Keywords: Fusion, Impurity transport, Gyrokinetics, Turbulent transport, Neoclassical transport

In a fusion plasma, the presence of impurities is inevitable. A thorough understanding of impurity transport is crucial for the achievement of the project ITER. In this work two gyrokinetic codes-GYSELA-5D and TERESA-4D-have been used to study impurity transport. Both turbulent and neoclassical impurity transport have been considered to address impurity pinch caused by trapped particle driven turbulence, as well as the transport of light to heavy impurities and the impact of the poloidal asymmetry of the impurity density on neoclassical transport, and nally the e ects of a toroidal rotation on impurity transport. The overall results obtained from nonlinear gyrokinetic simulations imply that heavy impurities tend to be accumulated in the core plasma. However, this accumulation should be avoided by using external heating of the plasma.

Sommaire

C Impurity density distribution from the parallel dynamics

Bibliographie Acknowledgement

The work discussed in this manuscript is the result of my three years of a PhD program, during which I have had an opportunity to meet a lot of people who made me personally and professionally a better person. Without them, this work would have been incomplete, and for this, I would like to express my sincere gratitude.

First of all, I am deeply indebted to my supervisors Etienne and Maxime, without whom this work would not have been possible. Their thoughtfulness and support helped me enormously when I got discouraged about not being productive enough. By watching them, I learned what it really means to become a good mentor and it was a great pleasure to have worked with them during the past three years. I also deeply appreciate their guidance and careful revision of all my manuscripts and their trust in me, which encouraged me to get through this journey.

I am also very thankful to Virginie, Yanick, and Xavier, who accompanied me through my PhD course. Having a regular video meeting with them was a huge motivation and it always ended up with a fruitful discussion and constructive advice for my work. Thank you for sharing your knowledge and your expertise. It was a great honor and a privilege to have worked with you.

I would like to thank also all the other current and former colleagues of the group plasma chauds in IJL. Alejandro, Anil, Daniele, Dimitri, Erwan, Frédéric, Guillaume, Homam, Jérôme, Julien, Liliane, Luigi, Sarah, Stéphane, Thierry and all the others, with whom I shared many unforgettable moments. Thank you also for my friends whom I met outside my o ce. Spending time with you made me forget about physics for a moment and gave me a lot of energy to get back to work again.

A special thank goes to Marie for being with me during all these years, for having listened to all my complaints. I cannot even imagine where I would be without you. I often saw the world as black and white, but you were the color to me. The only color that I had.

Finally, I would like to thank my mom and sister who have supported me during my entire studies on the other side of the world. Especially, to my father, who sacri ced everything for our family. As you know, I am not good at expressing my innermost thoughts to you, but just remember that I am so proud of being your son and having a father like you. You mean a lot to me.

Résumé

L'énergie des étoiles, et du Soleil en particulier, provient de la fusion nucléaire. Au centre du Soleil, la température atteint 15 millions de degrés. À une telle température, les électrons se libèrent des noyaux, générant un plasma. Le comportement du plasma est particulièrement complexe puisque chaque particule est in uencée par des interactions à longue portée avec toutes les autres particules. La physique des plasmas cherche à décrire l'évolution et le comportement de tels systèmes.

Au cours de la réaction de fusion entre les noyaux de deutérium et tritium, de l'énergie est libérée, de l'ordre de plusieurs MeV, sous forme d'énergie cinétique des noyaux formés (hélium, appelé également particule alpha, et neutron, voir Figure 1). F 1 -Représentation de la réaction de fusion du deutérium (D) et du tritium (T), qui produit un noyau d'hélium (ou particule alpha) et un neutron de haute énergie1 .

Alors qu'un champ gravitationnel puissant fournit les conditions idéales pour que la fusion se produise dans le Soleil, une température et une gravité insu santes rendent di cile la réalisation de la fusion nucléaire sur Terre. Il faut donc trouver une autre méthode. L'application d'un champ magnétique a été proposée pour con ner le plasma, car un champ magnétique peut retenir les particules chargées pour les faire tourner hélicoïdalement autour des lignes de champ. Le rayon de cette hélice est appelé rayon Résumé de Larmor. Si l'on parvient à concevoir une ligne de champ magnétique fermée en forme de tore, elle peut alors empêcher un plasma chaud de heurter la paroi. C'est dans les années 1960 que le réacteur à fusion magnétique, tokamak, est apparu. Actuellement, le tokamak ITER, qui sera le plus grand tokamak au monde, est en construction sur le site de Cadarache (Figure 2). Dans la fusion nucléaire, la présence inévitable d'impuretés est étroitement liée à la réaction de fusion elle-même et à la présence de parois. Une compréhension approfondie du transport des impuretés est cruciale pour la réalisation du projet ITER. Au cours d'une décharge de plasma, diverses impuretés peuvent être présentes simultanément, par exemple, de l'hélium (He) en tant que réactif des réactions de fusion Deutérium (D)-Tritium (T), de l'argon (Ar) ou de l'azote (Ni) qui peuvent être injectés volontairement pour réduire le ux de chaleur sur le divertor, et des impuretés -carbone (C) et tungstène (W) -provenant des interactions plasma-paroi. Il a été montré par exemple, (i) que l'accumulation d'impuretés lourdes dans le coeur du plasma peut entraîner une chute des performances du réacteur en rayonnant l'énergie (ii) que l'augmentation d'impuretés légères dans le tokamak peut diluer le combustible principale DT.

Il existe trois mécanismes di érents par lesquels le transport d'impuretés est généré : (i) la turbulence, (ii) les e ets de collisions (transport néoclassique), et (iii) les instabilités MHD. Au cours de cette thèse, nous nous sommes limités au cas électrostatique, ainsi seuls les deux premiers mécanismes ont été étudiés numériquement et théoriquement.

Pour cela, deux codes gyrocinétiques-(i) GYSELA-5D développé au CEA-IRFM et (ii) TERESA-4D développé à l'IJL-ont été utilisés pour étudier le transport des impuretés. La théorie gyrocinétique repose sur l'élimination du mouvement rapide de gy-ration autour des lignes de champ pour réduire la dimension du modèle. Les équations gyrocinétiques décrivant la dynamique de la fonction distribution des centres-guides peuvent alors être résolues à l'aide de codes numériques complexes. Pourtant, malgré cette réduction, résoudre l'équation 5D dans une simulation non-linéaire avec de multiples espèces (ions, electrons, impuretés) reste très coûteux (par exemple, une simulation bien résolue a besoin de 3 millions d'heures de calcul). Pour cette raison, le code gyrocinétique TERESA a également été utilisé pour compléter les études. C'est un modèle réduit de GYSELA qui ne prend en compte qu'une réponse adiabatique des particules passantes et seule la dynamique des particules piégées est prise en compte. Cette dernière hypothèse permet en e et de réduire la dimension du problème 6D à 4D en réalisant une moyenne supplémentaire sur le rebond des particules piégées. La fonction de distribution est alors celle des centres-bananes. Il est donc possible d'étudier de manière approfondie les phénomènes liés aux particules piégées avec un temps de calcul beaucoup moins important.

a) Le transport turbulent convectif

Le transport turbulent des impuretés peut être divisé en deux parties : une partie di usive et une partie convective. En général, dans les tokamaks, la partie di usive conduit à un transport d'impuretés vers l'extérieur (vers les parois). À l'inverse, la partie convective, en fonction de la nature de la turbulence et du pro l de la courbure magnétique, peut être dirigée vers l'intérieur et conduire à l'accumulation des impuretés au coeur du plasma. Une étude détaillée de la partie convective peut être réalisée grâce aux dépendances paramétriques de chaque terme, par exemple des termes dits de thermo-di usion et de pincement de courbure. Dans ce but, le code gyrocinétique TERESA a été utilisé dans le cas d'une turbulence liée aux particules piégées, et une approche analytique, la théorie quasi-linéaire, a été comparée aux résultats numériques, donnant un bon accord qualitatif. Il a été constaté que le pincement de thermo-di usion est dirigé vers l'intérieur lorsque la turbulence à mode d'électrons piégés (TEM) est dominante. De plus, un signe négatif du pincement de courbure a été constaté dans le cas d'un cisaillement magnétique négatif. Les résultats globaux obtenus à l'aide de TERESA suggèrent, tout d'abord, qu'une utilisation appropriée d'un système de chau age, tel que l'ICRH, pourrait privilégier une turbulence ITG plutôt que TEM a n d'expulser les impuretés du coeur vers l'extérieur, et également que la formation d'un cisaillement magnétique négatif près de l'axe magnétique pourrait être béné que pour empêcher l'accumulation d'impuretés au coeur des tokamaks.

b) La modélisation gyrocinétique des impuretés légères à lourdes en relation avec l'asymétrie poloïdale

Le principal mécanisme de transport des diverses impuretés di ère d'une espèce à l'autre en fonction de leur masse et de leur charge. Pour les impuretés légères, comme l'hélium (He), les particules sont transportées principalement par la turbulence alors que le transport des impuretés lourdes, comme le tungstène (W), est souvent dominé par des e ets néoclassiques. Le code gyrocinétique GYSELA, implémenté avec un opérateur de collision linéarisé, nous permet d'étudier le transport turbulent et néoclassique des impuretés de manière complète, en incluant les couplages entre les deux mécanismes. Les simulations non-linéaires réalisées dans ce manuscrit ont con rmé que GYSELA reproduit un fort ux turbulent pour l'hélium alors que le tungstène est principalement dominé par le ux néoclassique, ce qui correspond bien à nos prédictions analytiques. Récemment, il a été montré que l'asymétrie poloïdale de la densité des impuretés peut modi er substantiellement le transport néoclassique des impuretés. En l'absence de systèmes de chau age externes, l'asymétrie poloïdale est principalement due à la turbulence. L'asymétrie poloïdale générée par la turbulence s'est avérée plus forte pour les impuretés lourdes, ce qui conduit à une forte modi cation du transport néoclassique, notamment dans le régime de P rsch-Schlüter (PS). Les résultats globaux sont en bon accord avec nos descriptions analytiques, nous permettant de retrouver le coe cient du facteur d'écran thermique H GYS ∼ -0.5 proche de la prédiction théorique.

c) Les e ets de la rotation toroïdale sur le transport d'impuretés

La rotation toroïdale du plasma est un mécanisme qui peut accentuer l'asymétrie poloïdale de la densité des impuretés. Pendant la décharge du tokamak, la rotation toroïdale peut être générée de manière intrinsèque ou externe par le chau age par injection de faisceau neutre (NBI). La force centrifuge, en relation avec la rotation toroïdale, pousse les impuretés lourdes vers l'extérieur du tore, générant une forte asymétrie poloïdale 'in-out'. D'un point de vue analytique, cette forte asymétrie poloïdale peut augmenter ou réduire le transport néoclassique d'un ordre de grandeur et même changer sa direction. Dans le cadre de simulations gyrocinétiques non-linéaires réalisées avec GYSELA, des termes sources ajustables ont été appliqués pour étudier les e ets d'une forte rotation toroïdale. Les simulations non-linéaires réalisées avec une source de quantité de mouvement toroïdale supplémentaire ont montré que, dans le cas d'une forte rotation, le ux néoclassique vers le coeur a tendance à augmenter en raison d'une asymétrie poloïdale accrue, tandis que le ux turbulent est fortement affecté par la roto-di usion, ce qui conduit à un ux vers l'extérieur dans le cas d'une turbulence à gradient de température ionique (ITG). Ce terme, proportionnel à A/Z où A, Z sont respectivement la masse et la charge de l'impureté, devient dominant surtout pour les impuretés lourdes.

Dans ce manuscrit, nous avons étudié le transport turbulent et néoclassique des impuretés dans di érentes con gurations. Les résultats globaux obtenus tout au long de ce manuscrit impliquent que les impuretés lourdes ont tendance à s'accumuler dans la région centrale et sont très sensibles aux conditions externes. Heureusement, il semble qu'une utilisation appropriée d'un système de chau age par résonance cyclotronique ionique (ICRH) peut atténuer l'accumulation de tungstène au coeur en réduisant l'asymétrie poloïdale et en augmentant le ux turbulent vers l'extérieur. Il pourra être intéressant à l'avenir de prendre en compte le système de chau age ICRH dans les simulations GYSELA.

Bien que les connaissances actuelles sur le transport des impuretés soient comparables à celles des ions principaux (deutérium et tritium), il reste encore beaucoup de questions ouvertes. Par exemple, dans le cadre de GYSELA, les e ets du système de chau age de l'ICRH, des électrons cinétiques et du transport d'impuretés dans un tokamak à géométrie allongée semblent pouvoir être étudiés dans un avenir relativement proche. D'autre part, l'étude du transport d'impuretés à travers une barrière de transport (piédestal, fort gradient) est en cours à l'aide de GYSELA (thèse de G. Lo-Cascio). Il est pour l'instant impossible pour un code premier principe de prendre en compte tous les processus physiques impliqués dans un plasma de tokamak, tels que le transport dans le plasma de coeur, la physique du plasma de bord, la stabilité MHD, les chau ages externes et la génération du courant. C'est pour cette raison qu'une modélisation intégrée tenant compte de modèles réduits établis grâce aux résultats donnés par les codes "premier principe" de type GYSELA, sera essentielle pour une prédiction able des performances des tokamaks.

9

Fossil fuels started to t into the scheme of human history since the industrialization in the 1800s, and might end up in less than 100 years from now due to the depletion of available resources. Excessive usage of fossil fuels causes a steady increase in greenhouse gases, which is believed to be responsible for the rise of global average temperature. In the report of Intergovernmental Panel on Climate Change (IPCC) (IPCC, 2007), one of the scenarios predicts a doubling of CO 2 emissions and an increase of global temperature about 3-4 • C by 2100 (Figure 1.1).

F

1.1 -Projected surface temperature changes for the early and late 21st century relative to the period [START_REF] Goldstein | Classical mechanics[END_REF]-1999(IPCC, 2007)).

It is an undeniable fact that global warming is anthropogenic, which means originating in human activity, and we are currently living in the transition era for energy application from fossil fuels to renewable energy. However, although they can be replenished, known forms of renewable energy (e.g., wind, solar, hydro) will not be enough to sustain our living standards. An alternative solution, therefore, is necessary to replace the primary source and to mitigate upcoming climate change in the lifetimes of our children and grandchildren.

Two energy sources can ful ll the requirements of sustainable energy : (i) Nuclear ssion by splitting large nuclei into smaller ones, and (ii) Nuclear fusion by combining small nuclei into larger ones. Of these, nuclear fusion has been considered as the prospective green and safe energy.

Nuclear fusion : basic concepts

Nuclear fusion is the way the Sun and stars generate their energy. In the center of the Sun, the expected temperature amounts to 15 million degrees. At such temperature, 1.1. Nuclear fusion : basic concepts electrons are in some sense detached from nuclei, forming a cloud of charged ions and electrons, the plasma. The behavior of plasma is extremely complicated since each particle is in uenced by long-range interactions with other particles, and a new area of physics, namely plasma physics, was born from the e ort to understand such behavior.

For fusion to occur, light nuclei must possess su cient kinetic energy to overcome the electrostatic repulsion, the Coulomb barrier. In the case of deuterium ions, the necessary energy to overcome this barrier amounts to 0.4MeV(∼ 4.6 × 10 9 K). However, such extreme temperature is not necessary due to quantum tunneling e ect and signicant rates of fusion can be achieved even within the tens-of-keV range.

During the fusion reaction, energy is released when light elements are transformed into heavy elements which have higher binding energy3 . As a result, more tightly bounded nuclei emerge and the corresponding mass defect is transformed into tremendous energy, which is the order of MeV. If fusion starts with a light nucleus like hydrogen, two nuclei have to be fused to produce a higher atomic number element moving from left to right on Figure 1.2 while ssion goes from right to left with a heavy nucleus being separated into a lighter element.

F

1.2 -The relationship between the nuclear binding energy of an isotope and its atomic mass. Taken from (https://www.dlt.ncssm.edu/tiger/).

The most accessible nuclear fusion happens by combining the hydrogen isotopes-deuterium (D) and tritium (T) -producing helium and a neutron. The neutron carries 80% of re-leased energy and it has to be captured in order to convert its energy into heat to run a power plant.

2 1 D + 3 1 T --→ 4 2 He (3.5 MeV) + n (14.1 MeV) (1.1)
For this reaction, the fuel is a mix of deuterium and tritium. There is 5 × 10 16 kg deuterium in the oceans and if we consider the usual quantity consumed in reactors (around 100 kg per reactor per year), these will last for the foreseeable future. While an ample supply of deuterium can be extracted from the oceans, the total storage of tritium is relatively scarce, estimated at around 50 kg. A 1000 MW plant is expected to burn out 250 g of tritium daily, thus an alternative to breed tritium fuel is a paramount priority [START_REF] Tanabe | Tritium : Fuel of Fusion Reactors[END_REF].

There is no material that can withstand temperatures of millions of degrees. While a strong gravitational eld provides ideal conditions for fusion to occur in the sun, on Earth, an alternative method is necessary to shape the plasma and eventually con ne it. Applying a magnetic eld has been proposed to con ne plasma. Indeed, a magnetic eld can hold plasma particles in tight circular orbits around the magnetic eld lines, called Larmor orbits, preventing them from moving across the magnetic eld.

Magnetic con nement : Tokamaks

In order to obtain a closed set of magnetic eld lines, a torus-shaped machine called tokamak was invented by a group of Soviet scientists. In a con ned plasma, A toroidal surface covered by a contours of constant pressure is called ux surface which implies the magnetic eld lines must lie on this surface (Figure 1.3). It is instructive to analyse the global geometry of the magnetic eld since important issues, such as stability, transport, and particle trajectory, depend on it.

Magnetic eld con guration

The general form of the magnetic eld in an axisymmetric tokamak B can be expressed as [START_REF] Hazeltine | Plasma Con nement[END_REF])

B = B ϕ êϕ + B θ êθ = I(ψ)∇ϕ + ∇ϕ × ∇ψ (1.2)
where ψ is the opposite of the poloidal magnetic ux, which is a label of ux surface, I(ψ) is a ux label measuring the toroidal eld and ϕ is the toroidal angle (Figure 1.3). Some useful relations can be derived from Eq.(1.2) by using vectorial relations, which will be used extensively in this manuscript :

B × ∇ψ B 2 = I B b -R φ (1.3) B × R 2 ∇ϕ = ∇ψ (1.4)
1.2. Magnetic con nement : Tokamaks F 1.3 -The tokamak magnetic con guration and the toroidal coordinate system (r, θ, ϕ) (Grandgirard, 2016).

where we used the unit vector in the toroidal direction φ = R∇ϕ and

R 2 B 2 = I 2 + |∇ψ| 2 .
Another very useful notation is the ux surface average [START_REF] Helander | Collisional Transport in Magnetized Plasmas[END_REF]. When a quantity is given Q(ψ, θ), its ux surface average Q is the volume average of Q between two neighbouring ux surfaces ψ and ψ + dψ.

Q (ψ) = Q(ψ, θ)dV dV = Q(ψ, θ) B • ∇θ dθ dθ B • ∇θ (1.5)
and the volume element is de ned as

dV = g 1/2 dθdϕ = dθdϕ/|(∇ϕ × ∇ψ) • ∇θ| (1.6)
where

g 1/2 = 1/|(∇ϕ × ∇ψ) • ∇θ| = 1/|B • ∇θ| is the Jacobian 4 .
Note that the ux surface average annihilates the operator

B • ∇ = g -1/2 ∂ ∂θ . B • ∇Q(ψ, θ) = 0 (1.7)
Using the ux surface average, it is possible to introduce the safety factor as the average change in ϕ relative to the average change in θ along the magnetic eld.

q(ψ) = B • ∇ϕ B • ∇θ (1.8)
In general, the value of q is usually larger than unity for stability reasons.

4. When we switch to a new coordinate basis, also known as substitution of variables, we stretch things out by di erent amounts in di erent directions. The extent of this stretching is called the Jacobian. We use it so that when we integrate over our new coordinates, we can correct for the stretching and nd the actual area from our old coordinates.

Circulating & trapped particles

In general, the magnetic eld in tokamaks is composed of toroidal and poloidal magnetic elds respectively. A eld line forms a helix wound on a toroidal surface. In a strong, axisymmetric magnetic eld, the equilibrium motion of a charged particle can be characterized by three constants of motion [START_REF] Noether | Invariant variation problems[END_REF].

E = 1 2 mv 2 + µB + eφ eq µ = mv 2 ⊥ 2B P ϕ = mRv ϕ -Zeψ (1.9)
where µ is the magnetic moment and P ϕ is the toroidal canonical momentum.

Since the background magnetic eld is spatially inhomogeneous (stronger near the vertical axis of the tokamak), particles undergo poloidal variations of the magnetic eld while owing along the eld lines. As the magnetic eld B changes in the particle frame, v ⊥ varies to conserve the magnetic moment µ, thus v changes accordingly to conserve energy. Due to this e ect, particles are separated into 'passing' and 'trapped' particles. From the energy conservation Eq.(1.9), the parallel velocity of particles can be written

v = ± 2 m (E -µB) (1.10)
When E > µ s B, the parallel velocity has always positive sign even in the strong magnetic eld B max , completing its orbit without any interruption. However, when E < µ s B, the parallel velocity approaches to zero and at a certain point (θ = θ b ) particles bounce back. Since the energy is conserved regardless of angle E| θ=0 = E| θ=θ b , it is possible to express B = B b | θ=θ b as follows :

B b = B min(θ=0) 1 + v 2 v 2 ⊥ (1.11)
Knowing that in the limit of circular concentric surfaces, the magnetic eld is equal to

B(r, θ) = B 0 1 + cos θ (1.12)
where B 0 denotes the eld amplitude at the magnetic axis and = r/R 0 .

It is clear that the magnetic eld for a bouncing point B b must be smaller than B max and this relation gives the trapping condition which separates each species into passing and trapped populations (Figure 1

.4). v v ⊥ √ 2 (1.13)
The trapped population, with insu cient parallel velocity to overcome the strong magnetic eld, is con ned to the low-eld side with banana-shaped orbits and experiences toroidal precession, which is the average drift motion of the particles in the toroidal direction.

F 1.4 -Regions of trapped and circulating particles. Reconstructed from (Helander and Sigmar, 2005).

Particle transport in tokamaks

Understanding transport of matter, energy and momentum is one of the most important issues in the eld of controlled nuclear fusion. Transport across the magnetic eld which con nes the high-temperature plasma threatens the e ciency of magnetic fusion.

