A Gallo

A Sepetys

Y Marandet

H Bufferand

G Ciraolo

N Fedorczak

S Brezinsek

J Bucalossi

J Coenen

F Clairet

Y Corre

C Desgranges

P Devynck

J Gaspar

R Guirlet

J Gunn

C C Klepper

J.-Y Pascal

P Tamain

E Tsitrone

E A Unterberg

Interpretative transport modeling of the WEST boundary plasma. Part I: main plasma and light impurities

Understanding impurity transport in tokamak plasmas is crucial to control radiative losses and material migration in future magnetic fusion reactors. In this work we deploy the SolEdge2D-EIRENE code to model the boundary plasma in a WEST discharge, satisfactorily reproducing measurements of both upstream and divertor plasma conditions. The spatial distribution of oxygen, studied here as a representative light impurity, is compared to vacuum ultraviolet spectroscopy measurements acquired with an oscillating line of sight. The simulation captures a key feature of the experiment, namely a factor of 2 higher oxygen brightness in the inner divertor region compared to the outer one. This spatial asymmetry in oxygen concentration is interpreted by analyzing the balance of friction forces and thermal gradient forces that the light impurity exchanges with the main plasma.

1 Introduction: studying the impact of light impurities in the WEST boundary plasma

Future magnetic nuclear fusion reactors will have to rely on metallic plasmafacing components (PFCs) in order to sustainably handle long-term tritium retention and neutron-induced material damage [START_REF] Philipps | Key issues in plasma-wall interactions for ITER: a european approach[END_REF]. The operation of these devices will be limited by plasma dilution and radiative losses due to both low and high Z impurities. In fact, impurities play a key role since they control the magnitude and the location of radiative losses. These losses can be beneficial in the boundary plasma as they help dissipating power loads onto the PFCs, but also deleterious in the core plasma as they reduce its temperature and the fusion reaction rate. The latter is of particular concern for fusion devices equipped with PFCs made of high Z materials, such as tungsten (W): such heavy impurities, eroded from the PFCs, can penetrate in the core plasma while being incompletely ionised, causing strong radiative losses. Therefore, a thorough understanding of the impurity transport in metallic tokamaks is needed to optimize and analyze the plasma scenarios of existing machines, as previously done in ASDEX Upgrade [START_REF] Dux | Plasma-wall interaction and plasma behaviour in the nonboronised all tungsten asdex upgrade[END_REF] and JET [START_REF] Brezinsek | Erosion, screening, and migration of tungsten in the jet divertor[END_REF], as well as to prepare those of future ones. The full-W tokamak WEST [START_REF] Bucalossi | The west project: Testing iter divertor high heat flux component technology in a steady state tokamak environment[END_REF] is well fitted to study the impact of intrinsic light impurities, such as oxygen (O), on the erosion and transport of W. This can be done with the help of numerical models like the 2D multi-fluid plasma solver SolEdge2D [START_REF] Bufferand | Near wall plasma simulation using penalization technique with the transport code soledge2d-eirene[END_REF][START_REF] Bufferand | Numerical modelling for divertor design of the west device with a focus on plasmawall interactions[END_REF]. In this contribution SolEdge2D, coupled with the Monte Carlo neutrals code EIRENE [START_REF] Reiter | The eirene and b2-eirene codes[END_REF], is used to reproduce the deuterium (D) main plasma (section 3) and to study the spatial distribution of O as a representative light impurity (section 4) in a well characterized WEST plasma discharge (section 2). In particular, we compare our simulation with experimental data obtained via the WEST vacuum ultraviolet (VUV) spectroscopy system [START_REF] Schwob | Highresolution duomultichannel soft xray spectrometer for tokamak plasma diagnostics[END_REF] with a moving line of sight in the poloidal plane, allowing one to retrieve the angular distribution of O light emission. These findings are analyzed and discussed in terms of thermal gradient and friction forces exerted by main ions and electrons on light impurity ions. The latter play a dominant role in setting the poloidal distribution of incident impurity fluxes onto the various PFCs. These fluxes rule the sputtering of heavy impurities which can then sustantially increase their content due to self-sputtering and migrate towards the confined plasma. A companion paper [START_REF] Sepetys | In preparation for Nuclear Fusion[END_REF] studies the contamination pathways for heavy impurities for the very same WEST plasma discharge by comparing VUV measurements of molybdenum line emission with the results of numerical simulations performed with the ERO2.0 code [START_REF] Romazanov | Beryllium global erosion and deposition at jet-ilw simulated with ero2.0[END_REF]. These two papers should be seen as a unique effort showing that we can now provide valuable support to the interpretation of the role of impurities in present and future WEST operations. Our approach is enabled by the rapidly growing availability of WEST experimental data and by the synergistic deployment of state-of-the-art modeling tools, with the primary goal of understanding the core plasma contamination to improve the performance of the device.

Experimental assessment of oxygen spatial distribution in the WEST divertor plasma

To conduct this investigation we analyze a particular WEST plasma discharge (#54067) performed during its third experimental campaign. This discharge is well diagnosed and offers the opportunity for a thorough comparison with our numerical modeling tools, as later presented in section 3.

In general, WEST represents a good test bed for boundary plasma modeling in virtue of the simple open divertor geometry of its vacuum vessel and its set of dedicated edge plasma diagnostics [START_REF] Bucalossi | The west project: Testing iter divertor high heat flux component technology in a steady state tokamak environment[END_REF]. In particular, the considered plasma discharge is characterized by stable L-mode conditions and, during a 4 s flattop phase, benefits from a unique feature of its VUV spectroscopy system [START_REF] Schwob | Highresolution duomultichannel soft xray spectrometer for tokamak plasma diagnostics[END_REF], which can scan the lower half of the plasma thanks to a sweeping line of sight. This is done by moving the spectrometer mechanically up and down with a period of a few seconds. When used in this scanning mode, the system provides spatial profiles of spectral line emission from which information can be deduced about the emitting ion spatial distribution. In the VUV spectra of discharge #54067, several OIII and OIV lines are identified, thus allowing the comparison of the poloidal distribution of O 2+ and O 3+ lines found experimentally with the one simulated by SolEdge2D-EIRENE, as described in section 4. The main machine parameters during this discharge are: deuterium (D) as fuelling gas, lower single null (LSN) magnetic configuration (shown in figure 2.b), X-point to lower divertor distance dX = 115 mm, toroidal magnetic field B T = 3.8 T, plasma current I P = 500 kA throughout the entire 10 s duration of the discharge. In terms of power balance, during the steady state L-mode phase between 6 s and 9 s, the most impor-tant parameters, shown in figure 1.a, are: Ohmic power P ohm 0.1 MW, power from lower hybrid current drive (LHCD) P LHCD 4.2 MW, total heating power P heat 4.3 MW, power radiated in the bulk plasma (defined in section 3.3) measured by bolometers P bulk rad 1.6 MW, power losses due to magnetic ripple P ripple = 0.1P LHCD 0.4 MW. The power entering the scrape-off layer (SOL) can, therefore, be estimated from a power balance as P SOL = P heat -P bulk rad -P ripple 2.3 MW. The line integrated density measured by the interfero-polarimeter system [START_REF] Gil | Renewal of the interfero-polarimeter diagnostic for west[END_REF], shown in figure 1.b, is n main lid = 4 × 10 19 m -2 in the main plasma, whereas in the lower divertor it is n div lid = 3 -4 × 10 18 m -2 , with a 30% increase during this phase. This is a commonly observed behaviour in WEST discharges performed in LSN configuration, possibly due to the outgassing of D atoms from the PFCs as their temperature rises. Figure 1: a) time traces for the WEST plasma discharge #54067: Ohmic power (black), lower hybrid current drive power (green), total heating power (red), power radiate in bulk plasma (blue), power losses due to magnetic ripple (grey), power entering the scrape-off layer (magenta); b) time traces of the main plasma (red) and divertor (blue) line integrated density; the yellow shaded area corresponds to the VUV spectroscopy scan time interval.

