Validation of the model for ELM suppression with 3D magnetic fields using low torque ITER baseline scenario discharges in DIII-D Free

Operation of high performance tokamak plasmas requires some mechanism to control the periodic release of plasma stored energy (up to 25% of the pedestal stored energy at ITER pedestal collisionalities¹) on timescales fast enough (100 s of microseconds) to damage the divertor target plates^2,3 due to the MHD instability known as “Edge Localized Modes” (ELMs). In ITER and other next-step, burning plasma tokamaks, these ELMs can lead to excessive erosion of the divertor and possible contamination of the plasma with impurities,⁴ although recent work⁵ suggests that ELM control requirements may not be as severe as initially thought. Consequently, significant research effort has been expended to develop techniques for controlling the ELM-induced peak heat flux, including Resonant Magnetic Perturbations (RMPs),^6–8 Quiescent H-mode (QH mode) operation (inherently ELM-free steady state H-mode⁹), and ELM triggering (“pacing”) using deuterium^10,11 or lithium pellet^12,13 injection. At the DIII-D tokamak,¹⁴ resonant magnetic perturbations (RMPs) have been used to suppress ELMs in tokamak discharges with ITER-like pedestal conditions,^6,7,15 and have also been used to suppress ELMs in KSTAR,^16,17 EAST,¹⁸ and ASDEX-Upgrade.¹⁰ However, most of the existing database for RMP ELM suppression has been obtained in H-modes with significant torque injection T_inj (5–10 Nm) by neutral beams parallel to the plasma current (co-I_p), and high levels of pedestal toroidal rotation $v T$⁠. In contrast, T_inj will be limited in ITER, creating a need to extend the high T_inj RMP ELM suppression results to more ITER-relevant levels of T_inj ∼ 0–1 Nm.²⁰ Initial investigation of the dependence of RMP ELM suppression on input torque indicated that RMP ELM suppression was lost when $v T$ dropped below about 40 km/s.¹¹ 

To date, much of the research on RMP ELM suppression has focused on the dependence of ELM suppression on the amplitude^21,22 and/or spectrum^23–25 of the applied magnetic perturbations. In this paper, we report the results of experiments at DIII-D that explored the dependence of RMP ELM suppression on more ITER-relevant levels of neutral beam torque. RMP ELM suppression was readily obtained for $T i n j$ = 5 Nm, but lost when the input torque was reduced to 3.5 Nm, as shown in Fig. 1. These results are not only important for extrapolating RMP ELM suppression access to ITER, but also provide an excellent opportunity to test the hypothesis for RMP ELM suppression based on resonant field penetration at a resonant surface near the top of the H-mode pedestal.^26,27 In this hypothesis, ELMs are suppressed when the plasma transitions from an ideal screened response to a tearing response^21,22 at a resonant surface that prevents expansion of the pedestal to an unstable width. To test this hypothesis, we compare linear, single-helicity $n=3$ single-fluid and two-fluid simulations of the plasma response to the applied RMP using the resistive MHD code m3d-c1 (Ref. 28) to the measured electron perpendicular rotation and ELM behavior. We find that RMP ELM suppression is correlated with: (1) a zero-crossing in the electron perpendicular rotation $ω ⊥ e$ ∼ 0 at the 9/3 rational surface near the top of the pedestal at $Ψ N$∼ 0.93, and (2) a peak in the two-fluid linear tearing response that tracks the location of this $ω ⊥ e$ ∼ 0 point. This shift in the location of the $ω ⊥ e$ ∼ 0 point and the calculated tearing response are determined primarily by changes in the width of the electron diamagnetic flow frequency $ω e *$ profile and not changes in the $E ¯ × B ¯$ rotation frequency $ω E$ profile when the torque is changed.

This manuscript is organized as follows: Sec. II presents the hypothesis for RMP ELM suppression to which the experimental results will be compared; Sec. III presents the experimental approach; Sec. IV presents the loss of ELM suppression when the input torque is reduced in $ν *$ ∼ 0.15 ITER Baseline Scenario (IBS) discharges; Sec. V describes the variation in the simulated tearing response with input torque; Sec. VI describes the fluctuation response that confirms the $E ¯ × B ¯$ rotation frequency $ω E$ changes; Sec. VII investigates the possibility of collisionality or $q 95$ changes as the cause of loss of RMP ELM suppression at low torque; Sec. VIII compares electron rotation and ELM behavior in $ν *$ ∼ 1–2 IBS discharges with the ELM suppression hypothesis; Sec. IX describes a demonstration of the hypothetical prediction that RMP ELM suppression at low torque input can be achieved by also reducing the electron pedestal width and height. Finally, summary and discussion of the implications for ITER are presented in Sec. X.

The hypothesis for RMP ELM suppression^21,22,26,27 is deduced from the EPED1 H-mode pedestal physics model,²⁹ which was developed to predict pressure pedestal heights and widths in ELMing H-modes without magnetic perturbations. The EPED1 model consists of the ELITE linear MHD stability code for the peeling-ballooning modes responsible for large, Type I ELMs in tokamak H-modes,³⁰ and a scaling for local transport in the pedestal due to Kinetic Ballooning Modes (KBMs). In this model, the local transport due to kinetic ballooning modes (or some other mechanism) sets a maximum limit on the steepness of the H-mode pedestal pressure gradient, as shown in Fig. 2. As the ELM-free H-mode evolves, the pressure pedestal broadens into the plasma core, as indicated by the sequence (blue-green-red) of pressure profiles and pedestal tops (indicated by the vertical lines for the corresponding pedestal profile). Eventually, the pressure pedestal becomes broad enough (red profile) that the pedestal is peeling-ballooning unstable and an ELM occurs, which reduces the width of the pedestal below the stability limit and the cyclic process repeats. This H-mode pedestal model implies a mechanism for ELM suppression by RMPs in which transport increases in a narrow region at the top of the pedestal, preventing the pedestal from expanding to an unstable width, as indicated by the shaded box in Fig. 2.

