Prof Ian Jonathan Chapman

Graves

Maurizio Patrick T Marina

Prof Cassio Amador

Dr Jonathan Citrin

Prof Olivier Sauter

Dr Marco Sertoli

Tomas Odstrcil

Alexander Lessig

Le Mathieu

Victor Capitaine

Neel

Dr Jean-Marie Thiebaud

Hélène Lebrun

김상용 집사님

김미정 권사님

이윤수 집사님

최만식 집사님

임은희 권사님

한충희 장로님

김경희 권사님

백채훈 집사님

김신재 집사님

정욱 집사님

Deo Soli

Glori

alone, for You led and will lead my path throughout my life

Although only my name gures on the front page, I do admit that this manuscript is a fruit of a full

Fusion energy on Earth

The worldwide energy consumption has been increasing due to continuous industrialization and way of life sophistication. Statistics show that the vast majority of the world's energy is currently generated from combustion of fossil fuels (Figure 1.1). Indeed, abundance of resource and relative technological simplicity make this process an attractive energy source.

However, fossil fuel combustion produces large quantities of greenhouse gases such as carbon dioxide, resulting in acid rain and contributing to global warming. Drastic meteorological events related to climate change have been causing serious human and material damage regardless of country. Moreover, experts make more pessimistic predictions for the future.

This sense of urgency has led the government leaders around the world to agree and engage for actions to reduce greenhouse gas emission at the COP21 conference 1 .

Active research is ongoing to develop alternative energy sources. Renewable energy sources such as solar, wind and biofuel promise to generate power in a sustainable manner. However, these alternatives are currently far from providing large scale supply to replace fossil fuel consumption. Nuclear energy is interesting in terms of specic energy, which is the energy stored in a system per unit mass. Indeed, the specic energy for conventional nuclear fuels is about a million times larger than other energy sources relying on chemical reactions.

Nowadays, nuclear energy is generated by ssion which consists in breaking heavy atoms into lighter by-products. Despite its eciency, issues on radioactive waste and safety are still raised, combined with geopolitical concerns related to the non-proliferation of nuclear weapons. Eorts are concentrated on seeking plausible solutions for these issues.

Another way to generate nuclear energy is by fusing light nuclei to form a heavier atom.

Fusion reaction, by which stars are fuelled, is common in the universe. It is a very attractive option due to its inherent safety and the abundance of the fuels on Earth. For these reasons, plasma physicists have been focusing on studying the feasibility of sustained fusion reaction on Earth, in order to supply a signicant portion of the energy consumption. The rst magnetic connement device was built in the former USSR at the end of 1950s. An international collaboration is ongoing around the ITER project which is currently under construction at Cadarache (France). Seven member entities are participating in this project -China, the EU, India, Japan, Korea, Russia and the USA. Commercial fusion reactors are expected probably for the next century.

In this section, general principles of fusion are briey presented in Section 1.1.1. Among dierent fusion devices which have been developed, tokamaks are currently the most advanced concept in terms of performance. Section 1.1.2 is dedicated to a description of a tokamak, as this thesis deals only with the physics of tokamaks.

General principles

For a nucleus with A nucleons (Z protons and N neutrons, respectively noted p and n), its mass m is smaller than the mass of its constituents Zm p + N m n . The binding energy where c is the speed of light in vacuum, represents the energy which would be released if a nucleus were formed directly from A independent nucleons. If the energy balance of a nuclear reaction

∆E ≡ k∈products B(A k , Z k ) - k∈reactants B(A k , Z k ) (1.2)
is positive (i.e. if the reaction is exoenergetic), the energy can be extracted. Figure 1.2 shows that mass is converted into energy when light nuclei fuse together, just as it occurs when splitting heavy nuclei in ssion.

Choice of D-T fuel

For fusion, the reactions which convert hydrogen and its isotopes, deuterium and tritium, to helium have the greatest energy gain. Some examples of fusion reactions which could potentially be used as the basis for an energy source on Earth are : 3) to (1.6) as function of energy [START_REF] Huba | NRL plasma Formulary[END_REF].

the order of MeV. However, the temperature required for fusion reactions is roughly 10 keV ( 10 8 Kelvin), thanks to interactions of quantum nature such as the tunnelling eect. This far exceeds the ionization energies of hydrogen and the lighter elements which are of the order 10 eV. This means that matter must be in a plasma state.

The relevant quantity at microscopic scales to quantify the probability of a reaction is the cross section. It represents the eective surface area for interaction. Among the possible fusion reactions on Earth, the D-T fusion reaction (1.3) has by far the largest cross section at relatively low energies (Figure 1.3). As a potential source of energy, D-T fusion is attractive since the reaction generates helium (α-particle) as by-product which is a chemically inert gas with no threat to the environment. Also, deuterium is naturally abundant in nature. It is known that a cubic meter of sea-water contains more than 30 g of deuterium. However, tritium is extremely rare in nature and radioactive with a half-life time of approximatively 12.5 years. It can be produced by neutron bombardment of lithium, which is abundant on Earth, via the reactions 6 3 Li + n --→ 4 2 He + 3 1 T ∆E = 4.8 MeV

(1.7)

7 3 Li + n --→ 4 2 He + 3 1 T + n ∆E = -2.5 MeV (1.8)
Therefore, it is planned to produce tritium in situ from lithium covering around the vacuum vessel and neutrons from the D-T reaction. However, the breeding reaction is endoenergetic with the common isotope of lithium 7

3 Li, of which the abundance is 92.5% compared to 7.5% of 6 3 Li. Still, 7 3 Li can be used with beryllium which has a role of neutron multiplier.

In addition, recent studies show that a lithium wall in a fusion device improves plasma connement [START_REF] Krasheninnikov | On lithium walls and the performance of magnetic fusion devices[END_REF][START_REF] Kugel | The eect of lithium surface coatings on plasma performance in the National Spherical Torus Experiment[END_REF]. In terms of fuel abundance, there is enough deuterium and lithium on Earth to provide energy by fusion for several thousands of years [START_REF] Cowley | The quest for fusion power[END_REF].

1.1. FUSION ENERGY ON EARTH 5

Connement methods

Energy losses (e.g. due to radiation, escaping particles, ...) must be balanced by depositing the produced energy back into the plasma. The α-particle formed by the D-T reaction carries away 3.5 MeV, which is 20% of the energy generated. Helium heats the plasma by redistributing its energy through collisions with cooler particles. The rest of the energy cannot be recovered for the fusion reaction as plasmas are transparent to neutrons. Instead, neutrons are slowed down in the blanket where their kinetic energy is transformed into heat energy and collected by the water coolant. In a fusion power plant, this energy will be used for electrical power production. Ideally, the fusion reactions must be self-sustained thanks to the plasma heating by α-particles. This state is called ignition. In practice, a ratio of the fusion and injected power of about 30 is considered as sucient for a reactor.

The quality of the connement is quantied by the connement time τ E which measures the characteristic plasma energy decay time. It is dened as a ratio between the thermal energy content W th and the power loss P loss τ E ≡ W th P loss

(1.9)

For D-T fuel, the condition for ignition is summarized in the Lawson criterion [START_REF] Lawson | Some criteria for a power producing thermonuclear reactor[END_REF][ [START_REF] Wesson | Tokamaks[END_REF] N T τ E 3 × 10 21 keV.m -3 .s

(1.10)

where N is the particle density of the D-T fuel, T is the plasma ion temperature and τ E is the energy connement time. The connement time is dened as the ratio of stored energy in plasma to the power loss. The product N T τ E is called `triple product'. This condition on the triple product gives three possibilities to trigger thermonuclear fusion reactions.

• Gravitational connement has been the most natural connement method since the existence of the universe. Stars give good examples of gravitationally conned thermonuclear reactors. Conditions in the cores are such that all three parameters N , T and τ E have sucient values to satisfy the ignition criteria. Meanwhile, the rst step of the chain reaction in the Sun involves protons. Two protons fuse to form 2

2 He (or diproton), which then becomes 2 1 D through a beta-plus decay. This rst step has an extremely low cross section because one of the proton beta-plus decays into a neutron through weak interaction, and so is extremely slow. Otherwise, the Sun would have burnt its hydrogen much too fast. Some of the stars are even hot and dense enough to fuse elements much heavier than hydrogen isotopes. Nevertheless, it is impossible to perform gravitational connement in Earth environment. Hence, other solutions are explored by a trade-o between the three parameters N , T and τ E .

• The idea of inertial connement to compress a fuel target containing deuterium and tritium, so that temperature and density would be sucient to reach ignition, even if the connement time is very short (τ E ∼ 100 ps). The most advanced technology uses high-power lasers to try and compress the D-T target as quickly and homogeneously as possible. The main inertial fusion facilities in the world are the National Ignition Facility (USA) and the Laser Mégajoule (France) which is under construction. The main challenge of inertial connement fusion is to compress the target homogeneously.

Indeed, uid instabilities such as the Rayleigh-Taylor instability would occur if a direct impact was made by a nite number of multiplied laser beams. The idea to bypass this diculty is to insert the D-T fuel capsule inside a hohlraum (Figure 1.4). Technology on the homogeneous compression of target is related to the principle of thermonuclear bomb. Such military-level classication prevents the access to this process to most countries.

• So far, the most advanced concept for civil use is magnetic connement. It aims at maximising the energy connement time and plasma temperature at the expense of lower densities (N 10 20 m -3 ). Quasi-continuous operational use and intrinsic safety are among the advantages of the magnetic connement. Hence, this method is currently regarded as the best prospect for near-term fusion reactors and will be detailed in Section 1.1.2.

Magnetic connement

Twisted magnetic eld lines

Energy and charged particles of the plasma are conned by strong magnetic elds in a magnetic connement device. In this way, particles of mass m a and charge e a gyrate around Figure 1.5: Toroidal coordinate system (r,θ,ϕ). The major radius of this torus is R 0 .

the magnetic eld line at the cyclotron pulsation ω c = |e a |B/m a and with a Larmor radius ρ L = v ⊥ /ω c , where B is the magnetic eld strength, v ⊥ the velocity transverse to the magnetic eld.

The simplest magnetic conguration is a solenoid. Magnetic coils around a long cylinder produce a magnetic eld parallel to the cylinder axis. Such a magnetic conguration prevents ions and electrons from being lost radially, but not from being lost at the ends of the solenoid.

One solution to this problem is to use magnetic mirrors at the ends of the cylinder, in order to trap particles inside the linear device. In the past, this solution has never been demonstrated

to be eective enough. The other way is to eliminate the ends by bending the cylinder to form a torus. In this section, we will focus on toroidal closed congurations (Figure 1.5).

A containment device with a simple toroidal magnetic eld would appear to be sucient to conne a plasma. Nonetheless, closer consideration of the particle orbits reveals that the connement is poor in this case. In fact, the toroidal magnetic eld B t is proportional to the inverse of the major radius 1/R. Therefore, the Larmor radii of the particles dier as they gyrate, being always smaller in the region of higher eld, i.e. towards the centre of the torus.

This causes electrons and ions to drift along B × ∇B in opposite directions. The resulting vertical charge separation then produces an electric eld E. The plasma drifts horizontally out of the torus in the horizontal E × B direction, so the connement is poor [START_REF] Wesson | Tokamaks[END_REF].

The eects due to drifts can be attenuated with an additional poloidal magnetic eld B p , such that the eld lines are bent helically along the plasma column. As a consequence, the B × ∇B drift changes direction as particles move along the magnetic eld lines alternately on the low and high eld side. Consequently, the net drift eect is averaged to zero. The magnetic eld is twisted by dierent ways.

• The stellarators use shaped toroidal coils so that the two necessary components of the magnetic elds are generated. Large Helical Device (LHD) in Toki (Japan) and Wendelstein-7X in Greifswald (Germany) are currently the largest stellarators.

• In tokamaks, the poloidal magnetic eld is induced by a toroidal current in the range of a mega-Ampère (MA). Only tokamak congurations will be considered in this thesis.

Presentation of tokamaks

Tokamaks have been the predominant research line worldwide for magnetically conned fusion. It is an axisymmetric device using helical magnetic elds to conne a plasma in a torus shaped vessel. The conguration of a tokamak is shown in Figure 1.6-a. The magenta coloured doughnut shape represents the plasma conned in the vacuum vessel. The plasma chamber is surrounded by the toroidal eld coils by which the toroidal magnetic eld is produced. The poloidal magnetic eld is created by the plasma current generated by the central solenoid and poloidal coils. The combination of the toroidal and poloidal elds results in helical eld lines. An additional eld produced by an imbalance of poloidal eld coils is used to control the vertical position and shape of the plasma. Recent tokamaks are equipped with superconducting coils to prevent ohmic losses.

The induced current provides a heat source to the plasma thanks to the Joule eect. This process is called ohmic heating. The ohmic heating power density is given by

P Ohm = ηJ 2 (1.11)
where η is the plasma resistivity and J is the plasma current density. Hence, the plasma is heated when increasing the plasma current. However, the plasma current (or its density) is limited by the design of the superconducting coils. Also, the plasma resistivity is a decreasing function of the electron temperature T e , with η ∝ T -3/2 e . Therefore, this method has limitations and additional means of heating must be implemented.

4 Image from [START_REF] Ongena | Magnetic-connement fusion[END_REF] 1.1. FUSION ENERGY ON EARTH 9

Plasmas can be heated by injecting high-energy beams of neutral atoms into the core plasma, which is called neutral beam injection (NBI). The injected atoms are typically hydrogen isotopes. The energy of neutral particles is directly transferred to charged plasma particles by collisions. The overall plasma temperature is thus increased. Additional energy can be provided to the plasma by means of electromagnetic waves. Plasma heating by waves coupled to the resonant frequencies of the ion species and electrons are respectively called ion cyclotron resonance heating (ICRH) and electron cyclotron resonance heating (ECRH).

These electromagnetic waves can also be used to drive current in the plasma. Depending on the frequency range, one talks about ion cyclotron current drive (ICCD), electron cyclotron current drive (ECCD) and lower hybrid current drive (LHCD).

Parameters of geometry and stability

A tokamak plasma is characterized by a number of quantities. Geometrical parameters are dened by the plasma inside the last closed ux surface (LCFS) as shown in Figure 1.6b. The major radius R 0 and the minor radius a of the plasma dene the aspect ratio R 0 /a.

In tokamaks, it can be as low as 1.

δ = R 0 -R X a (1.13)
with R X the position of the X-point. In a general case, plasma is up-down asymmetric. Thus, upper triangularity δ u and lower triangularity δ l are separately dened as shown in Figure 1.6-b. The poloidal shape of the vacuum vessel is a strong constraint on the plasma geometry.

As far as TCV (Swiss Plasma Center) is concerned, more exotic plasma shapes can be studied thanks to a very broad range of elongation and triangularity possible (e.g. [START_REF] Martynov | The stability of the ideal internal kink mode in realistic tokamak geometry[END_REF][START_REF] Marinoni | The eect of plasma triangularity on turbulent transport: modeling TCV experiments by linear and non-linear gyrokinetic simulations[END_REF]).

In tokamaks, the magnetic eld lines at equilibrium wind around nested toroidal surfaces, called magnetic surfaces. For a given magnetic surface, the safety factor q denotes the number of turns made in the toroidal direction for each turn in the poloidal direction. The formal

denition of q is q = 1 2π B • ∇ϕ B • ∇θ dθ (1.14)
The magnetic surface is resonant if q is a rational number, i.e. the magnetic eld lines are closed back on to themselves. Then, q = m/n for a mode with poloidal and toroidal mode numbers (m,n).

CHAPTER 1. INTRODUCTION

The parameter β represents the eciency of the plasma connement by the magnetic eld. It is dened as

β = P V B 2 0 /2µ 0 (1.15)
where P V = P dV/ dV is the kinetic pressure P = N T averaged in plasma volume, B 0 is the magnetic eld on the geometrical axis and µ 0 is the vacuum permeability. Similarly, the poloidal beta β p is dened by normalising the kinetic pressure by the poloidal magnetic eld B p

β p = P V B 2 p /2µ 0 (1.16)
Usually, β ∼ 10 -2 and β p ∼ 1 in tokamaks. The values of β for practical reactor are below the Troyon limit, since high-n instabilities such as ballooning modes develop at high-β [START_REF] Troyon | MHD-limits to plasma connement[END_REF].

1.2 Impurities in tokamak plasmas

Main issues related to impurities

In general, all particles other than fuels for fusion reaction are called impurities. Impurities in tokamak plasmas are inevitable. For D-T fuel, helium ashes are produced by fusion reactions. In addition, other impurities from the edge region pollute the plasma. Either they come from erosion by sputtering of material surfaces in contact with the plasma, or injected in the plasma intentionally in the form of gas pu and pellet. Especially, tokamak divertors are important sources of impurities as they are exposed to high heat uxes. Consequently, tokamak plasmas rapidly become polluted by impurities from dierent sources.

The fusion reaction rate is proportional to deuterium density times tritium density. However, the presence of impurities in the core dilutes fusion reactants, and thus decreases the probability of fusion reaction. Apart from fuel dilution, the power radiated by the impurities is a major concern. Indeed, impurities produced by plasma-wall interaction are heavier than hydrogen isotopes, and will be mostly metallic in ITER. More power is radiated out as heavy ions get closer to the plasma center.

Once impurities penetrate the plasma, they get ionized because of the high temperature.

Nevertheless, the ions are fully stripped only for atoms lighter than argon (Z Ar = 18) in the temperature range of typical tokamak plasmas. Heavier impurities, e.g. tungsten (Z W = 74), are present in partially stripped states. Non-fully stripped ions radiate very signicant amounts of power with line emission and radiative recombination. Even fully stripped ions radiate more than hydrogen isotopes through bremsstrahlung which scales with the square of the charge. Dierent radiation mechanisms are detailed in Ref. [START_REF] Vezinet | Soft X-ray measurements and analysis on tokamaks in view of real-time control[END_REF].

From these properties, impurities even in very small quantities aect the connement by cooling down the core plasma and can lead to radiative collapse [START_REF] Noda | A review of recent experiments on W and high Z materials as plasma-facing components in magnetic fusion devices[END_REF].

For tungsten, its concentration has to be maintained below -4 [Pütterich et al., 2010].

Therefore, impurity transport in the core plasma is an important matter of concern for fusion performance.

Eects of magnetohydrodynamic instabilities on impurities

A plasma can be described as a uid of charged particles immersed in magnetic elds.

This magnetized uid is the framework of magnetohydrodynamics (MHD). Numerous plasma parameters such as current, pressure gradient or beta can trigger MHD instabilities which degrade the plasma performance and limit the operational domain. They appear at resonant magnetic surfaces where values of q are rational. In the core region, sawteeth and neoclassical tearing modes (NTMs) associated with low mode numbers (m,n) appear.

Sawtooth oscillations

Sawteeth are relaxation oscillations of the core temperature and density proles [START_REF] Yamada | Investigation of magnetic reconnection during a sawtooth crash in a high-temperature tokamak plasma[END_REF]][Sabot et al., 2006]. They are triggered by the internal kink mode with (m, n) = (1, 1). A sawtooth period is characterized by two main phases : the `ramp' phase and the `collapse' phase. Looking at the temperature prole, the ramp phase is characterized by a steady rise of the core temperature, followed by a collapse phase, so-called crash, where a sudden drop and a attening of the temperature occur in the core (Figure 1.7). The amplitude of the drop on core temperature can reach more than 1 keV. The sawtooth crash time is usually of the order τ crash ∼ 100 µs [START_REF] Edwards | Rapid collapse of a plasma sawtooth oscillation in the JET tokamak[END_REF]][Yamada et al., 1994]. It is very short compared to the sawtooth period ranging between 20 ms and 500 ms depending on tokamak experiments [START_REF] Hastie | Sawtooth instability in tokamak plasmas[END_REF]. Hence, the fusion reaction rate is expected to decrease signicantly at each crash. Figure 1.8: Evolution of Argon injected at time t = t 0 in presence of sawteeth [START_REF] Dux | Z dependence of the core impurity transport in ASDEX-Upgrade H mode discharges[END_REF]. The density evolution of Ar in the midplane is shown versus major radius R. The magnetic axis is at R 0 = 1.72 m.

The sawtooth mechanism involves magnetic reconnection at the q = 1 resonant surface which takes place on a time much smaller than the current resistive time scale. The rst reconnection model for sawteeth was proposed by Kadomtsev [START_REF] Kadomtsev | On disruptive instability in tokamaks[END_REF], based on the works of Sweet [START_REF] Sweet | The neutral point theory of solar ares[END_REF] and Parker [START_REF] Parker | Sweet's mechanism for merging magnetic elds in conducting uids[END_REF]. However, this reconnection model predicts a reconnection time : τ rec ∼ √ τ A τ R , where τ A and τ R are the Alfvén and resistive times respectively, which is much larger than the sawtooth crash time measured in tokamaks. Progress in theory [START_REF] Rosenbluth | Nonlinear properties of the internal m = 1 kink instability in the cylindrical tokamak[END_REF][START_REF] Waelbroeck | Current sheets and nonlinear growth of the m = 1 kink-tearing mode[END_REF][START_REF] Wesson | Sawtooth reconnection[END_REF][ [START_REF] Wang | Nonlinear dynamics of the m = 1 instability and fast sawtooth collapse in high-temperature plasmas[END_REF][START_REF] Zakharov | The theory of the early nonlinear stage of m = 1 reconnection in tokamaks[END_REF][START_REF] Rogers | Nonlinear ω * -stabilization of the m = 1 mode in tokamaks[END_REF][START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF] and numerical studies [START_REF] Aydemir | Nonlinear studies of m = 1 modes in high temperature plasmas[END_REF][START_REF] Biskamp | Dynamics of the sawtooth collapse in tokamak plasmas[END_REF][START_REF] Lütjens | Non-linear modeling of core MHD in tokamaks[END_REF]][Halpern et al., 2011a[START_REF] Sugiyama | Compressible magnetohydrodynamic sawtooth crash[END_REF] accounting for non-linear eects have been made to improve the description of the reconnection process. Still, the reconnection time found in simulations diers from measurements.

During sawtooth crashes, particles and energy are rapidly redistributed in the core region. Regarding impurities, it results in a very eective radial transport in the core region during sawtooth crashes. Figure 1.8 from Ref. [START_REF] Dux | Z dependence of the core impurity transport in ASDEX-Upgrade H mode discharges[END_REF] summarizes the impurity behaviour in an ASDEX-Upgrade experiment. A short argon gas pu was injected at time t = t 0 , and the Ar density prole is then hollow. It can be seen that the impurity penetration is accelerated during sawtooth crashes. However, as soon as the impurity prole becomes peaked, impurities are ushed out of the core region by the sawtooth crashes. The prediction based on the Kadomtsev reconnection model [Ödblom et al., 1996] is qualitatively in agreement with this observation. This process is benecial, for example, to remove helium ashes produced in the core region and impurities pushed inward by neoclassical transport (see Chapter 5). At the same time, it also allows impurities to penetrate in core plasmas and induce the deleterious eects discussed earlier.

IMPURITIES IN TOKAMAK PLASMAS 13

A more advanced impurity transport model is necessary to understand the impurity dynamics during the crash and to explore methods to control the impurity content in the core. This thesis is focused mainly on the impurity behaviour during sawtooth cycles.

Neoclassical tearing modes (NTMs)

Usually, neoclassical tearing modes (NTMs) in core plasmas are associated with (m, n) = (3, 2) or (2, 1) tearing modes. It is often observed that NTMs are seeded by sawtooth crashes [Sauter et al., 2002a[START_REF] Buttery | On the form of NTM onset scalings[END_REF][START_REF] Chapman | Empirical scaling of sawtooth period for onset of neoclassical tearing modes[END_REF]. These modes cause a reorganization of the magnetic eld conguration via the formation of magnetic islands at the corresponding resonant surfaces. Figure 1.9 shows a schematic representation of a chain of magnetic islands. Particles trapped in magnetic islands spans freely the secondary magnetic surfaces. Radial transport is eectively short-circuited across the islands so that the plasma pressure becomes at inside the island. As a consequence, plasma energy globally decreases and the bootstrap current is reduced. The resulting current hole near the island O-point is destabilizing. Furthermore, an overlap of neighbouring island chains causes the magnetic eld lines to become stochastic. This inevitably results in a loss of plasma connement. The aim of NTM theory consists in predicting the non-linear saturated state of islands. It has been developed to a considerable level of sophistication [START_REF] Sauter | Marginal β-limit for neoclassical tearing modes in JET H-mode discharges[END_REF][START_REF] Brennan | A categorization of tearing mode onset in tokamaks via nonlinear simulation[END_REF][START_REF] Popov | Nonlinear 3D MHD code NFTC for simulations of plasma instabilities[END_REF][START_REF] Maget | Bi-uid and neoclassical eect on a Double-Tearing mode in Tore Supra[END_REF], although discrepancies with experimental observations remain [Rosenberg et al., 2002][Lütjens and[START_REF] Lütjens | Saturation levels of neoclassical tearing modes in International Thermonuclear Experimental Reactor plasmas[END_REF].

A formation of magnetic islands in the core region also aects impurity transport. Indeed, it was shown that the impurity transport is enhanced inside the islands [START_REF] Hender | The role of MHD in causing impurity peaking in JET hybrid plasmas[END_REF]. Thus, accumulation of impurities is accelerated by the presence of NTMs compared to predictions with turbulent and neoclassical models.

Electromagnetic wave injection for impurity control

It has been shown that some heating schemes using radio-frequency (RF) waves are ecient to control impurity accumulation thanks to turbulent eects [START_REF] Sugai | Radio frequency control of impurity transport[END_REF] CHAPTER 1. INTRODUCTION 2002][Dux et al., 2003[START_REF] Carraro | Study of impurity transport in FTU ITB plasmas[END_REF][START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF][START_REF] Leigheb | Molybdenum transport in high density FTU plasmas with strong radio frequency electron heating[END_REF].

However, the presence of MHD instabilities make impurity transport properties more complex. Thus, a direct control of sawteeth and NTMs could prevent impurity inux by reducing transport processes due to MHD instabilities.

