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Nuclear Materials and Energy

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Physics of toroidal gap heat loading on castellated plasma-facingcomponentsR. Dejarnaca,⁎, J.P. Gunnb, P. Vondraceka,c, M. Komma, R. Paneka, R.A. Pittsd

a Institute of Plasma Physics, Czech Academy of Sciences, 182 00 Prague, Czech Republicb CEA, IRFM, F-13108 Saint-Paul-lez-Durance, Francec Faculty of Mathematics and Physics, Charles University, 121 16 Prague, Czech Republicd ITER Organisation, Route de Vinon-sur-Verdon, CS 90 046, F-13067 St Paul-lez-Durance Cedex, France

A B S T R A C T

Because the gaps between plasma-facing components in fusion devices are comparable in size to the ion Larmor radius (of the order of 1 mm), magnetic field linetracing, the so-called "optical approximation", cannot accurately predict the fine scale heat load distribution around the gap edges. Finite Larmor radius effectsdominate ion deposition. The poloidal component of the ion flux striking the surface is always in the diamagnetic/EXB drift direction, meaning that ions pre-ferentially load one side of the gap. Usually electrons can be described optically due to their smaller Larmor radius. Depending on the local inclination of magneticflux surfaces, it is possible that ions and electrons wet the same side of the gap, or opposite sides. Two-dimensional particle-in-cell simulations and dedicatedexperiments performed in the COMPASS tokamak are used to better understand processes responsible for plasma deposition on the sides of toroidal gaps betweencastellated plasma-facing components in tokamaks. The different contributions of the total incoming flux along a toroidal gap have been observed experimentally forthe first time in COMPASS. These experimental results confirm the model predictions, demonstrating that in specific cases the heat deposition does not necessarilyfollow the optical approximation. The role played by electric fields in the deposition pattern is marginal, contrary to local non-ambipolarity that can change theasymmetrical plasma deposition from one side of the toroidal gap to the other.

1. Introduction

For power handling reasons, the water cooled, plasma-facingcomponents in the ITER tungsten divertor will be castellated, com-prising tens of thousands of individual monoblocks (MB) [1]. Gaps inboth the toroidal (TG) and poloidal (PG) directions introduce leadingedges onto which plasma heat flux can be focused, even if misalign-ments between neighboring MBs are eliminated. Decomposing thetotal particle/heat flux flowing into TGs as the sum of the ion andelectron contributions and accounting for their 3D trajectories, thetheory says that, depending on the magnetic configuration, regionswhich would be shadowed in a purely optical picture can be accessedby ions with finite Larmor radius, a problem which can be particularlyacute during Edge Localized Modes (ELM) when energetic ions fromthe pedestal region impact the target during the transient heat pulse.Heat loads on the ITER MBs were recently assessed using 3D ion orbitcalculations [2,3], guiding efforts to find a shaping solution whichavoids gap edge over-heating. If toroidal beveling will hide leadingedges on each side of PGs from inter-ELM heat loads (at the expense ofhigher perpendicular loads on the top surface), no solution for hidingall the long TG edges on both inner and outer ITER vertical targets hasyet been found. The ion orbit simulations predict that even mitigated

ELMs can melt TG edges in the strike point regions, posing a potentialrisk to MB lifetime.

Although the consequences of TG loading may be significant forITER, the phenomenon had never actually been seen in a tokamak untila series of dedicated experiments, reported here, was performed on theCOMPASS tokamak. The low magnetic field in COMPASS makes theratio of the ion Larmor radius to the gap width relevant to ITER ELMingconditions. Using special instrumented central column graphite tilesbearing TGs, viewed with a high spatial resolution (0.5 mm/pixel)infra-red camera diagnostic, clear evidence for TG heating has beenobtained for the first time. By changing the direction of the poloidal andtoroidal magnetic fields, the four possible configurations of field lineinteractions with TGs have been examined. The experimental surfacetemperature profiles, proportional to heat load distributions, have beencompared qualitatively with heat flux profiles derived from both theoptical approximation (OA) and simulations using the 2D particle-in-cell (PIC) code SPICE2 [4], including self-consistent electric fields.

In two of the field configurations, plasma flux is observed to im-pinge on the unshadowed TG side only, as expected optically and asexpected at the ITER outer vertical target (OVT), but in the other two,heat is deposited on both gap sides, and therefore in the magneticallyshadowed regions as expected at the ITER inner vertical target (IVT), in

https://doi.org/10.1016/j.nme.2019.02.010Received 31 July 2018; Received in revised form 7 February 2019; Accepted 7 February 2019

⁎ Corresponding author.E-mail address: dejarnac@ipp.cas.cz (R. Dejarnac).

