External RMP effect on locked-mode-like instability in helical plasmas

The slowing-down mechanism of the locked-mode-like instabilities with and without an island structure is investigated through the effects of an external RMP (resonant magnetic perturbation) on the instabilities. For both instabilities, the slowing-down duration decreases with the increase in the external RMP, and the RMP dependence is consistent with the braking model of the j × B force due to the interaction between the instabilities and the external RMP. Moreover, the relationship between the amplitude and the frequency of both locked-mode-like instabilities during the slowing down is consistent with the force balance model between the j × B force due to the external RMP and a viscous force. These results suggest that the slowing down of both locked-mode-like instabilities with finite external RMP occurs due to the j × B force driven by the external RMP.


Introduction
In the LHD (large helical device), a reactor-relevant plasma with a volume-averaged beta value of 5% can be stably maintained for more than ten times as long as the confinement time [1]. On the other hand, collapse events due to various MHD instabilities observed in different operational regimes cause serious degradation of confinement property: the torusoutward magnetic axis shift configuration (R ax > 3.75 m) with a peaked pressure profile [2], the torus-inward shifted configuration (R ax < 3.60 m) [3] and the standard magnetic axis configuration (R ax = 3.60 m) with a low magnetic shear. In particular, in low magnetic shear discharges, the frequency of the precursor decreases, and the collapse occurs. This event is called the locked-mode-like instability [4]. In high beta discharges of the LHD, the plasma pressure gradient is main- * Author to whom any correspondence should be addressed. tained without the collapse events even if linear MHD analysis predicts that the interchange mode is unstable. In a helical plasma, investigating the physical mechanism of the collapse and the relationship between the index of linear MHD instability and the collapse is useful for reflecting MHD stability characteristics of the high beta LHD discharges on the design of a helical fusion plasma.
Two types of locked-mode-like instabilities whose precursors have different radial mode structures are observed in different regimes of a magnetic shear and a beta value of the LHD [4][5][6]. Regarding the radial profile of internal fluctuations, the precursor of the first type has the odd-type structure with respect to the resonant surface. This structure is similar to the tearing mode, which is considered to have the large magnetic island (type-I, tearing-type locked-mode-like instability). The other instability has the even-type structure, similar to the interchange mode which is considered to not have the large magnetic island (type-II, interchange-type locked-mode-like instability).
In locked mode discharges of tokamaks [7] and RFPs [8], it is demonstrated that the disruption is escaped by maintaining the rotation of a precursor of the locked mode by using a rotating external RMP [9] and/or tangential NBIs. These results suggest that the slowing down of the precursor could be related to the trigger mechanism of the collapse.
In the LHD, in order to establish the method of avoiding the collapse, the slowing-down mechanism of the MHD fluctuations due to the locked-mode-like instability has been investigated. From the comparison between the frequency of the precursor and the E × B flow at the resonant surface during the slowing down phase, the precursor's frequency is almost the same as the E × B flow around the resonant surface [5]. It is found that the decrease in the E × B flow around the resonant surface occurs through the following two stages [5]. The slowing-down of the first stage (Δt first ) is caused by the movement of the resonant surface to the plasma core region with a small E × B flow due to increasing the plasma current in the codirection because the plasma current in the codirection enhances the increase of the core rotational transform. The slowing down of the second stage (Δt second ) occurs due to the decrease of the plasma flow around the resonant surface but the resonant surface moves slightly. The above two stages are observed in both of the two types of instabilities [6]. Figure 1 shows the time evolution of (a) the amplitude of the magnetic fluctuation due to the m/n = 1/1 mode measured by a magnetic probe, (b) the mode frequency, (c) the radial magnetic fluctuation amplitude measured by saddle coils, (d) the plasma current and (e) the radial profile of an E × B flow and the radial location of the ι/2π = 1 resonant surface in the interchange-type locked-mode-like instability discharge. Here, m and n denote the poloidal and the toroidal mode number, respectively. Furthermore, it is found that the duration of the slowing-down phase (Δt first + Δt second ) decreases with the increase of the external RMP amplitude in interchange-type locked-mode-like discharges [6]. However, the reason for the decrease of an E × B flow in Δt second is unclear.