Sawteeth can be avoided in hybrid scenarios, but mostly exist in tokamak plasmas. The sawtooth instability can be controlled by modifying its period between crashes using electromagnetic wave injection. Indeed, fast ions driven by ICRH are shown to stabilize sawteeth [START_REF] Graves | Internal kink mode stabilization and the properties of auxiliary heated ions[END_REF][START_REF] Chapman | Controlling sawtooth oscillations in tokamak plasmas[END_REF]][Graves et al., 2011][Chapman et al., 2011]. Also, both ECRH [START_REF] Bibliography Pietrzyk | Behaviour of central plasma relaxation oscillations during localized electron cyclotron heating on the TCV tokamak[END_REF][START_REF] Angioni | Eects of localized electron heating and current drive on the sawtooth period[END_REF] and ECCD [START_REF] Mück | Sawtooth control experiments on ASDEX Upgrade[END_REF][START_REF] Lennholm | Demonstration of eective control of fast-ion-stabilized sawteeth by electron-cyclotron current drive[END_REF][START_REF] Chapman | Sawtooth control using electron cyclotron current drive in ITER demonstration plasmas in DIII-D[END_REF] are good means to control sawteeth. In particular, a change of sawtooth dynamics is observed when ECRH is deposited in the core region [START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF][START_REF] Yun | Appearance and dynamics of helical ux tubes under electron cyclotron resonance heating in the core of KSTAR plasmas[END_REF]. These observations may lead to methods for real time control of the sawtooth period and the crash amplitude for ITER [START_REF] Kim | Real-time sawtooth control and neoclassical tearing mode preemption in ITER[END_REF]. Magnetic islands associated with NTMs can be reduced in size or suppressed by using localised RF waves at electron cyclotron resonant frequencies (ECRF) [START_REF] Gantenbein | Complete suppression of neoclassical tearing modes with current drive at the electron-cyclotronresonance frequency in ASDEX Upgrade tokamak[END_REF][START_REF] Sauter | On the requirements to control neoclassical tearing modes in burning plasmas[END_REF]][Février et al., 2016]. A real-time control of NTMs with EC waves would improve connement and avoid impurity accumulation [START_REF] Felici | Integrated real-time control of MHD instabilities using multi-beam ECRH/ECCD systems on TCV[END_REF].

As a consequence, the interplay between impurity accumulation, radiation and MHD instabilities needs to be understood to design optimum scenarios for impurity control.

Motivation

Deuterium-Tritium fusion oers a promising alternative to fossil fuels, to produce electricity in large amounts. However, numerous unresolved issues still have to be addressed and solved in the following decades so that dream becomes reality. Among those, understanding impurity transport in fusion plasmas is a subject of utmost importance. Indeed, impurities in the core region degrade fusion performances due to dilution and radiative cooling. Particularly, heavy impurities produced by plasma-wall interaction radiate a signicant amount of power when they penetrate in the core region. In fact, even a small concentration of a metallic impurity may lead to a radiative collapse of the plasma.

Impurity transport results from a competition between dierent mechanisms : MHD instabilities, collisional and turbulent transport. A complex impurity behaviour is observed in sawtoothing discharges. So far, impurity transport has been mostly studied with neoclassical and turbulent transport codes, which do not take into account MHD instabilities. Indeed, a sudden radial transport is known to occur during sawtooth crashes, which aects signicantly the impurity transport dynamics. It appears that transport codes are not able to reproduce the impurity dynamics observed in sawtoothing plasmas. Therefore, it is necessary to develop a predictive model including the combined eects of transport and MHD instabilities. Such a tool should allow the investigation of underlying physical processes, and could be used to devise ecient methods to control impurity accumulation. As a preparation for modelling experimental discharges, a scan of rescaled plasma parameters is made in order to simulate sawtoothing bulk plasmas in an elongated geometry with the XTOR-2F code. It turns out that several simulations exhibit features that bear similarities with compound sawteeth reported in experiments. Thus, a detailed analysis of simulated compound sawteeth is made and presented in Chapter 3. XTOR-2F simulations show that, in some situations, the plasma hot core is radially displaced and rotates, but is not fully expelled out of the q = 1 surface. The behaviour of this so-called partial crash is consistent with experimental observations. A scan of the q = 1 radius shows that partial crashes occur when it exceeds a critical value, at xed poloidal beta β p . This critical value depends on the plasma elongation, which is the other control parameter in this work. It can be noticed that the partial crash time is larger than the collapse time of an ordinary sawtooth, likely due to a weaker diamagnetic stabilization. This suggests that partial crashes result from a competition between destabilizing eects such as the q = 1 radius and diamagnetic stabilization.

In this thesis, we will refer to experimental observations in ASDEX-Upgrade tokamak, where numerical modelling using the XTOR-2F code was requested for a better understanding of the physics at stake. energy, particles and impurities are rapidly redistributed in the core region [START_REF] Wesson | Transport in the sawtooth collapse[END_REF]. Also, oscillations of the central temperature are sometimes observed before and after the sawtooth crash [START_REF] Campbell | Sawtooth activity in ohmically heated JET plasmas[END_REF]. These are called `precursor' and `post-cursor' oscillations respectively.

More exotic shapes of sawtooth exist and are frequently observed in experiments.

• A drop of central temperature during the sawtooth ramp phase is frequently observed (Figure 2.1). This drop has a smaller amplitude than the `ordinary' collapse, and does not terminate the ramp phase. It is referred to as a `partial collapse'. The onset of these so-called `compound sawteeth' is discussed in Chapter 3.

• In cases where the ECRF deposition is near the q = 1 surface, it is observed that the sawtooth period increases [START_REF] Angioni | Eects of localized electron heating and current drive on the sawtooth period[END_REF]. At the same time, a change of the sawtooth shape has been revealed. Indeed, during the ramp phase, the hot core remains radially displaced and rotates in the poloidal direction. This is called `humpback' sawtooth [START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF]. Moreover, it has been observed in some cases that the hot core could split into multiple helical ux tubes [START_REF] Yun | Appearance and dynamics of helical ux tubes under electron cyclotron resonance heating in the core of KSTAR plasmas[END_REF].

• `Monster' sawteeth have a period much larger than the connement time and a larger crash amplitude than `ordinary' sawteeth. They are characterized by a quasi-stationary phase between the sawtooth ramp and crash. They are produced when the ICRH deposition layer is located inside the q = 1 surface [START_REF] Bhatnagar | ICRF power-deposition proles, heating and connement of monster sawtooth and peakeddensity prole discharges in JET[END_REF]] provided a sucient power is injected [START_REF] Chapman | Power requirements for electron cyclotron current drive and ion cyclotron resonance heating for sawtooth control in ITER[END_REF]. It has been reported that their large amplitude collapse is more likely to trigger NTMs [START_REF] Buttery | On the form of NTM onset scalings[END_REF]][Maget et al., 2005[START_REF] Chapman | Empirical scaling of sawtooth period for onset of neoclassical tearing modes[END_REF].

1 Image from [START_REF] Campbell | Sawtooth activity in ohmically heated JET plasmas[END_REF] 2.

MAGNETOHYDRODYNAMICS 19

The previous list is of course not complete. The physics of sawtooth is so vast that there

are still unanswered open questions, mainly concerning the fast reconnection time. Studying sawtooth physics allows one to investigate magnetic reconnection on a laboratory scale. It is also important for ecient tokamak operation as sawteeth regularly atten the temperature and redistribute particles in the core region, which aects plasma connement.

Magnetohydrodynamics

Magnetic conguration at equilibrium

Magnetic ux surfaces

In cylindrical coordinates (R,ϕ,Z) the magnetic eld B can be written as

B = B R êR + B ϕ êϕ + B Z êZ = ∇ × A (2.1)
where A is the vector potential. Here, the unit vectors are such that êϕ = êR × êZ , as shown in Figure 1.5. If we consider an axisymmetric magnetic eld, all derivatives with respect to the toroidal angle ϕ vanish, i.e. ∂ ϕ = 0. Thus, the poloidal component of the magnetic eld

B p = B R êR + B Z êZ is B p = ∂A ϕ ∂Z êR - 1 R ∂(RA ϕ ) ∂R êZ = ∇ϕ × ∇ψ (2.2)
where ψ is the poloidal ux function dened as

ψ(R, Z) ≡ -RA ϕ (R, Z) (2.3)
As a consequence, the magnetic eld B is written in the form

B = I(ψ)∇ϕ + ∇ϕ × ∇ψ (2.4)
as R∇ϕ = êϕ . The toroidal component I(ψ) represents also the poloidal current ux function.

It is clear that B • ∇ψ = 0, hence ψ is constant along magnetic eld lines. The magnetic eld vector B lies on surfaces of constant ψ, called magnetic ux surfaces. In tokamaks, these ux surfaces have a form of nested tori at equilibrium.

The Grad-Shafranov equation

In tokamaks, the magnetic eld is set up such that the pressure force is balanced at stationary phase, in order for the plasma to be well conned. Thus the basic equilibrium force balance is It can be seen that the magnetic axis is shifted to the LFS with respect to the geometric axis.

J × B = ∇P (2.5)

CHAPTER 2. SAWTOOTH OSCILLATIONS IN THE CORE REGION

where J = a N a e a V a is the total current density and P = a N a T a is the total plasma pressure. Consequently,

B • ∇P = 0 (2.6)
and the pressure is constant on ux surfaces, i.e. P = P (ψ).

Since P and I are functions of ψ, the force balance equation (2.5) can be rewritten as

R 2 ∇ • ∇ψ R 2 = -µ 0 RJ ϕ = -µ 0 R 2 P -II (2.7)
which is called the Grad-Shafranov equation. Knowing the pressure gradient P and II , the numerical solution for ψ gives the magnetic conguration at equilibrium. The CHEASE code [START_REF] Lütjens | The CHEASE code for toroidal MHD equilibria[END_REF] is used to solve the Grad-Shafranov equation in this thesis. Figure 2.2

shows magnetic surfaces computed for a circular poloidal cross-section.

The magnetic axis is shifted towards large R values with respect to the geometric center of the poloidal cross-section. This is called the Shafranov shift, and it is due to the toroidal geometry of the conguration.

From kinetic to uid description of plasmas

The kinetic equation A plasma is a mixture of charged particles (electrons and dierent ion species), each of them characterized by its position x and velocity v. Thus each particle can be represented as a point in 6-D phase space (x,v). The distribution function F a (x, v, t) of a species a is dened as the number of particles per unit volume in phase space near the point (x,v) at the time t. Hence, the integration of F a over the velocity space gives the particle density N a in real space

F a (x, v, t)d 3 v = N a (x, t) (2.8)
From the conservation equation of the distribution function, the Vlasov equation is deduced

∂ t F a + dx dt • ∇ x F a + dv dt • ∇ v F a = 0
(2.9)

Each particle with charge e a experiences the electromagnetic force e a (E + v × B), where E and B are the electric and magnetic elds respectively. Thus, the Vlasov equation can be written as

∂ t F a + v • ∇F a + e a m a (E + v × B) • ∇ v F a = 0 (2.10)
The eects of collisions which occur at length scales comparable to the Debye length, are regrouped into the collision operator C a . Consequently, the kinetic equation in presence of short-range electromagnetic uctuations reads

∂ t F a + v • ∇F a + e a m a (E + v × B) • ∇ v F a = C a (F a ) (2.11)
The evolution of the electric eld E and magnetic eld B is described by the Maxwell equations :

∇ • B = 0 ∇ • E = σ ε 0 ∂ t B+∇ × E = 0 ∇ × B = 1 c 2 ∂ t E + µ 0 J
where ε 0 , µ 0 and c are respectively the permittivity, the permeability and the speed of light in vacuum, σ is the charge density.

Solving these coupled equations is very challenging and requires important numerical resources. It is usually done to study small-scale phenomena like plasma turbulence.

Fluid equations

Phenomena such as MHD instabilities have larger space and time scales than those of gyro-motion. In other words, ρ * = ρ L /L 1 and ω/ω c 1 where L and ω -1 represent the characteristic length and time scales of MHD instabilities. Also, the uid closure is valid if Landau resonances are avoided. Indeed, kinetic eects have to be considered at Landau resonances as energy can be transferred between particles and MHD waves. For a MHD mode with wave vector k, the condition to avoid Landau resonances is written ω k et al., 2013]. In this case, a reduction of the system into 3-D real space is sucient to describe the macroscopic behaviour of the plasma. Therefore, the distribution function F a remains Maxwellian, i.e. there are no particular velocities or regions of the velocity space where the charged particles behave dierently from the typical thermal particles of that species.

• v [Belmont
The framework of a uid description of the plasma is known as the magnetohydrodynamics (MHD).

The density N a , macroscopic uid velocity V a and pressure tensor Pa

N a V a = vF a d 3 v (2.13) Pa = m a v v F a d 3 v (2.14)
Here, v ≡ v -V a is the dierence between the velocity v of a given particle and the uid velocity V a of the species a. The pressure tensor Pa is decomposed into a scalar pressure P a and a stress tensor Πa , which contains the anisotropic and o-diagonal terms of the pressure tensor.

Pa = P a 1 + Πa (2.15)
The moments of the Vlasov equation (2.10) yield the continuity equation

∂ t N a + ∇ • (N a V a ) = 0 (2.16)
the force balance (or momentum) equation

m a N a [∂ t V a + (V a • ∇)V a ] = N a e a (E + V a × B) -∇P a -∇ • Πa (2.17)
and the pressure equation

∂ t P a + V a • ∇P a + γP a ∇ • V a + (γ -1) ∇ • Q a + Πa : ∇V a = 0 (2.18)
where γ is the adiabatic index, the notation Π :

∇V ≡ Π ij ∂ j V i and Q a is the microscopic heat ux dened by Q a = m a v 2 2 v F a d 3 v (2.19)

MHD equations Orderings

The uid velocity V a of a species a can be decomposed into components parallel and perpendicular to the magnetic eld.

V a = V ,a + V ⊥,a (2.20)
The momentum equation (2.17) gives the following decomposition of the perpendicular velocity

V ⊥,a = V E + V * a + B × ∇ • Πa N a e a B 2 + B × m a N a [∂ t V a + (V a • ∇)V a ] N a e a B 2
(2.21)

where

V E = E × B B 2 (2.22)
is the electric drift and

V * a = B × ∇P a N a e a B 2 (2.23)
is the diamagnetic velocity.

The MHD theory assumes a large perpendicular electric eld, i.e. V E /V T ∼ 1, where V T is the thermal velocity. It can be noticed that the diamagnetic velocity is small compared to the thermal velocity (V * a /V T ∼ ρ * ), and the following terms of the expression (2.21) are of a higher order in ρ * . This implies that the only drift that we retain at the lowest order is the E × B drift.

Single uid ideal model

Let us consider a plasma with electrons and bulk ions only, whose respective quantities are denoted with the indices e and i. We assume that both ions and electrons verify the continuity equation (2.16), the momentum equation (2.17) and the pressure equation (2.18).

In the following, we assume quasineutrality of the plasma, i.e. the electron and ion densities are considered to be locally equal (N = N e = N i ). This assumption is valid if we consider only plasma scales whose characteristic length is larger than the Debye length

λ D = ε 0 T e N e e 2
(2.24)

If the plasma is considered as a single uid, we can dene

• The mass density ρ m = m e N e + m i N i m i N i • The mass ow velocity V = (m e N e V e + m i N i V i )/ρ m V i = V ,i + V E • The current density J = -N e(V e -V i )
• The plasma pressure P = P e + P i Contributions of the pressure anisotropy tensor are neglected since the viscous stress eects are neglected in ideal MHD.

In this framework, the equation for the mass density (2.16) is thus

∂ t ρ m + ∇ • (ρ m V) = 0 (2.25)
As the electron mass is very small compared to the ion mass, the electron inertia can be neglected in the momentum equation (2.17)

ρ m ∂ t V = -∇P + J × B (2.26)
and the sum of pressure equations (2.18) written for electron and ion gives

∂ t P + V • ∇P + γP ∇ • V = 0 (2.27)
Gauss' law becomes ∇ • E = 0 as plasmas are highly polarizable and quasineutral (thus σ 0) uids. The electric eld E is self-consistently determined from Ohm's law

E + V × B = 0 (2.28)
Replacing this relation in Faraday's law gives the time evolution of the magnetic eld B

∂ t B = ∇ × (V × B) (2.29)
Since the MHD theory considers the non-relativistic uid model of a magnetized plasma, the time variation of the electric eld E is negligible. Hence, Ampere's law becomes

∇ × B µ 0 J (2.30)

Resistive-diusive MHD (ηχMHD) model

Collisions which are a fundamental aspect of the plasma dynamics are neglected in the ideal MHD model. Now, if we consider the kinetic equation with a collision operator C a (F a ), the associated uid equations for the species a read

∂ t N a + ∇ • (N a V a ) = 0 (2.31) m a N a [∂ t V a + (V a • ∇)V a ] = N a e a (E + V a × B) -∇P a -∇ • Πa + R a (2.32) ∂ t P a + V a • ∇P a + γP a ∇ • V a + (γ -1) ∇ • Q a + Πa : ∇V a = (γ -1)H a (2.33)
where R a and H a are respectively the friction force and the energy exchange between the species a with other species. This framework, known as the resistive-diusive MHD or ηχMHD, adds dissipation in the form of resistivity, diusion, viscosity and anisotropic heat transport.

Ohm's law then reads

E + V × B = ηJ (2.34)
where η is the plasma resistivity. Therefore, Faraday's law gives

∂ t B = ∇ × (V × B) -∇ • (ηJ) (2.35)
The pressure equation becomes

∂ t P + V • ∇P + γP ∇ • V = -(γ -1)∇ • (Q e + Q i ) (2.36) with Q a = -χ ,a N a ∇ T a -χ ⊥,a N a ∇ ⊥ T a (2.37)
representing the anisotropic heat transport for the species a. Details of this model can be found in Ref. [START_REF] Braginskii | Transport processes in plasma[END_REF].

Two-uid MHD model

If we consider now a small electric eld so that V E /V T ∼ ρ * , the diamagnetic velocities are taken into account. Therefore,

V i = V + V * i (2.38) V e = V + V * e - J N e e b
(2.39) by writing V ≡ V ,i + V E .

Using this expansion, the continuity equation reads

∂ t ρ m + ∇ • (ρ m V a ) + m i e i Ξ • ∇P i = 0 (2.40) with Ξ ≡ ∇ × (B/B 2 ).
It is shown that the several terms on the left-hand side of the momentum equation are cancelled by a part of the stress tensor [START_REF] Hazeltine | Plasma Connement[END_REF]. Thus, the momentum equation reads

ρ m [∂ t V + (V.∇) V + (V * i • ∇)V ⊥ ] = J × B -∇P (2.41)
In this ordering, the cross-eld heat ux Q × a has to be considered in addition to a purely

diusive heat ux Q a Q × a = 5 2 P a B × ∇T a e a B 2
(2.42)

Therefore, the pressure equation becomes

∂ t P + V • ∇P + γP ∇ • V + γ e i Ξ • (T ∇P i + P i ∇T i + P e ∇T e ) = -(γ -1)∇ • (Q e + Q i ) (2.43)
where T = P/N .

Finally, Ohm's law reads

E + V × B = ηJ - ∇ P e N e e b
(2.44)

A more detailed derivation of the ηχMHD or two-uid MHD equations, as well as a discussion on the approximation done to derive them can be found in Ref. [START_REF] Nicolas | Sawtooth driven particle transport on tokamak plasmas[END_REF].

2.3

Sawtooth oscillations

The internal kink mode

Sawtooth oscillations are associated with the MHD instability called internal kink mode.

This mode has poloidal and toroidal mode numbers m = n = 1, which dene a q = 1 resonant surface. The energy principle is a powerful method for testing ideal MHD stability in 3-D magnetic connement congurations. Its derivation in toroidal geometry is discussed in e.g.

[ [START_REF] Bussac | Internal kink modes in toroidal plasmas with circular cross sections[END_REF][START_REF] Basu | Magnetic reconnection and m = 1 oscillations in current carrying plasmas[END_REF] and is summarized in [START_REF] Nicolas | Sawtooth driven particle transport on tokamak plasmas[END_REF].

Usually, q < 1 in the core region due to a peaked plasma current density prole. The internal kink mode is unstable when the minimum value of the safety factor q is below unity and the poloidal beta β p is above a certain critical value β p,c in the ideal model [START_REF] Bussac | Internal kink modes in toroidal plasmas with circular cross sections[END_REF]. It has been shown that the critical value β p,c depends strongly on the plasma geometry as it aects magnetic eld line bending [START_REF] Coppi | Stability of global modes in advanced plasma connement congurations[END_REF][START_REF] Lütjens | Ideal MHD stability of internal kinks in circular and shaped tokamaks[END_REF][START_REF] Martynov | The stability of the ideal internal kink mode in realistic tokamak geometry[END_REF]. Indeed, the internal kink mode is destabilized by the plasma elongation, and stabilized by the plasma triangularity. Also, the internal kink mode stability is sensitive to the shear prole i.e. the details of the q-prole [START_REF] Hastie | Stability of ideal and resistive internal kink modes in toroidal geometry[END_REF][START_REF] De Blank | Theory of the m = 1 kink mode in toroidal plasma[END_REF][START_REF] Lütjens | Ideal MHD stability of internal kinks in circular and shaped tokamaks[END_REF][START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF]. Particularly, the internal kink mode is destabilized when increasing the radial position of the q = 1 surface [Bussac et al., 1975][Turnbull and[START_REF] Turnbull | Toroidal eects on current driven modes in tokamaks[END_REF]. Indeed, Ref. [START_REF] Bussac | Internal kink modes in toroidal plasmas with circular cross sections[END_REF] shows that, for an equilibrium with a parabolic q-prole, the critical value β p,c decreases by a third, when the q = 1 radius is increased from r q=1 /a 1 to r q=1 /a 0.4.

The growth rate of the internal kink mode depending on the MHD model (ideal, resistive or two-uid) is well detailed in Ref. [START_REF] Basu | Magnetic reconnection and m = 1 oscillations in current carrying plasmas[END_REF]. As the internal kink mode instability grows, the plasma center is helically displaced.

Magnetic reconnection

The displacement of the hot core leads to a compression of magnetic surfaces towards the q = 1 surface, forming a current sheet [START_REF] Waelbroeck | Current sheets and nonlinear growth of the m = 1 kink-tearing mode[END_REF][START_REF] Biskamp | Magnetic reconnection in plasmas[END_REF]. The piled-up magnetic ux surfaces break due to the plasma resistivity or other non-ideal eects leading to a reorganization of the magnetic topology, called `magnetic reconnection'. The rst reconnection model for sawtooth crash was proposed by Kadomtsev [START_REF] Kadomtsev | On disruptive instability in tokamaks[END_REF], based on the works of Sweet [START_REF] Sweet | The neutral point theory of solar ares[END_REF] and Parker [START_REF] Parker | Sweet's mechanism for merging magnetic elds in conducting uids[END_REF]. It is based on helical ux conservation (for details see Refs. [START_REF] Kadomtsev | On disruptive instability in tokamaks[END_REF][START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF][START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF]), and predicts a complete expulsion of the hot core out of the q = 1 resonant surface during the sawtooth collapse (Figure 2.3). In this scheme, the core region is replaced by a cooler island formed during the magnetic reconnection process.

Indeed, the helical ux is maximum at the q = 1 radius, and the magnetic surfaces with the same helical ux value reconnect. Hence, magnetic reconnection during a sawtooth crash aects the plasma from the magnetic axis up to the `mixing' radius, which is larger than the inversion radius (cf. Figure 1.7). This model is also called `complete' reconnection model, as all the magnetic surfaces with safety factor q < 1 reconnect. As a consequence, the initially peaked temperature prole is attened in the core region. Also, the q-prole becomes at with values larger than unity in the core region when q ≤ 1 before the reconnection. This model is qualitatively consistent with the dynamics of the hot core observed during the sawtooth relaxation.

Kadomtsev's reconnection model predicts a Sweet-Parker reconnection time which scales as τ rec ∼ √ τ A τ R , where τ A and τ R are the Alfvén and resistive times respectively [START_REF] Sweet | The neutral point theory of solar ares[END_REF][START_REF] Parker | Sweet's mechanism for merging magnetic elds in conducting uids[END_REF]. However, these reconnection times computed from the plasma parameters are much larger than the sawtooth crash times measured in tokamaks. Non-linear eects, such as the compressibility [START_REF] Sugiyama | Compressible magnetohydrodynamic sawtooth crash[END_REF], the electron inertia [START_REF] Wesson | Sawtooth reconnection[END_REF] and twouid eects [START_REF] Aydemir | Nonlinear studies of m = 1 modes in high temperature plasmas[END_REF][START_REF] Zakharov | The theory of the early nonlinear stage of m = 1 reconnection in tokamaks[END_REF][ [START_REF] Wang | Nonlinear dynamics of the m = 1 instability and fast sawtooth collapse in high-temperature plasmas[END_REF] are shown to accelerate the reconnection process. Still, the reconnection time found in simulations differs from measurements. Dierent theories on the magnetic reconnection during sawtooth crash can be found in Ref. [START_REF] Biskamp | Magnetic reconnection in plasmas[END_REF].

2.4

Numerical methods

The XTOR-2F code

The initial equilibrium

The toroidal MHD equilibrium is computed with the CHEASE code [START_REF] Lütjens | The CHEASE code for toroidal MHD equilibria[END_REF] which solves the Grad-Shafranov equation (2.7) using a Hermite bicubic nite element discretization. The proles of P and II must be provided as inputs to the code. These proles can rather be analytical functions or read from experimental data. Dierent options are detailed in Ref. [START_REF] Lütjens | The CHEASE code for toroidal MHD equilibria[END_REF].

As far as the internal kink mode is concerned, it is important to determine an appropriate value of poloidal beta β p so that the mode is ideally stable. Therefore, it is necessary to start by determining the critical value β p,c for given P and II proles. This critical value is deduced from the study of the ideal linear growth rate of the n = 1 mode by varying the initial pressure prole at a xed current prole at equilibrium. By this, β p,c which corresponds to the marginal stability point of the internal kink mode in the ideal MHD can be deduced (Figure 3.3). For sawtooth simulations, the value of β p is scaled such that the initial equilibrium is linearly stable in the ideal MHD and unstable in presence of the resistivity [Halpern et al., 2011a][Ahn et al., 2016a].

The CHEASE code computes the magnetic equilibrium and the corresponding toroidal metrics and mesh-grids which are used in the XTOR-2F code as inputs.