Nuclear Materials and Energy 19 (2019) 19–27

Available online 14 February 20192352-1791/ © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

T

excellent agreement with the theory and the SPICE2 code results. Thissame code has been used to benchmark the 3D ion orbit calculations forITER [5], providing confidence both in the existence of the TG loadingphenomenon and in the validity of the simpler ion orbit approach. It isshown that the electric field plays an insignificant role in the plasmadeposition around leading edges and PG. Here, the effect of electricfields is assessed by comparing PIC simulations with ballistic simula-tions using the same code but forcing the electric field to zero. Thecomparison experiment vs. code has also emphasized the role played bylocal non-ambipolarity in determining the distribution of heat loading.In a separate experiment, time-varying bias voltage waveforms havebeen applied to a specially designed electrically insulated TG to selectbetween electron-dominated and ion-dominated regimes. In the case ofthe two-sided deposition and depending on the bias voltage, the heatload is observed to switch sides, following theoretical expectations.These unique experiments demonstrate not only that TG loading doesoccur, but that the physics of this phenomenon is understood and mustbe accounted for in the ITER divertor shaping design.

After presenting the theoretical description of particle deposition inTGs in Section 2, the experimental confirmation by dedicated experi-ments on COMPASS tokamak is presented in Section 3. The powerdeposition profiles around TGs calculated by the SPICE2 code for thedifferent field configurations are compared to experimental surfacetemperature in Section 4. The role of non-ambipolarity together withthe results of a dedicated TG biasing experiment on COMPASS, sup-ported by PIC simulations, are shown in Section 5. Finally, the mainconclusions are summarized in Section 6.

2. Theory of particle deposition in a toroidal gap

The total magnetic field B has two components, one toroidal Bϕ andone poloidal Bθ. Usually, magnetic flux surfaces are not aligned with theTG gap axis. In the ITER divertor, the cassettes holding the vertical tar-gets are tilted with respect to the magnetic flux surfaces with an in-cidence angle projected onto the plane parallel to the MBs top surfaceθ// = 3.7° at IVT and θ// = 5.6° at OVT (see Fig. 1 in [2]). In COMPASS,

the pitch angle of the magnetic surfaces, arctan(Bθ/Bϕ), with respect tothe purely toroidally oriented TGs at the experiment location (seeSection 3) is similar, with 3° < arctan(Bθ/Bϕ) < 6°, in the range ofplasma currents used in the experiments. Therefore, in a purely opticalpicture, the plasma flowing in the parallel direction impinges preferablyonly one side of the TG and the deposited heat flux on this side can bedescribed as: qdep = q//,0 sin(θ//), with q//,0 the energy flux densityparallel to B. However, if the helical trajectories around the magneticfield line of the incoming charged particles are taken into account, thisoptical picture is not valid anymore. The total particle flux Γtot is com-posed of ions and electrons and it can be decomposed around the TG asthe sum of the following contributions: (1) a parallel flux Γpar for parti-cles following the OA, (2) a flux due to the Larmor gyro-motion aroundthe field lines ΓLarmor and (3) an E×B driven flux ΓE × B due to theelectric field E forming at the MB top surface:

= + + ×tot par Larmor E B (1)

The electrons have a Larmor radius rL,e much smaller than themillimetric gap width (lgap), by roughly 2 orders of magnitude, so theirtrajectories mainly follow the B orientation and they can thus be con-sidered as the main contributors to Γpar.

Considering the ions which can have their Larmor radius rL,i ∼ lgap

(in ITER ELMy conditions [2] or in COMPASS L-mode conditions), theirhelical gyro-motion along the field lines must be taken into account andare the main contributors to ΓLarmor. This flux impinges preferentiallyone side of the TG according to the gyration motion around the fieldlines (left hand rule).

On the MBs top surface, electric fields pointing towards the surfacewith strong gradients form. Charges particles that can enter the gap areprone to an E× B drift that can drive them away from their initialtrajectories so that they impinge preferentially the TG side which islocated in the direction of that drift and contribute to ΓE × B.

