The rotating resonant magnetic perturbation (RMP) could lock the tearing mode (TM) and prevent disruption. It is found that the TM instantaneous frequency oscillates during TM locking on the RMP. In this work, the TM frequency oscillation was derived analytically based on the toroidal torque balance, which was verified experimentally. It is found that two peaks with higher frequency (2*f*_{TM0}-*f*_{RMP}) and lower frequency (*f*_{RMP}) occurred in the spectrum when the window width of FFT was shorter than the TM frequency oscillating period. Those two peaks were explained analytically based on the description of TM instantaneous frequency. The independence of the temporal information and the spatial information of the TM was explained analytically and verified by the spatial structure analysis of each peaks. This work is important for mode structure identification and provides a scheme for mode coupling control.

## I. INTRODUCTION

The major disruption, which is the rapid and complete loss of energy and particles on tokamak devices,^{1,2} not only terminates the plasma discharge^{3} but also causes high thermal and electromagnetic loads on plasma-facing materials, having a highly damaging effect on the machine.^{4,5} Hence, major disruptions have to be controlled and mitigated for large tokamak devices and future fusion reactors.^{6,7} Based on statistic investigation, (neoclassical) tearing mode [(n)TM] locking is one of the most common causes of major disruptions and occurs under a wide variety of plasma conditions near the ideal MHD stability limit.^{8,9} The modes with poloidal/toroidal mode numbers m/n = 2/1 are the most detrimental ones to plasma confinement and often lead to disruptions in DIII-D.^{10,11} The control of the m/n = 2/1 mode locking is an important criterion for disruption prevention.^{12,13}

One of the possible ways to prevent the locked mode is to drive the mode rotation by applying rotating RMPs. In J-TEXT, locked mode unlocking has been achieved by rotating RMP.^{14,15} Especially, the disruption with a low edge safety factor is even prevented by the 3 kHz rotating RMP in the experiments.^{16} In the high *β*_{N} (∼2.5) DIII-D discharges, the TM was excited and led to disruptions.^{17} The RMPs were used to keep the mode rotating at 17 Hz, and the disruption was prevented.^{18} The parameters of rotating RMPs, which have effects on the effectivity of mode locking control, were tested through experiments on the J-TEXT tokamak.^{16} Only when the TM is locked on the RMP, the disruption can be controlled. The electromagnetic torque generated by the RMP is required to overcome the viscous torques and lock mode.^{19} On the other hand, once the TM is locked on the rotating RMP, a further increase in RMP amplitude would increase the mode amplitude.^{20,22} The TM with a larger amplitude causes the destabilization of the discharge condition^{21} and decreases the effectivity of disruption control.^{16} Hence, the progress of mode locking in rotating RMP needs to be investigated.

The effects of static RMP on TM has been investigated sufficiently via experiments,^{22–24} simulations,^{25,26} and analytical theories.^{27,28} A oscillation of TM frequency has been observed in a DITE tokamak.^{29} After applying the rotating RMP, the TM rotation is oscillated by the electromagnetic torque, which could explain the harmonic in the spectrogram and the decreases the TM amplitude. A negative correlation between the amplitude of TM rotation oscillation and the relevant frequency difference was found in J-TEXT experiments.^{30} Besides, during the mode locking phase, the TM amplitude seems to be suppressed by rotating the RMP significantly based on the result of the wavelet. The phenomenon suggests a possible method to suppress the TM, which should be confirmed through experiments and theories. The analytical description of the progress of mode locking in rotating RMP and understanding of the relevant spectrum have not been studied in detail.

To solve this problem, experimental and analytical investigations of the process of mode locking with rotating RMP are carried out in this paper. An analytical calculation result for the TM rotation oscillation is presented. The analytical results show that the amplitude of the TM rotating oscillation is linear with the RMP amplitude. The relationship between the amplitude of the TM rotating oscillation and the frequency difference between the TM and RMP relies on the viscosity coefficient, which could be indirectly measured in experiments. The analytical result is verified by applying an RMP rotating for a long time and agreed with the outcome of experiments. Based on the analytical description of TM instantaneous frequency, an analytical description of the relationship between the spectrum performance and the window width was provided. The independent relationship between the temporal and spatial information of TM was found during the mode locking progress with the Fourier transform. Based on those relationships, a static component during mode locking on a static RMP and the understanding of the spectrum during the mode coupling were further explained. The analytical result could be used as a reference of the RMP amplitude for disruption control.

