Gas breakdown in radio-frequency field within MHz range: a review of the state of the art

1.   Introduction

The control of the heat flux to the divertor target is a great challenge in fusion devices. In ITER, it is required that the maximum steady-state heat flux deposited on the target must not exceed 15 MW/m² [1]. Advanced divertors, such as the snowflake divertor (SFD) [2] and the X divertor, have effective heat flux control capability. The topologically stable SFD, characterized by two X-points and a simplified geometry, offers advantages such as longer connection lengths and larger magnetic field expansions. The HL-3 [3, 4], a novel advanced tokamak, can operate with a standard divertor (SD), a snowflake divertor, and other divertor magnetic configurations flexibly, utilizing carbon as plasma-facing material. The use of tungsten (W) material is being considered in HL-3 following ITER. Due to their high atomic number, W impurities are incompatible with core plasma; therefore, it is necessary to assess the impact of magnetic configurations on W impurity core accumulation in HL-3.

The behavior of impurities in the boundary plasma can be effectively simulated using the kinetic code DIVIMP [5], which has been validated by comparison with experiments [6–8] and other boundary transport codes, such as SOLPS-ITER [6, 9] and EDGE2D [10]. The leakage of W impurities under various divertor conditions on EAST [11] has been studied with the aid of DIVIMP, which showed non-negligible effects of parallel gradient force, especially in the far scrape-off layer (SOL) region with detached plasma. Ma et al investigated the transport of W impurities on DIII-D [12], showing that a small angle slot (SAS) divertor has a small mean free ionization path, which leads to low W leakage. Sinclair et al and Abrams et al further studied W impurity transport in the V-shaped SAS (SAS-V) divertor, and demonstrated that the erosion of SAS-V was reduced significantly compared to the SAS in a forward toroidal magnetic field (B_t). Moreover, they found W impurity erosion and leakage are significantly influenced by the strike point location [13, 14]. Maeker et al simulated W transport with extended grids (to the first wall) [15], and found that impurities are mainly transported from the baffle to the near SOL. Zamperini et al simulated collection probe experiments on DIII-D using a combination of DIVIMP with 3DLIM, and the simulation region was extended from the first wall to the near SOL region [16, 17]. Fluid codes coupled with kinetic codes, such as SOLPS-ITER [18–21] and SOLEDGE-ERO2.0 [22–24], have also been employed to investigate the transport of W impurities in different devices. The contributions of drifts and first wall erosion on the W core accumulation were illustrated, respectively. Most of the related investigations focused on SDs. The SFD has the advantage of heat flux control. However, the larger plasma-wetted area and longer connection length change the source and transport of the W impurities, respectively. The behavior of W impurities in the SFD is poorly studied.

In this work, we study the effects of divertor magnetic configurations on the W core accumulation by comparing the SD and the SFD in HL-3. In section 2, the simulation setup is introduced. In section 3, the principal results of the simulations are presented, encompassing analyses of W transport in the two divertors under similar total W erosion flux and identical divertor conditions, as well as the effect of the W source location on the transport of W impurities. Finally, our conclusions are summarized.

2.   Simulation setup

We perform comparative simulations to analyze the transport of W impurities in the SFD (the second X-point is situated outside the dome) and SD configurations, as depicted in figure 1. The simulations are conducted using the DIVIMP with the aid of SOLPS-ITER. Figure 1 also exhibits the regions’ definition on physical and computational meshes as well as the locations of the inner divertor entrance (IDE), inner mid-plane (IMP), outer mid-plane (OMP) and outer divertor entrance (ODE) in the SD. Moreover, the regions’ definition of the SFD is the same as the SD in this work.

Figure  1.  (a) A sketch of the divertor magnetic configurations for the SFD (blue) and the SD (red), as well as the regions’ definition on (b) physical and (c) computational meshes of the SD in DIVIMP. The arrows indicate the positions of Ne seeding and the shaded regions correspond to the vessel walls in (a). The IDE, IMP, OMP, ODE and separatrix are shown on both grids. The SOL, PFR and CORE regions are labeled. The regions’ definition of the SFD is the same as the SD in this work.

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DIVIMP, a quasi-kinetic 2D Monte Carlo code, is usually employed to track the trajectories of test impurity particles on the edge plasma background until they are deposited, have left the simulation region or have reached the maximum tracing time. Then, the impurity parameters, such as impurity densities, are obtained by statistically average. In this work, a fixed radial diffusion coefficient ${D}_{\perp }$ = 0.5 m² s⁻¹ is employed [25–28], and the effects of ${D}_{\perp }$ on the W impurity transport can be found in [28]. It should be noted that the effects of self-sputtering, prompt redeposition, reflection and drifts, including E×B drift, diamagnetic drift and viscosity drift [29], are not included.