During the early stages, the classical transport theory had gained a famous reputation by successfully predicting the transport coe cients of plasma, such as di usion and heat conduction [START_REF] Braginskii | Transport processes in a plasma[END_REF]. However, such classical theory largely underestimates the transport coe cients when it comes to the high-temperature plasma in tokamaks, sometimes by several orders of magnitude.

To remedy this problem, special attention has been dedicated to the e ect of the magnetic eld in tokamaks, which is spatially inhomogeneous. It turns out that the toroidal magnetic con guration in tokamaks can exert a strong e ect on the transport of plasma combined with the collisions. When these e ects of the geometry were taken into account, one found transport coe cients closer to the experimental values than the classical ones. The birth of the so-called neoclassical transport theory has fascinated many other scientists and a complete and detailed analytical approach was elegantly established thanks to the pioneering works by [START_REF] Galeev | Transport phenomena in a collisionless plasma in a toroidai magnetic system[END_REF][START_REF] Hinton | Theory of plasma transport in toroidal connement systems[END_REF][START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF].

However, as the technology of plasma diagnostics advances, it has been reported that even with the neoclassical theory, the transport coe cients are still underestimated by an order of magnitude. The reason for this discrepancy comes from the fact that plasma in tokamaks is never in a quiescent state due to the background density and temperature gradients. Coupled with the Coulomb forces, the collective nature of the plasma makes the particles behave as waves, modes, vortices and streamers, which might also interact between themselves. These uctuations enhance the transport of matter and energy leading to the so-called anomalous or turbulent transport which largely dominates the classical and neoclassical transport.

The ultimate goal of transport theory is to gure out how the uxes are generated by thermodynamical forces-related to the density, pressure, temperature, and electric potential-and how they are a ected by the magnetic con guration and the tokamak geometry. Theoretically, the particle ux across the magnetic surfaces can be readily obtained from the continuity and momentum balance equation [START_REF] Helander | Collisional Transport in Magnetized Plasmas[END_REF])

∂N ∂t + ∇ • (N V ) = 0 (1.14) mN ∂V ∂t = -∇p + N Ze(E + V × B) + F (1.15)
where we neglected the viscosity term and F = mvC(f )d 3 v is the frictional force and V is the macroscopic uid velocity. Note that the velocity of a particular particle di ers from this average velocity by v = v -V . Applying Gauss's law to Eq.(1.14) gives

- ∂ ∂t V N d 3 r = S N V • dS = Γ • ∇ψ dθdϕ B • ∇θ (1.16)
where we used the surface element dS = ∇ψdθdϕ/B • ∇θ, particle ux Γ = N V with the ux surfaces assumed to be frozen. The important term in the above equation is Γ • ∇ψ which represents the particle ux across the ux surfaces. This term can be further simpli ed using the relation Eq.(1.4), which writes :

Γ s • ∇ψ = Γ s • (B × R φ) = (Γ s × B) • R φ = R φ • (N s V s × B) (1.17)
where the subscript s denotes ion species. To relate this equation with the momentum balance equation, we project Eq.(1.15) on the toroidal direction. Using the ux surface averaged toroidal projection and by assuming the stationary state (∂ t = 0) and an axisymmetric case (∂ ϕ = 0), one can write

e s Γ s • ∇ψ = -R φ • N s e s E -R φ • F = -N s e s RE ϕ -RF sϕ (1.18)
After some calculations and using Eq.(1.3), the nal form of particle ux can be recast as 5

e s Γ s • ∇ψ = B × ∇ψ B 2 • F Classical -I F s + N s e s E B Neoclassical + N s e s E × B B 2 • ∇ψ Turbulent (1.19)
The classical ux is related to the perpendicular friction forces while the neoclassical ux is linked to the parallel component of it. This di erence explains their underlying mechanism. Classical transport is generated by the frictional relaxation of the diamagnetic ow associated with gyromotion. On the other hand, neoclassical ux is generated through the friction force acting on the guiding center, which ows mainly along the magnetic eld line. The last term is related to E × B drift, and is called turbulent particle ux.

Turbulent transport

Understanding turbulent transport is a challenging task due to the nonlinear features of turbulence. In a linear analysis, where the amplitudes of waves are assumed to be small, the problem can be solved in algebraic form with the help of Fourier transformation, i.e. φ n (x, t) = φ 0 exp(ik • x -iΩt) where k is a wave vector and Ω is a complex frequency. In this simplest form, each mode is characterized by a dispersion relation D(Ω, k) = 0. Each solution of this relation de nes an eigenmode. In the linear approximation, the evolution of each mode is treated independently, thus the nonlinear terms are neglected. An eigenvalue of the dispersion relation can be written as :

Ω(k) = ω(k) + iγ(k) (1.20) 5. e s Γ s • ∇ψ = -N s e s RE ϕ -RF sϕ = -R φ • N s e s E -R φ • F s = - I B b -B×∇ψ B 2 • (N s e s E) - I B b -B×∇ψ B 2 • F s = -I esNsE B + N s e s E×B B 2 • ∇ψ -I B F s + B×∇ψ B 2 • F s = -I F a +NsesE B + N s e s E×B B 2 • ∇ψ + B×∇ψ B 2 • F
where the real part ω(k) is the frequency and the imaginary part γ(k) is the growth rate (and the mode is damped if γ < 0).

In fusion plasma, many modes are linearly unstable (γ > 0). For such a mode, the wave amplitude grows exponentially in time, increasing the nonlinear interaction among the waves and individual particles. Since the background density and temperature pro les are inhomogeneous, the radial pressure gradient plays a crucial role in driving unstable modes, which yield turbulent transport (Figure 1.5). In general, two main electrostatic micro-instabilities play a crucial role in turbulent transport : (i) ion temperature gradient (ITG) modes and (ii) trapped electron modes (TEM). These modes are unstable in the limit of large wavelength such as k ⊥ ρ i < 1 where k ⊥ is the perpendicular wave and ρ i is the ion Larmor radius. In tokamaks, these instabilities are induced by the magnetic drift (v ∇B ) combined with background density and temperature gradients (Figure 1.5). The initial perturbation (white line) is enhanced since the unbalanced v E×B ampli es the uctuations, given the temperature gradient. In non-linear regime, ITG/TEM modes produce particle, momentum, electron and ion heat transport. ITG-driven modes usually depend on the ion temperature logarithmic gradient and also on the ratio of electron to ion temperature T e /T i . This mode usually rotates in the ion diamagnetic rotation while TEMs, driven through a resonant interaction of the modes with trapped electrons at the precession frequency, rotates in the electron diamagnetic direction [START_REF] Garbet | Physics of transport in tokamaks[END_REF].

Neoclassical transport

Neoclassical transport broadens classical transport to include the e ects of magnetic eld and collisions, which requires the following ordering to be satis ed :

1. The macroscopic scale length L is assumed to be large compared to the Larmor radius δ = ρ c /L 1 which gives a homogeneous plasma over the gyro motion.

2. the gyrofrequency is much larger than the collision frequency ν/Ω 1. So, the associated time derivative can often be neglected

∂ ∂t ∼ D L 2 ∼ δ 2 ν (1.21)
where ν is the collision frequency and D ∼ νρ 2 is the di usion coe cient.

The dynamics of neoclassical transport depends on the relative magnitude of collisionality ν * which is the ratio of the detrapping frequency ν detrap = ν/ and the bounce frequency

ω b ≈ 1/2 v T /(qR). ν * = ν detrap ω b ≈ qR 3/2 ν v T (1.22)
In the weak collisional regime (ν * 1), also named banana regime, passing particles have su cient time to complete their orbit without any interruption. The radial displacement mainly happens by collisions among trapped particles (the radial displacement due to collisions of passing particles is negligible). Therefore, the typical radial displacement is equal to the width of banana orbit ∆l = δ b . Taking into account the fraction of trapped particles f t √ 2 , one can nd the perpendicular di usion coe cient

D banana = f t δ 2 b ν q 2 3/2 ρ 2 c ν (1.23)
In the intermediate collisional regime (1 ν * -3/2 ), named plateau, only a small fraction of trapped particles experience collisions before completing their orbit. Proportional to ω b /ν, the perpendicular di usion coe cient can be simply expressed as follows :

D plateau = ω b ν D banana qv T R ρ 2 c (1.24)
As the name plateau implies, the perpendicular di usion coe cient of this regime is constant regardless of collisionality (Figure 1.6).

In the strong collisional regime (ν * -3/2 ), so-called the P rsch-Schlüter (PS) regime, the orbit of passing / trapped particles is constantly interrupted by collisions.

The main transport in this regime, therefore, is usually governed by parallel motion with the di usion coe cient D ∼ v 2 T /ν. The displacements of particles ∆l due to collisions are caused by the curvature magnetic drift v D during the time τ . Then, the perpendicular di usion coe cient can be obtained readily

D PS = q 2 ρ 2 c ν (1.25) F 1.
6 -Dependence of neoclassical perpendicular di usion coe cient on collisionality. Reconstructed from [START_REF] Helander | Collisional Transport in Magnetized Plasmas[END_REF].

According to the collisionality de ned above, the neoclassical particle ux can be separated into di erent collisional regimes. Therefore, it is meaningful to decompose the neoclassical uxes (Γ neo = Γ BP + Γ PS ) from Eq.(1.19) as follows [START_REF] Helander | Collisional Transport in Magnetized Plasmas[END_REF] :

Γ s • ∇ψ BP = -I B(F s + N s e s E (A) ) e s B 2
(1.26)

Γ s • ∇ψ PS = -I F s + N s e s E (A) e s B   1 - B 2 B 2   (1.27)
The rst part relates to the banana-plateau ux which is mainly driven by the pressure anisotropy in the weakly collisional regime. The information of such anisotropy is contained in the frictional force term F s . The second part, corresponding to the P rsch-Schlüter ux, results from the poloidal variation of B on the magnetic surfaces. In this regime, the pressure tends to be isotropic due to collisions and its e ect is negligible. Note that the above analytical derivation does not take into account the poloidal asymmetry of impurity density. In the following chapter, the e ects of poloidal asymmetry on neoclassical transport will be addressed.

Impurities in tokamaks

Although an ideal fusion reaction requires only a pure mix of deuterium (D) and tritium (T), the generation of impurities is inevitable in a reactor. During a plasma discharge, di erent types of impurity can be produced :

1. Helium (He), as the ash of D-T burning plasma, is an inevitable impurity. The hot helium produced by the fusion reaction is often called as α-particle, which will be essential in future tokamak experiments for heating scenarios.

2. Tungsten (W) and carbon (C) are often used as plasma-facing components (PFC).

Sputtering of these rst wall components results in a substantial impurity source in a tokamak plasma, which yields radiation losses (Figure 1.7).

3. Low-Z or intermediate-Z impurities-argon (Ar), nitrogen (N), and neon (Ne)-are often injected, namely impurity seeding, in order to reduce direct heat loads on the divertor plates by radiating in the peripheral region.

In recent decades, a strong emphasis has been placed on the behavior and transport of impurities since even with a small tungsten concentration ∼ 10 -5 [START_REF] Pütterich | Calculation and experimental test of the cooling factor of tungsten[END_REF], the hot plasma can be completely extinguished by radiation loss. Especially, tungsten is not totally ionized even at the high fusion temperatures and can experience further radiation. Also, increasing impurity concentration naturally decreases the concentration of the main fuel which will degrade the fusion power e ciency (Figure 1.8).

Impurity accumulation

To achieve maximum fusion e ciency, it is mandatory to minimize and remove the impurities from the plasma. This can be achieved using impurity transport along the open magnetic eld lines of the Scrape-O Layer (SOL), which drives impurities to the divertor plate. It is therefore possible to remove impurities from the plasma while keeping good thermal insulation between the hot plasma and the cold boundary region. However, this task is far from easy to accomplish.

According to the standard neoclassical theory, which explains particle transport driven by collisional e ects, impurity ux can be calculated for the case when impurities (main ions) are in the PS (banana) regime and ion-electron collisions are neglected [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF] :

Γ z,neo = -N z D   ∂ r N z N z -Z ∂ r N i N i + Z 2 ∂ r T i T i   (1.28)
where N s is the density, T s is the temperature, D is the di usion coe cient, Z is the impurity charge and i(z) refers to the main ions (impurity). In the stationary state (∂ t Γ = 0), the nal impurity density pro le can be recast as :

N z (r) N z (r 0 ) = N i (r) N i (r 0 ) Z T i (r 0 ) T i (r) Z 2
(1.29)

According to neoclassical theory, the impurity density depends on the relative peaking of the pro les of main ion density N i and temperature T i . Peaked or hollow impu-rity density pro le at the magnetic axis r 0 becomes signi cant especially for high Z impurities. For example, the central accumulation of heavy impurities, dominated by neoclassical transport, has been frequently observed in many experiments [START_REF] Ran | Impurity accumulation in plasma regimes with high energy con nement[END_REF][START_REF] Ida | Impurity behaviour in PBX L-and H-mode plasmas[END_REF][START_REF] Fussmann | Impurity transport and neoclassical predictions[END_REF][START_REF] Pasini | Measurements of impurity transport in JET[END_REF][START_REF] Tokar | Nature of high-Z impurity accumulation in tokamaks[END_REF]. With a peaked temperature pro le, the temperature gradient dependency in Eq.(1.29) would be advantageous, to avoid core accumulation by generating a temperature screening. Therefore, a proper use of an external heating system could make it possible to control advantageously impurity transport, and more details about this topic will be discussed below.

E ects of a high impurity concentration in plasma are twofold. When it comes to on-axis accumulation, high Z impurities will experience further ionization by emitting ultraviolet or X radiation, which cools the plasma and reduces the fusion yield [START_REF] De Vries | The in uence of an ITER-like wall on disruptions at JET[END_REF]. When properly controlled, however, the presence of impurities can be bene cial to an improved con nement regime. It is now well known that a properly induced radiation by impurity seeding in the periphery and SOL improves the con nement by reducing localized overheating in the divertor [START_REF] Schweinzer | Con nement of 'improved H-modes' in the all-tungsten ASDEX Upgrade with nitrogen seeding[END_REF][START_REF] Giroud | Impact of nitrogen seeding on con nement and power load control of a high-triangularity JET ELMy H-mode plasma with a metal wall[END_REF][START_REF] Kallenbach | Impurity seeding for tokamak power exhaust : from present devices via ITER to DEMO[END_REF][START_REF] Xu | Divertor impurity seeding with a new feedback control scheme for maintaining good core con nement in grassy-ELM H-mode regime with tungsten monoblock divertor in EAST[END_REF].

For these reasons, it is of paramount importance that impurity density pro le and impurity concentration have to be controlled and predicted, which is the ultimate goal of the study of impurity transport. F 1.9 -(Left) SXR radiation pro le from AUG discharge [START_REF] Odstrcil | Optimized tomography methods for plasma emissivity reconstruction at the ASDEX Upgrade tokamak[END_REF]. (right) ICRH control of impurity accumulation at the core in JET-ILW discharge [START_REF] Lerche | Optimization of ICRH for core impurity control in JET-ILW[END_REF].

Poloidal asymmetry of impurity density

Neoclassical theory often applies the simplifying assumption of uniform density distribution of impurities over each ux surface [START_REF] Hinton | Theory of plasma transport in toroidal connement systems[END_REF]. However, with the progress of diagnostics, such as soft X-ray (SXR) cameras and bolometry, this assumption proved to be wrong, and a poloidal asymmetric distribution has been observed repeatedly in many tokamaks. Since the rst experimental observation [START_REF] Terry | Observation of poloidal asymmetry in impurity-ion emission due to ∇B drifts[END_REF], the non-uniform distribution of impurity density has received much attention because a strong poloidal asymmetry can modify considerably neoclassical particle ux and even change its direction [START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF].

To determine the poloidal variation of the impurity density, the momentum balance equation can be projected onto the parallel direction to the magnetic eld [START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF])

N z m z ω 2 2 ∇ R 2 + ZN z e∇ φ + T z ∇ N z = 0 (1.30)
where ∇ = b • ∇ and ω is the toroidal rotation frequency and the parallel friction and viscosity have been neglected. The rst term in Eq.(1.30) represents the centrifugal force induced by toroidal rotation. It can induce poloidal asymmetry, enhancing its impact, particularly for heavy impurities. The e ect of the centrifugal force was theoretically derived in [START_REF] Wong | Transport of impure plasma with arbitrary toroidal rotation[END_REF], showing that in case of a strong rotation, impurities are pushed away toward the wall, generating 'in-out' poloidal asymmetry. This situation often happens when neutral beam injection (NBI) is applied as plasma heating.

The second term is related to the parallel gradient of the electrostatic potential. In order to study its e ect, it is necessary to solve a quasi-neutrality equation for φ (r, θ). Although the poloidal variation will be small for electrons e φ/T e 1, this e ect will be strongly ampli ed for high Z impurities. External input of energy via waves-Ion Cyclotron Resonance Heating (ICRH)-is considered to be one of the main causes, as well as centrifugal e ects, for the poloidal electric potential to be modi ed. For example, when the cyclotron heating is applied, the perpendicular velocity of particles is increased and the number of trapped particles increases, which impact the electric potential. Fortunately, recent experiments indicate that poloidal asymmetry driven by ICRH tends to expel impurities from the core and largely reduce the radiation cooling factor (Figure 1.9).

Outline

Throughout this introduction, basic notions of nuclear fusion, turbulent and neoclassical particle transport and the importance of studying impurities are addressed. The main goal of this manuscript is to address key aspects of the following general question How to model impurity transport in the core of tokamaks ?

A complete identi cation of main mechanisms for impurity transport is too broad to be discussed in this manuscript. Hence, only a restricted number of topics will be discussed in the following chapters :

1. Impurity convection due to turbulence driven by trapped particles 2. Impurity neoclassical and turbulent transport of light to heavy impurities

Impact of the toroidal rotation on impurity transport

Note that in this manuscript we restrict ourselves to the electrostatic case where the e ects of MHD instabilities are neglected.

F 1.10 -The scope of this thesis. Two main transport channels : (i) turbulent transport (ii) neoclassical transport will be discussed in this manuscript.

Chapitre 2

Gyrokinetic model for impurity transport The study of impurity transport requires the understanding of both turbulent and neoclassical transport. Due to the strong interplay between the complex trajectory of charged particles and the uctuations of collective modes, a wide range of spatiotemporal scales needs to be considered and it necessitates an appropriate model which encompasses such multi-scale nature properly. According to various approaches, plasma models can be roughly classi ed into kinetic models and uid models.

The ultimate goal of plasma modelling is to predict the plasma behavior by solving kinetic or uid models coupled with Maxwell's equations. To this end, it is necessary to follow the time evolution of the global pro les, such as temperature, density and current, during the energy con nement time τ E . Although we are fully equipped with the laws of physics which describe the evolution of the plasma, it is untractable to obtain solutions for three-dimensional complex geometry and nonlinear problems. For this reason, plasma research strongly bene ts from progress in computational modelling and this aspect will be brie y discussed in this chapter.

Plasma models

A typical magnetic fusion plasma is composed of approximately 10 20 m -3 of ions and electrons. A complete description of all these particles require a solution of Newton-Maxwell equations for all individual particles. These equations consist of a set of 6N di erential equations which determine the N -body description of the system. This approach of tracking all particles is untractable with the current high performance computing and even in the any foreseeable future. Therefore, the number of equations must be reduced with reasonable assumptions in order to lower the dimension of the system.

Kinetic model

Any physical properties of microscopic detail can be obtained by its position and velocity vectors in phase space. Therefore, instead of calculating the dynamics of single particles, it is useful to introduce a statistical approach with the distribution function F (r, v, t), which is the density of particles in (r, v) space at time t. The evolution of the distribution function is described by the kinetic theory.

The Vlasov equation describes the evolution of the distribution function in a 6dimensional phase space (r, v) under the e ect of self-consistent electric and magnetic elds :

∂F s ∂t + v • ∂F s ∂r + q s m s (E + v × B) • ∂F s ∂v = 0 (2.1)
A great complexity to solve the Vlasov equation emerges from the third term of Eq.(2.1), which contains both the external elds and the self-consistent elds arising from the motion of all charged particles inside the plasma. These elds satisfy the Maxwell equations

∇ • E = ρ/ 0 (2.2) ∇ • B = 0 (2.3) ∇ × E = - ∂B ∂t (2.4) ∇ × B = µ 0 J + 1 c 2 ∂E ∂t (2.5)
where the plasma charge density ρ and the plasma current density J are obtained by taking the moments of F s :

ρ(x, t) = s q s dvF s (2.6) J (x, t) = s q s vdvF s (2.7)
The set of equations from Eq.(2.1) to Eq.(2.7) constitutes a complete and self-consistent equations which need to be solved simultaneously. The di erent processes to resolve these nonlinear equations in the linear, quasi-linear and nonlinear approximations form the basis of kinetic theory and one generally has to use numerical methods even for a linear analysis.

Fluid model

The main advantage in using a uid model is to alleviate the complexities of kinetic equations. The uid model is based on a set of equations which describes the macroscopic quantities such as density N , mean velocity V and temperature T . These quantities can be readily calculated from the distribution function by multiplying various power of v and then integrating over velocity space. The rst three moments of the Vlasov equation, multiplying it by 1, v and v 2 /2 and integrating over the velocity space, give the equations of conservation of mass, momentum and energy.

∂N ∂t + ∇ • (N V ) = 0 (2.8) ∂mN V ∂t + ∇ • Π = N Ze(E + V × B) (2.9) ∂ ∂t 3N T 2 + mN V 2 2 + ∇ • Q = ZeN E • V (2.10)
where the following notations for the ux of momentum and energy are used :

Pressure tensor :

Π jk = mnv j v k f (2.11) Heat ux : Q = mn v 2 v f 2 (2.
12)

The uid model governs the space-time evolution of macroscopic variables (N, T, V ) for each species in a fusion plasma. However, each moment equation requires a higher moment equation. For example, the evolution of density N contains the ow velocity V , the evolution of V contains the pressure Π and the evolution of temperature T contains the higher moment of heat ux Q. Therefore, it is necessary to truncate these consecutive equations at some point by introducing a simplifying assumption for the hierarchy.

Gyrokinetic model

Although the kinetic model is the most fundamental way to describe a plasma, a broad range of spatio-temporal scales makes it di cult to simulate micro-instabilities and collisional e ects via the Vlasov equation. In order to overcome this scale limit, a reduced model called gyrokinetic model has been developed. Given the fact that characteristic frequencies of micro-turbulence ω are smaller than the cyclotron frequency ω c , high frequency scales are averaged in gyrokinetic approach while keeping essential kinetic e ects.

