K Afonin

A Gallo

Y Marandet

Ph Moreau

G Bodner

H Bufferand

G Ciraolo

C Desgranges

P Devynck

A Diallo

J Gaspar

C Guillemaut

R Guirlet

J.-P Gunn

N Fedorczak

T Loarer

R Lunsford

P Manas

F Nespoli

N Rivals

P Tamain

E A Unterberg

Interpretive modeling of boron transport in the boundary plasma of WEST experiments with the impurity powder dropper

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Introduction: preliminary IPD experiments in WEST

The tungsten (W) Environment in Steady-state Tokamak (WEST) [1,2] utilizes almost exclusively W plasma-facing components (PFCs) to validate their use in ITER and future pilot plants. While W PFCs have a higher melting point, heat conductivity, and lower retention of the hydrogenic species than carbon PFCs, they can contaminate the plasma with high Z impurities. This leads to large radiative losses in the confined plasma, which degrades tokamak performance by lowering the central electron temperature. In a pilot plant, this loss of performance can lead to a lowered fusion reaction rate and possibly a disruption. One of the methods to mitigate W contamination is to condition the PFCs with low-Z coatings as is done in glow discharge boronization (GDB) [3,4], which deposits layers of boron (B) onto the PFCs with a glow discharge seeded with a B-enriched gas such as B 2 D 6 or B 2 H 6 . To ensure uniform distribution, the magnetic field coils have to be shut down during GDB, which presents a conflict for superconducting devices (such as ITER) that have constraints on their toroidal field duty cycle. Lastly, B 2 D 6 gas is toxic which adds additional safety concerns and costs. For these reasons, future reactors will need other methods of wall conditioning.

An alternative method of wall conditioning is currently being studied on a number of different tokamaks and stellarators: injection of low-Z powder during the plasma discharge. A device named the Impurity Powder Dropper (IPD) was developed by the Princeton Plasma Physics Laboratory [5] to test this real-time wall conditioning technique. Results of the first deployment of the IPD on WEST are detailed in [6]. Time traces of line-averaged 𝑛 𝑒 , input LH heating power 𝑃 𝐿𝐻 𝑡𝑜𝑡𝑎𝑙 , total radiated power 𝑃 𝑟𝑎𝑑 𝑡𝑜𝑡𝑎𝑙 and O-II visible line spectroscopy measurements are given in figure 1. In those experiments B powder was injected into ten lower single-null (LSN) L-mode discharges with the following characteristics: toroidal field 𝐵 T = 3.7 T, lower hybrid current drive (LHCD) power 𝑃 LHCD ∼ 4.5 MW, and volume-averaged electron density 𝑛 e ∼ 3.5 × 10 19 m -3 . For each discharge, the B drop rate was varied to determine the largest amount of B that could be dropped without triggering a disruption. The maximum tolerated drop rate was found to be 9 -17 mg/s for that series of discharges, with 310 mg of B injected over the whole commissioning experiment.

Measurements obtained in those experiments are used here for modelling analysis of B transport, both as a source of the input parameters to setup the simulated experimental conditions (e.g. outer midplane electron density 𝑛 𝑒 outer , 𝑛 𝑒 and 𝑇 𝑒 profiles at the divertor) as well as a reference for the simulated impact of B injection (e.g. bolometry and spectroscopy measurements).

Measurements from the following diagnostics were employed: embedded Langmuir probes mounted at the lower divertor for the measurements of electron density (𝑛 e ) and temperature (𝑇 e ) [7], midplane fast-sweep reflectometry for the boundary 𝑛 e radial profile [8], visible spectroscopy for impurity line emission at the lower outer divertor and at the midplane antenna limiter [START_REF] Meyer | First EPs Conference on Plasma Diagnostics[END_REF], and bolometry for radiated power (𝑃 rad ) [10,[START_REF] Vallet | EPS Conference on Plasma Physics[END_REF]. Notably, visible spectroscopy data for the upper divertor was not available for this experiment, so the ratio of B line emission from the upper divertor to the emission from the lower divertor, as well as any such ratio for any other element could not be measured.