SOLPS-ITER [25, 30] is a widely used boundary plasma simulation code package. In this simulation, a fixed input power P_CEI = 10 MW is applied to the core-edge interface (CEI), with the deuterium ion (D⁺) density at the CEI of the SFD and the SD configuration set at 4.0×10¹⁹ m⁻³ and 5.0×10¹⁹ m⁻³, respectively, corresponding to the same separatrix density of 2.2×10¹⁹ m⁻³. Neon (Ne) impurities are seeded to facilitate the plasma detachment, with seeding rates ranging from 3×10¹⁷ to 1×10¹⁹ Ne atoms/s for the SFD and 1×10¹⁹ to 4×10²⁰ Ne atoms/s for the SD.

The Ne puffing port is located near the outer strike point (OSP) of each divertor, as indicated by the arrows in figure 1(a). Both the DIII-D experiments and our previous simulations have demonstrated that impurity screening in the divertor can be improved by deuterium molecules (D₂) fueling from the mid-plane [30, 31]. In this study, the D₂ puffing rates for the SFD and SD are 5.0×10²¹ D₂ molecules/s and 1.0×10²² D₂ molecules/s, respectively.

The sputtering yield (Y) of W is influenced by several factors, including the type, energy, and angle of the incident particles, as well as the surface morphology of the target. For simplification, Y can be calculated using an experimentally fitted sputtering empirical formula. In this study, Eckstein fitting [32] is employed to calculate the sputtering yield of the W target by the incident of D and Ne, assuming an incidence angle of 45°, by considering the surface roughness [33–36]. The incident particles take only D and Ne ions into account, while the neutral fluxes are negligibly small: thus, they are not included. In the simulation, SOLPS-ITER provides the background plasma for DIVIMP. Additionally, it determines the incident particle flux and incident particle energy [32] gained by the average energy per ion leaving a Maxwellian plasma, along with energy acceleration in the presheath as well as the sheath to calculate the W source [37]. DIVIMP traces the trajectories of sputtered W atoms with Thompson energy and cosine angular distribution, using the calculated source and plasma background.

3.   Analysis of simulation results

In this work, the P_CEI in both the SD and SFD of HL-3 are the same. The Ne seeding in the vicinity of the OSP is applied to reduce the electron temperature (T_e) in the divertor region to vary the divertor operation regime. Over a relatively broad range of Ne seeding rates, both SFD and SD plasmas can vary from attachment to detachment, with detachment defined as the T_e at the strike point T_e,sp $\leqslant$ 5 eV [38]. The operation regime transition is indicated by the green dashed line in figure 2(a). Despite significant differences in Ne puffing rates between two configurations, the ranges of T_e,sp are similar, as depicted in figure 2(a). It provides optimal conditions to investigate the impact of divertor magnetic configurations on W impurity transport.

Figure  2.  The variations of (a) T_e and (b) Ne ion flux density at the strike point, and (c) the total W erosion flux against the Ne puffing rate in the SFD and SD. The markers indicate the selected cases with similar total W erosion flux for attached (“★”) or detached (“+”) plasma.

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As the Ne puffing rate increases, the Ne flux density to the targets of both the SD and the SFD rises, while the T_e at both the inner strike point (ISP) and the OSP of the SD as well as the OSP of the SFD decrease. The inner divertor of the SD (ISD) is easier to detach than its outer divertor (OSD); while the outer divertor of the SFD (OSFD) is more prone to detachment than its inner divertor (ISFD). The T_e at the ISP of the SFD initially decreases, then increases, and ultimately drops. This can be explained by the redistribution of the Ne density [20, 39], i.e. when the Ne seeding rate is increased above one particular level, the Ne ions’ accumulation region changes from the inner divertor to the outer divertor.

The divertor operation determines the incident flux and ion energy to the target which, in turn, affect target erosion. Moreover, the introduction of Ne impurities complicates the erosion process. The total W erosion flux (${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$) can be expressed as

where ${\varPhi }_{\mathrm{I}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ and ${\varPhi }_{\mathrm{O}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ denote the total W erosion flux of the inner target (IT) and outer target (OT), respectively. The W erosion flux density of the IT and OT is represented by ${\varGamma }_{\mathrm{W},\mathrm{I}\mathrm{T}}$ and ${\varGamma }_{\mathrm{W},\mathrm{O}\mathrm{T}}$, respectively, and ${A}_{\mathrm{I}\mathrm{T}}$ and ${A}_{\mathrm{O}\mathrm{T}}$ denote the erosion areas of the IT and OT. Here, ${\varGamma }_{i,j}$ represents the ion flux density of i to the j target, and ${Y}_{i,j}\left(E,\alpha \right)$ represents the corresponding sputtering yield, which depends on the incident ion energy E and angle $\alpha$. Here, ions i can represent either D or Ne ions, and the target j can be either IT or OT. To avoid confusion, we define the W ion flow (Φ_W) here as the integration of W ion flux density over the area traversed by the ions within a physical grid.

