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Effects of an external RMP (Resonant Magnetic Perturbation) on the slowing-down duration time ($\Delta t_{\rm slowing}$) of locked-mode-like instabilities with and without the island structure are investigated in LHD. For both instabilities, the $\Delta t_{\rm slowing}$ decreases with the increase of the external RMP and the RMP dependence is consistent with the braking force model due to the interaction between the instability and the external RMP, $F_{\rm RMP}$. In contrast, the braking force model due to the interaction between the instability and the resistive wall of the vacuum vessel is not consistent with the relationship between the amplitude and the frequency of both locked-mode-like instabilities during the slowing-down. These results suggest that the slowing-down of both locked-mode-like instabilities occurs due to the $F_{\rm RMP}$.

Locked-mode-like instabilities are observed in low magnetic shear regime of LHD operation as MHD instabilities with a collapse similar to the locked mode instabilities of tokamaks, which sometimes leads to the disruption. After a $m/n=1/1$ precursor appears, its frequency decreases with the increase of its amplitude and a non-rotating magnetic island rapidly grows, resulting in the minor collapse in the core region as shown in Fig.1. Here, $m$ ($n$) is poloidal (toroidal) mode number. In both tokamaks and LHD, the collapse occurs after the slowing-down of the instability, thus it is important to clarify the physical mechanism of the slowing-down. This research is expected to contribute not only to understanding the collapse occurrence mechanism in helical plasmas but also to understanding the disruption occurrence mechanism due to the locked mode instability in tokamaks.

In the previous study $[1]$, the mode frequency of the instability during the slowing-down is compared with the $\vec E \times \vec B$ rotation frequency, and when no external RMP is applied, these frequencies are almost the same, but when the external RMP is applied, even if the mode frequency become zero, it is found that the $\vec E \times \vec B$ rotation frequency has a finite value. In this study, the relationship between the $\Delta t_{\rm slowing}$ and the external RMP, and the relationship between the frequency and the amplitude of the instability during the slowing-down are compared with those predicted by the locked mode braking force models, and the slowing-down mechanism is considered.

Locked-mode-like instabilities are categorized into two types according to the internal structure of the precursor during the slowing-down $[1]$. One is the instability having the tearing-parity structure in the radial displacement, which is considered to have the island structure as well as the locked mode instability in tokamaks. The other is the instability having the interchange-parity structure, which is considered to be without the island structure. As shown by an arrow in Fig. 1, the $\Delta t_{\rm slowing}$ is defined as the time from the start of the slowing-down to the stop. Figure 2 shows the RMP dependence of the $\Delta t_{\rm slowing}$. Here it should be noted that the external RMP with $I_{\rm RMP}/B_{\rm t} \sim 120 \rm A/T$ almost compensates the intrinsic error field caused by coil misalignment ($I_{\rm RMP}/B_\rm t$; RMP coil current, $B_\rm t$; toroidal magnetic field). Therefore, the effective external RMP, which is the external RMP calibrated by the intrinsic error field, is proportional to the $I_{\rm RMP,eff}/B_{\rm t} \equiv 120 - \it I_{\rm RMP}/B_{\rm t}$. The $\Delta t_{\rm slowing}$ of both instabilities with and without island structure shortens as the effective external RMP increases. This result suggests that the effective external RMP affects the $\Delta t_{\rm slowing}$ regardless of the magnetic island size of the instability.

The slowing-down of the locked mode instability of tokamaks is considered to be caused by the electromagnetic breaking forces $[2]$. One is due to the interaction between the toroidal perturbed current of the instability ($j_\rm t$) and the effective external RMP proportional to $I_{\rm RMP,eff}/B_\rm t$, $F_{\rm RMP}$. The other is due to the interaction between the $j_\rm t$ and the external RMP induced by the instability through the eddy current in a resistive wall of a vacuum vessel, $F_{\rm rw}$. Since the $F_{\rm RMP}$ occurs as the $\vec j \times \vec B$ force, the amplitude of the $F_{\rm RMP}$ is proportional to $\delta b_{\rm meas} \times I_{\rm RMP,eff}/B_\rm t$ assuming that the $j_{\rm t} \propto \delta b_{\rm meas}$. Here the $\delta b_{\rm meas}$ is the time averaged value of the perturbation of the magnetic field measured outside a plasma during the slowing-down. Figure 3 shows the relationship between the $F_{\rm RMP}$ and the $\Delta t_{\rm slowing}$. The relationships represent a negative correlation for both instabilities with and without island structure, which suggests that the slowing-down occurs due to the $F_{\rm RMP}$ in both instabilities. Moreover, it is found that the relation curves between $F_{\rm RMP}$ and $\Delta t_{\rm slowing}$ for both instabilities with and without island structure appear to be almost the same.

The $F_{\rm rw}$ model $[2]$ shows that the relationship between the $\delta b_{\rm meas}$ and the frequency ($f$) is described as $f \propto 1 + \{1 - k (\delta b_{\rm meas})^2)\}^{0.5}$. Here, the $k$ is a free parameter. In the JT-60U discharges, the observed relationship is consistent with the above model equation $[3]$. Figure 4 shows the relationship of the locked-mode-like instability with the island structure during the slowing-down. It is noted that the instability without the island structure has a similar relationship $[1]$. The relationship is not consistent with the $F_{\rm rw}$ model shown by red curves in Fig.4, for both locked-mode-like instabilities with and without island structure.

$[1]$ Y. Takemura et al., Nuclear Fusion $\bf 59$ (2019) 066036.

$[2]$ R. Fitzpatrick, Nuclear Fusion $\bf 33$ (1993) 1049.

$[3]$ A. Isayama et al., Plasma and Fusion Research $\bf 5$ (2010) 037.

Affiliation | National Institute for Fusion Science |
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Country or International Organization | Japan |