Characteristics of MHD instabilities limiting the beta value in LHD

Effects of low-n magnetohydrodynamic instabilities on plasma performance have been assessed in the regime where an achieved beta value is limited by instabilities. The unstable regime of an ideal interchange mode is characterized by enhanced magnetic hill and reduced magnetic shear. Experiments have clarified that (i) low-n modes are significantly destabilized in the ideal-unstable configurations and lead to degradation of central beta by at most 60%, and (ii) the degree of their damages strongly depends on the mode rotation velocity. The occurrence of the minor collapse is independent of an existence of an error field.


Introduction
Suppression of magnetohydrodynamic (MHD) instabilities is one of key subjects to be solved for realizing the production of high-beta plasma in magnetic confinement systems. In tokamaks and reversed field pinches (RFPs), MHD instabilities driven by plasma current and/or plasma pressure obviously degrade plasma confinement and limit the achieved beta value, and techniques for controlling them have been investigated for a long time. The plasma rotation control is one of the candidates for the mode stabilization because experiments show that MHD modes such as tearing, external kink and so on, abruptly grow and lead to major disruption after the mode rotation breaking. The rotation of the tearing mode is decelerated and stopped by the error field, and it leads to major disruption as a 'locked mode' [1,2]. The restive wall mode, which is the instability to be suppressed for the sustainment of high-beta plasma, can be suppressed by the plasma rotation [3]. Thus, the plasma flow driving the mode rotation is key for stabilization of the modes, and effects of the toroidal flow driven by neutral beams and rotating magnetic perturbations on the mode stabilization has been verified through a variety of experiments in tokamaks [3].
In heliotron configurations, interchange instability is one of the major concerns for high beta plasma production because of magnetic hill at the plasma where a strong pressure gradient is formed with an increase in the beta value. The characteristics of the interchange mode strongly depend on the height of the magnetic hill and the strength of the magnetic shear. It is predicted theoretically that the linear growth rate of the interchange mode strongly depends on the beta (gradient) and magnetic Reynolds number, S [4,5], where S is defined as τ R /τ A , and τ R and τ A are magnetic diffusion time and Alfven transit time, respectively. Previous experiments in various magnetic configurations of heliotron-DR with low S (~10 5 ) indicate that the achieved beta value was limited by the stability boundary of the resistive interchange mode [6]. In compact helical system (CHS) experiments with low-S (~10 5 ), no strong MHD instability was observed, whereas amplitudes of modes observed in CHS are one order higher than those observed in the large helical device (LHD) with high S (~10 8 ) [7]. The S dependence of the mode amplitude was investigated in configurations with the constant beta value under the ideally-stable condition, which shows that the saturation level of the mode is proportional to S −1/3 and has the same dependence of the linear growth rate of the resistive interchange mode [4]. These results suggest that the interchange mode can be suppressed further in the reactor-relevant regime with high S. However, since the growth rate of the ideal interchange mode is independent of S, verification of the significance of the ideal stability boundary remains a major problem to be solved in helical devices. Here we focus on the effect of the mode on the confinement in the regime where the destabilization of the ideal interchange mode is predicted by linear theory.
The access to the ideal-unstable regime is possible by enhancing the magnetic hill and/or by reducing the magnetic shear. The former can be realized by shifting the magnetic axis inwards, while the latter is realized by increasing the plasma aspect ratio and plasma current in the co-direction. The inward shift configuration has an advantage for good transport properties because of a decrease of the helical ripple, which suggests that the configuration is favorable for not only reduction of thermal transport but also confinement of the alpha particle in the helical fusion reactor [8]. Therefore, to what extent MHD activity can be suppressed in the inward shifted configuration with enhanced magnetic hill should be clarified through actual experiments. The magnetic shear can be easily reduced by plasma currents flowing in the co-direction. It has been predicted that the bootstrap current flows in the co-direction and is enhanced by reduction of collisionality [9], which makes the magnetic shear weaker in the reactorrelevant plasmas with low collisionality compared to that in the high collisional plasmas of the present devices. Therefore, we should verify the actual effects of MHD activities in the ideal unstable regime for optimization of the magnetic configuration for reactor design.
