Effects of a shallow SAS divertor on detachment in KSTAR

For long pulse operation of fusion reactors, it is important to reduce sputter-erosion and power loading of the divertor target by means of plasma detachment. It has been reported that the small-angle-slot (SAS) divertor employed by the DIII-D tokamak can initiate detachment at a relatively low upstream plasma density as it can effectively dissipate heat by concentrating neutrals near the target. Motivated by these findings in DIII-D, we investigated the effects of a SAS-like divertor in KSTAR using SOLPS-ITER simulations without drifts. One remarkable feature revealed by our simulation study is that even a very shallow SAS can lead to a considerably lower heat load on the divertor target than the original flat, open divertor of KSTAR. Deuterium neutrals are concentrated along the divertor separatrix line in the shallow SAS, while deuterium density in the open divertor peaks in the far-scrape-off layer. Furthermore it was found that the neutral density and temperature-drop induced by SAS are both fairly incentive to the depth of the slot. The highest heat dissipation was obtained for a SAS depth of 10.3 cm.


Introduction
Advanced tokamaks and future reactors require long-pulse operation, which inevitably entails new divertor designs that can significantly reduce the heat flux and temperature at the divertor targets. To suppress the erosion and sputtering of the plasma facing components (PFCs) to acceptable levels, the heat flux and temperature should be maintained under 10 MW m −2 and 5 eV [1,2]. * Author to whom any correspondence should be addressed.
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Damage to the PFCs can be reduced effectively by heat dissipation through detachment at the divertor target [3,4]. Usually detachment is obtained by increasing the upstream density in the scrape-off layer (SOL). However in this case, the core density would also be high, possibly leading to deterioration of confinement of the core plasma. Hence, research on how to trigger detachment at low upstream SOL density, maintaining high quality confinement of the core plasma, is an important issue. From extensive studies of critical factors which determine the upstream density required for the onset of the detachment [5][6][7][8][9][10], it has been found that detachment can be affected by the shape of the divertor, PFC material and the magnetic configuration [11][12][13][14][15][16].
One direction of optimizing the divertor structures is controlling the path of neutral particles by shaping the divertor. Specifically, divertor closure restricts the paths of neutral particles, confining them near the divertor target and leading to heat dissipation through interactions between the neutral particles and the plasma. The effect of divertor target closure has been validated by a number of numerical simulations as well as experiments in several fusion devices [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The general conclusion of these studies is that the higher the closure is, the higher the neutral particle density at the divertor target is, and thus detachment can be obtained at lower upstream densities. The high correlation between neutral particle density and electron temperature at divertor targets has been reported in several studies (see [22] and references in [23]).
Based on these results and additional studies [21][22][23][24], DIII-D installed a small angle slot (SAS) divertor [25,26], to obtain more efficient heat dissipation than with conventional divertor shapes. The most critical feature of the SAS divertor is its small angle at the outboard side near the end of the slot. The SAS takes advantage of both closed slot structure and slanted ('vertical') target effects, significantly increasing the neutral density in the vicinity of the strike point even at a lower upstream electron density [24][25][26].
Generally volumetric loss processes which are important in reducing the heat flux to the target (i.e., via detachment), are directly affected by the divertor structure and wall materials as shown in the case of SAS in DIII-D. In this paper, through numerical simulations using the SOLPS-ITER package [27][28][29][30][31] (but without particle drifts), we examined SASlike divertors in the KSTAR tokamak geometry to identify the factors that dominate the volumetric heat loss near the target (see reference [32,33] for general characteristics of KSTAR). Due to the limited space around the divertor in KSTAR, the depth of the slot used in our study is much smaller than the original SAS in DIII-D (hence, named SAS-like). Even with a shallow SAS divertor, we obtained a considerable drop in heat flux and temperature near the target. From comparison of the original divertor of KSTAR and the SAS, we have verified that the high neutral deuterium density near the divertor separatrix line plays a critical role in detachment. Interestingly, we discovered that, as long as the slot angles are preserved, the detachment is weakly sensitive to the length of the divertor leg (slot depth). This paper is organized as follows. In section 2, we describe the setup of the simulation parameters for the SOLPS-ITER code. In section 3, simulation results and analysis are given for comparison between the original and SAS-like divertors in KSTAR geometry. We discuss the effect of the slot depth with other factors held unchanged. In section 4, discussion and conclusions are given.

