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# Effects of inter-ELM quasi-coherent modes on the dynamics of pedestal turbulence on HL-2A tokamak

J. Wen, Z.B. Shi, W.L. Zhong, X.L. Zou, M. Jiang, R. Ke, J.Q. Xu, W. Chen, Z.C. Yang, A.S. Liang, et al.

► To cite this version:

J. Wen, Z.B. Shi, W.L. Zhong, X.L. Zou, M. Jiang, et al. Effects of inter-ELM quasi-coherent modes on the dynamics of pedestal turbulence on HL-2A tokamak. Nuclear Fusion, 2024, 64 (7), pp.076054. 10.1088/1741-4326/ad52a8. cea-04788899

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Submitted on 18 Nov 2024

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# Effects of inter-ELM quasi-coherent modes on the dynamics of pedestal turbulence on HL-2A tokamak

J. Wen<fnref n="1" /> ID, Z.B. Shi<fnref n="1" />,*, W.L. Zhong<fnref n="1" />,*, X.L. Zou<fnref n="2" />, M. Jiang<fnref n="1" />, R. Ke<fnref n="1" /> ID, J.Q. Xu<fnref n="1" /> ID, W. Chen<fnref n="1" /> ID, Z.C. Yang<fnref n="1" />, A.S. Liang<fnref n="1" />, X. Yu<fnref n="1" />, N. Wu<fnref n="1" /> ID, X.X. He<fnref n="1" /> ID, P.W. Shi<fnref n="1" /> ID, G.L. Xiao<fnref n="1" />, R.H. Tong<fnref n="1" />, B.D. Yuan<fnref n="1" />, Y.R. Zhu<fnref n="1" />, Y.P. Zhang<fnref n="1" />, J.M. Gao<fnref n="1" /> ID, M.K. Han<fnref n="1" /> ID, J.Q. Dong<fnref n="1" />,<fnref n="3" />, Y. Shen<fnref n="1" /> ID and X.Q. Ji<fnref n="1" />

E-mail: shizb@swip.ac.cn and zhongwl@swip.ac.cn

Received 18 December 2023, revised 8 May 2024

Accepted for publication 31 May 2024

Published 12 June 2024

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CrossMark

## Abstract

The role of a series of quasi-coherent modes (QCMs) on the turbulence properties has been investigated during Type-III ELMs under HL-2A high confinement mode (H-mode) scenarios. The QCMs are essentially electrostatic and appear during the inter-ELM periods, with a frequency ranging from 20 kHz to 60 kHz. These QCMs are localized in the pedestal region and are related to the saturation of density gradient in the pedestal. Nonlinear couplings between QCMs and ambient turbulence have been observed and also verified through the envelope modulation of turbulence in density by the radial electrical field fluctuation of the modes. The presence of QCMs can increase the radial and poloidal turbulence correlation lengths, thereby modulating the turbulent transport. Experimental results show that QCMs significantly impact pedestal turbulence and transport by increasing the correlation length as well as the decorrelation time of turbulent eddies. The flow shearing rate in pedestal region is also enhanced to a level that surpasses the decorrelation frequency of turbulence, thus, the existence of QCMs has the ability to put off the ELM burst due to relative stronger stabilization of turbulence by sheared flows. The above results have demonstrated that the pedestal dynamics is largely determined by the complex interactions among QCMs, turbulence and the shear flow.

Keywords: high confinement mode, quasi-coherent mode, turbulence, flow shear

(Some figures may appear in colour only in the online journal)

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<footnote n="1">¹ Southwestern Institute of Physics, Chengdu 610041, China</footnote>

<footnote n="2">² CEA, IRFM, F-13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote n="3">³ Zhejiang University, Hangzhou 310027, China</footnote>

<footnote>* Authors to whom any correspondence should be addressed.</footnote>
# 1. Introduction

Tokamaks, known for their toroidal confinement configuration, are regarded as the most promising candidate for achieving controlled fusion reactions. One of the key factors that limit the confinement level is the transport process that strongly restricts the performance of the toroidal devices by causing losses of particles and energy across the magnetic field lines which can be attributed to various mechanisms [1] including diffusion resulted from random Coulomb collisions or turbulence mixing, convection driven by convective cells, avalanches, or magnetic stochasticity, as well as relaxation oscillations by sawteeth or edge localized modes (ELMs) [2, 3], among others. While neoclassical transport [4] due to collisional mechanisms is always present in toroidal geometry, the plasma transport is significantly influenced by anomalous transport induced by microinstabilities or turbulence, commonly referred to as turbulent transport which is typically 1–2 order of magnitude larger than the former [5]. Such turbulent transport can lead to a degradation of confinement, making the investigation of anomalous transport in magnetically confined plasmas a subject of intense research. Therefore, understanding and controlling this transport holds immense significance for the design as well as optimization of the performance in future fusion devices.