Gyrokinetic ordering

In fusion plasma, the presence of a strong magnetic eld allows the separation on the dynamics between a gyromotion perpendicular to the magnetic eld line with gyrofrequency ω c and a parallel motion along the magnetic eld line with parallel velocity v = v • b. Under an inhomogeneous magnetic eld, charged particles exhibit three types of quasi-periodic motion ; (i) a rapid gyromotion around a magnetic eld line with cyclotron frequency ω c and the Larmor radius ρ c ; (ii) an intermediate bounce motion with bounce frequency ω b and the banana width δ b ; (iii) a slow precession with precession frequency ω d and the major tokamak radius R 0 . The periodic time scales ω d ω b ω c are usually well respected for magnetically con ned plasma, allowing to decouple the fast gyromotion from the bounce motion and from the toroidal precession of charged particles.

For plasma turbulence, the characteristic frequencies ω are so much smaller than the cyclotron frequency that a reduction of the dimension can be performed via phasespace coordinate transformations. The modern gyrokinetic theory consists of the guidingcentre transform and a gyro-center transform based on the Hamiltonian or Lagrangian formalism with the Lie perturbation theory [START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF]. In order to apply this approach, a certain ordering should be satis ed. From experimental observations, it has been reported that tokamak micro-turbulence in the core obeys the following gyrokinetic ordering

ω ω c ∼ k k ⊥ ∼ q s φ T ∼ δN s N 0 ∼ δB B 0 ∼ ρ c L N ∼ O(ρ * s ) (2.13)
where ω is a characteristic frequency of microturbulence, k and k ⊥ are parallel and perpendicular components of the wave vector k, δA represents the perturbed quantity A, L N = |∇ ln N 0 | -1 is a characteristic scale length of N 0 and ρ * = ρ c /a is the dimensionless parameter which corresponds to the ratio between the Larmor radius and the tokamak minor radius. Within this gyro-ordering, a gyrokinetic model can be derived by eliminating the high frequency ω c .

In this framework, one can develop the gyrokinetic equation at an arbitrary order in ρ * while keeping the Hamiltonian structure of the Vlasov equation. The gyro-averaged Hamiltonian Hs can be written as

Hs = 1 2 m s v 2 G + µB + q s φ (2.14)
Note that gyro-averaging at this stage does not simply mean averaging out the cyclotron motion, but the the transform should retain the information of the elds at the Larmor radius scale.

F 2.1 -Reproduced from [START_REF] Garbet | Gyrokinetic simulations of turbulent transport[END_REF]. By applying gyro-centre transform, the dimensionality is reduced by one and the high frequency motion is averaged. Here guiding-centre represents the point around which particles circle while gyro-centre signi es a new coordinate obtained from gyro-centre transformation.

Numerical features of gyrokinetic codes

In this manuscript, we use two di erent gyrokinetic models, implemented in two gyrokinetic codes :

1. GYSELA-5D (GYrokinetic SEmi-LAgrangian) (see section 2.3) 2. TERESA-4D (Trapped Element REduction in Semi-lagrangian Approach) (see section 2.4)
These codes are developed at IRFM (Institute for Magnetic Fusion Research) and IJL (Institut Jean Lamour) with strong collaborations at both national and international levels. Before we proceed to the theoretical derivation of the two models, let us discuss key numerical features of both codes. Originated from the same Semi-Lagrangian family, GYSELA and TERESA share a lot of numerical common points, such as global geometry, full-f approach, Semi-Lagrangian numerical scheme and capability of treating kinetic electrons. The main di erences between two codes arise from ; (i) the way in which they drive plasma out of equilibrium, gradient driven in TERESA and ux driven in GYSELA (ii) species of interest ; passing and trapped particles in GYSELA and only trapped particles in TE-RESA, which means TERESA can be considered as a reduced model when compared to GYSELA. A brief description of these salient features is given in Table 2.1. Regarding more detailed information on numerical features in gyrokinetic code, the reader is referred to [START_REF] Garbet | Gyrokinetic simulations of turbulent transport[END_REF]. A brief overview of numerical schemes for the Vlasov equation will be discussed in the following section.

Semi-Lagrangian method

Since the pioneering work by [START_REF] Lee | Gyrokinetic approach in particle simulation[END_REF], known as a rst gyrokinetic code, three di erent types of numerical schemes have become mainstream over the last few decades :

1. Lagrangian method uses a discrete description of macro-particles in phase space.

Each macro-particle represents a large number of plasma particles. These macro particles follow the characteristics of the Vlasov equation, along which the distribution function is exactly conserved, while charge and current densities need to be calculated by extrapolation. The main drawback of the Lagrangian method is that not only the inherent numerical noise makes hard to describe the distribution function accurately, but it accumulates in time also.

2.

Eulerian method uses a xed grid in all directions of phase space in order to avoid numerical noise from Lagrangian method, and applying nite di erences or spectral methods to model the Vlasov equation. Although it is noise free, the integro-di erential equation imposes the Courant-Friedrichs-Lewy (CFL) stability condition, which constraints the maximum time step of numerical simulation.

3. To take advantage of both methods and avoid limitations, Semi-Lagrangian method has been applied in GYSELA & TERESA with an accurate description of the phase space. The xed grid point (Eulerian) is adopted in order to avoid numerical noise and the characteristics of the Vlasov equation are used for the time evolution of the distribution function. A more detailed information about Semi-Lagrangian method can be found in [START_REF] Sonnendrücker | The Semi-Lagrangian method for the numerical resolution of the Vlasov equation[END_REF].

Adiabatic electrons

• Electrons are assumed to follow a Boltzmann response in a ux surface

Kinetic electrons

• Electrons are treated kinetically in the same way as ions

Full-F

• The whole distribution function F = F eq + δF is numerically treated without any scale separation • The back reaction of turbulent transport is accounted for in the time evolution of the equilibrium • The mean gradients are somewhat maintained or can be relaxing in time by a prescribed heat source ( ux driven).

• More appropriate within a global description δF

• Only the deviation part δF from the Maxwellian is numerically treated with the assumption (δF F eq ) • Statistical noise is largely reduced due to the scale separation, contributing to the stability of numerical simulations • The mean gradients are naturally imposed in the assumed equilibrium (gradient driven)

• More appropriate within a local ( ux tube) description 2.3 GYSELA-5D : Gyrokinetic model GYSELA (GYrokinetic SEmi-LAgrangian) is a global, full-F, ux driven, electrostatic gyrokinetic code which treats electrons adiabatically 7 in a simpli ed concentric circular magnetic con guration. The time evolution of the full distribution function of each ion species (Deuterium + impurity) is governed self-consistently by a 5D non-linear gyrokinetic Vlasov equation coupled with 3D Poisson equation. In this section, the salient features of the code are presented in a concise manner. For a more detailed analytical derivation of GYSELA, the reader is referred to (Grandgirard et al., 2016). F 2.2 -3D visualization of the turbulent uctuations of the electric potential resulting from the GYSELA-5D simulations 8 .

Gyrokinetic Vlasov equation

Unlike δF -algorithm, GYSELA deals with the evolution of 5D gyro-center distribution function F for each species without any segregation between equilibrium and perturbation. In the Hamiltonian formulation [START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF], the 5D colli-7. Although kinetic treatment of electrons was implemented in GYSELA, it hasn't been fully tested. Therefore, we restrict ourselves solely to adiabatic electrons in this manuscript.

8. GYSELA-X git repository (https://gyselax.github.io/)

sional gyrokinetic equation can be recast as

d Fs dt = ∂ Fs ∂t + [ Fs , Hs ] = ∂ Fs ∂t + dx G dt • ∂ Fs ∂x G + dv G dt ∂ Fs ∂v G = C( Fs ) + S( Fs ) (2.15)
where C and S are respectively a collision operator and source terms and [F, G] is the gyrokinetic Poisson bracket operator, de ned as

[F, G] = ω s B * ,s ∂F ∂α ∂G ∂µ - ∂F ∂µ ∂G ∂α - b q s B * ,s • ∇F × ∇G + B * ,s m s B * ,s • ∇F ∂G ∂v G - ∂F ∂v G ∇G (2.16)
where

B * = B + (m s v G /q s )∇ × b and B * ,s = b • B *
is the volume element in the guiding-center velocity space. Within the electrostatic limit, the evolution of gyrocenter coordinates associated to Eq.(2.15) are described by

B * ,s B dx G dt = v G b + v E×B + v D (2.17) B * ,s B m s dv G dt = -q s b • ∇ φ -µb • ∇B + m s v G B v E×B • ∇B (2.18) B * ,s B m s dµ dt = 0 (2.19)
where the guiding center drifts are de ned respectively

v E×B = B × ∇ φ B 2
(2.20)

v D = m s v 2 + µB q s B B 2 × ∇B B (2.21)
Combining Eq.(2.17, 2.18) with Eq.(2.15) gives the non-linear Vlasov equation of the gyro-centers in the conservative form :

B * ,s ∂ Fs ∂t + ∇ • B * ,s dx G dt Fs + ∂ ∂v G B * ,s dv G dt Fs = C( Fs ) + S( Fs ) (2.22)
It can be readily seen from Eq.(2.22) that (i) the above equation doesn't contain any gyro-angle dependency, according to which the dimensionality is reduced by one and the high frequency motions are removed (ii) it doesn't have ∂/∂µ due to the conservation of magnetic moment, thus µ enters as a parameter.

Quasi-neutrality equation

In the electrostatic case where the value of β-the ratio of the kinetic energy to the magnetic energy-is assumed to be small, the electric potential can be calculated from the Poisson equation.

∇ 2 Φ = - 1 0 s N s q s (2.23)
where 0 is the permittivity of vacuum. The above equation can be further simplied using the electron Debye length λ e = 0 T e /N 0 e 2 where N 0 stands for a typical plasma density, which leads to

λ 2 e ∇ 2 eΦ T e = - 1 N 0 s N s q s (2.24)
Since the characteristic length of gyrokinetic model is much longer than the electron Debye length by several orders of magnitude, the term λ 2 e ∇ 2 in the above equation can be neglected. As a result, the plasma is considered to be in quasi-neutral state and the nal form of the Poisson equation satis es this condition as follows :

Ñe = i Ñi e i (2.25)
where the perturbed density is de ned as Ñ = N -N FS . Therefore, the quasineutrality equation can be closed if the perturbed density for each species is correctly obtained.

For ion species, a special attention has to be paid since the density in Poisson's equation is not the density of gyro-centers but the density of particles. Thus, one can show that the perturbed ion density is composed of the density of gyrocenter N Gs and the polarization density N pol,s

Ñs = ( Fs -Feq,s )d 3 v gyrocenter density + ∇ ⊥ • m s N eq,s q s B 2 ∇ ⊥ Φ polarization density (2.26)
where N eq,s is the equilibrium density of the species s and Feq,s is the equilibrium distribution of gyro-centers which takes the following form

Feq,s = N eq,s m s 2πT eq,s 3/2 exp - m s v 2 s 2T eq,s (2.27)
For electrons, the polarization density is often neglected due to their small mass (m e m i ). In this condition, the parallel dynamics of electrons are so fast that electrons can reach a Boltzmann equilibrium instantaneously leading to the so-called adiabatic electron response. In this approach, we assume Ñe = 0 for the modes which remain constant on a ux surface. Then, the adiabatic electron density reads :

N e = N eq,e exp e T e (φ -φ F S ) (2.28)
Using small value of φ, a perturbed electron density can be obtained by a Taylor expansion of Eq.(2.28)

Ñe = N eq,e e T e (φ -φ FS ) (2.29) Ñe (x G , t) = i q i Ñi (x G , t) (2.30)
Inserting Eq.(2.26, 2.29) into Eq.(2.25) gives the nal form of quasi-neutrality equation as follows :

N eq,e e T e (φ -φ FS ) = i e i ∇ ⊥ • N eq,i m i q i B 2 ∇ ⊥ φ + d 3 v( Fi -Feq,i ) (2.31)
With the Vlasov equation (2.22) and the quasi-neutrality equation (2.31), we now have a closed set of gyrokinetic equations in the electrostatic limit. Given the electric potential, the 5D gyro-center distribution function F is evolved by the Vlasov equation, which again can be used to calculate the electric potential by Poisson's equation.

Although the analytical derivations we have described seem rather short and simple, the theoretical formalism and its numerical implementations are sophisticated. For better understanding, the Hamiltonian and Lagrangian physics should be drawn from di erential geometry and Lie-transform perturbation methods. An exhaustive and detailed derivation can be found in [START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF].

Collisional operator

Although the main transport channel in core fusion plasma is turbulent transport, adding a collision operator might be advantageous for several reasons. First of all, collisional damping of turbulence generated large ow, such as zonal ow, can enhance turbulent transport [START_REF] Lin | E ects of collisional zonal ow damping on turbulent transport[END_REF]. Second, neoclassical transport can be dominant, especially for heavy impurities even in the core of fusion plasma and in certain regions where transport barriers are located, therefore synergy between turbulence and neoclassical e ects might play an important role on impurity transport [START_REF] Estève | Self-consistent gyrokinetic modeling of neoclassical and turbulent impurity transport[END_REF]. Third, the stabilization of trapped electron mode (TEM) depends sensitively on collisional e ects by which the resonance of trapped particles with the precession motion are weakened due to a modi cation of the perturbed distribution function [START_REF] Manas | Enhanced stabilisation of trapped electron modes by collisional energy scattering in tokamaks[END_REF]. Finally, from the numerical point of view, the relaxation process toward Maxwellian due to collisions can smooth small scale structures in velocity space, which can contribute to the stability of gyrokinetic simulations [START_REF] Abel | Linearized model Fokker-Planck collision operators for gyrokinetic simulations[END_REF].

For this purpose, a linearized model of collision operator has been implemented in GYSELA, satisfying the conservation of particles, momentum, energy and also Boltzmann's H-theorem (Donnel et al., 2019a). Furthermore, it has also been benchmarked against neoclassical theory by reproducing a good agreement. This operator was constantly employed in GYSELA simulations, so it is instructive to make a short description of the collision operator.

The linearized collision operator describing the collisions of species a colliding on species b can be expressed as :

C ab (F a , F b ) = C 0 ab (F M 0a , F M 0b ) + C 1 ab (F a , F b ) (2.32)
where F M 0s = F s -δF s is the local unshifted Maxwellian distribution function and C 0 ab represents the exchange of energy between the unshifted Maxwellians and the second term

C 1 ab = C ab (δF a , F M 0b ) + C ab (F M 0a , δF b ) consists of three terms C 1 ab (F a , F b ) = C ν,ab (F a ) + C d,ab (F a ) + C ,ab (F a , F b ) (2.33)
The rst term C ν,ab deals with the norm of the velocity (v , v ⊥ ) while the second term C d,ab modi es the direction of the velocity vector. The last term C ,ab ensures momentum exchange between species and the total conservation of parallel momentum. Given their di erent roles, it is customary to separate the evolution of the distribution function at equilibrium and its perturbed part, which reads

∂F M 0a ∂t = b C 0 ab (F M 0a , F M 0b ) ∂δF a ∂t = b C 1 ab (F a , F b ) (2.34)
Consequently, e ects of collisions on the distribution function can be computed by Eq.( 2.34) at each collisional time step ∆t coll and the modi ed distribution function evolves via the gyrokinetic Vlasov equation (2.15) by generating neoclassical transport of particles. A more detailed analytical derivation and numerical implementation can be found in (Donnel et al., 2019a).

External sources

In ux driven simulations, the equilibrium pro les should be maintained by a prescribed heat source in order to take into account the back reaction of turbulent trans-port. In the absence of heat source, the turbulent regime is evanescent by attening the temperature pro le and the system relaxes quickly towards a marginal state. Furthermore, in the context of the long time simulation, up to energy con nement time, the implementation of a heat source is essential. In GYSELA, a versatile source has been implemented by using projections of the velocity dependent part of the source onto the orthogonal basis of Hermite H h and Laguerre L l polynomials [START_REF] Sarazin | Predictions on heat transport and plasma rotation from global gyrokinetic simulations[END_REF].

S(r, θ, v G , µ) = +∞ l=0 +∞ h=0 c hl (r, θ)H h (v )L l (µ)e -v 2 -μ (2.35)
where we used the normalized parallel velocity vG = v G / 2T /m and the normalized magnetic moment μ = µB/T . A separate injection of particles, heat and parallel momentum can be accomplished by choosing a proper coe cient c hl . A more detailed analytical derivation of how to properly choose these coe cients can be found in [START_REF] Sarazin | Predictions on heat transport and plasma rotation from global gyrokinetic simulations[END_REF]. Here, we simply use the nal form of heat and momentum source. First, the pure heating source can be written as :

S heat = S 0 S r √ 2π 3/2 T 5/2 v2 G - 1 2 - J B 2 -J 2 B (2 -μ)(2v G -J B ) e -v 2 G -μ
(2.36)

where J B = √ 2T sce J /B 2 with the parallel current J . The radial envelope pro le S r is composed of the sum of two hyperbolic tangents and the term S 0 corresponds to the amplitude of source that we can choose arbitrarily. In a similar manner, the pure source of parallel momentum can be expressed as

S mom = S v G 0 S r 4π 3/2 T 2 [2v G (2 -μ) -J B (1 + 2v 2 G -μ)]e -v 2 G -μ
(2.37)

Prescribing heat and parallel momentum sources has numerous advantages in terms of mimicking experimental additional heating system, such as ICRH (Ion Cyclotron Resonance Heating) and NBI (Neutral Beam Injection). As has been previously reported by several authors, di erent sources of external heating, which sustain the hot plasma in tokamaks, can have a signi cant impact on impurity transport [START_REF] Hinton | Neoclassical ion transport in rotating axisymmetric plasmas[END_REF][START_REF] Lerche | Optimization of ICRH for core impurity control in JET-ILW[END_REF][START_REF] Angioni | A comparison of the impact of central ECRH and central ICRH on the tungsten behaviour in ASDEX upgrade H-mode plasmas[END_REF]. E ects of adding a toroidal momentum source on impurity transport in GYSELA is addressed in Chapter 5.

TERESA-4D : Gyro-bounce kinetic model

Low frequency turbulence driven by micro-instabilities is responsible for the anomalous heat and particle transport in magnetically con ned fusion plasma. Among these instabilities, trapped electron modes (TEM) might play an important role in explaining the anomalous heat and particle transport observed in tokamaks. In this section, a bounce averaged gyrokinetic model, named TERESA (Trapped Element REduction in Semi-lagrangian Approach) dedicated to instabilities driven by trapped particles will be presented. TERESA is a global, full-f gyro-bounce kinetic code which covers both TIM/TEM regimes simultaneously in 4D. Unlike GYSELA, the equilibrium pro le is maintained by thermal baths and only trapped particles, both kinetic ions and kinetic electrons, are treated self-consistently. A detailed derivation is available in [START_REF] Drouot | Étude de la turbulence liée aux particules piégées dans les plasmas de fusion[END_REF]. F 2.4 -By applying gyro-centre transform, the dimensionality is reduced by one and the high frequency motions are removed [START_REF] Drouot | A gyro-kinetic model for trapped electron and ion modes[END_REF].

Action and angle variables

In classical mechanics, Hamiltonian-Jacobi theory provides an useful framework by which dynamics of motion are solved with the help of a coordinate transformation. One powerful transformation often used is the transformation from a set of positionmomentum (p, q) coordinates to the action-angle (J , α) variables [START_REF] Goldstein | Classical mechanics[END_REF]. A dynamical system which shows integrable and quasi-periodic particle motions, such as tokamaks, is an ideal framework to use action and angular variables. Under this transformation, frequencies of periodic motions are obtained without solving the equations of motion ; and the Hamiltonian becomes function of J only.

dJ dt = - ∂H ∂α = 0 dα dt = ∂H ∂J = ω (2.38)
For trapped particles, the motion of a charged particle can be separated into three parts : (i) the cyclotron motion with the Larmor radius ρ c and the cyclotron frequency ω c (ii) the bounce motion characterized by the banana width δ b and the bounce frequency ω b (iii) the toroidal precession by the major tokamak radius R 0 and the precession frequency ω d . These periodic motions are related respectively to the action-angle variables (J , α), which can be expressed as [START_REF] Depret | Trapped-ion driven turbulence in tokamak plasmas[END_REF][START_REF] Sarazin | Kinetic features of interchange turbulence[END_REF] :

J =        J 1 = -ms qs µ J 2 = msv s 2π ds J 3 = q s ψ 0 α =        α 1 = ω c (J)t + α 10 α 2 = ω b (J)t + α 20 α 3 = ω d (J)t + α 30 (2.39)
where ψ 0 is the poloidal magnetic ux at the bounce point. It should be noted that ψ is used as a radial coordinate in this manuscript since dψ = -R 0 B θ dr where B θ is the poloidal magnetic eld.

In a tokamak con guration, the above characteristic time-space scales satisfy the following order, ω d ω b ω c and ρ c δ b R 0 . Given that the turbulence driven by trapped particles is characterized by the order of the precession frequency ω d , the disparate time scale allows the removal of the fast cyclotron phase θ c and the bounce angle θ b ; the resulting bounce-averaged kinetic description preserves the invariance of the magnetic moment J 1 and the bounce action J 2 .

The analytical derivation of the precession frequency ω d can be found in (?). For a large aspect ratio tokamak, the toroidal precession frequency for species s can be expressed as :

ω d = q a E eZB min R 0 ωd with ωd = 2E(κ 2 ) K(κ 2 ) -1 + 4s 0 E(κ 2 ) K(κ 2 ) + κ 2 -1 (2.40)
where q is the safety factor, E is the kinetic energy, κ = 1-λ 2 λ is the pitch-angle parameter with λ = µB min /E, s 0 is the magnetic shear and K(κ 2 ) and E(κ 2 ) are the rst and second kind elliptic functions respectively.

Note that the precession frequency does not depend on the mass of the particle but on its charge. Since the toroidal precession of ions and electrons are in opposite directions but have the same time scale, it enables the investigation of both TIM and TEM regimes with a minimum of numerical constraints.

Gyro-bounce Vlasov equation

As a result of the dimension reduction by averaging the kinetic equation over the bounce motion, our initial 6D model comes to 4D kinetic model. In order to obtain the nal form of the equations, let us consider now the Vlasov equation :

∂F s ∂t + [F s , H s ] = 0 (2.41)
where [A, B] is the Poisson bracket de ned in action-angle variables :

[A, B] = ∂A ∂α i ∂B ∂J i - ∂A ∂J i ∂B ∂α i (2.42)
Using the fact that the equilibrium Hamiltonian depends only on the action variable J (Eq 2.38) and by gyro-averaging the cyclotron motion and the bounce motion, one can further simplify Eq.(2.41) as follows :

∂ Fs ∂t + 1 q s ∂ H ∂ψ 0 ∂ Fs ∂α 3 = 0 (2.43)
where we used J 3 = q s ψ 0 and the bar notation represents an averaged quantity. The Hamiltonian at the equilibrium leads to

1 q s ∂ H ∂ψ 0 = ω d,s → H = E(1 + q s Ω d ψ 0 ) (2.44)
with Ω d = ω d Zs E which does not depend on the radial direction.

