the space potential is flat in the inner region of the plasma (r $\u2264$ 40 mm) and has gradient in the outer region (r > 40 mm) within the error. We note that because the space potential is evaluated by using a model, and thus, accuracy depends on the model. In addition, the evaluation of the radial electric field has a very large error due to the large error derived from the floating potential error. The LIF measurement supports weak *E _{r}* in the inner region, but measurement accuracy is poor in the outer region. We assume that the

*E*×

*B*flow is smaller than diamagnetic drift velocity around r = 40 mm. Precise potential or flow measurement is an issue left to be addressed in the future.

Because 64ch probe array is aligned at r = 40 mm,^{15} we estimated the axial wave number by the radial movable probe at r = 40 mm and one of the probe tips of the 64ch probe at the same azimuthal angle with the movable probe. The relative positions of probes were adjusted so that the cross-coherence between two points becomes the highest. Radius position of probes was measured from the center of the vacuum vessel, and absolute radius positions of probes agreed with the design value of the vacuum vessel with an error of less than 1 mm. The axial mode number is estimated as $n=k\u2225L=\delta \theta L/2\pi d$. Here, $k\u2225$ is axial wave number, $\delta \theta $ is cross-phase of the axially aligned probe tips, *d* = 0.5 m is interval of the axially aligned probe tips, and *L* = 4 m is the machine length of the PANTA.

## III. RESULTS

### A. Coexistence of fluctuations with different origins

In this experiment, three fluctuations with different origins were observed. Two-dimensional Fourier power spectrum in frequency (*f* ) and azimuthal wave number ($k\theta $) domain is calculated from the density of normalized ion saturation current $I\u0303is/I\xafis$ and of normalized floating potential fluctuation $V\u0303f/T\xafe$ as shown in Figs. 2(a) and 2(b), where $A\xaf$ and $A\u0303$ denote long time average and fluctuation component of *A*. Here, the azimuthal mode number *m* is calculated as $m=rk\theta $ and a positive sign of *m* denotes that the mode propagates in the electron diamagnetic direction. Two clear peaks at (*m*, *f* ) = (1, 1.6 kHz) and (*m*, *f* ) = (4, 9.4 kHz) are observed in the power spectrum of both $I\u0303is/I\xafis$ and $V\u0303f/T\xafe$. An azimuthal symmetric fluctuation at (*m*, *f* ) = (0, ∼300 Hz) is also observed in only floating potential fluctuation. Figure 2(d) shows the axial coherence and mode number spectrum of density fluctuation. Both the (*m*, *f* ) = (1, 1.6 kHz) and (*m*, *f* ) = (4, 9.4 kHz) modes have high coherence, and the axial mode number *n* is $n\u223c0$ for (*m*, *f* ) = (1, 1.6 kHz) mode is $n\u223c$ 1–2 for (*m*, *f* ) = (4, 9.4 kHz) mode. Coherence is also high, so the evaluated axial wave number may be reliable.

The apparent azimuthal speed of the (*m*, *f* ) = (4, 9.4 kHz) mode is 600 $m/s$, which is close to the electron diamagnetic velocity of ∼750 $m/s$. The (*m*, *f* ) = (4, 9.4 kHz) mode is strongly excited around r = 30–40 mm where density gradient scale length is at its shortest as shown in Fig. 2(c). The axial mode number of the (*m*, *f* ) = (4, 9.4 kHz) is finite ($n\u223c$ 1–2) as shown in Fig. 2(d). The amplitude of the normalized density is of the same order as that of the normalized potential. Observed features of the (*m*, *f* ) = (4, 9.4 kHz) mode are coincident with characteristic of drift waves, as shown in the previous study.^{14}

The characteristics of the (*m*, *f* ) = (1, 1.6 kHz) mode are summarized below. The phase velocity is estimated to be 400 m/s. The radial structure is broad, as shown in Fig. 2(c) and previous study.^{14} The axial mode number is $n\u223c0$, which means that the (*m*, *f* ) = (1, 1.6 kHz) mode has an axially symmetric (AS) structure. Such axially symmetric fluctuations with finite azimuthal wave number can be linearly excited by the Kelvin–Helmholtz instability, or nonlinearly excited convection cells. To identify specific instability, precise measurements of flow shear and/or space potential are necessary. Hereafter, based on the above results, the (*m*, *f* ) = (4, 9.4 kHz) is referred to as the drift wave (DW) and the (*m*, *f* ) = (1, 1.6 kHz) mode is referred to as the axially symmetric mode (AS).

Since the *m* = 0 fluctuation was obtained from the floating potential fluctuation, it is possible that the *m* = 0 fluctuation could be caused by electron temperature fluctuation or by the space potential fluctuation. In helicon plasmas, the plasma can fluctuate globally due to various external factors, e.g., variations of plasma production caused by changes in the coupling state with the helicon wave. The external factors are, however, not likely to be periodic. Furthermore, if the fluctuations are due to the global variation of discharge, then not only the electron temperature but also the electron density should be fluctuated. Nevertheless, the *m* = 0 mode has no density fluctuation. It is certainly possible that instabilities are excited that cause the temperature to fluctuate, but even in such a case, the density is expected to fluctuate as well because the main energy source of primary instabilities is the free energy derived from the density gradient in this plasma. If the *m* = 0 mode is due to space potential fluctuation, the zonal flow is a candidate for this *m* = 0 phenomena. Identification of the zonal flow, direct observation of the space potential and axial and radial structure of it is essential. In order to clarify whether the *m* = 0 mode is a fluctuation of the entire plasma or a zonal flow, precise measurement of space potential will be performed in the future work.