3 Modeling of the main plasma

Model, setup and assumptions

The WEST plasma discharge #54067 described in section 2 is numerically modeled with our edge and scrape-off layer code suite: SolEdge2D-EIRENE [START_REF] Bufferand | Numerical modelling for divertor design of the west device with a focus on plasmawall interactions[END_REF]. The model, in the following referred to as S2DE, consists in the coupling of the 2D multi-fluid plasma solver SolEdge2D [START_REF] Bufferand | Near wall plasma simulation using penalization technique with the transport code soledge2d-eirene[END_REF] and the kinetic Monte Carlo neutral code EIRENE [START_REF] Reiter | The eirene and b2-eirene codes[END_REF]. Thanks to its immersed boundary condition, the so-called penalization technique, S2DE can be used to calculate density, temperature and velocity of electrons and multiple ion species in realistic tokamak geometry. As sketched in figure 2.a, the simulation domain (green area) extends from an inner boundary located at a normalized minor radius ρ = 0.823 in the edge plasma region (blue solid line) down to the wall contour (black solid line) accounting for in-vessel plasma-facing components (PFCs) as, for instance, the baffle and the LHCD antenna. The latter, however, has a finite toroidal extent ( 30 • ) and a complex geometrical structure which cannot be accurately described by a 2D model. This antenna, which can be moved radially in experiments, is positioned in the simulation at a location where the level of recycling on its surface is low enough to be consistent with visible camera observations (i.e., no recycling detected there). The ability to calculate plasma fluxes on main chamber structures, such as the baffle, is a strong asset for S2DE in the panorama of interpretative 2D transport models, especially if the focus is on plasma-wall interactions and core contamination studies. The S2DE mesh is based on the magnetic equilibrium for discharge #54067 for t = 8 s obtained with the real time plasma shape reconstruction tool VacTH [START_REF] Faugeras | 2d interpolation and extrapolation of discrete magnetic measurements with toroidal harmonics for equilibrium reconstruction in a tokamak[END_REF][START_REF] Urban | Validation of equilibrium tools on the compass tokamak[END_REF]. For the simulation presented in this paper, the output of the VacTH calculation is modified to account for two experimental evidences. i) The top part of the magnetic equilibrium given by VacTH (figure 2.b, red contours) has a high upper triangularity. This feature leads to strong particle and heat fluxes at the upper divertor targets in S2DE simulations. Such fluxes are not backed by Langmuir probe measurements of ion saturation current j sat . Moreover, the high recycling at the upper divertor observed in simulations, stronger than at the lower divertor, is not supported by visible light imaging data. ii) The location of the strike points at the lower divertor targets given by VacTH shows a disagreement with respect to the one inferred from embedded diagnostics like Langmuir probes, thermocouples, fiber Bragg gratings and from infrared thermography. The modified magnetic equilibrium adopted for this study (figure 2.b, blue contours), therefore, has an X-point vertical position lowered by 15 mm and a distance between the strike points along the divertor targets reduced by 25 mm. We now discuss the main input parameters for the S2DE run, also summarized below in table 1. The D ion density at the inner simulation boundary n IB D + = 2.5 × 10 19 m -3 is chosen in order for the electron density at the same location to match the fast sweep reflectometer experimental data within the error bars (n IB e 3 × 10 19 m -3 ). We assume a total power entering the simulation domain at the inner boundary of P IB tot = 2.3 MW, using P SOL obtained from the power balance previously discussed in section 2; this total power is assumed to be equally shared between electrons and D ions, P IB D + = P IB e = 1.15 MW. The total O density at the inner simulation boundary, which represents the initial impurity source in this simulation, is

n IB O n+ = 0.02n IB D + = 5 × 10 17 m -3
, where the superscript n+ signifies sum over all O ionisation states from now on (excluding neutrals); this assumption is motivated by recent experimental studies of the visible spectroscopy data in the WEST lower divertor showing that an O content of 1 -2% is consistent with both effective W sputtering yields and residual gas analysis [START_REF] Van Rooij | [END_REF]. Typical L-mode transport coefficients [START_REF] Bufferand | Numerical modelling for divertor design of the west device with a focus on plasmawall interactions[END_REF][START_REF] Chankin | Simulation of ASDEX upgrade ohmic plasmas for SOLPS code validation[END_REF] are used for particles and momentum D = ν = 1 m 2 s -1 as well as for energy χ = 2 m 2 s -1 ; these values, uniform over the whole domain and equal for all species, ensure a very good match of the main plasma profiles in the lower divertor with embedded diagnostics, as shown below. We adopt a recycling coefficient uniform along the full-W wall contour of R = 0.99 and an albedo coefficient for the pump A = 0.9, both coefficients equal for D and O; these numbers represent the standard S2DE assumption for W PFCs and for the WEST pumping system, respectively. In steady state, recycled fluxes at the wall are the main sources of D and O.

It should be noted that magnetic drifts are turned off in this simulation: the potential consequences of this working hypothesis on the transport of light impurities in the divertor region are discussed at the end of the paper.

Quantity

Main plasma profiles along the outer midplane and poloidal distribution

In this section we discuss the results concerning the main plasma, whereas the light impurity modeling results are presented in section 4. Let us start with the outer midplane radial profiles for the main plasma density and temperature, shown in figure 3. In figure 3.a, the density profiles of electrons (blue line) and D ions (red line) simulated with S2DE are superimposed to experimental n e data from the midplane fast sweeping reflectometer [START_REF] Clairet | Fast sweeping reflectometry upgrade on tore supra[END_REF] (green dots with horizontal error bars) and the reciprocating Langmuir probe (RLP) [START_REF] Gunn | Evidence for a poloidally localized enhancement of radial transport in the scrape-off layer of the tore supra tokamak[END_REF] (cyan and magenta circles). In the edge plasma the profile of n e calculated by S2DE matches reflectometry data within the 1.2 cm radial uncertainty, whereas it is higher by a factor of 1.8 at the main separatrix, and by a factor of 2.1 at the secondary separatrix. A quantitative agreement with reflectometry data throughout the entire n e profile can be achieved by dividing the transport coefficient values by 2 (not shown). However, the simulation discussed here represents our best effort in simultaneously matching outer midplane and lower divertor target measurements, shown below, after scanning n IB D + , D, ν, χ. On the other hand, the agreement on n e from S2DE and RLP data (cyan circles) is poor. It is important to mention that this probe is located at the top of the machine, where the uncertainties affecting the magnetic equilibrium reconstruction are bigger: RLP data acquired in the vicinity of the secondary X-point are remapped at the outer midplane along magnetic flux surfaces, which can result in uncertainties in the radial direction. It is likely that the very low n e ≤ 2 × 10 17 m -3 measured by the RLP corresponds to a radial location further out in the SOL. By displacing the remapped RLP data by 5 cm outwards (magenta circles), the agreement with reflectometry data becomes quantitative, as shown in the inset. The same argument applies to the temperature profiles depicted in figure 3.b: although here only a factor of 2 to 4 difference is found between T e from S2DE (blue line) and T e from the RLP data (cyan circles), the agreement is greatly improved if the above mentioned 5 cm radial shift is applied (magenta circles), as can be seen in the inset. No other measurements are available at time of writing for further comparison with the S2DE profiles of T e or T i (red line). Nevertheless, the values of T sep e 80 eV and T sep i 120 eV found in the simulation, as well as temperature decay lengths of the order of few centimeters, are compatible with typical experimental findings in L-mode plasma conditions as well as with estimations from a simple two-point model [START_REF] Stangeby | The Plasma Boundary of Magnetic Fusion Devices[END_REF]. The corresponding poloidal maps for D ion density n D + and temperature T D + are shown in figure 4.a and 4.b, respectively. Despite the considered plasma discharge being in LSN configuration, it is clear that the secondary separatrix (black dashed lines), the X-point of which lies close to the upper divertor surface, has a strong impact on SOL profiles with the assumed transport coefficients: the n D + map shows values that are poloidally rather uniform in the edge and near SOL regions, with density in the lower and upper divertor regions of the same order of magnitude (10 19 m -3 ); the T D + map, instead, shows a D plasma which is hotter in the low field side (LFS) SOL when compared to the high field side (HFS) SOL, where the D neutral pressure is higher (not shown).

Radiation pattern and synthetic diagnosis

The total radiated power in the S2DE simulation (D + O) is P tot rad = 818 kW, of which the fraction radiated in the SOL (P SOL rad = 664 kW) is f SOL rad 81%. The power radiated by D accounts for 137 kW ( 17% of the total) and the D radiation pattern is more strongly localised in the SOL (f SOL rad 92%). As can be seen in the emissivity map found in figure 5.a, D radiates mainly in the immediate vicinity of the lower divertor targets (recycling) but important contributors are also the entire lower and upper divertor regions as well as the HFS near SOL. The power radiated by O is discussed in section 4. The total radiated power measured by the WEST bolometry system for discharge #54067 and averaged over the t = 6 -9 s plateau is P tot rad 2.3 MW. All bolometry measurements are characterized by an experimental uncertainty of 10%. Here the S2DE simulation cannot be directly compared to the bolometry measurement as the former does not include the contribution from the core plasma, which is the dominant one. Due to the geometry of the channels of the bolometry system, traced in figure 6.a, we can only separate the contributions of lower divertor, bulk plasma and upper divertor, which correspond to P updiv rad 0.2 MW, P bulk rad 1.6 MW and P lodiv rad 0.5 MW, respectively. It is found that the measured P lodiv rad + P updiv rad 0.7 MW is bigger than P SOL rad in S2DE, which accounts also for the contribution of the HFS and LFS edge and SOL. In this experimental setup, the power radiated in these regions cannot be disentangled from P bulk rad . Nevertheless, a quantitative comparison between the value measured along the channels comprised within the S2DE simulation domain is possible thanks to a newly developed SYNthetic DIagnostic: SYNDI is a Monte Carlo beam tracer for computing the power received by a synthetic detector due to the radiation emitted by the simulated plasma. The plasma radiation is prescribed as a 2D map of volume emissivity, assuming toroidal symmetry. The detector is defined as a section of plan (rectangle or circular), with arbitrary orientation and collects the plasma radiation through a diaphragm of arbitrary orientation. A light beam is defined from a random starting position on the detector, and a random passing position on the diaphragm. The beam path is then followed toward the plasma as a straight line, along which the volume emissivity is simply integrated. When the beam path reaches the first wall contour, it is either stopped or allowed to reflect with prescribed properties and probabilities. A prescribed number of beams are processed, line integrals of emissivity are averaged and finally multiplied by the geometrical extension of the detector/diaphragm system. The radiated power measured by the lower divertor bolometry channels 1, 2 and 3 (figure 6.c, black dots) is respectively 13.5, 3.5 and 2 times higher than the one found by SYNDI for the S2DE run (figure 6.c, red squares). This result suggests that the working assumptions behind our simulation lead to a strong underestimation of the baffle top and/or the far LFS SOL contributions (channel 1) but allow to reproduce reasonably well the power radiated in the lower outer divertor (channel 2) as well as the X-point and lower inner divertor region (channel 3). A better agreement is found for the upper divertor channels 14, 15 and 16 (figure 6.b) where S2DE matches the measurements within a factor of 2.8, 0.64 and 1.7, respectively.

