Drifts effect on the divertor W leakage mechanisms under different dissipative divertor conditions of EAST

A new drift module has been developed in the DIVIMP code, enabling two-dimensional simulations of tungsten (W) transport in the edge plasma with full drifts included. By using the SOLPS-DIVIMP code package, the impact of drifts on W transport and screening has been investigated for various levels of dissipative divertor conditions and different divertor geometric configurations in EAST. Simulation results reveal that E⃗×B⃗ drifts can enhance the W leakage by more than one order of magnitude. Under the favorable Bt direction, W erosion mainly occurs on the outer divertor target, making the W leakage from the outer divertor region the dominator. A leakage path from the near-scrape-off layer (SOL) region is revealed by the modeling results. In the leakage path, both the ion temperature gradient force and the reversed poloidal E⃗×B⃗ drift are pointing upstream. With the radial E⃗×B⃗ drift pushing W ions from the well-screened far-SOL region to the near-SOL region, the leakage from the near-SOL region becomes significant. As the divertor condition varies from the low-recycling regime to the deep detachment regime, the decrease of the ion temperature gradient velocity and poloidal E⃗×B⃗ drift velocity narrows the width of the near-SOL leakage tunnel and thus enhances W screening. While under the unfavorable Bt , W erosion and leakage from the inner divertor target matters, the leakage mechanism especially the leakage path from the near-SOL region is similar as the favorable Bt cases. Furthermore, the effect of different divertor geometries on the W screening has been investigated. The configuration with the outer strike point (OSP) on the horizontal divertor plate is proved to narrow the near-SOL leakage tunnel, and thus the unreversed poloidal E⃗×B⃗ drift pointing to the divertor target dominates and helps to enhance the divertor W screening. For the same D 2 puffing rate, the W leakage ability of cases with the OSP on the horizontal target can be more than 10 times weaker than the cases with the OSP on the vertical target, especially when the divertor is detached.

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Introduction
Tungsten (W) is widely used as the plasma-facing material (PFM) of current tokamaks [1][2][3][4] and is foreseen to be the PFM of ITER [5,6] and DEMO [7,8].However, due to the high radiation losses of W impurities, the W concentration in the core plasma is strictly limited [9].Therefore, it is important to minimize W erosion and understand the key physics that governs W leakage from the divertor to the core plasma.Divertor detachment is proposed as the most promising solution to reduce the divertor W erosion, but the W transport process in the boundary plasma for different dissipative divertor conditions is not well understood.
Previous studies prove that the parallel impurity transport in the edge plasma is dominated by the friction force and the temperature gradient force [10][11][12][13].Simulation results for JT-60U [14], EAST [15] and ITER [16] demonstrate that the divertor W screening is closely related to the edge plasma condition.Compared to the high-recycling regime, the detachment regime can decrease the parallel ion temperature gradient force on impurities and thus enhance the divertor W screening.
Recent experiments and simulations have shown that the ⃗ E × ⃗ B drifts have a significant influence on the particle flow of the edge plasma, which is crucial for divertor in-out asymmetry [17][18][19][20] and detachment bifurcations [21][22][23].Drifts also have a significant impact on the impurity transport.In the favorable B t direction (favorable for H-mode access, ion B×∇B drift towards the X-point), the ⃗ E × ⃗ B drifts facilitate the impurities transport from the outer divertor to the inner divertor, resulting in a higher radiative impurity concentration in the inner divertor region [24][25][26][27].W transport simulations for the DIII-D metal rings campaign have revealed that the impurity drift velocities can shift the impurity stagnation point towards the target, leading to significant W impurities leakage [28].The full-orbit edge impurity transport code IMPGYRO, which incorporates the drifts effect, has been used to investigate the transport of high-Z W impurities. Previous IMPGYRO simulation results indicate that the grad-B drift can play an important role in facilitating the inward penetration of W into the core region from the crown region of the scrape-off layer (SOL), where the parallel impurity velocity is relatively low [14,16].Additionally, it is found that the ⃗ E × ⃗ B drift velocity which can be much larger than the main ion velocity can dominate the W transport in the outer SOL region [29].
EAST is a superconducting tokamak with a full-metal wall.The first-wall material is molybdenum, and the PFMs of upper and lower divertors are W.As shown in figure 1, the lower W divertor consists of horizontal and vertical target plates.Based on EAST geometry, the Monte Carlo impurity transport code DIVIMP [11] with a newly developed drift module is employed to simulate W transport under different divertor operation conditions.The DIVIMP code can trace the leakage pathway of W impurities, and by comparing simulation results with and without the drifts effect included, the impact of drifts on W transport and leakage mechanisms will be discussed in this paper.
The rest of this paper is organized as follows.The SOLPS-DIVIMP simulation setup is introduced in section 2. The twodimensional simulation results of W impurity transport under different divertor conditions are presented in section 3, followed by the summary in section 4.

