Optimization of 3D controlled ELM-free state with recovered global confinement for KSTAR with n = 1 resonant magnetic field perturbation

The real-time adaptive approach in this study detects ELMs from a D_α emission measurement and finds the optimum RMP strength or coil current I_RMP sufficient to maintain the ELM-free state while small enough to maximize the confinement. The adaptive ELM control experiment (#26004) in KSTAR introduced here is outlined in figure 1. Figure 1 shows a H-mode plasma with fully suppressed ELMs via adaptive feedback RMP amplitude control. The relevant plasma parameters are plasma major radius R₀ = 1.8 m, minor radius a₀ = 0.45 m, the toroidal magnetic field B_T = 1.8 T at major radius R₀, Greenwald density fraction n_G ∼ 0.4, elongation κ ∼ 1.71, upper triangularity δ_up ∼ 0.37, lower triangularity δ_low ∼ 0.85, and pedestal collisionality ν_{e, ped}∼0.5. In this discharge, a hysteresis effect is utilized where ELM suppression can be maintained over long periods with a lower RMP strength than initially required for access to the ELM suppression regime [17]. Because reduction of the RMP amplitude leads to an increased pressure pedestal height, this enables global confinement recovery in an ELM-free state [30] by adjusting RMP levels. To avoid ELMs while maximizing the confinement, we use a pre-programmed low n = 1 RMP spectrum [8] with 90 degree phasing and apply real-time feedback to control its amplitude. During the plasma current flattop before applying RMP, with I_p = 0.51 MA and ∼3 MW of co-neutral beam injection heating, β_N ∼ 2.13, β_p ∼ 1.71, and H₉₈ ∼ 1.03, close to the targets of the proposed ITER baseline scenario. In this discharge, the plasma edge safety factor q₉₅ ∼ 5, which is higher than the target value of q₉₅ ∼ 3. Here, q₉₅ is defined as the pitch of the magnetic field line in the edge where the normalized poloidal flux (ψ_N) is 95%. However, after achieving the first stable ELM suppression through traditional means (7.1 s), the plasma performance significantly decreases to β_N ∼ 1.62, β_p ∼ 1.30, and H₉₈ ∼ 0.68. The 30% reduction in overall confinement by RMP mainly comes from degradation in density and temperature pedestal, as shown in figures 1(c) and (d). Such extensive confinement and H₉₈ degradation is a well-known general trend in low-n RMP experiments [31–33] and will not be acceptable in a future fusion reactor because this leads to a significant increase in fusion cost.

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After this initial degradation, the real-time adaptive ELM control scheme starts to recover the original performance before RMPs were introduced while maintaining stable ELM suppression. The controller leverages the D_α emission signal near the outer divertor target to calculate the frequency of ELMs (f_ELM) [34] in real-time and change I_RMP accordingly. To achieve ELM suppression, the RMP amplitude (or coil current, I_RMP) is raised until f_ELM decreases to 0, i.e. ELM suppression. Then, during the resulting ELM-free period, the controller lowers the RMP strength to raise the pedestal height until ELMs reappear, at which point the control again starts to ramp up the RMP amplitude until suppression is recovered (figure 1(a)). In the experiment presented in figure 1, there are 0.5 s of RMP flattop intervals between the RMP-ramp up and down phase to achieve saturated RMP response. Throughout this process, we adjust the lower bound of I_RMP to slightly higher value (by 0.1 kA) than where the most recent ELM returns. This adaptive constraint reduces the likelihood of ELM suppression loss and control oscillation. The feedback system leads the plasma to a converged operating point that optimizes both ELM-free operation and confinement, recovering most of the performance lost in the initial application of RMP.

In the selected discharge, this adaptive ELM control scheme achieves a stable ELM-free phase at 10.5 s with recovered global confinement, as shown in figure 1(b). Although a few ELMs occur before convergence, the controller successfully reaches a stable operating point with minimized ELMy periods. In the final state, the plasma performance shows β_N ∼ 1.91, β_p ∼ 1.53, and H₉₈ ∼ 0.9, recovering up to 68% of the original confinement degradation. Such increase in H₉₈ is especially important as this leads to the 60% recovery in Q_fus degradation, thus emphasizing the performance of adaptive control.

