Simulation study of the influence of E× B drift on tungsten impurity transport in the scrape-off layer

Tungsten (W) is used as the plasma-facing material in the divertor region of future fusion reactors, such as ITER; however, its concentration in the core plasma must be maintained at an extremely low level. W transport in the scrape-off layer (SOL), which is related to the source of core W contamination, has been extensively studied. In this study, the influence of E× B drift on the transport of W impurity in the SOL is studied via numerical simulations of a model case based on EAST upper single-null configuration with high recycling divertor plasma. W transport is simulated using DIVIMP on the background plasma obtained from scape-off layer plasma simulation-ITER simulation including drifts. The E× B drift of W ions is introduced based on the background electric field. Therefore, both the direct E× B drift effect of W ion and the indirect effect via background plasma on W transport in the SOL are studied. The influence on the flux of W impurities entering confined plasma across the last closed flux surface Γ enter is focused on, which is expected to be proportional to the core W concentration. Results reveal that Γ enter is mainly from the outer (inner) target under a favorable (unfavorable) toroidal field B T and can be increased by more than one order of magnitude compared with the case without drifts; this reflects the significant effect of E× B drift. The effects due to the background plasma and the poloidal and radial E× B drift of W ion, as well as the related mechanisms, are analyzed in detail for three stages of W transport in the SOL: effective sputtering from the target, leakage from the divertor, and entry into the confined plasma.

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Introduction
Advantages, such as high heat conductivity, high melting point, high physical sputtering threshold, and particularly low tritium retention [1], make tungsten (W) the most promising candidate for plasma-facing materials (PFMs) in the divertor region of future fusion devices, such as International Thermonuclear Experimental Reactor (ITER) [2,3], Demonstration Fusion Power Reactor (DEMO) [4], and China Fusion Engineering Test Reactor (CFETR) [5].To mitigate the critically high heat load on the divertor target, radiative gas seeding is required to exhaust the heat power [6][7][8].However, it increases the rate of W erosion and affects the lifetime of PFMs [9,10].The eroded W impurities possibly leak from the divertor region to the upstream through the last closed flux surface (LCFS) and enter into the confined plasma.Due to a high radiative cooling rate, these impurities can lead to a dramatic radiation energy loss in the core plasma region, causing serious performance degradation of the core plasma or even disruption.The concentration of W in the core plasma region must be maintained below ∼10 −5 to achieve the ignition condition for fusion reactors [11,12].
The restriction on the discharge due to the tungsten concentration has been widely studied in devices with tungsten walls.
In EAST experiments, it is usually observed that the longpulse steady-state operation of H-mode plasma is restricted by a considerable increase in the core radiation induced by accumulated W impurities [13], with the tolerance of W concentration in the core region as ∼10 −4 [14,15].ASDEX-Upgrade experiments revealed that the acceptable W concentration is about 2 × 10 −5 in the confined plasma with an H-factor of 1.2 [16].The hybrid scenario experiments of JET with the ITERlike wall (ILW) also indicated that W accumulation limits the duration of the discharge with good confinement [17].Thus, W concentration in the confined plasma must be controlled for the steady-state operation of future reactors.
Assuming a constant global residence time of core W impurities in core plasma, the W concentration will be proportional to the W flux entering the confined plasma across the LCFS (Γ enter ) [18].Therefore, it is of great importance to understand the transport of W impurities in the scrape-off layer (SOL), which determines the source of the tungsten contamination inside the confined plasma.Many efforts have been conducted on this subject for various existing tokamaks, such as EAST [19][20][21][22], DIII-D [23,24], JET [25][26][27], and ASDEX-Upgrade [28], and for future fusion reactors, such as ITER [29,30], CFETR [31][32][33], and DEMO [34].
In recent years, it was found that the E × B drift has significant effects on the SOL plasma by changing the in-out asymmetry [35,36], degree of detachment [37][38][39], characteristics of the target profile [40][41][42], distribution of low-Z radiation impurity [43][44][45], etc.Since the velocity of impurity ion caused by parallel forces is inversely proportional to the mass of impurity ion, the effect of E × B drift on high-Z impurity is expected to be more pronounced than that on low-Z impurity, especially in the divertor region, where the pitch angle is small [46].Therefore, it can be expected that the E × B drift have a considerable effect on the W transport from the target to confined plasma.The E × B drift can directly and indirectly affect W transport by affecting the W ion movement and background plasma, respectively.It is necessary to understand the direct and indirect effects of E × B drift on W transport in the SOL and its synergetic effect on Γ enter .
However, the effects of E × B drift on the transport of W impurity in the edge plasma have been scarcely reported.The erosion/deposition of W has been simulated by threedimensional Monte-Carlo code ERO with drifts included for EAST [47] and JET [48], whereas W transport toward the upstream has not been discussed.The drifts of W ions have been included in the recently developed full-orbit impurity transport code IMPGYRO in simulations for ASDEX-Upgrade [49,50], JT-60U [51,52], and ITER [53], contrary to the effect of E × B drift on the background plasma.Studies on the W erosion/deposition [46] and the influence on the W erosion and leakage of divertor geometries [54] have been conducted for DIII-D, where the effect of E × B drift on the background plasma and W ions are considered using OSM-EIRENE/scape-off layer plasma simulation (SOLPS)-ITER and DIVIMP codes, respectively.Although the comprehensive effects of the E × B drift on the W transport from the target to confined plasma are not focused on in the studies for DIII-D, it provides an effective way to study W transport including the E × B drift via numerical simulations.
In this work, the synergetic effect of E × B drift on Γ enter , as well as the detailed effects on the W transport process in the SOL are studied via numerical simulations.In a similar way as very recent studies conducted for DIII-D [46,54], W transport is simulated herein using DIVIMP on the background plasma from the SOLPS-ITER simulation.The direct E × B drift effect on W ions is introduced in the DIVIMP simulation, while the indirect effect via the background plasma can be also included.Results show that E × B drift can considerably increase Γ enter under both favorable and unfavorable toroidal fields B T (the direction of B T is favorable/unfavorable for Hmode access when the corresponding ion B × ▽B drift points to/away from the active X point).The detailed mechanisms of the direct and indirect E × B drift effects are analyzed for the W transport process from the target to the confined plasma, which is divided into three stages: effective sputtering from the target, leakage from the divertor, and entry into the confined plasma.
The remainder of this paper is organized as follows.Section 2 describes the simulation scheme, including the setups in SOLPS-ITER and DIVIMP and the method for introducing the E × B drift effect on the W ions in DIVIMP.The simulated Γ enter and W distributions are compared for the cases with (for both favorable and unfavorable B T ) and without drifts in section 3.In section 4, the influence of E × B drift on the transport process of W impurities from the target to the confined plasma is analyzed and detailed mechanisms are discussed.A conclusion is given in section 5.

