Experimental confirmation of the linear relation between plasma current and external vertical magnetic field in EXL-50 spherical torus energetic electron plasmas

1.   Introduction

The primary challenge addressed in the investigation of high-temperature plasma confinement in toroidal systems is the attainment of equilibrium. In a tokamak, equilibrium is reached when the extension force is balanced by the force arising from the interplay between the plasma current (I_p) and the external vertical magnetic field (B_V). In the framework of the single-fluid approximation, the required B_V to sustain the equilibrium of the plasma torus is determined by the Grad-Shafranov formula [1]:

where a and R are minor and major radii of the plasma column, l_i is the internal inductance and β_p = 8π²a²$\langle$p$\rangle$/μ₀I_p² is the poloidal beta. When I_p and B_V are given, the position of a plasma column depends on both the kinetic energy of the plasma particles (β_p) and current distribution over the column cross-section (l_i, R and a).

The Grad-Shafranov (G-S) equation [1–3] has been widely applied to reconstruct magnetic flux in magnetic devices [4–11] which contain a central solenoid (CS) to drive plasma current. The G-S equation is also employed in plasma non-inductive current start-up experiments utilizing RF waves [12–17] and helicity injection techniques [14, 18–21]. However, it is important to note that the G-S equation is based on the assumption of single-fluid equilibrium ($\nabla{p}-\boldsymbol{J}\times \boldsymbol{B}=0$, where p is pressure) [1–3]. This raises questions about their applicability in scenarios involving burning plasma [22–24], runaway discharges [25, 26], and multi-fluid plasma with RF wave heating, specifically involving energetic electrons [16]. It remains unclear whether the G-S equation and equation (1) are suitable in such cases. To address this concern, Ishida et al derived equilibrium equations for multi-fluid systems that involving energetic particles, enabling the reconstruction of multi-fluid equilibrium [27, 28]. These equations have been subsequently applied in the EXL-50 spherical torus for magnetic flux reconstruction purposes. This paper presents the resultsof the equilibrium experiments of the EXL-50 spherical torus, which is a medium-sized spherical torus with a large number of energetic electrons carrying the main plasma current [29–31]. According to previous experiment results [30, 32–35], the energetic electrons exist and play an important role in the EXL-50 plasma, making it no longer suitable to be treated as a single-fluid plasma. Given that the simulation code based on the multi-fluid model of EXL-50 is under development, we carried an experimental investigation of the relationship between plasma current and external vertical magnetic field. We hope it can shed some light on the theoretical equilibrium model in future. The result shows that the relationship between B_V, I_p and β_p in EXL-50 experiments coincides with the Shafranov formula. Firstly, the relationship between I_p and B_V was verified by conducting B_V scans under similar plasma shape and heating conditions. Secondly, an analysis was performed on the relationship between B_V/I_p and β_p for nearly 500 discharges in the equilibrium state, revealing that the data predominantly followed two distinct linear trends. Further investigation confirmed that these two lines corresponded to equilibrium states with and without electron cyclotron waves (ECW), respectively. Importantly, a linear relationship between RB_V/I_p and β_p was observed, consistent with the Shafranov formula, while the physics model underlying still remains unclear and needs to be further investigated.

The paper is split into the following parts: the EXL-50 energetic electron contribution to the plasma current is presented in the first part, and the B_V, B_V/I_p, RB_V/I_p and β_p statistics are analyzed in the second part. The summary and discussion are in the last part.

2.   Experimental setup and results

2.1   EXL-50 and energetic electron current

The EXL-50 is a spherical torus without CS, with major and minor radii of approximately 0.58 m and 0.41 m, respectively, toroidal magnetic field (B_T of approximately 0.5 T at r ~ 0.48 m, and aspect ratio of A > 1.45. At present, EXL-50 uses three ECW systems (28 GHz) to heat the plasma and drive the plasma current. System A (gyrotron source power of ~ 50 kW) is mainly used to produce the initial plasma and form closed flux surfaces [31], systems B and C (gyrotron source power of ~ 400 kW) are used to increase the plasma current and maintain the current flattop for multiple seconds. Discharges with plasma currents substantially above 100 kA with a 100 kW ECW are routinely obtained in EXL-50 [29], and the main current is identified as the contribution of energetic electrons [27], which are driven by stochastic ECW [36].

Figure 1 illustrates a typical discharge in the EXL-50 system, showcasing various plasma parameters. The sub-figures, arranged from top to bottom, depict the plasma current and B_V (z ~ 0 m, R ~ 0.3 m), integral density, hard X-ray intensity [37], loop voltage and electron temperature and density profiles. The bottom two sets of profiles are electron temperature and density profiles obtained through Thomson scattering diagnostic (TS) measurements during the 2–3 s interval [38]. The ECW injected power was approximately 140 kW and was switched off after 3 s. Throughout the discharge, a consistently low loop voltage was maintained, and the plasma current remained stable at around 118 kA during the 2–3 s time period, indicating that the ECW primarily drove the plasma current. Notably, an increase in plasma current corresponds to a significant rise in hard X-ray intensity, suggesting the presence of energetic electrons in the EXL-50 system and a correlation between the plasma current and hard X-ray emission [16, 31].

Figure  1.  Temporal evolutions of (a) plasma current and B_V, (b) integral density, (c) hard X-ray, and (d) loop voltage. TS results of (e) electron density and (f) electron temperature during 2–3 s.