Over the past several decades, great efforts have been made in achieving plasmas with improved performance through various confinement scenarios, such as internal transport barrier [6] and H-mode [7], also the latter is the reference baseline scenario for ITER. The H-mode confinement regime is obtained when a transport barrier at the edge develops, which is known as pedestal. It is crucial for plasma performance as it not only controls the confinement in that region but also sets the boundary conditions for core confinement [8]. However, it is worth noting that the pedestal is characterized by steep gradients which can also act as sources of free energy and trigger microinstabilities, including both electrostatic and electromagnetic ones. While the electromagnetic turbulence can be easily resolved in experiments through magnetic fluctuations, the situation for electrostatic microinstabilities and the corresponding turbulence is more complex in the edge plasmas. There exist strong nonlinear interactions between turbulence and several kinds of coherent modes such as zonal flows at L-mode edge and QCM in H-mode pedestal. Although the modulation and suppression effects on turbulence by the former are well characterized in the past several decades, the nonlinear interplay between the latter and turbulence is still an open question, which is important for understanding the physics of transport and confinement in the H-mode plasmas [9, 10]. At present, there is no exact definition for QCM, however, according to a lot of works focusing on its characteristics, it is strongly suggested that the mode is a phenomenon that arises from the interaction between coherent modes and broadband turbulence in frequency space, rather than being an absolute concept in physics [11–17]. It can be identified by its distinct feature in the frequency spectrum [18], i.e. it oscillates around a specific frequency and has a wide spectrum that approximately satisfies the condition $0.5 \lesssim \Delta f/f_{peak} \lesssim 1$. Here, $\Delta f$

represents the bandwidth which typically has a magnitude of tens of kilo hertz and $f_{peak}$ denotes the peak frequency, respectively. QCM is a kind of coherent structure that emerges from the nonlinear coupling in frequency and wavenumber spaces. While researches usually define the QCM based on microinstabilities, such definition by the linear properties lacks of rigor. The nonlinear interaction between QCMs and ambient turbulence (AT) is a key factor that determines the plasma behavior. As a result, the QCM exhibits characteristics that bridge the gap between coherent and broadband fluctuations thus the underlying mechanisms are essential for understanding the transport in the H-mode pedestal.

The presence of QCMs has been observed in many tokamak experiments over the past two decades, garnering significant attention among fusion researchers. For example, JET has reported the identification of broadband high-frequency oscillations both in carbon wall [19] and ITER-like wall discharges [20], which are commonly observed as washboard modes in H-mode discharges [21]. These oscillations were found to be modulated in phase with the beryllium-II line emission from the divertor [22]. Linear analysis shows that the microtearing modes serve as the source of magnetic fluctuations in the JET pedestal and nonlinear simulations have demonstrated their relevance in producing experimentally observed transport levels [23]. Moreover, QCMs are frequently observed in both neutral beam heated plasmas and Ohmic plasmas. DIII-D has reported a set of high-frequency modes in the range of 80–250 kHz that correlate with the saturation of the pedestal pressure in quiescent H-mode (QH mode) plasmas. These characteristics align with those of kinetic ballooning modes [24]. The onset of radial electromagnetic QCMs with frequencies above 200 kHz has been found to correlate with the saturation of the electron temperature pedestal gradient, resulting in a limitation of the maximum pedestal pressure gradient due to enhanced transport [25]. Additionally, nonlinear coupling of quasicoherent fluctuations associated with radial distortions pushing out of the pedestal has been proposed as a possible mechanism for triggering low-frequency ELMs relevant for future fusion devices [26]. The EAST tokamak has recently reported several types of interesting QCMs, including both electrostatic and electromagnetic fluctuations [15]. In the case of the HL-2A tokamak, QCMs are generally interpreted as trapped electron modes excited by a critical gradient threshold of the electron temperature [27, 28]. Moreover, QCMs have been found to trigger a series of Type-III ELMy bursts, with their excitation being closely related to the coupling with AT [29]. Recently, a new type of pedestal QCM has been observed, which is closely associated with the initiation of Type-I ELMs and has the potential to delay the occurrence of ELM bursts, thus contributing to continuous operation within an ELM-free regime [30]. In recent years, gyrokinetic and gyrofluid simulations [31, 32] have facilitated the integration of experimental and theoretical approaches. However, it must be noted that verifying the ambiguous instabilities remains a critical task in future research and may require multiscale nonlinear gyrokinetic analyses.

As discussed above, understanding the pedestal QCMs and turbulence is crucial for future fusion reactors like ITER. It
has already been known that turbulence can have both positive and negative effects on pedestal dynamics. On one hand, it can enhance transport, leading to increased heat and particle fluxes, which degrades energy confinement and overall plasma performance [33]. On the other hand, a proper increase in transport can avoid high pedestal pressure hence keep it below the critical pressure gradient for triggering ELMs [34]. However, although numerous results focus on the nature of the microturbulence of the QCMs, the pedestal is also largely controlled by the dynamics of sheared flows whose effects cannot be well resolved by linear or even nonlinear simulations. The localized equilibrium sheared flow creates transport barriers, which have proven to be an effective factor for reducing or even eliminating rapid turbulent transport in fusion devices which works through flow shearing decorrelation mechanism [35]. As a consequence, the interaction among QCMs, turbulence and flows remains largely mysterious. More attention should be paid on studying the dynamics of turbulence and flows in the presence of QCMs, together with the goal of achieving plasmas with high confinement performance.