Overall, the simulation seems to underestimate the radiated power suggesting that the assumption on the light impurity concentration might be too low with respect to the actual experimental conditions.

Main plasma profiles along the lower divertor

Finally, in figure 7, the background plasma quantities at the lower divertor targets calculated by S2DE are compared with experimental data measured at t = 8 s by two embedded diagnostic systems of WEST: divertor Langmuir probes (DLP) [START_REF] Dejarnac | Submitted to Review of Scientific Instruments[END_REF] and thermocouples (TC) [START_REF] Gaspar | Heat flux estimation in WEST divertor with embedded thermocouples[END_REF][START_REF] Gaspar | First heat flux estimation in the lower divertor of west with embedded thermal measurements[END_REF]. Within the experimental uncertainties of 30%, a nearly quantitative agreement of n e , T e and the parallel heat flux q is found between DLP measurements (black circles) and the S2DE simulation (blue line), as shown in figures 7.a, 7.b and 7.d, respectively. In particular, concerning n e , DLP data suggest that the lower divertor plasma is in the conduction limited regime as the peak at both targets (n div e = 3 to 4 × 10 19 m -3 ) is found to be higher than the average value measured at the outer midplane separatrix via reflectometry (n omp e 1.5 × 10 19 m -3 ), shown above in figure 3.a. This observation is backed by the significant T e gradient found by S2DE between the outer midplane (T omp e 80 eV) and the peak at both targets (T div e 40 eV). The ion saturation current density j sat (figure 7.c) measured by DLP is reproduced by S2DE within the error bars at the inner divertor target. On the other hand, while the outer target profile is satisfactorily matched in the far SOL, the j sat values in the near SOL sit below the DLP error bars. This discrepancy could be due to the absence of mean drift flows, which would recirculate ions from the inner to the outer target, leading to an in/out asymmetry of the j sat profile closer to the experimental one. The agreement in q between S2DE and DLP data is good, in terms of both absolute values and decay lengths. For the estimation of q from DLP data, a sheath heat transmission coefficient γ = 8 is adopted, therefore assuming hot ions (T i 2T e ) and a pure D plasma [START_REF] Stangeby | The Plasma Boundary of Magnetic Fusion Devices[END_REF]. Given the unavailability of T i measurements at the time of writing, such an assumption can be refined by tuning γ for each individual probe using the local T i /T e ratio calculated from S2DE. In fact, the simulated T i /T e at the lower divertor targets (not shown) is as low as 0.7 in the vicinity of the strike points but rises up to 1.2 in the far SOL, leading to γ S2DE values in the range 6.3 to 7.2. The q data from DLP are therefore recalculated for each probe (green circles in figure 7) using γ = γ S2DE /8, resulting in an even better agreement with S2DE, which is quantitative within error bars. Estimates of the same quantity coming from the TC system (red diamonds) further corroborate our observations at the outer divertor target, where TC, DLP and S2DE agree very nicely. On the other hand, at the inner divertor target, TC data present a peak q smaller by factor of 2 than both uncorrected DLP data and S2DE calculation. This discrepancy between the two diagnostics, which is anyway reduced when considering the S2DE-corrected data, is not yet understood at the time of writing. Overall, a good agreement is obtained for both outer midplane and lower divertor main plasma parameters between the WEST boundary plasma diagnostics and the S2DE simulation under the above mentioned assumptions.

Modeling of a generic light impurity 4.1 Impurity poloidal and outer midplane distribution

The multi-fluid nature of S2DE allows the modeling of light impurities without having to enforce the so-called trace approximation, and enables the calculation of the forces that main plasma and impurities exert on each other. This capability was recently enriched with the adoption of a new collisional closure [START_REF] Bufferand | Three-dimensional modelling of edge multi-component plasma taking into account realistic wall geometry[END_REF] using the equations proposed by Zhdanov in his book [START_REF] Zhdanov | Transport processes in multicomponent plasma[END_REF] to express collisional closure terms such as parallel heat conductivities, collisional forces or thermal equipartition coupling sources. In fact, while the radial, cross-field transport is ruled by mean drifts and turbulence, collisional processes are of great importance for the parallel transport, especially in SOL plasmas. In the case of this particular work we assume O as a proxy for various light impurities (C, N, O, B, ...): O is intrinsically present because of both native oxide layers and co-deposited layers on the metallic PFCs and, together with carbon (C), it is suspected to be one of the main sputtering species in early WEST L-mode plasma discharges [START_REF] Van Rooij | [END_REF]. As anticipated in section 3, n IB O n+ = 0.02n IB D + is fixed as condition at the inner boundary of the simulation domain. Then O n+ ions are transported under the abovementioned diffusion coefficients, the same as for D + , while interacting with the background plasma. The resulting poloidal map of n O n+ is shown in figure 8.a: globally, O n+ ions are found in the LFS SOL and in the lower inner divertor region, besides a fairly uniform distribution in the edge plasma. Two peculiar features are the clear effect of the second separatrix, connecting oxygen ions that reach the far SOL the upper divertor, and the depletion of the lower outer divertor region, discussed below. Equally important is to look at the poloidal map of the total O ion concentration, defined here as c O n+ = n O n+ /(n D + + n O n+ ) and shown, in logarithmic scale, in figure 8.b: it is eye catching how c O n+ increases moving radially from the LFS separatrix ( 5%) to more remote LFS areas (5% -> 10%) where less and less D + ions are found. Conversely, in the edge region as well in the HFS SOL and at the two divertors c O n+ < 5%. By zooming in on the lower and upper divertor (figures 8.c and 8.d, respectively) it is possible to appreciate poloidally non symmetric c O n+ patterns. In particular, the outer divertor region is characterized by c O n+ < 1%, whereas the inner divertor one exhibits c O n+ 1% -2%. Such an asymmetry, which for the upper divertor is reversed with overall higher c O n+ values at the targets, can be explained by analyzing the balance of the parallel forces acting on O n+ ions, as discussed below. In terms of radiated power, in our S2DE simulation O contributes with 681 kW ( 83% of the total) with a pattern that is slightly less strongly localised in the SOL (f SOL rad 79%) when compared to D. As the reader can see in figure 5.b, besides a non negligible contribution from the edge, the main radiating zones are the two divertors and the HFS SOL, where the pattern from the inner divertor extends way more poloidally compared to the LFS. In order to understand what is the most represented O ionisation state in

Parallel impurity distribution in the lower divertor

Due to the strong temperature gradients between the outer midplane and the lower divertor targets, the O density distribution in the latter region is significantly different with respect to the former one, as expected. In figures 10.a and 10.b we show parallel profiles of density for all O ionisation states along the main magnetic separatrix in the inner and outer divertor regions, respectively. The results are qualitatively the same for the two divertor regions, with the same high ionisation states that populate the outer midplane dominating at the entrance of the divertor (right hand side of figures 10). As the divertor targets are approached (left hand side of figures 10), the density of low ionisation states becomes comparable and ultimately higher, with all states in the range of 10 17 m -3 , except for O 7+ and O 8+ (orange and yellow curves, respectively), which barely exist at these temperatures. The total O density (black curve), peaks in the vicinity of the divertor targets and then drops moving towards the X-point (parallel distance 5 -10 m). Such drop is stronger for the outer divertor region, consistently with the O depletion region observed in figure 8.a and discussed below. The corresponding parallel profiles of temperature for all O ionisation states are presented in figures 10.c and 10.d, where T D + is added as a reference (red curve). In both divertor regions, which exhibit qualitatively comparable results, the high ionisation states are in thermal equilibrium with the main ion, as already observed at the outer midplane, whereas O + , O 2+ and O 3+ (grey, purple and green curves, respectively) are 5 -15 eV colder. This discrepancy is likely due to the fact that the time needed to these ionisation states to relax through collisions towards a Maxwellian distribution with the same T as the main ion is longer than their ionisation time in this temperature range.

