A brief review: effects of resonant magnetic perturbation on classical and neoclassical tearing modes in tokamaks

1.   Introduction

Plasma disruption is of great concern for tokamak operation. It mainly refers to the phenomenon that the plasma loses the confinement of energy, leading to the termination of discharge and even results in damage to the device. When disruption occurs, high-energy runaway electrons (REs) hit the plasma-facing component, which poses a great challenge to the avoidance and mitigation of disruption. The first wall will melt if it is hit by the REs, and the REs also shorten the life of the device. For ITER, up to 70% of the pre-disruption plasma is converted into runaway current, and several tens of mega-electron volt (MeV) energy can be generated during disruption [1, 2]. The generation mechanisms of REs include the Dreicer mechanism [3] (primary generation mechanism), avalanche mechanism [4, 5] (secondary generation mechanism), hot tail mechanism, etc [6].

Over the years, much effort has been devoted to exploring various ways to avoid disruption effectively, and some methods have been found in experiments to mitigate the disruption. Nowadays, the most popular method to suppress the runaway current is injecting a large number of impurities, such as massive gas injection (MGI) [7]. Suppression of REs by MGI on C-Mod, DIII-D, JET and TEXTOR have been extensively studied, and the suppression of REs has been successfully achieved on these tokamaks [8–10]. This is done by increasing the relativistic electron collision frequency ν_rel, which can be expressed as follows: ν_rel = n_ee⁴ ln Λ/(4πε₀²m_e²c³), ln Λ means the Coulombian logarithm, m_e is the electron mass, ε₀ is the vacuum permittivity and ε₀ = 8.85×10^-12 F m^-1. When ν_rel increases, it can effectively decelerate some high-energy electrons. However, the achievement of the complete suppression of REs must be based on increasing the plasma density to the Rosenbluth density [11]. It is difficult to meet the requirements by MGI on these tokamaks, thus causing a bottleneck. Furthermore, the application of resonant magnetic perturbations (RMPs) has a significant suppression effect on runaway current on ASDEX Upgrade and COMPASS [12, 13]. It is also found on J-TEXT that RMP can excite large magnetic islands, which have the ability to enhance runaway loss during disruption [14]. In contrast, the RMPs have failed to suppress the runway electrons for the large device JET [15]. Therefore, the suppression of RMP on runaway current is inconsistent on different tokamaks.

Some novel methods are being proposed to suppress the runaway current. Electrode biasing (EB) and limiter biasing (LB) are traditional auxiliary systems and they change the floating potential voltage directly, and some experiments with EB and LB have been carried out in recent years [16, 17]. On the IR-T1 tokamak [18] in Iran, the results show that the application of biased voltage during a major disruption can suppress the generation of REs. They also found that the EB applied in edge plasma can modify the radial electric field $E_{\mathrm{r}}$ and lead to the enhancement of particle confinement. On the ISTTOK tokamak in Portugal, improved particle confinement is clearly observed by negative bias associated with a large radial electric field [19], and it is also found that the average plasma density increases under this condition.

In this paper, we discuss the influence of both EB and LB applied to positive bias on runaway current on J-TEXT. The suppression of runaway current has been clearly observed, and it is found that as the voltage increases, the runaway current decreases, and the potential of plasma has been found to be sensitive to the biased voltage. It is also believed that the biased voltage can suppress the runaway current by affecting the radial transport of REs. The paper is arranged as follows. The experimental setup is presented in section 2. In section 3, the influence of positive biased voltage by both electrode and limiter on runaway current is introduced. A discussion is presented in section 4. A summary is given in section 5.

2.   Experimental setup

The experiments reported in this paper are carried out on J-TEXT, which has a circle plasma with an iron core, and some of its basic parameters are as follows: major radius R₀ = 1.05 m, minor radius a = 0.25‒0.29 m with a movable graphite limiter [20]. It usually operates in the following parameters: the center line-averaged electron density n_e is in the range of (1‒7) × 10¹⁹ m^-3, with toroidal magnetic field B_t of ~2.0 T, pre-disruption plasma current I_p = 120‒220 kA in typical discharges, and the edge safety parameter q_a changes with the change in minor radius, toroidal magnetic field and the plasma current.