Hs = E(1 + q s Ω d ψ 0 ) + eZ s φ(J , α 3 ) (2.45)
Applying the gyro-bounce average operator lters the fast frequencies ω c and ω b and the small radial scales ρ c and δ b by reducing the dimensionality of the model from 6D to 4D. The nal form of the gyro-bounce average operator J 0s reads [START_REF] Depret | Trapped-ion driven turbulence in tokamak plasmas[END_REF])

J 0s = 1 - E T eq,s δ 2 b,s 4 ∂ 2 ψ -1 1 - E T eq,s q 2 0 ρ 2 c,s 4a 2 ∂ 2 α -1 (2.46)
where ρ c,s and δ b,s are the Larmor radius and the banana width respectively and T eq,s is the equilibrium temperature of the species s.

Finally, the bounce-averaged Vlasov equation for species s can be expressed as :

∂F s ∂t -[J s φ, F s ] + Ω d E Z s ∂F s ∂α 3 = 0 (2.47)
where F s is the distribution function of the banana centers. The overbar notation will be omitted hereafter for the readability of the equation. It is important to note that the nal reduced model involves two parameters (E, κ) in energy space and two coordinates (ψ, α) (note that only α = α 3 is left at the end due to the gyro-bounce average operator) in real space.

Quasi-neutrality equation

As discussed in the previous section (2.3.2) where we assumed that the characteristic length of our numerical model is much longer than the electron Debye length λ e , the quasi-neutrality equation can be simply expressed as a sum of the perturbed density for each species :

s Ñs q s = 0 (2.48)
Consequently, it is essential to formulate properly the perturbed density, which consists of adiabatic passing particles and kinetic trapped particles.

Ñs = N s,passing + N s,trapped -N eq,s (2.49)
For the trapped particles density, the bounce-averaged distribution function should be integrated over velocity in angle-action variables. A more rigorous derivation can be found in [START_REF] Drouot | Étude de la turbulence liée aux particules piégées dans les plasmas de fusion[END_REF], and here we use the nal expression for the sake of simplicity.

Then, the trapped particles density can be written as :

N s,trapped = 4π √ 2m -3/2 s 1-2 1 dλ 4ω b +∞ 0 J 0s Fs E 1/2 dE + q s N eq,s f T T s ∆s (Φ) (2.50)
where we introduced the trapped-particle fraction f T = 2 √ 2 /π and the operator ∆s to describe the polarization density which arises from the di erence between the bounce-averaged banana center and real trapped particle densities across a eld line.

∆s (Φ) = qρ c,s a ψ 2 ∂ 2 α + δ 2 b,s ∂ 2 ψ (2.51)
For the passing particle density, the analytical form can be expressed in a relatively simple manner due to its adiabatic response. The density has a linear dependence on Φ and an ad-hoc coe cient s,φ governs the response of adiabatic passing particles.

N s,passing = -N eq,s (1 -f T ) q s T eq,s (Φ -s,φ Φ α ) -1 (2.52)
For φ = 1, the passing particles do not react to the zonal potential Φ , which is constant on a ux surface. Therefore, the zonal ows Φ = Φ ZF driven by trapped particles can freely grow without any interference of passing particles. In the opposite case φ = 0, the passing particles can respond to the zonal potential according to which the amplitude of zonal ows are strongly suppressed. In the TERESA model, the value of φ is not computed self-consistently, thus it will be used as a free parameter

0 ≤ φ ≤ 1.
Finally, by inserting Eq.(2.50, 2.52) into Eq.( 2.49), one can obtain the nal form in the normalized unit [START_REF] Idouakass | Impurity density gradient in uence on trapped particle modes[END_REF])

2 √ π T eq,i T 0 s q s C s +∞ 0 J 0,s F s E 1/2 dE = C ad (Φ -φ,s Φ α ) -C pol s C s τ s Z 2 s ∆ s Φ (2.53)
where

C s = N s /N eq , τ s = T eq,i /T s , C ad = C pol 1-f T f T s (Z 2 s C s τ s ), 1-2 1 dλ 4ω b = f T , C pol = eω 0 L ψ /T 0 with the ion precession frequency ω 0 = T 0 /(eR 2 0 B θ )
and the radial length of the simulation box L ψ = aR 0 B θ .

When compared to GYSELA, the major advantage of TERESA-4D is that an arbitrary number of particle species can be tackled with a reasonable computation time. Therefore, it is possible to deal with multiple impurities in a single simulation, which might be especially useful for the study of parametric dependencies.

Introduction

Central accumulation of heavy impurities in the past and present day devices has renewed interest for the study of neoclassical impurity transport [START_REF] Ida | Impurity behaviour in PBX L-and H-mode plasmas[END_REF][START_REF] Pasini | Measurements of impurity transport in JET[END_REF][START_REF] Rice | Impurity toroidal rotation and transport in Alcator C-mod ohmic high con nement mode plasmas[END_REF][START_REF] Pütterich | Observations on the W-transport in the core plasma of JET and ASDEX upgrade[END_REF][START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF][START_REF] Loarte | Tungsten impurity transport experiments in alcator c-mod to address high priority research and development for iter[END_REF][START_REF] Casson | Theoretical description of heavy impurity transport and its application to the modelling of tungsten in JET and ASDEX upgrade[END_REF][START_REF] Casson | Predictive multi-channel ux-driven modelling to optimise ICRH tungsten control and fusion performance in JET[END_REF]. It is now a well-known fact that poloidal asymmetry of impurity density driven by plasma rotation [START_REF] Wong | Transport of impure plasma with arbitrary toroidal rotation[END_REF][START_REF] Wesson | Poloidal distribution of impurities in a rotating tokamak plasma[END_REF][START_REF] Romanelli | E ects of density asymmetries on heavy impurity transport in a rotating tokamak plasma[END_REF][START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF][START_REF] Belli | P rsch-Schlüter neoclassical heavy impurity transport in a rotating plasma[END_REF][START_REF] Maget | Natural poloidal asymmetry and neoclassical transport of impurities in tokamak plasmas[END_REF] or by temperature anisotropies [START_REF] Ingesson | Comparison of basis functions in soft x-ray tomography and observation of poloidal asymmetries in impurity density[END_REF][START_REF] Reinke | Poloidal variation of high-Z impurity density due to hydrogen minority ion cyclotron resonance heating on Alcator C-mod[END_REF][START_REF] Kazakov | Poloidal asymmetries due to ion cyclotron resonance heating[END_REF][START_REF] Bilato | The impact of the ion-cyclotron-resonance location on the poloidal asymmetries of impurity density in an ICRF-heated rotating plasma[END_REF][START_REF] Goniche | Ion cyclotron resonance heating for tungsten control in various JET H-mode scenarios[END_REF] generated by external heating system, such as ICRH, can enhance or reduce neoclassical impurity transport signi cantly [START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF].

One of the possible objectives to avoid such core accumulation is to increase turbulent di usion and reduce the neoclassical convection. Furthermore, in future fusion reactors like ITER, the fraction of NBI heating is expected to be low, and also the speci cites of future reactors, such as high temperature and large machine size, will lead to a relatively weak neoclassical regime for impurities. Subsequently, over the last decades, the impact of the turbulent transport, particularly the turbulent convection, has attracted much attention, and some progress has been made in theoretical description [START_REF] Frojdh | Impurity e ects on η i mode stability and transport[END_REF][START_REF] Estrada-Mila | Gyrokinetic simulations of ion and impurity transport[END_REF][START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF][START_REF] Fülöp | Impurity transport in ITER-like plasmas[END_REF]Guirlet et al., 2006;[START_REF] Angioni | Particle and impurity transport in the Axial Symmetric Divertor Experiment Upgrade and the Joint European Torus, experimental observations and theoretical understanding[END_REF][START_REF] Bourdelle | A new gyrokinetic quasilinear transport model applied to particle transport in tokamak plasmas[END_REF][START_REF] Dubuit | Fluid simulations of turbulent impurity transport[END_REF][START_REF] Nordman | In uence of the radio frequency ponderomotive force on anomalous impurity transport in tokamaks[END_REF][START_REF] Camenen | Impact of the background toroidal rotation on particle and heat turbulent transport in tokamak plasmas[END_REF][START_REF] Fülöp | Impurity transport driven by ion temperature gradient turbulence in tokamak plasmas[END_REF][START_REF] Fülöp | E ect of poloidal asymmetry on the impurity density pro le in tokamak plasmas[END_REF][START_REF] Mollen | E ect of poloidal asymmetries on impurity peaking in tokamaks[END_REF]Angioni et al., 2012b,a;[START_REF] Skyman | E ects of the equilibrium model on impurity transport in tokamaks[END_REF]. In addition, centrifugal e ects (Angioni et al., 2012a), poloidal asymmetry of impurity density [START_REF] Fülöp | E ect of poloidal asymmetry on the impurity density pro le in tokamak plasmas[END_REF], electromagnetic e ects [START_REF] Hein | Electromagnetic e ects on trace impurity transport in tokamak plasmas[END_REF][START_REF] Moradi | Impurity transport due to electromagnetic drift wave turbulence[END_REF] and dependence on the main ions mass [START_REF] Pusztai | Turbulent transport of impurities and their e ect on energy con nement[END_REF], have been investigated both analytically and numerically.

A turbulent transport of impurities can be decomposed into multiple components, such as thermo-di usion, roto-di usion and curvature pinch, which can be characterized as o -diagonal transport components [START_REF] Angioni | O -diagonal particle and toroidal momentum transport : a survey of experimental, theoretical and modelling aspects[END_REF]. Furthermore, the investigation of parametric dependencies of these terms is a formidable task due to the nonlinear nature of instabilities driven by turbulence. To overcome this, a quasilinear model has been applied in case of ITG [START_REF] Fülöp | Impurity transport driven by ion temperature gradient turbulence in tokamak plasmas[END_REF] and TEM [START_REF] Mollen | Impurity transport in trapped electron mode driven turbulence[END_REF] instabilities. However, these models have been usually tested with their own assumptions for reduced models. Such assumptions sometimes preclude us from clearly apprehending characteristic features of convective velocity. Therefore, more integrated and/or complete models, such as full-F and global gyrokinetic model would be necessary to examine the impact of impurity pinch.

To this end, in this chapter, we investigate the parametric dependencies of impurity pinch driven by trapped particle turbulence using the code TERESA [START_REF] Depret | Trapped-ion driven turbulence in tokamak plasmas[END_REF][START_REF] Sarazin | Kinetic features of interchange turbulence[END_REF][START_REF] Cartier-Michaud | Staircase temperature pro les and plasma transport self-organisation in a minimum kinetic model of turbulence based on the trapped ion mode instability[END_REF][START_REF] Drouot | A gyro-kinetic model for trapped electron and ion modes[END_REF]. More ac-curately, e ects of trapped particle driven mode on turbulent impurity particle ux is studied by means of global non-linear full-F bounce-averaged gyrokinetic code. To restrict our study solely to a turbulent case, only light impurities in collisionless regime will be studied actively. Throughout this chapter, even though each species is treated self-consistently, the concentration of impurity C s will be kept at the trace limit, thereby the presence of impurities does not change the nature of the background plasma turbulence. The quasi-linear derivation of pinch terms from a non-linear full-F gyrokinetic model is also presented in this chapter. Especially, the thermo-di usion and the curvature pinch in trapped ion/electron dominant mode (TIM/TEM) are studied in detail.

Impurity Pinch

In general, impurity ux can be decomposed into two di erent parts ; (i) di usive part which depends on the density gradient and (ii) convective part :

Γ z = -D z ∇N z + N z V z (3.1)
where D z is a di usion coe cient and V z is a convective (pinch) velocity. This seemingly simple equation contains nonlinear turbulent e ects in its coe cient D z and pinch velocity V z , which make them hard to analyze. One useful way to study such nonlinear evolution of turbulent impurity ux is a quasi-linear theory, in which one assumes a linear response for the transported particles to the uctuating potential. Particular attention should be paid to the validity of the quasi-linear theory. For instance, the main assumption, the ordering of the characteristic times governing the plasma dynamics, should be veri ed. Within the framework of quasi-linear theory, an analytical expression of the pinch velocity V z can be derived and will be presented in Section 3.2.1. In addition, a useful method has been designed in this chapter from nonlinear simulations for the study of the convective part and this method will be introduced in Section 3.2.3.

Quasi-linear impurity transport

Turbulent particle transport can be addressed by means of the quasi-linear model in the framework of weak turbulence. The quasi-linear theory has been already widely addressed by several authors for the purpose of studying turbulent impurity ux [START_REF] Dubuit | Fluid simulations of turbulent impurity transport[END_REF][START_REF] Bourdelle | A new gyrokinetic quasilinear transport model applied to particle transport in tokamak plasmas[END_REF][START_REF] Futatani | Reversal of impurity pinch velocity in tokamaks plasma with a reversed magnetic shear con guration[END_REF]. Recently, TERESA code has also shown that particle/heat uxes obtained from nonlinear simulations were found to be in good agreement with quasi linear predictions [START_REF] Medina | Radial density and heat uxes description in the velocity space : Nonlinear simulations and quasi-linear calculations[END_REF][START_REF] Gravier | Di usive impurity transport driven by trapped particle turbulence in tokamak plasmas[END_REF]. In this section, as an extension of previous work, the quasi linear framework will be applied for impurity ux, and an analytical formula of impurity particle ux will be presented in detail based on the kinetic approach (the uid approach is also detailed in Appendix A).

First, we recall the Vlasov equation of the bounce-averaged distribution function F derived in Eq.(2.47) :

∂F ∂t -[J φ, F ] + Ω d E Z ∂F ∂α = 0 (3.2)
Here, the distribution function F and the electric potential φ can be split into perturbed and unperturbed parts with the assumptions F F eq and φ eq = 0.

F = F eq (ψ, E, t) + F φ = φ (3.3)
where the Maxwellian distribution function F eq is de ned as

F eq = N eq T 3/2 eq exp - H eq T eq (3.4)
where the equilibrium Hamiltonian [START_REF] Gravier | Di usive impurity transport driven by trapped particle turbulence in tokamak plasmas[END_REF] as

H eq = E(1 + eΩ d ψ) with Ω d = ω d Z s /E and ψ is the dimensionless radial coordinate (dψ = -R 0 B θ dr) is used (H eq is di erently used in
H eq = E).
By assuming small amplitude ( F F eq ) according to which frequency and growth rates of the modes can be described by linear theory, Eq.(3.2) can be rewritten by keeping only rst order terms of the perturbed part :

∂ F ∂t - ∂J φ ∂α ∂F eq ∂ψ + Ω d E Z ∂ F ∂α = 0 (3.5)
Now the above equation can be rewritten using the Fourier decomposition Fs = n F n e i(nα-ωt) and φ = n φ n e i (nα-ωt) where ω = ω r + iγ is composed of the real frequency and the growth rate. It is also important to note that in the framework of quasi-linear theory, the assumption of the distribution function close to the equilibrium Maxwellian is underlined for the validity of model. After some arrangement, we nally nd the relationship between the perturbed distribution function and the perturbed potential as follows :

F n = κ N + E Teq -3 2 κ T -eΩ d E Teq ω n -Ω d Z E F eq J φ n (3.6)
where

κ N = -1 Neq dNeq dψ , κ T = -1
Teq dTeq dψ are the logarithmic density and temperature pro le respectively.

By applying spatial averages of Eq.(3.2) over the toroidal angle α (< . >= 1 2π dα) and combining it with Eq.(3.5), the above equation can be expressed as :

∂F eq ∂t = ∂ ∂ψ

F ∂J φ ∂α (3.7)
where we neglected high order terms above the second order. Then, Eq.(3.7) yields :

∂< N > α ∂t + ∂Γ z ∂ψ = 0 (3.8)
where the nal form of impurity transport Γ z can be written as :

Γ = n (nJ φ n ) 2 γ ω r -n Ω d Z E 2 + γ 2 κ N + E T eq - 3 2 κ T -eΩ d E T eq F eq √ EdE (3.9)
The above equation can be summarized into three terms :

Γ z N z = D z κ Nz + C T κ Tz + C P (3.10)
The rst term in Eq.(3.10), proportional to κ N z , is a di usive part of the impurity ux. The second term is the thermo-di usion C T κ T z which stems from the inherent temperature gradient in tokamak [START_REF] Weiland | Di usive particle and heat pinch e ects in toroidal plasmas[END_REF][START_REF] Frojdh | Impurity e ects on η i mode stability and transport[END_REF]. It should be noted that the temperature gradient in the thermo-di usion term depends on the temperature gradient of the transported species, and not on the temperature gradient which drives the background turbulence. In addition, the sign of coe cient C T in Eq.(3.10) depends on the phase velocity of the order of any turbulent modes. For example, the direction of thermo-di usion is directed inwardly when turbulence is propagating in the electron magnetic drift direction while it is directed outwardly when turbulence is propagating in the ion magnetic drift direction (at high κ Nz , TIM modes can reverse frequency). Usually, the magnitude of C T tends to decrease for high-Z impurities [START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF]Guirlet et al., 2006;[START_REF] Dubuit | Fluid simulations of turbulent impurity transport[END_REF]. Thus in the case of high charge impurities such as tungsten (W), the e ect of thermo-di usion would decrease signi cantly.

The last term corresponds to the curvature pinch C p which is proportional to magnetic eld curvature. It originates from the perpendicular compression of E × B drift velocity. This term is proportional to ∇q/q (q is the safety factor) leading to an inward direction except for reversed shear plasma [START_REF] Futatani | Reversal of impurity pinch velocity in tokamaks plasma with a reversed magnetic shear con guration[END_REF]. Usually, another pinch exists, the parallel compression pinch [START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF], which is proportional to Z/A where A is the impurity mass number, such that m z = Am H and has a direction opposed to thermo-di usion (Table 3.1). Thus it is inward for ion dominated turbulence and outward for electron dominated turbulence. This term cannot be accounted for in TERESA simulations because the parallel electric force is neglected. T 3.1 -Impurity pinch direction. Reproduced from [START_REF] Bourdelle | A new gyrokinetic quasilinear transport model applied to particle transport in tokamak plasmas[END_REF].

Thermo-di usion Curvature pinch Trapped Ion Mode (TIM) Outward Inward (Outward if s 0) Trapped Electron Mode (TEM) Inward

Validity of the quasi-linear approximations

Before we move to the main results obtained by nonlinear simulations, it is crucial to discuss the validity of the quasi-linear theory. The two main approximations-(i) a weak uctuation level (δF F ) and (ii) no trapped particles in the saturated potential eld-are often used to justify the validity. The rst approximation which is equivalent to the condition where the distribution function F is close to the Maxwellian equilibrium is well satis ed in the core of tokamaks. As a result, the second approximation is often tested for veri cation.

To demonstrate the second approximation, we introduce a dimensionless number, named the Kubo number K de ned as follows [START_REF] Kubo | Stochastic Liouville equations[END_REF] :

K = τ ac τ f (3.11)
Here, τ ac represents the turbulence auto-correlation time, as experienced from the point-of-view of particles with zero energy. It is de ned as τ ac = φ(t)φ(t+τ ) dτ / φ 2 (t) where φ(t)φ(t + τ ) is the auto-correlation function of the potential. It is calculated at the location of the maximum saturated potential φ max of the simulation box (Figure 3.1). And τ f is the ight time of a particle along the electric potential structure. The ratio between these two times should be lower than 1 (K < 1) in order that the particles are not trapped in the potential well. In this thesis, an analytical approach to obtain τ f has been applied by assuming that particles follow a circle trajectory around the maximal electric potential φ max . The ight time τ f for a particle to travel along the circle can be obtained using the Hamiltonian relation in the angle-action variables dJ 3 dt = -∂H ∂α and dα dt = ∂H ∂J 3 and a simple form of electric potential φ = φ 0 cos (nα) cos (kψ) where each n, k represents the mode number in α, ψ direction and φ 0 is the maximal electric potential.

τ f = dt = dθ dt -1 dθ = 2π nkφ 0 (3.12)
In Figure 3.2, the Kubo number K is calculated from TERESA simulations by varying κ T i , κ Te simultaneously and using the most unstable modes n and k obtained in the linear phase. For given plasma parameters (κ n = 0.1 and κ T = 0.05, 0.1, 0.2 and 0.3), F 3.1 -Wave-particle interaction time τ ac obtained by an auto-correlation function g(τ ) (normalized to g(τ = 2)) of the electric potential at a location (ψ, α) where φ max is obtained during the relaxation state (2ω -1 0 < t < 4ω -1 0 ). For input parameters, plasma density κ n = 0.1 and temperature κ T = 0.05 pro les with numerical grids (N α , N ψ , N κ , N E ) = (1024, 256, 1, 96) are used. Here, τ ac is equivalent to the surface integral of g(τ ).

the Kubo number K satis es the quasi-linear ordering of the characteristic turbulence time. The auto-correlation time τ ac is found to be smaller than the ight time along the turbulence eddy τ f , which means the particles are not trapped in the saturated potential. The Kubo number tends to increase in case of high temperature gradient κ T . Note that the method used in [START_REF] Lim | Impurity pinch generated by trapped particle driven turbulence[END_REF] to calculate the Kubo number was evidenced to overestimate it, sometimes by an order of magnitude (Guillevic et al., prep) . Note also that more work needs to be done to estimate the Kubo number from the point-of-view of resonant particles (with E = 0).

Pinch velocity from nonlinear numerical simulations

To clarify e ects of impurity pinch in the particle ux, it is necessary to separate our numerical impurity ux into di usive and convective part. Accordingly, it is useful to express Eq.(3.1) as a linear equation of the logarithmic density gradient κ N z = -∇N z /N z as follows :

Γ z N z = -D z ∇N z N z + V z = D z κ N z + V z (3.13) F 3.2 -
The kubo number K is calculated at a location (ψ, α) where φ max is obtained for di erent background temperature pro les κ T while density pro le is xed at κ N = 0.1.

Given that the impurity particle ux Γ z and the density gradient κ N z can be measured from nonlinear numerical simulations, it is possible, if multiple simulations with different values of κ Nz are performed, to calculate an accurate value of V z from Eq.(3.13).