Impurity distribution at the wall contour

Away from the inner simulation boundary, where the transport of O is governed by its interactions with the background, the density ratio n O n+ /n D + can be different from 2%: at the outer strike point of our simulation, for instance, we find n OSP O n+ /n OSP D + = 0.84%, whereas at the inner divertor target n ISP O n+ /n ISP D + = 1.76%. These numbers indicate significant variations of the incident particle flux at the divertor plates, where light impurities govern the sputtered W flux in L-mode discharges as the one studied here. In figure 11 we present the corresponding incident particle flux along the WEST wall contour, where the key PFCs are highlighted by the grey areas: the total (ions plus neutrals) D particle flux Γ D (black line) is overall two orders of magnitude higher than the total O particle flux Γ O n (blue line). Locally, though, the latter can significantly differ from 0.02 × Γ D (red line). In particular, at the lower divertor targets, where both the incident particle flux and the sputtered particle flux (not shown) are peaking, one finds Γ ISP O n /Γ ISP D = 0.75% and Γ OSP O n /Γ OSP D = 0.5%, respectively. Therefore, the assumption of a spatially uniform impurity concentration throughout the whole simulation domain, as done both in previous studies [START_REF] Gallo | First efforts in numerical modeling of tungsten migration in west with soledge2d-eirene and ero2.0[END_REF] and in the companion paper [START_REF] Sepetys | In preparation for Nuclear Fusion[END_REF], represents a potentially important source of uncertainty on the calculation of sputtered fluxes using numerical models like ERO2.0 [START_REF] Romazanov | Beryllium global erosion and deposition at jet-ilw simulated with ero2.0[END_REF]. The above-mentioned strong in/out asymmetry in O concentration, and ,d) temperature for all oxygen ionisation states in inner and outer divertor, respectively. Calculation from SolEdge2D-EIRENE along the main magnetic separatrix. hence in particle flux, observed in the lower divertor of our S2DE simulation can be explained by the balance between two main forces acting on impurities as these are among the outputs of the Zhdanov model: i) friction forces between O n+ and both electrons and main ions, ii) forces due to the electron and main ion thermal gradients. Figure 12 represents these forces along a flux surface comprised between the main and the secondary separatrix, therefore in the near SOL. The zero of the horizontal axis corresponds to the inner divertor target, the other end to the outer divertor target. In the left portion of the viewgraph, if a force is positive it means it is directed towards the inner divertor target. In the right hand side, instead, a positive

Comparison with VUV spectroscopy measurements

The O spatial distribution obtained by S2DE can be compared with experimental data from the WEST vacuum ultraviolet spectrocopy system (VUV) [START_REF] Schwob | Highresolution duomultichannel soft xray spectrometer for tokamak plasma diagnostics[END_REF]. The VUV spectrometer is located at the LFS equatorial plane and has a single line of sight that can be scanned around the poloidal angle in such a way that its intersection with the WEST wall contour ranges between the inner midplane and the baffle. Further technical details concerning the VUV diagnostic system and the processing of the experimental data can be found in the companion paper [START_REF] Sepetys | In preparation for Nuclear Fusion[END_REF]. The following O 2+ and O 3+ lines (OIII and OIV in spectroscopic notation) are identified in the WEST VUV spectra of discharge #54067: OIII 599.6 Å, OIII 610.4 Å, OIV 609.8 Å, OIV 625.1 Å, OIV 625.9 Å. Since the VUV spectroscopy data are not absolutely calibrated, the experimental poloidal profiles can provide a relative information concerning the O light emission at the different poloidal angles and integrated along the line of sight, but cannot be used to quantify the emissivity. Therefore, both VUV data and S2DE results are normalised to their maximum value for the sake of this comparison. For a given impurity Z emitting a spectral line at the wavelength λ, the emissivity λ = n e n Z PEC λ is calculated over the whole simulation domain using poloidal maps of n e , n Z provided by S2DE and the photon emission coefficient of this particular line PEC λ . The latter is found in the ADAS database in the form of PEC λ (n e , T e ) matrices and is then interpolated over the poloidal maps of n e and T e from S2DE using a spline method. An example of such λ map for the OIII line at 599.6 Å is shown in figure 13.a. In this particular discharge, instead of scanning the full poloidal angle range (θ = 9 • -28 • ), only the region of the lower divertor (θ = 21 • -24.2 • , magenta lines) is scanned during the steady state plateau between 6 and 9 s in order to increase the spatial resolution with the aim of detecting the fine details of the OIII and OIV line emission patterns. By integrating the value of λ [ph m -3 s -1 ] along the VUV line of sight and dividing by 4π one can calculate the corresponding brightness B λ [ph m -2 s -1 ] profiles as a function of the poloidal angle. Such profiles are presented for OIII line at 599.6 Å, for the overlapping OIII and OIV lines at 610.4 Å and 609.8 Å, and for the OIV multiplet at 625.1 Å and 625.9 Å in figure 13.b, 13.c and 13.d respectively. In this set of figures the VUV experimental data are represented by the black diamonds, whereas the S2DE simulation is the solid blue line. The first striking feature is that, for all three cases, the peak of the normalised brightness B λ /B max λ calculated by S2DE occurs along the lines of sight going through the inner divertor target (θ = 21 • -21.5 • ). This finding agrees well with VUV measurements within the 0.3 • instrumental uncertainty (horizontal black lines). Another common feature between the simulations and VUV data is that B λ /B max λ is lower by a factor of 0.4 -0.6 along the lines of sight going through the outer divertor target (θ = 24 • -24.2 • ). This consistency concerning the in/out asymmetry of O in the lower divertor region give us confidence in our capability to efficiently model the spatial distribution of light impurities with S2DE and, therefore, to thoroughly estimate on which PFCs the peaking of the incident light impurity flux and of the consequent sputtered flux occurs. Nevertheless, our description requires to be further improved as the S2DE results clearly disagree with the VUV profiles in the poloidal angle range θ = 22 • -22.5 • , which corresponds to lines of sight going through the private flux region (PFR), where experimental data are characterized by a third, intermediate, peak. Here, for all three cases, the simulation exhibits a much lower B λ /B max λ . This disagreement, which is still not understood at the time of writing, could be related to the absence of E×B drifts in this particular simulation: such drifts are known to recirculate particles between the two divertor legs and might enhance the transport of O n+ ions in the PFR as well as around the X-point. Furthermore, the increase of the divertor density during the VUV scanning time window discussed in section 2 could represent an additional source of disagreement between the simulations, performed as a snapshot at t = 8 s, and the experimental data, acquired over a 3 s time window in which the background plasma is not exactly stationary. This hypothesis has to be tested with future simulations. On the other hand, an absolute calibration of the VUV spectroscopy data will enable a quantitative comparison with S2DE calculations, enabling not only the study of the spatial distribution of the identified light impurities but also the estimation of their fractional abundance in the WEST plasma.

Conclusion and future work

In this paper we studied the transport of light impurities in the boundary plasma of a WEST discharge by interpreting a number of experimental measurements from SOL and divertor diagnostic systems with the help of the SolEdge2D-EIRENE transport code. By making experiment-based hypotheses on input power, outer midplane density and transport coefficients, we modeled a main plasma whose parameters are in good qualitative or even quantitative agreement with the measurements of a diverse set of diagnostics. This analysis allows us to check the consistency between various measurements at different locations and based on different physical phenomena, while at the same time estimating the main plasma parameters also in regions where measurements are not available. The working assumption of a 2% O concentration at the inner simulation boundary, as a proxy for a generic light impurity, leads to an overall underestimation of the level of radiated power. This discrepancy suggests that the assumed light impurity content could be too low and that, moreover, a more complex impurity mix should be considered: visible spectroscopy measurements showed that C, for instance, was one of the dominant light impurities during the WEST C3 experimental campaign, as the damage caused by isolated transient events removed part of the W coating from some carbon-based PFCs. For this reason, future modeling efforts will fully exploit the multi-fluid capabilities of SolEdge2D-EIRENE to simulate a D/O/C WEST plasma with various concentrations of these impurities. Measurements of the line-integrated emission distribution of two O ionisation states were obtained by scanning the lower half of the plasma with the line of sight of the unique WEST VUV spectroscopy system: uncalibrated measurements of several OIII and OIV lines showed that the maximum of the O 2+ and O 3+ light emission occurs at the poloidal angle at which the line of sight goes through the inner divertor leg. The signal coming from the outer divertor leg is different by a factor between 0.4 and 0.6, while a third intermediate brightness peak appears at poloidal angles corresponding to the PFR. The application of a synthetic diagnostic to the SolEdge2D-EIRENE simulation highlighted a similar O pattern, with relative brightness between inner and outer divertor in agreement with the experiment, giving us confidence regarding our capabilities of modeling the light impuritiy transport. Nevertheless, the model failed to reproduce the intermediate PFR peak. This disagreement could potentially be linked to the absence of E × B drifts in the simulation, which would modify transport in the PFR and, therefore, will be accounted for in future efforts, as already performed in [START_REF] Bufferand | Implementation of drift velocities and currents in soledge2deirene[END_REF]. Another technical limitation of this work is the fact that the assumed O fraction was imposed at the inner boundary of the simulation which is found in the edge region. Coming mostly from the surface of the PFCs, O should rather be treated like a sputtered species and, therefore, generated at the wall contour. Future development efforts have to go in the direction of a more accurate mix of impurities, each of which originating from the relative PFC location. In this panorama, the effect of boron, an impurity which is now routinely and intentionally introduced, will have to be carefully analyzed. Another potential source of disagreement is the accuracy of the main plasma reproduced by the code: despite the overall satisfactory agreement presented in this work, further improvements will be enabled by the growing availability and reliability of measurements from the WEST diagnostic systems during more recent and future experimental campaigns.