In the case that the multi-fluctuations coexist, the fluctuations can interact with each other by nonlinear three-wave coupling. The nonlinear coupling strength is evaluated by squared bicoherence defined as

where $b2(f1,f2)$ is the squared bicoherence and *F* is the Fourier transform of the time series signal. The $A*$ denotes complex conjugate of *A*, and the angle bracket $\u27e8\xb7\u27e9$ denotes ensemble averaging. Figure 3(b) shows the result of the bicoherence analysis. The strong bicoherence between 1.6 and 9.4 kHz is observed, that is, the DW is modulated by the AS mode corresponding to the appearance of sideband modes of the DW, i.e., (*m*, *f* ) = (3, 7.8 kHz) and (*m*, *f* ) = (5, 11.0 kHz) modes, as shown in Fig. 2(a). The sideband modes satisfy the matching condition of the nonlinear coupling between the AS mode and the DW. Weak nonlinear couplings between the low frequency component and the AS mode, and between the low frequency component and the DW, are also visible. The associated phenomena of the low frequency component will be presented in Sec. III B.

### B. Dynamic interaction between the DW, the AS mode, and the *m* = 0 potential fluctuation

Since the DW and AS mode have different azimuthal mode number, we can trace time evolution of fluctuation power from instantaneous azimuthal mode power spectrum. Here, the Fourier decomposition is applied only in the azimuthal wave number domain and instantaneous time evolution of the wave number spectrum is obtained as shown in Fig. 4(a). The mode power of the DW (*m* = 4) and the AS mode (*m* = 1) is modulated in time. It is obvious that the AS mode quasi-periodically burst. Figure 4(b) displays the time evolution of the mode power of the AS mode and DW. It demonstrates that the AS mode is mutually exclusive with the DW during the intermittent burst. (As the power of the AS mode increases, the power of the DW decreases.) The probability density function (PDF) of the AS mode has a positive tail (skewness of 1.26) as shown in Fig. 4(c), that is, the AS mode burst is intermittent. The modulation spectrum of the instantaneous power spectrum is shown in Fig. 5. The modulation in the AS mode has a frequency of around 300 Hz, which is almost identical to the frequency of the *m* = 0 potential fluctuation while that of the DW has frequencies of about 300 Hz and 1.6 kHz. The 300 Hz component also corresponds to the period of intermittent burst of the AS mode, while the 1.6 kHz component corresponds to the modulation of the DW by the AS mode itself. The amplitude modulation is also indicated by the bicoherence analysis as shown in Fig. 3.

We focus on the dynamic interaction during the intermittent burst of the AS mode. Since the AS mode and the DW have different frequency, the spatiotemporal behavior during the intermittent burst can be extracted by envelope analysis. After applying bandpass filter to the $I\u0303is/I\u0303is$ signal obtained by radial movable probe, the envelopes are calculated by the Hilbert transform. Radial movable probe was scanned in a shot-by-shot manner. Applying conditional averaging, spatiotemporal structures of fluctuation component of the envelope ($E\u0303$) of the AS mode and the DW are reconstructed and shown in Figs. 6(a) and 6(b). The instantaneous mode power of the AS mode was used as the reference signal for the conditional averaging, and the intermittent burst was detected using the template method.^{14} The reference waveform of the instantaneous mode power obtained by conditional averaging is shown in Fig. 7(a). Cross-correlation between the instantaneous mode power and the reference waveform is used for the phase reference of the conditional sampling. We note that low-pass-filter ($f<$ 1 kHz) is applied to the envelope fluctuation obtained by Hilbert transform in order to cut the influence of the carrier wave (1.6 kHz). The envelope of the AS mode increases at r = 20–50 mm, while that of the DW decreases at r = 30–40 mm. The decrease in power of the DW is slightly delayed than the rise in power of the AS mode, and the time lag is ∼0.2 ms. The power of the AS mode increases at most by a factor of two and then breaks down, while that of the DW decreases by about 30%.