One hypothesis for the source of the transport enhancement that limits the width of the pedestal is penetration of the applied resonant magnetic perturbations (a tearing response) at the pedestal top.^26,27 This hypothesis requires four things:

• low electron perpendicular rotation, since electron perpendicular rotation screens the applied field [Fig. 3(a)]:^31–34 the frames in Fig. 3 are shaded for the level of electron perpendicular rotation $ω ⊥ e$⁠: green for high $ω ⊥ e$ resulting in screening of the applied RMP field, and yellow for $ω ⊥ e$ low enough to allow some tearing response to the RMP.

• a rational surface in the region of low electron perpendicular rotation, as indicated by the blue horizontal lines on the safety factor $q(Ψ)$ profile in Fig. 3(b),

• an applied RMP that drives modes resonant on this rational surface: in Fig. 3(b), the length of the blue lines at the rational surfaces indicates the width of the island formed by the applied RMP in the absence of the plasma response (“vacuum island width”); and

• the driven mode response must be close enough to the top of the pedestal to limit the pedestal width to a stable value, schematically indicated by the red ellipses representing the tearing response location leading to a magnetic island.

The results from the RMP ELM suppression experiments in low torque IBS discharges in DIII-D provide a good test of this hypothesis. The radius where the tearing response peaks can be varied by changing the injected torque $T i n j$⁠, the pedestal normalized pressure $β N P E D$⁠, or both, as shown schematically in Fig. 4. In two-fluid resistive MHD theory,^32,33 the tearing response to an applied magnetic perturbation peaks where the perpendicular electron rotation $ω ⊥ e$ approaches 0,³⁴ where $ω ⊥ e$ is given by the sum of the $E ¯ × B ¯$ and electron diamagnetic rotation $ω e *$⁠: $ω ⊥ e = ω E + ω e *$⁠. In the ELM suppression hypothesis, the transport change resulting from resonant field penetration at the top of the pedestal leads to ELM suppression, as discussed for n = 2 RMPs on DIII-D in Refs. 21 and 22.

In the IBS ELM suppression experiments, the input torque $T i n j$ was varied to change the pedestal toroidal rotation $v T$ and hence the $E ¯ × B ¯$ rotation frequency, given by

where $P C$ and $n C$ are the carbon pressure and density, $v T$ and $v P$ are the carbon toroidal and poloidal velocities, $B T$ and $B P$ are the toroidal and poloidal magnetic fields, and R is the major radius. The three terms on the right hand side of Eq. (1) are then the carbon diamagnetic rotation frequency $ω C *$⁠, the toroidal rotation frequency $ω T$⁠, and the poloidal rotation frequency $ω P$⁠, such that Eq. (1) can be written as

The carbon diamagnetic rotation frequency appears in Eqs. (1) and (2) because in DIII-D, these rotational terms are measured using charge exchange recombination (CER) spectroscopy on carbon impurity ions. While changing the input torque $T i n j$ indeed changed the toroidal rotation frequency $ω T$ in the results reported in this paper, it also changed the electron diamagnetic rotation frequency $ω e *$⁠, with both changes contributing to the variation in the radius where $ω ⊥ e ≈0$⁠.

The goal of these experiments was to determine the level of ELM control which could be achieved in the $Q f u s$ = 10 Baseline Scenario for ITER, where $Q f u s$ is the ratio of fusion power produced to input power. This ELM control could be due to an increase in the ELM frequency and concomitant reduction in the stored energy lost in an ELM $Δ W E L M$⁠, often referred to as mitigation, or due to elimination of ELMs, referred to as suppression. These experiments were conducted in previously developed ITER Baseline Scenario discharges in DIII-D,³⁵ which match the ITER aspect ratio [red equilibrium in Fig. 5(a)] and normalized plasma current $I n = I p / a B T ∼1.4$⁠. At this value of $I n$⁠, the safety factor at the top of the pedestal $q 95$ = 3.1 [red trace in Fig. 5(b)]. In order to maximize the resonant components of the applied magnetic perturbation field at the top of the pedestal, the DIII-D internal RMP coils (“I-coils”) were operated with the upper row only.²³ The normalized plasma pressure $β N$ in the $Q f u s$ = 10 ITER Baseline Scenario was held constant at $β N$ = 1.8 [red trace in Fig. 5(d)] using feed-forward or feedback control of the injected neutral beam power $P i n j$⁠. Holding $β N$ constant at 1.8 matches ITER Baseline Scenario conditions, and also provides a more accurate determination of the level of ELM mitigation achieved with the RMP by removing the contribution to the $Δ W E L M$ decrease due to the lower $β N$ that results from “density pump-out” with the RMP. The line average electron density $n e$ (solid red line) and pedestal electron density $n e p e d$ (dashed red line) for the IBS low torque discharge are shown in Fig. 5(e), where the combination of moderate pedestal collisionality $ν *$ ∼ 1 and $β N$ ∼ constant result in little density pump-out. Because the $Q f u s$ = 10 Baseline Scenario in ITER has limited torque input capabilities that scale in DIII-D to $T i n j$ = 0.5–1 Nm, the co- $I p$ torque injection $T i n j$ from the neutral beams was reduced to 1 Nm by exchanging a portion of the co- $I p$ neutral beams for counter- $I p$ neutral beams during the RMP pulse [red trace in Fig. 5(f)]. At these reduced $T i n j$ levels of 1 Nm, the disruptivity of IBS discharges at ITER-relevant collisionalities $ν *$∼ 0.15 in DIII-D increases significantly even in the absence of RMPs,³⁶ so these experiments were conducted at both $ν *$ ∼ 1 (red traces in Fig. 5), where stable discharges are readily and reproducibly obtained, and at $ν *$ ∼ 0.15, where the high disruptivity limited the reliability of the discharges.