Simulation scheme
SOLPS is a boundary plasma code composed of two coupled codes, i.e.B2.5 (a multi-fluid code for plasma transport calculation) and EIRENE (a Monte-Carlo code for neutral transport calculation) [55], and widely applied to many tokamaks to study the divertor physics and boundary plasma transport [8,10,39,[56][57][58][59][60][61][62][63][64][65][66][67][68][69].The latest version SOLPS-ITER introduces advanced models for the drift effect and neutral transport [70] and has been recognized as an effective tool for studying the effect of drift on the SOL plasma [42,43,45].Complete W species can be included in the SOLPS-ITER; however, the simulation is too time-consuming [71].Although approximate W transport results can be obtained using bundled charge states, this method is sometimes numerically unstable [71] and can not account for the prompt redeposition process [27].In this work, the background plasma is simulated using SOLPS-ITER for studying W impurity transport, and the indirect effect of E × B drift on W impurities via the background plasma is investigated.The plasma potential distribution is obtained for introducing the direct E × B drift effect on W impurity transport.
DIVIMP is a two-dimensional (2D) impurity transport code that uses the Monte-Carlo method [72,73] and is widely applied for modeling the impurity transport in the plasma edge owing to its simple structure, short calculation time, and applicability in plasma of any collisionality [19,23,27].In this work, DIVIMP is employed to simulate the transport of W, which is treated as a trace impurity in the plasma background provided by SOLPS-ITER simulation.To include the direct E × B drift effect, i.e. the E × B drift of W impurity, the 2D distributions of poloidal and radial E × B drift velocities, which are calculated according to the simulated plasma potential by SOLPS-ITER, are input into DIVIMP [46,54].These E × B drift velocities are applied to W ions at each time step for W transport simulation, where the radial E × B drift is treated as a cross-field pinch velocity and the poloidal E × B drift is treated as an effective parallel velocity in addition to the parallel velocity decided by parallel forces.
Using the SOLPS-ITER plus DIVIMP simulation scheme, the effect of E × B drift on W transport in the SOL is studied based on a typical upper single-null (USN) equilibrium of EAST tokamak [42].Based on the model case, the related physical mechanisms are investigated, without any attempt to reproduce experimental observations.For simplicity, we focus on the W transport in the high-recycling SOL plasma.The detailed simulation setups for SOLPS-ITER and DIVIMP are given below.

SOLPS-ITER simulation setups
SOLPS-ITER simulations are performed for the model case based on a typical USN equilibrium of EAST tokamak, which was used in our previous study for the double-peaked density profile at the divertor target [42].While the cross-field transport coefficients are kept, the boundary conditions and D 2 and neon seeding rates are adjusted.The desired highrecycling regime (section 3) is achieved using the same simulation setups for the cases with (for both favorable and unfavorable B T ) and without drifts, with similar upstream conditions.The details of the simulation setups are given below.
Figure 1(a) shows the computational grid.The plasma transports in a structured quadrilateral grid with the sizes of 98 and 28 in the x (poloidal) and y (radial) directions, respectively, and the neutrals transport in the unstructured triangular meshes.The poloidal grid index ix increases from the upper outer target to the upper inner target, and the radial grid index iy increases from the core-edge interface (CEI) or private flux region edge to the scape-off layer (SOL) edge.The step widths in x and y directions are non-uniform.In the poloidal direction, cells are refined in the divertor region, especially near the targets; in the radial direction, cells are refined near the separatrix.Regions in computational meshes are illustrated in figure 1(e), where OT, ODE, OMP, IMP, IDE, and IT denote the outer target, outer divertor entrance, outer mid-plane, inner mid-plane, inner divertor entrance, and inner target, respectively.In the simulation, the total power across the CEI (P SOL ) is fixed at 5.5 MW, equally divided by electrons and ions, and the density of D + at the CEI (n CEI D + ) is fixed at 2.8 × 10 19 m −3 .The particle diffusion coefficient D ⊥ , electron thermal diffusivity χ e , and ion thermal diffusivity χ i are set to be poloidally symmetric and have the radial profiles at the OMP shown in figure 1(d).With the setting of cross-field transport coefficients, the edge plasma profiles for a H-mode discharge (see figures 1(b) and (c)) can be obtained.Sheath boundary conditions are applied on both the OT and IT.D 2 and neon (Ne) are puffed from the slot indicated by the red arrow in figure 1(a) at a seeding rate of 8 × 10 19 and 2 × 10 19 s −1 , respectively.D and Ne species are included in the SOLPS-ITER simulation.All particles moving onto PFMs are set to be fully recycled.The surface albedo for the upper and lower cryopumps is set to 0.9.SOLPS-ITER simulations are performed for the three cases to provide the plasma background for further W transport simulation, including the case without drifts and the cases with full drifts and currents under favorable and unfavorable toroidal fields.The simulated electron density (n e ) and electron temperature (T e ) profiles at OMP are shown in figures 1(b) and (c), respectively, for the three cases.With dedicated setups for SOLPS-ITER simulation, the upstream conditions for the three cases are similar.Furthermore, the divertor plasma near the target (OT or IT), which has a dominated contribution to Γ enter , is in the high-recycling regime with peak T e ∼ 20 eV near the target for all three cases (see section 3).