By definition both the Larmor flux and the E× B flux always pointin the same direction in the TG since ions with a Larmor flux in theopposite direction do not exist; such orbits are absent from the dis-tribution function because they are scraped-off further upstream. It

Fig. 1. Theoretical representation of the different contributions of the total particle flux (parallel, E × B and Larmor) impinging on a TG in the four possiblecombinations of Bϕ and Bθ directions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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should be noted that since the electric field always points into thesurface, the direction of the E× B drift, and thus the TG side loaded byΓE × B, only depends on the direction of B, which is not the case for theparallel flux, which changes with the direction of Bθ (or of the plasmacurrent Ip). Therefore, it can happen that ΓLarmor and ΓE × B impinge adifferent side than Γpar for a specific configuration of Bϕ and Bθ. It has tobe noted that both ΓLarmor and ΓE × B always impinge the same side ofthe TG independently of the orientation (and not the direction) of B orindependently of the pitch angle. For the sake of clarity, the term'Larmor/E×B flux' will be used to refer to those two contributions toΓtot and will be noted ΓLarmor/ExB.

By changing the direction of the poloidal and toroidal magneticfields in the four possible configurations, the theoretical different in-teractions of a magnetic field line with a TG are examined following theabove reasoning and are represented in Fig. 1. The red arrow, goingfrom left to right, indicates the direction of the incident parallel flow.Within these four different cases, two types of interaction with a TG areobserved. The first type of interaction shows that only one side of theTG is loaded by all the different components of the total particle flux asrepresented in Fig. 1a and d. The Larmor/ExB flux points in the samedirection as the parallel flux and this is what is predicted to happen atthe ITER OVT by ion orbit calculations [2]. The second type of inter-action show that both sides of the TG are loaded as represented inFig. 1b and c. The Larmor/ExB flux loads the opposite side of the TGloaded by Γpar, thus by definition the magnetically shadowed side,where the OA predicts no flux. This type of interaction with TG ispredicted to happen at the ITER IVT [2]. These predictions are based on3D ion orbit calculations but evidence of such phenomena has neverbeen so far sought experimentally in any tokamak.

3. Experimental confirmation of theoretical predictions

3.1. The COMPASS experimental configuration

A dedicated set of experiments has been performed on theCOMPASS tokamak (R= 0.56 m, a= 0.2 m) to thoroughly study TGedge loading and to verify theoretical predictions described inSection 2. A specially designed inner-wall limiter (IWL) is equippedwith a 1 mm wide TG cut in a flat surface (see Fig. 2) and is in a directview of a high resolution infrared camera to monitor the TG edgesheating. The limiter is made of graphite and has a roof-shape with anapex in the middle protruding above the toroidally neighboring tiles by8 mm. In the region of interest, where the TG is located, the magneticfield incidence angle projected on the plane perpendicular to the IWLsurface and parallel to the toroidal direction is constant toroidally andis θ⊥ = 1.5°. Because the TG is located on the left side of the apex, whenviewing the central column from outside, the plasma is flowing in theparallel direction only from left to right independently of the B direc-tion. A poloidally running gap with the same width lgap = 1 mm is alsopresent in the middle of the TG at x = −30 mm, recreating the gapcrossing in between 4 MBs.

The experimental configuration is similar to that used in [6–8] tostudy plasma-wall interactions. The camera used is a mid-IR InSbcamera (with a 240 × 320 pixel resolution, 2 kHz acquisition fre-quency, 0.13 ms integration time) mounted on an outer mid-plane portdirectly viewing the special IWL in order to extract the surface tem-perature Tsurf,IR. A 50 mm lens was used, yielding a spatial resolution ofr= 0.5 mm/pixel. Ohmically heated and slightly elongated (κ = 1.1)inner-wall limited deuterium discharges were used with a magneticfield B = 0.9 T, a constant line-averaged density n̄e = 3 × 1019 m−3

and a plasma current Ip = 130 kA, for a total ohmic input powerPOhm ∼ 150 kW. For such discharges, the typical values of the localelectron density and temperature at the contact point from previousmeasurements with embedded Langmuir probes in a previous IWL [8]are ne = 5.1018 m − 3 and Te = 30 eV, respectively. Assuming Ti = Te,this yields an ion Larmor radius rL,i ∼ 0.7 mm ∼ lgap. This value is

comparable to that expected for ELM ions in the ITER divertor [2]. Thedischarge duration was ∼300 ms with a flat top phase of ∼150 ms. Theplasma column leans on the inner wall with the contact point at avertical position z = +35 mm on the upper edge of the TG. The stan-dard, forward B and Ip direction in COMPASS (clockwise looking fromthe top), noted (−B, −Ip) by the usual convention and in the rest ofthis manuscript, yields a pitch angle arctan(Bθ/Bϕ) ∼ +6° corre-sponding to the case a) in Fig. 1 and performed in the COMPASS dis-charge #11610. In order to reproduce the three other cases, the plasmacurrent direction was changed on a shot-to-shot basis to change the Bθ