The MGI system has been used to trigger disruption with argon and obtain stable runaway current, which is located at the bottom of port 9 [21]. The Langmuir–Mach probe (LMP) array, which is located at the top of port 13, measures the edge-plasma parameters [22]. There is one array of 24 Mirnov coils for the detection of MHD activities [23]. Three hard x-ray (HXR) detectors are distributed in the forward, radial and backward REs movement directions for measuring thick target bremsstrahlung radiation when the REs hit the first wall or the plasma. The layout of these systems is shown in figure 1(a), and the direction of the plasma current and toroidal magnetic field is counter-clockwise from the top view. In order to measure the energy spectrum of REs, a multi-channel analyzer (MCA) with 0.5 ms time resolution [24] and 1024 channel energy resolution is used.

Figure  1.  (a) Top view of the layouts of the EB system, HXR detector, LMP, MGI system and the limiter, (b) poloidal cross-section of the EB system and (c) poloidal cross-section of the LB system and power system.

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The EB system [17, 25], shown in figure 1(b), is installed at the mid-plane of port 8, and is equipped with a disc (4 cm in diameter and 1.3 cm in thickness) graphite head [26] and contains a pneumatic driving part. The electrode can reciprocate in the range of 23.5‒25.5 cm in a single discharge [16]. DC voltage and modulation voltage can be selected in the experiment. The LB system in figure 1(c) shares the same power system with the EB system. Three movable limiters are located at the top, bottom and low-field side of the vacuum, respectively. Voltages can be applied to all three limiters at the same time, or only one of them.

The corresponding U_EB–I_EB of the EB system and the U_LB–I_LB characteristic of the LB system are shown in figure 2, represented by dots and diamonds, respectively. The data in figure 2 represent the change in currents with voltages on the electrode or the limiter of the low-field side with undisruptive discharges. It is shown that both the EB current I_EB and the low-field-side limiter current I_LB are approximately proportional to the biased voltage under positive voltages, whereas I_EB tends to saturate at about -300 V with a value of -20 A.

Figure  2.  Volt–ampere characteristic of the EB and the LB systems.

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3.   Experimental results

3.1   Influence of EB on runaway current

The basic plasma parameters in this experiment are as follows: I_p = 160 kA, B_t = 1.8 T, n_e = (1‒2) × 10¹⁹ m^-3 and q_a ≈3.5. MGI injects argon at 0.4 s to trigger disruption, leading to a stable runaway current platform. The voltage is applied to the electrode during a time of 0.35‒0.42 s at a position of 25 cm, just 5 mm inside the last closed flux surface (r~25.5 cm). The injection of argon causes the temperature quench (TQ) phase. The sharp drop in the plasma temperature then results in an enhancement of the plasma resistivity, and it further leads to the decay of the plasma current in what is called the current quench (CQ) phase. A strong toroidal electric field is induced in the vacuum during CQ, accelerating the runaway seeds to a higher speed, and a stable runaway current is eventually formed.

In the experiment, different positive voltages have been applied to the electrode. The discharge results at +100, +270 and +450 V biased voltage are shown in figure 3. When the positive voltages are applied to the electrode, the runaway current is suppressed. The higher the voltage value, the smaller the runaway current, and the effect of complete suppression is achieved at +450 V. Figure 3(c) gives the details of the electrode current I_EB. A spike appears in the current, which is caused by the MGI valve injection of a large amount of argon impurity causing the TQ and CQ processes, and a strong loop voltage, as shown in figure 3(d), will be generated in the vacuum, resulting in a strong current spike on the electrode.

Figure  3.  Discharges when the electrode is applied to +100, +270 and +450 V, respectively. The figures give the time evolution of (a) the plasma current I_p, (b) MHD, (c) electrode current I_EB, (d) loop voltage V_loop and (e) H_α emission.

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In the J-TEXT tokamak, hydrogen is usually used for discharge, and most of the atoms and ions at the plasma boundary are in the ground state. When they collide with the electrons, the energy level of the ground-state particles changes, and the radiation generated in this process is represented by H_α emission. It can also be used to characterize the plasma confinement, since when the H_α increases, more particles are transported to the low-field side. During the runaway plateau phase, due to the deterioration of particle confinement, the particles are diffused and lost continuously on the electrode, the current shows a slight increase, as shown in figure 3(e), and the H_α emission also keeps rising. During the final runaway CQ phase, REs are rapidly lost, leading to a rise in H_α density and electrode current again. For discharge #1070545, no runaway current is generated and the electrode is accompanied by higher current. The electrode current even reaches saturation, and the H_α emission is maintained at a high level after disruption for several milliseconds during CQ, then decreases slowly at 0.406 s.