DIVIMP simulation setups
The computational grid for DIVIMP is the same as that for SOLPS-ITER.To simulate W transport, some background plasma information calculated by SOLPS-ITER, such as electron density n e , electron/ion temperature T e /T i , and plasma flow velocity V b , is input into DIVIMP.The W source along the targets is set according to the primary W sputtering flux Γ prim sput , which is calculated using the Eckstein-Preus formula [74,75] based on T e , T i , D, and Ne particle flux at the targets.Self-sputtering and prompt redeposition are included in the model, which is also one of the reasons for choosing DIVIMP for W transport simulation.The initial velocity and angular distributions of the launched W atoms are specified as Thompson energy distribution and cosine angle distribution, respectively [19].The launched W atoms travel in a straight line until ionization, and the W ions are traced further.
The diffusive cross-field transport of W ions is decided by the cross-field diffusion coefficient D ⊥,W , which is set to a constant value of 1 m 2 s −1 [19,25].The parallel transport of W ions depends on the net parallel force: The coefficients α e and β i are of the order Z 2 : where The terms from left to right in equation ( 1) are the impurity pressure gradient force, parallel electric field force (FE), friction force between background plasma and impurity ions (F fric ), and electron/ion thermal forces (F the /F thi ).s is the coordinate along the field line.m i is the mass of background ions.Z, m z , and n z are the charge, mass, and density of the impurity, respectively.dp z / ds, d(kT e )/ ds and d(kT i )/ ds are the gradient of impurity pressure, electron temperature, and ion temperature, respectively.v i and v z are the parallel velocities of main ions and impurity ions, respectively.τ s is the Spitzer stopping time inversely proportional to the collision frequency [76].E is the electric field.In general, the total parallel force is dominated by F fric and F thi [77].FE is neglected in our calculation.The ionization and recombination of particles are calculated based on the random number, timestep, and effective ionization/recombination rate coefficients from the ADAS database [78].
To include the E × B drift, the poloidal and radial E × B drift velocities of W ions are calculated according to the plasma potential distribution in background plasma from the SOLPS-ITER simulation and input into DIVIMP.The poloidal E × B drift velocity V E×B,pol and radial E × B drift velocity V E×B,rad of W ions are shown in figure 2 for both cases under favorable and unfavorable B T .The poloidal and radial E × B drift of W ions can be turned on individually in the code.
As listed in table 1, to understand the individual drift effect (via background plasma or radial/poloidal drift of W ion) on W transport, DIVIMP simulations are performed for nine cases as follows: the case without drifts (w/o drift), the case with only the effect of drift via the background plasma under favorable B T (favB T .BG) and unfavorable B T (unfavB T .BG), the cases with drift effect via background plasma and poloidal E × B drift of W ions under favorable (favB T .BG + W pol ) and unfavorable B T (unfavB T .BG + W pol ), the cases with the effect of drift via the background plasma and radial E × B drift of W ions under favorable (favB T .BG + W rad ) and unfavorable B T (unfavB T .BG + W rad) , and the cases with the effect of drift via the background plasma and full E × B drift of W ion under favorable (favB T .BG + W) and unfavorable B T (unfavB T .BG + W).
The simulated Γ enter and W concentration averaged inside LCFS C avg W are compared in section 3 for the cases w/o drift, favB T .BG + W, and unfavB T .BG + W to show the overall effect of the E × B drift.The detailed mechanism will be discussed in section 4 according to the comparison of the W fluxes of the nine cases.