Figure 2 :

 2 Figure 2: a) domain of the SolEdge2D-EIRENE simulation (green area), WEST wall contour with relevant plasma-facing components (black solid line), inner simulation boundary (blue solid line), main and secondary magnetic separatrix (solid and dashed red line); b) Magnetic flux contours for WEST discharge #54067 at t = 8 s as calculated by VacTH (red lines) and modified with input from plasma diagnostics and simulations (blue lines).

Figure 3 :

 3 Figure 3: outer midplane radial profiles of a) density and b) temperature with main (dashed line) and secondary (dotted line) separatrix, wall contour (thick solid line). Solid curves for SolEdge2D-EIRENE simulations (blue for electrons, red for ions), green dots with horizontal error bars for fast sweeping reflectometer data, cyan (uncorrected) and pink (corrected) dots for reciprocating Langmuir probe data (remapped at the outer midplane).

Figure 4 :

 4 Figure 4: poloidal maps of deuterium ion a) density and b) temperature calculated by SolEdge2D-EIRENE with main (solid line) and secondary (dashed line) magnetic separatrix. The assumptions on deuterium ion density and power at the inner simulation boundary are recalled in the cutouts.

Figure 5 :

 5 Figure 5: poloidal maps of emissivity for a) deuterium and b) oxygen calculated by SolEdge2D-EIRENE. The total radiated power P tot rad and the fraction radiated in the scrape-off layer f SOL rad are reported in the cutouts.

Figure 6 :

 6 Figure 6: a) poloidal map of emissivity calculated by SolEdge2D-EIRENE with bolometry channels (grey lines); b) power radiated along channels 14 to 16 and c) along channels 1 to 3 from bolometers (black dots) and calculated by SolEdge2D-EIRENE (red squares).

Figure 7 :

 7 Figure 7: Radial profiles of a) electron density, b) electron temperature, c) ion saturation current density and d) parallel heat flux along the lower divertor. Blue lines for SolEdge2D-EIRENE simulation, black circles for divertor Langmuir probes data and red diamonds for thermocouples data. Green circles for divertor Langmuir probes data corrected with ion-to-electron temperature ratio from SolEdge2D-EIRENE.

Figure 8 :

 8 Figure 8: Poloidal maps a) total oxygen density and b) total oxygen concentration with details of the c) lower and d) upper divertor zones. Calculation from SolEdge2D-EIRENE with main (solid black line) and secondary (dashed black line) magnetic separatrix.

Figure 9 :

 9 Figure 9: outer midplane radial profiles of a) density and b) temperature for oxygen high ionisation states with main (dashed black) and secondary (dotted black) separatrix and wall contour (thick black) vertical lines. Calculation from SolEdge2D-EIRENE.

Figure 10 :

 10 Figure 10: Parallel profiles of a),b) density and c),d) temperature for all oxygen ionisation states in inner and outer divertor, respectively. Calculation from SolEdge2D-EIRENE along the main magnetic separatrix.

Figure 11 :

 11 Figure 11: Wall contour profiles of the total deuterium flux (black line), 2% of the total deuterium flux (red line) and total oxygen flux (blue) with the most relevant WEST plasma-facing components (grey areas). Calculation from SolEdge2D-EIRENE.

Figure 12 :

 12 Figure 12: Parallel profile of the main forces acting on oxygen ions: friction with deuterium ions (red solid line), friction with electrons (red dashed line), deuterium ion thermal gradient (blue solid line) and electron thermal gradient (blue dashed line), net force (black thick line). Calculation from SolEdge2D-EIRENE.

Figure 13 :

 13 Figure 13: a) poloidal map of emissivity for the OIII line at 599.6 Å from SolEdge2D-EIRENE with outermost VUV spectroscopy lines of sight (magenta lines), main (solid white line) and secondary (dashed white line) magnetic separatrix; b) angular profile of normalised brightness for OIII 599.6 Å, c) 610.4 Å + OIV 609.8 Å and d) OIV 625.1 Å + OIV 625.9 Å with SolEdge2D-EIRENE simulation (blue lines) and VUV spectroscopy data (black diamonds with horizontal error bars).

Table 1 :

 1 Main input parameters for the SolEdge2D-EIRENE modeling.

		Value	Unit	Motivation
	n IB D + P IB tot	2.5 × 10 19 m -3 2.3 MW	n IB e to match fast sweeping reflectometer Power balance (heating and bolometry)
	n IB O n+ D, ν	5 × 10 17 1	m -3 m 2 s -1 Best match to divertor Langmuir probes Visible spectroscopy (previous studies)
	χ	2	m 2 s -1 Best match to probes and thermocouples
	R	0.99	-	Standard assumption for W PFCs
	A	0.9	-	Standard assumption for WEST pumps

Acknowledgements

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Work performed under EUROfusion WP PFC. Centre de Calcul Intensif dAix-Marseille is acknowledged for granting access to its high performance computing resources.


F Nespoli
email: fnespoli@pppl.gov

P Tamain

N Fedorczak

D Galassi

Y Marandet

A new mechanism for filament disconnection at the X-point: poloidal shear in radial E × B velocity

Plasma filaments generated by turbulence in the scrape-off layer are sometimes observed in experiments and simulations to disconnect from the target plate in the vicinity of the X-point, resulting in a "quiescent zone" with reduced fluctuations, possibly affecting the fluctuating part of the divertor heat loads. This phenomena is usually explained by the flux tubes squeezing, induced by flux expansion, down to scales where collisions dominate. In this work, we consider an additional mechanism spontaneously arising at the X-point: the poloidal shear in radial E × B velocity. Through a simple model, tested against 3D global turbulence simulations, we quantify the effectiveness of this disconnection mechanism and identify ways to control it.

Introduction

Plasma filaments, also referred to as blobs, are generated spontaneously at the edge of the toroidally confined plasmas by turbulence through charge separation due to charge-dependent drifts [START_REF] Krasheninnikov | Recent theoretical progress in understanding coherent structures in edge and SOL turbulence[END_REF]. In the commonly accepted description, they are generated by interchange type instabilities in the zone of bad curvature (typically the low field side close to the outboard midplane) and the perturbation extends along the field lines towards the limiter or divertor plates, to which they are in general electrically connected. Filaments are self-propelled radially outwards by an internal E × B drift. Under certain conditions, they can disconnect from the target plates increasing their radial velocity i.e. the aging of a reactor first wall. Furthermore, they are believed to impact and, under certain circumstances, even completely determine the scrape-off layer (SOL) width [START_REF] Carralero | Experimental validation of a filament transport model in turbulent magnetized plasmas[END_REF][START_REF] Militello | Scrape off layer profiles interpreted with filament dynamics[END_REF]. The understanding of filamentary transport is therefore crucial for the physics of the SOL and for the correct prediction of the heat loads on the first wall. Recently, experimental measurements involving fast framing imaging in the MAST [START_REF] Walkden | Quiescence near the X-point of MAST measured by high speed visible imaging[END_REF] and NSTX [START_REF] Maqueda | Intermittent divertor filaments in the National Spherical Torus Experiment and their relation to midplane blobs[END_REF][START_REF] Scotti | Divertor leg filaments in NSTX-U[END_REF] tokamaks gave indications that the filaments disconnect from the target plates close to the X-point. This results in MAST in a "quiescent zone" close to the X-point, where the plasma fluctuations are substantially reduced. A similar phenomena has been observed in turbulence simulations of the COMPASS [START_REF] Galassi | Drive of parallel flows by turbulence and largescale E B transverse transport in divertor geometry[END_REF] and WEST [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF] tokamaks using the flux-driven turbulence code TOKAM3X [START_REF] Tamain | The TOKAM3X code for edge turbulence fluid simulations oftokamak plasmas in versatile magnetic geometries[END_REF]. In NSTX, the disconnection of the midplane filament from the divertor results, in the proximity of the strike point, in a decrease of the fluctuation level with respect to rest of the SOL [START_REF] Scotti | Disconnection of SOL turbulence between the outer midplane and divertor target plate in NSTX[END_REF]. Decorrelation of turbulence across the X-point region has also been observed in simulations with both the BOUT [START_REF] Umansky | Simulation of turbulence in the divertor region of tokamak edge plasma[END_REF] and ArbiTER [START_REF] Baver | Penetration of filamentary structures into the divertor region of spherical tokamaks[END_REF] codes. The disconnection of filaments at the X-point is usually explained by the flux tubes squeezing, induced by flux expansion, down to scales where collisions dominate [START_REF] Farina | Effectcs of the magnetic field geometry on the flute-like perturbations near the divertor X point[END_REF]. In this work, we consider an additional mechanism for blob disconnection spontaneously arising at the X-point: the poloidal shear in radial E × B velocity. The rest of the paper is structured as follows: in section 2 we discuss the results of the TOKAM3X simulations showing blob disconnection at the X-point, resulting in the formation of a quiescent zone. In section 3 we discuss the disconnection of filaments at the X-point through flux expansion driven squeezing of the flux tubes. In section 4 we consider an additional mechanism for the disconnection of blobs at the X-point, namely the poloidal shear in radial E × B velocity, spontaneously arising in the presence of an X-point. In section 5 we derive, in steps of increasing complexity, a condition for the effectiveness of this disconnection mechanism. Finally, in section 6, we compare the two considered mechanisms for blob disconnection and we discuss the implications for plasma operation.

