Dynamics of the pedestal in the recovery phase in EAST type-I ELM plasmas

In the Experimental Advanced Superconducting Tokamak high-confinement mode plasmas, a low-frequency n = 1 magnetic coherent mode with frequency f = 20–50 kHz and a high-frequency mode with electromagnetic characteristics (HFEM, f ∼ 280 kHz) have been found between type-I edge localized modes (ELMs). Both the HFEM and the n = 1 mode are located in the pedestal region, but their radial locations seem to be somewhat different. It seems from the present data that the HFEM is closer to the maximum density gradient region, while the n = 1 mode may be closer to the separatrix. The experimental results demonstrate that the electron temperature recovers more rapidly than the pedestal density, and the n = 1 mode is excited in the pedestal after an ELM collapse. With the increase in the pedestal density, the HFEM appears and becomes dominant, while the amplitude of the n = 1 mode decreases significantly. The observations indicate that the HFEM may suppress the amplitude of the n = 1 mode. In the pre-ELM phase, the pedestal electron density and temperature are saturated, the characteristics of the HFEM show a significant change (a much broader frequency spectrum and reduced mode amplitude), and the n = 1 mode recovers again. Analysis using a wavelet bispectrum reveals that a nonlinear coupling between the n = 1 mode and the high-frequency magnetic fluctuations exists in the pre-ELM phase. The relations between the nonlinear mode coupling, the reappearance of the n = 1 mode and the ELM crash are discussed.


Introduction
The high-confinement mode (H-mode) [1] is characterized by spontaneous formation of edge transport barriers with steep temperature and density gradients, referred to as the * Authors to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 'pedestal', which plays a prominent role in the realization of high-performance plasmas [2]. However, energy accumulation in the narrow pedestal results in an explosive collapse of the gradients by the edge localized mode (ELM) [3][4][5], and the gradients will increase again under some physical mechanisms until the onset of the next ELM. The pedestal collapse and rebuilding in edge plasmas are very general phenomena, which are important for sustainment of H-mode plasmas. Therefore, understanding the pedestal dynamics in the recovery phase between ELMs is important for the development of fusion energy.
In recent years, the pedestal coherent fluctuations or quasicoherent modes (QCMs) have attracted much attention, and experimental studies have focused on a variety of coherent modes at the pedestal in the H-mode phase [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In DIII-D, a high-frequency coherent mode has been observed in the QH-mode pedestal region [6], which has the characteristics predicted for the kinetic ballooning mode. In the Alcator C-Mod tokamak, an electromagnetic mode of around 300 kHz was found in the steep gradient region, which could limit the growth of the pedestal temperature and enhance the edge transport [7,8]. In AUG, a saturated high-frequency magnetic mode was observed in the final phase of the ELM cycle and it appeared in the steep gradient region of the pedestal [9]. Similar observations were made in DIII-D [10]. In the edge region, the gradients of electron temperature and electron density are usually observed to be saturated late in the ELM cycle. In Experimental Advanced Superconducting Tokamak (EAST), an electrostatic coherent mode (ECM) with frequency of 20-90 kHz has been obtained in the enhanced D α H-mode, and experimental results demonstrated that the ECM provides a channel for maintenance of long-pulse Hmode plasmas, so that particles and heat can be continuously exhausted across the pedestal [11]. In HL-2A H-mode plasmas, inward particle flux driven by a low-frequency coherent mode ( f= 20−100 kHz) has been observed during the formation of transport barriers [12]. The QCM has also been observed in Wendelstein 7-X. These experimental results show that the frequency and amplitude of QCMs can be affected by the plasma parameters and edge magnetic topology [13]. In addition, recent results from DIII-D show that the nonlinear coupling of pedestal modes could be a possible mechanism for ELM triggering [21]. Therefore, more studies are needed on the effect of pedestal modes on pedestal evolution, which is our focus in this paper.