In Sec. II, the phenomenon of TM rotation oscillation is introduced. The theoretical mode of TM rotation oscillation during mode locking is presented in Sec. III A. The experimental verification of the mode was carried out in Sec. III B. The calculated results are generally consistent with the experimental results. The analytical explanation of the spectrum during mode locking is provided in Sec. IV. Finally, Sec. V gives the discussion and conclusion.

## II. THE EXPERIMENTAL SETUP AND TYPICAL MODE LOCKING PROGRESSION TO THE ROTATING RMP

The J-TEXT tokamak^{31} has a circular cross section with a limiter configuration. The J-TEXT has a major radius of *R* = 105 cm and a minor radius of *a* = 25.5 cm. The plasma is ohmically heated in this work. The electron temperature at the core of the plasma is ∼0.9 keV.^{32} On the J-TEXT, the rotating RMP field is generated by four sets of in-vessel saddle coils (3 rows × 4 columns × 1 turn)^{33} when they are supplied with AC current at a fixed frequency (from 1 to 5 kHz) during one shot.^{34} In this work, the spectrum of the rotating RMP field is chosen to have a dominant 2/1 component with an amplitude of ∼0.5 Gauss/kA at 4 kHz, calculated under the vacuum assumption and taking into account the wall-eddy current effect, as shown in Fig. 1.^{16}

Figure 2 depicts a typical example of TM locking by rotating RMP for discharge No. 1048266. The primary plasma parameters for the experiments are the toroidal magnetic field *B*_{T} = 1.8 T, the plasma current *I*_{p} = 160 kA, and the central line-averaged electron density *n*_{e} = 1.1 × 10^{19} *m*^{−3}. Typically, a saturated TM exists in this parameter, and its initial rotation frequency, as shown in Fig. 2, is 4.9 kHz.

Starting at 0.2595 s, a 4 kHz rotating RMP is applied and ramped up to a maximum of ∼0.35 Gauss at 0.292 s. Figure 2(a) depicts the curve of the 2/1 component RMP amplitude, *B*_{r,RMP}^{2/1}. During this time frame, the Mirnov signal [the black curve in Fig. 2(b)] displays a distinct beat-like modulation. When the amplitude of the Mirnov signal [the red curve in Fig. 2(b)] is increasing/decreasing, the frequency of the d*B*_{θ}/dt curve is also increasing/decreasing, indicating that the TM instantaneous frequency is changing. The time evolution of TM instantaneous frequency, *f*_{TM}, is depicted in Fig. 2(d) by the blue curve. Following the application of the rotating RMP, the *f*_{TM} oscillates with an amplitude proportional to the RMP amplitude. The oscillation could be detected in the frequency spectrum when the amplitude of the TM frequency oscillation is sufficiently significant between 0.27 and 0.285 s. The oscillating magnetic field caused by the TM has an amplitude of ∼12 Gs, as measured by the magnetic probe, *r*_{probe} = 0.315 m. The maximum oscillated magnetic field generated by the RMP coils is ∼0.4 Gs at the final closed surface, *r*_{LCS} = 0.255 m, which is significantly weaker than δ*B*_{θ}. Thus, the RMP has minimal effects on the TM amplitude. The evolution of TM amplitude over time is depicted in Fig. 2(d). During the progression from mode locking to rotating RMP, the amplitude of TM is nearly constant.

The Mirnov signal is linear with the amplitude and frequency of the oscillating poloidal magnetic field generated by the TM. During the TM locking process (0.26–0.285 s), the maximum magnetic signal is nearly linear with the TM frequency, *f*_{TM}, as shown in Fig. 3. The red dashed line corresponds to the blue circle points. The deviation from the fitting line may be attributable to the small oscillation on the TM amplitude, which is discussed in Sec. V. The linear relationship between the maximum magnetic signal and *f*_{TM} is further evidence that the amplitude of the TM remains nearly constant throughout this period.