In this study, the effects of low-n MHD instabilities on plasma performance have been investigated in two kinds of ideal unstable regimes realized by enhancing the magnetic hill and reducing the magnetic shear. The previous stability analyses of enhanced magnetic hill and reduced magnetic shear discharges show that the onset of the observed instabilities is qualitatively consistent with the ideal stability boundary [10,11]. Figure 1 shows a schematic view of how to approach the ideal unstable regime. In the experiments, we changed the configurational parameters (R ax and I p ) characterizing MHD instabilities based on the optimized configuration for highbeta plasma production (blue circle). The experimental setup is introduced in section 2. The experimental results in the enhanced magnetic hill and weak magnetic shear configurations are described in section 3. The experiments for investigating the effect of the error field on the modes also are shown in section 3. Section 4 gives the discussion and summary.

Experimental setup
The LHD is a heliotron device with a pair of helical coils and three pairs of poloidal coils. All coils are superconductive [12]. The typical major and minor radii are 3.6 m and 0.65 m, respectively. The available toroidal magnetic field at R ax , B t , is up to 3 T. The plasma aspect ratio, A p , can be changed by controlling the central position of the helical coil current and it can be selected from 5.8 to 8.3 when the magnetic axis position in vacuum, R v ax , is 3.6 m. The A p of 5.8 and R v ax of 3.6 m have been used as the standard configuration, whereas A p of 6.6 and R v ax of 3.6 m have been applied for the high-beta operation [13]. The R v ax can be set at 3.5-4.1 m and the temporal movement of R v ax is available during a discharge by changing the poloidal coil currents. The target plasmas were produced and maintained by neutral beams (NBs) and the total power is about 15 MW. The H 2 gas-puff was applied in the experiments.
In the experiments with enhanced magnetic hill, the R v ax was moved from 3.6 m to 3.5 m in 2 s in the configuration with B t of 0.425 T. The A p was set at 6.6. Figure 2 shows profiles of rotational transform and V″ in the configurations with R v ax of 3.6, 3.55 and 3.5 m, where V″ = d 2 V/dΦ 2 , and V and Φ are the volume and the toroidal magnetic flux, respectively. The negative V″ corresponds to the magnetic hill. The pressure profile is assumed as P = P 0 (1 − ρ 2 )(1 − ρ 8 ), where ρ is normalized minor radius. The central beta, β 0 , is 3.3% in all investigated configurations. In the R v ax = 3.6 m case, the resonant surface of the m/n = 2/1 mode is located at ρ of 0.28 where the magnetic well is formed, where m and n are the poloidal mode and the toroidal mode numbers, respectively. The 2/1 resonant surface is shifted inward when R v ax is changed from 3.6 to 3.55 m, and then the surface is moved from the magnetic well to magnetic hill regions. In the case of R v ax = 3.5 m, the resonance vanishes by increasing the central rotational transform. The m/n = 1/1 mode is predicted to be ideally stable because of the relatively strong magnetic shear.
In the reduced magnetic shear experiments, the I p was driven by two co-NBs in the configuration with R v ax = 3.6 m and A p of 7.1. The ι/2π = 1 resonant surface is located at ρ of 0.7 and the magnetic shear, 1/ι(dι/dρ), is 1.66 in the vacuum configuration. The increases of I p and A p contribute to the reduction of magnetic shear and change a key instability from m/n = 2/1 to the 1/1 mode because the ι/2π = 0.5 resonant surface vanishes due to the increase in central rotational transform and the ι/2π = 1 surface moves to the core (figure 3). This movement decreases V″ as shown in figure 3(b). In the experiments, it has been observed that the m/n = 1/1 mode appears with I p ramp-up and leads to minor collapse in the core [13].
The previous experiments clarified that the onset of the mode depends on beta, A p and I p [11].
The experiments for investigating the effect of the error field on MHD characteristics were done in the configuration R v ax = 3.6 m and A p = 5.8. The ι/2π = 1 surface is located at ρ of 0.88, and the 1/ι(dι/dρ) is about 2.6 in vacuum. In LHD, the natural error field producing the m/n = 1/1 magnetic island exists, which was identified by electron beam mapping experiments [14]. The size of the island is about 23 cm in the horizontally elongated cross section and the toroidal angle where the O-point is located at the low field side is about −126 degrees. The resonant magnetic perturbation (RMP) system on the LHD consists of ten pairs of coils located at the top and bottom of the LHD main body. The dominant Fourier component is m/n = 1/1. The magnetic island due to the error field corresponds to that produced by the RMP current of 0.1 kA when B t = 1 T, which was applied as the 'reduced' error field operation described in section 3.4. The studies on the interaction between RMP field and MHD characteristics are summarized in [11,15].