direction and therefore the B orientation. In discharge #11615, the Ipdirection was reversed, changing the orientation of the field lines with apitch angle arctan(Bθ/Bϕ) ∼ −6°, corresponding to the case Fig.1b). Bykeeping the Ip reversed and now reversing the toroidal field, the pitchangle changes back to arctan(Bθ/Bϕ) ∼ +6° but now the configurationcorresponds to the case Fig. 1c) where both TG sides should be loaded(discharge #11594). Finally, in the last discharge #11600, the plasmacurrent was changed back to the forward direction to reproduce theconfiguration Fig. 1d).

3.2. Experimental observations

Poloidal profiles of the surface temperature gradient,ΔTsurf = Tsurf,IR −Tsurf,base, across the TG taken 10 mm before(x = −40 mm) and 10 mm after (x = −20 mm) the PG are plotted inFig. 3, with each quadrant corresponding to the theoretical configura-tion of its counterpart in Fig. 1. Tsurf,base is the base top surface tem-perature of the IWL around the TG taken as the Tsurf,IR mean value at

Fig. 2. CAD view of the COMPASS IWL used in this experimental study with a1 mm wide TG on the left side of the limiter from the apex, where the magneticfield lines intersect the limiter with = 1. 5o Below, a scheme of the IWL radialcross-section is shown.

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the contact point (35 < Z < 40 mm) and is in the range 80–85 °C. Thepoloidal profiles have to be looked at from the right as an observersitting on the apex and correspond to the schematic insets in Fig. 1.

The theoretical predictions described in Section 2 have been con-firmed experimentally. Fig. 3 shows a good qualitative agreementconsistent with the theoretically expected loading.

In the two cases where the theory predicts that both Γpar and ΓLarmor/

ExB are directed towards the same side of the TG (cases a & d), only onepeak is observed on the Tsurf,IR profiles on the expected side of the gap.A sharp Tsurf,IR gradient (ΔTsurf = 12 °C) occurring over ∼5 mm isobserved at the gap corner, which is the signature of local heating of theTG edge. The other side of the TG shows no heating with a zero surfacetemperature gradient at the gap corner. It can thus be concluded thatthe magnetically shadowed side of the TG (z < 27 mm in Fig. 3a andz > 28 mm in Fig. 3d) receives zero flux as predicted by the theory. Theslow decrease of Tsurf,IR away from the bottom side of the gap in allcases for z < 27 mm is because the plasma contact point is locatedslightly above the TG and is representative of the power decay in theSOL.

In the two other cases (b & c) where both sides should be theore-tically loaded, two peaks on Tsurf,IR are observed. The side impacted bythe parallel flux is always hotter than the Larmor/E × B side, showingtwice larger temperature gradient (ΔTsurf,par ∼ 6 °C) at the corner than

on the Larmor/E × B side (ΔTsurf,ExB ∼ 3 °C), clearly indicating astronger effect of the parallel flux. To zero order, the result indicatesthat the heat load carried by Γpar is ∼2 times larger than the one carriedby ΓLarmor/E × B. Most importantly, these observations represent the firstexperimental evidence of plasma deposition on the magnetically sha-dowed side of a TG, where the OA predicts no flux. This effect waspredicted for the first time in [2] in the context of a physics study of therequired ITER W divertor MB front surface shaping. Until now, how-ever, such TG loading has remained a theoretical concept and has notpreviously been observed experimentally. The results presented heredemonstrate that the effect must now be taken seriously, even ifshaping design to avoid TG loading at all locations in the ITER divertoris unlikely to be possible.

