Non-linear modeling of the threshold between ELM mitigation and ELM suppression by resonant magnetic perturbations in ASDEX upgrade

Phys. Plasmas 26, 042503 (2019); https://doi.org/10.1063/1.5091843 26, 042503 © 2019 Author(s). Non-linear modeling of the threshold between ELM mitigation and ELM suppression by resonant magnetic perturbations in ASDEX upgrade Cite as: Phys. Plasmas 26, 042503 (2019); https://doi.org/10.1063/1.5091843 Submitted: 06 February 2019 • Accepted: 19 March 2019 • Published Online: 12 April 2019 F. Orain, M. Hoelzl, F. Mink, et al. COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Numerical study of tearing mode seeding in tokamak X-point plasma Physics of Plasmas 26, 042504 (2019); https://doi.org/10.1063/1.5086402 Kinetic modeling of ELM-induced tungsten transport in a tokamak plasma Physics of Plasmas 26, 042508 (2019); https://doi.org/10.1063/1.5092319 Edge localized modes and the pedestal: A model based on coupled peeling–ballooning modes Physics of Plasmas 9, 2037 (2002); https://doi.org/10.1063/1.1449463 Non-linear modeling of the threshold between ELM mitigation and ELM suppression by resonant magnetic perturbations in ASDEX upgrade Cite as: Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 Submitted: 6 February 2019 .Accepted: 19 March 2019 . Published Online: 12 April 2019 F. Orain, 1,2 M. Hoelzl, 2 F. Mink, 2 M. Willensdorfer, 2 M. B ecoulet, 3 M. Dunne, 2 S. G€ unter, 2 G. Huijsmans, 3,4 K. Lackner, 2 S. Pamela, 5 W. Suttrop, 2 E. Viezzer, 6 ASDEX Upgrade Team 2 and EUROfusion MST1 Team a) AFFILIATIONS 1 CPHT, Ecole Polytechnique, CNRS, 91128 Palaiseau, France 2 Max-Planck-Institute for Plasma Physics, 85748 Garching, Germany 3 CEA, IRFM, 13108 Saint-Paul-Lez-Durance, France 4 Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands 5 CCFE, Culham Science Centre, Abingdon, Oxon OX14 3DB, United Kingdom 6 Department of Atomic, Molecular and Nuclear Physics, University of Seville, 41012 Seville, Spain a) See author list in H. Meyer et al., Nucl. Fusion 57, 102014 (2017). ABSTRACT The interaction between Edge-Localized Modes (ELMs) and Resonant Magnetic Perturbations (RMPs) is modeled with the magnetohydrodynamic code JOREK using experimental parameters from ASDEX Upgrade discharges. According to the modeling, the ELM mitigation or suppression is optimal when the amplification of both tearing and peeling-kink responses results in a better RMP penetration. The ELM mitigation or suppression is not only due to the reduction of the pressure gradient but predominantly arises from the toroidal coupling between the ELMs and the RMP-induced mode at the plasma edge, forcing the edge modes to saturate at a low level. The bifurcation from ELM mitigation to ELM suppression is observed when the RMP amplitude is increased. ELM mitigation is characterized by rotating modes at the edge, while the mode locking to RMPs is induced by the resonant braking of the electron perpendicular flow in the ELM suppression regime. Published under license by AIP Publishing. https://doi.org/10.1063/1.5091843 I. INTRODUCTION Research in magnetically confined fusion plasmas mainly focuses on tokamak and stellarator devices. In the ITER tokamak under construction, the high-confinement regime (or “H-mode”) has been chosen as the operating regime, since it is characterized by an improved confinement induced by a transport barrier at the edge. 1 This transport barrier is associated with a large edge pressure gradient raising the whole pressure profile on the so-called “pedestal.” However, instabilities named Edge-Localized Modes (ELMs), observed in tokamaks and sometimes in stellarators, 2 are triggered beyond a certain threshold in the pressure gradient (ballooning modes) and/or plasma current density (peeling modes). ELMs induce the quasiperiodic relaxation of the plasma edge which leads to the expulsion of a large particle flux burst and the deposition of a large transient heat flux on plasmafacing components. The ELM-induced heat flux might be intolerably largefortheITERdivertortargets; 1 thus, reliable methods to control or suppress ELMs must be mastered. A promising control method consists of applying nonaxisymmetric magnetic perturbations, smaller than the toroidal magnetic field by a factor of around 10 4 . These perturbations are aimed at inducing magnetic reconnection on the resonant surfaces, characterized by a safety factor q¼m/n,wheremand nare, respectively, the poloidal and toroidal mode numbers. They are therefore called Resonant Magnetic Perturbations or RMPs. The original goal of RMPs, coming from Tore Supra’s ergodic divertor, 3 is to create magnetic island chains on