The shaded part of Figure 3.3 represents the beginning of the turbulent state from nonlinear simulation TERESA, and this region has been selected to extract a steady state value of ∇N z . In Figure 3.3(a), the time evolution of the 10 most unstable modes φ n is represented. In the rst period 0 < t < 2ω -1 0 is, the background turbulence is not entirely developed. The following shaded zone 2ω -1 0 < t < 4ω -1 0 indicates where turbulence is close to a stationary state. Since we focus mainly on the region where turbulence is already set up, the shaded region will be appropriate to compute the steady state value of ∇N z . It can be seen in Figure 3.3(b) that the time cumulative Γ z increases linearly, which means Γ z has a constant value after 2ω -1 0 , and this value will be used in Eq.(3.13) to nd impurity pinch velocity.

At this stage, special attention needs to be paid when determining the value of κ N z = -∇N z /N z since the background plasma turbulence can alter the given initial impurity density pro le in time. It is described in Figure 3.4 that the value of the impurity density gradient κ nal N z (t = 2ω -1 0 ) in turbulent regime is no longer identical to the initial impurity density gradient κ init N z (t = 0). It is therefore necessary to compute the modi ed κ nal N z value in the turbulent regime. Note that the early turbulent stage (2ω -1 0 < t < 4ω -1 0 ) was chosen on purpose with the validity of quasi-linear approach in mind. In this stage, the impurity density pro le does not reach its saturated shape. In our nonlinear simulations, numerical bu er regions are implemented for the purpose of numerical stability at both edge sides 0 < ψ < 0.15 and 0.85 < ψ < 1.0. Consequently, only the core region 0.2 < ψ < 0.8 will be taken into account for the computation of κ nal N z . It should be noted that ψ = 0 stands for the core plasma, and ψ = 1 for the plasma edge, whereas the opposite convention was sometimes used in previous papers using the same TERESA model for impurity study [START_REF] Drouot | Étude de la turbulence liée aux particules piégées dans les plasmas de fusion[END_REF][START_REF] Idouakass | Impurity density gradient in uence on trapped particle modes[END_REF][START_REF] Gravier | Di usive impurity transport driven by trapped particle turbulence in tokamak plasmas[END_REF].

By using the value of κ nal N z and the corresponding Γ z value from the nonlinear simulation, we can plot Γ z /N z against κ nal N z and draw a straight line by means of linear regression method. As an example of the method, Figure 3.5 shows that a straight line is drawn using 6 di erent values of κ nal N z with the corresponding Γ z impurity ux. The slope of the line is equal to the di usion coe cient D z and its x-intercept/y-intercept signi es respectively the peaking factor (PF) -RV z /D z and the pinch velocity V z . According to our notations, positive (negative) peaking factor represents a peaked (hollow) impurity density pro le and positive (negative) pinch velocity means particles moving towards the edge (the core).

Finding V z values using a linear regression method with a restricted number of points (κ N z , Γ z /N z ) provides only an approximate value. It is therefore necessary to run several simulations with various values of κ N z to increase accuracy of our results.

Parametric dependencies of impurity pinch

Determining the complete parametric dependencies of impurity ux by nonlinear gyrokinetic simulations is a formidable task. However, the reduced approach of the bounce-averaged TERESA model by considering only trapped particle species enables us to carry out an in-depth study for local parametric dependencies of convective terms. Among various pinch terms, thermo-di usion and curvature pinch terms will be covered in detail and their parametric dependencies will be presented in the following section. Note that the parallel compressible pinch plays also an important role in impurity convection [START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF]. However, our approach based on the trapped particle species precludes us from studying the parallel compressible pinch due to the lack of passing particles.

For the results shown in this thesis, the following parameters shown in Table 3.2-the numerical grids (N α , N ψ , N κ , N E ), the impurity concentration C z and the zonal ows control parameter φ -are xed during the entire chapter. The number of grid N κ = 1 with κ = 0 means that we consider strongly trapped particles only. T 3.2 -Plasma parameters

N α N ψ N κ N E C z φ 1024 256 1 96 10 -5 0.3
It should be noted that the impurity concentration C z = 10 -5 is kept in the trace limit (N z Z 2 /N i 1), so the presence of impurities doesn't have any impact on the background turbulence. E ects of impurity concentration on impurity transport is also one of the main issues in this eld, and recent works regarding the in uence on the background turbulence have been recently performed using the TERESA code [START_REF] Idouakass | Impurity density gradient in uence on trapped particle modes[END_REF][START_REF] Lesur | Validity limits of the passive treatment of impurities in gyrokinetic tokamak simulations[END_REF].

Thermo-di usion

First of all, the thermo-di usion term C T κ T z in Eq.(3.10) is studied with respect to impurity charge. In order to study the thermo-di usion exclusively, the curvature pinch C p term has been suppressed arti cially by putting the modi ed Hamiltonian H eq,z = E for impurities corresponding to Ω d = 0 in the distribution function F eq,z as used in [START_REF] Gravier | Di usive impurity transport driven by trapped particle turbulence in tokamak plasmas[END_REF]. The impurity mass number A = 20 is chosen while di erent charge numbers Z = 2, 4, 8 and 10 are tested. Initial density and temperature pro les used for TIM/TEM instability are listed in Table 3.3.

For impurity density pro les, 8 di erent values of κ init N z = ±0.05, ±0.1, ±0.3, ±0.5 were chosen and their modi ed κ nal N z value in turbulent region was used to calculate V z as we explained in the previous section. To trigger TIM and TEM instabilities independently, di erent temperature gradients for ions/electrons have been used. With uid model, it has been already reported in [START_REF] Dubuit | Fluid simulations of turbulent impurity transport[END_REF] that the sign of the coecient C T depends on the phase velocity of turbulence. It is described in Figure 3.6 that the direction of thermo-di usion is outward (positive velocity V z ) in the case of TIM T 3.3 -Initial density/temperature pro les

κ N i κ N e κ T i κ T e κ T z
TIM 0.1 0.1 0.2 0.05 0.05 TEM 0.1 0.1 0.05 0.2 0.05 whereas it is directed inwardly (negative velocity V z ) for TEM dominant case. This direction change can be understood according to the nature of the TIM/TEM. When TIM dominates, the most unstable mode resonates with the H eq /T eq larger than 3/2 while it is lower than 3/2 for TEM. According to Eq.(3.9), this di erent resonant location of the most unstable mode leads to a di erent direction of V z . Furthermore, C T tends to decrease with the charge number Z, which signi es that the e ect of thermo-di usion will be less signi cant in a high-Z impurity case such as tungsten (W). F 3.6 -Direction of thermo-di usion and its charge dependency in TIM/TEM dominated plasma for a xed impurity mass number A = 20. The coe cient C T decreases when the impurity charge Z increases. For TEM, the thermo-di usion pinch is found to be inward, whereas it is outward in TIM turbulence.

Moreover, it has already been highlighted using uid model that the direction of impurity particle ux depends directly on the sign of impurity temperature pro le κ T z [START_REF] Dubuit | Fluid simulations of turbulent impurity transport[END_REF]. To verify this local parametric dependence, the following density/temperature pro les are tested (Table 3. According to Eq.(3.10), the impurity particle ux Γ z is expected to be perfectly di usive when the impurity temperature pro le is at (κ T z = 0) if the curvature pinch is neglected (C P = 0). κ T z in the TIM/TEM modes. As we outlined before, the pinch velocity disappears when κ T z = 0, which signi es that the impurity particle ux is now perfectly di usive and the direction of V z is closely related to the sign of κ T z . A positive value of κ T z means that the plasma core is hotter than the edge which corresponds to a realistic temperature pro le for a future fusion scenario. In this case, the impurity pinch moves inward (V z < 0) in the TEM mode whereas it moves outward (V z > 0) in the TIM mode. This direction change according to TIM/TEM instabilities can be explained from Eq.(3.9). A detailed explanation of the thermo-di usion transport can also be found in [START_REF] Angioni | O -diagonal particle and toroidal momentum transport : a survey of experimental, theoretical and modelling aspects[END_REF].

By introducing a new function g n (Z, E) from Eq.(3.9),

g n (Z, E) = γ ω r -n Ω d Z E 2 + γ 2 E T eq - 3 2 exp - E T eq √ E (3.14)
we can rewrite the thermo-di usion as follows :

n ∞ 0 (nJ φ n ) 2 g n (Z, E)dE (3.15)
To see how this term varies with impurity charge Z, we de ne a new function X n (Z) ≡ ∞ 0 g n (Z, E)dE which integrates g n over the whole range of energy ; and we will verify how its derivative behaves when impurity charge gets higher.

dX n (Z) dZ = - ∞ 0 2γnΩ d E Z 2 ω r -nΩ d E Z [(ω r -nΩ d E Z ) 2 + γ 2 ] 2   E T eq - 3 2   exp - E T eq √ EdE (3.16)
Assuming T eq = 1 and ω r nΩ d , we can further simplify Eq.(3.16) into

dX n (Z) dZ = -1.33 2γnΩ d ω r 1 Z 2 (ω 2 r + γ 2 ) 2
(3.17)

From the above equation, we can readily see that the absolute value of X n (Z) decreases with Z in both cases TIM (ω r > 0) and TEM (ω r < 0). Note that this relation may fail to apply if the condition ω r nΩ d is not satis ed. To verify this limit in the case of our numerical simulations, linear analysis has been performed to nd the most unstable mode number n and the corresponding values of ω r , γ according to the plasma parameters. In Figure 3.9, X n (Z) is calculated with di erent impurity charges. The background pro les from Table 3.3 have been used to obtain n, ω r , γ in linear analysis and Ω d = 1 is used due to normalization. The values of n, ω r and γ are displayed in Figure 3.9, and we can clearly see that Eq.(3.17) produces satisfying results in both TIM/TEM cases even though ω r nΩ d is roughly assured (factor 2). Therefore, the magnitude of thermo-di usion is found to be decreasing with increasing impurity charge.

Here, the same order of magnitude for TIM/TEM convection is found in Figure 3.7(a). Although the TIM instability plays a minor role in impurity transport compared with other main instabilities, such as ITG and TEM, our investigations must be taken as a qualitative analysis of transition between ion-driven and electron-driven instabilities in general. Furthermore, our model considers only trapped particles where ITG cannot be investigated.

In Figure 3.7(b), the normalized peaking factors (PF) -RV z /D z are depicted as a function of κ T z . Since the PF sign inversely depends on the sign of V z , PF exhibits the F 3.8 -Linear analysis in case of TIM/TEM. Linear instability growth rate plotted against the mode n. From these calculations, the values of n, ω r , γ in TIM/TEM-cases can be found. F 3.9 -Thermo-di usion contribution X n (Z) from Eq.(3.17) in case of TIM (left) and TEM (right) using n, ω r , γ obtained by linear analysis. The thermo-di usion contribution X n (Z) decreases with increasing impurity charge. opposite behavior from V z . Here, a negative PF signi es hollow impurity density pro le and positive PF means peaked impurity density pro le, which needs to be avoided for the e ciency of fusion. Consequently, in the general plasma fusion scenario where the core is hotter than the edge (positive κ T,z in Figure 3.7), the ion dominant mode would be favorable to mitigate core accumulation in light of turbulent convection, especially by the e ect of thermo-di usion. Now, we aim to determine the magnitude of the C T coe cient with respect to the intensity of the background plasma turbulence. Since we are only interested in the absolute value of V z in connection with to turbulence intensity, ion and electron temperature pro les have been chosen arbitrarily without considering TIM or TEM dominance. Detailed density/temperature pro les are listed below in Table 3.5.

T 3.5 -Initial density/temperature pro les

κ ni , κ ne κ T i κ T e κ T z
0.1, 0.15, 0.2, 0.25 0.25 0.25 0.25

While keeping the temperature gradient constant for all species, main ion and electron density gradients have been gradually increased from κ N i = κ N e = 0.1 to 0.25. It is shown in Figure 3.10 that the absolute value of V z increases with the growth rate γ of the most linear unstable mode. The more turbulent is the plasma (larger growth rate γ), the stronger is the pinch velocity. This result corresponds well with our expectation. Because the electric potential φ is directly linked to the growth rate γ, a strong background turbulence will enhance both di usion and convection terms according to Eq.(3.9). F 3.10 -Pinch velocity V z plotted against the linear growth rate γ.

Curvature pinch

In the previous section, the parametric dependencies of the thermo-di usion term were studied by arti cially deactivating the magnetic curvature e ects in the Hamilto-nian H eq = E of the distribution function. To do likewise for the curvature pinch term, e ects of magnetic curvature are now included by employing the entire Hamiltonian H eq = E(1 + eΩ d ψ) in the distribution function F eq , while the thermo-di usion term is arti cially suppressed in this section by putting a at impurity temperature pro le κ T z = 0.

According to Eq.( 2.40), it should be noticed that the precession frequency ω d is expressed as a function of the trapping parameter κ and the magnetic shear s 0 . In general, the sign of ω d for trapped impurities is mostly positive except for strong negative magnetic shear (s 0 < 0) [START_REF] Kadomtsev | Reviews of Plasma Physics : Turbulence in Toroidal Systems[END_REF]. This means that, in other words, the e ect of curvature pinch term can be readily demonstrated by simply varying the sign of the normalized precession frequency

Ω d = Zsω d,s E
in Eq.(3.9). In this framework, we introduce an ad-hoc coe cient α in the Hamiltonian, in order to control both the sign and magnitude of the normalized precession frequency Ω d . The equilibrium Hamiltonian then reads H eq = E(1 + αeΩ d ψ). Note that α > 0 corresponds to the usual sign of Ω d for trapped particles, while α < 0 represents a strong negative magnetic shear. Then, we use the same linear approach as in Figure 3.5 in order to obtain the pinch velocity. The following density and temperature pro les are tested for numerical simulations with di erent values of α (Table 3.6) : T 3.6 -Initial density/temperature pro les

κ N i κ N e κ T i κ T e κ T z α 0.1 0.1 0.2 0.2 0 0, ±0.5
The direction of the curvature pinch velocity V z with di erent values of α is depicted on Figure 3.11. As we expected from Eq.(3.9), for the usual sign of precession frequency (positive α), the curvature pinch velocity V z is directed inward. On the other hand, the strong negative magnetic shear s 0 (negative α) can switch the sign of V z by pushing the impurity particles toward the edge. The α = 0 case corresponds to no magnetic curvature, hence the impurity ux becomes purely di usive and the linear line Γ z /N z passes exactly through the origin point (V z = 0).

Finally, to investigate the impact of α on the magnitude of V z , we run multiple simulations while keeping α positive. The magnitude of V z as a function of Ω d is shown on Figure 3.12. The enhancement of V z with respect to Ω d is consistent with our quasilinear model (Eq.3.9). It should be also noted that the curvature pinch term sometimes prevails over the thermo-di usion term. But in this work, we focused on the parametric tendency of impurity pinch terms when compared to the thermo-di usion.

Conclusion

In this chapter, the parametric dependencies of impurity pinch-caused by trapped particle driven turbulence-were studied using the bounce-averaged gyrokinetic code TERESA. Among various terms, the thermo-di usion and the curvature pinch were treated in detail, and their qualitative behavior was compared with the quasi-linear (QL) model. In order to justify the validity of our approach, the Kubo number was computed, and the quasi-linear approximation is found to be valid in case of a weakly turbulent plasma. The parametric dependencies of convective terms were studied with respect to various parameters such as ∇N i,e /N i,e , ∇T i,e /T i,e , ∇T z /T z , impurity charge Z and magnetic shear s 0 . For all simulations, light impurities in the trace limit have been kept without assuming any further impurity recombination and ionization.

First of all, the sign of thermo-di usion coe cient C T was investigated in TIM/TEM turbulence separately. The results obtained from nonlinear simulations TERESA were in agreement with the previous studies (Guirlet et al., 2006;[START_REF] Dubuit | Fluid simulations of turbulent impurity transport[END_REF]. Furthermore, the QL theory explained that impurities move inward in TEM and outward when TIM is dominant. This can be explained by di erent energy resonance positions in the case of TIM/TEM. It is also found that the magnitude of C T tends to decrease with impurity charge Z while the growth rate γ enhances the coe cient C T .

A more comprehensive understanding of impurity dynamics for further work would require more sophisticated model, including a collision operator, realistic tokamak geometry, magnetic perturbation etc. However, the limited computing performance of the current machine precludes us from reaching this level. In this context, the bounceaveraged gyrokinetic TERESA code allows us more detailed studies about impurity dynamics with a reasonable computation time, and most importantly, these results are in qualitative agreement with our quasi-linear model.

Introduction

Impurity transport is mainly composed of turbulent and neoclassical transport (in the electrostatic limit). And it has been often assumed that the approach of adding two di erent channels independently provides a reliable framework to compute impurity particle ux. However, recent works have emphasized that the synergy e ect between turbulent and neoclassical transport might be non-negligible when background plasma turbulence impacts on the plasma state, which in turn modi es neoclassical transport [START_REF] Vernay | Synergy between ion temperature gradient turbulence and neoclassical processes in global gyrokinetic particle-in-cell simulations[END_REF][START_REF] Estève | Self-consistent gyrokinetic modeling of neoclassical and turbulent impurity transport[END_REF][START_REF] Donnel | Neoclassical impurity ux in presence of turbulent generated poloidal asymmetries and pressure anisotropy[END_REF].

Since the rst evidences of central accumulation by neoclassical transport [START_REF] Connor | The neo-classical transport theory of a plasma with multiple ion species[END_REF][START_REF] Rutherford | Impurity transport in the P rsch-Schlüter regime[END_REF], numerous studies have been conducted to identify the source of such deleterious core accumulation, particularly in connection with the impact of poloidal asymmetries of impurity density [START_REF] Helander | Neoclassical transport in a rotating impure plasma[END_REF][START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF][START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF][START_REF] Maget | Natural poloidal asymmetry and neoclassical transport of impurities in tokamak plasmas[END_REF]. Unlike conventional neoclassical prediction which assumes almost uniform impurity distribution over the ux surface [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF], a strong poloidal asymmetry has been repeatedly observed in various tokamaks [START_REF] Lerche | Optimization of ICRH for core impurity control in JET-ILW[END_REF][START_REF] Odstrcil | Optimized tomography methods for plasma emissivity reconstruction at the ASDEX Upgrade tokamak[END_REF][START_REF] Goniche | Ion cyclotron resonance heating for tungsten control in various JET H-mode scenarios[END_REF] and mounting evidence suggests that accounting for such asymmetry is a crucial key for a predictive model [START_REF] Casson | Predictive multi-channel ux-driven modelling to optimise ICRH tungsten control and fusion performance in JET[END_REF].

In order to assess these e ects coming from both turbulent and neoclassical transport, nonlinear gyrokinetic simulations have been performed with the full-F gyrokinetic code GYSELA (Grandgirard et al., 2016). To this end, a linearized collision operator has been successfully implemented with numerical improvements in terms of energy conservation giving a better agreement with theoretical predictions (Donnel et al., 2019a). Therefore, in this chapter, as an extension of previous work (Donnel et al., 2019a), GYSELA numerical simulations are extended to investigate the impurity transport of various impurities in di erent collisional regimes-especially, helium (He), argon (Ar) and tungsten (W) representing low, intermediate, high collisional regimes respectively. To achieve this end, turbulent and neoclassical transports in presence of a poloidal asymmetry driven by turbulence are compared with an analytical model in each case and also the e ects of poloidal asymmetry on neoclassical transport are investigated.

Neoclassical impurity ux

In this section, two di erent analytical descriptions of neoclassical impurity ux are derived with di erent assumptions. First, neoclassical uxes are expressed as a function of the CGL-pressure tensor [START_REF] Chew | The boltzmann equation and the one-uid hydromagnetic equations in the absence of particle collisions[END_REF] assuming weak poloidal asymmetries of the impurity density. The purpose of such derivation is to clarify the main driving terms of neoclassical uxes i.e. Banana-Plateau (BP) and P rsch-Schlüter (PS). Second, a more detailed form of impurity ux is given including the impact of poloidal asymmetries of the density analytically, the e ect of which is signi cant over neoclassical particle uxes, changing an order of magnitude or even its sign [START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF]. This analytical formula will be used for comparison with numerical simulations afterward in Section 4.3. A more detailed derivation of the latter can be found in [START_REF] Donnel | Neoclassical impurity ux in presence of turbulent generated poloidal asymmetries and pressure anisotropy[END_REF].

Impurity ux with uniform density distribution

The analytical form of neoclassical impurity ux can be derived from the gyrokinetic equation,

∂ Fz ∂t + 1 B * ∇ z • (żB * Fz ) = C( Fz ) (4.1)
where Fz is the distribution function of impurity and

B * = B + mz eZ v b • (∇ × b)
is involved in the volume element in gyrokinetic velocity space and C is the collision operator.

In steady state, the conservative form of particle ux ∇ • Γ = 0 can be obtained by taking the rst moment of Eq.( 4.1). Splitting the ux as Γ = Γ b + Γ ⊥ , the perpendicular ow can be recast as :

Γ ⊥z = Γ E + Γ D + Γ mag = N z v E + N z v Dz -∇ × N z µ Ze b (4.2)
where v E is the E × B drift, v D is the magnetic drift and the magnetization ux

Γ mag = ∇ × M z with M z = -d 3 v
Fz µb is added to account for the correction associated with the cyclotron motion of the particles around its guiding-centre [START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF] and the bracket is an average over the distribution function.

.

.. = 1 N d 3 v Fz ... (4.3)
Applying an average of the distribution function over the magnetic drift v D leads to the impurity ux as a function of CGL-pressure tensor. After properly arranging all the terms in Eq.( 4.2), one nds an analytical expression of neoclassical particle BP and PS uxes (Appendix.B),

Γ BP • ∇ψ ψ = - I Ze B • ∇ • Π ψ B 2 ψ (4.4) Γ PS • ∇ψ ψ = - I Ze (N eb • ∇φ + b • ∇ • Π)   1 B - B B 2 ψ   ψ (4.5)
where I = RB ϕ is related to the poloidal current that ows through a magnetic surface, Π is the CGL-pressure tensor and the bracket ... ψ corresponds to a ux surface average.

The BP ux which is dominant in the long mean free path regime is mainly driven by the pressure anisotropy. The origin of pressure anisotropy is usually due to the magnetic drift driven by non-uniform magnetic eld. Since the e ect of collisions is less strong than that of the advection in this regime, pressure is not completely isotropized leading to perpendicular ux [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]. The main driving source of PS ux, conversely, arises from the poloidal variation of the friction force [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]. In this regime, the contribution of pressure anisotropy vanishes because of strong collisions.

