ELM sputter erosion modeling of a tungsten coated small angle slot divertor in DIII-D

We modeled plasma edge localized mode (ELM) sputter erosion for a Small Angle Slot divertor with a tungsten coated region (SAS-VW), designed for experiments in the DIII-D tokamak, and proposed for use in future advanced tokamaks. The simulations use a free-streaming, 1000 eV, C+6 and D+1 ELM impingement model, with SOLPS-ITER, ITMC-DYN, and REDEP/WBC code packages for background plasma, material response, and erosion/redeposition respectively. The results show ELM’ing plasma gross and net tungsten erosion fluxes of the mixed-material C/W surface peaking at the slot entrance region, and an order of magnitude higher than for non-ELMs. The per-pulse erosion, however, remains low, of order 0.5 nm, due to expected moderate ELM frequencies and duration in DIII-D. The ELMs result in a ∼25x higher peak sputtered W current leaving the divertor slot region, towards the core plasma, compared to the ELM-free plasma case. The time-integrated escape current, however, may not significantly affect core plasma high-Z contamination concerns, for a 1% ELM duty factor, but may be an issue for higher frequency ELMs. In general, the modeling results appear favorable for effective testing of the SAS-VW divertor in DIII-D, and extrapolation to innovative divertor designs in future ITER-like and DEMO fusion devices.


Introduction
We recently analyzed carbon/tungsten mixed surface evolution, sputtering erosion, and transport for the new tungsten coated region of a small angle slot V-shaped divertor (SAS-VW) in the DIII-D tokamak during quiescent plasma * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. operation [1]. This divertor design aims to achieve a closed slot dissipative plasma to minimize heat load and surface erosion, and to study high-Z material performance [2][3][4]. Limited experiments have been performed, and proposals for more extensive work are contemplated, in relation to the DIII-5 year operations plan. Erosion modeling is important to understand present and proposed DIII-D experiments, in terms of coating lifetime and high-Z material transport including core plasma contamination potential. Our more general purpose is to evaluate the potential for future applications of this innovative divertor concept. Study [1] showed a complex, mixed C/W equilibrium surface, forming in 30 s of exposure, with low net W erosion, but enough to permit testing of high-Z material response and plasma transport for the SAS-VW divertor in DIII-D. We now extend the analysis to the important case of plasma operation with ELMs.
Increases in particle flux and impact energy during ELM transients are expected to periodically increase W erosion. Previous DIII-D experiments on W coupon samples in the open lower divertor measured a 10-fold increase in the W erosion rate during ELMs [4]. The closed geometry of the SAS-VW divertor is expected to increase near-target electron densities compared to open geometries [5], which in turn is predicted to reduce the quiescent plasma net erosion rate. However, the effect of the divertor geometry on increased erosion during an ELM is yet to be determined. In addition, while the SAS-VW divertor is predicted to yield a low current of sputtered tungsten leaving the divertor towards the core plasma [1], it is also unclear how ELMs may impact that current. One issue for the SAS-VW type geometry is that the first several cm of the divertor slot constitutes a type of 'leading edge', being adjacent to the broader plasma divertor region and with an attached near-surface plasma. To address these issues this study seeks to identify general trends regarding ELM plasma SAS-VW tungsten sputter erosion, with focus on W transport in the critical slot region. Future modeling plans include comparisons with experiments and code/data validation.

Coupled code package ELM sputter erosion modeling
The SAS-VW design, located in the upper outer major radius part of the DIII-D vacuum vessel, is summarized in [3] and references within. Figure 1 shows the design geometry, poloidal magnetic flux lines, and the 10 cm divertor slot W coated region of interest here.
Based on previous DIII-D experiments we employ a model that should reasonably describe the key features of ELM phenomena for SAS-VW. Namely, the ELMs are characterized here by the free-streaming, experimentally based model of [4,6], with 1000 eV free-streaming D + ions, and 2%, 1000 eV C +6 ion impingement. This model uses a refined version of the Fundamenksi-Moulton free-streaming model (FSM) [7,8] for the dynamics of divertor density, particle flux, and heat flux during ELM's. The model depends only on inter-ELM pedestal and divertor conditions and incorporates a detailed particle recycling model. The FSM treats an ELM event as a detachment of a filament of plasma from the confined region, i.e. pedestal top, and its ejection into the scrape-off-layer (SOL) directly to the divertor. The FSM solves the Vlasov-Poisson equations for adiabatic expansion of this filament in the quasineutral limit. As discussed in [4,[6][7][8] the model predictions have been systematically tested against a DIII-D database of ELM ion and energy fluence measurements and are shown to be consistent with the model across a wide range of pedestal and divertor conditions. The model has also been shown to be consistent with spectroscopic measurements of W gross erosion for three different pedestal conditions. In terms of the ELM time parameters, it is widely predicted, e.g. [4,9], that about 10%-15% of the plasma core energy is released to the divertor plate in a duration of about 0.1-1.0 ms, as also predicted for ITER-like designs. The ELM frequency can vary between 1 and 100 Hz with roughly the corresponding scaling of the energy released during each ELM. This range is due to variations/uncertainties in the parameters that govern ELM behavior, e.g. plasma shape, recycling, divertor geometry, H-mode power threshold, etc. For our DIII-D analysis, for computation of several time-integrated results, we use typical parameters of 10 and 50 Hz ELM frequencies, and 1 ms ELM duration. We also consider some variations in free-streaming particle energies.