A reduction in the T_e at the target (corresponding to incident ion energy) leads to a decrease in sputtering yield. Meanwhile, an increase in the Ne ion flux density (${\varGamma }_{\mathrm{N}\mathrm{e}}$) raises the target erosion. The ${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ of both divertor geometries exhibits a first increasing and then decreasing trend [40] within the range of selected puffing rates, as illustrated in figure 2(c). To mitigate the impact of the W source strength on the W concentration in the core between two geometries, cases with similar ${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ in the same divertor regime (attachment or detachment) are chosen for analysis. The corresponding Ne puffing rates, inner and outer target erosion, as well as total erosion, are presented in table 1. In attachment, the low Ne seeding rate leads to small ${\varGamma }_{\mathrm{N}\mathrm{e}}$. Despite a large sputtering yield induced by high incident particle energy, ${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ is relatively small. In the SFD, the erosion of the IT is stronger than that of the OT, whereas IT erosion is weaker than that of the OT in the SD. In detachment, a higher Ne seeding rate leads to larger ${\varGamma }_{\mathrm{N}\mathrm{e}}$. The IT of the SFD and the OT of the SD are significantly eroded, resulting in a large ${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$. Additionally, the differences in erosion between the IT and OT are notable. It should be noted that each geometry has only one detached target.

Table  1.  The Ne puffing rate, total W erosion flux, and W erosion flux of the IT and OT in attached and detached regimes of the SFD and SD. The total W erosion flux of the SFD is similar to that of the SD in the same regime.

		Ne puffing rate (s−1)	${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ (s−1)	${\varPhi }_{\mathrm{I}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ (s−1)	${\varPhi }_{\mathrm{O}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ (s−1)
Attachment	SFD	3×1017	9.37×1018	8.09×1018	1.28×1018
	SD	3×1019	9.18×1018	2.01×1018	7.17×1018
Detachment	SFD	1×1019	1.66×1019	1.66×1019	4.64×1014
	SD	1.5×1020	1.31×1019	1.08×1018	1.20×1019

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The 2D distributions of W densities (n_W) in the selected cases of both SFD and SD configurations are shown in figure 3. The peak W concentration at the CEI of the SFD is significantly higher than that of the SD in the same divertor operation regime. In detachment, the peak W concentration is 8.93×10⁻⁵ and 3.78×10⁻⁴ in the SD and SFD, respectively. This suggests that the SFD exhibits relatively poor screening capability for W impurities. Regardless of the divertor plasma regime, the n_W of the ISFD is significantly higher than that of the OSFD, while the OSD has a higher n_W than the ISD. In the upstream region, the n_W is higher in the low field side (LFS) of the SD, while it is higher in the high field side (HFS) of the SFD. Overall, the divertor regime has neglectable effects on the distribution trend of the n_W.

Figure  3.  The distributions of n_W in ((a), (b)) attachment and ((c), (d)) detachment of the SD and SFD, respectively.

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The concentration of W impurities in the core region is one of the critical issues in fusion investigation due to their high power radiation, which may lead to disruption. This work focuses on the effects of the divertor magnetic configuration on the accumulation of W in the core region. Since the impact of the divertor regime on n_W distribution is slight and plasma detachment is the primary regime in future fusion reactors, we mainly investigate the W impurity accumulation in the detached regime. Figures 3(c) and (d) illustrate that the W concentration in the core region of the SFD is significantly higher than that of the SD. In the following, we mainly investigate why the SFD has worse W impurity shielding capability than the SD, and the main factors which influence the W core accumulation. Accordingly, to investigate the effect of a single factor, such as magnetic configuration or W source position, on the behavior of W impurities, a consistent plasma background is employed, and the magnetic configuration or W source position is varied.

3.1   The core W accumulation in detached plasma

In this section, we focus on the difference in W core accumulation between the SFD and SD by analyzing target erosion, W ion flux, and other relevant factors, to determine why the SFD has worse W shielding capability compared to the SD.