the resonant surfaces located at the plasma edge. If the islands are sufficiently large, they overlap and produce a chaotic (or stochastic) layer: in this case, the perpendicular transport is increased and the pressure gradient is therefore reduced. The initial aim of the RMPs is thus to slightly reduce the pressure gradient just below the ELM- Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-1 Published under license by AIP Publishing Physics of Plasmas ARTICLE scitation.org/journal/php triggering threshold, while keeping the pressure large enough to maintain a good confinement. RMPs successfully managed to fully suppress ELMs in DIII-D, ASDEX Upgrade, KSTAR, and EAST 4–7 and to mitigate them in JET and MAST. 8,9 However, the physics of the ELM mitigation and suppression is actually more complicated than this simple picture, in particular because the plasma response to the RMP application is capable of making them ineffective. The plasma response, intensively studied theoretically, 10–18 is understood as follows. The electron perpendicular flow generates currents on resonant surfaces, which can induce a magnetic field opposite to the applied perturbation, resulting in a zero net perturbation penetrating in the plasma: RMPs are then screened. More recently, it was found that RMPs can also excite marginally stable peeling-kink modes at the plasma edge, capable of improving the RMP penetration. 19–21 This paper describes the impact that both “resonant” and “peeling-kink” responses have on the ELM mitigation and suppression. The interaction between RMPs and ELMs was modeled with the magnetohydrodynamic (MHD) code JOREK 22 including two-fluid diamagnetic and neoclassical effects. 18 In Sec. II, the generic features of the ELM control by RMPs, obtained in modeling in close comparison with the experimental observations of ASDEX Upgrade discharges, are presented. In Sec. III, the threshold between ELM mitigation and ELM suppression is modeled above a given RMP amplitude: the mechanism inducing the ELM suppression is described. Finally, the conclusion and a discussion are provided in Sec. IV. II. ELM CONTROL BY RESONANT MAGNETIC PERTURBATIONS IN ASDEX UPGRADE: GENERIC FEATURES This section describes the main mechanisms determining how the plasma response affects the coupling between ELMs and RMPs in the nonlinear simulations. A. Experimental configuration and simulation parameters In order to compare with the experimental observations of ASDEX Upgrade discharges, realistic plasma parameters and geometry are used in the modeling. As a reference case, equilibrium reconstruction of the density, temperature, and magnetic profiles is performed from the discharge #31 128, where a strong ELM mitigation is observed in experiments. 5 Note that the temperature and density profiles used in modeling are the experimental pre-ELM profiles before RMPs are applied, in order to study a peeling-ballooning unstable plasma. These profiles, extracted at the time t¼2.4 s of the discharge #31 128, are plotted in Fig. 12. The time evolution of the main plasma parameters during this discharge is plotted in Fig. 2(a) in Ref. 5.Asin experiments, n¼2 static magnetic perturbations are applied in modeling, with a nominal current of about 6 kAt flowing in the RMP coils. In the experiments, the modification of the differential phase DU between the upper and lower coil currents allows changing the applied RMP spectrum and thus the plasma response to RMPs. In the discharge #31 128, a constant phase DU ¼þ90is steadily applied. During the discharge #30826 operated with similar plasma profiles as in the discharge #31 128, the time variation of DU from þ90to 90 showed that the strongest ELM mitigation is found for DU ¼þ90 (Ref. 23). During this discharge, when DU is reduced towards the opposite phase 90, a transient ELM-free phase is observed, suggesting that the stability limit previously reduced by effective RMPs is increased back to a threshold close to the stability limit without RMPs: 24 hence, in this plasma configuration, RMPs have the smallest effect on ELM for DU 90. TheimpactRMPshaveonELMsismodeledforthesetwoopposite phases (DU ¼þ90and 90), for a nominal applied RMP amplitude (i.e., the experimental amplitude) and for an amplitude increased by 50%. The modeling was done with the reduced MHD model of JOREK including the two-fluid diamagnetic rotation, the neoclassical poloidal friction, and a source of toroidal rotation reproducing the experimental profile. 