Simulation setup
For the EAST lower-single null (LSN) configuration, a series of background plasma is generated by SOLPS-ITER with full (E × B and diamagnetic) drifts included under both favorable B t and unfavorable B t directions.The drift viscosity terms and the ion-neutral current contribution are included in the simulation.The background plasma distribution with different divertor condition is simulated with D 2 puffing from the outer midplane (OMP) and Ne seeding from the outer divertor region, as shown in figure 1(a).For all these cases, the Ne seeding rate is set to be 1.0 × 10 18 s −1 , and a scan of the D 2 fueling rate from 2.0 × 10 20 s −1 to 5.0 × 10 21 s −1 is carried out to achieve different dissipative divertor conditions.To evaluate the divertor geometry effects, the magnetic topology is also changed with the outer strike point (OSP) located at the horizontal target (H cases), the corner (C cases), and the vertical target (V cases), as shown in figure 1(b).The total power across the core-edge interface is fixed to be P SOL = 2 MW.The radial particle transport coefficient and the electron/ion heat conductivity coefficient are set to be D ⊥ = 0.3 m 2 s −1 , and χ e = χ i = 1.0 m 2 s −1 respectively.For all simulated species, the surface albedo of the cryopumps is 0.9 (90% recycling), while the albedo for other parts of the first wall surface is 1.0.
In the SOLPS-ITER simulation, the two-dimensional Braginskii equations are solved on the poloidal and radial grid coordinates [30] of figure 1(a).In the two-dimensional simulation, the x-coordinate is along the poloidal direction, and the y-coordinate varies perpendicular to flux surfaces.By solving the two components of the Braginskii momentum conservation equation [30] for ions perpendicular to a magnetic surface, the velocities V x and V y of the ions can be evaluated by: The EAST first wall and 98 × 38 computational meshes for the simulation.The divertor region, main SOL region, core region, and private flux region (PFR) are marked in blue, green, magenta, and cyan respectively.The connecting line between the divertor region and the main SOL region is defined as the divertor entrance.(b) Three different magnetic configurations with OSP located at the horizontal target (blue line), corner (red line), and vertical target (green line) are used in this work.
Here the subscript '⊥' means the direction perpendicular both to magnetic field B and the y-axis.z is the toroidal direction.The poloidal velocity is thus a sum The metric coefficients h x and h y are cell lengths in poloidal and radial directions.T e and T i are electron and ion temperature.n and p = n(T e + T i ) are ion density and thermal pressure.D and D AN are the classical and anomalous diffusion coefficient.The anomalous pinch velocity v AN is an input parameter for SOLPS-ITER, representing the radial and poloidal motions of the plasma that are not included in the model.The non-ambipolar terms in equations ( 1) and ( 2) are caused by inertial, viscous and ion-neutral friction forces, which are contributed by currents: The ⃗ E × ⃗ B drift velocity v E×B and effective diamagnetic drift velocity v Dia are calculated by: where φ represents the electric potential.T and Z are the temperature and charge state of ions.The ⃗ E × ⃗ B drift velocities of all ions are the same, whereas the diamagnetic drift velocity is inversely proportional to the charge state of ions.
A new drift module is developed in DIVIMP code to evaluate drift effects on W transport.The ⃗ E × ⃗ B and diamagnetic drifts effect is included by importing poloidal and radial drift velocities from the SOLPS-ITER results, ensuring consistency of the drifts effect on the main ions and W impurities. Then DIVIMP is dedicatedly used to simulate the W source and edge transport based on background plasma generated by SOLPS-ITER simulations.According to the particle and energy flux impinging to the divertor target, W erosion distribution is calculated by using the empirical formula proposed by Eckstein [31].Then W particles are launched with a spatial distribution corresponding to the W erosion rates, and the processes of ionization, recombination, and transport of W particles are tracked by the DIVIMP code.By turning on and off the drifts in the DIVIMP simulation, the effect of drifts on W transport can be distinguished.

