Comparison of MHD stability properties between QH-mode and ELMy H-mode plasmas by considering plasma rotation and ion diamagnetic drift effects

Magnetohydrodynamic (MHD) stability at tokamak edge pedestal in a quiescent H-mode (QH-mode) and type-I ELMy H-mode plasmas in DIII-D experiment was analyzed by considering plasma rotation and ion diamagnetic drift effects. QH-mode plasma is marginally stable to kink/peeling mode (K/PM), but ELMy H-mode one is almost unstable to peeling-ballooning mode (PBM). It was identified that there are three physics features responsible for the difference in the MHD stability properties between QH-mode plasma and ELMy H-mode one. These are the distance of pedestal foot from the last closed flux surface (LCFS), the amount of the ion diamagnetic drift frequency at pedestal, and impact of coupled rotation and ion diamagnetic drift effects. These features were confirmed through the numerical experiments that the stability properties of the QH-mode plasma can be changed to that of the ELMy H-mode one by shifting the plasma profiles inward in the radial direction and halving the ion diamagnetic drift frequency. The reasons of the change in the stability properties are thought as that K/PM is stabilized due to the inward shift of the bootstrap current profile, and PBM is destabilized due to the reduction of the coupled rotation and ion diamagnetic drift stabilizing effect. Importance of these features was validated through numerical experiments with experimental data of other QH-mode plasmas in DIII-D. All the results show that MHD stability properties of QH-mode plasma can be obtained in case that pedestal foot is close to LCFS, ion diamagnetic drift frequency is large due to high ion temperature, and strong rotation shear exists near pedestal.


Introduction
In large tokamak fusion reactor with H-mode pedestal, edge localized modes (ELMs) are thought as one of the critical issues because of large ELM heat load. In ITER, for example, no ELMs will be acceptable from the viewpoint of divertor protection [1]. It has been widely accepted that the strongest candidates of instability triggering ELMs is a peeling-ballooning mode (PBM), which is a magnetohydrodynamic (MHD) mode destabilized due to large pressure gradient and current density near edge pedestal region [2], and hence, the ELM suppression can be achieved when the pedestal pressure gradient and current density are not large enough to destabilize PBM. Quiescent H-mode (QH-mode) [3,4] is one of desirable operation regimes which realize stationary ELM-free H-mode with high confinement performance, such as H 98y2 [5], with ITER and DEMO relevant plasma parameters [6].
One of the phenomena characterizing QH-mode is edge harmonic oscillations (EHOs), which are thought to play a role to sustain the edge transport barrier (pedestal) by enhancing edge particle transport instead of the ELMs [7]. The trigger of the EHOs has been considered as a current-driven kink/peeling mode (K/PM) whose stability is affected by plasma rotation, because the QH-mode can be obtained experimentally when plasma current density and rotation shear are large at edge pedestal region [8]. The interpretation has been validated with both linear and nonlinear MHD analysis with ideal, resistive and extended MHD models [9][10][11][12]. In addition, in QHmode plasmas, the ion diamagnetic drift frequency ω * i is large enough to be comparable to the plasma rotation frequency [9], and hence, the E × B rotation has been recognized as a strong candidate responsible for entering QH-mode; the rotation is composed of the sum of plasma fluid rotation and the ion diamagnetic drift [13]. Here E (B) is the electric (magnetic) field. These experimental conditions in QH-mode plasmas imply both plasma rotation and ω * i can affect the K/PM stability, and in fact, our recent works identified that a low-n (n being toroidal mode number) K/PM in QH-mode plasmas in DIII-D and JT-60U experiments can be stabilized by the coupled rotation and ω * i effects although the mode is destabilized when considering only rotation effect [14,15]. It was also found that the stabilizing effect by the coupled effects becomes more effective in case using the rotation profile of main ion species evaluated by assuming radial force balance with the profile of measured impurity ion species [15].
QH-mode in DIII-D is usually obtained under conditions of low pedestal collisionality (ν * e ⩽ 0.3), high edge safety factor (q 95 ⩾ 5), strong plasma shaping, and momentum input with neutral beam injection (NBI) in the direction counter to the plasma current I p [9]. Recent results have realized to relax such conditions; for example, QH-mode could be sustained with near zero NBI torque by producing E × B rotation with nonresonant magnetic field [13,16]. However, in case the QHmode phase is terminated, type-I ELMs are usually observed in the same discharge, and this trend could be a weakness of QH-mode from the viewpoint of robust divertor protection in large reactors. Hence, to understand the key physics responsible for the difference in the MHD stability properties between QH-mode and ELMy H-mode is important to realize robustly QH-mode operation in not only existing experiments but also future tokamak reactors.
