Inversion techniques to obtain local rotation velocity and ion temperature profiles for the x-ray crystal spectrometer on EAST

1.   Introduction

In tokamak plasmas, anomalous transport is usually attributed to micro-instabilities, or microturbulence [1, 2]. Some well-known electron channel micro-instabilities include trapped electron mode [3–5] (TEM, $0.5<k_{\theta} \rho_{\mathrm{s}}<2$) and electron temperature gradient mode [6–8] (ETG, $k_{\theta} \rho_{\mathrm{s}}>2$), where $k_{\theta}$ is the poloidal wavenumber of turbulence, $\rho_{\mathrm{s}}=c_{\mathrm{s}} / \Omega_{\mathrm{i}}$ is the ion sound gyroradius, $c_{\mathrm{s}}=\sqrt{T_{\mathrm{e}} / m_{\mathrm{i}}}$ is the ion sound speed, and $\Omega_{\mathrm{i}}=\mathrm{e} B / m_{\mathrm{i}}$ is the ion-gyro frequency. In D-T burning plasmas, slowing-down alpha-particles heat electrons primarily [9], causing a predictable T_e/T_i > 1 and low collisionality region in plasma cores. These conditions tend to destabilize the TEM mode according to theories and gyrokinetic simulations, so the study of TEM is important for understanding the transport behaviors of burning plasmas. While TEM might be unstable in the inner core, the ion temperature gradient (ITG) mode may coexist with TEM, and the mixed ITG-TEM regime [10] is then expected elsewhere. Recently, multi-scale simulations showed that the high-k TEM and ETG may have a non-negligible impact on the electron heat transport [11–19], but these results are still uncertain, and direct experimental measurements of density and temperature fluctuations at the electron scale are needed to ultimately verify the impact of these micro-instabilities.

TEM can be driven by the electron density gradient or electron temperature gradient, and these modes driven by different free energy sources can have distinct features, such as different scales (wavenumbers) and linear growth rates. Ernst et al [20] showed a new diagram for electron modes, including TEM and ETG mode. That study identified a boundary at η_e = 1, separating short- and long-wavelength TEMs, where $\eta_{\mathrm{e}}=\left(R / L_{T_{\mathrm{e}}}\right) /\left(R / L_{n_{\mathrm{e}}}\right)=L_{n_{\mathrm{e}}} / L_{T_{\mathrm{e}}}$ is the ratio of the normalized ETG and density gradient. For $\eta_{\mathrm{e}}<\eta_{\mathrm{e}, \text { crit }}=1$, the TEM is mainly driven by the density gradient $R / L_{n_{\mathrm{e}}}$ (density gradient-driven TEM; DGTEM), and may have a maximum linear growth rate at a relatively large scale. For $\eta_{\mathrm{e}}>\eta_{\mathrm{e}, \text { crit }}=1$, ETG and TEM can be destabilized by the ETG. Zonal flows are the dominant turbulence saturation mechanism for $\eta_{\mathrm{e}}<1$ and become weak for $\eta_{\mathrm{e}}>1$. A recent study in DIII-D showed that the DGTEM turbulence dominates the inner core of H-mode plasmas with $T_{\mathrm{e}} \sim T_{\mathrm{i}}$, moderate density peaking and low collisionality [21]. In this study, the simulation results show that the critical density gradient to destabilize DGTEM, $a / L_{n_{\mathrm{e}} \text {, crit }}$, is sensitive to other parameters, such as $T_{\mathrm{e}} / T_{\mathrm{i}}, a / L_{T_{\mathrm{i}}}$, and $a / L_{T_{\mathrm{e}}}$. Thus, the $\eta_{\mathrm{e}, \text { crit }}=L_{n_{\mathrm{e}}, \text { crit }} / L_{T_{\mathrm{e}}}$ is not necessarily equal to unity, and the DGTEM can be activated when $\eta_{\mathrm{e}}$ is relatively low $\left(\eta_{\mathrm{e}}<\eta_{\mathrm{e}, \text { crit }} \sim 3, T_{\mathrm{e}} / T_{\mathrm{i}} \sim 1, a / L_{T_{\mathrm{i}}} \sim 1\right)$, so the DGTEM may become unstable for the ramp-up/ramp-down states of ITER plasmas. The DGTEM turbulence measurements are provided by a Doppler back-scattering system in DIII-D, and one of the main features of DGTEM is broadband fluctuations in the turbulence spectrogram. Here, the broadband (BB) means that the spectrum width can be hundreds of kHz, which is a widely accepted feature of turbulent fluctuations. Nevertheless, the direct experimental measurements of DGTEM turbulence are still rare and need more focus.

Collective Thomson scattering (CTS) is a powerful technique to detect short-scale fluctuations [22–25], and was successfully used for demonstrating density gradient stabilization of ETG turbulence in NSTX [26] and studying turbulence characteristics in TEXT [27] and Tore Supra [28]. The probing beam source of CTS could be a high-frequency microwave source (f ~ hundreds of GHz) or far-infrared lasers (f ~ THz). In EAST, a four-channel poloidal CTS diagnostic system using a $\mathrm{CO}_{2}$ laser has been employed to detect short-scale density fluctuations in different regions (core region, $\rho=0-0.4$; gradient region, $\rho=0.4-0.8$), while the poloidal wavenumbers are $k_{\theta}=12 \mathrm{~cm}^{-1}, 22 \mathrm{~cm}^{-1}$ in each region. Studies on turbulence, poloidal rotation velocity and confinement improvement have been carried out using signals monitored by this diagnostic system [29, 30].

