Small resonant magnetic perturbations result in three-dimensional material transport in the fusion plasma edge

Erosion patterns in a 3D magnetic topology are significantly altered when compared to an axisymmetric scenario due to differences in the impurity transport in a plasma environment. When resonant magnetic perturbations are applied in L-Mode discharges, eroded impurities can buildup in regions where an axisymmetric plasma would otherwise lead to zones of net erosion across the full toroidal length of the divertor. The reduction on the local and integrated erosion observed across a parametric study of the anomalous diffusion and impurity content in a plasma, can lead to the extension of divertor lifetimes in low power scenarios when 3D fields are applied. By contrast, in axisymmetric scenarios, most of the carbon eroded from the divertor are carried away to the far scrape-off layer.


Introduction
Strong magnetic fields converge at the Earth's north pole, and the aurora borealis is the most well-known and visible example of charged particle interactions with magnetic fields [1,2].In a small scale laboratory setting, charged particles can erode wall material which can become charged, travel along field lines, and get redeposited due to the influence of external forces on particle motion.Solar flares can disrupt the Earth's 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.magnetosphere, and this results in the non-symmetric motion observed in the aurora, and by consequence, the charged particle trajectories [3].A similar effect is observed when we look at its analogue in a smaller scale in fusion reactors, where an axisymmetric field can be made highly three-dimensional by the application of external resonant magnetic perturbations (RMPs).In this scenario, the application of these external fields has a purpose: mitigate highly disruptive events in the plasma edge (Type-I edge localized modes), and create discrete regions of heat dissipation along critical wall components (i.e. the divertor) [4][5][6].However, significant insight on particle transport and erosion patterns can be determined from L-Mode discharges due to the elimination of the erratic radial electric fields present in H-Mode scenarios, the former being the focus of this study.
ITER is a nuclear fusion experiment device with the longterm goal to produce a net positive energy output from a burning plasma relative to the input energy, and to achieve this it is expected to dissipate heat loads to the divertor of up to 10 MW m −2 [7,8].Such power densities are expected while operating in the high confinement (H) mode due in part to edge localized modes (ELMs).The large pressure gradients at the plasma edge lead to large currents in the pedestal which provide free energy that can destabilize peeling and ballooning modes over a large range of toroidal mode numbers [9][10][11], leading to ELMs.One of the methods that has been proposed to mitigate the impact of ELMs on the divertor target is edge magnetic field ergodisation by resonant magnetic perturbation application.One of the added effects for this is the breaking apart and creating a highly three-dimensional scrapeoff layer due to the formation of ergodic flux tubes at the edge [12][13][14][15][16].
ELM mode suppression has been observed across multiple tokamak devices.DIII-D has achieved this by the application of n = 3 RMP application at low average triangularities and ITER similar shaped plasmas, with the latter requiring larger currents for suppression [17,18].While suppression/mitigation of Type-I ELMs is an achievable goal for ITER, there are still energetic events which cannot be fully addressed, or generalized to a global model, and such events can cause significant erosion of reactor components.In addition, the stochasticity introduced by RMPs on the plasma edge can affect the erosion/deposition fluxes due to plasma-wall interactions (PWIs).Therefore, to better understand the effects of 3D fields on PWIs, it is necessary to reconstruct standard operating scenarios in existing devices and model particle erosion and impurity transport with full 3D capability.
The DIII-D tokamak has one of the most comprehensive worldwide programs to improve the understanding of the behavior of high temperature plasmas and 3D physics in a plasma environment.This device is a mid-size scale reactor (R = 1.66 m, a = 0.67 m) with a maximum toroidal field of B T = 2.2 T and is able to produce highly elongated plasmas (κ ⩽ 2) [19].The device capability to utilize RMPs to suppress ELMs [17,18] and the knowledge that ITER is expected to operate with n = 3 and n = 4 RMP [20] scenarios makes DIII-D an ideal candidate to study erosion/deposition and impurity transport in 3D scenarios.
Previously, the ERO2.0 has been used to model PWI processes and particle transport across numerous devices.A detailed study on beryllium transport on ITER showed beryllium transport from beryllium migration from the first wall along the divertor targets, raising concerns to fuel retention due to build-up of co-deposited layers [21].Additional results show that the large uncertainties regarding the plasma conditions in the far-SOL are required to properly predict erosion rates at the first wall, as current extrapolations using the onionskin model can lead to large erosion rates on the order of mm's after a few hours of operation [22].This paper will highlight the first implementation of the ERO2.0 code for full device modeling with 3D fields applied and demonstrate the main transport mechanisms relevant to carbon erosion and deposition in the DIII-D open divertor.