The present work aims to explore the influence of inter-ELM QCMs on pedestal turbulence in HL-2A tokamak. The experimental setup and diagnostic arrangements are presented in section 2. Section 3 gives experimental results of interplay among inter-ELM QCMs, pedestal turbulence and flows. Finally, the conclusions and discussions are drawn in section 4.

## 2. Experimental setup and diagnostics

The experiments were performed in a series of H-mode deuterium plasmas with Type-III ELMs in the HL-2A tokamak [36], which is a medium-sized tokamak operating in a lower single-null configuration with major/minor radius of $R/a = 1.65$ m/0.4 m. Being an integral part of China's fusion research program, the HL-2A serves as an exceptional platform for investigations of both physics and technological aspects. It provides a solid foundation for supporting projects such as ITER and fusion reactors. Figure 1 illustrates the plasma cross-section and the schematic of two main diagnostic instruments employed in this study. The investigation involves characterizing the turbulence and flows in the pedestal region using advanced diagnostics, namely the Beam Emission Spectroscopy (BES) system [37] and the Doppler Backward Scattering (DBS) reflectometer [38], also known as Doppler Reflectometry (DR) [39]. The BES system consists of 32 channels and is capable of measuring local density fluctuations with relatively low wavenumbers of $|k_\theta| \le 2.6$ cm$^{-1}$ and $|k_r| \le 4$ cm$^{-1}$. To achieve high spatial resolution, customized in-vacuum optics are employed, enabling measurements with $\Delta r \le 1$ cm and $\Delta z \le 1.5$ cm. The BES system covers the radial range of $r/a = 0.8 \sim 1$ and detects the Doppler-shifted emission from a 50 kV deuterium neutral beam injection (NBI), which observes the red-shifted $D_\alpha$ emission at approximately $\lambda \sim 659$ nm.

To simultaneously measure poloidal flows and density fluctuations, an 8-channel DBS system is employed in this

experiment [40]. The DBS system is installed on the low field side, approximately 13 cm below the equatorial plane, and utilizes a novel multiplexer-based frequency array source. Operating in the Q-band frequency range (30–50 GHz), the DBS system provides eight simultaneous frequencies at 34–48 GHz with interval of 2 GHz. The DBS system is configured with extraordinary mode (X-mode) polarization for detecting the pedestal region based on the typical H-mode discharge parameters. The DBS is sensitive to the backscattered signal at a fixed wavenumber ($k_\perp$) determined by the Bragg selection rule $k_\perp = 2k_i \sin\theta_s$, where $k_i$ represents the incidental microwave wavenumber and $\theta_s$ is the tilt angle of the microwave beam relative to the normal of the cutoff layer, respectively. When turbulence rotates in the poloidal direction, it induces a Doppler shift ($f_D$) in the spectrum, which can be estimated using $f_D \approx k_\perp v_\perp / 2\pi$. Here, $v_\perp$ represents the turbulence velocity perpendicular to the magnetic field. The measured perpendicular velocity ($v_\perp$) comprises two components, $v_\perp = v_{E \times B} + v_{\text{phase}}$, where $v_{E \times B}$ denotes the equilibrium $E \times B$ drift velocity resulting from the radial electric field in the lab frame, $v_{\text{phase}}$ is the phase velocity of the turbulence moving in the plasma frame, respectively. In general, $v_{\text{phase}}$ is expected to be small or negligible, allowing $v_\perp$ to be interpreted as the radial electric field ($E_r$) using $E_r = -v_{E \times B} B_T$ [41], and $B_T$ represents the local toroidal magnetic field. Furthermore, the turbulent density fluctuations ($\tilde{n}_e$) from the DBS system are approximately estimated by integrating the spectral power. This is achieved by applying a Fourier transform to the complex DBS signal ($C = I + iQ = Ae^{i\phi}$) consisting of the two simultaneously measured $I/Q$ time series. An interesting aspect for DBS, the turbulence is measured with backscattering at a fixed wavenumber given by the Bragg selection

<figure><img src="image_5.png" /><figcaption>**Figure 1.** Sketch of the flux surfaces and arrangement of the BES and DBS systems.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 3. Time evolutions of (a) divertor Dα emission, (b) normalized electron density gradient R/Lne, (c) QCM mode intensity calculated by density fluctuations within frequency range of 20–60 kHz, (d) spectrogram of density fluctuation and (e) comparison of pedestal density profiles in the presence and absence of QCMs measured by FMCW reflectometer, corresponding to the red/blue dashed lines at 876.5/884.5 ms.</figcaption></figure>