In order to better interpret the experimental data and to help design future experiments it is necessary to model the transport of injected B in the WEST boundary plasma. The main goal of modelling is to study the distribution of B incident flux on the PFCs. For this reason, a novel modelling workflow was developed and tested, and the results are described in this paper. Section 2 outlines the architecture of the modelling workflow. Section 3 describes the modelling results for a series of simulations. Section 4 compares the impact of B injection in experiment to the simulated impact of the added B neutral source. Section 5 concludes and provides a plan for future work with the developed workflow.

Modelling workflow for IPD experiments

Previous attempts have been made to simulate the transport of impurities introduced through powder injection [12][START_REF] Feng | [END_REF][START_REF] Krashennikov | DPP05 Meeting of APS 15[END_REF][15][START_REF] Effenberg | Nuclear Materials and Energy 17[END_REF]. In [12], a pure deuterium (D) boundary plasma was modelled with the EMC3-EIRENE [START_REF] Feng | [END_REF] code suite, and then used as an input for the DUST Transport (DUSTT) code [START_REF] Krashennikov | DPP05 Meeting of APS 15[END_REF]. DUSTT in turn provided trajectories and evaporation rates for the B spherical particles. B transport and plasma-surface interactions were then modeled using ERO2.0 [15].

Another approach is described in [START_REF] Effenberg | Nuclear Materials and Energy 17[END_REF], where EMC3-EIRENE was used exclusively, with a point source of atomic B placed in the upper part of the boundary plasma. In both cases, the plasma was considered to have no impurities other than B, and EMC3 treated impurities under the trace approximation, i.e. while impurity radiative losses were taken into account when calculating plasma temperature, they did not contribute to charge balance. The effect of impurities on the main plasma was therefore not completely self-consistent.

In our study, we approached the problem in a similar way to [12], with SOLEDGE-EIRENE

[17] calculating the boundary plasma and modelling B transport, and DIS (Dust Injection Simulator) code [18] modelling powder dynamics and ablation. SOLEDGE-EIRENE allows us to model plasma up to the first wall everywhere in the poloidal cross-section, which enables us to calculate B incident flux on the entirety of the wall contour. DIS was recently used to model powder injection experiments on LHD using EMC3-EIRENE plasma backgrounds in ref. [29]. For the plasma before B injection (pre-drop phase), two elements were modelled: D as the main ion species, and oxygen (O) as a proxy for all light impurities present before B injection, which are usually O, carbon and nitrogen. Using an additional light impurity, like O, allows for a better match of the electron temperature and radiated power measurements from experiment. Eight species of O were modelled, one for each ionization state. The modelling workflow was separated into three steps: (i) modelling of the pre-drop phase of the discharge, (ii) modelling of B particle ablation and evaporation with plasma parameters from the pre-drop phase and (iii) modelling of the drop phase of the discharge with the neutral B volumetric source provided by the B particle simulation from step (ii). Pre-drop and drop phases of a simulation refer to steady-state SOLEDGE-EIRENE solutions, which corresponded to the experimental plasma conditions in the middle of the flat-top phase before and during B powder injection. The effects of B injection on the modelled plasma were then compared to the effects of B injection observed in experiment. While the most recent version of SOLEDGE-EIRENE code suite supports 3D modelling [19], only 2D simulations were carried out and analyzed in this preliminary study. Moreover, dynamics of plasma-wall interactions were not modelled, thus the reduction of D recycling due to B deposition on the PFCs and the erosion of W PFCs was not taken into account. Additionally, the B recycling coefficient was assumed to be zero (i.e. B was assumed to "stick" to the PFCs on which it was deposited). For D and O, the recycling coefficients were set to 0.95 for the pump under the baffle and 1.0 for the rest of the wall. No cross-field drifts were assumed in this analysis.