In the locked mode of tokamaks, several slowing-down models are proposed [10]. The well-known slowing-down force is the j × B force. Two types of j × B forces are considered depending on what induces B: F rw and F RMP . The former is the slowing-down force due to the interaction between the toroidal perturbed current (δj t ) due to the instability and the perturbed magnetic field due to the eddy current flowing in the resistive wall of the vacuum vessel. The latter is the force due to the interaction between δj t and the perturbed magnetic field due to the external RMP coils. In the JT-60U tokamak, the relationship between the magnetic fluctuation amplitude and frequency of the precursor during the slowingdown phase is consistent with the prediction of the F rw model, suggesting that the contribution of F rw to the slowing down is large [11].
In this paper, when an amplitude of the imposed external RMP is changed, the duration time of the slowing-down phase, and the relationship between the magnetic fluctuation amplitude and frequency of the precursor are obtained in lockedmode-like instability with different internal mode structures. The experiment results are compared with the F rw and F RMP models.
This paper is organized as follows. In section 2, the experimental setup is explained. In section 3, the experiment results regarding the locked-mode-like instability are analyzed. In section 3.1, the effects of the external RMP on the slowingdown duration time of the interchange-type instability are shown. In section 3.2, the RMP dependence of the slowingdown duration is compared with the slowing-down models considered in tokamaks. In section 3.3, the experiment results of the tearing-type instability are shown and the effect of the internal structure of the instability on the slowing down is discussed. In section 3.4, the relationship between the magnetic fluctuation amplitude and the frequency of a precursor of both instabilities is shown. The summary and discussion are presented in section 4.

Experimental setup
The locked-mode-like instability typically occurs in low magnetic shear plasmas of the LHD. When the plasma aspect ratio A p increases, the magnetic shear in the whole region of a plasma decreases [12]. In this experiment, A p is set to 7.1, which is higher than the configurations with A p = 5.8-6.6, where the reactor-relevant high beta discharges are achieved in the LHD.
For the production and heating of plasmas, two tangential NBIs are used. The plasma current increases during discharges due to two tangential NBIs with the same direction and the core rotational transform increases, leading to the decrease in the magnetic shear. Perpendicular NBIs are modulated for measurement of an E × B flow by a charge exchange spectroscopy.
The toroidal and poloidal mode number of magnetic fluctuations are identified by a toroidal array with six magnetic probes and a helical array with 15 probes outside a plasma [13]. For evaluation of the amplitude and frequency of a magnetic fluctuation, one of the toroidal array probes located ∼0.3 m away from the ι/2π = 1 surface is used. The slowly changing radial magnetic fluctuation amplitude is measured by two arrays of saddle loops with a large cross-sectional area. The effective area through which the radial magnetic flux passes is ∼0.4 m 2 . The averaged distance between the saddle loop and the ι/2π = 1 surface is ∼0.5 m.
The definition of Δt first is the duration when the ι/2π = 1 surface largely decreases, and that of Δt second is when the ι/2π = 1 surface does not largely change but the E × B flow velocity at the ι/2π = 1 surface decreases. Therefore, accurate evaluation of the radial location of the ι/2π = 1 surface is important. During the slowing-down phase in the locked-mode-like instabilities, the small flattening region in the radial electron temperature profile appears due to the m/n = 1/1 precursor. The Thomson scattering system with high spatial resolution can observe the time evaluation of the flattening. The RMP coil system in the LHD, which has ten vertical pairs of coils at the top and the bottom, can calibrate the intrinsic error field. According to measurement of the magnetic surface mapping in vacuum [14], there is an m/n = 1/1 magnetic island due to the intrinsic error field, which is almost corrected by the external RMP with an RMP coil current (I RMP /B t ) of 110 A/T. Namely, the magnetic island due to the intrinsic error field shrinks to a smaller size. In this paper, I RMP /B t is changed from 0 to 100 A/T. As I RMP /B t increases, the corrected error field amplitude decreases, which means that a positive sign of I RMP /B t corresponds to the opposite phase to the error field. It should be noted that I RMP /B t and the error field amplitude have a negative correlation. Figure 1 shows a typical waveform of the interchange-type locked-mode-like instability discharges where the intrinsic error field is almost cancelled by imposing an external static RMP of I RMP /B t = 100 A/T. Next, when the amplitude of the imposed external RMP is changed, behaviours of the m/n = 1/1 mode as the precursor are explained.  