Overall influence of E × B drift on W contamination inside LCFS
The radial profiles of n e and T e at the OMP with different drift options in the SOLPS-ITER are shown in figures 1(b) and (c), respectively.The radial profiles of n e , T e and T i at both targets are shown in figure 3. Note that W is treated as the trace impurity; thus, the plasma background solutions and the resulting primary sputtering flux of W are only related to the effect of drifts on the background plasma itself.Compared with the case under favorable B T , the plasma near the outer target becomes cooler and denser and the plasma near the inner target becomes hotter and sparser under unfavorable B T .These changes can be explained by the E × B drift, as indicated by the theory [76] and previous simulations [35,36].As reported in our previous work [42], the double-peaked feature in the n e profile at the outer target under unfavorable B T is induced due to the E × B drift of main ions, i.e. the poloidal E × B drift pointing toward the upstream obstructs the poloidal ion flow toward the target in the near-SOL, and the ions are transported by the radial E × B drift from the near-SOL to far-SOL and deposited there.
The contribution to W erosion on both targets is dominated by Ne ion flux, whereas that of incident D ion flux is negligible due to the high energy threshold.Note that, for the experiments, the intrinsic low-Z impurities, such as C, may have a notable contribution to W sputtering, which is not considered in the model case.For future fusion reactors with a full-tungsten wall, W sputtering is expected to be mainly contributed by seeded impurities, such as Ne and Ar.The profiles of the total incident Ne ion flux at the IT and OT are shown in figures 4(a) and (d), respectively.The primary W sputtering flux Γ prim sput (figures 4(c) and (f )) is calculated by summing over all the charge states of Ne ions: , where Z is the charge number of Ne ions, Γ Ne,Z is the incident flux of Z-charged Ne ions, and Y(E Z ) is the sputtering yield of Z-charged Ne ions.Assuming that the potential drop in the sheath is 3T e /e based on the basic sheath theory [76], the incident energy can be calculated by The sputtering yield Y is calculated using the Eckstein-Preus formula [75], which fits the experimental data well, assuming a smooth surface.As shown in figures 4(b) and (e), we calculated the mean W sputtering yield induced by Ne ion flux by dividing Γ prim sput with the total incident Ne ion flux.The primary W sputtering flux Γ prim sput from IT and OT are listed in table 2 for the cases w/o drift, favB T .BG + W, and unfavB T .BG + W. Γ prim sput is primarily observed in the target with higher plasma temperature, i.e.OT for the cases w/o drift and favB T .BG + W and IT for the case unfavB T .BG + W.
In figure 5, the charge-state dependent distributions of the incident Ne flux and the resulting W sputtering flux are drawn for the target where Γ prim sput mainly occurs; As one can see, the situations are similar for the case w/o drift and favB T .BG + W. The Ne ions with a low charge state of about 1-3 dominate the incident Ne flux, as shown in figures 5(a) and (b).As the effect of sheath acceleration is proportional to Z, W sputtering is mainly induced by Ne ions with Z of ∼3 (figures 5(d) and (e)), whereas the contribution of Ne 1+ is very small.For the case unfavB T .BG + W, the contributions of Ne ions with Z of ∼5 to the Ne incident flux and the W sputtering flux are also significant (figures 5(c) and (f )).Considering that the Ne atoms are seeded from the puffing slot near the OT, the Ne ions may be ionized to higher charge states along the longer transport distance before they bombard the IT, so that the mean sputtering yield at the IT for the case unfavB T .BG + W is larger than that at the OT for the case w/o drift and favB T .BG + W. In addition, the wetted area at the IT is larger; hence, the case unfavB T .BG + W has the largest total W sputtering flux.
In addition to the primarily sputtered W flux, self-sputtering is considered in DIVIMP.The self-sputtering factor f self is the averaged value calculated by dividing the number of selfsputtered W atoms with the number of primary sputtered atoms at the target.With f self (∼25%) for all three cases, as listed in table 2, the W sputtering flux Γ sput from the target can be calculated by The simulated W flux entering the confined plasma Γ enter,tot from IT and OT and the resulting W concentration averaged over the simulation region inside the LCFS C avg W are listed in table 2. To obtain the value of Γ enter,tot , which reflects the source strength of W contamination in the core plasma, the radial W particle flux density Γ rad = D ⊥,W ▽n W + n W V E×B,rad is first calculated; then, Γ enter,tot is obtained by integrating all inward Γ rad along the LCFS.It should be noted that, even though each test particle in DIVIMP is independent and undergoes its own random walk, with a large enough ensemble of    test particles the net cross-field particle flux can be effectively approximated as a diffusive flux of the form Γ rad = D ⊥,W ▽n W . Compared with the case w/o drift, Γ enter,tot is increased by 15.6 times under favorable B T and 62.4 times under unfavorable B T .The increase in C avg W is positively correlated with the increase in Γ enter,tot .C avg W increases by 6.6 times under favorable B T and 42.9 times under unfavorable B T .The distributions of W concentration for the three cases are shown in figure 6; W contamination inside the LCFS is mainly from the target with higher sputtering flux.Compared with the relatively moderate increase in the total W sputtering flux Γ sput,tot , i.e. 1.4 times under favorable B T and 4.8 times under unfavorable B T , the increases in Γ enter,tot and C avg W are much more significant, indicating the considerable effect of E × B drift on the W transport process from the target to confined plasma.