Previous simulation results showed that TEM could be unstable in the core of EAST H-mode plasmas [31], and it is interesting to see whether the core turbulence measured by diagnostics shows the nature of TEM or not. In this paper, we report a BB mode observed by a CTS system in reproducible H-mode shots of the EAST tokamak. The experimental identification methods of BB mode in the turbulence spectrogram are explained, and the correlation between turbulence and energy confinement is studied. The rest of this paper is organized as follows. Section 2 describes the experimental setup and the associated diagnostics. Section 3 shows the characteristics of BB turbulence observed in different confinement regimes, including L–H/H–H/H–L transitions and H-mode. Two representative shots, #75014 and #75015, are chosen to show the BB mode dependence on the plasma parameters. It is found that η_e is a critical value for motivating BB turbulence at ρ = 0.4‒0.8, and the frequency of BB is strongly dependent on the electron channel parameters. In section 4, the negative correlation between energy confinement and the intensity of the BB turbulence during H-mode is presented, and a possible mechanism to describe the confinement degradation in these H-mode plasmas is discussed. In section 5, the discussion and conclusions are offered.

2.   Experimental setup and diagnostics

All the experiments mentioned in this paper are performed in EAST, which is a fully superconducting tokamak with a noncircular cross-section. The main parameters of EAST are as follows: major radius $R_{0} \sim 1.88 \mathrm{~m}$, minor radius $a \sim 0.45 \mathrm{~m}$, plasma current $I_{\mathrm{p}} \leqslant 1 \mathrm{MA}$, and toroidal magnetic field $B_{\mathrm{T}} \leqslant 3.5 \mathrm{~T}$. EAST is equipped with lower hybrid wave heating (LHW), ion cyclotron range of frequencies, electron cyclotron resonance heating and neutral beam injection systems. In this study, the electron-dominated heating can cause a high electron temperature $\left(1<T_{\mathrm{e}} / T_{\mathrm{i}}<5\right)$ in L-mode in the core region, while in $\mathrm{H}$-mode plasmas the electron and ion temperature is nearly equal $\left(T_{\mathrm{e}} \sim T_{\mathrm{i}}\right)$ in the region of interest $(\rho=0.4-0.8)$. The collisionality in the region of interest $(\rho=0.4-0.8)$ is relatively low $\left(\nu_{\mathrm{eff}}=0.1 n_{\mathrm{e}} Z_{\mathrm{eff}} R / T_{\mathrm{ek}}^{2}<5\right.$, where $n_{\mathrm{e}}$ is the electron density in $10^{19} \mathrm{~m}^{-3}, Z_{\text {eff }}$ is the effective ion charge, $R$ is the major radius in $\mathrm{m}$ and $T_{\mathrm{ek}}$ is the electron temperature in $\mathrm{keV}$), while in the pedestal region the collisionality is much higher, at $v_{\text {eff }}$ $>10$gt;10$.

The turbulence measurements are achieved by a four-channel poloidal CTS diagnostic system using a CO₂ laser, which is similar to that of the HT-7 tokamak [32, 33]. Figure 1 shows the layout of the system. The CO₂ laser is divided into a main beam (MB) and four local oscillation beams (LOs). The power of the MB is 10 W, while the power of the LOs is about 0.1‒0.25 mW. The wave vector/frequency of the MB and LOs are the same, marked as $\left(\overrightarrow{k_{0}}, \omega_{0}\right)$. The MB $\left(\overrightarrow{k_{0}}, \omega_{0}\right)$ passes through the plasma from top to the bottom, and overlaps with LO1, 2 in region B (outer region) and LO3, 4 in region A (core region) separately. The scattering beam $\left(\overrightarrow{k_{\mathrm{s}}}, \omega_{\mathrm{s}}\right)$ from the detecting region (A, B) mixes with the corresponding LO in the HgCdTe detector. The signals of the detectors are determined by

Figure  1.  Layout of the four-channel poloidal CO₂ laser collective scattering diagnostic system.

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where $i_{\mathrm{s}}$ is the current signal of the detector, and its intensity is proportional to the square of the electric field intensity of the laser $\left(\left.E(t)\right|^{2}\right) . A_{\mathrm{s}}$ and $A_{\mathrm{LO}}$ are the intensity of the scattering laser and LO, respectively. Some expanded terms of $|E(t)|^{2}$ are neglected because their frequency is beyond the upper response range of the detector [28].

The energy and momentum conservation requires a relationship between the wave vector/frequency of drift wave turbulence $(\vec{k}, \omega)$, scattering beam $\left(\overrightarrow{k_{\mathrm{s}}}, \omega_{\mathrm{s}}\right)$ and the $\mathrm{MB}$ (or incident beam, $\overrightarrow{k_{0}}, \omega_{0}$), which is

Notably, the wavenumbers of both the MB $\left(k_{0} \approx 5927 \mathrm{~cm}^{-1}\right)$ and scattering beam $\left(k_{\mathrm{s}}\right)$ are much larger than that of the drift wave turbulence in the region of interest $(k \sim 10$ $30 \mathrm{~cm}^{-1}$), which leads to $k \ll k_{0}, k_{\mathrm{s}} \approx k_{0}$, so the Brag condition is satisfied:

where $\theta_{\mathrm{s}}$ is the scattering angle, which is always smaller than $0.3^{\circ}$. In order to improve the spatial localization, the angle between LO1, 2 and the poloidal plane is chosen to be $\alpha=5^{\circ}$ to make a good instrumental selective function, as has been calculated in [34]. Note that in this paper the main focus is on H-mode plasmas, and the high density with relatively low temperature can lead to a very high collisionality in the pedestal region $(\rho=0.8-1)$. TEMs are not expected to be unstable in this highly collisional region, so the detecting region of channels using LO1 and LO2 is $\rho \sim 0.4-0.8$. The $\mathrm{CO}_{2}$ laser is kept in pure $\mathrm{TEM}_{00}$ mode with a beam width of $w \sim 1 \mathrm{~cm}$ in the waist, thus achieving wavenumber resolution $\Delta k=2 / w=2 \mathrm{~cm}^{-1}$