SFD and SD cases with similar ${\varPhi }_{\mathrm{W},\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$, i.e. W impurity sources, are selected for comparison. It is clear that ${\varPhi }_{\mathrm{I}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ and ${\varPhi }_{\mathrm{O}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$ of both geometries are significantly different, as shown in table 1, which may influence the transport of W impurities. Therefore, we first investigate the erosion of the W target. Figure 4 displays the profiles of T_e, ${\varGamma }_{\mathrm{N}\mathrm{e}}$, and the corresponding W target erosion along targets of both configurations. In the SD, the T_e,sp of the IT and OT are 2.3 eV and 24.4 eV, respectively. Detachment predominantly takes place at the IT, resulting in the primary erosion of W at the OT. Meanwhile, in the SFD, the T_e,sp of the IT and OT are 16.6 eV and 2.17 eV, respectively. The OT achieves detachment and W erosion mainly occurs at the IT. Since the T_e along the target in these selected cases are relatively low, the incident energy of D⁺ is below the physical sputtering threshold of W (~ 209.6 eV). Therefore, W erosion is dominated by Ne ions, whose sputtering threshold of W is ~ 33.9 eV. The peak T_e and ${\varGamma }_{\mathrm{N}\mathrm{e}}$ of the OT in the SD are close to the strike point (SP). Thus, erosion mainly occurs in the vicinity of the SP. Meanwhile, in the SFD, the peak T_e of the IT locates in the middle of the common flux region (CFR), and there are two peaks of ${\varGamma }_{\mathrm{N}\mathrm{e}}$, locating at the SP and the far SOL, respectively. The difference in peak position results in less variation of W erosion along the target, with slightly higher erosion rates in the vicinity of the SP and far SOL. The disparities in erosion profiles between the two divertor configurations affect the distribution of W atoms, resulting in distinct W ionization distributions.

Figure  4.  The target profiles of ((a), (d)) T_e, ((b), (e)) ${\varGamma }_{\mathrm{N}\mathrm{e}}$, and ((c), (f)) W erosion flux density in the detached regime of the SD and SFD.

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Figure 5 illustrates the spatial distribution of the W ionization source (S_W0) in the divertor region of the SD and the SFD. The S_W0 in the OSD is higher than that in the ISD, whereas it is higher in the ISFD than the OSFD. Therefore, we focus on analysis of the HFS of the SFD and the LFS of the SD here. Due to low T_e, the S_W0 in the ISFD and OSD extend away from the target at the far SOL. The S_W0 at the IDE in the SFD is significantly higher than that at the ODE in the SD. Notably, the S_W0 is also high in the HFS above the IDE of the SFD. This phenomenon is attributed to the relatively short leg length and the small distance between the divertor entrance and the far SOL of the ISFD, which causes the W source near the divertor entrance. Most of the W impurities originate from the erosion of the IT. Because of the stronger W impurity source and the closer to the divertor entrance and separatrix, the W impurity finds it much easier to escape from the divertor of the SFD than from that of the SD, which increases the core W accumulation significantly.

Figure  5.  The spatial distributions of S_W0 in the physical grid of (a) the SD and (b) the SFD. The white dashed line indicates the divertor entrance.

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The differences in W impurity transport from the divertor to the core in the SFD and SD are analyzed using the Φ_W. The integration of W ion flux density over the area traversed by the ions within a physical grid is referred to as the Φ_W. The Φ_W on computational grids of the SD and SFD are shown in figure 6. As a simplified physical representation, W ions are transported along and across the magnetic field line. Then, the Φ_W can be divided into two distinct directions accordingly. The parallel flux can be obtained as ${\varPhi }_{\mathrm{W},\mathrm{p}\mathrm{o}\mathrm{l}}={\sum }_{i=1}^{Z}{n}_{i}{v}_{i}\mathrm{d}{A}_{\mathrm{p}\mathrm{o}\mathrm{l}}$, and the cross-field flux can be effectively approximated as a diffusive flux of the form ${\varPhi }_{\mathrm{W},\mathrm{r}\mathrm{a}\mathrm{d}}={D}_{\perp ,\mathrm{W}}\nabla {n}_{\mathrm{W}}\mathrm{d}{A}_{\mathrm{r}\mathrm{a}\mathrm{d}}$ [41]. The ${A}_{\mathrm{p}\mathrm{o}\mathrm{l}}$ and ${A}_{\mathrm{r}\mathrm{a}\mathrm{d}}$ are the areas which W ions have crossed. The direction of the impurity flow is expressed by the colored arrows shown in figure 6. W impurities can be transported to both the target and upstream in the OT of the SD and in the IT of the SFD at the position where the density-averaged parallel W velocity is zero. The flux below the magenta line flows to the target, while it points upstream in the position above the magenta line. In the SFD, the magenta line is closer to the divertor entrance, and ion flux flowing to the target is more pronounced than in the SD. The Φ_W patterns suggest that W ions entering the core in the SFD and SD primarily originate from the dominated eroded target. The radial ion flux along the separatrix is integrated to represent total W flux into the core (Φ_EC). The Φ_EC of the SD and SFD are 2.07×10¹⁷ s⁻¹ and 7.34×10¹⁷ s⁻¹, respectively. It can be seen that the Φ_EC of the SFD is significantly larger than that of the SD. In terms of transport, the ionization of W atoms in the SFD occurs closer to the divertor entrance, and the length from the ionization position to crossing the separatrix point in the parallel direction is longer in the SFD (~26.04 m), compared to that in the SD (~19.16 m). The total S_W0 is ~1.23×10²⁰ s⁻¹ in the SFD and ~6.24×10¹⁹ s⁻¹ in the SD. This indicates that, compared to the SD, the stronger S_W0 is an important factor contributing to a larger W influx entering the core of the SFD. Excluding the influence of the S_W0 strength on the W accumulation in the core, a simplified factor of the W ion influx to the core to the total ionization source of the divertor is employed. Specifically, the factors of the SFD and SD are ~ 0.00597 and ~ 0.0033, respectively. This difference may be caused by the ionization location and the total forces exerted on W ions.