These operating conditions vary in some important ways from the conditions in the majority of previously reported RMP ELM suppression experiments in the DIII-D tokamak. Most RMP ELM suppression results have been obtained in the so-called “ITER-similar-shape” (ISS), introduced by Evans et al., in Ref. 15. ISS discharges have only minor differences in shape from IBS discharges for largely historical reasons, as shown in Fig. 5(a). However, the normalized plasma current $I n$ in ISS discharges is significantly lower than in the IBS discharges reported here, resulting in a higher $q 95$ = 3.5 [black trace in Fig. 5(b)], within the so-called “resonant window” for RMP ELM suppression in DIII-D. At this higher $q 95$⁠, maximizing the resonant perturbation field components at the top of the pedestal requires the use of the DIII-D RMP coils in a 2-row even parity configuration, although for the comparison in Fig. 5 we have selected an ISS discharge 145419 which also has the RMP coils in single upper row only [black trace, Fig. 5(c)]. In most ISS discharges, such as 145419 in Fig. 5, the application of the RMP results in significant drops in the normalized plasma pressure $β N$ [black trace, Fig. 5(d)] and both the line average $n e$ and pedestal electron $n e p e d$ densities [Fig. 5(e)] The most significant change from previous RMP ELM suppression experiments is the level of co- $I p$ torque input from the neutral beams: $T i n j$ ∼ 4–6 Nm [black trace in Fig. 5(f)] versus the target $T i n j$ of 1 Nm in these experiments.

To illustrate the differences between these experiments in IBS discharges and most of the previous RMP ELM suppression experiments in ISS discharges, Fig. 5 compares a representative ISS RMP ELM suppression discharge (black traces) with one of the IBS RMP ELM control discharges with $T i n j$ = 1 Nm (red traces) in which the ELMs are mitigated, but not suppressed. RMP ELM suppression is readily achieved in the low collisionality ISS conditions with the upper row only of the RMP coils at input torque levels $T i n j$ ≥ 5 Nm (black curves in Fig. 5 for discharge 145419).²⁶ As an initial step toward extending RMP ELM control to IBS discharges with ITER-relevant values of $T i n j$⁠, the impact of the RMP on ELMs at $T i n j$ = 1 Nm [Fig. 5(f)] was explored in IBS H-modes with high H-mode pedestal collisionality $ν *$ ∼ 1–2 with an $n$ = 3 RMP coil current ramp [red curve, Fig. 1(c)] in the same upper single row RMP coil configuration. As the RMP coil current increased [Fig. 5(c)], the ELM amplitude decreased and the ELM frequency increased. However, ELM suppression was not obtained for the limited range of RMP currents available (≤4 kA). This result emphasizes the need for a careful study of the causes for the loss of RMP ELM suppression at low $T i n j$ presented in this paper. Extension of RMP ELM suppression to IBS discharges with ITER-relevant levels of $T i n j$ approaching 0–1 Nm was also attempted in IBS discharges with ITER pedestal collisionality $ν *$ ∼ 0.15. As shown in Fig. 1, when $T i n j$ was reduced to 3.5 Nm, the toroidal rotation $v T$ in the H-mode pedestal decreased [Fig. 1(a)], and RMP ELM suppression was lost [Figs. 1(b) and 1(c)]. In this manuscript, we use these results to validate the hypothesis for RMP ELM suppression, based on a tearing response on a resonant surface close enough to the top of the pedestal to limit the pedestal width to a stable value.^21,22,26,27

