Erosion and impurity transport for the edge localized mode suppression window in KSTAR Open Access

Three-dimensional effects are present in most currently operating plasma facilities due to uneven particle sourcing, asymmetric plasma heating, wall conditioning, or externally imposed fields.^1–3 Either type of these operational techniques can introduce changes in the wall composition, transient flows in the plasma, or in the total magnetic field that can affect how impurities travel along a reactor volume. Energetic particles striking the walls can erode material, and this can lead to the erosion and ejection of impurities, which travel along the magnetic field and can deposit primarily along the divertor or limiter targets present in current devices. The focus of this paper is on the latter of the mentioned operational techniques, i.e., the application of resonant magnetic perturbations (RMPs) used in KSTAR during H-Mode operation.^4,5 The application of these external fields mitigates highly disruptive events in the plasma edge (Type-I edge localized modes) and creates localized discrete regions of heat deposition along the divertor targets. Particle transport in H-Mode is unpredictable as the large increase in the radial electric field $Er→$ and, by extension, the $E→×B→$ drifts can significantly alter the poloidal and radial particle flows in the divertor. Due to the known complexity of the radial field at the scrape-off layer (SOL), the effects of $E→×B→$ drifts will not be considered in this study.

The goal of ITER is to produce a net positive energy output from a burning plasma (i.e., Q > 10). In addition to the divertor having strict dissipative heat load limitations, it must be robust enough to operate during high confinement (H) mode and the accompanying edge localized modes (ELMs) present. These occur only in H-mode and are due to large pressure gradients and subsequent current densities at the plasma edge, which can destabilize peeling and ballooning modes.^6–8 In order to mitigate the effects of ELMs, resonant magnetic perturbations (RMP) have been implemented and are part of the design of multiple tokamaks worldwide.^4,9–12 These external field coils are applied in such a way that the magnetic field in the edge becomes ergodic and the scrape-off layer becomes highly three dimensional at the edge, near the divertor targets.^13–17 

The effects of RMPs on ELM suppression have been well-studied across devices. KSTAR has been able to experimentally identify an ELM suppression window and validate their threshold model by extrapolating this window from an experimental threshold to achieve MHD-free operation⁴ for an n = 1 mode configuration. DIII-D pioneered the use of such coils and has been able to achieve suppression with n = 3 RMPs for both high and low triangularity scenarios, as well as ITER-like plasmas with a wide range of coil currents.^9,18 One important metric that has been used for predicting an ELM suppression window for ITER is the X-point displacement.¹⁹ This maximizes the radial displacement of the X-point to stabilize the peeling modes at the plasma edge.²⁰ Such schemes are important to the success of minimizing plasma–wall interactions as the lifetime of divertors and other wall components for ITER and any fusion pilot plant will be determined by stabilization of these high energy events, as these can introduce large concentrations of impurities by the erosion of these components and melting due to high heat loads at the targets.

As previously mentioned, at KSTAR, it has been possible to determine and experimentally validate the ELM-suppression window for an n = 1 RMP during H-Mode discharges. With the upcoming upgrades for a tungsten wall,^21,22 it will be an important step toward understanding the effects of a high-Z material on plasma response, erosion, and impurity transport. The device itself is a mid-scale reactor (R = 1.8 m, a = 0.5 m) and toroidal field with magnitudes greater than 3.5 T. Additionally, it has three sets of in-vessel control coils (upper, middle, lower), which can be phased in such a way that stable n =1 and n = 2 RMPs can be used to achieve ELM suppression. This device was equipped with carbon plasma-facing components, and an impactful amount of physics understanding of ELM suppression and plasma response has been developed over the years.^4,5 Because of this, this study will focus on carbon plasma facing components (PFCs) for determining significant trends in the erosion of the divertor targets and carbon buildup in the SOL.