The paper will present the experimental detection of a strong low-frequency n = 1 coherent mode with frequency f = 20-50 kHz and a high-frequency electromagnetic mode (HFEM) at a frequency of 270-300 kHz during the formation of edge transport barriers of the type-I ELM in EAST H-mode plasmas. The paper is organized as follows: section 2 is the experimental setup, section 3 presents the experimental results, including the dynamic evolution of the pedestal during the inter-ELM phase, the characteristics of the n = 1 mode and the HFEM, and the relation between the n = 1 mode and the crash of the type-I ELM. Finally, section 4 is a summary and discussion.

Experimental setup
EAST is a medium-sized superconducting tokamak (R 0 = 1.85 m, a = 0.45 m). The plasma parameters in this experiment are as follows: plasma current I P = 500 kA, toroidal magnetic field B T = 1.6 T, line-averaged density 2.5 × 10 19 m −3 , heating power of the lower hybrid wave P LHW ≈ 2.1 MW, neutral beam heating power P NBI ≈ 3.5 MW, safety factor q 95 ≈ 3.6 and the configuration is upper single null. Figure 1 displays a poloidal view of the principal diagnostics for the analysis, including an O-mode polarized poloidal correlation reflectometer (PCR) [22], a fast-sweeping density profile reflectometer (DPR) [23][24][25], magnetic probes (MPs) and a multi-channel midplane electron cyclotron emission (ECE) diagnostic [26]. The PCR was mounted on the lowfield side (LFS) midplane to measure the density fluctuations (ñ e ) and it had four fixed probing frequencies (20.4, 24.8, 33, 40 GHz) corresponding to different cutoff densities n cut = [0.5,0.76,1.35,2] ×10 19 m −3 . At the same radial position, two antennas with a poloidal separation of 6 cm were arranged to receive the reflected waves. The PCR sampling frequency is 2 MHz ( f s = 2 MHz). The DPR is approximately 3 cm above the midplane. In addition, the poloidal magnetic fluctuation (B θ ) was measured by the MPs at the vacuum wall. There are two sets of high-frequency arrays ( f s = 2 MHz) separated by 22.5 • toroidally and two sets of low-frequency arrays ( f s = 200 kHz) separated by 180 • toroidally on the vacuum wall. Each set of high-and lowfrequency arrays contains three and 24 MPs on the same poloidal cross-section, that is, the high-frequency MPs contain 2(toroidal) × 3(poloidal) positions and the low-frequency MPs contain 2(toroidal) × 24(poloidal) positions, respectively. The multi-channel ECE diagnostic at the midplane is used to measure the electron temperature with a sampling frequency of up to 1 MHz. In this study, we choose the hrs04h channel as the pedestal top temperature (T ped e ) and the radial location in the poloidal magnetic flux coordinate (ψ N ) is ψ N ∼ 0.9.  . Here, the in-phase (I) and quadrature (Q) signals are produced from an I/Q detection of reflectometry. The wave amplitude (a) and phase (ϕ ) can be directly obtained from the complex signal, S fluc = a iϕ t . Generally, both the amplitude and phase contain the information of density fluctuation, while the relation between a (or ϕ ) and density fluctuation is usually very complicated [27]. The complex signal is usually analyzed in order to obtain the characteristics of density fluctuation, e.g. in [28,29].