Figure 2 depicts the simulation domain for SOLEDGE-EIRENE. The simulation grid was built based on the magnetic equilibrium reconstruction of WEST discharge 56919 from the NICE reconstruction code [20]. The walls were considered to be made of pure W with the pump located under the baffle. Since the plasma in this study was modelled in 2D, the geometry of the wall contour was assumed to be axisymmetric. In reality, WEST has toroidally localized antennae that act as poloidal limiters, which cannot be accurately described by a 2D model. The antennae have a toroidal length of approximately 30°, so in order to have a compromise in terms of the recycling flux and cooling effect on the plasma, the portion of the wall contour that represents the antennae is placed at major radius 𝑅 LHCD = 3.1 m, further away from the separatrix with respect to its actual radial position in the modelled experiment. This working assumption is consistent with previous WEST modeling efforts [21,22,26]. The inner boundary of the simulation domain was placed far from the separatrix in the confined plasma to model B transport in edge plasma, since there was evidence from the VUV spectroscopy (increase of 24.3 nm B-V line intensity) that B may be transported into the confined plasma [6]. Input parameters described above were chosen to try to qualitatively match the simulated plasma with the measurements from the IPD experiment, for example plasma discharge #56919.

At the same time, the goal of the modelling analysis described in this paper is to perform scans of various input parameters to extract general trends that could inform future WEST experiments.

Therefore, rather than trying to precisely reproduce a given WEST discharge, we aim to estimate the impact of various plasma parameters on B distribution.

Modelling results

A set of nine pre-drop simulations were run with varying 𝑃 𝑖𝑛 and 𝑛 O core values, given in Table 1. To match simulations with experimental data, 𝑃 𝑖𝑛 in the simulation should equal 𝑃 heating -𝑃 𝑟𝑎𝑑 core from the experiment. Here, 𝑃 heating is the total heating power in the experiment and 𝑃 𝑟𝑎𝑑 core is the radiated power emitted by a portion of the plasma in the core region with the same boundary as the core boundary in the simulation. 𝑃 𝑟𝑎𝑑 core was approximated with 𝑃 𝑟𝑎𝑑 bulk = 𝑃 𝑟𝑎𝑑 total -𝑃 𝑟𝑎𝑑 div , where 𝑃 𝑟𝑎𝑑 div is the radiated power measured by bolometers targeting both divertors (chords 1-3, 14, 15 in figure 5 in [27]). In the case of the pre-drop phase in shot 56919 𝑃 𝑟𝑎𝑑 bulk ≈ 1.9 MW, therefore 𝑃 𝑖𝑛 ≈ 𝑃 heating -𝑃 𝑟𝑎𝑑 bulk = 2.6 MW. For the simulations, we chose three values of 𝑃 𝑖𝑛 above this estimation, since 𝑃 𝑟𝑎𝑑 bulk is an overestimation of 𝑃 𝑟𝑎𝑑 core due to midplane bolometry lines of sight (LOS) passing through midplane section of the scrape-off layer (SOL). The effect of the input parameter 𝐶 𝑂 𝑐𝑜𝑟𝑒 on the overall 𝐶 𝑂 spatial distribution is shown in figure 4: here 𝐶 𝑂 is normalized by the ratio between 𝐶 𝑂 𝑐𝑜𝑟𝑒 for a given run and the lowest 𝐶 𝑂 𝑐𝑜𝑟𝑒 used in the scan (2.36%), for improved comparability. If in the rest of the simulation domain (not shown) the normalized 𝐶 𝑂 is almost identical for the three runs, the only region showing a significant change is the far SOL of the inner divertor plasma, as highlighted by the magenta circle in panels a, b and c.