18 The parameters used are similar to the ones described in Ref. 21 and reproduce as accurately as possible the experimental situation. The main limitations of the modeling are the increased central resistivity g 0 ¼8.1 10 7 Xm (value one order of magnitude larger than the Spitzer value, but the T3=2 edependency is accounted for in the simulations, T e being the electron temperature) and the restriction of the simulations to the toroidal modes n8to limit their computational costs. The impact of these limitations is discussed in Sec. IV A. B. General plasma response to RMPs Before presenting the interaction between ELMs and RMPs, the plasma response to RMPs (without ELMs), described in more detail in Ref. 21, is summarized in order to make the following more understandable. In this paragraph, only the axisymmetric component (n¼0) and the n¼2 mode are included in the simulation, to prevent ELMs [generally constituted of modes n4(Ref.25)] from growing and to consider only the plasma response to RMPs. The n¼2 RMPs are applied in the simulation and increased towards their nominal amplitude within 1000 Alfv en times t A (0.5 ms), for different phases DU between upper and lower coil currents. For all phases, the 3Dequilibrium obtained presents similar main features: RMPs are screened by the flows in the plasma center, preventing magnetic islands from growing on central resonant surfaces. However, at the very edge, the large resistivity (related to the low temperature) allows island chains to form and a small chaotic layer is observed due to the overlapping of magnetic islands on the resonant surfaces q9/4 (for a normalized poloidal flux w0.97). Around the X-point, a typical lobe structure is induced by the destruction of the separatrix. The magnetic topology is plotted for the configuration DU ¼90in Fig. 1(a). The main differences depending on the phase DU actually appear only at the edge, as shown in Figs. 1(b) and 1(c). In the case when RMPs are strongly mitigated in experiments (DU ¼þ90), the chaotic layer found in the modeling is largest, covering the edge plasma for w 0.97, as indicated by the yellow arrow in Fig. 1(b). Near the X-point [highlighted by the red cross in Fig. 1(b)], the distortion of the field lines due to the amplification of peeling-kink modes is also maximum in this configuration. Since both the ergodic layer width and the peeling-kink amplitude are maximum, we call it the “resonant configuration.” In contrast, for the opposite phase DU ¼90when RMPs have least effect on ELMs in experiments, the smallest chaotic layer and the smallest distortion near the X-point are found in the modeling [Fig. 1(c)]: we call it “nonresonant configuration.” As explained in Ref. 21, in the optimal (resonant) configuration DU ¼þ90,thepoloidal coupling between the peeling-kink modes (m>nq) and the tearing modes (magnetic islands on q¼m/n) allows for the penetration and Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-2 Published under license by AIP Publishing amplification of the RMPs at the edge. In the opposite (“nonresonant”) configuration (DU ¼90), these modes remain at low amplitude because of the plasma screening. C. RMP effect on ELMs From now on, all modes from n¼0to8areincludedinthe modeling to study the effect of “penetrated” RMPs on ELMs in the two configurations described in the previous paragraph (Sec. II B). As a comparison, a simulation is also run without RMPs for the same plasma parameters. In a first step, in order to particularly highlight the saturation of the modes in the simulations, the mode coupling was enhanced by increasing the anomalous viscosity (the dynamic viscosity is set to 3.7 10 7 kg/(ms) corresponding to a kinematic viscosity of 2 m 2 /s): therefore, the anomalous viscous term dominates over the neoclassical friction term, resulting in an overestimated stabilization of the modes. In Sec. III, the reduction of the anomalous viscosity by a factor of 10 allows the perpendicular dissipation to be mainly ruled by the neoclassical friction and to reproduce more realistically the threshold between ELM mitigation and ELM suppression. In the simulation without RMPs, since the experimental profiles before RMP application are used as input for modeling, the plasma is unstable. As shown in Fig. 2(a), peeling-ballooning modes with dominant mode numbers n¼6, 7, and 8 grow exponentially. Their nonlinear coupling generates the growth of lower nmodes, 26 until all modes saturate and induce the ELM crash. Note that linear simulations show that n9 modes are also linearly unstable, with a growth rate smaller than n¼7 and 8. Since these two most unstable modes dominate the linear growth and the low n6 modes are nonlinearly dominant, 25 it was chosen to discard the n9 modes to reduce the computation time of the simulations. This “natural” ELM without RMPs is compared to the cases with RMPs. Figures 2(b) and 2(c) present, respectively, the time evolution of the magnetic energy of the different modes when nonresonant (DU ¼90) and resonant (DU ¼þ90) RMPs are applied. In both cases, we observe the growth of the n¼2 mode induced by the RMP application. In the nonresonant configuration [Fig. 2(b)], the n¼4mode growth follows the dynamics of n¼2, due to quadratic coupling (cross-terms [n¼2] [n¼2]). However, the n¼6and8modes grow exponentially with a growth rate close to the one without RMP, FIG. 1. Poincar e plot of the magnetic topology (a) in [R,Z] coordinates (b) and (c) in radial and poloidal [w,h] coordinates for the resonant (DU ¼þ90) and nonresonant (DU ¼90) configurations. (a) Is given for the nonresonant case. The red crosses present the X-point location and the red lines sketch the strong kinking of the field lines near the X- point in the resonant (DU ¼þ90) case, in opposition to the straight field lines observed near the X-point in the nonresonant (DU ¼90) case. The yellow arrows highlight the radial width of the chaotic layer. FIG. 2. Time evolution of the magnetic energy of the toroidal modes n¼1–8 in the following cases: (a) ELM without RMP. (b) ELM with nonresonant RMPs. (c) ELM with resonant RMPs. Units are arbitrary but the normalization is the same in all the figures of the paper. Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-3 Published under license by AIP Publishing leading to an ELM crash of similar amplitude as in the case without RMPs. Note that the odd modes present a growth delayed by the RMP application but they finally develop exponentially. In contrast, when resonant RMPs are applied [Fig. 2(c)], the n¼2modedrivenby RMPs has a larger amplitude, due to the stronger penetration. This amplitude is large enough to allow the toroidal coupling of all even modes with the n¼2 mode: these modes all follow the same dynamics driven by RMPs and saturate at an amplitude lower than the saturation level of the ELM without RMPs. This saturation induces the mitigation or the suppression of the ELMs. It is interesting to observe that the n¼2 symmetry imposed by RMPs also prevents the growth of odd modes in the resonant configuration. The mechanism of the ELM stabilization by nonlinear mode coupling, observed here, was first described for a JET case in Ref. 27. In the resonant case when ELM suppression is observed in modeling [Fig. 2(c)], RMPs induce an enhanced perpendicular transport due to the formation of a chaotic layer at the edge; therefore, the edge pressure gradient is reduced. In order to check if the ELM suppression is due to the reduction of the pressure gradient under the peeling-ballooning stability limit, the following test is done: a simulation is performed with a pressure gradient reduced to the same level as in the ELM suppression case, but without applying RMPs. The modified current density profile is also considered. In this simulation, the exponential growth of the n¼8 Edge-Localized mode is observed, leading to an ELM crash. As shown in Fig. 3, the growth rate is reduced by a factor of two as compared to the natural ELM with an unchanged pressure profile, but the ELM is still unstable and leads to a crash. It shows that the ELM suppression is indeed caused by the ELM saturation induced by the mode coupling, rather than by the reduction of the pressure gradient below the ELM stability limit. III. THRESHOLD BETWEEN ELM MITIGATION AND SUPPRESSION A. Bifurcation from mitigation to suppression In this section, the reduced viscosity allows revealing the fine mechanisms distinguishing ELM mitigation from ELM suppression. In order to characterize the ELM energy, the energy of the modes can be separated into two parts: the externally injected static energy induced by RMPs and the intrinsic (nonstatic) energy corresponding to the energy of the ELM, mitigated or not by RMPs. The sum of the intrinsic magnetic energy of all perturbations is plotted in Fig. 4 for different cases: natural ELM without RMP, ELM with nonresonant RMPs applied (DU ¼90) at nominal amplitude, and ELM with resonant RMPs applied (DU ¼þ90) at nominal and increased amplitude (by 50%). We observe that when nonresonant RMPs are applied at nominal amplitude (green line), the ELM reaches the same level of energy as for the uncontrolled ELM (blue dashed line), showing the inefficiency of nonresonant RMPs. However, when resonant RMPs are applied at nominal amplitude (red line), the ELM energy is reduced significantly (red line), as in ELM mitigation observed in experiments. Furthermore, when the resonant RMP amplitude is increased by 