W transport under the favorable B t direction
To investigate the impact of drifts on W transport for various dissipative divertor conditions, a series of LSN background plasma is generated by SOLPS-ITER with B t = −2.4T (favorable B t direction).In these cases, the OSP is located at the corner (C cases), and by adjusting the D 2 injection rate, the divertor operation condition changes from the low-recycling regime to the deep detachment regime.Figure 2 shows n e and T e profiles along inner and outer divertor targets.The inner and outer targets are detached with the D 2 injection rate higher than 1 × 10 21 s -1 and 3 × 10 21 s -1 respectively.Under the favorable B t , ⃗ E × ⃗ B drifts drive particles from the lower outer divertor region to the lower inner divertor region.Therefore, the inner divertor detaches at a lower D 2 injection rate.
The blue line in figure 3 represents the total W erosion rate integrated over the divertor targets for all cases.From the low-recycling regime to the deep detachment regime, the total W erosion rate is effectively suppressed by more than two orders of magnitudes.Due to the divertor in-out asymmetry, the inner and outer divertor targets are subject to different particle and heat loads.The red line in figure 3 shows the ratio of the outer target erosion to the total erosion: R ero = Γ ero outer /(Γ ero inner + Γ ero outer ), where Γ ero inner and Γ ero outer are the W gross erosion rates of the inner and outer targets accordingly.The W erosion mainly comes from the outer divertor target especially when the inner target is detached (Γ D2 > 1 × 10 21 s −1 ).Therefore, for the favorable B t cases, study of W leakage from the outer divertor target is more important.
In the DIVIMP simulations, the W leakage of eroded W particles from divertor targets into the core region r leak is quantitatively calculated by Γ core /Γ gross , where Γ gross and Γ core are the W fluxes from the gross erosion and to the core region respectively.As shown in figure 4(a), r leak decreases with the D 2 puffing rate increase, indicating an improvement of the divertor W screening. Figure 4(b) shows the ratio of the W leakage rate with drifts to that without drifts (r leak Drifts on /r leak Drifts off ) under the same background plasma conditions.According to figure 4(b), drifts can significantly increase W leakage by more than one order of magnitude.But generally, with the D 2 injection rate increases, r leak Drifts on /r leak Drifts off decrease, indicating that the impact of drifts on increasing W leakage is gradually diminished.
To illustrate the result more straightforward, we analyzed the W transport under three typical background plasmas with Γ D2 equal to 3.0 × 10 20 s −1 , 1.0 × 10 21 s −1 , and 3.0 × 10 21 s −1 .These three cases represent different outer divertor conditions of the low-recycling regime, high-recycling regime, and detachment regime, respectively.Figure 5 shows the two-dimensional W distribution for these cases with and without full drifts included, while figure 6 shows the W distribution separately including ⃗ E × ⃗ B drifts or diamagnetic drifts.The W distributions in figures 5 and 6 indicate that, for the selected three typical cases, the impact of drifts on W transport is primarily attributed to ⃗ E × ⃗ B drifts, while the influence of diamagnetic drifts is relatively negligible.Note that the velocity of diamagnetic drifts is inversely proportional to the charge state of ions.Therefore, the impact of diamagnetic drifts on high-Z W transport is insignificant.Therefore, we focus on investigating the impact of ⃗ E × ⃗ B drifts on W transport.
Figure 7 shows the main velocities of W along the outer divertor entrance for the selected cases.A positive value denotes a direction towards the outer target, and a negative value denotes a direction towards the upstream.As shown in figure 7(a), the total W impurity leakage velocity v tot decreases with the divertor condition varies from the low-recycling regime to the deep detachment regime.This is mainly due to the decrease of the ion temperature gradient velocity, as shown in figure 7 Under the favorable B t direction, the parallel ⃗ E × ⃗ B drift is pointing to the outer divertor target for most of the outer divertor region.Whereas a near-SOL ⃗ E × ⃗ B flow reversed region exists due to the shift of the T e peak along the radial directions [32], as shown in figure 2(b).The direction of the reversed poloidal E × B drift velocity in near-SOL is pointing upstream in the outer divertor region and towards the divertor target in the inner divertor region.Therefore, both the ion temperature gradient velocity and the reversed poloidal ⃗ E × ⃗ B drift velocity in the nearseparatrix SOL region help to push W ions upstream, while the unreversed poloidal ⃗ E × ⃗ B drift velocity contributes to W screening in the far-SOL region.
Overall, with the divertor operation condition varying from the low-recycling regime to the deep detachment regime, the absolute values of both v TiG and reversed v E×B decrease, which leads to a lower parallel leakage velocity of v tot .Consequently, the increase of the D 2 injection rate can reinforce the divertor W screening.

The low-recycling regime.
In the low-recycling regime (Γ D2 equal to 3.0 × 10 20 s −1 ), W source and leakage from both inner and outer divertor targets are significant.But compared to the outer divertor region, ⃗ E × ⃗ B drifts affect the W leakage in the inner divertor region more obviously, as shown in figures 5(a) and (d). Figure 8 plots the parallel profiles of  the absolute value of ion temperature gradient velocity v TiG and the ⃗ E × ⃗ B drifts velocity v para E×B , as well as the ion temperature T i and the electron density n e along a magnetic field line in the divertor region.Due to the divertor in-out asymmetry caused by the ⃗ E × ⃗ B drifts, the inner divertor region has a higher average n e and a lower average T i than that in the outer divertor region.As discussed in [15], the ion temperature gradient velocity v TiG ∝ T i 3/2 /n e , so v TiG in the inner divertor region is relatively smaller, and thus the parallel ⃗ E × ⃗ B drift has a greater impact on the W transport.The ratio of the W leakage rate with drifts to that without drifts.divertor region and pointing upstream in most of the inner divertor region, as shown in figure 9(c).Therefore, the poloidal ⃗ E × ⃗ B drift theoretically enhance the W screening of the outer divertor region and the leakage of the inner divertor region.However, there exist reversed poloidal ⃗ E × ⃗ B drift in the near-SOL region, as demonstrated by the red region adjacent to the separatrix in figures 9(c) and (d).The near-SOL reversed poloidal ⃗ E × ⃗ B velocity may play an important role on W leakage from the divertor to the core plasma.
By leveraging the Monte Carlo DIVIMP code, W particles transported across the divertor entrance to the main SOL region are recorded.In the DIVIMP simulation, all particles are launched as neutrals and followed until they deposit on a solid surface regardless of the charge state.Figure 10 shows the W leakage location along the inner and outer divertor entrances, which is defined as the number of W particles into the main SOL divided by the total number of W particles launched from the targets (4.9 × 10 5 and distributed according to the gross erosion rate).Note that some fraction of the W impurities from the inner divertor target can transport directly through the peripheral region outside the computational meshes and enter the main SOL.These particles are also recorded, as shown in the red dashed ellipse in figure 10.Compared to the results without drifts (figure 10(a)), the W leakage path is changed significantly by ⃗ E × ⃗ B drifts as shown in figure 10(b).In the inner divertor region, both the poloidal and radial ⃗ E × ⃗ B drifts enhance the W leakage across the far-SOL divertor entrance, and thus move the location of peak W leakage from the near-SOL to the far-SOL.Whereas in the outer divertor region, the W leakage from the far-SOL is reduced by the poloidal ⃗ E × ⃗ B drift.Meanwhile, the radial ⃗ E × ⃗ B drift pushes W ions from the far-SOL region to the near-SOL poloidal ⃗ E × ⃗ B reversed region.As illustrated by figure 11, due to the large parallel ion temperature gradient and the reversed ⃗ E × ⃗ B drift, the near-SOL W leakage velocity is much higher than that in the far-SOL.Therefore, the ⃗ E × ⃗ B drifts can dramatically increase the near-SOL W leakage of the outer divertor region and make this leakage path important.