4. PIC simulations of plasma deposition in a TG

4.1. Toroidal gap sides heat loading

The deposited particle and heat flux profiles along TG sides are cal-culated using the 2D-3V PIC code SPICE2 [4], which has been previouslyused in similar studies to calculate plasma deposition profiles inside gaps[5–6,9–10]. The choice of the PIC technique is relevant because it cal-culates the ion and electron trajectories using three-dimensional velocity

Fig. 3. Poloidal profiles of Tsurf,IR gradient across the TG taken 10 mm before (blue dashed line) and 10 mm after (red line) the PG located in the middle of the TG(Fig.12 from [6]). Each quadrant corresponds to the same quadrant in Fig. 1 showing the theoretical prediction of the different total flux contributions along the TGsides with directions: a) (−B, −IP), b) (−B, +IP), c) (+B, +IP), d) (+B, −IP). (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

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vectors in a realistic two-dimensional geometry and in a self-consistentelectric field. This allows separating the different contributions fromelectrons, ions and the ExB drift of the total particle flux. The calculationis made on a square grid having a spatial resolution of one Debye length,which for the conditions of the simulations is λD∼0.02 mm. SPICE2 si-mulates the collisionless electrostatic sheath forming around the surfaceand inside the TG. The magnetic field vector is defined in 3D and as-sumed constant, which is true to a high approximation over the veryshort distances (few mm) appropriate for this experiment. Ions are in-jected into the simulation domain with a velocity distribution functionsatisfying the Bohm criterion [11]. Electrons are assumed to be Max-wellian. The case considered here is a fully ionized magnetized plasmawith singly charged deuterium ions incident on a completely absorbing,conducting wall. The potential drop in the sheath and magnetic pre-sheath is fixed at −3kTe with respect to the plasma potential (thus ap-proximating the case of a floating surface). Due to the symmetry inducedby the semi-infinite 2D geometry of the PIC simulation box, the fourcases described in Section 2 are reduced to only two cases: (1) only oneside of the TG is loaded by all the components of the incoming flux (casesa & d in Fig. 1) and (2) both sides are loaded (cases b & c in Fig. 1). Forthe purpose of comparison with the experiment, the two configurationsa) and b) of Fig. 3 are considered. The low gradients in Tsurf,IR at the gapedges make the comparison with synthetic temperatures from forwardthermal simulations very difficult: a slight variation in the total fluxupstream (larger than the experimental resolution) will lead to largeerrors on the small absolute values of the measured temperatures.Therefore, only the spatial distribution of the heat fluxes calculated bythe PIC model will be presented and compared qualitatively with theexperimental Tsurf,IR distribution with the aim of identifying and loca-lizing the three processes at play. The temperature increase is propor-tional to the total heat flux deposited around the edge. The total powerdeposition profiles from SPICE2, noted qPIC,tot, as well as the contribu-tions coming from the ions, noted qi (mainly the Larmor/ExB heat flux,qLarmor/ExB), and electrons, noted qe (mainly the parallel flux qpar), arepresented in Fig. 4 for both cases of interest and assuming ambipolarconditions. Profiles are normalized to the total flux falling on the IWL topsurface far from the perturbation from the gap q⊥,0 = q//,0 sin(θ⊥).Floating conditions make the fraction of the total energy flux carried byions dominant with respect to that carried by electrons yieldingqPIC,tot ∼ qtheory = f*qi + (1 − f)*qe, with f= 5/7 [12], as shown inFig. 4 far from the gap (ion-dominated regime).

The first case shown in Fig. 4a is similar to the experimental finding(Fig. 3a) where all fluxes are deposited exclusively on the magneticallyaccessible gap side. Both ions and electrons contribute to this total flux,with the latter being totally focused inside the TG and the former, dueto their much larger Larmor gyration, spreading their power around thecorner onto the top surface. The deposited power inside the gap is madeon a distance ∼0.5 mm, half the gap width.

The second case shown in Fig. 4b exhibits two peaks similarly to theexperimental observations (Fig. 3b), with the peak on the magneticallywetted side of the TG due only to the electron component of qpar, andwith the peak on the magnetically shadowed side due to the Larmor/E × B flux, carried mostly by ions. The shadowed side also sees a smallcontribution from the electrons. This is an artefact due to the use of areduced ion-to-electron mass ratio of 200 in the code to accelerate thesimulations computational time [5]. Consequently, the electron Larmorradii are artificially enhanced but in reality, there is no electronreaching this side of the gap.

Although the experimental result is qualitatively reproduced, thesimulation is unable to reproduce the relative amplitudes of the twopeaks. The calculated Larmor/ExB flux at the TG edge is much largerthan the projected parallel flux (qLarmor/ExB>qpar sinθ// by a factor 2.5),in quantitative disagreement with the experimental observations. As inthe case of plasma deposition around a PG [6], the disagreement ori-ginates from an overestimate of the ion contribution due to the as-sumption of ambipolar conditions. A second set of PIC simulations withmore realistic non-ambipolar conditions corresponding to Vfl ∼ −1kTe,which was measured by fixed Langmuir probes in a previous similarCOMPASS experiment within 10 mm of the last closed flux surface,confirms this hypothesis. The resulting power deposition profiles for thetwo cases of interest are presented in Fig. 5, with here the total powermainly carried by the electrons, with the fraction f= 1.2/7 far from thegap (electron-dominated regime).