The two-point correlation technique is used to analyze the space-time statistical properties of a fluctuation field [44]. This technique assumes that the turbulent fluctuation is both stationary and homogeneous, which serves as a fundamental model in many turbulence theories. According to this assumption, fluctuations can be represented as a combination of wave packets, with each packet characterized by a random relationship between wavenumber, $k(f)$, and frequency, $f$. In this study, the local wavenumber-frequency spectrum is calculated using different spatial density fluctuations. Two adjacent radial/poloidal channels of the BES system are utilized, with a separation distance of $\Delta x_1/\Delta x_2 = 0.9/1.2$ cm. The wavenumber-frequency spectrum expressed by $S(k,f)$, is defined as the representation of the fluctuation field at different combinations of wavenumber and frequency

$$S(k,f) = \frac{1}{M} \sum_{j=1}^{M} I_{\Delta k} [k - k^j(f)] |S_1^j(f) + S_2^j(f)| / 2. \quad (1)$$

In the equation (1), $I_{\Delta k}$ represents the indicator function, $S_1^j$ and $S_2^j$ denote the power spectra measured at two different positions. The local wavenumber, $k^j(f) = \Delta\theta_{12}^j(f)/\Delta x$, is calculated based on the phase shift, $\theta_{12}^j(f)$, between the two points for each realization $j$ and $M$ refers to the total number of realizations. The range of $k^j(f)$ is given by $k^j(f) \le |\pm\pi/\Delta x|$. Figure 4 shows the spectral characteristics, including contour plots of the wavenumber-frequency spectrum $S(k,f)$, the wavenumber spectrum $S(k) = \sum_f S(k,f)$ and spectrum averaged wavenumber $\langle k \rangle = \sum_f \bar{k}(f)S(f)$. Here, $\bar{k}(f) = \sum_k k \cdot S(k|f)$ represents the statistic dispersion relation, $S(k|f) = S(k,f)/S(f)$ is the conditional wavenumber spectrum and $S(f) = \sum_k S(k,f)$ is cross power spectrum. For analyzing the density fluctuations with QCM phase in 875~876 ms, clear power peaks ranging from 20 kHz to 60 kHz are observed in both the radial wavenumber-frequency spectrum $S(k_r,f)$,

and the poloidal wavenumber-frequency spectrum $S(k_\theta, f)$, as shown in figures 4(a) and (b), respectively. The dashed curves in the wavenumber spectrum, $S(k)$, also exhibit a distinct peak due to the contributions from QCM. The averaged radial wavenumber is calculated to be $\langle k_r \rangle = -0.24 \text{ cm}^{-1}$, indicating a radial outward propagation of the mode. The averaged poloidal wavenumber is estimated as $\langle k_\theta \rangle = 1.20 \text{ cm}^{-1}$, with the positive sign corresponding to the electron diamagnetic drift direction in the laboratory frame.

The correlation length $L_c$ gives an indication of the spatial extent over which the fluctuations are correlated. It helps us understand how the presence of inter-ELM QCMs influences the correlation length, and consequently the transport properties in the plasma edge. The correlation length can be estimated from the inverse of the $k$ spectrum broadening as

$$L_c = 1 / \sqrt{\int \bar{\sigma}_k^2(f) \, df} \quad (2)$$

where $\bar{\sigma}_k^2(f)$ is the wavenumber spectral width estimated as

$$\bar{\sigma}_k^2(f) = \int [k - \bar{k}(f)]^2 \cdot S(k|f) \, dk \\ \qquad = \frac{\int [k - \bar{k}(f)]^2 \cdot S(k,f) \, dk}{\int S(k,f) \, dk} \quad (3)$$

This suggests that a larger $k$ spectrum broadening corresponds to a smaller correlation length, implying that the turbulence is more localized, and vice versa. Figures 5(a) and (b) depict the radial and poloidal correlation lengths ($L_{cr}$ and $L_{c\theta}$) of turbulence, respectively. The statistical values of correlation length are calculated using a typical data length of 2 ms for both plasma scenarios, one with QCMs (in red) and the other without QCMs (in blue). When QCMs are present during the inter-ELMs, the correlation lengths are approximately $L_{cr} \sim 2.8$ cm and $L_{c\theta} \sim 2.5$ cm. In contrast, in the absence of
<figure><img src="image_7.png" /><figcaption>**Figure 4.** (a) Radial and (b) poloidal wavenumber-frequency spectra $S(k,f)$ of the QCMs calculated by the density fluctuations from BES signals. The corresponding wavenumber spectra are plotted by dashed curves and their averaged values are $\langle k_r \rangle / \langle k_\theta \rangle = -0.24/1.20 \text{ cm}^{-1}$, respectively. Negative $k_r$ denotes radially outward propagation while positive $k_\theta$ corresponds to the electron diamagnetic drift direction, respectively.</figcaption></figure>

QCMs during the inter-ELMs, the correlation lengths notice-
ably decrease to around $L_{cr} \sim 1.0$ cm and $L_{c\theta} \sim 1.1$ cm. This
indicates that the presence of QCMs may significantly affect
turbulent transport, as the turbulence structures become larger
compared to the case without QCMs. Additionally, figure 5(c)
provides an example of the comparison of the $k_\theta$ wavenumber
spectral width with or without QCMs. It can be observed that
the wavenumber spectral width, $\bar{\sigma}_k^2(f)$, narrows when QCMs
are present, compared to the scenario without QCMs.

