2.1. The EXL-50 spherical tokamak

The EXL-50 is a medium-sized spherical tokamak without a central solenoid, the major radius of EXL-50 is approximately $0.58\ {\rm m}$, the minor radius is approximately $0.41\ {\rm m}$, $B_T$ (at $r \sim 0.58\ {\rm m}$) is approximately $0.48\ {\rm T}$, and aspect ratio of ${\rm A} >= 1.45$. Currently, the highest plasma current recorded in the experiment is $150$ kA. The line integral electron density is usually $2\sim 18\times 10^{17}\ {\rm m}^{-2}$, with an electron cyclotron resonance heating (ECRH) power of approximately $140\ {\rm kW}$. The EXL-50 uses two sets of 28 GHz ECW systems (O-mode) to heat the plasma and drive plasma current (Shi et al. Reference Shi, Liu, Song, Song, Song, Tong, Cheng, Liu, Wang and Sun2022; Wang et al. Reference Wang, Guo, Shi, Chen, Liu, Song, Zhao, Song, Liu and Guan2022). System #1 (source power of gyrotron 50 kW) is mainly used to produce the initial plasma and to form a closed flux surface, and system #2 (source power of gyrotron $400\ {\rm kW}$) is used to increase the plasma current and sustain the current flattop for several seconds. Referring to the work of (Ikegami et al. Reference Ikegami, Aihara, Hosokawa and Aikawa1973), EXL-50 uses a smooth metal wall, and observation windows are shielded with $28\ {\rm GHz}$ shielding materials, and the distribution of the ECWs in the vacuum chamber is approximately stochastic with cold plasma (Ikegami et al. Reference Ikegami, Aihara, Hosokawa and Aikawa1973). The plasma density is diagnosed using an interferometer (Li et al. Reference Li, Bai, Tao, Li, Lun, Liu, Liu, Liu and Deng2021). Energetic electron energy is diagnosed by the HX bremsstrahlung radiation from the plasma (figure 1a). The HX detectors with improved lead shielding are applied. Beside the original shielding (10 mm lead $+5\ {\rm mm}$ steel) (Cheng et al. Reference Cheng, Zhu, Chen, Li, Bai, Chen, Huang, Dai and Liu2021), the CdZnTe detectors have their own independent $50\ {\rm mm}$ lead shielding. For this improved shielding HX system, there are two detectors with a collimator and one blinded detector without a collimator (figure 1b). For the discharges with plasma current less than $100\ {\rm kA}$, the HX counts of the blinded detector are much small than the detectors with a collimator. A large number of energetic electron-driven instabilities have be observed in the discharge of EXL-50 (Wang et al. Reference Wang, Xiuchun, Xiaokun, Bing, Adi and Yuejiang2023d,Reference Wang, Shi, Dong, Gao, Lu, Wang, Chen, Liu, Zhang and Wang2023b,Reference Wang, Tan, Shi, Wang, Dong, Liu, Zhuang, Li, Song and Yuan2023c).

Figure 1. (a) Top view of the EXL-50. Lines of sight of the interferometer and HX diagnostics are indicated in the figure. The ECW beam is aimed at the centre of the machine when the toroidal injection angles are $0^\circ$. (b) Typical HX spectra in the forwards, and backwards directions and the blinded during the flat top phase. The boundary marked is the last closed flux surface (LCFS).

2.2. Density fluctuations

In EXL-50, the bulk electron density and temperature are typically in the range of $4$–$40 \times 10^{17}\ {\rm m}^{-2}$ and $10$–$200\ {\rm eV}$, respectively, while the energetic electron temperature is approximately $100$–$300\ {\rm keV}$ (Cheng et al. Reference Cheng, Zhu, Chen, Li, Bai, Chen, Huang, Dai and Liu2021; Ishida, Peng & Liu Reference Ishida, Peng and Liu2021; Li et al. Reference Li, Bai, Tao, Li, Lun, Liu, Liu, Liu and Deng2021; Guo et al. Reference Guo, Shi, Liu, Song, Sun, Liu, Li, Tian, Zhang and Xie2022; Li et al. Reference Li, Li, Xie, Liu, Bai, Tao, Lun, Li, Bo and Liu2022; Shi et al. Reference Shi, Liu, Song, Song, Song, Tong, Cheng, Liu, Wang and Sun2022; Wang et al. Reference Wang, Guo, Shi, Chen, Liu, Song, Zhao, Song, Liu and Guan2022, Reference Wang, Xiuchun, Xiaokun, Bing, Adi and Yuejiang2023d). Figure 2(a) illustrates the relationship between single-pass wave power absorption coefficient and electron energy, calculated using GENRAY (Smirnov & Harvey Reference Smirnov and Harvey2001) (with a bulk plasma temperature of $100\ {\rm eV}$). The efficiency of single-pass absorption of ECW power by the plasma is notably low. Moreover, as shown in figure 2(c), the interferometer often observes significant density fluctuations (Wang et al. Reference Wang, Li, Bai, Dong, Shi, Zou, Liu, Zhuang, Li and Li2023a). When the ECW passes through the plasma, the plasma and its fluctuations can modulate the ECW phase.

Figure 2. (a) Relationship between the energy of electrons and wave single passing absorption rate calculated by GENRAY. (b) Time evolution of HX intensity and (c) density fluctuations.