Normalization in the XTOR-2F code

The quantities in the XTOR-2F code are normalized according to the following convention. Here, B 0 and R 0 are respectively the magnetic eld magnitude and the major radius at the geometric axis.

• The basic length unit is the minor radius a.

• The basic time scale is the Alfvén time

τ A = R 0 /V A , where V A is the Alfvén velocity V A = B 0 / √ µ 0 ρ m = B 0 / √ µ 0 m i N i .
• The mass density is normalized to its value on axis ρm = N i /N i (0).

• The normalized pressure is P = R 2 0 µ 0 P/(a 2 B 2 0 ).

• The normalized magnetic eld is B = R 0 /(aB 0 )B.

NUMERICAL METHODS

29

• The velocity is normalized to V = (τ A /a)V

• The normalized current density is J = (µ 0 R 0 /B 0 )J.

The two-uid eects are scaled by the diamagnetic parameter di = R 0 /(aτ A ω c,i ) = V A /(aω c,i ). Note that di characterizes the amplitude of the diamagnetic terms. The Lundquist number S is dened as S = τ R /τ A = 1/η where τ R = µ 0 a 2 /η is the resistive time.

Non-linear two-uid MHD model

The XTOR-2F code [START_REF] Lütjens | XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks[END_REF]] is a non-linear and two-uid full 3D MHD code. The toroidal MHD equilibrium is computed with the CHEASE code [START_REF] Lütjens | The CHEASE code for toroidal MHD equilibria[END_REF], which solves the Grad-Shafranov equation. The dened equilibrium and the associated mesh-grids are used as inputs to the XTOR-2F code. The following set of normalized MHD equations is modelled in the XTOR-2F code

∂ t N + ∇ • (N V) + di Ξ • ∇P i = ∇ • (D ⊥ ∇N ) + S N (2.45) N ∂ t V + (V.∇) V + di (V * i • ∇)V ⊥ = J × B -∇P + µ∇ 2 (V + V * i ) (2.46) ∂ t P + V • ∇P + γP ∇ • V + di γΞ • (T ∇P i + P i ∇T i + P e ∇T e ) = (γ -1) ∇ • N χ ∇ T + ∇ • (N χ ⊥ ∇ ⊥ T ) + S H (2.47) ∂ t B = ∇ × (V × B) + di ∇ × ∇ P e N e -∇ × (ηJ) (2.48)
Here, S N and S H are the particle and heat sources, D ⊥ is the normalized perpendicular particle diusion coecient, χ and χ ⊥ are the normalized parallel and perpendicular heat diusivities and µ the normalized plasma dynamic viscosity. Also, N = N e = N i is considered for the quasi-neutrality constraint.

The resistivity η varies in time with the plasma temperature, following Spitzer's resistivity law η ∝ T -3/2 . The eect of the source terms S N and S H in Equations (2.45) and (2.47) is also to restore the density and pressure proles within their characteristic diusion times thanks to the perpendicular diusion operators.

Numerical scheme

The system of MHD equations can be summarized into a matrix dierential equation

∂ t x = F(x) (2.49)
where x regroups all physical quantities (N , V, P , B). With a time discretization denoted with an index n, this equation is solved using a fully implicit scheme

x n+1 -x n = ∆tF x n+1 + x n 2 + Θ(x n+1 -2x n + x n-1 ) (2.50)
where ∆t is the time step of the discretization and Θ is a constant for numerical stabilization.

This equation can be rewritten as

G(∆ n , xn ) ≡ ∆ n -∆tF 1 2 + Θ ∆ n + xn = 0 (2.51) with ∆ n ≡ x n+1 -x n and xn ≡ (1 -Θ)x n + Θx n-1 .
Finally, the equation ( 2.51) is solved at each iteration using a Newton-Krylov method

M -1 G (∆ k n , xn )(∆ k+1 n -∆ k n ) + M -1 G(∆ k n , xn ) = 0 (2.52)
where M is the preconditioner matrix, k is the number of iterations of the Newton-Krylov method and G is the derivative of G with respect to ∆ n . For the current version of XTOR-2F

the Newton-Krylov method is provided either by the NITSOL-library or the PETSc-library.

The preconditioning matrix M is applied to the system, in order to optimize the eciency of the method. This matrix is chosen from the linearized MHD equations, and contains the coupling of the n = 0 mode with the other toroidal modes. Also, the XTOR-2F code has been upgraded in order to improve the preconditioning matrix when additional equations to model the impurity transport [Ahn et al., 2016b] or the propagation of current sources [START_REF] Février | First principles uid modelling of magnetic island stabilization by electron cyclotron current drive (ECCD)[END_REF] have to be solved along with MHD equations that model bulk plasmas.

The XTOR-2F code uses a nite dierence method in the radial direction, and a pseudospectral method in the poloidal and toroidal directions. Time-steps are automatically adjusted for an optimum resolution of the Newton-Krylov solver. The plasma dynamics such as sawtooth crashes, which occur during a very short time scale compared to the connement time, can then be studied with an appropriate time resolution [START_REF] Nicolas | Non-linear magnetohydrodynamic simulations of density evolution in Tore Supra sawtoothing plasmas[END_REF][START_REF] Nicolas | Impurity behavior during sawtooth activity in tokamak plasmas[END_REF]][Ahn et al., 2016a][Ahn et al., 2016b].

Sawtooth simulations with the XTOR-2F code

Parameter rescaling

It is necessary to rescale the plasma parameters in the simulations to avoid excessive simulation times. In particular, large resistive times correspond to the Lundquist number in tokamaks of the order S ∼ 10 8 -10 9 , which is numerically demanding. Hence, the plasma resistivity is rescaled by orders of magnitude in the simulations. Usually, S ∼ 10 7 is used in the simulations presented in this thesis. Other plasma parameters are rescaled so as to preserve the ratios between dierent time scales with respect to experimental values. and di [Halpern et al., 2011b].

In addition, dierent regimes of the internal kink mode instability have already been identied by Halpern et al. for a circular plasma shape [Halpern et al., 2011a][Halpern et al., 2011b] : saturated helical equilibrium, kink cycling and sawtooth oscillations. For a dened initial equilibrium, the transitions between dierent regimes occur while varying the rescaled plasma parameter values such as the perpendicular heat diusivity χ⊥ , the poloidal beta β p , the Lundquist number S and the diamagnetic parameter di (Figure 2.4).

Grids and convergence

The width of the current sheet w during the reconnection roughly varies as w ∝ S -1 .

Therefore, the radial resolution is chosen such that the current sheet at the reconnection layer can be accurately computed, for a chosen Lundquist number.

The toroidal and poloidal mode resolutions used in the simulations are a sensitive subject. Increasing the mode resolution is numerically demanding as a spectral method is used for the poloidal and toroidal directions. A signature of an insucient resolution is a saturation of the mode energies at the same level [START_REF] Aydemir | On resistive magnetohydrodynamic studies of sawtooth oscillations in tokamaks[END_REF]. This is not observed in the XTOR-2F simulations of sawteeth. A discussion on the mode resolution used in the XTOR-2F simulations to describe sawtooth cycles can be found in Refs. [START_REF] Lütjens | XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks[END_REF]][Halpern et al., 2011b][Ahn et al., 2016a].

Summary

In this chapter, we have explored basic concepts and modelling tools necessary to understand the sawtooth phenomenon. Sawteeth are due to the unstable internal kink mode with poloidal and toroidal mode numbers (m, n) = (1, 1). The internal kink mode is characterized by a shift of the core region inside the resonant q = 1 surface in the radial direction. The inner core is rigidly pushed towards this surface, creating a current sheet where non-ideal eects cause magnetic reconnection.

The MHD framework is adopted to model sawteeth. The numerical tools presented in Section 2.4 are used throughout this thesis. The CHEASE code provides the MHD equilibrium, and the XTOR-2F code is used to solve the set of two-uid MHD equations. The plasma parameters are rescaled in order to prevent excessive numerical costs while maintaining physical aspects.

These tools are used in the following chapters to study compound sawteeth (Chapter 3)

and the transport of heavy impurities during sawtooth cycles (Chapter 5). Sustained sawtooth cycles are simulated with the XTOR-2F code in an elongated plasma geometry as the majority of tokamaks have a non-circular poloidal cross-section. This will allow one to model experimental scenarios and compare with measurements via synthetic diagnostics which have been implemented in the XTOR-2F code. An appropriate set of rescaled plasma parameters is necessary to simulate sawteeth. While performing a parameter scan, several simulations in an elongated plasma geometry contained sawtooth cycles which exhibit features similar to compound sawteeth reported in experiments. These simulations are analysed in this chapter, in order to understand the onset of the partial crash during the sawtooth ramp phase. The main results presented in this chapter are published in Ref. [Ahn et al., 2016a]

Partial crashes during sawtooth ramp phase

A drop of the central temperature or density value during the sawtooth ramp phase is frequently observed in plasma discharges [START_REF] Pfeier | Double-sawtooth oscillations in the Doublet III tokamak[END_REF]][Campbell et al., 1986][Nagayama 34 CHAPTER 3. NON-LINEAR DYNAMICS OF COMPOUND SAWTEETH et al., 1996]. This phenomenon is called `partial crash' as its amplitude is smaller than the one observed during a `full' sawtooth crash (Figure 2.1). It should be noted that the terminology `partial crash' is associated to the previously described phenomenon, and does not imply any hypothesis whether the reconnection is complete or not. These sawteeth are then designated as `compound' sawteeth.

It has been reported from ECE measurements that during the partial crash, the electron temperature drops only in an o-axis region [START_REF] Taylor | Evolution of the electron temperature prole of ohmically heated plasmas in TFTR[END_REF]][Campbell et al., 1986]. Also, SXR tomographic inversions of discharges in Tokamak de Varennes [START_REF] Janicki | Tomographic analysis of counpound sawteeth on the Tokamak de Varennes[END_REF] and in EAST [START_REF] Xu | Compound sawtooth in EAST LHCD plasma: An experimental study[END_REF], show that the hot core is preserved during the partial crash. Indeed, the hot core is radially displaced and rotates in the poloidal direction, inside the q = 1 surface.

At the end of the partial crash, the displaced hot core regains its equilibrium position before the precursor oscillations of the following sawtooth full crash appear. The partial crash is thus clearly dierent from the sawtooth crash described in the previous paragraph, which is characterized by a complete expulsion of the hot core out of the q = 1 surface (Figure 2.3).

It has also been reported that the partial relaxation is frequently observed after a sawtooth collapse followed by post-cursor oscillations [START_REF] Campbell | Sawtooth activity in ohmically heated JET plasmas[END_REF].

Theoretical models to understand the partial crash usually rely on `partial' (or incomplete) reconnection. In contrast to complete reconnection, the magnetic surfaces that reconnect are in a region o the initial magnetic axis during the partial reconnection [START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF]. Hence, q 0 remains below unity throughout the partial reconnection process for an initial q-prole with q < 1 in the core region. A possible explanation for partial reconnection is the existence of a critical island width [START_REF] Biskamp | Partial reconnection in the nonlinear internal kink mode[END_REF]]. In that model, the magnetic reconnection occurs as the Kadomtsev reconnection model in the beginning, until the cooler island reaches its critical width where the reconnection process stops. Also, the existence of multiple q = 1 surfaces was proposed to understand the partial crash [START_REF] Pfeier | Double-sawtooth oscillations in the Doublet III tokamak[END_REF][START_REF] Parail | Internal disruption in a tokamak[END_REF][START_REF] Denton | Skin currents and compound sawteeth in tokamaks[END_REF][START_REF] Aydemir | Toroidal studies of sawtooth oscillations in tokamaks[END_REF]]. The reconnecting magnetic surfaces are those around the region where q is below unity, which is located o the initial magnetic axis. Hence, multiple reconnection processes may explain the preserved hot core during the partial crash.

Resistive MHD simulations were able to reproduce compound sawteeth when using a heat diusivity that depends on the temperature gradient in cylindrical geometry [START_REF] Denton | Skin currents and compound sawteeth in tokamaks[END_REF], or on pressure in toroidal geometry [START_REF] Aydemir | Toroidal studies of sawtooth oscillations in tokamaks[END_REF]]. The heat transport coecients thus dened generate a locally higher current density in the o-axis region, which lead to the onset of multiple q = 1 surfaces. Hence, multiple reconnection processes oer a possible explanation for compound sawteeth. Still, the precise conditions which dictate the onset and dynamics of partial crashes remain poorly understood.

This chapter presents the main results published in Ref. [Ahn et al., 2016a

Physical parameters rescaling

For the present study, the Lundquist number is rescaled to S = 10 7 for the XTOR-2F sawtooth simulations. The diamagnetic parameter is then set to di = 0.15, which is about two or three times larger than the experimental values. Also, the normalized perpendicular heat diusivity of χ⊥0 = 3 × 10 -6 is chosen on axis and χ⊥ ∝ η. Thus, the products S di and S χ⊥ are about a third of those expected from the experiments. The characteristic time for particle diusion is chosen to be 10 times larger than the heat diusion time. As far as the viscosity is concerned, measurements of the plasma viscosity in experiments are not available yet. For the following simulations, the normalized plasma viscosity is set to μ = 7.5 × 10 -6 , so that the magnetic Prandtl number is P r m = S μ = 75.

The parameter τ ≡ P i /P e becomes τ = T i /T e under the quasi-neutrality constraint N e = N i . This parameter thus accounts for the dierence between the electron and ion temperatures. In the following simulations, electron and ion temperature are equal, i.e. τ = 1.

Choice of the initial equilibrium and its stability

The rescaling of the plasma parameters has to be performed for each initial equilibrium.

The toroidal MHD equilibrium is computed with the CHEASE code [START_REF] Lütjens | The CHEASE code for toroidal MHD equilibria[END_REF] with the chosen set of initial pressure and current proles. A number of parameters play an important role in the internal kink mode stability.

• First, for a given current prole, which determines the corresponding q-prole, the internal kink mode stability is determined by the value of the poloidal beta β p . In the following, the poloidal beta is a volume averaged quantity computed by the CHEASE code

β p = 8π I 2 tot R 0 Ω P J dψdθ (3.1)
where I tot is the total toroidal current, R 0 the major radius, Ω the plasma cross section, J the Jacobian and P the kinetic pressure.

• Second, the q-prole plays a very important role. Indeed, the internal kink mode stability depends on the current prole which determines the shear, the on-axis safety factor value q 0 and the radial position of the q = 1 surface [START_REF] Bussac | Internal kink modes in toroidal plasmas with circular cross sections[END_REF][START_REF] Lütjens | Ideal MHD stability of internal kinks in circular and shaped tokamaks[END_REF]. For instance, it has been shown that the internal kink mode is unstable even for a plasma pressure equal to zero for a suciently large inversion radius [START_REF] Turnbull | Toroidal eects on current driven modes in tokamaks[END_REF]. Experimental observations in Tokamak de Varennes [START_REF] Janicki | Tomographic analysis of counpound sawteeth on the Tokamak de Varennes[END_REF] clearly show that compound sawteeth are obtained when increasing the total current value i.e. decreasing the safety factor value q a at the edge (Figure 3.1). The attening of the q-prole is likely to push the inversion radius toward the edge. Hence, the inversion radius is chosen as a control parameter for this study.

• Finally, it is well known that the shape of the magnetic surfaces plays an important role in the internal kink mode stability threshold [START_REF] Lütjens | Ideal MHD stability of internal kinks in circular and shaped tokamaks[END_REF][START_REF] Coppi | Stability of global modes in advanced plasma connement congurations[END_REF][START_REF] Martynov | The stability of the ideal internal kink mode in realistic tokamak geometry[END_REF]. The internal kink mode is strongly destabilized by the elongation κ. Indeed, the plasma elongation κ signicantly lowers β p,c , and above a certain elongation κ c , the internal kink mode is unstable even for a vanishing plasma pressure [START_REF] Martynov | The stability of the ideal internal kink mode in realistic tokamak geometry[END_REF]. For this reason, the plasma elongation oers a second control parameter dierent from the radial position of the q = 1 surface.

Two cases of sawtooth simulations without partial crash are presented. One has an elongation at separatrix κ a = 1.6 and the q = 1 radius is at r s = 0.4 a, and the other has an elongation κ a = 1.8 and r s = 0.32 a. The case with an elongation κ a = 1.6 and the inversion radius at r s = 0.4 a will be the reference case throughout this work (Figure 3.2-a). Also, two cases of compound sawtooth simulations are obtained from the reference case : one by increasing the elongation to κ a = 1.8 with a xed q = 1 position at r s = 0.4 a (Figure 3.2-b), and the other by increasing the inversion radius to r s = 0.47 a at a xed geometry. The triangularity at separatrix is set to δ a = 0.4 for all the cases. The inverse aspect ratio for theses simulations is set to a/R 0 = 0.37, which is close to actual values in tokamaks.

The critical value β p,c is deduced from the study of the ideal linear growth rate of the n = 1 mode by varying the initial pressure prole at xed current prole at equilibrium. Indeed, β p,c corresponds to the marginal stability point of the internal kink mode in the ideal MHD (Figure 3.3). For the reference case with κ a = 1.6 and δ a = 0.4, it is found that β p,c 0.15. In the simulations, the value of β p is chosen to be approximately 2/3 β p,c in order to obtain an initial equilibrium linearly stable in the ideal MHD and unstable in presence of the resistivity [Halpern et al., 2011a]. From this equilibrium, sustained sawtooth oscillations are then obtained with the XTOR-2F code by rescaling the plasma parameters as described χ⊥ = 3 × 10 -6 , μ = 7.5 × 10 -6 are the same for all simulations.

in Section 3.2.1. The geometry and rescaled plasma parameters used in the XTOR-2F code for dierent plasma cross sections are summarized in Table 3.1.

It is stressed that the equilibrium during the sustained sawtooth oscillations diers from the initial equilibrium. Indeed, during the sustained sawtooth cycles, the internal kink mode is destabilized before the initial equilibrium is regained. Hence, the change of the equilibrium state is marked by the dierence of the q-proles corresponding to the respective equilibria (Figure 3.4). In the XTOR-2F code, the safety factor q is computed from the quantity B φ /(J B θ ), averaged along the initial magnetic surface. Here, B θ and B φ are respectively the poloidal and toroidal components of the magnetic eld. The computed q-prole gives q 0 0.94 and r s 0.35 a for the asymptotic equilibrium, whereas q 0 0.77 and r s = 0.4 a for the initial equilibrium. Also, this can be seen in Figure 3.2, where the rst sawtooth cycle has a larger period (τ ST 5×10 4 τ A ) than the sustained sawtooth cycles (τ ST 3.5×10 4 τ A )

for the reference case. q-proles of the initial (dashed line) and the asymptotic (solid line) equilibria for the geometry κ a = 1.6, δ a = 0.4 and the q = 1 surface at r s = 0.4 a. The q = 1 radius is changed from r s = 0.4 a to r s 0.35 a, as the initial equilibrium is not regained.

XTOR-2F simulation results

3.3.1 q-prole after a sawtooth full relaxation Before studying partial crashes, the plasma behaviour during the ramp phase of an ordinary sawtooth is investigated in order to understand the crash onset. For this purpose, sustained sawtooth oscillations without partial crash are simulated with the XTOR-2F code.

Sustained sawtooth oscillations are computed for the following sets of elongation κ a , triangularity δ a and normalized inversion radius r s /a : (κ a , δ a , r s /a)=(1.6, 0.4, 0.4) and (1.8, 0.4, 0.32). The following analysis focuses on the reference case with (κ a , δ a , r s /a)=(1.6, 0.4, 0.4).

A at q-prole slightly above unity after a sawtooth crash is predicted by the Kadomtsev complete reconnection model and is reproduced by the XTOR-2F simulations (Figure 3.5).

In the simulations, the q-prole remains at for a while after a sawtooth relaxation, during the ramp phase, before recovering the pre-crash equilibrium state. The safety factor in the core region after a crash is very close to unity (|q -1| < 3%). Also, non-monotonic q-proles are observed during the ramp phase, leading to the formation of two q = 1 surfaces at t = 1.53 × 10 5 τ A . The temperature after the sawtooth crash is locally higher between the inversion and mixing radii compared to the attened prole in the core region (Figure 3.6), which is also predicted by the complete reconnection model. As a consequence, the toroidal current density is locally enhanced in the o-axis region. That is observed due to a lower plasma resistivity which enhances the hollow prole of the safety factor.

It has been shown that the internal kink mode can be ideally unstable even with q > 1. Analytical studies show that the internal kink mode can be unstable for ultra-at (|q -1| 1) [START_REF] Hastie | Toroidal internal kink stability in tokamaks with ultra at q proles[END_REF] and non-monotonic q-proles [de Blank and Schep, 1991], although q > 1. Moreover, for a given MHD energy δW T , the growth rate is enhanced by low shear values [START_REF] Hastie | Stability of ideal and resistive internal kink modes in toroidal geometry[END_REF] and the internal kink mode is destabilized at the radius corresponding to the lowest shear value [START_REF] Lütjens | Stability thresholds for ballooning modes driven by high β internal kinks[END_REF]. However, the q-prole is dicult to determine after the sawtooth crash due to the remaining structures which take some time to reorganize. Poincaré plots are then more relevant in order to get indications on the evolution of the magnetic topology. Poincaré plots are used to identify magnetic surfaces, when they exist, by tracking eld lines during several hundreds of toroidal turns. This tool is useful when visualizing complex magnetic structures such as islands and stochastic regions.

The present analysis focuses on the time interval between the end of the sawtooth crash post-cursor and the time needed for recovering an asymptotic equilibrium. These cases are dened as sawteeth `without partial crash' as the time evolution of the pressure does not show any drop of central pressure value (Figures 3.7,. However, a reorganization of the magnetic surfaces is visible in the Poincaré plots during the sawtooth ramp phase from sawtooth simulations without partial crash (Figure 3.8). Indeed, the magnetic reorganization is mainly localized in the o-axis region where the shear is equal to zero (Figure 3 and the q = 1 surface at r s = 0.4 a, at t = 1, 50 × 10 5 τ A .

linear simulations with the reconstructed equilibria. These equilibria are reconstructed by the CHEASE code [START_REF] Lütjens | The CHEASE code for toroidal MHD equilibria[END_REF], which solves the Grad-Shafranov equation (2.7).

It is necessary to dene P and II as inputs of the CHEASE code in order to solve the Grad-Shafranov equation. However, the function II is unknown at the time of interest.

The m = n = 0 components of the pressure P and the toroidal component of current density J ϕ are extracted from the sawtooth simulations with the XTOR-2F code, at dierent moments. The CHEASE code then reconstructs the equilibria with precision (Figure 3.9), in order to be used as input for determining the plasma stability at the selected times. This computation supposes that the ux surfaces of the snapshot are identical to the initial ux surfaces. From the reconstructed toroidal equilibria, non-linear MHD simulations with the XTOR-2F code are performed to study the energy evolution of the n = 1 mode with the same mode resolutions and plasma parameters as Table 3.1.

Results

For the reference simulation of sawteeth without partial crash (κ a = 1.6, δ a = 0.4, r s = 0.4 a), the n = 1 mode energy behaviour is represented in Figure 3.10 from the reconstructed equilibria. The two time points, where the previously described method is applied, correspond respectively to t = 1.50 × 10 5 τ A and t = 1.58 × 10 5 τ A . It can be noticed that the magnetic surfaces, represented in the corresponding Poincaré plots , are close to the asymptotic equilibrium.

In Figure 3.10, it is observed that the n = 1 mode is unstable for the time slice t = 1.50 × 10 5 τ A just after the end of the sawtooth crash post-cursor. The instantaneous growth rate of the n = 1 mode amplitude reads γ 1.2 × 10 -4 before the non-linear saturation at t = 4.9 × 10 4 τ A . In contrast, the growth rate of the n = 1 mode instability is almost vanishing for the latter time t = 1.58 × 10 5 τ A . These growth rates are smaller than the n = 1 mode growth rate for a full sawtooth crash γ 4.7 × 10 -4 . Moreover, the energy saturates at W sat ∼ 10 -9 in normalized value, which is several orders of magnitude smaller than W sat ∼ 10 -4 for the reconstructed equilibrium just before the sawtooth full crash. The analysis of the poloidal mode spectra shows that the energy content of the n = 1 mode is dominant for the poloidal mode number m = 1. This means that the internal kink mode (m, n) = (1, 1) is unstable during the sawtooth ramp phase, although it does not impact the central pressure values (Figure 3.8-a) and the magnetic surfaces in the core region (Figures

3.8-b to 3.8-f ).
The origin of this destabilization can be explained by the q-proles in the core region, during the sawtooth ramp (Figure 3.5). It is shown that the internal kink mode can be unstable even for q > 1 when q is very close to unity [START_REF] Hastie | Toroidal internal kink stability in tokamaks with ultra at q proles[END_REF][START_REF] De Blank | Theory of the m = 1 kink mode in toroidal plasma[END_REF][START_REF] Hastie | Stability of ideal and resistive internal kink modes in toroidal geometry[END_REF]. Indeed, for at q-proles, the magnetic shear ŝ 1.

In the analytical expressions of the linear growth rate, small values of the magnetic shear ŝ enhance the growth rate [START_REF] Hastie | Stability of ideal and resistive internal kink modes in toroidal geometry[END_REF]. Also, it can be noticed that the radial position of the minimum safety factor value at t = 1.50×10 5 τ A (Figure 3.5) corresponds to the radius where the magnetic activity occurs (Figure 3.8-c). Hence, the internal kink mode is weakly unstable during the sawtooth ramp phase, and saturates to a low level. As a consequence, the instability is not sucient to trigger a pressure drop in the core region (Figure 3.7).

The partial crash during the sawtooth ramp phase

In this section, XTOR-2F simulation results of compound sawteeth are described. From the reference case (κ a = 1.6, δ a = 0.4, r s = 0.4 a), sustained sawtooth cycles with partial relaxations are obtained by increasing the elongation to κ a = 1.8 at xed q = 1 radius in one case, and increasing the inversion radius to r s = 0.47 a at xed geometry for the other case. In this section, the case with the geometry κ a = 1.8, δ a = 0.4 and the q = 1 position at r s = 0.4 a is analysed. Figure 3.2-b and Figure 3.11 show that a drop of the central pressure occurs during the sawtooth ramp phase, at about 10 4 τ A after a sawtooth full crash. The amplitude of this drop is much smaller than the ordinary sawtooth crash. This relaxation, which occurs during the sawtooth ramp phase, is characteristic of the experimental observations of compound sawteeth [START_REF] Pfeier | Double-sawtooth oscillations in the Doublet III tokamak[END_REF][START_REF] Janicki | Tomographic analysis of counpound sawteeth on the Tokamak de Varennes[END_REF][START_REF] Xu | Compound sawtooth in EAST LHCD plasma: An experimental study[END_REF]][Campbell et al., 1986].