In this paper, the data of H-mode plasmas are from 2017 EAST experiments. The four-channel poloidal CTS diagnostics system provides turbulence measurements in the core $(\rho \sim 0-0.4)$ and gradient region $(\rho \sim 0.4-0.8)$ covering two wavenumbers $k_{\theta}=12 \mathrm{~cm}^{-1}, 22 \mathrm{~cm}^{-1}\left(1<k_{\theta} \rho_{\mathrm{s}}<5\right)$, and the corresponding information can be found in table 1. Confinement degradation mainly due to the electron heat loss is observed in over $30 \mathrm{H}$-mode shots, and the steady-state BB at high frequency (200-2000 kHz) can be seen in the turbulence spectrogram of CTS channel 1, which is $k_{\theta}=12 \mathrm{~cm}^{-1}$ in $\rho \sim 0.4-0.8$. This BB mode is not observed in the other three channels. The line-averaged density is provided by a three-channel HCN interferometer, and the density profiles are from 11-channel Faraday effect polarimetry interferometry (POINT) [35, 36]. The electron temperature is measured by Thomson scattering (TS) [37] diagnostics and electron cyclotron emission diagnostics (ECE). The electron temperature measured by ECE is imprecise in the outer region $(\rho>0.4)$ because of the fast electrons driven by LHW, but the core temperature is less affected.

Table  1.  Detected region and wavenumber of turbulence.

Region B: ρ = 0.4‒0.8	Region A: ρ = 0‒0.4
$k_{1}=12 \mathrm{~cm}^{-1}$	$k_{3}=12 \mathrm{~cm}^{-1}$
$k_{2}=22 \mathrm{~cm}^{-1}$	$k_{4}=22 \mathrm{~cm}^{-1}$

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3.   Experimental identification of the BB mode

3.1   Overview of the BB mode

The BB density fluctuations in the spectrogram are usually considered to be types of micro-instabilities, such as DGTEM [20] or ETG [27]. Appropriate plasma parameters $\left(T_{\mathrm{e}} / T_{\mathrm{i}}>1, \nu_{\text {eff }}<5\right)$ are essential for destabilizing this turbulence, which applies to the BB mode observed by the EAST CTS system. Figure 2 shows a typical H-mode discharge #75011, which is heated by 1 MW 4.6 GHz LHW from 3 to 6 s. The time range of the confinement regimes is shown in table 2.

Figure  2.  A typical H-mode shot #75011; the time ranges of different confinement regimes are noted by dotted lines. (a) Turbulence spectrogram of CTS channel $k_{\theta}=12 \mathrm{~cm}^{-1}, \rho=0.4-0.8$. (b) Turbulence spectrogram of CTS channel $2, k_{\theta}=22 \mathrm{~cm}^{-1}, \rho=$ $0.4-0.8$. The color bars of both (a) and (b) are in arbitrary units. (c) Spectrum power of turbulence of channel $1\left(S_{k 1}\right.$, blue) and channel 2 $\left(S_{k 2}\right.$, red $)$. (d) Eigenfrequency of channel $1\left(f_{\text {avel }}\right.$, blue $)$ and channel 2 $\left(f_{\text {ave} 2}\right.$, red). (e) Core electron temperature (blue) and core ion temperature (red). (f) Line-averaged density. (g) $\mathrm{D}_{\alpha}$ light. The light gray rectangle crossing (c) and (d) indicates the time range when the $S_{k 1}$ and $S_{k 2}$ change oppositely from 3.78 to $3.98 \mathrm{~s}$.

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Table  2.  Time ranges of confinement regimes for shot #75011.

L-mode	L–H	H-mode	H–H	H–L
3‒3.65 s 5.38‒6 s	3.65‒4.02 s	4.02‒4.41 s 4.66‒5.27 s	4.41‒4.66 s	5.27‒5.38 s

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Limit cycle oscillations can be found in figure 2(g) $\mathrm{D}_{\alpha}$ light during the transition state. The BB mode can be seen during L–H/H–H /H–L transitions and H-mode from the turbulence spectrogram $S_{k}(t, f)$ in figure 2(a). This mode is clearly shown in the f ~ 200‒2000 kHz range, which is different from the low-frequency mode, so a high-pass filter with a cut-off frequency of $f_{\text {cut }}=200\; \mathrm{kHz}$ is adopted when calculating the BB turbulence power $S_{k}$ and eigenfrequency (weight-averaged frequency) $f_{\text {ave}}$. Here, $S_{k}$ and $f_{\text {ave}}$ are defined as

The BB mode appears in CTS channel 1 $\left(k_{\theta}=12 \mathrm{~cm}^{-1}, \rho=0.4-0.8\right)$ during the L–H transition, and the spectrum power $S_{k}$ and eigenfrequency $f_{\text {ave}}$ of BB can be found in figures 2(c) and (d), shown by the blue solid lines. An interesting phenomenon is that the BB intensity grows during L–H transition, while the intensity of CTS channel 2 $\left(k_{\theta}=22 \mathrm{~cm}^{-1}, \rho=0.4-0.8\right)$ decreases. In this discharge, the $k_{\theta} \rho_{\mathrm{s}}$ of channel 1 and channel 2 are 1‒2 and 2‒3, respectively. The opposite change of turbulence in different wavenumbers during the L–H transition, as shown with $S_{k}$ in figure 2(c), excludes the possibility that the high-k turbulence (channel 2) may be the result of larger scale turbulence (channel 1) cascading to smaller scales [38]. The ETG mode is a possible candidate of turbulence at $k_{\theta}=22 \mathrm{~cm}^{-1}$, while the $\mathrm{BB}$ mode at $k_{\theta}=12 \mathrm{~cm}^{-1}$ may be explained by the TEM because it is sensitive to the increase in density in H-mode. A widely accepted approximation for the ETG threshold can be expressed as