For the first case with all flux components loading the same gapside, the simulation is similar to the previous result with only one peakon the directly wetted side of the TG but now the total heat flux iscarried predominantly by electrons (Fig. 5a). Due to a smaller potentialdrop in the electrostatic sheath, the peak amplitude is 40% higher thanin the ambipolar case. Changes are more drastic in the second case(Fig. 5b) where the peak amplitude asymmetry observed in Fig. 4b hasnow switched sides and is in better qualitative agreement with ex-periment (Fig. 3b). Electrons continue to follow the OA, but with a

Fig. 4. PIC computed normalized deposited heat flux density profiles across a TG with ambipolar conditions (ion-dominated regime) corresponding to the two casesof interest in Fig. 3:a) (−B, −IP) and b) (−B, +IP), (Fig. 13 from [6]).

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higher energy since the less negative potential drop in the sheath allowsmore energetic electron to reach the wall, so that the parallel flux nowmakes a larger contribution to qtot leading to increased loading on themagnetically wetted side. Thanks to their gyro-motion, the ions are stillable to impact on the shadowed side of the TG, where the OA predictsno accessibility, but now with a smaller contribution. As in the ambi-polar case, the deposition of electrons on the magnetically shadowedside of the gap is not real due to the use of heavy electrons in the si-mulation.

4.2. Role of electric fields in the TG plasma deposition

In order to investigate the role of electric fields more in detail,ballistic simulations have been performed using SPICE2, i.e. PIC si-mulations where the electric field was forced to zero. Fig. 6 shows adirect comparison of ion power deposition profiles along the TG sides

with ambipolar conditions with electric fields (PIC) and without (bal-listic) for the two cases a) and b) of Fig. 4. Profiles have been nor-malized in each case to the ion heat flux falling on the IWL top surfacefar from the gap.

In the case b) of a 2-sided TG deposition, the presence of electricfield reduces the top surface loading at the magnetically wetted cornerdownstream and enhances heat loading at the magnetically shadowedside inside the TG (Fig. 6b).

Ions can penetrate deeper in the gap because in the presence ofelectric fields because of their acceleration in the sheath above the gap.Helix-like trajectories of incoming ions are stretched so that they canpenetrate deeper into the gap. A more positive structure in the negativepotential is consequently observed in the gap (Fig. 7b). In the ballisticcase, ion ‘streamlines’ or contours tangential to average velocity vectorsat every grid point (Fig. 7a), show a less deep penetration into the gapshadowed side. This is similar to what happens in PGs where ions can

Fig. 5. PIC computed normalized deposited heat flux density profiles across a TG assuming non-ambipolar conditions (electron-dominated regime) corresponding tothe two cases in Fig. 3:a) (−B, −IP) and b) (−B, +IP), (Fig. 14 from [6]).

Fig. 6. PIC and ballistic ion power deposition profiles around a TG normalized to the power flux falling on the IWL top surface far from the gap with ambipolarconditions for the cases a) and b) of Fig. 4. In the insets, a scheme of an incoming ion ballistic trajectory to illustrate the extra ion deposition on the magneticallywetted TG side.

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jump into the gap thanks to their Larmor gyration, whilst electronscannot, thus creating a positive structure in the gap potential capable ofreaching values higher than the plasma potential under certain condi-tions [5,13]. The extra downstream heat flux observed in the ballisticcase in Fig. 6b comes from ions that manage to make some full rotationswithin the gap allowing to strike the downstream top surface withoutbeing intercepted by the magnetically shadowed side, as illustrated in theinset of the graph which shows the ballistic trajectory of an incoming ion.This is possible because the ion Larmor radius (0.7 mm) in the COMPASSexperiment conditions is smaller than the gap width (1 mm). AdditionalPIC simulations show that this effect is not present for radii greater thanthe gap width. This extra heat flux is not present when the electric field ispresent because ion trajectories are not helix-like anymore and when anion enters the gap it does not go out but continues its trajectory towardsthe upstream, magnetically shadowed Larmor/ExB side.