3.2. *Impact of QCMs on pedestal turbulence*

In the following, we will explore the nonlinear interactions
between inter-ELM QCMs and AT, which refers to fluctu-
ations with frequencies larger than tens of kHz. The pres-
ence of QCMs has a significant impact on the turbulence cor-
relation lengths in both the radial and poloidal directions, as
illustrated in figure 5. To validate this effect, we employed
the bispectrum analysis technique [45] to directly confirm the
existence of nonlinear three-wave interactions among the fluc-
tuating quantities. The squared bicoherence is utilized as a cor-
relation measure for the Fourier transforms $X(f)$, $Y(f)$, and
$Z(f)$ of the fluctuating signals $x(t)$, $y(t)$, and $z(t)$, respectively

$$
\hat{b}_{XYZ}^2(f_1, f_2) = \frac{|\hat{B}_{XYZ}(f_1, f_2)|^2}{\langle |X(f_1) Y(f_2)|^2 \rangle \langle |Z(f_3 = f_1 + f_2)|^2 \rangle} \quad (4)
$$

the bispectrum, denoted as $\hat{B}_{XYZ}(f_1, f_2) = \langle X(f_1) Y(f_2) Z^*(f_3 = f_1 + f_2) \rangle$, is calculated as the correlation between the Fourier transforms $X(f_1)$, $Y(f_2)$, and the complex conjugate $Z^*(f_3)$, where $f_3 = f_1 + f_2$. When $X(f) = Y(f) = Z(f)$, it represents the auto-bispectrum, while in other cases, it represents the cross-bispectrum. The squared bicoherence, $\Sigma \hat{b}_{XYZ}^2(f_3) = \Sigma_{f_3=f_1+f_2} \hat{b}_{XYZ}^2(f_1, f_2)/N(f_3)$, sums up the squared bicoherence values for all $f_1$ and $f_2$ that satisfy the frequency matching condition $f_3 = f_1 + f_2$. This measure indicates the level of

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Statistical correlation length $L_c$ of turbulence as a function of QCM mode intensity $I_{QCM}$. The values with and without QCMs are shown by red and blue symbols, respectively. (a) Radial and (b) poloidal values. (c) An example of the comparison of $k_\theta$ spectrum broadening with or without QCM.</figcaption></figure>

coherent three-wave coupling and is normalized by the num-
ber of realizations, N(f<sub>3</sub>), for each f<sub>3</sub> in the summation. The
fast Fourier transform bispectrum analysis is employed in the
<figure><img src="image_9.png" /><figcaption>Figure 6. (a) Contour plot of squared auto-bicoherence $\hat{b}_{\tilde{n}_e, \tilde{n}_e, \tilde{n}_e^*}^2(f_1, f_2)$ of density fluctuations and (b) the corresponding summed squared bi-coherence $\sum \hat{b}_{\tilde{n}_e, \tilde{n}_e, \tilde{n}_e^*}^2(f_3)$.</figcaption></figure>

data processing with a typical data length of about 50 ms to ensure a sufficient amount of ensemble average, represented by angle brackets $\langle \cdot \rangle$ in equation (4). Figure 6(a) displays the contour plot of the auto-bicoherence $\hat{b}_{\tilde{n}_e, \tilde{n}_e, \tilde{n}_e^*}^2(f_1, f_2)$, using the density fluctuations measured by the DBS system with the 37 GHz channel. Figure 6(b) illustrates the corresponding sum of squared bicoherence $\sum \hat{b}_{\tilde{n}_e, \tilde{n}_e, \tilde{n}_e^*}^2(f_3)$. Notably, straight lines are observed within frequency ranges $f_1 + f_2 \approx 40-60$ kHz and $f_2 \approx \pm(40-60)$ kHz as shown in figure 6(a). Moreover, a significant amplitude of $\sum \hat{b}_{\tilde{n}_e, \tilde{n}_e, \tilde{n}_e^*}^2(f_3)$ is observed at the QCM frequency ranging from 20 kHz to 60 kHz in figure 6(b), which confirms the presence of substantial nonlinear coupling between QCMs and AT. This suggests that the QCMs may be driven by the AT and, in turn, modulate the turbulence with a distinct frequency peak around 40 kHz and a bandwidth of $\Delta f \approx 40$ kHz.