The enhanced confinement quality by adaptive RMP control occurs with the recovery of both the temperature and density pedestals. For the profile reconstruction, ion temperature is measured by charge exchange recombination system [35] for carbon (6+) impurities at outboard mid-plane. Electron temperature is measured by the Thomson scattering [36] and electron cyclotron emission [37] system. Core electron density is measured by the Thomson scattering and two-color interferometry system [38]. To obtain well-resolved profiles, the data are averaged over 100 ms. The pedestal height is obtained from hyperbolic tangent fits with edge profiles, where its location depends on the pedestal width. The equilibria from EFIT code [39] is used for the radial profile mapping and fitting. Kinetic equilibria are reconstructed for the plasma detailed analysis. This equilibrium is calculated with the pressure profile (summation of thermal pressure profile from radial profile reconstruction and fast ion pressure from NUBEAM code [40]) and current density profile (core current from motional Stark effect diagnostics [41] and edge current using NUBEAM, Ohmic and Sauter current models [42]) as a constraint. Figures 1(c) and (d) shows the time traces of fitted pedestal heights for all channels. As can be seen in the figure, all pedestals are significantly improved from the first ELM suppression phase (7.1 s). For example, electron (T_{e, ped}) and ion (T_{i, ped}) temperature pedestals increase by 22% and 50%, respectively. In addition, the electron density pedestal (n_{e, ped}) is also recovered by 10% at the same time. Interestingly, H₉₈ ∼ 0.9 at 10.5 s is much larger than H₉₈ ∼ 0.75 at 6.2 s, even with the same I_RMP = 3.6 kA. This indicates that the confinement recovery by adaptive approach is not solely attributable to decreased I_RMP, but rather that another contributor leads the plasma to a reinforced recovery to the high-confinement state.

We note that the ion temperature pedestal exhibits significant recovery compared to the other channels. This is mainly due to the rapid and significant increase of ion temperature pedestal height by decreasing RMP strength. The traces of pedestal height versus I_RMP before the first ELM reappearance (5.3–7.7 s) reveal this trend. Figure 2 shows the changes of ion, electron temperature and electron density with respect to the I_RMP during 5.3–7.7 s. In the figure, n_{e, ped} and T_{e, ped} have a similar dependence on I_RMP during the pedestal degradation (5.3–6.5 s) and recovery (7.1–7.7 s) phases, showing $\frac{{\Delta}{n}_{\text{e,}\;\text{ped}}}{{\Delta}{I}_{\text{RMP}}}\sim -1{0}^{15}\enspace {\text{m}}^{3}\enspace \text{A}$ and $\frac{{\Delta}{T}_{\text{e,}\;\text{ped}}}{{\Delta}{I}_{\text{RMP}}}\sim -0.06\enspace \;\text{eV}\enspace {\text{A}}^{-1}$. However, T_{i, ped} in the recovery phase shows a 50% larger response of −0.09 eV A⁻¹ compared to the degradation phase, −0.06 eV A⁻¹. The difference of responses in these phases leads to the faster and larger recovery of the ion temperature pedestal. Here, figure 2(d) shows that β_p exhibits similar trend with T_{i, ped}, where $\frac{{\Delta}{\beta }_{\text{p}}}{{\Delta}{I}_{\text{RMP}}}$ in the recovery phase has a 50% larger response of −0.14 kA compared to the degradation phase, −0.07 kA. Because such a boosted response of β_p leads to the reinforced confinement recovery, this similarities between T_{i, ped} and β_p responses indicates that T_{i, ped} dynamic can be considered as a key to the successful confinement optimization via adaptive RMP control.

