2. The EXL-50 tokamak

The EXL-50 is a medium-sized, spherical tokamak without a central solenoid. It has major and minor radii of approximately $0.58$ and$0.41m$, respectively. The toroidal magnetic field strength $({B_T})$ at a radius of approximately $0.58m$ is approximately $0.48T$, while the aspect ratio $(A)$ is greater than or equal to $1.45$. The EXL-50 is primarily a hydrogen discharge, and the main plasma ions are hydrogen ions with small amounts of impurity ions such as boron and helium. The highest plasma current recorded in an experiment is $150\;\textrm{kA}$ (Shi et al. Reference Shi2022). The line integral electron density ranges from 2 – $18 \times {\textrm{10}^{17}{m}^{ - 2}}$. The plasma is heated using an electron cyclotron wave (ECW) at approximately 140 kilowatts (kW). The EXL-50 employs two sets of 28 GHz ECW systems operating in the O-mode to heat the plasma and drive the plasma current. System #1, with a gyrotron source power of 50 kW, is used to produce the initial plasma and form a closed flux surface. System #2, with a gyrotron source power of 400 kW, sustains the plasma current during the flattop phase for several seconds.

The plasma density and fluctuations have been determined using a microwave interferometer near the midplane (Li et al. Reference Li, Bai, Tao, Li, Lun, Liu, Liu, Liu and Deng2021). This system measures the phase shift between two probing beams induced by the line-integrated plasma density. Density fluctuations are detected in the phase fluctuations of the signals. Localized electron temperature and density measurements are conducted using Thomson scattering (TS) (Li et al. Reference Li, Li, Xie, Liu, Bai, Tao and Deng2022b). The Bremsstrahlung emission generated by energetic electron losses to the wall is measured using hard x ray (HXR) detectors (Cheng et al. Reference Cheng, Zhu, Chen, Li, Bai, Chen and Liu2021). The rise and fall in the low-frequency magnetic field (1–200 kHz) are measured using a magnetic probe. High-frequency magnetic probes are used to detect coherent electromagnetic fluctuations such as whistler waves (Wang et al. Reference Wang, Cheng, Liu, Song, Gou, Song, Liu, Banerjee, Li, Sun, Gu, Li, Dong, Shi, Peng and Liu2023b). The ECW heating generates an energetic electron population through stochastic heating (Wang et al. Reference Wang, Li, Bai, Dong, Shi and Zou2023c). These energetic electrons can provide free energy for the instabilities (Wang et al. Reference Wang, Lun, Bo, Liu, Liu and Shi2023d, Reference Wang, Shi, Dong, Gao, Lu, Wang, Chen, Liu, Zhuang and Wang2024).

3. Experimental results

Figure 1 illustrates the typical plasma discharge in the EXL-50 device, wherein CMs are observed. The temporal evolutions of the plasma current and density of two discharges, #9193 (black) and #19737 (blue), are shown in figures 1(a) and 1(b), respectively. Figure 1(c) shows the temporal evolution of the core electron and ion temperature for discharge #19737, during which the ECW is injected at approximately $200\;\textrm{kW}$ in the O-mode. Here, the temperature of the bulk electrons exceeds that of the ions. Burst-like density fluctuations with chirping frequencies were observed using a microwave interferometer at specific time intervals of 2.8–3.5 and 1.8–3 s in shots #9193 and #19737, respectively. The level of density fluctuations $(\delta {n_{\textrm{CM}}}/{n_{\textrm{el}}})$ is between 0.1 % and 1 %. Moreover, the mean fluctuation frequency decreases as the electron temperature decreases.

Figure 1. Typical observations of CMs in EXL-50 plasma discharges. Panels (a) and (b) show the temporal evolution of the plasma current and line-averaged density, respectively, for two discharges: #9193 (black) and #19737 (blue). Panel (c) shows the time evolution of core electrons and ion temperatures (ion temperature multiplied by three) during discharge #19737. Panels (d) and (e) show the power spectra of plasma density fluctuations exhibiting burst CMs in the ion acoustic frequency range (10–100 kHz) for discharges #9193 and #19737.