Figure  6.  The Φ_W on the computational grid of (a) the SD and (b) the SFD, with the direction depicted using colored arrows. The magenta lines represent points where the density-averaged parallel W velocity is zero.

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To investigate the relation of W impurities in the core with the W ionization position, the initial ionization source of W ions entering the core is defined as ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$. Figures 7(a) and (c) show the ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ distribution, with the ISFD and OSD divided into three segments: near, middle, and far SOL, delineated by the blue and white solid lines. The majority of the W entering the core of both the SFD and SD originates from the middle and far SOL of the divertor. The ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ locates close to the target in the SD, and it covers the whole region from the target to the IDE in the SFD. The ionized W source above the IDE in the SFD also significantly contributes to W accumulation in the core. These regions cover nearly all the regions where the S_W0 are high in the OSD and ISFD; thus, a high W ionization source is essential for a strong ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$. The difference between the distribution of ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ and S_W0 indicates that ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ is linked to S_W0 as well as to the penetration capacity of W impurities into the core at various positions. To represent the penetration capacity, ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}/{S}_{\mathrm{W}0}$ is introduced. The simulation results indicate that W penetrations in the SFD and SD are significantly stronger at positions of the near and medium SOL away from the target, particularly approaching the separatrix, as shown in figures 7(b) and (d). Compared to the SD, the regions exhibiting pronounced penetration in the SFD are situated farther away from the target. The strong penetration capacity of W ions at different locations is related to the total forces applied to them.

Figure  7.  The distributions of ((a), (c)) ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ and ((b), (d)) ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}/{S}_{\mathrm{W}0}$ in the SD and SFD under detachment. The dashed line indicates the divertor entrance.

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Figures 8(a)–(d) show the total force (friction, thermal, electric and pressure gradient forces) exerted on W ions with density-weighted averaging in the SFD and SD, ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$=$\sum_Z^{ }n_{\mathrm{W}}^{Z+}\boldsymbol{F}_{\rm{T}\rm{O}\rm{T}}^{\mathbf{\boldsymbol{ }}Z+}/\sum_z^{ }n_{\mathrm{W}}^{Z+}$, where ${n}_{\mathrm{W}}^{Z+}$ and ${\boldsymbol{F}}_{\rm{T}\rm{O}\rm{T}}^{{Z}+}$ represent the density of the Zth W ion and the corresponding total force, respectively. We analyze the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ in the divertor region first. In the position away from the target of near and a portion of the middle SOL of the OSD, ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ points upstream, and the W ion closer to the separatrix can be transported to an upstream position closer to the core, which results in easier entry of the core via radial transport. Therefore, the closer to the separatrix, the stronger the W penetration rate. The ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ and W transport in the near SOL of the ISFD exhibit similarities to those in the near and a portion of the middle SOL of the SD. In the middle SOL, the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ near the divertor entrance has different directions. Accordingly, we define the region near the divertor entrance with ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ pointing towards upstream and the target in the middle SOL as region 1 and region 2, respectively. When we compare these regions with the S_W0 region (figures 8(c)–(f)), W atoms were ionized in regions 1 and 2 only in the SFD. The W ions ionized in region 1 are driven upstream by the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ there. Despite the direction of ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ towards the target in region 2, the W ions ionized there can be transported to region 1 via radial transport, resulting in potential leakage. Moreover, the overlap of S_W0 and ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ pointing to upstream regions promotes the core W accumulation of the SFD. Thus, the penetration capacity of W ions in the middle SOL of the ISFD is stronger than that of the OSD. Above the divertor entrances of the OSD and ISFD, ions are close to the core and can directly penetrate via radial transport; thus, W penetration is high there. Therefore, the significantly strong ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ in the middle and far SOL of the OSD is due to the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ pointing upstream and the high S_W0, respectively. Furthermore, the strong ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ in the far SOL of the ISFD is attributed to the high S_W0. Meanwhile, the strong ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ in the middle SOL of the ISFD is contributed by the high W ionization strength, and the ionization location near the IDE combined with the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ pointing upstream. Similarly, the strong W penetration capability and high S_W0 above the divertor entrance of the ISFD result in a strong ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$.