Impurity ux with poloidal asymmetries Determination of the poloidal variation of impurity ux Γ z

In what follows, neoclassical impurity ux in presence of poloidal density asymmetry is derived. As it is already treated in the previous section, the total impurity ux from gyrokinetic equation can be written

Γ z = Γ z b + N z B B 2 × ∇φ + B ZeB 2 × ∇ • Π (4.6) Using an axisymmetric notation ∇φ = ∂φ ∂ψ ∇ψ + ∂φ ∂θ ∇θ ∇P ⊥z = ∂P ⊥z ∂ψ ∇ψ + ∇P ⊥z ∂θ ∇θ (4.7)
Eq.( 4.6) can be reformulated as follows :

Γ z = K z B -N z Ω z R 2 ∇ϕ + Π z ZeB (b × κ) +   N z ∂φ ∂θ + 1 Ze ∂P ⊥,z ∂θ   b B × ∇θ (4.8)
where Π z = P z -P ⊥z is the pressure anisotropy, κ = -b × (∇ × b) is the magnetic curvature and we have introduced the following relations :

Ω z = ∂φ ∂ψ + 1 N z Ze ∂P ⊥z ∂ψ K z = Γ z B + I B 2 N z Ω z (4.9)
In the above equation, K z (ψ, θ) is a function of the poloidal angle and a more precise form of it can be obtained from the incompressibility of impurity ow ∇ • Γ z = 0. The detailed derivation to nd K z (ψ, θ) is described in [START_REF] Donnel | Neoclassical impurity ux in presence of turbulent generated poloidal asymmetries and pressure anisotropy[END_REF]). Here we use the value of K z = K z0 (ψ) + K z1 (ψ, θ) directly from the aforementioned paper.

K z0 (ψ) = K z ψ K z1 (ψ, θ) = - 1 B ∂ ∂ψ   IΠ z ZeB   + 1 B ∂ ∂ψ   IΠ z ZeB   (4.10)
Link between the friction force F and K z

Another analytical form of the neoclassical impurity ux can be derived from the momentum equation by taking its toroidal projection.

mN ∂V ∂t + ∇ • Π -eN (E + V × B) = F (4.11)
where the stress tensor Π = P + mN V V and F = mvC(f )d 3 v is the frictional force. By projecting the above equation in the R φ-direction and applying ux surface average, we can rewrite the above equation as follows :

R φ • ∂(m z N z V z ) ∂t + R φ • ∇ • Π -N z ZeR φ • (E + V × B) = RF zi (4.12)
Neoclassical theory for tokamaks usually assumes a steady state (∂ t = 0) and a toroidal axisymmetry (∂ ϕ = 0). Also, the stress tensor Π is a small term in neoclassical ordering compared to other components in case of strong collisions, thus it is often neglected. Using the relation B = I(ψ)∇ϕ + ∇ϕ × ∇ψ where I is related to the poloidal current inside a magnetic surface and φ = R∇ϕ, the particle ux across magnetic surfaces can be derived as follows [START_REF] Helander | Collisional Transport in Magnetized Plasmas[END_REF] (the detailed expression of F zi is explained in Appendix B.2) :

Γ z • ∇ψ neo = - IF zi ZeB (4.13)

Impurity ux with poloidal asymmetries

Finally, one can nd the analytical form of the impurity ux with poloidal asymmetry by properly arranging Eq. (4.10, B.19) and Eq.(4.13). As already done in the previous section, one can divide the total impurity ux into two di erent channels

Γ ψ neo = Γ ψ BP + Γ ψ PS . Γ BP = - I Ze B 2 N z -1 ψ B • ∇θ N z   ∂P ⊥z ∂θ + B ∂ ∂θ   Π z B     ψ (4.14) Γ P S = I Ze m z ν zi    T i e I L ψ,i   N z B 2 ψ - B 2 N z -1 ψ   -u   N z ψ - B 2 ψ B 2 Nz ψ   + T i I e   N z B 2 L ψ,z ψ - 1 L ψ,z ψ B 2 N z -1 ψ   - 1 B ∂ ∂ψ   IΠ z ZeB   ψ + B Nz ∂ ∂ψ IΠ z ZeB ψ B 2 z Nz ψ    (4.15)
where

1/L ψ = 1/L ψ,i + 1/L ψ,z with 1/L ψ,i = ∂ ψ ln P i -3 2 ∂ ψ ln T i and 1/L ψ,z = -1
T i ZNz ∂ ψ P ⊥z and u is the value related to the poloidal velocity of the main ion. In the trace limit of impurities

(N z Z 2 /N i ) << 1, u can be expressed as u =   k neo - 3 2   If c e B 2 ψ (4.16)
where the value of k neo ≈ 1.17 is obtained in the limit of large aspect ratio 1 with the main ion in the banana regime [START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF]. Thus, the following formula u ≈ -0.33 Ifc e B 2 ψ will be used for further analysis. Here, f c represents the fraction of circulating particles and can be obtained in a large aspect ratio with circular cross section

f c = 1 -f t = 3 B 2 ψ 4 λc 0 dλ √ 1 -λB ψ ≈ 1 -1.46 √ (4.17)
where λ = µ/E is the pitch angle and λ c = B -1 max is the critical λ for trapping. Comparing Eq. (4.4, 4.5) with Eq. (4.14, 4.15) shows how an uneven distribution of impurities yields a considerable e ect on their ux. Note that the main driving sources for BP/PS uxes-pressure anisotropy for BP ux and poloidal variation of frictional force for PS ux-are unchanged while additional terms have been added in Eq.( 4.15) arisen from the poloidal dependency of K z1 (ψ, θ) in Eq.(4.10). The presence of Π z term in Eq.( 4.15) may seem odd at rst sight since pressure tends to be isotropic in a highly collisional regime. However, it has been pointed out that even in a high collisional regime, pressure anisotropy can still have an in uence on PS ux [START_REF] Donnel | Neoclassical impurity ux in presence of turbulent generated poloidal asymmetries and pressure anisotropy[END_REF].

Numerical results

Various impurities coexist during tokamak discharge, and consequently the main transport channel can be di erent-turbulent or neoclassical-according to their mass (A) and charge (Z). Thus, the purpose of this section is, rst of all, to provide impurity ux of di erent species in presence of poloidal asymmetries generated by turbulence and to compare them with the analytical models discussed previously. Second, the effect of poloidal asymmetries on impurity transport is investigated in detail. To achieve this end, we selected 3 impurities-helium (Z=2, A=4), argon (Z=18, A=40) and tungsten (Z=40, A=184)-each representing di erent collisional regimes according to the normalized collisionality ν * a [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF] :

ν * a = ν detrap,a ω b,a ≈ qR 0 3/2 ν a v T a (4.18) = qR 0 3/2 4 √ π 3 e 4 ln Λ (4π 0 ) 2 Z 2 a T 2 a n a Z 2 a + b =a √ 2n b Z 2 b 1 + ma m b 1 + v 2 T b v 2 T a 3/2 (4.19)
While the main ions are in the banana regime with ν * D = 0.1 at mid radius, the relative ratio of the detrapping frequency ν detrap,a = ν a /2 over the bounce frequency ω b,a = v T a √ √ 2πqR determines three collisional regimes -ν * 1 for the banana regime, 1 ν * -3/2 for the plateau regime and ν * -3/2 for the P rsch-Schlüter regime (Figure 4.1).

For numerical simulations, the parameters on Table 4.1 are chosen for impurities as well as for main ions. With given density and temperature pro le for main ions (R/L ni = 2.2 and R/L T i = 6), the main instability is the ion temperature gradient (ITG) instability and electrons are treated adiabatically. An isotropic heat source is added close to the inner boundary of simulations [START_REF] Sarazin | Predictions on heat transport and plasma rotation from global gyrokinetic simulations[END_REF], allowing the convergence toward the steady state. Also a Krook operator is applied at the outer boundary region for an arti cial damping of the heat ux. Without any further assumptions of ionization or recombination, the impurity concentration was kept in the trace limit (C z = 10 -6 ) so that the background turbulence is not a ected by the presence of impurities. Also, all numerical simulations have been performed in the axisymmetric geometry with concentric magnetic surfaces. The magnetic eld is de ned as B = (B 0 R 0 /R)[ r q(r)R 0 e θ +e ϕ ] where B 0 is the magnetic eld on the axis and the safety pro le is given as q(r) = 1.5 + 1.3 r a 2.5

. Even though electrons are treated adiabati- cally, running a simulation with impurity requires considerable computation time since higher resolution is necessary for the impurity Larmor radius (ρ z = 1 Z mz m i ρ i ). To overcome this issue, the same strategy as in [START_REF] Donnel | Neoclassical impurity ux in presence of turbulent generated poloidal asymmetries and pressure anisotropy[END_REF]) has been applied : (i) the rst simulation runs without impurity until turbulence is well developed and steady. At this stage, the numerical grid 256,512,32,127,64) is used for main ions. (ii) once the main ions get to a steady state, the highly resolved numerical grid (N r , N θ , N ϕ , N v , N µ ) = (512, 1024, 32, 127, 64) is used without impurity in order to reorganize the system before adding impurity. (iii) after the reorganization, impurity is put into the simulation with the same numerical grid like previous step.

(N r , N θ , N ϕ , N v , N µ ) = (
The physical duration of each simulation lasted su ciently long for the background turbulence to be well developed and the nal stage (iii) of each impurities lasted for sufciently long time. However, according to the scaling law (ITER Physics Expert Group et al., 1999), the con nement time for impurities to reach the steady state is much longer than the ion con nement time, hence the radial plasma pro le of impurities, such as density and temperature, are not completely steady.

Impurity ux with poloidal asymmetry driven by turbulence

By convention, the total impurity ux can be expressed as a sum of two di erent contributions Γ tot = Γ turb + Γ neo . Various approaches have been suggested to de ne such uxes properly [START_REF] Estève | Self-consistent gyrokinetic modeling of neoclassical and turbulent impurity transport[END_REF]. In this work, we adopt the following de nitions :

< Γ turb z • ∇ψ > ψ = d 3 v Fz v n =0 E • ∇ψ ψ (4.20) T 4.1 -Numerical simulations parameters Parameters Normalization Time step ∆tω ci = 16 Normalized gyroradius ρ * = ρ i /a = 1/190 Impurity concentration C z = 10 -6 Inverse aspect ratio = a/R 0 = 1/4.4 Density R/L n = 2.2 Temperature R/L T = 6 Magnetic eld B = B 0 R 0 R r q(r)R 0 e θ + e ϕ Safety factor q(r) = 1.5 + 1.3 exp{[2.5 log(r/a)]} < Γ neo z • ∇ψ > ψ = d 3 v Fz (v D,s + v n=0 E ) • ∇ψ ψ (4.21)
where n is the toroidal mode number. The impurity uxes, averaged out over 4000ω ci , are shown respectively on Figure 4.2 where the black solid line stands for the GYSELA numerical ux which is the sum of the turbulent ux (red dashed) and the neoclassical ux (blue dashed).

Qualitative di erences between species are clearly observed on Figure 4.2. For Helium, the dominant ux is mainly coming from turbulent transport owing outward from the core (positive Γ z ) except in the region ρ ∼ 0.3 where the peak of neoclassical ux is presumably produced by poloidal density asymmetry. These results t well with our prior expectation that helium, as a light impurity, is in the banana regime (Figure 4.1) where collisional e ects are small when compared to turbulent uxes. Also, the outward ow of impurity usually gives rise to a hollow density pro le in Figure 4.3, in which He impurity density keeps decreasing along the time at ρ = 0.3. In recent works [START_REF] Bonanomi | Light impurity transport in JET ILW L-mode plasmas[END_REF][START_REF] Kappatou | Understanding helium transport : experimental and theoretical investigations of low-Z impurity transport at ASDEX upgrade[END_REF][START_REF] Manas | Light impurity transport in tokamaks : on the impact of neutral beam fast ions[END_REF], it was experimentally and numerically observed that light impurities, such as helium and boron, are less dangerous than heavy impurities for core accumulation since, particularly in the conditions of ITG dominated regime and high NBI fraction, the outward thermodi usion and roto-di usion terms are dominant.

The neoclassical contribution to the total impurity ux becomes larger when considering higher Z impurities : in the simulations we have performed, it is signi cant for argon and tungsten. Argon, whose collisionality is extended mostly over the Plateau region (Figure 4.1), shows two distinct regions where the total ux (black solid line) is roughly aligned (i) with neoclassical ux (ρ < 0.3) and (ii) with turbulent ux (ρ > 0.5). In terms of magnitude, turbulent ux is comparable to neoclassical ux giving nearly equal contributions to the total ux. Unlike helium, argon generates an inward ux (Γ z negative) leading to accumulation in the core in Figure 4.3, in which Ar impurity density keeps increasing at ρ = 0.3.

For tungsten, the heaviest of the analyzed impurities, turbulent ux is reduced by a factor of 2 while neoclassical transport still exerts an important e ect in the core. The overall impurity ux is directed inward driving its core accumulation due to neoclassical convection (Figure 4.3). A large number of experimental cases have already con rmed the core accumulation [START_REF] Meyer | Overview of physics studies on asdex upgrade[END_REF][START_REF] Jo Rin | Overview of the JET preparation for deuterium-tritium operation with the ITER like-wall[END_REF][START_REF] Park | Overview of KSTAR research progress and future plans toward ITER and K-DEMO[END_REF] and recent studies have proposed that a proper use of additional heating system might mitigate such accumulation by reducing the neoclassical convection and increasing the turbulent di usion [START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF][START_REF] Casson | Predictive multi-channel ux-driven modelling to optimise ICRH tungsten control and fusion performance in JET[END_REF]. In this work, no additional heating systems are used except for the heat source on the main ion species.

A comparison of the neoclassical contribution between numerical ux (blue dashed) and semi-analytical model (grey solid) from Eq. (4.14, 4.15) is depicted on Figure 4.4. A special attention has to be paid at this point with respect to how our analytical formula is applied numerically. Unlike other works which use a solution of drift-kinetic equation (NEO [START_REF] Belli | Full linearized Fokker-Planck collisions in neoclassical transport simulations[END_REF]) or a moment approach (NCLASS [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF][START_REF] Houlberg | Bootstrap current and neoclassical transport in tokamaks of arbitrary collisionality and aspect ratio[END_REF]), here we make direct use of GYSELA numerical data to estimate the neoclassical ux, which means every term in Eq. (4.14, 4.15) is substituted at each time step to obtain neoclassical transport. The goal is to provide direct test of the analytical formula It is clearly shown on Figure 4.4 that for all impurities, the analytical expressions including poloidal asymmetry of impurity density reproduce satisfying results with 4.15). Every single component in Eq. (4.14, 4.15) was calculated from non-linear GYSELA simulations, for instance density N , temperature T and pressure P ⊥ , P are computed from the GYSELA distribution function. The di erence between two uxes can be attributed to the assumptions made in Eq. (4.14, 4.15) which are based on purely neoclassical approach, such as Eq.( 4.16, 4.17) while our simulations are also in turbulent regime.

our gyrokinetic simulations. The neoclassical ux is directed inward for all impurities due to the inward neoclassical convection driven by main ion density pro le. In agreement with our previous results, helium generates a weaker neoclassical ux compared to other impurities. It is further instructive to analyze the neoclassical ux in detail by splitting its components. F 4.5 -Banana-Plateau (BP) ux and P rsch-Schlüter (PS) ux for di erent impurity species, calculated from numerical simulations GYSELA using Eq. (4.14, 4.15). On Figure 4.5, the radial BP, PS uxes are calculated from the numerical simulations GYSELA using the analytical formula Eq. (4.14, 4.15). As expected, tungsten produces the most important PS ux compared to other impurities due to its high collisionality while the PS ux coming from helium is almost negligible. For BP uxes, argon whose collisionality is extended over the plateau region produces a stronger BP ux than for the other species. Moreover, helium in the banana regime starts to generate a BP ux.

Poloidal asymmetry driven by background turbulence

A number of studies have pointed out the inhomogeneity of impurity density over ux surfaces can strongly modify the neoclassical transport [START_REF] Romanelli | E ects of density asymmetries on heavy impurity transport in a rotating tokamak plasma[END_REF][START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF][START_REF] Angioni | The impact of poloidal asymmetries on tungsten transport in the core of JET H-mode plasmas[END_REF] while its impact on turbulent transport seems less signi cant [START_REF] Casson | Theoretical description of heavy impurity transport and its application to the modelling of tungsten in JET and ASDEX upgrade[END_REF]. Although there are many factors which can trigger poloidal asymmetry, in this chapter, we restrict ourselves to the poloidal asymmetry driven by turbulence only. The poloidal asymmetries generated by turbulence are expected to be larger for high-Z impurity since the response to the electric potential is Boltzmann-like, thus it is proportional to impurity charge. According to Eq.( 4.15), PS ux is mostly driven by poloidal variation of frictional force which is a function of impurity density N z from Eq.(B.19). Therefore, it is straightforward that PS ux is modi ed through the poloidal asymmetry and its e ect is stronger in case of heavy impurities-tungsten, in this work. This fact is well shown on Figure 4.6 in which the perturbed density Ñz = Nz-Nz ψ Nz ψ of di erent impurities are presented on the poloidal section. As expected, tungsten shows the strongest poloidal asymmetry (∼ 20%) while helium presents a very weak asymmetry (∼ 3%). Such asymmetry of F 4.6 -Poloidal asymmetries of impurity density given by numerical simulations performed with GYSELA, for helium (left), argon (center), and tungsten (right). impurity density can be expressed as N z (ψ, θ) = N z (ψ)[1 + δ cos θ + ∆ sin θ] where δ and ∆ represent 'in-out' and 'up-down' asymmetry respectively. According to [START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF], it is usually an 'in-out' asymmetry which plays an important role on neoclassical transport since the geometrical factor in Eq.( 4.15) (corresponding to P A , P B in Eq.( 4.23)) can be recast as a function of δ and ∆ and the 'up-down' asymmetry ∆ appears only as a second order term in impurity ux.

B 2 ψ N z ψ   N z B 2 ψ - B 2 N z -1 ψ   = 2 ( + δ) + δ 2 + ∆ 2 2 B 2 ψ N z ψ   N z B 2 ψ - B 2 N z -1 ψ   = δ + δ 2 + ∆ 2 2 (4.22)
However, poloidal asymmetry driven by turbulence tends to generate an 'up-down' asymmetry (Fig. 4.7) through poloidal convective cells (Donnel et al., 2019c). Such an 'up-down' asymmetry would be less dramatic than neoclassical 'in-out' asymmetries and their di erent impact has been already treated in [START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF].

Poloidal asymmetry a ects neoclassical transport, but it can also change the socalled thermal screening factor H neo . As the name suggests, this term related to the main ion temperature pro le L T i prevents impurities from accumulating in the core. Derived from the main ion parallel heat ux in neoclassical theory, the factor is usually equal to H neo -1 2 when the main ion is in the banana regime [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF], but this screening term is also susceptible to be modi ed by poloidal asymmetry. To clarify this mechanism, it is helpful to rewrite Eq.( 4.15) in a di erent way, as it was F 4.7 -Impurity density distribution as a function of the poloidal angle given by the numerical simulations performed with GYSELA. done in [START_REF] Angioni | The impact of poloidal asymmetries on tungsten transport in the core of JET H-mode plasmas[END_REF] :

R Γ neo Z • ∇ψ ψ ∝   - R L ni -H neo R L T i + 1 Z R L nz   ∝     - R L ni -H sym R L T i + 1 Z R L nz   P A -0.33P B f c R L T i   (4.23)
where H neo = H sym + 0.33 P B P A f c with the factor H sym = -1 2 in the absence of poloidal asymmetries. This factor appears in the expression of 1

L ψ,i = ∂ ln N i ∂ψ + H sym ∂ ln T i
∂ψ in Eq. (4.15).

In this equation, the second term corresponds to the thermal screening term and f c is the fraction of passing particles. The geometrical factors P A , P B quantify the impact of poloidal asymmetry, de ned as follows :

P A = 1 2 2 B 2 ψ N z ψ   N z B 2 ψ - B 2 N z -1 ψ   P B = 1 2 2 B 2 ψ N z ψ   N z B 2 ψ - B 2 N z -1 ψ   (4.24)
Eq.( 4.23) reveals that the non-uniform distribution of impurities can modify the neoclassical ux or even reduce the temperature screening term by facilitating the accumulation of impurities at the core. From Eq.( 4.24), it readily appears that, if the impurity density is uniform over ux surfaces, i.e. if N z = N z , then P A = 1 and P B = 0. In this case, standard neoclassical results are recovered. The radial pro les of P A , P B for di erent impurities are presented in Figure 4.8, where the outer regions ρ < 0.1 and ρ > 0.9 are excluded to mitigate arti cial dissipation in bu er region. Although external heating systems are absent in our cases, the background turbulence generates a poloidal asymmetry of density and it is strongly enhanced for W due to its high charge and heavy mass. The impact of poloidal asymmetry on the thermal screening 4.9 -Neoclassical impurity ux of tungsten from the theoretical predictions Eq. (4.14,4.15) with two di erent values of H sym . With the thermal screening H sym = -1 2 (red solid line) and without the thermal screening H sym = 0 (green solid line). For comparison, the neoclassical impurity ux from GYSELA is also depicted (blue dashed line).

Conclusion

In this chapter, numerical modelling of turbulent and neoclassical transport has been done with various impurities-He, Ar and W-using the full-F gyrokinetic code GYSELA. Our results demonstrate that the main dominant particle ux and its direction depend sensitively on the type of impurities. Most of light impurity (He) transport is due to turbulence while heavy impurity (W) transport is dominated by neoclassical e ects. Moreover, the main two driving terms for neoclassical ux-BP, PS uxes-have been investigated. For He, both BP and PS uxes are found to be small due to its low collisionality. In contrast, signi cant inward BP and PS uxes are generated for Ar and W leading to core accumulation.

In the absence of additional heating system, such as NBI and ICRH, we have also veri ed that the poloidal asymmetry driven by turbulence can modify the neoclassical particle ux and its impact becomes signi cant in case of heavy impurities. Furthermore, these strong asymmetries for heavy impurities reduce the thermal screening factor which facilitates the deleterious core accumulation. The neoclassical ux given by our analytical model is shown to be in good agreement with our numerical simulations and GYSELA reproduces the thermal screening factor close to the theoretical prediction H sym -1 2 in the region where poloidal asymmetries induced by turbulence are small (ρ > 0.5).

The overall inward ux for heavy impurities observed in this chapter has already been reported in di erent tokamaks. To avoid such core accumulation, a proper use of ICRH/ECRH near magnetic axis is necessary. The impact of toroidal rotation on impurity transport is investigated in the next chapter.