Modeling setup
Plasma backgrounds from the fluid/kinetic 3D code EMC3-EIRENE [14,23,24] are used as input for the ERO2.0 simulations based on three discharges used during the 2016 DIMES campaign [25].Three scenarios are used to study the erosion during L-Mode discharges for a deuterium (D) plasma: the 0 • phasing (#164250), the 60 • phasing (#164251), and the no RMP reference scenario (#164264); such discharges are chosen in order to remove the uncertainty of ELM events during H-Mode.The external coil current applied is set at 4 kA to generate the n = 3 perturbation in the magnetic field.Upstream densities are fixed at 1.75 × 10 19 m −3 across scenarios, and relevant background information are provided by EFIT reconstructions for the discharges for the EMC3-EIRENE modeling [25].The open divertor on the lower leg of the DIII-D vessel is the region of interest for the study due to the helical lobe structures that form when RMP's are applied.A carbon wall is used for the present study as DIII-D utilized carbon tiles for the first wall and the divertor, though recent studies have demonstrated the effect of different ELM mitigation techniques on tungsten erosion in the open divertor [26].Sputtering (Y) and reflection (R) values are extracted from high resolution SDTrimSP matrices [27], for incoming plasma ion energies defined by T ≈ 2T i + 3ZT e .An average angle of incidence of 60 • to calculate the sputtering of the background plasma species [28,29].Chemical erosion for these scenarios is calculated via the Roth formulation for a carbon PFC for T surf = 300 K [30]. Figure 1 shows the sputtering yield for a 60 • as previously mentioned, and normal incidence.For traced particles, the incidence angle is calculated via the directional cosines with the particle velocity components and the result is a distribution of incident angles at the surface, which have been calculated via SDTrimSP and are in the ERO2.0 database.
Impurities are created as traced (i.e.eroded) neutral particles within the ERO2.0 code which have an ionization (and subsequently, recombination) probability, and their full gyro-motion is extracted from the Fokker-Planck equation as a six dimensional set of equations: where r and v are the displacement and velocity vectors in space and time, K is the drift or friction vector, B is the diffusion tensor with components in both the parallel and perpendicular direction of the magnetic field, and ∆W is the stochastic increment for the diffusive process for traced particles.Anomalous diffusion for the impurities is a free parameter in the code, defined as D ⊥ .An additional force is included to the force balance, the thermal force, denoted as F th,e/i , which acts in the direction of the thermal gradient in the plasma [31] and utilizes the conductive ion and electron heat fluxes calculated in EMC3-EIRENE for the impurity transport in ERO.Traced particles from a surface cell are created and ejected from the surface, and ionized based on the ionization cross section and the plasma background parameters.The impurity full gyro-motion orbit is calculated, and these particles will then have a probability to be reflected once it reaches the surface.If so, the test particle is reejected and carries a fraction of the flux (RΓ inc ), and the remaining fraction is deposited at that surface location (1-R)Γ inc .This process can repeat itself until the particle tracing time ends, or the particle recombines at the surface.Within a time-step, provided they have sufficient energy, these particles can re-erode material from the surface and this tracing process begins anew.However, the re-eroded particles are only traced within the subsequent step, so it is important to account for this contribution as these can change the erosion profiles quite significantly in these 3D scenarios.This will be discussed later for RMP scenarios at the open divertor, where zones of local net erosion and net deposition are observed along the length of the toroidal sector, demonstrating the asymmetry resulting from the discharge conditions.
The erosion rates along the open divertor target is dependent on the incident ion temperature (2T i ) sheath potential ( 3ZT e ), and the incidence flux is set by the Bohm criterion and is dependent on the density at the surface and the ion and the ion/electron temperatures Γ inc ∼ c s × n e .The data at the target plates are extracted from the EMC3-EIRENE post-processing, and written as an input in the ERO2.0 code. Figure 2 shows these plasma parameters at the surface for the no RMP and the RMP scenario.Of note, the effect of the 3D field on the density and electron temperature at the target surface is observed where the scrape-off layer is broken up into 3 lobes (n = 3) along the full toroidal direction.