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In addition to the changes of pedestal heights, the radial profiles during discharges are compared. Figure 3 illustrates the radial profiles of ion, electron temperature, and density at three important time slices during the recovery phase; first saturated ELM suppression state (7.1 s), first optimized ELM suppression state (7.7 s), and finally converged state (10.5 s). As shown in figures 3(a)–(c), all radial profiles in the core plasma are almost identical during the recovery phase. Therefore, the improved confinement by decreasing RMP strength results from increased n_{e, ped}, T_{e, ped}, and T_{i, ped}, with the last one dominant. Here, the statistical error bars of n_{e, ped}, T_{e, ped}, and T_{i, ped} are ∼12%, ∼11%, and ∼5%, respectively. It turns out that ∼67% of improvement comes from the ion temperature pedestal, while the contribution of n_{e, ped} and T_{e, ped} to the confinement recovery is 20% and 13% respectively. In this respect, the recovery of T_{i, ped} is responsible for reinforced recovery by adaptive control. The large growth of T_{i, ped} is mainly due to the simultaneously increased upper limit of T_{i, ped} before the loss of ELM suppression and its enhanced response to the RMP strength. In addition, n_{e, ped} shows a large increase near I_RMP ∼ 5 kA (figure 2(b)), which can be attributed to reduced particle pumping from ELMs. This occurs before 7 s and does not directly contribute to confinement recovery beginning at 7.1 s. However, it still strengthens the confinement recovery with increasing T_{i, ped}. Given that the profiles of 7.7 s and 10.5 s are very similar, control iterations after 7.7 s can be considered as a repeated cycles similar to first ELM suppression period (5.3–7.7 s) for the control convergence. Therefore, the following analysis is focused on the first control iteration for easier explanation.

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As mentioned earlier, the RMP induces additional transport process in the edge region, resulting the degradation of pedestal height and its gradient. However, RMP-induced pedestal transport can also facilitate the improvement of the T_{i, ped} upper limit in the ELM-free phase and its response to the RMP strength by broadening the ion temperature pedestal. Effect of RMP-induced transport on the ion temperature pedestal can be found from the analysis of the profiles in detail. Figures 5(a) and (b) illustrate ion temperature pedestal and E × B flow (ω_E) profiles for five times between 5.3 and 7.7 s. Before ELM suppression (5.3–6.3 s), T_{i, ped} decreases with I_RMP, while the pedestal gradient is well sustained (or even slightly increased). After ELM suppression (>6.5 s), however, the pedestal gradient starts to change. The transition from 6.6 to 7.1 s shows broadening of the ion temperature pedestal and decreasing of its gradient. This widening is maintained in the pedestal recovery phase up to 7.7 s. The decrease in pedestal height and gradient are both due to RMP-induced transport. However, the rapid broadening of the ion temperature pedestal after ELM suppression indicates that its gradient is not governed by the transport affecting the pedestal height but instead by an 'additional' transport source that occurs in the ELM suppression phase. For example, reduced ω_E profiles and its gradient with ion temperature pedestal broadening may indicate the change of turbulence and neoclassical transport, which is known to increase with smaller E × B well at the pedestal region [45, 46].

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The change in ion temperature pedestal width improves the ELM stability. In theory, pedestal pressure (P_ped) or pedestal poloidal beta (${\beta }_{\text{p,}\;\text{ped}}=\frac{{P}_{\text{ped}}}{{B}_{\text{p}}^{2}/2{\mu }_{0}}$) should stay under the stability limit to avoid the reappearance of ELM crashes. Although it is not yet theoretically revealed how low β_{p, ped} should be than this limit, the stability analysis confirms that experimental β_{p, ped} stays below $\sim 70\%$ of the stability limit during the ELM suppression phase. Therefore, in this work, we assumes the ELM suppression can be maintained under the 70% of β_{p, ped} limit imposed by stability constraint. Here, the pedestal stability is predicted using ideal peeling-ballooning (PBM) theory [3] and the EPED1 [47] algorithm. The fixed-boundary equilibrium code, CHEASE [48], is used for accurate equilibrium mapping, and the ideal MHD stability code, MISHKA1 [49], is employed for PBM stability calculation. All other required parameters are taken from the reconstructed radial profiles and plasma equilibrium.

This stability limit is known to improve with increased pedestal width [50]. Therefore, widened pressure pedestal via ion-pedestal broadening allows for higher β_{p, ped} during the ELM-free phase. Numerical analysis reveals that the β_{p, ped} limit increases by 53% due to ion temperature pedestal broadening. This change is presented in figure 6. In the figure, β_{p, ped} limits derived with (orange) and without (gray) broadened ion temperature pedestal are presented with experimental points (magenta). It can be seen that the limit is enhanced by pedestal widening. With the expansion of the β_{p, ped} limit illustrated as dotted lines, β_{p, ped} can further increase from 0.2 (gray dotted line) to 0.31 (orange dotted line). This enhanced β_{p, ped} limit allows access to higher T_{i, ped} in the ELM suppression phase. For example, ELM suppression can be maintained at 7.7 s where T_{i, ped} = 0.7 keV, which is higher than 0.6 keV in ELMy phase (6.3 s), as shown in figure 2(c).