Part of this chapter can be also found in [START_REF] Lim | Gyrokinetic modelling of light to heavy impurity transport in tokamaks[END_REF]))

Introduction

During a plasma discharge, toroidal rotation can be intrinsically generated or externally from NBI heating system. It is well known that the centrifugal force driven by toroidal rotation can generate a signi cant 'in-out' poloidal asymmetry, which leads to an enhancement of neoclassical transport. In contrast, the impact of such poloidal asymmetry on turbulent transport was found to be much smaller than on neoclassical transport [START_REF] Casson | Theoretical description of heavy impurity transport and its application to the modelling of tungsten in JET and ASDEX upgrade[END_REF]. However, the presence of a turbulent roto-di usion, i.e., a radial ux due to the presence of a radial gradient of the toroidal rotation, can play an important role in case of a strongly rotating plasma. Its magnitude even can match the di usive ux, especially for heavy impurities [START_REF] Camenen | Impact of the background toroidal rotation on particle and heat turbulent transport in tokamak plasmas[END_REF].

A non-negligible impact of strong toroidal rotation on both turbulent and neoclassical impurity transport has already been presented in [START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF][START_REF] Casson | Theoretical description of heavy impurity transport and its application to the modelling of tungsten in JET and ASDEX upgrade[END_REF]. The main approach therein was to calculate each transport channel independently ; (i) turbulent transport through the gyrokinetic formulation in the toroidally co-moving frame (GKW) and (ii) neoclassical transport by means of Hinton ordering [START_REF] Hinton | Neoclassical ion transport in rotating axisymmetric plasmas[END_REF], but without considering frictional e ects at the lowest order (NEO) [START_REF] Belli | P rsch-Schlüter neoclassical heavy impurity transport in a rotating plasma[END_REF].

In this chapter, as an extension of Chapter 4, the direct impact of toroidal rotation on turbulent and neoclassical impurity transport for tungsten (W), in connection with a poloidal asymmetry, is investigated (Section 5.2) by means of the full-F gyrokinetic code GYSELA (Grandgirard, 2016). To this end, adjustable heat and momentum sources are employed [START_REF] Sarazin | Predictions on heat transport and plasma rotation from global gyrokinetic simulations[END_REF], allowing the self-consistent investigation of both turbulent and neoclassical transports in the laboratory frame including frictional and synergetic e ects as well. Section 5.2 outlines the theoretical e ects of toroidal rotation on poloidal asymmetries and impurity transport and section 5.3 presents numerical results obtained from nonlinear simulations performed with GYSELA.

Theoretical approach

In GYSELA, the injection of heat and toroidal momentum can be accomplished by employing adjustable source terms S in the gyrokinetic Vlasov equation [START_REF] Sarazin | Predictions on heat transport and plasma rotation from global gyrokinetic simulations[END_REF] :

∂ Fs ∂t + 1 B * ∇ z • (żB * Fs ) = C( Fs ) + S heat + S mom (5.1)
where Fs is the gyro-averaged distribution function for species s, B * = B + ms eZs v b • (∇ × b) is the volume element in gyrokinetic velocity space, C( Fs ) is the collision operator and S heat , S mom (de ned in Eq.(2.36) and Eq.(2.37)) represent the heat and toroidal momentum sources respectively.

Distribution function with toroidal rigid rotation

According to the neoclassical theory, developed in [START_REF] Hinton | Neoclassical ion transport in rotating axisymmetric plasmas[END_REF][START_REF] Wong | Transport of impure plasma with arbitrary toroidal rotation[END_REF], at the lowest order in δ = ρ pi /L where ρ pi is the poloidal ion Larmor radius and L is the radial length scale, the distribution function is Maxwellian, and both the temperature T eq (ψ) and toroidal rotation frequency ω(ψ) are ux functions, which are constant on magnetic ux surfaces.

F M = N eq (ψ, θ) m 2πT eq (ψ) 3/2 exp - mu 2 2T eq (ψ)
(5.2)

with u = V -V 0 the velocity in the moving reference frame V 0 , and V is the velocity of particles in the reference frame. However, the density N eq (ψ, θ) is not a ux function due to the centrifugal force [START_REF] Hinton | Neoclassical ion transport in rotating axisymmetric plasmas[END_REF].

N eq (ψ, θ) = N 0 (ψ) exp - eZφ T eq (ψ) + mω 2 (R 2 -R 2 0 ) 2T eq (ψ)
( 5.3) where e is the positive elementary charge. Inserting the above equation Eq.( 5.3) into the Maxwellian function Eq.( 5.2) reads :

F M = N 0 (ψ) m 2πT eq (ψ) 3/2
exp -H T eq (ψ) (5.4) where

H = 1 2 mu 2 + µB + eZφ 0 (ψ) + eZ φ1 (ψ, θ) - mω 2 (R 2 -R 2 0 ) 2 (5.5)
The electric potential on the magnetic surface is approximately equipotential φ φ 0 (ψ). In a rotating plasma, however, ions are pushed towards the outboard region due to the centrifugal force. Consequently, an equilibrium electric potential φ1 (ψ, θ) is set up to ensure the quasi-neutrality constraint in the presence of centrifugal forces.

φ1 = ω 2 2 m i e T e T i + Z i T e (R 2 -R 2 0 ) (5.6)
Therefore, the density distribution in a toroidally rotating plasma can be written as :

N eq (ψ, θ) = N 0 (ψ) exp - eZφ 0 (ψ) T eq (ψ) - eZ φ1 (ψ, θ) T eq (ψ) + 1 2 mω 2 (R 2 -R 2 0 ) T eq (ψ)
(5.7) Under strong rotation, the centrifugal force tends to push impurities in the outboard region while an induced equilibrium potential φ1 , which varies poloidally, tends to balance the density asymmetry. These e ects are enhanced for heavy impurity ions due to both their high-Z and greater mass.

Poloidal asymmetries generated by toroidal rotation

Parallel impurity transport

In the case of high-Z impurities, they are usually placed in the highly collisional regime according to the de nition of collisionality. To derive the poloidal variation of the impurity density, rst of all, we start from the momentum balance equation in steady state. (5.8) where V z is the mean ow, π z is the viscosity tensor and F zi is the frictional force.

N z m z (V z • ∇)V z + ∇p z + ∇ • π z + N z Ze∇Φ -ZeN z (V z × B) = F zi
In the above equation, the viscosity tensor π z can often be neglected in a high collisionality limit, and the temperature can be assumed to be a ux function T (ψ) as a consequence of high parallel heat transport. Then, the nal expression of parallel impurity dynamic can be written by projecting Eq.( 5.8) in the parallel direction :

m z N z ω 2 2 ∇ R 2 + N z Ze∇ Φ + T z ∇ N z = F ,zi
(5.9)

The rst term corresponds to the centrifugal force related to a strong toroidal rotation, the second term is connected to the electrostatic force and the last term is the pressure gradient along the magnetic eld. In a rotating plasma, particles are pushed towards the outside due to the centrifugal force and consequently, an additional electric potential is set up to maintain the quasi-neutrality constraint in plasma.

An analytical formula for a non-uniform density distribution can be determined from the parallel impurity transport. The rst analytical derivation was done in [START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF] by including both inertial and frictional terms in Eq.(5.9) for main ions in Banana regime and impurities in P rsch-Schlüter regime. Here, we recall the nal expression of impurity poloidal asymmetry for the sake of concision. A detailed derivation is also available in Appendix C.

With the following assumptions, such as an axisymmetric magnetic eld and a small inverse aspect ratio ( 1), the poloidal asymmetry of impurity density to the lowest order of can be expressed as N z = 1 + N c cos θ + N s sin θ where N s and N c represents the 'up-down' and 'in-out' asymmetry factors respectively :

N s = 2 g 1 + (1 + γ)M 2 0 1 + (1 + γ) 2 g 2 ('up-down') (5.10) N c = 2 M 2 0 -(1 + γ)g 2 1 + (1 + γ) 2 g 2 ('in-out') (5.11) with M 2 0 = M 2 R 2 0 /R 2 .
The following dimensionless parameters are de ned,

M 2 = m z ω 2 R 2 2T z 1 - Zm i m z T e T e + T i (5.12) g = - m i N i I eτ iz N z B • ∇θ 1 N i ∂N i ∂ψ - 1 2 1 T i ∂T i ∂ψ (5.13) with γ = eL ⊥ B 2 u T i
where the poloidal rotation for main ions u ∼ -0.33 Ifc e B 2 with f c the fraction of circulating particles.

The rst parameter is the modi ed Mach number, which indicates the impact of toroidal rotation over impurity distribution through M 2 0 in Eq. (5.10, 5.11). The second dimensionless parameter g estimates the steepness of density and temperature gradient pro les for main ions. In the standard neoclassical theory, this parameter is assumed to be small, which corresponds to small frictional force. However, the impact becomes dominant when impurities are localized in the outboard region, especially for high-Z impurities (g ∝ Z 2 ) or in the pedestal region.

Neoclassical and turbulent impurity transport in rotating plasma

Turbulent ux

In the case of toroidally rotating plasma, the turbulent ux includes an additional term compared with Eq.(3.10) : 5.14) where u z = -R 2 dωz dr 1 v th,z is the normalized toroidal rotation gradient and the terms in the RHS represent turbulent di usion, thermo-di usion, roto-di usion and curvature pinch respectively .

RΓ z N z = D z R L N z + C T R L T z + C u u z + C p (
The rst evidence of the roto-di usion driven by the presence of a radial gradient of the toroidal plasma rotation has been pointed out using a uid approach [START_REF] Camenen | Impact of the background toroidal rotation on particle and heat turbulent transport in tokamak plasmas[END_REF]. Similarly to the mechanisms of momentum pinch [START_REF] Peeters | Toroidal momentum pinch velocity due to the coriolis drift e ect on small scale instabilities in a toroidal plasma[END_REF], the roto-di usion term is activated by a symmetry breaking mechanism within the toroidal momentum transport. Depending on the direction of the uctuation propagation, analogously to thermo-di usion, this term is directed outward (inward) in the case of ITG (TEM) turbulence. The magnitude of the roto-di usion increases with the ratio of impurity mass to charge, therefore its impact can be signi cant especially for heavy impurities. It was also shown that the centrifugal e ects, associated with toroidal rotation, can have a non-negligible impact on other components (Angioni et al., 2012a), such as thermo-di usion, curvature pinch, which indicates that the inclusion of toroidal rotation is necessary for the study of impurity transport.

Neoclassical ux

The importance of including poloidal asymmetries in neoclassical impurity transport is now well recognized and experimentally demonstrated. A strong non-uniform distribution in the outboard region can lead to core accumulation [START_REF] Casson | Theoretical description of heavy impurity transport and its application to the modelling of tungsten in JET and ASDEX upgrade[END_REF][START_REF] Angioni | The impact of poloidal asymmetries on tungsten transport in the core of JET H-mode plasmas[END_REF]. Here, we recall the analytical expression of heavy impurity transport which takes into account such inhomogeneous poloidal density derived in Eq. (4.23). 5.15) where H neo = H sym +0.33 P B P A f c is the temperature screening coe cient with the factor H sym = -1 2 in the absence of poloidal asymmetries and the geometric factors P A and P B (4.24) related to the poloidal asymmetry. A series of recent studies has shed light on the poloidal asymmetry caused by external heating system, such as NBI and ICRH heating, and it was found that both factors P A and P B can reach up to 100 in JET tokamak with heating systems. It was also evidenced that enhanced poloidal asymmetry tends to reduce the magnitude of temperature screening coe cient H neo leading to a core accumulation near the magnetic axis [START_REF] Casson | Predictive multi-channel ux-driven modelling to optimise ICRH tungsten control and fusion performance in JET[END_REF].

R Γ neo Z • ∇ψ ψ ∝   - R L ni -H neo R L T i + 1 Z R L nz   ∝     - R L ni -H sym R L T i + 1 Z R L nz   P A -0.33P B f c R L T i   (

Numerical results

E ects of toroidal rotation on impurity transport can be investigated using the full-F gyrokinetic code GYSELA by means of an additional toroidal momentum source. In general, the centrifugal force driven by toroidal rotation is dominant for heavy impurities, which are often observed to accumulate in the core. Accordingly, tungsten (W) is chosen for our simulations with impurity charge Z W = 40 and mass A W = 184. Impurities are kept in the trace limit (N W Z 2 1), and thereby their presence does not impact the background turbulence. The initial radial pro les of density and temperature are set to R/L ns = 2.2 and R/L T s = 6 for both ions and impurities generating ion temperature gradient (ITG) driven turbulence. According to Eq.(4.19), the main ions are in the Banana regime while tungsten impurities are placed in the collisional P rsch-Schlüter regime. Other parameters used for numerical simulations are listed on Table 5.1 as well as the magnetic eld pro le and security factor.

* = ρ i /a = 1/190 Impurity concentration C W = 10 -6 Inverse aspect ratio = a/R 0 = 1/4.4 Density R/L n = 2.2 Temperature R/L T = 6 Magnetic eld B = B 0 R 0 R r q(r)R 0 e θ + e ϕ Safety factor q(r) = 1.5 + 1.3 exp{[2.5 log(r/a)]}
To save on numerical computation time, each simulation is composed of two different steps : (i) the rst stage is only composed of main ions without any momentum source. This part lasts until the background turbulence is su ciently developed. (ii) In the second step, we inject tungsten impurities with a toroidal momentum source. It should be noted that we apply the momentum source for main ions, so that toroidal rotation is transferred by inter-species collisional e ects like it would be in a NBI heating scenario. Radial pro les of the applied source terms in GYSELA are displayed in Figure 5.1. The presence of a heat source is inherent to ux driven simulations allowing investi-gation of di erent heating power on the energy con nement time. In addition to the heat source, an adjustable momentum source is added to inject toroidal momentum for main ions. It is important to choose an appropriate radial pro le for momentum source, since the toroidal velocity saturation time is proportional to ρ -3 * , which makes untractable to get a steady state for our current simulation (ρ -1 * = 190). This problem can be overcome by properly choosing the initial toroidal velocity radial pro le, similar to the saturated toroidal velocity pro le which has already been obtained in [START_REF] Sarazin | Predictions on heat transport and plasma rotation from global gyrokinetic simulations[END_REF]. E ective injection of toroidal momentum can be veri ed by comparing radial proles of toroidal velocity described in Figure 5.2. For deuterium, Mach number M s = V ϕ,s /V th,s is often found to be small even with additional momentum injection, while it can largely exceed unity for tungsten. Such a high Mach number for tungsten can also be found in experiments. For example, Mach number for deuterium was found to be as high as M D ∼ 0.7 in JET operations by means of neutral beam injections (NBI) [START_REF] De Vries | Scaling of rotation and momentum con nement in JET plasmas[END_REF], which indicates that an appropriate treatment of toroidal rotation is necessary for impurity transport modelling.

Enhanced poloidal asymmetry by toroidal rotation

In Figure 5.3, a comparison of poloidally inhomogeneous impurity density ÑW obtained from non-linear gyrokinetic simulations are shown for di erent cases. In the absence of momentum injection (Figure 5.3 left), toroidal velocity of both deuterium and tungsten remains low and the main contribution for poloidal W asymmetry mainly arises from the background turbulence (Donnel et al., 2019c). The e ects of centrifugal force become dominant for higher toroidal rotation. For the case with impurity (W) Mach ∼ 1, approximately 20% excess (de cit) of W impurities is localized in the outboard (inward) region, and this 'in-out' asymmetry is found to be ampli ed for W Mach ∼ 2 up to 40%. Such enhanced 'in-out' asymmetry can also be found in the framework of an analytical approach. To this end, the poloidal map of density uctuations is reconstructed based on two analytical formulas derived in Eq.(5.7), Eq. (5.10, 5.11), and the result extracted from nonlinear GYSELA simulations. The results for the 3 cases are shown respectively from left to right in Figure 5.4. It should be noted that, although there are some discrepancies in terms of amplitudes, the overall 'in-out' enhanced poloidal asymmetries are found no matter which method of reconstruction is used. Recently, the analytical formula derived from parallel impurity transport in Eq. (5.10, 5.11) has already been compared with experimental data from the C-Mod tokamak, producing results in a qualitative agreement between them [START_REF] Reinke | Parallel transport studies of high-Z impurities in the core of Alcator C-Mod plasmas[END_REF].

Impurity particle ux

In general, neoclassical impurity transport can be expressed as a function of density, temperature and collisionality :

Γ neo,W ∝ f (N i , N W , T i , T W , ν z i).
In contrast to the assumption made by the standard neoclassical theory, a non-uniform distribution of impurity density has to be included for a correct prediction of neoclassical ux. In Figure 5.5, neoclassical W-ux is calculated from a reconstructed density Ñz = δ cos θ + ∆ sin θ, expressed as a function of the δ='in-out' / ∆='up-down' asymmetry parameters respectively. As the case δ = ∆ = 0 represents a uniform impurity density, the importance of including poloidal asymmetry can be clearly identi ed since neoclassical ux is strongly enhanced as δ and ∆ increase. In line with the previous result, neoclassical and turbulent ux can be directly computed from nonlinear simulations. The following de nitions are used for each type of particle transport :

< Γ turb z • ∇ψ > ψ = d 3 v Fz v n =0 E • ∇ψ ψ (5.16) < Γ neo z • ∇ψ > ψ = d 3 v Fz (v D,s + v n=0 E ) • ∇ψ ψ (5.17)
where n is the toroidal mode number.

Consistently with the previous results in Figure 5.4 and 5.5, non-linear simulations with GYSELA reproduce enhanced neoclassical ux as the magnitude of injected momentum source increases, leading to outboard localization of W impurities (Figure 5.6). While strong density inhomogeneity driven by the toroidal rotation has a direct impact on neoclassical transport, turbulent transport is directly linked to the roto-di usion in Eq. (5.14), which in turn leads to an outward turbulent ux in case of ITG-turbulence. This term, proportional to A/Z where A is the mass number of the impurity, can often match the di usive part especially in case of heavy impurities [START_REF] Camenen | Impact of the background toroidal rotation on particle and heat turbulent transport in tokamak plasmas[END_REF]. The central W accumulation in the region between ρ = 0 and ρ = 0.3 is shown in Figure 5.7 with di erent Mach numbers. As expected analytically and numerically from the previous cases, a stronger accumulation is found for higher Mach number (M W ∼ 2) due to the enhanced poloidal asymmetry by toroidal rotation (Figure 5.3). No signi cant di erence is observed between t = 0 and t = 7 × 10 4 ω ci t, which corresponds to the collisional transfer time of toroidal rotation from deuterium to tungsten. In Chapter 4, the overall W inward ux-mainly from dominant neoclassical ux-has been reported even in the absence of external toroidal rotation. Consistently with those F 5.7 -Central W accumulation between ρ = 0 and ρ = 0.3 is represented for two di erent Mach numbers (toroidal rotation of the plasma). results, it is evidenced in Figure 5.7 that a strong toroidal rotation gives rise to an inwardly peaked impurity density pro le. Fortunately, it has been reported that a proper use of Ion Cyclotron Resonance Heating (ICRH) / Electron Cyclotron Resonance Heating (ECRH) can be favorable for impurities removal [START_REF] Dux | In uence of the heating pro le on impurity transport in ASDEX upgrade[END_REF][START_REF] Leigheb | Molybdenum transport in high density FTU plasmas with strong radio frequency electron heating[END_REF][START_REF] Sertoli | Modi cation of impurity transport in the presence of saturated (m,n) = (1,1) MHD activity at ASDEX upgrade[END_REF][START_REF] Angioni | A comparison of the impact of central ECRH and central ICRH on the tungsten behaviour in ASDEX upgrade H-mode plasmas[END_REF][START_REF] Shen | Suppression of molybdenum impurity accumulation in the core using on-axis electron cyclotron resonance heating in EAST[END_REF]. ECRH is often considered to be more e cient than ICRH since it causes an increase of heat and main di usion leading to a attened main ion density pro le, which can reduce neoclassical inward convection.

Conclusion

In this chapter, we have studied the poloidal asymmetry of impurity density in toroidally rotating plasma and its impact on neoclassical and turbulent transports. For the neoclassical impurity transport, the rotation-induced poloidal asymmetry tends to increase inward impurity particle ux leading to deleterious core W accumulation, while turbulent transport is directed outwards as toroidal rotation increases due to the roto-di usion term. For our parameters, it was found that the rst (neoclassical) e ect dominates. Within the framework of non-linear gyrokinetic simulations, numerical results obtained with GYSELA are in qualitative agreement with the analytical description.

The generic inward impurity transport driven by toroidal rotation, mainly from dominant neoclassical transport, is an undesirable result for future fusion scenarios. To avoid this, high power ICRH heating near the magnetic axis has been proposed in order to mitigate W accumulation, since it can reduce impurity density asymmetry but also increase the outward turbulent ux by increasing main ions temperature pro le. This aspect will be soon tested within the GYSELA framework using an anisotropic heating source for the main ions.

Chapitre 6 Conclusion

In nuclear fusion, the presence of inevitable impurities is closely related to the operational performance of the fusion reactor. A thorough understanding of impurity transport is, therefore, crucial for the achievement of the project ITER. During a plasma discharge, various impurities can exist at the same time, for instance, Helium (He) ash as a reactant of DT fusion reactions, Argon (Ar) or Nitrogen (N) which can be voluntarily injected to reduce the heat ux loaded on the divertor and heavy impurities like tungsten (W) from the plasma-wall interactions. It is now a well-known fact that a high concentration of heavy (high Z) impurities might lead to complete disruptions of burning plasma by radiating a high fraction of energy, while light (low Z) impurities lead to a dilution of the main DT fuel.

There are three di erent mechanisms by which impurity transport is generated ; (i) turbulence, (ii) collisional e ects, and (iii) MHD instabilities. During this thesis, we restricted ourselves to the electrostatic cases, thereby only the rst two main causes are investigated both numerically and theoretically. To this end, two gyrokinetic codes-(i) GYSELA-5D developed in CEA-IRFM and (ii) TERESA-4D developed in IJL-have been used to study impurity transport. Despite the dimensional reduction via gyrokinetic approach from 6D to 5D, a high computational cost of GYSELA precludes a certain type of study, such as the investigation of parametric dependencies. For this reason, the bounce-averaged gyrokinetic code TERESA has been used to complete the studies.