The 2-D pressure evolution during the partial crash is represented in Figure 3.12. Dierent time points during the partial crash process are chosen and indicated with dashed vertical lines in Figure 3.12-a. The evolution of the pressure contours during the partial relaxation shows that the hot core is preserved during the whole process. It is radially displaced and rotates in the poloidal direction, in a core region located inside the q = 1 surface (Figures 3.12-b to 3.12-f ). The evolution of the magnetic topology during a partial crash at the time points marked in Figure 3.12-a is represented in the Poincaré plots (Figure 3.13). The dynamics of the hot core described in the contour plots of pressure, is also observed in the Poincaré plots. The magnetic structure corresponding to the hot core is radially displaced and rotates poloidally, without disappearing completely, even during the partial crash. The magnetic topology during the partial crash is modied only in the core region inside the q = 1 surface represented with a dashed curve in the Poincaré plots. Also, a signicant magnetic activity is observed in the very core region near the center of the plasma, while the hot core is radially displaced with the magnetic axis (Figure 3.13-c). These observations are reminiscent of a saturated helical state of an internal kink mode preserved in the core region.

The q-prole before the partial relaxation is at in the core region, similar to the case presented in Section 3.3.1. Again, the internal kink mode is unstable during the sawtooth Each time point corresponds respectively to the contour plots of Figures 3.12-b to 3.12-f.

The dashed lines in the Poincaré plots represent the q = 1 surface at equilibrium. ramp phase due to the at q-proles in the core region, but the growth rate is larger. In Figure 3.14, the time evolutions of the kinetic energy corresponding to the n = 1 mode for the reconstructed equilibria before and after the partial crash are represented with the same plot as Figure 3.10 in the case of a smaller elongation (κ a = 1.6). The growth rate of the n = 1 mode energy for the reconstructed equilibria before the partial crash (t = 1.22×10 5 τ A ) reads γ 8.5 × 10 -4 until the saturation at t = 4.9 × 10 4 τ A . This growth rate is about 6 times larger than the case without partial crash, represented with the green dotted line.

Also, the energy saturates at W sat ∼ 10 -5 , which is about 4 orders of magnitude larger than W sat ∼ 10 -9 for the case without kink with elongation κ a = 1.6. Still, the saturated level for the partial crash is an order smaller than the reconstructed equilibrium before a full sawtooth crash.

From the set of simulations described in Table 3.1, it is shown that the increase of the q = 1 radius triggers partial crashes. Also, the critical q = 1 position depends on the plasma elongation. Hence, the plasma elongation which is used as one of the control parameters, allows the internal kink mode to become suciently unstable to radially displace the hot core for the case of compound sawteeth at an elongation κ a = 1.8, in contrast to the case without partial crash at smaller elongation (κ a = 1.6). This radial displacement of the hot core leads to a drop of the core pressure value (Figure 3.11), which induces a partial crash very similar to what is observed in experiments (Figure 3.12a).

Discussion

The reconnection model proposed by Kadomtsev [START_REF] Kadomtsev | On disruptive instability in tokamaks[END_REF][START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF] addresses scenarios with monotonic q-proles. The model predicts that the q-prole is at-48 CHAPTER 3. NON-LINEAR DYNAMICS OF COMPOUND SAWTEETH tened in the core region after the sawtooth crash with q 1, then q decreases below unity during the sawtooth ramp phase while the plasma recovers its initial shape. However, the magnetic topology appears to be quite complex during the recovery phase. Several theoretical models have suggested in the past that the partial reconnection or the eect of multiple q = 1 surfaces are responsible for the partial crash [START_REF] Pfeier | Double-sawtooth oscillations in the Doublet III tokamak[END_REF][START_REF] Parail | Internal disruption in a tokamak[END_REF][START_REF] Denton | Skin currents and compound sawteeth in tokamaks[END_REF][START_REF] Aydemir | Toroidal studies of sawtooth oscillations in tokamaks[END_REF]].

In the XTOR-2F simulations of sawteeth without partial crash, the q-prole remains very at and close to unity in the core region during the sawtooth ramp phase (Figure 3.5).

Analytical studies of the ideal stability show that the internal kink mode can be unstable even for q > 1 in this case [START_REF] Hastie | Toroidal internal kink stability in tokamaks with ultra at q proles[END_REF][START_REF] De Blank | Theory of the m = 1 kink mode in toroidal plasma[END_REF][START_REF] Hastie | Stability of ideal and resistive internal kink modes in toroidal geometry[END_REF]. Indeed, the study of the n = 1 mode energy growth rate shows that the internal kink mode is unstable during the sawtooth ramp phase (Figure 3.10) although q > 1 at the times of interest. Also, more than one q = 1 surface appear due to the skin current formed at a nite radius during the ramp phase. This behavior was already observed in previous simulations of compound sawteeth using heat diusivity that depend on the temperature or its gradient [START_REF] Denton | Skin currents and compound sawteeth in tokamaks[END_REF][START_REF] Aydemir | Toroidal studies of sawtooth oscillations in tokamaks[END_REF]. This class of transport model allows the current prole to be locally peaked in an o-axis region. Multiple q=1 surfaces appear due to the development of a hollow current density prole, and lead to a partial crash. In the present XTOR-2F simulations, the perpendicular heat diusivity also depends on the temperature, χ ⊥ ∝ η, and produces a hollow current prole during the sawtooth simulations. Thus multiple q = 1 surfaces and reconnection are generic during the sawtooth ramp phase following a 'full' crash. A reorganization of the magnetic surfaces is observable in the Poincaré plots (Figures 3.8-b to 3.8-f ) in the region of the skin current sheet until the equilibrium is recovered. However, no pressure drop occurs in the core region during the ramp phase (Figures 3.7,. Hence, the presence of multiple q = 1 surfaces is a necessary but not sucient condition for a partial crash.

In the compound sawtooth cycles simulated with a more elongated cross section (κ a = 1.8, δ a = 0.4, r s = 0.4 a) or a larger inversion radius (κ a = 1.6, δ a = 0.4, r s = 0.47 a), the internal kink mode instability has a larger growth rate due to the destabilizing eects (Figure 3.14).

The internal kink mode then becomes suciently unstable to trigger a radial displacement of the hot core during the recovery phase. This radial displacement of the hot core leads to a substantial drop of the central pressure value (Figures 3.11 to 3.13). The amplitude of the drop is smaller than that of the full sawtooth crash, which characterizes a partial crash . In experiments, partial crashes with amplitudes smaller than the full sawtooth crash are frequently observed [START_REF] Pfeier | Double-sawtooth oscillations in the Doublet III tokamak[END_REF][START_REF] Nagayama | ECE image reconstruction of partial sawtooth crashes in ohmic plasmas[END_REF]. Also, SXR diagnostics show that the maximum emissivity point is radially displaced and remains in the core region inside the q = 1 surface at equilibrium during the partial crash [START_REF] Janicki | Tomographic analysis of counpound sawteeth on the Tokamak de Varennes[END_REF][START_REF] Xu | Compound sawtooth in EAST LHCD plasma: An experimental study[END_REF]. The SXR emissivity is a monotonic function of the plasma density and temperature in absence of highly radiating impurities in the plasma. This dynamics of the hot core is also observed in the 2D electron density reconstructions from the FSR measurements of the Tore Supra discharge #44634 (Figure 3.15-c). Thus, the pressure contours simulated with the XTOR-2F code are in qualitative agreement with the experimental observations using SXR tomographic inversions.

It can be remarked that the partial crash time is larger than the sawtooth full crash time. In the compound sawtooth of the Tore Supra discharge #44634 (Figure 3.15), the characteristic time of reconnection can be assimilated to the duration of the oscillating phase as measured by reectometry. The oscillating phase during the partial crash lasts τ rec 8 ms, which is larger than τ rec 5 ms for the sawtooth full crash. Also, it can be noticed that the sawtooth crash phase during the time interval t = 5.045-5.05 s corresponds to the `second kind' crash according to the denition in Ref. [START_REF] Udintsev | Experimental observation of m/n = 1/1 mode behaviour during sawtooth activity and its manifestations in tokamak plasmas[END_REF]. In the simulated compound sawteeth (Figure 3.16), the duration for the drop of the energy content inside the q = 1 surface during a partial crash is τ rec 5.5 × 10 3 τ A , which is about 1.5 times larger than τ rec 3.6 × 10 3 τ A for a full sawtooth crash. In fact, the characteristic times of sawtooth crashes τ rec represents the duration needed for the energy content inside the q = 1 surface to be expelled, which is quantied by the integral of the pressure inside the volume delimited by the q = 1 surface (Figure 3.16-a). For the diagnostics with xed measurement points the characteristic times for the crashes are deduced from the evolution of the central pressure. In this case, amplitudes of the collapse are more pronounced and the characteristic times are shorter (Figure 3.16-b) than the evolution of the energy content in the core region.

Hence, the characteristic time of the energy evolution in the core region is a more relevant parameter to quantify the energy exchange due to the reconnection.

Figure 3.17: Time evolution of the kinetic energy of the n = 1 mode from reconstructed equilibria before the partial crash at t = 1.22 × 10 5 τ A (dashed line, multiplied by 5 for visibility) and the full crash at t = 1.49 × 10 5 τ A (solid line). cf. Figure 3.16b

This dierence of the crash times can be explained by a weaker diamagnetic eect during the sawtooth ramp phase, where the pressure gradient is small due to the preceding sawtooth full crash. Indeed, diamagnetic eects play an important role in the internal kink mode stability, thus on the island growth rate. The two-uid eects are stabilizing the internal kink mode in the linear phase [START_REF] Basu | Magnetic reconnection and m = 1 oscillations in current carrying plasmas[END_REF]. However, they are strongly destabilizing in the non-linear phase [START_REF] Aydemir | Nonlinear studies of m = 1 modes in high temperature plasmas[END_REF][START_REF] Zakharov | The theory of the early nonlinear stage of m = 1 reconnection in tokamaks[END_REF][START_REF] Wang | Nonlinear dynamics of the m = 1 instability and fast sawtooth collapse in high-temperature plasmas[END_REF]. The scan of the diamagnetic parameter di with the XTOR-2F code also shows that at the sawtooth crash, which is a non-linear phenomenon, the collapse time decreases as diamagnetic eects increase [Halpern et al., 2011b]. The non-linear evolution of the n = 1 mode energy from the reconstructed equilibria corresponding to times before the partial crash (t 1.22 × 10 5 τ A ) and the ordinary crash (t 1.49 × 10 5 τ A ) is represented in Figure 3.17. A non-linear saturation of the n = 1 mode occurs for the equilibrium reconstructed from a time slice before the partial crash.

This is in contrast with the time evolution of an equilibrium reconstructed before a full crash. In that case, a collapse is observed. The various saturation scenarios were investigated in Ref. [Halpern et al., 2011b]. The rst evolution (partial crash) corresponds to a so-called `saturated helical equilibrium' while the second (full crash) is identied as a sawtoothing regime. Saturated helical states are typically observed when diamagnetic stabilization is weak (small values of di in Ref. [Halpern et al., 2011b]) while the sawtooth regime corresponds to strong diamagnetic stabilization. This picture is also consistent with a local Lundquist number that is lower before a partial crash then before a full crash.

As a consequence, a partial crash can be interpreted as a result of a competition between the unstable internal kink mode and the diamagnetic stabilization eect during the sawtooth ramp phase.

52 CHAPTER 3. NON-LINEAR DYNAMICS OF COMPOUND SAWTEETH

Summary

In this chapter, compound sawteeth have been studied with the XTOR-2F code. Nonlinear 3-D simulations of sawtoothing cycles show that q-proles are at in the core region after a sawtooth crash, with q 1. The q-prole remains at during the beginning of the ramp phase, and becomes hollow due to a plasma that is slightly hotter between the inversion and mixing radii right after the crash. This class of q-proles is unstable with respect to the internal kink mode, due to the low magnetic shear. The growth of the resulting weakly unstable internal kink mode leads to a localized and modest reorganization of the magnetic topology. However, no temperature drop is observed during this reorganization phase. This picture is changed when increasing the radial position of the q = 1 surface, which is known to be destabilizing. Above a critical q = 1 radius, a relaxation of the central pressure, i.e. a partial crash, is observed. The amplitude of the pressure drop is nevertheless smaller than a full crash. The pressure drop is triggered by the helical displacement of the hot core which rotates poloidally inside the q = 1 surface at equilibrium during the partial crash.

The displaced hot core is only partially expelled out of the q = 1 surface at equilibrium.

Partial crashes are also observed when increasing the elongation. Indeed, the critical value of the inversion radius depends on the plasma elongation. Hence, the onset of a partial crash is closely related to an overall more unstable internal kink mode. Also, it is found that the partial crash time is longer than a full crash time. This is attributed to a weaker diamagnetic stabilizing eect during the sawtooth ramp phase, due to a low pressure gradient. The onset of a partial crash is therefore understood as the conjunction of enhanced instability due to elongation and weak diamagnetic stabilization.

It is known that partial crashes are observed in circular devices such as Tokamak de

Varennes or Tore Supra. So far, no partial crashes have been obtained in circular geometry with the XTOR-2F code, setting a parabolic q-prole in the initial equilibrium. This is in contrast with previous MHD simulations [START_REF] Denton | Skin currents and compound sawteeth in tokamaks[END_REF][START_REF] Aydemir | Toroidal studies of sawtooth oscillations in tokamaks[END_REF]. Dierences are the initial current proles and the heat transport model, which plays an important role to control the temperature prole, and thus the current prole. Still, several features of the simulations agree with experimental observations: radial displacement and helical rotation of the hot core during the partial crash, smaller amplitude of the pressure drop compared to a full sawtooth crash. Partial crashes are observed in Tokamak de Varennes at large values of the plasma current, therefore presumably large values of the inversion radius r s . This picture is also consistent with the present XTOR-2F simulations. Thus, the predictive capability of the XTOR-2F code is once more veried. This will allow one to rely on the study of the impurity behaviour in sawtoothing plasmas simulated by the XTOR-2F code.

Hence, the main conclusion is that the onset of partial crash is related to the competition between the strength of the instability and diamagnetic stabilization, and is also very sensitive to the details of the q-prole after a full crash. for the ITER divertor thanks to its capability to handle large heat uxes and avoid tritium retention [START_REF] Kaufmann | Tungsten as rst wall material in fusion devices[END_REF]Neu, 2007][Pitts et al., 2011]. In addition to metallic impurities, gases such as Ar or Ne are intentionally seeded into the plasma as a potential solution to mitigate edge-localized modes (ELMs) at pedestal region [START_REF] Strachan | JET radiative mantle experiments in ELMy H-Mode[END_REF]] and reduce heat loads on PFCs.

Impurities in core plasmas not only dilute the D-T fuel, but may also radiate signicant amounts of energy. The total radiated power is one of the main components of the power balance of a tokamak discharge. It dictates the nal energy content for given heat sources and connement time. Severe limitations on tolerable concentrations are imposed for high Z impurities as they drastically cool down the core region even in a small concentration [Noda 54 CHAPTER 4. IMPURITY BEHAVIOUR IN TOKAMAK DISCHARGES et al., 1997]. When this limit is exceeded, the core plasma loses more energy by radiation than it gains from heating, and this leads to a radiative collapse [START_REF] Ongena | Towards the realization on JET of an integrated H-mode scenario for ITER[END_REF]]. In the particular case of tungsten, tokamak operation is jeopardized when its concentration N W /N e is above 10 - It is necessary to know the spatial distribution of the radiation in order to assess the impurity behaviour in the core plasma. Impurities are ionized in the plasma edge. Charged particles radiate in the plasma through bremsstrahlung, radiative recombination, dielectronic recombination, line radiation. Details on dierent radiative processes can be found in Ref. [START_REF] Vezinet | Soft X-ray measurements and analysis on tokamaks in view of real-time control[END_REF].

Light impurities (e.g. Be, C or N) are fully ionized in a tokamak core plasma. However, several tokamaks have now metallic plasma-facing components. Plasmas are thus contaminated by heavier impurities (e.g. Fe, Cu, Mo or W) which are not fully stripped in most of the plasma volume. The atomic physics for such ions is more complicated than for light ions.

Therefore, emission coecients associated with each process are calculated numerically and stored in databases (e.g. ADAS 1 ).

In general, the total emissivity of a species a of density N a can be written as

E a = N e N a Za k=0 f a,k K f f a,k + K f b a,k + K bb a,k (4.1)
Here, N a = k N a,k is the total density of species a in the plasma volume, with N a,k the 1 Atomic Data and Analysis Structure (see http://www.adas.ac.uk/) Ref. [START_REF] Pütterich | Calculation and experimental test of the cooling factor of tungsten[END_REF], where details on computations can be found.

density of its ionization state k. A fractional abundance f a,k is then dened as f a,k = N a,k /N a . The quantities K f f a,k , K f b a,k and K bb a,k are the emission rate coecients resulting from free-free, free-bound, bound-bound interactions respectively, for an ionization state k. Approximate analytic expressions of these coecients are summarized in Refs. [START_REF] Ingesson | Visible-light tomography of tokamak plasmas[END_REF][START_REF] Vezinet | Soft X-ray measurements and analysis on tokamaks in view of real-time control[END_REF].

Since numerous ionization states exist at the same temperature for heavy ions, it is more practical to regroup fractional abundances of dierent ionization states and their corresponding emission coecients in a total cooling factor L a . It has been shown that the eect of transport on the ionisation balance is small for heavy elements [START_REF] Vezinet | Impurity density derivation from bandpass soft X-ray tomography: applicability, perspectives and limitations[END_REF]. As an example, the cooling factor computed for tungsten [START_REF] Pütterich | Calculation and experimental test of the cooling factor of tungsten[END_REF] is shown in Figure 4.1. The total emissivity then has a simpler form

E a = N e N a L a (T e ) (4.2)
It can be noticed that the cooling factor is a function of the electron temperature T e . Therefore, the spatial distribution of a species a can be deduced from its measured emissivity distribution knowing the electron density and temperature. However, an important amount of impurities might cool down the plasma by radiation and aect the bulk plasma properties.

Therefore, only a trace of metallic impurities is injected by LBO.

Soft X-ray diagnostics

Measurements of radiation

In experiments, bolometer measurements are used to estimate the total plasma radiated power [START_REF] Huber | Upgraded bolometer system on JET for improved radiation measurements[END_REF][START_REF] Ingesson | Tomography diagnostics: bolometry and Soft X-Ray detection[END_REF]. However, they are often dominated by radiation at the plasma edge and near the limiter or divertor. Therefore, the radiated power 56 CHAPTER 4. IMPURITY BEHAVIOUR IN TOKAMAK DISCHARGES due to impurities in the core is dicult to extract from bolometer measurements. Otherwise, spectroscopy provides valuable information on ions from line-integrated measurements with high spectral resolution. In particular, charge-exchange recombination spectroscopy (CXRS) diagnostics provide local measurements from which light impurity (up to Ar) densities in the core can be deduced. It is usually used in connection with neutral beam injection [START_REF] Boileau | The deduction of low-Z ion temperature and densities in the JET tokamak using charge exchange recombination spectroscopy[END_REF]]. However, most spectroscopy diagnostics lack spatial sampling due to a limited number of lines of sight.

As a compromise, a diagnostic with poorer spectral resolution but high spatial and temporal resolution is a useful complement to the lack of spatial and/or temporal resolution of some spectroscopy diagnostics. Soft X-ray (SXR) diagnostics in tokamaks usually detect photons with energy in the approximate range from 1 keV to 20 keV. Its measurements are spectrallyintegrated quantities compensated by good time and spatial resolution. A tomographic reconstruction of the total emissivity can then be performed in a poloidal cross-section thanks to a large number of lines of sight (Figure 4.2). In addition, the SXR emissivity is not only dependent on electron temperature and density but also on impurity densities [Hutchinson, 1987][de Michelis and[START_REF] De Michelis | Soft X-rays spectroscopic diagnostics of laboratory plasmas[END_REF]. Therefore, tomographic inversions of SXR measurements can deliver local values of the plasma SXR emissivity which is proportional to impurity densities.

These features make the SXR diagnostics complementary to spectroscopy in that they provide information about MHD instabilities and impurity transport in core plasmas.

Soft X-ray detection method

The most common SXR detectors are semi-conducting diodes which are placed behind a pin hole to form just as many lines of sight. Each detector views a plasma volume with a certain solid angle, on which SXR emissivity is integrated. In tokamak tomography, the cone is commonly considered `thin' enough so that the volume integral can be approximated to a line integral. A discussion on limitations and justications of this approximation is detailed in Ref. [START_REF] Ingesson | Projection-space methods to take into account nite beam-width eects in two-dimensional tomography algorithms[END_REF]. More sophisticated algorithms are used in order to take into account the exact geometry of the lines of sight [START_REF] Vezinet | Soft X-ray measurements and analysis on tokamaks in view of real-time control[END_REF].

A beryllium lter is installed in front of each detector set acting as a high-pass lter. Saturation of signals is prevented by varying thickness of Be lters. It typically ranges between 25 µm and 200 µm. The measured signals are spectrally and spatially-integrated data. In a poloidal cross-section described by the (R,Z) coordinates, the signal measured by each line of sight l can be written as

y l = hν∈E l dν R,Z E ν (R, Z)ds l (R, Z) (4.3)
where E l is the energy range of the measured signal and ds l is the unit length of the line of sight. This method is widely used in dierent elds (e.g. geophysics, medical imaging, etc.), and particularly in fusion research, to diagnose tokamak plasmas. For SXR diagnostics, the `object' to be reconstructed is the spatial distribution of local emissivity. In a poloidal cross-section described by (R, Z) coordinates, the local emissivity can be represented as a function E 0 (R, Z). For an equipment with N L lines of sight, the tomography problem consists in resolving the system y l = E 0 (R, Z)ds l (4.4) for each 1 ≤ l ≤ N L , where y l is the line-integrated power detected, s l is the curvilinear abscissa of the line of sight. This system of equations which is known as the Fredholm 58 CHAPTER 4. IMPURITY BEHAVIOUR IN TOKAMAK DISCHARGES integral equations of the rst kind [START_REF] Twomey | On the numerical solution of fredholm integral equations of the rst kind by the inversion of the linear system produced by quadrature[END_REF], is always under-determined. Indeed, an innite number of y l would be needed to determine exactly the initial E 0 (R, Z). However, the number of lines of sight is usually limited to less than 200 in tokamaks, as a result of a trade-o between various operational factors (e.g. available space, price etc.).

One solution to the tomography problem is to reduce degrees of freedom by writing the solution as a series of orthogonal functions. An example of basis functions is found by expanding the solution's poloidal component into a Fourier series and its radial component into a series of Zernike polynomials [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications II[END_REF]. This method was used in the past for the SXR tomography, for example, in Alcator C-Mod [START_REF] Camacho | Soft X-ray tomography diagnostic of the Alcator C tokamak[END_REF] and JET [START_REF] Granetz | X-ray tomography on JET[END_REF]. Another possibility is to divide the poloidal cross-section into a nite element grid on which the emissivity is reconstructed. The size of the pixels has to be small enough so that the emissivity can be considered as constant, but at the same time, not too small as it denes the number of unknowns. Shapes for the pixels are not restricted to rectangular pixels. The method described in this section is the one used for analysing the ASDEX-Upgrade discharge that will be discussed in the following sections.

In a matrix format, the system of equations (4.4) is written as

X * E 0 = y (4.5)
where * denotes the usual matrix multiplication. The matrix X is an integral operator where information on the geometry of every line of sight in the poloidal cross-section grid is regrouped. The size of the matrix X is N L × N p , where N p is the total number of pixels used to subdivide the poloidal cross section. The emissivity distribution is represented by the vector E 0 of length N p . The vector y has N L elements which correspond to the line integrated values for each line of sight.

The inversion of the matrix X is in practice intractable. This is mainly due to the fact that the system has a lot more unknowns N p than equations N L . Even if there is a large number of lines of sight so that N L = N p , the matrix X is ill-conditioned. Hence, a solution E has to be chosen among the innity of solutions to the under-determined problem. The parameter χ 2 , dened as

χ 2 = (X * E -y) T * (X * E -y) = X * E -y 2 2 (4.6)
where the exponent T denotes a matrix transposition and . 2 denotes the Euclidean norm, gives the quadratic measure of the mismatch between the solution found E and the expected solution E 0 , which corresponds to the initial object.

Methods for regularization

An innite number of solutions E gives χ 2 = 0, but they are mostly erroneous solutions resulting from an over-t. Therefore, to obtain the closest solution E to the expected solution E 0 , the relevant quantity to minimize is the functional Φ, dened as

Φ = 1 2 χ 2 + λR (4.7)
Here, R is a regularizing functional reecting the expected properties of the physical solution E 0 (e.g. continuity, derivability, smoothness, etc.). The regularization parameter λ determines the weighting between the minimization of χ 2 and the satisfaction of conditions imposed by the functional R. Hence, to solve a tomography problem : rst, appropriate functional R and parameter λ are chosen, then, a solution E for the corresponding λ is computed.

The functional R can be built according to the Tikhonov regularization method [START_REF] Tikhonov | Solution of incorrectly formulated problems and the regularization method[END_REF]. A solution E of the Tikhonov regularization veries the relation

X T * X + λH * E = X T * y (4.8)
where H is a regularization matrix associated to R with the relation R = E T * H * E. As soon as the regularization matrix has been chosen, the L-curve method can be used to nd the optimum regularization parameter λ [START_REF] Hansen | The L-curve and its use in the numerical treatment of inverse problems[END_REF].

A linear regularization of n-th order consists in dening the matrix H as H = L T * L, where the matrix L represents a n-th order spatial derivative. For example, the simplest form of the matrix L for the linear regularizations of 0-th, 1-st and 2-nd order would be respectively the identity matrix I, the gradient matrix ∇ and the laplacian matrix ∆. Hence, the corresponding functional R requires a solution such that respectively the Euclidean norm, the gradient and the curvature are minimum.