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The broader ion-pedestal also can lead to a larger response of T_{i, ped} on RMP strength. Inspired from (Hu et al 2020) [51], the change of temperature pedestal height (ΔT_ped) by ΔI_RMP and magnetic islands can be described as equation (1),

$$\begin{equation}\frac{{\Delta}{T}_{\text{ped}}}{{\Delta}{I}_{\text{RMP}}}\approx \nabla {T}_{\text{ped}}\sum\limits _{m\geqslant {q}_{\text{ped}}}\frac{\partial {W}_{\mathit{\text{m, n}}}}{\partial {I}_{\text{RMP}}},\end{equation} \tag{ 1 }$$

where W_{m, n} and ∇T_ped are the (m, n) island width and pedestal gradient, respectively. q_ped is the edge safety factor on the pedestal top. This expression is based on the concept where the contribution of an island on the pedestal degradation (ΔT_ped) by RMP is the accumulation of profile flattening at the islands in the pedestal region. We note that constant ∇T_ped over the pedestal region is assumed to make interpretation easier. This expression addresses that pedestal height changes more rapidly with RMP strength as the pedestal gradient grows and q_ped decreases. With the given q profile monotonic, q_ped is reduced by increasing pedestal width. Because the summation term (∑) increases with q_ped and width, the broadened ion temperature pedestal can lead to a stronger response of T_{i, ped} despite the decrease of ion temperature pedestal gradient (∇T_ped). In addition, ion temperature pedestal is known to be heavily influenced by neoclassical transport [15, 46, 52]. Here, RMP can increase the neoclassical heat flux and the amount is roughly proportional to the square of perturbed field strength and ${I}_{\text{RMP}}^{2}$. Smaller edge E × B can increase the sensitivity of ion heat flux to RMP strength [53, 54]. Because a decreased ion temperature pedestal gradient reduces the ω_E [19, 55] at the pedestal (figure 5(b)), this change in radial electric field also contributes to increasing the response of T_{i, ped}.

On the other hand, the responses of n_{e, ped} and T_{e, ped} to RMP strength are almost identical whether or not the ELMs are fully suppressed. This means that additional RMP-induced transport in the ELM-free phase has a smaller effect on the electron density and temperature pedestal gradient. Although the electron pedestal width has considerable uncertainty due to limitations in the resolution of edge diagnostics, its value lies between 4%–6% in normalized poloidal flux without showing a considerable widening like ion temperature pedestal, suggesting that additional transport has only a relatively small effect on electron channels. We note that a large decrease in electron pedestal height still occurs without a clear change in its width, and this additional transport is expected to have little correlation with 'pump-out' commonly observed in RMP experiments.

Increased T_{i, ped} response by RMP-induced transport leads to an extensive recovery of T_{i, ped} during RMP ramp-down and makes an ion temperature pedestal higher than the RMP ramp-up phase (ELMy) even with the same RMP strength. In addition, enhanced pedestal stability allows for larger T_{i, ped} before the return of ELMs. The synergy between these effects boosts the pedestal recovery and enables adaptive control to maximize the confinement, resulting in a much higher pedestal than during the initial phase of ELM suppression, as shown in figure 7, which illustrates β_{p, ped} versus I_RMP. The changes to the pedestal from 5.3 to 7.8 s are shown, and the ELM suppressed states are marked with star points.

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Another advantage of RMP-induced transport is that it improves the control stability. Adaptive control can be unstable due to a bifurcation of the plasma state during transitions between ELMy and ELM-free regimes, which causes oscillation of the control system. In particular, it can take a long time or even become impossible for a controller to find the optimal solution because of the sudden jump in RMP strength required for re-entry (I_IN) to or exit (I_OUT) from ELM suppression. The schematic diagram in figure 8(a) illustrates how this characteristic will delay the control convergence. In practice, ELM control must be done quickly to minimize damage to the reactor, so an adaptive approach is generally hard to use in such a bifurcating system. However, RMP-induced transport eases these control difficulties by reducing I_IN during adaptive control, as shown in figure 8(b).