Statistics of the effects of E × B drift on the three stages of W transport
The 2D distributions of W ion particle flux Γ W (in s −1 ) for cases w/o drift, favB T .BG + W, and unfavB T .BG + W are shown in figure 7.For cases w/o drift and favB T .BG + W, the W ion particle flux into the confined plasma region mainly results from the outer target, which has a higher plasma temperature (figure 3) and thus a higher sputtering flux (table 2 and figure 4).This result is consistent with the distribution of W concentration shown in figure 6.For the case unfavB T .BG + W, the dominant side is the high-field side (HFS).For convenience, the side mainly contributing to W contamination inside the LCFS is referred to as the 'leakage side', which is the low-field side for the cases w/o drift and with drift under favorable B T , and HFS for the case with drift under unfavorable B T .W ions are produced via ionization at some distance off the target, then the transport pattern of W ion particle flux Γ W forms as figure 7 shows.A typical W transport process from the target to upstream can be understood from this figure.Near the target, the W ion particle flux appears toward the target due to the friction force, particularly in the near-SOL.Above a certain distance from the target, Γ W appears toward the upstream because the temperature gradient force is dominant.The W ions then leak along magnetic field lines from the divertor region to the main SOL, while a part of W ions diffuse radially outward and leave the SOL region simulated.In the main SOL, in addition to the outward diffusion, W ions tend to diffuse into the confined plasma before reaching the mid-plane.This typical impurity flux pattern from the target at the leakage side to the confined plasma holds for all cases.Thus, the effect of E × B drift on W transport can be understood by studying its influence on Γ W in detail.It should be noted that if D ⊥,W is much lower than the typical value of 1 m 2 s −1 used in this work, the flow pattern discussed below will be modified [33,79].
The W impurity flux mainly results from the leakage side; therefore, the W impurity flux entering the confined plasma across the LCFS Γ enter from the leakage side is studied.Further discussion focuses on the three stages of W impurity transport from the target to the confined plasma, including the effective sputtering from the target, leakage from the divertor, and entry into the confined plasma (figure 8).Thus, Γ enter can be expressed as follows: where f redep and Γ eff sput are the prompt redeposition fraction and the effective sputtering flux calculated by DIVIMP, respectively.f redep is the averaged value calculated by dividing the number of promptly redeposited W particles with the number of total sputtered atoms at the target.The fraction of the W flux leaking from the divertor region Γ leakdiv is denoted as f leakdiv , i.e.
The fraction of W flux further entering the confined plasma is denoted as f enter , i.e.
The calculation results of the W fluxes of the three stages at the leakage side and the corresponding fractions are listed in table 3 for all nine cases.The resulting C avg W is also listed, which is approximately proportional to Γ enter .The dramatic increase in Γ enter is mainly due to the indirect effect via background plasma and the radial E × B drift of W impurity.The effects of E × B drift at each stage are concluded below: (1) f redep is considerably decreased (thus a significant increase in Γ eff sput ) due to the effect of drift via the background plasma, particularly under unfavorable B T ; (2) f leakdiv is increased by the effect of drift via the background plasma and the radial E × B drift of W ions, while the poloidal E × B drift of W ion decreases f leakdiv ; (3) All drift effects increase f enter .
The physical mechanisms of the effects of E × B drift on f redep , f leakdiv , and f enter are investigated in sections 4.2-4.4,respectively.Because the effects of drifts are similar at the leakage side for both favorable and unfavorable B T , below we concentrate on the cases under favorable B T for convenience in illustrating the effects of E × B drift.A comparison of the 2D distributions of the W ion particle flux Γ W at the leakage side is shown in figure 9 for cases w/o drift, favB T .BG, fav.B T .BG + W pol , favB T .BG + W rad, and favB T .BG + W.

Effects of drift on f redep
The effect of drift via the background plasma decrease f redep by ∼21% under favorable B T .As a result, the Γ eff sput increases by ∼2 times because it is proportional to 1 − f redep .In the DIVIMP code, the prompt redeposition of W particles is assumed to occur when the ionization length of the impurity atom is less than the Larmor radius r gyro of the impurity ion in the first ionized state [80] or the width of magnetic pre-sheath (treated as the Larmor radius of background ion) [81].For W + , f redep is negatively related to the ratio of the mean ionization length r ion to r gyro : where V in is the initial velocity when launching the impurity atoms, R ion is the effective ionization coefficient, and q = 1 is the charge number of W + .Herein, T e of the high-recycling plasma is ∼20 eV near the target at the leakage side, indicating a weak change in R ion [31].Therefore, the decrease in n e due to the effect of drifts on the background plasma increases the r ion /r gyro, thereby decreasing f redep .It is consistent with the spectroscopic investigation conducted for ASDEX-Upgrade, which shows that f redep decreases with a decrease in n e [82].
As shown in table 3, due to the combined effect of the drifts on the primary sputtering flux Γ prim sput and prompt redeposition fraction f redep , the total effective W sputtering flux Γ eff sput,tot can be significantly increased by 3.4 times under favorable B T and by 15.0 times under unfavorable B T .This indicates that the effect of drifts on the background plasma can strongly enhance the erosion of the W target, shortening the lifetime of the divertor target.