As shown in Fig. 1, reducing the input torque $T i n j$ from 5 Nm to 3.5 Nm reduces the edge toroidal rotation and results in a loss of RMP ELM suppression in IBS discharges with ITER-relevant pedestal $ν *$∼ 0.15 and an $n$ = 3 RMP in DIII-D. This loss of ELM suppression was investigated using a $T i n j$ step to 5 Nm in a 3.5 Nm IBS discharge, as shown in Fig. 6. After establishing a stationary ELMing H-mode with 6 kA of $n$ = 3 RMP from the upper RMP coils [single upper row only; Fig. 6(a)] and $T i n j$ = 3.5 Nm, $T i n j$ is increased to 5 Nm between 4400 ms and 4700 ms, leading to an increase in toroidal angular momentum $L T$ [Fig. 6(c)]. Because the RMP is applied in the L-mode and stepped to full current 1.4 s before the torque ramp is applied at 4400 ms, there is very little decrease (pump-out) in either the line average density $n e$ or the pedestal density $n e p e d$ when the RMP coil current is stepped to the full value at 3000 ms [Fig. 6(d)]. The neutral beam injection was programmed for feed forward to hold $β N$ = 1.8 [Fig. 6(b)], but because the RMP had been at full current for 1.4 s prior to the torque ramp, there is little density pump-out (line average $n e$ drops by 10% while $n e p e d$ drops only after the ELMs are suppressed at the end of the $T i n j$ ramp) or degradation in the energy confinement {neutral beam power $P i n j$ is constant [Fig. 6(b)]} as the $T i n j$ rises. As expected, the increase in input torque $T i n j$ [Fig. 7(a)] steadily increases the toroidal rotation $v T$ at $Ψ N$= 0.8 [Fig. 7(b)], and at $Ψ N$ = 0.95 [Fig. 7(c)], where this increase is modulated by the ELMs. Here, $Ψ N$ is the normalized poloidal magnetic flux, which provides an effective radial coordinate in shaped plasmas. This increase in $v T$ across the outer portion of the discharge in the ELM suppressed state [Fig. 7(d)] results in a more positive $E ¯ × B ¯$ frequency $ω E$ in the core without a strong effect on the pedestal $ω E$ [Fig. 7(f)]. The increase in $ω E$ results in a small outward shift in the zero-crossing in the $ω E$ profile which contributes to an outward shift in the zero-crossing in the perpendicular electron rotation $ω ⊥ e$ profile as well.

The $T i n j$ increase also modifies the pedestal electron pressure $P e [ped]$ and diamagnetic flow $ω e *$ profiles, as shown in Fig. 8, but contrary to the $v T$ and $ω E$ changes, this change only occurs when the ELMs suppress, indicating that these changes are a result of ELM suppression and not a cause. The electron pressure pedestal width $Δ P e [ped]$ [Fig. 8(c)] drops when the ELMs are suppressed [Fig. 8(d)] after $T i n j$ has reached 5 Nm [Fig. 8(a)]. This temporal behavior in the electron pressure pedestal during the torque input ramp and across ELM suppression is consistent with previously reported results at constant torque which showed a sudden change in the edge $T e$^37,38 when the ELMs suppressed, as shown in Fig. 7 of Ref. 38. This change results in a significant steepening and narrowing of the $P e$ pedestal and leads to an outward shift in the peak in the $ω e *$ profile, which follows the peak of the electron pressure gradient. This result also contributes to an outward shift in the zero-crossing in the perpendicular electron rotation $ω ⊥ e$ profile.

The measured changes to the electron perpendicular rotation terms are plotted on the same scales in Fig. 9, showing that the indirect change in $ω e *$ has a much larger impact on the radius of the zero-crossing in $ω ⊥ e$ than the more direct change in the $E ¯ × B ¯$ rotation. The large change in the edge electron diamagnetic flow profile indicates that transport changes associated with accessing ELM suppression, such as those due to the turbulence changes induced by the RMP (Sec. VI),^39–44 may play an important role in the evolution from ELMing to RMP ELM suppressed H-mode. There is also a significant difference in the radius of the tearing response predicted by single-fluid resistive MHD, given by $ω E$ = 0, and by two-fluid resistive MHD, given by $ω E$ +  $ω e *$ =  $ω ⊥ e$ = 0. In Fig. 9, vertical bars have been added to (a) and (c) to indicate the radii of the resonant surfaces in the plasma edge, labeled by poloidal mode number $m$ for $n$ = 3. The change in the radius where $ω E$ = 0 is small, and remains between the q = 9/3 (⁠ $Ψ N$ = 0.93) and 10/3 (⁠ $Ψ N$ = 0.95) surfaces [Fig. 9(a)], but close enough to each to be consistent with the tearing response peaking there, as predicted in single-fluid theory and the forced magnetic reconnection model.⁴⁵ In contrast, the change in the radius where $ω ⊥ e$ = 0, indicated by the black and red vertical bands in Fig. 9(c), corresponds to a shift in $ω ⊥ e$ = 0 radius from the 8/3 (⁠ $Ψ N$ = 0.89) to the 9/3 (⁠ $Ψ N$ = 0.93) rational surface.

Linear, single-helicity $n$ = 3 two-fluid and single-fluid plasma response modeling has been completed for the $T i n j$ = 3.5 Nm and 5 Nm times in IBS discharge 160921 (Fig. 6) using the m3d-c1 resistive MHD code.²⁸ The radial profile of the total resonant field in Gauss per kA of RMP coil current is plotted for the two $T i n j$ values in Figs. 10(b) and 10(d), together with the experimentally measured electron perpendicular rotation profiles [ $ω E$ for the single-fluid model in Fig. 10(a) and $ω ⊥ e$ for the two-fluid model in Fig. 10(c)] from Fig. 9. For the 3.5 Nm ELMing case (blue lines in Fig. 10), the $ω E$ profile and the kinetic equilibrium used in the m3d-c1 simulation were obtained by coherently averaging the profiles from the last 20% of the ELM cycle just before an ELM crash during the time indicated by the grey vertical band in Figs. 7 and 8. For the 5 Nm ELM suppressed case, the corresponding $ω ⊥ e$ and equilibrium were obtained by averaging the profiles in the early ELM suppressed phase during the time indicated by the red vertical band in Figs. 7 and 8. The location of the resonant surfaces, with the corresponding poloidal mode number $m$ for the applied $n$ = 3 perturbation, is shown by the vertical bands in Fig. 9. The grey resonant surfaces are those located far from the zero-crossing in the corresponding electron perpendicular rotation profile, the blue corresponds to the resonant surface closest to the zero-crossing in the 3.5 Nm ELMing case, and the red corresponds to the resonant surface closest to the zero-crossing in the 5 Nm case. For comparison, the applied RMP field versus radius is shown by the dashed lines in Fig. 10(d).

