Statistical analysis of locked mode induced disruption in EAST

Disruption is ubiquitous and likely unavoidable in tokamak operations.^1–3 For future large fusion reactors such as ITER, disruption will have the greatest potential impact on the release of stored thermal and magnetic energy.^4–7 Therefore, predicting and mitigating plasma disruption, which threatens fusion reactors, is especially significant.

Disruption of plasma in tokamaks can be categorized into three main types. The first type is a vertical displacement event (VDE) that occurs due to a loss of vertical stability. The second type is the so-called minor disruption, which only results in thermal quench (TQ). The third type is a major disruption, which starts with the TQ process and is followed by the plasma current quench (CQ) process. There is a significant difference in the intensity of the interaction with the wall between major and minor disruptions. During major disruption, the interaction with the wall is more intense. Many simulations and experimental studies try to provide insights into the mechanisms and factors contributing to major disruption in tokamaks. Major disruptions in tokamaks are mainly characterized by the growth of a m = 2/n = 1 magnetic island. The formation of the radial magnetic topology, which is triggered by the growth of magnetic islands, can cause rapid thermal transport and lead to instability events.^8,9 Instabilities and transport dynamics are intertwined, with instabilities affecting plasma profiles and transport and vice versa.¹⁰ Besides, impurities at the plasma edge, cooling down the plasma and reducing the effective value of q₉₅, can also destabilize the system and trigger disruption.⁹ In addition, fast reconnection during MHD mode activity, facilitated by the stochastization of magnetic field lines, also contributes to disruption.¹¹ 

Locked mode (LM) is one of the most significant physical phenomena in disruption. A statistical analysis of disruption conducted in JET covering the operational period from 2000 to 2007 reveals that in many cases of disruption, multiple triggers occurred consecutively. Nonetheless, it was observed that LM served as the precursor to most disruptions.^12,13 It was also found that LM ranked first with 48.4% of all possible shutdown triggers for unintentional disruption.¹⁴ Recently, a detailed statistical analysis of LM was performed in DIII-D, and it is concluded that whether LM eventually leads to disruption is closely related to some physical quantities.¹⁵ It makes great progress on locked mode induced disruption (LMiD) and gives a potential chance to clarify the specific physical process of LMiD. Nevertheless, there is still a lack of specific physical processes for LMiD. Therefore, it is hard to evaluate whether the physical quantities obtained from these statistical analyses for the prediction and active control of LMiD for future reactors are sufficient or not. Evaluating the physics process and giving sufficient key physical quantities for LMiD are significant for the next step, accurate prediction and active control.

In this paper, the effects of key parameters on LMiD in EAST are investigated by using an intuitive physical model and statistical methods. In Sec. II, a database of disruption triggered by LM in EAST is established, and the time scale is discussed. In Sec. III, an intuitive physical model is proposed to understand the physical process of LMiD. In Sec. IV, an analysis of the influencing factors of LMiD is mentioned on the basis of this physics model. At last, a summary is given in Sec. V.

In this investigation, we focused on locked mode experiments in EAST from 2015 to 2022,^16–22 which are mainly triggered by externally applied resonant magnetic perturbation (RMP). Out of all 214 plasma discharges, 110 are locked mode induced disruption discharges. Figure 1 summarizes the composition of the database, containing L/H mode, heating methods, and RMP mode number parameters. The auxiliary heating techniques used in these experiments consist of lower hybrid wave (LHW), neutral beam injection (NBI), and some electron cyclotron heating (EC). These heating methods are strongly correlated with the occurrence frequency of H-mode at around 56%. Besides, the RMP system with toroidal mode numbers n = 1 is predominantly applied. The line-averaged plasma density n_e in the database is almost uniformly distributed and varies approximately in the interval 0.7–6.2 × 10¹⁹ m⁻³, which is much lower than the so-called Greenwald’s density limit. For shots with disruption, data were collected at about 100–200 ms before CQ. For shots without disruption, one to two data points were selected during mode locking. These are typical moments when the perturbed magnetic field or the RMP current reaches its peak value. In this database, 267 valid samples were obtained from these shots.