The high-recycling regime.
In the case of Γ D2 = 1.0 × 10 21 s −1 , with the ⃗ E × ⃗ B drifts carrying particles from the outer divertor region to the inner divertor region, the inner divertor is in a high-density and low-temperature detached condition, while the outer divertor remains in the high-recycling attached condition.The detachment of the inner divertor region results in a significant reduction of W erosion and makes the W source mainly from the outer divertor target, as shown in figure 3.   E×B , as well as (b) the ion temperature T i and the electron density ne along the middle ring of the SOL region (r-rsep equals 5.34 cm on the outer divertor target).v TiG points upstream in both the inner and outer divertor regions, while v E×B points upstream in the inner divertor region and towards the target in the outer divertor region.The inner divertor region is represented by the blue shadow from the inner target (IT) to the inner divertor entrance (IDE), while the outer divertor region is represented by the red shadow from ODE to the outer target (OT).
Figure 12 compares the total W velocity distribution without and with the drifts included for the selected case.In the outer divertor region, there exists a near-SOL poloidal ⃗ E × ⃗ B reversed region similar to the case of Γ D2 = 3.0 × 10 20 s −1 .Apart from this region, the poloidal ⃗ E × ⃗ B drift is directed to the outer divertor target, resulting in a reduction of the W leakage velocity and even changing the direction of total W velocity from pointing upstream to pointing to the divertor target, as shown in figures 12(a) and (b).
The W leakage path along the inner and outer divertor entrance is plotted in figure 13.Most of the W leakage comes from the outer SOL region due to the stronger W erosion on the outer target.Compared to the results without drifts, the ⃗ E × ⃗ B drifts are foreseen to drive the far-SOL W particles towards the near-SOL and increase the leakage from the near-SOL poloidal ⃗ E × ⃗ B drift reversed region.Note that there exists W leakage from the IDE with the drifts included, and the leaked W particles originate from the outer divertor target.
The W leakage velocity and the main components including v i , v TiG , and v para E×B along the ODE are plotted in figure 14.Compared to the low-recycling case (figure 11), the v TiG for this high-recycling case is much smaller, which makes v i and v para E×B more significant.Due to the existence of the near-SOL reversed poloidal ⃗ E × ⃗ B drift, there still exists a near-SOL leakage path (negative v tot ), but the width of the leakage path is narrower than the low-recycling case.

The detachment regime.
Figure 15 shows the two-dimensional velocities distribution for the case of Γ D2 = 3.0 × 10 21 s −1 , in which both the inner and outer divertor targets are detached.In this case, slight W erosion only exists on the outer divertor target.Compared to the previous attached cases, the maximum leakage velocity in the divertor region is the smallest, resulting in the best divertor W screening. Figure 16 shows the path of W leakage from the divertor region.Without the drifts, eroded W particles can barely be transported to the upstream region, but the ⃗ E × ⃗ B drifts help W to move from the outer divertor target to the PFR and enhance the leakage from the X-point region.The W leakage velocity and the main components along the ODE are shown in figure 17.The value of v TiG is small under the detachment condition, but there still exists a near-separatrix reversed poloidal v para E×B , which leads to a narrow W leakage pathway in the near SOL, as marked by the red shadow in figure 17(a).

Discussions on the drift effects.
As discussed in previous sections, ⃗ E × ⃗ B drifts can dramatically change the W transport process from the divertor target to the upstream.With the increase of the D 2 puffing rate, the divertor condition is changed from the low-recycling regime to the deep detachment regime, and the drift effects on W transport are changed simultaneously, as illustrated by figure 18.Under the favorable   B t direction, ⃗ E × ⃗ B drifts are foreseen to enhance the W leakage of the inner divertor region, while there exists a near-SOL parallel ⃗ E × ⃗ B drift reversed region, which can enhance the W leakage of the outer divertor region.For the case under the low recycling divertor condition (figure 18(a)), the W source from the inner divertor target is comparable to that from the outer divertor target, and therefore the W leakage from both inner and outer divertor regions is important.But when the inner divertor is detached (figure 18(b)), the W source from the inner divertor target is negligible, and W particles are mainly leaked from the near-SOL ⃗ E × ⃗ B reversed region of the outer divertor.The radial ⃗ E × ⃗ B drift also helps to push W particles from the far-SOL to the near-SOL leakage pathway and consequently increases the total W leakage ability.For the case with both detached divertors (figure 18(c)), W erosion is well suppressed, but ⃗ E × ⃗ B drifts can still drive W originated from the outer divertor target to the PFR and a small fraction of the W ions can enter the core region by diffusion through the near X-point region.
⃗ E × ⃗ B drifts can enhance the W leakage capabilities of both inner and outer divertor regions.However, as the divertor condition transits from a low-recycling regime to a detachment regime, the impact of W leakage from the inner divertor region (which is directly affected by ⃗ E × ⃗ B drifts) becomes less significant.Furthermore, the decrease of v TiG and v E×B leads to a narrow near-SOL leakage path in the outer divertor region, ultimately reducing the effects of ⃗ E × ⃗ B drifts, as illustrated in figure 4(b).