In order to further investigate the modulation relationship between QCMs and turbulence, the envelope analysis [46]

is used to identify interaction between the QCMs and AT. The envelope based on the analytic signal approach consists of a real signal of finite energy which is Fourier transformable and band-limited and the Hilbert transform of the original real signal. For a real signal $S_r(t)$, the corresponding analytic signal is defined as $H(t) = S_r(t) + i\hat{H}_f(t)$, where $\hat{H}_f(t) = \frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{S_r(\tau)}{t-\tau} d\tau$ is the Hilbert transform of signal $S_r(t)$. The envelope is then defined as the modulus of the analytic signal. Figure 7(a) displays the time evolutions of bandpass filtered radial electric field fluctuations of QCM ($\tilde{E}_{r,QCM}$) in the frequency range of $f = 20-60$ kHz and the envelope of AT ($\tilde{n}_{e,AT}^{\text{env}}$) represented by the high frequency density fluctuations in frequency range of $f = 200-600$ kHz. Both of which are obtained from DBS system during the inter-ELM phase in the presence of QCM. The result reveals that AT periodically appears with a time interval of approximately 0.025 ms, which matches well with the frequency of $\tilde{E}_{r,QCM}$ centered around 40 kHz. This indicates that the AT has been clearly modulated by QCMs, demonstrating that QCM energy originates from the inverse cascading of high-frequency turbulence. The cross-correlation function (CCF) analysis is also used to help us determine the time delay or phase shift between the two signals, providing information about the temporal relationship between the fluctuations $\tilde{E}_{r,QCM}$ and $\tilde{n}_{e,AT}^{\text{env}}$. The CCF is defined as the correlation at different time lags as

$$ \gamma_{xy}(\tau) = \frac{\langle [x(t + \tau) - \bar{x}] [y(t) - \bar{y}] \rangle}{\sqrt{\langle [x(t + \tau) - \bar{x}]^2 \rangle \langle [y(t) - \bar{y}]^2 \rangle}} \quad (5) $$

Here, $x(t)$ and $y(t)$ represents the time evolutions of signals $\tilde{E}_{r,QCM}$ and $\tilde{n}_{e,AT}^{\text{env}}$, while the variable $\tau$ represents the time lag applied to one of the signals. Figure 7(b) shows that a small phase difference exists between the fluctuations $\tilde{E}_{r,QCM}$ and $\tilde{n}_{e,AT}^{\text{env}}$. Particularly, the amplitude of $\tilde{E}_{r,QCM}$ in the range of 20–60 kHz has been adjusted to facilitate a comparison with AT fluctuations, $\tilde{n}_{e,AT}^{\text{env}}$, in the 200–600 kHz range. Remarkably, this adjustment does not impact the results obtained from the CCF analysis. Consequently, it can be inferred that $\tilde{E}_{r,QCM}$ leads $\tilde{n}_{e,AT}^{\text{env}}$ by approximately 3 µs, i.e. the background high frequency turbulence is strongly suppressed by the QCM-induced electric field in a short time.

The information regarding the nonlinear interaction between the QCM and turbulence is captured in the envelopes. Therefore, the modulation interaction can be determined by employing correlation techniques to examine the fluctuation envelopes. To clearly identify the modulation effect of QCM on turbulence energy, the cross spectrum is analyzed between the fluctuations of unfiltered $\tilde{E}_r$ and $\tilde{n}_{e,AT}^{\text{env}}$. Figure 8 shows the corresponding experimental results where significant peaks around the QCM frequency, ranging from 20 kHz to 60 kHz, are discovered in both the cross power and coherence. This indicates a strong correlation between the modulation of the turbulence envelopes and the QCM, as presented in figures 8(a) and (b), respectively. Moreover, the cross phase at the QCM frequency is approximately $-\pi/4$, as shown in figure 8(c). The negative sign across the QCM frequency range suggests that AT energy delays the $\tilde{E}_r$, in other words, the latter
<figure><img src="image_10.png" /><figcaption>**Figure 7.** (a) Time traces of radial electric field fluctuations $\tilde{E}_{r,QCM}$ bandpass filtered in the QCM frequency range of 20–60 kHz and the envelope of AT ($\tilde{n}_{e,AT}^{\text{env}}$) in frequency of $f = 200–600$ kHz. (b) The cross-correlation function between fluctuations $\tilde{E}_{r,QCM}$(20–60 kHz) and $\tilde{n}_{e,AT}^{\text{env}}$(200–600 kHz).</figcaption></figure>

modulates the former. Hence, it is likely that the modulation of ambient fluctuations is induced by the oscillation of the QCMs through both phase modulation (−π/2 radian) and amplitude modulation (π radian) simultaneously.