where $\frac{R}{L_{T_{\mathrm{e}}}}=-\frac{R}{T_{\mathrm{e}}} \frac{\mathrm{d} T_{\mathrm{e}}}{\mathrm{d} r}$ is the normalized ETG, $\frac{R}{L_{n_{\mathrm{e}}}}=-\frac{R}{n_{\mathrm{e}}} \frac{\mathrm{d} n_{\mathrm{e}}}{\mathrm{d} r}$ is the normalized density gradient, $Z_{\text {eff }}$ is the effective ion charge, $\hat{s}$ is the magnetic shear and q is the safety factor. A, B and C are constants, which can be calculated through sets of gyrokinetic simulations, as done in [7]. From equation (5), when when $R / L_{n_{\mathrm{e}}}$ is the first threshold to be crossed, as is believed to be the case in this experiment, a predictable critical value of $\eta_{\mathrm{e}, \mathrm{ETG}}=C$ is expected, and the ETG mode would be linearly stabilized when $\eta_{\mathrm{e}}<\eta_{\mathrm{e} \text {,ETG }}=C$. Another important term is $T_{\mathrm{e}} / T_{\mathrm{i}}$, the ratio of the electron and ion temperature. For shot #75011, the electron temperature decreases during the H-mode and the electron heat loss is the dominant mechanism that is responsible for the confinement degradation, so $T_{\mathrm{e}} / T_{\mathrm{i}}$ decreases during the H-mode and reduces the first critical term $Z_{\mathrm{eff}} T_{\mathrm{e}} / T_{\mathrm{i}}$ in equation (5), and $T_{\mathrm{e}} / T_{\mathrm{i}}>1$ for shot #75011. As mentioned above, DGTEM is sensitive to the value of $\eta_{\mathrm{e}}$ for $\eta_{\mathrm{e}}<1$. In brief, when $\eta_{\mathrm{e}}<\eta_{\mathrm{e} \text {,crit }}$ can be satisfied, it is possible to see ETG stabilization and DGTEM destabilization at the same time, which is similar to that observed in figure 2(c) during the L–H transition. However, the cascading effect of larger scale turbulence inducing small ones seems to be dominant in H-mode where the intensity and frequency of $k_{\theta}=12 \mathrm{~cm}^{-1}$ and $22 \mathrm{~cm}^{-1}$ show the same trend (figures 2(c) and (d)). In the next subsection, precise profiles for more shots with BB observed by CTS channel 1 and related confinement degradations are analyzed, and the BB dependence on $\eta_{\mathrm{e}}$ is discussed.

3.2   Critical threshold of η_e for BB mode

Finding the parameter dependence of BB is essential to identify the turbulence type, and η_e, as discussed in subsection 3.1, plays a crucial role in destabilizing the TEM and ETG. Shot #75014 is selected to show that the BB is sensitive to the change of η_e in the region $\rho \sim 0.4-0.8$, which is the detecting region of CTS channels 1 and 2. Figure 3 shows the time traces of shot #75014; the core electron and ion temperature are provided by ECE and x-ray crystal spectrometer (XCS), respectively. Three time points are shown by the vertical solid lines with different colors, and the precise electron temperature profile measured by TS is available at each time point. In figure 3(e), the averaged density gradient $\left\langle R / L_{n_{\mathrm{e}}}\right\rangle$ calculated by POINT line-integrated density is presented, and the definition is

Figure  3.  Time traces of shot #75014; the time ranges of different confinement regimes are noted by the dotted lines. (a) Turbulence spectrogram of CTS channel 1, $k_{\theta}=12 \mathrm{~cm}^{-1}, \rho=0.4-0.8$. (b) Turbulence spectrogram of CTS channel $2, k_{\theta}=22 \mathrm{~cm}^{-1}$, $\rho=0.4-0.8$. (c) Spectrum power of turbulence of channel 1 $\left(S_{k 1}\right.$, blue) and channel 2 ($S_{k 2}$, red). (d) Eigenfrequency of channel 1 $\left(f_{\text {ave} 1}\right.$, blue $)$ and channel $2\left(f_{\text {ave} 2}\right.$, red $)$. (e) Averaged density gradient $\left\langle R / L_{n_{\mathrm{e}}}\right\rangle$. (f) Core electron temperature (blue) and core ion temperature (red). The light gray crossing (c), (d), (e) and (f) indicates the period in which the BB is slightly suppressed.

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where the 〈 〉 means that the value is calculated by line-integrated density, not the precise profile. The subscript 'i' is the channel of the POINT system, i = 1‒6 is chosen here, and channel 6 goes across the magnetic axis. It can be seen that the averaged density gradient decreases during the L–H transition in most regions except in the very core region $(\rho<0.3)$. However, at $r / a \sim 0.47$ (blue line in figure 3(e)), $\left\langle R / L_{n_{\mathrm{e}}}\right\rangle$ reaches its maximum during the L–H transition (close to 3.11 s). The BB mode also appears during the L–H transition with the increasing frequency, which is relevant to the change of temperature and density, and the relevant profiles can be found in figure 4.

Figure  4.  Profiles of the electron temperature from TS (a), density from POINT system (b) and ion temperature from XCS [39] (c) of shot #75014. The spline method is adopted to fit the raw data. The legend reports the time at which the measurements were taken, following the colors used in figure 3. The width of the pale yellow rectangles indicates the region of interest $(\rho=0.4-0.8)$.