For the near-surface intra-ELM background plasma, we use inter-ELM plasma parameters from a SOLPS-ITER code solution [1,10], with electron density increased by the freestreaming ions, consistent with a charge-neutral flow model. An alternative intra-ELM density model [11] is also used for comparison. The intra-ELM background plasma temperatures and electron density are shown in figures 2 and 3. As shown in figure 2, the SOLPS-ITER solution (also using B2.5, a two-dimensional fluid plasma transport code, and EIRENE, a three-dimensional kinetic neutral transport code), shows an attached plasma with electron temperature of ∼40 eV at the  slot entrance, falling to ∼5 eV at the x = .07 m approximate transition point to a detached plasma, and falling further to ∼2-3 eV at/near the strike point. More information on the SOLPS-ITER background plasma used for this study can be found in [1].
The local electron density near a plasma/facing surface plays a major part in determining ionization and subsequent transport and redeposition rates for sputtered atoms. Applying the intra-ELM density increase models, in particular at the most-critical divertor surface area, viz. the first several cm along the slot entrance, gives electron density increases of about 2x and 5x respectively over the inter-ELM density. However, we note that the density is still relatively low at the slot entrance, even with the inter-ELM density enhancement. As will be discussed, this low density affects the sputtered flux of tungsten leaving the slot region towards the core plasma.
Also given by the SOLPS-ITER solution and REDEP/WBC code package calculations is the background plasma, carbon ion divertor-impinging flux and energies for both inter and intra ELM phases, as a function of carbon ion charge state. The results, for a 2% C/D plasma concentration [1] show dominant C ion states of +1 through +4 with trace C +5 flux.
The ITMC-DYN code package (Ion Transport in Materials and Compounds-Dynamics), e.g. [12]., is used to compute sputtering coefficients and sputtered atom velocity distributions for all incoming ion species, these species being carbon ions in charge states +1 to +5 from the background plasma; the 1000 eV free-streaming D +1 and C +6 ions; and redeposited C and W ions. (There is no W sputtering from the background D + ions, due to below-threshold energy everywhere in the divertor). This code package integrates all collisional and near surface thermal processes to study the effect of impurities, surface segregation, hydrogen isotope retention, and mixedmaterial surface evolution and erosion, in plasma facing materials under multiple mixed ions irradiation during steady state and transient events. ITMC-DYN is applied to the 30 s equilibrium mixed C/W surface of figure 5, [1], assumed unchanged during the ELMs due to the short exposure times.
Using the ITMC-DYN sputter yields and distributions, and the SOLPS-ITER plasma parameters with density modification, the REDEP/WBC code package (3-D, 3-V, fullkinetic, sub-gyro-orbit, oblique incidence magnetic field sheath, erosion/redeposition), e.g. [13], computes the individual sputtered W atom and resulting W ion trajectories, particle-by-particle, using Monte Carlo. For this study a tungsten atom is launched from the surface with velocity vector chosen from the ITMC-DYN distribution corresponding to the incident ion species, i.e. C, D, or W, and the incident ion energy. A sputtered W atom is ionized per local temperature-dependent electron impact ionization rate coefficients and electron density. The resulting W ion trajectory and charge state evolution is further computed in detail based on Lorentz force motion and full-kinetic collisions with the plasma electrons and ions. A particle history terminates upon redepositing anywhere on the divertor surface or leaving the divertor slot. Sputtering by redeposited particles is computed. A typical REDEP/WBC run uses 10 5 histories per incident particle species and per divertor spatial segment. Erosion/redeposition metrics are computed and convolved based on the respective incident species fluxes and ITMC-DYN sputter yields. Further description of the computational methods and code coupling is given in [1,12,13] and references within.