An analysis of the turbulence properties in terms of its decorrelation time has been done in order to further study the effects of QCM on transport in detail. The level of turbulent density fluctuation can be characterized by the root mean square (RMS) value, denoted as $\tilde{n}_e^{\text{RMS}}$. It is then normalized by the density fluctuation ratio $|\tilde{n}_e/\tilde{n}_e| = \tilde{n}_{e,QCM}^{\text{RMS}}(20–60 \text{ kHz})/\tilde{n}_e^{\text{RMS}}(1–600 \text{ kHz})$, where $\tilde{n}_{e,QCM}^{\text{RMS}}(20–60 \text{ kHz})$ denotes fluctuations in the frequency range of QCM and $\tilde{n}_e^{\text{RMS}}(1–600 \text{ kHz})$ represents broadband turbulence fluctuation amplitude. Figure 9(c) shows the radial profiles of the QCM, specifically data from both BES and DBS systems available in 874~877 ms. It reveals that the QCM is located in the region of $0.85 < \rho < 0.98$ with a clear peak just in the pedestal region, as can be inferred from figure 3(e). The excitation of QCM may be related to the steep pedestal gradients of pressures and currents. Figure 9(a) plots the distribution of correlation time $\tau_c$ of density fluctuations obtained from BES system, defined as the e-folding time of the auto-correlation function. The gray line shows the normal broadband turbulence $\tau_c$ profile without inter-ELM QCMs selected around the time $t \sim 884.5$ ms, while red line represents the $\tau_c$ with

<figure><img src="image_11.png" /><figcaption>**Figure 8.** Cross spectrum between $\tilde{E}_r$ and $\tilde{n}_{e,AT}^{\text{env}}$(200–600 kHz). (a) Cross power, (b) cross coherence and (c) cross phase.</figcaption></figure>

QCM at time slice $t \sim 876.5$ ms. The peak $\tau_c$ also appears in the QCM region, which further confirms that the turbulence is significantly enhanced because of the increased correlation time from typically $\langle\tau_c\rangle \approx 10 \ \mu\text{s}$ in HL-2A edge plasmas to $\langle\tau_c\rangle \approx 40 \ \mu\text{s}$ when QCMs are excited. As a consequence, the transport becomes stronger due to the enhanced turbulence. Furthermore, figure 9(b) shows the profiles of correlation times of both the QCM frequency range $\tau_{c,QCM}$(20–60 kHz) and AT frequency range $\tau_{c,AT}$(200–600 kHz) in the presence of QCM. In the QCM locations, $\tau_{c,QCM}$ becomes much larger, while $\tau_{c,AT}$ becomes much smaller. This suggests that the energy source of enhanced turbulence with QCM may originate from AT through the nonlinear coupling, which is also verified by previous bispectrum analysis.

According to the previous results, the QCMs will increase transport across the pedestal. In fact, we should note that the turbulent transport could also be significantly reduced due to equilibrium sheared flow [33]. This reduction is often localized at a specific region such as transport barriers, especially in the pedestal region in magnetically confined fusion plasmas where large flow shear plays a crucial role. The enhanced mean $E \times B$ flow shear in the pedestal suppresses the turbulence and transport through decorrelation effects on turbulence eddies [35]. Such diagram is also examined by calculating the flow shearing rate $\omega_{E \times B}$ and turbulent decorrelation rate $\tau_c^{-1}$ in the QCM region, as illustrated in figure 10. The shearing rate $\omega_{E \times B}$ is derived from multichannel velocity measurements obtained from DBS using $\omega_{E \times B} \equiv \partial v_{E \times B}/\partial r \approx \partial v_{\perp}/\partial r$, where $v_{E \times B} \approx v_{\perp}$ is derived under the indication from both numerical calculations and experiments that $E \times B$ velocity dominates over the velocity in the plasma frame at the plasma edge [47], while the decorrelation frequency $\tau_c^{-1} = 1/\tau_c$ is calculated based on
<figure><img src="image_12.png" /><figcaption>**Figure 9.** (a) The profiles of correlation time $\tau_c$ of density fluctuations with and without QCM. (b) The correlation time of QCM $\tau_{c,QCM}$ and AT $\tau_{c,AT}$ during the QCM period. (c) Radial profiles of the turbulent density fluctuations of QCM ($\tilde{n}_{e,QCM}^{\text{RMS}}/\tilde{n}_e^{\text{RMS}}$) measured by BES and DBS systems.</figcaption></figure>