The ERO2.0 has been used to model global erosion and deposition patterns for different magnetic configurations across several plasma devices.^23–25 One of the most recent ITER studies on beryllium transport for an axisymmetric scenario showed significant accumulation of beryllium from the first wall along the divertor targets, which raised concerns regarding fuel retention due to layer build-up.²⁴ A comparison of these results with those of the WALLDYN-3D code demonstrate a need for consideration of the ERO2.0 sputter yield database to take into account the scaling laws between hydrogen isotopes and beryllium, as significant differences were observed in the resulting deposition patterns in the main chamber.²⁶ A global transport study for erosion during RMP application for L-mode scenarios in DIII-D showed that carbon impurities can buildup thicker layers due to net deposition between the lobe structures that arise when the SOL breaks up during RMP application, whereas for a 2D configuration, the full length of the open divertor target would be subjected to net erosion.²⁵ This study will present results on erosion and impurity transport for the ELM suppression window for carbon PFCs in KSTAR.

The 3D fluid/kinetic code EMC3-EIRENE^27–30 provides the plasma backgrounds used as input for the ERO2.0 simulations based on the ELM suppression window for MHD stable plasmas in KSTAR (equilibrium #16586 at $t=2965$ ms) for a disconnected double null configuration.⁵ This EMC3 code is used to simulate the scrape-off layer by solving a plasma fluid model with particle, energy, and momentum sources, and is run iteratively with the EIRENE neutral particle tracing in order to account for all included neutral particle interactions until a stable solution is reached. The Generalized Perturbed Equilibrium Code (GPEC) was used as a way to calculate the perturbed equilibrium in the SOL for rational surfaces in the range $0.9<ψN<1$ and used as an input for the magnetic geometry in the EMC3-EIRENE modeling.³¹ The code calculates the perturbations for rational surfaces between $ψlow=0.1$ and $ψhigh$ for which the cutoff of rational surfaces can be defined by the user. The FLARE code uses inputs from GPEC and 2D magnetic equilibria to generate the magnetic mesh and grids for the fluid and neutral modeling of the plasma and resulting maps on the target surfaces.³² The ELM suppression window is a narrow region in current and phase space, and its guiding axis has a middle coil phasing dependence on the current $IM$ given by $ϕ=110−15·IM$ for fixed upper and lower coil currents of $IU=IL=5$ kA, and $ϕ$ is the relative current between the upper and middle coils, and the lower and middle coils: $ϕ=ϕU−ϕM=ϕL−ϕM$⁠. The upstream densities were fixed at 0.5 × 10¹⁹ m⁻³ across scenarios for an input power of 3 MW, with a carbon impurity chemical sputtering yield of 0.04 and an anomalous diffusion coefficient of $D⊥=1$ m²/s for the main ion species and impurities, and energy diffusivity in the perpendicular direction of $χi=χe=3$ m²/s for ions and electrons.

Because of the nature of the magnetic configuration, there are four regions of interest to determine potential erosion regions: the lower outer divertor target (LOD), the upper outer divertor target (UOD), the upper inner divertor target (UID), and the lower inner divertor target (LID). There are multiple plasma wall interactions (PWIs) and impurity transport processes accounted for in the ERO2.0 code such as physical sputtering, chemical erosion, material mixing, ionization, recombination, and deposition along the targets. A carbon wall is used for modeling of KSTAR and to determine the respective sputtering and reflection coefficients pre-calculated via SDTrimSP³³ for incoming plasma ion energies defined by $T≈2Ti+3ZTe$⁠. One previous study from KSTAR highlights that the angle of incidence of particles at the targets are assumed to be close to the angle of incidence by the magnetic field³⁴ and as such, this assumption was used in the ERO2.0 modeling as well. Chemical erosion for these scenarios is calculated via the Roth formula for a carbon PFC for $Tsurf=300$ K³⁵ and traced deposition is calculated for CD, $CD2$⁠, $CD3$⁠, and $CD4$⁠. Within the code, these molecules are then dissociated right at the target surface, and any subsequent erosion caused by them is calculated based on the individual species (i.e., C and D). On Fig. 1, some values for the sputtering yields used in the simulations are presented for a fine grid (i.e., 500 combinations for energy and angles of incidence). Figure 2 shows the corresponding reflection coefficients, noting the dependence on energy and up to grazing incidence where most particles will be reflected for large incident energies. The yields and reflection values are looked up within the ERO2.0 code by using the incident energy of particles at the surface cell, and angle of incidence, which in these simulations was taken to be the relative angle between the magnetic field and the surface cell normal (⁠ $θB$⁠).