Effects of divertor geometry on W screening
Previous studies demonstrate that a closed divertor is beneficial for the onset of detachment [33][34][35], but the geometric effect on W transport has not been studied.A scan of the OSP locations from the horizontal target to the vertical target is carried out in the simulation, as shown by the blue (H cases) and green (V cases) lines in figure 1(b).By increasing the D 2 puffing rate, various dissipative divertor conditions from low recycling to detachment are achieved with OSP on both the horizontal target (H cases) and the vertical target (V cases).Recycled neutrals are directed towards the SOL in the H case but towards the PFR in the V cases.Figure 19 shows T e profiles along inner and outer divertor targets for both H and V cases.At the same puffing rate, the target electron temperature T e is lower and the target electron density n e is higher in the H cases.The electron density at the OMP is also 50% to 70% higher in the H cases due to the higher fueling efficiency of the recycled main plasma from the divertor.Figure 20(a) illustrates the divertor W leakage rate for H cases and V cases calculated by the SOLPS-DIVIMP code package.For both sets of cases, the divertor W leakage rate decreases with the increase of the D 2 puffing rate.The W leakage rate of H cases is comparatively lower than that of V cases with the same D 2 puffing rate, indicating better divertor W screening with OSP on the horizontal target.Figure 20(b) shows the ratio of the W leakage rate with drifts included to that without drifts.With the increase of the D 2 puffing rate, the value of r leak Drifts on /r leak Drifts off decreases for the H cases but increases for the V cases.Moreover, unlike the V cases which will be explained later, the ⃗ E × ⃗ B drifts of H cases can enhance the divertor W screening (r leak Drifts on /r leak Drifts off < 1) under the high dissipative divertor conditions.
Two cases with the same D 2 puffing rate of Γ D2 = 7 × 10 20 s −1 but different OSP locations are selected to further investigate the effect of divertor geometry on the edge W transport. Figure 21 shows the two-dimensional W density distribution of the selected two cases with and without drifts.For the case with OSP on the horizontal target, the core W concentration is reduced with the drifts, as illustrated by figures 21(a) and (b).Whereas for the case with OSP on the vertical target, the core W concentration is increased with the effect of ⃗ E × ⃗ B drifts, as illustrated by figures 21(c) and (d).
Figure 22 provides the two-dimensional W velocity distribution in the divertor region.The average W leakage velocity in the outer divertor region of the V case is much higher than that of the H case.The parallel ⃗ E × ⃗ B drift velocity is also shown in figures 22(c) and (f ).It is noted that the near-SOL parallel ⃗ E × ⃗ B drift reversed regions exist for both H and V cases.However, compared to the V case, ⃗ E × ⃗ B drifts of the H case change the W leakage velocity distribution more significantly, as illustrated by figures 22(a) and (b).The velocities along the ODE are plotted in figure 23.The OSP on the horizontal target helps to obtain a relatively higher plasma density and a lower plasma temperature in the divertor region, resulting in a much smaller v TiG of the H case than that of the V case, which makes v para E×B more important.Meanwhile, the low v TiG and the narrow parallel v para E×B reversed region lead to a narrower near-SOL W leakage path as shown in figure 23(a).
The pathway of divertor W leakage across the divertor entrance and the first ionization location of the leaked W particles are recorded by the DIVIMP code.Figures 24 and 25 show the W leakage locations along the divertor entrance and the first ionization location of the leaked W particles for the H and V cases respectively.With the drift velocities included, both the W leakage from the near-SOL ⃗ E × ⃗ B reversed region and the screening of the far-SOL outer divertor region are    enhanced, and the W leakage from the inner divertor region is also enhanced, similar to the results discussed in section 3.2.1.For the V case, the W leakage originates from a large area of the outer divertor target, and the ⃗ E × ⃗ B drifts contribute to W leakage from the near-SOL of both inner and outer divertor.While for the H case, the main W leakage source is from the PFR near the OSP with the W source from the outer SOL region well screened by the ⃗ E × ⃗ B drifts, and the W particles leak to the core plasma from the near X-point region.
A schematic diagram of the W leakage path for the H and V cases is presented in figure 26.At the same D 2 puff rate, the H case shows a more effective W screening in the vicinity of   the divertor target and a narrower leakage tunnel in the nearseparatrix SOL region.Thus, it is hard for W to leak from the SOL region, although the ⃗ E × ⃗ B drifts can still lead to some X-point W leakage through the PFR.Under the same D 2 puffing rate, the V case has a relatively wider near-SOL leakage tunnel and the ⃗ E × ⃗ B drifts can enhance the divertor W leakage through the near SOL of the outer divertor region.Meanwhile, ⃗ E × ⃗ B drifts can also drive W particles to the inner divertor and enhance the W leakage of the inner divertor region.Therefore, configuration with OSP on the horizontal divertor plate is beneficial for the W screening.
Moreover, as illustrated in figure 20(b), the increase of the 2 puffing rate, the value of r leak Drifts on /r leak Drifts off of the V cases increases.Figure 27