In order to verify the physical mechanism, we need to examine the time scales and key parameters of this process. Next, we discuss these aspects of LMiD in detail. We use a significant parameter, decay time, to quantify the plasma disruption intensity. It is the ratio of the time duration for the plasma current I_p to drop from its maximum value at the flat-top phase to 40% of that value, divided by 0.6.¹⁵ This method is similar to the previous statistics and can effectively estimate the decay time.^23,24 The time when the current quench begins is considered the moment of plasma disruption. The current quench is determined by the L/R time of the plasma and is, hence, directly related to the temperature.²⁵ Therefore, a lower temperature after the thermal quench means a higher R and, therefore, a lower current quench time. The parameter decay time for all 110 disruptive events in our collected database is shown as a histogram in Fig. 2. The parameter decay time is mostly less than 50 ms, as it is often associated with the major disruption caused by LMiD. Survival time, which is the interval from mode locking to disruption, starts at the moment when radial magnetic perturbation δB_r suddenly grows.^26, Figure 3 illustrates the plasma current and the δB_r evolution of a LMiD process. It is obvious that the locked mode starts to grow at around 5 s, and then the disruption occurs at 5.81 s. Survival time, which reflects the early predictive capability of the plasma for mitigating the damage caused by disruption in fusion reactors, is a significant timescale and a topical research issue in the global community.^27–31  Figure 4 gives the distribution of the survival time. Most of the data points are below 1500 ms, with two distinct peaks at 400 and 900 ms. Although the condition of LM has some differences, these two parameters agree with the statistical distribution of extensive data on major disruptions in DIII-D.¹⁵ 

RMP can be employed to actively trigger locked mode (LM) as a way to gain insights into the mechanisms and factors contributing to locked mode induced disruption. Locked modes are produced by an electromagnetic force opposing the plasma’s rotation. When externally applied static RMP reaches or exceeds critical thresholds, it will induce LM.⁷,²⁶,³² The electromagnetic force mainly comes from the radially perturbed magnetic field multiplied by the non-ideal perturbed current at the rational surface, which results in magnetic islands primarily on the 2/1 rational surface.

The physical mechanisms of LMiD have been investigated in some previous research.³³ These studies have revealed that the major disruption involves energy loss and negative voltage spikes. During the nonlinear growth phase of LM, the magnetic islands overlap, resulting in the formation of a new magnetic topology. The new magnetic topology would enhance local radial heat and particle transport.³⁴  Figure 5 shows the electron density evolution given by reflectometry after LM. The locking moment is defined as a zero moment, and the 2/1 magnetic island is located in the gray area. It is observed that the particles in the core transport outward after locked mode. This leads to a decrease in the density of the core region and an increase in the density at the pedestal.

Based on the experimental analysis and previous studies, a physical process for LMiD can be proposed. Generally, the disruption has three stages: (1) the precursor phase, when MHD instabilities like the LM grow; (2) the thermal quench, when islands overlap and stochastization leads to a rapid quench of the thermal energy; and (3) the loss of the discharge, either due to loss of Ip control due to the very low Te after the thermal quench or due to loss of VS control due to the thermal quench (beta drop), which in turn leads to a current quench. The characteristics of LMiD that distinguish it from the general disruption process are mainly between stages 1 and 2. The process leading to disruption can be described as follows: during the LM growth stage, plasma can be transported outward along the magnetic field lines. This is similar to the onset of tearing modes and the flattening of the temperature profile due to the presence of a large magnetic island;^35–37 it can provide precursors from the changes in the electron temperature profile and potential disruption. These changes result in a lower temperature in the core plasma and a higher temperature at the pedestal region, as described for the disruption shot in Fig. 6(a), where temperature profiles are measured by Thomson scattering diagnostics. The temperature profile changes more significantly in Fig. 6(a), which has significantly higher temperature loss in the core and a higher temperature gradient at the boundary compared with the non-disruption shot in Fig. 6(b). As a consequence, the temperature (density) near the last closed surface (LCS) increases to a higher value, resulting in a larger temperature (density) gradient between the LCS and the first wall. Therefore, the energy (particle) from the plasma will be more easily expelled. Then, the heat flux strikes the first wall, producing more impurities in the plasma. Meanwhile, the drop in plasma temperature increases the plasma resistance, leading to an increase in the loop-voltage V_loop. The increase in loop-voltage causes an increase in the inward electric drift force (V_loop × B_P). This leads to more impurities entering the plasma, which further reduces the plasma temperature. It is also found that the most important reason for LMiD is the distance between the magnetic island outer boundary and the LCS (d_edge), which is defined as a − (r_s + w/2), where a is the minor radius, r_s is the radial position of the 2/1 surface, and w is the width of the magnetic island. The two examples in Fig. 6 have very different values of the parameter d_edge, while the calculated widths of the magnetic island w only have a 12% difference. This suggests that d_edge plays a significant role. When enough energy loss leads to the significant production of impurities, a thermal quench occurs. After this stage, the very low Te or the loss of VS control due to the thermal quench (β drop) causes a current quench.