We conducted an extensive analysis using data from more than 10 discharges to determine the relationship between the electron temperature and frequency of the mode. We observed a positive correlation between the frequency of the CM and electron temperature measured using the TS diagnostics at approximately 0.7 m from the centre of the plasma, as shown in figure 2. The frequency range of the fluctuations is comparable to frequency called CMs. Note that the TS is designed for a minimum density measurement of approximately $1 \times {10^{18}}\;{\textrm{m}^{ - 3}}$, and the temperature error of the TS measurement is large at low densities $( < 1 \times {10^{18}}\;{\textrm{m}^{ - 3}})$. The CMs frequency usually increases and then decreases during a burst cycle, and the CMs frequency chirping maintenance time in the experiment is usually more than $10\;\textrm{ms}$, which is much larger than the frequency chirping caused by the phase space variation.

Figure 2. Correlation between the frequency of CMs and electron temperature. The scatter plot shows the electron temperature measured using TS at approximately 0.7 m on the horizontal axis versus the frequency of CMs on the vertical axis. Here, a positive correlation is observed, with the mode frequency increasing with an increase in the electron temperature.

As shown in figure 3, the presence of CMs increases HXR intensity. This represents a degradation in the energetic electron confinement, as HXR intensity refers to the thick-target radiation caused by energetic electrons hitting the device wall. Therefore, CMs may lead to an increase in the loss of energetic electrons. Moreover, when energetic electrons hit the device wall, wall outgassing occurs, which increases the background plasma density and the electron collision rate. With a further increase in the background electron density as well as in the loss of energetic electrons, the free energy supporting the CM may decrease, and the frequency and intensity of the CM may decrease and eventually disappear. Once a certain level of parameter accumulation is reached, a new cycle of CM excitation begins.

Figure 3. Time evolution of line-averaged density (a i and b i), the rise and fall in the power spectra (a ii and b ii) of the CMs and HXR intensity over time (a iii and b iii), including the fitted curve represented by the blue dashed line (a iii and b iii). The HXR signal shows a considerable increase when CMs appear (represented by blue) and a decrease as the modes dissipate. This indicates a greater energetic electron loss from the plasma during CM driving.

Energetic electrons can initiate whistler waves (Spong et al. Reference Spong, Heidbrink, Paz-Soldan, Du, Thome, van Zeeland, Collins, Lvovskiy, Moyer, Austin, Brennan, Liu, Jaeger and Lau2018; Wang et al. Reference Wang, Shi, Dong, Gao, Lu, Wang, Chen, Liu, Zhuang and Wang2024). Theoretical predictions suggest interactions between low-frequency instabilities and whistler waves via wave–particle or wave–wave interactions. In shot #14 954, whistler waves and CMs were simultaneously observed (figure 4). The CMs were observed at 2.25 and 2.45 s. The presence of CMs was associated with a decrease in the frequency and intensity of the whistler wave, suggesting possible interactions between them.

Figure 4. Observation of CMs and whistler waves in EXL-50 discharge. Panel (a) shows the power spectra of high-frequency magnetic fluctuations exhibiting whistler wave signals, and (b) shows the power spectra of plasma density fluctuation signals. The concurrent detection of CMs and whistler activity in their respective frequency ranges is evident. An inverse correlation between the density fluctuation intensity of CMs and whistler wave magnetic fluctuation intensity suggests a competitive relationship.

Figure 5 shows the temporal evolution of plasma density, density fluctuation intensity of CMs and whistler wave magnetic field fluctuations for over 2.2 to 2.5 s. When the CMs emerge, the plasma density increases, whereas the whistler wave intensity decreases. Figure 5(d) shows the correlation between the intensity of the CMs and fluctuations in the whistler wave magnetic field. Here, the intensity of the CMs increases before that of the whistler waves. Although a slight decrease in the plasma density is observed at approximately 2.35 s, the whistler wave intensity does not exhibit a considerable decrease. This implies that the effect of density change on the whistler wave intensity is relatively weak. Hence, the impact of density changes on the CMs and whistler wave intensity can be disregarded in this procedure, while the depletion of energetic electrons may be responsible for reducing the whistler wave intensity.

Figure 5. Time evolution of: (a) line-averaged plasma density; (b) CM density fluctuation; and (c) whistler wave magnetic field fluctuation over time in an EXL-50 discharge. Following the appearance of CMs, the plasma density increases while the whistler wave intensity decreases. Panel (d) shows the relationship between the intensity of the CM and whistler wave magnetic fluctuation (with four time points), wherein the CM intensity rises before the whistler wave intensity does.