Quiescent zone in TOKAM3X simulations

Throughout this paper, to support our reasoning, we will use results from TOKAM3X simulations. TOKAM3X [START_REF] Tamain | The TOKAM3X code for edge turbulence fluid simulations oftokamak plasmas in versatile magnetic geometries[END_REF] is a flux-driven fluid code modeling turbulence in the plasma edge and SOL in realistic magnetic geometry. Throughout this paper we will focus mainly on the simulations described in details in Ref. [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF], modeling the edge and SOL of the WEST tokamak in isothermal approximation (T e = T i = T 0 ), with T e and T i the electron and ion plasma temperature respectively, and T 0 the reference value for plasma temperature. In these simulations, lengths and times are expressed in units of the sound speed ion Larmor radius ρ s and of ω -1 c respectively, with ω c the ion cyclotron frequency. In Ref. [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF] the blob 3D structure and dynamics have been investigated in detail complementing different analysis techniques such as 2D and 3D conditional average sampling and blob tracking. As a result, it was shown how the filaments connected to the outer midplane (OMP) terminate in correspondence of the X-point when close to the separatrix. One of such filaments, as detected from the 3D blob recognition and tracking algorithm (BRAT) is shown in Fig. 1, color Figure 1: Snapshot of a low-field side filament terminating in the vicinity of the X-point, as detected from the 3D blob recognition and tracking algorithm [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF] in TOKAM3X simulation of the WEST tokamak, color coded with the normalized density fluctuation δn/n 0 . The divertor plates are plotted in black. δn/n 0 is also shown for four different poloidal planes. coded with the density fluctuation δn/n 0 , also plotted on four different poloidal planes, with n 0 the reference value for density. In Fig. 2 we show the relative fluctuation level σ n / n , where σ n is the standard deviation over time of the plasma density n and the brackets denote average over toroidal direction and time, normalized to the maximum value at the OMP in order to compare the different configurations. As it emerges from the picture, as one approaches the X-point the fluctuation level is reduced by a factor 3 with respect to its value in the main SOL, resulting in a "quiescent zone" caused by the disconnection of the filaments. Below the X-point, the fluctuation level remains low along the divertor leg all the way to the divertor plate. Here, the relative fluctuation level σ n / n ≤ 0.39 (corresponding to a normalized value of 1.02 in Fig. 2)in the far SOL, and decreases monotonically as one moves radially inwards towards the strike point, where σ n / n =0.07 (normalized value 0.18). A similar decrease of the fluctuation level is observed in correspondence of the secondary X-point at the top of the plasma. In Fig. 2 the normalized relative fluctuation level is also shown for TOKAM3X isothermal simulations in COMPASS (analyzed in details in Ref. [START_REF] Galassi | Tokamak Edge Plasma Turbulence Interaction with Magnetic Xpoint in 3D Global Simulations[END_REF]) and TCV geometry, respectively. For the three simulations, with different size, resolution and grid geometry, a similar reduction of the fluctuation level in the vicinity of the X-point is observed, suggesting that this phenomena is not linked to the details of the magnetic configuration and/or the grid used for the simulations and could occur in all diverted plasmas.

Flux tube squeezing at the X-point

In the following, we discuss the possible mechanisms responsible for the disconnection of filaments at the X-point. We consider a blob at radial upstream position r 0 of radial and poloidal size at the midplane a r and a θ , respectively. The flux expansion in correspondence of the X-point at this r v w Figure 3: Deformation of the flux tube associated with a blob (in red) as resulting from field line tracing for the magnetic geometry used in the WEST TOKAM3X simualtion (coarser grid shown in grey). The poloidal coordinates (r, θ) and the one used in Ref. [START_REF] Farina | Effectcs of the magnetic field geometry on the flute-like perturbations near the divertor X point[END_REF] (v, w) are shown in black and blue respectively in correspondence of the X-point. radial position is f x . In tokamaks, a θ has been measured to vary from a few to a few hundreds ρ s [START_REF] Carralero | Experimental validation of a filament transport model in turbulent magnetized plasmas[END_REF], with the typical value of ∼ 20ρ s [START_REF] Zweben | Edge and SOL turbulence and blob variations over a large database in NSTX[END_REF][START_REF] Fedorczak | Width of turbulent SOL in circular plasmas: A theoretical model validated on experiments in Tore Supra tokamak[END_REF][START_REF] Kirk | L-mode filament characteristics on MAST as a function of plasma current measured using visible imaging[END_REF][START_REF] Tsui | Filamentary velocity scaling validation in the TCV tokamak[END_REF][START_REF] Boedo | Transport by intermittent convection in the boundary of the DIII-D tokamak[END_REF], and typically the blob poloidal cross section has elliptical shape with a θ > a r . The main cause of blob disconnection at the X-point is so far considered to be the squeezing and tilting of the flux tubes as they go through the high flux expansion and magnetic shear region in correspondence of the magnetic null, down to spatial scales where collisional processes dominate, finally decoupling the perturbations in the upstream SOL from the ones in the divertor leg [START_REF] Farina | Effectcs of the magnetic field geometry on the flute-like perturbations near the divertor X point[END_REF]. An example of the flux tube deformation is shown in Fig. 3. Following Ref. [START_REF] Farina | Effectcs of the magnetic field geometry on the flute-like perturbations near the divertor X point[END_REF], as the flux tube associated with the filament with poloidal size a θ at the midplane approaches the X-point region, it stretches in the direction parallel to the separatrix leg (v in Fig. 3) accordingly to the flux expansion f x ∼ B p,OM P /B p , where B p is the poloidal field and B p,OM P its value at the OMP for the same flux surface. In order to conserve its magnetic flux, the filament also squeezes in the direction perpendicular to the separatrix leg (w in Fig. 3) by the same factor f x . We can consider the flux expansion to effectively damp the density fluctuation associated to the OMP filament when its poloidal size becomes a w,X a min , where the X subscript denotes the X-point poloidal location, a min ∼ ρ s is the scale where collisional processes become important, and ρ s is the sound speed Larmor radius. This disconnection mechanism is therefore effective for

f x a θ /a min ( 1
)
The latter is always satisfied in the direct vicinity of the X-point as f x diverges by definition, and more generally on a layer whose thickness depends on the details of the magnetic configuration (radial profile of f x ) and on the size of the blobs at the OMP a θ . Though, this condition also means that at a given radial position, all filaments with large enough poloidal extension a θ /ρ s f x can survive the X-point squeezing and extend further towards the target plate.

Poloidal shear in radial E × B velocity at the X-point

We consider here an additional mechanism involved in the disconnection of the blobs at the X-point, namely the poloidal shear in radial E × B velocity. As we will show in this section, this shear layer arises spontaneously in diverted plasmas at the X-point as the topology of the region adjacent to the SOL changes from closed to open field lines, and it is therefore to be found universally in all diverted plasmas, with some exceptions that will be discussed later. Indeed, the commonly accepted description of the divertor leg below the X-point is the one depicted in Ref. [START_REF] Wagner | A study of the perpendicular particle transport properties in the scrapeoff layer of ASDEX[END_REF], and summarized in the following. The plasma enters the divertor at the X-point with an exponentially decaying density in the SOL n(x) = n X exp(-r/λ), with λ the SOL width and n X the density at the X-point, and diffuses along the divertor leg and into the private flux region. In the following we use the coordinate system (r, θ, z) such that r is the radial coordinate remapped at the outer midplane, r = 0 at the separatrix and increasing in the SOL; θ is the coordinate along the poloidal direction and z is the coordinate along the magnetic field. Both θ and z are null at the X-point location and increase towards the target. The plasma density is described by the continuity equation

∂ t n + ∇ • (n(v || + v ⊥ )) = ∇ • (D∇ ⊥ n) (2)
which, assuming steady-state condition, uniform parallel velocity v || and diffusion coefficient D and neglecting the drifts velocities v ⊥ can be rewritten in our coordinate system as D∇ 2 r n = v || ∇ z n. The plasma density n(r, z) is therefore solution of the classic heat diffusion equation with boundary condition n(r, z = 0) = n X exp(-r/λ) resulting, for z ≥ 0, in

n(r, z) = n X 2 exp S 2λ 2 - r λ erfc S 2λ - r S (3)
with n X = n(r = 0, z = 0) the density at the X-point and S the spreading factor