The β 0 and profiles of T e and electron density, n e , are measured with the Thomson scattering system. The line averaged electron density is measured with far infrared (FIR) interferometer. The MHD modes are identified by a toroidal array with 5 magnetic probes and a helical array with 16 probes. The two saddle loop arrays each with 12 loops were used for identifying the size and spatial location of the m/n = 1/1 magnetic island [16].

MHD activity in enhanced magnetic hill configurations
The experiments were done in the configuration with B t of 0.425 T. Figure 4 shows a typical discharge with real-time movement of R v ax inward in order to enhance the magnetic hill at the ι/2π = 1/2 surface. The R v ax is varied from 3.6 m to 3.5 m in 2 s. The central beta value starts to decrease when R v ax comes close to 3.53 m. The rotating m/n = 2/1 mode appears before the drop of the central beta, and the amplitude is increased with deceleration of the mode rotation. At 3.36 s, the mode rotation is completely stopped. Although the mode rotates and stops by turns, no recovery of the central beta is observed. Figure 5 shows the comparison of the T e profiles before and after the stop of the rotation. The T e profile is locally flattened at the ι/2π = 1/2 resonance (R = 3.4 m and 3.85 m) before the stop of the rotation (3.3 s), which is caused by rotating m/n = 2/1 activity. After the stop of the rotation, the flattening of the profile is extended to the core region, which drops the central beta by more than 30%. It has been found that the mode can be stabilized when the R v ax is moved outwards during the discharge if the mode rotation is maintained.   Figure 6 shows a typical discharge in the reduced magnetic shear configuration with B t of 1.25 T. The I p is induced by neutral beams in order to decrease the magnetic shear in the high aspect ratio plasma configuration with low shear. While the bursts of the m/n = 1/1 mode appear and drop the central beta, the mode rotation slows down when the I p exceeds about 28 kA. The minor collapse occurs after the stop of the rotation at 4.5 s, which is similar to the case of the m/n = 2/1 mode observed in the enhanced magnetic hill configuration described in section 3.1. Then the β 0 decreases by about 60%. The T e profiles before and after the collapse are shown in figure 7. The rotating m/n = 1/1 mode forms the profile flattening around the ι/2π = 1 resonance (3.2 m and 4.0 m). The central T e is significantly dropped by the extension of the flattening. Figure 8 shows changes of central beta and the formation of the temperature profile flattening as a function of the mode frequency. The m/n = 2/1 and 1/1 activities were obtained in the enhanced magnetic hill configuration and the reduced magnetic shear configuration, respectively. In the m/n = 2/1 case, the flattened width normalized by the minor radius is increased to 0.1 when the frequency is decreased to 0.4 kHz, and the central beta starts to decrease simultaneously. The m/n = 1/1 mode forms the profile flattening when the frequency falls to 0.9 kHz, and it linearly extends with the decrease in the frequency. When the frequency reaches zero, the flattened width of the temperature profile exceeds 0.5, which causes the degradation of central beta by about 60%. These results show that the mode rotation contributes to the stabilization.     Figure 9 shows the discharges in the cases of the natural and reduced error fields. In the reduced error field case, the RMP current normalized by B t , I RMP /B t , of 0.1 kA T −1 was applied and the phase is almost the same as that of the error field [14]. The R v ax of 3.6 m, A p of 5.8 and B t of 0.9 T were selected. The white and black arrows are the onset of the m/n = 1/1 mode causing the profile flattening and the time where the mode abruptly grows, respectively. The electron density is kept at about 1 × 10 19 m −3 in both discharges because the large beamdriven current is required to destabilize the mode in the A p = 5.8 ratio configuration with high shear. In the discharge with the natural error field (left figure), the strong m/n = 1/1 mode appears at the time where the I p /B t reaches about 115 kA T −1 . When the I p /B t exceeds about 130 kA T −1 , the mode abruptly grows and extends the flattening region to 50% of the minor radius, which consequently drops the central electron pressure. The I p /B t gradually decreases after the collapse, and the w/a is decreased simultaneously. The central pressure is recovered by the stabilization of the mode in addition to the increase in the n e . The mode is destabilized again at 5.5 s by increasing the I p /B t . The ion saturation currents are measured with two Langmuir probes. Both probes are located on the divertor plates at the same poloidal direction of the high field side and toroidally separated by 180 degrees. When the mode is enhanced at 4.9 s, an asymmetric structure of the ion saturation current starts to appear and extends as shown in figure 9(E). This structure is due to the growth of the m/n = 1/1 mode and is in phase with the island due to the error field. After the mode is completely suppressed at 5.26 s, the asymmetric structure disappears till the mode is strongly enhanced again at 5.28 s.