BES density fluctuations. Figure 10(a) shows the contrast of the flow shearing rate with QCM and without QCM. It indicates that the $\omega_{E \times B}$ in the QCM region is significantly greater when QCM is present, which is partially related to the outward movement of the pedestal. Figure 10(b) compares the decorrelation frequency $\tau_c^{-1}$ with or without QCM, demonstrating that turbulence decorrelation frequency decreases to a much lower level when QCM exists. The relative stabilization effect of the flow shear on turbulence eddies [48, 49] can be expressed by the ratio of shearing rate to turbulence decorrelation rate $|\omega_{E \times B}|/\tau_c^{-1}$, which is illustrated in figure 10(c). The $|\omega_{E \times B}|/\tau_c^{-1}$ in QCM region is several times larger in the case of QCM compared to without it, implying that the turbulence is also strongly influenced by the flow. It also suggests that the flow shear $|\omega_{E \times B}| > \tau_c^{-1}$ has already inhibited partial growth of coherent fluctuation that causes large transport; otherwise, radial transport would be stronger, resulting in poorer confinement. The enhancement of flow shear is essential for understanding the increased pedestal pressure in the presence of QCM, since the QCM induces large transport as indicated before. Thus, it is obvious that the pedestal dynamics is a result of the complex nonlinear interactions among coherent modes, AT and sheared flows, i.e. the QCMs induced by the nonlinear inverse cascade of the high frequency turbulence may serve as a moderator which prevents the fast increase of the pedestal pressure that helps to put off the ELM burst, although the critical pressure gradient is still reached finally because of the large flow shear. Nevertheless, a clear theoretical interpretation of the physical processes is not available

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Comparison of turbulence and flow with and without QCM. (a) Flow shearing rate $\omega_{E \times B}$ calculated from DBS, (b) turbulent decorrelation rate $\tau_c^{-1}$ from BES measurements and (c) their ratio $|\omega_{E \times B}|/\tau_c^{-1}$. The location of QCM is denoted by the shadow region.</figcaption></figure>

at present, making it an important topic that requires further attention in future researches.

## 4. Summary and discussion

In summary, a succession of QCMs highly localized within the pedestal region during the inter-ELM periods are observed on HL-2A H-mode plasmas with auxiliary heating power of $P_{\text{NBI}} \sim 650$ kW and $P_{\text{LHCD}} \sim 300$ kW. In this work, experimental investigations have been carried out on the spectrum features of inter-ELM QCMs and their interactions between pedestal turbulence and flows. The characteristics of QCMs can be summarized as follows:

1. The QCMs have central frequency varies from 20 kHz to 60 kHz and are mainly observed in density fluctuations, implying that they are in electrostatic nature. The pedestal density gradient saturates as long as the QCMs are excited which results in an increased stiffness level of density profiles in the pedestal region till the ELM burst. An outward shift of the pedestal location as well as a higher average electron density across the whole region are obtained in the presence of QCM. Analyses have shown that the modes rotate in the electron diamagnetic drift direction while propagates radially outward whose typical
averaged poloidal and radial wavenumbers are around $k_\theta \sim 1.20 \text{ cm}^{-1}$ and $k_r \sim -0.24 \text{ cm}^{-1}$, respectively.

2. In addition, it is shown that the radial and poloidal correlation lengths of AT, based on BES density fluctuations, increase to $L_{cr} \sim 2.8 \text{ cm}$ and $L_{c\theta} \sim 2.5 \text{ cm}$ in the presence of QCM which are 1–2 times larger than that in other H-mode plasmas without it, indicating that such quasi-coherent fluctuation is important in driving the turbulent transport in the pedestal region. A strong nonlinear coupling between QCMs and the background turbulence is identified, showing that the former has both amplitude and phase modulation effects on the latter which can also be inferred from the cross correlation between the turbulence energy and radial electrical field fluctuation of the QCMs, where the latter leads the former by several microseconds.

3. Despite the turbulence correlation time at QCM location increases, the significant enhancement of flow shearing rate is suggested to be responsible for the transport reduction thus a continuous increase in the pedestal pressure will be achieved until the critical pressure gradient for ELMs is reached. These findings emphasize the significance of understanding and controlling the QCM to improve tokamak performance and ELM bursts.

## Acknowledgments

The authors would like to express heartfelt gratitude to the entire HL-2A/HL-3 team for the invaluable support during the experiments. This work is supported by the National Key R&D Program of China (Grant Nos. 2019YFE03020004, 2022YFE03060003 and 2022YFE03060002), National Natural Science Foundation of China (Grant No. 12175055 and 12275096), Sichuan Science and Technology Program (Grant Nos. 2023ZYD0014 and 2021JDRC0010), and Innovation Program of Southwestern Institute of Physics (Grant No. 202301XWCX001). It is also partially supported within the framework of the cooperation between the French Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA) and the China National Nuclear Corporation (CNNC).

## ORCID iDs

J. Wen https://orcid.org/0000-0001-6645-3641

R. Ke https://orcid.org/0000-0003-0117-0098

J.Q. Xu https://orcid.org/0000-0003-4834-3005

W. Chen https://orcid.org/0000-0002-9382-6295

N. Wu https://orcid.org/0000-0001-7949-5330

X.X. He https://orcid.org/0000-0002-1088-4845

P.W. Shi https://orcid.org/0000-0002-2071-7741

J.M. Gao https://orcid.org/0000-0002-9546-5139

M.K. Han https://orcid.org/0000-0001-8735-8440

Y. Shen https://orcid.org/0000-0001-5676-0082

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