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The normalized gradient profiles, $R / L_{T_{\mathrm{e}}}, R / L_{n_{\mathrm{e}}}$ and $R / L_{T_{\mathrm{i}}}$, decrease during the $\mathrm{L}-\mathrm{H}$ transition and stay at relatively low levels in the region of interest $(\rho \sim 0.4-0.8)$ in the H-mode, as shown in figures 5(a)‒(c). At 3.11 s, during the L–H transition, the normalized ETG $R / L_{T_{\mathrm{c}}}$ decreases largely, while the normalized electron density gradient $R / L_{n_{\mathrm{e}}}$ has only a small decline, which leads to the reduction in η_e, as shown in figure 5(d). In this process, the density gradient gradually becomes dominant, and the BB mode is activated when the local η_e is smaller than a linear threshold, $\eta_{\mathrm{e} \text {, crit }}$, as presented in figure 5(i). A possible value of $\eta_{e, \text { crit }} \sim[2.2,2.8]$ is suggested in figure 6, including the data collected in shots #75014 and #75015. Limited by the resolution and quantities of profiles and turbulence diagnostics, the precise value of $\eta_{\text {e.crit }}$ is not confirmed. The $k_{\theta} \rho_{\mathrm{s}}$ of BB is 1‒1.6 at 3.11 s and 1.21‒1.34 at 4.11 s (red and blue solid profiles in figure 5(e), respectively), which is close to the scale of TEM. For smaller-scale turbulence observed by CTS channel 2 $\left(k_{\theta}=22 \mathrm{~cm}^{-1}, \rho=0.4-0.8\right.$, $k_{\theta} \rho_{\mathrm{s}} \sim 1.9-3$ for the L–H transition), no clear evidence of the existence of BB can be found. In brief, the BB turbulence, observed at the TEM scale, becomes unstable when $\eta_{\mathrm{e}}<\eta_{\mathrm{e}, \text { crit }} \sim[2.2,2.8]$, which fits the features of DGTEM in the previous work of DIII-D [21].

Figure  5.  Important parameters of shot #75014 for turbulence analysis. (a) $R / L_{T_{\mathrm{e}}}$, (b) $R / L_{n_{\mathrm{e}}}$, (c) $R / L_{T_{\mathrm{i}}}$, (d) $\eta_{\mathrm{e}}$, (e) $k_{\theta} \rho_{\mathrm{s}}$, (f) electron diamagnetic drift wave frequency $\omega_{*}$, (g) $T_{\mathrm{e}} / T_{\mathrm{i}}$, (h) collisionality, (i) PSD of CTS channel $1, k_{\theta}=12 \mathrm{~cm}^{-1}, \rho=0.4-0.8$.

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Figure  6.  $R / L_{T_{\mathrm{e}}}-R / L_{n_{\mathrm{e}}}$ in $\rho=0.4-0.8$, collected in shots #75014 and #75015. The gray circles show a typical L-mode condition without $\mathrm{BB}$, featuring high $\eta_{\mathrm{e}}$. As $\eta_{\mathrm{e}}$ decreases, the BB appears during $\mathrm{L}-\mathrm{H}$ and $\mathrm{H}$-mode, shown by the red and blue scatters. The $\mathrm{BB}$ is unstable when $\eta_{\mathrm{e}}<\eta_{\mathrm{e} \text {,crit }}$, and $\eta_{e, \text { crit }} \sim[2.2,2.8]$ is suggested by the light red area.

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According to a previous work, the high $T_{\mathrm{e}} / T_{\mathrm{i}}$ is in favor of destabilizing TEM [21]. For shot #75014, $T_{\mathrm{e}} / T_{\mathrm{i}}>1$ can be satisfied for all three confinement regimes, as shown in figure 5(g), which is beneficial for reducing the threshold of TEM. The normalized collisionality $\nu_{\text {eff }}$ is under 3.5 at $\rho=0.4-0.8$, and is much higher in the pedestal region $(\rho>0.8)$. Though collisions cannot stabilize all DGTEM completely [40–42], the high collisionality is still a drawback for BB in the edge. The BB turbulence grows when the edge transport barriers form, so BB is not likely the dominant mechanism of transport in the pedestal region during H-modes. The strongly sheared $\mathbf{E} \times \mathbf{B}$ flow would stabilize the large-scale turbulence (like ITG) in the edge and establish the edge transport barrier, as has been proved by theories and experiments. But ITG is linearly stable in the core at t = 4.11 s (figure 5(c)), where the ITG threshold is estimated by $R / L_{T_{i}, \text { crit }}=(2 / 3) R / L_{n_{\mathrm{e}}}+(20 / 9) T_{\mathrm{i}} / T_{\mathrm{e}}$, so the ITG is not likely to be the source of $\mathrm{BB}$ turbulence.

Three typical stages of BB can be observed in figure 3(a). (1) Broadband quasi-coherent (BB-QC) mode observed during L–H transition. In this stage, the BB grows from low frequency and shows quasi-coherent features compared with the low-frequency band while the turbulence power increases, and the fast variation of BB-QC frequency may be induced by the rapid change of density/temperature gradients during L–H transition, which is different to the steady-state H-mode; (2) BB suppression. After the L–H transition, the BB is suppressed. No electron temperature profiles are attainable in this stage, so the precise value of η_e is also unknown. However, from ECE measurements, the core electron temperature (solid blue line, figure 3(f)) falls quickly in this stage, which lags behind the decline of $\left\langle R / L_{n_{\mathrm{c}}}\right\rangle$. This time lag (width of light gray rectangle in figure 3) may cause a short increase of η_e at this stage, leading to a small suppression of BB; and (3) steady BB. η_e at $\rho=0.4-0.8$ stays at a relatively low and stable level, and the evolution of BB depends on the change in electron temperature and density. Note that in figures 3(c) and (d), the BB power increases while the BB frequency decreases, and the electron heat loss becomes the dominant channel of confinement degradation during the H-mode. The phenomenon of the BB intensity increasing while the BB frequency decreases is widely observed in EAST H-mode confinement degradation regimes, like in shots #75011 (figure 2), #75014 (figure 3) and another 30+ shots. As discussed before, the density gradient gradually plays a vital role in this period, which is beneficial for the growth of BB power. The characteristics of BB frequency will be analyzed in the next subsection to further understand the BB turbulence.