Based on the relevant parameters and some other potential key parameters in the physical process mentioned earlier, a large number of parameters for LMiD in EAST are statistically analyzed, which can reveal the interrelationships of these physical parameters during the LMiD process. We performed statistical analysis on the following key parameters to determine their impacts on the occurrence of disruption after locked mode in EAST experiments: the width of the magnetic island w, the distance between the magnetic island outer boundary and the wall d_edge, the loop-voltage V_loop, plasma electron core temperature T_e, plasma stored energy W_MHD, energy confinement time τ_E, and toroidal magnetic field B_t. These results indicate that the location and size of magnetic islands play crucial roles in thermal quench. Specifically, disruption is more likely to happen when the distance between the magnetic island boundary and the LCS is small. Additionally, experiments with lower energy and electron temperatures in the plasma core are more susceptible to disruption. The radial inward force (F), defined as V_loop × B_p, was also identified as a crucial parameter in determining the probability of disruption. Figure 7 shows that disruption is more likely to occur when both B_p and V_loop are relatively large. The product of these two quantities can be regarded as an equivalent force that acts toward the plasma core. The radial inward force may trigger an explosive increase in impurities, which can lead to thermal quench and even plasma disruption. This phenomenon well confirms the contribution of V_loop in the physical process mentioned earlier.

Magnetic islands close to the first wall are more likely to trigger thermal quench due to larger temperature gradients. The statistics in Fig. 8 also show that large islands approaching the plasma boundary are also more likely to trigger disruption. According to the island evolution in Fig. 3, the island grows linearly with time. It indicates that the island enters the Rutherford stage, which is proportional to resistivity^40, $(η∼Te−3/2)$⁠. This will accelerate the island’s growth and further reduce the d_edge value. Therefore, it is a comprehensive factor to induce the disruption. Figure 9 demonstrates that shots at higher temperatures and energies tend to be safer and trigger fewer disruptions. For this reason, the magnetic island grows in proportion to the resistance η, which continues to linearly increase.⁴⁰ Therefore, a lower temperature increases the likelihood of disruption. However, this does not mean that high energy and temperature are safe or that they are necessary conditions for non-disruption cases. This is because many other factors can cause a fast drop in temperature and energy, followed by disruption. Nevertheless, it is still effective to use low energy and temperature as precursors for distinguishing disruption or not. Figure 10 shows that there is no clear distinction between disruptions based on l_i like previous work.¹⁵ This could be explained by the fact that the data parameters in EAST are far from Greenwald’s density limit. Figure 10 also indicates that q₉₅ is strongly correlated with LMiD. The parameters q₉₅ and d_edge are both related to the position of the rational surface, so they have similar performance in distinguishing LMiD. Further d_edge couples the information of magnetic islands and may be related to the formation of thermal quench.

Heating power may also be a factor that will impact LMiD, as shown in Fig. 11. It is shown that the disruption in region (a) with higher auxiliary heating power is less than in region (b). Region (a) of Fig. 11 has higher plasma stored energy and a shorter energy confinement time. Shots in this region usually use a combination of auxiliary heating methods. On the other hand, Region (b) has lower plasma stored energy and a relatively longer energy confinement time. Shots in this region have low auxiliary heating power, primarily using neutral beam injection (NBI).

Eight key physical parameters have been identified that influence the occurrence of disruption after locked mode in EAST experiments, and the effects of each parameter were observed through a histogram analysis presented in Fig. 12. The histogram reveals that the plasma parameters V_loop × B_p are positively correlated with disruption, while d_edge, τ_E, q₉₅, T_e, B_t, and W_MHD are negatively correlated. This is consistent with the LMiD process mentioned in Sec. II. These parameters reveal some trends, but disruption is a complex process that is not affected by a single factor, and the coupling of parameters is hard to clarify. Therefore, more advanced methods such as machine learning are required to handle the related statistical data in the future.