Experimental results
It is observed that a high-frequency QCM ( f = 270−300 kHz) appears in all spectra of the electron density fluctuations (ñ e ) and magnetic fluctuations after the lowfrequency mode. Figure 2(d) shows the coherence spectrum betweenñ e andB θ , which indicates that there is a significant coherence around the frequency of the HFEM. In addition, we found that the appearance of the HFEM can suppress the amplitude of the n = 1 mode, while it can persist if there is no HFEM during the inter-ELM phase, as shown by the shaded areas in figure 2. The left red color shaded area has an n = 1 mode and also a signal in the vicinity of the HFEM. However, the intensity near 300 kHz is weaker than in some inter-ELM periods (black color shading). These phenomena imply that there is a competition between the two modes and that there is a general tendency for the n = 1 amplitude to be strong when the HFEM amplitude is weak and vice versa. The specific results will be discussed in the next section. Figure 3 displays the time evolutions of parameters in H-mode discharge #80667 during one inter-ELM phase. From top to bottom in the picture are the D α emission, the spectrum of B θ , the spectrum of density fluctuations from the PCR channel with S fluc = 33 GHz, the comparisons of the integrated spectral power over the different frequency bands of density and magnetic fluctuations, and the density and electron temperature evolutions in the pedestal top. Here, the pedestal top density (n ped e ) was calculated directly through the method of fitting the density profiles measured by DPR [30], the pedestal top temperature (T ped e ) at ψ N ∼ 0.9 measured by ECE and the position is close to the pedestal top of the density profile. During the inter-ELM phase, it can be clearly seen that there is a lowfrequency n = 1 coherent mode with frequency f = 20-50 kHz and a high-frequency QCM of 300 kHz alternately appearing in magnetic fluctuations before each onset of an ELM in the series, and the pedestal density also has different growth rates in the three phases as displayed in figure 3(e). In panel (e), the red dashed lines represent the rate of rise of the pedestal density.

The dynamics of the pedestal in the inter-ELM phase
For a more detailed study of the behaviors of the n = 1 mode and HFEM and a plausible link to the dynamics of the pedestal in the recovery phase, the time sequence in the inter-ELM phase may be divided into three phases as denoted by the shaded areas with different colors in figure 3. Phase 1, just after the ELM crash, is characterized by the faster recovery of the pedestal electron temperature than the pedestal density. The T ped e recovers faster than the n ped e for the 1 ms between 4157 and 4158 ms and then it slows dramatically. It is interesting here that there may be a threshold in the electron temperature above which the n = 1 mode amplitude increases dramatically. Figure 4 plots the relation between the amplitude of poloidal magnetic fluctuation from the n = 1 mode and the pedestal temperature in phase 1 from 4158 ms to 4160 ms. The results show that an abrupt increase in magnetic fluctuation amplitude is observed when T ped e increases above approximately 0.497 keV. The observation suggests that the pedestal electron temperature may be driving the instability that causes the n = 1 mode. And the n = 1 mode appears only in magnetic fluctuation, not in density fluctuation. Once the mode is excited, the pedestal temperature almost saturates (stops increasing) until the next ELM, which indicates that the fluctuations might be driving electron thermal transport and stopping the rise of T e . Phase 2 of figure 3 is dominated by the HFEM detected in both density and magnetic fluctuations. On the other hand, the high-frequency magnetic fluctuation (300-500 kHz) is stimulated as well in the pedestal recovery phase. A comparison of the HFEM magnitude estimated with the integrated spectral power of density fluctuation in 270-300 kHz and the pedestal density shows that the rate of rise of the pedestal density reduces as the HFEM amplitude abruptly increases at the transition from phase 1 to phase 2, as demonstrated in figures 3(d) and (e). It seems that the HFEM has the effect of reducing the pedestal density growth. However, there is also the possibility that the pedestal parameter in phase 2 only facilitates the emergence of the HFEM. Phase 3 of figure 3 is named the pre-ELM phase, which could routinely last about 2-3 ms, and during this phase the pedestal electron temperature and density remain invariable. It is observed that in the pre-ELM phase the characteristics of the HFEM seem to change significantly. The HFEM has a very coherent peak in phase 2, while it shows a broadband mode in phase 3 and the fluctuation amplitude shows a reduction, especially for the density fluctuation. Meanwhile, the n = 1 mode recovers again during the pre-ELM phase before the next ELM crash. Figure 5 shows the auto-power spectra of theB θ andñ e in different stages of the inter-ELM phase. From these figures we can see that the n = 1 mode can be observed clearly in the first and third phases, but in the second phase the mode is very weak. Significantly, the n = 1 mode is only observed in magnetic fluctuations but is weak or not present in density fluctuations. In contrast, the HFEM appears mainly in phase 2. Note that although the value of the power spectrum density is still strong at the frequency of the HFEM in the magnetic spectrum in phase 3 (red color), the amplitude of the HFEM in the density fluctuation spectrum is obviously reduced and there is no obvious peak, as shown in the inset of figure 5(b). Therefore, it can be considered that the HFEM mainly occurs in phase 2. As can be seen in figures 3(b)-(d) and 5, there exists a relation of 'as one falls, another rises' between the n = 1 mode and the HFEM in the inter-ELM phase. The n = 1 mode sharply increased just prior to the ELM crash, which implies a plausible relationship between the magnetic fluctuation and the onset of the ELM, which will be further discussed in section 3.4.