of the slowing-down phase, respectively. It should be noted that the timing/amount of gas puffing, the magnetic configuration and the heating condition are almost the same as those in figure 1 except for I RMP /B t . In figures 1 and 2, the error field amplitude increases when I RMP /B t decreases. From figures 1 and 2, it is found that Δt first is not largely changed, but Δt second decreases as the error field increases. Figure 3 shows the external RMP dependence of (a) Δt first and (b) Δt second . Circles display several discharges of the same I RMP /B t and cross symbols correspond to the averaged value of the discharges of each I RMP /B t . There is no clear dependence of Δt first on the amplitude of the external RMP in the 0 to 100 A/T region. On the other hand, Δt second decreases as I RMP /B t decreases. It is found that the external RMP dependence of the duration of the slowing-down phase (Δt slowing = Δt first + Δt second ) as reported in reference [6] reflects the external RMP dependence of Δt second . It is interesting that Δt first for I RMP /B t = 100 A/T is different from the other cases. The duration Δt first is determined by the change rate of the current profile and the time when the radial movement of the resonant surface stops. The effect of the external RMP on the change rate and the time as shown in the above is outside the scope of this paper and is a topic for future research.

Comparison between slowing-down models and experimental observation of locked-mode-like instability
In tokamaks, the slowing-down of the precursor of the locked mode is considered by j × B forces [10], as shown in section 1. There are two types of j × B forces, F rw and F RMP , due to a perturbed magnetic field induced by an eddy current on the resistive wall of a vacuum vessel (B eddy ) and by the external coils (B RMP ), respectively. Assuming that B eddy is proportional to a perturbed current due to the precursor, δj t , and its rotation angular frequency ω, (1) Here, δj t is assumed to be proportional to the radial magnetic fluctuation amplitude δb r measured outside a plasma and τ w is the wall time constant. Assuming that B RMP is proportional to I err /B t − I RMP /B t and the penetration of the external RMP into a plasma is shielded due to the rotation of the external RMP in the frame of the precursor, F RMP is expressed as Here, I RMP /B t of 110 A/T corresponds to the compensation coil current for the intrinsic error field, as shown in section 2. In addition, ω 0 is the slip frequency on the external RMP frame and τ rec is the typical reconnection timescale [10]. In this study, ω = ω 0 because the external RMP does not rotate in the laboratory frame.
As mentioned above, δb r plays an important role in both F rw and F RMP . Therefore, the dependence of I RMP /B t on δb r time-averaged during Δt second of the interchange-type lockedmode-like instability is shown in figure 4. It is clearly found that δb r increases with I RMP /B t in the 0 to 100 A/T region. Note that δb r is the amplitude of the minor radial component of the magnetic fluctuation due to the m/n = 1/1 mode measured by a magnetic probe and the time-averaged value of δb r during Δt second cannot be evaluated if Δt second = 0. Figure 5 shows relationships between Δt second and (a) F rw and (b) F RMP . The forces are the time-averaged value during the second stage of the slowing-down phase. When a force strongly contributes to the slowing down, the slowing-down time decreases with the increase of the force. In other words, the correlation between the force and the slowing-down time is negative. From figure 5, Δt second increases with F rw , but Δt second decrease with the increase in F RMP . These results suggest the contribution of F RMP to the slowing down is large against F rw . Here, τ w is assumed to be 1 ms. Since τ w is the resistive skin time of the vacuum chamber, it is not changed by the instability. If a different value for τ w is selected, the change does not affect the above qualitative behaviour.