In order to clarify the nonlinear dynamics in the intermittent burst, we combine conditional sampling and bicoherence analysis. Since the timescale of the burst is the same order of the period of the AS mode, wave number bicoherence is more suitable than frequency bicoherence.^{16} The summed bicoherence for *m* is defined as

each *s* is the number of summand for each segment. The summed bicoherence could be an indicator for total contribution to nonlinear coupling. For simplicity, we separate the intermittent burst event into four phases: (i) quasi-stationary phase before the burst; (ii) the *m* = 1 mode power increasing phase; (iii) the power decreasing phase; and (iv) quasi-stationary phase after the burst, as represented in Fig. 7(a). Then, we estimate the total bicoherence from conditionally sampled instantaneous mode power for each phase. Figure 7(b) shows wave number total bicoherence at each phase. The nonlinear coupling of the AS mode (*m* = 1) and its second harmonics (*m* = 2) increase at the phases (ii) and (iii). The nonlinear coupling of the DW mode (*m* = 4) slightly increases as well. Moreover, the daughter modes of the AS mode and DW also increase the nonlinear coupling. The *m* = 0 component also increases nonlinear coupling during the burst, and it may be associated with the background change in density or the *m* = 0 potential fluctuation even though the bicoherence is evaluated from the density fluctuation. Figure 7(c) shows the *m* = 0 component of conditionally averaged floating potential fluctuation. It indicates that the *m* = 0 potential fluctuation is synchronizing with the intermittent burst. The amplitude of the *m* = 0 potential is of the same order as observed in LMD-U.^{17,18}

Summed bicoherence could give us information for the total strength of nonlinear coupling, but no information for nonlinear energy transform direction. For further investigation, evaluation of nonlinear energy transfer is necessary. The method of nonlinear energy transfer estimation was developed by Ritz *et al.*^{19} Several modified method is also proposed^{20,21} and applied to various types of plasma turbulence.^{22–24} This method assumes that time evolution of wave number spectrum *F*(*t*, *m*) follows the nonlinear three wave coupling equation written as

where $\Lambda mL$ and $\Lambda m1,m2NL$ are the linear and nonlinear coupling coefficients, respectively. $\Lambda mL$ and $\Lambda m1,m2NL$ can be estimated from *F*(*t*, *m*) by means of the multiple linear regression.^{21} *F*(*t*, *m*) can be evaluated by instantaneous wave number spectrum. By multiplying Eq. (3) to $F*(t,m)$ and taking ensemble average, conservation equation of spectral power $Pm=\u27e8F(t,m)F*(t,m)\u27e9$ is obtained as

where $Tm(m1,m2)$ is the nonlinear spectral energy transfer function defined as

Here, the positive values of $Tm(m1,m2)$ mean the *m* mode gains energy from the *m*_{1} and *m*_{2} modes, whereas the negative value means losses to the *m*_{1} and *m*_{2} modes. Because plots of energy transfer $Tm(m1,m2)$ are not always easy to interpret, for more simple understanding, we calculate summed nonlinear energy transfer defined as

This nonlinear energy transfer analysis is applied to the conditional sampled instantaneous mode power with about 1000 ensembles at each phase of the intermittent burst as shown in Fig. 7(c). In phase (ii), the AS mode gains energy from the other modes; then, the AS mode releases energy in phase (iii). While the drift wave loses energy during phases (ii) and (iii) corresponding to the delay of the mode power decrease in the DW. Therefore, the intermittent competition could be related to nonlinear energy transfer. We note that the *m* = 0 component also gains energy from the other modes, which may be related to density corrugation due to the intermittent burst or the *m* = 0 potential fluctuation, but further verification is required.

## IV. DISCUSSION AND SUMMARY

In this study, different types of fluctuations coexist and intermittently interact with each other synchronizing with the low frequency potential fluctuation. Coexistence of the multiple fluctuations was observed in several linear plasmas.^{6–10} Intermittent bursts of nonlinear coupling coefficient of the drift waves are observed in experiment.^{16} Intermittent phenomena associated with Kelvin–Helmholtz type instability were reported in simulation studies.^{25,26} Dynamic phenomena synchronizing with the zonal flows are observed in linear plasmas^{13,17,25} and tokamaks.^{27}

One of the possible mechanisms of the intermittent phenomena observed in the present study is discussed below. In this system, the drift waves and the Kelvin–Helmholtz instability are linearly excited, and zonal flow is secondary driven by Reynolds stress due to the drift wave turbulence. Then, the Kelvin–Helmholtz instability is enhanced by zonal flow as tertiary instability. In the enhancement, the Kelvin–Helmholtz instability additionally gains energy from the drift wave turbulence via nonlinear three wave coupling. This kind of phenomenon is discussed in theoretical and simulation studies.^{5,25} In order to clarify it, precise flow or radial electric field measurements, such as laser induced fluorescence^{28} and advanced probe measurements,^{29,30} are required and left for future work.

In summary, we have discovered the dynamic interactions between three fluctuations with different origins in the PANTA plasma. Experimental results indicate that (i) the three fluctuations are characterized as the axial symmetric mode, the drift wave, and the azimuthal symmetric potential fluctuation by spatiotemporal structure, (ii) the intermittent competition between axial symmetric mode and the drift wave in the period id the azimuthal symmetric potential fluctuation, and (iii) the nonlinear energy transfer is dynamically changed during the intermittent competition. The present work could be worthwhile to understand how plasma turbulence forms the dissipative structure in the open system.

## ACKNOWLEDGMENTS

This work was supported by JSPS KAKENHI Grant Nos. JP21K03508, JP20J12625, JP17H06089, and JP17K06994; the collaboration programs of RIAM Kyushu University and of the National Institute for Fusion Science (Nos. NIFS17KOCH002 and NIFS18KNWP007); and JSPS Core-to-Core Program (PLADyS).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

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