Effects of drift on f leakdiv
The effect of drift via the background plasma increases f leakdiv by ∼48% under favorable B T .According to equation (1), background plasma can affect W ion transport through parallel forces.As pointed out by Stangeby and Moulton [77], instead of F fric , it is more useful to introduce the intrinsic friction force in the investigation.By substituting F fric with F fric,int , the net intrinsic force F net,int can be calculated according to equation (1).Thus, while it keeps F net,int = F net at the stagnation point of the parallel velocity V W,|| , the direction of the net force on the static W ion can be demonstrated.The impurity ion will leak from the divertor region when its first ionization occurs at a location where F net,int points to the upstream, i.e.F thi > F fric,int [43,77].
Considering that W has multiple ionized states, the densityweighted average values of F fric,int , F thi .and F net,int are calculated for further discussion and referred to as F avg fric,int , F avg thi , and F avg net,int , respectively.As shown in figures 10(a) and (b), strong F avg net,int in the near-SOL region near the target points to the target, which explains the Γ W pointing to the target (figure 7).When the drifts are included in the simulation of background plasma, the region where F avg net,int points to the target shrinks (figure 10(b)).As shown in figure 11, the change in F avg net,int is mainly caused by the decrease in the absolute value of F avg fric,int , which is due to the decrease in the collision frequency caused by the effect of drifts, i.e. decreasing n e and increasing T e in the divertor at the leakage side.As a result, the region where F avg net,int points to the target shrinks, particularly in the near-SOL region.Meanwhile, as discussed in section 4.2, the ionization length of impurity atom r ion increases due to the drift effects on the background plasma.As shown in figure 12(b), the distribution of ionization source of W + ions S ion,W+ obviously extends toward the upstream.Due to the effect of drift on F avg fric,int and S ion,W+ , the fraction of S ion,W+ located beyond the stagnation points increases, thereby increasing f leakdiv .Moreover, the combined effects are relatively strong in the near-SOL region, which makes the W flux toward upstream closer to the separatrix (figures 9(a) and (b)).Comparison   of the radial profiles of the poloidal particle flux of W ions Γ W,pol at the ODE between the cases w/o drift and favB T .BG (figure 13) shows that the leakage flux becomes stronger and closer to the separatrix due to the effect of drift via the background plasma.
Comparison of the cases favB T .BG and favB T .BG + W pol shows that the poloidal E × B drift of W ions decreases f leakdiv by ∼24% under favorable B T .The effect of poloidal E × B drift of W ions can be easily understood based on the distribution of V E×B,pol shown in figure 2. When the E × B drift of W impurity is included, the poloidal velocity of W V W,pol is determined by both the parallel velocity of W V W,|| and the poloidal E × B drift velocity V E×B,pol , i.e.V W,pol = V W,|| B θ /B + V E×B,pol , where B θ and B are the strength of the poloidal and total magnetic fields.As B θ /B is a small factor, V E×B,pol can affect V W,pol significantly.Since V E×B,pol points to the target at the leakage side, the stagnation points for the W poloidal velocity V W,pol moves away from the target than that for V W,|| , as shown by the magenta lines in figure 10(c).Figure 12(c) shows that the fraction of the ionization source of W + ions S ion,W+ located beyond the stagnation points decreases, thereby decreasing f leakdiv .According to the normalized Γ W,pol shown in figure 13(b), the effect of the poloidal E × B drift of W ions is relatively stronger in the far-SOL region, which is consistent with the difference between the stagnation points for V W,pol and V W,|| there.
When the radial E × B drift of W ions is included, f leakdiv further increases by ∼28% for the case favB T .BG + W rad under favorable B T .Because V E×B,rad points to the separatrix at the leakage side, the radial E × B drift pushes the W ions from the far-SOL region, i.e. the convective flow, toward the near-SOL region.As shown in figures 9(d) and (e), this effect is particularly strong in the region near the divertor target and is consistent with the larger V E×B,rad and slower V W,pol there.As a result, Γ W,pol at the divertor entrance becomes much stronger in the near-SOL region, as shown in figure 13(a).
The synergetic influence due to the indirect effect of drifts via the background plasma and the effect of E × B drift of the W ions increase f leakdiv by 1.5 times under favorable B T and by 1.7 times under unfavorable B T compared with that case w/o drift.Meanwhile, the leakage flux of W impurities reaches closer to the separatrix.

Effects of drift on f enter
The effect of drift via the background plasma increases f enter by ∼29% under favorable B T .The increase in f enter is related to two mechanisms.First, as discussed in section 4.3, the leakage flux is closer to the separatrix due to the effect of drift on the background plasma (figure 13).As a result, it can be seen in figures 9(a) and (b) that, the cross-field transport distance to the confined plasma is shortened.Second, as indicated by the red circle in figure 14(b), when the drifts of the background plasma are included, F avg thi is increased near the separatrix inside the LCFS.Thus, the W density n W is reduced at the position where W ions enter the confined plasma (figures 15(a) and (b)), thereby promoting the W impurity diffusion from the main SOL into the confined plasma.As a result, the poloidally averaged W density inside the LCFS is increased when the W flux entering the confined plasma increases.It should be noted that, the plasma distribution in the confined region is obtained by SOLPS-ITER simulations with a simplified mode at a mean-field level, which implies that the effect of F avg thi inside the LCFS should be considered qualitatively reasonable at a macroscopic level.Elaborate works are required to explore the existence of some neglected mechanisms due to the various modes and fluctuations that can considerably affect the results of this study.Furthermore, the results can be affected by many factors such as transport coefficients, divertor plasma state, and impurity radiation.For future fusion reactors, the impurity and radiation distribution are different from the existing experimental devices [44] and the much larger machine size is expected to reduce the influence of F avg thi inside the LCFS.The poloidal drift of W impurity further increases f enter by ∼67% under favorable B T .As shown in figure 9(c), the poloidal drift of W impurity induces a large poloidal flux inside the LCFS, which can further reduce the accumulation of W impurity at the position where W ions enter the confined plasma (figures 15(b) and (c)).Note that the poloidal E × B drift results from a large E r dip in the pedestal region, which can be attributed to the large pressure gradient there.Therefore, the drift benefits the W impurity diffusion from the main SOL to the confined plasma, in a similar way to the effect of F avg thi .
The radial drift of W impurity increases f enter by ∼33% under favorable B T .From figure 13, it can be seen that the radial drift of W impurity brings the leakage flux closer to the separatrix so that the cross-field transport distance to the confined plasma is reduced.Therefore, the radial drift of W impurity can promote the diffusion process from the upstream SOL to the confined plasma.
Compared with the case w/o drift, the synergetic influence due to the indirect effects of drifts via the background plasma and the effects of E × B drift of W ions increase   f enter by 2.5 times under favorable B T and by 2.7 times under unfavorable B T .