At $T i n j$ = 3.5 Nm, the tearing response (total resonant field) peaks at the 9/3 rational surface in the single-fluid simulation [Fig. 10(b)], but is screened by a factor of 13.3 relative to the applied 9/3 field shown in Fig. 10(d). This plasma response peak occurs at $Ψ N$= 0.93 where $ω E$ ∼ 0, consistent with single-fluid resistive MHD plasma response theory. However, when $T i n j$ = 5 Nm [red curves in Figs. 10(a) and 10(b)], the resonant plasma response remains small (screened by a factor of 50 relative to the applied 10/3 RMP field) except just inside the separatrix. The radius where $ω E$ ∼ 0 at $T i n j$= 5 Nm shifts out a small distance relative to the 3.5 Nm case, but remains closest to the 9/3 rational surface.

The plasma response from the two-fluid m3d-c1 simulations is compared to the electron perpendicular rotation $ω ⊥ e$ in Figs. 10(c) and 10(d). The screening of the applied resonant field [dashed lines in Fig. 10(d)] is much weaker than in the single-fluid simulation, and the plasma response at $T i n j$ = 3.5 Nm peaks at the 8/3 rational surface (⁠ $Ψ N$= 0.89), where $ω ⊥ e$∼ 0 as shown by the blue curve in Fig. 10(c). In the 5 Nm case, the resonant response in the two-fluid simulation increases significantly at the 9/3 rational surface at $Ψ N$ = 0.93, near where $ω ⊥ e$ ∼ 0. These simulations show that the tearing response peaks in the vicinity of the zero-crossing in the electron perpendicular rotation given by the two-fluid term $ω ⊥ e$⁠, and tracks the location of this $ω ⊥ e$ ∼ 0 point when $T i n j$ is changed. This shift in the tearing response radius brings the tearing response closer to the top of the pedestal at the same time that the ELMs are suppressed. In contrast, the tearing response in the single-fluid model is weak across the plasma edge in the 5 Nm ELM suppressed case, suggesting that the two-fluid model captures the tearing response and ELM suppression behavior better than the single-fluid model.

While the results above support the working RMP ELM suppression model described in Sec. II, they also indicate that when the RMP is first applied in a stationary ELMing IBS H-mode, there is a tearing response near $Ψ N$ ∼ 0.9, corresponding to $q$ = 8/3 in these IBS discharges. A similar behavior is found in ISS ELMing H-modes,¹⁵ except that given the higher $q 95$ = 3.5 in ISS H-modes versus $q 95$ = 3.1 in ISB H-modes, the initial tearing response when the RMP is first applied occurs again near $Ψ N$ ∼ 0.9, corresponding to the $q$ = 9/3 surface.⁴⁶ A tearing response located near $Ψ N$ = 0.9 is too deep in the plasma to stabilize ELMs, and the question arises how the discharge subsequently evolves to an RMP ELM suppressed state with the tearing response sufficiently far out in radius to limit the width of the pedestal and stabilize peeling-ballooning modes.

To explore these dynamics of RMP ELM suppression access, we compared the evolution of the $E ¯ × B ¯$ and electron rotation profiles at an intermediate level of input torque $T i n j$ = 4 Nm which did not suppress ELMs. The $E ¯ × B ¯$ rotation profiles in these two $v *$ ∼ 0.15 IBS H-modes are shown in Fig. 11 for input torques of $T i n j$ = 4 and 5 Nm. As expected, increasing $T i n j$ from 4 to 5 Nm increases the $E ¯ × B ¯$ rotation $ω E$ in the plasma core independent of whether or not the ELMs suppress, but the pedestal $ω E$ rotation does not change (blue and red curves in Fig. 11) until the ELMs suppress at $T i n j$ = 5 Nm (green curve in Fig. 11). The edge region of interest is expanded in Fig. 12 for the terms in the two-fluid electron perpendicular rotation,(⁠ $ω E$⁠, $ω e *$⁠, and $ω ⊥ e$⁠) using the same color key as in Fig. 11: red corresponds to an RMP ELMing IBS H-mode with $T i n j$ = 4 Nm; blue to an RMP ELMing IBS H-mode with $T i n j$ = 5 Nm, and green to an RMP ELM suppressed H-mode with $T i n j$ = 5 Nm. As shown in Fig. 12(a), the $ω E$ profile in the edge remains essentially unchanged as long as the ELMs are only mitigated, but not suppressed. In contrast to this behavior of the $E ¯ × B ¯$ rotation, the electron diamagnetic rotation $ω e *$ changes as $T i n j$ is increased from 4 Nm to 5 Nm [red and blue curves, Fig. 12(b)], even though the ELMs are not yet suppressed. There is an additional narrowing and outward shift in the $ω e *$ profile when the ELMs suppress [green trace, Fig. 12(b)]. As shown in Figs. 11 and 12, the $T i n j$ increase has a much larger impact on the width of the electron pressure pedestal and the corresponding electron diamagnetic flow than on the $E ¯ × B ¯$ rotation, as noted before in Fig. 9. These $ω e *$ changes scale with the input torque in the mitigated ELM phase of the discharge, and are not a consequence of ELM suppression. Together, these changes provide a dynamic picture for access to RMP ELM suppression in which a tearing response is formed as soon as the RMP is applied where $ω ⊥ e$ ∼ 0, too far into the plasma edge to suppress ELMs, but creating a resonant drag in the edge near the $ω ⊥ e$ ∼ 0 radius, which corresponds here to the 8/3 rational surface. As the torque is increased, the edge $E ¯ × B ¯$ rotation does not change due to the resonant braking from the penetrated magnetic perturbation fields at the 8/3 surface, but the tearing response drives increased transport which modifies the electron pressure pedestal, moving the $ω ⊥ e$ ∼ 0 point out to the 9/3 rational surface, where the plasma response bifurcates to a tearing response on this rational surface. Due to the nonlinear nature of the bifurcation from a screening to a tearing response, the result is a bistable system in which the tearing response is localized at the 8/3 rational surface unless enough torque is added to move this bifurcation to the 9/3 rational surface, which finally suppresses the ELMs.