For the present purposes of identifying general trends, we assume toroidal symmetry of the divertor. While our simulation packages are fully capable of treating 3D effects, and such effects, e.g. tile misalignments, are known to apply to DIII-D divertors, the present toroidal symmetry analysis is sufficient to establish the general trends identified in this study. Likewise, the results are relevant to assessing SAS divertor performance for high-power tokamak applications where good toroidal alignment will be a key requirement.

Tungsten erosion
The highest net sputter erosion is found to occur in the ∼2 cm long divertor slot entrance segment, but more importantly, due to the SAS geometry, this is the only segment that contributes to escaping tungsten ions from the divertor slot. Therefore, we focus on detailed results for this segment. Table 1 summarizes various W erosion/redeposition parameters for the three classes of incident particles on this segment of the mixed C/W divertor surface. Total erosion is given by the sum of the three contributions.
There are significant differences in the species-dependent sputtered particle transport, most importantly for the key parameters of redeposition fraction, and divertor slot-escape fraction. These differences are mainly due to variations in sputtered atom energy distributions. Since net erosion, to first order, scales with redeposition fraction, 'R', as (1-R), seemingly small differences in R shown in table 1 are important.
Most ELM plasma sputtering erosion is due to the freestreaming D + and C +6 ions as opposed to the background plasma carbon or self-sputtering. For all species, the less than unity SAS-VW average tungsten redeposition contrasts with a conventional divertor geometry such as for the DIII-D DiMES probe with a tungsten or molybdenum surface, e.g. [14]. For the DiMES open divertor geometry, with attached plasma, and extrapolating to a full W divertor surface, there is essentially complete predicted (average) redeposition with little or no divertor-escaping material. This high redeposition is due to the less oblique poloidal magnetic field angles with the surface compared to the SAS-VW geometry. A similar high-Z transport result is also predicted in other conventional divertor tokamaks, e.g. JET [15,16].
Summarizing net erosion during the inter-ELM phase, about half of the net tungsten erosion during ELMS, due directly to the impinging plasma, is from the carbon content of the plasma, with the remaining erosion due to the free-streaming deuterium ions. Self-sputtering (which depends on redeposition fractions and self-sputtering yields) constitutes about a 15% additional effect. Figure 4 summarizes and compares the ELM case erosion, for this worst-case segment, with ELM-free plasma results [1]. The ELM plasma has order of magnitude higher peak erosion fluxes and divertor slot-escaping tungsten current. However, the net erosion fluence and per-shot erosion (∼0.3 nm for a 3 s shot) is about the same and low in both cases, due to the low ELM duty factors, such as for 1%, based on an ELM frequency of 10 Hz, at 1 ms duration. Even for a 50 Hz ELM frequency, the erosion is still only about 0.7 nm shot −1 . As discussed in [1] these erosion magnitudes present no sputter erosion lifetime problem to SAS-VW experiments in DIII-D.
Extrapolating the results to use of this divertor concept in future devices, with much longer plasma operating times, there is little or no apparent ELM-plasma tungsten erosion lifetime problem for an ITER-like device with several percent operating factor (and noting that we have found computationally that beryllium behaves reasonably like carbon, in terms of mixedmaterial Be/W formation and response).
There is a potential concern for high operating time devices (e.g. DEMO), with extrapolated net erosion rate in the mm/year range, though erosion/redeposition differences due to D-T plasma operation and pure tungsten surfaces would need detailed analysis. One point regarding the latter issue is that based on the present analysis, the absence of a several percent impurity in the plasma, like carbon or beryllium, in post-ITER machines, would reduce inter-ELM plasma tungsten erosion by roughly a factor of two, and with greater reductions for the intra-ELM phase. We conclude with the general observation that net erosion rate considerations may potentially limit the tolerable range of ELM frequencies/duration, for DEMO type tokamaks using this type of divertor.

Slot escaping tungsten
As stated, the ELMs cause a major increase in the peak sputtered W current, I W , leaving (escaping) the divertor slot region, i.e. reaching x ⩽ 0. This is due to the high freestreaming particle fluxes, with smaller differences due to variations in transport/redeposition parameters from the quiescent plasma results. This current is important in terms of the potential for core plasma contamination. To reach the core plasma/SOL boundary the escaping W ions from the divertor slot would have to traverse the ∼12 cm-long-poloidally divertor plasma region (see figure 1). It is considered likely that a high fraction of I W would be transported back to the divertor or first wall, due to collisions with the incoming plasma (friction, thermal, E-field forces, etc.), and thus would not reach the core plasma/SOL boundary.