S = S(z) = 4Dz v || . ( 4
)
We recall that, in this description, S is a function of z and of the transport parameters D and v || . This is indeed a good description of the density distribution in the divertor leg in our simulations, as it is shown in Fig. 4a for the WEST case. Here, the parallel profile of plasma density n is plotted at different positions in the SOL along the parallel coordinate z, from the X-point location (z = 0) up to the target plate (thick black line in the picture). In the figure, the fit with Eq. ( 3) is shown with a red line for the parallel profile taken at the radial position the closest to the separatrix, leading to D = 10 -3 ρ 2 s ω c and λ = 1.1ρ s , resulting in a good agreement in the near SOL. Such density distribution along the divertor leg exhibits a strong gradient in the parallel direction. This induces in turn a parallel gradient in the plasma potential as well, as it is shown in Fig. 4b. Indeed, from parallel force balance

nηj || = ∇ || P e + 0.71n∇ || T e -n∇ || φ (5)
and neglecting the first term (typically the resistivity η < 10 -5 ) and in isothermal approximation (∇ || T e = 0, ∇ || P e = T e ∇ || n), we can write

∇ || n n ∼ ∇ || φ T e (6)
and the same relation holds for the gradients in the poloidal (θ) direction due to the toroidal symmetry of the system. This results in a radially directed where we used the approximation B t ∼ B for the toroidal magnetic field B t , valid in the vicinity of the X-point. The radial velocity given by Eq. ( 7) also decreases (in absolute value) along the divertor leg, resulting finally in a shear layer just below the X-point. Such result is confirmed by our simulations, how it emerges from Fig. 5, where we plot the radial component of the E × B velocity in the surroundings of the X-point for the WEST case. Here, the shear layer extends poloidally over approximately four simulation cells, corresponding to a poloidal span of l θ ∼ 53ρ s , corresponding to a segment of field line of parallel length 2143 ρ s , with a variation in radial velocity of approximately ∆v r /c s ∼ 0.02. We remark that this result is in first approximation independent of the direction of the magnetic field, whose reversal would result in the change of sign for the E × B velocity but not its absolute value and the shearing rate. Finally, we remark that this model of the divertor leg and therefore of the shear layer below the X-point is valid as long as the plasma density n is decreasing along the divertor leg, i.e. always in divertor sheath limited regime, and also in high-recycling and detached regimes, as long the X-point is sufficiently far from the divertor plates/detachment front so that in the vicinity of the X-point n is still decreasing towards the target. The other cases, where the density gradient along the divertor leg are reversed, could still result in shear layer at the X-point but can not be described by the model exposed here and would need to be addressed in detail in future works.

E × B drift v r = - ∇ θ φ × B B 2 = - T e B ∇ θ n n (7)

Effectiveness of disconnection through E × B shear

In this section we derive, in steps of increasing complexity, a condition for this disconnection mechanism to be effective. For each of these approximations, we compare with the simulation results. In this process, several assumptions and approximations are used, which are necessary to the final aim of having an analytical expression for the aforementioned condition. Nevertheless, the assumptions made are always conservative, providing an underestimation of the effectiveness of this disconnection mechanism, that has to be regarded as a lower limit. This approach is justified a posteriori by the comparison with the simulation results, showing how the condition derived under these restrictive assumptions is still satisfied, and provides a baseline for future works.

Local approximation

As a first approximation the poloidal shear of radial velocity can locally compete with parallel dynamics, and therefore help in disconnecting the filaments at the X-point, when

τ || τ sh,θ = ∆L || c s ∆v r ∆L θ = B B p ∆v r c s 1 ( 8
)
where we compare the shearing time τ sh,θ = (∆v r /∆L θ ) -1 with the characteristic time of the parallel dynamics for this problem τ || = ∆L || /c s , where we assumed that the perturbation associated with the blob travels along the field line at the sound speed c s . ∆v r is the variation of the radial velocity across the shear layer of poloidal width ∆L θ , corresponding to a portion of the field line of length ∆L || . This condition is satisfied in the vicinity of the X-point for our simulations, as it emerges from Fig. 5, where the radial component of the E × B velocity is shown, averaged over time and the toroidal direction, together with the contour levels of B p /B. Indeed, across the poloidal shear layer ∆v r /c s ∼ 0.02, which is bigger than B p /B in the shear layer region. The condition in Eq. ( 8) can also be applied locally to each simulation cell, with ∆v r the variation of the radial velocity in the poloidal direction across the simulation cell. The result is shown in

Blob radial displacement through a layer with uniform velocity

We can rewrite the approximate condition in Eq. ( 8) in a more quantitative way: we consider a blob of radial extension a r,X just above the X-point. As the density perturbation front travels along the field line across the radial velocity shear layer, it is displaced radially by the distance ∆r = t l 0 (v r -v 0 )dt where t l = l || /c s = (B/B p )l θ /c s is the time necessary to cross the shear layer of poloidal extension l θ and v 0 is the radial velocity above the X-point. We consider the blob to be effectively disconnected by the X-point shear layer if ∆r a r,X . As a first approximation, considering the radial velocity uniform across the shear layer, this condition results in

B B p ∆v r c s a r,X l θ (9)
where ∆v r = v r -v 0 . We remark that this condition recovers approximately our first estimate in Eq. ( 8), as the term at the right hand side is typically around 1. Indeed, for our simulation in Fig. 1 the poloidal extension of the shear layer is l θ ∼ 53ρ s and the radial blob size at the midplane computed in Ref. [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF] with a conditional average technique is a r ∼ 5ρ s , resulting in a radial blob size at the X-point a r,X = f x a r with f x ≤ 14 in the simulation and the RHS term is therefore a r,X /l θ = 1.32. Since in this simulation ∆v r ∼ 0.02c s , Eq. ( 9) is satisfied for B p /B 0.015, similarly to what estimated previously, which is still valid for the X-point velocity shear layer (Fig . 5). We remark that the latter provides only an underestimation, since we neglected the effect introduced by the magnetic shear that, as shown in Fig. 3, results in the tilting of the flux tube cross section with respect to the radial direction at the X-point of an angle α, facilitating the filament disconnection by the shear in radial E × B. Indeed, the radial projection of the cross section in correspondence of the X-point has then to be multiplied by the tilting factor

Θ = cos 2 α + sin 2 αf 4 x (a r /a θ ) 2 -1/2 1 ( 10
)
since typically α ∼ π/4.

Blob radial displacement through the poloidal shear layer

We proceed now to the evaluation of the radial displacement ∆r resulting from the velocity shear given by the density distribution along the divertor leg described by the model in section 4:

∆r = t l 0 (v r -v 0 )dt = - T e Bc s z l 0 ∇ z n n B B p dz (11)
where t l = z l /c s is the time necessary to cross the shear layer of width z l along the parallel coordinate z and we used ∇ θ = (B/B p )∇ z . Even though, as it will be discussed further in section 6, the radial velocity above the X-point is usually v 0 > 0, here we make the conservative simplifying assumption v 0 = 0. We remark that in the surrounding of the magnetic null the poloidal field B p can be expressed as B p = B p (r, z) ∝ re az [START_REF] Farina | Effectcs of the magnetic field geometry on the flute-like perturbations near the divertor X point[END_REF], where a is a proportionality constant. The integral in Eq. ( 11) diverges therefore for the flux surface coinciding with the X-point (r = 0), but it can be shown to be finite arbitrarily close to it (r > 0). In the following, for simplicity, we give a lower estimate of ∆r, by neglecting the z dependency of B p and substituting in the integral with its value just after the shear layer B p (z l ), since for z in the interval [0, z l ] we have

B p (z) ≤ B p (z l ) ∆r ≥ - T e c s B p (z l ) z l 0 ∇ z ln n n X dz = T e c s B p (z l ) F (12)
where we define F = -ln [n(r, z l )/n(r, 0)]. F is a function of the shear layer parallel width z l and of the radial coordinate r, and can be computed directly once the poloidal distribution of the plasma density n is known. Nevertheless, in order to obtain an analytical expression for F, we consider the density distribution across the divertor leg given in Eq. [START_REF] Walkden | Quiescence near the X-point of MAST measured by high speed visible imaging[END_REF][START_REF] Militello | Scrape off layer profiles interpreted with filament dynamics[END_REF]. F can then be expressed as a function of the transport parameters D and λ:

F = - Dz l v || λ 2 -ln 1 2 erfc Dz l v || λ 2 -r v || 4Dz l (13)
We can further simplify F by giving an estimate of the shear layer width resulting from our model. This can be done using an analytic approximation of erfc in in Eq. ( 3), e.g. erfc(χ) ∼ 1tanh(χ √ π ln 2), and considering the proximity of the divertor leg r ∼ 0, keeping in mind that F decreases approximately linearly with r for r ≤ 2λ. The approximated shear layer width along the field line is therefore z l ∼ π(ln 2) 2 v || λ 2 /D. By plugging this estimate in Eq. ( 13), the transport parameters simplify giving F = 1.68. The condition in Eqs. [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF][START_REF] Tamain | The TOKAM3X code for edge turbulence fluid simulations oftokamak plasmas in versatile magnetic geometries[END_REF], asserting when the poloidal shear of radial velocity can have a role in disconnecting the filaments at the X-point, can then be rewritten as a (weak) function of the transport parameters

B p B T e B F c s a r,X (14)
where we identified v || ≡ c s , and we remind that in general both B p and F depend on r, z l . The condition in Eq. ( 14), once again, is always satisfied at the X-point as B p = 0 by definition here, and is further satisfied on a region whose radial extension depends on the plasma and transport parameters and on the details of the magnetic equilibrium. For the WEST simulation using the values resulting form the fitting of the density profiles in the near SOL divertor leg with Eq.