The sensitivity of fluctuations to $E ¯ × B ¯$ rotation shear, and the dependence of the fluctuation frequency in the lab frame on the $E ¯ × B ¯$ rotation frequency $ω E$ make density fluctuation measurements a good diagnostic to confirm where and when the changes to $ω E$ and plasma response occur as the input torque $T i n j$ is changed. In these experiments, density fluctuations were measured in both ion and intermediate scales using Beam Emission Spectroscopy (BES)⁴⁷ and Doppler Back Scattering (DBS),⁴⁸ respectively.

Figure 13 shows the radial profiles of the density fluctuation intensity $n ̃ / n ∝ δ I / I$⁠, where I is the absolute intensity of the Doppler shifted Balmer $D α$ emission from the neutral beam injection, and the local Hahm-Burrell $E ¯ × B ¯$ shearing rate $Ω H B$⁴⁹ on the low field side midplane in $ν *$∼ 0.15 IBS discharge 160921 with (black) $T i n j$ = 3.5 Nm and (red) 5 Nm. Both time intervals have identical n = 3 single upper row RMP fields, but the $T i n j$ = 3.5 Nm interval is ELMing (black line and points) and the $T i n j$ = 5 Nm interval (red line and points) is ELM suppressed. At ion scales, the fluctuation amplitude (Fig. 13) and cross-power spectra (not shown) do not change much when $T i n j$ is changed; there is a small reduction in $n ̃ / n$ at the top of the pedestal (⁠ $Ψ N$ ∼ 0.95) when $T i n j$ is increased from 3.5 Nm to 5 Nm and ELMs are suppressed. This reduction in $n ̃ / n$ occurs in the range 0.92 <  $Ψ N$ < 0.97 where the local $E ¯ × B ¯$ shearing rate increases when $T i n j$ is increased to 5 Nm and ELMs are suppressed. This region corresponds to the locations of the 9/3 and 10/3 resonant surfaces for the n = 3 perturbation, and confirms the radial position of the measured changes to the $ω E$ profile, and the importance of these resonant surfaces for the plasma response. These BES results are based on analyses over 100 ms time windows before the start and after the end of the $T i n j$ ramp, and consequently cannot address the time at which the fluctuations change, only the radial location of the changes. To address the temporal evolution of the $ω E$ changes we use the density fluctuation measurements from the DBS system.

The density fluctuation response at intermediate scales measured by the DBS during the torque ramp is shown in Fig. 14. As the torque increases (the upper black trace), the ELMs (lower black trace) become more widely spaced. Although there is some increase in the fluctuation amplitude between the ELMs after the RMP is applied, the largest change in fluctuation amplitude occurs only at the end of the torque ramp when the ELMs suppress near 4680 ms, where an intermediate-k (⁠ $k ρ s ∼ 2.8–3.2$⁠) broadband feature (orange feature near 2 MHz) emerges following the last ELM-driven fluctuation burst in the range 1500 ≤  $f$≤ 2500 kHz that persists throughout the RMP ELM suppressed phase at $T i n j$ = 5 Nm. A second, lower frequency broadband feature (yellow feature centered near 0.6 MHz in Fig. 14) associated with lower wavenumber $k$ simultaneously forms in the range 0 <  $f$ <1000 kHz. Since the amplitude and mean frequency of density fluctuations depend on the $E ¯ × B ¯$ shear and $E r$⁠, respectively, the onset of density fluctuation changes only after the ELMs suppress, rather than throughout the torque input ramp, supports the “clamping” of the edge $ω E$ profile [Fig. 12(a)] during the $T i n j$ ramp until the ELMs suppress. The role of the turbulent transport associated with these ion and intermediate scale density fluctuations in the access to RMP ELM suppression (if any) is beyond the scope of this manuscript.