In the case a) of a 1-sided TG deposition (Fig. 6a), the two profilesare very similar indicating that the role of electric fields is marginal. Inthe ballistic case, the same mechanism as in the 2-sided deposition isresponsible for the slightly larger gap side loading. Incoming ions witha Larmor radius smaller than the gap width can make a full gyrationinside the gap before impinging on the TG side, as illustrated in theinset of Fig. 6a.

At the limiter top surface, incoming ions are gradually scraped-offfrom the plasma as they arrive closer to the IWL top surface as a resultof their Larmor gyro-motion and the grazing incidence angle of the fieldlines. This creates a local decrease of ion density and gives rise to anapparent velocity component pointing toward the surface, simply be-cause ions moving out the surface are missing in the distribution. It isillustrated in Fig. 7a by the change of direction of their streamlinesabove the surface in the magnetic presheath. In the presence of electricfields, this effect is enhanced by an ExB drift in the same direction (Bϕ ispointing towards the page in Fig. 7b) but this effect is reduced in theDebye sheath close to the top surface because of a polarization drift.The polarization drift is the direct consequence of ions passing throughthe sheath and experiencing a temporally evolving local electric field intheir frame of reference [5].

5. Role of non-ambipolarity: the TG biasing experiment

In order to investigate experimentally the role of non-ambipolaritydescribed in Section 4.1, a recent dedicated experiment has been

performed in COMPASS. A new IWL with the same geometry as theprevious one has been built with now two embedded graphite blockswhere TGs are cut. The blocks lie on boron nitride beds to insureelectrical insulation and are connected to a power supply allowingbiasing of the TGs. The insulated TGs are located on each side of theapex with a constant incidence angle θ⊥ = 2.5° for both wings. Theblock on the left side presents only one TG, whilst the block on the rightpresents both a TG and a PG (Fig. 8c). The results presented here are forthe right block including the gap crossing but identical results arepresent on the left block. Similar experimental configuration andplasma conditions as described in Section 3.1., where B and Ip direc-tions were changed to reach the 4 possible configurations of field linesinteractions, were used. Different voltage waveforms were applied(square or triangular) to sweep the biasing voltage from 0 V (grounded)to −180 V, in order to force conditions to go from an electron-domi-nated regime as in the original experiment (Fig. 3) to an ion-dominatedregime (Fig. 4). The value of −180 V has been chosen to be at least at−3Te below the floating potential estimated at Vfl = −30 to −50 Vfrom previous similar experiments. In total, 12 discharges were per-formed with the plasma contact point on the TG and all results areconsistent with the ones presented in the paragraph below, where onlythe most interesting theoretical case b) of Fig. 1 will be presented forthe sake of conciseness of this paper, corresponding to the two-sided TGheat loading case.

In the discharge of interest with forward B and Ip, the pitch angle isarctan(Bθ/Bϕ) ∼ +6° and since the observed TG is now located on theright side of the apex, the parallel flux comes only from the right andimpinges the lower side of the TG for the same interaction as describedin Fig. 1b with the Larmor/ExB side being the TG upper side. The ap-plied voltage has a triangular shape with a sweeping period of 120 ms(Fig. 8a). Similar temperature gradients as defined in Section 3.2. arepresented in Fig. 8b along the line of view located downstream the PGand indicated on the IR camera image in Fig. 8c. Now, since the plasmacontact point is on the TG (Z= 29 mm), Tsurf,base is a linear interpola-tion of the surface temperatures on both sides far from the TG(Z < 22 mm & Z > 34 mm) as a proxy of the Tsurf profile that would beon the graphite block without the presence of the gap.

The Tsurf,IR distribution around the TG when zero voltage is appliedcorresponds to the grounded case as presented in Fig. 3. When theapplied voltage is gradually decreased to −180 V, the temperaturegradient at the downstream, directly magnetically wetted side

Fig. 7. Computed ion streamlines for a) the ballistic case and b) PIC with ambipolar conditions case around the COMPASS TG for case b) of Fig. 6 (−B, +IP)corresponding to the two-sided TG heat loading. In the PIC graph background, the electric potential decays from the plasma potential (zero volt) to the floatingpotential (−3kTe) at the IWL surface.