For the SXR diagnostics in ASDEX-Upgrade, the minimum Fisher regularization (MFR) algorithm is used [START_REF] Odstrcil | SXR tomography of tungsten radiation patterns with and without MHD[END_REF]. The MFR algorithm was rst introduced in [Anton et al., 1996], and has a similar principle to the linear regularization, but the requirement for the solution E is the minimization of Fisher information I F dened as

I F = E (ζ) E(ζ) dζ (4.9)
where ζ is a spatial variable and the prime stands for the derivative with respect to ζ. This constraint allows the algorithm to nd a solution with inhomogeneous smoothing. Indeed, a stronger smoothing is applied in low emissivity regions. Thus, structures of high emissivity, usually located in the core plasmas, are recovered more eciently than with linear regularization methods. In a matrix format, the iterative process of the MFR algorithm has three main steps. Every p-th iteration :

• First, a diagonal matrix W (p) which contains preponderation of pixels for the Fisher information, is dened as

W (p) ij = δ ij for p = 0 δ ij /E (p) i
for p > 0 • Second, the regularizing matrix H (p) is calculated. Here, the linear regularization of 1-st order is chosen as an example.

H (p) = ∇ T * W (p) * ∇ (4.11)
• Finally, the system is solved with the Tikhonov regularization method

X T * X + λ (p) H (p) * E (p+1) = X T * y (4.12)
In general, there are little dierences between E (p) and E (p+1) after a few iterations (typically below 5 iterations). Hence, the MFR is adapted for real-time analysis of discharges [START_REF] Mazon | Soft X-ray tomography for real-time applications : present status at Tore Supra and possible future developments[END_REF]. Also, the MFR method is shown to be ecient, in terms of precision and numerical costs, compared to linear regularizations and the maximum entropy (ME)

method [START_REF] Weisen | Xray tomography on the TCV tokamak[END_REF].

In tokamaks, plasma emissivity gradients are expected to be low along the magnetic surfaces. Optimum solutions for tokamak plasmas are thus those obtained by smoothing emissivities preferentially along magnetic ux surfaces [START_REF] Ingesson | Soft X ray tomography during ELMs and impurity injection in JET[END_REF][START_REF] Odstrcil | Modern numerical methods for plasma tomography optimisation[END_REF]. For this method, the magnetic equilibrium is supposed to be known. The regularization matrix H then reads

H = S(ς)∇ T * W * ∇ + S(-ς)∇ T ⊥ * W * ∇ ⊥ (4.13)
where S(x) = 1/(1 + e -x ) for x ∈ IR, is a logistic sigmoid function. The matrices ∇ and ∇ ⊥ represent respectively the gradients along and across magnetic surfaces. A positive value of ς gives more weight to smoothness along magnetic surfaces. However, articial patterns along magnetic surfaces are obtained for a high value of ς. Hence, a trade-o on the value of ς is necessary to take into account the anisotropic smoothing. For the following analysis of ASDEX-Upgrade discharges, this tomographic inversion method is applied to the SXR diagnostic of which dierent lines of sight are represented in Figure 4.2.

ASDEX-Upgrade measurements

Motivation

The electron cyclotron resonance heating (ECRH) deposition in the core plasma is found to control central impurity accumulation and thus is routinely used in experiments [START_REF] Dux | Inuence of the heating prole on impurity transport in ASDEX Upgrade[END_REF][START_REF] Puiatti | Analysis of metallic impurity density proles in low collisionality Joint European Torus H-mode and L-mode plasmas[END_REF][START_REF] Leigheb | Molybdenum transport in high density FTU plasmas with strong radio frequency electron heating[END_REF]. Numerical computations show that the impurity pinch velocity is directed outwards when the electron temperature gradients are large [START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF]].

ASDEX-UPGRADE MEASUREMENTS 61

When a q = 1 surface is present in the central region of tokamak plasmas, ECRH triggers a strong MHD activity, typically an internal kink mode with mode numbers (m, n) = (1, 1).

The shape of the sawtooth cycles is then modied [START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF][START_REF] Bierwage | Dynamics of multiple ux tubes in sawtoothing KSTAR plasmas heated by electron cyclotron waves : II. Theoretical and numerical analysis[END_REF].

During the recovery phase, the hot core rotates in the poloidal direction until the next crash. It is usually called `saturated (1,1) mode'. Also, the relative positions of the ECRH deposition radius and the q = 1 radius are shown to change the sawtooth period [START_REF] Bibliography Pietrzyk | Behaviour of central plasma relaxation oscillations during localized electron cyclotron heating on the TCV tokamak[END_REF][START_REF] Angioni | Eects of localized electron heating and current drive on the sawtooth period[END_REF][START_REF] Chapman | Sawtooth control using electron cyclotron current drive in ITER demonstration plasmas in DIII-D[END_REF]. Thus, the ECRH located in the core plasma also modies the overall shape and period of sawteeth.

The modied sawtooth dynamics aects the impurity transport in the core region [Sertoli et al., 2015a]. The physical reasons for the eectiveness of ECRH in mitigating tungsten accumulation or its connection with MHD activity are still unclear. In view of ITER operation with a tungsten divertor, it is vital to understand if sawtooth tailoring to prevent impurity accumulation should be rethought. We have thus designed a series of experiments in the ASDEX-Upgrade tokamak aiming at explaining the various forms of sawtooth activity and their parametric dependences.

Experimental set-ups

A series of discharges in ASDEX-Upgrade is designed and performed in order to study the role of the EC power. In this section, 3 of these H-mode ASDEX-Upgrade discharges, #31114, #32409 and #32410, are analysed. A gyrotron can either heat electrons or drive current, depending on the angle of the EC beam with respect to the plasma current. Indeed, if the beam is injected perpendicularly to the plasma current, the gyrotron beam heats the electrons. Otherwise, it mainly drives current. An ECCD deposition is called `co-current' if the gyrotron beam is injected in the direction of the toroidal plasma current, and `countercurrent' if the injection is in the opposite direction. The main idea of these discharges was to use gyrotrons to heat the plasma (ECRH) near the magnetic axis (ρ RF 0.1) to prevent impurity accumulation and drive current (ECCD) in order to trigger the (1,1) mode by directly modifying the magnetic shear. Dierent sawtooth shapes are obtained by scanning the ECCD deposition radius ρ RF throughout the discharges. The impurity behaviour dependence on sawtooth shapes is studied. A summary of auxiliary power sources used in these discharges is shown in Table 4.1.

Detailed scenarios are presented in the following paragraphs.

ASDEX-Upgrade discharge #31114

In the discharge #31114, a total of 7.5 MW of NBI heating and 1.4 MW of EC waves is injected . Two gyrotrons are used in this scenario, each providing 0.7 MW.

One gyrotron is used for heating and the other gyrotron drives EC current (co-current). The time evolution of the deposition radius ρ RF for each EC beam is shown in prevent impurity accumulation in the plasma core. Meanwhile, the radial position ρ RF of the ECCD beam is scanned from the core to the outer region.

The inversion radius ρ inv can be measured from the evolution of the electron temperature prole during sawtooth crashes. In this discharge, it is located at ρ inv 0.35. This is close to the q = 1 radius computed by the equilibrium reconstruction with the CLISTE code [START_REF] Mccarthy | The CLISTE interpretive equilibrium code[END_REF]. Thus, the ECCD deposition radius is scanned from the interior to the outer region of the q = 1 radius.

ASDEX-Upgrade discharges #32409 and #32410

The eect of EC waves is illustrated by the discharges #32409 and #32410. These two scenarios are the same except for the injected gyrotron power. In both discharges, a total of 7.5 MW of NBI heating is injected . As shown in Figure 4.4-b, a sweep of the ECCD deposition radius ρ RF is made from the core to the edge, while crossing the q = 1 radius, similarly to the discharge #31114. A total of 1 MW of ECRH is injected in the core region at ρ RF 0.1 until t = 2.5 s, in order to prevent impurity accumulation in the center and to avoid a disruption before the ramp-up is completed. Then from t > 2.5 s on, heating is turned o and is replaced by 0.2 MW of co-ECCD for the discharge #32409, and 0.4 MW for #32410.

Evaluation of impurity density from measurements

It is necessary to evaluate a 2-D map of the tungsten density, in order to investigate its behaviour during MHD instabilities. The 2-D SXR emissivity map is reconstructed by the tomographic inversion method described in Section 4.1.3. The SXR emissivity of tungsten can be isolated by subtracting the background plasma contribution. However, the contributions of each ionisation state cannot be disentangled as the measured SXR signal is the sum of the emissivity of all species in the plasma. Therefore, in the method described above, tungsten is considered as an ion with an eective electric charge

Z W,ef f = Z W k=0 kf W,k
where f W,k is the fractional abundance of the ionization state k.

The 2-D electron density distribution is evaluated from fast-sweep reectometry measurements [START_REF] Sabot | A dual source D-band reectometer for density prole and uctuations measurements in Tore Supra[END_REF]. In addition, a method for the reconstruction of the time dependent 2-D electron temperature map from 1-D ECE measurements has been developed in ASDEX-Upgrade [START_REF] Sertoli | Characterization of saturated MHD instabilities through 2D electron temperature prole reconstruction from 1D ECE measurements[END_REF]. Thus, the overall concentration of tungsten is evaluated combining SXR, electron density and temperature measurements, using the cooling

Eect of ECCD deposition radius

Impurity ush out from the hot core in presence of a saturated (1,1) mode

The ASDEX-Upgrade discharge #31114 is analysed in this section. Especially, two time intervals corresponding to two dierent ECCD deposition radii ρ RF with respect to the inversion radius ρ inv 0.35 are studied :

• 4.85 s < t < 5.25 s, where ρ RF ρ inv • 7.02 s < t < 7.24 s, where ρ RF > ρ inv In Figure 4.5, the electron temperature and SXR measurements are displayed at dierent radii . The measurements made near the magnetic axis (ρ pol 0), inside the q = 1 surface (ρ pol 0.2) and outside the q = 1 surface (ρ pol 0.4) are shown in blue, red and green respectively. It should be noted that the SXR measurements shown in Figure A change of sawtooth shape depending on the position of ECCD is seen in the ECE measurements (Figure 4. 5-a,b). Indeed, so-called `humpback' sawteeth with a period of about 120 ms are observed when the ECCD deposition is located near the q = 1 surface. However, sawteeth have an ordinary shape with an approximate period of 70 ms when ρ RF > ρ inv .

These modications are visible in the SXR measurements d). Meanwhile, the dynamics of the impurity can be roughly estimated from integrated SXR signals.

The 2-D impurity density map is recovered from the SXR lines of sight, electron density and temperature measurements as explained in Section 4.2.2 and detailed in Ref. [Sertoli et al., 2015b]. This method is applied when the internal kink mode saturates at t 4.982 s, where ρ RF 0.38 . The ECCD deposition is then close to the inversion radius ρ inv 0.35. is correlated with the displacement of the hot core. It is consistent with the hollow tungsten density prole shown in Figure 4.7-d. This shows that tungsten is continuously expelled from the displaced hot core and redistributed in the core region. Before the next crash, a state is reached where tungsten is depleted in the displaced core.

Figure 4.8 shows the evolution of this dark spot during the sawtooth crash at t 5.00 s.

The dark spot which was continuously rotating in the poloidal direction is progressively expelled across the q = 1 surface. As soon as the initial dark spot is expelled, another dark spot appears near the magnetic axis. The complex behaviour observed in the 2-D SXR emissivity map during sawtooth crash is likely due to impurity redistribution. The probable explanation to this behaviour is that the impurities are transported by the convection due to the MHD instability while magnetic reconnection occurs [START_REF] Nagayama | Analysis of sawtooth oscillations using simultaneous measurement of electron cyclotron emission imaging and X-ray tomography on TFTR[END_REF][Ahn et al., [Odstrcil, private comm.]. The q = 1 surface is marked by dashed green lines. A dark spot marked by a cross, is progressively expelled out of the q = 1 surface. As soon as the initial dark spot is expelled, another dark spot is observed in the core. Tungsten density proles before and after sawtooth crash are displayed in Figure 4.9. As seen in Figure 4.7, the tungsten density prole is hollow during the (1,1) mode saturation.

The post-crash tungsten density prole is attened inside the inversion radius located at ρ pol 0.35. This picture is in a good agreement with Kadomtsev's model [Ödblom et al., 1996]. Indeed, impurity density prole is attened after a sawtooth crash. Therefore, it is essential to investigate why the impurity prole is hollow during the helical equilibrium.

Comparison with transport codes

The tungsten behaviour in the ASDEX-Upgrade discharge #31114 is modelled with transport codes [Sertoli et al., 2015b], using experimental data and neglecting the eects of MHD activity. The temperature and density proles from IDA (Integrated Data Analysis) used as inputs of the transport codes, correspond to the time t 4.97 s where the (1, 1) mode is saturated. The modelling was performed by combining the neoclassical and gyrokinetic codes NEO [START_REF] Belli | Full linearized Fokker-Planck collisions in neoclassical transport simulations[END_REF] and GKW [START_REF] Casson | Gyrokinetic simulations including the centrifugal force in a rotating tokamak plasma[END_REF]. Further details on the comparison are presented in Ref. [Sertoli et al., 2015b]. It can be inferred from this comparison that the sawtooth instability which is present throughout the discharge is responsible for the dierence of tungsten proles obtained between the measurements and the values predicted by transport codes. Indeed, during the (1,1) mode saturation, the usual assumption on an axisymmetric magnetic equilibrium to model neoclassical and turbulent transport is not valid. Moreover, neoclassical transport properties are shown to be strongly modied in a 3-D helical equilibrium [START_REF] Helander | Collisional transport in magnetized plasmas[END_REF]. Thus, the impurity transport has to be considered based on

Eect of injected ECCD power

The ASDEX-Upgrade discharges #32409 and #32410 are analysed in this section. As shown in Figure 4.4, 1MW of ECRH is injected near the magnetic axis (ρ RF 0.1) until t = 2.5 s, and then switched to ECCD afterwards. The scan of the radial position of ECCD starts at t = 3.6 s, starting from the core and moving towards the edge. Scenarios are identical in both discharges except for ECCD power, equal to 200 kW for #32409, and 400 kW for #32410. Figure 4.12-a. Also, it can be noticed that the tungsten prole is at during the time interval 2.5 s < t < 3.2 s and starts to peak after t 4 s. On the contrary, the tungsten density prole is hollow until t 4.2 s for discharge #32410 .

From these two discharges, two conclusions can be drawn concerning the eects of EC wave deposition on impurity control. First, the impurity proles are more likely to remain hollow for the cases with a higher gyrotron power injected. Second, the impurity accumulation is controlled more eciently when the ECCD is injected near the magnetic axis. Indeed, the tungsten density prole starts to peak as soon as the ECCD deposition radius is moved o-axis at t 4 s. The eciency of the on-axis EC wave deposition was also mentioned in previous studies [START_REF] Dux | Inuence of the heating prole 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]. In the discharge #31114, it can be noticed that impurity peaking is prevented by a continuous control with ECRH.

Looking at the electron temperature and the line-integrated SXR measurements (Figure 4.6-b) displayed in Figure 4.13, a correlation between the shape of sawteeth and the depletion of tungsten density proles can be deduced. In particular, signicant changes of the sawtooth dynamics are visible from the electron temperature at ρ pol 0.2 measured by the ECE channel 51. Indeed, a small bump is present at the beginning of the 5 sawtooth cycles between t 3.32 s and t 3.86 s in the discharge #32409 (Figure 4.13-a). Figure 4.12-a shows that the tungsten density prole remains hollow during these sawtooth cycles. Sawtooth cycles with similar shape, with a more visible bump, appear more frequently between t 3.22 s and t 4.21 s of the discharge #32410 (Figure 4.13-b). Again, the tungsten density prole is hollow during this time interval as shown in Figure 4.12-b. On the contrary, tungsten density proles are either at or peaked in both discharges if the `bump' is not visible.

The shapes of sawteeth previously described were analysed in detail. Two time windows are chosen for the analysis : 3.71 s < t < 3.99 s where sawtooth cycles with a `bump' in the ECE channel 51 signal are observed ( 14-e,f. They reveal that the tungsten prole is hollow when the amplitude of the mode at f 13 kHz is dominant. Also, the impurity prole starts to peak when the amplitude of the mode at f 13 kHz decays and the mode at f 14 kHz becomes dominant.

Therefore, from this analysis, it can be deduced that the tungsten density prole depletion is related to the mode frequency. Proles are hollower for lower frequencies.

Open questions

The ASDEX-Upgrade discharges display a complex impurity behaviour in presence of sawteeth. The resulting impurity transport is a competition between MHD driven transport, neoclassical and turbulent transport. Furthermore, the centrifugal force due to toroidal rotation and the radiative cooling increase the complexity of the phenomenon. A particular point in these discharges is that gyrotrons are used to heat electrons or drive current in the core region. The roles played by EC waves have to be clearly identied in order to understand the observations.

ECRH is shown to aect the impurity transport properties. Indeed, the impurity diusivity in the core region is increased by about a factor of 3 with respect to the neoclassical predictions when the deposition layer is located near the magnetic axis [START_REF] Dux | Inuence of the heating prole on impurity transport in ASDEX Upgrade[END_REF]].

The impurity pinch velocity which is initially directed inward can reverse direction when ECRH is added [START_REF] Scavino | Eects of plasma shape on laser blow-o injected impurity transport in TCV[END_REF]. It can be noticed that ECCD in the core region has the same eect on impurity transport coecients [START_REF] Dux | Inuence of the heating prole on impurity transport in ASDEX Upgrade[END_REF]. It has been suggested

that turbulent transport is the reason for such a reversal in the direction of the impurity pinch [START_REF] Angioni | Direction of impurity pinch and auxiliary heating in tokamak plasmas[END_REF] since the RF heating modies the temperature gradients and the current prole.

EC waves are also known to modify the MHD dynamics of plasmas. In particular, the sawtooth dynamics is strongly aected when the deposition layer is located in the core region.

Indeed, the stability of the internal kink mode is sensitive to the shear value at the q = 1 surface [START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF]. When the ECCD is located in the core region, the plasma current prole, thus the q-prole, is modied. ECRH induces a current as well, similarly to an ECCD mechanism, since electrons at the deposition layer are strongly heated. The socalled `humpback' sawteeth, the period of which is characterized by the poloidal rotation of the displaced hot core, are reported in TCV experiments [START_REF] Bibliography Pietrzyk | Behaviour of central plasma relaxation oscillations during localized electron cyclotron heating on the TCV tokamak[END_REF][START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF][START_REF] Angioni | Eects of localized electron heating and current drive on the sawtooth period[END_REF]. Therefore, it is necessary to investigate how the impurity behaves during the saturated (1,1) mode, in order to explain the dierence between ASDEX-Upgrade observations and calculations with transport codes.

• First, the presence of a displaced hot core breaks the assumption of toroidal symmetry. Thus, the neoclassical transport properties are strongly modied as the toroidal canonical momentum is no longer an invariant [START_REF] Helander | Collisional transport in magnetized plasmas[END_REF]. Therefore, the impurity transport has to be revised and computed with a 3-D equilibrium CHAPTER 5. TRANSPORT OF HEAVY IMPURITIES IN PRESENCE OF SAWTOOTH OSCILLATIONS The motion of a particle a with charge e a and mass m a in a magnetic eld B is described by the Newton equation

dv dt = e a B m a v × b (5.1)
where v is the velocity of the particle a and b ≡ B/B is the unit vector along the magnetic eld line. The velocity v can be decomposed into a velocity parallel to the magnetic eld line v = v b and a perpendicular velocity v ⊥ = b × (v × b) which corresponds to cyclotron rotation. The particle a gyrates around a magnetic eld line with a cyclotron pulsation ω c and Larmor radius ρ L dened as

ω c = |e a |B m a ρ L = m a v ⊥ |e a |B (5.2)
The center of the gyration motion is the guiding center and will be noted G.

The strong guide eld ordering can be used in tokamak plasmas. Here, a small parameter ρ * = ρ L /L 1 is considered, where L ∼ |∇ ln B| -1 . At this scaling, gradients of macroscopic elds are assumed to be small on the scale of the Larmor radius. Also, the macroscopic ow variations are considered to be slower than cyclotron motion. Thus, ω/ω c 1, where ω ∼ |∂ t ln B|.

At the lowest order in ρ * , only the cyclotron motion is taken into account, and the guiding center G follows the magnetic eld lines with a constant velocity. However, at the next order, the trajectory of the guiding center G is modied by drift velocities. These particle drifts are due to magnetic eld gradient ∇B and toroidal curvature κ ≡ (b • ∇)b. The magnetic drift velocity then reads

v d = v 2 ⊥ 2ω c B × ∇B B 2 - v 2 ω c κ × b (5.3)
In particular, κ ∇ ⊥ B/B in a low-beta plasma and

v d - v 2 + v 2 ⊥ /2 ω c R êZ (5.4)
The presence of an electric eld E yields an additional drift velocity. The E × B drift velocity V E dened as

V E = E × B B 2
(5.5) is independent of the particle charge and mass. As a consequence, this drift leads to a net ow of the plasma but not to a current. The polarization drift or inertia drift velocity dened

Trapping condition

An axisymmetric equilibrium with an adiabatic approximation can be characterized by three constants of motion : the magnetic moment µ a , the kinetic toroidal angular momentum p ϕ,a and the total energy E a

µ a = m a v 2 ⊥ 2B (5.7) p ϕ,a = m a Rv ϕ -e a ψ
(5.8)

E a = 1 2 m a v 2 + µ a B + e a φ (5.9)
In the absence of electrostatic potential φ, the particle energy of the species a is

E a = 1 2 m a v 2 + µ a B
(5.10) Thus, the parallel velocity reads

v = ± 2 m a (E a -µ a B) (5.11)
A particle is decelerated by the mirror force -µ a ∇ B as it moves to the high-eld side (HFS). As a consequence, the particle can be either trapped or passing, depending on its energy E a . At small inverse aspect ratio , the magnetic eld B varies as B(r, θ) B 0 1 + cos θ (5.12) where = r/R 0 is the inverse aspect ratio. Hence, B min ≤ B ≤ B max with

B min = B(r, θ = 0) = B 0 1 + B max = B(r, θ = π) = B 0 1 - (5.13)
If E a > µ a B max , particles are decelerated in the HFS, but still have sucient energy to explore the whole range of poloidal angles. The particle is then called passing or circulating.

If µ a B min < E a < µ a B max , a bounce angle θ b , where the parallel velocity v = 0, is found.

In this case, the particle moves along the magnetic eld line within a poloidal angle range |θ| < θ b (Figure 5.1).

Using the invariance of µ a and E a , one can deduce

B b = B(r, θ = ±θ b ) = B min 1 + v 2 v 2 ⊥ θ=0 (5.14)
As a consequence, the condition B min ≤ B b ≤ B max gives the trapping condition, for 1

v v ⊥ θ=0 ≤ 2 1 - √ 2
(5.15)

The fraction of trapped particles on a magnetic surface of minor radius r reads f t 1.46 √

(5.16)

Particle excursion from magnetic ux surfaces

As seen in the expression (5.4), the presence of the drift velocity v d shifts the guiding center trajectory in a direction perpendicular to the magnetic eld (Figure 5.1).

The invariance of the toroidal angular momentum p ϕ,a (5.8) for a passing particle gives m a r cos θv ϕ e a dψ dr δr c

(5.17)

As v ϕ v , dψ/dr R 0 B θ and q rB ϕ /(R 0 B θ )

δr c r R 0 B θ m a v e a cos θ q v ω c,a cos θ (5.18) For passing particles, v ⊥ v V T,a , where V T,a = T a m a (5.19)
is the thermal velocity of the particle a. The guiding center trajectory is shifted by δr c after one poloidal turn, where

δr c ∼ qρ L (5.20)
For trapped particles, v v ⊥ V T,a . The trajectory of its guiding center takes the form of a banana in the poloidal plane. The poloidal angle θ bounces between the angles ±θ b . Its characteristic width δr b is maximum at θ = 0, and is called banana width. Again, the invariance of the toroidal angular momentum p ϕ,a (5.8) at two points of the trajectory θ = 0 and θ = θ b gives m a Rv ϕ e a dψ dr δr b

(5.21) 1

As v ϕ v 1/2 v ⊥ δr b m a v e a B θ qρ L -1/2 (5.22)
Along its trajectory, a trapped particle moves over a typical distance qR 0 with a speed v . An estimation of the bounce time of a trapped particle travelling along the eld line is

τ qR 0 v (5.23)
The characteristic frequency of a particle oscillating between the two cusps of the banana orbit is called bounce frequency ω b ω b τ -1 V T,a qR 0 1/2 (5.24)

Coulomb collisions

Charged particles in tokamak plasmas interact through Coulomb collisions. A transfer of momentum due to the Coulomb force results in a deviation of particle velocities. Due to the long range of the Coulomb force, the velocity deection is usually small. Still, in tokamak plasmas, collisions with large scattering angle exist with a small probability. These collisions allow fusion reactions to occur. Nonetheless, large scattering collisions are rare enough to be neglected in the framework of transport and MHD studies of this thesis.

As an example, let us consider a simple model in order to evidence the main features of the process : a population of particles a with mass m a , charge e a and velocity where N e and T e are respectively the electron density and temperature of the plasma, and e is the elementary charge. Indeed, the Coulomb interaction is shielded for particles with L λ D . In the case of laboratory plasmas, λ D λ min . Thus, most collisions are well described by small velocity deections as illustrated in Figure 5.2. The Coulomb logarithm Λ = ln(λ D /λ min ) is in the range Λ = 10 -20 for tokamak plasmas with typical density N e 10 20 m -3 and temperature T 10keV.