In general, turbulent impurity transport can be separated into two di erent parts : A di usive part and a convective part. For a peaked density pro le in tokamaks, the diffusive part always drives outward impurity transport. The convective part, on the other hand, can be directed inward leading to core accumulation according to the nature of the background turbulence and the pro le of magnetic curvature. A detailed investigation of the convective part can be ful lled through the parametric dependencies of each term, such as thermo-di usion and curvature pinch. For this purpose, the bounceaveraged gyrokinetic code TERESA has been used for this study in case of trapped particle turbulence, and an analytical approach, namely quasi-linear theory, has been 99 compared with numerical results giving a qualitatively good agreement between them. It was found that thermo-di usion is directed inward when trapped electron mode (TEM) turbulence is dominant. In addition to that, a negative sign of curvature pinch has been reproduced by modifying the sign of the precession drift. Although negative magnetic shear could be favorable for the turbulent convective term, low or negative magnetic shear in the core is found to suppress turbulence by triggering transport barriers, which consequently leads to core impurity accumulation due to neoclassical transport.

The overall results obtained using TERESA suggest that, rst of all, a proper use of a heating system, such as ICRH, might be useful to induce TEM to ITG turbulence to expel impurities from the core, and also a formation of a negative magnetic shear near the magnetic axis might be bene cial, from the point of view of the turbulent convective term, to prevent impurities from accumulating in the core region. For a more complete and predictive model, however, it would be essential to take into account both neoclassical and turbulent transport e ects.

The main transport channel of various impurities di ers from species to species according to their mass and charge. For light impurities, such as Helium (He), particles are transported mainly by turbulence while the transport of heavy impurities, such as tungsten (W), are often dominated by neoclassical e ects. The full-F gyrokinetic code GYSELA, implemented with a linearized collision operator, allows us to study both turbulent and neoclassical impurity transport in a self-consistent manner. Recently, thanks to the development of diagnostics in tokamaks, poloidal asymmetry of impurity density has renewed interest in impurity transport since it can substantially modify the neoclassical transport. In the absence of external heating systems, poloidal asymmetry is mainly driven by background turbulence and parallel friction. Nonlinear simulations performed in this manuscript have con rmed that GYSELA reproduces a strong turbulent ux for helium while tungsten is primarily dominated by neoclassical ux, which corresponds well to our analytical predictions. Moreover, poloidal asymmetry generated by turbulence was found to be stronger for heavy impurities due to a Boltzmann type response, which leads to a strong modi cation of the neoclassical transport, especially in the P rsch-Schlüter (PS) regime. The overall results agree well with our analytical descriptions, recovering the coe cient of thermal screening factor H GYS ∼ -0.5 close to the theoretical prediction except in the region where strong poloidal asymmetries are generated by the turbulence.

Another physical mechanism for poloidally non-uniform impurity density is the toroidal rotation. During tokamak discharge, toroidal rotation can be generated intrinsically or externally by Neutral Beam Injection (NBI) heating. The centrifugal force, in connection with toroidal rotation, pushes heavy impurities toward the wall, generating a strong 'in-out' poloidal asymmetry. From an analytical standpoint, such enhanced poloidal asymmetry can increase or reduce neoclassical transport by an order of magnitude and even change its direction. Within the framework of nonlinear gyrokinetic simulations, adjustable source terms have been applied to investigate the e ects of a strong toroidal rotation. Indeed, nonlinear simulations GYSELA performed with an additional toroidal momentum source showed that, in the case of a strong rotation, the neoclassical ux tends to increase due to an enhanced poloidal asymmetry while turbulent ux is strongly a ected by the roto-di usion, leading to an outward ux in the case of an ion temperature gradient (ITG) turbulence (Figure 6.1). This term becomes dominant, especially for heavy impurities, due to its proportionality to A/Z where A, Z is impurity mass and charge respectively.

In this manuscript, we have studied both turbulent and neoclassical transport in di erent con gurations. The overall results obtained throughout this manuscript imply that heavy impurities are prone to be accumulated in the central region and are very sensitive to external conditions. Fortunately, an additional use of central Ion Cyclotron Resonance Heating (ICRH) / Electron Cyclotron Resonance Heating (ECRH) had shown to avoid impurity accumulation e ciently. Although ICRH can mitigate W core accumulation by reducing the poloidal asymmetry and by increasing the outward turbulent ux, ECRH is often considered to be more e cient since an increased heat and main ion di usion atten main ion density pro le, which, in turn, strongly reduces neoclassical inward convection for impurities. Applying the additional ICRH / ECRH heating system for impurities in GYSELA remains one of the future works.

Although the current knowledge of impurity transport is comparable to those of ions, there are still a lot of open questions. For instance, in the framework of GYSELA, numerical investigations of the e ects of the ICRH heating system, kinetic electrons, and impurity transport in elongated tokamak geometry seem achievable in a foreseeable future, and for instance, the study of impurity transport across a transport barrier (pedestal, steep gradients) is being investigated by means of GYSELA (PhD thesis of G. Lo-Casio). In the long run, an integrated modelling, capable of interpreting experimental observation as well as validating plasma theory against experiments, will be an essential tool of tokamak fusion research. For a current state-of-the-art numerical code, derived from the rst principle description, it is out of reach to take into account all the physical processes, such as core transport, edge physics, MHD stability, external heating, and current drive. It is for this reason that a more comprehensive integrated modelling will be key for a reliable prediction of the performance in tokamaks. Annexe A

Quasi-linear impurity transport by uid approach

In this appendix, the impurity particle ux Γ Z is derived by a uid approach of the quasi-linear theory. This approach gives the same results as those presented in Sec.3.2.1 and the presence of di usive and convective parts are expressed below. First, the bounce-averaged distribution function F s satis es the Vlasov equation Eq.(2.47) :

∂F s ∂t -[J φ, F s ] + EΩ d Z s ∂F s ∂α = 0 (A.1)
The rst and second moments of Eq.(A.1) then lead to the time evolution of the density and pressure terms respectively. Here, φ represents the gyro-bounce averaged potential. This notation will be omitted from now for clarity. Using the linearization in the Fourier space φ = l φ l (ψ, t)e ilα and Ñs = l N l (ψ, t)e ilα , Eq.(A.2 A.3) becomes :

∂N

-iωN l + il 3 2 Ω d Z s (T l N eq + N l T eq ) -ilφ l ∂N eq ∂ψ = 0 (A.4) (-iωT eq + il 5 2 Ω d Z s T 2 eq )N l + (-iωN eq + il 5Ω d Z s N eq T eq )T l -ilφ l ∂p eq ∂ψ = 0 (A.5)

where N eq , T eq and p eq are the equilibrium density, temperature and pressure respectively.

103

where we put N l = Neq (1+Ω d ψ) 3/2 and T l = Teq (1+Ω d ψ) for the perturbed density/temperature, so κ neq , κ Teq was replaced by κ neq = κ n + 3Ω d 2(1+Ω d ψ) and κ Teq = κ T + Ω d (1+Ω d ψ) . By putting ω = ω r + iγ in Eq.(A.11) to take only real part of above equation, and by binding all the terms with respect to κ n and κ T separately, the above impurity particle ux Γ z can be simpli ed as follows : .12) where the coe cients D z , C T and C P represents impurity turbulent di usion, thermodi usion and curvature pinch respectively. In steady state (∂ t = 0) and the axisymmetric system (∂ ϕ = 0), Eq.(B.9) can be rewritten

Γ z N z = D z κ Nz + C T κ Tz + C P (A
Γ neo • ∇ψ ψ = - I ZeB 2 (N z e • ∇φ + B • ∇ • Π) ψ (B.12)
It is still possible to decompose further Eq.(B.12) into 2 parts Γ neo = Γ BP + Γ PS by using the identity [START_REF] Helander | Collisional Transport in Magnetized Plasmas[END_REF] 1

B = 1 B - B B 2 ψ + B B 2 ψ (B.13)
Finally, we obtain the impurity BP, PS ux respectively. (B.17)

Γ BP • ∇ψ ψ = - I Ze B • ∇ • Π ψ B 2 ψ (B.

B.2 Parallel frictional force

In case of heavy impurities in the trace limit, the frictional force F zi can be written as a linear function of the parallel velocity V s and the parallel heat ux q s [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]. .18) where x ab = v T b /v T a It is worthwhile relating Eq.(4.10) with the frictional force F zi in order to show how the poloidal density asymmetry is linked to the impurity ux (For more complete derivation, see Appendix of Donnel et al. (2019a)).

F zi = -N z m z ν zi V z -V i - 3 5 1 1 + x 2 zi q z N z T z + 3 5 1 1 + x 2 iz q i N i T i (B
F zi = m z ν zi   -N z T i eB I L ψ + B(N z u -K z )   (B.19)
where 1/L ψ = 1/L ψ,i + 1/L ψ,z with 1/L ψ,i = ∂ ψ ln P i -3 2 ∂ ψ ln T i and 1/L ψ,z = -1 T i ZNz ∂ ψ P ⊥z and u is the value related to the poloidal velocity of the main ions.

Annexe C

Impurity density distribution from the parallel dynamics

In the case of the large aspect ratio ( 1) and for a circular cross section, the magnetic eld b 9 , the impurity density n and the Mach number M can be expanded in :

b 2 = 1 -2 cos θ + O( 2 ) (C.1) N = 1 + N c cos θ + N s sin θ + O( 2 ) (C.2) M 2 = M 2 0 (1 + 2 cos θ) + O( 2 ) (C.3)
By using the quasi-neutrality constraint and the modi ed Mach number Eq.( 5.12), the initial parallel impurity transport Eq.(5.9) can be expressed as [START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF] :

(1 + αN ) ∂N ∂ϑ = g(n -b 2 + γ(N -nb 2 )b 2 ) + N ∂M 2 ∂ϑ -N ∂M 2 ∂ϑ b 2 (C.4)
where we introduced α = Nz Z 2 T 0 2N 0 T i with 2N 0 /T 0 = N e0 /T e + N i0 /T i In order to nd the expression for 'up-down' and 'in-out' asymmetric parameters derived in Eq. (5.10, 5.11) where we have neglected all the higher order terms 2 , n c , n s ∼ O( 2 ). (1 + α) 2 + (1 + γ) 2 g 2 ('up-down') (C.9)

In the trace limit of impurities (N z Z 2 1), these equations can be further simpli ed since α 1 :

N c = 2 M 2 0 -(1 + γ)g 2 1 + (1 + γ) 2 g 2 ('in-out') (C.10) N s = 2 g 1 + (1 + γ)M 2 0 1 + (1 + γ) 2 g 2 ('up-down') (C.11)
Note that the toroidal rotation causes the impurities to accumulate on the outside of the ux surface, which will enhances the neoclassical impurity transport.

  fusion : basic concepts . . . . . . . . . . . . . . . . . . . . . . . 1.2 Magnetic con nement : Tokamaks . . . . . . . . . . . . . . . . . . . . . 1.2.1 Magnetic eld con guration . . . . . . . . . . . . . . . . . . . . 1.2.2 Circulating & trapped particles . . . . . . . . . . . . . . . . . . 1.3 Particle transport in tokamaks . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Turbulent transport . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Neoclassical transport . . . . . . . . . . . . . . . . . . . . . . . 1.4 Impurities in tokamaks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Impurity accumulation . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Poloidal asymmetry of impurity density . . . . . . . . . . . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Gyrokinetic model for impurity transport 2.1 Plasma models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Kinetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Gyrokinetic model . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical features of gyrokinetic codes . . . . . . . . . . . . . . . . . . 2.3 GYSELA-5D : Gyrokinetic model . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Gyrokinetic Vlasov equation . . . . . . . . . . . . . . . . . . . . 2.3.2 Quasi-neutrality equation . . . . . . . . . . . . . . . . . . . . . iii v 5.2.2 Poloidal asymmetries generated by toroidal rotation . . . . . . 5.2.3 Neoclassical and turbulent impurity transport in rotating plasma 5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Enhanced poloidal asymmetry by toroidal rotation . . . . . . . 5.3.2 Impurity particle ux . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion A Quasi-linear impurity transport by uid approach B Perpendicular ux and CGL pressure tensor B.1 Magnetic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Parallel frictional force . . . . . . . . . . . . . . . . . . . . . . . . . . .

F 2 -

 2 Le tokamak ITER 2 .

  1.3.1 Turbulent transport . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Neoclassical transport . . . . . . . . . . . . . . . . . . . . . 1.4 Impurities in tokamaks . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Impurity accumulation . . . . . . . . . . . . . . . . . . . . . 1.4.2 Poloidal asymmetry of impurity density . . . . . . . . . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F 1. 5 -

 5 (Left) Poloidal section of a tokamak (right) the basic mechanism of the drift kinetic type instability (Del Sarto and Ghizzo, 2017).

F

  Left) Nitrogen impurity seeding in the COMPASS tokamak[START_REF] Komm | Divertor impurity seeding experiments at the compass tokamak[END_REF] (right) Tungsten erosion during plasma discharge in the EAST tokamak 6 . 6. https://ceramics.org/ceramic-tech-today/nuclear-2/understanding-tungsten-erosion-intokamak-walls F 1.8 -Ignition curves in the presence of di erent W concentrations. When concentration goes high, the surface of achievable ignition condition diminishes[START_REF] Pütterich | Calculation and experimental test of the cooling factor of tungsten[END_REF].

Contents 2 . 1

 21 Plasma models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Kinetic model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Gyrokinetic model . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical features of gyrokinetic codes . . . . . . . . . . . . . 2.3 GYSELA-5D : Gyrokinetic model . . . . . . . . . . . . . . . . . . 2.3.1 Gyrokinetic Vlasov equation . . . . . . . . . . . . . . . . . . 2.3.2 Quasi-neutrality equation . . . . . . . . . . . . . . . . . . . 2.3.3 Collisional operator . . . . . . . . . . . . . . . . . . . . . . 2.3.4 External sources . . . . . . . . . . . . . . . . . . . . . . . . 2.4 TERESA-4D : Gyro-bounce kinetic model . . . . . . . . . . . . 2.4.1 Action and angle variables . . . . . . . . . . . . . . . . . . . 2.4.2 Gyro-bounce Vlasov equation . . . . . . . . . . . . . . . . . 2.4.3 Quasi-neutrality equation . . . . . . . . . . . . . . . . . . .

•

  Magnetic eld is prescribed externally Electromagnetic • Ampère's law and self-generated currents are taken into account Global • The whole plasma in toroidal geometry is treated numerically Local • Local region around the magnetic eld, i.e. ux tube, is treated numerically.

  F 2.3 -Schematic view of GYSELA-5D code.

F 2. 5 -

 5 Schematic view of TERESA-4D code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Impurity Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.1 Quasi-linear impurity transport . . . . . . . . . . . . . . . . 49 3.2.2 Validity of the quasi-linear approximations . . . . . . . . . 52 3.2.3 Pinch velocity from nonlinear numerical simulations . . . . 53 3.3 Parametric dependencies of impurity pinch . . . . . . . . . . . 56 3.3.1 Thermo-di usion . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.2 Curvature pinch . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 47

F

  (a) Time evolution of the 10 most unstable φ n modes. (b) Cumulative impurity particle ux Γ z as a function of time. The shaded region represents the beginning of the turbulent regime and this region is chosen for the study of impurity pinch.

F 3. 4 -

 4 Impurity density pro les N z with di erent initial values κ init N z = ±0.3, ±0.5. Initial density pro les at t = 0 (left) and measured density pro les at t = 2ω -1 0 (right) are shown. Both radial pro les are measured between ψ = 0.2 and ψ = 0.8, and normalized with a reference density N 0 at ψ = 0. F 3.5 -Example of linear relationship between density gradient and impurity particle ux. The value of y-interception corresponds to pinch velocity V z .

  4) :Five di erent values of impurity temperature pro les κ T z = 0.0, ±0.1modes have been tested to verify the direction change of thermo-di usion.

  Figure 3.7(a) shows the relation between the pinch velocity V z and F 3.7 -Thermo-di usion V z (left) and normalized peaking factor (-RV z /D z ) (right) plotted against κ T z , for both TEM and TIM turbulent cases.

F 3 .

 3 11 -Γ z /N z plotted against κ N z , with di erent values of the ad-hoc coe cient α in H eq = E(1 + αeΩ d ψ). The negative value of α represents the strong negative magnetic shear, which drives curvature pinch outwardly. It should be noted that the thermo-di usion term is arti cially suppressed by imposing κ T z = 0. F 3.12 -Pinch velocity V z plotted against Ω d . With α positive, the curvature pinch is directed inward and proportional to Ω d as we expected from Eq.(3.9).

F 4 .

 4 1 -Radial pro le of collisionality for D, He, Ar and W.

F 4 .

 4 2 -Impurity uxes of di erent impurities in presence of poloidal asymmetries obtained by numerical simulations GYSELA. Total impurity ux (black solid line) is separated into a turbulent part (red dashed line) and a neoclassical part (blue dashed line).

F 4 .

 4 3 -Time evolution of impurity density in the core (left) and at the edge (right).

F 4 . 4 -

 44 Comparison of neoclassical ux between numerical simulations GYSELA (blue dashed line) from Eq.(4.21) and semi-analytical model (grey solid line) from Eq.(4.14) and Eq.(

F 4 . 8 -

 48 The radial pro le of geometrical factors P A (left), P B (right) without any external heating system. is depicted on Figure4.9. The theoretical prediction of neoclassical uxes in two different cases : (i) with the thermal screening without any poloidal assymetry H sym = -1

F

F 5 . 1 -

 51 Raidal pro le of GYSELA sources.

F 5 . 2 -

 52 Radial toroidal velocity pro le averaged over the ux surface for deuterium (D) and tungsten (W) w and w/o momentum source.

F 5 .

 5 3 -Time averaged 2D poloidal section of perturbed W density. (Left) w/o momentum source (center) case with tungsten Mach ∼ 1 and (right) case with tungsten Mach ∼ 2.

F 5 . 4 -

 54 (Left) Poloidal asymmetry obtained from nonlinear simulations GYSELA. (Center) Reconstructed impurity density from Eq.(5.7). (Right) Reconstructed impurity density from Eq.(5.10, 5.11).

F 5 . 5 -

 55 Neoclassical W-ux as a function of the δ='in-out', and ∆='up-down' asymmetry parameters. The reconstructed density ÑW = δ cos θ + ∆ sin θ is used to calculate the neoclassical ux in Eq.(5.15) while other terms, i.e., N i , T i , T W , ν zi are taken from the GYSELA simulations. Flux is normalized to the symmetric case with δ = ∆ = 0.

F 5 .

 5 6 -W impurity ux averaged over the overall radial position between ρ = 0.2 and ρ = 0.8 with di erent magnitudes of the momentum source. (Left) neoclassical ux and (right) turbulent ux.

F 6. 1 -

 1 Di erent directions of convective terms in both turbulent and neoclassical impurity ux.

  using the following identities ∇(AB) = A∇B + B∇A and ∇ radial particle ux can be written as follows :Γ ⊥z • ∇ψ ψ =   B ZeB 2 × (N z e∇φ + ∇ • Π) of the magnetic eld B = I(ψ)∇ϕ + ∇ϕ × ∇ψ, one can nd the useful relationship

  The magnetic curvature is de ned κ = (b • ∇)b. By using the vectorial identityA × (∇ × B) = A • ∇B -(A • ∇)B, we can rewrite the magnetic curvature : κ = (b • ∇)b = -b × (∇ × b) = -b × (∇ × B/B) = -b × (∇( 1 B ) × B)b × (∇ × B)/B = -b × (b × ∇ ln B) + µ 0 J × B/B 2 = 1 B [∇b(b • ∇)]B + µ 0 J × B/B 2 (B.16) De ning ∇ ⊥ = ∇b(b • ∇) = -b × (b × ∇)as the gradient perpendicular to B, we can then rewrite the above equation asκ = ∇ ⊥ ln B + µ 0 J × B B 2 = ∇ ⊥ B B + µ 0 ∇ ⊥ P B 2

  , we insert Eq.(C.1 C.2 C.3) into Eq.(C.4), which writes :cos θ gN c (1 + γ) -N s (1 + α) + 2g + sin θ gN s (1 + γ) + N c (1 + α) -

9.

  The normalized magnetic eld b = 1/(1 -cos θ) 111 Annexe C. Impurity density distribution from the parallel dynamicsThis last equation should be satis ed with all the di erent angles θ, therefore each coe cient of cos θ and sin θ should be equal to zero.gN c (1 + γ) -N s (1 + α) + 2g = 0 (C.6) gN s (1 + γ) + N c (1 + α) -2 M 2 0 = 0 (C.7)By arranging both Eq.(C.6, C.7), we can nally obtain the 'up-down' asymmetry term N s and 'in-out' asymmetry term N c

https://www.energy.gov/science/doe-explainsnuclear-fusion-reactions

https://www.iter.org/

The atomic nucleus is composed of protons and neutrons. When these elements are assembled together, a nuclear force holds them together. This force is associated with a binding energy and this binding energy varies from element to element.

Part of this chapter can also be found in[START_REF] Lim | Impurity pinch generated by trapped particle driven turbulence[END_REF]

(red solid line) and (ii) without the thermal screening H sym = 0 (green solid line) are compared with the neoclassical ux obtained from GYSELA simulation (blue dashed line). The e ect of H sym on impurity transport can be easily understood by looking at the relative magnitude of each ux. When the thermal screening e ect is removed H sym = 0, the neoclassical ux shows a strong inward ux, leading to the strong core accumulation, while the accumulation is mitigated in the case of thermal screening (H sym = -1 2 ), as expected. In overall radius, GYSELA reproduces a value close to the theoretical prediction except in the region 0.1 < ρ < 0.5. The main reasons of such di erence can be summarized as follows. First, the e ect of poloidal asymmetries driven by turbulence can reduce the thermal screening e ect through the geometrical factor P B in Eq.(4.23). Second, the term u in Eq.(4.15), linked with the poloidal rotation of the main ion, is calculated in the neoclassical case only, thus the value of k neo in Eq.(4.16) di ers from the neoclassical prediction (k neo ∼ 1.17) due to the turbulence.

After some calculation, Eq.(A.4) and Eq.(A.5) can be rewritten :

(ω -ω T )T eq N l + lφ l ∂p eq ∂ψ + (ω -2ω T )N eq T l = 0

Here, we replaced 5lΩ d T eq /2Z s by ω T . By introducing

the logarithmic density and temperature pro le respectively with ψ the dimensionless radial coordinate (dψ = -R 0 B θ dr), further arrangement can be recast as :

Combining these equations gives :

(A.8) Averaging Eq.(A.2) over α gives the impurity particle ux Γ z according to the continuity equation.

Now, the particle ux Γ Z can be written :

Knowing the relation between the perturbed potential and the density, we insert Eq.(A.8) into Eq.(A.10). After some arrangement, the particle ux in the TERESA model cannally be written as follows :

Annexe B

Perpendicular ux and CGL pressure tensor

The perpendicular part of particle ux consists of three di erent drift terms :

where the magnetic drift consists of the curvature and ∇B drifts.