3.2   Influence of LB on runaway current

The influence of LB on the runaway current has also been explored. The parameters are as follows: plasma current I_p = 180 kA, toroidal magnetic field B_t = 2.2 T, electron density n_e~(1‒1.5) × 10¹⁹ m^-3. Since there are different vacuum conditions in each experiment, runaway current is hard to generate at some plasma parameters, so the basic plasma parameters of I_p and B_t are not the same in any two experiments, but this will not affect the qualitative results in both experiments. The voltage is applied to the low-field-side limiter only and the radial position of the limiter is fixed at 255 mm. Different voltages are applied to the limiter during a time of 0.35‒0.42 s. Typical experimental results are shown in figure 4.

Figure  4.  Discharges compared to the reference when positive voltages of +200 and +300 V, respectively, are applied to the low-field-side limiter. The figures give the time evolution of (a) plasma current I_p, (b) MHD, (c) limiter current I_LB, (d) loop voltage V_loop and (e) H_α emission.

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It can be seen from figure 4(c) that there is still a large limiter current in the TQ and CQ processes, for discharge #1076005, where the H_α emission and I_LB also increase with time during the runway plateau phase. However, in discharge with +300 V biased voltage, the H_α radiation remains for a long time during the CQ process, which is similar to the phenomenon observed in the discharge #1070545 in the EB experiment.

Figure 5 shows the runaway current as a function of the electrode current or the limiter current. This completely reveals that the runaway current is negatively correlated with the biased current, either with the EB or LB system. Due to the higher plasma parameters in the LB experiment, the runaway current of the reference discharge is higher, but the limiter requires a smaller threshold for the complete suppression of runaway current. The size of the limiter is 25 cm in length, 10 cm in width and 5 cm in thickness, and the electrode is 4 cm in diameter and 1.3 cm in thickness. Compared to the limiter, the electrode is small in size and has a smaller contact surface with the plasma. If a simple electric power calculation is performed, when the REs can be completely suppressed, the electric power required by the electrode and the limiter is 53.1 and 30.4 kW, respectively. It can be seen from figure 2 that the biased current difference between the electrode and the limiter is not particularly large under the same positive biased voltage. Due to the larger size of the limiter, the effective area for interaction with the plasma is also larger. Therefore, the suppression efficiency of the runaway current is higher, and the REs can be completely suppressed at a lower voltage. The slope of the straight line in figure 5 is assumed to represent the 'suppression efficiency' of the two systems on the runaway current, and it can be seen that the LB is more efficient on the runaway current.

Figure  5.  Relationship between the biased current and the value of the runaway current. Red squares represent the LB results and the blue triangles are the EB data.

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4.   Discussion

4.1   HXR energy spectrum

The HXR bremsstrahlung spectra of the REs reflect the relationship between the number of REs and their energy. It is an important way to analyze the behavior of the REs, and it also reflects information about the maximum energy max (E_RE) and radiation temperature T_HXR of the REs. Figure 6 shows the energy spectra during the runaway plateau phase.

Figure  6.  (a) Comparison of HXR spectra with +200 V bias (I_RE~62 kA) to the reference discharge (I_RE~123 kA) during the runaway current plateau phase. Maximum energy of the HXR is lower with the application of +200 V bias. (b) Comparison of HXR spectra with +200 V bias to the reference discharge with the same runaway current plateau (~62 kA).

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The maximum energy max (E_RE) is defined as the intersection of the linear fit of the high-energy tail of the spectrum with ${\log }_{10}{N}_{\gamma }/s=2$ (for the lowest count rate), as described in [27]. With the application of biased voltage, max (E_RE) is significantly lower than that in the discharge without biased voltage, which is about 1.2 MeV. The radiation temperature T_HXR is derived from the linear fitting of the high-energy tail of the energy spectrum [28, 29], T_HXR = -1/k, where k is the slope of the linear fit line and itself a negative value, which means that when the spectrum is more inclined, T_HXR is lower. However, when the slope of the spectrum is flatter, this means that more high-energy tail electrons exist, and the T_HXR is higher. It can be seen that the T_HXR is lower after the application of biased voltage, and the smaller the T_HXR, the lower the HXR emission energy. The defined T_HXR is similar to the variation of max (E_RE). However, the temperature of the measured spectrum is not completely equal to the temperature of REs, but it can be used to describe the possible evolution of REs qualitatively.