Figure  8.  In the detachment regime in the SD and SFD, the distributions of ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ exerted on the ions on ((a), (b)) the physical grid and ((c), (d)) the computational grid, and the regions of (e) pointing upstream as well as (f) of S_W0. The black and green solid lines divide the SOL into inner, middle and far SOL. The arrows in (a) represent the direction of the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ for both magnetic configurations. Regions 1 and 2 that are defined due to different directions of ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ in the middle SOL are labeled in (d).

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In summary, the higher concentration of W impurities in the core of the SFD than that of the SD in the detached regime can be attributed to several reasons: (1) W erosion predominantly occurs at the IT of the SFD. The shorter leg length of the ISFD, i.e. the distance between the X-point and the strike point of the IT in the poloidal plane, as well as the far SOL closer to the divertor entrance, leads to a stronger W source near the divertor entrance. Therefore, the S_W0 is much stronger, especially near the divertor entrance. (2) Due to the high S_W0 near the divertor entrance, W ions ionized in the region where ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ points to the target can escape via radial transport, resulting in potential leakage. Moreover, the region overlap of S_W0 and ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ pointing upstream promotes core W accumulation.

It should be noted that there are significant distinctions in the plasma parameters and target erosion between the SD and SFD in the detached cases. Consequently, the W source and W transport mechanisms are different. We obtained differences in W impurity accumulation in the core due to the combined effects in the SD and SFD. However, the specific effect of a single individual factor, such as W source distribution and W impurity transport, on W accumulation in the core is not yet fully understood.

3.2   The effect of magnetic configuration and W source position on W accumulation with the same plasma background

The analysis in the above section explains the primary factors contributing to higher concentrations of W impurities in the core of the SFD than in the SD. Nevertheless, differences in the divertor magnetic configuration, plasma parameters, and W source in the two cases make it challenging to separate the influence of each parameter. Thus, the same plasma background is required. In DIVIMP, since the SD and SFD grids have identical mesh resolution, we can apply the plasma solution of one magnetic configuration to both grids. In this section, we aim to ascertain the impact of individual parameters on the W accumulation in the core. Firstly, the plasma background and W source are fixed, and the detached plasma background of the SFD (with a puffing rate of 1×10¹⁹ Ne atoms/s) and the corresponding inner target erosion are adopted. By changing the magnetic field configuration (SD and SFD), the effect of the divertor magnetic configuration on W transport is studied. Since the plasma of the SD is not “real” by this treatment, the DIVIMP results are not realistic. However, it can show the direct influence of a magnetic configuration on temperature gradient force as well as W transport clearly. Subsequently, we focus on the SFD to investigate the impact of the W source position on W core accumulation. The W source along the IT is divided into near, middle, and far segments (the same as in figure 7(c)) and separate simulations with each W source considered are carried out to establish correlations between the W source position and ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$, as well as W accumulation in the core.

3.2.1   Impact of magnetic configuration on W accumulation

Figure 9 illustrates the n_W distributions of the SD and SFD, utilizing identical detached plasma backgrounds and W sources from the SFD. The n_W at the CEI of the OMP in the SFD (1.46×10¹⁶ m⁻³) is significantly higher than that in the SD (9.65×10¹⁴ m⁻³). Similarities in the distribution of the n_W are observed in the two divertor configurations, with the highest n_W located in the far SOL of the inner divertor. The corresponding S_W0 distributions are shown in figures 10(a) and (c). In comparison to the SFD, the S_W0 at the IDE is weaker in the SD, despite both configurations displaying strong ionization sources in the middle and far SOL. This difference is primarily attributed to the longer leg length and distance from the far SOL to the divertor entrance in the SD than those of the SFD. The n_W in the core is also influenced by the penetration capacity of W, which is determined by the total forces on W ions. Figures 10(b) and (d) show the ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ in the two divertor configurations. In the middle SOL of the SD, the strong S_W0 position is different from the region where ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ points upstream, leading to low n_W in the core.

Figure  9.  With identical detached plasma background and IT W erosion, n_W distributions in (a) the SD and (b) the SFD magnetic configurations.

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Figure  10.  With identical detached plasma backgrounds and IT W erosion from the SFD, ((a), (c)) S_W0 and ((b), (d)) ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ distributions in SD and SFD configurations. The positive direction of ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ is from the IT to the OT. The divertor entrances are indicated by dashed white lines.