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The TS measurements yielded peak values of electron density and temperature at approximately 1.2×10¹⁸ m⁻³ and 100 eV. It is worth mentioning that since the TS beam is close to the plasma boundary, the core density and temperature may be higher. The ECW-driven current of bulk plasma was calculated using CQL3D code [39] with T_e = 500 eV and n_e = 1.2×10¹⁸ m⁻³, resulting in a current of less than 10 kA. Only when the plasma temperature reaches around 100 keV [40, 41] can a current of nearly 50 kA be driven, indicating that the plasma current is mainly carried by energetic electrons.

The EXL-50 system exhibits a significant population of energetic electrons, whose contributions to plasma current and pressure cannot be disregarded [12, 17, 27]. Consequently, the conventional single-fluid assumption of the G-S equation does not hold for the EXL-50 plasma. Indeed the EFIT equilibrium construction, which assumes G-S single-fluid MHD equilibrium and the plasma current only inside the last closed flux surface (LCFS), does not fit the EXL-50 experiments well. Therefore, experiments were conducted to investigate the multi-fluid equilibrium in the system.

2.2   B_V, I_p and $\beta$_p statistical analysis

The linear relationship between B_V and I_p has been extensively established in steady-state tokamak plasma [12–17, 19, 42, 43]. In this study, the relationship between I_p and B_V in a multi-fluid steady-state plasma on EXL-50 was investigated. By progressively increasing B_V on a shot-by-shot basis, while maintaining a constant injection power ECW at 20 kW and 130 kW, we examined the correlation between B_V and I_p. Figure 2 illustrates that I_p exhibited a linear and positive relationship with B_V under similar heating conditions and plasma shapes, in agreement with the Shafranov formula.

Figure  2.  Dependence of the driven current by ECW on the vertical field B_V. The red dash lines are the fitting results.

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To further investigate the validity of the Shafranov formula in a multi-fluid plasma scenario, we conducted an analysis of the relationship between the ratio B_V/I_p and β_p in approximately 500 steady-state plasma discharges. Diamagnetic measurement, a widely employed technique in fusion devices [44–46], was utilized to derive the poloidal beta, β_p from the toroidal magnetic flux produced by the plasma, and using the equilibrium relation, given here in its simplified form:

where ∆φ is the diamagnetic flux.

In figure 3, the horizontal axis is β_p (equation (2)), the vertical axis is B_V/I_p and the intensity corresponds to the number of occurrences. Notably, two distinct bright regions are observed within the figure, each displaying a linear relationship between B_V/I_p and β_p along a specific line (referred to as line 1 and line 2). Six points (pentagons, a–f markers) are randomly selected from the figure to investigate the causes of the bright regions.

Figure  3.  Statistical analysis of about 500 EXL-50 discharges for the relation between the increasing range of B_V/I_p and β_p.

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Figure 4 presents visible light plasma images corresponding to the pentagon marked data in figure 3. The images were used to determine the brightness boundary and estimate the large and small radii of the plasma using a simple fitting calculation (indicated by the green dashed line). It should be noted that cases (a) and (b) represent scenarios where the ECW was turned off, and the effect of hard X-rays on the camera was negligible.

Figure  4.  Visible light plasma images for the pentagons in figure 3.

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Observing the plasma images, it is evident that the β_p increases significantly in line 1 and line 2, while the plasma shape in the CCD camera remains unchanged from figures 4(a) and (b) and from figures 4(c) and (d). Moreover, the plasma region varies considerably at similar β_p ((a), (c), (d) and (b), (e), (f)), which suggests that the two-line region in figure 3 may be affected by the R variation of the plasma.

The relationship between 4πB_VR/μ₀I_p and β_p is shown in figure 5. The β_p is approximately linear with 4πB_VR/μ₀I_p, which agrees with the Shafranov formula.

Figure  5.  Statistical analysis of over 100 EXL-50 discharges for the relation 4πB_VR/μ₀I_p versus β_p.

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3.   Discussion and conclusion

Equilibrium is a crucial aspect of tokamak devices, particularly for future magnetic confinement devices, where achieving fusion plasma necessitates the use of strong neutral beam injection (NBI) and microwave heating, as well as the presence of a large number of energetic alpha particles generated during fusion [47–50]. This indicates that the application of conventional single-fluid G-S equilibrium to combustion plasma equilibrium reconstruction [51] cannot be confidently relied upon.

In the above demonstration, we conducted multi-fluid equilibrium magnetic surface reconstruction and statistical analysis of equilibrium parameters on the EXL-50 device, yielding preliminary results. Firstly, the linear relationship between B_V and I_p was experimentally confirmed for similar discharge conditions. Secondly, statistical analysis found that there is also a linear relationship between B_V/I_p and β_p. Finally, the relationship between 4πB_VR/μ₀I_p and β_p under different plasma shapes was estimated using the CCD camera, revealing a positive correlation between 4πB_VR/µ₀I_p and β_p, in agreement with the Shafranov formula.

Though the experimental result of EXL-50 generally fits the G-S equation, there are still problems remaining unsolved on EXL-50 spherical torus. Despite the fact that multi-fluid plasma, which includes energetic particles, does not meet the assumption of single-fluid equilibrium, the Shafranov formula still exhibits some applicability. This finding holds significance for the control of plasma shape in fusion plasma and runaway discharge. However, it should be noted that the present study represents an initial phase in the pursuit of combustion plasma control and equilibrium reconstruction, and there is still much ground to cover in this field.