Should the particles get reflected based on their incidence parameters, the test particle is re-ejected while carrying a fraction of the flux (R $Γinc$⁠), and the remaining fraction is deposited at that surface location (1-R) $Γinc$⁠. The process iterates until the maximum particle tracing time (⁠ $τmax$⁠) is reached, the particle recombines and deposits at a surface, exits the volume or times out mid-path due to reaching $τmax$⁠. Should the test particles re-erode material, the newly eroded particles will only be created as a test particle and traced in the subsequent time step and can contribute significantly to the steady state fluxes on the target plates. All the relevant plasma information at the target plates is obtained via EMC3-EIRENE post-processing and are shown in Fig. 3. The most recent study for these scenarios has shown two main points: (1) that the radial cutoff for the equilibrium truncation for GPEC can significantly affect the magnetic footprint at the target, and (2) that increasing $IM$ for these configurations (⁠ $ψN=0.99$⁠) leaves little opportunity for peak heat load minimization (⁠ $qpk$⁠) and narrows the area where heat is deposited.⁵ This narrowing is indirectly observed in Fig. 3 as the increase from 1 to 5 kA in the coil current changes the electron and ion temperature maps. As the magnetic footprint for the n = 1 mode narrows, so do the heat fluxes, and by extension the regions of high plasma temperatures, with slightly increased values over significantly smaller areas.

The results presented here correspond to the resulting erosion, transport, and subsequent deposition of carbon impurities for the n = 1 resonant magnetic perturbation applied during the ELM suppression window in KSTAR. As seen on Fig. 3, the changes mentioned in the magnetic configuration due to the variation of the RMP coil current can have significant changes in the resulting maps for the plasma parameters along the targets, as increasing the coil current decreases the magnetic footprint at the divertor target. Most notably, there are narrow zones of high ion and electron temperatures for the outer divertor targets but an accompanying low density relative to the reference no RMP scenario. The application of the fields breaks up the scrape-off layer and distributes the particle fluxes along the lobes, as opposed to having a single poloidal region where $Γinc$ heavily concentrated. The resulting erosion/deposition patterns for these scenarios are shown on Fig. 4. The no RMP scenario reference demonstrates that the net erosion at the target is spread over a large area with larger rates where the plasma density peaks. By contrast, the 0 kA scenario, though providing comparable peak net erosion rates at the zones with highest electron and ion temperatures, is localized to the n = 1 lobe pattern, with relatively low net deposition in the far SOL. The 5 kA scenario increases strong suppression of carbon net erosion relative to the 0 kA case.

The toroidal average values are presented on Fig. 5 where the net erosion footprint of the no RMP scenario is seen to wet a large area of the divertor plate, whereas for the RMP scenarios, increasing the coil current has a strong reduction in the erosion across the target length with a relatively low net deposition rates for the distance along the divertor beyond 20 cm for the no RMP reference, and slightly higher deposition for the 3D scenarios. Additionally, the integrated net erosion at the lower outer divertor target is presented as a function of the coil current. For the configurations used here, this target is the limiting functional plate as it exhibits the largest amount of carbon erosion in the device. For the steady state erosion fluxes obtained, operating at the lowest coil current leads to the highest amount of erosion at the target and will be the main contributor to impurity concentrations in the plasma. Most of the analysis has been focused on the outer divertor target as it has been shown to receive the highest heat loads;⁵ however, it is necessary to understand the transport on a global scale, particularly when there is no periodicity (n = 1) in the plasma configuration, as for such scenarios, the full reactor volume needs to be modeled, as opposed to, e.g., a 90° for an n = 4 RMP.