The collision frequency ν ab between two species a and b for this example is

ν ab 4 √ 2π 3 e 2 a e 2 b Λ (4πε 0 ) 2 N b m 2 a V 3 a (5.27)
Note that there are dierent ways to dene ν ab . An expression of the collision frequency given by Ref. [START_REF] Hinton | Collisional transport in plasma[END_REF] reads

ν ab = 4 √ 2π 3 N b m a 1 m a + 1 m b e 2 a e 2 b Λ (4πε 0 ) 2 1 (V 2 a + V 2 b ) 3/2
(5.28)

This expression is consistent with the action-reaction principle given by

m a N a ν ab = m b N b ν ba (5.29)

Transport in tokamak plasmas

Collisional transport

Collisions lead to radial diusion by moving particles from one orbit to another. This occurs with a step-size of the order of the Larmor radius ρ L,a and a characteristic time ν -1 a .

The associated `classical' diusion coecient then reads

D CL a ρ 2 L,a ν a (5.30)
However, transport coecients predicted by classical transport theory are not relevant to tokamak plasmas as most eld geometric eects are omitted. Consequently, the `neoclassical' theory which takes into account geometric and resonant eects has been developed.

The condition (5.15) separates the velocity space (V ,a ,V c,a ) into trapping and circulating regions. The ratio between parallel and perpendicular velocities of a particle a is modied by collisions with a characteristic frequency ν a . Therefore, collisions can be considered as the cause of random walks in the velocity space. After a sucient number of collisions, a trapped particle is scattered out of the trapping region to become passing. An eective collision frequency ν eff is dened to characterize this process ν eff = ν a 2

(5.31)

The collisionality parameter ν * a is then dened as the ratio of this eective collision frequency ν eff and the bounce frequency ν b 5.32) This dimensionless number characterizes the capacity of a particle to complete an orbit of its trapped trajectory before it is detrapped.

ν * a = ν eff ν b qR 0 3/2 ν a V T,a (
Three regimes of neoclassical transport are identied according to the value of the collisionality ν * a .

• In the banana or Galeev-Sagdeev regime, trapped particles barely collide and usually complete a bounce or transit motion before experiencing a collision. The collisionality parameter is then small (ν * a 1). The radial diusion is mainly governed by the modication of trapped particle motion through collisions. The diusion coecient can be written as Ref. [START_REF] Wenzel | Neoclassical analysis of impurity transport following transition to improved particle connement[END_REF]] (a) in the limit of 1 ( = 10 -3 ) and (b) = 0.1. Each diusion coecient has its own expression dened on the entire domain of collisionality [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]Sigmar, 1981][Wenzel and[START_REF] Wenzel | Neoclassical analysis of impurity transport following transition to improved particle connement[END_REF]. Hence, they are summed to give the total contribution of collisional transport

D B a q 2 ρ 2 L,a 3
D coll a = D CL a + D N C a D N C a = D BP a + D P S a (5.36)
It can be noticed that the consideration of small inverse aspect ratio ( 1) is not really valid in tokamaks. As a consequence, the plateau regime almost vanishes for realistic geometrical parameters .

Turbulent transport

Particle and heat transport coecients measured in experiments are larger than neoclassical predictions by an order of magnitude for ions (two orders for electrons). The eects of turbulence have been studied to explain this anomalous transport. It is due to uctuations of the magnetic eld B and the electric eld E around their equilibrium values. These quantities can be written as B = B 0 + B 1 and

E = E 0 + E 1 , with | B 1 |/|B 0 | 1 and | E 1 |/|E 0 | 1.
The variation of B 1 around the equilibrium magnetic eld B 0 induces a perpendicular velocity V δB = V B 1 /B. This velocity depends on the parallel velocity, which is of the order of the thermal velocity. Hence, the radial transport driven by magnetic perturbations is negligible for ion species and impurities compared to electrons. The perturbation E 1 of the electric eld causes a radial transport directly via the electrical drift velocity V E which is the same for all species. These radial uxes modify the density and temperature gradients, which in turn sustain the turbulence via micro-instabilities.

Neoclassical transport is associated with small toroidal and poloidal mode numbers (n = 0, m 1), whereas turbulent transport process has much larger mode numbers m, n 1.

Hence, the total diusion coecient D tot and pinch velocity V tot p are then deduced by adding collisional and turbulent transport coecients

D tot = D coll + D turb V tot p = V coll p + V turb p (5.37)

Heavy impurity transport

Understanding the dynamics of heavy impurities in fusion plasmas is an important issue, especially since it has been decided to equip ITER with a tungsten (Z W = 74, A W = 184) divertor. The collisionality of the impurity has to be known in order to characterize its neoclassical regime. In general, the collision frequency of the species a in multi-species plasmas is a sum of collision frequencies with other species b : ν a = ν aa + b =a ν ab . A general expression of collision frequency is given in (5.28).

Here, quantities related to the main ion species and to an impurity will be labelled respectively with indexes i and z. Heavy impurities have high charge numbers in the core region (Z z Z i ). The relevant parameter to quantify the impurity content in plasmas is the impurity strength parameter 5.38) rather than its concentration N z /N i , as Coulomb collisions involve charge numbers. Heavy impurities are considered as trace (α z 1) and thermalized to ion temperature (T z T i ).

α z ≡ Z 2 z N z Z 2 i N i (
Under these assumptions,

ν iz = √ 2 Z 2 z N z Z 2 i N i ν ii α z ν ii ν zi α -1 z ν zz (5.39)
Therefore, ν i ν ii and ν z ν zi . Finally

ν z Z 2 z m i Z 2 i m z ν i (
* z Z 2 z Z 2 i m i m z ν * i (5.41)
As a consequence, heavy impurities are highly collisional due to their high charge number.

For tungsten, an approximate charge Z W 45 can be retained according to the tungsten ion charge distribution in the core region where the electron temperature ranges from 3 to 6keV (Figure 4.1).

At high collisionality, the relevant framework is the Prsch-Schlüter regime. However, the main ion species characterized by low charge number remains in the weakly collisional banana or plateau regime except in the pedestal region and near the magnetic axis. In this case, neoclassical theory predicts that the ion temperature gradient drives an outward impurity ux, and is thus responsible for a favourable thermal screening eect [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]Sigmar, 1981][Helander and[START_REF] Helander | Collisional transport in magnetized plasmas[END_REF]. Recent experiments and simulations with transport codes are consistent with this picture [Angioni et al., 2014[START_REF] Casson | Theoretical description of heavy impurity transport and its application to the modelling of tungsten in JET and ASDEX Upgrade[END_REF].

In addition, neoclassical transport for heavy impurities is shown to be enhanced due to the poloidal asymmetry of the impurity density caused by the centrifugal force [START_REF] Hinton | Neoclassical ion transport in rotating axisymmetric plasmas[END_REF][START_REF] Catto | Ion transport in toroidally rotating tokamak plasmas[END_REF][START_REF] Belli | PrschSchlüter neoclassical heavy impurity transport in a rotating plasma[END_REF] or RF heating [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] Bilato | Modelling the inuence of temperature anisotropies on poloidal asymmetries of density in the core of rotating plasmas[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]. Indeed, the eects due to rotation are stronger for heavy impurities. In a rotating plasma with toroidal angular frequency Ω, an ion species a has a Mach number

M a = m a Ω 2 R 2 2T i (5.42)
with T i the temperature of the main ion species. For heavy impurities, the Mach number M z can be of order unity (M z ∼ 1) even if M i 1 for the main ion species. As a consequence, neoclassical transport has to be considered along with turbulent transport for heavy impurities. In addition, strong toroidal rotation is shown to aect the impurity transport properties. Indeed, the impurity diusivity is enhanced by toroidal plasma rotation by a factor (1+M z ) 2 [Romanelli and Ottaviani, 1998]. Also, the thermal screening eect is weakened due to the eect of rotation, and can even change its sign so that the temperature gradient contribution drives an unfavourable inux of impurity [START_REF] Fülöp | Nonlinear neoclassical transport in a rotating impure plasma with large gradients[END_REF]Helander, 1999][Angioni and[START_REF] Angioni | Neoclassical transport of heavy impurities with poloidally asymmetric density distribution in tokamaks[END_REF].

5.2

Fluid description of heavy impurity transport

Stakes

Magneto-hydrodynamic (MHD) instabilities such as tearing modes and sawtooth oscillations add complexity to the impurity behaviour in core plasmas. It is well known that sawtooth crashes drive an ecient core radial transport of impurities [START_REF] Wesson | Transport in the sawtooth collapse[END_REF][START_REF] Dux | Z dependence of the core impurity transport in ASDEX-Upgrade H mode discharges[END_REF][START_REF] Yamada | Transport analysis of high-Z impurities including sawtooth eects in a tokamak system[END_REF]. If the pre-crash impurity prole is peaked, the prole is attened after a sawtooth crash in a region which is approximatively encompassed by the q = 1 surface. Sawteeth then tend to prevent impurity accumulation in the center. On the contrary, for a hollow impurity density prole, the impurities penetrate inside the q = 1 surface at each sawtooth crash. This behaviour is consistent with the Kadomtsev reconnection model [START_REF] Kadomtsev | On disruptive instability in tokamaks[END_REF][START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF] for dierent pre-crash impurity proles [Ödblom et al., 1996]. In addition, the presence of 2D structures of impurity density during sawtooth crashes has been evidenced in the XTOR-2F simulations [START_REF] Nicolas | Impurity behavior during sawtooth activity in tokamak plasmas[END_REF]. All in all, sawteeth tend to smooth the impurity density gradients in the core.

In the following, we will illustrate and analyse heavy impurity transport during sawtooth oscillations with the XTOR-2F code [START_REF] Lütjens | XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks[END_REF]. As described in Chapter 2, the XTOR-2F code has been upgraded to modulate impurity continuity and parallel momentum equations. Both are coupled self-consistently with the set of two-uid equations describing the bulk plasma, which is an important improvement made during this thesis, compared to the work in Ref. [START_REF] Nicolas | Impurity behavior during sawtooth activity in tokamak plasmas[END_REF]. For this study, let us consider a plasma containing only one type of impurity.

Relationship between radial particle ux and parallel forces

Orderings A species is in a highly collisional regime when the collisionality ν/ω t 1, where ν and ω t are the characteristic collision and transit frequencies respectively. The distribution function in the velocity space is then relaxed to a Maxwellian function due to the collisions.

Also, macroscopic scale lengths L (e.g. density or temperature gradients) are large compared to the Larmor radius ρ L : ρ * ≡ ρ L /L 1. Therefore, the uid description is legitimate for species in a collisional regime.

MHD equations are derived from the kinetic equation by supposing a large perpendicular electric eld, i.e. |V E |/V T ∼ 1. In the framework of the neoclassical theory, rapid uid motions are neglected by using the drift ordering where |V E |/V T ∼ ρ * . Thus, the plasma is supposed to be in a quiescent state, i.e. in a conguration of nested magnetic ux surfaces for long time scales. These assumptions are likely not to be fullled during a sawtooth crash.

However, this is not a strong limitation since particle and heat uxes are mainly tied to fast reconnection processes during the crash. For now, we restrict the analysis to conventional neoclassical theory.

Tokamak plasmas are usually magnetized with a eld such that the gyrofrequency ω c is large compared to the collision frequency ν. Then, the dimensionless small parameter δ ≡ ν/ω c is small and can be used as an expansion parameter. Each eld is then separated into a mean eld represented by a ux surface averaged quantity and small perturbations which correspond to poloidal asymmetries. For example, the density N in the toroidal coordinate system (ψ, θ, ϕ) is written as

N (ψ, θ) = N 0 (ψ) + N 1 (ψ, θ) + O(δ 2 ) with N 1 (ψ, θ) /N 0 (ψ) ∼ O(δ)
1.

The starting point is the continuity and momentum equations for the species a

∂ t N a + ∇ • (N a V a ) = 0 (5.43) N a m a (∂ t V a + (V a • ∇)V a ) = -∇P a + N a e a (E + V a × B) + R a (5.44)
The pressure anisotropy is neglected since only high collisionality (Prsch-Schlüter regime)

is considered for this study [START_REF] Hinton | Theory of plasma transport in toroidal connement systems[END_REF]. The uid velocity of the species a is a sum of the E × B, diamagnetic and parallel velocities

V a = V E + V * a + V ,a b (5.45)
where V * a = B × ∇P a /(N a e a B 2 ). In the force balance equation, the eect of collisions between the species a and other species b lies in the friction force R a = b R ab . The friction force between two species R ab is a sum of drag and thermal forces. A drag force is induced by a dierence of velocities between the two species. A thermal force appears due to the velocity dependence of collision frequencies and is proportional to the heat uxes. From the action-reaction principle, R ab = -R ba .

As neoclassical uxes are associated with parallel friction, the parallel components of the friction force are to be expressed in terms of dierent gradient lengths in the incompressible uid limit.

In the drift ordering where ow variations are slow in space and time, the force balance equation ( 5.44) reads -N a e a (E + V a × B) = -∇P a + R a (5.46) Through this study, the inertia term (V a • ∇)V a is supposed to be small compared to other terms in the expression (5.46). In other words, the toroidal plasma rotation is considered to be weak enough so that the centrifugal force can be neglected. This corresponds for example to plasmas with little momentum injection by NBI, which is not the case for ASDEX-Upgrade discharges presented in Chapter 4. Ω in the expression (5.42) is then considered small enough so that M a 1 for every ion species.

Friction force between impurity and main ion species

With the expression (2.4) of the magnetic eld in the toroidal coordinate system (ψ, θ, ϕ), the projection of Equation (5.46) along the toroidal direction R 2 ∇ϕ reads [START_REF] Hazeltine | Plasma Connement[END_REF] N a e a V a • ∇ψ = -R 2 R a • ∇ϕ (5.47) Here, the quantities are supposed to be axisymmetric : ∂ ϕ P a = 0 and ∂ ϕ φ = 0 with φ the electric potential. Also, the ion and impurity Ware uxes are negligible compared to neoclassical uxes (for more detail, see p.1155 of Ref. [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]) so that the contribution of the induction eld to E ϕ can be ignored. Thus, E ϕ 0 under these conditions.

Neoclassical transport is associated with poloidal asymmetries on magnetic ux surfaces.

The neoclassical impurity particle ux across the ux surfaces is then Γ ψ a = Γ a • ∇ψ , where (5.48) corresponds to a ux surface average of the quantity G.

G ≡ Gdθ B • ∇θ dθ B • ∇θ
By writing R a = R ,a b + R ⊥,a , Equation (5.47) becomes

Γ ψ a = - I(ψ) e a 1 B R ,a (5.49)
The perpendicular friction force R ⊥,a is responsible for the classical transport. It is neglected since the classical ux is small compared to the neoclassical ux at large aspect ratio [START_REF] Hazeltine | Plasma Connement[END_REF]. The parallel friction force R ,a is to be written as a function of gradient lengths to derive the expression of the neoclassical radial ux. For this study, the plasma is supposed to contain a single type of impurity ion, and the impurity is considered as a trace (α z ≡ N z e 2 z /(N i e 2 i ) 1). For heavy impurities (m z /m i 1), the friction force R z R zi and its projection along the parallel direction is then written in the form [START_REF] Helander | Collisional transport in magnetized plasmas[END_REF][START_REF] Braginskii | Transport processes in plasma[END_REF][START_REF] Rutherford | Eects of density asymmetries on heavy impurity transport in a rotating tokamak plasma[END_REF] R

,zi = -m z N z ν zi (V ,z -V ,i ) - 2 5 C 0 m z N z ν zi Q ,i P i (5.50)
Here, Q ,i = -χ ,i N i ∇ T i is the parallel heat ux of the main ion species and ν zi the collision frequency between the impurity and the main ion species.

It can be noticed that Ref. [START_REF] Rutherford | Eects of density asymmetries on heavy impurity transport in a rotating tokamak plasma[END_REF]] adopted Braginskii's procedure for electronion collision [START_REF] Braginskii | Transport processes in plasma[END_REF] to derive the impurity ux in the Prsch-Schlüter regime.

However, this computation considers both the impurity and the main ion species in the Prsch-Schlüter regime. In this case, the favourable thermal screening eect predicted by the neoclassical theory is not retrieved. This is why the adjustable coecient C 0 is dened as a uid closure. It is chosen in order to obtain the appropriate screening factor for the impurity Prsch-Schlüter ux with bulk ions in the low collisionality regime. In general, C 0 depends on the impurity strength parameter α z , and is thus a radial function which is deduced from kinetic computations accounting for collisions [START_REF] Estève | A multi-species collisional operator for full-F gyrokinetics[END_REF][START_REF] Estève | Etude gyrocinétique du transport multi-espèces néoclassique et turbulent dans un plasma de fusion[END_REF]. Since our study deals with trace impurities only (α z 1), C 0 can be chosen as a constant, and is calculated in the following section. CHAPTER 5. TRANSPORT OF HEAVY IMPURITIES IN PRESENCE OF SAWTOOTH OSCILLATIONS

Derivation of impurity radial uxes

Parallel uxes

It is shown in the expression (5.49) that the perpendicular Prsch-Schlüter ux is associated with the parallel friction force. Hence, parallel velocities and heat uxes have to be derived in order to deduce R ,zi according to the expression (5.50). For this, the assumption on uid incompressibility is crucial. The perpendicular particle and heat ows are then balanced by parallel ows to ensure particle and heat conservation.

At the lowest order in δ and using the expressions (2.4) and ( 5.45), the particle velocity of species a reads V a = K a (ψ, θ)B -T a,0 e a A P * a ,ψ R 2 ∇ϕ

(5.51)

where

A P * a ,ψ ≡ ∂ ψ ln P a,0 + e a T a,0 ∂ ψ φ 0 (5.52)
and K a is a priori a function of ψ and θ. In the neoclassical time scale, the continuity equation

∂ t N a + ∇ • Γ a = 0 (5.53)
gives a divergence-free particle ux ∇ • Γ a = 0 for each species a. This yields the uid incompressibility ∇ • V a = 0 as N a (ψ, θ) N a,0 (ψ) at the lowest order in δ. Under this constraint, K a needs to be a radial function. As a consequence, the parallel velocity of the species a is written as

V ,a = K a (ψ)B - T a,0 e a I(ψ) B A P * a ,ψ (5.54)
Unlike for the particle ux Γ a , the assumption ∇ • Q a = 0 for each species a is not general. Indeed, the energy equation for the species a is written in the form

∂ t (N a S a ) + ∇ • (N a S a V a ) = - 2 3 ∇ • Q a T a + b H ab (5.55)
where S a ≡ ln (P a N -γ a ) is the entropy, and H ab is the energy exchange with another species b. Hence, even if the convective term ∇ • (N a S a V a ) is negligible at lowest order in δ, only the total heat ux is divergence free (i.e. ∇ • a Q a = 0). Nevertheless, ∇ • Q a = 0 for each species a, if the energy exchanges due to collisions with other species b H ab are negligible. This is the case for species with large mass disparity since the energy transfer term in the collision operator is proportional to the mass ratio [START_REF] Helander | Collisional transport in magnetized plasmas[END_REF]. In the case of plasmas with heavy impurities (m z /m i 1), main ions and electrons, this assumption is valid.

The perpendicular heat ux reads

Q ⊥,a = 5 2 P a e a B × ∇T a B 2
(5.56)

Hence, at lowest order in δ, the heat ux can be written as

Q a = L a (ψ)B - 5 2 P a,0 e a ∂ ψ T a,0 R 2 ∇ϕ (5.57)
where L a is constant along magnetic ux surfaces because of the incompressible heat ux constraint ∇ • Q a = 0. Therefore, the parallel heat ux has the following form From the expressions (5.50), (5.54) and (5.58), the perpendicular impurity particle ux then reads 5.62) by considering the isothermal assumption T z T i = T .

Q ,a = L a (ψ)B - 5
Γ ψ z = -m z N z,0 T 0 ν zi I(ψ) 2 e 2 z 1 B 2 - 1 B 2 ∂ ψ ln P z,0 - e z e i ∂ ψ ln P i,0 + C 0 e z e i ∂ ψ ln T i,0 (
For a geometry of circular concentric magnetic surfaces, it is known [START_REF] Hinton | Theory of plasma transport in toroidal connement systems[END_REF] that

I(ψ) 2 1 B 2 - 1 B 2 2r 2
(5.63) in the limit of small inverse aspect ratio 1. Also, ∂ r ψ = B 0 r/q in this approximation.

CHAPTER 5. TRANSPORT OF HEAVY IMPURITIES IN PRESENCE OF SAWTOOTH OSCILLATIONS

For highly-charged heavy impurities, e z /e i 1 and the radial ux reads 5.64) with the Prsch-Schlüter diusion coecient D P S z = 2ρ 2 z q 2 ν zi .

Γ z • ∇r = q B 0 r Γ ψ z = -D P S z N z,0 ∂ r ln N z,0 - e z e i ∂ r ln N i,0 + (C 0 -1) e z e i ∂ r ln T i,0 (
Depending on the value of the coecient C 0 in the expression (5.64), the temperature gradient will reduce or enhance the impurity inux. For trace impurities (i.e. α z 1), computations from the neoclassical theory show that C 0 1 when the impurity and the main ion species are both in the Prsch-Schlüter regime [START_REF] Rutherford | Eects of density asymmetries on heavy impurity transport in a rotating tokamak plasma[END_REF]. In that case, the temperature gradient has no eect on the impurity radial ux. However, for the most common case of heavy impurities in the Prsch-Schlüter regime and the main ion species in the banana-plateau regime, the coecient C 0 is predicted to be C 0 = 3/2 in Ref. [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]. The ion temperature gradient then drives a favourable thermal screening eect. For this study, C 0 = 3/2 is set in the expression (5.50) of the collisional force to reproduce the appropriate neoclassical prediction.

Poloidal asymmetry of impurity density

The derivation of the Prsch-Schlüter uxes in this section is done under the assumption of small poloidal asymmetries compared to surface averaged quantities. With the expressions (5.50), (5.54), (5.58) and the constraint (5.61), the parallel friction force reads, in the small inverse aspect ratio limit ε 5.65) by writing B = B 0 /(1 + ε cos θ) and I = B 0 R 0 . In this limit, the parallel gradient can be written as ∇ = 1/(qR 0 )∂ θ .

= r/R 0 1, R ,zi = m z N z,0 ν zi T 0 e z ∂ ψ ln P z,0 - e z e i ∂ ψ ln P i,0 + C 0 e z e i ∂ ψ ln T i,0 2r cos θ (
If only the poloidal asymmetry of impurity density is considered, the parallel projection of the force balance equation (5.59) reads 5.66) where L Y ≡ |∂ r ln Y | -1 is the gradient length of the quantity Y . The integration of the expression (5.66) along θ for highly-charged heavy impurities (e z /e i 1) gives

1 qR 0 ∂ θ N z N z,0 = - 2q R 0 ν zi ω c,z R 0 L Nz + e z e i R 0 L N i -(C 0 -1) e z e i R 0 L T i cos θ (
N z,1 N z,0 = -2q 2 e z e i ν zi ω c,z R 0 L N i -(C 0 -1) R 0 L T i sin θ
(5.67)

NUMERICAL STUDIES WITH THE XTOR-2F CODE 91

Hence, the ordering used to derive the Prsch-Schlüter uxes in Equation 5.62 reads

2q 2 e z e i ν zi ω c,z R 0 L N i -(C 0 -1) R 0 L T i 1 (5.68)
It should be noted that the relative level of poloidal asymmetry of the impurity density N z,1 /N z,0 is proportional to the parameter δ 1. However, the assumption of small poloidal asymmetries is far from being satised. Indeed, N z,1 /N z,0 is also proportional to the square of the safety factor q 2 , the impurity charge and the ion density and temperature gradient lengths. Therefore, the assumption becomes easily invalid in the edge region, where the safety factor values can go up to q 5 or more, and the ion density and temperature gradients are steep. In the core region, q 1 and the ion density and temperature gradient lengths are usually large. Hence, the validity condition is fullled in the framework of this study.

5.3

Numerical studies with the XTOR-2F code 5.70) where V z is the impurity uid velocity as dened in (5.45), D ⊥,z and V p,z are the perpendicular impurity diusion coecient and radial pinch velocity, S z is the impurity density source, and R ,zi is the parallel friction force represented in (5.50). The impurity pressure is P z = N z T , by assuming T = T z = T i due to collisional thermalization.

∂ t N z + ∇ • (N z V z ) = ∇ • (D ⊥,z ∇N z -V p,z N z ) + S z (5.69) N z m z ∂ t V ,z + (V z • ∇)V ,z = -∇ P z + N z e z E + R ,zi (
It is necessary to have ∇ • Q i = 0 to derive the expression (5.58) for the main ion species. However, the energy equation (2.47) is written for the total pressure P = P e + P i in the current XTOR-2F model, which would only give ∇ • (Q e + Q i ) = 0. In this case, it is not straightforward to deduce the ion heat ux incompressibility from the current energy equation (2.47). Moreover, the contribution of the diamagnetic heat ux becomes negligible in the expression of Q e + Q i for T e T i and N e N i , because of opposite charge numbers.

Therefore, an additional equation for the main ion species is added 5.72) with τ = T i /T e , in order to have a heat source consistent with the total heat source S H of Equation (2.47).

∂ t P i + V • ∇P i + γ e i Ξ • ∇(P i T i ) = (γ -1) ∇ • (N χ ,i ∇ T i b) + ∇ • (N χ ⊥,i ∇ ⊥ T i ) + S H i ( 5
S H i = τ Z i + τ S H (
The mesh used in the following simulations is (M r , M θ , M φ ) = (255,32,12), where M r , M θ , M φ are the number of grid points in r, θ, φ directions respectively. A total of 4 toroidal modes (n = 0, ..., 3) and n + 8 poloidal modes (m = 0, ..., n + 7) for each toroidal mode, are evolved in the simulations.

Plasma parameter settings and initial conditions

As mentioned in Chapter 2, the Lundquist number is rescaled to S = 10 7 for the XTOR-2F sawtooth simulations. Other plasma parameters are rescaled so as to preserve the ratios between dierent characteristic time scales with respect to experimental values. The diamagnetic parameter is then set to di = 0.1, which is about two or three times larger than the experimental values. Also, the normalized perpendicular heat diusivity is set to χ⊥ = 3×10 -6 . The products S di and S χ⊥ are then about a third of those expected from the experiments.