The HXR spectra for the reference discharge #1075997 and the discharge with +200 V bias in #1076005 are shown in figure 6(a). The maximum energy of the HXR spectrum in the discharge with +200 V bias is lower than that of the reference discharge. This indicates that the +200 V bias enhances the radial transport loss of REs, which leads to a decrease in both the runaway confinement and the runaway energy. This is beneficial for the suppression of runaway current. The runaway energy is also lower when the +200 V bias is applied to a similar runaway current plateau (62 kA), as shown in figure 6(b). This illustrates that the positive bias has the effect of reducing the runaway confinement and the runaway energy by enhancement of the runaway radial transport loss.

4.2   The loss of REs

The loss of REs includes collisional transport, electrostatic perturbation and magnetic perturbation. Due to the low collision frequency of REs, neoclassical transport due to collisions is generally not considered. Therefore, the enhancement of the radial transport of REs is also one of the means for runaway loss, and the runaway current is greatly affected by the number of runaway seeds. The suppression of runaway current can also be achieved by increasing the radial transport of runaway seeds. In the experiment of LB, a Langmuir probe has been used. Figure 7 gives information about the radial electric field during the time when the limiter is applied to +250 voltage. By measuring the radial distribution of the floating potential V_f(r), the radial electric field can be obtained from the gradient of the potential E_r = -dV_f/dr. As can be seen from figure 7, the value of the radial electric field is about 2 kV m^-1 without LB. When +250 V limiter biased voltage is applied, the radial electric field increases significantly in the positive direction, the direction of the radial electric field measured is the component along the minor radius in the horizontal direction, and the maximum value of E_r can reach about 7 kV m^-1. In addition, the average radial electric field values vary with different voltages, which shows that as the voltage increases, E_r also increases linearly.

Figure  7.  Time evolution of radial electric field at the last-closed flux surface, discharge without biased voltage (the orange line) as a reference and discharge with +250 V limiter biased voltage (the blue line), respectively.

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The velocity disturbance caused by the electric field disturbance in the direction perpendicular to the magnetic field is $\tilde{v}\propto \tilde{E}/B,$ which further causes the disturbance of the radial displacement of the particles $\mathrm{∆}r\approx \tilde{v}{\tau }_{\mathrm{trans}},$ for the correlation time $\tau ,$$\tau =\mathrm{\pi }qR/v_{||}$ is substituted, where q and R are the safety factor and the major radius, respectively, and $v_{||}$ is the velocity parallel to the direction of the magnetic field lines. Since the probe measures the change of the floating potential V_f(r), the component of potential disturbance is included. Therefore, the diffusion coefficient of REs due to electrostatic disturbance can be expressed as [30],

Generally, $v_{||}$ is assumed to be one tenth of the speed of light before disruption, due to the inverse dependence on the $v_{||},$, which can be negligibly small, and it is believed that the loss of REs is dominated by magnetic field disturbance. However, the addition of biased voltage enhances the amount of electric field perturbation, and the increase in $\tilde{E}$ leads to the radial diffusion coefficient in equation (1) increasing proportionally, which further enhances the loss of REs due to electrostatic perturbation.

In the experiment, the effect of biased voltage on the plasma parameters at the boundary has also been studied. Figure 8 mainly describes the changes in electron density n_e, C-Ⅲ emission and H_α radiation at the plasma boundary, and HXR radiation with +450 V electrode biased voltage and +300 V limiter biased voltage, with the aim of observing the loss of REs at the boundary.

Figure  8.  Time evolution of plasma current I_p, electron density n_e, C-Ⅲ intensity, H_α emission at the boundary (r≈24 cm), and the HXR intensity after +450 V biased voltage is applied to the electrode (discharge #1070545) and +300 V is applied to the limiter (discharge #1076014).