50% (purple line), the intrinsic energy vanishes, corresponding to full ELM suppression. It means that above a certain level of energy injected by the RMP penetration, the toroidal coupling of the edge modes (normally unstable) with the n¼2staticmode induced by RMPs is large enough to force them to saturate. Note that when nonresonant RMPs are applied at a larger amplitude multiplied by 2, ELM suppression is still not observed, which confirms the importance of the resonant condition to obtain ELM suppression. The details of the energy evolution in ELM mitigation (for nominal resonant RMP amplitude) and ELM suppression regimes (for increased RMP amplitude) are provided in Figs. 5(a) and 5(b), respectively. The simulation is started with the n¼2modealone,when RMPs are applied at t0.1 ms and the other even modes n¼4, 6, and 8 are added at t0.6 ms. Note that the odd modes were discarded from the simulation in order to reduce the computational time, since they anyway remain at the noise level when resonant RMPs are applied at sufficient amplitude, as shown in Sec. II C. In the ELM mitigation case (a), when the modes n¼4, 6, and 8 are included in the simulation, an initial “linear” phase is observed FIG. 3. Magnetic energy of the n¼8 mode without RMP for the initial pressure profile [as in Fig. 2(a), dashed line] and for a reduced pressure profile [as in Fig. 2(c), full line]. The corresponding growth rate cis given. FIG. 4. The sum of the magnetic energies associated with the nonaxisymmetric components is plotted in the cases of an ELM without RMP (dash blue), with nonresonant RMP (green), with resonant RMP at nominal amplitude (red), and with resonant RMP at amplitude increased by 50% (dashed-dotted line in purple). The value corresponding to the magnetic energy perturbation of the RMPs alone is subtracted. In the nonlinear evolution, this difference can also drop slightly below zero. Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-4 Published under license by AIP Publishing (from t0.6ms to 1 ms). All along this phase, these modes grow exponentially, with a reduced growth rate as compared to the ELM without RMPs. Then, in the nonlinear phase (from t1ms),themodesreach a saturation amplitude below the amplitude of the ELM without RMPs. During this ELM-mitigated regime, which lasts until t1.9 ms, the n¼4, 6, and 8 modes are rotating in the EBor electron diamagnetic direction and plasma filaments are expelled in the opposite (ion diamagnetic) direction: these modes behave as peeling-ballooning modes in a similar way as in the natural ELM regime, but with a much smaller amplitude and hence they induce a much smaller exhaust. Note that the interaction between the rotating unstable modes and the n¼2 perturbation induced by RMPs forces the n¼2modetoco- rotate with the other modes, at the frequency fmV h /(2pr res ) nqV h /(2pr res )10–25 Hz. q,r res ,andV h are, respectively, the safety factor, minor radius, and poloidal rotation at the resonant surface where the amplitude is maximal (for q¼4); n¼2andm¼nq are, respectively, the toroidal and poloidal mode numbers. During ELM mitigation, the energy loss rate (respectively the particle loss rate) between t¼1.1 ms and 1.3 ms is reduced by one third (respectively by 40%) as compared to the natural ELM relaxation. In comparison, in the ELM suppression case [Fig. 5(b)], the amplitude of the static n¼2 mode is larger, since a larger current is applied in RMP coils (þ50%) and thus RMPs penetrate more significantly. During the initial phase when the modes n>2 are added in the simulation at t0.6ms,themodesn¼4, 6, and 8 start with an initial amplitude imposed by the coupling with n¼2andremainatthesamelevel, until they grow and saturate at an even smaller amplitude than in the ELM mitigation case (around one order of magnitude smaller). In the ELM suppression regime, the energy and particle loss rates are strongly reduced as compared to the natural ELM relaxation, by 60% and 80%, respectively. The remaining radial transport can be explained by the chaos induced by the RMP penetration, as discussed in Sec. III D. Still in the ELM suppression case [Fig. 5(b)], the edge modes n¼4, 6, and 8 are initially rotating (as in the ELM mitigation case) and also force the n¼2 mode to co-rotate with them, but the modes suddenly start to slow down around t1.6 ms, until they stop rotating and remain static around t1.75 ms. The perturbation of the electron temperature at the outboard midplane is plotted in Fig. 6 for 17 time steps between t¼1.652 ms and 1.802 ms. It highlights the propagation of the last rotating perturbation (pictures 1 to 6, from t¼1.652 ms to 1.70 