Effect of error field on onset of modes
In the case of the reduced error field, the mode forming the flattening appears when the I p /B t reaches about 118 kA T −1 , and it suddenly grows when I p /B t approaches about 133 kA T −1 , which are hardly different from the natural error field case. The extension of the flattening region and the drop of the central pressure are slightly larger than in the natural error field case when the collapse occurs, which may be due to the difference of the pressure (beta) just before the collapse. The mode is stabilized by the reduction of I p /B t as well as in the error field case.

Discussion and summary
The experiments in the enhanced magnetic hill and reduced magnetic shear configurations show that the modes with finite rotation appeared firstly when the configurational parameters related to the ideal stability exceeded the thresholds, and the modes significantly grew and led to the minor collapse when the mode rotation was stopped or greatly decelerated till some threshold. The previous studies in the enhanced magnetic hill configuration clarified that the m/n = 2/1 mode is destabilized when the ι/2π = 0.5 resonant surface moves from the magnetic well to the magnetic hill regions by the inward movement of the magnetic axis position [17]. Also, results of the numerical simulation show that the mode forms the flattening of the pressure profile around the resonant surface [10]. The experiments in reduced magnetic shear configurations clarified that the onset of the m/n = 1/1 mode strongly depends on beta (gradient), the plasma aspect ratio and the plasma current. The results are qualitatively consistent with theoretical prediction of low-n analysis [11]. However, the effect of the plasma flow on the modes is not included in these analyses.
The observation that the strong growth of the mode after the stop of the rotation degrades the plasma confinement is phenomenologically similar to the tearing locked mode observed in tokamaks and RFPs [2,[18][19][20]. It has been believed that the deceleration of the tearing mode is due to the error field and the wall. However, the deceleration of the mode rotation observed in the experiments on the LHD is independent of the existence of the error field, while the mode is stopped in phase with the error field [21]. Also, it has been found in the experiments of the configurations with different plasma aspect ratios that deceleration of the mode rotation rarely depends on the distance between the resonant surface and the vacuum vessel wall. The nominal rotation of the interchange mode is quantitatively consistent with that of the sum of E × B and electron diamagnetic flows [22]. We consider two possibilities for the deceleration of the mode. One is due to reductions of both E × B and electron diamagnetic flows, which is caused by the profile flattening formed by the growth of the mode [11]. The other is due to the movement of the resonant surface to core region where the plasma flows are relatively small. This movement is due to the increase in the rotational transform caused by the plasma current.
The experimental results suggest that the significant deterioration of the plasma confinement due to the modes may be avoided if the rotation can be maintained as well as in tokamaks. Also, as shown in figure 9, the plasma confinement can be recovered even if the plasma can return to the stable region during a discharge because of no major disruption. The effect of the plasma flow on the interchange instabilities remains an open subject. The nonlinear calculation including the plasma flow is required in order to deeply understand the physical mechanisms of the nonlinear growth of the interchange modes and the impact on the plasma confinement.
This study clarified the impact of low-n MHD activities on the plasma confinement in the ideal unstable regime in order to find the achievable beta regime for the helical reactor. The low-n instabilities deforming the plasma profile appeared when the magnetic shear was reduced or the magnetic hill was enhanced. The minor collapse decreasing the central beta by at most 60% occurred when their mode rotations were stopped. The onset of the modes is independent of the existence of the error field.