3.3   Parameter dependence of BB frequency

As shown in sections 3.1 and 3.2, the BB frequency decreases during H-mode degradation regimes, with both the electron temperature and density gradient decreasing, and various mechanisms can play potential roles in this process. The turbulence frequency is sensitive to the wavenumber, and both theoretical and experimental studies on turbulence suggest that the frequency is positively related to the wavenumber [27, 43]. Though the BB turbulence is observed at a constant wavenumber $k_{\theta}=12 \mathrm{~cm}^{-1}$ the value of $k_{\theta} \rho_{\mathrm{s}}$ still changes slightly due to the decline in electron temperature during H-mode degradation regimes. In experimental studies, the frequency features of TEM also depend on the confinement regimes, or plasma parameters. According to previous studies from different tokamaks, some quasi-coherent modes (QCMs) observed in Ohmic-heated plasmas are thought to be connected to TEM [44–46]. In these studies, QCMs tend to oscillate around a specific frequency with a relatively narrow frequency band ($\Delta f \sim$ tens of kHz). In H-mode plasmas, the TEM turbulence usually features $\mathrm{BB}(\Delta f \sim$ a few hundreds of $\mathrm{kHz}$) fluctuations [21], which implies that the TEM is driven by a larger gradient and has stronger nonlinearity compared with the Ohmic-heated (L-mode) case. Considering that the BB power increases when the confinement degrades, the change in electron channel parameters may also play an important role by downshifting the linear frequency of drift waves. Thus, comparing the values of electron drift wave frequency (linear results) with the frequency of BB turbulence (nonlinear results) is helpful to understand BB. Figure 5(f) shows the electron drift wave frequency $\omega_{*}$ of three time points for shot #75014, and the definition of $\omega_{*}$ is

The $\omega_{*}$ downshifts in the H-mode (blue line) compared to the L–H transition (red line), but no obvious change in BB frequency can be observed in figure 5(i). However, it is not rigorous to conclude that the BB frequency is independent of $\omega_{*}$, because in L–H transitions the η_e is known to be higher than in H-modes and the value of η_e is known to play a vital role in affecting both the intensity and frequency of BB. The establishment of the pedestal can bring in many other variables, which make the question more complicated. Also, the BB-QC grows from the low-frequency band, suggesting that during L–H transitions this micro-instability is still weaker than the steady BB (H-mode). Taking these situations into account, analyzing the correlation between $\omega_{*}$ and $f_{\text {ave}}$ during H-mode would give more comprehensive results because the electron channel parameters change slowly and the steady state of BB turbulence can be satisfied in this stage.

Figure 7(a) shows the $\omega_{*}$ downshifting during the H-mode in shot #75015. The BB frequency also downshifts and the power spectral density (PSD) spectra broaden, as presented in figure 7(b). The averaged frequency (or eigenfrequency) $f_{\text {ave}}$ of BB (f ~ 200‒2000 kHz) reduces from 612 kHz (vertical red dotted line, 3.11 s) to 465 kHz (vertical blue dotted line, 4.11 s), which is consistent with the trend of $\omega_{*} . f_{\text {ave}}$ is larger than $\omega_{*}$, suggesting that the free energy drives the linear drift wave into a turbulent state, featuring high frequency and wavenumber in the phase space [47]. In the laboratory frame, the turbulence frequency can be written as $f_{\text {tur }}=k_{\theta} v_{\theta} / 2 \pi=k_{\theta}\left(v_{\text {ph }}+v_{E \times B}\right) / 2 \pi$, where $v_{\text {ph }}$ is the phase velocity of turbulence and $v_{E \times B}$ is the $\mathbf{E} \times \mathbf{B}$ velocity. It should be noted that the frequency shift of BB turbulence is dominated by the $v_{\text {ph }}$ but not the $v_{E \times B}$ term, because the latter is expected to dominate in the low-frequency range (f ~ tens of kHz) with $v_{E \times B}<1 \mathrm{~km} / \mathrm{s}$ in the core region [29], while the BB frequency is much higher in this study.

Figure  7.  (a) $\omega_{*}$ of two time points for shot #75015. (b) PSD of two time points for shot #75015, monitored by CTS channel 1, $k_{\theta}=12 \mathrm{~cm}^{-1}, \rho=0.4-0.8$. (c) Electron temperature monitored by ECE, $T_{\mathrm{e}}(\rho \sim 0.4)$ versus $T_{\mathrm{e}}(\rho \sim 0)$. (d) $f_{\text {ave}}$ (in kHz) $T_{\mathrm{e}}(\rho \sim 0)<R / L_{n_{\mathrm{e}}}>(\rho \sim 0.66)$, the $T_{\mathrm{e}}$ is in $(\mathrm{keV})$, and the data are collected from shots #75010, #75013, #75014 and #75015. (e) Details of (c) for H-mode with $f_{\text {ave}}>450\; \mathrm{kHz}$ and the fitting result. Time interval between each point is $10 \mathrm{~ms}$. Note that the fitting error in (c) and (e) only represents the $95 \%$ confidence bound of the linear regression fitting method but not the error of the raw data.