The characteristics of the n = 1 mode and HFEM in the inter-ELM phase
The poloidal and toroidal structures of the low-frequency mode can be obtained through coherence analysis of the MPs. Here, the coherence is defined as γ = P xy /(P xx P yy ) 0.5 , where P xy represents the Fourier cross-power spectrum between two time series signals x (t) and y (t), and P xx (P yy ) is the Fourier auto-power spectrum of x (t) or y (t). The cross-phase (∆∅) between x (t) and y (t) signals can be calculated from P xy = |P xy | e i∆∅ and the toroidal mode number (n) is equal to the ratio between the cross-phase and the toroidal separation angle between MPs. Figure 6 plots the coherence spectra of theB θ measured by MPs with a toroidal separation angle of 180 • and 22.5 • , respectively. The results indicate that there is an obvious coherence value between 20 kHz and 30 kHz up to 0.9, which is the frequency range of the low-frequency mode. As shown in figure 6(c), the toroidal mode number of the low-frequency mode is n = 1 when the MPs have a toroidal separation of 180 • , and the same result is also obtained with the two highfrequency MPs with toroidal angle equal to 22.5 • , as depicted in figures 6(d)-( f ). Therefore, the toroidal mode number of the low-frequency mode can be determined to be n = 1.
To confirm the poloidal structure, we use the filtered signals ( f = 20-35 kHz) ofB θ with different poloidal positions from low-frequency MPs to estimate the cross-correlation function (CCF) [31]. The reference signal is CMP13T, just a little above the midplane. Figures 7(a) and (b) show respectively the results at the LFS and high-field side (HFS). The results imply that the poloidal mode number of the n = 1 mode is m = 10-15. It is noted that the q 95 value of this discharge is about 3.6. Therefore, the n = 1 mode with m = 10-15 seems to be at the very edge. This seems to be consistent with the radial distribution measured by PCR shown later. Based on the above analysis, the observed n = 1 mode is a boundary instability possibly driven by the electron temperature or electron temperature gradient, propagating in the electron diamagnetic direction in the laboratory frame, and the toroidal and poloidal mode numbers are n = 1, m = 10-15 respectively. It seems that this mode has similar magnetic coherent mode characteristics to those previously observed on EAST [32].
Similarly, the multi-channel density fluctuation measurement from the PCR diagnostic introduced in section 2 is applied to study the characteristics of the HFEM.   In order to gain a deeper insight into the radial location and characteristics of the n = 1 mode and the HFEM, figure 9 illustrates the correlation analysis between the magnetic fluctuation (B θ ) and density fluctuation (S fluc ) or density fluctuation amplitude (A fluc ). Here, the density fluctuation S fluc = I + iQ is measured by PCR, and the density fluctuation amplitude is calculated by the filter signals of I and Q with It is shown that the n = 1 mode can be observed in the spectrum of A fluc measured by PCR and has a significant coherence withB θ . Although the density fluctuation (S fluc ) by PCR cannot observe the n = 1 mode (figure 2(c)), the mode can be seen in A fluc , which could be attributed to disturbance of the reflectometry cutoff surface or flow induced by the n = 1 mode [33]. We will use the A fluc to study the radial distribution of the n = 1 mode. Figures 9(d)-( f ) display the density and density gradient profiles, the radial distributions of coherence and the cross-phases between the magnetic signals (B θ ) and the density fluctuation (S fluc ) or fluctuation amplitude (A fluc ) at different radial positions for the n = 1 mode or the HFEM. Here, the squares (blue color) represent the correlation analysis between A fluc andB θ for the n = 1 mode, the circles (red color) represent the correlation analysis between S fluc andB θ for the HFEM, and the crosses represent the error bars. From the radial distributions shown in figure 9(e), it is observed that both the HFEM and the n = 1 mode are located in the pedestal region, but their radial locations seem to be somewhat different. It seems from the present data that the HFEM is closer to the maximum density gradient region (ψ N ∼ 0.95) while the n = 1 mode may be closer to the separatrix (ψ N = 1). However, this point should be further confirmed by diagnostics with finer spatial resolution. In addition, the cross-phase between A fluc andB θ for the n = 1 mode is almost the same at different radial positions, which excludes the possibility that the n = 1 mode is a tearing mode, and so the n = 1 mode is likely to be a kink mode.