W transport under the unfavorable B t direction
W transport under EAST LSN magnetic configuration with the unfavorable B t direction (B t = 2.4 T) is also investigated by the SOLPS-DIVIMP code package.Various levels of dissipative divertor plasma conditions from low recycling to detachment are generated by SOLPS-ITER with a scan of the D 2 puffing rate from Γ D2 = 2 × 10 20 s −1 to Γ D2 = 1 × 10 21 s −1 .W erosion and transport for these unfavorable B t cases are then simulated by DIVIMP.The blue line in figure 28 shows the divertor W leakage rate of different D 2 injection cases under the unfavorable B t direction.The computational meshes for these cases are the same with cases under the favorable B t direction, with the OSP fixed on the horizontal target.Under the unfavorable B t direction, the divertor W leakage rates decrease from the low recycling condition to the detachment condition.The red line in figure 28 shows the ratio of the W leakage rate with drifts included to that without drifts.With the increase of the D 2 puffing rate, the value of r leak Drifts on /r leak Drifts off decreases, which means the enhancement of divertor W leakage by the drifts effect is diminished.However, unlike the cases under the favorable B t direction, the ⃗ E × ⃗ B drifts of the detachment cases cannot enhance the divertor W screening under the unfavorable B t direction (r leak Drifts on /r leak Drifts off ⩾ 1).Two cases with different D 2 puffing rates of Γ D2 = 5 × 10 20 s −1 and Γ D2 = 1 × 10 21 s −1 are selected to further investigate the W leakage mechanisms under the unfavorable B t direction.Figure 29 shows the n e and T e profiles along the inner and outer divertor targets.Due to the drifts effect, the outer divertor detached at a lower D 2 puffing rate.For the selected two cases with Γ D2 = 5 × 10 20 s −1 and Γ D2 = 1 × 10 21 s −1 , the outer divertor targets are detached (T e < 10 eV), while the inner divertor targets are in the high-recycling regime and the partially detached regime respectively.
The pathways of divertor W leakage into the main SOL region for the selected cases with and without drifts are illustrated in figure 30.In contrast to the results under the favorable B t direction, W erosion and leakage mainly occur in the inner divertor region.Note that there exists W leakage along the edge of the high-field side grids which is caused by neutral W transport outside of the simulation meshes.Owing to the open divertor geometry of the inner divertor, the neutral W leakage from the path outside of the meshes is imperative, especially when the W source from the inner divertor target is the dominator.Apart from the neutral W leakage outside of the meshes, W leakage from the divertor entrance shows a similar pattern to the results under the favorable B t direction.When the inner divertor is in the high-recycling regime (Γ D2 = 5 × 10 20 s −1 ), the ⃗ E × ⃗ B drifts can enhance the W leakage from the outer divertor region and the inner divertor near-SOL ⃗ E × ⃗ B reversed region.But when the inner divertor is in the detachment regime (Γ D2 = 1 × 10 21 s −1 ), the decrease of the ion temperature gradient velocity and poloidal ⃗ E × ⃗ B drift velocity narrows the width of the near-SOL leakage path, making the W leakage from the divertor entrance insignificant.
The first ionization locations of the leaked W particles are illustrated in figure 31.Note that the value in the contour plots is the number of recorded particles divided by the total number of eroded particles.Since the outer divertor is detached, the leaked W particles mainly originate from the inner divertor target.For the inner divertor detached case (Γ D2 = 1 × 10 21 s −1 ), W ions are well screened in the divertor region, but the neutral W leakage through the path outside of the meshes is still significant.The ⃗ E × ⃗ B drifts have no effect on the neutral particles and thus cannot further enhance the divertor W screening by reducing the neutral W leakage. Therefore, the value of r leak Drifts on /r leak Drifts off is above 1 for the unfavorable B t cases even though the divertor is in the high dissipative detached conditions, as shown in figure 28.Whereas for the favorable B t cases when the W source and leakage are mainly from the more closed outer divertor, the neutral W leakage from the path outside of the meshes is negligible.Then the enhancement of divertor screening on W ions by ⃗ E × ⃗ B drifts dominates, resulting in the value of r leak Drifts on /r leak Drifts off smaller than 1.

Summary
A new drift module has been developed in the DIVIMP code to import ⃗ E × ⃗ B and diamagnetic drift velocities from SOLPS-ITER into DIVIMP.Two-dimensional DIVIMP simulation is carried out to evaluate the impact of drifts on W transport and divertor screening for various dissipative divertor conditions and different divertor configurations in EAST.Simulation results reveal that ⃗ E × ⃗ B drifts play an important role in W transport and can enhance the divertor W leakage by more than one order of magnitude.Under the favorable B t direction, it is observed that the W erosion mainly occurs on the outer divertor target, making the W leakage from the outer divertor region the dominator.By comparing the W transport process with and without drifts in DIVIMP, the drift effects on W transport mechanisms are revealed.Modeling results prove that, in the outer divertor region, there exists a near-SOL W leakage path where both the ion temperature gradient force and the reversed poloidal ⃗ E × ⃗ B drift point upstream.With the radial ⃗ E × ⃗ B drift pushing W ions from the well-screened far-SOL region to the near separatrix region, the near-SOL parallel leakage path plays an important role in divertor W leakage.
Simulation results also prove that with the divertor condition varying from the low-recycling regime to the deep detachment regime, the decrease of the ion temperature gradient velocity and the reversed poloidal ⃗ E × ⃗ B drift velocity narrows the width of the near-SOL leakage path, leading to a decrease of the W leakage enhancement caused by ⃗ E × ⃗ B drifts.Moreover, when the divertor is in the detachment regime, ⃗ E × ⃗ B drifts will drive the W ions from the wellscreened outer divertor region to the PFR and cause W leakage from the X-point region.The configuration with the OSP on the horizontal divertor plate is proved to have a significant effect on narrowing the width of the near-SOL leakage path because plasma at the target is colder and denser.Therefore, the far-SOL unreversed poloidal drift dominates and helps to enhance the divertor W screening.Note that for the same D 2 puff rate, cases with OSP on the horizontal target exhibit a higher upstream electron density than cases with OSP on the vertical target, and a higher upstream electron density is also favorable for W screening [15].
Under the unfavorable B t direction, the W source and leakage from the inner divertor dominate.⃗ E × ⃗ B drifts have similar effects on the divertor W leakage as under the favorable B t direction.However, due to the open configuration of the inner divertor, some fraction of neutral W particles can transport into       the main SOL region directly through the peripheral region outside the computational meshes.The neutral W leakage from the peripheral region is important when the divertor is detached and the SOL W screening is good.Therefore, to precisely investigate the W transport processes in the high-filed side edge plasma of EAST, future simulations will require a computational mesh that covers the entire peripheral region.
/B and b z = B z /B are the ratio of the poloidal/toroidal magnetic field strength to the total magnetic field strength.V || is the ion velocity parallel to the magnetic field line.v (a) ⊥ and v (a) y are the sum of the ⃗ E × ⃗ B drifts and diffusive velocities, which are given by:

Figure 2 .
Figure 2. The electron temperature Te (a), (b) and electron density ne (c), (d) profiles along the inner (a), (c) and outer (b), (d) divertor targets for cases with different D 2 injection rates under the favorable Bt direction.