ms) followed by a sustained static perturbation. The mechanism of the mode braking and the possible correlation between ELM suppression and mode braking are described in Subsection III B. In comparison, Fig. 7 shows that for the same time slices, the edge modes keep on rotating in the ELM mitigation regime. Note that the modes rotate counterclockwise (i.e., in the EBdirection) with a larger speed than in the early phase of the rotation braking observed in ELM suppression, as shown by the black arrows following the perturbation in time. The qualitative comparison of the rotation with experimental measurements is provided in Subsection III B. Furthermore, it is interesting to notice that in the ELM mitigation regime, all the even modes (except n¼2) reach the same order of FIG. 5. Magnetic energy of the modes n¼2, 4, 6, and 8 in the ELM mitigation case obtained for nominal RMP amplitude (a) and in the ELM suppression case obtained at RMP amplitude increased by 50% (b). FIG. 6. Perturbation of the electron temperature at the outboard midplane in the ELM suppression regime for 17 time steps between t¼1.652 ms and 1.802 ms. A constant time Dt¼8.8 ls separates two consecutive steps. The arrows highlight the mode rotation at the beginning (pictures 1 to 6, from t¼1.652 ms to 1.70 ms), followed by a sustained static perturbation. Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-5 Published under license by AIP Publishing magnitude in the nonlinear phase. Indeed, when the perturbation of the magnetic energy is maximal (for t¼1.64 ms), the ratio between the amplitudes of the energies of the different modes is the following: E n¼4 /E n¼6 ¼2.5 and E n¼6 /E n¼8 ¼1.05. The amplitude of these modes is also nonnegligible as compared to the n¼2 mode amplitude: E n¼2 / E n¼4 ¼3.8. In contrast, in the ELM suppression regime, the n¼2 mode number clearly dominates over the other modes (at the maximal energy perturbation for t¼1.67 ms, E n¼2 /E n¼4 ¼18.9) and the next mode number n¼4 is also dominant over the larger mode numbers: E n¼4 /E n¼6 ¼4.5 and E n¼6 /E n¼8 ¼2.8. This is in stark contrast with the mode distribution in the natural ELM without RMP, where the medium modes n6dominateoverthelowermodes.Therefore, when RMPs are applied, the different energy redistribution during ELM mitigation and ELM suppression shows that the application of the n¼2 RMPs had imposed the modes n¼4, 6, and 8 to remain at a lower level in ELM suppression as compared to ELM mitigation. In other words, a stronger energy transfer has been operated from medium toroidal nmode numbers towards lower nmodes (principally towards n¼2). B. Mechanism of the mode braking during ELM suppression The rotation of the different modes can be directly compared with the experimental observations. In the discharge #33133, the bifurcation from ELM mitigation to ELM suppression is observed. 28 The toroidal mode spectrum is calculated from the measurements of the magnetic fluctuations, which means that only fluctuating modes can be observed in the spectrograms. In Fig. 8,negativenmode numbers correspond to modes rotating in the electron diamagnetic direction and indicates the movement of edge modes, while positive mode numbers describe core modes rotating in the ion diamagnetic direction. 25 The comparison of the ELM mitigation [Fig. 8(a)]andELM suppression [Fig. 8(b)] regimes shows that rotating modes are observed at the edge in the ELM mitigation phase. In contrast, in the ELM suppression phase, almost no fluctuating modes are present, which means that the ELMy regime characterized by rotating edgelocalized modes is replaced by a regime with either static modes or no mode at all. These observations corroborate the idea that the ELM suppression regime might consist of static saturated modes instead of rotating modes in natural ELMs or mitigated ELMs, as found in modeling. In the modeling of the ELM suppression case described above (Subsection III A), the braking of the edge modes until they become static seems to be induced by the braking of the electron perpendicular rotation. Theorized first in Ref. 10 and refined for RMPs in Ref. 12,the bifurcation from a screened configuration towards the penetration of RMPs is correlated with the sudden braking of the electron perpendicular rotation on the resonant surfaces induced by electromagnetic torque. In the simulation of the ELM suppression presented in Fig. 5(b), such a bifurcation is observed at the pedestal top, as plotted in Fig. 9. On the resonant surfaces q¼3 (for a normalized poloidal magnetic flux w¼0.82), the perpendicular electron velocity is already close to zero at the beginning, allowing the