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gradually decreases from ΔTsurf,IR = 8 °C to ΔTsurf,IR = 2 °C. This is thedirect consequence of the strong reduction of the electron contribution,or qpar, in ion-dominated conditions. At the transition from the electron-dominated to ion-dominated regime characterized by floating condi-tions (symbolized by the black curve in Fig. 8), one can note a sharptransition from the black curve to the magenta curve, followed by a lesssignificant effect, which is representative of the fraction f [12] changingvalue from 5/7 (floating conditions) to 7/7 (ion saturation). On themagnetically shadowed, Larmor/ExB side the effect is opposite with anincrease of ΔTsurf,IR as expected by the theory with the increase ofqLarmor/ExB with ambipolar, ion-dominated conditions. The increase ishowever not very significant ∼1.5 °C going from ΔTsurf,IR = 0.5 °C(electron-dominated) to ΔTsurf,IR = 2 °C (ion-dominated). It has to benoted that due to the combination of the short duration and low powerCOMPASS discharges and no leading edge, the TG edge heating is notsignificant but such effect on both sides is observed systematically in allinvestigated discharges. The change in the asymmetrical TG heatloading from one side to the other predicted by PIC simulations oc-curring when ambipolarity is broken has been recreated experimen-tally. It demonstrates the role of local non-ambipolarity in TG edgeheating and that this must be taken into account in predictive calcu-lations.

6. Summary and conclusion

This paper describes the physics of toroidal gap heat loading whenthe ion Larmor radius is of the same size or larger than the gap width.The total flux density impinging the gap sides comprises 1) a parallel-to-magnetic-field component mainly carried by electrons, 2) an ioncontribution resulting from the large Larmor gyro-motion and 3) anE × B drift due to the strong sheath electric fields. Finite Larmor radiuseffects dominate ion deposition with the poloidal component of the ionflux striking the surface always in the diamagnetic/E × B drift direc-tion. Depending on the directions of the poloidal and toroidal magneticfields, the different heat flux components will load either one side (ITEROVT case) or both sides (ITER IVT case) of the gap, thus in the lattercase loading the magnetically shadowed side where the optical ap-proximation predicts no flux. The theoretically predicted toroidal gapedge loading on the magnetically shadowed side was confirmed ex-perimentally for the first time on the COMPASS tokamak proving thatthis effect is real. Toroidal gap edge heating has been unequivocally

observed by measuring surface temperature increases across the gap bya high resolution IR camera in the four possible field line interactionswith a toroidal gap, depending on the plasma current and toroidal fieldorientations. Experimental results are qualitatively well reproduced by2D-3 V PIC simulations. However, the simulations must invoke non-ambipolarity in the gap vicinity, consistent with data obtained from tileembedded Langmuir probes under similar conditions, to fully explainthe observed relative peak loading amplitudes on each side of the gapacross all total magnetic field directions. A strong experimental con-firmation of the proposed local non-ambipolarity was achieved in arecent experiment by biasing the TG negatively with respect to thesurrounding limiter. In the case of the two-sided deposition and de-pending on the bias voltage, the heat load is observed to switch sides,following theoretical expectations. The role of the sheath electric fieldis assessed by comparing PIC and ballistic simulations. In the case of a1-sided deposition on the directly wetted side, the profiles are very si-milar indicating the role of electric fields is marginal. In the case of a 2-sided deposition, the presence of electric fields reduces the top surfaceloading at the downstream corner and enhances heat loading at themagnetically shadowed side inside the gap. This is caused by the pre-sence of a negative potential well that accelerates ions that enter thegap, instead of allowing them to make a full revolution before im-pinging the top surface downstream corner because of a Larmor radiussmaller than the gap width. However, it can be noted that for ITERELMy conditions, no discrepancy is observed between PIC and ballisticsimulations because of ion Larmor radii much larger than the gapwidth. The results from this unique set of experiments and PIC simu-lations demonstrate not only that TG loading does occur but that thephysics of this phenomenon is understood and must be accounted for inthe ITER divertor shaping design.

Acknowledgments

This work was supported by the project Czech Science FoundationGA16-14228S and has been co-funded by the MEYS projects 8D15001and LM2015045. This work has been carried out within the frameworkof the EUROfusion Consortium (WP PFC) and has received fundingfrom the Euratom research and training programme 2014–2018 undergrant agreement no 633053. The views and opinions expressed hereindo not necessarily reflect those of the European Commission or of theITER Organization.

Fig. 8. a) Applied voltage waveform from 0 to −180 V with a period of 120 ms, b) corresponding Tsurf,IR gradients around the TG for different voltages and c)snapshot of the IR camera with the line of view of Tsurf,IR profiles indicated in red and the pitch angle indicated by the yellow arrow. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

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Supplementary materials

Supplementary material associated with this article can be found, inthe online version, at doi:10.1016/j.nme.2019.02.010.

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