In this study, we perform MHD stability analysis including the coupled rotation and ω * i effects in a QH-mode plasma and a type-I ELMy H-mode one in DIII-D to identify such the key physics. After introducing briefly basic equations solved numerically for the MHD stability analysis in section 2, comparison of the results of the MHD stability analysis in QHmode and ELMy H-mode plasmas is shown in section 3. In this section 3, the key physics features responsible for triggering EHOs or ELMs are also discussed. With the understandings in the previous section, numerical experiments validating the impacts of the key physics features on MHD stability properties are performed with other QH-mode experimental results in DIII-D. These results are shown in section 4. Section 5 presents a summary and discussion of this study.

Model equations
In this study, plasma rotation in the toroidal direction is taken into account in both the equilibrium and stability calculations. For the equilibrium calculation, the Grad-Shafranov equation including the toroidal rotation is solved numerically with the isothermal assumption on each magnetic surface T 0 (ψ) = T i (ψ) + T e (ψ) due to strong parallel heat conductivity [17,18], where ψ is the poloidal magnetic flux normalized as 0 (1) on magnetic axis (plasma surface), and T i (T e ) is the ion (electron) temperature. For the stability calculation, the basic equation is the extended Frieman-Rotenberg equation with the definition of velocity vectors as the equation (1) which is the linearized equation of motion of the diamagnetic MHD model; the details are written in [19]. Here ρ 0 is the mass density, ξ = ξ ⊥ + ξ ∥ B/|B| is the Lagrangian displacement vector defined with the linearized Eulerian velocity as F MHD (F * i ) is the force operator coming from the ideal MHD (ion diamagnetic correction) part, e is the quantum of electricity,Z is the ion mean charge satisfying n e =Zn i , n e (n i ) is the electron (ion) number density, p i is the ion pressure, and the subscript 0 (1) indicates the equilibrium (perturbed) quantity. The diamagnetic MHD model was developed to investigate the impact of the ion diamagnetic drift on ideal MHD stability in both static and rotating plasmas, hence, the linear extended MHD code MINERVA-DI solving (1) and (2) allows to analyze the linear MHD stability with the rotation and ω * i effects, where ω * i ≡ V 0, * i · k is the ion diamagnetic drift frequency, and k is the wave number vector.

Comparison of MHD stability in QH-mode and
ELMy H-mode plasmas considering rotation and ion diamagnetic drift effects

Results of stability analysis in QH-mode and ELMy H-mode plasmas
In this section, we present the results of MHD stability analysis considering plasma rotation and ω * i effects in QH-mode and ELMy H-mode plasmas in DIII-D. In the discharge whose shot number is #163477, both QH-mode and ELMy H-mode plasmas were observed in a different time range as shown in figure 1; QH-mode was in 1.1 s ⩽ t ⩽ 3.0 s and ELMy Hmode was in t ⩾ 3.0 s. The main objective of the discharge was to study the E × B rotation shear threshold for EHO by a torque scan which was performed by adjusting a combination of NBIs. During the scan, the gas puff rate and the total NBI power were almost fixed but the fueling by NBIs was thought to be changed, hence the volume-averaged electron density ⟨n e ⟩ increased until 3.1 s as shown in figure 1. In this study, the MHD stability at 1.80 s (QH-mode) and that at 4.44 s (ELMy H-mode) are analyzed; the dominant toroidal mode number (n) of the EHO in QH-mode plasma is n EHO = 1. In the discharge, toroidal magnetic field B t and I p are clockwise from the top view of the torus, and the toroidal rotation of impurity (carbon) is in the counter-I p direction. The profiles of T i and rotation velocity of carbon were measured with charge exchange recombination spectroscopy, those of T e and n e were measured with Thomson scattering; the profiles of the ELMy H-mode were reconstructed by fitting the data from 4.1 s to 4.68 s; these times correspond to 80% and 99% of the ELM cycle. The current density near pedestal is determined based on the Sauter model [20,21] with the ACCOME code [22]. Note that the electron collisionality ν * e at ψ = 0.95 is around 0.2 (0.5) in the QH-mode (ELMy H-mode) plasmas, which are small enough to consider the Sauter model reliable. The shape of the last closed flux surface (LCFS), profiles