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It has been shown that the plasma enters the ELM suppression state above a certain δB_{r, edge} threshold [43], where δB_{r, edge} is the perturbed radial field strength at the pedestal. Again, δB_{r, edge} is calculated using IPEC code [44] and derived through radially averaging δB_r at ψ_N = 0.9–1.0. The thresholds of δB_r for RMP-induced ELM suppression is obtained from the reference discharge (#26004). This threshold (∼20 G) is shown as the red contour of figure 7. Here, β_{p, ped} amplifies the perturbed field [43], and the same δB_r can be obtained with a smaller I_RMP with larger β_{p, ped}. Because RMP-induced transport enhances β_{p, ped} in an ELM-free state, this leads to a lower I_IN, making access to the next ELM suppression regime easier. The ELM suppression of 7.8 s shown in figure 7 results from reduced I_IN compared to the former one at 6.5 s. Thus, I_IN for each suppression entry changes as 4.9 → 3.6 → 3.53 → 3.5 kA, as seen in figure 1(a), resulting in fast and stable system optimization. This interesting example shows uncommon positive effect [56, 57] of self-organized transport on pedestal confinement.

We note that such an RMP-induced hysteresis shown in figure 7 is not trivial to be produced in the experiment as it conventionally requires a delicate pre-programmed RMP waveform under the absence of real-time control. This leads to difficulties in investigating and exploiting the hysteresis, which is critical to optimize the ELM-free state. In this respect, adaptive RMP control is an effective methodology as it can automatically generate the hysteresis and utilize it. In addition, the adaptive scheme has been successfully operated for more than a 100 confinement times ($\sim 5$ s) of KSTAR, and therefore, this control is also expected to be applicable to long pulse plasma in ITER.

It is worth pointing out that successful adaptive control in these experiments is mainly due to a broadened ion temperature pedestal during the ELM suppression phase. In order to determine the change in ion heat transport, interpretive transport analysis is conducted using ASTRA 7 [58] code. The ion neoclassical heat diffusivity (χ_i, neo) is also calculated based on NCLASS [53] model to compare it with experimental ion heat diffusivity (χ_i). The results are shown in figure 9, where χ_i (a) and χ_i, neo (b) for 5.3–7.7 s are included. As shown in figure 9(a), the ion heat diffusivity (χ_i) of the pedestal region rapidly increases via additional transport after transitions to the ELM-free state. In addition, the pedestal heat diffusivity does not change much during 7.1–7.7 s, indicating that it is insensitive to the decreasing I_RMP. It has been reported that the neoclassical transport effect dominates ion heat transport under RMPs [46, 52]. However, this collisional transport strongly depends on the RMP strength. Therefore, the broadened ion temperature pedestal does not seem to be related to the neoclassical process. Here, it can be seen in figure 9(b) that χ_i at 5.3 s (gray) exceeds neoclassical level in all cases, supporting the existence of additional transport. We note that following analyses will focus on the center region of the pedestal (ψ_N = 0.96), where the change in ion heat diffusivity is clearly observed in time.

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Fluctuation measurements on KSTAR reveal significant edge turbulence triggered by RMPs [25, 26, 59] after ELM suppression. Figures 10(a) and (b) illustrate the spectrogram and the coherence strength of δT_e and δn_e fluctuations at ψ_N ∼ 0.96. Figure 10(c) shows the poloidal magnetic field fluctuations (δB_pol) at the inner wall. In this work, edge T_e and n_e fluctuations (k_y ρ_s ⩽ 0.3) are measured from electron emission image spectroscopy (ECEI) [60] and beam emission spectroscopy (BES) [61], respectively. The k_y is the bi-normal wave number, ${\rho }_{\text{s}}=\sqrt{2{m}_{\text{i}}{T}_{\text{e}}}/eB$ is the hybrid Larmor radius, and m_i is deuterium mass. Magnetic field perturbations are captured by the Mirnov coil (MC) signal [62]. The spectrogram of the measured fluctuation is derived using the Fourier transform. Coherence of the electron density and temperature fluctuation is calculated from a bi-spectrum analysis with two radially adjacent channels in ECEI and BES, respectively. The ELM peaks and core modes are statistically removed from the integrated amplitude of coherent fluctuations in all channels. Here, δT_e and δn_e have strong coherence over the frequency range of 10–70 kHz. The magnetic fluctuations in the 200–400 kHz range are also observed during the same period. As shown in figure 10(d), they show an immediate instigation of turbulence as ELM suppression begins followed by quick saturation within 200 ms. We note that coherence before 6.4 s comes from ELM noise, and a magnetic signal of <50 kHz is due to core modes. It is noteworthy that the strength of coherent fluctuations remains almost identical during 6.6–7.7 s. Here, the widening of the ion temperature pedestal coincides with the occurrence of edge fluctuations. Furthermore, they are both insensitive to RMP strength. Therefore, these similarities support the claim that the ion temperature pedestal is widened primarily due to increased heat diffusivity by edge turbulence.