Figure 3 .
Figure 3.The total W erosion rate (blue line) and the outer target erosion ratio (red line) for cases with different D 2 injection rates.

Figure 9
shows the 2-dimensional W velocity distribution and the ⃗ E × ⃗ B drift components in the divertor region.The W velocity distributions without and with the drifts are plotted in 9(a) and (b) respectively.The poloidal ⃗ E × ⃗ B drift velocity is pointing to the divertor target in most of the outer

Figure 4 .
Figure 4. (a) The total W leakage rate calculated by DIVIMP with full drifts included for cases with different D 2 injection rates.(b)The ratio of the W leakage rate with drifts to that without drifts.

Figure 7 .
Figure 7. Profiles of (a) the W parallel velocity vtot, (b) the main plasma velocity v i , (c) the ion temperature gradient velocity v TiG , and (d) the parallel ⃗ E × ⃗ B drift velocity v E×B along the outer divertor entrance (ODE).IR represents the number of flux tubes along the ODE, where IR = 20 represents the separatrix and IR = 38 represents the outmost ring of the grids.

Figure 8 .
Figure 8. Parallel profiles of the absolute value of (a) ion temperature gradient velocity v TiG and the ⃗ E × ⃗ B drifts velocity v paraE×B , as well as (b) the ion temperature T i and the electron density ne along the middle ring of the SOL region (r-rsep equals 5.34 cm on the outer divertor target).v TiG points upstream in both the inner and outer divertor regions, while v E×B points upstream in the inner divertor region and towards the target in the outer divertor region.The inner divertor region is represented by the blue shadow from the inner target (IT) to the inner divertor entrance (IDE), while the outer divertor region is represented by the red shadow from ODE to the outer target (OT).

Figure 9 .
Figure 9. Distribution of different velocities for the low-recycling regime (Γ D2 = 3.0 × 10 20 s −1 ).(a) W velocity distribution without drifts.(b) W velocity distribution with drifts included.The color represents the magnitude, and the arrow represents the direction.(c) Parallel ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing to the outer target, and a negative value means pointing to the inner target.(d) Radial ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing from the SOL to the PFR, and a negative value represents the opposite direction.

Figure 10 .
Figure 10.W leakage path to the main SOL with drifts excluded (a) and included (b) for the case of the low recycling regime (Γ D2 = 3.0 × 10 20 s −1).The higher W leakage regions are marked by a green ellipse, while the enhanced W screening region is marked by a purple ellipse.The W leakage from the peripheral region outside of the computational meshes is marked by a red dashed ellipse.

Figure 11 .
Figure 11.Profiles of total W velocity vtot, main plasma velocity v i , ion temperature gradient velocity v TiG , and parallel ⃗ E × ⃗ B drift velocity v para E×B along ODE with drifts included under the low-recycling regime (Γ D2 = 3.0 × 10 20 s −1 ).The red shadow in (a) shows the pathway for W leakage near the separatrix region, where the vtot is pointing upstream.

Figure 12 .
Figure 12.Distribution of different velocities for the high-recycling regime (Γ D2 = 1.0 × 10 21 s −1 ).(a) W velocity distribution without drifts.(b) W velocity distribution with drifts included.The color represents the magnitude, and the arrow represents the direction.(c) Parallel ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing to the outer target, and a negative value means pointing to the inner target.(d) Radial ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing from the SOL to the PFR, and a negative value represents the opposite direction.

Figure 13 .
Figure 13.W leakage path along the divertor entrance to the main SOL with drifts excluded (a) and included (b) for the case of Γ D2 = 1.0 × 10 21 s −1 .The higher W leakage region is marked by a green ellipse, while the W screening enhanced region is marked by a purple ellipse.

Figure 14 .
Figure 14.Profiles of total W velocity vtot, main plasma velocity v i , ion temperature gradient velocity v TiG , and parallel ⃗ E × ⃗ B drift velocity v para E×B along ODE for the case of Γ D2 = 1.0 × 10 21 s −1 .A negative value means pointing upstream.The red shadow in (a) shows the pathway for W leakage near the separatrix region, where the vtot is pointing upstream.

Figure 15 .
Figure 15.Distribution of different velocities for the detachment regime (Γ D2 = 3.0 × 10 21 s −1 ).(a) W velocity distribution without drifts.(b) W velocity distribution with drifts included.The color represents the magnitude, and the arrow represents the direction.(c) Parallel ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing to the outer target, and a negative value means pointing to the inner target.(d) Radial ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing from the SOL to the PFR, and a negative value represents the opposite direction.

Figure 16 .
Figure 16.W leakage path along the divertor entrance to the main SOL with drifts excluded (a) and included (b) for the case of Γ D2 = 3.0 × 10 21 s −1 .The higher W leakage region is marked by a green ellipse.