of T e and T i , plasma pressure p and n e , current density parallel to the magnetic field j ∥ and q are shown in figures 2(a)-(d), respectively. As discussed in [9,13], the shear of the toroidal rotation frequency associated with E × B drift, Ω E×B , is thought to be responsible for obtaining QH-mode, where the frequency is derived from the radial force balance E · ∇ψ = 0. In this study, the ion diamagnetic drift and the E × B drift by induction electric field, E ind. = −V MHD × B, are treated independently in (1), hence, we evaluate the plasma fluid rotation frequency Ω v×B as In the previous work [15], we found that the one-fluid rotation frequency Ω v×B,CD defined as has larger impact on stability of K/PM compare to that of measured impurity (carbon) Ω v×B,C Here the subscript C (D) indicates the quantity of carbon (deuterium), Ω ϕ (Ω θ ) is the measured rotation frequency in the toroidal (poloidal) direction, r is the plasma minor radius, B p is the poloidal magnetic field, respectively. Note that we assume that the deuterium number density can be evaluated based on the charge neutrality condition, and the temperature is the same as the measured one of carbon. The ω * i profile of the plasma is evaluated with both deuterium and carbon pressure profiles as The profiles of Ω v×B,CD and ω * i are shown in figure 2(e). MHD stability at edge pedestal in the QH-mode and ELMy H-mode plasmas is analyzed by scanning a range of n number of the MHD mode as 1 ⩽ n ⩽ 50, and the ω * i effect is always considered; the position of the plasma surface is assumed at ψ 0.5 = 0.995, and the position of ideal conducting wall surrounding the plasma is the same as the DIII-D vacuum vessel. The result of the stability analysis is summarized by drawing the MHD stability diagram on the (j ped,max , α max ) plane, the analysis which is performed with the equilibria having different pressure and current profiles near pedestal, where j ped,max is the maximum flux averaged current density near edge pedestal, α max is the maximum normalized pressure gradient defined as α ≡ −(µ 0 /2π 2 )(dp/dψ)(dV/dψ)(VR/2π) 0.5 , µ 0 is the permeability in the vacuum, and V(ψ) is the plasma volume in each magnetic flux surface. The j ped,max and α max are varied by adjusting the height of pressure pedestal and the amount of bootstrap current density near the pedestal; the details are written in [23]. Figure 3 shows the MHD stability diagrams on the ( j ped,max , α max ) plane in (a) QH-mode and (b) ELMy H-mode plasmas, where ⟨ j⟩ is the current density averaged in the plasma. From the figures, it is confirmed that there are two main differences in MHD stability properties between QHmode and ELMy H-mode plasmas. One is the kind of unstable mode determining the position of stability boundary close to the operation point on the diagrams. The (j ped,max , α max ) values of the points in these plasmas are almost the same as each other, but the QH-mode plasma is near the K/PM stability boundary although the ELMy H-mode one is near the PBM one. The result seems consistent with the current qualitative understanding; namely, the trigger of EHOs is K/PM and that of ELMs is PBM, as introduced in section 1. The other is that both K/PM and PBM are stabilized by plasma rotation in the QH-mode plasma although those stability boundaries change little in the ELMy H-mode plasma. Such the stabilization of both K/PM and PBM by rotation with the ion diamagnetic drift was also observed numerically in other QH-mode discharges [15]. It should be noted that we assume that a QH-mode can be obtained in case the operation point or its error bars exist in the region between the K/PM stability boundaries identified in the static and rotating conditions, as suggested in [14]; in this study, the error bars determined to be ±20% of j ped,max and α max are drawn on the operation point.

Analysis for identifying key physics responsible for EHO/ELM trigger
In this section, we investigate key physics responsible for the difference in MHD stability properties between QH-mode and ELMy H-mode plasmas. As shown in figure 3, there are two main differences; one is the MHD mode determining the stability boundary near the operation point, and the other is the impact of plasma rotation on the stability of both K/PM and PBM. To identify the key physics, we pay attention to the difference in plasma profiles between the QH-mode and ELMy H-mode plasmas.