The electromagnetic torque generated by the rotating RMP oscillates the TM rotation. The electromagnetic torque has no relationship with the exact values of the TM phase, *ξ*_{TM}, and the RMP phase, *ξ*_{RMP}, according to Fitzpatrick's theory.^{27} The electromagnetic torque is proportional to the RMP amplitude and −sin(Δ*ξ*), where Δ*ξ* is the phase difference between the TM and the RMP, Δ*ξ* = *ξ*_{RMP} − *ξ*_{TM}. Figure 4 depicts the mode locking progression in detail. The instantaneous frequency of the TM oscillates with the phase difference Δ*ξ*, as predicted by Fitzpatrick’s theory. The TM instantaneous frequency consists of two components: the period average component [*f*_{avg}, red dashed curve in Fig. 4(c)] and the oscillating component (δ*f*_{TM}). *f*_{avg} is the average TM frequency during a frequency oscillating cycle. The experimental results presented in Ref. 30 demonstrate that the mode locking threshold is related to the frequency difference between TM and RMP, Δ*f*_{RMP-TM}. In addition to the electromagnetism torque, the viscous torque also influences the rotation of the TM. The viscous torque generated by the bulk plasma accelerates the TM to its initial frequency, which is linear with δ*f*_{TM}. Simultaneously, δ*f*_{TM} led to the non-sinusoidal distortion of the *dB*_{θ}/*dt* signal. A phenomenological appreciation of the relationship between the distortion of the *dB*_{θ}/*dt* signal and the higher/lower frequency peak is presented in Sec. IV. The oscillation of TM rotation during mode locking is supported by prior research. When the instantaneous frequency of the TM reaches the frequency of the RMP, the TM locks onto the RMP. In other words, mode locking occurs when δ*f*_{TM} is greater than Δ*f*_{RMP-TM}, providing a criterion for calculating the mode locking threshold.

## III. DYNAMIC PROGRESS OF MODE LOCKING OF THE ROTATING RMP

### A. The theoretical model of mode locking

Effectively, the various magnetic islands in plasma are treated as solid body obstacles embedded in a phenomenological (incompressible) viscous single fluid. When there is no net plasma flow across the island separatrix, the “no-slip” constraint is applied.^{35} Torque balance, which includes electromagnetic torque and viscous torque, dominates the island rotation. In a plasma with a low beta (or a high collision rate), the neoclassical toroidal viscous NTV torque is insufficient to be considered for the J-TEXT. Due to the strong poloidal flow damping^{36,37} in a tokamak plasma, only the toroidal flow is investigated in this paper. As shown in the following equation, the toroidal torque balance governs the tearing mode rotation:^{27}

where *M* is the moment of inertia of the TM, with *M* = 4*π*^{2}*ρR*^{3}*r*_{s}*w*_{is}; *w*_{is} is the width of the magnetic island; *ρ* is the native mass density at the resonant surface; *R* is the major radius of the device; and *r*_{s} is the radial location of the resonant surface. *T*_{EM,φ} is the electromagnetic torque generated by RMP on TM. The following equation explains the electromagnetic toroidal torque:^{27}

where *B*_{r} is the radial magnetic field generated by RMP; *j*_{1} is the amplitude of the perturbed surface current density of the TM; *ξ*_{RMP} and *ξ*_{TM} are the helical phases of RMP and TM, respectively; and Δ*ξ* is the phase difference between them (Δ*ξ* = *ξ*_{RMP} − *ξ*_{TM}).

It was assumed that there was no net plasma flow across the various island separatrixes and that the plasma velocity profile was roughly constant across each portion of the inner region containing plasma. Due to deviations in the island frequencies from their natural values, the viscous torque is proportional to the flux-averaged change in the phenomenological (angular) single fluid velocity. Based on Fitzpatrick’s model,^{27} the viscous torque can be described as follows:

where *f*_{TM} and *f*_{TM0} are the toroidal instantaneous frequency and the toroidal natural frequency of the TM, respectively; and *μ*_{⊥} is the viscosity of plasma, assumed constant during the calculating region. To simplify the toroidal torque balance of tearing mode, the following equation may be used:

where *σ*_{⊥} = *μ*_{⊥}/*ρ* is the viscosity coefficient. The natural TM frequency (*f*_{TM0}) is slower than δ*f*_{TM} and is assumed constant in the calculation. It was also assumed that the difference in frequency between RMP and TM (Δ*f*_{RMP-TM} = *f*_{RMP} − *f*_{TM}) is significantly greater than the oscillation of TM frequency (δ*f*_{TM} = *f*_{TM} − *f*_{TM0}). Neglecting the effect of δ*f*_{TM}, the phase difference between RMP and TM can be presented as follows:

Then, the TM frequency oscillation (δ*f*_{TM}) can be calculated from Eq. (4). Consequently, the frequency oscillation of the TM could be represented by

When only the RMP-generated electromagnetic torque is applied to the TM, the TM frequency is linear with cos(Δ*ξ*). Due to the effect of the viscous torque, there exists a phase difference between the TM frequency oscillation and cos(Δ*ξ*), i.e., *ξ*_{vis}, in Eq. (6). Meanwhile, the viscosity coefficient can be calculated with *ξ*_{vis} on the experiments,

The initial condition-related term on the right-hand side of Eq. (6) decreases with time. The decay time is measured in milliseconds and is significantly shorter than the mode locking progress period, so this term can be disregarded. The TM frequency is therefore described as follows:

where *f*_{TM1} is the amplitude of TM frequency oscillation, determined as follows:

When *f*_{TM1} is larger than or equal to the frequency difference Δ*f*_{RMP-TM}, the TM is locked to the RMP. Consequently, the RMP amplitude threshold for mode locking, *B*_{r,LM}, can be calculated as follows:

### B. Experimental verification

Section III A presented the relationship between the TM frequency oscillation and the RMP parameters (frequency and amplitude), which needs to be validated in experiments. A discharge with a rotating RMP, No. 1048 269, depicted in Fig. 5, is used to illustrate the comparison between the calculated and experimental results of the TM frequency oscillation. The present parameters of No. 1048 269 are identical to those of No. 1048 266, resulting in a high degree of experimental similarity between the two discharges. At 0.26 s, a 3 kHz rotating RMP is applied. The RMP amplitude curve depicted in Fig. 5(a) has two flattops. During the lower flattop region, the TM frequency [shown as the solid blue curve in Fig. 5(c)] is oscillated by the rotating RMP for an extended period, ∼50 ms. When the amplitude of the RMP reaches 0.65 Gauss, mode locking occurs. Throughout the higher flattop region, the TM is locked onto the RMP and maintains the same frequency as the RMP. When the RMP amplitude falls below 0.3 Gauss, the TM is unlocked and accelerated to its initial frequency (4.5 kHz).

Calculating the TM frequency oscillation, δ*f*_{TM}, requires five parameters based on Eq. (9)—*j*_{1}, *σ*_{⊥}, Δ*ξ*, *ξ*_{vis}, and Δ*f*_{RMP-TM}—where *j*_{1} is the amplitude of oscillated surface current density generated by the tearing mode. These parameters are measurable based on the experimental data. The measurement progress is described in detail below.

At the TM separatrix, the gradient of plasma flow dominates the viscous force. The plasma flow at the edge region is represented by the toroidal rotation profiles of the impurity CV, *V*ϕ_{-CV}, as shown in Fig. 5(e). When the RMP amplitude is at the lower flattop, the *V*ϕ_{-CV} profile [the blue curve in Fig. 5(e)] completely changes from the initial region *V*ϕ_{-CV} profile [black curve in Fig. 5(e)] to the co-Ip direction. When mode locking occurred, the *V*ϕ_{-CV} profile [the red curve in Fig. 5(e)] continued to change in the direction of co-Ip. After the RMP amplitude reaches zero, the TM frequency recovers to the initial frequency, and the *V*ϕ_{-CV} profile [the cornflower blue curve in Fig. 5(e)] also recovers as the *V*ϕ_{-CV} profile before RMP application. The variation in the *V*ϕ_{-CV} profile influences the natural TM frequency, which is expressed as the average TM frequency, *f*_{avg} [the red curve in Fig. 5(c)]. Consequently, the frequency difference between TM and RMP, Δ*f*_{RMP-TM}, changes over time and is calculated as the difference between *f*_{avg} and the RMP frequency, *f*_{RMP} [the black curve in Fig. 5(c)], during the calculation of δ*f*_{TM}.