The generation of the n =1 mode in phase 3
In order to get a better understanding of the generation of the n = 1 mode and its influence in phase 3, the nonlinear coupling between three waves in a very short time is analyzed by the wavelet bicoherence, which combines the wavelet and bispectral analyses (Fourier bicoherence is more suitable for long time series), and is defined as [34,35]: The normalized squared bicoherence is defined as: ] .
The normalized squared bicoherence gives a method to measure the degree of three-wave coupling at frequencies f 1 , f 2 and f, with the frequency f meeting the summed rule of , t) is the wavelet coefficient, and t and T are the time and the finite time interval. In the analysis, we use the Cmor wavelet function as the mother function, the time length of the experimental data is only 2 ms, and the frequency resolution is 2 MHz. The Cmor is a basic function of wavelet transform, which has good resolution in both time and frequency. Figures 10(a) and 11 show the wavelet bicoherence spectra of magnetic fluctuations and the characteristics of threewave coupling in different phases of the inter-ELM cycle. As plotted in figure 10(a), the x-and y-axes are the frequencies f 1 and f 2 , and the color bar represents the intensity of the three-wave coupling in phase 3, which reveals that the bicoherence value can be up to 0.4 when f = f 1 + f 2 ≈ 25 kHz or f 2 = ±25 kHz, which is much higher than the values at other frequencies. The summed squared bicoherence shown in figure 10(b) also has a distinct peak at f ≈ 25 kHz. Therefore, strong nonlinear coupling is clearly observed in the frequency range f 1 + f 2 = 25 kHz and 400 kHz < f 2 < 800 kHz, which implies significant three-wave interaction between the n = 1 mode and the high-frequency magnetic turbulence. In addition, wavelet-bispectrum analysis was also applied to phase 1 and phase 2 and no nonlinear interaction was found, as shown in figures 11(a) and (b), respectively. Meanwhile, according to the integrated spectral power over the different frequency bands of density and magnetic fluctuations in figure 3(d), the n = 1 mode amplitude abruptly increases (green color) and the HFEM amplitude decreases (red color) at the transition from phase 2 to phase 3. Therefore, combined with the results of bicoherence analysis and fluctuation amplitude change as shown in figures 3(d) and 10, a possible mechanism of the reappearance of the n = 1 mode in phase 3 could be due to its nonlinear coupling with the high-frequency magnetic turbulence. However, it is impossible to confirm the causal relation and energy flow direction just from the bicoherence analysis.

Relation between the n=1 mode and the type-I ELM crash
In order to study the influence of the n = 1 mode on the type-I ELM crash in this scenario, further analysis was performed. Figure 12 demonstrates the time evolutions of the D α , the pedestal top density (n ped e ), the electron temperature at the top of the pedestal, and the poloidal magnetic fluctuation in the filtered frequency range of 20-35 kHz. It is clear that in phase 3 (pink shaded portion), the electron temperature and density reach saturation as shown in panels (b) and (c). Meanwhile, a more powerful proof is that the evolution of the density profile remained essentially unchanged within 2.5 ms before the ELM burst, measured by the profile reflectometry diagnostic system every 50 µs as demonstrated in figure 12(e). That is to say, the pedestal pressure remains constant at this stage. It should  be noted that the electron temperature (T ped e ) is measured by ECE, but due to the optical thin effect, the more external ECE channel cannot be used to represent the T e . Therefore, it is impossible to provide an accurate temperature profile of the pedestal. More importantly, the poloidal magnetic fluctuation of 20-35 kHz increases sharply in phase 3, which is consistent with the frequency of the n = 1 mode. It is plotted in figure 12(d).