It is then necessary to understand if it is solely at the LOD where net erosion has been shown here to occur, and where carbon impurities are being deposited. Although a full toroidal map of the wall is required to understand what the local rates are, the poloidal maps along a cross section demonstrate some features in Fig. 6. The distance measured along the wall contour direction begins at the lowest point at the lower outer divertor and runs counterclockwise. For these configurations, the no RMP reference scenario shows to have the widest erosion footprint as expected from Fig. 4 along the LOD. Additionally, the effect of the coil current is seen for the RMP scenarios, where the erosion is lower as the current is increased. In the far SOL for this target, there is significant deposition of impurities, as seen also in Fig. 4, though not as notably; this feature is lost in Fig. 5 due to averaging. Figure 6 also demonstrates that most of the eroded carbon is distributed and deposited along the three other plates, with most of the impurities depositing along the inner wall and the target plates. The trends observed are in agreement with the observations in Fig. 5, and the deposition rates are proportional to the decrease in particle erosion at the LOD.

For our purpose, Eq. (4) is used across the full simulation volume to calculate $Zeff$ by adding the contributions from all toroidal slices between the inner boundary (shown in Fig. 7 in black) and the wall. This then provides an effective charge state for the plasma as shown in Fig. 7 for all scenarios covered in this study. As previously observed, there is a strong dependence on erosion and subsequently $Zeff$ on the RMP coil current. By increasing the coil current and changing the phasing for the ELM suppression window, less carbon is eroded at the divertor targets, and less carbon impurities are available to significantly change the average charge state in the plasma. The analysis is not without faults, as there is no available core model in the codes used in this study that can accurately predict core impurity transport. The assumption used in ERO2.0 is that of a reflecting boundary at the core; any impurity that reaches it is ejected at a random location in the SOL to mimic core transport. It is important to note that the highest charge state for carbon contributes significantly to this measurement, which is why most CES systems only utilize $nC+6$ as an indicator of the effective charge in plasmas.

This study focused on the use of the ERO2.0 code to explore the effect of resonant magnetic perturbations on global impurity transport for the experimentally discovered ELM suppression window for the n = 1 perturbation in KSTAR. Output from GPEC was used as a way to include the perturbed field, sample the edge, build the EMC3-EIRENE grid, and use these as inputs to calculate the plasma wall interactions and global transport of impurities. Three main takeaways are summarized in the study, first, the magnetic footprint has a strong effect on the resulting erosion of the divertor targets, as the increase in the coil current narrows the edge topology,⁵ and has a beneficial effect on the net erosion (decreases in the lower divertor target). Second, global particle transport for these RMP configurations demonstrate similar zones of net erosion (i.e., lower outer divertor target) and deposition of impurities along the other three divertor targets as the edge plasma parameters dictate the resulting erosion rates. Finally, it is possible to determine $Zeff$ with the ERO2.0 code, provided there is enough information about the edge and core to be able to validate such an analysis. Anomalous diffusion is the main mechanism for particles to exit the core region, so core–edge integration will play a key role in the improvement of these global impurity transport models. As PWIs are one of the limiting factors in steady state fusion power operation, it is necessary to bring attention to multiphysics modeling studies, where impurity maps from codes such as ERO2.0 can be used in feedback to inform additional power losses and the plasma response and to solve for a self-consistent problem.

The authors would like to acknowledge the input provided by Hyungho Lee at the National Fusion Research Institute in South Korea at KSTAR for his input and revision of the work presented in this submission. This work was funded in part by the U.S. Department of Energy under Grant Nos. DE-SC0020284 and DE-SC0020357. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC award FES-ERCAP0020290.The authors gratefully acknowledge the computing time granted by the JARA Vergabegremium and provided on the JARA Partition part of the supercomputer JURECA³⁹ at Forschungszentrum Jülich.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

The authors have no conflicts to disclose.

Marcos X. Navarro: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Writing – original draft (lead). Jonathan Van Blarcum: Formal analysis (supporting); Software (supporting); Writing – review & editing (supporting). Heinke Gerd Frerichs: Formal analysis (supporting); Software (supporting); Validation (supporting). Juri Romazanov: Software (supporting); Writing – review & editing (supporting). Andreas Kirschner: Methodology (supporting); Writing – review & editing (supporting). Jong Kyu Park: Investigation (supporting). Seong-Moo Yang: Investigation (supporting). Oliver Schmitz: Conceptualization (supporting); Funding acquisition (lead); Project administration (equal); Resources (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.