First, we pay attention to the difference in the position of the MHD stability boundary in the stability diagram. It is well known that the destabilizing sources of ideal MHD modes are the pressure gradient and plasma current parallel to the magnetic field [24], and as shown in figures 2(c) and (d), the pressure pedestal and corresponding bootstrap current profiles in QH-mode are closer to the LCFS than those in ELMy H-mode. Hence, an impact of the difference in the profiles on MHD stability is analyzed by shifting all the T e , T i , n e and n i profiles of QH-mode plasma inward to align the pressure pedestal foot with that of ELMy H-mode one; the shifted profiles are shown in figure 4. As shown in the figure, the pressure pedestal profiles in ψ 0.5 > 0.97 become similar to each other; the shift was performed by subtracting 0.01 in ψ. Figure 5 shows the comparison of the ( j ped,max , α max ) stability diagrams of the QH-mode plasma between shifted and unshifted profiles; (a) without and (b) with rotation effects. In both static and rotating cases, K/PM becomes more stable  ELMy H-mode plasmas on the (j ped,max , αmax) plane. The target represents the equilibrium values observed experimentally, and the numbers on the diagram show the n number of the MHD mode determining the stability boundary. These diagrams show the following differences in MHD stability properties between QH-mode and ELMy H-mode plasmas. One is the position of the stability boundary. The current-driven K/PM boundary is close to the target in the QH-mode plasma, but the PBM boundary is near the target in the ELMy H-mode one although the targets in both plasmas exist on almost the same position in the diagram. The other is the impact of plasma rotation on MHD stability. Ω v×B,CD rotation stabilizes the MHD modes in the QH-mode plasma, but has little impact on MHD stability in the ELMy H-mode plasma.
when shifting inward the profiles, but PBM stability does not change much. The K/PM is mainly driven by plasma current parallel to magnetic field, hence, the reason why K/PM becomes stable is thought as that the current is far from the LCFS in association with the shift of pressure profile.
Next, as a parameter which can have large impact on PBM stability, we pay attention to the difference in the amount of ω * i ; the maximum value of |ω * i | in the QH-mode plasma is about twice of that in the ELMy H-mode, as shown in figure 2(e). Since the ω * i effect stabilizes short wavelength MHD modes regardless of plasma rotation [19,25], it should be worth investigating the impact of ω * i on the stability diagram. An impact of ω * i on the stability was analyzed by using a new ω * i profile estimated artificially by multiplying a coefficient C * i to the original profile of ω * i as ω * i = C * i ω * i,org. . As shown in (6), the change in ω * i affects the Ω v×B profile, hence, the comparison of the Ω v×B profiles is shown in figure 6(a). Figures 6(b) and (c) show the comparison of the ( j ped,max , α max ) stability diagrams between the QH-mode plasma with original profiles and that with shifted profiles and reduced (C * i = 0.5) ω * i ; figure (b) ((c)) shows the diagram without (with) rotation effects. Note that the modified rotation profile is taken into account in both equilibrium and stability calculations. It is clearly shown that PBM becomes more unstable by reducing ω * i in both static and rotating cases. In particular, the reduction of the ω * i stabilizing effect is visible in the rotating plasma case, and as the result, the stability diagram of the QH-mode plasma becomes similar to that of the ELMy H-mode one by shifting inward the plasma profiles and reducing ω * i by half. In other words, PBM can be stabilized effectively by coupled plasma rotation and the ω * i effects. This is thought as the reason why plasma rotation has   It should be also checked how the MHD stability diagram will change when shifting all the plasma profiles of ELMy H-mode plasma outward to align the pressure pedestal foot with that of QH-mode one and doubling ω * i ; the profiles of p, j ∥ and Ω v×B are shown in figure 7. The resultant (j ped,max , α max ) stability diagram of the rotating ELMy H-mode plasma with shifted profiles and doubled (C * i = 2.0) ω * i profile, shown in figure 7(d), indicates that the stability boundary position moves closer to that of the original QHmode plasma. From these results, it is identified that there are three strong candidates of key physics features responsible for the difference in MHD stability properties between QH-mode and ELMy H-mode plasmas; one is the distance of plasma pedestal (foot) from LCFS, another is the amount of ω * i , and the other is the coupled effects between rotation and ω * i .