(3) i.e. D = 10 -3 ρ 2 s ω c , λ = 1.1ρ s , with z l ∼ 2000ρ s , a r ∼ 5ρ s and a r,X = f x a r with f x ≤ 14 in the simulation, the condition in Eq. ( 14) results in B p /B 0.022, consistently with the value we estimated by applying the approximate condition Eq. ( 8) directly to the simulation output (B p /B ∆v r /c s ∼ 0.02). Once again, the latter is an underestimate since we neglected the tilting of the flux tube due to magnetic shear, reducing the radial projection of the filament at the X-point by a factor Θ < 1 (Eq.( 10)).

Discussion

Summarizing, the disconnection of filaments at the X-point observed in both experiments and turbulence simulations can be due to two mechanisms, namely the deformation of the flux tubes by the high flux expansion and magnetic shear at the X-point, and the poloidal shear of radial E × B velocity, discussed for the first time in this work. Both mechanisms arise naturally in the presence of an X-point, and can act synergistically to decorrelate the density fluctuations in the X-point region resulting in the formation of a quiescent zone, possibly affecting both the blob dynamics at the midplane (i.e. the particle flux to the first wall), and the fluctuating part of the heat loads on the divertor plates. Indeed, a blob of radial and poloidal size at the midplane a r and a θ , respectively, at at radial upstream position r and flux expansion in correspondence of the X-point f x , be squeezed down to scales where collisional processes dominate and will terminate at the X-point if a θ /ρ s f x . Otherwise, blob with bigger poloidal size a θ /ρ s > f x can survive the X-point squeezing and extend further towards the target plate. These filaments encounter, just below the X-point, the poloidal shear layer of radial E × B velocity. The latter causes the filament to lose coherence if

f x T e F B p c s a r Θ
, where we have rewritten Eq. ( 14) by recalling a r,X = f x a r Θ. The probability of the blob being disconnected by the shear layer decreases with their radial size at the outer midplane a r and with the poloidal field at the shear layer location (i.e., at fixed f x , Θ, with the plasma current I p ), and increases with the plasma temperature. This second mechanism can therefore help in suppressing the blobs of bigger poloidal size that would survive the X-point squeezing, resulting in an extended quiescent zone similarly to what has been observed in MAST.

The relative importance of the two disconnection mechanisms could be assessed in a dedicated experiment in which the magnetic configuration (f x profile, Θ) is kept fixed and i) the plasma temperature is changed scanning the input power, ii) the poloidal field is varied at fixed flux expansion. The quiescent zone could be monitored by fast framing imaging similarly to Ref. [START_REF] Walkden | Quiescence near the X-point of MAST measured by high speed visible imaging[END_REF], and the fluctuation level at the strike point could be measured by Langmuir probes embedded in the divertor. The size of the blob at the midplane, affecting the efficiency of both mechanisms and on which one has close to no control while planning a plasma discharge, needs to be monitored by gas puff imaging or through measurements from reciprocating Langmuir probes. Furthermore, in section 5.3 we considered the radial velocity above the X-point to be v 0 = 0. In general though v 0 > 0 due mainly to the naturally outwards motion of blobs the outer midplane, as derived from theoretical models and observed in experiments. As shown in Ref. [START_REF] Nespoli | 3D structure and dynamics of filaments in turbulence simulations of WEST diverted plasmas[END_REF], at the X-point the blob radial velocity is increased such that v r /f x ∼ const along the field line as the filament tends to keep its alignment with the field line through its motion. This is reflected in the WEST simulation also in the value of the average radial velocity above the X-point v 0 /c s = 2 • 10 -3 . The COMPASS simulation exposed in Ref. [START_REF] Galassi | Drive of parallel flows by turbulence and largescale E B transverse transport in divertor geometry[END_REF] exhibits the same feature. Therefore, the value of the shear across the X-point can vary with the direction of the magnetic field. In particular, for favorable drift direction (∇B drift directed towards the X-point), resulting in negative radial E × B below the X-point with v 0 > 0, the shear experienced by the blobs would be increased with respect to the reversed field case, where both v 0 and E × B are positive. An experimental evidence of the modification of the fluctuation level at the X-point upon field reversal could therefore be interpreted as the relevance of the poloidal shear in blob disconnection, as the other mechanism (squeezing of flux tubes by flux expansion) would not be influenced by the direction of the magnetic field.

If the trends described in this paper are confirmed by the envisaged experiments, whose realization is outside the scope of the present work, one could then think about acting on the magnetic configuration (f x profile, Θ) or the plasma parameters (I p , T e ) to enhance or diminish the fluctuation level in the divertor leg, possibly affecting and therefore controlling the fluctuations of the heat loads on the target. We remark that how the disconnection of filaments at the X-point from the target, together with the generation of divertor filaments along the divertor leg [START_REF] Scotti | Divertor leg filaments in NSTX-U[END_REF], affects the final heat loads on the targets is an open question and needs to be investigated extensively both through simulations and experiments in future works. Furthermore, both mechanisms, always present in the vicinity of the magnetic null, can become even more important in advanced magnetic configurations featuring a secondary X-point along the divertor leg, either in the main SOL (snowflake (SF) divertor, specially in its SF-variant [START_REF] Ryutov | The snowflake divertor[END_REF]), or at the strike point (X and super-X divertor [START_REF] Kotschenreuther | On heat loading, novel divertors, and fusion reactors[END_REF][START_REF] Kotschenreuther | The super X divertor (SXD) and a compact fusion neutron source (CFNS)[END_REF]). The extension of the present analysis to such configurations requires dedicated derivation and numerical modeling and is foreseen for future work.

Finally, we remark that the condition in Eq. ( 14) is the result of several approximations (e.g. the assumptions ∇T e = 0, ∇v || = 0 and v ⊥ = 0) and provides therefore an underestimate of the effectiveness of the poloidal shear of radial velocity in disconnecting the filaments. Dropping the isothermal assumption would result in a steeper plasma potential poloidal gradient (Eq. ( 6)) as T e decreases going towards the target, i.e. in a stronger velocity shear ∆v r . The inclusion of the effect of temperature gradients in this model through the coupling with the energy conservation equation and the comparison with anisothermal TOKAM3X simulations are foreseen for future works. The non-uniform parallel velocity along the divertor leg and the perpendicular drifts, helping in redistributing the plasma across the divertor leg below the X-point, which are included in our simulations but not in the analytical model borrowed from Ref. [START_REF] Wagner | A study of the perpendicular particle transport properties in the scrapeoff layer of ASDEX[END_REF], need to be accounted for in this derivation (starting from the continuity Eq. ( 2)). However, this would need to be done in an iterative way and no simple analytical solution might be possible.

Figure 2 :

 2 Figure 2: Relative fluctuation level σ n / n normalized to their maximum value for each simulation, in COMPASS, TCV, and WEST geometry (from left to right).

Figure 4 :

 4 Figure 4: Parallel profiles along the outer divertor leg of averaged a) plasma density n b) plasma potential for different values of the radial coordinate r in the SOL. In a), the fit of the profile the closest to the separatrix with Eq. (3) is shown with a red line. The thick black line shows the position of the divertor plate.

Figure 5 :

 5 Figure 5: Radial component of the E × B velocity in the surroundings of the X-point, averaged over time and the toroidal direction. The contour levels of B p /B are also shown with the relative values.

Fig. 6 ,

 6 Fig. 6, where log 10 B B p ∆v r c s is plotted for the WEST simulation. As it emerges from the figure,

Figure 6 :

 6 Figure 6: log 10 B B p ∆v r c s evaluated locally for each simulation cell for the WEST simulation.

Acknowledgments

This work was granted access to the HPC resources of CINES, under the allocations A0030506912 & A0050506912 made by GENCI, to the HPC resources of Aix-Marseille University financed by the project Equip@Meso (ANR-10-EQPX-29-01), and to the EUROfusion High Performance Computer (Marconi-Fusion) under the project HEAT. The project leading to this publication (TOP project) has received funding from Excellence Initiative of Aix-marseille University-A*MIDEX, a french "Investissements d'Avenir" programme. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053 (project WP19-ER/EPFL-02). The views and opinions expressed herein do not necessarily reflect those of the European Commission.

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