Results
This study focuses on modeling erosion and impurity transport in L-Mode discharges in the DIII-D open divertor through an RMP discharge.These low confinement modes are chosen because it removes any uncertainty in the physics understanding that H-Mode scenarios can introduce, such as erratic drifts, and large and chaotic density and temperature gradients at the plasma edge when not operating in an ELM-suppressed regime.Figure 3(top) includes the no RMP reference erosion patterns for the open divertor, for which a high baseline erosion rate is observed for the carbon wall.The effects of such 3D magnetic configuration on the erosion footprint is observed in figure 3(bottom).The application of the RMP's demonstrate an overall decrease in the erosion rate cross the open carbon divertor.In addition, regions of net deposition of carbon are observed between the lobe structures (in blue).This indicates two important features: (1) the impurity level from the background species sets a baseline deposition in low temperature regions, and (2) traced deposition from particles eroded along the strike zones contribute to buildup of carbon as the toroidally averaged profiles in figure 4 shows.The latter differs from previous observations [32,33] where impurities eroded in 2D scenarios are carried away to the far SOL and deposited along the main wall, or in private flux regions.As seen in figure 4, physical sputtering from the background deuterium for both scenarios occur along a similar width of the target, with slightly higher values for the RMP scenario.Chemical erosion from deuterium is low (strong dependence on D fluxes, for T surf = 300 K), and erosion from the carbon in the background is also negligible as it only contributes 1% of the composition.Re-erosion of carbon towards the far SOL is a significant contributor to the toroidal fluxes.When looking at the traced carbon deposition rates, for the no RMP scenario, this occurs at the far SOL which agrees with the observations mentioned, whereas the RMP traced deposition occurs within the lobe, and less so in the far SOL.In addition, the background carbon deposition is slightly higher in the RMP scenario due to the regions of lower temperature between the lobes.The traced particle contribution to the deposition can be observed in figure 5.
The two surface maps shown in figure 3 correspond to the no RMP reference scenario and the 60 • phasing.Traced particles are created at all locations but are weighted higher to locations of high erosion.For visualization purposes, only traced particles created along the full torus over a 4 cm poloidal width are shown in green; this is the starting locations in particles.The blue crosses show the end location for traced particles that are deposited along the divertor target.As seen in figure 3(bottom), the density of the particles in the RMP scenario is significantly higher in the regions between the lobe structures, than in the outer perimeter.In addition, the no RMP reference scenario shows a more homogeneous distribution of particles throughout the toroidal direction (i.e.no distinct localization features).Once the carbon is ionized in the plasma, the particles travel along the magnetic field, and follow the topology that forms the lobes.Anomalous diffusion then contributes to particles across the lobe and the friction force eventually overtakes the particles they deposit, leading to carbon buildup.This traced deposition in 3D L-Mode scenarios will significantly affect the toroidally averaged profiles as the more traced particles escape high temperature regions (i.e.along the ergodic flux tubes), they deposit within the lobes and are unable to contribute to particle re-erosion.Figure 6(top) demonstrates the effects of traced erosion and deposition on the open divertor.At the initial timestep, particle re-erosion is not considered, providing a baseline for the zones of net erosion and net deposition along the target.At steady state (t = 1.5 s), the reference scenario shows an increase in particle erosion along the strike zone, meaning that re-erosion plays an important role in this scenario.On the low field side, particle deposition is enhanced, meaning that impurities eroded are carried away from the strike line to the far SOL.Results for the 3D scenarios (0 • and 60 • phasing) show a similar behavior at the far SOL, with a smaller increase in particle deposition relative to the reference scenario.However, as previously mentioned, the rest of the traced deposition occurs between the lobes (figure 3) and is reflected in an upwards shift for these profiles along the lobe location (R = 1.47-1.62m).These modeling efforts show that RMP application in L-Mode discharges lead to enhanced impurity deposition, and an overall decrease in net erosion.
A parametric study was done for all three scenarios, to investigate the effect of anomalous diffusion and the level of carbon in the background species on the erosion of the divertor target.Figure 6(bottom) shows that increasing the carbon concentration decreases integrated erosion along the target, and while not shown here, increasing the level to 5% shifts the profiles from net erosion to net deposition, meaning that in these scenarios the background deposition significantly surpasses the traced erosion and deposition fluxes.The level of carbon in the background for DIII-D can vary between 1% and 6% [34] so it is important to keep this under consideration in such a study even though most of the contribution to erosion results from the deuterium background.Anomalous diffusion appears to have a linear effect on the erosion rates, though the effect is not too significant at high values of D ⊥ , and has a limited effect on the integrated erosion rates.The largest difference in the rates comes from the inclusion of diffusion in the simulations (i.e.D ⊥ = 0 to 0.3 m 2 s −1 ).In the absence of anomalous diffusion, the traced particles are carried away from the lobes and deposited in the far SOL leading to increased fluxes when R >1.62 m.However, it is difficult to isolate the complete effects of anomalous diffusion for extremely low values as it becomes very computationally expensive to do so.
In conclusion, a thorough analysis has been performed for L-Mode discharges in the DIII-D open divertor for two RMP scenarios and a no RMP reference case to determine the differences in impurity transport and their effect on the 3D erosion profiles along the vessel.The inclusion of 3D fields shows a decrease in the erosion along the divertor target relative to the  reference scenario, with a non-negligible fraction of impurities being deposited between the flux lobes in the presence of anomalous diffusion.The inclusion of such fields breaks up the SOL changing the plasma parameters at the target surfaces, and the resulting magnetic topology affects the particle transport as discussed here.Additionally, the traced deposition has a limited increase as a function of D ⊥ , and the impurity content in the background has an added effect to the total deposition (i.e.background+impurities).A similar approach to such a study can be performed on H-Mode discharges on similar devices provided appropriate E-field and neutral particle flux data is available, as their 3D distributions can have a significant impact on the first wall and divertor profiles at reactor relevant conditions.Additionally, by designing rotating coils that apply these external fields, the rotational speed can be set to erode the divertor evenly, as large erosion fluxes can introduce regions where the height and angular distribution on the material surface can enhance sputtering and flaking of additional impurities.