3.1. Discussion

Owing to the limitations imposed by the diagnosis, it is currently difficult to identify the type of instability in question. Nevertheless, existing theories can facilitate pertinent discussions regarding the probability of IAW, GAM and BAE.

The EXL-50 generally uses only an ECW to heat and drive the plasma current, which is transported by energetic electrons. The drift velocity of the bulk electrons $(5 \times {10^4}\;\textrm{m}\;{\textrm{s}^{ - 1}})$ is lower than the ion acoustic velocity $(\mathrm{\sim }{10^5}\;\textrm{m}\;{\textrm{s}^{ - 1}})$. Consequently, the IAW is difficult to stimulate under current conditions (Lesur, Diamond & Kosuga Reference Lesur, Diamond and Kosuga2014). Although the frequency of the CM is consistent with the dispersion relation of the GAM, according to the theory of GAMs, the GAM has a density fluctuation component with m = 1, whose fluctuation strength near the midplane is small (Conway et al. Reference Conway, Smolyakov and Ido2021). The EXL-50 microwave interferometer is closeted near the midplane of the device, and strong density fluctuations (0.1 %–1 %) have been measured. Therefore, the CM is unlikely to be GAM.

Note that BAE has a GAM-like dispersion relationship (Heidbrink et al. Reference Heidbrink, van Zeeland, Austin, Bierwage, Chen, Choi, Lauber, Lin, Mckee and Spong2020). However, no significant CM magnetic field rise and fall components were observed in the EXL-50 experiment. Although both LFM and BAE were observed in the DIII-D experiment, the magnetic field fluctuation component of LFM was also not observed (Heidbrink et al. Reference Heidbrink, van Zeeland, Austin, Bierwage, Chen, Choi, Lauber, Lin, Mckee and Spong2020). This suggests that the electromagnetic rise and fall are not necessarily measured by the magnetic probe. This could be due to (i) the relatively local mode structure, (ii) the weak fluctuation intensity and (iii) the cutoff of the fluctuation that propagates from the core to the edge.

According to Zonca et al. (Reference Zonca, Chen and Santoro1996), for $\varOmega _{{\ast} \textrm{pi}}^2 \ll \; ({\textstyle{7 \over 4}} + \tau ){q^2}$, only the BAE accumulation point may be unstable when ${\eta _i} > {\eta _{\textrm{ic}}}$, and $\textrm{lm(}\varOmega \textrm{)}$ increases linearly with ${\eta _i}$ and${\varOmega _{{\ast} \textrm{ni}}}$, where ${\eta _i} = {L_{\textrm{ni}}}/{L_{\textrm{Ti}}}$, $\tau = {T_e}/{T_i}$ and ${\eta _{\textrm{ic}}}\sim (2/\sqrt {7 + 4\tau \; } ){\omega _{\textrm{ti}}}/q{\omega _{{\ast} \textrm{ni}}}$. As shown in figure 1(c), ${T_e}/{T_i}\sim 3$ at R = 0.7 m. Assuming $R/{L_{\textrm{ni}}}\sim R/{L_{\textrm{ne}}}\sim 4$, $R/{L_{\textrm{ti}}}\sim R/{L_{\textrm{te}}}\sim 8$ (Wang et al. Reference Wang2023a). Here ${\eta _{\textrm{ic}}}$ and $\varOmega _{{\ast} \textrm{pi}}^2/({\textstyle{7 \over 4}} + \tau ){q^2}$ are shown in figure 6; ${\eta _i}\; > {\eta _{\textrm{ic}}}$ and $\varOmega _{{\ast} \textrm{pi}}^2/({\textstyle{7 \over 4}} + \tau ){q^2} < 0.2$ with q = 2, n = 4, and the BAE accumulation point may unstable on EXL-50.

Figure 6. Here ${\eta _{\textrm{ic}}}$ and $\varOmega _{{\ast} \textrm{pi}}^2/({\textstyle{7 \over 4}} + \tau ){q^2}$ versus ${T_i}$ with q = 2 and n = 4. On EXL-50, typical ${T_i}$ is between 20 and 50 eV, and the BAE accumulation point may be unstable.