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To make sure that the correlation between $\omega_{*}$ and $f_{\text {ave}}$ is statistically significant, more data are needed. Four shots (#75010, #75013, #75014 and #75015) are chosen to study the correlation between $f_{\text {ave}}$ and $\omega_{*}$ From equation (7), we can derive that $\omega_{*} \propto T_{\mathrm{e} *} R / L_{n_{\mathrm{e}}}$. The radial location $\rho \sim 0.66$ is selected for analysis due to the low η_e in this region. Here we make the assumption that the temperature in $\rho \sim 0.4-0.8$ is still proportional to the core temperature when no major changes of the inner or edge transport barriers can be observed. We can see that the assumption is satisfied at $\rho \sim 0.4$ as shown in figure 7(c). In other words, a linear correlation $f_{\text {ave}} \propto \omega_{*} \propto T_{\mathrm{e}}(\rho \sim 0.66) R /$ $L_{n_{\mathrm{e}}}(\rho \sim 0.66) \propto C T_{\mathrm{e}}(\rho \sim 0) R / L_{n_{\mathrm{e}}}(\rho \sim 0.66)$ is predicted in their H-modes with steady pedestal structures, where $C=T_{\mathrm{e}}(\rho \sim 0.66) / T_{\mathrm{e}}(\rho \sim 0)$ is a constant that depends on the profile of electron temperature.

In figure 7(d), a positive correlation between $f_{\text {ave}}$ and $T_{\mathrm{e}}(\rho \sim 0)\left\langle R / L_{n_{\mathrm{e}}}\right\rangle(\rho \sim 0.66)$ can be found in its H-mode with typical pedestal structure, as indicated by the dotted blue line. The fitting results are shown in figure 7(e). Note that in these L–H and H–L transitions the slope of the $f_{\text {ave}}-T_{\mathrm{e}}(\rho=0) R / L_{n_{\mathrm{e}}}(\rho=0.66)$ function is different from that in their H-modes. The turning point, marked as the yellow square, shows the moment when the pedestal structure changes, and the slope of the correlation function becomes steeper (from type 1 to type 2). The gradient and width of the pedestal can strongly affect the shape of profiles (electron temperature and density) in the region of interest ($\rho \sim 0.4-0.8$), thus changing the shape factor $C$ and the slope of the correlation function. The value of $\eta_{\mathrm{e}}$ and the driving force $R / L_{n_{\mathrm{e}}}$ are also responsible for the change in $\mathrm{BB}$ frequency dependence in the different confinement regimes, but a further demonstration will need more theoretical support and data. If we change the radial position from $\rho \sim 0.66$ to $\rho \sim 0.47$ (larger $R / L_{n_{\mathrm{e}}}$ and $\eta_{\mathrm{e}}$), the results are similar to that shown in figures 7(c)‒(e). The duration of the H–L transitions is usually short and the exact time of transition is unpredictable, so the electron temperature profile from TS is not available in the aforementioned four experiments.

In conclusion, the frequency of BB turbulence ($f_{\text {ave}}$) is strongly dependent on the electron temperature and electron density gradient from statistical results, implying that the BB is driven by trapped electrons with strong nonlinearity, which explains the BB frequency decrease during the H-mode (figure 3(d), section 3.2).

4.   The role of BB turbulence in confinement degradation of H-mode

The study of the η_e threshold and the frequency domain characteristics for BB turbulence mainly focuses on how the plasma parameters affect the BB. On the other hand, BB also plays an important role in the core transport. As mentioned before, the confinement degradation of these H-mode plasmas is mainly due to the electron thermal loss, while no large MHD bursts are observed. For instance, the stored energy of shot #75014 drops from 106.7 kJ (3.33 s) to 88.5 kJ (4.11 s), while the core electron temperature decreases from 1.9 to 1.0 keV and its BB power $S_{k}$ increases from 0.096 to 0.157 in the same time range. Note that no auxiliary heating change is performed in this process, so the confinement degeneration is a result of self-organizing plasmas.

Figures 8(a) and (b) show the operational regime of BB in five repeatable shots. The $H_{98}$ is negatively proportional to the edge safety factor $q_{95}$ and the slope of traces becomes steeper when $q_{95}>6.78$, which is a regime that is close to the $\mathrm{H}-\mathrm{L}$ transition. The line-averaged density $n_{\mathrm{e}}$ increases with increasing $q_{95}$ and this trend saturates when $q_{95}>6.7$. In general, the correlation between $H_{98}$ and $q_{95}$ is negative, as shown in these figures, but that between $n_{\mathrm{e}}$ and $q_{95}$ is positive. The inverse trend indicates that the particle transport is not the main reason for the confinement degradation.

Figure  8.  (a) $H_{98}$ versus $q_{95}$; (b) line-averaged density versus $q_{95}$; (c) $H_{98}$ versus intensity of BB; (d) $H_{98}$ versus intensity of low-frequency mode. The data are from shots #75010, #75011, #75013, #75014 and #75015. Time interval between each point is $10 \mathrm{~ms}$.

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A negative correlation between $H_{98}$ and the BB power $S_{k}$ can be found in figure 8(c). Here, we adopt the P-values and Pearson correlation coefficient to measure this linear dependence, and the definition of the Pearson correlation coefficient is

where μ and σ are the mean and standard deviations of variables A and B, and N is the scalar of variables. In this study, A and B are $H_{98}$ and the BB power $S_{k}$ of each shot, respectively, arranged by time sequence. For all five shots, the P-values are less than 0.005, which corresponds to a significant correlation in $R_{\text {pearson }}(A, B) \quad(P \in [0,1]$, and $P<0.005$ is a new standard to claim statistical significance [48]). The Pearson correlation coefficient R_pearson(A, B) can range from −1 to 1, with −1 representing a direct negative correlation, 0 no correlation, and 1 a direct positive correlation. The $R_{\text {pearson }}$ is −0.82, −0.96, −0.97, −0.90 and −0.92 for #75010, #75011, #75013, #75014 and #75015, respectively, which claims a negative correlation between $H_{98}$ and the BB power statistically. This negative correlation is not observed from $H_{98}$ and the low-frequency band (figure 8(d)), which is reasonable because the low-frequency band of turbulence is also modulated by poloidal rotation [29]. Up to now, the role of the low-frequency band is still not clear.