For a 1% ELM duty-factor (e.g. 1 ms ELM @ 10 Hz) the ELMs contribute only about 20% to the time-integrated W slot-escape current. However, the fraction of the slot-escaping inter-ELM and intra-ELM sputtered W currents reaching the core plasma/SOL boundary can differ substantially, due to different transport properties in the divertor plasma outside of the SAS slot region. This topic is under investigation for SAS-VW via coupled REDEP/WBC-SOLPS-ITER computations of W ion transport in the entire divertor region.
Preliminary divertor transport analysis was performed for sputtered W transport in the divertor plasma region outside of the SAS-VW slot. This calculation uses escaping W ions from the in-slot analysis as boundary conditions at x = 0, see figure 1. (This sub-analysis also uses the 3D, fullkinetic, REDEP/WBC package, and additionally uses a recent SOLPS-ITER plasma solution, with drifts, for plasma parameters outside of the slot region.) For the simulation a particle history terminates upon reaching the core-plasma or SOL boundary, roughly defined here as x = −0.12 m, Z = 1.0 m, in figure 1, or redepositing on the wall or divertor. Results show, as expected, major attenuation of W current reaching the core plasma/SOL boundary. Correspondingly seen are major differences in the transport of sputtered W in the x < 0 divertor plasma region, depending on the particles doing the sputtering. As with in-slot divertor transport, the transport differences are largely due to the higher sputtered tungsten atom energy distribution (see table 1) for intra-ELM C +6 plasma impingement. Preliminary results show order of 3% fraction of slot-escaping W, from intra-ELM plasma freestreaming C +6 ion sputtering, reaching the core-plasma/SOL boundary. The fraction is about 5x smaller for W resulting from lower energy inter-ELM carbon ion sputtering. (We also compute transport fractions for intra-ELM free-streaming D + , and inter-ELM C ion species tungsten sputtering). Per these preliminary results the shot-total number of W ions reaching the core plasma/SOL boundary would be about twice as much for an ELM'ing plasma, with a 1% ELM duty factor, compared to a quiescent plasma shot. While not minor, we deem a factor of two effect, in this context, probably not highly significant, due to low slot-escape current to begin with and expected in-plasma low inward W transport, but this obviously needs further detailed assessment.
In terms of the high-Z core-plasma contamination potential for a 5% ELM duty factor (50 Hz frequency), the shot-total W ions escaping the slot and reaching the core-plasma/SOL boundary (per the preliminary full-divertor volume transport results) would be about 5x higher than for a quiescent plasma. This substantial increase may or may not be a problem for DIII-D or future devices, depending on core plasma transport of the W.
We conclude at present that ELMs, with the modeled characteristics, and at the lower end of the assumed frequency range, probably do not significantly change the effects of high-Z core plasma contamination in DIII-D. If the contamination is acceptable for the ELM-free plasma it will likely be the same for a DIII-D ELM'ing plasma. However, this conclusion may clearly be different for higher ELM frequencies, and/or for higher than 1 ms durations.
A full analysis of core plasma high-Z contamination from an SAS-VW type divertor involves coupling of the REDEP/WBC-ITMC/DYN-SOLPS-ITER computations with a core-plasma/SOL transport code. This is a complex, time-consuming endeavor, among other things requiring a coupling between the full kinetic REDEP/WBC output with a fluid level core-plasma code. Such work is planned. Factor of ∼2 decrease from reference case due to higher sputtered W redeposition. Above, but with electron temp. = 1 /2 inter-ELM value 8.9 Increase from above case due to lower W atom ionization rate coefficients.
Detailed code/data validation of SAS-VW DIII-D experiments is also planned. We observe now, however, that DIII-D/SAS-VW operation with ELM frequencies of 30-100 Hz, and duration 1-2 ms, did not result in plasma termination, such as due to core-plasma tungsten contamination. While specific simulation parameters may be more demanding than the present experimental conditions, this result is obviously encouraging.

Other cases
Our results/conclusions regarding peak erosion/transport rate metrics apply to a broad range of ELM frequencies. The peak gross erosion rate, net erosion rate, and escape current, are independent of ELM frequency, except for very high frequencies. For very high ELM frequencies (assuming such plasma regimes would even be used for operation with any high-Z divertor surface) the change in mixed-material C/W surface composition during the ELMs due to the longer exposure times, with corresponding change in self-consistent sputter yields, would need to be re-calculated. Likewise, for effects of any significant frequency-dependent change in ELM freestreaming particle energies, fluxes, etc. This can be done with our modeling codes and methods and is planned for future studies. Table 2 shows the effect of some changes in intra-ELM electron density and temperature on I W . The effects are relatively small and do not alter the present conclusions. Also, since plasma conditions vary substantially across the 10 cm W slot region, selected erosion calculations we performed across the slot inherently yield information on effects of plasma density and temperature variations on the ELM erosion and transport. We do not see fundamental changes. For example, at the x = 07 m divertor slot point, the net W erosion flux is ∼35% less than at the entrance region. This small effect is primarily due to somewhat higher W redeposition, with the higher plasma density in this region tending to offset the lower temperature/lower W ionization rate coefficients.