Owing to the smooth metal wall and window shielding, the vacuum vessel of the EXL-50 can be approximated as a shielding cavity. Since the single passing absorption efficiency of an electron cyclotron wave is so weak in the present low temperature EXL-50 plasmas, the angle and mode of ECW are randomized during the multiple wall reflections. According to Ikegami et al. (Reference Ikegami, Aihara, Hosokawa and Aikawa1973), the distribution of ECWs in the vacuum vessel is approximately stochastic. The stochastic electron acceleration model on EXL-50 may hold.

2.3. Weak ECW stochastic heating of electrons

When ECW exclusively heat the energy component $w$ perpendicular to the magnetic field direction, the time evolution of the electron distribution function $(f(t,w))$ is described by the Fokker–Planck equation, employing the relationship derived by Sturrock (Reference Sturrock1966),

(2.1)\begin{equation} \frac{\partial f}{\partial t}=R\frac{\partial}{\partial w}\left(w\frac{\partial f}{\partial w}\right)-\frac{f}{\tau}+Q\delta(w), \end{equation}

where $R$ is the heating rate, which is positively correlated with the strength of the stochastic electromagnetic wave electric field and is assumed to be independent of $w$ (Sturrock Reference Sturrock1966). Here $Q$ is the electron source. For the electron energy distribution function obtained from (2.1), the photon number $({\eta }(\varepsilon,t))$ of the x ray bremsstrahlung emitted by the statistically accelerated energetic electrons is given by

(2.2)\begin{equation} {\eta}(\varepsilon,t)=\frac{c}{\varepsilon}\int_{\varepsilon}^{\infty}{\frac{{\rm d}w}{\sqrt {w}}f(t,w)G(w,\varepsilon)} . \end{equation}

Here $C$ is a numerical constant, $G(w,\varepsilon )$ is the energy-dependent part of the total cross-section for the bremsstrahlung of photons with energy $\varepsilon$ produced by the electrons with energy $w$. The time development of the x ray bremsstrahlung photon number due to the statistically heated (or accelerated) energetic electrons by the stochastic ECW is as follows (Ikegami et al. Reference Ikegami, Aihara, Hosokawa and Aikawa1973):

(2.3)\begin{equation} {\eta}(\varepsilon,t)\propto\varepsilon^{{-}0.5}/R\int_{0}^{t/\langle\tau\rangle}{\rm d}s\,s^{{-}1}{\rm e}^{{-}s} \times\int_{0}^{\infty}{\rm d}u\ln(u+\sqrt{u^2-1})\exp{\left(-\frac{\varepsilon}{R\langle\tau\rangle}\frac{u^2}{s}\right)}, \end{equation}

where $\langle \tau \rangle$ is the average electron energy decay time. Parameters such as $\langle \tau \rangle$ and ${\rm R}$ have specific ranges: for ‘$\tau$’ the reasonable range is several to several hundreds of milliseconds; for ‘$R$’ the reasonable range is $0.1$ to several tens of megaelectronvolts per second (Ikegami et al. Reference Ikegami, Aihara, Hosokawa and Aikawa1973).

To validate the generation of energetic electrons through stochastic heating, the time evolution of HX counts at different energy bands was analysed. Figure 3 illustrates the number of x ray bremsstrahlung photons (solid line) and theoretical curves (dotted line) at different energy bands. Numerical calculations (dotted lines) were performed to compare the experimental results with the theoretically derived photon numbers at a specific energy, as described in (2.3). The excellent agreement between the experimental and theoretical curves confirms the presence of stochastic heating and the generation of energetic electrons in the EXL-50 experiment. It is worth mentioning that the EXL-50 optimizes the shielding of the HX system to mitigate the effects of thick target radiation. As a result, the number of photons within the detection system is reduced, leading to a decrease in the system's signal-to-noise ratio. Nevertheless, the basic trend remains reliable and acceptable.

Figure 3. Time evolutions of x ray bremsstrahlung photon numbers of specified energy. The ECW injection power is approximately 105 kW (a) and 25 kW (b). The solid line is the measurement result by HX and the dotted line is the theoretical curves.

Furthermore, the relationship between ECW input power and heating rate was analysed under the same gas delivery conditions. The heating rate is plotted as a function of the input power in figure 4. The plasma heating rate shows a positive correlation with the applied ECW power, confirming the effectiveness of stochastic heating.

Figure 4. Relationship between heating rate and input ECW power; the heating rate is displayed as a function of the input power.

2.4. The ECW-driven plasma current

Figure 5 shows time traces of plasma current $I_p$ (figure 5a) and line-integral electron density $n_{{\rm el}}$ (figure 5b) under the same heating power ($50\ {\rm kW}$) and different ECWs incident angles on EXL-50. Although the toroidal and poloidal incidence angles of the ECWs change significantly, the direction and amplitude of the plasma current do not change, meaning that the direction and amplitude of the plasma current are not sensitive to the incidence angle. Meanwhile, the direction of plasma current is dominated by $B_V$.

Figure 5. Time traces of $I_p$ (a) and line-integral electron density (b) for different ECW incidence angles (shots #4946, #4947, #4950 and #4951) and for $B_V$ (#14802). The direction and amplitude of plasma current are not sensitive to the incidence angle. Meanwhile, the direction of plasma current is dominated by $B_V$.

Conventional analyses, such as ray tracing codes, reveal that the direction and amplitude of the plasma current are highly responsive to the angle of the ECW injection, especially when there is efficient absorption of the ECW through signal passage. The insensitivity of the ECW-driven plasma current to the incidence angle is in contrast to the conventional tokamak understanding of the ECW on EXL-50 and implies a requirement for new models.