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It is imperative to ascertain what induces the difference in ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ between the SD and SFD, to understand the underlying physical mechanisms. With identical plasma backgrounds, in the middle SOL of the SD and SFD, ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ points upstream and to the target, respectively. The primary reason for this lies in the fact that ion temperature gradient forces [42] are affected by the connection length. Figure 11 presents poloidal profiles of ion temperature gradient forces in the near (the magnetic flux tube index (ir) is 21), middle (ir = 28), and far SOL (ir = 34) of two divertor configurations. Compared to the SFD, the SD has a shorter connection length in the near SOL, resulting in larger electron and ion temperature gradients which, in turn, lead to larger electron and ion temperature gradient forces. Conversely, the SD has longer connection lengths in the middle and far SOL, corresponding to smaller electron and ion temperature gradient forces. Compared to the SFD, although the connection length has a small effect on the temperature gradient force in the middle SOL of the SD, it does alter the direction of ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$, thus weakening W penetration and accumulation in the core. Therefore, by fixing the plasma background and W source while changing only the magnetic configuration (SD versus SFD), it is found that the connection length can influence the temperature gradient force, which leads to ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ in the middle SOL of the SFD pointing upstream. As a result, the W penetration capability of the SFD is enhanced. The strong W penetration capability in conjunction with the high n_W at the divertor entrance of the SFD causes a higher W concentration in the core of the SFD than that of the SD.

Figure  11.  With identical detached plasma backgrounds and IT W erosion from the SFD, the ion temperature gradient force (FIG) profiles in the near (ir = 21), middle (ir = 28), and far SOL (ir = 34) in the SD (solid line) and SFD (dashed line) grid. The referenced ir values of different regions are shown in figure 10(c).

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3.2.2   Impact of W source position on W accumulation

The above subsection reveals that the S_W0 of the ISFD is higher in the middle and far SOL. Additionally, ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ is predominantly distributed in the middle and far SOL of the inner divertor and above the IDE. Nevertheless, the relationship between ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ and the position of the W source remains unclear. The W source located at the IT of the SFD is divided into three parts: near, middle, and far segments. In the simulation, W impurities generated by one segment erosion are tracked, while ignoring W sources from the other two positions. By using this approximation, the contribution of W sources in various positions to ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ and core W accumulation can be studied. Table 2 displays the proportion of each positional W source (${\varPhi }_{\mathrm{W},i}^{\mathrm{e}\mathrm{r}\mathrm{o}}$, where i represents the near, middle, and far segment) to the total erosion sources (${\varPhi }_{\mathrm{W},\mathrm{I}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$) in the IT and their contributions to the n_W in the core. The contribution of the W impurity source at location i to W impurity accumulation in the core is represented by ${n}_{\mathrm{W},i}^{\mathrm{C}\mathrm{E}\mathrm{I},\mathrm{O}\mathrm{M}\mathrm{P}}/{n}_{\mathrm{W},\mathrm{I}\mathrm{T}}^{\mathrm{C}\mathrm{E}\mathrm{I},\mathrm{O}\mathrm{M}\mathrm{P}}$, where ${n}_{\mathrm{W},i}^{\mathrm{C}\mathrm{E}\mathrm{I},\mathrm{O}\mathrm{M}\mathrm{P}}$ refers to W density at the CEI of the OMP corresponding to the W source location i, and ${n}_{\mathrm{W},\mathrm{I}\mathrm{T}}^{\mathrm{C}\mathrm{E}\mathrm{I},\mathrm{O}\mathrm{M}\mathrm{P}}$ represents W density corresponding to entire IT erosion. The W source is the weakest in the near segment (26.3%), the strongest in the middle segment, and the contributions of the middle and far segments to the total IT erosion are similar (37.35% and 35.17%), as shown in table 2. The W source in the near segment contributes only 2% to the W impurity content in the core, whereas the W source in the far segment makes the largest contribution (71.4%).

Table  2.  The proportions of the near, middle, and far segment W sources to the total W source of the IT in the SFD, as well as their corresponding contributions to the W density at the CEI of the OMP.

i	Near segment	Middle segment	Far segment
${\varPhi}_{\mathrm{W},i}^{\mathrm{e}\mathrm{r}\mathrm{o}}/{\varPhi}_{\mathrm{W},\mathrm{I}\mathrm{T}}^{\mathrm{e}\mathrm{r}\mathrm{o}}$	26.3%	37.35%	35.17%
${n}_{\mathrm{W},i}^{\mathrm{C}\mathrm{E}\mathrm{I},\mathrm{O}\mathrm{M}\mathrm{P}}/{n}_{\mathrm{W},\mathrm{I}\mathrm{T}}^{\mathrm{C}\mathrm{E}\mathrm{I},\mathrm{O}\mathrm{M}\mathrm{P}}$	2%	26.3%	71.4%