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Linear gyrokinetic simulations confirms that enhanced edge turbulence may occur in the ELM suppression phase. The gyrokinetic code, CGYRO [63], is used in the linear analysis of micro-instabilities. The linear initial value solver is employed to find the unstable mode in the target radial point with wavelength k_y ρ_s = 0.1–1.5. This simulation is based on a flux-tube approach with a full gyro-kinetic description for both electron and ion channels. The reconstructed radial profiles and kinetic equilibrium described above are included for the accurate modeling. This calculation is performed at ψ_N = 0.96, where the changes of experimental fluctuations are robust. The linear growth rate and real frequency are normalized by E × B shearing rate (γ_E) and Bohm sound speed (C_S).

As shown in figure 11(a), the normalized linear growth rates (γ) of turbulence mode exceed the onset limit (>1) after the transitions to the ELM-free state. This is mainly due to decreased stabilizing effect from the E × B shearing rate (γ_E) [45, 64], which comes from the degraded pressure pedestal (figure 5(b)) after entering ELM suppression (6.6 s). The real frequency and numerical testing indicates that the excited mode is an ITG/TEM hybrid mode, which mainly lies on ion direction as shown in figure 11(b). Here, the bi-normal wave length k_y ρ_s ∼ 0.3 and real frequency ∼51 kHz of the most unstable mode exhibits similar properties to the measured fluctuations of electron channels. The simulation results show that ion thermal diffusion can be increased with these unstable modes, supporting the idea of ion temperature pedestal broadening by turbulence. However, theoretical analysis on RMP-induced turbulence still has many missing pieces. Recent studies have shown that the characteristics of transport in the presence of RMP deviates significantly from linear gyrokinetic calculations, raising the importance of non-linearity [65] and non-locality [66] which is not included in this linear analysis. In addition, the reduced gradient of ion temperature pedestal during its broadening can be explained by introducing RMP-induced transport. However, it is still less clear how it can contribute to increased width. In the future, nonlinear gyrokinetic studies including these aspects will shed further light on the accurate description of edge turbulence under RMPs.

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The considerable effect of RMP-induced transport on ion heat diffusion might be inconsistent with the general trend of other devices [16, 17, 32], where such turbulence mainly affects electron channel and has a minor effect on ion transport. Although it is difficult to evaluate the turbulence effect on n_e and T_e due to limitations in the diagnostics, we still confirm that there is a clear correlation between edge fluctuation and ion temperature pedestal. Therefore, this observation suggests new possible role of turbulence in the ion temperature pedestal, where ELM-free state is achieved with the low-n (=1) RMP.

As discussed earlier, ion temperature pedestal widening is key to the fast and successful convergence of adaptive control. Because edge-turbulence can play important role on the ion-pedestal, the turbulence level should be well sustained to maintain such an favorable effect. However, figure 10(d) shows that the amplitude of edge fluctuation disappears as ELMs re-occur, and the favorable effects from widened ion temperature pedestal will also start to decrease. Here, the ion temperature pedestal will return to its initial ELMy state on an energy confinement time scale, so the advantageous turbulence effect can last a few 100 ms after returning to the ELMy phase. Thus, I_RMP must re-increase immediately after the loss of ELM suppression fully exploit this effect. In this respect, a real-time adaptive ELM control is a unique methodology both to utilize and control the edge turbulence and to uncover the novel beneficial effect of turbulence.