The oscillated current generated by the tearing mode is assumed to be primarily on the resonance surface. The oscillating magnetic flux, *ψ*_{2/1}, could then be calculated using the following equation under the vacuum field hypothesis and assumption of a large aspect ratio:

where *r*_{s} is the minor radius of the *q* = 2 surface and *ψ*_{0} is the magnetic flux at the resonant surface. *ψ*_{0} can be calculated with the amplitude of δ*B*_{θ},

where *r*_{coil} is the radial position of Mirnov coils. Then, *j*_{1} can be calculated using Ampere’s law,

There exists an oscillation in the time evolution curve of δ*B*_{θ} when the TM frequency is oscillated by the rotating RMP. The amplitude of the δ*B*_{θ} oscillation is significantly less than that of δ*B*_{θ}. Hence, *j*_{1} remains nearly constant throughout the calculation region. As shown by the blue curve in Fig. 6, the time evolution of the phases of TM and RMP contains some errors introduced by measurement, calculation, and the current of RMP. Typically, these errors are random or oscillate at a high frequency (>5 kHz). To reduce the error, the phase difference, Δ*ξ*, should be filtered by a low frequency pass filter with a cut-off frequency of 20 kHz.

The phase shift *ξ*_{vis} is dependent only on Δ*f*_{RMP-TM} and *σ*_{⊥}, according to Eq. (7). In the experiments, both *ξ*_{vis} and Δ*f*_{RMP-TM} can be directly measured. Figure 6 demonstrates the viscosity coefficient *σ*_{⊥} calculation in detail. When the TM frequency reaches its maximum or minimum value within each Δ*ξ* cycle, it is simple to measure *ξ*_{vis}. Thus, by using formula (14), the viscosity coefficient, *σ*_{⊥}, can be indirectly measured. Based on measurements from 0.3 to 0.35 s, *σ*_{⊥} was about 3.3 ± 0.6 m^{2}/s. The radial position of the resonant surface was 0.9 a, where the toroidal ion rotation was about 1.2 × 10^{4} m^{2}/s. The ion temperature was about 80–100 eV, and the ion density was about 1 × 10^{19} m^{−3} at the resonant surface. Based on the Braginskii equations,^{38} the viscosity coefficient was about 1.2–1.5 m^{2}/s. Then, the experimental value of the viscosity coefficient was reasonable. At this juncture, all parameters (Δ*f*_{RMP-TM}, Δ*ξ*, *ξ*_{vis}, *j*_{1}, and *σ*_{⊥}) required to calculate δ*f*_{TM} have been determined,

Figure 7(b) depicts a comparison between the calculated result, δ*f*_{CAL} (the red curve), and the experimental result, δ*f*_{EXP} (the blue curve). To illustrate the difference between δ*f*_{CAL} and δ*f*_{EXP}, Fig. 7(c) depicts the ratio between them, K_{Exp/Cal}, obtained by dividing δ*f*_{EXP} by δ*f*_{CAL}. In the calculation region, *ξ*_{vis} is determined based on Eq. (7). For the entire region, the phase of δ*f*_{CAL} is the same as that of δ*f*_{EXP}. From 0.3 to 0.35 s, K_{Exp/Cal} is close to 1, confirming the validity of the Sec. III A analytical result. There are significant differences between δ*f*_{EXP} and δ*f*_{CAL}, which will be discussed in Sec. V. In addition, the TM is locked on the RMP at 0.3591 s, when the amplitude of δ*f*_{CAL} is marginally greater than Δ*f*_{EXP}. Based on Eq. (10), the mode locking threshold can be calculated as 0.64 Gauss. The mode locking threshold could be calculated using formula (10) as 0.64 Gauss. The mode locking threshold in the experiment is ∼0.63 Gauss, which is the mean value between 0.358 and 0.36 s. Due to the significant effect of the TM frequency oscillation, the phase difference, Δ*ξ*, evolves nonlinearly with time just prior to mode locking, rendering Eq. (9) invalid. Hence, there is a difference between the analytical and experimental results.