RMP ELM suppression in H-modes with ITER-relevant pedestal collisionalities $ν *$ ∼ 0.15, including both ITER-similar-shape (ISS)¹⁵ and ITER Baseline Scenario (IBS),^35,38 generally requires reducing the plasma density or collisionality below a maximum value. Because either density or collisionality, which are not easily separated in DIII-D H-modes at ITER-relevant $ν *$ values, can affect access to RMP ELM suppression, it is important to verify that the loss of RMP ELM suppression when the input torque $T i n j$ is reduced from 5 Nm to 3.5 Nm is not a consequence of an unintended increase in either collisionality or density. A limited density and collisionality scan was performed at the $T i n j$ = 5 Nm conditions, as shown in Fig. 15. RMP ELM suppression was achieved for $T i n j$ = 5 Nm in discharges with both higher density [red traces, Fig. 15(b)] and collisionality $ν *$ [red traces, Fig. 15(c)], and lower density [blue traces, Fig. 15(b)] and collisionality $ν *$[blue traces, Fig. 15(c)] than the collisionality and density obtained in IBS discharges with $T i n j$ = 3.5 Nm. This indicates that the control parameter regulating the loss of RMP ELM suppression at reduced torque is not the density or collisionality.

RMP ELM suppression has also been well-established as a resonant effect which requires the safety factor at the top of the pedestal $q 95$ to be in the relatively narrow “resonant window for ELM suppression,” given by 3.45 ≤  $q 95$ ≤ 3.65, for ISS discharges with $n$ = 3 RMPs using both rows of the DIII-D RMP coil set. However, this $q 95$ dependence for ELM suppression has been obtained from discharges with relatively high levels of torque input parallel to the plasma current: $T i n j$ ≥ 5 Nm (such as ISS discharge 145419 in Fig. 1), and the possibility must be considered that this resonant window might depend on the input torque. This possibility was eliminated as a cause for the loss of ELM suppression at $T i n j$ = 3.5 Nm in a series of discharges with both varying flattop levels of $q 95$ and with $q 95$ ramps, as shown in Fig. 16. The pedestal safety factor $q 95$ was varied in each case by changing the plasma current $I p$ at constant $B T$⁠. RMP ELM suppression was obtained for $T i n j$ = 4–5 Nm and $q 95$ = 3.1 (black traces in Fig. 16). However, when $T i n j$ was reduced to 3.5 Nm, ELM suppression was lost for values of pedestal safety factor 3.2 ≤  $q 95$ ≤ 3.8 using both stationary and dynamically ramped $q 95$ (red, blue, and green traces in Fig. 16). ELM suppression was also not achieved in separate IBS discharges where $B T$ was reduced in order to maximize $δ b r / B T$⁠, and in IBS discharges in which $T i n j$ was reduced by replacing a portion of the co- $I p$ neutral beam injection with electron cyclotron heating (⁠ $T i n j$ ∼ 0 Nm), rather than counter- $I p$ neutral beam injection. These results suggest that the relevant control parameter in these IBS H-modes for obtaining RMP ELM suppression is the level of injected torque $T i n j$⁠, and not a $T i n j$ dependence of the ELM suppression window.

The RMP ELM suppression hypothesis was also tested at $T i n j$ = 1 Nm in IBS discharges with high pedestal $ν *$∼ 1–2 (red traces, Fig. 5). The high $ν *$ allows access to an established operating scenario that is free of disruptive tearing modes that occur in IBS discharges with both low torque $T i n j$ ∼ 1 Nm and pedestal $ν *$ ∼ 0.15 even without the RMP.⁵⁰ For RMP coil currents of 3–4 kA, the ELMs were mitigated, with a doubling of the ELM frequency and a drop in the ELM amplitude in $D α$ emission of about a factor of 2, as shown in Figs. 17(f)–17(g). This ELM mitigation behavior in high $ν *$> 1 discharges has been previously reported for ISS discharges in DIII-D³⁹ and is similar to initial RMP ELM control results from ASDEX-Upgrade.¹⁹ 

The pedestal response to the RMP in these $ν *$ ∼ 1–2, $T i n j$ = 1 Nm IBS discharges is similar to previous results in $ν *$ > 1 ISS discharges.^8,38,41 As the RMP current is increased [Fig. 17(a)], both the line average $n e$ and pedestal $n e P E D$ electron densities drop by 20% and 10%, respectively [Fig. 17(b)]. There is a modest increase in the pedestal electron temperature $T e P E D$ [Fig. 17(c)] as $n e P E D$ drops, such that the electron pedestal pressure $P e P E D$ [Fig. 17(d)] remains constant. However, as reported previously in RMP experiments at $ν *$ >1,³⁹ the core toroidal rotation $v T$ drops and the $v T$ profile flattens [Fig. 17(e)].

Toroidal rotation and frequency profiles for the discharges of Fig. 17 are plotted in Fig. 18. The toroidal rotation in the edge [Fig. 18(a)] drops by a factor of 2 when the RMP is applied, reaching $v T$ = 0 km/s at the top of the pedestal. This $v T$ change when the RMP is applied results in a decrease in the $E ¯ × B ¯$ rotation frequency across the plasma edge, with the $ω E =0$ point moving inward from $Ψ N =0.92$ to $Ψ N =0.89$ [Fig. 18(b)]. There is only a small change in the electron diamagnetic rotation frequency [Fig. 18(c)], which is not large enough to offset the change in $ω E$⁠, leading to a net inward shift in the $ω ⊥ e ∼0$ point from $Ψ N ∼0.9$ without the RMP to $Ψ N ∼0.86$⁠. This radial shift in $ω ⊥ e ∼0$ would move the tearing response further from the top of the pedestal, consistent with the lack of ELM suppression in these discharges. These results are consistent with the RMP ELM suppression hypothesis, and suggest a possible origin for the loss of ELM suppression at high collisionality seen in DIII-D and ASDEX-Upgrade: at $ν *$≥ 1, the RMP strongly brakes the toroidal rotation and reduces $ω E$⁠, while leaving the electron pressure pedestal largely unchanged. As a consequence of these changes, the zero-crossing in the electron perpendicular rotation remains too far inside the plasma core to allow the tearing response to limit the pedestal width to a stable value.