Effect of internal mode structure of locked-mode-like instabilities on slowing-down mechanism
The previous sections show experiment results of the interchange-type locked-mode-like instability. From here, the results of the tearing-type locked-mode-like instability are shown. The tearing-type instability appears in the region of a higher magnetic shear and a lower volume-averaged beta value than the parameter region where the interchange-type instability appears [6]. In order to clarify the slowing-down mechanism of the tearing-type locked-mode-like instability, the external RMP dependences of the slowing-down time of the tearing-type locked-mode-like instability are investigated. Figure 6 shows the external RMP dependences of Δt first and Δt second . Similar to the interchange-type locked-mode-like instability, Δt first of the tearing-type locked-mode-like instability does not depend on I RMP /B t , but Δt second increases with I RMP /B t . Figure 7 shows the dependence of I RMP /B t on δb r of the tearing-type locked-mode-like instability, together with the experimental data of the interchange-type instability as already shown in figure 4. For the tearing-type instability as well as the interchange-type instability, δb r increases with I RMP /B t . It is interesting that δb r of the interchange-type instability is larger than that of the tearing-type instability at the same I RMP /B t . The external RMP drives the locked mode in tokamaks, while it suppresses the tearing mode appearing before the mode locking. Similarly, in the locked-mode-like instability discharges,  the suppression of the precursor during the slowing-down by the external RMP is also observed. However, the suppression mechanism is a future work. Figure 8 shows (a) the F rw dependence and (b) the F RMP dependence of Δt second for the tearing-type locked-mode-like instability together with the interchange-type locked-modelike instability. Similar to the interchange-type locked-modelike instability, Δt second increases as F rw increases, but Δt second decreases as F RMP increases. This result suggests that F RMP has a large contribution to the slowing-down regardless of the internal mode structure of the instability, that is, the size of the magnetic island of the instability. However, quantitatively, it can be seen that Δt second of the tearing-type instability is shorter than that of the interchange-type instability at the same F RMP . In order to understand the reason for the quantitative difference, additional data and more systematic analyses are required in future.

Relationship between amplitude and frequency of precursor during slowing-down phase
Next, the relationship between the frequency (f ) and magnetic fluctuation amplitude (δb/B t ) of the precursor during the slowing-down phase is investigated. According to the JT-60U tokamak experiment study, the relationship between f and δb/B t of the tearing mode, which is the precursor of the locked mode, is almost consistent with the equation of the F rw model [11]. This result suggests that the contribution of F rw to the slowing-down of the precursor of the locked mode is large. The δb/B t -f relationship of the interchange-type instability during the sum of Δt first and Δt second has already been reported [6]. However, there was no conclusive result. Here, the δb/B t -f relationship only during Δt second is focused upon, and is compared with the relationship based on the following force balance models.
In the force balance model of F rw and a viscous force (F vc ), the relationship between the mode frequency and the magnetic fluctuation amplitude of the mode is expressed in [10]. The viscous force is expressed as where f 0 corresponds to the rotation frequency due to the neoclassical flow. The force balance between F rw and F vc is Substituting equations (1) and (3) into equation (4) yields In addition, the δb/B t -f relationship based on the force balance of F RMP and F vc is also derived.
Here, ωτ rec 1 is assumed since ω is several krad s −1 and τ rec ∼ 0.2 s in the typical LHD plasma parameters.
On the other hand, if the external RMP is completely penetrated, F RMP,no-slip and the δb/B t -f relationship based on the F RMP,no-slip model is derived as follows [10].