penetration of magnetic islands. On the q¼7/2 surface (for w¼0.88), located near the pedestal top, the abrupt reduction of the perpendicular electron flow occurs between t¼1.6 and 2ms. This rotation braking likely induces the braking of the modes: edge modes are forced to be in phase with the penetrated mode induced by RMPs and thus to become static. The FIG. 7. Perturbation of the electron temperature at the outboard midplane in the ELM mitigation regime, plotted for the same time steps as in the ELM suppression regime (Fig. 6). The arrows underline the mode rotation in the EBor electron diamagnetic direction. FIG. 8. Experimental mode spectrum in ELM mitigation (a) and ELM suppression (b) phases of the discharge #33 133: the intensity of the modes is plotted depending on their frequency and toroidal mode number. Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-6 Published under license by AIP Publishing mode penetration inducing the sudden braking of the electron perpendicular velocity was observed in ELM suppression regimes in the experiments of DIII-D 29,30 and KSTAR. 31,32 The reduction below 1 km/s of the velocity of saturated edge modes observed in nonbursting ELM suppression regimes of KSTAR, 31,32 as well as the phase-locking of edge peeling-kink modes with static RMPs in grassy-ELM suppression regimes of DIII-D 33 are consistent with the edge mode locking found in this modeling. In ASDEX Upgrade, the zero-crossing of the electron perpendicular flow at the pedestal top is observed in some ELM suppression discharges, but not all of them, 28 meaning that the RMP-induced mode penetration at the pedestal top may be one of the ELM suppression mechanisms, but other mechanisms may exist too. C. Impact of the electric field evolution It is interesting to notice that the expulsion of filaments through the separatrix can be characterized by the evolution of the radial electricfield.Indeed,inthemodeling,atthetimewhenfilamentsare expelled from the plasma edge during the ELM crash, the EBvelocity and thus the radial electric field E r in the pedestal evolve from a large negative value to a zero or even positive value at the very edge. The plot of the EBrotation close to the separatrix during the simulation of an ELM crash (Fig. 10, for the discharge #23 221 described further in Ref. 34) shows the transient reversal of the EBvelocity at theveryedge(w¼0.963) when the ELM filaments are expelled. The vanishing of the minimum E r during an ELM has also been observed in experiments using charge exchange recombination spectroscopy measurements: during the ELM crash, the edge E r collapses to very small L-mode like values and then recovers again to the pronounced E r well about 4 ms after the ELM crash. 35–37 The evolution of the E r profile is due to the fact that the nonlinear Maxwell stress induces a strong shear of the plasma filaments close to the separatrix, 22 which can also allow cold pulses to penetrate inside the separatrix. 38 For the discharge considered here (#31 128, Sec. III A), the natural ELM crash (without RMPs) is also characterized by the transient vanishing of the EBvelocity and the radial electric field E r at the edge when the filaments are expelled. Interestingly, in the ELM mitigation case, the plot of E r [Fig. 11(a)]showsthatE r vanishes in the pedestal at the moment when the magnetic activity of the edge modes is maximal (for t¼1.6 ms). This is in line with the fact that plasma filaments are crossing the separatrix at this time. In contrast, in the ELM suppression regime, the radial electric field does not vanish in the pedestal [Fig. 11(b)], reflecting the absence of filament exhaust in ELM suppression. Another feature displayed in Figs. 11(a) and 11(b) is the fact that the radial electric field in the pedestal is smaller in absolute value in the ELM suppression regime (10 to 20 kV/m) than during ELM mitigation (20 to 35 kV/m), because of the larger amplitude of the penetrated RMPs. The initial electric field profile before RMP application is also plotted in Figs. 11(a) and 11(b): it shows that compared to the pre-RMP application, the E r well was strongly reduced during ELM mitigation (factor 2) and even more during ELM suppression (factor 3–4). Turbulence cannot be observed in this modeling since turbulent transport is represented for simplicity by diffusive transport terms. Nevertheless, the reduction of the E r well (and thus also of the E r shear) observed in this MHD modeling may have a link with the increased broadband turbulence observed in ELM suppression states in ASDEX Upgrade experiments, 39 as suggested by turbulence modeling, 40,41 where enhanced fluctuations are induced by energy transfer from kinetic energy (EBflow) to magnetic energy (magnetic fluctuations). Furthermore, the evolution of the electron temperature and density profiles (Fig. 12) shows a very similar degradation of the pedestal density and temperature in both the ELM mitigation and the ELM suppression cases. A more realistic transport model might also be necessary to reproduce more accurately the large density pumpout observed in experiments. 