## IV. UNDERSTANDING THE SPECTRUM DURING MODE LOCKING TO THE ROTATING RMP

Based on Eq. (8), the Mirnov signals (*dB*_{θ}/*dt*) can be expressed as follows:

where *NS* is the effective area of the Mirnov probe, δ*B*_{θ} is the perturbed poloidal magnetic field generated by the TM, and *ξ*_{0} is the TM phase at *t* = 0. When δ*f*_{TM} is insignificant, *f*_{TM} can be substituted by *f*_{TM0} during the calculation. The Mirnov signal could be expressed as follows by substituting Eq. (8) for Eq. (15) and retaining only the first order frequency oscillation term:

Short-time Fourier analysis is used to calculate the frequency spectrum, and the window width is a crucial parameter. When the window width is greater than the cycling period of TM frequency oscillation (1/Δ*f*_{RMP-TM}), the frequency spectrum of the Mirnov signal closely resembles the result of Fourier analysis. This is how Eq. (16) could be modified,

where *A*_{mirnov} is just a coefficient, *A*_{Mirnov} = −2*πNSδB*_{θ}. It is evident that the Mirnov signal oscillating at the TM frequency can be separated into three sinusoidal components with respective frequencies of *f*_{TM0}, *f*_{TM} + Δ*f*_{RMP-TM}, and *f*_{TM} − Δ*f*_{RMP-TM}. Figure 8 displays the Fourier analysis results of a predetermined Mirnov signal to explain these results. As shown in Fig. 8(a), the TM frequency is calculated as *f*_{TM} = 4 + 0.2 cos(1000*πt*) kHz. Figure 8(e) identifies these three components in the frequency spectrum as the TM frequency (*f*_{TM0}), higher frequency (2*f*_{TM0}-*f*_{RMP}), and lower frequency (*f*_{RMP}) peaks. When the window width is significantly less than 1/Δ*f*_{RMP-TM}, only a single peak appears in the Fourier analysis of the Mirnov signal. In Fig. 8(c), the Fourier analysis results for the blue filled region and the red filled region are represented by the blue curve and the red curve, respectively, with comparable window widths. These two curves have different major peaks, whose frequencies equal the TM instantaneous frequency. Due to the linear relationship between the Mirnov signal and the TM frequency, the amplitude of the peaks in the spectrum increases with the TM frequency. In this instance, the frequency oscillation of TM was easily discernible in the frequency spectrum. When the window width is equal to 1/Δ*f*_{RMP-TM}, the Fourier analysis yields information regarding TM frequency oscillations. However, the frequency resolution is insufficient to differentiate between three peaks. In this instance, the Fourier analysis tends toward chaos. As depicted in Fig. 8(d)’s green curve, the result of the short-term Fourier analysis is a mixture of two phenomena, and the main peak is widened.

The results of a short-time Fourier analysis with varying window widths are shown in Fig. 9. As shown in Fig. 9(c), when the Hanning window width is 2 ms, the TM instantaneous frequency is presented clearly in the spectrum. As shown in Fig. 9(e), when the Hanning window width is 8 ms, three peaks can be distinguished in the spectrum during mode locking. As shown in Fig. 9(d), when the Hanning window width is 4 ms, all three cases are present in the spectrum.

Based on Eq. (17), the initial phase of the small peak is constant during the calculation, so the phase differences between the various magnetic probes are constant as well. As shown in Fig. 10, a poloidal array of Mirnov probes is used to measure the *δB*_{θ} distributions of each frequency component to explain this. These three components have the same spatial structure, m = 2, indicating that the non-uniform rotation of TM cannot alter the Fourier transform-calculated mode structure. During the Fourier transform, the spatial and the temporal information of the TM are independent of one another.