Settings of simulations
The numerical simulations have been performed with the SOLPS-ITER code package. In this code, the multi-fluid plasma transport code B2.5 is coupled to the EIRENE Monte Carlo neutral transport code [30,31]. B2.5 provides the plasma background on which the neutral trajectories are computed by EIRENE and then EIRENE gives B2.5 the source terms for momentum and energy of neutral particles to be used in the fluid equations of B2.5. Particle drifts are not included in our simulations. The species included in the simulations are plasma species (D, D 2 , D + and D + 2 ) and all ionization states of sputtered carbon impurity (C, C + , C 2+ , C 3+ , C 4+ , C 5+ , C 6+ ). Electrons, D + and all the carbon ions are treated by B2.5, while D + 2 and all the neutral species by EIRENE. We only activate physical sputtering in the simulations, based on the Roth-Bohdansky formula [34], to minimize the impact of carbon radiation on the onset of detachment. This led to small carbon concentration under high density, low temperature divertor conditions. In this way we could study the role of neutral deuterium in detachment onset in the SAS-like divertors. The total recycling rate for D is fixed to R = 1. The density of the main ion species (D + ) at the interface of core and edge is varied as a boundary condition of the simulation to obtain the scan of outer midplane density (n e,omp ). For the sheath boundary condition near the target plate, the standard Bohm sheath criterion is used, i.e. v = c s . The EIRENE grids are set to be smaller near the top and bottom targets for better resolution of the neutral trajectories near the targets (figure 1).
In the modelling, a computational mesh is constructed based on a lower single null magnetic field configuration of KSTAR as shown in figure 1. The width of B2.5 meshes is 15 mm at the outer midplane, which is wide enough to cover the heat decay length 4-5 mm obtained from simulations. This decay length is comparable to the value typically measured in KSTAR experiments (references [32,33]). We used the same mesh width for SAS and the original open divertors. The equilibrium magnetic field is provided by the equilibrium fitting code [35]. The lower central divertor of KSTAR is used for the comparison between the original shape and the SAS-like shape. The ion B × ∇B drift is directed toward the target from the X-point. P SOL = 1.2MW (input power at the inner core boundary of the grid), I p = 0.6MA and B t = 2.0 T were used. The list of reactions used in the EIRENE simulations can be found in table 1 in reference [36]. In common with most studies using plasma boundary codes, the cross-field transport of particles and heat is assumed to be diffusive with anomalous diffusion coefficients; D ⊥ = 1.0 m 2 s −1 for particles and χ e (χ i ) = 1.0 m 2 s −1 for heat. From these parameters, profiles of L-mode plasmas were formed. Simulations with smaller radial transport coefficients (e.g. χ e (χ i ) = 0.4 m 2 s −1 , D ⊥ = 0.12 m 2 s −1 ) and shorter radial decay length boundary conditions did not alter our results qualitatively. Pumping was not included in order to focus on the effect of the divertor structure. Neutral-neutral collisions were not included either.