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Figures 8(c)‒(h) show the evolution of the electron density n_e, C-Ⅲ and H_α emission at the plasma boundary, and all three signals display a significant increase after 0.35 s application of biased voltage. Since the electrode is equipped with a graphite head, some of impurities will occur during the discharge process, resulting in a stronger C-Ⅲ signal in discharge #1070545 than in discharge #1076014 after 0.35 s. Figures 8(i) and (j) give the evolution of HXR radiation. During the discharge of the biased electrode, the HXR radiation keeps rising after 0.35 s, reaches its peak when disruption occurs, and then decreases rapidly. However, during the discharge of the biased limiter #1076014, the HXR signal peaks at about 0.37 s, followed by a slow decline in radiation intensity. Since no runaway current is formed in both discharges, the HXR intensity is lower after disruption. This reveals that the biased voltage significantly enhances the RE losses, which means that a large number of REs are expelled from the plasma due to the biased voltage and the radial transport of the REs increases.

The effect of the magnetic turbulence and accelerating electric field on REs is also studied. Figure 9 shows the relationship between the ratio of I_RE/I_p and MHD integrated intensity, and I_RE/I_p as a function of E_acceleration/E_critical in the two experiments of the biased electrode and biased limiter, respectively. E_acceleration is calculated by the mean value of V_loop during the CQ process, and the MHD integrated intensity is measured by Mirnov coils in both the TQ and CQ processes. The blue diamonds and red circles represent the data for the biased electrode and biased limiter, respectively.

Figure  9.  (a) Relationship between the ratio of I_RE/I_p and MHD integrated intensity and (b) I_RE/I_p as a function of E_acceleration/E_critical.

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It can be inferred from figure 9(a) that larger runaway currents exist when the magnetic fluctuation is lower. This trend is well illustrated by the fitted line in figure 9(a), both with biased electrode and the biased limiter. It is worth mentioning that the loss of REs can be enhanced by magnetic fluctuation, and the runaway beam themselves may also have a stabilizing effect on MHD [31]. Figure 9(b) gives the ratio of I_RE/I_p as a function of E_acceleration/E_critical. Lower E_acceleration/E_critical results in smaller I_RE/I_p. It is worth noting that the direction of the arrow in the figure represents the trend of increasing biased voltages. It seems that when the biased voltage is larger, the MHD behavior during the CQ process is stronger, which on the one hand enhances the loss of runaway seeds, and on the other hand, the corresponding decrease in the ratio of E_acceleration/E_critical also affects the formation of REs.

5.   Summary

The suppression of REs is a serious issue for large-scale tokamak devices. The alternative runaway suppression by EB and LB has been carried out on the J-TEXT tokamak. Positive voltages of different amplitudes are applied to the electrode and limiter; 0–500 and 0–350 V are applied, respectively. Parameters, such as plasma current I_p, loop voltage V_loop, H_α radiation and HXR radiation, are measured. It is found that the runaway current decreases during the increase in the biased voltage in both experiments. Moreover, the runaway current can be completely suppressed at +450 V by EB and +300 V by LB.

By comparing the energy spectra of the REs during the runaway plateau phase, it is found that the number of REs decreases after the application of +200 V limiter biased voltage. Both the radiation temperature T_HXR and the maximum energy max (E_RE) of the REs decrease significantly, indicating that both the number and energy of the REs are reduced, and the suppression of runaway current is achieved by reducing the generation of runaway seeds.

The radial electric field E_r generated by the bias may be the main factor suppressing the runaway current, and it is calculated using the gradient from the floating potential, including the component of potential disturbance. In the limiter biased experiment, the electric field measured by the Langmuir probe rises significantly at about 0.35 s. The average values of the radial electric field vary with different voltages, and E_r increases linearly as the voltage increases. The diffusion coefficient of the REs is enhanced by electrostatic disturbance, which accelerates the loss of the runaway seeds, leading to a lower runaway plateau. The electron density n_e, H_α signal and C-Ⅲ radiation at the boundary are significantly improved after the biased voltage is applied. In addition, there is a significant enhancement in HXR intensity. As a consequence of all the above factors, the number of electrons at the plasma boundary increase, and the interaction between the electrons and the first wall is enhanced, resulting from the improvement in runaway loss.

The suppression of runaway current by bias is a reliable method in some small and medium-size tokamaks, and the effect of radial electric field on runaway confinement and maximum runaway energy is demonstrated. It is found that a positive radial electric field can reduce both the runaway current plateau and the maximum runaway energy. The electrode cannot be used on large-scale devices due to high edge-plasma temperature. This method of suppressing runaway current can be used for the protection of the first wall during disruption in large-scale devices by increasing the radial electric field. The radial electric field can be modulated by the limiter bias or active plasma rotation control.