Conclusion
In this study, the influence of E × B drift on the W impurity transport process from the divertor target to confined plasma is investigated via numerical simulations.The background plasma is simulated using SOLPS-ITER for a model case based on the EAST USN configuration.The divertor plasma at the side, which has a dominant contribution to the flux of W impurities entering confined plasma across the LCFS Γ enter , is in the high-recycling regime with peak T e ∼ 20 eV near the target.On the background plasma, W impurity transport is further simulated using DIVIMP, where the E × B drift of W ions is introduced.
Results indicate that Γ enter mainly results from the OT for the case w/o drift and with drifts under favorable B T and from the IT with drifts under unfavorable B T .The influence on W sputtering and transport at the leakage side are concluded in table 4. Compared with the case without drifts, Γ enter is increased by more than one order of magnitude for the cases with drifts.C avg W , which is approximately proportional to Γ enter , is also significantly increased.It indicates the important effect of E × B drift on the W contamination in the confined plasma.
In addition to the increase in the W sputtering flux, the synergetic E × B drift effect has a positive contribution to Γ enter at the three stages of W transport from the divertor target to the confined plasma, including the effective sputtering from the target, leakage from the divertor, and entry into the confined plasma.The mechanisms related to the indirect effect of drift via the background plasma and the poloidal and radial E × B drift of W ions for the three stages are schematized in figure 16 and summarized as follows: (1) 1-f redep increases by ∼2-3 times • The drift effects on the background plasma decrease the electron density, which increases the ionization length of sputtered W and decreases f redep (x in figure 16).
(2) f leakdiv increases by ∼1.5 times • The drift effects on the background plasma increase the ionization length of sputtered W; thus, the distribution of the ionization source of W + is extended toward the upstream (y in figure 16).Meanwhile, due to the decrease in plasma density, the Spitzer stopping time increases, thereby decreasing the friction force and shortening the distance between the stagnation point and the target (z in figure 16).Therefore, f leakdiv is increased.• The poloidal E × B drift of W ions increases the distance between the stagnation point and the target, particularly in the far-SOL region, and thus decreases f leakdiv ({ in figure 16).• The radial E × B drift of W ions leads to the convective flux from the far-SOL to the near-SOL region, particularly near the divertor target, which increases f leakdiv (| in figure 16).
(3) f enter increases by ∼2.5 times • The change in the position of the stagnation points and convective flux also makes the leakage flux at the divertor entrance closer to the separatrix.In the main SOL, due to the short distance between the leakage flux and LCFS, W impurities diffuse into the confined plasma easily, which increases f enter (} in figure 16).• The poloidal W flow is induced by both the ion thermal force due to the drift effects on background plasma and the poloidal E × B drift of W ions.Because of the poloidal W flow, the W density at the location where W flux enters the confined plasma is reduced, which makes the diffusion across the LCFS easier, increasing f enter (~in figure 16).
Due to the significant effect on W transport, the E × B drift must be considered when estimating the W contamination for future fusion reactors.In particular, due to the high plasma temperature in the far-SOL region at the outer divertor [10], the sign of the slope of the plasma temperature changes, implying that the direction of the poloidal E × B drift will point toward the upstream rather than the target.Therefore, the effect of poloidal E × B drift of W ions will increase the leakage flux of W, rather than decreasing it.This may cause a considerable contribution to Γ enter from the far-SOL region, where the W sputtering is hard to be suppressed.

Figure 1 .
Figure 1.(a) Computational meshes used in the simulation, comprising the quadrangles for plasma transport marked in blue and the triangles for neutral transport marked in gray.The red arrow represents the gas puffing location at the outer target.The simulated radial (b) ne and (c) Te profiles at OMP are shown for the case without the drift (black lines), the case with the drift under favorable B T (blue lines), and the case with the drift under unfavorable B T (red lines).(d) The radial profile of transport coefficients used in the simulation.The vertical black dashed lines in panels (b)-(d) represent the position of the separatrix.(e) Illustration of regions in computational meshes, where OT, ODE, OMP, IMP, IDE, and IT are the outer target, outer divertor entrance, outer mid-plane, inner mid-plane, inner divertor entrance, and inner target, respectively.Their positions are indicated in (a).ix/iy is the poloidal/radial grid index.

Figure 2 .
Figure 2. Two-dimensional distribution of the (a) plasma potential and (b) poloidal and (c) radial E × B drift velocities of W ions for the cases with drifts under favorable B T .Panels (d)-(f ) are the same for the case with drifts under unfavorable B T .A positive value of poloidal velocity means the direction is from the outer target to the inner target (clockwise in the core region).A positive value of radial velocity means the direction points out radially.

Figure 3 .
Figure 3. Profiles of (a) ne, (b) Te and (c) T i at the outer target simulated by SOLPS-ITER for the case without drift (black), with drifts under favorable B T (blue), and with drifts under unfavorable B T (red).Panels (d)-(f ) are the same for the inner target.The horizontal axes show the distance from the separatrix along the targets.

Figure 4 .
Figure 4. Profiles of (a) incident Ne ion flux, (b) mean W sputtering yield, and (c) primary W sputtering flux Γ prim sput at the inner target for the case without drift (black), with drifts under favorable B T (blue) and with drifts under unfavorable B T (red).Panels (d)-(f ) are the same for the outer target.The horizontal axes show the distance from separatrix along the targets.