Figure 1 .
Figure 1.Calculated sputtering yields for deuterium on carbon and carbon self-sputtering via SDTrimSP for normal incidence and 60 • incidence.

Figure 2 .
Figure 2. Comparison of the plasma parameters along the DIII-D open divertor for the no RMP and 60 • RMP scenarios.The left column shows the no RMP parameters along the plate and the right column shows the RMP parameters.The effect of the RMP field on the SOL is reflected in these along the divertor.Units for these are: ne(m −3 ), Te(eV), T i (eV).

Figure 3 .
Figure 3. Top: Carbon rates (nm/s) for the no RMP scenario; bottom: carbon rates for the 60 • phasing scenario.Parameters for both scenarios include 1% carbon as part of the main ion species, and D ⊥ = 1 m 2 s −1 .For the net rates sector (240-360 • ), a blue rate (positive) means net deposition, and a red rate (negative) indicates net erosion.The figure is divided into three sectors demonstrating the gross erosion fluxes (left), total deposition flux (middle) and net flux (right).

Figure 4 .
Figure 4. Top: Toroidally averaged erosion contributions to the total gross erosion.Bottom: Toroidally averaged deposition contributions to the total deposition.The solid lines correspond to the no RMP scenarios and the dashed lines correspond to the 60 • RMP scenario.These profiles correspond to a carbon concentration in the background of 1%, and D ⊥ = 0.3 m 2 s −1 .

Figure 5 .
Figure 5. Net profile (gross erosion-total deposition) for particle visualization for traced carbon: (a) no RMP and (b) 60 • phasing.Particle starts are shown in green and ends in blue.The simulation parameters are the same as described in figure3, as are the surface maps (net rates from all contributions).Of note is the clustering of traced particles to the regions between the lobes versus no distinct clustering of particles in the reference scenario.

Figure 6 .
Figure 6.(a) Toroidally averaged erosion/deposition profiles for the reference, 0 • and 60 • phasing configurations for the DIII-D open divertor.The solid lines represent the initial time step fluxes where no re-erosion is calculated, and the dashed lines correspond to the steady state fluxes after 15 time steps (dt = 0.1 (s).(b) Integrated net rates (net erosion regimes) for the parametric study of the erosion rates as a function of carbon concentration in the background (D:C mix), and anomalous diffusion of impurities D ⊥ .