5,28 However, the similar pressure reduction found for both regimes in the modeling highlights again the fact that the reduction of the pressure gradient cannot explain alone the ELM suppression: a strong mode coupling is needed to saturate the edge modes at a low level and to induce the edge-localized mode braking. D. Evolution of the stochasticity level Another important parameter characterizing the dynamics of the ELMs—controlled or not—is the level of chaos or stochasticity of the magnetic field lines. In the modeling of the crash of a natural ELM FIG. 9. Radial profile of the perpendicular electron rotation evolving in time between t¼0.6 ms (blue) and t¼2.05 ms (red) in the ELM suppression case. The resonant braking on the q¼7/2 surface at the pedestal top is highlighted by a black arrow. The radial pressure profile at the times t¼0.6 ms (dashed-dotted line, black) and t¼2.05 ms (full line, black) is overlaid in order to locate the pedestal top and the strong gradient region. FIG. 10. Time evolution of the EBvelocity in an ELM simulation (without RMP) at different radial positions: just inside the separatrix (w¼0.963, red), at the separatrix (w¼1, green), and just outside the separatrix (w¼1.091, blue). The black arrow shows the moment when filaments are expelled out of the separatrix. Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-7 Published under license by AIP Publishing without RMPs, a wide stochastization of the magnetic field at the plasma edge is observed from the magnetic surface characterized by q¼3 to the edge. 38,42 In the ELM considered here, the full stochastization from the q¼3 surface (located around w¼0.82) is observed in Fig. 13. In this Poincar e plot, field lines are followed for 100 toroidal turns and their end point is colored according to the temperature of the starting point. It shows that long radial patterns extend from w ¼0.8 to the edge, due to the enhanced transport following the broken field lines in the stochastic region. The evolution of the magnetic topology at the edge in ELM mitigation and ELM suppression cases is plotted, respectively, in Figs. 14 and 15,wherePoincar  e plots are presented for four different times: (a) just before the inclusion of n4 in simulation, so for the 3D- equilibrium induced by n¼2RMPs,att¼0.6 ms; (b) during the linear growth phase of the modes n4, at t0.8 ms; (c) during the “nonlinear” phase when the mode-mixing is close to the maximum, at t1.6 ms; (d) after the mitigated or suppressed ELM regime, at t2 ms. In these Poincar e plots, the connection length of the field lines allows the different field line structures to be highlighted. The dark brown structure around w¼0.82 underlines the q¼3 surface, and the black structure at the pedestal top (w0.88) exhibits the q¼7/2 surface. Before including the modes n4 in the simulation (a), a similar n¼2 structure induced by RMPs is observed in both cases: an (m¼6, n¼2) magnetic island chain appears on the q¼3 surface and very small islands are formed on the q¼7/2 surface. A stochastic layer is observed at the very edge, and the kinking of the field lines is maximal near the X-point. Logically, since a larger magnetic perturbation is applied in the ELM suppression case, the magnetic islands and the FIG. 11. Radial profiles of the radial electric field in the cases of ELM mitigation (a) and ELM suppression (b), for different times between 0.6 and 2.05 ms. In the ELM mitigation case, E r vanishes at the time when the magnetic activity of all modes is maximal (t¼1.6 ms). The initial E r profile is also plotted in a black dotted line in order to show the reduction of the E r well induced by the RMP application. FIG. 12. Radial profiles of electron density (a) and temperature (b) at the starting time t¼0 ms of the simulations (profiles before RMP application, black dotted line) and in the cases of ELM mitigation (red) and ELM suppression (blue) by RMPs, from the time when n>2 modes are included in simulation (t0.6 ms, dash) to the time after the mode activity (t2.05 ms, full line). Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 26, 042503 (2019); doi: 10.1063/1.5091843 26, 042503-8 Published under license by AIP Publishing