One of the goals of the pre-drop phase plasma modelling was to qualitatively match the simulated plasma parameters with those measured in the experiment. Figure 5 shows experimental data for shot 56919 with the parameters from a selected subset of simulations to show the sensitivity of different plasma parameters due to the variation of input parameters. (1.0 in all simulations). The feedback loop on the diffusion coefficient in SOLEDGE attempts to match profiles with a feedback algorithm, providing an approximate diffusion coefficient map, so there is no guarantee from the algorithm to reach the target 𝑛 𝑒 profile. Methods to improve this feedback mechanism and, therefore, the match between simulated and measured 𝑛 𝑒 will be investigated in future efforts. This discrepancy can affect the atomic processes of light impurities, which in turn would affect impurity distribution and synthetic diagnostic data in pre-drop and drop phase simulations. However, the goal of this study is to predict general trends of B distribution in the plasma and B influx distribution on the PFCs rather than trying to reproduce a given WEST discharge. Figure 5(b) and (c) compare experimental and modelled 𝑇 𝑒 𝑂𝑆𝑃 and 𝑛 𝑒 𝑂𝑆𝑃 . Here, the discrepancies are more pronounced. These can be resolved by lowering 𝑃 𝑖𝑛 to match the measured 𝑇 𝑒 𝑂𝑆𝑃 , which was done in [START_REF] Afonin | 64th APS DPP Spokane 24[END_REF], but that led to additional inconsistencies (e.g. in the outer midplane density). Figures 5(d,e) compare averaged bolometry signals 𝒮 from the pre-drop phase (from t=5.6 s to t=6.6 s) of the experiment with those from simulations calculated with the SYNDI synthetic diagnostic [24] for the same set of SOLEDGE-EIRENE simulations. Lines of sight of the bolometer are shown on the figure 6.

SYNDI calculates synthetic bolometry signals by multiplying 2D maps of emissivity

provided by a SOLEDGE-EIRENE mask with weight coefficients for each LOS from a given diagnostic. The resulting matrix is then integrated to obtain the synthetic measurement for a given LOS. Figure 6 shows the 𝑃 rad map for one of the simulations with five lines of sight from bolometry, three for the lower divertor and two for the upper divertor. To model the boron drop, varying amounts of B powder (mass rate 𝜇 𝐵 = 2.3, 4.6 mg/s) were injected into the simulations described above. Figure 7 shows the 2D maps of the relative B concentration 𝑛 𝐵 Σ ions /𝑛 𝑒 for three simulations, which remained relatively unchanged despite different levels of 𝐶 O core and 𝑃 𝑖𝑛 .

The first goal of the modelling analysis was to investigate the effect of modified plasma parameters on B flux distribution to the PFCs. To achieve this, the normalized total B incident flux was calculated for each drop phase simulation:

Γ B norm = Γ B (𝐿)/∫ Γ B (𝐿)𝑑𝐿,
Where

Γ B = ∑ Γ 𝐵 𝑖 5 𝑖=0
, 𝐿 is the wall contour length, Γ 𝐵 𝑖 is the wall incident flux of B in the i-th ionization state for 𝑖 > 0 and the neutral B wall incident flux for 𝑖 = 0. One-dimensional plots of Γ B norm (𝐿) along the wall are given in figure 8 for different sets of input parameters. In general, the Γ B norm distribution stayed approximately the same for the variations in 𝑃 𝑖𝑛 , 𝜇 B and 𝐶 𝑂 core , with B being spread more homogeneously at higher 𝑃 𝑖𝑛 and lower 𝜇 B , all other parameters being equal. Interestingly, in all of the drop phase simulations, the upper divertor received a substantial amount of B with respect to Γ 𝐵 𝑛𝑜𝑟𝑚 on the lower divertor. A difference of 1.5-4 times between the integrated incident fluxes of B ions and neutrals to the upper and lower divertors was estimated despite the LSN configuration. This ratio increased for smaller 𝑃 𝑖𝑛 and higher 𝜇 𝐵 . Experimental validation of this influx ratio remains unsolved due to the lack of visible spectroscopy data during the C5 campaign. Future experiments on WEST will feature larger visible spectroscopy coverage at the upper divertor to test this hypothesis.