Furthermore, to identify the possible factors affecting the locked mode disruption intensity, we first applied a statistical analysis to some parameters that are potentially related to the physical mechanism of the disruption, taking their possible interactions into account. Figure 13 reveals the correlation of LMiD with four parameters. When rapid disruption with greater harmfulness occurred, the key parameter intervals tended to be at lower plasma density (n_e), the lower distance between the magnetic island outer boundary and the LCS (d_edge), lower plasma pressure normalized to the magnetic pressure (β_n), and a higher ratio of internal inductance (l_i) and q₉₅. l_i was introduced to factor in the effect of the current density profile. The choice of experimental parameters that can prevent plasma disruption is of great relevance for the field, as it allows for a meaningful comparison with the statistical outcomes of other devices.

Using the experimental data from the EAST tokamak, we set up a database of locked mode induced disruption (LMiD) events. We have collected almost all the data from the error field penetration experiment in EAST, and we scanned the data in as wide a range as possible. Nevertheless, there are still some regions that we cannot scan, such as high beta plasmas in low q₉₅ (we cannot operate in this region steadily as the tungsten impurity) and higher density plasmas in higher q₉₅ (the limited RMP current cannot reach the penetration threshold). However, these regions can present the main LMiD characteristic. We have examined their time scales, which include the current quench time and the locked mode survival time. These two parameters indicate the disruption intensity and the warning threshold time for the fusion reactor, respectively, and are very important for reference. The formation of locked modes is closely related to the presence of magnetic islands and several other physical factors.

In order to evaluate the key parameters in LMiD, an intuitive physical process is proposed. The LMiD process can be described well, from the evolution of magnetic islands to the loss of plasma stored energy. The statistical analysis based on this mechanism is then performed. We then analyzed the importance of various parameters in relation to disruption, aiming to identify the key factors that trigger LMiD. The result reveals that the plasma parameter V_loop × B_p is positively correlated with disruption, while d_edge, τ_E, q₉₅, T_e, B_t, and W_MHD are negatively correlated. This is consistent with the intuitive LMiD process. We also used the frequency of disruption occurrences to investigate the different plasma parameters. It is found that n_e, d_edge, ratios of l_i and q₉₅, and β_n are related to the intensity of plasma disruptions.

Future work will focus on validating the proposed physical mechanism by expanding the sample size of the database, increasing the experimental data near the parameter limits, such as Greenwald’s density limit, β limit, etc., and using other methods such as machine learning to process data and conduct further research.

The authors acknowledge the collaboration of the EAST team. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12175276, 11875292, 12005261, and 12105323, the National Magnetic Confinement Fusion Science Program of China under Grant No. 2017YFE0301100, the Natural Science Foundation of Anhui Province under Grant No. (2208085J39), and the HFIPS Director’s Fund under Grant No. BJPY2022B05.

The authors have no conflicts to disclose.

Wei-Ran Zhou: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal). Guo-Hong Deng: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal). You-Wen Sun: Conceptualization (equal); Resources (equal); Software (equal). Hui-Hui Wang: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Deng Zhou: Resources (equal); Supervision (equal). Tong-Hui Shi: Data curation (equal); Resources (equal); Software (equal). Shuai Gu: Data curation (equal); Software (equal). Cheng Ye: Data curation (equal); Formal analysis (equal); Software (equal). Qun Ma: Data curation (equal); Software (equal). Qing Zang: Data curation (equal); Resources (equal); Software (equal). Kai-Yang He: Data curation (equal); Software (equal). Da-Long Chen: Data curation (equal); Resources (equal). Biao Shen: Data curation (equal); Resources (equal). Man-Ni Jia: Data curation (equal); Software (equal). Zheng-Ping Luo: Data curation (equal); Resources (equal). Hai-Qing Liu: Data curation (equal); Resources (equal). Zi-Qiang Zhou: Data curation (equal); Resources (equal). Tao Zhang: Data curation (equal); Resources (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.