Figure 17 .
Figure 17.Profiles of total W velocity vtot, main plasma velocity v i , ion temperature gradient velocity v TiG , and parallel ⃗ E × ⃗ B drift velocity v para E×B along ODE for the case of Γ D2 = 3.0 × 10 21 s −1 .A negative value means pointing upstream.The red shadow in (a) shows the pathway for W leakage near the separatrix region, where the vtot is pointing upstream.

Figure 18 .
Figure 18.Sketch illustrating W leakage for different outer divertor operation conditions of (a) low-recycling regime (Γ D2 = 3 × 10 20 s −1 ), (b) high-recycling regime (Γ D2 = 1 × 10 21 s −1 ), and (c) detachment regime (Γ D2 = 3 × 10 21 s −1 ).The divertor, PFR, and upstream regions are marked by green, yellow, and magenta respectively.The red elongated bars above divertor targets indicate the location of W erosion, and the blue arrows indicate the paths of W leakage.The red big arrow represents W leakage tunnel where the W leakage flux is high.

Figure 19 .
Figure 19.The profiles of electron temperature Te along the inner and outer divertor targets for H (a), (b) and V (c), (d) cases with different D 2 injection rates under the favorable Bt direction.

Figure 20 .
Figure 20.(a)The total W leakage rate with full drifts included and (b) the ratio of the W leakage rate with drifts included to that without drifts.The blue lines and red lines represent the results with OSP located at the horizontal target (H) and vertical target (V) respectively.

Figure 21 .
Figure 21.2D W density distribution simulated by using DIVIMP code for OSP located at the horizontal target (a), (b) and vertical target (c), (d) with Γ D2 = 7.0 × 10 20 s −1 .Full drifts are excluded in (a) and (c), and included in (b) and (d).
shows the pathway of divertor W leakage and the first ionization location of the leaked W particles for a detached V case (Γ D2 = 5 × 10 21 s −1 ).Under vertical target geometry, the ⃗ E × ⃗ B drifts can easily drive the W ions across the PRF to the inner divertor region and dramatically enhance the W leakage even for the detachment cases.Therefore, with the increase of the D 2 puffing rate, the r leak Drifts on cannot decrease too much as shown in figure 20(a), but the r leak Drifts off decreases dramatically due to the small v TiG and the large friction force, which in turn leads to a larger value of r leak Drifts on /r leak Drifts off .Whereas for H cases, the value of r leak Drifts on /r leak Drifts off decreases with the increase of the D 2 puffing rate, indicating the enhanced effects of ⃗ E × ⃗ B drifts on W screening becomes stronger, especially in the detachment regime.As discussed in section 3.2.4,with the divertor condition from a high-recycling regime to a deep detachment regime, the near-SOL W leakage path becomes narrower.The lower T e and higher n e of H cases leads to a narrower near-SOL leakage path and a stronger SOL W screening, making W leakage from the near-SOL less important, and on the grounds of which ⃗ E × ⃗ B drifts can enhance the divertor W screening.

Figure 22 .
Figure 22.Distribution of W velocities for OSP located at the horizontal (a)-(c) and vertical (d)-(f ) targets with Γ D2 = 7 × 10 20 s −1 .(a) and (d): total W velocity with drifts excluded.(b) and (e): total W velocity with drifts included.The color represents the magnitude, and the arrow represents the direction.(c) and (f ): parallel ⃗ E × ⃗ B drift velocity distribution.A positive value means pointing to the outer target, and a negative value means pointing to the inner target.

Figure 23 .
Figure 23.Profiles of total W velocity vtot, main plasma velocity v i , ion temperature gradient velocity v TiG and parallel ⃗ E × ⃗ B drift velocity v para E×B along ODE for OSP located at the horizontal target ((a) and (b)) and the vertical target ((c) and (d)) with Γ D2 = 7 × 10 20 s −1 .The red shadow in (a) and (c) shows the pathway for W leakage near the separatrix region, where vtot is pointing to upstream.

Figure 24 .
Figure 24.W leakage path along the divertor entrance to the main SOL with drifts excluded (a) and included (b) for the case with OSP on the horizontal target (Γ D2 = 7.0 × 10 20 s −1 ).The higher W leakage region is marked by a green ellipse, while the W screening enhanced region is marked by a purple ellipse.The first ionization positions of the leaked W particles are recorded without (c) and with drifts (d).

Figure 25 .
Figure 25.W leakage path along the divertor entrance to the main SOL with drifts excluded (a) and included (b) for the case with OSP on the vertical target (Γ D2 = 7.0 × 10 20 s −1 ).The higher W leakage region is marked by a green ellipse, while the W screening enhanced region is marked by a purple ellipse.The first ionization positions of the leaked W particles are recorded by (c) and (d).

Figure 26 .
Figure 26.Sketch illustrating W leakage with the OSP locations at the (a) horizontal and (b) vertical targets with the same D 2 injection rate of Γ D2 = 7.0 × 10 20 s −1 under the favorable Bt direction.

Figure 27 .
Figure 27.W leakage path along the divertor entrance to the main SOL with drifts excluded (a) and included (b) for the case with OSP on the vertical target (Γ D2 = 5.0 × 10 21 s −1 ).The higher W leakage region is marked by a green ellipse.The first ionization positions of the leaked W particles are recorded by (c) and (d).

Figure 28 .
Figure28.The total W leakage rate with full drifts included (blue circle line), and the ratio of the W leakage rate with drifts included to that without drifts (red triangle line).