In numerous devices, mode locking and TM control by a static RMP were investigated. According to Eq. (17), static components should exist in the Mirnov signals in this instance. However, this static component is not visible in the frequency spectrum of Mirnov signals, and it is difficult to extract because it shares the same (low) frequency range as the static RMP field, equilibrium field, etc. To illustrate the static component, a comparison between the low pass filter analysis result and empirical mode decomposition (EMD) is presented. In Fig. 11(a), the equilibrium fields extracted with a low pass filter and EMD are depicted as the red and black curves, respectively. The difference between the results of these two methods of analysis is depicted by the red curve in Fig. 11(c), which nearly coincides with the magenta curve, 5*I*_{RMP} + 1. During mode locking to a static RMP, this phenomenon suggests the presence of a static component. In addition, a high pass filter is typically applied during the signal analysis of rotating TMs, which eliminates this static component naturally. As a result, the filters disregard the static component in most experimental analyses.

Similar to the progression of mode locking to the rotating RMP, the mode coupling resembles the coupling of modes. The spectrum during mode coupling is depicted in Fig. 12(b). Before 0.31 s, the m/n = 2/1 mode frequency is nearly constant at 8 kHz throughout the mode coupling process. The m/n = 3/1 mode rotates asymmetrically, as indicated by the black curve in Fig. 12(c). Between the fundamental peak and the double frequency harmonic peak of the m/n = 3/1 mode, there is a fuzzy peak. This peak, whose frequency equals 2*f*_{3/1} − *f*_{2/1}, is generated by the oscillation of the m/n = 3/1 mode frequency. This phenomenon suggests that the 2/1 mode contributes a magnetic torque to the 3/1 mode during mode locking. Increasing the frequency difference between the 2/1 mode and the 3/1 mode could therefore prevent mode coupling and the corresponding disruption. The experimental findings for this subject will be presented in a separate paper.

## V. DISCUSSION AND SUMMARY

There are three phenomena that cannot be explained by the result of the analysis. The first is the decreasing progress of Δ*f*_{RMP-TM}. The second is the difference between δ*f*_{EXP} and δ*f*_{CAL}. The third is the time evolution of δ*B*_{θ}.

The RMP-generated TM frequency oscillation causes the TM to spend more time in the deceleration phase. As explained in Ref. 25, the time integral of the electromagnetic torque of RMP is negative and manifests as a net decelerate effect on TM. Therefore, if the amplitude of the TM frequency oscillation is low, the net braking effect is minimal, and the average TM frequency, *f*_{Avg}, should decrease gradually. However, *f*_{Avg} [red curve in Fig. 5(c)] decreases more rapidly during 0.27–0.28 s than the latter region in discharge No. 1048 269. Specifically, at 0.26 s, when the RMP amplitude *B*_{r,RMP}^{2/1} reaches 0.4 Gauss, the change rate of *f*_{Avg} during the mode locking process appears to be at its highest, while the amplitude of TM frequency oscillation is close to zero. The difference between the experimental result and the theoretical description may be due to the RMP response. It is hypothesized that the RMP could improve the transport of electrons in the edge region, resulting in a modification of the radial electric field. Typically, the radial electric field is the origin of plasma rotation in the edge region. Consequently, the natural TM frequency could be slowed down even in the absence of electromagnetic torque.

There exists a significant difference between δ*f*_{EXP} and δ*f*_{CAL} during 0.25–0.3 s and 0.359–0.465 s. In the experiments, δ*f*_{EXP} is less than δ*f*_{CAL} from 0.26 to 0.3 s, while K_{Exp/Cal} increases with decreasing Δ*f*_{RMP-TM}. This phenomenon could be the result of the flow-screen effect. According to Fitzpatrick's theory, the “no-slip” constraint should hold during the calculation. Plasma is in a high-slip, flow-screened metastable equilibrium when Δ*f*_{RMP-TM} is large.^{39} In this case, a non-time asymptotic theory should be used to calculate *B*_{r} at the resonant surface, *B*_{r,rs}. Based on function (7) in Ref. 37, the plasma response to the applied field is negatively correlated with plasma rotation and contributes to the weakening effect on *B*_{r,rs}. The electromagnetic torque is linear with *B*_{r,rs}. Then, the amplitude of δ*f* should be less than the result calculated by Eq. (9) and increase with decreasing Δ*f*_{RMP-TM}. The theoretical expectation matches the experimental findings.