W 2 .Figure 5 .
Figure 5. (a)-(c) incident flux profiles and (d)-(f ) resulting W sputtering flux profiles of different charge states of Ne ions at the target with larger Γ prim sput for the case w/o drift (a) and (d), with drifts under favorable B T (b) and (e) and with drifts under unfavorable B T (c) and (f ).Panels (g)-(l) are the same for UO target.The horizontal axes are the distances from the separatrix along the targets.

Figure 6 .
Figure 6.Two-dimensional distributions of W concentration for cases (a) w/o drift, (b) favB T .BG + W, and (c) unfavB T .BG + W. The solid blue lines represent the boundary of the plasma zone in the simulation.The dashed black lines represent the separatrix.C avg W in each plane is the corresponding W concentration averaged over the simulation region inside the LCFS.

Figure 7 .
Figure 7. Two-dimensional distributions of the W ion particle flux Γ W in computational meshes for cases (a) w/o drift, (b) favB T .BG + W, and (c) unfavB T .BG + W. The directions of Γ W are shown by black arrows.The horizontal solid black line represents the position of the separatrix.The white arrows indicate the overall trend of the W ion flux.

Figure 8 .
Figure 8. Schematic of the three stages of W transport from the target to the confined plasma.

Figure 9 .
Figure 9. Two-dimensional distributions of Γ W in computational meshes for the cases (a) w/o drift, (b) favB T .BG, (c) favB T .BG + W pol , (d) favB T .BG + W rad , and (e) favB T .BG + W. The directions of Γ W are shown by black arrows.The horizontal solid black line represents the position of the separatrix.The white arrows indicate the overall trend of the W ion flux.

Figure 10 .
Figure 10.Two-dimensional distributions of the value of density-weighted average intrinsic net force F avg net,int in computational meshes in the outer divertor region for cases (a) w/o drift and (b) favB T .BG + W. Positive values indicate that the force points from the outer target to the inner target.The black arrows represent the direction of Γ W .The solid black and magenta lines with squares represent the stagnation points for the W parallel velocity V W,|| and W poloidal velocity V W,pol , respectively.The horizontal solid black line represents the position of the separatrix.

Figure 11 .
Figure 11.(a) Poloidal profiles of the density-weighted average intrinsic friction force F avg fric,int (blue) and density-weighted average ion temperature gradient force F avg thi (red) in the outer divertor region for the cases w/o drift (dashed lines) and with drifts under favorable B T (solid lines).The poloidal profiles are for the radial position where iy = 15, corresponding to the flux surface with r − rsep of 0.86 cm at the outer mid-plane and r − rsep of 5.1 cm at the outer target.The positive value means that the force points from the outer target to the inner target.

Figure 12 .
Figure 12.Two-dimensional distributions of the values of the ionization source of W + ions S ion,W+ (s −1 ) in computational meshes for cases (a) w/o drift, (b) favB T .BG, and (c) favB T .BG + W. The solid black and magenta lines with square markers represent the stagnation points for V W,|| and V W,pol , respectively.The black arrows represent the direction of Γ W .The horizontal solid black line represents the position of the separatrix.

Figure 13 .
Figure 13.Radial profiles of the (a) poloidal particle flux of W ions Γ W,pol and (b) normalized value at the outer diverter entrance mapped to the target for the cases with different drift options.The vertical dashed black lines represent the location of the separatrix.The horizontal axis are the distances from the separatrix the along targets.

Figure 14 .
Figure 14.Two-dimensional distributions of the values of density-weighted average ion temperature gradient force F avg thi in computational meshes for the cases (a) w/o drift and (b) favB T .BG.The red circle in panel (b) indicates the region where W ions enter the confined plasma and F avg thi is notably increased for the case favB T .BG.The positive value indicates that the force points to the direction in which ix increases.The black arrows represent the direction of Γ W .The horizontal solid black line represents the position of the separatrix.

Figure 15 .
Figure 15.Two-dimensional distributions of the value of total density of W ions n W in computational meshes for the cases (a) w/o drift, (b) favB T .BG, and (c) favB T .BG + W. The black arrows represent the direction of Γ W .The horizontal solid black line represents the position of the separatrix.

Figure 16 .
Figure 16.Schematic of the basic flow pattern of W impurity during its transport from the target to the confined plasma (taking the LSN configuration under favorable B T as an example), and the physical mechanisms of the influences of drifts.V E×B,rad and V E×B,pol are the radial and poloidal E × B drift velocities of W ions. lstag is the distance between the stagnation points and the target.The yellow, blue, and orange rectangles refer to the divertor SOL, main SOL, and core region, respectively.The transparent and orange thick arrows indicate the basic flow pattern for the cases w/o drift and with drift under favorable B T , respectively.The red and blue thin arrows indicate the direction of V E×B,rad and V E×B,pol , respectively.The black and blue dashed lines denote the connection lines of stagnation points for V W,|| w/o drift and V W,pol with the drift.The sequence numbers represent the physical mechanisms of drifts on the W transport from the target to the core, and the process explanation of these mechanisms are listed on the right.

Table 1 .
List of the nine cases simulated by DIVIMP and the related drift effects.w/o drift fav B T unfav B T

Table 2 .
Calculation results of tungsten erosion (the subscript 'tot' represents the sum of inner and outer targets, and the values in brackets are the normalized values by the case w/o drift).

Table 3 .
Calculation results of the tungsten fluxes and the corresponding fractions for the nine cases (all fluxes are at the leakage side except for Γ eff sput,tot ).

Table 4 .
Ratios of the sputtering flux, the fractions related to W transport for the three stages, Γ enter (at the leakage side), and C avg W for cases with drifts under favorable and unfavorable B T to those for the case w/o drift.