We also performed a sensitivity study, running computer simulations with moderate (±50%) changes in ELM freestreaming particle energies. Although sputter rates obviously change, no major qualitative changes in the ELM plasma erosion/redeposition behavior were observed.
Finally, an initial analysis using a different SOLPS-ITER plasma solution for SAS-VW, with divertor region drifts, which involves about 50% lower slot entrance region plasma temperatures, does not show major changes in conclusions about ELM plasma tungsten erosion/redeposition performance. In particular, the ELM-plasma peak gross and net erosion fluxes, and peak slot-escape current, for this driftdependent plasma solution, are within 20% of the reference plasma solution values. The net erosion fluence for a 3 s shot is roughly half the reference value, due to lower quiescentplasma sputter yields. The ELM's contribute a higher percentage to the erosion fluence, but this is still a small effect.

Conclusions
We analyzed plasma transient ELM sputtering for the innovative SAS-VW divertor use in the DIII-D tokamak using advanced, coupled, simulation packages, with uncertain but reasonably anticipated ELM parameters and background plasma conditions. Sputtering by the ELM plasma has major differences from the non-ELM case, due to the high energy free-streaming ions including tungsten sputtering by the energetic D + ions. Transport and redeposition of the sputtered surface material depends in a complex way on the mixed C/W surface response to the types, energies, and charge states of the impinging ions; as influenced by sputtered atom velocity distributions, poloidal magnetic field structure, and attached and detached plasma areas along the divertor.
About half of the SAS-VW surface tungsten erosion with an ELM'ing plasma is due to the free-streaming D + ions, with the rest from free streaming C +6 and background plasma carbon, and with moderate self-sputtering. Resulting tungsten gross and net peak sputter erosion fluxes are much higher than for an ELM-free plasma. The total per-shot net erosion, however, is reasonably low, for a reference 1 ms ELM duration and 3 s plasma shot, being ∼0.3 nm and 0.7 nm, for 10 Hz and 50 Hz ELM frequencies respectively. Therefore, there is no apparent high-Z component sputter erosion lifetime problem for DIII-D SAS-VW operation, for the ELM plasma models studied. Sputter erosion may, however, may limit the range of tolerable ELM frequencies/durations, for this type of divertor used in long operating time devices such as DEMO.
In terms of potential high-Z contamination of the core plasma, the peak ELM-plasma sputtered W ion current leaving the divertor slot region, towards the core plasma, is much higher, ∼25x, than for an ELM-free plasma. This peak occurs during the inter-ELM periods, and is mostly due to the free streaming ELM C +6 and D +1 impingement, and the relatively low electron density at/near the divertor slot entrance. Again, because of the limited total ELM time, the time-integrated slot escape tungsten current is only ∼20% higher for 10 Hz frequency ELMs (1% duty-factor), compared to a quiescent plasma. This leads to a preliminary estimate of ∼100% higher value for the shot-total W ions leaving the slot and traversing the broader divertor plasma, and reaching the coreplasma/SOL boundary, compared to an ELM-free plasma shot. For 50 Hz ELM frequencies, the shot-integrated tungsten reaching the said boundary is about 5 times higher, thus much higher than for quiescent plasma shots. ELMs, therefore, may not materially increase concerns about core plasma high-Z contamination for 1% or so ELM duty factors, but may be a significant issue for higher ELM frequencies or durations.
The sputter erosion/redeposition results are similar (within factors of 2) for the intra-ELM background plasma density and temperature variations studied. Our analysis generally indicates that C/W material mixing and high-Z material erosion and transport in the SAS-VW divertor can be reliably studied in DIII-D and other devices, with ELM'ing plasmas, at least for moderate ELM frequencies. This study also highlights the potential effects of innovative divertor designs on reactor performance in future ITER-like and DEMO devices.
Future modeling plans include comparisons with experiments, selected 3D treatment, high frequency ELM effects analysis, full-coupled divertor/core-plasma sputtered tungsten transport analysis, and code/data validation.