As far as the viscosity is concerned, measurements of this quantity in experiments are not available. In this paper, the normalized plasma viscosity is set to μ = 5 × 10 -6 , so that the magnetic Prandtl number is P r m = S μ = 50. The parameter τ ≡ P i /P e becomes τ = T i /T e under the quasi-neutrality constraint N e = N i . In the following simulations, electron and ion temperatures are equal, i.e. τ = 1. It should be noted that this is usually not the case in experiments as dierent auxiliary heatings are added. The initial density and temperature proles used for the present study are shown in Figure 5.4-a. For the following simulations, a at initial impurity density prole is set. The main ion density is normalized to the on-axis density value N 0 = 3 × 10 19 m -3 . The impurity is considered as a trace in the simulations, with a concentration N z, 0/N 0 = 10 -5 such that the bulk MHD dynamics is not aected. The normalized impurity density is Nz = 1 with Nz ≡ N z /N z,0 .

In the present XTOR-2F model, the transport coecients D ⊥,z and V p,z on the right hand side of the impurity density equation (5.69) include the contributions of turbulent, classical and neoclassical transport. However, calculations of Section 5.2.3 show that a Prsch-Schlüter impurity ux is driven by poloidal asymmetries. Thus, it is important to notice that the Prsch-Schlüter particle ow is intrinsically present in the ∇ (5.69). Hence, the neoclassical part of the coecients D ⊥,z and V p,z contain only the classical and the banana-plateau coecients. In Figure 5.4-b, the red dashed line represents the diusion coecient D ⊥,z on the right hand side of Equation ( 5.69) which is set to reproduce the experimental radial shape. Indeed, this diusion coecient prole is based on uctuation measurements which show that the turbulence intensity is small inside the q = 1 surface [START_REF] Mattioli | Laser blow-o injected impurity transport in L mode Tore Supra plamsas[END_REF][START_REF] Guirlet | Particle transport in low core turbulence Tore-Supra plasmas[END_REF]. The blue solid line in Figure In the simulations, fully-stripped carbon (C 6+ ) is chosen as the impurity species. The impurity is present as a trace, without any feedback on MHD quantities. Using carbon ion allows one to avoid accounting for centrifugal force and radiation. As high-Z impurities are in a highly collisional regime, the collisionality of carbon ion is rescaled to a large value such that it is in the Prsch-Schlüter regime. The results are expected to be the same for tungsten provided it behaves as a tracer, and centrifugal and radiation eects are negligible.

• Γ z = ∇ • (N z V z ) term of Equation

Simulation results

Temperature screening eect from thermal force It was shown in Section 5.2.3 that the Prsch-Schlüter impurity ux is related to the collisional force R zi between the main ion species and the impurity. Moreover, the expression of the impurity radial ux in Equation (5.64) shows that the drag force and the thermal force give respectively the appropriate contributions of the main ion density gradient and the temperature gradient. In a rst step, simulations without the internal kink mode are performed in order to verify whether the implemented model reproduces expected features of the radial impurity ux.

To illustrate this point, cases with and without thermal force are simulated by setting appropriate values of the coecient C 0 in the expressions of the parallel friction force (5.50) and the impurity radial ux (5.64). With the density and temperature proles presented in Figure 5.4-a, an inux of impurity is obtained by considering only the drag force (C 0 = 0).

For the case with thermal force, the coecient C 0 is set to C 0 = 3/2, so that an outward ux of impurity appears due to the temperature screening eect, as predicted in Ref. [START_REF] Hirshman | Neoclassical transport of impurities in tokamak plasmas[END_REF]. In Figure 5.5, the time evolution of the ux surface average of the impurity density prole is presented for the two cases. These gures show that the impurity radial uxes behave in accordance with the analytical calculations.

Eects of sawtooth oscillations on impurity transport

With the rescaled plasma parameters mentioned in the previous paragraph, sustained sawtooth cycles are obtained with the XTOR-2F code (Figure 5.6). These simulations are used to assess the impurity behaviour in sawtoothing plasmas. The neoclassical time scales are very large compared to the duration of sawtooth crashes. Hence, the temperature screening eect, resulting from the collisions between the main ion species and the impurity, aects the impurity dynamics only during the sawtooth ramp phase.

The evolution of the surface averaged impurity proles throughout the sawtooth cycles is presented in Figure 5.7-a. The impurity proles are those between the post-cursor and precursor oscillations of a sawtooth cycle. It is noticed again that when the temperature screening eect is taken into account, an impurity outow is driven throughout the sawtooth cycles. However, a clear dierence on the impurity behaviour is seen between the simulations with and without sawteeth (Figure 5.7-b). The surface averaged impurity proles after t 3.2 × 10 5 τ A at a quasi-stationary state, show that the impurity proles get atter in the The presence of sawtooth cycles decreases the quantity of impurities expelled from the core region.

presence of sawteeth. Indeed, the central impurity density has decreased by about 2% for the case with sawteeth, which is less than a third of what is obtained for the case without sawteeth.

The weakened temperature screening eect is partly related to the time averaged temperature prole which is less peaked than the initial temperature prole. This is due to a periodic attening of the temperature prole at sawtooth crashes [START_REF] Porcelli | Model for the sawtooth period and amplitude[END_REF].

Hence, the overall radial pinch velocity of the impurity is less important than in the case without sawteeth, where the temperature prole remains unchanged. Also, the particle redistribution during the sawtooth crash tends to weaken the impurity outow. Indeed, sawtooth crashes drive an ecient radial transport of impurities. In particular, the Kadomtsev reconnection model [START_REF] Kadomtsev | On disruptive instability in tokamaks[END_REF][START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF] predicts an inux of impurity during sawtooth crashes for a hollow pre-crash impurity prole [Ödblom et al., 1996], regardless of its steepness (Figure 5.8). However, in the XTOR-2F simulations, a density utter of about 5% is found in the impurity proles after a sawtooth crash (blue dashed lines). Impurity penetration is not as visible as predicted by the Kadomtsev reconnection model, if the precrash impurity density prole is atter than the corrugations due to the crash (Figure 5.8-a).

On the contrary, the impurity prole after a sawtooth crash obtained by the XTOR-2F code is consistent with Kadomtsev's prediction for a hollow pre-crash prole (Figure 5.8-b). The radial redistribution of particles due to sawtooth crashes is sucient to weaken the outward impurity ux compared to the case without sawteeth, although the impurity proles in the present study are almost at.

Correlation of impurity dynamics with convective cells

In this paragraph, the mechanisms that underlie the impurity transport during a crash are investigated in more detail. For this purpose, dierent time points during a sawtooth crash are selected and marked by vertical lines in Figure 5.9-a. The corresponding magnetic congurations are displayed as Poincaré plots in Figures 5.9-b to 5.9-f. In Figure 5.10, the impurity density that corresponds to the time points marked in Figure 5.9-a are shown in color background. As the electron mass is very small compared to the ion mass, electron heat conduction is expected to be faster than the ion convection. Thus, the temperature contours It can be noticed that the impurity density structures shown in Figure 5.10 bear similarities with the patterns in the Poincaré maps of the eld line (Figure 5.10), with a slight poloidal shift. This is an indication of the role of the parallel dynamics, which tends to align the impurity density structures with the magnetic topology. At the same time, it can also be seen that the orientation of these structures is reminiscent of the E × B velocity eld. These structures are driven by the E × B ow. For example, in Figure 5.10-b, the impurity density bump and hole located respectively at θ π/2 and θ π, are more or less similar to magnetic surfaces in the island (Figure 5.9-c). These same structures are then rotated along the reconstructed streamlines c).

Since it is dicult to establish the relative roles of parallel and E × B ows using maps, another method is adopted here. It is based on the comparison of the two components of CHAPTER 6

Conclusion

The production of sustained electricity using Deuterium-Tritium fusion reactions oers a promising alternative to present sources of electricity. Yet, several issues must be solved before achieving a commercial fusion reactor. Impurity accumulation in the plasma core region is one of these issues as it may degrade fusion performances due to dilution and radiative cooling. Therefore, impurity transport has become a subject of utmost importance.

Impurity concentration is determined by turbulence and neoclassical physics on long time scales, of the order of the connement time. However, radial transport is also aected by MHD (Magneto-Hydro-Dynamic) events which occur on much faster time scales. In particular, sawtooth cycling plays a crucial role as it periodically redistributes the impurity density in the core region. Thus, the global impurity behaviour results from a complex combination of dierent transport processes : MHD instabilities, collisions and turbulence.

There is no code so far able to address these 3 transport processes at the same time.

However, it appears that collisional and MHD transport can be described in the same framework, bound on uid equations. Indeed, heavy impurities are in the high collisional (Prsch-Schlüter) regime due to high charge numbers. This allowed tackling this subject with one tool only, to address two classes of processes out of three. In addition, turbulent transport can be taken into account by adding appropriate transport coecients. This thesis work aimed at studying heavy impurity transport in sawtoothing plasmas using non-linear tridimensional MHD simulations with the XTOR-2F code.

The basic numerical tools used in this work are the CHEASE code and the XTOR-2F code, which model the magnetic equilibrium and non-linear two-uid MHD respectively. A set of rescaled plasma parameters is used to simulate sustained sawtooth cycles with the XTOR-2F code, since using an experimental plasma resistivity in simulations is numerically demanding. The impurity behaviour in sawtoothing bulk plasmas can now be simulated with the XTOR-2F code in an elongated cross-section, in order to compare simulations with experiments.

Features that bear similarities with compound sawteeth reported in experiments were noticed in several simulations in an elongated geometry. A detailed analysis on the simulated compound sawteeth showed that the plasma hot core is radially displaced and poloidally rotates during the partial crash. It is only partially expelled out of the q = 1 surface and returns back to the center after the partial crash. This behaviour is consistent with experimental observations. The unstable internal kink mode is shown to be responsible for the displacement of the hot core. Partial crashes occur when the q = 1 radius exceeds a critical value, at xed poloidal beta β p . This critical value is shown to depend on the plasma elongation, which was the other control parameter used in this work. It has been revealed that the model used for the heat diusivity plays an important role as it aects the temperature prole, thus the current prole via the resistivity. Also, a weaker diamagnetic stabilization during the ramp phase made the partial crash time larger than the collapse time of an ordinary sawtooth.

Thus, a partial crash is shown to be the result of a competition between destabilizing effects such as the q = 1 radius and diamagnetic stabilization. The consistency of simulated compound sawteeth with experimental observations again conrmed the capability of the XTOR-2F code to be used as a predictive model.

The behaviour of tungsten in ASDEX-Upgrade sawtoothing plasmas was evaluated using 2-D SXR (Soft X-Ray), electron density and temperature measurements. The objective of these discharges was to study the eects of core deposition of EC (Electron Cyclotron) waves on controlling tungsten accumulation in presence of sawteeth. The observations showed that the tungsten density prole is hollow when EC waves are located near the magnetic axis and is likely associated with the internal kink mode with a lower mode frequency. The predictions by turbulent and neoclassical transport models did not reproduce the observations. The global tungsten transport is indeed a result from complex physics combining MHD, collision and turbulence. Thus, a numerical tool which models MHD instabilities and includes appropriate impurity transport model has been developed during this thesis in order to investigate and understand experimental observations.

Since tungsten is in the Prsch-Schlüter regime, uid equations can be used to model its transport. Thus, the eects of sawteeth on impurity transport are studied by deriving appropriate uid equations for impurities in the high collisionality (Prsch-Schlüter) regime,

  Figure 1.1: World energy consumption since 1965 for dierent energy sources 2 . The units are in million tonnes for oil and million tonnes oil equivalent for other energy sources. A tonne oil equivalent equals approximatively 12 MWh of electricity.

Figure 1 . 2 :

 12 Figure 1.2: Aston's curve which represents binding energy per nucleon versus mass number.

  Figure 1.3: Cross sections of the fusion reactions (1.3) to (1.6) as function of energy [Huba,

Figure 1 . 4 :

 14 Figure 1.4: A schematic gure of a hohlraum 3 . The multiplied laser beams penetrate in the hohlraum by the Laser Entrance Hole (LEH) and hit the wall made of gold ( 197 79 Au). X-ray

  Figure 1.6: (a) A schematic view of a tokamak device 4 . (b) Geometric parameters of a plasma

5

 5 for spherical tokamaks, e.g. MAST-Upgrade (Culham Centre for Fusion Energy) and NSTX-Upgrade (Princeton Plasma Physics Laboratory), but typical values are near 3. The ellipticity of the plasma is quantied by the plasma elongation the height of the highest LCFS point. The triangularity δ is used to quantify the `D-shape' of the plasma. It is dened as the relative position of the X-point with respect to the geometrical axis

Figure 1

 1 Figure 1.7: Schematic representation of sawtooth oscillations 5 .

  Figure 1.9: Schematic representation of magnetic islands. Each line represents a particle orbit. The separatrix are shown in bold lines and O-points are marked by `+'.

  to study sawtooth cycles are presented in Chapter 2. The dierent sets of MHD equations used to study numerically sawteeth are derived from kinetic theory. The principles of the internal kink mode stability and magnetic eld reconnection during sawtooth crashes are briey presented. The basic numerical tools used in this work are the equilibrium code CHEASE and the non-linear MHD XTOR-2F code. Details on the numerical aspects of the XTOR-2F code are presented at the end of the chapter.

Figure 2 . 2 :

 22 Figure 2.2: Magnetic ux surfaces of a circular equilibrium computed by the CHEASE code.

Figure 2 . 3 :

 23 Figure 2.3: Schematic representation of the Kadomtsev reconnection model during sawtooth crash. After the collapse, the core region is replaced by a cooler island.

Figure 2 . 4 :

 24 Figure 2.4: (a) Dependence on β p and χ ⊥ of kink cycling and helical state regimes for a `low shear' equilibrium[Halpern et al., 2011a]. (b) Dierent regimes of cycling when varying S and di[Halpern et al., 2011b].

  crashes during sawtooth ramp phase . . . . . . . . . . . 33 3.2 Set up of parameters for XTOR-2F simulations . . . . . . . . . 35 3.2.1 Physical parameters rescaling . . . . . . . . . . . . . . . . . . . . after a sawtooth full relaxation . . . . . . . . . . . . . . . 38 3.3.2 Stability of the n = 1 mode during a sawtooth cycle . . . . . . . . 40 3.3.3 The partial crash during the sawtooth ramp phase . . . . . . . . . 43 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 3

 3 Figure3.1: Experimental conditions on the electron density and total current to obtain compound sawteeth in Tokamak de Varennes discharges[START_REF] Janicki | Tomographic analysis of counpound sawteeth on the Tokamak de Varennes[END_REF]. It shows that, for a xed value of electron density, compound sawteeth are observed when the total current value is above a threshold.

  Figure 3.2: Time evolution pressure at dierent radii during sawtooth cycles in the cross section geometry (a) κ a = 1.6, (b) κ a = 1.8. For both cases, δ a = 0.4 and the q = 1 surface is at r s = 0.4 a. The dierence between the initial equilibrium and the asymptotic equilibrium is marked by the dierence on the sawtooth period.

  κ

Figure 3

 3 Figure 3.3: The linear growth rate of the internal kink mode in ideal MHD for dierent values of β p in an elongated plasma cross section (κ a = 1.6, δ a = 0.4) and the q = 1 surface at r s = 0.4 a. In this case, β p,c 0.15.

Figure

  Figure 3.4: q-proles of the initial (dashed line) and the asymptotic (solid line) equilibria

  Figure 3.5: (a) Time evolution of the pressure at dierent radii for sawtooth simulationswith the case κ a = 1.6, δ a = 0.4 andr s = 0.4 a. (b) Evolution of q-proles after the sawtooth crash, each time point is marked by a vertical line in (a). The q-prole is at at the sawtooth crash and becomes non-monotonic during the ramp phase.

Figure 3

 3 Figure 3.6: Temperature proles before (dashed line) and after (solid line) the sawtooth crash. The inversion radius is represented by the vertical line.

  Figure 3.7: Time evolution of the pressure proles (at θ = 0) during the ramp phase for the case κ a = 1.6, δ a = 0.4 and r s = 0.4 a. The times corresponds to Poincaré plots in Figure 3.8. The inversion radius is represented by dashed vertical lines.

  .5). Magnetic structures with multiple helicities such as (2,2), (3,3) are also visible in the Poincaré plots (e.g. in Figure 3.8-d).

  Figure 3.8: (a) Time evolution of the pressure at dierent radii. (b)-(f ) Poincaré plots ofthe core region (r < 0.6 a) after post-cursor oscillations during the time interval marked by vertical lines in (a), for the case without partial crash in the geometry κ a = 1.6, δ a = 0.4 and the q = 1 surface at r s = 0.4 a. The dashed lines in the Poincaré plots represent the q = 1 surface at equilibrium.

Figure 3 .

 3 Figure 3.10: Time evolution of the kinetic energy of the n = 1 mode from reconstructed equilibria at t = 1.50 × 10 5 τ A (solid line) and t = 1.58 × 10 5 τ A (dashed line). The two times correspond respectively to the Poincaré plots in Figures 3.8-b and 3.8-f. For the latter case (dashed line), the energy is below the noise level of the solver.

Figure 3 .

 3 Figure 3.11: Time evolution of the pressure proles (at θ = 0) during the ramp phase for the case κ a = 1.8, δ a = 0.4 and r s = 0.4 a. The times correspond to pressure contour plots in Figure 3.12. The inversion radius is represented by dashed vertical lines.

  Figure 3.12: (a) Time evolution of pressure at dierent radii, zoomed at the time interval where the partial crash occurs. (b)-(f ) The pressure contours of the core region (r < 0.6 a)at dierent time points marked by vertical lines in (a). In the contour plots, some of the ux surfaces and the q = 1 surface at equilibrium are represented respectively by thin solid lines and dashed lines.

Figure 3 .

 3 Figure 3.14: In solid lines, the time evolution of the kinetic energy of the n = 1 mode from reconstructed equilibria before (t = 1.22 × 10 5 τ A , solid line) and after (t = 1.33 × 10 5 τ A , dashed line) the partial crash (κ a = 1.8, δ a = 0.4 and r s = 0.4 a). The two times correspond respectively to the Poincaré plots in Figures 3.13-a and 3.13-e. In dashed line, the same plot as Figure 3.10 (κ a = 1.6, δ a = 0.4 and r s = 0.4 a) is represented for comparison (+).

  Figure 3.15: (a) FSR and (b) ECE measurements in the equatorial plane during a compound sawtooth of the Tore Supra discharge #44634. (c) The 2D electron density reconstructions in the core region, which correspond to the time points marked by vertical lines in (a), againshows that the hot core is partially expelled during the partial crash.

  tracing in experiments . . . . . . . . . . . . . . . . . . 54 Metallic impurities resulting from plasma-wall interaction are always present even in small quantities. Indeed, they may be produced by plasma facing components (PFCs) due to sputtering (e.g. C, Mo or W), radio-frequency (RF) antennas (e.g. Fe, Ni or Cu) and diagnostic elements. Particularly, tungsten (Z W = 74) has been identied as the best candidate

Figure 4 . 1 :

 41 Figure 4.1: The cooling factor (blue dashed line) and fractional abundance (gray dashed lines) of tungsten as a function of electron temperature. Contributions of emission process other than line radiation is plotted for information (gray solid line). This gure is taken from

Figure 4 . 2 :

 42 Figure 4.2: Lines of sight of the SXR detectors in a) toroidal and b) poloidal cross-sections. In a), the lines of sight of SXR cameras H, I, J, K, L, M, F are shown. Only those of cameras I and J are represented in b) for visibility. An example of magnetic ux surfaces computed by EFIT is shown in b). The last closed ux surface (LCFS) is shown in bold solid line.

Figure 4 .

 4 Figure 4.2 shows the lines of sight of SXR detectors installed in ASDEX-Upgrade. Thanksto a sucient number of lines of sight and projections, a 2-D SXR emissivity map can be reconstructed with tomographic inversion. Such tomographic inversion is now routinely available on most tokamaks[START_REF] Mazon | Soft X-ray tomography for real-time applications : present status at Tore Supra and possible future developments[END_REF].

  BEHAVIOUR IN TOKAMAK DISCHARGESwith 1 ≤ i ≤ N L , 1 ≤ j ≤ N L and δ ij the Kronecker delta.

Figure 4

 4 Figure 4.3: a) Auxiliary power in the ASDEX-Upgrade discharge #31114. A total power of 7.5 MW of NBI heating and 1.4 MW of EC waves is injected in the plasma. b) Deposition radii of the 2 gyrotrons used during the discharge #31114. One gyrotron is used for heating and the other for current drive with 0.7 MW per gyrotron.

Figure 4

 4 Figure 4.4: a) Auxiliary power in ASDEX-Upgrade discharges #32409 and #32410. For both discharges, 7.5 MW of NBI heating is injected. The gyrotron power after t = 2.5 s is 0.2 MW and 0.4 MW for the discharges #32409 and #32410 respectively. b) ECCD deposition radius ρ RF during the ASDEX-Upgrade discharges #32409 and #32410.

Figure 4 . 5 : 4 (

 454 Figure 4.5: The a-b) electron temperature and c-d) line-integrated SXR measurements of the ASDEX-Upgrade discharge #31114 during the time intervals 4.85 s < t < 5.25 s and 7.02 s < t < 7.24 s, where the ECCD deposition is at ρ RF ρ inv and ρ RF > ρ inv respectively. The displayed measurements correspond to ρ pol 0 (blue), ρ pol 0.2 (red) and ρ pol 0.4 (green) (cf.Figure 4.6-a). Sawtooth crashes are marked with dashed vertical lines.

Figure 4 .

 4 6-a). Sawtooth crashes are marked with dashed vertical lines. factor as shown in Figure 4.1 N W (ρ, θ, t) ∝ E SXR (ρ, θ, t) N e (ρ, θ, t)L SXR W(T e (ρ, θ, t))(4.14)

4. 5 Figure 4 . 6 :

 546 Figure 4.6: Lines of sight of the SXR detectors (red lines) and ECE channels (blue dots) of which measurements are shown in Figures 4.5and 4.13. The equilibria which correspond to ASDEX-Upgrade discharges a) #31114 at t 5.0 s and b) #32409 at t 2.5 s are shown in the background.The displayed magnetic ux surfaces (dashed lines) from the magnetic axis (+) to the separatrix (bold lines) correspond to ρ pol = 0.2, 0.4, 0.6, 0.8 and 0.9 respectively.

Figure 4 .

 4 Figure 4.7: 2-D maps of a) the electron temperature T e , b) its variation ∆T e (t) ≡ T e (t) -T e (t = t 1 ) with respect to the start of the sawtooth cycle t = t 1 , c) SXR emissivity and d) tungsten density [Sertoli et al., 2015b]. The ECRH deposition at ρ RF 0.1 is represented by dashed black lines and the inversion radius is marked by thick white lines. Contours of ∆T e (t) are over-plotted on c)d) for correlation of the quantities with the mode structure.

Figure 4 .

 4 Figure 4.8: a)-h) Reconstructed 2-D SXR emissivity during a sawtooth crash at t 5.00 s of the ASDEX-Upgrade discharge #31114[Odstrcil, private comm.]. The q = 1 surface is

Figure 4 . 9 :

 49 Figure 4.9: Tungsten proles before (red line) and after (black line) a sawtooth crash when ρ RF ρ inv [Sertoli et al., 2015b]. The post-crash tungsten density prole is in a good agreement with Kadomtsev's model.

68 CHAPTER 4 .

 4 Figure 4.10: Comparison of the tungsten density distribution at t 4.97s of the ASDEX-Upgrade discharge #31114 with predictions from transport codes [Sertoli et al., 2015b]. a) The model predicts a peaked surface averaged tungsten density prole, whereas the measured prole is hollow. b)-c) The comparison of the 2-D tungsten density maps shows clear discrepancy.

Figure 4 .

 4 Figure 4.10 clearly shows that the transport models themselves are not able to reproduce the observed tungsten behaviour. The ux surface averaged tungsten density prole evaluated from the measurements is peaked at an o-axis radius ρ pol 0.45 and hollow inside the inversion radius ρ pol 0.35 (Figure 4.10-a). Meanwhile, the transport codes predict a tungsten prole peaked at the magnetic axis. Indeed, the computed impurity pinch velocity is directed inward. The discrepancy between the observation and the prediction is clearly visible in the 2-D tungsten density maps (Figure 4.10-b,c). The HFS/LFS asymmetry due to rotation is recovered by the transport model. However, the depletion of tungsten inside the inversion radius is not recovered.

Figure 4 .Figure 4 .

 44 Figure 4.11: Time evolution of the tungsten density proles during the discharges a) #32409 and b) #32410[Sertoli, private comm.] where the radial position of ECCD is scanned. The gyrotron power injected after t = 2.5 s is 0.2 MW for #32409 and 0.4 MW for #32410.

Figure 4 .Figure 4 .

 44 Figure 4.11 shows the evolution of the tungsten density proles during the discharges. It is clearly visible that the tungsten density prole is hollow at the beginning of both discharges until t = 2.5 s, when 1 MW of ECRH is injected near the magnetic axis. After the ECRH is switched o and replaced by 200 kW of ECCD at t = 2.5 s, the depletion of the tungsten density prole is barely visible in discharge #32409. The tungsten density prole is hollow only in the time interval 3.2 s < t < 3.8 s. This behaviour is illustrated by

Figure 4 .

 4 6-b).

Figure 4 .

 4 Figure 4.14: a)-b) The electron temperature measurement, c)-d) the spectrogram of the ECE channel [Sertoli, MST-1 TFM] located at ρ pol 0.1 and e)-f ) the tungsten density at dierent radii which correspond to the time intervals 3.71 s < t < 3.99 s and 4.05 s < t < 4.52 s of the discharge #32410.

Figure 4 .

 4 14-a), and 4.05 s < t < 4.52 s where the `bump' starts vanishing (Figure 4.14-b). Figures 4.14-c,d show the spectrograms of the ECE channel located at ρ pol 0.1. The spectrograms indicate that the `bumps' observed in the ECE channel 51 are characterized by a mode with a frequency f 13 kHz. Indeed, the change of sawtooth shapes observed at 4.05 s < t < 4.52 s is correlated with a lower amplitude of this mode. The tungsten density values at dierent radii during these time windows are shown in Figures 4.

Figure 5

 5 Figure 5.1: a) Schematic gures of passing and trapped particle trajectories projected in a poloidal cross section. b) A 3-D representation of a banana orbit.