The RMP ELM suppression hypothesis described in Sec. II predicts that, when the input torque $T i n j$ and $E ¯ × B ¯$ rotation are reduced, it should still be possible to obtain RMP ELM suppression by realigning the tearing response radius with the top of the pedestal by reducing the height and width of the electron pressure pedestal and electron diamagnetic rotation. This prediction has been verified in a separate experiment in which an $n$ = 3 RMP was used to suppress ELMs in zero torque (⁠ $T i n j$ = 0) ISS helium discharges.⁵¹ The temporal evolution of one of these discharges is shown in Fig. 19. The discharge begins with a high power phase [Fig. 19(a)] in which, following an L-H transition at 1050 ms [Fig. 19(c)], the plasma normalized pressure $β N$ reaches 1.8 [Fig. 19(b)] prior to reducing the auxiliary heating power and the injected torque $T i n j$ at 1800 ms. The ELMing H-mode persists (due to hysteresis in the H-mode transition) to 2200 ms [Fig. 19(c)], at which time, indicated by a vertical blue line in Fig. 19, there is an H-L back transition. Just before 2700 ms, there is a second L-H transition [D_α in Fig. 19(c)], as indicated by the reformation of a high pressure pedestal [β_N^PED in Fig. 19(b)] and an increase in the overall $β N$⁠. This subsequent RMP ELM suppressed H-mode continues to the end of the discharge without any ELMs while $T i n j$ = 0 Nm, demonstrating that RMP ELM suppression can be obtained in RMP H-modes with $T i n j$ = 0 Nm by also reducing the pedestal pressure and electron diamagnetic flow.

This discharge also demonstrates that it is possible to establish an RMP ELM suppressed H-mode without passing through a phase of ELMing. This last observation indicates that, although ELMs may seed small islands that grow and trigger ELM suppression as suggested by Callen et al.,⁴⁵ such seeding of islands by ELMs may not be a necessary condition for suppressing ELMs. This is potentially an important result for ITER, because ITER will need to apply the RMP in L-mode before the L-H transition, or early in the H-mode before ELMing starts, and this may enable ITER to avoid the strong electron pedestal pressure gradient and diamagnetic flow which screens the RMP and prevents a tearing response. Once ELM suppression is established, it would be necessary to use auxiliary heating to increase $β N$ while the RMP-island interaction maintains the narrow electron perpendicular rotation profile needed for limiting the pedestal width to stable values.

In Fig. 20, the radial profiles in the plasma edge of the $E ¯ × B ¯$ rotation frequency $ω E$⁠, the electron diamagnetic frequency $ω e *$⁠, and the two-fluid electron perpendicular rotation frequency $ω ⊥ e$ are plotted at 4300 ms in the RMP ELM suppressed phase of the helium ISS discharge 158490 shown in Fig. 19. The electron diamagnetic rotation profile in the edge, while quite narrow, still dominates the relatively shallow radial electric field $E r$ well. The resulting electron perpendicular rotation profile indicates that $ω ⊥ e$ ∼ 0 across the entire edge of the discharge from $Ψ N$ = 0.8 out to $Ψ N$ = 0.97, allowing a tearing response over a rather broad radial range of rational surfaces driven by the applied $n$ = 3 RMP.

A series of experiments have been conducted in ITER Baseline Scenario (IBS) discharges in DIII-D to extend RMP ELM suppression to lower levels of injected torque that are more relevant to the existing capabilities anticipated in ITER. When the torque input $T i n j$ was reduced below about 4 Nm, RMP ELM suppression was lost. These RMP ELM suppression results in low torque IBS discharges support the hypothesis for RMP ELM suppression based on a tearing response to the RMP which limits the pedestal width to a stable value. The tearing response occurs on the rational surface where $ω ⊥ e$ ∼ 0 as predicted by two-fluid theory. The tearing response as calculated from linear, single helicity $n$ = 3, two-fluid resistive MHD using the m3d-c1 code, moves from the 9/3 rational surface (⁠ $Ψ N$ = 0.93) to the 8/3 rational surface (⁠ $Ψ N$ = 0.89) when $T i n j$ is reduced from 5 Nm to 3.5 Nm. This shift in the radius of the calculated tearing response is consistent with the changes to the experimentally measured two-fluid electron perpendicular rotation $ω ⊥ e$ profile, and is consistent with the tearing response limiting the H-mode pedestal width, leading to stabilization of the peeling-ballooning modes responsible for Type I ELMs. Scans in input torque $T i n j$ indicate that the H-mode pedestal during the $n$ = 3 RMP exists in a bistable state in which the tearing response at the rational surface where $ω ⊥ e$ ∼ 0, initially corresponding to the 8/3 surface at $Ψ N$ = 0.89), produces a resonant braking that clamps the $E ¯ × B ¯$ rotation profile while initiating an evolution on the transport timescale in the width and height of the electron pressure pedestal. When the injected torque is large enough to overcome the braking on the resonant surface, the $ω ⊥ e$ ∼ 0 radius shifts abruptly to the next rational surface, here the 9/3 surface at $Ψ N$ = 0.93, which is then close enough to the separatrix to limit the H-mode pedestal width to a stable value and suppress the ELMs.