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Since W sources from the middle and far segments make dominant contributions to W core accumulation, we focus on the analysis of the W source from these two positions. Figures 12(a) and (b) show the ionization distributions of W atoms and the W stagnation point positions in the ISFD corresponding to the W source of the middle and far segments, respectively. By comparing the two W source cases, it is found that: (1) in the middle SOL erosion case, the W atom ionization source is concentrated in the near target region. Meanwhile, in the far SOL erosion case, the W atom ionization source is further away from the target, and it may even extend to outside the inner divertor region (with the divertor entrance as the boundary). This difference is primarily due to a higher T_e in the middle SOL compared to that in the far SOL at the target, as shown in figure 4(d). (2) In the far SOL erosion case, the stagnation point of W ions is closer to upstream. This is primarily attributed to the small FIG, as exhibited in figure 11. Consequently, more W atoms are ionized above the stagnation point, reducing the W impurity shielding properties of the divertor. The ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ distributions corresponding to the erosion of the middle and far SOL, respectively, are depicted in figures 10(c) and (d). It can be seen that the W source from the middle target erosion significantly influences ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ distribution in the middle SOL, while the W source from the far segment contributes to ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ distribution in the far SOL and above the IDE. Therefore, in the W source from the far segment, more W is ionized above the stagnation point (near or above the IDE). This leads to strong ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$ in the far SOL and above the IDE, promoting W impurity accumulation in the core. Hence, the control of W core accumulation requires the reduction of W target erosion at the far SOL target.

Figure  12.  ((a), (b)) The ionization distributions of W atoms and the stagnation point positions. ((c), (d)) Distributions of ${S}_{\mathrm{W}0}^{\mathrm{E}\mathrm{C}}$, with the W source located in the middle and far segments of the ISFD, respectively. The W stagnation point, X-point and IDE are labeled by the cyan solid line, red star, and dashed line, respectively.

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4.   Conclusion

In this work, DIVIMP combined with SOLPS-ITER is employed to simulate and analyze the behavior of W impurities in the SFD and SD with Ne impurities injected from the OSP on the HL-3 device, which has the advantage of a flexible divertor configuration. It is found that the erosion of the W target depends mainly on the T_e at the strike point, i.e. with a similar T_e range, the total W eroded flux is of the same order of magnitude, regardless of significant differences in the Ne puffing rate between the two configurations. In the cases with similar W erosion rates, under either attached or detached regimes, the W impurity density in the core of the SFD is notably higher than that of the SD. This indicates that the SFD has a more severe issue of W impurity accumulation in the core. The physical reasons for this can be explained as follows:

(1) The SFD is dominated by erosion of the IT, while the OT erosion is more severe in the SD. In the SFD, the inner divertor leg is short and the far SOL of the target is close to the divertor entrance; therefore, the W source is close to the divertor entrance, leading to poor W atom shielding of the divertor. Therefore, the S_W0 is much stronger, especially near the divertor entrance.

(2) In the SFD, the ionization rates are high near and above the divertor entrance. Due to the high S_W0 near the divertor entrance, W ions ionized in the region where ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ points to the target can escape via radial transport, thereby enhancing the leakage. The region overlap of high S_W0 and ${\boldsymbol{F}}_{\rm{W},\;\rm{T}\rm{O}\rm{T}}$ pointing upstream near and above the divertor entrance enhances W penetration and leads to more W entering the core.

Based on the above analysis, we focus on the erosion of the inner target of the SFD and further investigate the effect of the W source location on impurity accumulation in the core. It is found that W impurities in the core are primarily contributed by the erosion of the far SOL target. This is mainly due to the increased ionization of W impurities above the W stagnation point and it is close to the divertor entrance. Consequently, it facilitates pronounced accumulation of W impurities in the core. Therefore, mitigation of W erosion at the far SOL of the IT is imperative to effectively suppress the W density in the core of the SFD.

The simulations of this work neglect the effects of self-sputtering, prompt redeposition and drifts. Self-sputtering and prompt redeposition not only influence the W source strength, but also affect its distribution along the target. Moreover, the profile variation is notably prominent near the separatrix of both magnetic configurations. However, the variation of n_W at the OMP caused by self-sputtering and prompt redeposition is small in both magnetic configurations. Therefore, self-sputtering and prompt redeposition have a minimal impact on the conclusions drawn in this work. Since the reflection of W ions at the target is small compared to the total particles released, the reflection was not considered in DIVIMP. Drift can significantly affect impurity sources and transport. The effect of E×B drift on W ions is more pronounced than other drift components [43]. In forward B_t (ion $\boldsymbol{B}\times \nabla B$ drift points to X-point), the E×B drift drives W ions from the outer divertor to the inner divertor through a private flux region, and in reversed B_t, the influence on W ions of E×B drift is vice versa. Note that the direction of drift is the same for the D⁺ and impurity ions. Thus, the E×B drift also affects the location of impurity stagnation points [7, 21], the W source, leakage from the divertor, and transport into the confined plasma [41]. Moreover, the E×B drift affects the poloidal asymmetry of W distribution [21]. It is noteworthy to observe that the simulation domain only includes the core edge region and we take the W concentration at the CEI to represent the core concentration, which should be lower than that in the real core. In further work, we will couple the DIVIMP simulation to STRAHL simulation [44] to obtain the real W core concentration. Furthermore, only a single SFD configuration is selected in this work, while other SFD configurations may also influence plasma parameters and W impurity transport. All these aspects will be addressed in future work.