Figure 5 . 2 :e 2 (

 522 Figure 5.2: Schematic gure of Coulomb collision between two charged particles with the same sign.

Figure 5 . 3 :

 53 Figure 5.3: Diusion coecients of dierent collisional regimes computed according to

•Figure 5

 5 Figure 5.3-a shows classical and neoclassical diusion coecients as function of collisionality in the limit of

  5.40) CHAPTER 5. TRANSPORT OF HEAVY IMPURITIES IN PRESENCE OF SAWTOOTH OSCILLATIONS and the impurity collisionality reads ν

2 P

 2 With the expressions(5.54) and(5.58), the parallel friction force R ,zi can be expressed as a function of dierent radial derivatives. The force balance equation(5.46) projected along the parallel direction of the impurity z gives N z e z ∇ φ = -∇ P z + R ,zi function G, by denition of ux-surface average operator in Equation(5.48). Hence, averaging Equation (5.59) over ux surfaces, with N z N z,0 (ψ), gives the constraint BR ,zi = 0(5.61)

Figure 5 . 4 :

 54 Figure 5.4: (a) The initial density (blue line) and temperature (red line) proles used in the simulations. (b) The initial diusion prole set to represent the turbulent transport (red line), and the initial Prsch-Schlüter diusion prole (blue line) computed intrinsically from the impurity uid equations.

Figure 5 . 5 :

 55 Figure 5.5: Evolution of the impurity proles for the cases (a) without thermal force (C 0 = 0), and (b) with an appropriate coecient for the thermal force (C 0 = 3/2) as predicted by the neoclassical theory. The temperature screening eect appears in presence of the thermal force.

Figure 5 .Figure 5

 55 Figure 5.6: Evolution of the pressure at dierent radii. The time points which correspond to the surfaces averaged impurity proles in Figure 5.7-a are marked by the vertical lines. These time points are chosen during the quiescent phase when the plasma has reached a quasi-stationary equilibrium.

Figure 5 . 8 :

 58 Figure 5.8: Impurity proles after sawtooth crash for (a) shallow and (b) hollow pre-crash proles (+). Post-crash impurity proles computed by the XTOR-2F code (blue lines) and with Kadomtsev reconnection model (red lines) are shown. The inversion radius r inv and the mixing radius r mix are marked by vertical lines. For a pre-crash prole with more contrasted hollowness, the penetration of impurity is clearly visible and is in accordance with the Kadomtsev reconnection model.

Figure 5 Figure 5 .Figure 5 .

 555 Figure 5.9: (a) Time evolution of pressure at dierent radii during sawtooth crash and (b)-(f )Poincaré of the core region (r/a < 0.6) plots corresponding to the time points marked by vertical lines. The q = 1 surface is represented by dashed lines.

  which have been implemented in the XTOR-2F code. The simulations without MHD instabilities reproduced the eects of the main ion density and temperature gradients which are consistent with neoclassical theory. The presence of regular sawtooth crashes is shown to atten the impurity prole. Also, a spatial non-uniformity of 5% in post-crash impurity density proles is obtained due to 2-D structures of impurity density which appear during sawtooth crashes. They are shown to be mainly driven by the E × B drift velocity, and responsible for the sudden impurity transport in the core plasmas. Finally, sawtoothing bulk plasmas with a saturated helical state are simulated with the XTOR-2F code by adding an ECCD (Electron Cyclotron Current Drive) source located inside the q = 1 surface. These simulations which are reminiscent of the ASDEX-Upgrade discharge scenario show that the saturated internal kink mode is responsible for tungsten depletion in the core region. Indeed, the impurity density prole remains atter than what was expected for ordinary sawtooth cycles. However, hollow impurity density proles as observed in the ASDEX-Upgrade discharge are not reproduced. This could be due to limitations of the closure that is being used 110 APPENDIX A. RÉSUMÉ LONG DE LA THÈSE de-scie et son contrôle en injectant dans la région centrale des ondes dont la fréquence correspond à la fréquence cyclotronique électronique. Le diagnostic X-mou est l'outil principal qui permet de suivre l'évolution des impuretés lourdes dans le coeur du plasma. Des mesures expérimentales montrent que la dynamique des impuretés en présence des dents de scie est diérente des prédictions faites par les codes de transport qui ne tiennent pas compte de la MHD. En eet, le comportement des impuretés résulte d'une combinaison complexe des diérents processus de transport. Il est alors nécessaire d'avoir un outil numérique capable de modéliser les instabilités MHD en ayant un modèle approprié du transport d'impuretés, an de poursuivre et compléter les études expérimentales.En l'absence d'instabilités MHD, le transport des impuretés est porté par les collisions (transport néoclassique) et la turbulence. Il est établi que le transport néoclassique est important pour les impuretés lourdes dans la région centrale du plasma de tokamak. Les équations uides modélisant le transport des impuretés dans un régime de collisionalité élevée (Prsch-Schlüter) ont été implémentées et couplées à l'ensemble des équations de la MHD bi-uide.Le couplage des équations a conduit à une amélioration de la matrice du préconditionnement, ce qui est un progrès apporté dans le code XTOR-2F en terme numérique. Dans les simulations, les impuretés sont considérés comme des traceurs an de négliger les rétroactions sur le plasma de fond. Les résultats des simulations sans instabilités MHD sont cohérents avec le transport radial prédit par la théorie néoclassique. Les simulations avec des dents-de-scie montrent que la présence des relaxations pe¯iodiques aectent signicativement le transport des impuretés, en accord avec les observations expérimentales. Ceci résulte d'une compétition entre processus néoclassiques et relaxations dues aux dents de scie. Premièrement, l'eet favorable d'écrantage thermique est aaibli par rapport au cas sans dents-de-scie. Deuxièmement, une uctuation de 5% apparaît dans les prols de densité d'impuretés après un crash dents-de-scie. Elle est due aux structures de densité d'impuretés qui apparaissent pendant la phase du crash. Il est montré que ces structures sont principalement convectées par la vitesse E × B. Enn, un état hélicoïdal saturé, qui se rapproche des scénarios expérimentaux, est simulé avec le code XTOR-2F en ajoutant une source de courant. Il s'avère qu'une durée prolongée des oscillations du précurseur fait que les prols de densité d'impuretés restent plus plats que ceux prédits pour les cycles de dents-de-scie ordinaires.The impact of two-uid MHD instabilities on the transport of impurities in tokamak plasmas Keywords : MHD, sawteeth, impurity transport, signal analysis Abstract : Impurity accumulation can degrade the performance of magnetically conned fusion plasmas. In particular, heavy impurities accumulated in the core plasma dilute fusion reactants and may also lead to a radiative collapse of the plasma due to excessive cooling by radiation.Therefore, understanding the transport of heavy impurities produced by plasma-wall interaction has become a subject of utmost importance. The plasma core is likely to be aected by a magnetohydrodynamic (MHD) instability called `internal kink' that induces relaxation oscillations named `sawteeth'. Sawteeth are responsible for periodic relaxations of the core density and temperature and aect signicantly the radial transport. Especially, particles and heat are redistributed during the crash phase the duration of which is short compared to the connement time. In absence of MHD instabilities, impurity transport is governed by collisions (neoclassical transport) and turbulence. It is shown that neoclassical transport is important for heavy impurities in the core region of tokamak plasmas. Meanwhile, experimental measurements in the ASDEX-Upgrade tokamak show that the impurity dynamics in presence of sawteeth diers from the predictions made by transport codes. In this thesis, the numerical tool used to simulate sawteeth is the XTOR-2F code, which is a non-linear tridimensional code solving two-uid MHD equations. Fluid equations that model the transport of impurities in a highly collisional (Prsch-Schlüter) regime have been implemented and coupled to the set of two-uid MHD equations. The simulations show a dierence between the impurity proles with and without sawteeth, which is consistent with experimental observations. This results from a competition between neoclassical processes and sawtooth relaxations.I * * * Impact des instabilités MHD bi-uide sur le transport d'impuretés dans les plasmas de tokamaks Mots-clés : MHD, dents-de-scie, transport d'impuretés, analyses des signaux Résumé : Les performances des plasmas de fusion connés magnétiquement peuvent être dégradées par l'accumulation d'impuretés. Plus particulièrement, les impuretés lourdes accumulées au centre du plasma diluent les réactifs, et peuvent aussi conduire à un collapse radiatif du plasma par de fortes pertes par rayonnement. La compréhension du transport des impuretés lourdes produites lors de l'interaction plasma-paroi est donc devenue cruciale. Le coeur du plasma est sujet à une instabilité magnétohydrodynamique (MHD) appelée kink interne, conduisant à des oscillations de relaxation nommées dents-de-scie. Les dents-de-scie entraînent une relaxation périodique de densité et de température dans le coeur du plasma, et aectent signicativement le transport radial. Notamment, les particules et la chaleur sont redistribuées pendant un crash dont la durée est très courte par rapport au temps de connement. En l'absence des instabilités MHD, le transport des impuretés est porté par les collisions (transport néoclassique) et la turbulence. Il est établi que le transport néoclassique est important pour les impuretés lourdes dans la région centrale du plasma de tokamak. Cependant, des mesures expérimentales du tokamak ASDEX-Upgrade montrent que la dynamique des impuretés en présence des dents-de-scie est diérente des prédictions faites par les codes de transport. Dans cette thèse, l'outil numérique utilisé pour simuler les dents-de-scie est le code XTOR-2F, qui est un code non-linéaire tridimensionnel résolvant les équations de la MHD bi-uide. Les équations uides modélisant le transport des impuretés dans un régime de collisionalité élevée (Prsch-Schlüter) ont été implémentées et couplées avec l'ensemble des équations de la MHD bi-uide. Les simulations montrent que les prols de densité d'impuretés sont aectés par les dents-de-scie, en accord avec les observations expérimentales. Ceci résulte d'une compétition entre processus néoclassiques et relaxations dues aux dents-de-scie.I

  written for the electron. This mass disparity allows us to consider incompressible heat uxes ∇ • Q a = 0 for electrons and ions at the 24 CHAPTER 2. SAWTOOTH OSCILLATIONS IN THE CORE REGION lowest order. Hence, the sum of electron and ion momentum equations reads

  4 [Pütterich et al., 2010]. Hence, it is crucial to understand how impurities are

	transported in tokamak plasmas. In general, intrinsic impurities are stationary or evolve with
	the bulk plasma. It is then dicult to determine impurity transport coecients. Therefore,
	for impurity transport studies, trace impurities are injected in the plasma by gas pu for gas
	impurities, or laser blow-o (LBO) for metallic impurities.
	Dedicated experimental campaigns were held in ASDEX-Upgrade and TCV. The objec-
	tive of the discharges was to investigate heavy impurity transport in the presence of sawteeth
	and the eects of electron cyclotron (EC) waves on controlling its accumulation. In this chap-
	ter, the principle of soft X-ray (SXR) measurements and tomographic inversions, which are
	mainly used to trace impurities, are briey introduced in Section 4.1. Then, dierent dis-
	charge scenarios performed to investigate the impurity behaviour are presented in Section
	4.2.2. Several discharges among the performed scenarios are analysed in Section 4.2.3 to
	deduce dierent impurity behaviour depending on the experimental set-up. Finally, possible
	explanations to the observations are discussed in Section 4.2.5.
	4.1	Impurity tracing in experiments
	4.1.1 Radiation in the core plasmas

  at the next order in ω/ω c , if the electric eld E is time dependent with a time scale ω -1 .

	5.1. PARTICLE TRAJECTORIES AND COLLISIONS	77
	as	v P =	1 Bω c	∂E ⊥ ∂t	(5.6)
	must be added

United Nations Climate Change Conference (http://www.cop21.gouv.fr)

Data from BP statistical review of world energy 2016 (http://www.bp.com)

Image from the website of National Ignition Facility (http://lasers.llnl.gov/)

Image from[ITER Physics Expert Group on Disruptions, Plasma Control, and MHD and ITER Physics Basis Editors, 1999]

Images from a lecture of plasma physics, Australian National University

conguration [START_REF] Garcìa-Regaña | On neoclassical impurity transport in stellarator geometry[END_REF].

• Second, the underlying reconnection process related to the sawtooth shape modication by EC waves provides another option in an MHD framework. Indeed, the radially displaced hot core which remains inside the q = 1 surface is reminiscent of the partial crash described in Chapter 3. As Figure 3.16 shows, the energy content inside the q = 1 surface slightly and slowly decreases during the partial crash, which is a signature of magnetic reconnection. This may lead to the onset of a large impurity density gradient.

If it is the case, impurities are then slowly ushed out from the core region during the reconnection.

The observed impurity behaviour results from complex physics which combine MHD, neoclassical and turbulent transport. However, dierent transport processes must not be treated separately as synergistic eects exist between neoclassical and turbulent transport [START_REF] Estève | Etude gyrocinétique du transport multi-espèces néoclassique et turbulent dans un plasma de fusion[END_REF], MHD and 3-D transport. Consequently, a model including the transport theory and the sawtooth instability is necessary to investigate the underlying mechanism which expels tungsten from the hot core.

Summary

In this chapter, the impurity behaviour in experiments is described.

• The SXR diagnostic has a great advantage to give measurements proportional to the impurity density at a fast acquisition frequency. Combined with electron density and temperature measurements, the tomographic inversion of SXR measurements allows one to assess the 2-D impurity distribution in the core during a fast MHD events such as sawteeth. The impurity behaviour during ASDEX-Upgrade discharges is deduced from these measurements.

• The impurity proles observed in ASDEX-Upgrade discharges are dierent from the predictions with transport codes.

• Two main conclusions can be drawn from the detailed analysis of measurements. First, the impurity prole is hollow when EC waves are injected in the core region. Second, the depletion of tungsten in the core region is closely related to the internal kink mode characterized by a lower frequency.

As an intermediate step towards the understanding of the impurity behaviour observed in ASDEX-Upgrade discharges, the next chapter focuses on modelling heavy impurity transport in sawtoothing plasmas.

The impurity behaviour is a result of a complex combination of MHD driven transport, neoclassical and turbulent transport. However, no code is so far able to address the 3 transport processes at the same time. Since heavy impurities are highly collisional in the core, 74 CHAPTER 4. IMPURITY BEHAVIOUR IN TOKAMAK DISCHARGES appropriate uid equations can be used to model neoclassical transport. Therefore, the interaction between MHD and neoclassical transport can be addressed with the XTOR-2F code. Three main milestones have to be achieved in order to study such ASDEX-Upgrade scenarios using the XTOR-2F code.

• An appropriate heavy impurity transport model has to be implemented in the XTOR-2F code.

• A new set of rescaled parameters has to be found in order to simulate sustained sawtooth cycles with ASDEX-Upgrade equilibrium and geometry.

• A sawtoothing bulk plasma with saturated (1,1) mode has to be simulated with the implemented ECCD propagation model [START_REF] Février | Global modelling of magnetic island control in tokamaks[END_REF].

The geometrical aspect of sawtooth dynamics has already been discussed in Chapter 3.

Chapter 5 focuses on a particular point : impurity transport in sawtoothing plasmas. The study of the combined eect of neoclassical transport and sawteeth is presented. of the impurity ux along the magnetic eld line Γ z,δB (blue lines) and the ux driven by E × B velocity Γ z,E (red lines). The q = 1 radius is marked by vertical lines.

the impurity ux. This approach provides quantitative correlations of the impurity density structures with magnetic utter and E × B convection. From the expression (5.45) of the velocity, the impurity uxes parallel and perpendicular to the magnetic surfaces respectively reads

The impurity ux driven by the diamagnetic velocity is neglected since Z 1 for heavy impurities.

In Figure 5.11, the radial and poloidal components Γ ψ z,δB , Γ θ z,δB and Γ ψ z,E , Γ θ z,E in the poloidal section at ϕ = 0, are represented, by dening

These uxes are calculated at time t 1.205 × 10 5 τ A (Figure 5.11), when the magnetic reconnection process during sawtooth crash has almost ended. At this time, the expelled hot core is located outside the q = 1 surface, and the cooler island occupies a large part of the core region (Figure 5.9-e). Also, complex impurity density structures are seen inside the

Γ θ z,δB in most of the core region, i.e. the E × B ux overcomes the magnetic utter ux in all directions. However, in some specic locations, e.g. π/2 < θ < π and outside the inversion radii, Γ ψ z,E Γ ψ z,δB and (Figures 5.11, 5.12). Hence, the impurity structures during the sawtooth crash are driven by the two uxes, but the E × B velocity plays a prominent role in most locations during sawtooth crashes.

In Ref. [START_REF] Nagayama | Analysis of sawtooth oscillations using simultaneous measurement of electron cyclotron emission imaging and X-ray tomography on TFTR[END_REF], soft X-ray (SXR) and temperature measurements are 5.4. TOWARDS COMPARISON WITH EXPERIMENTS 101 analysed in detail during a sawtooth crash. The presence of the background metal impurities allows the impurity behaviour to be deduced from SXR emission. It is shown that a SXR emission peak is expelled out of the q = 1 surface during a sawtooth crash, along with the hot core. However, SXR and temperature contours become uncorrelated after the sawtooth crash. Particles escape from the reconnection layer during the sawtooth crash and are transported with their respective thermal velocity. Hence, the impurity transport time scale is large compared to the electron one which can be associated with the modications of the magnetic conguration [START_REF] Wesson | Transport in the sawtooth collapse[END_REF]. The impurity behaviour during a sawtooth crash obtained by the XTOR-2F code is consistent with this picture. Indeed, the impurity density structures are not representative of magnetic ux surfaces, and they are spread along streamlines associated with the E × B velocity.

As a consequence, the impurity density bump and hole result from the time lag between impurity and electron transport.

Towards comparison with experiments

In the experimental scenarios analysed in Chapter 4, the dynamics of sawteeth are modied and the saturated helical state is obtained by adding ECCD sources. XTOR-2F simulations of the saturated helical state is then necessary in order to investigate the eects of the modied bulk plasma dynamics on the impurity behaviour in the core region, as observed in ASDEX-Upgrade discharges.

A model of ECCD source has been developed and implemented in the XTOR-2F code for magnetic island control [START_REF] Février | Global modelling of magnetic island control in tokamaks[END_REF], in parallel to the present model of impurity transport [Ahn et al., 2016b]. Starting with a geometry of circular concentric magnetic surfaces [Halpern et al., 2011b], XTOR-2F simulation results show that the dynamics of sawteeth is modied by adding a radially localized ECCD source [START_REF] Février | MHD simulations of sawteeth response to ECCD deposition[END_REF]. These are consistent with the experimental observations previously reported [START_REF] Furno | Understanding sawtooth activity during intense electron cyclotron heating experiments on TCV[END_REF][START_REF] Angioni | Eects of localized electron heating and current drive on the sawtooth period[END_REF]. In particular, when the ECCD source is localized near the inversion radius (r RF 0.9 r inv ), the precursor phase lasts more than half a sawtooth period, as shown in Figure 5.13. Indeed, the time evolution of the pressure shows that the shape of sawteeth is changed as soon as the ECCD source is switched on at t 9.8 × 10 4 τ A . These long-living helical oscillations of the hot core simulated by the XTOR-2F code are reminiscent of the saturated internal kink mode in ASDEX-Upgrade scenarios described in Chapter 4. Here, the current driven by the ECCD source is set equal to 1% of the total plasma current, which is the order of magnitude necessary to control sawteeth [START_REF] Hender | Chapter 3: MHD stability, operational limits and disruptions[END_REF].

The impurity behaviour in presence of the saturated helical state is then studied using the model implemented in the XTOR-2F code and described in this chapter. In this simulation, a partially stripped tungsten ion 184 74 W 45+ is considered. Tungsten is supposed to be present as a tracer (N W /N e = 10 -5 ) and the eect of toroidal rotation is again neglected. density proles at dierent phases of the precursor of a single sawtooth period (1.6 ×10 5 τ A < t < 1.8 × 10 5 τ A ). These surface averaged proles are also time averaged during one helical cycle. Here, the initial tungsten prole is peaked in order to be consistent with the ASDEX-Upgrade experiment. It should be noted that the initial impurity density prole is almost recovered before the next sawtooth crash for ordinary sawtooth cycles. However, it can be seen in Figure 5.14 that the impurity density prole is at in the core region, in presence of long-living precursor oscillations. Therefore, these preliminary simulation results show that the overall impurity content is weaker in presence of the long-living helical state.

These simulations which are to some extent representative of the ASDEX-Upgrade discharges show that the saturated helical state is responsible for tungsten depletion in the core region. Still, hollow impurity density proles as observed in ASDEX-Upgrade (Figure 4.10)

have not been reproduced so far.

5.5. SUMMARY 103

Summary

Appropriate uid equations for impurities in the Prsch-Schlüter regime have been implemented in the XTOR-2F code in order to investigate the combined eect of neoclassical transport and sawtooth crashes. An analytical calculation shows that the neoclassical impurity ux in the Prsch-Schlüter regime is recovered by adjusting the friction between the impurity and the main ion species.

It has been numerically veried that the impurity radial uxes obtained with this model are consistent with neoclassical predictions. The simulations show that the presence of sawtooth crashes changes the impurity thermal screening. Indeed, the time averaged outward impurity ux is weaker in presence of sawteeth. This is mainly due to a sudden inux of impurity during the sawtooth crash for hollow pre-crash proles. It is consistent with the observations presented in Chapter 4, where tungsten density prole is regularly attened in the core region after each sawtooth crash. Also, periodic attening of the core temperature prole weakens the thermal screening eect during the recovery phase. Impurity density utter in the percent range persists after each sawtooth crash. This makes it dicult to assess the agreement with the predictions of the Kadomtsev model for weakly hollow impurity proles. Nevertheless, the long time scale behaviour is denitely consistent with a weaker thermal screening eect.

Impurities are transported across the q = 1 surface via the dynamics of density bump and hole which appear during sawtooth crashes. These structures are shown to be mainly driven by the E×B ow in the core region. Impurities are more sensitive to E×B ows than magnetic utter because their uid velocity along the eld lines is very small. This is to be contrasted with electrons which move fast along the magnetic eld so that magnetic utter is the dominant electron transport mechanism. Hence, impurity transport is controlled by E × B vortices that cross the q = 1 surface during the reconnection process.

Finally, the presence of a saturated helical state in the core region attens the impurity density prole during the whole duration of the precursor, i.e. ushes out more eciently than ordinary sawtooth cycles. However, the impurity transport model has to be further improved since hollow impurity density proles as observed in the ASDEX-Upgrade discharge are not reproduced.

for reproducing neoclassical transport.

The main conclusion of this work is that the impurity behaviour in sawtoothing plasmas results from a competition of sawtooth relaxations with turbulent and neoclassical transport processes. The radial transport of impurities during sawtooth crashes is governed by the impurity density structures which are driven by MHD ows. These structures are responsible for the uctuations of the post-crash impurity density proles in the core region, which is not predicted by Kadomtsev's model.

Future work

Now that the appropriate uid equations for heavy impurities are implemented in the XTOR-2F code, the eects of the saturated (1,1) mode on the impurity behaviour remain to be explored. For that purpose, sawtooth cycles with a helical equilibrium between two consecutive crashes (so-called `humpback' sawteeth) have to be simulated. An EC wave source and its propagation model developed and implemented in the XTOR-2F code for NTM (Neoclassical Tearing Mode) controls is used for this purpose.

The eects due to toroidal rotation on tungsten redistribution have to be taken into account to complete the model. On the one hand, preliminary simulations without MHD instabilities showed that the centrifugal force weakens the temperature screening eect. This is consistent with predictions made by analytical computations. On the other hand, the poloidal asymmetry of the impurity density was completely wiped out by the violent impurity redistribution during sawtooth crashes. In addition, the complex interaction between heavy impurity radiation and the sawtooth dynamics can be addressed. For instance, the radiative cooling due to heavy impurities can be high enough to modify the bulk plasma dynamics which leads to a reduced temperature gradient and resistivity. Simulations can be compared with experimental SXR measurements using synthetic SXR diagnostic modules which have been implemented in the XTOR-2F code.

Furthermore, the XTOR-K code which is an upgrade of the XTOR-2F code models a population of particles kinetically, resulting in a mixed uid-kinetic description. Modelling the main ion and impurity species in the kinetic framework allows one to address the impurity transport self-consistently. Indeed, the adjustment of the uid equations to recover the Prsch-Schlüter transport would be avoided. Also, the transport of impurities in the banana-plateau regime can be modelled with this new tool.

APPENDIX A

Résumé long de la thèse

Les performances des plasmas de fusion connés magnétiquement peuvent être dégradées par l'accumulation d'impuretés. Plus particulièrement, les impuretés lourdes accumulées au centre du plasma diluent les réactifs, et peuvent aussi conduire à un collapse radiatif du plasma lorsque les pertes par rayonnement sont élevées. La compréhension du transport des impuretés lourdes produites lors de l'interaction plasma-paroi est donc cruciale.

Le coeur du plasma est sujet à une instabilité magnétohydrodynamique (MHD) appelée kink interne caractérisée par des nombres de mode poloïdal et toroïdal (m, n) = (1, 1).

Cette instabilité conduit à des oscillations de relaxation nommées dents-de-scie. Les dentsde-scie entraînent une relaxation périodique de densité et de température dans le coeur du plasma. Un cycle de dents-de-scie est consitué principalement de deux phases : une phase de rampe et une phase de collapse. Pendant une durée du crash, de l'ordre de 100µs, qui est très courte par rapport au temps de connement, les particules et la chaleur sont redistribuées et le transport radial est signicativement aecté. Dans cette thèse, l'outil numérique utilisé pour simuler les dents-de-scie est le code XTOR-2F. XTOR-2F est un code non-linéaire tridimensionnel résolvant les équations de la MHD bi-uide dans une géométrie toroïdale.

An de préparer la modélisation des décharges expérimentales, une étude paramétrique a été conduite an de simuler des dents-de-scie avec le code XTOR-2F, dans une géométrie allongée. Il s'avère que certaines simulations présentent des caractéristiques similaires aux observations expérimentales des dents-de-scie composées. En eet, dans des cas particuliers, les simulations avec le code XTOR-2F montrent que le coeur chaud du plasma est déplacé dans la direction radiale et tourne dans le sens poloidal, tout en restant à l'intérieur de la surface résonante q = 1. Ce phénomène, nommé `crash partiel', correspond aux observations expérimentales. Le balayage en position de la surface q = 1 montre que le crash partiel Ces décharges sont destinées aux études du transport du tungstène en présence des dents-