It should be noted that the δb/B t -f relationship based on the no-slip model of F RMP has the same dependence with the one based on the slip model in the limit of ωτ rec 1. Here, α rw , β rw , α RMP , β RMP, α RMP,no-slip and β RMP,no-slip are free parameters. Note that α rw , α RMP, α RMP,no-slip (each β) are related with the proportionality constant of F rw , F RMP and F RMP,no-slip (F vc ) except δb and f , respectively. Figure 9(a) shows the δb/B t -f relationship of the interchange-type locked-mode-like instability with the external RMP of I RMP /B t = 50 A/T during Δt second . Black and grey symbols correspond to the typical behaviour and a transient behaviour of δb/B t and f , respectively. The transient behaviour is observed for a short time just after the precursor suddenly appears. The intermittent appearance of the precursor is often observed in discharges with external RMP as shown in [6]. The blue dash-dotted lines display the equations of the F rw model with (α rw , β rw ) = (1.0 × 10 8 , 0.8) and (2.5 × 10 7 , 0.8), and the red dashed line displays the equation of the F RMP model with (α RMP , β RMP ) = (6.6 × 10 1 , 1.7). The δb/B t -f relationship is consistent with F RMP , but it is not consistent with the F rw model. This result suggests that the F RMP model is closer to the experimental data when comparing the two models. Figure 9(b) shows the δb/B t -f relationship of the tearingtype instability with the external RMP of I RMP /B t = 50 A/T, which coincides with the F RMP model of (α RMP , β RMP ) = (6.6 × 10 1 , 1.4). These results suggest that the F RMP model is

Summary and discussion
In order to clarify the slowing-down mechanism of the precursor in the locked-mode-like instability of the LHD, the effect of the external RMP on the slowing down is investigated. The slowing-down phase can be divided into two stages depending on the slowing-down processes. In the first stage of the slowing-down phase, the E × B flow at the resonant surface decreases due to the movement of the resonance surface to a small E × B flow region, and in the second stage, the resonance surface is almost constant, but the E × B flow around the resonant surface itself decreases. This characteristic was already found in the previous research, but it is unclear how the duration of the second stage is determined. It is investigated how the duration of the second stage changes by imposing the external RMP, and the RMP dependence of the duration of the second stage is compared with the slowing-down models for a precursor of the locked mode in tokamaks: F rw and F RMP . The former is the j × B force caused by the interaction between a perturbed current due to the instability and an RMP field created by its eddy current, and the latter is the j × B force caused by the interaction between the perturbed current and an RMP field due to the external coils at the resonant surface. In addition, F RMP with the complete penetration of the external RMP into the resonant surface (F RMP,no-slip ) is also considered.
It is found that the duration of the second stage decreases as the effective external RMP amplitude increases. The duration time of the second stage has a negative correlation with F RMP , but a positive correlation with F rw . Thus, the above results suggest that F RMP make a large contribution to the slowing down of the precursor.
Furthermore, the relationship between the magnetic fluctuation amplitude of the precursor and its frequency during the second stage with the external RMP is not consistent with that predicted by the F rw model, but it is consistent with the F RMP model. This result supports the statement that the contribution of F RMP to the slowing down is larger than F rw .
The above are the experimental results of the interchangetype locked-mode-like instability, which has a magnetic island with a small width. The same qualitative results for the tearingtype locked-mode-like instability are obtained. These results suggest that the slowing down of the precursor is mainly caused by the j × B force driven by the external RMP regardless of the size of the magnetic island of the instability.
According to an early work, it is reported that the penetration of the external RMP into a plasma is shielded when the external RMP rotates in the frame of the precursor, and that the B RMP is reduced at the resonant surface and F RMP is smaller than that F RMP,no-slip in the complete penetration case. The Δt second dependence on F RMP shown in figure 5(b) and 8(b) is almost the same against the F RMP,no-slip in the limit of ωτ rec 1. The evaluation of experimental B RMP at the resonant surface during the shielding of the penetration of the external RMP is a future research topic.
From the quantitative aspect, the duration of the second stage of the tearing-type locked-mode-like instability looks shorter than that of the interchange-type instability at the same F RMP . Here, δj t is assumed to be proportional to δb r observed outside a plasma. The interchange-type locked-mode instability is expected to have an odd-function-type δj t profile, which would have larger |δj t | around the resonance surface than in the tearing-type instability for the same δb r . The improvement in the quantitative accuracy for F RMP of the interchange-type locked-mode-like instability is a future task.
In this study, it is found that the contribution of the electromagnetic force due to the external RMP to the slowing down of the precursor is large, from the dependence of the slowingdown time on the external RMP. On the contrary, even in the discharges with small external RMP (I RMP /B t = 100 A/T), the precursor frequency decreases for finite Δt second . According to a previous work on the external RMP effects on the locked-like mode instability, the locked location of the precursor is consistent with the phase of the error field when the finite error field exists. On the other hand, the locked locations of the precursor are not fixed when the error field is almost compensated [15]. The behaviour suggests that the slowing-down mechanism on the precursor of the locked-mode-like instability in the case without the error field is different from that with the error field. The slowing-down mechanism in the discharges with the small external RMP should be resolved in future.