Effect of the SAS-like divertor
Here we compare the original shape and the SAS-like shape of the divertor in KSTAR. The configurations of those two cases have the same magnetic equilibrium and the same SOLPS-ITER input settings except for the shallow slot at the outer divertor in the SAS-like case (see figure 1). The electron temperature (T e ), the parallel ion saturation current density (J para ) and the parallel heat flux density (Q para ) were calculated scanning the electron density at the outer midplane to compare the difference in the onset of detachment (figure 2). Note that Q para , the heat flux parallel to the total magnetic field (toroidal and poloidal) is obtained by multiplying the geometrical factor 1 cos θ B total B pol to Q perp , which is the energy flux of charged particles normal to the target. Hence Q para represents the volumetric heat along the field line toward the target. Compared to the SAS in the DIII-D divertor [25,26], the slot is very shallow. There is not much difference in the inner target (upper row of figure 2) as there is no modification of the inner target. However, from the data of the outer target (lower row in figure 2), it is clear that the SAS-like divertor has advantages over the original divertor. In SAS, the divertor plasma rapidly enters detachment with significant drop in T e (<3 eV) and parallel heat flux density (Q para ), followed by the roll-over of the particle flux density (J para /e) at a considerably lower density (by about 35%), compared to the original divertor. Note that the roll-over of the particle flux is usually used as an indicator of divertor detachment in experiments [23], due to a lack of accurate measurements of T e , e.g., by Langmuir probes, in particular, in low temperature plasmas near the detachment. We also have compared the energy flux transported by neutrals toward targets to the energy flux of plasma particles; for open divertors, it is 5 to 10% of Q perp and 15 to 20% for the SAS-like divertors. The ratio becomes similar for open and SAS-like divertors after the detachment onset.
According to existing research, the lowered onset density of detachment is caused by the increase of volumetric power loss due to increased interaction with neutral particles [10,22,23,36]. Figure 3 shows the molecular deuterium density along the target surface. In the SAS-like divertor (blue line with filled markers), the deuterium density is higher over most of the target compared to the original divertor (red line with filled markers). The temperature (empty markers) is strongly correlated with the deuterium density both in the original and the SAS-like divertors, with a higher neutral density resulting into a lower electron temperature. The same correlation between the deuterium density on the target and the temperature was found in previous studies [10,22,23,36]. Figure 4 represents the neutral deuterium distribution for upstream densities corresponding to figure 3. It is observed that the neutrals are highly concentrated across the entire target plate for the SAS-like cases. SAS leverages the closed slot structure to enhance neutral build-up inside the slot, and the V-shaped slant target plates near the strike point to direct recycling neutrals toward the strike point from both the common SOL and the private flux region, thus improving plasma cooling across the target plate. In contrast, for the original KSTAR divertor configuration, recycling neutrals are directed away from the strike point with strong leakage of neutrals from the SOL toward the confined  plasma, thus reducing power and momentum dissipation in the divertor. Figure 5 shows the distribution of the radiation for the same parameters as in figures 3 and 4. The power loss by radiation occurs over a larger volume in the SAS-like divertor (figures 5(c) and (d)) than in the original divertor (figures 5(a) and (b)). Especially in the case of higher n e,omp (figure 5(d)), the SAS-like divertor's temperature is lower than 1 eV at the target surface and the radiation loss zone is completely detached from the target, leaving a narrow non-radiative gap. This is because for electron temperatures below 1 eV, rates of D 2 dissociation and subsequent ionization or excitation are very low [23,36], resulting in reduced radiation [23].
One remarkable feature of the data in figures 3-5 is that the SAS-like divertor has high neutral deuterium density at the strike point, while the original open divertor has high neutral deuterium density in the far-SOL. The high concentration of neutrals near the separatrix is considered to be caused by the shape of the SAS target rather than just the closure; while closure can increase the total amount of neutrals in the entire divertor region, the specific shape of the targets can affect the path of the recycled neutrals and therefore, their spatial distribution in the divertor volume.
The density distribution of neutral deuterium differs not only in the vicinity of the target, but also throughout the entire divertor volume. Figure 6 shows the cumulative deuterium density and temperature along three magnetic flux tubes from the X-point to the target. Note that, since the temperature drop due to volumetric power loss is a cumulative effect, the integrated n D 2 can be more informative than n D 2 itself in understanding the relation between D 2 and T e . As can be seen, there is a correlation between the density of deuterium and the temperature change; along the line near to the separatrix (red line in figure 6(a)), the neutral deuterium density of the SAS is larger than the original divertor and hence the temperature is significantly lower ( figure 6(b)). In contrast, along the mid-apart (green in figure 6(a)) or far-apart (blue in figure 6(a)) from the separatrix, there is no difference in the neutral deuterium densities between the SAS and the original divertor, hence the temperature is comparable (figures 6(c) and (d)). Figure 6 shows that the beneficial effect of SAS, i.e. the higher neutral density resulting in lower temperature, extends from the X-point to the target. Furthermore, such an effect of SAS is limited to the near-SOL where the heat and particle fluxes are concentrated. Hence the result also implies that potentially the SAS effects could be more evident in scenarios with narrow SOL width (i.e. at high plasma current and/or in H-mode) than for the broad SOL (i.e. at low current and/or in L-mode).

Effect of the slot depth
A set of simulations were performed changing the depth of the slot, while maintaining the angle between the slot facets and the magnetic separatrix (∼ 43 • ). Figure 7(a) shows three different slot depths used in the simulations. The parallel current ( figure 7(b)) on the outer target increases as the slot depth (hence, the leg length) decreases, but interestingly, the onset of roll-over (hence, detachment) occurs at a similar upstream density in all three SAS-like cases. Also the temperature and the heat flux are not affected by the depth of the slot (figures 7(c) and (e)) and remain considerably lower than in the original divertor. As can be seen in figure 7(d), the neutral deuterium density is almost the same for all the slot depths (and significantly higher than for the open divertor). This simulation data strongly suggests that the slot angles are more important than the slot depth for the onset of detachment. This finding implies an advantage of SAS from the cost point of view: an easier access to divertor detachment might be obtained just by a minimal change of the divertor target shape, with no need of a deep slot structure. Note that the depth of the shortest slot in figure 7 is about 5.6 cm, which increases the leg length (i.e. distance from the X-point to the target) just by 10% from the original divertor.

Conclusion
In summary, using SOLPS-ITER simulations without particle drifts, we studied the effects of a SAS-like divertor in KSTAR. We observed that with the SAS-like divertor, detachment is accessed at lower plasma density at the outer midplane by 35% in agreement with a similar work at DIII-D [37]. The concentration of deuterium neutrals around the separatrix line was higher in the SAS-like divertor than in the open divertor, while there is little difference far from the separatrix. In addition, we found that detachment is very weakly sensitive to the depth of the slot. These observations suggest that the target shape and angle to the field line at the strike point are crucial elements to achieve high neutral density around the separatrix line. This result is compatible with the conclusion in references [38,39], where the optimal divertor structure is obtained from an optimization algorithm.
In addition to divertor geometry, particle drifts can also affect the onset of divertor detachment [40][41][42][43][44][45] and will be, therefore, included in future modelling. Another aspect that should be investigated is the sensitivity of the detachment in SAS to a small change in the outer strike point location [37] and the effects of SAS on core behavior [45,46] in KSTAR. In addition, the change of heat flux transported by neutral particles and radiations toward the target for different divertors should be analyzed. These issues are under investigation as a next step of our research.