Numerical experiments analyzing the impacts of shift of profiles and reduction of ω * i
The results in the previous section implies that MHD stability properties of QH-mode plasma, which is marginally unstable to K/PM, can be changed to those of ELMy H-mode plasma marginally unstable to PBM by shifting plasma profiles inward, reducing ω * i , and considering plasma rotation. To confirm this trend more quantitatively, we performed numerical experiments analyzing the impacts of these physics features on MHD stability properties with other QH-mode experimental results in DIII-D. The MHD stability in four QH-mode equilibria was analyzed in the same manner as performed in previous sections; the shot number/time are #153440@1.725 s, #157102@2.42 s, #157188@2.06 s, and #163518@2.35 s. In all the discharges, B t and I p are clockwise from the top view of the torus, and the toroidal rotation of the impurity carbon is in the counter-I p direction, as the #163477 discharge; the results of MHD stability analysis in these discharges are shown in [15]. Figure 8 shows the p and j ∥ profiles of these discharges, and figure 9 is for the Ω v×B and ω * i profiles; both the original and inward shifted profiles with C * i = 0.5 are plotted. The shift of the profiles was performed by adding 0.01 in ψ by following the value used in the previous section. Figure 10 shows the comparison of (j ped,max , α max ) stability diagrams between the original QH-mode rotating plasma and that with shifted profiles and reduced (C * i = 0.5) ω * i ; (a) #153440@1.725 s, (b) #157102@2.42 s, (c) #157 188@2.06 s, (d) #163588@2.35 s. In all the cases, the K/PM is stabilized and PBM is destabilized by shifting the profiles inward and reducing ω * i by half. As the result, the PBM stability boundary moves closer to the operation points but the K/PM one becomes far from the points with shifted profiles and half ω * i although the points were near the K/PM boundary but far from the PBM one in the original QH-mode plasmas.
The trends are the same as those identified in the previous section, hence, we consider that type-I ELM could appear in these QH-mode discharges in case the following conditions were satisfied; one is the inward shift of plasma profiles, and the other is the reduction of ω * i .

Summary and discussion
We analyzed MHD stability at tokamak edge pedestal by considering plasma rotation and ion diamagnetic drift (ω * i ) effects in QH-mode and ELMy H-mode plasmas in DIII-D. It was confirmed that the QH-mode plasma is near K/PM stability boundary, but the ELMy H-mode one is near the PBM boundary when analyzing the stability with static assumption; the results are consistent with present understanding that PBM (K/PM) triggers ELMs (EHOs). It was found that the difference in MHD stability properties is due to the following physics features. One is that the distance of pedestal (foot) from the LCFS is responsible for the stability of K/PM; the mode becomes more stable as pedestal foot becomes far from the LCFS due to shifting inward the bootstrap current profile. Another is the difference in the amount of ω * i is responsible for the stability of PBM; PBM becomes more unstable when reducing the amount of ω * i . The other is the coupled rotation and ω * i effects; both K/PM and PBM cannot be stabilized in case the amount of ω * i is small. These features imply that several key physics for EHO trigger can be regarded as 'pedestal (foot) close to LCFS', 'large ω * i due to high T i pedestal' and 'strong rotation shear near pedestal', which help to put K/PM stability boundary close to and PBM one far from the operation point. Impacts of these key physics features on MHD stability properties were reproduced in numerical experiments which were performed by shifting plasma profiles of other DIII-D QH-mode discharges inward and reducing ω * i by half.
The physics features responsible for EHO trigger identified in this study were confirmed with the results of MHD stability analysis only in one of each QH-mode plasma and ELMy H-mode one and those in virtual numerical experiments based on the experimental results in several QH-mode plasmas. Hence, to discuss whether or not QH-mode plasmas can always be realized in case satisfying them, it is necessary to perform a validation study with at least several pairs of QH-mode and ELMy H-mode plasmas; if possible, the study with the experimental data in not only DIII-D but also other experiments is desirable to make the results more reliable. For example, as discussed in [15,23], both PBM stability in ELMy H-mode plasmas in JT-60U and K/PM stability in QH-mode plasmas in DIII-D and JT-60U depend on the direction of toroidal rotation in case the coupled rotation and ω * i effects are considered. It is known that the QH-mode plasmas in both DIII-D and JT-60U favor the rotation direction counter to the plasma current, and the above validation study will help to clarify physics reasons of this trend. We will try to perform such the quantitative analysis in near future.
Another important future work is to identify the physics mechanisms determining the position of pedestal profiles. Though this is beyond the scope of the study because the mechanisms will be identified by studying not only MHD but also heat/particle/momentum transport physics at edge pedestal, we will try to discuss it in future.
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