From the experimental results above, the confinement degradation in the core of H-mode plasmas is mainly due to electron heat loss driven by the BB turbulence. As a density gradient-driven micro-instability, the BB turbulence is sensitive to the η_e threshold and is not observed in L-modes (featured high η_e at $\rho=0.4-0.8$). The low η_e at $\rho=0.4-0.8$ is in favor of destabilizing BB. In figure 9, a possible mechanism of confinement degeneration and evolution of BB is presented. In L–H transitions, the establishment of the edge pedestal leads to a higher $n_{\mathrm{e}}, T_{\mathrm{e}}, T_{\mathrm{i}}$ at $\rho>0.8$. The density $n_{\mathrm{e}}$ and ion temperature $T_{\mathrm{i}}$ also increase in the core region, but the core electron temperature $T_{\mathrm{e}, \text { core }}$ decreases in $\mathrm{L}-\mathrm{H}$ transitions because the increasing collisionality results in energy exchange between ions and electrons. Without inner transport barriers, the opposite change of $T_{\mathrm{e}}$ in the core and pedestal regions leads to a low $R / L_{T_{\mathrm{e}}}$ and $\eta_{\mathrm{e}}$ in the region of interest ($\rho=0.4-0.8$) to destabilize the BB turbulence. As discussed above, the nonlinear growth of BB is responsible for enhancing the transport level of electron channels in H-modes, which also flattens $T_{\mathrm{e}}$ at $\rho=0.4-0.8$, leading to a low η_e in a more radial position. This positive feedback dominates the transport in H-mode plasmas, as shown in figure 9. The BB frequency downshifts in this process because of the decreasing linear drift wave frequency. Due to the extensive use of lower hybrid waves, precise $T_{\mathrm{e}}$ profiles with high time resolution from ECE are not available in these H-mode shots with energy confinement degradation, so the H–L transition is not studied carefully and will be considered in future work.

Figure  9.  A possible mechanism to explain the process of BB turbulence destabilization and confinement degradation.

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5.   Discussion and conclusion

A BB mode is observed by CTS diagnostics in repeatable shots of EAST for the first time, characterized by (1) high frequency and $\mathrm{BB}(f \sim 200-2000\; \mathrm{kHz})$, electron scale $\left(k_{\theta} \rho_{\mathrm{s}}=1-2\right.$, not observed at $\left.k_{\theta} \rho_{\mathrm{s}}=2-5\right)$; (2) sensitivity to the η_e threshold, which implies a density gradient-driven microturbulence; (3) dependence of BB frequency on the electron diamagnetic drift frequency; (4) appearance in L–H/H–H /H–L transitions and H-modes but not in L-modes; and (5) negative correlation between BB power and energy confinement in EAST H-mode confinement degradation regimes. The previous simulation results using trapped-gyro-Landau-fluid (TGLF) code suggest that the TEM can be unstable in the wavenumber range $k_{\theta} \rho_{\mathrm{s}}=1-2$ for EAST H-mode plasmas [31, 49], so the TEM is the most possible candidate for BB mode. Considering that the BB is destabilized in the low-η_e regime, the main driven force of BB mode would be the density gradient. It should be stressed that the BB mode is usually observed at $\rho=0.4-0.8$ and rarely observed in the very core region $\rho=0-0.4$, even when η_e is also below $\eta_{\mathrm{e}, \text { crit }}$ One of the reasons is that the receivers of CTS channels 3 and 4 $(\rho=0-0.4)$ are close to some noise sources, which severely affect the CTS signals in 2017 EAST experiments. The larger $k_{\theta} \rho_{\mathrm{s}}$ in the core channels of CTS is also a potential drawback for the BB turbulence measurement because the TEM is usually expected to be unstable at a relatively low level of $0.5<k_{\theta} \rho_{\mathrm{s}}<2$ in gyrokinetic simulations.

The large-scale drift wave turbulence ITG mode is found to be linearly stabilized in the core of the H-mode plasmas presented in this paper. One of the theoretical approximations for the ITG threshold, $R / L_{T_{\mathrm{i}} \text {, rit }}=(2 / 3) R / L_{n_{\mathrm{e}}}+(20 / 9) T_{\mathrm{i}} / T_{\mathrm{e}}$, has been calculated in figure 5(c) (blue dotted line). The ion temperature is below the linear threshold in the region ρ=0 - 0.76, so ITG is not likely to be responsible for the flat profile of $T_{\mathrm{e}}$ and the degeneration of energy confinement in their H-modes. Another potential contributor is the ETG mode, which is on a smaller scale $\left(k_{\theta} \rho_{\mathrm{s}}>2\right)$. In this study, the ETG is not likely to be the main contributor to the electron heat transport because of the density gradient-dominating regime in H-modes. Identifying the ETG mode in the experimental turbulence spectra is difficult, especially when large-scale turbulent modes (TEM, ITG) exist and lead to cascading into smaller-scale ones. Applying gyrokinetic simulations would be helpful to understand the role of ETG in EAST H-mode plasmas, and will be considered in our future work.

The characteristics of BB turbulence are close to the DGTEM, and direct measurements of BB are helpful to understand the transport processes in EAST H-mode plasmas. Comparisons with TEM weak-turbulence theory/simulations would further enhance the understanding of BB features, such as the BB frequency evolution. The BB turbulence reported in this paper is not observed by other diagnostics in EAST and shows very different features to the coherent modes in the edge [50].