The general theory for the ELM instability is that they are coupled edge peeling-ballooning modes. There are also experimental observations that a causal relationship between the onset of the n = 1 mode and the ELM could exist [36]. In our present case, the n = 1 mode in the pre-ELM phase could be a kink mode very close to the separatrix [37]. It is not yet known from the present data analysis whether the ELMrelated fluctuation here is dominated by the n = 1 mode. In addition, it is supposed that the sharp increase of the n = 1 mode amplitude in the pre-ELM phase is due to the nonlinear interaction with the background high-frequency magnetic fluctuations. It has been observed in DIII-D that the nonlinear coupling of pedestal modes and the radial distortions pushing out of the pedestal is a possible mechanism for triggering the low-frequency ELMs [21]. Although the present paper focuses on the dynamics of pedestal recovery in the inter-ELM phase, it is interesting to further study the relation between the nonlinear coupling of the pedestal modes, the n = 1 mode and the ELM crash in the future.

Summary and discussion
In conclusion, this paper presents the experimental observation of the dynamics of pedestals in the recovery phase in EAST type-I ELMy H-mode plasmas. The experimental results indicate that a strong low-frequency n = 1 magnetic coherent mode with frequency f = 20-50 kHz and an HFEM at a frequency of 270-300 kHz are observed during the formation of transport barriers. Both of the modes are localized in the edge pedestal region, while their radial locations seem to be slightly different. The HFEM is more likely to be closer to the maximum density gradient region, while the n = 1 mode may be closer to the separatrix. A diagnostic with finer spatial resolution is needed to confirm this point in the future. Some characteristics of the HFEM are similar to the highfrequency quasi-coherent fluctuations observed in the Alcator C-Mod tokamak, which are consistent with expectations for kinetic ballooning modes [7]. Moreover, some features of the HFEM are also similar to the MTM observed in DIII-D [38] and JET [39]. However, the nature of the HFEM observed in our experiment is still unclear, and more experiments and simulations are needed to confirm the physical mechanisms of the mode.
After the ELM collapse, the electron temperature recovered rapidly and the initial rapid rise in pedestal electron temperature ceases at the onset of n = 1 magnetic fluctuation. This observation may indicate that the electron temperature drives the n = 1 mode. In addition, it clearly shows that the amplitude of the n = 1 mode decreases significantly in the transition from phase 1 to phase 2, while the HFEM appears and becomes dominant. The observations indicate that there is a general tendency for the n = 1 amplitude to be weak when the HFEM amplitude is strong, that is, the HFEM may suppress the amplitude of the n = 1 mode.
In phase 3, i.e. the pre-ELM phase, which could routinely last about 2-3 ms, the pedestal electron temperature, density and electron pressure remain invariable. It is also found that in this phase the characteristics of the HFEM show a significant change (a much broader frequency spectrum and reduced mode amplitude) and the n = 1 mode recovers again. By using bispectral analyses based on wavelets, it is shown that a nonlinear coupling between the n = 1 mode and the highfrequency magnetic fluctuations exists in the pre-ELM phase. It is supposed that the reappearance of the n = 1 mode in this phase could be due to its nonlinear coupling with the highfrequency magnetic turbulence, while the causal relation and energy flow direction still need further study. Similar observations of low-frequency magnetic fluctuations lasting for several milliseconds prior to an ELM crash have been reported in JET [36]. In addition, a nonlinear coupling of pedestal modes just before an ELM has also been observed in DIII-D and it is suggested that the radial distortions pushing out of the pedestal is a possible mechanism for triggering low-frequency ELMs [21]. For our present work, it would be interesting to further study the relation between the nonlinear coupling of pedestal modes, the n = 1 mode and the ELM crash in the future.