The modelling results also suggested that the OSP received little to no injected B when compared to the other affected PFCs such as the baffle region. In contrast, the experiment observed similar levels of B-II line brightness at the baffle and the OSP, show in figure 9.

Spectroscopy measurements from the experiment could not be compared with the synthetic diagnostic data from simulations provided by SYNDI as there was no inclusion of B recycling in the model. Therefore, there were practically no B+ ions at the lower divertor, as B + ions were only present in the vicinity of neutral B source. Future modelling should include a non-zero recycling coefficient for B, which might affect Γ B norm distribution. Lastly, the lack of B at the OSP might be an indication of incomplete transport dynamic of B such as a lack of radial electric field or assumption of axisymmetry of B source, which must be addressed in future modelling.

Comparative analysis of B injection in modelling and experiments

Having modeled both pre-drop and drop phases, we compared the impact of B injection in the experiments with the impact of adding a B source in the simulations. For example, we compared the difference between bolometry signals in the pre-drop and drop phases in the experiments to the same difference in the synthetic bolometry signals from the simulations. For bolometry, SYNDI was applied to the 𝑃 rad 2D maps. Figure 10 relatively to volume near ISP, an effect that was studied in [START_REF] Effenberg | Nuclear Materials and Energy 17[END_REF] and tied to the balance between friction and thermal terms in momentum equation. As for the channel 2 and 3, the increase of the bolometry signal in the simulation is caused by the increased radiance of the plasma due to the injected impurities.

Lack of the effect of B injection on the lower divertor bolometry measurements in the experiments can be attributed to the phenomena not taken into account in the modelling workflow:

the most likely candidate for this effect is reduced D and O recycling, which were not modeled here as the wall properties were assumed to be static. In the simulations, B was simply pumped by the wall without affecting the D recycling. A similar discrepancy was found between the 𝑇 𝑒 measured by the embedded lower divertor Langmuir probes and that estimated from the simulations. In every simulation, B injection resulted in a 2-10 times decrease of the peak 𝑇 𝑒 OSP , which was not consistent with the experimental results. In the experiment, B powder injection had no effect on 𝑇 𝑒 𝑂𝑆𝑃 , as shown in [6] and in figure 12. Additionally, 𝑛 𝑒 at the lower divertor increased in the simulations while the experimental measurements slightly decreased [6], further suggesting the reduction in recycling observed in experiment as a probable cause for the discrepancy. In order to improve B transport modelling, the workflow can be enhanced by switching to a 3D plasma model to account for non-axisymmetry of the B source and PFCs and/or turbulent transport, as SOLEDGE-EIRENE allows for 3D modelling in a multi-fluid approach [START_REF] Devynck | [END_REF]. Other potential ways to better match experimental data are to include cross-field drifts or to imitate effects, caused by the changing wall properties, e.g. to reduce D/O recycling in drop phase to match lower divertor temperature and 𝑃 𝑟𝑎𝑑 with the experimental values, or to reduce 𝐶 𝑂 core to match decrease of VUV spectroscopy measurements in drop phase in the experiment.

Figure 1 .

 1 Figure 1. Time traces from two shots (control and with B powder injection) of IPD experiment on WEST: (a) line-average 𝑛 𝑒 along equatorial line of sight; (b) total LH input power; (c) total radiated power; (d) O-II (𝜆 = 441.5 nm) visible line intensity at the outer strike point, lower divertor

Figure 2 .

 2 Figure 2. Quadrangular simulation mesh for SOLEDGE-EIRENE with the pump location shown in magenta and wall coordinate ticks shown in dark blue.

Figure 3 .

 3 Figure 3. Particle trajectories from DIS for B particles of varying diameters (15-150 um) with the evaporation flux Γ 𝑒𝑣𝑎𝑝 shown in color (a.u.)