In contrast, from 0.359 to 0.465 s, δ*f*_{CAL} is quite different from δ*f*_{EXP}, and K_{Exp/Cal} equals zero. In Sec. III, it is assumed that Δ*ξ* evolves linearly with time. When the TM is locked on the RMP, the phase difference Δ*ξ* is fixed, and the hypothesis is not satisfied. As a result, Eq. (10) fails to explain the observed phenomenon following mode locking.

The detailed time evolution of *f*_{TM} and δ*B*_{θ} is depicted in Fig. 13. The dashed lines represent the points in the experiment where the TM amplitude reaches its maximum value. Based on the theory by Fitzpatrick, the stabilizing effects of RMP are dominated by Δ*ξ*. The RMP contributes to a stabilizing/destabilizing effect on TM during π < Δ*ξ* < 2π/0 < Δ*ξ* < π; these regions are depicted in lilac and pink, respectively, in Fig. 9. As predicted by theory, the TM amplitude reaches its maximum/minimum value when Δ*ξ* equals 0.5 π and −0.5 π. The experimental increasing/decreasing region of the TM amplitude differs significantly from what is predicted by theory. In addition, the rate of change during the oscillating region of TM amplitude is much lower than the rate of TM amplitude increase after mode locking, which is attributed to the destabilizing effect of RMP. These phenomena suggest that the apparent cause of the TM amplitude oscillation is not the RMP. Meanwhile, the δ*B*_{θ} curve evolves almost reversely with the *f*_{TM} curve, according to Fig. 13. There may be strong correlations between the amplitude and frequency of the TM. On the J-TEXT tokamak, the TM amplitude is nearly negatively linear with its frequency. The statistical result provides additional support to this hypothesis.

The progress of mode locking with rotating RMP is investigated experimentally and theoretically in the summary. The instantaneous frequency curves of TM calculated with Eq. (10) and measured in the experiment are highly comparable. Based on the TM instantaneous frequency formula, the spectrum was analytically and experimentally well explained. Two peaks with higher frequency (2*f*_{TM0}-*f*_{RMP}) and lower frequency (*f*_{RMP}) emerge when the window width of FFT is shorter than 1/Δ*f*_{RMP-TM}. For the progression of mode locking by a static RMP, the existence of the static component (*f*_{RMP} = 0), which is accidently ignored by a high pass filter, has been confirmed. Therefore, the EMD is better suited for data analysis during mode locking. Analytical explanations were provided for the independence of the temporal information and spatial information of the TM. Important to the analysis of mode coupling, the phase difference between different probes can identify the mode structure.

## ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China, under Grant No. 2018YFE0309101, the National Natural Science Foundation of China (Contract Nos. 12075096, 12047526, and 51821005), and “the Fundamental Research Funds for the Central Universities,” under Grant No. 2020kfyXJJS003.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Da Li**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal). **Mao Li**: Conceptualization (equal); Data curation (equal); Investigation (equal); Writing – original draft (equal). **Yonghua Ding**: Conceptualization (equal). **Nengchao Wang**: Conceptualization (equal); Writing – review & editing (equal). **Bo Rao**: Funding acquisition (equal); Investigation (equal). **Ying He**: Data curation (equal); Investigation (equal). **Feiyue Mao**: Data curation (equal); Investigation (equal); Writing – original draft (equal). **Chengshuo Shen**: Data curation (equal); Investigation (equal). **Ruo Jia**: Data curation (equal); Formal analysis (equal); Investigation (equal). **Song Zhou**: Data curation (supporting); Investigation (supporting). **Zhengkang Ren**: Data curation (equal); Funding acquisition (equal). **Chuanxu Zhao**: Data curation (equal); Formal analysis (equal). **Abba Alhaji Bala**: Formal analysis (equal); Writing – original draft (equal). **Zhipeng Chen**: Supervision (equal); Writing – review & editing (equal). **Zhongyong Chen**: Supervision (equal); Writing – review & editing (equal). **Zhoujun Yang**: Data curation (supporting); Writing – review & editing (equal). **Lin Yi**: Supervision (equal); Writing – review & editing (equal). **Kexun Yu**: Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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