Figure 5 (

 5 a) shows 𝑛 𝑒 𝑜𝑢𝑡𝑒𝑟 from the pre-drop phase with the simulated densities. Modelled 𝑛 e outer changes minimally with 𝐶 O core and 𝑃 𝑖𝑛 and generally matches the measurements, especially in the plasma edge region. The discrepancy in the SOL region might be caused by the unitary recycling coefficient for D and O

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Normalized O concentration 𝐶 𝑂 × ( 𝐶 𝑂 𝑐𝑜𝑟𝑒 2.36% ) -1

Figure 7 .

 7 Figure 7. 2D map of relative concentration 𝑛 𝐵 Σ ions /𝑛 𝑒 for a set of simulations

Figure 8 .Figure 9 .

 89 Figure 8. Normalized distribution of B total incident flux summed over all B species Γ B norm for drop phase simulations with the variations of: (a) input power 𝑃 𝑖𝑛 , with O core concentration 𝐶 𝑂 core = 2.59% and B mass rate 𝜇 𝐵 = 4.6 mg/s (b) 𝐶 𝑂 core , 𝑃 𝑖𝑛 = 2.875 and 𝜇 𝐵 = 4.6 mg/s (c) 𝜇 𝐵 , 𝐶 𝑂 core = 2.59% and 𝑃 𝑖𝑛 = 2.875

  shows bolometry signals 𝒮 for plasma simulations as well as experimental values for the same signals from shot #56919 in the drop phase and the ratio between them and the signals from the pre-drop phase, that were shown in figures 4 (d, e). The rows of the figure 10 show 𝒮 low,up from lines of sight targeting lower and upper divertor respectively. The columns show 𝒮 for two drop rates in modelling as well as measured bolometry signals in the experiment, with average value and standard deviation calculated before and after B injection. The figure 11 shows ratios of the signals (

Figure 10 .Figure 11 .

 1011 Figure 10. Bolometry channel output signals 𝒮 from the experiment (red boxes) and calculated with SYNDI for studied simulations (dots): (a-b) signals at the lower divertor for the drop phase with B mass rate 𝜇 B = 2.3, 4.6 mg/s respectively; (c-d) same, for the upper divertor;

Figure 12 .

 12 Figure 12. Embedded Langmuir probe measurement of OSP target temperature profile over time

Figure 13 .Figure 14 . 5 .

 13145 Figure 13. Photon emissivity coefficient for O-II (𝜆 = 4415 Å) in one of the pre-drop phase simulations (input power 𝑃 𝑖𝑛 = 2.875 MW, O core concentration 𝐶 𝑂 core = 2.59%) and the position of the analyzed visible spectroscopy line of sight (width not to scale)

Table 1 .

 1 Parameters for pre-drop simulations in SOLEDGE-EIRENE

	all other parameters being equal (figure 5 f, g), which is caused by the decrease of O radiative
	cooling rate 𝐿 𝑍 𝑂 with the increase of 𝑇 𝑒 :	𝑑(𝐿 𝑍 𝑂 ) 𝑑(𝑇 𝑒 ) < 0, 𝑇 𝑒 ∈ [20 eV, 80 eV] [28], and 𝑇 𝑒 in SOL is in
	this range in the simulations.
	Input power 𝑃 𝑖𝑛 , MW	2.75	2.875	3.0

These lines of sight were chosen because they only go through the boundary plasma. Since O was the only impurity in the simulation, we chose to compare experimental and synthetic bolometry signals from the boundary plasma in order to match the light impurity content. The experimental trend in bolometry signals was not qualitatively reproduced by SOLEDGE, which may be due to different impurity profiles in the simulations and the experiment. Increasing 𝑃 𝑖𝑛 led to a decrease in SYNDI bolometry signal,

Acknowledgments

This research was supported by the U.S. DOE under Contract No. DE-AC02-09CH11466 with Princeton University. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 -EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission.

Neither the European Union nor the European Commission can be held responsible for them.